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We derive useful reduction formulae which express one-loop Feynman integralswith a large number of external momenta in terms of lower-point integralscarrying easily derivable kinematic coefficients which are symmetric in theexternal momenta. These formulae apply for integrals with at least two moreexternal legs than the dimension of the external momenta, and are presented interms of two possible bases: one composed of a subset of descendant integralswith one fewer external legs, the other composed of the complete set ofminimally-descendant integrals with just one more leg than the dimension ofexternal momenta. In 3+1 dimensions, particularly compact representations ofkinematic invariants can be computed, which easily lend themselves tospinor-helicity or trace representations. The reduction formulae have a closerelationship with D-dimensional unitarity cuts, and thus provide a path towardscomputing full (all-epsilon) expressions for scattering at arbitrarymultiplicity.

Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts