Reduction schemes for one-loop tensor integrals

Abstract

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.

One-loop calculations in quantum field theory: From Feynman diagrams to unitarity cuts

“…Before closing, we mention important research topics that we have not discussed in this review. First, we only briefly mentioned the recent developments in Passarino-Veltman reduction technology [138,63,119,61] that lead to efficient and numerically-stable ways of computing multi-leg one-loop Feynman diagrams [2,12]. While these methods are traditional, the results that have been achieved by applying them to realistic problems are spectacular and the potential of traditional computational methods does not seem to be anywhere near to being exhausted.…”

“…Those techniques minimize difficulties related to inverse Gram determinants and employ a cache system to recycle tensor integrals with common sub-topologies and avoid the computation of relevant scalar integrals more than once. For tensor N -point integrals with N ≥ 5, a procedure can be used that reduces the number of propagators and the rank of the corresponding integral at the same time [63]. Such a procedure does not introduce inverse Gram determinants.…”

“…Such a procedure does not introduce inverse Gram determinants. In the case of three-and four-point functions, where the reduction does introduce small Gram determinants, an expansion procedure around the limit of vanishing Gram determinants and other relevant kinematical structures is applied [63,64].…”

Techniques for one-loop tensor integrals in many-particle processes

“…In Refs. [2,5] we follow basically the same strategy to reduce tensor integrals, which has the advantage that no inverse Gram determinants appear in the reduction. Instead modified Cayley determinants occur in the denominator, but we did not find numerical problems with these factors.…”

“…All relevant formulas can be found in Refs. [2,5], here we only sketch their structure. Moreover, we illustrate the improvement of the new methods over the conventional PV reduction for some 4-point integrals in delicate kinematical limits.…”