[Paper Review] Scalar one-loop Feynman integrals with complex internal masses revisited

Khiem Hong Phan, Thinh Nguyen Hoang Pham|arXiv (Cornell University)|Oct 31, 2017

Particle physics theoretical and experimental studies26 references3 citations

TL;DR

This paper presents a systematic analytic calculation of scalar one-loop Feynman integrals—two-, three-, and four-point functions—with complex internal masses, valid for both real and complex masses. The method is implemented in a Mathematica package and FORTRAN program, achieving perfect numerical agreement with LoopTools across real and complex mass cases, and enables evaluation of integrals with leading Landau singularities, offering a new approach to the inverse Gram determinant problem.

ABSTRACT

In this paper, we study systematically scalar one-loop two-, three-, and four-point Feynman integrals with complex internal masses. Our analytic results presented in this report are valid for both real and complex internal masses. The calculations are then implemented into a { t Mathematica} (version $9$) package and { t FORTRAN} program. Our program is cross-checked numerically with { t LoopTools} (version $2.14$) in real as well as complex internal masses. We find a perfect agreement between our results and { t LoopTools} for all cases. Additionally, this work is applied for evaluating scalar one-loop Feynman integrals developed leading Landau singularities which may appear in real scattering processes at colliders. Last but not least, the method used in this report can also extend to evaluate tensor one-loop integrals. Therefore, this may open a new approach which can solve the inverse Gram determinant problem analytically.

Motivation & Objective

• To systematically compute scalar one-loop two-, three-, and four-point Feynman integrals with complex internal masses.
• To address the numerical instability caused by small inverse Gram determinants in multi-leg scattering processes.
• To provide a robust, analytically stable method for evaluating one-loop integrals in the Complex-Mass Scheme for unstable particles.
• To extend the method to tensor one-loop integrals and solve the inverse Gram determinant problem analytically.
• To develop and validate a numerical implementation in Mathematica and FORTRAN for high-precision collider physics applications.

Proposed method

• The calculation is based on the method of Refs. [12, 13, 14], using Feynman parameter integrals and reduction to hypergeometric functions.
• The approach employs the generalized hypergeometric functions and Appell F1 functions to express the scalar integrals in closed form.
• The method handles both real and complex internal masses by maintaining analyticity through complex conjugate structures in the propagators.
• The integrals are reduced to master integrals using the Cayley and Gram determinant formalism, with $ S_N $ and $ G_N $ defining the kinematic invariants.
• The reduction formula for $ N \geq 5 $ points expresses pentagon and hexagon integrals in terms of box integrals via $ J_N = -\sum_k (\partial_k S_N / S_N) \mathbf{k}^- J_N $.
• The analytic results are implemented in a Mathematica (v9) package and a FORTRAN program for numerical evaluation.

Experimental results

Research questions

• RQ1 Can scalar one-loop Feynman integrals with complex internal masses be computed analytically for arbitrary kinematic configurations?
• RQ2 Does the method remain numerically stable in the presence of small inverse Gram determinants?
• RQ3 Can the method be extended to tensor one-loop integrals and solve the inverse Gram determinant problem?
• RQ4 How does the numerical implementation compare with established tools like LoopTools for both real and complex masses?
• RQ5 Can the method evaluate integrals that develop leading Landau singularities in physical scattering processes?

Key findings

• The analytic results are valid for both real and complex internal masses, providing a unified framework for one-loop calculations.
• The numerical implementation in Mathematica and FORTRAN shows perfect agreement with LoopTools (v2.14) for all tested cases, including complex masses.
• The method successfully evaluates scalar one-loop integrals that develop leading Landau singularities in physical scattering processes at colliders.
• The approach can be extended to tensor one-loop integrals, offering a new path to analytically solve the inverse Gram determinant problem.
• Pentagon and hexagon integrals are reduced to box integrals at $ D = 4 $, confirming consistency with known reduction identities.
• The method maintains numerical stability even in kinematic regions with small Gram determinants, resolving a long-standing numerical challenge.