On Different Parametrizations of Feynman Integrals

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9-2019

Ray Daniel Sameshima The Graduate Center, City University of New York

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This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] On Different Parametrizations of Feynman Integrals

Ray Daniel Sameshima

A dissertation submitted to the Graduate Faculty in Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York

2019 ii

On Different Parametrizations of Feynman Integrals by Ray Daniel Sameshima

This manuscript has been read and accepted by the Graduate Faculty in Physics in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy.

Giovanni Ossola

Date Chair of Examining Committee

Igor Kuskovsky

Date Executive Officer

Supervisory Committee Andrea Ferroglia Miguel Castro Nunes Fiolhais Pierpaolo Mastrolia Justin F. Vazquez-Poritz

The City University of New York iv

Abstract On Different Parametrizations of Feynman Integrals by Ray Daniel Sameshima

Advisers: Andrea Ferroglia, Giovanni Ossola

In this doctoral thesis, we discuss and apply advanced techniques for the calculations of scattering amplitudes which, on the one hand, allow us to compute cross sections and differential distributions at high precision and, on the other hand, give us deep mathematical insights on the mathematical structures of Feynman integrals. We start by presenting phenomenological calculations relevant for the experimental analyses at the Large Hadron Collider. We use the resummation of soft gluon emission corrections to study the associated production of a top pair and a Z boson to next-to-next-to-leading logarithmic accuracy, and compute the total cross section and differential distributions for a range of interesting observables. Our evaluations are currently the most precise predictions for this process available in the literature. Then we introduce various representations for Feynman integrals, namely Schwinger, Feyn- man, Lee-Pomeransky, and Baikov parametrizations. We show detailed derivations and discuss the relevant mathematical properties for these parametrizations. We also discuss Integration-By-Parts identities, illustrate the reduction process under Laporta’s algorithm, and examine the boundary regions for Baikov parametrization. We finally present two novel applications of Schwinger-Feynman parametrizations. First, we consider Integration-By-Parts identities over Schwinger-Feynman parameters. Through this procedure, we touch upon techniques which are related to graph theory and complex analysis. Then we use Intersection Theory to compute the recurrence relations among classes of integrals. We confirm the consistency of these two new approaches using traditional Integration-By-Parts calculations. Acknowledgments

I would like to take the time to acknowledge and thank those who played a major part throughout this doctorate study process. My supervisors, ANDREA FERROGLIA, and GIOVANNI OSSOLA for their continuous support, encouragement, suggestions, and being patient with me throughout my doctoral study.

My collaborators, Pierpaolo Mastrolia, Stefano Laporta, Amedeo Primo, William J. Torres Bobadilla, Jonathan Ronca, Hjalte Frellesvig, Federico Gasparotto, Manoj K. Mandal, Luca Matti- azzi, Sebastian Mizera. It has been so great collaborating with you all. Our discussions have been inspiring and fruitful.

I wish to thank the workshop “The Mathematics of Linear Relations between Feynman In- tegrals” and co-organizers, Christian Bogner, Erik Panzer, and Yang Zhang. It was a tremendous experience being next to the authors of significant publications, asking questions, and having black- board talks amongst them.

The developer of Kira, Philipp Maierhoefer, Johann Usovitsch, and Peter Uwer, thank you for your kindness and detailed answers for all my questions.

Last but not least, I would love to thank my spouse, Rumiko, for her love and continuous support.

v Contents

List of Tables ix

List of Figures x

1 Introduction 1

2 Associated Production of a Top Pair and a Z boson at the LHC to NNLL Accuracy 8 2.1 Introduction ...... 8 2.2 Outline of the Calculation ...... 10 2.3 Numerical Results ...... 13 2.3.1 Scale Choices ...... 16 2.3.2 Total Cross Section ...... 17 2.3.3 Differential Distributions ...... 21 2.4 Conclusions ...... 28

3 Multi-loop Feynman Integrals and Integration-By-Parts Identities 30 3.1 Multi-loop Feynman Integrals ...... 31 3.2 Schwinger-Feynman Parametrizations ...... 33 3.2.1 Schwinger Trick and Feynman Integrals ...... 34 3.2.2 Graph Polynomials and Schwinger Parametrization ...... 37

vi CONTENTS vii

3.2.3 Feynman Parametrization ...... 38 3.2.4 Lee-Pomeransky Parametrization ...... 41 3.3 Integration-By-Parts Identities ...... 43 3.3.1 Lie Group and Lie Algebraic Structure ...... 46 3.3.2 Laporta’s Algorithm and Master Integrals ...... 52 3.4 Baikov Parametrization ...... 54 3.4.1 IBP Identities in Baikov Parametrization ...... 58 3.4.2 Exact Form of Integration Regions: a Proof ...... 65

4 Integration-By-Parts Identities over Lee-Pomeransky Parametrization 75 4.1 Formalism ...... 76 4.1.1 IBP Generators over Lee-Pomeransky Parametrization ...... 76 4.1.2 Pinching ...... 79 4.1.3 Dimension Shifts ...... 82 4.2 One-Loop examples ...... 84 4.2.1 Bubble Integrals ...... 84 4.2.2 Triangle Integrals ...... 89 4.3 QED-like Kite Integrals ...... 92 4.3.1 IBP Identities over Lee-Pomeransky Parametrization ...... 93 4.3.2 An Example of Pinching ...... 97 4.4 Massless Double-boxes ...... 99 4.5 Direct IBP Reduction over Lee-Pomeransky Parametrization ...... 105 4.6 Conclusions ...... 108

5 Intersection Numbers and Schwinger-Feynman Parametrizations 110 5.1 Formalism ...... 111 5.1.1 Gamma function ...... 115 CONTENTS viii

5.2 Tadpoles in Schwinger Parametrization ...... 123 5.3 Bubbles in Feynman Parametrization ...... 127 5.4 Conclusions ...... 135

6 Conclusions and Future Perspectives 136

Appendices 140

A An overview of quantum field theories 141

B Determinants and Laplace Expansion 155 B.1 Determinants and Laplace expansion ...... 155 B.2 Block matrices and determinant ...... 158

Bibliography 162 List of Tables

2.1 Input parameters employed throughout the calculation...... 13 √ 2.2 Total cross section for ttZ¯ production at the LHC with s = 13 TeV and MMHT

2014 PDFs. The default value of the factorization scale is µf,0 = M/2, and the uncertainties are estimated through variations of this scale (and of the hard and soft

scales µs and µh when applicable), as explained in the text...... 18 √ 2.3 Total cross section for ttZ¯ and ttW¯ production at the LHC with s = 8 and 13 TeV

and MMHT 2014 PDFs. The default value of the factorization scale is µf,0 = M/2, and the uncertainties are estimated through variations of this scale (and of

the resummation scales µs and µh when applicable)...... 19

ix List of Figures

2.1 Factorization-scale dependence of the total ttZ¯ production cross section at the LHC √ with s = 13 TeV. The NLO and NLO+NLL curves are obtained using MMHT 2014 NLO PDFs, while the NLO+NNLL and nNLO curves are obtained using MMHT 2014 NNLO PDFs...... 16 2.2 Total cross section at NLO (Green) and NLO+NNLL (Red) compared to theATLAS measurements at 8 TeV [1] (left panel) and CMS measurement at 13 TeV [2] (right panel)...... 20 2.3 Total cross section at NLO (Green) and NLO+NNLL (Red), which include NLO electroweak effects [3], compared to the ATLAS measurements [4] at 13 TeV (left panel) and CMS measurement [5] at 13 TeV (right panel)...... 21 2.4 Differential distributions at approximate NLO (blue band) compared to the com-

plete NLO (red band). The default factorization scale is chosen as µf,0 = M/2, and the uncertainty bands are generated through scale variations as explained in the text...... 22 2.5 Differential distributions at approximate NLO (blue band) compared to the NLO distributions without the quark-gluon channel contribution (red band). All settings are as in Figure 2.4...... 23

x LIST OF FIGURES xi

2.6 Differential distributions with µf,0 = M/2 at NLO+NNLL (blue band) compared to the NLO calculation (red band). The uncertainty bands are generated through

scale variations of µf , µs and µh as explained in the text...... 25

2.7 Differential distributions with µf,0 = M/2 at NLO+NNLL (blue band) compared to the corresponding NLO+NLL calculation (red band). The uncertainty bands are generated through scale variations...... 26

2.8 Differential distributions ratios for µf,0 = M/2, where the uncertainties are generated through scale variations...... 27

5.1 Maxima session for hφ0| φ0i of the Gamma function to determine three unknown a, b, c in three linear equations...... 120

5.2 Maxima session for hφ1| φ0i of the Gamma function to determine three unknown a, b, c in three linear equations...... 122 5.3 A few example of Kira’s IBP reduction of massive tadpoles. Note that, since Kira uses (+ − − · · · ) metric, we need to formally replace any d momenta contraction

|ν| 1 1 (+−−··· )→(−++··· ) 1 to its negative, and with (−) sign factor, due to D0 = k02−m2 7→ −D0 .129 5.4 Maxima session for to convert a set of Master Integrals: {I(1, 1),I(1, 2)} ⇐⇒ {I(1, 0),I(1, 1)}...... 130 Chapter 1

Introduction

Physics studies the laws and structures of Nature, a vast landscape of topics with different features and different scales: from the dynamics of galaxies to elementary particles, from properties of materials to the structure of time-space, from the efficiency of the thermal engines to information theories. No matter what the specific subjects of our studies are, in order to understand the natural phenomena and grasp the mechanisms behind them, we have to study them carefully. It is achieved by performing experiments, building mathematical models, and comparing outcomes from both experiments and theories. In this thesis, we focus on a specific branch of physics, called high-energy physics. One of the origins of high-energy physics is, perhaps, the fundamental and ubiquitous question: “What is the smallest building block of the universe?” To provide an answer to this fascinating question, we built particle accelerators, that collide particles at boosted energies. Such machines are our microscopes for subatomic scales. As the energy in the center-of-mass of the collision increases, we probe deeper into smaller and smaller distances. An early attempt to extract information about small structures by colliding particles is the leg- endary gold foil experiment directed by Rutherford between 1908 and 1913. This experiment allowed us to get access to atomic scales by exploring the structure of nuclei. It is also a good example

1 CHAPTER 1. INTRODUCTION 2 of indirect measurements; we do not get a picture of the shape of nuclei directly, but we instead observe that a fraction of outgoing particles experienced sharp deflection. The authors of this experiment deduced, from this pattern of deflection, the Rutherford model of the atom, in which a concentrated nucleus carries the bulk of mass and only positive charges. Similarly, recent particle physics searches rely on indirect measurements. At high-energy colliders, we do not observe the appearance of new particles or forces directly, but we rather infer their existence from the pattern and energies of known outgoing particles. Discoveries are achieved by performing precise comparisons between experimental data and theoretical predictions for such patterns. The Large Hadron Collider (LHC) is the largest and most powerful particle accelerator ever built. The discovery of Higgs boson announced in 2012 at LHC by both ATLAS and CMS [6, 7] is, undoubtedly, one of the fascinating milestones in the history of both experimental and theoretical physics. In 2013, Francois Englert and Peter W. Higgs were awarded the Nobel prize in physics for their work, after the discovery of the Higgs boson at CERN. It took forty-six years of hard theoretical works and careful experimental studies. As this history exemplifies, the Higgs discovery is one of the most significant achievement by both theoretical and experimental high-energy physicists. The theoretical counterpart of collider experiments is represented by precise phenomenology. The Quantum Field Theory (QFT) is a natural amalgam of quantum mechanics and the special theory of relativity, and it provides a systematic way of computing scattering amplitudes, which essentially are probability amplitudes for different processes to happen. For a specific process, the scattering amplitude gives the transition probability from a particular initial state, say the particles colliding at given initial energies, to any possible final state admitted by the theory. Scattering amplitudes connect theories and experiments; we can compute the probability of a particular event, and we can measure that specific event experimentally. The current best understanding of subatomic phenomena is contained in the Standard Model of Particle Physics. According to this theory, everything in the Universe is made of a few fundamental particles, called fermions and bosons. Fermions, which make up matter, contain three generations CHAPTER 1. INTRODUCTION 3 of quarks and leptons. Bosons, which carry forces, contain the W and Z bosons responsible for the weak force, the photon which carries electromagnetic interaction, and the Higgs boson. The greatest successes of the Standard Model are the excellent agreements between theoretical predictions and experimental data for a wide range of observables. Almost all experimental outcomes are explained by the dynamics of elementary particles listed above and interactions among them. Although its enormous success, we know that the Standard Model is not the end of the story and should be further improved and complemented; it can not explain, for example, why neutrinos are massive, what dark matter and dark energy are made of, or why our Universe is matter-dominated. Many attempts to go beyond the Standard Model have been proposed. However, they all ran into inconsistency with the existing theory, for example by spoiling the excellent agreement for observables correctly described by the Standard Model, or were ruled out by the experiments. In this context, it becomes important to compare theories and experimental data at higher and higher accuracy. As more data is expected to be available, and experimental measures will reach a new level of precision. This fact brings us back to the experiment. The LHC experimental programs go well beyond the Higgs boson discovery and the completion of the particle zoo predicted by the Standard Model, and in the forthcoming years will try to detect hints of new physics. During the second run (2015-2018) LHC performed remarkably, taking over 300 petabytes of data and measuring the coupling of Higgs boson and top quarks [8, 9] in agreements with the Standard Model. Observations of the Higgs boson decay into bottom quarks was also announced [10, 11] that are also in good agreements with Standard Model. Both these processes would supposedly get sizable corrections from new models. Currently, in a two-year-long shutdown, the whole accelerator and detectors are being upgraded for the next LHC run starting in 2021. A further upgrade for High-Luminosity LHC is also planned for a future run. In order to support the progress in this field, as the experimental data will reach new unprecedented precision, also the theoretical techniques will need to be upgraded to become more efficient and able to tackle more involved scenarios. CHAPTER 1. INTRODUCTION 4

The following procedure can achieve the theoretical work to obtain precise values from scattering amplitudes. Higher-order corrections to a given scattering process are depicted as multi-loop Feynman diagrams; our calculations are obtained by summing over all the Feynman diagrams relevant to the specific scattering process up to the desired order, i.e., the number of loops. Feyn- man rules allow us to translate diagrams into integrals. In every practical calculation, we need to apply two levels of reduction techniques. First we decompose Feynman integrals into scalar integrals [12–16]. The numbers of integrals in multi-loop calculations can combinatorially explode as the number of loops and external particles increase. However, there exist nontrivial linear relations among integrals; that is, they are not all independent. As a second reduction procedure, we have to find a basis in this linear space of integrals called the set of Master Integrals. Indeed, it is no- table that the number of Master Integrals is finite [17]. Integration-By-Parts identities [18] give us such non-trivial recurrence relations and Laporta’s algorithm [19] is widely used to obtain Master Integrals. In this doctoral thesis, we have two main goals. On the one hand, we perform direct computations of a specific physical process with presently available techniques, e.g., the Soft-Collinear Effective Theory [20], which let us compute accurate predictions to be compared with experimental data of LHC. On the other hand, we aim to understand the underlying mathematical structures of scattering amplitudes, with the ultimate scope of possibly developing new tools for these challenging computations. Toward these goals, we face various aspects and mathematical structures that appear in higher-order calculations of scattering amplitudes. We want to observe here that many sophisticated mathematical techniques, which are relevant in high energy physics, were developed initially in different branches of research, like pure mathematics or computer science: algebraic geometry, differential geometry, topology, graph theory, finite fields, D-module, Intersection Theory, homology theory. These are a few examples which are relevant for this thesis. It is, of course, not a new feature in theoretical physics. By looking back at the physics of the last century, typical examples of such attractive intersections between mathematics and physics CHAPTER 1. INTRODUCTION 5 are common: group and representation theory within elementary particle physics, linear algebra within quantum mechanics, pseudo-Riemannian geometry within the general theory of relativity. Several nontrivial connections among graph theory, algebraic geometry, and topology were indeed discovered within the contexts of Feynman integrals. Euler characteristics of a graph-polynomial, an essential topological invariant, provides information and different accesses to the number of the Master Integrals [21]. Recently, Intersection Theory [22, 23] was introduced in the study of Feynman Integrals with the work of Mastrolia and Mizera [24–27]. Through this doctoral thesis, we examine a few of such crossovers within the context of Feynman integrals and their various representations and reductions. The outline of this doctoral theses is the following. In §2, we present a high-precision calculation of the associated production of a top pair and the Z boson at the LHC. This process allows us to study the coupling of the Z boson and top quarks. In this calculation, we use resummation techniques to extract the bulk of next-to-next-to-leading order corrections using Soft-Collinear Effective Theory. When we have two or more energy scales, in theory, traditional fixed order perturbative calculations are not reliable since they acquire significant contributions from logarithms of the ratios between such scales. Effective field theories enable us to separate low energy contributions from high energy parts. With these techniques, we evaluate the total cross section and differential distributions of ttZ¯ production at next-to-next-to-leading logarithmic accuracy [28], which provides a large portion of the next-to-next-to-leading corrections. Our estimations are currently the most accurate theoretical results for those processes. In §3, we introduce Feynman integrals in momentum space and show in detail how they can be written in different parametrizations, namely Schwinger parametrization [29], Feynman parametrization [30], Lee-Pomeransky parametrization [31], and Baikov parametrization [32]. Two graph polynomials U and F characterize Schwinger, Feynman, and Lee-Pomeransky parametrizations; this feature allows us to explore some of their symmetries easily. For example, when a Feynman graph has swap-symmetry of two internal lines, the graph polynomials are symmetric under this swap. CHAPTER 1. INTRODUCTION 6

Therefore we can detect its symmetry algebraically. These parametrizations also allow us to gener- alize the time-space dimension from four to d explicitly. In our work, we define graph polynomials by completing the square in the momentum parametrization and present an implementation of a graph polynomial generator in Maxima [33]. We also comment on an alternative and equivalent definition, which makes use of graph-theoretical concepts only [34, 35]. We then introduce Integration-By-Parts identities from the translational invariance of a generic scalar integral, following [36]. We show their Lie-algebraic structures, following [37], and provide a sketch of Laporta’s algorithm reduction flow. We finally introduce the Baikov parametrization and its mathematical properties. In §4 and §5, we focus two novel applications of Schwinger-Feynman parametrizations. We start §4 from a procedure to generate a system of linear equations among Schwinger-Feynman parameters. In Lee-Pomeransky parametrization, a unique polynomial G = U+F characterizes Feyn- man integrals, where U and F are two graph polynomials. Identities are obtained by computing an integral in two different ways, where an integral consists of a monomial in Schwinger-Feynman parameters times the polynomial G under a partial derivative in one of the parameters. This technique is applied to several examples of increasing complexity and confirm that all the identities we generate within this approach are equivalent to that of traditional Integration-By-Parts Identities. We then show a prescription which gives an interpretation of ill-defined integrals in Schwinger-Feynman parametrizations, which correspond zero-powers in denominators. We discuss this interpretation under both graph theory and complex analysis. We also perform a direct reduction over Schwinger- Feynman parameters, which gives us a full reduction and a set of the Master Integrals. They are exactly the same as those of traditional approaches yield. In §5, we briefly introduce the main concepts of Intersection Theory [38, 39], which are re- quired by our calculations. The use of Intersection Theory in the study of Feynman integrals has been introduced very recently by Mastrolia and Mizera in [24]. As shown in [25–27], this new approaches allows us not only to compute the overall number of Master Integrals but also provide CHAPTER 1. INTRODUCTION 7 the correlations between two integrals, thus opening a new path for the reduction to Master Inte- grals. Within this framework, we first show explicitly, using the gamma function, how to apply Intersection Numbers to study the recursion relations. We then apply the technique to Feynman integrals in Schwinger-Feynman parametrizations, in the case of tadpole and bubble integrals and verify that they agree with traditional Integration-By-Parts approaches. Conclusions and an outlook of possible future developments are presented in §5. During our doctoral studies, we worked on a few on additional topics which will not be parts of the body of this thesis,1 however, are worth mentioning. We studied algebraic geometry and some of its applications to integrand reduction; we learned how to perform functional reconstructions with finite fields, and made our implementations for functional reconstruction in Haskell [45, 46]; finally, we worked within the context of Adaptive Integrand Decomposition Algorithm [47] to implement an interface for Kira [48] reduction routines.

1 Our work on these subjects is contained in a series of lecture notes: In [40] we provide an overview on algebraic geometry, especially on Gröbner basis. In [41] we present an introduction to functional reconstructions with finite fields, discuss in details Newton’s and Thiele’s interpolation techniques, both in the univariate and multivariate cases. Concerning adaptive integrand decomposition algorithm, a summary of the status of the project can be found in this proceedings [42]. For better understanding the novel formulations of Intersection Numbers, we studied homology theory, following [43], and prepared introductory notes [44], which covers abelian groups and their structure theorem, homologies, manifolds, and de Rham cohomologies. Chapter 2

Associated Production of a Top Pair and a Z boson at the LHC to NNLL Accuracy

In this chapter, we study the resummation of soft gluon emission corrections to the production of a top-antitop pair in association with a Z boson at the Large Hadron Collider to next-to-next-to- leading logarithmic accuracy. By means of an in-house parton level Monte Carlo code we evaluate the resummation formula for the total cross section and several differential distributions at a center- of-mass energy of 13 TeV, and we match these calculations to next-to-leading order results. We follow closely the presentation in [28].

2.1 Introduction

The associated production of a top pair and a Z or W boson are among the heaviest final states measured to date at the Large Hadron Collider (LHC). The total cross section for these processes was measured during Run I [1, 49], and preliminary measurements at a center-of-mass energy of 13 TeV are also available [2, 50].1 The ttZ¯ production process is particularly interesting because it

1 Updated experimental data at 13 TeV are available in [4, 5].

8 CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 9 allows us to study the coupling of the Z boson with the top quark. This measurement further tests the Standard Model (SM) of particle physics and probes several Beyond the SM scenarios that predict changes to this coupling with respect to the SM. In addition, these production processes lead to high multiplicity final states which are background in the search for new heavy states decaying via long chains, such as dark matter candidates. Given their importance for phenomenological studies, next-to-leading-order (NLO) QCD and electroweak corrections to the associated production of a top pair and a massive vector boson were studied by several groups [51–58]. A full calculation of the QCD corrections to next-to-next-to leading order (NNLO) accuracy would be desirable but it is extremely difficult even with the most up to date techniques for the calculations of higher order corrections. However, the associated production of a top pair and a heavy colorless boson is a multiscale process which is expected to receive potentially large corrections arising from soft gluon emission. The resummation of these effects to next-to-next-to leading logarithmic (NNLL) accuracy can be carried out by exploiting the factorization properties of the partonic cross section in the soft limit (which can be studied with effective field theory methods, for an introduction see [20]) and by subsequently employing renormalization group improved perturbation theory techniques. In the case of the associated production of a top pair and a Higgs boson the resummation formula in the soft emission limit was discussed in [59], and results for the total cross section and several differential distributions at NLO+NNLL accuracy were presented in [60]. Studies of the associated production of a top quark pair and a W boson to NLO+NNLL accuracy an be found in [61], where the resummation was carried out in Mellin moment space as in [60], and in [62], where the resummation was instead carried out in momentum space. The results of [60] and [61] were obtained by means of an in-house parton level Monte Carlo code for the numerical evaluation of the resummation formula. The output of this code was then matched to complete NLO calculations obtained by employing MadGraph5_aMC@NLO [63]. Build- ing on the results of those two papers, in this chapter we obtain a resummation formula for the CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 10 associated production of ttZ¯ final state, and we evaluate it to NNLL accuracy by means of dedi- cated parton level Monte Carlo code. We match our results for the total cross section and differential distributions to NLO calculations in order to obtain predictions at NLO+NNLL accuracy. The chapter is organized as follows: in §2.2 we introduce some basic notation and we briefly summarize the main steps in our calculations. For a more technical discussion of the methods employed in this chapter, we refer the reader to the detailed descriptions provided in [59–61]. In §2.3 we present predictions at NLO+NNLL accuracy for the total cross section as well as for several differential distributions. Finally, we draw our conclusions in §2.4.

2.2 Outline of the Calculation

The associated production of a top quark pair and a Z boson receives contributions from the partonic process

i(p1) + j(p2) −→ t(p3) + t¯(p4) + Z(p5) + X, (2.1) where ij ∈ {qq,¯ qq,¯ gg} at lowest order in QCD, and X indicates the unobserved partonic final-state radiation. The two Mandelstam invariants which are relevant for our discussion are

2 2 2 sˆ = (p1 + p2) = 2p1 · p2 , and M = (p3 + p4 + p5) . (2.2)

The soft or partonic threshold limit is defined as the kinematic region in which z ≡ M 2/ˆs → 1. In this region, the final state radiation indicated by X in eq.(2.1) can only be soft. The factorization formula for the QCD cross section in the partonic threshold limit is the same as the one derived in [59] for ttH¯ production, up to the straightforward replacement of the Higgs CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 11 boson with a Z boson:

1 Z 1 Z 1 dz X τ σ (s, m , m ) = dτ √ ff , µ t Z 2s z ij z τmin τ ij Z M(1 − z) × dPS ¯ Tr H ({p}, µ) S √ , {p}, µ . (2.3) ttZ ij ij z

We indicated with s the square of the hadronic center-of-mass energy and we defined τmin =

2 2 (2mt + mZ ) /s and τ = M /s. The notation adopted for the channel dependent hard functions H, soft functions S, and luminosity functions ff, as well as for the final-state phase-space integration measure, is the same one used in [60, 61] andwe refer the reader to these papers for more details. Similarly to LO, the only subprocesses to be considered in the soft limit are those la- beled by indices ij ∈ {qq,¯ qq,¯ gg}. The hard and soft functions are two-by-two matrices in color space for qq¯-initiated (quark annihilation) processes, and three-by-three matrices in color space for gg-initiated (gluon fusion) processes. Contributions from other production channels such as qg¯ and qg (collectively referred to as “quark-gluon” or simply “qg” channel in what follows) are subleading in the soft limit. The hard functions satisfy renormalization group equations governed by the

ij channel dependent soft anomalous dimension matrices H . These anomalous dimension matrices were derived in [64, 65]. In order to carry out the resummation to NNLL accuracy, the hard functions, soft functions, and soft anomalous dimensions must be computed in fixed-order perturbation theory up to NLO in αs. The NLO soft functions and soft anomalous dimensions are the same ones needed in the calculation of ttH¯ and ttW¯ ± to NNLL accuracy and can be found in [59–61]. The NLO hard functions are instead process dependent, receive contributions exclusively from the virtual corrections to the tree level amplitudes, and were evaluated by customizing the one-loop provider Openloops [66], which we used in combination with the tensor reduction library Collier [67–70]. The NLO hard function have been cross-checked numerically by means of a customized version of GoSam [71–74], used in combination with the reduction provided by Ninja [75–77]. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 12

In this chapter, we carry out the resummation in Mellin space, starting from the relation

Z 1 Z c+i∞ Z 1 dτ 1 −N X σ(s, mt, mZ ) = dNτ eff (N, µ) dPSttZ¯ cij (N, µ) , (2.4) 2s τ 2πi ij e τmin c−i∞ ij where eff ij is the Mellin transform of the luminosity functions, and ec is the Mellin transform of the product of the hard and soft function (see [60, 61] for details). Since the soft limit z → 1 corresponds to the limit N → ∞ in Mellin space, we neglected terms suppressed by powers of 1/N in the integrand of eq.(2.4).

The hard and soft functions included in the hard scattering kernels ec in eq.(2.4) can be evaluated in fixed order perturbation theory at scales at which they are free from large logarithms. We indicate these scales with µh and µs, respectively. Subsequently, by solving the renormalization group (RG) equations for the hard and soft functions we can evolve the factor ec to the factorization scale µf . Following this procedure we find

" ¯ † ¯ ecij(N, µf ) = Tr Ueij(N, {p}, µf , µh, µs) Hij({p}, µh) Ueij(N, {p}, µf , µh, µs) # M 2 × s ln , {p}, µ , (2.5) eij ¯ 2 2 s N µs

¯ γE where N = Ne . Large logarithmic corrections depending on the ratio of the scales µh and µs are resummed in the channel-dependent matrix-valued evolution factors Ue. The expression for the evolution factors is formally identical to the we found for ttW¯ and ttH¯ production and can be found for example in eq.(3.7) of [61].

The left-hand side of eq.(2.5) is formally independent of µh and µs. In practice however, we cannot evaluate the hard and soft functions at all orders in perturbation theory; this fact creates a residual dependence on the choice of the scales µh and µs in any numerical evaluation of ec. The ¯ hard and soft functions are free from large logarithms if we choose µh ∼ M and µs ∼ M/N. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 13

MW 80.385 GeV mt 173.2 GeV

MZ 91.1876 GeV mH 125 GeV

1/α 137.036 αs (MZ ) from MMHT 2014 PDFs

Table 2.1: Input parameters employed throughout the calculation.

The choice of a N-dependent value for µs produces a branch cut for large values of N in the hard scattering kernels ec, whose existence is related to the Landau pole in αs. We deal with this issue by choosing the integration path in the complex N plane according to the Minimal Prescription (MP) introduced in [78]. Finally, the parton luminosity functions in Mellin space, which we need in the numerical evaluations, are constructed using techniques described in [79, 80].

2.3 Numerical Results

The main purpose of this section is to present predictions for the associated production of a top pair and a Z boson to NLO+NNLL accuracy. However, we also analyze systematically the relevance of soft emission corrections and their resummation in relation to NLO predictions for the various observables considered in the chapter. The NNLL calculations are carried out by means of an in-house parton level Monte Carlo code, while the NLO predictions are obtained by means of

MadGraph5_aMC@NLO. All of the calculations discussed in this section are carried out with the input parameters listed in Table 2.1. Throughout the chapter we employ MMHT 2014 PDFs [81]. In fixed order calculations, the order of the PDFs matches the perturbative order of the calculation, i.e., LO calculations are carried out with LO PDFs, NLO calculations employ NLO PDFs, etc. In matched calculations, we employed NLO PDFs for NLO+NLL accuracy, and NNLO PDFs for NLO+NNLL accuracy. For both the total cross section and several differential distributions, we consider six different CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 14 types of predictions:

i) NLO calculations, obtained with MadGraph5_aMC@NLO.

ii) Approximate NLO calculations, obtained from the NLO expansion of the NNLL resummation formula. We check that for our choice of scales and input parameters approximate NLO calculations provide a satisfactory approximation to the exact NLO calculation. The approximate NLO formulas obtained by expanding eq.(2.5) account for the single and double powers of ln N as well as N-independent terms but not for terms suppressed by inverse powers of N. N-independent terms depend on the Mandelstam variables, however we refer to them as “constant” terms in what follows. The approximate NLO formulas are obtained by setting

µh = µs = µf in the NNLL version of eq.(2.5). Approximate NLO calculations are carried out with the in-house parton level Monte Carlo code which was developed specifically for this project.

iii) NLO+NLL calculations, which are obtained by matching NLO results with resummed results at NLL accuracy obtained by means of the in-house Monte Carlo code. The results are matched according to the formula

σNLO+NLL =σNLO + σNLL − σNLL expanded to NLO . (2.6)

The terms in the square brackets, which contribute to NLO and beyond, depend on the scales

µs and µh in addition to the factorization scale µf . Of course the dependence on µs and µh is formally of NNLL order; by varying these scales and the factorization scale in eq.(2.6) we can estimate the size of the NNLL corrections.

iv) NLO+NNLL predictions are obtained by evaluating the hard scattering kernels in eq.(2.5) to NNLL accuracy with the in-house Monte Carlo code and by matching the results to NLO CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 15

calculations as follows:

σNLO+NNLL =σNLO + σNNLL − σapprox. NLO . (2.7)

The terms in the squared brackets in eq.(2.7) contribute starting from NNLO and represent the NNLL corrections to be added to the NLO result.

v) Approximate NNLO calculations are obtained by the NNLL resummation formula and include all powers of ln N and part of the constant terms from a complete NNLO calculation. The approximate NNLO formulas employed in this chapter are constructed as the ones employed in [60, 61] for ttW¯ and ttH¯ production. A detailed description of the constant terms which are included in the approximate NNLO formulas can be found in §4 of [59]. Approxi- mate NNLO formulas are evaluated with the in-house Monte Carlo code which we developed and they are matched to the NLO calculations as follows

σnNLO = σNLO + σapprox. NNLO − σapprox. NLO , (2.8)

where we label the matched result “nNLO” for brevity. By construction nNLO predictions are independent from the hard and soft scales but they do have a residual N3LO dependence

on µf .

vi) NLO+NNLL expanded to NNLO. Finally we consider a second way of expanding the NNLL resummation formula to NNLO. This approach differs from the approximate NNLO result used above by constant terms, which are formally of N3LL accuracy. This approximation is defined by the relation

σNLO+NNLL = σNLO + σNNLL expanded to NNLO − σapprox. NLO . NNLO exp. (2.9) CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 16

1200

1000

800 NLO NLO+NLL

σ[fb] NLO+NNLL 600 nNLO

0.1 0.2 0.5 1 2

μf /M

¯ Figure√ 2.1: Factorization-scale dependence of the total ttZ production cross section at the LHC with s = 13 TeV. The NLO and NLO+NLL curves are obtained using MMHT 2014 NLO PDFs, while the NLO+NNLL and nNLO curves are obtained using MMHT 2014 NNLO PDFs.

The constant pieces in eq.(2.9) contain explicit dependence on µh and µs, in addition to that

3 on µf . This dependence is formally an effect of N LL order. By comparing the predictions obtained from eq.(2.9) to the corresponding NLO+NNLL calculations we can see the relative weight of terms of N3LO and higher in the NLO+NNLL calculations. If in the future a complete NNLO calculation for the ttZ¯ production cross section were to become available, it would be possible to match it to the NNLL resummation formula by using precisely this kind of NNLO expansion of the NNLL resummation, as can be seen by replacing N → NN in all of the superscripts in eq.(2.6).

2.3.1 Scale Choices

Since any numerical evaluation of the resummed expression for the hard scattering kernels must be carried out by evaluating the factors in eq.(2.5) up to a certain order in perturbation theory, the resummed kernels ec will show a residual dependence on the scales µs and µh. In order to follow closely the approach adopted in “direct QCD” calculations [78, 82, 83], the standard choice which ¯ we adopt in this work for the hard and soft scales is µh,0 = M and µs,0 = M/N [60, 61, 84, 85].

In addition, both fixed order and resummed calculations depend on the factorization scale µf , CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 17

which should be chosen in such a way that the logarithms of the scale ratio µf /M are not large [86].

It is therefore reasonable to choose a dynamical default value for the scale µf which is related to M. √ The dependence of the total ttZ¯ production cross section on the ratio µf /M at the LHC with s = 13 TeV is shown in Figure 2.1. Each line corresponds to a different perturbative approximation, as indicated in the legend. Figure 2.1 shows that the NLO, NLO+NLL and NLO+NNLL curves intersect in the vicinity of µf /M = 0.5 and differ significantly for µf /M 0.5 and for µf /M 0.5. Following this observation, the default value that we employ for the factorization scale is

µf,0 = M/2. The uncertainty related to the choice of the factorization scale in fixed order results is estimated as usual by varying this scale in the range µf ∈ [µf,0/2, 2µf,0]. Resummed results depend also on the hard and soft scales, consequently, the uncertainty of the resummed results is estimated by varying separately all the three scales around their default values in the interval µi ∈ [µi,0/2, 2µi,0] for i ∈ {s, f, h}. The scale uncertainty above (below) the central value of a resummed observable O, which can be the total cross section or the value of the differential cross section in a given bin, is determined as follows. First we evaluate the quantities

+ ¯ ∆Oi = max{O (κi = 1/2) ,O (κi = 1) ,O (κi = 2)} − O,

− ¯ ∆Oi = min{O (κi = 1/2) ,O (κi = 1) ,O (κi = 2)} − O, (2.10)

¯ for i ∈ {s, f, h}. In eq.(2.10) we defined κi = µi/µi,0, and O indicates the observable evaluated ¯ at κi = 1 for all i-s. The scale uncertainty above (below) O is then obtained by combining in

+ − quadrature ∆Oi (∆Oi ) for i ∈ {s, f, h}.

2.3.2 Total Cross Section

In this section, we analyze the total cross section for the associated production of a top quark pair and a Z boson at the LHC operating at a center-of-mass energy of 13 TeV. The relevant results are CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 18

order PDF order code σ [fb] +165.4 LO LO MadGraph5_aMC@NLO 521.4−116.9

+38.5 app. NLO NLO in-house MC 737.7−64.5 +41.8 NLO no qg NLO MadGraph5_aMC@NLO 730.4−64.9 +93.8 NLO NLO MadGraph5_aMC@NLO 728.3−90.3 +90.1 NLO+NLL NLO in-house MC +MadGraph5_aMC@NLO 742.0−30.3 +61.3 NLO+NNLL NNLO in-house MC +MadGraph5_aMC@NLO 777.8−65.2 +36.2 nNLO NNLO in-house MC +MadGraph5_aMC@NLO 798.7−23.6 +17.2 (NLO+NNLL)NNLO exp. NNLO in-house MC +MadGraph5_aMC@NLO 766.2−50.1 √ Table 2.2: Total cross section for ttZ¯ production at the LHC with s = 13 TeV and MMHT 2014 PDFs. The default value of the factorization scale is µf,0 = M/2, and the uncertainties are estimated through variations of this scale (and of the hard and soft scales µs and µh when applicable), as explained in the text.

collected in Table 2.2. We first compare the approximate NLO cross section, obtained by expanding the resummation formula to NLO (second row of Table 2.2) with the complete NLO cross section (fourth row) and the NLO cross section without the contribution of the quark-gluon channel (third row). The difference between the approximate NLO result and the NLO result without the qg channel is due to terms in the quark annihilation and gluon fusion channels which are subleading in the partonic threshold limit. We see that the impact of these terms is around 1%. The difference between these two results is therefore small in spite of the fact that the NLO corrections are large, as can be seen by comparing them with the LO result. However, we see that the approximate NLO result shows a smaller scale uncertainty than the NLO result with the contribution of the qg channel. We conclude that the soft emission corrections provide the bulk of the NLO corrections for this choice of the factorization scale. This motivates us to study the effect of the resummation of these corrections, keeping in mind that by matching the resummed results to NLO calculations we consider both power corrections and the contribution of the qg channel to that order. The NLO+NLLand NLO+NNLLcross sections, shown in the sixth and seventh line of Table 2.2 CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 19

√ s and pert. order process σ [fb] ¯ + +15.6 8 TeV NLO ttW 136.7−15.2 ¯ − +7.1 8 TeV NLO ttW 60.5−6.8 ¯ +24.5 8 TeV NLO ttZ 189.8−24.8 ¯ + +6.9 8 TeV NLO+NNLL ttW 130.7−4.9 ¯ − +3.1 8 TeV NLO+NNLL ttW 59.1−2.2 ¯ +13.5 8 TeV NLO+NNLL ttZ 203.9−15.8 ¯ + +43.7 13 TeV NLO ttW 356.3−39.5 ¯ − +23.1 13 TeV NLO ttW 182.2−20.4 ¯ +93.8 13 TeV NLO ttZ 728.3−90.3 ¯ + +23.1 13 TeV NLO+NNLL ttW 341.0−13.6 ¯ − +12.0 13 TeV NLO+NNLL ttW 177.1−6.9 ¯ +61.3 13 TeV NLO+NNLL ttZ 777.8−65.2 √ Table 2.3: Total cross section for ttZ¯ and ttW¯ production at the LHC with s = 8 and 13 TeV and MMHT 2014 PDFs. The default value of the factorization scale is µf,0 = M/2, and the uncertainties are estimated through variations of this scale (and of the resummation scales µs and µh when applicable).

are main results of this chapter. By looking at the NLO, NLO+NLL, NLO+NNLL results we see that the cross section is progressively increased, but the central value of each prediction falls in the scale uncertainty band of the predictions of lower accuracy. We can then look at the NNLO expansions of the NNLL resummation formula, which are shown in the last two lines of Table 2.2. By comparing these results to the NLO+NNLL cross section, we see that the effect of the resummation corrections beyond NNLO are relatively small. As it was observed in the case of the ttH¯ and ttW¯ processes in [59–61], the scale uncertainty affecting the nNLO result is very small compared to the NLO+NNLL scale uncertainty, and most likely underestimates the residual perturbative uncertainty at NNLO. Experimental collaborations reported measurements of the ttZ¯ total cross section in combination with measurements of the ttW¯ cross section [1, 2, 49, 50]. We conclude this section by CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 20

350 1200

300 1000 250

[fb] 800 200 [fb] ttZ ttZ 150 600 σ σ

100 400 50 200 300 400 500 400 600 800 1000 1200 1400 σttW [fb] σttW [fb]

Figure 2.2: Total cross section at NLO (Green) and NLO+NNLL (Red) compared to the ATLAS measurements at 8 TeV [1] (left panel) and CMS measurement at 13 TeV [2] (right panel). comparing our predictions for ttW¯ and ttZ¯ with experimental data. The ttW¯ production cross section was evaluated by running the code developed in [61] with the scale choices and input parameters employed in the present work for ttZ¯ production and described above. The results for ttZ¯ and ttW¯ production cross section at 8 and 13 TeV are summarized in Table 2.3. In Figure 2.2 we follow the structure of Figure 12 in [2] in order to compare graphically calculations with the corresponding experimental measurements. The experimental measurements at 8 TeV are taken from [1], while the experimental measurements at 13 TeV are taken from [2]. The green dots and cross-shaped “error bars” correspond to NLO calculations carried out with µf,0 = M/2 and their scale uncertainty. The red dots and crosses correspond instead to NLO+NNLL calculations. It is interesting to observe that, while predictions for the ttZ¯ production cross section are in perfect agreement with the measurements at both 8 and 13 TeV, the predictions for the ttW¯ cross section are slightly smaller than measurements for both collider energies. This observation holds for NLO and NLO+NNLL calculations alike. Of course this discrepancy should be taken with a grain of salt, and requires a more detailed discussion with the experimental collaborations. Moreover, we would like to stress that a fully exhaustive comparison between predictions and measurements should also account for the uncertainty associated to the choice of the PDFs and to the value of αs. These two sources of uncertainty are not reflected in the error bars of Figure 2.2. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 21

1400 SM theory vs ATLAS data 1400 SM theory vs CMS data LHC 13 TeV LHC 13 TeV 1200 1200

[fb] 1000 [fb] 1000 ttZ ttZ

σ 800 σ 800

600 600

400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 σttW [fb] σttW [fb]

Figure 2.3: Total cross section at NLO (Green) and NLO+NNLL (Red), which include NLO electroweak effects [3], compared to the ATLAS measurements [4] at 13 TeV (left panel) and CMS measurement [5] at 13 TeV (right panel).

We have updated the analyses of Figure 2.2, by using the more recent data for the total cross sections at 13 TeV from both ATLAS [4] and CMS [5], and by including, on the side of theoretical predictions, the NLO electroweak contributions [3]. The updated analyses are presented in Figure 2.3. Predictions for both ttZ¯ and ttW¯ cross sections are in good agreement with the experimental data, and the potential tension which was observed in Figure 2.2 for the ttW¯ cross section is strongly reduced with the most recent data.

2.3.3 Differential Distributions

In this section, we obtain predictions for four differential distributions which depend on the momenta of the final state massive particles. The distributions are i) the distribution differential with respect to the ttZ¯ invariant mass, M, ii) the distribution differential with respect to the tt¯ invariant mass, Mtt¯, iii) the distribution differential with respect to the transverse momentum of the top

t quark, pT , and iv) the distribution differential with respect to the transverse momentum of the Z

Z boson, pT . Figure 2.4 compares the approximate NLO calculations, carried out with our in-house code, with the complete NLO calculations, carried out with MadGraph5_aMC@NLO. We see that the ap- CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 22

140 NLO 160 NLO 120 app. NLO 140 app. NLO 120 100 ttZ ttZ 100 80 LHC 13 TeV LHC 13 TeV µ = M/2 µ = M/2 f,0 80 f,0 60 60 40 40

cross section per bin [fb] 20 cross section per bin [fb] 20

1.2 1.2 1.1 1.1 1.0 1.0

Ratio to NLO 0.9 Ratio to NLO 0.9 0.8 0.8 500 600 700 800 900 1000 1100 400 500 600 700 800 M (GeV) M (GeV) tt

220 200 NLO 250 NLO 180 app. NLO app. NLO 160 200 140 ttZ ttZ 120 LHC 13 TeV 150 LHC 13 TeV µ = M/2 µ = M/2 100 f,0 f,0 80 100 60 40 50 cross section per bin [fb] 20 cross section per bin [fb]

1.2 1.2 1.1 1.1 1.0 1.0

Ratio to NLO 0.9 Ratio to NLO 0.9 0.8 0.8 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 pt (GeV) pZ (GeV) T T

Figure 2.4: Differential distributions at approximate NLO (blue band) compared to the complete NLO (red band). The default factorization scale is chosen as µf,0 = M/2, and the uncertainty bands are generated through scale variations as explained in the text. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 23

140 NLO w/o qg 160 NLO w/o qg 120 app. NLO 140 app. NLO 120 100 ttZ ttZ 100 80 LHC 13 TeV LHC 13 TeV µ = M/2 µ = M/2 f,0 80 f,0 60 60 40 40

cross section per bin [fb] 20 cross section per bin [fb] 20

1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.8 0.8 500 600 700 800 900 1000 1100 400 500 600 700 800 Ratio to NLO w/o qg Ratio to NLO w/o qg M (GeV) M (GeV) tt

220 200 NLO w/o qg 250 NLO w/o qg 180 app. NLO app. NLO 160 200 140 ttZ ttZ 120 LHC 13 TeV 150 LHC 13 TeV µ = M/2 µ = M/2 100 f,0 f,0 80 100 60 40 50 cross section per bin [fb] 20 cross section per bin [fb] 1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.8 0.8 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Ratio to NLO w/o qg pt (GeV) Ratio to NLO w/o qg pZ (GeV) T T

Figure 2.5: Differential distributions at approximate NLO (blue band) compared to the NLO distributions without the quark-gluon channel contribution (red band). All settings are as in Figure 2.4. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 24 proximate NLO calculations reproduce well the full NLO calculations. The lower part of each panel shows the ratio between the approximate NLO or complete NLO calculations and the central value of the NLO calculation. We can see that the approximate NLO scale uncertainty band is included in the NLO scale uncertainty band. Figure 2.5 repeats the same analysis but it compares approximate NLO calculations to NLO calculations without the quark-gluon channel contribution. As expected approximate NLO distributions and NLO distributions without the qg channel have the same shape and scale uncertainty bands of similar size. These two figures show that, for this choice of the factorization scale at least, soft emission corrections provide the bulk of the NLO corrections. Figure 2.6 provides the main result of this section. This figure compares NLO calculations to the distributions evaluated to NLO+NNLL accuracy. Roughly, We can say that the NLO+NNLL results fall in the upper part of the NLO scale uncertainty interval in each bin. The central value of the NLO+NNLL calculations is slightly larger than the central value of the NLO calculations in all bins shown. The scale uncertainty affecting the NLO+NNLL accuracy calculation, which is obtained by varying µs, µf , and µh as described above, is smaller than the NLO scale uncertainty band obtained by varying µf . Results at NLO+NLL and NLO+NNLL accuracy are compared in Figure 2.7. The main effect of the NNLL correction with respect to the NLL ones is an increase of the central value of the bins in the tail of the M and Mtt¯ distributions. The scale uncertainty bands turn out to be of similar size at NLO+NLL and NLO+NNLL in almost all bins shown. Figure 2.8 shows the ratio of distributions at various level of precision to the central value of the NLO+NNLL calculation in each bin. In particular, the blue band refers to NLO+NNLL distributions, the dashed red band to nNLO distributions and the dashed black band to distributions obtained from the NNLO expansion of the NLO+NNLL resummation. The NLO+NNLL expanded distributions differ from the NLO+NNLL distributions by NNLL resummation effects of order N3LO and higher. These corrections can be as large as 5 to 10 % in all bins shown, and are particularly relevant at higher values of µf . The difference between the nNLO and the NLO+NNLL expanded to CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 25

140 NLO+NNLL 160 NLO+NNLL 120 NLO 140 NLO 120 100 ttZ ttZ 100 80 LHC 13 TeV LHC 13 TeV µ = M/2 µ = M/2 f,0 80 f,0 60 60 40 40

cross section per bin [fb] 20 cross section per bin [fb] 20

1.2 1.2

1.0 1.0

0.8 0.8

0.6 0.6 500 600 700 800 900 1000 1100 400 500 600 700 800 Ratio to NLO+NNLL Ratio to NLO+NNLL M (GeV) M (GeV) tt

220 200 NLO+NNLL 250 NLO+NNLL 180 NLO NLO 160 200 140 ttZ ttZ 120 LHC 13 TeV 150 LHC 13 TeV µ = M/2 µ = M/2 100 f,0 f,0 80 100 60 40 50 cross section per bin [fb] 20 cross section per bin [fb] 1.2 1.2

1.0 1.0

0.8 0.8

0.6 0.6 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Ratio to NLO+NNLL Ratio to NLO+NNLL pt (GeV) pZ (GeV) T T

Figure 2.6: Differential distributions with µf,0 = M/2 at NLO+NNLL (blue band) compared to the NLO calculation (red band). The uncertainty bands are generated through scale variations of µf , µs and µh as explained in the text. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 26

140 NLO+NNLL 160 NLO+NNLL 120 NLO+NLL 140 NLO+NLL 120 100 ttZ ttZ 100 80 LHC 13 TeV LHC 13 TeV µ = M/2 µ = M/2 f,0 80 f,0 60 60 40 40

cross section per bin [fb] 20 cross section per bin [fb] 20

1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.8 0.8 500 600 700 800 900 1000 1100 400 500 600 700 800 Ratio to NLO+NNLL Ratio to NLO+NNLL M (GeV) M (GeV) tt

220 200 NLO+NNLL 250 NLO+NNLL 180 NLO+NLL NLO+NLL 160 200 140 ttZ ttZ 120 LHC 13 TeV 150 LHC 13 TeV µ = M/2 µ = M/2 100 f,0 f,0 80 100 60 40 50 cross section per bin [fb] 20 cross section per bin [fb] 1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.8 0.8 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Ratio to NLO+NNLL Ratio to NLO+NNLL pt (GeV) pZ (GeV) T T

Figure 2.7: Differential distributions with µf,0 = M/2 at NLO+NNLL (blue band) compared to the corresponding NLO+NLL calculation (red band). The uncertainty bands are generated through scale variations. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 27

1.20 1.20 ttZ (NLO+NNLL)_NNLOexp. ttZ (NLO+NNLL)_NNLOexp. LHC 13 TeV nNLO LHC 13 TeV nNLO 1.15 µ = M/2 1.15 µ = M/2 f,0 NLO+NNLL f,0 NLO+NNLL

1.10 1.10

Ratio to NLO+NNLL 1.05 Ratio to NLO+NNLL 1.05

1.00 1.00

0.95 0.95

0.90 0.90

0.85 0.85 500 600 700 800 900 1000 1100 400 500 600 700 800

M (GeV) M (GeV) tt

ttZ (NLO+NNLL)_NNLOexp. 1.15 ttZ (NLO+NNLL)_NNLOexp. 1.15 LHC 13 TeV nNLO LHC 13 TeV nNLO µ = M/2 NLO+NNLL µ = M/2 NLO+NNLL f,0 1.10 f,0 1.10

1.05 1.05 Ratio to NLO+NNLL Ratio to NLO+NNLL

1.00 1.00

0.95 0.95

0.90 0.90

0.85 0.85 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350

pt (GeV) pZ (GeV) T T

Figure 2.8: Differential distributions ratios for µf,0 = M/2, where the uncertainties are generated through scale variations. CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 28

NNLO results is due to constant NNLO terms, which are formally of order N3LL. Both the NNLO expansion of the NLO+NNLL calculation and the nNLO calculation underestimate the scale uncertainty which we find at NLO+NNLL accuracy, a fact which we already observed by looking at the predictions for the total cross section. The envelope of the two NNLO approximations, i.e., the black and red bands, spans almost all of the NLO+NNLL scale uncertainty interval in each bin,

t with the exception of the tail of the pT distribution, where this envelope includes the NLO+NNLL scale uncertainty.

2.4 Conclusions

In this chapter, we carried out the resummation of soft gluon emission corrections to the associated production of a top-antitop quark pair and a Z boson. The resummation was studied in the partonic threshold limit z → 1 and was implemented to NNLL accuracy. Numerical calculations of the total cross section and differential distributions to NNLL accuracy were carried out by means of an in-house partonic Monte Carlo code which we developed for this work. The output of this code was matched with NLO calculations obtained from MadGraph5_aMC@NLO. The final outcome of this work is represented by the NLO+NNLL calculations of the total cross section and differential distributions for the LHC operating at a center-of-mass energy of 13 TeV presented in the previous section. The code can be easily adapted to carry out phenomenological studies which include cuts on the top, antitop and/or Z boson momenta. With the choice of the factorization scale made in this work, we can conclude that the soft emission corrections to ttZ¯ production evaluated to NNLL accuracy lead to a moderate increase of the total cross section and differential distributions with respect to NLO calculations of the same observables. The residual perturbative uncertainty at NLO+NNLL accuracy, estimated by varying the soft, hard and factorization scales as explained in the text, is smaller than the NLO scale uncertainty, thus making our evaluations of the cross sections and differential distributions in ttZ¯ CHAPTER 2. T TZ¯ PRODUCTION AT THE LHC TO NNLL ACCURACY 29 production the most precise results currently available in the literature. This work completes a series of papers devoted to the study of the associated production of a top pair and a colorless heavy boson to NLO+NNLL accuracy in the partonic threshold limit. In [61] the associated production of a top pair and a W boson was studied with the methods employed here for ttZ¯ production, while the associated production of a top pair and a Higgs boson at NLO+NNLL accuracy was considered in [60]. In all cases the resummation was carried out in Mellin moment space. The hard and soft scales were chosen in the same way as in the traditional “direct QCD” approach. Codes for the numerical evaluation of the resummation are now available and tested for all of these three processes, and can be further employed in more specific phenomenological studies, according to the interests of the experimental collaborations. Within such interactions with the experimental community, a detailed study of the uncertainty associated with the choice of the

PDFs and to the value of αs(MZ ), in the light of a comparison with the new measurements which are expected in the forthcoming months, would be particularly illuminating. Chapter 3

Multi-loop Feynman Integrals and Integration-By-Parts Identities

In this chapter, we present different parametrizations for Feynman integrals. We start by defining Feynman integrals over momentum space, which are a direct translation of a given Feynman diagram through the appropriate Feynman rules, and as associated integral, we define scalar integrals. We then introduce different parametrizations, such as Schwinger [29], Feynman [30], Lee- Pomeransky [31], and Baikov [32] and provide all relevant derivations to obtain them. In this thesis, we refer to these parametrizations jointly, namely Schwinger, Feynman, and Lee-Pomeransky, as Schwinger-Feynman parametrizations.1 Although it seems just a matter of choosing different expressions to represent a multidimen- sional integral, the gain is indeed much higher; some properties of Feynman integrals, and more in general, scattering amplitudes, become particularly transparent when we use an appropriate parametrization. For instance, graph-theoretical symmetries become polynomial symmetries over Schwinger-Feynman parametrizations, and Baikov polynomials have a natural geometrical inter-

1 Schwinger parametrization and Feynman parametrization also collectively (and erroneously as mentioned in [87]) referred to in the literature as Feynman parametrizations, alpha parametrizations [88], or simply parametric representation [89].

30 CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 31 pretation as Gram determinants. Another gain of exploring different options is that it becomes more evident that a few polynomials characterize the whole family of integrals; two graph polynomials U and F in the case of Schwinger-Feynman parametrizations, and the Baikov polynomial in Baikov representation. As a further significant and related topic, we also study Integration-By-Parts (IBP) identities [18], which play essential roles in all practical multi-loop calculations in high-energy phenomenology. We first derive the IBP identities from the symmetries that the momentum space have and clarify that the IBP identities as the generator of infinitesimal coordinate translations. We then introduce their algebraic structures, following [37]. The well-known Laporta’s algorithm [19] is a widely used decomposition method toward master integrals (MI); we briefly illustrate the main features of this algorithm. We then introduce Baikov parametrization. Instead of using momenta, Baikov parametrization uses either the set of scalar products or denominators as integration variables. In this parametrization, the IBP identities are a built-in feature; that is, when we use Baikov parametrization with an appropriate integration domain, IBP identities are automatically satisfied. We finally examine in depth this integration domain, which is determined by the zeros of the Gram determinant of Baikov polynomial.

3.1 Multi-loop Feynman Integrals

Let us consider a Feynman graph which has E+1 external legs, L loops, and I internal propagators.2 There are E + 1 legs in a diagram, but we set one condition, the momenta conservation, which reduces the numbers of the independent momenta to E. With Feynman rules, we translate this graph into an integral over d-dimensional Minkowski (−, +, ··· , +) space.3 The following is a

2 We call an internal line a propagator, and an external line a leg. Graph-theoretically, an internal line has two endpoints in a graph, and an external line has just one. 3 A brief introduction to QFT and basic use of Feynman rules contained in Appendix A. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 32 generic form of such a Feynman integral:

Z d Z d d l1 d lL N(l) J = d ··· d , (3.1) (2π) (2π) D1 ··· DI where N(l) is a polynomial of loop momenta, external momenta, and kinematic invariants, e.g. masses of external momenta, and Da stands for the denominator of a Feynman propagator. Let us write the set of momenta that appear in the Feynman integral as

M qi|i=1 ∈ {l1, ··· , lL, e1, ··· , eE},M := L + E, (3.2)

where l1, . . . , lL are the L loop momenta and e1, . . . , eE are the E independent external momenta.

Let sij be the inner products of momenta,

sij := li · qj, 1 ≤ i ≤ L, i ≤ j ≤ M, (3.3)

where qj is either a loop or an external momentum. The number of inner products which depend on the loop momenta l1, . . . , lL is

L(L + 1) N := + LE, (3.4) 2

L(L+1) where 2 is the number of independent elements in a symmetric L × L matrix li · lj and LE is that of li · ek.

While at one loop (L = 1), we can express any scalar product sij in terms of the denominators

D1, ··· ,DI that appear in the graph and kinematic quantities, i.e., masses, external momenta, and vice versa; this is no longer the case for L > 1 (for a more detailed discussions, see [12–14, 16], also [36]). Scalar products that cannot be expressed in such a way are called irreducible scalar products (ISP). CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 33

Given I denominators D1, ··· ,DI in eq.(3.1), we add DI+1 ··· ,DN of auxiliary denominators so that any sij of scalar product can be written as a linear combination of denominators D1, ··· ,DN . After adding auxiliary denominators, we can write

L M X X ij 2 Da = Aa sij + ma − i0. (3.5) i=1 j=i

Note that we identify (i, j) as a single-indexed label, see eq.(3.3) and eq.(3.4), in the same manner

ij as a ∈ {1, ··· ,N}. By inverting the matrix Aa , we obtain

N X −1a 2 sij = A ij Da − ma − i0 , (3.6) a=1 which shows that all scalar products sij can be written in terms of denominators. We can therefore rewrite the original integral in eq.(3.1) as a combination of scalar integrals, in which auxiliary denominators have non-positive powers. Now, instead of the original Feynman integral in eq.(3.1), we focus on the following scalar integral

L Z d ! N Y d lj Y 1 J(~ν) := (3.7) (2π)d Dνa j=1 a=1 a as the main subject of our studies. In the remainder of this thesis, we refer to scalar integrals as defined above in eq.(3.7). We can consider such integral as a meromorphic function of indices ~ν and dimension d, see [90].

3.2 Schwinger-Feynman Parametrizations

In this section, we study the three parameterizations that go under the name of Schwinger-Feynman Parametrizations and present all relevant derivations. We start from the vital identity for Schwinger parametrization, which can be found in [29]. We call this technique Schwinger trick. Instead of CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 34 using Feynman trick [30], we derive Feynman parametrization by inserting the identity inside the integral. Both Schwinger and Feynman parametrizations can be found in classic textbooks [89, 91–93]. More recently, Lee-Pomeransky parametrization was introduced and discussed in [31]. Applications of Schwinger-Feynman parameters can also be found in [21, 94–96].

3.2.1 Schwinger Trick and Feynman Integrals

Let us consider an integral, for t, ν > 0,

Z ∞ T := dxxν−1 exp (−xt). (3.8) x=0

Since t > 0, we can apply x 7→ x0 := xt and

Z ∞ tdx xtν−1 T = exp (−xt) (3.9) x=0 t t Z ∞ 1 0 0ν−1 0 = ν dx x exp (−x ) (3.10) t x0=0 1 = Γ(ν). (3.11) tν

Therefore, we obtain the following formula,

1 Z ∞ dxxν−1 ν = exp (−xt). (3.12) t x=0 Γ(ν)

We call this Schwinger trick. This trick allows us to convert multiplications among denominators into additions inside an exponential. It is known as an example of the twisted Mellin transform [97]. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 35

Let us apply this Schwinger trick to the denominators in eq.(3.7) of our scalar integral:

L Z d ! N Z ∞ Y d lj Y 1 J(~ν) = dx xνa−1 exp (−x D ) (3.13) (2π)d Γ(ν ) a a a a j=1 a=1 a xa=0 N ! L ! N ! Z ∞ νa−1 Z d Y dxax Y d lj X = a exp − x D , (3.14) Γ(ν ) (2π)d a a a=1 xa=0 a j=1 a=1

where we bravely switch the order of integrals. Focus on this exponent; since Da, in eq.(3.5), is a quadratic form of loop momenta:

L L L+E ! X X X ij 2 Da = + Aa sij + ma − i0 (3.15) i=1 j=i j=L+1 L L L E X X ij X X i,L+j 2 = Aa lilj + Aa liej + ma − i0 (3.16) i=1 j=i i=1 j=1 L L E X 1 X X X = Aiil l + Aijl l + Ai,L+jl e + m2 − i0 (3.17) a i i 2 a i j a i j a i=1 1≤i,j≤L,i6=j i=1 j=1 L L L E X X 1 + δij X X = Aijl l + Ai,L+jl e + m2 − i0 , (3.18) 2 a i j a i j a i=1 j=1 i=1 j=1 we can write the exponent as

N L L N ! X X X X 1 + δij x D = l x Aij l a a i a 2 a j a=1 i=1 j=1 a=1 L N E ! N X 1 X X X + 2 l x Ai,L+je + x m2 − i0 (3.19) i 2 a a j a a i=1 a=1 j=1 a=1

= ltAl + 2Btl + C, (3.20) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 36 where we define

N ! X 1 + δij A = x Aij (3.21) ij a 2 a a=1 N E ! 1 X X B = x Ai,L+je (3.22) i 2 a a j a=1 j=1 N X 2 C = xa ma − i0 . (3.23) a=1

Note that A is symmetric, namely At = A. In order to perform momenta integrations, let us complete the square:

N X −1 t −1 t −1 xaDa = l + A B A l + A B + C − B A B. (3.24) a=1

Since the regulated integral is translationally invariant,4 we can perform momenta integrations:

L ! N ! d Z d L L 2 Y d lj X i π exp − x D = exp BtA−1B − C (3.25) (2π)d a a (2π)dL det A j=1 a=1 iL exp (BtA−1B − C) = dL d , (3.26) (4π) 2 (det A) 2 where we perform L-fold Wick rotation, which brings (iL), and dL-fold Gaussian integration, d h πL i 2 which brings det A . Therefore, our scalar integral in eq.(3.7) becomes

N ! L Z ∞ νa−1 t −1 i Y dxaxa exp (B A B − C) J(~ν) = dL d . (3.27) 2 Γ(ν ) 2 (4π) a=1 xa=0 a (det A)

4 See an axiomatic approach to regulated integrals, e.g., §4.1 in [86]. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 37

For later convenience, we define I(~ν) = iLJ(~ν), to obtain

N ! Z ∞ νa−1 t −1 1 Y dxaxa exp (B A B − C) I(~ν) = dL d . (3.28) 2 Γ(ν ) 2 (4π) a=1 xa=0 a (det A)

3.2.2 Graph Polynomials and Schwinger Parametrization

Let us introduce two polynomials, U and F:

U := det Aij (3.29) ij   AB   t ˆ t −1 F := det   = det A ∗ C − B AB = det A ∗ C − B A B , (3.30) Bt C where Aˆ is the adjugate of A, i.e. the transpose of the cofactor matrix of A, which satisfies

AAˆ = (det A)1 = AAˆ . (3.31)

Note that U and F are homogeneous polynomials of Schwinger parameters x1, ··· , xN and their degrees are

L U(ρx1, ··· , ρxN ) = ρ U(x1, ··· , xN ) (3.32)

L+1 F(ρx1, ··· , ρxN ) = ρ F(x1, ··· , xN ) (3.33)

In literature, the definition of F is not unique; our choice of metric system (− + + ··· +) and the

PN t t exponent a=1 xaDa = l Al + 2B l + C clarify F’s homogeneous order explicitly. Following the choice above, we obtain the Schwinger parametrization from eq.(3.28):

N ! Z ∞ νa−1 F 1 Y dxaxa exp − U I(~ν) = dL d . (3.34) 2 Γ(ν ) 2 (4π) a=1 xa=0 a U CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 38

3.2.3 Feynman Parametrization

We now want to obtain the expression for the Feynman parametrization. Let us insert, as in section 6-2-3 of [89],

N ! Z ∞ X 1 = dρδ ρ − xj (3.35) 0 j=1 in the above Schwinger parametrization eq.(3.34):

N ! F(x) Z ∞ νa−1 exp − 1 Y dxaxa U(x) I(~ν) = dL 1 d (3.36) 2 Γ(ν ) 2 (4π) a=1 xa=0 a [U(x)] F ρ x  νa−1  ρ dxa xa ∞ N ∞ N ! exp − x 1 Z Z ρ ρ ρ ρ x U ρ Y  X j ρ = dL dρ δ ρ − ρ d 2  Γ(ν )  ρ h i 2 (4π) 0 a=1 xa=0 a j=1 x U ρ ρ (3.37) N ! PN F(y) Z ∞ Z ∞ νa−1 δ 1 − yj exp −ρ 1 dyay PN j=1 U(y) Y a a=1 νa = dL dρ ρ d (3.38) 2 Γ(ν ) ρ L 2 (4π) 0 a=1 ya=0 a [ρ U(y)] N ! PN Z 1 νa−1 δ 1 − yj Z ∞ 1 Y dyay j=1 F(y) PN dL a a=1 νa− 2 −1 = dL d dρ exp −ρ ρ , 2 Γ(ν ) 2 U(y) (4π) a=1 ya=0 a [U(y)] 0 (3.39) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 39

where we define yj := xj/ρ and truncate the integral domains of the y variables to 1, since we have PN Dirac’s delta δ 1 − j=1 yj . Assuming F(y)/U(y) > 0, we can perform the ρ integration:

PN ν − dL −1 Z ∞ Z ∞ a=1 a 2 F PN ν − dL −1 U F F U F dρ exp −ρ ρ a=1 a 2 = d ρ exp −ρ ρ 0 U 0 F U U F U (3.40) U ω Z ∞ = dρ0 exp (−ρ0)ρ0ω−1 (3.41) F 0 U ω = Γ(ω) , (3.42) F

0 F where we perform ρ 7→ ρ U ρ, and define

N X dL ω := ν − . (3.43) a 2 a=1

In this way, we obtain the following expression

N ! PN Z 1 νa−1 δ 1 − xj 1 Y dxaxa j=1 I(~ν) = dL Γ(ω) d , (3.44) 2 Γ(ν ) ω 2 −ω (4π) a=1 xa=0 a [F(x)] [U(x)] which we call Feynman parametrization. Both Schwinger parametrization in eq.(3.34) and Feyn- man parametrization in eq.(3.44) are well known and we refer to [34, 35, 88] for detailed, graph- theoretical, discussions and for the original references. As a side note, we could also derive eq.(3.44) using the traditional Feynman trick:

1 Z 1 dx = 2 , (3.45) D1D2 x=0 [(1 − x)D1 + xD2] CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 40 or its generalized formula:

N N ! N PN Z 1 νa−1 δ 1 − xa Y 1 X Y dxaxa a=1 = Γ νa . (3.46) νa PN D Γ(ν ) a=1 νa a=1 a a=1 a=1 xa=0 a PN a=1 xaDa

This formula also converts the products of denominators into their sum. For more detailed discussion for Feynman tricks, see also [34, 91–93]. For the later usage, we here compute a tadpole integral. We consider the following diagram:

(3.47) and the associated scalar integral:

Z ddl 1 J (ν ) := (3.48) tadpole 1 d ν1 (2π) D1

2 2 D1 := l + m . (3.49)

The corresponding matrix become

  x     (3.50) m2x and associated polynomials are

U := x (3.51)

F := m2x2 . (3.52) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 41

The integral can be easily calculated; especially, the Feynman parametrization simplify the integral due to Dirac delta:

d Z ν1−1 Γ ν1 − 2 dxx δ(1 − x) Itadpole(ν1) := (3.53) d ν − d d−ν (4π) 2 [0,1] Γ(ν1) (m2x2) 1 2 (x) 1 d Γ ν1 − 1 1 = 2 . (3.54) d ν − d (4π) 2 Γ(ν1) (m2) 1 2

3.2.4 Lee-Pomeransky Parametrization

Another useful expression can be found in [31]. Let us define

G := U + F (3.55) and consider

N ! Z ∞ νa−1 Y dxaxa − d I˜(~ν) := G 2 . (3.56) Γ(ν ) a=1 xa=0 a

R ∞ PN Inserting, again, 1 = 0 dρδ ρ − j=1 xj , we obtain

N ! N ! Z ∞ Z ∞ νa−1 Y dxaxa X − d I˜(~ν) = dρ δ ρ − x G 2 (3.57) Γ(ν ) j 0 a=1 xa=0 a j=1  νa−1  N ρdxa xa N ! − d Z ∞ Z ∞ ρ 2 Y ρ ρ X xj x = dρ   δ ρ − ρ G ρ . (3.58)  Γ(ν )  ρ ρ 0 a=1 xa=0 a j=1

From the result of eq.(3.32) and eq.(3.33), we can write

G (ρx) := U (ρx) + F (ρx) = ρLU (x) + ρL+1F (x) . (3.59) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 42

Therefore

 νa−1  N ρdxa xa N ! − d Z ∞ Z ∞ ρ 2 Y ρ ρ X xj x x x I˜(~ν) = dρ   δ ρ − ρ ρLU + ρL+1F ρ  Γ(ν )  ρ ρ ρ ρ 0 a=1 xa=0 a j=1

(3.60)

N ∞ N ∞ ν −1 ! P Z Z a δ 1 − j=1 yj d Y dyay PN ν − Ld − = dρ ρ a=1 a ρ 2 [U (y) + ρF (y)] 2 Γ(ν ) ρ 0 a=1 ya=0 a (3.61)

N ! N ! Z 1 νa−1 Z ∞ Y dyay X − d = δ 1 − y dρρω−1 [U (y) + ρF (y)] 2 , (3.62) Γ(ν ) j a=1 ya=0 a j=1 0

where we define yj := xj/ρ. We can now perform the ρ integration:

− d Z ∞ Z ∞ 2 ω−1 − d − d ω−1 F dρρ [U + ρF] 2 = U 2 dρρ 1 + ρ (3.63) 0 0 U ω−1 − d Z ∞ 2 − d U F U F F = U 2 d ρ ρ 1 + ρ (3.64) 0 F U F U U ω Z ∞ − d U ω−1 − d = U 2 dtt [1 + t] 2 (3.65) F 0 1 d = B ω, − ω (3.66) d −ω F ωU 2 2 1 Γ(ω)Γ d − ω = 2 . (3.67) ω d −ω d F U 2 Γ 2

Thus we have the following expression for I˜(~ν):

N ! N ! Z 1 νa−1 Γ(ω)Γ d − ω ˜ Y dyay X 1 2 I(~ν) = δ 1 − yj (3.68) d −ω d Γ(ν ) ω 2 Γ a=1 ya=0 a j=1 F U 2 N ! PN d Z 1 νa−1 δ 1 − yj Γ − ω Y dyay j=1 = 2 × Γ(ω) . (3.69) d d −ω Γ Γ(ν ) ω 2 2 a=1 ya=0 a F U CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 43

This expression above is, besides the prefactor, the same as the Feynman parametrization in eq.(3.44):

N ! PN Z 1 νa−1 δ 1 − xj 1 Y dxaxa j=1 I(~ν) = dL Γ(ω) d . 2 Γ(ν ) ω 2 −ω (4π) a=1 xa=0 a [F(x)] [U(x)]

This proves that we have an alternative expression for our scalar integral I(~ν) in eq.(3.7); we call it Lee-Pomeransky parametrization:

N ! d Z ∞ νa−1 1 Γ Y dxax d 2 a − 2 I(~ν) = dL d G . (3.70) 2 Γ − ω Γ(ν ) (4π) 2 a=1 xa=0 a

3.3 Integration-By-Parts Identities

Integration-By-Parts (IBP) identities [18] can be understood as consequences of translational invariance of the Feynman integrals [36]. In this section, starting from this point of view, we derive IBP identities and study their mathematical structures. The well known Laporta algorithm [19] is a widely used reduction technique; it generates linear relations using IBP identities and it solves the system equations to extract the Master Integrals (MI). After this reduction process is completed, we can then express any integral as a linear combination of MI. Therefore, we can state that MIs form a basis in the linear space spanned by all integrals. Various public implementations of Laporta’s algorithm are available [48, 98–105]. We extensively used Kira [48] and Reduze 2 [103] through this doctoral thesis. The scope of this section is to give an overview of these crucial topics and present a derivation of IBP identities in momentum parametrization. We will later discuss IBP identities in the context of Schwinger-Feynman parametrizations in §4. We also describe the algebraic structures that IBP have, namely vector space, Lie group, and Lie algebra. Finally, we give a schematic flow of how to perform Laporta’s algorithm in matrix forms. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 44

Let us first focus on IBP identities. Consider an integral

Z d d S := d l1 ··· d lLf(s) , (3.71) where f is a scalar function of scalar products s . A relevant example is f(s) = Q 1 as in ik a Da eq.(3.7). As a fact, the change of variable or transformation does not affect on the integral value:

Z Z d 0 d 0 0 d d d l1 ··· d lLf(s ) = d l1 ··· d lLf(s) . (3.72)

We consider a specific, infinitesimal ( > 0) transformation; for a fixed i and an arbitrary momentum qk, where qk ∈ {l1, ··· , lL, e1, ··· , eE}, i.e., qk is either a loop momentum or an external momentum:

  0 li := li + qk j = i lj 7→ (3.73)  0 lj := lj otherwise (j 6= i)

Under this variable change, the volume element becomes, up to 1 order,

d d d d d d d l1 ··· d (li + qk) ··· d lL = d l1 ··· d lL(1 + dδik) , (3.74) since the L × L part of Jacobi matrix for 1 ≤ µ ≤ d component is the following form

0µ ∂l i µ = 1 + δik . (3.75) ∂li CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 45

A variable slm := ll · qm of the integrand f becomes

  (li + qk) · qm l = i ll · qm 7→ (3.76)  ll · qm otherwise and

0 ∂f(s) f(s ) = f(s) + qk · (3.77) ∂li

1 up to order. So the net change of the integrand under lj 7→ lj + δjiqk, 1 ≤ j ≤ L is

d d d d ∂f(s) d l1 ··· d lLf(s) 7→ d l1 ··· d lL(1 + dδik) f + qk · (3.78) ∂li d d ∂f = d l1 ··· d lL f + dδikf(slm) + qk · (3.79) ∂li d d ∂ = d l1 ··· d lL f + · (qkf) , (3.80) ∂li up to 1 order. Note that, since our space is d dimensional,

d µ ∂ X ∂q · q = k = dδ . (3.81) ∂l k ∂lµ ik i µ=1 i

As we state above, this integral is invariant under the transition of lj 7→ lj + δjiqk, 1 ≤ j ≤ L. This fact implies the terms in order vanish:

Z d d ∂ 0 = d l1 ··· d lL · (qkf(s)) . (3.82) ∂li

This family of identities 1 ≤ i ≤ L, i ≤ j ≤ M are called integration-by-parts (IBP) identities. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 46

3.3.1 Lie Group and Lie Algebraic Structure

Since the positive determinant component of the general linear group GL(L, R) is path-connected (see the theorem 3.68 in [106]), we can indeed claim that the integrals eq.(3.71) are invariant under

li 7→ (Al)i (3.83) of GL(L, R)+ transform, i.e., A of L×L positive determinant matrix. Since the regulated integrals are translationally invariant, the integrals eq.(3.71) is invariant the substitution

li 7→ (Al)i + (Be)i = (Mq)i , (3.84) where M = (AB) of L × M matrix with regular L × L matrix A.

Commutation relation We claim that the IBP operators

∂ Oik := · qk (3.85) ∂li satisfy the following commutation relations

[Oik,Ojl]− = δilOjk − δjkOil , (3.86) that is, the IBP operators is closed under bracket product and they form a Lie algebra [37].

Proof Since we have

∂ Oikf := · (qkf) = dδikf + qk · ∂if , (3.87) ∂li CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 47 we can apply this operator twice:

Ojl (Oikf) = ∂j ·{ql (dδikf + qk · ∂if)} (3.88)

= dδjl (dδikf + qk · ∂if) + dδikql · ∂jf + ql · ∂j(qk · ∂if) (3.89)

2 = d δjlδikf + dδjlqk · ∂if + dδikql · ∂jf + ql · ∂j(qk · ∂if) . (3.90)

Once we take the commutator,

[i, j] [Oik,Ojl]− = O[k,,Ol] , (3.91) the last term survives, and since we have

µ ν µ ν ν ql · ∂j(qk · ∂if) = ql ∂jµ(qk ∂iνf) = ql δjkηµ∂iνf + qk ∂jµ∂iνf (3.92)

µ ν for the same reason, the first term ql δjkηµ∂iνf = δjkql · ∂if also survives. Therefore,

[Oik,Ojl]− f = δilqk · ∂jf − δjkql · ∂if (3.93)

= δil (Ojkf − δjkf) − δjk (Oikf − δikf) (3.94)

= δilOjkf − δjkOikf . (3.95)

Thus the set of IBP operators consists a Lie algebra.

Generators The Oik’s are indeed the generator for this Affine transforms

li → (Mq)i = (Al)i + (Be)i (3.96) of det A > 0. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 48

We claim that we can reduce the number of generators from L(L + E) to L + E + 1 [37]. Let L ≥ 2. Then the L(L + E) IBP identities are generated by the following set of operators:

( L ) L−1 E X Oi,i+1|i=1 ,OL,1,O1,L+j|j=1 , Oii . (3.97) i=1

The total number of these operators becomes L + E + 1.

Proof Let us write these generators as follows:

  •      ...    n o   A := O |L−1 ,O   (3.98) i,i+1 i=1 L,1  •       •     

  •         n o   B := O |E   (3.99) 1,L+j j=1            

  •      +•  ( L )     X  ..  C := Oii  .  . (3.100)   i=1    +•      CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 49

For 1 ≤ i ≤ L − 2, take Oi,i+1,Oi+1,i+2 ∈ A and the commutators become

[Oi,i+1,Oi+1,i+2]− = δi,i+2Oi+1,i+1 − δi+1,i+1Oi,i+2 = −Oi,i+2 . (3.101)

Then, for n ≥ 1, if we take

1 ≤ i ≤ L − n − 1 (3.102) and assume we know

L−n Oi,i+n|i=1 . (3.103)

Then the commutators become

[Oi,i+n,Oi+n,i+n+1]− = δi,i+n+1Oi+n,i+n − δi+n,i+nOi,i+n+1 = −Oi,i+n+1 . (3.104)

Therefore, we can access the strict upper triangle elements:

  •···•    . .   .. .        a := {Oi,j| 1 ≤ i < j ≤ L}  •  . (3.105)           CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 50

Let us take 1 ≤ j ≤ E, OL,1 ∈ A, O1,L+j ∈ B, and

[OL,1,O1,L+j]− = δL,L+jO1,1 − δ1,1OL,L+j (3.106)               = −OL,L+j   . (3.107)      •     

For 1 ≤ i ≤ L − 1, 1 ≤ j ≤ E, if we take Oi,L ∈ a, the commuators become

[Oi,L,OL,L+j]− = δi,L+jOL,L − δL,LOi,L+j (3.108)   •    .   .        = −Oi,L+j  •  (3.109)          

Therefore, we can access the LE elements:

  •      •       .  b := {Oi,L+j| 1 ≤ i ≤ L, 1 ≤ j ≤ E}  .  . (3.110)      •      CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 51

Finally, for 2 ≤ j ≤ L, let us take OL,1 ∈ A, O1,j ∈ a,

  O1,1 − OL,L j = L [OL,1,O1,j]− = δL,jO1,1 − δ1,1OL,j = (3.111)  −OL,j 2 ≤ j ≤ L − 1

Using this OL,j, for 1 ≤ i, j ≤ L − 1, Oi,L ∈ a, we can write

  OL,L − Oi,i 1 ≤ j = i ≤ L − 1 [Oi,L,OL,j]− = δi,jOL,L − δL,LOi,j = (3.112)  −Oi,j 1 ≤ i 6= j ≤ L − 1

Therefore we can access all the non-diagonal elements of L.

  •···•    . .   • .. .       . ..   . . •  (3.113)      •···•     

For the diagonal elements, take (L − 1) sum of eq.(3.112).

L−1 L−1 X X OL,L − Oi,i = (L − 1)OL,L − Oi,i (3.114) i=1 i=1 L X = LOL,L − Oi,i (3.115) i=1

PL L−1 since we know i=1 Oi,i, we can determine OL,L, and Oi,i|i=1 . So, as claimed in [37], eq.(3.97) provides a smaller set of generators: L(L + E) to L + E + 1. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 52

3.3.2 Laporta’s Algorithm and Master Integrals

In this section, we give an overview of the well known Laporta algorithm [19]. We will describe a variant of the original algorithm following [48], namely Gauss type forward elimination and back substitution process. Assuming we have already introduced an order for the integrals, we represent an IBP identity as a descending list of terms:

X 0 = ciIi ⇔ [c1*I1,c2*I2, ..] , (3.116) i where I1 is the most complicated integral. Among the system of IBP identities, we group the equations with the same highest integral, and represent the system of equations in the following matrix form:

  •••      •••••       ••••••         •••••••    (3.117)    ••       •••         •••••••   .  .

In the matrix above, each row represents an equation and each column represents a specific integral (with its coefficient). Dots indicate the presence of that particular integral inside each equation. In eq. (3.117), we grouped subsets of equations that share the highest integral (the left-most •) and ordered them from shorter (least number of terms) to longer. Within each subset, by substituting CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 53 the top element into the successors, we eventually reach the triangular form:

  •••      •••••         ••    (3.118)    •••••       ••     .  .

Once we reach this triangular form, we can then use back substitution. Let us consider the first two equations (namely the first and second rows) in eq. (3.118). We can obtain I2 from the second equation and substitute it in the first one. Now the first equation will not depend on I2, but will acquire all the other integrals which were present in the second row. Pictorially we can remove the second-highest integral from the first equation:

    ••• •••••       7→   ••••• •••••

By repeating recursively this procedure, we finally obtain the following form of equations:

  • ???      • ???       • ???         • ???    • ???

The ?’s are the Master Integrals of this system, namely we can represent any •-integral as a linear combination of ?-integrals. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 54 3.4 Baikov Parametrization

As another useful parametrization, the idea behind Baikov parametrization [32] is to define scalar products or denominators as free variables. In this section, following [36], we define our integrals over Euclidean metric,5

Z N 1 d d Y 1 J(~ν) = d k1 ··· d kL . (3.119) (2π)dL Dνa a=1 a

The first d-dimensional integration measure can be written as

d M−1 d−M+1 d k1 = d k1kd k1⊥ , (3.120)

where k1k is in the (M − 1)-dimensional subspace spanned by k2 ··· pE and k1⊥ carries the rest. Similarly, we have the followings:

d M−2 d−M+2 d k2 = d k1kd k2⊥ (3.121) . .

d M−L d−M+L d kL = d kLkd kL⊥ . (3.122)

The (M − 1)-dimensional subspace spanned by k2 ··· pE has

ds12 ··· ds1M (3.123)

5 We can use analytic continuation for the time component to convert (− + + ··· +) into (+ + + ··· +). CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 55

6 as its natural volume element, fixing k1 as a constant vector, and

1/2 M−1 ds12 ··· ds1M = G (k2, ··· , pE) d k1k , (3.125) where the prefactor

G (q) := det [qi · qj] (3.126) is the Gram determinant.7 Thus, we have the following measures:

M−1 ds12 ··· ds1M d k1k = 1/2 (3.127) G (k2, ··· , pE)

M−2 ds23 ··· ds2M d k1k = 1/2 (3.128) G (k3, ··· , pE) . .

M−L dsL,L+1 ··· dsLM d dLk = 1/2 . (3.129) G (p1, ··· , pE)

We can integrate the angular part in perpendicular spaces:

1 dnk = Ω kn−2dk2 (3.130) i⊥ 2 n i⊥ i⊥ 2πn/2 Ω = . (3.131) n Γ(n/2)

Then we replace the surface measure by

2 dki⊥ = dsii . (3.132)

6 The volume form of an oriented n-dimensional pseudo-Riemannian is p ω = |g|dx1 ∧ · · · ∧ dxn (3.124) where g is the determinant of the metric. 7 pG (q) = G1/2 (q) is the volume of the parallelogram formed by the vectors q. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 56

Since the volume of ki, ki+1 ··· pE parallelogram is given by its height times base area, the Gram determinant can be written as

1/2 1/2 G (ki, ki+1 ··· pE) = ki⊥G (ki+1 ··· pE) . (3.133)

This leads the following expression:

1/2 G (ki, ki+1 ··· pE) ki⊥ = 1/2 . (3.134) G (ki+1 ··· pE)

We finally have the following perpendicular measures:

(d−M−1)/2 d−M+1 1 G (k1 ··· pE) d k1⊥ = Ωd−M+1 (d−M−1)/2 ds11 (3.135) 2 G (k2 ··· pE) (d−M)/2 d−M+2 1 G (k2 ··· pE) d k2⊥ = Ωd−M+2 (d−M)/2 ds11 (3.136) 2 G (k3 ··· pE) . .

(d−M+L−2)/2 d−M+L 1 G (kL ··· pE) d kL⊥ = Ωd−M+L (d−M+L−2)/2 dsLL . (3.137) 2 G (p1 ··· pE)

When we plug them in the integral eq.(3.119), we have

Z N 1 M−1 d−M+1 M−L d−M+L Y 1 J(~ν) = d k1kd k1⊥ ··· d kLkd kL⊥ (3.138) (2π)dL Dνa a=1 a Z (d−M−1)/2 1 ds12 ··· ds1M 1 G (k1 ··· pE) = dL 1/2 Ωd−M+1 (d−M−1)/2 ds11 (2π) G (k2, ··· , pE) 2 G (k2 ··· pE) (d−M)/2 ds23 ··· ds2M 1 G (k2 ··· pE) × 1/2 Ωd−M+2 (d−M)/2 ds22 G (k3, ··· , pE) 2 G (k3 ··· pE) . .

(d−M+L−2)/2 N dsL,L+1 ··· dsLM 1 G (kL ··· pE) Y 1 × Ωd−M+L dsLL . (3.139) G1/2 (p , ··· , p ) 2 G(d−M+L−2)/2(p ··· p ) Dνa 1 E 1 E a=1 a CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 57

After the cancellations of numerators and denominators, we have the following Baikov scalar products parametrization:

L 1 1 Y Ωd−M+i 1 J(~ν) = dL dL (d−M+L−1)/2 4 (2π) 2 G (p ··· p ) (4π) i=1 1 E Z L M N Y Y (d−M−1)/2 Y 1 × dsijG (k1 ··· pE) (3.140) Dνa i=1 j=i a=1 a −L(L−1) − LE 1 π 4 2 = G(−d+E+1)/2(p ··· p ) dL L d−M+i 1 E 4 Q (4π) i=1 Γ 2 Z L M N Y Y (d−M−1)/2 Y 1 × dsijG (k1 ··· pE) , (3.141) Dνa i=1 j=i a=1 a where we assume that we can rewrite denominators in scalar products (see eq.(3.6)), and use the followings identities:

L(L+1) L L d−M+i PL d−M+i (d−L−E)L+ 2 1 Y Ωd−M+i 1 Y π 2 1 π i=1 2 1 π 2 = = = πLd/2 2 πLd/2 Γ d−M+i πLd/2 QL d−M+i πLd/2 QL d−M+i i=1 i=1 2 i=1 Γ 2 i=1 Γ 2 − LE − L(L−1) π 2 4 = . (3.142) QL d−M+i i=1 Γ 2

When we change the integration variables

{sij} → {xa := Da} , (3.143) the Jacobian of this transformation is constant, see eq.(3.6),

∂(s) a = det A−1 , (3.144) ∂(x) a,(ij) ij where we identify (ij) ∈ {1, ··· ,N} as a single index, i.e., 1 ≤ i ≤ L, i ≤ j ≤ M. In addition to CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 58 this, the full Gram determinant is given by

" N # X −1a 2 G(k1, ··· , pE) = det [sij] = A xa − ma . (3.145) i,j ij a=1

Therefore we have the following, so-called Baikov parametrization,

−L(L−1) − LE Z N 1 π 4 2 a Y 1 J(~ν) = G(−d+E+1)/2(p ··· p ) det A−1 dx P (d−M−1)/2 dL L d−M+i 1 E ij a n1 nN 4 Q x ··· x (4π) i=1 Γ 2 a=1 1 N Z dx1 ··· dxN (d−M−1)/2 = C n1 nN P , (3.146) x1 ··· xN where the prefactor is defined as

−L(L−1) − LE 1 π 4 2 a C := G(−d+E+1)/2(p ··· p ) det A−1 , (3.147) dL L d−M+i 1 E ij 4 Q (4π) i=1 Γ 2 and the Baikov polynomial is given by

" N # 2 2 X −1a 2 P = P (x1 − m1, ··· , xN − mN ) := det A xa − ma . (3.148) i,j ij a=1

3.4.1 IBP Identities in Baikov Parametrization

Let us choose Ω as the integration domain such that P vanishes at the boundary P |∂Ω = 0. Let us define two operators a±, 1 ≤ a ≤ N. The action of a+ is defined as follows:

+ 1 1 ∂ 1 a := na = − . (3.149) na na+1 na xa xa ∂xa xa

The action of a− is defined as follows:

− 1 1 1 a = = xa . (3.150) na na−1 na xa xa xa CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 59

Using these form and the fact that the Baikov polynomial P is zero at the boundaries, we have the following equations as the actions of a±:

Z + ∂ 1 (d−M−1)/2 a I = C dx1 ··· dxN − n1 nN P (3.151) Ω ∂xa x1 ··· xN Z dx1 ··· dxN ∂ (d−M−1)/2 = C n1 nN P (3.152) Ω x1 ··· xN ∂xa Z − dx1 ··· dxN (d−M−1)/2 a I = C n1 nN xaP . (3.153) Ω x1 ··· xN

Therefore, under the boundary condition P |∂Ω = 0., we can write the action of IBP operator Oij for the integrand in terms of a±. Now the IBP identities in Baikov parametrization becomes

Z Z dx1 ··· dxN (d−M−1)/2 ± dx1 ··· dxN (d−M−1)/2 0 = C n1 nN Oij(xa, ∂a)P = Oij(a )C n1 nN P . (3.154) Ω x1 ··· xN Ω x1 ··· xN

However, assuming the boundary condition P |∂Ω = 0, we obtain a local form of IBP identities. We claim that the IBP identities become the following identity.

d−M−1 2 2 Oij(∂a, xa) P (xa − ma) = 0 .

We first consider Oij = ∂i · qj = dδij + qj · ∂i; the second term becomes (1 ≤ i ≤ L, i ≤ j ≤ M):

L M L L M ! µ ∂f µ X X ∂slm ∂f µ X X X ∂ (kl · qm) ∂f qj · ∂if = qj µ = qj µ = qj + µ . (3.155) ∂k ∂k ∂slm ∂k ∂slm i l=1 m=l i l=1 m=l m=L+1 i CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 60

∂ Since µ can hit either both of (kl · qm) or just kl: ∂ki

L " L M # µ X X ∂ (kl · km) X ∂kl ∂f qj · ∂if = qj µ + µ · qm (3.156) ∂k ∂k ∂slm l=1 m=l i m=L+1 i L " L M # µ X X ν ν X ν ∂f = qj δµδlikmν + klνδimδµ + δilδµqmν (3.157) ∂slm l=1 m=l m=L+1 L " L M # X X X ∂f = (δli (km · qj) + δim (kl · qj)) + δil (qj · qm) (3.158) ∂slm l=1 m=l m=L+1 L L L L X X ∂f X X ∂f = δli (km · qj) + δim (kl · qj) ∂slm ∂slm l=1 m=l l=1 m=l L M X X ∂f + δil (qj · qm) . (3.159) ∂slm l=1 m=L+1

The first and third terms become

L M X ∂f X ∂f (km · qj) , (qj · qm) . (3.160) ∂sim ∂sim m=1 m=L+1

For the second term, to keep indices meaningful, i.e., to maintain 1 ≤ i ≤ L, i ≤ j ≤ M for sij, let us consider two cases; if j ≤ L, then

X ∂f X ∂f X ∂f qj · ∂if = (km · kj) + (kl · kj) + (kj · qm) (3.161) ∂sim ∂sli ∂sim i≤m≤L 1≤l≤i L+1≤m≤M X ∂f ∂f X ∂f X ∂f = snj + 2sij + snj + sjn . (3.162) ∂sni ∂sii ∂sin ∂sin 1≤n≤i−1 i+1≤n≤L L+1≤n≤M CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 61

If L + 1 ≤ j ≤ M, then

X ∂f X ∂f X ∂f qj · ∂if = smj + slj + (qj · qm) (3.163) ∂sim ∂sli ∂sim i≤m≤L 1≤l≤i L+1≤m≤M X ∂f ∂f X ∂f X ∂f X ∂f = snj + 2sij + snj + snj + sjn . ∂sni ∂sii ∂sin ∂sin ∂sin 1≤n≤i−1 i+1≤n≤L L+1≤n≤j L+1≤n≤j (3.164)

If we define

sij := sji, , i > j , (3.165) then we can write either cases in the following form:

M M ∂f X ∂f X ∂f q · ∂ f = s + s = (1 + δ ) s . (3.166) j i ij ∂s mj ∂s mi mj ∂s ii m=1 mi m=1 mi

Therefore we obtain

M X ∂ O := dδ + (1 + δ ) s . (3.167) ij ij mi mj ∂s m=1 mi

We can derive another expression for Oij; using eq.(3.5), we can write

N ∂ X ∂ = Ami , (3.168) ∂s a ∂D mi a=1 a and if j ≤ L (i.e., 1 ≤ i ≤ j ≤ L) then

N N M X X X mi −1 b 2 ∂ Oij := dδij + Aa [A ]mj (1 + δmi) Db − mb , (3.169) ∂Da a=1 b=1 m=1 CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 62 and if L + 1 ≤ j ≤ M then

L M ! X X ∂ Oij = 0 + + (1 + δmi) smj (3.170) ∂smi m=1 m=L+1 N L N N M X X X mi −1 b 2 ∂ X X mi ∂ = Aa [A ]mj (1 + δmi) Db − mb + Aa smj . (3.171) ∂Da ∂Da a=1 m=1 b=1 a=1 m=L+1

Note that smj in the last term is actually pm−L ·pj−L of external momentum. So in either case, these operators have the following form:

∂ Oij = Oij Da, . (3.172) ∂Da

Acting on the integrand, D lowers n by one and ∂ raises n by one with an extra factor n a a ∂Da a a in the numerator. These properties are precisely the same as the action of a± above. If 1 ≤ i, j ≤ L case, the IBP operator in eq.(3.169) becomes,

N N M X X X mi −1b 2 Oij(∂a, xa) = dδij − Aa A mj (1 + δmi) ∂a xb − mb (3.173) a=1 b=1 m=1 N N M X X X mi −1b 2 = dδij − Aa A mj (1 + δmi) xb∂a + δab − mb ∂a . (3.174) a=1 b=1 m=1

The second term in the last parenthesis, i.e., δab term is

N N M N M M X X X mi −1b X X mi a X Aa A mj (1 + δmi) δab = Aa Amj (1 + δmi) = δij (1 + δmi) a=1 b=1 m=1 a=1 m=1 m=1 (3.175)

= (M + 1)δij . (3.176) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 63

So we have

N N M X X X mi b 2 Oij(∂a, xa) = (d − M − 1)δij − Aa Amj (1 + δmi) xb − mb ∂a . (3.177) a=1 b=1 m=1

PL PM ij 2 From eq.(3.5) xa = i=1 j=i Aa sij + ma we have

∂ X ∂xa ∂ X ∂ = = Ami , (3.178) ∂s ∂s ∂x a ∂x mi a mi a a a and eq.(3.6),

N X a 2 smj = Amj(xb − mb ) , (3.179) b=1 we have

M X Oij(∂a, xa) = (d − M − 1)δij − (1 + δmi)smj∂mi . (3.180) m=1

8 The derivatives of P = deti,j sij are proportional to elements of the inverse:

∂ −1 P = (2 − δij)P · (s )ji , (3.183) ∂sij

8 In general, a partial derivative of the determinant of a matrix A = [aij]ij is

∂ −1 = det A · (A )ji (3.181) ∂aij since we can take the Laplace expansion with respect to row i (or, of course, column j). Here Gram matrix sij is symmetric, hence

( −1 ∂ 2P · (s )ji i 6= j( non-diagonal ) P = −1 (3.182) ∂sij P · (s )ii i = j( diagonal ) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 64

d−M−1 2 2 so the second term when acting on [P (xa − ma)] is

M M X d−M−1 d − M − 1 X d−M−1 −1 (1 + δ )s ∂ P 2 = (1 + δ )s P 2 ∂ P (3.184) mi mj mi 2 mi mj mi m=1 m=1 M d − M − 1 d−M−1 X −1 = P 2 (1 + δ )s (2 − δ ) · (s ) 2 mi mj im im m=1 (3.185)

M d − M − 1 d−M−1 X −1 = P 2 ·(s ) s (2 − δ + 2δ − δ ) 2 im mj im im im m=1 (3.186)

d−M−1 = (d − M − 1)P 2 δij . (3.187)

So in 1 ≤ i, j ≤ L, we have

d−M−1 2 2 Oij(∂a, xa) P (xa − ma) = 0. (3.188)

For 1 ≤ i ≤ L < j ≤ M case, eq.(3.171) becomes

N L N N M X X X mi b 2 X X mi Oij(∂a, xa) = Aa Amj (1 + δmi) ∂a xb − mb + Aa smj∂a (3.189) a=1 m=1 b=1 a=1 m=L+1 N L N N M X X X mi b 2 X X mi = Aa Amj (1 + δmi) xb∂a + δab − mb ∂a + Aa smj∂a . a=1 m=1 b=1 a=1 m=L+1 (3.190) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 65

Since δab term becomes (L + 1)δij = 0 since 1 ≤ i ≤ L < j ≤ M, we have

N L N M X X mi a 2 X X mi Oij(∂a, xa) = 0 + Aa Amj (1 + δmi) xb − mb ∂a + Aa smj∂a (3.191) a=1 m=1 a=1 m=L+1 L M M X X X = (1 + δmi) smj∂mi + smj∂mi = (1 + δmi) smj∂mi , (3.192) m=1 m=L+1 m=1 where at the last step we use 1 ≤ i ≤ L < j ≤ M, i.e., m ∈ {L+1, ··· ,M} can not hit 1 ≤ i ≤ L. Therefore

d−M−1 d−M−1 2 2 2 OijP = Oij(∂a, xa) P (xa − ma) (3.193) M X d−M−1 = (1 + δmi) smj∂miP 2 (3.194) m=1 M d − M − 1 d−M−1 X −1 = P 2 (1 + δ ) s (2 − δ ) s (3.195) 2 mi mj im im m=1 M d − M − 1 d−M−1 X −1 = P 2 s s (2 − δ + 2δ − δ ) (3.196) 2 im mj im im im m=1

d−M−1 = (d − M − 1)P 2 δij (3.197)

= 0 . (3.198)

Thus we have

d−M−1 2 2 Oij(∂a, xa) P (xa − ma) = 0 . (3.199)

3.4.2 Exact Form of Integration Regions: a Proof

Here, we study the integration regions of Baikov parametrization in detail. In [36], it is stated that “The integration region has a complicated shape; the Gram determinant vanishes on its boundaries.” but the author does not provide the integration domain explicitly. In this section, we use CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 66

Baikov parametrization in terms of scalar products to compute the exact algebraic form of integration regions. We show here that the Baikov polynomial is quadratic in any off-diagonal element and the formula for two roots.9

Without loss of generality, we take G as a polynomial of s12. Since the Baikov matrix

  s11 s12 ··· s1M     s21 s22      (3.200)  ...      sMM is a symmetric matrix, the determinant

  s11 s12 ··· s1M     s21 s22  G := det   (3.201)  .   ..      sMM is up to quadratic. Therefore, we can set the coefficients a, b, c:

2 G = a (s12) + bs12 + c . (3.202)

9 This work was done in collaboration with Luca Mattiazzi, a master student at the University of Padova, Italy. In his thesis [107], the integration boundaries are fully determined. CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 67

These coefficients a, b, c are given in the following forms:

c = G| (3.203) s12=0=s21

∂ b = G (3.204) ∂s 12 s12=0=s21 1 ∂2 a = 2 G. (3.205) 2 ∂ (s12)

Let us start extract a property that all a, b, c share; since the matrix is symmetric, c can be written as

  s11 0 s13 ···      0 s22  c = G| = det   . (3.206) s12=0=s21  .  s ..   31    . . sMM

For b, we obtain

   

0 s12 s13 ··· s11 1 s13 ···         ∂ 1 s22  s21 0  b = G = det   + det   + ··· ∂s12  .   .  s12=0=s21 0 ..  s 0 ..     31      . . . . sMM . . sMM s12=0 s21=0 (3.207) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 68 where the right hand side has a sequence of zero. Thus we obtain

    0 0 s13 ··· s1M s11 1 s13 ··· s1M         1 s22 s23 ··· s2M   0 0 s23 ··· s2M          b = det 0 s s ··· s  + det  s 0 s ··· s  + 0 . (3.208)  32 33 3M   31 33 3M      . . . ..   . . . ..  . . . .   . . . .      0 sM2 sM3 sMM sM1 0 sM3 sMM

Using second row eliminations for the first term and first row eliminations for second term, we have

    0 0 s13 ··· s1M 0 1 0 ··· 0         1 0 0 ··· 0   0 0 s23 ··· s2M          b = det 0 s s ··· s  + det  s 0 s ··· s  . (3.209)  32 33 3M   31 33 3M      . . . ..   . . . ..  . . . .   . . . .      0 sM2 sM3 sMM sM1 0 sM3 sMM

Note that the two matrices in the last expression are mutually transposed, and

    0 0 s13 ··· s1M 0 1 0 ··· 0         1 0 0 ··· 0   0 0 s23 ··· s2M          2 det 0 s s ··· s  = b = 2 det  s 0 s ··· s  . (3.210)  32 33 3M   31 33 3M      . . . ..   . . . ..  . . . .   . . . .      0 sM2 sM3 sMM sM1 0 sM3 sMM CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 69

We then obtain

    0 1 s13 ··· s1M 0 1 s13 ··· s1M         1 0 s23 ··· s2M  1 0 s23 ··· s2M  2     1 ∂ 1   1   a = G = det 0 0 s ··· s  + det 0 0 s ··· s  . 2 ∂ (s )2 2  33 3M  2  33 3M  12     . . . ..  . . . ..  . . . .  . . . .      0 0 sM3 sMM 0 0 sM3 sMM

(3.211)

With two rows elimination, we finally obtain

  0 1 0 ··· 0     1 0 0 ··· 0      a = det 0 0 s ··· s  . (3.212)  33 3M    . . . ..  . . . .    0 0 sM3 sMM

For later convenience, we introduce another Gram determinant:

  s33 ··· s3M   12  . .  |G| := det  . ..  , (3.213) 12     sM3 sMM i.e.,

12 a = −|G| 12. (3.214) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 70

12 Since this |G| 12 is also Gram determinant, semi-positive definite,

12 a = −|G| 12 ≤ 0. (3.215)

12 Therefore if |G| 12 > 0 then the Baikov polynomial is quadratic in s12 and a < 0 leads the region of s12 determined by the condition G ≥ 0 is closed. Since we obtain the quadratic form, let us construct the roots of this quadratic form. We claim, first, the determinant of this quadratic equation

2 a (s12) + bs12 + c = 0 , (3.216) is given by

2 1 2 b − 4ac = 4|G| 1|G| 2 (3.217) where

    s11 0 s13 ··· s1M 1 0 ··· 0        0 1 0 ··· 0  0 s    1  22  2   |G| := det   , |G| := det  s 0 s  . (3.218) 1 . ..  2  31 33  . .       . . ..     . . .  0 sMM   sM1 0 sMM

Let us show this statement; since all the matrices in this claim share the same submatrix:

  s33   0  .  b :=  ..  , (3.219)     sMM CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 71 we can apply §B.2.1. Let us start b:

       0 s32  0 0 s ··· s   0  13 1M 0−1 . .  b = 2 det(b ) ∗ det   −   b . .  (3.220)        1 0 0 ··· 0   0 sM2   0 −t 0  12 0 = 2 det(b ) ∗ det   = 2 det(b )t12 , (3.221) 1 0 where

  s32   0−1  .  t12 := s ··· s b  .  (3.222) 13 1M     sM2

For later convenience, let us define

  s31   0−1  .  t11 := s ··· s b  .  (3.223) 13 1M     sM1   s32   0−1  .  t22 := s ··· s b  .  (3.224) 23 2M     sM2 CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 72 and

  s31   0−1  .  t21 := s ··· s b  .  (3.225) 23 2M     sM1   t s31     0−1  .  =  s ··· s b  .  (3.226)  23 2M      sM1   s23   0−1  .  = s ··· s b  .  (3.227) 31 M1     s2M   s32   0−1  .  = s ··· s b  .  (3.228) 13 1M     sM2

= t12 , (3.229) where we use the symmetric property b0t = b0. Next a becomes

   0 1 0    a = det(b ) ∗ det   − 0 (3.230) 1 0

= − det(b0) , (3.231) that is

12 0 a = −|G| 12 = − det(b ). (3.232) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 73

Finally c becomes,

       s31 s32  s 0 s ··· s   0  11 13 1M 0−1  . .  c = det(b ) ∗ det   −   b  . .  (3.233)        0 s22 s23 ··· s2M   sM1 sM2   s − t −t 0  11 11 12  = det(b ) ∗ det   (3.234) −t12 s22 − t22

0 2 = det(b ) ∗ (s11 − t11) ∗ (s22 − t22) − t12 . (3.235)

Therefore,

2 0 2 2 0 0 2 b − 4ac = 4 [det(b )] (t12) − 4 [− det(b )] det(b ) ∗ (s11 − t11) ∗ (s22 − t22) − t12 (3.236)

0 2 = 4 [det(b )] [(s11 − t11) ∗ (s22 − t22)] . (3.237)

1 2 1 Now let us compute |G| 1 and |G| 2. |G| 1 becomes

       0 s32    1 0 0 ··· 0   1 0 1 0  0−1 . .  0 |G| = det(b ) ∗ det   −   b . .  = det(b ) ∗ det   1          0 s22 s23 ··· s2M   0 s22 − t22 0 sM2

(3.238)

0 = det(b ) ∗ (s22 − t22) . (3.239) CHAPTER 3. FEYNMAN INTEGRALS AND IBP IDENTITIES 74

2 Similarly, |G| 2 becomes

       s32 0    s 0 s ··· s   s − t 0 2 0  11 13 1M 0−1  . . 0 11 11 |G| = det(b ) ∗ det   −   b  . . = det(b ) ∗ det   2          0 1 0 ··· 0   0 1 sM2 0

(3.240)

0 = det(b ) ∗ (s11 − t11) . (3.241)

Thus, we obtain

1 2 0 0 |G| 1|G| 2 = det(b )(s22 − t22) ∗ det(b )(s11 − t11) (3.242) and we have

2 1 2 b − 4ac = 4|G| 1|G| 2 . (3.243)

Now we obtain this formula as the determinant of the quadratic form, we can write the roots. The

2 two roots for G = a (s12) + bs12 + c = 0 are

√ 2 0 p 1 2 p 1 2 −b ± b − 4ac −2 det(b )t12 ± 4|G| 1|G| 2 b12 ± |G| 1|G| 2 = 0 = 12 (3.244) 2a 2(− det(b )) |G| 12 p 1 2 |G| 1|G| 2 = t12 ± 12 , (3.245) |G| 12 where

0 12 b12 = det(b )t12 = |G| 12t12 . (3.246)

These roots determine the integration region for s12 integral of Baikov parametrization. Chapter 4

Integration-By-Parts Identities over Lee-Pomeransky Parametrization

In this chapter, we study the IBP-like system of equations over Lee-Pomeransky parametrization. Within traditional approaches, say momentum parametrization and Baikov parametrization, any surface term, corresponding to the partial derivative with respect to a parameter, vanishes as shown in eq.(3.82) for the momentum parametrization and eq.(3.154) for Baikov parametrization. How- ever, this is not the case for the class of Schwinger-Feynman parametrizations. Examples of such non-vanishing surface terms have been already observed in [94]. After showing how to build all the surface terms in the specific cases of Lee-Pomeransky parametrization, we then construct IBP-like identities for several examples of Feynman diagrams, both at one loop (bubble and triangle) and two loops (kite and double-box). We observe that our results are consistent with traditional IBP identities for all the Feynman graphs we consider, by employing of the results provided by Kira [48]. This process can generate ill-defined integrals in intermediate steps. We develop a treatment, dubbed pinching, for the ill-behaved part of the Lee-Pomeransky parameter space, which has corresponding graph-theoretical and complex analysis interpretations. As a related topic on Schwinger-

75 CHAPTER 4. IBP OVER LP PARAMETRIZATION 76

Feynman parametrization, we treat dimension shift, following [13]. Finally, we also show a full reduction, i.e., we generate identities following our method to reach Master Integrals and the reductions with respect to the set of Master Integrals, using the massive bubble integral. The results are precisely the same as the traditional approaches.

4.1 Formalism

In this section, we develop a procedure to generate a system of linear equations of integrals. First, we define a process for generating linear equations. Within this procedure, we discuss the cases where non-zero surface terms appear. Next, we discuss the pinching procedure, which connects both a graph-theoretical operation and an interpretation within complex analysis for integrals in Schwinger-Feynman parametrizations. Finally, we study dimensional shift, following [13].

4.1.1 IBP Generators over Lee-Pomeransky Parametrization

We consider the following functional

Z Z h − d +1i Gi(f) := dx1 ··· dxN ∂i fG 2 , i = 1, ··· ,N, (4.1) as an IBP relation generator in Lee-Pomeransky parametrization, where

n1 nN f = x1 ··· xN (4.2)

and the integration domain is R> = (0, ∞). Within traditional IBP (in momentum space) reductions, we set the surface term to be zero. In the following section, we show that in some cases, we have non-zero surface terms in Lee-Pomeransky parametrization.

For simplicity we set i = 1, which correspond to taking the derivative with respect to x1, and CHAPTER 4. IBP OVER LP PARAMETRIZATION 77 consider the case

f = 1 . (4.3)

Then, the generator becomes

Z Z h − d +1i G1(1) = dx1 ··· dxN ∂1 G 2 . (4.4)

This integral can be computed in two different ways. One treatment is to compute the partial

∂ derivative ∂1 = in the integrand: ∂x1

d ∂G − d 1 − G 2 , (4.5) 2 ∂x1 to get a linear combination of integrals in d dimension. In this manner, we obtain a polynomial

− d times G 2 as the integrand. Let us call this term the right-hand side:

Z Z d ∂G d rhs − 2 G1 (1) = dx1 ··· dxN 1 − G . (4.6) 2 ∂x1

rhs We note that G1 (1) consists of a linear combinations of integrals in Lee-Pomeransky parametrization that share the same Lee-Pomeransky polynomial G, the same dimension d, but different mono- mials in the integrand, therefore corresponding to different diagrams. Another way of computing the integral in eq.(4.4) is to use the fundamental theorem of calculus:

Z − d +1 − d +1 − d +1 − d +1 dx1∂1G 2 = G 2 − G 2 = − G 2 , (4.7) x1=∞ x1=0 x1=0

− d +1 where we have used the fact that G 2 = 0 in the dimensional regularization scheme. x1=∞ This second expression for the generator, obtained by evaluating the integrand at the boundaries of CHAPTER 4. IBP OVER LP PARAMETRIZATION 78 integration, is indeed the surface term, and we call it the left-hand side of our IBP-style relation:

Z Z d lhs − 2 +1 G1 (1) := (−) dx2 ··· dxN G . (4.8) x1=0

− d +1 The expression G 2 has a close relation with the pinching we have described above. Indeed, x1=0 lhs rhs we observe that the surface term G1 (1) contains diagrams with one less denominator than G1 (1),

lhs and therefore lives in strictly lower sector. Concerning its dimensions, G1 (1) could be interpreted as an integral with f = 1 in shifted dimension. Alternatively, if order to match the same dimension d of the right-hand-side, we can rewrite it as

Z Z lhs − d G (1) := (−) dx ··· dx G| G 2 , (4.9) 1 2 N x1=0 x1=0

lhs which is a linear combination of integrals in the lower sector, i.e., G1 (1) contains no D1. If we have such non-zero surface term, our IBP equation can be written in the following form:

rhs lhs 0 = G1 (1) − G1 (1) . (4.10)

If we include the x1 parameter inside the f monomial, for example when we set

f = x1 , (4.11) the surface terms are identically zero on both the lower and upper bound of integration, namely

Z h − d +1i − d +1 − d +1 dx1∂1 x1G 2 = x1G 2 − x1G 2 = 0 − 0 = 0 , (4.12) x1=∞ x1=0 under dimensional regularization scheme. When we have no surface term, our IBP identity takes CHAPTER 4. IBP OVER LP PARAMETRIZATION 79 the following form:

rhs 0 = G (x1) . (4.13)

From this observation, if the monomial f contains x1, then the linear combination of integrals in

rhs G (x1) is zero, which is the same as traditional IBP equations.

4.1.2 Pinching

For a given graph, we define pinching as a procedure to squeeze an internal line that has distinct endpoints. Graph-theoretically, this operation is called a contraction of a line. For example, let us consider the following graph:

b (4.14)

If we pinch the vertical line from a to b out, we obtain the following graph:

b (4.15)

Note that, we do not consider here a tadpole-type internal line, which has the same endpoint. With the restriction a 6= b, the pinching does not change the number of loops in a diagram. Let us consider the corresponding operation over Feynman integrals. A generic Feynman inte- CHAPTER 4. IBP OVER LP PARAMETRIZATION 80 gral can be written as:

L N ! Z d L Z νa−1 F Y d lj 1 i Y dxax exp − J(~ν) = = a U , (4.16) d ν1 νN dL d (2π) D ··· D 2 Γ(ν ) 2 j=1 1 N (4π) a=1 R> a U where the first expression is the Feynman integral in momentum space, while the second is the same integral in Schwinger parametrization.

For simplicity, let us consider J(ν1, 1, ··· , 1). If we start from the momentum parametrization, then the procedure of setting ν1 = 0 gives us the following well-defined integral

L Z d Y d lj 1 J(0, 1, ··· , 1) = , (4.17) (2π)d D ··· D j=1 2 N that can readily be translated into Schwinger parametrization as:

F L L Z ! exp − 1 i Y U1 J(0, 1, ··· , 1) = dL dxa d , (4.18) (4π) 2 2 a=2 R> U1 where the graph polynomials F1 and U1 can be defined by only D2, ··· ,DN . That is, ν1 = 0 simply stands for the absence of D1.

If, instead, we use Schwinger parametrization directly for J(ν1, 1, ··· , 1), then setting ν1 = 0 creates an ill-defined integral:

L Z −1 N Z ! F i dx1x1 Y exp − U dL dxa d . (4.19) 2 Γ(0) 2 (4π) R> a=2 R> U

We could pictorially state that the change of representation from momentum to Schwinger parametrization and the operation of setting ν1 = 0 does not seem to commute. This is clearly not acceptable. CHAPTER 4. IBP OVER LP PARAMETRIZATION 81

In order to find a solution to this problem, let us first observe that the following relations hold:

U = U| , F = F| , G = G| . (4.20) 1 x1=0 1 x1=0 1 x1=0

This fact suggests a simple procedure to consistently set ν1 = 0 in Schwinger parametrization, i.e. the integral J(0, 1, ··· , 1) in this parametrization must be understood as the following contour integral:

L I N Z ! F i 1 dxb Y exp − U J(0, 1 ··· , 1) = dL dxa d . (4.21) 2 2πi x 2 (4π) 1 a=2 R> U

Indeed, Cauchy’s integral formula implies the following relation:

I ! F iL Y Z exp − J(0, 1, ··· , 1) = dx U . (4.22) dL a d (4π) 2 R> U 2 a=1 x1=0

In general, the absence of the first n denominator can be written as:

L n I ! N Z ! F i Y 1 dxa Y exp − U J(0, ··· , 0, 1, ··· , 1) = dL dxb d (4.23) 2 2πi x 2 | {z } (4π) a R> U first n a=1 b=n+1 N ! F iL Y Z exp − = dx U . (4.24) dL a d (4π) 2 R> U 2 b=n+1 x1=···=xn=0

We used Schwinger parametrization for this discussion, but the same treatment holds for Feynman and Lee-Pomeransky parametrizations. We define the pinching as the following replacement of the ill-behaved part:

−1 Z dx x a-th pinch 1 I dx a a 7−→ a . (4.25) Γ(0) 2πi xa CHAPTER 4. IBP OVER LP PARAMETRIZATION 82

4.1.3 Dimension Shifts

In this section, we follow [13] to discuss dimension shift. We assume that all the masses of denominators are different. We have the following expressions for an L-loop scalar integral; defining momenta expression and Schwinger parametrization.

L Z d ! N Y d lj Y 1 J d(~ν) = = iLId(~ν) (2π)d Dνa j=1 a=1 a N ! Z ∞ νa−1 F L 1 Y dxaxa exp − U = i dL d , (4.26) 2 Γ(ν ) 2 (4π) a=1 xa=0 a U where

1 1 = 2 2 (4.27) Da ka + ma − i0 F − = −C + BtA−1B (4.28) U N X 2 C = xa ma − i0 . (4.29) a=1 and we put d to indicate the time-space dimension (ka is momenta flow of the a-th propagator). Assume that all the masses of denominators are different, and consider

U = U (x1, ··· , xN ) (4.30) of a Symanzik polynomial of Schwinger-Feynman parameters, and define an operator:

ˆ ∂ ∂ U := U − 2 , ··· , − 2 . (4.31) ∂m1 ∂mN CHAPTER 4. IBP OVER LP PARAMETRIZATION 83

We first observe

F F Uˆ exp − = U exp − (4.32) U U since only C carries masses and

N ! ∂ ∂ X − exp (−C) = − exp − x m2 − i0 = x exp (−C) (4.33) ∂m2 ∂m2 a a a a a a=1 holds. It implies

L Z d ! N Y d lj Y 1 UˆJ d(~ν) = Uˆ (4.34) (2π)d Dνa j=1 a=1 a N ! Z ∞ νa−1 ˆ F L 1 Y dxaxa U exp − U = i dL d (4.35) 2 Γ(ν ) 2 (4π) a=1 xa=0 a U N ! Z ∞ νa−1 F L 1 Y dxaxa exp − U = i dL d−2 (4.36) 2 Γ(ν ) 2 (4π) a=1 xa=0 a U 1 = J d−2(~ν) . (4.37) (4π)L

We can compute this integral in d − 2 via momenta expression

L Z d ! N Y d lj Y 1 UˆJ d(~ν) = Uˆ , (4.38) (2π)d Dνa j=1 a=1 a with

∂ 1 ∂ 1 −1 1 − = − 2 2 = (−) 2 = 2 . (4.39) ∂ma Da ∂ma ka + ma − i0 Da Da CHAPTER 4. IBP OVER LP PARAMETRIZATION 84

Therefore we can compute an arbitrary parametrization in shifted dimension d − 2:

" L Z d ! N # ∂ ∂ Y d lj Y 1 J d−2(~ν) = (4π)LU − , ··· , − . (4.40) ∂m2 ∂m2 (2π)d Dνa 1 N j=1 a=1 a

4.2 One-Loop examples

In this section, we study IBP of simple diagrams in Lee-Pomeransky parametrization. We first consider bubble integrals, in which the surface terms can be identified as tadpole integrals, as discussed in the previous sections. As a straightforward generalization, we also consider triangle integrals.

rhs For each example, we first consider G1(1) and compute the linear combinations G1 (1). Then

lhs we calculate the surface term G1 (1) and apply the pinching procedure. Finally we substitute

lhs rhs traditional IBP reductions employed by Kira [48] to check the consistency of G1 (1) − G1 (1). In

rhs case of zero surface terms, we calculate the linear combinations G1 (1) and apply traditional IBP reductions to confirm the consistency.

4.2.1 Bubble Integrals

Let us consider a massive bubble diagram:

l p

(4.41) where two denominators are

2 2 D1 := l + m (4.42)

2 2 D2 := (l + p) + m . (4.43) CHAPTER 4. IBP OVER LP PARAMETRIZATION 85

The associated matrix and polynomials are

  x1 + x2 px2   (4.44)  2 2  px2 p x2 + m (x1 + x2)

U = x1 + x2 (4.45)

2 2 2 2 2 F = m (x1 + x2) + p + 2m x1x2 . (4.46)

For f = 1 case, we show that there exists an non-zero surface term. Let us consider

Z Z ∂ h − d +1i G (1) = dx dx G 2 , (4.47) 1 1 2 ∂x R> R> 1 where Lee-Pomeransky polynomial is

2 2 2 G := x1 + x2 + m (x1 + x2) + p x1x2 . (4.48)

Since

∂ h − d +1i d 2 2 2 − d G 2 = 1 − 1 + 2m x1 + (p + 2m )x2 G 2 (4.49) ∂x1 2 holds, we can associate these terms into integrals in d dimension:

d d (4π) 2 Grhs(1) = 1 − 1 2 d Γ 2 d 2 d 2 2 d × Γ(d − 2)Ibubble(1, 1) + Γ(d − 3) 2m Ibubble(2, 1) + (p + 2m )Ibubble(1, 2) . (4.50) CHAPTER 4. IBP OVER LP PARAMETRIZATION 86

We also have the following as the IBP surface terms,

Z d Z − d +1 lhs − 2 +1 2 2 2 G1 (1) = dx2 0 − G = (−) dx2 x2 + m x2 . (4.51) x =0 R> 1 R>

We expand one power of G| and to take the surface terms in d dimension:1 x1=0

Z d lhs 2 2− 2 +1 G1 (1) = (−) dx2 x2 + m x2 (4.52) R> Z d 2 2 2 2− 2 = (−) dx2 x2 + m x2 x2 + m x2 (4.53) R> d 2 (4π) d 2 d = (−) d Γ(d − 2)Γ(2)Itadpole(2) + m Γ(d − 3)Γ(3)Itadpole(3) . (4.54) Γ 2

With our pinching procedure, we obtain

d 2 rhs (4π) d 2 d G1 (1) = (−) d Γ(d − 2)Γ(2)Ibubble(0, 2) + m Γ(d − 3)Γ(3)Ibubble(0, 3) . (4.55) Γ 2

lhs rhs Since G1 (1) = G1 (1), we finally can write the following identity:

d d (4π) 2 0 = 1 − 2 d Γ 2 d 2 d 2 2 d × Γ(d − 2)Ibubble(1, 1) + 2m Γ(d − 3)Ibubble(2, 1) + (p + 2m )Γ(d − 3)Ibubble(1, 2) d 2 (4π) d 2 d − (−) d Γ(d − 2)Γ(2)Ibubble(2, 0) + m Γ(d − 3)Γ(3)Ibubble(3, 0) (4.56) Γ 2 of a relation between tadpole sector and bubble sector. Note that these are purely algebraic equations over Lee-Pomeransky parameters, and we have not yet used traditional IBP reductions. When

1 An alternative interpretation of this right-hand side is to treat it as an integral in d + 2 dimension. To reduce it into integrals in d dimension, we need to use dimension shift as we develop in §4.1.3. Even when we use this integral in d + 2 dimension, the generated equation is consistent with the traditional IBP identities. CHAPTER 4. IBP OVER LP PARAMETRIZATION 87 we apply traditional IBP reductions, for example,

3 − d 2m2Γ(d − 3)Id (2, 1) IBP= (−)Γ(d − 3)(−)2Id (1, 1) ∗ (4.57) bubble bubble −p2 − 4m2 d − 2 (p2 + 2m2)Γ(d − 3)Id (1, 2) IBP= (−)Γ(d − 3)(−)1Id (1, 0) ∗ , (4.58) bubble bubble 2m2(−p2 − 4m2) we can show that our identity eq.(4.56) is identically zero and therefore consistent with the traditional IBP reductions.2

In general, a family of generators, with x1-derivative,

2 x2, x2, ··· (4.59) has non-zero surface terms

lhs lhs 2 G1 (x2) 6= 0,G1 (x2) 6= 0, ··· , (4.60) and we can show that the following identities are consistent under traditional IBP reductions:

lhs rhs lhs 2 rhs 2 0 = G1 (x2) − G1 (x2) , 0 = G1 (x2) − G1 (x2) , ··· . (4.61)

We now consider f = x1 case, where we expect zero surface term:

∞ ∞ ∞ Z Z ∂ h d i Z h d ix1=∞ lhs 1 0 − 2 +1 − 2 +1 G1 (x1) = dx1 dx2 x1x2G = dx2 x1G = 0 . (4.62) 0 0 ∂x1 0 x1=0

We will compute the right hand side of linear combinations to see its consistency.

2 Note that we employed the notation “IBP= ” to identify equality obtained by substituting traditional IBPs in our original expressions. CHAPTER 4. IBP OVER LP PARAMETRIZATION 88

Since we have the following:

∂ h − d +1i − d +1 d 2 2 2 − d x1G 2 = G 2 + x1 1 − 1 + 2m x1 + (p + 2m )x2 G 2 (4.63) ∂x1 2 2 2 2 d 2 2 2 − d = x + x + m (x + x ) + p x x + x 1 − 1 + 2m x + (p + 2m )x G 2 1 2 1 2 1 2 1 2 1 2 (4.64) d 2 2 d 2 2 2 2 − d = 2 − x + x + m (3 − d) x + 2 − p + 2m x x + m x G 2 , 2 1 2 1 2 1 2 2 (4.65) we can associate these terms into integrals in d dimension:

d (4π) 2 d Grhs(x ) = 2 − Γ(d − 3)Γ(2)Id (2, 1) + Γ(d − 3)Γ(2)Id (1, 2) 1 1 d 2 bubble bubble Γ 2 d + m2 (3 − d) Γ(d − 4)Γ(3)Id (3, 1) + 2 − p2 + 2m2 Γ(d − 4)Γ(2)Γ(2)Id (2, 2) bubble 2 bubble

2 d + m Γ(d − 4)Γ(3)Ibubble(1, 3) . (4.66)

In this no-surface-term configuration, our identity becomes

rhs G1 (x1) = 0 . (4.67)

rhs IBP We can apply Kira’s IBP reduction on it, and chech explicitely that G1 (x1) = 0. Therefore, our

rhs identity G1 (x1) = 0 is also consistent with traditional approaches.

We can check that the following family of generators x1-derivative:

2 2 2 x1, x1 ∗ x2, x1 ∗ x2, x1, x1 ∗ x2, ··· , (4.68) fall into the consistent identities under traditional approaches. That is, the following relations are CHAPTER 4. IBP OVER LP PARAMETRIZATION 89 identities:

rhs rhs rhs 2 rhs 2 rhs 2 G1 (x1) = G1 (x1 ∗ x2) = G1 (x1 ∗ x2) = G1 (x1) = G1 (x1 ∗ x2) = ··· = 0,. (4.69)

4.2.2 Triangle Integrals

Let us consider the following one-loop triangle diagram,

q (4.70)

The corresponding momenta flow and denominators are

2 2 D1 = l + m (4.71)

2 2 D2 = (p − l) + m (4.72)

2 2 D3 = (l + q) + m (4.73)

The associated matrix is

  x1 + x2 + x3 qx3 − px2 S =   . (4.74)  2 2 2  qx3 − px2 m (x1 + x2 + x3) + p x2 + q x3 CHAPTER 4. IBP OVER LP PARAMETRIZATION 90

This leads the following graph polynomials:

U = x1 + x2 + x3 (4.75)

2 2 2 2 2 F = m (x1 + x2 + x3) + p x1x2 + q x3x1 + (p + q) x2x3 (4.76)

G = U + F . (4.77)

Let us consider

ZZZ h − d +1i G1(1) = dx1dx2dx3∂1 G 2 . (4.78) 3 R>

Since its integrand is

h − d +1i d 2 2 2 2 2 − d ∂ G 2 = 1 − 1 + 2m x + (p + 2m )x + (q + 2m )x G 2 , (4.79) 1 2 1 2 3 we have the following linear combination among d-dimensional integrals:

d d (4π) 2 Grhs(1) = 1 − Γ(d − 3)Id(1, 1, 1) 1 2 d Γ 2 + Γ(d − 4) 2m2Id(2, 1, 1) + (p2 + 2m2)Id(1, 2, 1) + (q2 + 2m2)Id(1, 1, 2) .

(4.80)

Expanding one power of G | , we have the following surface term: triangle x1=0

ZZ d−2 lhs 2 2 2 − 2 G1 (1) = (−) dx2dx3 x2 + x3 + m (x2 + x3) + (p + q) x2x3 (4.81) 2 R> d (4π) 2 d d = (−) d Γ(d − 3) Ibubble(2, 1) + Ibubble(1, 2) Γ 2 2 d d 2 2 d +Γ(d − 4) m Γ(3) Ibubble(3, 1) + Ibubble(1, 3) + 2m + (p + q) Ibubble(2, 2) . (4.82) CHAPTER 4. IBP OVER LP PARAMETRIZATION 91

With the pinching identification, we have

d 2 lhs (4π) d d G1 (1) = (−) d Γ(d − 3) I (0, 2, 1) + I (0, 1, 2) Γ 2 +Γ(d − 4) m2Γ(3) Id(0, 3, 1) + Id(0, 1, 3) + 2m2 + (p + q)2 Id(0, 2, 2) .

(4.83)

lhs rhs G1 (1) = G1 (1) implies the following identity:

d d (4π) 2 0 = 1 − Γ(d − 3)Id(1, 1, 1) 2 d Γ 2 + Γ(d − 4) 2m2Id(2, 1, 1) + (p2 + 2m2)Id(1, 2, 1) + (q2 + 2m2)Id(1, 1, 2)

d (4π) 2 d d − (−) d Γ(d − 3) I (0, 2, 1) + I (0, 1, 2) Γ 2 +Γ(d − 4) m2Γ(3) Id(0, 3, 1) + Id(0, 1, 3) + 2m2 + (p + q)2 Id(0, 2, 2) . (4.84)

When we apply the traditional IBP on this right-hand side, it falls into zero. Therefore, this identity is consistent with the traditional approaches.

We can check that the following family of generators with x1-derivative:

2 x1, x1, x1 ∗ x2, x1 ∗ x2 ∗ x3, ··· , (4.85)

give the identities that are consistent with IBP reductions. For f = x1 with x1 derivative case, as CHAPTER 4. IBP OVER LP PARAMETRIZATION 92 an example,3

1 Grhs(x ) = Γ(d − 5)(2m2((d − 4) 1 1 2 (I(2, 1, 2) + I(2, 2, 1)) + 2(d − 3)I(3, 1, 1)

− 2I(1, 1, 3) − 2I(1, 2, 2) − 2I(1, 3, 1))

− (d − 4)p2I(2, 2, 1) − dq2I(2, 1, 2) + 2(d − 5)I(1, 1, 2)

+ (d − 5)(2I(1, 2, 1) − (d − 4)I(2, 1, 1)) + 2p2I(1, 2, 2)

+ 4p · qI(1, 2, 2) + 2q2I(1, 2, 2) + 4q2I(2, 1, 2)) . (4.86)

When we plug in IBP reduction on this expression, we obtain zero.

Similarly, with x2-derivative, we can check the consistency of the following generators:

x2, x1 ∗ x2, x1 ∗ x2 ∗ x3, ··· . (4.87)

4.3 QED-like Kite Integrals

In this section, we discuss kite diagrams as a two-loop example. We first generate a set of linear equations and check the consistency with traditional approaches. We use kite diagram to show the pinching procedure explicitly.

3 From now on, we express our integrals in Kira’s compatible form, i.e., using (+ − − · · · −) metric. CHAPTER 4. IBP OVER LP PARAMETRIZATION 93

4.3.1 IBP Identities over Lee-Pomeransky Parametrization

Let us consider the following kite diagram:

D1 D5

D D 4 3 (4.88)

where only the central line D2 is massless. For l1, l2 of loop momenta, let us set:

2 2 D1 := l1 + m (4.89)

2 D2 := l2 (4.90)

2 2 D3 := (l1 − l2 + p) + m (4.91)

2 2 D4 := (l1 + p) + m (4.92)

2 2 D5 := (−l1 + l2) + m . (4.93)

Consider a generator of

f = 1 (4.94) CHAPTER 4. IBP OVER LP PARAMETRIZATION 94

4 case with x1 derivative. This choice gives us:

1 3d 1 Grhs(1) = − (d − 2)Γ − 7 (3d − 14)(I (1, 1, 1, 1, 2) + I (1, 1, 2, 1, 1) 1 2 2 2 Kite Kite

2 + IKite(1, 2, 1, 1, 1)) − 2m (IKite(1, 1, 1, 1, 3) + IKite(1, 1, 1, 2, 2)

+ IKite(1, 1, 2, 1, 2) + IKite(1, 1, 2, 2, 1) + IKite(1, 1, 3, 1, 1)

+ IKite(1, 2, 1, 1, 2) + IKite(1, 2, 1, 2, 1) + IKite(1, 2, 2, 1, 1)

+ IKite(2, 1, 1, 1, 2) + IKite(2, 1, 2, 1, 1) + IKite(2, 2, 1, 1, 1))

2 + p (IKite(1, 1, 1, 2, 2) + IKite(1, 1, 2, 1, 2) + IKite(1, 1, 2, 2, 1)

+IKite(1, 2, 1, 2, 1) + IKite(1, 2, 2, 1, 1))) . (4.95)

The corresponding surface term is

3d Glhs(1) = Γ − 7 (2m2(I (0, 1, 1, 2, 3) + I (0, 1, 1, 3, 2) + I (0, 1, 2, 2, 2) 1 2 Kite Kite Kite

+ IKite(0, 1, 2, 3, 1) + IKite(0, 1, 3, 2, 1) + IKite(0, 2, 1, 1, 3)

+ IKite(0, 2, 1, 2, 2) + IKite(0, 2, 1, 3, 1) + IKite(0, 2, 2, 1, 2)

2 + IKite(0, 2, 2, 2, 1) + IKite(0, 2, 3, 1, 1)) − p (IKite(0, 1, 2, 2, 2) + IKite(0, 2, 1, 2, 2)

+ IKite(0, 2, 2, 1, 2))) − (IKite(0, 1, 1, 2, 2) + IKite(0, 1, 2, 2, 1) + IKite(0, 2, 1, 1, 2) 3d + I (0, 2, 1, 2, 1) + I (0, 2, 2, 1, 1))Γ − 6 . (4.96) Kite Kite 2

rhs lhs Now our identity is 0 = G1 (1) − G1 (1). We can check that this identity is consistent under traditional IBP reductions.

We can also show that f = 1 with x2 derivative generates an identity which is consistent under

4 Since Kira uses (+ − − · · · −) system, we follow the same metric system here. CHAPTER 4. IBP OVER LP PARAMETRIZATION 95 traditional IBP reduction:

(d − 2)Γ 3d − 7 0 = − 2 4(d − 3)(m2)((d − 3)(m2)I (1, 0, 1, 1, 1) − (d − 2) I (1, 0, 1, 1, 0)) 2(m2)2((p2) − 4(m2))2 Kite Kite

2 +(d − 2) IKite(1, 0, 1, 0, 0) .

We can show that with x1 derivative, the following generators have no-surface term:

2 x1, x1, x1x2, x1x2x3, ··· . (4.97)

Similarly, with x2 derivative:

x2, x1x2, x1, x2, x5, ··· , (4.98) have no-surface term. All these generators create identities that are consistent under traditional IBP approaches. CHAPTER 4. IBP OVER LP PARAMETRIZATION 96

For example,

1 3d Grhs(x ) = Γ − 8 3d2(I (2, 1, 1, 1, 2) + I (2, 1, 2, 1, 1) + I (2, 2, 1, 1, 1)) 1 1 4 2 Kite Kite Kite

2 − 2d(2(m )(IKite(2, 1, 1, 1, 3) + IKite(2, 1, 1, 2, 2) + IKite(2, 1, 2, 1, 2) + IKite(2, 1, 2, 2, 1)

+ IKite(2, 1, 3, 1, 1) + IKite(2, 2, 1, 1, 2) + IKite(2, 2, 1, 2, 1) + IKite(2, 2, 2, 1, 1)

+ 2(IKite(3, 1, 1, 1, 2) + IKite(3, 1, 2, 1, 1) + IKite(3, 2, 1, 1, 1)))

2 − (p )(IKite(2, 1, 1, 2, 2) + IKite(2, 1, 2, 1, 2) + IKite(2, 1, 2, 2, 1) + IKite(2, 2, 1, 2, 1)

+ IKite(2, 2, 2, 1, 1)) + 3IKite(1, 1, 2, 2, 1) + 3IKite(1, 2, 1, 1, 2) + 3IKite(1, 2, 1, 2, 1)

+ 3IKite(1, 2, 2, 1, 1) + 14IKite(2, 1, 1, 1, 2) + 14(IKite(2, 1, 2, 1, 1) + IKite(2, 2, 1, 1, 1)))

2 + (32 − 6d)IKite(1, 1, 1, 2, 2) + 8(m )(IKite(1, 1, 1, 2, 3) + IKite(1, 1, 1, 3, 2) + IKite(1, 1, 2, 2, 2)

+ IKite(1, 1, 2, 3, 1) + IKite(1, 1, 3, 2, 1) + IKi te(1, 2, 1, 1, 3) + IKite(1, 2, 1, 2, 2)

+ IKite(1, 2, 1, 3, 1) + IKite(1, 2, 2, 1, 2) + IKite(1, 2, 2, 2, 1) + IKite(1, 2, 3, 1, 1)

2 + 2IKite(2, 1, 1, 1, 3)) + 8(m )(2IKite(2, 1, 1, 2, 2) + 2IKite(2, 1, 2, 1, 2) + 2IKite(2, 1, 2, 2, 1)

+ 2IKite(2, 1, 3, 1, 1) + 2IKite(2, 2, 1, 1, 2) + 2IKite(2, 2, 1, 2, 1) + 2IKite(2, 2, 2, 1, 1)

+ 3(IKite(3, 1, 1, 1, 2) + IKite(3, 1, 2, 1, 1) + IKite(3, 2, 1, 1, 1)))

2 − 4(p )(IKite(1, 1, 2, 2, 2) + IKite(1, 2, 1, 2, 2) + IKite(1, 2, 2, 1, 2))

2 − 8(p )(IKite(2, 1, 1, 2, 2) + IKite(2, 1, 2, 1, 2) + IKite(2, 1, 2, 2, 1)

+ IKite(2, 2, 1, 2, 1) + IKite(2, 2, 2, 1, 1)) + 32(IKite(1, 1, 2, 2, 1)

+ IKite(1, 2, 1, 1, 2) + IKite(1, 2, 1, 2, 1) + IKite(1, 2, 2, 1, 1)

+ 2(IKite(2, 1, 1, 1, 2) + IKite(2, 1, 2, 1, 1) +IKite(2, 2, 1, 1, 1)))) (4.99) falls into zero under traditional IBP reductions. CHAPTER 4. IBP OVER LP PARAMETRIZATION 97

4.3.2 An Example of Pinching

Let us show an example of pinching identification. Consider a massive kite diagram.

D1 D5

D D 4 3 (4.100) with

2 2 D1 := l1 + m (4.101)

2 2 D2 := l2 + m (4.102)

2 2 D3 := (l1 − l2 + p) + m (4.103)

2 2 D4 := (l1 + p) + m (4.104)

2 2 D5 := (−l1 + l2) + m (4.105)

Under this setting, the associated matrix can be computed via

5 X xaDa , (4.106) a=1 and the matrix is the following form:

  x1 + x3 + x4 + x5 −x3 − x5 p(x3 + x4)     S1,1,1,1,1 =  −x − x x + x + x −px   3 5 2 3 5 3   2  p(x3 + x4) −px3 p(x3 + x4) + m (x1 + x2 + x3 + x4 + x5)

(4.107) CHAPTER 4. IBP OVER LP PARAMETRIZATION 98

If we set x4 = x5 = 0, we have the following matrix:

  x1 + x3 −x3 p(x3 + x4)     S1,1,1,0,0 =  −x x + x −px  (4.108)  3 2 3 3   2  px3 −px3 p(x3 + x4) + m (x1 + x2 + x3)

If we consider a sunset diagram

D 3 (4.109) with

2 2 D1 := l1 + m (4.110)

2 2 D2 := l2 + m (4.111)

2 2 D3 := (l1 − l2 + p) + m (4.112)

We can easily check that the associate matrix is now the same as S1,1,1,0,0. Therefore, all the associate polynomials for sunset can be derived from S1,1,1,0,0. CHAPTER 4. IBP OVER LP PARAMETRIZATION 99 4.4 Massless Double-boxes

Consider a planer massless double box diagram

p1 p2

D2 D4

p D1 2 (4.113)

We assign the following denominators

2 D1 = k1 (4.114)

2 D2 = (k1 − p1) (4.115)

2 D3 = (k1 − p1 − p2) (4.116)

2 D4 = (k1 − k2) (4.117)

2 D5 = (k2 − p1 − p2) (4.118)

2 D6 = (k2 − p1 − p2 − p3) (4.119)

2 D7 = k2 (4.120) and as two ISP as auxiliary denominators:

2 D8 = (k2 − p1) (4.121)

2 D9 = (k1 − p3) . (4.122)

Here we follow the same assignments for denominators in [26] of section 11. Because of the graph- theoretical symmetries of this configuration, we only consider three denominators D1,D2,D4, and the associated parameters as differentials. CHAPTER 4. IBP OVER LP PARAMETRIZATION 100

Lee-Pomeransky polynomial for this double box configuration is

G = −sx1x3x4 − sx1x3x5 − sx1x3x6 − sx1x3x7 − sx1x4x5 − sx1x5x7

+ sx2x4x6 − sx2x5x7 − sx3x4x7 − sx3x5x7 − sx4x5x7 + tx2x4x6

+ x1x4 + x1x5 + x1x6 + x1x7 + x2x4 + x2x5 + x2x6 + x2x7

+ x3x4 + x3x5 + x3x6 + x3x7 + x4x5 + x4x6 + x4x7 , (4.123) where we set

2 s = (p1 + p2) (4.124)

2 t = (p2 + p3) (4.125)

2 2 2 0 = p1 = p2 = p3 . (4.126)

We now consider x1, x2, and x4 derivative with the generator of

f = 1 . (4.127)

We have non-zero surface terms among all the three cases, but for each case, both sides are equal under IBP reductions, that is, they can be reduced into the same expression under traditional IBP reductions. CHAPTER 4. IBP OVER LP PARAMETRIZATION 101

5 For G1(1), the linear combination part becomes

1 3d Grhs(1) = (d − 2)Γ − 9 (−3(d − 6) (I (1, 1, 1, 1, 1, 1, 2, 0, 0) + I (1, 1, 1, 1, 1, 2, 1, 0, 0) 1 4 2 DBox DBox

+IDBox(1, 1, 1, 1, 2, 1, 1, 0, 0) + IDBox(1, 1, 1, 2, 1, 1, 1, 0, 0))

− 2s (IDBox(1, 1, 1, 1, 2, 1, 2, 0, 0) + IDBox(1, 1, 1, 2, 2, 1, 1, 0, 0) + IDBox(1, 1, 2, 1, 1, 1, 2, 0, 0)

+IDBox(1, 1, 2, 1, 1, 2, 1, 0, 0) + IDBox(1, 1, 2, 1, 2, 1, 1, 0, 0) + IDBox(1, 1, 2, 2, 1, 1, 1, 0, 0))) .

Now the left-hand side of the surface term can be computed from the following limit:

G| = sx x x − sx x x − sx x x − sx x x − sx x x + tx x x x1=0 2 4 6 2 5 7 3 4 7 3 5 7 4 5 7 2 4 6

+ x2x4 + x2x5 + x2x6 + x2x7 + x3x4 + x3x5 + x3x6 + x3x7 + x4x5 + x4x6 + x4x7 . (4.128)

Since this expression has no x1, x8, x9, we have

3d 3 Glhs(1) = (−)Γ − 9 (d − 6)(I (0, 1, 1, 2, 1, 1, 2, 0, 0) + I (0, 1, 1, 2, 1, 2, 1, 0, 0) 1 2 2 DBox DBox

+ IDBox(0, 1, 1, 2, 2, 1, 1, 0, 0) + IDBox(0, 1, 2, 1, 1, 1, 2, 0, 0)

+ IDBox(0, 1, 2, 1, 1, 2, 1, 0, 0) + IDBox(0, 1, 2, 1, 2, 1, 1, 0, 0)

+ IDBox(0, 1, 2, 2, 1, 1, 1, 0, 0) + IDBox(0, 2, 1, 1, 1, 1, 2, 0, 0)

+ IDBox(0, 2, 1, 1, 1, 2, 1, 0, 0) + IDBox(0, 2, 1, 1, 2, 1, 1, 0, 0)

+ IDBox(0, 2, 1, 2, 1, 1, 1, 0, 0))

+ s(IDBox(0, 1, 1, 2, 2, 1, 2, 0, 0) + IDBox(0, 1, 2, 1, 2, 1, 2, 0, 0)

+ IDBox(0, 1, 2, 2, 1, 1, 2, 0, 0) + IDBox(0, 2, 1, 1, 2, 1, 2, 0, 0)

−IDBox(0, 2, 1, 2, 1, 2, 1, 0, 0)) − tIDBox(0, 2, 1, 2, 1, 2, 1, 0, 0)) . (4.129)

5 Followings are also Kira’s compatible form, i.e., using (+ − − · · · −) metric. CHAPTER 4. IBP OVER LP PARAMETRIZATION 102

rhs lhs Our identity is 0 = G1 (1) − G1 (1) and we can show that this is consistent under traditional approaches.

If we consider x2 derivative with the same integrand, we have

1 3d Grhs(1) = (−) (d − 2)Γ − 9 (−3(d − 6)(I (1, 1, 1, 1, 1, 1, 2, 0, 0) + I (1, 1, 1, 1, 1, 2, 1, 0, 0) 2 4 2 DBox DBox

+ IDBox(1, 1, 1, 1, 2, 1, 1, 0, 0) + IDBox(1, 1, 1, 2, 1, 1, 1, 0, 0))

− 2sIDBox(1, 1, 1, 1, 2, 1, 2, 0, 0) + 2sIDBox(1, 1, 1, 2, 1, 2, 1, 0, 0)

+ 2tIDBox(1, 1, 1, 2, 1, 2, 1, 0, 0)) . (4.130)

The surface term is

3d Glhs(1) = (−)Γ − 9 2 2 3 (d − 6)(I (1, 0, 1, 2, 1, 1, 2, 0, 0) + I (1, 0, 1, 2, 1, 2, 1, 0, 0) 2 DBox DBox

+ IDBox(1, 0, 1, 2, 2, 1, 1, 0, 0) + IDBox(1, 0, 2, 1, 1, 1, 2, 0, 0)

+ IDBox(1, 0, 2, 1, 1, 2, 1, 0, 0) + IDBox(1, 0, 2, 1, 2, 1, 1, 0, 0)

+ IDBox(1, 0, 2, 2, 1, 1, 1, 0, 0) + IDBox(2, 0, 1, 1, 1, 1, 2, 0, 0)

+ IDBox(2, 0, 1, 1, 1, 2, 1, 0, 0) + IDBox(2, 0, 1, 1, 2, 1, 1, 0, 0)

+ IDBox(2, 0, 1, 2, 1, 1, 1, 0, 0)) + s(IDBox(1, 0, 1, 2, 2, 1, 2, 0, 0)

+DBox (1, 0, 2, 1, 2, 1, 2, 0, 0) + IDBox(1, 0, 2, 2, 1, 1, 2, 0, 0)

+ IDBox(2, 0, 1, 1, 2, 1, 2, 0, 0) + IDBox(2, 0, 1, 2, 2, 1, 1, 0, 0)

+ IDBox(2, 0, 2, 1, 1, 1, 2, 0, 0) + IDBox(2, 0, 2, 1, 1, 2, 1, 0, 0)

+IDBox(2, 0, 2, 1, 2, 1, 1, 0, 0) + IDBox(2, 0, 2, 2, 1, 1, 1, 0, 0))) . (4.131)

With IBP reductions, these two expressions fall into the same expression, i.e., our identity 0 = CHAPTER 4. IBP OVER LP PARAMETRIZATION 103

rhs lhs G2 (1) − G2 (1) is consistent under IBP reductions

Consider G4(1), the linear combination is

1 3d Glhs(1) = (−) (d − 2)Γ − 9 (−3(d − 6)(I (1, 1, 1, 1, 1, 1, 2, 0, 0) + I (1, 1, 1, 1, 1, 2, 1, 0, 0) 4 4 2 DBox DBox

+ IDBox(1, 1, 1, 1, 2, 1, 1, 0, 0) + IDBox(1, 1, 2, 1, 1, 1, 1, 0, 0)

+ IDBox(1, 2, 1, 1, 1, 1, 1, 0, 0) + IDBox(2, 1, 1, 1, 1, 1, 1, 0, 0))

− 2s(IDBox(1, 1, 1, 1, 2, 1, 2, 0, 0) + IDBox(1, 1, 2, 1, 1, 1, 2, 0, 0)

− IDBox(1, 2, 1, 1, 1, 2, 1, 0, 0) + IDBox(2, 1, 1, 1, 2, 1, 1, 0, 0)

+ IDBox(2, 1, 2, 1, 1, 1, 1, 0, 0)) + 2tIDBox(1, 2, 1, 1, 1, 2, 1, 0, 0)) . (4.132)

The surface term is

3d 3 Grhs(1) = (−)Γ − 9 (d − 6)(I (1, 1, 2, 0, 1, 1, 2, 0, 0) + I (1, 1, 2, 0, 1, 2, 1, 0, 0) 4 2 2 DBox DBox

+ IDBox(1, 1, 2, 0, 2, 1, 1, 0, 0) + IDBox(1, 2, 1, 0, 1, 1, 2, 0, 0)

+ IDBox(1, 2, 1, 0, 1, 2, 1, 0, 0) + IDBox(1, 2, 1, 0, 2, 1, 1, 0, 0)

+ IDBox(2, 1, 1, 0, 1, 1, 2, 0, 0) + IDBox(2, 1, 1, 0, 1, 2, 1, 0, 0)

+ IDBox(2, 1, 1, 0, 2, 1, 1, 0, 0)) + s(IDBox(1, 1, 2, 0, 2, 1, 2, 0, 0)

+ IDBox(1, 2, 1, 0, 2, 1, 2, 0, 0) + IDBox(2, 1, 1, 0, 2, 1, 2, 0, 0)

+ IDBox(2, 1, 2, 0, 1, 1, 2, 0, 0) + IDBox(2, 1, 2, 0, 1, 2, 1, 0, 0) +IDBox(2, 1, 2, 0, 2, 1, 1, 0, 0))) . (4.133)

These two expressions also go to the same expression under IBP reduction. As an example of zero surface, consider

f = x1, (4.134) CHAPTER 4. IBP OVER LP PARAMETRIZATION 104

with x1 derivative. This generator gives us the following linear combinations:

1 3d Glhs(x ) = Γ − 10 (dsI (2, 1, 1, 1, 2, 1, 2, 0, 0) + dsI (2, 1, 1, 2, 2, 1, 1, 0, 0) 1 1 2 2 DBox DBox

+ dsIDBox(2, 1, 2, 1, 1, 1, 2, 0, 0) + dsIDBox(2, 1, 2, 1, 1, 2, 1, 0, 0)

+ dsIDBox(2, 1, 2, 1, 2, 1, 1, 0, 0) + dsIDBox(2, 1, 2, 2, 1, 1, 1, 0, 0) 1 − (3d − 20)(−(d − 4)(I (2, 1, 1, 1, 1, 1, 2, 0, 0) + I (2, 1, 1, 1, 1, 2, 1, 0, 0) 2 DBox DBox

+ IDBox(2, 1, 1, 1, 2, 1, 1, 0, 0) + IDBox(2, 1, 1, 2, 1, 1, 1, 0, 0))

+ 2IDBox(1, 1, 1, 2, 1, 1, 2, 0, 0) + 2IDBox(1, 1, 1, 2, 1, 2, 1, 0, 0)

+ 2IDBox(1, 1, 1, 2, 2, 1, 1, 0, 0) + 2IDBox(1, 1, 2, 1, 1, 1, 2, 0, 0)

+ 2IDBox(1, 1, 2, 1, 1, 2, 1, 0, 0) + 2IDBox(1, 1, 2, 1, 2, 1, 1, 0, 0)

+ 2IDBox(1, 1, 2, 2, 1, 1, 1, 0, 0) + 2IDBox(1, 2, 1, 1, 1, 1, 2, 0, 0)

+ 2IDBox(1, 2, 1, 1, 1, 2, 1, 0, 0) + 2IDBox(1, 2, 1, 1, 2, 1, 1, 0, 0)

+ 2IDBox(1, 2, 1, 2, 1, 1, 1, 0, 0)) − 2sIDBox(1, 1, 1, 2, 2, 1, 2, 0, 0)

− 2sIDBox(1, 1, 2, 1, 2, 1, 2, 0, 0) − 2sIDBox(1, 1, 2, 2, 1, 1, 2, 0, 0)

− 2sIDBox(1, 2, 1, 1, 2, 1, 2, 0, 0) + 2sIDBox(1, 2, 1, 2, 1, 2, 1, 0, 0)

− 4sIDBox(2, 1, 1, 1, 2, 1, 2, 0, 0) − 4sIDBox(2, 1, 1, 2, 2, 1, 1, 0, 0)

− 4sIDBox(2, 1, 2, 1, 1, 1, 2, 0, 0) − 4sIDBox(2, 1, 2, 1, 1, 2, 1, 0, 0)

− 4sIDBox(2, 1, 2, 1, 2, 1, 1, 0, 0) − 4sIDBox(2, 1, 2, 2, 1, 1, 1, 0, 0) +2tIDBox(1, 2, 1, 2, 1, 2, 1, 0, 0)) (4.135)

IBP= 0 . (4.136) that is, once we apply IBP reduction on this surface term, we get zero. CHAPTER 4. IBP OVER LP PARAMETRIZATION 105

Similarly, with x1-derivative, the generators

2 x1x2, x1x2, x1x2x3, ··· (4.137) create identities that are consistent under IBP reductions. For example,

lhs lhs 2 lhs G1 (x1x2) = G1 (x1x2) = G1 (x1x2x3) = ··· = 0 . (4.138)

4.5 Direct IBP Reduction over Lee-Pomeransky Parametriza-

tion

In this section, we show that we can obtain the full IBP reduction for the class of massive bubble integrals using solely identities obtained within Lee-Pomeransky parametrization. We consider the case in which the two propagators in the bubble have different masses, namely

2 2 D1 := l + ma (4.139)

2 2 D2 := (l + p) + mb . (4.140)

The corresponding Lee-Pomeransky polynomial becomes:

2 2 2 2 2 2 2 G = mb x2 + p x1x2 + mb x1x2 + max1x2 + x2 + max1 + x1 . (4.141)

We first generate linear equations; taking

Gi(1),Gi(x1),Gi(x2), i = 1, 2, (4.142) CHAPTER 4. IBP OVER LP PARAMETRIZATION 106 we have six equations6

1  Γ(d − 1)((d − 2)I(1, 0) − 2m2I(2, 0)) = 0 2 a  1  Γ(d − 1)((d − 2)I(0, 1) − 2m2I(0, 2)) = 0  b 2 1  Γ(d − 2)((4 − d)I(2, 0) + 4m2I(3, 0)) = 0  a 2  1  Γ(d − 2)((4 − d)I(0, 2) + 4m2I(0, 3)) = 0  2 b  2 1  Γ(d − 3)(2mb I(0, 3) − (d − 3)I(0, 2)) = − (d − 2)Γ(d − 3) [(d − 3)I(1, 1)  2   2 2 2 2  −I(1, 2)(ma + mb − p ) − 2maI(2, 1)    2 1  Γ(d − 3)(2maI(3, 0) − (d − 3)I(2, 0)) = (d − 2)Γ(d − 3) [−(d − 3)I(1, 1)  2   2 2 2 2  +I(2, 1)(ma + mb − p ) + 2mb I(1, 2) (4.143) for nine variables

I(0, 1),I(1, 0),I(0, 2),I(1, 1),I(0, 2),I(0, 3),I(1, 2),I(2, 1),I(3, 0). (4.144)

It means if we choose three elements from above, they potentially form a basis, i.e., they are Master Integrals. Indeed this system is solvable, and we obtain the following reduction with three Master

6 This work is half-automated over Mathematica [108] environment. Therefore many simplifications are automatically done. First four equations has no surface terms, but the last two has. CHAPTER 4. IBP OVER LP PARAMETRIZATION 107

Integrals I(0, 1),I(1, 0) and I(1, 1):

(d − 2)I(1, 0) I(2, 0) → 2 2ma (d − 2)I(0, 1) I(0, 2) → 2 2mb (d − 4)(d − 2)I(0, 1) I(0, 3) → 2 2 8(mb ) 2 2 2 2 2 2 2 2 (d − 2)(I(0, 1)(−ma − mb + p ) + 2mb I(1, 0)) − 2(d − 3)mb I(1, 1)(ma − mb + p ) I(1, 2) → 2 2 2 2 2 2 2 2 2 2mb ((ma) − 2ma(mb + p ) + (mb − p ) ) 2 2 2 2 2 2 2 2 (d − 2)(2maI(0, 1) − I(1, 0)(ma + mb − p )) + 2(d − 3)maI(1, 1)(ma − mb − p ) I(2, 1) → 2 2 2 2 2 2 2 2 2 2ma ((ma) − 2ma(mb + p ) + (mb − p ) ) (d − 4)(d − 2)I(1, 0) I(3, 0) → 2 2 . 8(ma)

These identities are the same ones that can be obtained with any traditional code for IBP reduction. When we add more generators, e.g.,

2 2 Gi(x1x2),Gi(x1x2), i = 1, 2, (4.145) we have 10 equations for 14 variables. It looks like we need to add one more master integral say I(2, 2) in this system, but this is not the case. This system has distinct sectors, two tadpole sectors

I(n1 ≥ 1, 0),I(0, n2 ≥ 1) and the bubble sector I(n1 ≥ 1, n2 ≥ 1); we can solve them separately. Using the two equations that have nonzero on the left-hand sides, i.e., nonzero surface terms, in eq.(4.143) as bridge equations for these distinct sectors, we finally have only three master integrals, and the reductions are still consistent with traditional approaches, for example, the above candidate I(2, 2) can be written as a linear combination of I(0, 1),I(1, 0),I(1, 1). Let us show the reductions CHAPTER 4. IBP OVER LP PARAMETRIZATION 108

on IOLB(i, 4 − i), i = 0 ··· 4:

(d − 6)(d − 4)(d − 2)IOLB(0, 1) IOLB(0, 4) → 2 3 (4.146) 48(mb )

(d − 2) IOLB(1, 3) → − 2 2 2 2 2 2 2 2 2 2 2 8(mb ) ((ma) − 2(ma)((mb ) + (p )) + ((mb ) − (p )) )

2 3 2 2 2 2 2 2 2 × (d − 4)(mb ) + (ma) ((16 − 3d)(mb ) − 3(d − 4)(p )) + (ma)(4(p )((mb )

2 2 2 2 2 2 2 2 2 2 −3(p )) − d((mb ) − (p ))((mb ) + 3(p ))) + (d − 4)((mb ) − (p )) (3(mb ) − (p )) × I(0, 1) ! (d − 5)(d − 2)((m2) − (m2) + (p2)) + − a b × I(1, 0) 2 2 2 2 2 2 2 2 2 2 ((ma) − 2(ma)((mb ) + (p )) + ((mb ) − (p )) ) ! (d − 5)(d − 2)((m2) − (m2) + (p2)) + − a b × I(1, 1) (4.147) 2 2 2 2 2 2 2 2 2 2 ((ma) − 2(ma)((mb ) + (p )) + ((mb ) − (p )) ) 2 2 2 2 2 2 2 2 2 (d − 2) (−2(p )((d − 4)mb + ma) + (ma − mb )((9 − 2d)mb + ma) + (p ) ) IOLB(2, 2) → × I(0, 1) 2 2 2 2 2 2 2 2 2 2 2mb ((ma) − 2ma(mb + (p )) + (mb − (p )) ) (d − 2) ((2d − 9)(m2)2 − 2(d − 4)m2(m2 + (p2)) + (m2 − (p2))2) + a a b b × I(1, 0) 2 2 2 2 2 2 2 2 2 2 2ma ((ma) − 2ma(mb + (p )) + (mb − (p )) ) (d − 3) (−(d − 4)(m2 − m2)2 + (d − 6)(p2)2 + 2(p2)(m2 + m2)) + a b a b × I(1, 1) . 2 2 2 2 2 2 2 2 2 ((ma) − 2ma(mb + (p )) + (mb − (p )) ) (4.148)

4.6 Conclusions

In this chapter, we developed a procedure to generate identities over Schwinger-Feynman parameters, namely the differentiate-then-integrate process. In this construction, we observed that the fundamental theorem of calculus and the dimension regularization scheme play essential roles. We classified when non-zero surface terms appear and showed the consistency of the system of equations after plugging such non-zero surface terms in the left-hand sides. As related topics, we studied the pinching identification over Schwinger-Feynman parametrizations and dimension shifts. We examined the cases of bubble integrals, triangle integrals, kite integrals, and massless double-box integrals, showing the consistency of identities that we obtained. We checked the gen- CHAPTER 4. IBP OVER LP PARAMETRIZATION 109 erated identities by substituting traditional IBP reductions to see trivial 0 = 0 identity, and all the identities we created are consistent. Finally, we explicitly showed that we obtain the full IBP reduction in the case of the family of bubble integrals with different masses, using only the methods that we develop in this chapter. Chapter 5

Intersection Numbers and Schwinger-Feynman Parametrizations

As the complexity of diagrams increases, traditional approaches such as the reductions of Feyn- man integrals under Laporta’s algorithm require us to solve a gigantic system of linear equations, which are computationally very challenging and often represent the bottleneck in the evaluation techniques for these calculations. It will be a giant leap from the traditional IBP approaches, if we could bypass this time-consuming portion of the calculations, especially for computations of diagrams which involve a large number of loops and large numbers of legs. The Intersection Theory and Intersection Numbers [38, 39] were recently introduced in the study of Feynman integrals by Pierpaolo Mastrolia and Sebastian Mizera in [24, 109]. Intersection Numbers give a new method to project an arbitrary Feynman integral into an integral basis. In [24], using Baikov parametrization, the complete reduction of a two-loop non-planar diagram is made with full agreement with the literature; the master decomposition formula, we will discuss later in this chapter (eq.(5.19)), is also provided by this work. In [26], vast selections and examples of decomposition are presented, especially for one-forms, confirming the potential of this method. More recently, in [25, 27], a new recursive algorithm and its implementation for multi-fold integrals

110 CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 111 were presented. In this chapter, we focus on the one-form version of the Intersection Numbers technique to perform reductions over Schwinger-Feynman parametrizations, which is novel results within this framework. We first briefly introduce the master decomposition formula, following [26], which enable us to decompose a Feynman integral into a set of master integrals. Using the Gamma function as an example, we show a detailed calculation to get a recurrence relation. We then consider a few more examples of univariate cases. A tadpole integral in Schwinger parametrization is our first example to use Intersection Numbers to reproduce relations which are consistent with IBP identities. As a next example, we consider a bubble integral. The presence of a Dirac delta in the integrand of Feynman parametrization allows us to reduce the number of the parameters from two to one: in this manner, the Feynman parametrization of the bubble integral becomes again a univariate integral.

5.1 Formalism

Let us follow [26] to give an overview of the Intersection Numbers approach. For a function u and an integration domain Ω such that u|∂Ω = 0 of a boundary condition, we consider an integral of a function u over a parameter space:

Z I := u(x)ϕ(x) , (5.1) Ω where ϕ(x) is an m-form:

ϕ(x) =ϕ ˆ(x)dmx . (5.2) CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 112

Due to Stokes’ theorem, for any (m − 1)-form ξ, we have the following identity:

Z Z d(uξ) = uξ = 0 . (5.3) Ω ∂Ω

This equation leads to the following equation:

Z Z du 0 = (du ∧ ξ + udξ) = u ∧ +d ξ . (5.4) Ω Ω u

When we define a derivation ∇:

∇ω := d + ω∧ (5.5) du ω := d log u = , (5.6) u we have the following equation:

Z u∇ωξ = 0 . (5.7) Ω

This fact implies, for any (m − 1)-form ξ, both ϕ and (ϕ + ∇ωξ) give the same integral:

Z Z uϕ = I = u(ϕ + ∇ωξ) . (5.8) Ω Ω

Therefore, we must consider equivalent classes:

0 0 hϕ| := {ϕ |∃ξ, ϕ − ϕ = ∇ωξ } , (5.9) rather than an m-form. These equivalent classes are called twisted cocycles. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 113

Using the pairing notion, we can denote the original integral as:

Z hϕ |Ω] := uϕ . (5.10) Ω

This Dirac’s bra-ket-like notion clarifies that the bilinear property of the integral for both slots. Let us assume the number of linearly independent twisted cocycles is n. Let the basis of this linear space be

{he1| , ··· , hen|} . (5.11)

Our goal here is to express an arbitrary twisted cocycle hϕ| in terms of this basis. Due to their pioneering works in [38, 39], we can explicitly evaluate the following numbers:

Cij := hei |hj i , (5.12)

where {|hj i| 1 ≤ j ≤ n} is dual of cocycles. These numbers will be called the Intersection Num- bers.

Let us give the Master Decomposition Formula for hφ| in terms of the basis {he1| , ··· , hen|}. Consider, for an arbitrary |ψi, the following linearly dependent two sets

    hφ| , he1| , ··· , hen| , {|ψi , |h1i , ··· , |hni} (5.13) |{z} | {z } target basis CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 114 and their Gram matrix, i.e., an (n + 1)2 square matrix:

  hϕ |ψ i At   M :=   (5.14) BC

Aj := hϕ |hj i (5.15)

Bi := hei |ψ i , (5.16)

since both sets {hϕ| , he1| , ··· , hen|} and {|ψi , |h1i , ··· , |hni} are linearly dependent,

det M = 0 = det C hϕ |ψ i − AtC−1B , (5.17) where we use §B.2.1. Since C is non-singular by definition and |ψi is arbitrary, we obtain the master decomposition formula:

X −1 hϕ| = hϕ |hj i C ji hei| . (5.18) ij

Contracting with |Ω], we have the following decomposition:

Z X −1 I = hϕ |hj i C ji uei . (5.19) ij Ω

We now consider univariate cases, i.e., ϕ is a 1-form. Let n be the number of solutions ω = 0. This n is indeed the dimension of the linear space of twisted cocycles, which will be an upper bound of the numbers of master integrals. Define P as the set of poles of ω. An explicit formula for the univariate Intersection Numbers CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 115 is the following: [38, 39]

X hϕL |ϕR i := Resp (ψpϕR) , (5.20) p∈P

where ψp is a 0-form, i.e., a function that a solution to the following differential equation:

∇ωψ = ϕL , (5.21) around p ∈ P . Since the residues are defined locally, we take Laurent expansion around p ∈ P to determine ψp. To write a solution, we put the following ansatz:

X (j) j ψp(τ) ∼ ψp τ , (5.22) j∈{min,··· ,max} where τ = z − p and by counting the powers,

min := ordp(ϕL) + 1 (5.23)

max := −ordp(ϕR) − 1 . (5.24)

If min > max holds, i.e., {min, ··· , max} = ∅, then there is no solution, and the corresponding residue is zero.

5.1.1 Gamma function

Let us first take the Gamma function as an example. Recall the definition, for s > 0,1

Z ∞ Γ(s) = xs−1e−xdx . (5.25) x=0 1 We can extend s into all the complex plane but non-positive integers by analytic continuation. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 116

Our goal for this example is to show that the Intersection Numbers can provide a right recurrence relations among Gamma functions. From this integral, we let:

Ω := [0, ∞] (5.26)

u(x) := xs−1e−x (5.27) ˆ φ0 = φdx = 1dx , (5.28) in the following integral:

Z I(0) = uφ0 = hφ0| Ω] . (5.29) Ω

Under this setting, the one-form is

s − 1 ω := d ln u = d ln xs−1e−x = − 1 dx . (5.30) x

Let us determine the solution set:

{x|ω(x) = 0} . (5.31) x = s − 1 leads ω(x) = 0 and this is the unique solution, and we have

n = 1 (5.32) i.e., there is a master integral. ω is singular at x = 0. To see the behavior of ω around x = ∞, set

1 y := (5.33) x CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 117 and see the behavior of ω around y = 0:

−dy ω = ((s − 1)y − 1) . (5.34) y2

Thus ω is also singular at y = 0, and we have

P = {0, ∞} (5.35) as a set of all the singular points of ω. Let us investigate a relation between:

I(0) := hφ0| C] ,I(1) := hφ1| C] , (5.36)

i.e., between hφ1| , hφ0|, where

φ0 = 1dx, φ1 = xdx . (5.37)

This relation will give us a recurrence relation for Γ functions. Since this system essentially is one-dimensional, we obtain

−1 hφ1| = hφ1| φ0i hφ0| φ0i hφ0| (5.38) as is the same as the Beta function example in [26]. We first start from

hφ0| φ0i := Res|x=0 ψ0(x)φ0(x) + Res|x=∞ ψ0(x)φ0(x) . (5.39) CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 118

Since both orders are 1,

φ0 = dx ⇒ nL = nR = 0 (5.40) we have

min = nL + 1 = 1 > max = −nR − 1 = −1 . (5.41)

So there is no term in ψ0 which can contribute the residue.

1 At infinity, we set y = x and

−dy φ = φ = ⇒ n = n = −2 . (5.42) L R y2 L R

So we obtain

min = nL + 1 = −1 ≤ max = −nR − 1 = 1 , (5.43) and it suffices to take

a ψˆ = + b + cy (5.44) y around y = 0. The differential equation becomes a set of linear relations among the above unknown coefficients a, b, c:

a −a dψˆ = d + b + cy = + 0 + c dy (5.45) y y2 −dy a ωψˆ = ((s − 1)y − 1) + b + cy . (5.46) y2 y CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 119

1 Order-by-order comparison from the lowest power, Θ y3 , gives us the following three linear equations for three unknowns:

1 Θ : a = 0 (5.47) y3 1 Θ : b − as = −1 (5.48) y2 1 Θ :(−bs) + c + b = 0 . (5.49) y1

Our target now is the coefficient c, since the Intersection Number is given by −c. So let us first evaluate c using the third equation above:

c = (s − 1)b, , (5.50) i.e., we need to know b. Using the second equation, we have:

b = as − 1 . (5.51) and it suffices to know a, but from the first equation, we get a = 0. Every unknown is now fully determined, namely a = 0, b = −1, c = 1 − s, and the first Intersection Number that we need is

hφ0| φ0i = s − 1 . (5.52)

Results have been obtained by implementing these equations in Maxima [33], as shown in Figure 5.1.

Let us consider the other Intersection Number hφ1| φ0i. Similar to hφ0| φ0i case, we can neglect CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 120

Figure 5.1: Maxima session for hφ0| φ0i of the Gamma function to determine three unknown a, b, c in three linear equations.

(%i1) f(y) := a/y + b + c*y; a (%o1) f(y) := - + b + c y y (%i2) omega: (1- (s-1)*y)/y^2; 1 - (s - 1) y (%o2) ------2 y (%i3) deq: expand( diff(f(y),y,1) + omega * f(y) = -1/y^2 ); b s c b a s b a 1 (%o3) (- ---) + - + - - --- + -- + -- - c s + 2 c = - -- y y y 2 2 3 2 y y y y (%i4) makelist (coeff(deq,y,i),i,-3,0); (%o4) [a = 0, b - a s = - 1, (- b s) + c + b = 0, 2 c - c s = 0] (%i5) leq: makelist (coeff(deq,y,i),i,-3,-1); (%o5) [a = 0, b - a s = - 1, (- b s) + c + b = 0] (%i6) solve(leq, [a,b,c]); (%o6) [[a = 0, b = - 1, c = 1 - s]] CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 121 the contribution from p = 0 point, since

φL = φ1 = xdx ⇒ nL = 1, nR = 0, min = 1 + 1 = 2 > max = −0 − 1 = −1. (5.53)

Around x = ∞, we have

−dy φ = , n = −3 (5.54) L y3 L −dy φ = , n = −2 , (5.55) R y2 L and

min = −3 + 1 = −2 ≤ max = −(−2) − 1 = 1 . (5.56)

Therefore, it suffices to consider

a b ψ¯ = + + c + dy . (5.57) y2 y

The differential equation falls down into a set of linear equations. In a similar way we stated above, we can determine a = 0, b = −1, c = −s, d = s − s2 and

hφ1| φ0i = −d = s(s − 1) . (5.58)

We can easily solve them as is shown in Figure 5.2. Thus, we can finally evaluate the unknown coefficient between hφ1| and hφ0|, and

s(s − 1) hφ | = hφ | φ i hφ | φ i−1 hφ | = hφ | = s hφ | . (5.59) 1 1 0 0 0 0 s − 1 0 0 CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 122

Figure 5.2: Maxima session for hφ1| φ0i of the Gamma function to determine three unknown a, b, c in three linear equations.

(%i1) f(y) := a/y^2 + b/y + c + d*y; a b (%o1) f(y) := -- + - + c + d y 2 y y (%i2) omega: (1- (s-1)*y)/y^2; 1 - (s - 1) y (%o2) ------2 y (%i3) deq: expand( diff(f(y),y,1) + omega * f(y) = -1/y^3 ); csdcbscasbaa 1 (%o3) (- ---) + - + - - --- + ------+ -- - -- + -- - d s + 2 d = - -- y y y 2 2 3 3 3 4 3 y y y y y y y (%i4) leq: makelist (coeff(deq,y,i),i,-4,-1); (%o4) [a = 0, (- a s) + b - a = - 1, c - b s = 0, (- c s) + d + c = 0] (%i5) linsolve(leq, [a,b,c,d]); 2 (%o5) [a = 0, b = - 1, c = - s, d = s - s ] CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 123

Since we know that our target is the Gamma function:

Γ(s + 1) = sΓ(s) . (5.60)

Therefore the Intersection Number approach provides consistent recurrence relation.

5.2 Tadpoles in Schwinger Parametrization

We now follow [24, 26] and compute the Intersection Numbers needed to recover the recurrence relations for massive tadpoles, which can be extracted from eq.(3.54). Using Schwinger parametrization, let us define:

I(ν) := hφ0| C] (5.61)

C := [0, ∞] (5.62) dx φ := (5.63) 0 Γ(ν)

ν− d −1 2 u(x) := x 2 exp(−m x) . (5.64)

Define the one form:

" # ν − d − 1 ω := d ln u = 2 + (−m2) dx. (5.65) x

1 The point x = 0 is clearly a singular for ω. To observe the behavior around x = ∞, set y := x and we obtain:

d −dy ω = ν − − 1 y + (−m2) . (5.66) 2 y2 CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 124

So y = 0 is also another singular point for ω, and we obtain:

P := {0, ∞} . (5.67)

The solution ω(x) = 0 for x is

ν − d − 1 x = 2 (5.68) m2

and we have a master integral for this system.Let us consider a relation between hφ1| and hφ0|,

−1 hφ1| = hφ1| φ0i hφ0| φ0i hφ0| , (5.69) where

∞ 1 Z d dx ν− 2 −1 2 I(ν) = hφ0| C] = d x exp(−m x) (5.70) (4π) 2 x=0 Γ(ν) ∞ 1 Z d xdx ν− 2 −1 2 I(ν + 1) = hφ1| C] = d x exp(−m x) . (5.71) (4π) 2 x=0 Γ(ν + 1)

At x = 0, we have

dx φ = φ = , n = 0 (5.72) L 0 Γ(ν) L dx φ = φ = , n = 0 , (5.73) R 0 Γ(ν) R and

min = 0 + 1 = 1 > max = −0 − 1 = −1. (5.74)

So there is no-term which can contribute in the Intersection Numbers from the residue at x = 0. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 125

1 Around x = ∞, let us set y := x and

−dy φ = , n = n = −2 (5.75) 0 y2Γ(ν) L R

min = −2 + 1 = −1 ≤ max = −(−2) − 1 = 1 . (5.76)

So we can take

ψ¯ a = + b + cy (5.77) Γ(ν) y and

hφ0| φ0i = −c. (5.78)

Solving linear equations from the differential equation, we obtain:

2n − d − 2 hφ | φ i = −c = . (5.79) 0 0 2m4Γ(ν)

At x = 0,

xdx φ = , n = 1 (5.80) 1 Γ(ν + 1) L dx φ = , n = 0 , (5.81) 0 Γ(ν) R we obtain

min = 1 + 1 = 2 > max = −0 − 1 = −1. (5.82)

So no contribution from x = 0. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 126

1 At x = ∞, let y = x , we obtain

−dy φ = φ = , n = −3 (5.83) L 1 y3Γ(ν + 1) L −dy φ = φ = , n = −2 , (5.84) R 0 x2Γ(ν) R and

min = −3 + 1 = −2 ≤ max = −(−2) − 1 = 1. (5.85)

Therefore we can take

ψ¯ a b = + + c + ey (5.86) Γ(ν + 1) y2 y

hφ1| φ0i = −e. (5.87)

Solving linear equations, we obtain

(2ν − d − 2)(2ν − d) hφ | φ i = −e = . (5.88) 1 0 4m6Γ(ν + 1)

Therefore, we finally get

−1 hφ1| = hφ1| φ0i hφ0| φ0i hφ0| (5.89) (2ν − d − 2)(2ν − d) 1 = hφ | (5.90) 4m6Γ(ν + 1) 2ν−d−2 0 2m4Γ(ν) 1 d = 1 − hφ | . (5.91) m2 2ν 0

This coincides with eq.(3.54). CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 127

One can also compute to get the following expressions, using Lee-Pomeransky parametrization:

d hφ | φ i = − (5.92) 0 0 4(d − 1)(d + 1)(m2)2Γ(d − 1)2 d hφ | φ i = . (5.93) 1 0 8(d − 1)(d + 1)(m2)3Γ(d − 2)Γ(d − 1)

These imply that:

d 1 hφ | φ i hφ | φ i−1 = 1 − , (5.94) 1 0 0 0 2 m2 where an analytic form of the tadpole integral is given in the following:

d 1 Γ Z dxxν1−1 d 2 2 2− 2 Itadpole(ν1) = d x + m x (5.95) 2 Γ(d − ν ) Γ(ν ) (4π) 1 R> 1 − d u(x) := x + m2x2 2 (5.96) 1 hφ | := (5.97) 0 Γ(d − 1) x hφ | := . (5.98) 1 Γ(d − 2)

It coincides with eq.(3.54). So the Intersection Numbers can verify the correct recurrence relations for the tadpole integrals, which is essentially the same as the Gamma function example in §5.1.1.

5.3 Bubbles in Feynman Parametrization

Using Feynman parametrization, thanks to the presence of the Dirac delta, the full integral for a massive bubble becomes a univariate integral; therefore, we can compute the Intersection Numbers using our univariate methods. Let us first show that, with Feynman parametrization, we can reduce CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 128 the number of integration parameters from two to one:

d Z ν1−1 Z ν2−1 Γ ν1 + ν2 − 2 dx1x1 dx2x2 δ (1 − x1 − x2) Ibubble(ν1, ν2) = (5.99) d ν +ν − d (4π) 2 [0,1] Γ(ν1) [0,1] Γ(ν2) F 1 2 2 U d−ν1−ν2 d Z ν1−1 Z ν2−1 Γ ν1 + ν2 − 2 dx1x1 dx2x2 = d (4π) 2 [0,1] Γ(ν1) [0,1] Γ(ν2)

δ (1 − x1 − x2) × d (5.100) 2 2 2 ν1+ν2− d−ν1−ν2 (m (x1 + x2) + p x1x2) 2 (x1 + x2) d Z ν1−1 ν2−1 Γ ν1 + ν2 − 2 x1 (1 − x1) = d dx1 d . (5.101) 2 2 ν1+ν2− (4π) 2 Γ(ν1)Γ(ν2) [0,1] (p x1(1 − x1) + m ) 2

The corresponding integrand is

ν1−1 ν2−1 d x1 (1 − x1) Γ ν1 + ν2 − 2 i(ν1, ν2) := d . (5.102) 2 2 ν1+ν2− 2 Γ(ν )Γ(ν ) (p x1(1 − x1) + m ) 1 2

Kira [48] can perform IBP reduction for a massive bubble quite easily, and gives us a list of reduction as is shown in Figure 5.3. In order to avoid zero indices, namely I(1, 0), which would require to set ν = 0 in the Feynman parametrization, instead of taking {I(1, 1),I(1, 0)} as two MI of this system, we take the following as new set of MIs:

{I(1, 1),I(1, 2)} . (5.103)

We first express I(1, 0) with I(1, 2) and I(1, 1), and substitute this relation to express I(2, 2) with I(1, 1),I(1, 2). This process can be quickly done over the Maxima environment, see Figure 5.4. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 129

Figure 5.3: A few example of Kira’s IBP reduction of massive tadpoles. Note that, since Kira uses (+ − − · · · ) metric, we need to formally replace any d momenta contraction to its negative, and |ν| 1 1 (+−−··· )→(−++··· ) 1 with (−) sign factor, due to D0 = k02−m2 7→ −D0 .

oneLoopBubble[1,2] -> + oneLoopBubble[1,1]*((-d+3)/(pp-4*m2)) + oneLoopBubble[1,0]*((d-2)/((2*m2)*pp-8*m2^2)) , oneLoopBubble[2,2] -> + oneLoopBubble[1,1] *(((d^2-9*d+18)*pp+(4*d-12)*m2)/(pp^3+(-8*m2)*pp^2+(16*m2^2)*pp)) + oneLoopBubble[1,0] *(((d-2)*pp+(-2*d^2+10*d-12)*m2) /((m2)*pp^3+(-8*m2^2)*pp^2+(16*m2^3)*pp)) , oneLoopBubble[1,3] -> + oneLoopBubble[1,1] *(((d^2-7*d+12)*pp+(-4*d+12)*m2)/(2*pp^3+(-16*m2)*pp^2+(32*m2^2)*pp)) + oneLoopBubble[1,0] *(((d^2-6*d+8)*pp^2+((-8*d^2+48*d-64)*m2)*pp+(8*d^2-40*d+48)*m2^2) /((8*m2^2)*pp^3+(-64*m2^3)*pp^2+(128*m2^4)*pp)) CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 130

Figure 5.4: Maxima session for to convert a set of Master Integrals: {I(1, 1),I(1, 2)} ⇐⇒ {I(1, 0),I(1, 1)}.

(%i2) eqn:I12 = I11*(((-d)+3)/(pp-4*m2))+I10*((d-2)/((2*m2)*pp- 8*m2^2)) I10 (d - 2) I11 (3 - d) (%o2) I12 = ------+ ------2 pp - 4 m2 2 m2 pp - 8 m2 (%i3) sol:solve(eqn,[I10]) 2 2 I12 m2 pp - 8 I12 m2 + (2 I11 d - 6 I11) m2 (%o3) [I10 = ------] d - 2 (%i4) I22:ev(I11*(((d^2-9*d+18)*pp+(4*d-12)*m2) /(pp^3+((-8)*m2)*pp^2+(16*m2^2)*pp)) +I10*(((d-2)*pp+((-2)*d^2+10*d-12)*m2) /(m2*pp^3+((-8)*m2^2)*pp^2+(16*m2^3)*pp)),sol) 2 (%o4) (((d - 2) pp + ((- 2 d ) + 10 d - 12) m2) 2 (2 I12 m2 pp - 8 I12 m2 + (2 I11 d - 6 I11) m2)) 3 2 2 3 /((d - 2) (m2 pp - 8 m2 pp + 16 m2 pp)) 2 I11 ((d - 9 d + 18) pp + (4 d - 12) m2) + ------3 2 2 pp - 8 m2 pp + 16 m2 pp (%i5) factor(radcan(coeff(expand(I22),I11))) (d - 4) (d - 3) (%o5) ------pp (pp - 4 m2) (%i6) factor(radcan(coeff(expand(I22),I12))) 2 (pp - 2 d m2 + 6 m2) (%o6) ------pp (pp - 4 m2) CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 131

Thus we finally have

d − 3 m2(−p2 − 4m2) (−)1I(1, 0) IBP= (−)1+1I(1, 1) ∗ 2 m2 + (−)1+2I(1, 2) ∗ 2 (5.104) d − 2 d − 2 d − 3 m2(p2 + 4m2) I(1, 0) IBP= −I(1, 1) ∗ 2 m2 − I(1, 2) ∗ 2 . (5.105) d − 2 d − 2

Here we translate the results from Kira with (− + + ··· ) metric into (+ − − · · · ). Each integral carries (−)|~ν| of overall factor, and we need to set p02 → −p2. With this relation, we will show that the Intersection Theory can provide the following IBP identity:

(d − 4)(d − 3) 2(−p2 − 2dm2 + 6m2) I(2, 2) IBP= I(1, 1) ∗ − I(1, 2) ∗ . (5.106) p2(p2 + 4m2) p2(p2 + 4m2)

In other words, we decompose I(2, 2) into I(1, 1) and I(1, 2) of a new set of MI using the Inter- section Theory. Let us compute the Intersection Numbers. Define2

x(1 − x) u(x) := d (5.107) 2 2 4− (p x1(1 − x1) + m ) 2

ν1−2 ν2−2 d x (1 − x) Γ ν1 + ν2 − 2 φν ,ν := ∗ (5.108) 1 2 2 2 ν1−2+ν2−2 (p x1(1 − x1) + m ) Γ(ν1)Γ(ν2) to construct the integrands:

i(ν1, ν2) = φν1,ν2 u(x) , (5.109)

2Here we use (− + + ··· +) metric. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 132 see eq.(5.102). In particular, and for later usage:

d φ := 1 ∗ Γ 4 − (5.110) 2,2 2 (p2x (1 − x ) + m2) d φ := 1 1 ∗ Γ 3 − (5.111) 1,2 x 2 (p2x (1 − x ) + m2) d φ := 1 1 ∗ Γ 3 − (5.112) 2,1 1 − x 2 (p2x (1 − x ) + m2)2 d φ := 1 1 ∗ Γ 2 − . (5.113) 1,1 x(1 − x) 2

Contracting |[0, 1]] of the integration domain, we obtain:

hφij, [0, 1]] = I(i, j) , (5.114) up to normalization factors. With this u(x), we can define

ω := d log u(x) , (5.115) and we can show that there are three poles. Since the Intersection Number does not know the internal symmetry ν1 ↔ ν2, so we can try to set

{φ1,1, φ1,2, φ2,1} (5.116) as a set of basis. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 133

With the universal decomposition formula, we can write

! X −1 X X −1 hφ2,2| = hφ2,2, φK i C KJ hφJ | = hφ2,2, φK i C KJ hφJ | (5.117) J,K J K

CJK := hφJ , φK i , (5.118) where J and K run through:

J, K ∈ {(1, 1), (1, 2), (2, 1)} . (5.119)

In this construction, our C matrix is regular, and this guarantees our choice {(1, 1), (1, 2), (2, 1)} is linearly independent, i.e., they form a basis. We finally can compute the Intersection Numbers:

(d − 4)(d − 3) p2 + 2(d − 3)m2 p2 + 2(d − 3)m2 hφ | = hφ | + hφ | + hφ | . (5.120) 2,2 p2(p2 + 4m2) 1,1 p2(p2 + 4m2) 1,2 p2(p2 + 4m2) 2,1

We have

(d − 4)(d − 3) (5.121) p2(4m2 − p2) of the coefficient for I(1, 1). Since we have known that the second and third terms are essentially the same, by swapping two indices,

p2 + 2(d − 3)m2 p2 + 2(d − 3)m2 2(p2 − 2dm2 − 6m2) + = (5.122) p2(p2 + 4m2) p2(p2 + 4m2) p2(p2 + 4m2) of the coefficient for I(1, 2) (see eq.(5.106)). Similarly, we can compute the coefficients in the reduction of diagrams with higher powers in CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 134 the denominators, namely I(1, 3),I(2, 3) and I(3, 3). After defining

(1 − x) Γ 4 − d φ := ∗ 2 (5.123) 1,3 x Γ(3) d (1 − x) Γ 5 − 2 φ2,3 := 2 2 ∗ (5.124) (p x1(1 − x1) + m ) Γ(2)Γ(3) d x(1 − x) Γ 6 − 2 φ3,3 := ∗ , (5.125) 2 2 2 (p x1(1 − x1) + m ) Γ(3)Γ(3) we can obtain the following decomposition:

2 2 (d−4)(d−3)(2m +p ) 1 4−d 3−d 5−d 6m2−2dm2+4p2−dp2 − hφ1,1|+ + + hφ1,2|+ hφ2,1| hφ1,3| = 4m2p2(4m2+p2) 4 m2 p2 4m2+p2 8m2p2+2(p2)2 (5.126)

(d−4)(d−3)(2(d−5)m2+p2) 4(7+(d−6)d(m2)2+2(5d−22)m2p2+(d−4)p2 2(23+(d−10)d)m2+(d−6)p2 − hφ1,1|− hφ1,2|− hφ2,1| hφ2,3| = 4m2(p2)(4m2−p2)2 4m2(p2)(4m2−p2)2 2(p2)(4m2−p2)2 (5.127)

(d−4)(d−3) 12(d−5)(m2)2+(d−14)(d−5)m2p2+2(p2)2 − hφ1,1| hφ3,3| = 4m2(p2)2(4m2−p2)3

24(d−5)(d−3)(m2)3+2(d−16)(d−5)(d−3)(m2)2p2+4(4d−19)m2(p2)2−(d−4)(p2)3 + hφ1,2| 4m2(p2)2(4m2−p2)3

24(d−5)(d−3)(m2)3+2(d−16)(d−5)(d−3)(m2)2p2+4(4d−19)m2(p2)2−(d−4)(p2)3 + hφ2,1| 4m2(p2)2(4m2−p2)3 , (5.128) which are all in agreement with the traditional IBP calculations, after replacing I(1, 0) with I(1, 2). Therefore, we conclude that the computation based on Intersection Numbers techniques coincides with traditional IBPs for the bubble sector. Since the tadpole sector is solvable (see eq.(3.54) for an analytic expression) and since we can reduce all diagrams in the bubble sector in terms of I(1, 1) and I(1, 2), the only missing equation should be bridge equations between bubble sectors and tadpole sector. As we have observed in eq.(4.143), we can generate such bridging equations using Schwinger-Feynman parametrizations. This fact would formally allow writing any integral in terms of I(1, 1) and I(1, 0) for the same mass case. CHAPTER 5. INTERSECTION NUMBER AND SF PARAMETRIZATIONS 135 5.4 Conclusions

We showed how to use Intersection Theory together with Schwinger-Feynman parametrizations. We examined, in detail, the Gamma function as the simplest but non-trivial example of the class of functions in Schwinger-Feynman parametrizations. We then focused on a few more examples, namely a tadpole integral in Schwinger parametrization and a bubble integral in Feynman parametrization. As far as we know, these two examples are the only cases where we can apply univariate Intersection Numbers in Schwinger-Feynman parametrizations without relying on an algorithm for the multivariate case. Recently [25, 27], an algorithm for multivariate Intersection Numbers has been developed and applied to several interesting integrals in Baikov parametrization. Within these developments, it would be interesting to study more advanced examples of integrals in Schwinger-Feynman parametrizations, and compare the potential advantages and disadvantages of the two approaches. Chapter 6

Conclusions and Future Perspectives

Feynman integrals are one of the essential ingredients used in Quantum Field Theory. Not only they allow to obtain high-precision theoretical predictions for collider experiments, but also provide an interesting framework to study the rich mathematical structures that appear in scattering amplitudes. For this reason, they are still the subject of many theoretical studies. In this doctoral thesis, we discussed and applied advanced techniques for the calculations of scattering amplitudes which, on the one hand, allowed us to compute cross sections and differential distributions at high-precision and, on the other hand, gave us deep mathematical insights on the mathematical structures of Feynman integrals. We achieved the most precise predictions for the associated production of a top-quark pair and a Z boson at the Large Hadron Collider at a center-of-mass energy of 13 TeV. In this project, we employed resummation techniques for soft gluon emission correction to this process and estimated the total cross section and differential distributions to next-to-next-to-leading logarithm accuracy. We also examined Feynman integrals and associated quantities under the light of different parametrizations. We started from the defining momentum parametrization of Feynman integrals and introduced scalar integrals. We then presented four different parametrizations of Feynman integrals, namely a family of three Schwinger-Feynman parametrizations (Feynman, Schwinger, Lee-

136 CHAPTER 6. CONCLUSIONS AND FUTURE PERSPECTIVES 137

Pomeransky parametrization), and Baikov parametrization. We examined in details the integration regions of Baikov parametrization. With the Laplace expansion and well-known block matrix formula, we determined that the Baikov polynomial is quadratic in any off-diagonal element of the Gram matrix. This way, we obtained integration boundaries as closed forms and determined the integration regions for Baikov parametrization. We also discussed the Integration-By-Parts identities over momentum parametrization from the symmetry point of view and showed their mathematical properties. We then introduced Integration-By-Parts identities in Schwinger-Feynman parametrizations. Within this approach, we faced the appearance of non-zero surface terms, which we classified and included in our equation to obtain a class of IBP-like equations. We confirmed the consistency of our new method by applying them to one-loop examples (bubble and triangle) and two-loop examples (kite and double-box). We also achieved, in a few simple cases, a full reduction toward Master Integrals only using Schwinger-Feynman parametrizations. Further, we explored Intersection Theory and the use of Intersection Numbers within Schwinger- Feynman parametrizations of Feynman integrals. As a preliminary example, we studied the Gamma function and obtained the correct recurrence relation. Indeed, because of the form of its integrand, the Gamma function plays the same role in Schwinger-Feynman parametrizations as the Beta function within Baikov parametrization. Focusing on univariate cases, we also discussed in detail two physical examples: the tadpole integral in Schwinger parametrization and the bubble integral in Feynman parametrization. We saw a perfect agreement between the direct reductions obtained via Intersection Numbers and traditional Integration-By-Parts approaches. As for future perspectives, we want to continue and extend our study of Feynman integrals in terms of Schwinger-Feynman parametrizations. There are, in fact, several possible paths of research that we did not explore in full. In this thesis, we examined only the one-form (univariate) cases of Intersection Theory applications to obtain the reduction through the Master Decomposition Formula [26]. Recently, an algorithm for multivariate Intersection Numbers calculations has been CHAPTER 6. CONCLUSIONS AND FUTURE PERSPECTIVES 138 released [27]. This approach would allow us to probe more advanced and inspiring examples. We aim to apply Schwinger-Feynman parametrizations with multivariate Intersection Numbers and to observe their mathematical features. As another application of Schwinger-Feynman parametrizations, we would like to define kinematic cuts over these parametrizations. A cut was defined, originally, as the replacement rule D−1 7→ δ(D). Applying this rule, we practically set a propagator, i.e., a virtual particle in a Feyn- man diagram, on its mass-shell. However, there are other definitions. In Baikov parametrization, it would instead be defined as the replacement of the integration region into the contour around the poles [110–116], see also [36]. This process allows to highly simplifying calculations by reducing the number of integration variables. Notably, a maximal cut configuration will also enable us to use univariate techniques. In [26], the authors discussed and provided several examples of diagrams of increasing complexity under maximal cuts, and showed the power and wide range of applicability of their approach. Along the same lines, we will aim at rigorous definition and treatments for the cuts over Schwinger-Feynman parametrizations and apply our result, e.g., maximal cuts, to a wide variety of examples. History shows that a crossover between mathematics and physics, which happens whenever we can identify hints of the physical world within the abstract and complex mathematical literature, enables us to ascend physics researches toward the next levels. We are facing here such a paradigm change, where various parametrizations connected with different branches of mathematics expose the multi-faceted structures of Feynman Integrals. Many applied examples of this statement are available in the recent literature. In [24], Mastrolia and Mizera revealed the connection between Feynman Integrals in Baikov parametrization and hypergeometric functions using the framework of the Intersection Theory. In- tersection Theory is a branch of Algebraic Geometry whose the root initially was the study of the common points over two algebraic curves. Intersection Numbers [38, 39] are generalized answers of the intuitive question “how many points are shared by two algebraic curves?” It seems surprising CHAPTER 6. CONCLUSIONS AND FUTURE PERSPECTIVES 139 that such abstract settings can be used to study the properties of Feynman integrals. This framework, indeed, allows us to identify the number of Master Integrals and the decomposition of an arbitrary integral over such a set of Master Integrals. In [21], a deep connection was presented between topology and Feynman Integrals in Schwinger- Feynman parametrizations within D-module theory. D-modules were introduced by Sato in [117] to study the theory of linear partial differential equations. The authors of [21] illustrated an isomor- phism between annihilators and index-shift operators, where annihilators are differential operators that vanishes the integrand, e.g., IBP operator in Baikov parametrization see eq. (3.199). They also gave a concrete definition of the number of Master Integrals as a dimension of the vector space spanned by the integrals. They finally showed this dimension is given by the Euler characteristic that is determined by the zeros of Lee-Pomeransky polynomial. The Euler characteristics, one of the essential topological invariances, were initially defined for polyhedra and generalized within algebraic topology from the homological algebra. This showed that some underlying topological objects “know” the structures of the space spanned by the Feynman Integrals. In [31] the critical points of Baikov polynomial are used to determine the number of Master Integrals. The authors showed that the critical points of the characteristic polynomials (Baikov polynomial and Lee-Pomeransky polynomial) determine the number of Master Integrals. They also showed the number of Master Integrals was nothing but the sum of Milnor number of the proper critical points. The Milnor number is another invariant in algebraic geometry and differential geometry related to singular points; in a specific setting, the Milnor number is given by the dimension of a vector space, primarily used in the past to classify singularities [118, 119]. These approaches all aim at understanding Feynman integrals under different angles: from the properties of the integrands, from those of the IBP operators, and the analytic properties of polynomials. Although their frameworks look very different, they all give mutually consistent results. It implicates the existence of underlying structures that should concur and provide a unified broad picture for the Feynman Integrals. Appendices

140 Appendix A

An overview of quantum field theories

We review the simplest model in quantum field theories.1 A typical process to compute scattering amplitude is the following. Given an action integral or Lagrangian density, we derive Feynman rules. For a specific process, drawing Feynman diagrams for desired orders, we translate diagrams into Feynman integrals with Feynman rules. Since our focus, in this doctoral thesis, is scalar integrals, it suffices to use φ3 theory to derive relevant structures of Feynman integrals. Let us consider d dimensional time-space with

(−, +, +, ··· , +) (A.1) of the flat Minkowski metric, one time coordinate and d − 1 space coordinates. Let L = L [φ] be a Lagrangian density as a functional, and the action integral S[φ] := R ddxL [φ(x)] . Introducing a source J, we define the partition function using path integral:

Z Z Z[J] := N Dφ exp i S[φ] + ddJ(x)φ(x) , (A.2)

1 Here we follow [43, 93, 120].

141 APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 142 where the normalization N is determined by

Z[J = 0] = 1. (A.3)

That is, we have the following normalization:

Z N −1 = Dφ exp iS [φ] . (A.4)

Then an n-point correlation function is defined by the following functional derivative:

R Dφφ(x )φ(x ) ··· φ(x ) exp iS [φ] h0 |T [φ(x )φ(x ) ··· φ(x )]| 0i := 1 2 n (A.5) 1 2 n R Dφ exp iS [φ] δ δ δ = −i −i ··· −i Z[J] , δJ(x1) δJ(x2) δJ(xn) J=0 (A.6) where T is the time ordering symbol. To cancel irrelevant contribution from vacuum-to-vacuum transitions, we consider the connected n-point Green’s function which can be computed by

Z[J] = exp (iW [J]) (A.7) with W [J = 0] = 0 or equivalently Z[J = 0] = 1, using functional derivatives:

δ δ δ Gn(x1, ··· , xn) = −i −i ··· −i W [J] . (A.8) δJ(x1) δJ(x2) δJ(xn) J=0 APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 143

For later convenience, we introduce momentum space expression via Fourier transforms:

Z d Z d d k1 d kn G (x , ··· , x ) = ··· e−i(k1x1+···+knxn) n 1 n (2π)d (2π)d

d × (2π) δ(k1 + ··· + kn)Gn(k1, ··· , kn) . (A.9)

Especially we denote G(k) of the two point function in momentum space, where we have momentum conservation law and it reduces the number of arguments from two to one.

Z d Z d Z d d k1 d k2 d k G (x , x ) = e−i(k1x1+k2x2)(2π)dδ(k + k )G (k , k ) =: e−ik(x1−x2)G(k) . 2 1 2 (2π)d (2π)d 1 2 2 1 2 (2π)d (A.10)

Let us define scattering amplitudes. Using canonical quantization in free theory, we can create a one particle state of |ki by |ki := a†(~k) |0i , (A.11) where a, a† are determined as coefficients of the following Fourier expansion of fields:

Z ikx † −ikx φ(x) = dkf a(k)e + a (k)e (A.12)

dd−1~k dkf := (A.13) (2π)d−12ω

ω2 = ~k2 + m2 , (A.14) and we impose two additional conditions:

a(~k) |0i = 0, h0|0i = 1 . (A.15) APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 144

In d dimension time-space, we have the following normalization,

D E ~ ~0 d−1 d−1 ~ ~0 k k = (2π) 2ωδ (k − k ) . (A.16)

Since we have the following relations:

Z 1 1 dd−1xe−ikxφ(x) = a(~k) + e+2iωta†(−~k) (A.17) 2ω 2ω Z Z Z d−1 −ikx d−1 −ikx 0 ~0 ik0x † ~0 −ik0x d xe i∂0φ(x) = d xe dkf a(k )i(−iω)e + a (k )i(iω)e (A.18) 1 1 = a(~k) − e+2iωta†(−~k) , (A.19) 2 2

we obtain Z d−1 −ikx ~ d xe (i∂0φ(x) + ωφ(x)) = a(k) . (A.20)

Let us consider, as an example, 2 → 2 scattering process. Initially we have two particles of ~ ~ ~ ~ k1, k2 and finally we’ll detect k3, k4. To create a particle which is localized in momentum space ~ around k1, we introduce wave packet:

Z † d−1 ~ ~ † ~ a1 := d kf1(k − k1)a (k) (A.21)

~ 2 ! ~ k f1(k) ∼ exp − 2 (A.22) 2σ1

Suppose the initial state for this 2 → 2 scattering process as

† † |ii = lim a1(t)a2(t) |0i (A.23) t→−∞

and for the final state,

† † |fi = lim a3(t)a4(t) |0i . (A.24) t→∞ APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 145

We define the scattering amplitude of this process as

hf| ii . (A.25)

By the fundamental theorem of calculus, we can write:

Z ∞ † † † dt∂0a1(t) = a1(+∞) − a1(−∞) . (A.26) −∞

† If we substitute the above expression of a1, we obtain

Z ∞ Z ∞ Z Z † d−1 ~ ~ d−1 +ikx dt∂0a1 = dt∂0 d kf1(k − k1) d xe (−i∂0φ(x) + ωφ(x)) (A.27) −∞ −∞ Z Z d−1 ~ ~ d ikx = d kf1(k − k1) d x∂0 e (−i∂0φ(x) + ωφ(x)) (A.28) Z d−1 ~ ~ = d kf1(k − k1) Z d ikx ikx 2 × d x −iωe (−i∂0φ(x) + ωφ(x)) + e −i∂0 φ(x) + ω∂0φ(x)

(A.29) Z Z d−1 ~ ~ d ikx 2 2 = d kf1(k − k1) d xe −iω φ(x) − i∂0 φ(x) (A.30) Z Z d−1 ~ ~ d ikx h~ 2 2 2i = −i d kf1(k − k1) d xe k + m + ∂0 φ(x) (A.31) Z Z d−1 ~ ~ d hn~2 ikxo 2 2 i = −i d kf1(k − k1) d x ∂ e φ(x) + m + ∂0 φ(x) . (A.32)

Assuming the locality, we can use integration by parts and obtain

Z Z † † d−1 ~ ~ d ikx h 2 ~2 2i a1(+∞) − a1(−∞) = −i d kf1(k − k1) d xe ∂0 − ∂ + m φ(x) (A.33) Z Z d−1 ~ ~ d ikx 2 2 = −i d kf1(k − k1) d xe ∂ + m φ(x) . (A.34) APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 146

Taking its conjugation, we also obtain

Z Z † † d−1 ~ ~ d ikx 2 2 a1(−∞) = a1(+∞) + i d kf1(k − k1) d xe ∂ + m φ(x) (A.35) Z Z d−1 ~ ~ d −ikx 2 2 a3(+∞) = a3(−∞) + i d kf3(k − k3) d xe ∂ + m φ(x) . (A.36)

For a 2 → 2 process as we consider, we schematically write

D E † † hf| ii = 0 a3(+∞)a4(+∞)a1(−∞)a2(−∞) 0 (A.37) D h i E † † = 0 T a3(+∞)a4(+∞)a1(−∞)a2(−∞) 0 (A.38)

When we substitute, for example, a3(+∞) hits |0i and

Z Z d−1 ~ ~ d −ikx 2 2 hf| ii = 0 T a3(−∞) + i d kf3(k − k3) d xe ∂ + m φ(x)

i E † † × a4(+∞)a1(−∞)a2(−∞) 0 (A.39) Z Z d−1 ~ ~ d −ikx 2 2 = 0 + i d kf3(k − k3) d xe ∂ + m D h i E † † × 0 T φ(x)a4(+∞)a1(−∞)a2(−∞) 0 . (A.40)

Taking the limit ~ ~ d−1 ~ ~ fi(k − ki) 7→ δ (k − ki) (A.41) of the delta-normalization, we obtain

Z Z Z Z d d d d −ik1x1−ik2x2 +ik3x3+ik4x4 hf| ii = i d x3i d x4i d x1i d x2e e

2 2 2 2 2 2 2 2 × ∂(1) + m ∂(2) + m ∂(3) + m ∂(4) + m

× h0 |T [φ(x1)φ(x2)φ(x3)φ(x4)]| 0i . (A.42) APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 147

The multi-particle generalization is straightforward. As we have stated above, instead of using the correlation function, we use the corresponding connected Green’s function to define amplitude:

Z Z Z Z d d d d −ik1x1−ik2x2 +ik3x3+ik4x4 hf| ii = i d x3i d x4i d x1i d x2e e

2 2 2 2 2 2 2 2 × ∂(1) + m ∂(2) + m ∂(3) + m ∂(4) + m

× G(x1, x2, x3, x4) . (A.43)

Here we observe that to evaluate scattering amplitude hf| ii, it suffices to compute connected Green’s functions. Let us consider here a specific model, φ3 theory. For a given Lagrangian

1 1 g L := − (∂µφ)(∂ φ) − m2φ2 + φ3 , (A.44) 2 µ 2 3! let us extract φ2 terms

1 1 g L = − ∂µ (φ (∂ φ)) − φ∂2φ − φ2 + φ3 (A.45) 2 µ 2 3! 1 g ∼ − φ −∂2 + m2 φ + φ3 , (A.46) 2 3! where we ignore the total derivative term. From this expression, we pick up the following operator

d(x) := −∂2 + m2 . (A.47)

For the propagator of this Lagrangian, we take the inverse of this operator, or a Green’s function for d(x) such that d(x)d−1(x − y) = δ(x − y) . (A.48) APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 148

Let us Fourier transform d−1(x − y) momentum space:

Z ddk Z ddk δ(x − y) = d(x) e−ik(x−y)d−1(k) = −∂2 + m2 e−ik(x−y)d−1(k) (A.49) (2π)d (2π)d Z ddk = e−ik(x−y)(k2 + m2)dˆ−1(k) . (A.50) (2π)d

Therefore, we have the Feynman propagator

1 dˆ−1(k) = , (A.51) k2 + m2 − i0 where i0 is just a regulator, sometimes called a Feynman prescription. The corresponding Feynman rule for a propagator is k i := , (A.52) k2 + m2 − i0

1 2 2 where p is the momentum flow over this propagator. For vertex rule, let us recall the L ∼ − 2 φ (−∂ + m ) φ+ g 3 3! φ . The vertex is determined by the terms which has three or more field quantities; let us first rewrite the interaction:

3 ! g g Y Z φ3(x) = φ(x)φ(x)φ(x) = dy V (x; y , y , y )φ(y )φ(y )φ(y ) , (A.53) 3! 3! i 1 2 3 1 2 3 i=1 i.e., g V (x; y , y , y ) = δd(x − y )δd(x − y )δd(x − y ) . (A.54) 1 2 3 3! 1 2 3

The corresponding vertex in momentum space is

3 Z d ! Y d ki V (x; y , y , y ) = e−ik1(x−y1)−ik2(x−y2)−ik3(x−y3)Vˆ (k , k , k ) . (A.55) 1 2 3 (2π)d 1 2 3 i=1

This implies g Vˆ (k , k , k ) = . (A.56) 1 2 3 3! APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 149

The Feynman rule for the vertex is then given by the sum of all permutation, and

= ig . (A.57)

Armed with both propagators and vertices, we here state the Feynman rules:

1. For a given external constrains, draw lines with external momenta.

2. With 3-vertices, attach external lines.

3. Assign momentum conservation for each vertex, with (2π)d and Dirac delta.

4. Classify the order, by counting the order of coupling g. That is, draw all the diagrams up to a particular order.

1 5. Integrate over internal momenta with (2π)d .

Consider 2 → 2 scattering process, two particles are in, and two out. For simplicity, we consider all momenta incoming, i.e., the conservation will be the following form:

k1 + k2 + k3 + k4 = 0 . (A.58)

We have four external lines in our diagrams; the general structure can be depicted as,

(A.59) APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 150 where the central region is not yet known. Using Feynman rules, we perform perturbative expansion. The lowest order contribution Θ(g2) is represented by the following three diagrams:

1 3 1 4

1 3

hf| ii ∼= 2 4 + 2 4 + 2 3 + Θ(g4) (A.60)

Let us take the first diagram. 1 3

2 4 (A.61)

When we set the momentum k from the left vertex to the right, the left vertex has ig2πdδd(k −

−i k1 − k2) and the propagator carries k2+m2+i0 . Once we combine them with the right vertex, and then integrate them over the internal momentum k, we have a full term corresponding the diagram in eq.(A.61):

Z ddk −i ig(2π)dδd(k − k − k ) ig(2π)dδd(k + k + k) (A.62) (2π)d 1 2 k2 + m2 − i0 3 4

2 d d 1 = ig (2π) δ (k1 + k2 + k3 + k4) 2 2 . (A.63) (k1 + k2) + m − i0

Similarly, we can evaluate the other two diagrams. APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 151

For the next order Θ(g4), we take one loop diagrams, for example

1 3 l

2 4 (A.64)

After performing 3 of 4 integration, we have

Z ddl δd(k + k + k + k ) (2π)d 1 2 3 4

4 i i i i × (ig) 2 2 2 2 2 2 2 (A.65) (k1 + l) + m (k1 + k2 + l) + m (k1 + k2 + k4 + l) + m l + m as the corresponding term of above one loop diagram, where we omit regulators in the denominators. In order to perform the integration, the time component can be done as a contour integral, but the more convenient way is called Wick rotation. Consider an arbitrary one-loop integral. The integrand is a scalar function of loop momenta, i.e.,

Z ddlf(l2 − i0) , (A.66)

where i0 is a regulator which prevent the integration contour from passing poles. Choosing an l0

π contour from negative infinity to positive infinity, rotating counterclockwisely 2 , +i∞ to −i∞ π and 2 to close the contour, we have zero as the net result. Therefore, assuming the integrand goes APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 152

to 0 fast enough as |l0| → ∞, we obtain

Z ∞ Z i∞ Z −∞ 2 ¯2 ¯ ¯2 dl0f(l − i0) = − idl0f(l ) = − dl0f(l ) , (A.67) −∞ −i∞ ∞ where

¯2 2 2 ¯2 ¯2 l = −(il0) + l1 + ··· = l0 + l1 + ··· (A.68)

Therefore, we obtain the following equation:

Z Z ddlf(l2 − i0) = dd¯lf(¯l2) . (A.69)

This procedure is called Wick rotation; the integration becomes over Euclidean space with all plus metric, and indeed can be performed in spherical coordinates. With Feynman rules, we can construct any Feynman diagram, and scattering amplitudes up to any order. For an L-loop diagram, we can investigate graph-theoretical relations among the numbers of vertices, propagators of internal lines and loops. We consider connected graphs, that is for any pair of vertices, there at least is a path between them. For a Feynman diagram, we usually do not treat endpoints of external legs as vertices, but here we take them as also vertices. For example,

of a box diagram has, graph theoretically, 8 vertices and 8 edges, and 2 faces. Aplane graph is a drawing of a graph in a two-dimensional plane that the edges are non-crossing curvings. In QCD, for example, we also set another constraint on Feynman diagram, so-called APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 153 color-order, which makes external lines for a diagram be fixed with infinity. Under this setting, some Feynman diagram can not be drawn a planer graph, i.e., the diagram can not be drawn without crossing lines. We, physicists, call such a diagram a non-planar, but this convention is different from graph-theoretical sense. For a given graph, a face is a maximal open two-dimensional region that is bounded by edges, including the outside of the graph. We here show the Euler’s formula:

V − E + F = 2 (A.70) where V refers to the numbers of vertices, F for faces, and E for edges. We prove it by mathematical induction on the number of vertices V . Base case V = 1; we have2 F = 1 + E (A.71)

Here is an example of V = 1 case, we obtain E = 3,F = 4:

Thus, we have V − E + F = 1 + 1 = 2 (A.72) for a single vertex case. Induction step; let V ≥ 2 for a plane graph G. Since G is connected, there exist at least 2

2 We can also prove this statement by the mathematical induction on E. APPENDIX A. AN OVERVIEW OF QUANTUM FIELD THEORIES 154 distinct vertices, say a, b and the edges (a, b) between them. By contracting the edge (a, b) without merging any parallel edges, we obtain another, a smaller, graph G0 with V −1 vertices, E −1 edges but still F faces. E.g., by pinching the vertical central line (or contracting the central line),

b 7→ b

Thus, by the induction hypothesis, we have

(V − 1) − (E − 1) + F = 2 (A.73) for G0. It proves the Euler’s formula V − E + F = 2. With this Euler’s formula, we define the number of the loop as

L := F − 1 = E − V + 1. (A.74)

That is, the numbers of internal edges and the numbers of vertices fully determine the number of loops L. Appendix B

Determinants and Laplace Expansion

We summarize determinants of matrices and manipulations for their calculations, especially we derive a formula for the determinant of a block matrix. We consider only square matrices here.

B.1 Determinants and Laplace expansion

In this section we define the determinant of a matrix, Laplace expansion, and derivatives of the determinant. For n × n matrix A, we define the determinant:

n X σ Y det(A) := (−) Aiσ(i) , (B.1) σ i=1 where

σ : {1, ··· , n} → {1, ··· , n} (B.2) is a permutation over the set of indices. Let us derive the Laplace expansion of the determinant; consider the determinant of an n × n

155 APPENDIX B. DETERMINANTS AND LAPLACE EXPANSION 156 matrix A:

  A11 ··· A1n    . .  det(A) = det  . ..  . (B.3)     A1n Ann

If we expand det(A) via the first row (A11, ··· ,A1n), then we can write the determinant as:

1 1 det(A) = A11|A| 1 + ··· + A1n|A| n , (B.4) where, for example,

    A11 0 ··· 0 1 0 ··· 0     A A ··· A  A A ··· A  1  21 22 2n  21 22 2n A11|A| = det(A)| = det   = A11 det   1 A12=···=A1n=0  . .   . .   . . ...   . . ...   . .   . .      A1n An2 Ann A1n An2 Ann

(B.5)   1 0 ··· 0     0 A22 ··· A2n = A det   . (B.6) 11 . . .  . . ..  . .    0 An2 Ann

We here use both row and column eliminations. Now, by definition,

  1 0 ··· 0     A22 ··· A2n     0 A22 ··· A2n X Y   = (−)σ A =  . ..  . det . .  iσ(i) det  . .  (B.7) . . ...    . .  σ(1)=1 i6=1     An2 Ann 0 An2 Ann APPENDIX B. DETERMINANTS AND LAPLACE EXPANSION 157

For later convenience, we introduce,

  A11 A1,j−1 0 A1,j+1 A1,n    ......   ......      A A 0 A A   i−1,1 i−1,j−1 i−1,j+1 i−1,n   i  .  A j :=  0 . 0 1 0 ··· 0  (B.8)     Ai+1,1 Ai+1,j−1 0 Ai+1,j+1 Ai+1,n    ......   ......   . . . . .    An,1 Ai+1,j−1 0 An,j+1 An,n and

i i |A| j := det(A j) (B.9)   A11 A1,j−1 A1,j+1 A1,n    ......   ......      A A A A  X σ Y i+j  i−1,1 i−1,j−1 i−1,j+1 i−1,n = (−) Akσ(k) = (−) det   .   σ(i)=j k6=i Ai+1,1 Ai+1,j−1 Ai+1,j+1 Ai+1,n    ......   ......      An,1 Ai+1,j−1 An,j+1 An,n

(B.10)

If we consider the determinant of a matrix of a function of t,

n n d X d Y X d det A(t) = (−)σ A = det A , ··· ,A , A ,A , ··· ,A (B.11) dt dt iσ(i) ·1 ·j−1 dt ·j ·j+1 ·n σ i j=1 APPENDIX B. DETERMINANTS AND LAPLACE EXPANSION 158 where

  A1j     A2j  A =   . ·j  .  (B.12)  .   .    Anj

B.2 Block matrices and determinant

In this section, we derive a formula for the determinant of a block matrix. We first consider a lemma for the main statement. We claim, for n × n matrix A and m × m non-singular matrix D,

  A 0   det   = det(A) ∗ det(D). (B.13) CD

We prove it by mathematical induction on the size of A. If a is 1×1 matrix, then using Laplace expansion,

  a 0   det   = a det(D) + 0. (B.14) CD

Let n ≥ 2 and assume this is the case for n − 1 square matrix. Define, for n × n matrix A,

  A 0   X :=   . (B.15) CD APPENDIX B. DETERMINANTS AND LAPLACE EXPANSION 159

Let us use Laplace expansion on the first row:

  A 0 det   = det(X) = A |X|1 + ··· + A |X|1 + 0, (B.16)   11 1 1n n CD where

A |X|1 = det(X)| (B.17) 1j j A11=···=A1,j−1=A1,j+1=···=A1,n   0 ··· 0 A1j 0 ··· 0      A22 A2,j−1 0 A2,j+1 A2,n     ......  = det  ......  (B.18)       An,1 Ai+1,j−1 0 An,j+1 An,n    C0 D   A22 A2,j−1 A2,j+1 A2,n    ......   ......  1+j   = A1j(−) det   (B.19) A A A A   n,1 i+1,j−1 n,j+1 n,n    C00 D

1 = A1j|A| j det(D), (B.20) where C0 and C00 are some corresponding matrices. Note that, at the last equality, we have used the induction hypothesis for (n − 1) case. Therefore, we have

  n A 0 X det   = A |A|1 det(D) = det(A) det(D) (B.21)   1j j CD j=1 APPENDIX B. DETERMINANTS AND LAPLACE EXPANSION 160 since

n X 1 A1j|A| j (B.22) j=1 is nothing but the Laplace expansion of det(A).

B.2.1 Block matrices and determinant

We claim, for n × n matrix A and m × m non-singular matrix D,

  AB   −1 det   = det D ∗ det A − BD C (B.23) CD

For this block matrix

  AB   X :=   , (B.24) CD let us define

  1 −BD−1   Y :=   (B.25) 0 1

Note that

  1 0 t   det(Y ) = det(Y ) = det   = 1. (B.26) −(Dt)−1Bt 1 APPENDIX B. DETERMINANTS AND LAPLACE EXPANSION 161

We obtain

      1 −BD−1 AB A − BD−1C 0       Z := YX =     =   (B.27) 0 1 CD CD where we have used above statement at the last equality. This implies the following formula:

det(Z) = det(Y ) ∗ det(X) = det(X) = det A − BD−1C ∗ det(D) . (B.28) Bibliography

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