A complete reduction of one-loop tensor 5- and 6-point integrals
Th. Diakonidis, J. Fleischer, J. Gluza, K. Kajda, T. Riemann, J. B. Tausk
TL;DR
The paper addresses the challenge of analytically reducing massive one-loop five- and six-point tensor integrals. It presents a unified algebraic framework based on signed minors and dimensionally shifted scalar integrals I_n^{[d+]^l} to cancel Gram determinant factors, enabling complete reductions to scalar 2–4-point functions. The authors derive explicit formulas for pentagon (n=5) tensors up to rank 3 and hexagon (n=6) tensors up to rank 4, expressing higher-rank tensors in terms of lower-point functions, and implement the method in Fortran and Mathematica (hexagon.m). Numerical tests across massive, massless, and mixed cases demonstrate stability and consistency with existing tools, providing a practical, scalable tool for NLO calculations with high external multiplicities.
Abstract
We perform a complete analytical reduction of general one-loop Feynman integrals with five and six external legs for tensors up to rank R=3 and 4, respectively. An elegant formalism with extensive use of signed minors is developed for the cancellation of inverse Gram determinants. The 6-point tensor functions of rank R are expressed in terms of 5-point tensor functions of rank R-1, and the latter are reduced to scalar four-, three-, and two-point functions. The resulting compact formulae allow both for a study of analytical properties and for efficient numerical programming. They are implemented in Fortran and Mathematica.