Line shapes of theZ(4430)

Abstract

The Belle Collaboration recently discovered the first manifestly exotic meson: ${Z}^{+}(4430)$, which decays into ${\ensuremath{\psi}}^{\ensuremath{'}}{\ensuremath{\pi}}^{+}$ and therefore has quark content $c\overline{c}u\overline{d}$. The proximity of its mass to the ${D}_{1}{\overline{D}}^{*}$ threshold has motivated the interpretation of the ${Z}^{+}$ as a charm meson molecule whose constituents are an $S$-wave superposition of ${D}_{1}^{+}{\overline{D}}^{*0}$ and ${D}^{*+}{\overline{D}}_{1}^{0}$. If this interpretation is correct, the small ratio of the binding energy of the ${Z}^{+}$ to the width ${\ensuremath{\Gamma}}_{1}$ of its constituent ${D}_{1}$ can be exploited to predict properties of its line shapes. Its full width at half maximum in the channel ${\ensuremath{\psi}}^{\ensuremath{'}}{\ensuremath{\pi}}^{+}$ should be approximately $\sqrt{3}{\ensuremath{\Gamma}}_{1}\ensuremath{\approx}35\text{ }\text{ }\mathrm{MeV}$, which is consistent with the measured width of the ${Z}^{+}$. The ${Z}^{+}$ should also decay into ${D}^{*}{\overline{D}}^{*}\ensuremath{\pi}$ through decay of its constituent ${D}_{1}$. The peak in the line shape for ${D}^{*}{\overline{D}}^{*}\ensuremath{\pi}$ should be at a higher energy than the peak in the line shape for ${\ensuremath{\psi}}^{\ensuremath{'}}{\ensuremath{\pi}}^{+}$ by about ${\ensuremath{\Gamma}}_{1}/\sqrt{12}\ensuremath{\approx}6\text{ }\text{ }\mathrm{MeV}$. The line shape in ${D}^{*}{\overline{D}}^{*}\ensuremath{\pi}$ should also be broader and asymmetric, with a shoulder on the high energy side that can be attributed to a threshold enhancement in the production of ${D}_{1}{\overline{D}}^{*}$.