PH.D. THESIS

IN THEORETICAL PHYSICS

QCD Radiative Corrections to Higgs

Physics

by

Taushif Ahmed

Supervisor: V. Ravindran

Examiners: Asmita MukherjeeMatthias Steinhauser

Exam date: February 21, 2017

HOMI BHABA NATIONAL INSTITUTETHE INSTITUTE OF MATHEMATICAL SCIENCES

IV CROSS ROAD, CIT CAMPUS, TARAMANI, CHENNAI 600113, INDIA

taushif.ahmed@kit.edu

Last modified: April 5, 2017

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Dedicated to

My Parents and Sister

ABSTRACT

The central part of this thesis deals with the quantum chromodynamics (QCD) radiative correc-tions to some important observables associated with the Drell-Yan, scalar and pseudo-scalar Higgsboson productions at next-to-next-to-next-to-leading order (N3LO) aiming to uplift the accuracyof theoretical results. The Higgs bosons are produced dominantly at the Large Hadron Collider(LHC) via gluon fusion through top quark loop, while one of the subdominant ones takes placethrough bottom quark annihilation whose contribution is equally important and must be includedin precision studies. Here, we have computed analytically the inclusive cross section of the Higgsboson produced through this channel under the soft-virtual (SV) approximation at N3LO QCDfollowing an elegant formalism. Moreover, the differential rapidity distribution is another mostimportant observable, which is expected to be measured in upcoming days at the LHC. This im-mediately calls for very precise theoretical predictions. The analytical expressions of the SVcorrections to this observable at N3LO for the Higgs boson, produced through gluon fusion, andleptonic pair in Drell-Yan (DY) production are computed and the numerical impacts of these re-sults are demonstrated. In addition, the CP-odd/pseudo-scalar Higgs boson, which is one of themost prime candidates in BSM scenarios, is studied in great details, taking into account the QCDradiative corrections. We have computed the analytical results of the three loop QCD correctionsto the pseudo-scalar Higgs boson production and consequently, obtained the inclusive productioncross section at N3LO under SV approximation. These indeed help to reduce the theoretical un-certainties arising from the renormalisation and factorisation scales and undoubtedly, improve thereliabilities of the theoretical results.

LIST OF PUBLICATIONS (Included in this thesis)

1. Higgs boson production through bb annihilation at threshold in N3LO QCDTaushif Ahmed, Narayan Rana and V. RavindranJHEP 1410, 139 (2014)

2. Rapidity Distributions in Drell-Yan and Higgs Productions at Threshold to Third Or-der in QCDTaushif Ahmed, M. K. Mandal, Narayan Rana and V. RavindranPhys.Rev.Lett. 113, 212003 (2014)

3. Pseudo-scalar Form Factors at Three Loops in QCDTaushif Ahmed, Thomas Gehrmann, Prakash Mathews, Narayan Rana and V. RavindranJHEP 1511, 169 (2015)

4. Pseudo-scalar Higgs Boson Production at Threshold N3LO and N3LL QCDTaushif Ahmed, M. C. Kumar, Prakash Mathews, Narayan Rana and V. RavindranEur. Phys. J. C (2016) 76:355

LIST OF PUBLICATIONS (Not included in this thesis)

1. Two-loop QCD corrections to Higgs→ b + b + g amplitudeTaushif Ahmed, Maguni Mahakhud, Prakash Mathews, Narayan Rana and V. RavindranJHEP 1408, 075 (2014)

2. Drell-Yan Production at Threshold to Third Order in QCDTaushif Ahmed, Maguni Mahakhud, Narayan Rana and V. RavindranPhys.Rev.Lett. 113, 112002 (2014)

3. Two-Loop QCD Correction to massive spin-2 resonance→ 3-gluonsTaushif Ahmed, Maguni Mahakhud, Prakash Mathews, Narayan Rana and V. RavindranJHEP 1405, 107 (2014)

4. Spin-2 form factors at three loop in QCDTaushif Ahmed, Goutam Das, Prakash Mathews, Narayan Rana and V. RavindranJHEP 1512, 084 (2015)

5. Higgs Rapidity Distribution in bb Annihilation at Threshold in N3LO QCDTaushif Ahmed, M. K. Mandal, Narayan Rana and V. RavindranJHEP 1502, 131 (2015)

6. Pseudo-scalar Higgs boson production at N3LOA +N3LL’Taushif Ahmed, Marco Bonvini, M.C. Kumar, Prakash Mathews, Narayan Rana, V. Ravin-dran, Luca RottoliEur.Phys.J. C76 (2016) no.12, 663

7. The two-loop QCD correction to massive spin-2 resonance→ qqgTaushif Ahmed, Goutam Das, Prakash Mathews, Narayan Rana and V. RavindranEur.Phys.J. C76 (2016) no.12, 667

8. NNLO QCD Corrections to the Drell-Yan Cross Section in Models of TeV-Scale Grav-ityTaushif Ahmed, Pulak Banerjee, Prasanna K. Dhani, M.C. Kumar, Prakash Mathews, NarayanRana and V. RavindranEur.Phys.J. C77 (2017) no.1, 22

9. Three loop form factors of a massive spin-2 with non-universal couplingTaushif Ahmed, Pulak Banerjee, Prasanna K. Dhani, Prakash Mathews, Narayan Rana andV. RavindranPhys.Rev. D95 (2017) no.3, 034035

LIST OF PREPRINTS (Not included in this thesis)

1. RG improved Higgs boson production to N3LO in QCDTaushif Ahmed, Goutam Das, M. C. Kumar, Narayan Rana and V. RavindranarXiv:1505.07422 [hep-ph]

2. Konishi Form Factor at Three Loop in N = 4 SYMTaushif Ahmed, Pulak Banerjee, Prasanna K. Dhani, Narayan Rana, V. Ravindran andSatyajit SetharXiv:1610.05317 [hep-th] (Under review in PRL)

LIST OF CONFERENCE PROCEEDINGS

1. Pseudo-scalar Higgs boson form factors at 3 loops in QCDTaushif Ahmed, Thomas Gehrmann, Prakash Mathews, Narayan Rana and V. RavindranPoS LL2016 (2016) 026

CONFERENCE ATTENDED

1. 16 - 19 March 2016 : MHV@30: Amplitudes and Modern Applications, Fermi NationalAccelerator Laboratory, Chicago, USA.

2. 23 - 27 February 2016 : Multiloop and Multiloop Processes for Precision Physics at theLHC, Saha Institute of Nuclear Physics, Kolkata, India.

3. 11 - 12 October 2015 : Recent Trends in AstroParticle and Particle Physics, Indian Instituteof Science, Bangalore, India.

4. 15 - 26 June 2015 : ICTP Summer School on Particle Physics, International Centre forTheoretical Physics, Trieste, Italy.

5. 23 - 28 February 2015 : Workshop on LHC and Dark Matter, Indian Association for theCultivation of Science, Kolkata, India.

6. 2 - 5 December 2014 International Workshop on Frontiers of QCD, Indian Institute ofTechnology Bombay, Mumbai, India.

7. 5 - 10 March 2014 Discussion Meeting on Radiative Correction, Institute of Physics,Bhubaneswar, India.

8. 7 - 17 July 2013 CTEQ School on QCD and Electroweak Phenomenology, University ofPittsburgh, Pennsylvania, USA.

9. 7 - 10 January 2013 Lecture Workshop in High Energy Physics, Indian Institute of Tech-nology Bombay, Mumbai, India.

10. 10 - 13 December 2012 Frontiers of High Energy Physics IMSc Golden Jubilee Sympo-sium, The Institute of Mathematical Sciences, Chennai, India.

SEMINARS PRESENTED

1. March 2016 Pseudo-Scalar Form Factors at 3-Loop in QCD, University of Buffalo, USA.

2. July 2015 Threshold Corrections to DY and Higgs at N3LO QCD, Wuppertal University,Germany.

3. July 2015 Threshold Corrections to DY and Higgs at N3LO QCD, DESY Hamburg, Ger-many.

4. July 2015 Threshold Corrections to DY and Higgs at N3LO QCD, INFN Milan, Italy.

5. July 2015 Threshold Corrections to DY and Higgs at N3LO QCD, INFN Sezione Di Torino,Italy.

6. July 2015 Threshold Corrections to DY and Higgs at N3LO QCD, Johannes GutenbergUniversitat Mainz, Germany.

7. July 2014 Two-loop QCD Correction to Massive Spin-2 Resonance → 3-gluons, Harish-Chandra Research Institute, India.

Contents

Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

0.1 Soft-Virtual QCD Corrections to Cross Section at N3LO . . . . . . . . . . 16

0.2 Soft-Virtual QCD Corrections to Rapidity at N3LO . . . . . . . . . . . . 17

0.3 Pseudo-Scalar Form Factors at Three Loops in QCD . . . . . . . . . . . . 18

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1 Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Quantum Chromo-Dynamics . . . . . . . . . . . . . . . . . . . . 26

1.2.1 The QCD Lagrangian and Feynman Rules 27

1.3 Perturbative Calculations in QCD . . . . . . . . . . . . . . . . . . 30

1.3.1 Soft-Virtual Corrections To Cross Section at N3LO QCD 31

1.3.2 Soft-Virtual QCD Corrections to Rapidity at N3LO 33

1.3.3 Pseudo-Scalar Form Factors at Three Loops in QCD 34

2 Higgs boson production through bb annihilation at threshold in N3LO QCD . 35

2.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 The Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . 37

2.3 Theoretical Framework for Threshold Corrections . . . . . . . . . . . . 37

2.3.1 The Form Factor 40

2.3.2 Operator Renormalisation Constant 44

2.3.3 Mass Factorisation Kernel 44

2.3.4 Soft-Collinear Distribution 45

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2.4 Results of the SV Cross Sections . . . . . . . . . . . . . . . . . . . 47

2.5 Numerical Impact of SV Cross Sections . . . . . . . . . . . . . . . . 57

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LOQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Theoretical Framework for Threshold Corrections to Rapidity . . . . . . . . 61

3.3.1 The Form Factor 64

3.3.2 Operator Renormalisation Constant 68

3.3.3 Mass Factorisation Kernel 69

3.3.4 Soft-Collinear Distribution for Rapidity 70

3.4 Results of the SV Rapidity Distributions . . . . . . . . . . . . . . . . 72

3.5 Numerical Impact of SV Rapidity Distributions . . . . . . . . . . . . . 84

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.7 Outlook-Beyond N3LO . . . . . . . . . . . . . . . . . . . . . . 87

4 A Diagrammatic Approach To Compute Multiloop Amplitude . . . . . . . 89

4.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Feynman Diagrams and Simplifications . . . . . . . . . . . . . . . . 90

4.3 Reduction to Master Integrals . . . . . . . . . . . . . . . . . . . . 91

4.3.1 Integration-by-Parts Identities (IBP) 91

4.3.2 Lorentz Invariant Identities (LI) 93

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Pseudo-Scalar Form Factors at Three Loops in QCD . . . . . . . . . . . . 95

5.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Framework of the Calculation . . . . . . . . . . . . . . . . . . . . 98

5.2.1 The Effective Lagrangian 98

5.2.2 Treatment of γ5 in Dimensional Regularization 99

5.3 Pseudo-scalar Quark and Gluon Form Factors . . . . . . . . . . . . . . 99

5.3.1 Calculation of the Unrenormalised Form Factors 100

Contents 13

5.3.2 UV Renormalisation 102

5.3.3 Infrared Singularities and Universal Pole Structure 105

5.3.4 Results of UV Renormalised Form Factors 110

5.3.5 Universal Behaviour of Leading Transcendentality Contribution 115

5.4 Gluon Form Factors for the Pseudo-scalar Higgs Boson Production . . . . . . 115

5.5 Hard Matching Coefficients in SCET . . . . . . . . . . . . . . . . . 118

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6 Conclusions and Outlooks . . . . . . . . . . . . . . . . . . . . . . 123

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A Inclusive Production Cross Section . . . . . . . . . . . . . . . . . . 129

B Anomalous Dimensions . . . . . . . . . . . . . . . . . . . . . . . 131

C Solving Renormalisation Group Equation . . . . . . . . . . . . . . . . 133

D Solving KG Equation . . . . . . . . . . . . . . . . . . . . . . . . 137

E Soft-Collinear Distribution . . . . . . . . . . . . . . . . . . . . . . 141

E.0.1 Results 144

F Soft-Collinear Distribution for Rapidity . . . . . . . . . . . . . . . . 149

F.0.1 Results 153

G Results of the Unrenormalised Three Loop Form Factors for the Pseudo-Scalar. 159

H Harmonic Polylogarithms . . . . . . . . . . . . . . . . . . . . . . 165

H.0.1 Properties 166

SYNOPSIS

The Standard Model (SM) of particle physics is one of the most remarkably successful fundamen-tal theories of all time which got its finishing touch on the eve of July 2012 through the discoveryof the long-awaited particle, “the Higgs boson”, at the biggest underground particle research am-phitheater, the Large Hadron Collider (LHC). It would take a while to make the conclusive remarksabout the true identity of the newly-discovered particle. However, after the discovery of this SM-like-Higgs boson, the high energy physics community is standing on the verge of a very crucial erawhere the new physics may show up as tiny deviations from the predictions of the SM. To exploitthis possibility, it is a crying need to make the theoretical predictions, along with the revolutionaryexperimental progress, to a spectacularly high accuracy within the SM and beyond (BSM).

The most successful and celebrated methodology to perform the theoretical calculations withinthe SM and BSM are based on the perturbation theory, due to our inability to solve the theoryexactly. Under the prescriptions of perturbation theory, all the observables are expanded in powersof the coupling constants present in the underlying Lagrangian. The result obtained from the firstterm of perturbative series is called the leading order (LO), the next one is called next-to-leadingorder (NLO) and so on. In most of the cases, the LO results fail miserably to deliver a reliabletheoretical prediction of the associated observables, one must go beyond the wall of LO result toachieve a higher accuracy.

Due to the presence of three fundamental forces within the SM, any observable can be expandedin powers of the coupling constants associated with the corresponding forces, namely, electromag-netic (αEM), weak (αEW) and strong (αs) ones and consequently, perturbative calculations can beperformed with respect to each of these constants. However, at typical energy scales, at whichthe hadron colliders undergo operations, the contributions arising from the αs expansion dominateover the others due to comparatively large values of αs. Hence, to catch the dominant contributionsto any observables, we must concentrate on the αs expansion and evaluate the terms beyond LO.These are called Quantum Chromo-dynamics (QCD) radiative or perturbative QCD (pQCD) cor-rections. In addition, the pQCD predictions depend on two unphysical scales, the renormalisation(µR) and factorisation (µF) scales, which are required to introduced in the process of renormalisingthe theory. The µR arises from the ultraviolet (UV) renormalisation, whereas the mass factorisa-tion (removes collinear singularities) introduces the µF . Any fixed order results do depend onthese unphysical scales which happens due to the truncation of the perturbative expansion at anyfinite order. As we include the contributions from higher and higher orders, the dependence of anyphysical observable on these unphysical scales gradually goes down. Hence, to make a reliabletheoretical prediction, it is absolutely necessary to take into account the contributions arising fromthe higher order QCD corrections to any observable at the hadron colliders.

This thesis arises exactly in this context. The central part of this thesis deals with the QCD radia-tive corrections to some important observables associated with the Drell-Yan, scalar and pseudo

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scalar Higgs boson production at three loop or N3LO order. In the subsequent discussions, wewill concentrate only on these three processes.

0.1 Soft-Virtual QCD Corrections to Cross Section at N3LO

The Higgs bosons are produced dominantly at the LHC via gluon fusion through top quark loop,while one of the sub-dominant ones take place through bottom quark annihilation. In the SM, theinteraction between the Higgs boson and bottom quarks is controlled through the Yukawa couplingwhich is reasonably small at typical energy scales. However, in the minimal super symmetric SM(MSSM), this channel can contribute substantially due to enhanced coupling between the Higgsboson and bottom quarks in the large tan β region, where tan β is the ratio of vacuum expectationvalues of the up and down type Higgs fields. In the present run of LHC, the measurements of thevarious coupling constants including this one are underway which can shed light on the propertiesof the newly discovered Higgs boson. Most importantly, for the precision studies we must takeinto account all the contributions, does not matter how tiny those are, arising from sub-dominantchannels along with the dominant ones to reduce the dependence on the unphysical scales andmake a reliable prediction.

The computations of the higher order QCD corrections beyond leading order often becomes quitechallenging because of the large number of Feynman diagrams and, presence of the complicatedloop and phase space integrals. Under this circumstance, when we fail to compute the completeresult at certain order, it is quite natural to try an alternative approach to capture the dominant con-tributions from the missing higher order corrections. It has been observed for many processes thatthe dominant contributions to an observable often comes from the soft gluon emission diagrams.The contributions arising from the associated soft gluon emission along with the virtual Feynmandiagrams are known as the soft-virtual (SV) corrections. The goal of this section is to discuss theSV QCD corrections to the production cross section of the Higgs boson, produced through bottomquark annihilation.

The NNLO QCD corrections to this channel are already present in the literature. In addition, thepartial result for the N3LO corrections under the SV approximation were also computed long back.In this work, we have computed the missing part and completed the full SV corrections to the crosssection at N3LO.

The infrared safe contributions from the soft gluons are obtained by adding the soft part of the crosssection with the UV renormalized virtual part and performing mass factorisation using appropriatecounter terms. The main ingredients are the form factors, overall operator UV renormalizationconstant, soft-collinear distribution arising from the real radiations in the partonic subprocessesand mass factorization kernels. The computations of SV cross section at N3LO QCD require allof these above quantities up to 3-loop order. The relevant form factor becomes available veryrecently. The soft-collinear distribution at N3LO was computed by us around the same time. Thiswas calculated from the recent result of N3LO SV cross section of the Higgs boson productionsin gluon fusion by employing a symmetry (maximally non-Abelian property). Prior to this, thissymmetry was verified explicitly up to NNLO order. However, neither there was any clear reasonto believe that the symmetry would fail nor there was any transparent indication of holding itbeyond this order. Nevertheless, we postulate that the relation would hold true even at N3LOorder! This is inspired by the universal properties of the soft gluons which are the underlyingreasons behind the existence of this remarkable symmetry. Later, this conjecture is verified byexplicit computations performed by two different groups on Drell-Yan process. This symmetry

0.2 Soft-Virtual QCD Corrections to Rapidity at N3LO 17

plays the most important role in achieving our goal. With these, along with the existing resultsof the remaining required ingredients, we obtain the complete analytical expressions of N3LO SVcross section of the Higgs boson production through bottom quark annihilation. It reduces thescale dependence and provides a more precise result. We demonstrate the impact of this resultnumerically at the Large Hadron Collider (LHC) briefly. This is the most accurate result for thischannel which exists in the literature till date and it is expected to play an important role in comingdays at the LHC.

0.2 Soft-Virtual QCD Corrections to Rapidity at N3LO

The productions of the Higgs boson in gluon fusion and leptonic pair in DY are among the mostimportant processes at the LHC which are studied not only to test the SM to an unprecedentedaccuracy but also to explore the new physics under BSM. During the present run at the LHC, inaddition to the inclusive production cross section, the differential rapidity distribution is amongthe most important observables, which is expected to be measured in upcoming days. This imme-diately calls for very precise theoretical predictions.

In the same spirit of the SV corrections to the inclusive production cross section, the dominantcontributions to the differential rapidity distributions often arise from the soft gluon emission di-agrams. Hence, in the absence of complete fixed order result, the rapidity distribution under SVapproximation is the best available alternative in order to capture the dominant contributions fromthe missing higher orders and stabilise the dependence on unphysical scales. For the Higgs bosonproduction through gluon fusion, we work in the effective theory where the top quark is integratedout. This section is devoted to demonstrate the SV corrections to this observable at N3LO for theHiggs boson, produced through gluon fusion, and leptonic pair in Drell-Yan (DY) production.

For the processes under considerations, the NNLO QCD corrections are present, computed longback, and in addition, the partial N3LO SV results are also available. However, due to reasonablylarge scale uncertainties and crying demand of uplifting the accuracy of theoretical predictions,we must push the boundaries of existing results. In this work, we have computed the missing partand completed the SV corrections to the rapidity distributions at N3LO QCD.

The prescription which has been employed to calculate the SV QCD corrections is similar tothat of the inclusive cross section, more specifically, it is a generalisation of the other one. Theinfrared safe contributions under SV approximation can be computed by adding the soft part of therapidity distribution with the UV renormalised virtual part and performing the mass factorisationusing appropriate counter terms. Similar to the inclusive case, the main ingredients to performthis computation are the form factors, overall UV operator renormalisation constant, soft-collineardistribution for rapidity and mass factorisation kernels. These quantities are required up to N3LOto calculate the rapidity at this order. The three loop quark and gluon form factors were calculatedlong back. The operator renormalisation constants are also present. For DY, this constant isnot required or equivalently equals to unity. The mass factorisation kernels are also available inthe literature to the required order. The only missing part was the soft-collinear distribution forrapidity at N3LO. This was not possible to compute until very recently. Because of the universalbehaviour of the soft gluons, the soft-collinear distributions for rapidity and inclusive cross sectioncan be related to all orders in perturbation theory. Employing this beautiful relation, we obtainthis quantity at N3LO from the results of soft-collinear distribution of the inclusive cross section.Using this, along with the existing results of the other relevant quantities, we compute the completeanalytical expressions of N3LO SV correction to the rapidity distributions for the Higgs boson in

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gluon fusion and leptonic pair in DY. We demonstrate the numerical impact of this correctionfor the case of Higgs boson at the LHC. This indeed reduces the scale dependence significantlyand provides a more reliable theoretical predictions. These are the most accurate results for therapidity distributions of the Higgs boson and DY pair which exist in the literature and undoubtedly,expected to play very important role in the upcoming run at the LHC.

0.3 Pseudo-Scalar Form Factors at Three Loops in QCD

One of the most popular extensions of the SM, namely, the MSSM and two Higgs doublet modelhave richer Higgs sector containing more than one Higgs boson and there have been intensesearch strategies to observe them at the LHC. In particular, the production of CP-odd Higgsboson/pseudo-scalar at the LHC has been studied in detail, taking into account higher order QCDradiative corrections, due to similarities with its CP-even counter part. Very recently, the N3LOQCD corrections to the inclusive production cross section of the CP-even Higgs boson is com-puted. So, it is very natural to extend the theoretical accuracy for the CP-odd Higgs boson to thesame order of N3LO. This requires the 3-loop quark and gluon form factors for the pseudo-scalarwhich are the only missing ingredients to achieve this goal.

Multiloop and multileg computations play a crucial role to achieve the golden task of makingprecise theoretical predictions. However, the complexity of these computations grows very rapidlywith the increase of number of loops and/or external particles. Nevertheless, it has become areality due to several remarkable developments in due course of time. This section is devoted todemonstrate the computations of the 3-loop quark and gluon form factors for the pseudo-scalaroperators in QCD.

The coupling of a pseudo-scalar Higgs boson to gluons is mediated through a heavy quark loop.In the limit of large quark mass, it is described by an effective Lagrangian that only admits lightdegrees of freedom. In this effective theory, we compute the 3-loop massless QCD corrections tothe form factor that describes the coupling of a pseudo-scalar Higgs boson to gluons. The evalua-tion of this 3-loop form factors is truly a non-trivial task not only because of the involvement of alarge number of Feynman diagrams but also due to the presence of the axial vector coupling. Wework in dimensional regularisation and use the ’t Hooft-Veltman prescription for the axial vectorcurrent, The state-of-the-art techniques including integration-by-parts (IBP) and Lorentz invari-ant (LI) identities have been employed to accomplish this task. The UV renormalisation is quiteinvolved since the two operators, present in the Lagrangian, mix under UV renormalization dueto the axial anomaly and additionally, a finite renormalisation constant needs to be introduced inorder to fulfill the chiral Ward identities. Using the universal infrared (IR) factorization properties,we independently derive the three-loop operator mixing and finite operator renormalisation fromthe renormalisation group equation for the form factors, thereby confirming recent results, whichwere computed following a completely different methodology, in the operator product expansion.This form factor is an important ingredient to the precise prediction of the pseudo-scalar Higgsboson production cross section at hadron colliders. We derive the hard matching coefficient insoft-collinear effective theory (SCET). We also study the form factors in the context of leadingtranscendentality principle and we find that the diagonal form factors become identical to those ofN = 4 upon imposing some identification on the quadratic Casimirs. Later, these form factors areused to calculate the SV corrections to the pseudo-scalar production cross section at N3LO andN3LL QCD.

List of Figures

2.1 Higgs boson production in gluon fusion . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Higgs boson production through bottom quark annihilation . . . . . . . . . . . . 38

2.3 Drell-Yan pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Total cross section for Higgs production in bb annihilation at various orders in as

as a function of µR/mH (left panel) and of µF/mH (right panel) at the LHC with√

s = 14 TeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1 Higgs boson production in gluon fusion . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Higgs boson production through bottom quark annihilation . . . . . . . . . . . . 62

3.3 Drell-Yan pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Rapidity distribution of Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . 86

4.1 One loop box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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List of Tables

3.1 Relative contributions of pure N3LO terms. . . . . . . . . . . . . . . . . . . . . 85

3.2 Relative contributions of pure N3LO terms. . . . . . . . . . . . . . . . . . . . . 85

3.3 Contributions of exact NNLO, NNLOSV, N3LOSV, and K3. . . . . . . . . . . . . 85

3.4 Contributions of exact NNLO, NNLOSV, N3LOSV, and K3. . . . . . . . . . . . . 86

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1 Introduction

Contents1.1 Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . 251.2 Quantum Chromo-Dynamics . . . . . . . . . . . . . . . . . . . . 26

1.2.1 The QCD Lagrangian and Feynman Rules 27

1.3 Perturbative Calculations in QCD . . . . . . . . . . . . . . . . . . 301.3.1 Soft-Virtual Corrections To Cross Section at N3LO QCD 31

1.3.2 Soft-Virtual QCD Corrections to Rapidity at N3LO 331.3.3 Pseudo-Scalar Form Factors at Three Loops in QCD 34

The Standard Model (SM) of Particle Physics is one of most remarkably successful fundamentaltheories which encapsulates the governing principles of elementary constituents of matter andtheir interactions. Its development throughout the latter half of the 20th century resulting from anunprecedented collaborative effort of the brightest minds around the world is undoubtedly one ofthe greatest achievements in human history. Over the duration of many decades around 1970s, thetheoretical predictions of the SM were verified one after another with a spectacular accuracy andit got the ultimate credence through the announcement, made on a fine morning of 4th July 2012at CERN in Geneva:

“If we combine ZZ and γγ, this is what we get: they line up extremely well in a regionof 125 GeV with the combine significance of 5 standard deviation!”

The SM relies on the mathematical framework of quantum field theory (QFT), in which a La-grangian controls the dynamics and kinematics of the theory. Each kind of particle is described interms of a dynamical field that pervades space-time. The construction of the SM proceeds throughthe modern methodology of constructing a QFT, it happens through postulating a set of underlyingsymmetries of the system and writing down the most general renormalisable Lagrangian from itsfield content.

The underlying symmetries of the QFT can be largely categorized into global and local ones.The global Poincaré symmetry is postulated for all the relativistic QFT. It consists of the familiartranslational symmetry, rotational symmetry and the inertial reference frame invariance centralto the special theory of relativity. Being global, its operations must be simultaneously appliedto all points of space-time On the other hand, the local gauge symmetry is an internal symme-try that plays the most crucial role in determining the predictions of the underlying QFT. These

23

24 Introduction

are the symmetries that act independently at each point in space-time. The SM relies on thelocal SU(3)×SU(2)L×U(1)Y gauge symmetry. Each gauge symmetry manifestly gives rise to afundamental interaction: the electromagnetic interactions are characterized by an U(1), the weakinteractions by an SU(2) and the strong interactions by an SU(3) symmetry.

In its current formulation of the SM, it includes three different families of elementary particles.The first ones are called fermions arising from the quantisation of the fermionic fields. Theseconstitute the matter content of the theory. The quanta of the bosonic fields, which form thesecond family, are the force carriers i.e. the mediators of the strong, weak, and electromagneticfundamental interactions. In addition to the these, there is a third boson, the Higgs boson resultingfrom the quantum excitation of the Higgs field. This is the only known scalar particle that waspostulated long ago and observed very recently at the Large Hadron Collider (LHC) [1, 2]. Thepresence of this field, now believed to be confirmed, explains the mechanisms of acquiring massof some of the fundamental particles when, based on the underlying gauge symmetries controllingtheir interactions, they should be massless. This mechanism, which is believed to be one of themost revolutionary ideas of the last century, is known as Brout-Englert-Higgs-Kibble (BEHK)mechanism.

Two of the four known fundamental forces, electromagnetism and weak forces which appear verydifferent at low energies, are actually unified to so called electro-weak force in high energy. Thestructure of this unified picture is accomplished under the gauge group SU(2)L× U(1)Y. Thecorresponding gauge bosons are the three W bosons of weak isospin from SU(2) and the B bo-son of weak hypercharge from U(1), all of which are massless. Upon spontaneous symmetrybreaking from SU(2)L× U(1)Y to U(1)EM, caused by the BEHK mechanism, the three media-tors of the electro-weak force, the W±,Z bosons acquire mass, leaving the mediator of the elec-tromagnetic force, the photon, as massless. Finally, the theory of strong interactions, QuantumChromo-Dynamics (QCD) is governed by the unbroken SU(3) gauge group, whose force carriers,the gluons remain massless.

Although the SM is believed to be theoretically self-consistent with a spectacular accuracy and hasdemonstrated huge and continued successes in providing experimental predictions, it indeed doesleave some phenomena unexplained and it falls short of being a complete theory of fundamentalinteractions. It fails to incorporate the full theory of gravitation as described by general relativity,or account for the accelerating expansion of the universe (as possibly described by dark energy).The model does not contain any viable dark matter particle that possesses all of the required prop-erties deduced from observational cosmology. It also does not incorporate neutrino oscillations(and their non-zero masses).

Currently, the high energy physics community is standing on the verge of a crucial era where thenew physics may show up as tiny deviations from the prediction of the SM! To exploit this pos-sibility it is absolutely necessary to make the theoretical predictions, along with the revolutionaryexperimental progress, to a very high accuracy within the SM and beyond. The relevance of thisthesis arises exactly in this context.

The most crucial quantity in the process of accomplishing the job of making any prediction basedon QFT is undoubtedly the scattering amplitude. This is the fundamental building block of anyobservable in QFT. In the upcoming section, we will elaborate on the idea of scattering amplitudewhich will be followed by a brief description of QCD. We will close the chapter of introductionby introducing the concept of computing the observables under certain approximation, known assoft-virtual approximation.

1.1 Scattering Amplitudes 25

1.1 Scattering Amplitudes

The fundamental quantity of any QFT which encodes all the underlying symetries of the theory iscalled the action. This is constructed out of Lagrangian density, which is a functional of the fieldspresent in the theory, and integrating over all space time points:

S =

∫d4xL

[φi(x)

]. (1.1.1)

By construction the QFT is a probabilistic theory and all the observables calculated based onthis theory always carry a probabilistic interpretation. For example, an important observable isthe total cross section which measures the total probability of any event to happen in colliders.The computation of the cross section, and in fact, almost all the observables in QFT requires theevaluation of scattering matrix (S -matrix) elements which describe the evolution of the systemfrom asymptotic initial to final states due to presence of the interaction. The S -matrix elementsare defined as

〈 f |S |i〉 = δ f i + i(2π)4δ(4)(p f − pi

)Mi→ f (1.1.2)

where, the δ f i represents the unscattered forward scattering states, while the other part Mi→ f

encapsulates the “actual” interaction (For simplicity, we will call Mi→ f as scattering matrix el-ement.). So, the calculation of all those observables essentially boils down to the computationof the second quantity. However, the exact computation of this quantity is incredibly difficult inany general field theory. The only viable methodology is provided under the framework of per-turbation theory where the matrix elements as well as the observables are expanded in powers ofcoupling constants, c, present in the theory:

Mi→ f =

∞∑n=0

cnM(n)i→ f . (1.1.3)

If the coupling constant is small enough, the evaluation of only the first term of the perturbativeexpansion often turns out to be a very good approximation that provides a reliable prediction toany observable. However, in QFT, it is a well-known fact that the coupling constants are truly not‘constants’, their strength depends on the energy scale at which the interaction takes place. Thisevolution of the coupling constant may make it comparatively large at some energy scale. In caseof Quantum Electro-Dynamics (QED), quantum field theory of electromagnetism, the magnitudeof the coupling constant, c = αEM, increases with the increase of momentum transfer:

αEM(Q2 ≈ 0) ≈1

137, and αEM(Q2 ≈ m2

W) ≈1

128(1.1.4)

where, mW ≈ 80 GeV is the invariant mass of the W boson. The smallness of αEM at all typicalenergy scales which can be probed in all collider experiments guarantees very fast convergence ofthe perturbation series to what we expect to be real non-perturbative result. However, this pictureno longer holds true in case of QCD where the coupling constant, c = αs, may become quite largeat certain energy scales:

αs(m2p) ≈ 0.55, and αs(m2

Z) ≈ 0.1 (1.1.5)

where, mp ≈ 938MeV and mZ ≈ 90GeV are the masses of the proton and Z boson. Clearlythe magnitude 0.55 is far from being small! Hence, computation of only the leading term in

26 Introduction

perturbative series often turns out to be a very crude approximation which fails to deliver a reliableprediction. We must take into account the contributions arising beyond leading term.

In perturbation theory, the most acceptable and well known prescription to compute the termsin a perturbative series is provided by Feynman diagrams. Every term of a series is representedthrough a set of Feynman diagrams and each diagram corresponds to a mathematical expression.Hence, evaluation of a term in any perturbative series boils down to the computation of all thecorresponding Feynman diagrams. Given an action of a QFT, one first requires to derive a set ofrules, called Feynman rules, which essentially establish the correspondence between the Feynmandiagrams and mathematical expressions. With the rules in hand, we just need to draw all thepossible Feynman diagrams contributing to the order of our interest and eventually evaluate thoseusing the rules. Needless to say, as the perturbative order increases, the number of Feynmandiagrams to be drawn grows so rapidly that after certain order it becomes almost prohibitivelylarge to draw.

In this thesis, we will concentrate only on the aspects of perturbative QCD. We will start our dis-cussion of QCD by introducing the basic aspects of this QFT which will be followed by the writingdown the quantum action and corresponding Feynman rules. Then we will discuss how to com-pute amplitudes beyond leading order in QCD and eventually get reliable numerical predictions athadron colliders for any process.

1.2 Quantum Chromo-Dynamics

Quantum Chromo-dynamics, familiarly called QCD, is the modern theory of strong interactions, afundamental force describing the interactions between quarks and gluons which make up hadronssuch as the protons, neutrons and pions. QCD is a type of QFT called non-Abelian gauge theorythat has underlying SU(3) gauge symmetry. It appears as an expanded version of QED. Whereasin QED there is just one kind of charge, namely electric charge, QCD has three different kinds ofcharge, labeled by “colour”. Avoiding chauvinism, those are chosen as red, green, and blue. But,of course, the colour charges of QCD have nothing to do with optical colours. Rather, they haveproperties analogous to electric charges in QED. In particular, the colour charges are conserved inall physical processes. There are also photon-like massless gauge bosons, called gluons, that actas the mediators of the strong interactions between spin-1/2 quarks. Unlike the photons, whichmediate the electromagnetic interaction but lacks an electric charge, the gluons themselves carrycolor charges. Gluons, as a consequence, participate in the strong interactions in addition tomediating it, making QCD substantially harder to analyse than QED.

In sharp contrast to other gauge theories, QCD enjoys two salient features: confinement andasymptotic freedom. The force among quarks/gluons fields does not diminish as they are sep-arated from each others. With the increase in mutual distance between them, the mediating gluonfields gather enough energy to create a pair of quarks/gluons which forbids them to be found asfree particles; they are thus forever bound into hadrons such as the protons, neutrons, pions orkaons. Although literature lacks the satisfactory theoretical explanation, confinement is believedto be true as it explains the consistent failure of finding free quarks or gluons. The other in-teresting property, the asymptotic freedom [3–7], causes bonds between quarks/gluons becomeasymptotically weaker as energy increases or distance decreases which allows us to perform thecalculation in QCD using the technique of perturbation theory. The Nobel prize was awarded forthis remarkable discovery of last century.

1.2 Quantum Chromo-Dynamics 27

In perturbative QCD, the basic building blocks of performing any calculation are the Feynmanrules, which will be discussed in next subsection.

1.2.1 The QCD Lagrangian and Feynman Rules

The first step in performing perturbative calculations in a QFT is to work out the Feynman rules.The SU(N) gauge invariant classical Lagrangian density encapsulating the interaction betweenfermions and non-Abelian gauge fields is

Lclassical = −14

FaµνF

a,µν +

n f∑f =1

ψ( f )α,i

(i /Dαβ,i j − m f δαβδi j

)ψ

( f )β, j . (1.2.1)

In the above expression,

Faµν = ∂µAa

ν − ∂νAaµ + gs f abcAb

µAcν ,

/Dαβ,i j ≡ γµαβDµ,i j = γµ

(δi j∂µ − igsT a

i jAaµ

)(1.2.2)

where, Aaµ and ψ( f )

α,i are the guage and fermionic quark fields, respectively. The indices representthe following things:

a, b, · · · : color indices in the adjoint representation⇒ [1,N2 − 1] ,

i, j, · · · : color indices in the fundamental representation⇒ [1,N] ,

α, β, · · · : Dirac spinor indices⇒ [1, d] ,

µ, ν, · · · : Lorentz indices⇒ [1, d] . (1.2.3)

Numbers within the ‘[]’ signifies the range of the corresponding indices. d is the space-timedimensions. f is the quark flavour index which runs from 1 to n f . m f and gs are the mass ofthe quark corresponding to ψ( f ) and strong coupling constant, respectively. f abc are the structureconstants of SU(N) group. These are related to the Gellmann matrices T a, generators of thefundamental representations of SU(N), through[

T a,T b]

= i f abcT c . (1.2.4)

The T a are traceless, Hermitian matrices and these are normalised with

Tr(T aT b

)= TFδ

ab (1.2.5)

where, TF = 12 . They satisfy the following completeness relation

∑a

T ai jT

akl =

12

(δilδk j −

1Nδi jδkl

). (1.2.6)

In addition to the above three parent identities expressed through the Eq. (1.2.4, 1.2.5, 1.2.6), wecan have some auxiliary ones which are often useful in simplifying colour algebra:∑

a

(T aT a)i j = CFδi j ,

f acd f bcd = CAδab . (1.2.7)

28 Introduction

The CA = N and CF = N2−12N are the quadratic Casimirs of the SU(N) group in the adjoint and

fundamental representations, respectively. For QCD, the SU(N) group index, N = 3 and the flavornumber n f = 6.

The quantisation of the non-Abelian gauge theory or the Yang-Mills (YM) theory faces an imme-diate problem, namely, the propagator of gauge fields cannot be obtained unambiguously. This isdirectly related to the presence of gauge degrees of freedom inherent into theLclassical. We need toperform the gauge fixing in order to get rid of this problem. The gauge fixing in a covariant way,when done through the path integral formalism, generates new particles called Faddeev-Popov(FP) ghosts having spin-0 but obeying fermionic statistics. The absolute necessity of introducingthe ghosts in the process of quantising the YM theory is a horrible consequence of the Lagrangianformulation of QFT. There is no observable consequence of these particles, we just need themin order to describe an interacting theory of a massless spin-1 particle using a local manifestlyLorentz invariant Lagrangian. These particles never appear as physical external states but must beincluded in internal lines to cancel the unphysical degrees of freedom of the gauge fields. Some al-ternative formulations of non-Abelian gauge theory (such as the lattice) also do not require ghosts.Perturbative gauge theories in certain non-covariant gauges, such as light-cone or axial gauges,are also ghost free. However, to maintain manifest Lorentz invariance in a perturbative gauge the-ory, it seems ghosts are unavoidable and in this thesis we will be remained within the regime ofcovariant gauge and consequently will include ghost fields consistently into our computations.

Upon applying this technique to quantise the YM theory, we end up with getting the following fullquantum Lagrangian density:

LY M = Lclassical +Lgauge− f ix +Lghost (1.2.8)

where, the second and third terms on the right hand side correspond to the gauge fixing and FPcontributions, respectively. These are obtained as

Lgauge− f ix = −12ξ

(∂µAa

µ

)2,

Lghost =(∂µχa∗) Dµ,ab χ

b (1.2.9)

with

Dµ,ab ≡ δab∂µ − gs fabcAcµ . (1.2.10)

The gauge parameter ξ is arbitrary and it is introduced in order to specify the gauge in a covariantway. This prescription of fixing gauge in a covariant way is known as Rξ gauge. A typical choicewhich is often used is ξ = 1, known as Feynman gauge. We will be working in this Feynman gaugethroughout this thesis, unless otherwise mentioned specifically. However, we emphasize that thephysical results are independent of the choice of the gauges. The field χa and χa∗ are ghost andanti-ghost fields, respectively.

All the Feynman rules can be read off from the quantized Lagrangian LY M in Eq. 1.2.8. We willdenote the quarks through straight lines, gluons through curly and ghosts through dotted lines. Weprovide the rules in Rξ gauge.

• The propagators for quarks, gluons and ghosts are obtained as respectively:

1.2 Quantum Chromo-Dynamics 29

j, β i, α

p2 p1

i (2π)4 δ(4) (p1 + p2) δi j

(1

/p1 − m f + iε

)αβ

b, ν a, µ

p2 p1

i (2π)4 δ(4) (p1 + p2) δab1p2

1

−gµν + (1 − ξ)p1µp1ν

p21

b a

p2 p1

i (2π)4 δ(4) (p1 + p2) δab1p2

1

• The interacting vertices are given by:

p1p2

p3

i, αj, β

a, µ

igs (2π)4 δ(4) (p1 + p2 + p3) T ai j

(γµ

)αβ

p1p2

p3

a, µb, ν

c, ρ

gs

3!(2π)4 δ(4) (p1 + p2 + p3) f abc

×[gµν(p1 − p2)ρ + gνρ(p2 − p3)µ + gρµ(p3 − p1)ν

]

p1p2

p3

bc

a, µ

−gs (2π)4 δ(4) (p1 + p2 + p3) f abc pµ1

30 Introduction

p1p2

p3 p4

a, µb, ν

c, ρ d, σ −g2

s

4!(2π)4 δ(4) (p1 + p2 + p3 + p4){ (

f ac,bd − f ad,cb)

gµνgρσ +(

f ab,cd − f ad,bc)

gµρgνσ

+(

f ac,db − f ab,cd)

gµσgνρ}

with

f ab,cd ≡ f abx f cdx

In addition to these rules, we have keep in mind the following points:

• For any Feynman diagram, the symmetry factor needs to be multiplied appropriately. Thesymmetry factor is defined as the number of ways one can obtain the topological configura-tion of the Feynman diagram under consideration.

• For each loop momenta, the integration over the loop momenta, k, needs to be performedwith the integration measure ddk/(2π)d in d-dimensions (in dimensional regularisation).

• For each quark/ghost loop, one has to multiply a factor of (-1).

1.3 Perturbative Calculations in QCD

The asymptotic freedom of the QCD allows us to perform the calculations in high energy regimeusing the techniques of perturbative QCD (pQCD). In pQCD, we make the theoretical predictionsthrough the computations of the scattering matrix (S-matrix) elements. The S-matrix elements aredirectly related to the scattering amplitude which is formally expanded, within the framework ofperturbation theory, in powers of coupling constants. This expansion is represented through theset of Feynman diagrams and the Feynman rules encapsulate the connection between these thesetwo. Hence, the theoretical predictions boil down to evaluate the set of Feynman diagrams. Usingthe Feynman rules presented in the previous Sec. 1.2.1, we can evaluate all the Feynman diagrams.

Achieving precise theoretical predictions demand to go beyond leading order which consists ofevaluating the virtual/loop as well as real emission diagrams. However, the contribution arisingfrom the individual one is not finite. The resulting expressions from the evaluation of loop dia-grams contain the ultraviolet (UV), soft and collinear divergences. For simplicity, together we callthe soft and collinear as infrared (IR) divergence.

The UV divergences arise from the region of large momentum or very high energy (approachinginfinity) of the Feynman integrals, or, equivalently, because of the physical phenomena at veryshort distances. We get rid of this through UV renormalisation. Before performing the UV renor-malisation, we need to regulate the Feynman integrals which is essentially required to identify thetrue nature of divergences. There are several ways to regulate the integrals. The most consistentand beautiful way is the framework of dimensional regularisation [8–10]. Within this, we needto perform the integrals in general d-dimensions which is taken as 4 + ε in this thesis. Uponperforming the integrals, all the UV singularities arise as poles in ε. The UV renormalisation,which is performed through redefining all the quantities present in the Lagrangian, absorbs these

1.3 Perturbative Calculations in QCD 31

poles and gives rise a UV finite result. The UV renormalisation is done at certain energy scale,known as renormalisation scale, µR. On the other hand, the soft divergences arise from the lowmomentum limit (approaching zero) of the loop integrals and the collinear ones arise when anyloop momentum becomes collinear to any of the external massless particles. The collinear di-vergence is a property of theories with massless particles. Hence, even after performing the UVrenormalisation, the resulting expressions obtained through the evaluation the loop integrals arenot finite, they contain poles arising from soft and collinear regions of the loop integrals.

To remove the residual IR divergences, we need to add the contributions arising from the realemission diagrams. The latter contains soft as well as collinear divergences which have the sameform as that of loop integrals. Once we add the virtual and real emission diagrams and evalu-ate the phase space integrals, the resulting expressions are guaranteed to be freed from UV, softand final state collinear singularities, thanks to the Kinoshita-Lee-Nauenberg (KLN) theorem. Ananalogous result for quantum electrodynamics alone is known as Bloch?Nordsieck cancellation.However, the collinear singularities arising from the collinear configurations involving initial stateparticles remain. Those are removed at the hadronic level through the techniques, known as massfactorisation, where the residual singularities are absorbed into the bare parton distribution func-tions (PDF). So, the observables at the hadronic level are finite which are compared with theexperimental outcomes at the hadron colliders. Just like UV renormalisation, mass factorisationis done at some energy scale, called factorisation scale, µF . The µR as well as µF are unphysicalscales. The dependence of the fixed order results on these scale is an artifact of the truncation of theperturbative series to a finite order. If we can capture the results to all order, then the dependencegoes away.

The core part of this thesis deals with the higher order QCD corrections employing the method-ology of perturbation theory to some of the very important processes within the SM and beyond.More specifically, the thesis contains

• the soft-virtual QCD corrections to the inclusive cross section of the Higgs boson productionthrough bottom quark annihilation at next-to-next-to-next-to-leading order (N3LO) [11],

• the soft-virtual QCD corrections at N3LO to the differential rapidity distributions of theproductions of the Higgs boson in gluon fusion and of the leptonic pair in Drell-Yan [12],

• the three loop QCD corrections to the pseudo-scalar form factors [13, 14].

In the subsequent subsections, we will discuss the above things in brief.

1.3.1 Soft-Virtual Corrections To Cross Section at N3LO QCD

The Higgs bosons are produced dominantly at the LHC via gluon fusion through top quark loop,while one of the sub-dominant ones take place through bottom quark annihilation. In the SM, theinteraction between the Higgs boson and bottom quarks is controlled through the Yukawa couplingwhich is reasonably small at typical energy scales. However, in the minimal super symmetric SM(MSSM), this channel can contribute substantially due to enhanced coupling between the Higgsboson and bottom quarks in the large tan β region, where tan β is the ratio of vacuum expectationvalues of the up and down type Higgs fields. In the present run of LHC, the measurements of thevarious coupling constants including this one are underway which can shed light on the propertiesof the newly discovered Higgs boson [1, 15]. Most importantly, for the precision studies we must

32 Introduction

take into account all the contributions, does not matter how tiny those are, arising from sub-dominant channels along with the dominant ones to reduce the dependence on the unphysicalscales and make a reliable prediction.

The computations of the higher order QCD corrections beyond leading order often become quitechallenging because of the large number of Feynman diagrams and, presence of the complicatedloop and phase space integrals. Under this circumstance, when we fail to compute the completeresult at certain order, it is quite natural to try an alternative approach to capture the dominant con-tributions from the missing higher order corrections. It has been observed for many processes thatthe dominant contributions to an observable often comes from the soft gluon emission diagrams.The contributions arising from the associated soft gluon emission along with the virtual Feynmandiagrams are known as the soft-virtual (SV) corrections. The goal of the works published in thearticle [11] is to discuss the SV QCD corrections to the production cross section of the Higgsboson, produced through bottom quark annihilation.

The next-to-next-to-leading order (NNLO) QCD corrections to this channel are already present inthe variable flavour scheme (VFS) [16–21], while it is known to NLO in the fixed flavour scheme(FFS) [22–27]. In addition, the partial result for the N3LO corrections [28–30] under the SVapproximation were also computed long back. In both [28, 29] and [30], it was not possible todetermine the complete contribution at N3LO due to the lack of information on three loop finitepart of bottom anti-bottom Higgs form factor in QCD and the soft gluon radiation at N3LO level.In this work [11], we have computed the missing part and completed the full SV corrections to thecross section at N3LO.

The infrared safe contributions from the soft gluons are obtained by adding the soft part of the crosssection with the UV renormalized virtual part and performing mass factorisation using appropri-ate counter terms. The main ingredients are the form factors, overall operator UV renormalizationconstant, soft-collinear distribution arising from the real radiations in the partonic subprocessesand mass factorization kernels. The computations of SV cross section at N3LO QCD require allof these above quantities up to 3-loop order. The relevant form factor becomes available very re-cently in [31]. The soft-collinear distribution at N3LO was computed by us around the same timein [32]. This was calculated from the recent result of N3LO SV cross section of the Higgs bosonproductions in gluon fusion [33] by employing a symmetry (maximally non-Abelian property).Prior to this, this symmetry was verified explicitly up to NNLO order. However, neither there wasany clear reason to believe that the symmetry would fail nor there was any transparent indicationof holding it beyond this order. Nevertheless, we conjecture [32] that the relation would hold trueeven at N3LO order! This is inspired by the universal properties of the soft gluons which arethe underlying reasons behind the existence of this remarkable symmetry. Later, this conjecture isverified by explicit computations performed by two different groups on Drell-Yan process [34,35].This symmetry plays the most important role in achieving our goal. With these, along with the ex-isting results of the remaining required ingredients, we obtain the complete analytical expressionsof N3LO SV cross section of the Higgs boson production through bottom quark annihilation [11]employing the methodology prescribed in [28,29]. It reduces the scale dependence and provides amore precise result. We demonstrate the impact of this result numerically at the LHC briefly. Thisis the most accurate result for this channel which exists in the literature till date and it is expectedto play an important role in coming days at the LHC.

1.3 Perturbative Calculations in QCD 33

1.3.2 Soft-Virtual QCD Corrections to Rapidity at N3LO

The productions of the Higgs boson in gluon fusion and leptonic pair in Drell-Yan (DY) are amongthe most important processes at the LHC which are studied not only to test the SM to an unprece-dented accuracy but also to explore the physics beyond Standard Model (BSM). During the presentrun at the LHC, in addition to the inclusive production cross section, the differential rapidity dis-tribution is among the most important observables, which is expected to be measured in upcomingdays. This immediately calls for very precise theoretical predictions.

In the same spirit of the SV corrections to the inclusive production cross section, the dominantcontributions to the differential rapidity distributions often arise from the soft gluon emission di-agrams. Hence, in the absence of complete fixed order result, the rapidity distribution under SVapproximation is the best available alternative in order to capture the dominant contributions fromthe missing higher orders and stabilise the dependence on unphysical scales. For the Higgs bosonproduction through gluon fusion, we work in the effective theory where the top quark is integratedout. This work published in the article [12] is devoted to demonstrate the SV corrections to thisobservable at N3LO for the Higgs boson, produced through gluon fusion, and leptonic pair in DYproduction.

For the processes under considerations, the NNLO QCD corrections are present [36–38], com-puted long back, and in addition, the partial N3LO SV results [39] are also available. However,due to reasonably large scale uncertainties and crying demand of uplifting the accuracy of theo-retical predictions, we must push the boundaries of existing results. In this work [12], we havecomputed the missing part and completed the SV corrections to the rapidity distributions at N3LOQCD.

The prescription [39] which has been employed to calculate the SV QCD corrections is similar tothat of the inclusive cross section, more specifically, it is a generalisation of the other one. Theinfrared safe contributions under SV approximation can be computed by adding the soft part of therapidity distribution with the UV renormalised virtual part and performing the mass factorisationusing appropriate counter terms. Similar to the inclusive case, the main ingredients to perform thiscomputation are the form factors, overall UV operator renormalisation constant, soft-collinear dis-tribution for rapidity and mass factorisation kernels. These quantities are required up to N3LO tocalculate the rapidity at this order. The three loop quark and gluon form factors [40–43] were cal-culated long back. The operator renormalisation constants are also present. For DY, this constantis not required or equivalently equals to unity. The mass factorisation kernels are also availablein the literature to the required order. The only missing part was the soft-collinear distribution forrapidity at N3LO. This was not possible to compute until very recently. Because of the universalbehaviour of the soft gluons, the soft-collinear distributions for rapidity and inclusive cross sec-tion can be related to all orders in perturbation theory [39]. Employing this beautiful relation, weobtain this quantity at N3LO from the results of soft-collinear distribution of the inclusive crosssection [32]. Using this, along with the existing results of the other relevant quantities, we com-pute the complete analytical expressions of N3LO SV correction to the rapidity distributions forthe Higgs boson in gluon fusion and leptonic pair in DY [12]. We demonstrate the numericalimpact of this correction for the case of Higgs boson at the LHC. This indeed reduces the scaledependence significantly and provides a more reliable theoretical predictions. These are the mostaccurate results for the rapidity distributions of the Higgs boson and DY pair which exist in theliterature and undoubtedly, expected to play very important role in the upcoming run at the LHC.

34 Introduction

1.3.3 Pseudo-Scalar Form Factors at Three Loops in QCD

One of the most popular extensions of the SM, namely, the MSSM and two Higgs doublet modelhave richer Higgs sector containing more than one Higgs boson and there have been intensesearch strategies to observe them at the LHC. In particular, the production of CP-odd Higgsboson/pseudo-scalar at the LHC has been studied in detail, taking into account higher order QCDradiative corrections, due to similarities with its CP-even counter part. Very recently, the N3LOQCD corrections to the inclusive production cross section of the CP-even Higgs boson becomesavailable [44]. So, it is very natural to extend the theoretical accuracy for the CP-odd Higgs bo-son to the same order of N3LO. This requires the 3-loop quark and gluon form factors for thepseudo-scalar which are the only missing ingredients to achieve this goal.

Multiloop and multileg computations play a crucial role to achieve the golden task of makingprecise theoretical predictions. However, the complexity of these computations grows very rapidlywith the increase of number of loops and/or external particles. Nevertheless, it has become a realitydue to several remarkable developments in due course of time. These articles [13,14] are devotedto demonstrate the computations of the 3-loop quark and gluon form factors for the pseudo-scalaroperators in QCD.

The coupling of a pseudo-scalar Higgs boson to gluons is mediated through a heavy quark loop. Inthe limit of large quark mass, it is described by an effective Lagrangian [45] that only admits lightdegrees of freedom. In this effective theory, we compute the 3-loop massless QCD corrections tothe form factor that describes the coupling of a pseudo-scalar Higgs boson to gluons. The eval-uation of this 3-loop form factors is truly a non-trivial task not only because of the involvementof a large number of Feynman diagrams but also due to the presence of the axial vector coupling.We work in dimensional regularisation and use the ’t Hooft-Veltman prescription [8] for the axialvector current, The state-of-the-art techniques including integration-by-parts [46, 47] and Lorentzinvariant [48] identities have been employed to accomplish this task. The UV renormalisationis quite involved since the two operators, present in the Lagrangian, mix under UV renormal-ization due to the axial anomaly and additionally, a finite renormalisation constant needs to beintroduced in order to fulfill the chiral Ward identities. Using the universal infrared factorizationproperties, we independently derive [13] the three-loop operator mixing and finite operator renor-malisation from the renormalisation group equation for the form factors, thereby confirming recentresults [49,50], which were computed following a completely different methodology, in the opera-tor product expansion. This form factor [13,14] is an important ingredient to the precise predictionof the pseudo-scalar Higgs boson production cross section at hadron colliders. We derive the hardmatching coefficient in soft-collinear effective theory (SCET). We also study the form factors in thecontext of leading transcendentality principle and we find that the diagonal form factors becomeidentical to those of N = 4 upon imposing some identification on the quadratic Casimirs. Later,these form factors are used to calculate the SV corrections [14] to the pseudo-scalar productioncross section at N3LO and next-to-next-to-next-to-leading logarithm (N3LL) QCD.

2 Higgs boson production throughbb annihilation at threshold in N3LOQCD

The materials presented in this chapter are the result of an original research done incollaboration with Narayan Rana and V. Ravindran, and these are based on the publishedarticle [11].

Contents2.1 Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 The Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . 372.3 Theoretical Framework for Threshold Corrections . . . . . . . . . . . . 37

2.3.1 The Form Factor 402.3.2 Operator Renormalisation Constant 442.3.3 Mass Factorisation Kernel 442.3.4 Soft-Collinear Distribution 45

2.4 Results of the SV Cross Sections . . . . . . . . . . . . . . . . . . . 472.5 Numerical Impact of SV Cross Sections . . . . . . . . . . . . . . . . 572.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1 Prologue

The discovery of Higgs boson by ATLAS [1] and CMS [2] collaborations of the LHC at CERN hasnot only shed the light on the dynamics behind the electroweak symmetry breaking but also putthe SM of particle physics on a firmer ground. In the SM, the elementary particles such as quarks,leptons and gauge bosons, Z,W± acquire their masses through spontaneous symmetry breaking(SSB). The Higgs mechanism provides the framework for SSB. The SM predicts the existence of aHiggs boson whose mass is a parameter of the model. The recent discovery of the SM Higgs bosonlike particle provides a valuable information on this, namely on its mass which is about 125.5 GeV.The searches for the Higgs boson have been going on for several decades in various experiments.Earlier experiments such as LEP [51] and Tevatron [52] played an important role in the discovery

35

36 Higgs boson production through bb annihilation at threshold in N3LO QCD

by the LHC collaborations through narrowing down its possible mass range. LEP excluded Higgsboson of mass below 114.4 GeV and their precision electroweak measurements [53] hinted themass less than 152 GeV at 95% confidence level (CL), while Tevatron excluded Higgs boson ofmass in the range 162 − 166 GeV at 95% CL.

Higgs bosons are produced dominantly at the LHC via gluon gluon fusion through top quark loop,while the sub-dominant ones are vector boson fusion, associated production of Higgs boson withvector bosons, with top anti-top pairs and also in bottom anti-bottom annihilation. The inclu-sive productions of Higgs boson in gluon gluon [53–61], vector boson fusion processes [62] andassociated production with vector bosons [63] are known to NNLO accuracy in QCD. Higgs pro-duction in bottom anti-bottom annihilation is also known to NNLO accuracy in the variable flavourscheme (VFS) [16–21], while it is known to NLO in the fixed flavour scheme (FFS) [22–27]. Inthe MSSM, the coupling of bottom quarks to Higgs becomes large in the large tan β region, wheretan β is the ratio of vacuum expectation values of up and down type Higgs fields. This can enhancecontributions from bottom anti-bottom annihilation subprocesses.

While the theoretical predictions of NNLO [53–61] and next to next to leading log (NNLL) [64]QCD corrections and of two loop electroweak effects [65,66] played an important role in the Higgsdiscovery, the theoretical uncertainties resulting from factorization and renormalization scales arenot fully under control. Hence, the efforts to go beyond NNLO are going on intensively. Someof the ingredients to obtain N3LO QCD corrections are already available. For example, quark andgluon form factors [40–42, 67, 68], the mass factorization kernels [69] and the renormalizationconstant [70] for the effective operator describing the coupling of Higgs boson with the SM fieldsin the infinite top quark mass limit up to three loop level in dimensional regularization are knownfor some time. In addition, NNLO soft contributions are known [71] to all orders in ε for bothDY and Higgs productions using dimensional regularization with space time dimension beingd = 4+ ε. They were used to obtain the partial N3LO threshold effects [28,29,72–74] to Drell-Yanproduction of di-leptons and inclusive productions of Higgs boson through gluon gluon fusionand in bottom anti-bottom annihilation. Threshold contribution to the inclusive production crosssection is expanded in terms of δ(1 − z) andDi(z) where

Di(z) =

(lni(1 − z)

1 − z

)+

(2.1.1)

with the scaling parameter z = m2H/s for Higgs and z = m2

l+l−/s for DY. Here mH , ml+l− and s aremass of the Higgs boson, invariant mass of the di-leptons and square of the center of mass energyof the partonic reaction responsible for production mechanism respectively. The missing δ(1 − z)terms for the complete N3LO threshold contributions to the Higgs production through gluon gluonfusion are now available due to the seminal work by Anastasiou et al [33] where the relevantsoft contributions were obtained from the real radiations at N3LO level. The resummation ofthreshold effects [75,76] to infra-red safe observables resulting from their factorization propertiesas well as Sudakov resummation of soft gluons provides an elegant framework to obtain thresholdenhanced contributions to inclusive and semi inclusive observables order by order in perturbationtheory. In [32], using this framework, we exploited the universal structure of the soft radiations toobtain the corresponding soft gluon contributions to DY production, which led to the evaluationof missing δ(1 − z) part of the N3LO threshold corrections. In [35], relevant one loop double realemissions from light-like Wilson lines were computed to obtain threshold corrections to Higgsas well as Drell-Yan productions up to N3LO level providing an independent approach. In [34]the universality of soft gluon contributions near threshold and the results of [33] were used toobtain general expression of the hard-virtual coefficient which contributes to N3LO threshold aswell as threshold resummation at next-to-next-to-next-to-leading-logarithmic (N3LL) accuracy for

2.2 The Effective Lagrangian 37

the production cross section of a colourless heavy particle at hadron colliders. For the Higgsproduction through bb annihilation, till date, only partial N3LO threshold corrections are known[28–30] where again the framework of threshold resummation was used. In both [28,29] and [30],it was not possible to determine the δ(1 − z) at N3LO due to the lack of information on three loopfinite part of bottom anti-bottom higgs form factor in QCD and the soft gluon radiation at N3LOlevel. In [30], subleading corrections were also obtained through the method of Mellin moments.The recent results on Higgs form factor with bottom anti-bottom by Gehrmann and Kara [31] andon the universal soft distribution obtained for the Drell-Yan production [32] can now be used toobtain δ(1 − z) part of the threshold N3LO contribution. For the soft gluon radiations in the bbannihilation, the results from [32] can be used as they do not depend on the flavour of the incomingquark states. We have set bottom quark mass to be zero throughout except in the Yukawa coupling.

We begin by writing down the relevant interacting Lagrangian in Sec. 2.2. In the Sec. 2.3, wepresent the formalism of computing threshold QCD corrections to the cross-section and in Sec. 2.4,we present our results for threshold N3LO QCD contributions to Higgs production through bbannihilation at hadron colliders and their numerical impact . The numerical impact of thresholdenhanced N3LO contributions is demonstrated for the LHC energy

√s = 14 TeV by studying the

stability of the perturbation theory under factorization and renormalization scales. Finally we givea brief summary of our findings in Sec. 2.6.

2.2 The Effective Lagrangian

The interaction of bottom quarks and the scalar Higgs boson is given by the action

Lb = φ(x)Ob(x) ≡ −λ√

2φ(x)ψb(x)ψb(x) (2.2.1)

where, ψb(x) and φ(x) denote the bottom quark and scalar Higgs field, respectively. λ is theYukawa coupling given by

√2mb/ν, with the bottom quark mass mb and the vacuum expec-

tation value ν ≈ 246 GeV. In MSSM, for the pseudo-scalar Higgs boson, we need to replaceλφ(x)ψb(x)ψb(x) by λφ(x)ψb(x)γ5ψb(x) in the above equation. The MSSM couplings are given by

λ =

−

√2mb sinαν cos β , φ = h ,√

2mb cosαν cos β , φ = H ,√

2mb tan βν , φ = A

respectively. The angle α measures the mixing of weak and mass eigenstates of neutral Higgsbosons. We use VFS scheme throughout, hence except in the Yukawa coupling, mb is taken to bezero like other light quarks in the theory.

2.3 Theoretical Framework for Threshold Corrections

The inclusive cross-section for the production of a colorless particle, namely, a Higgs bosonthrough gluon fusion/bottom quark annihilation or a pair of leptons in the Drell-Yan at the hadroncolliders is computed using

σI(τ, q2) = σI,(0)(µ2R)

∑i, j=q,q,g

1∫τ

dx Φi j(x, µ2F) ∆I

i j

(τ

x, q2, µ2

R, µ2F

)(2.3.1)

38 Higgs boson production through bb annihilation at threshold in N3LO QCD

with the partonic flux

Φi j(x, µ2F) =

1∫x

dyy

fi(y, µ2F) f j

(xy, µ2

F

). (2.3.2)

In the above expressions, fi(y, µ2F) and f j

(xy , µ

2F

)are the parton distribution functions (PDFs) of

the initial state partons i and j with momentum fractions y and x/y, respectively. These are renor-malized at the factorization scale µF . The dimensionless quantity ∆I

i j

(τx , q

2, µ2R, µ

2F

)is called the

coefficient function of the partonic cross section for the production of a colorless particle frompartons i and j, computed after performing the UV renormalization at scale µR and mass factor-ization at µF . The quantity σI,(0) is a pre-factor of the born level cross section. The variable τ isdefined as q2/s, where

q2 =

m2H for I = H ,

m2l+l− for I = DY .

(2.3.3)

mH is the mass of the Higgs boson and ml+l− is the invariant mass of the final state dilepton pair(l+l−), which can be e+e−, µ+µ−, τ+τ−, in the DY production.

√s and

√s stand for the hadronic

and partonic center of mass energy, respectively. Throughout this chapter, we denote I = H forthe productions of the Higgs boson through gluon (gg) fusion (Fig. 2.1) and bottom quark (bb)annihilation (Fig. 2.2), whereas we write I = DY for the production of a pair of leptons in theDrell-Yan (Fig. 2.3).

g

g

H

Figure 2.1. Higgs boson production in gluon fusion

b

b

H

Figure 2.2. Higgs boson production through bottom quark annihilation

One of the goals of this chapter is to study the impact of the contributions arising from the softgluons to the cross section of a colorless particle production at Hadron colliders. The infrared safecontributions from the soft gluons is obtained by adding the soft part of the cross section with theUV renormalized virtual part and performing mass factorisation using appropriate counter terms.This combination is often called the soft-plus-virtual cross section whereas the remaining portionis known as hard part. Hence, we write the partonic cross section by decomposing into two partsas

∆Ii j(z, q

2, µ2R, µ

2F) = ∆I,SV

i j (z, q2, µ2R, µ

2F) + ∆I,hard

i j (z, q2, µ2R, µ

2F) (2.3.4)

2.3 Theoretical Framework for Threshold Corrections 39

q

q

γ∗/Z

l+

l−

Figure 2.3. Drell-Yan pair production

with z ≡ q2/s. The SV contributions ∆I,SVi j (z, q2, µ2

R, µ2F) contains only the distributions of kind

δ(1 − z) andDi, where the latter one is defined through

Di ≡

[lni(1 − z)

(1 − z)

]+

. (2.3.5)

This is also known as the threshold contributions. On the other hand, the hard part ∆I,hardi j contains

all the regular terms in z. The SV cross section in z-space is computed in d = 4 + ε dimensions, asformulated in [28, 29], using

∆I,SVi j (z, q2, µ2

R, µ2F) = C exp

(Ψ I

i j

(z, q2, µ2

R, µ2F , ε

) )∣∣∣∣ε=0

(2.3.6)

where, Ψ Ii j

(z, q2, µ2

R, µ2F , ε

)is a finite distribution and C is the convolution defined as

Ce f (z) = δ(1 − z) +11!

f (z) +12!

f (z) ⊗ f (z) + · · · . (2.3.7)

Here ⊗ represents Mellin convolution and f (z) is a distribution of the kind δ(1 − z) and Di. Theequivalent formalism of the SV approximation in the Mellin (or N-moment) space, where in-stead of distributions in z, the dominant contributions come from the meromorphic functionsof the variable N (see [75, 76]) and the threshold limit of z → 1 is translated to N → ∞.The Ψ I

i j

(z, q2, µ2

R, µ2F , ε

)is constructed from the form factors F I

i j(as,Q2, µ2, ε) with Q2 = −q2,the overall operator UV renormalization constant ZI

i j(as, µ2R, µ

2, ε), the soft-collinear distributionΦI

i j(as, q2, µ2, z, ε) arising from the real radiations in the partonic subprocesses and the mass fac-torization kernels ΓI

i j(as, µ2, µ2

F , z, ε). In terms of the above-mentioned quantities it takes the fol-lowing form, as presented in [11, 29, 32]

Ψ Ii j

(z, q2, µ2

R, µ2F , ε

)=

(ln

[ZI

i j(as, µ2R, µ

2, ε)]2

+ ln∣∣∣∣F I

i j(as,Q2, µ2, ε)∣∣∣∣2) δ(1 − z)

+ 2ΦIi j(as, q2, µ2, z, ε) − 2C lnΓI

i j(as, µ2, µ2

F , z, ε) . (2.3.8)

In this expression, as ≡ g2s/16π2 is the unrenormalized strong coupling constant which is related

to the renormalized one as(µ2R) ≡ as through the renormalization constant Zas(µ

2R) ≡ Zas as

asS ε =

µ2

µ2R

ε/2 Zasas , (2.3.9)

where, S ε = exp[(γE − ln4π)ε/2)

]and µ is the mass scale introduced to keep the as dimensionless

in d-dimensions. gs is the coupling constant appearing in the bare Lagrangian of QCD. Zas can beobtained by solving the underlying renormalisation group equation (RGE)

µ2R

ddµ2

R

ln Zas =1as

∞∑k=0

ak+2s βk (2.3.10)

40 Higgs boson production through bb annihilation at threshold in N3LO QCD

where, βk’s are the coefficients of the QCD β-function. The solution of the above RGE in terms ofthe βk’s and ε up to O(a4

s)comes out to be

Zas = 1 + as

[2εβ0

]+ a2

s

[4ε2 β

20 +

1εβ1

]+ a3

s

[8ε3 β

30 +

143ε2 β0β1 +

23εβ2

]+ a4

s

[16ε4 β

40 +

463ε3 β

20β1 +

1ε2

(32β2

1 +103β0β2

)+

12εβ3

]. (2.3.11)

Results beyond this order involve β4 and higher order βk’s which are not available yet in theliterature. The βk up to k = 3 are given by [77]

β0 =113

CA −23

n f ,

β1 =343

C2A − 2n f CF −

103

n f CA ,

β2 =285754

C3A −

141554

C2An f +

7954

CAn2f +

119

CFn2f −

20518

CFCAn f + C2Fn f ,

β3 = N2(−

403

+ 352ζ3

)+ N4

(−

1027

+889ζ3

)+ n f N

(649−

2083ζ3

)+ n f N3

(3227−

1049ζ3

)+ n2

f N−2(−

443

+ 32ζ3

)+ n2

f

(449−

323ζ3

)+ n2

f N2(−

2227

+169ζ3

)+ CFn3

f

(154243

)+ C2

Fn2f

(33827−

1769ζ3

)+ C3

Fn f

(23

)+ CAn3

f

(53243

)+ CACFn2

f

(4288243

+1129ζ3

)+ CAC2

Fn f

(−

210227

+1769ζ3

)+ C2

An2f

(3965162

+569ζ3

)+ C2

ACFn f

(7073486

−3289ζ3

)+ C3

An f

(−

39143162

+683ζ3

)+ C4

A

(150653

486−

449ζ3

)(2.3.12)

with the SU(N) quadratic casimirs

CA = N, CF =N2 − 1

2N. (2.3.13)

n f is the number of active light quark flavors.

In this chapter, we will confine our discussion on the threshold corrections to the Higgs bosonproduction through bottom quark annihilation and more precisely our main goal is to compute theSV cross section of this process at N3LO QCD. In the subsequent sections, we will demonstratethe methodology to obtain the ingredients, Eq. (2.3.8) for computing the SV cross section of scalarHiggs boson production at N3LO QCD.

2.3.1 The Form Factor

The quark and gluon form factors represent the QCD loop corrections to the transition matrix el-ement from an on-shell quark-antiquark pair or two gluons to a color-neutral operator O. For thescalar Higgs boson production through bb annihilation, we consider Yukawa interaction, encapsu-lated through the operator Ob present in the interacting Lagrangian 2.2.1. For the process under

2.3 Theoretical Framework for Threshold Corrections 41

consideration, we need to consider bottom quark form factors. The unrenormalised quark formfactors at O(an

s) are defined through

FH,(n)

bb≡〈M

H,(0)bb|M

H,(n)bb〉

〈MH,(0)bb|M

H,(0)bb〉, (2.3.14)

where, n = 0, 1, 2, 3, . . . . In the above expressions |MH,(n)bb〉 is the O(an

s) contribution to the un-renormalised matrix element for the production of the Higgs boson from on-shell bb annihilation.In terms of these quantities, the full matrix element and the full form factors can be written as aseries expansion in as as

|MHbb〉 ≡

∞∑n=0

ansS n

ε

(Q2

µ2

)n ε2

|MH,(n)bb〉 , F H

bb ≡

∞∑n=0

ansS n

ε

(Q2

µ2

)n ε2

FH,(n)

bb

, (2.3.15)

where Q2 = −2 p1.p2 = −q2 and pi (p2i = 0) are the momenta of the external on-shell bottom

quarks. The results of the form factors up to two loop were present for a long time in [21, 78] andthe three loop one was computed recently in [31].

The form factor F Hbb

(as,Q2, µ2, ε) satisfies the KG-differential equation [79–83] which is a directconsequence of the facts that QCD amplitudes exhibit factorisation property, gauge and renormal-isation group (RG) invariances:

Q2 ddQ2 lnF H

bb(as,Q2, µ2, ε) =12

KHbb

as,µ2

R

µ2 , ε

+ GHbb

as,Q2

µ2R

,µ2

R

µ2 , ε

. (2.3.16)

In the above expression, all the poles in dimensional regularisation parameter ε are captured in theQ2 independent function KH

bband the quantities which are finite as ε → 0 are encapsulated in GH

bb.

The solution of the above KG equation can be obtained as [28] (see also [11, 32])

lnF Hbb(as,Q2, µ2, ε) =

∞∑k=1

aksS

kε

(Q2

µ2

)k ε2LH

bb,k(ε) (2.3.17)

with

LHbb,1(ε) =

1ε2

{− 2AH

bb,1

}+

1ε

{GH

bb,1(ε)},

LHbb,2(ε) =

1ε3

{β0AH

bb,1

}+

1ε2

{−

12

AHbb,2 − β0GH

bb,1(ε)}

+1ε

{12

GHbb,2(ε)

},

LHbb,3(ε) =

1ε4

{−

89β2

0AHbb,1

}+

1ε3

{29β1AH

bb,1 +89β0AH

bb,2 +43β2

0GHbb,1(ε)

}+

1ε2

{−

29

AHbb,3 −

13β1GH

bb,1(ε) −43β0GH

bb,2(ε)}

+1ε

{13

GHbb,3(ε)

}. (2.3.18)

In Appendix D, the derivation of the above solution is discussed in great details. AHbb

’s are calledthe cusp anomalous dimensions. The constants GH

bb,i’s are the coefficients of ai

s in the followingexpansions:

GHbb

as,Q2

µ2R

,µ2

R

µ2 , ε

= GHbb

(as(Q2), 1, ε

)+

∫ 1

Q2

µ2R

dxx

AHbb(as(xµ2

R))

42 Higgs boson production through bb annihilation at threshold in N3LO QCD

=

∞∑i=1

ais(Q

2)GHbb,i(ε) +

∫ 1

Q2

µ2R

dxx

AHbb(as(xµ2

R)) . (2.3.19)

However, the solutions of the logarithm of the form factor involves the unknown functions GHbb,i

which are observed to fulfill [40, 84] the following decomposition in terms of collinear (BHbb

), soft( f H

bb) and UV (γH

bb) anomalous dimensions:

GHbb,i(ε) = 2

(BH

bb,i − γHbb,i

)+ f H

bb,i + CHbb,i +

∞∑k=1

εkgH,kbb,i

, (2.3.20)

where, the constants CHbb,i

are given by [29]

CHbb,1 = 0 ,

CHbb,2 = −2β0gH,1

bb,1,

CHbb,3 = −2β1gH,1

bb,1− 2β0

(gH,1

bb,2+ 2β0gH,2

bb,1

). (2.3.21)

In the above expressions, XHbb,i

with X = A, B, f and γHbb,i

are defined through the series expansionin powers of as:

XHbb ≡

∞∑i=1

aisX

Hbb,i , and γH

bb ≡

∞∑i=1

aisγ

Hbb,i . (2.3.22)

f Ii i

are introduced for the first time in the article [84] where it is shown to fulfill the maximallynon-Abelian property up to two loop level whose validity is reconfirmed in [40] at three loop level:

f Hbb =

CF

CAf Hgg . (2.3.23)

This identity implies the soft anomalous dimensions for the Higgs boson production in bottomquark annihilation are related to the same appearing in the Higgs boson production in gluon fusionthrough a simple ratio of the quadratic casimirs of SU(N) gauge group. The same property is alsoobeyed by the cusp anomalous dimensions up to three loop level:

AHbb =

CF

CAAH

gg . (2.3.24)

It is not clear whether this nice property holds true beyond this order of perturbation theory. More-over, due to universality of the quantities denoted by X, these are independent of the operatorsinsertion. These are only dependent on the initial state partons of any process. Moreover, theseare independent of the quark flavors. Hence, being a process of quark annihilation, we can makeuse of the existing results up to three loop which are employed in case of DY pair productions:

XHbb = XDY

qq = XIqq = Xqq . (2.3.25)

Here, q denotes the independence of the quantities on the quark flavors and absence of I representsthe independence of the quantities on the nature of colorless particles. f H

bbcan be found in [40,84],

AHbb

in [40, 69, 85, 86] and BHbb

in [40, 85] up to three loop level. For readers’ convenience we listthem all up to three loop level in the Appendix B. Utilising the results of these known quantitiesand comparing the above expansions of GH

bb,i(ε), Eq. (2.3.20), with the results of the logarithm

2.3 Theoretical Framework for Threshold Corrections 43

of the form factors, we extract the relevant gH,kbb,i

and γHbb,i

’s up to three loop. For soft-virtual

cross section at N3LO we need gH,1bb,3

in addition to the quantities arising from one and two loop.The form factors for the Higgs boson production in bb annihilation up to two loop can be foundin [21, 28, 29] and the three loop one is calculated very recently in the article [31]. These resultsare employed to extract the required gH,k

bb,i’s using Eq. (2.3.17), (2.3.18) and (2.3.20). The relevant

one loop terms are found to be

gH,1bb,1

= CF

{− 2 + ζ2

}, gH,2

bb,1= CF

{2 −

73ζ3

}, gH,3

bb,1= CF

{− 2 +

14ζ2 +

4780ζ2

2

},

(2.3.26)

the relevant two loop terms are

gH,1bb,2

= CFn f

{61681

+109ζ2 −

83ζ3

}+ CFCA

{−

212281−

1039ζ2 +

885ζ2

2 +1523ζ3

}+ C2

F

{8 + 32ζ2 −

885ζ2

2 − 60ζ3

},

gH,2bb,2

= CFn f

{712ζ2

2 −5527ζ2 +

13027

ζ3 −3100243

}+ CACF

{−

36524

ζ22 +

893ζ2ζ3 +

107954

ζ2

−2923

27ζ3 − 51ζ5 +

9142243

}+ C2

F

{965ζ2

2 − 28ζ2ζ3 − 44ζ2 + 116ζ3 + 12ζ5 − 24}

and the required three loop term is

gH,1bb,3

= C2ACF

{−

615263

ζ23 +

27389

ζ22 +

9769ζ2ζ3 −

342263486

ζ2 −1136

3ζ3

2 +19582

9ζ3

+1228

3ζ5 +

40952638748

}+ CAC2

F

{−

15448105

ζ23 −

363445

ζ22 −

25843

ζ2ζ3 +13357

9ζ2

+ 296ζ23 −

115709

ζ3 −1940

3ζ5 −

6133

}+ CACFn f

{−

106445

ζ22 +

3929ζ2ζ3 +

44551243

ζ2

−41552

81ζ3 − 72ζ5 −

61194374

}+ C2

Fn f

{77245

ζ22 −

1523ζ2ζ3 −

317318

ζ2 +15956

27ζ3

−368

3ζ5 +

32899324

}+ CFn2

f

{−

409ζ2

2 −89281

ζ2 +32081

ζ3 −273522187

}+ C3

F

{21584105

ζ23 −

16445

ζ22 + 624ζ2ζ3 − 275ζ2 + 48ζ2

3 − 2142ζ3 + 1272ζ5 + 603}.

(2.3.27)

The results up to two loop were present in the literature [28, 29], however the three loop result isthe new one which is computed in this thesis for the first time. The other constants γH

bb,i, appearing

in the Eq. (2.3.20), up to three loop (i = 3) are obtained as

γHbb,1 = 3CF ,

γHbb,2 =

32

C2F +

976

CFCA −53

CFn f ,

γHbb,3 =

1292

C3F −

1294

C2FCA +

11413108

CFC2A +

(− 23 + 24ζ3

)C2

Fn f

+

(−

27827− 24ζ3

)CFCAn f −

3527

CFn2f . (2.3.28)

These will be utilised in the next subsection to determine overall operator renormalisation constant.

44 Higgs boson production through bb annihilation at threshold in N3LO QCD

2.3.2 Operator Renormalisation Constant

The strong coupling constant renormalisation through Zas is not sufficient to make the form factorF H

bbcompletely UV finite, one needs to perform additional renormalisation to remove the residual

UV divergences which is reflected through the presence of non-zero γHbb

in Eq. (2.3.20). This ad-ditional renormalisation is called the overall operator renormalisation which is performed throughthe constant ZH

bb. This is determined by solving the underlying RG equation:

µ2R

ddµ2

R

ln ZHbb

(as, µ

2R, µ

2, ε)

=

∞∑i=1

ais(µ

2R)γH

bb,i . (2.3.29)

Using the results of γHbb,i

from Eq. (2.3.28) and solving the above RG equation following themethodology described in the Appendix C, we obtain the following overall renormalisation con-stant up to three loop level:

ZHbb = 1 +

∞∑k=1

aksS

kε

µ2R

µ2

k ε2

ZH,(k)bb

(2.3.30)

where,

ZH,(1)bb

=1ε

{6CF

},

ZH,(2)bb

=1ε2

{− 22CFCA + 18C2

F + 4n f CF

}+

1ε

{976

CFCA +32

C2F −

53

n f CF

},

ZH,(3)bb

=1ε3

{9689

CFC2A − 132C2

FCA + 36C3F −

3529

n f CFCA + 24n f C2F +

329

n2f CF

}+

1ε2

{−

488027

CFC2A +

2473

C2FCA + 9C3

F +139627

n f CFCA −103

n f C2F −

8027

n2f CF

}+

1ε

{11413162

CFC2A −

432

C2FCA + 43C3

F −55681

n f CFCA −463

n f C2F −

7081

n2f CF

− 16ζ3n f CFCA + 16ζ3n f C2F

}. (2.3.31)

2.3.3 Mass Factorisation Kernel

The UV finite form factor contains additional divergences arising from the soft and collinear re-gions of the loop momenta. In this section, we address the issue of collinear divergences anddescribe a prescription to remove them. The collinear singularities that arise in the massless limitof partons are removed by absorbing the divergences in the bare PDF through renormalisationof the PDF. This prescription is called the mass factorisation (MF) and is performed at the fac-torisation scale µF . In the process of performing this, one needs to introduce mass factorisationkernels ΓI

i j(as, µ2, µ2

F , z, ε) which essentially absorb the collinear singularities. More specifically,MF removes the collinear singularities arising from the collinear configuration associated with theinitial state partons. The final state collinear singularities are guaranteed to go away once the phasespace integrals are performed after summing over the contributions from virtual and real emissiondiagrams, thanks to Kinoshita-Lee-Nauenberg (KLN) theorem. The kernels satisfy the following

2.3 Theoretical Framework for Threshold Corrections 45

RG equation :

µ2F

ddµ2

F

ΓIi j(z, µ

2F , ε) =

12

∑k

PIik

(z, µ2

F

)⊗ ΓI

k j

(z, µ2

F , ε)

(2.3.32)

where, PI(z, µ2

F

)are Altarelli-Parisi splitting functions (matrix valued). Expanding PI

(z, µ2

F

)and

ΓI(z, µ2F , ε) in powers of the strong coupling constant we get

PIi j(z, µ

2F) =

∞∑k=1

aks(µ

2F)PI,(k−1)

i j (z) (2.3.33)

and

ΓIi j(z, µ

2F , ε) = δi jδ(1 − z) +

∞∑k=1

aksS

kε

µ2F

µ2

k ε2

ΓI,(k)i j (z, ε) . (2.3.34)

The RG equation of ΓI(z, µ2F , ε), Eq. (2.3.32), can be solved in dimensional regularisation in pow-

ers of as. In the MS scheme, the kernel contains only the poles in ε. The solutions [28] up to therequired order ΓI,(3)(z, ε) in terms of PI,(k)(z) are presented in the Appendix (C.0.20). The relevantones up to three loop, PI,(0)(z), PI,(1)(z) and PI,(2)(z) are computed in the articles [69, 85]. For theSV cross section only the diagonal parts of the splitting functions PI,(k)

i j (z) and kernels ΓI,(k)i j (z, ε)

contribute since the diagonal elements of PI,(k)i j (z) contain δ(1−z) andD0 whereas the off-diagonal

elements are regular in the limit z → 1. The most remarkable fact is that these quantities areuniversal, independent of the operators insertion. Hence, for the process under consideration, wemake use of the existing process independent results of kernels and splitting functions:

ΓHi j = ΓI

i j = Γi j and PHi j = PI

i j = Pi j . (2.3.35)

The absence of I represents the independence of these quantities on I. In the next subsection, wediscuss the only remaining ingredient, namely, the soft-collinear distribution.

2.3.4 Soft-Collinear Distribution

The resulting expression from form factor along with the operator renormalisation constant andmass factorisation kernel is not completely finite, it contains some residual divergences which getcancelled against the contribution arising from soft gluon emissions. Hence, the finiteness of ∆H,SV

bbin Eq. (2.3.6) in the limit ε → 0 demands that the soft-collinear distribution, ΦH

bb(as, q2, µ2, z, ε),

has pole structure in ε similar to that of residual divergences. In articles [28] and [29], it wasshown that ΦH

bbmust obey KG type integro-differential equation, which we call KG equation, to

remove that residual divergences:

q2 ddq2Φ

Hbb

(as, q2, µ2, z, ε

)=

12

KHbb

as,µ2

R

µ2 , z, ε + G

Hbb

as,q2

µ2R

,µ2

R

µ2 , z, ε . (2.3.36)

KHbb and G

Hbb play similar roles as those of KH

bband GH

bbin Eq. (2.3.16), respectively. Also,

ΦHbb

(as, q2, µ2, z, ε) being independent of µ2R satisfy the RG equation

µ2R

ddµ2

R

ΦHbb(as, q2, µ2, z, ε) = 0 . (2.3.37)

46 Higgs boson production through bb annihilation at threshold in N3LO QCD

This RG invariance and the demand of cancellation of all the residual divergences arising fromF H

bb,ZH

bband ΓH

bbagainst ΦH

bbimplies the solution of the KG equation as [28, 29]

ΦHbb(as, q2, µ2, z, ε) = ΦH

bb(as, q2(1 − z)2, µ2, ε)

=

∞∑i=1

ais

(q2(1 − z)2

µ2

)i ε2S iε

( iε1 − z

)ΦH

bb,i(ε) (2.3.38)

with

ΦHbb,i(ε) = LH

bb,i(ε)(AH

bb, j → −AHbb, j,G

Hbb, j(ε)→ G

Hbb, j(ε)

)(2.3.39)

where, LHbb,i

(ε) are defined in Eq. (2.3.18). In Appendix E, the derivation of this solution is de-

picted in great details. The z-independent constants GHbb,i(ε) can be obtained by comparing the

poles as well as non-pole terms in ε of ΦHbb,i

(ε) with those arising from form factor, overall renor-malisation constant and splitting functions. We find

GHbb,i(ε) = − f H

bb,i + CHbb,i +

∞∑k=1

εkGH,kbb,i , (2.3.40)

where,

CHbb,1 = 0 ,

CHbb,2 = −2β0G

H,1bb,1 ,

CHbb,3 = −2β1G

H,1bb,1 − 2β0

(G

H,1bb,2 + 2β0G

H,2bb,1

). (2.3.41)

However, due to the universality of the soft gluon contribution, ΦHbb

must be the same as that ofthe DY pair production in quark annihilation since this quantity only depends on the initial statepartons, it does not depend on the final state colorless particle:

ΦHbb = ΦDY

qq = ΦIqq

i.e. GH,kbb,i = G

DY,kqq,i = G

I,kqq,i . (2.3.42)

In the above expression, ΦIqq and G

I,kqq,i are written in order to emphasise the universality of these

quantities i.e. ΦIqq and G

I,kqq,i can be used for any quark annihilation process, these are independent

of the operators insertion. In the beginning, it was observed in [28,29] that these quantities satisfythe maximally non-Abelian property up to O(a2

s):

ΦIqq =

CF

CAΦI

gg and GI,kqq,i =

CF

CAG

I,kgg,i . (2.3.43)

Some of the relevant constants, namely, GI,1qq,1,G

I,2qq,1,G

I,1qq,2 are computed [28, 29] from the results

of the explicit computations of soft gluon emissions to the DY productions. However, to calculatethe SV cross section at N3LO, we need to have the results of G

I,3qq,1,G

I,2qq,2. These are obtained by

employing the above symmetry (2.3.43). In [71], the soft corrections to the production cross sec-tion of the Higgs boson through gluon fusion to O(a2

s) was computed to all orders in dimensional

regularisation parameter ε. Utilising this all order result, we extract GH,3gg,1 and G

H,2gg,2. These essen-

tially lead us to obtain the corresponding quantities for DY production by means of the maximally

2.4 Results of the SV Cross Sections 47

non-Abelian symmetry. The third order constant GH,1gg,3 is extracted from the result of SV cross

section for the production of the Higgs boson at N3LO [33]. We conjecture that the symmetry re-lation (2.3.43) holds true even at the three loop level! Therefore, utilising that property we obtainthe corresponding quantity for the DY production, G

DY,1qq,3 which was presented for the first time

in the article [32]. Later the result was reconfirmed through threshold resummation in [34] andexplicit computations in [35]. This, in turn, establishes our conjecture of maximally non-Abelianproperty at N3LO. Being flavor independent, we can employ all these constants to the problemunder consideration. Below, we list the relevant ones that contribute up to N3LO level:

GH,1bb,1 = CF

{− 3ζ2

},

GH,2bb,1 = CF

{73ζ3

},

GH,3bb,1 = CF

{−

316ζ2

2},

GH,1bb,2 = CFn f

{−

32881

+709ζ2 +

323ζ3

}+ CACF

{242881−

4699ζ2 + 4ζ2

2 −1763ζ3

},

GH,2bb,2 = CACF

{1140ζ2

2 −2033ζ2ζ3 +

141427

ζ2 +2077

27ζ3 + 43ζ5 −

7288243

}+ CFn f

{−

120ζ2

2 −19627

ζ2 −31027

ζ3 +976243

}G

H,1bb,3 = CFCA

2{

15263

ζ23 +

19649

ζ22 +

110009

ζ2ζ3 −765127

486ζ2 +

5363

ζ32 −

5964827

ζ3

−1430

3ζ5 +

71359818748

}+ CFCAn f

{−

5329

ζ22 −

12089

ζ2ζ3 +105059

243ζ2

+45956

81ζ3 +

1483

ζ5 −716509

4374

}+ C2

Fn f

{15215

ζ22 − 88 ζ2ζ3 +

6056

ζ2 +253627

ζ3

+1123

ζ5 −42727324

}+ CFn f

2{

329ζ2

2 −199681

ζ2 −2720

81ζ3 +

115842187

}. (2.3.44)

The above GH,kbb,i enable us to get the ΦH

bbup to three loop level. This completes all the ingredients

required to compute the SV cross section up to N3LO that are presented in the next section.

2.4 Results of the SV Cross Sections

In this section, we present our findings of the SV cross section at N3LO along with the resultsof previous orders. Expanding the SV cross section ∆H,SV

bb, Eq. (2.3.6), in powers of as(µ2

F), weobtain

∆H,SVbb

(z, q2, µ2F) =

∞∑i=1

ais(µ

2F)∆H,SV

bb,i(z, q2, µ2

F) (2.4.1)

where,

∆H,SVbb,i

= ∆H,SVbb,i|δδ(1 − z) +

2i−1∑j=0

∆H,SVbb,i|D jD j .

48 Higgs boson production through bb annihilation at threshold in N3LO QCD

Before presenting the final result, we present the general results of the SV cross section in termsof the anomalous dimensions AH

bb, BH

bb, f H

bb, γH

bband other quantities arising from form factor and

soft-collinear distribution below:

∆H,SVbb,1

= δ(1 − z)[{

2GH,1bb,1 + 2GH,1

bb,1

}+ ζ2

{3AH

bb,1

}+ log

q2

µ2F

{2BHbb,1 − 2γH

bb,1

}]+D0

[log

q2

µ2F

{2AHbb,1

}+

{− 2 f H

bb,1

}]+D1

[{4AH

bb,1

}],

∆H,SVbb,2

= δ(1 − z)[{G

H,1bb,2 + 2

(G

H,1bb,1

)2+ gH,1

bb,2+ 4gH,1

bb,1G

H,1bb,1 + 2

(gH,1

bb,1

)2}

+ β0

{2G

H,2bb,1 + 2gH,2

bb,1

}+ ζ3

{− 8AH

bb,1 f Hbb,1

}+ ζ2

{− 2

(f Hbb,1

)2+ 3AH

bb,2 + 6GH,1bb,1AH

bb,1

+ 6gH,1bb,1

AHbb,1

}+ ζ2β0

{3 f H

bb,1 + 6BHbb,1 − 6γH

bb,1

}+ ζ2

2

{3710

(AH

bb,1

)2}

+ log q2

µ2F

{2BHbb,2 + 4G

H,1bb,1BH

bb,1 + 4gH,1bb,1

BHbb,1 − 2γH

bb,2 − 4γHbb,1G

H,1bb,1 − 4γH

bb,1gH,1bb,1

}+ log

q2

µ2F

β0

{− 2G

H,1bb,1 − 2gH,1

bb,1

}+ log

q2

µ2F

ζ3

{8(AH

bb,1

)2}

+ log q2

µ2F

ζ2

{4AH

bb,1 f Hbb,1 + 6AH

bb,1BHbb,1 − 6γH

bb,1AHbb,1

}+ log

q2

µ2F

ζ2β0

{− 3AH

bb,1

}+ log2

q2

µ2F

{ + 2(BH

bb,1

)2− 4γH

bb,1BHbb,1 + 2

(γmmaHHbb,1

)2}

+ log2 q2

µ2F

β0

{− BH

bb,1 + γHbb,1

}+ log2

q2

µ2F

ζ2

{− 2

(AH

bb,1

)2}]

+D0

[β0

{− 4G

H,1bb,1

}+ ζ3

{16

(AH

bb,1

)2}

+ ζ2

{2AH

bb,1 f Hbb,1

}+ log

q2

µ2F

{ − 4BHbb,1 f H

bb,1 + 2AHbb,2 + 4G

H,1bb,1AH

bb,1 + 4gH,1bb,1

AHbb,1 + 4γH

bb,1 f Hbb,1

}+ log

q2

µ2F

β0

{2 f H

bb,1

}+ log

q2

µ2F

ζ2

{− 2

(AH

bb,1

)2}

+ log2 q2

µ2F

{4AHbb,1BH

bb,1

− 4γHbb,1AH

bb,1

}+ log2

q2

µ2F

β0

{− AH

bb,1

}+

{− 2 f H

bb,2 − 4GH,1bb,1 f H

bb,1 − 4gH,1bb,1

f Hbb,1

}]+D1

[β0

{4 f H

bb,1

}+ ζ2

{− 4

(AH

bb,1

)2}

+ log q2

µ2F

{ − 8AHbb,1 f H

bb,1 + 8AHbb,1BH

bb,1

− 8γHbb,1AH

bb,1

}+ log

q2

µ2F

β0

{− 4AH

bb,1

}+ log2

q2

µ2F

{4(AH

bb,1

)2}

+

{4(

f Hbb,1

)2

+ 4AHbb,2 + 8G

H,1bb,1AH

bb,1 + 8gH,1bb,1

AHbb,1

}]+D2

[β0

{− 4AH

bb,1

}+ log

q2

µ2F

{12(AH

bb,1

)2}

+

{− 12AH

bb,1 f Hbb,1

}]+D3

[{8(AH

bb,1

)2}],

∆H,SVbb,3

= δ(1 − z)[{

23G

H,1bb,3 + 2G

H,1bb,1G

H,1bb,2 +

43

(G

H,1bb,1

)3+

23

gH,1bb,3

+ 2gH,1bb,2G

H,1bb,1

2.4 Results of the SV Cross Sections 49

+ 2gH,1bb,1G

H,1bb,2 + 4gH,1

bb,1

(G

H,1bb,1

)2+ 2gH,1

bb,1gH,1

bb,2+ 4

(gH,1

bb,1

)2G

H,1bb,1 +

43

(gH,1

bb,1

)3}

+ β1

{43G

H,2bb,1 +

43

gH,2bb,1

}+ β0

{43G

H,2bb,2 + 4G

H,1bb,1G

H,2bb,1 +

43

gH,2bb,2

+ 4gH,2bb,1G

H,1bb,1

+ 4gH,1bb,1G

H,2bb,1 + 4gH,1

bb,1gH,2

bb,1

}+ β2

0

{83G

H,3bb,1 +

83

gH,3bb,1

}+ ζ5

{− 96

(AH

bb,1

)2f Hbb,1

}+ ζ5β0

{− 64

(AH

bb,1

)2}

+ ζ3

{−

83

(f Hbb,1

)3− 8AH

bb,2 f Hbb,1 − 8AH

bb,1 f Hbb,2

− 16GH,1bb,1AH

bb,1 f Hbb,1 − 16gH,1

bb,1AH

bb,1 f Hbb,1

}+ ζ3β0

{− 8

(f Hbb,1

)2− 16G

H,1bb,1AH

bb,1

}+ ζ2

3

{160

3

(AH

bb,1

)3}

+ ζ2

{− 4 f H

bb,1 f Hbb,2 + 3AH

bb,3 + 3GH,1bb,2AH

bb,1 − 4GH,1bb,1

(f Hbb,1

)2

+ 6GH,1bb,1AH

bb,2 + 6(G

H,1bb,1

)2AH

bb,1 + 3gH,1bb,2

AHbb,1 − 4gH,1

bb,1

(f Hbb,1

)2+ 6gH,1

bb,1AH

bb,2

+ 12gH,1bb,1G

H,1bb,1AH

bb,1 + 6(gH,1

bb,1

)2AH

bb,1

}+ ζ2β1

{3 f H

bb,1 + 6BHbb,1 − 6γH

bb,1

}+ ζ2β0

{6 f H

bb,2 + 12BHbb,2 + 6G

H,2bb,1AH

bb,1 − 2GH,1bb,1 f H

bb,1 + 12GH,1bb,1BH

bb,1 + 6gH,2bb,1

AHbb,1

+ 6gH,1bb,1

f Hbb,1 + 12gH,1

bb,1BH

bb,1 − 12γHbb,2 − 12γH

bb,1GH,1bb,1 − 12γH

bb,1gH,1bb,1

}+ ζ2β

20

{− 12gH,1

bb,1

}+ ζ2ζ3

{40

(AH

bb,1

)2f Hbb,1

}+ ζ2ζ3β0

{32

(AH

bb,1

)2}

+ ζ22

{−

385

AHbb,1

(f Hbb,1

)2+

375

AHbb,1AH

bb,2 +375G

H,1bb,1

(AH

bb,1

)2+

375

gH,1bb,1

(AH

bb,1

)2}

+ ζ22β0

{AH

bb,1 f Hbb,1 + 18AH

bb,1BHbb,1 − 18γH

bb,1AHbb,1

}+ ζ2

2β20

{− 3AH

bb,1

}+ ζ3

2

{−

28342

(AH

bb,1

)3}

+ log q2

µ2F

{2BHbb,3 + 2G

H,1bb,2BH

bb,1 + 4GH,1bb,1BH

bb,2

+ 4(G

H,1bb,1

)2BH

bb,1 + 2gH,1bb,2

BHbb,1 + 4gH,1

bb,1BH

bb,2 + 8gH,1bb,1G

H,1bb,1BH

bb,1 + 4(gH,1

bb,1

)2BH

bb,1

− 2γHbb,3 − 4γH

bb,2GH,1bb,1 − 4γH

bb,2gH,1bb,1− 2γH

bb,1GH,1bb,2 − 4γH

bb,1

(G

H,1bb,1

)2− 2γH

bb,1gH,1bb,2

− 8γHbb,1gH,1

bb,1G

H,1bb,1 − 4γH

bb,1

(gH,1

bb,1

)2}

+ log q2

µ2F

β1

{− 2G

H,1bb,1 − 2gH,1

bb,1

}+ log

q2

µ2F

β0

{− 2G

H,1bb,2 + 4G

H,2bb,1BH

bb,1 − 4(G

H,1bb,1

)2− 2gH,1

bb,2+ 4gH,2

bb,1BH

bb,1

− 8gH,1bb,1G

H,1bb,1 − 4

(gH,1

bb,1

)2− 4γH

bb,1GH,2bb,1 − 4γH

bb,1gH,2bb,1

}+ log

q2

µ2F

β20

{− 4G

H,2bb,1

− 4gH,2bb,1

}+ log

q2

µ2F

ζ5

{+ 96

(AH

bb,1

)3}

+ log q2

µ2F

ζ3

{8AH

bb,1

(f Hbb,1

)2

− 16AHbb,1BH

bb,1 f Hbb,1 + 16AH

bb,1AHbb,2 + 16G

H,1bb,1

(AH

bb,1

)2+ 16gH,1

bb,1

(AH

bb,1

)2

50 Higgs boson production through bb annihilation at threshold in N3LO QCD

+ 16γHbb,1AH

bb,1 f Hbb,1

}+ log

q2

µ2F

ζ3β0

{24AH

bb,1 f Hbb,1

}+ log

q2

µ2F

ζ2

{− 4BH

bb,1

(f Hbb,1

)2+ 4AH

bb,2 f Hbb,1 + 6AH

bb,2BHbb,1 + 4AH

bb,1 f Hbb,2

+ 6AHbb,1BH

bb,2 + 8GH,1bb,1AH

bb,1 f Hbb,1 + 12G

H,1bb,1AH

bb,1BHbb,1 + 8gH,1

bb,1AH

bb,1 f Hbb,1

+ 12gH,1bb,1

AHbb,1BH

bb,1 − 6γHbb,2AH

bb,1 + 4γHbb,1

(f Hbb,1

)2− 6γH

bb,1AHbb,2

− 12γHbb,1G

H,1bb,1AH

bb,1 − 12γHbb,1gH,1

bb,1AH

bb,1

}+ log

q2

µ2F

ζ2β1

{− 3AH

bb,1

}+ log

q2

µ2F

ζ2β0

{4(

f Hbb,1

)2+ 6BH

bb,1 f Hbb,1 + 12

(BH

bb,1

)2− 6AH

bb,2 − 4GH,1bb,1AH

bb,1

− 12gH,1bb,1

AHbb,1 − 6γH

bb,1 f Hbb,1 − 24γH

bb,1BHbb,1 + 12

(γH

bb,1

)2}

+ log q2

µ2F

ζ2β20

{− 6 f H

bb,1 − 12BHbb,1 + 12γH

bb,1

}+ log

q2

µ2F

ζ2ζ3

{− 40

(AH

bb,1

)3}

+ log q2

µ2F

ζ22

{+

765

(AH

bb,1

)2f Hbb,1 +

375

(AH

bb,1

)2BH

bb,1 −375γH

bb,1

(AH

bb,1

)2}

+ log q2

µ2F

ζ22β0

{−

(AH

bb,1

)2}

+ log2 q2

µ2F

{4BHbb,1BH

bb,2 + 4GH,1bb,1

(BH

bb,1

)2

+ 4gH,1bb,1

(BH

bb,1

)2− 4γH

bb,2BHbb,1 − 4γH

bb,1BHbb,2 − 8γH

bb,1GH,1bb,1BH

bb,1 − 8γHbb,1gH,1

bb,1BH

bb,1

+ 4γHbb,1γ

Hbb,2 + 4

(γH

bb,1

)2G

H,1bb,1 + 4

(γH

bb,1

)2gH,1

bb,1

}+ log2

q2

µ2F

β1

{− BH

bb,1 + γHbb,1

}+ log2

q2

µ2F

β0

{− 2BH

bb,2 − 6GH,1bb,1BH

bb,1 − 6gH,1bb,1

BHbb,1

+ 2γHbb,2 + 6γH

bb,1GH,1bb,1 + 6γH

bb,1gH,1bb,1

}+ log2

q2

µ2F

β20

{+ 2G

H,1bb,1 + 2gH,1

bb,1

}+ log2

q2

µ2F

ζ3

{− 8

(AH

bb,1

)2f Hbb,1 + 16

(AH

bb,1

)2BH

bb,1 − 16γHbb,1

(AH

bb,1

)2}

+ log2 q2

µ2F

ζ3β0

{− 12

(AH

bb,1

)2}

+ log2 q2

µ2F

ζ2

{8AH

bb,1BHbb,1 f H

bb,1

+ 6AHbb,1

(BH

bb,1

)2− 4AH

bb,1AHbb,2 − 4G

H,1bb,1

(AH

bb,1

)2− 4gH,1

bb,1

(AH

bb,1

)2− 8γH

bb,1AHbb,1 f H

bb,1

− 12γHbb,1AH

bb,1BHbb,1 + 6

(γH

bb,1

)2AH

bb,1

}+ log2

q2

µ2F

ζ2β0

{− 6AH

bb,1 f Hbb,1 − 9AH

bb,1BHbb,1

+ 9γHbb,1AH

bb,1

}+ log2

q2

µ2F

ζ2β20

{3AH

bb,1

}+ log2

q2

µ2F

ζ22

{−

385

(AH

bb,1

)3}

+ log3 q2

µ2F

{43

(BH

bb,1

)3− 4γH

bb,1

(BH

bb,1

)2+ 4

(γH

bb,1

)2BH

bb,1 −43

(γH

bb,1

)3}

+ log3 q2

µ2F

β0

{− 2

(BH

bb,1

)2+ 4γH

bb,1BHbb,1 − 2

(γH

bb,1

)2}

2.4 Results of the SV Cross Sections 51

+ log3 q2

µ2F

β20

{23

BHbb,1 −

23γH

bb,1

}+ log3

q2

µ2F

ζ3

{83

(AH

bb,1

)3}

+ log3 q2

µ2F

ζ2

{− 4

(AH

bb,1

)2BH

bb,1 + 4γHbb,1

(AH

bb,1

)2}

+ log3 q2

µ2F

ζ2β0

{2(AH

bb,1

)2}]

+D0

[β1

{− 4G

H,1bb,1

}+ β0

{− 4G

H,1bb,2 − 4G

H,2bb,1 f H

bb,1 − 8(G

H,1bb,1

)2− 4gH,2

bb,1f Hbb,1

− 8gH,1bb,1G

H,1bb,1

}+ β2

0

{− 8G

H,2bb,1

}+ ζ5

{192

(AH

bb,1

)3}

+ ζ3

{32AH

bb,1

(f Hbb,1

)2

+ 32AHbb,1AH

bb,2 + 32GH,1bb,1

(AH

bb,1

)2+ 32gH,1

bb,1

(AH

bb,1

)2}

+ ζ3β0

{48AH

bb,1 f Hbb,1

}+ ζ2

{4(

f Hbb,1

)3+ 2AH

bb,2 f Hbb,1 + 2AH

bb,1 f Hbb,2 + 4G

H,1bb,1AH

bb,1 f Hbb,1 + 4gH,1

bb,1AH

bb,1 f Hbb,1

}+ ζ2β0

{2(

f Hbb,1

)2− 12BH

bb,1 f Hbb,1 + 4G

H,1bb,1AH

bb,1 + 12γHbb,1 f H

bb,1

}+ ζ2ζ3

{− 80

(AH

bb,1

)3}

+ ζ22

{23

(AH

bb,1

)2f Hbb,1

}+ ζ2

2β0

{16

(AH

bb,1

)2}

+ log q2

µ2F

{ − 4BHbb,2 f H

bb,1 − 4BHbb,1 f H

bb,2 + 2AHbb,3 + 2G

H,1bb,2AH

bb,1 − 8GH,1bb,1BH

bb,1 f Hbb,1

+ 4GH,1bb,1AH

bb,2 + 4(G

H,1bb,1

)2AH

bb,1 + 2gH,1bb,2

AHbb,1 − 8gH,1

bb,1BH

bb,1 f Hbb,1 + 4gH,1

bb,1AH

bb,2

+ 8gH,1bb,1G

H,1bb,1AH

bb,1 + 4(gH,1

bb,1

)2AH

bb,1 + 4γHbb,2 f H

bb,1 + 4γHbb,1 f H

bb,2 + 8γHbb,1G

H,1bb,1 f H

bb,1

+ 8γHbb,1gH,1

bb,1f Hbb,1

}+ log

q2

µ2F

β1

{2 f H

bb,1

}+ log

q2

µ2F

β0

{4 f H

bb,2 + 4GH,2bb,1AH

bb,1

+ 8GH,1bb,1 f H

bb,1 − 8GH,1bb,1BH

bb,1 + 4gH,2bb,1

AHbb,1 + 8gH,1

bb,1f Hbb,1 + 8γH

bb,1GH,1bb,1

}+ log

q2

µ2F

β20

{8G

H,1bb,1

}+ log

q2

µ2F

ζ3

{− 64

(AH

bb,1

)2f Hbb,1 + 32

(AH

bb,1

)2BH

bb,1

− 32γHbb,1

(AH

bb,1

)2}

+ log q2

µ2F

ζ3β0

{− 48

(AH

bb,1

)2}

+ log q2

µ2F

ζ2

{− 12AH

bb,1

(f Hbb,1

)2+ 4AH

bb,1BHbb,1 f H

bb,1 − 4AHbb,1AH

bb,2

− 4GH,1bb,1

(AH

bb,1

)2− 4gH,1

bb,1

(AH

bb,1

)2− 4γH

bb,1AHbb,1 f H

bb,1

}+ log

q2

µ2F

ζ2β0

{− 6AH

bb,1 f Hbb,1 + 12AH

bb,1BHbb,1 − 12γH

bb,1AHbb,1

}+ log

q2

µ2F

ζ22

{− 23

(AH

bb,1

)3}

+ log2 q2

µ2F

{ − 4(BH

bb,1

)2f Hbb,1 + 4AH

bb,2BHbb,1

+ 4AHbb,1BH

bb,2 + 8GH,1bb,1AH

bb,1BHbb,1 + 8gH,1

bb,1AH

bb,1BHbb,1 − 4γH

bb,2AHbb,1 + 8γH

bb,1BHbb,1 f H

bb,1

− 4γHbb,1AH

bb,2 − 8γHbb,1G

H,1bb,1AH

bb,1 − 8γHbb,1gH,1

bb,1AH

bb,1 − 4(γH

bb,1

)2f Hbb,1

}

52 Higgs boson production through bb annihilation at threshold in N3LO QCD

+ log2 q2

µ2F

β1

{− AH

bb,1

}+ log2

q2

µ2F

β0

{6BH

bb,1 f Hbb,1 − 2AH

bb,2 − 6GH,1bb,1AH

bb,1

− 6gH,1bb,1

AHbb,1 − 6γH

bb,1 f Hbb,1

}+ log2

q2

µ2F

β20

{− 2 f H

bb,1

}+ log2

q2

µ2F

ζ3

{32

(AH

bb,1

)3}

+ log2 q2

µ2F

ζ2

{12

(AH

bb,1

)2f Hbb,1 − 4

(AH

bb,1

)2BH

bb,1 + 4γHbb,1

(AH

bb,1

)2}

+ log2 q2

µ2F

ζ2β0

{3(AH

bb,1

)2}

+ log3 q2

µ2F

{4AHbb,1

(BH

bb,1

)2− 8γH

bb,1AHbb,1BH

bb,1

+ 4(γH

bb,1

)2AH

bb,1

}+ log3

q2

µ2F

β0

{− 4AH

bb,1BHbb,1 + 4γH

bb,1AHbb,1

}+ log3

q2

µ2F

β20

{23

AHbb,1

}+ log3

q2

µ2F

ζ2

{− 4

(AH

bb,1

)3}

+

{− 2 f H

bb,3 − 2GH,1bb,2 f H

bb,1

− 4GH,1bb,1 f H

bb,2 − 4(G

H,1bb,1

)2f Hbb,1 − 2gH,1

bb,2f Hbb,1 − 4gH,1

bb,1f Hbb,2 − 8gH,1

bb,1G

H,1bb,1 f H

bb,1

− 4(gH,1

bb,1

)2f Hbb,1

}]+D1

[β1

{4 f H

bb,1

}+ β0

{8 f H

bb,2 + 8GH,2bb,1AH

bb,1 + 24GH,1bb,1 f H

bb,1 + 8gH,2bb,1

AHbb,1

+ 8gH,1bb,1

f Hbb,1

}+ β2

0

{16G

H,1bb,1

}+ ζ3

{− 160

(AH

bb,1

)2f Hbb,1

}+ ζ3β0

{− 96

(AH

bb,1

)2}

+ ζ2

{− 28AH

bb,1

(f Hbb,1

)2− 8AH

bb,1AHbb,2 − 8G

H,1bb,1

(AH

bb,1

)2− 8gH,1

bb,1

(AH

bb,1

)2}

+ ζ2β0

{− 24AH

bb,1 f Hbb,1 + 24AH

bb,1BHbb,1 − 24γH

bb,1AHbb,1

}+ ζ2

2

{− 46

(AH

bb,1

)3}

+ log q2

µ2F

{8BHbb,1

(f Hbb,1

)2− 8AH

bb,2 f Hbb,1 + 8AH

bb,2BHbb,1 − 8AH

bb,1 f Hbb,2 + 8AH

bb,1BHbb,2

− 16GH,1bb,1AH

bb,1 f Hbb,1 + 16G

H,1bb,1AH

bb,1BHbb,1 − 16gH,1

bb,1AH

bb,1 f Hbb,1 + 16gH,1

bb,1AH

bb,1BHbb,1

− 8γHbb,2AH

bb,1 − 8γHbb,1

(f Hbb,1

)2− 8γH

bb,1AHbb,2 − 16γH

bb,1GH,1bb,1AH

bb,1 − 16γHbb,1gH,1

bb,1AH

bb,1

}+ log

q2

µ2F

β1

{− 4AH

bb,1

}+ log

q2

µ2F

β0

{− 8

(f Hbb,1

)2+ 8BH

bb,1 f Hbb,1 − 8AH

bb,2

− 32GH,1bb,1AH

bb,1 − 16gH,1bb,1

AHbb,1 − 8γH

bb,1 f Hbb,1

}+ log

q2

µ2F

β20

{− 8 f H

bb,1

}+ log

q2

µ2F

ζ3

{160

(AH

bb,1

)3}

+ log q2

µ2F

ζ2

{56

(AH

bb,1

)2f Hbb,1 − 8

(AH

bb,1

)2BH

bb,1

+ 8γHbb,1

(AH

bb,1

)2}

+ log q2

µ2F

ζ2β0

{+ 24

(AH

bb,1

)2}

+ log2 q2

µ2F

{ − 16AHbb,1BH

bb,1 f Hbb,1 + 8AH

bb,1

(BH

bb,1

)2+ 8AH

bb,1AHbb,2 + 8G

H,1bb,1

(AH

bb,1

)2

+ 8gH,1bb,1

(AH

bb,1

)2+ 16γH

bb,1AHbb,1 f H

bb,1 − 16γHbb,1AH

bb,1BHbb,1 + 8

(γH

bb,1

)2AH

bb,1

}

2.4 Results of the SV Cross Sections 53

+ log2 q2

µ2F

β0

{12AH

bb,1 f Hbb,1 − 12AH

bb,1BHbb,1 + 12γH

bb,1AHbb,1

}+ log2

q2

µ2F

β20

{4AH

bb,1

}+ log2

q2

µ2F

ζ2

{− 28

(AH

bb,1

)3}

+ log3 q2

µ2F

{8(AH

bb,1

)2BH

bb,1 − 8γHbb,1

(AH

bb,1

)2}

+ log3 q2

µ2F

β0

{− 4

(AH

bb,1

)2}

+

{8 f H

bb,1 f Hbb,2 + 4AH

bb,3 + 4GH,1bb,2AH

bb,1 + 8GH,1bb,1

(f Hbb,1

)2

+ 8GH,1bb,1AH

bb,2 + 8(G

H,1bb,1

)2AH

bb,1 + 4gH,1bb,2

AHbb,1 + 8gH,1

bb,1

(f Hbb,1

)2+ 8gH,1

bb,1AH

bb,2

+ 16gH,1bb,1G

H,1bb,1AH

bb,1 + 8(gH,1

bb,1

)2AH

bb,1

}]+D2

[β1

{− 4AH

bb,1

}+ β0

{− 12

(f Hbb,1

)2− 8AH

bb,2 − 32GH,1bb,1AH

bb,1 − 8gH,1bb,1

AHbb,1

}+ β2

0

{− 8 f H

bb,1

}+ ζ3

{160

(AH

bb,1

)3}

+ ζ2

{60

(AH

bb,1

)2f Hbb,1

}+ ζ2β0

{36

(AH

bb,1

)2}

+ log q2

µ2F

{12AHbb,1

(f Hbb,1

)2− 24AH

bb,1BHbb,1 f H

bb,1 + 24AHbb,1AH

bb,2 + 24GH,1bb,1

(AH

bb,1

)2

+ 24gH,1bb,1

(AH

bb,1

)2+ 24γH

bb,1AHbb,1 f H

bb,1

}+ log

q2

µ2F

β0

{36AH

bb,1 f Hbb,1 − 8AH

bb,1BHbb,1

+ 8γHbb,1AH

bb,1

}+ log

q2

µ2F

β20

{+ 8AH

bb,1

}+ log

q2

µ2F

ζ2

{− 60

(AH

bb,1

)3}

+ log2 q2

µ2F

{ − 12(AH

bb,1

)2f Hbb,1 + 24

(AH

bb,1

)2BH

bb,1 − 24γHbb,1

(AH

bb,1

)2}

+ log2 q2

µ2F

β0

{− 18

(AH

bb,1

)2}

+ log3 q2

µ2F

{4(AH

bb,1

)3}

+

{− 4

(f Hbb,1

)3

− 12AHbb,2 f H

bb,1 − 12AHbb,1 f H

bb,2 − 24GH,1bb,1AH

bb,1 f Hbb,1 − 24gH,1

bb,1AH

bb,1 f Hbb,1

}]+D3

[β0

{803

AHbb,1 f H

bb,1

}+ β2

0

{+

163

AHbb,1

}+ ζ2

{− 40

(AH

bb,1

)3}

+ log q2

µ2F

{ − 32(AH

bb,1

)2f Hbb,1 + 16

(AH

bb,1

)2BH

bb,1 − 16γHbb,1

(AH

bb,1

)2}

+ log q2

µ2F

β0

{−

803

(AH

bb,1

)2}

+ log2 q2

µ2F

{16(AH

bb,1

)3}

+

{16AH

bb,1

(f Hbb,1

)2

+ 16AHbb,1AH

bb,2 + 16GH,1bb,1

(AH

bb,1

)2+ 16gH,1

bb,1

(AH

bb,1

)2}]

+D4

[β0

{−

403

(AH

bb,1

)2}

+ log q2

µ2F

{20(AH

bb,1

)3}

+

{− 20

(AH

bb,1

)2f Hbb,1

}]+D5

[{+ 8

(AH

bb,1

)3}]. (2.4.2)

Upon substituting the values of all the anomalous dimensions, beta functions and gH,kbb,i

, GH,kbb,i, we

obtain the results of the scalar Higgs boson production cross section at threshold in bb annihilation

54 Higgs boson production through bb annihilation at threshold in N3LO QCD

up to N3LO for the choices of the scale µ2R = µ2

F :

∆H,SVbb,1

= δ(1 − z)[CF

{− 4 + 8ζ2

}]+D0

[CF

{8 log

q2

µ2F

} +D1

[CF

{16

}],

∆H,SVbb,2

= δ(1 − z)[CFCA

{1669− 8ζ3 +

2329ζ2 −

125ζ2

2

}+ C2

F

{16 − 60ζ3 +

85ζ2

2

}+ n f CF

{89

+ 8ζ3 −409ζ2

}+ log

q2

µ2F

CFCA

{− 12 − 24ζ3

}+ log

q2

µ2F

C2F

{176ζ3 − 24ζ2

}+ log2

q2

µ2F

C2F

{− 32ζ2

}]+D0

[CFCA

{−

161627

+ 56ζ3 +176

3ζ2

}+ C2

F

{256ζ3

}+ n f CF

{22427−

323ζ2

}+ log

q2

µ2F

CFCA

{536

9− 16ζ2

}+ log

q2

µ2F

C2F

{− 32 − 64ζ2

}+ log

q2

µ2F

n f CF

{−

809

}+ log2

q2

µ2F

CFCA

{−

443

}+ log2

q2

µ2F

n f CF

{83

}]+D1

[CFCA

{1072

9− 32ζ2

}+ C2

F

{− 64 − 128ζ2

}+ n f CF

{−

1609

}+ log

q2

µ2F

CFCA

{−

1763

}+ log

q2

µ2F

n f CF

{+

323

}+ log2

q2

µ2F

C2F

{64

}]+D2

[CFCA

{−

1763

}+ n f CF

{323

}+ log

q2

µ2F

C2F

{192

}]+D3

[C2

F

{128

}],

∆H,SVbb,3

= δ(1 − z)[CFC2

A

{68990

81− 84ζ5 −

1421281

ζ3 −4003ζ2

3 − 272ζ2 −1064

3ζ2ζ3

+252827

ζ22 +

13264315

ζ32

}+ C2

FCA

{−

9823−

371449

ζ5 −10940

9ζ3 +

32803

ζ23

+22106

27ζ2 +

278729

ζ2ζ3 −62468135

ζ22 −

20816315

ζ32

}+ C3

F

{1078

3+ 848ζ5

− 1188ζ3 +10336

3ζ2

3 −5503ζ2 − 64ζ2ζ3 +

1525ζ2

2 −184736

315ζ3

2

}+ n f CFCA

{−

1154081

− 8ζ5 +255281

ζ3 +336881

ζ2 +2083ζ2ζ3 −

6728135

ζ22

}+ n f C2

F

{−

709

+5536

9ζ5 +

40889

ζ3 −260027

ζ2 −5504

9ζ2ζ3 +

12152135

ζ22

}+ n2

f CF

{1627−

112081

ζ3 −3281ζ2 +

12827

ζ22

}+ log

q2

µ2F

CFC2A

{−

11803

+ 80ζ5

−2576

9ζ3 +

1603ζ2 +

685ζ2

2

}+ log

q2

µ2F

C2FCA

{3883

+ 240ζ5 +27040

9ζ3

2.4 Results of the SV Cross Sections 55

−338027

ζ2 − 1120ζ2ζ3 −83ζ2

2

}+ log

q2

µ2F

C3F

{− 100 + 5664ζ5 − 568ζ3 + 132ζ2

− 2752ζ2ζ3 −3845ζ2

2

}+ log

q2

µ2F

n f CFCA

{1963

+2089ζ3 −

163ζ2 −

85ζ2

2

}+ log

q2

µ2F

n f C2F

{83−

45289

ζ3 +15227

ζ2 +11215

ζ22

}+ log

q2

µ2F

n2f CF

{649ζ3

}+ log2

q2

µ2F

CFC2A

{44 + 88ζ3

}+ log2

q2

µ2F

C2FCA

{− 880ζ3 −

34969

ζ2 + 128ζ22

}+ log2

q2

µ2F

C3F

{128ζ2

−1792

5ζ2

2

}+ log2

q2

µ2F

n f CFCA

{− 8 − 16ζ3

}+ log2

q2

µ2F

n f C2F

{160ζ3

+4969ζ2

}+ log3

q2

µ2F

C2FCA

{3523ζ2

}+ log3

q2

µ2F

C3F

{5123ζ3

}+ log3

q2

µ2F

n f C2F

{−

643ζ2

}]+D0

[CFC2

A

{−

594058729

− 384ζ5 +40144

27ζ3

+98224

81ζ2 −

3523ζ2ζ3 −

299215

ζ22

}+ C2

FCA

{646427

+32288

9ζ3 +

659227

ζ2

− 1472ζ2ζ3 +1408

3ζ2

2

}+ C3

F

{12288ζ5 − 1024ζ3 − 6144ζ2ζ3

}+ n f CFCA

{125252

729−

24809

ζ3 −29392

81ζ2 +

73615

ζ22

}+ n f C2

F

{8429

−5728

9ζ3 −

150427

ζ2 −147215

ζ22

}+ n2

f CF

{−

3712729

+32027

ζ3 +64027

ζ2

}+ log

q2

µ2F

CFC2A

{62012

81− 352ζ3 −

60169

ζ2 +3525ζ2

2

}+ log

q2

µ2F

C2FCA

{−

2723− 2880ζ3 −

104329

ζ2 +1824

5ζ2

2

}+ log

q2

µ2F

C3F

{128 − 480ζ3 + 512ζ2 −

71045

ζ22

}+ log

q2

µ2F

n f CFCA

{−

1640881

+ 192ζ2

}+ log

q2

µ2F

n f C2F

{−

923

+ 640ζ3

+1600

9ζ2

}+ log

q2

µ2F

n2f CF

{80081−

1289ζ2

}+ log2

q2

µ2F

CFC2A

{−

712027

+1763ζ2

}+ log2

q2

µ2F

C2FCA

{−

1123− 192ζ3 +

17603

ζ2

}+ log2

q2

µ2F

C3F

{2432ζ3 − 192ζ2

}+ log2

q2

µ2F

n f CFCA

{231227−

323ζ2

}+ log2

q2

µ2F

n f C2F

{−

83−

3203ζ2

}+ log2

q2

µ2F

n2f CF

{−

16027

}

56 Higgs boson production through bb annihilation at threshold in N3LO QCD

+ log3 q2

µ2F

CFC2A

{96827

}+ log3

q2

µ2F

C3F

{− 256ζ2

}+ log3

q2

µ2F

n f CFCA

{−

35227

}+ log3

q2

µ2F

n2f CF

{3227

}]+D1

[CFC2

A

{124024

81− 704ζ3 −

120329

ζ2 +7045ζ2

2

}+ C2

FCA

{−

5443− 5760ζ3

−20864

9ζ2 +

36485

ζ22

}+ C3

F

{256 − 960ζ3 + 1024ζ2 −

142085

ζ22

}+ n f CFCA

{−

3281681

+ 384ζ2

}+ n f C2

F

{−

1843

+ 1280ζ3 +3200

9ζ2

}+ n2

f CF

{160081−

2569ζ2

}+ log

q2

µ2F

CFC2A

{−

2848027

+7043ζ2

}+ log

q2

µ2F

C2FCA

{−

2470427

+ 512ζ3 +9856

3ζ2

}+ log

q2

µ2F

C3F

{11008ζ3

− 384ζ2

}+ log

q2

µ2F

n f CFCA

{+

924827−

1283ζ2

}+ log

q2

µ2F

n f C2F

{329627

−1792

3ζ2

}+ log

q2

µ2F

n2f CF

{−

64027

}+ log2

q2

µ2F

CFC2A

{1936

9

}+ log2

q2

µ2F

C2FCA

{8576

9− 256ζ2

}+ log2

q2

µ2F

C3F

{− 256 − 2048ζ2

}+ log2

q2

µ2F

n f CFCA

{−

7049

}+ log2

q2

µ2F

n f C2F

{−

12809

}+ log2

q2

µ2F

n2f CF

{649

}+ log3

q2

µ2F

C2FCA

{−

7043

}+ log3

q2

µ2F

n f C2F

{1283

}]+D2

[CFC2

A

{−

2848027

+7043ζ2

}+ C2

FCA

{−

108169

+ 1344ζ3 +11264

3ζ2

}+ C3

F

{10240ζ3

}+ n f CFCA

{9248

27−

1283ζ2

}+ n f C2

F

{1696

9−

20483

ζ2

}+ n2

f CF

{−

64027

}+ log

q2

µ2F

CFC2A

{3872

9

}+ log

q2

µ2F

C2FCA

{8576

3− 768ζ2

}+ log

q2

µ2F

C3F

{− 768 − 4608ζ2

}+ log

q2

µ2F

n f CFCA

{−

14089

}+ log

q2

µ2F

n f C2F

{−

12803

}+ log

q2

µ2F

n2f CF

{1289

}+ log2

q2

µ2F

C2FCA

{− 1056

}+ log2

q2

µ2F

n f C2F

{192

}+ log3

q2

µ2F

C3F

{256

}]+D3

[CFC2

A

{774427

}+ C2

FCA

{17152

9− 512ζ2

}+ C3

F

{− 512 − 3072ζ2

}

2.5 Numerical Impact of SV Cross Sections 57

+ n f CFCA

{−

281627

}+ n f C2

F

{−

25609

}+ n2

f CF

{25627

}+ log

q2

µ2F

C2FCA

{−

140809

}+ log

q2

µ2F

n f C2F

{2560

9

}+ log2

q2

µ2F

C3F

{1024

}]+D4

[C2

FCA

{−

70409

}+ n f C2

F

{1280

9

}+ log

q2

µ2F

C3F

{1280

}]+D5

[C3

F

{512

}]. (2.4.3)

The results at NLO(∆H,SV

bb,1

)and NNLO

(∆H,SV

bb,2

)match with the existing ones [21]. At N3LO level,

only ∆H,SVbb,3|D j were known [28, 29], remaining terms were not available due to absence of the

required quantities gH,2bb,2

, gH,1bb,3

from form factors and GH,2bb,2, G

H,1bb,3 from soft-collinear distributions.

The recent results of gH,2bb,2

, gH,1bb,3

from [31], GH,2bb,2 from [71] and G

H,1bb,3 from [32] are being employed

to compute the missing δ(1 − z) part i.e. ∆H,SVbb,3|δ which completes the full evaluation of the SV

cross section at N3LO(∆H,SV

bb,3

)and is presented for the first time in [11] by us. For the sake of

completeness, we mention the leading order contribution which is

∆Hbb,0 = δ(1 − z) (2.4.4)

and the overall factor in Eq. (2.3.1) comes out to be

σH,(0)bb

(µ2

F

)=πλ2

(µ2

F

)12m2

H

. (2.4.5)

The above results are presented for the choice µR = µF . The dependence of the SV cross sectionon renormalisation scale µR can be easily restored by employing the RG evolution of as from µF

to µR [87]:

as(µ2

R

)= as

(µ2

F

) 1ω

+ a2s

(µ2

F

) { 1ω2

(−η1 logω

) }+ a3

s

(µ2

F

) { 1ω2

(η2

1 − η2)

+1ω3

(−η2

1 + η2 − η21 logω + η2

1 log2 ω) }

(2.4.6)

where

ω ≡ 1 − β0as(µ2

F

)log

µ2F

µ2R

,ηi ≡

βi

β0. (2.4.7)

The above result of the evolution of the as is a resummed one and the fixed order result can beeasily obtained by performing the series expansion of this equation (2.4.6).

2.5 Numerical Impact of SV Cross Sections

The numerical impact of our results can be studied using the exact LO, NLO, NNLO ∆Hbb,i, i =

0, 1, 2 and the threshold N3LO result ∆H,SVbb,3

. We have used√

s = 14 TeV for the LHC, the Z

58 Higgs boson production through bb annihilation at threshold in N3LO QCD

boson mass MZ = 91.1876 GeV and Higgs boson mass mH = 125.5 GeV throughout. The strongcoupling constant αs(µ2

R) (as = αs/4π) is evolved using the 4-loop RG equations with αN3LOs (mZ) =

0.117 and for parton density sets we use MSTW 2008NNLO [88]. The Yukawa coupling is evolvedusing 4 loop RG with λ(mb) =

√2mb(mb)/ν and mb(mb) = 4.3 GeV.

The renormalization scale dependence is studied by varying µR between 0.1 mH and 10 mH keepingµF = mH/4 fixed. For the factorization scale, we have fixed µR = mH and varied µF between0.1 mH and 10 mH . We find that the perturbation theory behaves better if we include more andmore higher order terms (see Fig.2.4).

H/mR

µ-110 1 10

[pb

]σ

0.3

0.4

0.5

0.6

0.7

LO

NLO

NNLO

SVLO3N

/4H = MF

µ

H/mF

µ-110 1 10

[pb

]σ

0.1

0.9

1.7LO

NLO

NNLO

SVLO3N

H = MR

µ

Figure 2.4. Total cross section for Higgs production in bb annihilation at various orders in as as afunction of µR/mH (left panel) and of µF/mH (right panel) at the LHC with

√s = 14 TeV.

2.6 Summary

To summarize, we have systematically developed a framework to compute threshold contributionsin QCD to the production of Higgs boson in bottom anti-bottom annihilation subprocesses at thehadron colliders. This formalism is applicable for any colorless particle. Factorization of UV, softand collinear singularities and exponentiation of their sum allow us to obtain threshold correctionsorder by order in perturbation theory. Using the recently obtained N3LO soft distribution functionfor Drell-Yan production and the three loop Higgs form factor with bottom anti-bottom quarks, wehave obtained threshold N3LO corrections to Higgs production through bottom anti-bottom anni-hilation. We have also studied the stability of our result under renormalization and factorizationscales.

3 Rapidity Distributions ofDrell-Yan and Higgs Boson atThreshold in N3LO QCD

The materials presented in this chapter are the result of an original research done in collab-oration with Manoj K. Mandal, Narayan Rana and V. Ravindran, and these are based on thepublished article [12].

Contents3.1 Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Theoretical Framework for Threshold Corrections to Rapidity. . . . . . . . 613.3.1 The Form Factor 643.3.2 Operator Renormalisation Constant 683.3.3 Mass Factorisation Kernel 693.3.4 Soft-Collinear Distribution for Rapidity 70

3.4 Results of the SV Rapidity Distributions . . . . . . . . . . . . . . . 72

3.5 Numerical Impact of SV Rapidity Distributions . . . . . . . . . . . . . 84

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.7 Outlook-Beyond N3LO . . . . . . . . . . . . . . . . . . . . . . 87

3.1 Prologue

The Drell-Yan production [89] of a pair of leptons at the LHC is one of the cleanest processes thatcan be studied not only to test the SM to an unprecedented accuracy but also to probe physics be-yond the SM (BSM) scenarios in a very clear environment. Rapidity distributions of Z boson [90]and charge asymmetries of leptons in W boson decays [91] constrain various parton densities and,in addition, possible excess events can provide hints to BSM physics, namely R-parity violatingsupersymmetric models, models with Z′ or with contact interactions and large extra-dimensionmodels. One of the production mechanisms responsible for discovering the Higgs boson of theSM at the LHC [1, 2] is the gluon-gluon fusion through top quark loop. Being a dominant one, it

59

60 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

will continue to play a major role in studying the properties of the Higgs boson and its couplingto other SM particles. Distributions of transverse momentum and rapidity of the Higgs boson aregoing to be very useful tools to achieve this task. Like the inclusive rates [55–61, 64, 92–98],the rapidity distribution of dileptons in DY production and of the Higgs boson in gluon-gluonfusion are also known to NNLO level in perturbative QCD due to seminal works by Anastasiouet al. [36]. The quark and gluon form factors [40–42, 67], the mass factorization kernels [69],and the renormalization constant [70,99,100] for the effective operator describing the coupling ofthe Higgs boson with the SM fields in the infinite top quark mass limit up to three loop level indimensional regularization with space-time dimensions n = 4+ε were found to be useful to obtainthe N3LO threshold effects [28, 29, 72–74] to the inclusive Higgs boson and DY productions atthe LHC, excluding δ(1 − z) terms, where the scaling parameter is z = m2

l+l−/s for the DY processand z = m2

H/s for the Higgs boson. Here, ml+l− , mH and s are the invariant mass of the dileptons,the mass of the Higgs boson, and square of the center of mass energy of the partonic reactionresponsible for the production mechanism, respectively. Recently, Anastasiou et al. [33] made animportant contribution in computing the total rate for the Higgs boson production at N3LO result-ing from the threshold region including the δ(1− z) term. Their result, along with three loop quarkform factors and mass factorization kernels, was used to compute the DY cross section at N3LOat threshold in [32].

In this thesis, we will apply the formalism developed in [39] to obtain rapidity distributions ofthe dilepton pair and of the Higgs boson at N3LO in the threshold region using the availableinformation that led to the computation of the N3LO threshold corrections to the inclusive Higgsboson [33] and DY productions [32].

We begin by writing down the relevant interacting Lagrangian in Sec. 3.2. In the Sec. 3.3, wepresent the formalism of computing threshold QCD corrections to the differential rapidity dis-tribution and in Sec. 3.4, we present our results for the threshold N3LO QCD corrections to therapidity distributions of the dilepton pairs in DY and Higgs boson. The numerical impact in caseof Higgs boson is discussed in brief in Sec. 3.5. The numerical impact of threshold enhancedN3LO contributions is demonstrated for the LHC energy

√s = 14 TeV by studying the stability

of the perturbation theory under factorization and renormalization scales. Finally we give a briefsummary of our findings in Sec. 3.6.

3.2 The Lagrangian

In the SM, the scalar Higgs boson couples to gluons only indirectly through a virtual heavy quarkloop. This loop can be integrated out in the limit of infinite quark mass. The resulting effectiveLagrangian encapsulates the interaction between a scalar φ and QCD particles and reads:

LHeff = GHφ(x)OH(x) (3.2.1)

with

OH(x) ≡ −14

Gaµν(x)Ga,µν(x) ,

GH ≡ −25/4

3as(µ2

R)G12FCH

as(µ2R),

µ2R

m2t

. (3.2.2)

CH(µ2R) is the Wilson coefficient, given as a perturbative expansion in the MS renormalised strong

coupling constant as ≡ as(µ2R), evaluated at the renormalisation scale µR. This is given by [70,101,

3.3 Theoretical Framework for Threshold Corrections to Rapidity 61

102]

CH

as(µ2R),

µ2R

m2t

= 1 + as

{11

}+ a2

s

{2777

18+ 19Lt + n f

(−

676

+163

Lt

) }+ a3

s

{−

2892659648

+897943

144ζ3 +

34669

Lt + 209L2t

+ n f

(40291324

−110779

216ζ3 +

176027

Lt + 46L2t

)+ n2

f

(−

6865486

+7727

Lt −329

L2t

) }(3.2.3)

up to O(a3s) with Lt = log

(µ2

R/m2t

)and n f is the number of active light quark flavors. For the DY

process, we work in the framework of exact SM with n f = 5 number of active light quark flavors.

3.3 Theoretical Framework for Threshold Corrections to Rapid-ity

The differential rapidity distribution for the production of a colorless particle, namely, a Higgsboson through gluon fusion/bottom quark annihilation or a pair of leptons in the DY at the hadroncolliders can be computed using

ddY

σIY

(τ, q2,Y

)= σI,(0)

Y

(τ, q2, µ2

R

)W I

(τ, q2,Y, µ2

R

). (3.3.1)

In the above expression, Y stands for the rapidity which is defined as

Y ≡12

log(

P2.qP1.q

)(3.3.2)

where, Pi and q are the momentum of the incoming hadrons and the colorless particle, respectively.The variable τ equals q2/s with

q2 =

m2H for I = H ,

m2l+l− for I = DY .

(3.3.3)

mH is the mass of the Higgs boson and ml+l− is the invariant mass of the final state dilepton pair(l+l−), which can be e+e−, µ+µ−, τ+τ−, in the DY production.

√s and

√s stand for the hadronic

and partonic center of mass energy, respectively. Throughout this chapter, we denote I = H forthe productions of the Higgs boson through gluon (gg) fusion (Fig. 3.1) and bottom quark (bb)annihilation (Fig. 3.2), whereas we write I =DY for the production of a pair of leptons in the DY(Fig. 3.3). In Eq. (3.3.2), σI

Y is defined through

σIY

(τ, q2,Y

)=

σI(τ, q2,Y

)for I = H ,

ddq2σ

I(τ, q2,Y

)for I = DY .

(3.3.4)

where, σI(τ, q2

)is the inclusive production cross section. σI,(0)

Y is an overall prefactor extractedfrom the leading order contribution. The other quantity W I is given by

W I(τ, q2,Y, µ2

R

)=

(ZI(µ2

R))2

σI,(0)Y

∑i, j=q,q,g

1∫0

dx1

1∫0

dx2HIi j (x1, x2)

1∫0

dzδ(τ − zx1x2)

62 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

g

g

H

Figure 3.1. Higgs boson production in gluon fusion

b

b

H

Figure 3.2. Higgs boson production through bottom quark annihilation

×

∫dPS 1+X |M

Ii j|

2δ

(Y −

12

log(

P2.qP1.q

)),

≡∑

i, j=q,q,g

1∫0

dx1

1∫0

dx2HIi j (x1, x2)

1x1x2

∆IY,i j

(τ,Y, as, µ

2, q2, µ2R, ε

)(3.3.5)

where, we have introduced the dimensionless differential partonic cross section ∆IY,i j. ZI is the

overall operator UV renormalisation constant, xk (k = 1, 2) are the momentum fractions of theinitial state partons i.e. pk = xkPk and H I

i j stands for

H Ii j (x1, x2) ≡

fi (x1) f j (x2) for I = DY ,

fi (x1) f j (x2) for I = H through bb annihilation ,x1 fi (x1) x2 f j (x2) for I = H in gg fusion .

fi(xk) is the unrenormalised PDF of the initial state partons i with momentum fractions xk. X isthe remnants other than the colorless particle I, dPS 1+X is the phase space element for the I + Xsystem and MI

i j represents the partonic level scattering matrix element for the process i j → I.

The renormalised PDF, fi(x1, µ

2F

), renormalised at the factorisation scale µF , is related to the

q

q

γ∗/Z

l+

l−

Figure 3.3. Drell-Yan pair production

3.3 Theoretical Framework for Threshold Corrections to Rapidity 63

unrenormalised ones through Altarelli-Parisi (AP) kernel:

fi(xk, µ

2F

)=

∑j=q,q,g

1∫xk

dzzΓi j

(as, µ

2, µ2F , z, ε

)f j

( xk

z

)(3.3.6)

where the scale µ is introduced to keep the unrenormalised strong coupling constant as dimen-sionless in space-time dimensions d = 4 + ε. as ≡ g2

s/16π2 is the unrenormalized strong couplingconstant which is related to the renormalized one as(µ2

R) ≡ as through the renormalization con-stant Zas(µ

2R) ≡ Zas , Eq. (2.3.9). The form of e Zas is presented in Eq. (2.3.11). Expanding the AP

kernel in powers of as, we get

ΓIi j(as, µ

2, µ2F , z, ε) = δi jδ(1 − z) +

∞∑k=1

aksS

kε

µ2F

µ2

k ε2

ΓI,(k)i j (z, ε) . (3.3.7)

ΓI,(k)i j (z, ε) in terms of the Altarelli-Parisi splitting functions PI,(k)

i j

(z, µ2

F

)are presented in the Ap-

pendix (C.0.20). Employing the Eq. (3.3.6), we can write the renormalisedH Ii j

(x1, x2, µ

2F

)as

H Ii j

(x1, x2, µ

2F

)=

∑k,l

1∫x1

dy1

y1

1∫x2

dy2

y2ΓI

ik(as, µ2, µ2

F , y1, ε)H Ikl

(x1

y1,

x2

y2

)ΓI

jl(as, µ2, µ2

F , y2, ε) .

(3.3.8)

In addition to renormalising the PDF, the AP kernels absorb the initial state collinear singularitiespresent in the ∆I

Y,i j through

∆IY,i j

(τ,Y, q2, µ2

R, µ2F

)=

∫dy1

y1

∫dy2

y2

(ΓI

(as, µ

2, µ2F , y1, ε

))−1

ik∆I

Y,kl

(τ,Y, as, µ

2, q2, µ2R, ε

)×

(ΓI

(as, µ

2, µ2F , y2, ε

))−1

jl. (3.3.9)

The ∆IY,i j is free of UV, soft and collinear singularities. With these we can express W I in terms of

the renormalised quantities. Before writing down the renormalised version of the Eq. (3.3.5), weintroduce two symmetric variables x0

1 and x02 instead of Y and τ through

Y ≡12

log

x01

x02

, τ ≡ x01x0

2 . (3.3.10)

In terms of these new variables, the contributions arising from partonic subprocesses can be shownto depend on the ratios z j = x0

j/x j which take the role of scaling variables at the partonic level. Interms of these newly introduced variables, we get the renormalised W I as

W I(x0

1, x02, q

2, µ2R

)=

∑i, j=q,q,g

1∫x0

1

dz1

z1

1∫x0

2

dz2

z2H I

i j

x01

z1

x02

z2, µ2

F

∆IY,i j

(z1, z2, q2, µ2

R, µ2F

). (3.3.11)

The goal of this chapter is to study the impact of the contributions arising from the soft gluonsto the differential rapidity distributions of a colorless particle production at Hadron colliders. Theinfrared safe contributions from the soft gluons is obtained by adding the soft part of the distri-bution with the UV renormalized virtual part and performing mass factorisation using appropriatecounter terms. This combination is often called the soft-plus-virtual (SV) rapidity distribution

64 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

whereas the remaining portion is known as hard part. Hence, we write the rapidity distribution bydecomposing into two parts as

∆IY,i j(z1, z2, q2, µ2

R, µ2F) = ∆I,SV

Y,i j (z1, z2, q2, µ2R, µ

2F) + ∆I,hard

Y,i j (z1, z2, q2, µ2R, µ

2F) . (3.3.12)

The SV contributions ∆I,SVY,i j (z1, z2, q2, µ2

R, µ2F) contains only the distributions of kind δ(1 − z1),

δ(1 − z2) andDi,Di where the latter ones are defined through

Di ≡

[lni(1 − z1)

(1 − z1)

]+

, Di ≡

[lni(1 − z2)

(1 − z2)

]+

with i = 0, 1, 2, . . . . (3.3.13)

This is also known as the threshold contributions. On the other hand, the hard part ∆I,hardY,i j contains

all the regular terms in z1 and z2. The SV rapidity distribution in z-space is computed in d-dimensions, as formulated in [39], using

∆I,SVY,i j (z1, z2, q2, µ2

R, µ2F) = C exp

(Ψ I

Y,i j

(z1, z2, q2, µ2

R, µ2F , ε

) )∣∣∣∣ε=0

(3.3.14)

where, Ψ IY,i j

(z1, z2, q2, µ2

R, µ2F , ε

)is a finite distribution and C is the convolution defined as

Ce f (z1,z2) = δ(1 − z1)δ(1 − z2) +11!

f (z1, z2) +12!

f (z1, z2) ⊗ f (z1, z2) + · · · . (3.3.15)

Here, ⊗ represents the double Mellin convolution with respect to the pair of variables z1, z2and f (z1, z2) is a distribution of the kind δ(1 − z j), Di and Di. The Ψ I

Y,i j

(z1, z2, q2, µ2

R, µ2F , ε

)is constructed from the form factors F I

i j(as,Q2, µ2, ε) with Q2 = −q2, the overall operator UVrenormalization constant ZI

i j(as, µ2R, µ

2, ε), the soft-collinear distribution ΦIY,i j(as, q2, µ2, z1, z2, ε)

arising from the real radiations in the partonic subprocesses and the mass factorization kernelsΓI

i j(as, µ2, µ2

F , z j, ε). In terms of the above-mentioned quantities it takes the following form, aspresented in [12, 39, 103]

Ψ IY,i j

(z1, z2, q2, µ2

R, µ2F , ε

)=

(ln

[ZI

i j(as, µ2R, µ

2, ε)]2

+ ln∣∣∣∣F I

i j(as,Q2, µ2, ε)∣∣∣∣2) δ(1 − z1)δ(1 − z2)

+ 2ΦIY,i j(as, q2, µ2, z1, z2, ε) − C lnΓI

i j(as, µ2, µ2

F , z1, ε)δ(1 − z2)

− C lnΓIi j(as, µ

2, µ2F , z2, ε)δ(1 − z1) . (3.3.16)

In this chapter, we will confine our discussion on the threshold corrections to the Higgs bosonproduction through gluon fusion and DY pair productions. More precisely, our main goal is tocompute the SV corrections to the rapidity distributions of these two processes at N3LO QCD. Inthe subsequent sections, we will demonstrate the methodology to get the ingredients, Eq. (3.3.16)to compute the SV rapidity distributions at N3LO QCD.

3.3.1 The Form Factor

The quark and gluon form factors represent the QCD loop corrections to the transition matrixelement from an on-shell quark-antiquark pair or two gluons to a color-neutral particle. For theprocesses under consideration, we require gluon form factors in case of scalar Higgs boson pro-duction in gg fusion and quark form factors for DY pair productions from qq annihilation (happensthrough intermediate off-shell photon, γ∗ or Z-boson). The unrenormalised quark form factors atO(an

s) are defined through

FI,(n)

i i≡〈M

I,(0)i i|M

I,(n)i i〉

〈MI,(0)i i|M

I,(0)i i〉, n = 0, 1, 2, 3, · · · (3.3.17)

3.3 Theoretical Framework for Threshold Corrections to Rapidity 65

with

i i =

gg for H,qq for DY .

In the above expressions |MI,(n)i i〉 is the O(an

s) contribution to the unrenormalised matrix elementfor the production of the particle I from on-shell i i annihilation. In terms of these quantities, thefull matrix element and the full form factors can be written as a series expansion in as as

|MIi i〉 ≡

∞∑n=0

ansS n

ε

(Q2

µ2

)n ε2

|MI,(n)i i〉 , F I

i i ≡

∞∑n=0

ansS n

ε

(Q2

µ2

)n ε2

FI,(n)

i i

, (3.3.18)

where Q2 = −2 p1.p2 = −q2 and pi (p2i = 0) are the momenta of the external on-shell quarks or

gluons. Gluon form factors F Hgg up to three loops in QCD were computed in [40–42,68,104,105].

The quark form factors F DYqq up to three loops in QCD are available from [40–42, 67, 68, 95, 96,

105, 106].

The form factor F Ii i

(as,Q2, µ2, ε) satisfies the KG-differential equation [79–83] which is a directconsequence of the facts that QCD amplitudes exhibit factorisation property, gauge and renormal-isation group (RG) invariances:

Q2 ddQ2 lnF I

i i(as,Q2, µ2, ε) =12

K Ii i

as,µ2

R

µ2 , ε

+ GIi i

as,Q2

µ2R

,µ2

R

µ2 , ε

. (3.3.19)

In the above expression, all the poles in dimensional regularisation parameter ε are captured in theQ2 independent function K I

i iand the quantities which are finite as ε → 0 are encapsulated in GI

i i.

The solution of the above KG equation can be obtained as [28] (see also [11, 32])

lnF Ii i(as,Q2, µ2, ε) =

∞∑k=1

aksS

kε

(Q2

µ2

)k ε2LI

i i,k(ε) (3.3.20)

with

LIi i,1(ε) =

1ε2

{− 2AI

i i,1

}+

1ε

{GI

i i,1(ε)},

LIi i,2(ε) =

1ε3

{β0AI

i i,1

}+

1ε2

{−

12

AIi i,2 − β0GI

i i,1(ε)}

+1ε

{12

GIi i,2(ε)

},

LIi i,3(ε) =

1ε4

{−

89β2

0AIi i,1

}+

1ε3

{29β1AI

i i,1 +89β0AI

i i,2 +43β2

0GIi i,1(ε)

}+

1ε2

{−

29

AIii,3 −

13β1GI

i i,1(ε) −43β0GI

i i,2(ε)}

+1ε

{13

GIi i,3(ε)

}. (3.3.21)

In Appendix D, the derivation of the above solution is discussed in great details. AIi i

’s are calledthe cusp anomalous dimensions. The constants GI

i i,i’s are the coefficients of ai

s in the followingexpansions:

GIi i

as,Q2

µ2R

,µ2

R

µ2 , ε

= GIi i

(as(Q2), 1, ε

)+

∫ 1

Q2

µ2R

dxx

AIi i(as(xµ2

R))

=

∞∑k=1

aks(Q

2)GIi i,k(ε) +

∫ 1

Q2

µ2R

dxx

AIi i(as(xµ2

R)) . (3.3.22)

66 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

However, the solutions of the logarithm of the form factor involves the unknown functions GIi i,k

which are observed to fulfill [40, 84] the following decomposition in terms of collinear (BIi i

), soft( f I

i i) and UV (γI

i i) anomalous dimensions:

GIi i,k(ε) = 2

(BI

i i,k − γIi i,k

)+ f I

i i,k + CIi i,k +

∞∑l=1

εlgI,li i,k

, (3.3.23)

where, the constants CIi i,k

are given by [29]

CIi i,1 = 0 ,

CIi i,2 = −2β0gI,1

i i,1,

CIi i,3 = −2β1gI,1

i i,1− 2β0

(gI,1

i i,2+ 2β0gI,2

i i,1

). (3.3.24)

In the above expressions, XIi i,k

with X = A, B, f and γIi i,k

are defined through the series expansionin powers of as:

XIi i ≡

∞∑k=1

aksXI

i i,k , and γIi i ≡

∞∑k=1

aksγ

Ii i,k . (3.3.25)

f Ii i

are introduced for the first time in the article [84] where it is shown to fulfill the maximallynon-Abelian property up to two loop level whose validity is reconfirmed in [40] at three loop:

f Iqq =

CF

CAf Igg . (3.3.26)

This identity implies the soft anomalous dimensions for the production of a colorless particle inquark annihilation are related to the same appearing in the gluon fusion through a simple ratioof quadratic Casimirs of SU(N) gauge group. The same property is also obeyed by the cuspanomalous dimensions up to three loop level:

AIqq =

CF

CAAI

gg . (3.3.27)

It is not clear whether this nice property holds true beyond this order of perturbation theory. More-over, due to universality of the quantities denoted by X, these are independent of the operatorsinsertion. These are only dependent on the initial state partons of any process:

XIi i = Xi i . (3.3.28)

Moreover, these are independent of the quark flavors. Here, absence of I represents the inde-pendence of the quantities on the nature of colorless particles. f I

i ican be found in [40, 84], AH

i iin [40, 69, 85, 86] and BH

i iin [40, 85] up to three loop level. For readers’ convenience we list them

all up to three loop level in the Appendix B. Utilising the results of these known quantities andcomparing the above expansions of GI

i i,k(ε) with the results of the logarithm of the form factors,

we extract the relevant gI,li i,k

and γIi i,k

’s up to three loop level using Eq. (3.3.20), (3.3.21) and(3.3.23). The relevant one loop terms for I = H and i i = gg are found to be

gH,1gg,1 = CAζ2 , gH,2

gg,1 = CA

{1 −

73ζ3

}, gH,3

gg,1 = CA

{4780ζ2

2 −32

}, (3.3.29)

3.3 Theoretical Framework for Threshold Corrections to Rapidity 67

the relevant two loop terms are

gH,1gg,2 = CA

2{

673ζ2 −

443ζ3 +

451181

}+ CAn f

{−

103ζ2 −

403ζ3 −

172481

}+ CFn f

{16ζ3 −

673

},

gH,2gg,2 = CA

2{

671120

ζ22 +

53ζ2ζ3 −

1429ζ2 +

113927

ζ3 − 39ζ5 −141677

972

}+ CAn f

{25960

ζ22

+169ζ2 +

60427

ζ3 +24103486

}+ CFn f

{−

163ζ2

2 −73ζ2 −

923ζ3 +

202736

}, (3.3.30)

and the required three loop term is

gH,1gg,3 = CA

2n f

{−

12845

ζ22 −

889ζ2ζ3 −

14225243

ζ2 −11372

81ζ3 +

2723ζ5 −

50350092187

}+ CFn f

2{−

36845

ζ22 −

889ζ2 −

13769

ζ3 +650827

}+ CACFn f

{156845

ζ22 + 40ζ2ζ3

+50318

ζ2 +20384

27ζ3 +

6083ζ5 −

473705324

}+ CAn f

2{

23245

ζ22 +

10027

ζ2 +699281

ζ3

+9123014374

}+ C3

A

{−

12352315

ζ23 −

574445

ζ22 −

14969

ζ2ζ3 +221521

486ζ2 −

1043ζ3

2

−57830

27ζ3 +

30803

ζ5 +39497339

8748

}+ CF

2n f

{296ζ3 − 480ζ5 +

3043

}. (3.3.31)

Similarly for I = DY and i i = qq, we have for one loop

gDY,1qq,1 = CF {ζ2 − 8} , gDY,2

qq,1 = CF

{−

34ζ2 −

73ζ3 + 8

},

gDY,3qq,1 = CF

{4780ζ2

2 + ζ2 +74ζ3 − 8

}, (3.3.32)

for two loop we require

gDY,1qq,2 = CF

2{−

885ζ2

2 + 58ζ2 − 60ζ3 −14

}+ CACF

{885ζ2

2 −57518

ζ2 +2603ζ3 −

70165324

}+ CFn f

{379ζ2 −

83ζ3 +

5813162

},

gDY,2qq,2 = CF

2{

1085ζ2

2 − 28ζ2ζ3 −4374ζ2 + 184ζ3 + 12ζ5 −

10916

}+ CACF

{−

65324

ζ22

+893ζ2ζ3 +

7297108

ζ2 −12479

54ζ3 − 51ζ5 +

15477973888

}+ CFn f

{712ζ2

2 −42554

ζ2 +30127

ζ3 −1293891944

}(3.3.33)

and the only required three loop term is

gDY,1qq,3 = CF

3{

21584105

ζ23 − 534ζ2

2 + 840ζ2ζ3 − 206ζ2 + 48ζ32 − 2130ζ3 + 1992ζ5 −

15274

}

68 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+ CACF2{−

15448105

ζ23 +

243245

ζ22 −

34483

ζ2ζ3 +55499

18ζ2 + 296ζ3

2 −23402

9ζ3

−3020

3ζ5 +

2303

}+ CF

2n f

{−

70445

ζ22 −

1523ζ2ζ3 −

754118

ζ2 +19700

27ζ3 −

3683ζ5

+73271162

}+ CA

2CF

{−

615263

ζ23 +

3727190

ζ22 +

17869

ζ2ζ3 −1083305

486ζ2 −

11363

ζ32

+85883

18ζ3 +

6883ζ5 −

489027138748

}+ CFn f

2{−

409ζ2

2 −346681

ζ2 +53681

ζ3

−2584452187

}+ CACFn f

{−

129845

ζ22 +

3929ζ2ζ3 +

155008243

ζ2 −68660

81ζ3 − 72ζ5

+3702974

2187

}+ CFn f ,v

(N2 − 4

N

){−

65ζ2

2 + 30ζ2 + 14ζ3 − 80ζ5 + 12}. (3.3.34)

n f ,v is proportional to the charge weighted sum of the quark flavors [42]. The other constants γIi i,k

,appearing in the Eq. (3.3.23), up to three loop (k = 3) are obtained as

γHgg,1 = β0 , γH

gg,2 = 2β1 , γHgg,3 = 3β2

and γDYqq = 0 . (3.3.35)

βi are the coefficient of QCD-β function, presented in Eq. (2.3.12). These will be utilised in thenext subsection to determine the overall operator renormalisation constants.

3.3.2 Operator Renormalisation Constant

The strong coupling constant renormalisation through Zas may not be sufficient to make the formfactor F I

i icompletely UV finite, one needs to perform additional renormalisation to remove the

residual UV divergences which is reflected through the presence of non-zero γIi i

. Due to non-zero γH

gg in Eq. (3.3.35), overall UV renormalisation is required for the Higgs boson production ingluon fusion. However, for DY this is not required. This additional renormalisation is called theoverall operator renormalisation which is performed through the constant ZI

i i. This is determined

by solving the underlying RG equation:

µ2R

ddµ2

R

ln ZIi i

(as, µ

2R, µ

2, ε)

=

∞∑k=1

aks(µ

2R)γI

i i,k . (3.3.36)

Using the results of γIi i,k

from Eq. (3.3.35) and solving the above RG equation following themethodology described in the Appendix C, we obtain the following overall renormalisation con-stant up to three loop level:

ZIi i = 1 +

∞∑k=1

aksS

kε

µ2R

µ2

k ε2

ZI,(k)i i

(3.3.37)

where,

ZH,(1)gg =

1ε

{2β0

},

ZH,(2)gg =

1ε

{2β1

},

3.3 Theoretical Framework for Threshold Corrections to Rapidity 69

ZH,(3)gg =

1ε2

{− 2β0β1

}+

1ε

{2β2

}and ZDY

qq = 1 . (3.3.38)

3.3.3 Mass Factorisation Kernel

The UV finite form factor contains additional divergences arising from the soft and collinear re-gions of the loop momenta. In this section, we address the issue of collinear divergences anddescribe a prescription to remove them. The collinear singularities that arise in the massless limitof partons are removed by absorbing the divergences in the bare PDF through renormalisationof the PDF. This prescription is called the mass factorisation (MF) and is performed at the fac-torisation scale µF . In the process of performing this, one needs to introduce mass factorisationkernels ΓI

i j(as, µ2, µ2

F , z j, ε) which essentially absorb the collinear singularities. More specifically,MF removes the collinear singularities arising from the collinear configuration associated with theinitial state partons. The final state collinear singularities are guaranteed to go away once the phasespace integrals are performed after summing over the contributions from virtual and real emissiondiagrams, thanks to Kinoshita-Lee-Nauenberg theorem. The kernels satisfy the following RGequation :

µ2F

ddµ2

F

ΓIi j(z j, µ

2F , ε) =

12

∑k

PIik

(z j, µ

2F

)⊗ ΓI

k j

(z j, µ

2F , ε

)(3.3.39)

where, PI(z j, µ

2F

)are Altarelli-Parisi splitting functions (matrix valued). Expanding PI

(z j, µ

2F

)and ΓI(z j, µ

2F , ε) in powers of the strong coupling constant we get

PIi j(z j, µ

2F) =

∞∑k=1

aks(µ

2F)PI,(k−1)

i j (z) (3.3.40)

and

ΓIi j(z, µ

2F , ε) = δi jδ(1 − z) +

∞∑k=1

aksS

kε

µ2F

µ2

k ε2

ΓI,(k)i j (z, ε) . (3.3.41)

The RG equation of ΓI(z, µ2F , ε), Eq. (3.3.39), can be solved in dimensional regularisation in pow-

ers of as. In the MS scheme, the kernel contains only the poles in ε. The solutions [28] up to therequired order ΓI,(3)(z, ε) in terms of PI,(k)(z) are presented in the Appendix (C.0.20). The relevantones up to three loop, PI,(0)(z), PI,(1)(z) and PI,(2)(z) are computed in the articles [69, 85]. For theSV cross section only the diagonal parts of the splitting functions PI,(k)

i j (z) and kernels ΓI,(k)i j (z, ε)

contribute since the diagonal elements of PI,(k)i j (z) contain δ(1−z) andD0 whereas the off-diagonal

elements are regular in the limit z → 1. The most remarkable fact is that these quantities areuniversal, independent of the operators insertion. Hence, for the processes under consideration,we make use of the existing process independent results of kernels and splitting functions:

ΓHi j = ΓDY

i j = ΓIi j = Γi j and PH

i j = PDYi j = PI

i j = Pi j . (3.3.42)

The absence of I represents the independence of these quantities on I. In the next subsection, wediscuss the only remaining ingredient, namely, the soft-collinear distribution.

70 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

3.3.4 Soft-Collinear Distribution for Rapidity

The resulting expression from form factor along with the operator renormalisation constant andmass factorisation kernel is not completely finite, it contains some residual divergences which getcancelled against the contribution arising from soft gluon emissions. Hence, the finiteness of ∆I,SV

Y,i iin Eq. (3.3.14) in the limit ε → 0 demands that the soft-collinear distribution,ΦI

Y,i i(as, q2, µ2, z1, z2, ε),

has pole structure in ε similar to that of residual divergences. In article [39], it was shown thatΦI

Y,i imust obey KG type integro-differential equation, which we call KGY equation, to remove

that residual divergences:

q2 ddq2Φ

IY,i i

(as, q2, µ2, z1, z2, ε

)=

12

K IY,i i

as,µ2

R

µ2 , z1, z2, ε

+ GIY,i i

as,q2

µ2R

,µ2

R

µ2 , z1, z2, ε

.(3.3.43)

KIY,i i and G

IY,i i play similar roles as those of K I

i iand GI

i iin Eq. (3.3.19), respectively. Also,

ΦIY,i i

(as, q2, µ2, z, ε) being independent of µ2R satisfy the RG equation

µ2R

ddµ2

R

ΦIY,i i(as, q2, µ2, z1, z2, ε) = 0 . (3.3.44)

This RG invariance and the demand of cancellation of all the residual divergences arising fromF I

i i,ZI

i iand ΓI

i iagainst ΦI

Y,i iimplies the solution of the KGY equation as [39]

ΦIY,i i(as, q2, µ2, z1, z2, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

Y,i i,k(z1, z2, ε) (3.3.45)

with

ΦIY,i i,k(z1, z2, ε) =

{(kε)2 1

4(1 − z1)(1 − z2)[(1 − z1)(1 − z2)]k ε2

}ΦI

Y,i i,k(ε) ,

ΦIY,i i,k(ε) = LI

ii,k

(AI

l → −AIl ,G

Il → G

IY,i i,l(ε)

). (3.3.46)

where, LIi i,k

(ε) are defined in Eq. (3.3.21). In Appendix F, the derivation of this solution is de-

picted in great details. The z j-independent constants GIY,i i,l(ε) can be obtained by comparing the

poles as well as non-pole terms in ε of ΦIY,i i,k

(ε) with those arising from form factor, overallrenormalisation constant and splitting functions. We find

GIY,i i,k(ε) = − f I

i i,k + CIY,i i,k +

∞∑l=1

εlGI,lY,i i,k (3.3.47)

where

CIY,i i,1 = 0 ,

CIY,i i,2 = −2β0G

I,1Y,i i,1 ,

CIY,i i,3 = −2β1G

I,1Y,i i,1 − 2β0

(G

I,1Y,i i,2 + 2β0G

I,2Y,i i,1

). (3.3.48)

In-depth understanding about the pole structures including the single pole [84] of the form factors,overall operator renormalisation constants and mass factorisation kernels helps us to predict all the

3.3 Theoretical Framework for Threshold Corrections to Rapidity 71

poles of the soft-collinear distribution. However, to determine the finite part of the SV correctionsto the rapidity distribution, we need the coefficients of εk (k ≥ 1), G

I,kY,i i,l. Now, we address the

question of determining those constants. This is achieved with the help of an identity which hasbeen found:

1∫0

dx01

1∫0

dx02

(x0

1x02

)N−1 dσIi j

dY=

1∫0

dτ τN−1σIi j . (3.3.49)

In the largeN limit i.e. N → ∞ the above identity relates [39] the ΦIY,i i,k

(ε) with the corresponding

ΦIi i,k

(ε) appearing in the computation of SV cross section, Eq. (2.3.38):

ΦIY,i i,k(ε) =

Γ(1 + kε)Γ2(1 + k ε2 )

ΦIi i,k(ε) . (3.3.50)

Hence, the computation of soft-collinear distribution for the inclusive production cross sectionis sufficient to determine the corresponding one for the rapidity distribution. All the propertiessatisfied by ΦI

i i,k(ε) are obeyed by ΦI

Y,i i,k(ε) too, see Sec. 2.3.4 for all the details. Utilising the

relation (3.3.50), the relevant constants GI,lY,i i,k to determine ΦI

Y,i i,k(ε) up to N3LO level are found

to be

GDY,1Y,qq,1 = CF

{− ζ2

},

GDY,2Y,qq,1 = CF

{13ζ3

},

GDY,3Y,qq,1 = CF

{180ζ2

2

},

GDY,1Y,qq,2 = CACF

{− 4ζ2

2 −673ζ2 −

443ζ3 +

242881

}+ CFn f

{83ζ3 +

103ζ2 −

32881

},

GDY,2Y,qq,2 = CACF

{−

319120

ζ22 −

713ζ2ζ3 +

2029ζ2 +

46927

ζ3 + 43ζ5 −7288243

}+ CFn f

{2960ζ2

2 −289ζ2 −

7027ζ3 +

976243

},

GDY,1Y,qq,3 = CA

2CF

{17392

315ζ2

3 +1538

45ζ2

2 +4136

9ζ2ζ3 −

379417486

ζ2 +5363ζ3

2 − 936ζ3

−1430

3ζ5 +

71359818748

}+ CACFn f

{−

137245

ζ22 −

3929ζ2ζ3 +

51053243

ζ2

+12356

81ζ3 +

1483

ζ5 −716509

4374

}+ CFn f

2{

15245

ζ22 −

31627

ζ2 −32081

ζ3 +115842187

}+ CF

2n f

{15215

ζ22 − 40ζ2 ζ3 +

2756ζ2 +

167227

ζ3 +1123ζ5 −

42727324

}. (3.3.51)

The corresponding constants for the Higgs boson production in gluon fusion can be obtained byemploying the identity

GH,kY,gg,i =

CA

CFG

DY,kY,qq,i . (3.3.52)

72 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

The results up to O(a2s) were present in the literature [39] and the term at O(a3

s) is computed forthe first time by us in the article [12]. Using these, the ΦI

Y,i ican be obtained which are presented

up to three loops in the Appendix F.0.1. This completes all the ingredients required to computethe SV correction to the rapidity distributions up to N3LO that are provided in the next section.

3.4 Results of the SV Rapidity Distributions

In this section, we present our findings of the SV rapidity distributions at N3LO along with theresults of the previous orders. Expanding the SV rapidity distribution , Eq. (3.3.14), in powers ofas(µ2

F), we obtain

∆I,SVY,i i

(z1, z2, q2, µ2

F

)=

∞∑k=1

aks(µ

2F)∆I,SV

Y,i i,k

(z1, z2, q2, µ2

F

)(3.4.1)

where,

∆I,SVY,i i,k

= ∆I,SVY,i i,k|δδδ(1 − z1)δ(1 − z2) +

∞∑j=0

∆I,SVY,i i,k|δD jδ(1 − z2)D j

+

∞∑j=0

∆I,SVY,i i,k|δD j

δ(1 − z2)D j +∑jsl

∆I,SVY,i i,k|D jDlD jDl . (3.4.2)

The symbol jsl implies j, l ≥ 0 and j + l ≤ (2k − 2). Terms proportional to D j and/or D j inEq. (3.4.2) were obtained in [39] and the first term is possible to calculate as the results for thethreshold N3LO QCD corrections to the production cross section are now available for DY [32]and the Higgs boson [33] productions. By setting µ2

R = µ2F we present the results. For I = H and

i i = gg, we obtain [12]

∆H,SVY,gg,1 = δ(1 − z1)δ(1 − z2)

[CA

{12ζ2

}]+D0δ(1 − z2)

[log

q2

µ2F

CA

{4}]

+D0D0

[CA

{4}]

+D1δ(1 − z2)[CA

{4}]

+D0δ(1 − z1)[

log q2

µ2F

CA

{4}]

+D1δ(1 − z1)[CA

{4}],

∆H,SVY,gg,2 = δ(1 − z1)δ(1 − z2)

[C2

A

{93 − 44ζ3 +

2683ζ2 +

2525ζ2

2

}+ n f CA

{−

803− 8ζ3

−403ζ2

}+ n f CF

{−

673

+ 16ζ3

}+ log

q2

µ2F

C2A

{− 24 + 56ζ3 − 44ζ2

}+ log

q2

µ2F

n f CA

{8 + 8ζ2

}+ log

q2

µ2F

n f CF

{4}

+ log2 q2

µ2F

C2A

{− 16ζ2

}]+D0δ(1 − z2)

[C2

A

{−

80827

+ 60ζ3 +443ζ2

}+ n f CA

{11227−

83ζ2

}+ log

q2

µ2F

C2A

{2689

+ 8ζ2

}+ log

q2

µ2F

n f CA

{−

409

}+ log2

q2

µ2F

C2A

{−

223

}+ log2

q2

µ2F

n f CA

{43

}]+D0D0

[C2

A

{2689

+ 8ζ2

}+ n f CA

{−

409

}

3.4 Results of the SV Rapidity Distributions 73

+ log q2

µ2F

C2A

{−

443

}+ log

q2

µ2F

n f CA

{83

}+ log2

q2

µ2F

C2A

{16

}]+D0D1

[C2

A

{−

443

}+ n f CA

{83

}+ log

q2

µ2F

C2A

{48

}]+D0D2

[C2

A

{24

}]+D1δ(1 − z2)

[C2

A

{268

9+ 8ζ2

}+ n f CA

{−

409

}+ log

q2

µ2F

C2A

{−

443

}+ log

q2

µ2F

n f CA

{83

}+ log2

q2

µ2F

C2A

{16

}]+D1D0

[C2

A

{−

443

}+ n f CA

{83

}+ log

q2

µ2F

C2A

{48

}]+D1D1

[C2

A

{48

}]+D2δ(1 − z2)

[C2

A

{−

223

}+ n f CA

{43

}+ log

q2

µ2F

C2A

{24

}]+D2D0

[C2

A

{24

}]+D3δ(1 − z2)

[C2

A

{8}]

+D0δ(1 − z1)[C2

A

{−

80827

+ 60ζ3 +443ζ2

}+ n f CA

{11227−

83ζ2

}+ log

q2

µ2F

C2A

{2689

+ 8ζ2

}+ log

q2

µ2F

n f CA

{−

409

}+ log2

q2

µ2F

C2A

{−

223

}+ log2

q2

µ2F

n f CA

{43

}]+D1δ(1 − z1)

[C2

A

{2689

+ 8ζ2

}+ n f CA

{−

409

}+ log

q2

µ2F

C2A

{−

443

}+ log

q2

µ2F

n f CA

{83

}+ log2

q2

µ2F

C2A

{16

}]+D2δ(1 − z1)

[C2

A

{−

223

}+ n f CA

{43

}+ log

q2

µ2F

C2A

{24

}]+D3δ(1 − z1)

[C2

A

{8}],

∆H,SVY,gg,3 = δ(1 − z1)δ(1 − z2)

[C3

A

{215131

81+

13649

ζ5 −54820

27ζ3 +

16003

ζ23 +

4191427

ζ2

− 88ζ2ζ3 +40432135

ζ22 +

12032105

ζ32

}+ n f C2

A

{−

9805981

+1192

9ζ5 +

253627

ζ3

−710827

ζ2 − 272ζ2ζ3 +124027

ζ22

}+ n f CFCA

{−

6399181

+ 160ζ5 + 400ζ3 −2270

9ζ2

+ 288ζ2ζ3 +17645

ζ22

}+ n f C2

F

{6089− 320ζ5 +

5923ζ3

}+ n2

f CA

{251527

+1123ζ3

−643ζ2 −

20815

ζ22

}+ n2

f CF

{896281−

2243ζ3 −

1849ζ2 −

3245ζ2

2

}+ log

q2

µ2F

C3A

{−

82849

+ 224ζ5 +10408

9ζ3 −

2252827

ζ2 − 224ζ2ζ3 −1276

3ζ2

2

}+ log

q2

µ2F

n f C2A

{4058

9−

11209

ζ3 +8488

27ζ2 +

2323ζ2

2

}+ log

q2

µ2F

n f CFCA

{6163−

3523ζ3 + 72ζ2

}+ log

q2

µ2F

n f C2F

{− 4

}

74 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+ log q2

µ2F

n2f CA

{−

3709−

323ζ3 −

1609ζ2

}+ log

q2

µ2F

n2f CF

{−

1043

+643ζ3

}+ log2

q2

µ2F

C3A

{88 − 264ζ3 −

6929ζ2 −

3845ζ2

2

}+ log2

q2

µ2F

n f C2A

{−

1363

+ 48ζ3

−208

9ζ2

}+ log2

q2

µ2F

n f CFCA

{−

443

}+ log2

q2

µ2F

n2f CA

{163

+163ζ2

}+ log2

q2

µ2F

n2f CF

{83

}+ log3

q2

µ2F

C3A

{1283ζ3 +

1763ζ2

}+ log3

q2

µ2F

n f C2A

{−

323ζ2

}]+D0δ(1 − z2)

[C3

A

{−

297029729

+ 192ζ5 +27128

27ζ3

+18056

81ζ2 −

6083ζ2ζ3 +

885ζ2

2

}+ n f C2

A

{62626729

−3923ζ3 −

641681

ζ2 +165ζ2

2

}+ n f CFCA

{171127−

3049ζ3 − 8ζ2 −

325ζ2

2

}+ n2

f CA

{−

1856729

−3227ζ3 +

16027

ζ2

}+ log

q2

µ2F

C3A

{61138

81− 704ζ3 +

1049ζ2 −

645ζ2

2

}+ log

q2

µ2F

n f C2A

{−

1684481

+ 32ζ3 +643ζ2

}+ log

q2

µ2F

n f CFCA

{− 126 + 96ζ3

}+ log

q2

µ2F

n2f CA

{40081

−329ζ2

}+ log2

q2

µ2F

C3A

{−

615227

+ 352ζ3 −1763ζ2

}+ log2

q2

µ2F

n f C2A

{202027

+323ζ2

}+ log2

q2

µ2F

n f CFCA

{20

}+ log2

q2

µ2F

n2f CA

{−

8027

}+ log3

q2

µ2F

C3A

{48427− 64ζ2

}+ log3

q2

µ2F

n f C2A

{−

17627

}+ log3

q2

µ2F

n2f CA

{1627

}]+D0D0

[C3

A

{61138

81− 704ζ3 +

1049ζ2 −

645ζ2

2

}+ n f C2

A

{−

1684481

+ 32ζ3 +643ζ2

}+ n f CFCA

{− 126 + 96ζ3

}+ n2

f CA

{40081−

329ζ2

}+ log

q2

µ2F

C3A

{−

53929

+ 960ζ3 + 176ζ2

}+ log

q2

µ2F

n f C2A

{407227− 32ζ2

}+ log

q2

µ2F

n f CFCA

{24

}+ log

q2

µ2F

n2f CA

{−

16027

}+ log2

q2

µ2F

C3A

{292 − 192ζ2

}+ log2

q2

µ2F

n f C2A

{−

4969

}+ log2

q2

µ2F

n2f CA

{169

}+ log3

q2

µ2F

C3A

{−

1763

}+ log3

q2

µ2F

n f C2A

{323

}]+D0D1

[C3

A

{−

1681627

+ 976ζ3 +1232

3ζ2

}+ n f C2

A

{365627−

2243ζ2

}+ n f CFCA

{8}

+ n2f CA

{−

16027

}+ log

q2

µ2F

C3A

{7400

9

− 384ζ2

}+ log

q2

µ2F

n f C2A

{−

13129

}+ log

q2

µ2F

n2f CA

{329

}

3.4 Results of the SV Rapidity Distributions 75

+ log2 q2

µ2F

C3A

{− 264

}+ log2

q2

µ2F

n f C2A

{48

}+ log3

q2

µ2F

C3A

{64

}]+D0D2

[C3

A

{3700

9− 192ζ2

}+ n f C2

A

{−

6569

}+ n2

f CA

{169

}+ log

q2

µ2F

C3A

{−

8803

}+ log

q2

µ2F

n f C2A

{1603

}+ log2

q2

µ2F

C3A

{192

}]+D0D3

[C3

A

{−

8809

}+ n f C2

A

{1609

}+ log

q2

µ2F

C3A

{160

}]+D0D4

[C3

A

{40

}]+D1δ(1 − z2)

[C3

A

{61138

81− 704ζ3 +

1049ζ2 −

645ζ2

2

}+ n f C2

A

{−

1684481

+ 32ζ3

+643ζ2

}+ n f CFCA

{− 126 + 96ζ3

}+ n2

f CA

{40081−

329ζ2

}+ log

q2

µ2F

C3A

{−

53929

+ 960ζ3 + 176ζ2

}+ log

q2

µ2F

n f C2A

{407227− 32ζ2

}+ log

q2

µ2F

n f CFCA

{24

}+ log

q2

µ2F

n2f CA

{−

16027

}+ log2

q2

µ2F

C3A

{292 − 192ζ2

}+ log2

q2

µ2F

n f C2A

{−

4969

}+ log2

q2

µ2F

n2f CA

{169

}+ log3

q2

µ2F

C3A

{−

1763

}+ log3

q2

µ2F

n f C2A

{323

}]+D1D0

[C3

A

{−

1681627

+ 976ζ3 +1232

3ζ2

}+ n f C2

A

{365627−

2243ζ2

}+ n f CFCA

{8}

+ n2f CA

{−

16027

}+ log

q2

µ2F

C3A

{7400

9− 384ζ2

}+ log

q2

µ2F

n f C2A

{−

13129

}+ log

q2

µ2F

n2f CA

{329

}+ log2

q2

µ2F

C3A

{− 264

}+ log2

q2

µ2F

n f C2A

{48

}+ log3

q2

µ2F

C3A

{64

}]+D1D1

[C3

A

{7400

9− 384ζ2

}+ n f C2

A

{−

13129

}+ n2

f CA

{329

}+ log

q2

µ2F

C3A

{−

17603

}+ log

q2

µ2F

n f C2A

{3203

}+ log2

q2

µ2F

C3A

{384

}]+D1D2

[C3

A

{−

8803

}+ n f C2

A

{1603

}+ log

q2

µ2F

C3A

{480

}]+D1D3

[C3

A

{160

}]+D2δ(1 − z2)

[C3

A

{−

840827

+ 488ζ3 +6163ζ2

}+ n f C2

A

{182827−

1123ζ2

}+ n f CFCA

{4}

+ n2f CA

{−

8027

}+ log

q2

µ2F

C3A

{3700

9

− 192ζ2

}+ log

q2

µ2F

n f C2A

{−

6569

}+ log

q2

µ2F

n2f CA

{169

}+ log2

q2

µ2F

C3A

{− 132

}+ log2

q2

µ2F

n f C2A

{24

}+ log3

q2

µ2F

C3A

{32

}]

76 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+D2D0

[C3

A

{3700

9− 192ζ2

}+ n f C2

A

{−

6569

}+ n2

f CA

{169

}+ log

q2

µ2F

C3A

{−

8803

}+ log

q2

µ2F

n f C2A

{1603

}+ log2

q2

µ2F

C3A

{192

}]+D2D1

[C3

A

{−

8803

}+ n f C2

A

{1603

}+ log

q2

µ2F

C3A

{480

}]+D2D2

[C3

A

{240

}]+D3δ(1 − z2)

[C3

A

{370027− 64ζ2

}+ n f C2

A

{−

65627

}+ n2

f CA

{1627

}+ log

q2

µ2F

C3A

{−

8809

}+ log

q2

µ2F

n f C2A

{1609

}+ log2

q2

µ2F

C3A

{64

}]+D3D0

[C3

A

{−

8809

}+ n f C2

A

{1609

}+ log

q2

µ2F

C3A

{160

}]+D3D1

[C3

A

{160

}]+D4δ(1 − z2)

[C3

A

{−

2209

}+ n f C2

A

{409

}+ log

q2

µ2F

C3A

{40

}]+D4D0

[C3

A

{40

}]+D5δ(1 − z2)

[C3

A

{8}]

+D0δ(1 − z1)[C3

A

{−

297029729

+ 192ζ5 +27128

27ζ3 +

1805681

ζ2 −6083ζ2ζ3 +

885ζ2

2

}+ n f C2

A

{62626729

−392

3ζ3 −

641681

ζ2 +165ζ2

2

}+ n f CFCA

{1711

27−

3049ζ3 − 8ζ2

−325ζ2

2

}+ n2

f CA

{−

1856729

−3227ζ3 +

16027

ζ2

}+ log

q2

µ2F

C3A

{61138

81− 704ζ3

+1049ζ2 −

645ζ2

2

}+ log

q2

µ2F

n f C2A

{−

1684481

+ 32ζ3 +643ζ2

}+ log

q2

µ2F

n f CFCA

{− 126 + 96ζ3

}+ log

q2

µ2F

n2f CA

{40081−

329ζ2

}+ log2

q2

µ2F

C3A

{−

615227

+ 352ζ3 −1763ζ2

}+ log2

q2

µ2F

n f C2A

{202027

+323ζ2

}+ log2

q2

µ2F

n f CFCA

{20

}+ log2

q2

µ2F

n2f CA

{−

8027

}+ log3

q2

µ2F

C3A

{48427− 64ζ2

}+ log3

q2

µ2F

n f C2A

{−

17627

}+ log3

q2

µ2F

n2f CA

{1627

}]+D1δ(1 − z1)

[C3

A

{61138

81− 704ζ3 +

1049ζ2 −

645ζ2

2

}+ n f C2

A

{−

1684481

+ 32ζ3 +643ζ2

}+ n f CFCA

{− 126 + 96ζ3

}+ n2

f CA

{40081

−329ζ2

}+ log

q2

µ2F

C3A

{−

53929

+ 960ζ3 + 176ζ2

}+ log

q2

µ2F

n f C2A

{407227

− 32ζ2

}+ log

q2

µ2F

n f CFCA

{24

}+ log

q2

µ2F

n2f CA

{−

16027

}

3.4 Results of the SV Rapidity Distributions 77

+ log2 q2

µ2F

C3A

{292 − 192ζ2

}+ log2

q2

µ2F

n f C2A

{−

4969

}+ log2

q2

µ2F

n2f CA

{169

}+ log3

q2

µ2F

C3A

{−

1763

}+ log3

q2

µ2F

n f C2A

{323

}]+D2δ(1 − z1)

[C3

A

{−

840827

+ 488ζ3 +6163ζ2

}+ n f C2

A

{182827−

1123ζ2

}+ n f CFCA

{4}

+ n2f CA

{−

8027

}+ log

q2

µ2F

C3A

{3700

9− 192ζ2

}+ log

q2

µ2F

n f C2A

{−

6569

}+ log

q2

µ2F

n2f CA

{169

}+ log2

q2

µ2F

C3A

{− 132

}+ log2

q2

µ2F

n f C2A

{24

}+ log3

q2

µ2F

C3A

{32

}]+D3δ(1 − z1)

[C3

A

{370027− 64ζ2

}+ n f C2

A

{−

65627

}+ n2

f CA

{1627

}+ log

q2

µ2F

C3A

{−

8809

}+ log

q2

µ2F

n f C2A

{1609

}+ log2

q2

µ2F

C3A

{64

}]+D4δ(1 − z1)

[C3

A

{−

2209

}+ n f C2

A

{409

}+ log

q2

µ2F

C3A

{40

}]+D5δ(1 − z1)

[C3

A

{8}]

(3.4.3)

and for I = DY and i i = qq, we get [12]

∆DY,SVY,qq,1 = δ(1 − z1)δ(1 − z2)

[CF

{− 16 + 12ζ2

}+ log

q2

µ2F

CF

{6}]

+D0δ(1 − z2)[

log q2

µ2F

CF

{4}]

+D0D0

[CF

{4}]

+D1δ(1 − z2)[CF

{4}]

+D0δ(1 − z1)[

log q2

µ2F

CF

{4}]

+D1δ(1 − z1)[CF

{4}],

∆DY,SVY,qq,2 = δ(1 − z1)δ(1 − z2)

[CFCA

{−

153512

+1723ζ3 +

8609ζ2 −

525ζ2

2

}+ C2

F

{5114− 60ζ3 − 134ζ2 +

3045ζ2

2

}+ n f CF

{127

6+

83ζ3 −

1529ζ2

}+ log

q2

µ2F

CFCA

{1933− 24ζ3 −

443ζ2

}+ log

q2

µ2F

C2F

{− 93 + 80ζ3 + 48ζ2

}+ log

q2

µ2F

n f CF

{−

343

+83ζ2

}+ log2

q2

µ2F

CFCA

{− 11

}+ log2

q2

µ2F

C2F

{18 − 16ζ2

}+ log2

q2

µ2F

n f CF

{2}]

+D0δ(1 − z2)[CFCA

{−

80827

+ 28ζ3 +443ζ2

}+ C2

F

{32ζ3

}+ n f CF

{11227−

83ζ2

}+ log

q2

µ2F

CFCA

{2689− 8ζ2

}+ log

q2

µ2F

C2F

{− 64 + 16ζ2

}+ log

q2

µ2F

n f CF

{−

409

}+ log2

q2

µ2F

CFCA

{−

223

}+ log2

q2

µ2F

C2F

{24

}+ log2

q2

µ2F

n f CF

{43

}]

78 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+D0D0

[CFCA

{2689− 8ζ2

}+ C2

F

{− 64 + 16ζ2

}+ n f CF

{−

409

}+ log

q2

µ2F

CFCA

{−

443

}+ log

q2

µ2F

C2F

{24

}+ log

q2

µ2F

n f CF

{83

}+ log2

q2

µ2F

C2F

{16

}]+D0D1

[CFCA

{−

443

}+ n f CF

{83

}+ log

q2

µ2F

C2F

{48

}]+D0D2

[C2

F

{24

}]+D1δ(1 − z2)

[CFCA

{2689− 8ζ2

}+ C2

F

{− 64 + 16ζ2

}+ n f CF

{−

409

}+ log

q2

µ2F

CFCA

{−

443

}+ log

q2

µ2F

C2F

{24

}+ log

q2

µ2F

n f CF

{83

}+ log2

q2

µ2F

C2F

{16

}]+D1D0

[CFCA

{−

443

}+ n f CF

{83

}+ log

q2

µ2F

C2F

{48

}]+D1D1

[C2

F

{48

}]+D2δ(1 − z2)

[CFCA

{−

223

}+ n f CF

{43

}+ log

q2

µ2F

C2F

{24

}]+D2D0

[C2

F

{24

}]+D3δ(1 − z2)

[C2

F

{8}]

+D0δ(1 − z1)[CFCA

{−

80827

+ 28ζ3 +443ζ2

}+ C2

F

{32ζ3

}+ n f CF

{11227−

83ζ2

}+ log

q2

µ2F

CFCA

{2689− 8ζ2

}+ log

q2

µ2F

C2F

{− 64 + 16ζ2

}+ log

q2

µ2F

n f CF

{−

409

}+ log2

q2

µ2F

CFCA

{−

223

}+ log2

q2

µ2F

C2F

{24

}+ log2

q2

µ2F

n f CF

{43

}]+D1δ(1 − z1)

[CFCA

{2689− 8ζ2

}+ C2

F

{− 64 + 16ζ2

}+ n f CF

{−

409

}+ log

q2

µ2F

CFCA

{−

443

}+ log

q2

µ2F

C2F

{24

}+ log

q2

µ2F

n f CF

{83

}+ log2

q2

µ2F

C2F

{16

}]+D2δ(1 − z1)

[CFCA

{−

223

}+ n f CF

{43

}+ log

q2

µ2F

C2F

{24

}]+D3δ(1 − z1)

[C2

F

{8}],

∆DY,SVY,qq,3 = δ(1 − z1)δ(1 − z2)

[CF

(N2 − 4

N

)n f ,v

{8 −

1603ζ5 +

283ζ3 + 20ζ2 −

45ζ2

2

}+ CFC2

A

{−

1505881972

− 204ζ5 +125105

81ζ3 −

4003ζ2

3 +99289

81ζ2 − 588ζ2ζ3

−2921135

ζ22 +

24352315

ζ32

}+ C2

FCA

{74321

36−

76249

ζ5 −5972

3ζ3 +

12643

ζ23

−39865

27ζ2 +

107369

ζ2ζ3 +137968

135ζ2

2 −78272315

ζ32

}+ C3

F

{−

55996

+ 1328ζ5

− 460ζ3 +7363ζ2

3 +1403

3ζ2 − 160ζ2ζ3 −

31645

ζ22 +

90016315

ζ32

}

3.4 Results of the SV Rapidity Distributions 79

+ n f CFCA

{110651

243− 8ζ5 −

1988881

ζ3 −12112

27ζ2 +

2723ζ2ζ3 −

2828135

ζ22

}+ n f C2

F

{−

4213−

2249ζ5 + 360ζ3 +

584827

ζ2 −1472

9ζ2ζ3 −

19408135

ζ22

}+ n2

f CF

{−

7081243

−30481

ζ3 +281681

ζ2 +592135

ζ22

}+ log

q2

µ2F

CFC2A

{3082

3+ 80ζ5

−2296

3ζ3 −

1360027

ζ2 +1084

15ζ2

2

}+ log

q2

µ2F

C2FCA

{−

34392

+ 240ζ5 +19864

9ζ3

+30260

27ζ2 − 608ζ2ζ3 −

17925

ζ22

}+ log

q2

µ2F

C3F

{1495

2− 96ζ5 − 1504ζ3

− 348ζ2 + 384ζ2ζ3 + 192ζ22

}+ log

q2

µ2F

n f CFCA

{−

30529

+3043ζ3 +

519227

ζ2

−18415

ζ22

}+ log

q2

µ2F

n f C2F

{230 −

20329

ζ3 −536027

ζ2 +3045ζ2

2

}+ log

q2

µ2F

n2f CF

{2209−

44827

ζ2

}+ log2

q2

µ2F

CFC2A

{−

24299

+ 88ζ3 +4849ζ2

}+ log2

q2

µ2F

C2FCA

{551 − 496ζ3 −

33329

ζ2 + 64ζ22

}+ log2

q2

µ2F

C3F

{− 270

+ 480ζ3 + 328ζ2 −7045ζ2

2

}+ log2

q2

µ2F

n f CFCA

{8509− 16ζ3 −

1769ζ2

}+ log2

q2

µ2F

n f C2F

{− 92 + 64ζ3 +

5369ζ2

}+ log2

q2

µ2F

n2f CF

{−

689

+169ζ2

}+ log3

q2

µ2F

CFC2A

{2429

}+ log3

q2

µ2F

C2FCA

{− 66 +

1763ζ2

}+ log3

q2

µ2F

C3F

{36 +

1283ζ3 − 96ζ2

}+ log3

q2

µ2F

n f CFCA

{−

889

}+ log3

q2

µ2F

n f C2F

{12 −

323ζ2

}+ log3

q2

µ2F

n2f CF

{89

}]+D0δ(1 − z2)

[CFC2

A

{−

297029729

− 192ζ5 +14264

27ζ3 +

2775281

ζ2 −1763ζ2ζ3

−61615

ζ22

}+ C2

FCA

{12928

27+

2569ζ3 −

956827

ζ2 − 16ζ2ζ3 +176

3ζ2

2

}+ C3

F

{384ζ5

− 512ζ3 − 128ζ2ζ3

}+ n f CFCA

{62626729

−5369ζ3 −

776081

ζ2 +20815

ζ22

}+ n f C2

F

{− 3 −

9449ζ3 +

138427

ζ2 −25615

ζ22

}+ n2

f CF

{−

1856729

−3227ζ3 +

16027

ζ2

}+ log

q2

µ2F

CFC2A

{31006

81− 176ζ3 −

6803ζ2 +

1765ζ2

2

}+ log

q2

µ2F

C2FCA

{−

35033

+1363ζ3 +

43129

ζ2 −485ζ2

2

}+ log

q2

µ2F

C3F

{511 − 48ζ3 − 24ζ2 −

1925ζ2

2

}

80 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+ log q2

µ2F

n f CFCA

{−

820481

+5129ζ2

}+ log

q2

µ2F

n f C2F

{144 +

3203ζ3 −

5929ζ2

}+ log

q2

µ2F

n2f CF

{40081−

329ζ2

}+ log2

q2

µ2F

CFC2A

{−

356027

+883ζ2

}+ log2

q2

µ2F

C2FCA

{1660

3− 96ζ3 −

563ζ2

}+ log2

q2

µ2F

C3F

{− 372 + 448ζ3

}+ log2

q2

µ2F

n f CFCA

{115627−

163ζ2

}+ log2

q2

µ2F

n f C2F

{−

2683−

163ζ2

}+ log2

q2

µ2F

n2f CF

{−

8027

}+ log3

q2

µ2F

CFC2A

{48427

}+ log3

q2

µ2F

C2FCA

{− 88

}+ log3

q2

µ2F

C3F

{72 − 64ζ2

}+ log3

q2

µ2F

n f CFCA

{−

17627

}+ log3

q2

µ2F

n f C2F

{16

}+ log3

q2

µ2F

n2f CF

{1627

}]+D0D0

[CFC2

A

{31006

81− 176ζ3 −

6803ζ2 +

1765ζ2

2

}+ C2

FCA

{−

88939−

3683ζ3 +

35209

ζ2 −485ζ2

2

}+ C3

F

{511 − 240ζ3 − 24ζ2

−1925ζ2

2

}+ n f CFCA

{−

820481

+5129ζ2

}+ n f C2

F

{1072

9+

3203ζ3 −

4489ζ2

}+ n2

f CF

{40081−

329ζ2

}+ log

q2

µ2F

CFC2A

{−

712027

+1763ζ2

}+ log

q2

µ2F

C2FCA

{11644

27+ 128ζ3 +

5603ζ2

}+ log

q2

µ2F

C3F

{− 372 + 832ζ3

}+ log

q2

µ2F

n f CFCA

{2312

27−

323ζ2

}+ log

q2

µ2F

n f C2F

{−

198427−

1283ζ2

}+ log

q2

µ2F

n2f CF

{−

16027

}+ log2

q2

µ2F

CFC2A

{4849

}+ log2

q2

µ2F

C2FCA

{956

9

− 64ζ2

}+ log2

q2

µ2F

C3F

{− 184 − 128ζ2

}+ log2

q2

µ2F

n f CFCA

{−

1769

}+ log2

q2

µ2F

n f C2F

{−

1049

}+ log2

q2

µ2F

n2f CF

{169

}+ log3

q2

µ2F

C2FCA

{−

1763

}+ log3

q2

µ2F

C3F

{96

}+ log3

q2

µ2F

n f C2F

{323

}]+D0D1

[CFC2

A

{−

712027

+1763ζ2

}+ C2

FCA

{−

11209

+ 336ζ3 + 352ζ2

}+ C3

F

{640ζ3

}+ n f CFCA

{231227−

323ζ2

}+ n f C2

F

{1369− 64ζ2

}+ n2

f CF

{−

16027

}+ log

q2

µ2F

CFC2A

{9689

}+ log

q2

µ2F

C2FCA

{1880

3− 192ζ2

}+ log

q2

µ2F

C3F

{− 768 − 192ζ2

}+ log

q2

µ2F

n f CFCA

{−

3529

}

3.4 Results of the SV Rapidity Distributions 81

+ log q2

µ2F

n f C2F

{−

2723

}+ log

q2

µ2F

n2f CF

{329

}+ log2

q2

µ2F

C2FCA

{− 264

}+ log2

q2

µ2F

C3F

{288

}+ log2

q2

µ2F

n f C2F

{48

}+ log3

q2

µ2F

C3F

{64

}]+D0D2

[CFC2

A

{4849

}+ C2

FCA

{1072

3− 96ζ2

}+ C3

F

{− 384 − 96ζ2

}+ n f CFCA

{−

1769

}+ n f C2

F

{−

1603

}+ n2

f CF

{169

}+ log

q2

µ2F

C2FCA

{−

8803

}+ log

q2

µ2F

C3F

{144

}+ log

q2

µ2F

n f C2F

{1603

}+ log2

q2

µ2F

C3F

{192

}]+D0D3

[C2

FCA

{−

8809

}+ n f C2

F

{1609

}+ log

q2

µ2F

C3F

{160

}]+D0D4

[C3

F

{+ 40

}]+D1δ(1 − z2)

[CFC2

A

{31006

81− 176ζ3 −

6803ζ2 +

1765ζ2

2

}+ C2

FCA

{−

88939−

3683ζ3 +

35209

ζ2 −485ζ2

2

}+ C3

F

{511 − 240ζ3 − 24ζ2

−1925ζ2

2

}+ n f CFCA

{−

820481

+5129ζ2

}+ n f C2

F

{1072

9+

3203ζ3 −

4489ζ2

}+ n2

f CF

{40081−

329ζ2

}+ log

q2

µ2F

CFC2A

{−

712027

+1763ζ2

}+ log

q2

µ2F

C2FCA

{11644

27+ 128ζ3 +

5603ζ2

}+ log

q2

µ2F

C3F

{− 372 + 832ζ3

}+ log

q2

µ2F

n f CFCA

{2312

27−

323ζ2

}+ log

q2

µ2F

n f C2F

{−

198427−

1283ζ2

}+ log

q2

µ2F

n2f CF

{−

16027

}+ log2

q2

µ2F

CFC2A

{4849

}+ log2

q2

µ2F

C2FCA

{956

9

− 64ζ2

}+ log2

q2

µ2F

C3F

{− 184 − 128ζ2

}+ log2

q2

µ2F

n f CFCA

{−

1769

}+ log2

q2

µ2F

n f C2F

{−

1049

}+ log2

q2

µ2F

n2f CF

{169

}+ log3

q2

µ2F

C2FCA

{−

1763

}+ log3

q2

µ2F

C3F

{96

}+ log3

q2

µ2F

n f C2F

{323

}]+D1D0

[CFC2

A

{−

712027

+1763ζ2

}+ C2

FCA

{−

11209

+ 336ζ3 + 352ζ2

}+ C3

F

{640ζ3

}+ n f CFCA

{231227−

323ζ2

}+ n f C2

F

{1369− 64ζ2

}+ n2

f CF

{−

16027

}+ log

q2

µ2F

CFC2A

{9689

}+ log

q2

µ2F

C2FCA

{1880

3− 192ζ2

}+ log

q2

µ2F

C3F

{− 768 − 192ζ2

}+ log

q2

µ2F

n f CFCA

{−

3529

}+ log

q2

µ2F

n f C2F

{−

2723

}+ log

q2

µ2F

n2f CF

{329

}

82 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+ log2 q2

µ2F

C2FCA

{− 264

}+ log2

q2

µ2F

C3F

{288

}+ log2

q2

µ2F

n f C2F

{48

}+ log3

q2

µ2F

C3F

{64

}]+D1D1

[CFC2

A

{9689

}+ C2

FCA

{2144

3− 192ζ2

}+ C3

F

{− 768 − 192ζ2

}+ n f CFCA

{−

3529

}+ n f C2

F

{−

3203

}+ n2

f CF

{329

}+ log

q2

µ2F

C2FCA

{−

17603

}+ log

q2

µ2F

C3F

{288

}+ log

q2

µ2F

n f C2F

{3203

}+ log2

q2

µ2F

C3F

{384

}]+D1D2

[C2

FCA

{−

8803

}+ n f C2

F

{160

3

}+ log

q2

µ2F

C3F

{480

}]+D1D3

[C3

F

{160

}]+D2δ(1 − z2)

[CFC2

A

{−

356027

+883ζ2

}+ C2

FCA

{−

5609

+ 168ζ3 + 176ζ2

}+ C3

F

{320ζ3

}+ n f CFCA

{115627−

163ζ2

}+ n f C2

F

{689− 32ζ2

}+ n2

f CF

{−

8027

}+ log

q2

µ2F

CFC2A

{4849

}+ log

q2

µ2F

C2FCA

{9403− 96ζ2

}+ log

q2

µ2F

C3F

{− 384 − 96ζ2

}+ log

q2

µ2F

n f CFCA

{−

1769

}+ log

q2

µ2F

n f C2F

{−

1363

}+ log

q2

µ2F

n2f CF

{169

}+ log2

q2

µ2F

C2FCA

{− 132

}+ log2

q2

µ2F

C3F

{144

}+ log2

q2

µ2F

n f C2F

{24

}+ log3

q2

µ2F

C3F

{32

}]+D2D0

[CFC2

A

{4849

}+ C2

FCA

{1072

3− 96ζ2

}+ C3

F

{− 384 − 96ζ2

}+ n f CFCA

{−

1769

}+ n f C2

F

{−

1603

}+ n2

f CF

{169

}+ log

q2

µ2F

C2FCA

{−

8803

}+ log

q2

µ2F

C3F

{144

}+ log

q2

µ2F

n f C2F

{1603

}+ log2

q2

µ2F

C3F

{192

}]+D2D1

[C2

FCA

{−

8803

}+ n f C2

F

{160

3

}+ log

q2

µ2F

C3F

{480

}]+D2D2

[C3

F

{240

}]+D3δ(1 − z2)

[CFC2

A

{48427

}+ C2

FCA

{1072

9− 32ζ2

}+ C3

F

{− 128 − 32ζ2

}+ n f CFCA

{−

17627

}+ n f C2

F

{−

1609

}+ n2

f CF

{1627

}+ log

q2

µ2F

C2FCA

{−

8809

}+ log

q2

µ2F

C3F

{48

}+ log

q2

µ2F

n f C2F

{1609

}+ log2

q2

µ2F

C3F

{64

}]+D3D0

[C2

FCA

{−

8809

}+ n f C2

F

{1609

}+ log

q2

µ2F

C3F

{160

}]+D3D1C3

F

[{160

}]

3.4 Results of the SV Rapidity Distributions 83

+D4δ(1 − z2)[C2

FCA

{−

2209

}+ n f C2

F

{409

}+ log

q2

µ2F

C3F

{40

}]+D4D0

[C3

F

{40

}]+D5δ(1 − z2)

[C3

F

{8}]

+D0δ(1 − z1)[CFC2

A

{−

297029729

− 192ζ5 +14264

27ζ3 +

2775281

ζ2 −1763ζ2ζ3 −

61615

ζ22

}+ C2

FCA

{12928

27+

2569ζ3

−956827

ζ2 − 16ζ2ζ3 +1763ζ2

2

}+ C3

F

{384ζ5 − 512ζ3 − 128ζ2ζ3

}+ n f CFCA

{62626729

−536

9ζ3 −

776081

ζ2 +20815

ζ22

}+ n f C2

F

{− 3 −

9449ζ3 +

138427

ζ2

−25615

ζ22

}+ n2

f CF

{−

1856729

−3227ζ3 +

16027

ζ2

}+ log

q2

µ2F

CFC2A

{31006

81− 176ζ3

−6803ζ2 +

1765ζ2

2

}+ log

q2

µ2F

C2FCA

{−

35033

+1363ζ3 +

43129

ζ2 −485ζ2

2

}+ log

q2

µ2F

C3F

{511 − 48ζ3 − 24ζ2 −

1925ζ2

2

}+ log

q2

µ2F

n f CFCA

{−

820481

+5129ζ2

}+ log

q2

µ2F

n f C2F

{144 +

3203ζ3 −

5929ζ2

}+ log

q2

µ2F

n2f CF

{40081

−329ζ2

}+ log2

q2

µ2F

CFC2A

{−

356027

+883ζ2

}+ log2

q2

µ2F

C2FCA

{1660

3− 96ζ3

−563ζ2

}+ log2

q2

µ2F

C3F

{− 372 + 448ζ3

}+ log2

q2

µ2F

n f CFCA

{1156

27−

163ζ2

}+ log2

q2

µ2F

n f C2F

{−

2683−

163ζ2

}+ log2

q2

µ2F

n2f CF

{−

8027

}+ log3

q2

µ2F

CFC2A

{48427

}+ log3

q2

µ2F

C2FCA

{− 88

}+ log3

q2

µ2F

C3F

{72 − 64ζ2

}+ log3

q2

µ2F

n f CFCA

{−

17627

}+ log3

q2

µ2F

n f C2F

{16

}+ log3

q2

µ2F

n2f CF

{1627

}]+D1δ(1 − z1)

[CFC2

A

{31006

81− 176ζ3 −

6803ζ2 +

1765ζ2

2

}+ C2

FCA

{−

88939

−3683ζ3 +

35209

ζ2 −485ζ2

2

}+ C3

F

{511 − 240ζ3 − 24ζ2 −

1925ζ2

2

}+ n f CFCA

{−

820481

+512

9ζ2

}+ n f C2

F

{1072

9+

3203ζ3 −

4489ζ2

}+ n2

f CF

{40081

−329ζ2

}+ log

q2

µ2F

CFC2A

{−

712027

+1763ζ2

}+ log

q2

µ2F

C2FCA

{11644

27+ 128ζ3

+5603ζ2

}+ log

q2

µ2F

C3F

{− 372 + 832ζ3

}+ log

q2

µ2F

n f CFCA

{231227−

323ζ2

}+ log

q2

µ2F

n f C2F

{−

198427−

1283ζ2

}+ log

q2

µ2F

n2f CF

{−

16027

}

84 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

+ log2 q2

µ2F

CFC2A

{4849

}+ log2

q2

µ2F

C2FCA

{9569− 64ζ2

}+ log2

q2

µ2F

C3F

{− 184 − 128ζ2

}+ log2

q2

µ2F

n f CFCA

{−

1769

}+ log2

q2

µ2F

n f C2F

{−

1049

}+ log2

q2

µ2F

n2f CF

{169

}+ log3

q2

µ2F

C2FCA

{−

1763

}+ log3

q2

µ2F

C3F

{96

}+ log3

q2

µ2F

n f C2F

{323

}]+D2δ(1 − z1)

[CFC2

A

{−

356027

+883ζ2

}+ C2

FCA

{−

5609

+ 168ζ3 + 176ζ2

}+ C3

F

{320ζ3

}+ n f CFCA

{115627

−163ζ2

}+ n f C2

F

{689− 32ζ2

}+ n2

f CF

{−

8027

}+ log

q2

µ2F

CFC2A

{4849

}+ log

q2

µ2F

C2FCA

{9403− 96ζ2

}+ log

q2

µ2F

C3F

{− 384 − 96ζ2

}+ log

q2

µ2F

n f CFCA

{−

1769

}+ log

q2

µ2F

n f C2F

{−

1363

}+ log

q2

µ2F

n2f CF

{169

}+ log2

q2

µ2F

C2FCA

{− 132

}+ log2

q2

µ2F

C3F

{144

}+ log2

q2

µ2F

n f C2F

{24

}+ log3

q2

µ2F

C3F

{32

}]+D3δ(1 − z1)

[CFC2

A

{48427

}+ C2

FCA

{1072

9− 32ζ2

}+ C3

F

{− 128 − 32ζ2

}+ n f CFCA

{−

17627

}+ n f C2

F

{−

1609

}+ n2

f CF

{1627

}+ log

q2

µ2F

C2FCA

{−

8809

}+ log

q2

µ2F

C3F

{48

}+ log

q2

µ2F

n f C2F

{1609

}+ log2

q2

µ2F

C3F

{64

}]+D4δ(1 − z1)

[C2

FCA

{−

2209

}+ n f C2

F

{409

}+ log

q2

µ2F

C3F

{40

}]+D5δ(1 − z1)

[C3

F

{8}]. (3.4.4)

For sake of completeness, we mention the leading order contribution which is

∆IY,i i,0 = δ(1 − z1)δ(1 − z2) . (3.4.5)

The above results are presented for the choice µR = µF . The dependence of the SV rapiditydistributions on renormalisation scale µR can be easily restored by employing the RG evolution ofas from µF to µR [87] using Eq. (2.4.6). In the next Sec. 3.5, we discuss the numerical impact ofthe N3LO SV correction to the Higgs rapidity distribution at the LHC.

3.5 Numerical Impact of SV Rapidity Distributions

In this section, we confine ourselves to the numerical impact of the SV rapidity distributions of theHiggs boson production through gluon fusion. We present the relative contributions in percentageof the pure N3LO terms in Eq. (3.4.2) with respect to ∆H,SV

Y,gg,3, for rapidity Y = 0 in Table 3.1 and 3.2.

3.5 Numerical Impact of SV Rapidity Distributions 85

δδ δD0 δD1 δD2 δD3 δD4 δD5 D0D0 D0D1

% 73.3 16.0 9.1 31.4 1.0 -9.9 -23.1 -13.7 -10.7

Table 3.1. Relative contributions of pure N3LO terms.

D0D2 D0D3 D0D4 D1D1 D1D2 D1D3 D2D2

% -0.3 3.1 7.3 -0.2 3.8 8.6 4.2

Table 3.2. Relative contributions of pure N3LO terms.

The notation DiD j corresponds to the sum of the contributions coming from DiD j and D jDi.We have used

√s = 14 TeV for the LHC, GF = 4541.68 pb, the Z boson mass mZ = 91.1876

GeV, top quark mass mt = 173.4 GeV and the Higgs boson mass mH = 125.5 GeV throughout.For the Higgs boson production, we use the effective theory where top quark is integrated out inthe large mt limit. The strong coupling constant αs(µ2

R) is evolved using the 4-loop RG equationswith αN3LO

s (mZ) = 0.117 and for parton density sets we use MSTW 2008NNLO [88], as N3LOevolution kernels are not yet available. In [107], Forte et al. pointed out that the Higgs bosoncross sections will remain unaffected with this shortcoming. However, for the DY process, it isnot clear whether the same will be true. We find that the contribution from the δ(1 − z1)δ(1 − z2)part is the largest. Impact of the threshold NNLO and N3LO contributions to the Higgs bosonrapidity distribution at the LHC is presented in Fig. 3.4. The dependence on the renormalizationand factorization scales can by studied by varying them in the range mH/2 < µR, µF < 2mH .We find that the inclusion of the threshold correction at N3LO further reduces their dependence.For the inclusive Higgs boson production, we find that about 50% of exact NNLO contributioncomes from threshold NLO and NNLO terms. It increases to 80% if we use exact NLO andthreshold NNLO terms. Hence, it is expected that the rapidity distribution of the Higgs bosonwill receive a significant contribution from the threshold region compared to inclusive rate dueto the soft emission over the entire range of Y . Our numerical study with threshold enhancedNNLO rapidity distribution confirms our expectation. Comparing our threshold NNLO resultsagainst exact NNLO distribution using the FEHiP [38] code , we find that about 90% of exactNNLO distribution comes from the threshold region as can be seen from Table 3.3 and 3.4, inaccordance with [108], where it was shown that for low τ (m2

H/s ≈ 10−5) values the threshold termsare dominant, thanks to the inherent property of the matrix element, which receives the largestradiative corrections from the phase-space points corresponding to Born kinematics. Here we

Y 0.0 0.4 0.8 1.2 1.6NNLO 11.21 10.96 10.70 9.13 7.80NNLOSV 9.81 9.61 8.99 8.00 6.71NNLOSV(A) 10.67 10.46 9.84 8.82 7.48N3LOSV 11.62 11.36 11.07 9.44 8.04N3LOSV(A) 11.88 11.62 11.33 9.70 8.30K3 2.31 2.29 2.36 2.21 2.17

Table 3.3. Contributions of exact NNLO, NNLOSV, N3LOSV, and K3.

have used the exact results up to the NLO level. Because of an inherent ambiguity in the definition

86 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

Y-3 -2 -1 0 1 2 3

Fra

c. u

ncer

.

0.8

0.9

1

1.1

1.2-3 -2 -1 0 1 2 3

/dY

[pb

]σd

2

4

6

8

10LO

NLO

SVNNLO

SVLO3N

Figure 3.4. Rapidity distribution of Higgs bosonY 2.0 2.4 2.8 3.2 3.6NNLO 6.10 4.23 2.66 1.40 0.54NNLOSV 5.21 3.66 2.25 1.14 0.42NNLOSV(A) 5.90 4.24 2.69 1.42 0.56N3LOSV 6.27 4.33 2.70 1.40 0.53N3LOSV(A) 6.51 4.54 2.88 1.53 0.60K3 2.07 1.89 1.70 1.63 1.51

Table 3.4. Contributions of exact NNLO, NNLOSV, N3LOSV, and K3.

of the partonic cross section at threshold one can multiply a factor zg(z), where z = τ/x1x2 andlimz→1 g(z) = 1, with the partonic flux and divide the same in the partonic cross section for aninclusive rate. In [64, 109] this was exploited to take into account the subleading collinear logsalso, thereby making the threshold approximation a better one. Recently, Anastasiou et al. usedthis in [33] to modify the partonic flux keeping the partonic cross section unaltered to improvethe threshold effects. Following [33, 110], we introduce G(z1, z2) such that limz1,z2→1 G = 1 in

3.6 Summary 87

Eq. (3.5.1):

W I(x0

1, x02, q

2, µ2R

)=

∑i, j=q,q,g

1∫x0

1

dz1

z1

1∫x0

2

dz2

z2H I

i j

x01

z1

x02

z2, µ2

F

G(z1, z2)

× limz1,z2→1

∆IY,i j

(z1, z2, q2, µ2

R, µ2F

)G(z1, z2)

. (3.5.1)

We also find that with the choice G(z1, z2) = z21z2

2, the threshold NNLO results are remarkablyclose to the exact ones for the entire range of Y [see Table 3.3 and 3.4, denoted by (A)]. Thisclearly demonstrates the dominance of threshold contributions to rapidity distribution of the Higgsboson production at the NNLO level. Assuming that the trend will not change drastically beyondNNLO, we present numerical values for N3LO distributions for G(z1, z2) = 1, z2

1z22, respectively,

as N3LOSV and N3LOSV(A) in Table 3.3 and 3.3. The threshold N3LO terms give 6%(Y = 0) to12%(Y = 3.6) additional correction over the NNLO contribution to the inclusive DY production.Finally, in Table 3.3 and 3.4, we have presented K3 = N3LOSV/LO as a function of Y in order todemonstrate the sensitivity of higher order effects to the rapidity Y .

3.6 Summary

To summarize, we present the full threshold enhanced N3LO QCD corrections to rapidity distri-butions of the dilepton pair in the DY process and of the Higgs boson in gluon fusion at the LHC.These are the most accurate results for these observables available in the literature. We show thatthe infrared structure of QCD amplitudes, in particular, their factorization properties, along withSudakov resummation of soft gluons and renormalization group invariance provide an elegantframework to compute these threshold corrections systematically for rapidity distributions orderby order in QCD perturbation theory. The recent N3LO results for inclusive DY and Higgs bosonproduction cross sections at the threshold provide crucial ingredients to obtain δ(1 − z1)δ(1 − z2)contribution of their rapidity distributions for the first time. We find that this contribution numer-ically dominates over the rest of the terms in ∆H,SV

Y,gg,3 at the LHC. Inclusion of N3LO contributionsreduces the scale dependence further. We also demonstrate the dominance of the threshold contri-bution to rapidity distributions by comparing it against the exact NNLO for two different choicesof G(z1, z2). Finally, we find that threshold N3LO rapidity distribution with G(z1, z2) = 1, z2

1z22

shows a moderate effect over NNLO distribution.

3.7 Outlook-Beyond N3LO

The results presented above is the most accurate one existing in the literature. However, in comingfuture, we may need to go beyond this threshold N3LO in hope of making more precise theoreticalpredictions. The immediate step would be to compute the complete N3LO QCD corrections tothe differential rapidity distributions. No doubt, this is an extremely challenging goal! Presently,though we are incapable of computing this result, we can obtain the general form of the thresholdN4LO QCD corrections to the rapidity distributions! However, due to unavailability of the quanti-ties, namely, form factors, anomalous dimensions, soft-collinear distributions at 4-loop level, weare unable to estimate the contributions arising from this. Nevertheless, the general form of this

88 Rapidity Distributions of Drell-Yan and Higgs Boson at Threshold in N3LO QCD

contribution is available to the authors which can be utilised to make the predictions once themissing ingredients become available in future.

4 A Diagrammatic Approach ToCompute Multiloop Amplitude

Contents4.1 Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Feynman Diagrams and Simplifications . . . . . . . . . . . . . . . . 90

4.3 Reduction to Master Integrals . . . . . . . . . . . . . . . . . . . . 914.3.1 Integration-by-Parts Identities (IBP) 914.3.2 Lorentz Invariant Identities (LI) 93

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.1 Prologue

The scattering amplitudes play the most crucial role in any quantum field theory. These are thegateway to unveil the elegant structures associated with the quantum world. At the phenomenolog-ical level, they are the main ingredients in predicting the observables at high energy colliders forthe processes within and beyond the SM. Hence, the efficient evaluation of the scattering ampli-tudes is of prime importance at theoretical as well as experimental level. However, in perturbativeQFT, the theoretical predictions based on the leading order calculation happens to be unreliable.One must go beyond the leading order to make the predictions more accurate and reliable. Whileconsidering the effects arising from the higher orders, the contributions coming from the QCD ra-diations dominate substantially, in particular, at high energy colliders like Tevatron or LHC. In thisthesis, we are concentrating only on the corrections arising from the QCD sector. In the processof computing these higher order QCD corrections, one has to carry out three different types ofcontributions, namely, virtual, real and real-virtual processes. Upon clubbing together all the threecontributions appropriately, finite result for any observable is obtained. As very much expected,the complexity involved in the calculations grows very rapidly as we go towards higher and higherorders in perturbation theory, where more and more different pieces interfere with each other thateventually contribute to the final physical observables.

In this Chapter, we will confine our discussion only to the higher order QCD virtual or loopcorrections. There exists at least two different formalisms to compute these.

89

90 A Diagrammatic Approach To Compute Multiloop Amplitude

1. Diagrammatic approach: one directly evaluates all the relevant Feynman diagrams appear-ing at the perturbative order under consideration.

2. Unitary-based approach: the unitary properties of the scattering amplitudes are employedextensively to avoid the direct evaluation of all the Feynman diagrams.

Despite the spectacular beauty of the unitary based approach, its applicability to the computa-tion of the amplitudes remains confined mostly to one loop or only few multiloop problems. Itsgeneralisation to any multiloop computation is still unavailable in the literature. In these morecomplicated scenarios, the first methodology of directly evaluating the Feynman diagrams is moreeffective and is therefore employed more often.

4.2 Feynman Diagrams and Simplifications

For any generic scattering process in QFT, we can expand any observable in powers of all thecoupling constants present in the underlying Lagrangian. Feynman diagrams are the diagram-matic representations of this expansion. In this thesis, we confine our discussion into QCD. Let usconsider a scattering process involving E external particles with momenta p1, p2, · · · , pE . With-out loss of generality, we consider the cross-section which can be expanded in powers of strongcoupling constant:

σE (p1, p2, · · · , pE) =

∞∑l=0

alsσ

(l)E (p1, p2, · · · , pE) . (4.2.1)

For the sake of simplicity, we suppress all the dependence on quantum numbers of the externalparticles. The index l denotes the order of perturbative expansion. The cross section for l = 0 iscalled the leading order (LO), l = 1 is next-to-leading order (NLO) and so on. The cross sectionat at each perturbative order, σ(l)

E , is related to the scattering matrix elements through

σ(0)E = K

∫| |M

(0)E 〉 |

2 dΦE ,

σ(1)E = K

∫2 Re ( 〈M(0)

E |M(1)E 〉 ) dΦE + K

∫| |M

(0)E+1〉 |

2 dΦE+1 ,

σ(2)E = K

∫2 Re ( 〈M(0)

E |M(2)E 〉 )dΦE + K

∫2 Re ( 〈M(0)

E+1|M(1)E+1〉 ) dΦE+1

+ K∫| |M

(0)E+2〉 |

2 dΦE+2

and so on. (4.2.2)

In the above set of equations, |M(l)E 〉 is the scattering amplitude at lth order in perturbation theory

involving E number of external particles. The quantity dΦE is the phase space element. "Re"denotes the real part of the amplitude and K is an overall constant containing various factors. Theamplitudes with E number of external particles and l ≥ 1 represent the contributions arising fromthe virtual Feynman diagrams, whereas amplitudes with more than E number of external particlescome from the real emission diagrams. In this chapter, we address the issue of evaluating thevirtual diagrams. The scattering matrix element can also be expanded perturbatively in powers ofas as

|ME〉 =

∞∑l=0

als|M

(l)E 〉 . (4.2.3)

4.3 Reduction to Master Integrals 91

Each term in the right hand side can be represented through a set of Feynman diagrams of sameperturbative order. In this chapter, we will explain the prescription to evaluate the contribution tothe matrix element arising from the virtual diagrams.

The evaluation of the scattering matrix element at any particular order begins with the generationof associated Feynman diagrams. We make use of a package, named, QGRAF [111] to accomplishthis job. QGRAF does not provide the graphical representation of the Feynman diagrams, ratherit generates those symbolically. We use our in-house codes written in FORM [112] to convert theraw output into a format for further computation. Employing the Feynman rules derived from theunderlying Lagrangian, which are the languages establishing the connection between the diagramsand the corresponding formal mathematical expressions, we obtain the amplitude. The raw am-plitude contains series of Dirac gamma matrices, QCD color factors, Dirac and Lorentz indices.We simplify those using our in-house codes. We perform the color simplification in SU(N) gaugetheory and follow dimensional regularisation where the space-time dimension is considered to bed = 4 + ε. The amplitude, beyond leading order, consists of a set of tensorial Feynman integrals.Instead of handling the tensorial integrals, we multiply the amplitude with appropriate projectorsto convert those to scalar integrals. Hence, the problem essentially boils down to solving thosescalar integrals. Often, at any typical order in perturbation theory, this involves hundreds or thou-sands of different scalar loop integrals. Of course, start evaluating all of these integrals is not apractical way of dealing with the problem. Remarkably, it has been found that the appeared inte-grals are not independent of each other, they can be related through some set of identities! Thisdrastically reduces the independent integrals which ultimately need to be computed. In the nextsection, we will elaborate this procedure.

4.3 Reduction to Master Integrals

The dimensionally regularised Feynman loop integrals do satisfy a large number of relations,which allow one to express most of those integrals in terms of a much smaller subset of indepen-dent integrals (where ”independent” is to be understood in the sense of the identities introducedbelow), which are now commonly referred to as the Master Integrals (MIs). For a detailed reviewon this, see [48,113]. These identities are known as integration-by-parts and Lorentz invarianceidentities.

4.3.1 Integration-by-Parts Identities (IBP)

The integration-by-parts identities [46, 47] are the most important class of identities which estab-lish the relations among the dimensionally regularised scalar Feynman integrals. These can beseen as a generalisation of Gauss’ divergence theorem in d-dimensions. They are based on the factthat, given a Feynman integral which is a function of space-time dimensions d, there always existsa value of d in the complex plane where the integral is well defined and consequently convergent.The necessary condition for the convergence of an integral is the integrand be zero at the bound-aries. This condition can be rephrased as, the integral of the total derivative with respect to anyloop momenta vanishes, that is

∫ l∏j=1

ddk j

(2π)d

∂

∂kµi

vµs 1

Db11 · · · D

bββ

= 0 . (4.3.1)

92 A Diagrammatic Approach To Compute Multiloop Amplitude

q k1 p1

p2k1 − p1 − p2p3

k1 − p1k1 − p1 − p2 − p3

Figure 4.1. One loop box

In the expression, k j are the loop momenta, vµs can be loop or external momenta, vµs = {kµ1 , · · · , kµl ; pµ1, · · · , pµE}.

Di are the propagators that depend on the masses, loop and external momenta. To begin with, adiagram contains a set of propagators as well as scalar products involving the loop and externalmomenta. However, we can express all the scalar products involving loop momenta in terms ofpropagators. This is possible since any Lorentz scalar can be written either in terms of scalar prod-ucts or propagators. Both of the representations are equivalent. For our convenience, we choose towork in the propagator representation. Performing the differentiation on the left hand side of theabove Eq. (4.3.1), one obtains set of IBP identities. Let us demonstrate the role of IBP identitiesthrough an one-loop example.

• Example: We consider an one loop box diagram, depicted through Fig. 4.1 where, all theexternal legs are taken to be massless, for simplicity, and the momentum q = p1 + p2 +

p3. The corresponding dimensionally regularised Feynman integral can be cast into thefollowing form ∫

ddk(2π)d

1

Db11 D

b22 D

b33 D

b44

≡ I [b1, b2, b3, b4] (4.3.2)

with

D1 ≡ k1 ,

D2 ≡ (k1 − p1) ,

D3 ≡ (k1 − p1 − p2) ,

D4 ≡ (k1 − p1 − p2 − p3) . (4.3.3)

We can obtain 4-set of IBP identities for each choice of the set {b1, b2, b3, b4}. For a choiceof vµs = pµ1 in Eq. (4.3.1), we obtain the corresponding IBP identities as

0 =

∫ddk

(2π)d

[b1

(−1 +

D2

D1

)+ b2

(1 −D1

D2

)− b3

(D1

D3−D2

D3−

sD3

)− b4

(D1

D4−D2

D4−

sD4−

uD4

) ] 1

Db11 D

b22 D

b33 D

b44

. (4.3.4)

It can be symbolically expressed as

0 = b1(−1 + 1+2−) + b2(1 − 2+1−) − b3(3+1− − 3+2− − s 3+)

− b4(4+1− − 4+2− − s 4+ − u 4+) (4.3.5)

4.4 Summary 93

where, we have made use of the convention as 1+2−I[b1, b2, b3, b4] = I[b1 + 1, b2−1, b3, b4]and the associated Mandelstam variables are defined as s ≡ (p1 + p2)2 = 2p1.p2, t ≡(p2 + p3)2 = 2p2.p3, u ≡ (p1 + p3)2 = 2p1.p3. From the Eq. (4.3.5), it is clear that theIBP identities provide recursion relations among the integrals of a topology and/or its sub-topologies. Similarly, we can get the IBP identities corresponding to other external as wellas internal momenta. Upon employing all of these identities, it can be shown that thereexists only three MIs, which are I[1, 0, 1, 0], I[1, 0, 0, 1] and I[1, 1, 1, 1]. Hence, at the endwe need to evaluate only three independent integrals corresponding to the problem underconsideration. For higher loop and more number of external legs, the IBP identities oftenbecome too clumsy to handle manually. Hence, these identities are generated systemati-cally with the help of some computer algorithms in some packages, such as AIR [114],FIRE [115], REDUZE [116, 117], LiteRed [118, 119].

4.3.2 Lorentz Invariant Identities (LI)

The Lorentz invariance of the scalar Feynman integrals can be used in order to obtain more set ofidentities among the integrals, which are known as Lorentz invariant identities [48]:

p[µj pν]k

∑i

pi,[µ∂

∂pν]i

I(pi) = 0 . (4.3.6)

It has been recently found [120] that the LI identities are not independent from IBP ones, sincethese can be reproduced generating and solving larger systems of IBPs. However, use of LI identi-ties along with the IBP helps to speed up the solution. Hence, in almost all of the computer codesfor performing automated reduction to MIs, LI identities are therefore extensively used.

Employing the IBP and LI identities, we obtain a set of MIs which ultimately need to be evaluated.Upon evaluation of the MIs, we can obtain the final unrenormalised result of the virtual correc-tions. Often these contain UV as well as soft and collinear divergences. The UV divergencesare removed through UV renormalisation. The UV renormalised result of the virtual correctionsexhibit a universal infrared pole structures which serve a crucial check on the correctness of thecomputation. In the next chapter, we employ this methodology to compute the three loop quarkand gluon form factors in QCD for the production of a pseudo-scalar.

4.4 Summary

We have discussed the techniques largely used for the computations of the multiloop amplitudeswhich is mostly based on the IBP and LI identities. These are employed in the computer codes toautomatise the reduction process. Among some packages, we utilise LiteRed [118, 119] for ourcomputations. In these articles [13, 121–123], we have applied this methodology successfully tocompute the 2- and 3-loop QCD corrections. In the next chapter, we will present the computationof 3-loop QCD form factors for the pseudo-scalar production where we have essentially made useof the methodology discussed in this chapter.

5 Pseudo-Scalar Form Factors atThree Loops in QCD

The materials presented in this chapter are the result of an original research done in collabo-ration with Thomas Gehrmann, M. C. Kumar, Prakash Mathews, Narayan Rana and V. Ravin-dran, and these are based on the published articles [13, 14].

Contents5.1 Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Framework of the Calculation. . . . . . . . . . . . . . . . . . . . 98

5.2.1 The Effective Lagrangian 98

5.2.2 Treatment of γ5 in Dimensional Regularization 99

5.3 Pseudo-scalar Quark and Gluon Form Factors . . . . . . . . . . . . . . 99

5.3.1 Calculation of the Unrenormalised Form Factors 100

5.3.2 UV Renormalisation 102

5.3.3 Infrared Singularities and Universal Pole Structure 105

5.3.4 Results of UV Renormalised Form Factors 110

5.3.5 Universal Behaviour of Leading Transcendentality Contribution 115

5.4 Gluon Form Factors for the Pseudo-scalar Higgs Boson Production . . . . . . 115

5.5 Hard Matching Coefficients in SCET . . . . . . . . . . . . . . . . . 118

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.1 Prologue

Form factors are the matrix elements of local composite operators between physical states. In thecalculation of scattering cross sections, they provide the purely virtual corrections. For example,in the context of hard scattering processes such as Drell-Yan [97, 124] and the Higgs boson pro-duction in gluon fusion [44, 54–57, 59–61, 84, 125–128], the form factors corresponding to thevector current operator ψγµψ and the gluonic operator Ga

µνGa,µν contribute, respectively. Here ψ

is the fermionic field operator and Gaµν is the field tensor of the non-Abelian gauge field Aa

µ corre-sponding to the gauge group SU(N). In QCD the form factors can be computed order by order in

95

96 Pseudo-Scalar Form Factors at Three Loops in QCD

the strong coupling constant using perturbation theory. Beyond leading order, the UV renormali-sation of the form factors involves the renormalisation of the composite operator itself, besides thestandard procedure for coupling constant and external fields.

The resulting UV finite form factors still contain divergences of infrared origin, namely, soft andcollinear divergences due to the presence of massless gluons and quarks/ antiquarks in the theory.The inclusive hard scattering cross sections require, in addition to the form factor, the real-emissionpartonic subprocesses as well as suitable mass factorisation kernels for incoming partons. The softdivergences in the form factor resulting from the gluons cancel against those present in the realemission processes and the mass factorisation kernels remove the remaining collinear divergencesrendering the hadronic inclusive cross section IR finite. While these IR divergences cancel amongvarious parts in the perturbative computations, they can give rise to logarithms involving physicalscales and kinematic scaling variables of the processes under study. In kinematical regions wherethese logarithms become large, they may affect the convergence and reliability of the perturbationseries expansion in powers of the coupling constant. The solution for this problem goes back tothe pioneering work by Sudakov [79] on the asymptotic behaviour of the form factor in QuantumElectrodynamics: all leading logarithms can be summed up to all orders in perturbation theory.Later on, this resummation was extended to non-leading logarithms [81] and systematised fornon-Abelian gauge theories [82]. Ever since, form factors have been central to understand theunderlying structure of amplitudes in gauge theories.

The infrared origin of universal logarithmic corrections to form factors [83] and scattering ampli-tudes results in a close interplay between resummation and infrared pole structure. Working indimensional regularisation in d = 4 + ε dimensions, these poles appear as inverse powers in theLaurent expansion in ε. In a seminal paper, Catani [129] proposed a universal formula for theIR pole structure of massless two-loop QCD amplitudes of arbitrary multiplicity (valid throughto double pole terms). This formula was later on justified systematically from infrared factor-ization [130], thereby also revealing the structure of the single poles in terms of the anomalousdimensions for the soft radiation. In [84], it was shown that the single pole term in quark and gluonform factors up to two loop level can be shown to decompose into UV (γI , I = q, g) and universalcollinear (BI), color singlet soft ( fI) anomalous dimensions, later on observed to hold even at threeloop level in [40]. An all loop conjecture for the pole structure of the on-shell multi-loop multi-leg amplitudes in SU(N) gauge theory with n f light flavors in terms of cusp (AI), collinear (BI)and soft anomalous dimensions (ΓIJ , fI - colour non-singlet as well as singlet) was proposed byBecher and Neubert [131] and Gardi and Magnea [132], generalising the earlier results [129,130].The validity of this conjecture beyond three loops depends on the presence/absence of non-trivialcolour correlations and crossing ratios involving kinematical invariants [133] and there exists noall-order proof at present. The three-loop expressions for cusp, collinear and colour singlet softanomalous dimensions were extracted [73, 134] from the three loop flavour singlet [85] and non-singlet [69] splitting functions, thereby also predicting [40] the full pole structure of the three-loopform factors.

The three-loop quark and gluon form factors through to finite terms were computed in [31, 41,42, 135] and subsequently extended to higher powers in the ε expansion [43]. These results wereenabled by modern techniques for multi-loop calculations in quantum field theory, in particular in-tegral reduction methods. These are based on IBP [46,47] and LI [48] identities which reduce theset of thousands of multi-loop integrals to the one with few integrals, so called MIs in dimensionalregularisation. To solve these large systems of IBP and LI identities, the Laporta algorithm [136],which is based on lexicographic ordering of the integrals, is the main tool of choice. It has beenimplemented in several computer algebra codes [114–119]. The MIs relevant to the form fac-

5.1 Prologue 97

tors are single-scale three-loop vertex functions, for which analytical expressions were derived inRefs. [42, 68, 105, 137–139].

Recently, some of us have applied these state-of-the-art methods to accomplish the task of com-puting spin-2 quark and gluon form factors up to three loops [123] level in SU(N) gauge theorywith n f light flavours. These form factors are ingredients to the precise description of productioncross sections for graviton production, that are predicted in extensions of the SM. In addition, thespin-2 form factors relate to operators with higher tensorial structure and thus provide the oppor-tunity to test the versatility and robustness of calculational techniques for the vertex functions atthree loop level. The results [123] also confirm the universality of the UV and IR structure of thegauge theory amplitudes in dimensional regularisation.

In the present work, we derive the three-loop corrections to the quark and gluon form factorsfor pseudo-scalar operators. These operators appear frequently in effective field theory descrip-tions of extensions of the SM. Most notably, a pseudo-scalar state coupling to massive fermions isan inherent prediction of any two-Higgs doublet model [140–147]. In the limit of infinite fermionmass, this gives rise to the operator insertions considered here. The recent discovery of a Standard-Model-like Higgs boson at the LHC [1, 15] has not only revived the interest in such Higgs bosonsbut also prompted the study of the properties of the discovered boson to identify either with light-est scalar or pseudo-scalar Higgs bosons of extended models. Such a study requires precise pre-dictions for their production cross sections. In the context of a CP-even scalar Higgs boson,results for the inclusive production cross section in the gluon fusion are available up to N3LOQCD [44, 59–61], based on an effective scalar coupling that results from integration of massivequark loops that mediate the coupling of the Higgs boson to gluons [148–150]. On the other handfor the CP-odd pseudo-scalar, only NNLO QCD results [61, 151–154] in the effective theory [45]are known. The exact quark mass dependence for scalar and pseudo-scalar production is knownto NLO QCD [56, 155], and is usually included through a re-weighting of the effective theoryresults. Soft gluon resummation of the gluon fusion cross section has been performed to N3LL forthe scalar case [28, 29, 34, 64, 72, 74, 156–158] and to NNLL for the pseudo-scalar case [159]. Ageneric threshold resummation formula valid to N3LL accuracy for colour-neutral final states wasderived in [34], requiring only the virtual three-loop amplitudes as process-dependent input. Thenumerical impact of soft gluon resummation in scalar and pseudo-scalar Higgs boson productionand its combination with mass corrections is reviewed comprehensively in [160]. The three-loopcorrections to the pseudo-scalar form factors computed in this thesis are an important ingredient tothe N3LO and N3LL gluon fusion cross sections [14] for pseudo-scalar Higgs boson production,thereby enabling predictions at the same level of precision that is attained in the scalar case.

The framework of the calculation is outlined in Section 5.2, where we describe the effective the-ory [45]. Due to the pseudo-scalar coupling, one is left with two effective operators with samequantum number and mass dimensions, which mix under renormalisation. Since these opera-tors contain the Levi-Civita tensor as well as γ5, the computation of the matrix elements requiresadditional care in 4 + ε dimensions where neither Levi-Civita tensor nor γ5 can be defined unam-biguously. We use the prescription by ’t Hooft and Veltman [8, 49] to define γ5. We describe thecalculation in Section 5.3, putting particular emphasis on the UV renormalisation. Exploiting theuniversal IR pole structure of the form factors, we determine the UV renormalisation constantsand mixing of the effective operators up to three loop level. We also show that the finite renormal-isation constant, known up to three loops [49], required to preserve one loop nature of the chiralanomaly, is consistent with anomalous dimensions of the overall renormalisation constants. As afirst application of our form factors, we compute the hard matching functions for N3LL resumma-tion in soft-collinear effective theory (SCET) in Section 5.5. Section 5.6 summarises our results

98 Pseudo-Scalar Form Factors at Three Loops in QCD

and contains an outlook on future applications to precision phenomenology of pseudo-scalar Higgsproduction.

5.2 Framework of the Calculation

5.2.1 The Effective Lagrangian

A pseudo-scalar Higgs boson couples to gluons only indirectly through a virtual heavy quarkloop. This loop can be integrated out in the limit of infinite quark mass. The resulting effectiveLagrangian [45] encapsulates the interaction between a pseudo-scalar ΦA and QCD particles andreads:

LAeff = ΦA(x)

[−

18

CGOG(x) −12

CJOJ(x)]

(5.2.1)

where the operators are defined as

OG(x) = Gµνa Ga,µν ≡ εµνρσGµν

a Gρσa , OJ(x) = ∂µ

(ψγµγ5ψ

). (5.2.2)

The Wilson coefficients CG and CJ are obtained by integrating out the heavy quark loop, and CG

does not receive any QCD corrections beyond one loop due to the Adler-Bardeen theorem [161],while CJ starts only at second order in the strong coupling constant. Expanded in as ≡ g2

s/(16π2) =

αs/(4π), they read

CG = −as254 G

12Fcotβ

CJ = −

asCF

32− 3 ln

µ2R

m2t

+ a2sC

(2)J + · · ·

CG . (5.2.3)

In the above expressions, Gµνa and ψ represent gluonic field strength tensor and light quark fields,

respectively and GF is the Fermi constant and cotβ is the mixing angle in a generic Two-Higgs-Doublet model. as ≡ as

(µ2

R

)is the strong coupling constant renormalised at the scale µR which is

related to the unrenormalised one, as ≡ g2s/(16π2) through

asS ε =

µ2

µ2R

ε/2 Zasas (5.2.4)

with S ε = exp[(γE − ln 4π)ε/2

]and µ is the scale introduced to keep the strong coupling constant

dimensionless in d = 4 + ε space-time dimensions. The renormalisation constant Zas [77] is givenby

Zas = 1 + as

[2εβ0

]+ a2

s

[4ε2 β

20 +

1εβ1

]+ a3

s

[8ε3 β

30 +

143ε2 β0β1 +

23εβ2

](5.2.5)

up to O(a3s). βi are the coefficients of the QCD β functions which are given by [77] and presented

in Eq. (2.3.12).

5.3 Pseudo-scalar Quark and Gluon Form Factors 99

5.2.2 Treatment of γ5 in Dimensional Regularization

Higher order calculations of chiral quantities in dimensional regularization face the problem ofdefining a generalization of the inherently four-dimensional objects γ5 and εµνρσ to values ofd , 4. In this thesis, we have followed the most practical and self-consistent definition of γ5for multiloop calculations in dimensional regularization which was introduced by ’t Hooft andVeltman through [8]

γ5 = i14!εν1ν2ν3ν4γ

ν1γν2γν3γν4 . (5.2.6)

Here, εµνρσ is the Levi-Civita tensor which is contracted as

εµ1ν1λ1σ1 εµ2ν2λ2σ2 =

∣∣∣∣∣∣∣∣∣∣∣∣∣δµ2µ1 δν2

µ1 δλ2µ1 δσ2

µ1

δµ2ν1 δν2

ν1 δλ2ν1 δσ2

ν1

δµ2λ1

δν2λ1

δλ2λ1

δσ2λ1

δµ2σ1 δν2

σ1 δλ2σ1 δσ2

σ1

∣∣∣∣∣∣∣∣∣∣∣∣∣(5.2.7)

and all the Lorentz indices are considered to be d-dimensional [49]. In this scheme, a finiterenormalisation of the axial vector current is required in order to fulfill chiral Ward identitiesand the Adler-Bardeen theorem. We discuss this in detail in Section 5.3.2 below.

5.3 Pseudo-scalar Quark and Gluon Form Factors

The quark and gluon form factors describe the QCD loop corrections to the transition matrixelement from a color-neutral operator O to an on-shell quark-antiquark pair or to two gluons.For the pseudo-scalar interaction, we need to consider the two operators OG and OJ , defined inEq. (5.2.2), thus yielding in total four form factors. We define the unrenormalised gluon formfactors at O(an

s) as

FG,(n)g ≡

〈MG,(0)g |M

G,(n)g 〉

〈MG,(0)g |M

G,(0)g 〉

, FJ,(n)

g ≡〈M

G,(0)g |M

J,(n+1)g 〉

〈MG,(0)g |M

J,(1)g 〉

(5.3.1)

and similarly the unrenormalised quark form factors through

FG,(n)q ≡

〈MJ,(0)q |M

G,(n+1)q 〉

〈MJ,(0)q |M

G,(1)q 〉

, FJ,(n)

q ≡〈M

J,(0)q |M

J,(n)q 〉

〈MJ,(0)q |M

J,(0)q 〉

(5.3.2)

where, n = 0, 1, 2, 3, . . . . In the above expressions |Mλ,(n)β 〉 is the O(an

s) contribution to the un-renormalised matrix element for the transition from the bare operator [Oλ]B (λ = G, J) to a quark-antiquark pair (β = q) or to two gluons (β = g). The expansion of these quantities in powers of as

is performed through

|Mλβ〉 ≡

∞∑n=0

ansS n

ε

(Q2

µ2

)n ε2

|Mλ,(n)β 〉

and F λβ ≡

∞∑n=0

ans

(Q2

µ2

)n ε2

S nε F

λ,(n)β

. (5.3.3)

where, Q2 = −2 p1.p2 and p′i s (p2i = 0) are the momenta of the external quarks and gluons. Note

that |MG,(n)q 〉 and |MJ,(n)

g 〉 start from n = 1 i.e. from one loop level.

100 Pseudo-Scalar Form Factors at Three Loops in QCD

5.3.1 Calculation of the Unrenormalised Form Factors

The calculation of the unrenormalised pseudo-scalar form factors up to three loops follows closelythe steps used in the derivation of the three-loop scalar and vector form factors [31, 42]. TheFeynman diagrams for all transition matrix elements (Eq. (5.3.1), Eq. (5.3.2)) are generated usingQGRAF [111]. The numbers of diagrams contributing to three loop amplitudes are 1586 for|M

G,(3)g 〉, 447 for |MJ,(3)

g 〉, 400 for |MG,(3)q 〉 and 244 for |MJ,(3)

q 〉 where all the external particlesare considered to be on-shell. The raw output of QGRAF is converted to a format suitable forfurther manipulation. A set of in-house routines written in the symbolic manipulating programFORM [112] is utilized to perform the simplification of the matrix elements involving Lorentzand color indices. Contributions arising from ghost loops are taken into account as well since weuse Feynman gauge for internal gluons. For the external on-shell gluons, we ensure the summingover only transverse polarization states by employing an axial polarization sum:∑

s

εµ ∗(pi, s)εν(pi, s) = −ηµν +pµi qνi + qµi pνi

pi.qi, (5.3.4)

where pi is the ith-gluon momentum, qi is the corresponding reference momentum which is anarbitrary light like 4-vector and s stands for spin (polarization) of gluons. We choose q1 = p2 andq2 = p1 for our calculation. Finally, traces over the Dirac matrices are carried out in d dimensions.

The expressions involve thousands of three-loop scalar integrals. However, they are expressible interms of a much smaller set of scalar integrals, called master integrals (MIs), by use of IBP [46,47]and LI [48] identities. These identities follow from the Poincare invariance of the integrands, theyresult in a large linear system of equations for the integrals relevant to given external kinematicsat a fixed loop-order. The LI identities are not linearly independent from the IBP identities [120],their inclusion does however help to accelerate the solution of the system of equations. By em-ploying lexicographic ordering of these integrals (Laporta algorithm, [136]), a reduction to MIs isaccomplished. Several implementations of the Laporta algorithm exist in the literature: AIR [114],FIRE [115], Reduze2 [116, 117] and LiteRed [118, 119]. In the context of the present calculation,we used LiteRed [118, 119] to perform the reductions of all the integrals to MIs.

Each three-loop Feynman integral is expressed in terms of a list of propagators involving loopmomenta that can be attributed to one of the following three sets (auxiliary topologies, [42])

A1 : {D1,D2,D3,D12,D13,D23,D1;1,D1;12,D2;1,D2;12,D3;1,D3;12}

A2 : {D1,D2,D3,D12,D13,D23,D13;2,D1;12,D2;1,D12;2,D3;1,D3;12}

A3 : {D1,D2,D3,D12,D13,D123,D1;1,D1;12,D2;1,D2;12,D3;1,D3;12} . (5.3.5)

In the above sets

Di = k2i ,Di j = (ki − k j)2,Di jl = (ki − k j − kl)2,

Di; j = (ki − p j)2,Di; jl = (ki − p j − pl)2,Di j;l = (ki − k j − pl)2

To accomplish this, we have used the package Reduze2 [116, 117]. In each set in Eq. (5.3.5),D′sare linearly independent and form a complete basis in a sense that any Lorentz-invariant scalarproduct involving loop momenta and external momenta can be expressed uniquely in terms ofD′sfrom that set.

As a result, we can express the unrenormalised form factors in terms of 22 topologically differentmaster integrals (MIs) which can be broadly classified into three different types: genuine three-loop integrals with vertex functions (At,i), three-loop propagator integrals (Bt,i) and integrals which

5.3 Pseudo-scalar Quark and Gluon Form Factors 101

are product of one- and two-loop integrals (Ct,i). Defining a generic three loop master integralthrough

Ai,mi1mi

2···mi12

=

∫ddk1

(2π)d

∫ddk2

(2π)d

∫ddk3

(2π)d

1∏j D

mij

j

, i = 1, 2, 3 (5.3.6)

where D j is the jth element of the basis set Ai. We identify the resulting master integrals appearedin our computation to those given in [42] and they are listed in the figures below.

B6,1 ≡ A1,111000010101 B6,2 ≡ A1,011110000101 B8,1 ≡ A3,011111010101

B4,1 ≡ A1,001101010000 B5,1 ≡ A1,011010010100 B5,2 ≡ A1,001011010100

C6,1 ≡ A1,011100100101 C8,1 ≡ A2,111100011101 A5,1 ≡ A1,001101100001

A5,2 ≡ A1,001011011000 A6,1 ≡ A1,010101100110 A6,2 ≡ A1,001111011000

A6,3 ≡ A1,001110100101 A7,1 ≡ A2,011110011100 A7,2 ≡ A2,011011001101

These integrals were computed analytically as Laurent series in ε in [68, 105, 137–139] and arecollected in the appendix of [42]. Inserting those, we obtain the final expressions for the unrenor-malised (bare) form factors that are listed in Appendix G.

102 Pseudo-Scalar Form Factors at Three Loops in QCD

A7,3 ≡ A1,011011110100 A7,4 ≡ A2,011110001101

A7,5 ≡ A2,011011010101 A8,1 ≡ A2,001111011101

A9,1 ≡ A1,011111110110 A9,2 ≡ A2,011111011101 A9,4 ≡ A2,111011111100

5.3.2 UV Renormalisation

To obtain ultraviolet-finite expressions for the form factors, a renormalisation of the couplingconstant and of the operators is required. The UV renormalisation of the operators [OG]B and[OJ]B involves some non-trivial prescriptions. These are in part related to the formalism used forthe γ5 matrix, section 5.2.2 above.

This formalism fails to preserve the anti-commutativity of γ5 with γµ in d dimensions. In addition,the standard properties of the axial current and Ward identities, which are valid in a basic regular-ization scheme like the one of Pauli-Villars, are violated as well. As a consequence, one fails torestore the correct renormalised axial current, which is defined as [49, 162]

Jµ5 ≡ ψγµγ5ψ = i

13!εµν1ν2ν3ψγν1γν2γν3ψ (5.3.7)

in dimensional regularization. To rectify this, one needs to introduce a finite renormalisationconstant Z s

5 [161, 163] in addition to the standard overall ultraviolet renormalisation constant Z sMS

within the MS -scheme: [Jµ5

]R

= Z s5Z s

MS

[Jµ5

]B. (5.3.8)

By evaluating the appropriate Feynman diagrams explicitly, Z sMS

can be computed, however thefinite renormalisation constant is not fixed through this calculation. To determine Z s

5 one hasto demand the conservation of the one loop character [164] of the operator relation of the axialanomaly in dimensional regularization:[

∂µJµ5]R

= asn f

2

[GG

]R

i.e. [OJ]R = asn f

2[OG]R . (5.3.9)

5.3 Pseudo-scalar Quark and Gluon Form Factors 103

The bare operator [OJ]B is renormalised multiplicatively exactly in the same way as the axialcurrent Jµ5 through

[OJ]R = Z s5Z s

MS[OJ]B , (5.3.10)

whereas the other one [OG]B mixes under the renormalisation through

[OG]R = ZGG [OG]B + ZGJ [OJ]B (5.3.11)

with the corresponding renormalisation constants ZGG and ZGJ . The above two equations can becombined to express them through the matrix equation

[Oi]R = Zi j[O j

]B

(5.3.12)

with

i, j = {G, J} ,

O ≡[OG

OJ

]and Z ≡

[ZGG ZGJ

ZJG ZJJ

]. (5.3.13)

In the above expressions

ZJG = 0 to all orders in perturbation theory ,

ZJJ ≡ Z s5Z s

MS. (5.3.14)

We determine the above-mentioned renormalisation constants Z sMS,ZGG,ZGJ up toO

(a3

s

)from our

calculation of the bare on-shell pseudo-scalar form factors described in the previous subsection.This procedure provides a completely independent approach to their original computation, whichwas done in the operator product expansion [50].

Our approach to compute those Zi j is based on the infrared evolution equation for the form factor,and will be detailed in Section 5.3.3 below. Moreover, we can fix Z s

5 up to O(a2s) by demanding the

operator relation of the axial anomaly (Eq. (5.3.9)). Using these overall operator renormalisationconstants along with strong coupling constant renormalisation through Zas , Eq. (5.2.5), we obtainthe UV finite on-shell quark and gluon form factors.

To define the UV renormalised form factors, we introduce a quantity Sλβ, constructed out of barematrix elements, through

SGg ≡ ZGG〈M

G,(0)g |MG

g 〉 + ZGJ〈MG,(0)g |MJ

g〉

and

SGq ≡ ZGG〈M

J,(0)q |MG

q 〉 + ZGJ〈MJ,(0)q |MJ

q〉 . (5.3.15)

Expanding the quantities appearing on the right hand side of the above equation in powers of as :

|Mλβ〉 =

∞∑n=0

ans |M

λ,(n)β 〉 ,

ZI =

∞∑n=0

ansZ(n)

I with I = GG,GJ , (5.3.16)

104 Pseudo-Scalar Form Factors at Three Loops in QCD

we can write

SGg =

∞∑n=0

ansS

G,(n)g and SG

q =

∞∑n=1

ansS

G,(n)q . (5.3.17)

Then the UV renormalised form factors corresponding to OG are defined as

[F G

g

]R≡SG

g

SG,(0)g

= ZGGFGg + ZGJF

Jg〈M

G,(0)g |M

J,(1)g 〉

〈MG,(0)g |M

G,(0)g 〉

≡ 1 +

∞∑n=1

ans

[F

G,(n)g

]R,

[F G

q

]R≡

SGq

asSG,(1)q

=ZGGF

Gq 〈M

J,(0)q |M

G,(1)q 〉 + ZGJF

Jq 〈M

J,(0)q |M

J,(0)q 〉

as[〈M

J,(0)q |M

G,(1)q 〉 + Z(1)

GJ〈MJ,(0)q |M

J,(0)q 〉

]≡ 1 +

∞∑n=1

ans

[F

G,(n)q

]R

(5.3.18)

where

SG,(0)g = 〈M

G,(0)g |M

G,(0)g 〉 ,

SG,(1)q = 〈M

J,(0)q |M

G,(1)q 〉 + Z(1)

GJ〈MJ,(0)q |M

J,(0)q 〉 . (5.3.19)

Similarly, for defining the UV finite form factors for the other operator OJ we introduce

SJg ≡ Z s

5Z sMS〈M

G,(0)g |MJ

g〉

and

SJq ≡ Z s

5Z sMS〈M

J,(0)q |MJ

q〉 . (5.3.20)

Expanding Z sMS

and |Mλβ〉 in powers of as, following Eq. (5.3.16), we get

SJg =

∞∑n=1

ansS

J,(n)g and SJ

q =

∞∑n=0

ansS

J,(n)q . (5.3.21)

With these we define the UV renormalised form factors corresponding to OJ through

[F J

g

]R≡

SJg

asSJ,(1)g

= Z s5Z s

MSF J

g ≡ 1 +

∞∑n=1

ans

[F

J,(n)g

]R,

[F J

q

]R≡SJ

q

SJ,(0)q

= Z s5Z s

MSF J

q = 1 +

∞∑n=1

ans

[F

J,(n)q

]R

(5.3.22)

where

SJ,(1)g = 〈M

G,(0)g |M

J,(1)g 〉 ,

SJ,(0)q = 〈M

J,(0)q |M

J,(0)q 〉 . (5.3.23)

5.3 Pseudo-scalar Quark and Gluon Form Factors 105

The finite renormalisation constant Z s5 is multiplied in Eq. (5.3.20) to restore the axial anomaly

equation in dimensional regularisation. We determine all required renormalisation constants fromconsistency conditions on the universal structure of the infrared poles of the renormalised formfactors in the next section, and use these constants to derive the UV-finite form factors in Sec-tion 5.3.4.

5.3.3 Infrared Singularities and Universal Pole Structure

The renormalised form factors are ultraviolet-finite, but still contain divergences of infrared origin.In the calculation of physical quantities (which fulfill certain infrared-safety criteria [165]), theseinfrared singularities are cancelled by contributions from real radiation processes that yield thesame observable final state, and by mass factorization contributions associated with initial-statepartons. The pole structures of these infrared divergences arising in QCD form factors exhibitsome universal behaviour. The very first successful proposal along this direction was presented byCatani [129] (see also [130]) for one and two-loop QCD amplitudes using the universal subtractionoperators. The factorization of the single pole in quark and gluon form factors in terms of soft andcollinear anomalous dimensions was first revealed in [84] up to two loop level whose validity atthree loop was later established in the article [40]. The proposal by Catani was generalized beyondtwo loops by Becher and Neubert [131] and by Gardi and Magnea [132]. Below, we outline thisbehaviour in the context of pseudo-scalar form factors up to three loop level, following closely thenotation used in [28].

The unrenormalised form factors F λβ (as,Q2, µ2, ε) satisfy the so-called KG-differential equation

[79–82] which is dictated by the factorization property, gauge and renormalisation group (RG)invariances:

Q2 ddQ2 lnF λ

β (as,Q2, µ2, ε) =12

Kλβ (as,

µ2R

µ2 , ε) + Gλβ(as,

Q2

µ2R

,µ2

R

µ2 , ε) (5.3.24)

where all poles in the dimensional regulator ε are contained in the Q2 independent function Kλβ

and the finite terms in ε → 0 are encapsulated in Gλβ. RG invariance of the form factor implies

µ2R

ddµ2

R

Kλβ (as,

µ2R

µ2 , ε) = −µ2R

ddµ2

R

Gλβ(as,

Q2

µ2R

,µ2

R

µ2 , ε) = −Aλβ(as(µ2R)) = −

∞∑i=1

ais(µ

2R)Aλβ,i (5.3.25)

where, Aλβ,i on the right hand side are the i-loop cusp anomalous dimensions. It is straightforwardto solve for Kλ

β in Eq. (5.3.25) in powers of bare strong coupling constant as by performing thefollowing expansion

Kλβ

as,µ2

R

µ2 , ε

=

∞∑i=1

ais

µ2R

µ2

i ε2

S iεK

λβ,i(ε) . (5.3.26)

The solutions Kλβ,i(ε) consist of simple poles in ε with the coefficients consisting of Aλβ,i and βi.

These can be found in [28, 29]. On the other hand, the RGE of Gλβ,i(as,

Q2

µ2R,µ2

Rµ2 , ε) can be solved.

The solution contains two parts, one is dependent on µ2R whereas the other part depends only the

boundary point µ2R = Q2. The µ2

R dependent part can eventually be expressed in terms of Aλβ:

Gλβ(as,

Q2

µ2R

,µ2

R

µ2 , ε) = Gλβ(as(Q2), 1, ε) +

∫ 1

Q2

µ2R

dxx

Aλβ(as(xµ2

R)). (5.3.27)

106 Pseudo-Scalar Form Factors at Three Loops in QCD

The boundary term can be expanded in powers of as as

Gλβ(as(Q2), 1, ε) =

∞∑i=1

ais(Q

2)Gλβ,i(ε) . (5.3.28)

The solutions of Kλβ and Gλ

β enable us to solve the KG equation (Eq. (5.3.24)) and thereby facilitateto obtain the lnF λ

β (as,Q2, µ2, ε) in terms of Aλβ,i,Gλβ,i and βi which is given by [28]

lnF λβ (as,Q2, µ2, ε) =

∞∑i=1

ais

(Q2

µ2

)i ε2S iεL

λβ,i(ε) (5.3.29)

with

Lλβ,1(ε) =1ε2

{− 2Aλβ,1

}+

1ε

{Gλβ,1(ε)

},

Lλβ,2(ε) =1ε3

{β0Aλβ,1

}+

1ε2

{−

12

Aλβ,2 − β0Gλβ,1(ε)

}+

1ε

{12

Gλβ,2(ε)

},

Lλβ,3(ε) =1ε4

{−

89β2

0Aλβ,1

}+

1ε3

{29β1Aλβ,1 +

89β0Aλβ,2 +

43β2

0Gλβ,1(ε)

}+

1ε2

{−

29

Aλβ,3 −13β1Gλ

β,1(ε) −43β0Gλ

β,2(ε)}

+1ε

{13

Gλβ,3(ε)

}. (5.3.30)

All these form factors are observed to satisfy [40, 84] the following decomposition in terms ofcollinear (Bλβ), soft ( f λβ ) and UV (γλβ) anomalous dimensions:

Gλβ,i(ε) = 2

(Bλβ,i − γ

λβ,i

)+ f λβ,i + Cλ

β,i +

∞∑k=1

εkgλ,kβ,i , (5.3.31)

where the constants Cλβ,i are given by [29]

Cλβ,1 = 0 ,

Cλβ,2 = −2β0gλ,1

β,1 ,

Cλβ,3 = −2β1gλ,1

β,1 − 2β0(gλ,1β,2 + 2β0gλ,2

β,1

). (5.3.32)

In the above expressions, Xλβ,i with X = A, B, f and γλβ,i are defined through

Xλβ ≡

∞∑i=1

aisX

λβ,i , and γλβ ≡

∞∑i=1

aisγλβ,i . (5.3.33)

Within this framework, we will now determine this universal structure of IR singularities of thepseudo-scalar form factors. This prescription will be used subsequently to determine the overalloperator renormalisation constants.

We begin with the discussion of form factors corresponding to OJ . The results of the form factorsF Jβ for β = q, g, which have been computed up to three loop level in this article are being used

to extract the unknown factors, γJβ,i and gJ,k

β,i , by employing the KG equation. Since the F Jβ satisfy

KG equation, we can obtain the solutions Eq. (5.3.29) along with Eq. (5.3.30) and Eq. (5.3.31)to examine our results against the well known decomposition of the form factors in terms of the

5.3 Pseudo-scalar Quark and Gluon Form Factors 107

quantities XJβ . These are universal, and appear also in the vector and scalar quark and gluon form

factors [40, 84]. They are known [11, 84–86, 166] up to three loop level in the literature. Usingthese in the above decomposition, we obtain γJ

β,i. The other process dependent constants, namely,

gJ,kβ,i can be obtained by comparing the coefficients of εk in Eq. (5.3.30) at every order in as. We

can get the quantities γJg,i and gJ,k

g,i up to two loop level, since this process starts at one loop. Fromgluon form factors we get

γJg,1 = 0 ,

γJg,2 = CACF

{−

443

}+ CFn f

{−

103

}. (5.3.34)

Similarly, from the quark form factors we obtain

γJq,1 = 0 ,

γJq,2 = CACF

{−

443

}+ CFn f

{−

103

},

γJq,3 = C2

ACF

{−

357827

}+ C2

Fn f

{223

}−CFn2

f

{2627

}+ CAC2

F

{308

3

}+ CACFn f

{−

14927

}. (5.3.35)

Note that γJq,i = γJ

g,i which is expected since these are the UV anomalous dimensions associatedwith the same operator [OJ]B. The γJ

β,i are further used to obtain the overall operator renormalisa-tion constant Z s

MSthrough the RGE:

µ2R

ddµ2

R

ln Zλ(as, µ2R, ε) =

∞∑i=1

aisγλi . (5.3.36)

The general solution of the RGE is obtained as

Zλ = 1 + as

[1ε

2γλ1

]+ a2

s

[1ε2

{2β0γ

λ1 + 2(γλ1)2

}+

1εγλ2

]+ a3

s

[1ε3

{8β2

0γλ1 + 4β0(γλ1)2

+4(γλ1)3

3

}+

1ε2

{4β1γλ1

3+

4β0γλ2

3+ 2γλ1γ

λ2

}+

1ε

{2γλ33

}]. (5.3.37)

By substituting the results of γJβ,i in the above solution we get Z s

MSup to O(a3

s):

Z sMS

= 1 + a2s

[CACF

{−

443ε

}+ CFn f

{−

103ε

}]+ a3

s

[C2

ACF

{−

193627ε2 −

715681ε

}+ C2

Fn f

{449ε

}+ CFn2

f

{80

27ε2 −5281ε

}+ CAC2

F

{6169ε

}+ CACFn f

{−

8827ε2 −

29881ε

}],

(5.3.38)

which agrees completely with the known result in [49]. In order to restore the axial anomalyequation in dimensional regularization (see Section 5.3.2 above), we must multiply the Z s

MS[OJ]B

by a finite renormalisation constant Z s5, which reads [49]

Z s5 = 1 + as{−4CF} + a2

s

{22C2

F −1079

CACF +3118

CFn f

}. (5.3.39)

108 Pseudo-Scalar Form Factors at Three Loops in QCD

Following the computation of the operator mixing constants below, we will be able to verify ex-plicitly that this expression yields the correct expression for the axial anomaly.

Now, we move towards the discussion of OG form factors. Similar to previous case, we considerthe form factors Z−1

GG[F Gβ ]R, defined through Eq. (5.3.18), to extract the unknown constants, γG

β,i

and gG,kβ,i , by utilizing the KG differential equation. Since, [F G

β ]R is UV finite, the product of Z−1GG

with [F Gβ ]R can effectively be treated as unrenormalised form factor and hence we can demand that

Z−1GG[F G

β ]R satisfy KG equation. Further we make use of the solutions Eq. (5.3.29) in conjunctionwith Eq. (5.3.30) and Eq. (5.3.31) to compare our results against the universal decomposition of theform factors in terms of the constants XG

β . Upon substituting the existing results of the quantitiesAGβ,i, B

Gβ,i and f G

β,i up to three loops, which are obtained in case of quark and gluon form factors, we

determine the anomalous dimensions γGβ,i and the constants gG,k

β,i . However, it is only possible to

get the factors γGq,i and gG,k

q,i up to two loops because of the absence of a tree level amplitude in thequark initiated process for the operator OG. Since [F G

β ]R are UV finite, the anomalous dimensionsγGβ,i must be equal to the anomalous dimension corresponding to the renormalisation constant ZGG.

This fact is being used to determine the overall renormalisation constants ZGG and ZGJ up to threeloop level where these quantities are parameterized in terms of the newly introduced anomalousdimensions γi j through the matrix equation

µ2R

ddµ2

R

Zi j ≡ γikZk j with i, j, k = G, J (5.3.40)

This can be equivalently written as

γi j =

µ2R

ddµ2

R

Zik

(Z−1)k j. (5.3.41)

The general solution (See Example 2 in Appendix C) of the RGE up to a3s is obtained as

Zi j = δi j + as

[2εγi j,1

]+ a2

s

[1ε2

{2β0γi j,1 + 2γik,1γk j,1

}+

1ε

{γi j,2

}]+ a3

s

[1ε3

{83β2

0γi j,1

+ 4β0γik,1γk j,1 +43γik,1γkl,1γl j,1

}+

1ε2

{43β1γi j,1 +

43β0γi j,2 +

23γik,1γk j,2

+43γik,2γk j,1

}+

1ε

{23γi j,3

}](5.3.42)

where, γi j is expanded in powers of as as

γi j =

∞∑n=1

ansγi j,n . (5.3.43)

Demanding the vanishing of γGβ,i, we get

γGG = as

[113

CA −23

n f

]+ a2

s

[343

C2A −

103

CAn f − 2CFn f

]+ a3

s

[2857

54C3

A −141554

C2An f

−20518

CACFn f + C2Fn f +

7954

CAn2f +

119

CFn2f

],

γGJ = as

[− 12CF

]+ a2

s

[−

2843

CACF + 36C2F +

83

CFn f

]+ a3

s

[−

16073

C2ACF

5.3 Pseudo-scalar Quark and Gluon Form Factors 109

+ 461CAC2F − 126C3

F −1643

CACFn f + 214C2Fn f +

523

CFn2f + 288CACFn f ζ3

− 288C2Fn f ζ3

]. (5.3.44)

In addition to the demand of vanishing γGβ,i, it is required to use the results of γJJ and γJG, which

are implied by the definition, Eq. (5.3.40), up to O(a2s) to determine the above-mentioned γGG

and γGJ up to the given order. This is a consequence of the fact that the operators mix under UVrenormalisation. Following Eq. (5.3.40) along with Eq. (5.3.14), Eq. (5.3.38) and Eq. (5.3.39), weobtain

γJJ = as

[− ε2CF

]+ a2

s

[ε

{−

1079

CACF + 14C2F +

3118

CFn f

}− 6CFn f

](5.3.45)

and

γJG = 0 . (5.3.46)

As it happens, we note that γJJ’s are ε-dependent and in fact, this plays a crucial role in determin-ing the other quantities. Our results are in accordance with the existing ones, γGG and γGJ , whichare available up to O(a2

s) [49] and O(a3s) [50], respectively. In addition to the existing ones, here

we compute the new result of γGG at O(a3s). It was observed through explicit computation in the

article [49] that

γGG = −β

as(5.3.47)

holds true up to two loop level but there was no statement on the validity of this relation beyondthat order. In [50], it was demonstrated in the operator product expansion that the relation holdseven at three loop. Here, through explicit calculation, we arrive at the same conclusion that therelation is still valid at three loop level which can be seen if we look at the γGG,3 in Eq. (5.3.44)which is equal to the β2.

Before ending the discussion of γi j, we examine our results against the axial anomaly relation.The renormalisation group invariance of the anomaly equation (Eq. (5.3.9)), see [49], gives

γJJ =β

as+ γGG + as

n f

2γGJ . (5.3.48)

Through our calculation up to three loop level we find that our results are in complete agreementwith the above anomaly equation through

γGG = −β

asand γGJ =

(as

n f

2

)−1γJJ (5.3.49)

in the limit of ε → 0. This serves as one of the most crucial checks on our computation.

Additionally, if we conjecture the above relations to hold beyond three loops (which could bedoubted in light of recent findings [133]), then we can even predict the ε-independent part of theγJJ at O(a3

s):

γJJ |ε→0 = a2s

[− 6CFn f

]+ a3

s

[−

1423

CACFn f + 18C2Fn f +

43

CFn2f

]. (5.3.50)

110 Pseudo-Scalar Form Factors at Three Loops in QCD

The results of γi j uniquely specify Zi j, through Eq. (5.3.42). We summarize the resulting expres-sions of Zi j below:

ZGG = 1 + as

[223ε

CA −43ε

n f

]+ a2

s

[1ε2

{4849

C2A −

1769

CAn f +169

n2f

}+

1ε

{343

C2A

−103

CAn f − 2CFn f

}]+ a3

s

[1ε3

{10648

27C3

A −1936

9C2

An f +3529

CAn2f −

6427

n3f

}+

1ε2

{523627

C3A −

249227

C2An f −

3089

CACFn f +28027

CAn2f +

569

CFn2f

}+

1ε

{285781

C3A −

141581

C2An f −

20527

CACFn f +23

C2Fn f +

7981

CAn2f +

2227

CFn2f

}]and

ZGJ = as

[−

24ε

CF

]+ a2

s

[1ε2

{− 176CACF + 32CFn f

}+

1ε

{−

2843

CACF + 84C2F

+83

CFn f

}]+ a3

s

[1ε3

{−

38723

C2ACF +

14083

CACFn f −128

3CFn2

f

}+

1ε2

{−

95129

C2ACF +

22003

CAC2F +

22729

CACFn f −643

C2Fn f −

329

CFn2f

}+

1ε

{−

32149

C2ACF +

58949

CAC2F − 356C3

F −3289

CACFn f +1096

9C2

Fn f +1049

CFn2f

+ 192CACFn f ζ3 − 192C2Fn f ζ3

}]. (5.3.51)

ZGG and ZGJ are in agreement with the results already available in the literature up to O(a2s) [49]

and O(a3s) [50], where a completely different approach and methodology was used.

5.3.4 Results of UV Renormalised Form Factors

Using the renormalisation constants obtained in the previous section, we get all the UV renor-malised form factors

[F λβ

]R, defined in Eq. (5.3.18) and Eq. (5.3.22), up to three loops. In this

section we present the results for the choice of the scales µ2R = µ2

F = q2.

[F

G,(1)g

]R

= 2n f TF

{−

43ε

}+ CA

{−

8ε2 +

223ε

+ 4 + ζ2 + ε

(− 6 −

73ζ3

)+ ε2

(7 −

ζ2

2

+4780ζ2

2

)+ ε3

(−

152

+34ζ2 +

76ζ3 +

724ζ2ζ3 −

3120ζ5

)}, (5.3.52)

[F

G,(2)g

]R

= 4n2f T

2F

{169ε2

}+ C2

A

{32ε4 −

3083ε3 +

(629− 4ζ2

)1ε2 +

(278027

+113ζ2 +

503ζ3

)1ε

−3293

81+

1156ζ2 −

215ζ2

2 − 33ζ3 + ε

(−

114025972

−23518

ζ2 +1111120

ζ22 +

110354

ζ3

−236ζ2ζ3 −

7110ζ5

)+ ε2

(481970511664

−69427

ζ2 −2183240

ζ22 +

2313280

ζ32 −

745081

ζ3

−1136ζ2ζ3 +

90136

ζ23 −

34120

ζ5

)}+ 2CAn f TF

{563ε3 −

523ε2 +

(−

27227−

23ζ2

)1ε

5.3 Pseudo-scalar Quark and Gluon Form Factors 111

−29581−

53ζ2 − 2ζ3 + ε

(15035486

+ζ2

18+

5960ζ2

2 +38327

ζ3

)+ ε2

(−

1169871458

+583108

ζ2

−32972

ζ22 −

168881

ζ3 +6118ζ2ζ3 −

4910ζ5

)}+ 2CFn f TF

{−

2ε−

713

+ 8ζ3 + ε

(266536

−196ζ2 −

83ζ2

2 −643ζ3

)+ ε2

(−

68309432

+50536

ζ2 +649ζ2

2 +4559ζ3 −

103ζ2ζ3

+ 8ζ5

)}, (5.3.53)

[F

G,(3)g

]R

= 8n3f T

3F

{−

6427ε3

}+ 4CFn2

f T2F

{569ε2 +

(87427−

323ζ3

)1ε−

41827

+ 2ζ2 +165ζ2

2

−809ζ3

}+ 2C2

Fn f TF

{23ε

+4576

+ 104ζ3 − 160ζ5

}+ 2C2

An f TF

{−

3203ε5

+2848081ε4 +

(−

608243

+5627ζ2

)1ε3 +

(−

54088243

+67681

ζ2 +27227

ζ3

)1ε2

+

(−

6232932187

−7072243

ζ2 −94190

ζ22 −

794881

ζ3

)1ε

+634597913122

−42971729

ζ2 +68720

ζ22

+652

3ζ3 −

3019ζ2ζ3 +

451645

ζ5

}+ 4CAn2

f T2F

{−

272081ε4 +

7984243ε3 +

(56027

+827ζ2

)1ε2 +

(108892187

+14081

ζ2 +32881

ζ3

)1ε

+95156561

+1027ζ2 −

157135

ζ22 −

20243

ζ3

}+ 2CACFn f TF

{2729ε3 +

(440827−

6409ζ3

)1ε2 +

(−

6511081

+743ζ2 +

35215

ζ22

+6496

27ζ3

)1ε

+1053625

972−

3112ζ2 −

116815

ζ22 −

2487481

ζ3 + 48ζ2ζ3 +329ζ5

}+ C3

A

{−

2563ε6 +

17603ε5 −

6226481ε4 +

(−

176036243

−30827

ζ2 −1763ζ3

)1ε3 +

(207316

243

−8164

81ζ2 +

49445

ζ22 +

906427

ζ3

)1ε2 +

(2763800

2187+

36535243

ζ2 −12881180

ζ22 −

39889

ζ3

+170

9ζ2ζ3 +

175615

ζ5

)1ε−

8440640526244

+6177731458

ζ2 +1448631080

ζ22 −

22523270

ζ32

+44765243

ζ3 −1441

18ζ2ζ3 −

17669

ζ23 +

1388245

ζ5

}, (5.3.54)

[F

G,(1)q

]R

= CF

{−

8ε2 +

6ε−

334

+ ζ2 + ε

(2916

+2548ζ2 −

73ζ3

)+ ε2

(299192−

1327576

ζ2

+13872880

ζ22 +

14348

ζ3

)+ ε3

(−

137632304

+320956912

ζ2 −15593456

ζ22 +

616912

ζ32 −

1625576

ζ3

+377864

ζ2ζ3 −3120ζ5

)}+ 2n f TF

{−

445162

+ ε

(82311944

−2391944

ζ2 −23ζ3

)+ ε2

(−

505337776

+18357776

ζ2 +22903116640

ζ22 +

91255832

ζ3 +118ζ2ζ3

)+ ε3

(2754151279936

−3508393312

ζ2 −316343699840

ζ22 −

229031399680

ζ32 −

6112123328

ζ3 +2053

34992ζ2ζ3 −

1216

ζ22ζ3

112 Pseudo-Scalar Form Factors at Three Loops in QCD

−754ζ2

3 −76ζ5

)}+ CA

{7115324

−23ζ2 − 2ζ3 + ε

(−

1142413888

+73213888

ζ2 +5390ζ2

2

+133ζ3 +

16ζ2ζ3

)+ ε2

(69243515552

−5511715552

ζ2 −326369233280

ζ22 −

531080

ζ32 −

9023511664

ζ3

−41108

ζ2ζ3 −172ζ2

2ζ3 −718ζ2

3 − 5ζ5

)+ ε3

(−

37171073559872

+1013165186624

ζ2

+33990731399680

ζ22 +

3403766319595520

ζ32 +

5312960

ζ42 +

58543946656

ζ3 −5615969984

ζ2ζ3 +3223

12960ζ2

2ζ3

+1

864ζ3

2ζ3 +89ζ2

3 +7

108ζ2ζ

23 + 8ζ5 +

512ζ2ζ5

)}, (5.3.55)

[F

G,(2)q

]R

= 4n2f T

2F

{95051458

+ ε

(−

1461775832

+1241917496

ζ2 +389ζ3

)}+ 2CFn f TF

{8ε3 +

163681ε2

+

(−

12821243

−247243

ζ2 +163ζ3

)1ε

+20765324

+35486

ζ2 +85

2916ζ2

2 +6265729

ζ3 −49ζ2ζ3

+ ε

(−

145742534992

−11146729

ζ2 −232457174960

ζ22 −

8534992

ζ32 +

99071458

ζ3 −77234374

ζ2ζ3

+127ζ2

2ζ3 +2827ζ2

3 −209ζ5

)}+ C2

A

{2796445

5832−

58718

ζ2 +5330ζ2

2 −1852ζ3 −

103ζ2ζ3

+ 20ζ5 + ε

(−

3432115723328

+10420379

69984ζ2 +

58920

ζ22 +

79212520

ζ32 +

841124

ζ3 −32972

ζ2ζ3

+518ζ2

2ζ3 + 13ζ23 −

75718

ζ5 −53ζ2ζ5

)}+ 2CAn f TF

{−

1783611458

+449ζ2 −

7645ζ2

2

−449ζ3 + ε

(2357551

5832−

47817117496

ζ2 −137135

ζ22 +

19135

ζ32 −

162127

ζ3 −4027ζ2ζ3

+223ζ5

)}+ CACF

{−

44ε3 +

(−

1365481

+283ζ2 + 16ζ3

)1ε2 +

(186925

486

−3919486

ζ2 −21245

ζ22 −

2183ζ3 −

43ζ2ζ3

)1ε−

6161381

+59399

972ζ2 +

74951329160

ζ22

+53135

ζ32 +

2135171458

ζ3 +9127ζ2ζ3 +

19ζ2

2ζ3 +289ζ2

3 + ε

(35327209

34992−

215800323328

ζ2

−353264569984

ζ22 −

113077672449440

ζ32 −

531620

ζ42 −

10301692916

ζ3 +1919158748

ζ2ζ3 −817405

ζ22ζ3

−1

108ζ3

2ζ3 −1219ζ2

3 −1427ζ2ζ

23 −

436ζ5

)}+ C2

F

{32ε4 −

48ε3 +

(84 − 8ζ2

)1ε2

+

(−

1252−

616ζ2 +

1283ζ3

)1ε

+6881216

+19312

ζ2 −28124

ζ22 −

103718

ζ3

+ ε

(1664992592

−3761648

ζ2 +3451480

ζ22 −

31288

ζ32 +

10607108

ζ3 −1081108

ζ2ζ3 +32845

ζ5

)},

(5.3.56)[F

J,(1)g

]R

= 2n f TF

{−

43ε

}+ CA

{−

8ε2 +

223ε

+ 4 + ζ2 + ε

(−

152

+ ζ2 −163ζ3

)+ ε2

(28724− 2ζ2 +

12780

ζ22

)+ ε3

(−

5239288

+15148

ζ2 +19120

ζ22 +

ζ3

12+

76ζ2ζ3

5.3 Pseudo-scalar Quark and Gluon Form Factors 113

−9120ζ5

)}+ CF

{ε

(−

212

+ 6ζ3

)+ ε2

(1558−

52ζ2 −

95ζ2

2 −92ζ3

)+ ε3

(−

102532

+8316ζ2 +

2720ζ2

2 +203ζ3 −

34ζ2ζ3 +

212ζ5

)}, (5.3.57)

[F

J,(2)g

]R

= 4n2f T

2F

{169ε2

}+ CACF

{(84 − 48ζ3

)1ε− 232 + 20ζ2 +

725ζ2

2 + 80ζ3

+ ε

(17545

108− 58ζ2 − 24ζ2

2 −383ζ3 + 10ζ2ζ3 − 14ζ5

)+ ε2

(4026351296

−23336

ζ2

+725ζ2

2 +1770ζ3

2 +53512

ζ3 − 2ζ2ζ3 − 34ζ23 −

13556

ζ5

)}+ 2CAn f TF

{563ε3 −

523ε2

+

(−

27227−

23ζ2

)1ε−

13381− 3ζ2 + 2ζ3 + ε

(7153243

−7

18ζ2 −

1360ζ2

2 +59927

ζ3

)+ ε2

(−

1352391458

+1139108

ζ2 −16724

ζ22 −

314681

ζ3 +7318ζ2ζ3 −

13730

ζ5

)}+ 2CFn f TF

{−

2ε−

293

+ ε

(14989216

−256ζ2 −

415ζ2

2 − 32ζ3

)+ ε2

(−

6066612592

+2233

72ζ2 +

15815

ζ22 +

140918

ζ3 − 2ζ2ζ3 +823ζ5

)}+ C2

A

{+

32ε4 −

3083ε3

+

(629− 4ζ2

)1ε2 +

(310427−

133ζ2 +

1223ζ3

)1ε−

739781

+772ζ2 −

615ζ2

2 − 55ζ3

+ ε

(−

32269972

−99736

ζ2 +1049120

ζ22 −

2393108

ζ3 −536ζ2ζ3 +

36910

ζ5

)+ ε2

(456995511664

−15323432

ζ2 +2129180

ζ22 −

7591840

ζ32 −

40991296

ζ3 −60536

ζ2ζ3 +77536

ζ23

+2011

30ζ5

)}+ C2

F

{ε

(76312

+ 17ζ3 − 60ζ5

)+ ε2

(−

18857144

+313ζ2 −

7615ζ2

2

+120

7ζ3

2 − 145ζ3 + 4ζ2ζ3 + 30ζ23 +

4703ζ5

)}, (5.3.58)

[F

J,(1)q

]R

= CF

{−

8ε2 +

6ε− 6 + ζ2 + ε

(− 1 −

34ζ2

73ζ3

)+ ε2

(52

+ζ2

4+

4780ζ2

2 +74ζ3

)+ ε3

(−

134

+ζ2

8−

141320

ζ22 −

712ζ3 +

724ζ2ζ3 −

3120ζ5

)}, (5.3.59)

[F

J,(2)q

]R

= 2CFn f TF

{8ε3 −

169ε2 +

(−

6527− 2ζ2

)1ε−

3115324

+239ζ2 +

29ζ3 + ε

(129577

3888

−731108

ζ2 −ζ2

2

10+

11927

ζ3

)+ ε2

(−

305433746656

+209511296

ζ2 −145144

ζ22 −

2303324

ζ3

−109ζ2ζ3 −

5930ζ5

)}+ C2

F

{32ε4 −

48ε3 +

(66 − 8ζ2

)1ε2 +

(−

532

+1283ζ3

)1ε

−1218

+ζ2

2− 13ζ2

2 − 58ζ3 + ε

(340332

+278ζ2 +

17110

ζ22 +

5596ζ3 −

563ζ2ζ3

+925ζ5

)+ ε2

(−

21537128

−82532

ζ2 −45716

ζ22 +

22320

ζ32 −

420524

ζ3 +272ζ2ζ3

114 Pseudo-Scalar Form Factors at Three Loops in QCD

+652

9ζ2

3 −23110

ζ5

)}+ CACF

{−

44ε3 +

(649

+ 4ζ2

)1ε2 +

(96154

+ 11ζ2

− 26ζ3

)1ε−

30493648

−19318

ζ2 +445ζ2

2 +3139ζ3 + ε

(−

794037776

+133216

ζ2 −22920

ζ22

−4165

54ζ3 +

896ζ2ζ3 −

512ζ5

)+ ε2

(973232393312

+413632592

ζ2 +331511440

ζ22 −

809280

ζ32

+89929

648ζ3 −

809ζ2ζ3 −

56912

ζ23 +

280960

ζ5

)}, (5.3.60)

[F

J,(3)q

]R

= Z s,(3)5 + 4CFn2

f T2F

{−

70481ε4 +

64243ε3 +

(18481

+169ζ2

)1ε2 +

(−

48342187

+4027ζ2

+1681ζ3

)1ε

+53823113122

−68081

ζ2 −188135

ζ22 −

416243

ζ3

}+ C3

F

{−

2563ε6 +

192ε5

+

(− 336 + 32ζ2

)1ε4 +

(280 + 24ζ2 −

8003ζ3

)1ε3 +

(− 58 − 66ζ2 +

4265ζ2

2

+ 552ζ3

)1ε2 +

(−

41936

+ 83ζ2 −146110

ζ22 −

31423

ζ3 +4283ζ2ζ3 −

12885

ζ5

)1ε

+41395

24+

193312

ζ2 +10739

40ζ2

2 −9095252

ζ32 + 1385ζ3 − 35ζ2ζ3 −

18263

ζ23 −

5625ζ5

}+ 2C2

Fn f TF

{−

64ε5 +

5609ε4 +

(−

68027

+ 24ζ2

)1ε3 +

(518081−

2669ζ2 −

4409ζ3

)1ε2

+

(−

78863243

+2381

27ζ2 +

28718

ζ22 −

93827

ζ3

)1ε

+1369027

1458−

1661081

ζ2 −85031080

ζ22

+22601

81ζ3 +

353ζ2ζ3 −

3869ζ5

}+ C2

ACF

{−

2129681ε4 +

(−

22928243

+88027

ζ2

)1ε3

+

(23338

243+

650081

ζ2 −35245

ζ22 −

360827

ζ3

)1ε2 +

(139345

4374+

14326243

ζ2 +33215

ζ22

−7052

27ζ3 +

1769ζ2ζ3 +

2723ζ5

)1ε−

1065979752488

−207547

729ζ2 +

19349270

ζ22 −

6152189

ζ32

+361879

486ζ3 +

3443ζ2ζ3 −

11369

ζ23 −

25949

ζ5

}+ 2CACFn f TF

{+

774481ε4 +

(6016243

−16027

ζ2

)1ε3 +

(−

8272243

−190481

ζ2 +84827

ζ3

)1ε2 +

(173182187

−5188243

ζ2 −8815ζ2

2

+1928

81ζ3

)1ε−

415865913122

+81778729

ζ2 −17135

ζ22 −

588127

ζ3 +223ζ2ζ3 +

1763ζ5

}+ CAC2

F

{352ε5 +

(−

28889− 32ζ2

)1ε4 +

(443627− 108ζ2 + 208ζ3

)1ε3

+

(39844

81+

9839ζ2 −

3325ζ2

2 −1928

9ζ3

)1ε2 +

(−

97048243

−12361

54ζ2 +

297536

ζ22

+3227

3ζ3 −

4303ζ2ζ3 + 284ζ5

)1ε−

709847729

+36845324

ζ2 −5366832160

ζ22 −

186191260

ζ32

−31537

18ζ3 −

5183ζ2ζ3 +

16163

ζ23 +

17509

ζ5

}. (5.3.61)

5.4 Gluon Form Factors for the Pseudo-scalar Higgs Boson Production 115

5.3.5 Universal Behaviour of Leading Transcendentality Contribution

In [135], the form factor of a scalar composite operator belonging to the stress-energy tensor super-multiplet of conserved currents of N = 4 super Yang-Mills (SYM) with gauge group SU(N) wasstudied to three-loop level. Since the theory is UV finite in d = 4 space-time dimensions, it is anideal framework to study the IR structures of amplitudes in perturbation theory. In this theory, oneobserves that scattering amplitudes can be expressed as a linear combinations of polylogarithmicfunctions of uniform degree 2l, where l is the order of the loop, with constant coefficients. In otherwords, the scattering amplitudes in N = 4 SYM exhibit uniform transcendentality, in contrast toQCD loop amplitudes, which receive contributions from all degrees of transcendentality up to 2l.

The three-loop QCD quark and gluon form factors [42] display an interesting relation to the SYMform factor. Upon replacement [167] of the color factors CA = CF = N and T f n f = N/2, theleading transcendental (LT) parts of the quark and gluon form factors in QCD not only coincidewith each other but also become identical, up to a normalization factor of 2l, to the form factors ofscalar composite operator computed in N = 4 SYM [135].

This correspondence between the QCD form factors and that of theN = 4 SYM can be motivatedby the leading transcendentality principle [167–169] which relates anomalous dimensions of thetwist two operators inN = 4 SYM to the LT terms of such operators computed in QCD. Examiningthe diagonal pseudo-scalar form factors F G

g and F Jq , we find a similar behaviour: the LT terms of

these form factors with replacement CA = CF = N and T f n f = N/2 are not only identical to eachother but also coincide with the LT terms of the QCD form factors [42] with the same replacementas well as with the LT terms of the scalar form factors inN = 4 SYM [135], up to a normalizationfactor of 2l. This observation holds true for the finite terms in ε, and could equally be validated forhigher-order terms up to transcendentality 8 (which is the highest order for which all three-loopmaster integrals are available [170]). In addition to checking the diagonal form factors, we alsoexamined the off-diagonal ones namely, F G

q , F Jg , where we find that the LT terms these two form

factors are identical to each other after the replacement of colour factors. However, the LT termsof these do not coincide with those of the diagonal ones.

5.4 Gluon Form Factors for the Pseudo-scalar Higgs Boson Pro-duction

The complete form factor for the production of a pseudo-scalar Higgs boson through gluon fusion,F

A,(n)g , can be written in terms of the individual gluon form factors, Eq. (5.3.3), as follows:

F Ag = F G

g +

(ZGJ

ZGG+

4CJ

CG

ZJJ

ZGG

)F J

g〈M

G,(0)g |M

J,(1)g 〉

〈MG,(0)g |M

G,(0)g 〉

. (5.4.1)

In the above expression, the quantities Zi j(i, j = G, J) are the overall operator renormalization con-stants which are required to introduce in the context of UV renormalization. These are discussedin Sec. 5.3.2 in great detail. The ingredients of the form factor F A

g , namely, F Gg and F J

g have beencalculated up to three loop level by us [13] and are presented in the Appendix G. Using thoseresults we obtain the three loop form factor for the pseudo-scalar Higgs boson production throughgluon fusion. In this section, we present the unrenormalized form factors F A,(n)

g up to three loop

116 Pseudo-Scalar Form Factors at Three Loops in QCD

where the components are defined through the expansion

F Ag ≡

∞∑n=0

ans

(Q2

µ2

)n ε2

S nε F

A,(n)g

. (5.4.2)

We present the unrenormalized results for the choice of the scale µ2R = µ2

F = q2 as follows:

FA,(1)

g = CA

{−

8ε2 + 4 + ζ2 + ε

(− 6 −

73ζ3

)+ ε2

(7 −

ζ2

2+

4780ζ2

2

)+ ε3

(−

152

+34ζ2 +

76ζ3 +

724ζ2ζ3 −

3120ζ5

)+ ε4

(314−

78ζ2 −

47160

ζ22 +

9494480

ζ32 −

74ζ3 −

49144

ζ23

)+ ε5

(−

638

+1516ζ2 +

141320

ζ22 +

4924ζ3 −

748ζ2ζ3 +

3291920

ζ22ζ3 +

3140ζ5 +

31160

ζ2ζ5

−127112

ζ7

)+ ε6

(12716−

3132ζ2 −

329640

ζ22 −

9498960

ζ32 +

55779716800

ζ42 −

3516ζ3 +

732ζ2ζ3

+49288

ζ23 +

491152

ζ2ζ23 −

9380ζ5 −

217480

ζ3ζ5

)+ ε7

(−

25532

+6364ζ2 +

141256

ζ22 +

284717920

ζ32

+21796

ζ3 −49192

ζ2ζ3 −3293840

ζ22ζ3 +

94915360

ζ32ζ3 −

49192

ζ23 −

34310368

ζ33 +

217160

ζ5

−31320

ζ2ζ5 +145712800

ζ22ζ5 +

127224

ζ7 +127896

ζ2ζ7 −511576

ζ9

)},

FA,(2)

g = CFn f

{−

803

+ 6 ln(

q2

m2t

)+ 8ζ3 + ε

(2827

36− 9 ln

(q2

m2t

)−

196ζ2 −

83ζ2

2

−643ζ3

)+ ε2

(−

70577432

+212

ln(

q2

m2t

)+

103772

ζ2 −34

ln(

q2

m2t

)ζ2 +

649ζ2

2 +4559ζ3

−103ζ2ζ3 + 8ζ5

)+ ε3

(1523629

5184−

454

ln(

q2

m2t

)−

14975432

ζ2 +98

ln(

q2

m2t

)ζ2

−709974320

ζ22 +

2235ζ3

2 −3292

27ζ3 +

74

ln(

q2

m2t

)ζ3 +

809ζ2ζ3 + 15ζ2

3 −643ζ5

)+ ε4

(−

3048766162208

+938

ln(

q2

m2t

)+

43217648

ζ2 −2116

ln(

q2

m2t

)ζ2 +

199165951840

ζ22

−141320

ln(

q2

m2t

)ζ2

2 −176105

ζ32 +

6942312592

ζ3 −218

ln(

q2

m2t

)ζ3 −

9757432

ζ2ζ3 −1681180

ζ22ζ3

− 40ζ23 +

8851180

ζ5 − 2ζ2ζ5 −127

8ζ7

)}+ CAn f

{−

83ε3 +

209ε2 +

(10627

+ 2ζ2

)1ε

−159181−

53ζ2 −

749ζ3 + ε

(24107486

−2318ζ2 +

5120ζ2

2 +38327

ζ3

)+ ε2

(−

1461471458

+799108

ζ2 −32972

ζ22 −

143681

ζ3 +256ζ2ζ3 −

27130

ζ5

)+ ε3

(633306134992

−11531648

ζ2

+ +1499240

ζ22

2531680

ζ32 +

19415972

ζ3 −23536

ζ2ζ3 −1153108

ζ23 +

53536

ζ5

)+ ε4

(−

128493871419904

+1332373888

ζ2 −215332592

ζ22 +

6491440

ζ32 −

1561275832

ζ3 +21527

ζ2ζ3 +51780

ζ22ζ3 +

14675648

ζ23

−2204135

ζ5 +17140

ζ2ζ5 +229336

ζ7

)}+ C2

A

{32ε4 +

443ε3 +

(−

4229− 4ζ2

)1ε2 +

(89027

5.4 Gluon Form Factors for the Pseudo-scalar Higgs Boson Production 117

− 11ζ2 +503ζ3

)1ε

+383581

+115

6ζ2 −

215ζ2

2 +119ζ3 + ε

(−

213817972

−10318

ζ2 +77120

ζ22

+110354

ζ3 −236ζ2ζ3 −

7110ζ5

)+ ε2

(610274511664

−99127

ζ2 −2183240

ζ22 +

2313280

ζ32 −

883681

ζ3

−5512ζ2ζ3 +

90136

ζ23 +

34160

ζ5

)+ ε3

(−

142142401139968

+75881648

ζ2 +798192160

ζ22 −

2057480

ζ32

+6060351944

ζ3 −25172

ζ2ζ3 −1291

80ζ2

2ζ3 −5137216

ζ23 +

14459360

ζ5 +31340

ζ2ζ5 −316928

ζ7

)+ ε4

(2999987401

1679616−

19434297776

ζ2 −15707160

ζ22 −

3517720160

ζ32 +

504191600

ζ42 −

1659347923328

ζ3

+116927

ζ2ζ3 +227811440

ζ22ζ3 +

937311296

ζ23 −

1547144

ζ2ζ23 −

813754

ζ5 −100180

ζ2ζ5 +84524

ζ3ζ5

−332ζ5,3 +

56155672

ζ7

)},

FA,(3)

g = n f C(2)J

{− 2 + 3ε

}+ CFn2

f

{(−

6409

+ 16 ln(

q2

m2t

)+

643ζ3

)1ε

+790127

− 24 ln(

q2

m2t

)−

323ζ2 −

11215

ζ22 −

8489ζ3

}+ C2

Fn f

{457

6+ 104ζ3 − 160ζ5

}+ C2

An f

{643ε5 −

3281ε4 +

(−

18752243

−37627

ζ2

)1ε3 +

(36416243

−170081

ζ2 +207227

ζ3

)1ε2

+

(626422187

+22088243

ζ2 −2453

90ζ2

2 −3988

81ζ3

)1ε−

1465580913122

−60548729

ζ2 +91760

ζ22

−77227

ζ3 −4399ζ2ζ3 +

323845

ζ5

}+ CAn2

f

{−

12881ε4 +

640243ε3 +

(12827

+8027ζ2

)1ε2

+

(−

930882187

−40081

ζ2 −132881

ζ3

)1ε

+1066349

6561−

5627ζ2 +

797135

ζ22 +

13768243

ζ3

}+ CACFn f

{−

169ε3 +

(5980

27− 48 ln

(q2

m2t

)−

6409ζ3

)1ε2 +

(−

2037781

− 16 ln(

q2

m2t

)+

863ζ2 +

35215

ζ22 +

174427

ζ3

)1ε

+ 72 ln(

q2

m2t

)−

587705972

−5516ζ2

+ 12 ln(

q2

m2t

)ζ2 −

965ζ2

2 +12386

81ζ3 + 48ζ2ζ3 +

329ζ5

}+ C3

A

{−

2563ε6 −

3523ε5

+1614481ε4 +

(22864243

+206827

ζ2 −1763ζ3

)1ε3 +

(−

172844243

−163081

ζ2 +49445

ζ22

−83627

ζ3

)1ε2 +

(2327399

2187−

71438243

ζ2 +3751180

ζ22 −

8429ζ3 +

1709ζ2ζ3 +

175615

ζ5

)1ε

+16531853

26244+

9189311458

ζ2 +272511080

ζ22 −

22523270

ζ32 −

51580243

ζ3 +7718ζ2ζ3 −

17669

ζ23

+20911

45ζ5

}. (5.4.3)

The results up to two loop level is consistent with the existing ones [84] and the three loop result isthe new one. These are later used to compute the SV cross-section for the production of a pseudo-scalar particle through gluon fusion at N3LO QCD [14]. This is an essential ingredient to compute

118 Pseudo-Scalar Form Factors at Three Loops in QCD

all the other associated observables.

5.5 Hard Matching Coefficients in SCET

Soft-collinear effective theory (SCET, [171–177]) is a systematic expansion of the full QCD theoryin terms of particle modes with different infrared scaling behaviour. It provides a framework toperform threshold resummation. In the effective theory, the infrared poles of the full high energyQCD theory manifest themselves as ultraviolet poles [178–180], which then can be resummed byemploying the renormalisation group evolution from larger scales to the smaller ones. To ensurematching of SCET and full QCD, one computes the matrix elements in both theories and adjuststhe Wilson coefficients of SCET accordingly. For the on-shell matching of these two theories, thematching coefficients relevant to pseudo-scalar production in gluon fusion can be obtained directlyfrom the gluon form factors.

The UV renormalised form factors in QCD contain IR divergences. Since the IR poles in QCDturn into UV ones in SCET, we can remove the IR divergences with the help of a renormalisationconstant ZA,h

g , which essentially absorbs all residual IR poles and produces finite results. The resultis the matching coefficient CA,eff

g , which is defined through the following factorisation relation:

CA,effg

(Q2, µ2

h

)≡ lim

ε→0(ZA,h

g )−1(ε,Q2, µ2h)

[F A

g

]R

(ε,Q2) (5.5.1)

where, the UV renormalised form factor[F A

g

]R, is defined as

[F A

g

]R

=[F G

g

]R

+4CJ

CG

[F J

g

]R

asS J,(1)

g

S G,(0)g

. (5.5.2)

The parameter µh is the newly introduced mass scale at which the above factorisation is carriedout. For the UV renormalised form factors [F A

g ]R in Eq. (5.5.1), we fixed the other scales asµ2

R = µ2F = µ2

h. Upon expanding the ZA,hg and CA,eff

g in powers of as as

ZA,hg (ε,Q2, µ2

h) = 1 +

∞∑i=1

ais(µ

2h)ZA,h

g,i (ε,Q2, µ2h) ,

CA,effg

(Q2, µ2

h

)= 1 +

∞∑i=1

ais(µ

2h)CA,eff

g,i

(Q2, µ2

h

)(5.5.3)

and utilising the above Eq. (5.5.1), we compute the ZA,hg,i as well as CA,eff

g,i up to three loops (i = 3).

Demanding the cancellation of the residual IR poles of[F A

g

]R

against the poles of (ZA,hg,i )−1, we

compute ZA,hg,i which comes out to be

ZA,hg,1 = CA

{−

8ε2 +

(− 4L +

223

)1ε

}− n f

{43ε

},

ZA,hg,2 = CFn f

{−

2ε

}+ n2

f

{169ε2

}+ CAn f

{563ε3 +

(−

523

+ 8L)

1ε2 +

(−

12827

+209

L

+23ζ2

)1ε

}+ C2

A

{32ε4 +

(−

3083

+ 32L)

1ε3 +

(350

9− 44L + 8L2 + 4ζ2

)1ε2 +

(69227

−134

9L −

113ζ2 + 4Lζ2 − 2ζ3

)1ε

},

5.5 Hard Matching Coefficients in SCET 119

ZA,hg,3 = C2

Fn f

{23ε

}+ CFn2

f

{569ε2 +

2227ε

}− n3

f

{64

27ε3

}+ C2

An f

{−

3203ε5 +

(28480

81

− 96L)

1ε4 +

(−

18752243

+3152

27L −

643

L2 −44827

ζ2

)1ε3 +

(−

32656243

+713681

L

−809

L2 +100081

ζ2 −1049

Lζ2 +34427

ζ3

)1ε2 +

(−

307152187

+83681

L +2396243

ζ2 −16027

Lζ2

−32845

ζ22 −

71281

ζ3 +1129

Lζ3

)1ε

}+ CAn2

f

{−

272081ε4 +

(7984243

−35227

L)

1ε3 +

(36827

−40081

L −4027ζ2

)1ε2 +

(2692187

+1681

L −4081ζ2 +

11281

ζ3

)1ε

}+ CACFn f

{2729ε3

+

(−

70427

+403

L −649ζ3

)1ε2 +

(−

243481

+1109

L +43ζ2 +

3215ζ2

2 +30427

ζ3

−323

Lζ3

)1ε

}+ C3

A

{−

2563ε6 +

(1760

3− 128L

)1ε5 +

(−

7263281

+ 528L − 64L2

− 32ζ2

)1ε4 +

(−

29588243

−5824

27L +

3523

L2 −323

L3 +246427

ζ2 − 48Lζ2 + 16ζ3

)1ε3

+

(80764

243−

2549281

L +5369

L2 −148681

ζ2 +5729

Lζ2 − 16L2ζ2 −35245

ζ22 −

83627

ζ3

+ 8Lζ3

)1ε2 +

(1943722187

−4909

L −12218243

ζ2 +107227

Lζ2 +127645

ζ22 −

17615

Lζ22 −

2449ζ3

−889

Lζ3 +809ζ2ζ3 +

323ζ5

)1ε

}. (5.5.4)

After cancellation of the IR poles, we are left with the following finite matching coefficients:

CA,effg,1 = CA

{− L2 + 4 + ζ2

},

CA,effg,2 = C2

A

{12

L4 +119

L3 + L2(−

1039

+ ζ2

)+ L

(−

1027−

223ζ2 − 2ζ3

)+

480781

+916ζ2

+12ζ2

2 −143

9ζ3

}+ CAn f

{−

29

L3 +109

L2 + L(3427

+43ζ2

)−

94381−

53ζ2 −

469ζ3

}+ CFn f

{−

803

+ 6 ln

µ2h

m2t

+ 8ζ3

},

CA,effg,3 = n f C

(2)J

{− 2

}+ CFn2

f

{L(−

3209

+ 8 ln

µ2h

m2t

+323ζ3

)+

7499−

209ζ2 −

1645ζ2

2

−1123ζ3

}+ C2

Fn f

{4576

+ 104ζ3 − 160ζ5

}+ C2

An f

{29

L5 −827

L4 + L3(−

75281

−23ζ2

)+ L2

(51227−

1039ζ2 +

1189ζ3

)+ L

(129283

729+

419881

ζ2 −485ζ2

2 +289ζ3

)−

794627313122

−19292

729ζ2 +

7345ζ2

2 −276481

ζ3 −829ζ2ζ3 +

4289ζ5

}+ C3

A

{−

16

L6 −119

L5

+ L4(38954−

32ζ2

)+ L3

(220681

+113ζ2 + 2ζ3

)+ L2

(−

20833162

+75718

ζ2 −7310ζ2

2

120 Pseudo-Scalar Form Factors at Three Loops in QCD

+1439ζ3

)+

22229

ζ5 + L(−

5000111458

−16066

81ζ2 +

1765ζ2

2 +183227

ζ3 +343ζ2ζ3

+ 16ζ5

)+

4109153926244

+3169391458

ζ2 −1399270

ζ22 −

243891890

ζ32 −

176584243

ζ3 −6059ζ2ζ3

−1049ζ2

3

}+ CAn2

f

{−

227

L4 +4081

L3 + L2(8081

+89ζ2

)+ L

(−

12248729

−8027ζ2

−12827

ζ3

)+

2801456561

+49ζ2 +

427ζ2

2 +4576243

ζ3

}+ CACFn f

{−

23

L3 + L2(2156

− 6 ln

µ2h

m2t

− 16ζ3

)+ L

(9173

54− 44 ln

µ2h

m2t

+ 4ζ2 +165ζ2

2 −3769ζ3

)+ 24 ln

µ2h

m2t

− 726935972

−41518

ζ2 + 6 ln

µ2h

m2t

ζ2 −6445ζ2

2 +20180

81ζ3 +

643ζ2ζ3

+6089ζ5

}. (5.5.5)

In the above expressions, L = ln(Q2/µ2

h

)= ln

(−q2/µ2

h

). These matching coefficients allow to

perform the matching of the SCET-based resummation onto the full QCD calculation up to three-loop order.

Before ending the discussion of this section, we demonstrate the universal factorisation propertyfulfilled by the anomalous dimension of the ZA,h

g which is defined through the RG equation

µ2h

ddµ2

h

ln ZA,hg (ε,Q2, µ2

h) ≡ γA,hg (Q2, µ2

h) =

∞∑i=1

ais(µ

2h)γA,h

g,i (Q2, µ2h) . (5.5.6)

The renormalisation group invariance of the UV renormalised [FAg ]R(ε,Q2) with respect to the

scale µh implies

µ2h

ddµ2

h

ln ZA,hg + µ2

hd

dµ2h

ln CA,effg = 0 . (5.5.7)

By explicitly evaluating the γA,hg,i using the results of ZA,h

g (Eq. (5.5.4)) up to three loops (i = 3), wefind that these satisfy the following decomposition in terms of the universal factors Ag,i, Bg,i andfg.i:

γA,hg,i = −

12

Ag,iL +

(Bg,i +

12

fg,i

). (5.5.8)

This in turn implies the evolution equation of the matching coefficients as

µ2h

ddµ2

h

ln CA,effg,i =

12

Ag,iL −(Bg,i +

12

fg,i

)(5.5.9)

which is in complete agreement with the existing results [181] upon identifying

γV = Bg.i +12

fg,i . (5.5.10)

5.6 Summary 121

5.6 Summary

In this part of the thesis, we derived the three-loop massless QCD corrections to the quark andgluon form factors of pseudo-scalar operators. Working in dimensional regularisation, we used the’t Hooft-Veltman prescription for γ5 and the Levi-Civita tensor, which requires non-trivial finiterenormalisation to maintain the symmetries of the theory. By exploiting the universal behaviourof the infrared pole structure at three loops in QCD, we were able to independently determinethe renormalisation constants and operator mixing, in agreement with earlier results that wereobtained in a completely different approach [49, 50].

The three-loop corrections to the pseudo-scalar form factors are an important ingredient to preci-sion Higgs phenomenology. They will ultimately allow to bring the gluon fusion cross section forpseudo-scalar Higgs production to the same level of accuracy that has been accomplished mostrecently for scalar Higgs production with fixed order N3LO [44] and soft-gluon resummation atN3LL [34, 156, 158, 160].

With our new results, the soft-gluon resummation for pseudo-scalar Higgs production [159, 160]can be extended imminently to N3LL accuracy [14], given the established formalisms at this or-der [34, 156]. With the derivation of the three-loop pseudo-scalar form factors presented here,all ingredients to this calculation are now available. Another imminent application is the thresh-old approximation to the N3LO cross section [14]. By exploiting the universal infrared struc-ture [34], one can use the result of an explicit computation of the threshold contribution to theN3LO cross section for scalar Higgs production [33] to derive threshold results for other processesessentially through the ratios of the respective form factors (which is no longer possible beyondthreshold [44,182], where the corrections become process-specific), as was done for the Drell-Yanprocess [32] and for Higgs production from bottom quark annihilation [11].

6 Conclusions and Outlooks

No doubt, the whole particle physics community is standing on the verge of a crucial era, where themain tasks can be largely categorized into two parts: testing the SM with unprecedented accuracyand searching for the physics beyond SM. In achieving these golden tasks, precise theoreticalpredictions play a very crucial role. The field of precision studies at theoretical level is mostlycontrolled by the higher order corrections to the scattering amplitudes, that are the basic buildingblocks of constructing any observable in QFT. Among all the higher order corrections, the QCDones contribute substantially to any typical observable. This thesis deals with this higher orderQCD radiative corrections to the observables associated with the Drell-Yan, scalar and pseudo-scalar Higgs boson.

The Higgs boson is among the best candidates at hadron collider, and hence it is of utmost im-portance to make the theoretical prediction as precise as possible to the associated observables.In the first part of the thesis, Chapter 2, we have computed the N3LO QCD radiative corrections,arising from the soft gluons, to the inclusive production cross section of the Higgs boson producedthrough bottom quark annihilation [11]. Of course, this is not the dominant production channelof the scalar Higgs boson in the SM, nonetheless its contribution must also be taken into accountin this spectacular precision studies. In order to achieve this, we have systematically employedan elegant prescription [28, 29]. The factorisation of QCD amplitudes, gauge invariance, renor-malisation group invariance and the Sudakov resummation of soft gluons are at the heart of thisformalism. The recently available three loop Hbb QCD form factors [31] and the soft gluon con-tributions calculated [32] from the threshold QCD corrections to the Higgs boson at N3LO [33],enable us to compute the full N3LO soft-virtual QCD corrections to the production cross section ofthe Higgs boson produced through bottom quark annihilation. One of the most beautiful parts ofthis calculation is that even without evaluating all the hundreds or thousands of Feynman diagramscontributing to the real emissions, we have obtained the required contribution arising from the softgluons! The universal nature of the soft gluons are the underlying reasons behind this remarkablefeature. We have also demonstrated the numerical impact of this result at the LHC. This is themost accurate result for this production channel existing in the literature till date and it is expectedto play an important role in coming days.

In the second part of the thesis, Chapter 3, we have dealt with an another very important observ-able, namely, the rapidity distributions of the Higgs boson produced through gluon fusion andthe leptonic pair in Drell-Yan. The importance of these two processes are quite self-evident! Wehave computed the threshold enhanced N3LO QCD corrections [12] to these observables employ-ing the formalism developed in the article [39]. The skeleton of this elegant prescription whichhas been employed is also based on the properties, like, the factorisation of QCD amplitudes,

123

124 Conclusions and Outlooks

gauge invariance, renormalisation group invariance and the Sudakov resummation of soft gluons.With the help of recently computed inclusive production cross section of the Higgs boson [33]and Drell-Yan [32] at threshold N3LO QCD, we have computed the contributions arising fromthe soft gluons to the processes under consideration. These were the only missing ingredients toachieve our goal. Our newly calculated part of this distribution is found to be the most dominantone compared to the other contributions. We have demonstrated numerically the impact of thisresult for the Higgs boson at the LHC. Indeed, inclusion of this N3LO contributions does reducethe dependence on the unphysical renormalisation and factorisation scales. It is worth mentioningthat, this beautiful formalism not only helps us to compute the rapidity distribution at threshold,but also enhance our understanding about the underlying structures of the QCD amplitudes.

In the third part of the thesis, Chapter 4, we have discussed the relatively modern techniques ofthe multiloop computations which have been employed to get some of the results calculated inthis thesis. The backbone of this methodology is the integration-by-parts [46, 47] and Lorentzinvariant [48] identities. The successful implementation of these in computer codes revolutionizesthe area of multiloop computations.

The last part, Chapter 5, is dealt with a particle, pseudo-scalar, which is not included in particlespectrum of the SM, but is believed to be present in the nature. Intensive search for this particlehas been going on for past several years, although nothing conclusive evidence has been found.However, to make conclusive remark about the existence of this particle, we need to revamp theunderstanding about this particle and improve the precision of the theoretical predictions. Thiswork arises exactly at this context. In these articles [13, 14], we have computed one of the impor-tant ingredients to calculate the inclusive production cross section or the differential distributionsfor the pseudo-scalar at N3LO QCD which is presently the level of accuracy for the scalar Higgsboson, achieved very recently [44]. In particular, we have derived the three loop massless QCDcorrections to the quark and gluon form factors of the pseudo-scalar. Unlike the scalar Higgs bo-son, this problem involves the γ5 which makes the life interesting as well as challenging. We havehandled them under the ‘t Hooft-Veltman prescription for the γ5 and Levi-Civita tensor in dimen-sional regularisation. Employing this prescription, however, brings some additional complication,namely, it violates the chiral Ward identity. In order to rectify this, we need to perform an addi-tional and non-trivial finite renormalisation. By exploiting the universal behaviour of the infraredpole structure at three loops in QCD, we were able to independently determine the renormalisationconstants and operator mixing, in agreement with the earlier results that were obtained in a com-pletely different approach [49, 50]. We must emphasize the approach which we have employedhere for the first time is exactly opposite to the usual one: the infrared pole structures of the formfactors have been taken to be universal that dictates us to obtain the UV operator renormalisationconstants upon imposing the demand of UV finiteness. With our new results, the threshold ap-proximation to the N3LO inclusive production cross section for the pseudo-scalar through gluonfusion are obtained [14] by us. This is also extended to the N3LL resummed accuracy in [14].We have also computed the hard matching coefficients in the context of soft-collinear effectivetheory which are later employed to obtain the N3LL’ resummed cross section [183]. We have alsofound some interesting facts about the form factors in the context of Leading Transcendentalityprinciple [167–169]: the LT terms of the diagonal form factors with replacement CA = CF = Nand T f n f = N/2 are not only identical to each other but also coincide with the LT terms of theQCD form factors [42] with the same replacement as well as with the LT terms of the scalar formfactors in N = 4 SYM [135], up to a normalization factor of 2l. This observation holds true forthe finite terms in ε, and could equally be validated for higher-order terms up to transcendentality8 (which is the highest order for which all three-loop master integrals are available [170]). Inaddition to checking the diagonal form factors, we also examined the off-diagonal ones, where we

125

find that the LT terms these two form factors are identical to each other after the replacement ofcolour factors. However, the LT terms of these do not coincide with those of the diagonal ones.

The state-of-the-art techniques, which mostly use our in-house codes, have been employed exten-sively to carry out all the computations presented in this thesis. The prescription of computing thethreshold correction is applicable for any colorless final state particle. We are in the process of ex-tending this formalism to the case of threshold resummation of differential rapidity distributions.The methodology of calculating the pseudo-scalar form factors can be generalized to the casesinvolving any number of operators which can mix among each others under UV renormalisation.

In conclusion, it has been a while the Higgs-like particle has been discovered at the LHC andfinally, we are very close to having enough statistics for precision measurements of the Higgsquantum numbers and coupling constants to fermions and gauge bosons. This, along with theprecise results from theoreticians like us, hopefully, would help to explore the underlying natureof the electroweak symmetry breaking and possibly open the door of new physics.

Appendices

127

A Inclusive Production CrossSection

In QCD improved parton model, the inclusive cross-section for the production of a colorless par-ticle can be computed using

σI(τ, q2) =∑

a,b=q,q,g

1∫0

dx1

1∫0

dx2 fa(x1, µ2F) fb(x2, µ

2F)σI

ab

(z, q2, µ2

R, µ2F

)(A.0.1)

where, f ’s are the partonic distribution functions factorised at the mass scale µF . σIab is the par-

tonic cross section for the production of colorless particle I from the partons a and b. This isUV renormalised at renormalisation scale µR and mass factorised at µF . The other quantities aredefined as

q2 = m2I ,

τ =q2

S,

z =q2

s. (A.0.2)

In the above expression, S and s are square of the hadronic and partonic center of mass energies,respectively, and they are related by

s = x1x2S . (A.0.3)

By introducing the identity ∫dzδ(τ − x1x2z) =

1x1x2

=Ss

(A.0.4)

in Eq. (A.0.1), we can rewrite the Eq. (A.0.1) as

σI(τ, q2) = σI,(0)(µ2R)

∑ab=q,q,g

1∫τ

dx Φab(x, µ2F) ∆I

ab

(τ

x, q2, µ2

R, µ2F

). (A.0.5)

The partonic flux Φab is defined through

Φab(x, µ2F) =

1∫x

dyy

fa(y, µ2F) fb

(xy, µ2

F

)(A.0.6)

129

130 Inclusive Production Cross Section

and the dimensionless quantity ∆Iab is called the coefficient function of the partonic level cross

section. Upon normalising the partonic level cross section by the born one, we obtain ∆Iab i.e.

∆Iab ≡

σIab

σI,(0) . (A.0.7)

B Anomalous Dimensions

Here we present A [40, 69, 85, 86], f [40, 84], and B [40, 85] up to three loop level. The A’s aregiven by

Agg,1 = CA{4},

Agg,2 = C2A

{2689− 8ζ2

}+ CAn f

{−

409

},

Agg,3 = C3A

4903−

1072ζ2

9+

88ζ3

3+

176ζ22

5

+ CACFn f

{−

1103

+ 32ζ3

}+ C2

An f

{−

83627

+160ζ2

9−

112ζ3

3

}+ CAn2

f

{−

1627

}and

Aqq,i = Abb,i =CF

CAAgg,i . (B.0.1)

The f ’s are obtained as

fgg,1 = 0 ,

fgg,2 = C2A

{−

223ζ2 − 28ζ3 +

80827

}+ CAn f

{43ζ2 −

11227

},

fgg,3 = CA3{

3525ζ2

2 +1763ζ2ζ3 −

1265081

ζ2 −1316

3ζ3 + 192ζ5 +

136781729

}+ C2

An f

{−

965ζ2

2 +2828

81ζ2 +

72827

ζ3 −11842729

}+ CACFn f

{325ζ2

2 + 4ζ2 +3049ζ3 −

171127

}+ CAn f

2{−

4027ζ2 +

11227

ζ3 −2080729

}and

fqq,i = fbb,i =CF

CAfgg,i . (B.0.2)

Similarly the B’s are given by

Bgg,1 = CA

{113

}− n f

{23

},

Bgg,2 = C2A

{323

+ 12ζ3

}− n f CA

{83

}− n f CF

{2},

131

132 Anomalous Dimensions

Bgg,3 = CACFn f

{−

24118

}+ CAn2

f

{2918

}−C2

An f

{23318

+83ζ2 +

43ζ2

2 +803ζ3

}+ C3

A

{792− 16ζ2ζ3 +

83ζ2 +

223ζ2

2 +5363ζ3 − 80ζ5

}+ CFn2

f

{119

}+ C2

Fn f{1},

Bqq,1 = CF{3},

Bqq,2 = C2F

{32− 12ζ2 + 24ζ3

}+ CACF

{1734

+886ζ2 − 12ζ3

}+ n f CFTF

{−

23−

163ζ2

},

Bqq,3 = CA2CF

{− 2ζ2

2 +449627

ζ2 −1552

9ζ3 + 40ζ5 −

165736

}+ CACF

2{−

98815

ζ22

+ 16ζ2ζ3 −410

3ζ2 +

8443ζ3 + 120ζ5 +

1514

}+ CACFn f

{45ζ2

2 −133627

ζ2 +200

9ζ3

+ 20}

+ CF3{

2885ζ2

2 − 32ζ2ζ3 + 18ζ2 + 68ζ3 − 240ζ5 +292

}+ CF

2n f

{23215

ζ22 +

203ζ2 −

1363ζ3 − 23

}+ CFn f

2{

8027ζ2 −

169ζ3 −

179

},

and

Bqq,i = Bbb,i . (B.0.3)

C Solving Renormalisation GroupEquation

To demonstrate the methodoogy of solving RGE, let us consider a general form of an RGE withrespect to the renormalisation scale µR:

µ2R

ddµ2

R

ln M = N (C.0.1)

where, M and N are functions of µR. We need to solve for M in terms of N. Our goal is tosolve it order by order in perturbation theory. We start by expanding the quantities in powers ofas ≡ as(µ2

R):

M = 1 +

∞∑k=1

aks M(k) ,

N =

∞∑k=1

aksN(k) . (C.0.2)

The µR dependence of M and N on µR enters through as. All the coefficients M(k) and N(k) areindependent of µR. Hence, ln M can be written as

ln M =

∞∑k=1

aks Mk (C.0.3)

with

M1 = M(1) ,

M2 = −12

(M(1))2 + M(2) ,

M3 =13

(M(1))3 − M(1)M(2) + M(3) ,

M4 = −14

(M(1))4 + (M(1))2M(2) −12

(M(2))2 − M(1)M(3) + M(4) . (C.0.4)

Using the above expansions in RGE. (C.0.1) and using the RGE of as

µ2R

ddµ2

R

as =ε

2as −

∞∑k=0

βk ak+2s (C.0.5)

133

134 Solving Renormalisation Group Equation

we get Mk’s by comparing the coefficients of as as

M1 =2ε

N(1) ,

M2 =2ε2 β0N(1) +

1ε

N(2) ,

M3 =8

3ε3 β20N(1) +

1ε2

{43β1N(1) +

43β0N(2)

}+

23ε

N(3) ,

M4 =4ε4 β

30N(1) +

1ε3

{4β0β1N(1) + 2β2

0N(2)}

+1ε2

{β2N(1) + β1N(2) + β0N(3)

}+

12ε

N(4) . (C.0.6)

By equating the Eq. (C.0.3) and (C.0.6) we obtain,

M(1) =2ε

N(1) ,

M(2) =1ε2

{2β0N(1) + 2(N(1))2

}+

1ε

N(2) ,

M(3) =1ε3

{83β2

0N(1) + 4β0(N(1))2 +43

(N(1))3}

+1ε2

{43β1N(1) +

43β0N(2) + 2N(1)N(2)

}+

23ε

N(3) ,

M(4) =1ε4

{4β3

0N(1) +223β2

0(N(1))2 + 4β0(N(1))3 +23

(N(1))4}

+1ε3

{4β0β1N(1) +

83β1(N(1))2

+ 2β20N(2) +

143β0N(1)N(2) + 2(N(1))2N(2)

}+

1ε2

{β2N(1) + β1N(2) +

12

(N(2))2 + β0N(3)

+43

N(1)N(3)}

+12ε

N(4) . (C.0.7)

We have presented the solution up to O(a4s). However, this procedure can be easily generalised to

all orders in as.

• Example 1: RGE of Zas

µ2R

ddµ2

R

ln Zas =1as

∞∑k=0

ak+2s βk (C.0.8)

Comparing this RGE of Zas with Eq. (C.0.1) we get

N(k) = βk−1 k ∈ [1,∞) . (C.0.9)

By putting the values of N(k) in the general solution (C.0.7), we get the corresponding solu-tions of Zas as

Zas = 1 +

∞∑k=1

aksZ

(k)as (C.0.10)

where

Z(1)as =

2εβ0 ,

135

Z(2)as =

4ε2 β

20 +

1εβ1 ,

Z(3)as =

8ε3 β

30 +

143ε2 β0β1 +

23εβ2 ,

Z(4)as =

16ε4 β

40 +

463ε3 β

20β1 +

1ε2

(32β2

1 +103β0β2

)+

12εβ3 . (C.0.11)

The Zas can also be expressed in powers of as by utilising the

as = asS ε

µ2R

µ2

ε/2 Z−1as

(C.0.12)

iteratively. We get

Zas = 1 +

∞∑k=1

aksS

kε

µ2R

µ2

k ε2

Z(k)as (C.0.13)

where

Z(1)as =

2εβ0 ,

Z(2)as =

1εβ1 ,

Z(3)as = −

43ε2 β0β1 +

23εβ2 ,

Z(4)as =

2ε3 β

20β1 +

1ε2

(−

12β2

1 − 2β0β2

)+

12εβ3 . (C.0.14)

To arrive at the above result, we need to use the Z−1as

in powers of as:

Z−1as

= 1 +

∞∑k=1

aksS

kε

µ2R

µ2

k ε2

Z−1,(k)as (C.0.15)

where

Z−1,(1)as = −

2εβ0 ,

Z−1,(2)as =

4ε2 β

20 −

1εβ1 ,

Z−1,(3)as = −

8ε3 β

30 +

163ε2 β0β1 −

23εβ2 ,

Z−1,(4)as =

16ε4 β

40 −

583ε3 β

20β1 +

1ε2

(32β2

1 +143β0β2

)−

12εβ3 . (C.0.16)

• Example 2: Solution of the Mass Factorisation KernelThe mass factorisation kernel satisfies the RG equation (2.3.32)

µ2F

ddµ2

F

ΓIi j(z, µ

2F , ε) =

12

∑k

PIik

(z, µ2

F

)⊗ ΓI

k j

(z, µ2

F , ε)

(C.0.17)

where, PI(z, µ2

F

)are Altarelli-Parisi splitting functions (matrix valued). Expanding PI

(z, µ2

F

)and ΓI(z, µ2

F , ε) in powers of the strong coupling constant we get

PI(z, µ2F) =

∞∑k=1

aks(µ

2F)PI,(k−1)(z) (C.0.18)

136 Solving Renormalisation Group Equation

and

ΓI(z, µ2F , ε) = δ(1 − z) +

∞∑k=1

aksS

kε

µ2F

µ2

k ε2

ΓI,(k)(z, ε) . (C.0.19)

Following the techniques prescribed above, it can be solved. However, unlike the previouscases here we have to take care of the fact that PI and ΓI are matrix valued quantities i.e.they are non-commutative. Upon solving we obtain the general solution as

ΓI,(1)(z, ε) =1ε

{PI,(0)(z)

},

ΓI,(2)(z, ε) =1ε2

{− β0PI,(0)(z) +

12

PI,(0)(z) ⊗ PI,(0)(z)}

+1ε

{12

PI,(1)(z)},

ΓI,(3)(z, ε) =1ε3

{43β2

0PI,(0) − β0PI,(0) ⊗ PI,(0) +16

PI,(0) ⊗ PI,(0) ⊗ PI,(0)}

+1ε2

{−

13β1PI,(0) +

16

PI,(0) ⊗ PI,(1) −43β0PI,(1) +

13

PI,(1) ⊗ PI,(0)}

+1ε

{13

PI,(2)},

ΓI,(4)(z, ε) =1ε4

{− 2β3

0PI,(0) +116β2

0PI,(0) ⊗ PI,(0) −12β0PI,(0) ⊗ PI,(0) ⊗ PI,(0)

+124

PI,(0) ⊗ PI,(0) ⊗ PI,(0) ⊗ PI,(0)}

+1ε3

{43β0β1PI,(0) −

13β1PI,(0) ⊗ PI,(0)

+124

PI,(0) ⊗ PI,(0) ⊗ PI,(1) −712β0PI,(0) ⊗ PI,(1) +

112

PI,(0) ⊗ PI,(1) ⊗ PI,(0)

+ 3β20PI,(1) −

54β0PI,(1) ⊗ PI,(0) +

18

PI,(1) ⊗ PI,(0) ⊗ PI,(0)}

+1ε2

{−

16β2PI,(0) +

112

PI,(0) ⊗ PI,(2) −12β1PI,(1) +

18

PI,(1) ⊗ PI,(1)

−32β0PI,(2) +

14

PI,(2) ⊗ PI,(0)}

+1ε

{14

PI,(3)}. (C.0.20)

In the soft-virtual limit, only the diagonal parts of the kernels contribute. Our findings areconsistent with the existing diagonal solutions which can be found in the article [28].

D Solving KG Equation

The form factor satisfies the KG differential equation (See Sec. 2.3.1):

Q2 ddQ2 lnF I

i j(as,Q2, µ2, ε) =12

K Ii j

as,µ2

R

µ2 , ε

+ GIi j

as,Q2

µ2R

,µ2

R

µ2 , ε

. (D.0.1)

In this appendix we demonstrate the procedure to solve the KG equation. RG invariance of the Fwith respect to the renormalisation scale µR implies

µ2R

ddµ2

R

K Ii j

as,µ2

R

µ2 , ε

= −µ2R

ddµ2

R

GIi j

as,Q2

µ2R

,µ2

R

µ2 , ε

= −AIi j

(as(µ2

R))

(D.0.2)

where, AIi j’s are the cusp anomalous dimensions. Unlike the previous cases, we expand K I

i j inpowers of unrenormalised as as

K Ii j

as,µ2

R

µ2 , ε

=

∞∑k=1

aksS

kε

µ2R

µ2

k ε2

K Ii j,k(ε) (D.0.3)

whereas we define the components AIi j,k through

AIi j =

∞∑k=1

aks

(µ2

R

)AI

i j,k . (D.0.4)

Following the methodology discussed in Appendix C, we can solve for K Ii j,k(ε)

K Ii j,1(ε) =

1ε

{− 2AI

i j,1

},

K Ii j,2(ε) =

1ε2

{2β0AI

i j,1

}+

1ε

{− AI

i j,2

},

K Ii j,3(ε) =

1ε3

{−

83β2

0AIi j,1

}+

1ε2

{23β1AI

i j,1 +83β0AI

i j,2

}+

1ε

{−

23

AIi j,3

},

K Ii j,4(ε) =

1ε4

{4β3

0AIi j,1

}+

1ε3

{−

83β0β1AI

i j,1 − 6β20AI

i j,2

}+

1ε2

{13β2AI

i j,1 + β1AIi j,2 + 3β0AI

i j,3

}+

1ε

{−

12

AIi j,4

}(D.0.5)

137

138 Solving KG Equation

Due to dependence of GIi j on Q2, we need to handle it differently. Integrating the RGE of GI

i j,(D.0.2), we get

GIi j

as,Q2

µ2R

,µ2

R

µ2 , ε

−GIi j

(as, 1,

Q2

µ2 , ε

)=

µ2R∫

Q2

dµ2R

µ2R

AIi j

⇒ GIi j

as(µ2R),

Q2

µ2R

, ε

= GIi j

(as(Q2), 1, ε

)+

µ2R∫

Q2

dµ2R

µ2R

AIi j (D.0.6)

Consider the second part of the above Eq. (D.0.6)

µ2R∫

Q2

dµ2R

µ2R

AIi j =

µ2R∫

Q2

dµ2R

µ2R

∞∑k=1

aksAI

i j,k

=

∞∑k=1

µ2Rµ2∫

Q2

µ2

dX2

X2 aksS

kε

(X2

)k ε2(Z−1

as(X2)

)kAI

i j,k (D.0.7)

where we have made the change of integration variable from µR to X by µ2R = X2µ2. By using the

Z−1as

(X2) from Eq. (C.0.15) and evaluating the integral we obtain

µ2R∫

Q2

dµ2R

µ2R

AIi j =

∞∑k=1

aksS

kε

µ2R

µ2

k ε2Q2

µ2R

k ε2− 1

K Ii j,k(ε) . (D.0.8)

The first part of GIi j in Eq. (D.0.6) can be expanded in powers of as(Q2) as

GIi j

(as(Q2), 1, ε

)=

∞∑k=1

aks(Q

2)GIi j(ε) . (D.0.9)

By putting back the Eq. (D.0.3), (D.0.7) and (D.0.8) in the original KG equation (D.0.1), we solvefor lnF I

i j(as,Q2, µ2, ε):

lnF Ii j(as,Q2, µ2, ε) =

∞∑k=1

aksS

kε

(Q2

µ2

)k ε2LI

i j,k(ε) (D.0.10)

with

LIi j,1(ε) =

1ε2

{− 2AI

i j,1

}+

1ε

{GI

i j,1(ε)},

LIi j,2(ε) =

1ε3

{β0AI

i j,1

}+

1ε2

{−

12

AIi j,2 − β0GI

i j,1(ε)}

+1ε

{12

GIi j,2(ε)

},

LIi j,3(ε) =

1ε4

{−

89β2

0AIi j,1

}+

1ε3

{29β1AI

i j,1 +89β0AI

i j,2 +43β2

0GIi j,1(ε)

}+

1ε2

{−

29

AIi j,3 −

13β1GI

i j,1(ε) −43β0GI

i j,2(ε)}

+1ε

{13

GIi j,3(ε)

},

139

LIi j,4(ε) =

1ε5

{AI

i j,1β30

}+

1ε4

{−

32

AIi j,2β

20 −

23

AIi j,1β0β1 − 2β3

0GIi j,1(ε))

}+

1ε3

{34

AIi j,3β0 +

14

AIi j,2β1 +

112

AIi j,1β2 +

43β0β1GI

i j,1(ε) + 3β20GI

i j,2(ε))}

+1ε2

{−

18

AIi j,4 −

16β2GI

i j,1(ε) −12β1GI

i j,2(ε) −32β0GI

i j,3(ε)}

+1ε

{14

GIi j,4(ε)

}. (D.0.11)

This methodology can easily be generalised to all orders in perturbation theory.

E Soft-Collinear Distribution

In Sec. 2.3.4, we introduced the soft-collinear distribution ΦHbb

in the context of computing SVcross section of the Higgs boson production in bb annihilation. In this appendix, we intend toelaborate the methodology of finding this distribution. For the sake of generalisation, we use Iinstead of H and omit the partonic indices. To understand the underlying logics behind finding ΦI ,let us consider an example at one loop level. The generalisation to higher loop is straightforward.

As discussed in the Sec. 2.3, the SV cross section in z-space can be computed in d = 4 + ε

dimensions using

∆I,SV(z, q2, µ2R, µ

2F) = C exp

(Ψ I

(z, q2, µ2

R, µ2F , ε

) )∣∣∣∣ε=0

(E.0.1)

where,Ψ I(z, q2, µ2

R, µ2F , ε

)is a finite distribution andC is the convolution defined through Eq. (3.3.15).

The Ψ I is given by, Eq. (2.3.8)

Ψ I(z, q2, µ2

R, µ2F , ε

)=

(ln

[ZI(as, µ

2R, µ

2, ε)]2

+ ln∣∣∣∣F I(as,Q2, µ2, ε)

∣∣∣∣2) δ(1 − z)

+ 2ΦI(as, q2, µ2, z, ε) − 2C lnΓI(as, µ2, µ2

F , z, ε) . (E.0.2)

For all the details about the notations, see Sec. 2.3. Considering only the poles at O(as) withµR = µF we obtain,

ln(ZI,(1)

)2= as(µ2

F)4γI

1

ε,

ln |F I,(1)|2 = as(µ2F)

q2

µ2F

ε2−4AI

1

ε2 +1ε

(2 f I

1 + 4BI1 − 4γI

1

) ,2C lnΓI,(1) = 2as(µ2

F)2BI

1

εδ(1 − z) +

2AI1

εD0

(E.0.3)

where, the components are defined through the expansion of these quantities in powers of as(µ2F)

Ψ I =

∞∑k=1

aks

(µ2

F

)Ψ I,(k) ,

ln(ZI)2 =

∞∑k=1

aks

(µ2

F

)ZI,(k) ,

ln |F I |2 =

∞∑k=1

aks

(µ2

F

) q2

µ2F

k ε2ln |F I,(k)|2 ,

141

142 Soft-Collinear Distribution

ΦI =

∞∑k=1

aks(µ

2F)ΦI

k ,

lnΓI =

∞∑k=1

aks(µ

2F) lnΓI,(k) (E.0.4)

and

Di ≡

[lni(1 − z)

1 − z

]+

. (E.0.5)

Collecting the coefficients of as(µ2F), we get

Ψ I,(1)|poles =

−4AI

1

ε2 +2 f I

1

ε

δ(1 − z) −4AI

1

εD0

+ 2ΦI1 (E.0.6)

where, we have not shown the ln(q2/µ2F) terms. To cancel the remaining divergences appearing in

the above Eq. (E.0.6) for obtaining a finite cross section, we must demand that ΦI1 have exactly

the same poles with opposite sign:

2ΦI1|poles = −

−4AI

1

ε2 +2 f I

1

ε

δ(1 − z) −4AI

1

εD0

(E.0.7)

In addition, ΦI also should be RG invariant with respect to µR:

µ2R

ddµ2

R

ΦI = 0 . (E.0.8)

We make an ansatz, the above two demands, Eq. (E.0.7) and (E.0.8) can be accomplished if ΦI

satisfies the KG-type integro-differential equation which we call KG:

q2 ddq2Φ

I(as, q2, µ2, z, ε

)=

12

K Ias,

µ2R

µ2 , z, ε + G

Ias,

q2

µ2R

,µ2

R

µ2 , z, ε . (E.0.9)

KI

contains all the poles whereas GI

consists of only the finite terms in ε. RG invariance (E.0.8)of ΦI dictates

µ2R

ddµ2

R

KI

= −µ2R

ddµ2

R

GI≡ Y I (E.0.10)

where, we introduce a quantity Y I . Following the methodology of solving the KG equation dis-cussed in the Appendix D, we can write the solution of ΦI as

ΦI(as, q2, µ2, z, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

k(z, ε) (E.0.11)

with

ΦIk(z, ε) = LI

k

(AI

i → Y Ii ,G

Ii → G

Ii (z, ε)

). (E.0.12)

where we define the components through the expansions

Y I =

∞∑k=1

aks(µ

2F)Y I

k ,

143

GI(z, ε) =

∞∑k=1

aks(µ

2F)G

Ik(z, ε). (E.0.13)

This solution directly follows from the Eq. (D.0.10). Hence we get

2ΦI1(z, ε) =

1ε2

{− 4Y I

1

}+

2ε

{G

I1(z, ε)

}. (E.0.14)

By expressing the components of ΦI in powers of as(µ2F), we obtain

ΦI(as, q2, µ2, z, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

k(z, ε)

=

∞∑k=1

aks(µ

2F)

q2

µ2F

k ε2Zk

asΦI

k(z, ε)

≡

∞∑k=1

aks(µ

2F)

q2

µ2F

k ε2ΦI

k(z, ε) (E.0.15)

and at O(as(µ2F)), ΦI

1(z, ε) = ΦI1(z, ε) upon suppressing the terms like log(q2/µ2

F). Hence, bycomparing the Eq. (E.0.7) and (E.0.14), we conclude

Y I1 = −AI

1δ(1 − z) ,

GI1(z, ε) = − f I

1δ(1 − z) + 2AI1D0 +

∞∑k=1

εkgI,k1 (z) . (E.0.16)

The coefficients of εk, gI,k1 (z) can only be determined through explicit computations. These do not

contribute to the infrared poles associated with ΦI . This uniquely fixes the unknown soft-collineardistributionΦI at one loop order. This prescription can easily be generalised to higher orders in as.In our calculation of the SV cross section, instead of solving in this way, we follow a bit differentmethodology which is presented below.

Keeping the demands (E.0.7) and (E.0.8) in mind, we propose the solution of the KG equation as(See Eq. (E.0.11))

ΦIk(z, ε) ≡

{kε

11 − z

[(1 − z)2

]k ε2}ΦI

k(ε)

=

{δ(1 − z) +

∞∑j=0

(kε) j+1

j!D j

}ΦI

k(ε) . (E.0.17)

The RG invariance of ΦI , Eq. (E.0.8), implies

µ2R

ddµ2

R

KI

= −µ2R

ddµ2

R

GI≡ Y ′I (E.0.18)

where, we introduce a quantity Y ′, analogous to Y . Hence, the solution can be obtained as

ΦIk(ε) = LI

k

(AI

i → Y ′Ii ,GIi → G

Ii (ε)

). (E.0.19)

Hence, according to the Eq. (D.0.11), for k = 1 we get

2ΦI1(z, ε) =

{1ε2

(−4Y ′I1

)+

2εG

I1 (ε))

}δ(1 − z) +

{−

4Y ′I1

ε2 +2εG

I1(ε)

} ∞∑j=0

ε j+1

j!D j (E.0.20)

144 Soft-Collinear Distribution

where, Y ′I and GI

are expanded similar to Eq. (E.0.13). Comparison between the two solutionsdepicted in Eq. (E.0.7) and (E.0.20), we can write

Y ′I1 = −AI1

GI1 (ε) = − f I

1 +

∞∑k=1

εkGI,k1 . (E.0.21)

Explicit computation is required to determine the coefficients of εk, GI,k1 . This solution is used

in Eq. (2.3.38) in the context of SV cross section of Higgs boson production. The method isgeneralised to higher orders in as to obtain the results of the soft-collinear distribution. In the nextsubsection, we present the results of the soft-collinear distribution up to three loops.

E.0.1 Results

We define the renormalised components of the ΦIqq,k through

ΦIqq(as, q2, µ2, zε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

qq,k(z, ε)

=

∞∑k=1

aks

(µ2

F

)ΦI

qq,k

(z, ε, q2, µ2

F

)(E.0.22)

where, we make the choice of the renormalisation scale µR = µF . The µR dependence can beeasily restored by using the evolution equation of strong coupling constant, Eq. (2.4.6). Below, wepresent the ΦI

i i,kfor i i = qq up to three loops and the corresponding components for i i = gg can

be obtained using maximally non-Abelian property fulfilled by this distribution:

ΦIgg,k =

CA

CFΦI

qq,k . (E.0.23)

The results are given by

ΦIqq,1 = δ(1 − z)

[1ε2 CF

{8}

+1ε

log q2

µ2F

CF

{4}

+ CF

{− 3ζ2

}+ log2

q2

µ2F

CF

{1}]

+D0

[1ε

CF

{8}

+ log q2

µ2F

CF

{4}]

+D1

[CF

{8}],

ΦIqq,2 = δ(1 − z)

[1ε3 CFCA

{44

}+

1ε3 n f CF

{− 8

}+

1ε2 CFCA

{134

9− 4ζ2

}+

1ε2 n f CF

{−

209

}+

1ε2 log

q2

µ2F

CFCA

{443

}+

1ε2 log

q2

µ2F

n f CF

{−

83

}+

1ε

CFCA

{−

40427

+ 14ζ3 +113ζ2

}+

1ε

n f CF

{5627−

23ζ2

}+

1ε

log q2

µ2F

CFCA

{134

9− 4ζ2

}+

1ε

log q2

µ2F

n f CF

{−

209

}+ CFCA

{1214

81

−187

9ζ3 −

46918

ζ2 + 2ζ22

}+ n f CF

{−

16481

+349ζ3 +

359ζ2

}+ log

q2

µ2F

CFCA

{−

40427

+ 14ζ3 +443ζ2

}+ log

q2

µ2F

n f CF

{5627−

83ζ2

}

145

+ log2 q2

µ2F

CFCA

{679− 2ζ2

}+ log2

q2

µ2F

n f CF

{−

109

}+ log3

q2

µ2F

CFCA

{−

119

}+ log3

q2

µ2F

n f CF

{29

}]+D0

[1ε2 CFCA

{883

}+

1ε2 n f CF

{−

163

}+

1ε

CFCA

{2689

− 8ζ2

}+

1ε

n f CF

{−

409

}+ CFCA

{−

80827

+ 28ζ3 +883ζ2

}+ n f CF

{11227−

163ζ2

}+ log

q2

µ2F

CFCA

{268

9− 8ζ2

}+ log

q2

µ2F

n f CF

{−

409

}+ log2

q2

µ2F

CFCA

{−

223

}+ log2

q2

µ2F

n f CF

{43

}]+D1

[CFCA

{5369− 16ζ2

}+ n f CF

{−

809

}+ log

q2

µ2F

CFCA

{−

883

}+ log

q2

µ2F

n f CF

{163

}]+D2

[CFCA

{−

883

}+ n f CF

{163

}],

ΦIqq,3 = δ(1 − z)

[1ε4 CFC2

A

{21296

81

}+

1ε4 n f CFCA

{−

774481

}+

1ε4 n2

f CF

{70481

}+

1ε3 CFC2

A

{49064243

−88027

ζ2

}+

1ε3 n f CFCA

{−

15520243

+16027

ζ2

}+

1ε3 n f C2

F

{−

1289

}+

1ε3 n2

f CF

{800243

}+

1ε3 log

q2

µ2F

CFC2A

{1936

27

}+

1ε3 log

q2

µ2F

n f CFCA

{−

70427

}+

1ε3 log

q2

µ2F

n2f CF

{+

6427

}+

1ε2 CFC2

A

{−

8956243

+202427

ζ3 −69281

ζ2 +35245

ζ22

}+

1ε2 n f CFCA

{4024243

−56027

ζ3

−20881

ζ2

}+

1ε2 n f C2

F

{−

22027

+649ζ3

}+

1ε2 n2

f CF

{−

16081

+1627ζ2

}+

1ε2 log

q2

µ2F

CFC2A

{834481−

1769ζ2

}+

1ε2 log

q2

µ2F

n f CFCA

{−

267281

+329ζ2

}+

1ε2 log

q2

µ2F

n f C2F

{−

163

}+

1ε2 log

q2

µ2F

n2f CF

{16081

}+

1ε

CFC2A

{−

1367812187

− 64ζ5 +1316

9ζ3 +

12650243

ζ2 −1769ζ2ζ3 −

35215

ζ22

}+

1ε

n f CFCA

{118422187

−72881

ζ3

−2828243

ζ2 +325ζ2

2

}+

1ε

n f C2F

{1711

81−

30427

ζ3 −43ζ2 −

3215ζ2

2

}+

1ε

n2f CF

{20802187

−11281

ζ3 +4081ζ2

}+

1ε

log q2

µ2F

CFC2A

{4909

+889ζ3 −

107227

ζ2 +17615

ζ22

}+

1ε

log q2

µ2F

n f CFCA

{−

83681−

1129ζ3 +

16027

ζ2

}+

1ε

log q2

µ2F

n f C2F

{−

1109

+323ζ3

}+

1ε

log q2

µ2F

n2f CF

{−

1681

}+ CFC2

A

{521194926244

−4849ζ5 −

128966243

ζ3

+536

9ζ2

3 −5784791458

ζ2 + 242ζ2ζ3 +9457135

ζ22 +

152189

ζ32

}+ n f CFCA

{−

41276513122

146 Soft-Collinear Distribution

−83ζ5 +

985681

ζ3 +75155729

ζ2 −443ζ2ζ3 −

2528135

ζ22

}+ n f C2

F

{−

42727972

+1129ζ5

+228481

ζ3 +60518

ζ2 −883ζ2ζ3 +

15245

ζ22

}+ n2

f CF

{−

1286561

−1480243

ζ3 −40481

ζ2

+148135

ζ22

}+ log

q2

µ2F

CFC2A

{−

2970291458

− 96ζ5 +10036

27ζ3 +

2455681

ζ2 −883ζ2ζ3

−74815

ζ22

}+ log

q2

µ2F

n f CFCA

{31313

729−

6209ζ3 −

734881

ζ2 +18415

ζ22

}+ log

q2

µ2F

n f C2F

{1711

54−

1529ζ3 − 8ζ2 −

165ζ2

2

}+ log

q2

µ2F

n2f CF

{−

928729

+8027ζ3

+16027

ζ2

}+ log2

q2

µ2F

CFC2A

{15503162

− 44ζ3 −752

9ζ2 +

445ζ2

2

}+ log2

q2

µ2F

n f CFCA

{−

205181

+ 24ζ2

}+ log2

q2

µ2F

n f C2F

{−

556

+ 8ζ3

}+ log2

q2

µ2F

n2f CF

{10081−

169ζ2

}+ log3

q2

µ2F

CFC2A

{−

178081

+449ζ2

}+ log3

q2

µ2F

n f CFCA

{57881−

89ζ2

}+ log3

q2

µ2F

n f C2F

{23

}+ log3

q2

µ2F

n2f CF

{−

4081

}

+ log q2

µ2F

4

CFC2A

{12154

}+ log

q2

µ2F

4

n f CFCA

{−

2227

}+ log

q2

µ2F

4

n2f CF

{227

}]+D0

[1ε3 CFC2

A

{387227

}+

1ε3 n f CFCA

{−

140827

}+

1ε3 n2

f CF

{12827

}+

1ε2 CFC2

A

{16688

81−

3529ζ2

}+

1ε2 n f CFCA

{−

534481

+649ζ2

}+

1ε2 n f C2

F

{−

323

}+

1ε2 n2

f CF

{32081

}+

1ε

CFC2A

{980

9+

1769ζ3 −

214427

ζ2 +35215

ζ22

}+

1ε

n f CFCA

{−

167281−

2249ζ3 +

32027

ζ2

}+

1ε

n f C2F

{−

2209

+643ζ3

}+

1ε

n2f CF

{−

3281

}+ CFC2

A

{−

297029729

− 192ζ5 +20072

27ζ3 +

4911281

ζ2 −1763ζ2ζ3

−1496

15ζ2

2

}+ n f CFCA

{62626729

−1240

9ζ3 −

1469681

ζ2 +36815

ζ22

}+ n f C2

F

{171127

−304

9ζ3 − 16ζ2 −

325ζ2

2

}+ n2

f CF

{−

1856729

+16027

ζ3 +32027

ζ2

}+ log

q2

µ2F

CFC2A

{31006

81− 176ζ3 −

30089

ζ2 +1765ζ2

2

}+ log

q2

µ2F

n f CFCA

{−

820481

+ 96ζ2

}+ log

q2

µ2F

n f C2F

{−

1103

+ 32ζ3

}+ log

q2

µ2F

n2f CF

{40081−

649ζ2

}+ log2

q2

µ2F

CFC2A

{−

356027

+883ζ2

}

147

+ log2 q2

µ2F

n f CFCA

{1156

27−

163ζ2

}+ log2

q2

µ2F

n f C2F

{4}

+ log2 q2

µ2F

n2f CF

{−

8027

}+ log3

q2

µ2F

CFC2A

{48427

}+ log3

q2

µ2F

n f CFCA

{−

17627

}+ log3

q2

µ2F

n2f CF

{1627

}]+D1

[CFC2

A

{62012

81− 352ζ3 −

60169

ζ2 +3525ζ2

2

}+ n f CFCA

{−

1640881

+ 192ζ2

}+ n f C2

F

{−

2203

+ 64ζ3

}+ n2

f CF

{80081−

1289ζ2

}+ log

q2

µ2F

CFC2A

{−

1424027

+3523ζ2

}+ log

q2

µ2F

n f CFCA

{462427−

643ζ2

}+ log

q2

µ2F

n f C2F

{16

}+ log

q2

µ2F

n2f CF

{−

32027

}+ log2

q2

µ2F

CFC2A

{9689

}+ log2

q2

µ2F

n f CFCA

{−

3529

}+ log2

q2

µ2F

n2f CF

{329

}]+D2

[CFC2

A

{−

1424027

+352

3ζ2

}+ n f CFCA

{462427−

643ζ2

}+ n f C2

F

{16

}+ n2

f CF

{−

32027

}+ log

q2

µ2F

CFC2A

{1936

9

}+ log

q2

µ2F

n f CFCA

{−

7049

}+ log

q2

µ2F

n2f CF

{649

}]+D3

[CFC2

A

{387227

}+ n f CFCA

{−

140827

}+ n2

f CF

{12827

}]. (E.0.24)

F Soft-Collinear Distribution forRapidity

In Sec. 3.3.4, we introduced the soft-collinear distribution ΦIi i

in the context of computing SVcorrection to the differential rapidity distribution of a colorless particle at Hadron collider. Inthis appendix, we intend to elaborate the methodology of finding this distribution. For simplicity,we will omit the partonic indices for our further calculation. To understand the underlying logicsbehind findingΦI , let us consider an example at one loop level. The generalisation to higher loop isstraightforward. The whole discussion of this appendix is closely related to the Appendix E wherewe discussed the soft-collinear distribution for inclusive production cross section for a colorlessparticle.

As discussed in the Sec. 3.3, the SV cross section in z-space can be computed in d = 4 + ε

dimensions using

∆I,SVY (z1, z2, q2, µ2

R, µ2F) = C exp

(Ψ I

Y

(z1, z2, q2, µ2

R, µ2F , ε

) )∣∣∣∣ε=0

(F.0.1)

where, Ψ IY

(z1, z2, q2, µ2

R, µ2F , ε

)is a finite distribution and C is the double Mellin convolution de-

fined through Eq. (??). The Ψ I is given by, Eq. (3.3.16)

Ψ IY,i j

(z1, z2, q2, µ2

R, µ2F , ε

)=

(ln

[ZI

i j(as, µ2R, µ

2, ε)]2

+ ln∣∣∣∣F I

i j(as,Q2, µ2, ε)∣∣∣∣2) δ(1 − z1)δ(1 − z2)

+ 2ΦIY,i j(as, q2, µ2, z1, z2, ε) − C lnΓI

i j(as, µ2, µ2

F , z1, ε)δ(1 − z2)

− C lnΓIi j(as, µ

2, µ2F , z2, ε)δ(1 − z1) . (F.0.2)

For all the details about the notations, see Sec. 3.3. Considering only the poles at O(as) withµR = µF we obtain,

ln(ZI,(1)

)2= as(µ2

F)4γI

1

ε,

ln |F I,(1)|2 = as(µ2F)

q2

µ2F

ε2−4AI

1

ε2 +1ε

(2 f I

1 + 4BI1 − 4γI

1

) ,C lnΓI,(1)(z1) = as(µ2

F)2BI

1

εδ(1 − z1) +

2AI1

εD0

,C lnΓI,(1)(z2) = as(µ2

F)2BI

1

εδ(1 − z2) +

2AI1

εD0

(F.0.3)

149

150 Soft-Collinear Distribution for Rapidity

where, the components are defined through the expansion of these quantities in powers of as(µ2F)

Ψ IY =

∞∑k=1

aks

(µ2

F

)Ψ I,(k)

Y ,

ln(ZI)2 =

∞∑k=1

aks

(µ2

F

)ZI,(k) ,

ln |F I |2 =

∞∑k=1

aks

(µ2

F

) q2

µ2F

k ε2ln |F I,(k)|2 ,

ΦIY =

∞∑k=1

aks(µ

2F)ΦI

Y,k ,

lnΓI =

∞∑k=1

aks(µ

2F) lnΓI,(k) (F.0.4)

and

Di ≡

[lni(1 − z1)

1 − z1

]+

,

Di ≡

[lni(1 − z2)

1 − z2

]+

. (F.0.5)

Collecting the coefficients of as(µ2F), we get

Ψ I,(1)|Y,poles =

−4AI

1

ε2 +2 f I

1

ε

δ(1 − z1)δ(1 − z2) −2AI

1

ε

{δ(1 − z1)D0 + δ(1 − z2)D0

}+ 2ΦI

Y,1 (F.0.6)

where, we have suppressed the ln(q2/µ2F) terms. To cancel the remaining divergences appearing

in the above Eq. (F.0.6) for obtaining a finite rapidity distribution, we must demand that ΦIY,1 has

exactly the same poles with opposite sign:

2ΦIY,1|poles = −

−4AI

1

ε2 +2 f I

1

ε

δ(1 − z1)δ(1 − z2) −2AI

1

ε

{δ(1 − z1)D0 + δ(1 − z2)D0

} (F.0.7)

In addition, ΦIY also should be RG invariant with respect to µR:

µ2R

ddµ2

R

ΦIY = 0 . (F.0.8)

We make an ansatz, the above two demands, Eq. (E.0.7) and (E.0.8) can be accomplished if ΦIY

satisfies the KG-type integro-differential equation which we call KGY :

q2 ddq2Φ

IY

(as, q2, µ2, z1, z2, ε

)=

12

K IY

as,µ2

R

µ2 , z1, z2, ε

+ GIY

as,q2

µ2R

,µ2

R

µ2 , z1, z2, ε

. (F.0.9)

KIY contains all the poles whereas G

IY consists of only the finite terms in ε. RG invariance (F.0.8)

of ΦIY dictates

µ2R

ddµ2

R

KIY = −µ2

Rd

dµ2R

GIY ≡ XI

Y (F.0.10)

151

where, we introduce a quantity XIY . Following the methodology of solving the KG equation dis-

cussed in the Appendix D, we can write the solution of ΦIY as

ΦIY (as, q2, µ2, z1, z2, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

Y,k(z1, z2, ε) (F.0.11)

with

ΦIY,k(z1, z2, ε) = LI

k

(AI

i → XIY,i,G

Ii → G

IY,i(z, ε)

). (F.0.12)

where we define the components through the expansions

XIY =

∞∑k=1

aks(µ

2F)XI

Y,k ,

GIY (z1, z2, ε) =

∞∑k=1

aks(µ

2F)G

IY,k(z1, z2, ε). (F.0.13)

This solution directly follows from the Eq. (D.0.10). Hence we get

2ΦIY,1(z, ε) =

1ε2

{− 4XI

Y,1

}+

2ε

{G

IY,1(z1, z2, ε)

}. (F.0.14)

By expressing the components of ΦIY in powers of as(µ2

F), we obtain

ΦI(as, q2, µ2, z1, z2, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

Y,k(z1, z2, ε)

=

∞∑k=1

aks(µ

2F)

q2

µ2F

k ε2Zk

asΦI

Y,k(z1, z2, ε)

≡

∞∑k=1

aks(µ

2F)

q2

µ2F

k ε2ΦI

Y,k(z, ε) (F.0.15)

and at O(as(µ2F)), ΦI

Y,1(z, ε) = ΦIY,1(z, ε) upon suppressing the terms like log(q2/µ2

F). Hence, bycomparing the Eq. (F.0.7) and (F.0.14), we conclude

XIY,1 = −AI

1δ(1 − z1)δ(1 − z2) ,

GIY,1(z1, z2, ε) = − f I

1δ(1 − z1)δ(1 − z2) + AI1

{δ(1 − z1)D0 + δ(1 − z2)D0

}+

∞∑k=1

εkgI,kY,1(z) . (F.0.16)

The coefficients of εk, gI,kY,1(z) can only be determined through explicit computations. These do not

contribute to the infrared poles associated withΦIY . This uniquely fixes the unknown soft-collinear

distribution ΦIY at one loop order. This prescription can easily be generalised to higher orders in

as. In our calculation of the SV correction to rapidity distribution, instead of solving in this way,we follow a bit different methodology which is presented below.

Keeping the demands (F.0.7) and (F.0.8) in mind, we propose the solution of the KGY equation as(See Eq. (F.0.11)) (which is just the extension of the Eq. (E.0.17) from one variable z to a case oftwo variables z1 and z2)

ΦIY,k(z1, z2, ε) ≡

{(kε)2 1

4(1 − z1)(1 − z2)[(1 − z1)(1 − z2)]k ε2

}ΦI

Y,k(ε)

152 Soft-Collinear Distribution for Rapidity

=

{kε2

1(1 − z1)

[(1 − z1)2

]k ε4}{

kε2

1(1 − z2)

[(1 − z2)2

]k ε4}ΦI

Y,k(ε)

=

{δ(1 − z1) +

∞∑j=0

(kε/2) j+1

j!D j

}{δ(1 − z2) +

∞∑l=0

(kε/2)l+1

l!Dl

}ΦI

Y,k(ε) . (F.0.17)

The RG invariance of ΦIY , Eq. (F.0.8), implies

µ2R

ddµ2

R

KIY = −µ2

Rd

dµ2R

GIY ≡ X′IY (F.0.18)

where, we introduce a quantity X′IY , analogous to XIY . Hence, the solution can be obtained as

ΦIY,k(ε) = LI

k

(AI

i → X′IY,i,GIY,i → G

IY,i(ε)

). (F.0.19)

Hence, according to the Eq. (D.0.11), for k = 1 we get

2ΦIY,1(z1, z2, ε) = 2ΦI

Y,1(z1, z2, ε)

=

{δ(1 − z1) +

∞∑j=0

(kε/2) j+1

j!D j

}{δ(1 − z2) +

∞∑l=0

(kε/2)l+1

l!Dl

}{

1ε2

(−4X′IY,1

)+

2εG

IY,1 (ε))

}(F.0.20)

where, X′IY and GIY are expanded similar to Eq. (F.0.13). Comparison between the two solutions

depicted in Eq. (F.0.7) and (F.0.20), we can write

X′IY,1 = −AI1

GIY,1 (ε) = − f I

1 +

∞∑k=1

εkGI,kY,1 . (F.0.21)

Explicit computation is required to determine the coefficients of εk, GI,kY,1. This solution is used

in Eq. (3.3.45) in the context of SV correction to differential rapidity distribution of Higgs bosonproduction or leptonic pair in DY production. The method is generalised to higher orders in as toobtain the results of the soft-collinear distribution. Hence, the all order solution of ΦI

Y is

ΦI(as, q2, µ2, z1, z2, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

Y,k(z1, z2, ε) (F.0.22)

with

ΦIY,k(z1, z2, ε) =

{(kε)2 1

4(1 − z1)(1 − z2)[(1 − z1)(1 − z2)]k ε2

}ΦI

Y,k(ε) ,

ΦIY,k(ε) = LI

k

(AI

i → −AIi ,G

IY,i → G

IY,i(ε)

). (F.0.23)

Up to three loop, GIY,i(ε) are found to be

GIY,i(ε) = − f I

i + CIY,i +

∞∑k=1

εkGI,kY,i (F.0.24)

153

where

CIY,1 = 0 ,

CIY,2 = −2β0G

I,1Y,1 ,

CIY,3 = −2β1G

I,1Y,1 − 2β0

(G

I,1Y,2 + 2β0G

I,2Y,1

). (F.0.25)

These are employed in the computation of rapidity distributions in Chapter 3. In the next subsec-tion, we present the results of the soft-collinear distribution up to three loops.

F.0.1 Results

We define the renormalised components of the ΦIY,i i,k

through

ΦIY,i i(as, q2, µ2, z1, z2, ε) =

∞∑k=1

aksS

kε

(q2

µ2

)k ε2ΦI

Y,i i,k(z1, z2, ε)

=

∞∑k=1

aks

(µ2

F

)ΦI

Y,i i,k

(z1, z2, ε, q2, µ2

F

)(F.0.26)

where, we make the choice of the renormalisation scale µR = µF . The µR dependence can beeasily restored by using the evolution equation of strong coupling constant, Eq. (2.4.6). Below,we present the ΦI

Y,i i,kfor I = H and i i = gg up to three loops and the corresponding components

for I = DY and i i = qq can be obtained using maximally non-Abelian property fulfilled by thisdistribution:

ΦHY,gg,k =

CA

CFΦDY

Y,q q,k . (F.0.27)

The results are given by

ΦHY,gg,1 = δ(1 − z1)δ(1 − z2)

[1ε2 CA

{8}

+1ε

log q2

µ2F

CA

{4}

+ CA

{− ζ2

}+ log2

q2

µ2F

CA

{1}]

+D0δ(1 − z2)[1ε

CA

{4}

+ log q2

µ2F

CA

{2}]

+D0D0

[CA

{2}]

+D1δ(1 − z2)[CA

{2}]

+D0δ(1 − z1)[1ε

CA

{4}

+ log q2

µ2F

CA

{2}]

+D1δ(1 − z1)[CA

{2}],

ΦHY,gg,2 = δ(1 − z1)δ(1 − z2)

[1ε3 C2

A

{44

}+

1ε3 n f CA

{− 8

}+

1ε2 C2

A

{1349− 4ζ2

}+

1ε2 n f CA

{−

209

}+

1ε2 log

q2

µ2F

C2A

{443

}+

1ε2 log

q2

µ2F

n f CA

{−

83

}+

1ε

C2A

{−

40427

+ 14ζ3 +113ζ2

}+

1ε

n f CA

{5627−

23ζ2

}+

1ε

log q2

µ2F

C2A

{1349− 4ζ2

}+

1ε

log q2

µ2F

n f CA

{−

209

}

154 Soft-Collinear Distribution for Rapidity

+ C2A

{121481−

559ζ3 −

676ζ2 − 2ζ2

2

}+ n f CA

{−

16481

+109ζ3 +

53ζ2

}+ log

q2

µ2F

C2A

{−

40427

+ 14ζ3 +223ζ2

}+ log

q2

µ2F

n f CA

{5627−

43ζ2

}+ log2

q2

µ2F

C2A

{+

679− 2ζ2

}+ log2

q2

µ2F

n f CA

{−

109

}+ log3

q2

µ2F

C2A

{−

119

}+ log3

q2

µ2F

n f CA

{29

}]+ δ(1 − z2)D0

[1ε2 C2

A

{443

}+

1ε2 n f CA

{−

83

}+

1ε

C2A

{1349− 4ζ2

}+

1ε

n f CA

{−

209

}+ C2

A

{−

40427

+ 14ζ3 +223ζ2

}+ n f CA

{5627−

43ζ2

}+ log

q2

µ2F

C2A

{134

9− 4ζ2

}+ log

q2

µ2F

n f CA

{−

209

}+ log2

q2

µ2F

C2A

{−

113

}+ log2

q2

µ2F

n f CA

{23

}]+D0D0

[C2

A

{1349− 4ζ2

}+ n f CA

{−

209

}+ log

q2

µ2F

C2A

{−

223

}+ log

q2

µ2F

n f CA

{43

}]+D0D1

[C2

A

{−

223

}+ n f CA

{43

}]+D1δ(1 − z2)

[C2

A

{1349− 4ζ2

}+ n f CA

{−

209

}+ log

q2

µ2F

C2A

{−

223

}+ log

q2

µ2F

n f CA

{43

}]+D1D0

[C2

A

{−

223

}+ n f CA

{+

43

}]+D2δ(1 − z2)

[C2

A

{−

113

}+ n f CA

{23

}]+D0δ(1 − z1)

[1ε2 C2

A

{443

}+

1ε2 n f CA

{−

83

}+

1ε

C2A

{1349− 4ζ2

}+

1ε

n f CA

{−

209

}+ C2

A

{−

40427

+ 14ζ3 +223ζ2

}+ n f CA

{5627−

43ζ2

}+ log

q2

µ2F

C2A

{134

9− 4ζ2

}+ log

q2

µ2F

n f CA

{−

209

}+ log2

q2

µ2F

C2A

{−

113

}+ log2

q2

µ2F

n f CA

{23

}]+D1δ(1 − z1)

[C2

A

{1349− 4ζ2

}+ n f CA

{−

209

}+ log

q2

µ2F

C2A

{−

223

}+ log

q2

µ2F

n f CA

{43

}]+D2δ(1 − z1)

[C2

A

{−

113

}+ n f CA

{23

}],

ΦHY,gg,3 = δ(1 − z1)δ(1 − z2)

[1ε4 C3

A

{21296

81

}+

1ε4 n f C2

A

{−

774481

}+

1ε4 n2

f CA

{70481

}+

1ε3 C3

A

{49064

243−

88027

ζ2

}+

1ε3 n f C2

A

{−

15520243

+16027

ζ2

}

155

+1ε3 n f CFCA

{−

1289

}+

1ε3 n2

f CA

{800243

}+

1ε3 log

q2

µ2F

C3A

{193627

}+

1ε3 log

q2

µ2F

n f C2A

{−

70427

}+

1ε3 log

q2

µ2F

n2f CA

{6427

}+

1ε2 C3

A

{−

8956243

+202427

ζ3 −69281

ζ2 +35245

ζ22

}+

1ε2 n f C2

A

{4024243

−56027

ζ3

−20881

ζ2

}+

1ε2 n f CFCA

{−

22027

+649ζ3

}+

1ε2 n2

f CA

{−

16081

+1627ζ2

}+

1ε2 log

q2

µ2F

C3A

{8344

81−

1769ζ2

}+

1ε2 log

q2

µ2F

n f C2A

{−

267281

+329ζ2

}+

1ε2 log

q2

µ2F

n f CFCA

{−

163

}+

1ε2 log

q2

µ2F

n2f CA

{16081

}+

1ε

C3A

{−

1367812187

− 64ζ5 +1316

9ζ3 +

12650243

ζ2 −1769ζ2ζ3 −

35215

ζ22

}+

1ε

n f C2A

{118422187

−72881

ζ3 −2828243

ζ2 +325ζ2

2

}+

1ε

n f CFCA

{171181−

30427

ζ3

−43ζ2 −

3215ζ2

2

}+

1ε

n2f CA

{20802187

−11281

ζ3 +4081ζ2

}+

1ε

log q2

µ2F

C3A

{4909

+889ζ3 −

107227

ζ2 +17615

ζ22

}+

1ε

log q2

µ2F

n f C2A

{−

83681−

1129ζ3 +

16027

ζ2

}+

1ε

log q2

µ2F

n f CFCA

{−

1109

+323ζ3

}+

1ε

log q2

µ2F

n2f CA

{−

1681

}+ C3

A

{521194926244

−484

9ζ5 −

64886243

ζ3 +5369ζ2

3 −2994251458

ζ2 +2863ζ2ζ3 +

691135

ζ22

+17392945

ζ32

}+ n f C2

A

{−

41276513122

−83ζ5 +

292081

ζ3 +38237

729ζ2 − 4ζ2ζ3

−1064135

ζ22

}+ n f CFCA

{−

42727972

+112

9ζ5 +

163681

ζ3 +27518

ζ2 −403ζ2ζ3

+15245

ζ22

}+ n2

f CA

{−

1286561

−40243

ζ3 −6827ζ2 +

124135

ζ22

}+ log

q2

µ2F

C3A

{−

2970291458

− 96ζ5 +713227

ζ3 +13876

81ζ2 −

883ζ2ζ3 −

30815

ζ22

}+ log

q2

µ2F

n f C2A

{31313

729−

2689ζ3 −

388081

ζ2 +10415

ζ22

}+ log

q2

µ2F

n f CFCA

{1711

54−

1529ζ3 − 4ζ2 −

165ζ2

2

}+ log

q2

µ2F

n2f CA

{−

928729

−1627ζ3 +

8027ζ2

}+ log2

q2

µ2F

C3A

{15503162

− 44ζ3 −1703ζ2 +

445ζ2

2

}+ log2

q2

µ2F

n f C2A

{−

205181

+1289ζ2

}+ log2

q2

µ2F

n f CFCA

{−

556

+ 8ζ3

}

156 Soft-Collinear Distribution for Rapidity

+ log2 q2

µ2F

n2f CA

{10081−

89ζ2

}+ log3

q2

µ2F

C3A

{−

178081

+449ζ2

}+ log3

q2

µ2F

n f C2A

{57881−

89ζ2

}+ log3

q2

µ2F

n f CFCA

{23

}

+ log3 q2

µ2F

n2f CA

{−

4081

}+ log

q2

µ2F

4

C3A

{12154

}+ log

q2

µ2F

4

n f C2A

{−

2227

}

+ log q2

µ2F

4

n2f CA

{227

}]+D0δ(1 − z2)

[1ε3 C3

A

{1936

27

}+

1ε3 n f C2

A

{−

70427

}+

1ε3 n2

f CA

{6427

}+

1ε2 C3

A

{8344

81−

1769ζ2

}+

1ε2 n f C2

A

{−

267281

+329ζ2

}+

1ε2 n f CFCA

{−

163

}+

1ε2 n2

f CA

{16081

}+

1ε

C3A

{4909

+889ζ3 −

107227

ζ2 +17615

ζ22

}+

1ε

n f C2A

{−

83681−

1129ζ3 +

16027

ζ2

}+

1ε

n f CFCA

{−

1109

+323ζ3

}+

1ε

n2f CA

{−

1681

}+ C3

A

{−

2970291458

− 96ζ5 +7132

27ζ3 +

1387681

ζ2 −883ζ2ζ3

−30815

ζ22

}+ n f C2

A

{31313729

−2689ζ3 −

388081

ζ2 +10415

ζ22

}+ n f CFCA

{1711

54

−1529ζ3 − 4ζ2 −

165ζ2

2

}+ n2

f CA

{−

928729−

1627ζ3 +

8027ζ2

}+ log

q2

µ2F

C3A

{15503

81− 88ζ3 −

3403ζ2 +

885ζ2

2

}+ log

q2

µ2F

n f C2A

{−

410281

+2569ζ2

}+ log

q2

µ2F

n f CFCA

{−

553

+ 16ζ3

}+ log

q2

µ2F

n2f CA

{20081−

169ζ2

}+ log2

q2

µ2F

C3A

{−

178027

+443ζ2

}+ log2

q2

µ2F

n f C2A

{57827−

83ζ2

}+ log2

q2

µ2F

n f CFCA

{2}

+ log2 q2

µ2F

n2f CA

{−

4027

}+ log3

q2

µ2F

C3A

{24227

}+ log3

q2

µ2F

n f C2A

{−

8827

}+ log3

q2

µ2F

n2f CA

{827

}]+D0D0

[C3

A

{+

1550381

− 88ζ3 −3403ζ2 +

885ζ2

2

}+ n f C2

A

{−

410281

+2569ζ2

}+ n f CFCA

{−

553

+ 16ζ3

}+ n2

f CA

{20081−

169ζ2

}+ log

q2

µ2F

C3A

{−

356027

+883ζ2

}+ log

q2

µ2F

n f C2A

{115627−

163ζ2

}+ log

q2

µ2F

n f CFCA

{4}

+ log q2

µ2F

n2f CA

{−

8027

}+ log2

q2

µ2F

C3A

{2429

}+ log2

q2

µ2F

n f C2A

{−

889

}

157

+ log2 q2

µ2F

n2f CA

{89

}]+D0D1

[C3

A

{−

356027

+883ζ2

}+ n f C2

A

{115627−

163ζ2

}+ n f CFCA

{4}

+ n2f CA

{−

8027

}+ log

q2

µ2F

C3A

{4849

}+ log

q2

µ2F

n f C2A

{−

1769

}+ log

q2

µ2F

n2f CA

{169

}]+D0D2

[C3

A

{2429

}+ n f C2

A

{−

889

}+ n2

f CA

{89

}]+D1δ(1 − z2)

[C3

A

{15503

81− 88ζ3 −

3403ζ2 +

885ζ2

2

}+ n f C2

A

{−

410281

+2569ζ2

}+ n f CFCA

{−

553

+ 16ζ3

}+ n2

f CA

{20081−

169ζ2

}+ log

q2

µ2F

C3A

{−

356027

+883ζ2

}+ log

q2

µ2F

n f C2A

{115627−

163ζ2

}+ log

q2

µ2F

n f CFCA

{4}

+ log q2

µ2F

n2f CA

{−

8027

}+ log2

q2

µ2F

C3A

{2429

}+ log2

q2

µ2F

n f C2A

{−

889

}+ log2

q2

µ2F

n2f CA

{+

89

}]+D1D0

[C3

A

{−

356027

+883ζ2

}+ n f C2

A

{115627

−163ζ2

}+ n f CFCA

{4}

+ n2f CA

{−

8027

}+ log

q2

µ2F

C3A

{4849

}+ log

q2

µ2F

n f C2A

{−

1769

}+ log

q2

µ2F

n2f CA

{169

}]+D1D1

[C3

A

{484

9

}+ n f C2

A

{−

1769

}+ n2

f CA

{169

}]+D2δ(1 − z2)

[C3

A

{−

178027

+443ζ2

}+ n f C2

A

{57827−

83ζ2

}+ n f CFCA

{2}

+ n2f CA

{−

4027

}+ log

q2

µ2F

C3A

{242

9

}+ log

q2

µ2F

n f C2A

{−

889

}+ log

q2

µ2F

n2f CA

{89

}]+D2D0

[C3

A

{2429

}+ n f C2

A

{−

889

}+ n2

f CA

{89

}]+D3δ(1 − z2)

[C3

A

{24227

}+ n f C2

A

{−

8827

}+ n2

f CA

{827

}]+D0δ(1 − z1)

[1ε3 C3

A

{1936

27

}+

1ε3 n f C2

A

{−

70427

}+

1ε3 n2

f CA

{6427

}+

1ε2 C3

A

{8344

81−

1769ζ2

}+

1ε2 n f C2

A

{−

267281

+329ζ2

}+

1ε2 n f CFCA

{−

163

}+

1ε2 n2

f CA

{16081

}+

1ε

C3A

{4909

+889ζ3 −

107227

ζ2 +17615

ζ22

}+

1ε

n f C2A

{−

83681−

1129ζ3 +

16027

ζ2

}+

1ε

n f CFCA

{−

1109

+323ζ3

}+

1ε

n2f CA

{−

1681

}+ C3

A

{−

2970291458

− 96ζ5 +7132

27ζ3 +

1387681

ζ2 −883ζ2ζ3

−30815

ζ22

}+ n f C2

A

{31313729

−2689ζ3 −

388081

ζ2 +10415

ζ22

}+ n f CFCA

{1711

54

158 Soft-Collinear Distribution for Rapidity

−1529ζ3 − 4ζ2 −

165ζ2

2

}+ n2

f CA

{−

928729−

1627ζ3 +

8027ζ2

}+ log

q2

µ2F

C3A

{15503

81− 88ζ3 −

3403ζ2 +

885ζ2

2

}+ log

q2

µ2F

n f C2A

{−

410281

+2569ζ2

}+ log

q2

µ2F

n f CFCA

{−

553

+ 16ζ3

}+ log

q2

µ2F

n2f CA

{20081−

169ζ2

}+ log2

q2

µ2F

C3A

{−

178027

+443ζ2

}+ log2

q2

µ2F

n f C2A

{57827−

83ζ2

}+ log2

q2

µ2F

n f CFCA

{2}

+ log2 q2

µ2F

n2f CA

{−

4027

}+ log3

q2

µ2F

C3A

{24227

}+ log3

q2

µ2F

n f C2A

{−

8827

}+ log3

q2

µ2F

n2f CA

{827

}]+D1δ(1 − z1)

[C3

A

{15503

81− 88ζ3 −

3403ζ2 +

885ζ2

2

}+ n f C2

A

{−

410281

+2569ζ2

}+ n f CFCA

{−

553

+ 16ζ3

}+ n2

f CA

{20081−

169ζ2

}+ log

q2

µ2F

C3A

{−

356027

+883ζ2

}+ log

q2

µ2F

n f C2A

{115627−

163ζ2

}+ log

q2

µ2F

n f CFCA

{4}

+ log q2

µ2F

n2f CA

{−

8027

}+ log2

q2

µ2F

C3A

{2429

}+ log2

q2

µ2F

n f C2A

{−

889

}+ log2

q2

µ2F

n2f CA

{89

}]+D2δ(1 − z1)

[C3

A

{−

178027

+443ζ2

}+ n f C2

A

{57827

−83ζ2

}+ n f CFCA

{2}

+ n2f CA

{−

4027

}+ log

q2

µ2F

C3A

{2429

}+ log

q2

µ2F

n f C2A

{−

889

}+ log

q2

µ2F

n2f CA

{89

}]+D3δ(1 − z1)

[C3

A

{24227

}+ n f C2

A

{−

8827

}+ n2

f CA

{8

27

}]. (F.0.28)

G Results of the UnrenormalisedThree Loop Form Factors for thePseudo-Scalar

In this appendix, we present the unrenormalised quark and gluon form factors for the pseudo-scalar production up to three loops for the operators [OG]B and [OJ]B. Specifically, we presentF

G,(n)β and F J,(n)

β for β = q, g up to n = 3 which are defined in Sec. 5.3. One and two loop resultscompletely agree with the existing literature [84]. It should be noted that the form factors at n = 2for F G,(n)

q and F J,(n)g correspond to the contributions arising from three loop diagrams since these

processes start at one loop order.

FG,(1)g = CA

{−

8ε2 + 4 + ζ2 + ε

(− 6 −

73ζ3

)+ ε2

(7 −

ζ2

2+

4780ζ2

2

)+ ε3

(−

152

+34ζ2

+76ζ3 +

724ζ2ζ3 −

3120ζ5

)}, (G.0.1)

FG,(2)g = 2CAn f TF

{−

83ε3 +

209ε2 +

(10627

+ 2ζ2

)1ε−

159181−

53ζ2 −

749ζ3 + ε

(24107486

−2318ζ2 +

5120ζ2

2 +38327

ζ3

)+ ε2

(−

1461471458

+799108

ζ2 −32972

ζ22 −

143681

ζ3 +256ζ2ζ3

−27130

ζ5

)}+ C2

A

{32ε4 +

443ε3 +

(−

4229− 4ζ2

)1ε2 +

(89027− 11ζ2 +

503ζ3

)1ε

+383581

+1156ζ2 −

215ζ2

2 +119ζ3 + ε

(−

213817972

−10318

ζ2 +77120

ζ22 +

110354

ζ3

−236ζ2ζ3 −

7110ζ5

)+ ε2

(610274511664

−99127

ζ2 −2183240

ζ22 +

2313280

ζ32 −

883681

ζ3 −5512ζ2ζ3

+90136

ζ23 +

34160

ζ5

)}+ 2CFn f TF

{12ε−

1253

+ 8ζ3 + ε

(342136−

143ζ2 −

83ζ2

2

−643ζ3

)+ ε2

(−

78029432

+29318

ζ2 +649ζ2

2 +97318

ζ3 −103ζ2ζ3 + 8ζ5

)}, (G.0.2)

159

160 Results of the Unrenormalised Three Loop Form Factors for the Pseudo-Scalar

FG,(3)g = 4CFn2

f T2F

{16ε2 +

(−

7969

+643ζ3

)1ε

+8387

27−

383ζ2 −

11215

ζ22 −

8489ζ3

}+ 2C2

Fn f TF

{6ε−

3536

+ 176ζ3 − 160ζ5

}+ 2C2

An f TF

{643ε5 −

3281ε4 +

(−

18752243

−37627

ζ2

)1ε3 +

(36416

243−

170081

ζ2 +207227

ζ3

)1ε2 +

(626422187

+22088243

ζ2 −245390

ζ22

−3988

81ζ3

)1ε−

1465580913122

−60548

729ζ2 +

91760

ζ22 −

77227

ζ3 −4399ζ2ζ3 +

323845

ζ5

}+ 4CAn2

f T2F

{−

12881ε4 +

640243ε3 +

(12827

+8027ζ2

)1ε2 +

(−

930882187

−40081

ζ2

−1328

81ζ3

)1ε

+1066349

6561−

5627ζ2 +

797135

ζ22 +

13768243

ζ3

}+ 2CACFn f TF

{−

8809ε3

+

(6844

27−

6409ζ3

)1ε2 +

(−

1621981

+1583ζ2 +

35215

ζ22 +

174427

ζ3

)1ε−

753917972

−593

6ζ2 −

965ζ2

2 +493481

ζ3 + 48ζ2ζ3 +329ζ5

}+ C3

A

{−

2563ε6 −

3523ε5 +

1614481ε4

+

(22864243

+206827

ζ2 −1763ζ3

)1ε3 +

(−

172844243

−163081

ζ2 +49445

ζ22 −

83627

ζ3

)1ε2

+

(2327399

2187−

71438243

ζ2 +3751180

ζ22 −

8429ζ3 +

1709ζ2ζ3 +

175615

ζ5

)1ε

+16531853

26244

+918931

1458ζ2 +

272511080

ζ22 −

22523270

ζ32 −

51580243

ζ3 +7718ζ2ζ3 −

17669

ζ23 +

2091145

ζ5

},

(G.0.3)

FJ,(1)

g = CA

{−

8ε2 + 4 + ζ2 + ε

(−

152

+ ζ2 −163ζ3

)+ ε2

(28724− 2ζ2 +

12780

ζ22

)+ ε3

(−

5239288

+15148

ζ2 +19

120ζ2

2 +ζ3

12+

76ζ2ζ3 −

9120ζ5

)}+ CF

{4 + ε

(−

212

+ 6ζ3

)+ ε2

(155

8−

52ζ2 −

95ζ2

2 −92ζ3

)+ ε3

(−

102532

+8316ζ2 +

2720ζ2

2 +203ζ3

−34ζ2ζ3 +

212ζ5

)}, (G.0.4)

FJ,(2)

g = 2CAn f TF

{−

83ε3 +

209ε2 +

(10627

+ 2ζ2

)1ε−

175381−ζ2

3−

1109ζ3 + ε

(14902243

−10318

ζ2 +24160

ζ22 +

59927

ζ3

)+ ε2

(−

4119312916

+2045108

ζ2 −2353360

ζ22 −

312881

ζ3

+436ζ2ζ3 −

16710

ζ5

)}+ CACF

{−

32ε2 +

(208

3− 48ζ3

)1ε−

4519

+ 24ζ2 +725ζ2

2

− 8ζ3 + ε

(−

16385108

−523ζ2 +

125ζ2

2 + 32ζ3 + 10ζ2ζ3 − 14ζ5

)+ ε2

(1073477

1296

161

−8159ζ2 +

1920ζ2

2 +1770ζ3

2 −191536

ζ3 + 9ζ2ζ3 − 34ζ23 −

22796

ζ5

)}+ 2CFn f TF

{263ε

−70918

+ 16ζ3 + ε

(26149216

−656ζ2 −

7615ζ2

2 − 44ζ3

)+ ε2

(−

8280612592

+322972

ζ2

+21215

ζ22 +

172918

ζ3 − 4ζ2ζ3 +1663ζ5

)}+ C2

A

{32ε4 +

443ε3 +

(−

4229− 4ζ2

)1ε2

+

(1214

27− 19ζ2 +

1223ζ3

)1ε

+1513

81+

1436ζ2 −

615ζ2

2 +2099ζ3 + ε

(−

202747972

+5936ζ2 −

34924

ζ22 −

2393108

ζ3 −536ζ2ζ3 +

36910

ζ5

)+ ε2

(768192111664

−35255432

ζ2 +1711180

ζ22

−7591840

ζ32 −

56831296

ζ3 −40712

ζ2ζ3 +77536

ζ23 +

401330

ζ5

)}+ C2

F

{− 6 + ε

(25912

+ 41ζ3

− 60ζ5

)+ ε2

(−

7697144

+ζ2

3−

18415

ζ22 +

1207ζ3

2 − 163ζ3 + 4ζ2ζ3 + 30ζ23 +

4703ζ5

)},

(G.0.5)

FG,(1)q = 2n f TF

{43ε−

199

+ ε

(355108−ζ2

6

)+ ε2

(−

65231296

+1972ζ2 +

2518ζ3

)+ ε3

(11867515552

−355864

ζ2 −191480

ζ22 −

475216

ζ3

)}+ CF

{−

8ε2 +

6ε−

112

+ ζ2 + ε

(258−

34ζ2 −

73ζ3

)+ ε2

(−

1132−

2116ζ2 +

4780ζ2

2 +74ζ3

)+ ε3

(−

415128

+22364

ζ2 −141320

ζ22 −

15548

ζ3

+7

24ζ2ζ3 −

3120ζ5

)}+ CA

{−

223ε

+26918

+ ε

(−

5045216

+2312ζ2 + 3ζ3

)+ ε2

(908932592

−485144

ζ2 −45ζ2

2 −27536

ζ3

)+ ε3

(−

162034131104

+88611728

ζ2 +751320

ζ22 +

4961432

ζ3 +ζ2ζ3

8

+152ζ5

)}, (G.0.6)

FG,(2)q = 4n2

f T2F

{169ε2 −

15227ε

+1249−

49ζ2 + ε

(−

7426243

+3827ζ2 +

13627

ζ3

)+ ε2

(47108729

−319ζ2 −

4330ζ2

2 −1292

81ζ3

)}+ C2

A

{4849ε2 −

612227ε

+1865

3−

3199ζ2 − 66ζ3

+ ε

(−

702941486

+14969108

ζ2 +29920

ζ22 +

31441108

ζ3 + 5ζ2ζ3 − 30ζ5

)+ ε2

(18199507

5832

−586116

ζ2 −63233720

ζ22 −

691140

ζ32 −

9959151296

ζ3 +523ζ2ζ3 −

392ζ2

3 −134312

ζ5

)}+ 2CFn f TF

{−

403ε3 +

2809ε2 +

(−

141727

+ 2ζ2

)1ε

+22021324

−143ζ2 −

829ζ3

+ ε

(−

2387173888

−7312ζ2 +

2512ζ2

2 +39427

ζ3

)+ ε2

(−

29007546656

+6181144

ζ2 −499180

ζ22

162 Results of the Unrenormalised Three Loop Form Factors for the Pseudo-Scalar

−9751324

ζ3 +136ζ2ζ3 −

296ζ5

)}+ C2

F

{32ε4 −

48ε3 +

(62 − 8ζ2

)1ε2 +

(−

1132

+1283ζ3

)1ε

+58124

+272ζ2 − 13ζ2

2 − 58ζ3 + ε

(12275288

−33124

ζ2 +49330

ζ22 +

5876ζ3

−563ζ2ζ3 +

925ζ5

)+ ε2

(−

4567793456

−201196

ζ2 −127980

ζ22 +

22320

ζ32 −

1336372

ζ3

−52ζ2ζ3 +

6529ζ2

3 −19330

ζ5

)}+ 2CAn f TF

{−

1769ε2 +

197227ε

−1708

9+

809ζ2 + 4ζ3

+ ε

(104858

243−

85327

ζ2 −23ζ2

2 −162227

ζ3

)+ ε2

(−

53695015832

+144718

ζ2 +81745

ζ22

+31499162

ζ3 +73ζ2ζ3 + 19ζ5

)}+ CACF

{2203ε3 +

(−

18049

+ 4ζ2

)1ε2 +

(20777

54

− 19ζ2 − 50ζ3

)1ε−

397181648

+1613ζ2 +

765ζ2

2 +1333

9ζ3 + ε

(6604541

7776−

6698ζ2

−5519120

ζ22 −

839827

ζ3 +896ζ2ζ3 −

512ζ5

)+ ε2

(−

9377482193312

+20035288

ζ2 +33377360

ζ22

+1793840

ζ32 +

390731648

ζ3 −44512

ζ2ζ3 −42512

ζ23 +

64112

ζ5

)}, (G.0.7)

FJ,(1)

q = CF

{−

8ε2 +

6ε− 2 + ζ2 + ε

(− 1 −

34ζ2 −

73ζ3

)+ ε2

(52

+ζ2

4+

4780ζ2

2 +74ζ3

)+ ε3

(−

134

+ζ2

8−

141320

ζ22 −

712ζ3 +

724ζ2ζ3 −

3120ζ5

)}, (G.0.8)

FJ,(2)

q = 2CFn f TF

{−

83ε3 +

569ε2 +

(−

4727−

23ζ2

)1ε−

4105324

+149ζ2 −

269ζ3 + ε

(1425373888

−695108

ζ2 +4160ζ2

2 +18227

ζ3

)+ ε2

(−

325651346656

+211671296

ζ2 −287180

ζ22 −

2555324

ζ3

−1318ζ2ζ3 −

12130

ζ5

)}+ C2

F

{32ε4 −

48ε3 +

(34 − 8ζ2

)1ε2 +

(−

52

+1283ζ3

)1ε−

3618

+92ζ2 − 13ζ2

2 − 58ζ3 + ε

(327532

+38ζ2 +

17110

ζ22 +

5036ζ3 −

563ζ2ζ3 +

925ζ5

)+ ε2

(−

20257128

−79332

ζ2 −209780

ζ22 +

22320

ζ32 −

403724

ζ3 +272ζ2ζ3 +

6529ζ2

3 −23110

ζ5

)}+ CACF

{443ε3 +

(−

3329

+ 4ζ2

)1ε2 +

(254554

+113ζ2 − 26ζ3

)1ε−

18037648

−479ζ2

+445ζ2

2 +467

9ζ3 + ε

(−

2219637776

−263216

ζ2 −1891120

ζ22 −

242927

ζ3 +896ζ2ζ3 −

512ζ5

)+ ε2

(11956259

93312+

389872592

ζ2 +9451360

ζ22 −

809280

ζ32 +

92701648

ζ3 −39736

ζ2ζ3 −56912

ζ23

+349160

ζ5

)}, (G.0.9)

163

FJ,(3)

q = 4CFn2f T

2F

{−

12881ε4 +

1504243ε3 +

(−

169−

169ζ2

)1ε2 +

(−

734322187

+18827

ζ2

−27281

ζ3

)1ε

+881372

6561− 26ζ2 −

83135

ζ22 +

3196243

ζ3

}+ C3

F

{−

2563ε6 +

192ε5 +

(− 208

+ 32ζ2

)1ε4 +

(88 + 24ζ2 −

8003ζ3

)1ε3 +

(254 − 98ζ2 +

4265ζ2

2 + 552ζ3

)1ε2

+

(−

50456

+ 83ζ2 −1461

10ζ2

2 −2630

3ζ3 +

4283ζ2ζ3 −

12885

ζ5

)1ε

+38119

24+

188512

ζ2

+8659

40ζ2

2 −9095252

ζ32 + 1153ζ3 − 35ζ2ζ3 −

18263

ζ23 −

5625ζ5

}+ 2C2

Fn f TF

{643ε5

−5929ε4 +

(148027

+83ζ2

)1ε3 +

(7772

81−

2669ζ2 +

5849ζ3

)1ε2 +

(−

116735243

+263327

ζ2

−33718

ζ22 −

511427

ζ3

)1ε

+3396143

2916−

32329162

ζ2 +8149216

ζ22 +

3977381

ζ3 −3439ζ2ζ3

+27845

ζ5

}+ C2

ACF

{−

387281ε4 +

(52168243

−70427

ζ2

)1ε3 +

(−

117596243

−221281

ζ2

−35245

ζ22 +

668827

ζ3

)1ε2 +

(1322900

2187+

39985243

ζ2 −160415

ζ22 −

2421227

ζ3 +1769ζ2ζ3

+272

3ζ5

)1ε

+1213171

13122−

198133729

ζ2 +146443

540ζ2

2 −6152189

ζ32 +

970249486

ζ3 −9269ζ2ζ3

−1136

9ζ2

3 +772

9ζ5

}+ 2CACFn f TF

{140881ε4 +

(−

18032243

+12827

ζ2

)1ε3 +

(24620243

+1264

81ζ2 −

102427

ζ3

)1ε2 +

(212078

2187−

16870243

ζ2 +885ζ2

2 +12872

81ζ3

)1ε−

58076476561

+299915

1458ζ2 −

5492135

ζ22 −

4294181

ζ3 +4229ζ2ζ3 −

283ζ5

}+ CAC2

F

{−

3523ε5 +

(3448

9

− 32ζ2

)1ε4 +

(−

1694827

+283ζ2 + 208ζ3

)1ε3 +

(44542

81+

11279

ζ2 −3325ζ2

2

− 840ζ3

)1ε2 +

(149299

486−

1275754

ζ2 +983936

ζ22 +

54673

ζ3 −4303ζ2ζ3 + 284ζ5

)1ε

−15477463

5832+

21455324

ζ2 −1002379

2160ζ2

2 −186191260

ζ32 −

5178118

ζ3 +9109ζ2ζ3 +

16163

ζ23

−3394

45ζ5

}. (G.0.10)

H Harmonic Polylogarithms

The logarithms, polylogarithms (Lin(x)) and Nielsen’s polylogarithm (Sn,p(x)) appear naturally inthe analytical expressions of radiative corrections in pQCD which are defined through

ln(x) =

∫ x

1

dtt,

Lin(x) ≡∞∑

k=1

xk

kn =

∫ x

0

dtt

Lin−1(t) , e.g. Li1(x) = −ln(1 − x) ,

S n,p(x) ≡(−1)n+p−1

(n − 1)!p!

∫ 1

0

dtt

[ln(t)]n−1[ln(1 − xt)]p ,

e.g. S n−1,1(x) = Lin(x) . (H.0.1)

However, for higher order radiative corrections (2-loops and beyond), these functions are notsufficient to evaluate all the loop integrals appearing in the Feynman graphs. This is overcome byintroducing a new set of functions which are called Harmonic Polylogarithms (HPLs). Theseare essentially a generalisation of Nielsen’s polylogarithms. In this appendix, we briefly describethe definition and properties of HPL [184] and 2dHPL. HPL is represented by H(~mw; y) with aw-dimensional vector ~mw of parameters and its argument y. w is called the weight of the HPL. Theelements of ~mw belong to {1, 0,−1} through which the following rational functions are represented

f (1; y) ≡1

1 − y, f (0; y) ≡

1y, f (−1; y) ≡

11 + y

. (H.0.2)

The weight 1 (w = 1) HPLs are defined by

H(1, y) ≡ − ln(1 − y), H(0, y) ≡ ln y, H(−1, y) ≡ ln(1 + y) . (H.0.3)

For w > 1, H(m, ~mw; y) is defined by

H(m, ~mw; y) ≡∫ y

0dx f (m, x) H(~mw; x), m ∈ 0,±1 . (H.0.4)

The 2dHPLs are defined in the same way as Eq. (H.0.4) with the new elements {2, 3} in ~mw repre-senting a new class of rational functions

f (2; y) ≡ f (1 − z; y) ≡1

1 − y − z, f (3; y) ≡ f (z; y) ≡

1y + z

(H.0.5)

and correspondingly with the weight 1 (w = 1) 2dHPLs

H(2, y) ≡ − ln(1 −

y1 − z

), H(3, y) ≡ ln

(y + zz

). (H.0.6)

165

166 Harmonic Polylogarithms

H.0.1 Properties

Shuffle algebra : A product of two HPL with weights w1 and w2 of the same argument y is acombination of HPLs with weight (w1 + w2) and argument y, such that all possible permutationsof the elements of ~mw1 and ~mw2 are considered preserving the relative orders of the elements of~mw1 and ~mw2 ,

H(~mw1 ; y)H(~mw2 ; y) =∑

~mw = ~mw1 ] ~mw2

H(~mw; y). (H.0.7)

Integration-by-parts identities : The ordering of the elements of ~mw in an HPL with weight w andargument y can be reversed using integration-by-parts and in the process, some products of twoHPLs are generated in the following way

H(~mw; y) ≡ H(m1,m2, ...,mw; y) = H(m1, y)H(m2, ...,mw; y)

− H(m2,m1, y)H(m3, ...,mw; y)

+ ... + (−1)w+1H(mw, ...,m2,m1; y) . (H.0.8)

ACKNOWLEDGMENTS

Finally, we are very close to the moment! The moment, every graduate student aspires for! Themoment, I have been craving for past several years! Yes, the ship was sailed almost six years backand here, I am about to write the closing chapter with an intention of revealing the marvelousexperiences of that long journey! The journey which has played an important role to enrich menot only academically but also non-academically. Before joining to Harish-Chandra ResearchInstitute (HRI), situated at the bank of Ganges and Yamuna in Allahabad and later to The Instituteof Mathematical Sciences (IMSc), Chennai, I could hardly imagine the experiences which I wouldgather in those upcoming days. And none of these would have been possible without the peoplewhom I have come across during my PhD and before. Let me begin by expressing my deepestgratitude to my supervisor Prof. V. Ravindran for his unwavering support, collegiality and ex-traordinary mentorship throughout my PhD life! His deep insight about the subject has helped meto enhance my understanding about the world of QCD. Not only that, to me, he truly symbolisesthe say, ‘A Teacher is a Friend, a Philosopher and a Guide’! His unconditional support from thehappiest to the darkest moment of my life is what makes all the differences. I would like to extendmy gratitude to all those people I had the opportunity to work with. To my friend and colleagueNarayan Rana for being a wonderful collaborator. Working with you has made many things pos-sible over a short span of time. To Prof. Prakash Mathews for his guidance, collaboration andsupport throughout my PhD life. To Prof. Thomas Gehrmann for his wonderful collaborationand several helps about the master integrals. To Dr. M.C. Kumar for fruitful collaborations,discussions and many helps for academic as well as non-academic purposes. To my colleaguesDr. Manoj K. Mandal, Dr. Maguni Mahakhud and Goutam Das for fruitful collaborations. ToProf. Roman N. Lee for his insightful help regarding the reduction of Feynman integrals througha wonderful package LiteRed. To Dr. Lorenzo Tancredi for sharing his deep understanding aboutthe computation of multiloop amplitude through a set of fantastic lectures and many helpful dis-cussions. To two new youths in our group, Pulak Banerjee and Prasanna K. Dhani for fantasticcollaborations and many valuable discussions. To Dr. Marco Bonvini and Luca Rottoli for fruitfulcollaborations and sharing their nice ideas about resummation with us. The format of this thesisis inspired by looking at your thesis, Marco, thank you. To Prof. Ashoke Sen for his extraordinarycourse, advanced QFT at HRI. To all the professors of HRI for some wonderful courses duringmy PhD course work. To all the professors and members IMSc. To Prof. N. D. Hari Dass forhis constant encouragement starting from my B.Sc and indeed, you are the one who initiated theinitial ignition of research into my mind at a very early stage of my life, my deepest gratitude toyou Professor!

Special thanks to my friend Jahanur. You were the very first inspiration for choosing Physics asmy career! Our outstanding discussions, academic as well as non-academic, starting from theBSc really have helped us to explore and understand many things in a different way. I wouldlike to express my deepest appreciation to Satya da for your continuous motivations, help andvaluable advice in choosing PhD supervisor. I am thankful to my fellow HRI mates for their won-derful company during my stay at HRI for almost 5 years, in particular, Gupta, Joshi, Shankha,Avinanda, Roji, Ushoshi, Utkarsh, Swapnomoy, Abhishek, Masud, Avijit, Shrobona, Dibya, Titas,Arijit, Nabarun, Atri and ‘hopeless’ Anirban da. My gratitude goes out to all of my friends atIMSc, specifically, Rusa, Pinaki, Shilpa, Biswas, Trisha, Upayan, Anish, Arghya and my multi-talented officemates Atanu, Arnab, Rupam. I convey my heartiest thanks to my wonderful friends

Annwesha, Chitrak, Atreyee, Sundaram, Madhurima, Kawsar, Gazi, Shahnawaj, Poulami, Saheli,Khadiza and Rudra. I would like to extend my special thanks to my beloved friend Farha for hertrust, love and constant encouragement. I thank Kamrin, Moonfared and Abid for their trust andsupports.

Finally, I am thankful to my beloved parents, sister, brother-in-law and whole family for theirunconditional love, supports and sacrifices, without those it would have been impossible to achievemy PhD.

Taushif Ahmed

Bibliography

[1] ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B716(2012) 1–29, arXiv:1207.7214 [hep-ex].

[2] CMS Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125GeV with the CMS experiment at the LHC, Phys. Lett. B716 (2012) 30–61,arXiv:1207.7235 [hep-ex].

[3] D. J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys.Rev. Lett. 30 (1973) 1343–1346.

[4] D. J. Gross and F. Wilczek, Asymptotically Free Gauge Theories. 1, Phys. Rev. D8 (1973)3633–3652.

[5] H. D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30(1973) 1346–1349.

[6] D. J. Gross and F. Wilczek, ASYMPTOTICALLY FREE GAUGE THEORIES. 2., Phys. Rev.D9 (1974) 980–993.

[7] H. D. Politzer, Asymptotic Freedom: An Approach to Strong Interactions, Phys. Rept. 14(1974) 129–180.

[8] G. ’t Hooft and M. J. G. Veltman, Regularization and Renormalization of Gauge Fields,Nucl. Phys. B44 (1972) 189–213.

[9] G. M. Cicuta and E. Montaldi, Analytic renormalization via continuous space dimension,Lett. Nuovo Cim. 4 (1972) 329–332.

[10] C. G. Bollini and J. J. Giambiagi, Dimensional Renormalization: The Number ofDimensions as a Regularizing Parameter, Nuovo Cim. B12 (1972) 20–26.

[11] T. Ahmed, N. Rana, and V. Ravindran, Higgs boson production through bb annihilation atthreshold in N3LO QCD, JHEP 1410 (2014) 139, arXiv:1408.0787 [hep-ph].

[12] T. Ahmed, M. Mandal, N. Rana, and V. Ravindran, Rapidity Distributions in Drell-Yan andHiggs Productions at Threshold to Third Order in QCD, Phys. Rev. Lett. 113 (2014)212003, arXiv:1404.6504 [hep-ph].

[13] T. Ahmed, T. Gehrmann, P. Mathews, N. Rana, and V. Ravindran, Pseudo-scalar FormFactors at Three Loops in QCD, JHEP 11 (2015) 169, arXiv:1510.01715 [hep-ph].

169

170 Bibliography

[14] T. Ahmed, M. C. Kumar, P. Mathews, N. Rana, and V. Ravindran, Pseudo-scalar HiggsBoson Production at Threshold N3LO and N3LL QCD, arXiv:1510.02235 [hep-ph].

[15] CMS Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125GeV with the CMS experiment at the LHC, Phys. Lett. B716 (2012) 30–61,arXiv:1207.7235 [hep-ex].

[16] D. A. Dicus and S. Willenbrock, Higgs Boson Production from Heavy Quark Fusion,Phys. Rev. D39 (1989) 751.

[17] D. Dicus, T. Stelzer, Z. Sullivan, and S. Willenbrock, Higgs boson production inassociation with bottom quarks at next-to-leading order, Phys. Rev. D59 (1999) 094016,arXiv:hep-ph/9811492 [hep-ph].

[18] F. Maltoni, Z. Sullivan, and S. Willenbrock, Higgs-boson production via bottom-quarkfusion, Phys. Rev. D67 (2003) 093005, arXiv:hep-ph/0301033 [hep-ph].

[19] F. I. Olness and W.-K. Tung, When Is a Heavy Quark Not a Parton? Charged HiggsProduction and Heavy Quark Mass Effects in the QCD Based Parton Model, Nucl. Phys.B308 (1988) 813.

[20] J. F. Gunion, H. E. Haber, F. E. Paige, W.-K. Tung, and S. S. D. Willenbrock, Neutral andCharged Higgs Detection: Heavy Quark Fusion, Top Quark Mass Dependence and RareDecays, Nucl. Phys. B294 (1987) 621.

[21] R. V. Harlander and W. B. Kilgore, Higgs boson production in bottom quark fusion atnext-to-next-to leading order, Phys. Rev. D68 (2003) 013001, arXiv:hep-ph/0304035[hep-ph].

[22] L. Reina and S. Dawson, Next-to-leading order results for t anti-t h production at theTevatron, Phys. Rev. Lett. 87 (2001) 201804, arXiv:hep-ph/0107101 [hep-ph].

[23] W. Beenakker, S. Dittmaier, M. Kramer, B. Plumper, M. Spira, and P. M. Zerwas, Higgsradiation off top quarks at the Tevatron and the LHC, Phys. Rev. Lett. 87 (2001) 201805,arXiv:hep-ph/0107081 [hep-ph].

[24] S. Dawson, L. H. Orr, L. Reina, and D. Wackeroth, Associated top quark Higgs bosonproduction at the LHC, Phys. Rev. D67 (2003) 071503, arXiv:hep-ph/0211438[hep-ph].

[25] W. Beenakker, S. Dittmaier, M. Kramer, B. Plumper, M. Spira, and P. M. Zerwas, NLOQCD corrections to t anti-t H production in hadron collisions, Nucl. Phys. B653 (2003)151–203, arXiv:hep-ph/0211352 [hep-ph].

[26] R. Raitio and W. W. Wada, Higgs Boson Production at Large Transverse Momentum inQCD, Phys. Rev. D19 (1979) 941.

[27] Z. Kunszt, Associated Production of Heavy Higgs Boson with Top Quarks, Nucl. Phys.B247 (1984) 339–359.

[28] V. Ravindran, On Sudakov and soft resummations in QCD, Nucl.Phys. B746 (2006) 58–76,arXiv:hep-ph/0512249 [hep-ph].

[29] V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl.Phys.B752 (2006) 173–196, arXiv:hep-ph/0603041 [hep-ph].

Bibliography 171

[30] N. Kidonakis, Collinear and soft gluon corrections to Higgs production at NNNLO, Phys.Rev. D77 (2008) 053008, arXiv:0711.0142 [hep-ph].

[31] T. Gehrmann and D. Kara, The Hbb form factor to three loops in QCD, JHEP 09 (2014)174, arXiv:1407.8114 [hep-ph].

[32] T. Ahmed, M. Mahakhud, N. Rana, and V. Ravindran, Drell-Yan production at threshold inN3LO QCD, Phys.Rev.Lett. 113 (2014) 112002, arXiv:1404.0366 [hep-ph].

[33] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, and B. Mistlberger,Higgs boson gluon-fusion production at threshold in N3LO QCD, Phys. Lett. B737 (2014)325–328, arXiv:1403.4616 [hep-ph].

[34] S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini, Threshold resummation atN3LL accuracy and soft-virtual cross sections at N3LO, Nucl. Phys. B888 (2014) 75–91,arXiv:1405.4827 [hep-ph].

[35] Y. Li, A. von Manteuffel, R. M. Schabinger, and H. X. Zhu, N3LO Higgs boson andDrell-Yan production at threshold: The one-loop two-emission contribution, Phys. Rev.D90 no. 5, (2014) 053006, arXiv:1404.5839 [hep-ph].

[36] C. Anastasiou, L. J. Dixon, K. Melnikov, and F. Petriello, Dilepton rapidity distribution inthe Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002,arXiv:hep-ph/0306192 [hep-ph].

[37] C. Anastasiou, K. Melnikov, and F. Petriello, Higgs boson production at hadron colliders:Differential cross sections through next-to-next-to-leading order, Phys. Rev. Lett. 93(2004) 262002, arXiv:hep-ph/0409088 [hep-ph].

[38] C. Anastasiou, K. Melnikov, and F. Petriello, Fully differential Higgs boson productionand the di-photon signal through next-to-next-to-leading order, Nucl. Phys. B724 (2005)197–246, arXiv:hep-ph/0501130 [hep-ph].

[39] V. Ravindran, J. Smith, and W. L. van Neerven, QCD threshold corrections to di-leptonand Higgs rapidity distributions beyond N2 LO, Nucl. Phys. B767 (2007) 100–129,arXiv:hep-ph/0608308 [hep-ph].

[40] S. Moch, J. A. M. Vermaseren, and A. Vogt, Three-loop results for quark and gluonform-factors, Phys. Lett. B625 (2005) 245–252, arXiv:hep-ph/0508055 [hep-ph].

[41] P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Quarkand gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002,arXiv:0902.3519 [hep-ph].

[42] T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, Calculation of thequark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094,arXiv:1004.3653 [hep-ph].

[43] T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, The quark andgluon form factors to three loops in QCD through to O(ε2), JHEP 11 (2010) 102,arXiv:1010.4478 [hep-ph].

[44] C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, and B. Mistlberger, Higgs BosonGluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001,arXiv:1503.06056 [hep-ph].

172 Bibliography

[45] K. G. Chetyrkin, B. A. Kniehl, M. Steinhauser, and W. A. Bardeen, Effective QCDinteractions of CP odd Higgs bosons at three loops, Nucl. Phys. B535 (1998) 3–18,arXiv:hep-ph/9807241 [hep-ph].

[46] F. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization GroupFunctions, Phys.Lett. B100 (1981) 65–68.

[47] K. Chetyrkin and F. Tkachov, Integration by Parts: The Algorithm to Calculate betaFunctions in 4 Loops, Nucl.Phys. B192 (1981) 159–204.

[48] T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions,Nucl.Phys. B580 (2000) 485–518, arXiv:hep-ph/9912329 [hep-ph].

[49] S. A. Larin, The Renormalization of the axial anomaly in dimensional regularization,Phys. Lett. B303 (1993) 113–118, arXiv:hep-ph/9302240 [hep-ph].

[50] M. F. Zoller, OPE of the pseudoscalar gluonium correlator in massless QCD to three-looporder, JHEP 07 (2013) 040, arXiv:1304.2232 [hep-ph].

[51] OPAL, DELPHI, LEP Working Group for Higgs boson searches, ALEPH, L3Collaboration, R. Barate et al., Search for the standard model Higgs boson at LEP, Phys.Lett. B565 (2003) 61–75, arXiv:hep-ex/0306033 [hep-ex].

[52] CDF, D0 Collaboration, T. Aaltonen et al., Combination of Tevatron searches for thestandard model Higgs boson in the W+W- decay mode, Phys. Rev. Lett. 104 (2010)061802, arXiv:1001.4162 [hep-ex].

[53] Tevatron Electroweak Working Group, CDF, DELPHI, SLD Electroweak and HeavyFlavour Groups, ALEPH, LEP Electroweak Working Group, SLD, OPAL, D0, L3Collaboration, L. E. W. Group, Precision Electroweak Measurements and Constraints onthe Standard Model, arXiv:1012.2367 [hep-ex].

[54] A. Djouadi, M. Spira, and P. M. Zerwas, Production of Higgs bosons in proton colliders:QCD corrections, Phys. Lett. B264 (1991) 440–446.

[55] S. Dawson, Radiative corrections to Higgs boson production, Nucl. Phys. B359 (1991)283–300.

[56] M. Spira, A. Djouadi, D. Graudenz, and P. M. Zerwas, Higgs boson production at theLHC, Nucl. Phys. B453 (1995) 17–82, arXiv:hep-ph/9504378 [hep-ph].

[57] S. Catani, D. de Florian, and M. Grazzini, Higgs production in hadron collisions: Soft andvirtual QCD corrections at NNLO, JHEP 05 (2001) 025, arXiv:hep-ph/0102227[hep-ph].

[58] R. V. Harlander and W. B. Kilgore, Soft and virtual corrections to proton proton→ H + xat NNLO, Phys. Rev. D64 (2001) 013015, arXiv:hep-ph/0102241 [hep-ph].

[59] R. V. Harlander and W. B. Kilgore, Next-to-next-to-leading order Higgs production athadron colliders, Phys. Rev. Lett. 88 (2002) 201801, arXiv:hep-ph/0201206[hep-ph].

[60] C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLOQCD, Nucl. Phys. B646 (2002) 220–256, arXiv:hep-ph/0207004 [hep-ph].

Bibliography 173

[61] V. Ravindran, J. Smith, and W. L. van Neerven, NNLO corrections to the totalcross-section for Higgs boson production in hadron hadron collisions, Nucl. Phys. B665(2003) 325–366, arXiv:hep-ph/0302135 [hep-ph].

[62] P. Bolzoni, F. Maltoni, S.-O. Moch, and M. Zaro, Higgs production via vector-boson fusionat NNLO in QCD, Phys. Rev. Lett. 105 (2010) 011801, arXiv:1003.4451 [hep-ph].

[63] T. Han and S. Willenbrock, QCD correction to the p p —> W H and Z H totalcross-sections, Phys. Lett. B273 (1991) 167–172.

[64] S. Catani, D. de Florian, M. Grazzini, and P. Nason, Soft gluon resummation for Higgsboson production at hadron colliders, JHEP 07 (2003) 028, arXiv:hep-ph/0306211[hep-ph].

[65] U. Aglietti, R. Bonciani, G. Degrassi, and A. Vicini, Two loop light fermion contribution toHiggs production and decays, Phys. Lett. B595 (2004) 432–441,arXiv:hep-ph/0404071 [hep-ph].

[66] S. Actis, G. Passarino, C. Sturm, and S. Uccirati, NLO Electroweak Corrections to HiggsBoson Production at Hadron Colliders, Phys. Lett. B670 (2008) 12–17,arXiv:0809.1301 [hep-ph].

[67] S. Moch, J. A. M. Vermaseren, and A. Vogt, The Quark form-factor at higher orders,JHEP 08 (2005) 049, arXiv:hep-ph/0507039 [hep-ph].

[68] T. Gehrmann, T. Huber, and D. Maitre, Two-loop quark and gluon form-factors indimensional regularisation, Phys. Lett. B622 (2005) 295–302, arXiv:hep-ph/0507061[hep-ph].

[69] S. Moch, J. Vermaseren, and A. Vogt, The Three loop splitting functions in QCD: TheNonsinglet case, Nucl.Phys. B688 (2004) 101–134, arXiv:hep-ph/0403192 [hep-ph].

[70] K. G. Chetyrkin, B. A. Kniehl, and M. Steinhauser, Decoupling relations to O(alpha-s**3) and their connection to low-energy theorems, Nucl. Phys. B510 (1998)61–87, arXiv:hep-ph/9708255 [hep-ph].

[71] D. de Florian and J. Mazzitelli, A next-to-next-to-leading order calculation of soft-virtualcross sections, JHEP 12 (2012) 088, arXiv:1209.0673 [hep-ph].

[72] S. Moch and A. Vogt, Higher-order soft corrections to lepton pair and Higgs bosonproduction, Phys. Lett. B631 (2005) 48–57, arXiv:hep-ph/0508265 [hep-ph].

[73] E. Laenen and L. Magnea, Threshold resummation for electroweak annihilation from DISdata, Phys. Lett. B632 (2006) 270–276, arXiv:hep-ph/0508284 [hep-ph].

[74] A. Idilbi, X.-d. Ji, J.-P. Ma, and F. Yuan, Threshold resummation for Higgs production ineffective field theory, Phys. Rev. D73 (2006) 077501, arXiv:hep-ph/0509294[hep-ph].

[75] G. F. Sterman, Summation of Large Corrections to Short Distance HadronicCross-Sections, Nucl. Phys. B281 (1987) 310–364.

[76] S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for HardProcesses, Nucl. Phys. B327 (1989) 323–352.

174 Bibliography

[77] O. V. Tarasov, A. A. Vladimirov, and A. Yu. Zharkov, The Gell-Mann-Low Function ofQCD in the Three Loop Approximation, Phys. Lett. B93 (1980) 429–432.

[78] C. Anastasiou, F. Herzog, and A. Lazopoulos, The fully differential decay rate of a Higgsboson to bottom-quarks at NNLO in QCD, JHEP 03 (2012) 035, arXiv:1110.2368[hep-ph].

[79] V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics,Sov.Phys.JETP 3 (1956) 65–71.

[80] A. H. Mueller, On the Asymptotic Behavior of the Sudakov Form-factor, Phys.Rev. D20(1979) 2037.

[81] J. C. Collins, Algorithm to Compute Corrections to the Sudakov Form-factor, Phys.Rev.D22 (1980) 1478.

[82] A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys.Rev. D24 (1981)3281.

[83] L. Magnea and G. F. Sterman, Analytic continuation of the Sudakov form-factor in QCD,Phys. Rev. D42 (1990) 4222–4227.

[84] V. Ravindran, J. Smith, and W. L. van Neerven, Two-loop corrections to Higgs bosonproduction, Nucl. Phys. B704 (2005) 332–348, arXiv:hep-ph/0408315 [hep-ph].

[85] A. Vogt, S. Moch, and J. Vermaseren, The Three-loop splitting functions in QCD: TheSinglet case, Nucl.Phys. B691 (2004) 129–181, arXiv:hep-ph/0404111 [hep-ph].

[86] A. Vogt, Next-to-next-to-leading logarithmic threshold resummation for deep inelasticscattering and the Drell-Yan process, Phys.Lett. B497 (2001) 228–234,arXiv:hep-ph/0010146 [hep-ph].

[87] T. Ahmed, G. Das, M. C. Kumar, N. Rana, and V. Ravindran, RG improved Higgs bosonproduction to N3LO in QCD, arXiv:1505.07422 [hep-ph].

[88] A. Martin, W. Stirling, R. Thorne, and G. Watt, Parton distributions for the LHC,Eur.Phys.J. C63 (2009) 189–285, arXiv:0901.0002 [hep-ph].

[89] S. D. Drell and T.-M. Yan, Massive Lepton Pair Production in Hadron-Hadron Collisionsat High-Energies, Phys. Rev. Lett. 25 (1970) 316–320. [Erratum: Phys. Rev.Lett.25,902(1970)].

[90] CDF Collaboration, T. Affolder et al., Measurement of d(σ)/dy for high mass Drell-Yane+e− pairs from pp collisions at

√s = 1.8 TeV, Phys. Rev. D63 (2001) 011101,

arXiv:hep-ex/0006025 [hep-ex].

[91] CDF Collaboration, F. Abe et al., Measurement of the lepton charge asymmetry in Wboson decays produced in pp collisions, Phys. Rev. Lett. 81 (1998) 5754–5759,arXiv:hep-ex/9809001 [hep-ex].

[92] J. Kubar-Andre and F. E. Paige, Gluon corrections to the drell-yan model, Phys. Rev. D 19(Jan, 1979) 221–229. http://link.aps.org/doi/10.1103/PhysRevD.19.221.

[93] G. Altarelli, R. K. Ellis, and G. Martinelli, Leptoproduction and Drell-Yan ProcessesBeyond the Leading Approximation in Chromodynamics, Nucl. Phys. B143 (1978) 521.[Erratum: Nucl. Phys.B146,544(1978)].

Bibliography 175

[94] B. Humpert and W. L. van Neerven, Infrared and Mass Regularization in Af Field Theories2. QCD, Nucl. Phys. B184 (1981) 225–268.

[95] T. Matsuura and W. van Neerven, Second Order Logarithmic Corrections to the Drell-YanCross-section, Z.Phys. C38 (1988) 623.

[96] T. Matsuura, S. van der Marck, and W. van Neerven, The Calculation of the Second OrderSoft and Virtual Contributions to the Drell-Yan Cross-Section, Nucl.Phys. B319 (1989)570.

[97] R. Hamberg, W. L. van Neerven, and T. Matsuura, A Complete calculation of the order α2s

correction to the Drell-Yan K factor, Nucl. Phys. B359 (1991) 343–405. [Erratum: Nucl.Phys.B644,403(2002)].

[98] A. Djouadi, J. Kalinowski, and P. M. Zerwas, Higgs radiation off top quarks inhigh-energy e+ e- colliders, Z. Phys. C54 (1992) 255–262.

[99] T. Inami, T. Kubota, and Y. Okada, Effective Gauge Theory and the Effect of HeavyQuarks in Higgs Boson Decays, Z. Phys. C18 (1983) 69.

[100] K. G. Chetyrkin, B. A. Kniehl, and M. Steinhauser, Hadronic Higgs decay to orderalpha-s**4, Phys. Rev. Lett. 79 (1997) 353–356, arXiv:hep-ph/9705240 [hep-ph].

[101] Y. Schroder and M. Steinhauser, Four-loop decoupling relations for the strong coupling,JHEP 01 (2006) 051, arXiv:hep-ph/0512058 [hep-ph].

[102] K. G. Chetyrkin, J. H. Kuhn, and C. Sturm, QCD decoupling at four loops, Nucl. Phys.B744 (2006) 121–135, arXiv:hep-ph/0512060 [hep-ph].

[103] T. Ahmed, M. Mandal, N. Rana, and V. Ravindran, Higgs Rapidity Distribution in bbAnnihilation at Threshold in N3LO QCD, JHEP 1502 (2015) 131, arXiv:1411.5301[hep-ph].

[104] R. V. Harlander, Virtual corrections to g g —> H to two loops in the heavy top limit, Phys.Lett. B492 (2000) 74–80, arXiv:hep-ph/0007289 [hep-ph].

[105] R. N. Lee, A. V. Smirnov, and V. A. Smirnov, Analytic Results for Massless Three-LoopForm Factors, JHEP 04 (2010) 020, arXiv:1001.2887 [hep-ph].

[106] G. Kramer and B. Lampe, Two Jet Cross-Section in e+ e- Annihilation, Z.Phys. C34(1987) 497.

[107] S. Forte, A. IsgrÚ, and G. Vita, Do we need N3LO Parton Distributions?, Phys. Lett.B731 (2014) 136–140, arXiv:1312.6688 [hep-ph].

[108] T. Becher, M. Neubert, and G. Xu, Dynamical Threshold Enhancement and Resummationin Drell-Yan Production, JHEP 07 (2008) 030, arXiv:0710.0680 [hep-ph].

[109] M. Kramer, E. Laenen, and M. Spira, Soft gluon radiation in Higgs boson production atthe LHC, Nucl. Phys. B511 (1998) 523–549, arXiv:hep-ph/9611272 [hep-ph].

[110] F. Herzog and B. Mistlberger, The Soft-Virtual Higgs Cross-section at N3LO and theConvergence of the Threshold Expansion, in Proceedings, 49th Rencontres de Moriond onQCD and High Energy Interactions, pp. 57–60. 2014. arXiv:1405.5685 [hep-ph].https://inspirehep.net/record/1297500/files/arXiv:1405.5685.pdf.

176 Bibliography

[111] P. Nogueira, Automatic Feynman graph generation, J.Comput.Phys. 105 (1993) 279–289.

[112] J. Vermaseren, New features of FORM, arXiv:math-ph/0010025 [math-ph].

[113] M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod.Phys. A22 (2007) 4375–4436, arXiv:0707.4037 [hep-ph].

[114] C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher orderperturbative calculations, JHEP 07 (2004) 046, arXiv:hep-ph/0404258 [hep-ph].

[115] A. V. Smirnov, Algorithm FIRE – Feynman Integral REduction, JHEP 10 (2008) 107,arXiv:0807.3243 [hep-ph].

[116] C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun. 181(2010) 1293–1300, arXiv:0912.2546 [physics.comp-ph].

[117] A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction,arXiv:1201.4330 [hep-ph].

[118] R. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685[hep-ph].

[119] R. N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals,J.Phys.Conf.Ser. 523 (2014) 012059, arXiv:1310.1145 [hep-ph].

[120] R. N. Lee, Group structure of the integration-by-part identities and its application to thereduction of multiloop integrals, JHEP 07 (2008) 031, arXiv:0804.3008 [hep-ph].

[121] T. Ahmed, M. Mahakhud, P. Mathews, N. Rana, and V. Ravindran, Two-Loop QCDCorrection to massive spin-2 resonance→ 3 gluons, JHEP 05 (2014) 107,arXiv:1404.0028 [hep-ph].

[122] T. Ahmed, M. Mahakhud, P. Mathews, N. Rana, and V. Ravindran, Two-loop QCDcorrections to Higgs→ b + b + g amplitude, JHEP 08 (2014) 075, arXiv:1405.2324[hep-ph].

[123] T. Ahmed, G. Das, P. Mathews, N. Rana, and V. Ravindran, Spin-2 Form Factors at ThreeLoop in QCD, arXiv:1508.05043 [hep-ph].

[124] G. Altarelli, R. K. Ellis, and G. Martinelli, Large Perturbative Corrections to the Drell-YanProcess in QCD, Nucl. Phys. B157 (1979) 461.

[125] D. Graudenz, M. Spira, and P. M. Zerwas, QCD corrections to Higgs boson production atproton proton colliders, Phys. Rev. Lett. 70 (1993) 1372–1375.

[126] A. Djouadi, M. Spira, and P. M. Zerwas, QCD corrections to hadronic Higgs decays, Z.Phys. C70 (1996) 427–434, arXiv:hep-ph/9511344 [hep-ph].

[127] M. Spira, QCD effects in Higgs physics, Fortsch. Phys. 46 (1998) 203–284,arXiv:hep-ph/9705337 [hep-ph].

[128] R. Harlander and P. Kant, Higgs production and decay: Analytic results at next-to-leadingorder QCD, JHEP 12 (2005) 015, arXiv:hep-ph/0509189 [hep-ph].

[129] S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B427(1998) 161–171, arXiv:hep-ph/9802439 [hep-ph].

Bibliography 177

[130] G. F. Sterman and M. E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys.Lett. B552 (2003) 48–56, arXiv:hep-ph/0210130 [hep-ph].

[131] T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbativeQCD, Phys. Rev. Lett. 102 (2009) 162001, arXiv:0901.0722 [hep-ph]. [Erratum:Phys. Rev. Lett.111,no.19,199905(2013)].

[132] E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCDscattering amplitudes, JHEP 03 (2009) 079, arXiv:0901.1091 [hep-ph].

[133] O. Almelid, C. Duhr, and E. Gardi, Three-loop corrections to the soft anomalousdimension in multi-leg scattering, arXiv:1507.00047 [hep-ph].

[134] S. Moch, J. A. M. Vermaseren, and A. Vogt, Higher-order corrections in thresholdresummation, Nucl. Phys. B726 (2005) 317–335, arXiv:hep-ph/0506288 [hep-ph].

[135] T. Gehrmann, J. M. Henn, and T. Huber, The three-loop form factor in N=4 superYang-Mills, JHEP 03 (2012) 101, arXiv:1112.4524 [hep-th].

[136] S. Laporta, High precision calculation of multiloop Feynman integrals by differenceequations, Int.J.Mod.Phys. A15 (2000) 5087–5159, arXiv:hep-ph/0102033 [hep-ph].

[137] T. Gehrmann, G. Heinrich, T. Huber, and C. Studerus, Master integrals for masslessthree-loop form-factors: One-loop and two-loop insertions, Phys. Lett. B640 (2006)252–259, arXiv:hep-ph/0607185 [hep-ph].

[138] G. Heinrich, T. Huber, and D. Maitre, Master integrals for fermionic contributions tomassless three-loop form-factors, Phys. Lett. B662 (2008) 344–352, arXiv:0711.3590[hep-ph].

[139] G. Heinrich, T. Huber, D. A. Kosower, and V. A. Smirnov, Nine-Propagator MasterIntegrals for Massless Three-Loop Form Factors, Phys. Lett. B678 (2009) 359–366,arXiv:0902.3512 [hep-ph].

[140] P. Fayet, Supergauge Invariant Extension of the Higgs Mechanism and a Model for theelectron and Its Neutrino, Nucl. Phys. B90 (1975) 104–124.

[141] P. Fayet, Supersymmetry and Weak, Electromagnetic and Strong Interactions, Phys. Lett.B64 (1976) 159.

[142] P. Fayet, Spontaneously Broken Supersymmetric Theories of Weak, Electromagnetic andStrong Interactions, Phys. Lett. B69 (1977) 489.

[143] S. Dimopoulos and H. Georgi, Softly Broken Supersymmetry and SU(5), Nucl. Phys. B193(1981) 150.

[144] N. Sakai, Naturalness in Supersymmetric GUTS, Z. Phys. C11 (1981) 153.

[145] K. Inoue, A. Kakuto, H. Komatsu, and S. Takeshita, Aspects of Grand Unified Models withSoftly Broken Supersymmetry, Prog. Theor. Phys. 68 (1982) 927. [Erratum: Prog. Theor.Phys.70,330(1983)].

[146] K. Inoue, A. Kakuto, H. Komatsu, and S. Takeshita, Renormalization of SupersymmetryBreaking Parameters Revisited, Prog. Theor. Phys. 71 (1984) 413.

178 Bibliography

[147] K. Inoue, A. Kakuto, H. Komatsu, and S. Takeshita, Low-Energy Parameters and ParticleMasses in a Supersymmetric Grand Unified Model, Prog. Theor. Phys. 67 (1982) 1889.

[148] J. R. Ellis, M. K. Gaillard, and D. V. Nanopoulos, A Phenomenological Profile of theHiggs Boson, Nucl. Phys. B106 (1976) 292.

[149] M. A. Shifman, A. I. Vainshtein, M. B. Voloshin, and V. I. Zakharov, Low-EnergyTheorems for Higgs Boson Couplings to Photons, Sov. J. Nucl. Phys. 30 (1979) 711–716.[Yad. Fiz.30,1368(1979)].

[150] B. A. Kniehl and M. Spira, Low-energy theorems in Higgs physics, Z. Phys. C69 (1995)77–88, arXiv:hep-ph/9505225 [hep-ph].

[151] R. P. Kauffman and W. Schaffer, QCD corrections to production of Higgs pseudoscalars,Phys. Rev. D49 (1994) 551–554, arXiv:hep-ph/9305279 [hep-ph].

[152] A. Djouadi, M. Spira, and P. M. Zerwas, Two photon decay widths of Higgs particles,Phys. Lett. B311 (1993) 255–260, arXiv:hep-ph/9305335 [hep-ph].

[153] R. V. Harlander and W. B. Kilgore, Production of a pseudoscalar Higgs boson at hadroncolliders at next-to-next-to leading order, JHEP 10 (2002) 017, arXiv:hep-ph/0208096[hep-ph].

[154] C. Anastasiou and K. Melnikov, Pseudoscalar Higgs boson production at hadron collidersin NNLO QCD, Phys. Rev. D67 (2003) 037501, arXiv:hep-ph/0208115 [hep-ph].

[155] M. Spira, A. Djouadi, D. Graudenz, and P. M. Zerwas, SUSY Higgs production at protoncolliders, Phys. Lett. B318 (1993) 347–353.

[156] V. Ahrens, T. Becher, M. Neubert, and L. L. Yang, Renormalization-Group ImprovedPrediction for Higgs Production at Hadron Colliders, Eur. Phys. J. C62 (2009) 333–353,arXiv:0809.4283 [hep-ph].

[157] D. de Florian and M. Grazzini, Higgs production through gluon fusion: Updated crosssections at the Tevatron and the LHC, Phys. Lett. B674 (2009) 291–294,arXiv:0901.2427 [hep-ph].

[158] M. Bonvini and S. Marzani, Resummed Higgs cross section at N3LL, JHEP 09 (2014) 007,arXiv:1405.3654 [hep-ph].

[159] D. de Florian and J. Zurita, Soft-gluon resummation for pseudoscalar Higgs bosonproduction at hadron colliders, Phys. Lett. B659 (2008) 813–820, arXiv:0711.1916[hep-ph].

[160] T. Schmidt and M. Spira, Higgs Boson Production via Gluon Fusion: Soft-GluonResummation including Mass Effects, arXiv:1509.00195 [hep-ph].

[161] S. L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969)2426–2438.

[162] D. A. Akyeampong and R. Delbourgo, Dimensional regularization, abnormal amplitudesand anomalies, Nuovo Cim. A17 (1973) 578–586.

[163] J. Kodaira, QCD Higher Order Effects in Polarized Electroproduction: Flavor SingletCoefficient Functions, Nucl. Phys. B165 (1980) 129.

Bibliography 179

[164] S. L. Adler and W. A. Bardeen, Absence of higher order corrections in the anomalousaxial vector divergence equation, Phys. Rev. 182 (1969) 1517–1536.

[165] G. F. Sterman and S. Weinberg, Jets from Quantum Chromodynamics, Phys. Rev. Lett. 39(1977) 1436.

[166] S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl.Phys. B353(1991) 183–186.

[167] A. V. Kotikov and L. N. Lipatov, On the highest transcendentality in N=4 SUSY, Nucl.Phys. B769 (2007) 217–255, arXiv:hep-th/0611204 [hep-th].

[168] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko, and V. N. Velizhanin, Three loopuniversal anomalous dimension of the Wilson operators in N=4 SUSY Yang-Mills model,Phys. Lett. B595 (2004) 521–529, arXiv:hep-th/0404092 [hep-th]. [Erratum: Phys.Lett.B632,754(2006)].

[169] A. V. Kotikov and L. N. Lipatov, DGLAP and BFKL evolution equations in the N=4supersymmetric gauge theory, in 35th Annual Winter School on Nuclear and ParticlePhysics Repino, Russia, February 19-25, 2001. 2001. arXiv:hep-ph/0112346[hep-ph]. http://alice.cern.ch/format/showfull?sysnb=2289957.

[170] R. N. Lee and V. A. Smirnov, Analytic Epsilon Expansions of Master IntegralsCorresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-LoopTranscendentality Weight, JHEP 02 (2011) 102, arXiv:1010.1334 [hep-ph].

[171] C. W. Bauer, S. Fleming, and M. E. Luke, Summing Sudakov logarithms in B→ Xsγ ineffective field theory, Phys. Rev. D63 (2000) 014006, arXiv:hep-ph/0005275[hep-ph].

[172] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, An Effective field theory forcollinear and soft gluons: Heavy to light decays, Phys. Rev. D63 (2001) 114020,arXiv:hep-ph/0011336 [hep-ph].

[173] C. W. Bauer and I. W. Stewart, Invariant operators in collinear effective theory, Phys. Lett.B516 (2001) 134–142, arXiv:hep-ph/0107001 [hep-ph].

[174] C. W. Bauer, D. Pirjol, and I. W. Stewart, Soft collinear factorization in effective fieldtheory, Phys. Rev. D65 (2002) 054022, arXiv:hep-ph/0109045 [hep-ph].

[175] M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, Soft collinear effective theoryand heavy to light currents beyond leading power, Nucl. Phys. B643 (2002) 431–476,arXiv:hep-ph/0206152 [hep-ph].

[176] M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory withnonAbelian gauge symmetry, Phys. Lett. B553 (2003) 267–276, arXiv:hep-ph/0211358[hep-ph].

[177] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Hard scatteringfactorization from effective field theory, Phys. Rev. D66 (2002) 014017,arXiv:hep-ph/0202088 [hep-ph].

[178] G. P. Korchemsky and A. V. Radyushkin, Loop Space Formalism and RenormalizationGroup for the Infrared Asymptotics of QCD, Phys. Lett. B171 (1986) 459–467.

180 Bibliography

[179] G. P. Korchemsky and A. V. Radyushkin, Renormalization of the Wilson Loops Beyond theLeading Order, Nucl. Phys. B283 (1987) 342–364.

[180] G. P. Korchemsky, Double Logarithmic Asymptotics in QCD, Phys. Lett. B217 (1989)330–334.

[181] T. Becher and M. Neubert, Threshold resummation in momentum space from effective fieldtheory, Phys. Rev. Lett. 97 (2006) 082001, arXiv:hep-ph/0605050 [hep-ph].

[182] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, and B. Mistlberger,Higgs boson gluon-fusion production beyond threshold in N3LO QCD, JHEP 03 (2015)091, arXiv:1411.3584 [hep-ph].

[183] T. Ahmed, M. Bonvini, M. C. Kumar, P. Mathews, N. Rana, V. Ravindran, and L. Rottoli,Pseudo-scalar Higgs boson production at N3LOA+N3LL′, arXiv:1606.00837[hep-ph].

[184] E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A15(2000) 725–754, arXiv:hep-ph/9905237 [hep-ph].