The Threshold Expansion of the 2-loop Sunrise Selfmass Master Amplitudes

TL;DR

The paper addresses the analytic threshold expansion of the four master amplitudes for the two-loop sunrise self-mass diagram with arbitrary masses. It employs a system of differential equations in the external momentum squared, exploiting threshold as a Fuchsian point and performing an expansion around n=4 to derive leading and subleading threshold coefficients, including nontrivial constants fixed via equal-mass dispersion relations. The authors provide explicit closed-form expressions for the threshold data, notably ${ m G}^{(-2)}_eta$, ${ m G}^{(-1)}_eta$, and ${ m G}^{(0)}_0$, and extend the expansion to higher orders with ${ m H}^{(eta,i)}$ coefficients, cross-validating against established results (BDU, DS) in special mass configurations. This work yields analytic benchmarks for multi-loop sunrise integrals with general mass configurations and demonstrates a robust approach for threshold analyses using differential equations at Fuchsian points.

Abstract

The threshold behavior of the master amplitudes for two loop sunrise self-mass graph is studied by solving the system of differential equations, which they satisfy. The expansion at the threshold of the master amplitudes is obtained analytically for arbitrary masses.