Competition of $χ_{c}(2P)$ quarkonia and continuum in $e^+ e^- \to e^+ e^- D \bar{D}$

TL;DR

The paper analyzes $D\bar{D}$ production in $e^+e^-$ collisions via two mechanisms: non-resonant continuum through $t$/$u$-channel exchanges (notably $D^*$ for neutral channels and $D^{(*)}$ for charged channels) and resonant production through $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$. It develops a detailed framework for $\gamma^*\gamma^*$ fusion, including reggeized $D^*$ exchanges, NRQCD-based form factors for the $\chi_{c2}$ and $\chi_{c0}$ vertices, and phenomenological couplings for $\chi_{cJ}\to D\bar{D}$ decays. By comparing to Belle and BaBar data, the authors find a strong neutral-channel continuum contribution near $M_{D\bar{D}}\sim3.8$ GeV and a smaller charged-channel continuum, with interference between $\chi_{c0}(3860)$ and the continuum improving near-threshold descriptions. They estimate $\sigma(e^+e^-\to e^+e^-\chi_{c2}(3930))\sim1$ pb and extract $\Gamma_{\gamma\gamma}\times B(\chi_{c2}\to D\bar{D})\sim0.32$ keV, consistent with experimental constraints and enabling predictions for differential distributions in Belle II kinematics.

Abstract

We discuss the production of $D \bar{D}$ pairs in $e^+ e^-$ collisions, where $D$ refers to either $D^0$ or $D^+$. The continuum mechanism with the $t/u$-channel vector-meson $D^*$ exchanges are considered. The results of the calculation depend on the parameter of the off-shell form-factor for the virtual $D^*$ mesons. The $D^* D γ$ coupling constants are found from the $D^* \to D γ$ decays. We find relatively large contribution for the $D^0 {\bar D}^0$ channel and much smaller contribution in the $D^+ D^-$ channel. In the second case we consider also the $D^{\pm}$ exchanges. We conclude that the bump at $M_{D^0 {\bar D}^0} = 3.8$ GeV observed by the Belle and BaBar Collaborations has rather continuum origin than it corresponds to the broad resonance $χ_{c0}(3860)$. We discuss also production of the $χ_{c2}(3930)$ resonance which is a candidate for the $χ_{c2}(2P)$ state. This state can decay into both $D \bar{D}$ channels, however the branching fractions are not well known at present. From a comparison of our model results to the BaBar data we find $B(χ_{c2}(3930) \to D \bar{D}) = 0.58 \pm 0.13$ using the two-photon width $Γ_{γγ} = 0.544$ keV (obtained for the Buchmüller-Tye potential) evaluated within the light-front approach (NRQCD limit). Our finding of $Γ_{γγ} \times B(χ_{c2}(3930) \to D \bar{D}) = 0.32 \pm 0.07$ keV is close to the Belle and BaBar results. Realistic predictions of the differential distributions in several variables and integrated cross-sections are given for the Belle II kinematics.

Figures (4)

• Figure 1: Diagrams representing $D^{+}D^{-}$ production in $e^{+}e^{-}$ collisions. For the process $\gamma \gamma \to D^{+} D^{-}$ we consider the $t$- and $u$-channel $D^{\pm}$ and $D^{*}(2010)^{\pm}$ exchanges, the contact term, and the exchange of $\chi_{c0,2}$ mesons in the $s$ channel. The diagrams for the $D^0 {\bar{D}}^0$ production are similar, but in the case of non-resonant production we consider only the $D^{*}(2007)^{0}$$t/u$-channel exchanges.
• Figure 2: Distributions in $\log_{10}(p_{t1}/ 1\,\rm{GeV})$, $|t_{1}|$, and $p_{t, D\bar{D}}$ for the $e^+ e^- \to e^+ e^- (\chi_{c2}(3930) \to D^0 {\bar{D}}^0)$ reaction for $\sqrt{s} = 10.54$ GeV. We show the results without and with the cut on $p_{t, {\rm sum}} < 0.05$ GeV. The calculations were done for the BT potential and with $g_{\chi_{c2} D \bar{D}} = 10.1$.
• Figure 3: The invariant mass distribution for the $e^{+} e^{-} \to e^{+} e^{-} D \bar{D}$ reaction for the neutral and charged channels, and the combined $D \bar{D}$ final state. (a) Comparison of theoretical results to the Belle data Belle:2005rte. (b) Comparison of theoretical results to the BaBar data BaBar:2010jfn. (c) The differential cross sections $d\sigma/dM_{D \bar{D}}$ for the complete model (continuum, $\chi_{c0}(3860)$, and $\chi_{c2}(3930)$ contributions) and for the continuum contribution alone. For the $\chi_{c0}(3860)$ term we take (\ref{['4.0_fit']}). The red long-dashed lines correspond to the $D^0 {\bar{D}}^0$ channel and the blue dashed lines correspond to the $D^{+}D^{-}$ channel. The black solid line represents the sum of all contributions for two channels. In panel (d), we show the differential distributions in transverse momentum of the $D \bar{D}$ pair. The calculations were done for $\sqrt{s} = 10.54$ GeV and with the cut on $M_{D \bar{D}} \in (3.91, 3.95)$ GeV. The meaning of the lines is the same as in Fig. \ref{['fig:M34_BaBar']}(c).
• Figure 4: Comparison of theoretical results to the BaBar data BaBar:2010jfn. The meaning of the lines is as in Fig. \ref{['fig:M34_BaBar']}, but for the $\chi_{c0}(3860)$ term we used the middle values from (\ref{['4.0_pdg']}).