(Pseudo)spin symmetry in the single-neutron spectrum of Λ hypernuclei

Symmetries in the single-particle (s.p.) spectra of atomic nuclei are important concepts in nuclear structure and have been extensively discussed in the literature, e.g., the violation of spin symmetry (SS) and the approximate pseudospin symmetry (PSS) in nuclear s.p. spectra. The breaking of SS, i.e., the remarkable spin–orbit splitting for the spin doublets ($n,l,j=l\pm 1/2$) caused by the spin–orbit potential (SOP), lays the foundation for understanding the traditional magic numbers in nuclear physics [1, 2]. Meanwhile, a new symmetry, the so-called PSS, is introduced to explain the quasi-degeneracy between the two s.p. states with quantum numbers ($n,l,j=l+1/2$) and ($n-1,l+2,j=l+3/2$) by defining the pseudospin doublets ($\tilde{n}=n-1$, $\tilde{l}=l+1$, $j=\tilde{l}\pm 1/2$) [3, 4]. This symmetry has been used to explain a number of phenomena in nuclear structure such as deformation [5], superdeformation [6], magnetic moment [7], and identical rotational bands [8].

Since the recognition of PSS, comprehensive efforts have been made to understand its origin. Apart from the relabeling of quantum numbers, the explicit transformations from the normal scheme to the pseudospin scheme have been proposed in [9–11]. In 1997, Ginocchio made substantial progress, clearly showing that PSS is a relativistic symmetry in the Dirac Hamiltonian and becomes exact when the scalar and vector potentials are the same size but of opposite sign, i.e., ${\rm{\Sigma }}(r)\equiv S(r)+V(r)=0$ [12]. He also pointed out that the pseudo-orbital angular momentum $\tilde{l}$ is nothing but the orbital angular momentum of the lower component of the Dirac wave function [12] and the occurrence of approximate PSS in nuclei is connected with certain similarities in the relativistic single-nucleon wave functions of the corresponding pseudospin doublets [13]. Later, Meng et al pointed out a more general condition for PSS ${\rm{d}}{\rm{\Sigma }}(r)/{\rm{d}}{r}=0$, and to what extent PSS is conserved is related to the competition between the pseudo-centrifugal barrier (PCB) and the pseudospin-orbit potential (PSOP) [14, 15]. Since then, SS and PSS have been studied extensively, with research carried out on areas such as PSS in the deformed nuclei [16–21], SS in anti-nucleon spectra [22–25], SS in anti-Λ spectra [26–28], PSS in the s.p. resonate states [29–36], perturbative interpretation of SS and PSS [37, 38], and PSS in supersymmetric quantum mechanics [39, 40]. Readers are referred to [41, 42] for reviews.

Almost all the studies on SS and PSS mentioned above have been performed in ordinary nuclei. Nuclear physicists have been very interested in hypernuclei [43–45] since the first Λ hypernucleus was discovered in 1953 [46]. Experimentally, the single-Λ hypernuclei are studied most extensively and so far more than 30 single-Λ hypernuclei, ranging from ${}_{{\rm{\Lambda }}}^{3}{\rm{H}}$ up to ${}_{{\rm{\Lambda }}}^{208}$Pb, have been observed [43, 47]. Several double-Λ hypernuclei, such as ${}_{{\rm{\Lambda }}{\rm{\Lambda }}}^{6}$He, have also been found [48]. Theoretically, to shed light on baryon–baryon interaction, it is very important to study the structure of the hypernuclei. It is different from the nucleon; with an additional strangeness degree of freedom, a hyperon does not suffer from the nucleon's Pauli exclusion principle. And if hyperons penetrate into the nuclear interior as an impurity, there will be a series of responses, e.g., a shrinkage of nuclear size [49–51], a change in deformation properties [52, 53], a modification of the cluster structure [54], a shift of neutron drip line [55–57] and the occurrence of a nucleon and hyperon skin or halo [54, 56, 58]. The effects of Λ impurity on SS and PSS are fascinating, and we urgently require knowledge about the s.p. spectra in hypernuclei. It is therefore of great importance and interest to investigate SS and PSS in the s.p. spectrum of hypernuclei, and the effects of Λ impurity.

Over recent decades, the relativistic mean field (RMF) model has achieved great successes in ordinary nuclei [59–66] and hypernuclei [42, 67–79]. The relativistic approach is suitable for a discussion of spin–orbit splittings, as it is naturally built in the relativistic framework. With the RMF model, SS and PSS have been studied extensively with great success.

In the present work, SS and PSS in the single-neutron spectrum of Λ hypernuclei are studied within the framework of the RMF model. The effects of Λ impurity on these symmetries are discussed and the corresponding mechanisms are given. The paper is organized as follows. In section 2, we present the RMF model for Λ hypernuclei and the formalism of SS and PSS in the single-neutron spectrum. After the numerical details in section 3, we present the results and discussions in section 4. Finally, a summary is presented in section 5.

2.1. RMF model for Λ hypernuclei

The starting point of the meson-exchange RMF model for Λ hypernuclei is the covariant Lagrangian density

$$\begin{eqnarray}&&{ \mathcal L }={{ \mathcal L }}_{{\rm{N}}}+{{ \mathcal L }}_{{\rm{\Lambda }}},\end{eqnarray} \tag{ 1 }$$

where ${{ \mathcal L }}_{{\rm{N}}}$ is the standard RMF Lagrangian density for nucleons [59–66], in which the couplings with the scalar σ, vector ${\omega }_{\mu }$, vector-isovector ${\vec{\rho }}_{\mu }$ mesons, and the photon A_μ are included. It is written as

$$\begin{eqnarray}{{ \mathcal L }}_{{\rm{N}}} & = & {\bar{\psi }}_{{\rm{N}}}[{i}{\gamma }^{\mu }{\partial }_{\mu }-{M}_{{\rm{N}}}-{g}_{{\rm{N}}\sigma }\sigma -{g}_{{\rm{N}}\omega }{\gamma }^{\mu }{\omega }_{\mu }\\ & & \left.-{g}_{{\rm{N}}\rho }{\gamma }^{\mu }\vec{\tau }\cdot {\vec{\rho }}_{\mu }-e{\gamma }^{\mu }{A}_{\mu }\displaystyle \frac{1-{\tau }_{3}}{2}\right]{\psi }_{{\rm{N}}}\\ & & +\displaystyle \frac{1}{2}{\partial }^{\mu }\sigma {\partial }_{\mu }\sigma -{U}_{\sigma }(\sigma )\\ & & -\displaystyle \frac{1}{4}{{\rm{\Omega }}}^{\mu \nu }{{\rm{\Omega }}}_{\mu \nu }+{U}_{\omega }({\omega }_{\mu })\\ & & -\displaystyle \frac{1}{4}{\vec{R}}^{\mu \nu }\cdot {\vec{R}}_{\mu \nu }+{U}_{\rho }({\vec{\rho }}_{\mu })\\ & & -\displaystyle \frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu },\end{eqnarray} \tag{ 2 }$$

where M_N and ${m}_{\phi }({g}_{{\rm{N}}\phi })\,(\phi =\sigma ,\omega ,\rho )$ in the following are the masses (coupling constants) of the nucleon and the mesons, respectively, and the field tensors Ω, $\vec{R}$, and F are defined as

$$\begin{eqnarray}&&{{\rm{\Omega }}}^{\mu \nu }={\partial }^{\mu }{\omega }^{\nu }-{\partial }^{\nu }{\omega }^{\mu },\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&{\vec{R}}^{\mu \nu }={\partial }^{\mu }{\vec{\rho }}^{\nu }-{\partial }^{\nu }{\vec{\rho }}^{\mu },\end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}&&{F}^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }.\end{eqnarray} \tag{ 3c }$$

The nonlinear self-coupling terms ${U}_{\sigma }(\sigma )$, ${U}_{\omega }({\omega }_{\mu })$, and ${U}_{\rho }({\vec{\rho }}_{\mu })$ for the σ-, ω-, and ρ-mesons in ${{ \mathcal L }}_{{\rm{N}}}$, respectively, have the following forms,

$$\begin{eqnarray}&&{U}_{\sigma }(\sigma )=\displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }^{2}+\displaystyle \frac{1}{3}{g}_{2}{\sigma }^{3}+\displaystyle \frac{1}{4}{g}_{3}{\sigma }^{4},\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{U}_{\omega }({\omega }_{\mu })=\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }^{\mu }{\omega }_{\mu }+\displaystyle \frac{1}{4}{c}_{3}{({\omega }^{\mu }{\omega }_{\mu })}^{2},\end{eqnarray} \tag{ 4b }$$

$$\begin{eqnarray}&&{U}_{\rho }({\vec{\rho }}_{\mu })=\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\vec{\rho }}^{\mu }\cdot {\vec{\rho }}_{\mu }.\end{eqnarray} \tag{ 4c }$$

The Lagrangian density ${{ \mathcal L }}_{{\rm{\Lambda }}}$ for neutral isospin singlet hyperons with σ- and ω-meson couplings included has the form [68]

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\rm{\Lambda }}}={\bar{\psi }}_{{\rm{\Lambda }}}\left[i{\gamma }^{\mu }{\partial }_{\mu }-{{M}}_{{\rm{\Lambda }}}-{g}_{{\rm{\Lambda }}\sigma }\sigma -{g}_{{\rm{\Lambda }}\omega }{\gamma }^{\mu }{\omega }_{\mu }-\displaystyle \frac{{f}_{{\rm{\Lambda }}\omega }}{2{{M}}_{{\rm{\Lambda }}}}{\sigma }^{\mu \nu }{\partial }_{\nu }{\omega }_{\mu }\right]{\psi }_{{\rm{\Lambda }}},\end{eqnarray} \tag{ 5 }$$

where ${M}_{{\rm{\Lambda }}}$ is the mass of the Λ hyperon, ${g}_{{\rm{\Lambda }}\sigma }$ and ${g}_{{\rm{\Lambda }}\omega }$ are the coupling constants with the σ- and ω-meson, respectively. The last term represents the tensor coupling with the ω-meson, which is related to the s.p. spin–orbit splitting of Λ hyperons.

With the mean field and no-sea approximations, the s.p. Dirac equations for baryons and the Klein–Gordon equations for mesons and photons can be obtained through the variational procedure. The Dirac equations for baryons are

$$\begin{eqnarray}&&[{\boldsymbol{\alpha }}\cdot {\boldsymbol{p}}+\beta ({M}_{{\rm{b}}}+{S}_{{\rm{b}}}({\boldsymbol{r}}))+{V}_{{\rm{b}}}({\boldsymbol{r}})+{T}_{{\rm{b}}}({\boldsymbol{r}})]{\varphi }_{{\rm{b}}}({\boldsymbol{r}})\,=\,{\varepsilon }_{{\rm{b}}}{\varphi }_{{\rm{b}}}({\boldsymbol{r}}),\end{eqnarray} \tag{ 6 }$$

where ${\boldsymbol{\alpha }}$ and β are the Dirac matrices, ${\varepsilon }_{{\rm{b}}}$ (${\rm{b}}={\rm{N}},{\rm{\Lambda }}$) are the s.p. energies, and ${S}_{{\rm{b}}}({\boldsymbol{r}})$, ${V}_{{\rm{b}}}({\boldsymbol{r}})$, and ${T}_{{\rm{b}}}({\boldsymbol{r}})$ are the scalar, vector, and tensor potentials, respectively, and

$$\begin{eqnarray}&&{S}_{{\rm{b}}}={g}_{{\rm{b}}\sigma }\sigma ,\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&{V}_{{\rm{b}}}={g}_{{\rm{b}}\omega }{\gamma }^{\mu }{\omega }_{\mu }+{g}_{{\rm{b}}\rho }{\gamma }^{\mu }\vec{\tau }\cdot {\vec{\rho }}_{\mu }{\delta }_{{\rm{b}}{\rm{N}}}+\displaystyle \frac{1-{\tau }_{3}}{2}e{\gamma }^{\mu }{A}_{\mu }{\delta }_{{\rm{b}}{\rm{N}}},\end{eqnarray} \tag{ 7b }$$

$$\begin{eqnarray}&&{T}_{{\rm{b}}}=-\displaystyle \frac{{f}_{{\rm{b}}\omega }}{2{M}_{{\rm{b}}}}{\sigma }_{\mu \nu }{\partial }^{\mu }{\omega }^{\nu }{\delta }_{{\rm{b}}{\rm{\Lambda }}}.\end{eqnarray} \tag{ 7c }$$

Note that the last two terms in equation (7b) are zero for Λ hyperons and the tensor potential in equation (7c) is zero for nucleons.

The Klein–Gordon equations for mesons and photon are

$$\begin{eqnarray}&&-{\rm{\Delta }}\phi +{U}_{\phi }^{{\prime} }(\phi )={S}_{\phi },\end{eqnarray} \tag{ 8 }$$

where ${U}_{\phi }^{{\prime} }(\phi )={{\rm{d}}{U}}_{\phi }(\phi )/{\rm{d}}\phi$ are the first-order derivatives of the nonlinear terms for σ-, ω-, and ρ-mesons, and is zero for photons. The source terms ${S}_{\phi }$ read,

$$\begin{eqnarray}{S}_{\phi }=\left\{\begin{array}{ll}{\displaystyle \sum }_{{\rm{b}}}-{g}_{{\rm{b}}\sigma }{\rho }_{{\rm{bs}}}, & \phi =\sigma ;\\ {\displaystyle \sum }_{{\rm{b}}}{g}_{{\rm{b}}\omega }{j}_{{\rm{b}}}^{\mu }-\displaystyle \frac{{f}_{{\rm{b}}\omega }}{2{M}_{{\rm{b}}}}{\partial }_{\nu }{j}_{{\rm{b}}{T}}^{\mu \nu }{\delta }_{{\rm{b}}{\rm{\Lambda }}}, & \phi ={\omega }_{\mu };\\ {\displaystyle \sum }_{{\rm{b}}}{g}_{{\rm{b}}\rho }{\vec{j}}_{{\rm{b}}}^{\mu }{\delta }_{{\rm{\text{bN}}}}, & \phi ={\vec{\rho }}_{\mu };\\ {{ej}}_{{\rm{c}}}^{\mu }, & \phi ={A}_{\mu },\end{array}\right.\end{eqnarray} \tag{ 9 }$$

with the scalar density ${\rho }_{{\rm{bs}}}$, the baryon current ${j}_{{\rm{b}}}^{\mu }$, the tensor current ${j}_{{\rm{b}}{T}}^{\mu \nu }$, the isocurrent ${\vec{j}}_{{\rm{b}}}^{\mu }$, and the electromagnetic current ${j}_{{\rm{c}}}^{\mu }$ defined as,

$$\begin{eqnarray}&&{\rho }_{{\rm{bs}}}=\displaystyle \sum _{i=1}^{{A}_{{\rm{b}}}}{\bar{\varphi }}_{{\rm{b}}{i}}{\varphi }_{{\rm{b}}{i}},\end{eqnarray} \tag{ 10a }$$

$$\begin{eqnarray}&&{j}_{{\rm{b}}}^{\mu }=\displaystyle \sum _{i=1}^{{A}_{{\rm{b}}}}{\bar{\varphi }}_{{\rm{b}}{\text{i}}}{\gamma }^{\mu }{\varphi }_{{\rm{b}}{\text{i}}},\end{eqnarray} \tag{ 10b }$$

$$\begin{eqnarray}&&{j}_{{\rm{b}}{T}}^{\mu \nu }=\displaystyle \sum _{i=1}^{{A}_{{\rm{b}}}}{\bar{\varphi }}_{{\rm{b}}{i}}{\sigma }^{\mu \nu }{\varphi }_{{\rm{b}}{i}},\end{eqnarray} \tag{ 10c }$$

$$\begin{eqnarray}&&{\vec{j}}_{{\rm{b}}}^{\mu }=\displaystyle \sum _{i=1}^{{A}_{{\rm{b}}}}{\bar{\varphi }}_{{\rm{b}}{i}}{\gamma }^{\mu }\vec{\tau }{\varphi }_{{\rm{b}}{i}},\end{eqnarray} \tag{ 10d }$$

$$\begin{eqnarray}&&{j}_{{\rm{c}}}^{\mu }=\displaystyle \sum _{i=1}^{{A}_{{\rm{p}}}}{\bar{\varphi }}_{{\rm{p}}{i}}{\gamma }^{\mu }{\varphi }_{{\rm{p}}{i}},\end{eqnarray} \tag{ 10e }$$

where ${A}_{{\rm{b}}}\,({A}_{{\rm{p}}})$ are the particle numbers of baryons (protons).

In the RMF model, the Dirac equation (6) for baryons, Klein–Gordon equations (8) for mesons and photons, mean field potentials in equation (7), and densities and currents in equation (10e) are solved self-consistently and iteratively in the coordinate space.

2.2. SS and PSS in the neutron spectrum

In order to study SS and PSS with the RMF theory, we will examine the Dirac equation (6) governing the motion of baryons.

In the spherical case, the Dirac spinors of baryons can be expanded as

$$\begin{eqnarray}&&{\varphi }_{\kappa m}({\boldsymbol{r}})=\displaystyle \frac{1}{r}\left(\begin{array}{c}{{iG}}_{\kappa }(r){Y}_{{jm}}^{l}(\theta ,\phi )\\ -{F}_{\kappa }(r){Y}_{{jm}}^{\tilde{l}}(\theta ,\phi )\end{array}\right),\end{eqnarray} \tag{ 11 }$$

where ${G}_{\kappa }(r)/r$ and ${F}_{\kappa }(r)/r$ are the upper and lower components of the radial wave functions, ${Y}_{{jm}}^{l}(\theta ,\phi )$ is the spinor spherical harmonic, $\tilde{l}=l-\mathrm{sign}(\kappa )$ is the pseudo-orbital angular momentum, and the quantum number κ is defined as $\kappa ={(-1)}^{j+l+1/2}(j+1/2)$.

With the radial wave functions, Dirac equation (6) reduces to

$$\begin{eqnarray}\left(\begin{array}{cc}M+{\rm{\Sigma }} & -\tfrac{{\rm{d}}}{{\rm{d}}{r}}+\tfrac{\kappa }{r}+T\\ \tfrac{{\rm{d}}}{{\rm{d}}{r}}+\tfrac{\kappa }{r}+T & -M+{\rm{\Delta }}\end{array}\right)\left(\begin{array}{c}{G}_{\kappa }\\ {F}_{\kappa }\end{array}\right)={\varepsilon }_{\kappa }\left(\begin{array}{c}{G}_{\kappa }\\ {F}_{\kappa }\end{array}\right),\end{eqnarray} \tag{ 12 }$$

where the mean field potentials ${\rm{\Sigma }}=V+S$ and ${\rm{\Delta }}=V-S$. In the following, the neutron spectrum will be discussed, and in this case the tensor potential T = 0.

One can go a step further to reduce the Dirac equation (12) to a second-order Schrödinger-like equation for either the upper or the lower component. To study SS and PSS, the Schrödinger-like equation is solved for G(r) and F(r), respectively [41, 42].

By substituting

$$\begin{eqnarray}&&F(r)=\displaystyle \frac{1}{M-{\rm{\Delta }}({\boldsymbol{r}})+\varepsilon }\left(\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}+\displaystyle \frac{\kappa }{r}\right)G(r),\end{eqnarray} \tag{ 13 }$$

one can obtain for the upper component G(r)

$$\begin{eqnarray}&&\left\{-\displaystyle \frac{1}{{M}_{+}}\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{r}}^{2}}+\displaystyle \frac{1}{{M}_{+}^{2}}\displaystyle \frac{{{\rm{d}}{M}}_{+}}{{\rm{d}}{r}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}\right.\\ &&\quad \left.+\,\left[(M+{\rm{\Sigma }})+\displaystyle \frac{1}{{M}_{+}}\displaystyle \frac{\kappa (\kappa +1)}{{r}^{2}}+\displaystyle \frac{1}{{M}_{+}^{2}}\displaystyle \frac{{{\rm{d}}{M}}_{+}}{{\rm{d}}{r}}\displaystyle \frac{\kappa }{r}\right]\right\}G=\varepsilon G,\end{eqnarray} \tag{ 14 }$$

with the energy-dependent effective mass ${M}_{+}=M-{\rm{\Delta }}+\varepsilon$. Here and in the following, the subscript κ is omitted to simplify the form. For the above equation, in analogy with the Schrödinger equation, ${\rm{\Sigma }}(r)$ is the central potential in which particles move; the term proportional to $\kappa (\kappa +1)=l(l+1)$ corresponds to the centrifugal barrier (CB), and the last term, which is proportional to $\tfrac{{{\rm{d}}{M}}_{+}}{{\rm{d}}{r}}$, corresponds to SOP, which leads to the substantial spin–orbit splitting in the single-neutron spectrum. Namely,

$$\begin{eqnarray}&&{V}_{\mathrm{CB}}(r)=\displaystyle \frac{1}{{M}_{+}}\displaystyle \frac{\kappa (\kappa +1)}{{r}^{2}};\end{eqnarray} \tag{ 15a }$$

$$\begin{eqnarray}&&{V}_{\mathrm{SOP}}(r)=\displaystyle \frac{1}{{M}_{+}^{2}}\displaystyle \frac{{{\rm{d}}{M}}_{+}}{{\rm{d}}{r}}\displaystyle \frac{\kappa }{r}.\end{eqnarray} \tag{ 15b }$$

It is well known that there is no spin–orbit splitting if potential ${V}_{\mathrm{SOP}}$ vanishes, i.e., $\tfrac{{{\rm{d}}{M}}_{+}}{{\rm{d}}{r}}=-\tfrac{{\rm{d}}(V-S)}{{\rm{d}}{r}}=0$ corresponds to the SS limit.

By substituting

$$\begin{eqnarray}&&G(r)=\displaystyle \frac{1}{-M-{\rm{\Sigma }}(r)+\varepsilon }\left(-\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}+\displaystyle \frac{\kappa }{r}\right)F(r),\end{eqnarray} \tag{ 16 }$$

one can obtain the Schrödinger equation for the lower component F(r)

$$\begin{eqnarray}&&\left\{-\displaystyle \frac{1}{{M}_{-}}\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{r}}^{2}}+\displaystyle \frac{1}{{M}_{-}^{2}}\displaystyle \frac{{{\rm{d}}{M}}_{-}}{{\rm{d}}{r}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}\right.\\ &&\quad +\,\left.\left[(-M+{\rm{\Delta }})+\displaystyle \frac{1}{{M}_{-}}\displaystyle \frac{\kappa (\kappa -1)}{{r}^{2}}-\displaystyle \frac{1}{{M}_{-}^{2}}\displaystyle \frac{{{\rm{d}}{M}}_{-}}{{\rm{d}}{r}}\displaystyle \frac{\kappa }{r}\right]\right\}F=\varepsilon F,\end{eqnarray} \tag{ 17 }$$

with the energy-dependent effective mass ${M}_{-}=-M-{\rm{\Sigma }}+\varepsilon$. The term in the above equation that is proportional to $\kappa (\kappa -1)=\tilde{l}(\tilde{l}+1)$ is regarded as PCB and the last term, that is proportional to $\tfrac{{{\rm{d}}{M}}_{-}}{{\rm{d}}{r}}$, corresponds to PSOP, which leads to the substantial pseudospin–orbit splitting. Namely,

$$\begin{eqnarray}&&{V}_{\mathrm{PCB}}(r)=\displaystyle \frac{1}{{M}_{-}}\displaystyle \frac{\kappa (\kappa -1)}{{r}^{2}};\end{eqnarray} \tag{ 18a }$$

$$\begin{eqnarray}&&{V}_{\mathrm{PSOP}}(r)=-\displaystyle \frac{1}{{M}_{-}^{2}}\displaystyle \frac{{{\rm{d}}{M}}_{-}}{{\rm{d}}{r}}\displaystyle \frac{\kappa }{r},\end{eqnarray} \tag{ 18b }$$

and $\tfrac{{{\rm{d}}{M}}_{-}}{{\rm{d}}{r}}=-\tfrac{{\rm{d}}(V+S)}{{\rm{d}}{r}}=0$ corresponds to the PSS limit.

In the present work, (hyper)nuclei ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn, and the Sn isotopes with $0{\rm{\Lambda }}$, $1{\rm{\Lambda }}$, and $2{\rm{\Lambda }}$ ranging from neutron number N = 70 to 118, are taken as examples to study the behavior of SS and PSS in the single-neutron spectrum and the effects of Λ impurity on these symmetries. The Dirac equation (12) in the RMF model is solved in the coordinate space with a box size of $R=20\,\mathrm{fm}$ and a step size of $0.1\,\mathrm{fm}$. For the NN interaction, PK1 parameter set [80] is used. For the ${\rm{\Lambda }}{\rm{N}}$ interaction, the scalar coupling constant ${g}_{{\rm{\Lambda }}\sigma }=0.618{g}_{{\rm{N}}\sigma }$ is adjusted by reproducing the single-Λ binding energy in the $1{s}_{1/2}$ orbit of hypernucleus ${}_{{\rm{\Lambda }}}^{40}$Ca [81], the vector coupling constant ${g}_{{\rm{\Lambda }}\omega }=0.666{g}_{{\rm{N}}\omega }$ is determined from the näive quark model [82], and ${f}_{{\rm{\Lambda }}\omega }=-{g}_{{\rm{\Lambda }}\omega }$ is taken as in [68]. These NN and ${\rm{\Lambda }}{\rm{N}}$ interactions describe well the Λ s.p. spectra of hypernuclei ranging from ${}_{{\rm{\Lambda }}}^{12}{\rm{C}}$ to ${}_{{\rm{\Lambda }}}^{208}$Pb [79].

In figure 1, the calculated single-neutron spectra and the mean field potential in the hypernucleus ${}_{{\rm{\Lambda }}}^{121}$Sn are presented. The left and right panels correspond to spin doublets and pseudospin doublets, respectively. In the left panel, six sets of neutron spin doublets are obtained, i.e., $1p$ ($1{p}_{1/2}$ and $1{p}_{3/2}$), $1d$ ($1{d}_{3/2}$ and $1{d}_{5/2}$), $1f$ ($1{f}_{5/2}$ and $1{f}_{7/2}$), $1g$ ($1{g}_{7/2}$ and $1{g}_{9/2}$), $2p$ ($2{p}_{1/2}$ and $2{p}_{3/2}$), and $2d$ ($2{d}_{3/2}$ and $2{d}_{5/2}$). And the splitting of spin doublets with the same main quantum number n always increases with the angular momentum l. This is because the centrifugal barrier ${V}_{\mathrm{CB}}(r)=\tfrac{1}{{M}_{+}}\tfrac{l(l+1)}{{r}^{2}}$ keeps the particle away from the centre so that a big overlap between the wave function G(r) and the spin–orbit potential always occurs for larger l [83]. In the right hand panel, four sets of pseudospin doublets are obtained, i.e., $1\tilde{p}$ ($1{d}_{3/2}$ and $2{s}_{1/2}$), $1\tilde{d}$ ($1{f}_{5/2}$ and $2{p}_{3/2}$), $1\tilde{f}$ ($1{g}_{7/2}$ and $2{d}_{5/2}$), and $2\tilde{p}$ ($2{d}_{3/2}$ and $3{s}_{1/2}$). An obvious energy dependence of PSS can be seen and PSS for the pseudospin doublets near the threshold is preserved pretty well, e.g., the quasi-degeneracy $2\tilde{p}$ doublets. From those discussions, spin–orbit and pseudospin–orbit splittings in the single-neutron spectra of ${}_{{\rm{\Lambda }}}^{121}$Sn are found to be quite different, i.e., obvious violations of SS for most of the spin doublets versus approximate PSS for the pseudospin doublets. As a typical example, the $2p$ doublets with the smallest spin–orbit splitting is 1.411 MeV while the pseudospin–orbit splitting is only 0.307 MeV for the $2\tilde{p}$ doublets, which is an order of magnitude smaller than the former one. Note that both SS and PSS of the single-neutron spectra in Λ hypernuclei are similar to those found in ordinary nuclei.

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Next, the effects of Λ impurity on SS and PSS in the single-neutron spectrum are discussed. In figure 2, taking ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn as examples, the spin–orbit splittings defined by ${\rm{\Delta }}{E}_{\mathrm{SO}}=({\varepsilon }_{{j}_{\lt }}-{\varepsilon }_{{j}_{\gt }})/(2l+1)$, and the pseudospin–orbit splittings ${\rm{\Delta }}{E}_{\mathrm{PSO}}=({\varepsilon }_{{j}_{\lt }}-{\varepsilon }_{{j}_{\gt }})/(2\tilde{l}+1)$, versus the average s.p. energies ${E}_{\mathrm{av}}=({\varepsilon }_{{j}_{\lt }}+{\varepsilon }_{{j}_{\gt }})/2$ are plotted in the upper and lower panels, respectively. For SS, as the number of Λ hyperons is increased, the energy splittings between all the spin doublets become larger. Namely, adding more Λ hyperons makes SS in the single-neutron spectrum worse. However, a complete reversal of the trend is observed for the pseudospin doublets in the lower panel. Thus with the addition of the number of Λ hyperons, ${\rm{\Delta }}{E}_{\mathrm{PSO}}$ decreases and PSS is improved in the single-neutron spectrum.

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However, it should be noted that the pseudospin–obit splitting is not bound to decrease if the spin–obit splitting increases. For example, if there is an inversion of the s.p. levels of the pseudospin doublets and/or the s.p. energy of both doublets goes in the same direction but with different magnitudes, the pseudospin–obit splitting could also increase.

In table 1, the values of the energy splittings ${\rm{\Delta }}{E}_{\mathrm{SO}}$ for spin doublets, and ${\rm{\Delta }}{E}_{\mathrm{PSO}}$ for pseudospin doublets shown in figure 2 are listed for (hyper)nuclei ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn. From table 1, we can see clearly that adding more Λ hyperons has opposite effects on SS and PSS. Quantitatively, adding one Λ hyperon enlarges the splittings of spin doublets by $0.01\sim 0.03\,\mathrm{MeV}$ while reduce the splittings of pseudospin doublets by $0.02\sim 0.05\,\mathrm{MeV}$.

Table 1.  Comparison of the spin–orbit splitting ${\rm{\Delta }}{E}_{\mathrm{SO}}$ and pseudospin-orbit splitting ${\rm{\Delta }}{E}_{\mathrm{PSO}}$ for the single-neutron spectra of (hyper)nuclei ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn calculated in the RMF model. Energies are in MeV.

$\mathrm{SS}$	${}^{120}\mathrm{Sn}$	${}_{{\rm{\Lambda }}}^{121}\mathrm{Sn}$	${}_{2{\rm{\Lambda }}}^{122}\mathrm{Sn}$	$\mathrm{PSS}$	${}^{120}\mathrm{Sn}$	${}_{{\rm{\Lambda }}}^{121}\mathrm{Sn}$	${}_{2{\rm{\Lambda }}}^{122}\mathrm{Sn}$
$1p$	0.430	0.460	0.492	$1\tilde{p}$	1.084	1.034	0.982
$1d$	0.571	0.594	0.618	$1\tilde{d}$	0.633	0.608	0.582
$1f$	0.668	0.685	0.703	$1\tilde{f}$	0.350	0.331	0.311
$2p$	0.441	0.470	0.500	$2\tilde{p}$	0.121	0.102	0.082
$1g$	0.704	0.716	0.728
$2d$	0.364	0.381	0.400

To analyze the mechanism of the effects of Λ impurity on SS and PSS shown in figure 2 and listed in table 1, we take the case of pseudopsin doublets as an example and compare the pseudo-centrifugal barrier ${V}_{\mathrm{PCB}}(r)$ and pseudospin–orbit potential ${V}_{\mathrm{PSOP}}(r)$. It was pointed out in [14] that if

$$\begin{eqnarray}&&| {V}_{\mathrm{PSOP}}(r)| \ll | {V}_{\mathrm{PCB}}(r)| ,\end{eqnarray} \tag{ 19 }$$

the pseudospin approximation will be good. Unfortunately, it is difficult to compare them, as both of them have a singularity at $r={r}_{0}$ where ${M}_{-}(r)=0$. Since we are only interested in the relative magnitude of the PCB and the PSOP in equation (18), following [14], the effective PCB and the effective PSOP,

$$\begin{eqnarray}&&{V}_{\mathrm{PCB}}^{\mathrm{eff}}(r)={M}_{-}(r)\displaystyle \frac{\kappa (\kappa -1)}{{r}^{2}},\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}&&{V}_{\mathrm{PSOP}}^{\mathrm{eff}}(r)=-\displaystyle \frac{{{\rm{d}}{M}}_{-}(r)}{{\rm{d}}{r}}\displaystyle \frac{\kappa }{r}\end{eqnarray} \tag{ 20b }$$

are introduced for comparison by multiplying by a common factor ${M}_{-}^{2}(r)$, and in order to examine the effect of the Λ impurity more carefully, we compare the effective PCB and effective PSOP multiplied by the squares of the lower component wave function F(r). In figure 3, the comparison of the PCB (solid lines) and PSOP (dashed lines) for the $2\tilde{p}$ pseudospin doublets in (hyper)nuclei ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn are presented in arbitrary scales. The upper and lower panels are for the partner states $2{d}_{3/2}$ and $3{s}_{1/2}$ of pseudospin doublet $2\tilde{p}$, respectively.

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From figure 3, it is clearly seen that the effective PCB is much bigger than the effective PSOP in the coordinate space with $r\lt 4\,$ fm, and generally they differ by two orders of magnitude. In a semi-quantitative sense, this indicates that the condition in equation (19) is satisfied and PSS is preserved approximately. Moreover, with adding more Λ hyperons, the effective PCB obviously becomes larger while the effective PSOP remains almost constant, which results in better PSS as the Λ hyperon number increases. For the spin doublets, similar analysis is performed. We found that adding more Λ hyperons will obviously reduce the effective CB while keeping the effective SOP almost constant, which results in a worse SS.

Finally, to see the universality of the effects of Λ impurity on SS and PSS in the single-neutron spectrum obtained by studying ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn, we repeat our analysis for other Sn isotopes with $0{\rm{\Lambda }}$, $1{\rm{\Lambda }}$ and $2{\rm{\Lambda }}$. In figure 4, we take four spin doublets, i.e., $1p$, $1d$, $2p$, and $1f$, as examples and plot the spin–orbit splittings ${\rm{\Delta }}{E}_{\mathrm{SO}}$. From all panels in figure 4, we observe that SS becomes worse with the increasing Λ hyperon number for all the spin doublets. Besides, in all the isotopes, the splitting ${\rm{\Delta }}{E}_{\mathrm{SO}}$ of the spin doublets with the same main quantum number n increases with angular momentum l, which is consistent with the conclusion obtained in figure 1. Similarly, in figure 5 the pseudospin–orbit splittings ${\rm{\Delta }}{E}_{\mathrm{PSO}}$ are depicted for the pseudospin doublets $1\tilde{p}$, $1\tilde{d}$, $2\tilde{p}$, and $1\tilde{f}$ in the Sn isotopes. We observe that PSS becomes better with the increasing Λ hyperon number for all the pseudospin doublets. These observations indicate that the effects of Λ impurity on SS and PSS are universal.

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As seen in figure 4, the spin–orbit splitting ${\rm{\Delta }}{E}_{\mathrm{SO}}$ generally decreases with the increasing nucleon number. As indicated in [83], this mainly comes from the larger and larger surface diffuseness of the potentials or the outwards tendency of the potentials with the increasing neutron number. In figure 5, the energy splittings of deep neutron pseudospin doublets, e.g., $1\tilde{p}$ doublet, generally decrease with the increasing neutron number, but increase in some mass ranges. As discussed in [84, 85], the splittings depend on the complex competition of the radius, depth and diffuseness of the potentials, which have very different influences, i.e., a larger radius and deeper potential will increase the pseudospin splitting while a larger diffuseness make the splitting smaller. However, the splittings of pseudospin doublets near the threshold, e.g., $2\tilde{p}$ and $1\tilde{f}$ doublets, always remain at a small magnitude. One possible reason is the larger pseudo-centrifugal barrier for those orbits, as discussed in [14].

We have studied SS and PSS in the single-neutron spectrum of Λ hypernuclei, and the effects of Λ impurity on the these symmetries, within the RMF framework.

Firstly, in hypernucleus ${}_{{\rm{\Lambda }}}^{121}$Sn, the SS and PSS of the single-neutron spectra are studied. Obvious violations of SS and approximate PSS in the single-neutron spectrum are found. For the spin doublets with the same main quantum number n, the spin–orbit splittings always increase with the angular momentum l. For the pseudospin splittings, there is an obvious energy dependence and the splittings for the pseudospin doublets near the threshold are very small. All these conclusions are consistent with those in ordinary nuclei.

Secondly, taking (hyper)nuclei ¹²⁰Sn, ${}_{{\rm{\Lambda }}}^{121}$Sn, and ${}_{2{\rm{\Lambda }}}^{122}$Sn as examples, the effects of Λ impurity on SS and PSS in the single-neutron spectrum have been investigated. Adding more Λ hyperons makes SS worse while improving PSS. In more detail, for the pseudospin doublets, the Λ hyperons increase the effective pseudo-centrifugal barrier ${V}_{\mathrm{PCB}}(r)$ while changing the effective pseudospin–orbit potential ${V}_{\mathrm{PSOP}}(r)$ slightly, thus, in this case, PSS condition $| {V}_{\mathrm{PSOP}}(r)| \ll | {V}_{\mathrm{PCB}}(r)|$ is better satisfied. For the spin doublets, the Λ hyperons have the opposite effect, i.e., Λ hyperons decrease the centrifugal barrier ${V}_{\mathrm{CB}}(r)$.

Finally, we conclude that the extension of our investigation to Sn isotopic chains with $0{\rm{\Lambda }}$, $1{\rm{\Lambda }}$, and $2{\rm{\Lambda }}$ indicates that the effects of Λ impurity on SS and PSS in the single-neutron spectrum are universal.

The authors thank Dr H Z Liang for helpful discussions. This work was partly supported by the Physics Research and Development Program of Zhengzhou University (grant no. 32410017) and the National Natural Science Foundation of China (grant nos. 11175002, 11335002, 11505157, 11675148, and 11105042).