Resolving the puzzling decay patterns of charged $Z_c$ and $Z_b$ states

We investigate the ratio of the branching fractions of the molecular candidates decaying into the ground and radially excited states within the quark-interchange model. Our numerical results suggest that these molecular candidates are more likely to decay into the radially excited states than into ground states. In particular, the ratio $\mathrm{\Gamma }[Z_c(4430)\to \pi \psi (2S)]/\mathrm{\Gamma }[Z_c(4430)\to \pi J/\psi ]\sim 9.9$ is close to the experimental measurement, which supports the interpretation of $Z_c(4430)$ as the $\overline{D}{D}^{*}(2S)$ molecular state. The ratios of the branching fractions of $Z_b(10610)$ and $Z_b(10650)$ to $\pi \mathrm{\Upsilon }(2S,3S)$ and $\pi \mathrm{\Upsilon }(1S)$ agrees very well with the Belle Collaboration’s measurement. We also predict the similar ratios for $Z_c(3900)$, $Z_c(4020)$, $R_{Z_c(3900)}\approx 1.4$, and $R_{Z_c(4020)}\approx 4.8$. Hopefully, the $\pi \psi (2S)$ mode and the ratios $R_{Z_c(3900)}$ and $R_{Z_c(4020)}$ will be measured by the BESIII and Belle collaborations in the near future, which shall be very helpful for understanding the underlying dynamics of these exotic states.

Figure 1

Prior quark-interchange diagrams contributing to the anticharmed meson-charmed meson scattering into a pion and charmonia.

Figure 2

(a) Wave functions of $\pi$, $D$, $D^{*}(2S)$, $J/\psi$, and $\psi (2S)$ in coordinate space. (b) Wave functions of $B$, $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, and $\mathrm{\Upsilon }(3S)$ in coordinate space. (c) The $r$ dependence of the confining potential $V_{\text{conf}}=\frac{{\alpha }_s}{r}-\frac{3b}{4}r$, with $b=0.18{\mathrm{GeV}}^2$.