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Explaining the Ds(2317)

The recently observed Ds(2317) meson is explained as a scalar c+anti-s system which appears as a bound state pole, below threshold, in the DK scattering amplitude. The standard c+anti-s charmed scalar D_{s0} is found at about 2.9 GeV, with a width of some 150 MeV. For the scattering length of DK in S wave we predict 5+-1 1/GeV.

Of course, this happened before with the light scalar mesons, which can easily be handled within a unitarized model for all non-exotic mesons [22], without making any distinction between the heavier scalar mesons and the light ones. But seventeen years after the publication of these results [16], their implications are still not fully understood, nor applied in data analysis.
In the specific case of the D * sJ (2317) + state, just by applying the model parameters established twenty years ago [23], the unitarization scheme results in a bound state below the DK threshold, for a cs confinement spectrum with ground state at 2.545 GeV, almost 200 MeV above the DK threshold. This achievement is entirely similar to the extra states obtained by the same model for the light scalars. No additional interactions are necessary to explain the scalar mesons.

Unitarization
In the unitarization scheme, we describe meson-meson scattering by the wave equation The operators H f and H c respectively describe the dynamics of two free mesons and of a confined quark-antiquark system. The latter is supposed to generate an infinity of confinement states.
Transitions from confined quarks to free mesons, mediated by quark-antiquark-pair creation, is described by V t .
By comparison of Eq. (1) with the usual expressions for scattering wave equations, we conclude that the generalized potential V for meson-meson scattering is given by With the definitions given in Appendix A, in particular formulae (7), (17) and Table 1, we obtain for the partial-wave scattering phase shift generated by wave equation (1) the result cotg (δ ℓ (p)) = The expression (3), which is an exact solution of Eq. (1), is very compact and elegant. But the summations over the radial excitations of the confined system do not converge fast. Hence, for the numerical calculation of scattering quantities it is not suitable. A more practical method has been described in Ref. [24]. Nevertheless, because of its close contact with the physical situation, expression (3) is extremely comprehensible, and useful as a starting point for making approximations.
For example, when we are studying a limited range of energies, we may restrict the summation to only the few terms near in energy, characterized by E nℓc in Eq. Another, in practice very useful approximation is inspired by the radial dependence of the transition potential V t , which is depicted in Ref. [25] for meson-meson scattering in the cases J P = 0 − , J P = 1 − , and J P = 0 + , We find there that V t is peaked at short distances. If we, moreover, take into account that eigenfunctions of the confinement Hamiltonian H c must also be of short range, then we may just define a transition radius a and approximate the spatial integrals of formula (17) by choosing a spherical delta shell U(r) = δ(r − a). In this case we end up with the expression cotg (δ ℓ (p)) ≈ This way we have pulled the p dependence outside the infinite summation over the radial confinement spectrum, which makes it much easier to handle truncations, as now the rest term does not depend on p and may thus be chosen constant. Our recent detailed analyses of scalar mesons have actually been performed in the approximation of formula (4).
Our understanding of the scalar mesons is based on studying the corresponding pole positions, and their movements as a function of the coupling λ. A pole which stems from a confinement state moves towards the energy eigenvalue of that state when λ is changed to smaller values.
However, we observe that besides the poles stemming directly from the confinement spectrum, there exist poles in the scattering amplitude which disappear into the background for vanishing λ. These somehow originate in the bulk of the sum over the radial excitations in expression (3), and are not related to any state in particular.
Moreover, we observe that the lightest ℓ = 1 qq scalar mesons can only reside somewhere between 1.3 and 1.5 GeV in the confinement spectrum, with the model parameters of Ref. [23].
The latter have been adjusted to bottomonium, charmonium, the light vector states, and to P -wave meson-meson scattering data, whenever available in the early eighties.
Finally, we observe that the extra poles in the scattering amplitude come close enough to the real-energy axis only in the case of S-wave scattering. There, they show up somewhere close to the lowest threshold, for energies below the ground states of the confinement spectrum.
For the physical value of the model parameter λ, a specific value has been selected by Nature. But in our model we can vary λ. This way, we can find out where poles are moving for smaller or larger values of the coupling. 4 The D * sJ (2317) + state For DK elastic S-wave scattering, we adopt the same expression that has been used for Kπ as described in Ref. [31]. We only have to substitute the confinement masses of ds by those for cs, thereby using the parameters of Ref. [23]. For the ground state of the J P = 0 + cs confinement spectrum, the mass comes out at 2.545 GeV. The first radial excitation is then 380 MeV (the model's level spacing) higher in mass. The summation over the higher radial excitations in Eq. (4) we approximate by a constant, normalized to 1. The relative couplings of the ground state and the first radial excitation to the DK channel are 1.0 and 0.2, respectively [31]. Thus we obtain For expression (4) with the substitution (5) Notice from Fig. 2 that in the region around 2.55 GeV, where quark models predicted J P = 0 + cs states [17,21], we find no structure in the cross section, nor poles in the scattering amplitude.
On the contrary, employing confinement plus unitarization, we find structures in the cross section some 250 MeV below and 400 MeV above that value, which explains the observation of the (probable) D * s0 ground state below the DK threshold, and predicts for the first radial excitation a resonance somewhere between 2.8 and 3.0 GeV, with a width of some 100-200 MeV. Based on Fig. 2, we may also predict the scattering length for DK elastic S-wave scattering. Keeping in mind that the J P = 0 + cs ground state comes out a little bit too low in mass with our value of λ, we estimate the theoretical error by varying λ. For a conservative choice, we obtain scattering length DK in S wave = 5 ± 1 GeV −1 . (6)

Conclusions
The two most important aspects of strong interactions for hadrons are confinement and OZIallowed real or virtual decay, which effectively describe gluonic interactions and quark-pair creation/annihilation, respectively. These are the only ingredients entering the model defined in Eq. (1), which, with a fixed small set of parameters, can reasonably describe hadronic phenomena ranging in energy from the light scalars up to bottomonium.
Here, we have discussed the model's results for the J P = 0 + cs states coupled to the DK channel. For the DK S-wave scattering length we predict the value 5 ± 1 GeV −1 .

A The T matrix for meson-meson scattering
First, we introduce the operators H c , describing the confinement dynamics in the interaction region, H f , representing the dynamics of the scattered particles at large distances, and V t , which stands for the transitions between these two sectors. For an arbitrary spherically symmetric confinement potential V c , we define these operators in configuration space by The various mass parameters of Eq. (7) are defined in Table 1. The transition potential V t , which provides the communication between the confined channel and the scattering channel, is in agreement with the potentials [25] that may describe the breaking of the color string.
Before performing the calculations to all orders, we write the transition operator as symbol definition m q (mq) constituent (anti-)quark mass µ c reduced mass in confinement channel M 1,2 meson masses µ reduced mass in scattering channel where G f represents the free boson propagator associated with the Hamiltonian H f in Eq. (7), given by The matrix elements of the T operator follow from Eq. (8) by For the generalized potential (2), the Born term takes the form The total center-of-mass energy E and the linear momentum p are, through Eq. (7), related by We denote the properly normalized eigensolutions of the operator H c in Eq. (7), corresponding to the energy eigenvalue E nℓc , by So, by letting the self-adjoint operator H c act to the left in Eq. (11), we find The individual matrix elements in the sum of formula (14) result in where, in order to simplify the expressions, we define the quantities J nℓ (p) = 2 π r 2 dr j ℓ (pr) U (r) F * nℓc (r) N nℓ (p) = 2 π r 2 dr n ℓ (pr) U (r) F * nℓc (r) (16) At this stage we should pay some attention to the quantum numbers of the two coupled systems, since the operator V t mediates between systems with different quantum numbers. From parity conservation we deduce that the relative orbital angular momenta ℓ of a meson-meson pair and ℓ c of the confined quark-antiquark pair differ by at least one unit. Also the total intrinsic spins are different for the two systems, when C parity is taken into account. The conservation of J, P , C, isospin, flavor, and color can be formulated in a consistent way [25,70,71], but we do not intend to go into details here. We just should recall that there exists a relation between (ℓ c , m c ) and (ℓ, m), and moreover that the overlap integral expressing this relation is absorbed into λ.
For the full Born term (14), we then obtain Similarly, one may determine the N+1-st-order term p | (V G f ) N +1 k of Eq. (10), yielding From Eqs. (17) and (18), we find for the matrix elements of the product (V G f ) N V the expression The terms labelled with N in Eq. (19) can be summed up, resulting in a compact final formula for the matrix elements of the transition operator T in Eq. (10), reading