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.
Highlights
Feynman integrals are a cornerstone of perturbative computations in Quantum Field Theory, and so it is important to have a good knowledge of the mathematics underlying them, including efficient techniques for their computation and a solid understanding of the space of functions needed to express them
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
It is well known that Feynman integrals satisfy systems of coupled first-order differential equations [15–18], and multiple polylogarithms (MPLs) are closely connected to the concepts of pure functions [19] and canonical differential equations [20]
Summary
Feynman integrals are a cornerstone of perturbative computations in Quantum Field Theory, and so it is important to have a good knowledge of the mathematics underlying them, including efficient techniques for their computation and a solid understanding of the space of functions needed to express them. It was realised that in the equal-mass case the two-loop sunrise integral can be expressed as iterated integrals of modular forms [50, 51] This class of functions is of interest in pure mathematics [52–56], and it is understood how to manipulate and evaluate these integrals efficiently [57, 58]. The relevance of this class of differential equations for Feynman integrals stems from the fact that they cover in particular the two- and three-loop equal-mass banana integrals Their solution space can be described in terms of iterated integrals of meromorphic modular forms, introduced and studied by one of us in the context of the full modular group SL2(Z) [56]. In appendix A we present a rigorous mathematical proof the main theorem from section 4, and in appendix B we collect formulas related to the sunrise and banana integrals
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