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OneLOop: for the evaluation of one-loop scalar functions
A. van Hameren
TL;DR
Collider-physics NLO calculations need reliable one-loop scalar functions across complex-mass and divergent kinematic configurations. OneLOOP extends a public library with complex-mass support and numerical cross-check routines, covering 1- to 4-point functions within dimensional regularization. The program handles the stated collider-physics configurations and UV/IR divergences, while exceptional phase-space singularities remain outside complete treatment.
Problem
Collider-physics NLO calculations require one-loop scalar functions for complex internal masses, infrared divergences, and broad kinematical configurations.
Method
OneLOOP extends the library to complex masses and supplies straightforward numerical-integration routines for cross-checking analytic continuations.
Results
ONELOOP evaluates scalar 1-point through 4-point functions for relevant collider-physics configurations and handles UV and IR divergences within dimensional regularization.
Takeaways & Limitations
The package provides analytic routines plus independent numerical-integration routines for validating one-loop scalar-function evaluations.
Takeaways & Limitations
Complete treatment of isolated exceptional phase-space singularities is beyond the work’s scope, and four-point numerical routines can be slow and limited to the first few decimals.
Abstract
from arXiv · showhide
OneLOop is a program to evaluate the one-loop scalar 1-point, 2-point, 3-point and 4-point functions, for all kinematical configurations relevant for collider-physics, and for any non-positive imaginary parts of the internal squared masses. It deals with all UV and IR divergences within dimensional regularization. Furthermore, it provides routines to evaluate these functions using straightforward numerical integration.
1 Introduction
OneLOop addresses the need for universal one-loop scalar functions in collider-physics NLO calculations, including complex internal masses and numerical cross-checks.
- Motivation: NLO collider calculations require one-loop amplitudes, including at least 6-point functions for processes with four or more final-state particles.These calculations provide virtual contributions to full differential predictions.
- Universal functions: One-loop amplitudes can be expressed using universal functions whose process dependence enters through kinematical input parameters and non-universal coefficients.Tensor approaches reduce tensor functions to scalar functions, while unitarity approaches use scalar functions directly.
- Required functions: NLO calculations require scalar 1-point, 2-point, 3-point, and 4-point functions, including infrared-divergent cases handled with dimensional regularization.Higher-point scalar functions can be expressed as linear combinations of these functions.
- Complex masses: Complex internal masses are needed for the complex-mass scheme used in consistent calculations involving unstable particles.Existing implementations included the finite 4-point function with four complex masses.
- Contribution: The write-up extends ONELOOP to arbitrary complex masses, adds numerical-integration routines, and compares them with existing implementations and direct integration.The numerical routines avoid evaluating dilogarithms and support checks of delicate analytic continuations.
2 Program description
The program description specifies ONELOOP’s implementation, conventions, interfaces, and Laurent-expanded outputs for scalar 1-point and 2-point functions.
- Implementation: ONELOOP is implemented as FORTRAN 77 source that can be compiled directly or built as a library using the supplied makefile.Subroutine, function, and common-block names begin with avh_olo_.
- 1-point function: The 1-point function uses the renormalization scale µ and dimensional-regularization parameter ε, with a configurable default scale.The scale is set through avh_olo_mu_set and is specified in energy units.
- 1-point function: The 1-point output is a complex array containing ε0 and ε−1 Laurent coefficients, while the ε−2 coefficient is always zero.For zero squared mass, both returned coefficients are zero.
- 2-point functions: ONELOOP also provides Passarino–Veltman B1, B00, and B11 functions with outputs represented as Laurent-expansion coefficient arrays.Their outputs are returned through the avh_olo_b11c interface.
2.3 The 3-point function
The 3-point and 4-point sections define their scalar functions, interfaces, kinematic inputs, complex-mass conventions, and Laurent-expanded outputs.
- 3-point function: The scalar 3-point function is evaluated through avh_olo_c0c using three complex kinematic invariants and three complex squared masses.The kinematic invariants are related to the external momenta through the stated momentum-squared definitions.
- 3-point function: Three-point kinematic inputs must have zero imaginary parts, while squared masses may be complex but must not have positive imaginary parts.Invalid signs or imaginary parts trigger an error message and are replaced by the permitted convention.
- 3-point function: The 3-point output array contains ε0, ε−1, and ε−2 coefficients of the Laurent expansion.By default, pole coefficients are zero unless the input exactly represents an infrared-divergent configuration.
- 4-point function: The scalar 4-point function is evaluated through avh_olo_d0c using six complex kinematic inputs and four complex squared masses.The six kinematic inputs encode four external momentum squares and two channel invariants.
- 4-point function: The 4-point output contains ε0, ε−1, and ε−2 Laurent coefficients, with pole coefficients defaulting to zero outside exactly specified infrared-divergent configurations.Its kinematic and mass inputs follow the same imaginary-part conventions described for the 3-point function.
2.5 IR-divergent one-loop functions
ONELOOP handles infrared-divergent 3-point and 4-point functions in dimensional regularization and provides configurable numerical-integration routines for validation.
- IR divergences: ONELOOP treats all listed infrared-divergent 3-point and 4-point cases within dimensional regularization.Infrared-finite cases have identically zero ε−1 and ε−2 coefficients.
- IR divergences: By default, divergent cases are returned only when inputs numerically match the exact divergent configuration, with an optional threshold setting for near-equalities.Warnings are issued when inputs are close to a divergent case under the default behavior.
- IR divergences: The infrared regularization scale is the same as the renormalization scale µ.Message output can be redirected or disabled through configurable unit settings.
- Numerical integration: Numerical-integration routines are included primarily to check analytic continuations and are implemented for straightforwardness rather than sophistication.They use the CUBA library and avoid evaluating dilogarithms for functions involving at least two scales.
- Limitations: Numerical integration does not fully treat isolated exceptional phase-space singularities, which are beyond the work’s scope.Four-point numerical routines can be slow and generally provide only the first few decimals correctly.
- Numerical integration: The numerical routines form an independent library that largely mirrors the main ONELOOP library and adds controls for evaluation limits and message output.The independent library uses avh_oni_ names and requires separate compilation with CUBA access.
3 One-loop scalar functions via numerical integration
The section derives straightforward numerical integral representations for one-loop scalar functions, using contour deformation to handle deformable singularities and dimensional regularization for infrared divergences.
- Integral representations: The derivation targets a straightforward numerical implementation as an independent check of analytic continuations in the hardwired scalar-function routines.The approach deliberately avoids integrand-smoothing methods intended primarily to accelerate numerical integration.
- Integral representations: The n-point scalar integral is formulated in dimensional regularization with Feynman parameters, whose delta function confines the integration to the n-simplex.The variables are rescaled by one parameter to obtain a representation over a non-finite integration region.
- Contour deformation: The integration region is mapped from [0,1] to [0,∞) before deformation, allowing numerical integrands on the unit hypercube and avoiding singularities at infinity.The deformation uses a bounded, strictly increasing function τ with τ(0) = 0 and finite τ(∞).
- Contour deformation: Contour deformation is chosen to keep the denominator’s imaginary part negative where its real part vanishes, while fixing endpoints at x_j = 0.Infrared singularities can remain at these endpoints because they cannot be removed by contour deformation.
- IR-divergent integrals: For IR-finite integrals, the integrand may be expanded in ε and integrated term by term, whereas IR-divergent bubbles vanish within dimensional regularization.The implementation considers scalar integrals up to n = 4, including boxes and triangles.
- IR-divergent integrals: IR-divergent box integrals are decomposed into triangle integrals plus an IR-finite four-point remainder that can be integrated numerically.The triangle coefficients are determined from the established reduction, while separate numerical representations are derived for IR-divergent triangles.
4 Summary
The paper presents ONELOOP as a package for one-loop scalar functions through four points, including complex internal masses and UV and IR divergences in dimensional regularization. It also supplies straightforward numerical-integration routines.
- ONELOOP evaluates one-loop scalar functions through the 4-point function for collider-physics kinematics and non-positive imaginary parts of internal squared masses.
- The package handles UV and IR divergences within dimensional regularization and provides numerical-integration routines for the same scalar functions.The numerical methods are presented as part of the write-up.
A Coefficients for IR-divergent box integrals
This appendix specifies coefficients and kinematic conventions for IR-divergent box integrals, organized according to the singularity patterns of the numbered box topologies.
- Coefficients and conventions: The appendix gives coefficients c_j for the box reduction and α, β_j for the associated finite remainder, with γ vanishing in all cases.The coefficients use the external-momentum conventions defined earlier.
- Coefficients and conventions: The kinematic notation includes s_12 = (p_1 + p_2)^2 and s_23 = (p_2 + p_3)^2, alongside the matrices S and Q used in Eq.(36).
- Topology classification: The box numbering follows [15], while some separately labeled cases are specializations of general boxes obtained by setting one or more propagator masses to zero.
- Topology classification: Boxes (16) and (15) have one soft and no collinear singularity, while box (14) has two soft and no collinear singularities.
- Topology classification: Boxes (13), (12), (11), (10), (9), and (5) contain one collinear singularity with zero, one, or two soft singularities.
- Topology classification: Boxes (8), (7), (6), and (4) contain two collinear and at least one soft singularity, whereas boxes (3), (2), and (1) have more than two collinear or two collinear without soft singularities.
B IR-divergent triangle integrals
The appendix derives numerically suitable integral representations for IR-divergent triangles involving at least two mass scales and depicts the relevant topologies using the labeling of [15].
- Triangle representations: The derivation starts from the general triangle representation and minimizes algebraic manipulation to obtain forms suitable for numerical integration.Triangles with only one mass scale are excluded because their analytic formulas are comparatively simple.
- Triangle configurations: The considered configurations are IR-divergent scalar triangles with at least two mass scales, including the topologies depicted in Figure 1.
- Triangle configurations: Triangle (2) in Figure 1 is a special case of triangle (3) obtained by setting m_2 = 0.
Triangle (6)
Triangle (6) is expressed through a modified Cayley matrix and an integral representation. After direct x-integration, the remaining convergent integral can be expanded in ε and integrated numerically term by term.
- Triangle (6) uses a modified Cayley matrix in its integral formulation.
- The x-integral can be performed directly, reducing the representation to a remaining one-dimensional integral.
- The remaining integral is convergent for any value of ε.
- Its integrand can be Taylor-expanded in ε and integrated numerically term by term.
Triangle (3)
Triangle (3) is obtained by eliminating row and column 1 in the modified Cayley matrix. Its small-x divergence is isolated through a factorization and subtraction, yielding finite integrals for extracting the ε−1 and ε0 terms.
- Triangle (3) uses a modified Cayley matrix with row and column 1 eliminated.
- Unlike triangle (6), its integral diverges because of small-x behavior.
- Factoring x1+ε from the denominator enables a subtraction-based decomposition.
- Both resulting integrals are finite, allowing extraction of the ε−1 and ε0 terms by expansion and evaluation at ε=0.
- The first two-dimensional integral vanishes when m2 = 0, corresponding to triangle (2) in [15].
Triangle (4)
Triangle (4) is reduced through an integral representation and direct x-integration. Subtraction handles the resulting expression, leaving a finite integral that can be expanded in ε at the integrand level.
- Triangle (4) is formulated using an integral representation.
- Performing the x-integral produces a reduced expression for the triangle.
- The resulting expression can be handled using subtraction.
- The remaining integral is finite and can be expanded in ε directly at the integrand level.