Related papers: Star Integrals, Convolutions and Simplices
We uncover an unexpected connection between the physics of loop integrals and the mathematics of spline functions. One loop integrands are Laplace transforms of splines. This clarifies the geometry of the associated loop integrals, since a…
We derive an analytic expression for the scalar one-loop pentagon and hexagon functions which is convenient for subsequent numerical integration. These functions are of relevance in the computation of next-to-leading order radiative…
We propose a systematic approach to calculating $n$-point one-loop parametric conformal integrals in $D$ dimensions which we call the reconstruction procedure. It relies on decomposing a conformal integral over basis functions which are…
Motivated by recent work on the utility of Mellin space for representing conformal correlators in $AdS$/CFT, we study its suitability for representing dual conformal integrals of the type which appear in perturbative scattering amplitudes…
We compute the one-loop scalar massless pentagon integral I_5^{6-2 eps} in D=6-2\eps dimensions in the limit of multi-Regge kinematics. This integral first contributes to the parity-odd part of the one-loop N=4 five-point MHV amplitude…
We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling…
We explicitly construct a class of multivariate generalized hypergeometric series which is conjectured in our previous paper [Alkalaev & Mandrygin 2025] to calculate multipoint one-loop parametric conformal integrals in $D$ dimensions. Our…
We calculate the three-loop master integrals contributing to the three-loop five-point amplitude on the special Coulomb branch of $\mathcal{N}=4$ SYM theory. For the genuine pentagon integrals, we follow the approach of Ref. [JHEP 12 (2025)…
The state-of-the-art in current two-loop QCD amplitude calculations is at five-particle scattering. Computing two-loop six-particle processes requires knowledge of the corresponding one-loop amplitudes to higher orders in the dimensional…
We introduce and study the Wilson-loop ${\rm d}\log$ representation of certain Feynman integrals for scattering amplitudes in ${\cal N}=4$ SYM and beyond, which makes their evaluation completely straightforward. Such a representation was…
In this paper we consider multi-loop integrals appearing in MHV scattering amplitudes of planar N=4 SYM. Through particular differential operators which reduce the loop order by one, we present explicit equations for the two-loop…
Based on the method developed in [K.~H.~Phan and T.~Riemann, Phys.\ Lett.\ B {\bf 791} (2019) 257], detailed analytic results for scalar one-loop two-, three-, four-point integrals in general $d$-dimension are presented in this paper. The…
We apply the differential equation technique to the calculation of the one-loop massless diagram with five onshell legs. Using the reduction to $\epsilon$-form, we manage to obtain a simple one-fold integral representation exact in…
Two-loop corrections to scattering amplitudes are crucial theoretical input for collider physics. Recent years have seen tremendous advances in computing Feynman integrals, scattering amplitudes, and cross sections for five-particle…
We present a method to obtain analytic results in terms of multiple polylogarithms for one-loop triangle, box and pentagon integrals depending on an arbitrary number of scales and to any desired order in the Laurent expansion in the…
We describe in some detail the present features of an automatic loop calculation program as well as the integration techniques that go into it. The program, called XLOOPS 1.0, allows one to calculate massive one- and two-loop Feynman…
Recently in arXiv:2012.05599 Rudenko presented a formula for the volume of hyperbolic orthoschemes in terms of alternating polylogarithms. We use this result to provide an explicit analytic result for the one-loop scalar n-gon Feynman…
We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master…
We explore the Mellin space representation for the collinear limit of $N$-point energy correlators in ${\cal N}=4$ super-Yang-Mills theory. We show that these correlators can be written as integro-differential operators acting on star…
Multi-loop scattering amplitudes/null polygonal Wilson loops in ${\mathcal N}=4$ super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an $1+1$ dimensional subspace of Minkowski…