Understanding the nature of $D_s(2317)$ and $D_s(2460)$ through nonleptonic B Decays

We consider the nonleptonic B decays $ B \to D^{(*)} D_s(2317)$ and $ B \to D^{(*)} D_s(2460)$, involving the newly discovered $D_s(2317)$ and the $D_s(2460)$ states. We find that experiments indicate disagreement with model calculations of their properties and/or breakdown of the factorization assumption for these decays . We point out that decays involving $B_s$ mesons where the $D_s$ resonances can be produced via the weak decay of the $b$ quark can provide further information about the nature of these newly discovered states. We also propose a model to calculate the two body nonleptonic decays $ B \to D^{(*)} D_s(2317)(D_s(2460))$, if the $D_s(2317)$ and $D_s(2460)$ are interpreted as $DK$ and $D^*K$ molecules.


Introduction
There has been recent observations of an unexpectedly light narrow resonance in D + s π 0 with a mass of 2317M eV /c 2 by the BaBar collaboration [1], together with another second narrow resonance in D s π 0 γ with a mass 2460M eV /c 2 [2].
The smaller than expected masses and narrow widths of these states have led, among other explanations [3], to a multi-quark anti-quark or a DK molecule interpretation of these states [4], or to an interpretation as p-wave states where the light degrees of freedom are in an angular momentum state j q = 1 2 [5], or even some combination of these [6]. There are also conflicting lattice interpretations of these states [7].
The mass difference between the D s (2317) and the well established lightest charm-strange meson, D s , is ∆M = 350M eV /c 2 . This is less than the kaon mass, thus kinematically forbidding the decay D s (2317) → D u,d + K. The possible resonance at 2460M eV /c 2 also has such a mass difference when taken with the lighter D * state. The interpretation of these states as bound D ( * ) K molecules just below the D ( * ) K threshold is particularly interesting in the light of the recent discovery of a narrow resonance in the decay J/ψ → γpp [8] which has been interpreted as a zero baryon number, "deuteron-like singlet 1 S 0 " bound state of p andp [9].
In the heavy quark theory, the ground state heavy meson involving a heavy and a light quark has the light degrees of freedom in a spin-parity state j P q = 1 2 − , corresponding to the usual pseudoscalar-vector meson doublet with J P = (0 − , 1 − ). The first excited states involves a p-wave excitation, in which the light degrees of freedom have j P q = 1 2 + or 3 2 + . This leads to two heavy doublets, the first giving J P = (0 + , 1 + ) and the latter a heavy doublet with J P = (1 + , 2 + ). Heavy quark symmetry rules out any pseudoscalar coupling of this doublet to the ground state at lowest order in the chiral expansion [10] and so these states are expected to be narrow. Recent Belle analysis of B − → D (+ * ) π + π − decays [11] indicate the presence of the 1 + state in this multiplet at a mass of M = (45.6 ± 4.4 ± 6.5 ± 1.6) MeV . In the D s system the counterpart states to these are naively expected to be a 100 MeV heavier because of the strange quark mass and so these states can probably be identified with D s1 (2536) and D sJ (2573) [12]. This is in line with the experimental observations that in the ground state the D s mesons are about a 100 MeV heavier than their nonstange counterparts.
The other excited doublet has J P = (0 + , 1 + ). These states are expected to decay rapidly through s-wave pion emission in the D u,d system and by kaon emission in the D s system and have large widths [13]. Observation of the 1 + state in the D system was reported by CLEO [14] some time ago. The recent Belle analysis of B − → D (+ * ) π + π − decays [11] also find evidence for the states in this doublet at These numbers are consistent with quark model estimates [15] and we expect these states to be broad.
The recently observed D s resonances have masses below these expectations and are very narrow, decaying through isospin violating transitions to D ( * ) s π final states. This has generated speculations that these states may not be p-wave excited states but rather something exotic like D ( * ) K molecules.
While the spectroscopy of these newly discovered states can provide clues to their structure, decays involving these states can yield further clues to their exact nature. We first look at nonleptonic B decays involving the p-wave D s resonant states which we will denote by D s0 , corresponding to the p-wave, j q = 1 2 , 0 + state, and D * s1 corresponding to the p-wave, j q = 1 2 , 1 + state. In B factories that do not produce the B s mesons the D s p-wave states cannot be directly produced via the weak current involving the b quark but they can only be produced through the sc current in the weak decay effective Hamiltonian. It was suggested in Ref. [16,17] that these theoretically expected broad states may be discovered through the three body decays B → D ( * ) D ( * ) K decays, where D ( * ) refer to D or D * , if these states are above the D ( * ) K threshold. These three body decays can also be used to measure both sin 2β and cos 2β [18,19,16].
In hadron B factories the D s resonant states can be produced directly from the weak decay of the b quark in the B s meson.
In this work we concentrate on non leptonic decays of the type B → D ( * ) D s (2317) and B → D ( * ) D s (2460), which are accessible at current B factories, and we also study nonleptonic decays of the types B s →

Nonleptonic Decay
Let us first assume that we can identify the the newly discovered states D s (2317) with D s0 and D s (2460) with D * s1 . In the Standard Model (SM) the amplitudes for B → D ( * ) D s0 (D * s1 ), are generated by the following effective Hamiltonian [20]: where the superscript u, c, t indicates the internal quark, f can be u or c quark, q can be either a d or a s quark depending on whether the decay is a ∆S = 0 or ∆S = −1 process. The operators O q i are defined where R(L) = 1 ± γ 5 , and q ′ is summed over all flavors except t. O 1f,2f are the current-current operators that represent tree level processes. O 3−6 are the strong gluon induced penguin operators, and operators O 7−10 are due to γ and Z exchange (electroweak penguins), and "box" diagrams at loop level. The values of the Wilson coefficients can be found in Ref. [20].
In the factorization assumption the amplitude for B → D ( * ) D s0 (D * s1 ), can now be written as where where We have defined with In the above equations N c represents the number of colors. To simplify matters we neglect the small penguin contributions and so as a first approximation we will neglect M 2 . The currents involving the heavy b and c quarks, in terms of form factors [22]. In the heavy quark limit the various form factors are related to a universal Isgur-Wise function ξ(v · v 1 ) where v and v 1 are the four velocities of the B and the D ( * ) mesons. One can therefore write, and The matrix elements < D s0 |sγ µ (1 − γ 5 ) c | 0 > and < D * s1 |sγ µ (1 − γ 5 ) c | 0 > are written in terms of the decay constants that are defined as We can now define the following ratios Let us focus on the ratio R D0 which within factorization and the heavy quark limit can be written as where we have neglected phase space ( and other) effects that are subleading in the heavy quark expansion.
Similarly we have Now in the heavy quark limit f D s0 = f D * s1 and f Ds = f D * s and so one would predict R D0 ≈ R D1 . There have been various estimates of the decay constant f D s0 in quark models [23] and in QCD sum rule calculations [24]; these typically find the p-wave , j q = 1 2 states to have the similar decay constants as the ground state mesons. We therefore expect f D s0 ∼ f Ds giving in addition to the heavy quark predictions Experimentally Belle measures [11] BR[B → DD s (2317)]BR[D s (2317) → D s π 0 ] = (9.9 +2.8 −2.5 ± 3.0) × 10 −4 The dominant decay of the D s (2317) is expected to be through the D s π mode [25,26] and so Now using the measured branching ratio [12] BR one obtains a combined branching ratio This leads to R D0 ≈ 1 10 (or,f D s0 ∼ 1 3 f Ds ) which is a factor 10 smaller then theoretical expectations.
There are a few possible explanations that can be put forward to explain this discrepancy between experiment and theoretical expectation and we will consider them now.
It is possible that the estimate of the decay constants of the p-wave, j q = 1 2 states in the various models are incorrect just like the mass predictions of these states are incorrect. This would require a major revision of model calculations that predict the properties of these states. From the experimental data we have seen To check this we note that experimentally Belle measures [11] BR Taking the central values we find Using the measured branching ratio [12] BR one can obtain, using the measured central values This then leads to R D1 ≈ 1 3 which is in disagreement with Eq. 15 and Eq. 20.
One might argue that factorization is not applicable to B → D ( * ) D ( * ) decays. However recent analysis in Ref. [27] find that factorization works well for these decays. Moreover the quantities in Eq. One can now construct the ratios Now in the SU (3) (3) breaking effects in the ratios may cancel [28]. Hence any large deviation of T Ds(2317) and T Ds(2460) from unity would be inconsistent with the j q = 1 2 p-wave interpretation of the new D s states. Note that the further assumption of factorization leads to T Ds(2317) ≈ T Ds(2460) and T Ds ≈ T D * s in the heavy quark limit.
As indicated earlier, among various other suggestions for the nature of the new D s states is the idea that these states may be D ( * ) K molecules. There are no serious models of such meson molecules that one can use to calculate nonleptonic decays involving these states. Here we will attempt a rough qualitative estimate of nonleptonic decay rates assuming that the D s (2317) and D s (2460) states are really a DK molecule and a D * K molecule respectively. Consider the nonleptonic decay B → DD s (2317). We assume that the decay proceeds through two stages: the first stage is the decay B → DDK, followed by the state D(p 2 )K(p K ) forming the molecule D s (2317) with the probability given by f (p 2 , p K ) so that Without a model for f (p 2 , p K ) we cannot make predictions but nonetheless it is useful to define the average probability function f as Hence we have We can define a similar function f * and the average f * for nonleptonic decays involving the D s (2460) and so We can consider the ratios which are related as Using the measured three body branching ratios [29] BR  In other words, we can write where We can similarly define f * ′ as Note that the ratios T Ds(2317) and T Ds(2460) (Eq. 25) in the molecular model are no longer equal to unity in the SU (3) limit since that depended on the identification of these states as p-wave states. Therefore the measurement of these ratios can provide useful information on the nature of the D s (2317) and the D s (2460) states.

Summary and Conclusions
In summary, in this work, we have considered the nonleptonic B decays B → D ( * ) D s (2317)(D s (2460)), involving the newly discovered D s (2317) and the D s (2460) states. We have discussed the implication of the measured nonleptonic decays for the properties and the nature of these states. If these states are the p-wave multiplet with the light degrees of freedom in the j q = 1 2 state, then we find that experiments indicate disagreement with model calculation of their properties and/or breakdown of the factorization assumption. We have suggested further tests involving nonleptonic B s meson decays, that do not assume

Acknowledgments
We thank Tom Browder for discussions. This work is supported by the Natural Science and Engineering Council of Canada (NSERC) under grant number A3828.