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

TL;DR

The paper develops a framework to solve a class of multi-loop Feynman integrals to all orders in $\epsilon$ using iterated integrals of meromorphic modular forms, tying the available function space to the monodromy of a second-order differential operator. It generalises previous results from the full modular group to arbitrary genus-zero subgroups, providing a constructive decomposition of meromorphic quasi-modular forms into total derivatives and a basis of non-derivative forms and proving linear independence of the resulting iterated integrals. The approach is applied to the two- and three-loop equal-mass banana integrals, yielding for the first time complete analytic expressions for the higher orders in $\epsilon$ of the three-loop case in terms of these iterated integrals and identifying the relevant monodromy group as $\Gamma_1(6)$. The work clarifies why modular data for genus-zero subgroups arise in these Feynman integrals, offers algorithmic tools to build function bases, and points to future directions such as canonical forms and handling non-cusp poles within the same modular framework.

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.

Figures (3)

• Figure 1: Paths for the analytic continuation and the calculation of the monodromies for a differential operator with $q$ regular singular poles, one of which at zero and one at infinity. The (blue) reference point $x_\mathsf{ref}$ has been conveniently chosen in the (green) disc $D_{x_0}$ around $x_0=0$.
• Figure 2: Geometry associated to the sunrise differential operator $\mathcal{L}^\textsf{sun}_t$ in eq. \ref{['eq:sunriseDO']}. The coefficient functions have poles at $(t_0,\ldots,t_3)=(0,1,9,\infty)$. The corresponding radii of convergence are shaded in green.
• Figure 3: Geometry of the banana differential operator in eq. \ref{['eq:L_ban_3']}. The coefficient functions have poles at $(x_0,x_1,x_2,x_3)=(0,1/4,1,\infty)$. The corresponding radii of convergence are shaded in green.

Theorems & Definitions (27)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3: DDMS
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 4