  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.
