Lessons from Elliptic Leading Singularities
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A good choice of Feynman integral bases plays an essential role in the computation of loop scattering amplitudes. While such choice is well understood when the integrals can be expressed as iterated integrals with logarithmic kernels, the situation remains less clear when elliptic curves get involved. A common approach is to construct integrals that satisfy differential equations in which the dependence on dimensional regulator factorizes completely. Is it possible, instead, to define a good integral basis purely from geometric considerations at the integrand level? We propose a generalization of dlog integrands with unit leading singularities to the case of integration on elliptic curves, which provides such a definition. We observe that the corresponding Feynman integrals satisfy a previously unnoticed form of differential equations, whose solutions evaluate to pure functions. We further argue that various properties of this basis may offer advantages in multi-scale amplitude calculations.
Based on: arXiv:2504.20897.
