The two-loop sunrise graph with arbitrary masses

Abstract

We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.Comment: 24 page

The two-loop master integrals for $ q\overline{q} $ → VV

“…The issue of the existence of such a basis for any multi-loop problem remains, in particular for those cases which cannot be expressed in terms of GHPLs or general Chen Iterated integrals [53][54][55][56]. Moreover, even in those cases where it is known that the final result will contain only GHPLs, no algorithm for finding such basis is known, while only some general criteria have been pointed out recently [32,33,57,58].…”

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

“…At two or more loops many Feynman integrals can be likewise expressed in terms of GPLs [7][8][9][10][11][12][13][14][15][16][17][18][19] (for further references, see [20,21] and the references therein), but there are also integrals which are counter examples, such as notably that of the fully massive sunset graph [22][23][24][25][26]. Certain graphs without massive propagators are also believed to be counter examples [27].…”

The iterated structure of the all-order result for the two-loop sunrise integral

“…The simplest Feynman integral which cannot be expressed in terms of multiple polylogarithms is the two-loop sunrise integral with non-vanishing masses. This Feynman integral has already received considerable attention in the literature [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In this paper we study the two-loop sunrise integral with equal non-zero masses in D = 2 − 2ε space-time dimensions.…”