Possible ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ molecular states

We investigate the possibility of deuteronlike ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ bound states within the one-boson-exchange model and systematically analyze the effects of the contact range ${\delta }^3(\overrightarrow{r})$ potential, the tensor term from the vector-meson exchange, and nonlocal potentials due to the dependence on the sum of the initial and final state center-of-mass momenta. We find that the pion-exchange potential including the ${\delta }^3(\overrightarrow{r})$ term and the tensor term of the $\rho$-exchange potential exhibit comparable magnitudes but opposite signs for any $S$-wave baryon-antibaryon systems. For the ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ system, it is most likely to form bound states with mass around 3.7 GeV in the $I(J^P)=0(2^{-})$ and $1(2^{-})$ channels.

Figure 1

Feynman diagram for the ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}\to {\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ process with $t$-channel meson exchanges, where $P_1=(M_{{\mathrm{\Sigma }}_c^{*}},\overrightarrow{p})$, $P_2=(M_{\mathrm{\Sigma }},-\overrightarrow{p})$, $P_3=(M_{{\mathrm{\Sigma }}_c^{*}},{\overrightarrow{p}}^{\prime })$, $P_4=(M_{\mathrm{\Sigma }},-{\overrightarrow{p}}^{\prime })$, and $q=P_1-P_3$ represent the four momenta of the corresponding particles.

Figure 2

Contributions of the vector and tensor coupling terms, $V_{\text{vector}}$ and $V_{\text{tensor}}$, respectively, in comparison to the total $\rho$-exchange potential for the ${\mathrm{\Sigma }}_c^{*}{\overline{\mathrm{\Sigma }}}_c^{*}$ systems with total spin $J=0$ (top row), $J=2$ (middle row), and $J=3$ (bottom row). Because of the $\delta$ potential in the tensor term, the relative importance is sensitive the cutoff. Here the $\mathrm{\Lambda }$ values are those taken in Ref. [66].

Figure 3

Effective potentials for the $S$-wave ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ systems with $\mathrm{\Lambda }=1\mathrm{GeV}$.

Figure 4

Dependence of the binding energy $E$ on the cutoff $\mathrm{\Lambda }$ for the $S$-wave ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ systems with the potential in Eq. (37).

Figure 5

Diagrams for the $t$-channel (a) pion and (b) $\rho$-meson exchanges for $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }\to \mathbb{B}{\overline{\mathbb{B}}}^{\prime }$.

Figure 6

(a) Ratio of the tensor-term contribution in the $\rho$-exchange amplitude to the pion-exchange amplitude in the $S$-wave $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }\to \mathbb{B}{\overline{\mathbb{B}}}^{\prime }$ process, and (b) ratio of the tensor-term contribution in the $\rho$-exchange potential to the pion-exchange potential at $r=0\mathrm{fm}$ with ${\mathrm{\Lambda }}_{\rho }=1\mathrm{GeV}$ in the $S$-wave $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }$ system.

Figure 7

(a) Ratio of the tensor-term contribution in $\omega$-exchange amplitude to total $\eta$-exchange amplitude in the $S$-wave $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }\to \mathbb{B}{\overline{\mathbb{B}}}^{\prime }$ process, and (b) ratio of the tensor-term contribution in $\omega$-exchange potential to total $\eta$-exchange potential at $r=0\mathrm{fm}$ and ${\mathrm{\Lambda }}_{\omega }=1\mathrm{GeV}$ in the $S$-wave $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }$ system.

Figure 8

(a) Ratio of the vector-term contribution in $\omega$-exchange amplitude to the total $\sigma$-exchange amplitude in the $S$-wave $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }\to \mathbb{B}{\overline{\mathbb{B}}}^{\prime }$ process, and (b) ratio of vector-term contribution in $\omega$-exchange potential to total $\sigma$-exchange potential at $r=0\mathrm{fm}$ and ${\mathrm{\Lambda }}_{\omega }=1\mathrm{GeV}$ in the $S$-wave $\mathbb{B}{\overline{\mathbb{B}}}^{\prime }$ system.

Figure 9

Effective potentials for the $S$-wave ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ systems at $\mathrm{\Lambda }=1\mathrm{GeV}$ after dropping the pion-exchange potential and the contribution of the tensor term in the $\rho$-exchange potential.

Figure 10

Dependence of the binding energy $E$ on the cutoff $\mathrm{\Lambda }$ for the $S$-wave ${\mathrm{\Sigma }}_c^{*}\overline{\mathrm{\Sigma }}$ systems with the potential in Eq. (53) which has dropped the pion-exchange potential and the contribution of the tensor term in the $\rho$-exchange potential.

Figure 11

The different $S$-wave effective potential $V(r)$ from different amplitude $\mathcal{M}(\overrightarrow{q})$ (a) or the different $S$-wave effective potential $V_0(r)$ from different amplitude $\mathcal{M}(\overrightarrow{q},\overrightarrow{k})$ (b), where $\mathrm{\Lambda }=1\mathrm{GeV}$, $m=0.5\mathrm{GeV}$ and $\overrightarrow{A}·\overrightarrow{B}=1$. The curves of ${\mathcal{M}}_2$ and ${\mathcal{M}}_3$ align perfectly, as they should according to Eq. (a42); similarly the ${\mathcal{M}}_4$ and ${\mathcal{M}}_5$ curves are identical due to Eq. (a44).