Influence from different fields of mesons on the pseudospin symmetry in the single-neutron resonant states

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Hua-Ming Dai, Min Shi, Shou-Wan Chen and Quan Liu. Influence from different fields of mesons on the pseudospin symmetry in the single-neutron resonant states[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac23d4
Hua-Ming Dai, Min Shi, Shou-Wan Chen and Quan Liu. Influence from different fields of mesons on the pseudospin symmetry in the single-neutron resonant states[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac23d4 shu
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Influence from different fields of mesons on the pseudospin symmetry in the single-neutron resonant states

    Corresponding author: Quan Liu, quanliu@ahu.edu.cn
  • 1. School of physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
  • 2. School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Abstract: In the framework of the relativistic mean field theory combined with the complex momentum representation method, we study in detail the pseudospin symmetry in the single-neutron resonant states and its dependence on $\sigma$, $\omega$ and $\rho$ meson fields. Compared with the effect of the $\rho$ field, the $\sigma$ and $\omega$ fields gives the main contribution to the pseudospin energy and width splitting of the resonant pseudospin doublets. Especially, we compare quantitatively the pseudospin wave functions splittings in resonant doublets, and investigate their dependencies on different fields of mesons, which is consistent with that of energy and width splittings. The present researches are helpful to understand the mechanism and properties of pseudospin symmetry for resonant states.

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    I.   INTRODUCTION
    • Pseudospin symmetry (PSS) is an important discovery in nuclear physics in 1969, which is physically related to the shell model. In 1949, the spin-orbit potential was introduced by Mayer and Jensen [1, 2], respectively, which led to the correct reproduction of magic numbers and the establish of the famous nuclear shell model. Twenty years later, a quasi-degeneracy was observed in heavy nuclei between single-nucleon partners with quantum numbers $ (n,l,j = l+1/2) $ and $ (n-1, l+2, $$ j = l +3/2) $ where $ n,l,j $ are the radial, the orbital and the total angular momentum quantum numbers, respectively [3, 4]. The quasi-degenerate partners were suggested as the pseudospin partners $ (\tilde n = n,\tilde l = l+1,j = \tilde l\pm1/2) $, which has been discussed in a number of phenomena in nuclear structure, including nuclear superdeformed configurations [5], identical bands [6], quantized alignment [7], pseudospin partner bands [8], magnetic moments and transitions [9] and γ-vibrational states in nuclei [10], as well as in nucleon-nucleus and nucleonnucleon scatterings [11]. Moreover, the role of PSS in the structure of halo nuclei [12] and superheavy nuclei [13] was studied in detail.

      The discovery of PSS has attracted the attention of physicists, but its origin and destroying mechanism have not been clarified. Based on the shell model of harmonic oscillator potential, Bahri et al. [14] pointed out that the special ratio between the intensity of the spin-orbit interaction and the intensity of the orbit-orbit interaction is the reason for PSS, and this ratio can be explained by the relativistic mean field (RMF) theory [15]. Blokhin et al. [16] showed that the normal state can be transformed into the pseudospin state ($ \tilde l $, $ \tilde s $) by a spiral unitary transformation. Based on the RMF theory, in 1997, Ginocchio [17] presented that PSS was a relativistic symmetry by solving the Dirac equation describing a spherical nucleus, and the condition of strict PSS is that the sum of scalar potential S and vector potential V in the Dirac equation is zero, and the pseudoorbital angular momentum $ \tilde l $ is the orbital angular momentum in the spin of Dirac wave function. Subsequently, Meng et al. [18] proposed a more general strict symmetry condition, i.e. $ \Sigma = S + V = $$ constant $, and indicated that the PSS in the real nucleus is related to the pseudocentrifugal barrier (PCB) and pseudospin-orbital coupling potential (PSOP). Since then, some progress has been made on the PSS in various systems, including extensions of the PSS study from stable to exotic nuclei [19], from bound to resonant states [20], from nucleon to anti-nucleon spectra [21], from nucleon to hyperon spectra [22], and from spherical to deformed nuclei [23, 24]. More detailed progress in PSS can be found in the review literatures [25, 26].

      In recent years, there has been an increasing interest in the exploration of continuum and resonant states, especially in the studies of the exotic nuclei with unusual N/Z ratios. Considering that the exotic nuclei are weakly bound and their Fermi surfaces are very close to the continuum threshold, thus, the continuum, especially the resonances in the continuum play a key role in the formation of these exotic phenomena [27, 28]. For this reason, many methods were developed for nuclear single particle resonances, which include scattering phase shift method [29], Jost function method [30, 31], Green function method [32-35], the analytic continuation in the coupling constant (ACCC) method [37-39], real stabilization method [40, 41] and complex scaling method [42, 43].

      Based on these methods, the PSS in single-particle resonant states have been studied. In Ref. [20], the PSS for the resonant states in $ ^{208} $Pb is investigated by solving the Dirac equation with Woods-Saxon vector and scalar potentials in combination with the ACCC method. In Ref. [44], RMF theory is combined with the ACCC method to determine the energies and widths of single particle resonant states in the Sn isotopes, and an isospin dependence of PSS is clearly shown in the resonant states. In Ref. [45], the PSS in the single proton resonant states in $ ^{120} $Sn was discussed by examining the energies, widths and the wavefunctions. In 2012, Lu et al. gave a rigorous verification of the PSS in single particle resonant states [30]. In 2013, we applied the complex scaling method to study the resonances and PSS in the Dirac-Morse potential [46]. In 2019, Sun et al. investigated the spin and pseudospin symmetry in the single-particle resonant states by solving the Dirac equation containing a Woods-Saxon potential with Green's function method [47].

      Based on these theories, lots of excellent works have flourished. And the complex momentum representation (CMR) is also an effective method to study the PSS in resonant states. Considering that the RMF is one of the most successful microscopic theoretical models which has been widely used to study the weakly bound nuclei [15, 26, 27, 48-50], in 2016, we have proposed a new scheme [51] to explore the resonances in the RMF framework, where the Dirac equation is solved directly in the complex momentum representation, and the bound and resonant states are dealt with on an equal footing. By using the newly developed method, we obtain not only narrow resonance but also wide resonance that are difficult to obtain before, and the physical mechanism of halo structure in some exotic nuclei [52-55] is proposed. Recently, based on the RMF-CMR method, we investigated the PSS in the single-particle resonant states and its isospin dependence [56].

      In 2010, within the framework of the RMF theory [57], we obtained a detailed knowledge on the contribution from different field of mesons to pseudospin energy splitting in the single-particle bound state.However, for the resonant states, the influence from different fields of mesons on PSS has not been examined in detail for real nuclei. By using the RMF-CMR method [51], we can extract the splitting of the energies, widths and wavefunctions between the single-neutron resonant pseudospin partners and check the contributions from the different fields of mesons to the pseudospin splitting, which is our original intention of writing this article. In Sec. II, we will give the theoretical framework. Detailed calculation results and data analysis will be shown in Sec. III. Finally, a summary is given in Sec. IV.

    II.   FORMALISM
    • In order to study the resonant partners in $ ^{208} $Pb, the formalism of RMF-CMR is introduced as follows. The basic assumption of the RMF theory is that the Lagrangian density of this system is obtained through the exchange and interaction of various mesons and photons

      $ \begin{aligned}[b] \mathcal{L} =& \bar{\psi}\left(i \gamma_{\mu} \partial^{\mu}-M\right) \psi+\frac{1}{2} \partial_{\mu} \sigma \partial^{\mu} \sigma-U(\sigma)-\frac{1}{4} \Omega_{\mu \nu} \Omega^{\mu v} \\& +\frac{1}{2} m_{\omega}^{2} \omega_{\mu} \omega^{\mu}-\frac{1}{4} \vec{R}_{\mu v} \vec{R}^{\mu v}+\frac{1}{2} m_{\rho}^{2} \vec{\rho}_{\mu} \vec{\rho}^{\mu}-\frac{1}{4} F_{\mu v} F^{\mu v} \\ &-\bar{\psi}\left(g_{\sigma} \sigma+g_{\omega} \gamma_{\mu} \omega^{\mu}+g_{\rho} \gamma_{\mu} \vec{\tau} \vec{\rho}^{\mu}+e \gamma_{\mu} A^{\mu}\right) \psi, \end{aligned} $

      (1)

      where M is the nuclear mass, $ {m_\sigma }({g_\sigma }),{m_\omega }({g_\omega }),{m_\rho }({g_\rho }) $ are the mass and coupling constants of different mesons, respectively. And the non-linear couplings of $ \sigma $ read $ U(\sigma ) = \dfrac{1}{2}m_\sigma ^2{\sigma ^2} + \dfrac{1}{3}{g_2}{\sigma ^3} + \dfrac{1}{4}{g_3}{\sigma ^4} $. Based on Lagrange density, Dirac equation can be obtained as follows

      $ [\vec{\alpha} \cdot \vec{p}+\beta(M+S)+V] \psi_{i} = \varepsilon_{i} \psi_{i}, $

      (2)

      where S is scalar potential, V is vector potential. The expression is as follows

      $ \left\{ \begin{array}{l} S(\vec{r}) = g_{\sigma} \sigma(\vec{r}) ,\\ V(\vec{r}) = g_{\omega} \omega^{0}(\vec{r})+g_{\rho} \tau_{3} \rho^{0}(\vec{r})+e A^{0}(\vec{r}). \end{array} \right. $

      (3)

      For convenience, here we express $ {g_\sigma }\sigma (\vec r) $, $ {g_\omega }{\omega ^0}(\vec r) $, $ {g_\rho }{\tau _3}{\rho ^0}(\vec r) $ as $ {V_\sigma }(\vec r) $, $ {V_\omega }(\vec r) $ and $ {V_\rho }(\vec r) $.

      In order to obtain the resonant states, we turn Eq. (2) to momentum space, and we can obtain the bound states, resonant states and continuous spectrum simultaneously

      $ \int d \vec k'\langle \vec k|H|\vec k'\rangle \psi (\vec k') = E\psi (\vec k), $

      (4)

      where $ H = \vec{\alpha} \cdot \vec{p}+\beta(M+S(\vec{r}))+V(\vec{r}) $. In the case of spherical nuclei, the momentum wave function can be expressed by upper and lower components

      $ \psi(\vec{k}) = \left(\begin{array}{l} f(k) \phi_{l j m_{j}}\left(\Omega_{k}\right) \\ g(k) \phi_{\tilde{l} j m_{j}}\left(\Omega_{k}\right) \end{array}\right). $

      (5)

      Separate the angular part and bring the radial part into Eq. (2), so that Dirac equation can be expressed as

      $ \left\{ \begin{array}{l} M f(k)-k g(k)+\int k^{\prime 2} d k^{\prime} V_+\left(k, k^{\prime}\right) f\left(k^{\prime}\right) = \varepsilon f(k) ,\\ -k f(k)-M g(k)+\int k^{\prime 2} d k^{\prime} V_-\left(k, k^{\prime}\right) g\left(k^{\prime}\right) = \varepsilon g(k) ,\end{array} \right.$

      (6)

      with

      $ V_+\left(k, k^{\prime}\right) = \frac{2}{\pi} \int r^{2} d r \Sigma(r) j_{l}\left(k^{\prime} r\right) j_{l}(k r), $

      (7)

      $ V_-\left(k, k^{\prime}\right) = \frac{2}{\pi} \int r^{2} d r \Delta(r) j_{\tilde l}\left(k^{\prime} r\right) j_{\tilde l}(k r), $

      (8)

      where $ {\Sigma}(r) = V(r)+S(r) $, $ {\Delta} (r) = V(r)-S(r) $.

      The bound states and resonant states can be obtained by solving the above equations.Finally, according to the following formula

      $ \left\{ \begin{array}{l} f(r) = i^{l} \sqrt{\dfrac{2}{\pi}} \int k^{2} d k j_{l}(k r) f(k) ,\\ g(r) = i^{\tilde l} \sqrt{\dfrac{2}{\pi}} \int k^{2} d k j_{\tilde l}(k r) g(k). \end{array} \right. $

      (9)

      The upper and lower component wave functions can be transformed into coordinate space for processing, and the detailed of solution process can see in Ref. [51].

    III.   NUMERICAL DETAILS AND RESULTS
    • Based on the formalism presented above, we explore the single-neutron resonant states and their pseudospin for $ ^{208} $Pb. The interactions are adopted to be the NL3 parameters [58]. In the calculations, the single-neutron energies and wavefunctions for bound states and resonances can be obtained. For resonances, we can also calculate their widths simultaneously. The single neutron spectrum in $ ^{208} $Pb in form of pseudospin partners according to pseudoangular momentum $ {\tilde l} $ are displayed in Fig. 1 with the correspinding $ \Sigma(r) $ potential. In the single-neutron resonant states, three pairs of pseudospin partners have been found, and they are $ 2{\tilde g} $, $ 1{\tilde i} $, $ 1{\tilde j} $, respectively. In Fig. 1, it can be seen that for the the pseudospin partners with pseudoangular momentum $ {\tilde l}>0 $, there is always a state without a pseudospin partner, and these separate states are called intruder states. The appearance of intruder states can be explained by examining the zeros of Jost functions [31]. In addition, these intruder states can also be explained by the method of supersymmetric quantum mechanics [59-62]. We also notice that, pseudospin partners have obvious threshold effect, for example, pseudospin partners $ 3{\tilde p} $, $ 2{\tilde f} $, $ 2{\tilde g} $, $ 1{\tilde h} $ and $ 1{\tilde i} $ are very close to the continuum threshold and the energy splitting between them are less than 1 MeV. The pseudospin partners near the zero potential energy surface tend to show better symmetry. The study of symmetry in this area becomes more interesting.

      Figure 1.  (Color online) Pseudospin partners of the single-neutron spectra in $ ^{208} $Pb with RMF-CMR method. The line is mean-field potential $ \Sigma(r) $ for neutrons.

      When $ d{\Sigma}/dr = 0 $, the system presents exact pseudospin symmetry [18]. Unfortunately, the condition is not satisfied in realistic system. For the $ ^{208} $Pb studied in this paper, from the above discussion according to Fig. 1, we have seen that the exact pseudospin symmetry can not be achieved. Pseudospin splitting exists either large or small, but it can not be zero. Hence, to explore the pseudospin splittiing is helpful to understand the character of this system. In order to check the pseudospin splitting induced by the different fields of mesons, we set $ {V_X}(\vec r) \to {\lambda _X}V(\vec r) $ to investigate the dependencies of energy, width and wavefunction splitting of resonant pseudospin partners on the coupling constant $ \lambda _X $, where X denotes $ \sigma $, $ \omega $ and $ \rho $, respectively. We perform the RMF-CMR calculations to obtain the variation of $ \Sigma(r) $ potential under different meson fields, which is displayed in Fig. 2. It can be seen that the $ \Sigma(r) $ potential of subfigures (a)(b) changes obviously, while the $ \Sigma(r) $ potential of subfigures (c)(d) has a small range of variation with coefficient. In the Fig. 2(a), with the increase of $ \lambda _{\sigma} $ from 0.96 to 1.04, the depth of the potential decrease monotonically and the surface of the potential moves outwards. From Fig. 2(b), we can see that the variation of $ \Sigma(r) $ potential with the $ \lambda _{\omega} $ is completely opposite to that in Fig. 2(a), that is, with the increase of $ \lambda _{\omega} $, the depth of the potential increase monotonically and the surface of the potential moves inwards. To further explore the competitive relationship between $ {\sigma} $- field and $ {\omega} $- field, we fix the coefficient of $ {\rho} $ field, and uniform changes the strength of $ {\sigma} $ and $ {\omega} $ fields. We can clearly see that the change of $ \Sigma(r) $ potential is very small in Fig. 2(c), which confirmed that the effect of $ {\sigma} $-field and $ {\omega} $-field on potential is opposite, and the degree is almost identical. Considering that the $ \Sigma(r) $ potential does not change obviously with the $ \lambda _{\rho} $, we extend the range of $ \lambda _{\rho} $ from 0.6 to 1.4. Even so, we can see from Fig. 2(d) that the change of $ \Sigma(r) $ is not obvious, and the $ \Sigma(r) $ gradually becomes slightly shallow and the surface of the potential moves a little inwards, which indicates that $ {\rho} $-meson field has little influence on $ \Sigma(r) $ potential.

      Figure 2.  (Color online) The change of neutron $ \Sigma(r) $ potential when the intensity of the meson field is changed under different meson fields.

      To demonstrate the evolution of PSS in resonant states, we analyze the dependence of the pseudospin splittings on the strength of the fields of mesons. Keeping $ \lambda _{\omega} $ and $ \lambda _{\rho} $ fixed at 1.0, we vary $ \lambda _{\sigma} $ in order to see how the energies and widths of the pseudospin partners are sensitive to the strength of the $ {\sigma} $-field. This dependence is shown in Fig. 3, where the open and filled marks represent the resonant and the bound partners, respectively. With the increase of $ \lambda _{\sigma} $, the $ {\Sigma} $ potential well becomes deeper, and some single-neutron resonant states evolve into bound states. Since the relationship between the pseudospin energy splitting of the bound partners and the strength of the meson field has been discussed in detail in Ref. [57], the variation trend of the resonant pseudospin splittings with the strength of the meson fields is the focus of this paper. It can be seen from Fig. 3 that, for a given $ \lambda _{\sigma} $, the energy and width splitting between the different pseudospin partners is different for different pseudospin partners with no exception. It shows that the PSS is correlated with the quantum numbers of single-neutron states and preserves a dynamic character as reported in Refs. [20, 44, 63, 64] for bound states and resonances. From Fig. 3(a), for all the pseudospin partners $ (2g_{9/2},1i_{11/2}) $, $ (2h_{11/2},1j_{13/2}) $, $ (2i_{13/2},1k_{15/2}) $, $ (3d_{5/2},2g_{7/2}) $ and $ (3f_{7/2},2h_{9/2}) $, the tendency for the change of pseudospin energy splittings with $ \lambda _{\sigma} $ is in agreement with the case of bound states [57]. As $ \lambda _{\sigma} $ increases, the energy splittings of all the resonant pseudospin partners decrease monotonously. The same trend is found in Fig. 3(b) for width variation with increasing $ \lambda _{\sigma} $. For example, the energy splitting of $ (2g_{9/2},1i_{11/2}) $ at $ {\lambda _\sigma } = 0.9 $ is 6 times that of $ {\lambda _\sigma } = 0.98 $, and the width splitting of $ (2g_{9/2},1i_{11/2}) $ at $ {\lambda _\sigma } = 0.9 $ is 10 times that of $ {\lambda _\sigma } = 0.96 $.

      Figure 3.  (Color online) The variation of energy splitting and width splitting of pseudospin partners with $ {\sigma} $ meson field intensity in $ ^{208} $Pb. The open and filled marks represent the resonant and the bound partners, respectively

      In addition, the evolution of splitting is almost identical for the different resonant pesudospin partners, and doesn't present a connection with the quantum numbers of partners, which is different from that of the bound pesudospin partners [57]. By comparing the variation of splitting with $ {\lambda _\sigma } $ for the partners with fixed $ {\tilde n} $, the sensitivity of energy and width splitting to $ {\lambda _\sigma } $ will be disclosed. For example, the energy and width splittings between the $ (2h_{11/2},1j_{13/2}) $ partner is almost same as that between the partner $ (2i_{13/2},1k_{15/2}) $ with the coupling constant $ {\lambda _\sigma } $ increasing from 0.98 to 1.02. In Ref. [57], the dependence of level inversion on $ {\lambda _\sigma } $ is seen for several bound partners, but such a situation does not happen in the resonant partners. The occurrence of this phenomenon is related to pseudocentrifugal barrier and the pseudospinorbital potential [18]. The pseudospin partners in resonance state near the Fermi surface, which have better symmetry. The splitting between different pseudospin partners becomes smaller. And, with increasing $ {\lambda _\sigma } $, for all the resonant partners, $ \Delta E = {E_{n,l,j = l + 1/2}} - {E_{n - 1,l + 2,j = l + 3/2}} $ is always less than 0 and $ \Delta \Gamma = {\Gamma _{n,l,j = l + 1/2}}-{\Gamma _{n - 1,l + 2,j = l + 3/2}} $ remains positive over the range of $ {\lambda_\sigma} $ considered here. All these indicate that the $ {\sigma} $ meson field plays an important role in influencing the PSS of resonant partners. As far as this conclusion is concerned, it is similar to that in bound partners.

      A similar trend is seen when we let the $ {\lambda _\omega } $ vary, fixing all other coupling constants. The results is shown in Fig. 4. From Fig. 4(a), the tendency for the change of pseudospin splittings with $ {\lambda _\omega } $ is consistent with the case of bound state [57]. When the $ {\lambda _\omega } $ increase, the energy splitting dramatically increase for all the resonant partners . The same trend is found in Fig. 4(b) for width variation except for $ (2h_{11/2},1j_{13/2}) $ with the $ {\lambda _\sigma } $ increasing from 1.08 to 1.1. It can be found that the energy splitting of $ (2h_{11/2},1j_{13/2}) $ at $ {\lambda _\omega } = 1.1 $ is 31 times that of $ {\lambda _\omega } = 0.96 $, and the width splitting of $ (2h_{11/2},1j_{13/2}) $ partners at $ {\lambda _\omega } = 1.1 $ is 31 times that of $ {\lambda _\omega } = 0.96 $. Compared with the $ {\sigma} $-field, the splitting develops towards an opposite direction, but the sensitivity of the splitting to the coupling constant is similar, which shows that the $ {\sigma} $ and $ \omega $ fields have an opposite contribution of the PSS. These differences can be understood by the shape of the $ \Sigma(r) $ potential given in Figs. 1(a) and 1(b). Further, the inversion of energy splitting can be seen for the partner $ (2i_{13/2},1k_{15/2}) $. With increasing $ {\lambda _\omega } $, the pesudospin energy splitting varies from $ {E_{n,l,j = l + 1/2}} > {E_{n - 1,l + 2,j = l + 3/2}} $ to $ {E_{n,l,j = l + 1/2}} < $$ {E_{n - 1,l + 2,j = l + 3/2}} $, which similar as Ref. [65]. All these indicated that the $ {\omega} $-field also plays a significant role in influencing the PSS for resonant partners.

      Figure 4.  (Color online) The variation of energy splitting and width splitting of pseudospin partners with $ {\omega} $ meson field intensity in $ ^{208} $Pb. The open and filled marks represent the resonant and the bound partners, respectively

      Actually, from Fig. 3 and Fig. 4, we have found that $ {\lambda _\sigma } $ and $ {\lambda _\omega } $ are not completely independent parameters. Varying equally the strength of $ {\sigma} $ and $ {\omega} $ fields, the variations of pseudospin energy and width splittings are very small, as shown in Fig. 5. The results show that the $ {\sigma} $-field contribution to the pseudospin energy and width splitting has nearly the same magnitude as the one obtained by the $ {\omega} $-field, but opposite sign. This can be clarified from the change of the shape of the $ \Sigma(r) $ potential given in Fig. 1(c).

      Figure 5.  (Color online) The variation of energy splitting and width splitting of pseudospin partners with ${\sigma}$ meson field intensity in $^{208}$Pb. The open and filled marks represent the resonant and the bound partners, respectively

      Finally, we keep $ {\lambda _\sigma } $ and $ {\lambda _\omega } $ fixed at 1 but vary $ \lambda_\rho $ in order to study the sensitiveness of the resonant pseudospin partners with the strength of the $ \rho $-field. The results are presented in Fig. 6 and the following behavior is observed: as $ \lambda_\rho $ increases, the energy and width splitting increases slowly. Such characteristics are consistent with the case of bound state [57], which can be explained by Fig. 2(d). It is noted that the the coupling constant $ {\lambda _\rho} $ in Fig. 6 varies from 0.0 to 2.0, that is, compared with $ {\lambda _\sigma } $ and $ {\lambda _\omega } $, the value range of $ {\lambda _\rho} $ has been expanded. In contrast with the $ {\sigma} $ and $ {\omega} $ fields, the energy splitting is less sensitive to the $ \rho $-field. For example, from Fig. 6(a), the energy splitting of $ (3f_{7/2}, 2h_{9/2}) $ partner is only reduced by 1.8 MeV from $ {\lambda _\rho } = 0.0 $ to $ {\lambda _\rho } = 2.0 $. In Fig. 6 (b), with the $ \lambda_\rho $ increasing, the width splitting also increases slowly and its variation range is small, which is similar to the evolution of energy splitting. The increase of width splittings in $ (2h_{11/2},1j_{13/2}) $ partner and $ (3d_{5/2},2g_{7/2}) $ partner is less than 1 MeV with the $ \lambda _\rho $ changing from 0 to 2.0. For the resonant partners $ (3f_{7/2},2h_{9/2}) $ and $ (2i_{13/2},1k_{15/2}) $, the width splitting increase slightly, but the increasing scope is far smaller than that by the $ {\sigma} $ and $ \omega $ fields. These result indicate that the $ \rho $-field only provides a minor contribution to the PSS, and the PSS comes mainly from the cancellation of the $ {\sigma} $ and $ {\omega} $ fields.

      Figure 6.  (Color online) The variation of energy splitting and width splitting of pseudospin partners with $ {\rho} $ meson field intensity in $ ^{208} $Pb. The open and filled marks represent the resonant and the bound partners, respectively

      So far, the effect has not been examined on the wavefunction. As the PSS is a relativistic symmetry, the wave functions of the pseudospin partners satisfy certain relations. According to the PSS limit [17], the lower component wave functions of the pseudospin partners are completely consistent. In real nuclei, the lower component wave functions of pseudospin partners are very similar, which have been tested for the bound partners in both spherical and deformed nuclei [23, 66-69]. There has been some discussions about the pseudospin symmetry of wave function for resonant partners [45, 47, 56]. But these discussions on the PSS are only based on intuitive judgments of wave function similarity in the coordinate space in a finite range. It cannot give quantitative results on the degree of wave function similarity since the resonant states are non-local and their wave functions are divergent in the coordinate space. Hence, to compare the similarity of wave functions in the coordinate space is rough. Just like the real part about wave functions of $ (2h_{11/2}, 1j_{13/2}) $ in Fig. 7. With the increase of radius, the resonant wave functions will be diffuse. Whether the wave functions for resonant partners have similar features at large radius becomes confusing.

      Figure 7.  (Color online) Real parts of the lower components of Dirac spinors for pseudospin partner $(2h_{11/2}, 1j_{13/2})$ in the coordinate space.

      But for better understanding of the condition of the PSS, we should check the influences from different fields of mesons. According to the Heisenberg's uncertainty relation, the wave functions of resonant states in the momentum space are local and square integrable. It is appropriate to compare the pseudospin similarity of wave functions in the momentum representation. Especially, it is convenient to obtain quantitatively the pseudospin splittings of wave functions in the momentum space. The real parts of the lower components of wave functions for resonant pseudospin partners $ (2h_{11/2}, 1j_{13/2}) $ with the different fields of mesons are displayed in Figs. 8-11. The splitting of wave functions between the partners is defined as

      Figure 8.  (Color online) Real parts of the lower components of Dirac spinors for pseudospin partner $ (2h_{11/2}, 1j_{13/2}) $ in the momentum space with several different parameters of $ {\sigma} $ meson field.

      Figure 9.  (Color online) The same as Fig. 8, but for ${\omega}$ meson field.

      Figure 10.  (Color online) The same as Fig. 8, but uniform strength of ${\sigma}$ meson field and ${\omega}$ meson field.

      Figure 11.  (Color online) The same as Fig. 8, but for $ {\rho} $ meson field

      $ \Delta G = \int_{0}^{\infty}\left|k^{2}\left[g_{a}(k)-g_{b}(k)\right]^{2}\right| dk, $

      (10)

      where $ g_{a} $(k) and $ g_{b} $(k) are the lower components of Dirac spinors in the momentum space.

      Real parts of the lower components of Dirac spinors for the resonant pseudospin partner $ (2h_{11/2}, 1j_{13/2}) $ with several different values of $ {\lambda _\sigma } $ are plotted in Fig. 8. Similar to the wave functions in coordinate space, the wave function of $ 1j_{13/2} $ is below the $ 2h_{11/2} $ in the momentum space. Moreover, with the increase of k, the wave function gradually moves away from the origin. When k increases to $ 2.5fm^{-1} $, all the wave functions return to the horizontal axis, indicating that those are convergent. From Fig. 8, the calculated ratio of the wave function splitting corresponding to different values of $ {\lambda _\sigma } $ (from 0.98 to 1.02) is 3.68: 3.46: 3.31, i.e., the wave function splitting decreases with the increase of $ {\lambda _\sigma } $. This is consistent with the trend of energy and width splittings.

      The pseudospin wave function splitting with variable $ {\lambda _\omega } $ is shown in Fig. 9. For the resonant partner $ (2h_{11/2}, 1j_{13/2}) $, the ratio of the wavefunction splitting with the increase of $ {\lambda _\omega } $ is 3.32: 3.46: 3.64, which is also consistent with that of energy and width splittings. In addition, the change trend of wave function splitting caused by $ {\omega} $ meson field is opposite to that by $ {\sigma} $ meson field.

      When $ {\lambda _\sigma } = {\lambda _\omega } $ is set, the wave functions for the resonant partner is shown in Fig. 10. It is difficult to distinguish the wave functions with different strength of fields. When $ {\lambda _\sigma } = {\lambda _\omega } $ increasing from 0.98 to 1.02, the calculated ratio of wave function splitting is 3.47: 3.46: 3.44, which shows that the variations of wavefunction splitting here are very small.

      From Fig. 11, the same situation occurs in the $ {\rho} $ meson field, even if the parameter changes from 0.8 to 1.2, the change of wave function splitting is not obvious. Corresponding to $ {\lambda _\rho } = 0.8, 1.0, 1.2 $, the calculated ratio for wavefunction splitting in resonant partner $ (2h_{11/2}, 1j_{13/2}) $ is 3.39: 3.46: 3.54, which agree with that of the energy and width splitting with $ \lambda _\rho $. This indicates that, as far as the wavefunction is concerned, the $ {\rho} $ meson field has little influence on the PSS.

    IV.   SUMMARY
    • The RMF-CMR theory by combining relativistic mean field (RMF) theory with the complex momentum representation (CMR) is adopted to check the pseudospin symmetry in the single-neutron resonant states in $ ^{208} $Pb. Influences from different fields of mesons on the pseudospin symmetry are investigated, and the splitting of the energies, widths and wavefunctions between the resonant pseudospin partners is extracted. It is shown that the energy and width splittings between the resonant pseudospin partners decrease with increasing $ \sigma $-field strength, and increase with increasing $ \omega $-field strength. In comparison with the $ \sigma $ and $ \omega $ fields, the $ \rho $-field produce more minor influence on energy and width splitting for the resonant pseudospin partners, which indicates that $ \sigma $ and $ \omega $ fields are dominant influencing the pseudospin symmetry, and the pseudospin splitting comes from the competition of the $ \sigma $ and $ \omega $ fields. It should be noted that the trend for the change of the energy splitting in the resonant partners with the strength of the meson field is in agreement with the case in the bound partners. At the same time, there are some differences in the influence of various meson fields on the PSS in the bound and resonant partners, which are also discussed in detail.

      Moreover, considering that only the wavefunctions of resonant states in the momentum space converge, the pseudospin wavefunction splitting for resonant partners in momentum space and its dependence on the strength of meson field are calculated and analyzed. The trend of wave function splitting with $ {\lambda _\sigma } $, $ {\lambda _\omega } $ and $ \lambda_\rho $ is consistent with that of the energy and width splittings in resonant partners.

    ACKNOWLEDGMENTS
    • Helpful discussions with Prof. Jian-You Guo are acknowledged.

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