Charmonium spectrum with a Dirac potential model in the momentum space

We study the charmonium spectrum using a complete one gluon exchange approach based on a phenomenological relativistic $q\overline{q}$ potential model with Dirac spinors in momentum space. We use phenomenological screening factors to include vacuum quantum effects. Our formulation does not rely on nonrelativistic approximations. We fit the lowest-lying charmonia (below the $D\overline{D}$ threshold) and predict the higher-lying resonances of the spectrum. In general, we reproduce the overall structure of the charmonium spectrum and, in particular, we can reasonably describe the $X(3872)$ resonance mass as (mostly) a $c\overline{c}$ state. The numerical values of the free parameters of the model are determined taking into account also the experimental uncertainties of the resonance energies. In this way, we are able to obtain the uncertainties of the theoretical resonance masses and the correlation among the free parameters of the model.

Figure 1

Screening function, Eq. (11), for potentials I (dashed blue) and II (solid red). For $k_s$ and $p_s$ we use the central values in Table 2. We highlight the values of the screening momenta $p_{1/2}^{\mathrm{I}}$ (potential I) and $p_{1/2}^{\mathrm{II}}$ (potential II) defined in Sec. 3b.

Figure 2

Complete charmonium spectrum obtained with potential I (red boxes) compared to the experimental spectrum from PDG (blue boxes). The size of the red boxes accounts for error bars at $1\sigma$ (68%) C.L. In the same way, the size of the blue boxes stands for the experimental error of the resonance masses. The dashed lines stand for the open charm energies $D\overline{D}$ and $D{D}^{*}$. The calculation of the uncertainties takes into account all the correlations among the parameters (see Sec. 3b for details). In the horizontal axis the quantum numbers $J^{PC}$ are reported for the states above. In the right side of the figure, the arrows indicate the states (below the $D\overline{D}$ threshold) that are used in the fitting procedure and the predictions of the model with potential I. It is important to note that we have used the latest nomenclature by the PDG where the vector mesons are also named as $X$ instead of $Y$ as it is commonplace in the charmonia research field. The states in both naming schemes relate as follows: $Y(4260)\leftrightarrow X(4260)$, $Y(4360)\leftrightarrow X(4360)$, $Y(4660)\leftrightarrow X(4660)$, $Y(4140)\leftrightarrow X(4140)$ and $Y(3915)\leftrightarrow X(3915)$.

Figure 3

Charmonium spectrum obtained with potential II. Notation as in Fig. 2.