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We derive sufficient conditions for theories consisting of multiple vectorfields, which could also couple to external fields, to be multi-fieldgeneralised Proca theories. The conditions are derived by demanding that thetheories have the required structure of constraints, giving the correct numberof degrees of freedom. The Faddeev-Jackiw constraint analysis is used and iscross-checked by Lagrangian constraint analysis. To ensure the theory isconstraint, we impose a standard special form of Hessian matrix. The derivationbenefits from the realisation that the theories are diffeomorphism invariance(or, in the case of flat spacetime, invariant under Lorentz isometry). Thesufficient conditions obtained include a refinement of secondary-constraintenforcing relations derived previously in literature, as well as a conditionwhich ensures that the iteration process of constraint analysis terminates.Some examples of theories are analysed to show whether they satisfy thesufficient conditions. Most notably, due to the obtained refinement on some ofthe conditions, some theories which are previously interpreted as beingundesirable are in fact legitimate, and vice versa. This in turn affects theprevious interpretations of cosmological implications which should later bereinvestigated.

On sufficient conditions for degrees of freedom counting of multi-field generalised Proca theories