Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reduction method for dimensionally regulated one-loop N-point Feynman integrals

G. Duplancic, B. Nizic

TL;DR

The paper develops a general, algorithmic framework to reduce any massless one-loop N-point Feynman integral with generic 4D external momenta to a compact set of eight basic scalar integrals (six box integrals in D=6, a triangle in D=4, and a two-point integral in D). It overcomes the longstanding problem of vanishing kinematic determinants by classifying reduction strategies according to det(S_N) and det(R_N) and by solving appropriate linear systems, enabling full reduction for arbitrary kinematics, including collinear configurations relevant to PQCD. Tensor integrals are first decomposed into tensor structures built from external momenta and the metric, then scalar reductions proceed via IBP-based recursion relations, which dramatically facilitate computer implementation. An alternative basis using six-dimensional box integrals isolates IR divergences in the two-point sector, improving numerical stability and practical evaluation for high-multiplicity processes. The framework thus provides a robust, general, and implementable method for one-loop multi-leg computations in massless theories, with significant impact for NLO PQCD analyses of exclusive hadronic processes and multi-parton amplitudes.

Abstract

We present a systematic method for reducing an arbitrary one-loop N-point massless Feynman integral with generic 4-dimensional momenta to a set comprised of eight fundamental scalar integrals: six box integrals in D=6, a triangle integral in D=4, and a general two-point integral in D space time dimensions. All the divergences present in the original integral are contained in the general two-point integral and associated coefficients. The problem of vanishing of the kinematic determinants has been solved in an elegant and transparent manner. Being derived with no restrictions regarding the external momenta, the method is completely general and applicable for arbitrary kinematics. In particular, it applies to the integrals in which the set of external momenta contains subsets comprised of two or more collinear momenta, which are unavoidable when calculating one-loop contributions to the hard-scattering amplitude for exclusive hadronic processes at large momentum transfer in PQCD. The iterative structure makes it easy to implement the formalism in an algebraic computer program.

Reduction method for dimensionally regulated one-loop N-point Feynman integrals

TL;DR

The paper develops a general, algorithmic framework to reduce any massless one-loop N-point Feynman integral with generic 4D external momenta to a compact set of eight basic scalar integrals (six box integrals in D=6, a triangle in D=4, and a two-point integral in D). It overcomes the longstanding problem of vanishing kinematic determinants by classifying reduction strategies according to det(S_N) and det(R_N) and by solving appropriate linear systems, enabling full reduction for arbitrary kinematics, including collinear configurations relevant to PQCD. Tensor integrals are first decomposed into tensor structures built from external momenta and the metric, then scalar reductions proceed via IBP-based recursion relations, which dramatically facilitate computer implementation. An alternative basis using six-dimensional box integrals isolates IR divergences in the two-point sector, improving numerical stability and practical evaluation for high-multiplicity processes. The framework thus provides a robust, general, and implementable method for one-loop multi-leg computations in massless theories, with significant impact for NLO PQCD analyses of exclusive hadronic processes and multi-parton amplitudes.

Abstract

We present a systematic method for reducing an arbitrary one-loop N-point massless Feynman integral with generic 4-dimensional momenta to a set comprised of eight fundamental scalar integrals: six box integrals in D=6, a triangle integral in D=4, and a general two-point integral in D space time dimensions. All the divergences present in the original integral are contained in the general two-point integral and associated coefficients. The problem of vanishing of the kinematic determinants has been solved in an elegant and transparent manner. Being derived with no restrictions regarding the external momenta, the method is completely general and applicable for arbitrary kinematics. In particular, it applies to the integrals in which the set of external momenta contains subsets comprised of two or more collinear momenta, which are unavoidable when calculating one-loop contributions to the hard-scattering amplitude for exclusive hadronic processes at large momentum transfer in PQCD. The iterative structure makes it easy to implement the formalism in an algebraic computer program.

Paper Structure

This paper contains 13 sections, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and general properties
  3. Decomposition of tensor integrals
  4. Recursion relations for scalar integrals
  5. Case I: ${\rm det}(S_N)\neq 0$, ${\rm det}(R_N)\neq 0$
  6. Case II: ${\rm det}(S_N)\neq 0$, ${\rm det}(R_N)=0$
  7. Case III: ${\rm det}(S_N)=0$, ${\rm det}(R_N)\neq 0$
  8. Case IV: ${\rm det}(S_N)=0$, ${\rm det}(R_N)=0$
  9. On the fundamental set of integrals
  10. The general scalar integral for $N=2$
  11. The scalar integrals for $N=3$
  12. The scalar integrals for $N=4$
  13. Conclusion

Figures (2)

  • Figure 1: One-loop N-point diagram
  • Figure 2: One of the diagrams contributing to the hard scattering amplitude of the process $\gamma \, \gamma \rightarrow \pi^+ \, \pi^-$ at NLO.