# Observed (2317) and tentative (2100–2300)

as the charmed
cousins of the light scalar nonet

###### Abstract

The very recently observed meson is described as a quasi-bound scalar state in a unitarized meson model, owing its existence to the strong OZI-allowed coupling to the nearby -wave threshold. By the same mechanism, a scalar (2100–2300) resonance is predicted above the threshold. These scalars are the charmed cousins of the light scalar nonet , , , and , reproduced by the same model. The standard and charmed scalars and , cousins of the scalar nonet , , , and , are predicted to lie at about 2.64 and 2.79 GeV, respectively, both with a width of some 200 MeV.

The (or simply ) charmed meson, just discovered [1] by the BABAR collaboration (see also Ref. [2]), is claimed [3] to “send theorists back to their drawing boards,” in view of its low mass. If indeed the tentative assignment gets confirmed, there appears to be a discrepancy with typical quark potential models, which predict a mass of 2.48 [4] or 2.49 GeV [5] for this state.

The aim of this Letter is to demonstrate that there is no difficulty [6] to obtain a low-mass scalar , with a standard configuration, provided one takes into account its OZI-allowed coupling to the nearby but closed threshold at about 2.36 GeV. The strong coupling to this -wave threshold will turn out to conjure up a nonperturbative pole, not related to the confinement spectrum, which is forced to settle down on the real energy axis due to the lack of phase space. Conversely, the same threshold pushes the ground-state confinement pole to much higher energies, which may thus have precluded its detection so far. By the same token and as a spin-off, we shall also predict two additional charmed scalar mesons, i.e., with a ( or ) configuration, still needing experimental confirmation.

The mechanism responsible for the low-mass charmed scalar mesons is precisely the same that produces the light scalar-meson nonet, i.e., the , , [7], and [8]. The latter scalar mesons were predicted with great accuracy in Ref. [9], in the framework of a unitarized quark model for all mesons. In more detail, and employing a simpler yet less model-dependent formulation, it has been shown [10, 11, 12] how unitarization leads to structures in the scattering amplitude for -wave meson-meson scattering which are not and even cannot be anticipated by the naive quark model. Especially, from the behavior of the scattering poles near threshold [13, 14], we learn that, in the limit of decoupling from the meson-meson continuum, all members of the light scalar-meson nonet disappear into the background. On the other hand, all other mesons end up as genuine states in this limit.

Now, it is straightforward to apply the latter, simple model to charmed mesons: one just has to replace one of the model’s effective-quark-mass parameters by the charmed mass. If we then take the other parameters fixed, which e.g. yield an excellent fit to the -wave phase shifts up to 1.6 GeV [10], then we obtain the movement of complex-energy poles as depicted in Figs. 1 and 2, corresponding to the and states, respectively.

Next, we interpret with some care each of the two figures, in both of which the relevant parts of the complex plane are depicted, where represents the total invariant mass for elastic meson-meson scattering in an wave. Figure 1 refers to and Fig. 2 to scattering. Each figure displays two trajectories, representing the positions, as a function of the overall coupling parameter , of the lowest two of an infinity of singularities in the corresponding scattering amplitudes.

The matrix of our unitarized meson model contains, in principle, all possible two-meson scattering channels. In the model of Refs. [9, 10, 11, 12, 13, 14], one single set of parameters applies to all channels, from the light flavors to bottom. These parameters are: four constituent quark masses, , , , and , one confinement parameter, , one overall coupling constant, , and two shape parameters of the transition potential.

When studying elastic scattering, one could then select this particular channel from a larger matrix. For the purpose of the present investigation, we use here, as mentioned above, a simplified version of the model, discussed in Ref. [10], which contemplates just one scattering channel. Nevertheless, the parameters are kept unaltered with respect to the light scalar mesons, except for the quark masses, of course. By comparing the results of the full model [9] with those of the one-channel limit of Ref. [10], we verify that the higher, closed channels do not have much influence on the general scattering properties, but have some effect on the precise pole positions.

The two singularities studied in each of Figs. 1 and 2 are the two lowest-lying poles of the scattering amplitude. We study their positions as a function of the overall coupling constant . The here chosen physical value of equals 0.75, as in Refs. [10, 11, 12, 13, 14]. However, by just showing the respective pole positions at in Figs. 1 and 2, important information on their differences would be concealed. Moreover, the display of the pole trajectories reveals what could happen if Nature were to choose a somewhat different value for . We shall come back to this point further on.

As one observes from the two figures, the behavior upon decoupling () is completely different. Whereas the higher of the two singularities in each figure ends up at the genuine confinement ground state, which are at 2.44 GeV for and 2.55 GeV for , respectively, for our model parameters, the lower poles disappear into the background, with ever increasing width. A similar behavior has been observed for the -matrix poles of the light scalar mesons [13, 14].

The lower pole in scattering (Fig. 1) does not end up below threshold when the overall coupling is increased to the physical value , and settles at GeV. Hence, in experiment it will be observed as a structure from threshold upwards in the partial-wave scattering cross section. For the given pole position, this corresponds to a peak at about 2.1 GeV, with a width of 150 MeV, thus coming out some 70 MeV higher than the real part of the pole. This shift is manifest in our description of resonances, getting more and more sizable as the resonance width increases. For instance, in our fit of the -wave phase shifts, the cross-section peak shows up more than 100 MeV above the pole [10]. On the other hand, one should also realize from Fig. 1 that a very modest decrease of would give rise not only to a larger real part of the pole position, but also of the imaginary part, thereby amplifying the mentioned shift upwards. Such a decrease of can be justified on the basis of flavor symmetry [15]. Therefore, we expect a resonance somewhere in the energy interval 2.1–2.3 GeV, possibly with a width of several hundred MeV [15]. This may correspond to the preliminary resonance reported by the BELLE collaboration [16], with a mass of 2.29 GeV and a width of 305 MeV.

The lower pole in scattering (Fig. 2) settles on the real axis for . However, for the sake of clarity we have depicted its trajectory slightly away from the real axis. The pole trajectory for increasing ends up on the real axis at 2.21 GeV, i.e., well below threshold. Then it moves upwards as a virtual bound state, towards threshold, where for it turns into a real bound state. For we find the pole at 2.28 GeV. Such a behavior was already forecast in Ref. [17]. If we were to decrease a little, again owing to flavor symmetry, we would find [15] a bound-state pole even closer to the experimental value of 2.317 GeV.

The higher poles in and scattering, which stem from the
scalar radial ground states at 2.44 GeV for
and 2.55 GeV for , move upwards in the second Riemann sheet.
For the physical value of the overall coupling, ,
they constitute resonance poles. These poles come out some 200 MeV higher than
what at first sight would be expected,
not only from the naive quark model, but also from the central resonance
positions of and with respect to ,
and and with respect to .
However, one should just compare this with the following situations,
in order to understand why our findings are not unreasonable:

a) , with respect to and ,

b) , with respect to and .

Actually, in our full unitarized model, both the and
stem from coinciding poles, connected to the model’s
scalar ground state at 1.29 GeV in the decoupling limit.

Summarizing, we have demonstrated in the foregoing how a low-mass scalar meson can be easily obtained by including its coupling to the most relevant OZI-allowed two-meson channel, i.e., . However, this bound state is of a highly nonperturbative nature, which in no way can be obtained in single-channel quark models, no matter how sophisticated the used confinement mechanism. In particular, we have employed a simple unitarized model, which successfully reproduces the light scalar nonet, so as to obtain a value of 2.28 GeV for the lowest scalar state, when leaving the parameters unchanged with the exception of the quark masses. Therefore, we conclude that the recently discovered is probably a scalar meson of the latter type, being a cousin of the light scalar mesons, rather than belonging to the scalar confinement spectrum, such as, we believe, the and . Due to a similar coupling to the threshold, we predict a (2100–2300) resonance, which may have been found already [16]. As a final remark concerning the , we should note that it becomes a bound state — actually a quasi-bound state owing to the, here disregarded, isospin-violating decay mode — contrary to the (2100–2300) and the light scalar mesons, which are all resonances. This peculiarity is due to the high value of the lowest OZI-allowed threshold (), while the (2100–2300) as well as all light scalar mesons have a threshold involving at least one pion, which thus lie lower on a relative scale.

Besides the latter results, we have also obtained some additional predictions for charmed scalar mesons. For the standard charmed spectrum we expect the lowest -wave resonances to occur at about 2.64 GeV for , and at about 2.79 GeV for , both with a width of some 200 MeV. However, these values may be subject to a modest change when flavor-symmetry arguments are applied as above [15]. In any case, note that also these two states, though being perturbative in the sense that they can easily be linked to the confinement spectrum, turn out to suffer a drastic, quite nonperturbative effect from unitarization.

Acknowledgments: We thank F. Kleefeld for useful information on the distribution of -matrix poles in the complex and planes. This work was partly supported by the Fundação para a Ciência e a Tecnologia of the Ministério da Ciência e da Tecnologia of Portugal, under contract number POCTI/FNU/49555/2002.

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