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Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations
Ansgar Denner, Stefan Dittmaier, Lars Hofer
TL;DR
One-loop calculations for general scattering processes require stable treatment across difficult phase-space regions, complex masses, and infrared singularities. Collier addresses this with adaptive tensor reductions, direct tensor handling, and alternative regularization options. It provides stable results for almost all phase-space points and supplies integrals for traditional and unitarity-based calculations.
Problem
One-loop amplitude methods can encounter numerical problems in specific phase-space regions, while general processes may require complex masses and broad integral coverage.
Method
Collier numerically evaluates scalar and tensor one-loop integrals using adaptive reduction methods, direct tensor-level reduction, complex masses, and dimensional or mass regularization.
Results
Collier provides stable results for almost all phase-space points and delivers the relevant scalar and tensor one-loop integrals for Feynman-diagrammatic and unitarity-based calculations.
Takeaways & Limitations
Collier serves as an integral library and an essential building block in the automated one-loop generators OpenLoops and Recola.
Takeaways & Limitations
Non-vanishing imaginary parts of momentum invariants are not supported, and N-point coefficients with N≥8 require one-dimensional-array representation because Fortran arrays have rank limit seven.
Abstract
from arXiv · showhide
We present the library Collier for the numerical evaluation of one-loop scalar and tensor integrals in perturbative relativistic quantum field theories. The code provides numerical results for arbitrary tensor and scalar integrals for scattering processes in general quantum field theories. For tensor integrals either the coefficients in a covariant decomposition or the tensor components themselves are provided. Collier supports complex masses, which are needed in calculations involving unstable particles. Ultraviolet and infrared singularities are treated in dimensional regularization. For soft and collinear singularities mass regularization is available as an alternative.
PROGRAM SUMMARY
Collier is a Fortran95 library for numerical one-loop integral calculations, distributed under GNU GPL version 3 for common systems.
- Availability: The program is available from the Collier project website.
- Program characteristics: Collier is a Fortran95 program for evaluating one-loop integrals.It is intended for computers with a Fortran95 compiler.
- Program characteristics: The library is distributed under the GNU GPL version 3.
- Program characteristics: Collier runs on Linux and Mac OS X without external routines or libraries.
1. Introduction
The paper motivates Collier by the need for stable, general one-loop calculations in automated high-energy predictions, especially when scalar-integral reductions become numerically problematic.
- Motivation: NLO QCD and electroweak corrections are important ingredients of theoretical predictions for collider scattering processes.
- Existing approaches: Automated one-loop amplitude calculations use both Feynman-diagrammatic and generalized-unitarity approaches.
- Motivation: Direct amplitude reduction to scalar integrals can become numerically problematic in specific phase-space regions.Such regions are usually handled by resorting to quadruple precision.
- Collier’s scope: Collier is designed to provide the tensor and scalar integrals needed by processes with complex masses and unrestricted external-particle multiplicity.
- Applications: The library has been used in diverse NLO QCD and electroweak calculations and integrated into OpenLoops and Recola.It has also been used for one-loop integrals in NNLO calculations involving collinear or soft external momenta.
2. Conventions
Collier defines conventions for one-loop scalar and tensor integrals, their Lorentz-covariant decompositions, and the treatment of ultraviolet and infrared singularities.
- Integral definitions: One-loop tensor N-point integrals are expressed as loop-momentum integrals with propagator denominators carrying the Feynman prescription.
- Integral definitions: Scalar integrals correspond to rank-zero tensor integrals and are denoted T 1=A through T 7=G.
- Tensor decomposition: Tensor integrals are decomposed into Lorentz-covariant structures built from metric tensors and external momenta.Curly brackets denote symmetrization over non-equivalent Lorentz-index permutations with unit weight.
- Tensor decomposition: Tensor coefficients are totally symmetric in their momentum indices, with metric tensors represented by index pairs labeled “00”.
- Regularization: Ultraviolet- and infrared-singular integrals are represented in dimensional regularization with D=4−2ε and explicit pole terms.
- Regularization: Mass regularization is available for collinear singularities, with declared small masses retained in mass-singular logarithms.The corresponding mass values must remain exactly consistent between initialization and subroutine calls.
3. Implemented methods
Collier combines adaptive tensor-reduction strategies with direct tensor-level reduction and recursive tensor construction to obtain stable one-loop results across phase space.
- Calculation of tensor coefficients: For N=1,2 tensor integrals use explicit numerically stable expressions.
- Calculation of tensor coefficients: For N=3,4 Collier normally uses Passarino–Veltman reduction, switching to recursive expansions when Gram determinants become small.The implemented expansion methods support arbitrary order in the expansion parameter.
- Calculation of tensor coefficients: The reduction procedure estimates errors and selects among expansion methods according to expected precision and convergence.If no method reaches the target accuracy, the result with the smallest estimated error is returned.
- Calculation of tensor coefficients: Stable results are obtained for almost all phase-space points, supporting reliable Monte Carlo integrations.
- Calculation of tensor coefficients: For N≥5, higher-point tensor integrals are reduced recursively without inverse Gram determinants, with modified reductions for N≥7.
- Calculation of full tensors: Collier recursively constructs full tensor components from invariant coefficients, including metric-tensor contributions and their combinatorial factors.
- Calculation of full tensors: For N≥6, reduction can be performed directly at tensor level because tensor coefficients contain redundant structures and are not unique.
4. Structure of the library
Collier is organized around two independent scalar-integral implementations, COLI and DD, supplemented by tensor-construction routines and a global user interface. Users can select either branch or calculate integrals with both for cross-checking.
- The library core contains two independent implementations, COLI and DD, of scalar integrals and Lorentz-invariant tensor coefficients.
- The tensors component constructs tensor integrals from coefficients and directly reduces N-point integrals for N ≥6 at tensor level.
- Collier’s global interface exposes parameter-setting and extraction routines together with calls for tensor coefficients or tensor components.
- Users may choose the COLI or DD branch, or evaluate each integral with both branches to cross-check results.
- The external cache requires identical tensor-integral call order at every phase-space point, while internal parameters must remain unchanged during each event.
5. Usage of the library
Collier is installed as a standalone Fortran95 library built with CMake, then initialized and configured before integral evaluation. Its interface supports tensor coefficients or components, event-level caching, output control, and multiple reduction branches.
- 5.1. Installation: Collier is a standalone Fortran95 code requiring the package archive and CMake, with no additional libraries required.
- 5.1. Installation: The source tree includes separate COLI, DDlib, and tensors directories, alongside auxiliary files, modules, demos, and build infrastructure.
- 5.1. Installation: Running cmake .. followed by make generates the Collier library, either dynamically or statically, for linking with user programs.
- 5.2. General usage instructions: Initialization requires Nmax and optionally accepts the maximal rank, output-folder name, and noreset flag.
- 5.2. General usage instructions: Large Nmax or rank settings can increase allocated memory and integral computation time, while the DD branch currently supports N-point integrals only up to Nmax = 6.
- 5.2. General usage instructions: After initialization, users can change parameters and flags, calculate coefficients or tensor components, and use InitEvent to reset event-level status and caches.
5.3. Calculation of tensor integrals
Collier calculates tensor integrals through routines that provide either covariant tensor coefficients or full tensor components, supporting multiple array representations and generic N-point evaluation. Inputs include momentum invariants and complex squared masses, while optional outputs provide approximate error estimates and momentum derivatives of two-point coefficients.
- Collier provides routines for both the coefficients of the Lorentz-covariant decomposition and the components of tensor integrals.
- Tensor coefficients up to rank bP can be stored in multidimensional or one-dimensional arrays, with equal-rank coefficients ordered by their indices.For N ≥ 8, Fortran’s rank-7 array limitation requires the one-dimensional representation.
- Full tensor components up to rank bP are likewise available as four-dimensional or one-dimensional arrays, with permutation-identical components sharing one entry.The mapping is explicitly tabulated for tensor components through rank bP = 3.
- Momentum invariants and squared masses are supplied as complex-valued inputs, but non-vanishing imaginary parts of momentum invariants are not supported.Invariant squared masses of single external particles must be supplied at their exact numerical values in infrared-divergent integrals.
- Optional error arrays estimate absolute tensor-coefficient uncertainties, while Collier also supplies momentum derivatives of two-point coefficients for external-particle wave-function renormalization.The error estimates are approximate order-of-magnitude indicators rather than precise reliability specifications.
- The generic tensor routine supports, in principle, arbitrary N, while dedicated tensor-component routines compute N-point integrals for N = 1, . . . , 7.The generic routine accepts the number of loop propagators through the additional integer argument Nn.
5.4. Setting and getting parameters
Collier exposes regularization, precision, reduction-mode, and expansion-cutoff parameters that control integral results, accuracy monitoring, and computational cost. These settings are configured globally rather than supplied in each individual integral call.
- Parameter groups: Regularization parameters affect tensor-integral results in addition to explicit masses and momentum arguments, while technical settings are typically fixed across integral calls.The technical parameters govern reduction-scheme selection and expansion-method iterations.
- UV regularization: Dimensional regularization handles ultraviolet divergences through ∆UV and µUV, whose combination with the integral scale determines UV-divergent results.At fixed perturbative order, physical S-matrix elements do not depend on ∆UV and µUV.
- IR and collinear regularization: Infrared divergences are dimensionally regularized by default, while collinear divergences can alternatively be regulated with a list of mass regulators.The regulator masses are treated as infinitesimally small except inside otherwise singular logarithms.
- Calculation modes: Collier supports three calculation modes: COLI only, DD only, or both branches with the higher-precision result returned and discrepancies logged when they exceed ηcheck.In mode=3, comparisons use the threshold ηcheck; the parameter is irrelevant in modes 1 and 2.
- Precision control: The target precision ηreq influences both result precision and runtime because Collier can try alternative reduction schemes and expansions until the requested accuracy is reached.A critical precision ηcrit separately raises an accuracy warning when an estimate fails to meet it.
- Expansion cutoff: The maximal rank bP max sets the expansion cutoff, with bP max ≥7 for four-point integrals and internal maxima bP max + 2 and bP max + 4 for three- and two-point integrals.Its value affects precision and computing time and must satisfy bP max ≥ Pmax + 4 − Nmax for requested tensor ranges.
5.5. Using the cache system
Collier’s cache system avoids repeated tensor-integral calculations through local or global caching. Global caching requires a stable per-phase-space-point call sequence and can consume substantial memory.
- Cache operation: The cache detects identical integrals reached through different reduction-tree paths and avoids recalculating them.Caching is available in local mode for one subroutine call or global mode across phase-space points.
- Configuration: Cache capacity is specified by the number of caches and the maximal N-point integral rank cached, and additional caches can be added later.The cache level can be adjusted individually, but only before the first phase-space point for that cache is evaluated.
- Global mode: Global caching requires each phase-space point to initialize its cache and repeat assigned integral calls in the same order without resetting global parameters.Integrals are identified by their positions in the user-call sequence.
- Resource boundary: Global caching may require high memory because cache storage is dynamically allocated during initial phase-space points.Different caches can accommodate subprocesses whose integral-call orders differ.
- Cache control: The global cache can be switched off temporarily and later resumed, while individual caches can also be disabled and re-enabled.A resumed cache continues from its paused position unless a new event initialization resets its position.
5.6. Error treatment and output files
Collier reports errors and accuracy problems through persistent flags, output files, and configurable stopping behavior. Users can inspect, redirect, limit, suppress, and reset these diagnostics.
- Flags and diagnostics: Internal errors and precision failures set global flags for user inspection and are also recorded with problematic integral calls in output files.The error flag records the most severe encountered error until reinitialization or a more negative value replaces it.
- Error stopping: Execution stops automatically when σerr falls below σstop, whose default is σstop = −8, but this behavior can be suppressed or reconfigured.The default distinguishes phase-space-point-specific errors from systematic errors expected across points.
- Accuracy monitoring: The accuracy flag starts at σacc = 0, changes to σacc = −1 when ηreq is missed, and changes to σacc = −2 when ηcrit is missed.It is reinitialized automatically by InitEvent for a new phase-space point.
- Output files: By default, Collier writes separate error, accuracy, and informational files in an output directory created during initialization.Users can choose another folder, suppress file output, or switch file output on and off later.
- Error routing: Errors are routed to ErrOut.coli, ErrOut.dd, or ErrOut.cll according to their source, with output channels configurable and standard channel 6 available.The selected channels can be queried by dedicated subroutines.
- Output limits: Error messages are limited to nmaxerr = 100 by default, while informational output is limited to nmaxinf = 1000 messages and can be adjusted.Counters and limits are normally reset on reinitialization unless noreset=.true. is supplied.
5.7. Sample programs
Collier provides two sample programs: one for interactive single-integral calculations and one demonstrating cache usage across repeated tensor-integral computations.
- demo: The demo program lets users choose a calculation mode and an example N-point integral, then writes the result to a corresponding data file.The output identifies the source-code location of the selected integral call in demo.f90.
- democache: The democache program performs repeated series of 8 tensor-integral computations for 1000 phase-space points with and without caching.It compares the COLI and DD branches in separate cached and uncached runs.
6. Conclusions
Collier evaluates one-loop scalar and tensor integrals for unrestricted-multiplicity scattering processes and supports both traditional and unitarity-based calculations. Its stability features and independent-library merger support reliable use in automated amplitude generators.
- Collier numerically evaluates one-loop scalar and tensor integrals for scattering processes without an a-priori restriction on particle multiplicities.
- Dedicated techniques automatically optimize numerical stability in delicate phase-space regions.
- Complex internal masses support calculations involving unstable particles, while dimensional or mass regularization treats infrared divergences.
- As a merger of COLI and DD, Collier enables checks of result correctness and numerical stability and is used in OpenLoops and Recola.
- Collier supports both traditional Feynman-diagrammatic and modern unitarity-based calculations, delivering scalar and tensor one-loop integrals on demand.
Appendix A. Sets of momentum invariants PN for N = 1, . . . , 7
The appendix lists the complete momentum-invariant sets P_N used for N-point tensor integrals from N = 1 through 7. These sets provide the ordered invariant inputs associated with the propagator momenta.
- N-point tensor integrals depend on the complete set of momentum invariants P_N formed from the propagator momenta p_i.
- The appendix explicitly lists the ordered invariant sets P_N for N = 2, . . . , 7.
- For N = 2, the listed set includes 1, (p2 − p1)^2, and the relevant squared external momentum.
- For N = 4 through 7, the sets extend this pattern with further adjacent and nonadjacent momentum differences and squared momenta.