High-lying states in the charmonium family

Following the gradual discovery of the ground states of $D$-wave charmonium mesons, there is increasing optimism that the $2D$ states will be identified in future experimental efforts. Consequently, this work begins with an analysis of the masses and decay properties of the predicted $2D$ states.

Our results indicate that the masses of the $2D$ states are predicted to be around 4.12–4.14 GeV, which is approximately 70 MeV smaller than the values reported in Ref. [46]. From Table 3, we observe that the two primary decay channels for the $2^{1}D_{2}$ state are $DD^{*}$ and $D^{*}D^{*}$, with the branching fraction reaching up to 98%. Therefore, we suggest that the $2^{1}D_{2}$ state may be detected via these decay modes. When comparing our results to those in Ref. [46], we find that our decay widths are generally smaller. Notably, the $D_{s}D_{s}^{*}$ decay mode is an order of magnitude smaller in our work than in Ref. [46].

For the $J^{PC}=1^{--}$ states with $n=2$, the two dominant decay channels are $DD$ and $D^{*}D^{*}$, with the $DD^{*}$ process being suppressed. In our calculations, the total decay width is 48.73 MeV, with branching ratios of 39.81% and 56.78% for the $DD$ and $D^{*}D^{*}$ channels, respectively. The ratio of partial widths is $\Gamma(DD):\Gamma(D^{*}D^{*})=1.0:1.4$. Although the decay modes of this state reported in Ref. [46] show a similar trend, there is a notable discrepancy in the total decay width, which they estimate to be 74 MeV. This difference may be attributed to our calculated mass of 4125 MeV, which is lower than their value of 4194 MeV. Furthermore, the decay widths for the $D_{s}D_{s}$ and $D_{s}D_{s}^{*}$ modes are significantly different: 0.04 MeV and 1.08 MeV in our work, compared to 8.0 MeV and 14 MeV in Ref. [46].

For the $2^{3}D_{2}$ state, the dominant decay channels are $DD^{*}$ and $D^{*}D^{*}$, with decay widths of 22.88 MeV and 23.95 MeV, respectively. Their branching ratios are 47.05% and 49.25%. Compared to Ref. [46], our mass is 21 MeV lower, and our total decay width of 48.63 MeV is significantly smaller than their value of 92 MeV. Specifically, for the $D_{s}D_{s}^{*}$ decay mode, we predict a decay width of 1.8 MeV with a branching ratio of 3.7%, while Ref. [46] estimates a decay width of 26 MeV with a branching ratio of 28.26%.

For the $2^{3}D_{3}$ state, the two main decay channels are $DD^{*}$ and $D^{*}D^{*}$, with branching ratios of 40.1% and 53.0%, respectively. The $DD$ decay channel for the $2^{3}D_{3}$ state is notably smaller than the value reported in Ref. [46], showing a difference of about one order of magnitude. However, the decay modes involving strange quarks remain consistent with those in the other $2D$ states, and are similarly smaller than those reported in Ref. [46]. Despite slight differences in the calculated masses of the $2D$ states, the decay modes and branching ratios are largely consistent.

For the $2^{1}D_{2}$, $2^{3}D_{2}$, and $2^{3}D_{3}$ states, the dominant decay channels are $DD^{*}$ and $D^{*}D^{*}$. However, compared to Ref. [46], the branching ratios for these two channels are notably higher in our study. On the other hand, decay modes involving strange quarks are less pronounced in our results, which highlights the need for further experimental validation.

In the MGI model, the predicted masses of the four $3D$ states are approximately 4.34 GeV. The dominant decay channels for the $3^{1}D_{2}$ state remain $DD^{*}$ and $D^{*}D^{*}$. We predict the branching ratios for these channels to be 63.17% and 23.77%, respectively. Additionally, the $DD_{2}^{*}$ channel exhibits a greater decay width compared to the $DD_{0}^{*}$, $DD_{1}$, and $DD_{1}^{\prime}$ channels. The $3^{1}D_{2}$ state also shows three decay processes involving strange quarks, with the $D_{s}D_{s0}^{*}$ channel contributing the largest decay width (3.22%) due to its higher mass threshold.

The $3^{3}D_{1}$ state has three primary decay channels: $DD$, $D^{*}D^{*}$, and $DD_{1}^{\prime}$, with the ratios of partial widths predicted to be $\Gamma(DD):\Gamma(D^{*}D^{*}):\Gamma(DD_{1}^{\prime})=1.9:1.7:1.0$. Compared to the $2^{3}D_{1}$ state, the $DD^{*}$ and $D^{*}D^{*}$ channels remain dominant, but the new $DD_{1}^{\prime}$ channel contributes a branching ratio of 18.76%. For the $3^{3}D_{2}$ state, the $DD^{*}$ channel dominates, with a branching ratio higher than that of the $D^{*}D^{*}$ channel, which contrasts with the $2^{3}D_{2}$ state, where $D^{*}D^{*}$ dominates. This state primarily decays into $DD^{*}$, $D^{*}D^{*}$, $DD_{1}$, $DD_{1}^{\prime}$, and $DD^{*}_{2}$, with the ratio of partial widths predicted to be $\Gamma(DD^{*}):\Gamma(D^{*}D^{*}):\Gamma(DD_{1}):\Gamma(DD_{1}^{\prime}):\Gamma(DD^{*}_{2})=9.2:4.9:1.1:1.2:1.0$.

Finally, for the $3^{3}D_{3}$ state, the most significant decay channel is $DD^{*}$, with a branching ratio of 51.2%. Compared to the $2^{3}D_{3}$ state, the partial decay width of the $3^{3}D_{3}$ state into the $D^{*}D^{*}$ channel decreases significantly, primarily due to the node effect in the corresponding spatial wave function.

Considering that the $1D$ state has already been observed, the prospects for detecting the $1F$ states have increased significantly with advancements in experimental techniques. In Table 3, we present the decay information for the $F$ states with $n=1,2,3$.

The masses of the $1F$ states are approximately 4.07 GeV, about 20 MeV smaller than the estimates provided in Ref. [46]. For the $1^{1}F_{3}$ state, the dominant strong decay channel is $DD^{*}$, followed by $D^{*}D^{*}$, which also contributes notably. We predict the ratio of partial widths to be $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=6.2:1.0$. However, Ref. [46] reports that the $DD^{*}$ channel accounts for nearly the entire strong decay width in their calculations.

Among these states, the broadest is the $1^{3}F_{2}$ state, with a width of 112.48 MeV. The primary decay channels contributing to its width are $DD$ and $DD^{*}$, which together account for 95% of the strong decay width. This result is in agreement with Ref. [46], although our findings show that the partial width of the $DD^{*}$ channel is larger than that of the $DD$ channel, which differs from the calculation in Ref. [46]. Similar to the $1^{1}F_{3}$ state, the $1^{3}F_{3}$ state also exhibits $DD^{*}$ as the dominant strong decay channel, contributing 92% to the total width, with the $D^{*}D^{*}$ channel being relatively suppressed.

The $1^{3}F_{4}$ state is a narrow state, with a strong decay width of 37.97 MeV, which is larger than the estimated width of 8.6 MeV reported in Ref. [46]. The $1^{3}F_{4}$ state predominantly decays into the $DD$, $DD^{*}$, and $D^{*}D^{*}$ channels, with a partial width ratio of $\Gamma(DD):\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.4:1.0:4.0$. This ratio differs from that reported in Ref. [46], where smaller phase spaces in the $D^{*}D^{*}$ decay mode lead to narrower partial widths. Additionally, this mass discrepancy may influence the widths of other decay channels.

For the $2F$ states, we find masses around 4.29 GeV, which is lower than those predicted by the quenched model in Ref. [46]. We observe fewer decay modes compared to those reported in the quenched model, and the total decay width is smaller. This is likely due to unquenched effects in the MGI model, which reduce the masses of high-lying states, making some decay channels inaccessible. The strong decay width of the $2^{1}F_{3}$ state is approximately 38.78 MeV, and it predominantly decays into $DD^{*}$ and $D^{*}D^{*}$, with $DD^{*}_{0}$, $D_{s}D_{s}^{*}$, and $D^{*}_{s}D_{s}^{*}$ channels being relatively suppressed. The ratio between the main decay channels is predicted to be $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=2:1$.

The $2^{3}F_{2}$ state has its decay width primarily contributed by the $DD$, $DD^{*}$, and $D^{*}D^{*}$ modes, with a corresponding partial width ratio of $\Gamma(DD):\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.2:1.0:1.1$. The $2^{3}F_{3}$ state exhibits two dominant decay modes: $DD^{*}$ and $D^{*}D^{*}$, with a predicted partial width ratio of $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=2.4$. The narrowest of the $2F$ states is the $2^{3}F_{4}$ state, which primarily decays into the $D^{*}D^{*}$ channel, with the branching fraction reaching 85.12%. Similar to the $2D$ states, decay modes involving strange quarks in the $2F$ states are less prominent in our results.

The decay channels $DD^{*}$ and $D^{*}D^{*}$ account for nearly the entire decay width of the $2^{1}F_{3}$, $2^{3}F_{3}$, and $2^{3}F_{4}$ states, which contrasts sharply with the results from Ref. [46], where multiple decay channels exhibit significant branching ratios. This difference is likely due to unquenched effects in our model. For the $3F$ states, the MGI model predicts masses around 4.45 GeV. The primary decay channels for these states remain $DD^{*}$ and $D^{*}D^{*}$, but with increasing mass, new decay channels become accessible. The $3^{1}F_{3}$ state has two dominant decay channels, $DD^{*}$ and $D^{*}D^{*}$, with the $DD^{*}_{2}$ channel also contributing significantly. The predicted ratio of partial widths is $\Gamma(DD^{*}):\Gamma(D^{*}D^{*}):\Gamma(DD^{*}_{2})=2.8:1.6:1.0$. The decay width of the $3^{3}F_{2}$ state is governed by the $DD$, $DD^{*}$, $D^{*}D^{*}$, and $DD_{1}$ channels, with the branching fraction $\text{Br}[\chi_{c2}(3^{3}F_{2})\to DD,\,DD^{*},D^{*}D^{*},\,DD_{1}]$ reaching up to 92%. The $3^{3}F_{3}$ state has three dominant decay modes: $DD^{*}$, $D^{*}D^{*}$, and $DD^{*}_{2}$, with the ratio of partial widths predicted to be $\Gamma(DD^{*}):\Gamma(D^{*}D^{*}):\Gamma(DD^{*}_{2})=2.7:1.3:1.0$. Finally, the $3^{3}F_{4}$ state predominantly decays into $DD^{*}$ and $D^{*}D^{*}$, with the ratio between partial widths being approximately $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.0:4.1$. The decay pattern for the $3F$ states closely follows that of the $1F$ and $2F$ states, with the $DD^{*}$ channel remaining dominant.

We observe in Table 2 that for all $G$-wave states, the masses are nearly identical for states with the same $n$. In Table 3, we present the strong decay widths for the $1^{3}G_{3}$, $2^{3}G_{3}$, and $3^{3}G_{3}$ states. As a result, the strong decay widths of these states are similar.

For the $1G$ states, the masses are approximately 4.25 GeV, and the dominant decay channels are $DD^{*}$ and $D^{*}D^{*}$. Due to the near equivalence in mass among these states, their strong decay widths are similar, although the ratios of the partial widths differ. The total decay widths of the $1G$ states are approximately equal, while the ratios between the partial widths of $DD^{*}$ and $D^{*}D^{*}$ vary. Specifically, for the $1^{1}G_{4}$ state, the ratio is $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.4:1.0$, while for the $1^{3}G_{4}$ state, it is $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.9:1.0$, and for the $1^{3}G_{5}$ state, the ratio is $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.6:8.0$.

When compared to the results in Ref. [46], we find that the $DD^{*}$ and $D^{*}D^{*}$ channels remain the dominant decay modes across all $1G$ states. However, Ref. [46] reports that the $DD$ channel contributes more significantly to the decay width, particularly for the $1^{3}G_{3}$ and $1^{3}G_{5}$ states. In fact, the width of the $DD$ channel significantly exceeds that of the $D^{*}D^{*}$ channel for the $1^{3}G_{3}$ state, which contrasts with our findings. This discrepancy may stem from differences in the wave function for the $D^{*}$ meson used in our calculation, which is based on the MGI model, as opposed to the simple harmonic oscillator (SHO) wave function employed in Ref. [46].

The masses of the $2G$ states are approximately 4.43 GeV, and their decay behaviors closely resemble those of the $G$ states. The dominant decay channels for the $2^{1}G_{4}$ and $2^{3}G_{4}$ states remain $DD^{*}$ and $D^{*}D^{*}$. However, the open-charm decay channels $DD_{0}^{*}$, $DD_{1}$, $DD_{1}^{\prime}$, $DD^{*}_{2}$, and $D^{*}D^{*}_{0}$ are relatively suppressed.

The ratios of the main partial widths for the $2^{1}G_{4}$ and $2^{3}G_{4}$ states are $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.6$ and $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=2.5$, respectively. In addition to the $DD^{*}$ and $D^{*}D^{*}$ modes, the $DD$ channel exhibits a considerable branching ratio for the $2^{3}G_{3}$ state. The partial width ratios of the dominant channels for the $2^{3}G_{3}$ state are $\Gamma(DD):\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=2.2:3.3:1.0$.

For both the $2^{3}G_{3}$ and $3^{3}G_{3}$ states, the $DD^{*}$ decay channel exhibits the largest partial width. Among the $2G$ states, the $2^{3}G_{5}$ state is the narrowest. Its decays are dominated by the $DD$ and $D^{*}D^{*}$ channels, with a corresponding partial width ratio of $\Gamma(DD):\Gamma(D^{*}D^{*})=1.0:11.4$.

Additionally, channels involving strange quarks also contribute to the decay widths. For example, the $D_{s}D_{s}^{*}$ channel contributes to the widths of the $2^{1}G_{4}$, $2^{3}G_{3}$, and $2^{3}G_{4}$ states, while the $D_{s}D_{s}$ channel contributes to the $2^{3}G_{3}$ state. However, these contributions to the total widths are negligible.

Regarding the $3G$-wave state, their masses are all around 4.54 GeV. The channel with the largest decay widths is $DD^{*}$ for the $3^{1}G_{4}$, $3^{3}G_{3}$, and $3^{3}G_{4}$ states, while the $D^{*}D^{*}$ channel is dominant for the $3^{3}G_{5}$ state. Although the $DD^{*}$ and $D^{*}D^{*}$ channels are still the dominant decay channels for all $3G$ states, their branching ratios are smaller than those of the corresponding $2G$ states. The corresponding ratios of $3^{1}G_{4}$ and $3^{3}G_{4}$ states between main partial widths are $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=1.5$ and $\Gamma(DD^{*}):\Gamma(D^{*}D^{*})=2.3$ , respectively. The decay rates into $DD_{0}^{*}$, $DD_{2}^{*}$, $D^{*}D_{1}^{\prime}$ and $D^{*}D_{2}^{*}$ channels are also sizable for $3^{1}G_{4}$ and $3^{3}G_{4}$ states, with the branching fractions of $\text{Br}[\psi(3^{1}G_{4})/\psi(3^{3}G_{4})\to DD_{0}^{*},DD_{2}^{*},D^{*}D_{1}^{\prime},D^{*}D_{2}^{*}]$ around 20%. The $3^{3}G_{3}$ state mainly decays into $DD$,$DD^{*}$, $D^{*}D^{*}$, $DD_{1}$, and $D^{*}D_{1}$. Other open charm decay channels are relatively suppressed. The predicted ratio between the dominant partial widths in the MGI model is $\Gamma(DD):\Gamma(DD^{*}):\Gamma(D^{*}D^{*}):\Gamma(DD_{1}):\Gamma(D^{*}D_{1})=4.4:5.9:2.5:1.6:1.0$. The branching ratio of the $DD$ channel for $3^{3}G_{3}$ state is 24.38%, which is larger than that of the $D^{*}D^{*}$ channel, similar to the $2^{3}G_{3}$ state The $3^{3}G_{5}$ state is narrowest $3G$ state with the width 23 MeV. The branching ratios of the $DD$ and $DD^{*}$ decay channels are smaller, while the $D^{*}D^{*}$ and $D^{*}D_{2}^{*}$ channels dominate. The ratio between the main partial widths is $\Gamma(D^{*}D^{*}):\Gamma(D^{*}D^{*}_{2})=5:1$.

In Tables 4 and 5, we present the radiative decay information for the $D$-wave states. The numerical results indicate that the decay widths for transitions to $P$-wave and $F$-wave states are generally larger, whereas those to $S$-wave and $D$-wave states are comparatively smaller. For the $2^{1}D_{2}$ state, the dominant radiative decay channels are $2^{1}D_{2}\rightarrow 2^{1}P_{1}\gamma$ and $1^{1}P_{1}\gamma$, with decay widths of 167.10 keV and 58.45 keV, respectively. The $3^{1}D_{2}$ state primarily decays through the channels $1^{1}P_{1}\gamma$, $2^{1}P_{1}\gamma$, and $3^{1}P_{1}\gamma$. For the $2^{3}D_{1}$ state, the largest radiative decay width corresponds to the $2^{3}D_{1}\rightarrow 2^{3}P_{0}\gamma$ channel, with a value of 126.39 keV. Other notable decay channels for this state include $2^{3}D_{1}\rightarrow 2^{3}P_{1}\gamma$ and $1^{3}P_{0}\gamma$, with widths of 115.98 keV and 82.20 keV, respectively. For the $3^{3}D_{1}$ state, while transitions to both $P$-wave and $F$-wave states still yield large widths, the dominant decay process is $3^{3}D_{1}\rightarrow 3^{3}P_{0}\gamma$, rather than $3^{3}D_{1}\rightarrow 2^{3}P_{0}\gamma$.

Table 4 shows that the largest decay width for the $2^{3}D_{2}$ state occurs for the $2^{3}D_{2}\rightarrow 2^{3}P_{1}\gamma$ channel, with a width of 127.69 keV. In contrast, the largest decay width for the $3^{3}D_{2}$ state is found for the $3^{3}D_{2}\rightarrow 3^{3}P_{1}\gamma$ channel, with a relatively narrow width of 81.40 keV, which is smaller than that of other $3D$ states.

The dominant decay channel for the $2^{3}D_{3}$ state is $2^{3}D_{3}\rightarrow 2^{3}P_{2}\gamma$, with a decay width of 212.66 keV, while other decay channels for this state are significantly smaller. For the $3^{3}D_{3}$ state, the most significant decay channel is $3^{3}D_{3}\rightarrow 3^{3}P_{2}\gamma$, with a width of 93.97 keV.

Our results reveal that the broadest partial radiative widths correspond to transitions where both the initial and final states have the same quantum numbers $n$ and $S$ for $D$ states.