Revisiting the $Z_c(4025)$ structure observed by BESIII in $e^{+}{e}^{-}\to (D{}^{*}\overline{D}{}^{*}{)}^{\pm ,0}{\pi }^{\mp ,0}$ at $\sqrt{s}=4.26\mathrm{GeV}$

Within the framework of the local hidden gauge formalism, we have reanalyzed the hidden-charm vector-vector meson $D^{*}{\overline{D}}^{*}$, $K^{*}{\overline{K}}^{*}$, and $\rho \rho$ interactions with the quantum numbers $I(J^P)=1(1^{+})$. The estimation indicates that $D^{*}{\overline{D}}^{*}$ system could form a bound state around 4010 MeV. By comparing our theoretical estimation with the BESIII measurements of the $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^{\pm }{\pi }^{\mp }$ and $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^0{\pi }^0$ reactions at $\sqrt{s}=4.26\mathrm{GeV}$, we find that the enhancement structure of the $Z_c(4025)$ in the $D^{*}{\overline{D}}^{*}$ invariant mass distribution observed by BESIII could be interpreted by a shallow $D^{*}{\overline{D}}^{*}$ bound state.

Figure 1

The modulus squared of the amplitude $|t_{D^{*}{\overline{D}}^{*}\to D^{*}{\overline{D}}^{*}}{|}^2$ calculated with different ${\alpha }_1$. The red solid and blue dashed curves correspond to ${\alpha }_1=-1.0$ and ${\alpha }_1=-0.7$, respectively.

Figure 2

The diagrams for the $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^{\pm ,0}{\pi }^{\mp ,0}$ reaction. Left: the mechanism of the $D^{*}{\overline{D}}^{*}\pi$ direct production. Right: the mechanism of the $S$-wave $D^{*}{\overline{D}}^{*}$ interaction.

Figure 3

The $D^{*}{\overline{D}}^{*}$ invariant mass distributions of the $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^{\pm ,0}{\pi }^{\mp ,0}$ reactions measured by BESIII divided by phase space factor $k\tilde{q}/s\sqrt{s}$ of Eq. (18). (a) The $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^{\pm }{\pi }^{\mp }$ reaction [23], and (b) the $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^0{\pi }^0$ reaction [24].

Figure 4

The $D^{*}{\overline{D}}^{*}$ invariant mass distributions with the fitted parameters $V_p/V_p^{\prime }$ and ${\alpha }_1$ in Table 2. The curves labeled as “total” show the calculated results, the “$|T{|}^2$” curves stand for the contribution from the shallow $D^{*}{\overline{D}}^{*}$ bound state, and the “WS BKG” curves are the events of the wrong sing background taken from Refs. [23, 24]. The experimental data labeled as “BESIII” show the BESIII measurements [23, 24], and the error bands labeled as “68% C.L.” correspond to the 68% confidence level of our results due to the uncertainties of the fitted parameters. Diagrams (a) and (b) correspond to the $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^{\pm }{\pi }^{\mp }$ and the $e^{+}{e}^{-}\to (D^{*}{\overline{D}}^{*}{)}^0{\pi }^0$ reactions, respectively.

Figure 5

The $D^{*}{\overline{D}}^{*}$ invariant mass distributions using Eq. (17), with the fitted parameters of “Fit B-charge/Fit B-neutral” in Table 2. The explanations of the curves are the same as the ones in Fig. 4.