Towards a Basis for Planar Two-Loop Integrals

By J. Gluza, K. Kajda, David A. Kosower. 2010

242 citations recorded by the source corpus.

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 {epsilon}, and also when all integrals are truncated to O({epsilon}). 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.

Subfields: Nuclear and High Energy Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6f550207d583efa45142a519935e021b1ba3c7f0

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Citations in the corpus

One-loop calculations in quantum field theory: From Feynman diagrams to unitarity cuts (2011, background): “On the other hand, in non-supersymmetric field theories, even the first step – the identification of a suitable basis of master integrals at two-loops – has been attempted only recently [148].”
Scattering Amplitudes from Multivariate Polynomial Division (2012, background): “Badger, Frellesvig and Zhang [17] combined onshell conditions with Gram-identities [18] to limit the number of monomials appearing in the residues.”
Loop amplitudes in gauge theories: modern analytic approaches (2010, background): “Recently a framework has been outline for reducing two-loop integrals to a finite basis avoiding doubled propagators, whose singularities are more difficult to control [129].”
Cuts of Feynman Integrals in Baikov representation (2017, background): “In the recent years, progress has been made also towards the extension of these reduction methods for two-loop amplitudes at the integral [17, 18, 19, 20, 21, 22, 23, 24, 25, 26] as well as the integrand [27, 28, 29, 30, 31, 32] level.”
Integral reduction with Kira 2.0 and finite field methods (2020, methodology): “In the last few years, new techniques have been explored and applied to integral reduction problems, employing syzygy equations [7–12], algebraic geometry [13–15], intersection numbers [16–20], ﬁnite ﬁeld and interpolation techniques [21–26], or special integral representations [27–30].”
Reduze 2 - Distributed Feynman Integral Reduction (2012, methodology): “Alternative methods to exploit IBP and LI identities for reductions have been proposed [6–9], see also [10, 11] and references therein.”
Magnus and Dyson series for Master Integrals (2014, methodology): “Within the continuous dimensional regularization scheme, Feynman integrals can be related by using integration-by-parts identities (IBP-id’s) [6, 7], Lorentz invariance identities [3], Gram identities [8], and quasi-Shouten identities [9].”
Uniqueness of two-loop master contours (2012, unknown): “Using the integral identities which have been explicitly constructed at five points [54], we believe it will be possible to explicitly construct the (unique) master contours in this case using our results.”
Two-loop integrand decomposition into master integrals and surface terms (2015, methodology): “[49] which initially do not include integral topologies with doubled propagators.”
A two-loop five-gluon helicity amplitude in QCD (2013, background): “The IBP identities have been understood in a unitarity compatible form which has shed light on the integral basis [42].”
Feynman integrals and intersection theory (2018, background): “[56] J.”