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$η$ collective mode as A$_{1g}$ Raman resonance in cuprate superconductors

We discuss the possible existence a spin singlet excitation with charge $\pm2$ ($η$-mode) originating the $A_{1g}$ Raman resonance in cuprate superconductors. This $η$-mode relates the $d$-wave superconducting singlet pairing channel to a $d$-wave charge channel. We show that the $η$ boson forms a particle-particle bound state below the $2Δ$ threshold of the particle-hole continuum where $Δ$ is the maximum $d$-wave gap. Within a generalized random phase approximation and Bethe-Salpether approximation study, we find that this mode has energies similar to the resonance observed by Inelastic Neutron Scattering (INS) below the superconducting (SC) coherent peak at $2Δ$ in various SC cuprates compounds. We show that it is a very good candidate for the resonance observed in Raman scattering below the $2Δ$ peak in the $A_{1g}$ symmetry. Since the $η$-mode sits in the $S=0$ channel, it may be observable via Raman, X -ray or Electron Energy Loss Spectroscopy probes.