SCATTERING AMPLITUDES FOR ZZ PRODUCTION AT THE LHC AND
TOP-QUARK MASS EFFECTS
By
Bakul Agarwal
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics—Doctor of Philosophy
2022
ABSTRACT
SCATTERING AMPLITUDES FOR ZZ PRODUCTION AT THE LHC AND
TOP-QUARK MASS EFFECTS
By
Bakul Agarwal
With the Large Hadron Collider providing experimental data with unprecedented pre-
cision, theoretical predictions must improve similarly to keep up. Among a plethora of
processes being studied at the LHC, the production of a pair of vector bosons is of particular
importance. Consequently, precise theoretical predictions for these processes are necessary.
This thesis discusses primarily the calculation of ZZ production through gluon fusion at
2-loops with full top-quark mass dependence as well as the technological improvements re-
quired to successfully perform the calculation. Also discussed briefly is the quark initiated
production of γγ + jet at 2-loops where some of these technologies allowed to overcome prior
bottlenecks in the calculation of the helicity amplitudes.
The 2-loop corrections for ZZ production through massless quarks had been known;
in this work, the 2-loop corrections through the massive top quark are calculated . To
achieve this, a new algorithm to systematically construct linear combinations of integrals
with a convergent parametric integral representation is developed. This algorithm finds
linear combinations of general integrals with numerators, dots, and dimension shifts as well
as integrals from subsectors.
To express the amplitudes in terms of these integrals, Integration-By-Parts (IBP) reduc-
tion is performed making use of syzygies and finite field based methods. A new algorithm is
employed to construct these syzygies using linear algebra. The IBP reductions for gg → ZZ
are successfully performed using these techniques. Further improvements, including prede-
termining the structure of the coefficients in IBP reductions, are used to successfully perform
the reductions for γγ + jet. Multivariate partial fractioning is used to simplify the final ex-
pressions to more manageable forms and render them suitable for fast numerical evaluation.
In the case of gg → ZZ, due to the presence of structures beyond polylogarithms, sector
decomposition is employed to numerically evaluate the finite master integrals. Evaluating the
amplitudes, agreement is found with previously calculated expansions specifically in the limit
of large and small top mass. Improved results are presented for scattering at intermediate
energies and/or for non-central scattering angles. With this calculation, the last building
block required for the calculation of the full NLO cross-section for gg → ZZ is known.
Dedicated to my parents.
iv
ACKNOWLEDGEMENTS
The process of obtaining this Ph.D. has been long and gruelling and it would have been
impossible without the people who have assisted or influenced me during this journey. It is
hard to overstate the importance of a support group and I cannot thank enough everyone
who has been a part of mine.
My parents, without whom none of this would be possible, have been incredibly support-
ive throughout these past years and have always believed in me, even when I have not, and
all the sacrifices they have made through their lives have enabled me to make the choices
I have. My friends also deserve my gratitude for providing the support I needed during
difficult times and giving me the comfort and encouragement when I needed it the most.
Special thanks to my little cousin sisters Aanya and Tanya for being a source of cheer and
comfort throughout this time.
My journey in Physics wouldn’t have even started if not for Pankaj Jain; he has my
eternal gratitude. I would like to thank C-.P. Yuan for giving me the opportunity to work
with him in the initial years of my Ph.D. I have had a lot of discussions about physics with
him over the years and I have thoroughly enjoyed all of them, in addition to learning a lot.
He has been very influential during my time at MSU; his idea that Ph.D. students should
be able to teach their advisors something at the end of their program has been a guiding
principle for me. He puts strong emphasis on having a broad base of knowledge about other
areas of research, something I will continue to improve upon.
I would also like to thank the department and the High Energy Physics group. com-
munication between the theorists and the experimentalists in the group has always been
great. This has made sure that I am aware of the experimental side of things. My coworkers
throughout my Ph.D. have been immensely helpful, both concerning my immediate research
as well as topics of broader interest. I would especially like to thank Joshua Isaacson; as
a senior graduate student during the time I joined, his guidance and friendship have been
v
invaluable. He was always keen to discuss physics with me and point me to the right direc-
tion whenever I had any doubts, physics or otherwise. I would also like to thank Kirtimaan
Mohan, Robert Schabinger, and Jan Winter for always entertaining all the stupid questions
I had and providing me with answers, as well as providing valuable moral support, and I
cannot thank Rob enough for patiently teaching me things when I needed them. Stephen
Jones, my collaborator for a large portion of this work, has been of great help and it’d have
been much harder for me to finish this work without his involvement.
I would like to thank Kim Crosslan and Brenda Wenzlick for being amazing, and it is
hard to exaggerate how helpful they have been in tackling the MSU bureaucracy. My time
at MSU has been made so much easier because of them. Also, my thanks to the members
of my thesis committee for being a part of my Ph.D. and for all their helpful comments.
Finally, I would like to thank my advisor Andreas von Manteuffel for providing me with
all the knowledge and tools needed to perform my research. I consider myself incredibly
lucky to have him as my advisor; he has been an amazing mentor and a constant guide
throughout. His depth of knowledge and expertise of the subject always amazes me. I have
learned a lot from him during this time, and he has always been very receptive to all the
new ideas I’ve had. His attitude towards solving problems, consistent effort to improving
efficiency, and incredible attention to detail are some of the qualities I strive to emulate.
vi
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter 1 The Standard Model and perturbative calculations . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Discovery of the Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Standard model Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Higgs potential and width . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Higgs potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Measuring Higgs width . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.3 Experimental constraints . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Theoretical status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 ZZ production at LHC . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.2 Diphoton production at LHC . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.1 History of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.2 SU (N ) gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.3 Perturbative QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5.4 Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.5.5 UV renormalisation and IR subtraction . . . . . . . . . . . . . . . . . 36
1.5.6 Asymptotic freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5.7 Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.6 Electroweak sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6.1 Electroweak symmetry breaking . . . . . . . . . . . . . . . . . . . . . 43
1.6.2 Custodial symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.6.3 Electroweak interactions . . . . . . . . . . . . . . . . . . . . . . . . . 48
Chapter 2 Setup of the calculation . . . . . . . . . . . . . . . . . . . . . . . . 52
2.1 Form factor decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2 1-loop amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2.1 Generation of the unreduced amplitude . . . . . . . . . . . . . . . . . 60
2.2.2 Reduction to master integrals . . . . . . . . . . . . . . . . . . . . . . 64
2.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 2-loop amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3.1 Generation of diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3.2 Class A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3.3 Class B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
vii
Chapter 3 IBP reduction using syzygies . . . . . . . . . . . . . . . . . . . . . 80
3.1 Integration-by-parts reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2 Baikov representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 Syzygies for IBP reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4 Linear algebra based syzygy construction . . . . . . . . . . . . . . . . . . . . 93
3.5 IBP reduction using syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Chapter 4 Finite basis integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1 Evaluation of Feynman integrals . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.2 Feynman parametric representation . . . . . . . . . . . . . . . . . . . 103
4.2 Finite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.1 Divergences in Feynman parametric representation . . . . . . . . . . 105
4.2.2 Finite integrals with dimension shifts . . . . . . . . . . . . . . . . . . 107
4.2.3 Finite integrals with numerators . . . . . . . . . . . . . . . . . . . . . 109
4.2.4 Choice of finite integrals for gg → ZZ . . . . . . . . . . . . . . . . . 115
Chapter 5 Compiling the 2-loop amplitude . . . . . . . . . . . . . . . . . . . 119
5.1 Inserting reductions into the amplitude . . . . . . . . . . . . . . . . . . . . . 119
5.1.1 Multivariate partial fractioning . . . . . . . . . . . . . . . . . . . . . 119
5.1.2 Backsubstitution of IBPs . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Renormalisation, IR subtraction and checks . . . . . . . . . . . . . . . . . . 123
5.2.1 UV renormalisation and IR subtraction . . . . . . . . . . . . . . . . . 123
5.2.2 Checks of the calculation . . . . . . . . . . . . . . . . . . . . . . . . . 126
Chapter 6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1 Results for the 2-loop gg → ZZ amplitude . . . . . . . . . . . . . . . . . . . 128
Chapter 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
APPENDIX A Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
APPENDIX B Evaluation using Feynman parameters . . . . . . . . . . . . . . . 153
APPENDIX C Dirac algebra and γ 5 schemes . . . . . . . . . . . . . . . . . . . . 157
APPENDIX D UV renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . 163
APPENDIX E List of denominators . . . . . . . . . . . . . . . . . . . . . . . . . 170
APPENDIX F Numerical checks . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
viii
LIST OF TABLES
Table 1.1: Standard Model Particles. . . . . . . . . . . . . . . . . . . . . . . . . . 2
Table 1.2: Branching ratios for the Standard Model Higgs boson with mh = 125.25
GeV [1]. Numbers reproduced from [2]. . . . . . . . . . . . . . . . . . . 15
Table 2.1: List of integral families and their propagators for the 2-loop amplitude. 76
Table 4.1: Numerical performance of different non-planar integrals for a physical
phase-space point. Timings generated with pySecDec [3] using the QMC
algorithm [4, 5] on a single Nvidia Tesla V100S GPU, with number of
evaluations neval = 107 . Note that the divergent integrals are only
evaluated to O(0 ) since they start at −1 . Reproduced from [6]. . . . . 117
Table 6.1: 1-loop and 2-loop helicity amplitudes for gg → ZZ for the phase-space
point s/m2t = 142/17, t/m2t = −125/22, m2Z /m2t = 5/18, and mt = 1,
(1) (2)
with Mλ1 λ2 λ3 λ4 and Mλ1 λ2 λ3 λ4 defined in Eq. 6.1.2. Only the 8 indepen-
dent helicity amplitudes (See Eqs. 2.1.20, 2.1.21, and 2.1.22) are shown
here. Note that these include only the top-quark contributions from class
A diagrams defined in Sec. 2.3.2. The numbers in parentheses denote
the uncertainty in the last digit. Reproduced from [6]. . . . . . . . . . 131
Table 7.1: Numerical poles for the Euclidean phase-space point s/m2t = −191,
t/m2t = −337, m2Z /m2t = −853, mt = 1 compared against the pre-
dicted values. Also shown are the 0 terms before IR subtraction with
the digits in parentheses denoting the uncertainty in the last digit. . . . 175
Table 7.2: Numerical poles for the physical phase-space point s/m2t = 142/17,
t/m2t = −125/22, m2Z /m2t = 5/18, mt = 1 compared against the pre-
dicted values. Also shown are the 0 terms before IR subtraction with
the digits in parentheses denoting the uncertainty in the last digit. . . . 176
ix
LIST OF FIGURES
Figure 1.1: Schematic of a scattering process at a hadron collider. For the LHC, the
incoming particles are both protons. Reproduced from [7]. . . . . . . . 4
Figure 1.2: Diagrams showing ZZ production from (a) qq and (b) gg initial states.
The cross section for gg → ZZ process starts at a relative order O(αS2 )
compared to qq → ZZ for pp scattering. . . . . . . . . . . . . . . . . . 5
Figure 1.3: W W scattering with the 3 different subprocesses depicted. . . . . . . . 8
Figure 1.4: Scalar field Φ with two different vacuum configurations. For µ2 < 0 the
vacuum state is no longer symmetric and the field acquires a non-zero
expectation value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Figure 1.5: W W scattering including the interactions with the Higgs boson. . . . . 13
Figure 1.6: Plot of the Standard Model Higgs branching ratios against the Higgs
mass. Measured value of Higgs mass is mh = 125.25 ± 0.17 GeV. Repro-
duced from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Figure 1.7: Distribution of four lepton invariant mass for the h → ZZ ∗ → l+ l− l+ l−
decay channel for (a) ATLAS (Reproduced from [9]) and (b) CMS (Re-
produced from [10]) experiments. . . . . . . . . . . . . . . . . . . . . . 16
Figure 1.8: Distribution of the invariant mass of the photon pair for the h → γγ
decay channel for (a) ATLAS (Reproduced from [9]) and (b) CMS (Re-
produced from [10]) experiments. . . . . . . . . . . . . . . . . . . . . . 16
Figure 1.9: Plot of the measured Higgs boson couplings to other Standard Model
particles against particle mass. Reproduced from [11]. . . . . . . . . . . 17
Figure 1.10: Higgs pair production through gluon fusion at LO. . . . . . . . . . . . . 19
Figure 1.11: Perturbative corrections to qq → γ in QCD represented using Feynman
diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 1.12: QCD corrections to gg → h. The uncertainty bands represent scale
uncertainty for the range µ ∈ { m4h , mh } with the central value at µ =
mh /2. Reproduced from [12]. . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 1.13: 1-loop correction to qq →− γ. . . . . . . . . . . . . . . . . . . . . . . . . 33
Figure 1.14: A 3-point massless integral. The incoming momenta p1 , p2 are massless. 34
Figure 1.15: Infrared divergences corresponding to the emission of a real particle from
a massless particle. Fig. 1.15a shows emission of a gluon with very small
momentum i.e. a soft divergence while Fig. 1.15b shows the emission of
a gluon collinear with the external particle. . . . . . . . . . . . . . . . . 37
Figure 1.16: QCD corrections to qq → − γ. Fig. 1.16b shows the αS correction through
the exchange of a virtual gluon while Fig. 1.16c show the αS correction
through the emission of a real gluon. . . . . . . . . . . . . . . . . . . . 38
x
Figure 1.17: Plot showing theoretical prediction for the strong coupling αS against
experimental measurements. Reproduced from [1]. . . . . . . . . . . . . 41
Figure 2.1: Feynman diagrams for the process gg → − ZZ at 1-loop (LO). Figs. 2.1a
and 2.1b are the Higgs exchange diagrams, and Figs. 2.1c- 2.1h are the
diagrams for continuum production of Z-bosons. Note that out of the 6
continuum diagrams, only 2 are independent and the rest can be obtained
through various crossings of external legs. . . . . . . . . . . . . . . . . . 61
Figure 2.2: Decomposition of a general 1-loop Feynman integral in d = 4. . . . . . 64
Figure 2.3: Comparison of |M|2 for gg → ZZ for three different type of contri-
butions, summed over all helicities for the external particles. Without
including the Higgs mediated contributions, the top quark contribution
increases with energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 2.4: Comparison of |M|2 for gg → ZZ for different helicity configurations for
the top quark contribution. The bottom panel (Fig. 2.4b) also includes
the Higgs mediated diagrams. . . . . . . . . . . . . . . . . . . . . . . . 72
Figure 2.5: Ratio of contribution of individual helicity configurations to the total
|M|2 for gg → ZZ for (a) top quark contribution including the Higgs
diagrams, and (b) massless quark contribution. . . . . . . . . . . . . . . 73
Figure 2.6: Example Feynman diagrams representing the two classes of diagrams at
2-loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Figure 2.7: Representative Feynman diagrams in class A with irreducible topologies.
The number of master integrals in the above top-level topologies are 3,
4, 3, 3, 5, 5, and 4 respectively. . . . . . . . . . . . . . . . . . . . . . . 77
Figure 2.8: Representative Feynman diagrams in class A with reducible topologies. 78
Figure 3.1: A tadpole graph with the thick loop corresponding to the massive prop-
agator with mass m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 3.2: A two-point function with one massive propagator. The thick line cor-
responds to the massive propagator. . . . . . . . . . . . . . . . . . . . . 82
Figure 3.3: A two-point function with massless propagators. The dashed line corre-
sponds to the cut propagators. . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 3.4: The non-planar topologies for 5-point 2-loop γγ + jet production. . . . 98
Figure 4.1: 3-point integral with 1 massive propagator. . . . . . . . . . . . . . . . . 105
Figure 4.2: Examples of divergent and finite integrals in the limit → 0 for a non-
planar topology. Thick solid lines represent the top-quark while thick
dashed lines represent Z-bosons. Topology (b) contains an irreducible
numerator, where k is the difference of the momenta of the edges marked
by the thin dash lines. Reproduced from [6]. . . . . . . . . . . . . . . 109
xi
Figure 4.3: Integrals appearing in the linear combination in Eq. 4.2.20. I1,1 is the
corner integral of the topology under consideration. I2,1 is a second
integral in the topology, but with a numerator (k 2 − m2t ), where k is
equal to the difference of the momenta of the edges marked by the thin
dashed lines. Integrals I3,1 , I4,1 , I5,1 , I6,1 , I7,1 belong to subtopologies. All
integrals are defined in d = 4 − 2 dimensions. Reproduced from [6]. . 114
Figure 4.4: Integrals appearing in the linear combination in Eq. 4.2.21. I1,2 is the
corner integral of the topology under consideration but with a numerator
(k 2 − m2t ), identical to I2,1 from Eq. 4.2.20. I2,2 is I1,2 but with an extra
numerator (k 2 −m2t ) where k is equal to the difference of the momenta of
the edges marked by the thin dashed lines. Integrals I3,2 , I4,2 , I5,2 , I6,2 , I7,2
are the same as I3,1 , I4,1 , I5,1 , I6,1 , I7,1 but with an extra numerator (k 2 −
m2t ). All integrals are defined in d = 4 − 2 dimensions. Reproduced
from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 5.1: Mass counterterm diagrams required at 2-loops. The big dark cross in
Fig. 5.1b corresponds to the counterterm vertex insertion. . . . . . . . 125
√
Figure 6.1: Comparison of the s dependence of the unpolarised interference V (2)
with expansion for large and small top-quark mass [13] at fixed cos(θ) =
−0.1286. The large top-mass expansion is shown in colour red, the
small top-mass expansion in blue, and the Padé improved small top-
mass expansion in purple. The exact result is shown in black. Note that
the error bars have been plotted for the exact result, they are too small
to be visible on the plot, however. Reproduced from [6]. . . . . . . . . 132
Figure 6.2: Comparison of the cos(θ) dependence of the unpolarised interference
(2)
V
√ with the results expanded in the limit of large top-quark√mass for
s = 247 GeV (Top Left Panel) √ and small top-quark mass for s = 403
GeV (Top Right Panel) and s = 814 GeV (Bottom Panel). Reproduced
from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
√
Figure 6.3: The s dependence of 1-loop and 2-loop interferences for polarised ZZ
production in gluon fusion at cos(θ) = −0.1286. Reproduced from [6]. 134
Figure 6.4: The cos(θ) dependence of 1-loop and √ 2-loop interferences for polarised
ZZ production in gluon fusion at s/mt = 1.426. The large top-quark
mass expansion [13] (to order 1/m12 t ) is shown for comparison. Repro-
duced from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure 6.5: The cos(θ) dependence of 1-loop and √ 2-loop interferences for polarised
ZZ production in gluon fusion at s/mt = 2.331. The Padé improved
small top-quark mass expansion [13] is shown for comparison. Repro-
duced from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
xii
Figure 6.6: The cos(θ) dependence of 1-loop and √ 2-loop interferences for polarised
ZZ production in gluon fusion at s/mt = 4.703. The small top-quark
mass expansion (to order m32 t ) and Padé improved expansion [13] are
shown for comparison. Reproduced from [6]. . . . . . . . . . . . . . . 137
√
Figure 6.7: The s dependence of 1-loop and 2-loop interferences for polarised ZZ
production in gluon fusion at cos(θ) = −0.1286. Here, the top left and
bottom right panels of Fig. 6.3 are reproduced using Catani’s original
subtraction scheme [14]. Reproduced from [6]. . . . . . . . . . . . . . 138
Figure 6.8: The cos(θ) dependence of 1-loop and √ 2-loop interferences for polarised
ZZ production in gluon fusion at s/mt = 1.426. The large top-quark
mass expansion [13] (to order 1/m12 t ) is shown for comparison. Here,
the top left and bottom right panels of Fig. 6.4 are reproduced using
Catani’s original subtraction scheme [14]. Reproduced from [6]. . . . . 138
Figure 6.9: The cos(θ) dependence of 1-loop and √ 2-loop interferences for polarised
ZZ production in gluon fusion at s/mt = 2.331. The Padé improved
small top-quark mass expansion [13] is shown for comparison. Here, the
top left and bottom right panels of Fig. 6.5 are reproduced using Catani’s
original subtraction scheme [14]. Reproduced from [6]. . . . . . . . . . 139
Figure 6.10: The cos(θ) dependence of 1-loop and √ 2-loop interferences for polarised
ZZ production in gluon fusion at s/mt = 4.703. The small top-quark
mass expansion (to order m32 t ) and Padé improved expansion [13] are
shown for comparison. Here, the top left and bottom right panels of
Fig. 6.6 are reproduced using Catani’s original subtraction scheme [14].
Reproduced from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Figure 7.1: The integration contour to perform Wick rotation. Note that the poles
lie outside the contour. . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Figure 7.2: The triangle anomaly graph. Here the dark blob vertex represents the
axial-vector coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Figure 7.3: All the diagrams contributing to 1-loop correction to the gluon propaga-
tor, including the counterterm diagram. Requiring that the sum is finite
allows the calculation of the counterterm δG . . . . . . . . . . . . . . . 167
xiii
LIST OF ALGORITHMS
Algorithm 3.1 Syzygies for linear relations without dimension shifts or dots [6] . . . . 95
Algorithm 4.1 Finite Feynman integrals [6] . . . . . . . . . . . . . . . . . . . . . . . 112
xiv
Chapter 1
The Standard Model and perturbative cal-
culations
1.1 Introduction
The Standard Model of particle physics provides a fundamental description of the universe
at the smallest scales accessible to us. It provides a "unified" prescription of the electromag-
netic, weak nuclear, and strong nuclear forces and, along with General Relativity, describes
almost all known physical phenomena. Many excellent predictions have been made to date
using the Standard Model with one of the most successful being the anomalous magnetic
moment g − 2 of the electron which is calculated to 10 digits of relative precision [15, 16].
Incredibly, the experimentally measured value [17], which is also one of the most precise mea-
surements to date and of comparable precision, shows excellent agreement with the Standard
Model prediction.
The field of subatomic physics started with the discovery of the electron in 1897 using a
simple cathode ray tube. Since then, many more new particles have been discovered, most
of them composite. The last puzzle of the Standard Model was solved in July 2012 with the
discovery of the Higgs Boson [9, 10]. Predicted in 1964 [18–21], the Higgs Boson completed
the Standard Model and answered the longstanding question of the origin of gauge boson
masses.
The Standard Model roster can be divided into two groups: bosons and fermions. Bosons
are integer spin particles and usually appear as force carriers in gauge theories. The photon
1
Fermions Bosons
νe νµ ντ W ±, Z
Leptons
e− µ− τ− γ
u c t g
Quarks
d s b h
Table 1.1: Standard Model Particles.
is the force carrier for the electromagnetic force, gluon is the mediator of strong force, and
W ± , Z are the mediators for the weak force; all known force-carriers are spin-1 particles.
Fermions are half-integer spin particles; electrons, protons, and neutrons make up almost
all of the visible mass in the universe; protons and neutrons are composite though and
themselves composed of quarks and gluons bound together through the strong force. All
the fermions also have corresponding anti-particles e.g. positron is the anti-particle of the
electron. Anti-particles are characterised by one or more of their quantum numbers having
the opposite sign compared to the corresponding particles. The positron e.g. is positively
charged, opposite to the electron. Neutrons are neutral, but they posses another quantum
number called Baryon number which is +1 for the neutron and −1 for the antineutron.
Quarks interact through all three forces: strong, weak, and electromagnetic. Electrons,
muons, and tau particles do not possess the colour charge and only interact with the weak
and electromagnetic forces, while neutrinos are neutral and only interact weakly. The list of
Standard Model particles is given in Tab. 1.1.
However, despite its many successes, the Standard Model falls short on many accounts.
Perhaps the biggest deficiency is the lack of a viable dark matter [22–25] candidate in the
Standard Model. Dark matter accounts for ∼ 80% of the non-relativistic matter content in
the universe which is unexplained by the Standard Model. It also does not explain Dark en-
ergy [26] which makes up ∼ 70% of the energy density in the universe. The origin of neutrino
masses [27] is still a mystery, and while adding a mass term involving right-handed neutrinos
is not forbidden in the Standard Model, such a particle would never interact with the rest
of the Standard Model particles and is of little consequence. In any case, the extremely
2
small masses of the neutrinos compared to the rest of the Standard Model roster remain
unexplained. Other issues include the Hierarchy problem [28, 29], lack of CP violation in
the strong sector [30], and matter anti-matter asymmetry [31] which cannot be explained by
the existing CP violation in the weak sector. Consequently, it is of extreme importance to
have precise theoretical predictions for physical observables in the Standard Model. Compar-
ing these theoretical predictions to experimental measurements, any deviations would signal
existence of some new physics; as such, precision calculations are essential for the particle
physics program.
At collider experiments such as the Large Electron-Positron collider (LEP) and the Large
Hadron Collider (LHC), the physical observables of interest are scattering cross-sections for
various scattering processes such as e+ e− → µ+ µ− at the LEP or pp → µ+ µ− at the LHC.
These cross-sections are calculated in quantum field theory using scattering amplitudes. In
the case of lepton colliders such as LEP the scattering amplitudes are sufficient (ignoring for
the moment parton showers, jet algorithms, etc for hadronic final states). However for the
hadron colliders such as the LHC, the situation is not so straightforward.
Hadrons are composite particles composed of quarks and gluons, referred to as "par-
tons" [32]. At very high energies such as at the LHC the constituent particles of the hadrons
are undergoing the actual "collisions". This high-energy or short-distance physics is well
described using perturbative quantum field theory and is the focus of this work. The long-
distance physics inside the hadrons cannot be described using perturbative quantum field
theory since at such low energies, Quantum Chromo-Dynamics enters the non-perturbative
regime; the expansion parameter, in this case the strong coupling constant αS , becomes too
large. The general expression for a collision between two hadrons h1 , h2 producing particles
c, d in the final state can be written as [1]
Z 1 X
h1 h2 →cd
dσ = dx1 dx2 fa/h1 (x1 , µ2F )fb/h2 (x2 , µ2F ) dσ̂ ab→cd (Q2 , µ2F ) (1.1.1)
0 a,b
3
Figure 1.1: Schematic of a scattering process at a hadron collider. For the LHC, the incoming
particles are both protons. Reproduced from [7].
where the functions fa/h1 , fb/h2 are the so-called "Parton Distribution Functions" and encode
the long-distance physics of the hadrons, and the quantity dσ̂ ab→cd denotes the partonic
cross-section for the collision process ab → cd between the partons a, b originating from
the hadrons h1 , h2 respectively and contains the short-distance physics of point-like particle
collisions. The above formula splitting the calculation of the short and long distance physics
into two separate pieces is based on the factorisation theorem (see Sec. 1.5.7). While it is
not known if it is in general possible to treat these two parts of the calculation completely
independently, for some processes such as the Drell-Yan process it has been proven that the
short-distance physics of the partonic collisions can be "factored" out of the long-distance
physics governing hadrons according to the above formula.
Consider the scattering process pp → ZZ + X at the LHC where X could correspond
to any number of jets. The actual scattering process looks similar to the schematic in
Fig. 1.1 with P1 and P2 corresponding to the momenta of the incoming protons. Since the
protons are composite, and composed of quarks and gluons, the hard scattering depicted
in Fig. 1.1 can be initiated through qq, qg, and gg initial states. In fact, the quark-quark
initialised scattering, as shown in Fig. 1.2, starts at the lowest perturbative order in QCD.
The process gg → ZZ formally only starts at NNLO for the hadronic scattering pp → ZZ.
4
(b)
(a)
Figure 1.2: Diagrams showing ZZ production from (a) qq and (b) gg initial states. The
cross section for gg → ZZ process starts at a relative order O(αS2 ) compared to qq → ZZ
for pp scattering.
However, it still contributes significantly to the hadronic cross-section at the LHC; this
is due to the fact that the parton distribution functions for gluons, at momenta required
to produce a pair of Z-bosons, are larger than those for the quarks. Thus even though
the partonic hard-scattering cross-section is expected to be suppressed, the gluonic parton
distribution functions compensate for that. Also at higher energies, due to the Goldstone
boson equivalence theorem [33, 34], the contribution from top quarks in the loop for gg →
ZZ, in particular for longitudinally polarised Z bosons, becomes dominant over all other
contributions. This effect is completely absent for the quark initialised process.
In this work we primarily consider the production of a pair of Z-bosons through gluon
fusion i.e. gg → ZZ at 2-loops with exact dependence on the mass of the top quarks
propagating through the loops. As mentioned above, the gluonic channel is significant due
to large parton distribution functions for gluons at intermediate energies relevant for ZZ
production. This is an essential process to be studied at the Large Hadron Collider since it
serves as a background process to the Higgs signal process gg → h → ZZ. It interferes with
the Higgs signal process producing a sizeable effect (∼ 10% [35, 36]). A crucial property of
the Higgs boson to be measured at the LHC is its decay width. Direct measurement of the
5
decay width of the Higgs boson through the resonance peak is limited at the LHC due to
the detector resolution (∼ 1 GeV) being much larger compared to the Standard Model value
of the width (∼ 4 MeV). An alternate approach to constrain the Higgs width using both
on-shell and off-shell Higgs production was proposed in [37]; precisely measuring gg → ZZ
is a crucial component of that. ZZ production also serves as a channel for new physics
searches where heavy resonances decay to a pair of Z-bosons. Additionally, the longitudinal
modes of the Z-bosons are sensitive to their axial-vector couplings to the fermions; this can
be used to constrain anomalous ttZ couplings [38, 39]. It is clear that pair production of
Z bosons provides multiple avenues to better and more precisely understand the Standard
Model.
This work presents the first calculation of helicity amplitudes for gg → ZZ at 2-loops with
full dependence on the top-quark mass. Prior to this, the amplitudes were known exactly
only for massless internal quarks and the contributions for top quarks were calculated only
using certain approximations. The text describes the steps and techniques used to calculate
the scattering amplitudes for this process including some new techniques which facilitated
the calculation. The tensor decomposition of the amplitude is described followed by syzygy-
based Integration-By-Parts reduction techniques used to reduce the amplitudes. For efficient
numerical evaluation of the master integrals, a basis of finite integrals is chosen based on a
new technique of combining divergent integrals into finite linear combinations. A method
for multivariate partial fractioning to simplify the analytic expressions is described as well.
Note that in this work, only on-shell Z-bosons in the final state are considered; in the
future, off-shell Z-bosons as well as the W -bosons may be included in the final state to
increase the scope of the calculation. Also, the focus of this work is the calculation of
scattering amplitudes for gg → ZZ To provide meaningful physical predictions, cross sections
and distributions need to be calculated; this is left for future work.
6
1.2 Discovery of the Higgs Boson
1.2.1 Higgs mechanism
Late 1940s and 1950s saw tremendous progress in the field of particle physics. The Dirac
equation predicted the magnetic moment of the electron but the experimental measurements
indicated a slight deviation. Gauge theories [40] were eventually developed and were incred-
ibly successful in making precise theoretical predictions, especially with the development of
the theory of renormalisation. Quantum Electrodynamics successfully predicted the anoma-
lous magnetic moment of the electron as well as the Lamb Shift paving the way for future
work in precision calculations and marking another achievement for Quantum theory. Pions,
proposed as the force carriers in the Yukawa potential governing the strong nuclear force,
were discovered with the masses predicted by theory. A still unsolved puzzle, however, was
the origin of masses for force carriers of the weak nuclear force, which were known from
experiments to be massive. No consistent way to generate masses for the gauge bosons while
preserving the gauge symmetry was known at the time. Another issue was that the scatter-
ing of W and Z bosons violated unitarity. Scattering amplitude for the longitudinal modes
of W and Z bosons grows with energy indefinitely. Consider WL+ WL− scattering as depicted
in Fig. 1.3
M(WL+ WL− → WL+ WL− ) = Mt + M4pt + Ms . (1.2.1)
Calculating the individual contributions, the 4-point vertex amplitude (Fig. 1.3c) is
M4pt = igw2 (2gµρ gνσ − gµν gρσ − gµσ gνρ )µL (p1 )νL (p2 )ρL (p3 )σL (p4 )
2
g 2 (−5 − 12 cos θ + cos 2θ) g 2 (1 + 3 cos θ)
s s
= +
32 m2W 2 m2W
0
s
+O , (1.2.2)
m2W
7
(a) M (b) Mt
(c) M4pt (d) Ms
Figure 1.3: W W scattering with the 3 different subprocesses depicted.
2
where s = 4Ecm is the Mandelstam variable, the amplitude for s-channel γ, Z exchange (Fig.
1.3d) is
2 0
g 2 cos θ 7g 2 cos θ
s s s
Ms = − + +O , (1.2.3)
4 m2W 4 m2W m2W
and for the t-channel γ, Z exchange (Fig. 1.3b) is
2 0
g 2 (−3 + cos θ) (1 + cos θ) g 2 (3 − cos θ)
s s s
Mt = + +O , (1.2.4)
16 m2W 8 m2W m2W
where cos θ is the scattering angle. Adding the three contributions together,
0
g 2 (1 − cos θ)
s s
M(WL+ WL− → WL+ WL− )gauge = +O . (1.2.5)
8 m2W m2W
8
The E 4 /m4w terms drop out after adding all the different contributions. However, the re-
mainder still grows with energy as ∼ E 2 /m2W . Non-unitary behaviour of this kind suggested
a missing piece. The answer was found through the field of condensed matter physics in
spontaneous symmetry breaking [41].
As a consequence of spontaneous symmetry breaking, Nambu-Goldstone bosons [41, 42]
are produced as excitations of the quantum field. While these bosons are themselves massless,
they are "absorbed" by the gauge fields providing them with an additional longitudinal degree
of freedom and a mass term. In certain cases the symmetries are explicitly broken in addition
to being spontaneously broken. The resulting Goldstone bosons are no longer massless, but
acquire a mass depending on how softly the symmetry is broken. These are referred to
as "Pseudo Nambu-Goldstone bosons". An example is pions, which are produced through
explicit breaking of the chiral symmetry in QCD due to the quark masses; this results in
pions acquiring a small mass. Since the light quark masses are small (∼ 5 − 10 MeV), the
pions as a result have a mass that is small compared to the other hadrons.
The Higgs mechanism is best demonstrated through an example. Consider the La-
grangian for a simple theory with a fermion and a spin-1 boson
1
L = ψ(iγ µ Dµ − m)ψ − F µν Fµν . (1.2.6)
4
with the covariant derivative Dµ = ∂µ + igAµ . This Lagrangian is invariant under the gauge
transformation given by:
ψ → eiα(x) ψ ,
1
Aµ → Aµ − ∂µ α(x) . (1.2.7)
g
It is easy to see, however, that adding a mass term 21 m2 A2 will break the gauge symmetry.
9
(a) µ2 > 0 (b) µ2 < 0
Figure 1.4: Scalar field Φ with two different vacuum configurations. For µ2 < 0 the vacuum
state is no longer symmetric and the field acquires a non-zero expectation value.
To ameliorate this, a complex scalar field with the Lagrangian
LΦ = (Dµ Φ)∗ (Dµ Φ) − V (Φ) (1.2.8)
can be added with the potential V (Φ) given by
V (Φ) = µ2 Φ∗ Φ + λ (Φ∗ Φ)2 , λ > 0, (1.2.9)
with the covariant derivative Dµ ensuring that the new scalar terms of the Lagrangian are
gauge invariant with the field transformation Φ → eiα(x) Φ. This potential presents two
interesting possibilities: For the case of µ2 > 0, the minimum of the potential is still at
zero. Expanding the field Φ around the minimum point Φ = 0 produces essentially the
same Lagrangian in Eq. 1.2.8 and the U (1) symmetry remains unbroken (Fig. 1.4a). For
µ2 < 0, however, the minimum of the potential shifts away from zero (Fig. 1.4b) and the
field acquires a non-zero vacuum expectation value given by
r
µ2 iα
hΦi0 = − e (1.2.10)
2λ
10
where α is some arbitrary phase. We can define the absolute value of this minimum
r
v µ2
√ = − (1.2.11)
2 2λ
as the vacuum expectation value. Expanding the field around the new minimum
v+φ
Φ= √ eiη(x) , (1.2.12)
2
where φ corresponds to the physical mode and η corresponds to the Goldstone mode, the
scalar part of the Lagrangian becomes
1 ∗ v+φ
Dµ (v + φ)eiη(x) Dµ (v + φ)eiη(x) − V ( √ eiη(x) )
LΦ =
2 2
1 1 1
= (∂µ φ)2 + φ2 (∂µ η)2 + gφ2 Aµ ∂µ η + gv 2 Aµ ∂µ η + 2gvφAµ ∂µ η + g 2 vφA2 + g 2 φ2 A2
2 2 2
2 2
1 2 2 2 µ µ 1
+ g v A + µ2 φ2 − φ3 + 2 φ4 + vφ(∂µ η)2 + v 2 (∂µ η)2 + (constant terms) .
2 v 4v 2
(1.2.13)
This Lagrangian can be simplified greatly using the transformation
1
Aµ → Aµ − ∂µ η(x) . (1.2.14)
g
This amounts to fixing the gauge for the vector boson so that α(x) is fixed to α(x) = η(x).
With this choice of gauge, the Lagrangian becomes
1 1 −µ2 3 µ2
LΦ = (∂µ φ)2 − −2µ2 φ2 + φ + 2 φ4
2 |2 {z } |v {z 4v }
Scalar mass term Scalar self-interaction terms
1 2 2 2 1
+ g v A + g 2 vφA2 + g 2 φ2 A2 + (constant terms) . (1.2.15)
|2 {z } | {z2 }
Gauge boson mass term Gauge-Scalar interaction terms
Note that the field η(x) completely disappears. The remaining scalar field is actually physical
11
p
with mass mφ = −2µ2 . The gauge boson also acquires a mass mA = gv. This is colloquially
referred to as the gauge boson "absorbing" the Goldstone mode; the additional degree of
freedom required for the longitudinal mode of the now massive gauge boson appears through
the Goldstone boson. This is referred to as the "Unitary gauge"; only the physical fields
propagate in this gauge.
The Higgs mechanism, described above to generate masses for bosons in a gauge theory,
was proposed in three seminal papers in 1964 by Brout and Englert [20], Higgs [18, 19, 43],
and Guralnik, Hagen, and Kibble [21]. In the Abelian example above, only one Goldstone
boson was generated. For non-Abelian gauge theories the number of Goldstone bosons is
equal to the number of broken generators; for a spontaneously broken SU (N ), for example,
there will be N Goldstone modes. The spontaneously broken SU (2)L ×U (1)Y in the Standard
Model generates 3 Goldstone bosons that give mass to the three vector bosons W ± and Z.
The Higgs mechanism can also explain the origin of fermion masses [44]; this can be achieved
by adding a Yukawa-type interaction to the Lagrangian in Eq. 1.2.6
1
/ − yψ Φψψ − F µν Fµν .
L = iψ Dψ (1.2.16)
4
Expanding Φ around the minimum after symmetry breaking, the Lagrangian becomes
y 1
L =iψ Dψ/ − √ψ (v + φ)ψψ − F µν Fµν
2 4
y v yψ 1
=ψ iD / − √ψ ψ − √ φψψ − F µν Fµν . (1.2.17)
2 2 4
√
Identifying yψ v/ 2 = mψ as the fermion mass,
m 1
/ − mψ ψ − ψ φψψ − F µν Fµν .
L =ψ iD (1.2.18)
v 4
It’s evident from the above equation that the interaction strength between the fermion and
the scalar is proportional to the fermion mass.
12
Figure 1.5: W W scattering including the interactions with the Higgs boson.
As was seen in Eq. 1.2.5, the amplitude for WL+ WL− scattering increases indefinitely with
energy. This non-unitary behaviour is fixed after including the Higgs boson interactions
0
−g 2 (1 − cos θ)
s s
Mh = +O (1.2.19)
8 m2W m2W
which when added to the pure gauge part of the amplitude in Eq. 1.2.5 gives
M(WL+ WL− → WL+ WL− ) = Mt + M4pt + Ms + Mh
0
s
=O (1.2.20)
m2W
implying that the Higgs boson unitarises gauge boson scattering.
It was mentioned previously that the longitudinal modes of the gauge bosons are es-
sentially the Goldstone bosons appearing as a consequence of the spontaneously broken
symmetry when considering the unitary gauge. This implies that any amplitude involving
the longitudinal modes of the gauge bosons can be calculated by replacing the gauge boson
with the corresponding Goldstone boson, known as the Goldstone boson equivalence theorem
[33, 34].
13
Figure 1.6: Plot of the Standard Model Higgs branching ratios against the Higgs mass.
Measured value of Higgs mass is mh = 125.25 ± 0.17 GeV. Reproduced from [8].
1.2.2 Standard model Higgs
In the Standard Model, the Higgs boson appears as a part of the Higgs doublet which
transforms under the SU (2)L × U (1)Y symmetry of the electroweak sector of the Standard
model, details of which are provided in Sec. 1.6. Some of the the prominent decay modes for
the Standard Model Higgs are shown in Fig. 1.6. It is clear that for the measured mass value
mh = 125.25 GeV [1], h → bb is the dominant decay mode with a branching ratio of ∼ 57%
(Tab. 1.2). However, bb channel has a huge QCD background from production of two jets
and identifying those jets as originating from b-quarks (b-tagging) is rather challenging; as
such this wasn’t the ideal discovery channel. Indeed the discovery of Higgs was reported in
the subdominant h → ZZ ∗ → 4l and h → γγ channels [9, 10] with the resonances appearing
in the invariant mass spectra (Fig. 1.7 and 1.8). The diphoton channel, h → γγ, and the 4l
channel, h → ZZ ∗ → 4l, are particularly suited for discovery owing to their cleaner signature
and more manageable backgrounds. Decay in the bb channel was discovered only a few years
14
Decay Channel BR Uncertainty (%)
h → bb 0.574 +3.2 −3.3
h → WW 0.218 +4.2 −4.2
h → gg 0.0855 +10.2 −10.0
h → ττ 0.0629 +5.7 −5.6
h → cc 0.0289 +12.2 −12.2
h → ZZ 0.0269 +4.2 −4.2
h → γγ 0.00228 +4.9 −4.9
h → Zγ 0.00156 +9.0 −8.8
Table 1.2: Branching ratios for the Standard Model Higgs boson with mh = 125.25 GeV [1].
Numbers reproduced from [2].
back [45, 46]. More recently, the rare decay h → µ+ µ− has been observed [47, 48].
The primary production mode for Higgs boson at the LHC is through gluon fusion via
a top quark loop (ggF) which accounts for ∼ 88% of the total Higgs cross-section at the
LHC with σggF = 48.6+4.6%−6.7% pb [2, 8, 49, 50]. This is largely due to high gluon luminosity at
the LHC for low invariant mass (i.e. gluons carry most of the energy of a proton at lower
energies probed by the LHC) as well as the large Yukawa coupling of the top quark with the
Higgs. Other important production modes are Vector-boson-fusion (VBF) with cross-section
σV BF = 3.78+2.2%
−2.2% , "Higgstrahlung" where the Higgs is produced in association with a vector
boson with σV H = 2.25+4.8%
−4.4% , and production in association with a top-quark pair (tth) with
σtth = 0.50+6.8%
−9.9% .
So far the measured properties of the observed particle are consistent with those for the
Standard Model Higgs Boson. It is a electrically neutral, spin-0 particle which transforms as
a scalar under parity conjugation. It’s mass has been measured to be mh = 125.25 ± 0.17 [1]
GeV. The coupling of the Standard Model Higgs to other particles is proportional to the
particle mass. This is indeed what has been observed at the LHC as seen in Fig. 1.9; the
straight line through the experimentally measured values indicates that the couplings are
consistent with the SM prediction. However, to ascertain the true nature of the Higgs boson,
two key properties need to be determined precisely: Higgs potential and decay width.
15
(a) (b)
Figure 1.7: Distribution of four lepton invariant mass for the h → ZZ ∗ → l+ l− l+ l− de-
cay channel for (a) ATLAS (Reproduced from [9]) and (b) CMS (Reproduced from [10])
experiments.
(a)
(b)
Figure 1.8: Distribution of the invariant mass of the photon pair for the h → γγ decay chan-
nel for (a) ATLAS (Reproduced from [9]) and (b) CMS (Reproduced from [10]) experiments.
16
Figure 1.9: Plot of the measured Higgs boson couplings to other Standard Model particles
against particle mass. Reproduced from [11].
1.3 Higgs potential and width
1.3.1 Higgs potential
Higgs potential in the gauge-invariant form can be written as
2
V (Φ) = µ2 Φ† Φ + λ Φ† Φ , λ > 0, (1.3.1)
17
where Φ is a complex SU (2) doublet with the general form
φ2 + iφ3
Φ= . (1.3.2)
φ0 + iφ1
Choosing a global gauge and expanding the Higgs field around the vacuum
1 0
Φ= √ . (1.3.3)
2 v+h
the potential becomes
1 λ
V (h) = v λv 2 + µ2 h + 3v 2 λ + µ2 h2 + vλ h3 + h4 .
(1.3.4)
2 4
Using Eq. 1.2.11 to substitute µ for v, the potential becomes
1 λ
V (h) = (2λv 2 )h2 + vλ h3 + h4 . (1.3.5)
2 4
where v is the vacuum expectation value (vev). From the above equation, the mass of the
Higgs is m2h = 2λv 2 which is measured at the LHC. Finally, the potential can be written in
terms of Higgs mass and the vacuum expectation value as
1 m2 m2
V (h) = m2h h2 + h h3 + h2 h4 . (1.3.6)
2 2v 8v
which means that the Higgs trilinear and quartic couplings are
3m2h
λhhh =
v
3m2
λhhhh = 2h . (1.3.7)
v
The vacuum expectation value appears also in the electroweak sector and can be measured
18
(a) Continuum production of hh (b) hh production through single Higgs
Figure 1.10: Higgs pair production through gluon fusion at LO.
precisely through e.g. muon lifetime measurements. With both the mass and the vacuum
expectation value precisely measured, µ and λ and consequently the Higgs potential are
completely fixed. This is not the end of the story however since new physics can potentially
alter each of these values as well as introduce higher dimensions terms in the Lagrangian; as
such it is of extreme importance to measure the Higgs trilinear and quartic couplings and
compare them against their theoretically predicted values.
Measuring the Higgs trilinear and quartic couplings is challenging, though, at the LHC.
The trilinear coupling can be measured by studying double Higgs production. There are
two modes of production for a pair of Higgs bosons through gluon fusion shown in Fig. 1.10;
these two modes interfere destructively, with the magnitude of cancellation depending on
λhhh . However, the cross-section at the LHC for gg → hh in SM is only 31.18 fb [51] (NNLL
+ NNLO in large top mass approximation). The trilinear coupling can be constrained to
0.5 < λhhh < 4.5 [52] at the High Luminosity LHC while the quartic coupling λhhhh is
essentially out of reach at the LHC. The prospects look a lot better at the proposed 100 TeV
Future Circular Collider and the lepton colliders: ILC, CLIC, µ+ µ− collider. Best constraint
for the trilinear coupling comes from FCC [53] with an estimated precision of 3.4 − 7.8% at
68% CL. For the quartic coupling, achievable precision is 50% at a 14 TeV muon collider [54].
1.3.2 Measuring Higgs width
Another crucial property to be measured to ascertain the true nature of the observed Higgs
boson is its decay width. This is predicted in SM to be Γh = 4.07 MeV [1] and any deviation
19
from this value would imply presence of new physics. However, being such a small value, it
cannot be measured directly at the LHC since the detector resolution is too low to measure
it precisely. The best constraint obtained from a direct measurement (Γh < 1 GeV [55–57])
is orders of magnitude larger than the value predicted in Standard Model of Γh = 4.07
MeV. Instead, an alternate method was proposed in [35, 37, 58] to use the offshell Higgs
signal to constrain the width. The basic idea is as follows: the production cross-section for
gg → h → ZZ ∗ → 4l can be written as [37]
2 2
dσ ghgg ghZZ
∼ . (1.3.8)
dM4l2 (M4l2 − m2h )2 + m2h Γ2h
where ghgg and ghZZ are effective couplings of the Higgs to gluons and Z-bosons. In the above
equation, all the quantities independent of M4l and Γh have been suppressed for simplicity.
Integrating the above near the resonance peak gives
2 2
ghgg ghZZ
σon-shell ∼ . (1.3.9)
Γh
2 2
Using the formula for the on-shell cross section in the above equation, the ratio ghgg ghZZ /Γh
can be constrained from experimental measurement of the cross section. Away from the
resonance peak, however, the denominator in Eq. 1.3.8 is dominated by M4l so the cross-
section (after integrating over M4l ) depends only on the couplings as
2 2
ghgg ghZZ
σoff-shell ∼ . (1.3.10)
M4l4
Assuming now that the couplings ghgg and ghZZ vary only a little over the whole phase-space
and that there are no significant new physics contributions (e.g. a new resonance that alters
the couplings drastically), the off-shell cross section can be used to constrain the product
2 2
ghgg ghZZ . Using these two measurements, the width Γh can be constrained.
20
1.3.3 Experimental constraints
As mentioned previously, Γh cannot be measured directly at the resonance peak at the LHC
since it is limited by detector resolution. The off-shell method described above, however, can
provide stringent constraint on the width. An upper bound of γh < 88 MeV was calculated
in [37] with the possibility of the constraint being improved to Γh < 10 ΓSM h with more data.
In fact, recent analysis [59] with Run II data has constrained the width much further to
3.2+2.8
−2.2 MeV with the 95% CL constraint 0.08 < Γh < 9.16 MeV both of which are consistent
with the SM value.
1.4 Theoretical status
1.4.1 ZZ production at LHC
Higher-order perturbative calculations are an essential part of the modern particle physics
program. One of the most important processes in that regard is the production of a pair
of vector bosons where the vector boson pair could correspond to any combination V V =
ZZ, W W, γγ, W Z, Zγ, W γ. The specifics below refer only to gg → ZZ production since
that is the primary focus of this work.
ZZ production is key to many new physics searches as well as precision physics [60–64].
In addition, it is a significant background to the crucial Higgs signal process gg → h →
V V → leptons [59, 65–67]. It was also observed that it interfere quite significantly (∼ 10%)
with offshell Higgs production [35, 36] which is important for indirect width measurement as
discussed in Sec. 1.3.2. As such, precise theoretical predictions for this process are indispens-
able. Since gluons don’t couple directly to vector bosons, gluon initiated channels always
appear at a higher order in the perturbation series. E.g. qq → ZZ already starts at tree
level while gg → ZZ starts at 1-loop order which is formally NNLO for the hadronic process
pp → ZZ. Nevertheless, due to high gluon luminosity at LHC it accounts for most of the
21
NNLO correction (∼ 60%) [68] to pp → ZZ. The NLO corrections to gg → ZZ were also
found to be significant[69] increasing the total pp → ZZ cross-section by (∼ 5%) [70]. Con-
sequently, pair production of vector bosons has been one of the focal points of perturbative
QCD efforts.
The qq 0 → V V processes are all known at NLO in QCD [71–76] as well as NLO EW for
most processes [77–84]. This was possible in large part to the so-called "NLO revolution"
based on unitarity methods that automated 1-loop calculations [85–91]. Gluon initiated
processes are all also known at 1-loop [92–101]. The 2-loop amplitudes are not universally
known, however. Complexity at 2-loops increases enormously compared to 1-loop and as
such 2-loop processes are calculated on a case-by-case basis. So far no convenient and com-
putationally feasible way to automate the 2-loop processes has been developed. Nevertheless,
there has been significant progress in the past decade. In particular for gg → ZZ, the 2-loop
contributions including only massless quarks in the loops are known [102, 103]. This does
not provide the complete picture, however, and corrections due to massive quark loops are
deemed important as a consequence of the Goldstone boson equivalence theorem [33, 34].
This is especially true for loops involving the top quark whose large coupling to Higgs and
the longitudinal modes of the gauge bosons (as a consequence of the equivalence theorem)
could result in significant corrections in high invariant mass region. The longitudinal modes
for the Z-bosons are, moreover, particularly interesting since they provide a window into
potential new physics manifesting through loop effects as well as they provide a way to
measure the anomalous ttZ coupling [38, 39].
Calculating the 2-loop contributions to gg → ZZ with massive internal quarks provide
significant challenges. Naively, due to addition of one more variable to the problem compared
to the massless case, an increase in complexity is expected. However, the traditional methods
to calculate multiloop amplitudes are insufficient. The 2-loop contributions from top quark
loops were calculated in an approximation where the top quark was considered much heavier
than every other scale, colloquially referred to as the heavy top-mass Limit, in [104, 105].
22
Since the expansion has a finite radius of convergence, it cannot be used in all the regions
of phase space, particularly for higher invariant mass. This can be improved with the use
of Padeé approximants as in [106] to extend the expansion over a larger phase space region.
Another approximation using an expansion around the top quark pair production threshold
together with the heavy top limit, improved with Padé approximants, was used in [107] for
contributions relevant to Higgs production. In [13], the heavy top limit was used together
with small top-mass limit to access the regions of higher invariant mass and improved using
Padé approximants to access the intermediate region. The amplitudes involving Higgs as a
mediator have been computed with exact top mass dependence [108–111]. More recently,
contributions to W W production from third generation quark loops (b, t) were computed
with exact mass dependence in [112]. It must be pointed out that the 2-loop amplitudes
for gg → ZZ with exact top quark dependence were also calculated independently in [113]
which appeared a couple months after this work was completed.
1.4.2 Diphoton production at LHC
Another important class of processes at LHC is the production of a pair of photons. It
provides a crucial channel for new physics searches [114, 115] as well as Higgs production.
Similar to ZZ, continuum production of γγ is also an important background to Higgs signal
process. In addition, the transverse momentum of the diphoton pair is sensitive to potential
new physics whether through Higgs production or through new heavy resonances. It is,
thus, a crucial part of the particle physics program [116] and better theoretical predictions
are always desirable.
The NNLO QCD corrections to pp → γγ are known already [117–120]. The ingredients for
3-loop amplitudes for four particle scattering became accessible only in the recent years [121,
122] with the full amplitudes for qq → γγ appearing very recently [123]. Including 1-jet in
addition, pp → γγ + jet is known only at NLO [124, 125]. For both pp → γγ at N3LO and
pp → γγ +jet at NNLO, the missing component was the 2-loop amplitudes for diphoton plus
23
jet production where they are required to subtract the IR poles for the former. However,
2 → 3 particle scattering amplitudes at 2-loop provided a great challenge and required
the development of powerful new techniques. These advances, discussed in detail in this
work, led finally to the calculation of first full 2 → 3 amplitudes, first in leading colour
approximations [126–136], for simpler helicity configuration [137], and then very recently in
full colour for all helicities [138, 139].
Calculation of both gg → ZZ and qq(g) → γγ + jet 2-loop amplitudes required advances
in the technology and are the focus of this work along with the results.
1.5 Quantum Chromodynamics
1.5.1 History of QCD
The discovery of the atomic nucleus, and consequently the proton, in the early 20th century
led to an obvious problem: the nucleus was too heavy to be accounted for just by the protons.
Among the prevalent suggestions was the electrically neutral bound states of electrons and
protons, referred to as neutrons, residing inside the nucleus. It was shown, however, that an
electron cannot be confined inside the nucleus. This issue was resolved with the discovery of
the neutron in 1932. The neutron was found to have certain remarkable properties. Other
than the electric charge, it was very similar to the proton; its mass was extremely close to
that of the proton and it had the same spin of 1/2. Furthermore, it was found that the
strong nuclear force had the same interaction strength for protons and neutrons, i.e. the
interaction strength between a pair of protons was the same as between a pair of neutrons
or a proton-neutron pair. This led to the development of isospin symmetry, as an analogue
of spin for electrons. Protons and neutrons are treated as two "isospin" states of the same
particle, the "nucleon", with pions, carriers of the strong nuclear force, acting much like the
24
classic spin operators in a harmonic oscillator.
π + |ni =|pi
π − |pi =|ni
π 0 |p, ni =|p, ni . (1.5.1)
Here π ± act like the well-known raising and lowering operators J ± while π 0 acts like the
total spin operator J.
The "isospin" model was very successful in explaining the strong nuclear force in early
days especially after the discovery of pions. As more new particles were discovered, they
were formed into isospin multiplets e.g. nucleon doublet (p, n) with I = 1/2, Sigma triplet
(Σ+ , Σ0 , Σ− ) with I = 1, etc. A major success of the isospin model was the prediction of rho
mesons as the vector bosons which were eventually discovered experimentally. With the ob-
servation of the "strange" decays of Kaons, however, a new quantum number "Strangeness"
was introduced for the particles [140, 141] and the "Eightfold way" was proposed by Gell-
Mann [142] and Ne‘eman [143] to classify the known particles. The isospin symmetry was
extended to an SU (3) flavour symmetry with particles being grouped into octets and decu-
plets.
Eventually, the quark model was proposed by Gell-Mann [144] and Zweig [145] to explain
the observed spectrum of particles with hadrons being composed of smaller particles called
quarks. Three quarks were postulated to transform under the SU (3)F flavour symmetry.
However, while the quark model was successful in explaining the spectrum of hadrons, it
gave rise to a new question. Specifically, some of the hadrons seemed to be violating Fermi-
Dirac statistics e.g. the ∆++ baryon was proposed to be made up of three up quarks,
|∆++ i = |u↑ u↑ u↑ i , (1.5.2)
with all three up quarks having Jz = +1/2. Since the three up quarks are identical fermions,
25
they cannot occupy the same quantum state [146, 147]. The solution to this problem was
introduced as the "colour charge" [148] being an additional degree of freedom possessed by
quarks. The ∆++ baryon can then be expressed as an anti-symmetric combination of three
up quarks with different "colours",
1
|∆++ i = √ |u↑r u↑g u↑b i − |u↑r u↑b u↑g i + |u↑b u↑r u↑g i − |u↑b u↑g u↑r i + |u↑g u↑b u↑r i − |u↑g u↑r u↑b i , (1.5.3)
6
making the resulting combination "colourless". Analogous to quarks, anti-quarks possess
the anti-colour charges r, g, b. This results in mesons, which are combinations of quarks and
anti-quarks, being colourless, e.g.
1
|π + i = √ (|ur dr i + |ug dg i + |ub db i) . (1.5.4)
3
The quarks now form a colour triplet,
|ur i
|ui = |ug i ,
(1.5.5)
|ub i
that transforms under the new SU (3)C group. Unlike the SU (3)F flavour symmetry, however,
SU (3)C colour is an exact symmetry.
In [148], the SU (3)C colour symmetry of the quarks was first considered as a Yang-Mills
gauge theory [40]. Gauge bosons for this theory were called "Gluons", acting as glue holding
the quarks together in hadrons. An interesting feature of this theory was that gluons carried
the colour charge as well, unlike photons in Quantum Electrodynamics. This leads to the
phenomena of asymptotic freedom and confinement, discussed later in this section.
26
1.5.2 SU (N ) gauge theory
A general SU (N ) group has N 2 − 1 generators; for SU (3) the number of generators is 8
corresponding to the 8 different gluons. The generators of a general SU (N ) group can be
represented using T a where the index a ∈ {1, ..., N 2 − 1} refers to the SU (N ) charge in the
adjoint representation. These T a are traceless and hermitian.
T r(T a ) = 0 , (1.5.6)
(T a )† = T a , (1.5.7)
and satisfy the following commutation relation:
a b
T , T = if abc T c . (1.5.8)
f abc are the totally anti-symmetric structure constants
f abc = −2 i T r T a, T b T c .
(1.5.9)
Further details of the SU (N ) algebra are provided in Appendix A.3.
The Lagrangian for the gluon field can be written as
1
Lg = − Ga,µν Gaµν , (1.5.10)
4
where the field strength tensor is
Gaµν = ∂µ Gaν − ∂ν Gaµ + gs f abc Gbµ Gcν . (1.5.11)
It is straightforward to see that the Lagrangian in Eq. 1.5.10 is invariant under the gauge
27
transformation
Gaµ → Gaµ − ∂µ θa (x) − gs f abc θb (x)Gcµ . (1.5.12)
Expanding the Lagrangian in Eq. 1.5.10, we get the kinetic term
1
− (∂ µ Ga,ν − ∂ ν Ga,µ )(∂µ Gaν − ∂ν Gaµ ) , (1.5.13)
4
the 3-point self-interaction term
gs abc µ a,ν
− f (∂ G − ∂ ν Ga,µ )Gbµ Gcν , (1.5.14)
2
and the 4-point self-interaction term
gs2 abe cde a,µ b,ν c d
+ f f G G Gµ Gν . (1.5.15)
4
Unlike with photons in Quantum Electrodynamics, gluons interact with themselves which
leads to the phenomena of asymptotic freedom and confinement, discussed later in Sec. 1.5.6.
Details on the QCD Lagrangian and the Feynman rules are given in Appendix A.
The above Lagrangian for the gluon field poses a problem though. Specifically, the
propagator of the gluon cannot be derived from this Lagrangian using a naive approach.
This is evident from considering the kinetic term in the Lagrangian
1
Lkin = − (∂ µ Ga,ν − ∂ ν Ga,µ )(∂µ Gaν − ∂ν Gaµ )
4
1 a,ν
= G (gµν ∂ 2 − ∂µ ∂ν )δ ab Gb,µ + ... . (1.5.16)
2
28
The propagator Dab,µν (x − y) is defined using
δ ac (gµρ ∂ 2 − ∂µ ∂ρ )Dbc,νρ (x − y) = iδ ab δµν δ (4) (x − y) , (1.5.17)
which in momentum space becomes
δ ac (−gµρ p2 + pµ pρ )De bc,νρ (p) = iδ ab δ ν .
µ (1.5.18)
Here, (−gµρ p2 + pµ pρ ) is singular due to gauge invariance and hence cannot be inverted to
obtain the propagator. This can be resolved by adding a gauge fixing term to the Lagrangian
1
Lgauge−f ixing = − (∂µ Ga,µ )2 . (1.5.19)
2ξ
which, of course, renders the Lagrangian in Eq. 1.5.10 no longer gauge invariant. Eq. 1.5.18
now becomes
ac 2 1 e bc,νρ (p) = iδ ab δ ν
δ −gµρ p + 1 − p µ pρ D µ (1.5.20)
ξ
which, when inverted, gives
ab
pµ pν
De ab,µν (p) = iδ −g µν
+ (1 − ξ) 2 . (1.5.21)
p2 p
In the above equation ξ is a free parameter used to fix the gauge. This family of gauges is
known as the Rξ gauges. Some common choices are "Feynman-‘t Hooft gauge" ξ = 1 which
is the most popular choice in particular for higher-order corrections owing to the simplicity of
the resulting expressions, and "Landau gauge" (ξ = 0). The physical observables, however,
are gauge-invariant and as such ξ dependence must drop out at the end.
This procedure of gauge-fixing is made more transparent using the Path-integral formal-
ism. Note that gauge-fixing is a requirement for quantisation in this case. A consequence
29
of this specific gauge-fixing procedure is the appearance of Faddeev-Popov ghost fields [149]
which are required for preserving manifest Lorentz invariance as well as unitarity. The
Faddeev-Popov Lagrangian is given by
Lghost = (∂ µ c∗a )(∂µ ca ) − gs f abc (∂ µ c∗a )cb Gcµ . (1.5.22)
The ghosts live in the adjoint representation, like the gluons, and interact only with gluons
through a 3-point vertex, appearing as internal lines in Feynman diagrams. Such diagrams
are essential to obtain physical results e.g. the gluon self-energy correction at 1-loop (see
Appendix D). Feynman rules are given in Appendix A.2.
The name "ghost" arises from the fact that these fields are not physical and are simply a
tool to make sense of the path integral and preserve unitarity for the specific gauge choice;
they act to cancel the unphysical degrees of freedom appearing from ambiguity in gauge
choice. This is also evident from the fact that the ghosts violate the Spin-Statistics theorem;
they appear as spin-0 particles (scalar), however they are anti-commuting (fermions). Indeed
other choices such as the Axial gauge [150] do not require the presence of such fields. It must
be pointed out that the gauge-fixing procedure above, and consequently the appearance of
Faddeev-Popov ghosts, is not simply an artefact of a non-abelian gauge theory; while the
ghosts do appear for Quantum Electrodynamics as well, they do not couple to the photon due
to the abelian nature of the theory and hence are irrelevant for any meaningful calculations.
1.5.3 Perturbative QCD
Calculating scattering amplitudes and cross-sections in quantum field theory can be very
challenging. Very few theories are exactly solvable; for almost all phenomenological purposes,
perturbation theory is used. The observable or quantity of interest is expanded in a small
parameter, usually the coupling strength of the interaction. This is justified in most cases,
e.g. the coupling constant of QED, the fine structure constant, α ' 1/137 and the coupling
30
Figure 1.11: Perturbative corrections to qq → γ in QCD represented using Feynman dia-
grams.
constant of QCD αS ' 0.118. For QCD, the cross-section can be expanded in the strong
coupling constant αS as:
α α 2 α 3
S S S
σ = σ0 + σ1 + σ2 + σ3 + ... (1.5.23)
2π 2π 2π
where σ0 , σ1 , σ2 , ... are the contributions from each order in perturbative expansion (e.g.
Fig. 1.11). Standard technique to calculate these is by the use of Feynman diagrams which
provide a highly intuitive pictorial representation of the scattering process in consideration
in addition to providing a convenient tool to calculate scattering amplitudes.
While at first glance it might seem that the higher-order corrections are increasingly less
important given each successive term is multiplied by higher powers of the coupling constant,
they can be large in many cases, perhaps best demonstrated via Higgs boson production
through gluon fusion. Fig. 1.12 shows the QCD corrections to this process. It is clear that
the perturbative corrections here are extremely important; the Next-to-Leading Order (NLO)
correction is about as large as the Leading Order (LO) term and even the Next-to-Next-to-
31
Figure 1.12: QCD corrections to gg → h. The uncertainty bands represent scale uncertainty
for the range µ ∈ { m4h , mh } with the central value at µ = mh /2. Reproduced from [12].
Leading Order term is significant, only at N3LO does the calculation become stable. The
uncertainty bands in the figure are estimated by varying the scale µ; this dependence on scale
enters through renormalisation 1.5.5 and factorisation 1.5.7, and physical observables should
not depend on it. In a sense, then, the dependence on the scale µ encodes some information
about the missing higher order terms and can be used to estimate the uncertainty. However,
the uncertainty bands at LO utterly fail to capture the NLO correction; only at N3LO the
uncertainty bands become small enough and it lies entirely within the NNLO correction
which is suggestive of the series converging (note that the perturbative series is expected to
be asymptotic and not convergent).
32
Figure 1.13: 1-loop correction to qq → − γ.
1.5.4 Regularisation
The type of corrections shown in Fig. 1.11 are commonly referred to as the "virtual cor-
rections" owing to the presence of additional virtual particles. Considering the diagram
corresponding to the second term in the perturbative expansion in Fig. 1.11 and inserting
Feynman rules, it is found that the external momenta are insufficient to constrain the mo-
menta of the internal edges or the "loop". This loop momentum k in Fig. 1.13 is a free
quantity that parameterises intermediate states. Quantum Mechanics dictates that all the
intermediate states should be summed over; in this case the loop momentum k needs to be
integrated over all possible values leading to the expression
d4 k (k/ − p/1 )γ µ (k/ + p/2 )
Z
M∼ . (1.5.24)
(2π)4 (k 2 )(k − p1 )2 (k + p2 )2
A closer look at the above integral shows that it diverges for high values of loop momenta:
d4 k (k/ − p/1 )γ µ (k/ + p/2 ) Wick rot. k2 Λ
dkE
Z Z Z
−−−−−→ d kE E6 ∼
4
, (1.5.25)
(2π)4 (k 2 )(k − p1 )2 (k + p2 )2 k→∞ kE kE
where Wick rotation (see Appendix B) has been performed to transform to Euclidean space
and the integration domain has been cutoff with a parameter Λ to regulate the divergence.
This kind of divergence that manifests for high values of loop momenta is commonly referred
33
Figure 1.14: A 3-point massless integral. The incoming momenta p1 , p2 are massless.
to as Ultraviolet (UV) divergence. In this example a cutoff regulator was used; other choices
exist e.g. Pauli-Villars regularisation.
In fact this diagram has two different kinds of divergences originating from different
regions of loop momentum. The second kind of divergence becomes apparent in the region of
small loop-momentum. Considering only the scalar integral from Fig. 1.14 with p21 = p22 = 0,
d4 k 1
Z
I= . (1.5.26)
iπ (k )(k − p1 )2 (k + p2 )2
2 2
For loop momentum in the soft limit k → 0,
d4 k 1 dk
Z Z
I∼ ∼ . (1.5.27)
iπ (k )(2k · p1 )(2k · p2 )
2 2
k→0 k
This integral also has a logarithmic divergence similar to Eq. 1.5.25, albeit of a very different
nature. This kind of divergence for small loop momentum is called a "Soft divergence".
There is another possibility for this integral when the loop momentum goes collinear to
either p1 or p2 .
1
Z
I∼ d4 k . (1.5.28)
k→p1 (k 2 )(k − p1 )2
34
These are called "Collinear divergences". The soft and collinear divergences are collectively
referred to as Infrared divergences.
As shown in Eq. 1.5.25, one way to regulate the UV divergence is by cutting the integral
off at some large scale Λ; this is "Cutoff regularisation" scheme. The divergences then appear
in the scale Λ. This scheme can also be used to regulate soft divergences by cutting off the
loop momentum at some low energy scale. The most commonly used scheme, however, is
dimensional regularisation [151–155]. In this scheme, ordinarily 4-dimensional objects are
assumed to be d-dimensional where d = 4 − 2 is a complex valued quantity in general. At
the end, the limit → − 0 is taken with poles appearing as 1/n (n > 0). Consider the UV
divergent integral
dd k 1 dd k
Z Z
I= d/2
∼ . (1.5.29)
iπ (k − m )(k + p)2
2 2
k→∞ k4
This integral is finite for d < 4. We can then analytically continue the integral to d = 4 − 2
with the divergences now appearing as poles in 1/. IR divergences can be similarly regulated:
dd k 1 dd k
Z Z
I∼ d/2
∼ . (1.5.30)
iπ (k )(2k · p1 )(2k · p2 )
2
k→0 k4
This integral converges for d > 4. Again, we can analytically continue the integral to
d = 4 − 2.
This provides a unified prescription to deal with both UV and IR divergences and has
allowed for much of the progress in perturbative calculations. A huge advantage of using
dimensional regularisation is that it preserves gauge symmetries unlike cutoff regularisation.
In Conventional Dimensional Regularisation, all the momenta, external vector polarisa-
tions, and Dirac matrices are treated in d-dimensions. This has certain disadvantages e.g. it
leads to spurious structures that vanish in d = 4; this is discussed in more detail in sec. 2.
An alternative approach is the ‘t Hooft-Veltmann scheme [153] where the loop momenta are
treated in d-dimensions while the external vector polarisations are treated in d = 4 dimen-
35
sions. This leads to significant simplifications in many situations and prevents appearance
of such spurious structures [156, 157].
Another consequence of dimensional regularisation is that the integral measure in a loop
integral gains an additional mass dimension of (d − 4)L where L is the number of loops,
along with the couplings gaining a mass dimension as well. It is conventional to redefine the
coupling to render it dimensionless by introducing a dimensionful parameter µ such that, for
4−d
example, gs0 = gs µ 2 with µ known as the "‘t Hooft scale". This is done to ensure that all
the terms in the Lagrangian have proper dimensions d.
1.5.5 UV renormalisation and IR subtraction
Since the physical observables are finite, these UV and IR divergences need to cancel in
some way. The process of removing the UV singularities is called "Renormalisation". The
Lagrangian is modified such that the "bare" parameters are replaced by "renormalised"
parameters. E.g.
m0 ψ 0 ψ0 → (Zm Zψ )mR ψ R ψR (1.5.31)
with the renormalisation constants Zm and Zψ defined according to
1
mR = m0
Zm
1
ψR = p ψ0 . (1.5.32)
Zψ
The UV divergences are absorbed into the renormalisation constants during this redefinition.
The renormalised parameters are finite implying that the bare parameters are also divergent.
The renormalisation constants are written such that they have a trivial part and a countert-
erm e.g. Zm = 1 + δm where 1 refers to the tree level term and δm is the counterterm and
cancels against the poles order by order in perturbation theory. In principle there are an in-
36
(a) Soft singularity. (b) Collinear singularity.
Figure 1.15: Infrared divergences corresponding to the emission of a real particle from a
massless particle. Fig. 1.15a shows emission of a gluon with very small momentum i.e. a
soft divergence while Fig. 1.15b shows the emission of a gluon collinear with the external
particle.
finite number of renormalisation schemes; the divergences are fixed, however any finite term
can be added to the renormalisation constant. Some of the commonly used schemes are the
"On-shell" scheme where the renormalised parameters are chosen using on-shell properties of
the particles and the modified "Minimal Subtraction" (M S) scheme [158, 159]; these are the
two schemes used in this work. Details of the renormalisation procedure for this calculation
are discussed in Sec. 5.2 and Appendix D.
In general, new divergences appear at every order in perturbation theory. These diver-
gences must all be absorbed during the renormalisation procedure to render the result finite.
For many theories new "renormalisation constants" are required at every order implying
that an infinite number is required to renormalise the theory at all orders, which results in a
loss of predictive power. Such theories are called non-renormalisable. To show that a theory
is renormalisable and requires only a finite number of counterterms is highly non-trivial;
requirements for a renormalisable theory were given in [160–162] with the renormalisability
of non-abelian gauge theories with spontaneous symmetry breaking (e.g. Standard Model)
proven in [163].
The IR poles cancel against poles appearing in real emission diagrams, which have soft
and collinear poles similar to the loop amplitudes through the emission of a soft and collinear
particle respectively (see Fig. 1.15). For the case in Fig. 1.15, the propagator for the initial
37
(a) Leading order term.
(b) Virtual correction. (c) Real correction.
Figure 1.16: QCD corrections to qq → − γ. Fig. 1.16b shows the αS correction through the
exchange of a virtual gluon while Fig. 1.16c show the αS correction through the emission of
a real gluon.
particle is
1 1 1
= = (1.5.33)
(p + k) 2 2p · k 2|~p||~k|(1 − cos θ)
for a massless particle. For both k → 0 (soft) and k → xp (collinear) the propagator
has a singularity. This is a real physical particle, hence the name real corrections. These
divergences appear during phase-space integration over the external momenta and can be
regulated using dimensional regularisation just like UV poles. It must be noted that these
divergences don’t appear for massive particles. Considering the cut diagram in Fig. 1.16c, a
gluon is emitted from a quark line. Since both virtual and real corrections occur at the order
αS , it is necessary to add both to get the correct result. This is most readily seen through
Fig. 1.16; the amplitude-squared can be thought of as γ → γ with the real and virtual
corrections simply being different cut configurations. Bloch-Nordsieck theorem [164, 165]
38
showed that the IR divergences for QED must cancel once all configurations are included
i.e. real correction diagrams in addition to the virtual corrections. This was extended to
non-abelian gauge theories in [166–168]. Observables for which this is possible are called IR
safe observables. However, computing amplitude-squared as in Fig. 1.16 is highly impractical
for all but the simplest processes. The standard technique is to subtract the poles from the
virtual corrections and add them to the real corrections. Sum of both is finite and produces
the full cross-section:
Z Z
σN LO = σLO + dσV + dσR
N N +1
Z Z Z
= σLO + dσV − dσS + (dσR + dσS ) . (1.5.34)
N 1 N +1
Here dσS is the subtraction term; it must fulfill two conditions: it must have the same singular
behaviour as dσR (the real emission term) such that the sum dσR + dσS is finite for d → 4,
and it must be integrable in d-dimensions over the 1-particle phase-space and produce the
infrared divergences to cancel against the virtual corrections. The pole structure for real and
virtual corrections is fixed; there is freedom to choose the finite part of dσS , however. Many
subtraction schemes exist with the most widely used ones at NLO being the Catani-Seymour
dipole subtraction [14, 169], the Frixione-Kunzst-Signer (FKS) subtraction [170, 171], and
the Nagy-Soper subtraction [172], and have been automated successfully [173, 174]. However,
the situation at NNLO is far from settled; the reader is referred to [175] for a detailed review
of the schemes utilised for NNLO calculations.
1.5.6 Asymptotic freedom
Unlike Quantum Electrodynamics, Quantum Chromodynamics is a non-abelian gauge theory
i.e. the force carriers (gluons) also carry the charge and, as a consequence, can interact with
each other. This leads to the phenomenon of "asymptotic freedom" [176, 177] at high energies
i.e. the coupling strength of strong interaction becomes smaller with increasing energy. On
39
the flip-side, the coupling increases with decreasing energy leading to the phenomenon of
"confinement". This energy dependence of the renormalised coupling strength arises as a
consequence of the regularisation and renormalisation procedure. However, the bare coupling
should remain constant irrespective of any renormalisation procedure which leads to the
equation
dαS,0
= 0. (1.5.35)
dµ
Writing the bare coupling αS,0 in terms of the renormalised coupling αS results in
dαS (µ)
= β(αS ) . (1.5.36)
d log(µ2 )
The Beta-function can be expanded order-by-order in αS as
α 2
αS S
β(αS ) = −αS β0 + β1 + O(αS3 ) . (1.5.37)
2π 2π
The first few coefficients are given by
11CA − 4TF nf
β0 =
6
17CA2 − 10CA TF nf − 6CF TF nf
β1 = (1.5.38)
6
where nf is the number of active flavours. For QCD, CA = 3 and TF = 1/2 which implies
β0 > 0 for nf < 17 i.e. the coupling decreases with energy for low number of active flavours;
this is the origin of asymptotic freedom. This behaviour is experimentally confirmed as
shown in Fig. 1.17.
For QED, on the other hand, β0 = −2/3 which means that the coupling grows with
40
Figure 1.17: Plot showing theoretical prediction for the strong coupling αS against experi-
mental measurements. Reproduced from [1].
energy. Solving Eq. 1.5.36 to get the analytical expression for the running coupling,
α(µ0 )
α(µ) = α(µ0 ) 2 (1.5.39)
1+ 2π
β0 log µµ2
0
It is clear that the above equation has a pole at very high energy, referred to as Landau
pole [178], where the running coupling diverges. For QED, the coupling increases with en-
ergy and the divergence appears at a very high energy (∼ 10286 eV); for QCD this happens
as the energy decreases. It makes sense, however, to interpret this as the transition from
perturbative to non-perturbative regime when the coupling α > 4π. Past this point, per-
turbation theory and, as a consequence, Eq.1.5.39 are no longer valid. The scale at which
perturbation theory breaks down for QCD is λQCD ' 200 MeV; for energies lower than this,
41
non-perturbative techniques such as Chiral Perturbation Theory and Lattice Gauge Theory
need to be used for calculations.
1.5.7 Factorisation
In collider experiments, often composite particles like protons are collided at high energies
and the outgoing particles are studied to measure the observables e.g. proton-proton colli-
sions at the LHC. At such high energies, however, the constituent partons [32] are the ones
actually undergoing scattering governed by the short-distance physics which is modelled us-
ing perturbation theory. But the behaviour of these partons inside the proton, specifically
their momentum distribution (Parton Distribution Functions), cannot be described using
perturbative QCD. To perform a complete calculation of the cross-section, both short and
long distance physics is required. That the full calculation can be factorised into the per-
turbative and non-perturbative parts which can then be evaluated independently is crucial
to this. The scale separating the two is called the factorisation scale (µF ). A scattering
cross-section for the collision of two hadrons h1 h2 → cd (e.g. pp at the LHC) can be written
in a factorised form as [1]
Z 1 X
h1 h2 →cd
dσ = dx1 dx2 fa/h1 (x1 , µ2F )fb/h2 (x1 , µ2F )dσ̂ ab→cd (Q2 , µ2F ) (1.5.40)
0 a,b
where a, b are the colliding partons and Q is the scattering energy. The fa/h1 , fb/h2 in the
above equations are the Parton Distribution Functions (PDFs) and specify the probability of
finding the parton a, b in the hadron h1 , h2 with momentum fraction x1 , x2 respectively. All
the unresolved particles that lead to IR poles below the factorisation scale µF are absorbed
in the PDF definitions. PDFs are process-independent but cannot be calculated using the
regular perturbation theory methods. Traditionally they are determined using data [179–181]
based on the DGLAP evolution equations [182–184]. There has been significant progress,
however, in calculating PDFs through first principles using Lattice methods.
42
Factorisation has been proven for many processes in gauge theory [185–188]. Like the
renormalisation scale, the actual calculation must be independent of the factorisation scale.
However, a scale dependence still remains owing to truncation of the series, and should
disappear if all the orders are calculated. This scale dependence manifests itself in the
Parton Distribution Functions as well as the hard-scattering cross-section (Eq. 1.5.40).
The detectors have finite resolution which means that they cannot measure particles with
arbitrarily low energies, neither can they distinguish between two particles extremely close
together. Physical observables then need to be devised such that they respect such physical
restrictions and are infrared safe i.e. additional soft or collinear radiations do not affect
them. One such observable is a jet which is just a collection of final-state partons combined
together according to a specific algorithm. They are designed to be simple to use in both
experiments and theory as well as insensitive to non-perturbative effects. A way to imagine
a jet is to consider a final state parton which radiates additional partons with radiation
within a certain cone around the radiating particle being part of the jet. Two major kinds
of algorithms are cone algorithms and sequential recombination algorithms. An in-depth
discussion of jets is out of scope for this work and the reader is instead referred to [189–191]
for details.
1.6 Electroweak sector
1.6.1 Electroweak symmetry breaking
The electroweak sector of the Standard Model describes the electromagnetic and weak inter-
actions through the SU (2)L × U (1)Y gauge symmetry [192–195]. The L in SU (2)L referes to
"left", i.e. the gauge bosons associated with this symmetry interact only with left-handed
particles and Y refers to "hypercharge", the additional quantum number possessed by SM
fermions. Using the general SU (N ) formula for the number of generators, there are 3 SU (2)L
gauge bosons W1 , W2 , W3 along with B for U (1)Y .
43
The kinetic term for the gauge bosons can be written as
1 1
Lkin = − B µν Bµν − W i,µν Wµν i
(1.6.1)
4 4
with the field strength tensors given by
B µν = ∂ µ B ν − ∂ ν B µ
Wiµν = ∂ µ Wiν − ∂ ν Wiµ + g2 ijk Wjµ Wkν (1.6.2)
where g2 is the coupling strength of SU (2)L . To generate masses for these bosons, the
SU (2)L × U (1)Y symmetry must be spontaneously broken through Higgs mechanism. The
Higgs part of the Lagrangian can be written as
Lhiggs = (Dµ Φ)† (Dµ Φ) − µ2 (Φ† Φ) − λ(Φ† Φ)2 , (1.6.3)
where the covariant derivative Dµ is such that the above Lagrangian is invariant under the
SU (2)L × U (1)Y gauge symmetry
g2
Dµ = ∂µ + ig1 Y Bµ − i σi Wi,µ . (1.6.4)
2
which can be written as
∂µ 0 i −g1 Bµ + g2 W3,µ g2 (W1,µ − i W2,µ )
Dµ = − (1.6.5)
2 g (W + i W ) −g B − g W
0 ∂µ 2 1,µ 2,µ 1 µ 2 3,µ
with the Hypercharge Y set to 1/2. The gauge fields themselves transform under infinitesimal
44
transformations as
Bµ → Bµ − ∂µ θ(x)
Wi,µ → Wi,µ − ∂µ θi (x) − g2 ijk θj (x) Wk,µ , (1.6.6)
similar to Eq. 1.5.12. Inserting Eq. 1.6.5 into Eq. 1.6.3 and expanding
the Higgs field around
0
the vacuum (keeping only the physical Higgs field) Φ = √1 , the gauge bosons gain
2
v+h
masses:
v2 2 2
g2 (W1 + W22 ) + (g1 Bµ + g2 W3,µ )2
Lgauge−mass = (1.6.7)
8
It is clear that two of the SU (2)L bosons W1 , W2 acquire a mass. The third boson W3 does
not acquire a mass itself, but rather the combination g1 Bµ + g2 W3,µ does. At this point it
makes sense to define certain quantities. The electric charge of a particle can be defined as
Q = Y + I3 , (1.6.8)
and the electromagnetic coupling e can be defined as
g1 g2
e= p 2 . (1.6.9)
g1 + g22
Parameterising the couplings with an angle θW ,
g1 = g2 tan θW , (1.6.10)
45
the fields Bµ and W3,µ can be rotated as
g2 Bµ − g1 W3,µ
Aµ = p = cos θW Bµ − sin θW W3,µ
g12 + g22
g1 Bµ + g2 W3,µ
Zµ = p 2 = sin θW Bµ + cos θW W3,µ . (1.6.11)
g1 + g22
It is also useful to define the linear combinations W± = W1 ± iW2 . The mass terms then
become
g2v2
1
Lgauge−mass = 2 W+ W− + Z 2
(1.6.12)
8 cos2 θW
A couple things are clear from the above equation. Z boson is predicted to be more massive
than the W ± bosons since cos2 θW < 1. Also, only 3 bosons gain mass with Aµ remaining
massless. This can be seen in another way; the Q operator acting on the vacuum state gives
Q|0i =(Y + I3 )|0i
1
Y +2 0 0 0
= = (1.6.13)
0 Y −2 1 √v 0
2
since Y = 1/2 for the Higgs field, meaning that Q is conserved compared to, for instance, Y :
Y 0 0 0
Y |0i = = . (1.6.14)
0 Y √v √v
2 2
This implies that the SU (2)L ×U (1)Y gauge symmetry spontaneously breaks down to U (1)EM
with the gauge boson of the U (1)EM symmetry, photon, remaining massless. Using Eqs. 1.6.9
and 1.6.10, the masses of the gauge bosons can be written in terms of electric charge e:
e2 v 2
1 2
Lgauge−mass = W+ W− + Z . (1.6.15)
8 sin2 θW cos2 θW
46
The gauge boson masses are then
e2 v 2
m2W =
4 sin2 θW
e2 v 2
m2Z = . (1.6.16)
4 sin2 θW cos2 θW
Since e and v are known, measuring one of the gauge boson masses fixes the other.
1.6.2 Custodial symmetry
It is useful to define the ρ-parameter
m2W
ρ= . (1.6.17)
m2Z cos2 θW
At tree level, in SM, ρ = 1; this is guaranteed by the so-called "Custodial symmetry" [196,
197]. Before spontaneous symmetry breaking, the Higgs field has a global SO(4) symmetry
4
φ2 + iφ3 X 2
Φ† Φ = φ2 − iφ3 φ0 − iφ1 = φi . (1.6.18)
φ0 + iφ1 i=1
Alternatively, it can be referred to as a global SU (2)L × SU (2)R symmetry since the two are
isomorphic. After spontaneous symmetry breaking, φ0 = h + v and the SO(4) is broken to
SO(3)
X3
† 2
Φ Φ → (h + v) + φ2i (1.6.19)
i=1
with the Goldstone bosons corresponding to W ± , Z transforming as a triplet under the
residual SO(3), which is the custodial symmetry. This SO(3) isn’t exact however and is
broken by the mixing between the W3 and B fields implying that the corrections to the ρ
parameter must be proportional to the mixing angle sin2 θW , which is indeed found through
47
explicit calculations. On the other hand, non-degenerate fermion doublets coupling to the
Higgs also break the custodial symmetry where the symmetry breaking is proportional to
the difference in the Yukawa couplings e.g. for the t − b doublet, the correction to the ρ-
parameter vanishes in the limit yt = yb . Thus the custodial symmetry prevents the gauge
bosons from acquiring arbitrarily large radiative corrections for their masses.
1.6.3 Electroweak interactions
Consider again the gauge-boson kinetic terms in Eq. 1.6.1. Expanding the field strength
tensors gives the gauge boson interaction terms e.g. the 3-point vertices
− ig2 cos θW ∂ µ Z ν (Wµ+ Wν− − Wν+ Wµ− ) + Z ν (W +,µ ∂µ Wν− + ...)
− ie ∂ µ Aν (Wµ+ Wν− − Wν+ Wµ− ) + Aν (W +,µ ∂µ Wν− + ...) ,
(1.6.20)
and the 4-point vertices
g22 +,µ −,ν + −
+ W W Wµ Wν + g22 cos2 θW Z µ Zµ W +,ν Wν−
2
− e2 Aµ Aµ W +,ν Wν− − eg2 cos θW Z µ Aµ W +,ν Wν− . (1.6.21)
It is interesting to note that the all-W interactions involve purely the SU (2)L coupling g2
implying that this is a pure "weak" interaction while those involving the photon field Aµ
have the coupling e implying the electromagnetic nature of the interaction, the fact that the
photon is coupling to the electrically charged W ± -bosons. The Z-boson interactions also
depend purely on the weak coupling g2 but only the purely weak part of Zµ = − sin θW Bµ +
cos θW W3,µ contributes. Also, Z being electrically neutral, does not couple to the photon.
Considering now the fermions, the Lagrangian can be written as
Lf ermion = Ψ(iγ µ Dµf )Ψ (1.6.22)
48
ψ2
where Ψ is the doublet Ψ = and the covariant derivative for fermions Dµf is
ψ1
g2
Dµf = ∂µ − ig1 Y Bµ − i PL σi Wi,µ , (1.6.23)
2
νe,L
where PL = (1 − γ 5 )/2. Taking the example of the (left-handed) e − νe doublet Le =
eL
, the interaction term can be written as
g2
Le = iLe (∂/ − ig1 Ye B
/ −i / i )Le + ieR (∂/ − ig1 YeR B)e
σi W / R + iν R (∂/ − ig1 YνR B)ν
/ R.
2
(1.6.24)
Since the right-handed fields eR , νR do not form part of an SU (2)L doublet, I3 = 0 for both.
For the neutrino this implies that YνR = 0 from the relation Q = Y + I3 , meaning that the
right-handed neutrino does not interact with through weak interactions at all. And since
it is not charged under SU (3)C either, it is completely invisible in the Standard Model. In
principle nothing forbids its presence, however it is impossible to detect in SM and does
not interact with any other SM particles making it irrelevant to any meaningful physical
observable. This does not mean, though, that the right-handed neutrinos are pointless; in
fact many new physics models predict their interactions. Since neutrinos have been shown
to have masses through oscillation experiments [1, 198, 199], assuming that they gain their
masses through the Higgs field necessitates the presence of right-handed fields. The other
hypercharges can also be inferred similarly; since Qe = −1, Ye = −1/2 which can also be
inferred using Qν = 0 and I3 (νe,L ) = 1/2. For the right-handed electron, YeR = Qe = −1.
Expanding the above Lagrangian in terms of the rotated A−Z basis and using Q = Y +I3
49
gives
−
e 0 W
Le = iLe ∂/Le + ieR ∂/eR + Le Le
2 sin θW W +
0
/+ 1 2 /
Qe A 2 sin θW cos θW
(1 − 2Qe sin θW )Z 0
+ eLe Le
−1 /
0 2 sin θW cos θW
Z
/ R − e Qe tan θW eR Ze
+ e Qe eR Ae / R
e
= iLe ∂/Le + ieR ∂/eR + / + eL + eL W
ν e,L W / − νe,L −e (eR Ae / R + eL Ae / L)
2 sin θW | {z } | {z }
Electromagnetic interaction
Charged current weak interaction
e
(1 − 2Qe sin2 θW )eL Ze / L − 2Qe sin2 θW eR Ze
+ / R + ν e,L Zν
/ e,L . (1.6.25)
sin 2θW | {z }
Neutral current weak interaction
where, with abuse of notation, e is the positron charge. A few things to note from the above
equations: The W ± -bosons act as raising and lowering operators on an SU (2) doublet; Z-
boson acts like the I3 operator, however due to mixing between the B − W3 fields there is
also a photon like interaction that, unsurprisingly, depends on the strength of this mixing;
νe,L has no electromagnetic interactions, as expected, since it is electrically neutral.
The interactions above can be writtenin amore condensed form (omitting the pure EM
ψ2
term) for a general fermion doublet Ψ = as
ψ1
e e
Lweak = ψ2W / + ψ1 + ψ 1 W / − ψ2 + / f + af γ 5 )ψf .
ψ Z(v (1.6.26)
2 sin θW 2 sin θW cos θW f
with the purely EM interaction term being ignored for brevity. Here vf , af refer to the vector
and axial coupling of the Z-boson to the fermion f and are given by
vf =I3 − 2Qf sin2 θW
af = − I3 , (1.6.27)
50
where Qf is the electric charge of the fermion in the units of the positron charge.
51
Chapter 2
Setup of the calculation
2.1 Form factor decomposition
Let us consider pair production of Z-bosons through gluon fusion,
g(p1 ) + g(p2 ) → − Z(p3 ) + Z(p4 ) . (2.1.1)
The gluons (p1 , p2 ) are considered to be incoming and the Z-bosons (p3 , p4 ) considered to be
outgoing so that momentum conservation implies
p1 + p2 = p3 + p4 . (2.1.2)
The external particles are also considered to be on-shell with the momenta then satisfying
the on-shell conditions
p21 = p22 = 0, p23 = p24 = m2Z . (2.1.3)
Formally, the scattering amplitude for this process can be written as
a,µ b,ν ∗ρ ∗σ
M = Mab µνρσ (p1 , p2 , p3 , p4 ) λ1 (p1 ) λ2 (p2 ) λ3 (p3 ) λ4 (p4 ) (2.1.4)
stripping away the polarisation vectors λi (pi ) for the gluons and the Z-bosons with λi
denoting the particle helicity. For brevity, the colour and Lorentz indices on the polarisation
vectors are henceforth suppressed and the abbreviation i ≡ λi (pi ) is instead used. Note
that the amplitude M, or equivalently Mab µνρσ (p1 , p2 , p3 , p4 ), is the full scattering amplitude;
52
no perturbative expansion has been performed yet. In general, it is desirable to maintain the
full Lorentz structure of the amplitude since complete polarisation information can be useful
to include e.g. decays of the outgoing particles. However, for the perturbative expansion
coefficients of the full amplitude, the calculation can be immensely simplified in many cases
by the use of explicit representations for the polarisation vectors λi (pi ), for example using
the spinor-helicity formalism [90, 200–204]; the most well-known example is perhaps the
Parke-Taylor formula [205, 206] where the leading-order amplitude for n-gluon scattering
collapses to a remarkably simple 1-line expression.
In general, evaluating Mab µνρσ (p1 , p2 , p3 , p4 ) directly can be an extremely difficult task.
The expressions involve a large number of tensor loop integrals, and while techniques exist
to handle them in a systematic way, most of the powerful modern methods involve scalar
loop integrals where the open Lorentz indices are contracted with loop or external momenta.
As such, it is highly beneficial to separate the Lorentz structure from the amplitude. Lorentz
invariance dictates that the amplitude Mab µνρσ (p1 , p2 , p3 , p4 ) can be decomposed in terms of
all possible, in this case rank-4, Lorentz tensors. These Lorentz tensors can be composed
of the independent external momenta as well as general process-independent tensors such
as the metric tensor gµν and the totally anti-symmetric Levi-Civita tensor µνρδ . For the
process at hand, a total of 138 tensor structures appear in the decomposition [102]:
Mµνρσ (p1 , p2 , p3 , p4 ) = a1 g µν g ρσ + a2 g µρ g νσ + a3 g µσ g νρ
X 3
+ ( a1,ij g µν pρi pσj + a2,ij g µρ pνi pσj + a3,ij g µσ pνi pρj
i,j=1
+ a4,ij g νρ pµi pσj + a5,ij g νσ pµi pρj + a6,ij g ρσ pµi pνj )
X 3
+ aijkl pµi pνj pρk pσl . (2.1.5)
i,j,k,l=1
In the above, parity-odd tensors involving the Levi-Civita tensor have been dropped. This
is because of Bose symmetry and charge-parity conservation for this process [96]. Note that
53
the colour indices have been dropped in the above equation for brevity.
Fundamentally, the decomposition in Eq. 2.1.5 is completely independent of the under-
lying process and should hold for any 2 → 2 scattering process involving four vector bosons
as well as satisfying Bose symmetry and conserving charge-parity. All the physics is now
contained in the coefficients aij.. which are referred to as "form factors". It would be quite
difficult to calculate all 138 form factors; thankfully, that is not needed and certain identities
can be exploited to reduce this number. Transversality of the gluon polarisation vectors
(from Ward identities) implies
1 · p1 = 0 , 2 · p2 = 0 . (2.1.6)
Further reduction can be achieved by explicitly choosing a gauge for the external particles
such as
1 · p2 = 0 , 2 · p1 = 0 , 3 · p3 = 0 , 4 · p4 = 0 , (2.1.7)
which reduces the total number of independent form factors to 20. The above gauge choice
for the polarisation vectors corresponds to the following polarisation sums:
pµ1 pν2 + pµ2 pν1
X X
a,µ
1 1
∗b,ν
= a,µ
2 2
∗b,ν
= −g µν
+ δ ab ,
pol pol
p1 .p2
X pµ3 pν3
µ3 ∗ν
3 = −g
µν
+ ,
pol
p3 .p3
X pµ4 pν4
µ4 ∗ν
4 = −g µν
+ . (2.1.8)
pol
p4 .p4
It is useful to define the Mandelstam variables
s = (p1 + p2 )2 , t = (p1 − p3 )2 , u = (p2 − p3 )2 . (2.1.9)
54
Since they are Lorentz-invariant, it is convenient to express the amplitude in terms of these
quantities. Momentum conservation (Eq. 2.1.2) allows us to write down a relation between
the Mandelstam variables:
s + t + u = 2 m2Z (2.1.10)
The amplitude can then be written in terms of the remaining 20 form factors as
X20
Mµνρσ (p1 , p2 , p3 , p4 ) = Ai (s, t, m2Z , m21 , m22 , ...) Tiµνρσ , (2.1.11)
i=1
where the Ai are the form factors depending only on the Mandelstam variables (s, t), mass of
the Z-boson (mZ ), and masses of the internal particles denoted by m1 , m2 , .... The remaining
20 tensors Ti are as follows:
T1µνρσ = g µν g ρσ , T2µνρσ = g µρ g νσ , T3µνρσ = g µσ g νρ , T4µνρσ = pρ1 pσ1 g µν ,
T5µνρσ = pρ1 pσ2 g µν , T6µνρσ = pσ1 pρ2 g µν , T7µνρσ = pρ2 pσ2 g µν , T8µνρσ = pσ1 pν3 g µρ ,
T9µνρσ = pσ2 pν3 g µρ , µνρσ
T10 = pρ1 pν3 g µσ , T11 µνρσ
= pρ2 pν3 g µσ , µνρσ
T12 = pσ1 pµ3 g νρ ,
µνρσ
T13 = pσ2 pµ3 g νρ , µνρσ
T14 = pρ1 pµ3 g νσ , T15 µνρσ
= pρ2 pµ3 g νσ , µνρσ
T16 = pµ3 pν3 g ρσ ,
µνρσ
T17 = pρ1 pσ1 pµ3 pν3 , µνρσ
T18 = pρ1 pσ2 pµ3 pν3 , T19 µνρσ
= pρ2 pσ1 pµ3 pν3 , µνρσ
T20 = pρ2 pσ2 pµ3 pν3 .
(2.1.12)
To extract the form factors from the amplitude, projection operators Piµνρσ can be con-
structed such that they fulfill
0 0 0 0
X
Piµνρσ ∗1µ ∗2ν 3ρ 4σ 1µ0 2ν 0 ∗3ρ0 ∗4σ0 Mµ ν ρ σ = Ai . (2.1.13)
pol
To find these projection operators, they can be decomposed in terms of the Tiµνρσ :
X20
Piµνρσ = Bij (s, t, m2Z ) (Tjµνρσ )† , i = 1, ..., 20 . (2.1.14)
j=1
55
Note that the coefficients Bij are functions of only the external kinematic configuration,
independent of the internal particles. The actual expressions for Bij are too complicated
to be provided here and the reader is instead referred to the VVamp project website. To
calculate them is relatively straightforward, however. Inserting Eqs. 2.1.11 and 2.1.14 into
Eq. 2.1.13,
20
X 0 0 0 0 X
Bij (s, t, m2Z ) (Tjµνρσ )† Tkµ ν ρ σ ∗1µ ∗2ν 3ρ 4σ 1µ0 2ν 0 ∗3ρ0 ∗4σ0 = δik .
(2.1.15)
j=1 pol
Polarisation sums from Eq. 2.1.8 can then be inserted into the above equation and the result-
ing linear system solved to obtain the coefficients Bij . Not all 20 form factors are independent
though. With identical bosons in both the initial and final states, the process exhibits Bose
symmetry which implies that the amplitude must be invariant under the exchange of the
incoming gluons or the outgoing Z-bosons [102]:
1↔2 : p1 ↔ p2 , λ1 (p1 ) ↔ λ2 (p2 ) ,
3↔4 : p3 ↔ p4 , λ3 (p3 ) ↔ λ4 (p4 ) .
This symmetry results in certain relations between the form factors. First. there are the
identities:
A7 = A4 , A12 = −A11 , A13 = −A10 , A14 = −A9 , A15 = −A8 , A20 = A17 .
(2.1.16)
56
Second, the following relations under the swapping of the Mandelstam invariants t ↔ u:
A1 (s, t) = A1 (s, u) , A4 (s, t) = A4 (s, u) , A7 (s, t) = A7 (s, u) ,
A16 (s, t) = A16 (s, u) , A17 (s, t) = A17 (s, u) , A20 (s, t) = A20 (s, u) ,
A2 (s, t) = A3 (s, u) , A5 (s, t) = A6 (s, u) , A8 (s, t) = A13 (s, u) ,
A9 (s, t) = A12 (s, u) , A10 (s, t) = A15 (s, u) , A11 (s, t) = A14 (s, u) ,
A18 (s, t) = A19 (s, u) . (2.1.17)
It must be noted that everything to this point has been treated in general d-dimensions i.e.
the external momenta and consequently the form factors and the projection operators are all
in d-dimensions. This has the advantage of being compatible with dimensional regularisation
and all the techniques used for solving loop integrals.
At this point, it makes sense to switch to physically relevant 4-dimensional observables.
A convenient choice is the so-called "helicity amplitudes", derived by explicitly fixing the
helicities for the external particles. Note that such helicity amplitudes are inherently 4-
dimensional quantities. It is rather straightforward to derive the expressions for the helicity
amplitudes in terms of the form factors Ai . It must be pointed out that since the Z-bosons
are massive, their helicities are frame dependent; in this case, the momenta are considered
to be in the center-of-momentum frame for both the incoming and outgoing states. This
allows the momenta to be parameterised according to
√ √
s s
pµ1 = (1, 0, 0, 1) , pµ3 = (1, β sin θ, 0, β cos θ) ,
√2 √2
s s
pµ2 = (1, 0, 0, −1) , pµ4 = (1, −β sin θ, 0, −β cos θ) , (2.1.18)
2 2
p
where β = 1 − 4m2Z /s and θ is the angle between the direction of the incoming gluon (p1 )
and the outgoing Z-boson with momentum p3 . The polarisation vectors can then be written
57
as following [6]:
1
µ± (p1 ) = √ (0, ∓1, −i, 0) ,
2
1
µ± (p2 ) = √ (0, ±1, −i, 0) ,
2
√
1 s
µ± (p3 ) = √ (0, ∓ cos θ, −i, ± sin θ) , µ0 (p3 ) = (β, sin θ, 0, cos θ) ,
2 2mZ
√
1 s
µ± (p4 ) = √ (0, ± cos θ, −i, ∓ sin θ) , µ
0 (p4 ) = (β, − sin θ, 0, − cos θ) . (2.1.19)
2 2mZ
It can be easily shown that the above choice of polarisation vectors satisfies the polarisation
sums in Eqs. 2.1.6, 2.1.7, and 2.1.8.
A straightforward calculation would show that there are a total of 22 × 32 = 36 helicity
amplitudes for this process. However, parity invariance reduces the number of independent
helicity amplitudes by a factor of two [157]:
Mλ1 λ2 λ3 λ4 = (−1)δλ3 0 +δλ4 0 M−λ1 −λ2 −λ3 −λ4 . (2.1.20)
Above identity under parity transformation holds for general external kinematics. For the
special case at hand where the gluons and Z-bosons are on-shell, following symmetry relations
also hold [13, 96]:
M+++− = M++−+
M+−−− = M+−++
M++±0 = M++0±
M+−±0 = −M+−0∓ , (2.1.21)
where the identities in Eq. 2.1.16 have been used to derive the above relations. Another
set of relations can be derived if the variables s, θ are eliminated in favour of β, t using
58
θ = arccos((t − u)/(βs)) ∈ [0, π], u = 2m2Z − s − t, and s = 4m2Z /(1 − β 2 ):
M++++ (β, t) = M++−− (−β, t),
M+−+− (β, t) = M+−−+ (−β, t),
M+±+0 (β, t) = M+±−0 (−β, t) . (2.1.22)
Eqs. 2.1.20, 2.1.21, and 2.1.22 together reduce the total number of independent helicity
amplitudes to 8. The explicit expressions for the helicity amplitudes in terms of the form
factors Ai , as defined through Eq. 2.1.19, are provided in an ancillary file with [6].
Before taking all the symmetries into account, there are 20 form factors Ai , while there
are only 18 helicity amplitudes. In fact, looking at the analytic forms of the projection
1
operators, they have spurious poles in the space-time dimension d of the form d−4
. This
is clearly not physical and suggests some redundancy between the form factors. It is not
unimaginable that for d = 4 certain linear relations appear between the form factors. This
phenomenon was already observed in [102] where allowing the Z-bosons to decay to fermions
and considering specific fermion helicities, only 18 independent "helicity amplitudes" were
found. These helicity amplitudes, corresponding to the helicities of the final-state fermions,
were found to be gauge-invariant physically relevant quantities, unlike the form factors Ai ;
explicit calculation done as part of final checks in this work (see Sec. 5.2.2) showed that while
the form factors Ai are dependent on the specific regularisation scheme for γ 5 , the helicity
amplitudes written in [102] are not. Choosing specific helicities for the external particles also
forces them to be four-dimensional. This prescription where external particles are treated in
d = 4 while all the internal quantities are treated in general d is colloquially referred to as
the ’t Hooft-Veltmann scheme.
Another disadvantage of using the form factors defined as in Eq. 2.1.13 is that they are not
orthogonal. An alternate approach was used in [13] to remedy this using the Gram-Schmidt
orthogonalisation procedure. Starting from the 20 tensors as in Eq. 2.1.12, the Gram-Schmidt
59
procedure results in 20 new tensors that are orthogonal linear combinations of the original
tensors. Crucially, 2 of the linear combinations vanish in d → 4 limit with only 18 remaining,
further indicating the redundancy in the original decomposition. A general prescription to
construct physically relevant form factors in d-dimensions was provided in [156, 157]. The
authors used the ‘t Hooft-Veltmann prescription, considering all objects in d-dimensions
except the external momenta which are treated in 4-dimensions. Lastly, it must be pointed
out that in this work only the contribution from top quarks in the loop are considered.
2.2 1-loop amplitude
2.2.1 Generation of the unreduced amplitude
The form factor decomposition in the previous subsection was general and valid to all orders
in perturbation theory. Here, the amplitude in Eq. 2.1.4 is perturbatively expanded in the
strong coupling constant αS as
α
S
M = M0 + M1 + O(αS2 ) . (2.2.1)
2π
Here M0 corresponds to the leading-order amplitude. Z-bosons don’t have "colour" charge
and hence don’t directly couple to gluons. This process, thus, starts at 1-loop through closed
quark loops. Qgraf [207] is used to generate the Feynman diagrams. All the quarks other
than the top quark are treated as massless and only the Higgs is allowed to couple to the top
quark. A total of 10 diagrams are found out of which 2 are zero due to colour conservation
since a single gluon couples to a closed quark loop. Of the remaining 8, 2 are mediated via
offshell production of the Higgs boson (Figs. 2.1a and 2.1b); remaining 6 are the "box" type
contributions. Note that only 2 of these "box" diagrams are truly independent, the rest can
be obtained through simple crossings of the external legs. In particular, Figs. 2.1d and 2.1e
can be obtained by by crossing the incoming gluons and the outgoing Z-bosons, respectively,
60
in Fig. 2.1c. We can further exchange the Z-bosons in Fig. 2.1d to obtain Fig. 2.1f. However,
since the Z-bosons are onshell (Eq. 2.1.3), and have the same invariant mass, crossing both
the gluons and the Z-bosons leaves the diagram unchanged which implies the diagrams in
Figs. 2.1c and 2.1d are equal to Figs. 2.1f and 2.1e respectively.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 2.1: Feynman diagrams for the process gg → − ZZ at 1-loop (LO). Figs. 2.1a and 2.1b
are the Higgs exchange diagrams, and Figs. 2.1c- 2.1h are the diagrams for continuum pro-
duction of Z-bosons. Note that out of the 6 continuum diagrams, only 2 are independent
and the rest can be obtained through various crossings of external legs.
The next step is to insert Feynman rules in the diagrams to obtain the amplitudes. The
Feynman rules employed here are given in the appendix A. Custom FORM [208–210] scripts
are used to insert the Feynman rules into the diagrams to generate algebraic expressions.
The colour structure of the remaining non-zero diagrams is quite simple; it is essentially a
61
gluon propagator like structure, so the colour factor is simply given by
C = C0 δab . (2.2.2)
1
For the tree-level propagator, C0 is just 1 (see Appendix A.2). At 1-loop, C0 = TF = 2
for
the diagram involving only a closed quark loop (see Appendix A.4). At higher loops, more
complicated structures appear in C0 i.e. the quadratic Casimirs CF , CA ; see Appendix A.3
for a definition of the quadratic Casimir operators.
The Z-boson couples to the fermion through both a vector and an axial-vector coupling
with the vertex written as
1 − γ5
¯ 1 + γ5
VµV f f = ie LZff¯γµ + RfZf¯γµ
2 2
e
=i γµ (vt + at γ5 ) (2.2.3)
2 sin θW cos θW
from the EW couplings in Eq. 1.6.26, where the coupling to the left-handed fermion is LZff¯ =
(I3f −qf sin2 θW )/(sin θW cos θW ) and to the right-handed fermion is RfZf¯ = −qf sin θW / cos θW .
Here e is the positron charge and qf is the electric charge of the fermion in terms of e. For
the top quark, the vector and axial couplings (vt , at ) are given by θW by vt = 1
2
− 43 sin2 θW
and at = − 21 , respectively, in terms of the weak mixing angle θW , as defined in Eq. 1.6.10.
At this point it is necessary to discuss the γ-matrices, in particular the γ 5 matrix. γ-
matrices are an essential part of most scattering amplitude calculations in the Standard
Model. A brief description of their properties is provided in Appendix C. While the γ-
matrices that satisfy the anti-commutation relation {γ µ , γ ν } = 2gµν can be consistently
treated in d-dimensions, the γ 5 matrix, as defined, is inherently a 4-dimensional object.
As such, it is not clear how to incorporate it in Conventional Dimensional Regularisation
(CDR) [151–154] where objects are treated in d-dimensions, along with the usual γ-matrices.
There are several schemes used to work around this issue; see Appendix C for details. This
62
calculation employs Kreimer’s [211–213] anti-commuting γ 5 scheme to properly treat γ 5 in
d-dimensions. In this scheme, the anti-commuting property of γ 5 is kept while traces are no
longer assumed to be cyclic. In other schemes such as the ’t Hooft, Veltman, Breiten-Lohner,
and Maison scheme [153, 214–216], the anti-commutation relation is given up and cyclicity of
the trace is kept. This scheme necessarily treats 4 and d − 4 dimensional objects separately.
A fundamental disadvantage of this is that the Ward identities are not preserved requiring
the addition of extra counterterms to the Lagrangian. The reader is referred to Appendix C
for a more detailed discussion of γ-matrices and different γ 5 schemes.
The two Z-bosons coupled to a fermion line can in general generate 3 different contribu-
tions: vector-vector (vt2 ), axial-vector (at vt ), and axial-axial (a2t ), and it is useful to split the
amplitude into these 3 parts. The vector-vector and axial-axial contributions are identical
for massless quarks [96]. This can be understood by looking at the traces involved; for traces
involving an even number of γ 5 ’s, the γ 5 ’s disappear as a result of the anti-commutation re-
2
lations and the fact that (γ 5 ) = 1; the resulting trace is identical to that of the vector-vector
case. For the case of a massive quark the difference must be proportional to the quark mass.
E.g.
h i h i
T r (p/1 + m)γ µ (p/2 + m)γ ν − T r (p/1 + m)γ µ γ 5 (p/2 + m)γ ν γ 5 = 8m2 g µν (2.2.4)
The axial-vector (at vt ) contribution to this amplitude vanishes identically as a conse-
quence of Bose symmetry and charge-parity conservation for this process [96]. The absence
of any terms containing the -tensor in Eq. 2.1.5 can also be explained through this since the
Levi-Civita tensor violates parity and hence any such terms are forbidden through charge-
parity conservation.
The projection operators from Eq. 2.1.13 are then applied to obtain analytical expressions
for individual form factors.
63
= c4,i + c3,i
+ c2,i + c1,i + Rational terms
Figure 2.2: Decomposition of a general 1-loop Feynman integral in d = 4.
2.2.2 Reduction to master integrals
The form factors contain only scalar loop integrals (see Sec. 1.5.4 for a brief discussion on
loop integrals) without any open Lorentz indices. The list of scalar loop integrals isn’t the
minimal list of integrals that needs to be evaluated, however. Indeed it can be shown that at
1-loop, all the integrals can be related to a minimal set of integrals often referred to as the
basis integrals or "master integrals". In fact, any 1-loop integral can be written as a linear
combination of basis integrals [217–219]. In d = 4, this linear combination is simply
I = c1,i Ai + c2,i Bi + c3,i Ci + c4,i Di + O(d − 4) (2.2.5)
as depicted in Fig. 2.2, where Ai , Bi , Ci , Di are all possible 1-, 2-, 3-, and 4-propagator scalar
Feynman integrals
64
dd k 1
Z
A0 (m21 ) = , (2.2.6)
iπ d/2 D1
dd k 1
Z
B0 (p2 , m21 , m22 ) = , (2.2.7)
iπ d/2 D1 D2
dd k 1
Z
C0 (p21 , p22 , s12 , m21 , m22 , m23 ) = , (2.2.8)
iπ d/2 D1 D2 D3
dd k 1
Z
D0 (p21 , p22 , p23 , p24 , s12 , s23 , m21 , m22 , m23 , m24 ) = , (2.2.9)
iπ d/2 D1 D2 D3 D4
with
D1 = k 2 − m21 ,
D2 = (k + p1 )2 − m22 ,
D3 = (k + p1 + p2 )2 − m23 ,
D4 = (k + p1 + p2 + p3 )2 − m24 . (2.2.10)
The assertion in Eq. 2.2.5 is in fact more general. Any N -point function can be written as
a linear combination of 4− and lower point functions in d = 4 dimensions.
The first approach to perform the tensor decomposition to obtain scalar integrals was
proposed by Passarino-Veltmann in [220]. Consider the following 1-loop 2-propagator tensor
integral as an example:
dd k kµ
Z
µ
I (p, m) = . (2.2.11)
(2π)d (k 2 − m2 )(k + p)2
Here Conventional Dimensional Regularisation (CDR) is being employed to regulate the
divergences. The only rank-1 Lorentz tensor possible for this integral is pµ . Using this, the
integral can be written as
dd k kµ
Z
µ
I (p, m) = = pµ B1 (p2 , m) , (2.2.12)
(2π)d (k 2 − m2 )(k + p)2
where the scalar part has been defined as B1 (p2 , m), which can be related to the original
65
integral by contracting with pµ as
dd k k·p
Z
2 2 µ
p B1 (p , m) = pµ I (p, m) =
(2π)d (k 2 − m2 )(k + p)2
1 dd k (k + p)2 − (k 2 − m2 ) − (m2 + p2 )
Z
=
2 (2π)d (k 2 − m2 )(k + p)2
1 dd k 1 1 dd k 1
Z Z
= d
− d
2 (2π) (k − m ) 2
2 2 (2π) (k + p)2
1 2 dd k 1
Z
− (p + m2 ) d
2 (2π) (k − m )(k + p)2
2 2
1
A0 (m) − A0 (0) − (p2 + m2 )B0 (p2 , m, m) .
= (2.2.13)
2
A0 (0) can be shown to vanish identically in dimensional regularisation (see Sec. 3.1); A0 (m)
and B0 (p2 , m) are the standard 1-loop scalar integrals from Eqs. 2.2.6 and 2.2.7, also known
as Passarino-Veltmann functions. Higher rank tensor integrals can be similarly decomposed:
dd k kµkν
Z
µν
I (p, m) = = g µν B00 (p2 , m) + pµ pν B11 (p2 , m) . (2.2.14)
(2π)d (k 2 − m2 )(k + p)2
Note the appearance of g µν as a rank-2 tensor. The B00 , B11 functions in the above equa-
tions can be similarly written in terms of A0 (m) and B0 (p2 , m) after contracting with rank-2
tensors gµν and pµ pν and solving the resulting linear system. These, along with the 3 and
4 propagator Passarino-Veltmann functions C0 and D0 , form a basis of integrals at 1-loop
in 4-dimensions. The basis integrals can then be evaluated either analytically using a mul-
titude of methods (see Sec. 4.1), or numerically using several libraries e.g. LoopTools [221],
QCDLoop [222], OneLOop [223], Golem [224], Collier [225], Package-X [226]. Passarino-
Veltmann reduction has been used to calculate a number of 1-loop processes including the
1-loop amplitude for gg → ZZ in [93, 96, 97].
The conventional Passarino-Veltmann reduction, however, has certain limitations:
1. In Eq. 2.2.13, for instance, B1 cannot be solved for the case where p2 = 0. In particular,
the reduction procedure for integrals with ≥ 2 propagators involves inverting the so-
66
called "Gram matrices" and these Gram matrices become singular in certain regions of
phase-space. This can lead to severe numerical instabilities and makes it challenging
to write efficient numerical algorithms.
2. Applying the reduction procedure to higher tensor rank integrals is a rather cumber-
some procedure and leads to an explosion in the number of terms in intermediate
stages. This results in severe algebraic complexity in the case of analytic calculation
and loss of precision in the case of numerical calculation.
3. For processes with large number of diagrams and/or external legs the integral-by-
integral reduction procedure is extremely tedious.
Several new approaches have been developed and refined in the past few decades to
resolve the above issues; the reader is referred to [91, 227] for an overview of the modern
1-loop amplitude methods. By far the most popular method for 1-loop computations is
the OPP method proposed by Ossola-Papadopoulos-Pittau in [86], closely related to the
ideas of generalised unitarity cuts [85, 228]. Further progress regarding efficient numerical
implementations and automation [87, 229–232] lead to the so-called "NLO revolution" as a
result of complete automation of 1-loop amplitudes. There are a number of public libraries for
integral reduction based on OPP method: CutTools [233], Samurai [234], Ninja [235]. The
computation of the amplitudes and NLO corrections has been automated in many publicly
available codes such as BlackHat [236], HELAC-NLO [237], MadGraph [238], GoSam2.0 [239],
and OpenLoops2 [240].
Traditional Passarino-Veltmann reduction relies on step-by-step procedure to reduce high
rank tensor integrals to find the coefficients of the master integrals. Using the OPP method,
the coefficients can be calculated directly as follows. Consider the integrand of a general
4-point Feynman diagram
N (k)
I(k) = (2.2.15)
D1 D2 D3 D4
67
where Di are the propagators in d-dimensions Di = (k + qi )2 − m2i with k being the d-
dimensional loop momentum, k the 4-dimensional loop momentum, and qi , mi being the
momentum flowing through the edge and the mass of the edge respectively. Note that the
numerator N (k) is a strictly 4-dimensional polynomial in the loop momentum k; this creates
a mismatch between the numerator and the denominator that leads to the rational terms. For
now, it suffices to consider everything in d = 4 (except in the definition of the integrals where
regularisation is required); the rational terms are computed later. The result of Eq. 2.2.5
can be extended to integrands as shown in [241] with the numerator decomposed as
˜
N (k) = (d(1234) + d(1234; k))
+ (c(123) + c̃(123; k)) D4 + (c(124) + c̃(124; k)) D3 + ...
+ (b(12) + b̃(12; k)) D3 D4 + (b(13) + b̃(13; k)) D2 D4 + ...
+ (a(1) + a(1; k)) D2 D3 D4 + (a(2) + a(2; k)) D1 D3 D4 + ...
+ P̃ (k)D1 D2 D3 D4 . (2.2.16)
The coefficients d(1234), c(123), etc. are independent of the loop-momentum and drop out
of the integration. These are the true coefficients same as ci,j in Eq. 2.2.5. The other
loop-momentum dependent coefficients d(1234; ˜ k), etc. are "spurious" and vanish after in-
tegration; however, these are still required to compute the lower coefficients e.g. to solve
˜
for c(123) both d(1234) and d(1234; k) are needed. In principle the problem now reduces
to calculating the above equation for sufficient values of k and inverting the system to find
the coefficients. This approach is, however, very inefficient since it requires inverting large
systems and may lead to numerical instabilities. Instead, the values of loop-momentum can
be chosen such that some of the propagators go on-shell e.g.
D1 = D2 = D3 = D4 = 0 . (2.2.17)
68
Denoting the solution to the above constraints by k = k0± and substituting into Eq. 2.2.16
results in
˜
d(1234) + d(1234; k0± ) = N (k0± ) . (2.2.18)
˜
It turns out that d(1234; ˜
k0+ ) = −d(1234; ˜
k0− ) = d(1234) T (k0+ ) , where the general tensor
T (k) is known. This allows the computation of the coefficients:
1
N (k0+ ) + N (k0− )
d(1234) =
2
˜ 1
N (k0+ ) − N (k0− )
d(1234) = + (2.2.19)
2T (k0 )
Once the 4-point coefficients are known, the lower point coefficients can be calculated by
setting fewer propagators on shell. Effectively, the on-shell conditions convert the linear
system into a block-triangular form that can be solved much more easily. Details of the
above procedure as well as derivation can be found in [86].
The OPP and generalised unitarity methods resolve the issues with singular Gram deter-
minants as well as explosion of terms. A huge advantage is that the procedure can be applied
to sum of Feynman diagrams instead of each individual Feynman integral. The method is
not generally applicable beyond 1-loop, however. The decomposition in Eqs. 2.2.13 or 2.2.16
works because the resultant numerator can be written as a linear combination of the prop-
agator factors in the denominator [241]. This is not possible in general beyond 1-loop; such
numerators are called "Irreducible Scalar Products". A much more general approach called
"Integration-By-Parts" (IBP) reduction [242] is most commonly used for processes beyond
1-loop. Details of IBP reduction and its application to this work are provided in Sec. 3.
69
2.2.3 Results
To calculate the 1-loop amplitudes for gg → ZZ, first tensor decomposition of the unreduced
amplitudes, as described in Sec. 2.1, is performed to obtain the form factors. The scalar inte-
grals in the form factors are reduced to master integrals using Passarino-Veltmann reduction
with the numerator algebra performed using custom FORM scripts and Package-X [226]. The
master integrals are then evaluated using LoopTools [221]. Helicity amplitudes are calcu-
lated from the form factors and are used to present the results below. After the calculation,
our results are compared to published results in [13]; agreement is found for each form factor
and helicity amplitude.
In Fig. 2.3a, the 1-loop amplitude-squared |M|2 summed over all helicity configurations
is compared for 3-different contributions: Only the top-quark contribution, top-quark contri-
bution with Higgs exchange diagrams included, and the contribution of a massless up-type
quark. It is clear that the top-quark contribution (in black) grows with energy. After includ-
ing the Higgs mediated diagrams, however, the growth is reduced and the amplitude-squared
plateaus; this is the unitarising effect of the Higgs boson. Another important observation
is that even after including the Higgs mediated diagrams, the contribution from top-quark
dominates over the massless contribution for high energies. This further underlines the im-
portance of a full 2-loop computation for this process. Fig. 2.3b shows the same data but
zoomed in to the region of smaller invariant mass. While the massless contribution is fairly
uniform and dominates in the low-energy region, the massive (top quark) contribution starts
√
to increase past s = 2mt when the top quarks in the loop can be produced on-shell.
√
This unitarising behaviour and the threshold effect for s = 2mt is even more apparent in
the individual helicity amplitudes as shown in Fig. 2.4. Fig 2.4a shows the individual helicity
amplitudes without the Higgs mediated diagrams included. In this case the "+ + 00" mode
(shown in red) is dominant and increases with energy, much like Fig. 2.3a. This is easily
understood through the Equivalence theorem where the "00" mode corresponds to scattering
70
(a)
(b)
Figure 2.3: Comparison of |M|2 for gg → ZZ for three different type of contributions,
summed over all helicities for the external particles. Without including the Higgs mediated
contributions, the top quark contribution increases with energy.
of two gluons producing two Goldstone bosons where the coupling of the Goldstone bosons
to the top quark is proportional to the top mass. Including the Higgs diagrams (Fig. 2.4b)
it is clear that the "+ + 00" mode does not increase indefinitely with energy. In fact, at
very high energies, the "+ − 00" mode takes over and becomes the dominant mode. Another
interesting feature of the above plot is the stark threshold effect observed in the "+ + 00"
mode. This is absent in the "+ − 00" mode as a consequence of a mismatch in the helicities;
the top quark pair produced on-shell can have a total spin of J = 0 or J = 1 with J = 0
preferred for the "00" mode for the Z-bosons while the initial state has a total spin of 2.
71
(a)
(b)
Figure 2.4: Comparison of |M|2 for gg → ZZ for different helicity configurations for the
top quark contribution. The bottom panel (Fig. 2.4b) also includes the Higgs mediated
diagrams.
Fig. 2.5 shows the ratio of individual helicities to the total |M|2 for both the top
(Fig. 2.5a) and massless (Fig. 2.5b) contributions. For the massive case, the longitudinal
mode "00" has the largest contribution, unsurprisingly with the "+ − 00" helicity being the
dominant mode for high energies. For the massless case, however, the longitudinal modes
are almost negligible and the "+ + ++" and "+ − +−" helicities contribute the most.
72
(a)
(b)
Figure 2.5: Ratio of contribution of individual helicity configurations to the total |M|2 for
gg → ZZ for (a) top quark contribution including the Higgs diagrams, and (b) massless
quark contribution.
2.3 2-loop amplitude
2.3.1 Generation of diagrams
At 2-loops, the contributions to this process involving s-channel Higgs production are already
known [108–111] and those for the massless quarks were computed in [102, 103]. As such,
only the top-quark contribution is considered here. The diagrams for the 2-loop amplitude
are generated using Qgraf [207]. A total of 166 diagrams are generated that contain at
73
[A] [B]
Figure 2.6: Example Feynman diagrams representing the two classes of diagrams at 2-loops.
least 1 top-quark propagator. Additional diagrams containing only massless bottom quarks
have been generated as well; see the discussion for diagrams of Class B below. A total of
49 out of the 166 diagrams containing at least 1 top-quark propagator vanish from colour
conservation since they have a single gluon coupled to a closed quark loop. The remaining
diagrams can broadly be split into two different "classes" (see Fig. 2.6):
Class A: In diagrams of class A, both Z-bosons are attached to the same closed fermion
loop. As described previously, the anti-commuting, Kreimer’s γ 5 scheme is used to handle
γ 5 in d-dimensions. In general, a reading point prescription is used to ensure that all traces
are read from the same point since cyclicity is not preserved. For a closed fermion loop,
however, in case of an even number of γ 5 ’s, it is rather straightforward to eliminate the γ 5
using the anti-commutation identities. This allows for a trivial implementation of the γ 5
scheme for this class of diagrams.
Class B: In this class of diagrams, the Z-bosons couple to different closed fermion loops.
The vector-vector (vt2 ) contribution for these diagrams can be shown to vanish due to Furry’s
theorem. The vector-axial (at vt ) piece vanishes identically, as discussed previously, as a result
of charge-parity conservation for this process. However, the axial-axial (a2t ) piece vanishes
only for a sum over a degenerate SU (2)L doublet. Since the third generation quarks (t, b)
are not degenerate, a finite term remains from the top-bottom mass splitting. Hence, for
consistency, additional diagrams of this type containing only massless bottom quarks are also
included. For these diagrams, there is a single γ 5 in each fermion loop which is non-trivial
74
to handle and requires a careful application of the anti-commuting γ 5 scheme. It must be
pointed out that these diagrams are effectively 1-loop*1-loop type diagrams and as such are
a lot easier to reduce to master integrals than Class A.
2.3.2 Class A
After generating the diagrams in class A, the Feynman rules are inserted for which custom
FORM scripts are employed. The resulting algebraic expressions for the form factors are re-
quired to be interpreted as linear combinations of Feynman integrals. For this, Reduze 2 [243–
246] is employed to map them to the 4 different integral families shown in Tab. 2.1.
An "integral family" is essentially a set of Feynman propagators that can be used to
classify the integrals appearing in one or more topologies. The concept of an "integral family"
is discussed in a bit more detail in Sec. 3. For the purpose of this section it suffices to say that
an integral family is a set of Feynman propagators grouped together such that all possible
scalar products between the momenta in the integral family (both loop momenta and external
momenta) are uniquely determined in terms of the inverse propagators. E.g. considering an
integral family with L loop momenta and E independent external momenta, the number of
distinct scalar products (dependent on a loop momentum) is N = L(L + 1)/2 + EL. This
is the number of propagators that an integral family must contain. For a 4-point topology
at 1-loop, this number is 4; at 2-loops, it is 9. Note that for a 4-point topology at 2-loops,
only 7 propagators can be present assuming all the vertices connect 3 edges. The 2 "extra"
propagators, even though they do not form a part of the topology are required to specify all
the scalar products.
For simplification of the expressions, Feynman gauge is used (ξ = 1) for internal gluons.
After mapping the diagrams to the above integral families, a total of 29247 integrals are
found in the amplitude. In the worst case, the integrals have up to 4 inverse propagators
in the numerator. The algebraic expressions at this stage of the calculation are quite large,
occupying ∼ 50 GB of disk space.
75
A B C D
k12 − mt2 k12 k12 − mt2 k12
(k1 + p1 ) 2 − mt2 (k1 + p1 ) 2 (k1 + p1 ) 2 − mt2 (k1 + p1 ) 2
(k1 + p1 + p2 ) 2 − mt2 (k1 + p1 + p2 ) 2 (k1 + p1 − p3 ) 2 − mt2 (k1 + p1 + p2 ) 2
(k1 + p4 ) 2 − mt2 (k1 + p4 ) 2 (k1 + p4 ) 2 − mt2 k22 − mt2
k22 − mt2 k22 − mt2 k22 − mt2 (k2 + p1 + p2 ) 2 − mt2
(k2 + p1 ) 2 − mt2 (k2 + p1 − p3 ) 2 − mt2 (k2 + p1 − p3 ) 2 − mt2 (k2 + p4 ) 2 − mt2
(k2 + p1 + p2 ) 2 − mt2 (k2 − p3 ) 2 − mt2 (k2 + p4 ) 2 − mt2 (k2 − k1 ) 2 − mt2
(k2 + p4 ) 2 − mt2 (k2 + p4 ) 2 − mt2 (k2 − k1 ) 2 (k2 − k1 + p2 ) 2 − mt2
(k1 − k2 ) 2 (k1 + k2 + p4 ) 2 − mt2 (k1 − k2 + p1 ) 2 (k2 − k1 + p4 ) 2 − mt2
Table 2.1: List of integral families and their propagators for the 2-loop amplitude.
Usually, topologies that have many common denominators are combined together in one
integral family. Even so, there can still be common propagators, often related through a shift
of loop momentum, between topologies from different topologies. This, however, leads to
redundancy in the definition of integrals and certain symmetry relations can be written down
that relate integrals from different integral families. Such relations aren’t limited to different
families, though, and similar relations between different "sectors" of the same integral family
exist as well as within a sector. Using such relations generated using Reduze 2, the total
number of integrals in the diagrams can be reduced to 4504. Adding diagrams together and
considering only form factors, this number drops to 1584 due to further cancellations and
the total size of the expressions drastically reduces to < 1 GB. This really underlines the
importance of working with objects like form factors instead of individual diagrams as well as
the use of symmetry relations; the resulting algebraic expressions are a lot more manageable
and the number of Feynman integrals appearing in the expressions reduces drastically.
Unlike the case of 1-loop, at 2-loops the master integrals are not known a priori. Addi-
tionally, the methods traditionally applied at 1-loop e.g. Passarino-Veltmann reduction or
OPP method are not adequate beyond 1-loop. The conventional approach to reduce inte-
grals beyond 1-loop is Integration-By-Parts reduction [242]. Reduze 2 is used to perform a
numerical reduction (i.e. with exact rational numbers substituted for kinematic variables)
to find the list of masters. A total of 264 master integrals are found out of which 172 are not
76
(a) (b) (c)
(d) (e)
(f) (g)
Figure 2.7: Representative Feynman diagrams in class A with irreducible topologies. The
number of master integrals in the above top-level topologies are 3, 4, 3, 3, 5, 5, and 4
respectively.
related via any crossing i.e. exchange of external legs. In terms of topologies, there are 85 ir-
reducible ones i.e. the topologies containing master integrals and which cannot be completely
expressed in terms of integrals from subtopologies, with a 6 propagator topology having the
most number of master integrals (6). Of the 13 top-level topologies i.e. the topologies with
7 propagators, 7 are irreducible (Fig. 2.7) and the other 6 are reducible (Fig. 2.8).
For diagrams of Class A two different colour structures appear given by the two quadratic
77
(a) (b)
(c) (d)
(e) (f)
Figure 2.8: Representative Feynman diagrams in class A with reducible topologies.
Casimir invariants CF and CA i.e. the colour structure of a diagram can be written as
C = (a1 CF + a2 CA ) TF δab . (2.3.1)
Interestingly, it is observed that the diagrams contributing to the CF piece are purely planar
while a mix of planar and non-planar diagrams contributes to the CA piece. Since the planar
diagrams are expected to be simpler to reduce than the non-planar diagrams, the CF piece
is expected to be simpler.
Full IBP reduction for diagrams of class A is extremely challenging and the conventional
approach proved inadequate. For instance, Reduze 2 was used to perform the reduction
for this process for the case of massless internal quarks in [102]. However, we find in our
78
experiments that it was only able to solve reductions for the massive case only for integrals
with 1 inverse propagator while the full problem requires the solution for integrals with up
to 4 inverse propagators. The new developments required to solve this problem are discussed
in Chapter 3.
2.3.3 Class B
Diagrams in Class B have contributions with a single γ 5 in each loop which leads to a
non-trivial structure and requires careful application of the reading point prescription in
Kreimer’s scheme. Details of the γ 5 scheme and the reading point prescription are given in
Appendix C.
Since these diagrams are effectively 1-loop*1-loop, they are significantly simpler and
hence treated separately from Class A. As described previously, the only surviving terms
are of the axial-axial type (a2t ). These should also be proportional to m2t assuming all the
other quark flavours are massless. The colour factor for this class of diagrams is particularly
simple since this is just a product of two 1-loop diagrams.
C = TF2 δab . (2.3.2)
In the context of this process, the application of the anti-commuting scheme is relatively
simple since every trace must be read from the same point due to the presence of only 1 γ 5 in
each loop. Passarino-Veltmann reduction is used along with the 1-loop tool Package-X [226]
for manipulation of tensor objects to reduce the amplitudes to master integrals. Exact results
for these diagrams were previously presented in [106] with which full agreement is found at
the level of helicity amplitudes.
79
Chapter 3
IBP reduction using syzygies
3.1 Integration-by-parts reduction
A general L-loop scalar Feynman integral with N propagators can be represented by
L
! N
dd kl 1
Z Y Y
I(ν1 , ..., νN ) = ν (3.1.1)
l=1 iπ
d
2
i=1
(qi2 − m2i ) i
where k1 , ..., kL are the loop momenta, p1 , ..., pE are the independent external momenta, qi
are the momenta of the propagators (linear combinations of loop and external momenta),
mi are the masses of the propagators, νi ≤ 0 are (integer) exponents of the propagators,
and d = 4 − 2 is the number of space-time dimensions. Here, positive powers νi > 0
of the propagators are also allowed, that is, an integral family is considered allowing for
general propagators. Note that in an integral family, all possible scalar products between
the momenta (both loop and external) are fixed in terms of the kinematic invariants and
inverse propagators. This requires the total number of propagators in an integral family to
be equal to L(L + 1)/2 + LE. Feynman integrals often contain fewer propagators than this;
the remaining propagators can appear in the numerator as irreducible scalar products. E.g.
the 4-point topologies at 2-loops considered in this work (Figs. 2.7 and 2.8) have a maximum
of 7 distinct propagators with the remaining 2 being used to express the irreducible scalar
products.
The total derivative of an integral in dimensional regularisation vanishes. This allows the
80
Figure 3.1: A tadpole graph with the thick loop corresponding to the massive propagator
with mass m.
following linear relations between different integrals [242]
L
! N
!
dd kl ∂ 1
Z Y Y
0= vµ ν , (3.1.2)
l=1
iπ d/2 ∂kjµ i=1
(qi − m2i ) i
2
where vµ could be any linear combination of loop and external momenta. The surface term
from the integration of the above integrand can be shown to vanish for small values of d.
Extending the formula to general d using analytic continuation leads to the relations known
as "Integration-By-Parts" identities.
It is instructive to look at an example. Consider a Feynman integral with just one
propagator in d dimensions e.g. in Fig. 3.1
dd k 1
Z
I= d/2
(3.1.3)
iπ (k − m2 )
2
often referred to as a "tadpole" integral. Taking the derivative of the integrand with respect
to the loop momentum (and dropping the iπ d/2 factor for brevity), as in Eq. 3.1.2, yields
" #
∂ kµ ∂kµ 1 ∂ 1
Z Z
dd k = dd k + kµ (3.1.4)
∂kµ (k − m2 )
2 ∂kµ (k 2 − m2 ) ∂kµ (k 2 − m2 )
1 1
Z Z
d
= (d − 2) d k 2 − 2m 2
dd k 2 . (3.1.5)
(k − m ) 2 (k − m2 )2
The left hand side of the above equation, which is a total derivative, can be shown to be
81
Figure 3.2: A two-point function with one massive propagator. The thick line corresponds
to the massive propagator.
vanish for small d. This leads to the following linear relation between integrals
1 1
Z Z
d
(d − 2) d k 2 = 2m2 dd k . (3.1.6)
(k − m2 ) (k 2 − m2 )2
This relation can be generalised for an arbitrary exponent of the propagator. Denoting with
I(ν) the integral with a general exponent ν for the propagator,
1
Z
I(ν) = dd k , (3.1.7)
(k 2 − m2 )ν
the general relation is
(d − 2ν) I(ν) = 2m2 ν I(ν + 1). (3.1.8)
The above relation implies that in this tower of integrals I(ν), only one is independent and
the rest are simply related to it. It must be pointed out that any of these integrals can be
chosen to be the independent integral or, as is referred to more commonly, the "master"
integral. Some choices might offer certain advantages over others e.g. finiteness in d = 4
dimensions. We will explore this in more detail in chapter 4. It is interesting to note
that setting m = 0 in Eq. 3.1.8 gives zero i.e. ∀ ν 6= d/2, I(ν) = 0. In other words,
massless tadpole integrals vanish in dimensional regularisation. This is a part of the more
general statement: scaleless integrals i.e. integrals with no dependence on external kinematic
variables vanish in dimensional regularisation.
82
In the above example, it was possible to derive an analytical solution for the IBP relation;
this is not possible in practice for general IBP systems. Consider a slightly more complicated
example, the 2-point integral with one massive and one massless propagator (Fig. 3.2). An
integral with general exponents for this topology can be written as
1
Z
I(ν1 , ν2 ) = dd k (3.1.9)
(k 2 − m2 )ν1 ((k + p)2 )ν2
where p is the external momentum entering the loop. Taking derivative with respect to the
loop-momentum vector k µ and choosing v µ = k µ , pµ ,
∂ kµ
Z
0= dd k = (d − 2ν1 − ν2 ) I(ν1 , ν2 ) − 2m2 ν1 I(ν1 + 1, ν2 )
∂kµ (k − m ) ((k + p)2 )ν2
2 2 ν1
− ν2 I(ν1 − 1, ν2 + 1) + ν2 (p2 − m2 ) I(ν1 , ν2 + 1),
(3.1.10)
∂ pµ
Z
0= dd k = (ν1 − ν2 ) I(ν1 , ν2 ) − ν1 I(ν1 + 1, ν2 − 1)
∂kµ (k − m ) ((k + p)2 )ν2
2 2 ν1
+ ν1 (p2 + m2 ) I(ν1 + 1, ν2 ) + ν2 I(ν1 − 1, ν2 + 1)
− ν2 (p2 − m2 ) I(ν1 , ν2 + 1) . (3.1.11)
It is clear that a trivial analytical solution akin to Eq. 3.1.8 is not possible in this situation.
To make the equations a bit simpler, it helps to start with ν2 = 0 which gives
0 = (d − 2ν1 ) I(ν1 , 0) − 2m2 ν1 I(ν1 + 1, 0) , (3.1.12)
0 = ν1 I(ν1 , 0) − ν1 I(ν1 + 1, −1) + ν1 (p2 + m2 ) I(ν1 + 1, 0) . (3.1.13)
In the above equations, Eq. 3.1.12 is just the solution for the tadpole integrals (Eq. 3.1.8),
while Eq. 3.1.13 provides solutions for integrals with negative exponents for the second
propagator i.e. for the propagator appearing in the numerator. E.g. choosing I(1, 0) as the
83
master integral and setting ν1 = 1 results in
d−2
I(2, 0) = I(1, 0) , (3.1.14)
2m2
d(p2 + m2 ) − 2p2
I(2, −1) = I(1, 0) . (3.1.15)
2m2
The above equations provide solutions for I(2, 0) and I(2, −1). Repeated application of
Eqs. 3.1.12 and 3.1.13 will allow the solution of integrals I(ν, −1) ∀ ν ≥ 1.
Setting, instead, ν1 = 0,
0 = (d − ν2 ) I(0, ν2 ) − ν2 I(−1, ν2 + 1) + ν2 (p2 − m2 ) I(0, ν2 + 1) , (3.1.16)
0 = − ν2 I(0, ν2 ) + ν2 I(−1, ν2 + 1) − ν2 (p2 − m2 ) I(0, ν2 + 1) . (3.1.17)
Since scaleless integrals vanish in dimensional regularisation, I(0, ν2 ) = 0 ∀ ν2 ≥ 0. The
above equations then imply
I(−1, ν2 ) = 0 . (3.1.18)
Now the case where ν1 , ν2 6= 0 can be considered. Setting ν1 = ν2 = 1 results in the
following equations:
0 = (d − 3) I(1, 1) − 2m2 I(2, 1) − I(0, 2) + (p2 − m2 ) I(1, 2), (3.1.19)
0 = − I(2, 0) + (p2 + m2 ) I(2, 1) + I(0, 2) − (p2 − m2 ) I(1, 2) . (3.1.20)
Here I(0, 2) = 0 (since it is scaleless) and I(2, 0) is given in Eq. 3.1.14. This leaves I(1, 1),
I(2, 1), and I(1, 2) undetermined. With 3 remaining integrals and only 2 equations, it
appears that the above 3 integrals cannot all be expressed in terms of the tadpole integral.
Indeed, another master integral is required with both ν1 , ν2 ≥ 1. Choosing I(1, 1) as the
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master integral in addition to I(1, 0), the identities can be written as
0 = (d − 3) I(1, 1) − 2m2 I(2, 1) + (p2 − m2 ) I(1, 2) , (3.1.21)
0 = − I(2, 0) + (p2 + m2 ) I(2, 1) − (p2 − m2 ) I(1, 2) . (3.1.22)
Solving the above equations gives I(2, 1) and I(1, 2) in terms of the chosen master integrals
I(1, 0) and I(1, 1).
To summarise, the reduction procedure is started by choosing the lowest value of ν =
ν1 + ν2 + ... = 1. For ν1 = 1, ν2 = 0, the system gets reduced to that of a massive tadpole
while for ν1 = 0, ν2 = 1 the equations lead to scaleless integrals that vanish. In the next step,
the value of ν is increased to 2. Starting with ν1 = ν2 = 1, the solutions require another
master integral. However, it can be shown that for higher values of ν, all the other integrals
can be expressed in terms of these 2 master integrals. This systematic way of generating and
solving linear relations for integer exponents νi starting from lowest values of ν = ν1 + ν2 + ...
was first presented by Laporta [247]. Many public codes based on this algorithm are available
for this purpose [244, 248–251].
Reduze 2, based on Laporta’s algorithm was used to perform the reductions for gg → ZZ
amplitudes at 2-loops for the case of massless internal quarks in [102]. It was found that
it was insufficient for the process at hand, however. In fact, even for the planar topologies
which are expected to be significantly simpler than the non-planar topologies, the reductions
for integrals with up to one inverse propagator, denoted by s ≤ 1 (i.e. ν < 0 for some
propagator), were incredibly difficult and required several months of CPU time. The integrals
with s = 2 were totally out of reach with conventional methods.
In particular, conventional IBP solvers have the following major issue.
1. Perhaps the most significant drawback is the presence of integrals with doubled prop-
agators, also referred to as dots, in IBP identities. As is evident from Eqs. 3.1.21
and 3.1.22, IBP identities generated this way necessarily contain these doubled propa-
85
gators for the conventional choice of vectors v µ = (k1µ , ..., pµ1 , ...). These integrals don’t
often appear in the amplitude and as such their reductions aren’t required.
2. While it is possible to avoid generating solutions for such integrals with doubled prop-
agators, reduction procedure for just the integrals required for the amplitude still pro-
ceeds through row-reduction of linear systems containing these integrals. This means
that a much larger linear system than is optimal is being solved, further slowing down
the process.
A solution was first provided to the above issues in [252] to avoid generating such integrals
with doubled propagators with specific choices of the generating vectors vµ using a Gröbner
basis approach. An alternate approach using linear algebra was instead proposed in [253],
which forms the basis of the approach used in this work. Before discussing these solutions
in detail, however, it is useful to introduce an alternate representation for the Feynman
integrals.
3.2 Baikov representation
Consider the general L-loop integral family with E independent external momenta and N
edges (propagators) from Eq. 3.1.1, with only T ≤ N propagators appearing in the integral
i.e. νi > 0 for i ≤ T . Let Q denote the set of all loop momenta and independent external
momenta i.e. Q = (k1 , ..., kL , p1 , ..., pE ). This integral has a total of Ld integration variables.
Since the integral has no open Lorentz indices, however, it is easy to see that it depends only
upon the scalar products between the loop momenta and the external momenta appearing
in the propagators, Qi · Qj , as well as internal masses and d. Maximum number of such
scalar products is L(L + 1)/2 + LE for i ≤ j and i ≤ L, j ≤ L + E. For an integral
with T distinct propagators appearing in the integral, the integral only depends on T scalar
products, implying that only T ≤ L(L + 1)/2 + LE degrees of freedom are required to
characterise the integral. The rest of the Ld − T variables can simply be integrated over.
86
Figure 3.3: A two-point function with massless propagators. The dashed line corresponds
to the cut propagators.
This can be looked at in a slightly different way by considering the loop momentum k1 ,
for example, and dividing it’s components into k1,⊥ and k1,k where k1,k is the projection of
k1 onto the space spanned by (k2 , ..., kL , p1 , ..., pE ) and k1,⊥ is orthogonal to it. This allows
performing the integrations over k1,⊥ independent of the actual Feynman propagators; the
number of remaining integration variables left is often less than that in the momentum-space
representation.
The general L-loop N edge Feynman integral in Eq. 3.1.1 can be written in Baikov’s
representation as
1
Z
d−L−E−1
I(ν1 , ..., νN ) = N0 dz1 ...dzN QN νi
P 2 , (3.2.1)
i=1 zi
Here N0 is a normalisation factor depending on the spacetime dimensions d and kinematics,
P is the Jacobian of the transformation known as the Baikov polynomial, and z1 , ..., zN are
the integration variables called the Baikov parameters which are just the inverse propagators
zi = qi2 − m2i . The Baikov polynomial P in general depends on the Baikov parameters and
the kinematic invariants. This representation was first proposed in [254] and later clarified
in [255]; the reader is referred to the above references for further details.
It was shown in [256, 257] that it is in general highly non-trivial to write down the
integration domain explicitly. While it is rather challenging to perform the actual integration
in this representation, a huge advantage the Baikov representation allows over the traditional
87
momentum-space representation is in performing unitarity cuts [258]. Cutting propagators
in this representation is reduced to the simple task of taking the residue of the integrand at
0 for the corresponding Baikov parameter [259, 260].
Evaluating Feynman integrals on cuts is useful for many purposes. The discontinuity of
a Feynman integral across a branch cut is given by the cut integral
Discsi (I) = −Cutsi I (3.2.2)
where si corresponds to some Mandelstam invariant. This relation was derived in [261, 262]
and is a consequence of the optical theorem [263]:
XZ
∗
M(i → f ) − M (f → i) = i dΠx M(i → x)M∗ (f → x) . (3.2.3)
x
In the above equation, i refers to the initial state, f to final state, and x to all possible
intermediate states. dΠx is the n-particle phase space weight for the intermediate state x.
This simplifies to
XZ
2Im M(i → i) = dΠx |M(i → x)|2 . (3.2.4)
x
The imaginary part of the massless bubble integral in Fig. 3.3, for instance, can be
calculated by evaluating the integral on the s-channel cut denoted via the dashed line. The
integral in Fig. 3.3 is known to evaluate to (for d = 4 − 2)
1
I= + 2 − log(−p2 − i0) + O() , p2 < 0 (3.2.5)
where p is the incoming external momentum (see Appendix B.2). This integral is analytic
for p2 < 0 and can be analytically continued to the region p2 > 0 using the principal branch
88
of the logarithm. This results in the integral picking up an imaginary part
1
I= + 2 − log(p2 ) − iπ + O() , p2 > 0 . (3.2.6)
i.e. Im I = −iπ and the discontinuity is given by Discs (I) = −2iπ.
In Baikov representation, the integral can be written as
(sz1 − 41 (−(p2 + i0) − z1 + z2 )2 )1/2−
Z
I = N0 dz1 dz2 (3.2.7)
z1 z2
with the normalisation factor
3
2iπ 2 −
N0 = 3 (3.2.8)
Γ( 2 − )(p2 + i0)1−
The s-channel cut here corresponds to setting both the propagators on-shell i.e. computing
the residue at z1 = z2 = 0 [257], which results in
4iπ p2
Cuts I = + O() = 2iπ + O() . (3.2.9)
p2 2
It is clear that the above results satisfy the cut-discontinuity equation (Eq. 3.2.2).
In addition to specific kinematic cuts as in the above example, the integrals can also be
evaluated on maximal cuts i.e. setting all propagators on-shell (this is different from optical
theorem). This is of particular importance in the method of evaluating Feynman integrals
using differential equations since the integrals evaluated on maximal cuts correspond to
the homogeneous solution of the differential equations [255, 264]. Since the homogeneous
solution is expressed using the same class of functions as the full solution, it can be used
to determine if the differential equations admit a polylogarithmic solution. In addition, the
solution on maximal cut can be used to determine the "uniformly transcendental" basis of
master integrals [265, 266] which satisfy particularly simple differential equations. The reader
89
is referred to [256, 257] for a detailed discussion and more complicated examples of integrals
evaluated on maximal cuts, including integrals involving elliptic functions. However, the
representation in Eq. 3.2.1 is not the most convenient representation for performing the
integration explicitly. In fact, the number of integration variables can, in many cases, be
reduced by performing the variable transformation loop-by-loop [256, 257].
Similar to the momentum-space representation, the IBP relations can also be written
down in the Baikov representation [255]. For the general integral in Eq. 3.2.1, the Integration-
By-Parts identities in Baikov representation are given by
N
d − L − E − 1 ∂P
∂fi fi 1
Z
d−L−E−1
X
0= dz1 ...dzN + fi − νi QN νi P 2 . (3.2.10)
i=1
∂z i 2P ∂z i zi z
i=1 i
Here fi are arbitrary polynomials in the Baikov parameters and the kinematic invariants.
The freedom in choosing the polynomials fi reflects the freedom in choosing the vectors vµ
in the momentum-space representation. Certain choices of fi can provided advantages over
the others; this is discussed in the next section (Sec. 3.3).
Integration-By-Parts relations can also be generated on cuts [267, 268] using the general
formula in Eq. 3.2.10 which can be useful to simplify the system of equations by reducing
them into smaller and more manageable parts as well as potentially provide significant per-
formance improvement as argued in [268]. The IBPs generated on cuts can also be used to
obtain the differential equations on cuts e.g. on maximal cuts which can be used to solve for
the homogeneous solution of the differential equations.
3.3 Syzygies for IBP reduction
In the above general equation for IBP relations via Baikov representation (Eq. 3.2.10), the
simplistic choice of the polynomials fi = 1 leads to equations that have terms with 1/P . It
is straightforward to see that such terms generate integrals with shifted dimensions in the
IBP relations. In general it is highly desirable to avoid such integrals in the IBP systems
90
since they do not commonly occur in the amplitudes. In addition, presence of such integrals
leads to bloated linear systems which are relatively difficult to solve. To avoid such integrals
is relatively straightforward, however, by simply imposing the constraint [267–269]
N
!
X ∂P
fi + fN +1 P = 0 (3.3.1)
i=1
∂zi
on the polynomials fi . This is known, in algebraic geometry, as a syzygy constraint. Note
the presence of a new polynomial fN +1 ; together with the rest of the fi , it forms a syzygy.
The formal mathematical definition and details of syzygies is beyond the scope of this work
and the reader is referred to [270] for more details regarding syzygies in the context of
Feynman integral reduction. For the purposes here, a syzygy is basically a relation between
polynomials in certain variables. E.g. consider a set of polynomials ei = {x, x + y, y 2 }. The
equation
f1 e1 + f2 e2 + f3 e3 = 0 (3.3.2)
is satisfied by fi = {{y, −y, 1}, {x + y, −x, 0}}. This set of solutions forms a syzygy module.
Note that any linear combination of the two syzygies, with the variables (x, y) allowed in the
coefficients, also forms a syzygy. This is easy to see by replacing fi by a(x, y)f1,i + b(x, y)f2,i
in Eq. 3.3.2 where f1,i and f2,i are the two different solutions.
For the case at hand, syzygies are formed by polynomials in the Baikov parameters
with the kinematic invariants appearing in the coefficients. Explicit solutions to the "no
dimension-shift" constraint in Eq. 3.3.1 were pointed out in [271] in Baikov representation.
These fIno−dim are simply linear polynomials in the Baikov parameters and kinemat-
ics. Note that these fino−dim generate integration-by-parts relations identical to those in
the momentum-space representation [271]. E.g. for the bubble function with one massive
91
propagator in Fig. 3.2, the no dimension-shift syzygies are
fino−dim = {{−p2 − z1 + z2 , p2 − z1 + z2 , 0}, {2(m2 + z1 ), −p2 − z1 + 2(m2 + z1 ) + z2 , −2}} .
(3.3.3)
Substituting the above syzygies into Eq. 3.2.10, it can be seen that the IBP relations from
Eqs. 3.1.10 and 3.1.11 are recovered.
As seen in Sec. 3.1, IBP relations generated this way contain integrals with doubled
propagators (dots) and it is highly desirable to avoid generating such integrals. Looking at
the general IBP equation in Baikov representation (Eq. 3.2.10), it is clear that such doubled
propagators are generated by the term νi fi /zi . They can be avoided by imposing the so-called
"no dot" constraint
fino−dot = bi zi ∀ i = 1, . . . , N 0 with νi ≥ 1 and N 0 < N , (3.3.4)
Here N 0 is the set of positive indices in the sector while N is the total number of propagators
in the integral family. This is a rather trivial constraint. Essentially, the fi are required to
be proportional to zi so as to cancel the zi in the denominator of the offending term. For
the bubble function with one massive propagator in Fig. 3.2, the no-dot syzygies are simply
fino−dot = {{z1 , 0, 0}, {0, z2 , 0}, {0, 0, 1}} . (3.3.5)
To impose the two constraints individually is quite straightforward. Finding fi that
satisfy simultaneously both the constraints in Eqs. 3.3.1 and 3.3.4 is, however, a highly
non-trivial problem in general. The initial strategy, based on the syzygy approach using
Baikov representation, proposed in [268] was to simply replace the fi in the no dimension-
shift constraint in Eq. 3.3.1 with bi zi and solve for bi . This was found to be computationally
expensive and impractical for complicated topologies. The approach proposed in [270] to
92
simultaneously solve the two constraints by computing a module intersection of the two
syzygy modules was suggested to be better. Computing module intersections is a well-
known problem in Algebraic geometry and several computer algebra packages exist for this
task e.g. Singular [272].
The usual module intersection approach using Singular was found to be inadequate
for the purpose of this calculation, however. For the planar topologies, for example, which
are expected to be significantly simpler in complexity, even syzygies for sectors with 6-lines
(propagators) were found to be out of reach, using Singular out of the box. With a few
optimisations with the assistance of the author of [270], further improvement was found
rendering 6-line planar sectors solvable. However, the top-level (7-line) sectors in the planar
as well as non-planar topologies remained intractable. In addition, the approach was found
to be computationally expensive requiring several days of run-time with large amounts of
RAM.
3.4 Linear algebra based syzygy construction
In light of the inadequacy of the module intersection based method implemented through
Singular, it was clear that an alternate approach was required to compute the syzygies.
Specifically, it was observed that the module intersection method was computationally ex-
pensive and not feasible for the problem at hand. Instead, a custom linear algebra based
syzygy solver was employed to compute the syzygies required for this work.
The basic idea behind the algorithm is as follows. As mentioned in the previous section,
it is straightforward to write down the solutions to the no dimension-shift constraint as well
as the no dot constraint. The important point to note here is that if certain syzygies satisfy
a constraint, then so does any linear combination of those syzygies where the coefficients are
93
allowed to depend on the variables i.e.
N
!
X ∂P
Fjno−dim + FNno−dim
+1 P =0 (3.4.1)
j=1
∂zj
where Fjno−dim = no−dim
P
i ai (z1 , ..., zN , s12 , ...)fj,i is the linear combination and ai are the poly-
nomials. The linear combination Fjno−dim must now satisfy the no dot constraint Fjno−dim =
bj zj which provides constraints on the polynomials ai .
In principle, the above construction should provide exactly the same syzygies as the
module intersection method if solving for general polynomials ai . However, solving for general
ai would require a Gröbner basis computation which is identical to the module intersection
approach. Instead, this method can be used to algorithmically construct the solutions up
to a certain polynomial degree by choosing an ansatz for the ai , starting with degree 0
polynomials for ai . Substituting an ansatz for ai in Eq. 3.4.1 turns the problem into row
reduction of a matrix. A custom linear solver based on Finite field methods [273, 274] is
used to provide fast and efficient row reduction of the resulting linear system.
A brief description of the algorithm first implemented for the multivariate case in, and
reproduced from, [6] is provided below.
While the module intersection method provides all possible syzygy solutions, it was ob-
served that for amplitude calculations only syzygies up to a certain degree are needed.
Indeed, in contrast with the module intersection approach based on Singular, the linear al-
gebra approach presented above was able to generate the required syzygies. For the top-level
planar topologies, the method was relatively quick requiring only O(10) CPU hours while
the non-planar topologies turned out to be significantly more difficult requiring O(100) CPU
hours per topology. This approach was also used to generate the required syzygies for the
5-point 2-loop topologies needed for the calculation of γγ + jet amplitude in [134, 138].
While this method was able to solve the syzygies for calculations at the frontier of per-
turbative QCD, there are certain limitations. In particular:
94
Algorithm 3.1 Syzygies for linear relations without dimension shifts or dots [6]
Input: Syzygies of degree 1 solving Eq. 3.3.1, maximal required degree nmax .
Output: Syzygies S1 , . . . , Snmax up to degree nmax solving Eqs. 3.3.1 and 3.3.4.
1: Start with syzygies of degree n = 1. Let I1 be a complete set of solutions (fi ) to the
no-dimension-shift constraint from Eq. 3.3.1, which are linear in the Baikov parameters
zk . These can directly be written down [271]. Abbreviating the momenta squared with
variables zN +1 , . . . , the vectors in I1 are of homogeneous degree 1 in the variables zk .
2: At degree n, form a matrix Mn , where each element of (fi ) ∈ In corresponds to a row.
The columns enumerate both the component i of (fi ) and the power products of zi in
them; the entries of the matrix are the coefficients. A column is called admissible, if it
satisfies the no-doubled propagator constraint in Eq. 3.3.4, and non-admissible otherwise.
All admissible columns are ordered to the right of the non-admissible columns.
3: Perform a row reduction of Mn . In the row reduced form, select all rows, which have an
admissible pivot column and form the corresponding syzygies Sn from them. Sn forms a
complete set of linear combinations of the syzygies in In , which satisfy Eq. 3.3.4 for all
of their terms, and are therefore our solutions at degree n.
4: If n is the user-defined maximal degree, stop and return the solutions S1 , . . . , Sn . Oth-
erwise, proceed.
5: For each vector of polynomials (fi ) ∈ In and each zk , form the vector of polynomials
(zk fi ). This gives the set In+1 , which are solutions of Eq. 3.3.1 of degree n + 1 in the zk
but not necessarily solutions of Eq. 3.3.4.
6: Replace n → n + 1 and go to step 2.
1. Solutions are only produced up to a certain polynomial degree by construction. In case
higher degree solutions are required, the task can be challenging in certain cases. E.g.
For some of the non-planar topologies, syzygies for degree up to 5 were constructed,
but these were found to be insufficient. It was estimated that computing the degree
6 syzygies would be unfeasible with the available computing resources. Instead, some
95
identities involving integrals with doubled propagators were indeed required for a com-
plete and successful IBP reduction. It has been suggested that syzygies up to a certain
degree, dependent on the specific problem at hand, are sufficient to obtain reductions
for integrals with arbitrarily high numerators. This is not proven, however.
2. By the nature of the construction wherein the kinematic variables are treated as inde-
terminates of the ring similar to the Baikov parameters, often identities are generated
with "true" degree less than the expected degree. In other words, the kinematic vari-
ables sometimes inflate the polynomial degree of the syzygy so that a degree "n" syzygy
is effectively only "n-1" (or lower). This can happen in two ways, either by having a
kinematic variable as an overall factor or by having each and every individual term
containing a kinematic variable. This can in principle be fixed by considering the
kinematic variables in the coefficient field instead of as indeterminates like the Baikov
parameters. It was suggested that in this approach the degree of syzygies required can
be less than in the original formulation. This is not very well understood, however,
and requires further investigation.
3. This construction may also result in syzygies that are polynomials multiplied by a
Baikov variable as an overall prefactor. Such syzygies are redundant: the loop edge
corresponding to the Baikov variable present as the overall prefactor is either part of the
topology under consideration or appears as an inverse propagator (in the numerator).
In the first case the syzygies simply generate identities from a lower topology that
are presumably contained in the identities generated specifically for that topology;
these are indeed dropped [275]. In the second case, the prefactor simply increases the
"tensor rank" of every integral in the identity by 1 which can alternatively be achieved
by increasing the rank of seed integrals in the syzygies with one degree.
Despite the above limitations, the method has been very successful in solving syzygies
for challenging problems.
96
3.5 IBP reduction using syzygies
The syzygies generated using the approach outlined in the previous section are used to gen-
erate IBP templates to generate relations between integrals, similar to templates generated
through conventional IBP relations e.g. in Eqs. 3.1.10 and 3.1.11. Seed integrals with spe-
cific propagator exponents are then inserted into the templates to generate the required
IBP relations, similar to Laporta’s algorithm, which are subsequently solved using a row-
reduction procedure. An in-house custom linear solver, Finred [276], based on Finite field
methods [273, 274] is used for this. Instead of solving the symbolic linear system directly,
the equations are sampled over a finite field and the symbolic dependence is reconstructed
through rational reconstruction. This method allows a very high degree of parallelisation
and hence can be much faster than the traditional approach.
In the case of ZZ production, top quark and Z-boson masses are set to numbers: mt = 1
and m2Z = 5/18m2t . The chosen ratio for mZ corresponds to the mass of the Z-boson in
terms of top-quark mass. This is to further simplify the linear relations to facilitate solving
the system. Note that setting mt = 1 simply amounts to factoring out powers of mt from
the quantities of interest rendering the rest of the kinematic variables, and the form factors,
dimensionless. In this way, all the Feynman integrals were successfully reduced to master
integrals. Even with the above simplifications, the reductions proved to be rather challenging
nevertheless and required a significant amount of computational resources. The reductions
took months of run-time on the High Performance Computing Cluster (HPCC) at MSU.
The resulting reduction tables occupied over 200 GB of disk space and involved extremely
complicated rational functions with up to total degree 190 in the kinematic variables. As
somewhat expected, the non-planar topologies proved to be the most challenging accounting
for almost all of the CPU time as well as disk space.
Note that choice of basis integrals has not been mentioned yet; for ZZ the reductions
were performed in terms of the conventional Laporta basis of integrals where the integrals
97
(a) The hexagon-box topology (b) The double-pentagon topology
Figure 3.4: The non-planar topologies for 5-point 2-loop γγ + jet production.
with least numerator rank and no dots along with specific ordering prescription are simply
chosen as master integrals. This is optimised for evaluation later; the details are discussed in
the next section (Sec. 4). An interesting point to note is that within the planar topologies,
Figs. 2.7a, 2.7c, and 2.7b, with adjacent gluons are significantly simpler than the topology
in Fig. 2.7d with the gluons at the opposite vertices.
For the case of γγ + jet, however, the situation is a lot more complicated due to the
presence of an additional external particle. Indeed, while the planar topologies had been
reduced, complete IBP reductions for a 5-point amplitude at 2-loops (including the non-
planar topologies) had been out of reach until very recently. The solutions for the non-
planar topologies were facilitated with the assistance of two key improvements. First, a
canonical basis (see Sec. 4.1 for a brief description) was chosen for the reductions instead
of the conventional Laporta basis. This already led to a reduction by a factor of 2 in the
number of samples required for a full reconstruction, effectively reducing the required CPU
time to half as well. Second, a new algorithm was used to guess the complete symbolic form
of the denominators of the rational coefficients in the reduction identities [138, 277, 278].
This further led to a reduction in the number of samples by a factor of ∼ 10 and effectively
made the reductions computationally feasible for the extremely challenging hexagon-box and
double-pentagon topologies.
98
For both the processes, the reduction identities are generated with the rational coefficients
in front of the master integrals over a common denominator. This representation is not ideal
for a couple of reasons. First, reduction identities occupy a huge amount of disk space in this
representation and are rather difficult to deal with for complicated processes. Second, having
everything over a common denominator can often lead to numerical instabilities. Instead,
multivariate partial fractioning of the rational coefficients results in a substantially simpler
form of the reduction identities, in particular for the non-planar topologies. For the case of
ZZ, this resulted in a reduction from over ∼ 10 GB to < 500 MB of disk space. Similarly, a
reduction by a factor of ∼ 100 is achieved for the most complicated reduction identities for
γγ + jet. Multivariate partial fractioning is discussed in more detail in Sec. 5.1.
While such drastic simplifications are certainly impressive, a deeper understanding behind
this would be highly desirable. It has been suggested that the difference in complexity is
due to combining structures with different singular behaviours and limits. Ideally, it would
be extremely useful to generate reduction identities directly in the partial fractioned form.
While an efficient algorithm to do this is not yet known, recent progress in understanding this
partial fractioning procedure as well as general structure of the singularities holds promise
for the future.
99
Chapter 4
Finite basis integrals
4.1 Evaluation of Feynman integrals
Once the Feynman integrals are reduced to a set of master integrals, the next step is to evalu-
ate them. There are a number of methods used to evaluate master integrals, both analytically
and numerically. This section will focus on the methods of differential equations [279–284]
and sector decomposition [161, 285–287], used for γγ + jet and ZZ production respectively.
4.1.1 Differential equations
Consider the tadpole integral from Eq. 3.1.3. This integral is dependent only on the mass
m. Taking the derivative under the integral sign with respect to m2 ,
d dd k d 1
Z
I(m2 ) = d/2
dm2 iπ dm k − m2
2 2
dd k 1
Z
=− d/2
. (4.1.1)
iπ (k − m2 )2
2
The integral in the second line of the above equation, resulting from the derivative, can be
reduced to I(m2 ) using IBP relation from Eq. 3.1.6. This results in the following differential
equation for I(m2 ):
d d−2
2
I(m2 ) = − I(m2 ) . (4.1.2)
dm 2m2
100
The above differential equation can be solved given appropriate boundary conditions to
obtain I(m2 ).
Consider now a more general system with m master integrals denoted by I1 , ..., Im . Tak-
ing the derivative with respect to the dimensionless kinematic variable yα and using IBP
reduction to reduce the right hand side integrals to master integrals, the derivatives can be
written as
∂ ~
I = Aα I~ , (4.1.3)
∂yα
where I~ is the vector of master integrals (I1 , ..., Im ) and Aα is an m × m matrix whose
entries are rational functions in the invariants and the dimensional regulator . Repeating
the procedure for all the kinematic variables results in the following form
dI~ = AI~ , (4.1.4)
where A is an m × m matrix of one-forms
X
A= Aα (~y , ) dyα (4.1.5)
α
It turns out that in many cases, the above system can be drastically simplified following
an algebraic basis change I~ = T (~y , ) I~0 [283, 288] such that the differential equation in
Eq. 4.1.4 becomes
dI~0 = A0 (~y )I~0 . (4.1.6)
The above form is colloquially referred to as the canonical or -form. Note that the entries
of the transformation matrix T are rational functions in the kinematic variables ~y and . In
the canonical form, the differential equations in most known cases can be easily solved order
101
by order in in terms of iterated integrals that evaluate to multiple polylogarithms (MPLs)
(see [289–295]). Efficient numerical codes are available for evaluation of such multiple poly-
logarithms [296–303] allowing a fast and convenient numerical implementation of Feynman
amplitudes expressed in terms of such functions.
The method of differential equations is very powerful and has been used to evaluate a
large number of Feynman integrals for many different processes. E.g. this method was used
to calculate the master integrals for the massless ZZ amplitudes [304–307] as well as the
master integrals for massless 5-point scattering [308–311] required for the γγ +jet calculation.
Despite the large-scale applicability and success of this method, however, there are some
limitations. In particular, reducing the system to the canonical form through an algebraic
transformation matrix T (~y , ) is impossible in many cases; such systems evaluate to a class of
functions beyond MPLs. This is expected to be the case for gg → ZZ with massive internal
quarks. There has been considerable progress over the recent years in understanding such
functions beyond polylogarithms, mostly concerning their elliptic generalisations [312–317].
However, most of the progress has been achieved for simple topologies with very few mass
scales; for processes such as gg → ZZ with many different mass scales, evaluation in terms
of such functions remains extremely challenging. Furthermore, it isn’t clear if the so-called
elliptic multiple polylogarithms (eMPLs) are enough to represent the master integrals for
this process and functions beyond even eMPLs could appear. On the other hand, for certain
cases, even if the -form is achievable, the functions cannot be evaluated in terms of multiple
polylogarithms [318].
An alternative approach that has been recently used for many phenomenological appli-
cations is the solution of differential equations by expansions [112, 113, 319–323]. Instead
of solving the differential equations analytically, a series expansion is constructed around a
point and the equations are solved numerically term-by-term in the expansion parameters.
This method is extremely useful for problems involving elliptic functions or beyond given
the lack of understanding concerning these functions. Another advantage is that any given
102
system of differential equations can be solved, in principle, regardless of whether a canonical
form can be achieved or not. This is, again, especially beneficial for elliptic integrals since it
can be challenging to obtain the canonical form in many cases with only a few cases where
this has been successfully done [324, 325].
4.1.2 Feynman parametric representation
Consider the general L-loop integral in d dimensions with N distinct propagators in the
momentum space representation from Eq. 3.1.1
L
! N
dd kl 1
Z Y Y
I(ν1 , ..., νN ) = ν (4.1.7)
l=1
iπ d/2 j=1
(qj2 − m2j + i 0) j
with integer exponents νj ∈ Z. Note the i0 for the correct Feynman propagator prescription.
If all indices νj are positive, then the identity (see e.g. [326, 327])
∞
1 (−1)νj
Z
ν −1 xj D
= dxj xj j e for νj > 0, D < 0 (4.1.8)
D νj Γ(νj ) 0
can be used to write down the integral as
ν −1
N
!Z L
!
dxj xj j dd kl
Z PN
xj (qj2 +m2j −i 0)
Y Y
I(ν1 , ..., νN ) =(−1)ν e j=1 (4.1.9)
j=1
Γ(νj ) l=1
π d/2
P
where ν = j νj is the sum of all propagator exponents. The integral over loop momenta is
just a Gaussian integral and can be performed easily after combining the propagators and
performing Wick rotation; see e.g. [326, 328] for more details. The resulting expression is
the Feynman parametric representation of this integral:
ν −1
N
! N
!
dxj xj j
Z Y X
I(ν1 , ..., νN ) = (−1)ν Γ(ν − L d/2) δ 1− xj
j=1
Γ(νj ) j=1
ν−(L+1) d/2
U
(νj > 0) . (4.1.10)
F ν−L d/2
103
The U and F polynomials in the above equation are the Symanzik polynomials of the first
and second kind respectively. U and F are both homogeneous polynomials in the Feynman
parameters x1 , ..., xN of degrees L and L + 1 respectively. The U polynomial is strictly non-
negative while the F polynomial vanished for certain kinematic configurations which, in fact,
correspond to the branch cuts of the integral.
The Feynman parametric representation can be used to directly evaluate the Laurent
expansion in of the integrals in many cases. E.g. the calculation of the massless bubble
integral in Fig. 3.3 (without the s-channel cut) is shown in Appendix B. In particular,
the work on linear reducibiliy [329, 330] has expanded the applicability of the method to
much more complicated examples. A maple implementation of the integration algorithm,
HyperInt [331], is also available that performs direct symbolic integration for Feynman
integrals that evaluate to multiple polylogarithms. The integrals in Feynman parametric
representation can also be evaluated numerically using e.g. Monte-Carlo methods; this is
the method of choice for evaluating the master integrals for ZZ and is discussed later in
the chapter. A crucial roadblock, before evaluating integrals in the Feynman parametric
representation numerically, is the presence of divergences. It is discussed in the next section
on how to deal with the divergences.
4.2 Finite integrals
As discussed in Sec 1.5.4, Feynman integrals often have divergences. Those originating
from regions of large values of loop momentum are called Ultraviolet (UV) poles and those
originating from soft or collinear configurations causing the propagators in the denominators
to vanish are called Infrared (IR) poles. It is instructive to study the structure of integrands
as well as origin of divergences in the Feynman parametric representation.
104
Figure 4.1: 3-point integral with 1 massive propagator.
4.2.1 Divergences in Feynman parametric representation
Consider the 3-propagator massless integral with 1-mass in d-dimensions in Fig. 4.1:
dd k 1
Z
I= d/2
. (4.2.1)
iπ (k − m )(k + p1 )2 (k + p1 + p2 )2
2 2
This integral can be written in Feynman parametric representation as
3
!
∞
U −1+2
Z X
I = −Γ(1 + ) dx1 dx2 dx3 δ 1 − xj (4.2.2)
0 j=1
F 1+
where the Symanzik polynomials are given by
U = x1 + x2 + x3 ,
F =(m2 x1 + Q2 x3 )(x1 + x2 + x3 ) (4.2.3)
where Q2 = −q 2 = −(p1 + p2 )2 . This integral is known to be IR divergent and hence
the integrand cannot simply be expanded in . Naively expanding the integrand around
= 0 leads to an incorrect result; the divergence isn’t explicit in but rather hidden in the
integrand. To see this, consider the region Q2 > 0 where the U and F polynomials for the
105
above integral are both strictly non-negative. The only possible divergences then occur at
the endpoints of the integration region.
To simplify the expression, integration in x2 can be trivially performed using the delta
distribution resulting in
∞
1
Z
I = −Γ(1 + ) dx1 dx3 . (4.2.4)
0 (m2 x1 + Q2 x3 )1+
Under the scaling {x1 , x3 } → {x1 /λ, x3 /λ} and the limit λ → 0, the integrand behaves as
1/λ indicating a non-integrable divergence near the boundary. This is the origin of the IR
divergence.
For UV divergences, consider a different example, a 2-loop "double" tadpole integral:
dd k1 dd k2 1
Z
I= . (4.2.5)
(iπ ) (k1 − m )(k22 − m2 )
d/2 2 2 2
The above integral clearly has a UV divergence in both loops since this is simply a product
of two 1-loop tadpole integrals. In Feynman parametric representation, the integral becomes
∞
U −4+3
Z
I = −Γ(−2 + 2) dx1 dx2 δ (1 − x1 − x2 ) (4.2.6)
0 F −2+2
with U = x1 x2 and F = m2 (x1 + x2 ) U. This simplifies to
∞
(x1 + x2 )2−2
Z
2 2−2
I = −Γ(−2 + 2) (m ) dx1 dx2 δ (1 − x1 − x2 ) . (4.2.7)
0 (x1 x2 )2−
In this case, the Gamma function prefactor is divergent in the limit → 0:
1
Γ(−2 + 2) = + O() . (4.2.8)
4
Using the Cheng-Wu theorem [332], the variable x1 can be taken out of the delta distribution.
The integrand then diverges in the limit λ → 0 for the scaling x1 → x1 /λ; this is the other
106
UV divergence.
The above approach of determining the scaling behaviour of integrands near boundaries
was presented in [333, 334]. These boundary divergences prevent a naive Monte-Carlo im-
plementation since the integrand cannot be expanded around = 0. A universal approach
to resolve these singularities and enable an expansion in is sector decomposition [161, 285–
287, 335]. After performing sector decomposition, the resulting expressions can be inte-
grated numerically. Public codes that implement sector decomposition and perform numer-
ical integration for arbitrary loop integrals in physical kinematical region are available e.g.
Fiesta4 [336] and pySecDec [3].
4.2.2 Finite integrals with dimension shifts
The usual approach to Integration-By-Parts reduction using Laporta’s algorithm leads to a
conventional basis of master integrals which prefers master integrals with the lowest number
of propagators and irreducible scalar products in the numerator. This basis is, however,
rather difficult to evaluate numerically using sector decomposition due to the presence of
boundary singularities which can cause severe numerical instability during Monte-Carlo eval-
uation. A natural solution would be to use integrals that are finite in d = 4 to avoid such
issues. Indeed it was observed in [337, 338] that the use of finite integrals in the basis leads
to significantly improved numerical stability. In addition, finite integrals often require fewer
orders in their expansion which makes them easier to evaluate numerically requiring less
computing resources to achieve similar precision. Another advantage is that the poles in the
regulator drop out of the Feynman integrals and become explicit. The resulting analytic
expressions for the amplitude are a lot more convenient and easier to simplify (see Sec. 5.1).
These factors together improve the overall numerical performance significantly.
To construct such finite integrals, one approach is to use integrals in dimensions different
than d = 4. This was explored in [339] by considering finite integrals in d = 6. This approach
is often used in conjunction with increasing propagator exponents, with finite integrals often
107
requiring both dimension-shifts and higher powers of propagators. Consider the massless
scalar integral with 3-propagators in Eq. 1.5.26 in d-dimensions:
dd k 1
Z
I= . (4.2.9)
iπ d/2 (k 2 )(k − p1 )2 (k + p2 )2
In the limit of small loop-momentum,
dd k 1 dd k
Z Z
I∼ d/2 2
∼ . (4.2.10)
iπ (k )(2k.p1 )(2k.p2 ) k→0 k4
This integral has an IR divergence for k → 0 in d = 4. It was shown in Sec. 1.5.4 that this
integral will converge for d > 4. This is the general idea behind introducing dimensionally-
shifted integrals to cure IR divergences; for large enough values of d, the singular behaviour
in soft or collinear regions is cured. The opposite approach works for UV divergences. E.g.
the bubble integral with 1-mass,
dd k 1 dd k
Z Z
I= d/2
∼ , (4.2.11)
iπ (k − m )(k + p)2
2 2
k→∞ k4
is finite for d < 4. Alternatively, the exponent of one of the propagators can be increased to
cure the singular behaviour for large values of loop-momentum. For example,
dd k 1 dd k
Z Z
I= d/2
∼ (4.2.12)
iπ (k − m )2 (k + p)2
2 2
k→∞ k6
is now finite for d < 6. It is straightforward to see that increasing propagators tames the
behaviour for large loop-momentum.
It was shown in [334, 340] that a basis of integrals which are simultaneously finite in
both UV and IR regions, constructed using dimension-shifts and higher propagator powers,
can always be found. From the above examples it is clear that the dimension shifts act in
opposite way to IR and UV poles. Consequently, finite integrals often require a combination
108
(k 2 − m2t )
(b) Divergent integral in d = 4 − 2 with an
(a) Divergent integral in d = 4 − 2
irreducible numerator
(c) Finite integral in d = 6 − 2 (d) Finite integral with a dot in d = 6 − 2
Figure 4.2: Examples of divergent and finite integrals in the limit → 0 for a non-planar
topology. Thick solid lines represent the top-quark while thick dashed lines represent Z-
bosons. Topology (b) contains an irreducible numerator, where k is the difference of the
momenta of the edges marked by the thin dash lines. Reproduced from [6].
of both dimension shifts and higher propagator powers.
Examples of such finite integrals for a non-planar topology are shown in Fig. 4.2. Fig. 4.2a
shows the corner integral for the topology while 4.2b shows an integral with 1 irreducible
numerator; both are divergent in d = 4 − 2. The corner integral, as shown in Fig. 4.2c,
is in fact finite in d = 6 − 2. Another finite integral can be constructed by increasing the
exponent of one of the propagators (colloquially referred to as adding a "dot") as shown in
Fig. 4.2d.
4.2.3 Finite integrals with numerators
As described in the previous subsection, it is rather straightforward to find finite integrals of
such kind involving dimension shifts and dots, especially using e.g. Reduze 2. However, inte-
109
grals with dots typically do not appear in amplitudes which is in fact the primary motivation
for syzygy based IBP reduction as discussed in Sec. 3; this is especially true for the top-level
topologies. Furthermore, dimension-shifted integrals require reductions for L additional dots
for topologies with L-loops which can also be quite challenging. It is therefore desirable to
consider alternative approaches to construct finite integrals.
Here, a different approach to construct finite integrals by considering linear combinations
of divergent integrals is explored. The Feynman parametric representation of a general linear
combination is considered and the boundary regions are analysed to put constraints on the
coefficients [341]. This ensures that the non-integrable divergences of the individual integrals
cancel for the combined integrand resulting in a single Feynman parametric integral that is
finite. Before the algorithm to construct such integrals with numerators, first described in [6],
is presented, it is useful to define some relevant quantities and formulas.
Similar to Eq. 4.1.8, propagators with negative exponents i.e. numerators with νj < 0
can be included by employing the identity [328, 342]
" #
1 ∂ −νj xj D
= −ν e for νj ≤ 0 . (4.2.13)
Dνj ∂xj j x j =0
Consider an integral I(ν1 , ..., νN ). Let N+ be the set of all positive νj , N− the set of all
P
negative νj , and r = j∈N+ νj . Then an integral with positive or negative indices, using
Eqs. 4.1.8 and 4.2.13 respectively, can be written as
Z Y dxj xνj −1 X
I(ν1 , ..., νN ) = (−1)r Γ(ν − L d/2) δ 1 − xj
j∈N+
Γ(ν j ) j∈N+
Y ∂ |νj | U ν−(L+1)d/2
|νj |
ν−L d/2
(νj 6= 0). (4.2.14)
j∈N ∂x
F
− j
xj =0 ∀ j∈N−
Using the above formula, an integral with an arbitrary combination of numerators and dots
can be expressed in Feynman parametric representation.
110
The ultimate objective here is to combine different integrals sharing a common parent
topology into one merged parametric representation. To that end, it would be desirable to
include integrals from subtopologies as well to write down the most general ansatz for a
finite linear combination. However, integrals from subtpologies, if their Feynman parametric
representations are generated naively using the above formula, have U and F polynomials
different from the parent topology, and hence the linear combination cannot be combined
over a common "denominator" easily. It would be useful, then, to be able to express the
subsector integrals in a similar way to the numerator integrals using the same Symanzik
polynomials as the parent topology. This can be achieved by taking derivatives with respect
to the Feynman parameters corresponding to the pinched lines without setting them to zero,
∞
1 ∂ −νj +1
Z
=− dxj −ν +1 exj D for νj ≤ 0, D < 0. (4.2.15)
D νj 0 ∂xj j
Here line corresponds to a propagator with a positive index i.e. a propagator present in the
topology. Before writing down the general formula, it is necessary to specify some notation.
Let N = {1, . . . , N } be the set of all indices, NT the set of positive indices of the parent
topology (parent lines), Nt the set of positive indices νj of the current topology which could
either be the parent topology or a subtopology, N∆t = NT \ Nt i.e. the set of pinched lines,
N\T = N \ NT be the set of negative indices of the parent topology (parent numerators),
P
r = j∈Nt νj the sum of positive indices of the current integral, and ∆t = |N∆T | the number
of pinched lines. The general integral can then be written in the Feynman parametric
representation as
! ! !
Z Y Y xνj −1 X
I(ν1 , ..., νN ) = (−1)r+∆t Γ(ν − L d/2) dxj δ 1− xj
j∈NT j∈Nt
Γ(νj ) j∈NT
!
Y ∂ |νj | Y ∂ |νj |+1 U ν−(L+1)d/2
|νj |
|ν |+1
(νj ∈ Z).
j∈N\T ∂x j∈N∆t ∂xj j F ν−L d/2
j
xj =0 ∀ j∈N\T
(4.2.16)
111
Algorithm 4.1 Finite Feynman integrals [6]
Input: Dimensionally regularized multiloop integrals with a common parent sector, possibly
involving higher powers of propagators, irreducible numerators, or dimension shifts.
Output: Linear combinations of the input integrals which are finite, i.e. they have a conver-
gent Feynman parametric representation for = 0.
1: From the ns input or “seed” integrals, form a general linear combination
Xns
I = ai Ii , (4.2.17)
i=1
where Ii are the seed integrals and ai are the unknown coefficients. The ai are assumed
to depend on the kinematic invariants and the dimensional regulator .
2: Using Eq. 4.2.16, write the Feynman parametric representation for each seed integral
and bring their linear combination over a common denominator such that
!
U ν0 −(L+1) (d0 −2)/2
Z Y X
ν0
I = (−1) dxj δ(1 − xj ) P (4.2.18)
j∈NT j∈NT
F ν0 −L (d0 −2)/2
where NT is the set of distinct propagators in the parent sector, ν0 is the effective number
of propagators, and d0 ∈ Z the effective number of space-time dimensions to be expanded
around. The numerator P is a homogeneous polynomial in the Feynman parameters,
X
P = cj Mj (x1 , ..., xNT ), (4.2.19)
j
where the coefficients cj are polynomials in ai , the kinematic variables, and , and
Mj (x1 , ..., xNp ) are monomials in Feynman parameters. Note that the numerator poly-
nomial P in general depends on and it is crucial to keep this dependence to produce
correct results. It is sufficient, however, to set = 0 in the exponents of the U and F
polynomials for the convergence analysis in the following two steps.
112
3: Check the scaling behaviour of the integrand near an integration boundary using the
prescription outlined in [333, 334].
4: Make sure a convergent integration of Eq. 4.2.18 is not prevented by a rapid growth of
the integrand near the boundary. This can be achieved by requiring the coefficients of
the offending monomials in the numerator to vanish, which provides constraints on the
ai .
5: Repeat 3-4 until all boundaries are checked.
Pnfin P
ns
At the end of this exercise, we are left with I = i=1 ai j=1 bij Ij , where nfin ≥ 0 is
P
the number of finite integrals found, and bij Ij are the finite combinations.
This expression for a completely general integral was first presented in [6]. Note that
the pinched lines are allowed to appear as numerators i.e. νj ≤ 0 for j ∈ N∆t . Also, the
Symanzik polynomials U and F are calculated by taking all indices N into account.
With the definitions above and the formula for a general Feynman integral, the algo-
rithm 4.1 to construct general linear combinations that are finite in d = 4 can be presented.
This algorithm was first described in [6].
Applying this algorithm to a set of seed integrals from topology in Fig. 4.2a with the
constraint of integrals with only up to 1 numerator rank, the following linear combination is
obtained:
If in,1 = s (m2z − s − t) I1,1 + s I2,1 + s I3,1 − s I4,1 − s I5,1 − (m2z − s − t) I6,1 − (m2z − t) I7,1 .
(4.2.20)
The integrals I1,1 etc. are as defined in Fig. 4.3. Allowing for integrals with up to numerator
rank 2 in the list of seed integrals generates several new linear combinations. Amongst them,
the linear combination
If in,2 = s (m2z − s − t) I1,2 + s I2,2 + s I3,2 − s I4,2 − s I5,2 − (m2z − s − t) I6,2 − (m2z − t) I7,2 ,
(4.2.21)
113
with the constituent integrals given in Fig. 4.4, is quite similar to the linear combination in
I1,1 : I2,1 : (k 2 − m2t )
I3,1 : I4,1 :
I5,1 : I6,1 :
I7,1 :
Figure 4.3: Integrals appearing in the linear combination in Eq. 4.2.20. I1,1 is the corner
integral of the topology under consideration. I2,1 is a second integral in the topology, but
with a numerator (k 2 − m2t ), where k is equal to the difference of the momenta of the edges
marked by the thin dashed lines. Integrals I3,1 , I4,1 , I5,1 , I6,1 , I7,1 belong to subtopologies. All
integrals are defined in d = 4 − 2 dimensions. Reproduced from [6].
Eq. 4.2.20. In fact, this linear combination can be generated simply by appending an addi-
tional inverse propagator k 2 −m2t to all the constituent integrals. Indeed it is straightforward
to see that additional numerators can be added to any finite linear combination keeping them
finite in the IR limit. However, adding numerators can lead to UV divergences; it is clear
in the current example that this is not the case though through power counting, but adding
additional numerators might lead to UV poles. One way to ensure UV convergence is to
consider integrals with dots. In fact another linear combination can be generated by just
adding a dot (on a specific propagator) in all integrals of the combination in Eq. 4.2.20. It
must be pointed out, though, that such linear combinations obtained by adding numerators
114
I1,2 : (k 2 − m2t ) I2,2 : (k 2 − m2t )2
I3,2 : (k 2 − m2t ) I4,2 : (k 2 − m2t )
I5,2 : (k 2 − m2t ) I6,2 : (k 2 − m2t )
I7,2 : (k 2 − m2t )
Figure 4.4: Integrals appearing in the linear combination in Eq. 4.2.21. I1,2 is the corner
integral of the topology under consideration but with a numerator (k 2 − m2t ), identical to
I2,1 from Eq. 4.2.20. I2,2 is I1,2 but with an extra numerator (k 2 − m2t ) where k is equal
to the difference of the momenta of the edges marked by the thin dashed lines. Integrals
I3,2 , I4,2 , I5,2 , I6,2 , I7,2 are the same as I3,1 , I4,1 , I5,1 , I6,1 , I7,1 but with an extra numerator (k 2 −
m2t ). All integrals are defined in d = 4 − 2 dimensions. Reproduced from [6].
or dots to other finite linear combinations aren’t the only possibilities. It was observed that
the number of finite linear combinations increases, in general, with the numerator rank.
4.2.4 Choice of finite integrals for gg → ZZ
The algorithm presented above expands significantly the choice of integrals available for a
basis of finite integrals. In principle, the amplitude can be expressed only in terms of such
finite linear combinations defined in d = 4 − 2 containing integrals with only additional
numerators. In fact, such a basis can be constructed for gg → ZZ where only finite linear
115
combinations are used to replace IR divergent integrals. There are, however, certain limi-
tations to this approach and in practice it was found to be useful to consider also integrals
with dimension-shifts and dots. This is largely due to the following reasons:
1. In certain cases, the corner integral of a topology already has a UV divergence. This
cannot be cured using a subtopology/numerator subtraction. In particular, UV diver-
gences arising from the Γ prefactor in Eq. 4.2.16 cannot be cured through a subtraction
since they only affect the endpoint divergences. Additionally, integrals from subtopolo-
gies or with numerators will in general have worse UV behaviour, unless an integral
with many dots is considered. It is possible to consider an integral from a super-
topology instead but this often leads to increase in complexity both analytically and
numerically, and as such is not desirable.
2. While finite linear combinations exist for higher numerator ranks and, in fact, the
number of available finite integrals grows with numerator rank, choosing such integrals
leads to extreme proliferation in the number of terms in the numerator polynomial
from Eq. 4.2.19. E.g. starting at numerator rank 1, the number of terms is O(10) e.g.
for the linear combination in Eq. 4.2.20. For the linear combinations with numerator
rank 2 this increases to O(1000) e.g. for the integral in Eq. 4.2.21 while for rank
3 this becomes O(100000). Note that these numbers depend on the topology under
consideration and in general the numerator polynomials would be simpler for lower
topologies. A significant disadvantage of this is that the pySecDec libraries required to
compute such integrals can reach sizes of ∼ 1 GB on the disk and are, hence, difficult
to compile on GPUs. Trying to condense the numerator polynomials results in the
appearance of spurious poles which worsen the numerical stability further.
3. As mentioned previously, finite linear combinations containing both numerators and
dots can also be constructed. This approach was explored to avoid the linear combi-
nations with high numerator rank. While these integrals, as long as numerator ranks
116
Integral Max. order in Rel. error Time(s)
Divergent corner integral (Fig. 4.2a) 0 ∼ 2 · 10−3 45
Divergent numerator integral (Fig. 4.2b) 0 ∼ 4 · 10−2 63
Finite integral in d = 6 − 2, (Fig. 4.2c) 1 ∼ 8 · 10−6 60
Finite integral in d = 6 − 2 with dot (Fig. 4.2d) 1 ∼ 8 · 10−4 55
Finite linear combination in Eq. 4.2.20 1 ∼ 1 · 10−4 18
Finite linear combination in Eq. 4.2.21 0 ∼ 5 · 10−4 150
Table 4.1: Numerical performance of different non-planar integrals for a physical phase-
space point. Timings generated with pySecDec [3] using the QMC algorithm [4, 5] on a single
Nvidia Tesla V100S GPU, with number of evaluations neval = 107 . Note that the divergent
integrals are only evaluated to O(0 ) since they start at −1 . Reproduced from [6].
are small, have more manageable numerator polynomials, their numerical performance
in the physical region of kinematics, i.e. the region where contour deformation is re-
quired, suffers drastically. This is largely due to the higher power of the F polynomial
in the denominator. In general, integrals with lower exponent of the F polynomial are
preferred.
Table 4.1 shows a comparison for the numerical performance for different divergent and
finite integrals from the non-planar topology in Fig. 4.2a. The first two orders in the expan-
sion are evaluated for most integrals except the finite linear combination in Eq. 4.2.21. From
the table it is clear that finite integrals show significantly improved numerical performance
compared to divergent integrals. The dimensionally-shifted finite integral in Fig. 4.2c has
the lowest exponent for the F polynomial, specifically the integrand ∼ 1/F; unsurprisingly,
it shows the best numerical performance. What is indeed surprising is that the finite linear
combination in Eq. 4.2.20, which has the integrand ∼ 1/F 3 has a numerical performance on
par with the dimension-shifted integral without dots and better than the dimension-shifted
integral with a dot (which has the integrand ∼ 1/F 2 . The linear combination with an
additional numerator (Eq. 4.2.21) also has the integrand ∼ 1/F 2 . However, the numerical
performance is worse than the "simpler" finite linear combination due largely to significantly
more complicated numerator polynomial and consequently much larger pySecDec libraries
117
which are harder to evaluate on GPUs.
At the end, the best numerical performance for gg → ZZ is achieved using a combination
of both finite linear combinations and dimensionally-shifted integrals. Finite linear combina-
tions with low numerator ranks are supplemented with dimensionally-shifted integrals with
low number of dots. To cure the UV poles, however, especially the ones appearing in lower
topologies such as the double tadpole, integrals with dots are used. This does not affect the
numerical performance though since the integrals are rather trivial to evaluate numerically
as well as calculating IBP reductions for them is rather easy. The full list of finite master
integrals is given in the ancillary files provided with [6].
It has been mentioned previously that choosing a canonical basis of master integrals
leads to a simpler form of the IBP relations, in general, over the generic choice of basis.
For gg → ZZ, the canonical basis is not known and is expected to be rather challenging
to find. In the absence of a canonical basis, a d-factoring basis of master integrals is used
to simplify the IBP reductions. That is, in addition to choosing the basis integrals to be
finite, they are chosen in such a way that the d-dependence of the denominators appearing in
the reduction identities factors out. Simply speaking, there are no irreducible denominator
factors appearing in the IBP identities that are simultaneously polynomials in both the
kinematic variables and d for the basis of integrals used for final numerical evaluation. This
was done using the code provided in [343] (see also [344]). It is also worth mentioning that
provided that a canonical basis is found, the structure of IBP reductions as well as the
amplitude can be simplified further by the use of a canonical basis that is also finite, referred
to as a uniform weight finite basis [345].
118
Chapter 5
Compiling the 2-loop amplitude
5.1 Inserting reductions into the amplitude
Even with the d-factoring basis of finite master integrals, the IBP identitites are extremely
complicated. For gg → ZZ, the IBP identities occupy over 200 GB of disk space before any
processing. This would be a nightmare to deal with, and hence several techniques are used
to simplify them.
5.1.1 Multivariate partial fractioning
The reduction identities are first processed through Fermat [246] which performs a GCD
(Greatest Common Divisor) calculation to simplify the coefficients. This already results in
a reduction in size by about a factor of 10-20 depending on the topology. The form obtained
from Fermat where all the terms are put over a common denominator, however, is not ideal
for numerical evaluation. In addition, further simplifications can be obtained by employing
multivariate partial fractioning on the reduction coefficients. Partial fraction decomposition
is simply the expansion of a rational function to a sum of simpler rational functions, or rather
rational decomposition of poles. E.g.
x 1 1
= + . (5.1.1)
x2 −1 2(x − 1) 2(x + 1)
The simple example above demonstrates how partial fraction decomposition can be used
to simplify expressions. However, naive approach to partial fractioning, for instance using
119
Mathematica, can lead to appearance of new denominators in the multivariate case. E.g.
x2 y −x2 x2 (x2 − 1)
= + . (5.1.2)
(x2 + y − 1)(y + 1) (x2 − 2)(y + 1) (x2 − 2)(x2 + y − 1)
In the above example, a new denominator factor x2 − 2 appears. For scattering amplitudes,
such spurious denominators that weren’t present before can cause problems. They can
cause severe numerical instabilities especially if they have poles in the physical kinematical
region. They can also make the final expressions more complicated. Instead, a mutivariate
partial fractioning procedure based on polynomial reductions with respect to a Gröbner
basis [277, 278, 346, 347] is used. In this approach more complicated polynomials can be
related to simpler polynomials where "simpler" is defined according to a polynomial ordering.
E.g. for the above example, assuming an ordering where variable y is preferred over x, and
the denominator 1/(1 + y) is preferred over 1/(−1 + y + x2 ), a polynomial reduction with
respect to the Gröbner basis using Singular [272] yields
x2 y
1 1
=−2
(x2 + y − 1)(y + 1) −1 + y + x2 1+y
1 1 1
− y+2 − + 1. (5.1.3)
−1 + y + x2 −1 + y + x2 1+y
which has a maximum degree of 1 in the numerator compared to 3 for the left-hand-side
expression. The result is also much simpler than the naive partial fraction result in Eq. 5.1.2
where even degree 4 terms are present in the numerator. Note that this decrease in degree
is not guaranteed for any arbitrary ordering.
This approach to multivariate partial fractioning is used to simplify the reduction iden-
tities for both gg → ZZ and γγ + jet production. In both cases, a reduction in disk size by
a factor of 100 is observed.
120
5.1.2 Backsubstitution of IBPs
For gg → ZZ, the partial fractioning is implemented using Singular with a polynomial
ordering that prefers polynomials with lower degrees in kinematic variables and smaller
coefficients [278]. As mentioned in Sec. 3.5, the reduction identities are first calculated in
the traditional Laporta basis with a generic ordering preferring master integrals with lower
numerator ranks. Partial fractioning these IBP identities reduces their size drastically and
makes the backsubstitution procedure a lot more manageable. Custom FORM scripts are then
used to backsubstitute the reduction identities into the form factors. Partial fractioning
is again performed on the resulting expressions which simplify drastically at this step; the
total size of amplitudes reduces from ∼300 GB to ∼600 MB. Note that for this step, first
partial fractioning in only the denominators dependent on d (including the denominators
that depend on both d and kinematics) is performed. In a second step, partial fractioning
is performed in denominators that depend only on the kinematic variables. This is simply
to facilitate the calculation of the Gröbner basis since the traditional Laporta basis is not
d-factoring i.e. there are irreducible denominator factors that are polynomials in d as well
as kinematic variables, e.g.
−75 + 25 d + 540 t − 180 t d − 972 t2 + 324 t2 d + 450 s − 90 s d − 972 s t + 324 s t d ,
where s, t are the Mandelstam variables defined in Eq. 2.1.9. Such denominators make the
Gröbner basis computation extremely challenging.
In the next step, a basis change is performed to express the form factors in terms of
the basis of finite integrals as chosen in Sec. 4.2.4; this basis is d-factoring. The basis
change identities are themselves partial fractioned and then backsubstituted into the reduced
amplitude in terms of the traditional Laporta basis. The form factors, which are now in the
d-factoring finite basis, are then partial fractioned, again in two steps. First, a partial
fractioning in only d-dependent denominators is performed. However, this step is simpler
121
than for the traditional Laporta basis since there are no complicated denominators depending
on kinematic variables involved. Since the amplitude is now expressed in a basis of finite
integrals and the 2-loop amplitude is required to be expanded only to O(0 ), the expressions
can be simplified drastically by setting d = 4 everywhere except the poles 1/(d − 4). Note
that this is possible only in the case of a d-factoring finite basis since after partial fractioning
in d, the poles get isolated. Finally, partial fractioning in kinematic variables is performed on
the resulting expressions to produce the final reduced amplitude. In the final representation,
the size of the worst coefficients is brought down to less than 1 MB. The list of all surviving
denominator factors are given in Appendix E. A C++ library is created for quick and efficient
numerical evaluation of the coefficients. All the coefficients for a generic point in phase space
can be evaluated within half a minute using rational arithmetic or within 3s using floating
point arithmetic with a target precision of 15 digits on a single CPU core.
A slightly different approach is used for γγ + jet. Since the integrals are already re-
duced to a canonical basis, there are no d-dependent denominators that also depend on
kinematic variables. However, the basis is not finite, so it is not possible to set d = 4 after
partial fractioning in d. In addition, the amplitude in this case involves many crossings of
the integrals. Performing a partial fraction decomposition for each crossing would be quite
cumbersome and computationally wasteful. Instead, the uncrossed IBP identities, after sim-
plifying through Fermat, are partial fractioned using MultivariateApart [278] and then the
crossings are applied. The crossed IBPs are then inserted into the amplitude and the result-
ing expressions are partial fractioned again. At the end, the canonical master integrals are
expanded in terms of the "Pentagon functions" defined in [311] and the resulting coefficients
of the pentagon functions are partial fractioned again. The final expressions, as expected,
are drastically simpler with the largest helicity coefficient for the most complicated colour
factor only 4.5 MB in size.
122
5.2 Renormalisation, IR subtraction and checks
5.2.1 UV renormalisation and IR subtraction
For gg → ZZ, the bare form factors Ai can be expanded perturbatively according to
αs,0 (1) αs,0 2 (2)
Ai = A + Ai + O(αs3 ) , (5.2.1)
2π i 2π
where αs,0 is the bare QCD coupling. Since the LO process for gg → ZZ already starts at
one loop, the two-loop process is effectively an NLO correction.
First, UV renormalisation of αs is performed in the 5-flavour MS scheme i.e. nf = 5
using
µ2R
αs,0 = αs S−1 Zαs , (5.2.2)
µ0 2
where S = (4π) e−γE , γE ≈ 0.577 is Euler’s constant, µR is the renormalisation scale, and
µ0 is the ’t Hooft scale introduced in the bare amplitude through conventional dimensional
regularisation. Note that the top-quark contribution to the gluon self energy is subtracted
at zero momentum [348].
The renormalisation constant Zαs in the above equation is given by
αs
Zαs = 1 + δZαs + O(αs2 ), (5.2.3)
2π
where
µ2R
1 1 2
δZαs = − β0 + TF . (5.2.4)
3 m2t
In the above equation, β0 is the first coefficient of the beta function expanded in αs , given
123
in Eq. 1.5.38,
11 CA − 4 TF nf
β0 = , (5.2.5)
6
and CA , CF are the quadratic Casimir invariants
N2 − 1
CA = N, CF = , (5.2.6)
2N
with TF = 21 .
The top-quark mass is renormalised in the on-shell scheme. The renormalised top-quark
mass is related to its bare mass according to
m2t,0 = m2t Zm , (5.2.7)
with the renormalisation constant defined as
2
αs 3 µR
Zm = 1 + δZm , δZm = CF − −4 . (5.2.8)
2π m2t
In principle, the top-quark mass in the amplitude can simply be replaced by the renormalised
mass in Eq. 5.2.7. However, in practice it is a lot more convenient to instead explicitly
subtract 1-loop counterterm diagrams (See Fig.).
1/2
Lastly, the gluon wave function is renormalised by multiplying the amplitude with ZG
for each external gluon, where the gluon renormalisation constant ZG is defined as
2
αs 2 µR
ZG = 1 + − TF 2
+ O(αs2 ) . (5.2.9)
2π 3 mt
This results in renormalised form factors given by
αs (1),ren αs 2 (2),ren
Aren
i = Ai + Ai + O(αs3 ). (5.2.10)
2π 2π
124
(a) Divergent diagram. (b) Mass counterterm diagram.
Figure 5.1: Mass counterterm diagrams required at 2-loops. The big dark cross in Fig. 5.1b
corresponds to the counterterm vertex insertion.
The renormalisation constants in the above equations are derived in Appendix D.
After UV renormalisation, IR subtraction must be performed to obtain finite results. The
IR structure of NLO amplitudes was first predicted by Catani in [14]. IR subtraction for
this calculation is performed using the “qT scheme” described in [349]. The finite remainders
resulting from IR subtraction are given by
(2),fin (2),ren (1),ren
Ai = Ai − Ai I(1) () . (5.2.11)
The I-operators in the “qT scheme” are given by
soft collinear
I(1) () = I(1) () + I(1) (), (5.2.12)
eγ µ2R
soft 1 iπ (0)
I(1) () = − + + δqT CA , (5.2.13)
Γ(1 − ) s 2
2
collinear µR β0
I(1) () = − , (5.2.14)
s
(0)
where δqT = 0. UV and IR finite form factors are then given by
αs (1),fin αs 2 (2),fin
Afini = A + Ai + O(αs3 ). (5.2.15)
2π i 2π
Note that the renormalisation scale µ2R = s for all the results presented in this work.
125
5.2.2 Checks of the calculation
Multiple checks are performed to establish the correctness of this calculation and results, as
described below:
1. The 2-loop form factors are explicitly checked to satisfy the identities in Eq. 2.1.16.
These identities hold at the level of reduced amplitude with fully symbolic kinematic
dependence.
2. The crossing relations in Eq. 2.1.17, on the other hand, are highly non-trivial to check
analytically. Instead, these relations are verified numerically for a phase-space point
within numerical precision.
3. All the finite linear combinations used in the basis are checked by numerically evalu-
ating them and comparing them against their explicit definitions in terms of divergent
integrals for a phase space point.
4. Algebraic cancellation of the spurious 1/4 pole is seen when the amplitude is expressed
in the chosen basis of finite integrals; the spurious 1/3 pole remains, however, for the
amplitude reduced in terms of symbolic master integrals. The 1/3 pole is verified to
vanish to 15 digits by calculating the amplitude numerically for a phase-space point in
the Euclidean region.
5. The 1/2 and 1/ poles are verified to match Catani’s IR formula [14] numerically, for
a phase-space point in the Euclidean region, to 9 digits for the double pole and 7 digits
for the single pole. Numbers are shown explicitly in Appendix F.
6. The 1/2 and 1/ poles are also verified to match Catani’s IR formula [14] for a point
in the physical region with explicit numbers shown in Appendix F.
7. The amplitude is also evaluated using an alternate finite basis constructed along the
same lines as the primary basis. The numerical results from the two bases are compared
126
and found to agree within expected numerical error. It must be pointed out that this
is a very stringent check on the correctness of the calculation since it simultaneously
checks the IBP reductions, the basis change from the primary to the alternate basis,
definitions of the finite integrals chosen, and their numerical evaluation. An error in
any one of these steps would imply disagreement between the two bases. Note that
the alternate basis, while composed of finite integrals, is numerically not as stable as
the primary basis and although it is useful for validating the calculation, it is generally
unsuitable for evaluation at a large number of phase-space points.
8. The axial-axial (a2t ) piece of the 2-loop amplitude is evaluated using the Kreimer’s
anti-commuting γ5 scheme [211, 212]. In addition, a separate numerical calcula-
tion is performed by a collaborator on this project, Stephen Jones, using Larin’s γ5
scheme [350, 351]. Agreement is found at the level of helicity amplitudes (but not for
the form factors Ai ) defined using Eq. 2.1.19 between the two schemes for a physical
as well as Euclidean phase space point within numerical precision. This provides an-
other strong check of the calculation since only physical gauge-invariant observables
are expected to agree for two different schemes.
9. Lastly, the calculation is compared against several known approximations in the lit-
erature, specifically the heavy top approximation [104–106] below the top-quark pair
production threshold and the small top-mass expansion [13] above the threshold (at
high center of mass energies). Agreement is found with both approximations in the
relevant regions of phase-space with the results of the comparison presented in Sec. 6.
127
Chapter 6
Results
6.1 Results for the 2-loop gg → ZZ amplitude
In this section, the results for the 2-loop amplitude for gg → ZZ are presented and compared
against several approximations. Before presenting the results however, some quantities rel-
evant for presentation of the results need to be defined. For most of the results presented
below, the helicity basis defined in Eq. 2.1.19 is used. Similar to the bare form factors, UV
renormalised and IR subtracted helicity amplitudes can be written as
µ ν ∗ρ ∗σ
Mfin fin
λ1 λ2 λ3 λ4 = Mµνρσ λ1 (p1 )λ2 (p2 )λ3 (p3 )λ4 (p4 ). (6.1.1)
where λ1 , λ2 are the helicities of the incoming gluons and λ3 , λ4 are the helicities of the
αs
outgoing Z-bosons. The helicity amplitudes can be expanded in 2π similar to form factors
in Eq. 5.2.15 as
α α 2
s (1) s (2)
Mfin
λ1 λ2 λ3 λ4 = Mλ1 λ2 λ3 λ4 + Mλ1 λ2 λ3 λ4 + O(αs3 ). (6.1.2)
2π 2π
(1)
It is useful to consider the squared 1-loop helicity amplitudes Vλ1 λ2 λ3 λ4 as well as the inter-
(2)
ference between the 1-loop and 2-loop helicity amplitudes Vλ1 λ2 λ3 λ4 , which can be defined
128
as
(1) ∗(1) (1)
Vλ1 λ2 λ3 λ4 = Mλ1 λ2 λ3 λ4 Mλ1 λ2 λ3 λ4 , (6.1.3)
(2) ∗(1) (2)
Vλ1 λ2 λ3 λ4 = 2 Re Mλ1 λ2 λ3 λ4 Mλ1 λ2 λ3 λ4 . (6.1.4)
Most of the results shown below are obtained by averaging over the helicities of the incoming
gluons as
(i) 1 X (i)
Vλ3 λ4 = V , (i = 1, 2) , (6.1.5)
4 λ ,λ λ1 λ2 λ3 λ4
1 2
with λ1 , λ2 ∈ {+, −}, and by summing over the helicities of the outgoing Z-bosons as
X (i)
V (i) = Vλ3 λ4 , (i = 1, 2) , (6.1.6)
λ3 ,λ4
(i)
with λ3 , λ4 ∈ {+, −, 0}. Note that the 1-loop amplitudes used in calculating Vλ1 λ2 λ3 λ4 are
just the top-quark contributions. Contributions from massless quarks are ignored for the
purpose of presentation of the numerical results below.
For the numerical results shown, the electroweak couplings are chosen as
GF = 1.1663787 · 10−5 GeV−2 ,
mZ = 91.1876 GeV , (6.1.7)
where the Fermi constant GF and Z boson mass mZ are fixed according to [1]. Since the
ratio of top-mass and Z-boson mass is fixed to m2Z /m2t = 5/18, inserting the above value
of mZ = 91.1876 GeV implies the top-mass value mt = 173.016 GeV. The W -boson mass is
fixed according to the ratio
m2W /m2t = 14/65 . (6.1.8)
129
This fixes the mass of the W -boson to mW = 80.296 GeV. For this calculation, the so-called
Gµ scheme is used i.e. GF , mZ , and mW are chosen as input quantities for the electroweak
p
parameters. The weak mixing angle is then fixed according to sin(θW ) = 1 − m2W /m2Z .
To numerically evaluate the master integrals, pySecDec [3, 5] is used. Sector decom-
position is applied and the integration is performed using the quasi-Monte Carlo (QMC)
algorithm [4, 352]. The QMC algorithm has been observed to show a much better con-
vergence compared to the traditional Monte-Carlo algorithms [337]. The different colour
structures appearing in the form factors (CF and CA ) as well as the vector-vector (vt2 ) and
axial-axial (a2t ) pieces are all evaluated separately. The target precision for each form factor,
for each colour structure and coupling type (vt2 or a2t ), is set to percent level. This is achieved
using a variant of the optimisation algorithm presented in [337]. The helicity amplitudes are
then calculated from the form factors.
Using the optimisation algorithm, requiring percent level precision on all form factors,
the time taken for numerical evaluation varies between 1.5-24 hours on 2 Nvidia Tesla v100
GPUs; this usually results in most form factors being calculated to per mille or better.
Almost the entire runtime is spent on calculating master integrals; the time taken to evaluate
the coefficients is negligible in comparison (see Sec. 5.1). The independent helicity amplitudes
for a phase-space point in the physical region are shown in Tab. 6.1
Fig. 6.1 shows a comparison of the interference term V (2) calculated in this work against
the large top-mass and the small top-mass expansions (using the analytic expressions for the
√
expansions provided in [13]) for varying s/mt and fixed value of cos(θ) = −0.1286, with the
scattering angle θ defined in Eq. 2.1.18. Also compared is the Padé improved small top-mass
expansion. The plot shows excellent agreement for this calculation with the expansion results
in the regions of phase-space where the approximation is expected to hold. For the smallest
√
value of s = 235 GeV shown on the plot, the large top-mass expansion agrees within 0.1%
with our result. Similar agreement to sub-per mille level is observed for the point with the
√
largest s = 878 GeV when compared to the small top-mass expansion as well as the Padé
130
(1) (2)
λ1 , λ2 , λ3 , λ4 Mλ1 λ2 λ3 λ4 (1-loop) Mλ1 λ2 λ3 λ4 (2-loop)
+ + ++ 0.1337854(1) − 0.0286060(1) i 3.15549(8) + 0.47235(8) i
+ + +− 0.0015573(1) + 0.0052282(1) i 0.15950(7) + 0.14052(8) i
+ − +− −0.01512820(8) − 0.01060416(8) i −0.38609(7) + 0.10539(7) i
− + ++ −0.0291599(1) − 0.0062178(1) i −0.46990(8) + 0.40207(8) i
+ + +0 0.0292668(5) + 0.0212966(5) i 1.1248(2) − 0.0805(2) i
+ − +0 −0.0643073(5) − 0.0459584(5) i −1.4803(2) + 0.4940(2) i
+ + 00 0.910006(2) + 1.132536(2) i 17.2585(6) + 29.5669(6) i
+ − 00 0.355092(2) + 0.404469(2) i 10.2869(5) − 1.0571(6) i
Table 6.1: 1-loop and 2-loop helicity amplitudes for gg → ZZ for the phase-space point
(1)
s/m2t = 142/17, t/m2t = −125/22, m2Z /m2t = 5/18, and mt = 1, with Mλ1 λ2 λ3 λ4
(2)
and Mλ1 λ2 λ3 λ4 defined in Eq. 6.1.2. Only the 8 independent helicity amplitudes (See
Eqs. 2.1.20, 2.1.21, and 2.1.22) are shown here. Note that these include only the top-quark
contributions from class A diagrams defined in Sec. 2.3.2. The numbers in parentheses denote
the uncertainty in the last digit. Reproduced from [6].
improved result. Also observed is the fact that the Padé approximation improves drastically
the agreement with our exact result compared to the regular small top-mass expansion. Note
√
that the regular small-mass expansion result is only visible for the two highest values of s
and diverges rapidly from the exact result for smaller values. The Padé improved result, on
√
the other hand, agrees to much lower energies closer to the s = 2mt threshold. In fact, this
plot also shows that the expansions considered here can reproduce the exact result within a
√
few percent for most values of s except for the region near the top-quark pair production
√
threshold s = 2mt , for the value of cos θ considered here.
In Fig. 6.2, a comparison of the interference term V (2) is shown for varying cos θ for
√
several fixed values of s. The large top-mass expansion changes very little with variation
in cos θ, as is clear from the top-left panel. On the other hand, the small top-mass expansion
diverges rapidly away from central scattering angles (cos θ ∼ 0) as seen in the bottom panel.
In fact, for the points far away from the centre, the small top-mass expansion is way off the
plotting range. This can be understood from the way the expansion is performed i.e. in
the limit m2Z m2t s, |t|, |u|. In this limit, |t| ∼ |u| ∼ s/2 for central scattering which
131
1.06
cos(θ) = −0.1286
1.04
1.02
Vexp /Vexact
(2)
1.00
(2)
0.98
Exact
1/m12
t
0.96 m32 4
t , mz
Padé
0.94
1 2 3 4 5
√
s/mt
√
Figure 6.1: Comparison of the s dependence of the unpolarised interference V (2) with
expansion for large and small top-quark mass [13] at fixed cos(θ) = −0.1286. The large
top-mass expansion is shown in colour red, the small top-mass expansion in blue, and the
Padé improved small top-mass expansion in purple. The exact result is shown in black. Note
that the error bars have been plotted for the exact result, they are too small to be visible
on the plot, however. Reproduced from [6].
justifies the expansion. However for back-to-back scattering (| cos θ| ∼ 1), |t| and |u| are no
longer guaranteed to be large compared to m2t and hence the approximation breaks down. It
must be pointed out that Padé approximation significantly improves the convergence of the
√
expansion. For the high energy point ( s = 814 GeV) in the bottom panel, the agreement
of our result with the Padé improved result is almost perfect. Even for the intermediate
√
energy point in the top-right panel ( s = 403 GeV), the agreement is generally within a few
percent close to the central scattering region.
Fig. 6.3 shows comparisons for specific outgoing helicities averaged over the incoming
gluon helicities as defined in Eq. 6.1.5 against the expansions. Both the 1-loop amplitude
(1) (2) √
squared Vλ3 λ4 and the 2-loop interference with 1-loop Vλ3 λ4 is shown for varying s with a
fixed scattering angle cos(θ) = −0.1286. Note that the expansions are only plotted for the 2-
132
1.010 √ 1.2 √
s/mt = 1.426 s/mt = 2.331
1.1
1.005
Vexp /Vexact Vexp /Vexact
(2) (2) 1.0
1.000
(2) (2)
0.9
0.995
0.8
Exact Exact
1/m12
t Padé
0.990 0.7
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
1.20 √
s/mt = 4.703
1.15
1.10
Vexp /Vexact
1.05
(2)
1.00
(2)
0.95
0.90
Exact
0.85 m32 4
t , mz
Padé
0.80
−1.0 −0.5 0.0 0.5 1.0
cos(θ)
Figure 6.2: Comparison of the cos(θ) dependence of the unpolarised interference V (2) with
√
the results expanded in the limit of large top-quark mass for s = 247 GeV (Top Left Panel)
√ √
and small top-quark mass for s = 403 GeV (Top Right Panel) and s = 814 GeV (Bottom
Panel). Reproduced from [6].
loop interference term. Looking at the plots, there is good agreement with the best available
expansions in the relevant regions. Unsurprisingly, the small top-mass expansion lies far
outside the plotting range for most points. The Padé improved expansion, however, agrees
√ √
well all the way down to lower s near the s = 2mt threshold. The level of agreement of
our result with the expansions greatly depends on the specific helicity configuration under
(2)
consideration. E.g. good agreement is seen for the V00 configuration while agreement for
(2) (2)
V+− and V+0 clearly starts to appear worse closer to the threshold. Note that the longitudinal
(2)
configuration V00 is dominant over the others even at the 2-loop level, similar to what was
(1)
observed for 1-loop in Sec. 2.2.3 as well as in the above plots. A rapid increase in V00 and
133
0.004 0.4
0.12 0.012
0.10 0.010
0.003 0.3
0.08 0.008
V+−
(2) V+−
(1) (2) V++ V++
(1)
0.06 0.002 0.2 0.006
V (2) V (2)
m−12
t m−12
t
0.04 0.004
m32 4
t , mz m32 4
t , mz
0.001 0.1
Padé Padé
0.02 0.002
V (1) V (1)
0.00 0.000 0.0 0.000
1 2 3
√ 4 5 1 2 3
√ 4 5
s/mt s/mt
300
0.025 8
0.4 250
0.020
6
200
0.3
0.015
(2)
V+0 (1)
V+0 (2)
V00 150 (1)
V00
4
0.2 V (2) V (2)
0.010
m−12
t 100 m−12
t
m32 4
t , mz m32 4
t , mz 2
0.1 Padé 0.005 50 Padé
V (1) V (1)
0.0 0.000 0 0
1 2 3
√ 4 5 1 2 3 4 5
√
s/mt s/mt
√
Figure 6.3: The s dependence of 1-loop and 2-loop interferences for polarised ZZ produc-
tion in gluon fusion at cos(θ) = −0.1286. Reproduced from [6].
134
0.0005
0.008 0.0004 0.020
0.0004
0.006 0.0003 0.015
V+−
(2) V+−
(1) V++
(2) 0.0003 V++
(1)
0.004 0.0002 0.010
0.0002
0.002 0.0001 0.005 0.0001
0.000 0.0000 0.000 0.0000
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
0.006 0.00030 0.175
0.004
0.005 0.00025 0.150
0.004 0.125 0.003
0.00020
(2)
V+0 (1)
V+0 (2)
V00 0.100 (1)
V00
0.003 0.00015 0.002
0.075
0.002 0.00010 V (2)
0.050
1/m−12
t 0.001
0.001 0.00005 0.025 V (1)
0.000 0.00000 0.000 0.000
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
Figure 6.4: The cos(θ) dependence of 1-loop and 2-loop interferences for polarised ZZ pro-
√
duction in gluon fusion at s/mt = 1.426. The large top-quark mass expansion [13] (to
order 1/m12
t ) is shown for comparison. Reproduced from [6].
(2) √
V00 past the top-quark pair production threshold s = 2mt is also seen, similar to what
was observed for 1-loop in Sec. 2.2.3.
Figs. 6.4, 6.5 and 6.6 show the comparisons for interference terms with specific helicity
√
configurations for varying cos θ and 3 different fixed values of s. Fig. 6.4 shows the com-
parison against large top-mass expansion. It is clear from the plot that there is very good
agreement for all helicity configurations. Note that in the chosen scaling, the 1-loop points
(2) (2)
almost lie on top of the 2-loop point with an almost complete overlap for V+− and V+0 .
Fig. 6.5 shows the comparison against the Padé improved small top-mass expansion for
intermediate energy. While in general the Padé approximation agrees well with our result,
there are some deviations, notably for cos θ away from 0 and sub-dominant helicities.
√
Fig. 6.6 shows the comparison plots for a high energy point with s = 814 GeV. The
Padé improved small top-mass expansion again shows excellent agreement with our result
for even larger values of | cos θ|, away from the center. This is a drastic improvement over
135
0.008
0.20 0.4 0.008
0.006
0.15 0.3 0.006
V+−
(2) V+−
(1) V++
(2) V++
(1)
0.004
0.10 0.2 0.004
0.05 0.002 0.1 0.002
0.00 0.000 0.0 0.000
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
0.25
25
0.007 0.5
0.20
0.006 20
0.4
0.15 0.005
(2) (1) (2) 15 (1)
V+0 0.004 V+0 V00 0.3 V00
0.10 0.003 10 0.2
V (2)
0.002
0.05 Padé
5 0.1
0.001 V (1)
0.00 0.000 0 0.0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
Figure 6.5: The cos(θ) dependence of 1-loop and 2-loop interferences for polarised ZZ pro-
√
duction in gluon fusion at s/mt = 2.331. The Padé improved small top-quark mass expan-
sion [13] is shown for comparison. Reproduced from [6].
the "regular" small top-mass expansion which is highly divergent away from the center.
It must be pointed out that the relative level of agreement shown in the above plots
between our exact result and the various expansions depends significantly on the exact
scheme used for IR subtraction. The comparisons shown in the above plots are done in the
“qT scheme” [349]. Conversion to Catani’s original subtraction scheme [14] can be performed
using
(1),fin,Catani (1),fin
Ai = Ai , (6.1.9)
(2),fin,Catani (2),fin (1),fin
Ai = Ai + ∆I1 Ai , (6.1.10)
where
1
∆I1 = − π 2 CA + iπβ0 , (6.1.11)
2
136
0.6 0.020
0.5 0.0150
0.5 0.0125
0.015 0.4
0.4 0.0100
V+−
(2) V+−
(1) V++
(2) 0.3 V++
(1)
0.3 0.010 0.0075
0.2
0.2 0.0050
0.005
0.1 0.1
0.0025
0.0 0.000 0.0 0.0000
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
0.035
0.8 300 8
0.030
250
0.025
0.6 6
0.020 200
(2)
V+0 (1)
V+0 (2) V00 (1)
V00
0.4 0.015 150 4
V (2)
0.010 100 m32 4
t , mz
0.2 Padé 2
0.005 50 V (1)
0.0 0.000 0 0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
Figure 6.6: The cos(θ) dependence of 1-loop and 2-loop interferences for polarised ZZ pro-
√
duction in gluon fusion at s/mt = 4.703. The small top-quark mass expansion (to order
m32
t ) and Padé improved expansion [13] are shown for comparison. Reproduced from [6].
Similar transformation rules apply for the helicity amplitudes. This implies that conversion
from the “qT scheme” to Catani’s original scheme requires subtraction of π 2 CA V (1) where the
factor of 2 compared to Eq. 6.1.11 comes from interference (see Eq. 6.1.4) and the iπβ0 term
is ignored since it does not contribute to the interference. This corresponds to a difference
of ∼ 30V (1) which can be as large as V (2) . Consequently, the 2-loop results can show a very
different qualitative behaviour compared to the 1-loop results when using Catani’s original
scheme as can be seen in Figs. 6.7, 6.8, 6.9, and 6.10. Note that the relative agreement
between our results and the expansions is visibly better in the “qT scheme” than in Catani’s
original scheme.
The above discussion implies that the comparisons shown above are largely scheme de-
pendent, at least looking at relative agreement. As such, to better understand the deviation
of the expansions from the exact result and estimate their impact on the physically relevant
observables, real radiation contributions also need to be taken into account.
137
0.004 50 8
0.005
0.000 40 6
0.003
−0.005 30
(2) V+− V+−
(1) (2)
V00 4 (1)
V00
0.002
−0.010 V (2)
20
m−12
t
−0.015 m32 4
t , mz 2
0.001
10 Padé
V (1)
−0.020
0.000 0 0
1 2 3
√ 4 5 1 2 3
√ 4 5
s/mt s/mt
√
Figure 6.7: The s dependence of 1-loop and 2-loop interferences for polarised ZZ pro-
duction in gluon fusion at cos(θ) = −0.1286. Here, the top left and bottom right panels of
Fig. 6.3 are reproduced using Catani’s original subtraction scheme [14]. Reproduced from [6].
0.000
0.06
0.004
0.0004
−0.001 0.05
0.003
0.0003 0.04
−0.002 (2) (1)
V+−
(2) V+−
(1) V00 V00
0.03 0.002
0.0002
−0.003
0.02 V (2)
0.0001 1/m−12
t 0.001
−0.004 0.01 V (1)
0.0000 0.00 0.000
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
Figure 6.8: The cos(θ) dependence of 1-loop and 2-loop interferences for polarised ZZ pro-
√
duction in gluon fusion at s/mt = 1.426. The large top-quark mass expansion [13] (to
order 1/m12
t ) is shown for comparison. Here, the top left and bottom right panels of Fig. 6.4
are reproduced using Catani’s original subtraction scheme [14]. Reproduced from [6].
138
0.000
10 0.6
0.010
−0.005 0.5
0.008 8
0.4
V+−
(2) −0.010 0.006 V+−
(1) (2) V00 6 (1)
V00
0.3
0.004 4
−0.015 V (2) 0.2
Padé
0.002 2 0.1
V (1)
−0.020
0.000 0 0.0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
Figure 6.9: The cos(θ) dependence of 1-loop and 2-loop interferences for polarised ZZ pro-
√
duction in gluon fusion at s/mt = 2.331. The Padé improved small top-quark mass expan-
sion [13] is shown for comparison. Here, the top left and bottom right panels of Fig. 6.5 are
reproduced using Catani’s original subtraction scheme [14]. Reproduced from [6].
70
0.020
0.0125 60 8
0.0100 0.015 50
6
0.0075 40
V+−
(2) (1) V+− (2)
V00 (1)
V00
0.010
0.0050 30 4
V (2)
20 m32 4
t , mz
0.0025 0.005 Padé 2
10 V (1)
0.0000
0.000 0 0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
cos(θ) cos(θ)
Figure 6.10: The cos(θ) dependence of 1-loop and 2-loop interferences for polarised ZZ
√
production in gluon fusion at s/mt = 4.703. The small top-quark mass expansion (to
order m32
t ) and Padé improved expansion [13] are shown for comparison. Here, the top
left and bottom right panels of Fig. 6.6 are reproduced using Catani’s original subtraction
scheme [14]. Reproduced from [6].
139
Chapter 7
Conclusions
In this work, the calculation of the 2-loop corrections to the process gg → ZZ with internal
top quarks has been presented. The exact dependence on the mass of the top quark has
been kept. The amplitude is represented in terms of a basis of finite master integrals which
are evaluated numerically.
To perform the very challenging Integration-By-Parts reductions of the Feynman inte-
grals, the method of syzygies is used. This is to avoid the introduction of integrals with
dots in the linear systems; a new algorithm to construct such syzygies using linear algebra
is employed. The resulting system of equations is then reduced using finite field techniques.
A new algorithm is presented to construct finite integrals as linear combinations with
the building blocks being Feynman integrals with numerators, higher powers of propagators,
dimensionally shifted integrals, and subsector integrals. The resulting parametric integrand
for such integrals is integrable for d = 4 and admits an expansion around d = 4 allowing
for numerical integration. This approach allows the construction of finite integrals consid-
ered more natural since the constituent integrals in the linear combinations appear in the
amplitudes already. To evaluate the master integrals numerically, pySecDec, based on the
method of sector decomposition, is employed. Choice of such finite integrals is observed to
significantly improve the numerical performance.
Results are provided for our 2-loop amplitudes along with comparisons against various
approximations. Good agreement is observed with the large top-mass and small top-mass
expansions in the regions where the approximations are expected to perform well. In compar-
ison to the regular small top-mass expansion, the Padé improved small top-mass expansion
140
is found to work over a significantly larger region of phase space, in particular for moderate
energies and/or non-central scattering. The amplitudes presented in this paper provide the
building block required to include the full top-quark mass effects in the next-to-leading order
cross section for ZZ production in gluon fusion.
141
APPENDICES
142
APPENDIX A
Feynman Rules
A.1 QCD Lagrangian
In this appendix, the Feynman rules relevant to this calculation are provided. The QCD
Lagrangian can be written as:
i 1 1
L = ψ (iγ µ Dµij − δ ij m)ψ j − Ga,µν Gaµν − (∂ µ Gaµ )2 + (∂ µ c∗a )Dµab cb , (A.1)
4 2ξ
with the following definitions:
- ψ i : Quark field with colour index i in fundamental representation
- Gaµν : Gluon field strength tensor with colour index a in the adjoint representation
- ca : Faddeev-Popov ghost field with colour index a in the adjoint representation
- Dµij = δ ij ∂µ − igs T a,ij Gaµ : Covariant derivative in the fundamental representation
- Dµab = δ ab ∂µ − gs f abc Gaµ : Covariant derivative in the adjoint representation
- T a,ij : SU(N) generator
- f abc : Structure functions
Splitting the Lagrangian into free and interacting part,
i 1 1
Lf ree = ψ (iγ µ ∂µ − m)ψ i − (∂ µ Ga,ν − ∂ ν Ga,µ )(∂µ Gaν − ∂ν Gaµ ) − (∂ µ Gaµ )(∂ ν Gaν )
4 2ξ
| {z }
Gauge fixing term
+ (∂ µ c∗a )(∂µ ca ) , (A.2)
| {z }
Ghost field
143
i a i g g2
Lint = gs T a,ij ψ G/ ψ − s f abc (∂ µ Ga,ν − ∂ ν Ga,µ )Gbµ Gcν + s f abe f cde Ga,µ Gb,ν Gcµ Gdν
| {z } |2 } |4
Fermion-Gauge term
{z {z }
3-point Gauge interaction 4-point Gauge interaction
− gs f abc (∂ µ c∗a )cb Gcµ . (A.3)
| {z }
Gauge-Ghost interaction
This allows us to read the QCD Feynman rules. Similar considerations give us the EW
Feynman rules.
A.2 Feynman rules
The following Feynman rules are used in this work:
QCD Propagators:
1. Quark
δij
i
p/ − m
2. Gluon
δ ab pµ p ν
µν
−i 2 g + (ξ − 1) 2
p p
3. Faddev-Popov Ghost
δab
i
p2
144
QCD Vertices:
1. Quark-gluon vertex
i gs Tija γ µ
2. Ghost-gluon vertex
gs f abc pµ
3. 3-gluon vertex
145
gs f abc (g µν (p1 − p2 )ρ + g νρ (p2 − p3 )µ + g µρ (p3 − p1 )ν )
4. 4-gluon vertex
−igs2 f abe f cde (g µρ g νλ − g µλ g νρ )
+f ace f bde (g µν g ρλ − g µλ g νρ )
ade bce µν ρλ µρ νλ
+f f (g g − g g )
Electroweak vertices relevant for gg → ZZ:
1. Higgs-top vertex
146
m
iδij
v
2. Z-quark vertex
e
i γµ (vt + at γ5 )
2 sin θW cos θW
3. Higgs-Z vertex
e
i mZ g µν
sin θW cos θW
147
A.3 SU(N) algebra
For the SU (N ) group, the Lie generators are defined using the commutation relation
[T a , T b ] = −Fbca T c , (A.4)
where the structure constants Fbca themselves satisfy the commutation relation
a b
[Fde , Fef ] = −Fbca Fdfc . (A.5)
T a are the generators in the fundamental representation while Fbca are the generators in the
adjoint representation. The generators T a are normalised according to
T r T a T b = TF δ ab
(A.6)
where TF = 1/2 by convention. The quadratic Casimir Cr invariant for the representation r
is defined as
T a T a = C r 1n (A.7)
with n being the dimensions of the representation r of T a . For the fundamental representa-
tion,
N2 − 1
CF = . (A.8)
2N
This is easily derived by taking the trace of Eq. A.7.
The generators in the adjoint representation can be written in terms of the generators in
148
fundamental representation as
Fbca = −2 Tr [T a , T b ] T c .
(A.9)
This can be used, along with the Fierz identity
1 1
Tija Tkla = δil δjk − δij δkl , (A.10)
2 N
to write down trace relation for adjoint representation similar to Eq. A.6:
T r F a F b = N δ ab .
(A.11)
The Casimir invariant in the adjoint representation is
CA = N (A.12)
derived simply by taking the trace of F a F a = CA 1n with n = N 2 − 1 and Eq. A.11.
An explicit representation for the generators T a can be written in terms of the Gell-Mann
149
matrices T a = λa /2 where the Gell-Mann matrices are
0 1 0 0 −i 0 1 0 0
λ1 = 1 0 0 ,
λ2 = i 0 0 ,
λ3 = 0 −1 0 ,
0 0 0 0 0 0 0 0 0
0 0 1 0 0 −i 0 0 0
λ4 = 0 0 0 ,
λ5 = 0 0 0 ,
λ6 = 0 0 1 ,
1 0 0 i 0 0 0 1 0
0 0 0 1 0 0
1
λ7 = 0 0 −i , λ8 = √ 0 1 0 .
3
0 i 0 0 0 −2
The adjoint generators F a can be written in terms of the totally antisymmetric structure
constants f abc
Fbca = −i f abc . (A.13)
Explicit values of f abc for a given representation can be determined using Eq. A.9.
A.4 Colour factors of some simple diagrams
We obtain:
C = Tika δkl Tljb δ ab = Tila Tlja = CF δij
150
C = Tika δkl Tljb δji = Tila Tlib = T r(T a T b ) = TF δ ab
C = f acd f bcd = T r F a F b = CA δ ab
C = f ace f bce = T r F a F b = CA δ ab
At 2-loops, considering diagrams with 1 closed fermion loop:
C = f acd f bcd × TF = T r F a F b TF = TF CA δ ab
C = T r T a T c T d T b δ cd = T r T a T c T c T b
= T r T a T b CF = TF CF δ ab
151
C = T r T a T c T b T d δ cd = T r T a T c T b T c
1 1
=− TF δ = TF CF − CA δ ab
ab
2N 2
1 ab
C = T r T b T c T d f acd = i
δ
4N
1
= − i TF CF − CA δ ab
2
152
APPENDIX B
Evaluation using Feynman parameters
In this Appendix, a few simple examples of direct integration using Feynman parametric
representation are shown.
B.1 Massive tadpole
Consider the integral in Minkowski space
dd k 1
Z
d/2
. (B.1)
iπ k − m2 + i0
2
This integrand is dependent on k 2 = k02 − k12 − k22 − k32 . To perform the integration, Wick
rotation (Fig. 7.1) can be used to transform Minkowski space into Euclidean space using
k0 → ik0,E with spatial components remaining unchanged. Thus, the integral over k0 can be
replaced by the integral over k0,E as
Z ∞ Z ∞
dk0 F (k0 ) → i dk0,E F (ik0,E ) . (B.2)
−∞ −∞
This is easy to see using Cauchy’s theorem; the integrals over the arcs in Fig. 7.1 can be
shown to vanish near infinity leaving the above relation. Note that the i0 prescription is
essential to ensure that the poles lie outside the integration contour (see Fig. 7.1). The
integral over loop momentum then becomes
dd k 1 dd kE 1
Z Z
d/2
=i d/2 2
(B.3)
iπ k − m2 + i0
2 iπ −kE − m2 + i0
153
Figure 7.1: The integration contour to perform Wick rotation. Note that the poles lie outside
the contour.
where kE2 = k0,E
2 2
+ k1,E 2
+ k2,E 2
+ k3,E . The d dimensional Euclidean integral above can be
written in spherical coordinates using
dd kE = kEd−1 dkE dΩd . (B.4)
Integral over the d dimensional solid angle gives
d−2 Z
!Z
π 2π
2π d/2
Z Y
dΩd = dθi sind−1−i θi dθd−1 = . (B.5)
i=1 0 0 Γ(d/2)
The remaining integral in kE is straightforward:
∞
1 1 d−2 d d
Z
kEd−1 dkE 2 2
= m Γ 1− Γ . (B.6)
0 kE + m 2 2 2
Combining the two, the integral gives
154
dd k
1 d
Z
d−2
= −m Γ 1 − . (B.7)
iπ d/2 k 2 − m2 + i0 2
Note that the integral converges only for Re (d) < 2. Using d = 4 − 2 and expanding around
= 0,
dd k 1 1
Z
d/2
= m2 + m2 (1 − γE − log m2 ) + O() (B.8)
iπ k − m + i0
2 2
where γE is the Euler-Mascheroni constant. It is common to remove these factors of γE
appearing in expansions of Feynman integrals by multiplying the integrals with eγE . Note
that as expected, the divergence shows up as the 1/ pole.
Evaluating the tadpole integral with the propagator raised to power 3 instead gives
dd k
1 1 d
Z
d−6
= − (d − 4)(d − 2)m Γ 1 − (B.9)
iπ d/2 (k 2 − m2 + i0)3 8 2
which when expanded gives
dd k 1 1 1
Z
= − + (γE + log m2 ) + O(2 ) . (B.10)
iπ d/2 (k 2 − m2 + i0)3 2m2 2m2
Simple power counting shows that this integral should be finite in d = 4 as seen above.
B.2 Massless bubble
The massless bubble integral from Fig. 3.3 (without cuts) can be written as
dd k 1
Z
γE
I=e d/2
. (B.11)
iπ (k + i0)((k − p)2 + i0)
2
155
Applying Feynman’s trick, this becomes
1 1
dd k eγE Γ(2) dd k eγE Γ(2)
Z Z Z Z
I= dx = dx .
0 iπ d/2 ((1 − x)k 2 + x(k − p))2 0 iπ d/2 (k 2 + 2xk.p + xp2 + i0)2
(B.12)
Completing the square (k + xp)2 gives
1 1
dd k eγE Γ(2) dd k eγE Γ(2)
Z Z Z Z
I= dx = dx (B.13)
0 iπ d/2 ((k + xp)2 − (∆ − i0))2 0 iπ d/2 (k 2 − ∆ + i0))2
where ∆ = −p2 x(1 − x). Integrating over the loop momentum, we get
1
1
Z
γE
I =e Γ(2)Γ(2 − d/2) dx 4−d
0 (∆ − i0) 2
1
1
Z
γE
=e Γ(2)Γ(2 − d/2) dx 4−d 4−d . (B.14)
0 (−p2 − i0) 2 (x − x2 ) 2
The integral over x converges for Re(d) > 2. Integrating, and expanding in , we get
1
+ 2 − log(−p2 − i0) + O() .
I= (B.15)
156
APPENDIX C
Dirac algebra and γ 5 schemes
C.1 Gamma matrices and identities
Fermions in a free theory satisfy the Dirac equation
(iγ µ ∂µ − m) ψ = 0 . (C.1)
γ µ are referred to as the Dirac or Gamma matrices. They generate a Clifford algebra and
satisfy the following anti-commutation relations
{γ µ , γ ν }+ = 2η µν 1d . (C.2)
where 1d is the d-dimensional identity matrix and η µν is the metric tensor for Minkowski
space. An explicit representation can be written for the Gamma matrices in 4 dimensions
in the Dirac representation
12 0
γ0 = , (C.3)
0 12
i
0 σ
γi = , (C.4)
−σi 0
157
for i = 1, 2, 3. Here σ i are the Pauli matrices:
0 1
σ1 = ,
1 0
0 −i
σ2 = ,
i 0
1 0
σ3 = . (C.5)
0 −1
The Gamma matrices satisfy the following identities in d-dimensions following from the
anti-commutation rule in Eq. C.2
γ µ γµ = d 1d , (C.6)
γ µ γ ν γµ = (2 − d)γ ν , (C.7)
γ µ γ ν γρ γµ = 2 η νρ − (2 − d)γ ρ γ ν , (C.8)
as well as the following trace identities
T r(γ µ ) = 0 , (C.9)
T r(γ µ γ ν ) = 4η µν , (C.10)
T r(γ µ1 ...γ µm ) = T r(γ µm ...γ µ1 ) , (C.11)
T r(γ µ1 ...γ µ2m+1 ) = 0 , (C.12)
X
T r(γ µ1 ...γ µ2m ) = 4 Sgn(σ) gµi1 µj1 ...gµin µjn (C.13)
perms
158
with 1 = i1 < ... < in and ik < jk . σ refers to permutation of the indices i1 , j1 , ..., in jn and
Sgn(σ) is the sign of the permutation.
C.2 γ 5 in d dimensions
In d = 4 dimensions, we can define a fifth Gamma matrix γ 5 as
γ 5 = iγ 0 γ 1 γ 2 γ 3 . (C.14)
An explicit representation in the Dirac basis is given by
0 12
γ5 = . (C.15)
12 0
γ 5 can also be defined using the totally anti-symmetric Levi-Civita tensor µνρσ
i µνρσ
γ5 = γµ γν γρ γσ , (C.16)
4!
which leads to the trace identity
T r(γ µ γ ν γ ρ γ σ ) = −4iµνρσ (C.17)
in d = 4. An anti-commutation relation follows from Eq. C.2 and C.14,
{γ µ , γ 5 }+ = 0 , (C.18)
as well as the identity
(γ 5 )2 = 14 . (C.19)
159
Figure 7.2: The triangle anomaly graph. Here the dark blob vertex represents the axial-
vector coupling.
Since the anti-symmetric tensor µνρσ is a purely 4-dimensional object, γ 5 and, as a
consequence, the trace identity in Eq. C.17 are not defined for d 6= 4. Dimensional regulari-
sation is one of the most popular methods to perform higher-order calculations. It preserves
gauge invariance which makes it highly attractive and simpler to implement. However, for
processes involving parity violating interactions, the naive regularisation scheme with the
4-dimensional γ 5 cannot be used.
There are several methods to "extend" the trace relation of Eq. C.17 to d dimensions.
For a consistent definition of γ 5 in d-dimensions, both the cyclicity of the trace and Eq. C.18
cannot be preserved [153, 154]. This can be seen most easily through the triangle anomaly
graph in Fig. 7.2. The sum of the above diagram and its crossed diagram, where the
incoming legs with momenta p1 , p2 are exchanged, can be written as
dd k T r(γ ρ γ 5 k/γ µ (k/ + p/1 )γ ν (k/ + p/1 + p/2 ))
Z
µνρ
I = + [p1 ↔ p2 , µ ↔ ν] . (C.20)
iπ d/2 k 2 (k + p1 )2 (k + p1 + p2 )2
It is straightforward to show that assuming cyclicity of trace along with the anti-commuting
γ 5 (Eq. C.18) implies that the above integral, vanishes. This is of course incorrect and
demonstrates the erroneous approach to evaluating the anomaly. To appropriately deal with
this issue, there are two major classes of schemes:
1. Schemes where the anti-commutation relation in Eq. C.18 is violated in favour of
160
preserving cyclicity of trace e.g. the HVBM scheme [153, 214–216] and the Larin scheme [350,
351].
2. Schemes where the anti-commutation relation in Eq. C.18 is preserved while
cyclicity of the trace is violated [211–213]
C.3 Anti-commuting γ 5 scheme
In Kreimer’s anti-commuting γ 5 scheme [211–213], the cyclicity of traces is not preserved.
This also implies that the traces need to be read from a specific point and the result in
general depends on the "reading point", which is chosen to be the axial-vector vertex so as
to conserve vector currents. All diagrams, and their traces, must be read from the same
reading point and in case of multiple γ 5 ’s, the traces need to be symmetrised.
Assuming that trace is not cyclic, the triangle anomaly above can be calculated using
the following trace identities concerning γ 5 in addition to the trace identities in the previous
section for the "regular" γ-matrices:
T r(γ 5 ) = 0 , (C.21)
T r(γ µ1 ...γ µ2m+1 γ 5 ) = 0 , (C.22)
T r(γ µ γ ν γ 5 ) = 0 , (C.23)
T r(γ µ1 ...γ µ4 γ 5 ) = 4iµ1 µ2 µ3 µ4 , (C.24)
T r(γ µ1 ...γ µm γ 5 ) = T r(γ µm ...γ µ1 γ 5 ) , (C.25)
X
T r(γ µ1 ...γ µ2m γ 5 ) = 4i Sgn(σ)µin+1 µin+2 µjn+1 µjn+2 gµi1 µj1 ...gµin+2 µjn+2 (C.26)
σ
with 1 = i1 < ... < in+2 and ik < jk . σ refers to permutation of the indices i1 , j1 , ..., in+2 jn+2
and Sgn(σ) is the sign of the permutation. Note that any contractions with the -tensor
are treated in a 4-dimensional subspace of the d-dimensional space since the -tensor is a
4-dimensional object.
Using the above trace rules, along with the requirement of reading traces from the same
161
vertex in each diagram, the correct result for the anomalous triangle diagram can be recov-
ered.
162
APPENDIX D
UV renormalisation
D.1 Renormalised Lagrangian
The bare Lagrangian can be written as
i 1 1 µ a
Lbare = ψ 0 (iγ µ ∂µ − m0 )ψ0i − (∂ µ Ga,ν ν a,µ a
0 − ∂ G0 )(∂µ G0,ν − ∂ν G0,µ ) −
a
(∂ Gµ,0 )(∂ ν Gaν,0 )
4 2ξ
a,ij i µ a gS,0 abc µ a,ν
+ (∂ µ c∗a a
0 )(∂µ c0 ) + gS,0 T ψ 0 γ G0,µ ψ0i − f (∂ G0 − ∂ ν Ga,µ b c
0 )G0,µ G0,ν
2
2
gS,0
+ f abe f cde Ga,µ b,ν c d
0 G0 G0,µ G0,ν − gS,0 f
abc µ ∗a b c
(∂ c0 )c0 G0,µ (D.1)
4
where all the bare parameters and fields are denoted by the subscript "0" e.g. Ga0,µ . The
bare parameters can be replaced by the renormalised parameters:
p
ψ0 = Zψ ψR
Ga,µ ZG Ga,µ
p
0 = R
p
ca0 = Zc caR
m0 = Zm mR
gS,0 = Zg gS,R
αS,0 = ZαS αS,R
ξ0 = Zξ ξ (D.2)
where Zψ etc. are referred to as renormalisation constants. In what follows, the R sub-
script has been dropped for the sake of brevity and any quantity without the subscript 0 is
understood to be renormalised.
163
The renormalisation constants can be expanded order-by-order in αS /(2π) e.g.
αS
Zψ = 1 + δZψ + O(αS2 ) (D.3)
2π
keeping terms only up to O(αS ). For brevity, it is useful to define
δψ = αS /(2π) δZψ , δG = αS /(2π) δZG , δc = αS /(2π) δZc
δm = αS /(2π) δZm , δg = αS /(2π) δZg , δαS = αS /(2π) δZαS . (D.4)
Replacing the bare parameters in the Lagrangian with the renormalised parameters and
expanding the renormalisation constants gives
L = Lren + Lct (D.5)
where
i 1 1
Lren = ψ (iγ µ ∂µ − m)ψ i − (∂ µ Ga,ν − ∂ ν Ga,µ )(∂µ Gaν − ∂ν Gaµ ) − (∂ µ Gaµ )(∂ ν Gaν )
4 2ξ
i g S
+ (∂ µ c∗a )(∂µ ca ) + gS T a,ij ψ γ µ Gaµ ψ i − f abc (∂ µ Ga,ν − ∂ ν Ga,µ )Gbµ Gcν
2
gS2 abe cde a,µ b,ν c d
+ f f G G Gµ Gν − gS f abc (∂ µ c∗a )cb Gcµ ,
4
i 1
Lct = ψ (i(δψ )γ µ ∂µ − (δψ + δm )m)ψ i − δG (∂ µ Ga,ν − ∂ ν Ga,µ )(∂µ Gaν − ∂ν Gaµ )
4
1 i
− δG (∂ µ Gaµ )(∂ ν Gaν ) + δc (∂ µ c∗a )(∂µ ca ) + (δψ + δg + 1/2δG )gS T a,ij ψ γ µ Gaµ ψ i
2ξ
gS g2
− (δg + 3/2δG ) f abc (∂ µ Ga,ν − ∂ ν Ga,µ )Gbµ Gcν + 2(δg + δG ) S f abe f cde Ga,µ Gb,ν Gcµ Gdν
2 4
− (δg + δc + 1/2δG ) gS f abc (∂ µ c∗a )cb Gcµ . (D.6)
D.2 1-loop counterterms
From the above Lagrangian, counterterm diagrams can be written as follows:
164
iδG δ ab pµ pν − g µν p2
iδij δψ p/ − (δψ + δm )m
iδc δ ab p2
1
i(δψ + δg + δG ) gS Tija γµ
2
3 abc
(δg + δG ) gs f g µν (p1 − p2 )ρ
2
νρ µ µρ ν
+ g (p2 − p3 ) + g (p3 − p1 )
165
−i(2δg + 2δG ) gs2 f abe f cde (g µρ g νλ − g µλ g νρ )
+f ace f bde (g µν g ρλ − g µλ g νρ )
ade bce µν ρλ µρ νλ
+f f (g g −g g )
1
(δc + δg + δG ) gs f abc pµ γµ
2
The counterterms can be determined by requiring that the divergences in the 1-loop
contributions are cancelled by the counterterm vertices. E.g. for gluon field renormalisation,
all the diagrams contributing to the 2-point function at 1-loop and the 1-loop counterterm
must sum up to be finite as shown in Fig. 7.3. Note that in principle, any arbitrary finite term
can be added to the counterterm without affecting the divergent piece. This allows a lot of
freedom in choosing the exact form of the counterterms. There are multiple renormalisation
"schemes" that differ in the choice of these finite terms. The most popular is the so-called
modified minimal subtraction (M S) scheme. In this scheme, an additional −γE + ln 4π is
added to the counterterm.
Consider the 1-loop corrections to the gluon propagator in Fig. 7.3. The sum of all the
166
Figure 7.3: All the diagrams contributing to 1-loop correction to the gluon propagator,
including the counterterm diagram. Requiring that the sum is finite allows the calculation
of the counterterm δG .
contributions to the gluon self-energy can be written as
Σ(p) = nf Σf (p) + nh Σh (p) + Σc (p) + ΣG (p) + Σct (p) (D.7)
where Σf (p) is the contribution from light quarks, Σh (p) from heavy quarks, Σc (p) from ghost
fields, ΣG (p) from gluon self-interactions, and Σct (p) from the counterterm. For example,
the contribution from massless quarks can be written as
dd k T r k/γ µ (k/ + p/)γ ν
1 g2
Z
Σf (p) = (4πµ2 )2−d/2 s 2 δ ab , (D.8)
2 16π (iπ d/2 ) k 2 (k + p)2
167
with a factor of µ4−d added to render it dimensionless. The integral above is straightforward
to evaluate and yields
αS ab µ ν 1
Σf (p) = δ (p p − g µν p2 ) 0
− γE + ln 4π + O( ) . (D.9)
6π
Rest of the integrals can be evaluated similarly yielding, for the counterterm,
nh
13 − 3ξ µ2
2nf + 2nh 1 2 X
δG = C A − TF − γE + ln 4π − TF ln R2 (D.10)
12 3 3 i
mi
with the heavy quark diagrams subtracted at zero momentum. Note that the longitudinal
part of the gluon propagator does not receive any corrections. Ghost propagator counterterm
can be calculated similarly:
3−ξ
1
δc = −CA − γE + ln 4π . (D.11)
8
Above counterterms were calculated using the M S scheme. For massive quarks, the
on-shell renormalisation scheme is commonly used. In this scheme, the renormalisation
conditions are set as
d
δψ = − Σ(p)
dp/ p
/=mp
1
δm = Σ(mp ) . (D.12)
mp
Evaluating Σ(p) from the 1-loop quark self-energy diagram yields
2
1 3 −γE µR
δψ = δm = − CF + 4 + 3 ln 4πe . (D.13)
2 m2p
With δψ , δG , δc , δm known, δg can be calculated using the 3-point gluon vertex. In Feyn-
168
man gauge
1 1 1
δψ + δg + δG = − (CF + CA ) − γE + ln 4π . (D.14)
2 2
In the M S scheme, δg is
nh
11CA − 4(nf + nh )TF µ2
1 1 X
δg = − − γE + ln 4π + TF ln R2 . (D.15)
12 3 i
mi
D.3 QCD β-function
From the above result, considering only the top-quark as the heavy particle,
2
αS 1 1 2 −γE µR
ZαS =1 + 2δg = 1 + − β0 − γE + ln 4π + TF + ln 4πe
2π 3 m2t
(D.16)
where
11CA − 4nf TF
β0 = (D.17)
6
is the beta function at 1-loop.
169
APPENDIX E
List of denominators
In this appendix, the denominator factors which occur in the coefficients of the master
integrals in the helicity amplitudes for gg → ZZ are listed. In the calculation, the masses of
the top quark and the Z boson, mt and mZ , have been set to mt = 1 and (mz /mt )2 = 5/18.
The exact set of denominator factors depends on the choice of master integrals. In this work,
the basis is chosen such that the dependence on the space-time dimension d factorises from
that of the kinematic invariants s = (p1 + p2 )2 and t = (p1 − p3 )2 .
We find 9 d-dependent denominators
d − 5, d − 4, d − 3, d − 2, d, 2d − 7, 3d − 10, 3d − 8, 5d + 52
and the 48 d-independent denominators sorted according to the polynomial ordering used
for partial fractioning.
8 polynomials dependent only on s:
s, −4 + s, −5 + 9 s, −10 + 9 s, −5 + 18 s, 134 + 9 s, −335 + 324 s,
90 − 245 s + 324 s2 .
6 polynomials dependent only on t:
t, −4 + t, 10 + 13 t, −5 + 18 t, 5 + 18 t, 5 + 31 t .
170
9 degree 1 polynomials in s, t:
− 5 + 6 s + 6 t, −5 + 9 s + 9 t, 31 + 9 s + 9 t, −5 + 9 s + 18 t, −5 − 18 s + 18 t ,
− 5 + 18 s + 18 t, −5 + 36 s + 18 t, −155 + 117 s + 117 t, −200 + 279 s + 279 t .
13 degree 2 polynomials in s, t:
25 − 90 t + 324 s t, 25 + 180 s − 90 t + 558 s t, 5 s − 5 t + 18 s t + 18 t2
25 − 135 t + 162 s t + 162 t2 , 25 − 180 t + 324 s t + 324 t2 ,
25 − 180 t + 648 s t + 324 t2 , 25 − 270 s − 90 t + 324 s2 + 324 s t,
25 − 580 s − 90 t + 558 s2 + 558 s t, 25 + 180 s − 180 t − 324 s2 + 324 t2 ,
25 − 1296 s − 180 t + 324 s t + 324 t2 , 25 − 25 s − 180 t + 90 s2 + 90 s t + 324 t2
25 − 155 s − 180 t + 324 s2 + 558 s t + 324 t2 ,
3350 − 6030 s − 1675 t + 2916 s2 + 3015 s t + 3015 t2 .
2 degree 3 polynomials in s, t:
− 25 + 180 t − 324 s t − 324 t2 + 81 s t2 ,
− 25 + 25 s + 180 t − 90 s2 − 414 s t − 324 t2 + 81 s3 + 162 s2 t + 81 s t2 .
171
8 degree 4 polynomials in s, t:
400 − 3080 s − 2880 t + 4489 s2 + 11808 s t + 5184 t2 − 2772 s2 t − 2592 s t2 + 324 s2 t2 ,
400 − 200 s − 2880 t − 695 s2 + 1440 s t + 5184 t2 − 180 s3 − 2772 s2 t − 2592 s t2 + 324 s4
+ 648 s3 t + 324 s2 t2 ,
112225 − 997920 s − 808020 t + 1679616 s2 + 3841992 s t + 1454436 t2 − 839808 s2 t
− 898128 s t2 + 104976 s2 t2 ,
112225 + 51300 s − 808020 t − 144180 s2 + 64800 s t + 1454436 t2 − 174960 s3 − 1073088 s2 t
− 898128 s t2 + 104976 s4 + 209952 s3 t + 104976 s2 t2 ,
3125 + 173250 s − 45000 t + 145800 s2 − 1206900 s t + 243000 t2 + 3149280 s2 t
+ 1953720 s t2 − 583200 t3 + 1889568 s2 t2 + 524880 s t3 + 524880 t4 ,
− 8375 − 23400 s + 88200 t − 32400 s2 + 68040 s t − 301320 t2 + 29160 s3 − 244944 s2 t
+ 29160 s t2 + 303264 t3 − 104976 s3 t − 104976 s2 t2 + 104976 s t3 + 104976 t4 ,
+ 9625 − 64800 s − 106200 t + 599400 s t + 398520 t2 − 839808 s2 t − 1405512 s t2
− 536544 t3 + 209952 s2 t2 + 314928 s t3 + 104976 t4 ,
− 2750 + 1800 s + 22725 t + 2025 s2 − 52650 s t − 56700 t2 + 26973 s2 t + 64881 s t2
+ 37908 t3 + 13122 s3 t + 26244 s2 t2 + 13122 s t3 .
172
2 degree 6 polynomials in s, t:
+ 2500 − 18000 t + 64800 s t + 27900 t2 − 249480 s t2 + 32400 t3 + 419904 s2 t2
+ 116640 s t3 − 56295 t4 − 209952 s2 t3 − 224532 s t4 − 14580 t5 + 26244 s2 t4
+ 52488 s t5 + 26244 t6 ,
+ 105625 − 3676500 s − 468000 t + 25891650 s2 + 17309700 s t − 797850 t2 − 50490540 s3
− 73614420 s2 t − 19260180 s t2 + 3863700 t3 + 29452329 s4 + 80752788 s3 t
+ 75149694 s2 t2 + 25850340 s t3 + 2001105 t4 − 18187092 s4 t − 60466176 s3 t2
− 72275976 s2 t3 − 35901792 s t4 − 5904900 t5 + 2125764 s4 t2 + 8503056 s3 t3
+ 12754584 s2 t4 + 8503056 s t5 + 2125764 t6 .
Note that during the initial IBP reduction to the traditional Laporta basis, spurious
denominators with mixed dependence on d and kinematic invariants were indeed introduced.
Rotation to the current basis followed by the partial fractioning approach described in Sec. 5.1
allowed us to systematically eliminate those denominators.
173
APPENDIX F
Numerical checks
In this appendix, details of the pole cancellation are provided. The amplitudes are
evaluated in the chosen basis of finite integrals using Kreimer’s anti-commuting γ5 scheme.
For an L loop amplitude, at worst 1/2L could appear. However, since this process is only an
NLO correction, only the 1/2 and 1/ poles should remain before UV renormalisation and
IR subtraction. Indeed, for the spurious 1/4 and 1/3 poles, analytical and high precision
numerical cancellations are seen, respectively. For the Euclidean point with s/m2t = −191,
t/m2t = −337, m2Z /m2t = −853, mt = 1, 15 digits of cancellation is seen for the 1/3 pole
while 8 digit cancellation is seen for the point in the physical region: s/m2t = 142/17,
t/m2t = −125/22, m2Z /m2t = 5/18, mt = 1.
The tables below show our results for the 1/2 and 1/ poles, and the 0 term of the
UV renormalised form factors Ai , and compare them against the predicted IR poles as in
Eq. 5.2.13 and 5.2.14, for the Euclidean (Tab. 7.1) and the physical (Tab. 7.2) point. The
digits in parentheses for the 0 term denote the uncertainty in the last digit. It is clear from
the tables below that the calculated poles show the structure predicted in [349] with good
numerical precision for both the Euclidean and the physical point which serves as a strong
check for the calculation.
174
FF 1/2 1/ 0
A1 +2.436734851 · 10−1 +8.212518984 · 10−1 + 1.531045661 i −2.806661(2) + 4.18190980(3) i
Pred. +2.436734852 · 10−1 +8.212518977 · 10−1 + 1.531045662 i
A2 −1.760872097 · 10−1 −6.021429768 · 10−1 − 1.106388569 i +2.509969(1) − 3.07651654(4) i
Pred. −1.760872097 · 10−1 −6.021429781 · 10−1 − 1.106388569 i
A3 −3.815946068 · 10−2 −7.236587884 · 10−2 − 2.397629627 · 10−1 i +1.2102(3) · 10−2 − 3.015063(4) · 10−1 i
Pred. −3.815946069 · 10−2 −7.236587838 · 10−2 − 2.397629627 · 10−1 i
A4 −1.565000574 · 10−4 −5.374251500 · 10−4 − 9.833188615 · 10−4 i +2.18538(3) · 10−3 − 2.748510(3) · 10−3 i
Pred. −1.565000575 · 10−4 −5.374251489 · 10−4 − 9.833188622 · 10−4 i
A5 +7.608919171 · 10−4 +1.926944077 · 10−3 + 4.780824914 · 10−3 i −1.051486(4) · 10−2 + 9.052930(4) · 10−3 i
Pred. +7.608919168 · 10−4 +1.926944068 · 10−3 + 4.780824912 · 10−3 i
A6 +7.576619247 · 10−4 +2.735071357 · 10−3 + 4.760530273 · 10−3 i −7.02484(5) · 10−3 + 1.41435102(3) · 10−2 i
Pred. +7.576619247 · 10−4 +2.735071351 · 10−3 + 4.760530273 · 10−3 i
A7 −1.565000574 · 10−4 −5.374251500 · 10−4 − 9.833188615 · 10−4 i +2.18538(3) · 10−3 − 2.748510(3) · 10−3 i
Pred. −1.565000575 · 10−4 −5.374251489 · 10−4 − 9.833188622 · 10−4 i
A8 −3.055600405 · 10−4 −1.158849558 · 10−3 − 1.919890357 · 10−3 i +4.35036(1) · 10−3 − 6.0546699(5) · 10−3 i
Pred. −3.055600405 · 10−4 −1.158849559 · 10−3 − 1.919890357 · 10−3 i
A9 +2.001982671 · 10−4 +7.482078266 · 10−4 + 1.257882810 · 10−3 i −3.07299(1) · 10−3 + 3.897481(1) · 10−3 i
Pred. +2.001982671 · 10−4 +7.482078292 · 10−4 + 1.257882810 · 10−3 i
A10 +3.636573767 · 10−4 +1.390161598 · 10−3 + 2.284926686 · 10−3 i −4.77622(2) · 10−3 + 7.274828(2) · 10−3 i
Pred. +3.636573768 · 10−4 +1.390161596 · 10−3 + 2.284926686 · 10−3 i
A11 +5.388240322 · 10−6 −1.272166624 · 10−4 + 3.385531242 · 10−5 i +1.04254(1) · 10−3 − 8.20955(1) · 10−4 i
Pred. +5.388240348 · 10−6 −1.272166651 · 10−4 + 3.385531259 · 10−5 i
A12 −5.388240322 · 10−6 +1.272166624 · 10−4 − 3.385531242 · 10−5 i −1.04254(1) · 10−3 + 8.20955(1) · 10−4 i
Pred. −5.388240348 · 10−6 +1.272166651 · 10−4 − 3.385531259 · 10−5 i
A13 −3.636573767 · 10−4 −1.390161598 · 10−3 − 2.284926686 · 10−3 i +4.77622(2) · 10−3 − 7.274828(2) · 10−3 i
Pred. −3.636573768 · 10−4 −1.390161596 · 10−3 − 2.284926686 · 10−3 i
A14 −2.001982671 · 10−4 −7.482078266 · 10−4 − 1.257882810 · 10−3 i +3.07299(1) · 10−3 − 3.897481(1) · 10−3 i
Pred. −2.001982671 · 10−4 −7.482078292 · 10−4 − 1.257882810 · 10−3 i
A15 +3.055600405 · 10−4 +1.158849558 · 10−3 + 1.919890357 · 10−3 i −4.35036(1) · 10−3 + 6.0546699(5) · 10−3 i
Pred. +3.055600405 · 10−4 +1.158849559 · 10−3 + 1.919890357 · 10−3 i
A16 +1.898361362 · 10−4 +6.165488820 · 10−4 + 1.192775622 · 10−3 i −2.233448(2) · 10−3 + 3.11183978(6) · 10−3 i
Pred. +1.898361362 · 10−4 +6.165488809 · 10−4 + 1.192775622 · 10−3 i
A17 −4.235989659 · 10−8 −1.659620988 · 10−7 − 2.661550798 · 10−7 i +8.1249(2) · 10−7 − 8.72727(4) · 10−7 i
Pred. −4.235989677 · 10−8 −1.659621000 · 10−7 − 2.661550810 · 10−7 i
A18 −9.857950093 · 10−8 −9.594603102 · 10−7 − 6.193932718 · 10−7 i +4.4198(6) · 10−7 − 5.632743(5) · 10−6 i
Pred. −9.857950139 · 10−8 −9.594603103 · 10−7 − 6.193932747 · 10−7 i
A19 +8.932087549 · 10−7 +3.205282901 · 10−6 + 5.612196125 · 10−6 i −7.43447(5) · 10−6 + 1.6553816(4) · 10−5 i
Pred. +8.932087551 · 10−7 +3.205282889 · 10−6 + 5.612196126 · 10−6 i
A20 −4.235989659 · 10−8 −1.659620988 · 10−7 − 2.661550798 · 10−7 i +8.1249(2) · 10−7 − 8.72727(4) · 10−7 i
Pred. −4.235989677 · 10−8 −1.659621000 · 10−7 − 2.661550810 · 10−7 i
Table 7.1: Numerical poles for the Euclidean phase-space point s/m2t = −191, t/m2t = −337,
m2Z /m2t = −853, mt = 1 compared against the predicted values. Also shown are the 0 terms
before IR subtraction with the digits in parentheses denoting the uncertainty in the last
digit.
175
FF 1/2 1/ 0
A1 −5.726898 · 10−1 − 4.634791 · 10−1 i −6.75706 · 10−1 − 4.05460 i 6.87787(1) − 7.90340(1) i
Pred. −5.726897 · 10−1 − 4.634791 · 10−1 i −6.75704 · 10−1 − 4.05460 i
A2 +4.153857 · 10−1 + 1.097935 · 10−1 i +1.40864 + 2.02204 i −2.53566(2) + 7.06651(3) i
Pred. +4.153857 · 10−1 + 1.097934 · 10−1 i +1.40865 + 2.02204 i
A3 +2.003102 · 10−1 + 3.116062 · 10−1 i −5.02052 · 10−1 + 1.86425 i −3.99592(2) + 2.59711(2) i
Pred. +2.003101 · 10−1 + 3.116062 · 10−1 i −5.02053 · 10−1 + 1.86425 i
A4 +3.147592 · 10−2 + 9.237206 · 10−4 i +1.39272 · 10−1 + 1.16086 · 10−1 i −4.1039(4) · 10−2 + 5.40365(5) · 10−1 i
Pred. +3.147591 · 10−2 + 9.237121 · 10−4 i +1.39272 · 10−1 + 1.16086 · 10−1 i
A5 +1.041667 · 10−1 + 5.382124 · 10−2 i +2.44023 · 10−1 + 5.97453 · 10−1 i −8.96421(5) · 10−1 + 1.736695(6) i
Pred. +1.041667 · 10−1 + 5.382123 · 10−2 i +2.44022 · 10−1 + 5.97453 · 10−1 i
A6 +1.242527 · 10−1 + 6.941130 · 10−2 i +2.52191 · 10−1 + 7.24307 · 10−1 i −1.20930(2) + 1.93865(2) i
Pred. +1.242527 · 10−1 + 6.941131 · 10−2 i +2.52189 · 10−1 + 7.24307 · 10−1 i
A7 +3.147592 · 10−2 + 9.237206 · 10−4 i +1.39272 · 10−1 + 1.16086 · 10−1 i −4.1039(4) · 10−2 + 5.40365(4) · 10−1 i
Pred. +3.147591 · 10−2 + 9.237121 · 10−4 i +1.39272 · 10−1 + 1.16086 · 10−1 i
A8 −1.017708 · 10−2 + 8.808524 · 10−2 i −4.41618 · 10−1 + 2.61228 · 10−1 i −1.00384(5) − 4.4284(4) · 10−1 i
Pred. −1.017707 · 10−2 + 8.808519 · 10−2 i −4.41613 · 10−1 + 2.61225 · 10−1 i
A9 +7.168287 · 10−2 − 5.063902 · 10−2 i +5.37076 · 10−1 + 9.24698 · 10−2 i 3.07426(8) · 10−1 1.266108(9) i
Pred. +7.168286 · 10−2 − 5.063902 · 10−2 i +5.37075 · 10−1 + 9.24707 · 10−2 i
A10 +1.873343 · 10−2 − 8.497011 · 10−2 i +4.70733 · 10−1 − 2.17284 · 10−1 i +9.3643(1) · 10−1 + 6.3029(1) · 10−1 i
Pred. +1.873344 · 10−2 − 8.497010 · 10−2 i +4.70734 · 10−1 − 2.17286 · 10−1 i
A11 −7.675742 · 10−2 + 5.097567 · 10−2 i −5.57824 · 10−1 − 1.06514 · 10−1 i −3.1397(3) · 10−1 − 1.35727(4) i
Pred. −7.675741 · 10−2 + 5.097571 · 10−2 i −5.57827 · 10−1 − 1.06513 · 10−1 i
A12 +7.675742 · 10−2 − 5.097567 · 10−2 i +5.57824 · 10−1 + 1.06514 · 10−1 i +3.1397(3) · 10−1 + 1.35727(4) i
Pred. +7.675741 · 10−2 − 5.097571 · 10−2 i +5.57827 · 10−1 + 1.06513 · 10−1 i
A13 −1.873343 · 10−2 + 8.497011 · 10−2 i −4.70733 · 10−1 + 2.17284 · 10−1 i −9.3644(1) · 10−1 − 6.3029(1) · 10−1 i
Pred. −1.873344 · 10−2 + 8.497010 · 10−2 i −4.70734 · 10−1 + 2.17286 · 10−1 i
A14 −7.168287 · 10−2 + 5.063902 · 10−2 i −5.37076 · 10−1 − 9.24698 · 10−2 i −3.07426(8) · 10−1 − 1.266108(9) i
Pred. −7.168286 · 10−2 + 5.063902 · 10−2 i −5.37075 · 10−1 − 9.24707 · 10−2 i
A15 +1.017708 · 10−2 − 8.808524 · 10−2 i +4.41618 · 10−1 − 2.61228 · 10−1 i 1.00384(4) + 4.4283(4) · 10−1 i
Pred. +1.017707 · 10−2 − 8.808519 · 10−2 i +4.41613 · 10−1 − 2.61225 · 10−1 i
A16 −6.195421 · 10−2 − 9.197693 · 10−2 i +1.25592 · 10−1 − 6.06299 · 10−1 i 1.76383(3) − 9.4291(3) · 10−1 i
Pred. −6.195417 · 10−2 − 9.197695 · 10−2 i +1.25596 · 10−1 − 6.06299 · 10−1 i
A17 +9.152404 · 10−4 + 4.922399 · 10−3 i −1.47185 · 10−2 + 2.71477 · 10−2 i −8.6390(6) · 10−2 + 2.7504(7) · 10−2 i
Pred. +9.152368 · 10−4 + 4.922402 · 10−3 i −1.47187 · 10−2 + 2.71472 · 10−2 i
A18 +6.800443 · 10−3 + 5.687424 · 10−3 i +7.80438 · 10−3 + 4.98318 · 10−2 i −1.02182(8) · 10−1 + 1.37512(8) · 10−1 i
Pred. +6.800439 · 10−3 + 5.687435 · 10−3 i +7.80405 · 10−3 + 4.98315 · 10−2 i
A19 +4.208648 · 10−3 + 4.547692 · 10−3 i −3.01730 · 10−4 + 3.55035 · 10−2 i −7.895(10) · 10−2 + 7.980(11) · 10−2 i
Pred. +4.208616 · 10−3 + 4.547808 · 10−3 i −3.13880 · 10−4 + 3.55067 · 10−2 i
A20 +9.152403 · 10−4 + 4.922399 · 10−3 i −1.47185 · 10−2 + 2.71477 · 10−2 i −8.6391(6) · 10−2 + 2.7504(7) · 10−2 i
Pred. +9.152368 · 10−4 + 4.922402 · 10−3 i −1.47187 · 10−2 + 2.71472 · 10−2 i
Table 7.2: Numerical poles for the physical phase-space point s/m2t = 142/17, t/m2t =
−125/22, m2Z /m2t = 5/18, mt = 1 compared against the predicted values. Also shown are
the 0 terms before IR subtraction with the digits in parentheses denoting the uncertainty
in the last digit.
176
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