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QED Exponentiation for quasi-stable charged particles: the $e^-e^+\to W^-W^+$ process

All real and virtual infrared singularities in the standard analysis of the perturbative Quantum Electrodynamics (like that of Yennie-Frautschi-Suura) are associated with photon emissions from the external legs in the scattering process. External particles are stable, with the zero decay width. Such singularities are well understood at any perturbative order and are resummed. The case of production and decay of the semi-stable {\em neutral} particles like $Z$ boson or $τ$ lepton, with the narrow decay width, $Γ/M \ll 1$, is also well understood at any perturbative order and soft photon resummation can be done. For the absent or loose upper cut-off on the total photon energy $ω$ the production and decay process of the semi-stable (neutral) particles decouples approximately and can be considered quasi-independently. In particular soft photon resummation can be done separately for the production and decay process treating semi-stable (neutral) particle as stable. So far, consistent treatment of the soft photon resummation for semi-stable {\em charged} particles like $W^\pm$ boson is not available in the literature and the aim of this work is to present a solution for this problem. Generally, it should be feasible because the underlying physics is the same as in the case of the neutral semi-stable resonances -- in the limit $Γ/M \ll 1$ production and decay processes for charged particles also necessarily decouple due to long lifetime of the particles. The technical problems to be solved in this work are related to the fact that semi-stable charged particle are able to emit photons.