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Golem95: a numerical program to calculate one-loop tensor integrals with up to six external legs
T. Binoth, J. -Ph. Guillet, G. Heinrich, E. Pilon, T. Reiter
TL;DR
One-loop amplitudes with many external legs require numerical tensor-integral tools beyond earlier public limits. Golem95 reduces form-factor representations numerically to basis integrals and provides master-integral calculations, with tests including perfect agreement for four-photon amplitudes; its current basis-integral implementation is limited to massless internal particles.
Problem
Public tensor-reduction software was available only for infrared-finite integrals with up to five external legs, while NLO scattering calculations require broader multi-leg support.
Method
Golem95 represents tensor integrals through form factors and numerically reduces them to basis integrals, using one-dimensional parameter-integral representations for most basis integrals.
Results
The program evaluates tensor integrals up to rank-six six-point functions and its four-photon amplitude results show perfect agreement with well-known results.
Takeaways & Limitations
Golem95 provides both numerical tensor reduction and a master-integral library for calculations using unitarity-cut techniques.
Takeaways & Limitations
The current version implements basis integrals for massless internal particles; massive internal-particle master integrals are deferred to a forthcoming version.
Abstract
from arXiv · showhide
We present a program for the numerical evaluation of form factors entering the calculation of one-loop amplitudes with up to six external legs. The program is written in Fortran95 and performs the reduction to a certain set of basis integrals numerically, using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with massless internal particles in a fast and numerically stable way.
PROGRAM SUMMARY
Golem95 evaluates one-loop multi-leg tensor integrals for next-to-leading-order scattering calculations. It numerically reduces tensor integrals to basis integrals, avoiding large algebraic expressions, with massless internal-particle basis integrals in the current version.
- Golem95 is implemented in Fortran95 and runs on computers with a Fortran95 compiler.
- The program evaluates one-loop multi-leg tensor integrals arising in next-to-leading-order corrections to particle-scattering amplitudes.
- Tensor integrals are represented through form factors and reduced numerically to a set of basis integrals.This avoids generating large algebraic expressions.
- The current basis-integral implementation supports massless internal particles only.Basis integrals for massive internal particles are planned for a future version.
- A rank 6 six-point form factor takes 0.13 seconds at a randomly chosen kinematic point on an Intel Core 2 Q9450 2.66 GHz processor.
1 Introduction
The introduction situates Golem95 in the push toward automated NLO calculations for processes with many external legs. It presents numerical tensor-integral reduction up to six external legs, with master-integral support and outputs based on Laurent-series coefficients.
- NLO predictions require control of real-emission and virtual corrections for the many scattering processes studied at TeV-scale collider experiments.
- Existing public tensor-reduction programs were limited to infrared-finite integrals and at most five external legs.
- Golem95 extends numerical tensor-integral reduction to processes with up to six external legs, focusing initially on massless QCD applications.
- The program's output contains the real and imaginary parts of the Laurent-series coefficients of the 1/ε^2, 1/ε, and finite terms.
- Golem95 can also serve as a master-integral library for calculations whose integral coefficients are obtained using unitarity-cut techniques.
2 Theoretical background
The theoretical framework represents one-loop tensor integrals through shift-invariant form factors and Feynman-parameter integrals, reducing them numerically to basis integrals while avoiding unstable inverse Gram determinants.
- 2.1 Form Factors: One-loop tensor integrals are expressed using tensor products of the metric and shift-invariant difference vectors, with associated form factors.The formulation can instead use loop-momentum combinations directly, preserving invariance under shifts of the loop momentum.
- 2.1 Form Factors: For N ≥ 6, tensor reduction can require only form factors with N ≤ 5, avoiding additional metric tensors while retaining extra external vectors.The five-point reduction avoids inverse Gram determinants and spurious higher-dimensional five-point functions.
- 2.2 Feynman parameter representations: Tensor integrals are related to Feynman-parameter integrals with numerator parameters, while pinched propagators are represented by subsets of the propagator-label set S.This set-based representation covers the lower-point integrals appearing alongside maximal N-point integrals in amplitudes.
- 2.3 Basis integrals: The reduction basis contains dimension-shifted four-point functions and two- and three-point functions, separating infrared poles into triangle functions.Some basis functions retain Feynman parameters in their numerators rather than being scalar integrals.
- 2.3 Basis integrals: When further scalar reduction would introduce inverse Gram determinants, golem95 tests the dimensionless parameter ˆB and switches to direct numerical integration when it is too small.The default threshold is ˆBcut = 0.005; one-dimensional parameter representations enable deterministic integration with a default relative-error target of 10^-8.
- 2.3 Basis integrals: The program can also provide massless-internal-particle master integrals as a library, including scalar box integrals obtained through form factors.The appropriate box type is selected automatically according to the kinematics.
3 Overview of the software structure
The software is organized into source, demonstration, documentation, and test directories, with user-defined kinematics configured through steering files.
- 3 Overview of the software structure: The program has four main directories: src for source files, demos for demonstrations, doc for documentation, and test for user-defined kinematics tests.The test programs can generate form factors for a specified rank, numerator, numerical point, and related settings through a steering file.
4 Description of the individual software components
The software is organized into Fortran95 modules for form factors, kinematics, integrals, numerical integration, and auxiliary functionality. Users can configure integration methods and switching parameters, with a specific limitation for four-off-shell-leg boxes.
- The program is written in Fortran95 and is downwards compatible with Fortran90.
- The src directory separates form-factor, integral, kinematic, and numerical-integration components.Form-factor modules cover two-point through six-point functions, while kinematic modules compute the S matrix, its inverse, and reduction coefficients.
- The parametre.f90 module defines defaults and controls the switch between analytic scalar-basis reduction and numerical integral evaluation.It also includes a switch for calculating only rational parts instead of complete form factors.
- Users can change the one-dimensional integration method and the cuts that trigger numerical integration without recompiling the library.
- For boxes with four off-shell external legs, one-dimensional numerical-integration expressions are unavailable, so the program always reduces numerically to scalar basis integrals.
5 Installation instructions
Installation uses a Perl configuration script that installs the golem95 library into a chosen directory and can select an available Fortran95 compiler. The tested compiler set includes GNU, Intel, DEC, NAG, and Portland compilers.
- The program is distributed as a v1.0.tar.gz archive with installation instructions in the accompanying Readme.
- Running ./configure.pl with an installation path installs the library in the specified directory, while omitting the path creates a libgolem subdirectory in the current directory.
- The installation directory contains Fortran module files and the static library libgolem.a.
- If no compiler is specified, the installation script selects the first matching Fortran95 compiler it finds.
- The program has been tested with GNU, Intel, DEC, NAG, and Portland Fortran compilers.
6 Test run description
The test runs demonstrate form-factor calculations for five- and six-point functions and show how golem95 can be embedded in a complete four-photon amplitude calculation. The examples include configurable kinematics, numerators, output files, and numerical-stability demonstrations.
- The demos directory provides examples for three- through six-point functions, four-photon helicity amplitudes, and numerical stability near det G or det S equal to zero.
- The rank-five five-point example calculates a form factor A5,5 from predefined demo input and writes the result to test5point.txt.
- The five-point test specifies momentum-conservation kinematics through invariant values and factors out rΓ from the reported result.The output is expressed through the 1/ε2, 1/ε, and finite terms under the stated integral-measure convention.
- Users can provide the number of legs, rank, numerator, and numerical point through param.input, after which the program compiles and runs the corresponding functions.
- For rank-one six-point functions, selecting all numerators produces separate output files and a combined Mathematica-readable results file.The calculation assumes momentum conservation for five- and six-point functions.
- The four-photon example evaluates the amplitude from form factors and reports perfect agreement with well-known results.The calculation uses three independent photon-ordering topologies after identifying diagrams related by charge flow.
7 Conclusions and Outlook
Golem95 numerically reduces tensor integrals through rank six and six-point functions while avoiding problematic inverse determinants by switching to direct numerical evaluation. Its current scope emphasizes massless internal particles, with broader automation and massive master integrals identified as future work.
- Golem95 evaluates tensor integrals through rank six and six-point functions using numerical reduction to analytically implemented basis integrals.
- When an inverse determinant becomes small, the program switches from reduction to numerical evaluation of the tensor integral.Most basis integrals use one-dimensional parameter representations for fast and precise numerical integration.
- The four-photon helicity-amplitude representation is illustrative because spinor-helicity methods can evaluate those amplitudes more simply without tensor four-point functions.
- The program can also serve as a master-integral library, including infrared-divergent integrals, with form factors lacking Feynman-parameter labels corresponding directly to scalar integrals.
- The current version does not implement master integrals with massive internal particles, although massive external legs are unrestricted.
- A future version is intended to combine form-factor evaluation with amplitude generation toward full automation of one-loop amplitude calculations.
A.1 Landau singularities
Landau and scattering singularities are distinct from spurious inverse-Gram artifacts and can create genuine numerical difficulties. Near scattering singularities, diagram-level reduction may fail, while combining diagrams can introduce large cancellations even when amplitudes remain finite.
- Distinguishing singularities: Actual Landau singularities are distinct from spurious inverse Gram determinants introduced by tensor reduction.Scattering singularities are one type of actual kinematic singularity, with det G approaching zero.
- Singular configurations: At configurations where mass and scattering singularities coincide, individual diagrams become infinite.
- Reduction breakdown: At a scattering singularity, the diagram-level reduction algorithm breaks down because B is proportional to det G / det S and diverges.
- Amplitude-level behavior: Gauge cancellations can reduce or even bound the singularity in complete amplitudes, but numerical combinations of separate diagrams remain unstable.The six-photon amplitudes are cited as an example where the singularity can become bounded.
- Numerical instability: Large cancellations are inherent to reductions based on scalar master integrals, which may become linearly dependent near singularities.
- Possible remedies: Possible remedies include multiple precision or excluding and smoothly interpolating across a phase-space region, but these can fail when the observable is controlled by the singularity.
A.2 One-dimensional integral representations
The appendix constructs fast numerical representations for infrared-finite box and triangle integrals by analytically removing Feynman-parameter integrations and numerically integrating the remaining variables. Contour deformation and stable auxiliary functions address poles, logarithmic cuts, and problematic kinematic limits.
- Integral representations: Infrared-finite box and triangle integrals are represented as one-dimensional Feynman-parameter integrals for fast deterministic numerical integration.The program switches to this numerical evaluation in a specified implementation condition.
- Four-point reduction: A variable transformation removes the delta distribution, after which x and y are integrated analytically and z is evaluated numerically with adaptive Gauss-Kronrod integration.
- Four-point reduction: For massless internal lines, six- and eight-dimensional four-point functions use two basis integrals and recurrence relations for the analytic integrations.
- Contour treatment: For six-dimensional three-mass four-point functions, the remaining z integral is evaluated after integrating over x and y, with S-matrix elements understood with an infinitesimal imaginary part.
- Contour treatment: Contour deformation avoids logarithmic branch cuts by choosing the deformation sign from the logarithm parameters, and the two logarithmic terms are computed separately.The separate treatment is needed because each term can have a pole even though their sum does not, and the terms can require different contour choices.
- Three-point functions: Three-point functions are reduced to lower-dimensional numerical integrals, while the H_i functions are defined to remain stable when X approaches Y.