Complete the algebraic theory of modular-form integration kernels for elliptic Feynman integrals: algorithmic seed construction and equivalence with elliptic multiple polylogarithms

Elliptic Feynman integrals - multi-loop integrals whose maximal cuts are spanned by complete elliptic integrals - are solved through differential equations whose kernels must be cast into a usable function class. Broedel, Duhr, Dulat, Penante and Tancredi (2018) showed that every modular form for a genus-zero congruence subgroup of SL(2,Z) can be written as a power of the complete elliptic integral of the first kind times an algebraic function of a Hauptmodul, and used this to make the differential equations of the sunrise and kite integrals purely algebraic. The construction is nevertheless case-by-case and local: the seed modular form linking the congruence subgroup to the elliptic integral must be identified by hand (in practice from Picard-Fuchs equations), only genus-zero modular curves and single-variable kinematics are covered, the global analytic behaviour is uncontrolled, and the conditions under which the solutions coincide with elliptic multiple polylogarithms are unknown. The open problem is to complete this program: an a-priori algorithmic construction of the complete-elliptic-integral representation, valid for modular forms of arbitrary genus-zero congruence subgroups, that does not rely on case-by-case identification of the seed, together with precise necessary and sufficient conditions for equivalence with elliptic multiple polylogarithms; extension to positive-genus modular curves, extension to several kinematic variables, and global analytic control of the local representations are directions and limitations acknowledged by the authors rather than open questions they pose. Subsequent literature has resolved substantial special cases - analytic continuation of iterated Eisenstein integrals, extension to all genus-zero subgroups via meromorphic modular forms, multi-variable solutions in the alternative language of iterated integrals on moduli spaces, and generalizations to geometries beyond elliptic curves (hyperelliptic curves and K3 surfaces, where Siegel and orthogonal modular forms - the higher-dimensional analogues of modular forms - appear) - but the general framework remains open.

Background

Multi-loop Feynman integrals - the loop-momentum integrals of perturbative quantum field theory - are conventionally computed in dimensional regularization (analytic continuation of the integrals to a non-integer space-time dimension $d=4-2\varepsilon$) and organized by the differential-equation method: the master integrals of a family (a finite basis of independent integrals to which every integral of the family reduces) satisfy a linear system of differential equations in the kinematic variables, and whenever this system can be brought to an $\varepsilon$-form (also called canonical form), in which every derivative carries an explicit factor of $\varepsilon$ and the coefficient matrix is built from integrable one-forms, the solution is immediately a Chen iterated integral (the path-ordered iterated integral of a sequence of one-forms) over those one-forms. When the maximal cut (the integral restricted to its leading-singularity locus, which determines the homogeneous solutions) is spanned by algebraic or logarithmic functions, the kernels are rational and the solutions are multiple polylogarithms (MPLs), the class underlying most analytic two-loop results. For many integrals with several massive propagators - the two-loop equal-mass sunrise and kite integrals are the prototypes - the maximal cut is instead spanned by complete elliptic integrals, chiefly $K(\lambda)=\int_0^1 dt/\sqrt{(1-t^2)(1-\lambda^2 t^2)}$, and the results lie beyond MPLs ("elliptic" Feynman integrals).

Adams and Weinzierl showed that the equal-mass sunrise and kite integrals are, to all orders in $\varepsilon$, iterated integrals of modular forms for the congruence subgroup $\Gamma_1(6)$. A congruence subgroup is a subgroup of $\mathrm{SL}(2,\mathbb{Z})$ that contains, for some integer $N$, all matrices congruent to the identity modulo $N$; a modular form of weight $k$ for such a group $\Gamma$ is a holomorphic function on the upper half-plane transforming as $f((a\tau+b)/(c\tau+d))=(c\tau+d)^k f(\tau)$ under $\Gamma$. The quotient of the upper half-plane by $\Gamma$ compactifies to a Riemann surface $X_\Gamma$, the modular curve, whose genus is a key invariant. When $X_\Gamma$ has genus zero, its field of meromorphic functions is generated by a single function, the Hauptmodul $x(\tau)$. Elliptic multiple polylogarithms (eMPLs) are the iterated integrals on an elliptic curve built from the Kronecker-Eisenstein kernel - a canonical doubly-periodic function on the curve that plays the role the logarithm plays for ordinary polylogarithms - the class studied in the mathematics literature and in string-amplitude calculations; evaluated at torsion points (points of finite order in the elliptic curve's group law), they reduce to iterated integrals of Eisenstein series, and elliptic multiple zeta values (eMZVs) are the corresponding constants.

The source - a 2018 conference-proceedings contribution by Broedel, Duhr, Dulat, Penante and Tancredi - supplies the converse, physics-facing direction. For genus-zero congruence subgroups ($\Gamma(2)$, $\Gamma_0(N)$ for $N=2,4,6$, and $\Gamma_1(6)$) every modular form $f$ of weight $k$ admits an explicit algebraic representation

$$f(\tau) = \frac{B(\tau)^{k/p}\,\big[d_0 + d_1 x(\tau) + \dots + d_m x(\tau)^m\big]}{\prod_i \big(x(\tau)-x_i\big)^{n_i}}$$

where $B$ is a "seed" modular form of lowest weight $p$ expressible through the complete elliptic integral of the first kind $K(\lambda(\tau))$, $x$ is the Hauptmodul, the poles $x_i$ sit at the zeros of $B$, and the degree of the polynomial is bounded by holomorphy at the cusp. This turns the integration kernels of $\varepsilon$-factorized elliptic differential equations into purely algebraic data, and the paper demonstrates it by rewriting eMZVs as iterated integrals over powers of complete elliptic integrals and rational functions, and the sunrise and kite systems in terms of modular forms for $\Gamma_1(6)$.

The construction is, however, deliberately local and case-by-case, and the authors state the gaps in prose. The seed $B$ must be related to $K(\lambda)$ case by case ("we do not know any generic way of doing this a priori"; in practice $B$ is extracted from the Picard-Fuchs equation, i.e. the homogeneous differential equation of the maximal cut, of a specific graph); only genus-zero modular curves are treated (the restriction is called inessential and the extension straightforward, but it is not carried out); the framework uses a single Hauptmodul, while realistic multi-scale Feynman integrals depend on several kinematic variables; the conditions under which solutions in this language coincide with eMPLs are not characterized (the introduction asks "how and when" this is possible); and the representations are local, with branch cuts acknowledged but no systematic monodromy or analytic-continuation treatment. These gaps matter practically: three-loop massive vertex and self-energy corrections with several scales (for example analytic top-mass dependence in Higgs production, or the three-loop $\rho$ parameter) are the current frontier where such a function-class theory is missing.

Problem Statement

Complete the program initiated in the source into a systematic and general theory of the algebraic representation of the integration kernels of elliptic Feynman-integral differential equations. The common objective is: given the system of differential equations for an elliptic Feynman integral, a framework in which its kernels are represented - algorithmically, rather than by case-by-case identification - as products of complete elliptic integrals and algebraic functions, together with a precise characterization of the resulting function class and of its relation to elliptic multiple polylogarithms. The source explicitly identifies the following two gaps, and their resolution constitutes the required objective:

1. (Seed algorithm) An a-priori, non-case-by-case construction that, for a genus-zero congruence subgroup $\Gamma\subseteq \mathrm{SL}(2,\mathbb{Z})$ and a modular form $f$ of weight $k$ for $\Gamma$, produces the representation of $f$ as powers of complete elliptic integrals of the first (and, where needed, second) kind multiplied by algebraic functions - in particular an algorithmic determination of the seed modular form $B$ and of its expression through $K(\lambda)$, not relying on manual identification from the Picard-Fuchs equations of specific graphs. The source proves that such a representation exists for every modular form for a genus-zero congruence subgroup, while admitting no generic way of obtaining it a priori and noting that finding it in explicit cases can be rather hard.

2. (eMPL equivalence) Necessary and sufficient conditions under which solutions of the elliptic differential equations - iterated integrals over products of complete elliptic integrals and rational/algebraic functions - can be recast as elliptic multiple polylogarithms. This is the question the source poses in its introduction ("how and when").

The source further states limitations and generalization directions that frame the program but are not, on the source's own articulation, jointly mandatory completion conditions: its explicit constructions cover only genus-zero modular curves - the restriction is called inessential and straightforward to extend, and what remains for congruence subgroups whose modular curves have positive genus, where no single Hauptmodul exists, is the construction of explicit algebraic bases and ansätze together with the analytic (holomorphy) constraints that replace the genus-zero counting; its framework uses a single Hauptmodul, while realistic multi-scale Feynman integrals depend on several kinematic variables, and the purely algebraic formulation is presented as lending itself to such generalizations; and its representations are local, with branch cuts acknowledged but no systematic monodromy or analytic-continuation treatment. Progress on these directions constitutes partial progress toward completing the program; additional case-by-case treatments of further single-variable graphs do not by themselves constitute a solution.

A solution may be delivered within an extended or modified formalism (for example meromorphic modular forms, iterated integrals on moduli spaces of marked genus-one curves, or automorphic forms beyond genus one - the higher-dimensional analogues of modular forms), provided the algorithmic construction achieves an algebraic representation of the integration kernels - products of powers of complete elliptic integrals and algebraic functions, generalizing the representation displayed in the Background section - and the conditions for equivalence with elliptic multiple polylogarithms are characterized as theorems.

References

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