Symbology for elliptic multiple polylogarithms and the symbol prime

Abstract

Elliptic multiple polylogarithms occur in Feynman integrals and in particular in scattering amplitudes. They can be characterized by their symbol, a tensor product in the so-called symbol letters. In contrast to the non-elliptic case, the elliptic letters themselves satisfy highly non-trivial identities, which we discuss in this paper. Moreover, we introduce the symbol prime, an analog of the symbol for elliptic symbol letters, which makes these identities manifest. We demonstrate its use in two explicit examples at two-loop order: the unequal-mass sunrise integral in two dimensions and the ten-point double-box integral in four dimensions. Finally, we also report the result of the polylogarithmic nine-point double-box integral, which arises as the soft limit of the ten-point integral.

An Infinite Family of Elliptic Ladder Integrals

“…While the two-loop elliptic ladder diagram has already received a large amount of attention over the last decade [51,53,60,98,99,115,116], diagrams in this family involving three or more loops have not been previously studied. However, they correspond to multi-soft limits of the traintrack diagrams considered in ref.…”

“…Here we adopt the conventions of refs. [51,53], and refer the reader there for further details. We have performed this procedure explicitly for the three-loop case of I 2,1 and packaged it into a more compact notation, the so-called E 4 functions [49].…”

“…The two-loop case I 0,1,1 was calculated in ref. [53], while the higher-loop cases can be obtained by taking the soft limit of the integral representation in eq. (3.8).…”

“…These diagrams can generally be expressed in terms of elliptic multiple polylogarithms (eMPLs) [15,[44][45][46][47][48][49], which contain MPLs as special cases and share much of their mathematical structure. In particular, eMPLs are endowed with a coaction structure [16,[50][51][52][53], and their numeric value can be efficiently computed for rational values of their arguments [16,[54][55][56][57][58].…”

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

“…Mathematically, eMPLs can be thought of as iterated integrals over the rational functions on the elliptic curve [71][72][73][74] or as iterated integrals on a genus-1 Riemann surface [41,[75][76][77][78][79][80]. Much of the MPL technology seems to have elliptic analogues, such as symbol calculus [59,[81][82][83][84]. However, elliptic symbol letters are complicated functions that satisfy non-trivial identities.…”