Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals

Hiren H. Patel

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Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals

Hiren H. Patel

arXiv:1612.00009v2Nov 30, 2016hep-ph

TL;DR

Package-X 2.0 addresses the lack of complete analytic one-loop integral expressions by extending a Mathematica package to broader propagator, dimensional, divergence, and fermion-line cases. It adds analytic and series-expansion capabilities together with arbitrary-precision numerical functions, while retaining scope boundaries around certain singular kinematic expansions.

Problem
Existing public packages numerically evaluate one-loop integrals, but the paper identifies a lack of complete analytic expressions for applications where compact analytic results exist.
Method
Package-X 2.0 extends LoopIntegrate and LoopRefine to four-point and weighted tensor integrals, divergence-specific extraction, Taylor series, open fermion lines, and arbitrary-precision special functions.
Results
Version 2.0 significantly expands Package-X’s scope by supporting four-point integrals, Taylor expansions, and open fermion lines, while producing analytic expressions and numerically stable arbitrary-precision evaluations within stated coverage.
Takeaways & Limitations
The release provides analytic one-loop results and expansions across broader integral structures, with separate UV/IR information and open-fermion-line support for quantum field theory calculations.
Takeaways & Limitations
Automatic asymptotic expansions around Landau singularities remain unsupported, limiting automated small-mass, threshold, and related expansions.

Abstract

from arXiv · show

This article summarizes new features and enhancements of the first major update of Package-X. Package-X 2.0 can now generate analytic expressions for arbitrarily high rank dimensionally regulated tensor integrals with up to four distinct propagators, each with arbitrary integer weight, near an arbitrary even number of spacetime dimensions, giving UV divergent, IR divergent, and finite parts at (almost) any real-valued kinematic point. Additionally, it can generate multivariable Taylor series expansions of these integrals around any non-singular kinematic point to arbitrary order. All special functions and abbreviations output by Package-X 2.0 supports Mathematica's arbitrary precision evaluation capabilities to deal with issues of numerical stability. Finally, tensor algebraic routines of Package-X have been polished and extended to support open fermion chains both on and off shell. The documentation (equivalent to over 100 printed pages) is accessed through Mathematica's Wolfram Documentation Center and contains information on all Package-X symbols, with over 300 basic usage examples, 3 project-scale tutorials, and instructions on linking to FeynCalc and LoopTools.

I. INTRODUCTION

Package-X addresses the lack of complete analytic one-loop integral results by generating dimensionally regulated tensor-integral expressions for broad propagator and kinematic configurations. Version 2.0 extends this scope with four-point integrals, Taylor expansions, open fermion lines, and expanded documentation.

I. INTRODUCTION: Package-X generates analytic results for dimensionally regulated one-loop rank-P tensor integrals with up to four denominator factors and arbitrary integer weights.The integrals support arbitrary external momenta and real-valued internal masses.
I. INTRODUCTION: The package fills a gap left by publicly available tools that numerically evaluate one-loop integrals but do not provide complete analytic expressions.The stated target applications include pole masses, oblique parameters, particle moments, decay rates, threshold cross sections, counterterms, and Wilson coefficients.
I. INTRODUCTION: The release addresses limitations of the original 2015 version while adding further features and enhancements.The paper directs readers to the original publication for the underlying algorithms and program structure.
I. INTRODUCTION: Version 2.0 introduces four distinct denominator factors, Taylor series expansions, and support for integrals with open fermion lines.These are identified as the release’s most prominent features.

II. UPDATES TO LOOPINTEGRATE

LoopIntegrate performs covariant tensor decomposition and, in version 2.0, accepts broader propagator structures and open fermion-line objects. New syntax and algebraic preprocessing improve momentum-routing flexibility and reduce integral complexity before decomposition.

II. UPDATES TO LOOPINTEGRATE: LoopIntegrate now supports up to four distinct propagator factors with arbitrary integer powers and expresses results through Passarino-Veltman coefficient functions.The decomposition is based on Lorentz covariance of dimensionally regulated loop integrals.
II. UPDATES TO LOOPINTEGRATE: Version 2.0 accepts nonzero p0, allowing any momentum routing without strategically routing a denominator factor or shifting the integration variable.The earlier syntax required p0 = 0.
II. UPDATES TO LOOPINTEGRATE: Partial-fraction expansion and propagator cancellation can lower denominator count and tensor rank before covariant decomposition.These operations are controlled by the Apart and Cancel options.
II. UPDATES TO LOOPINTEGRATE: LoopIntegrate can decompose integrands containing DiracMatrix, FermionLine, and FermionLineProduct objects, enabling open fermion-line calculations.This extends the routine beyond purely tensor-integral inputs.

III. UPDATES TO LOOPREFINE

LoopRefine extends analytic reduction to four-point and weighted integrals, supports nearby even spacetime dimensions, tests power infrared divergences, and extracts UV or IR divergent parts separately. Its kinematic coverage remains incomplete in specific singular cases.

III. UPDATES TO LOOPREFINE: LoopRefine converts four-point coefficient functions PVD to elementary functions, including cases with vanishing Gram determinant, but complete kinematic coverage remains unavailable.The algorithms do not apply when the modified Cayley determinant vanishes, and the additional formulae provide only further incomplete coverage.
III. UPDATES TO LOOPREFINE: Arbitrary integer-weight loop integrals can now be converted to analytic expressions, extending support beyond the original unit-weight restriction.Weighted integrals arise in general covariant-gauge calculations and small-momentum expansions.
III. UPDATES TO LOOPREFINE: Coefficient functions near even spacetime dimensions other than four can be related to functions near four dimensions and converted analytically by the same strategy.The relation is obtained by equating Feynman-parameter integral representations.
III. UPDATES TO LOOPREFINE: LoopRefine can test for power infrared divergences by checking whether the +iε → 0 limit is well behaved after analytic conversion.This check allows cancellations of power infrared divergences among terms in the input expression.
III. UPDATES TO LOOPREFINE: The Part option restricts computation to ultraviolet or infrared divergent parts, providing a rapid way to obtain 1/ε poles and identify their origin.Separate acceleration strategies are described for UV and IR contributions.

IV. THE SCALAR D0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION

LoopRefine combines analytic and numerical treatment of the scalar D0 function, including special handling for divergent and finite four-point cases. Version 2.0 also implements numerical special functions with arbitrary precision to improve stability.

IV. THE SCALAR D0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION: For nonvanishing Gram determinants, reduction of PVD ends with the UV-finite scalar function D0, which LoopRefine replaces to complete the integral.The scalar D0 function is the final scalar component in this reduction path.
IV. THE SCALAR D0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION: Infrared-divergent cases are substituted with expressions consistent with the +iε prescription so numerical evaluation produces the correct imaginary-part sign.The substitutions cover expressions with massive and massless internal lines.
IV. THE SCALAR D0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION: Infrared-finite four-point functions without vanishing Gram determinants are represented by the numerically implemented ScalarD0 because compact analytic expressions are unavailable.The real part uses a representation involving sixteen numerically evaluated dilogarithm functions, and ScalarD0 is compiled for rapid evaluation.
IV. THE SCALAR D0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION: Every special numerical function, including ScalarD0, uses native Wolfram Kernel functions and supports Mathematica arbitrary-precision evaluation.The stated purpose is numerically stable results for configurations otherwise prone to severe precision loss.
IV. THE SCALAR D0 FUNCTION: ANALYTIC EXPRESSIONS AND NUMERICAL IMPLEMENTATION: ExpandD0 replaces ScalarD0 with a corresponding analytic formula, using compact RootSums over continued dilogarithms to limit output size.The analytic library includes formulas adapted or derived from cited results for simpler and more general cases.

V. LOOPREFINESERIES: TAYLOR SERIES EXPANSIONS OF LOOP INTEGRALS

Package-X 2.0 constructs multivariable Taylor series of one-loop integrals by differentiating and reducing Passarino-Veltman coefficient functions. The method generalizes to arbitrarily high order and multiple expansions, but not around Landau singularities.

Motivation: Widely separated scales can make exact analytic representations unwieldy and numerical evaluation vulnerable to precision loss from large cancellations.Series approximations using the first few terms are therefore desired.
Method: LoopRefineSeries replaces LoopRefine to construct Taylor series expansions of loop integrals.It is designed for tensor integrals and supports multiple series expansions.
Method: The algorithm differentiates coefficient functions, rewrites them with fewer 00 index pairs, and applies reduction formulae to obtain analytic expressions.The implementation uses standard coefficient functions and Cayley-determinant-based reduction for negative index-pair counts.
Capabilities: Multiple series expansions and arbitrarily high orders are supported, constrained only by available memory and computation time.This generalization follows from the coefficient-function differentiation and reduction procedure.
Limitation: Expansions cannot be constructed around Landau singularities because the necessary derivatives usually do not exist.Consequently, LoopRefineSeries alone cannot automatically produce small-mass, threshold, or other asymptotic expansions.

VI. SUPPORT FOR OPEN FERMION LINES

Package-X 2.0 adds direct support for integrals with open fermion lines, including off-shell DiracMatrix objects and on-shell FermionLine structures. Its algebraic routines canonicalize these expressions through dimensional and four-dimensional identities, with γ5-related manipulations restricted to exactly four dimensions.

Motivation: The original package handled closed fermion loops but lacked direct support for evaluating integrals with open fermion lines.Version 2.0 addresses this gap.
New objects: DiracMatrix represents products of Dirac matrices from off-shell fermion lines, while FermionLine and FermionLineProduct represent on-shell lines and their direct products.These objects support diagrams containing one or more open fermion lines.
Algebraic manipulation: The routines expand matrix products, simplify repeated Lorentz-index contractions, and use Dirac algebra to move /p to line ends for the Dirac equation.These are the initial canonicalization steps for open fermion structures.
Algebraic manipulation: FermionLineProduct objects additionally use Sirlin identities, reduce remaining products to the SVTAP basis, and apply Gordon identities to transition currents.The full algorithm is repeated until no further identities apply.
Scope: Because γ5 is implemented naively in dimensional regularization, manipulations involving γ5 are valid only in exactly 4 dimensions.The identities used in the later steps are likewise four-dimensional.

VII. EXPANSION OF THE DOCUMENTATION

Package-X 2.0 expands its Mathematica-accessible documentation with fuller symbol and option coverage, clearer examples, and three project-scale tutorials. It also documents links to FeynCalc and LoopTools.

Documentation: Package-X documentation is accessible from within Mathematica and covers all front-end functions and available options.Version 2.0 expands the documentation with more details and usage examples.
Tutorials: The Ward Identities and γ5 tutorial demonstrates enforcing chiral Ward identities with Alder’s method for the Z∗γγ Green function.It addresses the package’s naive anticommuting γ5 definition in dimensional regularization.
Tutorials: The µ →eγ tutorial explains Projector and shows how Package-X can verify decoupling, gauge invariance, and Ward identities.These applications are presented through form-factor extraction.
Tutorials: The light-by-light scattering tutorial computes the leading-order matrix element and illustrates capabilities and limitations of covariant methods.It provides a project-scale example of Package-X use.
Interoperability: The documentation gives instructions for linking Package-X to FeynCalc through FeynHelpers and to LoopTools.Both linked packages are publicly available.

VIII. SUMMARY AND AVENUES FOR FURTHER DEVELOPMENT

Package-X 2.0 expands the software to four-point integrals, Taylor expansions, and open fermion lines, while several important scope and automation gaps remain. Future development targets singular kinematics, automated integral generation, numerical coefficient evaluation, and noncovariant integrals.

Summary: Version 2.0 significantly expands Package-X through support for four-point integrals, Taylor expansions, and open fermion lines.The paper presents these additions as the major release’s central advances.
Future development: Package-X cannot automatically construct asymptotic expansions around Landau singularities, including expansions relevant to mass regularization, thresholds, and eikonal limits.An update adding this capability is identified as desirable.
Future development: Four-point reduction algorithms do not completely cover vanishing Cayley determinants, including cases that can correspond to IR-divergent scattering amplitudes at physical threshold.Complete kinematic coverage of these singular cases remains a desired improvement.
Future development: Manual input of all integrals becomes increasingly tedious and error-prone for larger calculations, motivating automatic generation through links such as FeynArts.The limitation concerns workflow automation rather than the analytic reduction itself.
Future development: Large problems may not yield compact analytic expressions, motivating direct numerical evaluation of coefficient functions and links to modern packages such as Collier.LoopTools integration is documented, but broader numerical interoperability is desired.
Future development: Package-X is limited to Lorentz-covariant Feynman integrals, excluding noncovariant cases encountered in Coulomb and axial gauges and in NRQED, HQET, and SCET.Automatic analytic results for such integrals are identified as a desirable extension.

Appendix A: Convention and package structure changes between versions 1.0 and 2.0

Version 2.0 reorganizes naming, integration normalization, Passarino–Veltman conventions, contexts, auxiliary functions, and fermion self-energy definitions.

Naming conventions: Version 2.0 renames several predefined symbols and the dimensional-regularization scale to follow Mathematica naming conventions.The renamed symbols include pvA, pvB, pvC, pvC0, pvC0IR6, and μR.
Integration normalization: LoopIntegrate now factors out e^γEε instead of the previous gamma-function normalization in its integration measure.This change prepares Package-X for computing two-loop integrals in the future.
Passarino–Veltman conventions: Passarino–Veltman C-function argument orders were changed to match conventions used by FeynCalc, LoopTools, Collier, and the literature.The revised ordering is illustrated for pvC and pvC0.
Package structure: The contexts X‘IndexAlg‘, X‘Spur‘, and X‘OneLoop‘ are deprecated because all package symbols now belong to the common X‘ context.The auxiliary Passarino–Veltman function pvb is also obsolete and covered by higher-weight PVB functions.
Fermion self-energy: The off-shell fermion self-energy form factor C(p2) was redefined in Projector without a factor of i.

Appendix B: New functions/symbols introduced in version 2.0

Version 2.0 introduces functions for Taylor expansion, four-point tensor and scalar integrals, kinematic expressions, analytic special functions, Lorentz-scalar declarations, and fermion-chain algebra.

Taylor expansions: A new function generates multiple Taylor-series expansions of one-loop tensor integrals.
Four-point integrals: New functionality represents Passarino–Veltman D coefficient functions and produces analytic expressions for them through LoopRefine.
Scalar four-point functions: ScalarD0 and related functions provide finite parts of scalar four-point integrals for real invariants and positive real masses.
Analytic functions: ScalarC0, ScalarD0, and related functions can be expanded into analytic expressions, including continued dilogarithms and kinematic polynomials.
Kinematic and symbolic utilities: Replacement rules express Lorentz scalar products in terms of Mandelstam invariants and masses, while another function allows symbols to be declared Lorentz scalars.
Fermion-chain algebra: FermionLine and FermionLineProduct represent Dirac-matrix products between on-shell spinors and their direct products, and routines perform the Dirac algebra.