$Y(3940)$ as a mixed charmonium-molecule state

Using the QCD sum rules approach, we study the mass and the decay widths of the $Y(3940)$ state, assuming that it can be described by a mixed charmonium-molecule scalar state, i.e., a mixture between the ${\chi }_{c0}$ charmonium and $D^{*}{\overline{D}}^{*}$ molecule. Using a current with $J^{PC}=0^{++}$, we estimate for the mixing angle, $\theta =(76.0\pm 5.0)°$, resulting in a mass value of $M_Y=(3.95\pm 0.11)\mathrm{GeV}$, which is in reasonable agreement with the experimental mass of the $Y(3940)$ state. For the decay width, we evaluate the channels $Y\to J/\psi \omega$ and $Y\to \gamma \gamma$. We find the values ${\mathrm{\Gamma }}_{Y\to J/\psi \omega }\approx (1.7\pm 0.6)\mathrm{MeV}$ and ${\mathrm{\Gamma }}_{Y\to \gamma \gamma }\approx (1.6\pm 1.3)\mathrm{KeV}$, respectively. We also study the decay process of this state into channels containing $D\overline{D}$ mesons in the final state. The result for the order of magnitude of the product ${\mathrm{\Gamma }}_{Y\to \gamma \gamma }\times {\mathrm{\Gamma }}_{Y\to J/\psi \omega }\sim \mathcal{O}(10^3){\mathrm{KeV}}^2$ is also in reasonable agreement with the experimental data. We thus conclude that the present description of the $Y(3940)$ as a mixed charmonium-molecule state is a possible scenario to explain the structure of such a state.

Figure 1

OPE convergence in the region $1.7\le M_B^2\le 3.8{\mathrm{GeV}}^2$ for $\sqrt{s_0}=4.40\mathrm{GeV}$ and $\theta =76.0°$. One plots the relative contributions starting with the perturbative contribution (line with circles), and each other line represents the relative contribution after adding of one extra condensate in the expansion: + $⟨\overline{q}q⟩$ (dashed line), $+⟨G^2⟩$ (dotted line), $+⟨\overline{q}Gq⟩$ (dot-dashed line), $+⟨\overline{q}q{⟩}^2+⟨G^3⟩$ (line with triangles), and $⟨\overline{q}q⟩·⟨\overline{q}Gq⟩$ (solid line).

Figure 2

The pole (solid line) and continuum (dotted line) contributions for $\sqrt{s_0}=4.40\mathrm{GeV}$ and $\theta =76.0°$.

Figure 3

The mass as a function of the sum rule parameter $M_B^2$ for $\sqrt{s_0}=4.30\mathrm{GeV}$ (dotted line), $\sqrt{s_0}=4.40\mathrm{GeV}$ (solid line) and $\sqrt{s_0}=4.50\mathrm{GeV}$ (dashed line). The respective parentheses indicate the valid Borel window.

Figure 4

The form factor $g_{Y\psi \omega }(Q^2)$ as a function of the momentum $Q^2$ and Borel mass parameter $M_B^2$.

Figure 5

QCDSR results for the form factor $g_{Y\psi \omega }(Q^2)$, for $\sqrt{s_0}=4.40\mathrm{GeV}$ (circles). The solid line gives the parametrization of the QCDSR results through Eq. (32). The cross is the value of the coupling constant.

Figure 6

The form factor $g_{Y\gamma \gamma }(Q^2)$ as a function of the momentum $Q^2$ and Borel mass parameter $M_B^2$.

Figure 7

QCDSR results for the form factor $g_{Y\gamma \gamma }(Q^2)$, for $\sqrt{s_0}=4.40\mathrm{GeV}$ (circles). The solid line gives the parametrization of the QCDSR results. The cross is the value of the coupling constant, at $Q^2=0$.