On an approach to canonicalizing elliptic Feynman integrals

TL;DR

The paper addresses the challenge of obtaining canonical bases for elliptic Feynman integrals with multiple scales directly at the integrand level under the maximal cut. It introduces a universal construction based on the Legendre normal form of elliptic curves and a Möbius transformation to map to univariate elliptic families, yielding integrand-level $\varepsilon$-factorized differential equations. The authors provide explicit canonical integrand formulas, demonstrate them on five integral families including two new results, and discuss sub-sector dependence and two-variable constructions under next-to-maximal cuts. The work suggests broad applicability to elliptic and potentially higher-genus geometries and lays groundwork for extensions to Jacobi normal form and constant-period methods.

Abstract

We present generic expressions for the integrands of canonical bases under maximal cut in elliptic Feynman integral families with multiple kinematic scales. Such integrals frequently arise in phenomenologically relevant scattering processes. The derivation of our results starts from the Legendre normal form of elliptic curves, where the geometric properties of the curves are simple and explicit, and further kinematic singularities are presented as marked points. The simplicity of the normal form allows a straightforward construction of canonical bases with an arbitrary number of marked points. They can then be mapped into any univariate elliptic integral families via an appropriate Möbius transformation, leading to universal expressions for the integrands. As a demonstration, we discuss the application of our method to several concrete examples, including two new integral families whose canonical bases were not available in the literature. In several examples, we derive canonical bases for the full integral families without any cuts, demonstrating the simplicity of the sub-sector dependence of our canonical bases.

Figures (5)

• Figure 1: The unequal-mass sunrise diagram is shown above, where different colors represent different masses for the propagators. The only elliptic sector in the family is the top sector. The sub-sectors are obtained by pinching the propagators, which are double tadpoles. The case is well-studied in the literature Bogner:2019lfa.
• Figure 2: A family contributing to $2$-loop $t \bar{t} W$ production Becchetti:2025qlu is shown in the left panel. There are two four-point elliptic sectors, one of which has 4 scales and its canonical basis has been given in Becchetti:2025oyb. The remaining elliptic sector is shown in the right panel, which is more complicated with 5 scales. In the diagrams, thick orange lines represent top quarks with mass $m_t$, the curvy lines represent $W$ bosons with mass $m_W$, and the thick blue line is an off-shell particle coming from pinching while all the others are massless.
• Figure 3: The ${\cal T}_{3F}$ branch of $2$-loop $W$-pair production via light quark-antiquark annihilation He:2024iqg is shown in the left panel. The sector is reducible, and a special sub-sector is shown in the right panel. This sub-sector is the only elliptic sector. In the diagrams, thick orange lines represent top quarks with mass $m_t$ and the curvy lines represent $W$ bosons with mass $m_W$ while all the others are massless.
• Figure 4: The diagram for the non-planar double box family A is shown in the left panel. The sub-sectors are obtained by pinching the propagators. A special case is the elliptic triangle sector, shown in the right panel. This sub-sector and the top sector are the only sectors that are elliptic, and they share the same elliptic curve. In the diagrams, thick blue lines are massive while all the others are massless, and we use the shorthand $p_{1,2,3}=p_1+p_2+p_3$ and $k_{1,2}=k_1+k_2$ hereafter.
• Figure 5: The diagram for the non-planar double box family B is shown above. It is similar to Fig. \ref{['npdb_fig']}, with two internal lines becoming massless.