Two nearby states in the $X(3872)$ region: Resolving the radiative-decay ratio tension with $η_{c2}$

Abstract

Recently, LHCb reported the radiative-decay ratio ${\cal R}^{ψγ}\equiv {\cal B}[X(3872)\to ψ'γ]/{\cal B}[X(3872)\to J/ψγ]=1.67\pm 0.25$ extracted from $B^+\to K^+(J/ψγ, ψ'γ)$. This result differs markedly ($\sim4.6σ$) from the BESIII value obtained from $e^+e^-\to γ(J/ψγ, ψ'γ)$, ${\cal R}^{ψγ}=-0.04\pm 0.28$. Such a significant tension suggests that more than one state in the $X(3872)$ region contributes to the processes. We therefore propose a two-state scenario: a shallow $D^{*0}\bar{D}^0$ bound state with $J^{PC}=1^{++}$ and a $2^{-+}$ charmonium candidate, $η_{c2}$, slightly above the $D^{*0}\bar{D}^0$ threshold. We show that this hypothesis consistently describes these ratios along with other branching fractions and lineshapes across multiple processes. By contrast, fits without the $η_{c2}$ component fail to reproduce the radiative-ratio data. We also predict helicity-angle distributions that motivates the future experiments to test the two-state hypothesis and search for the so-far missing $η_{c2}$.

Figures (4)

• Figure 1: $B^{+,0}$ and $Y(4230)$ decay mechanisms. Final states $\alpha$ include $J/\psi\rho, J/\psi\omega, \{D^{*0}\bar{D}^0\}, J/\psi\gamma$, and $\psi' \gamma$. The shaded blobs are the rescattering amplitude that generates the $X(3872)$ pole.
• Figure 2: (a)-(e) Invariant mass $M_f$ and (f) helicity angle $\theta_{X(3872)}$ distributions for $B\to Kf$ where final states $f$ are indicated in each panel; $B^+\to K^+f$ for (a)-(e), $B^0\to K^0f$ for (e), and an efficiency-weighted sum of $B^+\to K^+f$ and $B^0\to K^0f$ in (f). The units are events/bin. Histograms are obtained by smearing the corresponding theoretical curves with experimental resolutions and then averaging within each bin. In (c), the lower histograms and the data [orange triangle] are obtained by cutting off contributions from $M_{\pi^+\pi^-\pi^0}<0.7695$ GeV. The legend in (a) applies to all panels except (b), which has its own legend. The data are from Ref. x3872_lhcb_lineshape in (a); x3872_lhcb_pipi in (b); x3872_babar_omega in (c) and (d); x3872_belle_DstarD2 ($D^{*0}\to D^0\pi^0$) in (e); x3872_babar_DstarD in (f). The uncorrected data points [orange triangle] in (c) are scaled to be comparable with the corrected data [black circles].
• Figure 3: Invariant mass ($M_f$) distributions for $e^+e^-\to \gamma f$ where final states $f$ are indicated in each panel. The data are from Ref. x3872_bes3_jpsipipi in (a); x3872_bes3_jpsi-omega in (b); x3872_bes3_DstarDbar in (c).
• Figure 4: Helicity angle distributions predicted by the default model. The unit is arbitrary; however, within each of panels (a) and (b), the model predicts the relative strengths of different final states from the same initial state. In (b), $X_L$ and $X_H$ contributions correspond to the integrated $M_{D^{*0}\bar{D}^0}$ regions. See the main text for details on the resolution smearing and integration procedures.