K3 periods of banana integrals depending on 2 different masses

K3 periods of banana integrals depending on 2 different masses.

Abstract: Maximal cuts of 3-loop banana integrals compute the periods of some 4-parameter family of K3 surfaces, and the moduli are the values of the masses of the particles.

It is well known that in the case that all four masses are equal, the Picard-Fuchs operator describing the periods is a symmetric square, and the periods can be written as products of modular forms.

We show that in the case that only three masses are equal and the fourth one is different, the periods are still products of modular forms.

In particular, this implies that the relevant period and maximal cuts can be expressed as products of complete elliptic integrals depending on complicated algebraic function of the masses.