Implications of $B\to χ_{c1}πK$ 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 $B\toχ_{c1}πK$ 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 $χ_{c1}π$--$D^\ast\bar D^\ast$ coupled-channel rescattering. For each data set, the $χ_{c1}π$ and $Kπ$ 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 $-2.15^{+2.10}_{-6.41}$ MeV and $-18.40^{+8.58}_{-13.70}$ MeV, respectively, relative to the $D^\ast\bar D^\ast$ threshold. For $J^{PC}=2^{++}$, the corresponding virtual poles are located at $-3.42^{+2.37}_{-4.29}$ MeV and $-4.26^{+1.34}_{-1.69}$ 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 $\cosθ_{χ_{c1}π}$ distribution provides a direct spin discriminator, while the $\cosθ_{Kπ}$ 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.