Role of the state in the and decays
We study the role of the resonance in the decays of and . The theoretical approach is based on the results of chiral unitary theory where the resonance is dynamically generated from the interaction. In order to further test the dynamical nature of the state, we investigate the decay close to the threshold and make predictions for the ratio of the invariant mass distributions of the decay and the partial decay width with all the parameters of the mechanism fixed in previous studies. The results can be tested in future experiments and therefore offer new clues on the nature of the state.
The resonance  is an axial-vector state with mass MeV and total decay width MeV Agashe:2014kda (). This state is described as a state within the quark model Gavillet:1982tv (); Godfrey:1998pd (); Li:2000dy (); Vijande:2004he (); Klempt:2007cp (); Chen:2015iqa (). On the other hand, the is also suggested to be a dynamically generated state made from the single channel interaction in the chiral unitary approach Roca:2005nm (). As shown in Ref. Roca:2005nm (), because the resonance has positive parity, it cannot couple to other pseudoscalar–vector channels. For reasons of parity it can also not decay into two pseudoscalar mesons. Thus, since the resonance is located below the mass threshold, its observation is difficult in two body decays. Indeed, the main decay channels of the are (branching ratio ), (), and ().
While Nature is probably more complicated and the state might have components of either type (see discussions in Ref. Aceti:2015pma ()), two comments are in order. First, the fact that states of different nature are possible does not mean that there should be a duplication of states with the same quantum numbers corresponding to each type of structure. The different structures mix and at the end it is a particular mixture what gives rise to the observed states. These features were well described in Refs. vanBeveren:1986ea (); Tornqvist:1995ay (); Fariborz:2009cq (); Fariborz:2009wf () for the () meson. One starts with a seed of and lets it couple to components respecting unitarity of the interaction. At the end, a physical state develops in which the original seed has been eaten up by the meson cloud, which becomes the dominant component of the wave function. The other comment is that, depending on the reaction, one or the other component will evidence itself more clearly, and in the present case, where we have a produced at the end, it is quite clear that it is this component the one which will show up.
In Refs. Aceti:2015pma (); Aceti:2015zva (), the decays of and were studied using the picture in which the is dynamically generated from the single channel interaction. The theoretical predictions are compatible with the experimental measurements. Very recently, the production of the resonance in the reaction within an effective Lagrangian approach was studied in Ref. Xie:2015wja () based on the results obtained in chiral unitary theory. The theoretical calculations are in agreement with the experimental data which provides further support for the molecular structure of the state.
On the experimental side, in Refs. Falvard:1988fc (); Jousset:1988ni (), the decay of was studied from the and decays by the DM2 Collaboration, while in Ref. Ablikim:2007ev (), the branching fraction of was measured from the decay of by the BES Collaboration. Because the and the mesons have quantum numbers and , respectively, the decay constitutes the ideal reaction to look for the state, with quantum numbers , coupling to an wave pair. However, since the is located below the threshold, it will contribute to the region close to the threshold of .
In the present work, following the formalism of Ref. Roca:2005nm (), we study the decays of and with the picture that the resonance is dynamically generated from the single channel interaction.
We want to study the role of the state, which is dynamically generated by the and interaction, in the decay. In the chiral unitary approach of Ref. Roca:2005nm (), the resonance was obtained by solving the Bethe-Salpeter equation in the channel to obtain the scattering amplitude
where is the transition potential and is the loop function for the propagators of the and mesons given in Ref. Roca:2005nm (). The and depend on the invariant mass of the system, and hence the scattering amplitude is also dependent on . The loop function is divergent, and it can be regularized both with a cutoff prescription or with dimensional regularization in terms of a subtraction constant Oller:2000fj (). In this work we will make use of the cutoff regularization scheme, which introduces a cutoff parameter . The cut off is tuned to get a pole of the matrix at the mass (1281.3 MeV) of the . This provides the coupling MeV of the resonance to the channel (see more details in Ref. Aceti:2015pma ()). With the explicit expressions for and taken from Ref. Roca:2005nm (), we obtain a good description of the resonance using a cutoff MeV, as in Ref. Roca:2005nm ().
For , the decay mechanism is shown in Fig. 1. To take into account the final state interaction of the pair, we have to consider the resummation of the diagrams shown in the figure.
According to the diagrams in Fig. 1, the transition matrix for the process can be given by
where the last equality follows from Eq. (1). The and are the bare production vertex and the spin structure (the spin of together with the one of the must give the spin of : ) factor for . We assume that this bare vertex is of a short range nature, i.e., just a coupling constant in the field theory language.
The spin structure of the , , and coupling can be written as
Summing and averaging over final and initial polarizations of the vector mesons we find
where and and are the and momenta in the rest frame, respectively,
where is the invariant mass of system, and is the Kählen or triangle function.
For a given value of , the range of is defined as,
where and are the energies of and in the rest frame.
On the other hand, if we are interested in the production of the resonance, the relevant mechanism is depicted diagrammatically in Fig. 2 and we have
where the spin factor is easily obtained. We must recall that the coupling of to is given by . Contracting the two in the propagator in Fig. 2 we have
Then, the partial decay width of is given by
and is the meson momentum obtained in the rest frame which is
This ratio is relevant because it has no free parameters (all the parameters are fixed by previous works) and, thus, it is a prediction of the theory. The shape, as well as the absolute values of the ratio for the mass distribution, can be compared with the experimental measurements.
Iii Numerical results and discussion
In Fig. 3, the numerical results of as a function of the invariant mass of the system are shown. The solid curve stands for the theory prediction and the dotted curve stands for the phase space. For evaluating the contributions of the phase space, we replace of Eq. (2) by a constant, thus removing any effect of the dependence of the resonance. Then we tune this constant such that the integrated in the range of energies from the threshold to GeV is the same as the one evaluated with the explicit resonance formalism.
In addition, in Fig. 3 we also show the results which are obtained without considering the spin structure factor by the dashed curve in Fig. 3. We see that the structure factor gives a small effect to our predictions and could be neglected.
We see a clear threshold enhancement in Fig. 3 which is caused by the contributions of the state below threshold, which is dynamically generated by the interaction. The theoretical predictions can be tested by future experiments.
Actually, the range of the invariant mass of in the decay of is from the threshold of up to GeV ( MeV), however, we cannot go so far because the chiral theory works well about MeV from the threshold, hence we consider only the range of MeV above the threshold as shown in Fig. 3.
On the other hand, experimentally we have, from Ref. Agashe:2014kda (),
Note that we have corrected the branching ratio quoted in the PDG which we found is misquoted. In Ref. Jousset:1988ni (), from where the PDG information is obtained, the peak around MeV of the mass distribution was attributed to the with a width of MeV. They obtain . Taking now into account that from the PDG, we obtain as shown in Eq. (15).
Note that we do not use the value of presented in Ref. Jousset:1988ni (), which was obtained from the decay of . Very recently, in Ref. Ablikim:2014pfc (), the branching fraction was measured, with the result . Taking this value into account, and the used before, we get . This value is consistent within errors with what we have obtained before.
One might think we should compare our theoretical result, , to the experimental result in Eq. (16), but, as discussed before, we take the scattering amplitude from the chiral unitary approach, and we can not go too far from the threshold. Furthermore, there could be also other contributions from higher mass states with spin-parity and at higher invariant mass region of . These higher states will not contribute too much to the lower energy region and hence will not affect our predictions here. On the other hand, note that the experimental results of Ref. Jousset:1988ni () were obtained in the 1980s and only few signal events were observed. Further improvement can be done in the future at BESIII or BelleII. The future experimental observation of the mass distribution would provide very valuable information on the mechanism of the decay.
In summary, we have studied the decays of and with the theoretical approach which is based on results of chiral unitary theory where the resonance is dynamically generated from the interaction. The ratio as a function of invariant mass of is predicted. A clear threshold enhancement in Fig. 3 compared with the phase space appears, which is caused by the presence of the state below threshold . The experimental observation of this mass distribution would then provide very valuable information to check our predictions and the basic nature of the resonance.
One of us, E. O., wishes to acknowledge support from the Chinese Academy of Science in the Program of Visiting Professorship for Senior International Scientists (Grant No. 2013T2J0012). This work is partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under the contract number FIS2011-28853-C02-01 and FIS2011-28853-C02-02, and the Generalitat Valenciana in the program Prometeo II-2014/068. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity Study of Strongly Interacting Matter (acronym HadronPhysics3, Grant Agreement n. 283286) under the Seventh Framework Programme of EU. This work is also partly supported by the National Natural Science Foundation of China under Grant No. 11475227. This work is also supported by the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No.Y5KF151CJ1).
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