Open charm and bottom meson-nucleon potentials à la the nuclear force

Let us consider the $PN$-$P^{\ast}N$ states of $J^{P}=1/2^{-}$ with a total angular momentum $J$ and parity $P$. $PN$ and $P^{\ast}N$ components in those states are represented by

$\displaystyle PN(^{2}{S}_{1/2}),P^{\ast}N(^{2}{S}_{1/2}),P^{\ast}N(^{4}{D}_{1/2}).$

(1)

Here the notation ${}^{2S+1}L_{J}$ in the parentheses stands for the combination of the total spin $S$ and the relative angular momentum $L$ for a given $J$. In view of the HQS symmetry, the wave functions given above are decomposed into the product of a heavy antiquark $\bar{Q}$ and a light component “$l$”. Here “$l$” is nonperturbatively composed of the light quarks ($q$) and gluons ($g$) inside the $PN$-$P^{\ast}N$ state. Such a light component may be schematically denoted by $qqqq$, because it should be a composite state of the light quark $q$ in $P$ or $P^{\ast}$ and the three quarks $qqq$ in the nucleon $N$. This is the special case of the so-called brown muck which was introduced in the early days when the heavy quark effective theory (HQET) was constructed.¹¹1In the present setting, the brown muck is regarded to have the special component $qN$ in $qqqq$.

The idea of the light composite state leads to the mass degeneracy of the $PN$-$P^{\ast}N$ states with different $J^{P}$, such as $J^{P}=1/2^{-}$ and $3/2^{-}$ by taking the heavy quark limit, because the spin-dependent interaction between the heavy antiquark ($\bar{Q}$) and the brown muck ($qqqq$) is suppressed by $1/m_{Q}$ with the heavy quark mass $m_{Q}$. The mass degeneracy of the $PN$-$P^{\ast}N$ states have been studied in Refs. Yasui et al. (2013); Yamaguchi et al. (2015); Hosaka et al. (2017).

For the interaction in the $PN$-$P^{\ast}N$ systems, we adopt the meson-exchange potential between $P^{(\ast)}$ and $N$. We consider the one-pion exchange potential (OPEP) as the long-range force. We also consider the $\sigma$-meson exchange potentials and the $\rho$ and $\omega$-meson exchange potentials as the middle-range force.

Let us first explain the derivation of the OPEP in details as an illustration. In constructing the OPEP, we need the information of the interaction vertices of $\pi$ and $P^{(\ast)}$ and those of $\pi$ and $N$. For the $\pi PP^{\ast}$ and $\pi P^{\ast}P^{\ast}$ vertices, we employ the heavy meson effective theory (HMET) satisfying the HQS as well as chiral symmetry Manohar and Wise (2000); Casalbuoni et al. (1997). Notice the absence of the $\pi PP$ vertex due to the parity conservation.

For heavy mesons $P$ and $P^{\ast}$, we define the effective field $H_{a}$ being a superposition of a heavy pseudoscalar meson and a vector meson as

$\displaystyle H_{\alpha}=\bigl{(}P_{\alpha}^{\ast\mu}\gamma_{\mu}+P_{\alpha}\gamma_{5}\bigr{)}\frac{1-\ooalign{\hfil/\hfil\crcr$v$}}{2},$

(4)

where the subscripts $\alpha=\pm 1/2$ represent the isospin components (up and down) in the light quark components. $P_{\alpha}$ and $P_{\alpha}^{\ast\mu}$ denote the pseudoscalar and vector meson fields, respectively. The relative phase of $P_{\alpha}^{\ast\mu}$ and $P_{\alpha}$ is arbitrary, and the present choice is adopted for the convenience in representing the $PN$-$P^{\ast}N$ potential as it will be shown later. Here $v^{\mu}$ ($\mu=0,1,2,3$) is the four velocity of the heavy meson (heavy antiquark) satisfying $v_{\mu}v^{\mu}=1$ and $v^{0}>0$. We notice that $(1-\ooalign{\hfil/\hfil\crcr$v$})/2$ is the operator for projecting out the positive-energy component in the heavy antiquark $\bar{Q}$ and discarding the negative-energy component. The complex conjugate of $H_{\alpha}$ is defined by $\bar{H}_{\alpha}=\gamma_{0}H_{\alpha}^{\dagger}\gamma_{0}$. The effective field $H_{\alpha}$ transforms as $H_{\alpha}\rightarrow U_{\alpha\beta}H_{\beta}S^{\dagger}$ under the heavy-quark spin and chiral symmetries. Here $S\in\mathrm{SU}(2)_{\mathrm{spin}}$ represents the transformation operator for the heavy-quark spin and $U_{\alpha\beta}=U_{\alpha\beta}(L,R)$ is a function in the nonlinear representation of chiral symmetry with $L\in\mathrm{SU}(2)_{\mathrm{L}}$ and $R\in\mathrm{SU}(2)_{\mathrm{R}}$ for light up and down flavors.

In terms of $H_{\alpha}$ defined by Eq. (4), the interaction Lagrangian for the $\pi P^{(\ast)}P^{(\ast)}$ vertex is given by

$\displaystyle{\cal L}_{\pi HH}=ig_{\pi}\mbox{tr}\bigl{(}H_{\alpha}\bar{H}_{\beta}\gamma_{\mu}\gamma_{5}A^{\mu}_{\beta\alpha}\bigr{)},$

(5)

where the axial current $A^{\mu}_{\beta\alpha}$ by pions is defined by $A^{\mu}=\bigl{(}\xi^{{\dagger}}\partial^{\mu}\xi-\xi\partial^{\mu}\xi^{{\dagger}}\bigr{)}/2$ with the nonlinear representation

$\displaystyle\xi=\exp\biggl{(}i\frac{\boldsymbol{\tau}\!\cdot\!\boldsymbol{\pi}}{2f_{\pi}}\biggr{)},$

(6)

with the pion decay constant $f_{\pi}=94$ MeV. The pion field is defined by $\boldsymbol{\pi}=(\pi_{1},\pi_{2},\pi_{3})$ with $\pi^{\pm}=(\pi_{1}\mp i\pi_{2})/\sqrt{2}$ for charged pions and $\pi^{0}=\pi_{3}$ for a neutral pion. Notice that the matrix $A^{\mu}$ is transformed by $A^{\mu}\rightarrow UA^{\mu}U^{{\dagger}}$ in the nonlinear representation of chiral symmetry. Thus we confirm that the interaction Lagrangian (5) is invariant under both the HQS and chiral symmetries. The coupling constant $g_{\pi}=0.59$ in Eq. (5) is determined from the decay width of $D^{\ast-}\rightarrow D^{-}\pi^{0}$ observed by experiments Zyla et al. (2020). We note that $g_{\pi}$ is nothing but the quark axial coupling $g_{A}^{q}$ whose value looks smaller than what is naively expected, $g_{A}^{q}=1$ Weinberg (1990). The small value is understood by considering corrections due to quark’s relativistic motion inside hadrons as discussed in detail for baryon decays Arifi et al. (2022). There are uncertainties for choosing the signs of the coupling constants in $\bar{D}$ ($\bar{D}^{\ast}$) and $B$ ($B^{\ast}$). In the present study, we assume that the $\pi$, $\sigma$, $\rho$, and $\omega$ mesons couple to the light constituent quarks in the heavy mesons as well as in the nucleons. In this scheme, we can consider that the signs of these meson couplings for the light quarks in the heavy meson are the same as for the nucleon, because both have the same light (up and down) constituent quarks according to the conventional quark model.

Below we consider the frame in which the heavy meson is at rest and set $v^{\mu}=(1,\boldsymbol{0})$ in Eq. (5). Thus we obtain the $\pi P^{(\ast)}P^{(\ast)}$ vertices:

	$\displaystyle{\cal L}_{\pi P^{\ast}P^{\ast}}$	$\displaystyle=\frac{ig_{\pi}}{f_{\pi}}\varepsilon_{\nu\rho\mu\sigma}v^{\nu}P_{\beta}^{\ast\rho{\dagger}}\bigl{(}\boldsymbol{\tau}\!\cdot\!\partial^{\mu}\boldsymbol{\pi}\bigr{)}_{\beta\alpha}P_{\alpha}^{\ast\sigma},$		(7)
	$\displaystyle{\cal L}_{\pi P^{\ast}P}$	$\displaystyle=i\frac{ig_{\pi}}{f_{\pi}}P_{\beta\mu}^{\ast{\dagger}}\bigl{(}\boldsymbol{\tau}\!\cdot\!\partial^{\mu}\boldsymbol{\pi}\bigr{)}_{\beta\alpha}P_{\alpha},$		(8)
	$\displaystyle{\cal L}_{\pi PP^{\ast}}$	$\displaystyle=i\frac{ig_{\pi}}{f_{\pi}}P_{\beta}^{{\dagger}}\bigl{(}\boldsymbol{\tau}\!\cdot\!\partial^{\mu}\boldsymbol{\pi}\bigr{)}_{\beta\alpha}P_{\alpha\mu}^{\ast}.$		(9)

We introduce the interaction Lagrangian of a pion and a nucleon in the axial-vector coupling

$\displaystyle{\cal L}_{\pi NN}$

$\displaystyle=\frac{g_{A}^{N}}{2f_{\pi}}\bar{\psi}\gamma_{\mu}\gamma_{5}\boldsymbol{\tau}\!\cdot\!\partial^{\mu}\boldsymbol{\pi}\psi.$

(10)

Here $\psi=(\psi_{+1/2},\psi_{-1/2})^{T}$ with the isospin components $\psi_{+1/2}$ and $\psi_{-1/2}$ for a proton and a neutron, respectively. The value of $g_{A}^{N}$ is given by the Goldberger-Treiman relation

$\displaystyle\frac{g_{A}^{N}}{f_{\pi}}=\frac{g_{\pi NN}}{m_{N}},$

(11)

and $g_{\pi NN}^{2}/4\pi=13.6$ from the phenomenological nuclear potential in Ref. Machleidt (2001) (see also Ref. Machleidt et al. (1987)). We adopt the values of the coupling constants and the cutoff parameters by referring the parameters in the CD-Bonn potential. The nuclear potentials used in the present study are explained in Appendix A.

With the interaction vertices (5) and (10), we construct the OPEP between $P^{(\ast)}$ and $N$ Yasui and Sudoh (2009); Yamaguchi et al. (2011, 2012). We show the demonstration to derive the potential for the simple model in Appendix B. The OPEP includes three channels: $P^{\ast}N\rightarrow P^{\ast}N$, $P^{\ast}N\rightarrow PN$, and $PN\rightarrow P^{\ast}N$. We notice that the $PN\rightarrow PN$ process is absent as a direct process due to the prohibition of the $\pi PP$ vertex, and that the $PN$-$PN$ interaction is indirectly supplied by multi-step process stemming from the mixing of $PN$ and $P^{\ast}N$ Yasui and Sudoh (2009); Yamaguchi et al. (2011, 2012). The OPEPs for $P^{\ast}N$-$P^{\ast}N$, $P^{\ast}N$-$PN$, and $PN$-$P^{\ast}N$ are given by

	$\displaystyle V_{\pi}^{P^{\ast}N\text{-}P^{\ast}N}(\boldsymbol{r})$	$\displaystyle={G_{\pi}}\Bigl{(}T(r;m_{\pi})\bigl{(}3(\boldsymbol{T}\!\cdot\!\hat{\boldsymbol{r}})(\boldsymbol{\sigma}\!\cdot\!\hat{\boldsymbol{r}})-\boldsymbol{T}\!\cdot\!\boldsymbol{\sigma}\bigr{)}+C(r;m_{\pi})\boldsymbol{T}\!\cdot\!\boldsymbol{\sigma}\Bigr{)}\boldsymbol{\tau}^{H}\!\cdot\!\boldsymbol{\tau}^{N},$		(12)
	$\displaystyle V_{\pi}^{P^{\ast}N\text{-}PN}(\boldsymbol{r})$	$\displaystyle=-{G_{\pi}}\Bigl{(}T(r;m_{\pi})\bigl{(}3(\boldsymbol{\epsilon}^{\ast}\!\cdot\!\hat{\boldsymbol{r}})(\boldsymbol{\sigma}\!\cdot\!\hat{\boldsymbol{r}})-\boldsymbol{\epsilon}^{\ast}\!\cdot\!\boldsymbol{\sigma}\bigr{)}+C(r;m_{\pi})\boldsymbol{\epsilon}^{\ast}\!\cdot\!\boldsymbol{\sigma}\Bigr{)}\boldsymbol{\tau}^{H}\!\cdot\!\boldsymbol{\tau}^{N},$		(13)
	$\displaystyle V_{\pi}^{PN\text{-}P^{\ast}N}(\boldsymbol{r})$	$\displaystyle=-{G_{\pi}}\Bigl{(}T(r;m_{\pi})\bigl{(}3(\boldsymbol{\epsilon}\!\cdot\!\hat{\boldsymbol{r}})(\boldsymbol{\sigma}\!\cdot\!\hat{\boldsymbol{r}})-\boldsymbol{\epsilon}\!\cdot\!\boldsymbol{\sigma}\bigr{)}+C(r;m_{\pi})\boldsymbol{\epsilon}\!\cdot\!\boldsymbol{\sigma}\Bigr{)}\boldsymbol{\tau}^{H}\!\cdot\!\boldsymbol{\tau}^{N},$		(14)

with the coefficient

$\displaystyle{G_{\pi}}=\frac{1}{3}{\frac{1}{2}}\frac{g_{\pi NN}}{2m_{N}}\frac{g_{\pi}}{f_{\pi}}.$

(15)

We notice that the coefficient $1/2$ is necessary due to the normalization factor of the wave functions, which was missing in Refs. Yasui et al. (2013); Yamaguchi et al. (2015); Hosaka et al. (2017). The derivation of the OPEP is shown in Appendix C in details. The functions $C(r;m)$ and $T(r;m)$ are defined by

	$\displaystyle C(r;m)$	$\displaystyle=\frac{m^{2}}{4\pi}\frac{1}{r}$
		$\displaystyle\times\biggl{(}e^{-mr}+\frac{\Lambda_{H}^{2}-m^{2}}{\Lambda_{N}^{2}-\Lambda_{H}^{2}}e^{-\Lambda_{N}r}+\frac{\Lambda_{N}^{2}-m^{2}}{\Lambda_{H}^{2}-\Lambda_{N}^{2}}e^{-\Lambda_{H}r}\biggr{)},$		(16)
	$\displaystyle T(r;m)$	$\displaystyle=\frac{1}{4\pi}\Biggl{(}m^{2}\biggl{(}\frac{1}{r}+\frac{3}{mr^{2}}+\frac{3}{m^{2}r^{3}}\biggr{)}e^{-mr}$
		$\displaystyle\hskip 10.00002pt+\Lambda_{N}^{2}\biggl{(}\frac{1}{r}+\frac{3}{\Lambda_{N}r^{2}}+\frac{3}{\Lambda_{N}^{2}r^{3}}\biggr{)}\frac{\Lambda_{H}^{2}-m^{2}}{\Lambda_{N}^{2}-\Lambda_{H}^{2}}e^{-\Lambda_{N}r}$
		$\displaystyle\hskip 10.00002pt+\Lambda_{H}^{2}\biggl{(}\frac{1}{r}+\frac{3}{\Lambda_{H}r^{2}}+\frac{3}{\Lambda_{H}^{2}r^{3}}\biggr{)}\frac{\Lambda_{N}^{2}-m^{2}}{\Lambda_{H}^{2}-\Lambda_{N}^{2}}e^{-\Lambda_{H}r}\Biggr{)},$		(17)

with $m=m_{\pi}$, respectively, as functions of an interdistance $r=|\boldsymbol{r}|$ for $\boldsymbol{r}$ being the relative coordinate vector between $P^{(\ast)}$ and $N$. The detailed information to derive the potentials are presented in Appendix C. Notice that the values of the cutoff parameters $\Lambda_{H}$ ($H=\bar{D},B$) and $\Lambda_{N}$ are dependent on the species of the exchanged light meson, e.g., the $\pi$ meson. Originally, $C(r,m)$ and $V(r,m)$ are defined by

	$\displaystyle C(r;m)=\int\frac{d^{3}\boldsymbol{q}}{(2\pi)^{3}}\frac{m^{2}}{\boldsymbol{q}^{\,\,2}+m^{2}}e^{i\boldsymbol{q}\cdot\boldsymbol{r}}F(\boldsymbol{q};m)\,,$		(18)
	$\displaystyle S_{\cal O}(\hat{\boldsymbol{r}})T(r;m)=\int\frac{d^{3}\boldsymbol{q}}{(2\pi)^{3}}\frac{-\boldsymbol{q}^{\,\,2}}{\boldsymbol{q}^{\,\,2}+m^{2}}S_{\cal O}(\hat{\boldsymbol{q}})e^{i\boldsymbol{q}\cdot\boldsymbol{r}}F(\boldsymbol{q};m),$		(19)

for the central and tensor parts, respectively, with $\hat{\boldsymbol{q}}=\boldsymbol{q}/|\boldsymbol{q}|$. We note that the contact term in the central part is neglected. The dipole-type form factor is given by

$\displaystyle F(\boldsymbol{q};m)=\frac{\Lambda_{H}^{2}-m^{2}}{\Lambda_{H}^{2}+|\boldsymbol{q}|^{2}}\frac{\Lambda_{N}^{2}-m^{2}}{\Lambda_{N}^{2}+|\boldsymbol{q}|^{2}},$

(20)

which is normalized at $q^{2}=m^{2}$ with a four-momentum $q$. The cutoff parameters $\Lambda_{H}$ and $\Lambda_{N}$ would correspond to the inverse of the spatial sizes of hadrons. See the derivations in Appendix C for more details. In Eqs. (13) and (14), we define the polarization vectors $\boldsymbol{\epsilon}^{\,(\lambda)}$ ($\boldsymbol{\epsilon}^{\,(\lambda)*}$) for the incoming (outgoing) $P^{\ast}$ meson with the polarization $\lambda=0,\,\pm 1$. The explicit forms of $\boldsymbol{\epsilon}^{\,(\lambda)}$ can be represented by

$\displaystyle\boldsymbol{\epsilon}^{\,(\pm)}=\frac{1}{\sqrt{2}}\bigl{(}\mp 1,-i,0\bigr{)},\quad\boldsymbol{\epsilon}^{\,(0)}=(0,0,1),$

(21)

by choosing the positive direction in the $z$ axis for the helicity $\lambda=0$. As for the spin-one operator for the $P^{\ast}$ meson in Eq. (12), we define $\boldsymbol{T}=(T_{1},T_{2},T_{3})$ by $(T_{i})_{\lambda^{\prime}\lambda}\equiv-i\varepsilon_{ijk}\epsilon_{j}^{(\lambda^{\prime})\ast}\epsilon_{k}^{(\lambda)}$ ($i,j,k=1,2,3$):

	$\displaystyle T_{1}$	$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\ 1&0&1\\ 0&1&0\end{array}\right),\quad T_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\ i&0&-i\\ 0&i&0\end{array}\right),$		(28)
	$\displaystyle T_{3}$	$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\ 0&0&0\\ 0&0&-1\end{array}\right),$		(32)

satisfying the commutation relation $[T_{i},T_{j}]=i\varepsilon_{ijk}T_{k}$ as the generators of the spin symmetry. We define the tensor operators $S_{\boldsymbol{\epsilon}}(\hat{\boldsymbol{r}})$ and $S_{\boldsymbol{T}}(\hat{\boldsymbol{r}})$ by $S_{\boldsymbol{\cal O}}(\hat{\boldsymbol{r}})=3(\boldsymbol{\cal O}\!\cdot\!\hat{\boldsymbol{r}})(\boldsymbol{\sigma}\!\cdot\!\hat{\boldsymbol{r}})-\boldsymbol{\cal O}\!\cdot\!\boldsymbol{\sigma}$ with $\hat{\boldsymbol{r}}=\boldsymbol{r}/r$ for $\boldsymbol{{\cal O}}=\boldsymbol{\epsilon}$ and $\boldsymbol{T}$. Here $\boldsymbol{\sigma}$ are the Pauli matrices acting on the nucleon spin, and $\boldsymbol{\tau}^{H}_{\beta_{1}\alpha_{1}}$ and $\boldsymbol{\tau}^{N}_{\beta_{2}\alpha_{2}}$ with $\alpha_{i},\beta_{i}=\pm 1/2$ are the isospin Pauli operators for $P^{(\ast)}$ ($i=1$) and $N$ ($i=2$), respectively.

Using the basis of the $J^{P}=1/2^{-}$ channel in Eq. (1), we represent the OPEPs (12), (13), and (14) by the matrix forms,

$\displaystyle V_{1/2^{-}}^{\pi}$

$\displaystyle=\left(\begin{array}[]{ccc}0&\sqrt{3}\,{C}_{\pi}&-\sqrt{6}\,{T}_{\pi}\\ \sqrt{3}\,{C}_{\pi}&-2\,{C}_{\pi}&-\sqrt{2}\,{T}_{\pi}\\ -\sqrt{6}\,{T}_{\pi}&-\sqrt{2}\,{T}_{\pi}&{C}_{\pi}-2\,{T}_{\pi}\end{array}\right),$

(36)

where we define ${C}_{\pi}=G_{\pi}C(r;m_{\pi})$ and ${T}_{\pi}=G_{\pi}T(r;m_{\pi})$ for short notations. In Eq. (36), we confirm that the mixing between $PN$ and $P^{\ast}N$ are represented by the off-diagonal parts including the tensor potentials. These tensor potentials induce the strong mixing by different angular momenta, leading to the strong attractions at short-range scales. Thus, the mixing of $PN$ and $P^{\ast}N$ is important to switch on the strong attraction. This is analogous to the OPEP in the nucleon-nucleon interaction.

The interaction Lagrangian for a $\sigma$ meson and a $P^{(\ast)}$ meson is given by

$\displaystyle{\cal L}_{\sigma_{I}HH}=\,$

$\displaystyle-g_{\sigma_{I}}\mathrm{tr}\bigl{(}\bar{H}\sigma_{I}H\bigr{)},$

(37)

which leads to the $\sigma P^{(\ast)}P^{(\ast)}$ vertices,

	$\displaystyle{\cal L}_{\sigma_{I}PP}=\,$	$\displaystyle 2g_{\sigma_{I}}\left(P^{\dagger}\sigma_{I}P\right),$		(38)
	$\displaystyle{\cal L}_{\sigma_{I}P^{\ast}P^{\ast}}=\,$	$\displaystyle-2g_{\sigma_{I}}\left(P^{\ast\mu\dagger}\sigma_{I}P^{\ast}_{\mu}\right).$		(39)

Here we introduce the channel-dependent $\sigma_{I}$ meson for isospin-singlet $(I=0)$ and isospin-triplet $(I=1)$ channels for the $PN$-$P^{\ast}N$ scatterings, as introduced in the CD-Bonn potential Machleidt (2001). The parameter of the $\sigma$ exchange potential in the CD-Bonn potential Machleidt (2001) has the different value for each partial waves, i.e., isospin channels. Thus, $\sigma_{I}$ in the present work also has an channel-dependent mass ($m_{\sigma_{I}}$), coupling constant ($g_{\sigma_{I}}$), and cutoff parameter ($\Lambda_{\sigma_{I}}$). Using the $\sigma NN$ vertices given by

$\displaystyle{\cal L}_{\sigma_{I}NN}$

$\displaystyle=g_{\sigma_{I}NN}\bar{\psi}\sigma_{I}\psi,$

(40)

we find that the $\sigma$ potentials for $PN$ and $P^{\ast}N$ are obtained by

	$\displaystyle V_{\sigma_{I}}^{PN\text{-}PN}(r)$	$\displaystyle=-\frac{g_{\sigma_{I}NN}g_{\sigma_{I}}}{m_{\sigma_{I}}^{2}}C(r;m_{\sigma_{I}}),$		(41)
	$\displaystyle V_{\sigma_{I}}^{P^{\ast}N\text{-}P^{\ast}N}(r)$	$\displaystyle=-\frac{g_{\sigma_{I}NN}g_{\sigma_{I}}}{m_{\sigma_{I}}^{2}}C(r;m_{\sigma_{I}}),$		(42)

where we employ the values of $m_{\sigma_{I}}$ and $g_{\sigma_{I}NN}$ in the CD-Bonn potential, see Appendix A. Concerning the values of $g_{\sigma_{I}}$, we choose $g_{\sigma_{I}}=g_{\sigma_{I}NN}/3$ by assuming that the coupling of a $\sigma$ meson and a hadron $h=P^{(\ast)}$, $N$ is proportional to the number of the light quarks in the hadron $h$: one light quark in $P^{(\ast)}$ and three light quarks in $N$. The $\sigma$-exchange potentials are expressed explicitly by

$\displaystyle V_{1/2^{-}}^{\sigma_{I}}$

$\displaystyle=\left(\begin{array}[]{ccc}C_{\sigma_{I}}&0&0\\ 0&C_{\sigma_{I}}&0\\ 0&0&C_{\sigma_{I}}\end{array}\right),$

(46)

for the basis by Eq. (1), where we define the function

$\displaystyle C_{\sigma_{I}}=-\frac{g_{\sigma_{I}NN}g_{\sigma_{I}}}{m_{\sigma_{I}}^{2}}C(r;m_{\sigma_{I}})$

(47)

for short notations.

Finally, we consider the exchange of the vector mesons, $\rho$ and $\omega$, at shorter range. The $\rho$ and $\omega$ potentials can be constructed from the $vP^{(\ast)}P^{(\ast)}$ vertices for light vector meson $v$ ($v=\rho$, $\omega$). Following the previous papers Yasui and Sudoh (2009); Yamaguchi et al. (2011, 2012), we consider the interaction Lagrangian

	$\displaystyle{\cal L}_{vHH}=\,$	$\displaystyle i\beta\mathrm{tr}\bigl{(}\bar{H}_{\beta}v^{\mu}(\rho_{\mu})_{\beta\alpha}H_{\alpha}\bigr{)}$
		$\displaystyle+i\lambda\mathrm{tr}\bigl{(}\bar{H}_{\beta}\sigma^{\mu\nu}(F_{\mu\nu}(\rho))_{\beta\alpha}H_{\alpha}\bigr{)},$		(48)

by respecting the HQS symmetry. The vector meson field is defined by $\rho_{\mu}=ig_{V}\hat{\rho}_{\mu}/\sqrt{2}$ with $\hat{\rho}_{\mu}$,

$\displaystyle\hat{\rho}_{\mu}=\begin{pmatrix}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}\\ \rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}\end{pmatrix}_{\mu},$

(49)

and $g_{V}\simeq 5.8$ the universal vector-meson coupling. In Eq. (48), the tensor field is given by $F_{\mu\nu}(\rho)=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+[\rho_{\mu},\rho_{\nu}]$. The coupling constants are given by $\beta=0.9$ and $\lambda=0.56$ GeV^-1 by following Refs. Casalbuoni et al. (1997); Isola et al. (2003). In Ref. Isola et al. (2003), $\beta$ was determined by the vector-meson dominance, and $\lambda$ was evaluated by the long distance charming penguin diagrams in the $B$ meson decay process. The $vP^{(\ast)}P^{(\ast)}$ vertices are obtained by the Lagrangians (48) as

	$\displaystyle{\cal L}_{vP^{\ast}P^{\ast}}=\,$	$\displaystyle-\beta g_{V}v_{\mu}P^{\ast\dagger}_{\beta\nu}(\boldsymbol{\tau}\cdot\boldsymbol{\rho}\,^{\mu})_{\beta\alpha}P^{\ast\nu}_{\alpha}$
		$\displaystyle+2i\lambda g_{V}\left(P^{\ast\nu\dagger}_{\beta}(\boldsymbol{\tau}\cdot\partial_{\mu}\boldsymbol{\rho}_{\nu})_{\beta\alpha}P^{\ast\mu}_{\alpha}\right.$
		$\displaystyle\left.\quad-P^{\ast\mu\dagger}_{\beta}(\boldsymbol{\tau}\cdot\partial_{\mu}\boldsymbol{\rho}_{\nu})_{\beta\alpha}P^{\ast\nu}_{\alpha}\right),$		(50)
	$\displaystyle{\cal L}_{vP^{\ast}P}=\,$	$\displaystyle 2\lambda g_{V}\epsilon_{\sigma\rho\mu\nu}v^{\sigma}P^{\ast\rho\dagger}_{\beta}(\boldsymbol{\tau}\cdot\partial^{\mu}\boldsymbol{\rho}^{\nu})_{\beta\alpha}P_{\alpha},$		(51)
	$\displaystyle{\cal L}_{vPP^{\ast}}=\,$	$\displaystyle 2\lambda g_{V}\epsilon_{\sigma\rho\mu\nu}v^{\sigma}P_{\beta}^{\dagger}(\boldsymbol{\tau}\cdot\partial^{\mu}\boldsymbol{\rho}^{\nu})_{\beta\alpha}P^{\ast\rho}_{\alpha},$		(52)
	$\displaystyle{\cal L}_{vPP}=\,$	$\displaystyle\beta g_{V}v_{\mu}P^{\dagger}_{\beta}(\boldsymbol{\tau}\cdot\boldsymbol{\rho}^{\mu})_{\beta\alpha}P_{\alpha}.$		(53)

For the $vNN$ vertex, we use the interaction Lagrangian

	$\displaystyle{\cal L}_{vNN}$	$\displaystyle=g_{\rho NN}\bar{\psi}\gamma_{\mu}\boldsymbol{\tau}\!\cdot\!\boldsymbol{\rho}^{\,\mu}\psi+\frac{f_{\rho NN}}{2m_{N}}\bar{\psi}\sigma_{\mu\nu}\boldsymbol{\tau}\!\cdot\!\partial^{\mu}\boldsymbol{\rho}^{\,\nu}\psi$
		$\displaystyle\quad+g_{\omega NN}\bar{\psi}\gamma_{\mu}\omega^{\mu}\psi+\frac{f_{\omega NN}}{2m_{N}}\bar{\psi}\sigma_{\mu\nu}\partial^{\mu}\omega^{\nu}\psi,$		(54)

for $\boldsymbol{\rho}^{\,\mu}=(\rho^{\mu}_{1},\rho^{\mu}_{2},\rho^{\mu}_{3})$ with $\rho_{\pm}^{\mu}=(\rho_{1}^{\mu}\mp i\rho_{2}^{\mu})/\sqrt{2}$ and $\rho_{0}^{\mu}=\rho_{3}^{\mu}$. The coupling constants are given by $g_{\rho NN}^{2}/4\pi=0.84$, $g_{\omega NN}^{2}/4\pi=20.0$, $f_{\rho NN}/g_{\rho NN}=6.1$, and $f_{\omega NN}/g_{\omega NN}=0.0$ Machleidt (2001) (see also Ref. Machleidt et al. (1987)). We leave a comment that the coupling strengths in Eqs. (48) and (54) reflect the number of constituent quarks inside the hadrons. This can be easily checked by the nonrelativistic quark model. We should notice, however, that the tensor parts, $\lambda$ and $f_{vNN}$ ($v=\rho$, $\omega$), could be different by some factors from the naive expectations, which would be understood from the composite structures of the constituent quarks.

From Eqs. (48) and (54), the one-boson exchange potentials are obtained as

$\displaystyle V_{1/2^{-}}^{v}$

$\displaystyle=\begin{pmatrix}C_{v}^{\prime}&2\sqrt{3}C_{v}&\sqrt{6}T_{v}\\ 2\sqrt{3}C_{v}&C_{v}^{\prime}-4C_{v}&\sqrt{2}T_{v}\\ \sqrt{6}T_{v}&\sqrt{2}T_{v}&C_{v}^{\prime}+2C_{v}+2T_{v}\end{pmatrix},$

(55)

with $v=\rho$, $\omega$ for the $1/2^{-}$ state in Eq. (1). The functions $C_{v}^{\prime}$, $C_{v}$, and $T_{v}$ are defined by

	$\displaystyle C_{\rho}^{\prime}$	$\displaystyle=\frac{g_{V}g_{\rho NN}\beta}{2m_{\rho}^{2}}C(r;m_{\rho})\boldsymbol{\tau}^{H}\!\cdot\!\boldsymbol{\tau}^{N},$		(56)
	$\displaystyle C_{\rho}$	$\displaystyle=\frac{g_{V}(g_{\rho NN}+f_{\rho NN})\lambda}{2m_{N}}\frac{1}{3}T(r;m_{\rho})\boldsymbol{\tau}^{H}\!\cdot\!\boldsymbol{\tau}^{N},$		(57)
	$\displaystyle T_{\rho}$	$\displaystyle=\frac{g_{V}(g_{\rho NN}+f_{\rho NN})\lambda}{2m_{N}}\frac{1}{3}T(r;m_{\rho})\boldsymbol{\tau}^{H}\!\cdot\!\boldsymbol{\tau}^{N},$		(58)
	$\displaystyle C_{\omega}^{\prime}$	$\displaystyle=\frac{g_{V}g_{\omega NN}\beta}{2m_{\omega}^{2}}C(r;m_{\omega}),$		(59)
	$\displaystyle C_{\omega}$	$\displaystyle=\frac{g_{V}(g_{\omega NN}+f_{\omega NN})\lambda}{2m_{N}}\frac{1}{3}C(r;m_{\omega}),$		(60)
	$\displaystyle T_{\omega}$	$\displaystyle=\frac{g_{V}(g_{\omega NN}+f_{\omega NN})\lambda}{2m_{N}}\frac{1}{3}T(r;m_{\omega}),$		(61)

with $\boldsymbol{\tau}^{H}$ and $\boldsymbol{\tau}^{N}$ being the abbreviations of $\boldsymbol{\tau}^{H}_{\beta_{1}\alpha_{1}}$ and $\boldsymbol{\tau}^{N}_{\beta_{2}\alpha_{2}}$ for the isospin Pauli operators acting on $P^{(\ast)}$ and $N$, respectively.