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 $D^{\ast }\to D\pi$ 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 $\left(-160_{-74}^{+57}-125_{-38}^{+23}i\right)\mathrm{keV}$, relative to the nominal $D^0{\bar{D}}^{\ast 0}$ threshold. Moreover, we confirm the presence of an isovector partner state, $W_{c1}$. It is found as a virtual state at $\left(3.1\pm 0.7+1.3_{-0.6}^{+1.9}i\right)\mathrm{MeV}$ relative to the $D^{+}{D}^{\ast -}$ 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 $K^{0}J/\psi {\pi }^{+}{\pi }^{-}$ -- thus the scheme presented here can be tested further by improved measurements.

Comments: 14 pages, 7 figures and 4 tables