The Mathematics of Linear Relations between Feynman Integrals

18 Mar 2019, 08:00 → 22 Mar 2019, 18:00 Europe/Berlin

02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

For the precise prediction of observables at particle colliders such as the LHC, the computation of a large number of Feynman integrals is indispensable. A crucial step in these computations is the reduction to a small set of master integrals by exploiting linear relations.

In the standard approach such relations are derived from the famous integration-by-parts identities in momentum space and combined in Laporta's algorithm. Such reductions can easily grow too large to be executed with reasonable computer resources. Motivated by this problem, many new ideas on efficient reductions of Feynman integrals were presented in the recent literature. This workshop is dedicated to the mathematical techniques behind these developments, including finite field techniques, Gröbner bases, Milnor numbers, unitarity cuts and syzygy computations. In particular, techniques from algebraic geometry and the theory of D-modules have provided new insights. In an attempt to boost the exchange of ideas and the development of advanced methods, this workshop brings together mathematicians with relevant expertise, physicists with experience in Feynman integral reductions and specialists on computer algebra tools.

Confirmed speakers include:

• Thomas Bitoun
• Janko Böhm 
• Mikhail Kalmykov
• David Kosower
• Stefano Laporta
• Kasper Larsen
• Roman Lee
• Viktor Levandovskyy
• Andreas von Manteuffel
• Pierpaolo Mastrolia
• Toshinori Oaku
• Mathias Schulze
• Alexander Smirnov
• Oleg Tarasov

Executive Summary (PDF) report.pdf

Participants (PDF) Participants_Indico_MLR2019.pdf

Monday 18 March

Registration

Welcome

1 Laporta algorithm for multi-loop vs multi-scale problems

Speaker: Dr Johann Usovitsch (Trinity College Dublin)

usovitsch.pdf

Coffee Break

2 FIRE6: Feynman Integral REduction with Modular Arithmetic

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.

Speaker: Dr Alexander Smirnov (Moscow State University)

asmirnov.pdf

Lunch break

3 Four-loop quark form factor with quartic fundamental colour factor

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.

Speaker: Dr Vladimir Smirnov (Moscow State University)

vsmirnov.pdf

4 Constructing multi-loop scattering amplitudes with manifest singularity structure

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.

Speaker: Dr Robert Schabinger (Michigan State University)

schabinger.pdf

5 Integrand reduction for particles with spins

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.

Speaker: Dr Hui Luo

luo.pdf

Discussion

Coffee Break

Tuesday 19 March

6 Computation of holonomic systems for Feynman amplitudes associated with some simple diagrams

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.

Speaker: Prof. Toshinori Oaku (Tokyo Woman's Christian University)

oaku.pdf

Coffee Break

7 Computer algebra and ring theory help Feynman integrals

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.

Speaker: Dr Viktor Levandovskyy (RWTH Aachen)

levandovskyy.pdf

Lunch break

8 The algebraic Mellin transform, after Loeser and Sabbah

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.

Speaker: Dr Thomas Bitoun (University of Toronto)

9 Feynman integral relations from Parametric annihilators

Speaker: Dr Erik Panzer

panzer.pdf

10 Counting master integrals with PSLQ

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.

Speaker: Dr Stefano Laporta

laporta.pdf

Coffee Break

Wednesday 20 March

11 Integration-by-parts reductions via unitarity cuts and syzygies

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.

Speaker: Dr Kasper Larsen (University of Southampton)

larsen.pdf

Coffee Break

12 Massively Parallel Methods and Other Trends in the Design of Computer Algebra Systems - With a Focus on Feynman Integral Reduction

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.

Speaker: Dr Janko Böhm (Technische Universität Kaiserslautern)

boehm.pdf

Lunch break

13 Feynman Integrals and Intersection Theory

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.

Speaker: Dr Pierpaolo Mastrolia (Padua University)

mastrolia.pdf

14 Algorithmic approaches to syzygies for no-dot or no-numerator relations

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.

Speaker: Dr Andreas von Manteuffel (Michigan State University)

manteuffel.pdf

Coffee Break

Thursday 21 March

15 L-loop watermelon and sunrise graphs: differential equations, index and dimension shifting relations, and quadratic constraints

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 $d$.

Speaker: Dr Roman Lee (Budker Institute of Nuclear Physics, Novosibirsk)

lee.pdf

Coffee Break

16 Differential Reduction of Feynman Diagrams: status and perspective

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.

Speaker: Dr Mikhail Kalmykov

kalmykov.pdf

Lunch break

17 Functional reduction of Feynman integrals

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.

Speaker: Dr Oleg Tarasov (Joint Institute for Nuclear Research, Dubna)

tarasov.pdf

18 Symmetries of Feynman Integrals: a new theory for FI evaluation

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.

Speaker: Prof. Barak Kol (The Racah Institute of Physics)

kol.pdf

19 Kite diagram through Symmetries of Feynman Integrals

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.

Speaker: Dr Subhajit Mazumdar

mazumdar.pdf

Coffee Break

Friday 22 March

20 Polynomials associated to graphs, matroids, and configurations

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.

Speaker: Prof. Mathias Schulze (Technische Universität Kaiserslautern)

schulze.pdf

Coffee Break

21 direct solutions & conjugate polynomials

Speaker: Dr David Kosower (Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay)

kosower.pdf

Lunch break

22 Reduction of multiloop Feynman integrals: an O(N) algorithm

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).

Speaker: Prof. Yan-Qing Ma (Peking University, Beijing)

ma.pdf

23 Direct Reduction of Amplitude

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.

Speaker: Dr Najam ul Basat (Chinese Academy of Sciences, Beijing)

basat.pdf

Discussion

Coffee Break