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In this paper we continue our analysis of the interplay between the pairingand the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC)systems with an effective dynamical electron-electron interaction $V(\Omega_m)\propto 1/|\Omega_m|^\gamma$ (the $\gamma$-model). In previous papers westudied the cases $0<\gamma <1$ and $\gamma \approx 1$. We argued that thepairing by a gapless boson is fundamentally different from BCS/Eliashbergpairing by a massive boson as for the former there exists an infinite number oftopologically distinct solutions for the gap function $\Delta_n (\omega_m)$ at$T=0$ ($n=0,1,2...$), each with its own condensation energy $E_{c,n}$. Here weextend the analysis to larger $1< \gamma <2$. We argue that the discrete set ofsolutions survives, and the spectrum of $E_{c,n}$ gets progressively denser as$\gamma$ approaches $2$ and eventually becomes continuous at $\gamma \to 2$.This increases the strength of "longitudinal" gap fluctuations, which tend toreduce the actual superconducting $T_c$ and give rise to a pseudogap region ofpreformed pairs. We also detect two features on the real axis for $\gamma >1$which become critical at $\gamma\to 2$. First, the density of states evolvestowards a set of discrete $\delta-$functions. Second, an array of dynamicalvortices emerges in the upper frequency half-plane. These two features comeabout because on a real axis, the real part of the interaction, $V' (\Omega)\propto \cos(\pi \gamma/2)/|\Omega|^\gamma$, becomes repulsive for $\gamma >1$,and the imaginary $V^{''} (\Omega) \propto \sin(\pi \gamma/2)/|\Omega|^\gamma$,gets progressively smaller at $\gamma \to 2$. The features on the real axis areconsistent with the development of a continuum spectrum of $E_{c,n}$ obtainedusing $\Delta_n (\omega_m)$ on the Matsubara axis. We consider the case $\gamma=2$ separately in the next paper.

physical review. b./physical review. b

Interplay between superconductivity and non-Fermi liquid at a  quantum-critical point in a metal. IV: The $\gamma$ model and its phase  diagram at $1<\gamma <2$