One-loop tensor integrals in dimensional regularisation

Abstract

We show how to evaluate tensor one-loop integrals in momentum space avoiding the usual plague of Gram determinants. We do this by constructing combinations of nand (n − 1)-point scalar integrals that are finite in the limit of vanishing Gram determinant. These non-trivial combinations of dilogarithms, logarithms and constants are systematically obtained by either differentiating with respect to the external parameters -essentially yielding scalar integrals with Feynman parameters in the numerator -or by developing the scalar integral in D = 6 − 2ǫ or higher dimensions. An additional advantage is that other spurious kinematic singularities are also controlled. As an explicit example, we develop the tensor integrals and associated scalar integral combinations for processes where the internal particles are massless and where up to five (four massless and one massive) external particles are involved. For more general processes, we present the equations needed for deriving the relevant combinations of scalar integrals.

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

“…In section 3 we derive our tensor reduction formalism which combines Passarino-Veltman-like methods with Feynman parameter space techniques. We prove that for generic 4-dimensional kinematics all higher dimensional N -point functions drop out for arbitrary N ≥ 5 and generalize the reduction methods of [3,17] to arbitrary N . We construct a hierarchy of tensor formulas up to N = 6 and rank ≤ N which can be solved by iteration.…”

“…Although it is well known [3,12,17] that the N -point integral can be split into a finite, (6 − 2 )-dimensional integral and a part with less external legs containing the infrared poles, we want to present our derivation which later allows to deal with the problem of vanishing Gram determinants in a transparent manner. As an ansatz we write (1) as a sum of (one-propagator) reduced diagrams and a remainder.…”

“…(18). This fact can be used as a guideline to define nontrivial groupings of (N − 1)-and N -point functions which is helpful for numerical purposes [17]. The other dangerous determinant is det(S) which vanishes if N ≥ 7.…”

“…(34), differences of reduced integrals appear as defined in Eq. (17). In principle one has to add to (34) also the solutions stemming from the homogeneous equation which are present if G is not of maximal rank.…”

“…In [17] recursion relations for integrals with up to 4 Feynman parameters are given. The derivation is based on the approach of [3] which uses a projective transformation [20].…”