Linearly reducible elliptic Feynman integrals

Elliptic Feynman integrals involve integrations on an elliptic curve, and are not expressible in terms of multiple polylogarithms (MPLs). I will discuss linearly reducible elliptic Feynman integrals, which can be expressed in terms of a 1-dimensional integral over a polylogarithmic integrand with algebraic arguments, called the inner polylogarithmic part (IPP). This representation may in some cases be analytically continued to obtain precise numerical results in the physical region, such as for the unequal mass sunrise. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can also be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs), by direct integration of the Feynman parametrization. Alternatively, the IPP can be mapped to a generalized integral topology satisfying a set of differential equations in canonical form. I will show that these differential equations can be solved directly in terms of eMPLs.

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