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

Abstract

We present a method to compute the Laurent expansion of the two-loop sunrise integral with equal non-zero masses to arbitrary order in the dimensional regularisation ε. This is done by introducing a class of functions (generalisations of multiple polylogarithms to include the elliptic case) and by showing that all integrations can be carried out within this class of functions.

The kite integral to all orders in terms of elliptic polylogarithms

“…In this sub-system the right-hand side of the differential equation contains an ε 0 -term, which cannot be removed by a change of basis, indicating that the result for I 6 and I 7 cannot be expressed in terms of multiple polylogarithms. Here, elliptic generalisations of multiple polylogarithms occur [27][28][29][30][31][32].…”

“…More challenging are Feynman integrals, which cannot be expressed in terms of multiple polylogarithms. A prominent example is the two-loop sunrise integral with non-zero masses [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Evaluating this integral one encounters elliptic generalisations of (multiple) polylogarithms and every term of the Laurent expansion of the dimensionally regulated sunrise integral may be expressed in terms of functions from this class [32].…”

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].…”

Differential equations for Feynman integrals beyond multiple polylogarithms

“…The simplest example is given by the two-loop sunrise integral [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] with equal masses. A slightly more complicated integral is the two-loop kite integral [45][46][47][48][49], which contains the sunrise integral as a sub-topology.…”

“…Thus we find that the functions of eq. (2.7) together with the multiple polylogarithms are the class of functions to express the equal mass sunrise graph and the kite integral to all orders in ε [42,47]. The functions in eq.…”