Analytic results for planar three-loop integrals for massive form factors

TL;DR

The paper addresses analytic evaluation of planar three-loop Feynman integrals contributing to heavy-quark form factors with two on-shell legs ($p_1^2=p_2^2=m^2$). It employs the differential equations approach in a uniformly transcendental basis to obtain analytic results, using a four-letter alphabet in the differential equations and solving via Chen iterated integrals and Goncharov polylogarithms. The main results are 90 master integrals at general momentum transfer $q^2$ expressed in multiple polylogarithms, and 51 threshold master integrals at $q^2=4m^2$ expressed in terms of Goncharov polylogarithms at unity with indices drawn from a seven-letter alphabet including sixth roots of unity. Threshold matching is achieved by expansion by regions and a careful DE analysis near the singular point $x=-1$, including a naive derivative trick to access single-scale contributions. The work provides analytic data and ancillary files enabling precise three-loop computations in heavy-quark form factors and informs future extensions to related kinematics.

Abstract

We use the method of differential equations to analytically evaluate all planar three-loop Feynman integrals relevant for form factor calculations involving massive particles. Our results for ninety master integrals at general $q^2$ are expressed in terms of multiple polylogarithms, and results for fiftyone master integrals at the threshold $q^2=4m^2$ are expressed in terms of multiple polylogarithms of argument one, with indices equal to zero or to a sixth root of unity.

Figures (5)

• Figure 1: Graphs for the planar 3-loop heavy quark form factor integrals. The solid lines represent massive propagators, while the dashed lines represent massless ones.
• Figure 2: Feynman integral for $f_{61}$. The dot on the left line means the second power of the corresponding propagator.
• Figure 3: A triangle diagram.
• Figure 4: Feynman integral for $f_{76}$.
• Figure :