arXiv2026
Superconductivity in doped SrTiO$_3$ occurs over a wide range of carrier densities, including those for which the Fermi energy is below the polar longitudinal optical phonon scale. In this regime, the assumptions underpinning conventional implementations of Migdal-Eliashberg theory, including frequency cutoffs at the phonon scale and a Coulomb pseudopotential $μ^\ast$, are not valid. We solve the finite-temperature $GW$ equations with full momentum and frequency dependence, without cutoffs or $μ^\ast$, for polar one-band models of SrTiO$_3$, using effective masses and three-phonon dielectric functions parameterized from ab initio calculations. Comparing different self-consistency levels, namely $G_0W_0$, $GW_0$, and fully self-consistent $GW$, we find that the one-shot ($G_0W_0$) kernel overestimates the pairing-onset temperature by one to two orders of magnitude. The dominant suppression comes from replacing $G_0$ by $G$, thereby incorporating the phonon renormalization factor in the electron Green function. Using the self-consistently computed interaction $W$ further lowers and narrows the pairing-onset dome. In the dilute limit, our calculations identify the pairing channel as the Fröhlich phonon interaction screened by the incipient ferroelectricity of the material, with plasmonic and electronic screening effects negligible. The numerical solution of the full equations reveals a pairing-onset scale that remains non-zero as the density tends to zero, whereas Fermi-surface projection or Fermi-energy frequency truncation removes it. This work highlights the relevance of incipient ferroelectricity, the importance of self-consistency, and the need for a full momentum- and frequency-dependent treatment in modeling superconductivity in SrTiO$_3$-like doped polar semiconductors.