Motivated by the experimental observation that superconductivity in bulk doped SrTiO3 is enhanced as a putative ferroelectric quantum critical point (FE-QCP) is approached, we study the pairing instability of a cubic system in which electrons exchange low-energy ferroelectric fluctuations. Instead of the gradient coupling to the lattice distortion associated with ferroelectricity, we consider direct coupling between the electrons and the bosonic ferroelectric field that appears in the presence of spin-orbit coupling. Working in the weak-coupling regime, we find that the pairing interaction is dominated by the soft transverse optical (TO) mode, resulting in a Tc enhancement upon approaching the FE-QCP. Focusing on even-parity states, we find that although the s-wave state always wins, states with higher Cooper-pair angular momentum become close competitors as the TO mode softens. We show that the cubic anisotropy of the FE fluctuations mixes the s-wave and g-wave states, resulting in a characteristic anisotropy of the gap function. The gap anisotropy behaves nonmonotonically as the FE-QCP is approached: upon decreasing the TO mode frequency, the gap anisotropy first changes sign and then increases in magnitude. We discuss the possible applications of our results to the superconducting state of SrTiO3.
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We thank A. Balatsky, K. Behnia, A. Chubukov, A. Klein, V. Kozii, G. Lonzarich, J. Ruhman, and P. Woelfle for insightful discussions. M.N.G. and R.M.F. were supported by the U.S. Department of Energy through the University of Minnesota Center for Quantum Materials, under Award No. DE-SC-0016371. T.V.T. was supported by São Paulo Research Foundation (Fapesp, Brazil) via Fellowship No. 2015/21349-7.