Molecular states from $\bar{B}^{(*)}N$ interactions

Abstract

In 2019, two new structures $\Lambda _b(6146)$ and $\Lambda _b(6152)$ were observed by the LHCb Collaboration at the invariant mass spectrum of $\Lambda _b^0\pi ^{+}\pi ^{-}$, which aroused a hot discussion about their inner structures. The $\Lambda _b(6146)$ and $\Lambda _b(6152)$ might still be molecular states because their masses are close to threshold of a $\bar{B}$ meson and a nucleon. In this work, we perform a systematical investigation of possible heavy baryonic molecular states from the $\bar{B}N$ interaction. Since the $\bar{B}N$ channel strongly couples to the $\bar{B}^{*}N$ channel, the possible $\bar{B}N{-}\bar{B}^{*}N$ bound states are also studied. The interaction of the system considered is described by the t-channel $\sigma$, $\pi$, $\eta$,$\omega$, and $\rho$ mesons exchanges. By solving the non-relativistic Schrödinger equation with the obtained one-boson-exchange potentials, the $\bar{B}^{(*)}N$ bound states with different quantum numbers are searched. The calculation suggests that recently observed $\Lambda _b(6146)$ can be assigned as a P-wave $\bar{B}N$ molecular state with spin parity $J^P=3/2^{+}$ or a $\bar{B}N{-}\bar{B}^{*}N$ bound state. However, assignment of $\Lambda _b(6152)$ as an F-wave $\bar{B}N$ molecular is disfavored. The $\Lambda _b(6152)$ can be explained as meson–baryon molecular state with a small $\bar{B}N$ component. The calculation also predict the existence of two $\bar{B}N{-}\bar{B}^{*}N$ bound states with $J^P=1/2^{-}$ and $J^P=3/2^{-}$.

Appendix A

In this section, we collect the potential related to the Fig. 1 in the coordinate space. For the $\bar{B}N \rightarrow \bar{B}N$ reaction, we have

$$\begin{aligned} V_{a/b}(r)_{\rho ^0}&=\frac{\beta {}gg_v}{2\sqrt{2}}\Bigg [Y_1(\Lambda ,m_{\rho ^0},r)\nonumber \\&\quad +{{\mathcal {V}}}_0(m_{\rho ^0},\frac{\Bigg (m_{\overline{B}^0}+2m_p)}{{4m^2_p}m_{\overline{B}^0}},r\Bigg )\Bigg ], \end{aligned}$$

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$$\begin{aligned} V_{a/b}(r)_{\omega }&=\frac{3\beta gg_v}{2\sqrt{2}}\Bigg [Y_1(\Lambda ,m_{\omega },r)\nonumber \\ {}&\quad +{{\mathcal {V}}}_0(m_{\omega },\frac{\Bigg (m_{\overline{B}^0}+2m_p)}{{4m^2_p}m_{\overline{B}^0}},r\Bigg )\Bigg ], \end{aligned}$$

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$$\begin{aligned} v_{a/b}(r)_{\sigma }&=-g_{\sigma }g_{\sigma NN} \Bigg [Y_1(\Lambda ,m_{\sigma },r)\nonumber \\ {}&\quad -{{\mathcal {V}}}_0\Bigg (m_{\sigma },\frac{1}{4m^2_P},r\Bigg )\Bigg ], \end{aligned}$$

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$$\begin{aligned} V_{c/d}(r)_{\rho ^{\pm }}&=\frac{\beta {}g g_v}{\sqrt{2}}\Bigg [Y_1(\Lambda ,m_{\rho ^{\pm }},r)\nonumber \\&\quad +{{\mathcal {V}}}_0(m_{\rho ^{\pm }},\frac{\Bigg (m_{\overline{B}^0}+m_{B^{-}}+4m_p)}{{4m^2_p}(m_{\overline{B}^0}+m_{B^{-}})},r\Bigg )\Bigg ]. \end{aligned}$$

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The $\bar{B}N\rightarrow \bar{B}^{*}N$ reaction potentials can be shown

$$\begin{aligned} V_{e/f}(r)_\pi&=-\frac{g(D+F)}{2\sqrt{2}ff_\pi }{{\mathcal {V}}}_1(m_{\pi ^0},r), \end{aligned}$$

(51)

$$\begin{aligned} V_{e/f}(r)_\eta&=\frac{g(D-3F)}{6\sqrt{2}ff_\pi }{{\mathcal {V}}}_1(m_{\eta },r), \end{aligned}$$

(52)

$$\begin{aligned} V_{e/f}(r)_{\rho ^0}&=\frac{\lambda gg_v}{\sqrt{2}m_p}{{\mathcal {V}}}_2(m_{\rho ^0},r), \end{aligned}$$

(53)

$$\begin{aligned} V_{e/f}(r)_{\omega }&=\frac{3\lambda gg_v}{\sqrt{2}m_p}{{\mathcal {V}}}_2(m_{\omega },r), \end{aligned}$$

(54)

$$\begin{aligned} V_{g/h}(r)_{\pi ^{\pm } }&=\frac{g(D+F)}{\sqrt{2}ff_\pi }{{\mathcal {V}}}_1(m_{\pi ^{\pm }},r), \end{aligned}$$

(55)

$$\begin{aligned} V_{g/h}(r)_{\rho ^{\pm }}&=\frac{2\lambda gg_v}{\sqrt{2}m_p}{{\mathcal {V}}}_2(m_{\rho ^{\pm }},r). \end{aligned}$$

(56)

Last, we give the $\bar{B}^{*}N \rightarrow \bar{B}^{*}N$ reaction potentials

$$\begin{aligned} V_{i/j}(r)_{\pi ^0}&=-\frac{g(D+F)}{2\sqrt{2}ff_\pi }{{\mathcal {V}}}_3(m_{\pi ^0},r), \end{aligned}$$

(57)

$$\begin{aligned} V_{i/j}(r)_\eta&=\frac{g(D-3F)}{6\sqrt{2}ff_\pi }{{\mathcal {V}}}_3(m_{\eta },r), \end{aligned}$$

(58)

$$\begin{aligned} V_{i/j}(r)_{\rho ^0}&=-\frac{\beta {}gg_v}{2\sqrt{2}}\Bigg [(Y_1(\Lambda ,m_{\rho ^0},r)\nonumber \\&\quad +{{\mathcal {V}}}_0(m_{\rho ^0},\frac{\Bigg (m_{\overline{B}^{*(0/-)}}+2m_p)}{{4m^2_p}m_{\overline{B}^{*(0/-)}}},r\Bigg )\Bigg ]\vec {\epsilon }_{\lambda _1}\cdot \vec {\epsilon }_{\lambda _3}^{\dagger }\nonumber \\&\quad +\frac{\lambda {}gg_v}{\sqrt{2}}{{\mathcal {V}}}_4\left( m_{\rho ^0},\frac{1}{m_{\overline{B}^{*(0/-)}}},r\right) , \end{aligned}$$

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$$\begin{aligned} V_{i/j}(r)_{\omega }&=-\frac{3\beta {}gg_v}{2\sqrt{2}}\Bigg [(Y_1(\Lambda ,m_{\omega },r)\nonumber \\&\quad +{{\mathcal {V}}}_0(m_{\omega },\frac{\Bigg (m_{\overline{B}^{*(0/-)}}+2m_p)}{{4m^2_p}m_{\overline{B}^{*(0/-)}}},r\Bigg )\Bigg ]\vec {\epsilon }_{\lambda _1}\cdot \vec {\epsilon }_{\lambda _3}^{\dagger }\nonumber \\&\quad +\frac{3\lambda {}gg_v}{\sqrt{2}}{{\mathcal {V}}}_4\left( m_{\omega },\frac{1}{m_{\overline{B}^{*(0/-),}}},\right) , \end{aligned}$$

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$$\begin{aligned} V_{i/j}(r)_{\sigma }&=-g_{\sigma }g_{\sigma {}NN}\Bigg [Y_1(\Lambda ,m_{\sigma },r)\nonumber \\&\quad -{{\mathcal {V}}}_0\Bigg (m_{\sigma },\frac{1}{4m^2_P},r\Bigg )\Bigg ]\vec {\epsilon }_{\lambda _1}\cdot \vec {\epsilon }_{\lambda _3}^{\dagger }, \end{aligned}$$

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$$\begin{aligned} V_{k/l}(r)_{\pi ^+ }&=\frac{g(D+F)}{\sqrt{2}ff_\pi }{{\mathcal {V}}}_3(\pi ^{\pm },r), \end{aligned}$$

(62)

$$\begin{aligned} V_{k/l}(r)_{\rho ^{\pm }}&=\frac{\beta {}g_1g_v}{\sqrt{2}}\Bigg [Y_1(\Lambda ,m_{\rho ^+},r)\nonumber \\&\quad {+}{{\mathcal {V}}}_0(m_{\rho ^{\pm }},\frac{\Bigg (m_{\overline{B}^{*0}}{+}m_{B^{*-}}{+}4m_p)}{{4m^2_p}(m_{\overline{B}^{*0}}{+}m_{B^{*-}})},r\Bigg )\Bigg ]\vec {\epsilon }_{\lambda _1}\cdot \vec {\epsilon }_{\lambda _3}^{\dagger }\nonumber \\&\quad -\frac{2\lambda {}g_1g_v}{\sqrt{2}}{{\mathcal {V}}}_4\left( m_{\rho ^{\pm }},\frac{2}{(m_{B^{*-}}+m_{\overline{B}^{*0}})},r\right) .\nonumber \\ \end{aligned}$$

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In the above, the following equations are used

$$\begin{aligned} {{\mathcal {V}}}_0(m,\mu _1,r)&=\mu _1\left[ \frac{1}{4}\nabla _r^2Y_1(\Lambda ,m_{\rho ^0},r)\right. \left. -\frac{1}{2}\left\{ \nabla _r^2,Y_1(\Lambda ,m,r)\right\} \right] \nonumber \\&\quad +\frac{1}{16m_{p}^2}\nabla _r^2Y_1(\Lambda ,m,r)\nonumber \\&\quad +\mu _1\left( \vec {\sigma }\cdot \vec {L}\frac{1}{r}\frac{\partial }{\partial r}Y_1(\Lambda ,m,r)\right) , \end{aligned}$$

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$$\begin{aligned} {{\mathcal {V}}}_1(m,r)&=\frac{1}{3}(\vec {\sigma }\cdot \vec {\epsilon }_{\lambda _3}^\dagger )(-\nabla _r^2Y_1(\Lambda ,m,r)\nonumber \\&\quad +\frac{1}{3}S(\hat{r},\vec {\sigma },\vec {\epsilon }_{\lambda _3}^\dagger ) \left( -r\frac{\partial }{\partial r}\frac{1}{r}\frac{\partial }{\partial r}Y_1(\Lambda ,m,r)\right) , \end{aligned}$$

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$$\begin{aligned} {{\mathcal {V}}}_2(m,r)&=-\vec {\epsilon }_{\lambda _3}^\dagger \cdot \vec {L} \left( \frac{1}{r}(\frac{\partial }{\partial r})\right) Y_1(\Lambda ,m,r)\nonumber \\&\quad +\frac{1}{3}(\vec {\sigma }\cdot \vec {\epsilon }_{\lambda _3}^\dagger )(-\nabla _r^2Y_1(\Lambda ,m,r))\nonumber \\&\quad -\frac{1}{6}S(\hat{r},\vec {\sigma },\vec {\epsilon }_{\lambda _3}^\dagger )\left( -r\frac{\partial }{\partial r}\frac{1}{r}\frac{\partial }{\partial r}Y_1(\Lambda ,m,r)\right) , \end{aligned}$$

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$$\begin{aligned} {{\mathcal {V}}}_3(m,r)&=\frac{1}{3}(\vec {\sigma }\cdot (-i\vec {\epsilon }_{\lambda _3}^\dagger \times \vec {\epsilon }_{\lambda _1}))(-\nabla _r^2Y_1(\Lambda ,m,r))\nonumber \\&\quad +\frac{1}{3}S(\hat{r},\vec {\sigma },(-i\vec {\epsilon }_{\lambda _3}^\dagger \times \vec {\epsilon }_{\lambda _1}))\nonumber \\&\quad \times \left( -r\frac{\partial }{\partial r}\frac{1}{r}\frac{\partial }{\partial r} Y_1(\Lambda ,m,r)\right) , \end{aligned}$$

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$$\begin{aligned} {{\mathcal {V}}}_4(m,\mu _2,r)&=\frac{1}{m_p} \left[ \frac{1}{3}i(\vec {\epsilon }_{\lambda _1}\times \vec {\sigma })\cdot (\vec {\epsilon }_{\lambda _3}^\dagger ) (-\nabla _r^2Y_1(\Lambda ,m,r))\right. \nonumber \\&\quad \left. +\frac{1}{3}S(\hat{r},i(\vec {\epsilon }_{\lambda _1}\times \vec {\sigma }),\vec {\epsilon }_{\lambda _3}^\dagger ) \left( -r\frac{\partial }{\partial r}\frac{1}{r}\frac{\partial }{\partial r}Y_1(\Lambda ,m,r)\right) \right. \nonumber \\&\quad \left. -\frac{1}{3}i(\vec {\epsilon }_{\lambda _3}^\dagger \times \vec {\sigma })\cdot (\vec {\epsilon }_{\lambda _1})(-\nabla _r^2Y_1(\Lambda ,m,r))\right. \nonumber \\&\quad \left. -\frac{1}{3}S(\hat{r},i(\vec {\epsilon }_{\lambda _3}^\dagger \times \vec {\sigma }),\vec {\epsilon }_{\lambda _1})\left( -r\frac{\partial }{\partial r}\frac{1}{r} \frac{\partial }{\partial r}Y_1(\Lambda ,m,r)\right) \right] \nonumber \\&\quad -\mu _2\left( \frac{1}{3}(\vec {\epsilon }_{\lambda _1}\cdot \vec {\epsilon }_{\lambda _3}^\dagger )(-\nabla _r^2Y_1(\Lambda ,m,r))\right. \nonumber \\&\quad \left. +\frac{1}{3}S(\hat{r},\vec {\epsilon }_{\lambda _1},\vec {\epsilon }_{\lambda _3}^\dagger ) \left( -r\frac{\partial }{\partial r}\frac{1}{r}\frac{\partial }{\partial r}Y_1(\Lambda ,m,r)\right) \right) .\nonumber \\ \end{aligned}$$

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