  In conventional superconductors, the pairing energy gap (\Delta) and
superconducting phase coherence go hand-in-hand. As the temperature is lowered,
both the energy gap and phase coherence appear at the transition temperature
T_c. In contrast, in underdoped high-T_c superconductors (HTSCs), a pseudogap
appears at a much higher temperature T^*, smoothly evolving into the
superconducting gap at T_c. Phase coherence on the other hand is only
established at T_c, signaled by the appearance of a sharp quasiparticle (QP)
peak in the excitation spectrum. Another important difference between the two
types of superconductors is in the ratio of 2\Delta / T_c=R. In BCS theory,
R~3.5, is constant. In the HTSCs this ratio varies widely, continuing to
increase in the underdoped region, where the gap increases while T_c decreases.
Here we report that in HTSCs it is the ratio z_A\Delta_m/T_c which is
approximately constant, where \Delta_m is the maximum value of the d-wave gap,
and z_A is the weight of the coherent excitations in the spectral function.
This is highly unusual, since in nearly all phase transitions, T_c is
determined by an energy scale alone. We further show that in the
low-temperature limit, z_{\it A} increases monotonically with increasing doping
x. The growth is linear, i.e. z_A(x)\propto x, in the underdoped to optimally
doped regimes, and slows down in overdoped samples. The reduction of z_A with
increasing temperature resembles that of the c-axis superfluid density.
