Radiative decays of f1(1285) as the K*K̅ molecular state

With as a dynamically generated resonance from interactions, we estimate the rates of the radiative transitions of the meson to the vector mesons , and . These radiative decays proceed via the kaon loop diagrams. The calculated results are in a fair agreement with experimental measurements. Some predictions can be tested experimentally; their analysis will be valuable for decoding the strong coupling of the state to the channel.


Introduction
The radiative decay mode of the resonance is interesting because it is the basic element in the description of the photoproduction data [1,2]. It is also advocated as one of the observables most suitable for learning about the nature of the state [3][4][5][6][7]. Using the chiral unitary approach, appears as a pole in the complex plane of the scattering amplitude of the interaction in the isospin and channel [8][9][10]. In other words, the axial-vector meson can be taken as a molecular state. For brevity, we use to represent the positive C-parity combination of and in what follows.
The experimental decay width of is MeV [7], quite small compared with its mass. This is naturally explained in Ref. [8] using the molecular picture, implying that is a dynamically generated state. The channel is the only allowed and considered pseudoscalar-vector channel in the chiral unitary approach, and the pole of is below the threshold; therefore, the total width of the resonance was not obtained in Ref. [8]. If the convolution of the width was taken into account, the partial decay 1 (1285) width of the channel would be approximately MeV (see more details in Ref. [8]). In fact, the dominant decay modes contributing to the width are peculiar. For example, the channel accounts for 52% of the width, and the branching ratio of channel is 38%. The decay of has been well investigated in Ref. [11] within the molecular state picture for . These theoretical calculations in Ref. [11] have been confirmed in a recent BESIII experiment [12]. There is another important decay channel, i.e., the channel, the branching ratio of which is % [7]. This decay mode was investigated in Ref.
[13] with the same picture as in Ref. [11], and the theoretical predictions agree with existing experimental data. One could posit that the decay of should be much enhanced, owing to the strong coupling of to the channel. Actually, the mass of is below the mass threshold of ; hence, it is easy to see that the above mechanism is much suppressed owing to the highly off-shell effect of the propagator, which was already found and discussed in Ref.
[13] (see more details in that reference). Yet, all of the above tests have been performed for hadronic decay modes and not for radiative decays. In this work, we study the radiative decays of the resonance, assuming that it is a which leads to the partial decay width MeV and a ratio . There is currently no experimental data on the decay. On the other hand, the recent value of obtained by the CLAS collaboration at Jafferson Lab, utilizing the analysis of the reaction, is much smaller, at MeV [1]. These values were obtained with [7]; the measured branching ratio was and the width was MeV in Ref. [1]. The measured mass of the state was MeV, compatible with the known properties [7] of the resonance. On the theoretical side, the authors in Ref. [2] report MeV and MeV under the assumption that has a quark-antiquark nature. This value is compatible with that obtained by the CLAS collaboration, within the error range, but is much smaller than the above PDG averaged value. Within the picture of being a quark-antiquark state, another theoretical prediction for the radiative decay was reported in Ref. [14] using a covariant oscillator quark model. It predicted in the range of 0.5090 .565 MeV, in the range of 0.048~0.057 MeV, and in the range of 0.0056~0.02 MeV; these predictions depend on a particular mixing angle between the and components. Note that and are the members of the pseudovector nonet in the quark model [2,14], where is a mostly state and is an state. However, the study in Ref. [15] shows that is not a genuine resonance and it shows up as a peak because of the and decay modes of around MeV. In fact, as discussed by the PDG [7], although these two states are well known, their nature remains to be established. Thus, further investigations about them are needed [16].
Here, we extend the work in Refs. [11,13] for the hadronic decays of to the case of radiative decays. In the molecular state scenario, decays into ( , , and ) via kaon loop diagrams, and we can evaluate simultaneously these processes. It is shown that the theoretical results are in a good agreement with f 1 (1285)KK * experimental data, hence supporting the strong coupling of the state to the channel. The present paper is organized as follows. In Sec. 2, we discuss the formalism and the main ingredients of the model. In Sec. 3 we present our numerical results and conclusions. A short summary is given in the last section.

Formalism
We study the decays under the assumption that is dynamically generated from the interaction; thus, this decay can proceed via through triangle loop diagrams, which are shown in Fig. 1. In this mechanism, first decays into , then decays into , and interacts to produce the vector meson V in the final state. We use p, k, and q for the momentum of , and and in Figs. 1 (a, b) , respectively. Then, one can easily obtain that the momentum of the final vector meson is , and the momenta of and K are and , respectively. On the other hand, the decay of can also go with exchange, where one needs a vertex; then, interacts to produce the vector meson V. However, it is easy to see that, compared with the mechanism shown in Fig. 1, this mechanism is strongly suppressed owing to the highly off-shell effect of the exchanged propagator when the invariant mass is the mass of the vector meson V. In fact, as shown in Ref. [17], for the case of decays, the contribution of the exchange is rather small, on the order of 0.5%, compared with the one from the K exchange. Therefore, it is expected that the contributions from the exchange will be also small for the decays, as studied here, and those contributions can be safely neglected.

Effective interactions and coupling constants
To evaluate the radiative decay of , we need the decay amplitudes of these diagrams, shown in Fig. 1. As mentioned above, the resonance is dynamically generated from the interaction of . For the charge conjugate transformation, we take the phase conventions and , which are consistent with the standard chiral Lagrangians, and write Chinese Physics C Vol. 44, No. 11 (2020) 114104 1) The use of for the state is implied throughout this paper. 114104-2 and stand for the polarization vector of and ( ), respectively. We will take the value of the coupling constant of as obtained in the chiral unitary approach [8]. The factors account for the weight of each ( ) component of , corresponding to the vertex for each diagram shown in Fig. 1, and can be easily obtained from Eq. (3) as, KKV PPV For the vertices, we take the effective Lagrangian describing the pseudoscalar-pseudoscalar-vector ( ) interaction as [18][19][20][21], with and MeV. The symbol denotes the trace, while the pseudoscalar-and vector-nonets are collected in the P and V matrices, respectively. We can write them as and with , , and .

KKV
Thus, the vertex can be written as where is the polarization vector of the vector meson. From Eq. (6) and from the explicit expressions for the V and P matrices as shown in Eqns. (7) and (8), the for each diagram shown in Fig. 1 can be obtained, for ω production, In terms of Eqns. (5) and (10), it is easy to see that Figs. 1 (a, c) give the same contribution and Figs. 1 (b, d) also give the same contribution. We hence only consider Figs. 1 (a, b) in the following calculation.
In addition, according to the Lagrangian in Eq. (6), the decay width is given by and we can obtain the coupling with the averaged experimental value of MeV, MeV, and MeV as quoted by the PDG [7]. Hence, in this work, we will take as in Eq. (6).

K * Kγ
For the electromagnetic vertex , the effective interaction Lagrangian takes the form as in Refs. [22][23][24][25] where , and K denote the vector meson, photon, and the K pseudoscalar meson, respectively. The partial decay width of is given by The values of the coupling constants can be determined from the experimental data [7], keV and keV, which lead to where the small errors are determined with the uncertainties of as above. In addition, we fix the relative phase between the above two couplings, taking into account the quark model expectation [26].

Decay amplitudes
The partial decay width of the decay is given by where and are the decay amplitudes in Figs. 1 (a,  b), respectively, and the energy of the photon is . In the cases of and production, these can be obtained in a straightforward manner. The above amplitudes, and , can be easily obtained with effective interactions. Here, we give explicitly the amplitude for the production, where is the energy, and we have taken the positive energy part of the propagator into account, which is a good approximation, given the large mass of (see more details in Ref. [11]). In Eq. (15), the factors and read 1) , , and the spin polarizations of , photon, and meson, respectively. The amplitude corresponding to Fig. 1 (b) can be easily obtained through the substitutions , , and into . The decay amplitudes of and share the similar formalism as in Eq. (15). To calculate in Eq. (15), we first integrate over using Cauchy's theorem. For doing this, we take the rest frame of , in which one can write with and as the polar and azimuthal angles of along the direction. The energy of the final vector meson is . Then, we have for , , and , and , and . Notice that we have dropped those terms containing or , because after the integration over the azimuthal angle , they do not yield contributions. After integrating over in Eq. (15), we have where , with and the energies of and in the diagram of Fig. 1 (a). and will be obtained just by applying the substitution to and with , , and . Finally, the partial decay width takes the form Chinese Physics C Vol. 44, No. 11 (2020) 114104 1) Note that we have omitted the term in , which is came from the momentum term of the numerator of the propagator, because it has no contribution. 114104-4 with For production, the relative minus sign between and combined with the minus sign between the couplings and is positive, and hence the interference of the two diagrams and shown in Fig.  1 is constructive. However, it is destructive for and production, which make much larger compared with the other two partial decay widths. K * Again, we want to stress that, in this work, those contributions of the exchange via diagrams containing anomalous vector-vector-pseudoscalar (VVP) vertices are not taken into account. 1) Such contributions were extensively studied in Refs. [17,[33][34][35] for the low-lying scalar, axial vector, and tensor meson radiative decays. As discussed in Refs. [33,34], these contributions are very sensible to the exact value of the VVP coupling. Furthermore, including such diagrams, the decay amplitudes would become more complex, owing to additional model parameters, which cannot be exactly determined. Hence, we leave these contributions to further studies when more precise experimental measurements become available.
In this section, we explain how the large width contributions are implemented. We study with the decay. For this purpose we replace in Eq. (24) by : where is the invariant mass of the system. Then, has the form where is energy-dependent, and it can be written as [36][37][38][39][40][41][42], with MeV, MeV and MeV.

Numerical results and discussion
The partial decay width of the decay as a function of from 800 to 1500 MeV is illustrated in Fig. 2, where the black solid, dashed, and dotted curves stand for the theoretical results of the , , and production. It is worth mentioning that the results for and are multiplied by a factor of 100, while the red solid line stands for the results for the production but with the contributions of the mass as in Eq. (30). One can see that, from Fig. 2, the theoretical results have the same order of magnitude within the given range of the cutoff parameter values. In the considered range of cutoffs, varies from to MeV, which is consistent (color online) Partial decay width of the decay as a function of the cutoff parameter . The black solid, dashed, and dotted curves denote the results for the , , and production, while the results for and are multiplied by a factor of 100. The red solid line denotes the results for the production but with the contributions of the mass as in Eq. (30).
Chinese Physics C Vol. 44, No. 11 (2020) 114104 1) The effective interaction Lagrangian in Eq. (11) for the vertex is VVP like, it is gauge invariant. The coupling constant is obtained from the partial decay width of and the phases for charged and neutral are fixed as the quark model expectation. 114104-5 with the experimental result within the error range [1,7]. In addition, the contribution of the width is also important and it will reduce the numerical results of by a factor of 18%.
In Table 1 we show explicitly the numerical results for the decays with some particular cutoff parameters. We show also the theoretical calculations of Refs. [2,14] and the experimental results [1,7], for comparison.
In general, we cannot provide the value of the cutoff parameter; however, if we divide by or , the dependence of these ratios on the cutoff will be smoothed. Two ratios are defined as These two ratios are correlated with each other. With measured experimentally, one can fix the cutoff in the model and predict the ratio . We also show, in Table 1 , the explicit numerical results for and , for some particular cutoff parameters.
In Fig. 3, we show the numerical results for the above two ratios, where the solid line denotes the results for , while the dashed line denotes the results for . Indeed, one can see that the dependence of both ratios on the cutoff is rather weak. The ratio is in agreement with the experimental result [7]. On the other hand, the result for is approximately . We can conclude firmly that the partial decay width of is much larger than the ones to and channels. This is owing to the destructive interference between Figs. 1 (a, b) for and production. Our present conclusion agrees wtih quark model calculations [2,14]. However, from Table 1 one can see that the presently obtained ratios and are much different from the values obtained by the quark models, especially for . In the quark model calculations, is always 9 ρ 0 ω around , which is owing to the isospin difference of and mesons. We hope that future experimental measurements will help to clarify this issue.
It is worth mentioning that there is only one free parameter in the present work (all the other parameters were fixed in previous works). In addition, the dependence of and on the cutoff is rather weak; thus, these can be the model predictions, and they would be compared with future experimental measurements. In addition, we want to note that, although we have assumed that is a dynamically generated state, the numerical results here are not tied to the assumed nature of . The crucial point is that it couples strongly to the channel, whatever its origin.

Summary
We have evaluated the partial decay rates of the radi-