A complete algorithmic theory of modular transformations for elliptic Feynman integrals: invariant function classes, fibre transformation matrices, ansatz classification, and analytic boundary-data transport

Elliptic Feynman integrals - master integrals of multi-loop scattering amplitudes whose underlying geometry is an elliptic curve - are evaluated efficiently through nome ($\bar q$) expansions only where $|\bar q|$ is small, and a modular transformation $\gamma\in\mathrm{SL}_2(\mathbb Z)$ can always achieve this. Weinzierl (2021) showed that the natural class of iterated integrals involved (iterated integrals over modular-form letters and Kronecker-kernel letters on a covering of the moduli space of a punctured torus) is closed under such transformations only if the modular transformation of the variables is combined with a compensating redefinition $U(\varepsilon,z,\tau)$ of the master-integral basis, constructed by hand only for the equal-mass and unequal-mass sunrise integrals. The open problem is to turn this into a complete algorithmic theory supporting black-box evaluation of elliptic Feynman integrals for all values of the arguments: give an algorithm constructing $U$ from an arbitrary elliptic system's differential equation in $\varepsilon$-form (the setting of the source's general framework); classify the admissible ansatz letters under modularity constraints; and determine transformed boundary constants analytically, where the source's own method is numerical matching combined with recognition of integer relations via the PSLQ algorithm. The source also records as undetermined whether a strictly larger class of iterated integrals could be modular-invariant without any basis redefinition, while arguing that such an enlargement would not be natural. Follow-up literature through 2026 delivers major partial progress - universal fibre transformations within the canonical framework of pure elliptic multiple polylogarithms (2025), genus-zero meromorphic-modular-form decompositions (2022), and modular-bootstrap constructions of $\varepsilon$-forms for multiscale and two-curve systems - but each component remains open in general.

Background

Multi-loop Feynman integrals, the building blocks of perturbative scattering amplitudes in quantum field theory, are analytically continued to non-integer space-time dimension $d=2-2\varepsilon$ (dimensional regularization), where $\varepsilon$ is the expansion parameter. A finite basis of so-called master integrals spans all integrals of a given family; the masters satisfy a first-order system of differential equations in the kinematic variables. If a basis $J$ can be found for which the system takes the $\varepsilon$-form $dJ = \varepsilon\, A\, J$ with the whole connection matrix multiplied by $\varepsilon$, the $\varepsilon$-expansion is generated term by term and the solutions are iterated integrals of controlled (uniform) transcendental weight.

For many integrals the solutions are multiple polylogarithms (MPLs), i.e. iterated integrals of rational one-forms. Their numerical evaluation is a solved black-box problem: if an argument $x$ lies outside the region where the sum representation converges, the evaluation routine internally transforms it, e.g. $x\to 1/x$, into the convergent region, and the transformed function is again an MPL of the same class, so no redefinition of the master integrals is needed.

This fails for elliptic Feynman integrals (two-loop sunrise and banana graphs, double boxes with internal masses, ...), whose geometry is an elliptic curve. A choice of two independent periods of the curve fixes a modular parameter $\tau$ in the complex upper half-plane; the nome is $\bar q = e^{2\pi i \tau}$. A modular transformation $\gamma=\begin{pmatrix}a& b\\ c& d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z)$, $\tau\mapsto\tau'=(a\tau+b)/(c\tau+d)$, can always be chosen so that $|\bar q|\le e^{-\sqrt{3}\pi}\approx 0.0043$, which makes $\bar q$-expansion evaluation fast (routines are implemented in the computer-algebra library GiNaC). This is the modern incarnation of the classical conjugate-modulus problem for elliptic integrals. The relevant letters of the function class are differential one-forms on a covering of the moduli space $\mathcal M_{1,n}$ of a genus-one curve with $n$ marked points: (a) modular-form letters $\omega^{\mathrm{modular}}(\eta)=2\pi i\,\eta(\tau)\,d\tau/N$ with $\eta$ a (typically Eisenstein-series) modular form of weight $k$ and level $N$ for the principal congruence subgroup $\Gamma(N)$, and (b) Kronecker-type letters $\omega_k(L(z),K\tau)$ built from the coefficients $g^{(k)}(z,\tau)$ of the Kronecker function, the doubly quasi-periodic kernels on the torus, where $L(z)$ is a linear function of the marked-point coordinates $z_i$ and $K$ an integer. Letters of modular weights $0\dots k_{\max}$ span a filtrated vector space $F_{k_{\max}}\mathcal M^{\mathrm{elliptic}}_{\bullet}$.

Weinzierl (2021) showed that this class is not closed under modular transformations alone: a weight-$k$ letter picks up the automorphic factor $(c\tau+d)^{k-2}$, generating new integrands with additional (possibly negative) powers of $\ln\bar q$ that spoil the expansion properties. Closure is restored only by a combined transformation: a modular transformation of the variables of the base manifold together with a compensating redefinition of the master-integral basis in the fibre, $J' = U(\varepsilon,z,\tau)\,J$, whose matrix $U$ absorbs the automorphic factors and redistributes lower-weight terms into covariant combinations, after which the transformed system is again in $\varepsilon$-form within the same class. This fibre transformation was constructed explicitly only for the equal-mass sunrise integral (three masters) and the unequal-mass sunrise integral (seven masters, in coordinates on $\mathcal M_{1,3}$). Boundary constants in the transformed basis were obtained by numerical matching in overlap regions where both $|\bar q|$ and $|\bar q'|$ are small, combined with the PSLQ integer-relation algorithm to convert high-precision numbers into analytic constants. What is missing, and what this problem asks for, is the general theory: how to construct $U$ algorithmically, how to classify admissible letters consistent with modularity, and how to transport boundary data analytically - alongside the question, left undetermined in the source, whether larger modular-invariant function classes exist.

Problem Statement

The problem is to complete the algorithmic theory of modular transformations for systems of elliptic Feynman integrals, replacing the case-by-case treatment of Weinzierl (2021) by a systematic theory. The objective is a black-box algorithm that evaluates an arbitrary elliptic Feynman-integral system efficiently for all values of its arguments - the analogue of what argument transformations already provide for multiple polylogarithms. The source paper resolves this structurally through the combined modular-plus-fibre transformation, which its Conclusions identify as the natural solution, but demonstrates it only for the equal-mass and unequal-mass sunrise integrals. The open components of the general theory are:

(i) Class closure without fibre redefinition. The source records as undetermined whether there exists a strictly larger natural class of iterated integrals - beyond the modular-form letters $\omega^{\mathrm{modular}}(\eta)=2\pi i\,\eta(\tau)\,d\tau/N$ and the Kronecker-type letters $\omega_k(L(z),K\tau)$ spanning $F_{k_{\max}}\mathcal M^{\mathrm{elliptic}}_{\bullet}$ - that is closed under all modular transformations $\gamma\in\mathrm{SL}_2(\mathbb Z)$ with no redefinition of the master integrals and that still supports fast $\bar q$-expansion evaluation, while arguing that such an enlargement "would not be natural". Whether such an enlargement exists, or can be ruled out, remains open; for this question an enlargement counts as natural only if it is again generated by a finite set of letters of the same kind - modular-form one-forms and Kronecker-type letters $\omega_k(L(z),K\tau)$ on the covering of $\mathcal M_{1,n}$, carrying the modular-weight filtration - so that membership is fixed by this letter/weight structure rather than by an ad hoc construction, and any class that adjoins letters or functions outside this modular/Kronecker framework is excluded.

(ii) Algorithmic fibre transformation. Given an arbitrary elliptic Feynman-integral system via its differential equation in $\varepsilon$-form in some initial basis (arbitrary topology, arbitrary number of kinematic variables and marked points, arbitrary $k_{\max}$) - the assumption under which the source's general framework is stated - give an algorithm that constructs the compensating fibre transformation matrix $U(\varepsilon,z,\tau)$ such that the combined modular-plus-fibre transformation keeps the system within the same class of iterated integrals and in $\varepsilon$-form, together with a criterion for when such a $U$ exists. The source gives only a strategy (absorb automorphic factors, redistribute lower-weight terms into covariant combinations) and explicit constructions for the two sunrise cases.

(iii) Admissible ansatz letters. The source notes that, when a differential equation in $\varepsilon$-form is sought from an ansatz whose unknown coefficients are fixed by $\bar q$-expansion matching, modularity constraints can reduce the number of admissible terms; a classification of the admissible letters consistent with the modularity constraints, beyond the cases treated so far, remains open.

(iv) Analytic boundary data. In the source, the boundary constants in the transformed basis are determined by numerical matching in overlap regions where both $|\bar q|$ and $|\bar q'|$ are small, combined with the PSLQ integer-relation algorithm; an analytic determination - transporting the known boundary constants exactly under the combined transformation, or computing them ab initio in the transformed basis - remains open.

Components (ii)-(iv) concern the combined-transformation route endorsed by the source paper; component (i) concerns the alternative possibility of an enlarged invariant class, which the source raises while arguing that it would not be natural.

A complete resolution of the task requires all four components (i)-(iv): the class-closure question (i) and the combined-transformation route (ii)-(iv) are not substitutes for one another, and a partial result on any single component - however substantial - does not by itself complete the task.

References

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