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The quark level involves chromomagnetic interactions. Therefore, we use the color factor f to indicate whether the color force is attractive or repulsive.
Regarding the quark-quark color interaction, the color factor f is
$ \begin{eqnarray} f(ik \rightarrow jl) = \dfrac{1}{4}\sum_{a=1}^{8}\lambda_{ji}^{a}\lambda_{lk}^{a}, \end{eqnarray} $
(1) where
$ \lambda^a $ denotes Gell-Mann matrices, and the quark colors are labeled by$ i, j, k $ , and l. The potential is$ \begin{eqnarray} V_{qq}(r) \approx +f\dfrac{\alpha_s}{r}. \end{eqnarray} $
(2) Considering the quark-antiquark color interaction, the color factor
$ \widetilde{f} $ is$ \begin{eqnarray} \widetilde{f}(ik \rightarrow jl) = -\dfrac{1}{4}\sum_{a=1}^{8}\lambda_{ji}^{a}\lambda_{lk}^{a}. \end{eqnarray} $
(3) The potential is
$ \begin{eqnarray} V_{q\bar{q}}(r) \approx +\widetilde{f}\dfrac{\alpha_s}{r}. \end{eqnarray} $
(4) In Table 1, we list the color factors of the multiplet in the SU(3) color representation.
$ 3_{c}\ \otimes\ \bar{3}_{c} $ $ 1_{c}\ \oplus\ {8}_{c} $ color factor $ -\dfrac{4}{3}\;\; \dfrac{1}{6}$ $ 3_{c}\ \otimes\ 3_{c} $ $ 6_{c}\ \oplus\ \bar{3}_{c} $ color factor $ \dfrac{1}{3} \;\;-\dfrac{2}{3} $ $ 3_{c}\ \otimes\ {3}_{c}\otimes\ {3}_{c} $ $ 1_{c}\ \oplus\ {8}_{1c}\ \oplus\ {8}_{2c}\ \oplus\ {10}_{c} $ color factor $ -2\;\;-\dfrac{1}{2} \;\; -\dfrac{1}{2}\;\;1$ $ 3_{c}\ \otimes\ {3}_{c}\otimes\ \bar{3}_{c} $ $ 3_{1c}\ \oplus\ 3_{2c}\ \oplus\ \bar{6}_{c}\ \oplus\ {15}_{c} $ color factor $ -\dfrac{4}{3} \;\;-\dfrac{4}{3}\;\;-\dfrac{1}{3}\;\;2$ Table 1. Color factor values for color representation.
Color confinement implies that physical hadrons are singlets. Under this restriction, we divide the pentaquark states into the following three categories:
1. Molecular model
Each cluster of the molecular model forms a quasibound cluster. In other words, clusters of the molecular model tend to be color singlets. We observe that
$ f_{1_{c}} < f_{8_{c}} $ in the color representation of the quark and antiquark; hence, it is easier to form a singlet state than octet states. Similarly,$ f_{1_{c}} < f_{8_{1c}}/f_{8_{2c}}< f_{10_{c}} $ in the three-quark color representation, and thus it is easier to form a singlet state than other states. Therefore, from the molecular model, we have two configurations,$ (c\bar{c})(q_{1}q_{2}q_{3}) $ and$ (\bar{c}q_{1})({c}q_{2}q_{3}) $ , where q denotes the$ u,d,s $ quark.2. Diquark-Diquark-antiquark model
The diquark prefers to form
$ \bar{3}_{c} $ by comparing the color factors of$ 6_{c} $ and$ \bar{3}_{c} $ . Similarly,$ \bar{3}_{c}\ \otimes\ \bar{3}_{c} $ prefers to form$ 3_{c} $ . Hence, we have$ \bar{3}_{c}(\mathcal{D})\ \otimes\ \bar{3}_{c}(\mathcal{D})\ \otimes\ \bar{3}_{c}(\mathcal{A}) $ to form a color singlet, where$ \mathcal{D} $ and$ \mathcal{A} $ represent the diquark and antiquark, respectively. Thus, the pentaquark configuration is$ (cq_{1})(q_{2}q_{3})(\bar{c}) $ , represented by the diquark-diquark-antiquark model.3. Diquark-triquark model
The triquark involves two quarks and an antiquark, which distinguish it from the molecule model. In this case,
$ f_{3_{1c}} / f_{3_{2c}} < f_{\bar{6}_{c}} < f_{15_{c}} $ is the color representation of the triquark quark, and we have$ 3_{c}(\mathcal{T})\ \otimes \ \bar{3}_{c}(\mathcal{D}) $ to form a color singlet, where$ \mathcal{T} $ represents a triquark. Thus, the pentaquark configurations represented by the diquark-triquark model are$ (c\bar{c}q_{1})(q_{2}q_{3}) $ and$ (cq_{1})(\bar{c}q_{2}q_{3}) $ .The separation of c and
$ \bar{c} $ into distinct confinement volumes provides a natural suppression mechanism for the pentaquark widths [6]. Thus, we do not consider$ (\bar{c}c) (q_{1}q_{2}q_{3}) $ and$ (\bar{c}cq_{1})(q_{2}q_{3}) $ . -
In this study, we investigate pentaquark states in the
$S U(3)_{f}$ frame. The overall wavefunction for a bounded multiquark state, while accounting for all degrees of freedom, can be written as$ \begin{equation} \psi_{\rm wavefunction} = \phi_{\rm flavor}\chi_{\rm spin}\varepsilon_{\rm color}\eta_{\rm space}. \nonumber \end{equation} $
Owing to Fermi statistics, the overall wavefunction above must be antisymmetric.
The molecular model of the pentaquark is composed of mesons and baryons, which must be color singlets because of color confinement. The relationship between spin and flavor is
$\phi_{\rm flavor}\chi_{\rm spin}$ = symmetric because the color wavefunction is antisymmetric and the spatial wavefunction is symmetric in the ground state. We study the$ P_{cs} $ state in a$S U(3)_f$ frame. There are two configurations for$ q_{2}q_{3} $ , where$ q_{2}q_{3} $ forms the$ \bar{3}_f $ and$ 6_f $ flavor representations with the total spin S = 0 and 1, respectively. When$ q_{2}q_{3} $ forms$ 6_f $ , it is combined with$ q_1 $ to form the flavor representation$ 6_f\ \otimes\ {3}_{f} $ =$ 10_f\ \oplus\ 8_{1f} $ , whereas when$ q_{2}q_{3} $ forms$ \bar{3}_f $ , it is combined with$ q_1 $ to form the flavor representation$ \bar{3}_f\ \otimes\ {3}_{f} $ =$ 8_{2f}\ \oplus\ 1_f $ . After inserting$ [c\bar{c}] $ and the Clebsch-Gordan coefficients, we apply the same method to the$ (cq_{1})(q_{2}q_{3})(\bar{c}) $ and$ (cq_{1})(\bar{c}q_{2}q_{3}) $ configurations and obtain the flavor wave function of$ P_{cs} $ in$ 8_{1f} $ and$ 8_{2f} $ . The results are reported in Table 2.Model Multiplet $ (I,I_{3}) $ Wave function Molecular model $ 8_{1f} $ $ (1,0) $ $ \dfrac{1}{\sqrt6}[({\bar c}d)(c\{us\})+({\bar c}u)(c\{ds\})]-\sqrt{\dfrac{2}{3}}({\bar c}s)(c\{ud\}) $ $ (0,0) $ $ \dfrac{1}{\sqrt2}[({\bar{c}}u)(c\{ds\})-({\bar{c}}d)(c\{us\})] $ $ 8_{2f} $ $ (1,0) $ $ \dfrac{1}{\sqrt2}\{ ({\bar c}d)(c[us])+({\bar c}u)(c[ds]) \} $ $ (0,0) $ $ \dfrac{1}{\sqrt6}\{({\bar c}d)(c[us])-({\bar c}u)(c[ds])-2(\bar c s)(c[ud])\} $ Diquark-diquark-antiquark model $ 8_{1f} $ $ (1,0) $ $ \dfrac{1}{\sqrt6}[({c}d)\{us\}{\bar c}+({c}u)\{ds\}{\bar c}]-\sqrt{\dfrac{2}{3}}({c}s)\{ud\}{\bar c} $ $ (0,0) $ $ \dfrac{1}{\sqrt2}[({c}u)\{ds\}{\bar c}-({c}d)\{us\}{\bar c}] $ $ 8_{2f} $ $ (1,0) $ $ \dfrac{1}{\sqrt2}\{ (cd)[us]{\bar c}+(cu)[ds]{\bar c} \} $ $ (0,0) $ $ \dfrac{1}{\sqrt6} \{ (cd)[us]{\bar c}-(cu)[ds]{\bar c}-2(cs)[ud]{\bar c} \} $ Diquark-triquark model $ 8_{1f} $ $ (1,0) $ $ \dfrac{1}{\sqrt6}[({c}d)(\bar c\{us\})+({c}u)(\bar c\{ds\})]-\sqrt{\dfrac{2}{3}}({c}s)(\bar c\{ud\}) $ $ (0,0) $ $ \dfrac{1}{\sqrt2}[({c}u)(\bar c\{ds\})-({c}d)(\bar c\{us\})] $ $ 8_{2f} $ $ (1,0) $ $ \dfrac{1}{\sqrt2}\{ ({c}d)(\bar c[us])+({c}u)(\bar c[ds]) \} $ $ (0,0) $ $ \dfrac{1}{\sqrt6}\{({c}d)(\bar c[us])-({ c}u)(\bar c[ds])-2( c s)(\bar c[ud])\} $ Table 2. Flavor wave function of hidden–charm strange pentaquark states in different models.
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Because quarks are fundamental Dirac fermions, the operators of the total magnetic moments and z-component are
$ \begin{eqnarray} \hat{\mu} = \ Q\dfrac{e}{m}\hat{S}, \ \ \ \ \ \ \ \hat{\mu_{z}} = \ Q\dfrac{e}{m}\hat{S_{z}}. \end{eqnarray} $
(5) As mentioned above, we do not consider the orbital excitation in the bound state; hence, the orbital excitation lies between the meson and baryon. The total magnetic moment formula can be written as
$ \begin{eqnarray} \hat{\mu} = \ \hat{\mu}_{\mathcal{B}}+\hat{\mu}_{\mathcal{M}}+\hat{\mu}_{l}, \end{eqnarray} $
(6) where the subscripts
$ \mathcal{B} $ and$ \mathcal{M} $ represent the baryon and meson, respectively, and l is the orbital excitation between the meson and baryon. The specific forms of the magnetic moments can be written as$ \begin{eqnarray} \hat{\mu}_{\mathcal{B}} = \sum_{i=1}^{3} \mu_{i}g_{i}\hat{S}_{i}, \end{eqnarray} $
(7) $ \begin{eqnarray} \hat{\mu}_{\mathcal{M}} = \sum_{i=1}^{2} \mu_{i}g_{i}\hat{S}_{i}, \end{eqnarray} $
(8) $ \begin{eqnarray} \hat{\mu}_{l} = \mu_{l}\hat{l} = \dfrac{M_{\mathcal{M}}\mu_{\mathcal{B}}+M_{\mathcal{B}}\mu_{\mathcal{M}}}{M_{\mathcal{M}}+M_{\mathcal{B}}}\hat{l}, \end{eqnarray} $
(9) where
$ g_{i} $ is the Lande factor, and$ M_{\mathcal{M}} $ and$ M_{\mathcal{B}} $ are the meson and baryon masses, respectively. The$ (\bar{c}q_1)(cq_2q_3) $ specific magnetic moment formula of the pentaquark in the molecular model is$ \begin{aligned}[b] \mu =& \langle\ \psi\ |\ \hat{\mu}_{\mathcal{B}}+\hat{\mu}_{\mathcal{M}}+\hat{\mu}_{l}\ |\ \psi\ \rangle = \sum_{SS_z,ll_z}\ \langle\ SS_z,ll_z|JJ_z\ \rangle^{2} \\&\times \Bigg\{ \mu_{l} l_z + \sum_{\widetilde{S}_\mathcal{B},\widetilde{S}_\mathcal{M}}\ \langle\ S_\mathcal{B} \widetilde{S}_{\mathcal{B}},S_\mathcal{M} \widetilde{S}_{\mathcal{M}}|SS_z\ \rangle^{2} \Bigg [ \widetilde{S}_{\mathcal{M}}\bigg(\mu_{\bar{c}} + \mu_{q_1}\bigg ) \end{aligned} $
$ \begin{aligned}[b]\quad\quad &+ \sum_{\widetilde{S}_{c}}\ \langle\ S_c \widetilde{S}_{c},S_{r} \widetilde{S}_{\mathcal{B}}-\widetilde{S}_{c}|S_\mathcal{B} \widetilde{S}_{\mathcal{B}}\rangle^{2}\\&\times \bigg(g\mu_{c}\widetilde{S}_{c}+(\widetilde{S}_{\mathcal{B}}-\widetilde{S}_{c})(\mu_{q_{2}}+\mu_{q_{3}})\bigg ) \Bigg ]\Bigg\}, \end{aligned} $
(10) where ψ represents the wave function in Table 2,
$ S_\mathcal{M} $ ,$ S_\mathcal{B} $ , and$ S_r $ are the meson, baryon, and diquark spin inside the baryon, respectively, and$ \widetilde{S} $ is the third spin component.For example, the recently observed
$ P_{cs}(4459) $ state is supposed to be the$ \bar{D}^{*}\Xi_{c} $ molecular state in the$ 8_{2f} $ representation with$ (I,I_{3}) = (0,0) $ . Its flavor wave functions are$ \begin{equation} | P_{cs}\rangle = \dfrac{1}{\sqrt{6}}\{({\bar c}d)(c[us])-({\bar c}u)(c[ds])-2(\bar c s)(c[ud])\}. \end{equation} $
(11) Take
$ J^{p}={\dfrac{1}{2}}^{-} $ ($ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ ) as an example.$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}} $ correspond to the angular momentum and parity of the baryon, meson, and orbital, respectively.$ \begin{aligned}[b]\\[-3pt] \mu = & \langle\ P_{cs}\ |\ \hat{\mu}_{\mathcal{B}}+\hat{\mu}_{\mathcal{M}}+\hat{\mu}_{l}\ |\ P_{cs}\ \rangle \\=& \langle \dfrac{1}{2}\dfrac{1}{2},1 0 |\dfrac{1}{2}\dfrac{1}{2}\rangle^{2}\Bigg [ \langle \dfrac{1}{2}\dfrac{1}{2},0 0 |\dfrac{1}{2}\dfrac{1}{2}\rangle^{2} \Bigg (\dfrac{1}{6}*\dfrac{1}{2}g\mu_{c}+\dfrac{1}{6}*\dfrac{1}{2}g\mu_{c}+\dfrac{4}{6}*\dfrac{1}{2}g\mu_{c}\Bigg )\Bigg ] \\ & + \langle \dfrac{1}{2}-\dfrac{1}{2},1 1 |\dfrac{1}{2}\dfrac{1}{2}\rangle^{2}\Bigg [ \Bigg (\dfrac{1}{6}*(\dfrac{1}{2}g\mu_{\bar{c}}+\dfrac{1}{2}g\mu_{d})+\dfrac{1}{6}*(\dfrac{1}{2}g\mu_{\bar{c}}+\dfrac{1}{2}g\mu_{u})+\dfrac{4}{6}*(\dfrac{1}{2}g\mu_{\bar{c}}+\dfrac{1}{2}g\mu_{s})\Bigg ) \\ & + \langle \dfrac{1}{2}-\dfrac{1}{2},0 0 |\dfrac{1}{2}-\dfrac{1}{2}\rangle^{2} \Bigg (\dfrac{1}{6}*-\dfrac{1}{2}g\mu_{c}+\dfrac{1}{6}*-\dfrac{1}{2}g\mu_{c}+\dfrac{4}{6}*-\dfrac{1}{2}g\mu_{c}\Bigg )\Bigg ] \\ = & \dfrac{1}{9} (\mu_{u}+\mu_{d}+4\mu_{s}+6\mu_{\bar{c}}-3\mu_{c} ). \end{aligned} $ (12) In this study, we use the following constituent quark masses [45]:
$\begin{aligned}[b] m_u =& m_d = 0.336\ \rm{GeV}, \\ m_s =& 0.540\ \rm{GeV},\\ m_c =& 1.660\ \rm{GeV}. \end{aligned} $
The numerical results with isospin
$ (I,I_3) = (1,0) $ and$ (I,I_3) = (0,0) $ are shown in Tables 3 and 4, respectively.$8_{1f}: \dfrac{1}{\sqrt6}[({\bar c}d)(c\{us\})+({\bar c}u)(c\{ds\})]-\sqrt{\dfrac{2}{3} }({\bar c}s)(c\{ud\})$ $ ^{2}S_{\frac{1}{2}} $ ($J^P={\dfrac{1}{2} }^{-}$ )$ {^{4}S_{\frac{3}{2}}} $ (${J^P={\dfrac{3}{2} }^{-} }$ )$ ^{6}S_{\frac{5}{2}}^{-} $ ($J^P={\dfrac{5}{2} }^{-}$ )$ (Y, I, I_3) $ ${\dfrac{1}{2} }^{+}\otimes0^{-}\otimes0^{+}$ ${\dfrac{1}{2} }^{+}\otimes1^{-}\otimes0^{+}$ ${\dfrac{3}{2} }^{+}\otimes1^{-}\otimes0^{+}$ ${\dfrac{1}{2} }^{+}\otimes1^{-}\otimes0^{+}$ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes0^{+} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ (0,1,0) $ 0.263 −0.493 0.735 −0.345 0.959 0.460 0.352 $ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,1,0) $ −0.145 0.125 −0.289 0.564 −0.172 0.278 $ {^{2}P_{\frac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{+}} $ )$ {^{4}P_{\frac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{+}} $ )$ {^{6}P_{\frac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{5}{2}}\otimes1^{-} $ $ (0,1,0) $ 0.177 −0.551 0.669 0.666 −0.276 0.311 0.335 $ {^{4}P_{\frac{5}{2}}} $ ($ {J^P={\dfrac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\dfrac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{7}{2}}} $ ($ {J^P={\dfrac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{5}{2}}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ (0,1,0) $ −0.403 0.865 0.394 0.292 0.285 $ 8_{2f} $ :$ \dfrac{1}{\sqrt2}\{ ({\bar c}d)(c[us])+({\bar c}u)(c[ds]) \} $ $ ^2S_{\frac{1}{2}} $ ($J^P={\dfrac{1}{2} }^{-}$ )$ ^{4}S_{{\frac{3}{2}}} $ (${J^P={\dfrac{3}{2} }^{-} }$ )$ {^{2}P_{\frac{1}{2}}} $ (${J^P={\dfrac{1}{2} }^{+} }$ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,1,0) $ 0.377 −0.067 0.465 −0.167 −0.007 0.273 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\dfrac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\dfrac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ (${J^P={\dfrac{5}{2} }^{+} }$ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes0^{-}]_{\frac{3}{2}}\otimes1^{-} $ ${\dfrac{1}{2} }^{+}\otimes1^{-}\otimes1^{-}$ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ (0,1,0) $ 0.315 −0.110 0.324 0.422 Table 3. Magnetic moments of pentaquark states in the molecular model with the wavefunction
$\dfrac{1}{\sqrt6}[({\bar c}d)(c\{us\})+ $ $ ({\bar c}u)(c\{ds\})]-\sqrt{\dfrac{2}{3}}({\bar c}s)(c\{ud\})$ in$ 8_{1f} $ and$\dfrac{1}{\sqrt2}\{ ({\bar c}d)(c[us])+({\bar c}u)(c[ds]) \}$ in$ 8_{2f} $ with isospin$ (I,I_3) = (1,0) $ . They are in the$ 8_{1f} $ representation from$ 6_f \otimes 3_f = 10_f \oplus 8_{1f} $ and the$ 8_{2f} $ representation from$ \bar{3}_f \otimes 3_f = 1_f \oplus8_{2f} $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}} $ correspond to the angular momentum and parity of the baryon, meson, and orbital, respectively. The unit is proton magnetic moments.$ 8_{1f} $ :$\dfrac{1}{\sqrt2}[({\bar{c} }u)(c\{ds\})-({\bar{c} }d)(c\{us\})]$ $ ^{2}S_{\frac{1}{2}} $ ($J^P={\dfrac{1}{2} }^{-}$ )$ {^{4}S_{\frac{3}{2}}} $ (${J^P={\dfrac{3}{2} }^{-} }$ )$ ^{6}S_{\frac{5}{2}}^{-} $ ($J^P={\dfrac{5}{2} }^{-}$ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes0^{+} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ (0,0,0) $ −0.201 0.126 0.117 −0.113 0.263 0.228 0.352 $ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,0,0) $ 0.021 −0.076 −0.076 −0.046 0.171 0.145 $ {^{2}P_{\frac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{+}} $ )$ {^{4}P_{\frac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{+}} $ )$ {^{6}P_{\frac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{5}{2}}\otimes1^{-} $ $ (0,0,0) $ −0.270 0.075 0.061 −0.103 0.163 0.145 0.329 $ {^{4}P_{\frac{5}{2}}} $ ($ {J^P={\dfrac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\dfrac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{7}{2}}} $ ($ {J^P={\dfrac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\dfrac{3}{2}}^{+}\otimes1^{-}]_{\frac{5}{2}}\otimes1^{-} $ $ {\dfrac{3}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ (0,0,0) $ −0.164 0.189 0.172 0.295 0.296 $ 8_{2f} $ :$ \dfrac{1}{\sqrt6} \{ ({\bar c}d)(c[us])-({\bar c}u)(c[ds])-2(\bar c s)(c[ud]) \} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\dfrac{1}{2}}^{-} $ )$ ^{4}S{{\dfrac{3}{2}}} $ ($ {J^P={\dfrac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\dfrac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes1^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,0,0) $ 0.377 −0.531 −0.231 −0.161 0.152 −0.116 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\dfrac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\dfrac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\dfrac{5}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\dfrac{1}{2}}^{+}\otimes0^{-}\otimes1^{-} $ $ [{\dfrac{1}{2}}^{+}\otimes0^{-}]_{\frac{3}{2}}\otimes1^{-} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes1^{-} $ $ (0,0,0) $ 0.324 −0.568 −0.184 −0.268 Table 4. Magnetic moments of pentaquark states in the molecular model with the wavefunction
$\dfrac{1}{\sqrt2}[({\bar{c}}u)(c\{ds\})-({\bar{c}}d)(c\{us\})]$ in$ 8_{1f} $ and$\dfrac{1}{\sqrt6} \{ ({\bar c}d)(c[us])-({\bar c}u)(c[ds])-2(\bar c s)(c[ud]) \}$ in$ 8_{2f} $ with isospin$ (I,I_3) = (0,0) $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}} $ correspond to the angular momentum and parity of the baryon, meson, and orbital, respectively. The unit is proton magnetic moments. -
In the diquark-diquark-antiquark model, there are two P-wave excitation modes inside the three-body bound state: ρ and λ excitation. ρ mode P-wave orbital excitation lies between the diquark
$ (cq_1) $ and diquark$ (q_2q_3) $ , whereas λ mode P-wave orbital excitation lies between$ \bar{c} $ and the center of mass system of$ (cq_1) $ and$ (q_2q_3) $ .The total magnetic moment formula of the diquark-diquark-antiquark model can be written as
$ \begin{eqnarray} \hat{\mu} = \ \hat{\mu}_{H}+\hat{\mu}_{L}+\hat{\mu}_{\bar{c}}+\hat{\mu}_{l}, \end{eqnarray} $
(13) where the subscripts H and L represent heavy
$ (cq_1) $ and light diquarks$ (q_2q_3) $ , respectively, and l is the orbital excitation. In the diquark-diquark-antiquark model, the specific magnetic moment formula of the pentaquark$ (cq_1)(q_2q_3)\bar{c} $ is$ \begin{aligned}[b]\\[536pt] \mu =& \langle\ \psi \ |\ \hat{\mu}_{H}+\hat{\mu}_{L}+\hat{\mu}_{\bar{c}}+\hat{\mu}_{l}\ |\ \psi \ \rangle =\sum_{S_z,l_z}\ \langle\ SS_z,ll_z|JJ_z\ \rangle^{2} \left \{ \mu_{l} l_z + \sum_{\widetilde{S}_{\bar{c}}}\ \langle\ S_{\bar{c}} \widetilde{S}_{\bar{c}},S_{\mathcal{G}} \widetilde{S}_{\mathcal{G}}|SS_z\ \rangle^{2} \Bigg [g\widetilde{S}_{\bar{c}}\mu_{\bar{c}}\right.\\ &+\left.\sum_{\widetilde{S}_{H},\widetilde{S}_{L}}\ \langle\ S_{H} \widetilde{S}_{H},S_{L} \widetilde{S}_{L}|S_{\mathcal{G}} \widetilde{S}_{\mathcal{G}}\rangle^{2}\bigg(\widetilde{S}_{H}(\mu_{c}+\mu_{q_1})+\widetilde{S}_{L}(\mu_{q_2}+\mu_{q_3})\bigg ) \Bigg ]\right \},\\ \end{aligned} $
(14) where
$ S_{\mathcal{G}} $ represents the spin of$ (cq_1)(q_2q_3) $ . The diquark masses are [46]$ \begin{aligned}[b] [u,d]=& 710\;{\rm{ MeV}}, \quad \{u,d\} =909\;{ \rm{MeV}},\quad [u,s]=948\;{\rm{ MeV}},\\ \{u,s\} =&1069\;{\rm{ MeV}},\quad [c,q]= 1973\;{\rm{ MeV}},\;\;\{c,q\} =2036\;{\rm{ MeV}},\\ [c,s]=&2091\;{\rm{ MeV}},\quad\{c,s\} =2158\;{\rm{ MeV}}. \end{aligned} $
The numerical results for states with the ρ excitation mode and isospin
$ (I,I_3) = (1,0) $ and$ (I,I_3) = (0,0) $ are presented in Tables 5 and 6, respectively. The numerical results for states with the λ excitation mode and isospin$ (I,I_3) = (1,0) $ and$ (I,I_3) = (0,0) $ are presented in Tables 7 and 8, respectively.$ 8_{1f} $ :$ \frac{1}{\sqrt6}[({c}d)\{us\}{\bar c}+({c}u)\{ds\}{\bar c}]-\sqrt{\frac{2}{3}}({c}s)\{ud\}{\bar c} $ $ ^{2}S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ {^{4}S_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ ^{6}S_{\frac{5}{2}} $ ($ J^P={\frac{5}{2}}^{-} $ )$ (Y, I, I_3) $ $ 0^{+}\otimes1^{+} \otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{0} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0^{+}\otimes1^{+})\otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{2} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})\otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0,1,0) $ 0.514 −0.377 0.368 0.206 −0.013 0.881 0.352 $ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ (0,1,0) $ −0.035 0.046 0.260 0.012 −0.074 0.422 $ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^6 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ (0,1,0) $ 0.719 0.233 −0.175 0.570 0.005 0.727 0.174 $ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^6P_{\frac{7}{2}}} $ ($ {J^P={\frac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ 1^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-}\otimes1^{-} $ $ (0,1,0) $ 0.410 0.190 1.083 0.369 0.554 $ 8_{2f} $ :$ \frac{1}{\sqrt2}\{ (cd)[us]{\bar c}+(cu)[ds]{\bar c} \} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ ^{4}S{{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ 0^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes{1^{-}} $ $ (0,1,0) $ −0.377 0.687 0.465 0.137 −0.224 0.256 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} )_{\frac{1}{2}}\otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}} \otimes{1^{-}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ (0,1,0) $ −0.360 0.695 0.344 0.473 Table 5. Magnetic moments of pentaquark states in the diquark-diquark-antiquark model with the wave function
$ \frac{1}{\sqrt6}[({c}d)\{us\}{\bar c}+({c}u)\{ds\}{\bar c}]-\sqrt{\frac{2}{3}}({c}s)\{ud\}{\bar c} $ in$ 8_{1f} $ and$ \frac{1}{\sqrt2}\{ (cd)[us]{\bar c}+(cu)[ds]{\bar c} \} $ in$ 8_{2f} $ with isospin$ (I,I_3) = (1,0) $ . They are in the$ 8_{1f} $ representation from$ 6_f \otimes 3_f = 10_f \oplus 8_{1f} $ and the$ 8_{2f} $ representation from$ \bar{3}_f \otimes 3_f = 1_f \oplus8_{2f} $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}}\otimes J_{4}^{P_{4}} $ correspond to the angular momentum and parity of$ (cq_1) $ ,$ (q_2q_3) $ ,$ \bar{c} $ , and the orbital, respectively. ρ mode P-wave orbital excitation lies between the diquark$ (cq_1) $ and diquark$ (q_2q_3) $ . The unit is proton magnetic moments.$ 8_{1f} $ :$ \frac{1}{\sqrt2}[({c}u)\{ds\}{\bar c}-({c}d)\{us\}{\bar c}] $ $ ^{2}S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ {^{4}S_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ ^{6}S_{\frac{5}{2}} $ ($ J^P={\frac{5}{2}}^{-} $ )$ (Y, I, I_3) $ $ 0^{+}\otimes1^{+} \otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{0} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0^{+}\otimes1^{+})\otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{2} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})\otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0,0,0) $ 0.050 −0.377 0.368 −0.490 −0.013 0.881 0.352 $ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ (0,0,0) $ 0.013 −0.287 0.150 −0.098 −0.019 0.478 $ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^6 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ (0, 0,0) $ 0.094 −0.342 −0.340 0.405 0.197 0.661 0.273 $ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^6P_{\frac{7}{2}}} $ ($ {J^P={\frac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ 1^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-}\otimes1^{-} $ $ (0,0,0) $ −0.446 0.024 0.918 0.322 0.388 $ 8_{2f} $ :$ \frac{1}{\sqrt6} \{ (cd)[us]{\bar c}-(cu)[ds]{\bar c}-2(cs)[ud]{\bar c} \} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ ^{4}S{{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ 0^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes{1^{-}} $ $ (0,0,0) $ −0.377 0.223 −0.231 0.292 0.091 −0.211 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} )_{\frac{1}{2}}\otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}} \otimes{1^{-}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ (0,0,0) $ −0.126 0.470 −0.070 0.016 Table 6. Magnetic moments of pentaquark states in the diquark-diquark-antiquark model with the wave function
$ \frac{1}{\sqrt2}[({c}u)\{ds\}{\bar c}-({c}d)\{us\}{\bar c}] $ in$ 8_{1f} $ and$ \frac{1}{\sqrt6} \{ (cd)[us]{\bar c}-(cu)[ds]{\bar c}-2(cs)[ud]{\bar c} \} $ in$ 8_{2f} $ with isospin$ (I,I_3) = (0,0) $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}}\otimes J_{4}^{P_{4}} $ correspond to the angular momentum and parity of$ (cq_1) $ ,$ (q_2q_3) $ ,$ \bar{c} $ , and the orbital, respectively. ρ mode P-wave orbital excitation lies between the diquark$ (cq_1) $ and diquark$ (q_2q_3) $ .$ 8_{1f} $ :$ \frac{1}{\sqrt6}[({c}d)\{us\}{\bar c}+({c}u)\{ds\}{\bar c}]-\sqrt{\frac{2}{3}}({c}s)\{ud\}{\bar c} $ $ ^{2}S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ {^{4}S_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ ^{6}S_{\frac{5}{2}} $ ($ J^P={\frac{5}{2}}^{-} $ )$ (Y, I, I_3) $ $ 0^{+}\otimes1^{+} \otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{0} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0^{+}\otimes1^{+})\otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{2} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})\otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0,1,0) $ 0.514 −0.377 0.368 0.206 −0.013 0.881 0.352 $ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ (0,1,0) $ 0.217 −0.080 0.507 0.259 −0.198 0.299 $ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^6 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ (0,1,0) $ 1.096 0.384 0.196 0.941 0.220 0.875 −0.048 $ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^6P_{\frac{7}{2}}} $ ($ {J^P={\frac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ 1^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-}\otimes1^{-} $ $ (0,1,0) $ 0.788 0.560 1.454 0.475 0.924 $ 8_{2f} $ :$ \frac{1}{\sqrt2}\{ (cd)[us]{\bar c}+(cu)[ds]{\bar c} \} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ ^{4}S{{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ 0^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes{1^{-}} $ $ (0,1,0) $ −0.377 0.687 0.465 0.525 0.164 0.062 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} )_{\frac{1}{2}}\otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}} \otimes{1^{-}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ (0,1,0) $ 0.223 1.277 0.577 1.055 Table 7. Magnetic moments of pentaquark states in the diquark-diquark-antiquark model with the wave function
$ \frac{1}{\sqrt6}[({c}d)\{us\}{\bar c}+({c}u)\{ds\}{\bar c}]-\sqrt{\frac{2}{3}}({c}s)\{ud\}{\bar c} $ in$ 8_{1f} $ and$ \frac{1}{\sqrt2}\{ (cd)[us]{\bar c}+(cu)[ds]{\bar c} \} $ in$ 8_{2f} $ with isospin$ (I,I_3) = (1,0) $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}}\otimes J_{4}^{P_{4}} $ correspond to the angular momentum and parity of$ (cq_1) $ ,$ (q_2q_3) $ ,$ \bar{c} $ , and the orbital, respectively. λ mode P-wave orbital excitation lies between$ \bar{c} $ and the center of mass system of$ (cq_1) $ and$ (q_2q_3) $ .The unit is proton magnetic moments.$ 8_{1f} $ :$ \frac{1}{\sqrt2}[({c}u)\{ds\}{\bar c}-({c}d)\{us\}{\bar c}] $ $ ^{2}S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ {^{4}S_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ ^{6}S_{\frac{5}{2}} $ ($ J^P={\frac{5}{2}}^{-} $ )$ (Y, I, I_3) $ $ 0^{+}\otimes1^{+} \otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{0} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0^{+}\otimes1^{+})\otimes{\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{1} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})_{2} \otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (1^{+}\otimes1^{+})\otimes {\frac{1}{2}}^{-}\otimes0^{+} $ $ (0,0,0) $ 0.050 −0.377 0.368 −0.490 −0.013 0.881 0.352 $ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{2}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ ^{4}P_{\frac{1}{2}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ (0,0,0) $ 0.334 −0.448 0.469 0.221 −0.179 0.318 $ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^2 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^4 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^6 P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ (0, 0,0) $ 0.575 −0.150 0.139 0.884 0.072 0.853 −0.014 $ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^6P_{\frac{7}{2}}} $ ($ {J^P={\frac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ (0^{+}\otimes1^{+}\otimes {\frac{1}{2}}^{-})\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes1^{-} $ $ ((1^{+}\otimes1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes1^{-} $ $ 1^{+}\otimes1^{+}\otimes{\frac{1}{2}}^{-}\otimes1^{-} $ $ (0,0,0) $ 0.035 0.503 1.397 0.459 0.867 $ 8_{2f} $ :$ \frac{1}{\sqrt6} \{ (cd)[us]{\bar c}-(cu)[ds]{\bar c}-2(cs)[ud]{\bar c} \} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ ^{4}S{{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{0^{+}} $ $ 0^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes{1^{-}} $ $ (0,0,0) $ −0.377 0.223 −0.231 0.616 0.410 −0.370 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {0}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} )_{\frac{1}{2}}\otimes{1^{-}} $ $ ({1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}} \otimes{1^{-}} $ $ {1}^{+}\otimes0^{+}\otimes{\frac{1}{2}}^{-} \otimes{1^{-}} $ $ (0,0,0) $ 0.359 0.949 0.121 0.495 Table 8. Magnetic moments of pentaquark states in the diquark-diquark-antiquark model with the wave function
$ \frac{1}{\sqrt2}[({c}u)\{ds\}{\bar c}-({c}d)\{us\}{\bar c}] $ in$ 8_{1f} $ and$ \frac{1}{\sqrt6} \{ (cd)[us]{\bar c}-(cu)[ds]{\bar c}-2(cs)[ud]{\bar c} \} $ in$ 8_{2f} $ with isospin$ (I,I_3) = (0,0) $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}}\otimes J_{4}^{P_{4}} $ correspond to the angular momentum and parity of$ (cq_1) $ ,$ (q_2q_3) $ ,$ \bar{c} $ , and the orbital, respectively. λ mode P-wave orbital excitation lies between$ \bar{c} $ and the center of mass system of$ (cq_1) $ and$ (q_2q_3) $ . -
Considering the diquark-triquark model, the total magnetic moment formula is
$ \begin{eqnarray} \hat{\mu} = \ \hat{\mu}_{\mathcal{D}}+\hat{\mu}_{\mathcal{T}}+\hat{\mu}_{l}. \end{eqnarray} $
(15) where l is the orbital excitation between the diquark and triquark. The magnetic moment formula of the pentaquark with
$ (cq_1)(\bar{c}q_2q_3) $ in the diquark-triquark model is$ \begin{aligned}[b] \mu =& \langle\ \psi\ |\ \hat{\mu}_{\mathcal{D}}+\hat{\mu}_{\mathcal{T}}+\hat{\mu}_{l}\ |\ \psi\ \rangle= \sum_{S_z,l_z}\ \langle\ SS_z,ll_z|JJ_z\ \rangle^{2} \left \{ \mu_{l} l_z + \sum_{\widetilde{S}_{\mathcal{D}},\widetilde{S}_{\mathcal{T}}}\ \langle\ S_\mathcal{D} \widetilde{S}_{\mathcal{D}},S_\mathcal{T} \widetilde{S}_{\mathcal{T}}|SS_z\ \rangle^{2} \Bigg [ \widetilde{S}_{\mathcal{D}}\bigg(\mu_{c} + \mu_{q_1}\bigg )\right.\\ &+\left. \sum_{\widetilde{S}_{\bar{c}}}\ \langle\ S_{\bar{c}} \widetilde{S}_{\bar{c}},S_{r} \widetilde{S}_{\mathcal{T}}-\widetilde{S}_{\bar{c}}|S_{\mathcal{T}} \widetilde{S}_{\mathcal{T}}\rangle^{2}\bigg(g\mu_{\bar{c}}\widetilde{S}_{\bar{c}}+(\widetilde{S}_{\mathcal{T}}-\widetilde{S}_{\bar{c}})(\mu_{q_2}+\mu_{q_3})\bigg ) \Bigg ]\right \}. \end{aligned} $ (16) where
$ S_{\mathcal{D}} $ ,$ S_{\mathcal{T}} $ , and$ S_{r} $ represent the diquark, triquark, and light diquark spin inside the triquark, respectively. The triquark masses are roughly the sum of the masses of the corresponding diquark and antiquark. The numerical results with isospin$ (I,I_3) = (1,0) $ and$ (I,I_3) = (0,0) $ are shown in Tables 9 and 10, respectively.$ 8_{1f} $ :$ \frac{1}{\sqrt6}[({c}d)(\bar c\{us\})+({c}u)(\bar c\{ds\})]-\sqrt{\frac{2}{3}}({c}s)(\bar c\{ud\}) $ $ ^{2}S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ {^{4}S_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ ^{6}S_{\frac{5}{2}}^{-} $ ($ J^P={\frac{5}{2}}^{-} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ (0,1,0) $ 0.522 −0.078 0.051 0.666 0.178 0.188 0.352 $ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,1,0) $ −0.137 0.058 0.015 0.080 0.354 0.088 $ {^{2}P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^{4}P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^{6}P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{5}{2}}\otimes1^{-} $ $ (0,1,0) $ 0.577 −0.030 0.098 0.152 0.508 0.157 0.242 $ {^{4}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{7}{2}}} $ ($ {J^P={\frac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{5}{2}}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ (0,1,0) $ 0.714 0.233 0.236 0.299 0.370 $ 8_{2f} $ :$ \frac{1}{\sqrt2}\{ ({c}d)(\bar c[us])+({c}u)(\bar c[ds]) \} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ ^{4}S{{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,1,0) $ −0.377 0.687 0.465 0.199 −0.184 0.235 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ (0,1,0) $ −0.307 0.803 0.381 0.558 Table 9. Magnetic moments of pentaquark states in the diquark-triquark model with the wave function
$ \frac{1}{\sqrt6}[({c}d)(\bar c\{us\})+({c}u)(\bar c\{ds\})]-\sqrt{\frac{2}{3}}({c}s)(\bar c\{ud\}) $ in$ 8_{1f} $ and$ \frac{1}{\sqrt2}\{ ({c}d)(\bar c[us])+({c}u)(\bar c[ds]) \} $ in$ 8_{2f} $ with isospin$ (I,I_3) = (1,0) $ . They are in the$ 8_{1f} $ representation from$ 6_f \otimes 3_f = 10_f \oplus 8_{1f} $ and the$ 8_{2f} $ representation from$ \bar{3}_f \otimes 3_f = 1_f \oplus8_{2f} $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}} $ correspond to the angular momentum and parity of the triquark, diquark, and orbital, respectively. The unit is proton magnetic moments.$ 8_{1f} $ :$ \frac{1}{\sqrt2}[({c}u)(\bar c\{ds\})-({c}d)(\bar c\{us\})] $ $ ^{2}S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ {^{4}S_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ ^{6}S_{\frac{5}{2}}^{-} $ ($ J^P={\frac{5}{2}}^{-} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ (0,0,0) $ 0.033 0.574 −0.601 0.910 −0.555 −0.056 0.352 $ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,0,0) $ 0.062 −0.126 0.265 0.473 −0.345 −0.064 $ {^{2}P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^{4}P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ {^{6}P_{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{5}{2}}\otimes1^{-} $ $ (0,0,0) $ 0.143 0.671 −0.503 0.707 −0.363 −0.002 0.212 $ {^{4}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{5}{2}}} $ ($ {J^P={\frac{5}{2}}^{+}} $ )$ {^{6}P_{\frac{7}{2}}} $ ($ {J^P={\frac{7}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ [{\frac{3}{2}}^{-}\otimes1^{+}]_{\frac{5}{2}}\otimes1^{-} $ $ {\frac{3}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ (0,0,0) $ 1.008 −0.445 0.041 0.313 0.420 $ 8_{2f} $ :$ \frac{1}{\sqrt6}\{({c}d)(\bar c[us])-({ c}u)(\bar c[ds])-2( c s)(\bar c[ud])\} $ $ ^2S_{\frac{1}{2}} $ ($ J^P={\frac{1}{2}}^{-} $ )$ ^{4}S{{\frac{3}{2}}} $ ($ {J^P={\frac{3}{2}}^{-}} $ )$ {^{2}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ {^{4}P_{\frac{1}{2}}} $ ($ {J^P={\frac{1}{2}}^{+}} $ )$ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{1}{2}}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ (0,0,0) $ −0.377 0.223 −0.231 0.164 −0.035 −0.165 $ ^{2}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ ^{4}P_{\frac{3}{2}} $ ($ J^P={\frac{3}{2}}^{+} $ )$ {^{4}P_{\frac{5}{2}}^{+}} $ ($ {J^P={\frac{5}{2}}^{+}} $ $ (Y, I, I_3) $ $ {\frac{1}{2}}^{-}\otimes0^{+}\otimes1^{-} $ $ [{\frac{1}{2}}^{-}\otimes1^{+}]_{\frac{3}{2}}\otimes1^{-} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ {\frac{1}{2}}^{-}\otimes1^{+}\otimes1^{-} $ $ (0,0,0) $ −0.359 0.293 −0.165 −0.196 Table 10. Magnetic moments of pentaquark states in the diquark-triquark model with the wave function
$ \frac{1}{\sqrt2}[({c}u)(\bar c\{ds\})-({c}d)(\bar c\{us\})] $ in$ 8_{1f} $ and$ \frac{1}{\sqrt6}\{({c}d)(\bar c[us])-({ c}u)(\bar c[ds])-2( c s)(\bar c[ud])\} $ in$ 8_{2f} $ with isospin$ (I,I_3) = (0,0) $ . On the third line,$ J_{1}^{P_{1}}\otimes J_{2}^{P_{2}}\otimes J_{3}^{P_{3}} $ correspond to the angular momentum and parity of the triquark, diquark, and orbital, respectively. The unit is proton magnetic moments.The magnetic moments of
$ P_{cs}(4459) $ in three configurations are compared, as shown in Table 11. The magnetic moments and numerical results illustrate that the molecular model is distinguishable from the other two models with$ 0({\dfrac{1}{2}}^{-}) $ but is indistinguishable with$ 0({\dfrac{3}{2}}^{-}) $ . The diquark-diquark-antiquark and diquark-triquark models are completely indistinguishable with$ 0({\dfrac{1}{2}}^{-}) $ and$ 0({\dfrac{3}{2}}^{-}) $ . In addition, the magnetic moments of$ P_{cs}(4459) $ have been studied in other papers. In Ref. [44], the numerical value in the molecular model was obtained as$ \mu_{P_{cs}} = -0.062\mu_{N} $ with 0($ {\dfrac{1}{2}}^{-} $ ) and$ \mu_{P_{cs}} = 0.465\mu_{N} $ with 0($ {\dfrac{3}{2}}^{-} $ ). In Ref., the magnetic dipole moments of$ P_{cs}(4459) $ in the molecular and diquark-diquark-antiquark models were extracted as$ \mu_{P_{cs}} = 1.75\mu_{N} $ and$ \mu_{P_{cs}}=0.34\mu_{N} $ , respectively. These numerical results differ from our results of$ \mu_{P_{cs}} = -0.531\mu_{N} $ with 0($ {\dfrac{1}{2}}^{-} $ ) and$ \mu_{P_{cs}} = -0.231\mu_{N} $ with 0($ {\dfrac{3}{2}}^{-} $ ) in the molecular model and$ \mu_{P_{cs}}=0.223\mu_{N} $ in the diquark-diquark-antiquark model because of the wavefunction and quark mass. We compare the results in Table 12.$ P_{cs}(4459) $ Multiplet Spin-orbit coupling $ I(J^P) $ Magnetic moment Numerical results Molecular model $ 8_{2f} $ $ {\dfrac{1}{2}}^{+}\otimes1^{-}\otimes0^{+} $ $0({\dfrac{1}{2} }^{-})$ $ \dfrac{1}{9}(6\mu_{\bar{c}}-3\mu_{c}+\mu_{u}+\mu_{d}+4\mu_{s}) $ −0.531 $ 0({\dfrac{3}{2}}^{-}) $ $ \dfrac{1}{6}(6\mu_{c}+6\mu_{\bar{c}}+\mu_{u}+\mu_{d}+4\mu_{s}) $ −0.231 Diquark-diquark-antiquark model $ 8_{2f} $ $1^{+}\otimes0^{+}\otimes{\dfrac{1}{2} }^{-}\otimes0^{+}$ $ 0({\dfrac{1}{2}}^{-}) $ $ \dfrac{1}{9}(6\mu_{c}-3\mu_{\bar{c}}+\mu_{u}+\mu_{d}+4\mu_{s}) $ 0.223 $ 0({\dfrac{3}{2}}^{-}) $ $ \dfrac{1}{6}(6\mu_{c}+6\mu_{\bar{c}}+\mu_{u}+\mu_{d}+4\mu_{s}) $ −0.231 Diquark-triquark model $ 8_{2f} $ $ {\dfrac{1}{2}}^{-}\otimes1^{+}\otimes0^{+} $ $ 0({\dfrac{1}{2}}^{-}) $ $ \dfrac{1}{9}(6\mu_{c}-3\mu_{\bar{c}}+\mu_{u}+\mu_{d}+4\mu_{s}) $ 0.223 $ 0({\dfrac{3}{2}}^{-}) $ $ \dfrac{1}{6}(6\mu_{c}+6\mu_{\bar{c}}+\mu_{u}+\mu_{d}+4\mu_{s}) $ −0.231 Table 11. Magnetic moments of
$ P_{cs}(4459) $ in the molecular, diquark-diquark-antiquark, and diquark-triquark models in the$ 8_{2f} $ representation with isospin$ (I,I_3) = (0,0) $ .Cases A B C $ J^P $ $ {\dfrac{1}{2}}^{-} $ $ {\dfrac{3}{2}}^{-} $ $ {\dfrac{1}{2}}^{-} $ $ {\dfrac{3}{2}}^{-} $ $ {\dfrac{1}{2}}^{-} $ $ {\dfrac{3}{2}}^{-} $ Our results −0.531 −0.231 0.223 −0.231 0.223 −0.231 Ref. [44] −0.062 0.465 − − − − Ref. 1.75 − 0.34 − − − Table 12. Our results and other theoretical results for the magnetic moments of
$ P_{cs}(4459) $ .The unit is proton magnetic moments. A, B, and C correspond to the molecular, diquark-diquark-antiquark, and diquark-triquark models.
Magnetic moments of hidden-charm strange pentaquark states
- Received Date: 2022-05-31
- Available Online: 2022-12-15
Abstract: In this study, the magnetic moments of hidden-charm strange pentaquark states with quantum numbers