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Package-X: A Mathematica package for the analytic calculation of one-loop integrals
Hiren H. Patel
TL;DR
One-loop calculations need compact analytic results across singular and general kinematic configurations, while existing tools may require external scalar-function implementations. Package-X provides Mathematica-based tensor-integral reduction, explicit analytic expressions, and fermion utilities, with reported speedups and checks against known physical quantities.
Problem
Existing tools can require external analytic forms for scalar functions, and general-purpose analytic results covering all one-loop kinematic configurations are unavailable.
Method
Package-X computes dimensionally regulated high-rank one-loop tensor integrals with up to three propagators, refines coefficients into explicit expressions, and includes Dirac-trace and fermion-form-factor routines.
Results
Package-X provides compact analytic output for supported configurations, achieves up to a 200-fold speed increase in the physical region, and reproduces several known physical quantities in agreement with literature.
Takeaways & Limitations
The package is intended as a research and educational tool for symbolic and numerical one-loop calculations, including applications at singular kinematic points.
Takeaways & Limitations
Package-X currently supports loop integrals with up to only three denominator factors.
Abstract
from arXiv · showhide
Package-X, a Mathematica package for the analytic computation of one-loop integrals dimensionally regulated near 4 spacetime dimensions is described. Package-X computes arbitrarily high rank tensor integrals with up to three propagators, and gives compact expressions of UV divergent, IR divergent, and finite parts for any kinematic configuration involving real-valued external invariants and internal masses. Output expressions can be readily evaluated numerically and manipulated symbolically with built-in Mathematica functions. Emphasis is on evaluation speed, on readability of results, and especially on user-friendliness. Also included is a routine to compute traces of products of Dirac matrices, and a collection of projectors to facilitate the computation of fermion form factors at one-loop. The package is intended to be used both as a research tool and as an educational tool.
I. INTRODUCTION
Package-X addresses limitations of existing one-loop tools by providing compact analytic results across kinematic configurations, with integrated tensor, trace, and fermion-form-factor workflows.
- Motivation: Existing packages often use Passarino–Veltman reduction, while FeynCalc leaves scalar basis functions requiring external analytic forms or additional software.FeynCalc and FormCalc also target broader numerical cross-section workflows.
- Motivation: One-loop integrals are needed beyond cross sections, including counterterms, pole positions, residues, oblique parameters, and electromagnetic moments.These applications can require evaluations at physical thresholds or zero external momenta.
- Motivation: At singular kinematic points, Passarino–Veltman reduction typically breaks down even though compact analytic expressions can exist.This motivates a general-purpose analytic approach for thresholds and zero external momenta.
- Package-X design: Package-X computes dimensionally regulated rank-P one-loop tensor integrals with up to three denominator factors for arbitrary real external momenta and internal masses.Its output is designed for numerical evaluation and symbolic manipulation in Mathematica.
- Package-X design: The workflow decomposes tensors, applies on-shell and kinematic relations, and then refines coefficient functions into explicit expressions.LoopIntegrate performs decomposition, Mathematica rules apply relations, and LoopRefine performs the final conversion.
- Package-X design: Supplementary routines compute Dirac traces and project fermion self-energy and vertex form factors from one-loop integrals.The package organizes these capabilities alongside tensor-algebra and one-loop computation contexts.
IV. LOOPREFINE: REDUCTION TO ELEMENTARY FUNCTIONS
LoopRefine converts coefficient functions into explicit expressions through staged reductions, dimensional expansion, and logarithmic simplification tailored to kinematic configurations.
- Reduction workflow: LoopRefine processes each coefficient function by recording internal masses and selecting the corresponding reduction routine.The routines cover pvA, pvB, pvb, and pvC coefficient functions.
- Reduction workflow: C-function reductions terminate at C0, whose explicit form is substituted when it is infrared divergent or sufficiently compact under ExplicitC0.This keeps the resulting expressions explicit while avoiding unnecessarily large forms.
- Dimensional expansion: LoopRefine replaces d with 4 −2ϵ and expands in ϵ, representing UV divergences as 1/ϵ poles and IR divergences as 1/ϵ and/or 1/ϵ2 poles.The expansion retains the leading terms needed for the regulated result.
- Expression simplification: The final stage combines and simplifies logarithms, organizes the expression by logarithms, and groups the µ2-dependent logarithm with the 1/ϵ pole.This produces a more structured explicit result.
- A and B functions: A functions are obtained by direct integration, while B functions with 00 index pairs are reduced iteratively and the remaining B functions are integrated over one Feynman parameter.The implementation includes definitions for the Källén function and discriminant abbreviations.
- A and B functions: Explicit catalogued expressions handle B functions at singular points, avoiding runtime Limit calculations that can be slow and produce unwieldy outputs.These expressions are accessed through LoopRefine[pvB[0, n, s, m0, m1]].
B. Reduction of auxiliary bξ functions
Package-X reduces one-loop coefficient functions using kinematic-specific algorithms, including auxiliary bξ functions for gauge-propagator contributions and special C-function configurations. The procedures support compact analytic expressions across singular and non-singular kinematics, with explicit handling of UV-related finite terms.
- Auxiliary bξ functions: Package-X handles gauge terms in massless vector propagators inside bubble integrals through auxiliary bξ functions whose reductions mirror standard B functions.These functions are iteratively reduced when they contain paired indices and directly integrated when they do not.
- Auxiliary bξ functions: The auxiliary bξ function is finite for n ≥ 1, while the n = 0 case is IR-divergent and is not encountered in practice because the gauge propagator supplies two numerator momenta.
- C-function reduction: Cases 1, 3, 5, and 6 follow Denner-Dittmaier, while new Cases 2 and 4 provide more compact expressions and address reduction at physical thresholds.Case 4 applies when a physical threshold coincides with the boundary of the physical region and the standard cases are inapplicable.
- Exceptional configurations: When the relevant reduction elements vanish, additional formulae cover exceptional configurations provided at least one of p1^2, p2^2, or q^2 is non-vanishing; if all vanish, Case 5 is required.For scalar C0 with r = n1 = n2 = 0, the relations can reduce it to scalar B0 functions.
V. THE SCALAR C0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION
Package-X combines explicit analytic C0 expressions with specialized numerical evaluation for real masses and external momenta. Its implementation preserves compact output while substantially accelerating numerical computation.
- Analytic expressions: Package-X includes a catalog of analytic C0 expressions for cases with det Z ≠ 0, obtained by direct integration.The catalog completes the one-loop computation after coefficient reduction produces the UV-finite scalar C0 function.
- Analytic expressions: LoopRefine substitutes C0 expressions automatically when they are IR-divergent or sufficiently simple and compact.More complicated finite cases remain represented numerically as pvC0[…].
- Numerical implementation: The numerical C0 implementation uses Cutkosky’s rule for the imaginary part in the physical region, requiring only a single logarithm.A continuation is used in the unphysical region.
- Numerical implementation: The implementation separately evaluates real and imaginary parts of the dilogarithm and exploits conjugate or real arguments to reduce computation.These choices avoid unnecessary evaluation of both parts and can reduce the number of dilogarithm evaluations.
- Performance: 200-fold speedup is achieved in the physical region over brute-force Mathematica Kernel evaluation, while C compilation produces performance roughly twice as fast as LoopTools.The unphysical-region speedup reaches 20-fold, and C-target performance rivals the Fortran implementation in LoopTools.
VI. HANDLING THE +iε PRESCRIPTION AND SIMPLIFYING LOGARITHMS
Package-X handles branch-sensitive one-loop expressions under the +iε prescription and simplifies logarithms using assumptions about real positive masses and recorded positive scales.
- Branch prescriptions: LoopRefine output observes the +iε prescription and follows Mathematica’s analytic conventions for Log and PolyLog.The prescription determines the logarithm branches in one-loop computations.
- Logarithm simplification: Positive-real internal masses permit mass-ratio logarithms to be flipped into a canonical form, keeping logarithmic parts compact.This transformation relies explicitly on the assumption that internal masses are positive real.
- Logarithm simplification: Keeping µ2 in logarithm numerators preserves the +iε prescription when other scales are potentially time-like external invariants.The µ2 factors are combined using a variable known to be positive.
- Logarithm simplification: The coefficient of the µ2-dependent logarithm is matched to the coefficient of the 1/ϵ pole and grouped with it; for finite results, that coefficient cancels exactly.This organizes divergent and scale-dependent terms before output.
- Branch prescriptions: Package-X introduces two analytic functions for complicated cases where compact expressions cannot assume a universal infinitesimal-imaginary-part sign.Their second argument controls the side of the branch used for evaluation.
VII. SPUR: COMPUTATION OF TRACES OF DIRAC MATRICES
Spur provides rule-based Dirac-trace computation for closed fermion lines, with refinements that improve speed and support nearly instantaneous form-factor projections.
- Function and scope: Spur evaluates traces over products of Dirac gamma matrices in numerators, supporting one-loop integrals with internal closed fermion lines.Projector can also be used within Spur for open-line form-factor calculations.
- Algorithmic refinements: The trace calculation is rule-based and uses expression-management and specialized rewriting strategies to improve computation speed.Large term groups are temporarily held in Lists, while simpler gamma-matrix tasks use non-iterative rules.
- Algorithmic refinements: Contraction identities are applied in the direction with fewer interposed gamma matrices, and traces with γ5 are handled by separate rules.These choices reduce unnecessary rule searches and expression growth.
- Performance: 10 times faster results are generally obtained than with the Mathematica packages FeynCalc and Tracer.The comparison is reported for trace computations.
- Performance: Form-factor projections are nearly instantaneous on a modern computer despite the slowdown caused by the larger traces produced when Projector is included inside Spur.Timing measurements were recorded on a 2.93 GHz Intel i7 processor.
VIII. PROJECTOR: PROJECTION ONTO FERMION FORM FACTORS
Package-X supports open-fermion-line calculations through projectors that extract specified form factors and convert projected traces into ordinary tensor integrals.
- Scope and role: Package-X does not directly handle open fermion chains but supplies projectors for projecting loop integrals onto specific form-factor functions.The projectors provide support for fermion self-energy and form-factor computations.
- Self-energy projection: For an off-shell self-energy, parity conservation and Lorentz covariance express the integral through form factors depending only on p2 and m2.The numerator contains a Dirac matrix structure depending on loop momentum k and external momentum p.
- Self-energy projection: The projectors F[A] and F[B] are multiplied by the numerator and traced to obtain the corresponding form factors.The resulting traces are converted into ordinary tensor integrals.
- Available projectors: Pre-programmed projectors cover off-shell self-energies and on-shell scalar- and vector-vertex functions in L/R-chiral and Vector/Axial-vector bases.They are available through Projector and are documented comprehensively.
IX. CROSSCHECKS AND FURTHER DEVELOPMENT
Package-X was cross-checked against independent programs, explicit literature formulae, hand derivations, and numerical Feynman-parameter integrations. The authors also identify limitations guiding further development.
- Reduction routines and basis functions were compared with an independent computer program, while C-function cases were checked against explicit low-rank formulae.
- Each C0 scalar function was derived by hand and compared with literature formulae when available, or numerical parameter integrations otherwise.
- H →gg and γγ decay rates, electron g −2, and neutrino electric and magnetic moments agreed with the literature.
- Package-X lacks a generally available analytic series-expansion method, and Mathematica’s Series fails when outputs contain specially defined functions such as pvC0.
- Support is limited to loop integrals with at most three denominator factors, although compact higher-point expressions may exist at special kinematic points.
- Naive dimensional-regularization treatment of γ5 can prevent automatic Ward identities for VVA and AAA functions, while open fermion chains require projector-based form-factor extraction.
Appendix A: Conventions
The appendix establishes conventions for spacetime quantities and Passarino-Veltman functions, including a deliberately unconventional argument ordering. It also relates tensor integrals to scalar representations through momentum differentiation and covariant matching.
- Package-X uses a slightly unconventional Passarino-Veltman argument ordering that makes invariance under pairwise interchange explicit.
- Feynman-parameter representations cover Passarino-Veltman coefficient functions with polynomial integrands whose coefficients are ordered as displayed.
- Tables II and III summarize conventions for spacetime quantities and Passarino-Veltman-function arguments.
- Tensor integrals are represented as derivatives of scalar-integral representations with respect to external momenta, then matched to covariant tensor decompositions.
- UV-divergent contributions arise from 1/ϵ poles in the leading gamma function and can be obtained by integrating the resulting polynomial integrands.
Appendix C: Derivation of reduction formulae for C functions Case 2
The Case 2 derivation reduces C-function parameter integrals by changes of variables that factor nested integrations and identify resulting pieces with B functions. Beta-function behavior distinguishes finite and infrared-divergent cases.
- A sequence of variable changes factors the nested integrals, transforming them into forms suitable for identifying coefficient B functions.
- When n2 or r is nonzero, the Beta function is finite and its O(ϵ) expansion yields the reduction formula.
- When n2 = r = 0, the Beta function develops a 1/ϵ pole, requiring a separate decomposition involving C0 and B functions.
- The resulting identifications correspond to an infrared-divergent triangle classification by Ellis and Zanderighi.
- The derivation also handles permutations placing the off-shell momentum s in the third argument through variable and integration-order interchanges.
- The y′ integral is converted to a coefficient B function after expanding (1 −y′)^n2 as a binomial series.
Appendix D: Derivation of reduction formulae for C functions Case 4
Case 4 derives special C-function reduction formulae under singular kinematic conditions by factoring Feynman-parameter integrals through tailored variable changes. The resulting terms are expressed using incomplete Beta and coefficient B functions, with a separate case required when all external invariants vanish.
- For Case 4, the derivation begins under det Z = 0 and X̃0j = 0, with separate subcases determined by whether p2 vanishes.
- Choosing α = −c/(2b) eliminates the y′2 and y′z′ terms, while the singular kinematic relations eliminate the remaining y′ term.
- Termwise binomial expansions and incomplete Beta-function representations reduce the y integrals to compact expressions.
- Variable substitutions stretch integration ranges to 0 →1 and factor the integrals into three terms.
- The resulting quadratic polynomials match integrands for B functions associated with the triangle’s three cut channels.
- Equation (23) follows by identifying the transformed y′ integration with a coefficient B-function representation.
- When p1^2 = p2^2 = q^2 = 0, neither equations (22) nor (23) applies and Case 5 is required.
Appendix E: Coefficient B functions with r = −1
The appendix extends the basis functions needed by two new C-function reduction algorithms and supplies explicit expressions for general and singular kinematics. Threshold cases are handled by analytic continuation when the nominal Feynman-parameter integral diverges.
- Reduction basis: The new C-function reduction algorithms in Cases 2 and 4 require B functions with the index r continued to −1.These formulae may be related to prior reductions requiring scalar functions with repeated propagators.
- Explicit expressions: Explicit expressions are constructed for both general kinematics and singular points and included in the Package-X source file.The expressions are listed in Table I.
- Divergences: The functions are UV-finite for all n ≥ 0, while many kinematic configurations develop IR-divergent 1/ϵ poles.
- Threshold cases: For three physical-threshold cases, the Feynman-parameter integral nominally diverges even for finite but infinitesimal ϵ.
- Threshold cases: Threshold integrals are evaluated by taking ϵ sufficiently large and negative for convergence, then analytically continuing the result to ϵ → 0.