On hyperlogarithms and Feynman integrals with divergences and many scales

Abstract

Hyperlogarithms provide a tool to carry out Feynman integrals in Schwinger parameters. So far, this method has been applied successfully mostly to finite single-scale processes. However, it can be employed in more general situations.We give examples of integrations of three-and four-point integrals in Schwinger parameters with non-trivial kinematic dependence, including setups with off-shell external momenta and differently massive internal propagators. The full set of Feynman graphs admissible to parametric integration is not yet understood and we discuss some counterexamples to the crucial property of linear reducibility. In special cases we observe how a change of variables can restore this prerequisite for direct integration and thereby enlarge the set of accessible graphs.Working in dimensional regularization, we furthermore clarify how a simple application of partial integration can be used to convert divergent parametric integrands to convergent ones. In contrast to the subtraction of counterterms, this scheme is ideally suited for our method of integration.

Feynman integrals and iterated integrals on moduli spaces of curves of genus zero

“…It has been fully implemented by Panzer [38,39,40] and various parts of the algorithm have found applications in different contexts, as described below. A conceptual disadvantage of this method is that the underlying geometry of every Feynman diagram is different.…”

“…The case of massless, single-scale, primitively overall-divergent Feynman diagrams in a scalar field theory was detailed in [17]. Since then, the method was applied to the computation of integrals of hexagonal Feynman graphs, arising in N = 4 supersymmetric Yang-Mills theory [26,27,28], integrals with operator insertions contributing to massive matrix elements of quantum chromodynamics (QCD) [33,34,35], one-and two-loop triangular Feynman graphs with off-shell legs [29], phase-space contributions [22,23] to the cross-section for threshold production of the Higgs boson from gluon-fusion at N3LO QCD [24], coefficients in the expansion of certain hypergeometric functions, contributing to superstring amplitudes [36,37], massless multi-loop propagator-type integrals [38], and a variety of three-and four-point Feynman integrals depending on several kinematical scales [39]. These applications arise from very different contexts and the method is combined with various other computational techniques.…”

“…• Finally, it can also be combined with dimensional regularisation to treat UV and IR divergences by the method of [39].…”

“…In principle such an expansion can be computed by sector decomposition [21], however in this case, a use of our algorithms may be prohibited by the type of polynomials appearing in the integrands of the resulting c j . Recently, an alternative approach, where the latter polynomials are given by Symanzik polynomials of G and its minors was suggested in [39].…”

Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

“…If it turns out that there is an order in which the dependence of the denominator of the integrand on the Feynman parameters is linear then the whole integral can be solved in terms of multiple polylogarithms. This strategy was successfully applied for example in [20][21][22][23][24] and implemented as the computer code HyperInt in [25].…”