# Two-loop kite master integral for a correlator of two composite vertices

## TL;DR

This paper derives a comprehensive analytical expression for a complex two-loop massless correlator diagram, using hypergeometric functions, which serves as a generating function for related scalar Feynman integrals.

## Contribution

It provides the first explicit calculation of the two-loop kite diagram correlator in terms of hypergeometric functions, generalizing previous results and enabling broader applications.

## Key findings

- Expressed the correlator as a double hypergeometric series (Kampé de Fériet function)
- Reduced special cases to generalized hypergeometric functions $_3F_2$
- Derived relations and Mellin moments in terms of Lauricella functions

## Abstract

We consider the most general two-loop massless correlator $I(n_1,n_2,n_3,n_4,n_5; x,y;D)$ of two composite vertices with the Bjorken fractions $x$ and $y$ for arbitrary indices $\{n_i\}$ and space-time dimension $D$; this correlator is represented by a "kite" diagram. The correlator $I(\{n_i\};x,y;D)$ is the generating function for any scalar Feynman integrals related to this kind of diagrams. We calculate $I(\{n_i\};x,y;D)$ and its Mellin moments in a direct way by evaluating hypergeometric integrals in the $\alpha$ representation. The result for $I(\{n_i\};x,y;D)$ is given in terms of a double hypergeometric series -- the Kamp\'{e} de F\'{e}rriet function. In some particular but still quite general cases it reduces to a sum of generalized hypergeometric functions $_3F_2$. The Mellin moments can be expressed through generalized Lauricella functions, which reduce to the Kamp\'{e} de F\'{e}rriet functions in several physically interesting situations. A number of Feynman integrals involved and relations for them are obtained.