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We prove that vector fields described by the generalized Proca class of theories do not admit consistent coupling with a gravitational sector defined by a scalar–tensor theory of the degenerate type. Under the assumption that there exists a frame in which the Proca field interacts with gravity only through the metric tensor, our analysis shows that at least one of the constraints associated with the degeneracy of the scalar–tensor sector is inevitably lost whenever the vector theory includes coupling with the Christoffel connection.

We derive conditions which are sufficient for theories consisting of multiple vector fields, which could also couple to non-dynamical external fields, to have the required structure of constraints of multi-field generalised Proca theories, so that the number of degrees of freedom is correct. The Faddeev–Jackiw constraint analysis is used and is cross-checked by Lagrangian constraint analysis. To ensure the theory is constraint, we impose a standard special form of Hessian matrix. The derivation benefits from the realisation that the theories are diffeomorphism invariance. The sufficient conditions obtained include a refinement of secondary-constraint enforcing relations derived previously in literature, as well as a condition which ensures that the iteration process of constraint analysis terminates. Some examples of theories are analysed to show whether they satisfy the sufficient conditions. Most notably, due to the obtained refinement on some of the conditions, some theories which are previously interpreted as being undesirable are in fact legitimate, and vice versa. This in turn affects the previous interpretations of cosmological implications which should later be reinvestigated.

In this paper the Proca field equations for a massive gauge particle are obtained in the presence of a natural momentum cutoff “ p max ” based on a covariant generalization of a one-parameter extension of the Heisenberg algebra. The Yukawa potential for a static point source in the presence of p max (generalized Yukawa potential) is obtained analytically and it is shown that in contrast with the Yukawa potential for a static point source in Proca electrodynamics, the generalized Yukawa potential has a finite value at the location of the static point source. Our calculations demonstrate that the Coulomb potential, the Yukawa potential, and the Coulomb potential in the presence of p max can be derived from the generalized Yukawa poitential. We show that the free space solutions of Proca electrodynamics in the presence of p max describe a massive gauge particle with the effective mass m eff = m 1 - mc p max 2 , where m is the rest mass of the ordinary Proca particle. Numerical estimations in Sect. 5 , show that the lower bound for p max must take the value p max min = ( 91.187 ± 0.007 ) GeV c in order to avoid complex values for the effective mass m eff . This lower bound for p max is near to the momentum scale of the electroweak interactions. It should be mentioned that for the very large values of p max the results of this work reduce to the well-known results of standard Proca electrodynamics.