The soaring kite: a tale of two punctured tori

Authors: Mathieu Giroux, Andrzej Pokraka, Franziska Porkert, Yoann Sohnle Preprint number: UUITP-03/24 Abstract: We consider the 5-mass kite family of Feynman integrals and present a systematic approach for constructing an ε-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be obtained from integrals over maximal cuts. Given an appropriate boundary value, the differential equation is systematically solved in terms of iterated integrals over Kronecker-Eisenstein g-kernels and modular forms. Then, the numerical evaluation of the master integrals is discussed, and important challenges in that regard are emphasized. In an appendix, we introduce new relations between g-kernels.