We study pairing in an itinerant system which is close to a quantum critical point(QCP). The fermion interaction is mediated by soft bosons due to the order parameter fluctuations. This effective interaction can lead to superconductivity if it results in pairing instability in at least one pairing channel, and it can also lead to the non-Fermi liquid tendency which is detrimental to superconductivity state. We model this story by adopting the approximation method that Eliashberg first used in his study on electron-phonon coupling superconductivity, and by introducing a pure dynamical pairing interaction V(\Omega)~1/\Omega^\gamma, with \gamma>0 corresponding to different pairing models. We solve the pairing equations on both the Matsubara axis and the real axis. Interestingly, the spectral properties that we obtained from real axis solution agree well with that observed experimentally in pseudogap regime in some cuprate materials. Specifically, the temperature evolution of density of states shows a transition from “gap filling” to “gap closing”. I will also show another interesting aspect of our model that there actually exist an infinite set of solutions to the gap equation. We show that this is a unique feature of pairing at a QCP and does not exist once the system is moved away from a QCP. The set is discrete for \gamma<2 and becomes continuous at \gamma=2, which corresponds to strong coupling limit of electron-phonon interaction.
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