Unimodular proca theory: breaking the U(1) gauge symmetry of unimodular gravity via a mass term

We study the Hamiltonian structure of unimodular-like theories, where the cosmological constant (or other supposed constants of nature) are demoted from fixed parameters to classical constants of motion. No new local degrees of freedom are present as a result of a U(1) gauge invariance of the theory. Hamiltonian analysis of the action reveals that the only possible gauge fixing that can be enforced is setting the spatial components of the four-volume time vector T-i approximate to 0. As a consequence of this, the gauge-fixed unimodular path integral is equivalent to the minisuperspace unimodular path integral. However, should we break the U(1) gauge invariance, two things happen: a mass-less propagating degree of freedom appears, and the (gauge-invariant) zero-mode receives modified dynamics. The implications are investigated, with the phenomenology depending crucially on the target "constant".