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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 

( D ( ∗ $^{)}$K = D ( ∗ $^{)0}$K+ + D ( ∗ ) $^{+}$K 0 $_{S}$) can be computed, with results for the various assignments: 

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 

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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 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 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]. 

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