Production and Evaporation of
Higher Dimensional Black Holes
Marco Oliveira Pena Sampaio
Churchill College
A dissertation submitted to the University of Cambridge
for the degree of Doctor of Philosophy
March 2010
Production and Evaporation of Higher Dimensional
Black Holes
Marco Oliveira Pena Sampaio
Abstract
This thesis is a study of the theory and phenomenology of trans-Planckian black holes,
in TeV gravity extra-dimensional theories. The introduction starts with the motivation for
this beyond the Standard Model scenario (chapter 1), a summary of the theoretical tools
to formulate the theory, and a summary of the best bounds from experiment (chapter 2).
In chapter 3, after setting up some notation and describing well known solutions in
4 + n-dimensional general relativity, we construct an approximate effective background
for a brane charged rotating higher-dimensional black hole. This is achieved by solving
Maxwell’s equations perturbatively on the brane to obtain the electromagnetic field. A
brief study of the effect of rotation on the absorption of classical particles is also provided.
Chapter 4 is a review of methods to model black hole production focusing on the
trapped surface method. A model for the mass and angular momentum loss into gravita-
tional radiation is described.
A detailed study of the effects of particle mass and charge, for fermions and scalars on
the effective brane charged background, is presented in chapters 5 and 6. After coupling
the fields to the background, the separated radial wave equations for both perturbations
are obtained (chapter 5) and they are integrated using a detailed numerical method as
well as analytic approximations (chapter 6). Similarly, a method is described to obtain
high accuracy angular functions based on series expansions. We conclude the theoretical
study by evaluating the Hawking spectra for various combinations of spin, mass, charge
and rotation parameters, and discuss them comparatively.
The last part of the thesis is on the implementation of the theoretical results in the
new CHARYBDIS2 Monte Carlo simulation of black hole production and decay (chapter 7),
and on the analysis of the phenomenological consequences (chapter 8). The main new
features implemented in CHARYBDIS2 are: a full treatment of the spin-down phase using
the angular and energy distributions of the associated Hawking radiation; an improved
model for energy and angular momentum loss in the production process, and a wider
range of options for the Planck-scale termination of the decay. The main conclusions of
this thesis and an outlook on future directions are summarised in the final chapter.
iii
Declaration
This dissertation is the result of my own work, except where explicit reference is made
to the work of others, and has not been submitted for another qualification to this or any
other university.
The approximate charged rotating background constructed in chapter 3 was published
in [1]. The original study of charge and mass effects developed in chapters 5 and 6 was
published in [1, 2]. The event generator described in chapter 7 was released in [3] and
published in [4] with part of the phenomenological study of chapter 8.
This thesis does not exceed the 60,000 word limit prescribed by the Degree Committee
for Physics and Chemistry.
Marco Oliveira Pena Sampaio
v
Acknowledgements
I start by thanking my supervisor Prof. Bryan Webber for all his support, guidance,
patience and positive thinking throughout my PhD. I have learnt a lot about doing re-
search, and the process of building independent ideas. I must thank him in particular for
the incredible continuous feedback provided during the preparation of this document.
I am very grateful for the pleasure of having collaborated with: Marc Casals, Sam
Dolan, Jonathan Gaunt, and especially Prof. M. Andrew Parker and James Frost. I must
single out James for his dedication and support to the CHARYBDIS2 project, especially for
his good-humoured and friendly character, positiveness and loyalty.
I have been lucky to be surrounded by many other stimulating researchers in the
Cambridge high energy physics group and the Cambridge Supersymmetry working group.
I have learned from them through discussions about many interesting areas of particle
physics. In particular, I thank my near office mates Deirdre Black, Steve Kom, Are
Raklev and the lineage of PhD students Jenni Smillie, Seyi Latunde-Dada, Andreas Pa-
paefstathiou, Jo Gaunt, Lucian Harland-Lang and Eleni Vryonidou. With them I had the
most satisfying discussions, and shared problems, hopes and successes.
A special note must go to a collaborator from a different project, and friend, Carlos
Herdeiro for his wise advice and positiveness during some important periods of my PhD.
This project was funded by Fundac¸a˜o para a Cieˆncia e Tecnologia (FCT) - Portu-
gal, grant SFRH/BD/23052/2005 co-financed by POPH/FSE. I am grateful to Churchill
College for providing not only academic resources, but most importantly, a home.
I thank Bryan, James and Jo for proof-reading the document.
There are so many people outside physics who made these years the best. I especially
want to single out Martin Huarte-Espinosa, Johann von Kirchbach, Kasia Gilewicz, Frida
Weierud, Carlos Guedes, Chris Edge and Raquel Ribeiro for being my Cambridge family.
I was also lucky to spend many hours with the Cambridge Volleyball friends, who kept
me physically healthy and cheerful. In Portugal, I thank my friends from Neiva, Viana
do Castelo and Porto for always receiving me as if I had never left.
I am most grateful of all to my two siblings and my parents. They believe and support
everything I do, and they are the constants in my life.
vii
Contents
1 Introduction 1
1.1 Particle physics and the Standard Model . . . . . . . . . . . . . . . . . . . 1
1.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Comparing couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Hierarchies and other unexplained properties . . . . . . . . . . . . . . . . . 9
1.5 Extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Theories with extra dimensions 13
2.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 The ADD scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 The Randall-Sundrum scenario . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Further brane constructions and other scenarios . . . . . . . . . . . . . . . 20
2.5 Experimental bounds on extra dimensions . . . . . . . . . . . . . . . . . . 21
2.5.1 Laboratory bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.2 Astrophysics and cosmology . . . . . . . . . . . . . . . . . . . . . . 23
2.5.3 Summary of the bounds . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Strong gravity I: Charged rotating black holes 27
3.1 Black holes in general relativity . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 Some solutions in four dimensions . . . . . . . . . . . . . . . . . . . 30
3.1.3 Exact solutions in higher dimensions . . . . . . . . . . . . . . . . . 35
3.2 Construction of a brane charged background . . . . . . . . . . . . . . . . . 38
3.2.1 The brane Maxwell field as a perturbation . . . . . . . . . . . . . . 40
3.2.2 Comments on backreaction . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.3 Systems of units and orders of magnitude . . . . . . . . . . . . . . . 45
3.3 Geodesics and the geometrical cross section . . . . . . . . . . . . . . . . . . 47
3.3.1 The critical impact parameter . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 Perturbative and numerical minimisation . . . . . . . . . . . . . . . 54
4 Strong gravity II: Models for black hole production 61
4.1 Setting up the initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Gravitational collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
ix
x CONTENTS
4.3 Trapped surface and other analytic bounds . . . . . . . . . . . . . . . . . . 64
4.3.1 Theoretical studies of black hole production . . . . . . . . . . . . . 64
4.3.2 The model for CHARYBDIS2 . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Latest developments in numerical relativity . . . . . . . . . . . . . . . . . . 69
5 Black hole decay and Hawking radiation 71
5.1 Perturbation theory and approximate decoupling . . . . . . . . . . . . . . 71
5.2 Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Perturbations of a brane charged black hole . . . . . . . . . . . . . . . . . 78
5.3.1 Wave equations I: Coupling the background . . . . . . . . . . . . . 78
5.3.2 Wave equations II: Separability . . . . . . . . . . . . . . . . . . . . 80
5.3.3 Higher spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.4 Decomposition of spheroidal waves into plane waves . . . . . . . . . 84
6 Analytic and numerical study of perturbations 89
6.1 The angular equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 The radial equations I: Analytic methods . . . . . . . . . . . . . . . . . . . 94
6.2.1 Near horizon equation . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.2 Far field solution and low energy matching . . . . . . . . . . . . . . 96
6.2.3 High energy approximation based on WKB arguments . . . . . . . 98
6.3 The radial equations II: Numerical methods . . . . . . . . . . . . . . . . . 100
6.3.1 Near horizon expansions . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.2 Far field expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4.1 The effect of particle mass . . . . . . . . . . . . . . . . . . . . . . . 105
6.4.2 The effect of black hole charge on neutral particles . . . . . . . . . 108
6.4.3 The effect of particle charge . . . . . . . . . . . . . . . . . . . . . . 109
6.4.4 The effect of black hole rotation . . . . . . . . . . . . . . . . . . . . 113
6.4.5 Interplay between rotation and charge effects . . . . . . . . . . . . . 119
6.5 High energy absorption cross sections . . . . . . . . . . . . . . . . . . . . . 121
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7 CHARYBDIS2 127
7.1 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.1.1 Adding the intrinsic spin of the colliding particles . . . . . . . . . . 132
7.2 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.1 Back-reaction and spin-down . . . . . . . . . . . . . . . . . . . . . . 135
7.3 Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3.1 Termination of the black hole decay . . . . . . . . . . . . . . . . . . 140
7.3.2 Fixed-multiplicity decay model . . . . . . . . . . . . . . . . . . . . 141
7.3.3 Variable-multiplicity decay model . . . . . . . . . . . . . . . . . . . 142
7.3.4 Boiling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.3.5 Stable remnant model . . . . . . . . . . . . . . . . . . . . . . . . . 143
CONTENTS xi
7.3.6 Straight-to-remnant option . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Program structure and usage . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4.1 General structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.4.2 Initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.4.3 Event generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4.4 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.5 CHARYBDIS2 and other generators . . . . . . . . . . . . . . . . . . . . . . . 151
8 Phenomenological study 153
8.1 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.2 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.2.1 The effect of mass and angular momentum loss . . . . . . . . . . . 156
8.2.2 The effect of black hole angular momentum . . . . . . . . . . . . . 156
8.2.3 Mass reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.2.4 Angular momentum reconstruction . . . . . . . . . . . . . . . . . . 164
8.2.5 Parton level angular correlators . . . . . . . . . . . . . . . . . . . . 167
8.3 Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9 Conclusions and Outlook 177
A Conventions and mathematical tools 181
A.1 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Planck mass convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3 The induced vierbein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
B The production model 185
B.1 The Yoshino-Rychkov mass/angular momentum bounds . . . . . . . . . . . 185
B.2 Details of the ‘constant angular velocity’ bias . . . . . . . . . . . . . . . . 189
C Expansion coefficients and matrices for radial equations 193
C.1 Expansion coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
C.1.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
C.1.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
C.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
D Details of the CHARYBDIS2 implementation 197
D.1 How to set up CHARYBDIS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
D.2 Remnant decay generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
D.2.1 Momenta selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
D.2.2 Angular momenta selection . . . . . . . . . . . . . . . . . . . . . . 201
D.3 The UPINIT subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
D.4 The UPEVNT subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
D.5 Eternal black hole angular correlators . . . . . . . . . . . . . . . . . . . . . 206
xii CONTENTS
Chapter 1
Introduction
In this chapter we summarize the current formulation of the Standard Model of particle
physics (SM) by describing its particle content and interactions, as well as Einstein’s
theory of gravity. We then analyse the main puzzles arising when the two theories are
compared, and discuss ways of extending them. Finally we present some alternatives and
emphasize extra-dimensional models which are discussed in detail in the next chapter.
1.1 Particle physics and the Standard Model
Particle physics is dedicated to the search of ever more fundamental indivisible units which
constitute the building blocks of all known matter. The best picture available so far is one
where fundamental point particles are modelled by quantum fields. Experimental data
up to the present date provides extremely detailed and accurate verification of most of
the predictions of the so called Standard Model (SM) [5]. The first step in constructing
the Standard Model is to realize what are the fundamental degrees of freedom observed
in experiments. These can be organised in two classes:
1. The force carriers, bosonic spin-1 fields which are responsible for the electromagnetic
and strong forces through the massless fields Aα and G
i
α (i = 1 . . . 8) respectively,
and the weak forces through the massive fields W±α and Zα.
2. The matter fermionic spin-1/2 fields which can be of two sub-types (all fields are
Dirac spinors):
• Quarks – They are of up-type uia or down-type dia with electric charges +2/3
and −1/3 respectively. There are three families, a = 1 . . . 3 with increasing
1
2 Chapter 1. Introduction
masses (up, charm, top, for up-type; and down, strange, bottom, for down-
type). The index i = 1 . . . 3 is a label for the strong force colour charge.
• Leptons – Here the equivalent of the up-type particles are the neutrinos1 νa
which are electrically neutral and the down-type particles are electron-like, ea
with electric charge −1. The family index runs through a = 1 . . . 3 (electron,
muon and tau families respectively).
With this particle content, we can construct free Lagrangians plus interactions. The
main leading principles in the construction of the SM Lagrangian are: Lorentz invariance,
renormalisability, unitarity and gauge invariance. In particular to obtain charged current
or coloured current interactions (for theW±α and the gluons respectively) the gauge group
must be non-abelian. Since the W±α is a massive particle and there are no non-abelian
massive gauge theories which are unitary and renormalisable, a mechanism must be in-
voked to give mass to the gauge bosons. One of the simplest ways to achieve this is the
Higgs mechanism, where an extra scalar field H is used.
The model is constructed by postulating a gauge group, at the fundamental level,
which is SU(3)C × SU(2)L ×U(1)Y . The gauge fields are the gluon Giα, the weak isospin
boson Aiα and the weak hypercharge boson Bα. The coupling constants are denoted gi with
i = 1, 2, 3 respectively, and the coupling to matter is achieved through covariantisation
using the gauge covariant derivative
∂α → Dα = ∂α + ig1Gα ·T1 + ig2Aα ·T2 + ig3Y Bα , (1.1)
where the dot · denotes summation over the gauge group generators, and bold face fields
denote column vectors composed of the gauge field components. T1 and T2 are the
generators of the corresponding gauge groups in the fundamental representation and Y is
the hypercharge.
The Yang-Mills Lagrangian for the theory is (given a convenient choice of generators
in the adjoint representation)
Lpure gauge = −1
4
Gαβ ·Gαβ − 1
4
Aαβ ·Aαβ − 1
4
BαβB
αβ , (1.2)
1Note however that right-handed neutrinos have not been observed experimentally, although the non-
zero neutrino masses indicate their existence.
1.1. Particle physics and the Standard Model 3
where we denote the field strengths by the same letter as the corresponding gauge field.
For a generic gauge field F iα with coupling g the field strength is
Fαβ = ∂αFβ − ∂βFα + g [Fα,Fβ] . (1.3)
To generate masses for the weak bosons, the Higgs field is introduced. This is chosen to
be a singlet under SU(3)C and in the fundamental representations of SU(2)L × U(1)Y ,
i.e. (1, 2, 1/2), with Lagrangian
LHiggs = (DαH)†DαH − 1
2
λ
(
H†H − 1
2
v2
)2
. (1.4)
The real parameters λ > 0 and v are constants. At the ground state of the potential, the
Higgs develops a non-zero expectation value (VEV) which breaks the SU(2)L × U(1)Y
symmetry to U(1)Q. The three Nambu-Goldstone bosons, associated with the flat direc-
tions of the potential around the vacuum, are absorbed by three of the weak field degrees of
freedom giving them longitudinal components and therefore mass. The remaining U(1)Q
symmetry corresponds to the electromagnetic field. With an appropriate choice of gauge
it is possible to express the low energy physical degrees of freedom as
H =
1√
2
(v + h)
(
0
1
)
(1.5)
Wα =
1√
2
(
A1α − iA2α
)
(1.6)
Zα = cos θWA
3
α − sin θWBα (1.7)
Aα = sin θWA
3
α + cos θWBα (1.8)
where h(x) is the remaining real Higgs scalar field. We can switch from the set {g2, g3, λ}
to {e, cos θW , mh} where e is the QED coupling, mh is the Higgs mass and θW ∈ [0, π/2]
is the Weinberg angle. These are defined by
g2 sin θW = e
g3 cos θW = e . (1.9)
m2h = λv
2
4 Chapter 1. Introduction
The gauge/Higgs sector of the SM after symmetry breaking is then
Ls={0,1} = 1
2
∂αh∂αh− m
2
h
2
h2 − λ
8
h3 (h+ 4v)− 1
4
Gαβ ·Gαβ +
−1
2
W †αβW
αβ +m2WW
†
αW
α − 1
4
ZαβZ
αβ +
m2Z
2
ZαZ
α − 1
4
AαβA
αβ +
1
4
h (h+ 4v)
(
2e2
sin2(2θW )
ZαZ
α +
e2
sin2 θW
W †αW
α
)
+ (1.10)
+iWαW
†
β
(
eAαβ +
e
tan θW
Zαβ
)
+
e2
2 sin2 θW
(
|WαW α|2 −
(
W †αW
α
)2)
where we can identify the Higgs free Lagrangian and self interactions, the electroweak
vector bosons free Lagrangians and self interactions, the Higgs/vector interactions and
the vector/vector interactions respectively2. In this effective Lagrangian, the Nambu-
Goldstones are no longer identified as physical degrees of freedom at low energies, because
they are part of the W and Z fields. However it is known that at high energies, a
longitudinal state of a W or a Z in an external leg of a matrix element can be replaced
by a Nambu-Goldstone boson state. This is the so called Goldstone boson equivalence
theorem (for a discussion see section 21.2 of [6]).
The fermionic matter sector is constructed similarly by considering all possible renor-
malisable terms before symmetry breaking. The field content and gauge representations
are chosen as to reproduce the chiral structure of the interactions and the electric and
colour charges observed in experiments. The up-type and down-type left-handed compo-
nents of the Dirac fields are placed in SU(2)L doublets transforming in the fundamental
representation, and the right-handed parts on SU(2)L singlets. The leptons are placed in
singlets under SU(3)C , whereas the quarks transform in the fundamental representation.
Finally the hypercharges are chosen as to produce the correct U(1)Q electric charges after
symmetry breaking. The fermion content is then (leptons and quarks respectively)
La =
(
νˆaL
eˆaL
)
→ (1, 2,−1
2
)
, Ra = eˆaR → (1, 1,−1)
Qa =
(
uˆaL
dˆaL
)
→ (3, 2, 1
6
)
,
Ua = uˆaR →
(
3, 1, 2
3
)
Da = dˆaR →
(
3, 1,−1
3
)
(1.11)
2Here the field strengths Wαβ , Zαβ, Aαβ are defined as before for the abelian field Bα.
1.1. Particle physics and the Standard Model 5
where the representations are denoted in parenthesis and the hats are a reminder that
these are fields before electroweak symmetry breaking. Note that for simplicity we have
not included right-handed neutrinos, which must be singlets under the full SM gauge
group. This would allow a Majorana mass term which can explain the small left-handed
neutrino masses and the decoupling of the right-handed neutrino through the seesaw
mechanism. Nevertheless, for the processes we will consider, left-handed neutrino masses
are negligible, so we can safely ignore the mechanism of mass generation for neutrinos
and exclude right-handed neutrinos. Then the Lagrangian is3
L 1
2
= L¯aiDLa + R¯aiDRa + Q¯aiDQa + U¯aiDUa + D¯aiDDa
−
√
2
(
L¯aλRabHR
b + Q¯aλUabH
cU b + Q¯aλDabHD
b + c.c.
)
, (1.12)
where λiab are complex Yukawa matrices in general. After the Higgs boson acquires a VEV
and electroweak symmetry is broken:
Ls= 1
2
= e¯a (i∂ −mea) ea + ν¯ai∂νa + u¯a (i∂ −mua) ua + d¯a (i∂ −mda) da +
−h
v
(
mea e¯
aea +mua u¯
aua +mda d¯
ada
)
+
− e
2
√
2 sin θW
(JαWWα + c.c.)− eJαAAα −
e
sin(2θW )
JαZZα − g1JαG,iGiα(1.13)
with the free kinetic terms, the Higgs/fermion interactions and the vector/fermion inter-
actions.4 The various currents are
JαA = −e¯aγαea +
2
3
u¯aγαua − 1
3
d¯aγαda (1.14)
JαG,i = u¯
aγαTiu
a + d¯aγαTid
a (1.15)
JαW = ν¯
aγα (1− γ5) ea + u¯aγα (1− γ5)Vabdb (1.16)
JαZ =
1
2
[
ν¯aγα (1− γ5) νa − e¯aγα
(
1− γ5 − 4 sin2 θW
)
ea
]
+
+
1
2
[
u¯aγα
(
1− γ5 − 8
3
sin2 θW
)
ua − d¯aγα
(
1− γ5 − 4
3
sin2 θW
)
da
]
(1.17)
and now the Yukawa couplings λiab are replaced by the masses of the physical particles
and the CKM matrix Vab (after diagonalising the original mass matrices by rotating the
3We use the usual slash notation for contraction with the Dirac gamma matrices γαVα ≡ V .
4Note that we have removed the hats from the fermion fields after symmetry breaking and used the e
symbol both for the charged leptons and the electric coupling constant.
6 Chapter 1. Introduction
original weak states to the physical states).
1.2 Gravity
The remaining fundamental force which is known up to the present is gravity, and it is
described by Einstein’s theory of General Relativity (GR). It is also a gauge field theory
but with a spin-2 field gab
5. It has been probed only classically at very large scales
compared to the fundamental scale in the theory: the 4-dimensional Planck length L4.
Its dynamics and interaction with matter are governed by the Einstein-Hilbert action
together with the action for the matter fields. The latter is covariantised with respect
to gab by replacing all partial derivatives with curved spacetime covariant derivatives and
the Minkowski metric by gab. The full action is
S = SEH + Smatter
=
∫
d4x
√
|g|
[
M24
2
R + Lmatter
]
(1.18)
where R is the Ricci scalar and M4 ≡ 1/L4 the Planck mass. This prescription ensures
that gravity couples universally to all fields. Since all known particles are described by the
SM, the matter Lagrangian is simply the curved spacetime covariantised version of the
Lagrangian in section 1.1. When varied with respect to gab, (1.18) provides the Einstein
equations of motion
Gab = M−24 Tab ≡
2
M24
√|g| δSmatterδgab . (1.19)
Except for the need to invoke dark matter at cosmological scales or to explain galaxy
rotation curves, all the gravitational observations performed so far are consistent with
GR. Some of the most precise measurements include [7]:
• The weak equivalence principle, Lorentz invariance and local position invariance
have been confirmed with precisions of 10−13, 10−22 and 10−4 respectively,
• Weak field effects in solar system experiments with accuracies up to 10−3,
• Some tests of the strong field regime in binary pulsar systems with precision 10−4,
• and torsion-balance measurements which test the inverse square dependence of the
gravitational force between test masses (confirmed, down to 56µm [8,9]).
5We denote curved spacetime indices by lower case letters from the Latin alphabet.
1.3. Comparing couplings 7
These experiments support the validity of GR for classical gravitational phenomena and
more importantly the strong field regime, where for example black hole physics is included.
However, gravity has only been probed well aboveM4, so for example it is not known how
it behaves at the quantum level. In particular, straightforward quantisation of the GR
perturbations on a background is not renormalisable so it cannot provide a full theory of
quantum gravity. Some examples of theories of quantum gravity are string theory [10]
or canonical quantisation [11, 12]. We will restrict our study to phenomena which are
typically away from this regime.
1.3 Comparing couplings
To better understand the scales in the SM and GR and how they compare to each other
it is instructive to rewrite the previous Lagrangians using a common probe length scale
L. We rescale all lengths by this amount as well as all the SM fields and the Ricci scalar
according to their mass dimensions. In addition, we assume that any interaction occurs
at proper length scales which are small with respect to the typical curvature scale of
spacetime, so that we can work in a locally inertial frame where, to a first approximation,
the metric is expanded around Minkowski spacetime plus linear perturbations:
gab = ηab +
L4
L
hab . (1.20)
We have normalised hab such that the kinetic term from the Einstein-Hilbert action takes
a canonical form. It follows from straightforward dimensional analysis and linearisation
that no new factors of L4/L appear in the SM Lagrangian, except for new terms which
are related to the coupling to gravity through products with hab. Furthermore any di-
mensionful quantity comes divided by an appropriate number of powers of L. The action,
up to linear order, is then given by the zeroth order SM action in Minkowski space plus
the Fierz-Pauli Lagrangian which describes linearized gravity (see for example [11])
SFP =
∫
d4x
[
1
8
(
hab,chab,c − 2hab,chac,b + 2hac,ah,c − h,bh,b
)
+
L4
2L
Tabh
ab
]
, (1.21)
where the comma denotes partial differentiation. Another way to estimate the magni-
tude of the gravitational interaction is by using a Newtonian approximation. In the limit
of non-relativistic collisions we can use the classical expressions for the electrical (Fe)
8 Chapter 1. Introduction
Interaction Types of Operators Couplings at E ∼ 1 TeV
Matter/Gravity Tabh
ab E
M4
10−16
Fermions/Vectors f¯ f ′ {A,Z,W,G} e, e
sin(2θW )
,
√
2e
4 sin θW
, g1 0.3, 0.4, 0.2, 1.2
W/Vectors W †W {A,Z} e, e
tan θW
0.3, 0.6
W/W
1
2!
W †WW †W
e2
sin2 θW
0.4
Gluon/Gluon {G4, G2∂αG} g21, g1 1.4, 1.2
Higgs/Higgs
{
h4
4!
,
h3
3!
}
3
(mh
v
)2
,
v
E
0.8, 0.25
Higgs/Fermions hf¯f
mf
v
2.10−6 − 0.7
Higgs/Vectors
h2
2!
{
1
2!
ZZ,W †W
}
2e2
sin2(2θW )
,
e2
2 sin2 θW
0.2, 0.2
h
{
1
2!
ZZ,W †W
}
4e2
sin2(2θW )
v
E
,
e2
sin2 θW
v
E
0.1, 0.1
Table 1.1: Interaction couplings for all known fields: The various types of interactions among
SM particles and gravity are shown schematically. The couplings in the third column
and their approximate values at 1 TeV in the fourth column are in one-to-one
correspondence with the operators in the second column.
and gravitational (Fg) forces. For estimate purposes take a particle at rest and compute
the forces on another charged test particle of the same type. Using the Newtonian ap-
proximation and the Coulomb force, and writing all the lengths and masses in natural
units
Fg
Fe
=
(
m
M4
)2
.
(
L4
r
)2
α.
(
L4
r
)2 = 1α
(
m
M4
)2
=
1
α
(
L4
L
)2
, (1.22)
where m,M4, L4 are the mass of the particle, Planck mass and Planck length respectively,
α = e2/(4π) is the fine structure constant and we have identified L ∼ 1/m as the typical
length scale for the process. From these estimates we see that the gravitational coupling
constant for a process at a scale L is controlled by L4/L or equivalently E/M4 (E ∼ 1/L
1.4. Hierarchies and other unexplained properties 9
is the typical energy for the process). Table 1.1 summarizes the SM and gravitational
couplings. In the first line, we observe that for the typical energies achieved in laboratory
experiments today, gravity is truly negligible. This is due to the extreme smallness of the
Planck length or equivalently the extremely large Planck mass which sets the energy scale
for strong gravity. This large scale justifies why no quantum gravitational effects have
been observed so far. In particular, there seems to be no hope for near future observations
of quantum gravity effects. However, this statement should be taken with care, because
of the enormous extrapolation involved along sixteen orders of magnitude. There could
be some modification at smaller length scales, such that the conventional 4-dimensional
Planck length is an effective scale derived from a more fundamental high energy theory.
In fact the modern view of quantum field theories is that the low energy theories we
observe are effective theories, from which the cutoff scale is not obvious (for a discussion
on gravity see [13]).
1.4 Hierarchies and other unexplained properties
In section 1.3 we have seen how gravity is many orders of magnitude weaker than all
known forces in the microscopic domain of particle physics. This is a manifestation of the
hierarchy problem which is observed in Standard Model loop corrections as follows. If we
take the SM as an effective field theory, we have to decide on a cutoff energy scale. In
this interpretation, the predictions of the theory are made at energies below the cutoff,
which sets the limit where new physics starts to arise. For the theory to be natural, its
low energy limit couplings should be stable under small variations of the couplings at the
cutoff [14]. The largest energy scale we know in nature is the Planck mass which would be
the most natural first guess for a cutoff. In the SM, the low energy Higgs mass is expected
to be of order6 ∼ 102 GeV, but the bare mass (at the Planck scale) receives corrections
which are quadratic in the cutoff when renormalised to lower energies. This can be seen
by looking at the Higgs Lagrangian (1.4) with v solved for mh
LHiggs = (DαH)†DαH − 1
2
λ
(
H†H
)2
+
1
2
m2hH
†H + constant , (1.23)
and by considering the self coupling and the Yukawa coupling to fermions λfHf¯f which
give one loop corrections to the Higgs mass as in figure 1.1:
6Perturbative unitarity bounds imply mH . 870 GeV (see for example [15]).
10 Chapter 1. Introduction
f
H H
H H
H
Figure 1.1: Lowest order loop contributions to the Higgs mass quadratic divergence: Fermion
contributions (left) and scalar contributions (right).
δm2h =
(
|λf |2 − 1
2
λ
)
Λ2cutoff
8π2
+ . . . (1.24)
The largest contribution from fermion loops in the SM comes from the top Yukawa cou-
pling. So, if Λcutoff is the Planck mass (which is ∼ 17 orders of magnitude larger than the
Higgs mass) the bare mass will be at the Planck mass scale and a small relative variation
induces a very large variation of the low energy mass parameter. Therefore, the bare
Higgs mass squared has to be unnaturally fine tuned with a precision of one part in 1034.
All the other couplings of the SM have a better ultraviolet behaviour either due to gauge
symmetry or chiral symmetry [6]. However, since all the masses in the SM are generated
either through the Higgs mechanism (for vector bosons) or the Yukawa interactions (for
fermions) all masses are fine tuned indirectly through the Higgs VEV [16].
Some solutions of the hierarchy problem involve including new fields to improve the
ultra-violet (UV) behaviour of the loop correction so as to cancel the quadratic divergences
in the Higgs boson mass corrections. For example in Supersymmetry, each fermion has two
scalar partners, such that |λf |2 = λ and the relative minus sign in equation (1.24) takes
care of the cancellation [16]. In little Higgs models a similar cancellation occurs between
bosons or between fermions [17]. Alternatively, in strong dynamics theories, instead of
invoking a symmetry, the Higgs is replaced by a pion-like field of a new strong sector
and the small mass is explained by a (natural) exponential running [14]. It is important
to note that these solutions still do not explain why gravity is so much weaker than all
other forces, so though the fine tuning problem is resolved, strictly speaking the hierarchy
problem remains.
Finally, there are other properties of the SM which are not well understood such
as: family replication; the large number of free parameters; the flavour structure in the
Yukawa couplings (which induces large mass splittings among the fermions, and mixings
in the quark sector); the (apparent) insufficient CP violation in the SM to generate the
1.5. Extra dimensions 11
baryon/anti-baryon asymmetry in the early Universe; and the generation of small neutrino
masses (which appear to be at the eV scale) as well as the pattern of their mixings.
1.5 Extra dimensions
A very different, but still natural, way of solving the hierarchy problem is possible if we
introduce extra dimensions. These constructions will be the focus of the remainder of
this work. The simple idea behind such models is to take the TeV scale as the cutoff in
the effective theory. This is assumed to be the only short distance fundamental scale,
controlling in particular the strength of the gravitational force. This scenario was first
proposed by Antoniadis, Arkani-Hamed, Dimopoulos and Dvali [18–21] in models with
large compact extra dimensions (ADD scenario). The motivation arises from observing
that the only scale which is well established experimentally in the microscopic domain is
MEW ∼ 1 TeV. This solves the hierarchy problem trivially, because the fine tuning of the
Higgs mass was due to the large cutoff scale. By setting such a cutoff close to the scale
of the expected Higgs mass, the problem disappears.
Since in the ADD scenario the only fundamental scale is ∼ 1 TeV, in particular the
Planck mass must be of that order. This means that the true fundamental Planck mass is
much smaller than the conventional Planck mass measured in gravitational experiments
at large length scales. This difference in magnitude is explained by postulating the exis-
tence of compact extra dimensions, with sizes below the lengths probed in gravitational
experiments. At short distances (well below the size of the extra dimensions) gravity
is higher dimensional and controlled by the fundamental Planck mass, whereas at long
distances, an effective theory for gravity arises from integrating out the extra dimensions.
It is this procedure of integrating out the extra dimensions that raises the Planck mass
by a large extra dimensional volume making gravity weak at large scales. Heuristically
gravity gets diluted by the extra dimensional volume.
Another class of extra-dimensional models which solves the hierarchy problem by
assuming TeV scale gravity and one curved extra dimension has been suggested by Randall
and Sundrum (RS scenarios [22, 23]). In the original proposal two types of models were
possible. In the first one the large effective Planck mass is due to the large curvature
along a compactified extra dimension with a small radius, whereas the second contains
a semi-infinite extra dimension with a small curvature (or equivalently large curvature
radius). The latter is in some sense more similar to ADD. Both options (ADD and RS)
12 Chapter 1. Introduction
are presented in more detail in the next chapter.
A crucial feature of these models is that gravity becomes strong at energy scales
∼ 1 TeV (note for example that the suppressing factors in (1.22) become of order 1 at
the Planck scale). Therefore new phenomena arises above the TeV scale, in particular
we would expect quantum gravitational effects. Since we do not known in advance what
the new physics of quantum gravity above 1 TeV is, the only predictions we can trust
are in particular limits where we hope that a combination of classical GR and quantum
field theory holds. This is the semi-classical limit. It includes for example strong gravity
at length scales large compared to the Planck length. This does not mean however that
the process is bound to be at low energies because of the IR/UV correspondence in
gravitational theories where large length scales are actually related to large concentrations
of energy [24]. In particular black holes are examples of such systems where their size grows
with the energy scale, so they can be formed in high energy processes where the collapsing
matter interacts gravitationally at large separations. Thus the gravitational interaction
can be effectively described by the classical theory at length scales large compared to
the Planck length even though the amount of energy involved in the process is above
the Planck mass. This is exactly what happens in astrophysical systems such as stars
and black holes so we would not expect quantum gravity to be necessary. This limit is
the strong semi-classical trans-Planckian regime which is the focus of this work. Other
calculable regimes exist in the weak trans-Planckian limit at large distances [25] and
sub-Planckian limit [26, 27] which we will not explore in detail.
Chapter 2
Theories with extra dimensions
Theories with extra dimensions were first introduced in 1912 by Nordstro¨m and in the
1920s by Kaluza and Klein as an attempt to provide a unified description of gravity and
electromagnetism (see for example [28] where the original papers have been re-printed).
Recently, they have been motivated and explored intensively in the context of String
Theory which attempts to unify gravity and quantum mechanics. Regardless of the fun-
damental theory, extra dimensions may exist, and in fact the consequences of their exis-
tence can be explored without making many assumptions about a possible high energy
(or UV) completion. In the remainder of this thesis, we will consider the existence of
extra dimensions at low energies and adopt an effective theory approach where the model
is built with a minimal set of assumptions consistent with current experimental bounds.
In addition some extra assumptions are made which motivate the theory as a solution to
known problems. This minimal approach is very useful because it provides a way to test
different UV completions which reduce to the same type of low energy effective theory.
The purpose of this chapter is to present details on how to formulate extra dimensional
effective field theories in general, then introduce the main extra dimensional scenarios
that have been proposed as solutions to the hierarchy problem as well as an overview of
more elaborate constructions, and finally present the main experimental bounds on extra
dimensions.
13
14 Chapter 2. Theories with extra dimensions
2.1 General formulation
Extra-dimensional theories are formulated similarly to 4-dimensional theories. In general
we assume an action of the form
S =
∫
dDX
√
|G| (Lgrav + Lmatter) (2.1)
=
∫
dDX
√
G
(
−ΛD + 1
2
MˆD−2D RD + . . .+ Lmatter
)
(2.2)
where D = 4 + n is the number of spacetime dimensions and MˆD is the reduced Planck
mass (see A.2). The dots denote higher curvature gravitational terms which we assume are
suppressed compared to the leading Einstein-Hilbert action with a cosmological constant
ΛD. The minimal matter Lagrangian must contain the SM fields properly imbedded
into the higher-dimensional spacetime consistently with current experimental data. The
details of the imbedding are model dependent.
Most of the models studied at present can be formulated as effective theories on a
particular background gravitational field. In many realisations the SM fields are confined
to a lower dimensional space called the brane. The general formalism for writing down
an effective theory on a background Minkowski spacetime, with a brane containing SM
fields, was presented in [29]. In principle, most of the results can be applied to a curved
background with a brane, so we summarize them and point out when specialisation to
Minkowski spacetime simplifies the framework. The basic assumption is that the small
amplitude and low energy perturbations, can be expanded perturbatively around the
following background1:
EAM(X) = E
(0)A
M(X)
Mink.−→ δAM GMN(X) = G(0)MN(X) Mink.−→ δAMηMN
Y M(X) = Y(0)
M(X)
Mink.−→ δMa xa φ(x) = v
(2.3)
where X are the bulk coordinates, GMN(X) is the bulk metric, ηMN is the D-dimensional
1Uppercase Latin indices from the middle of the alphabet (M,N . . .) denote full bulk spacetime compo-
nents and uppercase Latin indices from the beginning of the alphabet (A,B . . .) denote bulk local Lorentz
frame indices. Lower case Latin indices from the beginning of the alphabet (a, b, . . .) denote brane compo-
nents and indices from the middle of the alphabet (m,n, . . .) denote extra dimensional components. Greek
indices are reserved for local Lorentz frame components from the beginning of the alphabet (α, β, . . .) for
the first four dimensional components and the middle (µ, ν, . . .) for extra-dimensional components.
2.1. General formulation 15
Minkowski metric, EAM(X) is the D-bein defined by
EAMηABE
B
N = GMN
EAMG
MN(X)EBN = η
AB , (2.4)
Y M(x) are the bulk coordinates of the brane, xa are coordinates on the brane and φ
is a scalar field which serves as Higgs (so it has a constant value on the brane but not
necessarily in the bulk). On the brane we can construct the induced metric
gab(x) = GMN(Y )∂aY
M∂bY
N (2.5)
and the induced vierbein
eαa (x) = R
α
A(Y )E
A
M(Y )∂aY
M , (2.6)
where R(Y ) is the solution of
RµB(Y )E
B
M(Y )∂aY
M = 0, ∀ µ, a (2.7)
and can be obtained through a perturbative series in terms of the fluctuation in Y . For
example in Minkowski spacetime, if we set Y to its background value in equation (2.3),
the solution is trivially RAB = δ
A
B and the induced vierbein is simply a projection of
the bulk D-bein. The same lowest order solution holds for a curved spacetime where
E(0)
µ
a(Y ) = 0 (see appendix A.3). With these ingredients it is straightforward to write
down the effective theory for the bulk and brane perturbations. For the pure gravity
sector, this is just the Lagrangian provided in Eq. (2.1). For brane degrees of freedom we
write down a 4-dimensional effective action
Sbrane =
∫
d4x
√
g
(
−f 4 + L0 + L 1
2
+ L1
)
(2.8)
where f is a brane tension and we have scalar, fermion and vector boson Lagrangians
respectively. The Lagrangians are obtained by covariantising the SM Lagrangian with
respect to gab by replacing all partial derivatives by space-time covariant derivatives ∇a,
i.e. for bosons using the metric connection (or Christoffel symbols)
Γabc =
1
2
gad (gbd,c + gcd,b − gbc,d) (2.9)
16 Chapter 2. Theories with extra dimensions
and for fermions the spin-connection
ωαβa =
1
2
gbce
[α
b e
β]
[c,a] +
1
4
gbcgdee
[α
b e
β]
d e
γ
[c,e]e
δ
aηγδ , (2.10)
where the brackets denote commutation over the indices involved. Furthermore gamma
matrices (γα) or Pauli matrices (σα) are lifted from the local Lorentz frame using the
inverse of the induced vierbein. Then for a fermion ψ and a vector V a
γa∂aψ → γa∇aψ = eaαγα
(
∂a +
1
8
ωαβa γ[αγβ]
)
ψ ≡ ∇ψ
∂aV
b → ∇aV b = ∂aV b + ΓbacV c . (2.11)
Note the generalised “slash” notation with the curved spacetime gamma matrices γa =
eaαγ
α. With this formalism, after the background has been fixed, we can study pertur-
bations (in particular quantum mechanically) by expanding all fields around their back-
ground values. This provides an effective Lagrangian with all the free parts for each degree
of freedom, plus interactions. In this construction the degrees of freedom available are:
• All the SM fields on the brane,
• The higher dimensional graviton HAB given by expanding [13]
GAB = G
(0)
AB +
1
M
D/2−1
D
HAB + . . . , (2.12)
which (due to gauge freedom) contains (n + 4)(n+ 1)/2 degrees of freedom,
• The fields ZM describing fluctuations of the brane around its background position
Y M(x) = Y M(0)(x) +
1
f 2
ZM (2.13)
where only n degrees of freedom are dynamical since there is a residual gauge free-
dom on the brane due to general coordinate invariance (only fluctuations transverse
to the brane are physical). From the brane point of view they are n scalar fields.
In this study we assume a fixed background and study perturbations for each field around
the background. Thus we neglect interactions between different perturbations or higher
order self-interactions. Nevertheless, when appropriate, we will assess the magnitudes of
the neglected terms to justify the approximations.
2.2. The ADD scenario 17
2.2 The ADD scenario
The main scenario we will consider is the large extra dimensions class of models of An-
toniadis, Arkani-Hamed, Dimopoulos and Dvali [19–21] (ADD). As mentioned in the
introduction, this was proposed as a solution to the hierarchy problem by assuming the
fundamental scale of electroweak physics to be the same as for gravitational physics. The
fundamental theory is assumed to be extra-dimensional and the fundamental Planck mass
MD is set to 1 TeV. Furthermore, the background spacetime manifold is chosen in the
formM4 ×N n where M4 corresponds to a 4-dimensional Minkowski manifold on which
the SM brane is placed, and N n corresponds to n extra-dimensional directions which are
typically chosen with some compactification. The compactification radii may or may not
be equal, and the curvature vanishes or is small (so that the extra-dimensional space is
close to being flat). The canonical example is N n = Tn a flat n-torus with common radii
R. In what follows, for simplicity, we consider this case, though some of the considerations
do not depend on this choice. Nevertheless this model is a good benchmark scenario to
obtain bounds from experiments. This choice of background metric corresponds to the
Minkowski limit of (2.3) with Xm compactified with radii R and Xa infinite in size. We
also assumed that the brane thickness is much smaller than R, typically at the Electroweak
scale 1/MD. We will see in section 2.5 how current experimental constraints impose this
condition.
In this scenario, at short distances (much smaller thanR), gravity is higher dimensional
with the generalised Newton force law
F (r) = GD
m1m2
r2+n
(2.14)
where the fundamental Newton constant is
GD =
1
Mˆ2+nD S(2+n)
, (2.15)
S(2+n) is the area of the (2+n)-sphere and m1, m2 are test masses. Equation (2.14) is ob-
tained using Gauss’ law or by working with the gravitational part of the Lagrangian (2.1)
coupled to test masses in the Newtonian limit [21]. On the other hand, for macro-
scopic objects on the brane (with sizes much larger than R), gravity will be effectively
4-dimensional. This can be seen by taking for example a macroscopic body on the brane.
With respect to variations on the brane, its gravitational field must be fairly constant
18 Chapter 2. Theories with extra dimensions
along the extra dimensions. Then, the gravitational action (2.1) can be integrated along
the extra dimensions to get an effective 4-dimensional action. The effective 4-dimensional
reduced Planck mass Mˆ4 obtained is just a rescaling of MˆD by an extra-dimensional
volume factor V(n) =
∫
dnX:
Mˆ24 = Mˆ
2+n
D V(n) . (2.16)
Similarly the Newton force at macroscopic distances is obtained either using Gauss’ law,
a Newtonian approximation, or the leading term in a Kaluza-Klein expansion of the
potential [21]
F (r) = G4
m1m2
r2
, (2.17)
which agrees with the usual 4-dimensional result.
Finally, let us define a characteristic length scale for the extra dimensions through
V(n) = (2πL)
n. The 2π factor is included, because the Kaluza-Klein (KK) mass splitting
for the toroidal compactification is L−1 so, for example, it describes better the deviations
from the Newtonian potential at short distances (see (2.27) below)2. The Planck mass
and length are
M2+nD ≡ (2π)nMˆ2+nD , LD ≡M−1D , (2.18)
so the following relation arises from (2.16)
M4
MD
=
(
L
LD
)n
2
. (2.19)
Now it becomes clear how the hierarchy problem is solved. The question of why the
4-dimensional Planck mass and the electroweak scale are separated by several orders of
magnitude is replaced by the dynamical problem of generating an extra dimensional space
with a large size/volume (large compared to the electroweak length LD).
2.3 The Randall-Sundrum scenario
In more general constructions, the extra dimensions do not have to be flat. An exam-
ple of such are the Randall-Sundrum models RS1 [22] and RS2 [23] which explain the
Planck/electroweak hierarchy by introducing a curved extra dimension. The question of
2The Kaluza-Klein expansion is obtained from a dimensional reduction of the theory by Fourier ex-
panding the fields in the Lagrangian along the extra-dimensions and integrate the the extra-dimensions
to get a four-dimensional theory with an infinite tower of fields (see for example equation (2.30).
2.3. The Randall-Sundrum scenario 19
how spacetime evolved to that state is again dynamical and remains open. The geometry
of the model is defined by the non-factorisable (4 + 1)-dimensional ground state metric
ds2 = e−2kyηµνdx
µdxν − dy2, (2.20)
so
EAM =


eky A = M = 0, . . . , 3
1 A =M = 4
0 otherwise
(2.21)
In these coordinates space-time looks like 4-dimensional Minkowski with the lengths
rescaled by e−ky as we vary y. It can be seen more explicitly that the geometry is actu-
ally that of the AdS5 space-time through a coordinate transformation ρ = e
ky/k. This
gives the usual form of the AdS5 metric ds
2 = (kρ)−1ηABdX
AdXB which emphasizes its
conformal flatness.
In RS1, spacetime is compactified on an S1/Z2 orbifold by letting y ∈ [−πR, πR] and
identifying y ↔ −y. Two branes are present, one at each of the fixed points y = 0 and
y = πR with (constant) positive and negative tensions respectively given by Vhidden =
−Vvisible = 6Mˆ3Dk. The bulk cosmological constant is Λ5 = 6Mˆ35k2. Using the formalism
in section 2.1 it is possible to show [22] that the Einstein field equations obtained from
the effective gravitational action (with brane matter perturbations set to zero)
S =
∫
d4x
∫ πR
−πR
dy
√
|G|
[
−Λ5 + 1
2
M35R− Vvisibleδ(y − πR)− Vhiddenδ(y)
]
(2.22)
are solved by the metric (2.20). Furthermore, the effective Planck scale at large distances
is again obtained from integration over the extra dimension and keeping the zero mode
massless graviton. Then
M24 =
M35
k
(
1− e−2πkR) , (2.23)
so for this large curvature case (i.e. kR large), the effective Planck mass depends little
on R. The way the hierarchy problem is solved in this case is through the exponential
scaling of the metric e−2πkR between the hidden brane, at y = 0, and the visible brane,
at y = πR, which rescales all masses on the hidden brane to the physical masses on the
visible brane. So in RS1, the fundamental mass scale is still at the usual four-dimensional
Planck scale3 and the hierarchy on the visible brane is generated through the exponential
3Similarly all fundamental scales in the model such as R and k can be chosen using such a scale.
20 Chapter 2. Theories with extra dimensions
scaling created by the mild hierarchy kR ∼ 10.
The RS2 scenario, on the other hand, can be seen as the limit R → +∞ of the RS1
scenario but with the visible brane placed at y = 0. So the extra dimension is semi-
infinite, y ∈ [0,+∞[ and the Standard Model fields are confined to the 3-brane at y = 0
with Y A = δAµ x
µ. The effective four dimensional gravitational action in this case is the
special limit of (2.23)
M24 =
M35
k
⇒
(
M5
M4
)2
=
k
M4
. (2.24)
L ≡ 1/k takes the role of an effective size for the infinite extra dimension which localises
the zero mode graviton on the brane, so (2.24) becomes
(
M4
M5
)2
=
L
L5
. (2.25)
This shows that the hierarchy is again due to the large typical (effective) size of the
infinite extra dimension, or equivalently the small curvature. In some sense RS2 is similar
to ADD in how it solves the hierarchy problem, since the fundamental Planck scale is set
to M5 = MEW , in contrast with RS1 where all scales are at M4. All the masses on the
brane are naturally of the order ofMEW . At very short distances, as long as the curvature
k is small enough, we can use a flat metric for RS2.
2.4 Further brane constructions and other scenarios
In general the situation could be more complicated, with a combination of curvature and
compactification. Furthermore, in a realistic situation we would not expect the SM fields
to be confined to an infinitely thin brane, but instead to propagate in a subspace with a
typical thickness along the extra dimensions.
So in a complicated scenario we could have several extra dimensions compactified with
a large radius and/or large curvature, combined with thin sub-spaces, with or without
large curvature, where the SM fields can propagate along some or several extra dimensions.
Such thin branes can be effectively described by the Universal Extra Dimensions (UED)
scenario of [30]. The upper bound on the extra dimensional width for a thin brane
is [31, 32] R ≃ (700GeV)−1. These bounds come mainly from considering electroweak
loop corrections to vector boson self energies, which receive contributions from virtual
Kaluza-Klein states, and comparing them to electron-positron collider data from the LEP
2.5. Experimental bounds on extra dimensions 21
experiments, and also rare decays such as B¯ → Xsγ .
This bound on R is many orders of magnitude below the scale needed in a large extra
dimensions scenario (for n . 15) to explain the size of the Planck mass in the spirit
of [19,20]. So for example, if the thin brane is imbedded in a thicker compactified space,
gravitational effects at distances larger than R can still be described by a large extra
dimensions (or small curvature) effective theory.
More elaborate setups have been proposed where different SM fields are placed on
different branes imbedded in a thicker brane of size R . 1 (TeV)−1 [33, 34]. Such con-
structions aim to suppress the effects of dangerous operators which, for example, might
allow fast proton decay. The solution is to confine the fermions to different branes, such
that for example the overlap of the wavefunction between quarks and leptons is small,
preventing fast proton decay. Furthermore such models are interesting on their own as an
explanation for the hierarchy of the fermion masses in the SM by adjusting the positions
of the sub-branes.
We should note that if we look at gravitational processes occurring at length scales
sufficiently above the maximum thickness of the thick brane where all the SM sub-branes
live, we can treat all SM fields as being effectively on a single brane. Furthermore, if at the
same time the process is well below the size of the large extra dimensions, then gravity is
stronger and higher dimensional. These considerations justify the use of a single brane first
approximation to study higher-dimensional strong trans-Planckian gravitational effects.
2.5 Experimental bounds on extra dimensions
In this section we provide a summary of the most important experimental results which
provide bounds on extra-dimensional models.
2.5.1 Laboratory bounds
Gravity at mesoscopic scales
At large lengths r just above L, the usual gravitational force law starts to get modified.
Bounds on the size of L can be found by looking at deviations from the Newtonian
force law at the threshold r & L, or by looking at dense systems where the inter-particle
distance is much smaller than L. The various limits for the classical gravitational effective
potential of the theories we consider are as follows [21, 23, 35]. At very short distances
22 Chapter 2. Theories with extra dimensions
there is a limit where the gravitational potential is higher dimensional for all the models.
V (r)
G4L−1m1m2
=
(
L
r
)n+1
SnΓ(n) + . . . , r ≪ L . (2.26)
This is true at distances which are much smaller than the compactification radius or the
curvature radius. At large distances, we obtain the Newtonian potentials with corrections:
V (r)ADD
G4L−1m1m2
=
∑
k1, ... ,kn
L
r
e−
|k|r
L ∼ L
r
+ 2n
L
r
e−
r
L + . . . , r ≫ L (2.27)
V (r)RS1
G4L−1m1m2
=
L
r
+
L
r
Φ21(0)e
−µ1r + . . . , r ≫ R (2.28)
V (r)RS2
G4L−1m1m2
=
L
r
+
2
3
(
L
r
)3
+ . . . , r ≫ L (2.29)
where k = (k1, . . . , kn) is a vector of integers, µ1 ∼ 1/R is the first Kaluza-Klein (KK)
mass for the RS1 scenario and Φ1(0) is the corresponding first KK wavefunction evaluated
at zero. The strongest experimental bounds for deviations from the Newton law come
from torsion-balance experiments [8, 9] and are summarised in table 2.1. Note that for
RS2, the first correction in the r ≫ L limit is less suppressed than the corresponding
correction for n = 1 ADD, so whenever ADD is excluded by these experiments, so will
the RS2 model with the same L. Note that at short distances RS2 behaves exactly the
same as n = 1 ADD. For RS1, since the compactification radius is of the same order as
the four dimensional Planck length the scenario is not excluded by these experiments.
Collider bounds
At sub-Planckian energies, the quantum fluctuations of the gravitational field can be
treated as linearised perturbations in Minkowski spacetime, to obtain a higher dimen-
sional graviton (see equation (2.12) for example). Since we are looking at linearised per-
turbations, the interaction with matter is given by the higher dimensional generalisation
of the Fierz-Pauli Lagrangian (1.21). Therefore, matter interacts linearly with the higher
dimensional graviton through its energy-momentum tensor. From the 4-dimensional point
of view on the brane, this can be expanded in a tower of Kaluza-Klein states with increas-
ingly larger masses controlled by the inverse compactification radius. We have already
seen a manifestation of the KK tower for example in the expression for the gravitational
2.5. Experimental bounds on extra dimensions 23
potential (2.27). In general (see [26, 36]), from the 4-dimensional perspective, it can be
shown that we have: a tower of massive spin-2 gravitons G
(i)
kkαβ that couple directly to
the energy momentum tensor of matter on the brane; a tower of spin-1 particles which do
not couple at linear order, and a tower of spin-0 particles which couple only to the trace
of the energy-momentum tensor (however this vanishes under the equations of motion at
tree level for a massless gauge theory). The effective Lagrangian for the relevant degrees
of freedom is
L(i) = −1
2
G
(i)
kk
αβ (
+m2(i)
)
G
(i)
kkαβ +
1
2
G
(i)
kk
α
α
(
+m2(i)
)
G
(i)
kk
β
β+
−G(i)kk
αβ
∂α∂βG
(i)
kk
γ
γ +G
(i)
kk
αβ
∂α∂γG
(i)
kk
γ
β −
1
Mˆ4
G
(i)
kk
αβ
Tαβ . (2.30)
The energy momentum tensor for a matter Lagrangian with a fermion field f coupled to a
gauge field Fα with strength Fαβ contains operators of the form (schematically) γα∂β f¯ f ,
γαf¯∂βf , γαFβ f¯f and F
2
αβ . Note that even though the coupling to each KK graviton is
controlled by the four dimensional Planck mass, the spacing between KK modes is very
small (∼ 1/R) so for processes at high energies there will be a large number of KK modes.
It can be shown that this enhances the cross-section to produce an overall coupling of
1/Mˆ2D instead of 1/Mˆ
2
4 .
With these interaction vertices, the main processes for collider searches are tree level
graviton production with a photon or a jet [37, 38] f f¯ → γG(i)kk or f f¯ → GG(i)kk, and tree
level dilepton or diphoton production through KK graviton [39,40] f f¯ → G(i)kk → f ′f¯ ′ and
f f¯ → G(i)kk → γγ, which were searched for at LEP and the Tevatron.
Similar considerations apply to the RS1 model, but however, the KK mass spacing
is now large due to the small extra dimension. Thus instead of a contribution from a
continuum of modes we have well separated resonances with TeV scale masses and a
coupling controlled by the TeV scale. The main effect is then resonant production of KK
gravitons which appear as peaks in the dilepton or diphoton invariant mass spectra [41].
The most restrictive bounds are summarised in table 2.1.
2.5.2 Astrophysics and cosmology
Another source of constraints, comes from astrophysical systems, where very high energies
are reached. In that case, KK gravitons provide a new competing channel in high energy
collisions of SM particles, so they may be produced and decay, changing the dynamics.
24 Chapter 2. Theories with extra dimensions
Note however, that bounds from these systems do not apply to RS1, since the first KK
resonance mass is supposed to be at the TeV scale.
Supernovae
The most restrictive bounds come from Supernovae [42]. In these systems, a large number
of KK gravitons can be produced in the new channel nn→ nnG(i)kk (n denotes a neutron).
The KK gravitons are either gravitationally trapped in a halo around the neutron star
formed after the supernova explosion or emitted as invisible energy. This new competing
channel reduces the neutrino signal emitted in the explosion. Furthermore, the cooling
of the neutron star is slower, because the KK gravitons trapped in the halo are an extra
heat source through their decay, G
(i)
kk → γγ, into two photons which can fall into the star.
On the other hand if the photons are emitted outwards, they contribute to an enhanced
gamma ray flux from the neutron star which would be visible.
Finally, the diffuse γ-ray background would be enhanced by the photons emitted in the
decay of KK gravitons produced in all the Supernovae formed throughout the cosmological
evolution [42].
Early Universe KK gravitons
Similarly, in the early Universe, KK gravitons can be produced. The dominant channels
are 2γ → G(i)kk, νν¯ → G(i)kk, e+e− → G(i)kk. This would lead to unacceptably large amounts
of KK gravitons surviving the cosmological evolution, which would overclose the Uni-
verse [43]. Note that even though the coupling of each KK graviton to brane particles is
suppressed by 1/Mˆ4, it can be shown that the lifetimes involved are of the order of the
Hubble time [43].
A further restriction [43] comes from the possible decay of the KK gravitons into
photons which would distort the diffuse γ-rays background observed at present.
2.5.3 Summary of the bounds
Table 2.1 provides a summary of the bounds on MD from various sources. The general
conclusion is that a Planck mass at the TeV scale for extra-dimensional models with n = 1
is definitely excluded. However, the laboratory bounds still allow n ≥ 2, whereas cosmol-
ogy and astrophysics prefer n > 3. Note however that the modelling of astrophysical and
cosmological systems often contains model dependent assumptions so the values should be
2.5. Experimental bounds on extra dimensions 25
n = D − 4 1 2 3 4 5 6 7
Newton law de-
viations [8, 9]
> 105 8.0 < 10−3 < 10−3 < 10−3 < 10−3 < 10−3
KK gravitons at
colliders [37–40]
— 1.4 1.1 1.0 0.98 0.94 —
KK gravitons in
Supernovae [42]
> 105 1.7× 103 77 9.4 2.1 0.67 0.28
KK gravitons in
the early Uni-
verse [43]
— 68− 155 5− 28 0.8− 7.7 0.2− 3.2 — —
Table 2.1: This table summarises the most restrictive lower bounds for MD (in TeV) for
various number of extra dimensions. In the last row an interval of values is given
for various choices of reheating maximum temperature.
taken as indicative (for example, for the early Universe production of KK gravitons, the
bounds vary strongly with the assumed maximum temperature reached during reheating).
Note also that the RS1 scenario is not excluded but instead there is a lower bound for
the first KK resonance to have a mass above 250 GeV [41].
26 Chapter 2. Theories with extra dimensions
Chapter 3
Strong gravity I: Charged rotating
black holes
We have seen in the previous chapters how physics can change at short distances and
how gravity can become stronger and higher dimensional. In particular, if we look at the
higher dimensional Newton force between test masses, now the coupling is controlled by
the higher dimensional Planck mass, i.e. E/M4 → E/MD. This means that gravity not
only becomes strong, but for energies above MD it is dominant compared to all other
SM interactions. In the remainder of this work we are mainly interested in such strong
trans-Planckian processes in D-dimensional general relativity.
A crucial question is to understand what occurs in such high energy processes. We are
interested in processes where a collision occurs between two SM particles with a centre
of mass energy
√
s ≫ MD. Well established results in classical general relativity say
that when a certain amount of mass/energy is trapped in a small compact volume a
region is formed from which nothing can escape. This is usually referred to as the hoop
conjecture [44] and it is based on intuition from the simplest black hole solution – the static
spherically symmetric Schwarzschild black hole. The latter has a typical size given by the
Schwarzschild radius which grows with the trapped mass. The hoop conjecture says that
if a certain amount of mass/energy is trapped within a hoop defined by its Schwarzschild
radius, a black hole is formed. This argument was first invoked in [24, 45, 46] to suggest
that a black hole is produced in trans-Planckian collisions with a small impact parameter
in extra dimensional models. An interesting feature of the Schwarzschild radius is that it
grows with
√
s. This implies that the gravitational interaction can be treated classically
since it occurs at large distances. This is an example of the ultraviolet/infrared connection
27
28 Chapter 3. Strong gravity I: Charged rotating black holes
in theories of gravity [24] (i.e. high energies are associated with large lengths).
In the next chapters we consider collisions occurring on the background models pre-
sented in chapter 2, at scales of the order & 1 TeV. So even though the extra dimensions
may be compactified, the black hole so formed can effectively be treated as a (4 + n)-
dimensional object because the compactification radius is very large. In general the black
hole is characterized by a mass M , angular momentum J and some Standard Model
charges inherited from the colliding particles. In particular, for proton-proton collisions,
the black hole can have an electric charge Q and colour charges from the colliding quarks
and/or gluons.
Before examining in detail the process of black hole production in chapter 4, in this
chapter we present an overview of black hole solutions. This is useful for several reasons.
Black hole solutions are good to model the gravitational field of realistic concentrations
of matter. An example of such is again the Schwarzschild metric which gives the exact
behaviour of the external gravitational field for a spherically symmetric star (Birkhoff’s
theorem). Even for rotating bodies, we would expect black hole metrics with rotation
to give a good approximation for the gravitational field away from the massive body.
Furthermore, since we have two particles in the collision interacting predominantly grav-
itationally, exact black hole solutions will be useful to model their interaction. Finally,
our final state after the collision contains a black hole, so a black hole solution with the
correct properties is necessary.
The outline of the chapter is the following. First we provide an overview of some useful
well known black hole solutions in (4 + n)-dimensions. In section 3.2 the approximate
brane metric for a brane charged rotating black hole (constructed in [1]) is described, and
finally in section 3.3 some geometrical properties of the solutions are presented through
the study of their geodesics.
3.1 Black holes in general relativity
Black holes are solutions of the Einstein field equations (coupled or not to matter) con-
taining a region which is invisible to an outside observer. The boundary of such a region is
called the event horizon. They are associated with a high concentration of matter/energy
in a small volume which creates a gravitational field configuration such that no signal can
escape. The first suggestion of such dense objects goes back to Michell and Laplace who
suggested dark stars within Newtonian theory. The first solutions in Einstein’s theory
3.1. Black holes in general relativity 29
of gravity appeared shortly after it was proposed. In this section we present some ba-
sic definitions regarding the causal structure of spacetimes and some specific black hole
metrics.
3.1.1 Definitions and properties
A rigorous definition of a black hole relies on the global causal structure of the spacetime.
For all cases of interest, we take the spacetime to be asymptotically flat. This is because
we are considering black hole production occurring on a flat Minkowski background.
The causal structure of a given spacetime can be understood schematically by con-
structing its Penrose-Carter conformal diagram. Intuitively, for example in Minkowski
spacetime, this is obtained by changing variables such that the radial and time coordi-
nates are squeezed down to a finite range and the metric is rescaled by a conformal factor.
Then, by omitting the angular part, a metric with the same causal structure is obtained.
This is true because null rays are invariant under conformal transformations. So if we
consider Minkowski spacetime in D-dimensions expressed in spherical coordinates1
ds2 = dt2 − dr2 − r2dΩ2D−2 (3.1)
and perform the transformation
ds2 → ds¯2 = 4 cos2 (U
2
)
cos2
(
V
2
)
ds2 = dUdV − sin2 (U−V
2
)
dΩ2D−2 (3.2)
where
t∓ r = tan (1
2
{U ,V}) −π ≤ U ≤ V ≤ π , (3.3)
the diagram in figure 3.1 is obtained (by suppressing the angular part). Null rays are
represented by straight lines at ±π/4, so at each point we get two perpendicular null rays
which define the light cone. The horizontal curves of constant t (regularly spaced in t)
map to spacelike geodesics and they all converge to spacelike infinity which is represented
by the point I0. On the other hand, all timelike curves (regularly spaced in r) originate
from timelike past infinity I− and terminate at future timelike infinity I+. Similarly, all
null curves originate at J − and terminate at J +. In this diagram the left-hand side
corresponds to extending r to negative values, so if for physical reasons we decide to allow
1dΩ2D−2 is the line element on a (D − 2)-sphere.
30 Chapter 3. Strong gravity I: Charged rotating black holes
.
.
I+
J + J +
I−
J − J −
I0 I0
Figure 3.1: Penrose-Carter diagram for Minkowski spacetime.
only r positive, the corresponding diagram is obtain by reflecting the left-hand side onto
the right-hand part of the diagram (or equivalently identifying the points which are mirror
symmetric with respect to the vertical axis).
We are interested in spacetimes which are asymptotically flat in the past before the
black hole is formed, and at large spacelike distances after formation. So when considering
Penrose diagrams for black hole spacetimes, we will always have a part which maps onto
a section of Minkowski with I0, I± and J ±. Then a natural definition of black hole is a
region of spacetime from which no timelike or null rays can escape to reach I+ and J +
respectively. The event horizon is the boundary of such a region which must be a null
surface. This definition is global in nature and it requires a knowledge of the full history
of the spacetime (or at least up to very late times after formation).
3.1.2 Some solutions in four dimensions
The Kerr-Newman family
Various 4-dimensional analytic solutions of the Einstein field equations coupled to mat-
ter have been constructed throughout the years to model systems with different matter
content. These exact solutions are useful to understand fundamental properties of the
theory and, more pragmatically, to model realistic systems. Typically they must have a
high degree of (at least approximate) symmetry so that the analytic system is completely
integrable. Various solutions have been constructed which are physically relevant in cos-
3.1. Black holes in general relativity 31
mological and astrophysical contexts (see for example chapter 5 of [47]). In this study,
we are interested in solutions describing localized concentrations of matter, in particular
black holes.
In four dimensions, the most general stationary axisymmetric solutions of the Einstein-
Maxwell equations are the Kerr-Newman family of metrics and Maxwell potentials (see
for example chapter 3.6 of [48]). This can be expressed in Boyer-Lindquist {t, r, θ, φ}
coordinates
ds2 =
(
1− r
2 + a2 −∆
Σ
)
dt2 +
(r2 + a2 −∆) 2a sin2 θ
Σ
dtdφ− Σ
∆
dr2−
− Σdθ2 −
(
r2 + a2 +
a2 (r2 + a2 −∆) sin2 θ
Σ
)
sin2 θdφ2 , (3.4)
where
∆ = r2 + a2 +Q2 − 2Mr , Σ = r2 + a2 cos2 θ , (3.5)
and
Aadx
a = −Q r
Σ
(
dt− a sin2 θdφ) . (3.6)
This spacetime is asymptotically flat, which can be seen more explicitly by separating the
metric into two contributions and making a change of variables


x =
√
r2 + a2 sin θ cosφ
y =
√
r2 + a2 sin θ sinφ
z = r cos θ
, (3.7)
which define a spheroid (through their relation to r) according to
x2 + y2
r2 + a2
+
z2
r2
= 1 . (3.8)
Then (3.4) takes the form
ds2 = dt2 − dx2 − dy2 − dz2
− 2Mr
3 −Q2r2
r4 + a2z2

(dt+ aydx− axdy
r2 + a2
)2
+
1
∆
(
xdx+ ydy +
√
r2 + a2
r
zdz
)2 , (3.9)
where the last term vanishes in the limit r → +∞. This spacetime contains two horizons
32 Chapter 3. Strong gravity I: Charged rotating black holes
defined by the roots of ∆ = 0. In the form (3.4) or (3.9), the metric has coordinate
singularities at the horizons which are removable through a change of coordinates.
This solution describes a black hole with a mass M , angular momentum J = Ma and
electric charge Q. The physical interpretation of these parameters can be made more
transparent by comparing the first few asymptotic terms of the metric and the Maxwell
strength when r → +∞, with those for an isolated non-relativistic weakly gravitating
system [49]. The general results are expressed in a coordinate independent way through
the Komar surface integrals on a sphere at r → +∞ and constant t
M =
1
4π
∮
∇akbtdSab
J =
1
8π
∮
∇akbφdSab (3.10)
Q =
1
4π
∮
FabdSab ,
where kt = ∂t and kφ = ∂φ are the asymptotic killing vectors associated with time trans-
lations and rotation symmetry respectively, and F is the electromagnetic field strength.
dSab = dS n
t
[an
r
b] is the surface element where n
t and nr are the time-like and radial nor-
mals suitably anti-symmetrised. Furthermore, it is clear from (3.8) that the surfaces of
constant r are spheroids with oblateness a2/r2, so the black hole horizon is squashed.
A maximal extension of the spacetime can be obtained using several coverings. We will
not present the technical details, but instead summarize the main properties by analysing
the Penrose-Carter diagrams of figure 3.2 (based on the discussion in [47]).
The simplest case is the Schwarzschild black hole (a = Q = 0) represented in the
bottom left diagram. This is the unique static, spherically symmetric solution which
represents the exterior field of a mass M . Its maximal extension contains a white hole
(region-II’, inside r < rH) with a singularity, at r = 0, to the past of the (disconnected)
asymptotically flat regions I and I’. For a realistic black hole, formed from collapsing
matter, the white hole and the second asymptotically flat region-I’ are not present. Note
that the singularity at r = 0 cannot be removed through a change of coordinates. This
can be seen by evaluating the curvature scalar invariant RabcdRabcd ∼ 1/r6. The black
hole (region-II inside r < rH) contains a singularity at r = 0, in the future of regions I
and I’. It can be reached by any causal observer, who once inside it will necessarily hit the
singularity. This property is related to the fact that any point inside region-II represents a
closed trapped surface, i.e. any future directed light ray propagates to a region of smaller
3.1. Black holes in general relativity 33
Schwarzschild
future singularity
(r=0) I+
I−
I+
I−
J+)
r = rH
)
r = const.
i
t = const.
J−
J+ q
r = rH
-r = const.
-t = const.
J−
I0II’
II
II’
I0
past singularity
(r=0)
Reissner-Nordstro¨m Q2 < M2
singularity (r=0)singularity (r=0)
I+
I−
I+
I−
J+)
r = rH+
)
r = const.
i
t = const.
J−
J+ q
r = rH+
-r = const.
-t = const.
J−
I0II’
II
II’
IIIIII’
)
r = rH−
)
r = const.
i
t = const.
I0
I∓
I∓
I∓
I∓
J+(r=+∞))
r = rH+
)
r = const.
i
t = const.
J−(r=+∞)
(r=+∞)J+ q
r = rH+
-r = const.
-t = const.
(r=+∞)J−
I0II’
II
II’
I0
I∓I∓
J+(r=−∞))
r = rH−
9
r = 0
i
t = const.
J−(r=−∞)
(r=−∞)J+ q
r = rH−
-r = const
-t = const
(r=−∞)J−
I0IIIIII’
II’
I0
Kerr a2 < M2
Figure 3.2: Penrose-Carter diagrams for some cases of the Kerr-Newman family. We show
three qualitatively different cases of the maximal analytic extensions with some
causal regions and their boundaries (such as horizons and singularities) indicated.
In all diagrams region-I corresponds to the exterior region where an asymptotic
observer propagates. The right diagram is for a direction along the axis of rotation
of the black hole.
34 Chapter 3. Strong gravity I: Charged rotating black holes
r (note that the horizontal curves in region-II are of constant r – decreasing upwards).
Turning on the charge parameter Q, we obtain the Reissner-Nordstro¨m black hole.
There are three qualitatively different cases. If Q2 > M2 there are no horizons and the
singularity at r = 0 is naked. So the Penrose-Carter diagram is similar to Minkowksi
(see figure 3.1) except that a singularity is placed at r = 0. The diagram for Q2 < M2
is the top left of figure 3.2. Again, it contains a white hole in the past (region-II’)
which is absent in a realistic collapse. Regions I and I’ are similar to the Schwarzschild
case. Regarding horizons, now there is an outer horizon r+ (causally similar to the
horizon of the Schwarzschild black hole) which separates the black hole region-II from
the outside observers in the asymptotically flat region. The main difference is the second
horizon at a smaller radius r = r−, which introduces two new causal regions-III and
III’. In those regions, the points no longer represent closed trapped surfaces, so a causal
observer can travel back to a larger r and emerge in a new white hole region. The latter
communicates with another asymptotically flat region (repeating the bottom part of the
diagram). Thus a timelike observer can avoid hitting the (timelike) singularity. For the
special case Q2 = M2 the horizons are degenerate and regions II and II’ do not exist.
Finally, if we set Q = 0 and turn on a, we obtain the Kerr metric2. Since the metric
is no longer spherically symmetric, now the singularity is a ring. This is particularly
intuitive in spheroidal coordinates where the ring is defined by x2+ y2 = a2. An observer
travelling along the axis passes through the disk at x2 + y2 < a2 without hitting the
singularity. The direction along the axis was used to draw the conformal diagram in
figure 3.2 (representing the maximal analytic extension). Regions I, I’, II and II’ are
similar to those of the Reissner-Nordstro¨m black hole (again we have an outer horizon
and an inner horizon). Once the observer crosses the inner horizon, he/she can proceed
to another white hole region-II’ as before, or go through r = 0 to negative r into a new
asymptotically flat region (III or III’) without hitting the singularity. More relevant for
an asymptotic physical observer in region-I is the so called ergoregion. This is the region
outside the black hole where the Killing vector kt (which defines the asymptotic static
observers) becomes spacelike. From the point of view of an outside observer, no causal
observer in the ergoregion can travel along an orbit of kt and remain at rest with respect
to infinity. Locally, it is still possible to find a timelike Killing vector by forming a linear
combination of kt and kφ, so the solution is stationary, though not with respect to infinity.
2Note that the causal structure of the full Kerr-Newman metric is very similar.
3.1. Black holes in general relativity 35
3.1.3 Exact solutions in higher dimensions
The Myers-Perry family
Since we are interested in theories with extra dimensions, we need some generalisations
of the 4-dimensional metrics in the previous section. The most general metric describing
a higher dimensional rotating black hole was found by Myers and Perry [50]. While in
four dimensions the angular momentum is described by a spatial pseudo-vector which
defines the axis of rotation, in (4 + n) dimensions there are more angular momentum
generators than spatial directions so in general we will have several axes of rotation.
More rigorously, if we consider massive representations of the Lorentz group in (4 + n)
dimensions, SO(3+n, 1), they can be labelled by the mass associated with the eigenvalue
of the Casimir operator constructed from the generators of translations P aPa = m
2 and
the eigenvalues of the [(3+n)/2] Casimirs3 of the little group SO(3+n) [50]. The latter are
the angular momentum invariants. In our study, we are mostly interested in the special
case of a single rotation axis, because we will study particles colliding on the brane so
their angular momentum is constrained. In Boyer-Lindquist coordinates the metric is
ds2 =
(
1− r
2 + a2 −∆
Σ
)
dt2 +
(r2 + a2 −∆) 2a sin2 θ
Σ
dtdφ− Σ
∆
dr2−
− Σdθ2 −
(
r2 + a2 +
a2 (r2 + a2 −∆) sin2 θ
Σ
)
sin2 θdφ2 − r2 cos2 θdΩ2n , (3.11)
where
∆ = r2 + a2 − µ¯
rn−1
, Σ = r2 + a2 cos2 θ . (3.12)
Note that {t, r, θ, φ} have the same properties as in (3.4). The only two independent
parameters in (3.11) are {µ¯, a} which are related to the physical mass M and angular
momentum J of the black hole obtained using the higher dimensional generalisations of
the Komar integrals (3.10)
M
MD
=
(n + 2)
2
S2+n(2π)
−n(n+1)
n+2 Mn+1D µ¯ , (3.13)
J = S2+n(2π)
−n(n+1)
n+2 Mn+2D a µ¯ =
2
n + 2
Ma . (3.14)
3The brackets denote the integer part.
36 Chapter 3. Strong gravity I: Charged rotating black holes
Furthermore, we can switch to a third pair of parameters, closely related to the geometrical
properties of the black hole: {rH , a∗}. The first parameter is defined by the location of the
horizon of the black hole at the largest positive root of ∆(rH) = 0. rH is directly related
to the surface curvature of the horizon and thus (in some sense) has a frame independent
meaning. The second is a∗ = a/rH which is the oblateness of the (spheroidal) horizon
(see equation (3.8)).
Brane tension and split branes
In section 2.1 we saw that in general we can place the SM fields on different branes (split
brane scenario) which may have a non-zero tension. When considering a black hole formed
in a SM brane these effects could in principle be important. The simplest case would be
a black hole formed on a single tense brane. For the case of various split branes which are
separated by small distances compared to the black hole size, from the gravitational point
of view, this should be similar to a single effective brane. As pointed out in section 2.4,
this is the case for the scenario we are considering where the black holes formed have
masses well above 1TeV. Thus the typical Schwarzschild radius is above 1 ∼ 2 (TeV)−1
and the minimum diameter should be 3 ∼ 4 (TeV)−1, which is already well above the
upper bound on the width of the thick brane discussed in section 2.4.
However, even for the case of a single tense brane, the relevant black hole solutions
known in the literature tend to be limited to the case of codimension-2 branes in six
dimensions [51–55]. An example is the singly rotating black hole background with tension,
where the main conclusion is that the brane projected metric remains unchanged up to
a rescaling of the Planck mass [51,53,54]. Thus processes occurring on the brane remain
qualitatively the same (with an effective brane Planck mass) and only processes in the
bulk are qualitatively different. These observations, together with the fact that there are
a large number of brane degrees of freedom, justify neglecting the effect of brane tension
in 6-dimensions, for study of the Hawking evaporation in chapter 5 (note that D = 6 is
anyway disfavoured from the bounds in [42,56,57]). For the phenomenologically favoured
cases of codimension larger than 2, there are virtually no detailed studies of brane tension
effects. We may hope that a similar effect of rescaling of the Planck mass will occur for
brane fields, in which case our model does not need to be adapted, but further work is
required to understand if such an assumption holds.
3.1. Black holes in general relativity 37
Boosted solutions
So far we have been looking at solutions that describe the gravitational field of a massive
object in its rest frame. In the next chapter we will consider the gravitational interaction
between two colliding particles. The gravitational field of a massive particle with no
spin and no charge can be obtained from boosting the Schwarzschild metric in isotropic
coordinates, which describes exactly the exterior gravitational field. This was first done
in four dimensions by Aichelburg and Sexl [58]. In general D-dimensions the result is
ds2 = (1 + A)4/(D−3)
(
dt2 − dx2 − dx2T
)−
[
(1 + A)4/(D−3) −
(
1−A
1 + A
)2]
(dt− vdx)2
(1− v2)
(3.15)
with
A ≡ GDE
√
1− v2
2
[(
x− vt√
1− v2
)2
+ ρ2
](D−3)/2 , ρ2 ≡
D∑
i=2
x2i , dx
2
T ≡
D∑
i=2
dx2i (3.16)
where the boost was along the ∂x direction with velocity v to give the particle an energy
E. It is easy to check that off the plane x = vt, for distances
ρ
G
1/(D−2)
D
≫
(
G
1/(D−2)
D E/2
)1/(D−3) (
1− v2)2/(D−3) , (3.17)
or
x− vt
G
1/(D−2)
D
≫
(
G
1/(D−2)
D E/2
)1/(D−3) (
1− v2)1+2/(D−3) ≡ ǫ , (3.18)
A≪ 1 so the metric becomes flat. Thus close to the speed of light the region of high cur-
vature is squeezed on the transverse plane x = vt with a width ǫ. This can be made more
explicit by taking the special limit of an infinite boost v → 1 and M → 0 while keeping
the energy of the particle fixed. The result describes the gravitational field of a massless
particle (or the approximate gravitational field of an ultra-relativistic particle) [59]
ds2 = du¯dv¯ − dρ¯2 − ρ¯2dΩ¯2D−3 − Φ(ρ¯)δ(u¯)du¯2 (3.19)
Φ(ρ¯) =


−8GDE log ρ¯ , D = 4
16πGDE
SD−3(D − 4)ρ¯D−4 , D > 4
, (3.20)
38 Chapter 3. Strong gravity I: Charged rotating black holes
where we have used a radial coordinate ρ¯ = ρ, the angular coordinates Ω¯D−3 on the
transverse hyper-plane and the null coordinates u¯ and v¯. Now it is clear that the spacetime
before and after the gravitational shock is flat. On the plane u¯ = 0, we have a singularity
which introduces a discontinuity on the geodesics and their tangent vectors. This can be
fixed by changing coordinates [59]
u¯ = u
v¯ = v + θ(u)Φ +
uθ(u)(∂rΦ)
2
4
ρ¯ = r
(
1− uθ(u)
2
∂rΦ
)
dΩ¯2D−3 = dΩ
2
D−3 (3.21)
where θ(u) is the Heaviside step function, to obtain
ds2 = dudv −
[
1 + (D − 3) u
rD−2
θ(u)
]2
dr2 − r2
[
1− u
rD−2
θ(u)
]2
dΩ2D−3 . (3.22)
3.2 Construction of a brane charged background
In the previous section we have described some solutions which can be used to model
realistic higher-dimensional brane black holes. However, we have focused mostly on vac-
uum solutions in (4 +n) dimensions or solutions with a Maxwell field in four dimensions.
In a more general scenario we want to study objects which are higher dimensional but
also contain several types of charges and intrinsic spin. Furthermore, given that we are
working with a brane world scenario, we may need to confine some of the matter fields to
the brane. For full consistency this would require finding exact solutions where the gravi-
tational field of the brane is also taken into account (an example of such a self-consistent
solution is the Randall-Sundrum model where the brane tensions are included). This is
generically difficult.
In what follows we adopt a simpler approach by assuming that the gravitational field
of the matter on the brane is a small perturbation. Then, the leading contribution to
the gravitational field is sourced by the mass trapped inside the black hole, and the other
contributions from matter fields propagating outside the black hole are treated as small
perturbations. This can be shown to be consistent by performing some estimates using the
formalism in section 2.1. Consider the higher-dimensional black hole background (3.11)
3.2. Construction of a brane charged background 39
and assume a brane is placed at a constant value of the higher dimensional coordinates
of the n-sphere. Clearly the line element takes the form
GMNdX
MdXN = GabdX
adXb +GmndX
mdXn (3.23)
with no cross terms between the first four coordinates and the extra dimensional coordi-
nates. Thus we can consider each line element independently and construct respectively a
vierbein eαb and an n-bein e
µ
m. It then follows that a D-bein for the full spacetime is given
by the combination of the two, i.e. Eαb = e
α
b and E
µ
m = e
µ
m with the extra constraints
Eµa = E
α
m = 0. The induced metric and induced vierbein on the brane is a simple pro-
jection of the corresponding higher-dimensional quantity. Thus the combination of the
Einstein-Hilbert action with the brane action for a matter perturbation is
S =
∫
dDX
√
|G|
(
1
2
MˆD−2D RD + Lbrane
δ(n)(Xm)√|h|
)
(3.24)
where hmn = Gmn. The Einstein equations become
GMN = 1
M2+nD
Tabδ
a
Mδ
b
N
δ(n)(Xm)√|h| (3.25)
where we have assumed that δgab ∼ δGab, δSmatter/δgmn ∼ δSmatter/δgan ∼ 0. In a
realistic situation, the δ function would be replaced by a smeared function with support
on the brane. From the constraints in section 2.4 we can estimate such a factor by
replacing it with a typical inverse volume for the thin brane ∼ 1/Rn, with R . 1 TeV−1.
Then
GMN ∼ 1
M2D
Tabδ
a
Mδ
b
N
(
LD
R
)n
. (3.26)
In the absence of matter, the Einstein tensor for the background we are considering
vanishes. However, we know that the cancellation occurs from a linear combination of
components of the Riemann tensor which are non-zero, so we can compare them with the
perturbation. Furthermore the components of the Riemann tensor only give a meaningful
magnitude of the tidal forces felt by a local observer in an orthonormal frame. This was
done in [50]. For simplicity we set the rotation to zero and obtain
RMNOP ∼ µ
r3+n
∼ M
M2+nD r
3+n
. (3.27)
40 Chapter 3. Strong gravity I: Charged rotating black holes
This dependence agrees for example with the
√
RabcdRabcd curvature invariant for the
Schwarzschild black hole. Finally for the energy momentum tensor, taking the example
of the Reissner-Nordstro¨m black hole we get (we will see this again in section 3.2.2)
Tab ∼ Q
2
r4
. (3.28)
Inserting this estimate on the right hand side of equation (3.26) and requiring it is small
compared to the Riemann tensor component (3.26) gives the condition
M
MD
≫ Q2
(
r
LD
)n−1(
LD
R
)n
. (3.29)
So provided Q is small and the black hole mass is large (as long as R is not too small
compared to LD and we are looking at distances not too large compared to LD) the
electromagnetic field can be treated as a perturbation. Similarly we can include the
brane tension term in the brane action and estimate
Tab ∼ f 4 , (3.30)
so the corresponding condition is
M
MD
≫
(
f
MD
)4(
r
LD
)3+n(
LD
R
)n
, (3.31)
which holds similarly for trans-Planckian black holes as long as the tension is small com-
pared to MD.
In the remainder of this section, we will focus on a Maxwell field confined to the 4-
dimensional brane rather than a higher dimensional Maxwell field propagating in the full
bulk. The background brane metric used in the construction is the projection of (3.11).
For brane degrees of freedom, this suffices as an effective metric to describe how the
gravitational field affects them. In the remainder of this section we use Planck units
MD = 1 unless stated otherwise.
3.2.1 The brane Maxwell field as a perturbation
The simplest perturbation of the background field, which is relevant for the phenomeno-
logical scenario we are considering, is a classical abelian gauge field. This describes elec-
3.2. Construction of a brane charged background 41
tromagnetism and it is relevant because several Standard Model fields are electrically
charged.
We want to solve Maxwell’s equations for the vector potential Aa using the met-
ric (3.11) as the gravitational background. The combined gravitational plus electromag-
netic background can then be coupled to other fields to study the quantum propagation
of the corresponding perturbations on the background. In chapter 5 we will be interested
in the Hawking effect.
We want the solution to retain the symmetries of the effective four dimensional back-
ground. This has exactly the same symmetries as the Kerr-Newman solution so we use
the type of ansatz in equation (3.6)
Aadx
a = −Q r
Σ
(
dt− a sin2 θdφ) , (3.32)
where Aa is the vector potential. It can be checked that (3.32) solves the sourceless
Maxwell equations on the brane
DaA
ab =
1√|g|∂a
(√
|g|Aab
)
= 0 . (3.33)
This result follows because
√|g| = Σsin θ is exactly the same as for the Kerr-Newman
metric. In addition the identities
D[aA bc] = 0 (3.34)
where the brackets denote cyclic permutation of indices, are also satisfied. Note how the
modified r1−n term in ∆, which gives a 1/r2+n gravitational force law away from the
black hole, does not affect the stationary brane Maxwell field. Thus a brane charged
particle propagating outside the black hole feels an electric force that scales like 1/r2 and
a gravitational force that scales like 1/r2+n.
Gauss’ theorem applied to equation (3.33) allows the matching of Q to the physical
charge of the black hole4
∫
DaA
abdΣb = 4π
∫
dΣc
√
|g|Jc ⇒ Q =
∫
d3x
√
|g|J0 . (3.35)
Here we have integrated over spatial hypersurfaces of constant t with normal hypervolume
dΣb = d
3xδ0b , applyed Gauss’ theorem and integrated the left side on a sphere at r → +∞.
4Q coincides with the Komar charge defined in (3.10).
42 Chapter 3. Strong gravity I: Charged rotating black holes
3.2.2 Comments on backreaction
From the equivalence principle, we know that any SM field should source the right hand
side of Einstein’s equations through its energy-momentum tensor. In other words all fields
gravitate and generate a correction to the metric.
In section 3.2.1 we have found a consistent solution of the brane Maxwell field equations
on the background of a Myers-Perry projected black hole. Now we try to find self-
consistent corrections to the gravitational field. Ideally we would have to solve the coupled
Einstein/brane-Maxwell equations in the full (4+n)-dimensional space with the Maxwell
field confined to the brane, so it would involve finding a specific mechanism to confine
the field. This is too difficult in general and we are mostly interested in justifying the
smallness of the backreaction. Thus, we analyse the Einstein equations and introduce a
self-consistent ansatz to approximate corrections on the brane.
It can be checked, by direct computation, that the brane Einstein tensor for the
background projected metric is not vacuum like. This is not surprising since the actual
vacuum black hole solution lives in (4 + n)-dimensions. The non-zero components are
G(0)rr =
nµ¯r1−n
Σ2
G(0)θθ = −
G(0)rr
2r2
[
(n + 1)r2 + (n− 1)a2 cos2 θ]
G(0)φφ = −
G(0)rr
2r2Σ
[
(n + 1)r4 + (n+ 3)r2a2 + (n− 3)r2a2 cos2 θ + (n− 1)a4 cos2 θ]
G(0)tt =
G(0)rr
2r2Σ
[
2r4 + (n+ 3)r2a2 − (n + 1)r2a2 cos2 θ + (n− 1)a4 cos2 θ sin2 θ]
G(0)φt =
aG(0)rr
2r2Σ
[
(n+ 1)r2 + (n− 1)a2 cos2 θ]
G(0)tφ = −G(0)
φ
t Σ0 sin
2 θ , (3.36)
where Σ0 = r
2+a2. So from the brane point of view, an observer performing gravitational
measurements sees a black hole space-time together with an effective fluid due to the
embedding into the extra dimensions.
Before suggesting corrections to the metric it is useful to note some properties. We
expect such a corrected metric to reduce to the projected metric (3.11) in the Q = 0
limit and to the Kerr-Newman solution when n = 0. Furthermore, it should exhibit the
same symmetries as the Kerr-Newman metric if we want the Maxwell field to be of the
same form as in equation (3.32). Compared to the Kerr (Q = 0) limit, the Kerr-Newman
3.2. Construction of a brane charged background 43
metric is modified by a shift of the mass term µ¯r in ∆ to µ¯r−Q2. The term µ¯r is related
to the gravitational potential which in the chargeless (4 + n)-dimensional case is simply
replaced by µ¯r1−n. Similarly, we adopt an ansatz where µ¯r1−n is shifted to µ¯r1−n − Q2
(or equivalently ∆ → ∆ + Q2)5. Then the effective brane metric ansatz takes the same
form as the projected metric (3.11) but with
∆ = r2 + a2 +Q2 − µ¯r1−n . (3.37)
Remarkably, explicit evaluation of the Einstein tensor for this brane metric yields
Gba = G(0)
b
a + 8πT
b
a , (3.38)
where
8πT ba =


−Q
2(Σ0 + a
2 sin2 θ)
Σ2
0 0 −2aQ
2
Σ3
0 −Q
2
Σ2
0 0
0 0
Q2
Σ2
0
2aQ2Σ0a
2 sin2 θ
Σ3
0 0
Q2(Σ0 + a
2 sin2 θ)
Σ3


(3.39)
is the energy momentum tensor for the Maxwell field as computed from the definition
T ba =
1
4π
(
AacA
bc − 1
4
δbaAcdA
cd
)
. (3.40)
Equation (3.38) shows how the brane metric ansatz we have chosen reproduces exactly
the gravitational field generated by the Maxwell field while keeping the extra contribution
from the embedding into the bulk untouched. This indicates that we can consistently add
the Maxwell field on the brane and correct the brane metric accordingly. Furthermore,
we can see that the components of the zeroth order contribution to the Einstein tensor
has exactly the form anticipated from the estimate (3.26), so we see explicitly that the
correction from the Maxwell field is a perturbation.
Even though the effective metric (3.37) can not be the full solution it can be regarded
as a first approximation which is physically consistent (for a rigorous study in the second
5This substitution has been noted in a Randall-Sundrum context [60] where Q2 is interpreted as a
tidal charge.
44 Chapter 3. Strong gravity I: Charged rotating black holes
Randall-Sundrum model see [60, 61]). To solve the problem of the backreaction exactly,
we would have to construct a bulk energy momentum tensor for the Maxwell field, with
some typical thickness, and solve the bulk Einstein equations. This would give the effect
of the four dimensional brane Maxwell field on the bulk geometry as well as on the brane.
Keeping in mind the ansatz above it is tempting to assume that the physical metric will
have a form
∆→ r2 + a2 − µ¯r1−n +Q2(Ωn) . (3.41)
where now Q2 is a function of the transverse bulk coordinates Ωn such that
Q(Ωn) =
{
Q if Ωn on the brane
0 otherwise
.
If we imagine a brane with thickness ǫ such that the charge function Q2(Ωn) drops sud-
denly where the brane ends, then this choice ensures the vacuum Einstein equations are
obeyed in the bulk, as well as on the brane (together with the Maxwell field). The only
addition is a sharp δ function like energy momentum tensor where the brane ends. This
can be checked explicitly in the 5-dimensional case by using a generic function Q(χ) (χ is
the fifth dimensional coordinate) and applying the Gauss-Codazzi equations to obtain
brane Einstein equations at each hyperslice parallel to the χ = 0 brane. If the profile
chosen is flat inside the brane (Q(χ) = Q) and drops suddenly to zero at some χǫ, then
we obtain terms which are proportional to derivatives of Q(χ) at χǫ. These extra contri-
butions at χǫ spoil the construction but nevertheless we can ignore them or assume they
are somehow related to the mechanism that keeps the fields confined to the 4-dimensional
brane.
Regardless of these issues, note that the charge introduced in (3.37) consistently re-
duces the size of the black hole event horizon on the brane as we would expect for a
charged black hole. Furthermore the Maxwell field, which is independent of ∆, produces
terms in the geodesics which reproduce exactly the usual 4-dimensional electric force.
This is certainly a feature we want to keep. Finally, for the TeV gravity scenario of black
hole production we are considering we will see that the Q2 term in the metric is actually
a small perturbation. So the charge should not disturb the bulk geometry much and to
first order this effective brane metric should be a good approximation.
3.2. Construction of a brane charged background 45
3.2.3 Systems of units and orders of magnitude
In this section we find the relation between the black hole parameters and the corre-
sponding physical quantities in terms of well known constants, as well as the coupling of
charged test particles. In the last section we worked in a natural system of units where
all dimensionful quantities are in fact divided by the appropriate Planck unit factor. For
example lengths are divided by M−1D and masses by MD. Similarly any field comes di-
vided by the appropriate “Planck quantity”. Then the charge Q becomes a dimensionless
quantity describing the strength of the electric field compared to some reference charge.
The precise value of this parameter is found by matching to a known limit. Anticipating
the result we write Q = Z
√
α where
√
α is the fundamental charge and Z is the charge
of the black hole in units of
√
α. For the purpose of matching Q, the rotation parameter
can be set to zero.
Let’s start by looking at geodesics for charged particles. They are obtained from
varying the action
S =
∫
dλ
(
1
2
dxa
dλ
dxa
dλ
+ q
dxa
dλ
Aa
)
(3.42)
where q = z
√
α is the charge of the test particle. The coupling
√
α is found by taking the
non-relativistic limit. If we define the generalised momentum
Pa =
dL
dx˙a
=
dxa
dλ
+ qAa , (3.43)
conservation of the Hamiltonian H ≡ L − Pax˙a reduces to the 4-momentum constraint
pap
a = m2 where pa = dxa/dλ. The geodesic equation coupled to electromagnetism is
d2xa
dλ2
+ Γabc
dxb
dλ
dxc
dλ
+ qAab
dxb
dλ
= 0 . (3.44)
We consider radial geodesics dθ/dλ = dφ/dλ = 0. In four dimensions, the non-trivial
equations are
d2r
dλ2
+
1
2
m2U ′ − αU zZ
r2
E = 0 (3.45)
d2t
dλ2
+
(
U ′E + α
zZ
r2
)
1
U
dr
dλ
= 0 , (3.46)
46 Chapter 3. Strong gravity I: Charged rotating black holes
with U = ∆/r2. In the non-relativistic limit dt/dλ = E ∼ m, therefore dt ∼ mdλ and
m
d2r
dt2
= −mM
r2
+ α
zZ
r2
+ αm
Z2
r3
+O(r−4) , (3.47)
giving respectively the Newtonian and Coulomb force laws and the first relativistic correc-
tion due to the gravitational effect of the Maxwell field. To match α in four dimensions,
put back all length scales in terms of Planck units explicitly
m
d2r
dt2
1
M24
=
m
M4
(
−M
M4
L24
r2
+ α
M4
m
zZ
L24
r2
+ αZ2
L34
r3
+ . . .
)
, (3.48)
where the extra relativistic correction is suppressed by one more power of L4/r. Rewriting
the previous equation and setting the masses to electron masses and charges z, Z = 1 (i.e.
the unit is the electron charge)
m
d2r
dt2
1
M24
= −me
M4
(
me
M4
− αM4
me
)
L24
r2
+ . . . . (3.49)
The ratio of electric to gravitational force between electrons gives
α =
Fe
Fg
(
me
M4
)2
=
e2
4πǫ0
Gm2e
(
me
M4
)2
=
e2
4πǫ0
≃ 1
137
(3.50)
as expected. Equation (3.50) emphasizes how the electric force Fe = αFg(M4/me)
2 in
4-dimensions is orders of magnitude stronger than the gravitational force. This is simply
a statement of the hierarchy problem. However the same cannot apply in TeV gravity
scenarios where all forces are controlled by the same scale, so gravity becomes stronger
at short distances. Thus it is crucial to determine the relative strength of the (4 + n)-
dimensional gravitational force and the electric force.
Now let us rewrite equation (3.48) using MD
m
d2r
dt2
1
M2D
=
m
MD
[
−
(
MD
M4
)2
M
MD
L2D
r2
+ α
MD
m
zZ
L2D
r2
+
(
MD
M4
)2
αZ2
L3D
r3
+ . . .
]
(3.51)
The first and third contributions, which are due to the gravitational fields of the mass
3.3. Geodesics and the geometrical cross section 47
M and the charge Q, are suppressed by the same power of MD/M4. However, as we
approach short distances, the gravitational coupling must become higher dimensional,6
gravity becomes strong and the suppression factors must disappear. Note however, that
since the Maxwell field is confined to the brane, the r-power in the third term (which
is associated with the gravitational effect of the charge) must remain the same. As for
the second term, it is associated with the electric force between the test particle and the
charged body so it must remain the same, again because the Maxwell field is confined to
the brane and the magnitude of the electric force cannot change at shorter distances.
This qualitative discussion agrees with the short distance geodesic equation obtained
from the metric (3.37)
m
d2r
dt2
1
M2D
=
m
MD
[
−(n + 1)µ¯Mn+1D
Ln+2D
rn+2
+ α
MD
m
zZ
L2D
r2
+ αZ2
L3D
r3
+ . . .
]
. (3.52)
The first term is correctly modified to a higher dimensional force law, the second term
remains the same and the third term is controlled by the same power or r but without
the suppression factor (MD/M4)
2.
It is worth noting that for TeV gravity black holes in proton-proton collisions where
the black hole is formed from two partons (so the maximum charge is |Z| = 4/3), the fine
structure constant factor of 1/137 makes the Q2 contribution to the metric very small
(unless the black hole happens to charge up to |Z| ∼ 10 after production).
3.3 Geodesics and the geometrical cross section
In chapter 5, we will study the propagation of quantum perturbations of several fields out-
side the charged rotating black hole background presented in section 3.2. It is well known
that in the high energy limit wave propagation can be treated geometrically. Therefore,
in this limit, the study of the classical trajectories of test particles outside the black hole
provides useful information on wave propagation. The aim of this section is to obtain the
shapes of the absorptive disks as seen by an observer away from the black hole. This also
helps to understand how the black hole is perceived at infinity.
A first study in higher dimensions was done for some special cases in [62]. In this
section we generalise their arguments to an arbitrary trajectory and include particle charge
and mass.
6This can be checked explicitly by using the brane metric (3.37).
48 Chapter 3. Strong gravity I: Charged rotating black holes
3.3.1 The critical impact parameter
A classical particle stuck on the brane follows a geodesic curve Xµ(λ) determined by
varying the action (3.42). In this formulation, the conserved quantities are identified
by looking at the symmetries of the Lagrangian, or equivalently, the Killing vectors of
the metric. The brane metric (3.37), has the analytic form of the Kerr-Newman metric.
This type of metric produces two obvious conserved quantities associated with its time
and azimuthal Killing vectors and a third one related to a Killing tensor. For example
in [62] the three conserved quantities were combined with the Hamiltonian (which is also
conserved since the Lagrangian (3.42) does not depend on λ) to obtain a radial equation
of motion for a particle with mass µ. We can apply the same reasoning to our case and
obtain more general equations. For simplicity we switch to horizon radius units where
rH = 1 so that µ¯→ 1 + a2 +Q2 and we have the following mapping of parameters
r
rH
→ r a⋆ = a
rH
→ a
ωrH → ω µrH → µ
qrH → q Q
rH
→ Q
, (3.53)
(where ω will be the energy of the particle) so ∆ becomes
∆ = r2 + a2 +Q2 − (1 + a2 +Q2)r1−n . (3.54)
The equations for the geodesics are then
Σ
dt
dλ
=
r2 + a2
∆
[(r2 + a2)ω − aℓz − qQr] + aℓz − a2ω sin2 θ
Σ
dφ
dλ
=
aω (r2 + a2)− a2ℓz − aqQr
∆
− aω + ℓz
sin2 θ[
Σ
dθ
dλ
]2
= Q− cos2 θ
[
a2 (µ2 − ω2) + ℓ
2
z
sin2 θ
]
[
Σ
dr
dλ
]2
= [ω(r2 + a2)− ℓza− qQr]2 −∆
[
µ2r2 + (ℓz − aω)2 +Q
]
(3.55)
where ω, ℓz,Q are the constants of motion associated with time translations, azimuthal
translations and the Killing tensor respectively. Next we redefine λ → λω, express the
remaining constants of integration in terms of the impact parameter b (in units of rH), the
3.3. Geodesics and the geometrical cross section 49
ϑ
ζb
Incident particle
Direction parallel
to the equatorial plane
Direction aligned with J
J
Black hole
Figure 3.3: Diagram defining the ζ angle and the impact parameter b on the plane transverse
to the direction of incidence.
polar angle of incidence ϑ, the angular momentum magnitude ℓ, the angular momentum
orientation at infinity ζ , and the reduced mass ν = µ/ω and charge ǫ = q/ω:
Σ
dt
dλ
=
r2 + a2
∆
[r2 + a2 − abz − ǫQr] + abz − a2 sin2 θ
Σ
dφ
dλ
=
a (r2 + a2)− a2bz − aǫQr
∆
− a + bz
sin2 θ[
Σ
dθ
dλ
]2
= P − cos2 θ
[
a2 (ν2 − 1) + b
2
z
sin2 θ
]
[
Σ
dr
dλ
]2
= [r2 + a2 − abz − ǫQr]2 −∆
[
ν2r2 + (bz − a)2 + P
] ≡ R
(3.56)
where
b ≡ ℓ
ω
, bz ≡ ℓz
ω
= b cos ζ sinϑ, P ≡ b2 − b2z − a2 cos2 ϑ . (3.57)
The impact parameter b corresponds to the distance of closest approach to the origin, if the
spacetime was flat. ζ is the polar angle of the impact parameter on the plane perpendicular
to the direction of incidence (see figure 3.3) or equivalently the angle between the angular
momentum of the black hole and the angular momentum of the incident particle.
50 Chapter 3. Strong gravity I: Charged rotating black holes
The most general trajectory is parametrised by the set {b, ζ, ϑ, a,Q, ν, ǫ}. Our goal is
to determine whether a particle is absorbed or not for a given set. This can be achieved by
looking at the radial equation of motion in (3.56), and noting thatRmust be non-negative
over the trajectory. Let’s start by defining the functions A,B,C through
R = 1
rn
[−Ab2 + 2Bb+ C] . (3.58)
To investigate whether a particle is absorbed or not, we need to know if R takes negative
values at any point of the trajectory. If so, there is a region which is inaccessible. A way
to guarantee absorption is to choose the value of b in an interval such that R is positive
over the whole space. This requires determining its zeros with respect to b at all points
r and finding the intervals of b such that such zeros do not exist over the whole radial
domain. In principle this may produce complicated disks or ring-like regions on the {b, ζ}
polar plane at infinity.
Q = 0 case
In this case
A = rn+2 + rna2
(
1− cos2 ζ sin2 ϑ)− (1 + a2)r
B = −a cos ζ sinϑ (1 + a2) r
C = rn+4(1− ν2) + rn+2a2(1− ν2 + cos2 ϑ) + rna4 cos2 ϑ+
+r3ν2
(
1 + a2
)
+ ra2(1 + a2) sin2 ϑ (3.59)
where C ≥ 0 since ν ≤ 1. For a fixed b, it can be shown that A has exactly one zero
in the domain r > 0 so it changes sign only once7. So A starts from positive values at
infinity, decreases, goes through zero and takes negative values inwards. Since the signs
of B and C are fixed we only need to analyse two regions.
Let’s start with the outermost region. We want to determine the impact parameters
for which the particle is not able to penetrate completely through the region of A ≥ 0,
thus being scattered back at some point, or equivalently the range of parameters bmin <
b < bmax for which absorption is guaranteed. Then it is certain that any particle with
impact parameters within this range will reach the second innermost region. For any
7This is done by looking at the sign of the function at infinitesimal r and at infinity, using continuity
and the positivity of d2A/dr2 > 0 for r > 0.
3.3. Geodesics and the geometrical cross section 51
solution of the radial equation we must ensure that R ≥ 0 to allow the particle through
the region. As a function of b, R has two zeros
b± =
B ±√B2 + AC
A
(3.60)
Since, AC is positive, there is a negative and a positive root. Thus R is a parabola with
a maximum. R is positive only for 0 < b < b+. If we take b > bmax ≡ minr {b+}, where
the minimisation is over the radial region we are considering, there will always be a point
where R goes through zero and changes sign. This means that the particle is scattered
back at that point. Similarly, in the complementary case, there will never be such a zero
so the particle reaches the second interior radial region. Thus we have obtained the first
upper bound.
Regarding the remaining interior radial region the situation is a bit more complicated.
In that case, A < 0 so the roots are
b± =
−B ∓√B2 − |A|C
|A| . (3.61)
For B > 0 (or equivalently bz < 0, see (3.57)), the previous equation has no positive real
roots, so R > 0 and any particle reaching the region is absorbed. Thus, for this sign of
B, the absorptive disk is defined by the interval obtained in the outer region. For B ≤ 0,
we can have two real positive roots. Those are
b∓ =
|B| ∓√B2 − |A|C
|A| (3.62)
so R is a parabola with a minimum and two zeros. If b takes any value below minr{b−}
(again the minimisation is in the radial region we are considering), there is no zero, because
any r has associated critical b’s which are necessarily larger. The same occurs for b above
maxr{b+}, since all the b’s that allow a zero are smaller. Conversely, for b between the
last two values, there is always a point where R goes through zero and becomes negative,
so the particle is scattered back. Therefore, the impact parameters for absorption must
be smaller than minr {b−} or larger than maxr {b+}. However, note that at r = rA, (rA
defined such that A(rA) = 0) b+ diverges, so there is no finite b > maxr {b+} → +∞.
52 Chapter 3. Strong gravity I: Charged rotating black holes
The range of impact parameters for absorption can be summarised as
0 < b < bc ≡ min
{r≥rlow}
{
C√
B2 + AC − B
}
(3.63)
with
rlow ≡
{
rA, bz ≤ 0
1, bz > 0
. (3.64)
The expression to be minimised in (3.63) is rearranged to remove the apparent singularity
at rlow when B < 0 and to make it explicit when B ≥ 0. This form is more suitable for
numerical minimisation because the only singularities are at the extreme points of the
interval.
Q 6= 0 and ǫ 6= 0 cases
More generally when the charges are non-zero, the signs of B and C can also change
throughout the radial domain. The expressions for the coefficients are now more compli-
cated
A = rn+2 + rna2
(
1− cos2 ζ sin2 ϑ+Q2)− (1 + a2 +Q2)r
B = a cos ζ sinϑ
[
ǫQrn+1 +Q2rn − (1 + a2) r]
C = rn+4(1− ν2)− 2ǫQrn+3 + rn+2 [a2(1− ν2 + cos2 ϑ) + ǫ2Q2 − ν2Q2]+
+a2
(
a2 cos2 ϑ−Q2 sin2 ϑ) rn+1 + rna2 (a2 cos2 ϑ−Q2 sin2 ϑ)+
+r3ν2
(
1 + a2 +Q2
)
+ ra2(1 + a2 +Q2) sin2 ϑ . (3.65)
Nevertheless we can still write a general algorithm which scans the functions A,B,C from
infinity, through a finite number of regions where the signs of the various functions remain
fixed, and computes bounds on b. Also it is easy to show that inside the black hole horizon
R is positive, so at most we have to scan down to r = 1. Furthermore, A still contains
only one zero so we have two regions for A. For B and C, in general it is not possible to
say how many zeros we have, but since they are polynomials, at most we have n+ 1 and
n + 4 zeros respectively. Thus there are at most 2(n + 2)(n + 5) qualitatively different
regions for a fixed n. If we combine this with the two possible combinations of signs for
B at infinity8 the possible number of cases is doubled to 4(n+ 2)(n+ 5).
8Note A and C are always positive at infinity.
3.3. Geodesics and the geometrical cross section 53
We start with the first region where {A,B,C} have their signs as at infinity. Then
A > 0 and C > 0 so the roots of R are
b± =
B ±√B2 + AC
A
. (3.66)
Following arguments similar to those for the first case when Q = 0, in this region we need
0 < b < bmax ≡ min
{r}
{
C√
B2 + AC −B
}
. (3.67)
Now we have two possibilities as we enter a new region. Either A changes sign and B,C
keep their signs or A keeps its sign.
Let us start with the case when B,C change signs before A. We have four possible
cases for the signs s1, s2 of B,C: (s1, s2) = (+,+), (+,−), (−,+), (−,−) so
b± =
|B| ±√B2 + A|C|
A
b± =
|B| ±√B2 −A|C|
A
b± =
−|B| ±√B2 + A|C|
A
b± =
−|B| ±√B2 − A|C|
A
, (3.68)
respectively. Now R (as a function of b) is a parabola with a maximum. For the first and
third cases we have a positive and a negative zero so the condition is exactly the same as
in the previous region. For the second case we have two positive zeros, so
max
{r}
{ −C√
B2 + AC +B
}
≡ bmin < b < bmax ≡ min
{r}
{
C√
B2 + AC − B
}
. (3.69)
Finally the fourth case has no positive zeros, so R < 0 and no b is allowed (the procedure
stops).
The next cases are inside the region where A changes to negative. The four cases are
54 Chapter 3. Strong gravity I: Charged rotating black holes
(s1, s2) = (+,+), (+,−), (−,+), (−,−), with the respective roots
b± =
−|B| ±√B2 − |A||C|
|A|
b± =
−|B| ±√B2 + |A||C|
|A|
b± =
|B| ±√B2 − |A||C|
|A|
b± =
|B| ±√B2 + |A||C|
|A| . (3.70)
Since A < 0, R is a parabola with a minimum. For the first case both zeros are negative
so R is always positive for r > 0 and no constraint arises. For the second and fourth
cases where C < 0, we have a positive and a negative zero regardless of the sign of B so
we need
b > bmin ≡ max
{r}
{ −C√
B2 + AC +B
}
. (3.71)
For the third case, we have two positive roots so
b < bdown ≡ min
{r}
{
C√
B2 + AC − B
}
∨ b > bup ≡ max
{r}
{
C
−√B2 + AC − B
}
. (3.72)
This method can be implemented numerically for the general case where all parameters
are nonzero. In particular we can check that it reduces to the procedure described for
Q = 0. However, note that ν = µ/ω and ǫ = q/ω, so since the geometrical description is
good at high energies, these parameters should be small even when non-zero. Furthermore
as pointed out in 3.2.3 the charges in the scenario we are considering are typically small
and the same applies to the masses of the particles. Thus in the next sections we present
results for the simpler case where the charges and the particle mass are neglected.
3.3.2 Perturbative and numerical minimisation
We want to obtain the absorptive disks as seen from infinity in various directions of obser-
vation. The most efficient way is to numerically determine the critical impact parameter
for each set of parameters through the minimisation of (3.63).
Another useful approach is to solve the problem analytically in a particular case and
expand around it perturbatively. This method is useful as a check of the numerical analysis
3.3. Geodesics and the geometrical cross section 55
and shows most of the features of the result. Equation (3.63) is easy to minimise when
a = 0 (the Schwarzschild case), leading to
r0 ≡
(
n+ 3
2
) 1
n+1
b0 ≡
(
n+ 3
2
) 1
n+1
√
n+ 3
n+ 1
, (3.73)
r0, b0 are the minimiser and minimum respectively. For small rotation parameters, it
should be possible to find a good approximate solution of (3.63), by expanding perturba-
tively around (3.73), i.e.
rc ≡
+∞∑
p=0
rp
p!
ap . (3.74)
Since rc, bc are the minimiser and minimum when a 6= 0, rc must obey
∂b
∂r
∣∣∣∣
r=rc
= 0 , (3.75)
so that
bc = b|r=rc . (3.76)
Note that from now on, b is given by the expression to be minimised in (3.63) and all
other parameters ζ, ϑ, n are omitted. Expanding (3.75) in powers of a and imposing
each coefficient to vanish independently, we obtain conditions on the rp coefficients (see
equation (3.74))
dm
dam
(
∂b
∂r
∣∣∣∣
r=rc
)∣∣∣∣
a=0
= 0 . (3.77)
The total derivatives can be expanded in the following manner
dm
dam
=
dm−1
dam−1
(
∂
∂a
+
drc
da
∂
∂rc
)
=
dm−1
dam−1
∂
∂a
+
drc
da
dm−1
dam−1
∂
∂rc
+
dmrc
dam
∂
∂rc
=
1∑
k=0
(
drc
da
)k
dm−1
dam−1
∂
∂a1−k∂rkc
+
dmrc
dam
∂
∂rc
.
(3.78)
56 Chapter 3. Strong gravity I: Charged rotating black holes
Successive iterations of (3.78), suggest the following identity for any order p < m
dm
dam
=
p∑
q=0
p!
(p− q)!q!
(
drc
da
)q
dm−p
dam−p
∂p
∂ap−q∂rqc
+
+
p−1∑
q=0
dm−qrc
dam−q
q∑
k=0
q!
(q − k)!k!
(
drc
da
)k
∂q+1
∂aq−k∂rk+1c
, (3.79)
which can be checked by induction on p. Finally, setting p = m− 1 (valid for any m > 0)
dm
dam
=
m−1∑
q=0
(m− 1)!
(m− 1− q)!q!
(
drc
da
)q
∂m
∂am−q∂rqc
+
+
m−1∑
q=0
dm−qrc
dam−q
q∑
k=0
q!
(q − k)!k!
(
drc
da
)k
∂q+1
∂aq−k∂rk+1c
. (3.80)
Using (3.80) in (3.77), relabelling rc → r in the partial derivatives, and defining
βi,j ≡ ∂
i+jb
∂ai∂rj
∣∣∣∣
a=0,r=r0
(3.81)
gives the recursion relations (m > 1) for all the corrections
r1β0,2 = −β1,1
rmβ0,2 = −βm,1 −
m−1∑
q=1
[(
m− 1
q
)
rq1βm−q,q+1 + rm−q
q∑
k=0
rk1
(
q
k
)
βq−k,k+2
]
.
(3.82)
The first correction is
r1 = − 2 cos ζ sinϑ√
(n+ 1)(n+ 3)
. (3.83)
Furthermore, a similar expansion for bc can be found by using (3.80). Define
bc ≡
∞∑
m=0
bm
m!
am (3.84)
where
bm+1 ≡ d
m+1bc
dam+1
∣∣∣∣
a=0
=
[
dm
dam
(
∂
∂a
+
drc
da
∂
∂rc
)
bc
]
a=0
=
[
dm
dam
∂bc
∂a
]
a=0
. (3.85)
3.3. Geodesics and the geometrical cross section 57
b
ζ
n = 2, a = 0.1, θ = π2
21.510.50-0.5-1-1.5-2
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
n = 4, a = 0.5, θ = π2
21.510.50-0.5-1-1.5-2
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
Pert.
Num.
n = 10, a = 0.5, θ = π2
21.510.50-0.5-1-1.5-2
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
Figure 3.4: The perturbative result (dashed blue) agrees with the numerical minimisation
(solid red), for a . 1 (some combinations of (n, a, ϑ) indicated). The polar system
(b, ζ) used throughout to draw the absorptive disks (defined on the plane transverse
to the radial direction of observation at infinity) is indicated on the left plot.
The last step follows from (3.77). The result is (m ≥ 1)
b1 = β1,0
bm+1 =
m−1∑
q=0
[(
m− 1
q
)
rq1βm+1−q,q + rm−q
q∑
k=0
rk1
(
q
k
)
βq+1−k,k+1
]
.
(3.86)
The first correction is actually independent of the perturbation in r
b1 = −2 cos ζ sinϑ
n+ 1
. (3.87)
This perturbative method was used to check the exact numerical results obtained from
numerical minimisation. Figures 3.4 and 3.5 show a perfect agreement for small a and
they are good up to a ∼ 1. For larger a, the perturbative expansion seems to hold
as an asymptotic series. The inclusion of higher order corrections degrades the result
and the closest we get from the exact numerical result is by keeping O(2) corrections (see
figure 3.5). However, even though the perturbative result fails for large a, it is consistently
larger than the numerical one, which is supposed to be the true minimum. The lowest
order correction shows that for small a, the distortion is suppressed for large number
of extra dimensions. Furthermore, for incidence along the vertical axis, the distortion
is independent of ζ (this follows trivially from the azimuthal symmetry). For incidence
along the equatorial plane (or intermediate angles), the disk of absorption is distorted into
an oval, indicating the orientation of the angular momentum of the black hole. The exact
58 Chapter 3. Strong gravity I: Charged rotating black holes
n = 4, a = 1, θ = π2 , O(6)
210-1-2-3
2
1
0
−1
−2
n = 4, a = 1, θ = π2 , O(2)
210-1-2-3
2
1
0
−1
−2
n = 4, a = 2, θ = π2 , O(2)
210-1-2-3
2
1
0
−1
−2
Figure 3.5: The perturbative result starts to fail compared to the numerical minimisation, for
a & 1 (same color scheme as figure 3.4). n = 4, a = 1 (left) starts to disagree if we
include O(6) corrections. However, truncating at O(2) (centre) the result is still
good. Increasing a = 2 (right) further degrades the perturbative result.
1
4/5
3/5
2/5
1/5
0
ϑ/(π2 )
n = 2, a = 4
6420-2-4-6-8
6
4
2
0
−2
−4
−6
2
1.6
1.2
0.8
0.4
0
a
n = 3, ϑ = π2
210-1-2-3-4
2
1
0
−1
−2
4
3
2
1
n
a = 4, ϑ = π4
6420-2-4-6-8-10-12
8
6
4
2
0
−2
−4
−6
−8
Figure 3.6: The plots shows the variation of the absorptive disk by varying one parameter
with all the others fixed.
numerical results9 in figure 3.6 confirm the non-perturbative validity of these qualitative
features. The only exception is the conclusion regarding the suppression with n (actually
the opposite occurs for a⋆ & 1, see right plot in figure 3.6 where the distortion is greater
for larger n when a = 4). This figure also shows that for larger rotation parameter, the
distortion becomes much stronger.
The main conclusion of this analysis is that if the black hole is between a distant
observer and a source which radiates ultra-relativistic particles, it is possible to infer its
angular momentum by looking at the shape of the black disk in different directions. The
egg-shape disk is a clear indication of a rotating rather than a spherically symmetric black
hole and the details of the shape contains information on n and the angle of observation.
9These were obtained by running a code in Maple which was written using the Maple inbuilt minimi-
sation routine.
3.3. Geodesics and the geometrical cross section 59
Besides providing a more intuitive idea of the properties of the black hole as seen by
a distant observer, these disks can be used to compute the interaction cross section for
a beam of classical particles incident on the black hole along a given direction, which is
simply the area of the critical disk. At high energies, wave propagation can be described
geometrically, so these results will be useful in chapter 6 to check the absorption cross
sections. The latter are closely related to the transmission factors which determine the
spectrum of Hawking radiation.
60 Chapter 3. Strong gravity I: Charged rotating black holes
Chapter 4
Strong gravity II: Models for black
hole production
In the beginning of chapter 3 we have motivated the study of black hole solutions in
higher dimensions from the point of view of strong gravitational interactions in D dimen-
sions which lead to black hole production. In particular we are interested in collisions of
two highly relativistic particles at impact parameters of the order of the Schwarzschild
radius for the centre of mass energy
√
s. The kinematics of the colliding system in the
centre of mass frame for two particles of masses m1 and m2 at an impact parameter b is
in general
pM1 =
(√
m21 − p2, 0, 0, p, 0, . . . , 0
)
(4.1)
pM2 =
(√
m22 − p2, 0, 0,−p, 0, . . . , 0
)
(4.2)
with
p ≡ M
2
√
1− 2(m
2
1 +m
2
2)
M2
+
(
m21 −m22
M2
)2
. (4.3)
We have defined the invariant mass of the system M ≡ √s and assumed a collision of
particles on the brane along the z-axis. For the gravitational interaction to be described
classically we need the typical wavelength for each particle to be small compared to the
typical gravitational interaction length. The Heisenberg principle for a wave packet with
61
62 Chapter 4. Strong gravity II: Models for black hole production
minimal uncertainty then gives the condition [63]
∆x ∼ 1
p
≪ rH , (4.4)
which for highly-relativistic particles1 and a large enough invariant mass M is always
possible (since rH grows with M). Furthermore, in the trans-Planckian regime where
M is well above MD, the gravitational interaction is much stronger than all other SM
interactions, and at high energies ∼ M it occurs at large length scales, whereas high
energy SM interactions occur at short distances. Thus the interaction is gravitationally
dominated.
From this setup, the first estimates for the production cross section using the hoop
conjecture were obtained in [24, 45, 46]
σ ∼ πr2H (4.5)
where a higher dimensional Schwarzschild black hole2 with mass M and horizon radius
rH was assumed to be formed. The cross section is simply the area of the black disk with
radius rH , and b is the impact parameter. In this chapter, we review some models which
improve this early simplistic picture.
4.1 Setting up the initial state
To model the gravitational interaction between the particles, in addition to the kinematical
information of the initial state, we need to provide a spacetime metric which describes
their initial gravitational field when they are far apart. For the simplest case of two
particles with no spin or charges, this can be achieved in their centre of mass frame by
superposing two boosted metrics of the form (3.15) boosted in opposite directions. As
long as the initial positions of the particles are well separated, we can sum the two flat
parts of each metric with a shock centered at the two initial points (say x → x ± x0
respectively and set for one of the shocks a transverse coordinate x2 → x2 − b where b is
the impact parameter). This can be used to fix the initial conditions for the collision.
For a more general configuration where the colliding particles may have charges and
spin, in principle, the initial state is prepared similarly. As long as a metric is known
1And even moderately relativistic particles as long as their masses are not too close to the threshold
∼M/2.
2Also known as the Schwarzschild-Tangherlini solution.
4.2. Gravitational collapse 63
which describes the gravitational field of the particle in its rest frame, a simple boost and
a translation will suffice.
4.2 Gravitational collapse
In the introduction of this chapter, we have argued that the hoop conjecture indicates
that a black hole must form as long as the impact parameter b between the colliding
particles is small enough. Before presenting the model that we will use for production, it
is instructive to discuss what we would expect based on some order of magnitude estimates
and some known results in general relativity.
Assuming a black hole does form in the collision, in general it will start by having a
complex shape horizon. Furthermore, the production may be accompanied by emission of
gravitational and gauge radiation from the gravitational interaction between the energetic
particles and the gauge interactions sourced by their SM charges. In four-dimensional
general relativity, according to the ‘no hair’ theorem [64], the solution should quickly
relax to one of the axisymmetric stationary black hole solutions of Einstein’s equations.
We would expect a higher dimensional black hole on the brane to be formed similarly,
however other axisymmetric solutions are known in higher dimensions such as the ‘black
ring’3. Furthermore the Myers-Perry black hole is known to be unstable for large angular
momentum (see e.g. [67,68]). Nevertheless, at least for not too large angular momentum,
we would expect the black hole formed to be of Myers-Perry type.
It has been argued [24] that the typical time scale for loss of asymmetries related to the
production shock is of the order of the horizon radius rH (note we are using natural units
– see appendix A.2). This is physically reasonable because the asymmetries are related
to a distortion of the geometry (with respect to the stationary solution). Regardless
of the details of the interaction responsible for removing them, it will involve a signal
propagating over a region of typical size rH . On the other hand the time scale ∆t for
evaporation of a trans-Planckian black hole is controlled by the Hawking energy flux.
Since we are only interested in the order of magnitude, we can estimate it by using the
D-dimensional Schwarzschild case combined with dimensional analysis (see for example
section 3 of [24]) to obtain
∆t
rH
∝
(
M
MD
)D−2
D−3
. (4.6)
3This is known to exist in five dimensions [65], and there is strong evidence for D ≥ 6 [66].
64 Chapter 4. Strong gravity II: Models for black hole production
The prefactor is a constant of order unity containing the dimensionally reduced energy flux
and other convention-dependent constants, andMD is the Planck mass in D dimensions
4.
ForM ≫MD the time for evaporation will be much longer than for the loss of asymmetries
(usually called the balding phase).
Another issue is that of quantum gravity corrections to the classical production de-
scription. It has been shown, within some quantum gravity approximation schemes, that
these are small provided that the parton collision energy is sufficiently far above the
Planck scale [69–72]. This agrees with the qualitative considerations we have presented
in the introduction of this chapter.
In principle, the problem of evolving the Einstein equations classically to obtain the
final black hole solution and the radiation released in the collision, given the initial setup
described in section 4.1, can be solved numerically. The two pieces of information which
are most relevant to the theoretical modelling of the black hole production phase in parton-
parton collisions are: the maximum allowed impact parameter b, and the fractions of the
initial centre of mass energy and angular momentum that are trapped within the rotating
black hole. This has not yet been fully achieved for higher dimensional black holes (see
section 4.4). In the next section we comment on analytic or semi-analytic methods used
to study the production and present a model based on trapped surface bounds.
4.3 Trapped surface and other analytic bounds
The trapped surface inspired model presented in this section and in appendix B is due to
Jonathan R. Gaunt and was published as part of [4]. It is included here for completeness
because it is used in the event generator described in chapter 7.
4.3.1 Theoretical studies of black hole production
The main approach to studying higher dimensional black production has been the trapped
surface method [59,73–76]. This method utilises the fact that the black hole horizon begins
forming in the spacetime region outside the future lightcone of the collision event, where
we can solve for the geometry. By studying apparent horizons in spacetime slices of this
region, and in particular looking at certain areas associated with the apparent horizon
as b is varied, one can set bounds on the parton-level cross section and the mass M and
4See Appendix A.2 for our convention for the definition of MD.
4.3. Trapped surface and other analytic bounds 65
angular momentum J trapped for a given b.
Few trapped surface calculations conducted thus far have attempted to obtain results
for nonzero b. The one offering best bounds is that of Yoshino and Rychkov [76] who
considered dimensions D = 4 . . . 11. They produced (M,J) bounds up to the maximum
impact parameters which give rise to apparent horizons in their method. The results of
this calculation are the primary theoretical input to the model of production we use. A
more detailed discussion of the Yoshino-Rychkov method is given in appendix B.1.
The Yoshino-Rychkov calculation models the colliding partons as boosted Schwarzschild
black holes (equation (3.21)), and so neglects the effects of the spin, charge, and finite size
of the colliding partons. For each effect individually, trapped surface calculations have
been carried out [77–79] - but all are for b = 0. One important observation from those
calculations is that charge effects may be significant.
Alongside the trapped surface method, alternative techniques have been developed
which use a perturbative approach and/or other approximations to estimate directly the
mass lost in the production phase. In one setup [80], the collision is modelled as an ultra-
relativistic particle falling into a Schwarzschild black hole, and the gravitational emission
is calculated by assuming that the gravitational effects of the in-falling particle may be
treated as a perturbation on top of the Schwarzschild metric. Another calculation [81]
also uses a perturbative approach and assumes that the collision is instantaneous. As
a final example, D’Eath and Payne [82–84] have estimated the mass loss in the D = 4
axisymmetric collision case by finding the first two terms of Bondi’s news function, and
then extrapolating off axis. Here some assumptions about the angular dependence of the
radiation are made.
The results produced so far by these methods have been limited to b = 0 and certain
values of D. The b = 0 results from different techniques are compared with the trapped
surface bound in figure 4.1. The general indication from these is that much less mass is
lost during the production phase than the Yoshino-Rychkov upper limits indicate.
4.3.2 The model for CHARYBDIS2
Cross section
In earlier versions of CHARYBDIS [85], parton-level cross sections for different D values
were calculated according to the simple formula σ = πr2s(
√
s) which is based on Thorne’s
hoop conjecture [44]. Here rs(
√
s) is the radius of the D-dimensional Schwarzschild black
66 Chapter 4. Strong gravity II: Models for black hole production
Particle falling into BH
Instantaneous collision
News function
Trapped surface bound
Simulation
D
Fractional mass lost during production
1110987654
0.5
0.4
0.3
0.2
0.1
0
Figure 4.1: A comparison of various theoretical results for the production phase mass loss
in the b = 0 case, with the average mass loss produced for b = 0 by our model
for CHARYBDIS2. Black squares: ‘Particle falling into black hole’ results from [80].
Open squares: ‘Instantaneous collision’ results from [81]. Asterisk: ‘Bondi’s news
function’ result from [82–84]. Crosses: ‘Trapped surface method’ upper bound on
mass loss. Points with error bars: average b = 0 mass loss from our simulation.
The error bars represent the standard deviation in the b = 0 output.
hole with mass
√
s. Incorporation of the Yoshino-Rychkov cross section results simply
requires multiplying these σ values by the ‘formation factors’ given in Table II of [76].
The increase in σ ranges from a factor of 1.5 at D = 5 to 3.2 at D = 11. The maximum
impact parameters for black hole production, bmax, to be generated in CHARYBDIS2, is
adjusted accordingly (the two are related through σ = πb2max).
Mass and angular momentum loss
Following Yoshino and Rychkov, we denote the fractions of the initial state mass and
angular momentum, trapped after production, by ξ and ζ respectively. For a given number
of dimensions D and impact parameter b, the Yoshino-Rychkov bound on these quantities
is a curve in the (ξ, ζ) plane. Examples of such curves for various D and b values are given
as the solid lines in figure 4.2. A boundary curve ξb(ζ) always possesses the following two
key properties. First, it always passes through ζ = 0 with a ξ value between 0 and 1,
where ξb(0) = ξlb(b,D) in the language of [76]. Second, it increases monotonically after
this, passing through ζ = 1 with a value satisfying ξlb < ξb(1) ≤ 1. The allowed region is
then delimited by this curve and the lines ζ = 0, ζ = 1, and ξ = 1.
4.3. Trapped surface and other analytic bounds 67
ζ
ξ
D = 4, b = 0.6
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 4, b = 0.8
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 5, b = 0.5
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 5, b = 1.0
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 5, b = 1.145
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 6, b = 0.5
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 6, b = 1.0
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 6, b = 1.3
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 9, b = 0.5
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 9, b = 1.0
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
ζ
ξ
D = 9, b = 1.5
10.80.60.40.20
1
0.8
0.6
0.4
0.2
0
Figure 4.2: Plots displaying the Yoshino-Rychkov bound (solid line) and some output (ξ, ζ)
points from our simulation of the production phase (dots), for selected D and b
values (these are given above the plots in each case). Each plot contains 2000
sample output points, which have been generated with CVBIAS set to .TRUE..
68 Chapter 4. Strong gravity II: Models for black hole production
The new simulation of mass and angular momentum loss based on these curves consists
of a point being generated at random in the square 0 ≤ ξ ≤ 1, 0 ≤ ζ ≤ 1. The probability
distribution for generating this point goes to zero along the Yoshino-Rychkov bound
corresponding to the D and b values of the event, such that the generated point is always
inside the bound. The ξ and ζ coordinates of this point are then taken as the fractional
mass and angular momentum trapped during the production phase for that event.
The precise rules for the generation of the (ξ, ζ) point are as follows. First, the ζ value
for the point, ζ∗, is generated, according to a linear ramp distribution. This distribution
extends between ζ = 0 and ζ = 1, with value 0 at ζ = 0 and value 2 at ζ = 1. The ξ
value for the point is then also generated. The distribution in this case is similar, except
that now it extends between ξ = ξb(ζ
∗) and ξ = 1, ensuring that the point ends up inside
the Yoshino-Rychkov bound. The details of how the program calculates ξb(ζ
∗) for the D
and b appropriate to the event are given in Appendix B.1.
The decision to implement a probability distribution favouring smaller mass losses
than the Yoshino-Rychkov upper bound was made based on the results from the direct
calculations given in figure 4.1. In this figure we have plotted the mean mass lost in an
event with b = 0 using the above probability distribution. The error bars represent the
standard deviation in the b = 0 mass loss. We observe a reasonably good agreement
between the mean values obtained with our chosen probability distribution and the es-
timation method results, especially in the important D = 4 case where the estimation
method results agree closely. Given that we favour smaller mass losses, it then seems sen-
sible to ensure that the probability distribution also favours smaller angular momentum
losses – hence the ramp distributions in ξ and ζ .
One possible picture of the production phase is one in which the production phase
radiation is ‘flung out’ radially in a frame co-rotating with the forming event horizon.
In this scenario, the angular velocity of the event horizon does not change during the
production process (where we have to regard the forming black hole as a pseudo-Myers-
Perry solution at all points during the production phase for this statement to have any
meaning).
Based on this picture, we consider the option to bias the above probability distribution
such that (ξ, ζ) points corresponding to smaller changes in the horizon angular velocity
are more likely. An additional condition is added to this bias – for the points to have their
chance of being picked enhanced, they must also have an associated value of the oblateness
parameter a∗ = a/rH , which is sufficiently close to that of the initial state. This is to
4.4. Latest developments in numerical relativity 69
remedy the problem that, for D > 5, there are two curves with the same angular velocity
as the initial state in the square 0 ≤ ξ ≤ 1, 0 ≤ ζ ≤ 1, but only one is connected to the
initial state. In the CHARYBDIS2 generator described in chapter 7 the bias may be turned
on or off using the user switch CVBIAS. The details of its implementation are discussed in
appendix B.2.
In each of the plots in figure 4.2, there are 2000 (ξ, ζ) points generated for the ap-
propriate D and b using the model we have described, with the bias applied. One can
see in each case that there is an increased density of points around the ‘constant horizon
angular velocity’ curve.
4.4 Latest developments in numerical relativity
The model in the previous section gives a physically reasonable estimate for the maximum
impact parameter which allows black hole production, and to the amount of mass and an-
gular momentum lost. However a complete solution within classical general relativity will
give a point instead of a distribution on the (ξ, ζ) plane. This is in general a complicated
and computationally intensive problem in D = 4 + n dimensional numerical relativity.
The most well studied cases up to the date are in four dimensions. For example
some recent results of simulations of ultra-relativistic collisions of black holes [86–88] or
solitons [89] in four dimensions indicate that ∼ 25% of mass and ∼ 65% of angular mo-
mentum are lost in collisions at the maximum impact parameter for black hole formation.
However, note that the value obtained for the maximum impact parameter in this case is
∼ 50% above the Yoshino-Rychkov lower bound, corresponding to an initial-state angular
momentum that is more than double the maximum value possible for the black hole that
is formed. Therefore an angular momentum loss greater than 50% is inevitable in this
case. In D > 5 dimensions there is no upper limit on the angular momentum of a black
hole and such a large loss is not required.
Results for higher dimensions are not yet known but should be available in the near
future [90] to implement in our modelling for production.
70 Chapter 4. Strong gravity II: Models for black hole production
Chapter 5
Black hole decay and Hawking
radiation
In this chapter we discuss theoretical aspects of the Hawking effect [91] for the various
fields propagating in the black hole background constructed in section 3.2. The Hawking
effect is part of the more generic problem of instabilities of a black hole background. It
arises only at the quantum level, i.e. from the quantum fluctuations of field modes which
are classically stable. The reason it is relevant for the scenario we are studying is because
we assume that a classically stable black hole solution is formed during the production.
Since the black hole is produced in a vacuum, the only way to decay is then through
quantum effects.
We start with a general discussion of the SM fields, to assess the order of magnitude of
the various interactions in their equations of motion. In section 5.2 we summarize the key
results known from the theory of Hawking radiation which are relevant to this study. In
section 5.3 we provide the rules to couple various brane fields to the charged background
constructed in chapter 3. Separated wave equations are provided for scalars and fermions,
and some comments are made for higher spin fields (this study was published as part
of [1,2]). We end the chapter with a discussion of how to decompose the spheroidal wave
modes constructed for the fields, in terms of plane wave modes (published in [4]).
5.1 Perturbation theory and approximate decoupling
Before discussing the theory of Hawking fluxes emitted by the black hole, it is impor-
tant to summarise the various field perturbations which are allowed to propagate in the
71
72 Chapter 5. Black hole decay and Hawking radiation
background spacetime. Similarly to the construction of the brane charged metric in sec-
tion 3.2, we adopt a perturbative approach where all fields are expanded linearly around
their background value. Typically, in our study, the only fields which have a non-zero
background value are the gravitational field and the electromagnetic field, but more gen-
eral configurations are in principle possible.
First, we consider the bulk higher dimensional background given by the Myers-Perry
solution with one angular momentum (3.11). By expanding the bulk metric as in equa-
tion (2.12) a generalised version of the Fierz-Pauli Lagrangian (1.21) to a curved back-
ground vacuum solution is obtained
SH =
∫
dDx
√
|G(0)|
[
1
8
(
HAB;CHAB;C − 2HAB;CHAC;B + 2HAC ;AH;C −H;BH ;B
)
+
+
1
2M
D
2
−1
D
TABH
AB
]
, (5.1)
to describe the gravitational perturbations of the background. Note that we have denoted
the background covariant derivative ∇A by ;A for notational simplicity. Equation (5.1)
can be thought of as a covariantised version of the flat spacetime action for the gravita-
tional perturbations. This is similar to the covariantisation of the action for matter fields
by fixing it to its flat spacetime form in the locally inertial frame. TAB is the higher di-
mensional energy momentum tensor for the matter fields which, for brane confined fields,
must contain a delta function like factor.
As discussed in section 3.2, a simple projection of the metric (3.11) gives the brane
metric. In that section we have introduced a brane metric correction to approximate the
effect of a small black hole charge. On the brane, we use the corrected brane metric (3.37)
and the Maxwell field (3.32). The brane coupled SM low energy effective action is
Ss={0,1} =
∫
d4x
√
|g|
[
1
2
∇ah∇ah− m
2
h
2
h2 − λ
8
h3 (h + 4v)− 1
4
Gab ·Gab+
−1
2
W †abW
ab +m2WW
†
aW
a − 1
4
ZabZ
ab +
m2Z
2
ZaZ
a − 1
4
AabA
ab + (5.2)
+
1
4
h (h+ 4v)
(
2e2
sin2(2θW )
ZaZ
a +
e2
sin2 θW
W †aW
a
)
+
+iWaW
†
b
(
eAab +
e
tan θW
Zab
)
+
e2
2 sin2 θW
(
|WaW a|2 −
(
W †aW
a
)2)]
5.1. Perturbation theory and approximate decoupling 73
and
Ss= 1
2
=
∫
d4x
√
|g| [e¯ (i∇−me) e+ ν¯i∇ν + u¯ (i∇−mu)u+ d¯ (i∇−md)d+
−h
v
(
mee¯e+muu¯u+mdd¯d
)
+ (5.3)
− e
2
√
2 sin θW
(JaWWa + c.c.)− eJaAAa −
e
sin(2θW )
JaZZa − g1JaG ·Ga
]
where we have suppressed the family indices to avoid confusion with curved spacetime
indices, and made the fermion fields bold face. We are using the induced effective brane
metric with appropriate curved spacetime covariant derivatives and gamma matrices as
described in section 2.1. To leading order in the gravitational field, this introduces a
coupling to the background. Note that now the field strengths have the general form
Fab = ∇aFb −∇bFa + g [Fa,Fb] , (5.4)
and the various currents are
JaA = −e¯γae+
2
3
u¯γau− 1
3
d¯γad (5.5)
JaG =
∑
i
(
u¯γaTiu+ d¯γ
aTid
)
Ti (5.6)
JaW = ν¯γ
a (1− γ5) e+ u¯γa (1− γ5)Vd (5.7)
JaZ =
1
2
[
ν¯γa (1− γ5) ν − e¯γa
(
1− γ5 − 4 sin2 θW
)
e
]
+
+
1
2
[
u¯γa
(
1− γ5 − 8
3
sin2 θW
)
u− d¯γa
(
1− γ5 − 4
3
sin2 θW
)
d
]
, (5.8)
where we have denoted the CKM matrix acting on family space by V to avoid using family
indices. Note that in addition we need to expand Aa = A
(0)
a +A
(1)
a around the background
value A
(0)
a given in equation (3.32).
Now that we know the action, we want to assess the magnitude of the couplings to
the background fields, and among the SM perturbations. This is to justify keeping only
the coupling to the background. For example consider the Higgs field. Ignoring the
74 Chapter 5. Black hole decay and Hawking radiation
perturbation of the graviton and the brane displacements, the equation of motion is
(
∂a∂
a −m2h
)
h+ Γaab∂
bh
− λ
2
h2 (h+ 3v) +
1
2
(h+ 2v)
(
2e2
sin2(2θW )
ZaZ
a +
e2
sin2 θW
W †aW
a
)
− 1
v
(
mee¯e+muu¯u+mdd¯d
)
= 0 , (5.9)
where we have singled out the coupling of the scalar field to the gravitational background
connection in the first line. If we take the neutral non-rotating limit of the background
and look at the term ∂rh which couples to Γaar = ∂r log
√|g| ∼ 1/r, then the coupling in
front is ∼ pr/r ∼ ω/r (we assume pr is a typical radial momentum). In our problem the
typical length scale is the horizon radius. If we look at processes at r ∼ rH and energies
ω ∼ 1/rH , then the coupling to gravity is of order 1/r2H . Table 1.1, shows that all the other
dimensionless couplings appearing in (5.9) are typically small. Furthermore, the operators
are all quadratic (or higher) in the perturbations, so if we assume that the perturbations
have small amplitudes of the form A ∼ ǫ/rH for bosons or A ∼ ǫ/r3/2H for fermions (ǫ≪ 1)
the higher order interactions are naturally suppressed. Note however that though they
are small compared to the gravitational interaction with the background close to the
horizon, they become dominant away from the black hole due to the 1/r dependence of
the gravitational field. In general if we have a SM coupling with a number p of fields and
a reduced dimensionless coupling α (close to the horizon), and we use a typical reduced
amplitude ǫ for all perturbations, then equating the leading term with the power p term
gives
rH
r
ǫ ∼ αǫp ⇒ r ∼ rH
αǫp−1
. (5.10)
So in general if the higher order coupling and the amplitude of the perturbation is small,
such terms are only relevant far away from the black hole. Similar arguments can be used
for all the other perturbations, to motivate neglecting interactions between perturbations
of order higher than 2 (at the level of the Lagrangian). This also holds for the n massless
scalar fields Zm which describe brane fluctuations. Note that though massless, those
fields couple to other perturbations through derivative terms which further suppresses
their excitation at low energy.
These considerations justify treating each channel of the evaporation at linear order
in the equations of motion. In the remainder of this work we consider only the kinetic
parts of the Lagrangian covariantised with respect to the background for each type of
5.2. Hawking radiation 75
perturbation.
Another important question in the semi-classical decay is whether the emitted particles
can interact at longer distances (away from the black hole) before they are detected
by an observer at infinity. In general this depends on the typical length scale for SM
interactions in flat spacetime and on the angular separation between the quanta emitted
from the black hole radially. Furthermore, any such interaction is bound to occur before
the hadronisation of strongly interacting particles, which occurs at a typical length scale
1/ΛQCD ∼ (100 MeV)−1. This scale is orders of magnitude larger than the typical black
hole radii we will consider. Another process occurring before hadronisation at a length
scale of ∼ (1 GeV)−1 is parton showering (also a large distance process). Similarly, it has
been shown [92] that if secondary scattering occurs, it does mostly at the 1 GeV−1 scale.
The two former effects can be taken into account separately with existing showering and
hadronisation programs [93–95], whereas the latter can be safely neglected [92].
5.2 Hawking radiation
Since the pioneering work of Hawking [91] several studies in the literature have examined
the quantisation of various fields in black hole backgrounds which are analytically sim-
ilar to the one we are using [96–100]. In particular the metric (3.37) and the Maxwell
field (3.32) are similar in form to Kerr-Newman, so the quantisation procedure is formally
the same and we will not repeat the technical details [96–100].
For illustrative purposes consider a massless real scalar field Φ. The basic procedure
consists of choosing a vacuum state which corresponds to no incoming particles from
past null infinity J − (i.e. the collapse occurs in a vacuum with no extra particles other
than the collapsing matter). It is well known (see for example the discussion in [48] and
Hawking’s original paper [91]) that the details of the collapse are not important for the
Hawking radiation generated at late times, so alternatively the calculation can be carried
out in the eternal version of the black hole, with the Unruh vacuum [48]. The latter needs
an extra condition: no up-going particles from the past event horizon H− (with respect
to the Kruskal time). Formally the calculation is done by constructing a complete basis
of modes to expand the field operator1 on J − ∪H− (schematically)
Φ =
∑
α
ainΦ
in
α + a
†
in(Φ
in
α )
† +
∑
α
aupΦ
up
α + a
†
up(Φ
up
α )
† , (5.11)
1α denotes a complete set of quantum numbers.
76 Chapter 5. Black hole decay and Hawking radiation
where Φinα and Φ
up
α are the modes corresponding to incoming particles on J − and up-
going particles on H−. Then the boundary condition is implemented by requiring that the
annihilation operators, ain and aup, associated with modes which are positive frequency
with respect to the timelike killing vector kt and the Kruskal time respectively, vanish
when acting on the vacuum. With these boundary conditions, the field operator can
be constructed by scattering the expansion modes through spacetime. At late times (in
J +), the modes decompose into outgoing modes (corresponding to outgoing particles) and
down-going modes (corresponding to particles falling into the black hole). The outgoing
modes become populated in the chosen vacuum state, because part of the initial negative
frequency mode functions (Φin)†α gets absorbed by the black hole and part gets reflected.
The reflected part is decomposed (at J +), as a sum of a wave with negative frequency, and
a non-zero wave with positive frequency. This corresponds to part of a negative energy
mode being absorbed by the black hole (which reduces its mass) and a positive energy
mode escaping to infinity.
Once the field operator is known, the expectation value of any other operators (such
as the energy momentum tensor) can be computed. The physically relevant fluxes at J +
can be obtained from specific components of those expectation values. The main result
of Hawking’s original paper is then that black holes emit a continuous flux of particles.
In horizon radius units, the fluxes of particle number, energy, angular momentum and
charge are respectively [91]
d2N
dtdω
=
1
2π
∞∑
j=|s|
j∑
m=−j
1
exp(ω˜/TH)± 1T
(4+n)
k (ω, µ, a, q, Q) , (5.12)
d2E
dtdω
=
1
2π
∞∑
j=|s|
j∑
m=−j
ω
exp(ω˜/TH)± 1T
(4+n)
k (ω, µ, a, q, Q) , (5.13)
d2J
dtdω
=
1
2π
∞∑
j=|s|
j∑
m=−j
m
exp(ω˜/TH)± 1T
(4+n)
k (ω, µ, a, q, Q) , (5.14)
d2Q
dtdω
=
1
2π
∞∑
j=|s|
j∑
m=−j
q
exp(ω˜/TH)± 1T
(4+n)
k (ω, µ, a, q, Q) , (5.15)
where ω˜ = ω −mΩH − qΦH , k = {j,m} are the angular momentum quantum numbers,
5.2. Hawking radiation 77
s is the helicity of the particle
TH =
(n+ 1) + (n− 1)(a2 +Q2)
4π(1 + a2)rH
, (5.16)
and the signs ± are for fermions and bosons respectively. ΩH = a/(1 + a2) and ΦH =
Q/(1 + a2) are the angular velocity and electric potential of the horizon respectively. ΦH
can be defined using the timelike Killing vector close to the horizon. For metric (3.37),
we can pick a Killing vector field which is timelike at a given point, using the two Killing
vector fields kt = et and kφ = eφ. We denote such a vector
kp = et + Ωpeφ , (5.17)
where the subscript p labels a space-time point. Then if kp is timelike at p, Ω− < Ωp < Ω+
where
Ω± =
−gtφ ±
√
g2tφ − gttgφφ
gφφ
. (5.18)
At the horizon Ω+ = Ω− = ΩH , so in some sense there is a natural vector field which
defines the timelike direction close to the horizon. The electric potential at the horizon
ΦH is defined as the projection of the Aa field along kH . It can also be shown that ΩH
corresponds to the angular velocity of a physical observer close to the horizon whose frame
is dragged by the gravitational field of the rotating black hole [101].
Finally T
(4+n)
k are the so called transmission factors defined as the fraction of an
incident wave from infinity which is absorbed by the black hole. The boundary conditions
are such that close to the horizon the wave is purely ingoing for the above physical
observers [101].
Later on we will be particularly interested in the effect of electric charges. Before
going into a detailed calculation of the fluxes, it is instructive to look at an estimate from
the Schwinger formula for fermions [102]:
dN
dV dt
=
q2E2
π2
+∞∑
n=1
e−
nπµ2
qE
n2
≃ q
2E2
6
, (5.19)
where we took the small mass limit and E is the electric field. Equation (5.19) is valid in
flat space for a uniform electric field, and it gives the rate of production of opposite charge
pairs due to the electric field only. For (3.32), we know that the electric field drops like
78 Chapter 5. Black hole decay and Hawking radiation
1/r2 so strictly speaking this formula is not valid. Nevertheless we can still use (5.19) to
estimate the contribution of the background electric field to particle production and com-
pare it with the contribution from the gravitational field alone (i.e. the typical Hawking
flux for a neutral black hole). A rough estimate is obtained by considering the electric
field at the horizon and a volume of order (2rH)
3 around the black hole. Using our system
of units and noting that the electric field at the horizon is EH ∼ Q/r2H we get
dN
dt
rH ∼ q2Q2 = z2Z2α2 ≃ z2Z210−5 . (5.20)
So for order z, Z ∼ 1 charges we get a very small rate when compared to the typical
Hawking fluxes for a neutral black hole (which are of order ∼ 1). This indicates that pair
production due to the gravitational field is much stronger than pair production due to the
electric field. So the common claim that TeV-scale black holes lose their charges earlier
in their lifetime, is not necessarily true on the basis of Schwinger discharge alone [24]. We
will confirm this result with a more detailed calculation in chapter 6.
5.3 Perturbations of a brane charged black hole
Previous attempts to model the evaporation focused on neutral rotating black holes [103–
113]. In this section we use the background constructed in section 3.2 to generalise the
treatment to the case when both the background and some fields on the brane are charged.
In addition we also introduce particle mass which is necessary for heavy SM particles such
as the W,Z bosons and the top quark.
5.3.1 Wave equations I: Coupling the background
In section 5.1 we have argued that the dominant interaction comes from the coupling
of the various SM fields to the background. We start by writing the action for a brane
charged complex scalar field
SΦ =
∫
d4x
√
|g|
(
1
2
(DaΦ)
†DaΦ− 1
2
µ2Φ†Φ
)
, (5.21)
5.3. Perturbations of a brane charged black hole 79
where we have grouped together the covariant derivative with the electromagnetic coupling
in the gauge covariant derivative
Da = ∇a + iqAa . (5.22)
The mass of the particle is µ and we have anticipated the matching of the coupling q to
the one introduced in section 3.2.3. Variation of (5.21) gives the wave equation
(
DaDa + µ
2
)
Φ = 0 . (5.23)
To check the normalisation of the coupling is correct we take the classical limit in flat
space-time. Consider a slowly varying vector potential Aa, set
Φ ∼ eiS , (5.24)
and identify the mechanical 4-momentum of the classical particle with pa = −∇aS− qAa.
Then to leading order, equation (5.23) gives the mass-shell condition
pap
a = µ2 .
Conversely, Pa = −∇aS is the usual canonical momentum of the classical particle so we
match q to z
√
α as in equation (3.43). This coupling agrees with other well studied cases
(see for example [97,114–116]). Note that this differs by a factor of
√
4π (z
√
α = ze/
√
4π)
compared to (5.2) and (5.3) which is due to the different normalisation of the background
electromagnetic field (3.32). The latter is given by the SM electomagnetic field multiplied
by
√
4π. In the remainder we use this new normalisation.
For fields with higher spin the procedure is similar. For a Dirac field the action is
SΨ =
∫
d4x
√
|g|Ψ¯ (iD − µ)Ψ , (5.25)
with the gauge covariant derivative
D = ∇+ iqA . (5.26)
80 Chapter 5. Black hole decay and Hawking radiation
For a massive complex vector field V a we have the action
SΨ =
∫
d4x
√
|g|
(
−1
2
V †abV
ab + µ2V †a V
a + iqVaV
†
b A
ab
)
, (5.27)
for the W a field action (q =
√
α), or Za/
√
2 action (q = 0), or the electromagnetic field
Aa/
√
2 when q = µ = 0.
5.3.2 Wave equations II: Separability
In this section we present the separated wave equations on the background of section (3.2).
5.3.2.1 The scalar field
Using equation (3.37), the separation ansatz Φ = e−iωt+imφR(r)S(θ) and inserting in (5.23)
we obtain the radial equation
∆
d
dr
(
∆
dR
dr
)
+
(
K2 −∆U)R = 0 , (5.28)
where
K = ω(r2 + a2)− am− qQr (5.29)
U = µ2r2 + Λc,j,m + ω
2a2 − 2aωm . (5.30)
The boundary condition at the horizon is [101]
R = x
−iK⋆
δ0 (1 + . . .) (5.31)
with x = r − 1, K⋆ = ω(1 + a2)− am− qQ, and δ0 = n+ 1 + (n− 1)(1 + a2 +Q2) is the
leading order coefficient of the expansion of ∆ in powers of x. The angular equation has
exactly the same form as for the chargeless case studied for example in [105]:
1
sin θ
d
dθ
(
sin θ
dS
dθ
)
+
(
c2 cos2 θ − m
2
sin2 θ
+ Λc,j,m
)
S = 0 , (5.32)
but now c2 = a2(ω2 − µ2) and Λc,j,m is the corresponding angular eigenvalue obtained by
imposing regularity of the solution at cos θ = ±1. For a = 0 we have the closed form
Λ0,j,m = j(j + 1) with j a non-negative integer [117].
5.3. Perturbations of a brane charged black hole 81
Equation (5.28) is similar to the chargeless case with the additional terms:
1. Q2 in ∆, which changes the location of the horizon and therefore the Hawking
temperature of the black hole.
2. qQr in K which is related to the electric potential.
In K, ω is shifted by
− am
r2 + a2
− qQr
r2 + a2
. (5.33)
Evaluated at the horizon, both quantities in (5.33) are associated with the well known
phenomenon of superradiance [101, 118–121], i.e. for ω˜ < 0 a wave incident from infinity
is scattered back with a larger amplitude. This factor is also present in the expressions
for the fluxes such as (5.12) where the Planckian suppression factor in the denominator
becomes negative for superradiant modes.
5.3.2.2 The Dirac field
For a fermion field the standard procedure to separate the wave equation is to use the
Newman-Penrose formalism2. The method has been developed for the Kerr metric by
Chandrasekhar [123] and applied to the Kerr-Newman background by Page [115]. Page
points out how a simple substitution of some of Chandrasekhar’s quantities suffices to
obtain separated equations for the fermion field with charge. Below we state the final
result and refer the technical details to references [115, 122,123].
The separated wave equation for the massive charged Dirac field relies on the ansatz
Ψ = e−i(ωt−mφ)χ(r, θ) where
χ =


(r − ia cos θ)−1P−1/2(r)S−1/2(θ)√
2∆−1/2P+1/2(r)S+1/2(θ)√
2∆−1/2P+1/2(r)S−1/2(θ)
−(r + ia cos θ)−1P−1/2(r)S+1/2(θ)

 . (5.34)
The radial and angular equations are
∆
1
2
(
d
dr
− 2siK
∆
)
P−s = (λ+ 2isµr)Ps (5.35)
2Technical details on this formalism can be found in several textbooks – see for example [48, 122].
82 Chapter 5. Black hole decay and Hawking radiation
[
d
dθ
+ 2s
(
aω sin θ − m
sin θ
)
+
1
2
cot θ
]
S−s = (2sλ+ aµ cos θ)Ss (5.36)
where λ is the angular eigenvalue. To make contact with well know limits, it is useful to
obtain second order radial and angular equations by elimination (the prime denotes d
dr
):
d2Ps
dr2
+
(
(1− |s|)∆
′
∆
+
2isµ
λ− 2isµr
)
dPs
dr
+
+
[
K2
∆2
− isK
∆
∆′
∆
− 4s
2µ
λ− 2isµr
K
∆
+
2isK ′ − λ2 − µ2r2
∆
]
Ps = 0 (5.37)
1
sin θ
d
dθ
(
sin θ
dSs
dθ
)
+
aµ sin θ
−2sλ + aµ cos θ
(
d
dθ
− 2s
(
aω sin θ − m
sin θ
)
+
cot θ
2
)
Ss+
+
[
a2(ω2 − µ2) cos2 θ − 2saω cos θ − (m+ s cos θ)
2
sin2 θ
+ λ2 − a2ω2 + 2aωm− |s|
]
Ss = 0 .
(5.38)
Here s = ±1/2. In the zero mass limit we recover a radial equation with the same analytic
form as in references [111,112] except for the extra term in K ′. Similarly, in that limit, the
angular equation is exactly the same as for the spin-half spheroidal functions. In general,
the eigenvalues are obtained by imposing regularity of the solution at cos θ = ±1.
Finally setting the rotation parameter a to zero, note that the angular equation is
the same with or without mass and charge. Then the angular eigenvalue takes a closed
form [117] (λ = j + 1/2 with j a positive semi-integer) and we do not need to integrate
the angular equation to study the effects of both mass and charge. This simplification
was explored for example in Page’s paper in four dimensions [114].
5.3.3 Higher spins
For higher spins the procedure is similar and it has been studied in the literature for some
special cases of the background metric we are using. Most notably the massless vector
perturbations [103, 108] and the tensor gravitational perturbations [124, 125] in the zero
charge limit with non-zero angular momentum.
When the background charge is present there are several extra complications to con-
sider. Regarding electromagnetic perturbations, it is known in four dimensions that they
couple linearly to gravitational perturbations (at the level of the equations of motion) for
5.3. Perturbations of a brane charged black hole 83
the Kerr-Newman black hole (see for example Sect. 111, Chapter 11 of [122] and refer-
ences therein). Similarly, in the higher dimensional case, we would expect them to couple
to gravitational modes on the brane. However, in the limit of small charge, the pertur-
bations should approximately decouple. This is indeed the case and an approximation
scheme was developed by Dudley and Finley [126]. It amounts to considering separately
one perturbation (either electromagnetic or gravitational) while setting the other to zero
on the fixed background. This approximation was used for example in [127] and [128] to
compute quasinormal modes. In [128] this was compared to other methods to confirm
the validity of the approximation for small Q (the special case J = 0 was used). The
approximate second order wave equation for a perturbation of spin s is [128]
∆1−s
d
dr
[
∆1+s
dR
dr
]
+
[
K2 − isd∆
dr
K +∆
(
2is
dK
dr
− λ2
)]
R = 0 , (5.39)
where K is the same as in (5.29), but does not contain the particle charge q-term. This
correctly reduces to the exact result for scalars and fermions when a background charge
is present (if K contains the q-term in (5.29)) and it describes the electromagnetic or
gravitational perturbations in four dimensions, approximately, for small Q, or exactly for
Q = 0. Note also that the angular eigenvalue is determined by the angular equation (5.38)
which is valid for s = 0, 1/2, 1 when µ = Q = 0.
An important feature of electromagnetic and gravitational perturbations, compared
to scalars and fermions, is that they are electrically neutral. No electric coupling means
that qualitatively not much will change compared to the case of no background charge.
Specially for small charges we see from (5.39) that the charge of the background only
enters through the Q2 term in ∆. This affects mostly the Hawking temperature which in
the small Q limit will simply rescale the flux curves without much difference in shape. In
fact we will see later on in chapter 6 (figure 6.4) that for scalars and fermions, the effect
of a small background charge on neutral particles is indeed small on the transmission
factors and the flux curves are simply rescaled by the different Hawking temperature in
the thermal factor. So qualitatively nothing changes for neutral scalars and fermions
which obey (5.39) and we would expect the same for higher spins.
Furthermore, if we assume electroweak symmetry is not restored outside the black hole
and that the electrically charged weak vector boson W and the neutral Z provide a good
effective description of the weak degrees of freedom, it is tempting to guess that (5.39)
holds similarly for those perturbations (with K containing the electric q-coupling). This
84 Chapter 5. Black hole decay and Hawking radiation
is because the black hole background can only be electrically charged (or colour charged)
so the background values of the weak field perturbations vanish. Then we would ex-
pect (5.39) to be exact since there is no reason for the weak field perturbations to couple
to the linearized gravitational perturbations. This is in contrast with the equations for
electromagnetic perturbations where terms linear in the gravitational perturbations arise
from linearising bilinears in the gravitational/electromagnetic fields around their back-
ground values. For weak W and Z field perturbations (as for scalars and fermions) such
gravitational terms can not be present because even if they exist before linearisation,
when evaluated on the background for the W and Z fields they are identically zero.
Furthermore, because theW and Z fields are massive, they are described by a complex
or a real Proca field respectively (see equation (5.27)), which is an extra complication.
Alternatively, if electroweak symmetry is restored in the region outside the black hole3,
then we have to use the fundamental weak gauge fields associated with the SU(2)L×U(1)Y
sector of the Standard Model (instead of the electromagnetic, the W and the Z fields).
Due to these extra complications, the detailed study of massive charged vector pertur-
bations will be left to future work and the charge of the background will be neglected for
electromagnetic perturbations, since the former are neutral and the effect is negligible.
5.3.4 Decomposition of spheroidal waves into plane waves
For the purpose of developing the event generator in chapter 7 it will be important to
know how to express the quantum states associated with the spheroidal waves used to
expand the fields, in terms of plane waves. The latter are related to the natural basis
of quantum states for an observer in flat spacetime who typically measures a track in a
detector with a certain well defined 4-momentum. We work out in detail the result for
a spin-1/2 field which is the first relatively non-trivial case (for a scalar field a simplified
version of the reasoning is straightforward). Since the effect of particle mass is a small
perturbation (which will be implemented in a future release of the generator) we restrict
the argument to the massless limit.
It is well known (see for example [6]), that in the massless limit the Dirac field de-
composes into two independent fields with helicities h = ±1. If we denote each of those
3This should be the case if the black hole size is smaller than the electroweak breaking scale which is
typically 1/mW , the inverse mass of the W .
5.3. Perturbations of a brane charged black hole 85
2-component spinors by Ψh, then their equations of motion in flat space become
(∂t + hσ · ∂) Ψh = 0 , (5.40)
where σ = (σ1, σ2, σ3) are the Pauli matrices. Then whichever spatial coordinates x we
decide to use, the field operator will have the expansion (in terms of positive and negative
energy solutions)
Ψˆh =
∑
λ
1√
2ωλ
[
aˆh,λψh,λ(x)e
−iωλt + bˆ†h,λψ−h,λ(x)e
iωλt
]
, (5.41)
where aˆh,λ, bˆ
†
h,λ are respectively, the usual fermionic annihilation and creation operators
for particles and anti-particles; λ is an unspecified complete set of quantum numbers; ψh,λ
are spinorial normal modes (ω + ihσ · ∂)ψh,λ = 0, normalised according to∫
d3xψ†h,λψh′,λ′ = δh,h′δ(λ, λ
′) ; (5.42)
δ is a Dirac delta function such that
∑
λ′
f(λ′)δ(λ, λ′) = f(λ) ; (5.43)
and the sum sign represents integration if the quantum numbers are continuous. Similarly
to Klein-Gordon theory, it is possible to define a scalar product between 2-spinors ψ, χ
using a bilateral derivative:
(ψ, χ) = i
∫
d3x
(
ψ†∂tχ− ∂tψ†χ
)
. (5.44)
Then we find an expression for the operators
aˆh,λ =
(
ψh,λe
−iωλt, Ψˆh
)
(5.45)
bˆ†h,λ =
(
ψ−h,λe
iωλt, Ψˆh
)
. (5.46)
Ψh can also be expanded in a basis of spinors with a different set of quantum numbers γ,
which may be associated with a different choice of coordinates. The previous expressions
86 Chapter 5. Black hole decay and Hawking radiation
will then give a Bogoliubov transformation between operators in one basis and the other:
aˆh,λ =
∑
γ
1√
2ωγ
[(
ψh,λe
−iωλt, ψh,γe
−iωγt
)
aˆh,γ +
(
ψh,λe
−iωλt, ψ−h,γe
iωγt
)
bˆ†h,γ
]
(5.47)
bˆ†h,λ =
∑
γ
1√
2ωγ
[(
ψ−h,λe
iωλt, ψh,γe
−iωγt
)
aˆh,γ +
(
ψ−h,λe
iωλt, ψ−h,γe
iωγt
)
bˆ†h,γ
]
. (5.48)
The second term of the first expansion, and the first term in the second one will be zero.
This is because they are responsible for particle creation, which does not occur in this
change of basis, given that we are in the same Lorentz frame in Minkowski space-time.
This can be seen explicitly for our case of a transformation between plane wave states
and spheroidal states. First note that in the Kinnersley basis the plane wave spinors take
the form [129]
χ+
p
=
(
ei
φ˜
2 cos θp
2
cos θ
2
− e−i φ˜2 sin θp
2
sin θ
2
ei
φ˜
2 cos θp
2
sin θ
2
+ e−i
φ˜
2 sin θp
2
cos θ
2
)
, (5.49)
χ−
p
=
(
ei
φ˜
2 cos θp
2
sin θ
2
+ e−i
φ˜
2 sin θp
2
cos θ
2
ei
φ˜
2 cos θp
2
cos θ
2
− e−i φ˜2 sin θp
2
sin θ
2
)
, (5.50)
where Ωp = (θp, φp) defines the orientation of the momentum vector p, φ˜ = φ − φp
and we have omitted the eip·x dependence. As long as we are looking at a fixed plane
wave, φ can be shifted by choosing a new origin, which amounts to setting φp = 0. The
upper/lower component of χ±
p
has the interesting property of being invariant under the
exchange θp ↔ θ. Using the asymptotic form (3.7) for the Cartesian coordinates in terms
of spheroidal coordinates we obtain
p · x = cos θp cos θ
√
(ar)2 + c2 + ar sin θp sin θ cosφ , (5.51)
which is again symmetric under the exchange of θ’s. On the other hand, the spheroidal
spinors in the Kinnersley basis have the form [112]
χ±λ = e
imφ
(
−R
±
j,m(r)S− 1
2
,j,m(c, cos θ)
+R
±
j,m(r)S 1
2
,j,m(c, cos θ)
)
. (5.52)
We are seeking for a relation between plane waves and spheroidal waves. This can be
5.3. Perturbations of a brane charged black hole 87
achieved by writing down the general decomposition
χ±
p
eip.x =
∑
j,m
cλ(p)χ
±
λ (x) , (5.53)
where now λ = {ω, j,m}. But we know that the upper/lower component of the left hand
side spinor with helicity ± is invariant under exchange of θ’s, so the cλ prefactor must be
proportional to the upper/lower spheroidal function with argument θp. Furthermore we
put back the φp dependence by shifting φ to obtain
χh
p
eip.x =
∑
j,m
c˜λ · S⋆−h,j,m(c,Ωp)χhλ(x) , (5.54)
where the φp dependence in S
⋆
−h,j,m(c,Ωp) is implicit. An integral relation is obtained
when multiplying by an appropriate spinor, integrating over x and using the normalisation
condition (5.42)
∫
d3xχh
′
λ (x)
†χh
p
eip.x = c˜λ · S⋆−h,j,m(c,Ωp)δh,h′δ(ω − ωp) . (5.55)
Finally, we can use this in the Bogoliubov transformations to obtain
aˆ†h,λ ∝
∫
dΩp · S−h,j,m(c,Ωp)aˆ†h,p (5.56)
bˆ†h,λ ∝
∫
dΩp · S⋆h,j,m(c,Ωp)bˆ†h,p , (5.57)
where the prefactor is independent of the angular variables. These expressions show how
a state of a particle/anti-particle with helicity h decomposes into plane wave states with
the same helicity and momentum orientations Ωp. The probability of a certain orientation
is given by the square modulus of the spheroidal function with spin weight s = ∓h.
Since massless scalar, spinor and vector perturbations all follow the same master equa-
tions, these conclusions apply similarly to the remaining cases.
88 Chapter 5. Black hole decay and Hawking radiation
Chapter 6
Analytic and numerical study of
perturbations
In this chapter we perform a numerical study of the perturbations discussed in the previous
chapter. We start by analysing the angular equations using a series expansion method,
which was described in [4]. In sections 6.2, 6.3 and 6.4 we discuss in detail the methods
used to solve the radial equations for massive charged scalar and fermion fields. We
use first an approximate method which is valid for low energies [62,130–132] and a WKB
approximation in the high energy limit (they reduce in some limits to the results in [62,97,
130–132] which can be used as a check). The full numerical result is obtained in section 6.3
and a discussion of the effects of charge and mass is presented in section 6.4. This study
was published in [1, 2]. Section 6.4.4 is a review of the known effects of rotation which
will be useful to the implementation in the CHARYBDIS2 event generator. In section 6.5 we
make contact between the geometrical cross-sections from the analysis of geodesic motion
and the high energy limit of wave propagation. We end the chapter with a summary of
the main conclusions of the studies presented.
6.1 The angular equations
Before integrating the radial equations, the angular eigenvalues must be computed because
they enter the radial potential. Furthermore to study angular distributions the angular
functions are necessary, so we need to find the full solution. In general this must be done
numerically.
The most important cases we are studying are massless brane degrees of freedom. For
89
90 Chapter 6. Analytic and numerical study of perturbations
massless particles up to spin-1 the spheroidal wave functions Sk(c, x = cos θ), satisfy the
differential equation
[
d
dx
(
(1− x2) d
dx
)
+ c2x2 − 2hcx− (m+ hx)
2
1− x2 +Ak(c) + h
]
Sk(c, x) = 0 , (6.1)
where c = aω and Ak(c) = λ2 − a2ω2 + 2aωm − |h| − h is the angular eigenvalue and
for notational convenience we have set1 s → h. Note however that this equation also
works for massive scalar fields with c = a
√
ω2 − µ2. In the event generator presented in
chapter 7, we will use the method of Leaver [133], so we present this in detail. In [133]
the spheroidal wave functions for arbitrary spin are expanded as a series in x = cos θ
around x = −1. In that paper there is an extra parameter p which can take either sign.
Alternatively, we can expand around x = +1. So in general, we can construct three more
expansions. Even though all four of them converge uniformly in x ∈ [−1, 1], the numerical
errors behave differently if we consider a fixed region. Therefore it is useful to look at all
the options and use different expansions for different regions. The only extra complication
will be to match them in a common region. The possible expansions are
Ss,pk (c, x) = e
px (1 + x)α (1− x)β
+∞∑
n=0
bn (1 + sx)
n (6.2)
where α, β are chosen as to reproduce the correct behaviour around the regular singular
points x = ±1,
α =
∣∣∣∣m− h2
∣∣∣∣ , β =
∣∣∣∣m+ h2
∣∣∣∣ , (6.3)
and s = ±1 for expansions around x = −1 and x = 1 respectively. Substituting into (6.1)
we obtain the recurrence relation (for n > 0),
bn =
(
1
2
+
ǫ1
n+ σ
− ǫ2
n(n+ σ)
)
bn−1 +
(
1 +
γ
n
) sp
n+ σ
bn−2 (6.4)
and the simplifying condition p2 = c2 ⇒ p = ±c. We can set b−1 = 0 and choose the
1This is just because the s symbol is used in this section for another purpose.
6.1. The angular equations 91
normalisation b0 = 1. The parameters above are given by the following expressions:
σ = α(s+ 1) + β(1− s)
ǫ1 =
α(1− s) + β(1 + s)− 1− 4sp
2
ǫ2 =
Ak + h(h+ 1) + c2 − (α + β)(α+ β + 1)
2
+ α + β
+p [α(s+ 1) + β(s− 1)− s+ sh sign(p)]
γ = α + β − 1 + h sign(p) . (6.5)
For almost all values of Ak, the ratio bn+1/bn in (6.4) goes to 1/2 as n → ∞. So as the
order increases, the remainder will behave as (when N →∞)
+∞∑
n=N
bn (1 + sx)
n → bN (1 + sx)N 2
1− sx , (6.6)
which diverges at x = s and goes exponentially faster to zero as we move closer to x = −s.
So if we do not tune to particular values ofAk this is how the numerical errors will behave.
However, from the analytical point of view we still want to have an expansion which
converges uniformly, which is not the case in general as suggested by (6.6). Therefore we
need to know how the ratio bn+1/bn behaves for large n. Following [134] we determine
this by assuming that for large n
bn+1
bn
∼ bn
bn−1
∼ k(n) + . . . (6.7)
where . . . denotes sub-leading contributions. Inserting this in (6.4) and solving for k we
get two possible behaviours
k(n) ∼


1
2
+ 2sp
n
−2sp
n
→


1
2
0
. (6.8)
It is straightforward to check that uniform convergence for x ∈ [−1, 1] is only achieved
in the second case. A solution with these convergence properties is called a minimal
sequence solution [133]. Furthermore, there is a theorem (see Theorem 1.1 of [135] and
the reasoning in [134]) which ensures that the sequence obtained from the eigenvalue
problem in (6.4) with the initial conditions given above is a minimal sequence. Therefore,
92 Chapter 6. Analytic and numerical study of perturbations
after determining the eigenvalue we can compute the solution (up to a normalisation) by
fixing b0 = 1.
However, from the numerical point of view, a small error in the eigenvalues implies that
bn+1/bn will fail to go to zero when numerically evaluated through (6.4). So the remainder
will actually behave as (6.6) since in practice we are approximating the eigenfunction by
a nearby function, for which the sequence of expansion coefficients behaves as a dominant
solution of (6.4) when n is large2. Thus the numerical radius of convergence will be a bit
smaller and the expansion will fail at x = s.
We avoid the above convergence problem by using two expansions, one around each
singular point x = ±1, and then matching the normalisation in a region where both
converge appropriately. A simple procedure for matching follows from the observation
that
S+ = AS−
⇒ log |A| = log |S+| − log |S−| , (6.9)
where A is a constant and S± denotes expansion around x = ±1 respectively. We can find
|A| by averaging the quantity on the right-hand side over an overlap region. Furthermore
we can estimate the relative matching error through the quadratic deviation,
∆|A|
|A| = ∆ log |A|
∼
√〈
log2 |A|〉− 〈log |A|〉2 (6.10)
where 〈. . .〉 denotes average over the points used. In practice we use points in the overlap
region x ∈ [−0.25, 0.25].
For the expansions above to be useful we need to estimate an order of truncation,
ntrunc. A simple condition is that the ratio of consecutive terms bn+1/bn is approximately
constant above the order of truncation. This is equivalent (in the region sx < 0 where
there is strong convergence) to an exponentially suppressed upper bound on the remainder
when ntrunc is large (see (6.6)). From this and (6.4)
ntrunc ∼ max{[j + |h|+ 4c], [
√
|Ak|+ |h|(|h|+ 1) + (c+ |m|+ 2|h|+ 3)2]} . (6.11)
2See for example [135] for a discussion of numerical issues when generating minimal sequences which
are solutions of three-term recursion relations.
6.1. The angular equations 93
An initial estimate of the truncation error is obtained from (6.6). Another criterion for
truncation is that the first neglected higher order term is small compared to the truncated
sum.
In practice, after the spheroidal function is calculated with a certain truncation, then
ten more terms in the series are included and the two estimates are compared. Simul-
taneously, the first neglected term is compared with the estimate for the sum. If these
errors are still large then ten more terms are calculated. This procedure is repeated until
the desired accuracy is obtained.
To compute the eigenvalues Ak efficiently, the recurrence relation can be put in a
symmetric tridiagonal form by performing a change of basis,
an = xnbn (6.12)
such that the coefficient of the bn−2 term for a certain order n is the same as the coefficient
of the bn−1 term at order n− 1. This is possible if
xn =
√
n(n + σ)
−sp(n + 1 + γ)xn−1 =
√
n(n+ σ)
c(n+ 1 + γ)
xn−1 , (6.13)
were we have taken (without loss of generality) x0 = 1 and the convenient choice p = −sc.
Furthermore we checked that xn 6= 0 ∀n ≥ 1, so the transformation is well defined. Thus
the recurrence relation takes the form
√
cn(n+ σ)(n + 1 + γ)an − n
(
n+ σ
2
+ ǫ1
)
an−1
+
√
c(n− 1)(n− 1 + σ)(n+ γ)an−2 = −ǫ2an−1. (6.14)
This is a tridiagonal symmetric eigenvalue problem which can be solved numerically very
efficiently, by starting with a truncation order (as estimated above) and checking for
precision by repeating the calculation with some more corrections.
Various other methods exist in the literature to integrate the angular equation (6.1)
and obtain the angular eigenvalue. A particularly useful check of the numerical expansions
developed above that was used, were the series expansions of the angular eigenvalue
provided in [117]. In addition, the spheroidal functions were checked numerically against
a publicly available code for scalars [136]. For higher spins, several properties of the
spheroidal functions as well as some special cases were checked against the results in [117].
94 Chapter 6. Analytic and numerical study of perturbations
6.2 The radial equations I: Analytic methods
In this section we present analytic approximations to solve the radial equations and com-
pute transmission factors. These solutions are interesting because they provide the main
qualitative features in a good part of the interesting range of parameters where they are
valid. Furthermore, they can be used as a check of the results obtained numerically and
provide simple analytic expressions which can be quickly evaluated. We are going to focus
mostly on a low energy approximation, but we also present a high energy approximation.
The low energy approximation has been shown, in earlier studies, to give good results
even in the intermediate energy regime for spins up to one [130] so it will provide a good
overall qualitative picture of how the transmission factors behave in the full range.
We present the main steps of the calculation using the method in [130] adapted to
our problem. It consists of writing down approximations for the radial equation in two
regions: one near the horizon (Near Horizon solution) and the other one far from it (Far
Field solution). This provides two analytic approximations which hold exactly close to
the horizon and far from it respectively. The final step is to extrapolate them into a
common intermediate region to be matched. Below we summarize the solutions and keep
track of the conditions of validity. Since the new cases we are studying are for non-zero
electric charge and particle mass for spins s = 0, 1/2 we present the details for these fields
only (though the procedure is similar for other spins).
6.2.1 Near horizon equation
Equations (5.37) and (5.38) are valid for both spin-zero and spin-half fields. The analytic
approximations we use are valid for the massive charged scalar field, but however it turns
out they only work for the massless limit of charged fermions. Therefore we work with
the radial equation (5.37) but set µ = 0 for fermions. Following [130] close to the horizon
6.2. The radial equations I: Analytic methods 95
define the quantities
f ≡ ∆
r2 + a2 +Q2
A ≡ n + 1 + (n− 1)a
2 +Q2
r2
≃ A|r=1 ≡ A∗
B ≡ 1− |s|+ 2|s|+ n(r
2 + a2 +Q2)
r2A
− 4(a
2 +Q2)
r2A2
≃ B|r=1 ≡ B∗
P ≡ K
2
r2A2
≃ ω(1 + a
2)− am− qQ
A∗
≡ p
D ≡ r
2 + a2 +Q2
r2A2
[
λ2 + µ2r2δs,0 + 2isqQ− 4isωr
] ≃ D|r=1 ≡ D∗ . (6.15)
Then equation (5.37) is equivalent to
f(1− f)d
2R
df 2
+ (1−Bf)dR
df
+
[
P 2 − isP
f
+
P 2 − isP −D
1− f
]
R = 0 , (6.16)
and the approximations on the right hand side of each line of (6.15) can be used. The
approximations require the condition r − 1≪ 1. Equation (6.16) can be solved in terms
of hypergeometric functions. The general solution is a combination of two linearly inde-
pendent hypergeometric functions. The wave must be purely ingoing at the horizon. This
implies R ∼ e−ipr∗ with r∗ the tortoise coordinate defined by dr∗ = dr/f (up to a con-
stant). Then, it can be shown that the convergent solution with this boundary condition
is [137]
RNH = f
α(1− f)βF (a, b, c; f) , (6.17)
where
α =
|s| − s
2
− ip
β = 1− |s|+B∗
2
−
√(
1− |s|+B∗
2
)2
− p2 + isp+D∗
a = α + β − 1 +B∗
b = α + β
c = 1− |s|+ 2α . (6.18)
In the next section an extrapolation of this solution away from the horizon will be needed,
i.e. around f → 1 ⇒ 1 − f ≃ (1 + a2 + Q2)/rn+1. Note that the larger the value of n,
96 Chapter 6. Analytic and numerical study of perturbations
the more consistent this condition is with r − 1 ≪ 1 so the terms neglected in approxi-
mations (6.15) become less important3. Using some identities that relate hypergeometric
functions with argument f to argument 1−f and expanding around f = 1 we obtain [137]
(up to an overall normalisation constant)
R→ A1r−(n+1)β + A2r−(n+1)(2−β−B∗) , (6.19)
with
A1 =
(1 + a2 +Q2)βΓ(c− a− b)
Γ(c− a)Γ(c− b)
A2 =
(1 + a2 +Q2)2−|s|−β−B∗Γ(a+ b− c)
Γ(a)Γ(b)
. (6.20)
When matching powers of r in the next section we will have to make the approximations
ω, a,Q, µ≪ 1. Then
−(n + 1)β ≃ −1
2
+
√
1
4
+ λ2
−(n + 1)(2− β − B∗) ≃ −1
2
−
√
1
4
+ λ2 . (6.21)
For a≪ 1 the neglected terms are of order ω2 or µ2. So taking into account the leading
behaviour of λ2 when s = |s| the approximation is equivalent to
ω, µ≪
√
j(j + 1) + 2|s| . (6.22)
So the larger the ℓ and |s| the wider the energy range where the approximations work.
6.2.2 Far field solution and low energy matching
Away from the black hole r → +∞ we approximate ∆ ≃ r2 and
K
∆
≃ ω − qQ
r
+
ω(1 + a2 +Q2)
r
δn,0 . (6.23)
Equation (6.23) contains: the energy; a long range electric potential, and a long range
gravitational potential in four dimensions. Keeping terms up to order 1/r2 in equa-
3This improvement of the approximation for large n has been noted in [130].
6.2. The radial equations I: Analytic methods 97
tion (5.28)
d2R
dy2
+
2
y
dR
dy
+
[
1 +
ǫ
y
− γ
y2
]
R = 0 (6.24)
with
y = kr
k2 = ω2 − µ2δs,0
ǫ =
2isω − 2ωqQ+ (2ω2 − µ2) (1 + a2 +Q2)δn,0
k
(6.25)
γ = λ2 − q2Q2 − ω(1 + a2 +Q2) [2qQ− ω(1 + a2 +Q2) + 2is] δn,0 .
Note again that we are not studying the massive case for fermions. The δs,0 factor in k
is emphasizing this – it does not mean the µ2 term is absent for s = 1/2. The general
solution of equation (6.24) is given in terms of Kummer functions [137]
RFF = e
−iyyσ [B1M(u, v, 2iy) +B2U(u, v, 2iy)] , (6.26)
where
σ = |s| − 1
2
+
√(
|s| − 1
2
)2
+ γ
u = σ + 1− |s|+ i ǫ
2
v = 2(σ + 1− |s|) . (6.27)
Equation (6.26) can be matched to the near horizon solution in the limit y ≪ 1. This
conditions implies r ≪ 1/k, so for consistency with the limit r ≫ 1 we need k small.
Using the asymptotic expansion for the Kummer functions, the stretched solution is [137]
RFF → kσ
(
B1r
σ +B2
Γ(v − 1)
Γ(u)
(2ik)1−vrσ+1−v
)
. (6.28)
It can be easily shown that within the same approximations as in equation (6.21) the
r-powers match with those in equation (6.28). Then, up to an overall common constant
B1 = A1
B2 = A2
Γ(u)(2ik)v−1
Γ(v − 1) . (6.29)
98 Chapter 6. Analytic and numerical study of perturbations
Finally we expand in the far field limit y → +∞ to obtain (up to an overall common
constant)
RFF → Y (in)s
e−ikr
r1−|s|−s−iϕ
+ Y (out)s
eikr
r1−|s|+s+iϕ
, (6.30)
where
ϕ =
ωqQ
k
−
(
ω2 − µ2
2
)
(1 + a2 +Q2)
k
δn,0 , (6.31)
and
Y (in)s = (2ik)
−u
(
B1Γ(v)e
iπu
Γ(v − u) +B2
)
Y (out)s = (2ik)
u−vB1Γ(v)
Γ(u)
. (6.32)
Equation (6.30) contains a combination of incoming and outgoing waves. However for the
spin-half case the incoming/outgoing wave is dominant for s = ±1/2 respectively. Using
the conserved number current it is possible to show [130] that the transmission factor is
T
(4+n)
s,j,m = 1−
∣∣∣∣∣
Y
(out)
−|s|
Y
(in)
|s|
∣∣∣∣∣
2
. (6.33)
For fermions, to find out the relative normalisation between P1/2 and P−1/2 we insert back
the expansion (6.30) in the first order system (5.35), equate order by order and obtain
the relation
Y
(out)
−1/2 =
2iω
λ
Y
(out)
1/2 . (6.34)
Since the relative normalisation between incoming and outgoing coefficients for the same
s is fixed, now we can insert equation (6.34) in (6.33) to obtain the transmission factor.
For scalars, equation (6.33) is also valid if we set |s| = 0.
6.2.3 High energy approximation based on WKB arguments
To complete the analytic picture, we present some arguments for a useful approximation
in the high energy limit for scalars. This will give the leading asymptotic form for the
transmission factors.
The matching procedure in the previous section doesn’t work in the high energy limit
for two reasons. On one hand the powers in (6.19) and (6.28) no longer match at high
6.2. The radial equations I: Analytic methods 99
energy, rotation, charges and masses. Secondly we are stretching the near horizon solution
into r ≫ 1 and the far field solution into r ≪ 1/k. If k is large, then these conditions
are incompatible and we are effectively stretching the far field solution too close to the
horizon.
To understand this problem we look into the WKB approximation for the scalar radial
equation4. Some earlier works which have used the WKB approximation to compute
transmission factors are [138–142]. First note that the radial equation can be written in
a Schro¨dinger-like form through a change of independent variable. Start by choosing
dy =
dr
∆
, (6.35)
to obtain (
d2
dy2
− V
)
R = 0 , (6.36)
where V ≡ ∆U −K2 contains a leading term −k2r4 corresponding to the highest power
of r (all the other terms are suppressed). In the high energy limit, this term dominates
the solution. In fact, we can formally write an infinite WKB series [143]
R ∼ A+ exp
(
k
∞∑
n=0
S+n (y(r))
kn
)
+ A− exp
(
k
∞∑
n=0
S−n (y(r))
kn
)
. (6.37)
It is easy to check [143] that the leading correction reproduces the asymptotic form at
infinity consistent with (6.30). A necessary condition for the approximation to be valid is
∣∣∣∣dVdy
∣∣∣∣≪
∣∣∣V 32 ∣∣∣⇔
∣∣∣∣dVdr ∆
∣∣∣∣≪
∣∣∣V 32 ∣∣∣ , (6.38)
which (to leading order in r) is just r ≫ 1/k. This condition indicates that for large
k the field will start to take a WKB form not far from the horizon. Such a result is
not surprising if we note that these modes have very short wavelength, so the potential is
almost constant along many wavelengths (except very close to the horizon). Furthermore,
4Here we focus on the scalar case because the potential is real. A similar treatment can be applied to
fermions using the method in Chandrasekhar’s book [122] to reduce the complex potential to a real one.
However note that we would expect the spin of the particle to become irrelevant at high energies as we
reach the geometrical optics limit.
100 Chapter 6. Analytic and numerical study of perturbations
the WKB corrections obey
S+n =


−S−n , n even
S−n , n odd
(6.39)
so the odd terms (which are purely real [143]) only contribute with an overall common
factor. As for the even terms, they are products of
√
V times polynomial terms in V . In
general there is an imaginary and a real part for each even order correction, but since our
potential is real then it will either be real or imaginary. If
√
V is imaginary, the relative
amplitude between incoming and outgoing waves does not depend on r. But in the limit
of k large, the dominant term in the potential is −k2r4 which is negative, so the square
root is purely imaginary and the even order corrections only introduce a phase difference
between incoming and outgoing waves. This means that in the region where the WKB
solution is valid5, the relative amplitude between incoming and outgoing modes stays
fixed. The transmission coefficient can then be calculated at any point in such a region
provided we have a suitable analytic expansion in terms of incoming and outgoing waves.
Thus, in the high energy limit, the propagation of the field along a thin region outside
the horizon determines the behaviour of the transmission factors.
This behaviour can be seen explicitly in (6.19). There the scalar r-powers have a
common factor r−(n+1)(1−B∗/2) multiplied by
r±
√
(1−B∗)2/4−p2+D∗ . (6.40)
In the high energy limit the argument of the square root in (6.40) becomes negative and
we obtain a relative phase between the two modes which are respectively outgoing and
incoming. The transmission coefficient follows under the single approximation k ≫ 1
T
4+n
0,j,m = 1−
∣∣∣∣A+A−
∣∣∣∣
2
= 1−
∣∣∣∣A1A2
∣∣∣∣
2
. (6.41)
6.3 The radial equations II: Numerical methods
To obtain the transmission factor in the full range of parameters a numerical approach
must be taken. In this section we complete the study of the scalar and fermion fields with
mass and charge by developing the numerical method to solve the radial equations.
Starting with scalars, equation (5.28) can be written as a first order system of differen-
5This is the region connected to infinity such that V < 0.
6.3. The radial equations II: Numerical methods 101
tial equations. This is useful to perform the numerical integration using a similar method
as for fermions. Since there isn’t a unique way of reducing the second order equation to a
first order system, we take advantage of the extra freedom to construct a spinor-like ob-
ject with a conserved Wronskian and, simultaneously, an asymptotic behaviour at infinity
which gives the transmission factor straightforwardly. It is then possible to show that a
convenient choice is
P±0 =
∆
1
2
2
(
kR∓ idR
dr
)
. (6.42)
So the second order equation (5.28) is replaced by the first order coupled system6
dPs
dr
=Ms(r)Ps (6.43)
where
M0(r) =
∆′
2∆
σˆ1 − 1
2
(
V
k
− k
)
σˆ2 +
i
2
(
V
k
+ k
)
σˆ3 , (6.44)
V =
K2
∆2
− U
∆
, (6.45)
and σˆi are the Pauli matrices. Now, using (6.43), conservation of the Wronskian is easily
checked:
d
dr
(
P†sσˆ3Ps
)
=
d
dr
(|P+|s||2 − |P−|s||2) = 0 . (6.46)
We have chosen k =
√
ω2 − µ2 so that P±0 picks respectively the outgoing/incoming part
of the wave at infinity (see section 6.3.2). For fermions, the radial equation (5.35) is
already in the form (6.43) with
M 1
2
(r) =
λ
∆
1
2
σˆ1 − µr
∆
1
2
σˆ2 + i
K
∆
σˆ3 . (6.47)
So P1/2 obeys the same Wronskian relation (6.46) as (6.43). Again, the incoming solution
at the horizon takes the form
P 1
2
∼ x−iK⋆δ0 (a0 + . . . ) (6.48)
with a0 a constant spinor.
In the next sections we present the methods used to reduce the linear systems of
6Here s is the spin which we leave arbitrary since the same type of equation will hold for fermions.
102 Chapter 6. Analytic and numerical study of perturbations
equations at hand to initial value problems which are more convenient for numerical
integration.
6.3.1 Near horizon expansions
The boundary condition at the horizon is most easily implemented through a series ex-
pansion. This allows for a high precision initialisation of the radial functions slightly away
from the horizon to avoid numerical difficulties associated with the coordinate singularity.
The expansions we need are
R = xα
+∞∑
m=0
αmx
m
P 1
2
= xα
+∞∑
m=0
am
(√
x
)m
. (6.49)
Note that R can be used to initialise P0. By inserting into the wave equations (5.28)
and (6.43) respectively we obtain the following recurrence relations


α = −iK⋆
δ0
α0 = 1, αm =
−1
m(m+ 2α)δ20
[
(m+ α)δ0γ¯m +
m−1∑
k=0
(γk(k + α)δm−k + αkσm−k)
]
a0 =
(
0
1
)
, am = (N0 − δ0(m+ 2α))−1
[
bm −
m−1∑
j=0
Nm−jaj
]
m ≥ 1 .
(6.50)
where a choice of normalisation was made, when setting α0 and a0. The various coefficients
are defined in appendices C.1.1 and C.1.2. Using expansions (6.49) we have initialised Ps
at x = 0.1 by truncating the series at eighteenth order. A first estimate of the numerical
error can be made by modifying this choice (we have used x = 0.05 and x = 0.01 as a
check).
6.3.2 Far field expansions
Once the radial function is initialised, numerical integration routines can be used to
propagate the solution away from the horizon according to (6.43). When sufficiently away
6.3. The radial equations II: Numerical methods 103
from the horizon, the transmission factor can be evaluated by comparing the numerically
propagated solution with its asymptotic form at large r. An asymptotic expansion can
be found in the form
Ps = e
qrr−γ
+∞∑
m=0
qsmr
−m , (6.51)
if we expand
Ms =
+∞∑
m=0
Msmr
−m (6.52)
and equate (6.43) order by order. The leading behaviour is
Ps = Y
(out)
s e
iyyiϕd+s + Y
(in)
s e
−iyy−iϕd−s , (6.53)
where y = kr, Y
(out)
s and Y
(in)
s are arbitrary constants,
ϕ = ǫ
ω
k
− σµ
k
, (6.54)
ǫ = −qQ+ ω(1 + a2 +Q2)δn,0 σ = µ
2
(1 + a2 +Q2)δn,0 , (6.55)
and
d+0 =
(
1
0
)
d−0 =
(
0
1
)
d+1
2
=
(
1
− µ
ω+k
)
d+1
2
=
(
− µ
ω+k
1
)
. (6.56)
We can now factor out the dependence at infinity so that the leading asymptotic form
for the upper(lower) component of the spinor becomes Y
(out)
s (Y
(in)
s ) respectively. This
is achieved by performing a rotation on the spinor Ps such that it eliminates a fixed
number of sub-leading terms in the asymptotic expansion (6.52) (in practise we have
eliminated the first two sub-leading terms). Then the new spinor Qs is related to Ps
through Qs = RsPs and the the system to integrate becomes
dQs
dy
= AsQs . (6.57)
The explicit forms for As and Rs are given in appendix C.2.
104 Chapter 6. Analytic and numerical study of perturbations
Finally, the transmission factor is computed from the definition by taking the limit
Ts = lim
r→+∞
(
1−
∣∣∣∣Q+sQ−s
∣∣∣∣
2
)
= 1−
∣∣∣∣∣Y
(out)
s
Y
(in)
s
∣∣∣∣∣
2
(6.58)
(±s for upper/lower component respectively) and an estimate of the error is obtained by
varying the large r used in the limit. Furthermore, with the normalisation chosen in (6.50)
we can evaluate the Wronskian (6.46) at the horizon and use its conservation to obtain a
second expression
Ts = lim
r→+∞
kWs
|Q−s|2 =
kWs
|Y (in)s |2
, (6.59)
where
W0 = K⋆ W 1
2
=
ω + k
2
. (6.60)
By comparing the results from (6.58) and (6.59), we obtain another estimate of the nu-
merical errors. Equation (6.59) is particularly useful since it contains explicitly the zeros
of the transmission factor in the numerator.
To integrate (6.57), a code was written in C++ using the Gnu Standard Library
(GSL) numerical integration routines. This was checked against an independent code in
Maple11.
6.4 Numerical results
In this section we plot various quantities, using the approximations developed in sec-
tion 6.2 and the numerical method of section 6.3. The physically most relevant are those
in (5.12), (5.13), (5.14) and (5.15). When integrated over ω and summed over particle
type they give the rates of emission of particle number, energy, angular momentum and
charge. Nevertheless we still plot the transmission factors to keep track of where the new
effects enter. Most of the samples obtained with the numerical method for the study of
mass and charge effects in spin 1, 1/2 fields, were generated up to7 ω = 10, but some
up to ω = 5 to save computing time. We show plots with ω < 5 since the curves are
very quickly stabilised for large ω (either to a constant or a suppressed tail). Except for
sections 6.4.4 and 6.4.5, we focus on a = 0.
7Note we are using horizon radius units rH = 1. Sometimes we emphasize the rH dependence by
explicitly re-introducing rH .
6.4. Numerical results 105
n = 6
µrH
j = 2
j = 1
j = 0
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
0
,j
,m
1.210.80.60.40.20
0.2
0.15
0.1
0.05
0
0.4
0.2
0.0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
0
,j
,m
µrH = 0.4
n
j = 2
j = 1
j = 0
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(4
+
n
)
0
,j
,0
1.21.110.90.80.70.60.50.4
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
6
5
4
3
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(4
+
n
)
0
,j
,0
j = 2
j = 1
j = 0
ωrH
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
1.210.80.60.40.20
0.02
0.015
0.01
0.005
0
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
j = 2
j = 1
j = 0
ωrH
F
lu
x
,
d
N
(4
+
n
)
0
,j
,m
d
td
ω
1.21.110.90.80.70.60.50.4
0.01
0.008
0.006
0.004
0.002
0
F
lu
x
,
d
N
(4
+
n
)
0
,j
,m
d
td
ω
Figure 6.1: Scalar transmission factors and fluxes for n = 6 (left) and variable n (right) using
the analytic approximation. The left plots show variation with particle mass µ in
natural units r−1H for n = 6 and the right plots show variation with n for µ fixed.
The top plots show the transmission factors T
(4+n)
k and the bottom plots show
number fluxes, for a range of j modes. The curves are naturally grouped by j,
rotation is off and the line colour/type is the same for top and bottom plots. In
the top left plot we have indicated the exact numerical result in thinner lines.
The plots for the review of the effects of rotation (section 6.4.4) were produced using
the data files in [144] and spheroidal functions computed with the method in section 6.1.
6.4.1 The effect of particle mass
The effect of particle mass is important if the energy of the particle emitted during the
evaporation is comparable to its mass. For Standard Model heavy particles, such as the
top quark (mt ∼ 170 GeV) the Z (mZ ∼ 91 GeV), the W (mW ∼ 80 GeV) and the
Higgs boson, the effect will not be negligible in TeV gravity scenarios. This is because
their masses are of the same order of magnitude as the typical energy scale of a Hawking
emission (ω ∼ 1/rH). Typical values of µ for these heavy particles in TeV gravity scenarios
(relevant for the LHC), range from 0.1 to 0.5.
In figure 6.1 we present some representative curves for the transmission factor and
the number flux in the low energy limit using the analytic approximation. As mentioned
before the approximation becomes better with larger n so most of our plots will be for
106 Chapter 6. Analytic and numerical study of perturbations
n = 6, except for when we focus on the n dependence where we use n ≥ 3.
The most prominent property of the left-hand-side plots is a smooth drop close to
the mass µ. The higher the partial wave the less steep this is but there is always a
horizontal shift (see for example the j = 1 mode). This effect is quite important close to
the mass threshold where the probability of emission is suppressed. This is in contrast
with the simplified approach for the evaporation in current black hole event generators
where the spectrum is cut off sharply at ω = µ. Note that in the top left plot we
have included the exact result obtained with the numerical method using thinner lines.
At low energies the curves are indistinguishable, and at intermediate energies they are
qualitatively concordant. We have checked that this observation holds for the remaining
plots presented below, however we will not include further direct comparisons between
the analytic approximation and the exact numerical method.
Furthermore, for example the number flux for the j = 1 mode shows how increasing
the mass of the particle not only suppresses the flux around ω ∼ µ but also the total area
under the curve. Massive particles are therefore less likely to be produced. This effect
was previously studied in four dimensions, for example numerically, in Page’s paper for
leptons [114].
The right hand side plots show how the transmission factor is very mildly dependent
on n (at least in the limit of small µ). However the flux plot displays a strong variation
with n which is due to the strong dependence of the Hawking temperature appearing in
the thermal factor.
In figures 6.2 and 6.3 we confirm the scalar analytic results in the full energy range
and extend them to fermions with the exact numerical method. The top plots of figure 6.2
are for n = 2 and a range of masses, whereas for the bottom plots we fix µ and vary n.
We use a range ω ∈ [0, 2] because the transmission factors asymptote to one quickly, so
this is the interesting region.
The top plots show the first three partial waves for scalars and fermions, µ = 0, 0.5
and 1. For scalars we confirm the strong suppression at the mass threshold for the j = 0
partial waves, and the shift and suppression for higher partial waves. For fermions the
behaviour is similar, except that the first partial waves are not so sharply suppressed at
threshold.
The bottom plots show the variation with n for fixed µ = 0.5. At higher energies
we see a strong n dependence compared to the low energy mild dependence of the scalar
transmission factors in the range n = 3, . . . , 6 (figure 6.1). The general tendency is for
6.4. Numerical results 107
n = 2
j
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(6
)
0
,j
,0
21.510.50
1
0.8
0.6
0.4
0.2
0
2
1
0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(6
)
0
,j
,0
n = 2
j
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(6
)
1 2
,j
,1 2
21.510.50
1
0.8
0.6
0.4
0.2
0
5/2
3/2
1/2
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(6
)
1 2
,j
,1 2
µ = 0.5
j
n = 6
n = 2
n = 1
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(n
)
0
,j
,0
21.81.61.41.210.80.6
1
0.8
0.6
0.4
0.2
0
2
1
0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(n
)
0
,j
,0
µ = 0.5
j
n = 6
n = 2
n = 1
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(n
)
1 2
,j
,1 2
21.81.61.41.210.80.6
1
0.8
0.6
0.4
0.2
0
5/2
3/2
1/2
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(n
)
1 2
,j
,1 2
Figure 6.2: Scalar (left) and fermion (right) transmission factors for n = 2 (top) and variable
n (bottom) using the numerical method : The plots show the first three partial
waves (note that a = 0, so waves with different m for the same j are degenerate).
The top plots contain three masses µrH = 0, 0.5, 1 corresponding to the points
where each curve starts.
the transmission factor to be suppressed at intermediate energies with n, but note that
the larger the n the smaller is the separation between curves of different n.
In figure 6.3 we present examples of the number flux for non-zero mass, when the
charge and the rotation are set to zero. The top plots show n = 4 and µ = 0.5. We also
indicate the contributions from the first few j values to the total flux curve. Similarly to
the transmission factors, the main feature is a sharp suppression at threshold. The area
under the curves is larger for scalars than fermions, which agrees with earlier studies (see
for example [107, 112]). The bottom plots show three values of the mass and various n
values. The error from using the µ = 0 curve with a sharp cut at the mass is therefore large
(most notably for fermions). Regarding variation with n, it is opposite to the tendency for
the transmission factors so the n dependence of the Planckian factor [exp(ω˜/TH) ± 1]−1
dominates the magnitude.
108 Chapter 6. Analytic and numerical study of perturbations
4
3
2
1
0
Total
µ = 0.5
n = 4
j
ωrH
N
u
m
b
e
r
F
lu
x
,
∑ m
d
N
d
ω
d
t0
,j
,m
43.532.521.510.5
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
9/2
7/2
5/2
3/2
1/2
Total
µ = 0.5
n = 4
j
ωrH
N
u
m
b
er
F
lu
x
,
∑ m
d
N
d
ω
d
t
1 2
,j
,m
43.532.521.510.5
0.025
0.02
0.015
0.01
0.005
0
Q = 0
q = 0
µ
n = 5
n = 4
ωrH
N
u
m
b
e
r
F
lu
x
,
d
N
d
ω
d
t0
43.532.521.510.50
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1.0
0.5
0.0
N
u
m
b
e
r
F
lu
x
,
d
N
d
ω
d
t0
Q = 0
q = 0
µ
n = 5
n = 4
ωrH
N
u
m
b
e
r
F
lu
x
,
d
N
d
ω
d
t
1 2
43.532.521.510.50
0.05
0.04
0.03
0.02
0.01
0
1.0
0.5
0.0
N
u
m
b
e
r
F
lu
x
,
d
N
d
ω
d
t
1 2
Figure 6.3: Scalar (left) and fermion (right) number fluxes for n = 4 and zero charges: The
top plots show µ = 0.5 and the contributions from each partial wave to the total
flux. The bottom plots show variable µ and variable n = 5, . . . , 2. For each µ the
curves are naturally order in n from top to bottom, n = 5 and n = 4 are indicated
for µ = 0.
6.4.2 The effect of black hole charge on neutral particles
The next effect we consider is black hole charge. Neutral particles simply feel a different
gravitational field around the black hole. So by studying neutral particles we disentangle
the gravitational effect from the electromagnetic effect (since q = 0).
In figure 6.4 we present plots for transmission factors and fluxes using the low energy
analytic approximation. Here we focus on n = 6 for scalars and fermions. We should
note that some of the plots for fermions will display extrapolated results beyond the
small energy limit. This turns out to be quite well behaved, which is due to the better
matching of r-powers as pointed out in equation (6.22).
From the gravitational point of view, the main effect of Q is to decrease the hori-
zon radius and consequently increase the Hawking temperature. This is clearly seen in
the transmission factors of figure 6.4, where all the curves are pushed up with increas-
ing Q. The same happens with the fluxes where the effect is even larger, due to the
strong dependence of the thermal factor on Q through the Hawking temperature – see
equation (5.12).
6.4. Numerical results 109
n = 6
Q
j = 2
j = 1
j = 0
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
0
,j
,m
1.210.80.60.40.20
0.25
0.2
0.15
0.1
0.05
0
0.8
0.4
0.0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
0
,j
,m
n = 6
Q
j = 52
j = 32
j = 12
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
1 2
,j
,m
21.510.50
1
0.8
0.6
0.4
0.2
0
0.8
0.4
0.0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
1 2
,j
,m
j = 2
j = 1
j = 0
ωrH
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
1.210.80.60.40.20
0.03
0.025
0.02
0.015
0.01
0.005
0
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
j = 52
j = 32
j = 12
ωrH
F
lu
x
,
d
N
(1
0
)
1 2
,j
,m
d
td
ω
21.510.50
0.04
0.03
0.02
0.01
0
F
lu
x
,
d
N
(1
0
)
1 2
,j
,m
d
td
ω
Figure 6.4: Transmission factors and fluxes for neutral scalars and fermions using the analytic
approximation. The left plots show spin 0 and the right plots show spin 1/2. The
top plots show transmission factors Tk and the bottom plots show number fluxes,
for a range of j modes and different black hole charges Q. Rotation is off, µ = 0
and the line colour/type is the same for top and bottom plots.
6.4.3 The effect of particle charge
For particles with non-zero charge, in addition, we have a Coulomb repulsion/attraction
according to whether the particle has same/opposite sign charge compared to the black
hole. For definiteness we take the black hole charge to be positive.
In figure 6.5 we plot transmission factors and fluxes for scalars and fermions, various
charges, n = 6 and Q = 0.6, using the low energy analytic approximation. It is important
to note here that the Coulomb type coupling appearing in the radial equation is
qQ = (
√
αz)(
√
αZ) ≃ (0.1z)(0.1Z) (6.61)
For a TeV gravity black hole produced from proton-proton collisions, |Z| ≤ 4/3 and
|z| ≤ 1. So the figures we have chosen are above their typical values. However it is
easier to see the differences in the curves. Furthermore there may be stages during the
evaporation where the black hole charges up so this region of parameters is still important.
The main features of figure 6.5 are as follows. For scalars we can see clearly the phe-
nomenon of superradiance in the top plot, for particles with the same charge as the black
110 Chapter 6. Analytic and numerical study of perturbations
Q = 0.6
n = 6
q
j = 2
j = 1
j = 0
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(1
0
)
0
,j
,m
1.210.80.60.40.20
0.25
0.2
0.15
0.1
0.05
0
0.3
0.0
-0.3
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(1
0
)
0
,j
,m
Q = 0.6
n = 6
q
j = 52
j = 32
j = 12
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(1
0
)
1 2
,j
,m
21.510.50
1
0.8
0.6
0.4
0.2
0
0.3
0.0
-0.3
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(1
0
)
1 2
,j
,m
j = 2
j = 1
j = 0
ωrH
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
1.210.80.60.40.20
0.03
0.025
0.02
0.015
0.01
0.005
0
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
j = 52
j = 32
j = 12
ωrH
F
lu
x
,
d
N
(1
0
)
1 2
,j
,m
d
td
ω
21.510.50
0.04
0.03
0.02
0.01
0
F
lu
x
,
d
N
(1
0
)
1 2
,j
,m
d
td
ω
Figure 6.5: Transmission factors and fluxes for charged scalars and fermions using the low
energy analytic approximation. The left plots show spin 0 and the right plots show
spin 1/2. The top plots show transmission factors Tk and the bottom plots show
number fluxes, for a range of j modes and different particle charge q with Q = 0.6.
Rotation is off, µ = 0 and the line colour/type is the same for top and bottom
plots.
hole, where T
(10)
k < 0. This means that the associated incident wave gets reflected with a
larger amplitude. However, it does not favour the emission of positively charged particles
because the negative charge transmission factors are greatly enhanced. This is clear in the
flux plot where all the curves at low energies are higher for negative charge. We can un-
derstand this physically by recalling that the transmission factor describes the probability
of a wave incident from infinity to be transmitted down the black hole. Since negatively
charged particles are attracted by the Coulomb potential and positively charged parti-
cles are repelled, we would expect negative charges to have higher transmission factors.
This is confirmed for fermions in a wider range of energies. The other main feature is
that at higher energies, the Planckian factor (which favours discharge) dominates and the
tendency is inverted, i.e. positively charged particles are favoured. This is confirmed for
the scalar case using the high energy limit approximation shown in figure 6.6, where the
transmissions factors are still larger for negatively charged particles, but since they are
close to their asymptotic value T
(10)
k = 1, the thermal factor dominates.
Figure 6.7 shows the variation with n. Here the transmission factors for scalars are
6.4. Numerical results 111
j = 2
j = 1
j = 0
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
0
,j
,m
5.554.543.5
1
0.999
0.998
0.997
0.996
0.995T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(1
0
)
0
,j
,m
Q = 0.6
n = 6
q
ωrH
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
5.554.543.5
0.0014
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0
0.3
0.0
-0.3
F
lu
x
,
d
N
(1
0
)
0
,j
,m
d
td
ω
Figure 6.6: Asymptotic high energy transmission factors and fluxes for charged scalars using
the analytic approximation. The left plot shows transmission factors T
(4+n)
k and
the right plot shows fluxes for a range of j modes. Rotation is off, µ = 0 and the
line colour/type is the same for both plots.
weakly dependent on n whereas for fermions we have a stronger effect. This is due to extra
n dependent factors in the wave equation as for example the term 2|s|/A ∼ 1/(n + 1)
in B – equation (6.15). For the fluxes the separation is larger due to their stronger n
dependence through the Hawking temperature. In general, similarly to neutral black
holes, the effect of n is to increase the total fluxes.
Figures 6.8 and 6.9 show several cases of non-zero charges in the full energy range
using the exact numerical method. In figure 6.8 we have kept Q = 0.4 and in figure 6.9
we have kept Q = 0.6. q ranges between [−1, 1]. We use n = 2 and n = 4 as representative
cases.
The plots of figure 6.8 confirm the qualitative features in figures 6.5 and 6.7. They
also emphasize the general tendency for the transmission factor to be suppressed with n
except for the inverted behaviour for fermions at low energies and a small variation for
scalars in the supperradiant region (the latter agrees with figure 6.7). Furthermore, the
top plots confirm that the transmission factor favours negative charges at all energies.
The top plots of figure 6.9 show the total flux for the two extreme cases q = 1 and q =
−1 together with the first few partial waves contributing. The first striking observation is
the confirmation that for all partial waves there is a first region in energy where charging
up is favoured (i.e. the curve corresponding to negative charge is higher) and another
(dominant) region where discharge is favoured (the curve with positive charge is higher).
It is also clear that if we integrate over the curves discharge is always favoured as expected.
The bottom plots show a similar behaviour for a range of intermediate charges. Another
interesting point is that the charge splitting is larger for fermions than for scalars.
The inverted splitting at low energies is a direct consequence of the extra dimensions.
112 Chapter 6. Analytic and numerical study of perturbations
j = 1
Q = 0.6
n
q = 0.3
q = 0.0
q = −0.3
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(4
+
n
)
0
,j
,m
0.20.150.10.050
0.0001
5e− 05
0
−5e− 05
−0.0001
6
5
4
3
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(4
+
n
)
0
,j
,m
j = 12
Q = 0.6
n
q = 0.3q = 0.0q = −0.3
ωrH
T
ra
n
sm
is
si
on
F
ac
to
r,
T
(4
+
n
)
1 2
,j
,m
0.80.70.60.50.40.30.20.10
0.4
0.3
0.2
0.1
0
6
5
4
3
T
ra
n
sm
is
si
on
F
ac
to
r,
T
(4
+
n
)
1 2
,j
,m
q = 0.3
q = 0.0
q = −0.3
ωrH
F
lu
x
,
d
N
(4
+
n
)
0
,j
,m
d
td
ω
1.51.2510.750.50.250
0.03
0.02
0.01
0
F
lu
x
,
d
N
(4
+
n
)
0
,j
,m
d
td
ω
q = 0.0q = −0.3
ωrH
F
lu
x
,
d
N
(4
+
n
)
1 2
,j
,m
d
td
ω
0.80.60.40.20
0.04
0.03
0.02
0.01
0
F
lu
x
,
d
N
(4
+
n
)
1 2
,j
,m
d
td
ω
Figure 6.7: Transmission factors and fluxes for charged scalars and fermions using the analytic
approximation. The left plots show spin 0 and the right plots show spin 1/2. The
top plots show transmission factors Tk and the bottom plots show number fluxes,
for a range of charges q and various numbers of extra dimensions n with Q = 0.6
and j = 1 for scalar and j = 1/2 for fermions. Rotation is off, µ = 0 and the line
colour/type is the same for top and bottom plots.
In figure 6.10 we show the difference in number flux of positively and negatively charged
scalars and fermions when n = 0, . . . , 6. The left plot shows a typical QED coupling of
|q| = Q = 0.1 and the right plot a QCD like coupling of |q| = Q = 0.3. Note however that
we are dealing with an abelian theory so the latter is only indicative of the magnitude
of the effect for QCD. From this figure it is now clear that the splitting is controlled by
an interplay between the magnitude of the transmission factor (which prefers negative
charges) and the magnitude of the Planckian factor (which prefers positive charges).
For n = 0 and n = 1, the splitting is always positive so the Planckian factor dominates.
However as n increases, the transmission factor starts dominating at low energies and for
all n ≥ 2 we have the observed inverted region (where the curves are negative). Another
interesting feature of figure 6.10 is that the plots on the left have exactly the same shape
as the ones on the right. This is not surprising if we note that for qQ small we can expand
the fluxes perturbatively around qQ = 0 and since |qQ| is 0.01 and 0.09 respectively, we
would expect the perturbation to be dominated by the linear term so the difference is
proportional to |qQ|.
6.4. Numerical results 113
Q = 0.4
n = 2
j
q = 1
q = 0.4
q = −0.4
q = −1
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(6
)
0
,j
,0
21.510.50
1
0.8
0.6
0.4
0.2
0
2
1
0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(6
)
0
,j
,0
Q = 0.4
n = 2
jq = 1
q = 0.4
q = −0.4
q = −1
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(6
)
1 2
,j
,1 2
21.510.50
1
0.8
0.6
0.4
0.2
0
5/2
3/2
1/2
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(6
)
1 2
,j
,1 2
Q = 0.4
q = 1
j
n = 6
n = 2
n = 1
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(n
)
0
,j
,0
21.510.50
1
0.8
0.6
0.4
0.2
0
2
1
0
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(n
)
0
,j
,0
Q = 0.4
q = 1
j n = 6
n = 1
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(n
)
1 2
,j
,1 2
21.510.50
1
0.8
0.6
0.4
0.2
0
5/2
3/2
1/2
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(n
)
1 2
,j
,1 2
Figure 6.8: Scalar (left) and fermion (right) transmission factors for variable charge q (top)
and variable n (bottom): The first three partial waves are presented (note that
a = 0, so waves with different m for the same j are degenerate).
6.4.4 The effect of black hole rotation
The effect of rotation was studied before in detail for several fields in [103, 105–108,111–
113] with Q = µ = 0 limit. Data files for transmission factors were built by authors in
these studies to cover as much as possible the parameter space, for spins s = 0, 1/2, 1,
and were made publicly available in [144]. In this section we provide a short review of
the main results for this special case using the data [144], following [4]. This is important
because in chapter 7 we implement black hole rotation in the CHARYBDIS2 generator, so
a good understanding of the features of the various fluxes reveals how the simulation of
the physics is affected.
Energy dependence of the fluxes
When rotation is turned on, the partial waves for a given j which contribute to the
fluxes (5.12), (5.13) and (5.14) are no longer degenerate. This splitting is such that modes
with m > 0 (co-rotating) dominate the fluxes. Figures 6.11 and 6.12, illustrate how this
arises from the combination of the transmission factor Tk (TF) and the Planckian factor
[exp(ω˜/TH)±1]−1 (PF), for each partial wave contribution. In figure 6.11 we have plotted
transmission factors for scalar and vector fields with the corresponding Planckian factors,
114 Chapter 6. Analytic and numerical study of perturbations
2
1
0
q = 1 Total
2
1
0
q = −1 Total
µ = 0.0
n = 4
Q = 0.6
j
ωrH
N
u
m
b
e
r
F
lu
x
,
∑ m
d
N
d
ω
d
t0
,j
,m
43.532.521.510.50
0.09
0.06
0.03
0
5/2
3/2
1/2
q = 1 Total
5/2
3/2
1/2
q = −1 Total
µ = 0.0
n = 4
Q = 0.6
j
ωrH
N
u
m
b
er
F
lu
x
,
∑ m
d
N
d
ω
d
t
1 2
,j
,m
43.532.521.510.50
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1.0
0.8
0.4
0.2
-0.2
-0.4
-0.8
-1.0
Q = 0.6
µ = 0
q
n = 4
ωrH
N
u
m
b
e
r
F
lu
x
,
d
N
d
ω
d
t0
43.532.521.510.50
0.09
0.06
0.03
0
1.0
0.8
0.4
0.2
-0.2
-0.4
-0.8
-1.0
Q = 0.6
µ = 0
q
n = 4
ωrH
N
u
m
b
e
r
F
lu
x
,
d
N
d
ω
d
t
1 2
43.532.521.510.50
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Figure 6.9: Scalar (left) and fermion (right) number fluxes for n = 4, variable q, and Q = 0.6:
The top plots show two opposite and large |q| = 1 cases to illustrate the charge
splitting, together with the first three partial wave contributions. The bottom
plots show the variation of the curves between these two large charges.
µ = a = 0.0
Q = |q| = 0.1
n = 6
n = 6
n = 0
ωrH
d
N
d
ω
d
t+
|q
|
−
d
N
d
ω
d
t−
|q
|
21.510.50
0.0015
0.001
0.0005
0
−0.0005
−0.001
s = 1/2
s = 0
d
N
d
ω
d
t+
|q
|
−
d
N
d
ω
d
t−
|q
|
µ = a = 0.0
Q = |q| = 0.3
n = 6
n = 6
n = 0
ωrH
d
N
d
ω
d
t+
|q
|
−
d
N
d
ω
d
t−
|q
|
21.510.50
0.015
0.01
0.005
0
−0.005
−0.01
s = 1/2
s = 0
d
N
d
ω
d
t+
|q
|
−
d
N
d
ω
d
t−
|q
|
Figure 6.10: Scalar and fermion number flux asymmetries: Both plots show curves for the
difference in number fluxes between positively charged and negatively charged
particles for two values of |q|Q. The curves are naturally ordered in n (some
cases are labelled) from n = 0 (curve with the lowest maximum) to n = 6
(highest maximum).
6.4. Numerical results 115
n = 6
a∗ = 0.8m=3
m=−3
j = 4
m=−2
m=1
m=−1
j = 2
j = 1
j = 0
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(6
)
0
,j
,m
2.521.510.50
1
0.8
0.6
0.4
0.2
0
n = 6
a∗ = 0.8
m=3
m=−3
m=2
m=−2
m=1
m=−1
j = 3j = 2
j = 1
ωrH
T
ra
n
sm
is
si
o
n
F
a
ct
o
r,
T
(6
)
−
1
,j
,m
2.521.510.50
1
0.8
0.6
0.4
0.2
0
−0.2
3
2
1
0
−1
−2
−3 m
ωrH
B
o
se
F
a
c
to
r,
( ex
p
ω
−
m
Ω
T
H
−
1
) −1
2.521.510.50
6
4
2
0
−2
−4
n = 6
a∗ = 0.8
m=−52
m=32
m=−32
m=12 m=−12
j = 52
j = 32
j = 12
ωrH
T
ra
n
sm
is
si
o
n
F
a
c
to
r,
T
(6
)
−
1 2
,j
,m
2.521.510.50
1
0.8
0.6
0.4
0.2
0
5/2
3/2
1/2
−1/2
−3/2
−5/2 m
ωrH
F
e
rm
i
F
a
c
to
r,
( ex
p
ω
−
m
Ω
T
H
+
1
) −1
2.521.510.50
1
0.8
0.6
0.4
0.2
0
Figure 6.11: Transmission factors and Planckian factors for a∗ = 0.8 and n = 6. The left
plots show spin 0, 1 and the right plots show spin 1/2. The top plots show
the transmission factors Tk and the bottom plots show the Planckian factors
[exp(ω˜/TH)± 1]−1, for a range of j and m modes. Lines with the same m have
the same colour and line type.
as well as for fermions8. At the level of the transmission factors, the overall tendency
is for counter-rotating (m < 0) modes to be dominant. Recalling the interpretation of
the transmission factor as the fraction of a wave incident from infinity which is absorbed,
this means that counter-rotating modes are more easily absorbed by the black hole. It is
interesting to note that this agrees with the absorptive disks in figure 3.6, where as we
increase a, the left side of the disks (which is associated with geodesics with ℓz < 0 – see
central plot of figure 3.6) is larger than the right side. So in some sense, particles with
m < 0 are more easily absorbed in the geometrical optics approximation as well. For
8These are the types of fields implemented in the evaporation phase in CHARYBDIS2
116 Chapter 6. Analytic and numerical study of perturbations
n = 2
a∗ = 1.4
s = 0
ωrH
P
o
w
e
r
543210
0.03
0.02
0.01
0
ωrH
T
F
v
s
P
F
543210
1
0
−1
−2
−3
n = 2
a∗ = 1.4
s = 1
ωrH
P
o
w
e
r
543210
0.12
0.08
0.04
0
ωrH
T
F
v
s
P
F
543210
1
0
−1
−2
−3
n = 2
a∗ = 1.4
s =
1
2
ωrH
P
o
w
e
r
543210
0.04
0.03
0.02
0.01
0
ωrH
T
F
v
s
P
F
543210
1
0.8
0.6
0.4
0.2
0
Figure 6.12: Power spectra for a rotating six-dimensional black hole with a∗ = 1.4. For each
set of two plots, the top plot shows the transmission coefficients (red, solid) and
the Planckian factors (blue, dotted) as a function of ωrh, for the m = j modes
up to j = 10 for scalars (top left pair), j = 10 for vectors (top right pair) and
j = 19/2 for fermions (bottom pair). The bottom plots of each pair display
the power emission spectrum (thick red curve) containing contributions from all
modes, together with the curves for the leading m = j modes. The region of
overlap between different modes is small, leading to sharply-peaked oscillations
in the power emission spectrum.
6.4. Numerical results 117
the Planckian factors, the tendency is exactly opposite so Hawking radiation is generated
preferentially in co-rotating modes. The Planckian factor turns out to dominate so we
end up with a spectrum dominated by co-rotating modes. This is particularly striking for
lower n. For example in figure 6.12 we show the power spectrum for various spins and the
contributions from the m = j partial waves, together with the TFs and PFs for the latter.
The oscillations in the spectrum coincide with the m = j peaks showing that they give
the largest contribution. The same dominance for co-rotating modes occurs for larger
n, however other m > 0 modes also contribute which results in a smoother spectrum
(see figure 6.13). A preferential emission of co-rotating modes implies an efficient loss of
black hole angular momentum through Hawking radiation. Figure 6.13 also shows that in
general, rotation increases the area under the curves. This means a larger total emission
rate, despite the mild dependence of the Hawking temperature on a∗ for a fixed black hole
mass. On the other hand, the Hawking temperature grows roughly linearly with n, which
increases the flux with n as seen by comparing the vertical axis of the left plots with the
right plots.
Another general feature of the spectra is a shift towards emission at higher energies.
This is related to the explicit shift ω → ω˜ = ω−mΩH in the argument of the exponential of
the Planckian factor. For the m = j modes, which dominate the spectrum, this produces
a positive shift of the exponential suppression towards larger energies. In particular, for
bosonic fields (see figure 6.11 and 6.12) it introduces a superradiant region [120,121,145]
ω˜ < 0 where the Planckian factor becomes negative, together with the transmission
factor9. It has been shown [106, 108, 146] that this phenomenon increases the emission
of bosonic fields on the brane largely, with increasing a∗ as well as with n. This feature
is particularly striking for spin-1 fields – figure 6.13. It has been suggested [147] that
the presence of superradiance might increase dramatically the emission of bulk gravitons
in a rotating black hole which would compete with the brane channel, disproving the
assumption that brane emission is dominant. However, results for the ratio of bulk to
brane emission for the scalar field [113, 148, 149] indicate that bulk emission is typically
below 35% for n = 1, 2, . . . , 6 (though a full study of bulk gravitons would be necessary
and it is still lacking).
9Note that the denominator in equation (5.12) is negative for superradiant modes in the bosonic
case, thus cancelling out the negativity of the superradiant transmission factor T
(D)
k , and so contributing
positively to the fluxes.
118 Chapter 6. Analytic and numerical study of perturbations
1.4
1.2
1.0
0.8
0.4
0.0
a∗
n = 2, s = 0
ωrH
P
o
w
e
r
F
lu
x
543210
0.03
0.02
0.01
0
1.4
1.2
1.0
0.8
0.4
0.0
a∗
n = 6, s = 0
ωrH
P
ow
er
F
lu
x
543210
0.4
0.3
0.2
0.1
0
1.4
1.2
1.0
0.8
0.4
0.0
a∗
n = 2, s =
1
2
ωrH
P
o
w
e
r
F
lu
x
543210
0.04
0.03
0.02
0.01
0
1.4
1.2
1.0
0.8
0.4
0.0
a∗
n = 6, s =
1
2
ωrH
P
ow
er
F
lu
x
543210
0.3
0.2
0.1
0
1.4
1.2
1.0
0.8
0.4
0.0
a∗
n = 2, s = 1
ωrH
P
o
w
e
r
F
lu
x
543210
0.15
0.12
0.09
0.06
0.03
0
1.4
1.2
1.0
0.8
0.4
0.0
a∗
n = 6, s = 1
ωrH
P
ow
er
F
lu
x
543210
0.8
0.6
0.4
0.2
0
Figure 6.13: Power Spectrum of spin-zero, spin-half and spin-one fields on the brane. The
plots show the power spectrum for a range of rotations, a∗ = 0, . . . , 1.4 for a 6D
hole (left) and a 10D hole (right). Note the order-of-magnitude difference in the
total power between n = 2 and n = 6 as well as between s = 0, 1/2 and s = 1.
Angular fluxes
Some important properties of the angular distributions for our analysis follow from the
observation of figures 6.14 and 6.15. We have verified that in general, for any mode,
higher rotation tends to make the spheroidal functions more axial (figure 6.15). This
means that at low energies (where the modes are departing from being degenerate), the
angular distribution of Hawking radiation will tend to become more axial. However, for
higher energies, the effect of rotation on the emission spectrum is to favour emission of
modes with m = j in order to spin down the black hole. This will produce a more
equatorial angular distribution, as we can see from figure 6.15 where the m = j mode is
6.4. Numerical results 119
cos θωrH
n = 2 , a = 1
dN0
dωdt dcos θ
0.006
0.004
0.002
0 10.5
0−0.5−143
2
1
0
cos θωrH
n = 2 , a = 1
dN−1/2
dωdt dcos θ
0.016
0.012
0.008
0.004
0 10.5
0−0.5−143
2
1
0
cos θωrH
n = 2 , a = 1
dN−1
dωdt dcos θ
0.12
0.08
0.04
0 10.5
0−0.5−143
2
1
0
cos θωrH
n = 6 , a = 1
dωdt dcos θ
0.04
0.03
0.02
0.01
0 10.5
0−0.5−143
2
1
0
cos θωrH
n = 6 , a = 1
dωdt dcos θ
0.04
0.03
0.02
0.01
0 10.5
0−0.5−143
2
1
0
cos θωrH
n = 6 , a = 1
dωdt dcos θ
0.4
0.3
0.2
0.1
0 10.5
0−0.5−143
2
1
0
Figure 6.14: Angular dependence of the flux for states of positive helicity h = −s = 0, 1/2, 1,
non-zero rotation a = 1 and two different n = 2, 6. The negative helicity plots
are obtained by the reflection cos θ → − cos θ
always more central (in x = cos θ) than the m = 0 mode (for j 6= 0). So as we increase the
rotation parameter we have a competition between the increase in the angular function’s
axial character and the increase in probability of emission of more equatorial modes (which
are those with larger j). At low (high) energies the former (latter) wins as seen from the
energy dependence of the angular profiles shown in figure 6.14
Low-energy vector bosons are more likely to be emitted close to the rotation axis,
whereas high energy vector bosons are more likely to be emitted in the equatorial plane.
A similar but far less pronounced effect exists for spin-half particles.
In figure 6.14 we see that particles with a single helicity, such as the neutrino, will be
emitted asymmetrically by a rotating black hole [150–152]. For example, if the black hole’s
angular momentum vector is pointing north, then positive(negative) helicity states will
be preferentially emitted in the northern (southern) hemisphere. The unstable W and Z
vector bosons will be emitted from the black hole in two possible transverse polarisations
states, with similar north/south asymmetry. This should lead to asymmetries in their
decay products.
6.4.5 Interplay between rotation and charge effects
Though the new results for the charge effects described in previous sections have not yet
been implemented in the CHARYBDIS2 generator, it is interesting to observe how the effect
of rotation couples to the effect of charge. Figure 6.16 shows some cases with a rotation
parameter a = 0.9 (typical order of magnitude for a TeV gravity scenario rotating black
120 Chapter 6. Analytic and numerical study of perturbations
aω = 6
aω = 0
x = cos θ
S0,0,0(x, aω)
10.50-0.5-1
2
1.5
1
0.5
0
aω = 6
aω = 0
x = cos θ
S0,4,0(x, aω)
10.50-0.5-1
1.6
1.2
0.8
0.4
0
−0.4
−0.8
aω = 6
aω = 0
x = cos θ
S0,4,4(x, aω)
10.50-0.5-1
0.8
0.6
0.4
0.2
0
aω = 6
aω = 0
x = cos θ
S− 1
2
, 1
2
, 1
2
(x, aω)
10.50-0.5-1
2.5
2
1.5
1
0.5
0
aω = 6
aω = 0
x = cos θ
S− 1
2
, 9
2
, 1
2
(x, aω)
10.50-0.5-1
1.6
1.2
0.8
0.4
0
−0.4
−0.8
aω = 6
aω = 0
x = cos θ
S− 1
2
, 9
2
, 9
2
(x, aω)
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
aω = 6
aω = 0
x = cos θ
S−1,1,0(x, aω)
10.50-0.5-1
1.5
1
0.5
0
aω = 6
aω = 0
x = cos θ
S−1,5,0(x, aω)
10.50-0.5-1
1
0.5
0
−0.5
−1
aω = 6
aω = 0
x = cos θ
S−1,5,5(x, aω)
10.50-0.5-1
1
0.5
0
Figure 6.15: The figure shows various spheroidal wave functions Sh,j,m(x, aω) as a function
of cos θ. Plots for aω = 0, 1, 2, 3, 4, 5, 6 are shown (only the first and last are
indicated, since the curves are regularly ordered). The title of each plot indicates
{h, j,m}.
6.5. High energy absorption cross sections 121
µ = 0.0
n = 4
Q = 0.3
a = 0.9
q
ωrH
N
u
m
b
er
F
lu
x
,
d
N
d
ω
d
t
543210
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−0.01
diff (s=1/2)
-0.3 (s=1/2)
0.3 (s=1/2)
diff (s=0)
-0.3 (s=0)
0.3 (s=0)
N
u
m
b
er
F
lu
x
,
d
N
d
ω
d
t
n = 5
n = 4
n = 3
n = 2
µ = 0.0
|q| = Q = 0.3
s = 0
a = 0.9
ωrH
d
N
d
ω
d
t+
|q
|
−
d
N
d
ω
d
t−
|q
|
543210
0.006
0.004
0.002
0
−0.002
−0.004
−0.006
−0.008
−0.01
d
N
d
ω
d
t+
|q
|
−
d
N
d
ω
d
t−
|q
|
Figure 6.16: Scalar and fermion asymmetries for a = 0.9: The left plot shows number fluxes
for scalars and fermions with positive and negative |q| = 0.3 for n = 4. The
difference between positive and negative |q| curves is also shown in the same
plot. The right plot shows the difference for scalars and a range of n’s.
hole) and the typical QCD charges |q| = Q = 0.3. The left plot shows the split flux curves
for the QCD case both for scalars and fermions and the difference between the two. The
right plot shows the difference curves for scalars and a range of n’s. Qualitatively, the
splitting of the curves when a 6= 0 follows the same pattern as figure 6.10. The main
differences are: the oscillations which are due to the contribution of higher partial waves
when a 6= 0; and the shift of the spectrum towards higher energies, which is again due to
the contribution from the partial waves with larger j which are shifted to higher energies.
It is interesting to note that the oscillations persist for large n in the right plot, which is
not true for the left flux plots where they tend to be smoother.
6.5 High energy absorption cross sections
To better understand the contributions to the fluxes at high energy, it is instructive to
look into an approximation based on the geometrical cross section obtained from the disks
computed in section 3.3.
In [153], Unruh proved that the absorption cross section for a plane wave, incident on
a Schwarzschild black hole from infinity is10
σ =
∑
j,m
π
ω2
Tj,m ≡
∑
j,m
σj,m. (6.62)
This suggests the interpretation of σj,m as the contribution from a partial absorption
10We are using horizon radius units.
122 Chapter 6. Analytic and numerical study of perturbations
cross section for a wave with quantum numbers {j,m}. These cross sections are directly
related to the transmission factors so they are usually named greybody factors, because
they are responsible for distorting the black body spectrum of Hawking radiation. In this
non-rotating limit the flux becomes
dN
dtdω
=
1
2π2
ω2
exp(ω/TH)− (−1)2s
∑
j,m
σj,m. (6.63)
so the Planckian term factors out of the sum. In the limit ω → +∞ we would ex-
pect an incident plane wave to be well described by a beam of classical particles. Then
the absorption cross section is simply the area of the absorptive disk at infinity (which
in the non-rotating case is a circular disk). This type of approximation was noted by
DeWitt [154], who replaced the transmission coefficient by a theta function cutting the
j-sum in (6.63) at the maximum angular momenta allowed by the absorptive disk radius
(j = bmaxω). The success of his approximation, reinforces the interpretation of σ as the
total cross section for a classical beam of particles.
Expression (6.63) is more complicated for a rotating black hole (equation (5.12)) be-
cause the Planckian factor depends on m and can not be taken out of the sum. However,
it is still possible to prove that for a wave incident at an angle θ, the cross section is [129]
(we are using the scalar case)
σ(θ) =
4π2
ω2
∑
k={j,m}
|Sk(c, cos θ)|2Tc,k ≡
∑
k={j,m}
σc,j,m(θ) (6.64)
if we average over the solid angle we get that
σ ≡
∫
dΩ
4π
σ(θ) =
∑
k={j,m}
π
ω2
Tc,k =
∑
k={j,m}
σc,j,m . (6.65)
So for the rotating case, the same relation between partial wave cross-sections and trans-
mission factors exists. But in the high energy limit, assuming the geometric description is
good, we know how to compute the absorption cross-sections for a given angle of incidence
by using the disks obtained in section 3.3. Furthermore, note that at high energies we
would expect this result to be independent of the spin of the particle. In figure 6.17 we
compare the geometrical result obtained by numerically integrating the absorptive disks,
with the asymptotic value of the sum over transmission factors computed for the scalar
6.5. High energy absorption cross sections 123
σ
pir2H
n
Averaged geometrical cross-section
a
21.510.50
20
15
10
5
0
6
5
4
3
2
1
Figure 6.17: The plot shows a perfect agreement between the averaged geometrical cross sec-
tion (solid lines) and the asymptotic value of the sum of the greybody factors
(the points). We show the cases from n = 1 (red upper line) to n = 6 (bottom
line). The curves are organise in order from the top to the bottom.
field. We find an excellent agreement between the points taken from Table I of [62] and
our geometrical calculation.
Furthermore, we can extend DeWitt’s argument to compute the Hawking flux. If we
go back to the scalar angular flux before integration over the solid angle dΩ we have
dN
dtdωdΩ
=
1
4π2
∑
k
1
exp(ω˜/TH)− 1T
(4+n)
k |Sk(c, cos θ)|2
=
ω2
(2π)4
∑
k
σc,j,m(θ)
exp(ω˜/TH)− 1 . (6.66)
Since we know the partial absorption cross sections as a function of the angle by using the
geometrical disks, we can find a high energy approximation for the spectrum by cutting
off the sums according to the allowed regions on the (b, ζ) plane. A particularly interesting
limit is when we approximate the exponential in equation (5.12) as
ω˜ ≃ ω (6.67)
124 Chapter 6. Analytic and numerical study of perturbations
if
ω ≫
∣∣∣∣ am1 + a2
∣∣∣∣ ≤
∣∣∣∣ aj1 + a2
∣∣∣∣ ≤
∣∣∣∣ ajmax1 + a2
∣∣∣∣ =
∣∣∣∣abmaxω1 + a2
∣∣∣∣ (6.68)
which holds for a small or large or for small enough j. Furthermore jmax is large, when ω →
+∞, so for most of the modes contributing to the the sum in (5.12) this approximation
should work. This implies a similar factorisation in the high energy limit for rotating
black holes:
dN
dtdω
≈ 1
2π2
ω2
exp(ω/TH)− 1σ . (6.69)
6.6 Conclusions
In this chapter we have presented various interesting effects which are important to the
modelling of black hole events in theories with extra dimensions. We have used the approx-
imate charged rotating background geometry studied in chapter 3 and the wave equations
separated in chapter 5 to model charged massive scalars and fermions. Transmission fac-
tors were obtained using approximate analytic techniques, and numerical methods which
give the result in a wider range of energies. The angular functions, which are particularly
relevant for non-zero black hole angular momentum, were obtained using series expan-
sions and checked against well known approximations. The transmission factors for vector
bosons were included in the discussion of the effect of rotation. Here the data files for
s = 0, 1/2, 1 available in [144] were used. The most important effects are as follows:
• For massive particles, our analysis shows a damping of the spectrum close to the
threshold ω ≃ µ as well as an overall reduction of the area under the flux curves.
For fermions, the suppression is not so sharp at the threshold energy. The main
consequence for LHC phenomenology is that production of massive particles such as
the top,W±, Z and Higgs boson (which have masses of the same order of magnitude
as the typical 1/rH ∼ 100 GeV) is highly suppressed at low energies. The typical
mass parameters of these heavy Standard Model particles can go up to ∼ 0.5 (in
horizon radius units), so this is an important effect.
• Black hole discharge is sub-dominant. This is another important point for LHC
phenomenology and the development of event generators which tend to enforce quick
discharge. Nevertheless, black hole events at the LHC will have non-zero charge,
so statistically we would expect a fraction of them to charge up. For charged black
holes our plots show that the flux spectra for positive and negative charges are
6.6. Conclusions 125
split. Thus negatively charged particles are biased towards low energies whereas
positively charged particles are biased towards higher energies. So the dynamical
model of discharge should still be incorporated since it will produce an asymmetry
in the energy spectrum of positive/negative charged particles.
In particular, the inverted charge splitting at low energies is a new effect due to the
extra dimensions for n ≥ 2. So, even though electric discharge may be small in TeV
gravity black hole events, this splitting will still be present and it may be possible
to reconstruct it if such events occur in future experiments. For QCD charges, it
should be even larger but a non-abelian analysis will be necessary to determine
which observables will display it.
• Rotation splits the azimuthal symmetry such that the co-rotating modes, which are
responsible for spinning down the black hole, are dominant. The energy spectrum
receives contributions from partial waves up to a larger j, which results in a shift
towards higher energies.
The non-uniform angular functions correlate with the helicity of the particle. They
give low energy axial peaks in the angular spectra both for fermions and vector
bosons, but especially for the latter. These peaks are towards the upper/lower
hemisphere according to whether the helicity of the particle is positive/negative. At
high energies all angular distributions become equatorial.
To summarize, the effects of mass, charge and rotation are important for improving
the modelling of black hole events from high energy collisions in large extra dimensions
scenarios. They may provide further signatures of black hole events such as charge and
angular asymmetries, as well as a shift of the energy spectra towards higher energies. Two
points we haven’t discussed which deserve further attention are those of QCD charges and
the possible restoration of electroweak symmetry close to the black hole. Both may be
treated using an improved model based on some of the ideas we have discussed.
In the next chapter, we implement the effect of rotation (which is the dominant one)
in the event generator CHARYBDIS2.
126 Chapter 6. Analytic and numerical study of perturbations
Chapter 7
CHARYBDIS2
To study the phenomenology of microscopic black hole production and decay in scenarios
with extra dimensions, similarly to other electroweak scale processes, it is very convenient
to have a Monte Carlo (MC) program. This simulates collision events with frequency
and properties as they would occur in a real experiment. Such processes occurring at
short distances below the hadronisation scale are usually called hard processes. Several
factorisation theorems [155] indicate that we can treat the hard process without worrying
about non-perturbative QCD effects. In this chapter we develop a Monte Carlo event
generator which simulates black hole events. This can then be interfaced to a general
purpose showering and hadronisation program which simulates the physics occurring at
larger distances. Figure 7.1 contains a schematic diagram for a proton-proton collision.
A parton level event generator simulates the hard process indicated in the centre of the
P
P
proton remnant
initial state radiation
hard process
secondary decays
parton showers
hadronisation
Figure 7.1: A schematic description of a typical proton-proton collision with a hard process
occurring at short scales from two colliding partons (centre of the figure). Other
processes simulated by a showering and hadronisation program are also indicated.
Note that the relative length scales are qualitative, not accurate.
127
128 Chapter 7. CHARYBDIS2
figure, where the black dot represents the interaction and the outgoing lines the various
outgoing products. In the MC approach any observable O can be computed using the
approximation [156]
OΣ ≡
∫
Σ
dΦO(Φ)
≈
N∑
i=1
VΣ
N
O(Φi) , (7.1)
where we have defined the observable through an integral over the phase space variables
Φ (which denote the set of kinematical quantities labelling the initial state incoming
particles), subject to a generic set of constraints (or cuts) Σ on the final state outgoing
particles. VΣ is the volume of the phase space allowed by the cuts and N is the number
of events generated uniformly in phase space. The main advantages of the MC method
are as follows. First, if we are dealing with a higher dimensional phase space, most of the
integration methods based on quadrature by discretisation on a grid and interpolation,
converge more slowly than the Monte Carlo method. The errors for the former typically
behave like N−q/Dim (where N is the number of times the integrand is evaluated, q is
an order 1 integer and Dim is the dimension of the integral), whereas for the latter the
errors behave like N−1/2. So typically, for high dimensionalities, the MC method will
converge faster. On the other hand, a complicated set of cuts is easy to implement in the
MC method since they require only a check of whether the proposed phase space point is
allowed, whereas methods based on interpolation on a grid would require a complicated
parametrisation of the boundaries. Finally, an MC generator typically provides a sample
of events with weights which can be used to create samples of events with unit weight
(or unweighted), so as to mimic actual collisions at the rates they would occur in an
experiment. This is also useful because the final state particles propagating away from
the interaction region can be interfaced to experimental software to simulate the inter-
actions with a detector. Then predictions for realistic distributions of any experimental
observables can be computed easily.
Earlier black hole event generators [46,85,157] used models which are special cases of
the theory we have presented in the previous chapters. In particular, their production
cross-section was given by hoop conjecture arguments (which are not very rigorous) and
the effect of black hole rotation was not taken into account in the evaporation. In this
chapter we present an upgraded version of the widely-used CHARYBDIS event generator [85,
7.1. Production 129
158,159], to take into account the recent theoretical work on the production and decay of
rotating black holes. This new version is named CHARYBDIS21 and it is publicly available
in [3]. This is used in the next chapter to study the effects of rotation on experimentally
observable quantities, assuming that black hole production does indeed take place.
In section 7.1 we start by describing some specifics of the implementation of the model
for production which was presented in section 4.3.2. In section 7.2, the implementation
of the spin-down Hawking decay of the black hole through the emission of SM particles
confined to the physical 3-brane is presented. Here the simulation is refined to take full
account of the theoretical results of rotation presented in chapters 5 and 6. Some options
are included for the modelling of aspects that are not well understood, such as back-
reaction effects. Note however that the new effects of particle mass and charge described
in chapter 6 have not yet been incorporated. In section 7.3 we explain the extended range
of options we have included for dealing with the final Planck mass remnant. Finally in
section 7.4 we present some details of the structure of the program and how to use it,
emphasizing the new features compared to the earlier CHARYBDIS.
7.1 Production
For the production phase, we use the model described in chapter 4.3.2 which gives the
(parton level) hard process cross section. In general, for a hard process occurring at
length scales below the hadronisation scale, asymptotic freedom and several factorisation
theorems [155] indicate that the differential cross section for a hadron-hadron collision
(which describes the rate at which the process occurs) takes the form
dσhihj→X
dx1dx2
(s) = fi(x1)fj(x2)σˆij→X(sˆ) . (7.2)
We are working in the hadronic centre of mass frame and the collision is along the z-
axis. Here {x1, x2} are the momentum fractions of the partons {i, j} (quarks or gluons)
participating in the hard process. The function fi(x) is the parton density function (PDF)
which encodes all the non-perturbative physics of the hadron. It is independent of the
hard process and it gives the number density of partons of type i with a momentum
fraction x of the original hadron, which can participate in the collision. The partonic
1We will refer from now on to the new release as CHARYBDIS2 and will reserve CHARYBDIS for earlier
versions. The particular version described here is CHARYBDIS2.0.
130 Chapter 7. CHARYBDIS2
cross-section σˆij→X is the short range cross-section for the two partons to collide with
(partonic) centre of mass energy squared sˆ = x1x2s. The hadronic centre of mass energy
squared is s. Note that we are considering processes where the two hadrons are protons
so the two PDFs are of the same type. In this case the particles produced in the hard
process are X = BH+radiation, the black hole plus radiation released in the production.
The differential cross section provides a distribution function for the process to occur with
the given kinematical configuration of the partons {i, j}. The total cross section σhh→X is
obtained by integrating over the momentum fractions and summing over types of partons
σhh→X(s) =
∑
ij
∫
0≤x1,x2≤1
dx1dx2fi(x1)fj(x2)σˆij→X(sˆ) . (7.3)
The total rate of events for the process is
dN(s)
dt
= Lσhh→X(s) (7.4)
where L is the luminosity of the beams. The latter is defined as the number of pairs of
particles (one from each beam) crossing the transverse plane of collision per unit area and
unit time. The cross-section is a useful quantity which is independent of the experimental
details of the beams.
Since the partonic cross-section is the short scale counterpart of the hadronic cross-
section it can be similarly defined as
σˆ =
1
Lˆ
dNˆ
dt
(7.5)
where Lˆ is the partonic luminosity. If Lˆ is constant it is possible to show (assuming beams
with constant density) that
σˆ =
∫
dSP (b) (7.6)
where P (b) is the probability of interaction with an impact parameter b and dS is the
transverse element of area. Thus we can define a differential partonic cross-section
dσˆ
db
= 2πbP (b) (7.7)
7.1. Production 131
So in general, we can write a more general hadronic differential cross section
dσhihj→X
dx1dx2db
(s) = fi(x1)fj(x2)
dσˆij→X
db
(sˆ) . (7.8)
In our study of black holes, the b dependence is important because the formation and
decay of the black hole depends on b. In particular since the formation is dominated
classically P (b) is unity for all allowed impact parameters, i.e. P (b) = θ(bmax − b) so the
partonic cross-section is simply the area of the disc with radius bmax.
In CHARYBDIS2, we have implemented the distribution (7.8) to select: the partons
involved in the collision; their momentum fractions, and the impact parameter, given the
centre of mass energy
√
s. For the partonic cross-section σˆ(
√
sˆ) we used the lower bound on
the cross-section given by the maximum impact parameter from the model in section 4.3.2.
This improved bound can be switched back to the hoop conjecture geometrical estimate
by setting the value of the variable YRCSEC to .FALSE.. Furthermore, for each event, we
reduce the mass and angular momentum of the black hole using the model in section 4.3.2
when MJLOST=.TRUE.. The mass/energy lost during the production phase is distributed
between radiation and the kinetic energy of the formed black hole. The production phase
simulation must account for this. On the basis of several calculations [160–162], which
indicate that gauge radiation is negligible compared to gravitational radiation in the
production phase, we assume that all of the radiation is in the form of gravitons. Given
that gravitons are missing energy, it is sufficient for the simulation to represent the entire
radiation pattern using a ‘net graviton’ with a four-momentum equal to the sum of those
of the emitted particles.
The net graviton has an invariant mass µg, which may potentially lie anywhere be-
tween 0 and 1 − ξ (in units of the initial state mass), where ξ is the fraction of the
partonic system invariant mass that constitutes the black hole. An invariant mass of
1 − ξ corresponds to a completely symmetric emission of gravitons, whilst lower values
correspond to steadily more antisymmetric emissions (which might result if a small num-
ber of gravitons is released, and by chance they are emitted in similar directions). In
CHARYBDIS2, the invariant mass is randomly generated per event from a power distribu-
tion, P (µg) ∝ µpg. The mean of this distribution is set equal to FMLOST×(1 − ξ) by the
quantity FMLOST= (p + 1)/(p + 2) (default value 0.99, corresponding to p = 98). The
simulation of the production phase emission is then a two body decay from initial state
object into formed black hole plus net graviton, which is isotropic in the centre of mass
132 Chapter 7. CHARYBDIS2
frame of the initial state object.
7.1.1 Adding the intrinsic spin of the colliding particles
The model for the angular momentum of the black hole after formation used in the
previous section is based on using incoming particles with zero spin. Angular momentum
conservation requires us to include the intrinsic spin of the incoming particles falling into
the black hole. Since the results in the literature taking this effect into account are limited
to special cases (see for example [78]) we assume a simple model where first we combine
the spin states of the incoming particles into a state
|s1, h1〉z ⊗ |s2, h2〉z = |s, sz〉z . (7.9)
The collision axis is denoted by z, si, hi are the spin and helicity of the particles
2 and s, sz
are the angular momentum quantum numbers of the combined state in the rest frame.
Since we have unpolarised beams we give equal weight to each helicity combination. Then
this angular momentum state is combined with the orbital contribution obtained from the
model for angular momentum loss. We denote it by
|L,L〉z′ , (7.10)
where L is the nearest integer to J . Note that z′ is an axis in the plane perpendicular to the
beam axis (z-axis) chosen with uniform probability. Finally, using a Wigner rotation [163]
followed by a tensor product decomposition using Clebsch-Gordan coefficients, we obtain
|L,L〉z′ ⊗ |s, sz〉z = |L,L〉z′ ⊗
s∑
s′z=−s
d
(s)
s′z,sz
(
cos
π
2
)
|s, sz′〉z′
=
s∑
s′z=−s
d
(s)
s′z,sz
(0)
L+s∑
J=|L−s|
CJ,L,L,s,sz′ |J, L+ sz′, L, L, s, sz′〉z′ ,(7.11)
where d
(s)
sz′ ,sz is a Wigner function and CJ,L,L,s,ss′ is a Clebsch-Gordan coefficient for the
tensor product decomposition of |L,L〉z′ ⊗ |s, sz′〉z′ . From (7.11) it is straightforward
to determine the probabilities for all possible combinations of helicities and incoming
partons.
2Note we are assuming the massless limit where hi = ±si
7.2. Evaporation 133
This model introduces a spread in the orientation of the initial black hole angular
momentum axis around the plane perpendicular to the z-axis. Note that even though
the model for angular momentum loss in the previous sections does not include such an
effect, in a realistic situation we would not expect the angular momentum to be exactly
perpendicular to the beam axis after the production phase.
7.2 Evaporation
After formation, the black hole is allowed to decay semi-classically by emitting Hawking
radiation. We assume that the black hole remains stuck on the brane and that emission
occurs mostly in the form of SM fields on the brane [164]. The possibility of ejection would
come from graviton emission into the bulk [110,132]. Since the black hole is formed from
SM particles which are themselves confined, we assume that even if gravitons are emitted,
the extra-dimensional recoiling momentum is absorbed by the brane, avoiding ejection.
One could argue that whichever charges keep the black hole confined to the brane, they
are lost at the start of the evaporation through Schwinger emission. We have seen in
chapter 6 that discharge is not necessarily large on the basis of the Schwinger effect alone.
Furthermore, even if discharge does occur, it will be very unlikely that all the different
gauge charges are simultaneously neutralized at any stage during the evaporation, if we
assume that the black hole decays by emitting one quantum at a time. Furthermore, there
are a lot more SM degrees of freedom than gravitational ones so even if the unlikely event
of exact neutralisation occurs, it will still be unlikely that a graviton is emitted during
the brief period of neutrality. Thus we would expect the number of events in which the
black hole is ejected into the bulk to be at most a small fraction of the total.
In chapter 6 we argued that though electric discharge is indeed favoured, the effect
on the flux spectra is not large. In the generator we have kept the early simplified model
such that whenever a charged field is selected for emission, the electric charge of the
state is selected so as to reduce the total charge of the black hole (unless the BH is
neutral, in which case equal probabilities for particles and anti-particles are used). To
avoid complications in hadronisation, baryon number conservation is also assumed, and
colours are assigned to ensure that colour singlet formation is possible. This model will
be improved in future works.
The assumption that brane emission is dominant is supported by early studies of the
ratio of brane to bulk emission such as [54,113]. Furthermore we have a very large number
134 Chapter 7. CHARYBDIS2
of brane degrees of freedom compared to bulk gravitons. However it is important to note
that the fluxes for all the modes for bulk graviton emission in a higher dimensional singly
rotating background are not yet known in full (only tensor modes have been considered
recently [124, 125]). Nevertheless even if there is an enhancement due to superradiance
similar to that for vector fields it is unlikely that it will be enough to overcome the large
number of brane degrees of freedom.
In the generator we also assume that the emission of Hawking radiation can be treated
semi-classically and that the black hole has time to re-equilibrate between emissions.
These assumptions are valid as long as the mass of the black hole is much larger than the
(higher-dimensional) Planck mass MD, in which case the typical time between emissions
is large compared to rH (see for example (4.6)) and as long as the Hawking temperature,
which gives the typical energy scale of the emissions, is below MD.
The numerical methods employed to determine the transmission factors were described
in chapter 6. We have used (and checked) the high precision publicly available data [144]
for n = 2, . . . , 6 and completed it for n = 1 using the methods described in chapter 6.
The ranges for the parameters are3 n = 1, 2, . . . 6, ωrH = 0.05, 0.10, . . . 5.0 and a∗ =
0.0, 0.2, . . . 5.0 for the angular modes j = |h|, |h| + 1, . . . |h| + 12 and m = −j . . . j. For
each point we have computed the flux spectrum using (5.12). This quantity is used in
CHARYBDIS2 as a probability distribution function for the quantum numbers of a particle
with a given spin and to determine the relative probability of different spins (through
integration of equation (5.12)). For convenience in the Monte Carlo, we have computed
the following cumulative distributions from the transmission factors:
Ch,j,m,a∗,D(ωrH) =
∫ ωrH
0
dx
1
exp(x˜/τH)± 1T
(D)
k (x, a∗) (7.12)
Ch,a∗,D(K) =
K∑
Q=1
Ch,jm,a∗,D(ωrH →∞), (7.13)
where x is energy in units of r−1H , τH = THrH , x˜ = x −mΩ/rH and Q is an integer that
counts modes. The modes are ordered with increasing j and within equal j modes they
are ordered with increasing m.
Cumulative functions are more convenient since they allow for high efficiency when
selecting the quantum numbers. This is done by generating a random number in the
range [0, C(∞)], followed by inversion of the corresponding cumulant. In CHARYBDIS2,
3Note we have restored rH in these expressions.
7.2. Evaporation 135
when values of a∗ between those mentioned above are called, linear interpolation is used.
When a∗ is larger than 5, we use the cumulative functions for a∗ = 5. We have checked
that for most of the evaporation such large values are very unlikely. The exception is
the final stage, when the black hole mass approaches the Planck mass. Here one of the
remnant models takes over, as described in section 7.3.1.
7.2.1 Back-reaction and spin-down
The difficult problem of studying the back-reaction is interesting, both from the theoretical
and the phenomenological point of view, since it should start to influence the evaporation
as the mass of the black hole is lowered.
On the theory side, there is no well established framework to study the evolution of
a Hawking-evaporating black hole over the full range of possible initial conditions. The
usual approach [91, 114, 165, 166] is to write down mean value differential equations for
the variation of the parameters (M and J) such as (5.12) and integrate them with appro-
priate initial conditions. However, this is only valid for a continuous process of emission
where the variation of the parameters is very slow and the symmetry of the background
spacetime is kept. Thus, it ignores the momentum recoil of the black hole and the change
in orientation of the angular momentum axis between emissions, which are certainly neg-
ligible for an ultra-massive black hole, but will start to become important as we approach
the Planck mass. Furthermore, since the Hawking spectra at fixed background parame-
ters are used, it also neglects the effect of the backreaction on the metric by the emitted
particle. This point has been explored in simplified cases of j = 0 waves for fields of
several spins using the method in [167] with some results regarding the modification of
the thermal factors, but a full treatment is still lacking.
In the program, we have included two possible models for the momentum recoil of
the black hole set by the switch RECOIL, which takes the values 1 or 2. The orientation
of the momentum vector is always computed using the square modulus of the spheroidal
function (6.1). This is due to the decomposition of spheroidal one-particle states into plane
wave one-particle states,4 which is analogous to the decomposition of spherical waves into
plane waves, for the usual case of scattering off a spherical potential, as presented in
section 5.3.4.
RECOIL = 1 interprets the selected energy as the energy of the particle in the rest
frame of the initial black hole. The momentum orientation is computed in this frame
4Note that in the convention of equation (6.1) the physical helicity of the particle is actually −h.
136 Chapter 7. CHARYBDIS2
with probability distribution given by the square modulus of the spheroidal function
(6.1) and the momentum of the final black hole is worked out from conservation. The
argument for this model comes from the observation that particles in the decay are highly
relativistic. They propagate close to the speed of light, so the background they see is that
of the initial black hole, since no signal of the back-reaction on the metric can propagate
outwards faster than light. Thus the momentum of the emission is determined by the
background metric in this picture.
RECOIL = 2 takes the energy of the emission as being the loss in mass of the black
hole. This corresponds to the usual prescription for computing the rate of mass loss.
The orientation of the momentum in the rest frame of the initial black hole is computed
as before with a probability distribution given by the square modulus of the spheroidal
function (6.1) and the 4-momentum of the emission as well as that of the black hole are
worked out.
Note that for any of the previous options, full polarization information of the emission
is kept, as it is generated with the correct angular distribution. This will potentially pro-
duce some observable angular asymmetries and correlations, which would not be present
if angular distributions averaged over polarizations had been used.
The other quantity we need to evolve is the angular momentum of the black hole.
We have two options, controlled by the switch BHJVAR. The default BHJVAR = .TRUE.
uses Clebsch-Gordan coefficients to combine the state of angular momentum Mz = J
of the initial black hole (i.e. taking as quantisation axis the rotation axis of the black
hole), with the emitted j,m state for the particle. The probability of a certain polar
angle and magnitude for the angular momentum of the final black hole is given by the
square modulus of the corresponding Clebsch-Gordan coefficient, and the azimuthal angle
is chosen with uniform probability. If BHJVAR=.FALSE., the orientation of the axis remains
fixed even though the magnitude will change by subtraction of them value of the emission.
From previous versions of CHARYBDIS, we have kept the switch TIMVAR which allows
one to fix the parameters of the black hole used in the spectrum (such as the Hawking
temperature) throughout the evaporation. This option corresponds to a model where the
evaporation is no longer slow enough for the black hole to re-equilibrate between emission,
so in effect it represents a simultaneous emission of all the final state particles from the
initial black hole without any intermediate states.
In figure 7.2 we plot the evolution of the physical parameters M and J for BH events
with fixed initial M , in the non-rotating case and the highly rotating case, using the
7.2. Evaporation 137
Figure 7.2: Probability maps for physical parameters, constructed from 104 trajectories for
different BH events with fixed initial conditions M,J (for each horizontal line).
Each trajectory contributes with weight 1 to the bins it crosses on the {P, t/ttotal}
plane where P is the relevant parameter. Note that the time is normalised to the
total time for evaporation ttotal. The horizontal lines for the plots on the right are
due to the discretisation of J in semi-integers.
138 Chapter 7. CHARYBDIS2
Figure 7.3: Probability maps for the geometrical parameters A and a∗ which characterize re-
spectively the size and oblateness of the BH. Note that for the a∗ plot in the last
t/ttotal = 1 line there is very often a jump to very large a∗. In these plots we
have put all such points in the bin on the upper right corner to avoid squash-
ing the interesting region. Each horizontal line has the same initial M,J as the
corresponding one in figure 7.2.
7.2. Evaporation 139
default values for the evaporation model (see table 7.1 in section 7.4.2). In figure 7.3 we
plot the horizon area and oblateness for the same cases as in figure 7.2.
Events were generated for two possible initial masses, 10 TeV (reachable at the LHC)
and 50 TeV. The latter serves as a check of the semi-classical limit. We focus on n = 2
and n = 6. An important quantity necessary to produce these plots, is the time between
emissions. Since our model for the evolution relies on the mean value equation (5.12),
before each emission, an average time can be computed (i.e. δt for δN = 1):
δN =
dN
dt
δt ⇒ δt =
[
dN
dt
]−1
,
where a sum over all species is assumed (see section 7.3.1 for further details).
Each plot contains 104 trajectories (one per event generated), each contributing with
weight 1 to the density plot. The darker areas correspond to higher probability and in all
the plots we can discern a tendency line which is sharper for the 50 TeV case and more
diffuse for 10 TeV, reflecting the magnitude of the statistical fluctuations.
The left columns of figures 7.2 and 7.3 show respectively the evolution of the mass
parameter and horizon area for non-rotating black holes. The centre and right columns
show the evolution of the mass and angular momentum, or the horizon area and oblateness,
for the highly rotating case (a∗ ≃ 3). The main features are as follows:
• Non-rotating case: BothM and A decrease approximately linearly with time except
for the last ∼ 10 − 20 % when they drop faster. This is directly related to the
behaviour of the temperature which increases slowly (approximately linearly) for
most of the evaporation and rises sharply near the end. The rates tend to be faster
for higher n which is in agreement with the increase in Hawking temperature with n.
• Highly-rotating case: Here the statistical fluctuations tend to smear out the plots
for the case of lowest mass. However, the same tendency can be seen as for the
M = 50 TeV black holes; the latter display better a true semi-classical behaviour.
There is in general an initial period of roughly 10− 15 % of the total time when M
drops faster to about 60−70%. At the same time the angular momentum also drops
sharply to 20 %. This corresponds to the usual spin-down phase [166]. Note that
the fluctuations are quite large for the low-mass n = 2 plots. As for the geometrical
parameters, they follow a similar tendency if we make the correspondences M ↔ A
and J ↔ a∗. Again for the low-mass plots the statistical fluctuations smear out
140 Chapter 7. CHARYBDIS2
the sharper initial drop in area, in particular for n = 2 in which it can occasionally
increase substantially. The a∗ plots show how the black hole tends to become more
spherical in this spin-down phase. The remainder of the evolution resembles the
non-rotating case and can be identified with a Schwarzschild phase. Note however
that by the end of the evaporation (when M approaches the Planck mass) this
description breaks down and a∗ rises again, since even only one unit of angular
momentum has a very large effect on this quantity at the Planck scale. This means
that a∗ ceases to have a well defined geometrical meaning, as we reach the Planck
phase. Similarly to the non-rotating case, we have checked that the temperature
increases slowly and approximately linearly for most of the evaporation except for
a sharp rise as the Planck mass is approached.
These observations agree with the usual results in four dimensions5 and the results
of [111] in D dimensions.
7.3 Remnants
Our model for black hole decay relies heavily on the assumptions that we are in the semi-
classical regime and the evaporation is slow (i.e. there is enough time for re-equilibration
between emissions) [24, 168]. However, as the evaporation evolves, we will reach a point
where neither of these assumptions will be true as the mass and/or temperature of the
black hole become comparable with the Planck scale, so a complete theory of quantum
gravity is required. In the absence of such a theory, various models for the termination
of black hole decay have been suggested [169–174]. In the generator we introduce some
remnant models based on different physical assumptions. These are discussed in what
follows.
7.3.1 Termination of the black hole decay
First of all, we need a criterion to decide whether or not the remnant stage has been
reached. The various options in the program are connected to a departure from semi-
classicality. This occurs when the expectation value 〈N〉 for the number of emissions
becomes small, which is a sign of the low number of degrees of freedom associated with
the black hole. Together with the drop in 〈N〉, the Hawking temperature will rise sharply.
5see for example [166] or chapter 10.5.3 of [48]
7.3. Remnants 141
This is all related to the approach of the black hole mass to the Planck mass. The options
are:
• NBODYAVERAGE=.TRUE.: An estimate for the multiplicity of the final state is com-
puted at each step during the evaporation, according to the Hawking spectrum:
〈N〉 ≃ dN
dt
δt ≃ dN
dt
M
(
dE
dt
)−1
= MrH
∑
i gi
(
1
rH
dN
dt
)
i∑
j gj
(
dE
dt
)
j
. (7.14)
The sums are over all particle species with appropriate degeneracies gi. The inte-
grated flux and power are computed using (5.12). A natural criterion for stopping
the evaporation is when this estimate drops below some number close to 1. In the
generator we use 〈N〉 ≤ NBODY − 1 where NBODY gives the average multiplicity of
the remnant decay final state (see section 7.3.3 for further comments). Varying the
parameter NBODY will give a measure of uncertainties in the remnant model. In
addition, if we choose a remnant model that decays, (7.14) gives an estimate of the
final state multiplicity for such a decay. When NBODYAVERAGE = .FALSE., one of the
options below, inherited from earlier versions of CHARYBDIS, is used.
• KINCUT=.TRUE.: Terminate evaporation if an emission is selected which is not kine-
matically allowed. This is closely related to the rapid increase in temperature as we
approach the Planck mass and consequently the generation of kinematically disal-
lowed energies for the emission. Otherwise if KINCUT = .FALSE. the kinematically
disallowed emissions are rejected and the evaporation terminates when the mass of
the black hole drops below the Planck mass6.
7.3.2 Fixed-multiplicity decay model
The default remnant decay option is a fixed-multiplicity model similar to that in earlier
versions of CHARYBDIS. At the end of the BH evaporation, the remaining object is decayed
isotropically in its rest frame, into a fixed number NBODY of primary particles, where the
parameter NBODY is an integer between 2 and 5. The decay products are chosen with
6The Planck mass used in CHARYBDIS2 to decide on the termination is always the internal one, INTMPL,
which is obtained by converting the Planck mass input by the user (in a given convention), to the Giddings-
Thomas convention – see Appendix A.2.
142 Chapter 7. CHARYBDIS2
relative probabilities appropriate to the final characteristics of the black hole (i.e. weighted
according to the integrated Hawking fluxes for each spin).
The selection of the outgoing momenta of the decay products may be chosen either
using pure phase space (NBODYPHASE=.TRUE.) or by using the following probability density
function in the rest frame of the black hole (NBODYPHASE=.FALSE.):
dP ∝ δ(4)
(∑
i
pi − PBH
)∏
i
ρi (Ei,Ωi) d
3pi , (7.15)
which amounts to the usual phase space momentum conservation with an extra weight
function for each particle
ρi (Ei,Ωi) =
T
(D)
k (ErH , a∗)
exp(E˜/TH)± 1
|Sk(cos θi)|2 , (7.16)
where k = {j,m} are chosen according to the cumulants (7.13) combined with angu-
lar momentum conservation. Here Ei,Ωi are the energy and momentum orientation of
the emission in the rest frame of the remnant. The method for generating the phase
space (7.15) is described in appendix D.2. This choice treats the final state particles on
an equal footing, keeping a gravitational character for the decay (since it uses Hawking
spectra), as well as some correlations with the the axis of rotation through the spheroidal
function factor. Furthermore, at this stage, slow evaporation should no longer be valid,
so it makes sense to perform a simultaneous decay at fixed black hole parameters. This
remnant option can be used with any of the criteria for termination.
7.3.3 Variable-multiplicity decay model
In addition to a fixed multiplicity final state, an option has been introduced to select the
multiplicity of the final state on an event-by-event basis. We follow an idea in [175], which
has been used for example in the case of 2→ 2 sub-processes in [172]. Here we implement
a more general model for arbitrary multiplicity, which is invoked by setting the parameter
NBODYVAR=.TRUE..
As argued previously, when the remnant stage is reached, the black hole should no
longer have time to re-equilibrate between emissions. Under this assumption, the prob-
ability distributions should become time independent. It is relatively straightforward to
prove that under these conditions for a time interval δt, the multiplicity follows a Poisson
7.3. Remnants 143
distribution [175]:
Pδt(n) = e
−αδt (αδt)
n
n!
, (7.17)
with α some constant. From the Hawking flux, we have computed an estimate for the
average number of particles emitted during δt (i.e. the time interval until all mass disap-
pears), so α is determined from this condition. The final result is
Pδt(n) = e
−〈N〉 〈N〉n
n!
, (7.18)
where 〈N〉 is the estimate in (7.14). This expression gives us an estimate for the prob-
ability of emission of n particles from the remnant, so we choose to interpret n + 1 as
the multiplicity of the final system. In the generator we have removed the n = 0 case
(i.e. multiplicity 1 final state) since the probability of the remnant to have all the correct
quantum numbers and mass of a standard model particle will be vanishingly small.
After the multiplicity is chosen, either the pure phase space decay or the model de-
scribed in the previous section is used, according to the value of NBODYPHASE.
7.3.4 Boiling model
The boiling remnant model, activated by setting RMBOIL=.TRUE., is loosely motivated by
the expectation that at the Planck scale the system becomes like a string ball [169, 176],
which has a limiting temperature due to the exponential degeneracy of the string spec-
trum [173]. In this model, evaporation of the BH proceeds until the Hawking temperature
for the next emission would exceed a maximum value set by the parameter THWMAX. From
that point on, the temperature is reset to THWMAX and the oblateness is frozen at the cur-
rent value. The remaining object evaporates like a BH with those characteristics, until its
mass falls below a value set by the parameter RMMINM. It then decays into a fixed number
NBODY of primary particles, as in the fixed-multiplicity model, or a variable number if the
variable-multiplicity model is on.
7.3.5 Stable remnant model
A number of authors have proposed that the endpoint of black hole evaporation could
be a stable remnant [170, 171, 174]. This option is activated by setting RMSTAB=.TRUE..
In order for the cluster hadronisation model of HERWIG to hadronise the rest of the final
state successfully, the stable remnant must be a colourless object essentially equivalent to
144 Chapter 7. CHARYBDIS2
a quark-antiquark bound state. Therefore it is required to have baryon number BR = 0
and charge QR = 0 or ±1.
The stable remnant appears in the event record as Remnant0, Remnant+ or Remnant-,
with PDG identity code 50, 51 or –51, respectively, according to its charge. This object
will behave as a heavy fundamental particle with conventional interactions in the detector.
If a remnant with BR 6= 0 or |QR| > 1 is generated, the whole BH evaporation is
repeated until BR = 0 and |QR| ≤ 1. This can make the stable remnant option much
slower than the other options, depending on the length of the black hole decay chain.
7.3.6 Straight-to-remnant option
Recently, there has been discussion of the possibility that the formation of a semi-
classical black hole may lie beyond current experimental reach, with low-multiplicity
gravitational scattering more likely at the TeV scale [172]. To simulate this scenario,
CHARYBDIS2 provides the option of bypassing the evaporation phase by setting the switch
SKIP2REMNANT=.TRUE. and skipping directly to one of the remnant models presented in
the previous sections. This permits the study of a wide range of qualitatively different pos-
sibilities, from simple 2→ 2 isotropic scattering (fixed multiplicity) to more complicated
variable-multiplicity 2→ N sub-processes.
The 2 → N model is particularly flexible, allowing either a phase-space distribution
or one using the Hawking energy and angular spectra (see section 7.3.3). Apart from
this, all particle species are treated on an equal footing consistent with conservation laws.
Alternatively the quantum-gravity motivated boiling model can be used. Further work
will be presented in future publications exploring the phenomenological consequences of
these scenarios.
7.4 Program structure and usage
In this section we describe in detail the structure of the CHARYBDIS2 program. We will
focus on the new features, however a brief description of some features that were kept or
derived from CHARYBDIS is presented. The code for the original program was developed
in fortran 77 so we kept the same programing language.
Section 7.4.1 provides an overall description of the main logical building blocks of the
program and how they interact. In the following three sections the main blocks of the
run are explained in further detail.
7.4. Program structure and usage 145
7.4.1 General structure
Figure 7.4 describes the general structure of the code and its interface to a general purpose
parton showering and hadronisation generator. This can be split in three types of units
of code organised in three horizontal layers in the diagram:
• The main program: This is the code which contains the main program and con-
trols the run. It reads the input and initialises all variables which are used in the
run; it runs the program by calling the relevant routines of the event generator;
it analyses the events, and finally it produces output files. There are three pos-
sible main programs available to the user, maincharybdis.f, mainherwig.f and
mainpythia.f. The first runs CHARYBDIS2 standalone without the second layer in
the diagram (dashed boxes), so only the hard process in figure 7.1 gets generated.
The other two modes interface to HERWIG [93, 94] and PYTHIA [95] respectively for
the parton showering, hadronisation and secondary decays.
• General purpose event generator: This is either the herwig*.f code or pythia*.f
code7 which contains the subroutines responsible for: generating the showering of
the initial and final state partons involved in the hard process; performing secondary
decays of unstable particles, and hadronising the final state quarks and gluons (see
figure 7.1). These routines are called from the main program unit when the respec-
tive HERWIG or PYTHIA interfaces are enabled.
• Black Hole event generator (CHARYBDIS2): This is the charybdis2*.F code which
contains the subroutines responsible for generating hard process events with the
corresponding weights (according to the differential cross-section (7.8)). In the stan-
dalone CHARYBDIS2 implementation, they are called directly from the main program
unit (in maincharybdis.f), whereas for the HERWIG or PYTHIA implementation they
are called indirectly by the initialisation and event generation routines of HERWIG or
PYTHIA in the main program unit (mainherwig.f or mainphythia.f).
On the other hand, the running of the program can be described in three stages (three
columns connected by vertical lines in the diagram) which are called in sequence from the
main program:
• Initialisation: The input conditions for the incoming beams are set, together with
the parameters for the model for black hole production and decay. If HERWIG or
7The asterisk represents the version number.
146 Chapter 7. CHARYBDIS2
MAIN PROGRAM maincharybdis.f
or
mainherwig.f
or
mainpythia.f
Runs through the 3
stages below (con-
nected by arrows)
?
?
?
?
6
........... ...
...
...
..
ff
-?
6
?
?
?
?
6
........... ...
...
...
..
ff
-?
6
6
?
?
INITIALISATION
- Beam
- CHARYBDIS2
- HERWIG or PYTHIA
GENERATE EVENTS
- Production
- Decay/Evaporation
-Hadronisation, show-
ering and decays
TERMINATE RUN
- Finalise output files
- Produce histograms
- . . .
GENERAL PURPOSE EVENT GENERATOR
This can be herwig*.f or pythia*.f, or be absent
charybdis2*.F
UPINIT
Generates weights
and computes the
cross section
UPEVNT
Generates events and
puts them in the
common blocks
Terminate HERWIG or
PYTHIA and finish
their output files
END OF RUN
Figure 7.4: Diagram illustrating the general code structure of CHARYBDIS2. The run is con-
trolled by a main program unit which calls in sequence three sets of routines,
denoted above by initialisation, generate events and terminate run. Each of these
may or may not be interfaced to a general purpose event generator which is repre-
sented by the dashed box. The flow of arrows shows the order in which the various
blocks of code are executed and returned to.
7.4. Program structure and usage 147
PYTHIA are used, some initialisation routines are called which perform a weight
search and compute the cross-section by calling the CHARYBDIS2 subroutine UPINIT.
Otherwise, UPINIT is called directly from the main program. Some routines which
perform some user defined initial calculations are called and output files are prepared
for the run.
• Event generation: The routine UPEVNT which generates the hard process event is
called either directly from the main program (standalone mode) or through the
event generation routines in HERWIG or PYTHIA. Parton showering, hadronisation and
secondary decay routines are also called at this point. The (user-defined) analysis
routines are called, and the event is stored in appropriate output files.
• Termination of the run: Event generation is stopped, the (user-defined) final calcu-
lations are performed and the output files are finalised.
In the next sections we describe in detail each stage. Instructions on how to set up the
latest release of the code [3] are provided in appendix D.1.
7.4.2 Initialisation
The diagram in figure 7.5 represents the initialisation routines for the three possible main
programs (mainpythia.f, maincharybdis.f and mainherwig.f) divided in three blocks
(dashed boxes).
The first block is common to all implementations and consists of two subroutines which
set the values of the parameters for the run which control: the characteristics of the beam,
the model used for the black hole events, and some conventions. This can be done either
by setting the values in the subroutine CHDEFAULTS or by editing the charybdis2.init
file provided with the code (this is read by the subroutine CHREADINITFILE). A summary
of all the variables that are set in the initialisation is provided in table 7.1.
In the second block the (CHARYBDIS2 defined) UPINIT subroutine which is responsible
for searching for the maximum weight of the distribution (7.2) and computing the to-
tal cross-section (7.3) is called either directly, in the maincharybdis.f implementation,
or through PYINIT or HWIGIN, for the mainpythia.f and mainherwig.f implementa-
tions respectively. The UPINIT subroutine is explained in appendix D.3. Note that the
mainherwig.f implementation contains other initialisation code, which for example de-
fines the black hole and stable remnant particle codes, and computes other quantities to
be used internally by HERWIG.
148 Chapter 7. CHARYBDIS2
mainpythia.f maincharybdis.f mainherwig.f
CHDEFAULTS
?
CHREADINITFILE
?
??
?
PYINIT
?
-
9 UPINIT
?
HWUIDT
?
HWIGIN
ff
z
Other HERWIG init
?
?
CHARYBEG
...........?
...........?
HWABEG
...........?
Figure 7.5: Diagram illustrating the run of the initialisation block for the three possible im-
plementations (running vertically, the PYTHIA, CHARYBDIS2 and HERWIG implemen-
tations respectively). Top block: Read input parameters. Middle block: Initialise
general purpose generator and weight search. Bottom block: User-defined Initial
calculations routines. The three bottom dotted arrows indicate the event genera-
tion block follows (figure 7.6).
Finally, in the last block, user-defined initial calculations and output files are prepared
for the run. For example headers are prepared for the Les Houches Event files (LHE) [177]
where the events are stored, and the histories file where the black hole histories are stored.
The subroutines CHARYBEG and HWABEG (for HERWIG) can be edited according to the needs
of the user.
7.4.3 Event generation
The diagram in figure 7.6 is divided in two blocks (dashed boxes) to describe the event
generation loop.
In the first block the subroutine UPEVNT is called to generate the black hole and its
evaporation, either directly (maincharybdis.f) or through the PYEVNT (mainpythia.f)
7.4. Program structure and usage 149
Variable name Description Default
EBMUP(I) Beam energies (I=1,2) 7 TeV
IDBMUP(I) Beam identity 2212 (p)
PDFGUP(I) Codes for PDF group —
PDFSUP(I) Codes for PDF set —
LHAPDFSET Code for PDF set when using LHAPDF 10000
MINMSS Minimum partonic centre of mass energy 5 TeV
MAXMSS Maximum partonic centre of mass energy 14 TeV
NRN(I) Seeds for the pseudo-random number generator —
TOTDIM Total number of dimensions 6
MPLNCK Higher dimensional Planck mass 1 TeV
MSSDEF Convention for Planck mass 3
GTSCA Use Giddings-Thomas scale for PDFs .FALSE.
YRCSC Use Yoschino-Rychov cross-section enhancement .TRUE.
MJLOST Simulation of M , J lost in production/balding .TRUE.
CVBIAS ‘Constant angular velocity’ bias .FALSE.
FMLOST Isotropy of gravitational radiation lost 0.99
MSSDEC Allowed decay products (3=all SM particles) 3
GRYBDY Include grey-body factors .TRUE.
TIMVAR Allow TH to evolve with BH parameters .TRUE.
BHSPIN Simulate rotating black holes .TRUE.
BHJVAR Allow black hole spin axis to vary .TRUE.
BHANIS Non-uniform angular functions for the evaporation .TRUE.
RECOIL Recoil model for evaporation 2
NBODY Number of particles in the remnant decay 2
KINCUT Use a kinematic cut-off in the evaporation .FALSE.
THWMAX Maximum Hawking temperature 1 TeV
RMSTAB Stable remnant model .FALSE.
NBODYAVERAGE Use flux criterion for remnant – see equation (7.14) .TRUE.
NBODYVAR Variable-multiplicity remnant model .FALSE.
NBODYPHASE Use uniform phase space for remnants .FALSE.
SKIP2REMNANT Bypass evaporation phase .FALSE.
RMBOIL Use boiling remnant model .FALSE.
RMMINM Minimum mass for boiling model 100 GeV
LHEFILENAME Name for the les houches *.xml output file lhouches
HISFILENAME Name for the black hole histories *.xml output file histfile
Table 7.1: List of parameters for the run set in the initialisation, and some of their default
values. The first group defines the beam, PDFs and the hard process centre of mass
energy, the second defines the extra-dimensional model, the third the parameters
of the model used for the production phase, the fourth the model for evaporation,
the fifth the criterion for terminating the evaporation and performing the remnant
model, and the last one defines names for two output files.
150 Chapter 7. CHARYBDIS2
ff
6
6
6
.........?
.........?
.........?
- - -......?
......?
......?
?
PYEVNT
zff
?
PYLIST
?
?
UPEVNT
?
?
HWUINE
?
HWEPROz
ff
?
Other HERWIG
?
CHANAL
? ? ?
HWANAL
CHPRINT
? ? ?
Figure 7.6: Diagram illustrating the event generation loop which follows after initialisation.
Top block: Generate one hard process event and perform parton showering, sec-
ondary decays and hadronisation (for the PYTHIA and HERWIG implementations).
Bottom block: Event analysis routines (CHANAL and HWANAL) and black hole history
print-out (CHPRINT).
or HWEPRO (mainherwig.f) subroutines. Note that in the last two implementations other
internal subroutines are called which are responsible for the parton showers, secondary
decays, hadronisation and printing the event information on screen. A detailed descrip-
tion of the UPEVNT and how the model described in the beginning of this chapter is
implemented, is provided in appendix D.4. Note that UPEVNT generates weighted events.
However either in the main program or in the subroutines PYEVNT or HWEPRO the event is
accepted or rejected according to the weight to maximum weight ratio so as to generate
unweighted events.
In the second block, the event is analysed by the user-defined subroutines CHANAL or
HWANAL. By default they write the event information in the LHE file lhouches.xml and
the black hole decay history in the histories file histfile.xml. The subroutine CHPRINT
prints the black hole history on screen.
The arrows around the box in the diagram indicate that the loop is iterated until a
maximum number of unweighted events (set by the user) has been generated. The dotted
arrows indicate continuation to the termination block when the loop is finished.
7.5. CHARYBDIS2 and other generators 151
......?
......?
......?
PYSTAT
? ? ?
HWEFIN
CHAEND HWAEND
Figure 7.7: Diagram illustrating the termination of the run where the general purpose event
generators are terminated and the event analysis is finalised.
7.4.4 Termination
After the maximum number of events for the run (CHNMAXEV) is generated, the termi-
nation subroutines are called (see figure 7.7). For the mainpythia.f and mainherwig.f
implementations the subroutines PYSTAT and HWEFIN perform final calculations, obtain
the total cross-section and print out some other information for the run. The user-defined
subroutines CHAEND and HWAEND terminate the event analysis and finish the output files.
7.5 CHARYBDIS2 and other generators
In this chapter we have presented an improved black hole event generator to take into
account some of the latest theoretical developments, in particular the important effect
of rotation in the evaporation phase. The only other current simulation program which
models this effect is the BlackMax generator [178, 179]. Both programs take black hole
angular momentum fully into account, but they have other features and emphases that
are complementary. In the formation phase, BlackMax uses a geometrical approximation
for the cross section and parametrizes the loss of energy and angular momentum as fixed
fractions of their initial-state values, whereas CHARYBDIS2 incorporates a more detailed
model based on the Yoshino-Rychkov bounds and comparisons with other approaches.
The treatment of the evaporation phase in the two programs appears broadly similar,
but BlackMax has options for brane tension and split branes, and for extra suppression
of emissions that would spin-up the black hole, while CHARYBDIS2 includes treatment of
the polarisation of emitted fermions and vector bosons. The conservation of quantum
numbers is also treated somewhat differently. At the Planck scale, BlackMax emits a final
burst of particles with the minimal multiplicity needed to conserve quantum numbers,
whereas CHARYBDIS2 has a wider range of options.
A deficiency of both programs is the absence of gravitational radiation in the evap-
152 Chapter 7. CHARYBDIS2
oration phase. This is because the transmission factors have not yet been computed
for this case, due to extra theoretical difficulties in the separation of variables. Unlike
Standard-Model particles, gravitons will necessarily be emitted into the bulk, giving rise
to a new source of lost energy and the possibility of recoil off the brane. In the non-
rotating case, it is known [110,132] that bulk graviton emission is small for low numbers
of extra dimensions, but increases rapidly in higher dimensions due to the growing number
of polarisation states. However, the large number of Standard-Model degrees of freedom
ensures that brane emission remains dominant. Clearly a full treatment of the rotating
case is desirable, but there is hope that the effects will not be too significant, taking into
account the uncertainties in energy loss already allowed for in the formation phase.
Chapter 8
Phenomenological study
The work in this chapter was done in collaboration with James A. Frost. The plots labelled
with CHARYBDIS2.0 and the associated discussion are mostly based on his discussion in
section 5 of [4]. The complementary discussion in sections 8.2.4 and 8.2.5 was done
separately using a different parton level sample.
We present a study of the phenomenological aspects of the black hole events simulated
with CHARYBDIS2 keeping in mind the LHC experiment with proton beam energies of
7 TeV. For the experimental plots in this chapter, a range of CHARYBDIS2 samples were
produced using HERWIG 6.510 [93, 94] to do the parton showering, hadronisation and
Standard Model particle decays. The results of which were then passed through a generic
LHC detector simulation, AcerDET 1.0 [180]. CHARYBDIS2 parameter defaults are shown
in Table 7.1. In all following discussion, the number of extra spatial dimensions is n =
TOTDIM− 4. Samples were generated with a 1 TeV Planck mass (in the PDG convention,
i.e. MSSDEF = 3) so as to investigate the phenomenologically preferred region accessible at
the LHC. Black holes events were generated with a lower partonic centre of mass energy
of 5 TeV such that the semi-classical approximations for production are valid.
Our settings for AcerDET 1.0 are as follows: we select electrons and muons with
transverse momentum PT > 15 GeV and |η| < 2.5. The transverse momentum is defined
as the projection of the spatial momentum of the particle on the plane perpendicular
to the beam direction. The pseudo-rapidity is η = − log (tan θ
2
)
, where θ is the angle
of the spatial momentum with the beam axis. The electrons and muons are considered
isolated if they lie at a distance ∆R =
√
(∆η)2 + (∆φ)2 > 0.4 from other leptons or
jets and if less than 10 GeV of energy was deposited in a cone of ∆R = 0.2 around the
central cluster. The same prescription is followed for photons. Jets are reconstructed
153
154 Chapter 8. Phenomenological study
 [TeV]BHM
3 4 5 6 7 8 9 10
 
[p
b/T
eV
]
B
H
/d
M
σd
-310
-210
-110
1
10
210
310
410
CHARYBDIS 2.0
MPL=1TeV, MJLOST=FALSE
MPL=1TeV, MJLOST=TRUE
MPL=2TeV, MJLOST=TRUE
Figure 8.1: Differential cross-sections for black hole production with n = 4. The differential
cross-sections are shown for different Planck masses and MJLOST (simulation of
mass and angular momentum loss in production/balding).
from clusters using a cone algorithm of ∆R = 0.4, with a lower PT cut of 20 GeV.
Lepton momentum resolutions were parametrised from ATLAS full simulation results
published in [181].1 Where reference is made to reconstructed multiplicities or spectra,
the reconstructed objects are either electrons, muons, photons or jets from AcerDET.
8.1 Production
The black hole production process we are considering produces typically large hadronic
cross-sections for the LHC σ(14 TeV) & 100 pb. Though not affecting the total black
hole cross-section for a certain partonic centre of mass energy range, simulating the mass
and angular momentum loss during black hole formation does have a large effect on the
cross-section for a particular black hole mass range. The differential cross-section will be
reduced, for the same input state will produce a black hole of lesser mass, as is illustrated
in figure 8.1. This figure also shows how the total cross-section is a strong function of
the Planck mass, since the normalisation of the curve is reduced almost by an order of
magnitude from 1 TeV to 2 TeV.
Most collisions with sufficient energy to create a black hole are between two (valence)
quarks, however a minority occur in collisions between a quark and a gluon (see top plots
1Electrons are smeared according to a pseudo-rapidity dependent parametrisation; for muons, we take
the resolutions from |η| < 1.1.
8.1. Production 155
BHspin
0 2 4 6 8 10 12 14 16 18 20
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
n=2 qq/gg
n=2 qg
n=4 qq/gg
n=4 qg
n=6 qq/gg
n=6 qg
CHARYBDIS 2.0
BHspin
0 2 4 6 8 10 12 14 16 18 20
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
n=2 qq/gg
n=2 qg
n=4 qq/gg
n=4 qg
n=6 qq/gg
n=6 qg
CHARYBDIS 2.0
BHmass [GeV]
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Pr
ob
ab
ili
ty
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
CHARYBDIS 2.0 n=2
n=4
n=6
Figure 8.2: Effects of mass and angular momentum loss in the formation/balding phase: The
left (right) top plot shows the angular momentum distribution before (after) this
phase, whilst the bottom plot shows the distribution of black hole masses after the
losses. Note how black holes with mass below the partonic centre of mass energy
lower cut of 5 TeV are allowed due to the losses.
of figure 8.2). CHARYBDIS2 adds the spins of the colliding partons when forming a black
hole; the initial black hole angular momentum is either integer or half-integer accordingly.
An integer loss of orbital angular momentum in the formation process is simulated by the
Yoshino-Rychkov model described in section 4.3.2.
At high n the angular momentum of the colliding partonic system tends to be larger,
as seen in figure 8.2. The average spin of the produced black hole rises from 5.0 units for
n = 2, to 8.1 for n = 4 and 10.6 for n = 6. Setting MJLOST=.TRUE. decreases the angular
momentum of the produced black hole by an average of 30% for n = 2, 4, 6, whilst the
mass drops by 18% (n = 2) to 30% (n = 6). This can be seen qualitatively in figure 8.2
by comparing the top left plot (angular momentum distribution before the loss) and the
top right plot (angular momentum distribution after the loss).
156 Chapter 8. Phenomenological study
8.2 Evaporation
In this section we discuss how the angular momentum affects the phenomenology of the
final state particles expected to be observed in the black hole evaporation.
8.2.1 The effect of mass and angular momentum loss
The simulation of loss of mass and angular momentum during the production changes the
number and distributions of the final state particles in the decay. For example, the mass
of the black hole is the main factor determining the multiplicity of the final state, since it
determines the amount of energy available for the evaporation. However, the black hole
mass and angular momentum also determine the Hawking temperature and consequently
the typical energy of the particles emitted in the decay – a highly rotating, or higher
temperature black hole will emit more energetically. These effects result in a decrease of
the multiplicity of the final state as seen in the three top plots of figure 8.3. The decrease
in the number of Hawking emissions (see the parton level top left plot), follows the drop
in mass and is greatest for higher numbers of extra dimensions, with an average of two
fewer emissions (or 30%) manifest for n = 6.
The reduction in the black hole mass and number of Hawking emissions leads to a
decrease in the number of particles observed experimentally and to a reduced differenti-
ation between samples with different numbers of dimensions. The decrease of the initial
black hole mass and angular momentum also leads to a softening of the emitted particle
spectrum, with the high energy and transverse momentum tail of the distribution being
reduced (bottom plots of figure 8.3). This is due to the combined effect of having a smaller
amount of energy available, which suppresses Hawking emissions at large energy, and the
decrease in angular momentum which shifts the Hawking emission spectrum towards lower
energies.
8.2.2 The effect of black hole angular momentum
The inclusion of black hole angular momentum has several large effects upon the spectra
of the emitted particles.
8.2. Evaporation 157
No. Hawking emissions
0 2 4 6 8 10 12
Pr
ob
ab
ili
ty
0
0.05
0.1
0.15
0.2
0.25
No MJLost n=2
M&J Lost n=2
No MJLost n=4
M&J Lost n=4
No MJLost n=6
M&J Lost n=6
CHARYBDIS 2.0
Primary Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.05
0.1
0.15
0.2
0.25
no MJLost n=2
no MJLost n=4
no MJLost n=6
M&J Lost n=2
M&J Lost n=4
M&J Lost n=6
CHARYBDIS 2.0
Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1
0.12
no MJLost n=2
no MJLost n=4
no MJLost n=6
M&J Lost n=2
M&J Lost n=4
M&J Lost n=6
CHARYBDIS 2.0
| [GeV]
T
Primary Particle |P0 500 1000 1500 2000 2500
N
um
be
r /
   
50
 G
eV
 / 
Ev
en
t
-410
-310
-210
-110
no MJLost n=2
no MJLost n=4
no MJLost n=6
M&J Lost n=2
M&J Lost n=4
M&J Lost n=6
CHARYBDIS 2.0
| [GeV]
T
Particle |P0 500 1000 1500 2000 2500
N
um
be
r /
   
50
 G
eV
 / 
Ev
en
t
-210
-110
1
no MJLost n=2
no MJLost n=4
no MJLost n=6
M&J Lost n=2
M&J Lost n=4
M&J Lost n=6
Figure 8.3: Effect of simulating the mass and angular momentum lost in black hole production:
on particle multiplicity distributions, at parton level (top left), generator level (top
right) and after AcerDET detector simulation (center); and PT spectra generator
level (bottom left) and after AcerDET detector simulation (bottom right). A fixed
2-body remnant decay with the criterion M < INTMPL to stop the evaporation was
used, with all switches set to their default values.
158 Chapter 8. Phenomenological study
Primary Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.05
0.1
0.15
0.2
0.25
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1
0.12
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
| [GeV]
T
Primary Particle |P0 500 1000 1500 2000 2500
N
um
be
r /
   
50
 G
eV
 / 
Ev
en
t
-410
-310
-210
-110
1
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
| [GeV]
T
Particle |P0 500 1000 1500 2000 2500
N
um
be
r /
   
50
 G
eV
 / 
Ev
en
t
-410
-310
-210
-110
1
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
Figure 8.4: Particle multiplicity distributions and PT spectra at generator level (left) and
after AcerDET detector simulation (right) for non-rotating and rotating black hole
samples, with n = 2, 4 and 6 extra dimensions and MJLOST = .FALSE..
Energy spectra of the final state products
As discussed in section 6.4.4, the Hawking emission spectrum for a rotating black hole is
dominated by partial waves with large m > 0. The spectrum for these waves is shifted
towards higher energies, due to the −mΩH term in the exponential of equation (5.12),
which reduces the high energy Planckian suppression. Consequently, the particle energy
and transverse momentum (PT ) distributions for emissions from a rotating black hole are
harder. The number of primary emissions is correspondingly reduced. Figure 8.4 shows
the emitted particle multiplicity and PT spectra for different numbers of extra dimensions.
The effects of black hole rotation are largest for fewest number of extra dimensions, for
which the term ΩH has greater magnitude. This more than compensates for their slightly
lower Hawking temperature.
The effect of black hole rotation on the pseudorapidity distribution (figure 8.5) is
more subtle. Assuming no strong angular momentum recoil during the balding phase,
8.2. Evaporation 159
Primary Particle Pseudorapidity
-5 -4 -3 -2 -1 0 1 2 3 4 5
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1
0.12
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
Pseudorapidity
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Pr
ob
ab
ili
ty
0.02
0.04
0.06
0.08
0.1
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
Figure 8.5: Normalised particle η distributions at generator level (left) and after AcerDET
detector simulation (right) from black hole samples with n extra dimensions and
MJLOST = .FALSE..
the initial black hole formed will have an angular momentum axis perpendicular to the
beam direction. Since emission in the equatorial plane is favoured, particularly for scalars
and fermions, one would expect the component along the beam direction, and hence at
high η, to be enhanced, at least for initial emissions. This effect is seen experimentally
in figure 8.5, but is slight. Another reason why this effect is so washed out is due to the
boost between the black hole centre of mass frame and the laboratory frame where the
pseudorapidity is defined. In section 8.2.4 we discuss this further at parton level.
Similar trends can be seen in event variables such as missing transverse energy (MET)
and Σ|PT | (figure 8.6). The reduced particle multiplicity increases the probability of
minimal or no MET, where no neutrinos are present in the event (neither directly emitted
by the black hole, nor in weak decays of other primary emissions). The greater energy
of the Hawking emissions increases the very high MET tail: a neutrino emitted by a
rotating black hole is likely to have higher energy and momentum. The result is a flatter,
longer tail for the rotating case, extending further beyond 1 TeV, as shown in figure 8.6.
The broadening of the Σ|PT | curves is due to the lower multiplicity which implies bigger
fluctuations.
When compared to the number of primary emissions from the evaporation, a greater
number of detector objects (leptons, photons, hadronic jets) are observed following fast
detector simulation. Neutrinos emitted by the black hole will not be seen experimen-
tally, whereas a single heavy quark or vector boson will result in the detection of multiple
particles or jets of hadrons. Equally, the transverse momentum spectrum observed exper-
imentally will be slightly softer in general than that of the primary particles emitted by
160 Chapter 8. Phenomenological study
MET [GeV]
0 200 400 600 800 1000 1200 1400 1600 1800 2000
N
um
be
r /
   
50
 G
eV
 / 
Ev
en
t
-310
-210
-110
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
| [GeV]
T
Sum |P0 1000 2000 3000 4000 5000 6000 7000 8000
Pr
ob
ab
ili
ty
 / 
20
0 
G
eV
-410
-310
-210
-110
Non-Rot. n=2
Non-Rot. n=4
Non-Rot. n=6
Rotating n=2
Rotating n=4
Rotating n=6
CHARYBDIS 2.0
Figure 8.6: Missing transverse energy and scalar PT sum for rotating and non-rotating
black hole samples after AcerDET fast detector simulation. Samples used the
NBODYAVERAGE criterion for the remnant phase and include a simulation of the
mass and angular momentum lost during production and balding.
the black hole, due to secondary emissions, decays and radiation.
Particle emission probabilities
Black hole rotation has a large effect on the particle production probabilities (figure 8.7).
The most dramatic is the enhanced emission coefficient for vector particles. This is due
to the larger fluxes and agrees with the greater differential fluxes per degree of freedom
shown for example in figure 6.12 (see vertical axis of the power plots).
The greater proportion of vector emissions would provide strong evidence of rotating,
rather than Schwarzschild, black holes. However such measurements are difficult to make
in practice – at the LHC it will not be possible to distinguish gluon jets from quark ones.
Though highly boosted vector bosons provide experimental challenges, Z bosons can often
be studied via their leptonic decay modes. Perhaps the most accessible other means to
investigate black hole rotation might be the study of the photon multiplicity or its ratio to
other particles, TeV-energy photons being one manifestation of black holes reproduced by
neither other new physics scenarios nor SM backgrounds. Another experimental difficulty
for the detection and the isolation of the black-hole signal is that rotation decreases the
probability of producing a lepton – often useful in reducing jet-like SM backgrounds to
black hole events [182].
The emission probabilities for each particle species are largely independent of the
number of extra dimensions, which primarily affects the emission energy and multiplicity,
so that a reproduction of the distribution of particle species would be powerful evidence
8.2. Evaporation 161
PdgId Code
-40 -30 -20 -10 0 10 20 30 40
Pr
ob
ab
ili
ty
0
0.05
0.1
0.15
0.2
0.25
0.3
Non-Rot. n=4
Rotating n=4
CHARYBDIS 2.0
PdgId Code
-40 -30 -20 -10 0 10 20 30 40
Pr
ob
ab
ili
ty
 R
at
io
 (R
ot
ati
ng
/N
on
-R
ot
ati
ng
)
0
0.5
1
1.5
2
2.5
n=2
n=4
n=6
CHARYBDIS 2.0
Figure 8.7: Particle emission probabilities for rotating and non-rotating black holes with n = 4
(left) and the enhancement factor for each species and different n (right). The
horizontal axis shows the PDG code number for the SM particles, as defined in [5].
Quarks codes are 1-6 (down, up, strange, charm, bottom and top), leptons are
11-16 (electron and its neutrino; muon and its neutrino; tau and its neutrino),
gauge bosons 21-24 (gluon, photon, Z and W+) and the Higgs boson 25 (with the
HERWIG default mass of 115 GeV). Anti-particle states have a negative code.
of black holes. The particle-antiparticle imbalance in figure 8.7 is chiefly caused by the
(usually positively charged) input state. According to the model described in section 7.2,
up-type quarks and down-type antiquarks are favoured, so as to meet the constraints of
charge balance. Similarly, the net positive baryon number of the input state and the
need to conserve baryon number for hadronisation leads to a preference for quarks over
antiquarks. The apparent increase in this with rotation is a reflection of the reduced
particle multiplicity: with fewer particles amongst which to share the charge imbalance,
the effect is magnified. This is potentially a source of uncertainty since, unlike charge,
black holes do not have to conserve baryon or lepton quantum numbers. At present we are
constrained to conserve baryon number by the needs of hadronisation generators. Note
that baryon number is in any case not directly observable in experiment (though it may
have some subtle effects on the decay), whereas lepton number is. Large lepton number
violation could be another signature of a TeV gravity interaction.
Black hole evolution
As the Hawking emission proceeds, the black hole evolves, becoming lighter, hotter and
losing angular momentum as detailed in figure 8.8. The higher ΩH term in the Planckian
factor causes there to be fewer, more energetic emissions for few extra dimensions. Half
the mass is lost in the first 3 emissions for n = 2, compared with 4 (n = 4) and 5 for
162 Chapter 8. Phenomenological study
Hawking Emission Number
0 2 4 6 8 10 12 14 16 18 20
Fr
ac
tio
n 
of
 B
Hs
 R
em
ai
ni
ng
0
0.2
0.4
0.6
0.8
1
n=2
n=4
n=6
CHARYBDIS 2.0
TOTALt / t
0 0.2 0.4 0.6 0.8 1
B
la
ck
 H
ol
e 
M
as
s 
[G
eV
]
0
1000
2000
3000
4000
5000
6000
CHARYBDIS 2.0 n=2
n=4
n=6
TOTALt / t
0 0.2 0.4 0.6 0.8 1
B
la
ck
 H
ol
e 
Sp
in
0
2
4
6
8
10
12
CHARYBDIS 2.0 n=2
n=4
n=6
Figure 8.8: The evolution of black hole parameters during the Hawking evaporation phase The
top plot shows the fraction of black holes which are still evaporating at the Nth
emission. The lower two rows show the mean value tendency curves of the black
hole mass and angular momentum as a function of fractional time t/tTotal through
the evaporation. The bars indicate the standard deviation of the distribution.
n = 6. The distribution does have a substantial tail however, with 1% of black hole events
producing more than 11 primary emissions (top plot of figure 8.8).
Without the simulation of losses in production/balding (MJLOST), the black hole spins
down more quickly than it loses mass because its initial angular momentum is larger
(emissions at high m ∼ j are highly favoured). Turning on MJLOST suppresses initial
states with high black hole angular momentum. Consequently this effect is reduced in
magnitude, though the black hole angular momentum still tends to be reduced in the be-
ginnig of the evaporation (see bottom right plot of figure 8.8). Black holes with high initial
angular momentum tend to lose much of it during their first few emissions, whereafter
further emissions decrease the black hole mass more smoothly, whilst its angular momen-
tum stays relatively low, but non-zero. Thus rotation persists throughout the black hole
decay – only a small proportion of black holes settling into a Schwarzschild, non-rotating
8.2. Evaporation 163
Reco Mass - True Mass [GeV]
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1 Fit Mean = -86 GeV
Fit Sigma = 210 GeV
CHARYBDIS 2.0
Reco Mass - True Mass [GeV]
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000
Pr
ob
ab
ili
ty
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fit Mean = -50 GeV
Fit Sigma = 173 GeV
CHARYBDIS 2.0
Figure 8.9: Sample black hole mass resolutions after AcerDET detector simulation with n = 2
and no balding simulation for all events (left) and after a cut of MET< 100 GeV
(right). The fits are indicative of the resolution in the peak and do not model the
non-Gaussian tails which remain.
state. This is in direct agreement with the theoretical plots in figure 7.2 and 7.3, where a
small non-zero angular momentum persists after the majority of the angular momentum
has been lost.
As the black hole becomes lighter, its temperature rises, as does its oblateness (a∗)
and the typical time interval between emissions drops. These effects are gradual except
when the black hole mass becomes very low, at the end of the Hawking radiation phase.
At this point we have reached the remnant phase.
8.2.3 Mass reconstruction
In principle, it is possible to reconstruct the black hole mass by combining the 4-momenta
of all particles observed in the event and missing transverse energy (MET). Mass reso-
lutions of 200-300 GeV can be achieved for some samples as shown in figure 8.9, but
there is significant variation with different samples and black hole parameters. Events
with large amounts of MET (particularly from multiple sources) tend to be more poorly
reconstructed. Invoking a 100 GeV cut on MET results in better reconstruction at the
cost of some signal efficiency. Such a cut may not be entirely conservative however, for
there may be additional sources of MET neglected in our simulation, such as that from
Hawking emission of gravitons.
164 Chapter 8. Phenomenological study
8.2.4 Angular momentum reconstruction
The problem of determining the angular momentum axis and/or magnitude is much more
difficult than reconstructing the mass. This is because the angular momentum axis, which
controls the angular distributions, evolves during the evaporation, changing direction and
magnitude. Furthermore, all the decay products in the laboratory frame are in general
boosted with respect to the black hole centre of mass frame, and the black hole recoils
between each emission. Nevertheless, we may hope to see some of the effects by boosting
each event back to the partonic centre of mass frame. In principle we can determine
the latter reliably if there is little missing energy emitted in the evaporation. The only
missing energy in the SM comes from neutrinos and they represent a small part of the total
number of degrees of freedom. However missing energy from neutrinos can arise from the
secondary decay of SM heavy particles. Nevertheless the fraction of missing energy from
the evaporation should not be so large (for most of the events) as to degrade the results
much more compared to other factors such as the recoil. Note however that including
gravitons in the evaporation may degrade the reconstruction further. Alternatively to
reduce this effect we may cut on events with a small amount of missing energy as we did
for the mass reconstruction.
We start our study by considering some distributions which are not observables but
help to understand the angular correlations involved. The aim is to understand how the
momenta of the particles emitted correlate with the true angular momentum axis by using
knowledge of the black hole history.
The first distribution we study is the angle of a particle of a given spin with the true
initial angular momentum in the centre of mass frame of the initial black hole
cos θJ0 ≡
ps · J0
|ps||J0| . (8.1)
The first plot in figure 8.10 shows a spin dependent behaviour as expected. The sample
used to produced the plots used all the defaults values except for MJLOST=.FALSE.. This
produces black hole events with larger angular momentum and mass. This is not essential
since we could use a cut on the visible invariant mass for the event to select heavier black
holes if MJLOST=.TRUE.. We know from the theoretical plots in figure 6.14 that scalars and
fermions tend to be more equatorial (though fermions at low energies also have a small
axial peak) and vector bosons are very axial at low energies. This is consistent with the
larger probability for vector bosons at larger | cos θ| in the top left plot (though the effect
8.2. Evaporation 165
1
1/2
0
s
Any ωrH,0
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.12
0.1
0.08
0.06
0.04
0.02
0
1
1/2
0
s
ωrH,0 < 0.2
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
1/2
0
s
0.2 < ωrH,0 < 0.5
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.12
0.1
0.08
0.06
0.04
0.02
0
1
1/2
0
s
0.5 < ωrH,0 < 1.0
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.12
0.1
0.08
0.06
0.04
0.02
0
1
1/2
0
s
1.0 < ωrH,0 < 1.5
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
1/2
0
s
ωrH,0 > 1.5
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Figure 8.10: Distribution of angles of particles with various spins with the angular momentum
axis. The top left plot shows the distributions for cos θ for various spins for an
n = 3 sample with black hole masses above 5 TeV. The other plots show the
distribution for various ranges of energies as indicated in the title of each plot.
Note that the error bars for s = 0 are due to the small number of pure scalar
degrees of freedom in the SM (only the Higgs particle).
is not very large). We can improve this correlation by selecting particles in particular
ranges of energy for each initial black hole. For example if our cut requires high energy
particles for all spins we would expect an equatorial correlation, whereas at low energies
we expect a flatter distribution for scalars and fermions and an axial distribution for
vector particles. This is confirmed in the two top right plots and the bottom plots of
figure 8.10 where we have chosen ranges of energy in units of the horizon radius of the
initial black hole. This result suggests using soft vector particles as a guess for the axis,
to plot angular distributions. This will fail a considerable part of the time and smear out
the true correlation.
High energy particles tend to be emitted perpendicularly to the initial black hole an-
gular momentum. This is seen in the bottom right plot of figure 8.10 where the correlation
166 Chapter 8. Phenomenological study
Hardest emission reconstruction
cos θJ0
P
ro
b
a
b
il
it
y
10.80.60.40.20
0.3
0.25
0.2
0.15
0.1
0.05
0
Sphericity reconstruction
cos θJ0
P
ro
b
a
b
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it
y
10.80.60.40.20
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Figure 8.11: Axis reconstruction: Using the hardest emission (left) and the eigenvector of the
smallest eigenvalue of the sphericity tensor (right).
is stronger. If in addition we assume that the direction of the angular momentum vector
is perpendicular to the direction of the black hole momentum (which is true in the limit
where the angular momentum does not recoil during the production) we obtain another
guess for the axis. The left plot in figure 8.11 shows that this method works better.
However it relies on the assumption that the initial angular momentum is perpendicular
to the black hole momentum.
A third method to estimate the angular momentum axis is to consider the shape of
the event, i.e. to use all the momenta in the decay. For a rotating black hole we expect
most of the particles to be emitted equatorially (since only low energy vector bosons are
axial). Thus the event should have a disc like distribution of momenta indicating the
orientation of the axis. This axis should minimise the amount of momentum projected
along its direction. Another advantage of this reasoning is that it gives lower weight to
low energy particles which we want to eliminate since low energy vector bosons are more
axial, and low energy scalars and fermions are more uniform. If we denote the direction
of the angular momentum by n, and use projections of momenta squared, then we want
to minimise
minn
∑
i(pi · n)2∑
i |pi|2
=
minn
∑
i
∑
αβ n
αpαi p
β
i n
β∑
i |pi|2
= min
n
∑
αβ
nα
(∑
i p
α
i p
β
i∑
i |pi|2
)
nβ
= min
n
∑
αβ
nαSαβnβ (8.2)
8.2. Evaporation 167
where we are use Greek letters for spatial indices and the index i runs over all particles
in the event. We have defined the sphericity tensor Sαβ as usual [183]. The sphericity
tensor has the properties that all eigenvalues are non-negative and their sum is one. In
the eigenbasis it is clear that the direction which minimises the quantity in (8.2) is the
eigenvector associated with the smallest eigenvalue. In the right plot of figure 8.11 the
distribution for the angle between the guessed axis (the eigenvector associated with the
smallest eigenvalue) and the angular momentum axis is shown. This method is not as
good as the the left plot but it has the advantage of not relying on the assumption that
the angular momentum is on the plane transverse to the collision axis.
8.2.5 Parton level angular correlators
An alternative to reconstructing the black hole angular momentum to study angular
distributions, is to explore the property that polarised angular distributions are strongly
dependent on the helicity of the particle. For example from figure 6.14 we know that
there is a strong preference for vector bosons to be emitted in different hemispheres. This
motivates defining angular correlators of the form (in the frame of the initial black hole)
xi,j =
pi · pj
|pi||pj | (8.3)
which are cosines of angles between particles i and j. So for the case of particles with the
same helicity, we would expect the distribution to be higher at xi,j ∼ 1 and reduced at
xi,j ∼ −1, and the opposite to happen for particles with opposite helicities. Figure 8.12
shows the expected behaviour for pairs of emissions in a fixed black hole background (no
recoil). The probability density function used to determine the distribution of (8.3) is
derived in appendix D.5. The effect at fixed black hole parameters grows quickly with
a∗, especially for particles of helicity h = ±1. The bottom right plot shows that there is
some variation with n though not very strong (the curves are qualitatively similar).
For an evolving black hole, we expect these effects to get smeared, due to the mo-
mentum and angular momentum recoil. Again the best we can do is to compute similar
quantities in the rest frame of the initial black hole assuming a small amount of the
missing energy during the evaporation.
Figure 8.13 shows the correlators defined in (8.3) for various helicity combinations.
The top left plot shows the distribution for same helicity (red) and opposite helicity
(blue) fermions (solid lines) and vector particles (dashed lines) and the non-rotating case
168 Chapter 8. Phenomenological study
4.0
3.0
2.0
1.5
1.0
0.5
0.0
a∗
ρ
(
x 1
2
, 1
2
)
for n = 4
x 1
2
, 1
2
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
4.0
3.0
2.0
1.5
1.0
0.5
0.0
a∗
ρ (x1,1) for n = 4
x1,1
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
4.0
3.0
2.0
1.5
1.0
0.5
0.0
a∗
ρ (x0,0) for n = 4
x0,0
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
4.0
3.0
2.0
1.5
1.0
0.5
0.0
a∗
ρ
(
x 1
2
,− 1
2
)
for n = 4
x 1
2
,− 1
2
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
4.0
3.0
2.0
1.5
1.0
0.5
0.0
a∗
ρ (x1,−1) for n = 4
x1,−1
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
6
5
4
3
2
1
n
ρ (x1,1) for a∗ = 1
x1,1
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
Figure 8.12: Angular correlators for an eternal black hole. The top plots show the probability
density functions for the angular correlator between particles of same helicity for
fermions, vector bosons and scalars (left to right), for a range of a∗ values. The
two left bottom plots are similar but between particles with opposite helicities.
The bottom right plot shows the dependence with n for a∗ = 1 and two vector
bosons with the same helicity.
(black) for comparison. The black curve is not constant due to the recoil of the black
hole between emissions. The recoil tends to make different pairs of emissions more back
to back (especially subsequent emissions) which causes the rise at xi,j ∼ −1 and the fall
off at xi,j ∼ 1.
We can see (particularly for vector particles) that the asymmetry predicted in fig-
ure 8.12 persists, with an enhancement at xi,j = −1 for opposite helicity correlators,
compared to the same helicity correlators. It is also clear that the effect of the recoil
between emissions is larger in the rotating case. This is because the spectrum is harder,
hence the relative boost when the black hole recoils is also larger. Additionally, in the
8.2. Evaporation 169
a = 0 (any)
hi = −hj = ±1
hi = hj = ±1
hi = −hj = ±1/2
hi = hj = ±1/2
RECOIL=1
Angular correlators (all final state)
xi,j
P
ro
b
ab
il
it
y
10.50-0.5-1
0.05
0.04
0.03
0.02
0.01
0
5th6th
4th5th
3rd4th
2nd3rd
1st2nd RECOIL=1
h1 = h2 = +1/2 various nemission
xi,j
P
ro
b
a
b
il
it
y
10.50-0.5-1
0.09
0.06
0.03
0
3rd8th
3rd7th
3rd6th
3rd5th
3rd4th RECOIL=1
h1 = h2 = +1/2 various nemission
xi,j
P
ro
b
a
b
il
it
y
10.50-0.5-1
0.06
0.05
0.04
0.03
0.02
0.01
0
a = 0 (any)
hi = −hj = ±1
hi = hj = ±1
hi = −hj = ±1/2
hi = hj = ±1/2
RECOIL=1
Angular correlators (all ωrH0 < 0.2)
xi,j
P
ro
b
ab
il
it
y
10.50-0.5-1
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
a = 0 (any)
hi = −hj = ±1
hi = hj = ±1
hi = −hj = ±1/2
hi = hj = ±1/2
RECOIL=1
Angular correlators (all 0.2 < ωrH0 < 0.4)
xi,j
P
ro
b
ab
il
it
y
10.50-0.5-1
0.08
0.06
0.04
0.02
0
a = 0 (any)
hi = −hj = ±1
hi = hj = ±1
hi = −hj = ±1/2
hi = hj = ±1/2
RECOIL=1
Angular correlators (all ωrH0 > 1.5)
xi,j
P
ro
b
ab
il
it
y
10.50-0.5-1
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
hi = −hj = ±1
hi = hj = ±1
hi = −hj = ±1/2
hi = hj = ±1/2
RECOIL=2
Angular correlators (all final state)
xi,j
P
ro
b
ab
il
it
y
10.50-0.5-1
0.04
0.03
0.02
0.01
0
5th6th
4th5th
3rd4th
2nd3rd
1st2nd RECOIL=2
h1 = h2 = +1/2 various nemission
xi,j
P
ro
b
a
b
il
it
y
10.50-0.5-1
0.12
0.1
0.08
0.06
0.04
0.02
0
3rd8th
3rd7th
3rd6th
3rd5th
3rd4th
RECOIL=2
h1 = h2 = +1/2 various nemission
xi,j
P
ro
b
a
b
il
it
y
10.50-0.5-1
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
Figure 8.13: Parton level angular correlators evaluated in the frame of the initial black hole.
The sample used was the same as in figure 8.10. The top left plot shows correlators
between any two final state fermions or any two final state vector bosons for some
helicity combinations. The correlator for any two particles when rotation is off
is shown for comparison. The top centre and top right plots show the same
distributions using various pairs of particles according to their ordering during
the evaporation. The central row of plots shows the same distributions using
specific intervals of energy (compared to the horizon radius for the initial black
hole – rH0). Twice the bin size was used to compensate for the lower statistics.
The bottom row of plots contains the same plots as the top row, but with a
different recoil model RECOIL=2.
170 Chapter 8. Phenomenological study
rotating case, we have a forward peak at xi,j = 1. This is due to the possibility of having
an accumulated boost in a certain direction for the particles emitted later in the decay,
so pairs of those particles will tend to be more collinear. Since in the rotating case the
multiplicity of the event is reduced and the spectrum is harder, this effect tends to be
important (practically all events end up with a large accumulated boost). This is verified
in the top centre and top right plots where for the first pair in the beginning of the evap-
oration (top centre plot) no forward peak is observed, whereas for later consecutive pairs,
the forward peak tends to increase rapidly. For pairs which are increasingly separated in
the order of emission, the forward peak also disappears (top right plot). The reason why
no forward peak appears in the non-rotating case can be justified as follows. For events
with multiplicity N , the number of consecutive pairs which can contribute to a forward
peak is N − 2, whereas the total number of pairs is N(N − 1)/2. So the fraction of pairs
contributing to a forward peak is at most 2(N −2)/(N(N −1)). Typically, when rotation
is turned off, the multiplicity increases from 5 ∼ 8 to 10 ∼ 15 so the fraction of pairs
contributing to the peak is reduced roughly by a factor of ∼ 2. Furthermore, a larger
multiplicity means a smaller magnitude for the boost in the recoil by another factor of
∼ 3 (since the same energy is distributed among more particles which are softer). So
overall, we have a suppression factor of at least ∼ 6. This explains the absence of the
forward peak for the non-rotating sample.
The central row of plots shows that by selecting particular ranges of energy we recover
the strong asymmetry for low energy vector bosons (left and centre plots). For high
energy particles (right), all helicities are equivalent. This is because at high energies all
the angular spectra become equatorial regardless of the spin.
The bottom row plots are the same as in the first row, except that RECOIL=2 was
used (see section 7.2.1). Similar conclusions are obtained with this option (particularly
for the ranges of energy selected in the middle row of plots). The only difference is that
the forward peak due to the recoil is somewhat sharper and the backward effect is smaller.
It can be shown that the mass reduction for a particle with a small energy E ≪ M (i.e.
in the beginning of the evaporation) in the frame of the initial black hole is
Mfinal =


M − E
(
1 +
E
2M
)
+ . . . , RECOIL = 1
M − E , RECOIL = 2
. (8.4)
This explains the larger backward effect (at x = −1) in the top plots when RECOIL=1,
8.2. Evaporation 171
RECOIL=2
RECOIL=1
BH frame energy vs Hawking energy
ω/M
E
/M
10.80.60.40.20
0.5
0.4
0.3
0.2
0.1
0
Figure 8.14: Energy of the particle emitted: The energy of the particle (E/M) in the frame
of the black hole is shown as a function of the energy selected from the Hawking
spectrum (ω/M).
since in that case the black hole mass reduction is a bit larger in the beginning of the
evaporation, enhancing the effect of the back to back recoil. The sharper forward peak
when RECOIL=2 can be explained by expressing E in terms of the selected Hawking energy
ω (neglecting particle mass)
E
M
=


ω
M
,
ω
M
∈ [0, 1
2
] , RECOIL = 1
ω
M
(
1− 1
2
ω
M
)
,
ω
M
∈ [0, 1] , RECOIL = 2
. (8.5)
It is easy to see (figure 8.14) that at high energies E close to the kinematic limit (which are
more important in the last part of the evaporation), the range of energies ω contributing
to a range of energies E is always much wider for RECOIL=2. So there will be more
hard particles selected at the end of the evaporation for the latter, contributing to the
accumulated boost and hence the forward peak.
Note that the plots in figure 8.13 were produced assuming knowledge of the helicities
of the outgoing particles. This is usually not an observable at hadron colliders. However,
as mentioned at the end of section 6.4.4, for example the decay modes of the theW and Z
bosons are dependent on the helicity of the intermediate states. For example in the decay
W− → ℓ−ν¯ℓ the charged lepton tends to be collinear with theW− for negative helicity, and
anti-collinear for positive helicity. For Z decays a similar argument holds, although the
correlation should be weaker. The only disadvantage of W decays is the neutrino which
makes re-construction difficult, whereas for Z decays the disadvantage is the mixture of
172 Chapter 8. Phenomenological study
Table 8.1: Parameters used for remnant comparison. All samples have n = 2 and
MJLOST=.FALSE.
Legend Remnant Criterion Fixed/Variable Remnant No./Mean
Kincut on M < INTMPL (KINCUT=.TRUE.) Fixed 2
Kincut off M < INTMPL (KINCUT=.FALSE.) Fixed 2
Nbody2 Flux (NBODYAVERAGE=.TRUE.) Fixed 2
Nbody3 Flux (NBODYAVERAGE=.TRUE.) Fixed 3
Nbody4 Flux (NBODYAVERAGE=.TRUE.) Fixed 4
Nvar2 Flux (NBODYAVERAGE=.TRUE.) Variable 2
Nvar3 Flux (NBODYAVERAGE=.TRUE.) Variable 3
Nvar4 Flux (NBODYAVERAGE=.TRUE.) Variable 4
Boiling RMMINM < M < INTMPL Variable 2
left-handed and right-handed couplings. Nevertheless with a more sophisticated set of
cuts it may be possible to keep some of the asymmetry at the experimental level. This
study will be completed in a future work.
8.3 Remnants
CHARYBDIS2 includes several models for the remnant phase. Both fixed multiplicity and
variable multiplicity decays have parameter switches to enable the systematics to be
studied, as detailed in section 7.3.1.
The fixed multiplicity model, present in CHARYBDIS and optional in CHARYBDIS2, is
linked to the choice of the variable KINCUT. If KINCUT=.FALSE., proposed decays that
are kinematically disallowed are ignored; if KINCUT=.TRUE., their proposal terminates
the evaporation phase. The former choice will give a greater number of less energetic
particles, as evidenced by figure 8.15 which contrasts a range of remnant models defined
in Table 8.1.
CHARYBDIS2 uses the NBODYAVERAGE remnant criterion as a default, where the fluxes
are used to calculate the expected number of further emissions. This provides a physically
motivated model. Using this criterion with either a fixed 2-body (“Nbody2”) or a variable
multiplicity remnant model (“Nvar2”) gives a distribution lying between the upper and
lower values obtained using the older model switches (upper plots of figure 8.15), indicating
good control over the uncertainties mentioned in section 7.3.1. The string-motivated
boiling model gives a slightly higher multiplicity, since successive emissions are produced
until the remnant mass drops below the remnant minimum mass, resulting in a greater
8.3. Remnants 173
Primary Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.05
0.1
0.15
0.2
0.25
KinCut off
Nbody2
KinCut on
NVar2
Boiling
CHARYBDIS 2.0
| [GeV]
T
Primary Particle |P0 500 1000 1500 2000 2500
N
um
be
r /
   
50
 G
eV
 / 
Ev
en
t
-210
-110
KinCut off
Nbody2
KinCut on
NVar2
Boiling
CHARYBDIS 2.0
Primary Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.05
0.1
0.15
0.2
0.25
0.3
Nbody2
Nbody3
Nbody4
CHARYBDIS 2.0
Primary Particle Multiplicity
0 5 10 15 20 25
Pr
ob
ab
ili
ty
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22 NVar2
NVar3
NVar4
CHARYBDIS 2.0
Figure 8.15: Primary particle multiplicity and |PT | distributions for black hole samples with
n = 2, using a wide range of remnant options, as defined in Table 8.1.
number of softer particles produced in the remnant phase.
The new NBODYAVERAGE model is also more robust with respect to changes in the
number of particles produced in the remnant phase. This is because the flux calculation
allows the spin-down phase to be terminated whenever the expected number of further
emissions is fewer than that selected for the remnant phase. This is illustrated in the lower
plots of figure 8.15, where changing the number of particles produced in the remnant phase
results in similar multiplicities and spectra; events with 4-body remnant decays do not
always have two more particles than their 2-body analogues.
Another advantage of the NBODYAVERAGE method is that by using the integrated power
and flux, the spin-down phase is terminated at a point that allows a smoother transition
to the remnant phase, as shown by their spectra in figure 8.16, where the NBODYAVERAGE
(lower) method gives a more concordant distribution of particle transverse momenta.
Performing a remnant decay only when the mass drops below the Planck mass gives
a much softer momentum spectrum, in contrast to the high energies favoured by light
rotating black holes. The option to start the remnant decay based on the drop of 〈N〉
174 Chapter 8. Phenomenological study
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r /
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ve
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0.2
0.4
0.6
0.8
1
1.2
CHARYBDIS 2.0 Hawking Radiation
Remnant Particles
| [GeV]
T
Particle |P0 500 1000 1500 2000 2500
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um
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r/ 
50
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eV
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Ev
en
t
-210
-110
Hawking Radiation
Remnant Particles
CHARYBDIS 2.0
PdgId Code
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0.2
0.4
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1
CHARYBDIS 2.0 Hawking Radiation
Remnant Particles
| [GeV]
T
Particle |P0 500 1000 1500 2000 2500
N
um
be
r/ 
50
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eV
 / 
Ev
en
t
-210
-110
Hawking Radiation
Remnant Particles
CHARYBDIS 2.0
Figure 8.16: Particle type and PT plots for 2-body remnant decays using the old model “Kin-
cut off” (top) and the new model “Nbody2” (bottom) as defined in Table 8.1.
Distributions are normalised per event.
provides a smoother transition, since the final decay particles will have a harder spectrum,
more similar to the Hawking phase. Emissions in the remnant phase are predominantly
coloured, with positive baryon number favoured, so as to meet the constraints of baryon
number conservation.
8.4 Conclusions
In this and the previous chapter, we have presented in detail the physics content of the
new black hole event generator CHARYBDIS2, together with some results illustrating im-
portant features of the simulation of the different phases of black hole production and
decay. The main new features compared to most earlier generators, including CHARYBDIS,
are: detailed modelling of the cross section and the loss of energy and angular momentum
during formation of the black hole (the so-called balding phase), based on the best avail-
able theoretical information; full treatment of angular momentum during the evaporation
8.4. Conclusions 175
phase, including spin of the incoming partons, rotation of the black hole, and anisotropy
and polarisation of all Standard Model fields emitted on the brane; and finally a variety
of options for the Planck-scale termination phase, ranging from a stable remnant to a
variable-multiplicity model connecting smoothly with the evaporation phase.
Our main finding is that angular momentum has strong effects on the properties
of the final state particles in black hole events. Even after allowing for a substantial
loss of angular momentum in the balding phase, the isotropic evaporation of a spinless
Schwarzschild-Tangherlini black hole is not a good approximation, nor is the notion of a
rapid spin-down phase followed by mainly isotropic evaporation at foreseeable energies.
Although the Hawking temperature does not depend strongly on the angular momentum
of a spinning black hole of a given mass, there is a strong bias in the emission spectra
towards higher-energy emissions into higher partial waves, which help the black hole to
shed its angular momentum. The resultant spectra are flatter, with substantial tails
beyond 1 TeV. As a consequence of these more energetic emissions, rotating black holes
emit with reduced multiplicity relative to their non-rotating counterparts. However the
absolute multiplicity can still be large.
The preferential equatorial emission of scalar, fermionic and high energy vector par-
ticles leads to slightly less central distributions at detector level. This effect is reduced
by the evolution of the spin axis during evaporation (away from the initial orientation
perpendicular to the beam direction). We have seen that this can be improved by working
in the frame of the reconstructed black hole. The reconstruction of the black hole angular
momentum is in general difficult due to its recoil. However we have shown that angular
asymmetries survive if, instead of plotting angular distributions with respect to the re-
constructed axis of rotation, we consider angular correlators between pairs of particles.
This seems particularly promising for low energy vector particles.
The emission of polarised higher-spin fields is favoured, compared to the spinless case,
leading to increased vector emission and marking a further departure from a purely demo-
cratic distribution of particle species. This shows little dependence upon the number of
dimensions.
These findings will complicate the interpretation of black-hole events, should they oc-
cur at the LHC or future colliders. While the basic signature of energetic, democratic
emission of all Standard Model species and large missing energy remains valid, the deduc-
tion of the fundamental Planck scale and the number of extra dimensions will be more
difficult than was anticipated in earlier studies [158]. On the other hand, many interesting
176 Chapter 8. Phenomenological study
new and potentially observable features emerge, such as the different angular distribu-
tions and polarisation of particles of different spins. Further analysis strategies will be
investigated in future work.
Chapter 9
Conclusions and Outlook
In this thesis we have studied the theoretical modelling of black hole production and
evaporation in extra-dimensional theories with TeV gravity, and some phenomenological
consequences for near future collider experiments.
After motivating extra dimensions as a solution to the hierarchy problem and the
possibility of black hole production in particle collisions in those scenarios, and after
reviewing the formalism of extra-dimensional effective theory, we started by investigating
some of the properties of the extra-dimensional black holes (chapter 3). Using the Myers-
Perry metric for a singly rotating black hole, we found that it is possible to construct
a more general effective background, so as to include (approximately) the effect of the
black hole electric charge on the brane, in a consistent way. The result was an effective
brane metric which is a modified version of the Kerr-Newman metric. An important
observation, which was relevant in later chapters to study black hole discharge and to
justify the perturbative treatment, was that the electric force at distances below the
extra-dimensional radius, becomes weaker as the gravitational force becomes dominant.
To help in characterising the black hole geometrically, we described the method to compute
classical absorption cross-sections for an arbitrary direction of incidence, and presented
some examples of the shapes of the absorptive disks seen by an observer at infinity (in the
rotating case). As expected we found a strong correlation with the angular momentum axis
and magnitude as we varied the direction of incidence and number of extra dimensions.
In chapter 4, we considered the problem of modelling the production. This was largely
a review of the best known arguments and bounds, on the amount of energy and angular
momentum trapped in the black hole, and a description of a model to include those bounds
in the CHARYBDIS2 generator.
177
178 Chapter 9. Conclusions and Outlook
A theoretical study of the Hawking evaporation (which is central to modelling the
decay of the black holes which may be produced at colliders) was detailed in chapters 5
and 6. After justifying a perturbative approach and summarising the basic theory of
Hawking radiation, we used the brane charged effective metric to study the new cases
of massive and charged fermionic and scalar perturbations in detail. We then found and
separated the wave equations to obtain the general radial and angular equations for those
perturbations. In chapter 6 we developed analytic and numerical methods which allowed
us to solve the radial equations to obtain transmission factors, the angular functions, and
hence the Hawking spectra. A detailed numerical evaluation showed several interesting
new features, most notably: i) the large suppression of the spectrum for massive particles
at threshold, ii) the sub-dominance of discharge in the evaporation (unlike common claims
regarding Schwinger discharge) and iii) the inverted low energy charge splitting of the
spectrum for more than one extra dimension. We also performed a comparison with
known results concerning the effect of rotation, emphasizing the importance of including
the non-trivial spheroidal functions. The latter are helicity dependent and introduce axial
peaks in the angular spectrum. Finally we showed that the charge splitting effect survives
when considered simultaneously with rotation.
The theoretical study in the first part of the thesis showed that there are still several
interesting and relevant effects to consider in the modelling of the evaporation of brane
black holes. A very interesting problem to pursue in the future is the construction of an
effective background for a black hole spacetime with non-abelian charges, and the study
of various non-singlet Standard Model perturbations.
In the second part of the thesis we have implemented several theoretical results into
the new event generator CHARYBDIS2, and analysed the phenomenological consequences.
Concerning the model for mass and angular momentum loss at production, the main
conclusion was that there is a considerable reduction of the differential cross-section for
a given black hole mass range. We have also implemented a more complete model for
the evaporation including the effect of rotation for all Standard Model fields with polar-
isation information. This has several important effects, namely: the shift of the energy
spectra of the final state particles towards higher energies and consequently a reduction
of the average multiplicity; non-uniform angular distributions, which may be observable
by plotting observables with respect to the reconstructed rotation axis or by analysing
angular correlators in the reconstructed black hole frame; and finally a large enhancement
of vector emission. We found that all these effects do not change dramatically the classi-
179
cal signatures of black hole events such as high multiplicity events with many QCD jets
and leptons (the average multiplicity is reduced but it is still large), relatively democratic
emission of all Standard Model degrees of freedom, and large amounts of transverse mo-
mentum. Instead the new effects add new signatures to the scenario. In the future, it
would be interesting to include the charge and mass effects we have discussed in the first
part of the thesis, which are likely to add up new signatures relatively smoothly on top
of the signatures of rotation. Another important open question to address is the effect of
the gravitons on a rotating background.
Extra-dimensional black holes are definitely very interesting and theoretically rich ob-
jects. They involve a combination of general relativity, quantum mechanics and particle
physics, so their modelling is complicated and there are still many open problems and
interesting avenues to pursue. With the recent start of the LHC experiment it is par-
ticularly timely to provide an increasingly better modelling to either exclude or observe
them. The study in this thesis will hopefully help with performing this task and motivate
further work in this unusual field.
180 Chapter 9. Conclusions and Outlook
Appendix A
Conventions and mathematical tools
Throughout this thesis, unless stated otherwise, all theoretical expressions are in natural
units where the Planck constant and speed of light are respectively ~ = c = 1.
A.1 Differential geometry
We use the mostly minus convention for the signature of the metric in D dimensions
(+,−, . . . ,−).
For the components of the (torsion free) metric connection components in a coordinate
basis we use (GMN is the metric)
ΓMNP =
1
2
GMQ (∂NGPQ + ∂PGNQ − ∂QGNP ) . (A.1)
To avoid confusion with the higher-dimensional metric tensor or other quantities,
we express geometric objects such as the Riemann tensor, Ricci tensor and scalar, and
Einstein tensor by calligraphic letters RMNPQ, RMN , R and GMN respectively. The sign
convention for the Riemann tensor is
RMNPQ = ∂QΓMNP − ∂PΓMNQ + ΓMRQΓRNP − ΓMRPΓRNQ (A.2)
and the Einstein tensor is defined
GMN = RMN − 1
2
GMNR . (A.3)
181
182 Chapter A. Conventions and mathematical tools
A.2 Planck mass convention
We keep the dependence on the (4+n)-dimensional Planck mass explicit in all expressions
and adopt as reference convention, the PDG definition1 which uses the Einstein-Hilbert
action
SEH =
1
2
Mˆ2+nD
∫
dDx
√
|G|RD = 1
16πGD
∫
d(4+n)x
√
|G|RD , (A.4)
to set the reduced Planck mass MˆD. The Planck mass MD is then defined as
M2+nD = (2π)
nMˆ2+nD . (A.5)
An alternative convention is obtained by defining
M2+n4+n = 2M
2+n
D . (A.6)
This is the Giddings-Thomas convention [24] used internally in the CHARYBDIS2 generator
described in chapter 7.
A.3 The induced vierbein
Following the appendix of [29], we can generalise the arguments for a curved background
as follows. The matrix R(Y ) is defined as
R = exp
(
iθαµJ
(αµ)
)
(A.7)
with θαµ a function of Y and J
(αµ) are part of the generators of the D-dimensional Lorentz
group in the vector representation
J (AB)
C
D = iη
CE
(
δAEδ
B
D − δADδBE
)
. (A.8)
The particular components involved in (A.7) are
J (αµ)
ν
β = −iηµνδαβ
J (αµ)
β
ν = iη
αβδµν (A.9)
J (αµ)
ν
σ = J
(αµ)γ
δ = 0 .
1See for example the extra dimensions section of the PDG review [5].
A.3. The induced vierbein 183
To solve (2.7) expand
EBM (Y )∂aY
M = E(0)
B
a (Y(0))− ǫBa (Y ) (A.10)
where ǫBa is a perturbative parameter which is zero on the background and we choose
Y M(0) = δ
M
a x
a. We expand at order O(ǫi)
θαµ =
∑
i=0
θ(i)αµ
R =
∑
i=0
R(i) . (A.11)
The zeroth order solution must obey
R(0)
µ
BE
(0)B
a (x) = 0 . (A.12)
For the particular case where E(0)
ν
a(Y0) = 0 the solution is the identity R
(0) = I and
the induced vierbein is simply e(0)
α
a = E
(0)α
a (x). The perturbative corrections can be
constructed iteratively using the same arguments as in [29]. Using (A.9) in the (i+ 1)th
order expansion of (A.7), we can solve for
θ(i+1)αµ = ηµνe
(0)a
α
[
R(i)
ν
Bǫ
B
a − e(0)
β
a R
(i+1)ν
β
∣∣∣
θ(i+1)=0
]
, (A.13)
where we have defined the inverse of the zeroth order induced vierbein
e(0)
a
αe
(0)α
b = δ
a
b (A.14)
e(0)
α
b e
(0)b
β = δ
α
β (A.15)
and the second term in R(i+1) is evaluated with the θ(i+1) term deleted.
184 Chapter A. Conventions and mathematical tools
Appendix B
The production model
B.1 The Yoshino-Rychkov mass/angular momentum
bounds
This appendix is divided into two sections. The first is a brief review of the method in
the Yoshino-Rychkov paper [76], focusing on the key equations we used to implement
the Yoshino-Rychkov boundary curves in CHARYBDIS2. The second section explains how
CHARYBDIS2 calculates the boundary curve ξb(ζ) for a given D and b.
Summary of the Yoshino-Rychkov method
In the spacetime outside the future lightcone of the collision event (i.e. regions I, II
and III of figure B.1), the metric for the complete system is obtained by combining two
higher dimensional Aichelburg-Sexl metrics (equation (3.21)) corresponding to partons
travelling in opposite directions (in the centre of mass frame). This gives the correct
spacetime outside region IV of figure B.1, because the colliding partons are taken as
travelling at the speed of light, so there can be no interaction between their gravity waves
before the collision.
The next stage in the calculation is the selection of a spacetime slice somewhere in
the union of regions I, II and III, and the determination of an apparent horizon (AH). An
AH is a surface whose outgoing null geodesic congruence has zero expansion. Assuming
the cosmic censorship hypothesis [184], an event horizon (EH) must be present outside
any AH; thus finding an AH is sufficient to show that a black hole forms. Furthermore,
using the fact that the EH must lie outside the AH, combined with the area theorem [185]
185
186 Chapter B. The production model
I
-
6
I
II III
IV
Left parton and
accompanying shock
Right parton and
accompanying shock
z
t
Collision
event
*
Figure B.1: Spacetime regions in a parton-parton collision. In this diagram, the z axis is
defined to lie along the direction of motion of the left parton, and D− 2 spacelike
dimensions are suppressed.
(which states that the EH area never decreases) is used to set bounds on the mass and
angular momentum of the formed black hole.
The slice used by Yoshino and Rychkov is the future-most slice outside region IV -
i.e. its boundary. This slice gives the most restrictive, and therefore best, bounds on the
maximum impact parameter for black hole formation bmax. It also gives the best bounds
on the mass and angular momentum trapped in the black hole following production.
To obtain the maximum impact parameter for black hole formation, bY Rmax, for a given
D, the impact parameter b is increased at the given D until it is no longer possible to find
an AH. Note that this method gives a lower bound, since there may be impact parameters
greater than bY Rmax for which an AH forms to the future of the slice considered.
Next we discuss the calculation of the mass and angular momentum bounds for a
fixed b and D. In a trapped surface method, this is achieved by calculating the D − 2
dimensional area corresponding to the AH, AAH . Since the true black hole EH is outside
the AH, and the black hole EH area can never decrease according to the area theorem, it
is usually the case that the D − 2 dimensional area of the final produced black hole EH,
AEH , is greater than AAH :
AAH ≤ AEH . (B.1)
Now, we expect the horizon area of a black hole to be linked to its mass, so we can
express the above formula in terms of the mass. The AH mass is defined as the mass of
B.1. The Yoshino-Rychkov mass/angular momentum bounds 187
a Schwarzschild black hole with area AAH :
MAH =
(D − 2)ΩD−2
16πGD
(
AAH
ΩD−2
)(D−3)/(D−2)
. (B.2)
where the (D − 2)-area of a unit sphere is
Ωp =
2π
p+1
2
Γ [(p+ 1)/2]
. (B.3)
Then (B.1) implies
MAH ≤Mirr , (B.4)
where Mirr is the irreducible mass of the produced Myers-Perry black hole – this is the
mass of a Schwarzschild black hole having the same horizon area as the Myers-Perry black
hole. Since it is defined in terms of the area of the Myers-Perry black hole it is a function
of bothM and J . Equation (B.4) then represents the trapped surface bound on the mass
and angular momentum trapped in the black hole during production, with the equation
of the boundary being characterised by
MAH =Mirr . (B.5)
To convert (B.5) into a boundary line in the (M,J) plane (which can be scaled to a
boundary line in the (ξ, ζ) plane, using notation from section 4.3.2) we first need an
equation for the irreducible mass of a Myers-Perry black hole with mass M and angular
momentum J . This may be extracted from the definition of Mirr:
AMyers−Perry(M,J) = ASchwarzschild(Mirr) = ΩD−2r
D−2
S (Mirr) . (B.6)
Computing the left hand side of (B.6) using the Myers-Perry metric [50], we find the link
between M , J and Mirr of a Myers-Perry black hole:
rD−2S (Mirr) = r
D−3
S (M)rH(M,J) , (B.7)
where rH(M,J) is the Myers-Perry horizon radius, given by
r2H(M,J) +
[
(D − 2)J
2M
]2
= rD−3S (M)r
5−D
H (M,J) , (B.8)
188 Chapter B. The production model
whilst rS(M) is the horizon radius of a Schwarzschild black hole of mass M , given by
rS(M) =
[
16πGDM
(D − 2)ΩD−2
]1/(D−3)
. (B.9)
We now combine equations (B.5) and (B.7) to show that a point on the bound with mass
M has horizon radius rHb(M) given by
rHb(M) =
rD−2S (MAH)
rD−3S (M)
. (B.10)
Using this and (B.9) in (B.8), we obtain that a point on the bound with mass M has
angular momentum Jb where
Jb(M) =
2MAH
(D − 2)
[
16πGDMAH
(D − 2)ΩD−2
]1/(D−3)√
(M/MAH)D−2 − 1 , (B.11)
which gives an explicit equation for the trapped surface boundary line in the (M,J) plane.
Solving for M , the boundary line becomes
Mb(J) = MAH
{
1 +
[
(D − 2)J
2MAH
]2 [
(D − 2)ΩD−2
16πGDMAH
]2/(D−3)}1/(D−2)
. (B.12)
To produce valid bounds using (B.1) it is necessary that an arbitrary surface outside the
AH has a larger area. Unfortunately, for the Yoshino-Rychkov slice, this does not hold.
However, Yoshino and Rychkov found a different area, Alb, which they demonstrated is a
true lower bound on AEH . Alb is equal to twice the area of the intersection of the AH with
the transverse collision plane using a flat metric. The calculation of the (M,J) bounds
then proceeds as described above, with Mlb and Alb replacing MAH and AAH in equations
(B.1) - (B.12).
Calculation of the mass/angular momentum boundary
After replacing all MAH symbols in the equation by Mlb we rewrite equation (B.12) in
terms of the fractions of initial state mass and angular momentum ξ and ζ
ξb(ζ) = ξlb
{
1 +
[
(D − 2)bζ
4ξlb
]2 [
(D − 2)ΩD−2
32πGDµξlb
]2/(D−3)}1/(D−2)
, (B.13)
B.2. Details of the ‘constant angular velocity’ bias 189
where µ =
√
s/2 is the energy of each colliding parton in the centre of mass frame. This
equation can be used to calculate the Yoshino-Rychkov bound ξb(ζ) for a given b and D,
provided one is able to obtain ξlb for the b and D values used.
The problem of implementing the Yoshino-Rychkov bound in CHARYBDIS2 is then one
of ensuring that the program has a means of obtaining ξlb for all values of D and b
(5 ≤ D ≤ 11, 0 ≤ b ≤ bY Rmax(D)). In the program, we use the data files ξlb vs. b for
D = 5 to D = 11 that were generated by Yoshino and Rychkov (which they used to
produce the D = 5 to D = 11 plots in figure 10 of [76]). The value of ξlb for a fixed b and
D is obtained by linear interpolation between points in these data files, which provides
sufficient accuracy due to the close spacing in b.
B.2 Details of the ‘constant angular velocity’ bias
This section describes the implementation of the constant angular velocity bias in the
simulation of the black hole production phase. With the bias off (CVBIAS=.FALSE.), the
values of (ξ,ζ)1 are simply those generated from the linear ramp distributions described
in section 4.3.2. When the bias is turned on, the (ξ, ζ) point to be passed to the Yoshino-
Rychkov boundary routine is obtained in a more complex fashion, which is outlined below.
First, a point is generated using the linear ramp distributions as before. The horizon
angular velocity ΩH and the a∗ = a/rH value corresponding to the point, ΩH(ξ, ζ) and
a∗(ξ, ζ), are calculated using the standard equations (5.18) and (3.13), and compared to
those of the initial state, ΩH(1, 1) and a∗(1, 1). In particular, the quantities |ΩH(ξ, ζ)−
ΩH(1, 1)|/ΩH(1, 1) and | log[a∗(ξ, ζ)/a∗(1, 1)]| are calculated, and compared to the values
of some constants ∆ and Λ respectively (whose values will be discussed shortly). Note that
|ΩH(ξ, ζ)−ΩH(1, 1)|/ΩH(1, 1) and | log[a∗(ξ, ζ)/a∗(1, 1)]| are essentially both measures of
the differences between the values at the point and the initial state values.
The point is then assigned a number α(ξ, ζ) between 0 and 1, whose value largely
depends on whether both |ΩH(ξ, ζ)−ΩH(1, 1)|/ΩH(1, 1) ≤ ∆ and | log[a∗(ξ, ζ)/a∗(1, 1)]| ≤
Λ or not (i.e. whether ΩH(ξ, ζ) and a(ξ, ζ) are sufficiently close to Ω(1, 1) and a(1, 1) or
not). If one or both of the conditions are not satisfied, then the point is assigned a constant
k < 1 (as defined below). If both conditions are satisfied, the point is assigned the value
of a function χ(ξ, ζ). The function χ(ξ, ζ) has the key properties k < χ(ξ, ζ) ≤ 1, and
1ξ, ζ are the trapped mass and angular momentum fractions passed to the routine which imposes the
Yoshino-Rychkov boundary condition.
190 Chapter B. The production model
approaches 1 as ΩH(ξ, ζ) gets closer to ΩH(1, 1). The details of our choices for k and
χ(ξ, ζ) will be discussed shortly.
A random number β is generated according to a uniform distribution between 0 and 1.
If α(ξ, ζ) > β then the point is accepted, otherwise it is rejected. If the point is rejected,
further (ξ, ζ) points have to be generated by the ramp distributions, and put through
the above procedure, until a point is accepted. The final point is passed to the Yoshino-
Rychkov boundary routine.
It is reasonably clear that this procedure for generating a (ξ, ζ) point (to be passed
to the Yoshino-Rychkov boundary routine) is equivalent to a procedure which generates
a point from a biased probability distribution of the form asserted in section 4.3.2. To
be specific, the biased probability distribution resembles the basic ramp probability dis-
tribution, but all points whose ΩH and a∗ values are sufficiently close to those of the
initial state have had their probabilities enhanced. The enhancement is greater the closer
ΩH(ξ, ζ) is to ΩH(1, 1).
We now discuss our choices for the function and the parameters used in the above
procedure. A suitable choice for the function χ(ξ, ζ), which has the properties stated, is
based on the Breit-Wigner form (note that we introduce a further ’width’ parameter Γ):
χ(ξ, ζ) =
Γ2/4
([ΩH(ξ, ζ)− ΩH(1, 1)]/ΩH(1, 1))2 + Γ2/4 . (B.14)
The constant k may then be fixed by imposing continuity on α(ξ, ζ), such that the biased
probability distribution represented by the above procedure does not possess any sudden
jumps. Note that we hope the dividing curves between the enhanced region and the unen-
hanced regions to be |ΩH(ξ, ζ)−ΩH(1, 1)|/ΩH(1, 1) = ∆ on either side of the ‘connected’
curve ΩH(ξ, ζ) = ΩH(1, 1) (which is connected to the point ξ = 1, ζ = 1). As explained
in the main text, the a∗ condition is only present to remove probability enhancement
around the other ‘disconnected’ ΩH(ξ, ζ) = ΩH(1, 1) curve, and should not interfere with
the angular velocity based enhancement around the right curve.
On the basis of these assumptions, continuity of α(ξ, ζ) is assured by taking
k = χ(ξ, ζ)||ΩH(ξ,ζ)−ΩH(1,1)|/ΩH(1,1)=∆ =
Γ2/4
∆2 + Γ2/4
. (B.15)
Note that with this choice of χ(ξ, ζ) and k, Γ is a constant which sets the ‘width’ of the
probability peak around the connected ΩH(ξ, ζ) = ΩH(1, 1) curve, and the ‘height’ of this
peak. The smaller Γ is, the sharper and stronger the probability enhancement around the
B.2. Details of the ‘constant angular velocity’ bias 191
appropriate curve. The default value of Γ is 0.4 for a probability enhancement which is
not too strong - however, the value of this variable could potentially be changed.
The values of ∆ and Λ were chosen by looking at a large number of individual (b,D)
cases used by CHARYBDIS2, and trying to find a suitable combination of values that gave
enhancement of a suitable region only around the connected curve. This procedure re-
sulted in the choice ∆ = 0.2 and Λ = 0.4 (these values should not be changed, as small
changes can cause drastic changes in the regions where the probability is enhanced).
A final comment is appropriate explaining the slightly peculiar form of the a∗ condi-
tion for probability enhancement – | log[a∗(ξ, ζ)/a∗(1, 1)]| ≤ Λ. The reason for this form is
because, when trying to find a condition that discriminated points around the connected
curve from those around the disconnected curve, we noticed that those around the dis-
connected curve, had a∗ values that were one order of magnitude (or more) away from the
a∗ values of the points around the connected curve. To remove the enhancement around
the disconnected curve, a condition based on the logarithm of the ratio a∗(ξ, ζ)/a∗(1, 1)
is then more appropriate.
192 Chapter B. The production model
Appendix C
Expansion coefficients and matrices
for radial equations
C.1 Expansion coefficients
C.1.1 Scalars
The expansion coefficients we need are defined by
∆ = x
+∞∑
m=0
δmx
m
K2 −∆U =
+∞∑
m=0
σmx
m
γ¯m =
m−1∑
k=0
(k + α)αkδm−k
γm = (m+ α)αmδ0 + γ¯m .
(C.1)
It can be shown then that
δ0 = n+ 1 + (n− 1)
(
a2 +Q2
)
(C.2)
δ1 = 1− n(n− 1) (1 + a
2 +Q2)
2
(C.3)
δ2 =
n(n2 − 1) (1 + a2 +Q2)
6
(C.4)
δm+1 = −(1 + ρm+1)δm , m ≥ 3 (C.5)
193
194 Chapter C. Expansion coefficients and matrices for radial equations
where
ρ2 =
n− 2
3
(C.6)
ρm+1 =
(
1− 1
m+ 2
)
ρm (C.7)
and
σ0 = K
2
⋆ (C.8)
σ1 = 2K⋆(2ω − qQ)− U0δ0 (C.9)
σ2 = 2K⋆ω + (2ω − qQ)2 − U0δ1 − U1δ0 (C.10)
σ3 = 2ω(2ω − qQ)− U0δ2 − U1δ1 − U2δ0 (C.11)
σ4 = ω
2 − U0δ3 − U1δ2 − U2δ1 (C.12)
σm = −U0δm−1 − U1δm−2 − U2δm−3 , m ≥ 5 (C.13)
where
U0 = Λ + ω
2a2 − 2aωm+ µ2 (C.14)
U1 = 2µ
2 (C.15)
U2 = µ
2 (C.16)
C.1.2 Fermions
Similarly to the scalar case define
2∆M 1
2
(r) =
+∞∑
m=0
Nm
(√
x
)m
∆
1
2 =
√
x
+∞∑
m=0
δ¯mx
m
b2m =
m−1∑
j=0
2δm−j(j + α)a2j
b2m+1 =
m−1∑
j=0
δm−j(2j + 2α + 1)a2j+1 ,
(C.17)
C.2. Matrices 195
The matrices we need are
N0 = 2iK⋆σˆ3 (C.18)
N1 = 2λδ¯0σˆ1 − 2µδ¯0σˆ2 (C.19)
N2 = 2i(2ω − qQ)σˆ3 (C.20)
N3 = 2λδ¯1σˆ1 − 2µ
(
δ¯1 + δ¯0
)
σˆ2 (C.21)
N4 = 2iωσˆ3 (C.22)
N2m = 0 , m > 2 (C.23)
N2m+1 = 2λδ¯mσˆ1 − 2µ(δ¯m + δ¯m−1)σˆ2 , m ≥ 1 (C.24)
where δ¯i are obtained from the following expansion
∆
1
2 = =
√
xδ
1
2
0
(
1 +
+∞∑
m=1
δm
δ0
xm
) 1
2
(C.25)
by fixing a certain order of truncation and expanding the square root in powers of x up
to the given order.
C.2 Matrices
In the main text we have used the following matrices:
R0 =

 e
iyyiϕ 0
0 e−iyy−iϕ

 (C.26)
R 1
2
=

 e
iyyiϕ 0
0 e−iyy−iϕ

 1k(ω + k)

 ω + k −µ
−µ ω + k

 (C.27)
As =

 iBs (Xs + iYs) e
−iΦ
(Xs − iYs) eiΦ −iBs

 (C.28)
196 Chapter C. Expansion coefficients and matrices for radial equations
with
Φ = 2
(
y + ϕ log y −
j∑
m=1
cm
mym
)
(C.29)
Bs =


V
2k2
− 1
2
− ϕ
y
−
j∑
m=2
cm
ym
, s = 0
ω
k2
K
∆
− µ
k2
µr
∆
1
2
− 1− ϕ
y
−
j∑
m=2
cm
ym
, s = 1/2
. (C.30)
cm are coefficients such that the corresponding powers in the asymptotic expansion of (C.30)
are cancelled;
Xs =


1
∆
(
y +
(n− 1) (1 + a2 +Q2) kn+1
2yn
)
, s = 0
λ
∆
1
2
, s = 1/2
(C.31)
Ys =


V
2k2
− 1
2
, s = 0
1
∆
(
µωy
k2
(
y −∆ 12
)
+ ωµa2 − aµm− qQy
k
)
, s = 1/2
(C.32)
and now
∆ = y2 + k2
(
a2 +Q2
)− (1 + a2 +Q2) kn+1
yn−1
. (C.33)
Appendix D
Details of the CHARYBDIS2
implementation
D.1 How to set up CHARYBDIS2
The current release of CHARYBDIS2 [3] contains the following code files (the symbol *
denotes the version number):
• maincharybdis.f, mainherwig.f and mainpythia.f – These are the three possible
main programs for stand-alone parton level CHARYBDIS2, interface with HERWIG and
interface with PYTHIA respectively as described in section 7.4.
• charybdis2*.F – Contains the main event generation code.
• charybdis2*.inc – Contains the declaration of global common blocks.
• charybdis2.init – List of all input variables which can be changed by the user
according to the format specified in the header instructions.
• Makefile – This contains different flags to select different main program options
(CHARYBDIS2, HERWIG or PYTHIA) and libraries, and the compilation instructions to
build the executable file.
In addition, the data files with the cumulative functions constructed from the Hawking
fluxes and the model for mass and angular momentum loss have to be placed in a sub-
directory called data files. Further instruction can be found in the Makefile, the main
programs, the charybdis2.init file, the README file and the project webpage [3].
197
198 Chapter D. Details of the CHARYBDIS2 implementation
D.2 Remnant decay generation
D.2.1 Momenta selection
Equation (7.15) assumes that we have a massive object (in this case a black hole), which
decays into N objects at once and where the phase space distribution for each of the
emissions is independent of the others. To generate the phase space, we will use a decay
chain structure as in figure D.1
qN = P
z
pN
:
qN−1 j
pN−1
:
qN−2
^
(. . .)
W
q2
z
p2

p1
Figure D.1: N-body decay as a chain
The 2-body case
For the special case of a 2-body decay, the phase space is
dP ({E1,Ω1} , {E2,Ω2}) ∝ ρ (E1,Ω1) ρ (E2,Ω2) δ(4) (P − p1 − p2) dE1dΩ1dE2dΩ2 (D.1)
the delta function produces the following constraints

 E{1,2} =
M
2
[
1 +
(m{1,2}
M
)2
−
(m{2,1}
M
)2]
Ω2 = −Ω1
(D.2)
so we can eliminate 4 parameters by integrating dE1dE2dΩ2. The integration over E1 is
straightforward and we eliminate the δ(M − E1 − E2) factor. The remaining integration
is done by changing from variables dE2dΩ2 to d
3p2. After calculating the appropriate
D.2. Remnant decay generation 199
Jacobian factor we get
dP (Ω1) ∝ ρ1 (E1,Ω1) ρ2 (E2,Ω2)
E2 |p2| dΩ1 ∝ ρ1 (E1,Ω1) ρ2 (E2,Ω2) dΩ1 (D.3)
where E1, E2,Ω2 are determined by the expressions above.
Therefore, to generate a 2-body decay we only need to generate the orientation of the
momentum vector in the rest frame of the initial object. The energies and momentum
magnitudes are fixed by the kinematically allowed values (D.2). Since the weight is
independent of φ1 it is generated uniformly and accepted with probability 1. To generate
cos θ1 we note that for all ρi
ρ(E,Ω) = n(E)A(E, cos θ) . (D.4)
where
n(E) =
T
(D)
k (ErH , a∗)
exp(E˜/TH)± 1
(D.5)
is just a constant and
A = |Sk(cos θ)|2 (D.6)
is the angular probability function. Thus we can construct the following cumulative
function
c(cos θ1) =
∫ cos θ1
−1
A1(E1, y)A2(E2, y)dy . (D.7)
Then the phase space probability density function becomes (up to a constant prefactor)
dP (Ω1) ∝ dc dφ1 . (D.8)
So to generate cos θ1 we generate c uniformly, and obtain cos θ1 by inverting the function
c(cos θ1). For this special case the procedure has efficiency 1.
The N-body case
When we have more particles, the same initial manipulations apply in the rest frame of
q2 ≡ P −
N∑
i=3
pi . (D.9)
200 Chapter D. Details of the CHARYBDIS2 implementation
The only change is that the initial probability distribution was written in the rest frame
of P , so we need to perform an initial boost. The Lorentz transformation gives
Ei =
Eq2E
q2
i − ~q2.~piq2
Mq2
(D.10)
where Mq2 is the invariant mass of the 4-momentum q2. The phase space factors become
dEidΩi =
|~pq2|
|~p| dE
q2
i dΩ
q2
i . (D.11)
Thus the integration of dE1dE2dΩ2 is exactly as before. Furthermore it is useful to perform
Lorentz transformations of the remaining phase space variables to the rest frames of the
following momentum transfers
qj ≡ P −
N∑
i=j+1
pj . (D.12)
It is useful to define in addition
µj ≡
j∑
i=1
mi . (D.13)
The phase space manipulations are the following
dP ∝ ρ1 (E1,Ω1) ρ2 (E2,Ω2) . . . δ(4) (q2 − p1 − p2) dE1dΩ1dE2dΩ2 . . . dENdΩN
⇔ dP ∝ ρ1ρ2 . . . δ(4) (q2 − p1 − p2) |~p1
q2|
|~p1|
|~p2q2|
|~p2| dE
q2
1 dΩ
q2
1 dE
q2
2 . . .
| ~pNqN |
| ~pN | dE
qN
N dΩ
qN
N
⇔
∫
dP ∝ ρ1ρ2 . . . ρN |~p1
q2|
|~p1||~p2|Eq22
dΩq21
|~p3q3|
|~p3| dE
q3
3 dΩ
q3
3 . . . 1.dE
qN
N dΩ
qN
N (D.14)
where in the second line we boost each momentum pi to the frame of qi and in the third
line we integrate over dE1dE2dΩ2 and use the fact that qN = P . Note again that


Eq2{1,2} =
Mq2
2
[
1 +
(
m{1,2}
Mq2
)2
−
(
m{2,1}
Mq2
)2]
Ωq22 = −Ωq21
. (D.15)
Following [186], we can perform a change of variable and an ordering of the Mj ’s to
generate them uniformly in the allowed region
Mi ≡ µi + ri(M − µN) (D.16)
D.2. Remnant decay generation 201
with
0 ≤ r2 ≤ r3 . . . rN−1 ≤ 1 . (D.17)
Then it is possible to show (neglecting the masses of the outgoing particles pi and overall
constant factors) that the probability distribution becomes
∫
dP ∝ ρ1ρ2 . . . ρN |~p1
q2 |
|~p1||~p2|Eq22
dΩq21
|~p3q3|
|~p3| dr2dΩ
q3
3 . . .
|~piqi|
|~pi| dri−1dΩ
qi
i . . . 1.drN−1dΩ
qN
N .
(D.18)
Note that now we have a set of weights for each phase space point, so the efficiency
is reduced since we have to perform an acception/rejection procedure according to the
weight.
D.2.2 Angular momenta selection
Selecting the angular momentum quantum numbers with the choice of phase space (7.15)
would require an integration over the momenta, to obtain a probability distribution for
the {ji, mi}. This is in general complicated. Furthermore, for consistency, we would have
to impose angular momentum conservation in a way that the partial waves add up to
the angular momentum of the remnant before decaying. Since the model for the remnant
is only supposed to be a rough description of the decay (possibly constrained by some
physically reasonable assumptions) we adopt a method where we simply impose angular
momentum conservation combined with a product of independent probabilities for each
partial wave
P ({j1, m1} , . . . , {jN , mN}) ∼
N∏
i=1
Pi (ji, mi) δJrem,
P
jmj
, (D.19)
where we took the axis of the angular momentum of the remnant as quantisation axis, i.e.
Mrem = Jrem. To simplify the method we do not generate exactly this distribution, but
apply the following algorithm: First we select the partial wave numbers for the first N−1
waves sequentially by using, for each wave, the cumulative functions (7.13). In the process
we start by combining the angular momentum of the remnant with the negative of the first
wave selected (according to the usual rules). Then the resulting angular momentum is
combined with the next partial wave selected, similarly. This is repeated until the N−1th
partial wave. In this sequence, the range of allowed values for the next ji is kept. For the
last Nth partial wave, mN is chosen so as to add up the total M = Mrem −
∑
imi = 0.
202 Chapter D. Details of the CHARYBDIS2 implementation
UPINIT
? -
Weight Search
?
N = NCHSRCH times CHEVNT(.FALSE.)
?
R1(sˆ) =
sˆ
α
2 − MINMSSα
MAXMSSα − MINMSSα ∈ [0, 1]
?
R2(x1) =
log(x−11 )
log
(
s
sˆ
) ∈ [0, 1]
?
Evaluate x1fi(x1) and x2fj(x2)
?
Wn =
∑
ij
F (sˆ)x1fi(x1)x2fj(x2)σˆ(sˆ)ff
6
6
6
Save σ(s) ≈ 〈W 〉 = 1
N
N∑
n=1
Wn
?
Save ∆σ(s) ≈
√
〈W 2〉 − 〈W 〉2
N
?
Save maximum Wmax
?
Write input information on screen
Figure D.2: Diagram illustrating the implementation of the UPINIT subroutine.
Finally, the last jN is selected among the allowed values using the relative probabilities
for waves with m = mN (this is obtained from (7.13)).
D.3 The UPINIT subroutine
Figure D.2 contains a diagrammatic representation of the UPINIT subroutine in CHARYBDIS2.
This is largely the same as the implementation in CHARYBDIS, with the only difference
being the Yoschino-Rychkov factors which enhances the cross-section. This subroutine
consists of a weight search which calls the CHEVNT subroutine with the option not to gen-
erate events (i.e. .FALSE.). The scan of weights is done NCHSRCH times (the default value
is 105). The aim is to compute the total cross section (7.3). To make the integrand flatter,
in the early version of CHARYBDIS the change to the variables {R1, R2} (as defined in the
figure) was performed. The Jacobian times the phase space volume factor that arises by
applying the transformation to (7.3) and applying the MC approximation (7.1) is
F (sˆ) =
2(MAXMSSα − MINMSSα)
αsˆ
α
2
log
(s
sˆ
)
(D.20)
where
α =
2
D − 3 − 7 . (D.21)
D.4. The UPEVNT subroutine 203
CHEVNT(.TRUE.)
?
Production
Same initial block for calculating the weight and the PDFs as in fig. D.2
?
Store weight and select parton-parton combination using PDFs
?
CHYRMJLS Obtain b, and MBH , JBH after losses
?
CHINTRINSICSPIN Combine orbital angular momentum with parton spins........?
Figure D.3: Diagram illustrating the production in CHEVNT.
Then in each call of CHEVNT(.FALSE.), R1 and R2 are randomly generated uniformly
in their allowed ranges, the PDFs are evaluated, and finally the weight is computed
by multiplying the partonic cross section by the PDF factors and the F factor for the
kinematic variables corresponding to {R1, R2}. In this loop the maximum weight Wmax
is determined.
Finally, the sum of the weights and the sum of the squares of the weights is used to
compute the total cross section and its error ∆σ. The full set of input parameters for the
run (table 7.1) are also printed on screen.
D.4 The UPEVNT subroutine
The UPEVNT subroutine simply calls the CHEVNT with argument .TRUE. to generate weighted
events. The CHEVNT subroutine can be split into three stages which are responsible for
black hole production, evaporation and remnant final decay, respectively.
Figure D.3 shows the production schematically. The first part consists of one step in
the weight search of figure D.2. This provides the weight for the event selected. The exact
combination of incoming partons is then chosen by generating the following cumulant
uniformly
C(i, j) =
∑i
i′=1
∑j
j′=1 fi′(x1)fj′(x2)∑13
i′=1
∑13
j′=1 fi′(x1)fj′(x2)
∈ [0, 1] , (D.22)
where i = 1, . . . , 13 runs over all quarks anti-quarks and the gluon. The weight for the
event is then given by the sum over possible parton-parton combinations. The incoming
204 Chapter D. Details of the CHARYBDIS2 implementation
...............?
Evaporation – Loop until close to Planck scale
CHSELECT
?
Sel. particle type using relative cumulants Ch,a∗,D(+∞)
CHEMIT
?
-
(or CHNOGRYBODY)
CHFINDK
?
Sel. K using Ch,a∗,D(K)
CHFINDE Sel. ω using Ch,K,a∗,D(ωrH)
ff
6
Recoil + CHEMOM -
?
CHSPHRDLINI
?
Initialise spheroidal series
CHSPHRDLMATCH
?
Match two spheroidal series
CHSPHRDLCUMULATIVE
?
Cumulative spheroidal func.
CHSPHRDLXRAND
?
Generate X = cos θ
CHRAZM Generate φ uniformlyff
6
CHNEWJ+ New J-axis
?
Add particle to event record with polarisation and continue evaporation -
6
6
6
ffffff
.......?
.......?
.......?
Figure D.4: Diagram illustrating the evaporation in CHEVNT
parton information is stored in the Les Houches common block event record.
Next, the subroutine CHYRMJLS is called to select the impact parameter for the collision
and obtain the mass and angular momentum of the black hole produced. Then the orbital
angular momentum is combined with the intrinsic spin of the colliding partons by calling
CHINTRINSICSPIN. The missing energy associated with gravitational radiation is placed
in a collective graviton in the event record.
In figure D.4 the evaporation is described as a loop over the main routines and pieces
of code in CHEVNT. For each call of the loop, a Hawking emission is produced.
The first subroutine CHSELECT selects the particle type by generating the following
cumulant uniformly
C(i) =
∑i
j=1 gjCh,a∗,D(+∞)∑7
j=1 gjCh,a∗,D(+∞)
∈ [0, 1] (D.23)
where j runs over the following collective sets of particles (organised by spins): leptons,
D.4. The UPEVNT subroutine 205
..........?
Remnant
CHVARNBODY Select N (optional)
?
..........?
or
Stable Remnant
?
CHSELECT Select N remnant particles
?
CHJSELECTNBODY Select Js
?
-
or pure phase space
?
CHPSELECTNBODY Select momenta
?

Save remnant particles information in the event record
? ?
Sort out colour connections for the full event (including evaporation)
Figure D.5: Diagram illustrating the remnant models in CHEVNT.
neutrinos, quarks, gluon, photon,W , Z and Higgs (note that Ch,a∗,D(+∞) is proportional
to the amount of particles of spin h that can be emitted – see equation (7.13)). The
longitudinal modes of the W and Z are actually treated as scalars. The gj count the
number of degrees of freedom for each particle from the possible helicities and gauge
charges (the latter are generated with uniform probability).
After the particle type has been selected we need to find its momentum vector. The
energy is obtained with the subroutine CHEMIT. This starts with the selection of a partial
wave number K by calling CHFINDK, which uses the cumulant in equation (7.13). The
energy is obtained similarly (after K is fixed) by calling CHFINDE. Next a recoil model
according to section 7.2.1 is chosen and the subroutine CHEMOM is called which generates
the orientation of the momentum vector. The latter consists of a call of CHSPHRDLINI
and CHSPHRDLMATCH to initialise the series for the spheroidal wave functions and match
the two expansions (see section 6.1). CHSPHRDLCUMULATIVE generates the cumulant
c(cos θ) =
∫ cos θ
−1
|Sk(x)|2 dx ∈ [0, 1] (D.24)
uniformly, and cos θ is obtained by inverting c(cos θ). The azimuthal angle is generated
uniformly by calling CHRAZM. The final step checks if the emission is kinematically allowed,
stores the emission in the event record and the black hole history, and checks if the
evaporation loop should continue.
Figure D.5 shows the final step of the event. The remnant models are those in sec-
206 Chapter D. Details of the CHARYBDIS2 implementation
tion 7.3 so there is either a decay to SM particles, or the final black hole remnant is made
stable. The N -body decay starts with CHVARNBODY (if a variable multiplicity model is
used), which selects the multiplicity of the decay according to equation (7.18) and the
text below it. Then the N particle types are chosen using the subroutine CHSELECT at
fixed black hole parameters. The momenta of the particles can be chosen using the phase
space described in section D.2 by calling CHJSELECTNBODY and CHPSELECTNBODY, or a pure
phase space which was kept from CHARYBDIS. Finally, the remnant decay information is
stored in the event record and the colour connections for the full event are determined.
D.5 Eternal black hole angular correlators
We define the probability density function for the correlator xi,j (up to a normalisation
constant)
ρ(xi,j) ∝
∫
dxidxjdφidφjρi(xi)ρi(xj)δ
(
xi,j −
√
1− x2i
√
1− x2j cos(φi − φj)− xixj
)
(D.25)
where we have defined the spatial momenta of particle i (or j)
pi =
(√
1− x2i cosφi,
√
1− x2i sinφi, xi
)
|p| (D.26)
and
ρi(xi) =
+∞∑
K=0
∫ +∞
0
dω
T
(D)
k (x, a∗)
exp(ω˜rH/τH)± 1 |Sk(aω, xi)|
2 (D.27)
is the probability density of having a particle of type i (with any energy) emitted with
direction xi with respect to the angular momentum axis. Due to the azimuthal symmetry,
the distribution is uniform in the φi direction. This can be written in a more convenient
form by using the definitions in equations (7.13)
ρi(xi) =
+∞∑
K=0
∫ +∞
0
dω
dCh,K,a∗,D(ωrH)
dω
|Sk(aω, xi)|2 . (D.28)
D.5. Eternal black hole angular correlators 207
Then making the change of variable y = fK(ω) ≡ Ch,K,a∗,D(ωrH)/Ch,K,a∗,D(+∞)
ρi(xi) =
+∞∑
K=0
∫ 1
0
dy Ch,K,a∗,D(+∞)|Sk(af (−1)K (y), xi)|2
=
∫ 1
0
dy
+∞∑
K=0
Ch,K,a∗,D(+∞)|Sk(af (−1)K (y), xi)|2 (D.29)
where f (−1) is the inverse function (not 1/f). Now if we define
Ci(xi) ≡
∫ xi
−1
dx ρi(x)
=
∫ 1
0
dy
+∞∑
K=0
Ch,K,a∗,D(+∞)
∫ xi
−1
dx|Sk(af (−1)K (y), x)|2 (D.30)
⇒ ρi(xi) = dCi(xi)
dxi
. (D.31)
Going back to equation (D.25) we perform the changes of variables
Φ = φi + φj (D.32)
φ = φi − φj (D.33)
wi = gi(xi) = Ci(xi)/Ci(1) (D.34)
to obtain
ρ(xi,j) ∝
∫ 1
0
∫ 1
0
∫ 4π
0
∫ 2π
−2π
dwi dwj dΦ dφ δ
(
xi,j −
√
1− x2i
√
1− x2j cosφ− xixj
)
⇒ ρ(xi,j) = 1
2π
∫ 1
0
∫ 1
0
∫ 2π
0
dwi dwj dφ δ
(
xi,j −
√
1− x2i
√
1− x2j cosφ− xixj
)
(D.35)
where xi = g
(−1)(wi) and we have normalised the distribution. The histogram for ρ(xi,j)
is obtained by generating the phase space wi, wj, φ uniformly and adding a unit weight
to the bin for the corresponding
xi,j =
√
1− x2i
√
1− x2j cosφ+ xixj . (D.36)
208 Chapter D. Details of the CHARYBDIS2 implementation
Bibliography
[1] M. O. P. Sampaio, Charge and mass effects on the evaporation of higher-
dimensional rotating black holes, JHEP 10 (2009) 008, [arXiv:0907.5107].
[2] M. O. P. Sampaio, Distributions of charged massive scalars and fermions from
evaporating higher-dimensional black holes, JHEP 02 (2010) 042,
[arXiv:0911.0688].
[3] J. A. Frost, J. R. Gaunt, M. O. P. Sampaio, and B. R. Webber, CHARYBDIS2
project, April, 2009. http://projects.hepforge.org/charybdis2/.
[4] J. A. Frost, J. R. Gaunt, M. O. P. Sampaio, M. Casals, S. R. Dolan, M. A. Parker,
and B. R. Webber, Phenomenology of production and decay of spinning extra-
dimensional black holes at hadron colliders, JHEP 10 (2009) 014,
[arXiv:0904.0979].
[5] Particle Data Group Collaboration, C. Amsler et al., Review of particle physics,
Phys. Lett. B667 (2008) 1.
[6] M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory.
Reading, USA: Addison-Wesley, 1995. 842 p.
[7] C. M. Will, The confrontation between general relativity and experiment, Living
Reviews in Relativity 9 (2006), no. 3.
[8] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D.
Hoyle, and H. E. Swanson, Tests of the gravitational inverse-square law below the
dark-energy length scale, Physical Review Letters 98 (2007), no. 2 021101,
[hep-ph/0611184].
209
210 BIBLIOGRAPHY
[9] L.-C. Tu, S.-G. Guan, J. Luo, C.-G. Shao, and L.-X. Liu, Null test of Newtonian
inverse-square law at submillimeter range with a dual-modulation torsion
pendulum, Physical Review Letters 98 (2007), no. 20 201101.
[10] M. B. Green, J. H. Schwarz, and E. Witten, Superstring theory. Vol. 1:
Introduction. Cambridge, UK: Cambridge University Press, 1987. 469 pages
(Cambridge monographs on mathematical physics).
[11] C. Kiefer, Quantum gravity, Int. Ser. Monogr. Phys. 124 (2004) 1–308.
[12] C. Rovelli, Loop quantum gravity, Living Rev. Rel. 11 (2008) 5.
[13] J. F. Donoghue, Introduction to the effective field theory description of gravity,
gr-qc/9512024.
[14] L. Susskind, Dynamics of spontaneous symmetry breaking in the Weinberg-Salam
theory, Phys. Rev. D20 (1979) 2619–2625.
[15] A. Djouadi, The anatomy of electro-weak symmetry breaking. I: The Higgs boson
in the standard model, Phys. Rept. 457 (2008) 1–216, [hep-ph/0503172].
[16] S. P. Martin, A Supersymmetry primer, hep-ph/9709356.
[17] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Electroweak symmetry breaking
from dimensional deconstruction, Phys. Lett. B513 (2001) 232–240,
[hep-ph/0105239].
[18] I. Antoniadis, A possible new dimension at a few TeV, Phys. Lett. B246 (1990)
377–384.
[19] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, The hierarchy problem and
new dimensions at a millimeter, Phys. Lett. B429 (1998) 263–272,
[hep-ph/9803315].
[20] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, New dimensions
at a millimeter to a Fermi and superstrings at a TeV, Phys. Lett. B436 (1998)
257–263, [hep-ph/9804398].
[21] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phenomenology, astrophysics
and cosmology of theories with sub-millimeter dimensions and TeV scale quantum
gravity, Phys. Rev. D59 (1999) 086004, [hep-ph/9807344].
BIBLIOGRAPHY 211
[22] L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension,
Phys. Rev. Lett. 83 (1999) 3370–3373, [hep-ph/9905221].
[23] L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett.
83 (1999) 4690–4693, [hep-th/9906064].
[24] S. B. Giddings and S. D. Thomas, High energy colliders as black hole factories:
The end of short distance physics, Phys. Rev. D65 (2002) 056010,
[hep-ph/0106219].
[25] G. F. Giudice, R. Rattazzi, and J. D. Wells, Transplanckian collisions at the LHC
and beyond, Nucl. Phys. B630 (2002) 293–325, [hep-ph/0112161].
[26] G. F. Giudice, R. Rattazzi, and J. D. Wells, Quantum gravity and extra dimensions
at high-energy colliders, Nucl. Phys. B544 (1999) 3–38, [hep-ph/9811291].
[27] E. A. Mirabelli, M. Perelstein, and M. E. Peskin, Collider signatures of new large
space dimensions, Phys. Rev. Lett. 82 (1999) 2236–2239, [hep-ph/9811337].
[28] T. Appelquist, A. Chodos, and P. G. O. Freund, Introduction to ‘Modern
Kaluza-Klein theories.’. Redwood City, Calif ; Wokingham : Addison-Wesley Pub.
Co, 1987.
[29] R. Sundrum, Effective field theory for a three-brane universe, Phys. Rev. D59
(1999) 085009, [hep-ph/9805471].
[30] T. Appelquist, H.-C. Cheng, and B. A. Dobrescu, Bounds on universal extra
dimensions, Phys. Rev. D64 (2001) 035002, [hep-ph/0012100].
[31] T. Flacke, D. Hooper, and J. March-Russell, Improved bounds on universal extra
dimensions and consequences for LKP dark matter, Phys. Rev. D73 (2006)
095002, [hep-ph/0509352].
[32] U. Haisch and A. Weiler, Bound on minimal universal extra dimensions from
B¯ → Xsγ, Phys. Rev. D76 (2007) 034014, [hep-ph/0703064].
[33] N. Arkani-Hamed and M. Schmaltz, Hierarchies without symmetries from extra
dimensions, Phys. Rev. D61 (2000) 033005, [hep-ph/9903417].
212 BIBLIOGRAPHY
[34] N. Arkani-Hamed, Y. Grossman, and M. Schmaltz, Split fermions in extra
dimensions and exponentially small cross-sections at future colliders, Phys. Rev.
D61 (2000) 115004, [hep-ph/9909411].
[35] P. Callin and F. Ravndal, Higher order corrections to the Newtonian potential in
the Randall-Sundrum model, Phys. Rev. D70 (2004) 104009, [hep-ph/0403302].
[36] T. Han, J. D. Lykken, and R.-J. Zhang, On Kaluza-Klein states from large extra
dimensions, Phys. Rev. D59 (1999) 105006, [hep-ph/9811350].
[37] CDF Collaboration, T. Aaltonen et al., Search for large extra dimensions in final
states containing one photon or jet and large missing transverse energy produced
in pp¯ collisions at
√
s = 1.96-TeV, Phys. Rev. Lett. 101 (2008) 181602,
[arXiv:0807.3132].
[38] LEP Exotica Working Group Collaboration, W. Adam et al., Combination of
LEP results on direct searches for large extra dimensions, .
http://lepexotica.web.cern.ch/LEPEXOTICA/notes/2004-03/ed note final.ps.gz.
[39] D0 Collaboration, V. M. Abazov et al., Search for large extra spatial dimensions in
dimuon production at D0, Phys. Rev. Lett. 95 (2005) 161602, [hep-ex/0506063].
[40] D0 Collaboration, V. M. Abazov et al., Search for large extra spatial dimensions
in the dielectron and diphoton channels in pp¯ collisions at
√
s = 1.96-TeV, Phys.
Rev. Lett. 102 (2009) 051601, [arXiv:0809.2813].
[41] D0 Collaboration, V. M. Abazov et al., Search for Randall-Sundrum gravitons in
dilepton and diphoton final states, Phys. Rev. Lett. 95 (2005) 091801,
[hep-ex/0505018].
[42] S. Hannestad and G. G. Raffelt, Supernova and neutron-star limits on large extra
dimensions reexamined, Phys. Rev. D67 (2003) 125008, [hep-ph/0304029].
[43] S. Hannestad, Strong constraint on large extra dimensions from cosmology, Phys.
Rev. D64 (2001) 023515, [hep-ph/0102290].
[44] K. S. Thorne, Magic without magic: John Archibald Wheeler. San Francisco, USA:
Freeman, 1972. edited by J. Klauder.
BIBLIOGRAPHY 213
[45] P. C. Argyres, S. Dimopoulos, and J. March-Russell, Black holes and
sub-millimeter dimensions, Phys. Lett. B441 (1998) 96–104, [hep-th/9808138].
[46] S. Dimopoulos and G. L. Landsberg, Black holes at the LHC, Phys. Rev. Lett. 87
(2001) 161602, [hep-ph/0106295].
[47] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time.
Cambridge, UK: Cambridge University Press, 1973.
[48] V. P. Frolov and I. D. Novikov, Black hole physics: Basic concepts and new
developments, . Dordrecht, Netherlands: Kluwer Academic (1998) 770 p.
[49] S. W. Hawking and W. Israel, General relativity. An Einstein century survey.
Cambridge, UK: Cambridge University Press, 1979. 919 p.
[50] R. C. Myers and M. J. Perry, Black holes in higher dimensional space-times, Ann.
Phys. 172 (1986) 304.
[51] N. Kaloper and D. Kiley, Exact black holes and gravitational shockwaves on
codimension-2 branes, JHEP 03 (2006) 077, [hep-th/0601110].
[52] D. Kiley, Rotating black holes on codimension-2 branes, Phys. Rev. D76 (2007)
126002, [arXiv:0708.1016].
[53] D.-C. Dai, N. Kaloper, G. D. Starkman, and D. Stojkovic, Evaporation of a black
hole off of a tense brane, Phys. Rev. D75 (2007) 024043, [hep-th/0611184].
[54] T. Kobayashi, M. Nozawa, and Y.-i. Takamizu, Bulk scalar emission from a
rotating black hole pierced by a tense brane, Phys. Rev. D77 (2008) 044022,
[arXiv:0711.1395].
[55] S. Chen, B. Wang, R.-K. Su, and W. Y. P. Hwang, Greybody factors for rotating
black holes on codimension-2 branes, JHEP 03 (2008) 019, [arXiv:0711.3599].
[56] S. Hannestad and G. G. Raffelt, Stringent neutron-star limits on large extra
dimensions, Phys. Rev. Lett. 88 (2002) 071301, [hep-ph/0110067].
[57] V. H. Satheeshkumar and P. K. Suresh, Bounds on large extra dimensions from
photon fusion process in SN1987A, JCAP 0806 (2008) 011, [arXiv:0805.3429].
214 BIBLIOGRAPHY
[58] P. C. Aichelburg and R. U. Sexl, On the gravitational field of a massless particle,
Gen. Rel. Grav. 2 (1971) 303–312.
[59] D. M. Eardley and S. B. Giddings, Classical black hole production in high-energy
collisions, Phys. Rev. D66 (2002) 044011, [gr-qc/0201034].
[60] A. N. Aliev and A. E. Gumrukcuoglu, Charged rotating black holes on a 3-brane,
Phys. Rev. D71 (2005) 104027, [hep-th/0502223].
[61] A. Chamblin, H. S. Reall, H.-a. Shinkai, and T. Shiromizu, Charged brane-world
black holes, Phys. Rev. D63 (2001) 064015, [hep-th/0008177].
[62] S. Creek, O. Efthimiou, P. Kanti, and K. Tamvakis, Greybody factors for brane
scalar fields in a rotating black-hole background, Phys. Rev. D75 (2007) 084043,
[hep-th/0701288].
[63] S. B. Giddings and V. S. Rychkov, Black holes from colliding wavepackets, Phys.
Rev. D70 (2004) 104026, [hep-th/0409131].
[64] S. W. Hawking, Information loss in black holes, Phys. Rev. D72 (2005) 084013,
[hep-th/0507171].
[65] R. Emparan and H. S. Reall, A rotating black ring in five dimensions, Phys. Rev.
Lett. 88 (2002) 101101, [hep-th/0110260].
[66] D. Ida and K.-i. Nakao, Isoperimetric inequality for higher-dimensional black
holes, Phys. Rev. D66 (2002) 064026, [gr-qc/0204082].
[67] M. Shibata and H. Yoshino, Nonaxisymmetric instability of rapidly rotating black
hole in five dimensions, Phys. Rev. D81 (2010) 021501, [arXiv:0912.3606].
[68] O. J. C. Dias, P. Figueras, R. Monteiro, H. S. Reall, and J. E. Santos, An
instability of higher-dimensional rotating black holes, arXiv:1001.4527.
[69] T. Banks and W. Fischler, A model for high energy scattering in quantum gravity,
hep-th/9906038.
[70] S. N. Solodukhin, Classical and quantum cross-section for black hole production in
particle collisions, Phys. Lett. B533 (2002) 153–161, [hep-ph/0201248].
BIBLIOGRAPHY 215
[71] S. D. H. Hsu, Quantum production of black holes, Phys. Lett. B555 (2003) 92–98,
[hep-ph/0203154].
[72] A. Jevicki and J. Thaler, Dynamics of black hole formation in an exactly solvable
model, Phys. Rev. D66 (2002) 024041, [hep-th/0203172].
[73] E. Kohlprath and G. Veneziano, Black holes from high-energy beam-beam
collisions, JHEP 06 (2002) 057, [gr-qc/0203093].
[74] H. Yoshino and Y. Nambu, High-energy head-on collisions of particles and hoop
conjecture, Phys. Rev. D66 (2002) 065004, [gr-qc/0204060].
[75] H. Yoshino and Y. Nambu, Black hole formation in the grazing collision of
high-energy particles, Phys. Rev. D67 (2003) 024009, [gr-qc/0209003].
[76] H. Yoshino and V. S. Rychkov, Improved analysis of black hole formation in
high-energy particle collisions, Phys. Rev. D71 (2005) 104028, [hep-th/0503171].
[77] D. M. Gingrich, Effect of charged partons on black hole production at the large
hadron collider, JHEP 02 (2007) 098, [hep-ph/0612105].
[78] H. Yoshino, A. Zelnikov, and V. P. Frolov, Apparent horizon formation in the
head-on collision of gyratons, Phys. Rev. D75 (2007) 124005, [gr-qc/0703127].
[79] H. Yoshino and T. Shiromizu, Collision of high-energy closed strings: Formation of
a ring-like apparent horizon, Phys. Rev. D76 (2007) 084021, [arXiv:0707.0076].
[80] E. Berti, M. Cavaglia, and L. Gualtieri, Gravitational energy loss in high energy
particle collisions: Ultrarelativistic plunge into a multidimensional black hole,
Phys. Rev. D69 (2004) 124011, [hep-th/0309203].
[81] V. Cardoso, E. Berti, and M. Cavaglia, What we (don’t) know about black hole
formation in high-energy collisions, Class. Quant. Grav. 22 (2005) L61–R84,
[hep-ph/0505125].
[82] P. D. D’Eath and P. N. Payne, Gravitational radiation in high speed black hole
collisions. 1. Perturbation treatment of the axisymmetric speed of light collision,
Phys. Rev. D46 (1992) 658–674.
216 BIBLIOGRAPHY
[83] P. D. D’Eath and P. N. Payne, Gravitational radiation in high speed black hole
collisions. 2. Reduction to two independent variables and calculation of the second
order news function, Phys. Rev. D46 (1992) 675–693.
[84] P. D. D’Eath and P. N. Payne, Gravitational radiation in high speed black hole
collisions. 3. Results and conclusions, Phys. Rev. D46 (1992) 694–701.
[85] C. M. Harris, P. Richardson, and B. R. Webber, CHARYBDIS: A black hole event
generator, JHEP 08 (2003) 033, [hep-ph/0307305].
[86] U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, and J. A. Gonzalez, The
high-energy collision of two black holes, Phys. Rev. Lett. 101 (2008) 161101,
[arXiv:0806.1738].
[87] M. Shibata, H. Okawa, and T. Yamamoto, High-velocity collision of two black
holes, Phys. Rev. D78 (2008) 101501, [arXiv:0810.4735].
[88] U. Sperhake et al., Cross section, final spin and zoom-whirl behavior in high-
energy black hole collisions, Phys. Rev. Lett. 103 (2009) 131102,
[arXiv:0907.1252].
[89] M. W. Choptuik and F. Pretorius, Ultra relativistic particle collisions,
arXiv:0908.1780.
[90] M. Zilhao et al., Numerical relativity for D dimensional axially symmetric
space-times: formalism and code tests, arXiv:1001.2302.
[91] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975)
199–220.
[92] C. Alig, M. Drees, and K.-y. Oda, QCD effects in the decays of TeV black holes,
JHEP 12 (2006) 049, [hep-ph/0610269].
[93] G. Corcella et al., HERWIG 6.5: an event generator for hadron emission reactions
with interfering gluons (including supersymmetric processes), JHEP 01 (2001) 010,
[hep-ph/0011363].
[94] G. Corcella et al., HERWIG 6.5 release note, hep-ph/0210213.
BIBLIOGRAPHY 217
[95] T. Sjostrand, S. Mrenna, and P. Z. Skands, PYTHIA 6.4 Physics and manual,
JHEP 05 (2006) 026, [hep-ph/0603175].
[96] W. G. Unruh, Second quantization in the Kerr metric, Phys. Rev. D10 (1974)
3194–3205.
[97] G. W. Gibbons, Vacuum polarization and the spontaneous loss of charge by black
holes, Commun. Math. Phys. 44 (1975) 245–264.
[98] P. Candelas, P. Chrzanowski, and K. W. Howard, Quantization of electromagnetic
and gravitational perturbations of a Kerr black hole, Phys. Rev. D24 (1981)
297–304.
[99] A. C. Ottewill and E. Winstanley, The renormalized stress tensor in Kerr
space-time: General results, Phys. Rev. D62 (2000) 084018, [gr-qc/0004022].
[100] M. Casals and A. C. Ottewill, Canonical quantization of the electromagnetic field
on the Kerr background, Phys. Rev. D71 (2005) 124016, [gr-qc/0501005].
[101] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Rotating black holes: Locally
nonrotating frames, energy extraction, and scalar synchrotron radiation,
Astrophys. J. 178 (1972) 347.
[102] J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82
(1951) 664–679.
[103] D. Ida, K.-y. Oda, and S. C. Park, Rotating black holes at future colliders:
Greybody factors for brane fields, Phys. Rev. D67 (2003) 064025,
[hep-th/0212108].
[104] C. M. Harris and P. Kanti, Hawking radiation from a (4+n)-dimensional black
hole: Exact results for the Schwarzschild phase, JHEP 10 (2003) 014,
[hep-ph/0309054].
[105] C. M. Harris and P. Kanti, Hawking radiation from a (4+n)-dimensional rotating
black hole, Phys. Lett. B633 (2006) 106–110, [hep-th/0503010].
[106] D. Ida, K.-y. Oda, and S. C. Park, Rotating black holes at future colliders. II:
Anisotropic scalar field emission, Phys. Rev. D71 (2005) 124039,
[hep-th/0503052].
218 BIBLIOGRAPHY
[107] G. Duffy, C. Harris, P. Kanti, and E. Winstanley, Brane decay of a
(4+n)-dimensional rotating black hole: Spin-0 particles, JHEP 09 (2005) 049,
[hep-th/0507274].
[108] M. Casals, P. Kanti, and E. Winstanley, Brane decay of a (4+n)-dimensional
rotating black hole. II: Spin-1 particles, JHEP 02 (2006) 051, [hep-th/0511163].
[109] V. Cardoso, M. Cavaglia, and L. Gualtieri, Black hole particle emission in
higher-dimensional spacetimes, Phys. Rev. Lett. 96 (2006) 071301,
[hep-th/0512002].
[110] V. Cardoso, M. Cavaglia, and L. Gualtieri, Hawking emission of gravitons in higher
dimensions: Non-rotating black holes, JHEP 02 (2006) 021, [hep-th/0512116].
[111] D. Ida, K.-y. Oda, and S. C. Park, Rotating black holes at future colliders. III:
Determination of black hole evolution, Phys. Rev. D73 (2006) 124022,
[hep-th/0602188].
[112] M. Casals, S. R. Dolan, P. Kanti, and E. Winstanley, Brane decay of a
(4+n)-dimensional rotating black hole. III: Spin-1/2 particles, JHEP 03 (2007)
019, [hep-th/0608193].
[113] M. Casals, S. R. Dolan, P. Kanti, and E. Winstanley, Bulk emission of scalars by a
rotating black hole, JHEP 06 (2008) 071, [arXiv:0801.4910].
[114] D. N. Page, Particle emission rates from a black hole. 3. charged leptons from a
nonrotating hole, Phys. Rev. D16 (1977) 2402–2411.
[115] D. N. Page, Dirac equation around a charged, rotating black hole, Phys. Rev. D14
(1976) 1509–1510.
[116] T. Nakamura and H. Sato, Absorption of massive scalar field by a charged black
hole, Phys. Lett. B61 (1976) 371–374.
[117] E. Berti, V. Cardoso, and M. Casals, Eigenvalues and eigenfunctions of
spin-weighted spheroidal harmonics in four and higher dimensions, Phys. Rev.
D73 (2006) 024013, [gr-qc/0511111].
BIBLIOGRAPHY 219
[118] S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations
for gravitational electromagnetic and neutrino field perturbations, Astrophys. J.
185 (1973) 635–647.
[119] W. H. Press and S. A. Teukolsky, Perturbations of a rotating black hole. II.
Dynamical stability of the Kerr metric, Astrophys. J. 185 (1973) 649–674.
[120] A. A. Starobinskii, Amplification of waves during reflection from a rotating black
hole, Zh. Eksp. Theor. Fiz. 64 (1973) 48.
[121] A. A. Starobinskii and S. M. Churilov, Amplification of electromagnetic and
gravitational waves scattered by a rotating black hole, Zh. Eksp. Theor. Fiz. 65
(1973) 3.
[122] S. Chandrasekhar, The mathematical theory of black holes. Oxford, UK:
Clarendon, 1992. 646 p.
[123] S. Chandrasekhar, The solution of Dirac’s equation in Kerr geometry, Proc. Roy.
Soc. Lond. A349 (1976) 571–575.
[124] J. Doukas, H. T. Cho, A. S. Cornell, and W. Naylor, Graviton emission from
simply rotating Kerr-de Sitter black holes: Transverse traceless tensor graviton
modes, Phys. Rev. D80 (2009) 045021, [arXiv:0906.1515].
[125] P. Kanti, H. Kodama, R. A. Konoplya, N. Pappas, and A. Zhidenko, Graviton
emission in the bulk by a simply rotating black hole, Phys. Rev. D80 (2009)
084016, [arXiv:0906.3845].
[126] A. L. Dudley and J. D. Finley, Separation of wave equations for perturbations of
general type-D space-times, Phys. Rev. Lett. 38 (1977) 1505–1508.
[127] K. D. Kokkotas, Quasinormal modes of the Kerr-Newman black hole, Nuovo Cim.
B108 (1993) 991–998.
[128] E. Berti and K. D. Kokkotas, Quasinormal modes of Kerr-Newman black holes:
Coupling of electromagnetic and gravitational perturbations, Phys. Rev. D71
(2005) 124008, [gr-qc/0502065].
[129] S. R. Dolan, Scattering, absorption and emission by black holes. Cambridge Ph.D.
thesis., 2006.
220 BIBLIOGRAPHY
[130] S. Creek, O. Efthimiou, P. Kanti, and K. Tamvakis, Greybody factors in a rotating
black-hole background-ii : fermions and gauge bosons, Phys. Rev. D76 (2007)
104013, [arXiv:0707.1768].
[131] S. Creek, O. Efthimiou, P. Kanti, and K. Tamvakis, Scalar emission in the bulk in
a rotating black hole background, Phys. Lett. B656 (2007) 102–111,
[arXiv:0709.0241].
[132] S. Creek, O. Efthimiou, P. Kanti, and K. Tamvakis, Graviton emission in the bulk
from a higher-dimensional Schwarzschild black hole, Phys. Lett. B635 (2006)
39–49, [hep-th/0601126].
[133] E. W. Leaver, An analytic representation for the quasi normal modes of Kerr black
holes, Proc. Roy. Soc. Lond. A402 (1985) 285–298.
[134] W. G. Baber and H. R. Hasse´, The two centre problem in wave mechanics, Proc.
Camb. Phil. Soc. 25 (1935) 564.
[135] W. Gautschi, Computational aspects of three-term recurrence relations, SIAM
Review 9 (1967) 24.
[136] S. Zhang and J. Jin, Computation of special functions, . John Wiley & sons, see
http://jin.ece.uiuc.edu/routines/routines.html.
[137] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions. Dover
Publications, Inc., 1965.
[138] S. Iyer and C. M. Will, Black hole normal modes: A WKB approach. 1.
Foundations and application of a higher order WKB analysis of potential barrier
scattering, Phys. Rev. D35 (1987) 3621.
[139] H. T. Cho and Y. C. Lin, WKB analysis of the scattering of massive Dirac fields
in Schwarzschild black hole spacetimes, Class. Quant. Grav. 22 (2005) 775–790,
[gr-qc/0411090].
[140] A. S. Cornell, W. Naylor, and M. Sasaki, Graviton emission from a
higher-dimensional black hole, JHEP 02 (2006) 012, [hep-th/0510009].
[141] J. Grain and A. Barrau, A WKB approach to scalar fields dynamics in curved
space- time, Nucl. Phys. B742 (2006) 253–274, [hep-th/0603042].
BIBLIOGRAPHY 221
[142] H. T. Cho, A. S. Cornell, J. Doukas, and W. Naylor, Bulk dominated fermion
emission on a Schwarzschild background, Phys. Rev. D77 (2008) 016004,
[arXiv:0709.1661].
[143] C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and
engineers. McGraw-Hill, Inc., 1978. page 487.
[144] S. R. Dolan, M. Casals, P. Kanti, and E. Winstanley, Greybody factors for
higher-dimensional black holes, . http://mathsci.ucd.ie/∼sdolan/greybody/.
[145] Y. B. Zel’dovich, The generation of waves by a rotating body, ZhETF Pis. Red. 14
(1971) 270.
[146] D. Ida, K.-y. Oda, and S. C. Park, Anisotropic scalar field emission from TeV
scale black hole, hep-ph/0501210.
[147] V. P. Frolov and D. Stojkovic, Quantum radiation from a 5-dimensional rotating
black hole, Phys. Rev. D67 (2003) 084004, [gr-qc/0211055].
[148] H. Nomura, S. Yoshida, M. Tanabe, and K.-i. Maeda, The fate of a
five-dimensional rotating black hole via Hawking radiation, Prog. Theor. Phys. 114
(2005) 707–712, [hep-th/0502179].
[149] E. Jung and D. K. Park, Bulk versus brane in the absorption and emission: 5D
rotating black hole case, Nucl. Phys. B731 (2005) 171–187, [hep-th/0506204].
[150] D. A. Leahy and W. G. Unruh, Angular dependence of neutrino emission from
rotating black holes, Phys. Rev. D19 (1979) 3509–3515.
[151] M. Casals, S. R. Dolan, P. Kanti, and E. Winstanley, Angular profile of emission
of non-zero spin fields from a higher-dimensional black hole, Phys. Lett. B680
(2009) 365–370, [arXiv:0907.1511].
[152] A. Flachi, M. Sasaki, and T. Tanaka, Spin polarization effects in micro black hole
evaporation, JHEP 05 (2009) 031.
[153] W. G. Unruh, Absorption cross-section of small black holes, Phys. Rev. D14
(1976) 3251–3259.
222 BIBLIOGRAPHY
[154] B. S. DeWitt, Quantum field theory in curved space-time, Phys. Rept. 19 (1975)
295–357.
[155] J. C. Collins and D. E. Soper, The theorems of perturbative QCD, Ann. Rev. Nucl.
Part. Sci. 37 (1987) 383–409.
[156] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical recipes in C:
The art of scientific computing. Cambridge University Press, 1992.
[157] M. Cavaglia, R. Godang, L. Cremaldi, and D. Summers, Catfish: A Monte Carlo
simulator for black holes at the LHC, Comput. Phys. Commun. 177 (2007)
506–517, [hep-ph/0609001].
[158] C. M. Harris et al., Exploring higher dimensional black holes at the large hadron
collider, JHEP 05 (2005) 053, [hep-ph/0411022].
[159] C. M. Harris, Physics beyond the standard model: Exotic leptons and black holes at
future colliders, hep-ph/0502005.
[160] V. Cardoso, J. P. S. Lemos, and S. Yoshida, Electromagnetic radiation from
collisions at almost the speed of light: an extremely relativistic charged particle
falling into a Schwarzschild black hole, Phys. Rev. D68 (2003) 084011,
[gr-qc/0307104].
[161] R. Casadio, B. Harms, and O. Micu, Microscopic black holes as a source of
ultrahigh energy gamma-rays, astro-ph/0202513.
[162] R. Casadio, S. Fabi, and B. Harms, Electromagnetic waves around dilatonic stars
and naked singularities, Phys. Rev. D70 (2004) 044026, [gr-qc/0307022].
[163] M. Chaichian and R. Hagedorn, Symmetries in quantum mechanics: From angular
momentum to supersymmetry. Bristol, UK: IOP, 1998. 304 p.
[164] R. Emparan, G. T. Horowitz, and R. C. Myers, Black holes radiate mainly on the
brane, Phys. Rev. Lett. 85 (2000) 499–502, [hep-th/0003118].
[165] D. N. Page, Particle emission rates from a black hole: Massless particles from an
uncharged, nonrotating hole, Phys. Rev. D13 (1976) 198–206.
BIBLIOGRAPHY 223
[166] D. N. Page, Particle emission rates from a black hole. 2. massless particles from a
rotating hole, Phys. Rev. D14 (1976) 3260–3273.
[167] M. K. Parikh and F. Wilczek, Hawking radiation as tunneling, Phys. Rev. Lett. 85
(2000) 5042–5045, [hep-th/9907001].
[168] J. Preskill, P. Schwarz, A. D. Shapere, S. Trivedi, and F. Wilczek, Limitations on
the statistical description of black holes, Mod. Phys. Lett. A6 (1991) 2353–2362.
[169] S. Dimopoulos and R. Emparan, String balls at the LHC and beyond, Phys. Lett.
B526 (2002) 393–398, [hep-ph/0108060].
[170] B. Koch, M. Bleicher, and S. Hossenfelder, Black hole remnants at the LHC,
JHEP 10 (2005) 053, [hep-ph/0507138].
[171] H. Stoecker, Stable TeV - black hole remnants at the LHC: Discovery through
di-jet suppression, mono-jet emission and a supersonic boom in the quark-gluon
plasma, Int. J. Mod. Phys. D16 (2007) 185–205, [hep-ph/0605062].
[172] P. Meade and L. Randall, Black holes and quantum gravity at the LHC, JHEP 05
(2008) 003, [arXiv:0708.3017].
[173] O. Lorente-Espin and P. Talavera, A silence black hole: Hawking radiation at the
Hagedorn temperature, JHEP 04 (2008) 080, [arXiv:0710.3833].
[174] F. Scardigli, Glimpses on the micro black hole Planck phase, arXiv:0809.1832.
[175] J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole,
Phys. Lett. B360 (1995) 7–12, [gr-qc/9505012].
[176] D. M. Gingrich and K. Martell, Study of highly-excited string states at the large
hadron collider, Phys. Rev. D78 (2008) 115009, [arXiv:0808.2512].
[177] J. Alwall et al., A standard format for Les Houches event files, Comput. Phys.
Commun. 176 (2007) 300–304, [hep-ph/0609017].
[178] D.-C. Dai et al., BlackMax: A black-hole event generator with rotation, recoil, split
branes and brane tension, Phys. Rev. D77 (2008) 076007, [arXiv:0711.3012].
[179] D.-C. Dai et al., Manual of BlackMax, a black-hole event generator with rotation,
recoil, split branes, and brane tension, arXiv:0902.3577.
224 BIBLIOGRAPHY
[180] E. Richter-Was, AcerDET: A particle level fast simulation and reconstruction
package for phenomenological studies on high p(T) physics at LHC,
hep-ph/0207355.
[181] ATLAS Collaboration, G. Aad et al., The ATLAS experiment at the CERN large
hadron collider, JINST 3 (2008) S08003.
[182] The ATLAS Collaboration, G. Aad et al., Expected performance of the ATLAS
experiment - Detector, trigger and physics, arXiv:0901.0512.
[183] J. D. Bjorken and S. J. Brodsky, Statistical model for electron-positron
annihilation into hadrons, Phys. Rev. D1 (1970) 1416–1420.
[184] R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cim.
1 (1969) 252–276.
[185] S. W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett.
26 (1971) 1344–1346.
[186] E. Byckling and K. Kajantie, Particle kinematics. John Wiley & sons, 1973. page
162.