Meromorphic modular forms and the three-loop equal-mass banana integral

Abstract

We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.

Feynman Integrals

“…Calabi-Yau (l − 1)-folds) enter the computation of Feynman integrals. The banana integrals have been studied in [329,330,366,403,[411][412][413][414][415][416], other Feynman integrals related to Calabi-Yau manifolds have been studied in [326][327][328].…”

Functions Beyond Multiple Polylogarithms for Precision Collider Physics

“…Certain Feynman integrals have also been found to be expressible in terms of more specialized spaces of functions. For instance, diagrams depending on just a single kinematic variable, such as the two-and three-loop banana diagrams with all equal internal masses and the three-loop contributions to the ρ parameter, can be expressed in terms of iterated integrals of modular forms [126,225,278,279]. The finite part of the two-loop banana integral with distinct masses has also been expressed in terms of functions that generalize the infinite sum representation of classical polylogarithms, namely ELi n;m (x; y…”

The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals

“…[185-188, 206, 207] for alternative definitions of elliptic generalisations of polylogarithmic functions that are closely related to eMPLs). Closely related to eMPLs are iterated integrals of modular forms [134,208], including meromorphic modular forms [135], which have also appeared in the context of Feynman integrals [103,[209][210][211][212][213]. Iterated integrals of holomorphic modular forms and eMPLs are examples of pure functions [201], and the corresponding Feynman integrals satisfy differential equations in canonical form [102,103,127] (though it is not possible to bring them into canonical dlog-form).…”