The Mathematics of Linear Relations between Feynman Integrals

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Monday, 18 March 201909:0009:00 - 09:2009:2009:20 - 09:30Room: 02.43009:30 Laporta algorithm for multi-loop vs multi-scale problems - Johann Usovitsch (Trinity College Dublin)Laporta algorithm for multi-loop vs multi-scale problems
- Johann Usovitsch (Trinity College Dublin)

09:30 - 10:30Room: 02.43010:3010:30 - 11:00Room: 02.43011:00 FIRE6: Feynman Integral REduction with Modular Arithmetic - Alexander Smirnov (Moscow State University)FIRE6: Feynman Integral REduction with Modular Arithmetic- Alexander Smirnov (Moscow State University)

11:00 - 12:00Room: 02.430 FIRE is a program performing reduction of Feynman integrals to master integrals. The C++ version FIRE was presented in 2014. There have been multiple changes and upgrades since then including the possibility to use multiple computers for one reduction task and to perform reduction with modular arithmetic. The goal of this talk is to present the current version of FIRE that is 6.2.12:0012:00 - 14:00Room: 02.43014:00 Four-loop quark form factor with quartic fundamental colour factor - Vladimir Smirnov (Moscow State University)Four-loop quark form factor with quartic fundamental colour factor- Vladimir Smirnov (Moscow State University)

14:00 - 14:30Room: 02.430 An analytical result for the four-loop QCD corrections for the colour structure (dabcdF)^2 to the massless non-singlet quark form factor is obtained. The evaluation involves non-trivial non-planar integrals diagrams with master integrals in the top sector. To evaluate the master integrals second mass scale is introduced and the corresponding differential equations are solved with respect to the ratio of the two scales. Analytical results for the cusp and collinear anomalous dimensions, and the finite part of the form factor are presented.14:30 Constructing multi-loop scattering amplitudes with manifest singularity structure - Robert Schabinger (Michigan State University)Constructing multi-loop scattering amplitudes with manifest singularity structure- Robert Schabinger (Michigan State University)

14:30 - 15:00Room: 02.430 The infrared exponentiation properties of dimensionally-regularized multi-loop scattering amplitudes are typically hidden at the level of the integrand, materializing only after integral evaluation. In this talk, we provide evidence that this long-standing problem may be addressed by introducing an appropriate integral basis which is simultaneously finite and uniform weight.15:00 Integrand reduction for particles with spins - Hui LuoIntegrand reduction for particles with spins- Hui Luo

15:00 - 15:30Room: 02.430 Scattering amplitudes with spinning particles are shown to decompose into multiple copies of simple building blocks to all loop orders, which can be used to efficiently reduce these amplitudes to sums over scalar integrals. For example, analytic results could be obtained for the five gluon, two loop, and four gluon, three loop planar scattering amplitudes in pure Yang-Mills theory as well as for leading singularities to even higher orders.15:3015:30 - 16:00Room: 02.43016:0016:00 - 17:00Room: 02.430 -
Tuesday, 19 March 201909:30 Computation of holonomic systems for Feynman amplitudes associated with some simple diagrams - Toshinori Oaku (Tokyo Woman's Christian University)Computation of holonomic systems for Feynman amplitudes associated with some simple diagrams
- Toshinori Oaku (Tokyo Woman's Christian University)

09:30 - 10:30Room: 02.430 Holonomic systems are a class of systems of linear (partial or ordinary) differential equations. One of the most fundamental properties of a holonomic system is that its solution space is finite-dimensional. A Feynman amplitude is the integral of a rational function, or more generally, the product of complex powers of polynomials. Hence we can, in principle, apply the following two facts in (computational) $D$-module theory: 1. For (multivariate) polynomials $f_1,\dots,f_d$ and complex numbers $\lambda_1,\dots,\lambda_d$, the multi-valued analytic function $f_1^{\lambda_1}\cdots f_d^{\lambda_d}$ satisfies a holonomic system, which can be computed algorithmically. 2. If a function satisfies a holonomic system, its integral with respect to some of its variables also satisfies a holonomic system, which can be computed algorithmically. In the integration, it would also be natural to ragard the integrand as a local cohomology class associated with $f_1,\dots,f_d$, which roughly corresponds to the 'residue' of $f_1^{\lambda_1}\cdots f_d^{\lambda_d}$ at $\lambda_1 = \cdots = \lambda_d = -1$ (at least in positive mass case for external diagrams). There are also algorithms for computing a holonomic system for such a cohomology class. However, actual computation, especially of integration, is hard in general because of the complexity. I shall present some worked out examples together with an interpretation based on microlocal analysis.10:3010:30 - 11:00Room: 02.43011:00 Computer algebra and ring theory help Feynman integrals - Viktor Levandovskyy (RWTH Aachen)Computer algebra and ring theory help Feynman integrals- Viktor Levandovskyy (RWTH Aachen)

11:00 - 12:00Room: 02.430 Scientists, dealing with Feynman integrals, use plenty of algebraic techniques. We will give an overview of appearing algebras, morphisms between them and localizations in a structured way, also giving details on the arising algorithmic questions and possible realizations of the latter in modern computer algebra systems. If the time allows, Gel'fand-Kirillov dimension for algebras and modules (clarifying the overused notion of "holonomic module") as well as the notion of a holonomic rank (of a localized module) will be explained.12:0012:00 - 14:00Room: 02.43014:00 The algebraic Mellin transform, after Loeser and Sabbah - Thomas Bitoun (University of Toronto)The algebraic Mellin transform, after Loeser and Sabbah- Thomas Bitoun (University of Toronto)

14:00 - 14:45Room: 02.430 We will present the essentials of Loeser and Sabbah's theory of the Mellin transform for D-modules. It is a fundamental ingredient in our work with Bogner, Klausen and Panzer on Feynman integral relations from parametric annihilators. In particular, for a holonomic D-module M, we will express the dimension of its Mellin transform as the Euler characteristic of M. We will not assume any prior knowledge of D-module theory, so the talk should be accessible to non-specialists.14:45 Feynman integral relations from Parametric annihilators - Erik PanzerFeynman integral relations from Parametric annihilators- Erik Panzer

14:45 - 15:15Room: 02.43015:15 Counting master integrals with PSLQ - Stefano LaportaCounting master integrals with PSLQ- Stefano Laporta

15:15 - 16:15Room: 02.430 In this talk I describe the use of the integer relation algorithm PSLQ to determine with high reliability the number of independent (master) integrals for some troublesome topologies, using the Baikov representation.16:1516:15 - 17:15Room: 02.430 -
Wednesday, 20 March 201909:30 Integration-by-parts reductions via unitarity cuts and syzygies - Kasper Larsen (University of Southampton)Integration-by-parts reductions via unitarity cuts and syzygies
- Kasper Larsen (University of Southampton)

09:30 - 10:30Room: 02.430 In this talk I will discuss a new approach for generating integration-by-parts reductions of Feynman integrals. The approach makes use of ideas from modern unitarity and from algebraic geometry, which I will explain. I then demonstrate the power of the approach by performing fully analytically IBP reductions relevant for two-loop five-gluon scattering in QCD. I will also discuss Azurite, a publicly available code for providing bases of loop integrals.10:3010:30 - 11:00Room: 02.43011:00 Massively Parallel Methods and Other Trends in the Design of Computer Algebra Systems - With a Focus on Feynman Integral Reduction - Janko Böhm (Technische Universität Kaiserslautern)Massively Parallel Methods and Other Trends in the Design of Computer Algebra Systems - With a Focus on Feynman Integral Reduction- Janko Böhm (Technische Universität Kaiserslautern)

11:00 - 12:00Room: 02.430 With a view toward Feynman Integral Reduction, I discuss fundamental algorithms in commutative algebra as implemented in the open source computer algebra system Singular. I introduce a highly efficient framework for massively parallel computations in computer algebra, which combines Singular with the workflow management system GPI-Space and is based on the idea of separating coordination and computation. I also give an outlook on the potential of cross border methods in the next generation computer algebra system OSCAR, which integrates Singular with the systems GAP for group theory, polymake for polyhedral geometry, and Antic for number theory. Using commutative algebra methods, I address the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals with degree-four numerators.12:0012:00 - 14:00Room: 02.43014:00 Feynman Integrals and Intersection Theory - Pierpaolo Mastrolia (Padua University)Feynman Integrals and Intersection Theory- Pierpaolo Mastrolia (Padua University)

14:00 - 15:00Room: 02.430 I will discuss how intersection theory controls the algebra of Feynman integrals, and show how their direct decomposition onto a basis of master integrals can by achieved by projection, using intersection numbers. After introducing a few basics concepts of intersection theory, I will show the application of this novel method, first, to special mathematical functions, and, later, to Feynman integrals on the maximal cuts, also explaining how differential equations and dimensional recurrence relations for master integrals can be directly built by means of intersection numbers. The presented method exposes the geometric structure beneath Feynman integrals, and offers the computational advantage of bypassing the system-solving strategy characterizing the reduction algorithms based on integration-by-parts identities.15:00 Algorithmic approaches to syzygies for no-dot or no-numerator relations - Andreas von Manteuffel (Michigan State University)Algorithmic approaches to syzygies for no-dot or no-numerator relations- Andreas von Manteuffel (Michigan State University)

15:00 - 16:00Room: 02.430 The calculation of syzygies for the generation of linear relations between Feynman integrals can be challenging for state-of-the-art calculations in quantum chromodynamics. I will discuss how basic linear algebra methods can be employed to efficiently compute the relevant syzygies for different applications: relations in the Baikov representation which avoid squared propagators (“dots”) and relations in the Lee-Pomeransky representation which avoid numerators. In the latter case I will discuss applications for relations involving also higher order differential operators.16:0016:00 - 17:00Room: 02.430 -
Thursday, 21 March 201909:30 L-loop watermelon and sunrise graphs: differential equations, index and dimension shifting relations, and quadratic constraints - Roman Lee (Budker Institute of Nuclear Physics, Novosibirsk)L-loop watermelon and sunrise graphs: differential equations, index and dimension shifting relations, and quadratic constraints
- Roman Lee (Budker Institute of Nuclear Physics, Novosibirsk)

09:30 - 10:30Room: 02.430 We derive reduction formulae, differential equations and dimensional recurrence relations for $L$-loop two-point massive tadpoles and sunrises with arbitrary masses and regularized both dimensionally and analytically. The differential system obtained has a Pfaff form and can be turned into $(\epsilon+\tfrac{1}{2})$-form when the analytic regularization is removed. For odd $d$ this form allows us to present coefficients of $\epsilon$-expansion explicitly in terms of Goncharov's polylogarithms. Using the symmetry properties of the matrix in the right-hand side of the differential system, we obtain quadratic constraints for the solutions of the obtained differential system near any integer $d$.10:3010:30 - 11:00Room: 02.43011:00 Differential Reduction of Feynman Diagrams: status and perspective - Mikhail KalmykovDifferential Reduction of Feynman Diagrams: status and perspective- Mikhail Kalmykov

11:00 - 12:00Room: 02.430 We point out that the Mellin-Barnes representation of Feynman Diagrams can be used for getting homogeneous system of linear differential equations of hypergeometric types. This set of equations are enough for reduction of original diagrams to some basis with the following construction of coefficients of epsilon-expansion.12:0012:00 - 14:00Room: 02.43014:00 Functional reduction of Feynman integrals - Oleg Tarasov (Joint Institute for Nuclear Research, Dubna)Functional reduction of Feynman integrals- Oleg Tarasov (Joint Institute for Nuclear Research, Dubna)

14:00 - 15:00Room: 02.430 A method for reducing Feynman integrals, depending on several kinematic variables and masses, to a combination of integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle, box and pentagon integrals with massless internal propagators to simpler integrals will be described. The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the $_2F_1$ Gauss hypergeometric function and the logarithmic function was derived. By using the functional equations, the one-loop box integral with massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the $F_1$ Appell and $_2F_1$ Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension $d$. The reduction equations for the box integral with some kinematic variables equal to zero are considered. With the help of three step functional reduction the one-loop pentagon integral with massless propagators, which depends on 10 variables was reduced to a combination of 60 integrals depending on 4 variables. The on-shell pentagon integral depending on 5 variables was reduced to 15 integrals depending on 3 variables. For the integrals with 3 variables an analytic result in terms of the $F_3$ Appell and $_2F_1$ Gauss hypergeometric functions was obtained.15:00 Symmetries of Feynman Integrals: a new theory for FI evaluation - Barak Kol (The Racah Institute of Physics)Symmetries of Feynman Integrals: a new theory for FI evaluation- Barak Kol (The Racah Institute of Physics)

15:00 - 15:45Room: 02.430 Symmetries of Feynman Symmetries (SFI) is a general theory for the evaluation of Feynman diagrams, which is related to both IBP and DE. For any given diagram topology it defines a set of partial differential equations in terms of the most general parameters - masses and kinematical invariants. The equation system is associated with a group G whose orbits foliate the parameter space. The general integral is shown to reduce to its value at conveniently chosen parameters on the same G-orbit plus a line integral over simpler diagrams - diagrams with one propagator contracted. Additional aspects of the theory and some applications will be discussed.15:45 Kite diagram through Symmetries of Feynman Integrals - Subhajit MazumdarKite diagram through Symmetries of Feynman Integrals- Subhajit Mazumdar

15:45 - 16:15Room: 02.430 The Symmetries of Feynman Integrals (SFI) is a method for evaluating Feynman Integrals which exposes a novel continuous group associated with the diagram which depends only on its topology and acts on its parameters. Using this method we study the kite diagram, a two-loop diagram with two external legs, with arbitrary masses and spacetime dimension. Generically, this method reduces a Feynman integral into a line integral over simpler diagrams. We identify a locus in parameter space where the integral further reduces to a mere linear combination of simpler diagrams, thereby maximally generalizing the known massless case.16:1516:15 - 17:15Room: 02.430 -
Friday, 22 March 201909:30 Polynomials associated to graphs, matroids, and configurations - Mathias Schulze (Technische Universität Kaiserslautern)Polynomials associated to graphs, matroids, and configurations
- Mathias Schulze (Technische Universität Kaiserslautern)

09:30 - 10:30Room: 02.430 Kirchhoff (Symanzik) polynomials are obtained from a graph as a sum of monomials corresponding to (non-)spanning trees. They are of particular importance in physics in the case of Feynman graphs. One can consider them as special cases of matroid (basis) polynomials or of configuration polynomials. However the latter two generalizations differ in case of non-regular matroids. This more general point of view has the advantage that the classes of matroids and configurations are stable under additional operations such as duality and truncation. In addition there are important configuration polynomials, such as the second graph polynomial, which are not of Kirchhoff type. I will give an introduction to the topic and present new results relating the algebro-geometric structure of the singular locus of configuration hypersurfaces to the structure of the underlying matroid. Their proofs make essential use of matroid theory and, in particular, rely on duality.10:3010:30 - 11:00Room: 02.43011:00 direct solutions & conjugate polynomials - David Kosower (Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay)direct solutions & conjugate polynomials- David Kosower (Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay)

11:00 - 12:00Room: 02.43012:0012:00 - 14:00Room: 02.43014:00 Reduction of multiloop Feynman integrals: an O(N) algorithm - Yan-Qing Ma (Peking University, Beijing)Reduction of multiloop Feynman integrals: an O(N) algorithm- Yan-Qing Ma (Peking University, Beijing)

14:00 - 15:00Room: 02.430 We modify Feynman integrals by introducing an auxiliary parameter \eta into each Feynman propagator. The modified Feynman integrals are analytical functions of \eta, and physical Feynman integrals can be obtained by taking \eta -> 0^+. Asymptotic expansions of the modified Feynman integrals at \eta -> \infinity can be very easily calculated, which contain only equal-mass vacuum integrals. Due to uniqueness theorem of analytical functions, the asymptotic series determine both values of physical Feynman integrals and their linear relations. Based on the asymptotic expansion, we construct an algorithm to generate linear relations that can reduce arbitrary multi-loop Feynman integral to master integrals. Computation complexity of numerically solving these linear relations is O(N) when the number of linear equations N is large. Our method may overcome the difficulty of IBP method, where linear equations are coupled and thus computation complexity is O(N^3).15:00 Direct Reduction of Amplitude - Najam ul Basat (Chinese Academy of Sciences, Beijing)Direct Reduction of Amplitude- Najam ul Basat (Chinese Academy of Sciences, Beijing)

15:00 - 15:30Room: 02.430 We propose an alternative approach based on series representation to directly reduce multi-loop multi-scale scattering amplitude into set of freely chosen master integrals. And this approach avoid complicated calculations of inverse matrix and dimension shift for tensor reduction calculation. During this procedure we further utilize the Feynman parametrization to calculate the coefficients of series representation and obtain the form factors. Conventional methodologies are used only for scalar vacuum bubble integrals to finalize the result in series representation form. Finally, we elaborate our approach by presenting the reduction of a typical two-loop amplitude for W boson production.15:3015:30 - 16:00Room: 02.43016:0016:00 - 17:00Room: 02.430