Exploring the vacuum structure of gravitationally induced neutrino masses

In this work, we explore the proposed mechanism in which the gravitational $\theta$ anomaly generates neutrino masses. We highlight that the leading renormalizable interactions of the neutrino condensate forbid the possibility of generating hierarchical masses consistent with observation. This conclusion still holds when Standard Model loop corrections are accounted for. We show that higher-dimensional operators can alleviate this problem. The higher-dimensional operators could be generated from the gravitational anomaly itself, but there is no clear way to know without a deeper understanding of the low-energy description of this mechanism. Because of that, we explore the possibility of new particles generating neutrino mass splittings. We show that both new particles that alter the scalar potential of the condensate or new particles in loops for the neutrino self-energy can solve this problem.