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New Developments in FeynCalc 9.0
Vladyslav Shtabovenko, Rolf Mertig, Frederik Orellana
TL;DR
Higher-order QFT calculations require software, but existing tools do not cover every specialized workflow, particularly for unusual theories and semi-automatic manipulation. This note presents FeynCalc 9.0’s improved tensor and loop-integral handling, interfaces, and EFT applications. The authors conclude that further development should target multi-loop calculations, effective-field-theory matching, software interfaces, and non-relativistic theories.
Problem
QFT practitioners need specialized software because no universal package covers every calculation, while unusual theories can be difficult to implement in fully automatic tools.
Method
The note develops and demonstrates FeynCalc 9.0 features for tensor decomposition, loop-integral extraction, IBP-tool interfacing, and non-relativistic EFT matching.
Results
FeynCalc 9.0 presents improved tensor reduction and loop-integral manipulation, including decomposition for one-loop tensors of any rank and multiplicity and conversion to an IBP-tool-compatible form.
Takeaways & Limitations
FeynCalc remains a flexible semi-automatic platform whose future directions include multi-loop calculations, EFT matching, software interfaces, and non-relativistic theories.
Takeaways & Limitations
The presented explicit non-covariant NRQCD approach may remain useful for higher quarkonium Fock states because projector techniques have not yet been generalized to them.
Abstract
from arXiv · showhide
In this note we report on the new version of FeynCalc, a Mathematica package for symbolic semi-automatic evaluation of Feynman diagrams and algebraic expressions in quantum field theory. The main features of version 9.0 are: improved tensor reduction and partial fractioning of loop integrals, new functions for using FeynCalc together with tools for reduction of scalar loop integrals using integration-by-parts (IBP) identities, better interface to FeynArts and support for SU(N) generators with explicit fundamental indices.
1. Introduction
FeynCalc 9.0 is presented as a specialized Mathematica tool for semi-automatic QFT calculations amid growing reliance on software. The note introduces its development history, compares its role with other packages, and previews the new version’s features and applications.
- Motivation: Computer tools have become increasingly important for higher-order perturbative calculations in quantum field theory.Recent achievements would have been difficult to complete within a reasonable time using only pen and paper.
- Motivation: Because no universal package meets every particle theorist’s needs, different programs should be used for specific tasks.FeynCalc is introduced as one such specialized tool.
- FeynCalc’s role and history: FeynCalc is a Mathematica package for algebraic QFT calculations and semi-automatic evaluation of Feynman diagrams.Its development began in 1991 and later incorporated TARCER, PHI, and FeynArts interfacing.
- Organization: The note compares FeynCalc with other packages, surveys version 9.0 improvements, presents an NRQCD matching-coefficient example, and concludes with broader development directions.The planned sections cover package comparisons, new features, an effective-field-theory application, and conclusions.
2. Comparison to similar tools
FeynCalc occupies a semi-automatic niche between fully automated packages and pen-and-paper work, prioritizing flexible user control over end-to-end automation. Its strengths include arbitrary valid-expression input and manipulable intermediate results, while numerical evaluation and large-scale automation require other tools.
- Limitations: FeynCalc cannot generate diagrams, numerically evaluate master integrals, or perform phase-space integration, so these steps require other tools.This limits its ability to compute cross sections or decay rates by itself.
- Limitations: Mathematica-based algebra makes FeynCalc slower than tools relying on FORM and unsuitable for evaluating hundreds, thousands, or millions of diagrams.The authors note that linking FeynCalc with FORM is possible but does not remove this practical boundary.
- Input and workflow: Users can choose the order of Dirac, color, and loop-integral manipulations instead of following a fixed automated workflow.Intermediate expressions remain accessible for selective simplification, expansion, sorting, abbreviation, and collection.
- Limitations: Fully automatic tools can become difficult to adapt when exotic theories require custom models, especially nonrenormalizable or non-covariant theories.Manual amplitude entry may be more straightforward than implementing such models in an automated framework.
- Input and workflow: FeynCalc accepts standalone valid objects such as Dirac traces, Lorentz vectors, and loop integrals, functioning as a calculator for QFT expressions.Users may also derive rules manually or evaluate diagrams generated by FeynArts.
- Positioning: Unlike highly automated packages, FeynCalc does not provide numerical evaluation or a fully automatic scattering-process workflow.It is therefore not presented as a direct competitor to tools such as FormCalc.
- Comparison with related packages: FeynCalc, HEPMath, and Package-X share a semi-automatic philosophy but differ in features such as γ5 schemes, tensor-rank limits, color algebra, interfaces, and analytic libraries.FeynCalc supports general SU(N) generators, while HEPMath is limited to SU(3) and Package-X lacks color-structure routines.
- Positioning: FeynCalc targets calculations too specific for fully automatic tools but too challenging to perform solely by hand.This defines its practical niche as semi-automatic QFT evaluation.
3. New features in FeynCalc 9.0
FeynCalc 9.0 broadens loop-integral reduction and manipulation, adds interfaces for IBP workflows and FeynArts, and enables more controlled expression handling. These changes extend tensor decomposition, partial fractioning, loop extraction, and algebraic input support while retaining user control over intermediate expressions.
- Improved tensor decomposition: FeynCalc 9.0 rewrites TID to support one-loop tensor decompositions of any rank and multiplicity.Earlier routines were limited in rank, multiplicity, or required manually constructed tensor bases.
- Improved tensor decomposition: TID selects Passarino-Veltman coefficient functions when Gram determinants vanish and scalar one-loop integrals otherwise.Scalar integrals can be converted to Passarino-Veltman scalar functions with ToPaVe, while UsePaVeBasis provides more compact coefficient-function expressions.
- Improved tensor decomposition: FCMultiLoopTID makes multi-loop tensor reduction available without the additional Mathematica code normally required by Tdec.The underlying algorithm decomposes integrals into symmetry-allowed tensor structures and determines coefficients using modified Gaussian elimination.
- Partial fractioning and IBP interfaces: FCLoopBasisIncompleteQ detects incomplete propagator bases, while FCLoopBasisFindCompletion identifies missing irreducible propagators for IBP preparation.Together with ApartFF, these functions can bring scalar multi-loop integrals into a form that can be forwarded to an IBP tool.
- Interfaces and expression control: FCLoopSplit, FCLoopIsolate, and FCLoopExtract simplify identifying unique loop integrals so they can be evaluated externally and substituted back into expressions.FCFAConvert also converts FeynArts CreateFeynAmp output into valid FeynCalc expressions with configurable momentum, polarization, and chirality options.
- Interfaces and expression control: ExpandScalarProduct and DiracGammaExpand now support selective expansion through momentum-based options, leaving other scalar products untouched.This avoids the unnecessary increase in terms caused by expanding every scalar product.
- Color algebra: FeynCalc 9.0 adds support for SU(N) generators in the fundamental representation with explicit fundamental and adjoint indices.This extends the package’s color-algebra input beyond expressions without explicit fundamental indices.
4. Using FeynCalc with non-relativistic EFTs
FeynCalc can support non-relativistic EFT calculations despite their non-covariant tensor structure, requiring only a relatively small amount of additional Mathematica code. The paper demonstrates this by using FeynCalc to match QCD onto NRQCD for χc0,2 → γγ at leading order.
- Motivation: Non-relativistic EFTs require separate treatment of temporal and spatial tensor components because their Lagrangians need not be manifestly Lorentz covariant.The explicit component dependence becomes important when expanding amplitudes in the relative velocity of heavy quarks.
- NRQCD matching: NRQCD matching compares perturbative QCD and NRQCD quantities at finite order in v to extract the matching coefficients multiplying NRQCD operators.The long-distance matrix elements are non-perturbative, whereas the matching coefficients can be calculated perturbatively under the stated scale assumptions.
- Application and scope: The example extracts leading-order-in-αs matching coefficients for χc0,2 → γγ at leading order in v, while χc1 → γγ is excluded by the Landau–Yang theorem.The implementation is presented as a pedagogical, quick, naive example rather than an optimized or flexible production tool.
- Calculation workflow: The workflow generates the two tree-level diagrams with FeynArts, converts them to FeynCalc notation, simplifies the amplitudes, imposes on-shell conditions, and performs the SPVAT decomposition.The calculation expands the QCD amplitude in |q|/m through second order and retains the vector and axial-vector spinor structures relevant to the process.
- Handling non-covariant structures: FeynCalc avoids introducing Cartesian or mixed Lorentz–Cartesian tensors by encoding the relevant structures through its built-in TensorFunction.This allows standard Lorentz-tensor routines to remain applicable during the non-relativistic expansion.
5. Summary
FeynCalc 9.0 adds features and demonstrates applications for obtaining new results, while its development remains ongoing toward broader theoretical-physics support. Continued demand for flexible semi-automatic tools persists despite newer general-purpose packages and fully automatic one-loop systems.
- FeynCalc 9.0 introduces new features and improvements and is used in cases that obtain new results.
- Future development directions include multi-loop calculations, matching coefficients in effective field theories, interfaces to other software, and non-relativistic theories.
- New general-purpose packages with comparable flexibility show that user-friendly, semi-automatic tools like FeynCalc remain in demand for research projects.