3 effective Lagrangian terms analogous to the one followed above. In the case of light vector mesons in the final state, the method has been developed in Ref.[13] on the basis on the hidden gauge symmetry idea [14]. We refer to the review [15] for a detailed description; here we only mention that in this approach the octet of light vector mesons is described by the field ρ$_{µ}$ = i g$_{V}$ √$_{2}$ ˆ ρ$_{µ}$ , where ˆ ρ$_{µ}$ is a hermitian 3 × 3 matrix analogous to (5), containing the light vector meson fields ρ ± , $^{0}$, K $^{∗±}$, K ∗ $^{0}$, ¯ K ∗ $^{0}$, ω$_{8}$ . The constant g$_{V}$ can be fixed to the value g$_{V}$ = 5 . 8 by the Kawarabayashi-Sukuzi, Riazuddin-Fayyazuddin rela- tions [16]. In Ref.[15] the effective Lagrangian terms describing the transitions S → HV and T → HV are reported, S and H representing the heavy doublets S$_{a}$ and H$_{a}$ in Eq. (4) and V denoting a generic light vector meson. They read: where A µba = i 2 ( ξ $^{†}$∂$_{µ}$ ξ − ξ∂$_{µ}$ξ $^{†}$) $_{ba}$. Λ$_{χ}$ is the chiral symmetry-breaking scale: we use Λ$_{χ}$ = 1 GeV. L S and L T describe transitions of positive parity heavy mesons with the emission of light pseudoscalar mesons in s − and d − wave, respectively, and g, h and h ′ are the effective coupling constants. On the other hand, L X and L X ′ de- scribe the transitions of higher mass mesons of negative parity with the emission of light pseudoscalar mesons in p − and f − wave, with coupling constants k $^{′}$, k$_{1}$ and k$_{2}$ . At the same order in the expansion in the light me- son momentum, the structure of the Lagrangian terms for radial excitations of the H , S and T doublets does not change, being only dictated by the spin-flavour and chiral symmetries, but the coupling constants g, h and h ′ must be replaced by ˜ g, ˜ h and ˜ h $^{′}$. The advantage of this formulation is that meson transitions into final states ob- tained by SU (3) and heavy quark spin rotations can be related in a straightforward way. With the effective Lagrangians in Eq.(7) we can evalu- ate the strong decays of D$_{sJ}$ (3040) to a charmed meson and a light pseudoscalar one in correspondence to the four classifications in Table I. In particular, the ratio L SHV = iζ$_{S}$Tr [ ¯ S$_{a}$H$_{b}$γ$_{µ}$ ( V µ − ρ $^{µ}$)$_{ba}$ ] + iµ$_{S}$Tr ¯ S$_{a}$H$_{b}$σ $^{λν}$F$_{λν}$ ( ρ )$_{ba}$ ] L T HV = iζ$_{T}$ Tr [ ¯ H$_{a}$T µ b ( V µ − ρ$_{µ}$ )$_{ba}$ (9) + iµ$_{T}$ Tr ¯ H$_{a}$T µ b v $^{ν}$F$_{µν}$ ( ρ )$_{ba}$ ] R$_{1}$ = Γ( D$_{sJ}$ (3040) → D ∗ $_{s}$η ) Γ( D$_{sJ}$ (3040) D $^{∗}$K ) (7) R$_{1}$ ( ˜ D ′ s $_{1}$) = 0 . 34 R$_{1}$ ( ˜ D$_{s}$$_{1}$ ) = 0 . 20 R$_{1}$ ( D$_{s}$$_{2}$ ) = 0 . 245 (8) R$_{1}$ ( D ∗′ s $_{2}$) = 0 . 143 . ( D ( ∗ $^{)}$K = D ( ∗ $^{)0}$K+ + D ( ∗ ) $^{+}$K 0 $_{S}$) can be computed, with results for the various assignments: where V µba = i 2 ( ξ $^{†}$∂$_{µ}$ ξ + ξ∂$_{µ}$ξ $^{†}$) $_{ba}$, and ζ$_{S,T}$ , µ$_{S,T}$ are effective coupling constants, and the field strength tensor F$_{µν}$ is defined as F$_{µν}$ ( ρ ) = ∂$_{µ}$ρ$_{ν}$ − ∂$_{ν}$ρ$_{µ}$ + [ ρ$_{µ}$, ρ$_{ν}$ ]. The calculation of all possible decay modes for each one of the four classifications of D$_{sJ}$ (3040) would require the values of several coupling constants, which are unknown. Nevertheless, some conclusions can be drawn. One can estimate the widths of the decays into DK ∗ and D$_{s}$φ for the two J P = 1 + states by the effective Lagrangians in (10) using the values of ζ and µ : ζ$_{S}$ = 0 . 10 and µ$_{S}$ = − 0 . 10 GeV − 1 computed for the analogous transitions involving the heavy mesons belonging to the lowest-lying doublet [15]. This provides us with hints about the contribution of such modes to the total width of D$_{sJ}$ (3040) in the case it is either ˜ D ′ s 1 or ˜ D$_{s}$$_{1}$ (in the case of D$_{s}$$_{2}$ and D ∗′ s 2 these decays proceed in p -wave and hence they are expected to contribute less significantly): It is important to notice that the dependence on the ef- fective couplings cancels in the ratio. The spread among the various predictions permits to discriminate among the assignments, in particular between ˜ D ′ s 1 and D ∗′ s $_{2}$. The mass of D$_{sJ}$ (3040) is large enough to allow other decay modes. Decays to the members of the doublets with s P ℓ = 1 2 + or s P ℓ = 3 2 + and a pseudoscalar meson are possible: these are the modes ( D ∗ $_{0}$, D ′ $_{1}$) K and ( D$_{1}$, D ∗ $_{2}$) K . As for channels with the η in the final state, the only kine- matically allowed mode is D ∗ s $_{0}$η (for D ′ s $_{1}$η the available phase space is tiny). The features of such decay modes are different for the four considered assignments. At the leading order in 1 m$_{c}$ expansion, the states ˜ D ′ s 1 and ˜ D$_{s}$$_{1}$ can decay to D ∗ $_{0}$K , D ∗ s $_{0}$η , D ′ $_{1}$K , D$_{1}$K and D ∗ $_{2}$K in p − wave; D$_{s}$$_{2}$ and D ∗′ s 2 both decay to D ∗ $_{0}$K , D ∗ s $_{0}$η , D ′ $_{1}$K in d − wave; D ∗′ s 2 can decay to D$_{1}$K and D ∗ $_{2}$K in d − wave, while in the case of D$_{s}$$_{2}$ the decay to D ∗ $_{2}$K proceeds in s − wave and the decay into D$_{1}$K is allowed at O ( 1 m$_{c}$ ) in d − wave. Other kinematically allowed modes of D$_{sJ}$ (3040) are those with the emission of a light vector meson, specif- ically decays into DK ∗ or D$_{s}$φ [19]. It is possible to describe such decay modes using an approach based on It can be noticed that the contribution of these modes to the full width should be sizable only in the case of ˜ D ′ s $_{1}$; therefore, this assignment can be disentangled through the experimental analysis of the DK ∗ and D$_{s}$φ decay channels. After having discussed four possible classifications for D$_{sJ}$ (3040), it is interesting to consider some features of its spin partner in the various cases [20]. Γ( ˜ D ′ s 1 → DK $^{∗}$) ≃ 95 MeV Γ( ˜ D ′ s 1 D$_{s}$φ ) 44 MeV (10) Γ( ˜ D$_{s}$$_{1}$ → DK $^{∗}$) ≃ 15 MeV Γ( ˜ D$_{s}$$_{1}$ D$_{s}$φ ) 6 MeV .