Heavy quark physics on the lattice
with improved nonrelativistic actions
Stefan Meinel
Dissertation submitted for the degree of
Doctor of Philosophy
at the University of Cambridge
Department of Applied Mathematics and Theoretical Physics
and
St John’s College
21 December 2009
ii
Declaration
This dissertation is the result of my own work and includes nothing which is the outcome
of work done in collaboration, except where specifically indicated in the text. No part of
this dissertation has been previously submitted for a degree or any other qualification.
Stefan Meinel
21 December 2009
iv
Abstract
Hadrons containing heavy quarks, in particular b quarks, play an important role in high
energy physics. Measurements of their electroweak interactions are used to test the Stan-
dard Model and search for new physics. For the comparison of experimental results with
theoretical predictions, nonperturbative calculations of hadronic matrix elements within
the theory of quantum chromodymanics are required.
Such calculations can be performed from first principles by formulating QCD on a
Euclidean spacetime grid and computing the path integral numerically. Including b quarks
in lattice QCD calculations requires special techniques as the lattice spacing in present
computations usually can not be chosen fine enough to resolve their Compton wavelength.
In this work, improved nonrelativistic lattice actions for heavy quarks are used to
perform calculations of the bottom hadron mass spectrum and of form factors for heavy-
to-light decays. In heavy-to-light decays, additional complications arise at high recoil,
when the momentum of the light meson reaches a magnitude comparable to the cutoff
imposed by the lattice. Discretisation errors at high recoil can be reduced by working in
a frame of reference where the heavy and light mesons move in opposite directions. Using
a formalism referred to as moving nonrelativistic QCD (mNRQCD), the nonrelativistic
expansion for the heavy quark can be performed around a state with an arbitrary velocity.
This dissertation begins with a review of the fundamentals of lattice QCD. Then,
the construction of effective Lagrangians for heavy quarks in the continuum and on the
lattice is discussed in detail. A highly improved lattice mNRQCD action is derived and
its effectiveness is demonstrated by nonperturbative tests involving both heavy-heavy and
heavy-light mesons at several frame velocities.
This mNRQCD action is then used in combination with a staggered action for the
light quarks to calculate hadronic matrix elements relevant for rare B decays, including
B → K∗γ and B → K``. A major contribution to the uncertainty of the results also
comes from statistical errors. The effectiveness of random-wall sources to reduce these
errors is studied.
As another application of a nonrelativistic heavy quark action, the spectrum of bot-
tomonium is calculated and masses of several bottom baryons are predicted. In these
computations, the light quarks are implemented with a domain wall action.
vi
Contents
Declaration iii
Abstract v
1 Introduction 1
2 Lattice QCD 5
2.1 Euclidean path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Real scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Extracting observables from Euclidean correlation functions . . . . . . . . . 11
2.3 Gauge fields and naive fermions on the lattice . . . . . . . . . . . . . . . . . 12
2.4 More about lattice fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Naive fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Staggered fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.4 Fermion doubling and chiral symmetry . . . . . . . . . . . . . . . . . 18
2.4.5 The Ginsparg-Wilson relation . . . . . . . . . . . . . . . . . . . . . . 20
2.4.6 Overlap fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.7 Domain wall fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Renormalisation and continuum limit . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Lattice artefacts and effective field theories . . . . . . . . . . . . . . . . . . 25
2.7.1 Non-zero lattice spacing effects . . . . . . . . . . . . . . . . . . . . . 25
2.7.2 Unphysical quark masses . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7.3 Finite-volume effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
viii CONTENTS
2.8 Relativistic heavy quarks on the lattice . . . . . . . . . . . . . . . . . . . . . 29
3 Effective Lagrangians for heavy quarks 31
3.1 Foldy-Wouthuysen-Tani transformation . . . . . . . . . . . . . . . . . . . . 32
3.2 Power counting and heavy-quark symmetries . . . . . . . . . . . . . . . . . 36
3.2.1 Heavy-light hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.2 Heavy-heavy mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Generalisation to a moving frame of reference . . . . . . . . . . . . . . . . . 38
3.4 Moving NRQCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 O(1/m3) relativistic correction in moving NRQCD . . . . . . . . . . 46
3.5 Euclidean mNRQCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.1 Euclidean quark and antiquark Green functions in mNRQCD . . . . 48
4 Lattice HQET 49
4.1 Relativistic corrections as operator insertions . . . . . . . . . . . . . . . . . 49
4.2 Continuum HQET Green functions . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Lattice discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Lattice moving NRQCD 55
5.1 Continuum evolution equation . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Lattice evolution equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Symanzik improvement . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3.1 Tadpole improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3.2 Energy shift and external momentum renormalisation . . . . . . . . 62
5.3.3 Reparametrisation invariance . . . . . . . . . . . . . . . . . . . . . . 64
5.3.4 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.5 High-β methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Nonperturbative tests of moving NRQCD 67
6.1 Tests with bottomonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1.1 Calculation of the two-point functions . . . . . . . . . . . . . . . . . 67
6.1.2 Fitting of the two-point functions . . . . . . . . . . . . . . . . . . . . 70
6.1.3 Kinetic mass, energy shift and external momentum renormalisation . 71
CONTENTS ix
6.1.4 Decay constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.5 Energy splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Tests with Bs mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.1 Calculation of the two-point functions . . . . . . . . . . . . . . . . . 80
6.2.2 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.3 Kinetic mass, energy shift and external momentum renormalisation . 82
6.2.4 Decay constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2.5 Energy splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Comparison of nonperturbative and perturbative results . . . . . . . . . . . 86
7 Heavy-hadron spectroscopy 89
7.1 Bottomonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.1.1 Lattice details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.1.2 Calculation of the two-point functions . . . . . . . . . . . . . . . . . 92
7.1.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.1.4 Tuning of the b quark mass . . . . . . . . . . . . . . . . . . . . . . . 94
7.1.5 Speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1.6 Radial/orbital energy splittings and the lattice spacing . . . . . . . . 98
7.1.7 Spin-dependent energy splittings . . . . . . . . . . . . . . . . . . . . 102
7.2 Bottom hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2.1 Quark propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2.2 B mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2.3 Singly-bottom baryons . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2.4 Doubly-bottom baryons . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2.5 The Ωbbb baryon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8 Rare B decays 121
8.1 Weak Decays of B Mesons in the Standard Model . . . . . . . . . . . . . . . 122
8.1.1 Electroweak interactions of quarks . . . . . . . . . . . . . . . . . . . 122
8.1.2 Effective weak Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 123
8.1.3 The decay B → pi`ν . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.1.4 The decay B → K∗γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 More about b→ s decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3 General definition of semileptonic form factors . . . . . . . . . . . . . . . . 129
8.4 Extraction of form factors from correlation functions . . . . . . . . . . . . . 129
8.4.1 The form factors f0 and f+ . . . . . . . . . . . . . . . . . . . . . . . 131
x CONTENTS
8.4.2 The form factor fT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.4.3 The form factor V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.4.4 The form factors A0, A1 and A2 . . . . . . . . . . . . . . . . . . . . 133
8.4.5 The form factors T1, T2 and T3 . . . . . . . . . . . . . . . . . . . . . 134
8.5 Matching of heavy-light currents . . . . . . . . . . . . . . . . . . . . . . . . 135
8.6 Reference frame choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.7 Heavy-quark expansion of the current . . . . . . . . . . . . . . . . . . . . . 140
8.8 Two-point and three-point functions with random-wall sources . . . . . . . 142
8.8.1 Light meson two-point functions . . . . . . . . . . . . . . . . . . . . 142
8.8.2 Heavy-light meson two-point functions . . . . . . . . . . . . . . . . . 157
8.8.3 Heavy-light meson three-point functions . . . . . . . . . . . . . . . . 164
8.9 Size of the 1/m corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.10 Preliminary form factor results . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.10.1 The form factors f0, f+, and fT . . . . . . . . . . . . . . . . . . . . 181
8.10.2 The form factors T1 and T2 . . . . . . . . . . . . . . . . . . . . . . . 184
9 Conclusions 187
9.1 Renormalisation of the lattice mNRQCD action . . . . . . . . . . . . . . . . 187
9.2 Heavy-hadron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.3 Calculation of form factors for B decays . . . . . . . . . . . . . . . . . . . . 188
A Conventions 191
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.2 Lattice derivatives and field strength . . . . . . . . . . . . . . . . . . . . . . 193
B Form factors and decay rate for B → K∗γ 195
B.1 Derivation of form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
B.2 Calculation of decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
C Data analysis methods 201
C.1 Correlated least-squares fitting . . . . . . . . . . . . . . . . . . . . . . . . . 201
C.2 Simultaneous fitting of multiple correlation functions . . . . . . . . . . . . . 202
C.3 Bayesian fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
C.4 Bootstrap method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
C.5 Autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Acknowledgements 207
CONTENTS xi
List of publications 209
Bibliography 210
xii CONTENTS
Chapter 1
Introduction
The existence of a third generation of quarks was predicted by Kobayashi and Maskawa
in 1973 to explain the phenomenon of CP violation [1]. Their prediction was confirmed
by the discovery of the Υ meson [2], a bound state of a bottom quark and its antiquark,
in 1977 at Fermilab.
The Υ(4S) resonance can decay through the strong interaction into pairs of B mesons,
which were first observed at the electron-positron storage ring CESR in 1983 [3]. B mesons
allow the study of weak interactions of the b quark, and thereby provide some of the most
stringent tests of the Standard Model of elementary particle physics.
A wealth of experimental results for decays and mixing processes of B mesons has
been obtained, most recently by the experiments BaBar at SLAC and Belle at KEK.
World-averages of heavy-hadron properties can be found in [4].
In many cases, the experimental results for exclusive decays, where a definite final
state is considered, are more precise than those for inclusive decays. The comparison of
theory and experiment for exclusive processes requires knowledge of the structure of the
hadrons, described by decay constants and form factors. These quantities are governed by
the non-perturbative dynamics of the strong interaction at low energies.
The theory of the strong interactions is quantum chromodynamics (QCD). Nonper-
turbative calculations in QCD can be performed from first principles by formulating the
theory on a Euclidean spacetime grid [5] and computing the path integral numerically.
With several improvements in the lattice actions and numerical algorithms, and with the
enormous growth in available computer power, lattice QCD now allows accurate predic-
tions of many hadronic properties. One major step towards this success was the inclusion
of vacuum polarisation effects from the light quarks [6]. Reviews of recent progress in
lattice QCD with dynamical quarks can be found in [7, 8].
2 CHAPTER 1. INTRODUCTION
A lattice with side length L and lattice spacing a can accommodate physics at energy
scales between approximately 1/L and 1/a. To minimise finite-volume effects, it is required
that mpiLÀ 1, where mpi is the mass of the lightest propagating particle in the theory, the
pion. With the limited resources available, it is usually not possible to achieve mb a ¿ 1
at the same time. In other words, the lattices are not fine enough to accurately resolve
the Compton wavelength of the b quark.
For this reason, special techniques are required when the b quark is to be included in
lattice QCD calculations. One possibility is the non-relativistic treatment of the b quark,
which is based on the observation that fluctuations in the b-quark’s momentum are of order
ΛQCD ¿ mb in heavy-light hadrons and mbvorb. ¿ mb in bottomonium. Here, ΛQCD is
the intrinsic scale of QCD, which governs the dynamics of the light degrees of freedom
(light quarks and gluons) in a heavy-light hadron, and vorb. is the internal speed of the b
quarks inside bottomonium (v2orb. ≈ 0.1).
One particular nonrelativistic technique for b quarks is lattice NRQCD [9, 10], which
can be applied to both heavy-heavy and heavy-light systems. Lattice NRQCD has been
used in [11] to calculate form factors for the decay B → pi`ν. The form factors are func-
tions of q2 = (p − p′)2, where p and p′ are the four-momenta of the B meson and pion,
respectively. In [11], the B meson was taken to be at rest in the lattice frame. The momen-
tum p′ of the pion in the lattice calculation is limited by the requirement that |p′|a¿ 1,
which means that only the low-recoil region (large q2) is accessible. However, experimental
data cover the entire range of recoil momenta [12, 13, 14]. The problem is more severe
for the radiative decay B → K∗γ [15, 16, 17, 18], which has q2 = 0, corresponding to
|p′| close to MB/2 in the B rest frame. This decay is rare because it proceeds through
a loop-suppressed flavour-changing neutral-current b → s transition, which also makes
it very sensitive to possible contributions from physics beyond the Standard Model. In
contrast to B → pi`ν, for rare three-body decays like B → K`+`− and B → K∗`+`− the
shape of the form factors is not known well from experiment, and lattice QCD calculations
for a wide range of q2 are desirable.
For a given q2, the momentum p′ of the final meson and hence discretisation errors
can be reduced by giving the B meson a non-zero momentum in the direction opposite to
p′. To substantially reduce |p′|, the momentum of the B meson has to be very large, so
that NRQCD would no longer be able to describe the b quark inside it due to relativistic
and lattice spacing errors. However, fluctuations of the momentum of the heavy quark
inside the B meson are much smaller than the momentum of the meson itself. Therefore, to
reduce errors, instead of discretising the momentum of the b quark itself, one can discretise
CHAPTER 1. INTRODUCTION 3
its fluctuations inside the moving B meson. The formalism which achieves this is known
as lattice moving NRQCD (mNRQCD). In mNRQCD, the nonrelativistic expansion is
performed around a state where the heavy quark is moving with a velocity v, the frame
velocity.
The idea of mNRQCD goes back to [19] and was developed further in [20, 21, 22].
Numerical tests of mNRQCD with a simple O(1/mb) action and with quenched gauge
field configurations (i.e. without dynamical quarks) were reported in [22, 23, 24, 25]. In
this dissertation I extend this work as follows:
• I independently derive a moving NRQCD action of order O(1/m2b , v
4
orb.). My deriva-
tion gives a result that differs slightly from the one reported in [21, 22, 23].
• I perform a wide range of numerical tests of the full O(1/m2b , v
4
orb.) lattice mNRQCD
action in realistic lattice QCD with 2+1 flavours of light dynamical quarks. The
accuracy through order v4orb. is crucial as I use calculations with bottomonium mesons
to determine renormalisation parameters nonperturbatively.
I then apply moving NRQCD to the calculation of form factors for semileptonic and
radiative B decays:
• I compute the heavy-light three-point and two-point correlation functions necessary
to extract all semileptonic and radiative B/Bs decay form factors, using the full
O(1/m2b , v
4
orb.) lattice mNRQCD action for the b quark and an improved staggered
action for the light quarks.
• I test a “random-wall source” method for the reduction of statistical errors.
• I analyse the data and obtain preliminary results for form factors describing B → K
and B → K∗ decays.
In this dissertation I also present my work on a second project, the calculation of the heavy-
hadron spectrum in lattice QCD with dynamical domain wall fermions. Here I implement
the b quark with standard lattice NRQCD. The domain wall fermion action [26, 27, 28]
used here for the light quarks has an approximate chiral symmetry that becomes exact,
even at finite lattice spacing, when the extent of an auxiliary fifth dimension is taken to
infinity. My work on the heavy-hadron spectrum includes the following:
• I first calculate the spectrum of the bottomonium system, including radial/orbital
excitations and spin-dependent energy splittings. These calculations serve as tests
of the lattice methods and also allow precise determinations of the lattice scale.
4 CHAPTER 1. INTRODUCTION
• I then calculate the masses of B/Bs mesons, singly-bottom baryons and doubly-
bottom baryons, using a domain-wall action for the light valence quarks. I also
perform the first lattice QCD calculation of the mass of the Ωbbb baryon. Currently,
bottom baryons are the focus of intense experimental studies at the Tevatron, and
further discoveries are possible with the Large Hadron Collider. Therefore, lattice
QCD predictions of their masses are highly desirable.
This dissertation is organised as follows. In Chapter 2, I give an introduction to the the-
oretical foundations of lattice QCD, including the fascinating topic of chiral symmetry
on the lattice. At the end of this chapter, I briefly review relativistic lattice methods for
heavy quarks. Chapter 3 contains derivations of effective Lagrangians for heavy quarks
in the continuum and a discussion of the power-counting. In particular, I derive the
O(1/m2b , v
4
orb.) mNRQCD Lagrangian and show how the Euclidean actions for quarks and
antiquarks are related to each other. Two different approaches exist for the lattice discreti-
sation of nonrelativistic heavy-quark Lagrangians. The first, known as lattice HQET, is
only applicable to hadrons containing a single heavy quark, but has the advantage that the
continuum limit can be taken. It is discussed briefly in Chapter 4. The second approach,
lattice (moving) NRQCD, is applicable to systems containing any number of heavy quarks.
In Chapter 5, I construct a highly improved lattice discretisation of the full O(1/m2b , v
4
orb.)
moving NRQCD action. I also discuss the mNRQCD energy shift and the renormalisation
of the external momentum. I then report on my nonperturbative tests of lattice moving
NRQCD in Chapter 6. These tests include the calculation of renormalisation parameters,
meson energy splittings and decay constants at several frame velocities. Chapter 7 is an
interlude; it contains my work on the calculation of the heavy-hadron spectrum. I return
to the subject of moving NRQCD in Chapter 8, which deals with the calculation of form
factors for rare B decays. There, I briefly review the physics of electroweak decays of B
mesons, define the form factors and show how they can be extracted from Euclidean cor-
relation functions. I discuss the necessary operator matching of the heavy-light currents
and estimate the optimal choice for the frame velocity as a function of q2. I then give
the details of the calculation of the two-point and three-point functions and compare nu-
merical results obtained with random-wall sources and point sources. I examine different
fitting strategies and present preliminary results for the form factors. Chapter 9 contains
the conclusions and an outlook to future work.
Chapter 2
Lattice QCD
In this chapter, the fundamentals of lattice QCD will be reviewed. I used the textbooks
[29, 30, 31, 32, 33, 34] in preparing this chapter; additional references will be given where
appropriate.
Lattice QCD is based on the nonperturbative regularisation of the Euclidean path
integral obtained by defining the fields only on the points or links of a spacetime grid.
This corresponds to a momentum cutoff of the order of the inverse lattice spacing, without
any reference to perturbation theory. Furthermore, as we will see in Sec. 2.3, exact gauge
invariance can be implemented in a beautiful way on the lattice [5].
It turns out that the lattice discretisation of fermions is more involved than for gauge
fields, due to the so-called doubling problem. This is related to chiral symmetry and the
axial anomaly. Section 2.4 is devoted to this rich topic.
When considering a lattice of finite volume, the path integral turns into an ordinary
multidimensional integral, which, thanks to the Euclidean weighting factor e−SE and the
positivity of SE , becomes calculable numerically by Monte-Carlo methods.
As usual for interacting quantum field theories, to obtain physical results, renormal-
isation is required. This will be discussed briefly in Sec. 2.6. In practice, the number of
lattice points is limited by the available computing resources, and one has to make careful
choices for the lattice spacings, volumes, and quark masses. Usually, extrapolations in all
these parameters are required to extract physical results, and these extrapolations can be
guided by predictions from effective field theories, as reviewed in Sec. 2.7.
Finally, in Sec. 2.8 I briefly discuss relativistic lattice methods for heavy quarks.
6 CHAPTER 2. LATTICE QCD
2.1 Euclidean path integral
As the Euclidean path integral is the starting point for lattice QCD, we will discuss it
in some detail in the following sections. We begin with a derivation of the Euclidean
path integral for a scalar field in Sec. 2.1.1, starting from the canonical formalism. This
clearly exposes the relation to the operators on the Hilbert space, gives us a prescription
for obtaining the Euclidean action, and shows how a non-zero temperature is introduced.
Remarks on fermions and gauge fields will be made in Secs. 2.1.2 and 2.1.3.
2.1.1 Real scalar field
We consider a real scalar field φ(t,x) with Minkowski space action
S =
∫
dt d3x L with L =
1
2
(∂tφ)
2 −
1
2
(∇φ)2 −
1
2
m2φ2 − V (φ), (2.1)
where V (φ) is a polynomial. The conjugate momentum field is then Π = ∂L/∂(∂tφ) = ∂tφ,
and the Hamilton functional becomes
H =
∫
d3xH with H =
1
2
Π2 +
1
2
(∇φ)2 +
1
2
M2φ2 + V (φ) . (2.2)
We denote the Heisenberg field operators associated with φ, Π by Φ and Π, respectively.
These satisfy the canonical commutation relations [Φ(0,x),Π(0,y)] = i δ3(x − y). We
also introduce zero-time “position space” and “momentum space” eigenstates |ϕ〉, |pi〉,
satisfying
Φ(0,x)|ϕ〉 = ϕ(x)|ϕ〉, Π(0,x)|pi〉 = pi(x)|pi〉. (2.3)
Note that their scalar product is 〈ϕ|pi〉 = exp
{
i
∫
d3x ϕ(x)pi(x)
}
.
The Hamilton operator H is obtained by replacing the classical fields φ, Π in (2.2) by
the corresponding field operators. We are interested in the quantum-mechanical transition
amplitude between two field eigenstates, 〈ϕ′|e−i t H|ϕ〉. The transition to Euclidean space
is obtained by inserting t = −iτ with τ > 0. To derive the path integral representation
for the Euclidean matrix element 〈ϕ′|e−τ H|ϕ〉, we split the Euclidean time interval τ into
N pieces of length a = τ/N , and insert complete sets of position space and momentum
space eigenstates after each step:
〈ϕ′|e−τH|ϕ〉 =
∫ N−1∏
k=1
d[ϕk]
∫ N∏
l=1
d[pil]
2pi
× 〈ϕ′|e−aH|piN 〉〈piN |ϕN−1〉〈ϕN−1|e
−aH|piN−1〉 · · · 〈pi1|ϕ〉. (2.4)
2.1. Euclidean path integral 7
Next, note that for a→ 0, we have
〈ϕk|e
−aH|pik〉 =
a→0
(1− aH[ϕk, pik]) 〈ϕk|pik〉
=
a→0
exp
{
−a
∫
d3x
[
H(ϕk, pik)− i
ϕk pik
a
]}
. (2.5)
Thus, for N →∞, Eq. (2.4) becomes
〈ϕ′|e−τH|ϕ〉 = lim
N→∞
∫ N−1∏
k=1
d[ϕk]
∫ N∏
l=1
d[pil]
2pi
× exp
−a
N∑
j=1
∫
d3x
[
H(ϕj , pij)− i
pij(ϕj − ϕj−1)
a
]
(2.6)
with ϕ0 = ϕ and ϕN = ϕ′. The limit N → ∞ replaces the sum in (2.6) by an integral
over [0, τ ] and we can write
〈ϕ′|e−τH|ϕ〉 =
∫
D[ΠE ]
∫ ϕ′
ϕ
D[φE ] exp
{
−
∫ τ
0
dτ ′
∫
d3x
[
H(φE ,ΠE)− iΠE ∂τ ′φE
]
}
. (2.7)
The functional integrals
∫
D[ΠE ]
∫
D[φE ] in (2.7) are over Euclidean fields ΠE(τ ′,x) and
φE(τ ′,x), subject to the boundary conditions φE(0,x) = ϕ(x) and φE(τ,x) = ϕ′(x). The
integral over ΠE is Gaussian and can be performed explicitly by completing the square.
This gives1
〈ϕ′|e−τ H|ϕ〉 =
∫ ϕ′
ϕ
D[φE ] e
−SE [φE ] (2.8)
with the Euclidean action
SE [φE ] =
∫
dτ
∫
d3x
[
1
2
(∂τφE)
2 +
1
2
(∇φE)
2 +
1
2
m2φ2E + V (φE)
]
. (2.9)
By comparing (2.1) and (2.9), we find that the Euclidean action SE [φE ] can be obtained
from the Minkowski space action S[φ] in the following formal way: first, replace φ(t,x) by
φE(τ,x). Subsequently, replace every remaining t by −iτ , so that the integration measure
and derivatives become dt→ (−i)dτ, ∂t → i∂τ . Finally, multiply the action by (−i).
Having derived Eq. (2.8), the next step is to obtain a path integral representation
for correlation functions. For simplicity we only consider the two-point function; the
generalisation to higher correlation functions is straightforward.
1In (2.8) we have absorbed the constant factor from the Π-integration by redefining the inte-
gration measure D[φE ].
8 CHAPTER 2. LATTICE QCD
Let |0〉 be the vacuum state. For τ > 0, the Euclidean two-point function (Schwinger
function) is
C(τ,x) = 〈0| eτH Φ(0,x) e−τH Φ(0,0) |0〉. (2.10)
First, note that C(τ,x) may be written as
C(τ,x) = lim
Lτ→∞
Tr
[
e−LτH eτH Φ(0,x) e−τH Φ(0,0)
]
Tr [e−LτH]
. (2.11)
To show (2.11), we use an orthonormal basis of energy eigenstates |n〉:
C(τ,x) = lim
Lτ→∞
∑
n,m〈n|e
−LτH|m〉〈m| eτH Φ(0,x) e−τH Φ(0,0)|n〉
∑
n〈n|e
−LτH|n〉
. (2.12)
Provided that the theory has a mass gap, in the limit Lτ → ∞ only the lowest-energy
state |0〉 remains, and the factor of 〈0|e−LτH|0〉 cancels.
We may also evaluate the traces in (2.11) using field eigenstates:
C(τ,x) = lim
Lτ→∞
∫
d[ϕ] 〈ϕ| e−(Lτ−τ)H Φ(0,x) e−τH Φ(0,0) |ϕ〉
∫
d[ϕ] 〈ϕ| e−LτH |ϕ〉
= lim
Lτ→∞
∫
d[ϕ]
∫
d[ϕ′] 〈ϕ| e−(Lτ−τ)H |ϕ′〉ϕ′(x)〈ϕ′| e−τH |ϕ〉ϕ(0)
∫
d[ϕ] 〈ϕ| e−LτH |ϕ〉
. (2.13)
Finally, using Eq. (2.8) to express the matrix elements in (2.13) as path integrals, we
obtain
C(τ,x) =
〈
φE(τ,x) φE(0,0)
〉
≡ lim
Lτ→∞
∫
D[φE ] φE(τ,x) φE(0,0) e−SE [φE ]
∫
D[φE ] e−SE [φE ]
(2.14)
where the functional integral is over the Euclidean field φE with the periodic boundary
conditions φE(Lτ ,x) = φE(0,x). The generalisation of the above derivation to n-point
correlation functions is straightforward.
Note that for finite Lτ , Eq. (2.11) is a thermal expectation value of the operator
eτH Φ(0,x) e−τH Φ(0,0) at the temperature 1/Lτ . Thus, for the Euclidean path integral
with periodic boundary conditions in the time direction a finite temporal extent corre-
sponds to a non-zero temperature 1/Lτ . The limit Lτ →∞ is a zero-temperature limit.
2.1.2 Dirac field
The Minkowski space action for a Dirac field with mass m is
S =
∫
dt d3x Ψ(iγˆµ∂µ −m)Ψ (2.15)
2.1. Euclidean path integral 9
with {γˆµ, γˆν} = 2 gµν . Note that iΨγˆ0 is the conjugate momentum field for Ψ, and the
Hamilton functional reads
H =
∫
d3x Ψ
(
−iγˆj∂j +m
)
Ψ. (2.16)
Because the field operators Ψ, Ψ obey the canonical anticommutation relations
{
Ψα(0,x), Ψβ(0,y)
}
= γˆ0αβ δ
3(x− y), (2.17)
the fields in the fermionic path integral have to be Grassmann-algebra-valued. The parti-
tion function can be written as a Berezin integral
Tr
[
e−LτH
]
=
∫
D[ΨE ,ΨE ] e
−SE [ΨE ,ΨE ] (2.18)
over Euclidean Grassmann-valued fields with the antiperiodic boundary conditions
ΨE(0,x) = −ΨE(Lτ ,x),
ΨE(0,x) = −ΨE(Lτ ,x). (2.19)
The Euclidean action in (2.18) is
SE [ΨE ,ΨE ] =
∫
dτ
∫
d3x ΨE
[
γˆ0∂τ − iγˆ
j∂j +m
]
ΨE . (2.20)
Eq. (2.20) suggests the definition of Euclidean gamma matrices γ0 ≡ γˆ0, γj ≡ −iγˆj which
satisfy {γµ, γν} = 2 δµν . Euclidean correlation functions can be computed from2
〈
Ψ(x1)...Ψ(xn) Ψ(y1)...Ψ(yn)
〉
= lim
Lτ→∞
∫
D[Ψ,Ψ] Ψ(x1)...Ψ(xn) Ψ(y1)...Ψ(yn) e−S[Ψ,Ψ]
∫
D[Ψ,Ψ] e−S[Ψ,Ψ]
. (2.21)
Writing the Euclidean action as
S[Ψ,Ψ] =
∫
d4x
∫
d4y
∑
α,β
Ψα(x)Kα β(x, y) Ψβ(y), (2.22)
where the kernel K satisfies the correct boundary conditions, one obtains the following
results from the rules of Grassmann integration:
∫
D[Ψ,Ψ] e−S[Ψ,Ψ] = detK (2.23)
2In the following, we drop the subscript E
10 CHAPTER 2. LATTICE QCD
and
〈
Ψα1(x1)...Ψαn(xn) Ψβ1(y1)...Ψβn(yn)
〉
= (−1)n(n−1)/2
∑
P∈Sn
(−1)σ(P )K−1α1 βP (1)
(
x1, yP (1)
)
... K−1αn βP (n)
(
xn, yP (n)
)
(2.24)
whereK−1 denotes the inverse matrix ofK with respect to both position and Dirac indices.
2.1.3 Gauge fields
Consider Nc copies of the Dirac field with Lagrangian (2.15). The resulting theory has a
global symmetry
Ψa(x) 7→ Gab Ψb(x), Ψa(x) 7→ Ψb(x)G
†
ba. (2.25)
with G ∈ SU(Nc). In the following we shall not write out the indices a, b, ... explicitly.
The symmetry (2.25) can be promoted to a local gauge symmetry with position-dependent
G(x),
Ψ(x) 7→ G(x)Ψ(x), Ψ(x) 7→ Ψ(x)G†(x), (2.26)
through replacing the derivative ∂µΨ(x) in the Lagrangian by
DµΨ(x) ≡ ∂µΨ(x) + igAµ(x)Ψ(x). (2.27)
The covariant derivative (2.27) contains a new field iAµ(x) ∈ su(Nn) that changes under
gauge transformations in the following way:
Aµ(x) 7→ G(x)Aµ(x)G
†(x) +
i
g
[∂µG(x)]G
†(x). (2.28)
The transformation law (2.28) ensures that DµΨ transforms in the same way as Ψ. To
obtain the complete theory, the following kinetic term for the gauge field is added to the
action (in Minkowski space):
SYM = −
1
2
∫
d4x Tr [FµνF
µν ] , (2.29)
where
Fµν ≡
1
ig
[Dµ, Dν ] = ∂µAν − ∂νAµ + ig[Aµ, Aν ]. (2.30)
For a gauge theory, the derivation of the path integral is more complicated since the
fields are only determined up to gauge transformations. The definition of a Hamiltonian
and canonical quantisation require gauge fixing. However, in the nonperturbative lattice
2.2. Extracting observables from Euclidean correlation functions 11
formulation of the path integral the integration over redundant degrees of freedom does
not cause problems. We shall therefore not derive the path integral in the continuum, but
rather define it directly on the lattice in Sec. 2.3.
Note that the Euclidean action can be obtained from the Minkowski space action in
the same way as described in Sec. 2.1.1, except that because ∂t becomes i∂τ one also needs
to replace A0 by iAE0 in order to achieve consistency with the gauge transformation law
(2.28). The Euclidean action is
SE =
∫
d4xE
[
ΨE(xE)(γµDµ +m)ΨE(xE) +
1
2
Tr [Fµν(xE)Fµν(xE)]
]
. (2.31)
2.2 Extracting observables from Euclidean correlation func-
tions
When Euclidean correlation functions are calculated numerically in lattice field theory, it
is usually not possible to perform an analytic continuation back to Minkowski space. Thus,
one has to extract results for physical observables directly from Euclidean correlators.
The energy spectrum of the theory can be extracted straightforwardly from two-point
functions, as can already be seen from Eq. (2.12). Consider a set of Euclidean fields Oi
associated with the Minkowski space operators Oi. Using the Heisenberg equations of
motion and H|0〉 = P|0〉 = 0 we have for τ > 0 and Lτ = ∞
Cij(τ,p) ≡
∫
d3x 〈Oi(τ,x)O
†
j(0)〉 e
−ip·x
=
∫
d3x 〈0|Oi(0)e−Hτ+iP·xO
†
j(0)|0〉 e
−ip·x. (2.32)
Now, let |X〉 denote an eigenstate of H with zero momentum, and let |Xk〉 denote the
boost of |X〉 to momentum k. Then we have the completeness relation
I = |0〉〈0|+
∑
X
∫
d3k
(2pi)3
1
2E(Xk)
|Xk〉〈Xk|, (2.33)
where
∑
X denotes the sum/integration over all zero-momentum eigenstates (except the
12 CHAPTER 2. LATTICE QCD
vacuum) of H. Inserting (2.33) into (2.32), we get3
Cij(τ,p) =
∑
X
∫
d3k
(2pi)3
1
2E(Xk)
∫
d3x 〈0|Oi(0)|Xk〉〈Xk|O
†
j(0)|0〉e
−E(Xk)τe−i(p−k)·x
=
∑
X
∫
d3k
1
2E(Xk)
〈0|Oi(0)|Xk〉〈Xk|O
†
j(0)|0〉e
−E(Xk)τδ3(p− k)
=
∑
X
1
2E(Xp)
〈0|Oi(0)|Xp〉 〈0|Oj(0)|Xp〉∗ e−E(Xp)τ . (2.34)
Thus, the energies of the states |Xp〉 with non-zero overlap 〈0|Oi(0)|Xp〉 can be extracted
from a multi-exponential fit to (2.34). For large τ , the lowest-energy state with a non-zero
overlap dominates, and a single-exponential fit will be sufficient to obtain its energy.
From such a fit, one also obtains the amplitudes 〈0|Oi(0)|Xp〉. Consider the case of
QCD, with O1 = J
µ
5 = uγˆ
µγˆ5d and O2 = uγˆ5d. Then the lowest-energy state contributing
to (2.34) is a charged pion at momentum p and we obtain
〈0| u(0)γˆµγˆ5d(0) |pi−(p)〉 = −ifpi pµ. (2.35)
Thus, the pion decay constant fpi, which is relevant for the leptonic decay pi → `ν, can be
extracted. Similarly, from Euclidean three-point correlation functions, one can straight-
forwardly extract form factors for electroweak decays with a single hadron in both the
initial and final state, such as B → pi`ν. This will be explained in Sec. 8.4.
In contrast, hadron-hadron scattering processes can not be calculated directly from
Euclidean correlation functions in infinite volume, except at kinematic thresholds [35].
There is a way around this: by studying the volume-dependence of multi-particle energy
levels in a finite box one can compute elastic scattering phase shifts [36, 37].
2.3 Gauge fields and naive fermions on the lattice
We will now discretise the theory with action (2.31) on a hypercubic lattice with points
x = an, n ∈ Z4. Since we will work exclusively with the Euclidean theory, we drop the
subscript “E” in the following.
Let us consider the Dirac action without the gauge field first. The derivative ∂µΨ must
be replaced by a difference quotient4:
1
2a
[Ψ(x+ aµˆ)−Ψ(x− aµˆ)] , (2.36)
3We assume 〈0|Oi|0〉 = 0. Otherwise, take the connected 2-point function.
4The symmetric difference is chosen to give a Hermitian Hamiltonian
2.3. Gauge fields and naive fermions on the lattice 13
where µˆ denotes the unit-vector in µ-direction. Expression (2.36) is not covariant under
gauge transformations, since Ψ(x+ aµˆ) and Ψ(x− aµˆ) transform differently under (2.26).
Thus, the fields must be parallel-transported to the same lattice point x first. The covariant
derivative may be written as
∆(±)µ Ψ(x) ≡
1
2a
[Uµ(x)Ψ(x+ aµˆ)− U−µ(x)Ψ(x− aµˆ)] , (2.37)
where Uµ(x), U−µ(x) ∈ SU(Nc) and U−µ(x) = U
†
µ(x − aµˆ). These link variables must
transform under gauge transformations as
Uµ(x) 7→ G(x)Uµ(x)G
†(x+ aµˆ) (2.38)
in order to guarantee the exact gauge invariance of the lattice Dirac action
Slat.F [Ψ,Ψ, U ] = a
4
∑
x∈aZ4
Ψ(x)
[
γµ∆(±)µ +m
]
Ψ(x). (2.39)
When defining Alat.µ by
Uµ(x) = exp
(
iagAlat.µ (x)
)
= 1+ iagAlat.µ (x) +O(a
2), (2.40)
we see that for sufficiently smooth fields in the limit a→ 0 the lattice expressions approach
the continuum expressions,
lim
a→0
∆(±)µ Ψ(x) = DµΨ(x),
lim
a→0
Alat.µ (x) = Aµ(x), (2.41)
and the gauge transformation law for the link variables, (2.38), becomes equivalent to
(2.28).
What is still missing is a lattice discretisation of the kinetic term for the gauge field.
For this, a lattice definition of the gauge field strength tensor is needed. Mathematically
speaking, the field strength is the curvature associated with the gauge connection. Thus, it
can be obtained from parallel transport around a small closed loop. The smallest possible
loop on the lattice is the plaquette
Uµν(x) ≡ Uµ(x)Uν(x+ aµˆ)U
†
µ(x+ aνˆ)U
†
ν (x). (2.42)
Inserting (2.40), we find
Uµν(x) = 1+ iga
2Fµν(x) +O(a
3). (2.43)
14 CHAPTER 2. LATTICE QCD
Thus, a possible choice for the lattice field strength would be
i
ga2
(1− Uµν(x)). (2.44)
Since the continuum Fµν is Hermitian, one may also choose the Hermitian conjugate of
(2.44) to represent Fµν . To obtain a real and positive lattice action, the natural choice is
to take the product of (2.44) and its Hermitian conjugate as the lattice discretisation of
FµνFµν in the action:
Slat.G [U ] = a
4
∑
x∈aZ4
∑
µ,ν
1
2
Tr
{
i
ga2
(
1− Uµν(x)
)
[
i
ga2
(
1− Uµν(x)
)
]†
}
=
∑
x∈aZ4
∑
µ,ν
1
2g2
Tr
[
2 · 1− Uµν(x)− U
†
µν(x)
]
= β
∑
x∈aZ4
∑
µ<ν
[
1−
1
Nc
Re Tr Uµν(x)
]
(2.45)
with β = 2Nc/g2. Eq. (2.45) is the plaquette action, as introduced by Wilson in [5]. For
smooth fields, the deviation to the continuum action (2.31) is of order O(a2).
The total lattice action Slat.[Ψ,Ψ, U ] for the Dirac field coupled to the gauge field is
of course the sum of (2.39) and (2.45). The path integral for an observable O[Ψ,Ψ, U ]
depending on the fermion and gauge fields is then defined as
〈O〉 =
∫
D[U ]
∫
D[Ψ,Ψ] O[Ψ,Ψ, U ] e−S
lat.[Ψ,Ψ,U ]
∫
D[U ]
∫
D[Ψ,Ψ] e−Slat.[Ψ,Ψ,U ]
, (2.46)
where
∫
D[Ψ,Ψ] stands for the Grassmann integrations over all colour/spin components
of Ψ and Ψ at all lattice points, and
∫
D[U ] =
∫ ∏
x
∏
µ dUµ(x) denotes the integrations
over all link variables Uµ(x) with the Haar measure. The group-multiplication invariance
of the latter leads to the gauge invariance of the path integral. In local coordinates {ωa}
on SU(Nc) the Haar measure is given by
dU = C
√
detA
N2c−1∏
a=1
dωa, (2.47)
where C is a normalisation factor so that
∫
dU = 1, and A is the metric on SU(Nc),
Aab = −2 Tr
[
(U−1∂aU)(U
−1∂bU)
]
= 2 Tr
[
∂aU ∂bU
†]. (2.48)
(From [30]). The fermion measure D[Ψ,Ψ] is gauge-invariant because det G(x) = 1 for
gauge transformations G(x) ∈ SU(Nc).
2.4. More about lattice fermions 15
Using the results (2.23) and (2.24), the Grassmann integrations can be performed
explicitly. This gives
〈O〉 =
∫
D[U ] O
[
K−1[U ], U
]
det (K[U ]) e−S
lat.
G [U ]
∫
D[U ] det (K[U ]) e−S
lat.
G [U ]
(2.49)
where K is the kernel of the lattice Dirac action. For the naive action (2.39), the kernel
is
K(x, x′) =
3∑
µ=0
1
2a
γµ
[
Uµ(x)δx′, x+aµˆ − U−µ(x)δx′, x−aµˆ
]
+m δx′, x. (2.50)
On a finite lattice, (2.49) is an ordinary multidimensional integral over real numbers (local
coordinates on SU(Nc)), and can be calculated numerically by Monte-Carlo methods, as
explained in Sec. 2.5.
2.4 More about lattice fermions
2.4.1 Naive fermions
The lattice fermion action defined in (2.39) is called the naive fermion action, for rea-
sons that will become clear when considering the resulting free (= without gauge fields)
propagator in momentum space:
K−1(p) =
[
iγµ sin(a pµ)/a+m
]−1
=
−iγµ sin(a pµ)/a+m
∑3
µ=0 sin
2(a pµ)/a2 +m2
. (2.51)
Note that the lattice momentum p is restricted to the periodic Brillouin zone, which is a
4-dimensional torus, with −pi < apµ ≤ pi. The sine functions in (2.51) vanish at the 16
points p(1) = (0, 0, 0, 0), p(2) = (pi, 0, 0, 0)/a, ... p(16) = (pi, pi, pi, pi)/a of the Brillouin zone,
and for momenta close to these points one can expand them in a Taylor series to obtain
the continuum limit a→ 0. Writing p = p(n) + k (modulo 2pi/a), this gives
K−1(p(n) + k) =
−iγ(n)µ kµ +m
k2 +m2
+O(a) (2.52)
with
γ(n)µ = γµ cos(a p
(n)
µ ) = ±γµ. (2.53)
For a → 0, (2.52) has a pole at k0 = i
√
k2 +m2, and hence (2.51) has 16 such poles in
the continuum limit. The matrices γ(n)µ in (2.53) can be written as
γ(n)µ = S
†
(n)γµS(n) (2.54)
16 CHAPTER 2. LATTICE QCD
with unitary S(n), see [31]. The transformation
Ψ′(n)(x) = e
−ip(n)·xS(n)Ψ(x), Ψ
′
(n)(x) = Ψ(x)S
†
(n)e
ip(n)·x (2.55)
brings the action for the n-th doubler to its standard form.
The interpretation of the doublers as corresponding to individual Dirac fields called
tastes can be made more explicit by transforming to a spin-taste basis, as will be explained
in Sec. 2.4.3. It turns out that the fermion doubling is related to chiral symmetry. This is
discussed in Sec. (2.4.4).
2.4.2 Wilson fermions
One way to remove the 15 unwanted doublers is to modify the lattice action by a term
that gives them infinite “mass” in the continuum limit, so that they decouple from the
theory. Wilson suggested the following action:
SWF [Ψ,Ψ, U ] = a
4
∑
x∈aZ4
Ψ(x)
[
γµ∆
(±)
µ −
1
2
a∆(+)µ ∆
(−)
µ +m
]
Ψ(x), (2.56)
so that
KW (x, x
′) = −
3∑
µ=0
1
2a
[
(1− γµ)Uµ(x)δx′, x+aµˆ + (1 + γ
µ)U−µ(x)δx′, x−aµˆ
]
+
(
m+
4
a
)
δx′, x.
(2.57)
This gives the free propagator
K−1W (p) =
−iγµ sin(a pµ)/a+ m˜(a, p)
∑3
µ=0 sin
2(a pµ)/a2 + m˜2(a, p)
, (2.58)
with m˜(a, p) = m + 2a
∑3
µ=0 sin
2(apµ/2). The behaviour near p = 0 is unaltered but
the doublers disappear. However, the new term in the action explicitly breaks the chiral
symmetry that one would have form = 0. This results in an additive mass renormalisation
and more severe complications related to operator mixing.
2.4.3 Staggered fermions
Starting from the naive fermion action (2.39), we perform a unitary transformation Ψ(x) =
Ω(x)Ψ′(x), Ψ(a) = Ψ
′
(x)Ω†(x) with5
Ω(x) = γx00 γ
x1
1 γ
x2
2 γ
x3
3 . (2.59)
5From now on, we use lattice units with a = 1.
2.4. More about lattice fermions 17
Then, using
Ω†(x)γµΩ(x+ µˆ) = (−1)
x0+...+xµ−1
︸ ︷︷ ︸
≡ηµ(x)
1, (2.60)
the action becomes
SF =
∑
x∈Z4
Ψ
′
(x)
[
ηµ(x)∆
(±)
µ +m
]
Ψ′(x). (2.61)
This is now diagonal in spin-space, and we see that the four components of Ψ′ interact
independently and equally.
The staggered fermion action is then constructed by keeping only one of the compo-
nents of Ψ′, e.g. the first, Ψ′1. The physical content of the resulting action can be analysed
by grouping the values of the one-component field Ψ′1 from the 16 corners of each hyper-
cube of the lattice into a new field Qαβ , living on a lattice with the double lattice spacing.
Let us consider the free case without gauge fields for simplicity. The coordinates of the
original lattice can be written as x = 2y+ ρ with y ∈ Z4 and ρµ = 0, 1. Then, one defines
Qαβ(y) =
1
8
∑
ρ
Ωαβ(ρ)Ψ
′
1(2y + ρ), Qαβ(y) =
1
8
∑
ρ
Ψ
′
1(2y + ρ)Ω
†
αβ(ρ). (2.62)
The index α is the spinor index, while the index β turns out to label the tastes. Re-
introducing the lattice spacing, the action becomes
Sstag.F = (2a)
4
∑
y ∈ 2aZ4
3∑
µ=0
Q(y)
[
(γµ ⊗ 1)∆
(±)
µ + a(γ5 ⊗ γ
T
µ γ
T
5 )∆
(−)
µ ∆
(+)
µ
]
Q(y)
+ (2a)4
∑
y ∈ 2aZ4
mQ(y)
[
1⊗ 1
]
Q(y), (2.63)
where the notation is “spin ⊗ taste” and derivatives are now understood with the double
lattice spacing:
∆(+)µ Q =
1
2a
[Q(y + 2aµˆ)−Q(y)] etc. (2.64)
In the new basis, (2.63) gives the following propagator (with b = 2a):
K−1stag.(p) =
1
b
∑3
µ=0
[
−i(γµ ⊗ 1) sin(b pµ) + (γ5 ⊗ γTµ γ
T
5 ) sin
2 b pµ
2
]
+m1⊗ 1
∑3
µ=0
4
b2 sin
2 b pµ
2 +m
2
. (2.65)
Since the lattice spacing forQαβ is b = 2a, the momenta are restricted to−pi/b < pµ ≤ pi/b,
and the sine function in the denominator vanishes only at p = 0. In the continuum limit,
(2.65) reduces to the correct propagator describing four Dirac fields of the same mass.
At non-zero lattice spacing, the term a(γ5 ⊗ γTµ γ
T
5 )∆
(−)
µ ∆
(+)
µ in (2.63) breaks the taste
symmetry.
18 CHAPTER 2. LATTICE QCD
As we have seen at the beginning of this section, a naive fermion field is equivalent to
be four identical copies of staggered fermion fields. Thus, the above calculations also show
that the naive action corresponds to 16 tastes in the continuum limit.
Rooted staggered fermions
In the continuum limit, the staggered fermion action describes four identical copies of a
Dirac field, which means that the corresponding fermion operator can be written as
K = ( /D +m)⊗ 14×4, (2.66)
where /D is the continuum Dirac operator. This was shown in the previous section for
the free case without gauge fields. The path integral for a continuum theory with action
(2.66) contains the determinant
det
[
( /D +m)⊗ 14×4
]
=
[
det( /D +m)
]4
. (2.67)
The determinant is responsible for the loops of virtual fermions, i.e. the sea quarks in
QCD. Hence, for such a continuum theory, one can reduce the number of sea quarks to
one by taking the positive 4-th root of the fermion determinant in the path integral6:
det( /D +m) =
(
det
[
( /D +m)⊗ 14×4
])1/4
. (2.68)
For rooted staggered fermions, the same procedure is done at non-zero lattice spacing. At
non-zero lattice spacing, the taste-symmetry is broken and the resulting theory is non-
local [38]. However, there is evidence that the continuum limit is still correct [39, 40, 41].
It is crucial to take this limit before the chiral limit, so that the taste splittings always
remain smaller than the Goldstone pion mass.
The use of rooted staggered fermions has been criticised in particular by M. Creutz
[42, 43, 44], and this criticism has been refuted in [45, 46, 47, 48].
2.4.4 Fermion doubling and chiral symmetry
The naive fermion action (2.39), which suffers from doubling, has a U(1) chiral symmetry
at m = 0,
Ψ′ = eiαγ5Ψ, Ψ
′
= Ψeiαγ5 . (2.69)
Equivalently, the naive lattice fermion matrix (2.50) with m = 0 satisfies {K, γ5} = 0. On
the other hand, in the Wilson action (2.56), the fermion doublers are removed by including
the operator 12a∆
(+)
µ ∆
(−)
µ , which breaks the chiral symmetry.
6the determinant is positive for m > 0, since
{
/D, γ5
}
= 0 and /D is anti-Hermitian
2.4. More about lattice fermions 19
An important observation can be made about the chirality of the doublers. Starting
from the naive lattice action, let us try to restrict the theory to, say, the left-handed
components
ΨL(x) = 12(1− γ5)Ψ, ΨL(x) = Ψ
1
2(1 + γ5). (2.70)
Due to the chiral symmetry, the left-handed components are not coupled to the right-
handed components, and so we can set the latter to zero. However, for half of the tastes,
the transformation (2.55) which was used to obtain the correct form of the action, changes
the handedness:
S(n)
1
2(1 + γ5)S
†
(n) =
1
2(1± γ5) (2.71)
with a minus sign for 8 of the 16 tastes. Thus, by doing the projection (2.70) one obtains
a theory of 8 left-handed and 8 right-handed fermions.
The number of doublers in the naive action, 24 = 16, is due to the hypercubic sym-
metry. Wilczek [49] suggested the following action that breaks hypercubic symmetry at
non-zero lattice spacing
K(p) = i
3∑
j=1
γj
1
a
sin(a pj) + iγ0
1
a
sin(a p0) + λ
3∑
j=1
1
a
sin2
(
1
2a pj
)
. (2.72)
This is chirally symmetric and has zeros only at p = (0, 0, 0, 0) and p = (pi/a, 0, 0, 0).
By doing the projection (2.70) one obtains a theory of a single left-handed and a single
right-handed fermion, i.e. a theory with the degrees of freedom of a single Dirac fermion7.
In [50], actions of the general form K(x, y) =
∑3
µ=0 γµ Fµ(x, y) that satisfy translation
invariance (i.e. F (x, y) = F (x − y)), Hermiticity (i.e. F ∗(z) = F (−z)), and |z|4F (z) → 0
for |z| → ∞, are considered. These properties imply that the Fourier transform Fµ(p) is a
continuous real vector field on the Brillouin zone, which is a 4-dimensional torus T 4. The
Poincare´-Hopf theorem states that for a compact manifold the sum of the indices at the
zeros of Fµ is equal to the Euler characteristic of the manifold, which is zero for T 4. This
implies that there is always an equal number of left-handed and right-handed fermions.
Very general proofs of the existence of an equal number of left-handed and right-handed
fermions were given by Nielsen and Ninomiya in [51, 52, 53].
The fermion doubling is also related to the axial anomaly. In the continuum theory
of a massless Dirac field coupled to a gauge field, the axial current JµA = Ψγ
µγ5Ψ is no
longer conserved after quantisation:
〈 ∂µJ
µ
A(x) 〉 =
g2
16pi2
²µνρλ
〈
Tr[Fµν(x)F ρλ(x)]
〉
. (2.73)
7However, breaking hypercubic invariance leads to other complications.
20 CHAPTER 2. LATTICE QCD
As Fujikawa has shown [54], this can be explained by the non-invariance of the fermion
path integral measure, when properly regulated in a gauge-invariant way, under the trans-
formation (2.69). However, on the lattice the fermion measure is in fact invariant under
(2.69) due to Tr γ5 = 0. The naive lattice action (2.39) at m = 0 is also invariant, and
hence with naive fermions the lattice axial current
Jµ (lat.)A =
1
2
[
Ψ(x)Uµ(x)γµγ5Ψ(x+ µˆ) + Ψ(x+ µˆ)γµγ5U
†
µ(x)Ψ(x)
]
(2.74)
is still exactly conserved in the quantum theory: 〈∆(−)µ J
µ (lat.)
A 〉 = 0. This can be viewed
as a cancellation between the contributions to the axial anomaly from the left-handed and
right-handed doublers [55]. Note that in the spin-taste basis introduced in Sec. 2.4.3, the
transformation (2.69) reads
Q 7→ eiα(γ5⊗γ5)Q, Q 7→ Qeiα(γ5⊗γ5), (2.75)
and it is therefore identified as a taste-non-singlet symmetry. The taste-singlet axial
transformation generated by γ5 ⊗ 1 is not an exact symmetry of the action at non-zero
lattice spacing, and the associated current is in fact anomalous [56, 57].
For Wilson fermions, the action is not invariant under (2.69) due to the Wilson term.
In this case the current (2.74) has the correct anomaly [55].
2.4.5 The Ginsparg-Wilson relation
In the last section we have established the following: on the lattice, the path integral
measure is invariant under the transformation (2.69). With the naive lattice action, which
is also invariant under (2.69), unphysical fermion doublers appear, turning this transfor-
mation into a non-singlet symmetry which therefore is non-anomalous.
On the other hand, the Wilson action is not invariant under (2.69), so that the doublers
disappear and the axial anomaly is generated. However, for a theory with multiple physical
quark flavours, the Wilson term also breaks all non-singlet chiral symmetries, which makes
renormalisation of the lattice theory very difficult.
There is, however, a more continuum-like way of generating the axial anomaly on the
lattice, which avoids doublers but leaves the non-singlet chiral symmetries for multiple
physical flavours intact. In this approach, the definition of the chiral transformation is
modified for non-zero lattice spacing, but reduces to the usual form for a → 0. The
simplest definition of the infinitesimal transformation reads [58]
δΨ = iαγ5
(
1−
a
2r0
K
)
Ψ, δΨ = iαΨ
(
1−
a
2r0
K
)
γ5, (2.76)
2.4. More about lattice fermions 21
where K is the lattice Dirac operator. Flavour non-singlet transformations can be defined
by including a flavour-group generator. The action is invariant under (2.76) if K satisfies
the relation
{γ5, K} =
a
r0
K γ5 K. (2.77)
Eq. (2.77) had already been derived by Ginsparg and Wilson [59] as the remnant of
continuum chiral symmetry obtained after a blocking transformation. Explicit solutions
for K will be given in Secs. 2.4.6 and 2.4.7. By multiplying (2.77) from both the left and
right with K−1 we find the following relation for the propagator K−1:
{
γ5, K
−1(x, x′)
}
=
a
r0
γ5 δx, x′ , (2.78)
that is, the propagator is chirally invariant in the usual sense at all non-zero distances.
It turns out that the path integral measure transforms under the flavour-singlet chiral
rotation (2.76) in precisely the right way to generate the axial anomaly [58]. As it should
be, the measure is invariant under flavour-non-singlet transformations.
Starting from a lattice Dirac operator K satisfying the Ginsparg-Wilson relation, a
mass m can be introduced in the following way:
K(m) =
(
1−
am
2r0
)
K +m. (2.79)
2.4.6 Overlap fermions
An explicit example of a lattice Dirac operator satisfying (2.77) is the overlap operator
[60, 61, 62]
K =
r0
a
[1 + γ5 sgn(H)] (2.80)
where the simplest choice is H = γ5 KW with the Wilson fermion operator (2.57) with
negative mass −r0/a. In (2.80), the function sgn(H) is the matrix sign function: for
Hermitian H, one has sgn(H) = H/
√
H2. In numerical simulations, the sign function is
implemented by some iterative approximation.
2.4.7 Domain wall fermions
Chiral fermions in 4 dimensions can also be obtained as zero-modes localised at domain
walls in 5-dimensional theories. Following [63], let us consider the 5-dimensional continuum
Dirac equation with a position-dependent mass term,
[
/D + γ5∂s +M(s)
]
Ψ(x, s) = 0, (2.81)
22 CHAPTER 2. LATTICE QCD
where s is the 5-th coordinate, /D is the 4-dimensional massless Dirac operator and
M(s) =
{
+M, s > 0
−M, s < 0
(M > 0). (2.82)
Since (2.81) is separable, we can write Ψ(x, s) as a linear combination of products of the
form8
Ψ(x, s) =
∑
n
[fn(s)PR + gn(s)PL]ψn(x), (2.83)
where PR/L =
1
2(1± γ5) and
[∂s +M(s)] fn(s) = µn gn(s),
[−∂s +M(s)] gn(s) = µn fn(s),
/D ψn(x) = −µn ψn(x). (2.84)
The functions f0 and g0 with eigenvalue µ = 0 have the form
f0(s) = F e
−
R s
0M(s
′) ds′ = F e−M |s|,
g0(s) = G e
+
R s
0M(s
′) ds′ = G e+M |s|, (2.85)
with constants F and G. Since g0(s) is not normalisable for G 6= 0, we must have G = 0.
Thus, the massless mode is right-handed (for M < 0, a left-handed mode is obtained).
One expects that all other eigenvalues are of order |µn| & O(M), and so the low-energy
effective theory consists of a 4-dimensional right-handed massless fermion.
For a compact fifth dimension with periodic boundary conditions (and hence with
a kink and an anti-kink in M(s)), in the absence of gauge fields one finds a massless
left-handed mode localised at one of the domain walls and a massless right-handed mode
localised at the other domain wall. However, interactions with gauge fields lead to a
coupling of the left-handed and right-handed modes and hence a breaking of exact chirality
and the appearance of a residual mass. This coupling vanishes as the length of the fifth
dimension is taken to infinity, and exact chirality is restored.
The appearance of chiral modes bound to domain walls can be used to construct chiral
lattice fermion actions [26]. A domain wall action commonly used in current simulations
was introduced in [27] and [28]. In lattice units, it is given by
SDW =
∑
x
Ls∑
s=1
Ψ(x, s)
[
Ψ(x, s)− PLΨ(x, s+1)− PRΨ(x, s−1) +
∑
x′
KW (x, x
′)Ψ(x′, s)
]
(2.86)
8This may also include integration over a continuous part of the spectrum.
2.5. Numerical calculations 23
where KW is the Wilson fermion operator (2.57) with negative mass −M5 = −r0 and the
gauge fields in KW are independent of the 5-th coordinate s. The 5-dimensional fermion
fields are subject to the boundary conditions
PL Ψ(x, Ls + 1) = −mPL Ψ(x, 1),
PR Ψ(x, 0) = −mPR Ψ(x, Ls). (2.87)
The Dirac mass turns out to be proportional to m. Left-handed and right-handed four-
dimensional fermion fields can be defined as
qL(x) = ΨL(x, 1), qR(x) = ΨR(x, Ls),
qL(x) = ΨL(x, 1), qR(x) = ΨR(x, Ls), (2.88)
where ΨL/R = PL/RΨ and ΨL/R = ΨPR/L. For m = 0 and in the limit Ls → ∞ the
left-handed and right-handed fields decouple completely.
Note that in order to cancel the contributions from the heavy fermion modes in the
bulk, one has to introduce additional 5-dimensional bosonic pseudofermion fields (also
referred to as Pauli-Villar fields) in the partition function. A commonly used choice for
their kernel is [64]
KPV = KDW
∣
∣
∣
m=1
, (2.89)
where KDW is the 5-dimensional domain wall operator defined by (2.86, 2.87). Performing
the Gaussian integrals over the fermion and pseudofermion fields in the partition function,
one obtains a factor of
detKDW
detKPV
. (2.90)
As shown in [65], this ratio of determinants can be written as a single determinant of a
4-dimensional truncated overlap operator that becomes an exact overlap operator in the
limit Ls →∞.
At finite Ls, a small residual chiral symmetry breaking remains. See Ref. [66] for a
detailed discussion of the choice of Ls in practical simulations.
2.5 Numerical calculations
Numerical calculations in lattice QCD usually start from Eq. (2.49), where the Grassmann
integrals over the fermion fields have been performed explicitly. A set of gauge field
configurations {U1, U2, ..., UN} is created via a Markov process such that for large N this
24 CHAPTER 2. LATTICE QCD
set is distributed with the probability density
p[Ui] ∝
(∏
f
detKf [Ui]
)
e−SG[Ui], (2.91)
where Kf are the lattice fermion actions for the various flavours of dynamical quarks9.
The expectation value of an observable O is then approximated as the ensemble-average
〈O〉 ≈
1
N
N∑
i=1
O
[{
K−1f [Ui]
}
, Ui
]
. (2.92)
For this to make sense, the product of the fermion determinants in (2.91) must be positive.
This is satisfied e.g. for Ginsparg-Wilson fermions. More generally, when a fermion action
satisfies γ5-Hermiticity, it follows that the corresponding determinant is real:
(detK)∗ = detK† = det(γ5Kγ5) = detK. (2.93)
Then, if mass-degenerate pairs of flavours are used, (2.91) will be positive. In many
numerical calculations, the u- and d- quarks are taken to have equal mass.
With rooted staggered fermions (see Sec. 2.4.3), one can take
[
det K†K|ee
]1/4
for each
flavour, where the subscript “ee” denotes the restriction of the matrix K†K to the even
sublattice (K†K is block-diagonal in terms of even and odd sites).
The algorithms used to update the gauge fields according to the distribution (2.91)
usually require the repeated (iterative) solution of equations like K G = S, which makes
the inclusion of dynamical fermions computationally expensive.
2.6 Renormalisation and continuum limit
The asymptotic freedom of QCD (for a sufficiently small number of flavours) implies that
the continuum limit of lattice QCD corresponds to the limit where the bare gauge coupling
is taken to zero.
In numerical computations, one chooses a value for the bare gauge coupling and then
“measures” the corresponding lattice spacing. This is done by computing some physical
quantity (in lattice units) and comparing the result to the experimental value. The bare
quark masses can be set by requiring that the lattice results for a few “input” hadron
masses agree with experiment (this may not be directly feasible; see Sec. 2.7.2).
9For heavy flavours, depending on the problem considered it can be a good approximation to
neglect the effects of pair creation and set the corresponding determinant to 1.
2.7. Lattice artefacts and effective field theories 25
When reducing the lattice spacing, the number of lattice sites in each direction has to be
increased accordingly to keep the physical box size fixed. One also has to take the infinite
volume limit, by doing the calculations with several physical box sizes. Extrapolations
in the lattice spacing, the volume and the quark masses can be performed with the help
of effective field theories [67]; this will be discussed in Sec. 2.7. It is also possible to
reduce cutoff effects at non-zero lattice spacing by including additional higher-dimension
operators to the lattice action (cf. Sec. 2.7.1).
While hadron masses can be obtained directly from the exponential decay of correlation
functions, the values of matrix elements of certain operators require further renormalisa-
tion. They may depend on the scale and renormalisation scheme, and it may be necessary
to convert the results to a continuum scheme like MS in order to make contact with ex-
periment. The same applies for the definition of the renormalised gauge coupling and
renormalised quark masses.
In general, the renormalisation is simplified when symmetries are preserved exactly by
the regulator. As an example, for flavour- and chirally symmetric lattice actions, one finds
ZV = ZA = 1 for the conserved vector- and axial vector currents.
One method to relate lattice and continuum schemes is lattice perturbation theory,
which has the advantage that analytic results can be obtained. The mismatch between
the lattice and the continuum theories lies in the ultraviolet where the coupling is small, so
that perturbation theory may be a good approximation. However, a systematic uncertainty
due to neglected higher-order terms and nonperturbative effects remains.
Methods for nonperturbative renormalisation include the regularisation-independent
(RI) scheme [68] and the Schro¨dinger functional method. Applications of the Schro¨dinger
functional method include the renormalisation of the gauge coupling [69, 70] and quark
masses [71], as well as the matching of heavy-quark effective theory (see Sec. 4) to QCD
[72].
2.7 Lattice artefacts and effective field theories
2.7.1 Non-zero lattice spacing effects
Errors due to a non-zero lattice spacing can be analysed be means of the Symanzik effective
field theory, where the lattice theory is modelled by a continuum theory with Lagrangian
LSym = LQCD + LI , (2.94)
26 CHAPTER 2. LATTICE QCD
where LI describes the lattice artefacts. It contains all higher-order local operators Oi
that are not forbidden by the symmetries of the lattice action,
LI =
∑
i
adimOi−4 ci Oi, (2.95)
with dimensionless short-distance coefficients ci depending on the couplings in the lattice
action and the renormalisation scale. Using this framework, one can predict the functional
dependence of lattice results on a, allowing a controlled continuum extrapolation. Note
that for the fermionic part of the action, the possible operators of dimension 5, like the
Wilson term (2.56), break chiral symmetry. Thus, if the lattice action has an exact
chiral symmetry, these dimension-five operators will be forbidden in the Symanzik effective
Lagrangian, so that chiral actions can not have O(a) errors.
Using the Symanzik effective theory, one can not only analyse non-zero lattice spacing
errors, but can also reduce them by adjusting the couplings in the lattice action so that
some of the coefficients ci in (2.95) get smaller or vanish. This requires the inclusion of
the corresponding higher-order operators in the lattice action.10 As the CPU time needed
to generate statistically independent gauge configurations with a fixed physical volume
grows like a−(4+z), where z > 0 accounts for critical slowing down,11 it can be much more
cost-effective to use Symanzik-improvement rather than reduce a [76].
Consider for instance a lattice derivative operator
∆(±)µ =
exp(aDµ)− exp(−aDµ)
2a
= Dµ +
a2
6
(Dµ)
3 +
a4
120
(Dµ)
5 + ... (2.96)
For smooth fields, the order a2 error in (2.96) can be removed by taking the improved
derivative
∆˜(±)µ = ∆
(±)
µ −
a2
6
∆(+)µ ∆
(±)
µ ∆
(−)
µ (2.97)
instead. However, at one-loop level, (2.97) still leads to O(αsa2) errors due to radiative
corrections. These errors can be reduced by adjusting the coefficient in (2.97), but it is
difficult to find the correct value. A simple method of reducing radiative corrections is
tadpole improvement [77], which is based on the following observation. In lattice pertur-
bation theory, the gauge link Uµ(x) is given in terms of the vector potential Alatµ as an
expansion in powers of a:
Uµ(x) = 1 + iagA
lat
µ (x) +
1
2
(iag)2
[
Alatµ (x)
]2
+ ... (2.98)
10For off-shell quantities, also the interpolating fields in correlation functions must be improved.
11Critical slowing down is particularly severe for topological modes, where the autocorrelation
time grows rapidly [73], possibly exponentially [74, 75], when a is reduced.
2.7. Lattice artefacts and effective field theories 27
Consider the expectation value
u0 =
1
Nc
〈Tr Uµ〉 , (2.99)
(the mean link) defined in Landau gauge. At tree-level, u0 = 1, and the one-loop result is
still close to 1, while the nonperturbative result for u0 is significantly smaller. It turns out
that higher-order terms in the expansion (2.98) are actually not suppressed by powers of
a, since the contractions of Alatµ with itself (leading to “tadpole” diagrams) produce UV-
divergences proportional to the same power of 1/a. The large tadpole renormalisations
in lattice operators can be taken care of through rescaling the links by a factor of 1/u0,
where u0 can be measured nonperturbatively.
Finally, note that for heavy quarks (am ? 1) the usual Symanzik effective theory
breaks down. Following [67], we consider operators like
∑
µ(−γ
µDµ)n (these describe
deviations from Lorentz/Euclidean invariance). The time derivative is of the order of the
heavy-quark mass,
an(−γ0D0)
nΨ ∼ (am)nΨ, (2.100)
and hence powers of a are accompanied by powers of m, spoiling the convergence of the
expansion. The physical interpretation is that the lattice can not resolve the Compton
wavelength of heavy quarks when the grid is too coarse. Hence, the standard relativistic
fermion actions develop large discretisation errors for am ? 1. The discussion of relativistic
heavy quarks in lattice QCD will be continued in Sec. 2.8.
2.7.2 Unphysical quark masses
The CPU time in dynamical lattice QCD calculations also grows like some inverse power
of the light quark mass, as the condition number of the fermion matrix is proportional
to (amq)−1. In addition, lighter quark masses require larger volumes to keep finite-size
effects under control (see Sec. 2.7.3). Thus, most present computations are done at larger-
than-physical u- and d- quark masses, and extrapolations are required.
For many observables, the functional form of the dependence on the light quark masses
can be predicted using chiral perturbation theory (χPT); see e.g. [78, 79] for an introduc-
tion. In QCD with Nf massless quarks, the SU(Nf )L×SU(Nf )R chiral flavour symmetry
turns out to be spontaneously broken to SU(Nf )V by a non-zero vacuum expectation value
〈qfRq
f ′
L 〉 = vδ
ff ′ . Fluctuations in the vacuum expectation value along the broken symmetry
directions, which do not change the potential energy, correspond to dim[SU(Nf )] = N2f −1
massless Goldstone bosons (pions). One can introduce an effective SU(Nf )-valued field
28 CHAPTER 2. LATTICE QCD
Σ(x) = exp(2iΦ/f) (with traceless and Hermitian Φ) for them, with Lagrangian
LχPT =
f2
8
Tr(∂µΣ∂
µΣ†) + higher order terms. (2.101)
Since the operators in the chiral effective Lagrangian contain derivatives, the contributions
from higher-order operators and from loops are suppressed by factors of p2/f2. This gives
the theory its predictive power.
In nature, the u-, d- and possibly s- quark masses are small enough so that one has an
approximate SU(3)L × SU(3)R symmetry. For Nf = 3, the field Φ can be parametrised
as
Φ =
pi0/
√
2 + η/
√
6 pi+ K+
pi− −pi0/
√
2 + η/
√
6 K0
K− K¯0 −2η/
√
6
. (2.102)
To lowest order, non-zero quark masses can be included in the chiral Lagrangian through
the term
−
f2
4
B0Tr
[
Mq(Σ + Σ
†)
]
, (2.103)
where B0 is a new low-energy coefficient and Mq = diag(mu,md,ms). Expanding Σ =
exp(2iΦ/f) around Φ = 0, Eq. (2.103) then gives a trivial constant term and, to quadratic
order,
B0Tr
[
MqΦ
2] = B0
[
(mu +md)pi
+pi− + (mu +ms)K
+K− + (md +ms)K
0K¯0
+ 16(4ms +md +mu) η η +
1√
3
(mu −md) η pi
0 + 12(mu +md)pi
0pi0
]
,
(2.104)
from which we can read off the leading-order masses of the 8 pseudo-Goldstone bosons,
m2pi± = B0(mu +md), etc. (2.105)
This is a first prediction from χPT. Calculations to order p4 (this includes one-loop cor-
rections) have been performed in [80]. At order p4, there are new low-energy coeffi-
cients and the results for masses, decay constants etc. include logarithms of the form
m2PS
(4pif)2 ln(m
2
PS/µ
2) where mPS is the mass of a Goldstone boson12. Like the chiral logs,
the low-energy-coefficients also depend on the scale µ, so that the overall scale-dependence
is cancelled.
12When dimensional regularisation is used, no power-law divergences appear. When a momen-
tum cutoff (or a lattice) is used as a regulator, the power-law divergences can still be absorbed
into the couplings of the given order.
2.8. Relativistic heavy quarks on the lattice 29
With many lattice actions, chiral symmetry is broken explicitly by lattice artefacts, so
that there will be additive terms in (2.105) that do not vanish for mq = 0. The effects of
non-zero lattice spacing can be systematically included in the χPT Lagrangian.
Note that it is also possible to study the interactions of (pseudo-)Goldstone bosons
with heavy particles (e.g. baryons) in χPT, as long as the momenta of the Goldstone
bosons involved are small compared to the scale of chiral symmetry breaking.
2.7.3 Finite-volume effects
As we have seen in Sec. 2.1.1, a finite temporal box size Lτ (with the correct boundary
conditions) corresponds to a non-zero temperature. If one is interested in zero-temperature
physics, one typically chooses Lτ about 2 - 4 times bigger than the spatial extent L. The
dominant finite-volume errors then arise from the finite spatial size.
In a simplistic picture, a single hadron in a periodic box much larger than the hadron
itself interacts with the infinite set of its mirror images. The interaction proceeds through
the exchange of the lightest excitations, the pions (in computations with dynamical quarks),
and so we expect the effect to be suppressed by e−mpiL.
Finite-volume effects can be studied systematically in chiral perturbation theory, us-
ing the same chiral Lagrangian as in infinite volume but performing the effective-theory
calculations in the finite box [81, 82]. The pion propagator then has to satisfy the periodic
boundary conditions, and can be written as an infinite sum over shifted infinite-volume
propagators.
As already mentioned in Sec. 2.2, the effects of finite volume can also be expressed
in terms of scattering parameters, and can in turn be used to compute these parameters
[36, 37].
2.8 Relativistic heavy quarks on the lattice
As seen in section 2.7.1, the usual relativistic lattice actions develop large discretisation
errors when am ? 1. A possible remedy would be to simply reduce the lattice spacing a.
For the charm quark, which hasmc(µ = mc) = 1.27
+0.07
−0.11 GeV in the MS scheme [83], this is
already feasible today, in particular with the help of Symanzik improvement. An example
is the Highly Improved Staggered Quark (HISQ) action [84], which allows calculations up
to about am ∼ 0.5. In [85], the HISQ action has been used to make precision calculations
of the D and Ds leptonic decay constants. With this approach, the same action can be
30 CHAPTER 2. LATTICE QCD
used for the charm quarks and the light quarks, and a partially conserved current can be
employed to calculate the decay constants.
In [86], the application of the HISQ action to bottom quarks is explored. The b quark
mass is mb(µ = mb) = 4.20
+0.17
−0.07 GeV [83]. The MILC collaboration [87] have generated
gauge configurations with lattice spacings in the range from a ≈ 0.18 fm (a−1 ≈ 1.1
GeV) to a ≈ 0.045 fm (a−1 ≈ 4.4 GeV). Even with HISQ, this is still not fine enough to
accommodate the b quark, and extrapolations in mb are required which lead to additional
systematic errors. Given that autocorrelation times grow dramatically as a is reduced
[75, 73], it appears that in the near future fine enough lattices with low pion masses and
large volumes will not be available.
One option for the inclusion of relativistic b quarks in lattice QCD is to use anisotropic
lattices, choosing the temporal lattice spacing at much finer than the spatial lattice spacing
as so that atm¿ 1 [88, 89]. From Eq. (2.100) one might expect that for heavy-light mesons
at rest the errors only appear as powers of atm. However, it turns out that errors governed
by asm can still appear [90]. See also [91] for a more recent paper.
Alternatively, one can work with isotropic lattices, but tune the couplings of the tem-
poral and spatial terms in the heavy-quark action separately so that the dispersion relation
for the heavy quark is corrected [92]. This is known as the “Fermilab method”. Additional
higher-dimension operators need to be included in order to achieve good accuracy. The
kernel for the simplest action has the form
KFNAL(x, x
′) = δx′, x − κt
[
(1− γ0)U0(x)δx′, x+0ˆ + (1 + γ
0)U−0(x)δx′, x−0ˆ
]
− κs
3∑
j=1
[
(rs − γ
j)Uj(x)δx′, x+jˆ + (rs + γ
j)U−j(x)δx′, x−jˆ
]
+ 2 cE κt [γ
0, γj ]Ej(x) δx′, x + cB κs ²jkl[γ
j , γk]Bl(x) δx′, x
(2.106)
where for convenience the fields are normalised such that the coefficient of the first
Kronecker-delta is 1. In (2.106), Ej and Bj are the chromoelectric and chromomagnetic
components of a lattice discretisation of Fµν . The bare mass is am0 = 1/(2κt)− 1− 3rsζ
with ζ = κs/κt. The parameters rs, ζ, cE , cB must be tuned as functions of am0. Note
that for rs = ζ = 1, cE = cB = 0, the Fermilab action reduces to the standard Wilson
action.
Chapter 3
Effective Lagrangians for heavy
quarks
Effective theories for heavy quarks are based on the separation of scales |k| ¿ m, where
m is the mass of the heavy quark and |k| is the typical magnitude of the spatial momenta
relevant for the system, defined in the rest frame of the hadron containing the heavy
quark(s). One important consequence of this separation of scales is that the effects of
heavy-quark pair production can be neglected or described by local effective operators.
The power counting will be discussed in more detail in Sec. 3.2. It turns out to be different
for hadrons containing more than one and hadrons containing only one heavy quark.
Heavy-quark effective theories can be used to analyse the behaviour of Wilson-like
lattice fermion actions in the large-mass limit [93, 94, 95]. Alternatively, one can directly
use a discretisation of a nonrelativistic effective Lagrangian to describe heavy quarks on
the lattice. The latter method is used in this dissertation.
In this chapter, I shall review the derivation of the standard heavy-quark effective
Lagrangians in the continuum. I will then perform an independent derivation of a “moving
nonrelativistic QCD” (mNRQCD) Lagrangian, obtaining a new result slightly different
from that reported previously in the literature. Two approaches to the lattice discretisation
of heavy-quark effective theories will then be discussed in Chapters 4 and 5.
One way to derive an effective Lagrangian is to write down all local operators com-
patible with the symmetries and required for the physical system under consideration.
The short-distance coefficients of the operators are adjusted so that the effective theory
and QCD give the same results for on-shell physical observables to the given order. This
approach is applied to heavy quarkonia in [9, 10] and goes back to [96]. If performed cor-
32 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
rectly, it has the advantage that no operators are missed out, but the method is technically
not so straightforward.
Another derivation of a continuum effective Lagrangian, appropriate to heavy-light
systems, is to “integrate out” the antiquark components of the theory in the path integral
approach [97]. This is usually done at tree-level only, and a matching to full QCD is
performed afterwards.
Here I will use the method of the Foldy-Wouthuysen-Tani (FWT) transformation,
following [98], where the effective Lagrangian is obtained via a field transformation. This
is a systematic procedure, which gives all operators bilinear in the heavy-quark fields
together with their tree-level coefficients. By construction, it also gives the tree-level
relation between the heavy quark fields in QCD and the ones in the effective Lagrangian.
3.1 Foldy-Wouthuysen-Tani transformation
In this section we derive a classical (tree-level) continuum effective Lagrangian for heavy
quarks order by order in 1/m via the Foldy-Wouthuysen-Tani (FWT) transformation,
which decouples the particle- and antiparticle components in the Dirac Lagrangian to a
given order1. The particle/antiparticle projectors are
P± =
1
2
(1± γˆ0), (3.1)
and hence the transformation should remove all terms from the Lagrangian that do not
commute with γˆ0. In the following, we work in Minkowski space. Writing the Dirac
Lagrange density as
L = Ψ(−m+ iγˆ0D0 + iγˆ
jDj)Ψ, (3.2)
we see that we must aim to cancel the iγˆjDj term. To this end, we introduce the field
redefinition
Ψ = exp
(
1
2m
iγˆjDj
)
Ψ(1),
Ψ = Ψ(1) exp
(
1
2m
iγˆjDj
)
= Ψ(1) exp
(
−
1
2m
iγˆj
←
Dj
)
, (3.3)
1The expansion parameter 1/m is purely formal at this stage. It turns out that for heavy-light
hadrons, the expansion really is in ΛQCD/m (cf. Sec. 3.2), and consequently the FWT method is
particularly convenient for these. The method is also correct for heavy-heavy mesons, but there
the power counting is different.
3.1. Foldy-Wouthuysen-Tani transformation 33
which does indeed cancel iγˆjDj , but also results in an infinite set of additional terms with
higher powers of 1/m. In terms of the new field Ψ(1), the Lagrangian reads
L = Ψ(1)(−m+ iγˆ
0D0)Ψ(1) +
∞∑
n=1
1
mn
Ψ(1) O(1)n Ψ(1). (3.4)
The new terms O(1)n (n ≥ 1) still contain pieces with odd powers of spatial gamma
matrices and thus do not commute with γˆ0. The order 1/m term is given by
O(1)1 = −
1
2
DjD
j −
ig
8
[γˆµ, γˆν ]Fµν
= −
1
2
DjD
j −
ig
8
[γˆj , γˆk]Fjk
︸ ︷︷ ︸
=OC(1)1
−
ig
2
γˆj γˆ0Fj0
︸ ︷︷ ︸
=OA(1)1
. (3.5)
Here, we wrote O(1)1 = O
C
(1)1 + O
A
(1)1, with
[
OC(1)1 , γˆ
0
]
= 0 and
{
OA(1)1 , γˆ
0
}
= 0. The
anticommuting part OA(1)1 can now be cancelled by a second field redefinition, similar to
(3.3),
Ψ(1) = exp
(
1
2m2
OA(1)1
)
Ψ(2),
Ψ(1) = Ψ(2) exp
(
1
2m2
OA(1)1
)
, (3.6)
which results in
L = Ψ(2)(−m+ iγˆ
0D0)Ψ(2) +
∞∑
n=1
1
mn
Ψ(2) O(2)n Ψ(2) (3.7)
with
OC(2)1 = O
C
(1)1,
OA(2)1 = 0,
OC(2)2 = −
g
8
γˆ0
(
Dadj Fj0 −
1
2
[γˆj , γˆk] {Dj , Fk0}
)
,
OA(2)2 = −
i
3
γˆj γˆkγˆlDjDkDl −
g
4
γˆj [D0, Fj0]. (3.8)
Next, to cancel OA(2)2, we write
Ψ(2) = exp
(
1
2m3
OA(2)2
)
Ψ(3),
Ψ(2) = Ψ(3) exp
(
1
2m3
OA(2)2
)
. (3.9)
34 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
The Lagrangian becomes
L = Ψ(3)(−m+ iγˆ
0D0)Ψ(3) +
∞∑
n=1
1
mn
Ψ(3) O(3)n Ψ(3) (3.10)
with
OC(3)1 = O
C
(2)1,
OA(3)1 = 0,
OC(3)2 = O
C
(2)2,
OA(3)2 = 0. (3.11)
Clearly, this procedure can be repeated to remove the γˆ0-anticommuting terms to arbitrar-
ily high but finite order in 1/m. Note that no additional time derivatives are introduced.
We will only keep the terms up to order 1/m2 now.
As these terms commute with γˆ0 and there are no time derivatives other than iγˆ0D0,
we can now remove the mass term2 from the Lagrangian by redefining
Ψ(3) = exp
(
−imx0γˆ0
)
Ψ˜,
Ψ(3) = Ψ˜ exp
(
imx0γˆ0
)
. (3.12)
The resulting Lagrangian is
L = Ψ˜
[
iγˆ0D0 −
1
2m
DjD
j −
ig
8m
[γˆj , γˆk]Fjk
−
g
8m2
γˆ0
(
Dadj Fj0 −
1
2
[γˆj , γˆk] {Dj , Fk0}
)]
Ψ˜ + O(1/m3). (3.13)
Using [γˆj , γˆk] = −2i²jklΣl with
Σj ≡
(
σj 0
0 σj
)
, (3.14)
and defining the chromoelectric and chromomagnetic components of the gluon field strength
tensor as Ej = F0j and Bj = −12²jklFkl , we obtain
L = Ψ˜
[
iγˆ0D0 +
D2
2m
+
g
2m
Σ·B+
g
8m2
γˆ0
(
Dad ·E+ iΣ · (D×E−E×D)
)]
Ψ˜
+ O(1/m3). (3.15)
2This step is not really necessary, since the mass term only corresponds to a trivial shift in the
energy. In fact, a mass term is generated as a counterterm through renormalisation.
3.1. Foldy-Wouthuysen-Tani transformation 35
For later reference, we also summarise the field transformation:
Ψ(x) = TFWT e
−imx0γˆ0 Ψ˜(x) (3.16)
with
TFWT = exp
[
1
2m
(iγˆ ·D)
]
× exp
[
1
2m2
(
−
ig
2
γˆ0γˆ ·E
)]
× exp
[
1
2m3
(
g
4
γˆ · (Dad0 E) +
1
3
(iγˆ ·D)3
)]
×
[
1 +O(1/m4)
]
. (3.17)
Next, introducing two-component fields ψ and ξ as
Ψ˜ =
(
ψ
ξ
)
, Ψ˜ =
(
ψ†, −ξ†
)
, (3.18)
we find that the Lagrangian (3.15) can be written as
L = ψ†
[
iD0 +
D2
2m
+
g
2m
σ ·B+
g
8m2
(
Dad ·E+ iσ · (D×E−E×D)
)]
ψ
+ ξ†
[
iD0 −
D2
2m
−
g
2m
σ ·B+
g
8m2
(
Dad ·E+ iσ · (D×E−E×D)
)]
ξ
+ O(1/m3). (3.19)
As desired, the quark field ψ and the antiquark field ξ are separated. In our conventions,
the field operator related to ψ annihilates a quark, whereas the field operator related to ξ
creates an antiquark. Note that the antiquark action Sξ =
∫
Lξ d4x can be obtained from
the quark action Sψ =
∫
Lψ d4x by making the following replacements:
ψ 7→ (ξ†)T ,
ψ† 7→ (ξ)T ,
iAµ 7→ (iAµ)
∗ = − (Aµ)
T ,
iEj 7→ (iEj)
∗ = − (Ej)
T ,
iBj 7→ (iBj)
∗ = − (Bj)
T ,
iσj 7→
(
iσj
)∗
= −
(
σj
)T
, (3.20)
where T denotes the transpose (in colour and/or spin space) and ∗ complex conjugation,
i.e. the antiquark field transforms under the complex conjugate representations in colour
36 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
and spin space. To see this, integrate by parts and take the transpose of the entire action
(the latter also introduces an overall minus sign, since the Grassmann-valued fields ξ and
ξ† are interchanged).
It is important to make the following remark: In the quantum field theory functional
integral, one integrates over arbitrary non-smooth fields. However, the FWT transforma-
tions e.g. contain derivative operators acting on these fields. Of course, the path integral
is only defined with an ultraviolet regulator such as the lattice, but one can not expect
that the field transformations commute with the UV regularisation and renormalisation.
The form of the local operators obtained through the FWT transformation is still correct,
but after (lattice) regularisation their coefficients must be renormalised away from their
tree-level values to take into account the short-distance (UV) effects.
The derivation given here also misses out operators which are not bilinear in the heavy-
quark fields, like four-fermion operators of the form
ψ† ξ ξ† ψ. (3.21)
In perturbation theory, these operators arise at 1-loop level and therefore have coefficients
suppressed by an additional factor of α2s [10]. Hence, these operators are far less important
than the bilinear operators, and can be neglected here.
3.2 Power counting and heavy-quark symmetries
3.2.1 Heavy-light hadrons
For mesons and baryons containing only a single heavy quark, the light degrees of freedom
are governed by the nonperturbative gluon dynamics and are thus characterised by the
scale ΛQCD. It follows that the typical momentum transfer between the heavy quark and
the light degrees of freedom is also of order ΛQCD. Thus, if the hadron is at rest, the
covariant derivative acting on the effective heavy-quark field is of order
|D0| ∼ |D| ∼ ΛQCD. (3.22)
The gluon gauge potential is also determined by the light degrees of freedom, and hence,
in a smooth gauge,
|gA0| ∼ |gA| ∼ ΛQCD (3.23)
and
|g E| ∼ Λ2QCD, |g B| ∼ Λ
2
QCD. (3.24)
3.2. Power counting and heavy-quark symmetries 37
By simple dimensional analysis it follows that for a heavy-light hadron at rest a term in
the Lagrangian that comes with 1/mn will be suppressed by a factor of
(
ΛQCD
m
)n
(3.25)
relative to the leading term. Consequently, (3.19) is accurate up to corrections of order
(ΛQCD/m)3. In the static limit
m→∞ (3.26)
the effective Lagrangian (3.19) therefore reduces to
L = ψ† iD0 ψ, (3.27)
(here we have set ξ = 0) and the heavy quark only acts as a static colour source. It is now
obvious that the physics of the heavy-light hadron in this limit does not depend on the
flavour or the spin of the heavy quark. This leads to the heavy-quark flavour symmetry
and the heavy-quark spin symmetry [99, 97], which are broken by corrections proportional
to ΛQCD/m (the flavour symmetry breaking is proportional to ΛQCD/m1 − ΛQCD/m2 for
two heavy-light hadrons with heavy quark masses m1 and m2).
3.2.2 Heavy-heavy mesons
The situation is different for heavy-heavy mesons (quarkonia). Naively speaking, the
heavy quark and antiquark orbit around their centre of mass with some velocity vorb.. For
bottomonium one has v2orb. ≈ 0.1 and for charmonium v
2
orb. ≈ 0.3 [9]. These estimates are
obtained by assuming that the average kinetic energy is of the order of the 2S−1S energy
splitting; the values for v2orb. calculated in this way are consistent with predictions from
potential models.
The spatial momentum and the kinetic energy of the orbital motion are of order
|k| ∼ m vorb., Ekin ∼ m v
2
orb.. (3.28)
It follows that the spatial covariant derivative acting on the heavy-quark field is of size
|D| ∼ m vorb.. (3.29)
At the short distances relevant for quarkonium, the quark-antiquark potential is similar
to the Coulomb potential. Thus, the kinetic energy is of the same order of magnitude as
the potential energy, and so we have (in Coulomb gauge)
|g A0| ∼ |D0| ∼ Ekin ∼ m v
2
orb.. (3.30)
38 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
Using the Yang-Mills equations, one can then show [10] that the vector potential is of
order
|gA| ∼ m v3orb., (3.31)
and hence
|g E| ∼ m2v3orb., |g B| ∼ m
2v4orb.. (3.32)
We now see that the individual terms in the effective Lagrangian are suppressed by various
powers of the internal velocity vorb., and this is the correct expansion parameter. All terms
in (3.15) are of order v4orb. or lower powers of vorb., but one term of order v
4
orb. is missing.
By expanding the expression for the relativistic kinetic energy in powers of the momentum
k,
Ekin =
√
k2 +m2 −m
=
k2
2m
−
k4
8m3
+
k6
16m5
− ... (3.33)
we see that we must include the term
Lkin,m−3 = Ψ˜
D4
8m3
Ψ˜ (3.34)
into the Lagrangian to obtain accuracy to order v4orb.. For more details of the heavy-heavy
power counting, see [10].
3.3 Generalisation to a moving frame of reference
The effective Lagrangian obtained in section 3.1 can be used to describe a heavy quark
with small spatial momentum k, that is |k| ¿ m. Note that the total 4-momentum pµ
can be written in terms of the residual momentum kµ as
pµ = m uµ + kµ (3.35)
with the trivial 4-velocity u = (1, 0, 0, 0). We will now work in a Lorentz-boosted frame
of reference moving with an arbitrary velocity v so that u = (γ, γv) with γ = 1/
√
1− v2.
Under the coordinate transformation
x′µ = Λµνx
ν , (3.36)
the original Dirac field in (3.2) transforms as follows:
Ψ′(x′) = S(Λ)Ψ(x), (3.37)
3.3. Generalisation to a moving frame of reference 39
where S(Λ) is the Dirac-spinor representation of Λ. Writing Ψ(x) in terms of the effective
field Ψ˜(x) using (3.16), we get
Ψ′(x′) = S(Λ) exp
(
1
2m
iγˆjDj
)
... exp
(
−imx0γˆ0
)
Ψ˜(x), (3.38)
where the ellipsis indicates the higher-order parts of the FWT transformation. Using
S(Λ) γˆµ = (Λ−1)µν γˆ
ν S(Λ), (3.39)
the matrix S(Λ) in (3.38) can be moved to the right:
Ψ′(x′) = exp
(
1
2m
i(Λ−1)j µγˆ
µDj
)
... exp
(
−imx0(Λ−1)0 ν γˆ
ν) S(Λ) Ψ˜(x). (3.40)
Next, we define a new field
Ψ˜′(x′) ≡ S(Λ) Ψ˜(x), (3.41)
and express (3.40) in terms of the coordinates x′. This finally gives
Ψ′(x′) = exp
(
1
2m
i(Λ−1)j µγˆ
µΛν jD
′
ν
)
... exp
(
−imΛ0λx
′λ(Λ−1)0 ργˆ
ρ
)
Ψ˜′(x′)
= exp
(
1
2m
i( /D′ − /u u ·D′)
)
... exp
(
−im u · x′ /u
)
Ψ˜′(x′), (3.42)
where /u ≡ uµγˆµ. In terms of the new field Ψ˜′(x′), the effective Lagrangian (3.13) reads,
through order 1/m,
L = Ψ˜
(
iγˆ0D0 −
1
2m
DjD
j −
ig
8m
[γˆj , γˆk]Fjk
)
Ψ˜
= Ψ˜′ S(Λ)
(
iγˆ0Λµ0D
′
µ −
1
2m
ΛµjΛ
j
νD
′
µD
′ν −
ig
8m
[γˆj , γˆk]ΛµjΛ
ν
kF
′
µν
)
S−1(Λ) Ψ˜′
= Ψ˜′
(
i /u u ·D′ −
D′2⊥
2m
−
1
4m
[γˆµ, γˆν ]D
′µ
⊥D
′ν
⊥
)
Ψ˜′
= Ψ˜′
(
i /u u ·D′ −
/D′2⊥
2m
)
Ψ˜′. (3.43)
Here we have used the notation
D′µ⊥ ≡ D
′µ − uµ u ·D′. (3.44)
To separate the particle- and antiparticle components in (3.43), we now have to use the
boosted projectors
P (u)± = S(Λ) P± S
−1(Λ) =
1± /u
2
. (3.45)
40 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
Defining Ψ˜′± ≡ P
(u)
± Ψ˜
′, we obtain
L = Ψ˜′+
(
+i u ·D′ −
/D′2⊥
2m
)
Ψ˜′+ + Ψ˜′−
(
−i u ·D′ −
/D′2⊥
2m
)
Ψ˜′− + O(1/m). (3.46)
The part with Ψ˜′+ in (3.46) is the standard continuum HQET (heavy-quark effective
theory) Lagrangian for a non-zero frame velocity [97].
3.4 Moving NRQCD Lagrangian
As a consequence of the Lorentz boost, the terms in the Lagrangian (3.43) commute
with /u′ rather than γ0 and the spatial derivatives are now accompanied by temporal
derivatives. For applications in lattice QCD, it is more convenient to introduce a different
form of the boosted Lagrangian that retains commutativity with γ0 and does not contain
time derivatives in the 1/m correction terms [19, 20]. This formalism is called Moving
nonrelativistic QCD (mNRQCD).
The commutativity with γ0 can be restored by simply removing the factor S(Λ) in the
definition of the field Ψ˜′, Eq. (3.41), so that
Ψ˜′(x′) ≡ Ψ˜(x). (3.47)
The relation between Ψ′ and Ψ˜′ then reads
Ψ′(x′) = S(Λ) exp
(
1
2m
iγjΛµjD
′
µ
)
... exp
(
−im u · x′ γ0
)
Ψ˜′(x′), (3.48)
and the Lagrangian becomes
L = Ψ˜′
[
iγ0u ·D′ +
(u ·D′)2 −D′2
2m
−
ig
8m
[γj , γk]Λρ jΛ
σ
kF
′
ρσ
−
g
8m2
γ0
(
Λν jΛ
ρ
ju
σ(D′adν F
′
ρσ)−
1
2
[γj , γk]Λν jΛ
ρ
ku
σ {D′ν , F
′
ρσ
}
)]
Ψ˜′
+ O(1/m3). (3.49)
Since we will work exclusively in the x′ frame from now on, it is convenient to rename the
coordinates such that
x′ → x
x → x′. (3.50)
3.4. Moving NRQCD Lagrangian 41
This means that in (3.49) we rename Ψ˜′ → Ψ˜, Ψ˜′ → Ψ˜, D′µ → Dµ, F
′
µν → Fµν . With the
new notation and after some additional algebraic steps we have
L = Ψ˜
[
iγˆ0u ·D +
(u ·D)2 −D2
2m
+
g
2m
Σ ·B′
+
g
8m2
γˆ0
(
Dadµ uνF
µν + i²jklΣ
jΛµk
{
Dµ, E
′
l
})
]
Ψ˜
+ O(1/m3). (3.51)
Here, we wrote B′j = −
1
2²jklΛ
µ
kΛ
ν
lFµν and E
′
j = Λ
µ
ju
νFµν where convenient to simplify
the notation. In terms of the fields Bj and Ej , these relations read
B′ = γ
(
B− v ×E−
γ
1 + γ
v (v ·B)
)
,
E′ = γ
(
E+ v ×B−
γ
1 + γ
v (v ·E)
)
. (3.52)
We now need to eliminate the time derivatives in the operators of order 1/m and 1/m2 in
(3.51). This can be done via further field redefinitions [19, 22].
It is convenient to write the Lagrangian (3.51) in the following form,
L = γ Ψ˜
[
O0 +
1
γm
O1 +
1
(γm)2
O2
]
Ψ˜ +O(1/m3), (3.53)
with
O0 = iγˆ
0(D0 + v ·D),
O1 =
1
2
(
(u ·D)2 −D2
)
+
g
2
Σ ·B′,
O2 =
g
8
γ γˆ0
(
Dadµ uνF
µν + i²jklΣ
jΛµk
{
Dµ, E
′
l
})
. (3.54)
We start by removing the time derivatives in O1. To see how this can be done, we note
that any field redefinition of the form
Ψ˜ = exp
(
1
γm
U
)
Ψ˜(1),
Ψ˜ = Ψ˜(1) exp
(
1
γm
U
)
, (3.55)
will result in
L = γ Ψ˜(1)
[
O0 +
1
γm
O(1)1 +
1
(γm)2
O(1)2
]
Ψ˜(1) +O(1/m
3), (3.56)
42 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
with the new operators
O(1)1 = O1 + {U, O0} ,
O(1)2 = O2 + {U, O1}+ UO0U +
1
2
{
U2, O0
}
. (3.57)
Thus, we need to write O1 = O(1)1 − {U, O0} with some operator U such that O(1)1 does
not contain time derivatives. This is indeed possible,
O1 =
1
2
[
γ2D20 + γ
2 {D0, v ·D}+ γ2(v ·D)2 −D20 +D
2]+
g
2
Σ ·B′
=
1
2
[
D2 − (v ·D)2
]
+
g
2
Σ ·B′
︸ ︷︷ ︸
≡ O(1)1
+
1
2
[
(γ2 − 1)D20 + γ
2 {D0, v ·D}+ (γ2 + 1)(v ·D)2
]
︸ ︷︷ ︸
= − {U, O0}
, (3.58)
and we can now read off the operator U :
U =
i
4
γˆ0
[
(γ2 − 1)D0 + (γ
2 + 1)v ·D
]
. (3.59)
The next step is to remove the time derivatives (other than the adjoint time derivative,
which acts on the gluon field strength only) in the new operator O(1)2, given in (3.57).
Similarly to before, we use a field redefinition
Ψ˜(1) = exp
(
1
(γm)2
V
)
Ψ˜(2),
Ψ˜(1) = Ψ˜(2) exp
(
1
(γm)2
V
)
, (3.60)
now with an extra power of 1/(γm), so that the lower order terms are unaffected. The
transformation (3.60) results in
L = γ Ψ˜(2)
[
O0 +
1
γm
O(2)1 +
1
(γm)2
O(2)2
]
Ψ˜(2) +O(1/m
3) (3.61)
with
O(2)1 = O(1)1,
O(2)2 = O(1)2 + {V, O0} . (3.62)
We need to write O(1)2 = O(2)2 − {V, O0} with some operator V such that O(2)2 does not
contain time derivatives. We will treat the terms in O(1)2 (Eq. (3.57)) individually. Note
3.4. Moving NRQCD Lagrangian 43
that the last term, −
{
−12U
2, O0
}
, is already in the desired form. The time-derivative in
the original O2, defined in (3.54), can be treated as follows:
ig
8
γ γˆ0²jklΣ
jΛ0 k
{
D0, E
′
l
}
= −
ig
8
γ γˆ0²jklΣ
jΛ0 k
{
v ·D, E′l
}
−
{
−
g
8
γ ²jklΣ
jΛ0 kE
′
l, O0
}
. (3.63)
Next, using
U =
1
4
(γ2 − 1)O0 +
i
2
γˆ0 v ·D (3.64)
we obtain
UO0U =
1
2
{
U2, O0
}
+
1
2
[U, [O0, U ]]
=
1
2
{
U2, O0
}
+
1
2
[
U,
[
O0,
1
4
(γ2 − 1)O0 +
i
2
γˆ0 v ·D
]]
=
1
2
{
U2, O0
}
+
1
2
[
U,
[
iγˆ0(D0 + v ·D),
i
2
γˆ0 v ·D
]]
=
1
2
{
U2, O0
}
−
1
4
[U, [D0,v ·D]]
=
1
2
{
U2, O0
}
−
i
16
[
γˆ0
(
(γ2 − 1)D0 + (γ
2 + 1)v ·D
)
, igv ·E
]
= −
{
−
1
2
U2, O0
}
+
g
16
γˆ0
(
(γ2 − 1)Dad0 + (γ
2 + 1)v ·Dad
)
(v ·E) (3.65)
and
{U, O1} =
{
1
4
(γ2 − 1)O0 +
i
2
γˆ0 v ·D, O1
}
=
{
i
2
γˆ0 v ·D, O1
}
−
{
−
1
4
(γ2 − 1)O1, O0
}
=
{
i
2
γˆ0 v ·D, O(1)1
}
−
{
i
2
γˆ0 v ·D, {U, O0}
}
−
{
−
1
4
(γ2 − 1)O1, O0
}
=
{
i
2
γˆ0 v ·D, O(1)1
}
+
[
U,
[
i
2
γˆ0 v ·D, O0
]]
−
{{
i
2
γˆ0 v ·D, U
}
−
1
4
(γ2 − 1)O1, O0
}
. (3.66)
Let us now consider the nested commutator in (3.66):
[
U,
[
i
2
γˆ0 v ·D, O0
]]
=
[
U,
[
i
2
γˆ0 v ·D, iγˆ0D0
]]
=
[
i
4
γˆ0
(
(γ2 − 1)D0 + (γ
2 + 1)v ·D
)
,
ig
2
v ·E
]
= −
g
8
γˆ0
(
(γ2 − 1)Dad0 + (γ
2 + 1)v ·Dad
)
(v ·E) . (3.67)
44 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
The adjoint time derivative Dad0 E acts only on the chromoelectric field and can be kept.
We conclude from (3.57), (3.63), (3.65) and (3.66) that
V = −
g
8
γ ²jklΣ
jΛ0 kE
′
l +
{
i
2
γˆ0 v ·D, U
}
−
1
4
(γ2 − 1)O1 − U
2 (3.68)
and
O(2)2 =
g
8
γ γˆ0
(
Dadµ uνF
µν + i²jklΣ
jΛmk
{
Dm, E
′
l
}
− i²jklΣ
jΛ0 k
{
v ·D, E′l
}
)
−
g
16
γˆ0
(
(γ2 − 1)Dad0 + (γ
2 + 1)v ·Dad
)
(v ·E)
+
i
4
γˆ0
{
v ·D, D2 − (v ·D)2 + gΣ ·B′
}
=
g
8
γ2γˆ0
(
Dad ·E− v · (Dad ×B)
)
+
ig
8
γγˆ0Σ ·
(
D×E′ −E′ ×D
)
−
igγ2
8(1 + γ)
γˆ0
{
v ·D, Σ · (v ×E′)
}
+
i
4
γˆ0
({
v ·D, D2
}
− 2(v ·D)3
)
+
ig
4
γˆ0
{
v ·D, Σ ·B′
}
+
(2− v2)gγ2
16
γˆ0
(
Dad0 − v ·D
ad
)
(v ·E) . (3.69)
We have now achieved the goal of removing the time derivatives through O(1/m2). Finally,
we rescale the fields
Ψ˜(2) =
1
√
γ
Ψv,
Ψ˜(2) =
1
√
γ
Ψv, (3.70)
to remove the factor of γ in front of L. We arrive at the following result for the tree-level
moving NRQCD Lagrangian in Minkowski space:
L = Ψv
[
iγˆ0D0 + iγˆ
0v ·D+
D2 − (v ·D)2
2γm
+
g
2γm
Σ ·B′
+
i
4γ2m2
γˆ0
({
v ·D, D2
}
− 2(v ·D)3
)
+
g
8m2
γˆ0
(
Dad ·E− v · (Dad ×B)
)
+
ig
8γm2
γˆ0 Σ ·
(
D×E′ −E′ ×D
)
−
ig
8(γ + 1)m2
γˆ0
{
v ·D, Σ · (v ×E′)
}
+
(2− v2)g
16m2
γˆ0
(
Dad0 − v ·D
ad
)
(v ·E) +
ig
4γ2m2
γˆ0
{
v ·D, Σ ·B′
}
]
Ψv
+ O(1/m3). (3.71)
As before, all terms commute with γˆ0. We can therefore introduce 2-component fields
ψv(x) and ξv(x),
Ψv =
(
ψv
ξv
)
, Ψv =
(
ψ†v, −ξ
†
v
)
, (3.72)
3.4. Moving NRQCD Lagrangian 45
to explicitly separate the Lagrangian into the quark and antiquark pieces (terms that
do not contain a factor of γˆ0 in (3.71) appear with the opposite sign in the antiquark
Lagrangian):
L = ψ†v
[
iD0 + iv ·D+
D2 − (v ·D)2
2γm
+
g
2γm
σ ·B′
+
i
4γ2m2
({
v ·D, D2
}
− 2(v ·D)3
)
+
g
8m2
(
Dad ·E− v · (Dad ×B)
)
+
ig
8γm2
σ ·
(
D×E′ −E′ ×D
)
−
ig
8(γ + 1)m2
{
v ·D, σ · (v ×E′)
}
+
(2− v2)g
16m2
(
Dad0 − v ·D
ad
)
(v ·E) +
ig
4γ2m2
{
v ·D, σ ·B′
}
]
ψv
+ ξ†v
[
iD0 + iv ·D−
D2 − (v ·D)2
2γm
−
g
2γm
σ ·B′
+
i
4γ2m2
({
v ·D, D2
}
− 2(v ·D)3
)
+
g
8m2
(
Dad ·E− v · (Dad ×B)
)
+
ig
8γm2
σ ·
(
D×E′ −E′ ×D
)
−
ig
8(γ + 1)m2
{
v ·D, σ · (v ×E′)
}
+
(2− v2)g
16m2
(
Dad0 − v ·D
ad
)
(v ·E) +
ig
4γ2m2
{
v ·D, σ ·B′
}
]
ξv
+ O(1/m3). (3.73)
A moving NRQCD Lagrangian of this order had already been presented in [22, 21, 23],
but my result (3.71, 3.73), which is published in [100], is slightly different.
For later reference, I now summarise the tree-level relation between the full QCD field
Ψ(x) and the 4-component moving NRQCD field (3.72). We have
Ψ(x) = S(Λ) TFWT e
−im u·x γˆ0 ADt
1
√
γ
Ψv (3.74)
(and correspondingly for Ψ) where TFWT is the FWT transformation (3.17) expressed in
the new frame of reference,
TFWT = exp
(
iγˆjΛµjDµ
2m
)
exp
(
igγˆ ·E′γˆ0
(2m)2
)
× ... , (3.75)
and ADt removes the unwanted time derivatives in the Lagrangian,
ADt = exp
(
U
γm
)
exp
(
V
(γm)2
)
× ... . (3.76)
The operators U and V in (3.76) were defined in Eqs. (3.59) and (3.68), respectively.
46 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
3.4.1 O(1/m3) relativistic correction in moving NRQCD
As explained in section 3.2.2, for heavy-heavy mesons one needs to include an additional
O(1/m3) term in order to achieve accuracy to order O(v4orb.). Instead of performing all
the field redefinitions to this order, we again obtain this term by expanding the kinetic
energy expression in powers of the residual momentum k,
Ekin =
√
(γmv + k)2 +m2 − γm
= v · k+
1
2γm
(
k2 − (v · k)2
)
+
1
4γ2m2
(
−{v · k, k2}+ 2(v · k)3
)
+
1
8γ3m3
(
−k4 + 3
{
k2, (v · k)2
}
− 5(v · k)4
)
+ ... (3.77)
Here, we ordered the terms with products of (v · k) and k2 in form of anticommutators,
since we are now going to replace k by the operator −iD, and the anticommutator-ordering
is what one would have obtained from field redefinitions. Indeed, we get the same results
up to O(1/m2) and the following O(1/m3) term:
Lkin,m−3 = −Ψv
1
8γ3m3
(
−D4 + 3
{
D2, (v ·D)2
}
− 5(v ·D)4
)
Ψv. (3.78)
Note that for heavy-heavy mesons the power counting in mNRQCD is modified when the
frame velocity is close to the speed of light (i.e. |v| → 1 in our units). The details are given
in [100]. For the moderate velocities used in the numerical calculations in the following
chapters this effect is negligible.
3.5 Euclidean mNRQCD Lagrangian
The Euclidean moving NRQCD Lagrangian can be obtained by following the general
procedure discussed in Sec. 2.1. It is also convenient to define the relation between the
3-dimensional chromoelectric field E and the 4-dimensional field strength tensor Fµν with
a different sign in Euclidean space, i.e.
Ej = −F0j . (3.79)
With this definition, Eq. (3.52) turns into the symmetric form
B′ = γ
(
B+ iv ×E−
γ
1 + γ
v (v ·B)
)
,
E′ = γ
(
E+ iv ×B−
γ
1 + γ
v (v ·E)
)
. (3.80)
3.5. Euclidean mNRQCD Lagrangian 47
The Euclidean Lagrangian, to which we now also include the O(1/m3) relativistic correc-
tion term (3.78), reads
L = ψ†v
[
D0 − iv ·D−
D2 − (v ·D)2
2γm
−
g
2γm
σ ·B′
−
i
4γ2m2
({
v ·D, D2
}
− 2(v ·D)3
)
+
g
8m2
(
iDad ·E+ v · (Dad ×B)
)
−
g
8γm2
σ ·
(
D×E′ −E′ ×D
)
+
g
8(γ + 1)m2
{
v ·D, σ · (v ×E′)
}
−
(2− v2)g
16m2
(
Dad0 + iv ·D
ad
)
(v ·E)−
ig
4γ2m2
{
v ·D, σ ·B′
}
−
1
8γ3m3
(
D4 − 3
{
D2, (v ·D)2
}
+ 5(v ·D)4
)]
ψv
+ ξ†v
[
D0 − iv ·D+
D2 − (v ·D)2
2γm
+
g
2γm
σ ·B′
−
i
4γ2m2
({
v ·D, D2
}
− 2(v ·D)3
)
+
g
8m2
(
iDad ·E+ v · (Dad ×B)
)
−
g
8γm2
σ ·
(
D×E′ −E′ ×D
)
+
g
8(γ + 1)m2
{
v ·D, σ · (v ×E′)
}
−
(2− v2)g
16m2
(
Dad0 + iv ·D
ad
)
(v ·E)−
ig
4γ2m2
{
v ·D, σ ·B′
}
+
1
8γ3m3
(
D4 − 3
{
D2, (v ·D)2
}
+ 5(v ·D)4
)]
ξv
+ O(1/m3). (3.81)
Naturally, the field redefinition (3.74) also needs to be continued to imaginary time. This
must be done independently for Ψ and Ψ.
Note that, as in (3.20), the antiquark action can be obtained from the quark action
through the replacements
ψv 7→ (ξ
†
v)
T ,
ψ†v 7→ (ξv)
T ,
iAµ 7→ (iAµ)
∗ ,
iEj 7→ (iEj)
∗ ,
iBj 7→ (iBj)
∗ ,
iσj 7→
(
iσj
)∗
. (3.82)
48 CHAPTER 3. EFFECTIVE LAGRANGIANS FOR HEAVY QUARKS
Remarkably, in Euclidean space there is also a much simpler prescription:
replace ψv 7→ (ξ
†
v)
T ,
ψ†v 7→ (ξv)
T ,
v 7→ (−v),
and take the complex conjugate of the whole action kernel. (3.83)
To see that the new prescription (3.83) is equivalent to (3.82), it is convenient to insert
factors of −(i2) = 1 into the Euclidean Lagrangian where appropriate such that it is
written explicitly in terms of the anti-Hermitian quantities iAµ, iE, iB and iσ. Then one
sees that (3.82) is equivalent to taking the complex conjugate of the whole action kernel,
apart from the remaining factors of i. Now, the crucial point is that precisely those terms
with additional factors of i come with an odd power of v. Thus, changing also the sign of
the velocity makes the new prescription work.
3.5.1 Euclidean quark and antiquark Green functions in mNRQCD
As shown in the previous section, in Euclidean space the antiquark action kernel equals
the complex conjugate of the quark action kernel with the opposite boost velocity,
Sψv =
∫
d4x
∫
d4x′
[
ψ†v(x)
]
cs
[
K(+v)(x, x′)
]
cs c′s′
[
ψv(x
′)
]
c′s′
, (3.84)
Sξv =
∫
d4x
∫
d4x′
[
ξv(x)
]
cs
[
K(−v)(x, x′)∗
]
cs c′s′
[
ξ†v(x
′)
]
c′s′
=
∫
d4x
∫
d4x′
[
ξv(x
′)
]
c′s′
[
K(−v)(x′, x)∗
]
c′s′ cs
[
ξ†v(x)
]
cs
=
∫
d4x
∫
d4x′
[
ξ†v(x)
]
cs
[
−K(−v)(x′, x)∗
]
c′s′ cs
[
ξv(x
′)
]
c′s′
. (3.85)
Here, we explicitly introduced the colour and spin indices c, c′, s, s′. We define the quark
and antiquark Green functions as
G(+v)ψv (x, x
′) = 〈 ψv(x) ψ
†
v(x
′) 〉,
G(+v)ξv (x, x
′) = 〈 ξv(x) ξ
†
v(x
′) 〉. (3.86)
From the relation between the actions Sψv and Sξv it follows that
[
G(+v)ξv (x, x
′)
]
cs c′s′
= −
[
G(−v)ψv (x
′, x)∗
]
c′s′ cs
= −
[
G(−v)ψv (x
′, x)†
]
cs c′s′
, (3.87)
that is, the Euclidean antiquark Green function can be obtained from the Euclidean quark
Green function with the opposite boost velocity.
Chapter 4
Lattice HQET
Recall from Sec. 3.2.1 that for hadrons containing only a single heavy quark of mass m
the contributions from the operators with factors of 1/mp are suppressed by (ΛQCD/m)p.
The leading-order Euclidean Lagrangian in the rest frame is simply
L(0)ψ = ψ
†D0 ψ. (4.1)
Eichten and Hill [101] suggested a simple lattice discretisation of (4.1), referred to as
lattice heavy-quark effective theory (LHQET). In LHQET, higher-order corrections to
(4.1) are treated as insertions in correlation function, so that the resulting theory may be
renormalisable. Even though LHQET is not used for the calculations in this dissertation,
we will discuss it briefly in this chapter. It will be instructive to compare it to lattice
nonrelativistic QCD, which will be introduced in Chapter 5.
4.1 Relativistic corrections as operator insertions
We first discuss the continuum theory, working to order 1/m and setting v = 0 for sim-
plicity. The effective Lagrangian has the form
Lψ = δm L
(−1)
ψ + L
(0)
ψ +
1
m
L(1)ψ (4.2)
where L(−1)ψ = ψ
†ψ is a mass counterterm, L(0)ψ is given by (4.1), and L
(1)
ψ reads
L(1)ψ = ψ
†
[
−
1
2
D2 −
g
2
σ ·B
]
ψ. (4.3)
Let us denote by SQCD the relativistic QCD action including the gluons Aµ and the light
quarks Ψ, Ψ. The full path integral with action S = SQCD+
∫
d4xLψ for the expectation
50 CHAPTER 4. LATTICE HQET
value of some observable O is
〈O〉 =
1
Z
∫
D[A,Ψ,Ψ, ψ, ψ†] O exp
(
−SQCD − δm S
(−1)
ψ − S
(0)
ψ −
1
mS
(1)
ψ
)
. (4.4)
However, in HQET one does not work with (4.4), but rather expands the exponential to
the given order in 1/m:
exp (−SQCD − Sψ) ≈ exp
(
−SQCD − δm S
(−1)
ψ − S
(0)
ψ
) (
1− 1mS
(1)
ψ
)
. (4.5)
Thus, one effectively calculates the expectation value
〈(
1− 1mS
(1)
ψ
)
O
〉(0)
=
1
Z
∫
D[A,Ψ,Ψ, ψ, ψ†]
(
1− 1mS
(1)
ψ
)
O
× exp
(
−SQCD − δm S
(−1)
ψ − S
(0)
ψ
)
, (4.6)
where the action used in the path integral is
SQCD + δm
∫
d4x ψ†ψ +
∫
d4x ψ†D0 ψ. (4.7)
Since this action contains all possible operators of dimension ≤ 4 compatible with the
symmetries, and no higher-order operators, it is renormalisable according to Weinberg’s
power-counting theorem [102]. The question then is whether the composite operators
in
(
1− 1mS
(1)
ψ
)
O can be renormalised. It is shown in [103] that the renormalisation of a
composite operator requires as counterterms only operators of the same or lower dimension.
When all these are included, the theory is formally renormalisable. It has been pointed
out in [104] that the renormalisation must be performed nonperturbatively. This shows
up in power-law divergences when using a momentum cut-off or the lattice regularisation,
but is also true when using dimensional regularisation where apparently no power-law
divergences exist.
In lattice HQET, nonperturbative renormalisation can be performed using Schro¨dinger
functional methods; see e.g. [72].
4.2 Continuum HQET Green functions
In the following, we consider the continuum HQET propagator to order 1/m on a given
background gauge-field. For x = (τ,x), x′ = (τ ′,x′) and τ > τ ′, we have, according to
Eq. (4.6),
G(1)ψ (x, x
′) =
1
Zψ
∫
D[ψ,ψ†] ψ(x)
{
1 +
∫
d4x′′ ψ†(x′′)
[
D2
2m
+
g σ ·B
2m
]
x′′
ψ(x′′)
}
ψ†(x′)
× exp
{
−
∫
d4y ψ†(y) (D0 + δm) ψ(y)
}
. (4.8)
4.2. Continuum HQET Green functions 51
Let us now perform the functional integral over the heavy-quark fields ψ,ψ†. This gives
G(1)ψ (x, x
′) = Gψ(τ,x, τ
′,x′)
+
∫ τ
τ ′
dτ ′′
∫
d3x′′Gψ(τ,x, τ
′′,x′′)
[
D2
2m
+
g σ ·B
2m
]
τ ′′,x′′
Gψ(x
′′, τ ′′, τ ′,x′),
(4.9)
where Gψ denotes the propagator according to the action
∫
d4x ψ†(x) (D0 + δm) ψ(x).
Now the question arises what boundary conditions to use. In Minkowski space, the field
operator related to ψ† creates heavy quarks and the field operator related to ψ annihilates
heavy quarks. It follows that for ψ the time-ordered Green function equals the retarded
Green function. In order for Wick rotation to give the correct result, one must also take
the retarded Green function in Euclidean space, i.e. consider forward propagation. The
propagator Gψ satisfies the initial condition
Gψ(τ = τ
′,x, τ ′,x′) = δ3(x− x′) (4.10)
and for τ > τ ′ we have
(D0 + δm)Gψ(τ,x, τ
′,x′) = 0, (4.11)
or
∂τGψ(τ,x, τ
′,x′) = −
[
ig A0(τ,x) + δm
]
Gψ(τ,x, τ
′,x′). (4.12)
The solution of this differential equation is
Gψ(τ,x, τ
′,x′) = e−δm(τ−τ
′)δ3(x− x′) U(x′, τ, τ ′), (4.13)
where U(x′, τ, τ ′) the time-ordered exponential of −ig A0 along the line with constant x′:
U(x′, τ, τ ′) = T exp
(
−ig
∫ τ
τ ′
A0(τ
′′,x′) dτ ′′
)
. (4.14)
Eq. (4.14) is the parallel transporter along the straight line from (τ ′,x′) to (τ,x′). By
inserting the solution (4.13) into Eq. (4.9), we finally obtain the following result for the
heavy quark propagator with the 1/m correction:
G(1)ψ (x, x
′) = e−δm(τ−τ
′)δ3(x− x′) U(x′, τ, τ ′)
+ e−δm(τ−τ
′)
∫ τ
τ ′
dτ ′′ U(x, τ, τ ′′)
[
D2
2m
+
g σ ·B
2m
]
τ ′′,x
δ3(x− x′) U(x′, τ ′′, τ ′).
(4.15)
52 CHAPTER 4. LATTICE HQET
4.3 Lattice discretisation
The simplest lattice discretisation of the action
(
δm S(−1)ψ + S
(0)
ψ
)
is, in lattice units,
∑
x
ψ†(x)
[
(1 + δm)ψ(x)− U †0(x− 0ˆ)ψ(x− 0ˆ)
]
. (4.16)
The corresponding propagator is
Gψ(x, x
′) = δx, x′(1 + δm)
−(τ−τ ′+1)Ulat(x′, τ, τ ′), (4.17)
where Ulat(x′, τ, τ ′) is given by
Ulat(x′, τ, τ ′) =
τ−τ ′−1∏
n=0
U †0(x
′ + n0ˆ). (4.18)
The lattice expression for the propagator with the 1/m correction analogous to (4.15) is
then
G(1)ψ (x, x
′) = (1 + δm)−(τ−τ
′+1)
×
{
δx, x′ Ulat(x
′, τ, τ ′)
+
τ∑
τ ′′=τ ′
Ulat(x, τ, τ ′′)
[
D2lat
2m
+
g σ ·Blat
2m
]
τ,x′′
δx, x′ Ulat(x
′, τ ′′, τ ′)
}
,
(4.19)
whereD2lat andBlat are lattice discretisations ofD
2 andB (examples are given in Sec. A.2).
Of course the operators need to be multiplied by renormalisation factors.
4.4 Signal-to-noise ratio
The simple form of the static action (4.16) leads to a rather bad signal-to-noise ratio
in heavy-light correlation functions, exponentially falling as the continuum limit is taken.
This has been explained in [105]. In general, the statistical variance in a two-point function
C(τ) = 〈Φ(τ)Φ†(0)〉 (4.20)
is given by
σ2(τ) = 〈[Φ(τ)Φ†(0)][Φ(τ)Φ†(0)]†〉 − |〈Φ(τ)Φ†(0)〉|2. (4.21)
4.4. Signal-to-noise ratio 53
Now, if Φ is an interpolating field for a heavy-light meson, then for large τ we have in the
static theory
C(τ)|static ∝ e
−EstQq¯τ , (4.22)
where EstQq¯ is the energy of the heavy-light ground state, which is dominated by a linearly
divergent contribution from the static propagator, Est ∝ g2/a.
On the other hand, it turns out that in the variance correlator (4.21) the contributions
from the two static propagators cancel each other and hence
σ2(τ)|static ∝ e
−mpiτ . (4.23)
It follows that the signal-to noise ratio is
C(τ)
σ(τ)
∣
∣
∣
∣
∣
static
∝ e
−
“
EstQq¯−
1
2mpi
”
τ
. (4.24)
This decays rapidly with τ . In contrast, when lattice NRQCD (see Chapter 5) is used,
the heavy quarks in σ2(τ) form a QQ¯ (quarkonium) bound state, so that
C(τ)
σ(τ)
∣
∣
∣
∣
∣
NRQCD
∝ e
−
“
EQq¯−
1
2 (EQQ¯+mpi)
”
τ
. (4.25)
The signal-to-noise ratio (4.25) still decays exponentially with τ , but with a much smaller
exponent [106].
Note that the signal-to-noise ratio in lattice HQET can be improved significantly by
replacing the gauge link U †0(x − 0ˆ) in (4.16) with a “smeared” link that depends also on
the neighbouring gauge links [107].
54 CHAPTER 4. LATTICE HQET
Chapter 5
Lattice moving NRQCD
In this chapter I will explain how the moving NRQCD Lagrangian (3.81) can be formulated
on the lattice such that the resulting theory works for both heavy-heavy and heavy-light
hadrons. The method is analogous to non-moving lattice NRQCD, which has been formu-
lated in [9, 10] and used with great success in recent dynamical lattice QCD calculations
[108, 109, 11, 110].
According to the power-counting rules derived in Sec. 3.2.2, for heavy-heavy mesons
the lowest-order moving NRQCD Lagrangian is
L(0) = ψ†v
[
D0 − iv ·D−
D2 − (v ·D)2
2γm
]
ψv+ξ
†
v
[
D0 − iv ·D+
D2 − (v ·D)2
2γm
]
ξv. (5.1)
The operators
(
D2− (v ·D)2
)
/(2γm) and
(
D0− iv ·D
)
are both of order O(v2orb.). Thus,
the operator
(
D2 − (v · D)2
)
/(2γm) can not be treated as an operator insertion like
in lattice HQET (cf. Chapter 4), but must be kept in the exponential e−S in the path
integral. As this operator has dimension 5, the resulting theory will be nonrenormalisable
in the standard sense, and power-law divergences occur. However, this does not cause any
problems as long as one works with a sufficiently low cutoff, like a sufficiently coarse lattice.
In terms of the lattice spacing a, one must have ma > 1, the opposite of the condition for
relativistic heavy quarks on the lattice (cf. Sec. 2.8). The contributions from higher-order
operators are then suppressed according to the power-counting rules. Instead of taking
the continuum limit, discretisation errors can be reduced using Symanzik improvement.
In lattice moving NRQCD, a lattice version of the full action (3.81) is used in the path
integral, in contrast to lattice HQET where only the static action is used. For this reason,
lattice mNRQCD can be applied to both heavy-heavy and heavy-light hadrons, and with
the action (3.81) it will be accurate to order O(v4orb.) for heavy quarkonium mesons and
O(Λ2QCD/m
2) for hadrons containing only a single heavy quark.
56 CHAPTER 5. LATTICE MOVING NRQCD
5.1 Continuum evolution equation
As in Chapter 4, we start by considering the continuum Green function on a given back-
ground gauge-field Aµ. Since the antiquark Green function can be obtained from the quark
Green function using Eq. (3.87), we consider only the quark field ψv(x) in what follows.
Recall from Eq. (3.81) that the Euclidean Lagrangian for ψv has the form
Lψv = ψ
†
v (D0 +H)ψv. (5.2)
As explained above, the full action is used in the weight factor e−S in the path integral.
Therefore, the Green function is the inverse of the full heavy-quark action kernel, i.e.
(D0 +H)Gψv(x, x
′) = δ4(x− x′), (5.3)
with x = (τ,x) and x′ = (τ ′,x′). As already discussed in 4.1 one must take the retarded
Green function in Euclidean space:
Gψv(τ,x, τ
′,x′) = 0 for τ < τ ′, (5.4)
and for τ = τ ′
Gψv(τ = τ
′,x, τ ′,x′) = δ3(x− x′). (5.5)
For τ > τ ′, Eq. (5.3) becomes
(D0 +H)Gψv(τ,x, τ
′,x′) = 0, (5.6)
or
∂0 Gψv(τ,x, τ
′,x′) = −(H + ig A0)Gψv(τ,x, τ
′,x′). (5.7)
Since H does not contain time derivatives acting on ψv, we can integrate this differential
equation, so that for τ2 > τ1 > τ ′
Gψv(τ2,x, τ
′,x′) = Gψv(τ1,x, , τ
′,x′)−
∫ τ2
τ1
(H + ig A0)Gψv(τ,x, τ
′,x′) dτ. (5.8)
The solution of (5.8) can be written as a Euclidean-time-ordered exponential function,
Gψv(τ2,x, τ
′,x′) = T exp
(
−
∫ τ2
τ1
(H + ig A0) dτ
)
Gψv(τ1,x, τ
′,x′). (5.9)
5.2. Lattice evolution equation 57
5.2 Lattice evolution equation
On the lattice, the quark Green function evolution Eq. (5.9) is conveniently approximated
by
Gψv(τ,x, τ
′,x′) =
(
1−
δH|τ
2
)(
1−
H0|τ
2n
)n
U †0(τ − 1,x)
×
(
1−
H0|τ−1
2n
)n(
1−
δH|τ−1
2
)
Gψv(τ − 1,x, τ
′,x′), (5.10)
which corresponds to the lattice action
Sψv =
∑
x,τ
ψ†v(τ,x)
[
ψv(τ,x)−
(
1−
δH|τ
2
)(
1−
H0|τ
2n
)n
U †0(τ − 1,x)
×
(
1−
H0|τ−1
2n
)(
1−
δH|τ−1
2
)n
ψv(τ − 1,x)
]
. (5.11)
Here, H0 contains the leading-order kinetic terms and δH contains all higher-order oper-
ators including Symanzik improvement terms. The leading evolution due to H0 from one
lattice time slice to the next is effectively divided into 2n smaller steps to avoid instabilities
from high-momentum modes [10].
The time-reversal symmetric split into H0 and δH (albeit with the opposite ordering)
was introduced in [10] for non-moving NRQCD. In the following, lattice versions of H0
and δH are constructed for the full moving NRQCD action (3.81), satisfying the following
requirements:
• for v = 0, the action reduces to the standard lattice NRQCD action currently in use
by the HPQCD collaboration (see e.g. [111]).
• the same order of Symanzik improvement as in the standard HPQCD lattice NRQCD
action is retained at non-zero velocity.
Lattice versions of earlier moving NRQCD actions were presented or used in [19, 21, 22,
112, 23, 24, 25, 113]; the following lattice formulation is the result of work done by myself
and Eike Mu¨ller and is published in [100].
The lattice H0 and δH are defined as
H0 = −iv ·∆± −
∆(2) −∆(2)v
2γm
, (5.12)
58 CHAPTER 5. LATTICE MOVING NRQCD
δH = −
g
2γm
σ ·B˜′
−
i
4γ2m2
({
∆(2), v ·∆±
}
− 2∆(3)v
)
+
g
8m2
(
i(∆± · E˜− E˜ ·∆±) + v · (∆ad × B˜)
)
−
g
8γm2
σ ·
(
∆˜± × E˜′ − E˜′ × ∆˜±
)
+
g
8(γ + 1)m2
{
v · ∆˜±, σ · (v × E˜′)
}
−
(2− v2)g
16m2
(
∆ad0 + iv ·∆
ad
)(
v · E˜
)
−
ig
4γ2m2
{
v · ∆˜±, σ · B˜′
}
−
1
8γ3m3
((
∆(2)
)2
− 3
{
∆(2), ∆(2)v
}
+ 5∆(4)v
)
+ δHcorr . (5.13)
The lattice derivative operators and field strength are defined in Appendix A.2. The “∼”
is used to denote Symanzik-improved quantities; the boosted fields E˜′ and B˜′ are related
to the fields E˜ and B˜ in the lattice frame through Eq. 3.80.
Note that in the continuum the Leibniz rule Dad ·E = D ·E−E ·D holds. In order to
get agreement with the standard HPQCD lattice NRQCD action at v = 0, the right-hand
side of this expression is discretised on the lattice. However, the other adjoint derivatives
in the action, which enter only at v 6= 0, are discretised as lattice adjoint derivatives
(defined in Eq. (A.14)). This is more efficient and for the term Dad0 (v ·E) it is crucial
since it avoids a time derivative acting on the quark field.
Note that in the static limit (m → ∞) one has H0 = −iv · ∆±. The symmetric
derivative ∆± is used to ensure Hermiticity of the Hamiltonian. It does however lead
to doublers, as we have seen for the naive lattice Dirac action in Sec. 2.4. With a finite
mass, these doublers are shifted to higher energy due to the second-order derivatives in
H0. However, the second-order derivatives are suppressed by a factor of 1/(2γm) and
hence γm must not be too large.
The terms in δHcorr provide the spatial and temporal lattice spacing improvement for
the leading evolution due to H0. We perform tree-level Symanzik improvement to order
O(a2,k4), as explained in the next section. This means that the we expect the leading
errors to be of order O(αsa2).
5.2. Lattice evolution equation 59
5.2.1 Symanzik improvement
An O(a2,k4)-improved version of H0 is given by
H˜0 = −iv · ∆˜± −
∆˜(2) − ∆˜(2)v
2γm
(5.14)
with the improved derivatives defined in Eq. (A.13) in Appendix A.2. However, we do not
simply replace H0 by H˜0. Let us first consider the time derivative in the lattice action.
Improving it in the standard way would introduce next-to-nearest neighbour couplings,
preventing the use of an evolution equation like (5.10). Instead, we try to find an operator
H˜?0 such that (explicitly re-introducing the lattice spacing a)
(
1−
aH˜?0
2n
)n
= exp
(
−
a
2
H˜0
)
, (5.15)
which yields a more continuum-like behaviour [10]. We obtain
aH˜?0 = 2n
[
1− exp
(
−
aH˜0
2n
)]
. (5.16)
One could now replace H0 → H˜?0 in the lattice action. However, for consistency with
previous work, we choose to put all correction terms into δH. We consider the operator
on the right-hand side of the temporal link in the lattice action (5.11); the operator acting
in the time slice at time τ − a. Then δHcorr, the lattice spacing improvement term in
(5.13), is defined by
(
1−
aH˜?0
2n
)n
=
(
1−
aH0
2n
)n(
1−
a δHcorr
2
)
. (5.17)
This gives
a δHcorr = 2
[
1−
(
1−
aH0
2n
)−n
(
1−
aH˜?0
2n
)n ]
= 2
[
1−
(
1−
aH0
2n
)−n
exp
(
−
aH˜0
2
) ]
, (5.18)
60 CHAPTER 5. LATTICE MOVING NRQCD
and, expanding in powers of a,
a δHcorr = a(H˜0 −H0)
+
a2
4n
(
−(1 + n)H20 − nH˜
2
0 + 2nH0H˜0
)
+
a3
24n2
(
−(2 + 3n+ n2)H30 + (3n+ 3n
2)H20 H˜0 − 3n
2H0H˜
2
0 + n
2H˜30
)
+
a4
192n3
(
−(6 + 11n+ 6n2 + n3)H40 + (8n+ 12n
2 + 4n3)H30 H˜0
−(6n2 + 6n3)H20 H˜
2
0 + 4n
3H0H˜
3
0 − n
3H˜40
)
+O(a5) . (5.19)
The term C ≡ H˜0 −H0 is of order |k|3, while H0 is of order |k|. Neglecting all operators
of order |k|5 and higher, we obtain
a δHcorr = a C −
a2
4n
(
H20 + n[C, H0]
)
−
a3H30
12n2
−
(2 + n)a4H40
64n3
. (5.20)
Had we considered the operators on the left-hand side of the temporal link in the lattice
action (5.11) instead, the ordering of H0 and H˜0 would be interchanged, and this would
change the sign of the commutator [C, H0] in (5.20), thereby cancelling the term in the
lattice action up to operators of order |k|5 and higher. We therefore remove this term on
both sides.
Let us go back to lattice units now. Writing H0 = A+B with
A = −iv ·∆±,
B = −
∆(2) −∆(2)v
2γm
, (5.21)
we obtain
δHcorr = H˜0 −H0 −
1
4n
(
A2 + {A , B}+B2
)
−
1
12n2
(
A3 +
{
A2 , B
}
+ABA
)
−
(2 + n)
64n3
A4. (5.22)
For performance reasons, we replace some 3rd- and 4th-order derivatives in (5.22) by more
5.3. Renormalisation 61
local expressions (the resulting change is of order |k|5 or higher):
(v ·∆±)3 → ∆(3)v ,
{
v ·∆±, ∆(2)v
}
→ 2∆(3)v ,
(v ·∆±)4 → ∆(4)v ,
(∆(2)v )
2 → ∆(4)v ,
{
(v ·∆±)2 , ∆(2)
}
→
{
∆(2)v , ∆
(2)},
{
(v ·∆±)2 , ∆(2)v
}
→ 2∆(4)v ,
(v ·∆±)∆(2)v (v ·∆
±) → ∆(4)v ,
(v ·∆±)∆(2)(v ·∆±) → 12(v ·∆
−)∆(2)(v ·∆+)
+ 12(v ·∆
+)∆(2)(v ·∆−) . (5.23)
This finally gives
δHcorr = H˜0 −H0
−
1
4n
[
− (v·∆±)2 +
{
iv·∆±, ∆(2)
}
− 2i∆(3)v
2γm
+
(∆(2))2 −
{
∆(2), ∆(2)v
}
+∆(4)v
4γ2m2
]
−
1
12n2
[
i∆(3)v +
{
∆(2), ∆(2)v
}
− 3∆(4)v
2γm
+
(v·∆−)∆(2)(v·∆+) + (v·∆+)∆(2)(v·∆−)
4γm
]
−
(2 + n)
64n3
∆(4)v . (5.24)
The result (5.24) can of course be simplified further but I show it in this form for clarity.
Also, most operators in δHcorr are already in the Hamiltonian. Note that for v = 0, the
correction term (5.24) reduces to the standard NRQCD improvement terms as in [111],
δHcorr
∣
∣
∣
v=0
=
1
24m
3∑
j=1
∆+j ∆
−
j ∆
+
j ∆
−
j −
(
∆(2)
)2
16n m2
. (5.25)
5.3 Renormalisation
In principle, all terms in the lattice moving NRQCD action must be multiplied by indi-
vidual renormalisation coefficients ci, to be determined by matching the effective theory
to continuum QCD.
62 CHAPTER 5. LATTICE MOVING NRQCD
Nonperturbative adjustment of all coefficients is forbiddingly complicated due to the
large number of couplings which have to be tuned simultaneously. A simple yet powerful
method of including the biggest part of the radiative corrections is tadpole improvement,
which was introduced in Sec. 2.7.1. In the present calculations, I use the tree-level values
for the couplings, ci = 1, but perform tadpole improvement.
With this choice, the remaining free parameters in the action are the heavy-quark mass
and the boost velocity. The bare heavy-quark mass can easily be adjusted nonperturba-
tively so that the lattice result for a quarkonium mass matches the experimental value.
Along with the heavy-quark mass, the boost velocity determines the size of the external
momentum. We will discuss this in more detail in Sec. 5.3.2.
When lattice moving NRQCD is used to compute matrix elements of certain currents,
these currents must also be matched to continuum QCD. This will be discussed in Sec. 8.5
of Chapter 8.
5.3.1 Tadpole improvement
As explained in Sec. 2.7.1, tadpole improvement is achieved by dividing the gauge links
Uµ(x) in the lattice action by u0, the mean link in Landau gauge. Before that, any
factors of U †µ(x)Uµ(x) = 1 should be cancelled. However, expanding out the whole action
(5.11) in terms of products of U ’s would be too expensive for numerical calculations, and
therefore link-pair cancellations are only taken into account separately within H0 and δH.
Also, no extra cancellations are made when derivative operators act on field strengths
in δH. These conventions are consistent with the ones in the standard HPQCD lattice
(non-moving) NRQCD action. Note that the majority of the link pair cancellations is
still correctly accounted for. In appendix D of [100] it is shown that the difference to full
cancellation is negligible.
5.3.2 Energy shift and external momentum renormalisation
The factor of e−im u·x γˆ
0
in the moving NRQCD field redefinition (3.74) corresponds to a
shift in the 4-momentum of a heavy quark by P = mu = (γm, γmv). However, the shifts
in energy and momentum will get renormalised.
Consider a hadron containing nQ heavy quarks (here we also count antiquarks posi-
tively). On the lattice, an interpolating field for the hadron is constructed from ψv and
ξv (see Chapter 6). This field is given a momentum k by inserting a phase eik·x. The
5.3. Renormalisation 63
physical momentum of the hadron is then
p = nQ Zp P+ k with P = γmv, (5.26)
where
Zp = ZγZmZv (5.27)
is the renormalisation of the bare external momentum P. Here and in the following
we assume that v points in one of the axis directions, so that only its magnitude is
renormalised. Note that Zγ and Zv are not independent renormalisation constants; they
are related through γ = 1/
√
1− v2.
The full (physical) energy of the hadron is given by
E = Ev(k) + nQ Cv, (5.28)
where Ev(k) is the energy obtained from the large-τ behaviour of the hadron two-point
function and Cv is the renormalised energy shift
Cv = ZmZγγm+ E0. (5.29)
All renormalisation parameters introduced up to here can be written in terms of the three
independent quantities Zm, Zv, E0. At tree-level, one has Zm = Zv = 1 and E0 = 0.
The combinations Zp and Cv can be determined nonperturbatively from the dispersion
relation of a hadron. Given expression (5.26) for the full (physical) momentum, we expect
that, up to lattice artifacts,
E =
√
p2 +M2kin
=
√
(nQ Zp P+ k)2 +M2kin , (5.30)
whereMkin is the kinetic mass of the hadron. Using (5.30) one can extract Cv, Zp andMkin
from the energies at various non-zero lattice momenta in combination with the energy at
k = 0:
Cv =
k2⊥ −
[
E2v(k⊥)− E
2
v(0)
]
2 nQ [Ev(k⊥)− Ev(0)]
, (5.31)
Zp =
E2v(k‖)− E
2
v(−k‖) + 2 nQ Cv
[
Ev(k‖)− Ev(−k‖)
]
4 nQ k‖ ·P
, (5.32)
Mkin =
√
(Ev(k) + nQ Cv)2 − (nQ Zp P+ k)2. (5.33)
Here, k‖ (k⊥) is a lattice momentum parallel (perpendicular) to v. Equations (5.31) and
(5.32) were first derived in [22].
My nonperturbative results for Cv, Zp, and Mkin for the full action (3.81) will be
presented in Chapter 6.
64 CHAPTER 5. LATTICE MOVING NRQCD
5.3.3 Reparametrisation invariance
The renormalisation parameter Zp is expected to be close to 1 due to an approximate
reparametrisation invariance [22], a symmetry that is exact in continuum HQET [97].
The HQET Lagrangian was derived in Sec. 3.3. Denoting the field Ψ˜+ used there by Qu,
we have1
L = Qu
(
i u ·D −
/D2⊥
2m
)
Qu
= Qu
(
i u ·D −
D2⊥
2m
−
ig
8m
[γˆµ, γˆν ]Fµν
)
Qu. (5.34)
The Lagrangian (5.34) is invariant under the transformation
u 7→ u+ ²,
Qu 7→ e
i m ²·x
(
1 +
/²
2
)
Qu,
Qu 7→ e
−i m ²·x Qu
(
1 +
/²
2
)
, (5.35)
where ² is an infinitesimal change in the 4-velocity, satisfying ² · u = 0 so that u2 = 1
is preserved. The symmetry (5.35) requires the coefficient of the operator −D2⊥/(2m) to
be 1. Absorbing an overall wave function renormalisation into the fields Qu, Qu, there
can be no radiative correction to this coefficient provided that the regulator respects
reparametrisation invariance.
In moving NRQCD, time and space are not treated on an equal basis, and the trun-
cation of the action at a given order in 1/m or vorb. breaks reparametrisation invariance
[100]. However, when only selected terms are included the continuum Lagrangian may
still possess an exact symmetry. This is the case for the incomplete Lagrangian
L = ψ†v
(
iD0 + iv ·D+
D2
2γm
)
ψv, (5.36)
which is invariant under
v 7→ v + ²,
ψv 7→ e
−iγm·x ψv,
ψ†v 7→ e
iγm·x ψ†v, (5.37)
with infinitesimal ² satisfying ² · v = 0. The symmetry (5.37) implies that the two terms
iv ·D and D2/(2γm) in (5.36) must renormalise by the same factor, so that Zp = 1.
1No ⊥ labels are needed in the [γˆµ, γˆν ]Fµν term, since Quu
µ[γˆµ, γˆν ]Qu = 0.
5.3. Renormalisation 65
The operator −(v ·D)2/(2γm), which was omitted in (5.36), breaks the reparametri-
sation invariance. An additional breaking in lattice moving NRQCD arises from lattice
artefacts. The remaining approximate symmetry still guarantees that Zp is close to 1 for
moderate boost velocities, as is confirmed by my nonperturbative calculations (Chapter
6) and by perturbation theory.
5.3.4 Perturbation theory
The heavy-quark renormalisation parameters Zψ (the wave function renormalisation), Zm,
Zv, E0, and the corresponding Zp and Cv have also been calculated in automated one-
loop lattice perturbation theory. There, one computes the inverse heavy-quark propagator
G−1(k) = G−10 (k) − Σ(k), where G0(k) is the tree-level propagator and Σ(k) is the self-
energy at one-loop order. The renormalisation parameters can then be extracted from
the expansion of Σ(k) in powers of the momentum; the details can be found in [100].
Results for a simple action at order 1/m have been obtained in [112]; these calculations
have been extended to an action similar to (5.12, 5.13) by Lew Khomskii [113]. New
calculations for exactly the action (5.12, 5.13) have then been performed by Eike Mu¨ller
[114] and are published in [100]. I will compare the perturbative results for Cv and Zp to
my nonperturbative results in Sec. 6.3 of Chapter 6.
Perturbation theory is thought to be a reasonably good approximation as the momenta
giving the dominant contributions to the radiative corrections are of the order pi/a, and the
running coupling αs is small at this scale. Note that this requires ma > 1 and ΛQCDa¿ 1
simultaneously. In order to obtain a good convergence, tadpole improvement is crucial
[77].
Perturbative calculations of the couplings ci in the action are also underway. So far,
results have been obtained for the coefficients of the 4-th-order derivative operators at
v = 0 [115, 116].
5.3.5 High-β methods
Lattice perturbation theory for highly improved actions is already hard at one-loop level
due to the extremely complicated Feynman rules, and automated methods are used to
derive them (see e.g. [117]). Two-loop calculations for the full lattice moving NRQCD
action seem to be out of reach currently.
An alternative approach has been introduced in [118]. There, nonperturbative Monte-
Carlo simulations are performed at very weak coupling (high β = 2Nc/g2) and in Coulomb
66 CHAPTER 5. LATTICE MOVING NRQCD
gauge. Results for a short-distance observable are computed at several values of β and
then fitted by a polynomial in αs = g2/(4pi). The fit results then give the perturbative
coefficients, provided that non-perturbative effects are negligible. The linear term can be
constrained by the 1-loop perturbative result, which stabilises the fit. Certain nonper-
turbative finite-volume effects can be suppressed by using twisted boundary conditions
[119].
As the inclusion of dynamical fermions is computationally very expensive, most high-
β calculations are done in the quenched limit, i.e. in pure gauge theory. However, at
order α2s only a handful of diagrams contain fermion loops and these may be accessible
in perturbation theory. The contribution from these diagrams can be combined with the
gluonic two-loop coefficient from the high-β simulation to give the full two-loop result.
High-β simulations have been carried out for a simple O(1/m) lattice moving NRQCD
action in [25] and further calculations are currently underway [116].
5.4 Signal-to-noise ratio
The signal-to-noise ratio for correlation functions involving heavy quarks implemented
with lattice mNRQCD can be estimated using the same general methods as in Sec. 4.4.
Note that for a hadronic correlation function at arbitrary lattice momentum k and
boost velocity v the variance correlator always contains contributions from hadrons at
rest (k = 0, v = 0). For example, for a B meson two-point function in moving NRQCD
one obtains
C(τ)
σ(τ)
∝ e
−
“
EQq¯(p)−
1
2 [EQQ¯(0)+mpi]
”
τ
, (5.38)
where
p = k+ Zpγmv. (5.39)
Thus, the noise is expected to grow exponentially as the boost velocity is increased at a
fixed k. I do indeed see this behaviour in my numerical calculations (cf. Chapter 6). Note
that this is not a disadvantage of moving NRQCD compared to standard NRQCD. The
amount of noise depends on p; for a comparable momentum in non-moving NRQCD one
gets a similar amount of noise (and higher discretisation and relativistic errors).
Chapter 6
Nonperturbative tests of moving
NRQCD
In this chapter, I report on a range of nonperturbative tests of the full lattice moving
NRQCD action (5.12, 5.13) that I performed by computing heavy-heavy (bottomonium)
and heavy-light (Bs) two-point functions in dynamical 2+1 flavour lattice QCD. These
tests include the calculation of renormalisation parameters, decay constants and energy
splittings at several values for the boost velocity. Lattice units are used throughout this
chapter.
Previous nonperturbative tests of mNRQCD for a simple O(1/m) action and with
quenched gauge field configurations (i.e. without dynamical quarks) were reported in [22,
23, 24, 25].
6.1 Tests with bottomonium
6.1.1 Calculation of the two-point functions
We begin by constructing “smeared” interpolating fields for quarkonium. To demonstrate
the effect of the moving NRQCD field redefinition, we start the construction with the
QCD fields Ψ, Ψ. A meson with momentum p can be obtained from
OΓ(τ,p) =
1
√
V
∑
x,y
Ψ(τ,x)Γ(x− y)Ψ(τ,y)e−ip
x+y
2 (6.1)
where Γ(r) is a Dirac-matrix-valued smearing function, and V = L3 is the spatial volume.
I do not include gauge links in Γ(r); instead I work with gauge configurations fixed to
68 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
Name Γ(r)
ηb(1S) exp[−|r|/rs] γˆ5
ηb(2S) [1− |r|/(2rs)] exp[−|r|/(2rs)] γˆ5
Υ(1S) exp[−|r|/rs] γˆj
Υ(2S) [1− |r|/(2rs)] exp[−|r|/(2rs)] γˆj
χb1(1P ) exp[−|r|/(2rs)] (r× γˆ)
j/rs
Table 6.1: The bottomonium states and corresponding smearing functions used in the
nonperturbative tests of moving NRQCD. More details on bottomonium can be found in
Sec. 7.1 of Chapter 7.
Coulomb gauge1. The bottomonium states and corresponding functions Γ(r) used here
are listed in Table 6.1. In addition to these, we also consider the local function
Γ(r) = δr,0 γˆ5γˆ0, (6.2)
which corresponds to the temporal axial current
J05 (x) = Ψ(x) γˆ5γˆ
0 Ψ(x). (6.3)
The lowest state above the vacuum that couples to (6.3) is also the ηb(1S) meson.
We now express Ψ and Ψ through the Euclidean version of the tree-level moving
NRQCD field redefinition (3.74). To lowest order we have
Ψ(x) =
1
√
γ
S(Λ) e−iγm(−iτ−v·x)γˆ
0
Ψv(x),
Ψ(x) =
1
√
γ
Ψv(x) e
iγm(−iτ−v·x)γˆ0 S(Λ). (6.4)
Note that for calculations of the spectrum no higher-order terms or radiative corrections
are needed in the field redefinition; it suffices that the state of interest has a non-vanishing
overlap with the interpolating field.
Let us, for example, consider the Υ states with polarisation j = 1, 2, 3. We allow
different radial smearing at source and sink, so that Γsc(r) = γˆj fsc(r) and Γsk(r) =
1On the lattice, Coulomb gauge is obtained by minimising the functional F [U ] =
−
∑
x
∑3
j=1 Tr
[
Uj(x) + U
†
j (x)
]
through gauge transformations (2.38) of the link variables Uj .
6.1. Tests with bottomonium 69
γˆj fsk(r). Using
S(Λ) γˆj S(Λ) = Λj µγˆ
µ, (6.5)
we obtain
OΓsk(τ,p)O
†
Γsc(τ
′,p) =
1
V
1
γ2
e−2γm(τ−τ
′)
∑
x,y,x′,y′
e−ik
x+y
2 fsk(x− y)e
ikx
′+y′
2 fsc(x′ − y′)
× Λj lΛ
j
mξ
†
v(τ,x)σ
lψv(τ,y)ψ†v(τ
′,y′)σmξv(τ ′,x′) + ... (6.6)
(no summation over j here) where the ellipsis denotes terms that do not contribute to
the connected meson correlator for τ > τ ′. In Eq. (6.6) we have k = p − 2γmv. This
is the tree-level mNRQCD momentum shift, as expected. On the lattice the momentum
k rather than p has a definite value; kj = 2pi nj/L where L is the spatial extent of the
lattice. The full momentum p will be given by (5.26) with nQ = 2.
According to Eqs. (2.24)and (2.92), the expectation value of (6.6) can be obtained as
〈OΓsk O
†
Γsc 〉 =
1
N
∑
U
1
V
1
γ2
e−2γm(τ−τ
′)
∑
x,y,x′,y′
e−ik
x+y
2 fsk(x− y)e
ikx
′+y′
2 fsc(x′ − y′)
× Λj lΛ
j
mTr
(
σl
[
G(+v)ψv (τ,y, τ
′,y′)
]
σm
[
G(−v)ψv (τ,x, τ
′,x′)
]†
)
,
(6.7)
where we average over N gauge configurations U . The trace is in (6.7) is over colour and
spin indices. We have also used Eq. (3.87) to express the antiquark green function G(+v)ξv
in terms of the quark green function G(−v)ψv with the opposite boost velocity.
The summations over all quark and antiquark source locations would render the lat-
tice computation too expensive. Therefore, using translation invariance, we remove the
summation over the antiquark source location x′, and correspondingly remove the factor
of 1/V .
Note that the mNRQCD field redefinition also introduced the factor of e−2γm(τ−τ
′)
in the two-point function, which corresponds to the tree-level energy shift. In the later
analysis, only energy differences will be measured, and so we remove this trivial factor.
Hence, the quantity
C(Γsk,Γsc,k, τ, τ
′) =
1
N
∑
U
1
γ2
∑
x,y
e−ik
x+y
2 fsk(x− y)
× Λj lΛ
j
m Tr
(
σl
[
G˜(+v)ψv (τ,y, τ
′,x′)
]
σm
[
G(−v)ψv (τ,x, τ
′,x′)
]†
)
(6.8)
70 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
with
G˜(+v)ψv (τ,y, τ
′,x′) =
∑
y′
eik
x′+y′
2 fsc(x′ − y′)G
(+v)
ψv
(τ,y, τ ′,y′) (6.9)
is computed on the lattice. The convoluted Green function (6.9) can be computed effi-
ciently by using the function
eik
x′+y′
2 fsc(x′ − y′) (6.10)
as the initial condition in the mNRQCD evolution Eq. (5.10). The momentum-dependent
phase factor exp(ik · (x′ + y′)/2) at the source improves the overlap with the momentum
considered. I used this factor in my computations; however, note that since there is no
sum over x′, one may also omit this factor to allow the calculation of correlators with
different momenta from the same source.
To maintain the periodic boundary conditions, I set f(r) to zero for |r| > Rs with
some cut-off radius Rs smaller than half the length of the lattice.
I performed the computations using 400 MILC gauge configurations (fixed to Coulomb
gauge) of size 203 × 64 with 2+1 flavors of rooted staggered light quarks, at β = 6.76
[120, 121, 87]. The light quark masses were mu = md = 0.007 and ms = 0.05 (in the
MILC convention for lattice masses). The Goldstone pion mass is about 300 MeV, and
the pion taste splittings range from (mpiA −mpi5) ≈ 110 MeV to (mpiI −mpi5) ≈ 240 MeV
[121].
The Landau gauge mean link, used for tadpole improvement in the mNRQCD action,
was u0 = 0.836. The inverse lattice spacing of these “coarse” MILC configurations is
known to be approximately 1.6 GeV [108].
The bare heavy quark mass was set to m = 2.8, which gave the correct Υ kinetic
masses using non-moving NRQCD [108]. The boost velocity v was always pointing in the
x-direction. The stability parameter was set to n = 2.
In order to increase statistics, on each gauge configuration I computed between 16 and
120 correlators with different origins (x′, τ ′) spread over the lattice. These origins were
also shifted randomly to reduce autocorrelations. The smearing parameter rs (see Table
6.1) was set to 1 for the S wave states and 0.5 for the P wave states.
6.1.2 Fitting of the two-point functions
In accordance with Eq. (2.34), I fitted the two-point correlators C(Γsk,Γsc,k, τ, τ ′) for a
given momentum k by a function of the form
nexp−1∑
n=0
An(Γsk)A
∗
n(Γsc) e
−Ev,n(τ−τ ′), (6.11)
6.1. Tests with bottomonium 71
where Ev,n is the (shifted) energy of n-th state and An(Γ) is, up to the normalisation
factor of
√
2En, the (real) amplitude for this state to be created by the operator with
smearing function Γ(r). For the excited states (n > 0) I actually used the quantities
∆Ev,n ≡ Ev,n+1 − Ev,n,
Bn(Γ) ≡ An(Γ)/A0(Γ) (6.12)
as parameters in the fit. In general, a matrix of correlators with different smearings at
source and sink was fitted simultaneously. The Bayesian method from [122] was used to
stabilise the fits. The number of exponentials nexp in (6.11) was increased until the fit
results for the states of interest (the ground state, or the first few low-lying states) become
independent of nexp. More details on the fitting are given in Appendix C.
6.1.3 Kinetic mass, energy shift and external momentum renormalisa-
tion
Results for the ηb(1S) kinetic mass Mkin and the renormalisation parameters Zp, Cv,
calculated with Eqs. (5.33), (5.32), and (5.31), are shown in Table 6.2. The energies were
obtained from 6-exponential fits to 2 × 2 matrix correlators with the ηb(1S) smearing
(cf. Table 6.1) and the local operator given by (6.2). Sample fits at v = 0 and v = 0.6 are
shown in Fig. 6.1.
For the calculation of Cv using (5.31), I averaged the results over the 4 different
perpendicular lattice momenta
k⊥ ∈
{
2pi
L
(0,±1, 0),
2pi
L
(0, 0,±1)
}
. (6.13)
The momentum parallel to the boost velocity in (5.32) was chosen to be k‖ =
2pi
L (1, 0, 0),
and in (5.33), for the measurement of Mkin, I used k = 0.
In order to fully take into account correlations in the energies at different momenta, I
used the bootstrap method. This is described in Appendix C.
Because the lattice is of finite extent, L = 20 in this case, the estimates for Cv and
Zp are affected by the choice of momenta in (5.31) and (5.32) since the formulae are
accurate only in the limit that the momenta are infinitesimal. Note that the uncertainty
due to using non-infinitesimal momenta will decrease for larger lattices for which smaller
momenta are available.
To estimate the size of the resulting systematic error I also performed the calculations
72 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
0 5 10 15 20 25 30 35 40
time
1e-05
0.0001
0.001
0.01
0.1
1
10
100
valu
e of
cor
rela
tor
local - localsmeared - locallocal - smearedsmeared - smeared
0 2 4 6 8
time
0.1
1
10
100
valu
e of
cor
rela
tor
local - localsmeared - locallocal - smearedsmeared - smeared
Figure 6.1: Fits to ηb matrix correlators (local and 1S smearing) at k = 0 and v = 0
(upper panel), |v| = 0.6 (lower panel). The fits have nexp = 6 and τmin = 1. At non-zero
momentum p = k + 2Zpγmv the noise in the two-point function increases exponentially
with the Euclidean time. For |v| = 0.6, the errors for the points beyond τ ∼ 8 are so large
that these points do not give any useful information and hence are not shown here.
6.1. Tests with bottomonium 73
|k⊥| = |k‖| = 2pi/L |k⊥| = |k‖| = 4pi/L
|v| Zp Mkin Cv/(γm) Zp Mkin Cv/(γm)
0 — 5.974(48) 1.0182(86) — 5.979(37) 1.0190(65)
0.2 1.008(19) 5.95(10) 1.015(18) 1.009(12) 5.969(62) 1.017(11)
0.4 0.9953(78) 5.931(44) 1.0084(77) 0.9830(65) 5.954(40) 1.0101(70)
0.6 0.898(27) 6.22(18) 1.010(28 ) 0.843(27) 6.37(15) 1.011(21)
Table 6.2: Nonperturbative results (using the ηb(1S)) for Mkin, Zp, Cv.
with the larger momenta
k⊥ ∈
{
2pi
L
(0,±2, 0),
2pi
L
(0, 0,±2)
}
, k‖ =
2pi
L
(2, 0, 0). (6.14)
For Cv, the results from |k⊥| = 2pi/L agree with those obtained from |k⊥| = 4pi/L
within statistical errors, indicating that the systematic error is small and does not increase
significantly when increasing the momentum perpendicular to v in the measurement. For
the measurement of Zp at |v| = 0.6 I find a 6% (2σ) change in Zp when going from
|k‖| = 2pi/L to |k‖| = 4pi/L. At |v| = 0.4 and smaller boost velocities the results are
equal within statistical errors. For the kinetic mass, which depends on both Cv and Zp,
I again find agreement within statistical errors between the results from the two different
momenta for all boost velocities considered. At small velocities, I find that both Zp and
Cv/(γm) are close to their tree-level value of 1, demonstrating that the renormalisations
are small.
6.1.4 Decay constant
Moving NRQCD is designed for the calculation of form factors (see Chapter 8). Thus, it
is important to perform tests not only for spectral quantities but also for decay constants.
For the ηb(1S) meson, I calculated the decay constant f defined by
〈0|Jµ5 (0)|ηb(1S),p〉 = if p
µ. (6.15)
For simplicity, I only considered the temporal component. Following the discussion in
Sec. 2.2, the matrix element (6.15) can be computed from the 2 × 2 matrix correlators
already used in Sec. 6.1.3. These include the local operator given by (6.2), which corre-
sponds to the lowest-order tree-level axial current. With the notation from Sec. 6.1.2, we
74 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
define
A ≡ A0(Γ)
∣
∣
∣
Γ(r)=δr,0 γˆ5γˆ0
. (6.16)
Then, according to Eq. (2.34) we have
A2 =
1
2E
|〈0|Jµ5 (0)|ηb(1S),p〉|
2 =
1
2E
f2E2, (6.17)
and hence
f =
2A
E
, (6.18)
where E is the physical energy of the ηb(1S) meson. Using Eq. (5.28) with nQ = 2, we
finally obtain
f = A
√
2
Ev(k) + 2 Cv
. (6.19)
For the energy shift Cv in (6.19), I used Eq. (5.31) with |k⊥| = 2pi/L.
The momentum of the meson is given by p = 2Zpγmv + k. The decay constant is a
Lorentz scalar and should ideally be independent of the value of p used to compute it.
In the following I compare two methods of reaching large |p|. First, at v = 0, i.e. with
standard NRQCD, I computed the decay constant at large non-zero lattice momentum k;
the results are shown in Table 6.3. Second, I computed the decay constant with k = 0
and three different boost velocities v; the results are shown in Table 6.4. In this case the
uncertainty in Zp (determined nonperturbatively from Eq. (5.32)) leads to an uncertainty
in the meson momentum.
A plot of the decay constant against the total momentum (with Zp from (5.32) with
|k‖| = 2pi/L) for the two methods is shown in Fig. 6.2. With NRQCD we see a strong
dependence on p due to both relativistic and discretisation errors. With moving NRQCD
the p-dependence is very small, giving evidence that the formalism works very well. Small
deviations from the constant behaviour are still expected here, since only the leading-order
current was used; i.e. TFWT and ADt were set to unity in (3.74) for this calculation.
6.1. Tests with bottomonium 75
|p|L/(2pi) |p| f
0 0 0.4724(23)
1 0.31416 0.4731(23)
2 0.62832 0.4755(24)
3 0.94248 0.4772(43)
4 1.25664 0.4835(77)
5 1.57080 0.4971(78)
6 1.88496 0.5209(46)
7 2.19911 0.5527(44)
8 2.51327 0.6006(45)
9 2.82743 0.6740(49)
10 3.14159 0.715(29)
Table 6.3: ηb(1S) decay constant with standard NRQCD (i.e. v = 0) computed with
several values of meson momentum |p| by varying |k|.
|v| |p| f
0 0 0.4724(23)
0.2 1.152(22) 0.4739(38)
0.4 2.433(19) 0.4810(36)
0.6 3.77(11) 0.499(11)
Table 6.4: ηb(1S) decay constant with mNRQCD at k = 0 computed with several values
of meson momentum |p| by varying |v|.
76 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
0 1 2 3 4 5
p
0
0.2
0.4
0.6
0.8
1
f
NRQCD
mNRQCD
Figure 6.2: Heavy-heavy decay constant in NRQCD and mNRQCD for different values
of the meson’s momentum, |p|/(2pi/L) = 0 . . . 10 (NRQCD) and p = Zp 2γmv for |v| =
0.2, 0.4, 0.6 (mNRQCD). The horizontal line indicates the value at p = 0.
6.1. Tests with bottomonium 77
|v| ∆Ev(0)
∆Ev(0)
∆E0(0)
0.0 0.3334(68) 1
0.2 0.329(10) 0.986(37)
0.4 0.320(15) 0.958(48)
0.6 0.20(11) 0.59(33)
Table 6.5: Υ(2S)−Υ(1S) energy splitting as a function of the boost velocity.
|v| ∆Ev(0)
∆Ev(0)
∆E0(0)
0.0 0.2703(89) 1
0.2 0.264(12) 0.976(56)
0.4 0.270(23) 0.998(91)
0.6 0.227(57) 0.84(21)
Table 6.6: χb1(1P )−Υ(1S) energy splitting as a function of the boost velocity.
6.1.5 Energy splittings
I also studied the velocity-dependence of various energy splittings between the bottomo-
nium states listed in Table 6.1. For the Υ and ηb states, I used 6-exponential 2× 2 matrix
fits with the 1S and 2S smearings; for the χb1 states a 6-exponential single-correlator fit
with the 1P smearing at both source and sink was used. The results for the Υ(2S)−Υ(1S),
χb1(1P ) − Υ(1S) and Υ(1S) − ηb(1S) splittings are listed in Tables 6.5, 6.6 and 6.7, re-
spectively.
Note that the energy splittings are not Lorentz scalars. Using (5.30), we expect that
the splitting between two states A and B at zero lattice momentum is given by
∆Ev(0)|v ≡ E
A
v (0)− E
B
v (0) =
√
(2Zpγmv)2 + (MAkin)
2
−
√
(2Zpγmv)2 + (MBkin)
2. (6.20)
If we set Zp = 1 and expand the splitting at velocity v relative to v = 0 in powers of the
78 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
|v| ∆Ev(0)|v
∆Ev(0)|v
∆Ev(0)|0
0.0 0.031469(98) 1
0.2 0.03039(20) 0.9656(71)
0.4 0.02837(85) 0.901(27)
0.6 0.0281(28) 0.894(88)
Table 6.7: Υ(1S)− ηb(1S) energy splitting as a function of the boost velocity.
0 0.2 0.4 0.6v
0
0.5
1
1.5
∆E v
/ ∆E
0
Υ(2S) − Υ(1S)χb1(1P) − Υ(1S)Υ(1S) − ηb(1S)
1 − 0.5v 2
Figure 6.3: Bottomonium energy splittings relative to v = 0 as a function of the boost
velocity. Points are offset horizontally for legibility. The data agree with an estimate for
the leading v2 dependence (see text).
6.2. Tests with Bs mesons 79
|v| ∆Ev(0)|1 ∆Ev(0)|2 ∆Ev(0)|3
0 −0.000009(63) −0.000039(68) 0.000053(73)
0.2 −0.00012(26) −0.00005(28) 0.00017(30)
0.4 −0.00046(56) 0.00055(62) −0.00010(57)
0.6 −0.0176(96) 0.0107(62) 0.0069(75)
Table 6.8: Dependence of the Υ(1S) energy on the polarisation direction. ∆Ev(0)|j is the
difference between Ev(0)|j and the polarisation-averaged energy.
boost velocity, we obtain
∆Ev(0)|v
∆Ev(0)|0
= 1−
(
2m2
MAkin M
B
kin
)
︸ ︷︷ ︸
≈0.5
v2 +O(v4), (6.21)
that is, we expect a quadratic decrease like 1 − 0.5|v|2. The numerical results, shown in
Fig. 6.3, are consistent with this estimate as desired.
Finally, for the Υ(1S) meson, I studied the dependence of the energy on the po-
larisation direction. If moving NRQCD works well, then there should be no difference
for polarisations parallel and perpendicular to the boost velocity. In Table 6.8 I show
the difference between the energy with definite polarisation direction, Ev(0)|j and the
polarisation-direction-averaged energy 13(Ev(0)|1 + Ev(0)|2 + Ev(0)|3). No significant de-
pendence on the polarisation direction can be seen (except maybe at |v| = 0.6, where a
1.8σ deviation in the energies was found).
6.2 Tests with Bs mesons
The results given in Sec. 6.1 show that lattice mNRQCD works very well as far as bot-
tomonium is concerned. From bottomonium correlators one can extract mNRQCD renor-
malisation parameters with high precision.
The main application of mNRQCD is the calculation of heavy-light form factors, and
it is therefore important to perform tests of mNRQCD also for heavy-light mesons. In the
following, I report on such tests involving Bs mesons.
80 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
6.2.1 Calculation of the two-point functions
Starting with the standard Dirac fields, we construct interpolating fields for the Bs and
B∗s mesons with momentum p from
OΓ(τ,p) =
1
√
V
∑
x,y
Ψl(τ,x)Γ(x− y)ΨH(τ,y)e
−ip·y , (6.22)
where Ψl is the Dirac spinor for the valence strange quark and ΨH is the Dirac spinor
for the b quark. I used Γ(r) = γˆ5 f(r) for the Bs pseudoscalar meson, Γ(r) = γˆj f(r)
with j = 1, 2, 3 for the B∗s vector meson and Γ(r) = γˆ
5γˆ0 f(r) for the computation of the
decay constant fBs . I compute 2×2 matrix correlators with Gaussian and local smearing,
f(r) = e−|r|
2/r2s and f(r) = δr,0.
In terms of the standard Dirac propagators, the two-point function reads
〈OΓsk(τ,p)O
†
Γsc(τ
′,p)〉 =
1
N
∑
U
1
V
∑
x,y,x′,y′
× Tr
[
Γsk(x− y)Gl
(
x′, x
)
Γ†sc(x
′ − y′)GH
(
y, y′
) ]
× e−ip·yeip·y
′
, (6.23)
with x = (τ,x), y = (τ,y), x′ = (τ ′,x′), y′ = (τ ′,y′). The tree-level leading-order
mNRQCD field redefinition (6.4) leads to the following expression for the b propagator:
GH
(
y, y′
)
= θ(τ − τ ′)
1
γ
e−γm(τ−τ
′)+iγmv·(y−y′) S(Λ)
(
Gψv(y, y
′) 0
0 0
)
S(Λ)
−θ(τ ′ − τ)
1
γ
e+γm(τ−τ
′)−iγmv·(y−y′) S(Λ)
(
0 0
0 Gξv(y, y
′)
)
S(Λ).
(6.24)
For the light quark, I used the ASQTAD (order a2, tadpole improved) staggered fermion
action [123, 124, 125]. The 4-component naive light quark propagator [111] can be obtained
from the 1-component staggered propagator Gχ(x′, x) via
Gl(x
′, x) = Gχ(x
′, x) Ω(x′)Ω†(x) (6.25)
with Ω(x) as defined in Eq. (2.59),
Ω(x) = (γ0)x0(γ1)x1(γ2)x2(γ3)x3
= (γˆ0)x0(−iγˆ1)x1(−iγˆ2)x2(−iγˆ3)x3 . (6.26)
6.2. Tests with Bs mesons 81
We also employ γˆ5-Hermiticity
Gl(x
′, x) = γˆ5G†l (x, x
′)γˆ5, (6.27)
to interchange the points x and x′ for the light quark propagator. As before, we remove
the factor of e−γm(τ−τ
′) and the summation over x′.
In the case where Γsk and Γsc contain the same Dirac matrix, we arrive at the following
expression (for τ > τ ′):
C(Γsk,Γsc,k, τ, τ
′) =
1
N
∑
U
1
γ
∑
x,y
fsk(x− y)e
−ik·yη(x, x′)
× Tr
[
G†χ(x, x
′) S(Λ)Ω(x′)Ω†(x)S(Λ)
(
G˜ψv(y, x
′) 0
0 0
)]
,
(6.28)
with k = p− γmv and
G˜ψv(y, x
′) =
∑
y′
f(x′ − y′)eik·y
′
Gψv(y, y
′). (6.29)
The phase factor η(x, x′) in (6.28) depends on the Dirac matrix in Γsk and Γsc. It is given
by
η(x, x′) =
1 for γˆ5,
(−1)x
′
j−xj for γˆj ,
(−1)
P
j(xj+x
′
j) for γˆ5γˆ0.
(6.30)
As before, I set f(r) to zero for |r| > Rs with some cut-off radius Rs smaller than half the
length of the lattice.
I performed the heavy-light simulations with the same gauge configurations as the
heavy-heavy simulations described in Sec. 6.1), using the same heavy-quark action and
parameters. Again, the boost velocity was always pointing in x-direction. The valence
strange quark mass for the Bs and B∗s mesons was set to 0.040. I used four staggered
propagators with source times τ ′ = 0, 16, 32, 48 for each gauge configuration. I computed
both forward- and backward-propagating meson correlators to increase statistics. The
smearing parameter rs was set to 2.5.
6.2.2 Fitting
The staggered/naive light quark action used here suffers from the doubling problem. As
shown in [111] for non-moving NRQCD, the spatial doublers do not contribute to the
heavy-light correlators2. However, the temporal doubler leads to a coupling to additional
2 This remains true with moving NRQCD for moderate values of γm; see the remarks in Sec. 5.2
82 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
|v| Zp Mkin Cv/(γm)
0 3.37(15) 1.002(52)
0.2 1.05(15) 3.72(47) 1.13(16)
0.4 1.05(18) 3.66(68) 1.10(23)
Table 6.9: Bs results for Mkin, Zp, Cv.
opposite-parity states, which manifest themselves as oscillating exponentials in the correla-
tors. I therefore fitted the heavy-light correlators C(Γsk,Γsc,k, τ, τ ′) for a given momentum
k with a function of the form
nexp−1∑
n=0
An(Γsk)A
∗
n(Γsc)e
−Ev,n(τ−τ ′)+(−1)τ−τ
′+1
enexp−1∑
n=0
A˜n(Γsk)A˜
∗
n(Γsc)e
− eEv,n(τ−τ ′), (6.31)
where the parameters denoted with a “∼” correspond to the opposite-parity states.
6.2.3 Kinetic mass, energy shift and external momentum renormalisa-
tion
Results for the Bs kinetic massMkin and the renormalisation parameters Zp, Cv computed
with Eqs. (5.33), (5.32), (5.31) for nQ = 1 are shown in Table 6.9. The energies and
the amplitude required for the calculation of the decay constant were obtained from 8-
exponential (4 of which are oscillating) fits to 2× 2 matrix correlators with the Gaussian
smearing and the local axial current. Examples of fits to these correlators at v = 0 and
|v| = 0.4 are shown in Fig. 6.4. These also demonstrate the worsening of the signal-
to-noise ratio as the boost velocity increases, in accordance with Eq. (5.38) (for the Bs
mesons considered here, one has to replace mpi in (5.38) by mss¯).
For the calculation of Cv, I again averaged the results over the 4 different lattice
momenta perpendicular to v
k⊥ =
2pi
L
(0,±1, 0),
2pi
L
(0, 0,±1), (6.32)
and the momentum parallel to the boost velocity required for the determination of Zp was
chosen to be k‖ =
2pi
L (1, 0, 0).
As expected, the statistical errors are larger than for the heavy-heavy mesons. In ad-
dition to the generally worse signal-to-noise ratio for heavy-light mesons, a much smaller
6.2. Tests with Bs mesons 83
0 5 10 15 20 25
time
1e-08
1e-06
0.0001
0.01
1
100
valu
e of
cor
rela
tor
local - localsmeared - locallocal - smearedsmeared - smeared
0 5 10 15 20 25
time
1e-08
1e-06
0.0001
0.01
1
100
valu
e of
cor
rela
tor
local - localsmeared - locallocal - smearedsmeared - smeared
Figure 6.4: Fits to Bs matrix correlators (local and Gaussian smearing) at k = 0 and
v = 0 (upper panel), |v| = 0.4 (lower panel). The fits have nexp = n˜exp = 4 and τmin = 2.
The noise in the two-point function increases exponentially with the Euclidean time as
expected.
84 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
|v| |p| fBs
0 0 0.1626(27)
0.2 0.576(11) 0.1608(52)
0.4 1.2163(96) 0.1634(94)
0.6 1.885(57) 0.174(17)
Table 6.10: Bs decay constant (unrenormalised, and in lattice units) with mNRQCD at
k = 0. The total momentum p is computed using the nonperturbative result for Zp from
the ηb(1S) dispersion relation.
number of origins (four) per gauge configuration was used to save computer time (the com-
putation of light-quark propagators is more expensive than the computation of mNRQCD
propagators). The results for Zp and Cv agree with those obtained using heavy-heavy
mesons in section 6.1.3.
6.2.4 Decay constant
The Bs decay constant is defined by
〈0|Jµ5 (0)|Bs,p〉 = ifBs p
µ, (6.33)
where Jµ5 (0) = s(0)γˆ
µγˆ5b(0). I computed fBs in a similar way as the ηb decay constant in
Sec. 6.1.4, using the temporal component of the current. As there is only one heavy quark
here, the equation analogous to (6.19) reads
fBs = A
√
2
Ev(k) + Cv
. (6.34)
Here I used Cv determined from the ηb(1S) dispersion relation since this is more precise
than the result from the Bs dispersion relation.
The results for the decay constant fBs at k = 0 and |v| = 0, 0.2, 0.4, 0.6 are listed
in Table 6.10 and plotted against the total momentum in Fig. 6.5. I find that the decay
constant is independent of the boost velocity within statistical errors, which demonstrates
that the combination of moving NRQCD and ASQTAD works well at these values of v. I
could not obtain reliable fits at large k and v = 0, and therefore do not show results with
non-moving NRQCD here.
6.2. Tests with Bs mesons 85
0 0.5 1 1.5 2p
0
0.05
0.1
0.15
0.2
0.25
f
Figure 6.5: The Bs decay constant at k = 0 and |v| = 0, 0.2, 0.4, 0.6 plotted against the
total momentum p = Zpγmv + k. The horizontal line indicates the value at v = 0.
6.2.5 Energy splittings
I also computed the B∗s − Bs energy splitting as a function of |v|; the results are shown
in Table 6.11. The statistical errors are so large that no definite statement can be made
about the velocity dependence.
|v| ∆Ev(0)
∆Ev(0)
∆E0(0)
0.0 0.0261(35) 1
0.2 0.0262(65) 1.00(28)
0.4 0.0310(80) 1.18(34)
Table 6.11: B∗s −Bs energy splitting as a function of the boost velocity.
86 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
6.3 Comparison of nonperturbative and perturbative re-
sults
In the following I compare my nonperturbative results for Zp (the renormalisation of the
external momentum) and Cv (the energy shift) given in Secs. 6.1.3 and 6.2.3 to predictions
from one-loop lattice perturbation theory [114, 100].
The perturbative results have the form
Zp = 1 + αs δZp, Cv = 1 + αs δCv, (6.35)
where δZp and δCv depend on the form of the heavy-quark and gluon actions as well as
the heavy-quark mass and the boost velocity. Results for δZp and δCv for the setup used
here (full moving NRQCD action (5.12, 5.13) with m = 2.8, n = 2, Lu¨scher-Weisz gluon
action) can be found in [114, 100].
To make use of (6.35), a numerical value of the strong coupling constant αs is needed.
In [114, 100], the strong coupling constant was defined in the potential scheme [77], and the
momentum scale q? was calculated with the Brodsky-Lepage-Mackenzie procedure [126]
for each quantity and each value of v. The q? values range approximately between 0.5/a
and 3/a. The coarse MILC configurations used here have a−1 ≈1.6 GeV [108]. Using the
running of the strong coupling constant αV (q) [127] this gives, for example, αV (2/a) ≈ 0.3.
Figures 6.6 and 6.7, which are taken from [100], show the perturbative and nonper-
turbative results for Zp and Cv. Nonperturbative results from both bottomonium and Bs
two-point functions are shown, with a slight horizontal offset for legibility. The uncer-
tainties shown on the data points for the perturbative results are purely statistical due to
the stochastic integration over the loop momentum. The error band on the perturbative
results is obtained by varying the scale in the range [q?/2, 2q?].
The heavy-light results have error bars so large that they agree with the perturbative
predictions. In the following, I focus on the statistically much more precise heavy-heavy
results. For Zp, these agree with the perturbative predictions at |v| = 0.2 and |v| = 0.4,
while there is a deviation at |v| = 0.6. For Cv/(γm), the (small) correction to the tree-
level result Cv/(γm) = 1 is in fact found to have the opposite sign for the nonperturbative
results compared to the perturbative results at all values of v.
This indicates that there may be sizable higher-loop or nonperturbative contributions.
To investigate the two-loop contribution, high-β simulations (cf. Sec. 5.3.5) are currently
being performed [116].
6.3. Comparison of nonperturbative and perturbative results 87
0.8 1
1.2
1.4
0
0.2
0.4
0.6
Z
p
fram
e ve
locit
y v
pertu
rbati
on th
eory
non
pertu
rbati
ve, h
eavy
-hea
vy
non
pertu
rbati
ve, h
eavy
-ligh
t
Figure 6.6: Comparison of nonperturbative and perturbative results for the external mo-
mentum renormalisation factor Zp. (Plot by Eike Mu¨ller [114], my nonperturbative data.)
0.8 1
1.2
1.4
1.6
0
0.2
0.4
0.6
C
v
/(γm)
fram
e ve
locit
y v
pertu
rbati
on th
eory
non
pertu
rbati
ve, h
eavy
-hea
vy
non
pertu
rbati
ve, h
eavy
-ligh
t
Figure 6.7: Comparison of nonperturbative and perturbative results for the energy shift
Cv, relative to the tree-level value γm. (Plot by Eike Mu¨ller [114], my nonperturbative
data.)
88 CHAPTER 6. NONPERTURBATIVE TESTS OF MOVING NRQCD
Chapter 7
Heavy-hadron spectroscopy
Before coming to the main application of lattice moving NRQCD, the calculation of B
decay form factors in Chapter 8, I will report in the following on a different project: the
calculation of masses and excited-state energies of hadrons containing b quarks. Here,
the hadrons considered are at rest, and so I use the standard non-moving lattice NRQCD
action which is obtained from the action described in Chapter 5 by setting v = 0.
The calculations described in the following make use of gauge field configurations gen-
erated by the RBC and UKQCD collaborations [128]. These configurations were created
with the renormalisation-group improved Iwasaki gluon action [129, 130] and 2+1 flavours
of light quarks described by a domain wall action (see Sec. 2.4.7).
The first part of this chapter (Sec. 7.1) is about bottomonium, i.e. the bound states of
bottom quark-antiquark pairs. This work is published in my paper [131]. The spectrum of
bottomonium is known very well from experiment [83], as can be seen in the level scheme
shown in Fig. 7.1. There are many narrow states and the energy splittings show little
dependence on the light quark masses once these are light enough. Thus, the calculation
of the bottomonium spectrum is an excellent way of testing the lattice methods used. It
also allows independent determinations of the lattice spacing of the RBC/UKQCD gauge
field ensembles.
The second part of this chapter (Sec. 7.2) then describes mass-calculations of b-
flavoured hadrons, using the domain wall action for the light valence quarks. This is still
work in progress; I published my preliminary results in the proceedings of the Lattice 2009
conference [132]. The hadrons considered include B mesons, singly- and doubly-bottom
baryons, and for the first time also the triply-bottom baryon Ωbbb. There is currently con-
siderable interest in mass predictions for b-flavoured baryons. A few singly-bottom baryons
have been found so far, and more results are expected from the Large Hadron Collider
90 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
=
BB threshold
(4S)
(3S)
(2S)
(1S)
(10860)
(11020)
hadrons
hadrons
hadrons
γ
γ
γ
γ
ηb(3S)
ηb(2S)
χb1(1P)
χb2(1P)
χb2(2P)
hb(2P)
ηb(1S)
JPC 0−+ 1−− 1+− 0++ 1++ 2++
χb0(2P)
χb1(2P)
χb0(1P)
hb (1P)
Figure 7.1: Level scheme of bb states, from the Particle Data Group [83]
7.1. Bottomonium 91
aml mpi [MeV] u0L MD range (step) nconf
0.005 330 0.8439 915 - 8665 (25) 311
0.01 420 0.8439 1475 - 8525 (25) 283
0.02 560 0.8433 1800 - 3600 (25) 73
0.03 670 0.8428 1275 - 3050 (25) 72
Table 7.1: The ensembles of RBC/UKQCD gauge configurations used for my calculation
of the bottomonium spectrum.
at CERN. Most recently, the Ωb baryon was discovered at Fermilab. There are now two
incompatible results for its mass, obtained by the D/0 [133] and CDF [134] collaborations.
Lattice QCD can contribute to resolve this discrepancy.
7.1 Bottomonium
7.1.1 Lattice details
The details of the domain wall fermion and Iwasaki gauge actions adopted by the RBC
and UKQCD collaborations are given in [135]. For the calculation of the bottomonium
spectrum I used the gauge configurations of size 243 × 64 as described in [128]. The
size of the fifth dimension in the domain wall action is Ls = 16, leading to a residual
mass of amres = 0.00315(2) [128]. The strange quark mass is ams = 0.04 and there are
ensembles with four different values for the degenerate light (up and down) quark mass
aml corresponding to pion masses in the range from ≈ 330 MeV to ≈ 670 MeV, as shown
in Table 7.1.
I started the “measurements” at the same molecular dynamics (MD) time as in [128]
to ensure complete thermalisation. Note however that the aml = 0.005 and aml = 0.01
ensembles have since been extended and I included the additional configurations. I per-
formed the measurements every 25 steps of MD time. This is approximately two times
the integrated autocorrelation time found in [128] for the 12th time slice of the pion cor-
relator on the aml = 0.005 ensemble. I confirmed that this separation gives sufficiently
independent measurements for the observables calculated here by doing a binning analysis.
As discussed in Chapter 5, the lattice NRQCD action is tadpole-improved, using the
mean link in Landau gauge u0L. I measured u0L for the different ensembles by transforming
subsets of configurations to Landau gauge (using the Chroma software [136]); the results
are listed in Table 7.1.
92 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
Name L S J P C RPC Γ(r)
ηb(nS) 0 0 0 − + A
−+
1 φnS(r)
Υ(nS) 0 1 1 − − T−−1 φnS(r) σ
j
hb(nP ) 1 0 1 + − T
+−
1 φnP (r) r
j/rs
χb0(nP ) 1 1 0 + + A
++
1 φnP (r) (r · σ)/rs
χb1(nP ) 1 1 1 + + T
++
1 φnP (r) (r× σ)
j/rs
χb2(nP ) 1 1 2 + + T
++
2 φnP (r) (r
jσk + rkσj)/rs
ηb(nD) 2 0 2 − + T
−+
2 φnD(r) r
jrk/r2s
Υ2(nD) 2 1 2 − − E−− φnD(r) (rjrkσl − rkrlσj)/r2s
Table 7.2: The smearing functions Γ(r) (with j 6= k, l 6= k) and the corresponding
irreducible representations of the octahedral group.
7.1.2 Calculation of the two-point functions
In Sec. 6.1.1 the computation of bottomonium two-point functions was discussed for mov-
ing NRQCD. Here, I use non-moving NRQCD, which is just a special case. The two-point
functions are computed as
C(Γsk,Γsc,k, τ, τ
′) =
1
N
∑
U
∑
x,y
Tr
[
G†ψ(τ,x, τ
′,x′) Γ†sk(x− y) G˜sc(τ,y, τ
′,x′)
]
e−ik
x+y
2
(7.1)
where τ > τ ′ and
G˜sc(τ,y, τ ′,x′) =
∑
y′
Gψ(τ,y, τ
′,y′) Γsc(x′ − y′) eik
x′+y′
2 . (7.2)
In Eqs. (7.1) and (7.2), the functions Γsc/sk are the smearing functions at source and sink,
respectively, which are now expressed directly in the non-relativistic spinor basis, i.e. they
are (2 × 2)-matrix-valued in spinor space. The gauge field configurations were fixed to
Coulomb gauge using Chroma [136]. In Table 7.2 the bottomonium states considered here
are listed, together with their continuum quantum numbers, smearing functions Γ(r) and
representations of the octahedral group [137].
As can be seen in Table 7.2, all representations are chosen to be different, so that no
mixing is expected here. As in [108], the radial functions φnS(r), φnP (r) and φnD(r) for
7.1. Bottomonium 93
State φ(r)
1S exp[−|r|/rs]
2S [1− |r|/(2rs)] exp[−|r|/(2rs)]
3S
[
1− 2|r|/(3rs) + 2|r|2/(27r2s)
]
exp[−|r|/(3rs)]
1P exp[−|r|/(2rs)]
2P [1− |r|/(6rs)] exp[−|r|/(3rs)]
1D exp[−|r|/(3rs)]
Table 7.3: The radial functions φ(r)
the n-th radially excited S-wave (L = 0), P -wave (L = 1) and D-wave (L = 2) states are
taken from the corresponding hydrogen atom wave functions and are given in Table 7.3.
The same lattice representations are used at source and sink but the radial smearing
functions are allowed to be different. I set the smearing parameters rs (in lattice units) to
1.0 (1S), 0.8 (2S), 0.6 (3S), 0.5 (1P ), 0.4 (2P ) and 0.5 (1D), respectively.
As in Sec. 6.1.1, I set the smearing functions to zero outside a ball with radius R smaller
than half the spatial lattice dimension, so that the wrapping around the lattice boundaries
does not cause any problems. Since the smearing functions decay exponentially with the
separation between quark and antiquark, R can be chosen such that the important features
remain. To ensure symmetry, the same cut-off radius must be taken at source and sink.
In order to increase statistics, I averaged the correlators (7.1) over eight different spatial
origins x′ located at the corners of a cube with side length L/2 = 12. In addition, I used
four different source time slices τ ′ with an equal spacing of 16, so that there are 32 origins
per configuration in total. Furthermore, I shifted the locations of the origins on the lattice
randomly from configuration to configuration in order to decrease autocorrelations.
7.1.3 Data analysis
After choosing a set of smearing functions Γ(r) with equal lattice representations but
different radial functions (e.g. 1S, 2S and 3S), I computed the square matrix of correlators
obtained by taking all combinations for source and sink.
As in Sec. 6.1.2, I fitted the matrix of correlators C(Γsk,Γsc,k, τ − τ ′) simultaneously
by a function of the form (6.11). Here I took the logarithms of the energy splittings as
94 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
0 5 10 15 20
time
0
5
10
15
20
25
30
cor
rela
tor
1S, 1S2S, 1S3S, 1S1S, 2S2S, 2S3S, 2S1S, 3S2S, 3S3S, 3S
Figure 7.2: Fit of a 3×3 matrix of correlators with the {Υ(1S), Υ(2S), Υ(3S)} smearings.
Note that both axes have a linear scale. The fit has nexp = 10 and τmin = 1.
the fit parameters to ensure the correct ordering of the states.
As before, I used the Bayesian method described in Sec. C.3, where the number of
exponentials in the fit is increased until the results for the low-lying states stabilise. This
is demonstrated for a 3× 3 matrix correlator in the Υ channel in Fig. C.1. A plot of the
corresponding fit with nexp = 10 is shown in Fig. 7.2.
I also tested for the presence of autocorrelations in the measurements using the binning
method (cf. Sec. C.5). In most cases I did not find significant autocorrelations, but for
some quantities the error estimates were slightly corrected upwards. The details can be
found in my paper [131].
7.1.4 Tuning of the b quark mass
The bare b quark mass, which is a free parameter in the NRQCD action, was tuned non-
perturbatively. I adjusted it such that the kinetic mass of the ηb(1S) meson as calculated
on the lattice matches the experimental value of 9.389(5) GeV [138]. The tuning was done
on the most chiral (aml = 0.005) ensemble of gauge configurations.
I computed the kinetic mass from
Mkin =
k2 − [E(k)− E(0)]2
2 [E(k)− E(0)]
, (7.3)
7.1. Bottomonium 95
amb aMkin(ηb)
Υ(2S)−Υ(1S)
splitting
2.30 4.988(12) 0.3258(47)
2.45 5.281(13) 0.3242(46)
2.60 5.575(13) 0.3231(54)
Table 7.4: Results for the tuning of the bare b quark mass in lattice units. Errors are
statistical/fitting only.
where I used the smallest possible lattice momentum a|k| = 1 · 2pi/L. As shown in the
next section, the kinetic mass is very stable and shows no significant dependence on k
even for much larger momenta. In order to increase statistics I averaged the results over
the different possibilities for the direction of k.
The comparison with experiment of course requires the knowledge of the lattice spac-
ing, which I determined as the ratio of the experimentally measured Υ(2S)−Υ(1S) mass
splitting, 0.56296(40) GeV [83], to the dimensionless lattice result. This will be discussed
in more detail in Sec. 7.1.6.
The lattice results for aMkin and the Υ(2S) − Υ(1S) splitting at the three different
bare quark masses amb = 2.30, 2.45 and 2.60 are shown in Table 7.4. As can be seen,
the Υ(2S)−Υ(1S) splitting is very insensitive to the value of the b quark mass. It is also
expected to have much smaller lattice discretisation errors than the 1P − 1S splitting as
discussed in the next section.
I found that in the range considered here, the dependence of the kinetic mass on the
bare heavy quark mass is described very well by the linear relation
aMkin = A+B · amb. (7.4)
A plot of aMkin as a function of amb is shown in Fig. 7.3. I performed fits of Eq. (7.4)
with the parameters A and B on 500 bootstrap samples for the kinetic masses at amb =
2.30, 2.45 and 2.60. The resulting average fit parameters are
A = 0.489(25),
B = 1.956(11). (7.5)
To obtain a first result for the lattice spacing of the aml = 0.005 ensemble, I used the
Υ(2S) − Υ(1S) mass splitting at amb = 2.45, giving a−1 = 1.736(25) GeV (the error is
96 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
4.8
5
5.2
5.4
5.6
5.8
2.2 2.3 2.4 2.5 2.6 2.7
aM
ki
n
amb
Figure 7.3: The kinetic mass of the ηb(1S) meson plotted against the bare heavy quark
mass. Errors are statistical/fitting only. The line shows the average over the bootstrap
ensemble of linear fit results.
statistical/fitting only). Of course the b quark mass was not yet tuned, but given the
relative independence of the Υ(2S)−Υ(1S) splitting on mb, the value of amb = 2.45 was
sufficiently close to the physical value. The final results for the lattice spacing obtained
with the correct b quark mass will be presented in Sec. 7.1.6.
Using the preliminary result for a−1, it follows that the ηb(1S) mass in lattice units
must be tuned to be aMkin = 5.407(77). Inserting this into (7.4) and solving for amb gives
amb = 2.514(36). (7.6)
The error quoted here is statistical/fitting only and is dominated by the uncertainty in
the lattice result for the Υ(2S)−Υ(1S) splitting.
I actually performed all remaining calculations for bottomonium with amb = 2.536.
This was an earlier result and the fits have been improved slightly since then. However it
is still inside the range of the new value (7.6).
For amb = 2.536 the results were aMkin = 5.449(13) and a−1 = 1.740(25) GeV. This
gives Mkin = 9.48(14) GeV which is compatible with the experimental result of 9.389(5)
GeV, confirming the successful tuning of the heavy quark mass.
7.1. Bottomonium 97
n2 aMkin(ηb) aC c2
1 5.450(17) 2.5913(84) -
2 5.450(17) 2.5912(85) 1.00003(85)
3 5.450(18) 2.5911(92) 1.0001(16)
4 5.461(22) 2.597(11) 0.9981(21)
5 5.457(20) 2.595(10) 0.9987(24)
6 5.452(20) 2.592(10) 0.9997(27)
8 5.454(22) 2.593(11) 0.9993(35)
9 5.447(20) 2.590(10) 1.0005(35)
12 5.445(21) 2.589(11) 1.0009(42)
Table 7.5: Kinetic mass, NRQCD energy shift and the square of the “speed of light”
for various lattice momenta k = n · 2pi/L, calculated on the aml = 0.005 ensemble with
amb = 2.536.
7.1.5 Speed of light
In order to examine how well the lattice data approximates the relativistic continuum
dispersion relation, I computed the kinetic mass of the ηb(1S) meson, defined by (7.3),
also for larger lattice momenta k = n · 2pi/L up to n2 = 12. For these calculations, I used
the local smearing function Γ(r) = δr,0 at source and sink so that multiple lattice momenta
can be obtained with little computational cost. For each value of n2, I averaged the results
over the possible directions of the vector n, and all components of n were chosen to be
less than or equal to 2.
The results are given in Table 7.5, where also the NRQCD energy shift, calculated as
C =
Mkin(k)− E(0)
2
, (7.7)
and for n2 > 1 the square of the “speed of light”
c2 ≡
[E(k)− E(0) +Mkin,1]
2 −M2kin,1
k2
(7.8)
are shown. In Eq. (7.8), Mkin,1 denotes the kinetic mass calculated with n2 = 1. In
the units used here, one should have c2 = 1. Deviations of c2 from 1 can be caused by
discretisation errors in the NRQCD, gluon and sea quark actions and also by missing
higher order relativistic corrections in the NRQCD action. The NRQCD action is highly
improved at tree level, and so the most significant errors one expects here are those caused
by missing radiative corrections.
98 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
As can be seen in the table, in the momentum range considered here the kinetic
mass shows no significant dependence on k within the small statistical/fitting errors.
Correspondingly, c2 remains compatible with 1, with statistical/fitting errors less than
0.5%, indicating that the effect of the errors mentioned above is small.
Analogous calculations for the Υ(1S) meson have been performed in [108] with the
same NRQCD action but with the Lu¨scher-Weisz gluon and the ASQTAD sea quark
action. There, the deviation of c2 from 1 in the same momentum range was also found to
be compatible with 1 within statistical errors of less than 1%.
7.1.6 Radial/orbital energy splittings and the lattice spacing
The lattice results for the various radial and orbital energy splittings are listed in Table
7.6. Systematic errors are known to be smallest for the spin-averaged masses, defined as
〈M〉 =
∑
J(2J + 1)MJ∑
J(2J + 1)
. (7.9)
However, in most cases not all of the states entering Eq. (7.9) are known from experiment.
For the 1S, 2S and 3S masses in this section I consider the J = 1 states (Υ) instead of the
spin-averages. Note that the J = 0 S-wave states (ηb) enter the spin-averaged masses only
with a weight of 1/4, and so the influence of systematic errors in the hyperfine splittings
is negligible here. For the 1P and 2P masses, I use the spin-averages over the χb triplet
(J = 0, 1, 2) states. The only experimentally known D-wave state [139] is Υ2(1D) with
J = 2, and therefore I consider this state here.
In terms of the NRQCD power counting (cf. Sec. 3.2.2), radial and orbital energy
splittings are of order O(v2orb.), where vorb. is the internal speed of the b quarks inside the
heavy-heavy meson. Recall that for bottomonium one has v2orb. ≈ 0.1. The lattice NRQCD
action in use includes all relativistic corrections of order O(v4orb.) (at tree-level), and hence
the missing relativistic corrections are of order O(v6orb.). Naively this leads to relativistic
errors for the radial and orbital splittings of O(v4orb.) = 1%. However, as discussed in [108],
for energy splittings one has to consider the difference between the expectation values of
the missing operators for the two states. This leads to a reduction of the relativistic errors
for the 2S − 1S splitting to about 0.5%.
Additional systematic errors for the NRQCD action are due to discretisation errors and
missing radiative corrections (beyond tadpole improvement). Estimates of these errors for
the 2S − 1S and 1P − 1S splittings are given in Table 7.7. They are taken to be equal to
the estimates obtained in [108] for exactly the same lattice NRQCD action on the “coarse”
MILC gauge configurations, which have a lattice spacing (a−1 ≈ 1.6 GeV) very similar to
7.1. Bottomonium 99
aml = 0.005 aml = 0.01 aml = 0.02 aml = 0.03
Υ(2S)−Υ(1S) 0.3236(46) 0.3270(73) 0.330(18) 0.327(23)
Υ(3S)−Υ(1S) 0.517(21) 0.537(23) - -
〈χb(1P )〉 −Υ(1S) 0.2589(30) 0.2572(22) 0.2628(57) 0.2613(61)
〈χb(2P )〉 −Υ(1S) 0.478(30) 0.502(26) 0.511(39) 0.516(37)
〈χb(2P )〉 − 〈χb(1P )〉 0.219(29) 0.245(24) 0.248(35) 0.255(33)
Υ2(1D)−Υ(1S) 0.4080(46) 0.4194(42) 0.417(12) 0.426(12)
Table 7.6: Results for the radial and orbital energy splittings in lattice units. Errors are
statistical/fitting only.
2S − 1S 1P − 1S
relativistic 0.5% 1.0%
radiative 0.5% 1.7%
discretisation 0.8% 3.2%
total 1.1% 3.8%
Table 7.7: Estimates of the systematic errors due to the lattice NRQCD action for the
2S − 1S radial and 1P − 1S orbital splittings [108].
100 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
aml = 0.005 aml = 0.01 aml = 0.02 aml = 0.03
a−12S−1S (GeV) 1.740(25)(19) 1.722(38)(19) 1.708(92)(19) 1.72(12)(2)
a−11P−1S (GeV) 1.698(19)(65) 1.709(15)(65) 1.673(36)(64) 1.682(40)(64)
Table 7.8: Results for the inverse lattice spacing obtained from the Υ(2S) − Υ(1S) and
〈χb(1P )〉 − Υ(1S) splittings. The first error given is statistical/fitting and the second
is an estimate of the systematic errors (relativistic, radiative and discretisation) due to
the NRQCD action. Systematic errors due to the gluon and sea quark actions are not
included.
the ensembles considered here. The reader is referred to [108] and [140] for the details.
As can be seen in the table, systematic errors are much smaller for the 2S − 1S splitting
compared to the 1P − 1S splitting. This is due to the smaller difference in the wave
functions for the 2S and 1S states. The 2S − 1S splitting thus allows a more reliable
determination of the lattice spacing.
Note that there are also discretisation errors due to the gluon and sea quark actions.
These are difficult to quantify at this stage as only data from one lattice spacing are
available. Gauge configurations with a smaller lattice spacing are currently being gen-
erated by the RBC and UKQCD collaborations so that a more systematic analysis will
become possible in the future. In [128], a preliminary error estimate of (aΛQCD)2 ≈ 4%
for the calculations of light hadron properties on the current ensembles was given. The
calculations for bottomonium are different in that the domain wall action only enters via
the sea quarks. The Iwasaki gluon action [129, 130] is renormalisation-group-improved
and is therefore expected to have a better scaling behaviour than the unimproved Wilson
action. However, this depends on the observable considered; see e.g. [141] for a scaling
study of the critical temperature and glueball masses. The stability of the “speed of light”
demonstrated in Sec. 7.1.5 provides some evidence for the smallness of the effect of gluon
discretisation errors for bottomonium.
For reference, the discretisation errors in the 2S − 1S and 1P − 1S splittings on the
coarse MILC lattices due to the Lu¨scher-Weisz gluon action were estimated in [108] to be
0.5% and 1.7%, respectively. These errors are proportional to the difference in the square
of the wave function at the origin, which is smaller between the 2S and 1S states.
My results for the inverse lattice spacings of the four ensembles from both the Υ(2S)−
Υ(1S) and the 〈χb(1P )〉 − Υ(1S) splittings are listed in Table 7.8. For the most chiral
ensemble the 2S − 1S splitting gives a−1 = 1.740(25)stat(19)syst GeV. No significant de-
7.1. Bottomonium 101
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
E
(G
eV
)
1S
2S
3S
1P
2P
1D
aml=0.005
aml=0.01
aml=0.02
aml=0.03
Figure 7.4: Radial and orbital energy splittings compared to the experimental results
(indicated by lines). Errors are statistical/fitting only and include the uncertainty in the
determination of the lattice spacing. The 1S and 2S masses, for which no error bars are
shown, are not predictions of the lattice calculation as these states are used to determine
the lattice scale and the overall energy shift.
pendence on the sea quark mass can be seen within the statistical errors, and therefore I
did not attempt an extrapolation. For comparison, the RBC and UKQCD collaborations
have obtained a−1 = 1.729(28)stat in the chiral limit, using the Ω− baryon mass [128].
This is consistent with my results obtained here.
Next, I used the lattice spacing determinations from the 2S − 1S splitting to convert
the other radial and orbital splittings from Table 7.6 to physical units. The results are
plotted in Fig. 7.4. Note that I used the individual results for the lattice spacings of the
different ensembles.
Overall, good agreement with experiment is seen, as in [108]. The dependence on the
light sea quark mass is found to be weak. This is expected since the typical gluon momenta
inside the bottomonium are much larger than all the values for the light quark masses
used here. However, note that large deviations between lattice results and experiment
were previously seen in quenched simulations (nf = 0), so the inclusion of 2+1 flavors of
dynamical light quarks is in fact very important. A comparison between quenched and
unquenched results can be found in [6].
102 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
aml = 0.005 aml = 0.01 aml = 0.02 aml = 0.03
Υ(1S)− ηb(1S) 0.03017(14) 0.03033(16) 0.03102(36) 0.03145(38)
Υ(2S)− ηb(2S) 0.0137(30) 0.0120(48) 0.013(12) 0.018(16)
χb0(1P )− 〈χb(1P )〉 −0.0207(20) −0.0206(18) −0.0231(36) −0.0175(70)
χb1(1P )− 〈χb(1P )〉 −0.0049(14) −0.0027(19) −0.0059(22) −0.0049(41)
χb2(1P )− 〈χb(1P )〉 0.0071(11) 0.0058(12) 0.0082(17) 0.0064(29)
hb(1P )− 〈χb(1P )〉 −0.0026(18) −0.0002(21) −0.0014(27) −0.0058(42)
χb1(1P )− χb0(1P ) 0.0158(18) 0.0176(25) 0.0173(40) 0.0126(77)
χb2(1P )− χb1(1P ) 0.0120(23) 0.0088(31) 0.0137(38) 0.0113(68)
hb(1P )− χb1(1P ) 0.0023(16) 0.0027(16) 0.0044(35) −0.0009(61)
Υ2(1D)− ηb(1D) 0.0011(21) −0.0012(18) −0.0086(70) −0.0050(61)
Table 7.9: Spin-dependent energy splittings in lattice units. Errors are statistical/fitting
only. Large systematic errors are expected as discussed in the text.
7.1.7 Spin-dependent energy splittings
The spin-dependent energy splittings in bottomonium, i.e. the fine and hyperfine structure,
are of order O(v4orb.) and hence any sub-leading corrections are missing in the NRQCD
action used here. Therefore, the relativistic errors in these splittings are expected to
be of order O(v2orb.) ≈ 10%. The spin-dependent energy splittings also receive radiative
corrections of order O(αs), the strong coupling constant at the scale set by the lattice
spacing. This may in principle lead to further systematic errors of the order of 20%,
although the tadpole improvement used here (see Sec. 2.7.1) reduces the problem. Finally,
discretisation errors are also expected to be larger than for the radial and orbital splittings,
especially for the S-wave hyperfine splitting as discussed below.
My results for the spin-dependent energy splittings in lattice units are summarised in
Table 7.9, where the errors given are statistical/fitting only.
S-wave hyperfine structure
Figure 7.5 shows a plot of the Υ(1S)−ηb(1S) and Υ(2S)−ηb(2S) energy splittings, where
I used the previous lattice spacing determinations from the 2S − 1S splittings to convert
to physical units.
The errors shown are statistical/fitting only but include the uncertainty in the de-
termination of the lattice spacing. The latter in fact enters with a factor of 2 here, as
7.1. Bottomonium 103
-80
-60
-40
-20
0
20
∆
E
(M
eV
)
Υ(1S)
ηb(1S)
Υ(2S)
ηb(2S)
aml=0.005
aml=0.01
aml=0.02
aml=0.03
Figure 7.5: S-wave hyperfine splittings (energies relative to the Υ(1S) and Υ(2S) states,
respectively) compared to experiment. Errors are statistical/fitting only and include the
uncertainty in the determination of the lattice spacing, which enters with a factor of 2.
Large systematic errors are expected as discussed in the text.
discussed in [140], due to the resulting uncertainty in the physical heavy quark mass (the
hyperfine splitting is approximately proportional to the inverse of that mass). The sta-
tistical error in the 1S hyperfine splitting is then dominated by far by this uncertainty,
while the 2S hyperfine splitting has an intrinsically higher statistical error as the state is
radially excited.
The Υ(1S)− ηb(1S) splitting has recently been measured by the BaBar collaboration
[138], who found 71.4+2.3−3.1(stat) ± 2.7(syst) MeV. This value is indicated in Fig. 7.5. The
lattice result in physical units for the aml = 0.005 ensemble is 52.5±1.5(stat) MeV, which
is too small by about 25%, in line with the large systematic errors expected. Similarly
to the radial and orbital splittings, little dependence on the light sea quark mass is seen,
which is expected for the same reason as discussed there.
Note that in [108] and [140] a significant dependence on the lattice spacing was found,
with the result increasing toward finer lattices, indicating that a substantial part of the
deviation is due to discretisation errors. The hyperfine splitting is indeed expected to be
sensitive to very short distances, as the spin-spin interaction potentials in simple models
contain a delta function at the origin (see e.g. [142]).
104 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
-60
-40
-20
0
20
40
∆
E
(M
eV
)
χb0(1P)
χb1(1P)
χb2(1P)
hb(1P)
aml=0.005
aml=0.01
aml=0.02
aml=0.03
Figure 7.6: P-wave spin splittings (energies relative to the spin-average of the χb(1P )
states) compared to experiment. Errors are statistical/fitting only and include the uncer-
tainty in the determination of the lattice spacing. Large systematic errors are expected as
discussed in the text.
Finally, note that in [143], where a relativistic heavy-quark action was used, the
Υ(1S) − ηb(1S) splitting on the same RBC/UKQCD gauge configurations was found to
be only 23.7± 3.7(stat) MeV. My result obtained with lattice NRQCD is much closer to
the experimental value.
P -wave spin-dependent splittings
A plot of the 1P spin-dependent splittings, converted to physical units using the previous
2S − 1S lattice spacing results, is given in Fig. 7.6.
I show the energy differences of the χb0(1P ), χb1(1P ), χb2(1P ) and hb(1P ) states to
the spin-average of the triplet 〈χb(1P )〉. The experimental results [83] for the triplet states
are also indicated in the plot; the hb states have not yet been observed.
The lattice results are found to be in relatively good agreement with experiment, even
within the purely statistical/fitting errors shown in the plot (those include the uncertainty
in the lattice spacing). This indicates that discretisation errors may be smaller than for
the S-wave hyperfine splittings. Note that in simple potential models the wave function
at the origin is zero (cf. the smearing functions in Table 7.2) and hence the P -wave spin
7.2. Bottom hadrons 105
splittings are expected to be not as sensitive to very short distances as the S-wave hyperfine
splittings.
My result for the experimentally unknown hb(1P ) − 〈χb(1P )〉 splitting on the aml =
0.005 ensemble is −4.5 ± 3.1 MeV, where the error quoted is statistical/fitting only and
includes the uncertainty from the determination of the lattice spacing.
D-wave spin-dependent splittings
Here, I only calculated the Υ2(1D)− ηb(1D) splitting, using the E−− and T
−+
2 represen-
tations as these two states do not mix and can be computed from the same heavy-quark
propagators.
My lattice results for the different ensembles are listed in Table 7.9. On the aml = 0.005
ensemble, I find the splitting in physical units to be 1.8±3.7 MeV where the error given is
statistical/fitting only and includes the uncertainty from the determination of the lattice
spacing. No experimental results are available.
7.2 Bottom hadrons
The results of the previous section demonstrate that the combination of lattice NRQCD
for the b quarks and the Iwasaki gluon / domain wall sea quark actions works well at this
lattice spacing. The next step is the calculation of heavy-light hadron masses on the same
gauge field configurations, using the domain wall action for the light valence quarks. Here,
the good chiral properties of the domain wall action will be very useful.
Presently, singly-bottom baryons are the focus of intense experimental and theoretical
study. The Λb is the first b-baryon discovered [144], followed more recently by the Σb and
Σ∗b [145] and the Ξb [146, 147]. As already mentioned at the beginning of this chapter,
there are two incompatible results for the mass of the most recent discovery, the Ωb baryon
[133, 134]. No baryons with more than one b quark have been observed so far, but they
may be discovered at the Large Hadron Collider.
A number of unquenched lattice calculations of bottom baryon masses have been done
recently [148, 149, 150, 151, 152]. It is important to perform independent determinations
of the same quantities with different lattice formulations in order to test universality, and
the combination of NRQCD heavy- and domain wall light quarks has not been used by
others. Compared to the static heavy-quark action, which was used in [149, 151, 152],
NRQCD has the advantage that it is not restricted to systems containing only a single
106 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
b quark. Also, heavy-light spin splittings which would vanish in the static limit can be
calculated.
7.2.1 Quark propagators
For the calculation of heavy-light meson and baryon masses, I reuse the u/d and s valence
quark domain wall propagators that were computed (using Chroma [136]) and saved during
the static-light calculation in [151].
These light-quark propagators have gauge-invariant Gaussian-smeared sources, where
APE-smeared gauge links [153, 154] were used in constructing the source. For each of
these propagators I also compute a second propagator by smearing the sink in the same
way as the source1.
All calculations are done in the exact isospin limit mu = md ≡ ml. So far, I performed
the spectrum calculations only with the propagators on the aml = 0.005, ams = 0.04
ensemble, with valence quark masses equal to the sea quark masses. These quark masses
correspond to pion and kaon masses of about 330 and 580 MeV, respectively. Note that
not only the u/d quark mass but also the strange quark mass is too large; the physical
point corresponds to ams ≈ 0.034 [128].
I compute NRQCD bottom quark propagators for point sources (a Kronecker-delta in
position-, colour- and spin space) and Gaussian-smeared sources (without link smearing).
A Gaussian-smeared source is obtained by applying the operator
(
1 +
σ
nS
∆(2)
)nS
(7.10)
to a point source. In (7.10), ∆(2) is the covariant lattice Laplacian defined in (A.12). I used
nS = 10 steps of smearing and set the width to σ = 1.0. I also computed sink-smeared
b-quark propagators, obtained by applying the operator (7.10) at the sink. The b-quark
mass is set to amb = 2.514, as obtained in Sec. 7.1.4.
In order to increase statistics, I compute hadron correlators directed both forward and
backward in time. The four-component NRQCD b-quark propagator in the nonrelativistic
Dirac gamma matrix-basis is given by
G(x, x′) = θ(τ − τ ′)
(
Gψ(x, x′) 0
0 0
)
− θ(τ ′ − τ)
(
0 0
0 Gξ(x, x′)
)
. (7.11)
Here, Gψ and Gξ may include the smearing operator (7.10) at source/and or sink.
1The Chroma input XML file for this procedure was provided by W. Detmold.
7.2. Bottom hadrons 107
Note that the light-quark propagators generated by the Chroma software are stored in
a chiral gamma matrix basis (the “DeGrand-Rossi” basis). Before combining them with
(7.11) to form hadron correlation functions, I convert the light-quark propagators to the
nonrelativistic Dirac gamma matrix basis (A.3) by means of a similarity transformation.
In the following, I shall denote the generic light-quark field (in the nonrelativistic Dirac
basis) by q and the bottom-quark field by
Q =
(
ψ
ξ
)
. (7.12)
7.2.2 B mesons
I use interpolating fields of the form q γ5 Q for the pseudoscalar B/Bs mesons and q γj Q
for the vector mesons B∗/B∗s . These fields may include the smearing types discussed in
Sec. 7.2.1.
Since all light-quark propagators Gq have a smeared source, there are two possibilities
for the meson source, which I denote as (L,s) and (S,s). Here, the capital letter denotes a
local (L) or smeared (S) source for the heavy quark; the lower-case letter similarly denotes
the source type for the light quark. At the sink, there are the four possibilities (L,s),
(S,s), (L,l), (S,l).
Thus, I perform (2 × 4)-matrix fits of the form (6.11), with four sets of amplitude
parameters corresponding to the smearings (L,s), (S,s), (L,l), (S,l). An example of
such a fit for the B meson is shown in Fig. 7.7. There, I also show an “effective-energy”
plot, where the effective-energy function is defined as
Eeff(τ + 1/2) = ln
(
C(τ)
C(τ + 1)
)
(7.13)
for a correlator C(τ). For large Euclidean time separation τ , this approaches the (un-
physical, i.e. shifted) ground-state energy. As can be seen in the figure, the various types
of smearing all behave similarly. Recall that no completely un-smeared correlators are
included here; these would behave very differently (with more contamination from excited
states).
My results for the hyperfine splittings in the B mesons are listed in Table 7.10. For
the conversion to physical units I used the result for the lattice spacing from Table 7.8.
The predicted hyperfine splittings agree well with the experimental values.
To compute the full hadron masses in a way that leads to only weak dependence on
the bare b quark mass, one can use the experimental value for the mass of e.g. the Υ(1S)
108 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
a∆M
lattice
∆M (MeV)
lattice
∆M (MeV)
experiment
B∗ −B 0.0277(49) 48(9) 45.78(35)
B∗s −Bs 0.0280(21) 49(4) 46.1(1.5)
Table 7.10: Spin splittings in B mesons for aml = 0.005, ams = 0.04. Errors are statisti-
cal/fitting only. Experimental values from [83].
meson as an input parameter in the following way,
M = Elat. +
nb
2
(
MΥexp. − E
Υ
lat.
)
, (7.14)
where nb denotes the number of b quarks in the hadron (nb = 1 for the B mesons considered
here) and Elat. denotes the energy in the lattice calculation.
The results for the full B meson masses computed using (7.14) are plotted together
with the experimental values in Fig. 7.8. As expected for the unphysical light-quark
masses aml = 0.005, ams = 0.04, the lattice results are found to be slightly above the
experimental results. Chiral extrapolations to the physical quark masses will be performed
in the future.
7.2.3 Singly-bottom baryons
For baryons containing a single b-quark and two light quarks q, q′, I use interpolating
fields of the form
OΓ q q
′
α = ²abc (CΓ)βγ q
a
β q
′b
γ Q
c
α , (7.15)
where a, b, ... are colour indices running from 1 to 3 and α, β, ... are spinor indices running
from 1 to 4. In (7.15), C = γ0γ2 is the Euclidean charge-conjugation matrix and Γ is a
Dirac matrix that determines the total spin of the light degrees of freedom: Γ = γ5 for
Sl = 0 and Γ = γj for Sl = 1. The total spin of the light degrees of freedom becomes a
conserved quantum number in the static limit mb →∞.
Note that (Cγ5)T = −Cγ5 and (Cγj)T = Cγj . This implies
OΓ q
′ q
α =
{
−OΓ q q
′
α for Γ = γ5,
OΓ q q
′
α for Γ = γj ,
(7.16)
and hence for q = q′ one must have Γ = γj and Sl = 1.
The singly-bottom baryons in the isospin limit and their interpolating operators are
listed in Table 7.11. Note that the operators with Γ = γj have an overlap with both the
7.2. Bottom hadrons 109
0 5 10 15 20
time
1e-09
1e-06
0.001
cor
rela
tor
(L,s)-(L,s)(S,s)-(L,s)(L,l)-(L,s)(S,l)-(L,s)(L,s)-(S,s)(S,s)-(S,s)(L,l)-(S,s)(S,l)-(S,s)
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
1.2
effe
ctiv
e en
ergy
(L,s)-(L,s)(S,s)-(L,s)(L,l)-(L,s)(S,l)-(L,s)(L,s)-(S,s)(S,s)-(S,s)(L,l)-(S,s)(S,l)-(S,s)
Figure 7.7: Fit to a 2×4 matrix of correlators for the B meson, with nexp = 6 and τmin = 3.
The upper panel shows the correlator data and the fit functions. The lower panel shows
an effective-energy plot with the ground-state fit result.
110 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
5.2
5.3
5.4
5.5
M
(G
eV
)
B
B*
Bs
Bs
*
energy shift from Υ
Figure 7.8: Masses of B mesons for aml = 0.005, ams = 0.04. The lines indicate the
experimental values [83].
Hadron(s) JP Sl Operator
Λb 12
+
0 ²abc (Cγ5)βγ qaβ q
′b
γ Q
c
α
Σb, Σ∗b
1
2
+
, 32
+
1 ²abc (Cγj)βγ qaβ q
′b
γ Q
c
α
Ξb 12
+
0 ²abc (Cγ5)βγ qaβ s
b
γ Q
c
α
Ξ′b, Ξ
∗
b
1
2
+
, 32
+
1 ²abc (Cγj)βγ qaβ s
b
γ Q
c
α
Ωb, Ω∗b
1
2
+
, 32
+
1 ²abc (Cγj)βγ saβ s
b
γ Q
c
α
Table 7.11: Interpolating fields for baryons containing a single b quark. Sl denotes the
total spin of the light degrees of freedom, which becomes a conserved quantum number
for mb →∞.
7.2. Bottom hadrons 111
J = 32 and J =
1
2 states. For the correlator Cjk(τ) with Γ = γj at the source and Γ = γk
at the sink one has at zero momentum and large τ > 0
Cjk(τ) = Z
2
3/2 e
−E3/2 τ 1
2(1 + γ0)(δjk −
1
3γjγk) + Z
2
1 e
−E1/2 τ 1
2(1 + γ0)
1
3γjγk (7.17)
(see [155]). The J = 32 and J =
1
2 contributions can be disentangled by multiplying with
the projectors (δij − 13γiγj) and
1
3γiγj , respectively:
(δij − 13γiγj) Cjk(τ) = Z
2
3/2 e
−E3/2 τ 1
2(1 + γ0)(δik −
1
3γiγk),
1
3γiγj Cjk(τ) = Z
2
1 e
−E1/2 τ 1
2(1 + γ0)
1
3γiγk. (7.18)
Note that for τ < 0, one has 12(1− γ0) instead of
1
2(1 + γ0) in (7.18).
For the singly-bottom baryons, I use the same type of smearing on both light quarks.
Thus, the baryon operators are classified in terms of their smearing in the same way as
the B meson operators in Sec. 7.2.2. Again, I perform (2× 4)-matrix fits. An example of
such a fit for the Λb baryon is shown in Fig. 7.9.
Table 7.12 lists the results for the heavy-light spin splittings. An experimental value
is known only for Σ∗b − Σb; my lattice result agrees with this but the statistical errors
are rather large. It is planned to increase statistics in the future by computing more
domain-wall propagators.
Results for the mass differences of the J = 12 singly-bottom baryons to the B/Bs/Λb
hadrons are listed in Table 7.13. To compute the absolute baryon masses, I compare two
different methods: (I) using the Υ(1S) mass via Eq. 7.14 with nb = 1 and (II) using the
B mass via
M = Esim. + nb
(
MBexp. − E
B
sim.
)
(7.19)
with nb = 1. The results are plotted in Fig. 7.9. The hadron masses at the present unphys-
ically large values for the light quark masses tend to be slightly above the experimental
results. Taking into account the overall picture, my lattice result for the Ωb mass appears
to favour the CDF measurement [134] over the D/0 measurement [133]. Definitive conclu-
sions can only be made after chiral extrapolation, and, eventually, after the inclusion of
different lattice spacings and volumes (finite-volume effects are also known to increase the
energy).
Note that the Υ mass shows little dependence on the sea quark masses, while the B
has a light valence quark. Thus, (7.14) and (7.19) lead to very different chiral behaviour
of M , which likely explains the discrepancies between the two methods seen at the present
quark masses.
112 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
0 5 10 15 20
time
1e-20
1e-16
1e-12
1e-08
0.0001
cor
rela
tor
(L,s)-(L,s)(S,s)-(L,s)(L,l)-(L,s)(S,l)-(L,s)(L,s)-(S,s)(S,s)-(S,s)(L,l)-(S,s)(S,l)-(S,s)
0 5 10 15 20
time
0
0.4
0.8
1.2
1.6
effe
ctiv
e en
ergy
(L,s)-(L,s)(S,s)-(L,s)(L,l)-(L,s)(S,l)-(L,s)(L,s)-(S,s)(S,s)-(S,s)(L,l)-(S,s)(S,l)-(S,s)
Figure 7.9: Fit to a 2 × 4 matrix of correlators for the Λb baryon, with nexp = 6 and
τmin = 3. The upper panel shows the correlator data and the fit functions. The lower
panel shows an effective-energy plot with the ground-state fit result.
7.2. Bottom hadrons 113
a∆M
lattice
∆M (MeV)
lattice
∆M (MeV)
experiment
Σ∗b − Σb 0.014(14) 25(25) 21.2(2.0)
Ξ∗b − Ξ
′
b 0.0105(94) 18(16) −
Ω∗b − Ωb 0.0108(59) 19(10) −
Table 7.12: Heavy-light spin splittings in singly-bottom baryons for aml = 0.005, ams =
0.04. Errors are statistical/fitting only. The experimental value for Σ∗b − Σb is from [83].
a∆M
lattice
∆M (GeV)
lattice
∆M (GeV)
experiment
Λb −B 0.235(20) 0.408(36) 0.3410(16)
Σb −B 0.350(17) 0.608(31) 0.5286(27)
Ξb −B 0.320(14) 0.556(25) 0.5132(30)
Ξ′b −B 0.407(10) 0.708(21) −
Ωb −B 0.4589(79) 0.798(18) 0.7752(69)
Λb −Bs 0.190(21) 0.330(36) 0.2539(17)
Σb −Bs 0.305(17) 0.530(31) 0.4415(28)
Ξb −Bs 0.275(14) 0.478(25) 0.4261(31)
Ξ′b −Bs 0.362(10) 0.630(20) −
Ωb −Bs 0.4141(68) 0.720(16) 0.6881(69)
Σb − Λb 0.115(23) 0.200(40) 0.1876(31)
Ξb − Λb 0.085(15) 0.148(27) 0.1722(34)
Ξ′b − Λb 0.172(22) 0.300(38) −
Ωb − Λb 0.224(21) 0.390(37) 0.4342(71)
Table 7.13: Mass differences of the J = 12 singly-bottom baryons to the B/Bs/Λb for
aml = 0.005, ams = 0.04. Errors are statistical/fitting only. Experimental values from
[83] and [134].
114 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
5.6
5.8
6
6.2
M
(G
eV
)
Λb
Σb Σb
*
Ξb
Ξb´ Ξb
*
Ωb Ωb
*
energy shift from Υ
energy shift from B
Figure 7.10: Masses of singly-bottom baryons for aml = 0.005, ams = 0.04. The lines
indicate the experimental values [83, 133, 134]. The D/0 result [133] for the Ωb mass is
indicated in black; the CDF result [134] in red.
7.2. Bottom hadrons 115
Hadrons JP SQQ Operator
Ξbb, Ξ∗bb
1
2
+
, 32
+
1 ²abc (Cγj)βγ Qaβ Q
b
γ q
c
α
Ωbb, Ω∗bb
1
2
+
, 32
+
1 ²abc (Cγj)βγ Qaβ Q
b
γ s
c
α
Table 7.14: Interpolating fields for baryons containing two b quarks. SQQ denotes the
total spin of the heavy quarks.
a∆M
lattice
∆M (MeV)
lattice
∆M (MeV)
experiment
Ξ∗bb − Ξbb 0.0140(75) 24(14) −
Ω∗bb − Ωbb 0.0219(54) 38(9) −
Table 7.15: Heavy-light spin splittings in doubly-bottom baryons for aml = 0.005, ams =
0.04. Errors are statistical/fitting only.
7.2.4 Doubly-bottom baryons
For the doubly-bottom baryons, the role of the light and heavy quarks in (7.15) is inter-
changed. The interpolating fields for the doubly-bottom baryons are listed in Table 7.14.
As for the singly-bottom baryons, I use the spin projection (7.18) to separate the J = 12
and J = 32 states. The energy splittings between these are listed in Table 7.15.
I also use the same smearing for both heavy quarks. A (2 × 4)-matrix fit for the Ωbb
is shown in Fig. 7.11. As can be seen in the effective-energy plot, there is now a more
significant difference between the correlators with and without smearing for the heavy
quarks. This is expected, as the two heavy quarks can be excited in a similar way as in
bottomonium.
Again, I use both methods (7.14) and (7.19) to compute the absolute hadron masses.
The results are shown in Fig. 7.12. Here, the differences between the two methods (7.14)
and (7.19) for the unphysical quark masses are enhanced since nb = 2.
116 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
0 5 10 15 20
time
1e-16
1e-12
1e-08
0.0001
1
cor
rela
tor
(L,s)-(L,s)(S,s)-(L,s)(L,l)-(L,s)(S,l)-(L,s)(L,s)-(S,s)(S,s)-(S,s)(L,l)-(S,s)(S,l)-(S,s)
0 5 10 15 20
time
0
0.4
0.8
1.2
1.6
effe
ctive
ene
rgy
(L,s)-(L,s)(S,s)-(L,s)(L,l)-(L,s)(S,l)-(L,s)(L,s)-(S,s)(S,s)-(S,s)(L,l)-(S,s)(S,l)-(S,s)
Figure 7.11: Fit to a 2 × 4 matrix of correlators for the Ωbb baryon, with nexp = 6 and
τmin = 3. The upper panel shows the correlator data and the fit functions. The lower
panel shows an effective-energy plot with the ground-state fit result.
7.2. Bottom hadrons 117
10.1
10.2
10.3
10.4
M
(G
eV
)
Ξbb
Ξbb*
Ωbb
Ωbb*
energy shift from Υ
energy shift from B
Figure 7.12: Masses of doubly-bottom baryons for aml = 0.005, ams = 0.04.
7.2.5 The Ωbbb baryon
For the Ωbbb baryon, I use interpolating operators of the form
²abc (Cγj)βγ Q
a
β Q
b
γ Q
c
α. (7.20)
The only physical state is the one with J = 32 .
As the Ωbbb baryon does not contain light valence quarks, the dependence on the
light sea quark masses is expected to be weak once these are light enough, similarly to
bottomonium. Thus, Eq. (7.14) is the better method for computing the absolute mass of
the Ωbbb, and no chiral extrapolation is required.
Also, since NRQCD is computationally cheap, one can go to very high statistics with
little cost. A plot of a (2× 2)-matrix correlator from about 105 NRQCD propagators on
the aml = 0.005, ams = 0.04 ensemble is shown in Fig. 7.13. As can be seen, the signal is
very good. The (unphysical) energy obtained from the fit is
aEΩbbb = 0.5527(12). (7.21)
I also computed a high-statistics Υ correlator from the same propagators. Fitting it gives
aEΥ(1S) = 0.29786(20). (7.22)
118 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
0 10 20 30
time
1e-12
1e-08
0.0001
1
cor
rela
tor
local - localsmeared - locallocal - smearedsmeared - smeared
0 10 20 30time
0
0.25
0.5
0.75
1
effe
ctive
ene
rgy
local - localsmeared - locallocal - smearedsmeared - smeared
Figure 7.13: Fit to a 2 × 2 matrix of correlators for the Ωbbb baryon, with nexp = 6 and
τmin = 3. The upper panel shows the correlator data and the fit functions. The lower
panel shows an effective-energy plot with the ground-state fit result.
7.2. Bottom hadrons 119
Reference Method MΩbbb (GeV)
This work lattice QCD 14.3748(33)
[156] bag model 14.30
[157] variational method 14.37(08)
[158] bag model 14.276
[159] relativistic quark model 14.569
[160] sum rules 13.28(10)
Table 7.16: Comparison of my lattice QCD result for the Ωbbb mass to various results
from continuum models. The error on my lattice result is statistical/fitting only. See the
references given in the table for the details of the errors quoted for the continuum results.
Using the bootstrap method to properly take into account correlations between the Ωbbb
and the Υ energies, Eq. (7.14) then leads to
MΩbbb = 14.3748(33) GeV, (7.23)
where the error is statistical only and includes the uncertainty in the lattice spacing (the
latter was taken from Table 7.8). To my knowledge, this is the first lattice QCD calculation
of the Ωbbb mass.
The Ωbbb mass has been estimated before using various continuum methods. In Table
7.16 I compare my result to these estimates. I find excellent agreement with the value from
the variational method given in [157]. The other results are also close, with the notable
exception of [160], where QCD sum rules were used.
The production of the Ωbbb at hadron colliders has been studied in [161, 162]. For the
LHC at
√
s = 14 TeV, the integrated cross section for direct fragmentation production
b→ Ωbbb was estimated to be about 8 pb.
120 CHAPTER 7. HEAVY-HADRON SPECTROSCOPY
Chapter 8
Rare B decays
The tests described in Chapter 6 demonstrate the good properties of moving NRQCD. In
the following I show how mNRQCD can be applied to the calculation of form factors for
B decays and present numerical results.
The interactions with W and Z bosons as well as top quarks are effects at very short
distances and can be described by an effective weak Hamiltonian. This method is intro-
duced in Sec. 8.1, where the “tree-level” decay B → pi`ν and the rare decay B → K∗γ
are considered as examples. The phenomenology of rare decays is rather complex and a
complete discussion is beyond the scope of this work. Certain long-distance contributions
to rare B decays, which are mentioned in Sec. 8.2, can not be calculated directly in lattice
QCD. However, one can choose regions of q2 where these effects are suppressed. The dom-
inant contributions to the semileptonic and radiative decays then come from operators
whose matrix elements can be computed directly in lattice QCD.
These matrix elements are conveniently parametrised in terms of Lorentz-invariant
functions called form factors. After giving the definitions of the complete set of B semilep-
tonic form factors in Sec. 8.3, I show how they can be extracted from Euclidean correlation
functions in Sec. 8.4.
For the lattice calculation, the heavy-light current in the weak Hamiltonian has to
be expressed in terms of the lattice quark fields. This requires the inclusion of several
operators multiplied by appropriate matching factors, as discussed in Sec. 8.5.
For a given q2, the boost velocity v in the moving NRQCD action should be chosen
such that discretisation errors are minimised. Estimates of the optimal choice for v are
shown in Sec. 8.6. Independently of discretisation errors, the heavy-quark effective field
theory description of the decay becomes less accurate at low q2, when the momentum
transfer is no longer small compared to mb. This is discussed in Sec. 8.7.
122 CHAPTER 8. RARE B DECAYS
I then describe in detail the calculation of the two-point and three-point functions with
mNRQCD heavy- and staggered light quarks in Sec. 8.8. All constructions are performed
for both point- and random-wall sources. The aim of the random-wall source method is
to reduce statistical errors. I show in extensive numerical studies to what extent this is
possible for the calculation of heavy-to-light decays. Particular emphasis is placed on the
analysis of different fitting methods.
Finally, after briefly demonstrating in Sec. 8.9 the contributions of the various operators
in the heavy-light current, I present some preliminary form factor results in Sec. 8.10.
8.1 Weak Decays of B Mesons in the Standard Model
8.1.1 Electroweak interactions of quarks
The Lagrange density for the interactions between the quarks and the electroweak gauge
bosons W±, Z and A in the Standard Model has the following form (see e.g. [163, 164]):
Lq.,e.w. = −
g2
2
√
2
(
JµWµ + J
µ†W †µ
)
− g2 sin θW J
µ
e.m.A
e.m.
µ −
g2
2 cos θW
Jµn.Zµ , (8.1)
with the charged weak current
Jµ =
(
u c t
)
γˆµ(1− γˆ5) V
d
s
b
, (8.2)
the electromagnetic current,
Jµe.m. = −
1
3
(
d s b
)
γˆµ
d
s
b
+
2
3
(
u c t
)
γˆµ
u
c
t
, (8.3)
and the weak neutral current
Jµn. = −
1
2
(
d s b
)
γˆµ
(
1− γˆ5 −
4
3
sin2 θW
)
d
s
b
+
1
2
(
u c t
)
γˆµ
(
1− γˆ5 −
8
3
sin2 θW
)
u
c
t
. (8.4)
8.1. Weak Decays of B Mesons in the Standard Model 123
In the charged weak current (8.2), the Cabibbo-Kobayashi-Maskawa matrix
V =
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
, (8.5)
gives rise to a mixing between the quark flavours. In the Standard Model, V is unitary,
V † V = 1. After absorbing unobservable phase factors into the definition of the quark
fields, 4 free real parameters remain in V . These are three mixing angles and one CP-
violating phase. It is known from experiments that V is close to unity; see [83] for the
currently known values of the CKM matrix elements and their uncertainties.
Note that the electromagnetic and weak neutral currents (8.3), (8.4) do not change the
quark flavour, and therefore transitions within up-type or within down-type quarks like
b→ s do not appear at tree-level.
8.1.2 Effective weak Hamiltonians
When describing weak decays of hadrons, it is very useful to separate the short-distance
weak interactions from the non-perturbative long-distance dynamics of the hadrons. The
heavy W and Z bosons as well as the top quark can be integrated out of the theory, and
their effects are then described by new local operators On involving the lighter degrees of
freedom. The resulting effective Hamiltonian has the general structure
Heff =
4GF√
2
∑
n
V nCKM Cn(µ)On, (8.6)
where Cn(µ) are the scale- and renormalisation scheme dependent short-distance coeffi-
cients (Wilson coefficients), and V nCKM stands for the product of CKM matrix elements
involved in the weak interactions which yield the effective operator On.
The general method to derive the effective Hamiltonian is the operator product ex-
pansion. In addition, one must use renormalisation-group methods to evolve the Wilson
coefficients from the scale µ ∼ mW to the scale appropriate to the process under consid-
eration, e.g. µ = mb for B decays. For the details, see for example [165, 166].
The operator product expansion can be done in perturbation theory, since asymptotic
freedom implies that the QCD coupling is weak at µ ∼ mW . The non-perturbative
aspect of the hadron decay then appears in the calculation of the low-energy QCD matrix
element 〈On〉µ, and this is what can be done on the lattice. Note that the scale- and
scheme dependence of this matrix element is cancelled by the corresponding dependence
of the Wilson coefficients Cn(µ), provided that the same renormalisation scheme and scale
124 CHAPTER 8. RARE B DECAYS
is used for both calculations. This means that the lattice results must be converted to the
continuum scheme by an appropriate matching calculation (cf. Sec. 8.5).
8.1.3 The decay B → pi`ν
We will now consider the semileptonic decay B → pi`ν, from which the magnitude of
the CKM matrix element Vub can be determined. Recent experimental results have been
published in [12, 13, 14].
The effective Hamiltonian for b → u e− ν¯e results from the Feynman diagrams shown
in Fig. 8.1.
Figure 8.1: Tree-level Feynman diagram contributing to b→ u e− ν¯e
Integrating out the W boson gives the well-known current-current interaction
Heff = Vub
GF√
2
C(µ) uγˆµ(1− γˆ5)b eγˆµ(1− γˆ5)νe (8.7)
where GF =
√
2g22/(8m
2
W ), and up to small loop corrections one has C(µ) = 1. Since the
mesons in the initial and final state are both pseudoscalars, only the vector current uγˆµb
contributes to the hadronic matrix element. The latter can be parametrised by two form
factors f+ and f−, or equivalently f+ and f0, as follows:
〈pi(p′)| uγˆµb |B(p)〉 = f+(q
2)
[
pµ + p′µ
]
+ f−(q
2) qµ
= f+(q
2)
[
pµ + p′µ −
M2B −M
2
pi
q2
qµ
]
+ f0(q
2)
M2B −M
2
pi
q2
qµ,
(8.8)
with q ≡ p− p′. Neglecting the lepton masses, the differential decay rate is given by [167]
dΓ
dq2
=
G2F |Vub|
2
192pi3M3B
[
(M2B +M
2
pi − q
2)2 − 4M2BM
2
pi
]3/2
|f+(q
2)|2. (8.9)
Hence, to determine |Vub| from the exclusive measurement of this decay rate, the non-
perturbative hadronic matrix element 〈pi(p′)| uγˆµb |B(p)〉 must be calculated. Two recent
unquenched lattice QCD calculations are [11] and [168]. Both works make use of the
ASQTAD action for the light quarks. For the b quark, lattice NRQCD was used in [11],
while the Fermilab action (cf. Sec. 2.8) was used in [168].
8.1. Weak Decays of B Mesons in the Standard Model 125
8.1.4 The decay B → K∗γ
Among the rare decays, the decay B → K∗γ (first observed by the CLEO collaboration
[15]) is the one with the most precise experimental results available [17, 16, 18]. As a
loop-mediated flavour-changing neutral current process it is suppressed in the Standard
Model, and therefore very sensitive to possible new physics contributions (see for example
[169, 170, 171, 172, 173, 174, 175, 176]).
Figure 8.2: Example of a one-loop diagram contributing to O7
The effective weak Hamiltonian relevant to this radiative decay can be written in the
following form,
Heff = −VtbV
∗
ts
4GF√
2
8∑
n=1
Cn(µ)On, (8.10)
where O1, ..., O6 are 4-quark operators, O7 is an electromagnetic and O8 is a chromomag-
netic dipole operator [177]. The CKM-matrix dependence has been factored out globally,
neglecting Vub ∼ 0 and using unitarity to replace VcbV ∗cs by −VtbV
∗
ts.
The dominant short-distance contribution to B → K∗γ is due to the electromagnetic
dipole operator
O7 =
e
16pi2
mb JµνF
µν
(e.m.) with Jµν = s¯σµν
1 + γˆ5
2
b, (8.11)
where σµν = i2 [γˆ
µ, γˆν ]. An example of a one-loop Feynman diagram contributing to O7 is
shown in 8.2.
As derived in Sec. B.1, the hadronic matrix element of the tensor current Jµν in O7
can be parametrised as follows:
qν〈K∗(p′, ε)|Jµν |B(p)〉 = 2 T1(q
2) ²µνρσ ε
∗ν pρ p′σ
+i T2(q
2)
[
ε∗µ (M
2
B −M
2
K∗)− (ε
∗ · q)(p+ p′)µ
]
+i T3(q
2) (ε∗ · q)
[
qµ −
q2
M2B −M
2
K∗
(p+ p′)µ
]
(8.12)
126 CHAPTER 8. RARE B DECAYS
where q = p−p′ and ε is the polarisation vector of the K∗ meson. For the physical on-shell
photon, one has q2 = 0. Note that T1(0) = T2(0).
In Sec. B.2 I give a derivation of the decay rate, with the result
Γ(B → K∗γ) =
α G2F
8pi4
C7(µ)
2|VtbV
∗
ts|
2m2bM
3
B
(
1−
M2K∗
M2B
)3
|T1(0)|
2. (8.13)
Here, the contributions from the operators other than O7 are neglected. These contribu-
tions will be discussed in Sec. 8.2.
So far, there are no unquenched lattice QCD calculations of the form factors for B →
K∗γ by other groups. Quenched calculations can be found [178, 179, 180, 181, 182, 183,
184]. In these works, the b quark was implemented with Wilson-like actions, so that mb
had to be unphysically small. The form factors were then extrapolated in mb to the
physical value. The precise form of the functional dependence on mb is unknown, which
leads to an additional systematic uncertainty.
8.2 More about b → s decays
Including also s `+`− final states, the effective Hamiltonian for b→ s transitions has the
form
Heff = −VtbV
∗
ts
4GF√
2
10∑
n=1
Cn(µ)On (8.14)
with
O1 = (c¯
b
L γˆ
µ baL) (s¯
a
L γˆµ c
b
L),
O2 = (c¯
a
L γˆ
µ baL) (s¯
b
L γˆµ c
b
L),
O3 = (s¯
a
L γˆ
µ baL)
∑
q(q¯
b
L γˆµ q
b
L),
O4 = (s¯
a
L γˆ
µ bbL)
∑
q(q¯
b
L γˆµ q
a
L),
O5 = (s¯
a
L γˆ
µ baL)
∑
q(q¯
b
R γˆµ q
b
R),
O6 = (s¯
a
L γˆ
µ bbL)
∑
q(q¯
b
R γˆµ q
a
R),
O7 =
e
16pi2
mb s¯
a
L σµν b
a
R F
µν
(e.m.),
O8 =
g
16pi2
mb s¯
a
L σµν F
µν
ab b
b
R,
O9 =
e2
16pi2
(s¯aL γˆ
µ baL)
∑
`(
¯`γˆµ `),
O10 =
e2
16pi2
(s¯aL γˆ
µ baL)
∑
`(
¯`γˆµγˆ5 `). (8.15)
8.2. More about b→ s decays 127
Figure 8.3: Example of a Feynman diagram contributing to O2.
(see e.g. [177, 165, 185, 166]). Here,
∑
q denotes a sum over the quark flavours u, d, c, s, b
and
∑
` denotes a sum over the lepton flavours e, µ, τ . The indices a, b are colour indices
and are summed over from 1 to 3. Under renormalisation-group running from µ = mW
down to µ = mb, the operators mix and their coefficients change significantly.
The operator O7 was already discussed in Sec. 8.1.4. The operators O9 and O10 give
additional short-distance contributions for b→ s`+`−; they arise from box diagrams (with
two W boson propagators) and penguin diagrams (with a photon or Z decaying to the
lepton pair).
At leading order, the operator O2 is generated by the Feynman diagram shown in
Fig. 8.3, so that at this order C2(mW ) = 1. The operator O1, where the colour structure
is changed, is generated from this by QCD corrections and is thus loop-suppressed. In
principle, there are also operators analogous to O1 and O2, with the replacement c 7→ u.
However, since Vub ≈ 0 is neglected here, these are not included. For the same reason, we
also neglect weak annihilation in B± → K(∗)± decays.
The hadronic matrix elements of O2 can be sizable when the c¯c pair forms charmonium
resonances (see Figs. 8.4 and 8.5). This is a non-local effect and can not be calculated
directly on the lattice. The dominant resonances are the J/ψ(1S) and ψ(2S); their squared
masses are 9.591 GeV2 and 13.587 GeV2, respectively [83]. Thus, one should keep q2 away
from these values when comparing the theoretical results with experimental data. For
B → K∗γ, one has q2 = 0, which is far away from the resonances. The remaining effect
for B → K∗γ is small [186]. For B → K(∗)`` decays, one can work at large q2 (close to
q2max), which is also far away from the resonances, so that the long-distance effects are also
under control [187]. The lattice calculation works best at large q2 anyway.
The operators O3 ... O6 have very small Wilson coefficients and can be neglected. The
chromomagnetic tensor operator O8 may also lead to non-local contributions like the one
shown in Fig. 8.6. The contributions from O8 to B → K∗γ were estimated in [188] and
128 CHAPTER 8. RARE B DECAYS
Figure 8.4: Long-distance contribution from charmonium resonances via O2.
Figure 8.5: Effect of charmonium resonances on the differential decay rate (schematically).
Figure 8.6: Example for a spectator-quark contribution to B → K∗γ due to O8.
8.3. General definition of semileptonic form factors 129
found to be suppressed relative to O7 by a factor of (C8/C7)ΛQCD/MB ≈ 0.05. See also
[189] for a discussion of O8 for B → K∗`+`− decays.
Finally, note that the K∗ meson decays through the strong interaction into a pseu-
doscalar kaon and a pion. The width of the K∗ is about 50 MeV [83]. For the unphysically
large u- and d-quark masses used in the initial lattice calculations, the K∗ is stable. How-
ever, extrapolations to the physical quark masses will be complicated by threshold effects.
8.3 General definition of semileptonic form factors
As discussed in the previous section, the short-distance contributions for B → K∗γ and
B → K(∗)`` decays are due to the operators O7, O9 and O10. The hadronic matrix
elements of the bilinear heavy-light quark currents contained in these operators can be
parametrised by Lorentz-invariant form factors as follows (see e.g. [190]):
〈P (p′)|q¯γˆµb|B(p)〉 = f+(q
2)
[
pµ + p′µ −
M2B −M
2
P
q2
qµ
]
+f0(q
2)
M2B −M
2
P
q2
qµ, (8.16)
qν〈P (p
′)|q¯σµνb|B(p)〉 =
ifT (q2)
MB +MP
[
q2(pµ + p′µ)− (M2B −m
2
P )q
µ] , (8.17)
〈V (p′, ε)|q¯γˆµb|B(p)〉 =
2iV (q2)
MB +MV
²µνρσε∗ν p
′
ρpσ, (8.18)
〈V (p′, ε)|q¯γˆµγˆ5b|B(p)〉 = 2MVA0(q
2)
ε∗ · q
q2
qµ
+(MB +MV )A1(q
2)
[
ε∗µ −
ε∗ · q
q2
qµ
]
−A2(q
2)
ε∗ · q
MB +MV
[
pµ + p′µ −
M2B −M
2
V
q2
qµ
]
, (8.19)
qν〈V (p′, ε)|q¯σµνb|B(p)〉 = 4 T1(q
2)²µρκσε
∗ρpκp′σ, (8.20)
qν〈V (p′, ε)|q¯σµν γˆ5b|B(p)〉 = 2iT2(q
2)
[
ε∗µ(M
2
B −M
2
V )− (ε
∗ · q)(p+ p′)µ
]
+ 2iT3(q
2)(ε∗ · q)
[
qµ −
q2
M2B −M
2
V
(p+ p′)µ
]
. (8.21)
A derivation of (8.20) and (8.21) is given in Sec. B.1.
8.4 Extraction of form factors from correlation functions
The matrix element 〈F (p′)|J |B(p)〉, where F denotes the final pseudoscalar (P ) or vector
(V ) meson, and J ∼ q¯ ΓJ b is the heavy-light quark current in the effective electroweak
130 CHAPTER 8. RARE B DECAYS
operator, can be extracted from the combination of the Euclidean 3-point function
CFJB(p
′, p, x0, y0, z0) =
∑
y
∑
z
〈
ΦF (x) J(y) Φ
†
B(z)
〉
e−ip
′·(x−y)e−ip·(y−z) (8.22)
with the Euclidean two-point functions
CBB(p, x0, y0) =
∑
x
〈
ΦB(x) Φ
†
B(y)
〉
e−ip·(x−y), (8.23)
CFF (p
′, x0, y0) =
∑
x
〈
ΦF (x) Φ
†
F (y)
〉
e−ip
′·(x−y). (8.24)
Here, ΦB ∼ q¯′γˆ5b and ΦF ∼ q¯′γˆ5q (F = P ), ΦF ∼ q¯′γˆjq (F = V ).
In the following we write τ = |x0− y0| and T = |x0− z0|. As in Sec. 2.2, one can show
by inserting complete sets of states that at large τ , T , and T − τ , the correlation functions
become
CFJB(p
′, p, τ, T ) → A(FJB)e−EF τ e−EB(T−τ), (8.25)
CFF (p, τ) → A
(FF ) e−EF τ , (8.26)
CBB(p, τ) → A
(BB) e−EBτ , (8.27)
where
A(FJB) =
√
ZV
2EV
√
ZB
2EB
∑
s
εj(p
′, s) 〈V
(
p′, ε(p′, s)
)
| J |B(p)〉, F = V,
√
ZP
2EP
√
ZB
2EB
〈P
(
p′
)
| J |B(p)〉, F = P
(8.28)
A(BB) =
ZB
2EB
, (8.29)
A(FF ) =
∑
s
ZV
2EV
ε∗j (p
′, s)εj(p
′, s), F = V (no sum over j),
ZP
2EP
, F = P.
(8.30)
Thus, the matrix elements 〈P (p′)|J |B(p)〉 and
∑
s εj(p
′, s) 〈V (p′, ε(p′, s)) |J |B(p)〉 can be
extracted from (8.28), once the factors ZB, ZF have been extracted from the two-point
functions (the energies EB, EF can be obtained from either the two-point or three-point
functions). Note that in Eqs. (8.28) and (8.29), EB denotes the full, physical energy of the
B meson; this is not equal to the energy obtained from the exponential decay in (8.25) or
(8.27) when an effective theory like mNRQCD is used for the b quark.
In the next sections I discuss briefly how the form factors can be extracted from the
matrix elements. I will only consider the case where all momenta point in x1-direction.
8.4. Extraction of form factors from correlation functions 131
Then, only certain combinations of operator indices and final state polarisations give non-
zero contributions (were appropriate, after contraction with qν . There are 21 combinations
in total (modulo the antisymmetry of σµν). They are listed in Table 8.1 together with the
form factors to which they contribute.
8.4.1 The form factors f0 and f+
First, at p = q = 0, we have from Eq. (8.16)
〈P (p′)|q¯γˆ0b|B¯(p)〉 = f0(q
2
max) [MB +MP ] , (8.31)
and hence
f0(q
2
max) =
〈P (p′)|q¯γˆ0b|B¯(p)〉
MB +MP
. (8.32)
For non-zero momentum transfer, we define r ≡ (q1, q0, 0, 0), so that r · q = 0. Then,
rµ〈P (p
′)|q¯γˆµb|B¯(p)〉 = f+(q
2) r · (p+ p′). (8.33)
Hence, if q1 6= 0, the form factor f+(q2) can be calculated in the following way,
f+(q
2) =
q1〈P (p′)|q¯γˆ0b|B¯(p)〉 − q0〈P (p′)|q¯γˆ1b|B¯(p)〉
q1(p0 + p′0)− q0(p1 + p′1)
. (8.34)
Once f+(q2) has been extracted, we obtain f0(q2) from Eq. (8.16) with µ = 0:
〈P (p′)|q¯γˆ0b|B¯(p)〉 = f+(q
2)
[
p0 + p′0 −
M2B −M
2
P
q2
(p0 − p′0)
]
+ f0(q
2)
M2B −M
2
P
q2
(p0− p′0).
(8.35)
This gives
f0(q
2) =
〈P (p′)|q¯γˆ0b|B¯(p)〉 − f+(q2)
[
p0 + p′0 −
M2B−M
2
P
q2 (p
0 − p′0)
]
M2B−M
2
P
q2 (p
0 − p′0)
, (8.36)
where p0 = EB, p′0 = EF .
8.4.2 The form factor fT
For momenta pointing in 1-direction, it is clear from Eq. (8.17) that only the matrix
element with σ10 = −σ01 is non-zero. Thus, we obtain
fT (q
2) = −i(MB +MP )
(p0 − p′0)〈P (p′)|q¯σ10b|B¯(p)
q2(p1 + p′1)− (M2B −m
2
P )(p
1 − p′1)
. (8.37)
132 CHAPTER 8. RARE B DECAYS
Gamma matrix in ΦF Gamma matrix in J 0) as An = BnA0, A˜n = B˜n A˜0, and use the relative amplitudes
Bn, B˜n as the fit parameters.
Examples of Bayesian fits (cf. Sec. C.3) to the kaon two-point functions are shown
in Fig. 8.13 (for p′ = 0) and Fig. 8.14 (for p′ = 2pi/L · (−2, 0, 0)). Note that at zero
momentum, the oscillating contributions are not detectable and one can set n˜exp = 0. At
non-zero momentum, I always set n˜exp = nexp.
I chose the priors for the relative amplitudes Bn to be 1 with a width of 2, and the
priors for the logarithms of the energy splittings to be (−1) with a width of 1. The fit
results from the ground-state energy and amplitude for range of values for nexp are listed
in Tables 8.3, 8.5, and 8.7. All fits are performed in the range τ = τmin ... (Lτ − τmin) with
τmin = 2. As can be seen in the tables, the results are stable for nexp & 5. I computed the
central values and errors given here using bootstrap, but they agree with those obtained
directly from the fit.
For comparison, I also show the results from unconstrained fits with nexp = 1 in Tables
8.2, 8.4, and 8.6 for a range of τmin. Here, one has to choose τmin sufficiently large so that
8.8. Two-point and three-point functions with random-wall sources 147
the contributions from the excited states are negligible. However, the statistical uncer-
tainty of the fit results increases with τmin (especially at non-zero momentum). Usually,
one increases τmin until the change in the ground-state fit results becomes smaller than
the statistical errors. This is satisfied for τmin & 10.
At zero momentum, I find that the error for the ground-state amplitude from the
RWcd correlator is about 4 times smaller than the error from the point source correlator.
For the ground-state energy, the improvement factor is only about 1.5. Note that for
the point-source correlator, the central values for the ground-state amplitude obtained
from Bayesian and unconstrained fits differ by about 2 standard deviations. Additionally,
the central values for the ground-state amplitude differ between the two types of sources
(RWcd and point), most significantly in the Bayesian fits (3.3 standard deviations). This
is despite the apparent stability observed individually for the two methods4.
I find that the improvement in the ground-state fit results from the RWcd source
increases with the momentum. At p′ = 2pi/L · (−2, 0, 0), the improvement factors are
about 7 for both the ground-state amplitude and energy. This is somewhat surprising,
given Figs. 8.11 and 8.12 for the relative errors in the correlation functions. However, note
that the fits take into account correlations between the data points at different τ , which
can be significant. Note that an increase with momentum in the improvement factor for
the light pseudoscalar mesons was also observed in [193] (with Bayesian fitting).
4Note that (inadvertently) different temporal boundary conditions for the light-quark propa-
gators were used for the two types of sources (periodic for the point source, antiperiodic for the
RWcd source). However, we expect the difference caused by this to be suppressed by e−mpiLτ . For
the quark masses in use, mpiLτ ≈ 12 so that e−mpiLτ ≈ 6 · 10−6. Thus, it is unlikely that this can
explain the difference seen.
148 CHAPTER 8. RARE B DECAYS
0 10 20 30 40 50 60
time
1e-06
0.0001
0.01
1
cor
rela
tor
K meson, p=0
Figure 8.13: Bayesian fit of the K meson two-point function (RWcd source) at p′ = 0.
The fit has τmin = 2 and nexp = 10, n˜exp = 0.
0 10 20 30 40 50 60
time
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
cor
rela
tor
K meson, p=2·2pi/L
Figure 8.14: Bayesian fit of the K meson two-point function (RWcd source) at p′ =
2pi/L · (−2, 0, 0). The fit has τmin = 2 and nexp = n˜exp = 10.
8.8. Two-point and three-point functions with random-wall sources 149
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
6 0.33989(23) 0.2759(18) 2.59 0.34030(15) 0.27980(32) 4.66
7 0.33951(23) 0.2753(19) 1.41 0.33979(16) 0.27869(34) 2.24
8 0.33931(23) 0.2745(19) 0.89 0.33949(16) 0.27783(36) 1.34
9 0.33924(24) 0.2745(19) 0.82 0.33935(17) 0.27733(38) 1.04
10 0.33924(24) 0.2745(19) 0.83 0.33926(17) 0.27705(41) 1.01
11 0.33914(25) 0.2744(19) 0.79 0.33922(18) 0.27688(44) 1.00
12 0.33910(25) 0.2742(19) 0.79 0.33923(19) 0.27692(48) 1.05
Table 8.2: Results from unconstrained fits to the K meson two-point function at p′ = 0.
All fits have nexp = 1, n˜exp = 0. Central values and errors directly from fit.
point source RWcd source
nexp E A E A
4 0.33900(27) 0.2702(20) 0.33919(19) 0.27669(47)
5 0.33898(28) 0.2701(21) 0.33918(19) 0.27667(48)
6 0.33898(27) 0.2701(20) 0.33918(19) 0.27668(50)
7 0.33898(29) 0.2701(21) 0.33918(19) 0.27668(47)
8 0.33897(29) 0.2700(21) 0.33918(19) 0.27666(48)
9 0.33898(28) 0.2701(20) 0.33918(19) 0.27666(48)
10 0.33897(28) 0.2700(21) 0.33918(19) 0.27665(50)
Table 8.3: Results from Bayesian fits to the K meson two-point function at p′ = 0. All
fits have τmin = 2 and n˜exp = 0. Central values and errors from bootstrap.
150 CHAPTER 8. RARE B DECAYS
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
6 0.4690(13) 0.2421(19) 2.18 0.46866(51) 0.24758(48) 3.63
7 0.4642(15) 0.2362(22) 1.43 0.46531(60) 0.24340(63) 1.71
8 0.4626(15) 0.2347(22) 1.34 0.46396(65) 0.24155(71) 0.99
9 0.4625(17) 0.2346(24) 1.38 0.46304(75) 0.24019(90) 0.90
10 0.4602(19) 0.2307(28) 1.21 0.46263(85) 0.2395(11) 0.90
11 0.4601(21) 0.2305(32) 1.33 0.46177(99) 0.2381(14) 0.87
12 0.4599(23) 0.2302(37) 1.22 0.4619(11) 0.2384(17) 0.91
Table 8.4: Results from unconstrained fits to the K meson two-point function at p′ =
2pi/L · (−1, 0, 0). All fits have nexp = n˜exp = 1. Central values and errors directly from fit.
point source RWcd source
nexp = n˜exp E A E A
4 0.4571(37) 0.2206(68) 0.4617(10) 0.2374(17)
5 0.4570(40) 0.2200(75) 0.4615(12) 0.2369(19)
6 0.4570(41) 0.2201(81) 0.4616(11) 0.2372(17)
7 0.4569(42) 0.2197(80) 0.4615(11) 0.2368(17)
8 0.4571(42) 0.2202(78) 0.4616(11) 0.2371(18)
9 0.4571(42) 0.2201(80) 0.4616(11) 0.2373(18)
10 0.4569(43) 0.2195(86) 0.4615(11) 0.2369(19)
Table 8.5: Results from Bayesian fits to the K meson two-point function at p′ = 2pi/L ·
(−1, 0, 0). All fits have τmin = 2. Central values and errors from bootstrap.
8.8. Two-point and three-point functions with random-wall sources 151
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
6 0.732(14) 0.219(10) 1.10 0.7261(42) 0.2159(28) 1.10
7 0.693(21) 0.186(16) 1.01 0.7125(64) 0.2043(49) 0.97
8 0.722(40) 0.214(40) 1.02 0.716(10) 0.2081(90) 0.98
9 0.728(66) 0.221(78) 1.06 0.721(14) 0.213(14) 0.98
10 − − − 0.726(22) 0.219(25) 1.00
Table 8.6: Results from unconstrained fits to the K meson two-point function at p′ =
2pi/L · (−2, 0, 0). All fits have nexp = n˜exp = 1. Central values and errors directly from fit.
point source RWcd source
nexp = n˜exp E A E A
4 0.682(23) 0.172(23) 0.7150(47) 0.2021(39)
5 0.677(30) 0.166(33) 0.7134(47) 0.2005(39)
6 0.677(32) 0.166(30) 0.7130(50) 0.2002(41)
7 0.676(32) 0.165(33) 0.7126(49) 0.1999(39)
8 0.676(32) 0.165(32) 0.7123(50) 0.1996(41)
9 0.674(34) 0.164(34) 0.7120(48) 0.1994(40)
10 0.677(29) 0.166(29) 0.7125(50) 0.1998(41)
Table 8.7: Results from Bayesian fits to the K meson two-point function at p′ = 2pi/L ·
(−2, 0, 0). All fits have τmin = 2. Central values and errors from bootstrap.
152 CHAPTER 8. RARE B DECAYS
Numerical results: K∗-meson two-point functions
Comparisons of the relative errors in the K∗ meson two-point functions from point sources
and RWcd sources are shown in Fig. 8.15 (for p′ = 0) and Fig. 8.16 (for p′ = 2pi/L ·
(−1, 0, 0)). Note that the results are averaged over the polarisation directions j = 1, 2, 3
(for p′ = 0) and j = 2, 3 (for p′ = 2pi/L · (−1, 0, 0)). Recall that all polarisation direc-
tions (and momenta) can be obtained from the same point-source propagators, while new
random-wall propagators are required for every polarisation direction (and momentum).
Thus, at zero momentum, the CPU time used to compute the RWcd two-point function
was four times larger for the RWcd source compared to the point source. Despite the
additional CPU time, the improvement at zero momentum is smaller than that for the
pseudoscalar kaon.
Bayesian fits of the K∗ correlators using the function (8.67) with nexp = n˜exp = 10 are
shown in Fig. 8.17 (for p′ = 0) and Fig. 8.17 (for p′ = 2pi/L · (−1, 0, 0)). Note that in
contrast to the K meson, the oscillations are strong even at p′ = 0. The priors for the
energy splittings and the relative excited-state amplitudes were set to the same values as
in Sec. 8.8.1.
The results for the ground-state amplitude and energy from Bayesian fits with various
values of nexp and τmin = 1 are listed in Tables 8.9 and 8.11. In addition, results from
unconstrained fits with nexp = n˜exp = 1 for a range of τmin are shown in Tables 8.8 and
8.10.
From the Bayesian fits, at zero momentum I find improvement factors of about 1.5
in the ground-state amplitude and about 1.5 in the ground-state energy. At p′ = 2pi/L ·
(−1, 0, 0), these factors are about 2.5 and 4, respectively. For the Bayesian fits I used
bootstrap to compute the central values and errors. The errors directly from the fit (not
shown) tend to be larger (for the non-Bayesian fit, they agree with the bootstrap errors).
I found that the bootstrap results are more stable under changes of the prior widths than
the results directly from the fit.
8.8. Two-point and three-point functions with random-wall sources 153
0 10 20 30 40 50 60
time
0.0001
0.001
0.01
0.1
1
rela
tive
err
or point sourceRWcd source
K* meson, p=0
Figure 8.15: Relative errors in the K∗ meson two-point functions at p′ = 0.
0 10 20 30 40 50 60
time
0.0001
0.001
0.01
0.1
1
rela
tive
err
or point sourceRWcd source
K* meson, p=1·2pi/L
Figure 8.16: Relative errors in the K∗ meson two-point functions at p′ = 2pi/L · (−1, 0, 0).
154 CHAPTER 8. RARE B DECAYS
0 10 20 30 40 50 60
time
1e-12
1e-09
1e-06
0.001
1
cor
rela
tor
K* meson, p=0
Figure 8.17: Bayesian fit of the K∗ meson two-point function (RWcd source) at p′ = 0.
The fit has τmin = 1 and nexp = n˜exp = 10.
0 10 20 30 40 50 60
time
1e-12
1e-09
1e-06
0.001
1
cor
rela
tor
K* meson, p=1·2pi/L
Figure 8.18: Bayesian fit of the K∗ meson two-point function (RWcd source) at p′ =
2pi/L · (−1, 0, 0). The fit has τmin = 1 and nexp = n˜exp = 10.
8.8. Two-point and three-point functions with random-wall sources 155
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
4 0.7091(47) 0.1217(14) 2.13 0.7183(19) 0.12476(53) 10.30
5 0.6675(65) 0.1060(21) 0.85 0.6689(28) 0.10703(88) 2.65
6 0.6508(80) 0.0992(28) 0.69 0.6504(34) 0.0998(11) 1.38
7 0.648(11) 0.0983(43) 0.70 0.6328(51) 0.0926(18) 1.02
8 0.655(15) 0.1017(68) 0.71 0.6244(63) 0.0890(24) 0.87
9 0.652(23) 0.100(11) 0.84 0.6183(88) 0.0862(36) 0.86
10 0.649(27) 0.098(14) 0.72 0.617(12) 0.0856(52) 0.91
Table 8.8: Results from unconstrained fits to the K∗ meson two-point function at p′ = 0.
All fits have nexp = n˜exp = 1. Central values and errors directly from fit.
point source RWcd source
nexp = n˜exp E A E A
4 0.638(10) 0.0924(39) 0.641(11) 0.0950(42)
5 0.636(10) 0.0912(41) 0.6257(67) 0.0887(29)
6 0.635(10) 0.0903(42) 0.6253(68) 0.0886(30)
7 0.634(10) 0.0899(39) 0.6246(76) 0.0882(35)
8 0.633(11) 0.0895(45) 0.6248(66) 0.0883(27)
9 0.633(10) 0.0894(38) 0.6249(58) 0.0886(23)
10 0.633(10) 0.0892(40) 0.6252(55) 0.0889(20)
Table 8.9: Results from Bayesian fits to the K∗ meson two-point function at p′ = 0. All
fits have τmin = 1. Central values and errors from bootstrap.
156 CHAPTER 8. RARE B DECAYS
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
4 0.8114(78) 0.1272(23) 1.99 0.8132(34) 0.12855(98) 3.15
5 0.765(11) 0.1098(34) 1.56 0.7685(54) 0.1126(17) 1.56
6 0.716(13) 0.0913(41) 1.14 0.7367(75) 0.1005(25) 1.08
7 0.728(20) 0.0962(75) 1.14 0.729(11) 0.0974(42) 1.08
8 0.723(31) 0.094(12) 1.18 0.699(15) 0.0850(55) 0.99
9 0.676(43) 0.074(16) 1.18 0.686(23) 0.0795(85) 1.01
10 0.712(68) 0.091(33) 1.18 0.669(30) 0.072(11) 1.03
Table 8.10: Results from unconstrained fits to the K∗ meson two-point function at p′ =
2pi/L · (−1, 0, 0). All fits have nexp = n˜exp = 1. Central values and errors directly from fit.
point source RWcd source
nexp = n˜exp E A E A
4 0.703(21) 0.0846(72) 0.7220(11) 0.0931(35)
5 0.700(21) 0.0827(77) 0.7070(83) 0.0879(27)
6 0.699(19) 0.0824(69) 0.7060(80) 0.0877(25)
7 0.697(20) 0.0818(72) 0.7045(85) 0.0869(27)
8 0.696(20) 0.0814(71) 0.7049(83) 0.0872(24)
9 0.695(22) 0.0810(76) 0.7052(82) 0.0872(25)
10 0.697(21) 0.0816(74) 0.7055(84) 0.0873(24)
Table 8.11: Results from Bayesian fits to the K∗ meson two-point function at p′ = 2pi/L ·
(−1, 0, 0). All fits have τmin = 1. Central values and errors from bootstrap.
8.8. Two-point and three-point functions with random-wall sources 157
8.8.2 Heavy-light meson two-point functions
For B mesons, the point-source correlator (without smearing) is
CB(τ,k) =
∑
y
Tr
[
G†q′(y, x)Gb(y, x)
]
e−ip·(y−x)
=
1
γ
∑
y
Tr
[
G†χq′ (y, x) Ω
†(y) S(Λ)
(
Gψv(y, x) 0
0 0
)
S(Λ) Ω(x)
]
e−ik·(y−x),
(8.68)
for τ = y0 − x0 > 0. Here we have used Eq. (6.24) to express Gb(y, x) in terms of the
moving-NRQCD propagator.
In order to obtain the random-wall correlator, we define
G˜pH(y, x0,k) =
∑
x
(
Gψv(y, x) 0
0 0
)
S(Λ) eik·x Ω(x) ξp(x). (8.69)
We then combine this with the zero-momentum random-wall light-quark propagator from
the same ξp to obtain the B meson random-wall correlator:
CB,RW(τ,k) =
1
γ
1
L3
1
nZ
nZ∑
p=1
∑
y
G˜pχq′ (y, x0,0)
∗ · tr
[
Ω†(y)S(Λ)G˜pH(y, x0,k)
]
e−ik·y. (8.70)
In (8.70), “tr” denotes a trace over spinor indices only.
I also compute correlators with gauge-invariant Gaussian smearing for the heavy quark
at the source and/or sink. The smearing is performed via the operator
(
1 +
σ
nS
∆(2)
)nS
, (8.71)
where ∆(2) is a covariant lattice Laplacian and σ, nS are the smearing parameters. The
operator (8.71) is inserted in Eq. (8.69) to the left of Ω(x) ξp(x) (for source smearing)
and/or to the left of Gψv(y, x) (for sink smearing).
Numerical results: B meson two-point functions
Figures 8.19 and 8.20 show comparisons of the relative errors (RWcd source vs point
source) in the B meson two-point functions (without smearing) at k = 0, v = 0 and at
k = 0, v = (0.4, 0, 0). As can be seen in the figures, the improvement depends strongly
on the separation τ between source and sink, and becomes small at large τ .
I performed fits to (2× 1)-matrix correlators with smeared sources and both local and
smeared sinks using a function of the form (6.31) with nexp = n˜exp. As for the light-meson
158 CHAPTER 8. RARE B DECAYS
two-point functions, here I also used the logarithms of the energy splittings and the relative
amplitudes for the excited states as the fit parameters. I set the priors for the former to
(−1) with a width of 1 and for the latter to 0.5 with a width of 2.
Plots of Bayesian fits with τmin = 2 and nexp = n˜exp = 10 are shown in Figs. 8.21 and
8.22. For the Bayesian fits of the B meson correlators, I show results both directly from
the fit and from bootstrap in Tables 8.14, 8.15, 8.16, and 8.17.
At v = 0, the errors from the Bayesian fits are larger for the RWcd source compared
to the point source, while the errors from the unconstrained fits are slightly smaller for
the RWcd source.
At v = (0.4, 0, 0), the situation is very different, albeit strongly dependent on the data-
analysis method used. With Bayesian fitting and error estimates from bootstrap, I find
improvement factors of about 3 for the ground-state energy and about 2 for the ground-
state amplitude from the RWcd source. However, from the same fits I see no improvement
when considering the errors directly from the fits.
With unconstrained fits, where I see agreement between the direct and the bootstrap
results, no significant improvement is observed for the RWcd source compared to the point
source, neither at v = 0 nor at v = (0.4, 0, 0).
8.8. Two-point and three-point functions with random-wall sources 159
0 10 20 30
time
0.0001
0.001
0.01
0.1
1
rela
tive
err
or
point sourceRWcd source
B meson, v=0.0
Figure 8.19: Relative errors in the B meson two-point functions (without smearing) at
k = 0, v = 0.
0 10 20 30
time
0.0001
0.001
0.01
0.1
1
rela
tive
err
or
point sourceRWcd source
B meson, v=0.4
Figure 8.20: Relative errors in the B meson two-point functions (without smearing) at
k = 0, v = (0.4, 0, 0).
160 CHAPTER 8. RARE B DECAYS
0 5 10 15 20
time
1e-07
1e-06
1e-05
0.0001
0.001
0.01
cor
rela
tor
smeared - smearedlocal - smeared
B meson, v=0
Figure 8.21: Bayesian fit of the B meson (2×1)-matrix correlator (RWcd source) at k = 0,
v = 0. The fit has τmin = 2 and nexp = n˜exp = 10.
0 5 10 15 20
time
1e-07
1e-06
1e-05
0.0001
0.001
0.01
cor
rela
tor
smeared - smearedlocal - smeared
B meson, v=0.4
Figure 8.22: Bayesian fit of the B meson (2×1)-matrix correlator (RWcd source) at k = 0,
v = (0.4, 0, 0). The fit has τmin = 2 and nexp = n˜exp = 10.
8.8. Two-point and three-point functions with random-wall sources 161
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
4 0.5776(18) 0.2386(12) 17.70 0.58702(98) 0.24617(58) 64.20
5 0.5603(21) 0.2242(15) 9.58 0.5565(14) 0.22229(90) 17.90
6 0.5396(26) 0.2068(19) 1.87 0.5430(18) 0.2095(13) 3.93
7 0.5426(32) 0.2092(26) 1.33 0.5315(24) 0.1990(19) 1.98
8 0.5362(41) 0.2022(36) 1.11 0.5292(31) 0.1963(27) 1.42
9 0.5260(53) 0.1911(49) 0.87 0.5223(42) 0.1891(40) 1.34
10 0.5243(67) 0.1890(68) 0.90 0.5184(54) 0.1842(53) 0.87
11 0.5308(84) 0.1975(96) 0.85 0.5152(75) 0.1802(79) 0.87
12 0.528(10) 0.193(13) 0.67 0.517(10) 0.183(12) 0.91
Table 8.12: Results from unconstrained fits to the B meson (2 × 1)-matrix correlator at
k = 0, v = (0.0, 0, 0). All fits have nexp = n˜exp = 1. The local amplitude A is shown.
Central values and errors directly from fit.
point source RWcd source
τmin E A χ2/d.o.f. E A χ2/d.o.f.
4 0.5436(37) 0.2398(22) 4.69 0.5382(20) 0.2390(12) 9.28
5 0.5261(52) 0.2258(35) 4.30 0.5163(33) 0.2215(22) 7.11
6 0.5031(76) 0.2029(53) 1.09 0.5082(53) 0.2094(38) 0.96
7 0.499(11) 0.2001(88) 1.13 0.4994(87) 0.2019(68) 0.96
8 0.500(18) 0.199(16) 1.15 0.511(15) 0.212(14) 0.95
Table 8.13: Results from unconstrained fits to the B meson (2 × 1)-matrix correlator at
k = 0, v = (0.4, 0, 0). All fits have nexp = n˜exp = 1. The local amplitude A is shown.
Central values and errors directly from fit.
162 CHAPTER 8. RARE B DECAYS
point source RWcd source
nexp = n˜exp E A E A
4 0.5248(56) 0.1890(60) 0.512(11) 0.174(16)
5 0.5253(55) 0.1897(58) 0.512(11) 0.174(16)
6 0.5260(57) 0.1902(61) 0.513(10) 0.175(14)
7 0.5256(58) 0.1897(63) 0.513(10) 0.175(14)
8 0.5258(59) 0.1899(64) 0.5145(89) 0.178(12)
9 0.5259(58) 0.1900(63) 0.5136(96) 0.177(13)
10 0.5259(59) 0.1900(64) 0.5149(87) 0.178(11)
Table 8.14: Results from Bayesian fits to the B meson (2× 1)-matrix correlator at k = 0,
v = 0. All fits have τmin = 2. The local amplitude A is shown. Central values and errors
directly from fit.
point source RWcd source
nexp = n˜exp E A E A
4 0.5245(55) 0.1866(72) 0.5113(90) 0.171(13)
5 0.5243(59) 0.1860(78) 0.5112(88) 0.172(12)
6 0.5251(55) 0.1876(71) 0.5117(77) 0.173(11)
7 0.5243(61) 0.1857(90) 0.5110(88) 0.171(12)
8 0.5248(55) 0.1867(75) 0.5107(88) 0.171(13)
9 0.5245(61) 0.1865(85) 0.5101(94) 0.170(14)
10 0.5248(58) 0.1871(74) 0.5109(72) 0.170(14)
Table 8.15: Results from Bayesian fits to the B meson (2× 1)-matrix correlator at k = 0,
v = 0. All fits have τmin = 2. The local amplitude A is shown. Central values and errors
from bootstrap.
8.8. Two-point and three-point functions with random-wall sources 163
point source RWcd source
nexp = n˜exp E A E A
4 0.491(13) 0.187(12) 0.498(10) 0.1955(95)
5 0.4918(99) 0.1883(82) 0.498(10) 0.1954(98)
6 0.4918(99) 0.1883(83) 0.498(11) 0.1954(99)
7 0.491(14) 0.187(12) 0.511(12) 0.196(12)
8 0.4919(99) 0.1883(83) 0.498(12) 0.196(12)
9 0.491(14) 0.187(12) 0.498(12) 0.196(12)
10 0.491(14) 0.187(12) 0.498(12) 0.196(12)
Table 8.16: Results from Bayesian fits to the B meson (2× 1)-matrix correlator at k = 0,
v = (0.4, 0, 0). All fits have τmin = 2. The local amplitude A is shown. Central values and
errors directly from fit.
point source RWcd source
nexp = n˜exp E A E A
4 0.482(20) 0.193(11) 0.4966(72) 0.1943(66)
5 0.480(24) 0.195(14) 0.4971(77) 0.1949(66)
6 0.479(24) 0.197(16) 0.4974(73) 0.1951(69)
7 0.480(24) 0.196(14) 0.4973(78) 0.1955(74)
8 0.480(24) 0.197(15) 0.4973(76) 0.1955(74)
9 0.481(23) 0.195(14) 0.4973(78) 0.1957(76)
10 0.478(27) 0.195(16) 0.4973(81) 0.1949(74)
Table 8.17: Results from Bayesian fits to the B meson (2× 1)-matrix correlator at k = 0,
v = (0.4, 0, 0). All fits have τmin = 2. The local amplitude A is shown. Central values and
errors from bootstrap.
164 CHAPTER 8. RARE B DECAYS
Figure 8.23: Contractions for the three-point functions with point sources.
8.8.3 Heavy-light meson three-point functions
In terms of the standard Dirac propagators, the point-source three-point function at τ =
|x0 − y0|, T = |x0 − z0| is given by
CFJB(τ, T, p, p
′) =
∑
y,z
e−ip
′·xe−i(p−p
′)·yeip·z Tr
[
ΓF Gq(x, y) ΓJ Gb(y, z) γˆ5 Gq′(z, x)
]
,
(8.72)
where ΓF = γˆ5 for F = P and ΓF = γˆj for F = V . See Fig. 8.23 for a diagram showing
the contractions. In (8.72) we used the simple form of the heavy-light current J = q¯ ΓJb.
When replacing the b quark propagator by the lattice mNRQCD propagator, the current
has to be replaced by the lattice current derived in Sec. 8.5. It is convenient to compute
and fit the three-point functions for the various terms in the lattice current individually.
Inserting the lattice current, the three-point function becomes
CFJB(τ, T, k, p
′) =
1
γ
∑
y,z
e−ip
′·xe−i(k−p
′)·yeik·z Tr
[
G†χq(y, x) F (x) Ω
†(y) γˆ5
× J
(
Gψv(y, z) 0
0 0
)
S(Λ) γˆ5 Ω(z)Gχq′ (z, x)
]
(8.73)
(for x0 > y0 > z0). In (8.73), we have F (x) = 1 for a pseudoscalar meson in the final
state and F (x) = (−1)xj γˆj for a vector meson in the final state. The symbol J in (8.73)
denotes the gamma matrix / derivative operator content of the heavy-light current:
J ∈
{
ΓS+(Λ), ΓS−(Λ), Γ (−iγˆ0v + iγˆ ± iv/γ) ·∆(±)S+(Λ)
}
. (8.74)
The three-point function (8.73) can be computed by using the spectator-quark (q′) prop-
agator as a source for the heavy-quark propagator, so that only the sum over y remains
8.8. Two-point and three-point functions with random-wall sources 165
Figure 8.24: Contractions for the three-point functions with random-wall sources.
when the contractions are performed. This is the sequential-source method. More pre-
cisely, one needs to compute moving-NRQCD propagators from the following source at
the time slice z0,
eik·z S(Λ) γˆ5 Ω(z)Gχq′ (z, x), (8.75)
evolving them to the sink time-slice y0. In contrast, both light-quark propagators are
computed from the same point source at x.
For smeared three-point functions, I insert the operator (7.10) in the source (8.75), to
the left of Ω(z)Gχq′ (z, x).
With the method described here, the point-source three-point functions for arbitrary
current operators J , arbitrary polarisations (which enter as F (x)) and arbitrary final-
state momenta p′ can be obtained with little computational cost, since no new propagators
need to be computed when changing any of these.
The situation is somewhat different for the random-wall three-point functions. To
obtain these, we now define the sequential-source heavy-quark random-wall propagator:
G˜pH(y, z0, x0,k,p
′) =
∑
z
(
Gψv(y, z) 0
0 0
)
S(Λ) γˆ5 e
ik·z Ω(z) G˜pχq′ (z, x0,−p
′). (8.76)
See Fig. 8.24 for a diagram illustrating the method. In (8.76), the spectator-quark random-
wall propagator G˜pχq′ (z, x0,−p
′) is defined as in (8.61).
For a pseudoscalar meson in the final state, the random-wall three-point correlator is
then given by
C5JB, RW(τ, T, k, p
′) =
1
γ
1
L3
1
nZ
nZ∑
p=1
∑
y
e−i(k−p
′)·y
× G˜pχq(y, x0,0)
∗ · tr
[
Ω†(y) γˆ5 J G˜
p
H(y, z0, x0,k,p
′)
]
.
(8.77)
166 CHAPTER 8. RARE B DECAYS
In (8.77), the quantity G˜pχq(y, x0,0) is defined as in (8.61) for p
′ = 0.
Similarly, for a vector meson in the final state, the random-wall three-point correlator
is
CjJB, RW(τ, T, k, p
′) =
1
γ
1
L3
1
nZ
nZ∑
p=1
∑
y
e−i(k−p
′)·y
× G˜pχq(y, x0, j)
∗ · tr
[
γˆj γˆ5 Ω
†(y) γˆ5 J G˜
p
H(y, z0, x0,k,p
′)
]
,
(8.78)
where the quantity G˜pχq(y, x0, j) is defined as in (8.65).
For smeared three-point functions, I insert the operator (7.10) in the equation for the
sequential-source heavy-quark propagator (8.76), to the left of Ω(z) G˜pχq′ (z, x0,−p
′). As
can be seen from Eqs. (8.77) and (8.78), with the random-wall method, the number of
heavy-quark propagators required is proportional to (#k)× (#p′). The number of light-
quark propagators required is proportional to (#p′)+(#j). Here, “#” stands for “number
different values used”.
Simultaneous fitting of two-point and three-point functions
I fit the three-point correlators with functions of the form
CFJB(τ, T, k, p
′) =
n(F )exp−1∑
n′=0
n(B)exp−1∑
n=0
A(ee)n′ n e
−E(F )
n′
τ e−E
(B)
n (T−τ)
+ (−1)τ
en(F )exp−1∑
n′=0
n(B)exp−1∑
n=0
A(oe)n′ n e
− eE(F )
n′
τ e−E
(B)
n (T−τ)
+ (−1)T−τ
n(F )exp−1∑
n′=0
en(B)exp−1∑
n=0
A(eo)n′ n e
−E(F )
n′
τ e−
eE(B)n (T−τ)
+ (−1)T
en(F )exp−1∑
n′=0
en(B)exp−1∑
n=0
A(oo)n′ n e
− eE(F )
n′
τ e−
eE(B)n (T−τ), (8.79)
again using the logarithms of the energy splittings and the relative amplitudes as the fit
parameters for the excited states. In (8.79), the labels “e” and “o” on the amplitude
parameters stand for “even” and “odd”, respectively.
I find that the results for the form factors are most accurate when the B(k,v) → F (p′)
three-point functions are fitted simultaneously with B(k,v) and F (p′) two-point functions,
sharing the energy parameters E(B)n , E˜
(B)
n , E
(F )
n′ , and E˜
(F )
n′ .
Furthermore, I include additional three-point functions to improve the fit as follows:
8.8. Two-point and three-point functions with random-wall sources 167
• ForB(k,v) → K(p′) at non-zero p′, I also include the three-point functionB(k,v) →
K(0) in the fit, sharing the energy parameters E(B)n and E˜
(B)
n .
• For B(k,v) → K∗(p′) at any p′, I also include the three-point function B(k,v) →
K(0) in the fit, sharing the energy parameters E(B)n and E˜
(B)
n .
In both cases, the B(k,v) → K(0) three-point function has a much better signal and
strongly constrains the B-meson energy and amplitude in the simultaneous fit. As demon-
strated in the next section, the results for EB andAB are up to about 5 times more accurate
compared to equivalent fits of the B-meson two-point function alone.
Numerical results: B → K three-point functions
So far, I have only analysed the three-point functions without smearing.
Comparisons (RWcd source vs point source) of the relative errors in the B → K
three-point functions with the operator J = γˆ0S+(Λ) are shown in Fig. 8.25 (for v = 0,
k = 0,p′ = 0) and Fig. 8.26 (for v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0)). In both
figures, the three-point functions are evaluated at T = 12. The improvement from the
RWcd source is found to decrease with increasing momentum transfer; it is also found to
decrease when increasing T .
For the B → K three-point functions, I performed unconstrained fits including a wide
range of T , setting n(B)exp = n˜
(B)
exp = n
(K)
exp = 1 and n˜
(K)
exp = 0 (for p′ = 0) or n˜
(K)
exp = 1 (for
p′ 6= 0). The fitting ranges for τ are chosen such that contaminations from excited states
are negligible.
In Fig. 8.27 I show a fit of the B → K three-point function withJ = γˆ0S+(Λ) at zero
recoil, i.e. v = 0, k = 0,p′ = 0. This is the case with the best signal-to-noise ratio. The
fit was performed simultaneously with the K-meson two-point function and the (2 × 1)-
matrix B-meson two-point function. I chose the following fitting ranges: T = 18 ... 26
and τ = 6 ... (T − 12) in the three-point function; τ = 12 ...32 in the B-meson two-point
function and τ = 10 ... 54 in the K-meson two-point function. To confirm that for these
fitting ranges the contaminations from excited states are negligible, I varied the values for
τ (B 2pt)min , τ
(K 2pt)
min , τ
(B 3pt)
min and τ
(K 3pt)
min in the fit. The latter two specify the range of τ in
the three-point function, which is
τ = τ (K 3pt)min ...
[
T − τ (B 3pt)min
]
. (8.80)
The fit results for the non-oscillating states are shown in Table 8.18 (for the point-source
data) and Table 8.19 (for the RWcd-source data).
168 CHAPTER 8. RARE B DECAYS
Note in particular that the results for the B-meson energy and amplitude with the
simultaneous fits are about 5 times more accurate than the results from fits to the B
meson two-point function alone (with the same τmin). The results from unconstrained fits
to the two-point functions with τmin = 12 (see Table 8.12) are
point source : EB = 0.528(10), AB = 0.193(13),
RWcd source : EB = 0.517(10), AB = 0.183(12).
(8.81)
When the fit is performed simultaneously with the B → K three-point function at zero
recoil (also setting τ (B 3pt)min = 12) and the K-meson two-point function, the results are
point source : EB = 0.5177(27), AB = 0.1822(35),
RWcd source : EB = 0.5158(19), AB = 0.1803(26).
(8.82)
The results for the three-point amplitude are
point source : AKJB = 0.0747(24),
RWcd source : AKJB = 0.0761(17).
(8.83)
The improvement factor in the statistical error of AKJB (RWcd source compared to point
source) is about 1.4. The findings for the other operators J are similar.
Figure 8.28 shows a fit of the B → K three-point function with J = γˆ0S+(Λ) at
v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0). This corresponds to the lowest value in q2
(the highest recoil) considered here. As can be seen in the plot, the signal-to noise ratio
is much smaller than at zero recoil. The fit shown here is simultaneous with the B → K
three-point function at v = (0.4, 0, 0), k = 0,p′ = 0, the K meson two-point functions
at both p′ = 2pi/L · (−1, 0, 0) and p′ = 2pi/L · (0, 0, 0), as well as the (2 × 1)-matrix
two-point function for the B meson at v = (0.4, 0, 0), k = 0. The fitting ranges are
τ (B 2pt)min = τ
(B 3pt)
min = 10, τ
(K 2pt)
min = 10, τ
(K 3pt)
min = 6.
The fit results for several fitting ranges are shown in Table 8.20 (for the point-source
data) and Table 8.21 (for the RWcd-source data). As can be seen in the tables, the results
are not as stable under changes of the fitting ranges as at zero recoil, and the results for
the three-point amplitudes from the point- and RWcd sources with the same fitting ranges
do not quite agree. It is planned for the future to perform Bayesian fits, which might be
more stable.
8.8. Two-point and three-point functions with random-wall sources 169
0 12
τ
0.001
0.01
0.1
1
10
rela
tive
erro
r
point sourceRWcd source
B K, v=0, k=0, p’=0 T = 12
Figure 8.25: Relative errors in the B → K three-point function with J = γˆ0S+(Λ) at
v = 0, k = 0,p′ = 0.
0 12
τ
0.001
0.01
0.1
1
10
rela
tive
erro
r
point sourceRWcd source
B K, v=0.4, k=0, p’= -1·2pi/L T = 12
Figure 8.26: Relative errors in the B → K three-point function with J = γˆ0S+(Λ) at
v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0).
170 CHAPTER 8. RARE B DECAYS
18
20
22
24
26
T
6
8
10
12
14
Τ
10
-7
10
-6
10
-5
10
-4
CHT,ΤL
Figure 8.27: Unconstrained fit of the B → K three-point function with J = γˆ0S+(Λ) at
v = 0, k = 0,p′ = 0. The numbers of exponentials are n(B)exp = n˜
(B)
exp = n
(K)
exp = 1, n˜
(K)
exp = 0.
The fitting range is T = 18 ... 26 and τ = 6 ... (T − 12). The fit is simultaneous with the
B and K two-point functions. The data shown are from the RWcd source. The fit has
χ2/d.o.f. = 0.92.
8.8. Two-point and three-point functions with random-wall sources 171
16
17
18
19
20
T
6
7
8
9
10
Τ
10
-7
10
-6
10
-5
10
-4
CHT,ΤL
Figure 8.28: Unconstrained fit of the B → K three-point function with J = γˆ0S+(Λ)
at v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0). The numbers of exponentials are n(B)exp =
n˜(B)exp = n
(K)
exp = n˜
(K)
exp = 1. The fitting range is T = 16 ... 20 and τ = 6 ... (T − 10).
The fit is simultaneous with the B and K two-point functions and with the B → K
three-point function at p′ = 0. The data shown are from the RWcd source. The fit has
χ2/d.o.f. = 1.04.
172
C
H
A
P
T
E
R
8.
R
A
R
E
B
D
E
C
A
Y
S
τ (B 2pt)min τ
(B 3pt)
min τ
(K 2pt)
min τ
(K 3pt)
min AKJB EK AK EB AB χ
2/d.o.f.
12 12 10 10 0.0747(37) 0.33926(24) 0.2739(18) 0.5164(40) 0.1807(47) 0.88
12 12 10 8 0.0754(28) 0.33930(24) 0.2726(18) 0.5181(31) 0.1828(39) 1.00
12 12 10 6 0.0747(24) 0.33935(24) 0.2723(17) 0.5177(27) 0.1822(35) 0.94
12 12 10 4 0.0747(22) 0.33945(23) 0.2719(17) 0.5180(24) 0.1824(32) 0.97
12 12 10 2 0.0734(20) 0.33956(22) 0.2716(16) 0.5165(22) 0.1807(29) 0.95
12 12 10 10 0.0747(37) 0.33926(24) 0.2739(18) 0.5164(40) 0.1807(47) 0.88
12 12 9 9 0.0725(31) 0.33927(23) 0.2737(18) 0.5141(35) 0.1785(42) 0.85
12 12 8 8 0.0755(28) 0.33938(23) 0.2726(17) 0.5184(31) 0.1836(39) 1.03
12 12 7 7 0.0750(25) 0.33961(22) 0.2732(17) 0.5177(28) 0.1839(36) 1.22
12 12 10 6 0.0747(24) 0.33935(24) 0.2723(17) 0.5177(27) 0.1822(35) 0.94
11 11 10 6 0.0720(16) 0.33942(23) 0.2721(17) 0.5146(19) 0.1797(24) 1.00
10 10 10 6 0.0709(12) 0.33952(23) 0.2710(17) 0.5140(14) 0.1787(18) 1.06
12 12 10 6 0.0747(24) 0.33935(24) 0.2723(17) 0.5177(27) 0.1822(35) 0.94
12 11 10 6 0.0720(16) 0.33940(23) 0.2723(17) 0.5145(19) 0.1784(28) 0.96
12 10 10 6 0.0711(13) 0.33951(23) 0.2711(17) 0.5141(14) 0.1778(23) 1.04
12 12 10 6 0.0747(24) 0.33935(24) 0.2723(17) 0.5177(27) 0.1822(35) 0.94
11 12 10 6 0.0747(24) 0.33937(24) 0.2722(17) 0.5178(27) 0.1834(32) 0.98
10 12 10 6 0.0740(23) 0.33935(24) 0.2722(17) 0.5170(25) 0.1817(27) 0.98
9 12 10 6 0.0747(23) 0.33936(24) 0.2721(17) 0.5181(25) 0.1847(25) 1.04
8 12 10 6 0.0793(20) 0.33942(23) 0.2722(17) 0.5233(21) 0.1912(18) 1.14
Table 8.18: Results from unconstrained simultaneous fits of the B → K three-point function with J = γˆ0S+(Λ) at v = 0,
k = 0,p′ = 0. Data from point source. n(B)exp = n˜
(B)
exp = n
(K)
exp = 1, n˜
(K)
exp = 0. Central values and errors directly from fit.
8.8.
T
w
o-p
oint
and
three-p
oint
functions
w
ith
random
-w
all
sources
173
τ (B 2pt)min τ
(B 3pt)
min τ
(K 2pt)
min τ
(K 3pt)
min AKJB EK AK EB AB χ
2/d.o.f.
12 12 10 10 0.0764(28) 0.33922(17) 0.27701(41) 0.5163(31) 0.1808(37) 0.96
12 12 10 8 0.0764(21) 0.33919(17) 0.27695(40) 0.5161(23) 0.1806(29) 0.94
12 12 10 6 0.0761(17) 0.33921(17) 0.27698(40) 0.5158(19) 0.1803(26) 0.92
12 12 10 4 0.0756(14) 0.33922(17) 0.27693(40) 0.5151(17) 0.1790(23) 0.99
12 12 10 2 0.0750(13) 0.33923(17) 0.27697(39) 0.5141(15) 0.1774(21) 1.18
12 12 10 10 0.0764(28) 0.33922(17) 0.27701(41) 0.5163(31) 0.1808(37) 0.96
12 12 9 9 0.0769(24) 0.33930(16) 0.27728(38) 0.5167(26) 0.1814(32) 0.93
12 12 8 8 0.0772(21) 0.33941(16) 0.27772(35) 0.5168(23) 0.1813(29) 1.09
12 12 7 7 0.0772(19) 0.33972(15) 0.27863(33) 0.5165(21) 0.1813(27) 1.49
12 12 10 6 0.0761(17) 0.33921(17) 0.27698(40) 0.5158(19) 0.1803(26) 0.92
11 11 10 6 0.0752(11) 0.33920(17) 0.27700(40) 0.5149(14) 0.1799(17) 0.92
10 10 10 6 0.07328(71) 0.33917(17) 0.27697(39) 0.5126(10) 0.1783(12) 1.07
12 12 10 6 0.0761(17) 0.33921(17) 0.27698(40) 0.5158(19) 0.1803(26) 0.92
12 11 10 6 0.0751(11) 0.33920(17) 0.27700(40) 0.5147(14) 0.1791(21) 0.93
12 10 10 6 0.07307(72) 0.33917(17) 0.27695(39) 0.5123(10) 0.1761(17) 1.07
12 12 10 6 0.0761(17) 0.33921(17) 0.27698(40) 0.5158(19) 0.1803(26) 0.92
11 12 10 6 0.0763(17) 0.33921(17) 0.27698(40) 0.5161(19) 0.1812(22) 0.91
10 12 10 6 0.0764(16) 0.33920(17) 0.27698(40) 0.5161(18) 0.1818(18) 0.92
9 12 10 6 0.0771(16) 0.33917(17) 0.27683(40) 0.5171(18) 0.1852(17) 1.24
8 12 10 6 0.0786(14) 0.33918(17) 0.27686(40) 0.5188(15) 0.1874(13) 1.25
Table 8.19: Results from unconstrained simultaneous fits of the B → K three-point function with J = γˆ0S+(Λ) at v = 0,
k = 0,p′ = 0. Data from RWcd source. n(B)exp = n˜
(B)
exp = n
(K)
exp = 1, n˜
(K)
exp = 0. Central values and errors directly from fit.
174
C
H
A
P
T
E
R
8.
R
A
R
E
B
D
E
C
A
Y
S
τ (B 2pt)min τ
(B 3pt)
min τ
(K 2pt)
min τ
(K 3pt)
min AKJB EK AK EB AB χ
2/d.o.f.
10 10 10 6 0.0280(34) 0.4607(16) 0.2279(24) 0.466(11) 0.169(10) 1.16
10 10 10 5 0.0292(30) 0.4599(16) 0.2272(23) 0.4713(95) 0.1741(94) 1.14
10 10 10 4 0.0317(27) 0.4597(16) 0.2261(23) 0.4775(82) 0.1767(86) 1.21
11 11 10 6 0.0303(78) 0.4602(16) 0.2279(24) 0.470(21) 0.181(22) 1.07
10 10 10 6 0.0280(34) 0.4607(16) 0.2279(24) 0.466(11) 0.169(10) 1.16
9 9 10 6 0.0339(21) 0.4604(16) 0.2277(24) 0.4778(63) 0.1794(59) 1.18
10 11 10 6 0.0310(81) 0.4603(16) 0.2280(24) 0.472(21) 0.175(21) 1.08
10 10 10 6 0.0280(34) 0.4607(16) 0.2279(24) 0.466(11) 0.169(10) 1.16
10 9 10 6 0.0328(21) 0.4606(16) 0.2278(24) 0.4745(65) 0.1758(76) 1.16
11 10 10 6 0.0321(42) 0.4605(16) 0.2277(24) 0.479(12) 0.190(15) 1.18
10 10 10 6 0.0280(34) 0.4607(16) 0.2279(24) 0.466(11) 0.169(10) 1.16
9 10 10 6 0.0281(35) 0.4605(16) 0.2278(24) 0.467(11) 0.1693(99) 1.17
Table 8.20: Results from unconstrained simultaneous fits of the B → K three-point function withJ = γˆ0S+(Λ) at v = (0.4, 0, 0),
k = 0,p′ = 2pi/L · (−1, 0, 0). Data from point source. n(B)exp = n˜
(B)
exp = n
(K)
exp = n˜
(K)
exp = 1. Central values and errors directly from fit.
8.8.
T
w
o-p
oint
and
three-p
oint
functions
w
ith
random
-w
all
sources
175
τ (B 2pt)min τ
(B 3pt)
min τ
(K 2pt)
min τ
(K 3pt)
min AKJB EK AK EB AB χ
2/d.o.f.
10 10 10 6 0.0395(36) 0.46280(79) 0.2395(10) 0.4849(84) 0.1799(82) 1.09
10 10 10 5 0.0402(31) 0.46289(79) 0.2396(10) 0.4856(72) 0.1807(73) 1.07
10 10 10 4 0.0387(26) 0.46291(78) 0.2397(10) 0.4807(63) 0.1768(65) 1.11
11 11 10 6 0.050(11) 0.46280(79) 0.2395(10) 0.503(18) 0.211(23) 1.06
10 10 10 6 0.0395(36) 0.46280(79) 0.2395(10) 0.4849(84) 0.1799(82) 1.09
9 9 10 6 0.0384(18) 0.46270(79) 0.2393(10) 0.4783(45) 0.1832(42) 1.13
10 11 10 6 0.050(11) 0.46278(79) 0.2395(10) 0.503(18) 0.210(21) 1.05
10 10 10 6 0.0395(36) 0.46280(79) 0.2395(10) 0.4849(84) 0.1799(82) 1.09
10 9 10 6 0.0371(18) 0.46276(79) 0.2395(10) 0.4750(46) 0.1720(53) 1.09
11 10 10 6 0.0394(38) 0.46281(79) 0.2395(10) 0.4845(89) 0.181(11) 1.10
10 10 10 6 0.0395(36) 0.46280(79) 0.2395(10) 0.4849(84) 0.1799(82) 1.09
9 10 10 6 0.0395(36) 0.46273(79) 0.2394(10) 0.4853(84) 0.1898(80) 1.15
Table 8.21: Results from unconstrained simultaneous fits of the B → K three-point function withJ = γˆ0S+(Λ) at v = (0.4, 0, 0),
k = 0,p′ = 2pi/L · (−1, 0, 0). Data from RWcd source. n(B)exp = n˜
(B)
exp = n
(K)
exp = n˜
(K)
exp = 1. Central values and errors directly from fit.
176 CHAPTER 8. RARE B DECAYS
Numerical results: B → K∗ three-point functions
The relative errors in the B → K∗ three-point functions with J = σ0j γˆ5S+(Λ) from
the RWcd and point sources are shown in Fig. 8.29 (at zero recoil) and Fig. 8.30 (at
v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0)). The improvement from the RWcd source is
small (and gets even smaller at larger T ), despite the higher computational cost.
Here I performed simultaneous Bayesian fits with the range τ = 1 ... (T − 2) and
T = 14 ... 16 or T = 15 ... 16 in the the B → K∗ three-point functions, τ = 2 ... 32 in the
B-meson (2× 1)-matrix two-point function and τ = 1 ... 63 in the K∗ two-point function.
For all these correlators I used the same number of exponentials, n(B)exp = n˜
(B)
exp = n
(K∗)
exp =
n˜(K
∗)
exp = nexp, varying nexp in the range 4 ... 10. I set the priors for the logarithms of the
energy splittings to (−1) with a width of 1, and the priors for the relative excited-state
amplitudes to 1 with a width of 2 (for the K∗) and 0.5 with a width of 2 (for the B
two-point function and for the B → K∗ three-point function).
As already mentioned in Sec. 8.8.3, I also included the B → K three-point function
(with J = γˆ0S+(Λ)) and the K two-point function for p′ = 0 in the fit in order to
improve the results for the B meson. For these correlators I used n(B)exp = n˜
(B)
exp = n
(K)
exp = 1,
n˜(K)exp = 0 and the same fitting ranges as given in Sec. 8.8.3, so that the contamination
from excited states is negligible.
A plot of a fit to the B → K∗ three-point function with J = σj0γˆ5S+(Λ) at v = 0,
k = 0,p′ = 0 is shown in Fig. 8.31. Fit results are given in Table 8.22 (point source data)
and Table 8.23 (RWcd source data). For the point source, I used T = 14 ... 16. For the
RWcd source, the range T = 15 ... 16 gave more reliable fits.
As can be seen in the tables, the errors in AK∗JB are comparable for the RWcd and
point source data, while the RWcd source gives more precise results for EK∗ and AK∗ ,
compared to the fits of the K∗ two-point function alone (cf. Sec. 8.8.1).
The findings for non-zero p′ and for the other operators J are similar.
8.8. Two-point and three-point functions with random-wall sources 177
0 12
τ
0.001
0.01
0.1
1
10
rela
tive
erro
r
point sourceRWcd source
B K*, v=0, k=0, p’=0 T = 12
Figure 8.29: Relative errors in the B → K∗ three-point function with J = σ0j γˆ5S+(Λ)
at v = 0, k = 0,p′ = 0.
0 12
τ
0.001
0.01
0.1
1
10
rela
tive
erro
r
point sourceRWcd source
B K*, v=0.4, k=0, p’= -1·2pi/L T = 12
Figure 8.30: Relative errors in the B → K∗ three-point function with J = σ0j γˆ5S+(Λ)
at v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0).
178 CHAPTER 8. RARE B DECAYS
14
15
16
T
1
3
5
7
9
11
13
15
Τ
10
-7
10
-6
10
-5
10
-4
CHT,ΤL
Figure 8.31: Bayesian fit of the B → K∗ three-point function with J = σj0γˆ5S+(Λ) at
v = 0, k = 0,p′ = 0. The fitting range is T = 14 ... 16 and τ = 1 ... (T − 2). The numbers
of exponentials are n(B)exp = n˜
(B)
exp = n
(K∗)
exp = n˜
(K∗)
exp = 8. The fit is simultaneous with the B
and K∗ two-point functions and with the B → K three-point function and K two-point
function. The data shown are from the point source.
8.8. Two-point and three-point functions with random-wall sources 179
nexp AK∗JB EK∗ AK∗ EB AB
4 0.01261(88) 0.6330(58) 0.0906(23) 0.5172(19) 0.1803(15)
5 0.0124(12) 0.6363(91) 0.0915(30) 0.5167(25) 0.1793(25)
6 0.0125(15) 0.6361(87) 0.0915(38) 0.5168(22) 0.1791(27)
7 0.0123(12) 0.6360(72) 0.0914(31) 0.5166(26) 0.1789(31)
8 0.0127(14) 0.6363(74) 0.0916(33) 0.5167(22) 0.1792(30)
9 0.0124(14) 0.6355(76) 0.0915(36) 0.5165(27) 0.1793(35)
10 0.0120(16) 0.6348(91) 0.0906(39) 0.5167(17) 0.1794(23)
Table 8.22: Results from simultaneous Bayesian fits of the B → K∗ three-point function
with J = σj0γˆ5S+(Λ) at v = 0, k = 0,p′ = 0. The fitting range in the B → K∗
three-point function is T = 14 ... 16 and τ = 1 ... (T − 2). The numbers of exponentials are
n(B)exp = n˜
(B)
exp = n
(K∗)
exp = n˜
(K∗)
exp = nexp. Data from point source. Central values and errors
from bootstrap.
nexp AK∗JB EK∗ AK∗ EB AB
4 0.0164(20) 0.680(33) 0.1067(99) 0.5166(13) 0.18011(65)
5 0.01342(98) 0.6302(64) 0.0908(17) 0.5155(20) 0.1790(26)
6 0.01314(79) 0.6282(51) 0.0902(15) 0.5158(16) 0.1789(23)
7 0.01316(80) 0.6274(47) 0.0898(11) 0.5156(17) 0.1782(27)
8 0.01305(85) 0.6262(44) 0.0895(14) 0.5152(21) 0.1775(30)
9 0.0131(12) 0.6257(49) 0.0894(17) 0.5153(17) 0.1779(29)
10 0.0130(12) 0.6255(40) 0.0893(10) 0.5155(19) 0.1784(24)
Table 8.23: Results from simultaneous Bayesian fits of the B → K∗ three-point function
with J = σj0γˆ5S+(Λ) at v = 0, k = 0,p′ = 0. The fitting range in the B → K∗
three-point function is T = 15 ... 16 and τ = 1 ... (T − 2). The numbers of exponentials are
n(B)exp = n˜
(B)
exp = n
(K∗)
exp = n˜
(K∗)
exp = nexp. Data from RWcd source. Central values and errors
from bootstrap.
180 CHAPTER 8. RARE B DECAYS
J AKJB(ΓJ = γˆ0) AKJB(ΓJ = γˆ1) AKJB(ΓJ = σ10)
Γ S+(Λ) 0.0280(34) 0.0063(11) 0.0218(25)
Γ S−(Λ) 0.0368(43) 0.0218(25) 0.0065(11)
i
2mb
γˆ ·∆(±) Γ S+(Λ) −0.00177(26) 0.00177(31) −0.00148(29)
−
i
2mb
γˆ0 v ·∆(±) Γ S+(Λ) −0.000084(55) 0.000139(59) 0.000189(57)
±
i
2γmb
v ·∆(±) Γ S+(Λ) 0.000143(51) −0.000168(51) −0.000123(53)
Table 8.24: Fit results for the B → K ground-state amplitudes of the individual terms
in the heavy-light lattice vector and tensor currents. The boost velocity is v = (0.4, 0, 0)
and the momenta are k = 0, p′ = 2pi/L · (−1, 0, 0). Data from the point source. Central
values and errors directly from fit.
8.9 Size of the 1/m corrections
The construction of the heavy-light lattice currents was discussed in Sec. 8.5. Now I give
examples for the contributions of the individual terms in the current at v = (0.4, 0, 0). I
consider the pseudoscalar kaon in the final state as the results are more precise compared
to the vector-meson.
Fit results for AKJB from the different operators J are shown in Table 8.24. Results
are given for ΓJ = γˆ0, ΓJ = γˆ1, and ΓJ = σ10.
It should be noted that the O(1/mb) corrections considered here are the unsubtracted
operators that mix with the lower-dimension operators. For the vector current in non-
moving NRQCD it was found in [11] that the matrix elements of the subtracted operators
are significantly smaller.
8.10 Preliminary form factor results
In the following I give some preliminary results for the form factors f0, f+, fT and T1, T2
obtained with the fitting methods described in Sec. 8.8. I computed the form factors from
the formulae in Sec. 8.4, using bootstrap to take into account the correlations between
the different quantities entering the formula. For the hadron masses and energies I also
8.10. Preliminary form factor results 181
used the lattice results. I computed the full B-meson energies using the non-perturbative
results for the mNRQCD energy shift Cv obtained from the ηb dispersion relation (see
Sec. 6.1.3). I set Zp = 1, which is compatible with the result obtained in Sec. 6.1.3. For
the lattice spacing I used a−1 = 1.6 GeV [108]. The values of q2 given in the tables
correspond to the experimental values for the hadron masses.
All form factor results shown in the following include the one-loop radiative corrections
to the leading-order heavy-light currents, i.e. the lattice currents are given by (8.55). I
used the matching coefficients calculated by Eike Mu¨ller [114, 116] and set the value
of the strong coupling constant to αs = 0.3, which is approximately equal to αV (2/a)
(cf. Sec. 6.3).
8.10.1 The form factors f0, f+, and fT
The preliminary results for f0, f+, and fT for B → K are listed in Table 8.25 and plotted
against q2 in Figs. 8.32, 8.33. At |v| = 0.4 I added to the error estimates a systematic
uncertainty resulting from the choice of τ (B 3pt)min , taken as the shift in the central value of
the form factor when τ (B 3pt)min is changed by 1.
At zero recoil I find that the statistical error in f0 is smaller for the RWcd-source data
compared to the point-source data by a factor of about 1.4. At the largest recoil considered
here, corresponding to v = (0.4, 0, 0), k = 0,p′ = 2pi/L · (−1, 0, 0), the estimated error for
the RWcd-source result is larger than that for the point-source data.
Overall, the central values of the results from the RWcd-source data tend to be above
those from the point-source data, sometimes outside the one-standard-deviation interval.
As already discussed in Sec. 8.8, the fit results are not very stable and systematic uncer-
tainties may be underestimated. It is planned to perform new Bayesian fits of the B → K
data in the future.
I find that the results for fT and f+ are very close to each other. Calculations using
light-cone sum-rules [197], albeit applicable only at low q2, give fT > f+ and show that
the difference between fT and f+ vanishes as the mass of the pseudoscalar meson in the
final state is taken to zero.
Unquenched lattice QCD results for f0 and f+ for B → pi can be found in [11] and
[168]. For B → K, preliminary results for f0 and f+ computed using the Fermilab action
for the b quark can be found in [198]. No previous lattice results for fT are available.
182 CHAPTER 8. RARE B DECAYS
point source RWcd source
q2 (GeV2) f0 f+ fT f0 f+ fT
22.91 0.869(17) − − 0.889(12) − −
20.68 0.746(19) 1.951(78) 1.852(82) 0.780(15) 1.980(55) 1.858(61)
18.22 0.617(45) 1.21(12) 1.37(20) 0.715(61) 1.40(19) 1.58(29)
Table 8.25: Preliminary results for the form factors f0, f+, fT for B → K decays, obtained
from simultaneous unconstrained fits. The error estimates are statistical/fitting only.
15 17.5 20 22.5 25
q2 / GeV2
0
0.5
1
1.5
2
2.5
form
fact
or
f+ pointf+ RWcdf0 pointf0 RWcd
preliminary
B K
Figure 8.32: Preliminary results for the form factors f0, f+ for B → K decays, obtained
from simultaneous unconstrained fits. The left-most points have v = (0.4, 0, 0), k = 0 and
p′ = 2pi/L · (−1, 0, 0). The error estimates are statistical/fitting only.
8.10. Preliminary form factor results 183
15 17.5 20 22.5 25
q2 / GeV2
0
0.5
1
1.5
2
2.5
form
fact
or
fT pointfT RWcd
preliminary
B K
Figure 8.33: Preliminary results for the form factor fT for B → K decays, obtained from
simultaneous unconstrained fits. The left-most points have v = (0.4, 0, 0), k = 0 and
p′ = 2pi/L · (−1, 0, 0). The error estimates are statistical/fitting only.
184 CHAPTER 8. RARE B DECAYS
point source RWcd source
q2 (GeV2) T1 T2 T1 T2
19.25 − 0.285(26) − 0.305(19)
17.85 0.65(16) 0.246(38) 0.69(12) 0.272(50)
Table 8.26: Preliminary results for the form factors T1, T2 for B → K∗ decays, obtained
from simultaneous Bayesian fits with with T = 15, 16 (RWcd source), T = 14, 15, 16 (point
source). The error estimates are statistical/fitting only.
Reference Result
[179] T2(q2max) = 0.269
+0.017
−0.009 ± 0.011
[182] T2(q2max) = 0.325± 0.033± 0.065
[183] T2(q2max) = 0.25± 0.02, 0.22± 0.02
[181] T2(q2max) = 0.25± 0.1
[184] T2(18.3 GeV2) = 0.38± 0.05
+0.00
−0.03
Table 8.27: Quenched lattice results for T2 from the literature, obtained by extrapolation
in the heavy-quark mass. The two results cited from [183] correspond to two different
extrapolation methods. For [184], the result from the finest lattice spacing is shown. See
the original works for the meaning of the error estimates.
8.10.2 The form factors T1 and T2
Preliminary results for the form factors T1 and T2 for B → K∗ are listed in Table 8.26
and are plotted against q2 in Fig. 8.34. The points with the lower value of q2 have v = 0,
k = 0 and p′ = 2pi/L · (−1, 0, 0). The data at significantly lower values of q2 are very
noisy and have not yet been analysed.
For T2(q2max) and T1 at the lower value of q
2 the errors for the RWcd-source data are
smaller by a factor of about 1.3; however, recall that the computational cost for the RWcd
method is higher.
For comparison with the literature, Table 8.27 shows results for T2 at or near q2max from
quenched lattice calculations. All of these calculations were performed with Wilson-like
actions for the b quark with unphysically light values for mb, and were extrapolated in mb.
8.10. Preliminary form factor results 185
17 18 19 20
q2 / GeV2
0
0.25
0.5
0.75
1
form
fact
or
T1 pointT1 RWcdT2 pointT2 RWcd
preliminary
B K*
Figure 8.34: Preliminary results for the form factors T1, T2 for B → K∗ decays, obtained
from simultaneous Bayesian fits with T = 15, 16 (RWcd source), T = 14, 15, 16 (point
source). The left-most points have v = 0, k = 0 and p′ = 2pi/L · (−1, 0, 0). The error
estimates are statistical/fitting only.
186 CHAPTER 8. RARE B DECAYS
Chapter 9
Conclusions
Moving NRQCD is a method for the lattice computation of heavy-to-light form factors. It
enables the calculation in a reference frame where the momentum of the light hadron in the
final state is reduced, and thereby leads to a reduction of lattice discretisation errors. In
this dissertation, I have independently derived an O(1/m2b , v
4
orb.) mNRQCD action in the
continuum, and performed nonperturbative tests of a lattice action with the same accuracy
for the first time. These tests confirm the reduction of systematic errors compared to non-
moving NRQCD (cf. Fig. 6.2). I have then carried out the first calculations of form factors
with mNRQCD. The computer programs I have written for this purpose will be used in
future large-scale computations. This work will allow further tests of the Standard Model
of elementary particles, and will help in searches for new physics.
In the course of this dissertation I have also performed calculations of the heavy-hadron
spectrum in lattice QCD with dynamical domain wall fermions, including the spectrum
of excited states in bottomonium and the ground-state masses of baryons containing one,
two, or three b quarks. Several of these heavy baryons have not yet been observed ex-
perimentally but might be discovered in the future at the Tevatron or the Large Hadron
Collider.
In the following I shall discuss some open issues and give directions for future work
related to the various parts of this dissertation.
9.1 Renormalisation of the lattice mNRQCD action
In Sec. 6.3 I compared my nonperturbative determinations of Zp (the renormalisation of
the external momentum) and Cv (the mNRQCD energy shift) to predictions from one-loop
lattice perturbation theory. In both cases I considered the deviations from the tree-level
188 CHAPTER 9. CONCLUSIONS
values. For Zp good agreement is seen between the perturbative and nonperturbative cor-
rections at moderate boost velocities. However, for Cv the full nonperturbative correction
was found to have a sign opposite to the sign of the one-loop correction. Note that this
does not impose any practical problems, as the corrections are both small and the non-
perturbative results for Cv and Zp can be used. Assuming that no mistakes were made in
either calculation, these findings do however demonstrate that one-loop perturbation the-
ory is somewhat limited. To go beyond one-loop perturbation theory one can use high-β
methods, as discussed in Sec. 5.3.5.
In the present calculations, all couplings in the lattice mNRQCD action are set to their
tree-level values, and tadpole-improvement is used to account for radiative corrections. As
demonstrated by the numerical results, this is already a good approximation. However, if
very high precision is to be achieved, eventually one has to go beyond this approximation.
9.2 Heavy-hadron spectroscopy
The heavy-light spectrum calculations described in Sec. 7.2 are not yet complete. Other
light-quark masses need to be included, and chiral extrapolations to the physical point need
to be performed. The statistical errors can be reduced by computing more propagators.
All calculations presented in Chapter 7 are only for one lattice spacing. A more
systematic analysis of discretisation errors will be performed once new ensembles of gauge
field configurations with a finer lattice spacing are made available by the RBC and UKQCD
collaborations.
Using NRQCD I have achieved high statistical accuracy for the Ωbbb baryon, similarly
to bottomonium. I am therefore planning to study excited states also for the Ωbbb. Given
how much insight into the strong interaction has been gained by analysing the spectrum
of bottomonium, it will be highly interesting to perform similar work for the Ωbbb, where
three heavy quarks form a colour singlet.
9.3 Calculation of form factors for B decays
In Chapter 8 I have developed and tested the methods necessary for the numerical cal-
culation of B decay form factors with lattice mNRQCD. Now, further computer time is
required to consider more points in q2, perform calculations with different light-quark
masses and also different lattice spacings. After more points in q2 have been included,
the q2-dependence of the form factors can be fitted by suitable models [199, 184]. In
9.3. Calculation of form factors for B decays 189
particular, for B → K∗γ the form factors need to be extrapolated to q2 = 0.
The statistical errors must be reduced further. I found that the reduction in statistical
errors when comparing the random-wall-source method to the point-source method is small
for the final form factor results. Due to the lower computational cost, the point-source
method may lead to more precise results than the random-wall-source method for the
same amount of computer time.
The fitting methods for the three-point functions have not yet been finalised and there
is still room for optimisations. In particular, the effect of using a “smeared” interpolating
field for the B meson in the three-point function has not yet been analysed, and Bayesian
fits need to be performed where only unconstrained fits have been done so far.
The contributions of the tree-level 1/mb corrections in the heavy-light currents have
not yet been included in the form factor results as the necessary mixing coefficients are
not yet known. A calculation of these coefficients would be useful. So far, the matching
calculation for the heavy-light currents was performed in one-loop lattice perturbation
theory. It would be desirable to go beyond this approximation, using for example high-β
methods.
The study of rare B decays is complicated by long-distance effects and, for B → K∗
transitions, by the instability of the K∗ under the strong interactions. These issues require
further investigation.
190 CHAPTER 9. CONCLUSIONS
Appendix A
Conventions
A.1 Notation
• I use units in which c = ~ = 1.
• Space-time coordinates: x = (t,x) = (x0, x1, x2, x3) in Minkowski space, x =
(τ,x) = (x0, x1, x2, x3) in Euclidean space
• Indices µ, ν, ... run from 0 to 3; indices j, k, ... run from 1 to 3.
• Indices appearing twice (and only twice) in a single term are summed over all allowed
values, except where otherwise stated.
• The Minkowski metric is (gµν) = diag(1,−1,−1,−1).
• The Lorentz boost of the coordinates is defined in terms of the 3-velocity v as:
Λ =
γ γ vk
γ vj δjk + γ
2
1+γ v
jvk
, (A.1)
with γ = (1− v2)−1/2.
• Gamma matrices in Minkowski space are denoted with a hat, to distinguish them
from γ = (1− v2)−1/2. I use
γˆ0 =
(
σ0 0
0 −σ0
)
, γˆj =
(
0 σj
−σj 0
)
,
γˆ5 = iγˆ
0γˆ1γˆ2γˆ3 =
(
0 σ0
σ0 0
)
, (A.2)
192 APPENDIX A. CONVENTIONS
with the Pauli matrices σj . I define σ0 = 12×2. Note that in calculations with
moving NRQCD, the Minkowski space gamma matrices are also used in Euclidean
space for convenience.
• Euclidean gamma matrices are denoted without a hat:
γ0 = γˆ0 =
(
σ0 0
0 −σ0
)
, γj = −iγˆj =
(
0 −iσj
iσj 0
)
,
γ5 = γ
0γ1γ2γ3 = γˆ5 =
(
0 σ0
σ0 0
)
. (A.3)
• The spinorial Lorentz boost is defined in terms of the 3-velocity v as:
S(Λ) =
1
√
2(1 + γ)
[
(1 + γ)1− γ v · γˆ γˆ0
]
=
1
√
2(1 + γ)
(
1 + γ γ σ · v
γ σ · v 1 + γ
)
.
(A.4)
• Feynman slash notation:
/u = uµγˆµ. (A.5)
• Covariant derivatives and field strength tensor:
Dµ =
∂
∂xµ
+ igAµ, (A.6)
[Dµ, Dν ] = igFµν , (A.7)
DµFρσ = D
ad
µ Fρσ + FρσDµ with D
ad
µ Fρσ = (∂µFρσ) + ig[Aµ, Fρσ]. (A.8)
• Chromoelectric and chromomagnetic fields in Minkowski space:
Ek = F0k, Bj = −
1
2
²jklFkl. (A.9)
• Chromoelectric and chromomagnetic fields in Euclidean space:
Ek = −F0k, Bj = −
1
2
²jklFkl. (A.10)
A.2. Lattice derivatives and field strength 193
A.2 Lattice derivatives and field strength
In this section I give explicit expressions for the discretised derivatives I use in the lattice
action, Eqs. (5.12, 5.13). All expressions are constructed from the elementary forward,
backward and symmetric derivatives
∆+µψ(x) = Uµ(x)ψ(x+ µˆ)− ψ(x),
∆−µψ(x) = ψ(x)− U−µ(x)ψ(x− µˆ),
∆±µψ(x) =
1
2
[Uµ(x)ψ(x+ µˆ)− U−µ(x)ψ(x− µˆ)] . (A.11)
For performance reasons, I construct higher-order operators to be maximally local by
balancing the occurrence of these three types. I also symmetrise the expressions.
• Unimproved derivatives:
∆(2) =
3∑
j=1
∆+j ∆
−
j ,
∆(2)v =
1
2
3∑
j,k=1
vjvk
(
∆+j ∆
−
k +∆
−
j ∆
+
k
)
,
∆(3)v =
1
2
3∑
j,k,l=1
vjvkvl
(
∆+j ∆
±
k ∆
−
l +∆
−
j ∆
±
k ∆
+
l
)
,
∆(4)v =
1
2
3∑
j,k,l,m=1
vjvkvlvm
(
∆+j ∆
−
k ∆
+
l ∆
−
m +∆
−
j ∆
+
k ∆
−
l ∆
+
m
)
. (A.12)
• Improved derivatives:
∆˜±j = ∆
±
j −
1
6
∆+j ∆
±
j ∆
−
j ,
∆˜(2) = ∆(2) −
1
12
3∑
j=1
∆+j ∆
−
j ∆
+
j ∆
−
j ,
∆˜(2)v = ∆
(2)
v +
1
4
3∑
j,k=1
vjvk∆+j ∆
−
j ∆
+
k ∆
−
k
−
1
12
3∑
j,k=1
vjvk
(
∆+j ∆
−
j ∆
+
j ∆
−
k +∆
−
j ∆
+
j ∆
−
j ∆
+
k
+∆+j ∆
−
k ∆
+
k ∆
−
k +∆
−
j ∆
+
k ∆
−
k ∆
+
k
)
. (A.13)
194 APPENDIX A. CONVENTIONS
• Unimproved adjoint derivative:
∆adµ F˜ρσ(x) =
1
2
[
Uµ(x)F˜ρσ(x+ µˆ)U
†
µ(x)− U−µ(x)F˜ρσ(x− µˆ)U
†
−µ(x)
]
. (A.14)
• Improved field strength tensor:
F˜µν(x) =
5
3
Fµν(x)
−
1
6
(
Uµ(x)Fµν(x+ µˆ)U
†
µ(x) + U−µ(x)Fµν(x− µˆ)U
†
−µ(x)− (µ↔ ν)
)
,
(A.15)
where
Fµν(x) =
−i
2g
(
Ωµν(x)− Ω
†
µν(x)
)
,
Ωµν(x) = 14
∑
{(α,β)}µν
Uα(x)Uβ(x+ αˆ)U-α(x+ αˆ+ βˆ)U-β(x+ βˆ), (A.16)
with
{(α, β)}µν = {(µ, ν), (ν, -µ), (-µ, -ν), (-ν, µ)} for µ 6= ν . (A.17)
Appendix B
Form factors and decay rate for
B → K∗γ
B.1 Derivation of form factors
The invariant matrix element for the exclusive decay B → K∗γ is
M = −〈γ(q, r), K∗(p′, s) |Heff |B(p)〉, (B.1)
where r = 1, 2 denotes two possible photon polarisations and s = 1, 2, 3 denotes three
possible K∗ polarisations. The relevant part of the effective Hamiltonian is given by
(cf. Sec. 8.1.4)
Heff = −VtbV
∗
ts
4GF√
2
C7(µ)
e
16pi2
mb JµνF
µν
(e.m.) with Jµν = b¯σµν
1 + γ5
2
s, (B.2)
where C7(µ) is the Wilson coefficient at the renormalisation scale µ.
Upon contracting the external photon with Fµν(e.m.), we obtain
〈γ(q, r), K∗(p′, s) |JµνF
µν
(e.m.)|B(p)〉 = i
[
qµε∗ν(γ)(q, r)− q
νε∗µ(γ)(q, r)
]
〈K∗(p′, s) |Jµν |B(p)〉
= 2i qµε∗ν(γ)(q, r) 〈K
∗(p′, s) |Jµν |B(p)〉, (B.3)
where ε(γ) = ε(γ)(q, r) is the polarisation vector of the photon. We shall now derive the
general structure of the matrix element 〈K∗(p′, s) |Jµν |B(p)〉. The operator Jµν contains
both a tensor and a pseudotensor part, and it is convenient to consider them separately.
Let us start with the tensor part of Jµν , so we need to look at
〈K∗(p′, ε) |b¯σµνs|B(p)〉, (B.4)
196 APPENDIX B. FORM FACTORS AND DECAY RATE FOR B → K∗γ
where ε = ε(p′, s) now denotes the K∗ polarisation vector. Under parity and time reversal,
the states transform as follows:
P|B(p)〉 = −|B(pP )〉,
P|K∗(p′, ε)〉 = |K∗(p′P , εP )〉,
T|B(p)〉 = −|B(pT )〉,
T|K∗(p′, ε)〉 = |K∗(p′T , εT )〉 (B.5)
with pP = pT = (p0,−p) etc. Since QCD is invariant under these transformations, we
must have for i, j = 1, 2, 3
〈K∗(p′, ε) |b¯ σ0j s|B(p)〉 = +〈K
∗(p′P , εP ) |b¯ σ0j s|B(pP )〉,
〈K∗(p′, ε) |b¯ σij s|B(p)〉 = −〈K
∗(p′P , εP ) |b¯ σij s|B(pP )〉,
〈K∗(p′, ε) |b¯ σ0j s|B(p)〉
∗ = +〈K∗(p′T , εT ) |b¯ σ0j s|B(pT )〉,
〈K∗(p′, ε) |b¯ σij s|B(p)〉
∗ = −〈K∗(p′T , εT ) |b¯ σij s|B(pT )〉. (B.6)
Here, we have made a choice of phase.
There are three Lorentz-vectors available: the momenta p, p′ and the polarisation
vector ε = ε(p′, s). Since the K∗ meson is in the final state, ε must appear as its complex
conjugate. Let us use the basis vectors p+ ≡ p + p′, p− ≡ p − p′ and ε∗. From the
transformation law under parity, Eq. (B.6), it follows that there are three possible basis
pseudotensors:
²µνρσε
∗ρpσ+, ²µνρσε
∗ρpσ− and ²µνρσp
ρ
−p
σ
+. (B.7)
Each of them can come with a function of Lorentz-scalar quantities as a prefactor. We
can build the following basic Lorentz scalars:
p2+, p
2
−, ε
∗2, p− · p+, ε
∗ · p− and ε
∗ · p+. (B.8)
However, since ε is a polarisation vector, we have ε2 = −1 and ε · p′ = 0. The latter
implies that both ε∗ ·p− and ε∗ ·p+ reduce to the single quantity ε∗ ·p. Furthermore, since
p2 =MB and p′2 =MK∗ , we have
p− · p+ = M
2
B −M
2
K∗ ,
p2+ = M
2
B +M
2
K∗ + 2p · p
′,
p2− = M
2
B +M
2
K∗ − 2p · p
′. (B.9)
B.1. Derivation of form factors 197
Hence, it suffices to consider functions of ε∗ · p and p2− only. Another restriction is that
the matrix element must be linear in the polarisation vector ε∗. It follows that we can
write the matrix element in the form
〈K∗(p′, ε) |b¯σµνs|B(p)〉 = f+(p
2
−)²µνρσε
∗ρpσ+ + f−(p
2
−)²µνρσε
∗ρpσ−
+ g(p2−)(ε
∗ · p)²µνρσp
ρ
−p
σ
+ , (B.10)
with unknown functions f+, f− and g. The transformation law under time reversal,
Eq. (B.6), implies that these functions are real.
Let us now consider the part of Jµν containing γ5. The identity
σµνγ5 =
i
2
²µνρσσ
ρσ (B.11)
allows us to relate the matrix element to (B.10):
〈K∗(p′, ε) |b¯ σµνγ5 s|B(p)〉 =
i
2
²µνρσ〈K
∗(p′, ε) |b¯σρσs|B(p)〉
=
i
2
²µνρσ
[
f+(p
2
−)²
ρσαβε∗αp+β + f−(p
2
−)²
ρσαβε∗αp−β
+ g(p2−)(ε
∗ · p)²ρσαβp−αp+β
]
. (B.12)
Using
²µνρσ²
ρσαβ = −2
(
δαµδ
β
ν − δ
β
µδ
α
ν
)
, (B.13)
Eq. (B.12) simplifies to
〈K∗(p′, ε) |b¯ σµνγ5 s|B(p)〉 = if+(p
2
−)
[
ε∗νp+µ − ε
∗
µp+ν
]
+ if−(p
2
−)
[
ε∗νp−µ − ε
∗
µp−ν
]
+ig(p2−)(ε
∗ · p)
[
p+νp−µ − p+µp−ν
]
. (B.14)
Next, recall that the full matrix element (B.3) contains a contraction with the photon
momentum q = p−. We have
qµ〈K∗(p′, ε) |b¯σµνs|B(p)〉 = f+(q
2)²µνρσp
µ
−ε
∗ρpσ+
= 2 f+(q
2)²νµρσε
∗µpρp′σ (B.15)
and
qµ〈K∗(p′, ε) |b¯σµνγ5s|B(p)〉 = if+(q
2)
[
ε∗ν(M
2
B −M
2
K∗)− (ε
∗ · q)(p+ p′)ν
]
+ if−(q
2)
[
ε∗νq
2 − (ε∗ · q)qν
]
+ ig(q2)(ε∗ · q)
[
q2 (p+ p′)ν − (M
2
B −M
2
K∗)qν
]
.
(B.16)
198 APPENDIX B. FORM FACTORS AND DECAY RATE FOR B → K∗γ
In order to disentangle (B.15) and (B.16), we now define the new form factors
T1(q
2) = −
1
2
f+(q
2),
T2(q
2) = −
1
2
[
f+(q
2) +
q2
M2B −M
2
K∗
f−(q
2)
]
,
T3(q
2) =
1
2
[
f−(q
2) + (M2B −M
2
K∗)g(q
2)
]
, (B.17)
so that
f+(q
2) = −2 T1(q
2),
f−(q
2) =
2
[
T1(q2)− T2(q2)
]
(M2B −M
2
K∗)
q2
,
g(q2) =
2 T3(q2)
M2B −M
2
K∗
−
2
[
T1(q2)− T2(q2)
]
q2
. (B.18)
This results in
qµ〈K∗(p′, ε) |b¯σµνs|B(p)〉 = −4 T1(q
2)²νµρσε
∗µpρp′σ (B.19)
and
qµ〈K∗(p′, ε) |b¯σµνγ5s|B(p)〉 = −2iT2(q
2)
[
ε∗ν(M
2
B −M
2
K∗)− (ε
∗ · q)(p+ p′)ν
]
−2iT3(q
2)(ε∗ · q)
[
qν −
q2
M2B −M
2
K∗
(p+ p′)ν
]
,
(B.20)
so that the tensor- and pseudotensor contributions are now separated. Note that at the
physical point q2 = 0, the second line of (B.17) implies
T1(0) = T2(0). (B.21)
B.2 Calculation of decay rate
We shall now express the decay rate
dΓ =
1
2MB
d3p′
(2pi)32EK∗
d3q
(2pi)32Eγ
2∑
r=1
3∑
s=1
|M|2(2pi)4δ(4)(p− p′ − q) (B.22)
in terms of the form factor at q2 = 0.
B.2. Calculation of decay rate 199
Equation (B.3) becomes
〈γ(q, r), K∗(p′, s) |JµνF
µν
(e.m.)|B(p)〉 = i T1(0)
{
− 4 ²νµρσε
∗ν(q, r)ε∗µ(p′, s)pρp′σ
− 2iε∗(q, r) · ε∗(p′, s) (M2B −M
2
K∗)
+ 2i
[
ε∗(p′, s) · q
] [
ε∗(q, r) · (p+ p′)
] }
.
(B.23)
Working in the B rest frame and using momentum conservation, we have
p = (MB,0),
q = (|q|,q),
p′ = (MB − |q|,−q),
|q| =
M2B −M
2
K∗
2MB
. (B.24)
We can also set ε0(q, r) = 0, which implies
ε∗(q, r) · (p+ p′) = ε∗(q, r) · q = 0. (B.25)
Therefore, (B.23) reduces to
〈γ(q, r), K∗(p′, s) |JµνF
µν
(e.m.)|B(p)〉 = i T1(0)
[
4MB ²jklε
∗j(q, r)ε∗k(p′, s)ql
+ 2iε∗(q, r) · ε∗(p′, s)(M2B −M
2
K∗)
]
.
(B.26)
The spin-summed absolute square of the amplitude becomes
∑
r,s
|〈γ(q, r), K∗(p′, s) |JµνF
µν
(e.m.)|B(p)〉|
2
= 16 |T1(0)|
2M2B
∑
r,s
²jkl²mno ε
∗j(q, r)εm(q, r)ε∗k(p′, s)εn(p′, s)qlqo
+ 4 |T1(0)|
2(M2B −M
2
K∗)
2
∑
r,s
ε∗j(q, r)εk(q, r)ε∗j(p′, s)εk(p′, s)
= 16 |T1(0)|
2M2B²jkl²mno
(
δjm −
qjqm
|q|2
)(
δkn +
qkqn
M2K∗
)
qlqo
+ 4 |T1(0)|
2(M2B −M
2
K∗)
2
(
δjk −
qjqk
|q|2
)(
δjk +
qjqk
M2K∗
)
= 16 |T1(0)|
2(M2B −M
2
K∗)
2. (B.27)
200 APPENDIX B. FORM FACTORS AND DECAY RATE FOR B → K∗γ
Finally, the total decay rate is
Γ =
1
2MB
∫
d3q
(2pi)32|q|
×
1
(2pi)32
√
M2K∗ + |q|
2
(2pi)4δ
(
MB −
√
M2K∗ + |q|
2 − |q|
)
∑
r,s
|M|2
=
1
2MB
∫
4pi|q|2d|q|
(2pi)22|q|
1
2
√
M2K∗ + |q|
2
δ
(
MB −
√
M2K∗ + |q|
2 − |q|
)
∑
r,s
|M|2
=
1
2MB
4pi|q|2
(2pi)22|q|
1
2
√
M2K∗ + |q|
2
1 +
|q|
√
M2K∗ + |q|
2
−1
∑
r,s
|M|2
∣
∣
∣
∣
∣
∣
|q|=
M2B−M
2
K∗
2MB
=
M2B −M
2
K∗
16piM3B
∑
r,s
|M|2
=
M2B −M
2
K∗
16piM3B
∣
∣
∣
∣VtbV
∗
ts
4GF√
2
C7(µ)
e
16pi2
mb
∣
∣
∣
∣
2
16 |T1(0)|
2(M2B −M
2
K∗)
2
=
α G2F
8pi4
|C7(µ)|
2|VtbV
∗
ts|
2m2bM
3
B
(
1−
M2K∗
M2B
)3
|T1(0)|
2. (B.28)
Appendix C
Data analysis methods
C.1 Correlated least-squares fitting
Suppose that we have real-valued “observables” yi where i = 1, 2, ... is a generic label; an
example will be given in Sec. C.2. In this notation, the total number of observables equals
the total number of degrees of freedom in the fit.
Suppose further that we have N statistically independent measurements (for example
data from N independent gauge field configurations) for all of these observables. We
denote a data point by yni , where n = 1 ... N . We define the average of the i-th observable
as
yi =
1
N
N∑
n=1
yni , (C.1)
and the data correlation matrix as
Cij =
1
N(N − 1)
N∑
n=1
(yni − yi)(y
n
j − yj). (C.2)
We would like to describe the data by smooth model functions fi(a) depending on P real-
values parameters a1, a2, ..., aP . The aim is to find those values of the parameters that
minimise the following χ2 function1:
χ2(a) =
∑
i,j
(C−1)ij
[
yi − fi(a)
][
yj − fj(a)
]
. (C.3)
1This definition, which contains C−1, requires the number of measurements N to be much larger
than the number of degrees of freedom, so that the data correlation matrix is well-determined and
invertible. If this is not the case, one can work with a pseudo-inverse of C instead, which is
computed using singular value decomposition.
202 APPENDIX C. DATA ANALYSIS METHODS
This is the standard least-squares fitting procedure for correlated data points [33, 34].
An efficient method for finding the minimum of (C.3) numerically is the Levenberg-
Marquardt algorithm [200, 201, 202]. A good fit should have χ2/dof ≈ 1, where “dof”
denotes the number of degrees of freedom.
Let a(0) = (a(0)1 , a
(0)
2 , ..., a
(0)
P ) be the point in parameter space (we assume that it
exists and is unique) that minimises χ2(a). The error estimate for a parameter ap in the
quadratic approximation is then given by
ap = a
(0)
p ±
√[
α−1(a(0))
]
p p , (C.4)
(no sum over the repeated index p here) where the matrix αp q is defined as follows:
αp q(a) =
1
2
∂2χ2(a)
∂ap∂aq
. (C.5)
Note that in the definition (C.5), second derivatives of the functions fi are usually ne-
glected. The inverse of αp q is the parameter covariance matrix.
When evaluating some smooth function g(a) of the parameters, the error estimate in
the quadratic approximation is
g(a) = g(a(0))±
√
√
√
√
∑
p, q
[
α−1(a(0))
]
p q
∂g(a(0))
∂ap
∂g(a(0))
∂aq
. (C.6)
In this work, I refer to results and error estimates computed with (C.4) or (C.6) as “ob-
tained directly from the fit”, as opposed to results and error estimates from bootstrap (see
Sec. C.4).
C.2 Simultaneous fitting of multiple correlation functions
The general principle introduced in Sec. C.1 also allows simultaneous fits of multiple
correlation functions in lattice QCD. As an example, we consider a simultaneous fit of the
following model functions:
• a three-point function with model CFJB(T, τ) and data in the range T = 15, 16 and
τ = 6 ... (T − 6),
• a two-point function with model CFF (τ) and data in the range τ = 10 ... 54,
• a two-point function with model CBB(τ) and data in the range τ = 12 ... 32.
C.3. Bayesian fitting 203
The model functions CFJB(T, τ), CFF (τ), and CBB(τ) depend on a set of parameters,
here various energies and amplitudes. In this example, CFJB(T, τ) could share the en-
ergy parameters describing the state F and its excitations with the two-point function
CFF (τ), and CFJB(T, τ) could share the energy parameters describing the state B and its
excitations with the two-point function CBB(τ).
To perform the simultaneous fit we would use the following model functions fi in the
notation of Sec. C.1:
f1 = CFJB(T = 15, τ = 6),
...
f4 = CFJB(T = 15, τ = 9),
f5 = CFJB(T = 16, τ = 6),
...
f9 = CFJB(T = 16, τ = 10),
f10 = CFF (τ = 10),
...
f54 = CFF (τ = 54),
f55 = CBB(τ = 12),
...
f75 = CFF (τ = 32). (C.7)
The simultaneous fit in this example has 75 degrees of freedom.
C.3 Bayesian fitting
Bayesian (constrained) fitting of lattice QCD data has been introduced in [122, 203]. The
use of the Bayesian method is best illustrated for the fitting of a Euclidean two-point
function to extract energies and amplitudes. It is known from the theory (see Sec. 2.2)
that a two-point function has the form of an infinite sum of exponentials. For a matrix of
two-point functions with a (finite) set of operators labelled by Γ, we have
C(Γsk,Γsc, τ) =
∞∑
n=0
An(Γsk)A
∗
n(Γsc) e
−En τ , (C.8)
where we assume that the energies are ordered so that En > En−1. We would like to
extract the amplitudes An(Γ) and energies En from a fit of (C.8) for a range of points
204 APPENDIX C. DATA ANALYSIS METHODS
τmin ≤ τ ≤ τmax. However, there is an infinite number of parameters but only a finite
number of data points. Therefore, when unconstrained fits are used, one has to truncate
the sum in (C.8) and keep only a finite number of exponentials: 0 ≤ n < nexp. Since the
contributions from higher excited states fall exponentially with τ , one has to choose τmin
large, so that the contributions from the states with n ≥ nexp are negligible. However,
when τmin is chosen too large, statistical errors in the fit results are too big; when τmin is
too small, the contributions from excited states introduce systematic errors in the results.
This behaviour can be seen for example in Table 8.10. The precise value one should choose
for τmin is not well-determined.
Ideally, we would like to include the data at small τ , since it is statistically very
precise. We would like to increase nexp until the systematic errors in the fits results for
the low-lying states are negligible. However, the fit becomes soon unstable when nexp is
increased. If the fit converges at all, one typically obtains unphysically large values for
the amplitudes at large n, such as 100 times the amplitude for the ground state, which
also destabilises the results for the low-lying states. From quark models it is expected
that the amplitudes of the excited states are smaller than or of a similar magnitude as the
ground-state amplitude.
The solution for this problem is to constrain the parameters to physically reasonable
ranges. This can be achieved through augmenting the χ2 function (C.3) by
χ2 → χ2 + χ2prior (C.9)
with a Gaussian prior
χ2prior =
∑
p
(ap − a˜p)2
σ˜2ap
. (C.10)
The term χ2prior favours parameter values in the ranges ap = a˜p ± σ˜ap ; the values for a˜p
and σ˜ap are chosen based on prior knowledge.
In the following discussion of the prior choices {a˜p, σ˜ap} we will distinguish between
parameters for low-lying and high-lying states. For example, the 3 × 3 matrix fit shown
in Fig. 7.2 contains operators optimised for the Υ(1S), Υ(2S) and Υ(3S) states. These
three states are referred to as low-lying for this fit, since their energies and amplitudes are
well-determined by the data. Higher excitations are referred to as high-lying.
I typically choose the prior widths for the parameters of the low-lying states to be
about 10 times larger than the resulting error estimates (C.4) from the fit. This ensures
that the influence of the priors on these parameters is negligible. I obtain initial guesses for
the central values from unconstrained fits including only a small number of exponentials
C.4. Bootstrap method 205
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
aE
nexp
Υ(1S)
Υ(2S)
Υ(3S)
χ2
/d
of
:
45
.8
2.
83
1.
83
0.
86
0.
80
0.
79
0.
79
0.
79
Figure C.1: Bayesian fit results for a 3 × 3 matrix correlator with the
{Υ(1S), Υ(2S), Υ(3S)} smearings as a function of the number of exponentials. The values
of χ2 divided by the number of degrees of freedom are also shown.
at large Euclidean time.
For the fit shown in Fig. 7.2, I used the logarithms of the energy splittings ln ≡
ln(En+1 − En) and the relative amplitudes Bn(Γ) ≡ An(Γ)/A0(Γ) as the fit parameters
for n > 0. I set l˜n = −1.4 and σ˜ln = 1. This corresponds to energy splittings of about
400 MeV, with a prior uncertainty of about 100 percent. For the relative amplitudes I set
B˜n = 0 and σ˜Bn = 5.
With the Bayesian method, one can increase nexp arbitrarily without destabilising the
fit. One has to increase nexp until the results and error estimates for the low-lying states
of interest become independent of nexp. This is demonstrated in Fig. C.1.
C.4 Bootstrap method
Equation (C.6) provides a method for taking into account correlations when computing
quantities that depend on more than one parameter of a single fit. However, in many
cases one also needs to compute quantities that depend on parameters from multiple non-
simultaneous fits of data from the same gauge field configurations.
An alternative technique for taking into account correlations that is applicable in both
206 APPENDIX C. DATA ANALYSIS METHODS
cases is the statistical bootstrap method [33, 34]. Recall from Sec. C.1 that we considered
N independent measurements. A single bootstrap sample is obtained by picking N out of
these N measurements randomly (with allowed repetition). One generates a large number
of such bootstrap samples, and performs the fits as in Sec. C.1 for each of these bootstrap
samples.
The function of the fit parameters that one is interested in is then computed for the
fit results from each bootstrap sample. In this way, a distribution for the function is
obtained. One computes the average and takes the 68% width of the distribution as the
error estimate.
Note that the bootstrap method must be modified for Bayesian fitting [122] such that
not only the data sets are resampled randomly, but also the central values a˜p of the priors
in (C.10) are drawn from Gaussian random distributions with widths σ˜ap for every fit.
C.5 Autocorrelations
The methods described above all require that the N measurements are statistically inde-
pendent. However, in lattice QCD the gauge field configurations are generated using a
Markov process with a non-zero autocorrelation time. In addition, when different measure-
ments are performed on a single gauge field configuration, these are also not completely
independent. In general, autocorrelations lead to underestimates of statistical errors. A
detailed discussion can be found in [34].
A simple method to reduce autocorrelations is the binning method. “Neighbouring”
measurements (in terms of Markov-chain time and/or position on a single gauge field
configuration) are grouped into bins of size B, and the average is computed for each bin.
The N/B bin-averages are then taken as the new measurements, which are statistically
more independent. The bin size B is increased until the error estimates become stable.
Acknowledgements
I would like to thank my supervisor Matt Wingate for his continued support and en-
couragement throughout this work. I thank all those who contributed through useful
discussions, including Christine Davies, Will Detmold, Alistair Hart, Ron Horgan, Lew
Khomskii, Andrew Lee, Peter Lepage, David Lin, Zhaofeng Liu, Eike Mu¨ller, Heechang
Na, Junko Shigemitsu and Laurent Storoni. I also thank the lecturers of the two summer
schools where I have learned so much: the school on “Lattice QCD and its applications”
in 2007 at the Institute for Nuclear Theory in Seattle, USA, and the school on “Modern
perspectives in lattice QCD: Quantum field theory and high performance computing” in
2009 at the E´cole de Physique in Les Houches, France.
I am deeply grateful to my parents for all their support. I would also like to thank my
friends, who made my stay in Cambridge very enjoyable.
I thank St John’s College Cambridge, the Cambridge European Trust and the Engineering
and Physical Sciences Research Council for financial support.
This work has made use of high performance computing resources provided by the
Fermilab Lattice Gauge Theory Computational Facility (http://www.usqcd.org/fnal),
the University of Cambridge High Performance Computing Service (http://www.hpc.
cam.ac.uk), the National Energy Research Scientific Computing Center (http://www.
nersc.gov/), the National Center for Supercomputing Applications (http://www.ncsa.
illinois.edu/) and Teragrid (http://www.teragrid.org/).
List of publications
1. “Form factors for rare B decays: strategy, methodology, and numerical study”
Z. Liu, S. Meinel, A. Hart, R. R. Horgan, E. H. Mu¨ller and M. Wingate
PoS LAT2009 (2009) 242, arXiv:0911.2370 [hep-lat]
2. “Radiative corrections to the m(oving)NRQCD action and heavy-light operators”
E. H. Mu¨ller, C. T. H. Davies, A. Hart, G. M. von Hippel, R. R. Horgan, I. Kendall,
A. Lee, S. Meinel, C. Monahan and M. Wingate
PoS LAT2009 (2009) 241, arXiv:0909.5126 [hep-lat]
3. “Bottom hadrons from lattice QCD with domain wall and NRQCD fermions”
S. Meinel, W. Detmold, C.-J. D. Lin and M. Wingate
PoS LAT2009 (2009) 105, arXiv:0909.3837 [hep-lat]
4. “Moving NRQCD for heavy-to-light form factors on the lattice”
R. R. Horgan, L. Khomskii, S. Meinel, M. Wingate, K. M. Foley, G. P. Lepage,
G. M. von Hippel, A. Hart, E. H. Mu¨ller, C. T. H. Davies, A. Dougall and K. Y. Wong
Phys. Rev. D80 (2009) 074505, arXiv:0906.0945 [hep-lat]
5. “Bottomonium spectrum from lattice QCD with 2+1 flavors of domain wall fermions”
S. Meinel
Phys. Rev. D79 (2009) 094501, arXiv:0903.3224 [hep-lat]
6. “Rare B decays with moving NRQCD and improved staggered quarks”
S. Meinel, E. H. Mu¨ller, L. Khomskii, A. Hart, R. R. Horgan and M. Wingate
PoS LAT2008 (2008) 280, arXiv:0810.0921 [hep-lat]
7. “Moving NRQCD and B → K∗γ”
S. Meinel, R. Horgan, L. Khomskii, L. C. Storoni and M. Wingate
PoS LAT2007 (2007) 377, arXiv:0710.3101 [hep-lat]
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