Vertical temperature boundary of the pseudogap under the superconducting dome in the phase diagram of $Bi_2$$Sr_2$$CaCu_2$$O_{8+\delta}$

Abstract : Combining electronic Raman scattering experiments with cellular dynamical mean field theory, we present evidence of the pseudogap in the superconducting state of various hole-doped cuprates. In $Bi_2$$Sr_2$$CaCu_2$$O_{8+\delta}$ we track the superconducting pseudogap hallmark, a peak-dip feature, as a function of temperature $T$ and doping $p$, well beyond the optimal one. We show that, at all temperatures under the superconducting dome, the pseudogap disappears at the doping $p_c$, between 0.222 and 0.226, where also the normal-state pseudogap collapses at a Lifshitz transition. This demonstrates that the superconducting pseudogap boundary forms a vertical line in the $T-p$ phase diagram.