Riassunto analitico
The Bern-Kosower master formula is a fundamental tool within the so-called "Worldline" formalism, in particular for the evaluation of one-loop effective vertices associated to scattering processes in particle physics. In more recent years, similar relations have been developed with the purpose of dealing with "open" tree-level dressed propagators, even with the addition of a constant electromagnetic field. In this article, we will find a mathematical "linking" relationship which will enable us to obtain effective vertices starting from the dressed propagators, both for scalar and spinor quantum electrodynamics (QED). We will see that this procedure will require a mass integration applied on the untruncated propagators, eventually leading to the same results obtained from the usual Bern-Kosower relations and solving a consistency problem associated with the emergence of spurious symmetry coefficients as one approches the task by an apparently intuitive "cut and close" method. We will also treat the constant electromagnetic field perturbatively, in particular by calculating the corrections to the two-photon untruncated propagator and, by using the linking relation, to the vacuum polarization in scalar QED up to the second order in the electromagnetic tensor F.
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