Radiative decays of $f_1(1285)$ as the $ K^*\bar K$ molecular state

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Ju-Jun Xie, Gang Li and Xiao-Hai Liu. Radiative decays of $f_1(1285)$ as the $ K^*\bar K$ molecular state[J]. Chinese Physics C.
Ju-Jun Xie, Gang Li and Xiao-Hai Liu. Radiative decays of $f_1(1285)$ as the $ K^*\bar K$ molecular state[J]. Chinese Physics C. shu
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Radiative decays of $f_1(1285)$ as the $ K^*\bar K$ molecular state

  • 1. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
  • 2. School of Physics and Engineering, Qufu Normal University, Shandong 273165, China
  • 3. Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China
  • 4. School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
  • 5. School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

Abstract: Within a picture of the $f_1(1285)$ being a dynamically generated resonance from the $ K^*\bar K$ interactions, we estimate the rates for the radiative transitions of the $f_1(1285)$ meson to the vector mesons $\rho^0$, $\omega$ and $\phi$. These radiative decays proceed via the kaon loop diagrams. The calculated results are in fair agreement with the experimental measurements. Some predictions can be tested by experiments and their implementation and comparison with these predictions will be valuable to decode the strong coupling of $f_1(1285)$ state to the $\bar{K}K^*$ channel.

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    1.   Introduction
    • The radiative decay mode of the $ f_1(1285) $ resonance is interesting because it is the basic element in the description of the $ f_1(1285) $ photoproduction data [1, 2]. It is also advocated as one of the observables most suited to learn about the nature of the $ f_1(1285) $ state [3-7]. By means of a chiral unitary approach, the $ f_1(1285) $ appears as a pole in the complex plane of the scattering amplitude of the $ K^* \bar K+c.c. $ interaction in the isospin $ I = 0 $ and $ J^{PC} = 1^{++} $ channel [8-10]. Or in another word, the axial-vector meson $ f_1(1285) $ can be taken as a $ K^*\bar{K} $ molecular state. For brevity, we use $ K^* \bar K $ to represent the positive C-parity combination of $ K^* \bar K $ and $ \bar{K}^* K $ in the following parts.

      The experimental decay width of the $ f_1(1285) $ is $ 22.7 \pm 1.1 $ MeV [7], quite small compared with its mass. This can be naturally explained in Ref. [8] with the molecular picture that the $ f_1(1285) $ is a dynamically generated state. The $ K^* \bar{K} $ channel is the only allowed and considered pseudoscalar-vector channel in the chiral unitary approach and the pole for $ f_1(1285) $ lies below the $ K^* \bar{K} $ threshold, therefore the total width for the $ f_1(1285) $ resonance is not obtained in Ref. [8]. If the convolution of the $ K^* $ width was taken into account, the partial decay width to the $ K^*\bar{K} $ channel is about $ 0.3 $ MeV (see more details in Ref. [8]). In fact, the dominant decay modes contributing to the width are peculiar. For example, the $ \eta \pi \pi $ channel accounts for 52% of the width, and the branching ratio of $ \pi a_0(980) $ channel is 38%. The decay of $ f_1(1285) \to \pi a_0(980) $ has been well investigated in Ref. [11] within the $ K^*\bar{K} $ molecular state picture for the $ f_1(1285) $. These theoretical calculations of Ref. [11] have been confirmed in a recent BESIII experiment [12].

      There is another important decay channel, i.e. the $ K \bar K \pi $, of which the branching ratio is $ (9.1\pm 0.4) $% [7]. This decay mode is investigated in Ref. [13] with the same picture as in Ref. [11], and the theoretical predictions are in agreement with the experimental data. One might thought that the decay of $ f_1(1285)\to \bar{K}K^* \to K\bar{K}\pi $ should be much enhanced, since the strong coupling of $ f_1(1285) $ to the $ \bar{K}K^* $ channel. Actually, the mass of $ f_1(1285) $ is below the mass threshold of $ \bar{K}K^* $, hence, it is easy to see that the above mechanism is much suppressed due to the highly off-shell effect of the $ K^* $ propagator, which is already found and discussed in Ref. [13] (see more details in that reference). Yet, all the above test have been done in the hadronic decay modes and not in the radiative decays. In this work, we will study the radiative decays of the $ f_1(1285) $ resonance with the assumption that it is a $ K^*\bar{K} $ state.

      On the experimental side, the Particle Data Group (PDG) averaged values on the radiative decays of $ f_1(1285) $ are [7]

      $ {\rm{Br}}(f_1 \to \gamma \rho^0) = (5.3 \pm 1.2) \%,$

      (1)

      $ {\rm{Br}}(f_1 \to \gamma \phi) = (7.5 \pm 2.7) \times 10^{-4}, $

      (2)

      which lead to the partial decay width $ \Gamma_{f_1 \to \gamma \rho^0} = 1.2 \pm 0.3 $ MeV and a ratio $ R_1 = {\rm{Br}}(f_1 \to \gamma \rho^0)/{\rm{Br}}(f_1 \to \gamma \phi) = 71 \pm 30 $. There is currently no experimental data about the $ f_1(1285) \to \gamma \omega $ decay. While the recent value of $ \Gamma_{f_1 \to \gamma \rho^0} $ obtained by the CLAS Collaboration at Jafferson Lab from the analysis of the $ \gamma p \to p f_1(1285) $ reaction is much smaller, which is $ 0.45 \pm 0.18 $ MeV [1]. These values are obtained with $ {\rm{Br}}(f_1 \to \eta \pi \pi) = 0.52 \pm 0.02 $ [7] and the measured branching ratio $ {\rm{Br}}(f_1 \to \gamma \rho^0)/{\rm{Br}}(f_1 \to \eta \pi \pi) = 0.047 \pm 0.018 $ and the width $ \Gamma_{f_1} = 18.4 \pm 1.4 $ MeV in Ref. [1]. The measured mass of the $ f_1(1285) $ state is $ M_{f_1} = 1281.0 \pm 0.8 $ MeV, which is compatible with the known properties [7] of the $ f_1(1285) $ resonance. On the theoretical side, the authors in Ref. [2] give $ \Gamma_{f_1 \to \gamma \rho^0} = 0.311 $ MeV and $ \Gamma_{f_1 \to \gamma \omega} = 0.0343 $ MeV under the assumption that $ f_1(1285) $ has a quark-antiquark nature. This $ \Gamma_{f_1 \to \gamma \rho^0} $ value is compatible with that of CLAS Collaboration within errors, but much smaller than the above PDG averaged value. Within the picture of $ f_1(1285) $ being a quark-antiquark state, another theoretical prediction for the $ f_1(1285) $ radiative decay is done in Ref. [14] using a covariant oscillator quark model. It predicts the $ \Gamma_{f_1(1285) \to \gamma \rho^0} $ is in the range of $ 0.509\sim 0.565 $ MeV, $ \Gamma_{f_1(1285) \to \gamma\omega} $ in the range of $ 0.048 \sim 0.057 $ MeV, and $ \Gamma_{f_1(1285) \to \gamma\phi} $ in the range of $ 0.0056 \sim 0.02 $ MeV, which depend on a particular mixing angle between the $ (u\bar{u} + d\bar{d})/\sqrt{2} $ and $ s\bar{s} $ components. Note that the $ f_1(1285) $ and $ f_1(1420) $ are the members of the pseudovector nonet in the $ q\bar{q} $ quark models [2, 14], where the $ f_1(1285) $ is a mostly $ u\bar{u} + d \bar{d} $ state and the $ f_1(1420) $ is a $ s\bar{s} $ state. However, the study of Ref. [15] shows that the $ f_1(1420) $ is not a genuine resonance and it shows up as a peak because of the $ K^*\bar{K} $ and $ \pi a_0(980) $ decay modes of the $ f_1(1285) $ around $ 1420 $ MeV. In fact, as discussed in the PDG [7], even these two states are well known, their nature remains to be established. Thus, further investigations about them are needed [16].

      In this work, we extend the works of Refs. [11, 13] for the hadronic decays of $ f_1(1285) $ to the case of the radiative decays. In the molecular state scenario, the $ f_1(1285) $ decays into $ \gamma V $ ($ V = \rho^0 $, $ \omega $, and $ \phi $) via the kaon loop diagrams, and we can evaluate simultaneously these processes. It is shown that the theoretical results are in good agreement with the experiment, hence supporting the strong coupling of the $ f_1(1285) $ state to the $ \bar{K} K^* $ channel.

      The present paper is organized as follows: In sec. II, we discuss the formalism and the main ingredients of the model; In sec. III we present our numerical results and conclusions; A short summary is given in the last section.

    2.   Formalism
    • We study the $ f_1(1285) \to \gamma V $ decays with the assumption that the $ f_1(1285) $ is dynamically generated from the $ K^* \bar{K} + c.c. $ interaction, thus this decay can proceed via $ f_1(1285) \to K^* \bar{K} \to \gamma V $ through the triangle loop diagrams, which are shown in Fig. 1. In this mechanism, the $ f_1(1285) $ first decays into $ K^* \bar{K} $, then the $ K^* $ decays into $ K \gamma $, and the $ K\bar K $ interact to produce the vector meson V in the final state. We use p, k, and q for the momentum of $ f_1(1285) $, $ \gamma $ and $ K^- $ and $ \bar{K}^0 $ in Figs. 1 A) and B), respectively. Then one can easily get the momentum of final vector meson is $ p-k $, and the momenta for $ K^* $ and K are $ p-q $ and $ p-q-k $, respectively. On the other hand, the decay of $ f_1(1285) \to \gamma V $ can also go with $ K^* $ exchange, where one needs a $ K^*K^*\gamma $ vertex, and then the $ K^*\bar{K} $ interact to produce the vector meson V. However, it is easy to see that, comparing with the mechanism shown in Fig. 1, this mechanism is much suppressed due to the highly off-shell effect of the exchanged $ K^* $ propagator when the $ K^*\bar{K} $ invariant mass is the mass of the vector meson V. In fact, as shown in Ref. [34], for the case of $ a_1/b_1 \to \gamma \pi $ decays, the contribution of $ K^* $ exchange is rather small, in the order of 0.5%, compared with the one from the K exchange. Therefore, it is expected that the contributions from $ K^* $ exchange are also small for the $ f_1 \to \gamma V $ decays, as studied here, and those contributions can be safely neglected.

      Figure 1.  Triangle loop diagrams representing the process $ f_1(1285) \to \gamma V $ with V being the $ \rho^0 $, $ \omega $, or $ \phi $ meson.

    • 2.1.   Effective interactions and coupling constants

    • To evaluate the radiative decay of $ f_1(1285) \to \gamma V $, we need the decay amplitudes of these diagrams shown in Fig. 1. As mentioned above, the $ f_1(1285) $ resonance is dynamically generated from the interaction of $ K^* \bar{K} $. For the charge conjugate transformation, we take the phase conventions $ \mathcal{C}K^* = - \bar{K}^* $ and $ \mathcal{C}K = \bar{K} $, which are consistent with the standard chiral Lagrangians, and write

      $ \begin{split}& |f_1(1285)> = \frac{1}{\sqrt{2}} (K^* \bar{K} - \bar{K}^* K) \\ =& - \frac{1}{2} (K^{*+} K^- + K^{*0}\bar{K}^0 - K^{*-}K^+ - \bar{K}^{*0} K^0)\ . \end{split} $

      (3)

      Then we can write the $ f_1(1285)\bar{K}K^* $ vertex as

      $ -it_{f_1 \to \bar{K}K^*} = -i g_{f_1}C_1 \epsilon^{\mu}(f_1) \epsilon_{\mu}(K^*), $

      (4)

      where $ \epsilon^{\mu}(f_1) $ and $ \epsilon_{\mu}(K^*) $ stand for the polarization vector of $ f_1(1285) $ and $ K^* $ ($ \bar{K}^* $), respectively. The factors $ C_1 $ account for the weight of each $ \bar{K}K^* $ ($ K\bar{K}^* $) component of $ f_1(1285) $, crrosponding to the $ f_1 \bar{K}K^* $ vertex for each diagram shown in Fig. 1, and can be easily obtained from Eq. (3) as,

      $ C^{A,B}_1 = - \frac{1}{2};\ \ C^{C,D}_1 = \frac{1}{2}. $

      (5)

      For the $ \bar K K V $ vertices, we take the effective Lagrangian describing the pseudoscalar-pseudoscalar-vector ($ PPV $) interaction as [17-20],

      $ {\cal L}_{PPV} = - i g <V^{\mu}[P,\partial_{\mu}P]>\ , $

      (6)

      where $ g = M/{2f} = 4.2 $ with $ M \approx (m_{\rho} + m_{\omega})/2 $ and $ f = 93 $ MeV. The symbol $ <> $ stands for the trace, while the pseudoscalar- and vector-nonet are collected in the P and V matrices, respectively. We can write them as

      $ V_\mu = \left( \begin{array}{*{20}{c}} \dfrac{\omega + \rho^0}{\sqrt{2}} & \rho^+ & K^{*+} \\ \rho^- & \dfrac{\omega - \rho^0}{\sqrt{2}} & K^{*0} \\ K^{*-} & \bar{K}^{*0} & \phi \end{array} \right)_\mu ,$

      (7)

      and

      $ P = \left( \begin{array}{*{20}{c}} \xi_1 & \pi^+ & K^+ \\ \pi^- & \xi_2 & K^0 \\ K^- & \bar{K}^0 & \xi_3 \end{array} \right),$

      (8)

      with $ \xi_1 = \frac{1}{\sqrt{2}} \pi^0 + \frac{1}{\sqrt{3}} \eta + \frac{1}{\sqrt{6}} \eta' $, $ \xi_2 = - \frac{1}{\sqrt{2}} \pi^0 + \frac{1}{\sqrt{3}} \eta + \frac{1}{\sqrt{6}} \eta' $, and $ \xi_3 = - \frac{1}{\sqrt{3}} \eta + \frac{2}{\sqrt{6}} \eta' $.

      Thus, the $ \bar K K V $ vertex can be written as

      $ -i t_{\bar K K \to V} = i g C_2 (2q + k - p)^{\mu} \varepsilon_{\mu}(p-k,\lambda_V), $

      (9)

      where $ \varepsilon_{\mu}(p-k,\lambda_V) $ is the polarization vector of the vector meson. From Eq. (6) and from the explicit expressions of the V and P matrices as shown in Eqs. (7) and (8), the factors $ C_2 $ for each diagram shown in Fig. 1 can be obtained,

      $ \begin{split} &{C_2^{A,C} = - \frac{1}{{\sqrt 2 }};\;\;C_2^{B,D} = \frac{1}{{\sqrt 2 }};\;\;{\rm{for}}\;\rho \;{\rm{production}},}\\& {C_2^{A,C} = - \frac{1}{{\sqrt 2 }};\;\;C_2^{B,D} = - \frac{1}{{\sqrt 2 }};\;\;{\rm{for}}\;\omega \;{\rm{production}},}\\& {C_2^{A,C} = 1;\;\;C_2^{B,D} = 1;\;\;{\rm{for}}\;\phi \;{\rm{production}}.} \end{split}$

      (10)

      In terms of Eqs. (5) and (10), it is easy to know that Figs. 1 A) and C) give the same contribution and Figs. 1 B) and D) also give the same contribution. We hence only consider Figs. 1 A) and B) in the following calculation.

      Besides, according to the Lagrangian of Eq. (6), the $ \phi \to K\bar K $ decay width is given by

      $\Gamma_{\phi \to K\bar K} = \frac{g^2m_{\phi}}{48\pi} \left( 1- \frac{4m^2_K}{m^2_{\phi}} \right)^{3/2}, $

      and we can obtain the coupling $ g \simeq 4.5 $ with the averaged experimental value of $ \Gamma_{\phi \to K \bar K} = 1.77 \pm 0.02 $ MeV, $ m_\phi = 1019.46 $ MeV, and $ m_K = (m_{K^+} + m_{\bar{K}^0})/2 = 495.6 $ MeV as quoted in PDG [7]. Hence, in this work, we will take $ g = 4.2 $ as in Eq. (6).

      For the electromagnetic vertex $ K^* K \gamma $, the effective interaction Lagrangian takes the form as in Refs. [21-24]

      $ {\cal L}_{K^*K\gamma} = \frac{eg_{K^*K\gamma}}{m_{K^*}} \varepsilon^{\mu \nu \alpha \beta} \partial_\mu K^*_\nu \partial_\alpha A_\beta K, $

      (11)

      where $ K^*_\nu $, $ A_\beta $ and K denote the $ K^* $ vector meson, photon, and the K pseudoscalar meson, respectively. The partial decay width of $ K^* \to K \gamma $ is given by

      $ \Gamma_{K^* \to K\gamma} = \frac{e^2g^2_{K^*K\gamma}}{96\pi} \frac{(m^2_{K^*} - m^2_K)^3}{m^5_{K^*}}.$

      (12)

      The values of the coupling constants $ g_{K^*K\gamma} $ can be determined from the experimental data [7], $ \Gamma_{K^{*+} \to K^+ \gamma} = 50.3 \pm 4.6 $ keV and $ \Gamma_{K^{*0} \to K^0 \gamma} = 116.4 \pm 10.2 $ keV, which lead to

      $ g_{K^{*+}K^+ \gamma} = 0.75 \pm 0.03, \; \; \; \; g_{K^{*0}K^0 \gamma} = -1.14 \pm 0.05, $

      (13)

      where the small errors are determined with the uncertainties of $ \Gamma_{K^* \to K\gamma} $ as above. In addition, we fix the relative phase between the above two couplings taking into account the quark model expectation [25].

    • 2.2.   Decay amplitudes

    • The partial decay width of the $ f_1(1285) \to \gamma \rho^0 $ decay is given by

      $ \Gamma_{f_1(1285) \to \gamma \rho^0} = \frac{E_\gamma}{12\pi M^2_{f_1}} \sum\limits_{\lambda_{f_1}, \lambda_\gamma, \lambda_\rho} |M_A + M_B|^2, $

      (14)

      where $ M_A $ and $ M_B $ are the decay amplitudes of Fig. 1 A) and B), respectively, and the energy of photon $ E_\gamma = |\vec{k}\; | = ({M^2_{f_1} - m^2_{\rho^0}})/{2M_{f_1}} $. While for the cases of $ \omega $ and $ \phi $ production, they can be obtained straightforwardly.

      The above amplitudes, $ M_A $ and $ M_B $, can be easily obtained with the effective interactions. Here we give explicitly the amplitude $ M_A $ for the $ \rho^0 $ production,

      $ \begin{split} M_{A} =& - \frac{e g g_{f_1}g_{K^{*+}K^+\gamma}}{2\sqrt{2}m_{K^{*+}}} \int \frac{d^4q}{(2\pi)^4} \frac{1}{q^2 - m^2_{K^-} + i \epsilon} \\ &\times \frac{1}{2\omega^*(q)} \frac{D_1}{M_{f_1} -q^0 - \omega^*(q) + i \Gamma_{K^{*+}}/2} \\ &\times \frac{D_2}{(p-q-k)^2-m^2_{K^{+}} + i \epsilon} , \end{split} $

      (15)

      where $ \omega^*(q) = \sqrt{|\vec{q}\; |^2 + m^2_{K^{*+}}} $ is the $ K^{*+} $ energy, and we have taken the positive energy part of the $ K^* $ propagator into account, which is a good approximation given the large mass of the $ K^* $ (see more details in Ref. [11]). In Eq. (15), the factors $ D_1 $ and $ D_2 $ read

      $ D_1 = \varepsilon_{\mu \nu \alpha \beta} (p-q)^{\mu} \varepsilon^{\nu}(p,\lambda_{f_1}) k^{\alpha} \varepsilon^{*\beta}(k,\lambda_\gamma), $

      (16)

      $ D_2 = (2q+k-p)^\sigma \varepsilon^*_{\sigma}(p-k,\lambda_\rho)\ ,$

      (17)

      with $ \lambda_{f_1} $, $ \lambda_\gamma $, and $ \lambda_{\rho} $ the spin polarizations of $ f_1(1285) $, photon and $ \rho^0 $ meson, respectively. The amplitude $ M_B $ corresponding to Fig. 1 B) can be easily obtained through the substitutions $ m_{K^{*+}} \to m_{K^{*0}} $, $ m_{K^+} \to m_{K^0} $, and $ m_{K^-} \to m_{\bar{K}^0} $ in $ M_A $. The decay amplitudes for $ f_1(1285)\to\gamma\phi $ and $ f_1(1285)\to\gamma\omega $ share the similar formalism as Eq. (15).

      To calculate $ M_A $ in Eq. (15), we first integrate over $ q^0 $ using Cauchy's theorem. For doing this, we take the rest frame of $ f_1(1285) $, in which one can write

      $ p = (M_{f_1},0,0,0), \; \; k = (E_\gamma,0,0,E_\gamma), $

      (18)

      $ q = (q^0,|\vec{q}\; |sin\theta cos\phi,|\vec{q}\; |sin\theta sin\phi,|\vec{q}\; |cos\theta), $

      (19)

      with $ \theta $ and $ \phi $ the polar and azimuthal angles of $ \vec{q} $ along the $ \vec{k} $ direction. The energy of final vector meson is $ E_V = (M^2_{f_1} + m^2_V)/2M_{f_1} $. Then we have

      $ V_1 = D_1D_2 = \mp iE_\gamma |\vec{q}\; |^2 sin^2\theta, $

      (20)

      for $ \lambda_{f_1} = 0 $, $ \lambda_\gamma = \pm 1 $, and $ \lambda_{\rho} = \mp1 $, and

      $ \begin{split} V_2 =& D_1D_2 = \pm i\frac{2E^2_\gamma}{m_{\rho^0}} \big ( q^0 - M_{f_1} - |\vec{q}\; |cos\theta \big ) \\ &\times \left(q^0 + \frac{E_V}{E_\gamma} |\vec{q}\; |cos\theta\right), \end{split} $

      (21)

      for $ \lambda_{f_1} = \pm 1 $, $ \lambda_\gamma = \pm 1 $, and $ \lambda_{\rho} = 0 $. Notice that we have dropped those terms containing $ sin\phi $ or $ cos\phi $, because after the integration over azimuthal angle $ \phi $, they do not give contributions.

      After integrating over $ q^0 $ in Eq. (15), we have

      $ F^A_1 =\frac{|\vec{q}\; |^4 (1-cos^2\theta)}{\omega\omega'\omega^*} \big ( X^A_1 + X^A_2 + X^A_3 \big ), $

      (22)

      $ \begin{split} F^A_2 = & \frac{|\vec{q}\; |^2}{\omega\omega'\omega^*} \left [\left(M_{f_1} -\omega^* - \frac{E_V}{E_\gamma}|\vec{q}\; |cos\theta\right) (\omega^* + |\vec{q}\; |cos\theta) X^A_1\right. \\ & + (\omega - M_{f_1} - |\vec{q}\; |cos\theta)\left(\omega + \frac{E_V}{E_\gamma}|\vec{q}\; |cos\theta\right)X^A_2 \\ & \left.+ (\omega' - E_\gamma - |\vec{q}\; |cos\theta) \left(E_V + \omega' + \frac{E_V}{E_\gamma}|\vec{q}\; |cos\theta\right) X^A_3 \right ], \end{split} $

      (23)

      where

      $ \begin{split} X^A_1 \!\! & = \!\! \frac{1}{\left(M_{f_1} - \omega^* - \omega + i\dfrac {\Gamma_{K^{*+}}} {2}\right)\left(E_\gamma - \omega^* - \omega' +i\dfrac {\Gamma_{K^{*+}}} {2}\right)}, \\ X^A_2 \!\! & = \!\! \frac{1}{\left(M_{f_1} - \omega^* - \omega + i\dfrac {\Gamma_{K^{*+}}} {2}\right)(E_V - \omega - \omega' + i\epsilon)}, \\ X^A_3 \!\! & = \!\! \frac{1}{\left(\omega + \omega^* - E_\gamma - i\dfrac {\Gamma_{K^{*+}}}{2}\right)(E_V + \omega + \omega' - i\epsilon)}, \end{split} $

      with $ \omega^\prime = \sqrt{|\vec{q}\; |^2 + E^2_\gamma +2E_\gamma |\vec{q}\; |cos\theta +m^2_{K^+}} $ and $ \omega = \sqrt{|\vec{q}\; |^2 + m^2_{K^-}} $ the energies of $ K^- $ and $ K^+ $ in the diagram of Fig. 1 A). $ F^B_1 $ and $ F^B_2 $ will be obtained just applying the substitution to $ F^A_1 $ and $ F^A_2 $ with $ m_{K^{*+}} \to m_{K^{*0}} $, $ m_{K^-} \to m_{\bar{K}^0} $, and $ m_{K^+} \to m_{K^0} $.

      Finally, the partial decay width takes the form

      $ \begin{split} \Gamma_{f_1 \to \gamma V} = & \frac{e^2g^2g^2_{f_1}E^5_\gamma}{192\pi^2M^2_{f_1}m^2_V} \sum\limits_{i = 1,2} |\int ^\Lambda_0 d|\vec{q}\; | \int ^{1}_{-1}dcos\theta \\&\times\big(C_A F^A_i + C_B F^B_i \big )|^2, \end{split} $

      (24)

      with

      $ C_A = -\frac{\sqrt{2}}{4} \frac{g_{K^{*+} K^+ \gamma}}{m_{K^{*+}}}, \; \; {\rm{for}} \; V = \rho^0 , \omega, $

      (25)

      $C_A = \frac{1}{2} \frac{g_{K^{*+} K^+ \gamma}}{m_{K^{*+}}}, \; \; {\rm{for}} \; V = \phi, $

      (26)

      $C_B = \frac{\sqrt{2}}{4} \frac{g_{K^{*0} K^0 \gamma}}{m_{K^{*0}}}, \; \; {\rm{for}} \; V = \rho^0, $

      (27)

      $ C_B = -\frac{\sqrt{2}}{4} \frac{g_{K^{*0} K^0 \gamma}}{m_{K^{*0}}}, \; \; {\rm{for}} \; V = \omega,$

      (28)

      $ C_B = -\frac{1}{2} \frac{g_{K^{*0} K^0 \gamma}}{m_{K^{*0}}}, \; \; {\rm{for}} \; V = \phi. $

      (29)

      For $ \rho^0 $ production, the relative minus sign between $ C_A $ and $ C_B $ combined with the minus sign between the couplings $ g_{K^{*+} K^+ \gamma} $ and $ g_{K^{*0} K^0 \gamma} $ is positive, and hence the interference of the two diagrams $ A) $ and $ B) $ shown in Fig. 1 is constructive. However, it is destructive for $ \omega $ and $ \phi $ production, which will make the $ \Gamma_{f_1(1285) \to \gamma \rho^0} $ is much lager than the other two partial decay widths.

      In Eq. (24), we have introduced a momentum cutoff $ \Lambda $ to kill the ultraviolet divergence and to compensate the off-shell effects that appears in the triangle loop integral. It is also can be done by introducing form factors to the intermediate particles, as has been done in Refs. [26-31].

      Again, we want to stress that, in this work, those contribution of $ K^* $ exchange via diagrams containing anomalous vector-vector-pseudoscalar (VVP) vertices are not taken into account. Such contributions are well studied in Refs. [32-35] for the low lying scalar, axial vector, and tensor mesons radiative decays. As discussed in Refs. [32, 33] these contributions are very sensible to the exact value of the VVP couplings. Furthermore, including such diagrams, the decay amplitudes would be more complex due to additional model parameters, and we cannot exactly determine these parameters. Hence, we leave these contributions to further studies when more precise experimental measurements become available.

    • 2.3.   The $ \rho^0 $ width contributions

    • In this section we explain how the large $ \rho^0 $ width contributions are implemented. We study the $ f_1(1285) \to \gamma \rho^0 $ with the $ \rho^0 \to \pi^+\pi^- $ decay. For this purpose we replace $ \Gamma_{f_1 \to \gamma \rho^0} $ in Eq. (24) by $ \overline{\Gamma}_{f_1 \to \gamma \rho^0} $:

      $ \overline{\Gamma}_{f_1 \to \gamma \rho^0} = \!\!\! \int^{(m_{\rho^0} + 2 \Gamma^0_{\rho^0})^2}_{(m_{\rho^0} - 2 \Gamma^0_{\rho^0})^2} \!\!\! d\tilde{m}^2 {\cal S}(\tilde{m}) \Gamma_{f_1 \to \gamma \rho^0} (m_{\rho^0} \!\! \to \!\! \tilde{m}), $

      (30)

      where $ \tilde{m} $ is the invariant mass of the $ \pi^+\pi^- $ system. The $ {\cal S}(\tilde m) $ has the form

      $ {\cal S}(\tilde{m}) = -\frac{1}{\pi} {\rm{Im}}\left( \frac{1}{\tilde{m}^2 - m^2_{\rho^0} + i m_{\rho^0}\Gamma_{\rho}(\tilde{m})} \right), $

      (31)

      where $ \Gamma_{\rho}(\tilde{m}) $ is energy dependent, and it can be written as [36-42],

      $ \begin{array} \Gamma_\rho(\tilde{m}) = \Gamma^0_{\rho^0} \left( \dfrac{\tilde{m}^2 - 4m^2_\pi}{m^2_{\rho^0} - 4m^2_\pi} \right)^{3/2}, \end{array} $

      (32)

      with $ m_{\rho^0} = 775.26 $ MeV, $ \Gamma^0_{\rho^0} = 149.1 $ MeV and $ m_\pi = m_{\pi^+} = m_{\pi^-} = 139.57 $ MeV.

    3.   Numerical results and discussion
    • The partial decay width of $ f_1(1285) \to \gamma V $ decay as a function of the $ \Lambda $ from 800 to 1500 MeV is illustrated in Fig. 2, where the black solid, dashed and dotted curves stand for the theoretical results of $ \rho^0 $, $ \omega $, and $ \phi $ production. It is worth to mention that the results of $ \omega $ and $ \phi $ are multiplied by a factor of 100, while the red solid line stands for the results for the $ \rho^0 $ production but with the contributions of the $ \rho^0 $ mass as in Eq. (30). One can see that, from the Fig. 2, the theoretical results have the same order of magnitude with the given cutoff parameter $ \Lambda $ range. In the range of cutoff we consider, the $ \Gamma_{f_1 \to \gamma \rho^0} $ varies from $ 0.4 $ to $ 0.9 $ MeV, which is consistent with the experimental result within errors [1, 7]. Besides, the contribution of the $ \rho^0 $ width is also important and it will reduce the numerical results of $ \Gamma_{f_1 \to \gamma \rho^0} $ by a factor of 18%.

      Figure 2.  Partial decay width of $ f_1(1285) \to \gamma V $ decay as a function of the cutoff parameter $ \Lambda $. The black solid, dashed and dotted curves stand for the results of $ \rho^0 $, $ \omega $, and $ \phi $ production, while the results of $ \omega $ and $ \phi $ are multiplied by a factor of 100. The red solid line stands for the results for the $ \rho^0 $ production but with the contributions of the $ \rho^0 $ mass as in Eq. (30).

      In table 1 we show explicitly the numerical results of the $ f_1(1285) \to \gamma V $ decays with some particular cutoff parameters. We show also the theoretical calculations of Refs. [2, 14] and the experimental results [1, 7] for comparison.

      $ \Lambda $ $ f_1 \to \gamma \rho^0 $ $ \Gamma $ ($ \overline{\Gamma} $) $ f_1 \to \gamma \omega $ [$ \times 10^{-2} $] $ f_1 \to \gamma \phi $ [$ \times 10^{-2} $] $ R_1 $ $ R_2 $
      $ 800 $ $ 0.42 $ ($ 0.34 $) $ 1.36 $ $ 0.71 $ $ 59 $ $ 31 $
      $ 1000 $ $ 0.56 $ ($ 0.46 $) $ 1.87 $ $ 0.93 $ $ 60 $ $ 30 $
      $ 1500 $ $ 0.88 $ ($ 0.72 $) $ 3.01 $ $ 1.41 $ $ 62 $ $ 29 $
      Ref. [2] $ 0.311 $ $ 3.43 $ $ 9 $
      Ref. [14] (set I) $ 0.509 $ $ 4.8 $ $ 2.0 $ $ 25 $ $ 11 $
      Ref. [14] (set II) $ 0.565 $ $ 5.7 $ $ 0.56 $ $ 101 $ $ 10 $
      Exp. [7] $ 1.2 \pm 0.3 $ $ 1.7 \pm 0.6 $ $ 71 \pm 30 $
      Exp. [1]a $ 0.45 \pm 0.18 $
      aThe measured width of $ f_1(1285) $ is about 6 MeV smaller than the previous word average [7].

      Table 1.  Partial decay width for $ f_1(1285) \to \gamma V $. All units are in MeV.

      In general we cannot provide the value of the cutoff parameter, however, if we divide $ \Gamma_{f_1(1285) \to \gamma \rho^0} $ by $ \Gamma_{f_1(1285) \to \gamma \omega} $ or $ \Gamma_{f_1(1285) \to \gamma \phi} $, the dependence of these ratios on the cutoff will be smoothed. Two ratios are defined as

      $ R_1 = \frac{\Gamma_{f_1(1285) \to \gamma \rho^0}}{\Gamma_{f_1(1285) \to \gamma \phi}}, \; \; \; \; \; \; \; \; \; R_2 = \frac{\Gamma_{f_1(1285) \to \gamma \rho^0}}{\Gamma_{f_1(1285) \to \gamma \omega}} . $

      (33)

      These two ratios are correlated with each other. With $ R_1 $ measured by the experiment, one can fix the cutoff in the model and predict the ratio $ R_2 $. We show also in table 1 the explicitly numerical results of $ R_1 $ and $ R_2 $ with some particular cutoff parameters

      In Fig. 3, we show the numerical results for the above two ratios, where the solid line stands for the results for $ R_1 $, while the dashed line stands for the results for $ R_2 $. Indeed, one can see that the dependence of both ratios on the cutoff $ \Lambda $ is rather weak. The ratio $ R_1 \simeq 60 $ is in agreement with the experimental result $ 71 \pm 30 $ [7]. On the other hand, the result of $ R_2 $ is about $ 30 $. It is a firm conclusion that the partial decay width of $ f_1(1285) \to \gamma \rho^0 $ is much larger than the ones to $ \gamma \omega $ and $ \gamma \phi $ channels. This is because the destructive interference between Fig. 1 A) and B) for $ \omega $ and $ \phi $ production. Our conclusion here is same with these quark model calculations [2, 14]. However, from table I one can see that the obtained ratios $ R_1 $ and $ R_2 $ here, are much different with the values obtained from the quark models, especially for $ R_2 $. In the quark model calculations, $ R_2 $ is always around $ 9 $, which is come from the isospin difference of $ \rho^0 $ and $ \omega $ meson. We hope that the future experimental measurements can clarify this issue.

      Figure 3.  The $ \Lambda $ dependence of the ratios $ R_1 $ (solid line) and $ R_2 $ (dashed line) defined in Eq. (33). The error band correspond the experimental result for $ R_1 $

      It is worth to mention that there is only one free parameter $ \Lambda $ (all the other parameters are fixed by previous works) in this work. In addition, the dependence of $ R_1 $ and $ R_2 $ on the cut off $ \Lambda $ is rather weak, thus, they can be predictions of the model, and they would be compared with the future experimental measurements.

      Besides, we want to note that, though we have assumed that the $ f_1(1285) $ is a dynamically generated state, the numerical results here are not tied to the assumed nature of the $ f_1(1285) $. The crucial point is that it couples strongly to $ \bar{K}K^* $ channel, whatever its origin.

    4.   Summary
    • We have evaluated the partial decay rates of the radiative decays $ f_1(1285) \to \gamma V $ with the assumption that the $ f_1(1285) $ is a dynamically generated state from the strong $ \bar K^* K $ interaction, and in this picture the $ f_1(1285) $ state has a strong coupling to the $ \bar{K}K^* $ channel. The theoretical results we obtained for the partial widths are sensitive to the free parameter $ \Lambda $, but they are compatible with experimental data within errors. Furthermore, the ratios $ R_1 = \frac{\Gamma_{f_1 \to \gamma \rho^0}}{\Gamma_{f_1 \to \gamma \phi}} $ and $ R_2 = \frac{\Gamma_{f_1 \to \gamma \rho^0}}{\Gamma_{f_1 \to \gamma \omega}} $, which are not sensitive to the only free parameter $ \Lambda $, are predicted. It is found that the values of $ R_1 $ and $ R_2 $ obtained here are different from other theoretical predictions using quark models. The precise experimental observations of those radiative decays would then provide very valuable information on the relevance of the strong coupling of the $ f_1(1285) $ to the $ \bar{K}K^* $ channel.

Reference (42)

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