The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven to be a powerful tool for the computation of Feynman integrals. It has been observed that the differential equation of Feynman integrals can in many instances be transformed into a so-called canonical form, which significantly simplifies its integration in terms of iterated integrals. The main result of this thesis is an algorithm to compute rational transformations of differential equations of Feynman integrals into a canonical form. Apart from requiring the existence of such a rational transformation, the algorithm needs no further assumptions about the differential equation. In particular, it is applicable to problems depending on multiple kinematic variables and also allows for a rational dependence on the dimensional regulator. First, the transformation law is expanded in the dimensional regulator to derive differential equations for the coefficients of the transformation. Using an ansatz in terms of rational functions, these differential equations are then solved to determine the transformation. This thesis also presents an implementation of the algorithm in the Mathematica package CANONICA, which is the first publicly available program to compute transformations to a canonical form for differential equations depending on multiple variables. The main functionality and its usage are illustrated with some simple examples. Furthermore, the package is applied to state-of-the-art integral topologies appearing in recent multi-loop calculations. These topologies depend on up to three variables and include previously unknown topologies contributing to higher-order corrections to the cross section of single top-quark production at the LHC.
In a recent paper \cite{ft} a new powerful method to calculate Feynman diagrams was proposed. It consists in setting up a Taylor series expansion in the external momenta squared. The Taylor coefficients are obtained from the original diagram by differentiation and putting the external momenta equal to zero. It was demonstrated that by a certain conformal mapping and subsequent resummation by means of Padé approximants it is possible to obtain high precision numerical values of the Feynman integrals in the whole cut plane. The real problem in this approach is the calculation of the Taylor coeff…
Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct $d\log$-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales…
The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to $d$-dimensional $d\log$-form integrands. In this work, we introduce the concept of generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct $d$-dimensional $d\log$-form integrands, and discuss how to convert them to Feynman integral…
We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of Feynman integrals, we present some selected new results for their expansions around $4-2\varepsilon$ dimensions.
We suggest a mathematical definition of the notion of master integrals and present a brief review of algorithmic methods to solve reduction problems for Feynman integrals based on integration by parts relations. In particular, we discuss a recently suggested reduction algorithm which uses Groebner bases. New results obtained with its help for a family of three-loop Feynman integrals are outlined.
The reduction of Feynman integrals to master integrals is an algebraic problem that requires algorithmic approaches at the modern level of calculations. Straightforward applications of the classical Buchberger algorithm to construct Groebner bases seem to be inefficient. An essential modification designed especially for this problem has been worked out. It has been already applied in two- and three-loop calculations. We are also suggesting to combine our method with the Laporta's algorithm.
The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical…
The integration of differential equations of Feynman integrals can be greatly facilitated by using a canonical basis. This paper presents the Mathematica package CANONICA, which implements a recently developed algorithm to automatize the transformation to a canonical basis. This represents the first publicly available implementation suitable for differential equations depending on multiple scales. In addition to the presentation of the package, this paper extends the description of some aspects of the algorithm, including a proof of the uniqueness of canonical forms up to constant transformati…
We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts iden…
Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This novel technique is based on a method due to Hoschele et al. and relies only on the knowledge of a single integral of uniform transcendental weight. As a corollary, the algorithm can also be used to test the uniform transcendentality of a given integral. We discuss the application to several cutting-edge examples, including non-planar four-loop HQET and non-p…
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. nontrivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer implementation of our algorithm in a simple example of four-loop generalized sun-set integrals with three equal non-zero masses. Our code provides values of the ma…
In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which rationalizes the square roots. In this paper, we give an algorithm for rationalizing roots. The algorithm is applicable whenever the algebraic hypersurface associated with the root has a point of multiplicity $(d-1)$, where $d$ is the degree of the algebraic hypersurface. We show that one can use the algorithm iteratively to rationalize multiple roots simulta…
Over recent years we have witnessed tremendous progresses in our understanding on the angular momentum decomposition. In the context of the proton spin problem in high energy processes the angular momentum decomposition by Jaffe and Manohar, which is based on the canonical definition, and the alternative by Ji, which is based on the Belinfante improved one, have been revisited under light shed by Chen et al. leading to seminal works by Hatta, Wakamatsu, Leader, etc. In chiral physics as exemplified by the chiral vortical effect and applications to the relativistic nucleus-nucleus collisions, s…
We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a simplified Feynman integral topology which depends on a Feynman parameter that should be integrated over. For each integral family, we set up a system of differential equations which we solve in terms of a piecewise collection of generalized series expansions in the Feynman parameter. These generalized series expansions can be efficiently integrated term by term, a…
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be …
We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel'fand-Kapranov-Zelevinsky (GKZ) system. In order to define such GKZ system, we consider the polynomial obtained from the Symanzik polynomials $g=\mathcal{U}+\mathcal{F}$ as having indeterminate coefficients. Noncompact integration cycles can be determined from the coamoeba---the argument mapping---of the algebraic variety associated with $g$. In general, we add a deformation to $g$ in order to define integrals of generic graphs as linear combinations of their canonical…
In this paper, we present the universal structure of the alphabet of one-loop Feynman integrals. The letters in the alphabet are calculated using the Baikov representation with cuts. We consider both convergent and divergent cut integrals and observe that letters in the divergent cases can be easily obtained from convergent cases by applying certain limits. The letters are written as simple expressions in terms of various Gram determinants. The knowledge of the alphabet enables us to easily construct the canonical differential equations of the $ d\log $ form and aids in bootstrapping the symbo…
Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms
Technical University of Munich（慕尼黑工业大学）；University of Oxford（牛津大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））
Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $ε$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $ε$. In a first step, the integrals are expressed by hypergeometric multi-sums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested product-sum expressions. In particular, we enhance the known multi-sum algorithms to derive recur…
Two-loop corrections to scattering amplitudes are crucial theoretical input for collider physics. Recent years have seen tremendous advances in computing Feynman integrals, scattering amplitudes, and cross sections for five-particle processes. In this paper, we initiate the study of the function space for planar two-loop six-particle processes. We study all genuine six-particle Feynman integrals, and derive the differential equations they satisfy on maximal cuts. Performing a leading singularity analysis in momentum space, and in Baikov representation, we find an integral basis that puts the d…
In 2013, Henn proposed a special basis for a certain class of master integrals, which are expressible in terms of iterated integrals. In this basis, the master integrals obey a differential equation, where the right hand side is proportional to $ε$ in $d=4-2ε$ space-time dimensions. An algorithmic approach to find such a basis was found by Lee. We present the tool epsilon, an efficient implementation of Lee's algorithm based on the Fermat computer algebra system as computational backend.
Chebyshev Approximations of Feynman Integrals for Collider Physics
CERN（欧洲核子研究中心）；Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh（爱丁堡大学物理与天文学学院希格斯理论物理中心）；Instituto Superior Técnico (IST), Universidade de Lisboa（里斯本大学高级技术研究所）；LIP（葡萄牙粒子与核物理研究所）；Department of Physics and Astronomy, Ghent University（根特大学物理与天文学系）
We explore the idea to bootstrap Feynman integrals using integrability. In particular, we put the recently discovered Yangian symmetry of conformal Feynman integrals to work. As a prototypical example we demonstrate that the D-dimensional box integral with generic propagator powers is completely fixed by its symmetries to be a particular linear combination of Appell hypergeometric functions. In this context the Bloch-Wigner function arises as a special Yangian invariant in 4D. The bootstrap procedure for the box integral is naturally structured in algorithmic form. We then discuss the Yangian …
We study the symbology of planar Feynman integrals in dimensional regularization by considering geometric configurations in momentum twistor space corresponding to their leading singularities (LS). Cutting propagators in momentum twistor space amounts to intersecting lines associated with loop and external dual momenta, including the special line associated with the point at infinity, which breaks dual conformal symmetry. We show that cross-ratios of intersection points on these lines, especially those on the infinity line, naturally produce symbol letters for Feynman integrals in $D=4-2ε$, wh…
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that allows based on properties of the space-time…
We report a calculation of the perturbative matching coefficients for the transverse-momentum-dependent parton distribution functions for quark at the next-to-next-to-next-to-leading order in QCD, which involves calculation of non-standard Feynman integrals with rapidity divergence. We introduce a set of generalized Integration-By-Parts equations, which allows an algorithmic evaluation of such integrals using the machinery of modern Feynman integral calculation.
Two-loop Six-point Planar Massless Feynman Integrals to Higher $ε$ Orders
University of Science and Technology of China（中国科学技术大学）；Johannes Gutenberg-Universität Mainz（美因茨约翰内斯古腾堡大学）；Universitá di Bologna e INFN, Sezione di Bologna（博洛尼亚大学及意大利国家核物理研究所博洛尼部分所）；Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
We construct a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules. The off-shell continuation appears as a natural consequence of using light-front quantisation surfaces. The construction extends to include massless fermions.
Following the work of Brown, we can canonically associate a family of motivic periods -- called the motivic Feynman amplitude -- to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown's "small graphs principle". This serves as motivation for explicitly computing the motivic Galois action, or, dually, the coaction of the Hopf algebra of functions on the motivic Galois group. In this paper, we study…
In recent three--loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short $S$-sums) arise. They are characterized by rational (or real) numerator weights also different from $\pm 1$. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In thi…
Complete computation of all three-loop five-point massless planar integrals
LAPTh, Université Savoie Mont Blanc, CNRS（拉普特实验室，萨瓦蒙布朗大学，法国国家科学研究中心）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（中国科学技术大学理论交叉中心）；State Key Laboratory of Nuclear Physics and Technology, Institute of Quantum Matter, South China Normal University（华南师范大学量子物质研究所核物理与技术国家重点实验室）；Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Nuclear Science（广东省物质结构与基本相互作用卓越基础研究中心，广东省核科学重点实验室）；Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences (HIAS, UCAS)（中国科学院大学杭州高等研究院）；Max-Planck-Institut für Physik, Werner-Heisenberg-Institut（马克斯·普朗克物理研究所，维尔纳·海森堡研究所）；Peng Huanwu Center for Fundamental Theory（彭桓武理论基础研究中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of penta…
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut（马克斯·普朗克物理研究所）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数理科学研究所）；Leung Center for Cosmology and Particle Astrophysics（梁林宇宙学与粒子天体物理中心）；INFN, Sezione di Milano（意大利国家核物理学院米兰分部）
We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this rang…
The diagrammatic coaction underpins the analytic structure of Feynman integrals, their cuts and the differential equations they admit. The coaction maps any diagram into a tensor product of its pinches and cuts. These correspond respectively to differential forms defining master integrals, and integration contours which place a subset of the propagators on shell. In a canonical basis these forms and contours are dual to each other. In this talk I review our present understanding of this algebraic structure and its manifestation for dimensionally-regularized Feynman integrals that are expandabl…
