In this talk I show recent results on how many excited baryon resonances appear as systems of one meson and one baryon, or two mesons and one baryon, with the mesons being either pseudoscalar or vectors. Connection with experiment is made including a discussion on old predictions and recent results for the photoproduction of the resonance, as well as the prediction of one baryon state around 1920 MeV which might have been seen in the reaction.
keywords:dynamically generated resonances, chiral dynamics, hidden gauge formalism for vector meson interaction.
Pacs:13.75.Lb, 12.40.Vv, 12.40.Yx, 14.40.Cs
The realization that the dynamics of QCD in the hadron world can be addressed at low energies by means of effective theories in which the building blocks are the ground state mesons and baryons Weinberg:1978kz () has produced tools to address the interaction of mesons or mesons and baryons, mainly through chiral Lagrangians, which have had a tremendous impact in our understanding of the spectrum of mesons and baryon resonances. We all accept that ground states of mesons and baryons are made of or three respectively. Yet, the spectrum of excited hadronic states can be much richer as we shall see.
The building blocks in these chiral theories are the low energy hadrons, such as the proton and baryons of its SU(3) octet. To these one adds also the decuplet of the , considered as spin realignments of the three quark ground state. The basic mesons are the pion and mesons of its octet, to which one also adds the nonet of the , which also corresponds to spin realignments of the ground state .
What about baryon resonances? The logical answer is that they are excitations of the quarks, which is the essence of quark models. This is plausible, but things could be more complicated.
Let us recall basic facts from the baryon spectrum. The first excited states are the and the . In quark models this will require quark excitation of around 500-600 MeV. If this is the case, one may think that it takes less energy to create one pion, or two (140-280 MeV). The question is whether they can be bound or get trapped in a resonant state. How do we know if this can occur? We need dynamics, a potential for the interaction of mesons with ground state baryons and then solve the Schroedinger equation (Bethe Salpeter equation with mesons treated relativistically) in coupled channels. This information can be extracted from chiral Lagrangians: the effective theory of QCD at low energies. This is the philosophy behind the idea of dynamically generated baryons: Many resonances are generated in this way, like the states from meson baryon: Kaiser:1995eg (), two cola () or the states from two mesons and a baryon, like the and others alberto1 (); alberto2 ().
From the pseudoscalar-baryon octet interaction there are many states generated and one sees them as peaks in the scattering matrices or poles in the complex plane Kaiser:1995eg (); angels (); inoue (); carmenjuan (); hyodo (); borasoyweise (); borasoyulf (); Oller:2006jw (). A feature of the chiral unitary approach is its great predictive power, with the risk that some of the predictions might not be fulfilled in Nature. But, so far, predictions are corroborated in production reactions, partial decay rates, meson baryon scattering amplitudes, helicity amplitudes, transition form factors review (); puri (). As an example we recall in the next section some of the early calculations on photoproduction of the , taking advantage that two experiments are now reported in this Conference ten years after the predictions were made.
2 Photoproduction of the
Ten years ago, with Spring8/Osaka in its initial stage, an application of the chiral unitary approach was made to predict cross sections for photoproduction of the Nacher:1998mi () in the , , reactions. The cross sections found were within measurable range, the could be clearly seen in the mass distribution, as usual, and the signal was much larger than the background. The experiment was early started at Spring8 and preliminary results were shown in 2003 ahn (). Five years later final results were published in Niiyama:2008rt (), which have been reported in this Conference ahnhere (). Also Jeff Lab has carried out the experiment with a different set up and the results have also been presented in this Conference moriya (). The model used in Nacher:1998mi () was a minimal model, in which only the vertex coming from minimal coupling, with P and B a pseudoscalar meson from the octet of the and B a baryon of the octet of the , was used. Possible effects of baryon resonances in the entrance channels were neglected. In spite of this, the agreement of data with the predictions is quite fair. A large signal is seen with small background. The size of the cross section is within 50 % of the predictions. In adidtion, the observed , distributions are not equal; they are shifted by a few MeV as predicted. This is a consequence of the effect of an isospin I=1 amplitude, which acts constructively in one case and destructively in the other. The was predicted to be roughly the average of the other two cross sections. The details of the recent experiments can be seen in the devoted talks ahnhere (); moriya () and are rather interesting. A remarkable thing is that the peak positions for , production seem to be reversed in moriya () than predicted, and in ahnhere () they seem to be angle dependent, with opposite trends at forward and backward angles.
The new experimental information obtained calls for a theoretical revival of the theory to the light of the findings made in chiral unitary approaches in the last decade. At stake are issues like the nature of the resonance, the existence of the two resonances for which experimental evidence has been claimed magas (); jidoseki (), and the possibility that the I=1 amplitude, which is clearly visible in the different , cross sections, could be of resonant character, evidencing a new resonance around 1400 MeV, for which hints were seen in cola (); ollerulf (). Recent claims for this resonance have been made in wuzou () from the study of the reaction.
3 Resonances from the interaction of vector mesons with baryons
This is a very novel development since, as we shall see, some of the high mass baryon resonances can be represented like bound states of vector mesons and baryons, either from the octet of stable baryons or the decuplet.
We follow the formalism of the hidden gauge interaction for vector mesons of hidden1 (); hidden2 (); hidden3 () (see also hidekoroca () for a practical set of Feynman rules). The Lagrangian involving the interaction of vector mesons amongst themselves is given by
where the symbol stands for the trace in the space and is given by
where is , with MeV the pion decay constant. The magnitude is the ordinary matrix of the vectors of the octet of the
The lagrangian gives rise to a contact term coming from , as well as to a three vector vertex
In this case one finds an analogy to the coupling of vectors to pseudoscalars given in the same theory by
where is the SU(3) matrix of the pseudoscalar fields.
where is now the ordinary SU(3) matrix of the baryon octet
With these ingredients we can construct the Feynman diagrams that lead to the and interaction, by exchanging a vector meson between the pseudoscalar or the vector meson and the baryon, as depicted in Fig. 1.
From the diagram of Fig. 1(a), and under the low energy approximation of neglecting in the propagator of the exchanged vector, where is the momentum transfer, one obtains the same amplitudes as obtained from the ordinary chiral Lagrangian for pseudoscalar-baryon octet interaction, namely the Weinberg-Tomozawa terms. The approximation of neglecting the three momenta of the vectors implies that in eq. (3) corresponds to the exchanged vector and the analogy with eq. (4) is more apparent. Note that becomes and the signs of the Lagrangians also agree.
A small amendment is in order in the case of vector mesons, which is due to the mixing of and the singlet of SU(3), , to give the physical states of the and the mesons. The practical rule is simple and can be found in angelsvec (). Upon the approximation consistent with neglecting the three momentum versus the mass of the particles (in this case the baryon), we can just take the component of Eq. (5) and then the transition potential corresponding to the diagram of 1(b) is given by
where are the energies of the incoming and outgoing vector meson. The coefficients of eq. (6) can also be found in angelsvec (), where one can see that the cases with , and , the last two corresponding to exotic channels, have a repulsive interaction and do not produce poles in the scattering matrices. However, the sectors , , and are attractive and one finds bound states and resonances in these cases.
with being the loop function of a vector meson and a baryon. This function is convoluted with the spectral function of the vector mesons to take into account their width as done in nagahiro ().
In this case the factor , appearing in the potential , factorizes also in the matrix for the external vector mesons. This trivial spin structure is responsible for having degenerate states with spin-parity for the interaction of vectors with the octet of baryons and for the interaction of vectors with the decuplet of baryons.
What we have done here for the interaction of vectors with the octet of baryons can be done for the interaction of vectors with the decuplet of baryons, and the interaction is obtained directly from that of the pseudoscalar-decuplet of baryons studied in kolodecu (); Sarkar:2004jh (). The study of this interaction in vijande (); souravbao () leads also to the generation of many resonances which are described below.
We search for poles in the scattering matrices in the second Riemann sheet, as defined in previous works luisaxial ().
|pole position||real axis|
For the states there is the with , which could correspond to the state we find with the same quantum numbers around the same energy. We also find in the PDG the , which could be the near degenerate spin parter of the that we predict in the theory. It is interesting to recall that in the study of Ref. mishajuelich () a pole is found around 1700 MeV, with the largest coupling to states. Around 2000 MeV, where we find another resonance, there are the states and , with and respectively, showing a good approximate spin degeneracy.
For the case there is in the PDG one state, the with , remarkably close to the energy were we find a state. The state obtained around 1900 MeV could correspond to the cataloged in the PDG with unknown spin and parity.
The case of the states having is rather interesting. The state that we find around 1830 MeV, could be associated to the with . More interesting seems to be the case of the state obtained around 1990 MeV that could be related to two PDG candidates, again nearly degenerate, the and the , with spin and parity and respectively.
Finally, for the case of the cascade resonances, , we find two states, one around 2040 MeV and the other one around 2080 MeV. There are two cascade states in the PDG around this energy region with spin parity unknown, the and the . Although the experimental knowledge of this sector is relatively poor, a program is presently running at Jefferson Lab to improve on this situation Nefkens:2006bc ().
|pole position||real axis||name||status||mass||width|
We also can see that in many cases the experiment shows the near degeneracy predicted by the theory. Particularly, the case of the three resonances around 1920 MeV is very interesting. One observes a near degeneracy in the three spins , as the theory predicts. It is also very instructive to recall that the case of the is highly problematic in quark models since it has a excitation and comes out always with a very high mass vijande (); pedro ().
The association of states found to some resonances reported in the PDG for the case of , and states looks also equally appealing as one can see from the table.
The reasonable results reported here produced by the hidden gauge approach should give a stimulus to search experimentally for the missing spin partners of the already observed states, as well as possible new ones.
4 States of two mesons and a baryon
There are two specific talks on this issue in the Workshop albertotalk (); jidotalk (). I will summarize a bit the important findings in this area by different groups. In alberto1 (); alberto2 () a formalism was developed to study Faddeev equations of systems of two mesons and a stable baryon. The interaction of the pairs was obtained from the chiral unitary approach, which proves quite successful to give the scattering amplitudes of meson-meson and meson-baryon systems in the region of energies of interest to us. The spectacular finding is that, leaving apart the Roper resonance, whose structure is far more elaborate than originally thought Krehl:1999km (); Dillig:2004rh (), all the low lying excited states are obtained as bound states or resonances of two mesons and one baryon in coupled channels.
Particularly relevant to this Conference is the issue of a possible bound state of . In jidothree (), using variational methods, the authors found a bound state of , with the being in the state jidothree (). The system was studied a posteriori in albertopheno () and it was found to appear at the same energy and the same configuration, although with a mixture of , see fig. 2. This state appears around 1920 MeV with . In a recent paper albertoulf () some arguments were given to associate this state with the bump that one sees in the reaction around this energy, which is clearly visible in recent accurate experiments Bradford:2005pt (); Sumihama:2005er (). If this association was correct there would be other experimental consequences, as an enhanced strength of the cross section close to threshold, as well as a shift of strength close to the threshold in the invariant mass distribution of the kaon pair. This experiment is right now under study nakanotalk (). Another suggestion of albertoulf () is to measure the total spin and amplitudes, the -direction along the photon momentum, since this would discriminate the cases where the peak around 1920 MeV is due to a or a resonance.
This work is partly supported by the EU contract No. MRTN-CT-2006-035482 (FLAVIAnet), by the contracts FIS2006-03438 FIS2008-01661 from MICINN (Spain) and by the Generalitat de Catalunya contract 2005SGR-00343. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU.
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