Four-fermion condensates in curved spacetimes: A functional approach

Four-fermion interactions appear in effective descriptions of particle physics, many-body systems, and gravitational theories with fermions. Although the same local operator may enter perturbative scattering, a vacuum Nambu-Jona-Lasinio (NJL) instability, or finite-density Bardeen-Cooper-Schrieffer (BCS) pairing, these regimes are distinguished by their interaction channels, quadratic kernels, and quantum states. We give a pedagogical functional account of these distinctions and compute the local one-loop contribution of a scalar-channel NJL mean field to the energy-momentum tensor in curved spacetime. We first display the state data in the in-out functional and construct the closed-time-path functional required for an in-in expectation value. For the NJL saddle, a Hubbard-Stratonovich field shifts the fermion mass, and the parity-even Dirac determinant generates local volume, curvature, and curvature-squared operators. We find the covariant quantum effective action before specializing to a spatially flat FLRW background and derive the corresponding energy density and pressure. The constant-condensate limit agrees with the direct flat-space mean-field calculation. We also explain which additional state and channel data are required for finite-density BCS pairing and comment on renormalization conditions in curved spacetimes.