Reduction formalism for dimensionally regulated one-loop N-point integrals

TL;DR

This work develops a dimensionally regulated approach to reduce massless one-loop N-point integrals. It introduces a scalar reduction formula valid for arbitrary N by decomposing into finite and divergent parts and using Gram-matrix pseudo-inverses to handle singular configurations, ensuring four-dimensional external kinematics throughout. The authors prove that for N≥5, higher-dimensional contributions cancel in tensor reductions, enabling an iterative scheme that expresses any rank-L N-point tensor in terms of scalar lower-point integrals, with explicit results for N up to 6. They also provide a parallel reduction in Feynman parameter space and demonstrate how the method yields a practical algorithm for calculating IR-divergent one-loop integrals with many external legs, with scope for future extension to massive particles.

Abstract

We consider one-loop scalar and tensor integrals with an arbitrary number of external legs relevant for multi-parton processes in massless theories. We present a procedure to reduce N-point scalar functions with generic 4-dimensional external momenta to box integrals in (4-2ε) dimensions. We derive a formula valid for arbitrary N and give an explicit expression for N=6. Further a tensor reduction method for N-point tensor integrals is presented. We prove that generically higher dimensional integrals contribute only to order εfor N>=5. The tensor reduction can be solved iteratively such that any tensor integral is expressible in terms of scalar integrals. Explicit formulas are given up to N=6.

Figures (1)

• Figure 1: $N$-point graph.