Realizing the phantom-divide crossing with vector and scalar fields

In generalized Proca theories, characterized by a vector field with broken $U(1)$ gauge invariance, late-time cosmic acceleration can be realized with a dark energy equation of state in the regime $w_{\rm DE} < -1$. In such scenarios, however, a phantom-divide crossing, as recently suggested by DESI observations, is not achieved without encountering theoretical inconsistencies. We incorporate a canonical scalar field with a potential, in addition to the vector field, and show that the phantom-divide crossing from $w_{\rm DE} < -1$ to $w_{\rm DE} > -1$ can occur at low redshifts. We propose a minimal model that admits such a transition and identify the region of parameter space in which all dynamical degrees of freedom in the scalar, vector, and tensor sectors are free from ghost and Laplacian instabilities. We further investigate the evolution of linear cosmological perturbations by applying the quasi-static approximation to modes well inside the Hubble radius. The dimensionless quantities $\mu$ and $\Sigma$, which characterize the growth of matter perturbations and the bending of light rays, respectively, depend on the sound speed $c_\psi$ of the longitudinal scalar perturbation associated with the vector field. Since $c_\psi$ is influenced by the transverse vector mode, the model exhibits sufficient flexibility to yield values of $\mu$ and $\Sigma$ close to 1. Consequently, unlike theories such as scalar Galileons, the present model can be consistent with observations of redshift-space distortions and integrated Sachs-Wolfe-galaxy cross-correlations.