Novel Isotope Effects on the Pairing Pseudogap in High-$T_c$ Cuprates: Evidences for Polaronic Metal and Precursor BCS-Like Pairing of Large Polarons

We have studied the novel isotope effects on the pairing pseudogap in underdoped and optimally doped cuprates within the large-polaron model and two non-standard BCS-like approaches. We have shown that in the intermediate-coupling regime the precursor pairing of large polarons occurs at a mean-field temperature $T^{\ast }>T_c$ and the near-absent, sizable and very large oxygen and copper isotope effects on $T^{\ast }$ exist in cuprates with small and large Fermi surfaces. Our results for $T^{\ast }$, isotope shifts and exponents in slightly underdoped and optimally doped cuprates are in quantitative agreement with existing experiments and explain the discrepancy between various experiments.