Feynman Diagrams and a Combination of the Integration by Parts (IBP) and the Integration by Fractional Expansion (IBFE) Techniques

IBP methods at finite temperature

We demonstrate the applicability of integration-by-parts (IBP) identities in finite-temperature field theory. As a concrete example, we perform 3-loop computations for the thermodynamic pressure of QCD in general covariant gauges, and confirm earlier Feynman-gauge results.

Module Intersection for the Integration-by-Parts Reduction of Multi-Loop Feynman Integrals

In this manuscript, which is to appear in the proceedings of the conference "MathemAmplitude 2019" in Padova, Italy, we provide an overview of the module intersection method for the the integration-by-parts (IBP) reduction of multi-loop Feynman integrals. The module intersection method, based on computational algebraic geometry, is a highly efficient way of getting IBP relations without double propagator or with a bound on the highest propagator degree. In this manner, trimmed IBP systems which are much shorter than the traditional ones can be obtained. We apply the modern, Petri net based, wo…

参数空间中的IBP与微分方程

IBPs and differential equations in parameter space

本研究提出一种在费曼参数空间中构建IBP恒等式与费曼积分微分方程的射影框架，将早期成果推广至现代语言，适用于任意圈阶，经单双圈简单图验证可行，为现有相关结果提供统一视角。

使用Gröbner基进行Feynman积分约化

Feynman integral reduction using Gröbner bases

本文提出利用有理双移位代数中的Gröbner基系统化处理Feynman积分的IBP约化问题，并引入一阶正规形IBP关系及基于函数域线性代数的替代方法以应对复杂情形。

用于费曼积分的符号齐次约束约简规则及LoopIn框架

Symbolic syzygy-constrained reduction rules for Feynman integrals and the LoopIn framework

本文提出了一种高效处理高次分子或传播子费曼积分的IBP约简算法，并介绍了LoopIn框架以自动化多环计算。

用于Feynman积分约简的强化学习与元启发式算法

Reinforcement Learning and Metaheuristics for Feynman Integral Reduction

本文提出利用强化学习和模拟退火等元启发式算法优化Feynman积分的IBP约简过程，在无需人工指导的情况下，其性能可媲美最高效的人工设计算法。

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficient…

费曼积分约化：稀疏有理函数的平衡重构及协同设计方法下的超级计算机实现

Feynman integral reduction: balanced reconstruction of sparse rational functions and implementation on supercomputers in a co-design approach

本文提出结合平衡重构与Zippel方法的稀疏有理函数重构技术，并采用多探针并行策略优化有限域IBP约化，通过协同设计实现超级计算机高效运行，以FIRE基准测试验证了其性能。

含参费曼积分的分部积分恒等式与微分方程

Integration-by-parts identities and differential equations for parametrised Feynman integrals

本研究在费曼参数空间利用射影几何，调整推广早期成果推导适用于任意圈阶的通用IBP恒等式，通过单双圈简单图验证方法，为费曼积分计算相关现有成果提供统一视角。

Symmetries of Feynman integrals and the Integration By Parts method

Integration By Parts (IBP) is an important method for computing Feynman integrals. This work describes a formulation of the theory involving a set of differential equations in parameter space, and especially the definition and study of an associated Lie group G. The group acts on parameter space and foliates it into G-orbits. The differential equations essentially provide the gradient of the integral within the orbit in terms of integrals associated with degenerate diagrams. In this way the computation of a Feynman integral at a general point in parameter space is reduced to the evaluation of …

The diamond rule for multi-loop Feynman diagrams

An important aspect of improving perturbative predictions in high energy physics is efficiently reducing dimensionally regularised Feynman integrals through integration by parts (IBP) relations. The well-known triangle rule has been used to achieve simple reduction schemes. In this work we introduce an extensible, multi-loop version of the triangle rule, which we refer to as the diamond rule. Such a structure appears frequently in higher-loop calculations. We derive an explicit solution for the recursion, which prevents spurious poles in intermediate steps of the computations. Applications for…

利用SAILIR学习解费曼环形积分

Learning to Unscramble Feynman Loop Integrals with SAILIR

SAILIR通过自监督学习解费曼环形积分，相比传统方法更节省内存，能处理高权重积分。

Azurite: An algebraic geometry based package for finding bases of loop integrals

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package {\sc Azurite} ({\bf A ZUR}ich-bred method for finding master {\bf I}n{\bf TE}grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems {\sc Singular} and {\sc Mathematica}. It can moreov…

Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections

We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of $73$ master integrals.