Unified description of the hidden-charm tetraquark states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3985</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Z</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3900</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>4020</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>
Zhi-Hui Guo, J. A. Oller
Abstract
The newly observed hidden-charm tetraquark state ${Z}_{cs}(3985)$, together with ${Z}_{c}(3900)$ and $X(4020)$, are studied in the combined theoretical framework of the effective range expansion, compositeness relation, and the decay width saturation. The elastic effective-range-expansion approach leads to sensible results for the scattering lengths, effective ranges, and the compositeness coefficients, i.e., the probabilities to find the two-charm-meson molecule components in the tetraquark states. The coupled-channel formalism by including the $J/\ensuremath{\psi}\ensuremath{\pi}$ and $D{\overline{D}}^{*}/\overline{D}{D}^{*}$ to fulfill the constraints of the compositeness relation and the decay width confirms the elastic effective-range-expansion results for the ${Z}_{c}(3900)$, by using the experimental inputs for the ratios of the decay widths between $D{\overline{D}}^{*}/\overline{D}{D}^{*}$ and $J/\ensuremath{\psi}\ensuremath{\pi}$. With the results from the elastic effective-range-expansion study as input for the compositeness, we generalize the discussions to the ${Z}_{cs}(3985)$ by including the $J/\ensuremath{\psi}{K}^{\ensuremath{-}}$ and ${D}_{s}^{\ensuremath{-}}{D}^{*0}/{D}_{s}^{*\ensuremath{-}}{D}^{0}$ and predict the partial decay widths of the $J/\ensuremath{\psi}{K}^{\ensuremath{-}}$. Similar calculations are also carried out for the $X(4020)$ by including the ${h}_{c}\ensuremath{\pi}$ and ${D}^{*}{\overline{D}}^{*}$, and the partial decay widths of the ${h}_{c}\ensuremath{\pi}$ is predicted. Our results can provide useful guidelines for future experimental measurements.