Reduction schemes for one-loop tensor integrals

TL;DR

This work delivers a comprehensive, numerically stable framework for evaluating one-loop tensor integrals up to six external legs. It introduces multiple reduction schemes that overcome Gram-d determinant instabilities: (i) expansions around small Gram and Cayley determinants, (ii) a numerical evaluation of a logarithmic tensor coefficient with modified Cayley determinants, and (iii) explicit rank-reducing relations reducing 5- and 6-point tensors to lower-point functions. The authors provide detailed formulas for 3-, 4-, 5-, and 6-point cases, discuss UV/IR regularization in dimensional space-time, and demonstrate practical viability by applying the methods to complete electroweak corrections in multi-fermion final states. The resulting toolkit is designed to be robust across complex masses and general kinematics, with direct applicability to high-precision collider phenomenology at the LHC and ILC. Overall, the paper delivers implementable, stable, and scalable tensor-integral reductions that extend PV-type methods to multi-leg one-loop processes.

Abstract

We present new methods for the evaluation of one-loop tensor integrals which have been used in the calculation of the complete electroweak one-loop corrections to e+ e- -> 4 fermions. The described methods for 3-point and 4-point integrals are, in particular, applicable in the case where the conventional Passarino-Veltman reduction breaks down owing to the appearance of Gram determinants in the denominator. One method consists of different variants for expanding tensor coefficients about limits of vanishing Gram determinants or other kinematical determinants, thereby reducing all tensor coefficients to the usual scalar integrals. In a second method a specific tensor coefficient with a logarithmic integrand is evaluated numerically, and the remaining coefficients as well as the standard scalar integral are algebraically derived from this coefficient. For 5-point tensor integrals, we give explicit formulas that reduce the corresponding tensor coefficients to coefficients of 4-point integrals with tensor rank reduced by one. Similar formulas are provided for 6-point functions, and the generalization to functions with more internal propagators is straightforward. All the presented methods are also applicable if infrared (soft or collinear) divergences are treated in dimensional regularization or if mass parameters (for unstable particles) become complex.

Figures (5)

• Figure 1: Schematic illustration of conventional Passarino--Veltman reduction.
• Figure 2: Schematic illustration of alternative Passarino--Veltman reduction.
• Figure 3: Schematic illustration of the reduction with modified Cayley determinants.
• Figure 4: Schematic illustration of the iteration for small Gram determinants, where thin arrows indicate that the relation involves a suppression factor $\Delta^{(3)}$. In each step the boxed coefficients are calculated in the order indicated by the labels "$a$", "$b$", etc. The $n$th iteration consists of the following $(n+1)$ steps: $n{\to}(n-1){\to}\dots{\to}1{\to}0$.
• Figure 5: Schematic illustration of the iteration for small Gram and modified Cayley determinants, where thin arrows indicate that the relation involves a suppression factor $\Delta^{(3)}$, $\tilde{X}^{(3)}_{k0}$, or $\tilde{X}^{(3)}_{0j}$. In each step the boxed coefficients are calculated in the order indicated by the labels "$a$", "$b$", etc. The $n$th iteration consists of the following $(n+1)$ steps: $n{\to}(n-1){\to}\dots{\to}1{\to}0$.