  We derive sufficient conditions for theories consisting of multiple vector
fields, which could also couple to external fields, to be multi-field
generalised Proca theories. The conditions are derived by demanding that the
theories have the required structure of constraints, giving the correct number
of degrees of freedom. 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
(or, in the case of flat spacetime, invariant under Lorentz isometry). 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.
