5 state seems less probable, this assignment can be dis- carded/confirmed studying the D ∗ $_{2}$K s − wave final state. Search of the spin partner in each doublet would provide further information, enriching the c ¯ s spectrum recently disclosed by experiments. We thank A. Palano for discussions. This work was sup- ported in part by the EU contract No. MRTN-CT-2006- 035482, ”FLAVIAnet”. Acknowledgments tion: Γ( D$_{sJ}$ (3040) → DK ∗ + D$_{s}$φ ) Γ( D$_{sJ}$ (3040) D $^{∗}$K + D ∗ $_{s}$η ) ≃ 0 . 79. In our dis- cussion we have examined other information that can be exploited. To summarize the results of our considerations and compare the distinctive features of the four proposed as- signments for D$_{sJ}$ (3040) (illustrated in Fig.1 in which all the known c ¯ s states have been included), we collect our findings and observations in Table II. In particular, we emphasize the role of the final states DK ∗ and D$_{s}$φ , which deserve an experimental investigation. Moreover, although the identification with a J P = 2 − s P ℓ = 3 2$^{−}$ [1] For reviews see: M. Neubert, Phys. Rept. 245 , 259 (1994); F. De Fazio, in At the Frontier of Par- ticle Physics/Handbook of QCD , ed. by M. Shif- man (World Scientific, Singapore, 2001), page 1671, arXiv:hep-ph/0010007. [2] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 90 , 242001 (2003); Y. Mikami et al. [Belle Collab- oration], Phys. Rev. Lett. 92 , 012002 (2004); D. Besson et al. [CLEO Collaboration], Phys. Rev. D 68 , (2003) 032002; L. Benussi [FOCUS Collaboration], Int. J. Mod. Phys. A 20 , 549 (2005). [3] For reviews see: P. Colangelo, F. De Fazio and R. Ferran- des, Mod. Phys. Lett. A 19 , 2083 (2004); E. S. Swanson, Phys. Rept. 429 , 243 (2006); P. Colangelo, F. De Fazio, R. Ferrandes and S. Nicotri, Prog. Theor. Phys. Suppl. 168 , 202 (2007). [4] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 97 , 222001 (2006). [5] J. Brodzicka et al. [Belle Collaboration], Phys. Rev. Lett. 100 , 092001 (2008). [6] P. Colangelo, F. De Fazio and S. Nicotri, Phys. Lett. B 642 , 48 (2006). [7] E. van Beveren and G. Rupp, Phys. Rev. Lett. 97 , 202001 (2006); F. E. Close, C. E. Thomas, O. Lakhina and E. S. Swanson, Phys. Lett. B 647 , 159 (2007). [8] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 80 , 092003 (2009). [9] P. Colangelo, F. De Fazio, S. Nicotri and M. Rizzi, Phys. Rev. D 77 , 014012 (2008). [10] Among the various possibilities, one has to investigate the size of 1 /m$_{c}$ corrections to this ratio, waiting for a higher statistics confirmation of the measurement. As an alternative, the possibility that D$_{sJ}$ (2860) is a mixing of J P = 0 + and J P = 2 + components has been suggested by E. van Beveren and G. Rupp, arXiv:0908.1142 [hep- ph]. [11] M. Di Pierro and E. Eichten, Phys. Rev. D 64 , 114004 (2001). [12] M.B. Wise, Phys. Rev. D 45 , R2188 (1992); G. Burdman and J.F. Donoghue, Phys. Lett. B 280 , 287 (1992); P. Cho, Phys. Lett. B 285, 145 (1992); H.-Y. Cheng et al., Phys. Rev. D 46 , 1148 (1992). [13] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B 292 , 371 (1992). [14] M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B 259 , 493 (1985); Phys. Rept. 164 , 217 (1988). [15] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281 , 145 (1997). [16] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16 , 255 (1966); Riazuddin and Fayyazuddin, Phys. Rev. 147 , 1071 (1966). [17] B. Chen, D. X. Wang and A. Zhang, Phys. Rev. D 80 , 071502 (2009). [18] Z. F. Sun and X. Liu, Phys. Rev. D 80 , 074037 (2009). [19] The decay into D $^{∗}$K ∗ is severely phase space suppressed. [20] The allowed decay modes are listed assigning the mass M = 3040 MeV to the spin partner. For a slightly differ- ent mass, some channels could no more be allowed. For example, some of the modes listed here for the J P = 3 − state have not been considered in [6], since such a state has a lower mass, M = 2860 MeV.