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D0C : A code to calculate scalar one-loop four-point integrals with complex masses

Dao Thi Nhung, Le Duc Ninh

TL;DR

The paper introduces D0C, a Fortran 77 code that computes the scalar one-loop four-point integral with complex internal masses, extending the traditional 't Hooft–Veltman approach to handle unstable particles in collider processes. It develops a robust reduction of D0 to a two-variable T-function framework, accommodating both real and complex masses while addressing Landau singularities and numerical stability. The method covers all physically relevant external-momentum configurations (including at least one lightlike or timelike momentum) and provides explicit formulas for general and special cases, with careful treatment to avoid cancellations. The implementation, integrated into LoopTools-2.4, offers reliable, cross-checkable results and includes guidance on numerical behavior near singularities, making it practical for precision radiative-correction calculations in high-energy physics.

Abstract

We present a new Fortran code to calculate the scalar one-loop four-point integral with complex internal masses, based on the method of 't Hooft and Veltman. The code is applicable when the external momenta fulfill a certain physical condition. In particular it holds if one of the external momenta or a sum of them is timelike or lightlike and therefore covers all physical processes at colliders. All the special cases related to massless external particles are treated separately. Some technical issues related to numerical evaluation and Landau singularities are discussed.

D0C : A code to calculate scalar one-loop four-point integrals with complex masses

TL;DR

The paper introduces D0C, a Fortran 77 code that computes the scalar one-loop four-point integral with complex internal masses, extending the traditional 't Hooft–Veltman approach to handle unstable particles in collider processes. It develops a robust reduction of D0 to a two-variable T-function framework, accommodating both real and complex masses while addressing Landau singularities and numerical stability. The method covers all physically relevant external-momentum configurations (including at least one lightlike or timelike momentum) and provides explicit formulas for general and special cases, with careful treatment to avoid cancellations. The implementation, integrated into LoopTools-2.4, offers reliable, cross-checkable results and includes guidance on numerical behavior near singularities, making it practical for precision radiative-correction calculations in high-energy physics.

Abstract

We present a new Fortran code to calculate the scalar one-loop four-point integral with complex internal masses, based on the method of 't Hooft and Veltman. The code is applicable when the external momenta fulfill a certain physical condition. In particular it holds if one of the external momenta or a sum of them is timelike or lightlike and therefore covers all physical processes at colliders. All the special cases related to massless external particles are treated separately. Some technical issues related to numerical evaluation and Landau singularities are discussed.

Paper Structure

This paper contains 11 sections, 52 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The method
  3. At least one lightlike momentum
  4. General case
  5. The $\mathcal{T}$ function
  6. General case: $A\neq 0$ and $B\neq 0$
  7. Special cases: $A=0$ or $B=0$
  8. The code D0C
  9. Conclusions
  10. Four-point leading Landau singularity
  11. Basic integral and three-point Landau singularity

Figures (2)

  • Figure 1: The box diagram.
  • Figure 2: The beta triangle $[0,1-\beta,-\beta]$ in the complex plane.