Adaptive integrand decomposition in parallel and orthogonal space

We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, d = d∥ + d⊥, being d∥ the dimension of the parallel space spanned by the legs of the diagrams. When the number n of external legs is n ≤ 4,thecorrespondingrepresentationofmultiloopintegralsexposesasubsetofintegration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external ones. We revisit the one-loop decomposition, which turns out to be controlled by t...