The key ingredients in any superconductor are the Cooper pairs, in which two electrons combine to form a composite boson. In all conventional superconductors the pairing strength alone sets the majority of the physical properties including the superconducting transition temperature T$_c$. In the cuprate high temperature superconductors, no such link has yet been found between the pairing interactions and T$_c$. Using a new variant of photoelectron spectroscopy we measure both the pair-forming ($Delta$) and a self energy/pair-breaking term ($Gamma_s$) as a function of sample type and sample temperature, and we make the measurements over a wide range of doping and temperatures within and outside of the pseudogap/competing order doping regimes. In all cases we find that T$_c$ is approximately set by a crossover between the pair-forming strength $Delta$ and 3 times the self-energy term $Gamma_s$ - a new paradigm for superconductivity. In addition to departing from conventional superconductivity in which the pairing alone sets T$_c$, these results indicate the zero-order importance of the near-nodal self-energy effects compared to competing order/pseudogap effects.
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