Gap structure and mean-field $T_c$ in HTS cuprates

We show that phase and amplitude fluctuations set in simultaneously in high-Tc cuprates and determine the mean-field transition temperature $T_c^{mf}$ which is found to increase substantially above $T_c$ in optimal and underdoped cuprates (by up to 60 or 70K). We find $\Delta/k_BT_c^{mf}$=2.5, little more than the weak-coupling BCS $d$-wave value. On the other hand the pseudogap $T^*$ has a distinct doping dependence from $T_c^{mf}$ and correlates with the pseudogap energy $E_g$. The gap structure for $\Delta$ and $E_g$ are characterised and shown to be distinct.