Renato M. Fonseca
GroupMath is a Mathematica package which performs several calculations related to semi-simple Lie algebras and the permutation groups, both of which are important in particle physics as well as in other areas of research.
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We present the Mathematica package E6Tensors, a tool for explicit tensor calculations in E6 gauge theories. In addition to matrix expressions for the group generators of E6, it provides structure constants, various higher rank tensors and expressions for the representations 27, 78, 351 and 351'. This paper comes along with a short manual including physically relevant examples. I further give a complete list of gauge invariant, renormalisable terms for superpotentials and Lagrangians.
AMFlow is a Mathematica package to numerically compute dimensionally regularized Feynman integrals via the recently proposed auxiliary mass flow method. In this framework, integrals are treated as functions of an auxiliary mass parameter and their results can be obtained by constructing and solving differential systems with respect to this parameter, in an automatic way. The usage of this package is described in detail through an explicit example of double-box family involved in two-loop $t\bar{t}$ hadroproduction.
We present FormTracer, a high-performance, general purpose, easy-to-use Mathematica tracing package which uses FORM. It supports arbitrary space and spinor dimensions as well as an arbitrary number of simple compact Lie groups. While keeping the usability of the Mathematica interface, it relies on the efficiency of FORM. An additional performance gain is achieved by a decomposition algorithm that avoids redundant traces in the product tensors spaces. FormTracer supports a wide range of syntaxes which endows it with a high flexibility. Mathematica notebooks that automatically install the packag…
We present a Mathematica package for doing computations with gamma matrices, spinors, tensors and other objects, in any dimension and signature. The approach we use is based on defining the commutation relations of the relevant matrices, and is thus general and flexible. As examples, we reproduce torsion conditions for $AdS_3$ compactification of type IIB supergravity from literature, and check the vanishing of supersymmetry variations of 10-dimensional supersymmetric Yang-Mills theory.
We present FaRe, a package for Mathematica that implements the decomposition of a generic tensor Feynman integral, with arbitrary loop number, into scalar integrals in higher dimension. In order for FaRe to work, the package FeynCalc is needed, so that the tensor structure of the different contributions is preserved and the obtained scalar integrals are grouped accordingly. FaRe can prove particularly useful when it is preferable to handle Feynman integrals with free Lorentz indices and tensor reduction of high-order integrals is needed. This can then be achieved with several powerful existing…
DiffExp is a Mathematica package for integrating families of Feynman integrals order-by-order in the dimensional regulator from their systems of differential equations, in terms of one-dimensional series expansions along lines in phase-space, which are truncated at a given order in the line parameter. DiffExp is based on the series expansion strategies that were explored in recent literature for the computation of families of Feynman integrals relevant for Higgs plus jet production with full heavy quark mass dependence at next-to-leading order. The main contribution of this paper, and its asso…
We have developed a Mathematica package capable of performing gamma-matrix algebra in arbitrary (integer) dimensions. As an application we can compute Fierz transformations.
We present the Mathematica package QMeS-Derivation. It derives symbolic functional equations from a given master equation. The latter include functional renormalisation group equations, Dyson-Schwinger equations, Slavnov-Taylor and Ward identities and their modifications in the presence of momentum cutoffs. The modules allow to derive the functional equations, take functional derivatives, trace over field space, apply a given truncation scheme, and do momentum routings while keeping track of prefactors and signs that arise from fermionic commutation relations. The package furthermore contains …
Slavnov-Taylor 1.0 is a Mathematica package which allows us to perform automatic symbolic computation in BRST formalism. This article serves as a self-contained guide to prospective users, and indicates the conventions and approximations used.
I present a Mathematica package designed for manipulations and evaluations of triple-K integrals and conformal correlation functions in momentum space. Additionally, the program provides tools for evaluation of a large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators. The package is accompanied by five Mathematica notebooks containing detailed calculations of numerous conformal 3-point functions in momentum space.
We give a simple tutorial introduction to the Mathematica package STRINGVACUA, which is designed to find vacua of string-derived or inspired four-dimensional N=1 supergravities. The package uses powerful algebro-geometric methods, as implemented in the free computer algebra system Singular, but requires no knowledge of the mathematics upon which it is based. A series of easy-to-use Mathematica modules are provided which can be used both in string theory and in more general applications requiring fast polynomial computations. The use of these modules is illustrated throughout with simple exampl…
Multivariate residues appear in many different contexts in theoretical physics and algebraic geometry. In theoretical physics, they for example give the proper definition of generalized-unitarity cuts, and they play a central role in the Grassmannian formulation of the S-matrix by Arkani-Hamed et al. In realistic cases their evaluation can be non-trivial. In this paper we provide a Mathematica package for efficient evaluation of multidimensional residues based on methods from computational algebraic geometry. The package moreover contains an implementation of the global residue theorem, which …
We present the Mathematica package MultivariateResidues, which allows for the efficient evaluation of multivariate residues based on methods from computational algebraic geometry. Multivariate residues appear in several contexts of scattering amplitude computations. Examples include applications to the extraction of master integral coefficients from maximal unitarity cuts, the construction of canonical bases of loop integrals and the construction of tree amplitudes from scattering equations.
The "Spinors" software is a "Mathematica" package which implements 2-component spinor calculus as devised by Penrose for General Relativity in dimension 3+1. The "Spinors" software is part of the "xAct" system, which is a collection of "Mathematica" packages to do tensor analysis by computer. In this paper we give a thorough description of "Spinors" and present practical examples of use.
We present WiLE, a Mathematica package designed to perform the weak coupling expansion of any Wilson loop in ABJ(M) theory at arbitrary perturbative order. For a given set of fields on the loop and internal vertices, the package displays all the possible Feynman diagrams and their integral representations. The user can also choose to exclude non planar diagrams, tadpoles and self-energies. Through the use of interactive input windows, the package should be easily accessible to users with little or no previous experience. The package manual provides some pedagogical examples and the computation…
We present a further extension of the HYPERDIRE project, which is devoted to the creation of a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations. Specifically, we present the implementation of the differential reduction for the Lauricella function $F_C$ of three variables.
HYPERDIRE is a project devoted to the creation of a set of Mathematica-based programs for the differential reduction of hypergeometric functions. The current version allows for manipulations involving the full set of Horn-type hypergeometric functions of two variables, including 30 functions.
In the Lee-Pomeransky representation, Feynman integrals can be identified as a subset of Euler-Mellin integrals, which are known to satisfy Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. Here we present an automated package to derive the associated GKZ system for a given Feynman diagram and solve it in terms of hypergeometric functions using two equivalent algorithms, namely the triangulation method and the Gröbner deformation method. We present our code in the form of a Mathematica package FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, an…
CasimirRFM: A Mathematica package for Riemann-flat compactifications with Casimir energies
CasimirRFM是一款Mathematica软件包，可实现黎曼平坦流形紧化的卡西米尔能量计算，支持多种场论与超引力场景，通过IIB型超引力紧化示例验证其功能。
We present the tensor computer algebra package xTras, which provides functions and methods frequently needed when doing (classical) field theory. Amongst others, it can compute contractions, make Ansätze, and solve tensorial equations. It is built upon the tensor computer algebra system xAct, a collection of packages for Mathematica.
In this paper we present a short overview of the new Wolfram Mathematica package intended for elementary "in-basis" tensor and differential-geometric calculations. In contrast to alternatives our package is designed to be easy-to-use, short, all-purpose, and hackable. It supports tensor contractions using Einstein notation, transformations between different bases, tensor derivative operator, expansion in basis vectors and forms, exterior derivative, and interior product.
We present STR (Star-Triangle Relations), a Mathematica package designed to solve Feynman diagrams by means of the method of uniqueness in any Euclidean spacetime dimension. The method of uniqueness is a powerful technique to solve multi-loop Feynman integrals in theories with conformal symmetry imposing some relations between the powers of propagators and the spacetime dimension. In our algorithm we include both identities for scalar and Yukawa type integrals. The package provides a graphical environment in which it is possible to draw the desired diagram with the mouse input and a set of too…
Lambda: A Mathematica-package for operator product expansions in vertex algebras
本文介绍Mathematica软件包Lambda，用于计算顶点代数及SUSY顶点代数中的λ-括号，即二维共形场论中的算子乘积展开，并展示语法及示例。
We present bipartiteSUSY, a Mathematica package designed to perform calculations for physical theories based on bipartite graphs. In particular, the package can employ the recently developed arsenal of techniques surrounding on-shell diagrams in N=4 SYM scattering amplitudes, including those for non-planar diagrams, with particular attention to computational speed. It also contains a host of tools for computations in N=1 Bipartite Field Theories, which utilize the same bipartite graphs. Through the use of an interactive graphical tool, it is possible to draw the desired diagrams on the screen …
A Mathematica Package for Computing N=2 Superfield Operator Product Expansions
本文描述了一个计算亚纯$N=2$超共形场论中超场算子乘积展开的通用Mathematica软件包。给定一组“基本”超场的SOPE，任意复杂复合场的SOPE都可以自动计算。正常订购的产品总是简化为标准形式。可以检查Jacobi恒等式，并计算Poisson括号（“经典SOP”）。我们提出了两个明确的例子：一个建设的“小”的$N=4$超共形代数的$N=2$超领域，和实现的$N=2$超共形代数的手征和反手征费米子超领域。
We describe an additional module for the Mathematica package FeynRules that allows for an easy building of any N=1 supersymmetric quantum field theory, directly in superspace. After the superfield content of a specific model has been implemented, the user can study the properties of the model, such as the supersymmetric transformation laws of the associated Lagrangian, directly in Mathematica. While the model dependent parts of the latter, i.e., the soft supersymmetry-breaking Lagrangian and the superpotential, have to be provided by the user, the model independent pieces, such as the gauge in…
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant for manipulations of hypergeometric functions_{p+1}F_p, and the second one, AppellF1F4, for manipulations with Appell hypergeometric functions F_1,F_2,F_3,F_4 of two variables.
Olsson.wl : a Mathematica package for the computation of linear transformations of multivariable hypergeometric functions
本文提出了用于计算多变量超几何函数线性变换的 Olsson.wl Mathematica 包，基于 Olsson 方法推导 Appell F1 级数的解析延拓，并提供推导收敛域的配套包 ROC2.wl。
We provide here all the procedures in \texttt{Mathematica} which are needed for the computation of the analytic images of the strong coupling constant powers in Minkowski (${\bar{\mathfrak A}_ν(s;n_f)}$ and ${\mathfrak A_ν^\text{glob}(s)}$) and Euclidean (${\bar{\mathcal A}_ν(Q^2;n_f)}$ and ${\mathcal A_ν^\text{glob}(Q^2)}$) domains at arbitrary energy scales (${s}$ and ${Q^2}$, correspondingly) for both schemes --- with fixed number of active flavours ${n_f=3, 4, 5, 6}$ and the global one with taking into account all heavy-quark thresholds. These singularity-free couplings are inevitable elem…
SpinorHelicity4D: a Mathematica toolbox for the four-dimensional spinor-helicity formalism
本文介绍了 Mathematica 软件包 SpinorHelicity4D，用于四维无质量和有质量旋量-螺旋度形式的解析与数值计算，提供表达式简化、Dirac 迹计算和复动理学生成等功能，兼顾易用性与高级定制需求。
In this paper we present Affine.m - program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. Algorithms are based upon the properties of weights and Weyl symmetry. The most important problems for us are the ones, concerning computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in popular computer algebra system…
We present a package for Mathematica that facilitates the numerical computation of the quasinormal mode (QNM) spectrum of a black hole/black brane. Requiring as input only the QNM equation(s), the application of a single Mathematica function will compute the spectrum efficiently, by discretizing the equation(s) and solving the resulting generalized eigenvalue equation. It is applicable to a large variety of black holes, independent of their asymptotics. The package comes fully documented and with several tutorials. Here we present a self-contained review of the method and consider several appl…
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions $F_D$ of $r$ variables; and the second one, FsFunction, for manipulations with Lauricella-Saran hypergeometric functions $F_S$ of three variables. Both functions are related with one-loop Feynman diagrams. The published version includes also Chapter 5 with two theorems about structure of coefficients of epsilon-expansion of the Ho…
Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians. MGFs satisfy numerous non-trivial algebraic- and differential relations which have been studied extensively in the literature and lead to significant simplifications. In this paper, we systematically combine these relations to obtain basis decompositions of all two- and three-point MGFs of total modular weight $w+\bar{w}\leq12$, starting from just two well-known identit…
$\texttt{MultiHypExp}$: A Mathematica Package For Expanding Multivariate Hypergeometric Functions In Terms Of Multiple Polylogarithms
本文提出 Mathematica 程序包 MultiHypExp，通过微分算子与 Pfaffian 系统方法将多元超几何函数在维数正规化参数中展开为多重多对数，以处理多圈多尺度 Feynman 积分。
GRAMA. A Mathematica program for analytical calculations in 10-dimensional supergravity. Version 1.0
“GRAMA”是一个Mathematica软件包，用于在10维（D=10）简单（N=1）超引力中进行符号张量计算和复杂的代数操作。该软件包在通用Mathematica环境中的主要新成分是计算Dirac矩阵的复杂乘积和协变导数的处理：旋量和向量。原则上，只要稍加修改，GRAMA也可以用于四维超引力的计算。在这个软件包的帮助下，我们能够获得10维超引力的运动方程和拉格朗日量，包括超弦修正（见hep-th/9507033）-否则不可能进行的计算。“GRAMA”被设计成一个社区和用户友好的程序。
We introduce the MathGR package, written in Mathematica. The package can manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices to partially or completely explicit indices and convert partial derivatives into total derivatives. Frequently used GR tensors and a model of FRW universe with ADM type perturbations are predefined. The package is built around the philosophy to "keep it simple", and makes use of latest tensor technologies of Mathematica.
We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The latter may form a prototype for regularized quantum field theory. We first generalize the method of Bender-Wu, and derive exact recursion relations which allow the determination of the perturbative wave-function and energy corrections to an arbitrary order, at least in principle. For 1D systems, we implement these equations in an easy to use Mathematica package we call BenderWu. Our package e…
The many intricate connections between scattering amplitudes, on-shell diagrams, and the positroid stratification of the Grassmannian has recently been described in great detail. In order to facilitate the exploration of this rich correspondence, we have prepared a public Mathematica package called "positroids" which includes an array of useful tools including those for the construction of canonical coordinates for positroid configurations, the drawing of representative on-shell (plabic) graphs, and the evaluation of on-shell differential forms. This note documents the functions made available…
HyperPrecision: A Mathematica package for High-Precision Numerical Evaluation of Multivariate Hypergeometric Functions
提出 HyperPrecision Mathematica 包，通过自动构造 Pfaffian 系统并沿一维等高线求解常微分方程，实现 Horn 型多元超几何函数及其 Laurent 展开的高精度数值计算。
FiniteFieldSolve: Exactly Solving Large Linear Systems in High-Energy Theory
本文提出通用工具包FiniteFieldSolve，通过有限域方法与素数自动重编译技术，在高能理论中精确且高效地求解大型线性系统，性能显著优于Mathematica原生求解器。
SubTropica
SubTropica 是一个 Mathematica 包，利用热带几何最新进展进行多多项式对数积分的符号积分，重点处理线性可约的欧拉积分，如费曼积分，并通过热带减法方案扩展，同时提供 AI 驱动的费曼积分库。
We present a new interface called FeynHelpers that connects FeynCalc, a Mathematica package for symbolic semi-automatic evaluation of Feynman diagrams and calculations in quantum field theory (QFT) to Package-X and FIRE. The former provides a library of analytic results for scalar 1-loop integrals with up to 4 legs, while the latter is a general-purpose tool for reduction of multi-loop scalar integrals using Integration-by-Parts (IBP) identities. With this add-on many types of calculations that were difficult or hardly feasible with FeynCalc previously, can now be done in a much simpler way. T…
We present version 3.0 of the Mathematica package DoFun for the derivation of functional equations. In this version, the derivation of equations for correlation functions of composite operators was added. In the update, the general workflow was slightly modified taking into account experience with the previous version. In addition, various tools were included to improve the usage experience and the code was partially restructured for easier maintenance.
We present a package to perform partial fraction decompositions of multivariate rational functions. The algorithm allows to systematically avoid spurious denominator factors and is capable of producing unique results also when being applied to terms of a sum separately. The package is designed to work in Mathematica, but also provides interfaces to the Form and Singular computer algebra systems.
DiracQ: A Quantum Many-Body Physics Package
DiracQ是一个用于量子多体物理的软件包，通过内置的交换和反交换规则扩展Mathematica，支持多体物理中的代数运算和计算示例。
Analytic continuations of the Horn $H_1$ and $H_5$ functions
该研究首次利用自动化符号计算软件包推导了双变量Horn $H_1$和$H_5$函数的解析延拓，并借助Pfaff-Euler变换扩展了收敛区域，公开了相关软件包。
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