Towards a basis for planar two-loop integrals

Abstract

The existence of a finite basis of algebraically independent one-loop integrals has underpinned important developments in the computation of one-loop amplitudes in field theories and gauge theories, in particular. We give an explicit construction reducing integrals with massless propagators to a finite basis for planar integrals at two loops, both to all orders in the dimensional regulator , and also when all integrals are truncated to OðÞ. We show how to reorganize integration-by-parts equations to obtain elements of the first basis efficiently, and how to use Gram determinants to obtain additional linear relations reducing this all-orders basis to the second one. The techniques we present should apply to nonplanar integrals, to integrals with massive propagators, and beyond two loops as well.

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

“…On one hand, we do know that this is possible in highly-supersymmetric theories, where calculations are carried out to a very high order in the perturbative expansion. On the other hand, in non-supersymmetric field theories, even the first stepthe identification of a suitable basis of master integrals at two-loops -has been attempted only recently [148]. Hence, it will be interesting to watch in the future how new ideas developed in the context of N = 4 SUSY Yang-Mills will penetrate into more phenomenological research related to perturbative computations in the Standard Model.…”

Manifesting enhanced cancellations in supergravity: integrands versus integrals

“…This has the unwanted side effect of introducing spurious infrared singularities even in D = 5. With more modern approaches [36][37][38][39][40][41][42][43] we can avoid the appearance of such integrals. This is achieved by imposing…”

“…In this section we show how one can rearrange integrands into a form where all terms are manifestly finite by power counting, except those that integrate to zero. We do so using modern integration-by-parts (IBP) technology [30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In our discussion we will be using the language of integrands and integrals interchangeably.…”

“…At two loops we use unitarity cuts to argue that cancellations cannot be made manifest at the integrand level. To further investigate this case, we use integration-by-parts (IBP) technology [30][31][32][33][34][35][36][37][38][39][40][41][42][43] to reorganize the integrand into pieces that are finite by power counting and pieces that are divergent by power counting, yet integrate to zero. Although this re-arrangement of the complete integrand is successful, it requires detailed knowledge of the specific integrals and their relations, making it difficult to generalize to higher loops.…”

Simplified Differential Equations Approach for Master Integrals

“…In the recent years, a lot of progress has been made towards the extension of these reduction methods to the two-loop order at the integral [8,9]as well as the integrand [10,11,12,13] level. The master equation at the integrand level can be given schematically as follows [13] N (l 1 , l 2 ; {p i }) where an arbitrary contribution to the two-loop amplitude (left), can be reduced to a sum of terms (right) of all partitions S m;n , with up to eight denominators; l 1 , l 2 are the loop momenta, D i are the inverse scalar Feynman propagators, N (l 1 , l 2 ; {p i }) is a general numerator polynomial and…”