HQET vertex diagram: ϵ expansion

Andrey G Grozin

doi:10.1103/physrevd.102.054022

Abstract

Differential equations for the one-loop HQET vertex diagram with arbitrary self-energy insertions and arbitrary residual energies are reduced to the $\varepsilon$ form and used to obtain the $\varepsilon$ expansion in terms of Goncharov polylogarithms.

Highlights

We consider the one-loop vertex diagram (Fig. 1) with arbitrary degrees of all three denominators: In1;n2;n3 ðθ; ω1; ω2Þ
Differential equations for the one-loop vertex diagram in heavy quark effective theory (HQET) with arbitrary self-energy insertions and arbitrary residual energies are reduced to the ε form and used to obtain the ε expansion in terms of Goncharov polylogarithms
All integrals with a given set li can be reduced [1], using integration by parts (IBP), to three master integrals with mi 1⁄4 ð0; 1; 1Þ, (1, 0, 1), and (1, 1, 1)

Summary

Published by the American Physical Society

The initial condition is fð1; 1Þ 1⁄4 ð1; 1; 1ÞT. If l1 1⁄4 0, f1 is trivial (4); if l2 1⁄4 0, f2 is trivial; if l1 1⁄4 l2 1⁄4 0, there is only one nontrivial master integral f3. If l1 1⁄4 l2, f2ðx; yÞ 1⁄4 f1ðx; y−1Þ (10), and there are only two unknown functions f1 and f3. We shall use the method of differential equations [3]

Using sinh

The symmetry properties of the canonical master integrals are