Jumpstarting (elliptic) symbol integrations for loop integrals

Abstract: We derive an algorithm for computing the total differentials of multi-loop integrals expressed as one-fold integrals of multiple polylogarithms, which can involve square roots of polynomials up to degree four and may evaluate to (elliptic) multiple polylogarithms ((e)MPL). This gives simple algebraic rules for computing the $(W{-}1, 1)$-coproduct of the resulting weight-$W$ functions up to period terms, and iterating it gives the symbol without actually performing any integration. In particular, our algorithm generalizes existing MPL integration rules and sidesteps the complicated rationalization procedure in the presence of square roots. We apply our algorithm to conformal double-$D$-gon integrals in $D$ dimensions with generic kinematics and possibly massive circumferential propagators. We directly compute, for the first time, the total differential and symbol (up to period terms) of the $D{=}3$ double-triangle and the $D{=}4$ double-box, which in the special case with massless propagators represent the first appearance of eMPL functions in (two-loop) scattering amplitudes of ${\cal N}{=}6$ Chern-Simons-matter theory and ${\cal N}{=}4$ super-Yang-Mills, respectively.