The Master Differential Equations for the 2-loop Sunrise Selfmass Amplitudes

TL;DR

The paper derives master first-order differential equations in $p^2$ for the four sunrise self-mass master integrals in $n$ dimensions with arbitrary internal masses, using integration-by-parts (Tarasov) identities. It analyzes the analytic structure, providing explicit initial conditions at $p^2=0$, and develops systematic expansions around $p^2=0$ and in $(n-4)$ for arbitrary $p^2$, including a complete analytic result for the special case $m_2=m_3=0$ and a comprehensive large-$p^2$ asymptotic framework built from multiple independent solutions. The results yield both analytic expressions (where possible) and robust procedures for numerical evaluation of the sunrise amplitudes, including threshold behavior through the common denominator $D$ and dispersive representations for cross-checks. Collectively, the master equations encode the full analytic structure and provide practical tools for precise computations of two-loop sunrise amplitudes in varied dimensions and mass configurations.

Abstract

The master differential equations in the external square momentum p^2 for the master integrals of the two-loop sunrise graph, in n-continuous dimensions and for arbitrary values of the internal masses, are derived. The equations are then used for working out the values at p^2 = 0 and the expansions in p^2 at p^2 =0, in (n-4) at n to 4 limit and in 1/p^2 for large values of p^2 .

Figures (2)

• Figure 1: The 2-loop sunrise self-mass graph.
• Figure 2: The 2-loop sunrise vacuum graph.