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

TL;DR

The paper addresses the precise determination of the $X(3872)$ properties and its isovector partner $W_c1$ by building a coupled-channel framework that includes the $D^0\bar{D}^{*0}$ and $D^+D^{*-}$ channels, three-body $D\bar{D}\pi$ effects, and explicit inelastic channels. It solves the $D\bar{D}^*$ scattering with a Lippmann–Schwinger equation, incorporating a potential with contact, one-pion exchange, and inelastic pieces, and fits to BESIII and LHCb data to extract pole positions and compositeness. The analysis finds a $X(3872)$ pole consistent with a quasi-bound, molecular state at $E_X={(-160^{+57}_{-74}-125^{+23}_{-38} i) ext{ keV}}$ relative to the $D^0\bar{D}^{*0}$ threshold, corresponding to $(3871.53^{+0.06}_{-0.08}-0.13^{+0.02}_{-0.04} i)$ MeV, with a $2.7\sigma$ significance, and identifies a novel isovector partner $W_{c1}^0$ at $(3.1\pm0.7+1.3^{+1.9}_{-0.6})$ MeV relative to the $D^+D^{*-}$ threshold; the isospin breaking ratio is $R_X=0.26(2)$ and the $X(3872)$ compositeness is $X=0.97(2)$. These results support a molecular picture for both states and yield testable predictions for lineshapes in $B^0\to K^0X(3872)$ decays and related channels, with the potential to map a SU(3)-flavor multiplet of hidden-charm hadronic molecules.

Abstract

We perform a simultaneous fit to BESIII data on $e^+e^-\to γ(D^0{\bar{D}^{0}}π^0/J/ψπ^+π^-)$ and LHCb data on $B^+\to K^+(J/ψπ^+π^-)$ to precisely determine the properties of the $X(3872)$, with full consideration of three-body effects from $D^*\to Dπ$ decay, respecting both analyticity and unitarity. The $X(3872)$ is determined to be a quasi-bound state with a significance of $2.7\,σ$, 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 $B^0\to K^0 X(3872)\to K^0 D^0\bar D^0π^0$ and $K^0J/ψπ^+π^-$ -- thus the scheme presented here can be tested further by improved measurements.

Figures (7)

• Figure 1: Best fit of BESIII data on $e^+e^-\to \gamma (D^0\bar{D}^{0}\pi^0/J/\psi\pi^+\pi^-)$BESIII:2023hml (first line) and LHCb data on $B^+\to K^+(J/\psi\pi^+\pi^-)$ with $J/\psi\pi^+\pi^-$ distribution from Ref. LHCb:2020fvo and $\pi^+\pi^-$ distribution from Ref. LHCb:2022jez (second line) in Scheme-I0. The green dashed and gray dash-dotted curves represent $X(3872)$, described by a Flatté formula as detailed in the Supplemental Materials supp, and the non-$X(3872)$ contributions, respectively. The line shapes in Scheme-I1 are almost indistinguishable from those in Scheme-I0.
• Figure 2: Pole positions of $X(3872)$ and $W_{c1}^0$, relative to the $D^0\bar{D}^{*0}$ threshold, from our analysis in Scheme-I0 with $1\sigma$ and $2\sigma$ statistical(sta.) uncertainties. Pole locations in Scheme-I1 are similar. The $X(3872)$ pole position with total uncertainties determined in this work is compared with that from the BESIII analysis BESIII:2023hml.
• Figure 3: Predicted $D^0\bar{D}^0\pi^0$ (left), in comparison with the Belle data for $B^0\to K^0D^0\bar{D}^0\pi^0$Belle:2023zxm, and $J/\psi\pi^+\pi^-$ (right) distributions using $P_\pm/P_0=2$ in Scheme-I0. Convolutions with the Belle Belle:2023zxm and LHCb LHCb:2020fvo energy resolutions are considered in the left and right panels, respectively.
• Figure 4: Comparison of the best fitted lineshapes in different schemes.
• Figure 5: Diagrams for $D^0\bar{D}^0\pi^0$ production. The gray square represents the $D\bar{D}^*$ scattering in $J^{PC}=1^{++}$ channels and $\otimes$ represents the production of $D\bar{D}^*$ from the given $1^{++}$ source.