BARI-TH/618-10 arXiv:1001.1089v1 [hep-ph] 7 Jan 2010 Open charm meson spectroscopy: Where to place the latest piece of the puzzle Pietro Colangelo and Fulvia De Fazio Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy We discuss how to classify the c ¯ s meson D$_{sJ}$ (3040) recently discovered by the BaBar Collabora- tion. We consider four possible assignments, together with signatures useful to distinguish among them. PACS numbers: 13.25.Ft, 12.39.Fe, 12.39.Hg cludes the assignment J P = 0 + for D$_{sJ}$ (2860). Other information comes from the measurement of the ratios In the infinite heavy quark mass limit, heavy-light Q ¯ q mesons can be conveniently classified in doublets labeled by the value of the total angular momentum ⃗s$_{ℓ}$ = ⃗s$_{¯}$ q + ⃗ ℓ of the light degrees of freedom (light antiquark and glu- ons) with respect to the heavy quark Q [1]. ⃗s$_{¯}$ q is the spin of ¯ q and ⃗ ℓ the orbital angular momentum. For ℓ = 0 ( s -wave states in the constituent quark model), the doublet has s$_{ℓ}$ = 1 2 and consists of two states (de- noted as ( P, P $^{∗}$), with P a generic heavy meson) having spin-parity J P s$_{ℓ}$ = (0 $^{−}$, 1 $^{−}$)$_{1}$$_{/}$$_{2}$ . p -wave states, with ℓ = 1, form two doublets: ( P ∗ $_{0}$, P ′ $_{1}$) with J P s$_{ℓ}$ = (0 $^{+}$, 1 $^{+}$)$_{1}$$_{/}$$_{2}$ , and ( P$_{1}$, P ∗ $_{2}$) with J P s$_{ℓ}$ = (1 $^{+}$, 2 $^{+}$)$_{3}$$_{/}$$_{2}$ . For ℓ = 2, d -wave states give rise to other two doublets: ( P ∗ $_{1}$, P$_{2}$ ) with J P s$_{ℓ}$ = (1 $^{−}$, 2 $^{−}$)$_{3}$$_{/}$$_{2}$ , and ( P ′∗ 2 , P$_{3}$ ) with J P s$_{ℓ}$ = (2 $^{−}$, 3 $^{−}$)$_{5}$$_{/}$$_{2}$ . where D ( ∗ $^{)}$K is the sum over the final states D ( ∗ $^{)0}$K+ and D ( ∗ ) $^{+}$K 0 S [8]. Comparing these data with the pre- dictions in [9], it seems very likely that D$_{sJ}$ (2710) is the first radial excitation of D ∗ $_{s}$(2112), while the case of D$_{sJ}$ (2860) requires further study since the measured ratio in (1) is larger than the theoretical prediction The construction proceeds further, and has been suc- cessfully applied to classify beauty mesons, as well as mesons with a charmed quark, albeit in the latter case the finite heavy quark mass corrections can be important. In the same analysis, the BaBar Collaboration ob- served another broad structure in the D $^{∗}$K distribution, D$_{sJ}$ (3040), with [8] In the case of c ¯ s mesons, both the lowest lying states, both the radial excitations (whose members of the var- ious doublets we denote by a tilde) can be described using this classification scheme. Among the known c ¯ s hadrons, the two mesons D$_{s}$ (1969) and D ∗ $_{s}$(2112) fill the doublet with J P s$_{ℓ}$ = (0 $^{−}$, 1 $^{−}$)$_{1}$$_{/}$$_{2}$ , while one can identify ( D ∗ s $_{0}$(2317) , D ′ s $_{1}$(2460)) and ( D$_{s}$$_{1}$ (2536) , D ∗ s $_{2}$(2573)) with the doublets J P s$_{ℓ}$ = (0 $^{+}$, 1 $^{+}$)$_{1}$$_{/}$$_{2}$ and J P s$_{ℓ}$ = (1 $^{+}$, 2 $^{+}$)$_{3}$$_{/}$$_{2}$ respectively (the discovery of D ∗ s $_{0}$(2317) and D ′ s $_{1}$(2460) is rather recent [2], and their classification has been the subject of discussions [3]). Studies of angular distributions for this state have not been attempted, due to the limited statistics. Here we discuss possible classifications for this new c ¯ s meson using the only information available, the measured mass, width and decay mode, together with the full set of information concerning the other levels/doublets. More recently, other candidates have been added to this list: D$_{sJ}$ (2860), discovered by the BaBar Collabo- ration [4], and D$_{sJ}$ (2710), discovered by the Belle and BaBar Collaborations [4, 5], both observed in the DK final state. In the case of D$_{sJ}$ (2710), it has been possi- ble to determine the spin-parity: J P = 1 $^{−}$, studying its production in B decays. As for D$_{sJ}$ (2860), the assign- ments J P = 3 $^{−}$, with radial quantum number n = 1 [6], and J P = 0 $^{+}$, with n = 2 (i.e. the first radial excita- tion of D$_{sJ}$ (2317)) [7] have been proposed soon after the discovery. Since D$_{sJ}$ (3040) decays to D $^{∗}$K and not to DK , it has unnatural parity, with possible J P = 1 $^{+}$, 2 $^{−}$, 3 $^{+}$, · · · . The lightest not yet observed states with such quantum numbers are the the two J P = 2 − states belonging to the ℓ = 2 doublets, with s$_{ℓ}$ = 3 2 and s$_{ℓ}$ = 5 2 : we denote them as D$_{s}$$_{2}$ and D ′∗ s $_{2}$, respectively (the case J P = 3 + corre- sponds to a doublet with s$_{ℓ}$ = 7 2 , which is expected to be heavier). In the case of radial excitations, the identifica- tion with the states with n = 2, J P = 1 $^{+}$, and s$_{ℓ}$ = 1 2 (the meson ˜ D ′ s $_{1}$) or s$_{ℓ}$ = 3 2 (the meson ˜ D$_{s}$$_{1}$ ) is possible. These four assignments are indicated in Table I which reports the classification of all the other known c ¯ s mesons The latest piece of information comes from the BaBar Collaboration [8], thanks to a new analysis of DK and D $^{∗}$K final states. D$_{sJ}$ (2710) and D$_{sJ}$ (2860) are seen de- caying to both DK and D $^{∗}$K , hence they have natural parity J P = 1 $^{−}$, 2 $^{+}$, 3 $^{−}$, · · · . The decay into D $^{∗}$K ex- BR ( D$_{sJ}$ (2860) → D $^{∗}$K ) BR ( D$_{sJ}$ (2860) DK ) ≃ 0 . 39 [10]. M ( D$_{sJ}$ (3040)) = 3044 ± 8$_{stat}$( +30 − 5 )$_{syst}$ MeV Γ( D$_{sJ}$ (3040)) = 239 35$_{stat}$( +46 − $_{42}$)$_{syst}$ MeV . (2) BR ( D$_{sJ}$ (2710) → D $^{∗}$K ) BR ( D$_{sJ}$ (2710) DK ) = 0 . 91 ± 0 . 13$_{stat}$ ± 0 . 12$_{syst}$ BR ( D$_{sJ}$ (2860) → D $^{∗}$K ) BR ( D$_{sJ}$ (2860) DK ) = 1 . 10 ± 0 . 15$_{stat}$ ± 0 . 19$_{syst}$ (1)