Radiative decay into baryon of dynamically generated resonances from the vectorbaryon interaction
Abstract
We study the radiative decay into and a baryon of the SU(3) octet and decuplet of nine and ten resonances that are dynamically generated from the interaction of vector mesons with baryons of the octet and the decuplet respectively. We obtain quite different partial decay widths for the various resonances, and for different charge states of the same resonance, suggesting that the experimental investigation of these radiative decays should bring much information on the nature of these resonances. For the case of baryons of the octet we determine the helicity amplitudes and compare them with experimental data when available.
Institute of Theoretical Physics, College of Applied Sciences,
Beijing University of Technology, Beijing 100124, China
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de ValenciaCSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain
1 Introduction
In a recent paper [1], the interaction was studied within the local hidden gauge formalism for the interaction of vector mesons. The results of the interaction gave a natural interpretation for the as a bound state, which otherwise is extremely problematic in quark models since it involves a excitation and appears with much higher mass. At the same time two states with were obtained, degenerate with the , which could be accommodated with two known states in that energy range. Also, three degenerate states with were obtained, which were more difficult to identify with known resonances since that sector is not so well established. The work of [1] was extended to the SU(3) sector in [2] to account for the interaction of vectors of the octet with baryons of the decuplet. In this case ten resonances, all of them also degenerate in the three spin states, were obtained, many of which could be identified with existing resonances, while there were predictions for a few more. At the same time in [2] the poles and residues at the poles of the resonances were evaluated, providing the coupling of the resonances to the different vectorbaryon of the decuplet components.
One of the straightforward tests of these theoretical predictions is the radiative decay of these resonances into photon and the member of the baryon decuplet to which it couples. Radiative decay of resonances into is one of the observables traditionally calculated in hadronic models. Work in quark models on this issue is abundant [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. For resonances which appear as dynamically generated in chiral unitary theories there is also much work done on the radiative decay into [19, 20, 21, 22, 23]. Experimental work in this topic is also of current interest [25, 26, 27].
In the present work we address the novel aspect of radiative decay into a photon and a baryon of the decuplet of the , since the underlying dynamics of the resonances that we study provides this as the dominant mode of radiative decay into photon baryon. This is so, because the underlying theory of the studies of [1, 2] is the local hidden gauge formalism for the interaction of vector mesons developed in [28, 29, 30, 31], which has the peculiar feature, inherent to vector meson dominance, that the photons couple to the hadrons through the conversion into a vector meson. In this case a photon in the final state comes from either a . Thus, the radiative decay of the resonances into is readily obtained from the theory by taking the terms with in the final state and coupling the to any of the final vector mesons. This procedure was used in [32] and provided good results for the radiative decay into of the and mesons which were dynamically generated from the interaction within the same framework [33]. This latter work was also extended to the interaction of vectors with themselves within SU(3), where many other states are obtained which can be also associated with known resonances [34]. The radiative decay of the latter resonances into or a and a vector has been studied in [35], with good agreement with experiment when available. Given the success of the theory in its predictions and the good results obtained for the decay of the , and mesons, the theoretical framework stands on good foot and the predictions made should be solid enough to constitute a test of the theory by contrasting with experimental data. The extension of the work of [1, 2] to the interaction of vector mesons with baryons of the octet of the proton has also been successful [36] and nine resonances, degenerated in spinparity and , appear dynamically generated in the approach, many of which can be naturally associated to know resonances in the PDG [37]. We also extend the present work to study the radiative decay of these resonances into a photon and a baryon of the octet. In this case we can also evaluate helicity amplitudes and compare them with experimental results when available.
The experimental situation in that region of energies is still poor. The PDG [37] quotes many radiative decays of resonances, and of the helicity amplitudes for decay of resonances into , with N either proton or neutron. However, there are no data to our knowledge for radiative decay into , with B a baryon of the decuplet. The reason for it might be the difficulty in the measurement, or the lack of motivation, since there are also no theoretical works devoted to the subject. With the present work we hope to reverse the situation offering a clear motivation for these experiments since they bear close connection with the nature invoked for these resonances, very different to the ordinary three quark structure of the baryons.
The numbers obtained for the radiative widths are well within measurable range, of the order of 1 MeV, and the predictions are interesting, with striking differences of one order of magnitude between decay widths for different charges of the same resonance.
The work will proceed as follows. In the next two Sections we present the framework for the evaluation of amplitudes of radiative decay. In Section 4 we show the results obtained for the different resonances generated with the baryon decuplet. Section 5 introduces the equations for the baryon octet, which are used in Section 6 to obtain results for the deacy width of the resonances dynamically generated with a vector and the baryon octet. In Section 7 we present the results for the helicity amplitudes of some resonances used in the previous section, and in Section 8 we finish with some conclusions.
2 Framework
In Ref. [1, 2], the scattering amplitudes for vectordecuplet baryon are given by
(1) 
where , refer to the initial and final vector polarization and the matrix is diagonal in the third component of the baryons of the decuplet. The transition is diagonal in spin of the baryon and spin of the vector, and as a consequence in the total spin. To make this property more explicit, we write the states of total spin as
(2) 
and
(3) 
where are the ClebschGordan coefficients and the polarization vectors in spherical basis
(4) 
We can write Eq. (1) in terms of the projectors as
(5) 
Since the ClebschGordan coefficients satisfy the normalization condition
(6) 
we have then
We can depict the contribution of a specific resonant state of spin to the amplitude described by means of Fig. 1.
Then the amplitude for the transition of the resonance to a final vectorbaryon state is depicted by means of Fig. 2.
As shown is Ref. [1, 2], the scattering amplitudes develop poles corresponding to resonances and a resonant amplitude is written as Eq. (1) with given by
(8) 
with and the couplings to the initial and final states. Accordingly, the amplitude for the transition from the resonance to a final state of vectorbaryon is given by
(9)  
When calculating the decay width of the resonance into we will sum over the vector and baryon polarization, and average over the resonance polarization . Thus, we have
where in the first step we have permuted the two last spins in the ClebschGordan coefficients and in the second we applied their orthogonality condition.
We observe that the normalization of the amplitudes is done in a way such that the sum and average of is simply the modulus squared of the coupling of the resonance to the final state. The width of the resonance for decay into is given by
(11) 
where is the momentum of the vector in the resonance rest frame and , the masses of the baryon and the resonance. We should note already that later on when the vector polarizations are substituted by the photon polarizations in the sum over in Eq. (2) we will get a factor two rather than three, because we only have two transverse polarizations, and then Eq. (11) must be multiplied by the factor .
3 Radiative decay
Next we study the radiative decay into of the resonances dynamically generated in Ref. [2] with a baryon of the decuplet. Recalling the results of [2] we obtained there ten resonances dynamically generated, each of them degenerated in three states of spin, , , . As we have discussed in the former section, the radiative width will not depend on the spin of the resonance, but only on the coupling which is the same for all three spin states due to the degeneracy. This would be of course an interesting experimental test of the nature of these resonances.
In order to proceed further, we use the same formalism of the hidden gauge local symmetry for the vector mesons of [28, 29, 30, 31]. The peculiarity of this theory concerning photons is that they couple to hadrons by converting first into a vector meson, , , . Diagrammatically this is depicted in Fig. 3. This idea has already been applied with success to obtain the radiative decay of the , , and resonances into in Ref. [32, 35]. In Ref. [32] the question of gauge invariance was addressed and it was shown that the theory fulfills it. In Ref. [38], it is also proved in the case of radiative decay of axial vector resonances.
The amplitude of Fig. 3 requires the convertion Lagrangian, which comes from Refs. [28, 29, 30] and is given by
(12) 
with the photon field, the SU(3) matrix of vector fields
(13) 
and the charge matrix
(14) 
In Eq. (12), is the vector meson mass, for which we take an average value , the electron charge, , and
The sum over polarizations in the intermediate vector meson, which converts the polarization vector of the vector meson of the amplitude into the photon polarization of the amplitude, leads to the equation
(15) 
with
(16) 
Thus, finally our amplitude for the transition, omitting the spin matrix element of Eq. (9), is given by
(17) 
As discussed in the former section, the radiative decay width will then be given by
(18) 
The couplings for different resonance and with and different baryon of the decuplet can be found in Ref. [2] and we use them here for the evaluation of . The factor in eq. (18) additional to eq. (11) appears because now we have only two photon polarizations and the sum over in eq. (2) gives 2 instead of 3 for the case of vector mesons.
4 Results for radiative decays into and baryon decuplet
The couplings of the resonances to the different channels are given in Ref. [2] in the isospin basis. For the case of and , there is no change to be done, but for the case of , one must project over the component. Since this depends on the charge of the resonance , the radiative decays will depend on this charge, as we will see. We recall that in our phase convention of isospin. The information on the resonances and their couplings to different baryons of decuplet and vector mesons ,, for different channels is listed in Table 1. We detail the results below.
S, I  Channel  
0, 1/2  
0, 3/2  
1, 0  
1,1  
2, 1/2  
3, 0  
4.1 channel
A resonance is obtained at which couples to . We have in this case
(19) 
and
(20) 
The coupling of the resonance to is obtained multiplying the coupling of Table 1 by the corresponding ClebschGordan coefficient for of Eqs. (19, 20). Then by means of Eqs. (17, 18), one obtains the decay width. In this case since the component is the same for and , one obtains the same radiative width for the two channels, which is .
4.2 channel
One resonance is obtained at which couples to , and . The isospin states for can be written as
(21) 
(22) 
(23) 
(24) 
Since all the ClebschGordan coefficients to are now different, we obtain different radiative decay width for each charge of the state. The results are for , for , for and for . It is quite interesting to see that there is an order of magnitude difference between for and , and it is a clear prediction that could be tested experimentally.
4.3 channel
We get a resonance at , which couples to . In this case
(25) 
and the radiative decay obtained is .
4.4 channel
Here we find three resonances at , and , which couple to , and . The relevant isospin states are
(26) 
4.5 channel
Here we also find three states at , and , which couple to , and . The isospin states for are written as
(29) 
(30) 
The radiative decay widths in this case are shown in Table 3.
4.6 channel
Here we have only one state at , which couples to and . The radiative decay width obtained in this case is .
As one can see, there is a large variation in the radiative width of the different states, which should constitute a good test for the model when these widths are measured.
In Table 4 we summarize all the results obtained making an association of our states to some resonances found in the PDG[37].
Theory  PDG data  Predicted width for  

pole position  name  
722  722  
1582  203  143  1402  
583  
20  199  561  
2029  206  399  
537  277  182  
54  815  
1902  320  
165  44  
330 
5 Extension to the dynamically generated states from vector meson  baryon octet interaction
In this section we take the states dynamically generated in [36] from the interacion of vector mesons and baryons of the octet. The generalization of the equations is rather obvious: eq. (9) becomes now ()
(31) 
and equations (17) and (18), which determine the radiative decay width, are identical. Once again one has to obtain the projection of the coupling from the isospin basis to the case, which we detail below.
6 Results for radiative decays into and baryon octet
The couplings of the resonances to the different channels are given in Ref. [2] in the isospin basis. For the case of and , there is no change to be done, but for the case of , one must project over the component. Since this depends on the charge of the resonance , the radiative decays will depend on this charge, as we will see. We recall that in our phase convention of isospin. The information on the resonances and their couplings to different baryons of octet and vector mesons ,, for different channels is listed in Table 5. We detail the results below.
S, I  Channel  
0, 1/2  
1, 0  
1,1  
2, 1/2  
6.1 channel
Two resonances are obtained at and which couple to , and . We have in this case
(32) 
and
(33) 
6.2 channel
We get three resonances at , and respectively, which couple to , and . In this case
(34) 
6.3 channel
Here we find two resonances at and , which couple to , and . The relevant isospin states are
(35) 
(36) 
and
(37) 
6.4 channel
Here we also find two states at and , which couple to , and . The isospin states for are written as
(38) 
(39) 
In Table 6 we summarize all the results obtained, making an association of our states to some resonances found in the PDG[37].
Theory  PDG data  Predicted width for  

pole position  name  
334  253  
196  79  
65 (166)  
321 (21)  
0 (17)  

363  69 (240)  7  
307  27 (90)  426  
400  89  
212  84 
As one can see, there is a large variation in the radiative width of the different states, which should constitute a good test for the model. For the case of the vectorbaryon octet states which decay into and a baryon of the octet, it is customary to express the experimental information in terms of helicity amplitudes and . We evaluate these amplitudes below to facilitate the comparison with experiment.
7 Helicity amplitudes
Recalling eq. (31) for the dynamically generated states from a vector and a baryon of the octet, we have the two cases and . The helicity amplitudes are defined as
(40)  
(41) 
where , is the CM photon momentum and . To acomodate these amplitudes to our eq. (31) we rewrite them taking , as
(42) 
where is given by eq. (17), with , , which fixes , and similarly for the other amplitudes. Hence, we obtain
(43)  
(44)  
(45) 
The ordinary formula to get the radiative decay width in terms of and is given in the PDG [37] as