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We present a formal derivation of the many-body perturbation theory for asystem of electrons and bosons subject to a nonlinear electron-boson coupling.The interaction is treated at an arbitrary high order of bosons scattered. Theconsidered Hamiltonian includes the well-known linear coupling as a speciallimit. This is the case, for example, of the Holstein and Fr\"{o}hlichHamiltonians. Indeed, whereas linear coupling have been extensively studied,the scattering processes of electrons with multiple bosonic quasiparticles arelargely unexplored. We focus here on a self-consistent theory in terms ofdressed propagators and generalize the Hedin's equations using the Schwingertechnique of functional derivatives. The method leads to an exact derivation ofthe electronic and bosonic self-energies, expressed in terms of a new family ofvertex functions, high order correlators and bosonic and electronic mean-fieldpotentials. In the electronic case we prove that the mean-field potential isthe $n$th-order extension of the well-known Debye-Waller potential. We alsointroduce a bosonic mean-field potential entirely dictated by nonlinearelectron-boson effects. The present scheme, treating electrons and bosons on anequal footing, demonstrates the full symmetry of the problem. The vertexfunctions are shown to have purely electronic and bosonic character as well asa mixed electron-boson one. These four vertex functions are shown to satisfy ageneralized Bethe-Salpeter equation. Multi bosons response functions are alsostudied and explicit expressions for the two and the three bosons case aregiven.

A functional approach to the electronic and bosonic dynamics of many-body systems perturbed with an arbitrary strong electron-boson interaction