New form of kernel in equation for Nakanishi function

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, 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 no 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.

Automated summary

The Bethe-Salpeter amplitude Phi(k, p) is expressed via a smooth function g(gamma, z) using the Nakanishi integral representation. The calculations of the kernel N have been restricted to one-boson exchange, but an unambiguous expression for the kernel N in terms of real functions has been derived in this work. This method can be generalized to any kernel given by irreducible Feynman graph, illustrated by the example of the cross-ladder kernel.