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Condensation mechanism of high-$T_c$ cuprates: the key role of pairon excitations

In this article we show that the condensation mechanism in cuprates involves the strong coupling of the condensate to pairon excited states. We present an accessible formalism that significantly extends our previous work, providing a theoretical basis for the energy-dependent gap function $Δ(E)$. The latter is proportional to the effective spin exchange energy, $J_{eff}$, with no retardation effects, such as the case of spin-fluctuation or phonon mediated couplings. The fundamental parameters of the superconducting (SC) state are the condensation energy per pair, $β_c$, and the antinodal energy gap, $Δ_p$, which are quantitatively extracted by fitting the cuprate quasiparticle spectrum from tunneling experiments. An explicit formula for the critical temperature is also derived in the model. Valid for any doping, we find $T_c$ to be proportional to $β_c$, and not the gap $Δ_p$, in sharp contrast to conventional SC. The numerical factor $β_c/k_BT_c\simeq 2.24$ originates from pair excitations of the condensate, following Bose statistics, with a mini-gap $δ_M \simeq 1\,$meV in the excitation spectrum. These results strongly suggest that the same `all-electron' mechanism is at work all along the $T_c$-dome.