Strong decays of the charmonium-like state $Y(4230)$ and radiative transitions of low-lying charmoniums

The recently reported strong decays of the charmonium-like state $Y(4230)$ by the BESIII collaboration and the dominant radiative transitions of the charmonium low-lying excited states are studied within a covariant quark model by involving the analytic confinement concept. The $Y(4230)$ resonance is interpreted as a four-quark state of molecular type. We use the compositeness condition to eliminate any constituent degrees of freedom from the space of physical states and rigorously fix the renormalized couplings of the hadron states. The helicity amplitudes, or Lorentz structures, are used to express the gauge-invariant transition amplitudes of the processes under consideration. The fractional strong decay width of $Y\to J/\psi+f_0(980)$ is calculated, as is the branching ratio of strong decays (via the $f_0(980)$ resonance) with hidden charms, $\mathcal B(Y\to K^{+}K^{-}J/\psi)/\mathcal B(Y\to\pi^{+}\pi^{-}J/\psi)$, as measured by the BESIII Collaboration in 2022. Only one common adjustable parameter for the charmonium states $\eta_c({}^1S_0)$, $J/\psi({}^3S_1)$, $\chi_{c0}({}^3P_0)$, $\chi_{c1}({}^3P_1)$, $h_c({}^1P_1)$, and $\chi_{c2}({}^3P_{2})$ is introduced to describe the quark distribution inside the hadron. We calculate the fractional widths of one-photon radiative decays for the states $J/\psi({}^3S_1)$, $\chi_{cJ}({}^3P_{J})$, $J=\{0,1,2\}$, and $h_c({}^1P_1)$. Our estimates of the decay widths of the processes under consideration are in reasonable agreement with the latest experimental data.