Abstract
The approach based on the Nakanishi integral representation of n-leg transition amplitudes is extended to the treatment of the self-energies of a fermion and an (IR-regulated) vector boson, in order to pave the way for constructing a comprehensive application of the technique to both gap- and Bethe-Salpeter equations, in Minkowski space. The achieved result, namely a 6-channel coupled system of integral equations, eventually allows one to determine the three Källén–Lehman weights for fully dressing the propagators of fermion and photon. A first consistency check is also provided. The presented formal elaboration points to embed the characteristics of the non-perturbative regime at a more fundamental level. It yields a viable tool in Minkowski space for the phenomenological investigation of strongly interacting theories, within a QFT framework where the dynamical ingredients are made transparent and under control.
Highlights
Tion of the interaction kernel needs in turn the knowledge of other key ingredients: (i) self-energies of both intermediate particles and quanta and (ii) vertex functions, as pointed out by Gell–Mann and Low [2]
The approach based on the Nakanishi integral representation of n-leg transition amplitudes is extended to the treatment of the self-energies of a fermion and an (IRregulated) vector boson, in order to pave the way for constructing a comprehensive application of the technique to both gap- and Bethe-Salpeter equations, in Minkowski space
Those 2- and 3point functions are quantities to be determined through the infinite tower of Dyson–Schwinger equations (DSEs) [3,4,5] that govern the whole set of N -point functions
Summary
Significant progresses have been done for implementing the approach based on BSE plus truncated DSEs, and a high degree of sophistication has been achieved, mainly in Euclidean space (see Refs. [6,7,8,9,10,11,20] and Refs. [21,22,23,24,25,26,27]). Significant progresses have been done for implementing the approach based on BSE plus truncated DSEs, and a high degree of sophistication has been achieved, mainly in Euclidean space [6,7,8,9,10,11,20] and Refs. [21,22,23,24,25,26,27]) As it is well-known, DSEs can be obtained in a very efficient way by using the path-
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