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FeynCalc 10: Do multiloop integrals dream of computer codes?
Vladyslav Shtabovenko, Rolf Mertig, Frederik Orellana
TL;DR
FeynCalc 10 addresses the difficulty of accessible multiloop calculations by extending a widely used Mathematica package with optimized multiloop routines and related tools. It broadens support for topology handling, integral representations, color algebra, numerical comparison, documentation, and light-cone expressions, while remaining a supplementary component of a larger framework rather than a standalone solution for unrestricted multiloop calculations.
Problem
Multiloop calculations remain difficult and expertise-intensive, while the original Passarino–Veltman-centered functionality was not efficient or general-purpose enough beyond one loop.
Method
The paper extends FeynCalc’s one-loop functionality with optimized multiloop routines for topology identification, reduction, integral representations, algebraic simplification, and related workflow tasks.
Results
FeynCalc 10 provides high-level access to multiloop techniques within a flexible framework that can structure, reduce, and help check calculations without replacing specialized tools.
Takeaways & Limitations
The release lowers the barrier to using multiloop techniques in daily research while allowing users to retain existing codes and inspect intermediate calculation stages.
Takeaways & Limitations
FeynCalc alone is not intended to complete unrestricted multiloop calculations, and Mathematica performance limits the size and complexity of calculations it can fully evaluate.
Abstract
from arXiv · showhide
In this work we report on a new version of FeynCalc, a Mathematica package widely used in the particle physics community for manipulating quantum field theoretical expressions and calculating Feynman diagrams. Highlights of the new version include greatly improved capabilities for doing multiloop calculations, including topology identification and minimization, optimized tensor reduction, rewriting of scalar products in terms of inverse denominators, detection of equivalent or scaleless loop integrals, derivation of Symanzik polynomials, Feynman parametric as well as graph representation for master integrals and initial support for handling differential equations and iterated integrals. In addition to that, the new release also features completely rewritten routines for color algebra simplifications, inclusion of symmetry relations between arguments of Passarino--Veltman functions, tools for determining matching coefficients and quantifying the agreement between numerical results, improved export to LaTeX and first steps towards a better support of calculations involving light-cone vectors.
PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY
FeynCalc 10 is a new release of the Mathematica package, superseding the previous version with routines required for multiloop calculations.
- PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY: FeynCalc 10 adds new routines required for multiloop calculations and supersedes the previous version.The release is implemented in the Wolfram Language and distributed under GPLv3.
1. Introduction
Multiloop calculations remain difficult because their diagrams and integrals grow rapidly in complexity, while existing one-loop methods do not generalize efficiently. FeynCalc 10 addresses this gap by extending its one-loop functionality with optimized multiloop algorithms as part of a broader semi-automatic framework.
- 1. Introduction: Multiloop diagrams remain challenging and require considerable expertise despite widely available tree-level and one-loop automation.Processes can involve thousands or tens of thousands of diagrams, increasing algebraic time and computational-resource demands.
- 1. Introduction: At two loops, the one-loop strategy of reducing integrals to a Passarino–Veltman basis becomes unfeasible, especially for complete massive master-integral bases.The difficulty arises alongside the proliferation and increasing complexity of diagrams.
- 1. Introduction: The paper extends FeynCalc’s established one-loop functionality with modern algorithms and optimized routines for multiloop calculations.The stated aim is to make analytic multiloop calculations accessible to a broader range of particle theorists.
- 1. Introduction: This work is the first of three publications developing a semi-automatic multiloop framework around FeynCalc, FeynHelpers, and FORM.The later papers are intended to connect FeynCalc to other QFT tools and introduce a FORM-based symbolic-evaluation framework.
- 1. Introduction: The package is presented as a flexible, documented extension rather than a claim that FeynCalc alone should perform complete multiloop calculations.The paper emphasizes high-level functions, examples, and a comprehensive manual while keeping code excerpts minimal.
2. Context and state of the art
The state of the art combines specialized tools and private or partially supported codes, leaving room for flexible public software that connects multiloop calculation stages. FeynCalc contributes topology handling, algebraic simplification, tensor reduction, and supplementary multiloop support within this ecosystem.
- 2. Context and state of the art: The ecosystem includes public and private tools, but several previously available packages have been abandoned or remain limited in support and documentation.The paper contrasts this situation with newer open-source efforts and regularly maintained tools.
- 2. Context and state of the art: Semi-automatic workflows remain valuable because users can inspect intermediate expressions and apply high-level functions sequentially with flexibility.This approach may be more efficient for some quantities than more automated tools that expose fewer intermediate stages.
- 2. Context and state of the art: FeynCalc 10 can derive multiloop amplitudes as linear combinations of integrals from specified topologies, but Mathematica limits the size and complexity of fully evaluable calculations.The package is positioned as a supplementary tool for structuring, reducing, and checking multiloop calculations.
- 2. Context and state of the art: Existing semi-automatic tools divide responsibilities across diagram generation, topology processing, FORM code generation, interfaces, and integral reduction.Alibrary, tapir, FeAmGen.jl, and HepLib provide different combinations of these capabilities, with some lacking tensor reduction or non-trace Dirac simplification.
- 2. Context and state of the art: FeynCalc identifies and minimizes topologies, simplifies Dirac and color algebra, and performs tensor reduction, while FORM-based tools generally evaluate diagrams faster.Feynman diagram generation is handled through FeynArts, with an experimental QGRAF interface noted.
3. Installation
FeynCalc can be installed automatically from a fresh Mathematica kernel, with stable and development versions available through the installer.
- 3. Installation: The recommended installation evaluates the installer imported from the FeynCalc GitHub repository in a freshly started Mathematica kernel.The automatic installer supports Mathematica versions 10.0 and above.
- 3. Installation: The installer provides the stable package by default and can install the development version by enabling InstallFeynCalcDevelopmentVersion.The development version may contain bugs but includes newer features.
4. Topologies and loop integrals
FeynCalc 10 introduces a structured representation for multiloop topologies and loop integrals, together with routines that automate their validation, manipulation, differentiation, and reduction preparation.
- 4.1. Three main building blocks: FeynCalc represents multiloop integral families with FCTopology and their integrals with GLI, while FCFeynmanPrepare derives Symanzik polynomials.FCTopology stores propagators forming a basis, GLI identifies the family and propagator powers, and FCFeynmanPrepare also supports explicit propagator representations.
- 4.3. Feynman parametrization: FeynCalc can derive Symanzik quantities for Minkowskian or Euclidean integrals and supports quadratic, eikonal, and tensor-integral inputs.FCFeynmanPrepare can compute U, F, and related matrices, with Euclidean handling enabled explicitly through an option.
- 4.1. Three main building blocks: FeynCalc supports constructing, validating, and completing propagator bases, including checks for overdetermined or incomplete topology definitions.Kinematics and additional topology information can be stored alongside propagators, loop momenta, and external momenta.
- 4.2. Basic operations: New routines generate partial-fractioning rules for overdetermined denominators and select relevant topologies from larger lists of loop integrals.FCLoopCreatePartialFractioningRules returns replacement rules and the new topologies appearing on their right-hand sides.
- 4.2. Basic operations: The package converts GLIs to propagator form, identifies equivalent master-integral mappings, and differentiates loop integrals for IBP relations, differential equations, and asymptotic expansions.Differentiated integrals still require IBP reduction to form a proper differential equation.
5. Master integrals
The master-integral workflow reduces large collections of loop integrals, identifies duplicate masters, represents them graphically, and provides tools for parametric and differential-equation methods.
- 5. Master integrals: The workflow remains computationally bounded: sufficiently massive or high-leg two-loop amplitudes can make even the reduction unfeasible.Numerical evaluation can be interfaced to FIESTA or pySecDec, while analytic evaluation is more difficult.
- 5.1. Unique master integrals: FeynCalc uses Pak’s method to find one-to-one mappings between master integrals that may be identical despite different propagator representations.The corresponding routine is FCLoopFindIntegralMappings.
- 5.2. Graph representations: FCLoopIntegralToGraph reconstructs directed graph representations of master integrals, while FCLoopGraphPlot styles them for visualization and FCGraphCuttableQ checks possible cuts.Graph reconstruction can return edge line momenta and additional information; plotting works best with Mathematica 12.2 or newer and can also use GraphViz after adjustments.
- 5.3. Feynman parametrization: FCFeynmanParametrize supports quadratic and eikonal propagators, Euclidean or tensor integrals, and Cartesian integrals in D−1 dimensions.FCFeynmanParameterJoin, FCFeynmanProjectiveQ, and FCFeynmanProjectivize support parameter joining and projective transformations for parametric integration.
- 5.4. Differential equations: FeynCalc provides initial differential-equation and iterated-integral support, including one-variable changes, ε expansions, and conversion toward harmonic or Goncharov polylogarithms.FCGPL objects are currently placeholders, with more GPL-related routines planned for future versions.
6. Features and improvements unrelated to multiloop calculations
FeynCalc 10 broadens non-multiloop support through more capable color and Passarino–Veltman algebra, nonrelativistic operators, convenience routines, light-cone expressions, and improved documentation.
- 6.1. Improved color algebra simplifications: SUNSimplify was rewritten with more color-algebra relations and improved handling of SUNTrace expressions.SUNTrace remains unevaluated by default unless SUNTraceEvaluate is enabled; SUNSimplify provides the preferred evaluation route.
- 6.2. Passarino–Veltman functions: PaVeOrder now canonicalizes Passarino–Veltman arguments using symmetry relations through rank 10 for B-functions and lower ranks for higher-point functions.The supported ranks are 9 for C-functions, 8 for D-functions, 7 for E-functions, and 6 for F-functions.
- 6.2. Passarino–Veltman functions: PaVeLimitTo4 simplifies infrared-finite PaVe expressions by expanding D-dependent prefactors so the result becomes O(ε0).The absence of infrared poles is assumed rather than checked.
- 6.3. Lagrangians and operators: Lagrangian manipulation now supports Cartesian nabla operators, derivative reshuffling by integration by parts, and utilities for vector products and free or dummy indices.Shift-PartialD assumes surface terms vanish, while FCTripleProduct, FCGetFreeIndices, and FCGetDummyIndices support custom symbolic workflows.
- 6.4–6.6. Algebra, convenience, and light-cone features: FeynCalc 10 adds GordonSimplify, improves Larin-scheme Dirac traces, supports light-cone vectors and Dirac matrices, and introduces FCCompareNumbers for controlled numerical comparisons.FCCompareNumbers compares numerical or semi-numerical expressions using a user-specified number of significant digits.
- 6.7. Up-to-date documentation using continuous integration: Documentation is maintained from text-based sources that generate synchronized HTML and PDF manuals, including usage descriptions and a tutorial.Changes to the source files trigger updates of the public PDF manual.
7. Examples
The examples show FeynCalc 10’s multiloop workflow from topology identification through tensor reduction and GLI rewriting, with checks against known two-loop results.
- Multiloop workflow: The examples use GLIs in identified FCTopology families rather than expressing one-loop-style amplitudes directly through Passarino–Veltman functions.This illustrates the package’s shift toward integral-family-based multiloop representations.
- Multiloop workflow: FeynCalc identifies common topology mappings, performs tensor reduction, rewrites loop-momentum scalar products as inverse denominators, and expresses amplitudes as GLI combinations.These operations are coordinated through topology-mapping and tensor-reduction routines.
- Two-loop self-energies: The massless-QED electron self-energy reduces to two distinct master integrals after identifying an equivalence among three initial masters.The final result agrees fully with the cited two-loop reference.
- Two-loop self-energies: The gluon self-energy example reproduces the reference result at O(ε0) after projector-based extraction and explicit master-integral insertion.The calculation uses Feynman gauge for 18 diagrams and finds complete agreement with the cited reference equations.
- Topology minimization: A 251-topology form-factor example minimizes externally generated topologies containing quadratic, eikonal, and mixed quadratic-eikonal propagators.The workflow handles mixed propagators, partial fractions, topology mappings, basis completion, and GLI rewriting.
8. Summary
FeynCalc 10 consolidates multiloop manipulation techniques in a flexible, modular framework intended to make these methods easier to use in routine research. Its broader workflow remains connected to external tools rather than being performed entirely within FeynCalc.
- Release goals: FeynCalc 10 integrates high-level functions for manipulating loop integrals and topologies, lowering the barrier to multiloop techniques.The release emphasizes convenient access to established algorithms within one framework.
- Release goals: Users can apply the new functions modularly alongside existing codes after converting integral families into FCTopology notation.The required conversion can usually be implemented with a few replacement rules.
- Scope and future framework: The authors do not intend FeynCalc alone to perform complete multiloop calculations, instead targeting a FORM-based framework that delegates selected steps to FeynCalc.Planned work includes an improved FeynHelpers interface and a public FORM-based setup.
Appendix A. Pak’s algorithm
Pak’s algorithm derives a canonical naming of Feynman parameters from the characteristic polynomial, allowing equivalent integral topologies to be recognized through polynomial and propagator-power comparisons.
- Polynomial representation: The characteristic polynomial P is constructed from the Symanzik polynomials U and F and records the integral family together with denominator powers.For loop integrals, the powers of each denominator are also retained.
- Canonicalization procedure: Pak’s algorithm iteratively permutes and sorts polynomial-matrix columns and rows to select a canonical parameter ordering.The procedure repeatedly retains matrices with the selected column vectors until the final permutations are obtained.
- Canonicalization procedure: The first surviving permutation provides the canonical naming of Feynman parameters and a list of symmetries under parameter renamings.The resulting permutation set is the algorithm’s symmetry information.
- Canonicalization procedure: The illustrative algorithm keeps the matrix associated with the selected maximal column vector through successive permutations until termination at i = n −1.The example ends with the surviving permutation used as the canonical representation.
- Topology mappings: Comparing canonical characteristic polynomials and propagator powers identifies one-to-one mappings between integrals related by loop-momentum shifts.Equivalent polynomial representations correspond to the same propagator structure under the stated mapping criterion.
Appendix B. Mapping of smaller topologies into larger topologies
Mapping smaller topologies into larger ones is handled by generating nonvanishing subtopologies of suitable parent topologies and applying Pak-style mappings at matching propagator counts.
- Scope and strategy: Pak’s algorithm alone maps topologies only when they contain the same number of propagators, so smaller-to-larger relations require parent-topology substructures.The method addresses cases where a smaller topology fits into a larger one.
- Subtopology generation: The workflow selects parent topologies with enough propagators, preferably complete bases suitable for IBP reduction, then enumerates their subtopologies.Subtopologies are generated by removing one or more propagators and checking whether the result is scaleless.
- Subtopology generation: Only nonvanishing subtopologies receive markers linking them to their parent topology and are passed as preferred candidates for topology mapping.Mappings are then found between smaller topologies and nonvanishing subtopologies with equal propagator counts.
- Computational cost: Generating and processing all nonvanishing subtopologies can be time-consuming and may increase the topology count by several orders of magnitude.The authors therefore recommend using this feature cautiously for large sets of complicated topologies.
Appendix C. Limitations of the current approach to topology minimization
Topology minimization uses canonical propagator ordering and momentum-shift equations, but eikonal and mixed propagators introduce ambiguities that require pragmatic handling and may leave pathological failures.
- Momentum-shift construction: The Pak algorithm can identify identical topologies, but finding explicit momentum shifts that realize their mapping is not always straightforward.The difficulty is converting topology equivalence into a concrete shift assignment.
- Quadratic propagators: For standard quadratic propagators, ordered propagator sets yield an overdetermined linear system whose solutions provide the required loop-momentum shifts.The system is typically overdetermined because there are fewer loop momenta than propagators.
- Eikonal propagators: Eikonal propagators do not uniquely determine momentum flow, so the implementation removes them before solving for momentum shifts.This avoids systems beyond linear equations, but relies on handling the remaining quadratic propagators.
- Solving the system: Squaring the equations permits solutions with sign changes of loop momenta, while removing suitable equations preserves solvability in the usually overdetermined system.The authors acknowledge pathological cases where this approach may fail and note the lack of a better solution without major performance costs.
- Mixed propagators: Propagators of the form l2 + c1l · s + c2 cannot be naively discarded because doing so may make the momentum-flow system unsolvable.Automatic momentum-flow determination for these interpolating propagators is described as tricky.