The Application of Automated
Perturbation Theory to Lattice QCD
Christopher John Monahan
Trinity College, Cambridge
Supervisor R.R. Horgan
This dissertation is submitted for the Degree of Doctor of Philosophy
at the
Department of Applied Mathematics and Theoretical Physics
Centre for Mathematical Sciences
Wilberforce Road, Cambridge CB3 0WA, UK
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 thesis has been, or is being, submitted for any
degree other than that of the Doctor of Philosophy at the University of
Cambridge.
Acknowledgements
First and foremost I would like to thank my supervisor Ron Horgan for his
enthusiasm, support and encouragement throughout my PhD. I would also
like to thank my collaborators, Rachel Dowdall, Georg von Hippel, Ali Naji,
Andrew Lee, Eike Mu¨ller and Junko Shigemitsu. In particular, I would like
to mention Alistair Hart for his early encouragement and endless patience
and Tom Hammant for our many discussions and our battles with FORTRAN. I
would also like to thank Matt Wingate, Ian Drummond and Christine Davies
for contributing their time and help.
My family and friends have provided continuous support, for which I am
deeply grateful. Without them, this dissertation would not have been possible
nor my time in Cambridge so enjoyable. Finally I would like to thank Kelcie
Ralph for her kindness and understanding and for always being there.
I thank Trinity College Cambridge and the Science and Technology Fa-
cilities Council for financial support.
This work has made use of high performance computing resources pro-
vided by the University of Cambridge High Performance Computing Ser-
vice (http://www.hpc.cam.ac.uk), CSC - IT Center for Science Ltd (http:
//www.csc.fi/english) and the PDC Center for High Performance Com-
puting (http://www.pdc.kth.se/).
ii
Abstract
Predictions of heavy quark parameters are an integral component of precision
tests of the Standard Model of particle physics. Experimental measurements
of electroweak processes involving heavy hadrons provide stringent tests of
Cabibbo-Kobayashi-Maskawa (CKM) matrix unitarity and serve as a probe
of new physics. Hadronic matrix elements parameterise the strong dynamics
of these interactions and these matrix elements must be calculated nonper-
turbatively.
Lattice quantum chromodynamics (QCD) provides the framework for
nonperturbative calculations of QCD processes. Current lattices are too coarse
to directly simulate b quarks. Therefore an effective theory, nonrelativistic
QCD (NRQCD), is used to discretise the heavy quarks. High precision simu-
lations are required so systematic uncertainties are removed by improving the
NRQCD action. Precise simulations also require improved sea quark actions,
such as the highly-improved staggered quark (HISQ) action. The renormal-
isation parameters of these actions cannot be feasibly determined by hand
and thus automated procedures have been developed. In this dissertation I
apply automated lattice pertubartion theory to a number of heavy quark
calculations.
I first review the fundamentals of lattice QCD and the construction of
lattice NRQCD. I then motivate and discuss lattice perturbation theory in
detail, focussing on the tools and techniques that I use in this dissertation.
I calculate the two-loop tadpole improvement factors for improved gluons
with improved light quarks. I then compute the renormalisation parameters
of NRQCD. I use a mix of analytic and numerical methods to extract the
one-loop radiative corrections to the higher order kinetic operators in the
NRQCD action. I then employ a fully automated procedure to calculate
the heavy quark energy shift at two-loops. I use this result to extract a
new prediction of the mass of the b quark from lattice NRQCD simulations
by the HPQCD collaboration. I also review the calculation of the radiative
corrections to the chromo-magnetic operator in the NRQCD action. This
computation is the first outcome of our implementation of background field
gauge for automated lattice perturbation theory.
Finally, I calculate the heavy-light currents for highly-improved NRQCD
heavy quarks with massless HISQ light quarks and discuss the application of
these results to nonperturbative studies by the HPQCD collaboration.
ii
Contents
1 Introduction and overview 1
2 Lattice QCD 5
2.1 Gluons on the lattice . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Light quarks on the lattice . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Circumventing the fermion doubling problem . . . . . . 9
2.2.2 Staggered fermions . . . . . . . . . . . . . . . . . . . . 11
2.3 Improving lattice QCD . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Symanzik improvement . . . . . . . . . . . . . . . . . . 13
2.3.2 Tadpole improvement . . . . . . . . . . . . . . . . . . . 14
2.3.3 Improved gluon actions . . . . . . . . . . . . . . . . . . 17
2.3.4 Improved staggered quark actions . . . . . . . . . . . . 19
2.4 Heavy quarks on the lattice . . . . . . . . . . . . . . . . . . . 22
2.4.1 Building the NRQCD action . . . . . . . . . . . . . . . 23
2.4.2 Perturbative improvement for NRQCD . . . . . . . . . 28
3 Lattice perturbation theory 30
3.1 Why use lattice perturbation theory? . . . . . . . . . . . . . . 30
3.2 Perturbation theory and renormalons . . . . . . . . . . . . . . 32
3.2.1 Mathematical preliminaries . . . . . . . . . . . . . . . 32
3.2.2 Renormalons and NRQCD . . . . . . . . . . . . . . . . 33
3.3 The tools of lattice perturbation theory . . . . . . . . . . . . . 36
3.3.1 Automated lattice perturbation theory . . . . . . . . . 37
3.3.2 Twisted boundary conditions . . . . . . . . . . . . . . 40
3.4 The quantum effective action . . . . . . . . . . . . . . . . . . 43
3.4.1 Symmetries of the effective action . . . . . . . . . . . . 45
3.4.2 Background field gauge . . . . . . . . . . . . . . . . . . 48
3.4.3 Background field gauge in HIPPY and HPSRC . . . . . . 52
4 Perturbative improvement 56
4.1 Tadpole improvement . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Landau tadpoles . . . . . . . . . . . . . . . . . . . . . . . . . 56
i
Contents
4.2.1 Landau tadpole results . . . . . . . . . . . . . . . . . . 58
4.2.2 Comparison to the literature . . . . . . . . . . . . . . . 60
4.2.3 Three-loop estimate . . . . . . . . . . . . . . . . . . . . 61
4.3 Plaquette tadpoles . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Plaquette tadpole results . . . . . . . . . . . . . . . . . 63
4.3.2 Three-loop estimate . . . . . . . . . . . . . . . . . . . . 65
4.4 NRQCD kinetic operator improvement . . . . . . . . . . . . . 67
4.4.1 Simple NRQCD . . . . . . . . . . . . . . . . . . . . . . 68
4.4.2 Calculating the kinetic correction coefficients . . . . . . 71
4.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . 73
4.4.4 Infrared divergences . . . . . . . . . . . . . . . . . . . 75
4.4.5 Comparison to the literature . . . . . . . . . . . . . . . 77
4.4.6 Observations . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 NRQCD chromo-magnetic operator improvement . . . . . . . 79
5 Perturbative renormalisation 83
5.1 NRQCD renormalisation parameters . . . . . . . . . . . . . . 84
5.1.1 Wavefunction renormalisation . . . . . . . . . . . . . . 84
5.1.2 Zero point energy shift and mass renormalisation . . . 86
5.1.3 One-loop results . . . . . . . . . . . . . . . . . . . . . . 87
5.1.4 Tadpole improvement . . . . . . . . . . . . . . . . . . . 88
5.1.5 Comparison to the literature . . . . . . . . . . . . . . . 90
5.2 Renormalisation parameters at two-loops . . . . . . . . . . . . 91
5.2.1 The fermionic contributions . . . . . . . . . . . . . . . 92
5.2.2 Energy shift results . . . . . . . . . . . . . . . . . . . . 92
5.2.3 Quenched high-β simulations . . . . . . . . . . . . . . 98
5.2.4 Tadpole improvement . . . . . . . . . . . . . . . . . . . 101
5.3 The b quark mass . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.1 Extracting the pole mass . . . . . . . . . . . . . . . . . 103
5.3.2 Matching the pole mass to the MS mass . . . . . . . . 104
5.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.4 The error budget . . . . . . . . . . . . . . . . . . . . . 107
5.3.5 Comparison to the literature . . . . . . . . . . . . . . . 109
ii
Contents
5.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 112
6 Heavy-light currents 115
6.1 Heavy-light decays and the CKM matrix . . . . . . . . . . . . 115
6.2 The continuum current . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Relativistic quark renormalisation parameters . . . . . . . . . 121
6.4 HISQ renormalisation parameters . . . . . . . . . . . . . . . . 123
6.5 The lattice current . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5.1 The lattice operator mixing matrix . . . . . . . . . . . 129
6.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.6.1 Relativistic wavefunction renormalisation . . . . . . . . 133
6.6.2 Comparison to the literature . . . . . . . . . . . . . . . 137
6.6.3 HISQ renormalisation parameters . . . . . . . . . . . . 138
6.6.4 Lattice operator matching . . . . . . . . . . . . . . . . 141
6.6.5 Comparison to the literature . . . . . . . . . . . . . . . 143
6.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 145
7 Conclusions and outlook 147
7.1 Tadpole improvement . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Operator improvement . . . . . . . . . . . . . . . . . . . . . . 148
7.3 The b quark mass . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4 Heavy-light currents . . . . . . . . . . . . . . . . . . . . . . . 150
A Conventions 152
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A.2 Lattice derivatives and field strength . . . . . . . . . . . . . . 153
A.3 Lattice fields in Fourier space . . . . . . . . . . . . . . . . . . 154
B Lattice Feynman rules 156
B.1 NRQCD: Davies and Thacker action . . . . . . . . . . . . . . 156
B.1.1 Two-point vertices . . . . . . . . . . . . . . . . . . . . 157
B.1.2 Three-point vertices . . . . . . . . . . . . . . . . . . . 158
B.2 NRQCD: “onlyH0” action . . . . . . . . . . . . . . . . . . . . 159
B.2.1 Two-point vertices . . . . . . . . . . . . . . . . . . . . 160
iii
Contents
B.2.2 Higher order vertices . . . . . . . . . . . . . . . . . . . 161
B.3 Quenched lattice QCD in background field gauge . . . . . . . 161
B.3.1 Two-point vertices . . . . . . . . . . . . . . . . . . . . 162
B.3.2 Three-point vertices . . . . . . . . . . . . . . . . . . . 164
B.3.3 Four-point vertices . . . . . . . . . . . . . . . . . . . . 166
B.4 Current insertions . . . . . . . . . . . . . . . . . . . . . . . . . 169
C Gluon selfenergy 172
C.1 Fermionic contributions . . . . . . . . . . . . . . . . . . . . . . 173
D Kinetic renormalisation parameters 178
D.1 Kinetic renormalisation parameters . . . . . . . . . . . . . . . 178
D.1.1 Extension to p6 . . . . . . . . . . . . . . . . . . . . . . 179
E HISQ renormalisation parameters 181
E.1 Mass renormalisation . . . . . . . . . . . . . . . . . . . . . . . 181
E.2 Wavefunction renormalisation . . . . . . . . . . . . . . . . . . 184
iv
List of Tables
3.1 Phase factors for twisted boundary conditions. . . . . . . . . . 43
4.1 Fermionic contributions to landau tadpole. . . . . . . . . . . . 60
4.2 Fermionic contributions to plaquette tadpole. . . . . . . . . . 64
4.3 Quenched contributions to plaquette tadpole. . . . . . . . . . 65
4.4 Renormalisation paramaters for simple NRQCD . . . . . . . . 74
4.5 Comparison of results from Mu¨ller for simple NRQCD . . . . 77
5.1 Heavy quark renormalisation parameters . . . . . . . . . . . . 88
5.2 Heavy quark renormalisation parameters . . . . . . . . . . . . 89
5.3 Tadpole corrected NRQCD renormalisation parameters . . . . 90
5.4 Heavy quark renormalisation parameters from Mu¨ller . . . . . 90
5.5 Heavy quark renormalisation parameters from Gulez et al. . . 91
5.6 Light quark contributions to heavy quark selfenergy . . . . . . 95
5.7 Light quark contributions to heavy quark selfenergy . . . . . . 96
5.8 Light quark contributions to heavy quark selfenergy . . . . . . 96
5.9 Light quark contributions to heavy quark selfenergy . . . . . . 97
5.10 Quenched one-loop energy shift . . . . . . . . . . . . . . . . . 100
5.11 One-loop coefficients in the NRQCD action . . . . . . . . . . . 100
5.12 Quenched E0 results . . . . . . . . . . . . . . . . . . . . . . . 101
5.13 Perturbative data required to extract MS mass . . . . . . . . 106
5.14 Simulation data required to extract MS mass . . . . . . . . . 106
5.15 Error budget for the b quark mass. . . . . . . . . . . . . . . . 110
6.1 Wavefunction renormalisation in continuum QCD . . . . . . . 134
6.2 Wavefunction renormalisation subtraction function . . . . . . 136
6.3 Finite rainbow diagram contribution to Zq . . . . . . . . . . . 136
6.4 Finite contributions to wavefunction renormalisation . . . . . 137
6.5 ASQTad wavefunction renormalisation from Gulez et al. . . . 138
6.6 HISQ wavefunction renormalisation from Mu¨ller . . . . . . . . 138
6.7 Massive HISQ renormalisation parameters . . . . . . . . . . . 139
6.8 Massive HISQ renormalisation parameters . . . . . . . . . . . 140
v
List of Tables
6.9 Vertex corrections for heavy-light matching . . . . . . . . . . . 143
6.10 Vertex corrections for heavy-light matching . . . . . . . . . . . 143
6.11 Vertex corrections for heavy-light matching . . . . . . . . . . . 144
6.12 Mixing coefficient for heavy-light matching . . . . . . . . . . . 144
6.13 Vertex corrections from Gulez et al. . . . . . . . . . . . . . . . 145
C.1 Unimproved light quark contributions to gluon selfenergy. . . . 176
C.2 Light quark contributions to gluon selfenergy. . . . . . . . . . 177
vi
List of Figures
2.1 The plaquette. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Six-link lattice operators. . . . . . . . . . . . . . . . . . . . . . 19
2.3 The “matching square” . . . . . . . . . . . . . . . . . . . . . . 29
4.1 One-loop contribution to tadpole correction factor . . . . . . . 57
4.2 Two-loop contribution to tadpole correction factor . . . . . . . 58
4.3 Light quark mass-dependence of Landau tadpoles . . . . . . . 59
4.4 Light quark mass-dependence of plaquette tadpoles . . . . . . 63
4.5 Quenched plaquette tadpole fit . . . . . . . . . . . . . . . . . 66
4.6 One-loop NRQCD selfenergy contributions . . . . . . . . . . . 72
5.1 Logarithmic fit to Zsubψ . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Two-loop fermionic contributions to E0 and ZM . . . . . . . . 92
5.3 Light quark contributions to heavy quark selfenergy . . . . . . 98
5.4 Two-loop selfenergy contributions to quenched E0 . . . . . . . 99
5.5 Recent b quark mass determinations . . . . . . . . . . . . . . 113
6.1 CKM unitarity triangle . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Continuum QCD contributions to heavy-light current . . . . . 121
6.3 Additional lattice NRQCD contributions to heavy-light current 127
6.4 Infrared behaviour of Zq . . . . . . . . . . . . . . . . . . . . . 135
6.5 Plot of infrared subtracted wavefunction renormalisation . . . 135
6.6 Infinite volume extrapolation of Zψ and ZM for HISQ quarks . 141
6.7 Infrared behaviour of ζ00 subtraction function . . . . . . . . . 142
C.1 Continuum QCD gluon selfenergy diagrams . . . . . . . . . . 172
C.2 Additional lattice QCD gluon selfenergy diagrams . . . . . . . 173
C.3 Plot of unimproved light quark contributions to gluon selfenergy175
C.4 Plot of ASQTad light quark contributions to gluon selfenergy . 175
vii
Chapter 1
Introduction and overview
Lattice quantum chromodyanmics (QCD) is the reigning approach for non-
perturbative computations in QCD, the gauge theory of the strong inter-
action. Two other fundamental forces, the weak and electromagnetic inter-
actions, are unified with QCD into the hugely successful Standard Model
of particle physics. This theory describes the subatomic constituents of the
Universe within the mathematical framework of a relativistic quantum field
theory with SU(3)× SU(2)× U(1) local gauge symmetry.
So far the Standard Model has largely withstood the challenge of exper-
imental verification. There are, however, a number of theoretical reasons to
expect the Standard Model to be an incomplete description of the fundamen-
tal degrees of freedom of the observable universe and it will almost certainly
fail at energy scales near the Planck scale. One of the great projects of 21st
Century theoretical physics is the understanding of possible Beyond the Stan-
dard Model (BSM) physics. An important aspect of this endeavour is a more
complete understanding of QCD.
QCD exhibits two features that complicate the calculation of hadronic
processes. The first is asymptotic freedom, the decreasing strength of QCD
interactions as the energy scale increases. At high energies the coupling con-
stant is much smaller than unity and is a good expansion parameter for
perturbative calculations of QCD processes. At the low energies of typical
hadrons, the coupling constant is too large for perturbation theory to con-
verge and nonperturbative techniques must be used. Secondly, quark con-
finement ensures that quarks do not occur singly, but only in colour singlet
bound states. Thus weak interaction processes cannot be measured without
first disentangling the QCD dynamics of the bound states in which they
occur. This too requires nonperturbative calculations.
Direct searches for new BSM particles have so far failed. The Large
Hadron Collider (LHC) is now collecting data and in a few years will run
1
Chapter 1. Introduction and overview
with a centre of mass energy of approximately 14 TeV. At these energies the
LHC may be able to observe new particles in the TeV regime, but, despite
hints of new particles at the Tevatron, no discovery has been announced. An
alternative approach to uncovering BSM physics studies the effect of BSM
particles on known Standard Model processes. These indirect searches require
both precise experimental measurements and precise theoretical predictions
from the Standard Model.
One particularly promising avenue for discovering BSM physics is the
weak interaction of heavy quarks. The Cabibbo-Kobayashi-Maskawa (CKM)
matrix parameterises the weak eigenstates of quarks in terms of the mixing
of the mass eigenstates. In the Standard Model the CKM matrix is unitary.
Deviations from unitarity may indicate new physics. Current tests of the
unitarity of the CKM matrix suggest that the measurements from a wide
range of observables are mutually consistent. However, the constraints on
many of these processes are poorly known and the dominant uncertainties
are theoretical
One of these dominant theoretical uncertainties comes from the CKM ele-
ment |Vub|, extracted from B meson decays. To compute this matrix element,
nonperturbative hadronic form factors characterising the strong dynamics of
the interaction must be calculated. The HPQCD collaboration has used a
nonrelativistic formulation for heavy quarks on the lattice, NRQCD, to pre-
dict a value of |Vub|= (3.55 ± 0.25 ± 0.50) × 10−3, where the first error is
experimental and the second is theoretical [1]. This result is in good agree-
ment with a number of other lattice calculations of comparable or larger error
size, collected in [2].
Theoretical predictions of nonperturbative quantities have improved vastly
over the last twenty years as a result of increased computer power and im-
proved simulation algorithms. Calculations of ‘gold-plated’ hadronic quan-
tities involving only one final state hadronic particle are now of a precision
that may be compared to experiment [3]. The development and refinement
of automated lattice perturbation theory has contributed considerably to the
improved precision of lattice predictions. Next to leading order perturbative
calculations have now been carried out for perturbative quantities in both
2
Chapter 1. Introduction and overview
the light [4] and heavy quark sectors [5].
Whilst progress over the last three decades has been impressive, further
work needs to be done. A recent update by the HPQCD and UKQCD collab-
orations [6] found good agreement between the lattice prediction and exper-
iment results for the decay constant fD, but a 3σ difference for fDs . Recent
updates by the Fermilab/MILC collaboration [7], on the other hand, obtained
good agreement with experiment for both fD and fDs [8].
Moreover, determinations of both of the CKM matrix elements |Vub| and
|Vcb| from inclusive and exclusive decays are currently in tension. Recent
studies of the B → Dτντ and B → D∗τντ decays, on which some deter-
minations of |Vcb| are based, have suggested an almost 2σ excess above the
expected Standard Model behaviour. These results, if confirmed, would hint
at the presence of new BSM physics such as a charged Higgs boson [9].
The significance of these discrepancies is currently not clear, highlighting
the need for continued progress in heavy quark physics on the lattice (for a
review of recent B andD meson physics on the lattice, see [10]). In particular,
continued improvement in the precision of lattice predictions is required to
tighten the unitarity constraints on the CKM matrix.
Lattice perturbation theory will play an important role in this improve-
ment programme. Lattice perturbation theory can be effectively employed to
calculate lattice renormalisation parameters, match lattice and continuum
calculations and compute the radiative corrections to lattice actions, all of
which are required to further improve lattice simulation results. In this disser-
tation I present a number of related lattice perturbation theory calculations
under the aegis of this improvement programme. I organise this dissertation
as follows.
In Chapter 2 I outline the lattice formulation of QCD, first for the gauge
sector and then for the fermionic sector. I briefly explore some of the problems
associated with marrying light quarks and the lattice, before describing heavy
quark effective theories for the lattice, and in particular lattice NRQCD.
I describe lattice perturbation theory in more detail in Chapter 3. I mo-
tivate the use of lattice perturbation theory as part of the programme to
extract precise results from lattice simulations. I justify the use of lattice
3
Chapter 1. Introduction and overview
perturbation theory with renormalisation group arguments. I introduce a
number of tools used in this dissertation: the HIPPY and HPSRC implementa-
tion of automated lattice perturbation theory; twisted boundary conditions;
and background field gauge.
In Chapter 4 I calculate the Landau and plaquette tadpole improvement
factors. I determine these factors at one-loop and reproduce results in the
literature. I then extend my calculation to the fermionic contributions at
two-loops. I combine my results with data from high-β simulations carried
out by Lee [11] and with results from [12] to present the first determination
of the full two-loop Landau and plaquette tadpole improvement factors for
Symanzik-improved gluons with ASQTad light quarks. I then calculate the
one-loop corrections to the O(a4p4) kinetic operators in the NRQCD action.
I perform this calculation using both analytic and numerical integration and
confirm results obtained by Mu¨ller using a fully automated procedure [13]. I
also review the calculation of the one-loop correction to the chromo-magnetic
operator in the NRQCD action. This calculation, undertaken by Hammant
[14], is the first outcome of our implementation of background field gauge in
the automated perturbation routines.
In Chapter 5 I compute the heavy quark renormalisation parameters in
NRQCD for a number of different lattice NRQCD discretisations. I calcu-
late the fermionic contributions to the two-loop heavy quark energy shift. I
combine my results with data extracted from quenched high-β simulations,
undertaken by Lee [11], to calculate the full two-loop energy shift. I then use
this result to obtain a new prediction of the mass of the b quark from lattice
NRQCD simulations by the HPQCD collaboration.
In Chapter 6 I calculate the matching coefficients for the temporal coeffi-
cient of the axial-vector and vector heavy-light currents in continuum QCD
and NRQCD with massless HISQ light quarks. I confirm previous results us-
ing NRQCD with massless ASQTad light quarks as a check of my calculation.
Finally, I summarise my results in Chapter 7.
4
Chapter 2
Lattice QCD
Lattice QCD is a natural approximation to QCD: we replace continuum
spacetime with a hypercubic lattice, with quarks on the lattice sites and
gluons on the link variables joining those sites. Spacetime points become
lattice nodes, x → xi = nia, separated by lattice spacing a. For a lattice of
side length L, the physical volume of the lattice is (aL)4. The lattice spacing
directly introduces an ultraviolet cutoff of pi/a in momentum space and serves
as a nonperturbative, gauge invariant regulator of the theory.
We relate the Euclidean hypercubic lattice to Minkowski spacetime by
a Wick rotation of the temporal coordinate from Minkowski time, t, to Eu-
clidean time, τ :
t→ −iτ. (2.1)
This coordinate change introduces an extra factor of i into the action,
SM =
∫
d4xM L → SE = i
∫
d4xE L, (2.2)
which in turn propagates into the path integral exponent,
ΓM =
∫
Dφ exp(iSM) → ΓE =
∫
Dφ exp(−SE). (2.3)
We can now interpret the path intergal exponent as the Boltzmann weight
factor for a field configuration with Euclidean action SE. Moreover, lattice
fields are defined at only a discrete set of lattice nodes and links. Therefore we
can replace the functional integral over field configurations by ordinary mul-
tidimensional integrals over the set of sites and links. Thus the expectation
value of an operator O is
〈O〉 =
1
Z
∫
Dφ OeiS[φ] →
1
Z
∫ ∏
xi
dφOe−S[φ]. (2.4)
5
Chapter 2. Lattice QCD
Here Z is the path integral normalisation factor,
Z =
∫
Dφ eiS[φ] →
∫ ∏
xi
dφ e−S[φ]. (2.5)
With our intepretation of the path integral exponent as a Boltzmann
weight factor, the path integral normalisation becomes the partition func-
tion for the theory. We can assign each field configuration a probability,
exp(−SE), and consequently quantum field theory is accessible to the pow-
erful techniques of statistical field theory.
Most importantly, we can exploit the discrete nature of the lattice formu-
lation to perform lattice simulations on a computer. In particular, numerical
computations can take advantage of the statistical interpretation of the path
integral to use Monte Carlo sampling approaches and other tools imported
from statistical mechanics.
2.1 Gluons on the lattice
Our first attempt at formulating gauge fields on the lattice might proceed
by simply placing the fields themselves at each lattice site. Unfortunately
this method fails to preserve gauge invariance. Without gauge invariance,
radiative corrections renormalise each of the quark-gluon, three-gluon and
four-gluon couplings differently. Each vertex renormalisation factor would
then have to be independently tuned using different physical quantities. This
tuning would incur a computation cost much greater than that required to
maintain gauge invariance and reduces the predictive power of the lattice
theory.
Instead of transcribing the gauge fields directly onto the lattice, we build
the gauge action from link variables that run between lattice sites. I denote
a link variable between lattice sites x and x+aµˆ as Uµ(x). The link variables
are elements of the SU(N) gauge group and given in terms of the gauge field,
Aµ, by
Uµ(x) = P
[
exp
(
iag
∫ x+aµˆ
x
dz τ ·Aµ(z)
)]
. (2.6)
6
Chapter 2. Lattice QCD
Figure 2.1: The plaquette operator, Pµν(x).
Here P is the path-ordering operator, g the bare coupling constant, and τ a
generator of the gauge group.
Under a gauge transformation, Λ(x), the link variables transform as
UΛµ (x) = Λ(x)Uµ(x)Λ
−1(x+ aµˆ). (2.7)
Gauge invariant quantities can thus be constructed from the trace of closed
loops of link variables – Wilson loops – or from a line of link variables ter-
minating in fermion fields at each endpoint lattice site.
Wilson’s original gluon action [15] is built from closed loops of link vari-
ables, called plaquettes,
Pµν(x) = Uµ(x)Uν(x+ aµˆ)U
†
µ(x+ aνˆ)U
†
ν(x). (2.8)
The plaquette Pµν is a one-by-one loop in the (µ, ν)-th plane, as shown in
Figure 2.1. We can conveniently express the Wilson gauge action, SG, in
terms of the inverse coupling β = 2N/g2 as
SG = β
∑
P
(
1−
1
2N
Tr
[
P + P †
]
)
= β
∑
P
(
1−
1
2N
ReTrP
)
. (2.9)
This action reproduces the correct Yang-Mills action in the continuum
limit. We can study the continuum limit by writing the link variables in
7
Chapter 2. Lattice QCD
terms of exponentials of the gauge potential,
Uµ(x) ' exp
[
iagτ ·Aµ
(
x+
a
2
µˆ
)]
, (2.10)
and then using the Baker-Campbell-Hausdorff formula
eAeB = eA+B+1/2[A,B]+1/12[A,[A,B]]+···. (2.11)
The derivation of the continuum action is straightforward, but slightly alge-
braically involved. See, for example, the detailed exposition in [16].
2.2 Light quarks on the lattice
We have now seen that we can transcribe gluon fields onto the lattice rel-
atively easily, by replacing gauge fields with link variables. Fermionic fields
do not translate to the lattice as simply. To formulate a relativistic lattice
quark action our starting point is the continuum Dirac action:
SD =
∫
d4x ψ¯(x) (iD/−m)ψ(x). (2.12)
We can rewrite this action as a lattice quark action that reproduces the
correct continuum Dirac action by directly replacing the covariant derivative
with the symmetrised difference operator
∆±µψ(x) =
1
2a
(
Uµ(x)ψ(x+ µˆ)− U
†µ(x− µˆ)ψ(x− µˆ)
)
, (2.13)
and introducing Euclidean gamma matrices that obey
{γ̂µ, γ̂ν} = 2δµν . (2.14)
The result is
SF =
∑
x
ψ¯(x)
(
∑
µ
γ̂µ∆
+
µ + am
)
ψ(x). (2.15)
Easy! Unfortunately this is too easy. To understand why this naive action
8
Chapter 2. Lattice QCD
fails, we can consider the form of the quark propagator, which is
G(p)−1 =
(
i
∑
µ
γ̂µ sin(apµ) + am
)
. (2.16)
Whilst G(p)−1 reproduces the correct (Euclidean) propagator in the con-
tinuum limit within one sixteenth (in four dimensions) of the Brillouin zone,
we immediately see that the presence of the sin(apµ) term introduces ad-
ditional unphysical degrees of freedom. The dominant contributions to the
propagator occur at the zeroes of G(p)−1. The expected continuum-like zero
occurs at pµ = 0, but there are also zeroes of sin(apµ) at pµ = ±pi/a. These
extra contributions to the quark propagator represent 15 spurious fermion
“doublers” that, even if they are projected out in external states, can still
occur in virtual states by absorption of gluons with momenta O(pi/a).
The presence of troublesome doublers is not a consequence of the naive
choice of discretisation procedure. Doublers arise from the invariance of the
fermion action under
ψ(x) →
∏
ν
(iγ̂5γ̂ν)
nνeix ·npi/aψ(x) (2.17)
where nν = {0, 1} (see, for example, [17]). In fact, solving the doubler prob-
lem is only possible by breaking chiral symmetry [18]. Under some general
assumptions about the action, such as locality and translational invariance,
the Nielsen and Ninomiya no-go theorem states that it is not possible to
formulate a chirally invariant and doubler-free fermion action on the lattice.
This celebrated theorem is one of the principal difficulties of lattice QCD.
2.2.1 Circumventing the fermion doubling problem
A number of distinct approaches exist to combat the fermion doubling prob-
lem. Wilson’s original solution was to add an irrelevant term to the action
to lift the mass of the doublers to O(pi/a). In the continuum limit the mas-
sive doublers decouple from the theory. Wilson’s extra term introduces O(a)
errors, which can be removed by including yet another operator of the same
9
Chapter 2. Lattice QCD
dimension, the “clover” term [19]. Whilst Wilson’s solution overcomes the
doubling problem, it does so at the expense of explicitly breaking chiral sym-
metry. Preserving chiral symmetry forbids additive mass renormalisations,
prevents operators with different chirality from mixing (which simplifies the
calculations of matrix elements for weak interactions) and allows the Ward-
Takahashi identities to be fully exploited in computations [17].
A distinct approach evades the no-go theorem by modifying the definition
of chiral rotation. The usual continuum prescription for chiral transforma-
tions is δψ = i²γ5ψ, δψ¯ = i²ψ¯γ5. The chiral symmetry of the massless Dirac
operator, D, under this rotation can be expressed by {D, γ5} = 0. If we
redefine chiral rotation on the lattice for quark and antiquark fields as
δψ = i²γ5
(
1−
a
2
D
)
ψ, δψ¯ = i²ψ¯
(
1−
a
2
D
)
γ5, (2.18)
then we can retain a remnant of the original symmetry. The operator D now
satisfies the Ginsparg Wilson relation [19, 20]
{D, γ5} = aDγ5D. (2.19)
One implementation of this relation is Neuberger’s “overlap” operator
[19], which necessarily preserves the modified chiral symmetry on the lat-
tice, but at the cost of being nonlocal and consequently computationally
very expensive. A second realisation extends the hypercubic lattice to five
dimensions. Gauge fields are constant in the fifth dimension, whilst massless
fermions of opposite chirality are bound to the four-dimensional “domain
walls”. Chiral symmetry breaking terms die off exponentially in the fifth
dimension and the full theory reduces to an overlap-like four-dimensional ef-
fective theory as the fifth dimension grows large. This domain wall realisation
is also computationally expensive and only preserves exact chiral symmetry
as the size of the fifth dimension tends to infinity.
10
Chapter 2. Lattice QCD
2.2.2 Staggered fermions
An alternative approach to reducing the number of doublers that does not
attempt to implement an action satisfying the Ginsparg-Wilson relation is
the “staggered fermion” formulation. In this approach, the fermion fields are
locally transformed under
ψ(x) → Γ(x)χ(x), ψ¯(x) → χ¯(x)Γ(x)†, (2.20)
to give a spin-diagonalised action in terms of the four-component field χ(x).
We can write the lattice action as
S =
∑
x
χ¯(x)Γ(x)†γµΓ(x+ µˆ) [Uµ(x)χ(x+ µˆ)− Uµ(x− µˆ)χ(x− µˆ)]
+m
∑
x
χ¯(x)χ(x)
=
∑
x
χ¯(x)αµ(x) [Uµ(x)χ(x+ µˆ)− Uµ(x− µˆ)χ(x− µˆ)] +m
∑
x
χ¯(x)χ(x),
where, for the choice of transformation Γ(x) = γx00 γ
x1
1 γ
x2
2 γ
x3
3 , the vector αµ(x)
is
αµ(x) =
{
1 if µ = 0,
(−1)x0+x1+···+xµ−1 otherwise.
(2.21)
We now interpret the sixteen degrees of freedom as four copies of four-
component Dirac spinors. By simulating only one component of the field we
reduce the degeneracy of the degrees of freedom to only four “tastes”1. These
tastes are not independent, but can transform into each other by exchanging
high momentum gluons with momenta O(pi/a) [20]. The staggered formu-
lation has a remnant of the original chiral symmetry and the spin degrees
of freedom have been been removed, leaving only colour degrees of freedom
at each site. Generating gauge configurations with staggered sea quarks is
therefore very cost efficient and this allows simulations to be performed near
the physical light quark masses.
There are a number of formal objections to the use of staggered quarks
1An obvious kin to flavour.
11
Chapter 2. Lattice QCD
as an ab initio method for calculating nonperturbatice QCD quantities. In
staggered quark simulations, the remaining four tastes are removed by divid-
ing each quark loop by four, equivalent to taking the fourth root of the quark
determinant. Without taste-changing interactions this procedure is formally
valid, as the determinant is block-diagonal in taste-space. However, taste-
changing processes do occur and it is not yet clear if the “rooting trick”
leads to a lattice action with the correct continuum limit. At finite lattice
spacing rooted staggered fermions are nonlocal and not unitary.
The issues are reviewed in some detail in [21] and more recently, albeit
more briefly, in [22]. Current numerical evidence, combined with some theo-
retical justification, seems to suggest that rooted staggered fermions have
the correct continuum limit at zero temperature and chemical potential
[21, 22, 23, 24]. This conclusion is far from undisputed and is unlikely to
be fully resolved without progress towards a formal proof that the rooted
staggered fermions have the correct physical continuum limit.
The dispute over the formal validity of rooted staggered fermions notwith-
standing, staggered fermions have been widely used in precision lattice simu-
lations (see for example, [3], [25] and [26]) and are amenable to the techniques
of lattice perturbation theory. I use the staggered quark discretisation for rel-
ativistic quarks thoughout this dissertation, in Chapters 4, 5 and 6.
2.3 Improving lattice QCD
Using a lattice with finite lattice spacing to perform QCD calculations in-
troduces discretisation errors that occur because the lattice regularisation
scheme removes all momenta larger than pi/a.
In principle, we could simply reduce the lattice spacing until we’ve de-
creased the errors to our desired level of precision. Unfortunately this method
is currently prohibitively expensive. We must fix the physical lattice size to
be larger than the correlation length of the lightest particles in the simulation
or the improvement of the discretisation errors is swamped by unmanageable
uncertainties from finite size effects. In general, finite size effects are esti-
mated to be < 1% if the lattice size, L, is large enough for the pion mass
12
Chapter 2. Lattice QCD
to obey mpiL ≥ 4. For current lattice pion masses of ∼ 300 MeV, this cor-
responds to lattice sizes of L ∼ 2.5 fm. Thus we need ever-increasing lattice
sizes as the continuum limit is approached. Since simulation cost scales as
approximately 1/a6 [27, 28], this solution remains computationally unfeasible
in practice.
A more sophisticated approach is to “improve” lattice actions by including
new terms in the action that directly remove discretisation errors. I outline
this process, generally referred to as Symanzik improvement, below.
2.3.1 Symanzik improvement
The Symanzik improvement procedure systematically removes both discreti-
sation and cutoff errors at finite lattice spacing by adding irrelevant operators
to the lattice action. These operators are arranged according to their mass
dimension, using power counting rules. For example, we can remove the dis-
cretisation errors at O(a2) in Wilson’s gauge action by introducing operators
built from six-link loops. I explain the construction of improved gauge actions
in more detail in Section 2.3.3.
We replicate the high-momentum contributions removed by the lattice
cutoff by adding local operators that give the same contribution to physical
amplitudes as modes with momenta larger than pi/a in continuum QCD [27].
These operators are nonrenormalisable but do not diverge at finite lattice
spacing, because the lattice spacing serves as an ultraviolet regulator. These
new terms in the lattice action are local in the sense that they are polynomials
in the lattice fields and in the derivatives of the fields. We can always Taylor
expand any nonlocal contributions in the external momentum, which is much
less than the lattice cutoff, leaving only local interactions. Our use of local
operators makes sense physically, because the additional terms correct for
high momentummodes, These additional operators therefore represent highly
virtual interactions, which, owing to the uncertainty principle, must be local
in extent [27]. I review the construction of such a Symanzik-improved effective
theory, NRQCD, in Section 2.4.1.
We fix the coupling constant for each additional operator by perturba-
13
Chapter 2. Lattice QCD
tively matching the lattice theory with its continuum counterpart order-by-
order in the strong coupling constant. Radiative corrections induce errors at
O(αsa2) and these higher order contributions modify the tree-level improve-
ment coefficients. We can calculate these corrections using nonperturbative
tuning or with lattice perturbation theory. I discuss lattice perturbation the-
ory in greater detail in Chapter 3 and present a perturbative improvement
calculation – for the kinetic operators in the NRQCD action – in Section 4.4.
2.3.2 Tadpole improvement
In an important paper from 1993, Lepage and Mackenzie [29] showed that
a significant proportion of the radiative corrections to operator coefficients
can be removed by dividing each link variable by its mean field value. Until
the early 1990s lattice perturbation theory had been plagued by unexpect-
edly slow convergence. Lepage and Mackenzie demonstrated that this was a
byproduct of a poor choice of expansion parameter and that by dividing the
link variables by their mean field value, the convergence of lattice perturba-
tion theory was greatly improved.
We formulate gluon actions in terms of link variables, which give rise to
vertices with an arbitrary number of gluons. These vertices are suppressed by
powers of the lattice spacing, but the gluon fields can be contracted together
to give tadpole loops with ultraviolet divergences that precisely cancel the
powers of the lattice spacing. Tadpole diagrams are therefore only suppressed
by powers of the coupling constant, not the lattice spacing, and lead to
large scale-independent renormalisations of lattice operators relative to their
continuum counterparts.
Divergent tadpole contributions arise from the high momentum modes
and are largely process independent, so we can remove these contributions
by integrating out the high energy modes. We split the link variables into
low momentum and high momentum modes
Uµ(x) = e
iagAµ(x) ≡ eiag[A
IR
µ (x)+A
UV
µ (x)] → u0e
iagAIRµ (x) ≡ u0U˜µ(x). (2.22)
In lattice simulations, each bare link variable in the action is divided
14
Chapter 2. Lattice QCD
by u0. There are a number of choices for the calculation of u0, the only
condition being that the process chosen is dominated by high momentum
modes. Two common options are the expectation value of the link variable
in Landau gauge or the fourth root of the plaquette. We fix the value of u0
by requiring that the value measured in simulations agrees with the value
of the parameter in the action. Unfortunately the numerical tuning that
this self-consistency requires is computationally expensive. Alternatively we
can use lattice perturbation theory to calculate the tadpole factor to a given
order in the coupling constant. I calculate the two-loop Landau and plaquette
tadpoles in Section 4.1.
Without tadpole improvement the operator renormalisations lead to large
logarithmic shifts in the coupling constant. Consequently the bare lattice cou-
pling constant is a crude choice of expansion parameter. Perturbative series
convergely poorly because the bare coupling constant is anomalously small
at the scales ' pi/a that dominate the renormalisation constants (which ac-
count for momenta excluded by the lattice cutoff). The bare lattice coupling,
αL(pi/a), deviates considerably from, for example, αMS(pi/a).
Instead of using the bare lattice coupling, we define an improved cou-
pling constant. We first fix the scheme, for example the “V -scheme”, defined
through the heavy quark potential, and then choose a characteristic scale
at which to evaluate the coupling [30]. It is natural and convenient to tie
the scale to the coupling of a gluon with momentum equal to the loop mo-
mentum flowing through the graphs. Brodsky, Lepage and Mackenzie (BLM)
developed a systematic procedure for determining the appropriate scale in
[30], which was then extended to higher orders in [31]. I briefly discuss the
BLM procedure as my work in Chapter 5 employs both the V -scheme and
scale setting using the BLM procedure.
We first must choose a scheme in which to define our coupling constant.
We typically choose a coupling constant defined via the heavy quark potential
[29, 30]. Defining the coupling constant in terms of a physical process is
equivalent to imposing a renormalisation scheme. In the MS scheme, the
15
Chapter 2. Lattice QCD
heavy quark potential is [30]
V (q2) = −
4picfαMS(q)
q2
[
1 +
αMS
pi
(
5
12
β0 − 2
)
+ · · ·
]
. (2.23)
This leads naturally to a choice of scheme for defining the coupling constant:
the effective potential,
V (q2) ≡ −
4picfαV (q)
q2
, (2.24)
defines the V-scheme coupling constant, αV , which is related to the MS
scheme by αV (q) = αMS(e
−5/6q)(1−2αMS/pi+ · · · ) [30]. The relation between
αMS and αV has been calculated to two-loops in [32, 33] and [34].
Our next task is to set the scale. At one-loop the natural choice for the
scale q? is a value that replicates the effect of a fully dressed gluon within
the diagram. For a one-loop contribution, with gluon momentum q,
I = αV (q
?)
∫
dq f(q). (2.25)
In the V-scheme, the apparently natural choice of q? would be
αV (q∗)
∫
dq f(q) =
∫
dq αV (q)f(q) (2.26)
except that αV (q) is singular [29, 31].
We can see why the right-hand side of Equation (2.26) diverges by con-
sidering the one-loop solution of the renormalisation group equation for the
running of the coupling, which is
αV (q) =
αV (µ)
1 + αV (µ)β0 log(q2/µ2)
. (2.27)
This solution has a pole at
q = ΛV ≡ µ exp
(
−
1
2β0αV (µ)
)
. (2.28)
This singularity arises because we are implicitly attempting to sum the
16
Chapter 2. Lattice QCD
perturbative logarithms in αV (q) to all-orders. Perturbative QCD is an asymp-
totic series, as I discuss in Section 3.2.2, and consequently it is incorrect to
undertake an all-orders summation of the series. The divergence in the cou-
pling constant can be removed by keeping only a finite number of terms in
the series.
We choose q? by setting µ = q? in Equation (2.27) and expanding the
right-hand side to give
αV (q) ∼ αV (q
?)− α2V (q
?)β0 log
(
q2
(q?)2
)
+ · · · , (2.29)
which has solution
log((q?)2) =
∫
d qf(q) log(q2)
∫
d q
≡
〈f(q) log(q2)〉
〈f(q)〉
. (2.30)
So far this discussion has been strictly first order. There are instances in
which 〈f(q)〉 vanishes or is anomalously small and q? is undefined. Hornbostel,
Lepage and Morningstar generalised the BLM method to cover such cases,
which are properly second order processes and require contributions from
higher moments of f(q) [31]. The scale is then set by
log((q?)2) =
〈f log(q2)〉 ±
[
〈f log(q2)〉2 − 〈f〉〈f log2(q2)〉
]1/2
〈f〉
. (2.31)
The sign is chosen to ensure q? is continuous and physically sensible, though
it can be obtained unambiguously using higher moments [31].
I now review some of the applications of Symanzik and tadpole improve-
ment to the gauge and light quark actions I introduce in Sections 2.1 and
2.2.2. My discussion of improved gauge actions is largely taken from [17] and
[19].
2.3.3 Improved gluon actions
As I discuss in the previous section, the Wilson gauge action has discretisation
errors at O(a2). Gauge invariance restricts the possible operators at a given
17
Chapter 2. Lattice QCD
dimension. There is one dimension four operator,
O(4) =
∑
µν
Tr (FµνFµν) , (2.32)
so the leading-order term in the expansion of the plaquette is O(4). There are
no gauge invariant operators of dimension five, so the lattice artifacts begin
at O(a2). We can remove these artifacts by including dimension six operators
in the gauge action. There are three operators of dimension six, which are
O(6)1 =
∑
µ,ν
Tr (DµFµνDµFµν) , (2.33)
O(6)2 =
∑
µ,ν,ρ
Tr (DµFνρDµFνρ) , (2.34)
O(6)3 =
∑
µ,ν,ρ
Tr (DµFµρDνFνρ) . (2.35)
On the lattice there are three six-link operators, the planar loop P(6)1 ,
the twisted loop P(6)2 , and the L-shaped loop P
(6)
3 . I show these operators in
Figure 2.2. We can then write the lattice gauge action as
Sg = β
(
c(4)L(4)1 +
∑
i
c(6)i L
(6)
i
)
, (2.36)
where L(j)i = ReTr(1 − P
(j)
i )/N and P
(4)
1 is the usual plaquette. Note that
the usual convention, unlike the Wilson action, is that β is related to the bare
coupling, g2, by β = 10/g2. At tree-level we can normalise the coefficients
c(j)i so that in the continuum limit the action reduces to FµνFµν/4. The
normalisation condition was first obtained by Lu¨scher and Weisz [35, 36]:
c(4)1 + 8c
(6)
1 + 8c
(6)
2 + 16c
(6)
3 = 1. (2.37)
We fix these coefficients by enforcing the absence of O(a2) errors in, for
example, the static charge potential. At tree-level, a convenient choice is the
18
Chapter 2. Lattice QCD
(c)(b)(a)
Figure 2.2: The three six-link lattice operators. From left to right: (a) the
planar loop P(6)1 , (b) the twisted loop P
(6)
2 and (c) the L-shaped loop P
(6)
3 .
“tree-level Lu¨scher-Weisz” action, with coefficients
c(4)1 = 1, c
(6)
1 = −
1
20
, c(6)2 = c
(6)
3 = 0. (2.38)
The one-loop improvement coefficients were obtained by Lu¨scher and
Weisz in [37] and then extended to include tadpole-improvement by Alford et
al. in [28]. The fermionic contributions were included by Hao et al. using the
full set of lattice perturbation theory techniques that I describe in the next
chapter [38]. The unquenched tadpole-improved one-loop coefficients are
c(4)1 = 1, c
(6)
1 = −
1
20u20
(1 + 0.4805αs − 0.3637(14)nf ) (2.39)
c(6)2 = −
1
u20
(0.03325αs − 0.009(1)nfαs) , c
(6)
3 = 0, (2.40)
where nf is the number of sea quarks.
Throughout this dissertation I use the term Symanzik-improved gluons to
denote the tree-level Lu¨scher and Weisz action with improvement coefficients
defined in Equation (2.38). The one-loop coefficients are not required for my
calculations as the corresponding improvements are always at non-leading
order.
2.3.4 Improved staggered quark actions
Staggered quarks suffer from taste-changing interactions that arise from the
exchange of highly virtual gluons. We can suppress these high momenta pro-
cesses by adding irrelevant four-quark operators to the action in much the
19
Chapter 2. Lattice QCD
same way that Wilson corrected his original fermion action by adding a higher
dimension operator. This method, however, is not the most efficient solution
[27].
Rather than adding irrelevant operators to improve the action, we replace
the simple link variables by sums of link variables over different paths, usually
known as “fat link variables”. This modification changes the quark-gluon
coupling to suppress gluon momenta near O(pi/a). A number of choices of
“smearing” operators exist, but the most heavily used choice of fat link is
the “Fat7” smearing [19]. Each link variable Uµ is replaced by FµUµ where
Fµ =
∏
ν 6=µ
(
1 +
a2∆(2)ν
4
)∣
∣
∣
∣
∣
sym
, (2.41)
and a2∆(2)ν is the lattice Laplacian acting on the link Uµ [39]. The fattened
link variables now have discretisation errors at O(a2), which we remove by
first improving the smearing operator:
FASQµ = Fµ −
∑
ν 6=µ
a2(∆ν)2
4
. (2.42)
We must also correct the covariant derivative by including a third-order
derivative
∆µ → ∆
ASQ
µ = ∆
F
µ −
a2
6
(∆µ)
3. (2.43)
The additional term is usually referred to as the “Naik” term. Here the ex-
tra superscript F in ∆Fµ indicates that the fattened link variables are used
in the covariant derivative instead of the simple link variables. Tadpole im-
provement of the link variables leads finally to the ASQTad (a-SQuared,
TADpole-improved) action [39],
SASQTad =
∑
x
ψ¯(x)
(
γµ∆
ASQTad
µ +m
)
ψ(x), (2.44)
where ∆ASQTadµ is the ∆
ASQ
µ operator with all link variables replaced by
tadpole-improved links. The ASQTad action has tree-level errors at O(a4)
20
Chapter 2. Lattice QCD
and errors at O(αsa2) are strongly reduced.
The ASQTad action has been used extensively by the HPQCD and MILC
collaborations; see for example [3] for a review of precision results obtained
using ASQTad sea quarks. I calculate the fermionic contributions to both the
two-loop tadpole improvement factor, u0, and the heavy quark renormalisa-
tion parameters using ASQTad quarks in Chapters 4 and 5.
Whilst the ASQTad action has negligible tree-level errors for light quarks,
this is not true for c quarks. Charm quarks are generally nonrelativistic in
typical mesons, so the rest energy of the quark is much larger then its mo-
mentum. The dominant errors are therefore O(a4m4). These errors can be
removed by tuning the coefficient of the Naik term
a2
6
(∆µ)
3 →
a2
6
(1 + ²)(∆µ)
3. (2.45)
We can further reduce taste-exchange interactions by repeatedly smearing
the links. One such action is the HISQ (Highly Improved Staggered Quarks)
action [23], which sandwiches a reunitarisation operation, U , between two
smearing operations,
FHISQµ = F
ASQTad
µ UF
ASQTad
µ . (2.46)
The reunitarisation operator projects the smeared link variables back to
SU(3). The resulting HISQ action is
SASQTad =
∑
x
ψ¯(x)
(
γµ∆
HISQ
µ +m
)
ψ(x), (2.47)
where
∆HISQµ = ∆µ
[
FHISQµ Uµ(x)
]
−
a2
6
(1 + ²)(∆µ)
3
[
UFHISQµ Uµ(x)
]
. (2.48)
In the first derivative, the HISQ-smeared link variables are used, whilst in
the second, only one-level of smearing is used.
The HISQ action is particularly suited to the simulation of charmed
mesons (see, for example, [6] for some recent c quark results). HISQ smear-
21
Chapter 2. Lattice QCD
ing reduces taste-splittings by approximately another factor of three. The
relative simplicity and low computational cost of the HISQ action allows the
simulation of sea quarks near their physical masses [6, 23, 25].
2.4 Heavy quarks on the lattice
Current HPQCD and MILC lattice simulations use physical lattice spacings
from around 0.18 fm down to 0.05 fm, generally referred to as “extracoarse”
and “ultrafine” respectively [22]. Unfortunately for heavy quark physics,
these are too large to reliably simulate the b quark, which has a Comp-
ton wavelength of ∼0.01 fm. We can overcome this difficulty by introducing
an effective theory for the heavy quarks that separates the long and short
distance physics. Recently, the HISQ action has been used to simulate b
quarks on the lattice, but this result incorporated an extrapolation up to
the physical b quark mass, because simulaion at the b quark mass is not yet
possible [25]. Effective theories remain the prevalent method for heavy quark
physics on the lattice.
There are two commonly used effective theories: heavy quark effective
theory (HQET) and nonrelativistic QCD (NRQCD). HQET requires non-
perturbative renormalisation, as I discuss in greater detail in Section 3.2.2,
so the role of lattice perturbation theory in HQET is very limited. In this
dissertation I use the NRQCD action for perturbative calculations involving
heavy quarks.
Heavy quark bound states are characterised by the small relative velocity
of their constituent quarks. In heavy quarkonium bound states, such as the
ηb and Υ mesons, the relative velocity is approximately v2 ' 0.1. The relative
velocity naturally induces three well-separated energy scales in heavy bound
states. These are the mass, O(M), the momentum, O(Mv), and the kinetic
energy, O(Mv2). In NRQCD we decouple the quark and antiquark fields and
then expand the quark fields using a nonrelativistic expansion. We then inte-
grate out the high energy modes of O(M) to obtain an effective theory with
nonrelativistic, low energy degrees of freedom. We can systematically correct
discretisation and nonrelativistic errors at each order in 1/M by adding ir-
22
Chapter 2. Lattice QCD
relevant operators to the action using power counting rules. Power counting
rules are presented in detail in [40] for NRQCD and in [41] for a discretisation
of NRQCD appropriate for a moving reference frame (mNRQCD).
NRQCD is not renormalisable, even at leading order, because theO(1/M)
corrections are already included in the propagator and the non-leading terms
are all higher order in the inverse heavy quark mass. There are consequently
an infinite number of divergences that cannot be absorbed into the low en-
ergy constants. However, at finite lattice spacings, the theory is regularised
by the lattice cutoff and the cutoff dependence of physical quantities can
be reduced by including terms of sufficiently high order in the action. In
practice, NRQCD simulations are restricted to a range of lattice spacings
given by 1/M . a . 1/ΛQCD, where ΛQCD is the scale parameter of QCD
characterising nonperturbative behaviour.
My presentation of the derivation of the NRQCD action largely follows the
development laid out in the excellent review in [40]. Historically, NRQCD was
developed from nonrelativistic QED [42] by Peter Lepage and others in [43]
and [44]. Improved NRQCD followed shortly after in [45], with the first full
analysis of heavy quarkonium annihilation and production using improved
NRQCD given in [46]. An alternative derivation of NRQCD, more suited to
the extension to mNRQCD, is presented in [41] and [47].
2.4.1 Building the NRQCD action
To build NRQCD we start by constructing a nonrelativistic Lagrangian de-
scribing free quark fields. We then add interaction terms to the desired order
of accuracy, using power counting rules to systematically organise the extra
terms as an expansion in the relative quark velocity v.
We first decouple the quark and antiquark fields using the Foldy-Wouthuy-
sen-Tani (FWT) transformation (see, for example, [48]). In the continuum,
the lowest order action is just the nonrelativistic Schro¨dinger action
S0 =
∫
d4xψ†(x)
(
iDt +
D2
2MQ
)
ψ(x). (2.49)
23
Chapter 2. Lattice QCD
One possible discretisation of the Schro¨dinger action is the Davies and
Thacker action [49]
SDT = a
3
∑
x
ψ†(x, t)
(
∆(+)4 −
3∑
j=1
∆(+)j ∆
(−)
j
2aM
)
ψ(x, t), (2.50)
where the elementary forward and backward difference operators are
∆+µψ(x) = Uµψ(x+ aµˆ)− ψ(x), (2.51)
∆−µψ(x) = ψ(x)− U
†
µψ(x− aµˆ). (2.52)
I define the link variable Uµ in Equation (2.7). Alternative discretisations are
possible, such as
SH = a
3
∑
x,t
ψ†(x)
[
ψ(x, t)− U †4(x, t− 1) (1 +H0)ψ(x, t− 1)
]
, (2.53)
where the leading order kinetic term is
H0 =
3∑
j=1
∆(+)j ∆
(−)
j
2aM
. (2.54)
I use the Davies and Thacker action in the calculation of kinetic renor-
malisation coefficients outlined in Section 4.4 and derive Feynman rules for
both actions in Appendix B.
So far so good. We now have a leading-order NRQCD action, SH , which we
systematically improve by specifying the desired symmetries of NRQCD. A
partial list of these symmetries includes: gauge invariance; parity; rotational
and translational symmetry; locality and unitarity (a more complete list is
given in [40]). These symmetries severely constrain our choice of additional
irrelevant operators at a given order in v. Power counting rules for NRQCD
are discussed in detail in [40] and [45] (and for mNRQCD in [41]) so I will not
repeat them here. Rather, I will discuss the origin of each of the correction
terms in the full action, which is correct to O(a2, v4, 1/M2).
The nonrelativistic expansion introduces relativistic errors that we can
24
Chapter 2. Lattice QCD
correct by including the next-to-leading order contribution from the nonrel-
ativistic expansion of the relativistic energy-momentum relation
δHkin = −c1
(∆(2))2
8a3M3
, (2.55)
where
∆(2) =
3∑
j=1
∆(+)j ∆
(−)
j . (2.56)
We first need to remove the leading order discretisation errors. In principle
we could reduce discretisation errors by improving ∆(2) to ∆˜(2) in the leading
order kinetic operator H0, but in practise it is preferable for performance
reasons to leave H0 alone and to include this improvement in a term δH that
includes all correction operators [47].
We therefore include the correction term
δHs = c5
∆(4)
24aM
(2.57)
in δH. This correction removes discretisation errors up to O(a4p4) in the
spatial direction. The operator ∆(4) is defined by
∆(4) =
3∑
j=1
(
∆(+)j ∆
(−)
j
)2
. (2.58)
Similarly, we improve the temporal derivative by including the correction
term
δHt = −c6
(∆(2))2
16na2M2
. (2.59)
Our formulation of the NRQCD action uses the fact that we are chiefly
interested in onshell quantities. We can therefore use the Shro¨dinger equation
to replace the temporal derivative with the approximate expression
iDt ≈ −
D2
2aM
, (2.60)
through a field redefinition. This field redefinition method is a general ap-
25
Chapter 2. Lattice QCD
proach applicable to all field theories and allows us to remove operators that
vanish by the equations of motion.
Removing the temporal derivatives from the action using this expression
greatly reduces the complexity of the numerical evaluation of quark prop-
agators. Since the Schro¨dinger equation is linear in the time derivative the
quark propagator obeys an evolution equation that can be specified as an
initial-value problem. The initial conditions on a single timeslice of the lat-
tice are specified and the system simply evolved time step by time step.
This configuration update method is computationally far cheaper than the
relativistic case. For the relativistic Dirac equation, the temporal difference
operator is constrained by Hermiticity to be the symmetric forward and back-
ward operator, which means that the evolution of the quark propagator is
a boundary-value problem, requiring multiple sweeps through the lattice for
each configuration update.
Finally, we include the spin-dependent terms, the “Darwin” term,
δHD = c2
ig
8a2M2
(
∆˜(±) · E˜− E˜ · ∆˜(±)
)
, (2.61)
the “spin-orbit” interaction
δHSO = −c3
g
8a2M2
σ ·
(
∆˜(±) × E˜− E˜× ∆˜(±)
)
, (2.62)
and the chromo-magnetic contribution,
δHσ ·B = −c4
1
2aM
σ · B˜. (2.63)
The improved chromo-electric and chromo-magnetic fields are given in terms
of the improved field-strength tensor by
E˜i = F˜i4, B˜i = −
1
2
²ijkF˜jk. (2.64)
I give an expression for the full improved field-strength tensor, taken from
26
Chapter 2. Lattice QCD
[50], in Appendix A. The O(a4v4) improved derivatives are
∆˜i = ∆i −
1
6
∆3j . (2.65)
The δHD, δHSO and δHσ ·B corrections are the O(v2) chromo-electric
and chromo-magnetic interactions. The terms containing the Pauli matrices
σi break the spin symmetry of the unimproved NRQCD actions in Equation
(2.50) and (2.53), which are invariant under the SU(2) mixing of the heavy
quark spin components [40]. These Pauli matrix terms induce spin splittings
that scale like Mv2. Splittings of radial and orbital-angular-momentum ex-
citations occur with the introduction of higher order O(v4) correction terms
[40].
Thus the fully O(a2, v4, 1/M2)-improved lattice NQRCD action is
SNRQCD =
∑
x,τ
ψ+(x, τ) [ψ(x, τ)− κ(τ)ψ(x, τ − 1)] , (2.66)
where
κ(τ) =
(
1−
δH
2
)(
1−
H0
2n
)n
U †4
(
1−
H0
2n
)n(
1−
δH
2
)
, (2.67)
and the δH contribution includes all interaction terms and higher order op-
erators
δH = δHkin + δHs + δHt + δHD + δHSO + δHσ ·B. (2.68)
We introduce the stability parameter n to stabilise the time evolution of the
propagator for small quark masses. This requires |1−H0/2n| < 1 [45] and
typically I use n = 2 or n = 4 in this dissertation.
Throughout this dissertation I refer to the action in Equation (2.66),
without radiative corrections to the operator coefficients, as “highly-improved
NRQCD”. I do not include radiative corrections in the operator coefficients in
the lattice perturbation theory calculations in this dissertation because these
improvements always contribute at the next order in the strong coupling
constant.
27
Chapter 2. Lattice QCD
2.4.2 Perturbative improvement for NRQCD
In general the operator coefficients, ci, are free parameters of the NRQCD
action. At tree-level they are equal to unity, but at higher orders they are
modified by radiative corrections. The ultraviolet behaviour of NRQCD dif-
fers from QCD and so these radiative corrections will produce O(αs) errors
in predictions of physical observables.
In principle we could tune the parameters of the action nonperturbatively,
but this process is time-consuming, as it requires independent tuning for
each simulation, and reduces the predictive power of the theory. For lattice
NRQCD, lattice perturbation theory is a more effective way to calculate the
radiative corrections.
The operator coefficients are calculated by matching an onshell quantity
in NRQCD to a corresponding onshell quantity in QCD, order-by-order in
the coupling constant. The general procedure for the calculation of one-loop
improved operator coefficients is:
1. Calculate the one-loop radiative corrections cLatti = 1 + αc
(1),Latt
i + · · ·
to an onshell lattice quantity.
2. Determine the one-loop expansion of the corresponding continuum co-
efficients cConti = 1 + αsc
(1),Cont
i + · · · .
3. Correct the coefficient of the operator in the lattice action by replacing
cLatti → c
Latt
i
[
1 + αs
(
c(1),Conti − c
(1),Latt
i
)
+ · · ·
]
.
I illustrate the conceptual basis for the matching procedure schematically
in Figure 2.3. There are two methods for extracting nonrelativistic opera-
tors at one-loop, corresponding to the the clockwise and counter clockwise
directions indicated by the arrows. The starting point for both methods is
continuum QCD in the top left.
28
Chapter 2. Lattice QCD
QCD Lattice NRQCD
NR expansion
Difference ⇒ matchingOne− loop process
Improved difference
Improvement One− loop calculations
Figure 2.3: Schematic representation of the matching procedure. The starting
point is continuum QCD in the top left. There are two methods for extracting
nonrelativistic operators at one-loop, corresponding to the the clockwise and
anticlockwise directions indicated by the arrows. I discuss each method in
more detail in the text in Section 2.4.2.
Following the arrows counter clockwise, we first calculate a process at
one-loop in continuum QCD. We then expand the result in a nonrelativistic
expansion. The second method follows the arrows clockwise. We first expand
(and discretise) the continuum QCD action to obtain lattice NRQCD. We
then calculate the same process as in the first method. The two results exhibit
different ultraviolet behaviour, represented by the failure of the “square” to
close. By improving operators we reduce the difference between the results
and bring the lattice NRQCD result, obtained via method two, closer to
the continuum result. This corresponds to reducing the gap between the two
methods, schematically illustrated by the “square” on the right of Figure 2.3.
In the next chapter I discuss lattice perturbation theory in more detail
and introduce some of the techniques required to undertake such matching
calculations.
29
Chapter 3
Lattice perturbation theory
3.1 Why use lattice perturbation theory?
Lattice QCD is primarily a tool for understanding the nonperturbative regime
of QCD—so why develop lattice perturbation theory? Beyond serving as
an explicit test of simulations in the weak-coupling regime, there are three
main reasons to use lattice perturbation theory when studying heavy quark
systems.
Firstly, we can use perturbative calculations to determine the renormali-
sation of the bare parameters of lattice actions, such as the mass and wave-
function renormalisation. The lattice serves as an ultraviolet regulator by
excluding all momenta greater than pi/a and therefore we can view lattice
QCD as simply another regularisation scheme. The renormalised parameters
of the bare lattice action can be calculated relatively easily in perturbation
theory. I present this procedure in more detail for the example of the heavy
quark energy shift and the mass and wavefunction renormalisations for lattice
NRQCD in Chapter 5.
Secondly, lattice perturbation theory provides a method for systemat-
ically matching regularisation schemes. Experimental results are typically
expressed in the Modified Minimal Subtraction (MS) scheme. As with any
other regularisation scheme, the effects of excluded momenta must be in-
cluded via the introduction of counterterms in the lattice action.
To compare lattice simulations with experimental results, lattice param-
eters, such as the coupling constant and renormalisation constants, must be
related to the equivalent parameters in the MS scheme. This matching pro-
cess is often most easily performed perturbatively. In Chapter 5 I discuss the
example of the two-loop heavy quark mass renormalisation, which relates
the bare lattice mass to the MS mass. I use the mass renormalisation and
heavy quark energy shift to extract a prediction of the b quark mass from
30
Chapter 3. Lattice perturbation theory
lattice NRQCD simulations. As another example, I compute the matching
coefficients for heavy-light currents with NRQCD heavy quarks in Chapter
6.
Thirdly, lattice perturbation theory allows us to perturbatively improve
the bare operator coefficients of the lattice action. This forms an integral
part of the Symanzik improvement program, which I discuss in more detail
in Section 2.3.1. For example, the lattice NRQCD actions of the previous
chapter define effective theories for heavy quarks at finite lattice spacing.
NRQCD is an expansion in irrelevant operators with coefficients chosen by
matching to continuum QCD so that continuum results can be determined
from simulations on the lattice. At tree-level, these coefficients are defined to
be unity, but beyond tree-level, radiative corrections modify the coefficients
and the ultraviolet behaviour of the nonrelativistic action differs from the
behaviour of continuum QCD. This difference introduces O(αs) errors. In
principle we could tune the coefficients of these additional operators nonper-
turbatively to reproduce the correct continuum results. In practise, however,
nonperturbative tuning is time-consuming and reduces the predictive power
of the lattice theory. Lattice perturbation theory is therefore often a better
method for determining the lattice action parameters. In Section 4.4 I de-
scribe the improvement process for the specific case of higher order kinetic
operators in the NRQCD action.
Whilst these three reasons motivate lattice perturbation theory, they do
not justify its use. Lattice QCD is essentially a nonperturbative tool. We
may justifiably ask: is it reasonable or appropriate to apply perturbative
techniques to a nonperturbative formulation of QCD? A justification for the
use of lattice perturbation theory is provided in [27]: the perturbative renor-
malisation factors account for the momenta excluded by the lattice cutoff.
This hard ultraviolet cutoff is typically at least 6 GeV for current lattice
spacings of around 0.1 fm. The coupling constant is certainly small at these
energies, αs(pi/a) ≈ 0.2, and perturbation theory is likely to be valid. We
therefore expect the effects of the excluded momenta are to be well described
by perturbation theory.
In fact, we can view lattice perturbation theory as a classic example of the
31
Chapter 3. Lattice perturbation theory
application of the renormalisation group to separate the physics of different
energy scales. Long distance phenomena are modelled nonperturbatively on
the lattice, whilst the effects of the asymptotically free short distance modes
are simply “integrated out” and rolled up into the renormalisation parame-
ters. Lattice perturbation theory thus provides the connection between the
low and high energy regimes of QCD.
3.2 Perturbation theory and renormalons
Lattice perturbation theory is not the only method for determining renor-
malisation parameters for lattice actions. Perturbative renormalisation can,
in fact, introduce ambiguities, “renormalons”, into lattice quantities. For ef-
fective theories such as HQET, nonperturbative renormalisation is required
to avoid these difficulties. Before reviewing some of the tools of lattice per-
turbation theory that I use in this dissertation, I briefly describe some of
the issues associated with renormalons and lattice perturbation theory. In
particular I discuss the interplay between NRQCD, renormalons, and lattice
perturbation theory.
3.2.1 Mathematical preliminaries
Renormalons are singularities in the Borel transformation of an asymptotic
series [51]. If a function has the power series expansion
f(x) =
∞∑
n=0
anx
n, (3.1)
then we define the Borel transformation f˜ as
f˜(t) =
∞∑
n=0
an
tn
n!
. (3.2)
Provided the Borel transform doesn’t have any singularities on the real pos-
itive axis of the Borel plane (the plane of the Borel parameter t) and is
32
Chapter 3. Lattice perturbation theory
sufficiently convergent at large t, then we can recover the original function
from the inverse Borel transform,
f(x) =
∫ ∞
0
dt exp(−t/x)f˜(t). (3.3)
The inverse Borel transform has the same series expansion as the original
function and defines the Borel sum of the original function, but is not guar-
anteed to exist.
If the Borel integral has singularities on the real axis, then the inverse
Borel transform is not well-defined. There is no unique prescription for de-
forming the contour along the real axis – do we close the contour above or
below the real axis? – to avoid the renormalon poles, so an inherent ambiguity
in the Borel sum arises. The divergent behaviour of the original series is en-
coded in these singularities, referred to as renormalons. Instantons also give
rise to singularities in the Borel plane, but they are higher order ambiguities
in the perturbation series [52].
The contribution to the function f(x) from a pole at t = t?, with residue
R?, is
f(x) ∼ −R?
∑
n
n!
x
t?
n+1
. (3.4)
Renormalons are therefore associated with n! growth in the power series
coefficients of the original function [53].
3.2.2 Renormalons and NRQCD
Perturbative QCD is generally believed to be an asymptotic series approxi-
mation to nonperturbative QCD, as first argued by Dyson [54]. Purely per-
turbative calculations of nonperturbative processes are therefore likely to suf-
fer from renormalon ambiguities, though the observables themselves are of
course well-defined. We may worry, then, that renormalon ambiguities may
afflict lattice perturbation theory calculations. In particular, perturbative
parameters, such as the pole mass, suffer from such renormalon ambigui-
ties [55, 56] and we must be sure that we understand any subtleties involving
33
Chapter 3. Lattice perturbation theory
renormalons when extracting precise results from lattice perturbation theory.
I use the pole mass in my calculation of the mass of the b quark in Chapter
5, but I prove there that any ambiguities in the pole mass must vanish in the
final result.
In fact, renormalon ambiguities necessarily disappear from physical ob-
servables computed in effective field theories such as NRQCD and HQET.
This was first shown by Neubert and Sachrajda [57] and Luke et al. [58]
for HQET and for NRQCD by Bodwin and Chen [59]. The domain of inte-
gration of full QCD is split between the short-distance coefficients and the
long-distance matrix elements, controlled by a ultraviolet regulator that is
imposed on the matrix elements. The choice of ultraviolet regulator, which
serves as a factorisation scale, is known as the “factorisation scheme”. The
consistency of the effective field theory ensures that observables are factori-
sation scale independent, provided sufficient accuracy in the effective field
theory is maintained.
Bodwin and Chen observed that hard dimensionful ultraviolet regulators
(for example momentum cutoff and lattice regularisations) guarantee that
short-distance coefficients are infrared renormalon ambiguity free (provided,
again, sufficient accuracy in the effective field theory is maintained). Corre-
spondingly, because matrix elements are completely determined in principle
by observables and short-distance coefficients, the matrix elements are renor-
malon ambiguity free in a cutoff-like factorisation scheme. This is consistent
with the unambiguous nonperturbative definition of matrix elements in terms
of Euclidean path integrals in the lattice formulation of the effective field the-
ory. In such factorisation schemes, ultraviolet divergences appear as power
dependencies on the cutoff in the matrix elements. Dimensional regularisa-
tion, on the other hand, does not guarantee the short-distance coefficients
are infrared ambiguity free, as the infrared finite parts of the coefficients are
integrated down to zero momentum (this point was first raised in [60] and
[61]).
A hard ultraviolet cutoff introduces divergences in the matrix elements
in powers of the cutoff [62]. In general we cannot subtract these divergences
perturbatively as this reintroduces renormalon ambiguities. In lattice formu-
34
Chapter 3. Lattice perturbation theory
lations of HQET and NRQCD, the hard ultraviolet cutoff is provided by the
inverse lattice spacing a−1 and the continuum limit corresponds to a diverg-
ing ultraviolet cutoff. At leading order, HQET is renormalisable, so HQET
calculations usually involve extrapolation to the continuum limit. Thus the
divergence of HQET matrix elements is a real issue and requires nonpertur-
bative renormalisation of HQET for precise results. However, this argument
does not apply to NRQCD. NRQCD is not renormalisable, so we do not take
the continuum limit, a → 0. The NRQCD matrix elements do not diverge,
but are finite and scale as a known function of a−1, and therefore have errors
that can be quantified at a given lattice spacing.
Braaten and Chen [53] studied the large-order behaviour of the short-
distance coefficients for two different quarkonium decays, J/ψ → e+e− and
ηc → γγ. The renormalon ambiguities of the decay coefficients are hypothe-
sised to cancel with corresponding ambiguities in the nonperturbative matrix
elements at relative order v2. To put this more explicitly, infrared renormalons
at the Borel parameter u = k/2 (with k an integer) give rise to ambiguities
of order (ΛQCD/M)k. Braaten and Chen hypothesised that these ambigui-
ties can be absorbed into NRQCD matrix elements that are suppressed by
v2k or less. They confirmed this hypothesis for the individual cases of the
J/ψ → e+e− and ηc → γγ decays [53].
Bodwin and Chen [59] presented a method to explicitly calculate the in-
frared renormalon ambiguities in the NRQCD matrix elements. Combined
with the results of Braaten and Chen, this explicitly demonstrated the can-
cellation of renormalon ambiguities in physical decay rates calculated with
NRQCD [59].
Thus, whilst we should be aware that renormalon ambiguities may be
present in perturbative calculations, we generally do not need to nonpertur-
batively renormalise NRQCD to avoid renormalon ambiguities and we are
free to use lattice perturbation theory for NRQCD calculations. I now dis-
cuss some of the tools and techniques that we use in lattice perturbation
theory.
35
Chapter 3. Lattice perturbation theory
3.3 The tools of lattice perturbation theory
Lattice perturbation theory is, unfortunately, rather more complicated than
its continuum cousin. In principle the perturbative techniques are analogous
to those in the continuum. We extract Feynman rules from the action by
splitting the action into the free-field terms – quadratic in the fields – and
the interacting terms – everything else. We build Feynman diagrams from
these rules and evaluate them by integrating over phase space. This simple
prescription rather hides the details, wherein lies the devil.
Georg von Hippel suggested that a useful “rule of thumb is that lattice
perturbation theory is ‘one-loop’ more complicated than continuum [pertur-
bation theory]” [63]. This statement indeed gives a fair reflection of the diffi-
culties involved. Lattice actions are almost always considerably more compli-
cated than continuum actions: deriving the Feynman rules by hand is rarely
possible for improved actions and automation is usually required. The result-
ing Feynman integrals are accordingly more arduous. The lattice explicitly
breaks Poincare´ symmetry and consequently there are few of the handy tricks
and assorted variable changes that are available to the continuum perturba-
tion theorist. Multidimensional numerical integration is the order of the day.
VEGAS, an adaptive Monte Carlo algorithm developed by Lepage [64], is
particularly suited to evaluating lattice integrals.
Beyond the obstacle of extra lattice action terms, lattice perturbation
theory is complicated by additional vertices with an arbitrary number of glu-
ons. These interactions are generated by the use of link variables rather than
gauge fields. Finally there are lattice artifacts arising from gauge-fixing us-
ing the Faddeev-Popov determinant and integration over the invariant Haar
measure.
I now discuss the automated lattice perturbation theory routines that I
use throughout this dissertation, HIPPY and HPSRC. Calculations with these
routines proceed in two steps. We first extract the Feynman rules from the
action using the HIPPY routines. We then construct the construct Feynman
diagrams using the HPSRC routines and evaluate the resulting diagram numer-
ically with VEGAS.
36
Chapter 3. Lattice perturbation theory
3.3.1 Automated lattice perturbation theory
Lu¨scher and Weisz developed the original idea for automated extraction of
Feynman rules from lattice gauge actions in [65]. HIPPY is an implementation
of the Lu¨scher-Weisz algorithm, extended to include quark actions, written in
PYTHON [66, 67]. Whilst automated methods for evaluating Feynman diagrams
exist [68, 69, 70], these tend to be symbolic algebra packages, and do not gen-
erate the Feynman rules automatically. This method undoubtedly speeds up
perturbative calculations, but the manual derivation of Feynman rules for
highly-improved actions is cumbersome. To my knowledge, the HIPPY and
HPSRC routines are the only automated lattice perturbation theory packages
that allow fully automated calculations using modern, highly-improved lat-
tice actions such as HISQ and mNRQCD.
The gauge action is built from closed loops of link variables, or “Wilson
lines”. We expand the link variables in powers of the bare coupling constant,
g, as
Uµ>0(x) = exp
(
agAµ
(
x+
a
2
µˆ
))
=
∞∑
r=0
1
r!
(
agAµ
(
x+
a
2
µˆ
))
, (3.5)
and define U−µ(x) = U †µ(x− aµˆ). By convention, the generators of the gauge
field are anti-Hermitian. In momentum space the Wilson lines are given by
L(x, y;U) =
∑
r
(ag)r
r!
∑
k1,µ1,a1
. . .
∑
kr,µr,ar
A˜a1µ1(k1) . . . A˜
ar
µr(kr)
× Vr(k1, µ1, a1; . . . ; kr, µr, ar), (3.6)
where the Vr are referred to as “vertex functions”. These vertex functions
are further decomposed into a matrix encoding the colour structure, Cr, and
a “reduced vertex”, Yr, that depends only on the momenta and positions of
the links:
Vr(k1, µ1, a1; . . . ; kr, µr, ar) = Cr(a1, . . . , ar)Yr (k1, µ1; . . . ; kr, µr) . (3.7)
37
Chapter 3. Lattice perturbation theory
Finally we write the reduced vertices as products of exponentials,
Yr (k1, µ1; . . . ; kr, µr) =
nr∑
n=1
fn exp
[
i
2
(
k1 · v
(n)
1 + . . . kr · v
(n)
r
)]
, (3.8)
where the fn are the amplitudes associated with each term and the v(n) are
the locations of each of the r factors of the gauge potential. Thus for each
combination of r Lorentz vertices there are nr terms, each with an associated
amplitude fn and location v
(n)
r .
Written in this form, the Feynman rules can be encoded in ordered lists
of “entities”, E, which have the form
E = (µ1, . . . , µr;x, y; v1, . . . , vr; f) . (3.9)
The complete set of entities at all orders in the expansion of a path is referred
to as a “field”, F{E}.
The reduced vertices are stored as list of entities in the “vertex file”
generated by HIPPY. HIPPY generates the vertex file by repeatedly applying
the convolution formulae derived in [71]. For example, for an action of the
form ψABψ, the Feynman rule for one-gluon emission is the convolution of
the Feynman rules for one-gluon emission from operator A with no-gluon
emission from B plus the corresponding convolution for no-gluon emission
from A with one-gluon emission from B. In some cases, for very complicated
actions, such as the highly-improved mNRQCD action, the convolution is
performed within the HPSRC code to avoid very large vertex files, which would
otherwise lead to memory leak issues.
The colour structure is handled in the HPSRC routines, when the Feynman
diagrams are constructed. All derivatives can be obtained by algebraic ma-
nipulation of the reduced vertices, so no numerical derivatives are required.
This avoids possible numerical instabilities that might arise when using finite
derivatives to approximate derivatives and ensures the same vertex files can
be used for both periodic and twisted boundary conditions. I discuss twisted
boundary conditions in more detail in the next section.
I construct and evaluate all Feynman diagrams using HPSRC, a set of
38
Chapter 3. Lattice perturbation theory
FORTRAN routines described in [66, 67]. The routines build the fundamental
vertices from the vertex files. For example, the function vert_ggg(k_i,a_i)
returns the vertex function associated with the three-gluon vertex. In this
case, the arguments of the routine are the set of initial and final momenta,
k_i, and the set of colour indices, a_i, associated with the gluons. The func-
tion vert_ggg returns a TaylUR object, which carries the derivatives of the
function with respect to the particle momenta [72, 73].
One subtlety of the HIPPY /HPSRC routines that is pertinent to the cal-
culations in this dissertation is the inversion of the gluon propagator. The
HPSRC module vert_gg constructs the gluon propagator by inverting the two-
point function from the vertex files. In Feynman gauge, this inversion can be
carried out directly. For singular gauges, such as Coulomb or Landau gauge,
the two-point function is not invertible. To overcome this difficulty, an in-
termediate gauge correction added, the two-point function is inverted and
then the correction is subtracted. The inversion routines in HPSRC can be
cumbersome for a large number of iterations, so to speed up the code hard-
wired propagators exist for a variety of gauge actions, including Wilson and
Symanzik-improved gluons. I use the hardwired propagators in my calcula-
tions of the Landau tadpole parameter in Section 4.1 and of the wavefunction
renormalisation in Sections 5.1.4 and 6.3.
Finally, I evaluate the Feynman diagrams using the adaptive Monte Carlo
algorithm VEGAS. VEGAS generates points in the interval (0, 1) with a distribu-
tion that is inversely proportional to the magnitude of the integrand to be
evaluated. Initially the distribution is uniform, and a new distribution is iter-
atively refined to reflect the structure of the integrand. For this algorithm to
work, the integrand must be neither too peaked nor exactly uniform. These
exceptions notwithstanding, the VEGAS algorithm is applicable to a very wide
range of integrands. Infrared divergences in the integrand can be handled by
introducing an infrared subtraction function, which I discuss in more detail
in Sections 5.1 and 6.3. Alternatively, we can handle the infrared behaviour
by introducing twisted boundary conditions, to which I now turn.
39
Chapter 3. Lattice perturbation theory
3.3.2 Twisted boundary conditions
The inverse lattice spacing serves as an ultraviolet regulator in lattice pertur-
bation theory, but infrared divergences may still occur. In fact, in effective
theories, such as NRQCD, the infrared divergences are guaranteed to match
those in continuum QCD. In many calculations, such as the vector current
matching computation of Chapter 6, we can introduce a gluon mass as an in-
frared cutoff. However, if there are ghost contributions, then this is no longer
possible. In this case twisted boundary conditions, introduced by ’t Hooft in
[74] and first applied to lattice calculations in [65], can serve as an infrared
regulator for lattice Feynman diagrams. For example, Hao et al. used twisted
boundary conditions to understand the effect of including fermionic loops
on the improvement coefficients of the Lu¨scher and Weisz gluon action (see
Section 2.3.3). My presentation is mostly based on [16] and [75].
For twisted boundary conditions we replace the standard link variables
with periodic boundary conditions,
Uµ(x+ Lνˆ) = Uµ(x), (3.10)
with twisted link variables that satisfy
Uµ(x+ Lνˆ) = ΩνUµ(x)Ω
†
ν . (3.11)
The twist matrices are field-independent SU(N) matrices that obey
Ω1Ω2 = zΩ2Ω1, (3.12)
where z = e2pii/N is an element of the centre of the group, Z(N). This relation
only fixes the twist matrices up to a unitary transformation, which guarantees
that any choice of twist results in the same physical amplitude. A second
useful property of the twist matrices is that any matrix that commutes with
both Ω1 and Ω2 is a multiple of the identity matrix.
The twisted gauge fields satisfy the same twist algebra as the twisted
links. Introducing twisted link variables corresponds to expanding the gauge
40
Chapter 3. Lattice perturbation theory
fields in a new basis of twisted periodic plane waves
Aµ(x) =
1
L3TN
∑
k
Γ(k)eik ·xeiakµ/2A˜µ(k) (3.13)
The Γ(k) are complex N ×N matrices in SU(N) that obey the condition
ΩνΓ(k)Ω
†
ν = e
ikνLΓ(k). (3.14)
Condition (3.14) has a nonzero solution if
kν =
(
2pi
NL
)
nν = mnν . (3.15)
Here the twist vector nν is defined modulo N . We can conveniently split the
momentum into an untwisted component `, which is a multiple of 2pi/L and
a twisted component (2pi/NL)nν . For two twisted directions, we can write
the twist vector as
nν = (n1, n2, 0, 0), (3.16)
where n1 and n2 take values between 0 and N − 1. The gauge fields must
be traceless and this imposes the condition that (n1, n2) 6= (0, 0). Twisted
boundary conditions therefore act as an infrared regulator by removing the
zero momentum modes. This condition also ensures that the number of de-
grees of freedom is unchanged. The sum over the colour indices of the gluons,
in the range a ∈ [1, . . . , N2 − 1], is replaced by the sum over the (N2 − 1)
possible twist vectors.
The matrices Γ(k) depend only on the twist vector nν so I follow the usual
convention of writing Γ(k) as Γn. The general solution of Equation (3.14) is
Γn = cΩ
−n2
1 Ω
n1
2 , (3.17)
where c is a complex phase. There are a number of choices for the phase
factor and I present them in Table 3.1.
Three useful properties that are independent of the phase factor choice
41
Chapter 3. Lattice perturbation theory
are
Γn′ = Γn if n
′
1 = n1 andn
′
2 = n2 (modN),
Γn=0 = I,
TrΓn = 0 unlessn = 0.
Two more phase-dependent properties are important: the “dagger phase”
relating Γ†n to Γ−n,
Γ†n = dnΓ−n, (3.18)
and the “composition phase” defined by
Γn′Γn = znΓn′+n. (3.19)
These phase factors may be conveniently written in terms of the symmet-
ric and antisymmetric products
(n,m) ≡ n1m1 + n2m2 + (n1 +m1)(n2 +m2), (3.20)
〈n,m〉 ≡ n1m2 − n2m1. (3.21)
I present a number of values for the dagger and composition phases for various
choices of c in Table 3.1. I updated the twisted colour factor routines in the
HPSRC code to use the phase factor given in the bottom row of Table 3.1.
Twists can be extended to more than two directions. In general the twist
matrices then satisfy
ΩµΩν = zµνΩνΩµ, (3.22)
with zµν = e2piinµν/N an element of the centre of the SU(N) gauge group [47].
We can also include fermionic variables by introducing an additional SU(N)
“smell” group. Quark fields are represented by replacing the usual SU(N)
spinors with N ×N matrices in colour-smell space.
Automated lattice perturbation theory and twisted boundary conditions
are two characteristic tools of lattice perturbation theory. Before I introduce
one final piece of lattice perturbation theory machinery, background field
42
Chapter 3. Lattice perturbation theory
Phase factor c Dagger phase dn Composition phase zn
1 zn1n2 zn1m2
z−n1n2 1 z−〈n,m〉
z[n1+n2][(n1+n2−1]/2 z−(n,n)/2 z[〈n,m〉−(n,m)]/2
z−[n1+n2−n1n2]/2 1 z〈n,m〉
Table 3.1: Phase factors for twisted boundary conditions. The first three rows
are taken from [75]. I give the dagger phase (Equation (3.18)) in column two
and the composition phase (Equation (3.19)) in column three. The symmet-
ric and antisymmetric products are defined in Equations (3.20) and (3.21)
respectively.
gauge, I set the scene and motivate the use of background field gauge by
surveying its conceptual underpinnings, grounded in the quantum effective
action.
3.4 The quantum effective action
My discussion of the quantum effective action is taken primarily from [48] and
[76]. For a quantum field theory defined by the action S[φ], the complete set
of vacuum-vacuum amplitudes of fields, φr(x), coupled to external classical
currents, Jr(x), is
Z[J ] =
∫
Dφφ exp
(
−S[φ] +
∫
d4xφr(x)Jr(x)
)
. (3.23)
We relate the sum of connected vacuum-vacuum amplitudes, W [J ], to Z[J ]
by
Z[J ] = exp (iW [J ]) . (3.24)
I denote the vacuum expectation value of the operator Φr(x) in the pres-
ence of a current J as φrJ(x). We obtain the vacuum expectation value by
43
Chapter 3. Lattice perturbation theory
functional differentiation of W [J ]:
φrJ(x) =
δW [J ]
δJr(x)
. (3.25)
To define the quantum effective action, Γ[φ], we use the Legendre trans-
form of W [J ],
Γ[φ] = −
∫
d4xφr(x)Jφr(x) +W [Jφ], (3.26)
where Jφr(x) is the current for which φ
r
J(x) has the value φ
r(x), that is
φrJ(x) = φ
r(x) if Jr(x) = Jφr(x). (3.27)
In the absence of external currents, the external fields are given by the
stationary points of Γ[φ], since
δΓ[φ]
δφr(x)
= −Jφr(x). (3.28)
We can conveniently express the effective action as
iΓ[φ0] =
∫
1PI
D φ exp (S[φ+ φ0]) . (3.29)
The “1PI” indicates that only one-particle irreducible connected diagrams are
included in the integration. This notation is schematic. There is no way to
calculate such a path integral beyond perturbation theory. If the one-particle
reducible diagrams were included in the path integral then all dependence on
φ0 would vanish, by simple change of variable in the integration. We interpret
the field φ0 as a background field, conjugate to an external source.
We can calculate the functionalW [J ] using the tree-level expansion of the
vacuum-vacuum amplitude if we replace the quantum action, S, by the quan-
tum effective action, Γ. This replacement is possible because any diagram
can be constructed from a tree-level expansion of one-particle irreducible
diagrams.
The quantum effective action carries all the quantum information of a
quantum field theory. Therefore, when we match different effective theories,
44
Chapter 3. Lattice perturbation theory
we are interested in comparing the quantum effective action associated with
each theory. If two quantum theories have the same quantum effective action,
they will give the same prediction for physical processes. But what are the
symmetries of the effective action?
3.4.1 Symmetries of the effective action
The symmetries of the original quantum action are also symmetries of the
quantum effective action, provided the transformations are linear in the fields.
Under an infinitesimal field transformation
φ(x) → φ(x) + ²F [x;φ] (3.30)
that leaves the original action invariant (that is, S[φ + ²F ] = S[φ]) the
quantum effective action is invariant under the associated transformation
φ(x) → φ(x) + ²〈F [x;φ]〉Jφ . (3.31)
The angled brackets 〈. . .〉J indicate the quantum average in the presence of
source J . For linear transformations, we have
〈F [x;φ]〉Jφ = F [x;φ], (3.32)
so the quantum effective action is invariant under the same transformation
as the original action.
Perturbative calculations require gauge-fixing. This naturally obscures
the underlying gauge symmetry of the theory, which we can no longer use to
constrain the form of the effective action. However, if we include new, non-
linear transformations of the ghost fields, we can preserve a reduced form
of gauge symmetry in the theory – BRST invariance (named after Becchi,
Rouet, Stora and Tyutin). The quantum effective action is not automatically
invariant under the BRST symmetry, because of the nonlinear ghost field
transformations. However, we can overcome this obstacle by introducing new
45
Chapter 3. Lattice perturbation theory
auxiliary sources, K, that couple to the BRST-transformed fields, s ∗ φ,
exp (iW [J,K]) =
∫
Dφ exp
(
− S[φ] +
∫
d4xφr(x)Jr(x)
+
∫
d4x s ∗ φr(x)Kr(x)
)
(3.33)
In this form, the effective action is constrained by the Zinn-Justin equa-
tion
(Γ,Γ) = 0, (3.34)
where we define the Zinn-Justin anti-bracket as
(F,G) =
∫
d4x
(
δF [φ,K]
δφr
δG[φ,K]
δKr
−
δF [φ,K]
δKr
δG[φ,K]
δφr
)
. (3.35)
The Zinn-Justin equation does not constrain the effective action to a
simple form. However, the Zinn-Justin equation is sufficient, when coupled
with power-counting arguments and various other symmetries such as Lorentz
and global gauge invariance, to show that the divergent terms in the quantum
effective action are related by Ward identities. These Ward identities imply
that the divergent terms are in fact gauge invariant.
Furthermore, the renormalised quantum effective action, ΓR = Γ − Γdiv,
also satisfies
(ΓR,ΓR) = 0. (3.36)
Thus Ward identities relating the renormalisation parameters will also hold
for the renormalised quantum effective action. This logic reverses the usual
argument, which is presented in, for example, [48]. This argument requires the
renormalised quantum effective action to be repackaged in a gauge invariant
manner, from which requirement we derive the Ward identities.
We obtain the renormalised quantum effective action from the original
action by multiplicative field and coupling constant renormalisation. This is
equivalent to the traditional view of renormalisation as adding counter-terms
46
Chapter 3. Lattice perturbation theory
to the original action. The renormalised fields are
AR = Z
−1/2
3 A0, ηR = Z˜
−1/2
3 η0, ηR = Z˜
−1/2
3 η0,
ψR = Z
−1/2
2 ψ0, ψR = Z
−1/2
2 ψ0, gR = Z
−1
1F Z2Z
1/2
3 . (3.37)
Here I denote the gauge fields by A; the quark and antiquark fields by ψ and
ψ respectively; the ghost and antighost by η and η; and the coupling constant
by g. We define the renormalisation constants through the quantum effective
action, which we write, before imposing Ward identities, as
Γ[A0, η0, η0] =
1
2
Tr
(
Z−13 (∂µA0ν − ∂νA0µ) (∂
µAν0 − ∂νA
µ
0) + λf
2[A0]
− 2gZ−11 (∂µA0ν − ∂νA0µ) [A
µ
0 , A
ν
0] + g
2Z−14 [A0µ, A0ν ][A
µ
0 , A
ν
0]
)
− Z−12 ψ0D/ψ0 + gZ
−1
1F ψ0A/0ψ0 − Z
−1
m mψ0ψ0
+ Z˜−13 ∂µη0∂
µη0 + gZ˜
−1
1 ∂µη
b
0A
µ,a
0 η
c
0f
abc. (3.38)
The gauge-fixing functional in this expression is f [A].
The Ward Identities for the divergent terms in the quantum effective
action are
Z1F
Z2
=
Z1
Z3
=
Z4
Z1
=
Z˜1
Z˜3
= constant. (3.39)
The value of the constant is theory-dependent. In QED, the constant is equal
to unity, so that only the photon field renormalisation renormalises the cou-
pling constant g. The combination gA is automatically renormalised and we
need only subtract the wavefunction renormalisation to render the quark-
antiquark-gluon vertex finite.
Thus, for the case of quantum electrodynamics (QED) and nonrelativistic
QED (NRQED), we can match relativistic lattice calculations with numerical
NRQED calculations in the continuum limit to determine matching coeffi-
cients without ultraviolet divergences causing a problem. In QCD, however,
the coupling constant is gauge dependent and the cancellation of ultraviolet
divergences is not guaranteed. The extra constraints provided by background
field gauge simplify such calculations in QCD.
47
Chapter 3. Lattice perturbation theory
3.4.2 Background field gauge
Background field gauge is a choice of gauge that preserves a form of gauge
invariance in such a way that the divergences in the unrenormalised quantum
effective action are tied together in a gauge invariant manner. This ensures
that the constant in the Ward identity is equal to unity, reducing the com-
plexity of many calculations, such as the computation of the quark-antiquark-
gluon vertex renormalisation factor. In a general gauge, three independent
calculations must be carried out to separate Z1F from Z1, Z2 and Z3, as is
done, for example, in [48]. On the other hand, in background field gauge, only
one renormalisation constant is required, the wavefunction renormalisation,
Z2, as demonstrated in [76].
For the particular case of NRQCD, we require background field gauge to
constrain the number of operators that appear in the quantum effective ac-
tion. In lattice perturbation theory, background field gauge greatly simplifies
calculations such as the perturbative improvement of the chromo-magnetic
operator in the NRQCD action [14], which I discuss in greater detail in Sec-
tion 4.5.
In background field gauge we only require the renormalisation factor Z3
to calculate the coupling constant renormalisation for the chromo-magnetic
operator. This ensures that all the relevant diagrams are finite, since we
calculate only one-particle irreducible diagrams. The continuum QCD pro-
cess can therefore be calculated in any scheme, which allows us to perform
the continuum calculation numerically on fine lattices. If the diagrams were
not finite, then the matching calculation would introduce logarithmic terms
log(µa) that involve the continuum regularisation scale, µ, and the lattice
spacing, a.
More generally, the Zinn-Justin equation guarantees that only gauge in-
variant operators appear in the effective action for any operator of engineering
dimension four or less. NRQCD is an expansion in irrelevant operators with
dimension greater than four, so without background field gauge we cannot
guarantee that we can expand the NRQCD action on a gauge invariant oper-
ator basis. In matching calculations, such as the computation of the radiative
48
Chapter 3. Lattice perturbation theory
corrections to the chromo-magnetic operator in the NRQCD action, we would
need to include gauge non-invariant operators with scale-dependent logarith-
mic coefficients. This would seriously complicate the matching procedure.
We introduce background field gauge by decomposing the gauge fields
into a smooth external source field, B, and a quantum field, q,
Aµ(x) = Bµ(x) + gqµ. (3.40)
We calculate the quantum effective action Γ[B,ψ, η, η] by expanding the
original action, S[B + q, ψ + ψ′, η + η′, η + η′] up to quadratic terms in the
fields and integrating over the primed quantum fields.
We are free to choose the gauge-fixing function as we wish, so we take
fa = Dµqaµ, (3.41)
where D is the background field covariant derivative,
Dµq
a
ν = ∂µq
a
ν + f
abcBbµq
c
ν . (3.42)
We can view this gauge as parameterised by the background field, B. The
gauge-fixing term, fafa, is invariant under the formal transformation
δBaµ = ∂µ²
a − fabc²bBcµ
δqaµ = − f
abc²bqc, (3.43)
where ² parameterises the transformation. This formal transformation is not
a true gauge transformation, which would not act on the classical field B.
However the original action is only a functional of the sum of fields, (B+ q),
which does transform as a true gauge transformation
δ(Baµ + q
a
µ) = ∂µ²
a − fabc²b(Bcµ + q
c
µ). (3.44)
The sum of the matter fields, (ψ + ψ′), also transforms according to a true
gauge transformation, so the original action is invariant under our formal
49
Chapter 3. Lattice perturbation theory
gauge transformation.
We assume the path integral measure is invariant under the formal gauge
transformation and since this formal transformation is linear in the fields, the
quantum effective action is also invariant. Thus we can expand the unrenor-
malised quantum effective action on a basis of gauge invariant operators.
This is the key to the success of background field gauge. We demand gauge
invariance in the quantum effective action before renormalisation and this
ties the renormalisation constants together in such a way that the constant
in the Ward identities expressed in Equation (3.39) is just unity. Background
field gauge is the only gauge that guarantees all higher order terms are gauge
invariant, which strongly restricts the number of terms that need to be in-
cluded in the action. Such higher order terms are necessarily generated in an
effective theory such as NRQCD.
The lattice formulation of an SU(3) gauge action in background field
gauge is much the same as the continuum case and was first presented in
two companion papers, [77] and [78]. In the lattice formulation of lattice
background field gauge, we split the link variable into the external field,
Bµ(x), and the quantum field, qµ(x) and define the link variable as
Uµ(x) = e
g0qµ(x+µˆ/2)eBµ(x+µˆ/2). (3.45)
We then build the gauge action from plaquettes as usual.
On the lattice, we replace the background field gauge covariant derivatives
with lattice operators that act on (Lie algebra valued) lattice functions, f(x)
as
Dµf(x) =
1
a
{
eBµ(x)f(x+ aµˆ)e−Bµ(x) − f(x)
}
D?µf(x) =
1
a
{
f(x)− e−Bµ(x)f(x− aµˆ)eBµ(x)
}
. (3.46)
The full lattice QCD action in background field gauge is the sum of the
usual Wilson gauge action, defined in terms of background field gauge pla-
50
Chapter 3. Lattice perturbation theory
quettes
SBFW [U ] =
1
g20
∑
x
∑
P
ReTr {1− Pµν(x)} , (3.47)
a gauge-fixing term,
SBFGF[B, q] = −λ0a
4
∑
x
Tr
{
D?µqµ(x)D
?
νqν(x)
}
(3.48)
and the ghost contribution,
SBFFP [B, q, c, c] = −2a
4
∑
x
Tr
{
Dµc(x)
[
J (g0qµ(x))
−1Dµ + g0Ad[qµ](x)
]
c(x)
}
.
(3.49)
In the above expressions I define the adjoint representation of a vector
field X, which takes values in the Lie algebra su(N), as
Ad[X] ·Y = [X, Y ], (3.50)
for all Y in su(N). Equivalently, in terms of the structure constants, the
matrix (Ad[X])ab associated with a given basis T a is
(Ad[X])ab = −fabcXc (3.51)
or
(Ad[X] ·Y )a = fabcXbY c. (3.52)
The functional J(X) is the exponential of the differential mapping, which
we define through the linear mapping J(X) : su(N) 7→ su(N), with
J(X) ·Y = e−X
d
dt
eX+tY
∣
∣
∣
∣
t=0
(3.53)
for all Y in su(N), and is given explicitly by
J(X) = 1 +
∞∑
k=1
(−1)k
(k + 1)!
Ad[X]k. (3.54)
51
Chapter 3. Lattice perturbation theory
Thus the complete lattice QCD action in background field gauge is
SBF[B, q, c, c] = SBFW [U ] + S
BF
GF[B, q] + S
BF
FP [B, q, c, c]. (3.55)
3.4.3 Background field gauge in HIPPY and HPSRC
We have extended the HIPPY and HPSRC code to include background field
gauge [79]. This was necessary to calculate the perturbative improvement
of the chromo-magnetic operator in the NRQCD action, which I discuss in
greater detail in Section 4.5.
To implement background field gauge in HIPPY routines, the identification
and ordering of the quantum and background fields is important. The routine
vert_ggg(k,a) that I introduce in Section 3.3.1 now includes an additional
boolean argument that specifies whether gluons are background or quantum
gluons. We use the convention that the value .TRUE. represents background
fields. Thus we specify the interaction of two background fields with a quan-
tum field by calling the three-gluon vertex with the additional argument
bfg=(/.TRUE.,.TRUE.,.FALSE./), that is, we would call the function as
vert_ggg(k,a,bfg).
The HPSRC routines now handle the symmetrisation of the vertices sepa-
rately for both background and quantum fields. The Feynman rules for back-
ground field gauge are given in the Appendix to [78] and I independently
rederive them in Appendix B as part of the development of our implemen-
tation of background field gauge. The vertices with background fields have
gauge-fixing terms that we have hardwired in the appropriate HPSRC vertex
functions. These additions were extensively checked in the calculation of the
radiative corrections to the chromo-magnetic operator in the NRQCD action.
I also hardwired the ghost and antighost vertices in background field
gauge in an HPSRC module as part of our implementation of background field
gauge. I present the Feynman rules for the ghost and antighost vertices in
Appendix B. I extended the ghost routines to include an additional boolean
argument, analogous to that of the vert_ggg(k,a,bfg) function. In this
case, however, the argument bfg is a scalar. This argument is used to deter-
52
Chapter 3. Lattice perturbation theory
mine whether to implement background field gauge via a simple IF construct
within the routine. An array is not required for the ghost and antighost ver-
tices, because calculations planned for the foreseeable future will need vertices
with either background fields or quantum fields but not both simultaneously.
No vertices involving ghosts, quantum fields and background fields occur at
one-loop.
We plan to reproduce the results given in the collection of papers by
Lu¨scher and Weisz [77, 78, 80, 81] for the relation between the QCD scale
parameter in theMS scheme, ΛMS, and on the lattice, ΛL. The QCD scale pa-
rameter characterises the energy scale of the strong coupling constant. Whilst
the energy dependence of the coupling constant can be determined from the
renormalisation group equations, the exact value of the coupling constant at
a given energy must be obtained from experimental measurements. The cur-
rent world average is αMS(M
0
Z) = 0.1184(7), which corresponds to a four-loop
scale parameter of ΛMS = 213 MeV for five flavours of sea quarks [82].
We define the beta function for a running coupling constant, α(µ2) =
g2(µ)/(4pi2), that depends on the scale µ, as
β(g(µ)) = µ
∂g(µ)
∂µ
(3.56)
We can write the beta function perturbatively as
β(g(µ)) = −g3(µ)
∞∑
n=0
bng
2n(µ). (3.57)
Here the first two coefficients for N colours and nf fermion flavours are
b0 =
1
(4pi)2
(
11N
3
−
4nfTf
3
)
, (3.58)
b1 =
1
(4pi)4
(
34
3
N2 −
20
3
NTf − 4CfTf
)
, (3.59)
and the Casimir operators are Cf = (N2−1)/(2N) and Tf = 1/2 for SU(N).
53
Chapter 3. Lattice perturbation theory
We then define the QCD scale parameter, Λ, by
Λ = lim
µ→∞
µ(b0g
2(µ))−b1/2b
2
0 exp
(
−
1
2b0g2(µ)
)
. (3.60)
The scale introduced by renormalisation, µ, and the coupling constant, g,
are both scheme dependent and hence the scale parameter is also scheme
dependent, but renormalisation group invariant. For example, ΛMOM/ΛMS =
5.73, where ΛMS is the scale parameter in the minimal subtraction scheme and
ΛMOM is the scale parameter defined in the momentum subtraction scheme
[83]. On the lattice, the scale parameter also depends on the choice of gauge
and fermion actions.
We may relate the scale parameters, Λ˜ and Λ, associated with two cou-
pling constants, g and g˜, via
Λ˜ = Λ exp
(
a1
2b0
)
, (3.61)
where the coefficient a1 is the first coefficient in the expansion of g˜ in terms
of g:
g˜2 = g2 + a1g
4 + . . . (3.62)
We can extract the coefficient a1, and hence the ratio between the scale
parameters in different regularisation schemes, from the gluon two-point func-
tion in background field gauge. In general the two-point functions are related
by the gluon and ghost wavefunction renormalisations but in background
field gauge the product of the renormalisation constants is unity and the
two-point functions are equal, up to O(a) corrections.
This computation will extend my work on the Ward identities of the gluon
selfenergy that I review in Appendix C and will serve as a cross-check of our
implementation of background field gauge for the ghosts. This calculation is
currently underway. The routines developed for this calculation can then be
easily extended to include the effects of sea quarks on the ratio of the QCD
scales.
I have now reviewed the machinery employed in lattice perturbation the-
54
Chapter 3. Lattice perturbation theory
ory calculations: automation, twisted boundary conditions and background
field gauge. In the next three chapters I apply these techniques to several dif-
ferent calculations at both one- and two-loops. I first calculate the two-loop
tadpole improvement factors, then the perturbative improvement of higher
order kinetic coefficients the NRQCD action in Chapter 4. I calculate the
heavy quark renormalisation parameters at one- and two-loops in Chapter 5
and use my two-loop results to extract a new prediction of the b quark mass
from lattice NRQCD simulations. In Chapter 6 I compute the matching co-
efficients between heavy-light vector and axial-vector currents in continuum
QCD and lattice NRQCD.
55
Chapter 4
Perturbative improvement
In the previous chapter I reviewed the three main reasons for developing
and implementing automated lattice perturbation theory: perturbative im-
provement, perturbative renormalisation and operator matching. With the
precision of lattice simulations now reaching the level of a few percent, the
role of lattice perturbation theory in constructing highly-improved actions,
and implementing them effectively in calculations, is increasingly important.
In this Chapter I apply the techniques of lattice perturbation theory to
three calculations that fall under the aegis of this improvement programme. I
use fully automated lattice perturbation theory to compute tadpole improve-
ment factors to two-loops. I then employ a mixed analytic and numerical
approach for the radiative corrections to higher order kinetic terms in simple
NRQCD actions. Finally I review the calculation of the radiative corrections
to the chromo-magnetic operator in the highly-improved NRQCD action.
4.1 Tadpole improvement
I first present my calculation of the two-loop fermionic contributions to the
tadpole improvement factor, u0, for both Landau and plaquette tadpoles.
I combine my results with quenched perturbative data from [12] and from
quenched high-β simulations undertaken by Lee [11] to extract the first deter-
mination of unquenched two-loop tadpole improvement factors for Symanzik-
improved gluons with ASQTad light quarks.
4.2 Landau tadpoles
In Section 2.3.2 I introduce two commonly-used definitions for the tadpole
improvement factor, the Landau and plaquette tadpoles. We define the Lan-
dau tadpole, uL0 , to be the expectation value of the link operator in Landau
56
Chapter 4. Perturbative improvement
Figure 4.1: Contributions to the Landau and plaquette tadpole parameter
at one-loop. Green wiggly lines indicate gluons and the large green blob
represents the definition dependent part of the diagram.
gauge:
uL0 =
1
N
ReTr〈Uµ〉Landau. (4.1)
We choose Landau gauge because it is the axis-symmetric gauge that max-
imises the value of u0 and therefore minimises tadpole contributions [27].
At one-loop there is a single diagram, given in Figure 4.1, that contributes
to the Landau tadpole. I calculated the corresponding Feynman integral us-
ing HPSRC and evaluated the integral numerically with VEGAS. The inversion
routines in HPSRC do not cope well with singular gauges, so for this calculation
I used hardwired gluon propagators. However, I generated the quark propa-
gators and quark-gluon vertices required for the two-loop contributions using
HIPPY. I discuss the combined HIPPY and HPSRC implementation of automated
lattice perturbation theory in more detail in Section 3.3.1.
I show the two-loop fermionic contributions in Figure 4.2. With the code
for the one-loop calculation in hand, the extension to two-loops is straightfor-
ward: I replace the bare gluon propagator by the “dressed” gluon propagator,
which includes the one-loop fermionic contributions. In all other respects the
calculation is the same. I calculate the one-loop gluon propagator using the
HPSRC module gluon_sigma, based on code written by Alistair Hart. I ex-
tended the module to include computation of derivatives of the selfenergy
using the TaylUR derived-type [73], implemented numerical evaluation with
VEGAS and compatibility with background field gauge. I made extensive use
of these additions to the gluon_sigma module in the calculations set out in
Chapter 5.
The gluon_sigma module evaluates the gluon selfenergy for a given set
of external gluon momenta, Lorentz indices and colours. A namelist input
57
Chapter 4. Perturbative improvement
(b)(a)
Figure 4.2: Contributions to the Landau and plaquette tadpole parameter at
two-loops: (a) is the fermion “bubble” and (b) the fermion “tadpole”. Green
wiggly lines indicate gluons, red lines are light quarks and the large green
blob represents the definition dependent part of the diagram.
selects which contributions, for example the fermion “tadpole” or “bubble”
diagrams (see Figure 4.2 for notation), are included in the calculation. The
full two-loop calculation is an eight-dimensional integral over the double loop
momenta.
I use zero gluon mass in all calculations that I describe in this section and
regulate the low momenta behaviour with the external gluon momentum, aq.
I confirmed that all results were independent of the gluon momentum for a
wide range of values between aq = 0.01 and aq = 0.9.
Throughout this chapter I use the convention for both Landau and pla-
quette tadpoles that
u0 = 1− u
(2)
0 αL − u
(4)
0 α
2
L −O(α
3
L). (4.2)
4.2.1 Landau tadpole results
I present my results for the one- and two-loop Landau tadpole parameters
in Table 4.1 for both unimproved Wilson gluons and for Symanzik-improved
gluons. I obtained the one-loop results with 105 function evaluations and 10
VEGAS iterations, which took about five minutes wall-clock time on six pro-
cessors. In this dissertation, whenever I refer to the time elapsed for a given
calculation this should be understood to refer to wall-clock time and not CPU
time. I carried out these calculations on the Cambridge High Performance
58
Chapter 4. Perturbative improvement
-0.074-0.0735-0.073-0.0725-0.072-0.0715
-0.071
0 0.02 0.04 0.06 0.08 0.1u0 am
Figure 4.3: The Light quark mass-dependence of the two-loop Landau tad-
poles. I plot the fit to a function quadratic in the light quark mass in blue.
I use Symanzik-improved gluons with ASQTad light quarks. Note the small
scale on the y-axis.
Computing Cluster, which consists of 2094 3.0 GHz Intel Woodcrest cores.
I calculated the two-loop contributions on 128 processors, with 107 func-
tion evaluations and 15 VEGAS iterations, which took approximately 2.5 hours
for the unimproved fermions and five hours for the ASQTad fermions. I
present results for six different light quark masses, from amq = 0.005 to
amq = 0.10 and extrapolate these results to zero light quark mass using a
function quadratic in the light quark mass. Including quartic terms did not
alter the result at zero quark mass within the quoted errors. I plot my data
and the quadratic fit for the light quark mass extrapolation, shown by the
blue line, in Figure 4.3.
I quote uncertainties for each result that are the statistical errors from
numerical integration of the Feynman diagrams, except in the massless light
quark two-loop result. In this case, I give the uncertainty from the fit to the
light quark mass.
I present a large number of tables of data in this dissertation and, unless
otherwise stated, the uncertainties I quote for my results are the statistical
errors from numerical integration.
59
Chapter 4. Perturbative improvement
Action u(2),L0 amq
u(4),L0
Unimproved ASQTad
Wilson 0.9739(2)
0.005 -0.0703(3) -0.1238(3)
0.01 -0.0704(3) -0.1238(3)
0.03 -0.0701(3) -0.1238(4)
0.05 -0.0698(3) -0.1241(6)
0.08 -0.0692(3) -0.1230(4)
0.10 -0.0688(3) -0.1226(4)
0.20 -0.0651(3) -0.1195(3)
0.0 -0.070287(56) -0.1238(1)
Symanzik 0.7503(2)
0.005 -0.0461(3) -0.0726(3)
0.01 -0.0460(3) -0.0727(3)
0.03 -0.0458(3) -0.0725(4)
0.05 -0.0456(3) -0.0727(5)
0.08 -0.0450(3) -0.0720(3)
0.10 -0.0447(3) -0.0716(3)
0.20 -0.0422(3) -0.0694(3)
0.0 -0.0460(1) -0.0727(1)
Table 4.1: Landau tadpole corrections at one- and two-loops for Wilson and
Symanzik-improved gluons. The two-loop results are the fermionic contri-
butions only, with unimproved staggered light quark results presented in
column four and ASQTad quarks in column five. The two-loop result at zero
light quark mass result is an extrapolation; one-loop results are fermion mass
independent. The uncertainties quoted are statistical errors from numerical
integration, except in the massless light quark two-loop result. In this case,
the uncertainty is from the fit to the light quark mass.
4.2.2 Comparison to the literature
We can compare the one-loop fermionic results given in Table 4.1 with pre-
viously published results for the Landau tadpole. In [12], Nobes et al. report
a value of
uL0 |nf=0 = 1− 0.9738(2)αL − 3.33(1)α
2
L (4.3)
60
Chapter 4. Perturbative improvement
for the quenched Landau tadpole for Wilson gluons and
uL0 |nf=0 = 1− 0.7501(1)αL − 2.06α
2
L (4.4)
for Symanzik-improved gluons. My one-loop results are in perfect agreement
with the one-loop values given in [12]. In common with the quenched results,
my two-loop coefficients are smaller for the Symanzik-improved action than
for Wilson gluons. This is what we would expect: the improved action is
designed to be more continuum-like, with smaller lattice artifacts, and hence
requires less tadpole improvement.
I combine the fermionic contributions given in Table 4.1 with the two-loop
quenched results in Equations (4.3) and (4.4) to present the first perturbative
calculation of the unquenched Landau tadpole at two-loops. I find
uL0 = 1− 0.7503(2)αL − (2.06(1)− 0.0733(4)nf )α
2
L (4.5)
for Symanzik-improved gluons with ASQTad sea quarks of mass amq = 0.01,
which corresponds, for example, to the light quark mass on the MILC “fine”
ensembles [84].
4.2.3 Three-loop estimate
I estimate the three-loop contribution to the Landau tadpole by subtracting
the full two-loop perturbative results in Equation (4.5) from the nonper-
turbative Landau tadpoles measured on the MILC collaboration’s ASQTad
ensembles. Data for ensembles with (2+1) flavours of sea quark are pre-
sented in [84] and [85] . For the “fine” ensemble, the Landau tadpole was
measured to be uL0 = 0.8461(1). This ensemble has light quark bare masses
amu,d = 0.0124, strange quark mass ams = 0.031 and bare coupling β = 7.09.
I use the relation between the bare lattice coupling and the β values listed
in Table 1 of [85],
αL =
5
2piβ
, (4.6)
to compute a value of the coupling constant. In this case I obtain αL = 0.1122.
61
Chapter 4. Perturbative improvement
Using this value of the coupling constant, I obtain
u(6),L0 α
3
L ' u
L
0 −
(
1− u(2),L0 αL + u
(4),L
0 α
2
L
)
= −0.057(1). (4.7)
My estimate of the three-loop perturbative contribution to the Landau tad-
pole for Symanzik-improved gluons and ASQTad light quarks is therefore
u(6),L0 ' −40.4(7). (4.8)
In [86], Hart et al. found that measurements of the Landau tadpole from
quenched high-β Monte Carlo simulations were well described by the two-
loop Landau tadpole calculated in perturbation theory. By fitting the high-β
results to a polynomial in the coupling constant, they inferred the size of the
three-loop contribution to the Landau tadpole. Direct comparison with my
results is not possible because the action used in [86] is anisotropic, but the
O(α3L) contributions to the Landau tadpole were found to be u
(6),L
0 ∼ O(1).
This is an order of magnitude smaller than my estimate, but, since α3L '
0.001, both results support the claim in [86] that the two-loop perturbative
calculation of the Landau tadpole provides a very good estimate of the full
nonperturbative Landau tadpole.
4.3 Plaquette tadpoles
The plaquette tadpole, uP0 , is defined to be the fourth root of the expectation
value of the plaquette:
uP0 =
〈
1
N
TrPµν
〉1/4
, (4.9)
where the plaquette Pµν is the one-by-one gauge invariant loop defined in
Equation (2.8).
My calculation of the plaquette tadpole closely parallelled the method I
used for the Landau tadpole factor. At one-loop there is again a single con-
tributing diagram, which I give in Figure 4.1. I calculated the corresponding
Feynman integral using the HPSRC routines and evaluated the integral numer-
62
Chapter 4. Perturbative improvement
ically using VEGAS. I show the two fermionic contributions to the two-loop
factor in Figure 4.2.
4.3.1 Plaquette tadpole results
I present my results for the one- and two-loop plaquette tadpole parameters
in Table 4.2 for both Wilson and Symanzik-improved gluons. I extrapolate
the results to zero light quark mass. I obtained results with 105 function
evaluations and 10 VEGAS iterations on six processors, which took about 40
seconds for the one-loop contributions. For the two-loop contributions I used
106 function evaluations and 15 VEGAS iterations on 128 processors, taking
approximately 40 minutes. I verified that both one- and two-loop results
were gauge-independent for three different choices of gauge parameter. I plot
my data and the quadratic fit for the light quark mass extrapolation, shown
by the blue line, in Figure 4.4.
-0.0698-0.0697-0.0696-0.0695-0.0694-0.0693
-0.0692
0 0.02 0.04 0.06 0.08 0.1u0 am
Figure 4.4: The light quark mass-dependence of the two-loop plaquette tad-
pole. I plot my fit to a function quadratic in the light quark mass in blue.
I use Symanzik-improved gluons with ASQTad light quarks. Note the small
scale on the y-axis. The smaller errors for the results at amq = 0.01 and
amq = 0.03 are due to better statistics.
In Table 4.3 I present results for the quenched plaquette tadpole from
63
Chapter 4. Perturbative improvement
Action u(2),P0 amq
u(4),P0
Unimproved ASQTad
Wilson 1.04717(2)
0.005 -0.06451(4) -0.13583(5)
0.01 -0.06450(4) -0.13576(1)
0.03 -0.06443(4) -0.13570(1)
0.05 -0.06427(4) -0.13563(4)
0.08 -0.06389(4) -0.13532(4)
0.10 -0.06355(4) -0.13502(4)
0.0 -0.0645121(21) -0.135769(69)
Symanzik 0.76708(2)
0.005 -0.03674(2) -0.06976(3)
0.01 -0.03674(2) -0.06971(1)
0.03 -0.03667(2) -0.06967(1)
0.05 -0.03659(3) -0.06964(3)
0.08 -0.03637(3) -0.06946(3)
0.10 -0.03617(3) -0.06930(3)
0.0 -0.0367294(55) -0.0697147(72)
Table 4.2: Plaquette tadpole corrections at one- and two-loops for Wilson
and Symanzik-improved gluons. The two-loop results are the fermionic con-
tributions only, with unimproved staggered light quark results presented in
column four and ASQTad quarks in column five. The two-loop result at zero
light quark mass result is an extrapolation; one-loop results are fermion mass
independent. The uncertainties quoted are statistical errors from numerical
integration, except in the massless light quark two-loop result. In this case,
the uncertainty is from the fit to the light quark mass. The smaller errors
for the ASQTad results at amq = 0.01 and amq = 0.03 are due to better
statistics.
Lee’s high-β simulations [11]. I extract the two-loop correction by fitting
these results to a polynomial in αL, including terms up to α4L, with the one-
loop coefficient constrained to the value obtained using perturbation theory.
I plot the quenched results in Figure 4.5 and the fit to a function quadratic in
the light quark mass. Including higher order terms in the fit did not improve
the extrapolation to zero light quark mass. For the two-loop coefficient I
64
Chapter 4. Perturbative improvement
obtain
u(4),P0 |nf=0 = 2.12(6). (4.10)
The two-loop coefficient is, coincidentally, in agreement with the Landau
tadpole coefficient. Whilst we would expect the one-loop terms to be close,
because both methods attempt to smooth the gauge field, there is no partic-
ular reason to expect the two-loop values to be similar.
I incorporate these results with my fermionic two-loop values to present
the first perturbative calculation of the unquenched plaquette tadpole at two-
loops. For Symanzik-improved gluons with ASQTad sea quarks with mass
amq = 0.01, I find
uP0 = 1− 0.76708(2)αL − (2.12(6)− 0.06971(1)nf )α
2
L. (4.11)
β αL uP0 |nf=0 β αL u
P
0 |nf=0
9 0.08841941 0.90818796 32 0.02486796 0.97970155
10 0.07957757 0.92121955 38 0.02094144 0.98306826
11 0.07234316 0.93072830 46 0.01729945 0.98615808
12 0.06631456 0.93807324 54 0.01473657 0.98827436
15 0.05305165 0.95283073 62 0.01283508 0.98984548
16 0.04973592 0.95627124 70 0.01136821 0.99103890
20 0.03978874 0.96610563 80 0.00994718 0.99217890
24 0.03315728 0.97230232 92 0.00864973 0.99322698
27 0.02947314 0.97561108 120 0.00663146 0.99482774
Table 4.3: Plaquette tadpole contributions extracted from quenched high-β
simulations by Lee [11]. The bare coupling constant is given by αL = 5/(2β).
4.3.2 Three-loop estimate
I estimate the three-loop contribution to the plaquette tadpole by subtracting
the full two-loop perturbative results in Equation (4.11) from the nonpertur-
bative plaquette tadpoles measured on the MILC collaboration’s ASQTad en-
sembles. Results for ensembles with (2+1) flavours of sea quark are presented
65
Chapter 4. Perturbative improvement
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
1
0 0.02 0.04 0.06 0.08 0.1u0 Bare coupling
Figure 4.5: Plot of the fit to a function quadratic in the light quark mass for
the quenched plaquette tadpole parameter.
in [22]. For the “coarse” ensemble, the plaquette tadpole was measured to be
uP0 = 0.8677(1). This ensemble has light quark bare masses amu,d = 0.010,
strange quark mass ams = 0.050 and bare coupling β = 6.760.
I use the relation between the bare lattice coupling and the plaquette
tadpole
αL = −1.303615 log u
P
0 , (4.12)
to compute a value of the coupling constant from the measured values of uP0
listed in Table 1 of [22]. This relation arises from the one-loop definition of the
coupling constant in terms of the one-by-one Wilson loop, i.e. the plaquette.
I obtain αL = 0.1850. Using this value for the strong coupling constant, I
obtain
u(6),Pα3L ' u
P
0 −
(
1− u(2),P0 αL − u
(4),P
0 α
2
L
)
= 0.080(1). (4.13)
Thus my estimate of the three-loop perturbative contribution to the plaquette
tadpole for Symanzik-improved gluons and ASQTad light quarks is
u(6),P ' 12.6(2). (4.14)
66
Chapter 4. Perturbative improvement
For the fermionic contribution, I use my value of u(4),P0 at light quark mass
amq = 0.01. I estimate the error from the dominant quenched contribution
at two-loops, multiplied by α2L. This result is the same order of magnitude
as my estimate of the three-loop Landau tadpole, Equation (4.8).
Equations (4.5) and (4.11) are the first determination of the unquenched
Landau and plaquette tadpoles at two-loops for Symanzik-improved gluons
with ASQTad light quarks. These new results allow me to estimate the three-
loop contributions to both tadpoles. State-of-the-art lattice simulations by
HPQCD now use HISQ light quarks, for which the two-loop tadpoles must
be calculated. Although the tadpole calculation is complicated by the reuni-
tarisation of the fattened links in the HISQ action, my code can be relatively
easily extended to include HISQ sea quarks. This extension is currently un-
derway.
In the next section, I report on my second perturbative improvement
calculation, for the higher order kinetic terms in simple NRQCD.
4.4 NRQCD kinetic operator improvement
The higher order kinetic terms in the NRQCD action correct errors arising
from the nonrelativistic expansion of the heavy quark fields. At tree-level,
the coefficients of the kinetic terms are equal to unity. Quantum fluctuations
renormalise the coefficients and these radiative corrections must be calculated
to achieve high precision in lattice NRQCD simulations.
Nonperturbative lattice studies of heavy-heavy mesons using NRQCD
have found large energy splittings between states with spatial momenta of
equal magnitude but different individual components, for example, states
with momenta p = (0, 0, 3) and p = (2, 2, 1) [47]. One contribution to these
energy splittings is the renormalisation of cubic symmetry breaking terms
in the NRQCD action, which first occur at O(a4p4). Based on the work
of [71], [87] and [88], Mu¨ller computed the renormalisation parameters of
the O(a4p4) kinetic energy terms for the full NRQCD action in Equation
(2.66) using automated lattice perturbation theory [47]. My calculation of
the corresponding renormalisation parameters for a simple NRQCD action
67
Chapter 4. Perturbative improvement
served as a cross-check of Mu¨ller’s numerical code.
In the course of my calculation I discovered that the radiative corrections
are infrared divergent unless the action is improved to include higher order ki-
netic corrections. Whilst the justification for this result is clear in retrospect,
we did not anticipate the divergent behaviour initially. Before presenting my
results, I derive expressions for the radiative corrections to the kinetic oper-
ators of a simple NRQCD discretisation. My derivation largely follows that
given in [47] and [87], tailored to my choice of NRQCD action.
4.4.1 Simple NRQCD
I use the simple discretisation of NRQCD given in [49], the “Davies and
Thacker” action,
SDT = a
3
∑
x
ψ†(x, t)
(
∆(+)4 −
3∑
j=1
∆(+)j ∆
(−)
j
2aM
)
ψ(x, t). (4.15)
In momentum space the tree-level quark propagator for this action is
G−10 (w,p) = e
−aw − 1 +
4
2aM
∑
j
sin2
(apj
2
)
, (4.16)
where w = −ip4 is the energy in Minkowski space. I derive the Feynman
rules associated with this action in Appendix B. I expand G−10 (w,p) for
small momenta as
G−10 (w,p) ≈ −aw +
4
2aM
∑
j
((apj
2
)2
−
1
3
(apj
2
)4
)
+ . . . (4.17)
The position of the tree-level quark pole, awT (p), is the zero of G−10 (w,p).
In this case the pole is given by
awT (p) =
a2p2
2aM
−
a4p4
24aM
+O(a6p6). (4.18)
The one-loop propagator is related to the tree-level propagator via the
68
Chapter 4. Perturbative improvement
selfenergy Σ(w,p):
G−11 (w,p) = G
−1
0 (w,p)− αsΣ(w,p), (4.19)
and so the dispersion relation at one-loop is
w(p) = wT (p)− αsΣ(wT (p),p). (4.20)
The NRQCD quark selfenergy is invariant under interchange of any two
spatial momentum components api ↔ apj, under spatial momentum reflec-
tions apj → −apj and transforms into its complex conjugate under ap4 →
−ap?4 [87]. Hence the small-momentum expansion of the selfenergy is
aΣ(w,p) = Σ0(w) + Σ1(w)
a2p2
2aM
+ Σ2(w)
a4(p2)2
8a2M2
+ Σ3(w)a
4p4 + . . . (4.21)
We can compute the expansion coefficients, Σi(w), which are functions of the
energy, w, by taking appropriate combinations of partial derivatives of the
selfenergy:
Σ0(w) = aΣ(p = 0), (4.22)
Σ1(w) = aM
∂2(aΣ)
∂(apz)2
∣
∣
∣
∣
∣
p=0
, (4.23)
Σ2(w) = a
2M2
∂4(aΣ)
∂(apy)2∂(apz)2
∣
∣
∣
∣
∣
p=0
, (4.24)
Σ3(w) =
1
24
(
∂4(aΣ)
∂(apz)4
− 3
∂4(aΣ)
∂(apy)2∂(apz)2
)
p=0
. (4.25)
I expand the Σi(w) functions as a Taylor series in w,
Σi =
∞∑
j=0
Σ(j)i w
j, (4.26)
and combine this expansion with the expression for the selfenergy in Equation
(4.21) and for the dispersion relations in Equations (4.18) and (4.20) to obtain
69
Chapter 4. Perturbative improvement
the one-loop dispersion relation:
aw(p) =
a2p2
2aM
(
1− αs
(
Σ(1)0 + Σ
(0)
1
))
−
a4p4
24aM
(
1− αsΣ
(1)
0
)
− αs
(
Σ(0)0 +
(
2Σ(2)0 + 2Σ
(1)
1 + Σ
(0)
2
) a4(p2)2
8a2M2
Σ(0)3 a
4p4
)
. (4.27)
I extract the renormalisation parameters by requiring that the one-loop
dispersion relation has the renormalised form
aw(p) =
a2p2
2aMr
−
a4p4
24aMr
− αs (E0 + aδw(p)) . (4.28)
We can immediately read off the mass renormalisation from the coefficient of
the O(a2p2) term by comparing Equations (4.27) and (4.28). Thus the mass
renormalisation is given by
Z−1M = 1− αs
(
Σ(1)0 + Σ
(0)
1
)
. (4.29)
We obtain the heavy quark energy shift from the one-loop coefficient:
E0 = Σ
(0)
0 . (4.30)
I discuss the mass renormalisations in more detail in Chapter 5, where I
also derive an expression for the NRQCD wavefunction renormalisation. For
now we need only the energy shift and the kinetic correction coefficients. The
kinetic corrections are:
aδw(p) = W1
a4(p2)2
8a2M2
+W2a
4p4, (4.31)
W1 = 2
(
Σ(2)0 + Σ
(1)
1
)
+ Σ(0)2 , (4.32)
W2 = Σ
(0)
3 −
Σ(0)1
24aM
. (4.33)
For consistency with, and comparison to, the literature (see, for example,
[47], [71] and [89]), I also introduce the Ωi parameterisation functions, which
70
Chapter 4. Perturbative improvement
are related to the Σ(j)i (w) functions by
Ω0 = ReΣ
(0)
0 = Σ(0), (4.34)
Ω1 = − ReΣ
(1)
0 = Im
∂Σ
∂p4
∣
∣
∣
∣
∣
p=0
, (4.35)
Ω2 = ReΣ
(0)
1 = aMRe
∂2Σ
∂p2z
∣
∣
∣
∣
∣
p=0
. (4.36)
I use the relative simplicity of the Davies and Thacker action to compute
the correction coefficientsWi using a mix of analytic and numerical methods.
I now outline these calculations.
4.4.2 Calculating the kinetic correction coefficients
In continuum QCD only one Feynman diagram contributes to the selfenergy
at one-loop, whereas in lattice NRQCD there are contributions from four di-
agrams. On top of the lattice artifact tadpole diagrams, the non-relativistic
formulation distinguishes space and time and this distinction introduces two
different Feynman rules, one for temporal gluons and one for spatial gluons.
The Feynman rules associated with the Davies and Thacker action, Equation
(4.15), are given in [49] and I have checked these Feynman rules by hand.
I have also computed the Feynman rules for an alternative discretisation of
simple NRQCD. My derivation of both sets of Feynman rules are presented
in Appendix B. I show the four diagrams that must be evaluated in Figure
4.6. Each diagram represents a four-dimensional Euclidean integral that, in
principle, can be evaluated numerically. However, the integrands are strongly
peaked around poles in the complex k0 plane. Numerical Monte Carlo inte-
gration methods, such as VEGAS, generally do not cope well with strongly
peaked integrands. To speed up the calculation, I evaluated the k0-integral
analytically and then computed the less strongly peaked three-dimensional
k-integral using VEGAS.
I illustrate the basic steps of the calculation using the example of the
temporal gluon “rainbow” diagram, illustrated in the top left corner of Figure
71
Chapter 4. Perturbative improvement
(d)(c)
(a) (b)
Figure 4.6: The four selfenergy diagrams contributing to the renormalisation
of the NRQCD kinetic operators. Double blue lines are heavy quarks, wiggly
green lines indicate spatial gluons and green dotted lines are temporal gluons.
The commonly used names, “rainbow” for the upper pair of diagrams and
“tadpole” for the lower, arise from the shape of the gluon propagators.
4.6, diagram (a). The full expression for this rainbow diagram with arbitrary
external quark spatial momenta p (all vertex momenta defined as incoming)
is
Σrbow = a3g2C2
∫
d4(ak)
a3(2pi)4
1
k̂2 + a2λ2
G0(we
−iap0 ,k− p), (4.37)
where C2 = 4/3 is the colour factor and I have introduced the conventional
lattice shorthand
k̂µ = 2 sin
(
akµ
2
)
. (4.38)
The heavy quark propagator for the Davies and Thacker action, G0, is given
in Equation (4.16).
I transform the k0 integral with the standard change of coordinates ak0 →
z = eiak0 , so that the integral becomes
Σrbow = −g2C2
i
2pi
∫
d3(ak)
(2pi)3
∮
dz
z
1
(z − z−)(z − z+)(z − zQ)
, (4.39)
and the contour of integration is now the unit circle. The gluon poles lie at
z± =
κ±
√
κ2 − 4
2
. (4.40)
72
Chapter 4. Perturbative improvement
where
κ = kˆ2 + a2λ2 + 2, (4.41)
and the heavy quark pole occurs at
zQ = 1−
1
2aM
3∑
j=1
̂(k − p)
2
j . (4.42)
I evaluate this integral by closing the contour at infinity: the condition
that zf(z) → 0 as z → ∞ ensures that the contour at infinity gives no
contribution. In this case we have
[(z−z−)(z−z+)(z−zQ)]
−1 = z−3[(1−z−z
−1)(1−z+z
−1)(1−zQz
−1)]−1, (4.43)
which indeed tends to zero in the z → ∞ limit. I close the contour to the
right to isolate the pole at z = z+. The result is just
Σrbow = −g2C2
∫
d3(ak)
(2pi)3
1
(z+ − z−)(z+ − zQ)
. (4.44)
I extracted the temporal rainbow contribution to W1 and W2 by differen-
tiating this expression according to Equations (4.32) or (4.33) respectively. I
evaluated the result at p = 0 and computed the remaining three-dimensional
integral numerically with VEGAS. I present all results in Section 4.4.3. In par-
ticular I demonstrate that the O(a4p4) renormalisation parameters for an
NRQCD action correct only to O(a2p2) are infrared divergent.
4.4.3 Numerical results
In Table 4.4 I show results for the energy shift and the renormalisation param-
eters for the higher order kinetic corrections in simple NRQCD. I give results
for the Davies and Thacker action, in Equation (4.15), and a minimally-
improved action given by
S˜DT = a
3
∑
x
ψ†(x, t)
(
∆(+)4 −
∆˜(2)
2aM
)
ψ(x, t). (4.45)
73
Chapter 4. Perturbative improvement
This action is the Davies and Thacker action with the unimproved Laplacian
replaced by the improved Laplacian,
∆˜(2) = ∆(2) −
a2
12
∑
j
[
∆(+)j ∆
(−)
j
]2
. (4.46)
I outline the derivation of the renormalisation parameters for the im-
proved action in Appendix D. I used a gluon mass of a2λ2 = 10−6 and fixed
the gauge to Feynman gauge. Extrapolation to zero gluon mass is not re-
quired [47]. In Table 4.4 I present my results for the heavy quark energy
shift, E0, and the correction coefficients W1 and W2 for five different heavy
quark masses between aMQ = 4.00 and aMQ = 5.40. I do not present re-
sults for masses smaller than aM = 4.0 as this would require introducing the
stability parameter that I discuss in Section 2.4.1.
For W2 I give only the finite contribution for the Davies and Thacker ac-
tion. For the improved action,W2 is infrared finite. I obtained all results with
104 function evaluations and 20 VEGAS iterations, which took approximately
10 minutes on a single processor for E0 and 40 minutes for W1 and W2.
NRQCD action aMQ E0 W1 W2
SDT
5.40 -2.277(2) 0.0546(1) -0.00122(1)
5.00 -2.332(2) 0.0551(1) -0.00118(1)
4.70 -2.381(2) 0.0555(1) -0.00116(1)
4.00 -2.498(2) 0.0562(1) -0.00110(1)
S˜DT
5.40 -2.145(2) 0.0398(1) 0.00153(1)
5.00 -2.190(2) 0.0391(1) 0.00161(1)
4.70 -2.230(2) 0.0385(1) 0.00167(1)
4.00 -2.320(2) 0.0372(1) 0.00182(1)
Table 4.4: Higher-order kinetic improvement factors for two different simple
NRQCD actions. I give results for the finite part of the Davies and Thacker
action in the second column and the full results for the improved NRQCD
action of Equation (4.45) in the third column. Errors for the uncertainties
associated with the finite part of W2 for the Davies and Thacker action are
from the fit to the gluon mass.
74
Chapter 4. Perturbative improvement
I extracted the infrared divergent contribution to W2 using four different
heavy quark masses between aM = 4.00 and aM = 5.40. I obtained
W div2 = 0.00875(10)
1
aM
log(a2λ2), (4.47)
in good agreement with expected analytic result
1
36piaM
log(a2λ2) ' 0.008842
1
aM
log(a2λ2). (4.48)
I explain the origin of this infrared divergence in the next section.
4.4.4 Infrared divergences
The infrared divergent behaviour exhibited in the results of Section 4.4.3
is not unexpected: the renormalisation parameter for the O(a4p4) term in
the selfenergy gains a infrared divergent contribution from the wavefunction
renormalisation, which is removed by including O(a4p4) improvement in the
action. We can understand the occurrence of this divergence in general terms
by examining the form of the selfenergy. At tree-level we can express the
selfenergy, at small external momenta, as
ΣT ∼ iap0 +
a2p2
2aM
+
a4p4
2aM
+O(a6p6). (4.49)
The O(a4p4) term arises from the Taylor expansion of the kinetic term
sin(ap). The quark self-interaction induces O(a4p4) contributions to the self-
energy at one-loop and each term is multiplicatively renormalised. Therefore
we can write the one-loop selfenergy as
Σ(1) ∼ Zψ
[
ip0 + ZM
(
a2p2
2aM
+
a4p4
2aM
)
+ Zp4a
4p4
]
+O(a6p6)
= (1 + αsδZψ)
[
ip0 + (1 + αsδZM)
(
a2p2
2aM
+
a4p4
2aM
)
+ αsδZp4a
4p4
]
.
(4.50)
75
Chapter 4. Perturbative improvement
We can read off the O(a4p4) renormalisation coefficient from this expression
for Σ(1). The one-loop contribution is
αs (δZψ + δZp4) . (4.51)
The wavefunction renormalisation is infrared divergent in Feynman gauge,
so this expression is manifestly infrared divergent. In fact, we can obtain the
infrared divergence analytically. The long wavelength behaviour of NRQCD
must match that of continuum QCD by construction, so the infrared di-
vergence in the wavefunction renormalisation is just that of the continuum
wavefunction renormalisation for massive quarks, which I discuss in more
detail in Chapter 6. The logarithmic infrared divergence is given by
ZIRQ = −
2αs
3pi
log
(
λ2
M2
)
. (4.52)
To obtain the analytic result forW div2 in Equation (4.48), we must also include
the factor of 1/24aM from the derivative definition ofW2 in Equation (4.33).
My results for the infrared divergence in Equation (4.47), obtained using
the mixed analytic and numerical approach described in this chapter, are
in full agreement with my results presented for the NRQCD wavefunction
renormalisation in the next chapter. These results were obtained using a
fully automated process and their agreement with both each other and the
analytic result provides confidence in both methods.
We now consider the selfenergy in the case of the simple action, but with
an improved derivative. Here the tree-level selfenergy takes the form
Σ˜T ∼ iap0 +
a2p2
2aM
+O(a6p6). (4.53)
Notice that the inclusion of an improved derivative removes the O(a4p4)
contributions from Σ˜T . Thus Σ˜(1) is
Σ˜(1) ∼ (1 + αsδZψ)
[
ip0 + (1 + αsδZM)
a2p2
2aM
+ αsδZp4a
4p4
]
(4.54)
76
Chapter 4. Perturbative improvement
and we can again easily identify the one-loop renormalisation of the O(a4p4)
contribution. In this case the renormalisation of the p4-term is simply given
by
αsδZp4 . (4.55)
We see this parameter is no longer infrared divergent or, at least, not mani-
festly so.
4.4.5 Comparison to the literature
In Table 4.5 I compare my results to those obtained by Mu¨ller using a fully
automated procedure. I compare the individual diagram contributions to
the energy shift, E0. I used a heavy quark mass of aM = 4.0 and a gluon
mass of a2λ2 = 10−4. All calculations were performed in Feynman gauge.
All the results are in good agreement with each other. As a further cross-
check, Mu¨ller reproduced the results for the correction coefficient in [87]. The
mutual agreement between my results and those of [47] and between [47] and
[87] support our confidence in the validity of both calculations.
Rainbow Tadpole
Temporal Spatial Temporal Spatial
E0 -3.1194(15) 0.2965(1) 1.2984(7) -0.9735(3)
Ref. [47] -3.1218(15) 0.2964(2) 1.2986(6) -0.9740(4)
Table 4.5: A comparison of results for the heavy quark energy shift E0 for the
Davies and Thacker NRQCD action. I give my results in the first row and
Mu¨ller’s results in the second row [47]. All results obtained with aM = 4.0
and gluon mass a2λ2 = 10−4.
4.4.6 Observations
Simulations of heavy-heavy mesons using lattice NRQCD have found large
energy splittings between mesons with equal spatial momentum, p, but with
different
∑3
i=1 p
4
i , for example states with p = (0, 0, 3) and p = (2, 2, 1)
77
Chapter 4. Perturbative improvement
[47]. One contribution to these energy splittings is the radiative corrections
to the leading cubic symmetry breaking term, which is c5∆(4)/(24aM). At
tree-level c(0)5 = 1. The coefficient c
(1)
5 cancels the radiative corrections at one-
loop and is related to W2 by c
(1)
5 = 24aMW2. The energy splittings can be
removed by tuning c5 nonperturbatively to a value of c5 = 2.6. Unfortunately
this large renormalised value is inconsistent with the relatively small values
obtained using automated lattice perturbation theory in [47] and [90], which
give a one-loop value of c(1)5 ' 0.4 for a range of heavy quark masses between
aM = 2.0 and aM = 3.4. My computation of the correction coefficients,
and extraction of the correct - but unpredicted - infrared divergence, for
the simple Davies and Thacker action confirms that the automated lattice
perturbation theory was implemented correctly. Therefore we must conclude
that the one-loop radiative corrections to c5 are not the sole source of the
mesonic energy splittings.
There are a number of other sources that could contribute to the dis-
crepancy between the correction coefficient calculated using nonperturbative
tuning and using automated lattice perturbation theory. The first possibility
is a large two-loop correction to the perturbative result. A correct choice of
scale should reduce second order contributions, but may come at the expense
of large higher order contributions. Two-loop perturbative results would be
required to confirm this hypothesis, but unfortunately the complexity of two-
loop lattice perturbation theory is formidable.
A second possibility is that O(a6p6) corrections strongly influence the
correction coefficient c5. Mu¨ller found that including the tree-level O(a6p6)
improvement term ∆(6)/(180aM) did indeed significantly reduce the renor-
malisation of c5, by a factor of 5–10. This is of particular concern in light of
the fact that I found the O(a4p4) renormalisation parameters were not only
non-negligible but infrared divergent for an action that wasn’t fully improved
to O(a4p4). Obtaining higher order kinetic corrections may prove necessary
in the future and in Appendix D I briefly discuss extending this work to
O(a6p6) to investigate the feasibility of further radiative improvement. How-
ever, this may not prove necessary. In the nonperturbative simulation, the
O(a6p6) improvement term was included with the tree-level value of the co-
78
Chapter 4. Perturbative improvement
efficient. The fitting process, whereby the O(a6p6) was fixed to its classical
value and the coefficient of the O(a4p4) term nonperturbatively tuned, may
not be optimal and indeed could be another source of the difference between
the perturbative and nonperturbative results for c(1)5 .
One final source of error may be the relatively large lattice spacing of
the lattice used in the nonperturbative simulations. NRQCD is not the only
constituent of the lattice simulation and other contributions to the energy
splittings might be reduced on finer lattices.
Further work is clearly needed to clarify this discrepancy. Nonpertur-
bative simulations on finer lattices are being explored and these may shed
light on the issue. Extending the perturbative renormalisation calculation to
O(αsa6p6) and O(α2sa
4p4) could also illuminate the problem more brightly.
However, the computational complexity of this extension is impressive and
not to be underestimated. Implementing the sixth-derivatives numerically is
highly nontrivial and the two-loop perturbative lattice calculation would re-
quire the evaluation of around 30 diagrams. Whilst calculation is possible in
principle, it is not yet feasible in practice.
In the final section of this chapter I review the calculation of the radia-
tive correction to the chromo-magnetic operator in highly-improved NRQCD.
This calculation was undertaken by Hammant [14], and represents the first
outcome of our implementation of background field gauge in the HIPPY and
HPSRC routines.
4.5 NRQCD chromo-magnetic operator improvement
The calculation of the radiative corrections to the chromo-magnetic oper-
ator in the highly-improved NRQCD action is an important component of
the improvement programme for lattice NRQCD simulations. The chromo-
magnetic contribution and four-fermion spin-spin interaction both contribute
to the hyperfine structure of heavy quarkonium states in NRQCD.
Gray et al. determined the Υ − ηb hyperfine splitting using NRQCD va-
lence quarks, with Symanzik-improved gluons and ASQTad sea quarks in [85].
This calculation did not include radiative corrections to the chromo-magnetic
79
Chapter 4. Perturbative improvement
operator in the NRQCD action and obtained results that were dependent on
the lattice spacing and in disagreement with the world experimental average
of 69.3(2.8) MeV [8].
Updated nonperturbative simulation results using a radiatively-corrected
NRQCD action have not yet been completed, but early indications suggest
that including the chromo-magnetic corrections and four-fermion contact
terms in the NRQCD action remove all lattice spacing dependence in the
Υ− ηb hyperfine splitting and bring the result into line with the experimen-
tal average [14].
As I discuss in Section 3.4.2 background field gauge greatly simplifies the
matching calculation for the chromo-magnetic operator. In continuum QCD,
the renormalised effective action includes the fermion terms
ΓQCD[ΨR,ΨR, A] = ΨRD/ΨR + bσΨR
σµνFµν
2mR
Ψ+ . . . , (4.56)
where σµν is the anticommutator [γµ, γν ]/2, Fµν is the field strength tensor
and mR is the renormalised mass. The coefficient bσ is given by
bσ = δZσZ2ZM , (4.57)
where δZσ is the renormalisation of the second term on the right-hand side of
Equation (4.56). We can expand bσ as a power series in the strong coupling
constant. The leading correction to this coefficient occurs atO(αS) and comes
from δZσ.
To match this term to the relevant contribution in lattice NRQCD, we
first perform an FWT transformation to obtain
(1 + bσ)ψ
†
R
σ ·B
2mR
ψR, (4.58)
where ψR is a two-component Pauli spinor. There are two diagrams that
contribute to this term in continuum QCD (see [14]) and the result is
bσ =
(
3
2pi
log
µ
m
+
13
6pi
)
αS, (4.59)
80
Chapter 4. Perturbative improvement
where µ is an infrared cutoff scale.
In lattice NRQCD, the quantum effective action contains the spin-dependent
term
ΓNRQCD[ψR, ψR, A] = c4Z
NRQCD
σ Z
NRQCD
2 Z
NRQCD
m ψ
†
R
σ ·B
2mR
ψR. (4.60)
We require the anomalous chromo-magnetic moment to be equal in both
QCD and NRQCD, which gives us, by comparing Equations (4.58) and (4.60),
the matching condition
(1 + bσ) = c4Z
NRQCD
σ Z
NRQCD
2 Z
NRQCD
m . (4.61)
The tree-level and one-loop coefficients are therefore
c(0)4 =1, (4.62)
c(1)4 =b
(1)
σ − δZ
NRQCD,(1)
σ − δZ
NRQCD,(1)
2 − δZ
NRQCD,(1)
m . (4.63)
There are four extra diagrams that contribute to the lattice NRQCD
calculation. Both NRQCD and QCD contributions to c(1)4 are logarithmically
divergent and the overall divergent contribution is
c(1),div4 = −
3α
2pi
log(aM). (4.64)
Including tadpole corrections, the final expression for the coefficient c(1)4 is
c(1)4 =
13
6pi
− δZ˜NRQCD,(1)σ − δZ˜
NRQCD,(1)
2 − δZ˜
NRQCD,(1)
m
− δZu0,(1)M − δZ
u0,(1)
σ + c
(1),div
4 , (4.65)
where the tildes indicate that the infrared divergence has been subtracted off.
I denote the tadpole correction factors with the superscript u0· The tadpole
correction factors are
δZu0,(1)M = −
(
2
3
+
3
a2M2
)
u(2)0 (4.66)
81
Chapter 4. Perturbative improvement
and
δZu0,(1)σ =
(
13
3
+
13
4aM
−
3
8n(a2M2
−
3
4a3M3
)
u(2)0 . (4.67)
In [14] we use Landau tadpoles, with the value of u(2)0 given in Table 4.1 for
Symanzik-improved gluons.
The results for each contribution to c(1)4 in Equation (4.65) are collected
in Table 1 of [14], with data for three different heavy quark masses. We also
provide an estimate of the effect of the radiative correction to the chromo-
magnetic operator and four fermion interactions on the Υ − ηb hyperfine
splitting. Although these estimates are no replacement for full nonperturba-
tive simulations, the results are encouraging and indicate that using a fully-
improved action will bring lattice NRQCD results for the hyperfine splitting
into line with the current experimental world average.
82
Chapter 5
Perturbative renormalisation
The precise theoretical and experimental determination of quark masses is
an important component of high-precision tests of the Standard Model. One
current focus for tests of the Standard Model is the unitarity of the CKM
matrix, which describes flavour-changing quark transitions. Quark masses
serve as an input into the tests of CKM matrix unitarity; for example, the
mass of the b quark is used in the extraction of the CKM matrix element
|Vub| from inclusive semileptonic decays of B mesons [91].
Recent high-precision calculations of the b quark mass using realistic
lattice QCD simulations [25] and perturbative QCD combined with experi-
mental results [92] are in good agreement, obtaining values of mb(mb, nf =
5) = 4.165(23) GeV and mb(mb) = 4.163(16) GeV respectively.
The lattice result was obtained using relativistic HISQ valence quarks
with ASQTad sea quarks. However, most current lattice studies of b quarks
use an effective field theory, such as NRQCD. Simulating both valence and sea
quarks with relativistic actions allows a much greater precision, but is only
now becoming possible with the advent of finer lattices and highly improved
actions. Lattices are not yet fine enough to directly simulate at the physical
b quark mass: even on the very finest lattices with HISQ heavy quarks an
extrapolation to the heavy quark mass is required [25]. Precise results from
effective theories for heavy quarks therefore still serve an important role in
precision tests of the Standard Model.
In this chapter I discuss the derivation and calculation of the heavy quark
renormalisation parameters in NRQCD. I calculate the wavefunction renor-
malisation, energy shift and mass renormalisation at one-loop for several
different NRQCD actions. These results are a useful cross-check of my code.
I then calculate the energy shift for highly-improved NRQCD at two-loops.
I use the two-loop results to extract a precise prediction of the mass of the
b quark from lattice NRQCD simulations. This result improves on a previ-
83
Chapter 5. Perturbative renormalisation
ous determination of mb(mb) = 4.4(3) GeV from unquenched lattice QCD
simulations using NRQCD valence b quarks [85]. The dominant error in
that calculation arose from the use of one-loop perturbation theory in the
matching between lattice quantities and the continuum result. I calculate
the fermionic contribution to the two-loop energy shift and mass renormali-
sation and discuss the quenched contributions calculated by Lee using weak
coupling (high-β) simulations [11]. This mixed strategy, incorporating high-β
quenched simulations and automated lattice perturbation theory, is the first
ever two-loop calculation in NRQCD.
5.1 NRQCD renormalisation parameters
I first derive an expression for the NRQCD wavefunction renormalisation. I
derive the mass renormalisation and energy shift in Section 4.4.1 and so do
not repeat the analysis here. An equivalent derivation of the wavefunction
renormalisation for mNRQCD is given in [47] and [88]. In this section I set
the lattice spacing to unity, that is a = 1.
5.1.1 Wavefunction renormalisation
I calculate the wavefunction renormalisation by comparing the tree-level and
one-loop quark propagators. At tree-level the heavy quark propagator is given
by Equation 4.16, which I write, for small momenta, as
G−10 (w,p) = wT − w, (5.1)
where w = −ip4 is the energy in Minkowski space. The tree-level onshell value
of w is wT . Following the derivation outlined in Chapter 4, I include one-
loop quantum corrections to the tree-level propagator via the heavy quark
selfenergy, αsΣ(w,p). I show the one-loop contributions in Figure 4.6.
I write the one-loop propagator as
G−11 (w,p) = G
−1
0 (w,p)− αsΣ(w,p), (5.2)
84
Chapter 5. Perturbative renormalisation
and define the wavefunction renormalisation, Zψ, such that the renormalised
quark propagator has the same form as the tree-level propagator, multiplied
by the wavefunction renormalisation. The new onshell value of w is w1, the
root of G(w,p) = 0. Thus G(w,p) at one-loop is
G−11 (w,p) = Z
−1
ψ (w1 − w), (5.3)
where the onshell value w1 is obtained by setting G(w1,p) = 0.
I expand the selfenergy around the new onshell value w1,
Σ(w,p) = Σ(w1,p) + (w1 − w)
∂Σ
∂w
∣
∣
∣
w=w1
+ · · · , (5.4)
and substitute this expression into G(w,p) to give
G−11 (w,p) = (w1 − w)
(
1− αs
[
Σ(w1,p) +
∂Σ
∂w
∣
∣
∣
w=w1
])
+ · · · . (5.5)
We compare Equations (5.3) and (5.5) to read-off the wavefunction renor-
malisation and use the relation Σ(w1,p) = Σ(wT ) +O(αs) to obtain
Zψ = 1 + αs
[
Σ(wT ,p) +
∂Σ
∂w
∣
∣
∣
w=wT
]
= 1 + αs (Ω0 + Ω1) . (5.6)
The wavefunction renormalisation is infrared divergent in Feynman gauge,
so I extract the gluon mass dependence of the wavefunction renormalisation
using an infrared subtraction function, Zsubψ :
Zsubψ =
4
3
∫
d4k
(2pi)4
M2
(k̂2 + iMk0)2(k̂2 + λ2)
. (5.7)
Here M is the heavy quark mass and λ2 the infrared cutoff scale.
I fit Zsubψ to a logarithmic function of the gluon mass squared to obtain
Zsubψ = −0.05259(10) + 0.10614(19) log λ
2, (5.8)
using VEGAS with 5×107 function evaluations and 20 iterations for 12 different
85
Chapter 5. Perturbative renormalisation
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01Subtraction function Gluon mass^2
Figure 5.1: Logarithmic fit of the gluon mass dependence for the wavefunction
renormalisation subtraction function. I plot the fit in blue and the data from
VEGAS as red crosses.
values of the gluon mass between λ2 =10−3 and λ2 =10−8. I plot the results
in Figure 5.1, with the fit shown by the blue line. We expect the long-distance
behaviour of the NRQCD wavefunction renormalisation to match that of the
massive wavefunction renormalisation in continuum QCD. The logarithmic
divergence of the subtraction function is in good agreement with the expected
behaviour, which is given by
ZIRψ = −
2
3pi
log λ2 ' −0.212207 log λ. (5.9)
5.1.2 Zero point energy shift and mass renormalisation
I derive expressions for the heavy quark energy shift and the mass renormal-
isation in Section 4.4.1, by examining the dispersion relation for the heavy
quark. I do not repeat the analysis here, but rather collect the resulting
expressions for convenience. The mass renormalisation is
Z−1M = 1− αs
(
Σ(1)0 + Σ
(0)
1
)
= 1− αs (Ω2 − Ω1) , (5.10)
86
Chapter 5. Perturbative renormalisation
and the heavy quark energy shift is
E0 = αsΣ
(0)
0 = αsΩ0. (5.11)
I compute the heavy quark energy shift, wavefunction renormalisation
and mass renormalisation using a fully automated procedure. I wrote new
HPSRC routines to evaluate the one-loop heavy quark selfenergy. The rou-
tines specify the external quark momentum and action type and select which
diagrams and parameters to evaluate using a namelist, which is read at run-
time. I implemented the TaylUR derived-type for the vertices and propagators
in order to extract the necessary derivatives of the diagrams. I generated all
the necessary vertex function files using HIPPY, which I discuss in greater
detail in Section 3.3.1.
5.1.3 One-loop results
I present numerical results for the one-loop energy shift and mass renor-
malisation for the “onlyH0” action in Table 5.1. This action corresponds to
the NRQCD action of Equation (2.66) without improvement, that is, with
δH = 0. I derive the Feynman rules for this action in Appendix B. I use
unimproved Wilson gluons and evaluate the results with 106 VEGAS iterations
and 10 function evaluations, which took approximately 55 minutes on 12
processors.
87
Chapter 5. Perturbative renormalisation
aMQ E0 ZM
5.00 -2.6233(35) 0.6439(16)
4.70 -2.6619(39) 0.6484(18)
4.00 -2.7535(39) 0.6594(18)
2.80 -3.0316(37) 0.8996(19)
2.50 -3.1414(38) 0.9291(19)
2.35 -3.2112(39) 0.9433(19)
1.95 -3.4376(46) 0.9933(20)
1.88 -3.4867(41) 1.0010(20)
1.80 -3.5444(39) 1.0146(20)
1.50 -3.8305(41) 1.0619(20)
Table 5.1: Heavy quark energy shift and mass renormalisation for the “on-
lyH0” action. Results for masses aM ≥ 4.0 use stability parameter n = 1,
whilst all other results use n = 2.
I give results for highly-improved NRQCD with Symanzik-improved glu-
ons in Table 5.2. I use 12 VEGAS iterations and 5×104 function evaluations
(15×104 for ZM), which required about 20 minutes on 12 processors.
I use a gluon mass of a2λ2 = 10−6 as an infrared regulator for the in-
frared finite mass renormalisation and energy shift. Extrapolation to zero
gluon mass is not required, as my results exhibit no dependence on the gluon
mass for a2λ2 . 10−4. I show only the infrared finite contributions to the
wavefunction renormalisation in Table 5.1.
5.1.4 Tadpole improvement
The one-loop analytic results for the tadpole corrections to the heavy quark
energy shift, Eu00 , are given in [47] and [89]:
Eu0,(1)0 = −
[
1 +
6 + c5
2aM
−
3
2
( c1
a3M3
+
c6
2na2M2
)]
u(2)0 ,
Zu0,(1)M = −
[
1−
3
2naM
−
c5
3
+ 3
( c1
a2M2
+
c6
2naM
)]
u(2)0 . (5.12)
88
Chapter 5. Perturbative renormalisation
aMQ n E0 ZM Zfiniteψ
5.40 1 -2.128(3) 0.551(3) -0.015(2)
5.00 1 -2.167(3) 0.585(3) -0.019(2)
4.00 1 -2.224(3) 0.719(3) -0.145(2)
4.00 2 -2.222(3) 0.730(3) -0.144(2)
2.80 2 -2.366(3) 1.030(3) -0.333(3)
2.50 2 -2.424(3) 1.150(3) -0.400(3)
2.50 4 -2.421(3) 1.163(3) -0.394(4)
1.95 2 -2.562(3) 1.522(4) -0.605(4)
1.95 4 -2.550(3) 1.524(4) -0.599(4)
1.88 2 -2.580(3) 1.584(4) -0.636(4)
1.88 4 -2.574(3) 1.592(4) -0.629(4)
1.60 4 -2.670(3) 1.920(4) -0.800(4)
Table 5.2: NRQCD renormalisation parameters: E0, ZM and the infrared
finite contribution to the wavefunction renormalisation, Zψ. I show results
for highly-improved NRQCD action with Symanzik-improved gluons.
The tadpole improvement factor vanishes for the wavefunction renormalisa-
tion.
I show tadpole improved results for highly-improved NRQCD in Table
5.3.
89
Chapter 5. Perturbative renormalisation
aMQ n E0 ZM
5.40 1 -0.918(3) -0.026(3)
5.00 1 -0.904(3) -0.032(3)
4.00 1 -0.870(3) 0.078(3)
4.00 2 -0.850(3) 0.089(3)
2.80 2 -0.765(3) 0.243(3)
2.50 2 -0.741(3) 0.290(3)
2.50 4 -0.715(3) 0.303(3)
1.95 2 -0.691(3) 0.430(4)
1.95 4 -0.642(3) 0.432(4)
1.88 2 -0.682(3) 0.447(4)
1.88 4 -0.637(3) 0.455(4)
1.60 4 -0.608(3) 0.541(4)
Table 5.3: Tadpole corrected NRQCD renormalisation parameters, E0, ZM .
The tadpole correction vanishes for the wavefunction renormalisation. I show
results for highly-improved NRQCD action with Symanzik-improved gluons
and include tadpole improvement corrections.
5.1.5 Comparison to the literature
In Table 5.4 I show results for the heavy quark energy shift, mass renormal-
isation and infrared finite contribution to the wavefunction renormalisation,
taken from [47]. The data in Table 5.4 use the “onlyH0” and highly-improved
NRQCD actions. We can see that my results, in Tables 5.1 and 5.2 are in
excellent agreement with the results of [47].
Action E0 ZM Zfiniteψ
“onlyH0” -2.9851(24) -0.1232(34) 1.1348(38)
highly-improved -2.36685(40) -0.3364(74) 1.0183(14)
Table 5.4: NRQCD renormalisation parameters from [47] for comparison. I
show only the infrared finite contribution to the wavefunction renormalisa-
tion, Zfiniteψ . All results use aM = 2.8 and n = 2.
90
Chapter 5. Perturbative renormalisation
Mass n E0 ZM Zfiniteψ
5.40 1 0.917(1) -0.026(4) -0.016(3)
4.00 2 0.850(1) 0.082(4) -0.142(3)
2.80 2 0.767(1) 0.235(4) -0.338(3)
2.10 4 0.666(1) 0.387(4) -0.539(3)
1.95 2 0.689(1) 0.421(4) -0.611(3)
1.95 4 0.646(1) 0.428(4) -0.603(3)
1.60 4 0.609(1) 0.543(4) -0.797(3)
1.20 6 0.609(1) 0.732(4) -1.184(3)
1.00 6 0.758(1) 0.859(4) -1.545(3)
Table 5.5: NRQCD renormalisation parameters from [89] for comparison. I
show only the infrared finite contribution to the wavefunction renormalisa-
tion, Zfiniteψ . All results use the fully improved NRQCD action with Symanzik-
improved gluons and include tadpole corrections to E0 and ZM .
In Table 5.5 I show results for the heavy quark energy shift, mass renor-
malisation and infrared finite contribution to the wavefunction renormalisa-
tion from [89]. These results use highly-improved NRQCD with Symanzik-
improved gluons and include tadpole corrections. We can see that my results,
in Tables 5.1 and 5.2, are also in excellent agreement with the results of [89].
I use the wavefunction renormalisation results as part of my matching
calculation of the heavy-light current in Chapter 6.
5.2 Renormalisation parameters at two-loops
The one-loop results presented above are a preliminary cross-check of the
routines I wrote to obtain the main result of this chapter: the heavy quark
energy shift at two-loops. Combined with my verification of the Ward iden-
tity for the fermionic contributions to the gluon selfenergy, which I discuss
in Appendix C, these two tests provide a stringent check of my two-loop rou-
tines. This computation is, to the best of my knowledge, the first two-loop
calculation undertaken in NRQCD.
91
Chapter 5. Perturbative renormalisation
(a) (b)
(c) (d)
Figure 5.2: Fermionic contributions to E0 and ZM , calculated using auto-
mated lattice perturbation theory. Blue lines are heavy quarks, green are
gluons and red are sea quarks. I give numerical results for each of these
diagrams in Tables 5.6 to 5.8. I denote each contribution using column head-
ings in those tables as follows: diagram (a) is the Rainbow-Bubble column;
diagram (b) the Rainbow-Tadpole; diagram (c) the Tadpole-Bubble; and di-
agram (d) the Tadpole-Tadpole.
5.2.1 The fermionic contributions
I show the two-loop fermionic contributions to the heavy quark selfenergy in
Figure 5.2. Applying the same method as the two-loop tadpole improvement
factor calculation in Chapter 4, I extended the one-loop routines to two-loops
using the HPsrc module gluon_sigma. I describe this module in more detail
in Section 4.1 and review the gluon selfenergy more fully in Appendix C.
I calculated the heavy quark energy shift, E0, at three different heavy
quark masses, aM = 2.50, aM = 1.88 and aM = 1.72. The heaviest mass
corresponds to the “coarse” lattices used by the HPQCD collaboration [93]
and the lighter two masses correspond to the “fine” lattices used by the
HPQCD collaboration in [85] and [93] respectively.
5.2.2 Energy shift results
I used three different sets of actions to calculate the energy shift, with either
five or nine different light quark masses, between amq = 0.01 and amq = 0.30,
92
Chapter 5. Perturbative renormalisation
for each set of actions. I extrapolated the results to zero light quark mass
using a function quadratic in the light quark mass. I plot the light quark mass
dependence and the extrapolation to zero mass for each set of actions listed
below in Figure 5.3. I plot results for a single heavy quark mass aM = 2.50
to serve as an illustration of the extrapolations. I use the ABCBA NRQCD
operator ordering for the results with ASQTad and HISQ light quarks. The
uncertainties are smaller than the size of the data points in Figure 5.3. I also
show the light quark mass fits for corresponding to each set of actions.
I used a gluon mass as the infrared regulator, which, in general, should
be avoided for two-loop calculations. However, in this case a gluon mass
regulator is appropriate because there are no ghost contributions. The sets
of actions I used are as follows:
• Davies and Thacker NRQCD action, Equation (4.15), with unimproved
Wilson gluons and unimproved staggered light quarks, Table 5.6. I used
107 function evaluations for the tadpole results in columns three and
four, which took approximately 1.5 hours on 128 processors, and 108
function evaluations for the rainbow results in columns five and six,
which took about eight hours on 128 processors. I describe my notation,
that is, my use of the terms “rainbow”, “tadpole” and “bubble”, in the
caption to Table 5.6.
• Highly-improved NRQCD, Equation (2.66), with Symanzik-improved
gluons and ASQTad light quarks. I used 107 function evaluations for
the tadpole diagram results, in columns two and three of Table 5.7
and three and four of Table 5.9. I used 25×108 function evaluations
for the rainbow diagrams in columns four and five of Table 5.7 and fix
and six of Table 5.9. The tadpole diagram contributions took about six
hours on 32 processors and the rainbow contributions approximately 23
hours on 64 processors. I give results for both the kernel ordering shown
in Equation (2.66), which I denote ABCBA, where A = (1 − δH/2),
B = (1−H0/2n)n and C = U †, in Table 5.7 and an alternative ordering
BACAB in Table 5.9. The ordering BACAB corresponds to the action
used by the HPQCD collaboration in their lattice NRQCD simulations
93
Chapter 5. Perturbative renormalisation
[93].
• Highly-improved NRQCD with Symanzik-improved gluons and HISQ
light quarks, Table 5.8. I used 108 function evaluations for the tadpole
diagram results, in columns two and three, and 25×108 function evalu-
ations for the rainbow diagrams in columns four and five. The tadpole
diagram contributions took around three hours on 128 processors and
the rainbow contributions about 18 hours on 256 processors. The action
kernel ordering was ABCBA.
94
Chapter 5. Perturbative renormalisation
aM amq Tadpole Rainbow Total
Tadpole Bubble Tadpole Bubble
2.50
0.30 0.4985(3) -0.40691(8) 11.934(3) -11.860(3) 0.166(4)
0.20 0.5127(3) -0.41332(8) 12.280(3) -12.194(3) 0.185(4)
0.15 0.5182(3) -0.4156(3) 12.408(3) -12.315(3) 0.196(4)
0.10 0.5221(3) -0.4167(3) 12.505(3) -12.406(3) 0.204(4)
0.05 0.5244(3) -0.4176(3) 12.560(3) -12.455(3) 0.212(4)
0.01 0.5251(3) -0.41728(8) 12.579(3) -12.468(4) 0.219(4)
0.0 0.5250(2) -0.4176(2) 12.575(3) -12.470(2) 0.2217(8)
1.88
0.30 0.5845(3) -0.4766(3) 11.866(2) -11.799(2) 0.176(3)
0.20 0.6012(3) -0.4841(3) 12.210(2) -12.131(3) 0.196(3)
0.15 0.6077(3) -0.4863(3) 12.337(2) -12.251(3) 0.207(3)
0.10 0.6123(3) -0.4878(3) 12.430(2) -12.373(3) 0.212(3)
0.05 0.6149(3) -0.4886(3) 12.488(2) -12.389(3) 0.225(4)
0.01 0.6158(3) -0.4886(3) 12.506(2) -12.402(3) 0.231(4)
0.0 0.6157(2) -0.4891(2) 12.502(3) -12.418(1) 0.233(1)
Table 5.6: Light quark rainbow and tadpole diagram results for the heavy
quark selfenergy at heavy quark mass aM = 1.88 and aM = 2.50 for six light
quark masses. Uncertainties for the individual diagrams in columns three to
six are statistical errors from numerical integration, whilst uncertainties for
the totals given in column seven are the errors from each contribution added
in quadrature. The results with amq = 0.0 are extrapolations to zero light
quark mass. All results use the “onlyH0” NRQCD action with unimproved
staggered light quarks. I give the diagrams in Figure 5.2. The main headings
“Tadpole” and “Rainbow” refer to the one-loop structure of the Feynman
diagrams, whilst the secondary headings “Tadpole” and “Bubble” indicate
the structure of the fermionic insertions in the dressed gluon propagator.
Thus the four contributions in Figure 5.2 correspond to the column head-
ings as follows: diagram (a) is the Rainbow-Bubble column; diagram (b)
the Rainbow-Tadpole; diagram (c) the Tadpole-Bubble; and diagram (d) the
Tadpole-Tadpole.
95
Chapter 5. Perturbative renormalisation
amq Tadpole Rainbow Total
Tadpole Bubble Tadpole Bubble
0.30 0.5528(3) -0.3900(1) 11.498(2) -11.426(2) 0.235(3)
0.25 0.5593(3) -0.3930(2) 11.630(2) -11.553(2) 0.243(3)
0.20 0.5649(3) -0.3953(2) 11.741(2) -11.658(2) 0.253(3)
0.15 0.5693(3) -0.3970(2) 11.828(2) -11.740(2) 0.260(3)
0.10 0.5726(3) -0.3980(3) 11.891(1) -11.797(2) 0.269(3)
0.08 0.5734(3) -0.3982(3) 11.911(2) -11.814(3) 0.272(3)
0.05 0.5744(3) -0.3984(2) 11.931(2) -11.830(3) 0.277(3)
0.03 0.5748(3) -0.3984(2) 11.939(2) -11.836(3) 0.279(3)
0.01 0.5750(3) -0.3984(2) 11.943(2) -11.838(3) 0.282(3)
0.0 0.5750(5) -0.3988(2) 11.9416(8) -11.8422(8) 0.285(1)
Table 5.7: Light quark rainbow and tadpole diagram results for the heavy
quark selfenergy at heavy quark mass aM = 2.50 for nine light quark masses.
All results use the full NRQCD action, but with operator ordering ABCBA,
with ASQTad light quarks. See the caption beneath Table 5.6 for full expla-
nation of the data.
amq Tadpole Rainbow Total
Tadpole Bubble Tadpole Bubble
0.30 0.3674(3) -0.3766(3) 11.377(3) -11.434(1) -0.066(3)
0.25 0.3715(3) -0.3798(3) 11.505(3) -11.557(1) -0.060(3)
0.20 0.3750(3) -0.3823(3) 11.609(3) -11.658(1) -0.056(3)
0.15 0.3777(3) -0.3841(3) 11.692(3) -11.736(1) -0.050(3)
0.10 0.3795(3) -0.3850(3) 11.751(3) -11.790(1) -0.046(3)
0.08 0.3802(3) -0.3855(3) 11.769(3) -11.809(1) -0.045(3)
0.05 0.3809(3) -0.3857(3) 11.783(3) -11.822(1) -0.044(3)
0.03 0.3811(3) -0.3857(3) 11.794(3) -11.826(1) -0.037(3)
0.01 0.3812(3) -0.3857(3) 11.799(3) -11.828(1) -0.034(3)
0.0 0.38119(4) -0.3860(1) 11.7961(9) -11.8346(9) -0.035(1)
Table 5.8: Light quark rainbow and tadpole diagram results for the heavy
quark selfenergy at heavy quark mass aM = 2.50 for nine light quark masses.
All results use the full NRQCD action with HISQ light quarks. See the cap-
tion beneath Table 5.6 for full explanation of the data.
96
Chapter 5. Perturbative renormalisation
aM amq Tadpole Rainbow Total
Tadpole Bubble Tadpole Bubble
2.50
0.30 0.5600(3) -0.3905(2) 11.493(2) -11.425(2) 0.237(3)
0.25 0.5626(3) -0.3935(2) 11.625(2) -11.552(2) 0.242(3)
0.20 0.5681(3) -0.3958(2) 11.735(2) -11.657(2) 0.250(3)
0.15 0.5726(3) -0.3974(2) 11.823(2) -11.738(2) 0.260(3)
0.10 0.5758(3) -0.3984(2) 11.886(2) -11.797(3) 0.266(4)
0.08 0.5767(3) -0.3986(2) 11.905(2) -11.813(3) 0.270(4)
0.05 0.5777(3) -0.3988(2) 11.925(2) -11.830(3) 0.274(4)
0.03 0.5782(3) -0.3988(2) 11.934(2) -11.835(2) 0.278(3)
0.01 0.5784(3) -0.3988(2) 11.938(2) -11.837(2) 0.281(3)
0.0 0.5779(6) -0.3991(1) 11.9363(8) -11.8411(8) 0.2823(6)
1.88
0.30 0.6477(4) -0.4549(3) 11.413(2) -11.356(2) 0.250(3)
0.25 0.6554(4) -0.4584(3) 11.543(2) -11.483(2) 0.257(3)
0.20 0.6618(4) -0.4610(3) 11.625(2) -11.558(2) 0.268(3)
0.15 0.6670(4) -0.4629(3) 11.740(2) -11.669(2) 0.275(3)
0.10 0.6708(4) -0.4640(3) 11.803(2) -11.727(3) 0.283(4)
0.08 0.6718(4) -0.4643(3) 11.822(2) -11.743(3) 0.287(4)
0.05 0.6730(4) -0.4645(3) 11.842(2) -11.760(3) 0.291(4)
0.03 0.6735(4) -0.4645(3) 11.850(2) -11.765(3) 0.294(4)
0.01 0.6738(4) -0.4645(3) 11.854(2) -11.767(3) 0.296(4)
0.0 0.6737(1) -0.4649(2) 11.852(4) -11.767(6) 0.2991(7)
1.72
0.30 0.6814(4) -0.4787(3) 11.382(2) -11.331(2) 0.254(3)
0.25 0.6895(4) -0.4823(3) 11.512(2) -11.457(2) 0.262(3)
0.20 0.6963(4) -0.4852(3) 11.622(2) -11.562(2) 0.271(3)
0.15 0.7017(4) -0.4871(3) 11.709(2) -11.644(3) 0.280(4)
0.10 0.7038(4) -0.4883(3) 11.772(2) -11.701(3) 0.288(4)
0.08 0.7056(4) -0.4886(3) 11.790(2) -11.717(3) 0.290(4)
0.05 0.7080(4) -0.4888(3) 11.810(2) -11.733(3) 0.296(4)
0.03 0.7088(4) -0.4888(3) 11.818(2) -11.739(3) 0.299(4)
0.01 0.7086(4) -0.4888(3) 11.822(2) -11.740(3) 0.302(4)
0.0 0.7082(3) -0.4892(2) 11.821(1) -11.745(1) 0.3041(3)
Table 5.9: Light quark rainbow and tadpole diagram results for the heavy
quark selfenergy at heavy quark mass aM = 1.88 and aM = 2.50 for nine
light quark masses. All results use the full NRQCD action, with operator
ordering BACAB, with ASQTad light quarks. See the caption beneath Table
5.6 for full explanation of the data.
97
Chapter 5. Perturbative renormalisation
-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3E0 Light quark mass UnimprovedASQTadHISQ
Figure 5.3: Plot of results for the light quark contributions to the heavy quark
energy shift. The solid lines are the light quark mass fits corresponding to
each set of results. I plot results for each set of actions, using the NRQCD
operator ordering for the ASQTad and HISQ light quark data. I use heavy
quark mass aM = 2.5 and gluon mass a2λ2 = 10−6. Errorbars are smaller
than the size of the data points. See the text in Section 5.2.2 for a full
explanation of the sets of actions used in the plot.
5.2.3 Quenched high-β simulations
Calculating higher order loop corrections grows ever more difficult with each
order, owing to the increasing number of diagrams and complicated vertex
structure. In [94], Dimm et al. suggested an alternative method for extracting
radiative corrections that avoids the difficulties associated with higher order
perturbation theory. By measuring the heavy quark two-point function in the
weak coupling regime and fitting the renormalisation parameters to a poly-
nomial in αs, the radiative contributions at each order can be obtained. This
technique has been successfully implemented for a range of calculations [47],
and in particular [86] and [90] demonstrate that the results obtained from
the automated perturbative procedure match those from high-β simulations.
To evaluate the quenched contributions shown in Figure 5.4 Lee per-
formed quenched simulations on L3×T lattices with temporal extent T = 3L
and spatial extents running from L = 3 to L = 10 [11]. He implemented
98
Chapter 5. Perturbative renormalisation
Figure 5.4: Quenched contributions to the heavy quark energy shift, extracted
from high-β simulation. Blue lines are heavy quarks, green are gluons and
crosses are counterterms. Large brown blobs represent the five gluon selfen-
ergy diagrams. Feynman diagrams based on [95].
twisted boundary conditions to reduce finite size effects and tunnelling be-
tween QCD vacua. He used Coulomb gauge with a range of coupling values,
from β = 10 up to β = 120. The number of independent configurations for
each (β, L) was about 300. The energy shift and mass renormalisation were
extracted using an exponential fit to the two point function parameterised
by
G(p) = Zψ exp
(
−
[
E0 +
a2p2
ZlattaM
+ . . .
]
t
)
, (5.13)
where the ellipsis indicates higher order terms included in the fits. The one-
loop coefficient was extracted by fitting to a free parameter to compare with
the perturbative result obtained on finite lattices by exact mode summation
[96]. Table 5.10 demonstrates that the simulation fits reliably reproduces
the exact one-loop perturbative results. To obtain the two-loop parameter,
the one-loop coefficient was then constrained to its exact value from lattice
perturbation theory.
Lee ran quenched simulations with and without tadpole improvement.
Tadpole improvement significantly reduced the magnitude of the coefficients,
from∼ 2.3 to∼ 0.7 at one-loop for E0. Results from the quenched simulations
were extrapolated to L = ∞ using a simultaneous multi-polynomial fit to the
99
Chapter 5. Perturbative renormalisation
L Esim0 E
pert. th.
0
4 0.5295(16) 0.5312
6 0.5988(16) 0.6020
8 0.6369(12) 0.6362
10 0.6560(11) 0.6565
∞ 0.7380(63) 0.7348(3)
Table 5.10: Sample quenched one-loop results from Lee, in column one, com-
pared with the exact perturbative results computed by Ron Horgan in column
two [5, 11, 96]. These results use a heavy quark mass of aM = 2.8 with stabil-
ity parameter n = 2. There is no error on the perturbation theory calculation
as it was obtained by mode summation. The error on the perturbation theory
extrapolation to L = ∞ is estimated from the fit.
coupling constant, αV , and the inverse lattice spatial extent, L−1.
He used a fully-improved NRQCD action with radiatively improved co-
efficients in the action [11]. I present the one-loop values of the operator
coefficients, ci, in the fully-improved NRQCD action in Table 5.11. I define
the coefficients in Section 2.4.1. I do not include c2 or c3 in Table 5.11, because
current simulations do not include one-loop corrections for these coefficients.
aM c1 and c6 c4 c5
2.50 1 + 0.95αV (2.0/a) 1 + 0.78αV (pi/a) 1 + 0.41αV (2/a)
1.72 1 + 0.766αV (1.8/a) 1 + 0.691αV (pi/a) 1 + 0.392αV (1.4/a)
Table 5.11: One-loop corrected coefficients in the fully-improved NRQCD
action used for high-β simulations by Lee [11]. I define the coefficients in
Section 2.4.1. I include both the explicit one-loop corrections and the BLM
scale of the coupling constant in the V-scheme, αV .
The coefficients in Table 5.11 correspond to the values chosen for the
NRQCD lattice simulations undertaken by Dowdall and the HPQCD Col-
laboration [93]. The values for c(1)1 , c
(1)
5 and c
(1)
6 were estimated from [47] and
the value of c(1)4 from [14]. There is no contradiction in choosing two different
100
Chapter 5. Perturbative renormalisation
scales for the coupling constants in the perturbative corrections to the action.
Each coefficient removes leading order errors induced by quantum fluctua-
tions and so we choose a scale that best characterises those fluctuations. The
scale of q? ∼ 2.0 for the kinetic improvement operators was chosen based on
[87], whilst the higher scale of q? = pi for the chromo-magnetic correction
reflects the short-distance nature of that operator.
For the fully-improved NRQCD action, the selfenergy is
E0 = E
(1)
0 αV (3.33/a) + E
(2),q
0 α
2
V (3.33/a) + E
(3),q
0 α
2
V (3.33/a), (5.14)
where I give the tadpole-improved quenched coefficients, indicated by a su-
perscript q, in Table 5.12. I give all results at a characteristic scale given by
q? = 3.33, determined using the BLM procedure in [97].
aM E(1),q0 E
(2),q
0 E
(3),q
0
2.50 0.6864(5) 1.35(10) 2.2(8)
1.72 0.5873(6) 1.56(11) 2.2(8)
Table 5.12: Quenched results from high-β simulations [11, 96]. The one-loop
results are the exact perturbative results. Two- and three-loop results are
extracted from a simultaneous polynomial fit to the strong coupling constant
and lattice size. Uncertainties are from the fit. I evaluate all results at a
characteristic scale of q? = 3.33.
5.2.4 Tadpole improvement
The one-loop analytic results for the tadpole corrections to the heavy quark
energy shift, Eu00 , are given in Section 5.1.4. The effects of these one-loop
contributions are included in the tadpole subtracted results from the high-β
simulation, as are the quenched two-loop tadpole contributions. However, the
two-loop fermionic contributions are not included in the high-β results, so I
101
Chapter 5. Perturbative renormalisation
add these in explicitly. The tadpole correction is
Eu0,(2),f0 = −
[
1 +
7
2aM
−
3
2
(
1
a3M3
+
1
2na2M2
)]
u(4),P0 , (5.15)
where I take the value of the fermionic contribution to the plaquette tad-
pole at two-loops, u(4),P0 , from the extrapolation to zero light quark mass for
the Symanzik-improved gluons with ASQTad light quarks in Table 4.2. The
quenched contribution to the energy shift without tadpole improvement, Eq0 ,
is
Eq0 = 2.264(7) (5.16)
for the heavy quark mass aM = 2.50 and
Eq0 = 2.648(10) (5.17)
for the heavy quark mass aM = 1.72.
5.3 The b quark mass
Quark confinement ensures that quark masses are not physically measurable
quantities, so the notion of quark mass is a theoretical construction. A wide
range of quark mass definitions exist, often tailored to exploit the physics
of a particular process. One common choice of quark mass is the pole mass,
defined as the pole in the renormalised heavy quark propagator. However,
the pole mass is a purely perturbative concept and suffers from infrared am-
biguities known as renormalons [55, 56]. I discuss the role of renormalons
in NRQCD in more detail in Section 3.2. To avoid these ambiguities, ex-
perimental results are usually quoted in the modified Minimal Subtraction
(MS) scheme, which is renormalon ambiguity free. Lattice calculations use
the renormalon-free bare lattice mass. These different quark mass definitions
must be matched to enable meaningful comparison. I match bare lattice
quantities to the MS mass using the pole mass as an intermediate step. Any
renormalon ambiguities cancel in the full matching procedure between the
lattice quantities and the MS mass as I explicitly demonstrate in Section
102
Chapter 5. Perturbative renormalisation
5.3.2.
I extract theMS mass from lattice simulation data in a two-stage process.
I first relate lattice quantities to the pole mass and then match the pole mass
to the MS mass evaluated at a scale equal to the b quark mass.
5.3.1 Extracting the pole mass
I determine the pole mass by relating the heavy quark pole mass, Mpole,
to the experimental Υ mass, M exptΥ = 9.46030(26) GeV [8], using the heavy
quark energy shift, E0:
2Mpole = M
expt
Υ − (Esim(0)− 2E0). (5.18)
Here Esim(0) is the energy of the Υ meson at zero momentum, extracted from
lattice NRQCD simulations. The quantity (Esim(0)−2E0) corresponds to the
“binding energy” of the meson in NRQCD.
I use values of Esim(0) obtained from lattice NRQCD simulations run by
the HPQCD collaboration on two different MILC ensembles. The first en-
semble, with a tuned heavy quark mass of aM = 2.50, is a “coarse” MILC
ensemble, with lattice spacing a = 0.605(5) GeV−1. The second ensemble,
with a tuned heavy quark mass of aM = 1.72, is a “fine” MILC ensemble,
with lattice spacing a = 0.4292(32) GeV−1. These heavy quark masses corre-
spond to the kinetic mass, defined through the nonrelativistic kinetic energy,
for the spin-average of the Υ and ηb mesons.
In these simulations, the lattice spacing was tuned using the HPQCD
value for r1, given by r1 = 0.3133(23) [93, 98]. Both ensembles use Symanzik-
improved gluons with 2+ 1 flavours of ASQTad sea quarks. Electromagnetic
effects were ignored. For further details of the configurations see [22, 98].
In principle we could extract the quark mass by directly matching the
pole mass to the bare lattice mass in physical units, M0(a), via the heavy
quark mass renormalisation, ZM ,
Mpole = ZM(µa,M0(a))M0(a). (5.19)
103
Chapter 5. Perturbative renormalisation
However, it proved difficult to reliably extract the quenched two-loop mass
renormalisation from high-β simulations and so I do not extract the b quark
mass in this way.
5.3.2 Matching the pole mass to the MS mass
The mass renormalisation relating the pole mass to the MS mass, MMS,
evaluated at some scale µ, is given by
MMS(µ) = Z
−1
cont(µ,Mpole)Mpole, (5.20)
and has been calculated to three-loops in [99]. The result is
Z−1cont(Mpole,Mpole) = 1−
4
3pi
αMS +
1
pi2
(1.0414nf − 14.3323)α
2
MS (5.21)
+
1
pi3
(
−0.65269n2f + 26.9239nf − 198.7068
)
α3MS,
where the coupling constant is evaluated at a scale equal to the pole mass.
Although the pole mass is plagued by renormalon ambiguities, these am-
biguities cancel when lattice quantities are related to the MS mass. We can
see that the renormalon ambiguities cancel when determining the pole mass
from the energy shift, by equating Equations 5.18 and 5.19 and rearranging
them to obtain
2(ZMM0(a)− E0) = M
expt
Υ − Esim(0). (5.22)
The two quantities on the right hand side of the equation are renormalon
ambiguity free: M exptΥ is a physical quantity and Esim(0) is determined non-
perturbatively from lattice simulations. Any renormalon ambiguities in the
two power series, ZM and E0, on the left-hand side of the equation must
therefore cancel.
This renormalon cancellation is also evident in the direct matching of the
bare lattice mass to the MS mass,
MMS(µ) = ZM(µa,M0(a))Z
−1
cont(µ,Mpole)M0(a), (5.23)
104
Chapter 5. Perturbative renormalisation
as both MMS and M0 are renormalon-free.
I combine Equations (5.18), (5.20) and (5.21) to obtain the final expres-
sion for the full matching between lattice quantities and the MS mass as
MMS(µ) =
1
2
Z−1cont(µ,Mpole)
[
M exptΥ − (Esim(0)− 2E0)
]
. (5.24)
5.3.3 Results
I now have all the pieces in place to extract the b quark mass and here I
collect them together to obtain my final result. First off, the perturbative
results for the heavy quark energy shift, which is given by
E0 = E
(1)
0 αV (3.33/a) +
(
E(2),q0 + nfE
(2),f
0
)
α2V (3.33/a) + E
(3),q
0 α
3
V (3.33/a),
(5.25)
where the coefficients E(i),j0 are given in Table 5.12 for the quenched results
and in Table 5.9 for the fermionic contributions.
To extract the b quark mass, I first express the coupling constant in the V -
scheme in terms of the coupling constant in theMS scheme, evaluated at the
same scale q?, using the three-loop relation in [32, 33, 34]. I used the world-
average value of αMS(M
2
Z) = 0.1184(7) GeV, with MZ = 91.1876(21) GeV,
both taken from the Particle Data Group [8], to set the absolute scale for
the coupling constant. I then run the coupling constant to the scale of the
heavy quark mass, which is yet to be determined. Finally I use Mathematica
to self-consistently solve the equation
MMS(MMS) =
1
2
Z−1cont(MMS,Mpole)
[
M exptΥ − (Esim(0)− 2a
−1E0)
]
, (5.26)
where I express the energy shift, E0, in terms of the coupling constant in the
MS scheme, evaluated at MMS.
I summarise all the data necessary to extract the b quark mass in theMS
scheme in Tables 5.13 and 5.14. In Table 5.13 I combine the quenched energy
shift results in Table 5.12 with the fermionic contributions in Table 5.9. I
include the fermionic contribution to the tadpole improvement at two-loops
in column five. This contribution is given in Equation (5.15). In Table 5.14 I
105
Chapter 5. Perturbative renormalisation
aM E(1)0 E
(2)
0 E
(3),q
0 E
u0,(2),f
0
2.50 0.6864(5) 1.35(10) + 0.2823(6)nf 2.2(8) −0.158532(16)
1.72 0.5873(6) 1.56(11) + 0.3041(3)nf 2.2(8) −0.166056(17)
Table 5.13: Perturbative data required to extract MS mass. The quenched
results, indicated by superscript q, are from high-β simulations [11, 96]. I cal-
culate fermionic contributions using automated lattice perturbation theory.
The one-loop data are the exact perturbative results extrapolated to infinite
lattice size. The two-loop results include both quenched and fermionic con-
tributions. The three-loop values include only quenched results. I evaluate
all results at a characteristic scale of q? = 3.33.
present the nonperturbative simulation results for Esim and the corresponding
inverse lattice spacing, both expressed in GeV.
aM Esim a−1
2.50 0.7397(66) 1.652(14)
1.72 0.96417(13) 2.330(17)
Table 5.14: Simulation data required to extract MS mass. All values ex-
tracted from nonperturbative simulations by the HPQCD collaboration on
MILC ensembles [93].
I obtain
MMS (MMS) = 4.185(26) GeV, (5.27)
for a heavy quark mass of aM = 2.50, and
MMS (MMS) = 4.154(27) GeV, (5.28)
for a heavy quark mass of aM = 1.72. I quote only the uncertainty associ-
ated with statistical and perturbative errors in these results. I discuss these
uncertainties in the next section and, in particular, describe two approaches
to estimating the systematic uncertainties due to lattice artifacts.
106
Chapter 5. Perturbative renormalisation
5.3.4 The error budget
Broadly, there are three main sources of uncertainty in my result for the
b quark mass: systematic errors, statistical errors and perturbative errors.
I expect the O(α3s) perturbative contributions to dominate the error in my
result. In this section I discuss each of these sources of error in turn. I tabulate
my estimated error budget in Table 5.15. I estimate all sources of uncertainty
by carrying out the analysis discussed in the previous section with a range
initial parameters specified by the uncertainties in those parameters. I take
the resulting range of b quark masses as the estimate of the uncertainty
due to the individual parameter concerned. Thus, for example, I calculate
the b quark mass with the inverse lattice spacing equal to its quoted value,
a−1 = 1.652 GeV. This is my final result for the mass of the b quark. I then
recalculate the mass with the inverse lattice spacing equal to its maximum
value a−1 = 1.666 GeV. The difference between the new result that I obtain
for the b quark mass and my original value for the b quark mass provides an
estimate of the uncertainty due to the error in the lattice spacing.
Statistical errors Statistical errors arise in the nonperturbative and experi-
mental input values, Esim and M
expt
Υ , and in the contributions at each order
in the expansion of the heavy quark energy shift, E0. I do not provide a fur-
ther breakdown of the errors in M exptΥ , because the associated uncertainties
are negligible. I do not have a breakdown of the errors associated with Esim
and so include the total uncertainty in this quantity as simply “statistical”
error. The corresponding relative error is ∼3 MeV, which is less than 0.1% of
my final result for the b quark mass. The uncertainty in the lattice spacing
for the nonperturbative simulations leads to an error in the b quark mass of
less than 0.1%. In Table 5.15 I denote uncertainty due to the error in the
value of the lattice spacing as ∆a−1. This should not be confused with the
systematic error due to lattice artifacts.
Statistical errors arise from the numerical evaluation of each contributions
to the heavy quark energy shift. The uncertainty in the one-loop coefficient
is from the extrapolation to infinite volume and the associated error in the
107
Chapter 5. Perturbative renormalisation
b quark mass is ∼1 MeV or 0.1% of the final result. The uncertainties in the
two-loop and three-loop quenched coefficients of E0 arise from the simulta-
neous fit to the lattice size and the strong coupling constant. The statistical
error in the two-loop fermionic coefficient is due to the numerical evaluation
of the Feynman diagrams and the extrapolation to zero light quark mass.
Systematic errors Systematic uncertainties arise from the nonzero lattice
spacing. With data only available for two lattice spacings, there are two
methods to estimate the lattice artifact errors.
The two results in Equations (5.27) and (5.28) agree within the quoted
uncertainties. At this level of precision, then, I assume that lattice artifact
errors are negligible. In this case, I simply average over the two results, and
use their standard deviation as my estimate of the systematic errors. I obtain
MMS (MMS) = 4.170(27)(22) GeV, (5.29)
where the first uncertainty is statistical/perturbative and the second is sys-
tematic. I assume that the statistical/perturbative errors are correlated be-
tween the results in Equations (5.27) and (5.28) and therefore do not combine
the two errors in quadrature, but quote a single value.
The NRQCD quantum effective action is an expansion in the lattice spac-
ing, the strong coupling constant and the inverse heavy quark mass. As an
alternative method for estimating systematic uncertainties, I assume only
that the NRQCD action is improved to sufficient order in the lattice spacing
and coupling constant that the leading order correction is a polynomial in the
inverse heavy quark mass. The leading order in this expansion is 1/(a2M2).
I then estimate the lattice artifact errors by fitting the results in Equations
(5.27) and (5.28) to a function of the form
F (M) = M0 +
A
a2M2
. (5.30)
I then take my final result to be equal to the value ofM0 from my fit. I obtain
MMS (MMS) = 4.213(27) GeV. (5.31)
108
Chapter 5. Perturbative renormalisation
In this case I do not quote a systematic uncertainty from the fitting procedure
because I fit to two parameters with two data points.
A more systematic discussion of the leading contributions missing from
the expansion of the quantum effective action is also required. I have not
demonstrated rigorously that the leading order term in the expansion is
quadratic in the inverse heavy quark mass and a more refined analysis would
be preferable and may include non-leading terms. This, of course, requires
results from different ensembles, that is, at different heavy quark masses.
This work is underway.
Perturbative errors I include all known three-loop contributions in my cal-
culation of the b quark mass. These contributions come from:
• the quenched heavy quark energy shift;
• matching αV to αMS;
• matching the pole mass to the MS mass;
• perturbative running of the MS mass to the desired scale.
The three-loop fermionic contribution to the energy shift is unknown, so I
estimate the error due to this contribution as O(1×α3
MS
). The corresponding
error in the b quark mass is ∼19 MeV, that is, a relative uncertainty of
approximately 0.5%.
The unknown fermionic contributions are the only unknown source of
uncertainty at three-loops in my result. In principle these effects can be cal-
culated using automated lattice perturbation theory. However, there are a
large number of diagrams to evaluate, many of which are likely to have com-
plicated pole structures and possible divergences (the energy shift is infrared
finite, but individual diagrams may have divergences that ultimately cancel).
The complexity of such a calculation would be considerable.
5.3.5 Comparison to the literature
My results for the b quark mass improves on the previous, preliminary, result
that we presented in [5] in a number of ways. In that work, we obtained a
109
Chapter 5. Perturbative renormalisation
Class of error Error
Estimated Estimated
uncertainty uncertainty
(aM = 2.50) (aM = 1.72)
MeV % MeV %
Statistical
M exptΥ 0.1 ¿ 0.1 0.1 ¿ 0.1
Esim 3.0 < 0.1 0.1 ¿ 0.1
∆a−1 1.2 < 0.1 1.5 < 0.1
Perturbative
E(1)0 0.2 ¿ 0.1 0.3 ¿ 0.1
E(2),q0 8.3 ∼ 0.2 11.5 ∼ 0.3
E(2),f0 0.3 ¿ 0.1 0.2 ¿ 0.1
E(3),q0 15.4 ∼ 0.4 15.0 ∼ 0.4
E(3),f0 ∼19.3 ∼ 0.5 ∼18.8 ∼ 0.5
Systematic a 31 ∼ 0.7 31 ∼ 0.8
Table 5.15: The b quark mass error budget: estimated uncertainties in my
calculation of the b quark mass from lattice NRQCD and lattice perturbation
theory. The entries for the “systematic” error are independent of the heavy
quark mass. I describe each source of uncertainty in more detail in Section
5.3.4.
value of
MMS (MMS) = 4.25(12) GeV (5.32)
for the b quark mass.
Firstly, the new results remove a number of discrepancies between the
nonperturbative data and the perturbation theory results. I used a value for
Esim based on simulations using an NRQCD action with a stability param-
eter of n = 4. The perturbation theory results, on the other hand, used an
NRQCD action with a stability parameter n = 2. I had previously calcu-
lated the fermionic contributions with HISQ light quarks, which significantly
reduced the fermionic contribution. Furthermore, the lattice NRQCD simu-
lation used Landau tadpoles [85, 98] whilst the quenched high-β simulations
110
Chapter 5. Perturbative renormalisation
used plaquette tadpoles [11]. With the new results, plaquette tadpoles were
used in both simulations and I include the fermionic contributions to the
two-loop tadpole correction for the heavy quark energy shift, which had not
previously been computed. Although the error arising from these discrep-
ancies is likely to be negligible, and certainly much less than the dominant
O(α3s) error, removing as many sources of error as possible is obviously de-
sirable.
The NRQCD simulations from which the value of Esim was extracted
have also been improved. Previous results used an NRQCD action without
radiative improvements to the coefficient of the chromo-magnetic operator,
c4, which is included in the new result. The lattice spacing determination has
also been improved [85, 98]. The new results use higher statistics and deter-
mine the value of r1/a from a combination of methods to reduce systematic
uncertainties [93].
In my new results I use simulation data from two different ensembles, al-
lowing me to estimate the systematic uncertainty arising from lattice artifact
errors under two different sets of assumptions. Both results are in agreement
within errors. This analysis was not possible with the preliminary result pre-
sented in [5].
Finally, I have analysed the uncertainties in my new result more rigor-
ously and provided an error budget for individual contributions. I had not
previously analysed the individual sources of error in detail.
My results are a significant improvement on the previous determination
of the b quark mass from lattice NRQCD simulations by the HPQCD col-
laboration [85], as can be seen in Figure 5.5. The dominant source of error
in that calculation was the contribution from the two-loop heavy quark en-
ergy shift and the total error quoted for the result was approximately seven
percent. All two-loop contributions have been included in the values given
in Equations (5.27) and (5.28) and the errors are reduced to less than one
percent, in line with other recent determinations.
In Figure 5.5, I also present a number of recent determinations of the
b quark mass. Two of these are from perturbative QCD sum rules, given
in [100, 101] and [92]. These results are in excellent agreement with quoted
111
Chapter 5. Perturbative renormalisation
uncertainties of very similar magnitude, although the errors are about 30%
larger than the errors in my result. Of course, these values are not extracted
from nonperturbative simulations, but do use input from experimental data.
I show results from three different lattice simulations in Figure 5.5. All
results are in agreement within the quoted uncertainties. The most precise
determination is by the HPQCD collaboration, using HISQ quarks for the b
quark and extrapolating up to the physical b quark mass [25]. This simulation
used MILC configurations with (2 + 1) flavours of ASQTad sea quarks.
I also present the result from the ETM collaboration using twisted mass
Wilson quarks at four different lattice spacings and two species of sea quarks
[102]. The final result is from the ALPHA collaboration [103]. They quote a
preliminary result with a larger uncertainty than the other results in Figure
5.5. The ALPHA collaboration use HQET with O(1/M) corrections and two
species of sea quarks, but do not give an indication of the errors from lattice
artifacts.
The vertical line in Figure 5.5 is a weighted average of results from other
work. This line serves to guide the eye only and does not include the results
of this work from [5] or from this dissertation. I do not account for any
correlations between results in this average. We can see that my new results
are in good agreement with this average.
5.4 Conclusions and outlook
I have calculated the two-loop heavy quark energy shift in highly-improved
NRQCD using a mixed approach combining quenched high-β simulations
with lattice perturbation theory. This calculation was the first determination
of any heavy quark parameter at two-loops in NRQCD and demonstrated
the effectiveness of such an approach. My predictions of the b quark mass
improves on the previous determination from lattice NRQCD simulations
by the HPQCD collaboration. I significantly reduce the uncertainties in the
final result by removing the dominant source of uncertainty in the previous
calculation: the two-loop heavy quark energy shift. My results include an
estimate of the systematic uncertainties arising from lattice artifact errors.
112
Chapter 5. Perturbative renormalisation
3.
9
4
4.
1
4.
2
4.
3
4.
4
4.
5
4.
6
4.
7
4.
8T
hi
s
w
or
k
(a
ve
ra
ge
)
Th
is
w
or
k
(fit
)
Th
is
w
or
k
(2
01
0 p
re
lim
ina
ry)
H
P
Q
CD
N
RQ
CD
(2
00
5)
H
P
Q
CD
H
IS
Q
(2
01
0)
A
LP
H
A
(2
01
0)
E
TM
(2
01
1)
N
ar
is
on
e
t a
l.
(2
00
9)
C
he
ty
rk
in
e
t a
l.
(2
00
9)
M
(G
eV
)
Figure 5.5: Recent results for the b quark mass. From top to bottom: this
work, using an averaging procedure (see Section 5.3.4); this work, using a fit-
ting procedure (see Section 5.3.4); the preliminary determination from this
work [5]; HPQCD collaboration with NRQCD valence quarks [85]; HPQCD
collaboration using HISQ valences quarks [25]; ALPHA collaboration using
HQET [103]; ETM collaboration [102]; and QCD sum rules from Narison et
al. [100, 101] and from Chetyrkin et al. [92]. The vertical line is to guide
the eye and is my determination of the weighted average of the results, not
including data from our preliminary determination in [5] or from this disser-
tation.
113
Chapter 5. Perturbative renormalisation
By including all two-loop perturbative effects I have brought the uncertainty
in the final result into line with a number of other recent determinations of
the b quark mass.
A complete analysis of the systematic uncertainty would require further
simulation data from ensembles at different lattice spacings. This work is
currently underway. The analysis of the high-β simulations will also be im-
proved by using a larger number of ensembles and this development will
improve further the precision of the three-loop contribution to the quenched
heavy quark energy shift and thereby reduce the uncertainty in the b quark
mass. Ultimately the three-loop fermionic contributions to the heavy quark
energy shift would be required for a complete three-loop calculation, but this
would require extensive developments of my code.
114
Chapter 6
Heavy-light currents
Electroweak processes are an important tool in understanding the Standard
Model, serving as an input into tests of CKM matrix unitarity and CP vi-
olation, and as a probe for new physics. Hadronic matrix elements, which
characterise the strong interaction dynamics of these processes, are a crucial
ingredient in the calculation of the electroweak parameters.
Lattice QCD simulations enable nonperturbative determinations of had-
ronic matrix elements from first principles. These simulations require us to
construct the electroweak lattice currents in terms of the appropriate lattice
operators. We then match the lattice currents to their continuum counter-
parts order-by-order in the strong coupling constant to extract the correct
continuum behaviour at the desired precision.
In this chapter I perturbatively match the temporal components of the
heavy-light axial and vector currents in lattice NRQCD with massless HISQ
light quarks to the corresponding currents in continuum QCD. I perform this
matching at zero external quark momentum and include only leading order
terms in the inverse heavy quark mass expansion. This calculation updates
the matching calculation of [89], which used highly-improved NRQCD with
massless ASQTad quarks for the lattice currents. My results are directly
relevant for the nonperturbative determination of the fB, fBs and fDs decay
constants by the HPQCD collaboration using MILC lattices with HISQ sea
quarks.
Before describing the matching calculation in detail, I first briefly review
some of the background and motivation for this calculation.
6.1 Heavy-light decays and the CKM matrix
The three-generation Cabibbo-Kobayashi-Maskawa (CKM) matrix parame-
terises the weak interaction quark eigenstates in terms of the mixing of the
115
Chapter 6. Heavy-light currents
mass eigenstates [104, 105]. The CKM matrix,
VCKM =
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
, (6.1)
is unitary in the Standard Model and deviations from unitarity may indi-
cate new physics. Tests of the unitarity of the CKM matrix are therefore of
paramount importance in understanding the Standard Model. We can pa-
rameterise the CKM matrix using three real angles and one CP-violating
phase. If this phase is non-zero, then weak interactions violate simultaneous
charge (C) and parity (P) invariance. The standard choice of parameterisa-
tion for the CKM matrix is the Wolfenstein parameterisation [106], which
uses λ = |Vus| ≈ 0.22 as a small expansion parameter:
VCKM =
1− λ
2
2 λ Aλ
3(ρ− iη)
−λ 1− λ
2
2 Aλ
2
Aλ3(1− ρ− iη) −Aλ2 1
. (6.2)
Unitarity imposes constraints on the parameters of the CKM matrix that
we can interpret geometrically as triangles in the complex plane. Only one
of these triangles is not near-degenerate and the corresponding normalised
constraint is
VudV ∗ub
VcdV ∗cb
+
VtdV ∗tb
VcdV ∗cb
+ 1 = 0. (6.3)
I show this CKM triangle in Figure 6.1. The angle β is known to 1◦ preci-
sion from B → J/ψK decays and precise determinations of |Vub| and |Vcb|
are therefore an integral component in over-constraining the CKM unitarity
triangle [8]. The ultimate goal is to measure the three angles and the lengths
of each of the sides and thus determine the apex of the CKM triangle (ρ, η)
in as many ways as possible. Over-constraining the CKM triangle provides a
stringent test of the three-generation Standard Model.
The CKM matrix elements |Vub| and |Vcb| are experimentally determined
from a number of different electroweak processes, including inclusive and ex-
116
Chapter 6. Heavy-light currents
γ
∣
∣
∣
∣
∣
∣
∣
VudV ∗ubVcdV ∗cb
∣
∣
∣
∣
∣
∣
∣
(0, 0)
∣
∣
∣
∣
∣
∣
∣
VtdV ∗tbVcdV ∗cb
∣
∣
∣
∣
∣
∣
∣
(1, 0)
(ρ, η)
β
α
Figure 6.1: The CKM unitarity triangle in the complex plane.
clusive B decays and B meson mixing [8]. The CKM matrix elements must
be extracted from strongly bound states and lattice QCD is required to cal-
culate the hadronic matrix elements that characterise the strong interaction
dynamics of the decays. There is currently a 2σ tension between the inclusive
and exclusive determinations of |Vcb| and the dominant uncertainties in these
measurements are theoretical, so an improved understanding of the theo-
retical inputs is increasingly important. This requires high-precision lattice
simulations.
The matrix element |Vub| is extracted experimentally from the exclusive
semileptonic decay B → pi`ν`, which has differential decay rate
dΓ
dq2
=
G2F
24pi3
|Vub|
2|ppi|
3|f+(q
2)|2 (6.4)
in the limit that the light quark is massless. Here pB is the momentum of the
B meson and ppi the momentum of the pion in the B meson rest frame. The
momentum transfer is q = pB − ppi. The form factors f+ and f0 parameterise
heavy-light matrix elements of the vector current, V µ,
〈pi(ppi)|V
µ|B(pB)〉 = f+(q
2)
[
pµB + p
µ
pi −
m2B −m
2
pi
q2
qµ
]
+ f0(q
2)
m2B −m
2
pi
q2
qµ. (6.5)
The axial vector current does not contribute to pseudoscalar-to-pseudoscalar
decays [1].
117
Chapter 6. Heavy-light currents
The HPQCD collaboration has calculated these form factors using lattice
QCD [1]. These simulations were run with (2 + 1) flavours of ASQTad sea
quarks and highly-improved NRQCD valence b quarks. Using experimental
input from the BaBar, Belle and CLEO experiments, the authors obtained
|Vub| = (3.55 ± (0.25)exp ± (0.50)theory) × 10−3. The uncertainties are exper-
imental and theoretical respectively and are dominated by the theoretical
input.
Leptonic decays of heavy mesons also provide experimental methods for
constraining the CKM matrix. For example, the ratio |Vtd/Vts| involves the
the decay constants fB and fBs . These decays proceed via a virtual charged
W boson and the decay width of such a process is given by
Γ(X → `ν) =
G2F
8pi
f 2Xm
2
`mX
(
1−
m2`
m2X
)2
|Vq1q2|
2, (6.6)
where X is a heavy hadron, such as a B, Bs or Ds meson and |Vq1q2| is the
relevant CKM matrix element. In fact, leptonic decay modes occur for pi± and
K± mesons as well, but I restrict my discussion here to heavy quark mesons.
We define the decay constant, fX , through the hadronic matrix element
〈0|Aµ|X(p)〉 = pµfX . (6.7)
The HPQCD collaboration has computed the decay constants fBs and
fDs using NRQCD heavy quarks and (2 + 1) flavours of ASQTad sea quarks
[50, 107, 108, 109]. There is currently tension between the value of |Vcb|
determined from experimental measurements of B± → τ±ν, presently in
good agreement with the Standard Model, and from determinations using
global fits to the CKM parameters [110].
Recent measurements of the branching ratios of the semileptonic decay
B → D(∗)τν, from which |Vub| can be determined, report a preliminary result
nearly 2σ above the Standard Model prediction [9]. This decay proceeds via
a W± boson, so if this discrepancy were to be confirmed, the excess would
hint at the presence of a Beyond the Standard Model charged Higgs boson.
These results, however, are very much preliminary findings.
118
Chapter 6. Heavy-light currents
Results from LHCb and the proposed SuperB factory will improve the
experimental resolution of B decays by an order of magnitude and corre-
sponding improvements in the theoretical inputs are required. The HPQCD
collaboration are therefore calculating the decay constants fBs and fDs using
NRQCD with HISQ sea quarks, which should improve the precision of the
final results [111].
The calculations of the form factors, f+ and f0, and the decay constants,
fB and fBs , using HISQ light quarks both require new matching calculations
to relate heavy-light currents on the lattice and in the continuum. In this
chapter I discuss this calculation and present results for massless HISQ light
quarks using currents at leading order in the inverse heavy quark mass.
6.2 The continuum current
The continuum axial-vector current is given by
Aµ(x) = q(x)γ̂5γ̂µh(x), (6.8)
where q(x) is the light quark field and h(x) the heavy quark field. I denote
the Euclidean Dirac matrices by γ̂µ. They are related to the Minkowski Dirac
matrices, γµ, by
γ̂0 = γ̂
0 = γ0 = γ
0, γ̂j = γ̂
j = iγj = −iγ
j, (6.9)
and satisfy
{γ̂µ, γ̂ν} = 2δµν . (6.10)
The continuum vector current is
Vµ(x) = q(x)γ̂µh(x). (6.11)
Morningstar and Shigemitsu calculate the heavy-light continuum QCD
current analytically, for the temporal component of the axial vector current
in Reference [112] and for both vector and axial vector currents in [113]. They
119
Chapter 6. Heavy-light currents
use massless light quarks, onshell mass and wavefunction renormalisation in
Feynman gauge and expand the current in powers of the inverse heavy quark
mass. For an incoming heavy quark of momentum, p, and an outgoing light
quark of momentum p′, the one-loop current is
〈q(p′)|Jµ|h(p)〉QCD = uq(p
′)W Jµ (p
′, p)uh(p), (6.12)
where Jµ indicates either the axial-vector or vector current and
W Jµ (p
′, p) = a1Γ
J
µ − a2
ipµ
M
ΓJ0γ0 − a3
p · p′
M2
ΓJµ − a4
ip′µ
M
ΓJ0γ0
+ a5
p · p′
M2
ipµ
M
ΓJ0γ0 +O(1/M
2). (6.13)
Here the operator ΓJ stands for either the vector current operator ΓVµ = γ̂µ
or the axial vector current ΓAµ = γ̂5γ̂µ. The spinors, uh and uq, are the stan-
dard Dirac heavy quark and light quark spinors respectively. The expansion
coefficients are given by
a1 = 1−
αs
pi
(
ln
λ
M
+
11
12
)
,
a2 =
2αs
3pi
,
a3 =
αs
pi
(
1
6
−
8pi
9
M
λ
− 2 ln
λ
M
)
a4 =
αs
3pi
(
1
2
+ 2 ln
λ
M
)
,
a5 =
αs
3pi
(
5 + 4 ln
λ
M
)
. (6.14)
Here λ is the gluon mass. Ultraviolet divergences are regulated using dimen-
sional regularisation.
In the Appendix to Reference [47], Mu¨ller quotes a slightly different in-
frared finite contribution to the coefficient a3:
aEM3 = −
αs
pi
(
−
29
18
−
8pi
9
M
λ
− 2 ln
λ
M
)
. (6.15)
120
Chapter 6. Heavy-light currents
(a) (b) (c)
Figure 6.2: Contributions to the continuum QCD axial vector and vector
heavy-light currents. The diagram on the left, diagram (a), represents the
vertex correction and the other diagrams, from left to right, are (b) the heavy
quark and (c) the light quark wavefunction renormalisation contributions re-
spectively. Double blue lines represent heavy quarks, single red lines indicate
light quarks, wiggly green lines are gluons and the black dots represent the
current insertion.
This discrepancy does not affect the results that I present in this dissertation,
but will need to be clarified in future work that requires a3.
Three diagrams contribute to the continuum QCD currents at one-loop,
which I illustrate in Figure 6.2. The diagram on the left represents the one-
loop vertex correction and the other two contributions are the one-loop wave-
function renormalisation diagrams. Before reviewing the lattice current cal-
culation, I study the light quark renormalisation parameters. I require the
wavefunction renormalisation for massless relativistic quarks for the match-
ing calculation laid out in this dissertation. Ultimately the mass renormali-
sation for HISQ quarks will also be required for the extension of this work
to massive light quarks. As the derivation of the renormalisation parameters
for relativistic quarks is different to that for NRQCD, which I present in
Chapter 5, I collect the necessary ingredients for both massive and massless
relativistic quarks here. I start by deriving an expression for the wavefunction
renormalisation.
6.3 Relativistic quark renormalisation parameters
I extract the relativistic quark wavefunction renormalisation by taking ap-
propriate derivatives of the one-loop quark selfenergy. For massive relativistic
121
Chapter 6. Heavy-light currents
quarks in Euclidean space, the tree-level propagator is
G−10 (p) = ip/+m0, (6.16)
where m0 is the bare quark mass. I write the one-loop propagator in terms
of the one-loop selfenergy, Σ, as
G−11 (p) = ip/+m0 − αsΣ. (6.17)
I now write the selfenergy as
Σ = ip/Σ1 +m0Σ0, (6.18)
so that the propagator becomes
G−11 (p) = (1− αsΣ1) ip/+ (1− αsΣ0)m0. (6.19)
I multiply the numerator and denominator in the propagator by
(1− αsΣ0)m0 − (1− αsΣ1) ip/ (6.20)
and then divide by (1 + αsΣ1) to obtain
G(p) =
−ip/+ (1− αs(Σ0 + Σ1))m0
(1− αsΣ1) p2 + (1− αs(2Σ0 − Σ1))m20
. (6.21)
The mass renormalisation is given by
m =
(
1− αsΣ0
1− αsΣ1
)
m0
∣
∣
∣
∣
p0=im0
= (1− αs(Σ0 − Σ1))m0 = Zmm0, (6.22)
that is,
Zm = 1− αs(Σ0 − Σ1). (6.23)
We define the wavefunction renormalisation through the residue of the pole
at p0 = im0. Denoting the denominator of G(p) by Γ(p), the wavefunction
122
Chapter 6. Heavy-light currents
renormalisation is given by
Z−1ψ =
dΓ
dp2
∣
∣
∣
∣
p0=im0
=
[
1 + αsΣ1 − 2m
2
0αs
d
dp2
(Σ0 − Σ1)
]
p0=im0
. (6.24)
I now apply the chain rule to take derivatives with respect to p0, and write
the wavefunction renormalisation as
Z−1ψ = 1 + αs
[
Σ1 + im
2
0
d
dp0
(Σ1 − Σ0)
]
p0=im0
. (6.25)
For massless relativistic quarks, the corresponding expression is just
Z−1ψ = 1 + αsΣ1|p0=0. (6.26)
I calculate the massless relativistic wavefunction renormalisation numeri-
cally, using both unimproved staggered quarks and ASQTad quarks. For mas-
sive HISQ quarks, the derivation and calculation is more involved. I briefly
discuss the derivation of the HISQ renormalisation parameters before outlin-
ing the calculation of the lattice current operators. My presentation for the
HISQ parameters is based on [114] and I provide a more detailed derivation
in Appendix E
6.4 HISQ renormalisation parameters
I write the inverse relativistic quark propagator as
G(p)−1 = G−10 (p)− Σ(p) (6.27)
where
G−10 (p) =
∑
µ
i sin (pµ)Kµ(p)γµ +m0, (6.28)
and
Kµ(p) = 1 +
1 + ²
6
sin2 (pµ) . (6.29)
123
Chapter 6. Heavy-light currents
Here I parameterise the selfenergy as
Σ(p) =
∑
µ
i sin (pµ)Bµ(p)γµ + C(p), (6.30)
to be consistent with the notation of [114]. We can see, by comparing this pa-
rameterisation with Equation (6.18), that Bµ corresponds to Σ1 in Equation
(6.18) and C(p) to Σ0.
To calculate the pole mass, I set p = 0 and p0 = iE, where E is the rest
energy. Then the pole condition,
∑
µ
sin2 (pµ) (Kµ −Bµ)
2 + (m0 − C)
2 = 0 (6.31)
becomes
− sinh2E
(
1−
1 + ²
6
sinh2E −B0
)2
+ (m0 − C)
2 = 0. (6.32)
Taking the positive energy solution, and expanding the rest energy E and
the parameter ² in the strong coupling constant, the tree-level mass, mtree,
at fixed bare mass m0, is given by
sinhmtree
(
1−
1 + ²tree
6
sinh2mtree
)
= m0. (6.33)
The parameter ²tree is fixed by requiring the mass to be equal to the
kinetic mass, defined as
m−1kin =
∂2E
∂p2z
∣
∣
∣
∣
pz=0
, (6.34)
for an onshell particle with momentum pµ = (iE, 0, 0, pz). The resulting
solution is [114]
²tree =
(
4−
√
4 +
12mtree
coshmtree sinhmtree
)
/ sinh2mtree − 1. (6.35)
We obtain the solution for mtree by plugging in this expression for ²tree
into Equation (6.33) and solving iteratively.
124
Chapter 6. Heavy-light currents
The one-loop mass is given by
αs
1
Z(0)2
m1 − αs²1
sinh3mtree
6
−B0 sinhmtree = −C, (6.36)
where Z(0)2 is the tree-level wavefunction renormalisation:
1
Z(0)2
=
(
1−
1 + ²tree
2
sinh2mtree
)
coshmtree. (6.37)
At non-zero spatial momentum, we define the kinetic mass,mkin, by Equa-
tion (6.34). The tree-level pole condition is
−
(
sinhE −
1 + ²tree
6
sinh3E
)2
+
(
sin pz +
1 + ²tree
6
sin3 pz
)2
+m20 = 0.
(6.38)
and the tree-level kinetic mass is
mkin =
(
sinhmtree −
1 + ²tree
6
sinh3mtree
)
×
(
1−
1 + ²tree
2
sinh2mtree
)
coshmtree. (6.39)
Setting mkin = mtree gives us the expression for ²tree in Equation (6.35).
We define the wavefunction renormalisation through the residue of the
quark propagator at the single particle pole:
∫ pi
−pi
dp0
2pi
eip0tG(p0,0) = Z2e−mt
1 + γ0
2
+ . . . , (6.40)
where the ellipses stand for poles corresponding to more complicated bound
state and multi-particle states. The final expression for the wavefunction
renormalisation is
Z−12 =
(
1−
1 + ²
2
sinh2m
)
coshm+ i
d
dp0
(iB sin p0 + C) . (6.41)
At tree-level, this expression reduces to the tree-level wavefunction renor-
malisation, Z(0)2 , given in Equation (6.37). The one-loop wavefunction renor-
125
Chapter 6. Heavy-light currents
malisation, Z(1)2 , is
1
Z(1)2
=
1
Z(0)2
{
1− αs²1
Z(0)2
2
coshmtree sinh
2mtree + i
d
dp0
(iB sin p0 + C)Z
(0)
2
+ αsm1 sinhmtree
[
1−
1 + ²tree
2
(
2 cosh2mtree + sinh
2mtree
)
]
Z(0)2
}
.
(6.42)
For massless HISQ with zero spatial momentum, we immediately see that
the one-loop wavefunction renormalisation becomes
Zψ = 1 + αsB0, (6.43)
since mtree = m0 = 0, ²tree = 0 and Z
(0)
2 = 1. As we would expect, this is just
Equation (6.26) rewritten in the notation of [114], that is, with B0 = Σ1.
6.5 The lattice current
I now derive the lattice operators corresponding to the continuum operators
in Equation (6.13), largely following the derivation in [112]. I illustrate the
extra lattice contributions in Figure 6.3. As I discuss in Section 2.4, the
lattice NRQCD heavy quark field is defined in terms of a two-component
Pauli spinor, which I here denote Ψ(x). We obtain this spinor from the Dirac
heavy quark field, h(x), using the FWT transformation, which decouples the
quark and antiquark fields.
We transform the external spinor uh(p) into a nonrelativistic Pauli spinor,
uQ(p),
uh(p) =
(
1−
1
2M
γ̂ ·p
)
uQ(p) +O(1/M
2). (6.44)
Then we can decompose the continuum current in Equation (6.12) as
〈q(p′)|Jµ|h(p)〉NRQCD = uq(p
′)ΩJµ(p
′, p)uQ(p), (6.45)
126
Chapter 6. Heavy-light currents
(b)
(d) (e)
(a) (c)
Figure 6.3: Additional lattice NRQCD contributions to the axial vector and
vector heavy-light currents. The three upper diagrams, (a), (b) and (c), rep-
resent the vertex corrections at next-to-leading order in the inverse heavy
quark mass. The lower diagrams, from left to right, are the tadpole diagrams
for (d) the heavy quark and (e) the light quark wavefunction renormalisation
contributions respectively. Double blue lines represent heavy quarks, single
red lines indicate light quarks, wiggly green lines are gluons and the black
dots represent the current insertion.
where in this case, the temporal component of ΩJµ is given by
ΩJ0 (p
′, p) = ηt0Γ
J
0 − iη
t
1Γ
J
0
γ̂ ·p
2M
+ iηt2
γ̂ ·p′
2M
γ0Γ
J
0 +O(α
2
s, 1/M
2), (6.46)
and the spatial components are
ΩJk (p
′, p) = ηs0Γ
J
k − iη
s
1Γ
J
k
γ̂ ·p
2M
+ iηs2
γ̂ ·p′
2M
γ0Γ
J
k − iη
s
3
pk
2M
ΓJ0
− ηs4
ip′k
2M
ΓJ0 +O(α
2
s, 1/M
2). (6.47)
Here the temporal coefficients are
ηt0 = a1 + a2 = 1 + αsB
t
0,
ηt1 = a1 − a2 = 1 + αsB
t
1,
ηt2 = 2(a3 + a4 + a5) = αsB
t
2, (6.48)
127
Chapter 6. Heavy-light currents
with
Bt0 = −
1
pi
(
ln
λ
M
+
1
4
)
,
Bt1 = −
1
pi
(
ln
λ
M
+
19
12
)
,
Bt2 =
1
3pi
(
12 +
16pi
3
M
λ
)
. (6.49)
The spatial coefficients are given by
ηs0 = a1 = 1 + αsB
s
0,
ηs1 = η0,
ηs2 = 2a3 = αsB
s
2,
ηs3 = 2a2 = αsB
s
3,
ηs4 = 2a4 = αsB
s
4, (6.50)
where
Bs0 =
1
pi
(
ln
λ
M
−
11
12
)
,
Bs2 =
1
pi
(
1−
16pi
3
M
λ
− 12 ln
λ
M
)
,
Bs3 =
4
3pi
Bs4 =
1
3pi
(
1 + 4 ln
λ
M
)
. (6.51)
The lattice NRQCD currents corresponding to the nonrelativistic contin-
128
Chapter 6. Heavy-light currents
uum operators in Equations (6.46) and (6.47) are
J (0)µ,lat(x) = q(x)Γ
J
µQ(x),
J (1)µ,lat(x) = −
1
2aM
q(x)ΓJµγ˜ ·∇Q(x),
J (2)µ,lat(x) = −
1
2aM
q(x)γ˜ ·
←
∇ γ0Γ
J
µQ(x),
J (3)k,lat(x) = −
1
2aM
q(x)ΓJ0∇kQ(x),
J (4)k,lat(x) = −
1
2aM
q(x)
←
∇k Γ
J
0Q(x). (6.52)
Here q(x) is the light quark field in the lattice theory, M is the bare heavy
quark mass and Q(x) is given in terms of the NRQCD heavy quark field,
Ψ(x), by
Q(x) =
(
Ψ(x)
0
)
. (6.53)
In this dissertation I am concerned only with the leading order contribu-
tion to the heavy-light current in the inverse heavy quark mass expansion.
This term is the J (0)µ,lat current. However, as I discuss in the next section,
renormalisation of the lattice operators causing mixing-down from the cur-
rent J (1)µ,lat. Therefore, at zero external momentum, there are two vertex cor-
rection diagrams that I must calculate for the leading order current. The first
diagram is diagram (a) in Figure 6.2 with the J (0)µ,lat current insertion. The
second is the same diagram, (a) in Figure 6.2, but including the J (1)µ,lat current
insertion. In the language of [89], these contributions are ζ00 and ζ10. I now
discuss these contributions more fully.
6.5.1 The lattice operator mixing matrix
The lattice operators in Equation (6.52) mix under renormalisation and the
elements of the mixing matrix can be calculated using lattice perturbation
theory. I write the mixing matrix, ZJij, as
〈q(p′)|J (i)µ,lat|h(p)〉lat =
∑
j
ZJijuq(p
′)Ω(j),Jµ (p
′, p)uQ(p). (6.54)
129
Chapter 6. Heavy-light currents
I further decompose the mixing matrix at one-loop as
Zij = δij + αs
(
1
2
(Zq + ZQ) δij + δi1δj1 + ζij
)
. (6.55)
In this expression, Zq is the light quark wavefunction renormalisation, ZQ
the heavy quark wavefunction renormalisation, ZM the heavy quark mass
renormalisation and ζij the vertex corrections.We can separate out the in-
frared divergences and express the renormalisation constants in terms of the
known logarithmic divergences, Zdivi , the infrared finite piece, Z˜i, and, where
appropriate, tadpole improvement corrections, Zu0i . here the tildes indicate
infrared finite contributions. Thus we write
Zq = Z˜q + Z
div
q + Z
u0
q
ZQ = Z˜Q + Z
div
Q
ZM = Z˜M + Z
u0
M
ζij = ζ˜ij + ζ
div
ij + ζ
u0
ij . (6.56)
The divergent contributions to the wavefunction renormalisation are
Zdivq =
αs
3pi
ln a2λ2, (6.57)
and
ZdivQ =
2αs
3pi
(
ln aM − ln
a2λ2
a2M2
)
. (6.58)
The mass renormalisation arises because I have expressed the continuum
pole mass M in terms of the bare lattice mass, M0:
M = (1 + ZM)M0 +O(α
2
s). (6.59)
The pole mass must appear in the continuum currents, but the lattice cur-
rents are defined in terms of the bare lattice mass. I present results for the
heavy quark mass renormalisation in NRQCD in Section 5.1.4 and do not
repeat that part of my analysis here.
We are interested in constructing the continuum operators from their
130
Chapter 6. Heavy-light currents
lattice counterparts, so we invert this relation and remove the dependence
on the external states, giving
Ω(j),Jµ (p
′, p) =
∑
j
(
ZJji
)−1
J (i)µ,lat. (6.60)
In the rest of this chapter I focus on the temporal component of the
axial-vector current. For this case, the mixing relation becomes
A0 = (1 + αsρ˜0)J
(0)
0,lat + (1 + αsρ1)J
(1),sub
0,lat
+ αsρ2J
(2),sub
0,lat +O(α
2
s, 1/M
2, a2αs). (6.61)
Here I have introduced the improved, more physical currents, J (i),sub0,lat , which
are defined by
J (i),sub0,lat = J
(i)
0,lat − αsζ10J
(0)
0,lat. (6.62)
The coefficients, ρ˜0, ρ1 and ρ2 are given by
ρ˜0 = B˜0 −
1
2
(Z˜q + Z˜Q)− ζ˜00,
ρ1 = B˜1 −
1
2
(Z˜q + Z˜Q)− Z˜M − ζ˜01 − ζ˜11,
ρ2 = B˜2 − ζ˜02 − ζ˜22, (6.63)
with
B˜0 =
1
pi
(
ln(aM0)−
1
4
)
,
B˜1 =
1
pi
(
ln(aM0)−
19
12
)
,
B˜2 =
4
pi
. (6.64)
These improved matrix elements have the O(αs/(aM0)) power law contri-
butions subtracted out [107]. We can understand the origin of these power-
law divergences as follows. The nonleading currents, J (i6=0)lat , mix with the
other NRQCD currents, renormalising the leading order current J (0)lat . We
131
Chapter 6. Heavy-light currents
can determine the form of the constant of proportionality using dimensional
arguments: we must have
J (1)lat =
C10
aM
J (0)lat +O
(
1
M
)
+ . . . (6.65)
The leading term on the right hand side of Equation (6.65) is the power
law contribution to the matrix element. Our matching calculation should
remove these artifacts of the effective theory. We match perturbatively, so
we can only remove power law contributions at a given order in the coupling
constant. If we write Equation (6.61) in terms of the uncorrected currents,
we have
A0 = (1 + αsρ0)J
(0)
0,lat + (1 + αsρ1)J
(1)
0,lat + αsρ2J
(2)
0,lat, (6.66)
where
ρ0 = B0 −
1
2
(Zq + ZQ)− ζ00 − ζ10. (6.67)
Concentrating for the moment on the mixing of J (1)0,lat with J
(0)
0,lat, the rel-
evant terms are
(1 + αsρ0)J
(0)
0,lat + J
(1)
0,lat. (6.68)
We can rewrite the J (1)0,lat current using Equation (6.65) to obtain
(1 + αsρ˜0)J
(0)
0,lat +
[
C10
aM
− αsζ10
]
J (0)0,lat +
1
aM
+ . . . (6.69)
The mixing coefficient ζ10 represents the projection of J
(1)
0,lat onto J
(0)
0,lat at one-
loop, which is precisely the one-loop contribution to C10/aM in Equation
(6.65). We have thus reduced the power law term, [C10/aM − αsζ10] J
(0)
0,lat,
from anO(αs/aM) contribution toO(α2s/aM). For we convenience, we there-
fore work with the subtracted current J (1),sub0,lat , which is equal to the last two
terms in Equation (6.69) up to higher order corrections.
In this work I calculate the leading order contribution in the inverse heavy
132
Chapter 6. Heavy-light currents
quark mass expansion. This is just
A0 = (1 + αsρ0)J
(0)
0,lat +O(α
2
s, 1/aM, a
2αs). (6.70)
I now present all my results necessary to extract the matching coefficient
ρ0. I start with the relativistic wavefunction renormalisation.
6.6 Numerical results
I computed all contributions to the matching calculation using the HIPPY and
HPSRC routines. I introduce these routines in Section 3.3.1. I wrote each contri-
bution as a TaylUR valued FORTRAN function in an HPSRC module. This allowed
me to easily extract different derivatives of the selfenergy and to specify
massless or massive quarks with a single flag.
6.6.1 Relativistic wavefunction renormalisation
Infrared behaviour I extract the logarithmic infrared behaviour of the mass-
less wavefunction renormalisation by evaluating the wavefunction renormal-
isation at different gluon masses and fitting to a function logarithmic in the
gluon mass.
I present my results for the divergent contribution to the wavefunction
renormalisation in Table 6.1. I quote the known analytic result, given in
Equation (6.57), in the first column of Table 6.1 as comparison. My results
are in excellent agreement with expected analytic behaviour.
I use Symanzik-improved gluons with three different relativistic staggered
quark actions, unimproved, ASQTad and HISQ. I show results obtained
with 12 different gluon masses, with values between a2λ2 = 1 × 10−3 and
a2λ2 = 10−8. I used exactly massless light quarks for the massless wavefunc-
tion renormalisation.
In Figure 6.4 I plot the massless wavefunction renormalisation for unim-
proved staggered quarks, ASQTad quarks and HISQ quarks. I include the
fits to logarithmic functions of the gluon mass for each action. The data in-
133
Chapter 6. Heavy-light currents
Analytic Unimproved ASQTad HISQ
0.106103 0.10611(11) 0.10587(30) 0.106095(8)
Table 6.1: Numerical results for the infrared divergent contributions to the
massless light quark wavefunction renormalisations, Z(IR)q , for three differ-
ent actions. I compare my results to the coefficient of the known analytic
divergence in column two, given to six significant places.
dicate that the fit is excellent and that my HPSRC routines are correctly able
to handle the logarithmic behaviour of the wavefunction renormalisation.
Finite contributions I use two independent methods to extract the finite con-
tributions to the massless wavefunction renormalisation. For the first method,
I calculate the finite contribution by computing the wavefunction renormal-
isation at different gluon masses and fitting to a function logarithmic in the
gluon mass. For the second method, I remove the logarithmic behaviour using
an infrared subtraction function, given by
Zsubq = −
32pi
3
∫
d4k
(2pi)4
k2 − k20
k2(k2 + λ2)2
. (6.71)
I evaluate the subtraction function numerically and extract the logarith-
mic behaviour by fitting to a function of the gluon mass. The subtraction
function is very quick to evaluate: I use 108 function evaluations and 20
VEGAS iterations for each gluon mass, which took only five minutes on six
processors. In Table 6.2 I show my results for the finite contribution to the
infrared subtraction function for unimproved staggered, ASQTad and HISQ
quarks. In Figure 6.5 I plot my results for the wavefunction renormalisation
with infrared behaviour subtracted out. This plot clearly shows the infrared
divergences are cleanly handled by the subtraction function.
In Table 6.3 I compare my results for the finite part of the rainbow dia-
gram contribution to the massless and massive wavefunction renormalisations
using the two different methods described above. I constrained the coefficient
of the logarithmic divergence to its known analytic value. The data in Table
134
Chapter 6. Heavy-light currents
-3-2.8-2.6-2.4-2.2-2-1.8-1.6-1.4-1.2-1
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01Wavefunction renormalisation Gluon mass^2 UnimprovedASQTadHISQ
Figure 6.4: Plot of the infrared behaviour of the massless wavefunction renor-
malisation for three light quark actions. I plot results for unimproved stag-
gered quarks with red crosses, for ASQTad quarks with green triangles and
for HISQ quarks with blue diamonds. Errorbars are smaller than the data
points. I show the corresponding fits to logarithmic functions in the gluon
mass as solid lines. I include only contributions from the infrared divergent
rainbow diagram, not the infrared finite contributions from the tadpole dia-
gram.
-0.609-0.608-0.607-0.606-0.605-0.604-0.603-0.602-0.601-0.6
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01Wavefunction renormalisation Gluon mass^2
Figure 6.5: Plot of the massless ASQTad wavefunction renormalisation with
infrared divergence subtracted. I plot a fit to a constant function. Note the
very small scale on the vertical axis.
135
Chapter 6. Heavy-light currents
Zsubq Z
sub,div
q
-0.191617(5) 0.106099(3)
Table 6.2: Numerical results for the light quark wavefunction renormalisa-
tion subtraction function, Zq. I show both the coefficients of the logarithmic
divergence, Zsub,divq , and the finite contribution, Z
sub
q .
6.3 show that, whilst both methods handle the infrared structure well, the
subtraction function method is most effective for the unimproved staggered
quark action. However, this may simply reflect the need for greater statistics
with ASQTad and HISQ quark calculations. Comparison with results in the
literature, which I review in the next section, suggests that I can more reliably
extract the infrared finite contributions using the constrained fit method.
Coefficient Unimproved ASQTad HISQ
Zrbowq via subtraction -0.60394(75) 0.79615(29) -0.8169(48)
Zrbowq via fit -0.60316(20) -0.79726(61) -0.81598(24)
Table 6.3: Numerical results for the finite rainbow diagram contributions to
the light quark wavefunction renormalisation, Zq. I show unimproved stag-
gered quarks in column two, ASQTad quarks in column three and HISQ
quarks in column four.
I show my results for all infrared finite contributions to the wavefunction
renormalisation in Table 6.4. I use three different sets of actions: unimproved
staggered quarks with unimproved Wilson gluons; ASQTad light quarks with
Symanzik-improved gluons; and HISQ light quarks with Symanzik-improved
gluons.
The individual contributions are the infrared finite contribution from the
rainbow diagram, Zrbowq ; the tadpole contribution, Z
tad
q ; and the tadpole cor-
rection factor, Zu0q . I also show the total infrared finite result, Z˜q, in Table
6.4.
The tadpole correction for the ASQTad wavefunction renormalisation is
136
Chapter 6. Heavy-light currents
Action ξ Zrbowq Z
tad
q Z
u0
q Z˜q
Unimproved
1 -0.6032(2) 1.296(1) − 0.693(1)
0 0.0001(3) 0.973(1) − 0.973(1)
ASQTad
1 -0.7973(6) 1.600(1) -1.72593(5) -0.923(1)
0 0.0001(5) 1.309(1) -1.72593(5) -0.417(1)
HISQ
1 -0.8160(2) 0.424(1) − -0.392(1)
0 0.0001(3) 0.134(1) − 0.134(1)
Table 6.4: Numerical results for all three finite contributions to the light quark
wavefunction renormalisation, Z˜q, for three different actions. I use Feynman
gauge, with ξ = 1, and Landau gauge, with ξ = 0.
given in [89] as
Zu0q = −
(
4−
1
4
−
3
2
)
u(2)0 = −
9
4
u(2)0 , (6.72)
where, in this case, I use the value of u(2)0 for Landau tadpoles from Equation
(4.5). I do not include tadpole corrections for the unimproved light quark
action. The tadpole correction is not required for HISQ quarks, because the
high energy modes that contribute to tadpole diagrams are reduced by con-
struction as part of the improvement procedure that reduces taste-changing
interactions.
I show results for both Feynman gauge, with gauge parameter ξ = 1, and
Landau gauge, with ξ = 0. For the rainbow diagram in Feynman gauge I use
12 different gluon masses and fit to logarithmic function of the gluon mass.
In Landau gauge, the rainbow contribution is infrared finite.
6.6.2 Comparison to the literature
Gulez et al. calculate the massless ASQTad wavefunction renormalisation in
[89]. We can see, by comparing my ASQTad results for the wavefunction
renormalisation in Table 6.4, with their values in Table 6.5, that I am able to
reproduce their results correctly. This agreement provides good confidence in
137
Chapter 6. Heavy-light currents
both my HPSRC routines and my fits for the logarithmic infrared behaviour.
As I discuss in the previous section, this agreement also justifies my choice of
ξ Zrbowq Z
tad
q Z
u0
q Z˜q
1 -0.798(3) 1.600 -1.726 -0.924(3)
0 0.000(3) 1.310 -1.726 -0.416(3)
Table 6.5: Numerical results for the finite contributions to the massless ASQ-
Tad wavefunction renormalisation, Z˜q. I take all results from [89]. The au-
thors of [89] use different notation, with Cq representing the wavefunction
renormalisation and Cregq specifying the infrared finite contribution from the
rainbow diagram, that is, Zrbowq in my notation.
fitting to the gluon mass to handle the infrared structure, because my results
obtained using the subtraction function method do not reproduce the data
in Table 6.5 so reliably.
In [47], Mu¨ller calculates the massless HISQ wavefunction renormalisa-
tion. I show his results in Table 6.6. We can see, by comparing my results in
Zrbowq Z
tad
q Z˜q
-0.8155(12) 0.4250(10) -0.3905(16)
Table 6.6: Numerical results for the finite contributions to the massless HISQ
wavefunction renormalisation from [47].
Table 6.4 with those in Table 6.6, that my results are in excellent agreement
with those of [47]. This establishes that my wavefunction renormalisation
routines work correctly and handle the infrared structure appropriately.
6.6.3 HISQ renormalisation parameters
I give the HISQ renormalisation parameters for three different masses in
Table 6.7 and 6.8. I calculate the massive HISQ renormalisation parameters
138
Chapter 6. Heavy-light currents
using exact modesummation on L4 lattices, where L is the spatial extent.
I implement twisted boundary conditions, with two-dimensional twists, to
regulate the infrared behaviour. I discuss twisted boundary conditions in
more detail in Section 3.3.2. I extrapolate the results to infinite lattice volume
using a polynomial fit to the inverse lattice side length.
In Figure 6.6 I present example results for the infinite volume extrapola-
tion. I show result for both the mass renormalisation and the wavefunction
renormalisation in Feynman gauge and use light quark mass amq = 0.1.
amq L Zm Zψ
ξ = 1 ξ = 0 ξ = 1 ξ = 0
0.01
4 0.03051297 0.04613782 -3.06231058 -4.50669680
6 0.02070912 0.03069662 -2.06558281 -2.94254494
8 0.01604188 0.02342252 -1.58606227 -2.19846734
12 0.01140586 0.01627413 -1.10484608 -1.45983861
16 0.00911651 0.01275412 -0.86284384 -1.09014261
24 0.00567171 0.00929034 -0.61847086 -0.71718699
28 0.00624375 0.00831570 -0.54751444 -0.60899295
∞ 0.0142(53) 0.010085(95) -0.1159(19) 0.1791(71)
0.05
4 0.03511824 0.05076104 -0.70002402 -0.89478086
6 0.02586957 0.03586130 -0.49326654 -0.57131307
8 0.02161898 0.02900166 -0.39027728 -0.41227203
12 0.01749555 0.02236487 -0.28070367 -0.24625620
16 0.01552171 0.01916007 -0.21982997 -0.15613470
24 0.01365601 0.01607513 -0.14889542 -0.05413055
28 0.01315297 0.01522533 -0.12501221 -0.02074331
∞ 0.010088(9) 0.002392(62) 0.0175(11) 0.0459(16)
Table 6.7: Renormalisation parameters for massive HISQ quarks. I extrapo-
late the results to L = ∞. Results at finite lattice spacing use modesumma-
tion and are therefore exact. Uncertainties for the L = ∞ results are from
the polynomial fit.
139
Chapter 6. Heavy-light currents
amq L Zm Zψ
ξ = 1 ξ = 0 ξ = 1 ξ = 0
0.10
4 0.04021517 0.05588301 -0.38471270 -0.42329209
6 0.03144242 0.04144233 -0.27003185 -0.24824283
8 0.02754374 0.03493137 -0.20866792 -0.15688533
12 0.02382350 0.02869585 -0.13745787 -0.05434727
16 0.02208393 0.02572454 -0.09374485 0.00631186
24 0.02046735 0.02288796 -0.03806222 0.08087924
28 0.02002501 0.02209865 -0.01821517 0.10676488
∞ 0.017421(17) 0.017416(10) 0.1026(68) 0.2642(79)
0.30
4 0.05594294 0.07173053 -0.12182111 -0.05634316
6 0.04825177 0.05831329 -0.05900989 0.02925742
8 0.04514575 0.05257860 -0.01968736 0.08121890
12 0.04203358 0.04693684 0.03140938 0.14692902
16 0.04030510 0.04396936 0.06536116 0.18964292
24 0.03810375 0.04054048 0.11094060 0.24600307
28 0.03727723 0.03936478 0.12778618 0.26658286
∞ 0.0.32560(58) 0.033352(23) 0.2310(11) 0.3926(74)
Table 6.8: Renormalisation parameters for massive HISQ quarks. I extrapo-
late the results to L = ∞. Results at finite lattice spacing use modesumma-
tion and are therefore exact. Uncertainties for the L = ∞ results are from
the polynomial fit.
140
Chapter 6. Heavy-light currents
-1-0.8-0.6-0.4-0.2 0 0.2 0.4
0 5 10 15 20 25 30Renormalisation parameter L Mass renormalisationWavefunction renormalisation
Figure 6.6: Plot of the infinite volume extrapolation of HISQ renormalisation
parameters in Feynman gauge. I plot results for the mass renormalisation, in
red, and the wavefunction renormalisation in green. I use a light quark mass
of amq = 0.1.
6.6.4 Lattice operator matching
I now present my results for the lattice vertex corrections. I used three dif-
ferent sets of actions to calculate the vertex corrections, with seven different
combinations of heavy quark mass and stability parameter for each set of
actions. I used a gluon mass as the infrared regulator. The vertex correction
ζ00 is infrared divergent in Feynman gauge, so I extracted the finite contribu-
tion using a subtraction function and a gluon mass of a2λ2 = 10−6. I use the
subtraction function method to speed up the code because the computation
of the vertex correction is relatively computationally expensive.
The subtraction function depends on the heavy quark mass, but as an
example I plot a fit to the logarithmic infrared behaviour of the subtraction
function with heavy quark mass aM = 2.8 in Figure 6.7. This plot demon-
strates that the infrared structure is correctly handled by the subtraction
function. Using a constrained fit to the logarithmic divergence, which must
match the known analytic continuum result, I obtain
F sub,finiteζ00 = −0.221626(75) (6.73)
141
Chapter 6. Heavy-light currents
for the finite contribution to the subtraction function. In Landau gauge ζ00
-2.4-2.2-2-1.8-1.6-1.4-1.2-1-0.8-0.6
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01Subtraction function Gluon mass^2
Figure 6.7: Plot of the infrared behaviour of the subtraction function, with
a constrained fit to the data. I use a heavy quark mass of aM = 2.8. Uncer-
tainties are smaller than the data points.
is infrared finite, whilst the vertex correction ζ10 is infrared finite and gauge-
independent. The sets of actions I used are as follows:
• The “onlyH0” NRQCD action with unimproved staggered quarks and
unimproved Wilson gluons, Table 6.9. I used 107 function evaluations
and 20 VEGAS iterations, which took about one hour on six processors
for each gluon mass.
• Highly-improved NRQCD with ASQTad light quarks and Symanzik-
improved gluons, Table 6.10. I used 5×105 function evaluations and 20
VEGAS iterations, which took about five minutes on twelve processors
for each gluon mass. I give results for the kernel ordering BACAB for
the NRQCD action (I review this issue in more detail in Section 5.2.2).
• Highly-improved NRQCD with HISQ light quarks and Symanzik-imp-
roved gluons, Table 6.11. I used 107 function evaluations and 20 VEGAS it-
erations, which took about one hour on six processors for each gluon
mass. I use the NRQCD kernel ordering BACAB.
I am now able to extract the mixing coefficient ρ˜0, which I define in
Equation (6.63). I present my final results for massless HISQ light quarks,
142
Chapter 6. Heavy-light currents
aM n ζ00 ζ10
ξ = 1 ξ = 0
5.40 1 0.622(1) 0.1194(3) -0.1436(1)
4.00 2 0.592(1) 0.0865(2) -0.1813(1)
2.80 2 0.544(1) 0.0381(2) -0.2270(1)
2.10 4 0.524(1) 0.0158(1) -0.2746(2)
1.95 2 0.508(1) 0.0016(1) -0.2726(2)
1.95 4 0.517(1) 0.0093(1) -0.2853(2)
1.60 4 0.503(1) -0.0048(1) -0.3133(2)
Table 6.9: Results for the vertex corrections at seven different quark mass
and stability parameter combinations. All results use the “onlyH0” NRQCD
action with massless unimproved staggered quarks.
aM n ζ00 ζ10
ξ = 1 ξ = 0
5.40 1 0.685(1) 0.1784(1) -0.0952(1)
4.00 2 0.744(1) 0.2371(2) -0.1235(1)
2.80 2 0.835(1) 0.3279(2) -0.1658(1)
2.10 4 0.932(2) 0.4242(3) -0.2100(1)
1.95 2 0.960(2) 0.4528(3) -0.2227(1)
1.95 4 0.961(2) 0.4513(3) -0.2189(1)
1.60 4 1.048(2) 0.5391(3) -0.2599(2)
Table 6.10: Results for the vertex corrections at seven different quark
mass and stability parameter combinations. All results use highly-improved
NRQCD and massless ASQTad quarks.
with highly-improved NRQCD heavy quarks and Symanzik-improved gluons
in Table 6.12.
6.6.5 Comparison to the literature
In Table 6.13 I show the results for the leading order vertex corrections
calculated for highly-improved NRQCD with massless ASQTad quarks in
143
Chapter 6. Heavy-light currents
aM n ζ00 ζ10
ξ = 1 ξ = 0
5.40 1 0.622(1) 0.1147(1) -0.0615(1)
4.00 2 0.657(1) 0.1505(1) -0.0806(1)
2.80 2 0.712(1) 0.2054(2) -0.1104(1)
2.10 4 0.771(1) 0.2624(2) -0.1417(1)
1.95 2 0.786(1) 0.2791(2) -0.1510(1)
1.95 4 0.787(1) 0.2785(2) -0.1511(1)
1.60 4 0.838(1) 0.3291(3) -0.1789(1)
Table 6.11: Results for the vertex corrections at seven different quark
mass and stability parameter combinations. All results use highly-improved
NRQCD and massless HISQ quarks.
aM n ρ˜0 ζ10 ρ0
ξ = 1 ξ = 0 ξ = 1 ξ = 0
5.40 1 0.243(2) -0.284(2) -0.0615(1) 0.305(2) 0.858(2)
4.00 2 0.239(2) -0.291(2) -0.0806(1) 0.319(2) -0.210(2)
2.80 2 0.266(2) -0.262(2) -0.1104(1) 0.377(2) -0.152(2)
2.10 4 0.317(2) -0.210(2) -0.1417(1) 0.458(2) -0.068(2)
1.95 2 0.350(2) -0.175(2) -0.1510(1) 0.501(2) -0.024(2)
1.95 4 0.341(2) -0.186(2) -0.1511(1) 0.492(2) -0.034(2)
1.60 4 0.421(2) -0.105(2) -0.1789(1) 0.600(2) 0.074(2)
Table 6.12: Results for the mixing coefficient ρ˜0 for massless HISQ light
quarks with highly-improved NRQCD and Symanzik-improved gluons.
[89]. We can directly compare the values in Table 6.13 with my results in
Table 6.10. We see my results are in excellent agreement with those of [89].
This agreement provides strong confidence in the fully automated method
that I use to extract my results.
144
Chapter 6. Heavy-light currents
aM n ζ00 ζ10
5.40 1 0.687(4) -0.095
4.00 2 0.746(4) -0.123
2.80 2 0.836(4) -0.166
2.10 4 0.932(4) -0.210
1.95 2 0.958(4) -0.222
1.95 4 0.961(4) -0.219
1.60 4 1.048(4) -0.259
1.20 6 1.207(4) -0.329
1.00 6 1.341(4) -0.378
Table 6.13: Results for the vertex corrections from [89] at 10 different quark
mass and stability parameter combinations. All values use highly-improved
NRQCD with massless ASQTad quarks in Feynman gauge. The authors of
[89] do not provide explicit values for ζ00, but these values are easily extracted
from their data.
6.7 Conclusions and outlook
In this chapter I have computed the massless relativistic quark wavefunc-
tion renormalisation, the massive HISQ renormalisation parameters and the
vertex corrections to the leading order heavy-light lattice currents. These
results, which I obtained for three different sets of actions, enabled me to
extract the leading order matching coefficient between the temporal compo-
nent of the heavy-light current in continuum QCD and on the lattice. My
results for ASQTad light quarks serve as a check of my code and are in perfect
agreement with the results given in [89].
My results using HISQ light quarks will serve as an input into the HPQCD
collaboration’s determination of the fB and fBs decay constants. The HPQCD
collaboration has previously calculated these decay constants using the MILC
ensembles with ASQTad light quarks [50, 107, 108, 109]. However, there is
currently tension between the value of |Vub| determined from experimental
measurements of B± → τ±ν and from determinations using global fits to
the CKM parameters [110]. Experimental data from the LHCb and the pro-
145
Chapter 6. Heavy-light currents
posed SuperB factory will improve the experimental resolution by an order
of magnitude and corresponding improvements in the theoretical inputs are
required. The HPQCD collaboration are therefore calculating the decay con-
stants fBs and fDs using NRQCD with HISQ light quarks, which should
improve the precision of the the decay constants [111].
My work can be progressively extended in a number of steps. As part of
the HPQCD collaboration’s calculation of the form factors for semileptonic
B decays, such as B → pi`ν`, the spatial components of the vector current,
including non-leading 1/M corrections, will need to be determined. For the
B → pi, B → K and Bs → K decays, massless HISQ light quarks are
sufficient. Massive light quarks will be required for the B → D and Bs → Ds
decays.
To facilitate this final extension, the massless continuum calculation will
also need to be calculated numerically using automated lattice perturbation
theory and extrapolated to the continuum limit. The analytic calculation is
complicated by the presence of two different scales and this is where the use of
a numerical calculation on fine lattices will help. The relativistic calculation
can be extended very simply by using massive light quark vertex files, with
little additional modification required.
146
Chapter 7
Conclusions and outlook
Lattice perturbation theory plays an important role in the extraction of pre-
cise predictions of physical processes from lattice simulations. Modern ac-
tions, such as HISQ and NRQCD, are highly-improved, which greatly reduces
the discretisation effects and systematic errors associated with nonperturba-
tive simulations. The price, however, is that calculating perturbative quan-
tities is no longer feasible by hand. In this dissertation I have applied auto-
mated lattice perturbation theory to a number of perturbative calculations
involving highly-improved actions at both one- and two-loops.
In the first chapter I introduce the basic formalism of lattice QCD and
highlight the important role that lattice perturbation theory plays in extract-
ing precise results from lattice simulations. In the second chapter I motivate
and justify lattice perturbation theory in more detail and introduce several
tools of the trade: automation algorithms, twisted boundary conditions and
background field gauge.
In the course of this dissertation I performed a number of calculations
using the automated perturbation theory routines HIPPY and HPSRC. I review
and summarise these calculations in the following sections. I also discuss
some open issues and indicate directions for future work.
7.1 Tadpole improvement
In Section 4.1 I calculate the tadpole improvement factors for both Landau
and plaquette tadpoles. I confirm previous one-loop results and obtain, for the
first time, the fermionic contributions to the two-loop tadpole improvement
factors. I combine my results with data from high-β simulations carried out by
Lee [11] to present the first calculation of the plaquette tadpole at two-loops
for Symanzik-improved gluons with ASQTad light quarks. I also connect my
results with those of [12] to extract the full two-loop Landau tadpole, again
147
Chapter 7. Conclusions and outlook
for Symanzik-improved gluons with ASQTad light quarks. I then use data
from simulations by the HPQCD collaboration [93] to estimate the three-loop
contribution to both Landau and plaquette tadpoles. I find agreement with
the study in [86] that the nonperturbative Landau tadpoles are well-described
by the two-loop approximation.
Current simulations by the HPQCD collaboration use HISQ light quarks,
for which the two-loop tadpole improvement factors need to be calculate.
This calculation is complicated by the reunitarisation operators in the HISQ
action, but the modular nature of the HPSRC modules that I wrote to compute
the tadpole improvement factors ensures that my results can extended to
HISQ with relative ease.
7.2 Operator improvement
I calculate the one-loop corrections to the higher order kinetic operators in
the NRQCD action in Section 4.4. I use a mix of analytic and numerical in-
tegration to extract the one-loop corrections for two simple NRQCD actions.
I demonstrate that these corrections are infrared divergent unless the action
is improved to the appropriate order. I compare my results to those obtained
by Mu¨ller using a fully automated procedure in [47] and the agreement of
our results provides confidence in both methods.
This work was motivated by the HPQCD collaboration’s discovery of large
spin splittings between mesonic states with the same total spatial momentum,
but different components of the momentum, in nonperturbative NRQCD
simulations. These spin splittings were removed by nonperturbatively tuning
the coefficient of theO(a4p4) kinetic operator in the NRQCD action to a value
of 2.6 [47]. Unfortunately this nonperturbative result is inconsistent with the
one-loop corrections, which were found to be relatively small, approximately
0.4 for a range of heavy quark masses between aM = 2.0 and aM = 3.4 [47].
There are a number of other sources that could contribute to the discrep-
ancy between these results. The first possibility is a large two-loop correction
to the perturbative result. Two-loop perturbative results would be required
to confirm this, but the complexity of two-loop lattice perturbation theory
148
Chapter 7. Conclusions and outlook
in this case would be formidable.
A second possibility is that O(a6p6) corrections strongly influence the cor-
rection coefficient. Obtaining higher order kinetic corrections may prove nec-
essary in the future and in Appendix D I briefly discuss extending my work
to O(a6p6) to investigate the feasibility of further radiative improvement.
However, this may not prove necessary. In the nonperturbative simulations,
the O(a6p6) improvement term was included with the tree-level value of the
coefficient. The fitting process, whereby the O(a6p6) was fixed to its classical
value and the coefficient of the O(a4p4) term nonperturbatively tuned, may
not be optimal and indeed could be another source of the difference in pertur-
bative and nonperturbative results. Finally, there may be errors introduced
by the relative coarseness of the lattice used in these simulations.
Further work is needed to clarify this discrepancy. Nonperturbative sim-
ulations on finer lattices are being explored and these may shed light on the
issue. Extending the perturbative renormalisation calculation to O(αsa6p6)
and O(α2sa
4p4) could also illuminate the problem more brightly. However,
the computational complexity of this extension is impressive and not to be
underestimated.
I also briefly review the calculation of the one-loop radiative correction to
the chromo-magnetic operator in the highly-improved NRQCD action. This
calculation was undertaken by Hammant [14] and is the first outcome of our
development and implementation of background field gauge in the HPSRC and
HIPPY routines.
As part of the development of our implementation of background field
gauge I derived background field gauge Feynman rules. I also hardwired the
ghost-gluon vertices in an HPSRC routine. A calculation of the known ratio
between the QCD scale parameter on the lattice and in the MS scheme
using the gluon selfenergy in background field gauge will serve as a further
check of our background field gauge implementation. This computation will
build on my analysis of the Ward identities for the gluon selfenergy.
149
Chapter 7. Conclusions and outlook
7.3 The b quark mass
In Chapter 5 I calculate the one-loop renormalisation parameters for a variety
of NRQCD actions. The modules I wrote for these calculations were the basis
of my computation of the fermionic contributions to the two-loop heavy quark
energy shift. To my knowledge this calculation is the first two-loop calculation
carried out for lattice NRQCD.
I combine my results with data extracted from high-β simulations data
undertaken by Lee [11] to determine the full heavy quark energy shift at
two-loops. With this result, I then extract a new prediction of the mass
of the b quark from nonperturbative studies by Dowdall and the HPQCD
collaboration [93]. My results are in good agreement with recent studies by
three other lattice collaborations and improves significantly on the previous
determination by the HPQCD collaboration using lattice NRQCD. I reduced
the uncertainty in the prediction to approximately one percent by removing
the dominant contribution to the uncertainty in the b quark mass, which
came from the two-loop energy shift.
The dominant sources of uncertainty in my results are the three-loop
heavy quark energy shift and the systematic uncertainties arising from lattice
artifacts. Results from Lee provide an estimate of the quenched three-loop
contribution [11], but the fermionic contribution is yet to be determined. In
principle, this is a possible extension of my work. However, there are a large
number of diagrams to compute at three-loops and in practice, it may be
more feasible to extract higher order contributions from high-β simulations.
New simulations at different heavy quark masses are required for a more
complete understanding of the lattice artifact errors.
7.4 Heavy-light currents
Finally, in Chapter 6, I compute the matching coefficients between heavy-
light currents in continuum QCD and NRQCD with massless HISQ light
quarks. I include only the leading order contributions in the inverse heavy
quark mass expansion. As part of this process, I confirm the results of [89],
150
Chapter 7. Conclusions and outlook
which used highly-improved NRQCD with massless ASQTad light quarks.
The temporal component of the axial-vector and vector currents serve as
inputs into the HPQCD collaboration’s determination of the fB and fBs
decay constants.
There are a number of stages to the extension of this work. As part
of the HPQCD collaboration’s calculation of the semi-leptonic B decays,
the spatial components of the vector current, including non-leading 1/M
corrections, will need to be determined. For the B → pi, B → K and Bs → K
decays, massless HISQ light quarks are sufficient. However, for the B → D
and Bs → Ds decays, non-zero mass light quarks will be required. This
final extension will require the continuum massive light quark calculation.
Rather than extending the analytic continuum calculation, my HIPPY and
HPSRC code can be developed to compute the continuum results numerically,
using relativistic quarks on fine lattices with high statistics. This will simplify
the extension to massive light quarks, which would otherwise be technically
involved if performed analytically.
151
Appendix A
Conventions
In this appendix I summarise the conventions and notation that I employ in
this dissertation.
A.1 Notation
Units I use units in which ~ = c = 1.
Indices Greek indices run from 0 to 3; latin indices run from 1 to 3.
Metric The Minkowski metric is gµν = diag(1,−1,−1,−1); the Euclidean
metric is δµν = diag(1, 1, 1, 1).
Wick rotation I relate Euclidean four-vectors, denoted (τ,x), and derivatives
to their Minkowski counterparts, denoted (t,xM), as follows:
τ = it, xj = x
j = xjM = −x
M
j . (A.1)
I relate derivatives via
∂0 = ∂
0 = −i∂0M = −i∂
M
0 , ∂j = ∂
j = −∂jM = ∂
M
j . (A.2)
Gauge fields I define the gauge field as
A0 = A
0 = −iA0M = −iA
M
0 , Aj = A
j = −AjM = A
M
j . (A.3)
Chromo-electric and -magnetic fields I relate the Euclidean chromo-electric
and chromo-magnetic fields to the Euclidean field strength tensor
Ei = −F0i, Bi = −
1
2
²ijkFjk. (A.4)
152
Appendix A. Conventions
Dirac algebra Euclidean gamma matrices, γ̂µ, satisfy
{γ̂µ, γ̂ν} = 2δµν . (A.5)
I relate the Euclidean gamma matrices to the Minkowski gamma matrices,
γµ, as follows:
γ̂0 = γ̂
0 = γ0 =
(
σ0 0
0 −σ0
)
, γ̂j = −iγ
j =
(
0 −iσj
iσj 0
)
, (A.6)
γ̂5 = γ̂0γ̂1γ̂2γ̂3 = γ5 =
(
0 σ0
σ0 0
)
. (A.7)
A.2 Lattice derivatives and field strength
I now give explicit expressions for the derivatives that I use in the lattice
actions throughout this dissertation. I denote link variables by Uµ(x).
Elementary forward, backward and symmetric difference operators
∆+µψ(x) = Uµ(x)ψ(x+ µˆ)− ψ(x), (A.8)
∆−µψ(x) = ψ(x)− U−µ(x)ψ(x− µˆ), (A.9)
∆±µψ(x) =
1
2
[
Uµ(x)ψ(x+ µˆ)− U−µ(x)ψ(x− µˆ)
]
. (A.10)
Unimproved derivative
∆(2n) =
3∑
i=1
(∆+i ∆
−
i )
n. (A.11)
Improved derivatives
∆˜±µ = ∆
±
µ −
1
6
∆+i ∆
±
i ∆
−
i , (A.12)
∆˜(2n) =
1
12
3∑
i=1
∆+i ∆
−
i ∆
+
i ∆
−
i . (A.13)
153
Appendix A. Conventions
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)
− Uν(x)Fνµ(x+ νˆ)U
†
ν(x)− U−ν(x)Fνµ(x− νˆ)U
†
−ν(x)
]
, (A.14)
where
Fµν(x) = −
i
2g
(
Ωµν(x)− Ω
†
µν(x)
)
,
Ωµν(x) =
1
4
∑
{(α,β)}
Uα(x)Uβ(x+ α̂)U−α(x+ α̂+ β̂)U−β(x+ β̂), (A.15)
and
{(α, β)} = {(µ, ν), (ν,−µ), (−µ,−ν), (−ν, µ)} for µ 6= ν. (A.16)
A.3 Lattice fields in Fourier space
For the purposes of deriving Feynman rules, I give the Fourier series repre-
sentation of lattice fields.
Gauge fields Lattice gauge fields are defined on the links between lattice
sites:
Aaµ(x) =
∫ pi/a
−pi/a
d4p
(2pi)4
eip(x+µˆ/2)A˜aµ(p), (A.17)
with inverse
A˜aµ(p) =
∑
x
e−ip(x+µˆ/2)Aaµ(x). (A.18)
Fermion fields Lattice fermions live on the lattice sites themselves:
ψ(x) =
∫ pi/a
−pi/a
d4p
(2pi)4
eipxψ˜(p), (A.19)
154
Appendix A. Conventions
where
ψ˜(p) =
∑
x
e−ipxψ(x). (A.20)
Ghost fields Ghosts have Fourier series representation
ca(x) =
∫ pi/a
−pi/a
d4p
(2pi)4
eipxc˜a(p), (A.21)
where
c˜a(p) =
∑
x
d4p
(2pi)4
e−ipxca(x). (A.22)
155
Appendix B
Lattice Feynman rules
In this appendix I derive Feynman rules for two simple discretisations of
NRQCD, for a quenched gauge action in background field gauge and for the
current operators used in Chapter 6. I give the Fourier series representations
of the lattice fields in Appendix A.
B.1 NRQCD: Davies and Thacker action
One choice of discretisation for an unimproved NRQCD heavy quark action
is given in Davies and Thacker [49]:
SDT =
∑
x,t
ψ†(x)
(
∆4 −
3∑
j=1
∆(2)
2M
)
ψ(x).
Here I implicitly trace over SU(3) colour generators. For all order three
vertices (and higher), there is an overall minus sign factor that must be
included because the Euclidean path integral weight is e−S, rather than
the Minkowskian eiS. For these higher order vertices, I consider S → −S.
Throughout this appendix, I set the lattice spacing to unity, a = 1. I take
colour factors directly from [49].
The Davies and Thacker action has Green function evolution equation
U4(x, t)G(x, t+ 1)−
(
1 +
∆(2)
2M
)
G(x, t) = δ(t)δ(x), (B.1)
with solution
G(x, t+ 1) = U †4(x, t)
(
1 +
∆(2)
2M
G(x, t− 1) + δ(t)δ(x)
)
(B.2)
and initial conditions G(x, t) = 0 for t ≤ 0.
Explicitly writing out the link variables in the covariant difference oper-
156
Appendix B. Lattice Feynman rules
ators, the Davies and Thacker action becomes
SDT =
∑
x,t
{
ψ(x)†U4(x)ψ(x, t+ 1)− ψ(x)†ψ(x)−
1
2M
3∑
j=1
[
− 2ψ(x)†ψ(x)
+ ψ(x)†Uj(x)ψ(x+ j, t) + ψ(x)†U
†
j (x− j)ψ(x− j, t)
]}
. (B.3)
In this appendix, I employ a variant of the vertex notation of [77] and
[78]. A general vertex V (i,j,k,l)(m, p, q, r, s)abcdµνρτ has i fermion legs, j back-
ground field legs, k quantum gluon legs and 2l ghost legs (one ghost and one
antighost). All momenta are incoming, with δ(m+p+q+r+s) = 0 and I take
the colour indices to be associated with the ordering (i, j, k, l). The Lorentz
indices are associated with the ordering (j, k, l), that is, the quarks do not
carry Lorentz indices. For example, for the vertex V (0,1,1,1)(p, q, r, s)abcdµνρσ the
associated fields are Baµ(p), q
b
ν(q), c
c
ρ(r) and c
d
σ(s).
B.1.1 Two-point vertices
We obtain the inverse quark propagator from the leading order terms in the
continuum expansion of the link variables (with no sum over µ),
Uµ(x) = 1− igT
aAaµ(x) +
g2
2
T aAaµT
bAbµ +O(g
3).
We Fourier expand the heavy quark fields to give
S(0)DT =
∑
x,t
∑
p,p′
ψ˜†(p′)ψ˜(p)ei(p0−p
′
0)tei(p−p
′) ·x
×
[
eip0 − 1 +
1
2M
∑
j
(e−ipj + eipj − 2)
]
=
∑
p,p′
ψ˜†(p′)ψ˜(p)δ(p− p′)
[
eip0 − 1 +
4
2M
∑
j
sin2
(pj
2
) ]
.
157
Appendix B. Lattice Feynman rules
Hence the corresponding Feynman rule is
V (2,0,0,0)(p,−p)ab = δab
(
eip0 − 1 +
1
2M
∑
j
p̂2j
)
(B.4)
for the inverse heavy quark propagator. In this expression I have used the
common lattice notation
p̂µ = 2 sin
(pµ
2
)
. (B.5)
B.1.2 Three-point vertices
The nonrelativistic expansion for NRQCD breaks the symmetry between
temporal and spatial directions and introduces separate Feynman rules for
temporal and spatial gluon vertices. For the temporal gluon three-point ver-
tex, only one term contributes:
ψ(x)†U4(x)ψ(x, t+ 1). (B.6)
We expand the link variable in terms of the gluon fields
S(1,T )DT = −ig
∑
x,t
ψ(x)†A0(x)ψ(x, t+ 1), (B.7)
which has Fourier series representation
S(1)DT = −ig
∑
x,t
∑
p′,p,k
ψ˜†(p′)A˜0(k)ψ˜(p)e
i(p0−p′0+k0)tei(p−p
′+k) ·xeik0/2eip0
= −ig
∑
p′,p
ψ˜†(p)ψ˜(p)A˜0(k)δ(p− p
′ + k)ei(p
′
0+p0)/2,
so the corresponding Feynman rule is
V (2,0,1,0)(p, q, r)abc0 = −ig(T
c)abei(p0+q0)/2. (B.8)
158
Appendix B. Lattice Feynman rules
The O(g) spatial terms in Equation (B.3) are
S1,SDT = −ig
∑
x,t
(
ψ(x)†Aj(x)ψ(x+ j, t) + ψ(x)†Aj(x− j)ψ(x− j, t)
)
,
which have Fourier space representation
S0,SDT = −
ig
2M
∑
j
∑
p′,p
ψ˜†(p′)ψ˜(p)A˜(k)δ(p− p′ + k)
(
ei(p+p
′)j/2 − e−i(p+p
′)j/2
)
,
(B.9)
and consequently an associated Feynman rule
V (2,0,1,0)(p, q, r)abcj = −
g
2M
(T c)ab(p̂+ q)j. (B.10)
The derivations of the four-point vertices follow a similar pattern. I have
derived these expression, but only state them here. The four-point vertices
are
V (2,0,2,0)(p, q, r, s)abcd00 =
g2
2
(T cT d)ab exp
(
i(p0 + q0)
2
)
(B.11)
and
V (2,0,2,0)(p, q, r, s)abcdjk =− δjk
g2
2M
(T cT d)ab cos
(
pj + qj
2
)[
1−
1
2M
∑
j
p̂2
]
.
(B.12)
B.2 NRQCD: “onlyH0” action
An alternative discretisation of an unimproved NRQCD heavy quark action
is
SH =
∑
x,t
ψ†(x)
[
ψ(x, t)− U †4(x, t− 1)
(
1 +
∆(2)
2M
)
ψ(x, t− 1)
]
(B.13)
159
Appendix B. Lattice Feynman rules
where I again implicitly trace over SU(3) colour generators. In this case the
action has an associated Green function evolution equation given by
G(x, t)− U †4(x, t− 1)
(
1 +
∆(2)
2M
)
G(x, t− 1) = δ(t)δ(x), (B.14)
with solution
G(x, t) = U †4(x, t− 1)
(
1 +
∆(2)
2M
)
G(x, t− 1) + δ(t)δ(x) (B.15)
and initial conditions G(x, t) = 0 for t < 0.
To clarify the following derivations I again write out the link variables
explicitly:
SH =
∑
x,t
{
ψ†(x)ψ(x)− ψ†(x)U †4(x, t− 1)ψ(x, t− 1)
−
1
2M
∑
j
[
ψ†(x)U †4(x, t− 1)Uj(x)ψ(x+ jˆ, t− 1) + ψ
†(x)U †4(x, t− 1)
× Uj(x− jˆ)
†ψ(x− jˆ, t− 1)− 2ψ†(x)U †4(x, t− 1)ψ(x, t− 1)
]}
.
(B.16)
B.2.1 Two-point vertices
For the action of Equation (B.13), we Fourier expand the heavy quark fields
and use the leading order terms in the continuum expansion of the link
variables to give
S(0)H =
∑
x,t
∑
p,p′
ψ˜†(p′)ψ˜(p)ei(p0−p
′
0)tei(p−p
′) ·x
×
[
1− e−ip0 −
e−ip0
2M
∑
j
(e−ipj + eipj − 2)
]
=
∑
p,p′
ψ˜†(p′)ψ˜(p)δ(p− p′)
[
1− e−ip0 −
e−ip0
2M
∑
j
p̂2j
]
.
160
Appendix B. Lattice Feynman rules
Hence the Feynman rule for the inverse heavy quark propagator is
V (2,0,0,0)(p,−p)ab = δab
[
1− e−ip0
(
1 +
1
2M
∑
j
p̂j
)]
. (B.17)
B.2.2 Higher order vertices
The derivation of the three- and four-point vertices follows exactly the scheme
of the derivation for the Davies and Thacker action, so I simply state the
results here.
The three-point vertices are
V (2,0,1,0)(p, q, r)abc0 = −ig(T
c)abe−i(p0+q0)/2
[
1−
1
2M
∑
j
p̂2j
]
(B.18)
and
V (2,0,1,0)(p, q, r)abcj = −
g
2M
(T c)abe−ip0(p̂+ q)j. (B.19)
The four-point vertices are
V (2,0,2,0)(p, q, r, s)abcd00 =−
g2
2
(T cT d)ab exp
(
−
i(p0 + q0)
2
)[
1−
1
2M
∑
j
p̂2j
]
(B.20)
and
V (2,0,2,0)(p, q, r, s)abcdjk =− δjk
g2
2M
(T cT d)ab cos
(
pj + qj
2
)[
1−
1
2M
∑
j
p̂2j
]
.
(B.21)
B.3 Quenched lattice QCD in background field gauge
I discuss background field gauge in more detail in Section 3.4.2. In this section
I derive the Feynman rules. The complete quenched lattice QCD action is
SBF[B, q, c, c] = SBFW [U ] + S
BF
GF[B, q] + S
BF
FP [B, q, c, c], (B.22)
161
Appendix B. Lattice Feynman rules
where SBFW is given in Equation (3.47), S
BF
GF in Equation (3.48) and S
BF
FP in
Equation (3.49).
In the continuum limit we can expand the action in terms of the four
fields, B, q, c and c. The Wilson term becomes
SW =
1
g20
Tr
∑
x
∑
µ,ν
Re
(
∆+µ (B + g0q)ν −∆
+
ν (B + g0q)µ
+ [(B + g0q)µ, (B + g0q)ν ]
)2
, (B.23)
whilst the gauge fixing term is
SGF = −λ0Tr
∑
x
(
qµ(x)− e
−Bµ(x−aµˆ)qµ(x− aµˆ)e
Bµ(x−aµˆ)
)
×
(
qν(x)− e
−Bν(x−aνˆ)qν(x− aνˆ)e
Bν(x−aνˆ)
)
(B.24)
and the Faddeev-Popov ghost term becomes
SFP = − 2Tr
∑
x
(
eBµ(x)c(x+ aµˆ)e−Bµ(x) − c(x)
) [
g0Ad[qµ(x)]c(x)
+
(
eBµ(x)c(x+ aµˆ)e−Bµ(x) − c(x)
) ]
(
1 +
1
2
g0Ad[qµ(x)]−
1
6
g20(Ad[qµ(x)])
2
)
.
(B.25)
B.3.1 Two-point vertices
There are four relevant two-point vertices: the inverse quantum and ghost
field propagators and two vertices with background field external legs. The
fifth two-point vertex is the measure term, which will not be discussed here.
The measure term is derived in, for example, [115].
162
Appendix B. Lattice Feynman rules
The V (0,0,2,0)(p,−p)abµν vertex
The contributing second order terms in the Wilson and gauge fixing parts of
the action are
S(0,0,2,0) = −
δab
2
Tr
∑
x
2
(
∆+µ qν∆
+
µ qν + (λ0 − 1)∆
+
ν qµ∆
+
µ qν
)
. (B.26)
We expand Equation (B.26) to first order in the coupling constant g0 and
use the Fourier series representation of the fields to give
S(0,0,2,0)W,1 = − δ
ab
∫
d4p
(2pi)4
d4q
(2pi)4
δ(p+ q)q˜aµ(p)q˜
b
ν(q)
(
δµν p̂
2 + (λ0 − 1)p̂µp̂ν
)
.
(B.27)
The corresponding Feynman rule is
V (0,0,2,0)(p,−p)abµν = δ
ab (Pµν(p) + λ0p̂µp̂ν) , (B.28)
where
Pµν = δµν p̂
2 − p̂µp̂ν . (B.29)
The derivation of the two–point vertices involving background gluons
both follow exactly the calculation for the inverse gluon propagator, without
the contribution from the gauge–fixing term. The results are
V (2,0,0)(p,−p)abµν = g
−2
0 δ
abPµν(p) (B.30)
and
V (1,1,0)(p,−p)abµν = g
−1
0 δ
abPµν(p). (B.31)
163
Appendix B. Lattice Feynman rules
B.3.2 Three-point vertices
The V (0,1,2,0)(p, q, r)abcµνρ vertex
The contributing term from the gauge fixing action is
S(0,1,2,0)GF = − 2λ0Tr
∑
x
(
qµ(x)− e
−Bµ(x−aµˆ)qµ(x− aµˆ)e
Bµ(x−aµˆ)
)
× [Bν(x− aνˆ), qν(x− aνˆ)]. (B.32)
For the purposes of updating the HIPPY /HPSRC routines to include the back-
ground field gauge, we need only consider the gauge fixing terms arising from
S(0,1,2,0)GF . The gauge fixing terms are hardwired into the HPSRC code, whereas
the other contributions are generated from the python vertex files.
The colour structure of the gauge fixing term is
C(0,1,2,0) = Tr
{
T b[T a, T c]
}
= Tr
{
facdT bT d
}
= −
1
2
fabc. (B.33)
The gauge fixing term is, after permuting the indices,
S(0,1,2,0)GF = λ0f
abc
∫
d4p
(2pi)4
d4q
(2pi)4
d4r
(2pi)4
[
δµρ
(
1− e−iqν
)
eiqν/2e−irρ/2
− δµν
(
1− e−irρ
)
e−iqν/2eirρ/2
]
e−ipµ/2δ(p+ q + r)B˜µ
a
(p)q˜ν
b(q)q˜ρ
c(r)
= iλ0f
abc
∫
d4p
(2pi)4
d4q
(2pi)4
d4r
(2pi)4
δ(p+ q + r)B˜µ
a
(p)q˜ν
b(q)q˜ρ
c(r)
×
[
eiqµ/2q̂ν − e
iqν/2r̂ρ
]
(B.34)
implying an associated Feynman rule
V (0,1,2,0)GF (p, q, r)
abc
µνρ = iλ0f
abc
(
δµρq̂νe
iqµ/2 − δµν r̂ρe
irµ/2
)
. (B.35)
This Feynman rule differs from equation [A.11] in [78] by a minus sign in
the exponential factors. The Feynman rules for background field gauge have
been independently checked by Tom Hammant. The analytical expressions
have been implemented in the HPSRC code, which has been used in several
164
Appendix B. Lattice Feynman rules
calculations and provides a further cross-check of these Feynman rules. I
therefore believe this discrepancy is an error in [78].
The V (0,1,0,1)(p, q, r)abcµ vertex
The background field ghost three–point interaction is induced via two terms
in SFP
S(0,1,0,1) = −2
∑
x
Tr
{
[Bµ(x), c(x+ aµˆ)]∆
+
µ c(x) + ∆
+
µ c(x)[Bµ(x), c(x+ aµˆ)]
}
(B.36)
The colour factor of the first term is fabc/2. For the second term the
colour factor is −fabc/2. So in momentum space S(0,1,0,1) is
S(0,1,0,1) = −fabc
∫
d4p
(2pi)4
d4q
(2pi)4
d4r
(2pi)4
[
eirµ
(
eiqµ − 1
)
− eiqµ
(
eirµ − 1
) ]
× e−i(q+r)µ/2B˜aµ(−q − r)c˜
b(q)˜c
c
(r)
= −fabc
∫
d4p
(2pi)4
d4q
(2pi)4
d4r
(2pi)4
δ(p+ q + r)B˜aµ(p)c˜
b(q)˜c
c
(r)i(̂q − r)µ.
(B.37)
Hence the Feynman rule associated with this vertex is
V (0,1,0,1)(p, q, r)abcµ = −if
abc(̂q − r)µ, (B.38)
in agreement with equation [A.12] in [78].
The V (0,0,1,1)(p, q, r)abcµ vertex
This three-point vertex arises from two terms in SFP in equation (B.25):
S(0,0,1,1) = −2g0
∑
x
Tr
{
1
2
∆+µ c(x)Ad[qµ(x)]∆
+
µ c(x) + ∆
+
µ c(x)Ad[qµ(x)]c(x)
}
.
(B.39)
The colour factor for these terms is
Tr {cAd[qµ]c} = Tr
{
caT a(Ad[qµ]c)
dT d
}
=
fabc
2
qaµc
bcc, (B.40)
165
Appendix B. Lattice Feynman rules
where I have used the property (Ad[X] ·Y )a = fabcXbY c.
Expanding equation (B.39) in momentum space gives
S(0,0,1,1) =− g0f
abc
∑
x
∫
d4p
(2pi)4
d4q
(2pi)4
d4r
(2pi)4
(
eiqµ − 1
)
eiqxeip(x+µˆ/2)
×
[
1
2
(
eirµ − 1
)
+ 1
]
eirxq˜aµ(p)c˜
b(q)˜c (r)
=− g0f
abc
∫
d4p
(2pi)4
d4q
(2pi)4
d4r
(2pi)4
δ(p+ q + r)q˜aµ(p)c˜
c(q)˜c (r)
× iqˆµ
1
2
(
eirµ/2 + e−irµ/2
)
. (B.41)
The Feynman rule is therefore
V (0,0,1,1)(p, q, r)abcµ = −ig0f
abcqˆµ cos
(rµ
2
)
. (B.42)
This Feynman rule disagrees with [A.8] of [78]. Using calculations per-
formed with HPSRC , and comparing the analytic expression with that derived
independently by Tom Hammant, I believe there is an error in the expression
of [78].
B.3.3 Four-point vertices
The four-point vertices are not listed in [78].
Unpicking the structure of the double commutator terms (those of the
form [X, Y ][Z,W ]), the associated colour factor is
C(4)A = Tr(T
aT c − T cT a)(T bT d − T dT b) = −
1
2
facef ebd. (B.43)
The other contributions to the four-point vertices have the general structure
({XY,Z} − 2XZY )W or W ({XY,Z} − 2XZY ), and have colour factors
given by
C(4)B = Tr(T
aT bT cT d + T aT bT dT c − 2T aT cT bT d), (B.44)
using the cyclic properties of the trace. However, since we are free to exchange
166
Appendix B. Lattice Feynman rules
the colour indices a and b on the background field legs, we have
C(4)A = −C
(4)
B ≡ C
(4). (B.45)
The V (0,2,0,1)(p, q, r, s)abcdµν vertex
The interaction of two background fields with a ghost anti–ghost pair is
generated by three terms in SFP.
These contributions have Fourier decomposition
S(0,2,0,1) =− 2C(4)
∫
d4p
(2pi)4
. . .
d4s
(2pi)4
δ(p+ q + r + s)B˜aµ(p)B˜
b
ν(q)c˜
c(r)˜c
d
(s)
× cos((r + s)µ/2). (B.46)
Hence the corresponding Feynman rule is
V (0,2,0,1)(p, q, r, s)abcdµν = −2δµνC
(4) cos
(
(r − s)µ
2
)
. (B.47)
The V (0,0,2,1)(p, q, r, s)abcdµν vertex
The interaction of two quantum fields with a ghost anti–ghost pair is gener-
ated by a single term in SFP:
S(0,0,2,1) = 2
∑
x
Tr
{
−
1
12
g20∆
+
µ c(x)(Ad[qµ(x)])
2∆+µ c(x)
}
. (B.48)
The colour factor is given by
Tr
{
ccT cAd[qµ](Ad[qµ]c)
eT e
}
=f bdeTr
{
T cT f (Ad[qµ])
ef
}
qbµc
ccd
=
1
2
f bdef ecaqaµq
b
µc
ccd = C(4)qaµq
b
µc
ccd. (B.49)
167
Appendix B. Lattice Feynman rules
Therefore the momentum expansion of S(0,0,2,1) is
S(0,0,2,1) =
C(4)
6
∫
d4q
(2pi)4
. . .
d4s
(2pi)4
δ(p+ q + r + s)q˜aµ(p)q˜
b
µ(q)c˜
c(r)c˜ (s)r̂µŝµ.
(B.50)
Hence the corresponding Feynman rule for the complete vertex is
V (0,0,2,1)(p, q, r, s)abcdµν =
1
6
g20C
(4)δµν r̂µŝν . (B.51)
The V (0,2,2,0)(p, q, r, s)abcdµνρσ vertex
The interaction of two background and two quantum gluons arises from the
fourth order terms in the gauge fixing action.
The first relevant term from the gauge fixing action is
S(0,2,2,0)GF,A = −λ0C
(4)
∑
x
δµρδνσB
a
µ(x− aµˆ)B
b
ν(x− aνˆ)q
c
ρ(x− aρˆ)q
d
σ(x− aσˆ),
(B.52)
with corresponding Feynman rule
V (0,2,2,0)GF,A (p, q, r, s)
abcd
µνρσ = −λ0C
(4)δµρδνσe
−i(q+s)ν/2ei(q+s)µ/2. (B.53)
The Fourier series representation of the other gauge fixing contributions
is
S(0,2,2,0)GF,B = λ0C
(4)
∫
d4p
(2pi)4
. . .
d4s
(2pi)4
(
δµσδνσ
(
1− e−irρ
)
eirρ/2e−isσ/2
+ δµρδνρe
−irρ/2
(
1− e−isσ
)
eisσ/2
)
e−ipµ/2e−iqν/2
× δ(p+ q + r + s)B˜aµ(p)B˜
b
µ(q)q˜
c
ρ(r)q˜
d
σ(s)
= iλ0C
(4)
∫
d4p
(2pi)4
. . .
d4s
(2pi)4
δ(p+ q + r + s)B˜aµ(p)B˜
b
ν(q)q˜
c
ρ(r)q˜
d
σ(s)
×
(
eirσ/2r̂ρ + e
isρ/2ŝσ
)
, (B.54)
168
Appendix B. Lattice Feynman rules
with associated Feynman rule
V (0,2,2,0)GF,B (p, q, r, s)
abcd
µνρσ = −iλ0C
(4)
(
δµσδνσe
irσ/2r̂ρ + δµρδνρe
isρ/2ŝσ
)
(B.55)
Finally, then, the overall Feynman rule is
V (0,2,2,0)GF (p, q, r, s)
abcd
µνρσ = −λ0C
(4)
[
δµρδνσe
−i(q+s)ν/2ei(q+s)µ/2
+ i
(
δµσδνσe
irσ/2r̂ρ + δµρδνρe
isρ/2ŝσ
)]
. (B.56)
B.4 Current insertions
In this section I derive the Feynman rules associated with the heavy-light
current operators, J (i)µ [89, 112, 113]. There are three NRQCD operators,
given in Equation (6.52). In the following, all momenta are incoming and the
current insertion is Γµ = γ̂µ for the vector current and Γµ = γ̂5γ̂µ for the
axial-vector current.
Feynman rules for J (0)µ
The Feynman rule associated with J (0)µ is simply Γµ.
Feynman rules for J (1)µ
The Feynman rules for the J (1)µ current are slightly more involved. The tree-
level Feynman rule, corresponding to a vertex insertion, is given by the lead-
ing order contribution to ∇Q(x) in the strong coupling expansion of the link
variable. The Fourier series representation is
J (1),treeµ =
(
−
1
2aM
)
∑
p,p′
q˜(p′)eip
′xΓµ
∑
j
γ̂j∇je
ipxQ˜(p)
= −
1
2aM
∑
p,p′
q˜(p′)eip
′xΓµ
3∑
j=1
γ̂j (i sin pj) e
ipxQ˜(p). (B.57)
169
Appendix B. Lattice Feynman rules
Therefore the associated Feynman rule is
J (1),treeµ = −
i
2aM
Γµ
3∑
j=1
γ̂j sin pj. (B.58)
The one-loop vertex insertion is given by the O(g2) contributions to
∇Q(x). In this case we have
J (1),g
2
µ =
(
−
1
2aM
)
(
−ig2
)
T a
∑
p,p′,k
q˜(p′)A˜a(k)eip
′xΓµ
×
3∑
j=1
γ̂j
1
2
(
eik(x+jˆ/2)eip(x+jˆ) + eik(x−jˆ/2)eip(x−jˆ)
)
Q˜(p)
=
ig2
2aM
T a
∑
p,p′,k
q˜(p′)A˜a(k)ei(p
′+p+k)xΓµ
3∑
j=1
γ̂j cos (p+ k/2)j Q˜(p).
(B.59)
The plus sign in the parenthesis on the second line (and hence the cosine)
arise because the gauge fields are anti-Hermitian and so −A† = +A. The
corresponding Feynman rule is
J (1),g
2
µ =
ig2
2aM
T aΓµ
3∑
j=1
γ̂j cos (p+ k/2)j . (B.60)
The two-gluon vertex insertion is
Jµ
(1),g4 =
(
−
1
2aM
)
(−ig2)2
2
{T a, T b}
∑
p,p′,k,k′
q˜(p′)A˜a(k)A˜b(k′)eip
′xΓµ
×
3∑
j=1
γ̂j
1
2
(
eik(x+jˆ/2)eik
′(x+jˆ/2)eip(x+jˆ) − eik(x−jˆ/2)eik
′(x−jˆ/2)eip(x−jˆ)
)
Q˜(p)
= −
g4
4aM
{T a, T b}
∑
p,p′,k,k′
q˜(p′)A˜a(k)A˜b(k′)ei(p
′+p+k′+k)xΓµ
×
3∑
j=1
γ̂j sin (p+ k/2 + k
′/2)j Q˜(p). (B.61)
170
Appendix B. Lattice Feynman rules
In this case, the minus sign in the second line occurs because −(A†)2 = −A2.
Finally, then, we have the Feynman rule
J (1),g
4
µ = −
g4
4aM
{T a, T b}Γµ
3∑
j=1
γ̂j sin (p+ k/2 + k
′/2)j . (B.62)
171
Appendix C
Gluon selfenergy
In continuum QCD four diagrams contribute to the gluon selfenergy at one-
loop. These diagrams, which comprise the fermionic bubble diagram, two
gluon diagrams (the bubble and the tadpole) and a single ghost contribution,
are shown in Figure C.1. The Ward identity, which is ultimately an expression
(c) (d)
(a) (b)
Figure C.1: Contributions to the gluon selfenergy in continuum QCD. From
top left, the diagrams are (a) the gluon bubble, (b) the gluon tadpole, (c)
the light quark bubble and (d) the ghost bubble. Green wiggly lines indi-
cate gluons, red solid lines represent light quarks and brown dotted lines are
ghosts.
of gauge invariance, takes the form
qµΠµν(q
2) = 0 (C.1)
and guarantees that the selfenergy, Πµν(q2), must be transverse in the exter-
nal momenta qµ. The selfenergy must therefore have the Lorentz structure
Πµν(q
2) = i(q2gµν − qµqν)Π(q
2), (C.2)
where Π(q2) is a regular function of q2 at q2 = 0 (see, for example, [116]) and
I have neglected colour indices for clarity. Thus the only divergences present
in the selfenergy are ultraviolet logarithmic divergences.
172
Appendix C. Gluon selfenergy
For lattice QCD, the situation is complicated by the presence of three
extra diagrams, the fermion and ghost tadpoles and the measure term, which
I illustrate in Figure C.2. Without the extra diagrams, the selfenergy is not
gauge invariant and therefore the Ward identity does not hold. Computing
all seven lattice diagrams to ensure the selfenergy obeys the relevant Ward
identity is a nontrivial check of the HPSRC and HIPPY code.
(c)(b)(a)
Figure C.2: Additional lattice QCD contributions to the gluon selfenergy.
From left, the diagrams are (a0 the light quark tadpole, (b) the ghost tadpole
and (c) the measure term. Green wiggly lines are gluons, red solid lines are
light quarks and brown dotted lines are ghosts. The black cross represents
the measure insertion.
The two fermionic contributions to the gauge boson selfenergy satisfy the
Ward identity independently of the remaining gluon, ghost and counterterm
diagrams, which I collectively call the quenched contributions. The fermion
diagrams are exactly those that contribute to the heavy quark selfenergy,
discussed in Chapter 5, and checking the validity of the Ward identity there-
fore serves as one cross-check of the code used in the heavy quark selfenergy
calculation.
In this appendix I confirm that the fermionic contributions to the gluon
selfenergy obey the Ward identity, Equation (C.1). This serves as an im-
portant check of the two-loop tadpole improvement factor and heavy quark
selfenergy calculations in Chapters 4 and 5.
C.1 Fermionic contributions
I calculated the light quark contributions to the gluon selfenergy using HPSRC
code based on a routine, gluon_sigma, written by Alistair Hart. This routine
takes the momentum, Lorentz and colour indices of the external gluons as
173
Appendix C. Gluon selfenergy
arguments and evaluates one or more of the seven lattice contributions by
exact modesummation according to a parameter list that specifies which
diagrams to calculate. I adapted the routine to work with the heavy quark
selfenergy code, discussed in Chapter 5, and implemented VEGAS evaluation
of the integrands. I updated the code to use the TaylUR derived type to extract
derivatives of the gluon selfenergy. I have also extended the routine so that
it is compatible with our implementation of background field gauge.
For the gluon selfenergy to satisfy the Ward identity, the light quark
bubble diagram – diagram (c) in Figure C.1 – must depend on the exter-
nal momentum in such a way that the overall contribution vanishes as the
momentum tends to zero. I find that this is indeed the case, as I illustrate
in Figure C.3 for unimproved staggered light quarks and in Figure C.4 for
ASQTad light quarks.
In Tables C.1 and C.2 I show results for unimproved staggered quarks
and ASQTad quarks respectively I use three different light quark masses,
mq = 0.05 and mq = 0.3. I obtained the results for the bubble diagrams
using six cores with 5×107 function evaluations and 15 iterations, taking ap-
proximately 1.5 hours each for the unimproved staggered quarks. The ASQ-
Tad computations took approximately 35 minutes on 32 processors. For the
tadpole diagrams I used six cores with 105 function evaluations and 15 iter-
ations, taking approximately two minutes for unimproved staggered quarks
and seven minutes for ASQTad quarks.
I found that, as expected, the light quark tadpole contribution is indepen-
dent of the external momenta. Tests of the Ward identity for the quenched
contributions are currently underway.
174
Appendix C. Gluon selfenergy
-0.12-0.1-0.08-0.06-0.04-0.02 0
0.02
0.001 0.01 0.1 1Gluon selfenergy aqam = 0.05am = 0.15am = 0.30
Figure C.3: Light quark bubble and rainbow contributions to the gluon self-
energy using unimproved staggered quarks. I plot three light quark masses:
amq = 0.05 with red crosses, amq = 0.15 with purple triangles and
amq = 0.30 with blue stars. The horizontal line indicates vanishing self-
energy. Uncertainties are smaller than the data points shown.
-0.12-0.1-0.08-0.06-0.04-0.02 0
0.02
0.001 0.01 0.1 1Gluon selfenergy aqam = 0.05am = 0.15am = 0.30
Figure C.4: Light quark bubble and rainbow contributions to the gluon selfen-
ergy using ASQTad light quarks. I plot three light quark masses: amq = 0.05
with red crosses, amq = 0.15 with purple triangles and amq = 0.30 with
blue stars. The horizontal line indicates vanishing selfenergy. Uncertainties
are smaller than the data points shown.
175
Appendix C. Gluon selfenergy
amq
aq 0.05 0.15 0.3
0.001 -0.00002(1) 0.000000(7) 0.000014(3)
0.002 -0.00002(1) 0.000000(7) 0.000015(3)
0.003 -0.00002(1) 0.000000(7) 0.000011(3)
0.004 -0.00002(1) 0.000000(7) 0.000011(3)
0.005 -0.00002(1) -0.000003(7) 0.000010(3)
0.006 -0.00003(1) -0.000007(7) 0.000007(3)
0.007 -0.00003(1) -0.000010(7) 0.000005(3)
0.008 -0.00003(1) -0.000013(7) 0.000005(3)
0.009 -0.00003(1) -0.000017(7) 0.000001(3)
0.01 -0.00003(1) -0.000023(7) 0.000001(3)
0.02 -0.00013(1) -0.000092(7) -0.000050(3)
0.03 -0.00029(1) -0.000207(7) -0.000126(3)
0.04 -0.00054(1) -0.000369(7) -0.000233(3)
0.05 -0.00084(1) -0.000577(7) -0.000373(3)
0.06 -0.00121(1) -0.000830(7) -0.000546(3)
0.07 -0.00163(1) -0.001130(7) -0.000744(3)
0.08 -0.00209(4) -0.001467(7) -0.000979(3)
0.09 -0.00264(4) -0.001849(7) -0.001243(3)
0.1 -0.00318(4) -0.002277(7) -0.001533(3)
0.2 -0.01102(4) -0.008643(7) -0.006039(3)
0.3 -0.02166(4) -0.018108(7) -0.013146(3)
0.4 -0.03410(4) -0.029730(7) -0.022360(3)
0.5 -0.04775(4) -0.042770(7) -0.033195(3)
0.6 -0.06216(4) -0.056687(7) -0.045159(3)
0.7 -0.07691(4) -0.071087(7) -0.057954(3)
0.8 -0.08344(4) -0.085673(7) -0.070962(3)
0.9 -0.10664(4) -0.100220(7) -0.084218(3)
Table C.1: Light quark bubble and tadpole diagram results for the gluon
selfenergy at three different light quark mass. The uncertainties quoted are
the errors from both bubble and tadpole contributions added in quadrature.
All results use unimproved staggered light quarks.
176
Appendix C. Gluon selfenergy
amq
aq 0.05 0.15 0.3
0.001 -0.00001(3) -0.000015(3) -0.003794(3)
0.002 -0.00001(3) -0.000015(3) -0.003795(3)
0.003 -0.00002(3) -0.000015(3) -0.003795(3)
0.004 -0.00002(3) -0.000015(3) -0.003795(3)
0.005 -0.00002(3) -0.000019(3) -0.003798(3)
0.006 -0.00002(3) -0.000020(3) -0.003798(3)
0.007 -0.00002(3) -0.000024(3) -0.003803(3)
0.008 -0.00002(3) -0.000028(3) -0.003804(3)
0.009 -0.00003(3) -0.000034(3) -0.003810(3)
0.01 -0.00004(3) -0.000037(3) -0.003811(3)
0.02 -0.00013(3) -0.000106(3) -0.003858(3)
0.03 -0.00029(3) -0.000219(3) -0.003934(3)
0.04 -0.00054(3) -0.000379(3) -0.004044(3)
0.05 -0.00084(3) -0.000582(3) -0.004184(3)
0.06 -0.00117(3) -0.000832(3) -0.004358(3)
0.07 -0.00159(3) -0.001129(3) -0.004558(3)
0.08 -0.00205(3) -0.001465(3) -0.004789(3)
0.09 -0.00260(3) -0.001849(3) -0.004053(3)
0.1 -0.00310(4) -0.002269(3) -0.005351(3)
0.2 -0.01089(3) -0.008626(3) -0.009938(3)
0.3 -0.02157(3) -0.018249(3) -0.017338(3)
0.4 -0.03447(3) -0.030370(3) -0.027269(3)
0.5 -0.04913(3) -0.044553(3) -0.039467(3)
0.6 -0.06535(3) -0.060431(3) -0.053682(3)
0.7 -0.08294(3) -0.077813(3) -0.069716(3)
0.8 -0.10183(3) -0.096532(3) -0.083690(3)
0.9 -0.12177(3) -0.116410(3) -0.106410(3)
Table C.2: Light quark bubble and tadpole diagram results for the gluon
selfenergy at three different light quark mass. The uncertainties quoted are
the errors from both bubble and tadpole contributions added in quadrature.
All results use ASQTad light quarks.
177
Appendix D
Kinetic renormalisation parameters
D.1 Kinetic renormalisation parameters
In Chapter 4 I discuss the calculation of the renormalisation coefficients
for a simple NRQCD action - the Davies and Thacker action. In this ap-
pendix I outline the derivation of the correction coefficients for the minimally-
improved NRQCD action given by
S˜0NRQCD = a
3
∑
x
ψ†(x, t)
(
∆+4 −
∆˜(2)
2aM
)
ψ(x, t) (D.1)
where I define ∆˜(2) in Appendix A.
The quark propagator now takes the form
G˜−10 (w,p) = e
−aw − 1 +
4
2aM
∑
j
(
sin2
(apj
2
)
+
1
3
sin4
(apj
2
))
(D.2)
and the tree-level pole lies at
aw˜T (p) =
a2p2
2aM
+O(a6p6). (D.3)
Note that there are no lattice artifacts at O(a4p4), in contrast with the unim-
proved Davies and Thacker action.
Obtaining the renormalisation parameters for this action parallels the
scheme set out in Chapter 4. However, in this case, the renormalised disper-
sion relation is
w˜(p) = w˜T (p)− αsa(E0 + δw˜(p)) (D.4)
178
Appendix D. Kinetic renormalisation parameters
are the correction terms are now
aδw˜(p) = W˜1
a4(p2)2
8a2M2
+ W˜2a
4p4, (D.5)
W˜1 = 2
(
Σ(2)0 + Σ
(1)
1
)
+ Σ(0)2 , (D.6)
W˜2 = Σ
(0)
3 . (D.7)
The exact expression for W˜2 differs from Equation (4.33): the improve-
ment term has removed the Σ(0)1 contribution to W2. Further improvement
has no further effect on the mathematical expression for W2. Equation (D.7)
holds for even the highly-improved NRQCD action of Equation (2.66) [47].
D.1.1 Extension to p6
Future work may require calculating radiative corrections to O(a6p6) kinetic
operators. Here I give the expressions required for such a calculation. I expand
the selfenergy to O(a6p6) as
aΣ(w, p) = Σ0(w) + Σ1(w)
a2p2
2aM
+ Σ2(w)
a4(p2)2
8a2M2
+ Σ3(w)a
4p4
+ Σ4(w)
a6(p2)3
48a3M3
+ Σ5(w)a
6(p2)p4 + Σ6(w)a
6p6. (D.8)
Σ5(w) and Σ6(w) are the coefficients of the new cubic symmetry breaking
terms at O(a6p6) and can be extracted using the following combinations of
partial derivatives:
Σ4(w) = a
3M3
∂6(aΣ)
∂(apx)2∂(apy)2∂(apz)2
∣
∣
∣
∣
∣
p=0
, (D.9)
Σ5(w) =
1
48
(
∂6(aΣ)
∂(apy)2∂(apz)4
− 3
∂6(aΣ)
∂(apx)2∂(apy)2∂(apz)2
)
p=0
, (D.10)
Σ6(w) =
1
48
(
∂6(aΣ)
∂(apz)6
−
∂6(aΣ)
∂(apy)2∂(apz)4
− 143
∂6(aΣ)
∂(apx)2∂(apy)2∂(apz)2
)
p=0
.
(D.11)
179
Appendix D. Kinetic renormalisation parameters
The minimally-improved action of Equation (D.1) can be further im-
proved to O(a6p6) by replacing the Laplacian with an operator correct to
O(a6p6):
∆
(2)
= ∆(2) −
a2
12
∆(2) +
a4
90
∑
j
[
∆+j ∆
−
j
]3
. (D.12)
In this case I find the correction coefficients to be
W3 = Σ
(0)
4 + 3Σ
(1)
2 + 6
(
Σ(3)0 + Σ
(2)
1
)
, (D.13)
W4 = Σ
(0)
5 +
Σ(1)3
2aM
, (D.14)
W5 = Σ
(0)
6 . (D.15)
Equation (D.15) holds for highly-improved NRQCD (Equation (2.66)),
though the actual numerical value will clearly differ.
I do not give results for the O(a6p6) kinetic coefficients. The computation
is not tractable with the mix of analytic and numerical methods that I employ
to calculate the radiative corrections for the Davies and Thacker action. Full
automation in the future will be required if O(a6p6) radiative corrections are
desired.
180
Appendix E
HISQ renormalisation parameters
In this appendix I derive the mass and wavefunction renormalisation param-
eters for the HISQ action. My derivation largely follows [114].
E.1 Mass renormalisation
I write the inverse propagator as
G(p)−1 = G−10 (p)− Σ(p) (E.1)
where
G−10 (p) =
∑
µ
i sin (pµ)Kµ(p)γµ +m0, (E.2)
Kµ(p) = 1 +
1 + ²
6
sin2 (pµ) , (E.3)
Σ(p) =
∑
µ
i sin (pµ)Bµ(p)γµ + C(p). (E.4)
To calculate the pole mass, I set p = 0 and p0 = iE, where E is the rest
energy. Then the pole condition,
∑
µ
sin2 (pµ) (Kµ −Bµ)
2 + (m0 − C)
2 = 0, (E.5)
becomes
− sinh2E
(
1−
1 + ²
6
sinh2E −B0
)2
+ (m0 − C)
2 = 0. (E.6)
Taking the positive energy solution, and expanding the rest energy E and
181
Appendix E. HISQ renormalisation parameters
the parameter ² as
E = mtree + αsm1 (E.7)
² = ²tree + αs²1, (E.8)
the tree-level mass, mtree, at fixed bare mass m0, is given by
sinhmtree
(
1−
1 + ²tree
6
sinh2mtree
)
= m0. (E.9)
The parameter ²tree is fixed by requiring the mass to be equal to the
kinetic mass, defined as
m−1kin =
∂2E
∂p2z
∣
∣
∣
∣
pz=0
, (E.10)
for an onshell particle with momentum pµ = (iE, 0, 0, pz). The resulting
solution is
²tree =
(
4−
√
4 +
12mtree
coshmtree sinhmtree
)
/ sinh2mtree − 1. (E.11)
We obtain the solution for mtree by plugging in this expression for ²tree
into Equation (E.9) and solving iteratively.
Substituting the expansions in Equations (E.7) and (E.8) and keeping
terms up to α∫ , the one-loop mass is given by
αs
1
Z(0)2
m1 − αs²1
sinh3mtree
6
−B0 sinhmtree = −C, (E.12)
where Z(0)2 is the tree-level wavefunction renormalisation:
1
Z(0)2
=
(
1−
1 + ²tree
2
sinh2mtree
)
coshmtree. (E.13)
At non-zero spatial momentum, the kinetic mass,mkin, is defined by Equa-
182
Appendix E. HISQ renormalisation parameters
tion (E.10). The tree-level pole condition is
−
(
sinhE −
1 + ²tree
6
sinh3E
)2
+
(
sin pz +
1 + ²tree
6
sin3 pz
)2
+m20 = 0.
(E.14)
We now need to calculate the onshell derivative
d2
dp2z
∣
∣
∣
∣
pz=0
=
[
∂2
∂p2z
+ 2
∂E
∂pz
∂2
∂pz∂E
+
(
∂E
∂pz
)
∂2
∂E2
]
pz=0
. (E.15)
If we define
F (E) = sinhE −
1 + ²tree
6
sinh3E, (E.16)
H(pz) = sin pz +
1 + ²tree
6
sin3 pz, (E.17)
and note that H(0) = 0 and
∂E
∂pz
∣
∣
∣
∣
pz=0
= 0,
∂H
∂pz
∣
∣
∣
∣
pz=0
= 1, (E.18)
then we obtain
[(
−F
∂F
∂E
)(
∂2E
∂p2z
)
+
(
∂H
∂pz
)2
]
pz=0
= 0. (E.19)
Thus the tree-level kinetic mass is
mkin =
1
Z(0)2
m0 =
∂F
∂E
∣
∣
∣
∣
pz=0
, (E.20)
or, writing F explicitly,
mkin =
(
sinhmtree −
1 + ²tree
6
sinh3mtree
)
×
(
1−
1 + ²tree
2
sinh2mtree
)
coshmtree. (E.21)
Requiringmkin = mtree gives us the expression for ²tree in Equation (6.35).
183
Appendix E. HISQ renormalisation parameters
E.2 Wavefunction renormalisation
The wavefunction renormalisation is defined through the residue of the quark
propagator at the single particle pole
∫ pi
−pi
dp0
2pi
eip0tG(p0,0) = Z2e−mt
1 + γ0
2
+ . . . , (E.22)
where the ellipses stand for poles corresponding to more complicated bound
state and multi-particle states. I use the variable transformation z = eip0 and
write the propagator as G(p) = g(z)/f(z), so that the integral becomes
∮
dz
2pii
zt−1
g(z)
f(z)
. (E.23)
The residue of the pole at p0 = im, or z = z1, is
e−mt
(
g(z)
zf ′(z)
)
z=z1
. (E.24)
Using the chain rule, the denominator in this expression is
zf ′ = z
df
dz
= −i
df
dp0
= −
df
dE
. (E.25)
Now, for the HISQ propagator,
f(z) = ((B0 −K0) sinhE +m0 − C) ((K0 −B0) sinhE +m0 − C) , (E.26)
so
zf ′|z=z1 = 2(m0 − C)
(
coshE −
1 + ²
2
sinh2E coshE
+ i
d
dp0
(iB sin p0 + C)
)
. (E.27)
184
Appendix E. HISQ renormalisation parameters
The residue is therefore
Res =
e−mt
2
(1 + γ0)
[(
1−
1 + ²
2
sinh2m
)
coshm+ i
d
dp0
(iB sin p0 + C)
]−1
,
(E.28)
and so the wavefunction renormalisation is
Z−12 =
(
1−
1 + ²
2
sinh2m
)
coshm+ i
d
dp0
(iB sin p0 + C) . (E.29)
At tree-level, this expression reduces to Z(0)2 , given in Equation (6.37).
The one-loop wavefunction renormalisation, Z(1)2 , is given by
1
Z(1)2
=
1
Z(0)2
{
1− αs²1
Z(0)2
2
coshmtree sinh
2mtree + i
d
dp0
(iB sin p0 + C)Z
(0)
2
+ αsm1 sinhmtree
[
1−
1 + ²tree
2
(
2 cosh2mtree + sinh
2mtree
)
]
Z(0)2
}
.
(E.30)
For massless HISQ with zero spatial momentum, we see that the one-loop
wavefunction renormalisation becomes
1
Z(1)2
= −B0 + i
dC
dp0
, (E.31)
since mtree = m0 = 0, ²tree = 0 and Z
(0)
2 = 1.
185
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Publications
l “Radiative improvement of the lattice NRQCD action using the background field
method and application to the hyperfine splitting of quarkonium states”
T.C. Hammant, A.G. Hart, G.M. von Hippel, R.R. Horgan and C.J. Monahan
Phys. Rev. Lett 107 (2011) 112002. arXiv:1105.5309
l “Improved automated lattice perturbation theory in background field gauge”
T.C. Hammant, R.R. Horgan, C.J. Monahan, A.G. Hart, E.H. Mu¨ller, A. Gray,
K. Sivalingham and G.M. von Hippel
PoS LAT2010 arXiv:1011.2696
l “The b quark mass from lattice nonrelativistic QCD”
A.G. Hart, G.M. von Hippel, R.R. Horgan, A. Lee and C.J. Monahan
PoS LAT2010 arXiv:1011.2696
l “Radiative corrections to the m(oving)NRQCD action and heavy-light operators”
E.H. Mu¨ller, C.T.H. Davies, A.G. Hart, G.M. von Hippel, R.R. Horgan, I.D. Kendall,
A. Lee, S. Meinel, C.J. Monahan and M. Wingate
PoS LAT2009 arXiv:0909.5126
196