Plausible explanation of the puzzle
From a Faddeev calculation for the system we show the plausible existence of three dynamically generated baryon states below 2.3 GeV whereas only two resonances, and are cataloged in the Particle Data Book Review. Our results give theoretical support to data analyses extracting two distinctive resonances, and from which the mass of is estimated. We propose that these two resonances should be cataloged instead of This proposal gets further support from the possible assignment of the other baryon states found in the approach in the with sectors to known baryonic resonances. In particular, is naturally interpreted as a bound state.
Baryon spectroscopy (see Ref. Kle2010 () for a recent general review) is an essential tool to analyze the baryon structure. Data on baryon masses and transitions, regularly compiled in the Particle Data Book Review (PDG) pdg2010 (), allow us when confronted with theoretical calculations to learn about the effective constituent degrees of freedom and their interactions inside the baryon. From the experimental point of view the information on baryonic resonances mainly comes from pion-nucleon ( scattering experiments. The photon nucleon ( reactions have led to advancement in the field, reconfirming many known resonances and claiming evidence for new ones. From the theoretical point of view it has become clear in the last years that the primitive quark model view of a baryon as formed by three effective valence quarks ( may require the implementation of higher Fock space terms, in the form of … or meson-baryon, meson-meson-baryon… components to provide a satisfactory explanation of some baryonic resonances. Paradigmatic cases are the and the For the relevance of was first pointed out in 1977 Jon77 () (more recently the role of has been also emphasized Mag05 ()). For the important role of has been recognized Pej09 (). These are particular examples of a more general situation where a model calculation (providing a reasonable overall description of the whole spectrum) overestimates the mass of a resonance so that a meson-baryon threshold lies in between the calculated value and the experimental data Gon07 (). As a consequence, the meson-baryon component may be dominant when the meson-baryon interaction is attractive, and the dynamical generation of the resonance from these hadronic degrees of freedom may be more efficient than a quark model description which would require and/or higher Fock space terms (note that the meson and baryon of the threshold might also correspond to dynamically generated states). This argument can be extended to resonances where the meson-baryon thresholds are above the masses if the meson-baryon interaction is sufficiently attractive as to provide the binding required by data.
As a matter of fact, meson-baryon components are present in all baryonic resonances. In some cases the contribution of these components to the masses may be properly taken into account by making use of the description with effective parameters for the quark-quark interaction. In other cases, as explained above, this may not be possible. The intermediate situation corresponds to the case of resonances for which both approximations may reasonably reproduce their masses. In such a case the two descriptions may be at least to some extent equally valid alternatives, the values of their effective parameters taking implicitly into account the non-explicit ( or meson-baryon) component contribution.
In this article we take these considerations into account to analyze and resonances with sectors. The motivation for this study comes mainly from the puzzle concerning the ( since the nominal mass of this resonance does not correspond in fact to any experimental analysis but to an estimation based on the value of the masses ( MeV and MeV) extracted from different data analyses pdg2010 (). This makes feasible the existence of a hidden resonance which could not be reasonably accommodated within a framework description what might be indicating its dynamically generated character. The theoretical examination of such a possible character is the main objective of this article. For this purpose we shall follow a procedure based on the combination of chiral Lagrangians with nonperturbative unitary techniques in coupled channels of baryons and/or pseudoscalar and/or vector mesons. This scheme has been very fruitful in the description of other baryonic resonances through the analysis of the poles of the meson-baryon or meson-meson-baryon scattering amplitudes (see for instance Ref. meba09 () and references therein).
If existing, the resonance could be generated from as suggested in Ref. Gon07 (). Since the has been dynamically generated as a bound state of in the sector (the same interaction generating the for ) Pej09 (), we shall investigate the three-body -- system but keeping the strong correlations of the system which generate the . In such a situation the use of the Fixed Center Approximation (FCA) to the Faddeev equations is justified Gal:2006cw (). For the sake of consistency, and resonances which can be dynamically generated altogether with will be also analyzed.
The contents of the article are organized as follows. In Section II, we revisit the cataloged resonances and comment on their description. In Section III, we present the FCA formalism to analyze the system. The analysis of the scattering amplitude is extracted in Section IV and a tentative assignment peaks in the amplitudes to baryonic resonances is proposed. Finally, in Section V we summarize our approach and main findings.
Ii The puzzle
In the PDG pdg2010 () there is only a well established resonance, ( and fair evidence of the existence of another one, (. However, a careful look at this last resonance shows that its nominal mass is in fact estimated from and , respectively, extracted from three independent analyses Man92 (); Vra00 (); Cut80 () of different character: in Ref. Man92 (), multichannel in Ref. Vra00 () and in Ref. Cut80 (). Moreover a recent new data analysis has reported a with a pole position at MeV Suz10 (). In this last analysis, incorporating data, the resonance is obtained from a bare state at 2162 MeV through its coupling to meson-baryon channels. This bare state represents the quark core component of the resonance within this calculation framework.
It is important to remark that i) all the analyses extract the and ii) the non extraction of in most of the mentioned analyses may be related to the restricted range of energy examined (typically below 2200 MeV).
From a description the is naturally accommodated as the lowest state in the second energy band of a double harmonic oscillator potential (one oscillator for each Jacobi coordinate of the system) that provides (up to perturbative terms) a reasonable overall description of the whole baryon spectrum capstick (). Actually quark models predict two states close in energy for the lowest symmetric and mixed symmetric orbital configurations in the second energy band. The is then assigned to the orbitally symmetric state. Experimental evidence for the mixed symmetric one has also been reported near MeV manleyprl1984 (). Similarly, the reported , with a more uncertain mass ( MeV), may be reasonably located in the fourth energy band. On the contrary, lying far below the energy of the lowest state in the second energy band (the first available band by symmetry to a state) could not be accommodated as a state without seriously spoiling the overall spectral description.
The same kind of problem was tackled in Ref. Pej09 () regarding the description of with a mass much lower than the corresponding to the third energy band, the first available band for such a state. There the consideration of the channel whose threshold ( MeV) lies close above the experimental mass of the resonance and far below the mass ( MeV) allowed for an explanation of and its partners, and as bound states in the sector. In addition and were also well described as bound states in the sector, although the bigger sensitivity in this case to the cutoff parameter employed left some room for alternative assignments of these bound states to nucleonic resonances Sar10 (). It should be pointed out that, contrary to the and its partners, these nucleon resonances around MeV can also be reasonably described as states in the first energy band capstick (). Therefore a more reliable explanation of data should include the contribution of the states as well as of the possible bound states.
Back to one can easily identify a meson-baryon threshold, MeV, in between the mass ( MeV) and the data. Then one can wonder about the possibility that the system may give rise to a bound state which could provide theoretical support to the fair evidence of the existence of . Actually this bound state nature could explain why this resonance is extracted in some data analyses but not in others. It turns out that only analyses reproducing the production cross section data extract it. Let us note that this would be a necessary condition to extract if corresponding to a state (let us recall that decays to and to with branching fractions of 40% and 55% respectively).
To examine this possibility we perform next an analysis of the system by assuming that is a bound state. Although according to our discussion above, a combined ( description of would be more appropriate we shall consider only the bound state option (adequate to our formalism) and assume that the value of the parameter (cutoff or subtraction constant) involved in the dynamical generation of from takes implicitly into account the component. Furthermore, the same consideration is extended to the dynamical generation of resonances from
We should finally notice that may couple to other wave meson-baryon channel, like We do not expect this channel to play any relevant role in the generation of since its threshold is far above in energy. However, the channel could influence the possible generation of higher mass resonances. For the sake of simplicity we shall not include it in our calculation. We should then keep in mind that the calculated masses for the higher resonances have a higher degree of uncertainty.
The interaction of a particle with a bound state of a pair of particles at very low energies or below threshold can be efficiently and accurately studied by means of the fixed center approximation (FCA) to the Faddeev equations for the three-particle system chan62 (). We shall extend this formalism to include states above threshold and apply it to The analysis of the scattering amplitude will allow us to identify dynamically generated resonances with states listed in the PDG.
The FCA to the Faddeev equations has been used with success recently in similar problems of bound three-body systems and contrasted with full Faddeev or variational calculations. In this sense, the system has been studied with the FCA in Ref. Xie:2010ig (), with very similar results as found in the full Faddeev calculations in Refs. alberto1920 () and in the variational estimate in Ref. jido1920 (). Similarly, the study Bayar:2011qj () of the system within the FCA has led to very similar results as the variational calculations of Dote:2008hw () when the chiral amplitudes are used, or the Faddeev calculation of Ikeda:2010tk (), when the energy dependence of the amplitude is used in agreement with chiral dynamics.
The important ingredients in the calculation of the total scattering amplitude for the -- system using the FCA are the two-body -, - and - unitarized wave interactions from the chiral unitary approach. Although the form of these interactions have been detailed elsewhere Pej09 (); Sar05 (); Roc05 (), we shall briefly revisit in Subsection A the - case. This will allow us to remind the general procedure of calculating the two-body amplitudes entering the FCA equations. Then in Subsections B and C the calculation of the -( amplitude will be detailed.
iii.1 Unitarized interaction
The interaction has been analyzed in the framework of the hidden gauge formalism gsl () in terms of the exchange of a meson in the channel between the and the Pej09 (); Sar10 (). Under the low energy approximation of neglecting in the propagator of the exchanged vector meson, where is the momentum transfer, and also the three momentum of the vector meson, one obtains for the potential the form
where MeV is the pion decay constant, and is the energy (polarization) of the incoming/outgoing rho meson and an isospin dependent coefficient with values
Then one can solve the Bethe-Salpeter equation with the on-shell factorized potential and, thus, the -matrix will be given by
with the potential of Eq. (1) in the isospin basis without the polarization factor . is the loop function for intermediate states that can be regularized both with a cutoff prescription as done in Ref. Pej09 (), or with dimensional regularization in terms of a subtraction constant as done in Ref. Sar10 (). Here we shall make use of the dimensional regularization scheme better suited to analyze the sensitivity of our results against variations of the parameter (small changes of the subtraction constant translate into significant changes in the values of the cutoff Oll01 ()). The expression for is then
where is the total incident momentum, which in the center of mass frame is being the invariant mass of the system. In Eq. (6), is the scale of dimensional regularization and the subtraction constant. Note that the only parameter dependent part of is ln Due to renormalization group invariance any change in is reabsorbed by a change in through ln so that the amplitude is scale-independent. In Eq. (6), is the momentum of the or the in the center of mass frame, which is given by
However, since the baryon and meson have large total decay widths and , they should be taken into account. For this purpose we replace the function in Eq. (5) by
with the triangle function. We shall take MeV and MeV.
In addition, one issue worth mentioning is that the spin dependence comes from the factor of the meson. The spin of the baryon does not appear in the present formalism due to the approximations done. The scalar structure indicate wave interaction of , therefore, one has degeneracy for the states for both and .
In order to evaluate the value of the scattering amplitude we have to fix the parameter ln As explained above the choice of is rather arbitrary since a change in it is reabsorbed by a change in Values of from 630 MeV to 1000 MeV have been employed in the literature. We choose MeV, a value rather close to the cutoff employed in Ref. Pej09 () ( MeV), and fix according to our comment at the end of the Section II, to get the ( bound state at 1675 MeV as corresponding to the estimated mass of in Ref. pdg2010 () We get (if instead we had used MeV we would have obtained
In Fig. 1 the modulus squared of the scattering amplitude as a function of the invariant mass of the system for is shown. Note that in the sector the bound state is located at MeV, a little bit lower than its location in our previous study Pej09 () as a consequence of the fine tuning of the parameter to get the bound state at MeV. Notice anyhow that the assignment of the states at 1887 MeV to and remains unambiguous.
iii.2 Single-scattering contribution for the interaction with the system
The FCA to the Faddeev equations for the three body -- system is depicted diagrammatically in Fig. 2. The external meson interacts successively with the baryon and meson which form the (). In terms of two partition functions and , the FCA equations are
where is the total three-body scattering amplitude and () 111In the present work, the label represents baryon of the compound system, while represents the meson. account for the diagrams starting with the interaction of the external particle with particle of the compound system. Hence, represent the and unitarized scattering amplitudes whose forms were derived in Refs. Sar05 () and Roc05 () respectively to which we refer for details. In the above equations, is the loop function for the meson propagating inside the resonance which will be discussed later on.
More specifically is the appropriate combination of the and unitarized two-body scattering amplitudes (, , and ) whereas stands for the corresponding combination of the two-body scattering amplitudes(, , ). For example, let us consider a cluster of in isospin , the constituents of which we call and and the external meson we call number . The isospin states are written as
where the kets on the right hand sides indicate the components of the particles and , .
The scattering potential for the single scattering contribution (Fig. 2 (a) + term with interaction initiated on can be easily obtained in terms of the two body potentials and derived in Refs. Sar05 () and Roc05 ().
Here we write explicitly the case of and total isospin ,
where the notation for the states in the third equality is for the matrix element, and for the one. This leads to the following amplitudes for the single scattering contribution,
Proceeding in a similar way, we can get all the amplitudes for the single scattering contribution required in the present calculation which are shown in Table 1.
It is worth noting that the argument of the total scattering amplitude is a function of the total invariant mass squared , while the argument in is and in is , where and are the invariant masses squared of the external meson with momentum and () inside the with momentum (), which are given by
where stands for the volume of a box where we normalize to unity our plane wave states. In Eq. (19), is the form factor of the as a bound state of . This form factor was taken to be unity neglecting the momentum in Ref. fcarhorho () since only states below threshold were considered. To consider states above threshold, we project the form factor into s-wave, the only one that we consider. Thus
is the module of the momentum of meson in the center of mass frame when is above the threshold of the system, otherwise, equals zero. The expression of is given in the next section 222The form factor that we use is suited to a molecule with two components with equal masses. Some different recoil corrections are needed when the two masses are different YamagataSekihara:2010yz (), but the results only affect moderately the peak around MeV..
In Fig. 3, we show the projection over s-wave of the form factor for the single scattering contribution as a function of the total invariant mass of system.
iii.3 Double-scattering and resummation contribution
In order to obtain the amplitude of the double-scattering contribution (Fig. 2 (b) + term with interaction initiated on ) one can proceed in the same way as in the case of the multi-rho meson interaction in Ref. fcarhorho (). The expression for the matrix for the double scattering is ()
where is the form factor, and we will take in the center of mass frame, .
Following the approach of Ref. danielprd81 (), we can get the expression for the form factor ,
where the normalization factor is
with and the total decay width of the baryon and the meson, respectively, taken as in Subsection A equal to MeV and MeV. Since the effect of the widths of baryon and meson is not very important.
To connect with the dimensional regularization procedure we choose the cutoff such that the value of the function of Eq. (6) at threshold coincides in both methods. Thus for MeV we get MeV as required.