Precise determination of the properties of $X(3872)$ and of its isovector partner $W_{c1}$

Abstract

We perform a simultaneous fit to BESIII data on $e^+e^-\to \gamma (D^0{\bar{D}^{0}}\pi^0/J/\psi\pi^+\pi^-)$ and LHCb data on $B^+\to K^+(J/\psi\pi^+\pi^-)$ to precisely determine the properties of the $X(3872)$, with full consideration of three-body effects from $D^*\to D\pi$ decay, respecting both analyticity and unitarity. The $X(3872)$ is determined to be a quasi-bound state with a significance of $2.7\,\sigma$, representing the most precise determination to date. Its pole is located at $\left(-160^{+57}_{-74}-125^{+23}_{-38}\,i\right) \rm keV$, relative to the nominal $D^0\bar{D}^{*0}$ threshold. Moreover, we confirm the presence of an isovector partner state, $W_{c1}$. It is found as a virtual state at $\left(3.1\pm0.7+ 1.3^{+1.9}_{-0.6}\,i\right)\ \rm MeV$ relative to the $D^+ D^{*-}$ threshold on an unphysical Riemann sheet, strongly supporting a molecular nature of both $X(3872)$ and $W_{c1}$. As a highly nontrivial prediction we show that the $W_{c1}$ leads to nontrivial lineshapes around 3.88 GeV in $B^0\to K^0 X(3872)\to K^0 D^0\bar D^0\pi^0$ and $K^0J/\psi\pi^+\pi^-$ -- thus the scheme presented here can be tested further by improved measurements.