Implications of $B\to {\chi }_{c1}\pi K$ data for an isovector $G$-odd $D^{\ast }{\bar{D}}^{\ast }$ molecular virtual state

Abstract

Establishing the near-threshold spin-isospin multiplet spectrum of $D^{(\ast )}{\bar{D}}^{(\ast )}$ systems is central to testing molecular interpretations of the $X(3872)$, $Z_c(3900)$, and $Z_c(4020)$. The isovector channels are particularly important in this context, as the associated structures would have a clear exotic character. In this work, we analyze the BaBar and Belle data on $B\to {\chi }_{c1}\pi K$ to search for possible isovector $D^{\ast }{\bar{D}}^{\ast }$ molecules with $J^{PC}=0^{++}$ and $2^{++}$. The three-body decay amplitude includes an effective nonresonant term, intermediate kaon resonances, and ${\chi }_{c1}\pi$--$D^{\ast }{\bar{D}}^{\ast }$ coupled-channel rescattering. For each data set, the ${\chi }_{c1}\pi$ and $K\pi$ invariant-mass distributions are fitted simultaneously under three scenarios: without rescattering and with either $J=0$ or $J=2$ rescattering. We then analytically continue the fitted coupled-channel $T$ matrices to search for poles. For the case of $J^{PC}=0^{++}$, the BaBar and Belle fits yield virtual poles at $-2.15_{-6.41}^{+2.10}$ MeV and $-18.40_{-13.70}^{+8.58}$ MeV, respectively, relative to the $D^{\ast }{\bar{D}}^{\ast }$ threshold. For $J^{PC}=2^{++}$, the corresponding virtual poles are located at $-3.42_{-4.29}^{+2.37}$ MeV and $-4.26_{-1.69}^{+1.34}$ MeV, respectively. The tensor virtual-state pole is consistent with a prediction from chiral effective field theory, which also predicts the existence of $W_{c1}$, an isospin partner of $X(3872)$. Since the invariant-mass distributions alone cannot distinguish total spin $J$, we also predict angular distributions. The $\cos {\theta }_{{\chi }_{c1}\pi }$ distribution provides a direct spin discriminator, while the $\cos {\theta }_{K\pi }$ distribution further tests the rescattering contribution. The measurements of these predictions would help establish the $D^{(\ast )}{\bar{D}}^{(\ast )}$ molecular multiplet spectrum in the future.

Comments: 15 pages, 5 figures