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In our paper (Wolfle and Balatsky, Phys. Rev. B 98, 104505 (2018)) we presented a microscopic theory of superconductivity for doped SrTiO$_{3}$ by proposing two pairing mechanisms acting simultaneously with relative strength depending on the closeness to the ferroelectric quantum critical point. The first mechanism rests on the dynamically screened Coulomb interaction, and the second assumed a coupling to the soft transverse optical phonon. In their comment Ruhman and Lee point out an error in our estimate of the deformation potential coupling to the soft mode. We agree that this type of coupling cannot explain the gigantic isotope effect observed experimentally, so that a different coupling mechanism needs to be found. As for the first pairing mechanism, Ruhman and Lee maintain the view expressed in their paper (Ruhman and Lee, Phys. Rev. B 94, 224515 (2016)) that the energy range over which the usual longitudinal optical phonon mediated interaction operates is limited by the Fermi energy. We object to this view and in this reply present evidence that the cutoff energy is much larger. In a weak coupling system such as SrTiO$_{3}$ the cutoff is given by the energy beyond which quasiparticles cease to be well defined.
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Recent experiments on electron- or hole-doped SrTiO$_{3}$ have revealed a hitherto unknown form of superconductivity, where the Fermi energy of the paired electrons is much lower than the energies of the bosonic excitations thought to be responsible for the attractive interaction. We show that this situation requires a fresh look at the problem calling for (i) a systematic modeling of the dynamical screening of the Coulomb interaction by ionic and electronic charges, (ii) a transverse optical phonon mediated pair interaction and (iii) a determination of the energy range over which the pairing takes place. We argue that the latter is essentially given by the limiting energy beyond which quasiparticles cease to be well defined. The model allows to find the transition temperature as a function of both, the doping concentration and the dielectric properties of the host system, in good agreement with experimental data. The additional interaction mediated by the transverse optical soft phonon is shown to be essential in explaining the observed anomalous isotope effect. The model allows to capture the effect of the incipient (or real) ferroelectric phase in pure, or oxygen isotope substituted SrTiO$_{3}$ .
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