One-loop pentagon integral in $d$ dimensions from differential equations in $ε$-form

TL;DR

The paper tackles the massless one-loop pentagon integral with five on-shell legs in arbitrary space-time dimension and provides a representation exact in $d$ via differential equations in $\epsilon$-form. The authors derive an $\epsilon$-form basis and obtain a unified $d\log$-type system, leading to a compact one-fold integral representation for $P^{(6-2\epsilon)}$ with the $\epsilon$-dependence isolated as $t^{\epsilon}$. They perform a careful analytic continuation from the Euclidean region to all physical regions, including a region-dependent extra term, and confirm consistency with numerical results from Fiesta while ensuring cancellation of $1/\epsilon$ poles. The result enables straightforward $\epsilon$-expansion (to arbitrary order) and can be rewritten in terms of Goncharov polylogarithms; it suggests potential extensions to more complex one-loop topologies such as the hexagon and informs NLO calculations in QCD and related theories.

Abstract

We apply the differential equation technique to the calculation of the one-loop massless diagram with five onshell legs. Using the reduction to $ε$-form, we manage to obtain a simple one-fold integral representation exact in space-time dimensionality. The expansion of the obtained result in $ε$ and the analytical continuation to physical regions are discussed.

Figures (2)

• Figure 1: Pentagon, box and bubble integrals.
• Figure 2: Motion of the branching points of the integrand in Eq. \ref{['eq:example_integral']} and the corresponding deformation of the integration contours. Upper (lower) half corresponds to the $+i0$ ($-i0$) prescription in the denominator of the argument of $f$. Left half: $s_5<0$ ($\phi=\pi$), right half: $s_5>0$ ($\phi=0$). Dashed arrows denote the movement of the branching points upon varying $\phi$ from $\pi$ to $0$. Notation $(n\pm)$ stands for the argument lying on the $n$-th sheet on the upper/lower bank of the cut.