From the via the (2317) to the :
connecting light and heavy scalar mesons
Abstract
Pole trajectories connecting light and heavy scalar mesons, both broad resonances and quasibound states, are computed employing a simple coupledchannel model. Instead of varying the coupling constant as in previous work, quark and meson masses are continuously changed, so as to have one scalar meson evolve smoothly into another with different flavor(s). In particular, it is shown, among several other cases, how the still controversial (800) turns into the established , via the disputed (2317). Moreover, a (3946) is predicted, which may correspond to the recently observed (3943) resonance. These results lend further support to our unified dynamical picture of all scalar mesons, as unitarized states with important twomeson components.
After more than four decades, understanding the scalar mesons continues to pose serious difficulties to theorists as well as experimentalists. Still today, no consensus exists about the lightest and oldest structures in the scalarmeson sector, namely the ((600) [1]) [2] and the ((800) [1]) [3, 4]. But also the discovery of the surprisingly light charmed scalar (2317) [5], though giving a new boost to meson spectroscopy in general, has not contributed to the understanding of scalar mesons, as can be seen from the many different approaches to the (2317) in the literature (see Ref. [6] for a representative, albeit not totally exhaustive, list of references). Here, we shall focus on a formalism which successfully describes all mesonic resonances, including the scalar mesons.
In Ref. [7] it was shown that the (2317) meson can be straightforwardly explained as a normal state, but strongly coupled to the nearby channel, which is responsible for its low mass. The framework for this calculation was a simple coupledchannel model, which had been employed previsously [8] to fit the wave phase shifts, and predict the now listed[1] (800), besides reproducing the established (1430). Furthermore, another charmed scalar meson was predicted in Ref. [7], i.e., a broad resonance above the threshold, somewhere in the energy region 2.1–2.3 GeV, which may correspond to the (2300–2400) [9, 1]. Also highermass and resonances were foreseen [7], which have not been observed so far.
The purpose of this Letter is to show the interconnection of the scalar mesons (800), (2300–2400), (2317) with one another, and also with the established (3415) [1]. Moreover, the same interconnection will be demonstrated for the highermass recurrences of these scalars, thereby finding a candidate for the very recently observed (3943) charmonium state [10]. For that purpose, we shall employ the abovementioned coupledchannel model, but now for fixed, physical coupling, while quark and threshold masses will be varied. Thus, a continuous and smooth transition can be achieved from one scalar meson to another. Crucial here will be a mass scaling [11, 12] of the two parameters modeling the offdiagonal potential that couples the confined and decay channels. This way, these two parameters, identical to the ones used in Refs. [8, 7, 11], suffice to reasonably describe a vast range of distinct scalar mesons. On the other hand, the confinement and quarkmass parameters are taken at their usual published values.
Starting point is a simple, intuitive coupledchannel model, describing a confined system, coupled to one mesonmeson channel accounting for the possibility of real or virtual decay via the mechanism. If the transition potential is taken to be a spherical delta function, the inverse matrix can be solved in closed form, reading [8]
(1) 
where are spherical Bessel and Neumann functions, respectively, is the coupling, is the deltashell radius, are the energies of the bare confinement spectrum, are the corresponding weight factors, is the onshell relative momentum in the twomeson channel, given by the kinematically relativistic expression
(2) 
and is the ensuing relativistic reduced mass
(3) 
As the present paper deals with scalar mesons, we have and in Eq. (1). Moreover, since only ground states and first radial excitations are considered here, we shall approximate the infinite sum in Eq. (1) by two confinementspectrum states plus one rest term, also sticking to the numerical values used in Refs. [8, 7, 11], namely , , and . As for the two confinement levels, we parametrize them by a harmonic oscillator [7, 11], i.e., , with
(4) 
where GeV, GeV (), GeV, and GeV, as in previous work [13, 8, 7, 11, 14]. Finally, we assume a mass scaling of the parameters and given by [11, 12]
(5) 
where the labels refer to a particular combination of quark flavors, and is the corresponding reduced quark mass. This procedure ensures flavor invariance of our equations. Using then the values GeV and GeV from the fit to the wave phase shifts in Ref. [8], we have fixed all our parameters,^{1}^{1}1Note that we use here somewhat shifted confinement levels as compared to Ref. [8], namely the ones following from Eq. (4). This gives rise to a slightly lighter and broader meson, and a heavier (1430). which allows to show the predictive power of our approach. For the required input mesons masses, we take the isospinaveraged values [1] GeV, GeV, and GeV.
Now we can compute pole trajectories in the complex energy plane for scalar resonances and (virtual) bound states, by searching the values of for which . However, instead of freely varying as in previous work, we shall keep fixed at its physical value of 0.75 GeV, while changing instead one of the quark masses, as well as one of the meson masses in the decay channel. This way we can make one scalar meson turn into another. For instance, by letting
(6) 
we smoothly change the () meson, coupling to the channel, into the (2317) (), coupling to . The poles themselves are numerically found and checked with two independent methods, i.e., the MINUIT package of CERN [15], and MATHEMATICA [16].
In Fig. 1, one sees in one glimpse the nine trajectories
(7) 
where the numbers between parentheses are the real parts (in MeVs) of the respective resonance/boundstate poles, the corresponding imaginary parts being
(8) 
Before discussing the actual trajectories, a few remarks are due concerning the precise values found for the pole positions. Clearly, for such a simple model without any fitting freedom, moreover covering a vast energy range, a very accurate reproduction of the masses and widths of all experimentally observed mesons cannot, and should not even be expected. In particular, the inclusion of only the lowest, dominant decay channel for each state will certainly reflect itself in one way or another. For instance, the much too small width of our (1788), which should correspond to the observed [17] (1820), is probably owing to the neglect of the important channel. Furthermore, the somewhat too large mass of our (3472), as compared to the established [1] (3415), may very well be due to the omission of vectorvector decay channels, which are relevant for charmonium ground states [18]. Note, however, that the latter discrepancy of 57 MeV is quite insignificant when compared to the huge coupledchannel shifts in charmonium recently found in Refs. [19, 20]. Notwithstanding, a clear identification can be made of our broad (704), (2114), and (1522) states with the listed [1] (800), (2300–2400), and (1430) resonances, respectively. Here, one should also notice that we give the real parts of the pole positions of our resonances, which usually do not coincide with the experimental masses resulting from BreitWigner fits when the widths are large. As for the remaining observed mesons, our (2327) is very close to the (2317), while our (3946), with a width of about 60 MeV, seems a good candidate for the brandnew [10] charmonium state (3943). Finally, we predict the two mediumbroad charmed mesons (2841) and (2923), so far undetected, as well as the very broad states (2673), (2840), and (4015), which will be extremely hard to observe at all. In any case, the predictions for the latter highermass states may change significantly when additional decay channels are taken into account.
Turning now to the trajectories themselves, it is remarkable to observe that physical states with radically disparate widths can be continuously connected to one another in flavor. This is one of the reasons why scalarmeson spectroscopy is so intricate. Moreover, as we shall see below, states on the same mass trajectory can have different origins when viewed as states distorted by meson loops, which point will become clearer when we study Fig. 2. Anyway, the first radial excitations of the , , , and systems are all on the same trajectory in Fig. 1, i.e., the one connecting the (1788) and (3946).
In Fig. 2, the lowest states for the various flavor combinations
are displayed again, but now also showing how the corresponding poles move when the coupling is reduced from its fixed value. We see that the (704) and the (2114) appear to find their origin in the continuum, corresponding to infinitely negative imaginary parts of their pole positions, while the (2327) and (3472) are connected to the confinement spectrum, with poles on the real axis. This is quite surprising for the nearby pair (2114)–(2327). However, even the physical (2317) itself can be either interpreted as a “confinement” state [11, 21], or a “continuum” state [7], depending on tiny changes in e.g. the parameter . What this figure also shows is an extremely delicate balance of coupling effects. With a small decrease of , the (2300–2400) and especially the meson would become even broader and thus almost impossible to observe experimentally, while the (2317) would be a resonance or a virtual state instead of a quasibound state.
Finally, in Fig. 3 a direct transition of the (704) into the (2327) is displayed, by letting , as in Eq. (6), and moreover in a different fashion. Namely, instead of giving the pole positions in the complex energy plane, we now plot the corresponding real and imaginary parts as a function of the varying quark mass, as well as the proportionally changing threshold value. It is striking to see how the (704) resonance quickly turns into a virtual bound state, while its real part remains almost constant. Here, we probably see the kinematical Adler zero [21] at work, which rapidly moves away as one of the decay masses increases from , thus allowing the pole to approach the real axis. Then, the pole moves along the real axis as a virtual state, until it touches the threshold at about 1.76 GeV, after which it becomes a bound state. Notice again the tiny margin, at least on this scale, by which the (2327) is bound.
To conclude, in the present paper we have shown how several light and heavy scalar mesons can be linked to one another, by continuously varying some of the involved flavor and decay masses. This way, the common dynamical nature of the studied — and probably all — scalar mesons, as ordinary states but strongly distorted due to coupled channels, is further substantiated. Thus, tetraquarks and other exotic configurations are not needed in this context. Moreover, we deduce that labeling scalar mesons as states as opposed to dynamical mesonmeson resonances makes no sense, in view of the tiny parameter variations needed to turn one kind of pole into another. Rather, scalar mesons should be considered nonperturbatively dressed systems, with large mesonmeson components, no matter if one uses a coupledchannel quark model [22] or e.g. the quarklevel linear sigma model [23]. As a consequence, the spectroscopy of scalar mesons is much more complex than for ordinary mesons, with the total number of potentially observable states being different from the number of confined, bare states.
In the course of this analysis, we have also found a candidate for the new charmonium state (3943) [10]. It is true that such a resonance, if indeed a scalar, should dominantly decay to , a mode which has not been observed yet. However, the reported decay is OZIforbidden, so that it cannot account for the measured sizable width of MeV.
We thank D. V. Bugg for enlightening discussions about scalar mesons in general, and the new charmonium state (3943) in particular. This work was supported in part by the Fundação para a Ciência e a Tecnologia of the Ministério da Ciência, Tecnologia e Ensino Superior of Portugal, under contract POCTI/FP/FNU/50328/2003 and grant SFRH/BPD/9480/2002.
References
 [1] S. Eidelman et al. [Particle Data Group Collaboration], Phys. Lett. B 592, 1 (2004).
 [2] J. E. Augustin et al. [DM2 Collaboration], Nucl. Phys. B 320, 1 (1989); D. M. Asner et al. [CLEO Collaboration], Phys. Rev. D 61, 012002 (2000) [arXiv:hepex/9902022]; E. M. Aitala et al. [E791 Collaboration], Phys. Rev. Lett. 86, 770 (2001) [arXiv:hepex/0007028]; D. V. Bugg, Phys. Lett. B 572, 1 (2003) [Erratumibid. B 595, 556 (2004)]; M. Ablikim et al. [BES Collaboration], Phys. Lett. B 598, 149 (2004) [arXiv:hepex/0406038].
 [3] E. M. Aitala et al. [E791 Collaboration], Phys. Rev. Lett. 89, 121801 (2002) [arXiv:hepex/0204018]; D. V. Bugg, Phys. Lett. B 572, 1 (2003) [Erratumibid. B 595, 556 (2004)]; M. Ablikim et al. [BES Collaboration], arXiv:hepex/0506055.
 [4] E. M. Aitala et al. [E791 Collaboration], arXiv:hepex/0507099; J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B 621, 72 (2005) [arXiv:hepex/0503043].
 [5] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 90, 242001 (2003) [arXiv:hepex/0304021]; D. Besson et al. [CLEO Collaboration], Phys. Rev. D 68, 032002 (2003) [arXiv:hepex/0305100]; P. Krokovny et al. [Belle Collaboration], Phys. Rev. Lett. 91, 262002 (2003) [arXiv:hepex/0308019].
 [6] M. Nielsen, R. D. Matheus, F. S. Navarra, M. E. Bracco, and A. Lozea, arXiv:hepph/0509131; T. Mehen and R. P. Springer, Phys. Rev. D 72, 034006 (2005) [arXiv:hepph/0503134]; K. Terasaki and B. H. J. McKellar, Prog. Theor. Phys. 114, 205 (2005) [arXiv:hepph/0501188]. I. W. Lee, T. Lee, D. P. Min, and B. Y. Park, arXiv:hepph/0412210; C. Quigg, J. Phys. Conf. Ser. 9, 1 (2005) [arXiv:hepph/0411058]; D. S. Hwang and D. W. Kim, Phys. Lett. B 606, 116 (2005) [arXiv:hepph/0410301]; Y. A. Simonov and J. A. Tjon, Phys. Rev. D 70, 114013 (2004) [arXiv:hepph/0409361]; M. A. Nowak and J. Wasiluk, Acta Phys. Polon. B 35, 3021 (2004); A. Zhang, Phys. Rev. D 72, 017902 (2005) [arXiv:hepph/0408124]; Y. Q. Chen and X. Q. Li, Phys. Rev. Lett. 93, 232001 (2004) [arXiv:hepph/0407062]; D. Becirevic, S. Fajfer and S. Prelovsek, Phys. Lett. B 599, 55 (2004) [arXiv:hepph/0406296]; P. Bicudo, Nucl. Phys. A 748, 537 (2005) [arXiv:hepph/0401106]; E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B 582, 39 (2004) [arXiv:hepph/0307133]; C. H. Chen and H. n. Li, Phys. Rev. D 69, 054002 (2004) [arXiv:hepph/0307075]; Y. B. Dai, C. S. Huang, C. Liu and S. L. Zhu, Phys. Rev. D 68, 114011 (2003) [arXiv:hepph/0306274]; G. S. Bali, Phys. Rev. D 68, 071501 (2003) [arXiv:hepph/0305209]; P. Colangelo and F. De Fazio, Phys. Lett. B 570, 180 (2003) [arXiv:hepph/0305140]; S. Godfrey, Phys. Lett. B 568, 254 (2003) [arXiv:hepph/0305122]; H. Y. Cheng and W. S. Hou, Phys. Lett. B 566, 193 (2003) [arXiv:hepph/0305038]; T. Barnes, F. E. Close, and H. J. Lipkin, Phys. Rev. D 68, 054006 (2003) [arXiv:hepph/0305025].
 [7] E. van Beveren and G. Rupp, Phys. Rev. Lett. 91, 012003 (2003) [arXiv:hepph/0305035].
 [8] E. van Beveren and G. Rupp, Eur. Phys. J. C 22, 493 (2001) [arXiv:hepex/0106077].
 [9] K. Abe et al. [Belle Collaboration], Phys. Rev. D 69, 112002 (2004) [arXiv:hepex/0307021]; J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B 586, 11 (2004) [arXiv:hepex/0312060].
 [10] K. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 94, 182002 (2005) [arXiv:hepex/0408126].
 [11] E. van Beveren and G. Rupp, Mod. Phys. Lett. A 19, 1949 (2004) [arXiv:hepph/0406242].
 [12] F. Kleefeld, AIP Conf. Proc. 717, 332 (2004) [arXiv:hepph/0310320].
 [13] E. van Beveren, G. Rupp, T. A. Rijken, and C. Dullemond, Phys. Rev. D 27, 1527 (1983).
 [14] E. van Beveren and G. Rupp, Phys. Rev. Lett. 93, 202001 (2004) [arXiv:hepph/0407281].

[15]
http://wwwasdoc.web.cern.ch/wwwasdoc/minuit/
minmain.html  [16] The Mathematica Book, 5th edition, Wolfram Media, 2003, ISBN 1579550223.
 [17] A. V. Anisovich and A. V. Sarantsev, Phys. Lett. B 413, 137 (1997) [arXiv:hepph/9705401].
 [18] E. van Beveren, C. Dullemond, and G. Rupp, Phys. Rev. D 21, 772 (1980) [Erratumibid. D 22, 787 (1980)].
 [19] T. Barnes, J. Phys. Conf. Ser. 9, 127 (2005) [arXiv:hepph/0412057].
 [20] Y. S. Kalashnikova, Phys. Rev. D 72, 034010 (2005) [arXiv:hepph/0506270].
 [21] G. Rupp, F. Kleefeld, and E. van Beveren, AIP Conf. Proc. 756, 360 (2005) [arXiv:hepph/0412078].
 [22] E. van Beveren, T. A. Rijken, K. Metzger, C. Dullemond, G. Rupp, and J. E. Ribeiro, Z. Phys. C 30, 615 (1986); E. van Beveren and G. Rupp, Eur. Phys. J. C 10, 469 (1999) [arXiv:hepph/9806246].
 [23] R. Delbourgo and M. D. Scadron, Mod. Phys. Lett. A 10, 251 (1995) [arXiv:hepph/9910242].