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Pairing and polarization in systems with retarded interactions

In a system where a boson (e.g, a phonon) of finite frequency $ω_0$ is coupled to electrons, two phenomena occur as the coupling is increased: electron pairing and polarization of the boson field. Within a path integral formalism and a Dynamical Mean-Field approach, we introduce {\it ad hoc} distribution function which allow us to pinpoint the two effects. When $ω_0$ is smaller than the bandwidth $D$, pairing and polarization occur for fairly similar couplings for all considered temperatures. When $ω_0 > D$, the two phenomena tend to coincide only for $T \gg ω_0$, but are no longer tied for low temperatures so that a state of paired particles without finite polarization is stabilized.