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FeynHelpers: Connecting FeynCalc to FIRE and Package-X
Vladyslav Shtabovenko
TL;DR
FeynCalc supports symbolic QFT computations, but tool conversions can require user-written code and robust maintenance. FeynHelpers connects it to Package-X and FIRE, simplifying calculations that were difficult or hardly feasible with FeynCalc. The authors demonstrate the interface with four examples from QED, QCD, and Higgs physics.
Problem
FeynCalc is mostly limited to algebraic manipulations, and connecting it to tools beyond FeynArts often requires users to convert inputs themselves.
Source: §1 · p. 1Method
FeynHelpers connects FeynCalc with Package-X's analytic one-loop library and FIRE's integration-by-parts reduction capabilities.
Source: §1 · p. 2, §2.2 · p. 6Results
The interface makes calculations that were difficult or hardly feasible with FeynCalc simpler, as demonstrated in four examples from QED, QCD, and Higgs physics.
Source: §6 · p. 35Takeaways & Limitations
The open-source interface code can be modified to accommodate specific requirements.
Source: §6 · p. 35FeynCalc does not automatically recognize multi-loop integrals from the same topology, so the current FIREBurn implementation evaluates each integral separately and is best suited to smaller, low-loop calculations.
Source: §4.3 · p. 22
Abstract
from arXiv · showhide
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. This is demonstrated with four different examples from QED, QCD and Higgs physics.
1. Introduction
FeynCalc supports flexible symbolic QFT calculations, but connecting it to other tools often requires user-written conversions. FeynHelpers provides a stable public interface to Package-X and FIRE, targeting semi-automatic calculations that are difficult to handle with FeynCalc alone.
1. Introduction
- FeynCalc is an established, flexible tool for small and medium-sized symbolic QFT computations, but connections to tools other than FeynArts usually require manual conversion.Source: §1 · p. 1
- Although users can write simple converters for individual projects, a community-ready interface must be robust, tested, maintained, and updated as tools and Mathematica change.Source: §1 · pp. 1–2
- FeynHelpers connects FeynCalc to Package-X and FIRE, enabling analytic results for most one-loop amplitudes with up to four external legs and reductions of many multi-loop amplitudes to master integrals.The authors present it as a stable public interface and a long-term solution while Package-X and FIRE remain actively developed.Source: §1 · p. 2
- The interface is intended to improve FeynCalc for semi-automatic calculations that are not easily handled by fully automatic all-in-one tools, rather than compete with those frameworks.Source: §1 · p. 2
2. Package-X and FIRE
Package-X provides analytic one-loop scalar-function results, while FIRE supplies general IBP reduction for loop integrals. The section outlines their scope, conventions, reduction approach, and practical limits.
2.1. Package-X
- Package-X provides analytic Passarino–Veltman functions for up to four legs and almost arbitrary kinematics, including cases with zero Gram determinants or vanishing or coinciding masses.Source: §2.1 · p. 3
- Package-X results use D = 4 − 2ϵ, omit the overall factor i/16π², and abbreviate 1/ϵ − γE + log(4π) as 1/˜ϵ.Source: §2.1 · p. 4
2.2. FIRE
- IBP relations follow from the divergence theorem, and reduction algorithms combine many such relations to express complicated integrals through a small set of master integrals.Source: §2.2 · pp. 5–6
- FIRE is a general-purpose IBP-reduction package with configurable procedures, Fermat and KyotoCabinet utilities, and either Mathematica or a faster C++ back end.Source: §2.2 · p. 6
- For thousands or millions of complicated multi-loop integrals, Mathematica is usually insufficient and FIRE's C++ engine becomes mandatory.The paper does not detail these large-scale workflows, which are mostly pursued with in-house FORM codes.Source: §2.2 · p. 9
- FeynCalc's built-in IBP reduction is limited to two-loop propagator-type integrals, motivating FIRE as a broader reduction back end for feasible small and medium-sized calculations.Source: §2.2 · p. 9
3. Implementation
FeynHelpers is designed to connect Package-X and FIRE to an existing FeynCalc session while aligning their syntax, conventions, and normalizations. It uses separate-kernel execution for FIRE and handles interface-specific constraints, including propagator-basis requirements.
3. Implementation
- FeynHelpers focuses on useful Package-X and FIRE functions for FeynCalc users, rather than passing through every function or duplicating FeynCalc’s algebra modules.For example, it omits Package-X’s Lorentz and Dirac algebra modules because similar functionality is already present in FeynCalc.Source: §3 · p. 9
- FIRE runs on a parallel Mathematica kernel while FeynCalc and FeynHelpers remain in the main kernel, avoiding symbol shadowing and FIRE source changes.The interface writes FIRE input and replacement-rule files for each integral; starting and stopping parallel kernels adds time penalties.Source: §3 · pp. 9–10
- FeynHelpers automatically handles convention differences when evaluating Package-X results, including the distinct one-loop integral normalizations used by the packages.FeynCalc’s PaVe functions and direct coefficient-function forms are among the ways to enter one-loop integrals; Table 1 summarizes the packages’ entry forms.Source: §3 · pp. 10–11, Table 1 · p. 11
- FIRE 5.2 cannot reduce integrals with linearly dependent propagators or propagators that do not form a basis, so FeynHelpers checks basis completeness and can complete it automatically.Users may also supply extra propagators; integrals with linearly dependent propagators require partial fractioning with ApartFF first.Source: §3 · pp. 11–12
- Unlike APart and FaRe, FeynHelpers automates the broader workflow of converting integrals, preparing input files, running FIRE, retrieving results, and returning them to FeynCalc.The paper notes that APart and FaRe provide some conversion-related functions but do not automate that entire workflow.Source: §3 · p. 12
4. Installation and Usage
FeynHelpers can be installed with its online installer, which can also offer to install Package-X and FIRE. Its PaXEvaluate and FIREBurn functions provide Package-X evaluation and FIRE reduction from FeynCalc, with documented scope boundaries for each interface.
4.1. Installation
- The online installer can download and install Package-X and FIRE when they are not found, and FeynHelpers should be loaded before FeynCalc.FeynHelpers is an add-on for FeynCalc 9.2 and above.Source: §4.1 · pp. 12–13
4.2. PaXEvaluate
- PaXEvaluate handles scalar loop integrals without loop-momentum-dependent numerator products and Passarino–Veltman functions, with related functions for UV, IR, or split results.Options include argument ordering, PaVe reduction, analytic expansions, and control of prefactors and epsilon expansion.Source: §4.2 · pp. 13, 15, 16
- PaXEvaluateUVIRSplit can distinguish UV and IR poles in dimensionally regulated one-loop scaleless integrals, which need not vanish under this prescription.FeynCalc’s handling is limited to one-loop integrals, and FIREBurn should not be used for consistent application because FIRE sets scaleless integrals to zero by default.Source: §4.2 · pp. 18–19
4.3. FIREBurn
- FIREBurn reduces scalar multi-loop integrals using IBP techniques and accepts the expression, loop momenta, and external momenta as its three arguments.It can create configuration files without starting FIRE, or run the reduction by default.Source: §4.3 · pp. 19, 21, 22
- In the example, FIREBurn reduces the supplied one-loop integral to an expression with a propagator denominator and a polynomial in D, m2, and pp.The output is shown in the supplied result passage.Source: §4.3 · p. 22
- FIREBurn evaluates each loop integral separately and lacks automatic recognition of integrals belonging to the same topology, limiting its practical use to smaller, low-loop calculations.The paper describes this per-integral evaluation as inefficient.Source: §4.3 · p. 22
5. Examples
Four examples show how FeynCalc and FeynHelpers support QED renormalization, QCD matching, Higgs decay, and two-loop self-energy calculations. The examples use analytic one-loop evaluation and multi-loop reduction to obtain results that can be checked against known results.
5. Examples
- The examples provide working Mathematica notebooks and present key steps and final results rather than reproducing the full code in the paper.The notebooks cover QED renormalization, NRQCD, Higgs decay, and two-loop QED self-energies.Source: §5 · p. 23
5.1. Renormalization of QED at 1-loop
- The QED example completes one-loop renormalization in MS, modified MS, and on-shell schemes, recovering F2(0) = g−2 and confirming Z_e = 1/√Z_A in the minimal subtraction schemes.With ΔUV = ΔIR, the on-shell fermion wave-function renormalization is gauge independent.Source: §5.1 · pp. 28–30
5.2. Quark-gluon vertex expanded in the relative momentum squared at 1-loop
- In the one-loop QCD-to-NRQCD matching workflow, amplitudes are expanded in q2 and form factors extracted; the results precisely recover Eqs. (24)–(25) and (29)–(30) from [37].The workflow generates diagrams, decomposes tensor integrals, applies Gordon decomposition, expands in q2, and extracts form factors.Source: §5.2 · pp. 30–32
5.3. Higgs decay to two gluons
- Package-X reproduces the Higgs-to-two-gluons form factor for both τ ≤ 1 and τ > 1, although only τ ≤ 1 applies to the top quark and known Higgs mass.Figure 1 compares the real and imaginary parts of A(τ) from the literature and FeynHelpers.Source: §5.3 · p. 33, Figure 1 · p. 33
5.4. 2-loop self-energies in massless QED
- The two-loop massless-QED self-energies reduce to two master integrals; vacuum polarization is gauge invariant, the electron self-energy depends on ξ, and the results agree with literature [46].The calculation uses FCMultiLoopTID for tensor decomposition and FIREBurn for IBP reduction.Source: §5.4 · pp. 34–35
6. Summary
FeynHelpers integrates Package-X’s one-loop library and FIRE’s IBP-reduction mechanism into FeynCalc, simplifying calculations that were previously difficult or hardly feasible. Four examples from QED, QCD, and Higgs physics demonstrate the interface.
6. Summary
- FeynHelpers integrates Package-X’s one-loop library and FIRE’s IBP-reduction mechanism into FeynCalc.The paper presents this as its first stable public version.Source: §6 · p. 35
- Four examples from QED, QCD, and Higgs physics demonstrate calculations simplified by the add-on.The paper says these calculations were difficult or hardly feasible with FeynCalc previously.Source: §6 · p. 35
- The open-source interface can be modified for specific requirements, with future development aimed at better integration and additional HEP-tool interfaces.Potential future interfaces include LiteRed and FormTracer.Source: §6 · p. 35