Understanding $D_{sJ}(2317)$

We analyze the hadronic and radiative decay modes of the recently observed $D_{sJ}(2317)$ meson, in the hypothesis that it can be identifield with the scalar $s_\ell^P={1\over 2}^+$ state of $c \bar s$ spectrum ($D_{s0}$). The method is based on heavy quark symmetries and Vector Meson Dominance ansatz. We find that the hadronic isospin violating mode $D_{s0} \to D_s \pi^0$ is enhanced with respect to the radiative mode $D_{s0} \to D_s^* \gamma$. The estimated width of the meson is $\Gamma(D_{s0})\simeq 7$ KeV.


Introduction
The BaBar Collaboration has reported the observation of a narrow peak in the D + s π 0 invariant mass distribution, corresponding to a state of mass 2.317 GeV [1]. The observed width is consistent with the resolution of the detector, thus Γ ≤ 10 MeV. In the same analysis no significant signals are found in the D s γ and D s γγ mass distributions. The meson has been denoted as D sJ (2317); the announcement has immediately prompted different interpretations [2,3].
A possible quantum number assignment to D sJ (2317) is J P = 0 + , as suggested by the angular distribution of the meson decay with respect to its direction in the e + − e − center of mass frame. This assignment can identify the meson with the D s0 state in the spectrum of the cs system. Considering the masses of the other observed states belonging to the same system, D s1 (2536) and D sJ (2573), the mass of the scalar D s0 meson was expected in the range 2.45−2.5 GeV, therefore ∼ 150 MeV higher than the observed 2.317 GeV. A D s0 meson with such a large mass would be above the threshold M DK = 2.359 GeV to strongly decay by S-wave Kaon emission to DK, with a consequent broad width. For a mass below the DK threshold the meson has to decay by different modes, namely the isospin-breaking D s π 0 mode observed by BaBar, or radiatively. The J P = 0 + assignment excludes the final state D s γ, due to angular momentum and parity conservation; indeed such a final state has not been observed. On the other hand, for a scalar cs meson the decay D s0 → D * s γ is allowed. However, no evidence is reported yet of the D s γγ final state resulting from the decay chain D s0 → D * s γ → D s γγ. In order to confirm the identification of D sJ (2317) with the scalar D s0 , one has at first to understand whether the decay modes of a scalar particle with mass of 2317 GeV can be predicted in agreement with the experimental findings presently available. In particular, the isospin violating decay to D s π 0 should proceed at a rate larger than the radiative mode D s0 → D * s γ, though not exceeding the experimental upper bound Γ ≤ 10 MeV. This letter is devoted to such an issue.
In order to analyze the isospin violating transition D s0 → D s π 0 one can use a formalism that accounts for the heavy quark spin-flavour symmetries in hadrons containing a single heavy quark, and the chiral symmetry in the interaction with the octet of light pseudoscalar states.
In the heavy quark limit, the heavy quark spin s Q and the light degrees of freedom total angular momentum s ℓ are separately conserved. This allows to classify hadrons with a single heavy quark Q in terms of s ℓ by collecting them in doublets the members of which only differ for the relative orientation of s Q and s ℓ .
The doublets with J P = (0 − , 1 − ) and J P = (0 + , 1 + ) (corresponding to s P ℓ = 1 2 − and s P ℓ = 1 2 + , respectively) can be described by the effective fields where v is the four-velocity of the meson and a is a light quark flavour index. In particular in the charm sector the components of the field H a are P ( * ) a = D ( * )0 , D ( * )+ and D ( * ) s (for a = 1, 2, 3); analogously, the components of S a are P 0a = D 0 0 , D + 0 , D s0 and P ′ 1a = D ′0 1 , D ′+ 1 , D ′ s1 . In terms of these fields it is possible to build up an effective Lagrange density describing the low energy interactions of heavy mesons with the pseudo Goldstone π, K and η bosons [4,5,6,7]: In (3) H a and S a are defined as H a = γ 0 H † a γ 0 and S a = γ 0 S † a γ 0 ; all the heavy field operators contain a factor √ M P and have dimension 3/2. The parameter ∆ represents the mass splitting between positive and negative parity states. The π, K and η pseudo Goldstone bosons are included in the effective lagrangian (3) through the field ξ = e iM f that represents a unitary matrix describing the pseudoscalar octet, with and f ≃ f π . In eq.(3) Σ = ξ 2 , while the operators D and A are given by: The strong interactions between the heavy H a and S a mesons with the light pseudoscalar mesons are thus governed, in the heavy quark limit, by three dimensionless couplings: g, h and g ′ . In particular, h describes the coupling between a member of the H a doublet and one of the S a doublet to a light pseudoscalar meson, and is the one relevant for our discussion. Isospin violation enters in the low energy Lagrangian of π, K and η mesons through the mass term with m q the light quark mass matrix: In addition to the light meson mass terms, L mass contains an interaction term between π 0 (I = 1) and η (I = 0) mesons: L mixing =μ 2 (7)  ) accounting for isospin violation, so that the width Γ(D s0 → D s π 0 ) reads: As for h, the result of QCD sum rule analyses of various heavy-light quark current correlators is |h| = 0.6 ± 0.2 [6]. Using the central value, together with the factor (m d − m u )/(m s − m d +mu 2 ) ≃ 1 43.7 [9] and f = f π = 132 MeV we obtain: Eq.(9) can receive SU(3) F corrections: a hint on their size comes from the use of f = f η = 171 MeV instead of f π in (9), which gives Γ(D s0 → D s π 0 ) ≃ 4 KeV . On the other hand, we neglect corrections related to the finite charm quark mass. The analogous calculation for D * s → D s π 0 involves the coupling g in (3). Since h and g have similar sizes (0.3 ≤ g ≤ 0.6), it turns out that the transitions D s0 → D s π 0 is enhanced with respect to D * s → D s π 0 essentially due to kinematics, being 1 Electromagnetic contributions to D s0 → D s π 0 are expected to be suppressed with respect to the strong interaction mechanism considered here.

Radiative D s0 → D * s γ decay
Let us now turn to D s0 → D * s γ, the amplitude of which has the form: where p is the D s0 momentum, ǫ the D * s polarization vector, and k and η the photon momentum and polarization. The corresponding decay rate is: The parameter d gets contributions from the photon couplings to the light quark part e ss γ µ s and to the heavy quark part e cc γ µ c of the electromagnetic current, e s and e c being strange and charm quark charges in units of e. Its general structure is: where Λ a (a = c, s) have dimension of a mass. Such a structure is already known from the constituent quark model. In the case of M1 heavy meson transitions, an analogous structure predicts a relative suppression of the radiative rate of the charged D * mesons with respect to the neutral one [10,11,12,13], suppression that has been experimentally confirmed [14]. From (12,13) one could expect a small width for the transition D s0 → D * s γ, to be compared to the hadronic width D s0 → D s π 0 which is suppressed as well.
In order to determine the amplitude of D s0 → D * s γ we follow a method based again on the use of heavy quark symmetries, together with the vector meson dominance (VMD) ansatz [11,13]. We first consider the coupling of the photon to the heavy quark part of the e.m. current. The matrix element D * s (v ′ , ǫ)|cγ µ c|D s0 (v) (v, v ′ meson four-velocities) can be computed in the heavy quark limit, matching the QCDcγ µ c current onto the corresponding HQET expression [15]: where h v is the effective field of the heavy quark. For transitions involving D s0 and D * s , and for v = v ′ (v · v ′ = 1), the matrix element of J HQET µ vanishes. The consequence is that d (h) is proportional to the inverse heavy quark mass m Q and presents a suppression factor since in the physical case v · v ′ = (m 2 D s0 + m 2 D * s )/2m D s0 m D * s = 1.004. Therefore, we neglect d (h) in (13).
To evaluate the coupling of the photon to the light quark part of the electromagnetic current we invoke the VMD ansatz and consider the contribution of the intermediate φ(1020): describes the strong interaction of a light vector meson (φ) with two heavy mesons (D * s and D s0 ). It can also be obtained through a low energy lagrangian in which the heavy fields H a and S a are coupled, this time, to the octet of light vector mesons. 2 The Lagrange density is set up using the hidden gauge symmetry method [5], with the light vector mesons collected in a 3 × 3 matrixρ µ analogous to M in (4). The lagrangian 3 reads as [16]: λ , g V being fixed to g V = 5.8 by the KSRF rule [17]. The couplingμ in (16) is constrained toμ = −0.13 ± 0.05 GeV −1 by the analysis of the D → K * semileptonic transitions induced by the axial weak current [16].
The resulting expression for 1 Λs is: The parameters are obtained from independent channels; we use their central values.
The numerical result for the radiative width: shows that the hadronic D s0 → D * s π 0 transition is more probable than the radiative mode D s0 → D * s γ. In particular, if we assume that the two modes essentially saturate the D s0 width, we have and at odds with the case of the D * s meson, where the radiative mode dominates the decay rate.
The same conclusion concerning the hierarchy of D s0 → D s π 0 versus D s0 → D * s γ is reached in [3] using the quark model. Since our calculation is based on a different method, the s P ℓ = 1 2 − and s P ℓ = 1 2 + doublets being treated as uncorrelated multiplets, we find the agreement noticeable.

Conclusions and perspectives
We have found that the observed narrow width and the enhancement of the D s π 0 decay mode are compatible with the identification of D sJ (2317) with the scalar state belonging to the s P ℓ = 1 2 + doublet of the cs spectrum. However, this conclusion does not avoid other questions raised by the BaBar observation, one being the low mass of the state. We believe that such a particular issue requires additional investigations. A second point is that the radiative mode, although suppressed, is not negligible, and should be observed at a level typically represented by the ratios in (20).
The quantum number assignment has two main and rather straightforward consequences. The first one is the existence of the axial vector partner D ′ s1 belonging to the same spin doublet s P ℓ = 1 2 + . Even in the case where the hyperfine splittings between positive and negative parity states are similar: M D ′ s1 − M D s0 ≃ M D * s − M Ds , this meson is below the D * K threshold. Therefore, its hadronic decay to D * s π 0 , at the rate would produce a narrow peak in the D * s π 0 mass distribution. The confirmation of such a state, the existence of which is suggested by the analysis of the D s γπ 0 mass distribution [1], will support the interpretation.
The second consequence concerns the doublet of scalar and axial vector mesons in the bs spectrum. Since the mass splitting between B and D states is similar to the corresponding mass splitting between B s and D s states, such mesons should be below the BK and B * K thresholds, thus producing narrow peaks in B s π 0 and B * s π 0 mass distributions, with rates resulting from expressions analogous to (9)-(21).
Note added. When this work was completed, the CLEO Collaboration announced the observation of a narrow resonance with mass 2.46 GeV in the D * + s π 0 final state and the confirmation of D sJ (2317) [18]. Moreover, a theoretical analysis based on the quark model was posted on the Los Alamos arXive, with the same conclusions presented here [19].