New form of kernel in equation for Nakanishi function

TL;DR

This paper derives a real-valued, unambiguous kernel expression for the Nakanishi integral representation of the Bethe-Salpeter amplitude, simplifying calculations and enabling generalization to complex kernels like the cross-ladder.

Contribution

The authors present a new real-valued kernel formula for the Nakanishi equation, facilitating easier and more unambiguous calculations of Bethe-Salpeter amplitudes.

Findings

Explicit kernel formula for one-boson scalar exchange

Reproduction of previously calculated binding energies

Method generalizable to complex Feynman graph kernels

Abstract

The Bethe-Salpeter amplitude $\Phi(k,p)$ is expressed, by means of the Nakanishi integral representation, via a smooth function $g(\gamma,z)$. This function satisfies a canonical equation $g=Ng$. However, calculations of the kernel $N$ in this equation, presented previously, were restricted to one-boson exchange and, depending on method, dealt with complex multivalued functions. Although these difficulties are surmountable, but in practice, they complicate finding the unambiguous result. In the present work, an unambiguous expression for the kernel $N$ in terms of real functions is derived. For the one-boson scalar exchange, the explicit formula for $N$ is found. With this equation and kernel, the binding energies, calculated previously, are reproduced. Their finding, as well as calculation of the Bethe-Salpeter amplitude in the Minkowski space, become not more difficult than in the Euclidean one. The method can be generalized to any kernel given by irreducible Feynman graph. This generalization is illustrated by example of the cross-ladder kernel.