Abstract
We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equations with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic multiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.
Highlights
(b) integrals for the family associated with graph (a) of figure 1 were evaluated in terms of multiple polylogarithms [17,18,19] (MPLs) in the framework of the method of differential equations [20,21,22] with the help of the strategy based on canonical bases [23]
We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equations with canonical bases
Integrals for the family associated with graph (a) of figure 1 were evaluated in terms of multiple polylogarithms [17,18,19] (MPLs) in the framework of the method of differential equations [20,21,22] with the help of the strategy based on canonical bases [23]
Summary
The Feynman integrals for the family of figure 1(b) can be organised in an integral family with nine propagators, where the first seven propagators correspond to the edges of the graph and the last two are so-called irreducible numerators, Fa1,a2,...,a9 (s, t, m2; D). By choosing integrals with unit leading singularities at the level of the maximal cuts, one can often bring the initial differential equations into a so-called precanonical form, where the corresponding matrices depend linearly on. Once this is achieved, the prescriptions of ref. When expanding eq (2.9) in , at each order we can write f in terms of Chen iterated integrals [36], defined in the following way: consider a path γ and a collection of one-forms ωi. We discuss sufficient criteria for when Chen iterated integrals over d log-forms with algebraic arguments, as the ones above, can be expressed in terms of other classes of functions
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