A lowlying scalar meson nonet in a unitarized meson model

published in
Zeitschrift für Physik C  Particles and Fields 30, 615–620 (1986)
Abstract
A unitarized nonrelativistic meson model which is successful for the description of the heavy and light vector and pseudoscalar mesons yields, in its extension to the scalar mesons but for the same model parameters, a complete nonet below 1 GeV. In the unitarization scheme, real and virtual mesonmeson decay channels are coupled to the quarkantiquark confinement channels. The flavordependent harmonicoscillator confining potential itself has bound states (1.3 GeV), (1.5 GeV), (1.3 GeV), (1.4 GeV), similar to the results of other boundstate models. However, the full coupledchannel equations show poles at (0.5 GeV), (0.99 GeV), (0.97 GeV), (0.73 GeV). Not only can these pole positions be calculated in our model, but also cross sections and phase shifts in the mesonscattering channels, which are in reasonable agreement with the available data for , and in wave scattering.
1 Introduction
The rich structure in mesonmeson scattering at intermediate energies has stimulated many theoreticians to fit the existing quark models to the experimental results [1, 2, 3, 4, 5, 6, 7]. Especially wave mesonmeson scattering shows structures which are very intriguing [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Detailed phaseshift analyses reveal two pronounced scalar mesonic resonances below 1 GeV, namely the (975) resonance in [8, 9, 10, 11, 12, 14] and the (980) in [15, 16] wave scattering. The other relevant resonances that appear nowadays in the tables of particle properties [17] are the (1300) in and (l350) in [18, 19] wave scattering.
It is well known that these particles cause severe problems if one wants to understand them as quark+antiquark () states. For instance, confronted with the quark model, the resonance positions do not fit the quadratic or linear GellMann–Okubo mass relations [4] (see, however, [3]). A possible resonance in wave scattering at 600 MeV, which was poorly recognized in early analyses [8], disappeared from the tables of particle properties in the seventies. Nevertheless, some years later this resonance revived within the bag model, due to a solution for the scalarmeson problem presented by Jaffe [6], who pointed out that these resonances stem from states. The large binding energy which is assumed for such configurations makes the low masses required by experiment possible. All kinds of quark configurations [6, 20, 21] and gluongluon bound states [7] might exist, other than the standard for mesons and for baryons. This probably no one doubts, but there is no experimental evidence that they should couple significantly to hadronhadron scattering [2, 5].
To select the (1300) resonance as the isosinglet partner of the (975), rather than the (600), is probably the result of bagmodel interference with the analysis of the wave scattering data, because in the bag model, and also in other boundstate hadron models, the lowest isospinzero object fits better with a total mass of about 1.3 GeV [20, 22].
In this paper we will show that we have no difficulties to explain the scalar mesons within our unitarized quark model, and to interpret them as states with a mesonmeson admixture. However, neither the model nor the data exclude poles in the scattering matrix which do not appear in the tables but nevertheless might be interpreted as resonances.
2 The Model
The unitarized quark model is described in many articles. We will therefore confine ourselves to only briefly discuss the main features here and to give a complete list of references, [23, 24, 25, 26, 27, 28]. In our treatment, mesonmeson scattering processes couple to quark configurations or mesons via the annihilation and creation of a pair out of the vacuum. The reverse coupling describes the decay process of a meson or the coupling of a meson to its virtual decay channels. In [23] and [24], the explicit form of a multichannel Schrödinger description of such a system is given (see also Appendix A). Several mesonmeson scattering channels are, via the QPC mechanism [29], coupled to permanently closed channels with the same quantum numbers. It is also shown in [23] how to account for relativistic effects, and for the effects of onegluon exchange in the channels.
Scattering matrices, phase shifts, cross sections and wave functions can be calculated from the Schrödinger equation by an approximative method [25, 26] which leads to an matrix that is explicitly analytic in the complex energy plane, and unitary.
Phase shifts and cross sections can be checked to be in good agreement with the data if available. In other cases, the pole positions of the scattering matrix can be compared with the boundstate and resonance positions found by experiment. Wave functions may be compared with those expected from leptonic decays.
It has been our observation [23] that the properties of the and mesonic resonances are reasonably well described with a few model parameters: the effective quark masses, where the effective up and down masses can be taken equal, one universal harmonicoscillator frequency which describes the confining force in the permanently closed channels for all possible flavor configurations, and two or three parameters to describe the coupling of the scattering sector to the confinement sector.
In this investigation, we apply the model to wave mesonmeson scattering. The quark and the antiquark in the permanently closed channel(s) move in relative waves, whereas the mesons in the scattering channels are in relative  and waves. For the and we use one Schrödinger equation with two permanently closed channels, one for the pair and one for the pair. The mixing occurs in our model quite naturally via the coupling to scattering channels which contain strange mesons. We will discuss the results furtheron.
In the first place, we do not alter the effective quark masses, nor the universal harmonic oscillator frequency. The only place where we allow some minor changes, if necessary, is in the potential which couples the confinement and the decay sectors. For the vector and pseudoscalar mesons, the socalled color splitting could be accounted for by a component of this potential. As a result of our calculations for the scalar mesons, we conclude that in their case other possible interactions seem to compensate the effects of color splitting. So we decide to set to zero the parameter which regulated the color splitting in the case of the vector and pseudoscalar particles, and not to take other contributions into account. The only legitimations of this procedure are the facts that the results came out reasonable and that it is not very relevant for the point we want to make in the present paper. The other two parameters in the coupling potential remain unaltered with respect to the corresponding parameters in the case of vector and pseudoscalar mesons.
3 Results
Let us first discuss wave scattering. The lowest bound state of our confining potential for pairs has a mass of about 1.3 GeV, which is at precisely the same place as the ground state of other boundstate meson models. If we turn on the overall coupling constant of the transition potential, bound states show up as resonances in scattering. At the model value of the overall coupling constant, which was obtained from the analysis of pseudoscalar and vector mesons [23], a pole shows up with a real part of about 1.3 GeV, which accidentally equals the abovementioned boundstate mass. Naively we might expect that one would only find a resonating structure in scattering in that energy domain. However, Fig. (1) shows that the calculated phase shifts have structures at much lower energies, which indicates that lowlying resonance poles have been generated.
We can scan the complex energy plane for these poles in the scattering matrix, finding one pole at about 450 MeV with a roughly 250 MeV imaginary part, and another pole at the (980) position. The imaginary part of the first pole is so large that a simple BreitWigner parametrization is impossible, and large differences between the “mass” of the resonance and the real part of the pole position will occur. How these poles are connected to the harmonicoscillator bound states is a very technical story, which is beyond the scope of this paper; suffice it to state that such a connection exists. As we have discussed in [27], these poles are special features of wave scattering and do not show up in  and higherwave scattering, which explains quite naturally why they are not found there.
Figure (1) also shows that the new structures at low energies are in reasonable agreement with the experimental situation. A criticism which might come up if one inspects this figure in more detail is that the theoretical phase shifts do not fit the data to a high precision, but only roughly follow the experimental slope in the data. However, this is not a fair criticism, since we are comparing our calculations with the raw data, with unsubtracted background, which are presently the only available data.
In our model we left out of consideration all possible finalstate interactions in the scattering channels, like meson exchange, Pomeron exchange, quark interchange, etc.. Moreover, the form of the transition potential may be too simple, since, for example, we have taken a local transition potential, and it could also have a more complicated dependence due to more sophisticated mesondecay form factors. So our calculations better do not follow the data very accurately. It remains, however, a pity that no analysis exists for mesonmeson scattering which subtracts the known effects and leaves us with the consequences of the remaining interactions, a strategy that is nowadays popular in analyzing nucleonnucleon scattering data [30], because then we could really see how good the remaining interactions are accounted for in our approach. From our present calculations, we must conclude that finalstate interactions will probably alter the phase shifts a bit in the region around 600 MeV in order to change the slope of the curve towards the data. Note that the phase shifts for low energies are almost completely accounted for by the coupling to the permanently closed channels.
In wave scattering, the data are limited to cross sections in the energy domain of the . Here we find that straightforward calculations lead to problems with the position of the resonance. We suspect that these difficulties are connected to the problem. Our strategy in this case is discussed below, and the result is depicted in Fig. (2), where we shifted the calculated cross section by 20 MeV in order to get the peak values of the experimental and the theoretical curves on top of each other.
The results for are depicted in Fig. (3). We see there that the phase shifts for low energies are rather well produced by the model, but at higher energies only a rough description of the data is given: the number of resonances at some energies agrees with the data but the detailed structure is not reproduced at all. Also here we have difficulties which are presumably related to the problem.
In previous investigations [23], we took a nonstrange quark content for the meson and a strange one for the meson, which is called ideal mixing. The data in the case of scalar mesons are, however, more sensitive to the quark contents of the ’s and we found the following: the channel in the isodoublet case is much less coupled than what would follow from our general approach as described in [28], because the data show that there is not much inelasticity below . The best result is obtained if the channel is completely decoupled and enhanced to compensate.
Something similar appears to be necessary in the isoscalar case: if we take for the and for the system, we find the best results for the pole, although the complete system is not very sensitive to these changes. The , however, is very sensitive to our approach, since the lowest threshold is . In this case we have to reduce the coupling by a factor 1/6 and to enhance the coupling to compensate. This would be the case if the mixing angle between and was . The plus sign is puzzling, because Törnqvist [2] claimed that he needs the standard [17] in order to fit the data. A more rigorous study on this point is in preparation, which shows that part of the problem might stem from our choice of transition potential.
4 Conclusions
If we allow only minor differences in the transition potential and take some action with respect to the mesons, we find that the scattering matrices for wave mesonmeson scattering agree reasonably with the data. An inspection of the complex energy plane shows the following poles in the various scattering matrices: ( MeV), ( MeV), ( MeV), and ( MeV). Preliminary results show that the and poles are rather sensitive to the transition potential. The is, however, always somewhere around 500 MeV central value, with a large width. The may vary more and even become slightly heavier than 1 GeV [32], with a rather large width. Nevertheless, the main conclusion is that these poles occur as normal configurations with mesonmeson admixtures.
Our calculations show clearly that there is no need to incorporate channels in our meson model. This indicates that there is no phenomenological reason why genuine configurations should couple strongly to mesonmeson channels.
Appendix A Appendix
The resonances are described by the set of Schrodinger equations
(1) 
which consist of confined (permanently closed) channels and free (open or closed scattering) channels. So , , are diagonal matrices containing the orbital angular momenta, the reduced masses, and the momenta of the various channels. The latter two quantities are determined from
(2) 
Here, and stand for the masses of the quarks in the confined channels or the meson masses in the scattering channels. The latter are taken to be the experimental values, while the quark masses were determined [23], for the pseudoscalars and vectors, to be
(3) 
The nonrelativistic limit of (2) is used in the confined channels, and for energies lower than the threshold also in the free channels.
The potential matrix reads
(4) 
The confining potential is a diagonal matrix containing the massdependent harmonic oscillators
(5) 
The matrix describes possible finalstate interactions. The transition potential is taken to be
(6) 
where is the universal coupling constant, the transition radius, and a phenomenological factor, with . The abovementioned parameters were also fixed [23] by the pseudoscalars and vectors to
(7) 
In comparison to the and case, we left out the color term, which was necessary for the pseudoscalar/vector splitting. This is, of course, not a very serious intervention, since for wave systems the color splitting is anyhow much smaller than for wave systems [32].
The numbers are the relative decay couplings. If we pursue the concepts behind the confining potential further to the phase of the pair creation, we are led to the assumption that all the interquark forces are harmonicoscillator forces during this process. This idea provides us a scheme in which the possible decay channels and their relative strengths can be calculated [28]. Their values are listed in Table 1.
Decay Products  Initial Mesons  

Channel  Spin  and  
0  3/40  
0  1/40  
0  1/16  
0  1/40  1/16  1/24  
0  1/40  
2  1/2  
0  1/120  
2  1/6  
0  1/120  1/48  1/72  
2  1/6  5/12  5/18  
0  1/48  
2  5/12  
0  1/12  
0  
0  1/36  
2  5/9  
0  1/16  
0  1/48  
0  1/24  
0  1/48  
2  5/12  
0  1/144  
2  5/36  
0  1/72  
2  5/18 
The and denote the pure and states, respectively. Of course, these are not the physical and . See the Conclusions for a discussion about this point. The (wave) scattering is described in one system of equations, with two confined channels containing an and an pair. These systems are linked to each other via decay in which strange mesons occur.
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