Molecular pentaquark states with open charm and bottom flavors

Abstract

We study the possibly-existing molecular pentaquark states with open charm and bottom flavors, i.e., the states with the quark contents $c\bar{b}qqq$ and $b\bar{c}qqq$ ($q=u,d,s$). We investigate the meson-baryon interactions through the coupled-channel unitary approach within the local hidden-gauge formalism, and extract the poles by solving the Bethe-Salpeter equation in coupled channels. These poles qualify as molecular pentaquark states, which are dynamically generated from the meson-baryon interactions through the exchange of vector mesons. Our results suggest the existence of the $\varSigma _c^{(*)} B^{(*)}$ and $\varSigma _b^{(*)} \bar{D}^{(*)}$ molecular states with isospin $I=1/2$, the $\varXi _c^{(\prime ,*)} B^{(*)}$ and $\varXi _b^{(\prime ,*)} \bar{D}^{(*)}$ molecular states with isospin $I=0$ and $I=1$, as well as the $\varOmega _c^{(*)} B^{(*)}$ and $\varOmega _b^{(*)} \bar{D}^{(*)}$ molecular states with isospin $I=1/2$.

Appendices

Baryon wave functions

We summarize all the relevant baryon wave functions in this appendix. The wave functions for the $J^P=1/2^+$ baryons are:

$$\begin{aligned} \big | \varXi _c^+ \big \rangle= & {} \Big | {1\over \sqrt{2}} c(us-su) \Big \rangle \big | \chi _{MA} \big \rangle , \nonumber \\ \big | \varXi _c^0 \big \rangle= & {} \Big | {1\over \sqrt{2}} c(ds-sd) \Big \rangle \big | \chi _{MA} \big \rangle , \nonumber \\ \big | \varXi _c^{\prime +} \big \rangle= & {} \Big | {1\over \sqrt{2}} c(us+su) \Big \rangle \big | \chi _{MS} \big \rangle , \nonumber \\ \big | \varXi _c^{\prime 0} \big \rangle= & {} \Big | {1\over \sqrt{2}} c(ds+sd) \Big \rangle \big | \chi _{MS} \big \rangle , \nonumber \\ \big | \varOmega _c^{0} \big \rangle= & {} \Big | css \Big \rangle \big | \chi _{MS} \big \rangle , \nonumber \\ \big | \varXi _b^{0} \big \rangle= & {} \Big | {1\over \sqrt{2}} b(us-su) \Big \rangle \big | \chi _{MA} \big \rangle , \nonumber \\ \big | \varXi _b^{-} \big \rangle= & {} \Big | {1\over \sqrt{2}} b(ds-sd) \Big \rangle \big | \chi _{MA} \big \rangle , \nonumber \\ \big | \varXi _b^{\prime 0} \big \rangle= & {} \Big | {1\over \sqrt{2}} b(us+su) \Big \rangle \big | \chi _{MS} \big \rangle , \nonumber \\ \big | \varXi _b^{\prime -} \big \rangle= & {} \Big | {1\over \sqrt{2}} b(ds+sd) \Big \rangle \big | \chi _{MS} \big \rangle , \nonumber \\ \big | \varOmega _b^{-} \big \rangle= & {} \Big | bss \Big \rangle \big | \chi _{MS} \big \rangle , \nonumber \\ \big | \varLambda ^{0} \big \rangle= & {} {1\over \sqrt{2}} \left( \big | \phi _{MS} \rangle \big | \chi _{MS} \big \rangle + \big | \phi _{MA} \big \rangle \big | \chi _{MA} \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MS} \rangle = {1\over 2} \left( \big | dus \big \rangle + \big | dsu \big \rangle - \big | uds \big \rangle - \big | usd \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MA} \rangle = {1\over 2\sqrt{3}} \left( \big | u(ds-sd) \big \rangle + \big | d(su-us) \big \rangle \right. \nonumber \\{} & {} \left. - 2\big | s(ud-du) \big \rangle \right) , \nonumber \\ \big | \varSigma ^{+} \big \rangle= & {} {1\over \sqrt{2}} \left( \big | \phi _{MS} \rangle \big | \chi _{MS} \big \rangle + \big | \phi _{MA} \big \rangle \big | \chi _{MA} \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MS} \rangle = -{1\over \sqrt{6}} \left( \big | u(us+su) \big \rangle - 2\big | suu \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MA} \rangle = {1\over \sqrt{2}} \big | u(su-us) \big \rangle , \nonumber \\ \big | \varSigma ^{0} \big \rangle= & {} {1\over \sqrt{2}} \left( \big | \phi _{MS} \rangle \big | \chi _{MS} \big \rangle + \big | \phi _{MA} \big \rangle \big | \chi _{MA} \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MS} \rangle = {1\over 2\sqrt{3}} \left( \big | u(ds+sd) \big \rangle + \big | d(su+us) \big \rangle \right. \nonumber \\{} & {} \left. - 2\big | s(du+ud) \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MA} \rangle = {1\over 2} \left( \big | u(ds-sd) \big \rangle - \big | d(su-us) \big \rangle \right) , \nonumber \\ \big | \varXi ^{0} \big \rangle= & {} {1\over \sqrt{2}} \left( \big | \phi _{MS} \rangle \big | \chi _{MS} \big \rangle + \big | \phi _{MA} \big \rangle \big | \chi _{MA} \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MS} \rangle = {1\over \sqrt{6}} \left( \big | s(us+su) \big \rangle - 2\big | uss \big \rangle \right) , \nonumber \\{} & {} \big | \phi _{MA} \rangle = -{1\over \sqrt{2}} \big | s(us-su) \big \rangle . \end{aligned}$$

(A.1)

The wave functions for the $J^P=3/2^+$ baryons:

$$\begin{aligned} \big | \varXi _c^{*+} \big \rangle= & {} \Big | {1\over \sqrt{2}} c(us+su) \Big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varXi _c^{*0} \big \rangle= & {} \Big | {1\over \sqrt{2}} c(ds+sd) \Big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varOmega _c^{*0} \big \rangle= & {} \big | css \big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varXi _b^{*0} \big \rangle= & {} \Big | {1\over \sqrt{2}} b(us+su) \Big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varXi _b^{*-} \big \rangle= & {} \Big | {1\over \sqrt{2}} b(ds+sd) \Big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varOmega _b^{*-} \big \rangle= & {} \big | bss \big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varSigma ^{*+} \big \rangle= & {} {1\over \sqrt{3}}\big | u(su+us)+suu \big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varSigma ^{*0} \big \rangle= & {} {1\over \sqrt{6}} \big ( \big | s(du+ud)+d(su+us) \big \rangle , \nonumber \\{} & {} + \big | u(sd+ds) \big \rangle \big ) \big | \chi _S \big \rangle \nonumber \\ \big | \varXi ^{*0} \big \rangle= & {} {1\over \sqrt{3}}\big | s(us+su)+uss \big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varDelta ^{++} \big \rangle= & {} \big | uuu \big \rangle \big | \chi _S \big \rangle , \nonumber \\ \big | \varOmega ^{-} \big \rangle= & {} \big | sss \big \rangle \big | \chi _S \big \rangle . \end{aligned}$$

(A.2)

Discussion on the dimensional and cut-off regularizations

Fig. 6

Re[$G^{\mathcal{M}\mathcal{B}}$] (Im[$G^{\mathcal{M}\mathcal{B}}$]) and Re[$G^{\prime \mathcal{M}\mathcal{B}}$] (Im[$G^{\prime \mathcal{M}\mathcal{B}}$]) stand for the real (imaginary) parts of the loop function for various coupled channels, calculated through the dimensional and cut-off regularizations, respectively. The dash black lines indicate the poles positions

Table 7 The poles extracted from the $PB_{1/2}$ sector of the $b\bar{c}sud$ system with $I=0$, obtained using the dimensional regularization with $a(\mu =1 \, \mathrm GeV)=-3.2$ and the cut-off regularization with $q_{\max }=450\text { MeV}$. Pole positions, binding energies ($E_B$), widths, and threshold masses of various coupled channels are in units of MeV. The couplings $g_i$ have no dimension

During the present study, we find that the results obtained using the dimensional and cut-off regularizations are not always the same. In this appendix we use the $PB_{1/2}$ sector of the $b\bar{c}uud$ system with $I={1/2}$ and the $PB_{1/2}$ sector of the $b\bar{c}sud$ system with $I=0$ as two examples to discuss this difference.

The dimensional regularization for the meson-baryon loop function $G^{\mathcal{M}\mathcal{B}}(s)$ has been given in Eq. (44), while the cut-off regularization for this loop function is

$$\begin{aligned} G_{ll}^{\prime \mathcal {M}\mathcal {B}}(s)= & {} i \int \frac{\textrm{d}^4 q}{(2\pi )^4}\frac{M_l}{E_l(\textbf{q})} \frac{1}{k^0_i + p^0_i - q^0-E_l(\textbf{q})+i\epsilon } \nonumber \\{} & {} \times \frac{1}{q^2-m_l^2+i\epsilon } \nonumber \\= & {} \int _{|\textbf{q}|<q_{\textrm{max}}} \frac{\textrm{d}^3 q}{(2\pi )^3} \frac{1}{2\omega _l(\textbf{q})} \frac{M_l}{E_l(\textbf{q})} \nonumber \\{} & {} \times \frac{1}{k^0_i + p^0_i - \omega _l(\textbf{q})-E_l(\textbf{q})+i\epsilon }, \end{aligned}$$

(B.3)

where l denotes the intermediate channel; $p^0_i$ and $k^0_i$ are the energies of the initial meson and baryon, respectively; $m_l$ and $M_l$ are the masses of the intermediate meson and baryon, respectively; $\omega _l=\sqrt{m_l^2 + \textbf{q}^2}$ and $E_l=\sqrt{M_l^2 + \textbf{q}^2}$ are the energies of the intermediate meson and baryon, respectively.

For the $PB_{1/2}$ sector of the $b\bar{c}uud$ system with $I={1/2}$, we consider the $N \bar{B}_c$, $\varLambda _b \bar{D}$, and $\varSigma _b \bar{D}$ coupled channels. The results are summarized in Table 6, where we choose the subtraction constant for the dimensional regularization to be $a(\mu =1 \text { GeV})=-3.2$, and we choose the three-momentum cutoff for the cut-off regularization to be $q_{\max }=450 \text { MeV}$. Using these two parameters, the loop functions $G_{\varSigma _b \bar{D}}^{\mathcal{M}\mathcal{B}}(s)$ and $G_{\varSigma _b \bar{D}}^{\prime \mathcal{M}\mathcal{B}}(s)$ have the same value at the $\varSigma _b \bar{D}$ threshold. Compared to the cut-off regularization, there exists an extra bound state below the $\varLambda _b \bar{D}$ threshold when using the dimensional regularization. This pole should be discarded, as already discussed in Ref. [97]. The reason can be clearly seen in Fig. 6, where we show the real and imaginary parts of the $G^{\mathcal{M}\mathcal{B}}(s)$ and $G^{\prime \mathcal{M}\mathcal{B}}(s)$ loop functions. As shown in Fig. 6c and d, their imaginary parts are exactly the same within the effective range. In the absence of coupled channel effects, the scattering matrix is given by $T=(V^{-1} - G)^{-1}$, and a pole is expected when the real part of the loop function equals the inverse of the interaction. As shown in Fig. 6a, we use the dash line to indicate the extra bound state located at $7372.6 \text { MeV}$, where the real part of $G_{\varLambda _b \bar{D}}^{\mathcal{M}\mathcal{B}}(s)$ is positive and far below the $\varLambda _b \bar{D}$ threshold. Moreover, the potential $V_{\varLambda _b \bar{D} \rightarrow \varLambda _b \bar{D}}^{P\mathcal {B}}(s)$ is also positive around this energy region, indicating the interaction to be repulsive. Therefore, this pole can not be a bound state and should be discarded. Oppositely, as shown in Fig. 6b, the real part of $G_{\varLambda _b \bar{D}}^{\prime \mathcal{M}\mathcal{B}}(s)$ is negative below the $\varLambda _b \bar{D}$ threshold, so that the above bound state does not appear at all.

For the $PB_{1/2}$ sector of the $b\bar{c}sud$ system with $I=0$, we consider the $\varLambda \bar{B}_c$, $\varLambda _b \bar{D}_s$, $\varXi _b \bar{D}$, and $\varXi _b^\prime \bar{D}$ coupled channels. The results are summarized in Table 7, where we choose the subtraction constant for the dimensional regularization to be $a(\mu =1 \text { GeV})=-3.2$, and we choose the three-momentum cutoff for the cut-off regularization to be $q_{\max }=450 \text { MeV}$. Similarly, compared to the cut-off regularization, there exists an extra bound state below the $\varLambda _b \bar{D}_s$ threshold when using the dimensional regularization, and this pole should also be discarded. Moreover, in this case the results obtained using the dimensional regularization are significantly different from those obtained using the cut-off regularization, which needs further investigations in the future studies.