Feynman amplitude and Meijer $G$-functions. Unified representation for divergent and convergent graphs

A parametric a representation of the Feynman amplitude for spinor graphs is obtained; it is expressed in terms of Meijer $G$-functions, and is valid for both divergent and convergent graphs. The Chisholm–Nakanishi–Symanzik representation for scalar convergent graphs is shown to be a special case of the representation. In addition, this expression has a number of helpful properties: 1) in such a representation it is clear that the mathematical nature of the infrared divergences, which are usually associated only with the photon zero mass, are of the same kind as the singularities of a Feynman amplitude for which there are no photons; 2) the expression has a form in which the scale-invariant terms are separated out from the terms that break this invariance; 3) the expression preserves gauge invarianee and the Ward identity for renormalized amplitudes (this is demonstrated for the example of the simplest graphs in quantum lectrodynamics).