Elliptic Leading Singularities and Canonical Integrands

In the well-studied genus zero case, bases of d log integrands with integer leading singularities define Feynman integrals that automatically satisfy differential equations in canonical form. Such integrand bases can be constructed without input from the differential equations and without explicit involvement of dimensional regularization parameter ε. We propose a generalization of this construction to genus one geometry arising from the appearance of elliptic curves. We argue that a particular choice of algebraic 1-forms of the second kind that avoids derivatives is crucial. We observe that the corresponding Feynman integrals satisfy a special form of differential equations that has not been previously reported, and that their solutions order by order in ε yield pure functions. We conjecture that our integrand-level construction universally leads to such differential equations.