Constraints on the nuclear Schiff moment from its correlation with electromagnetic properties

In this study, we apply the LSSM calculations performed with the effective interactions SNV [27] and SN100PN [28] in order to compute $s_{N}$ and the third-order contribution induced by the nucleon EDM, $a_{N}$, for ¹²⁹Xe. Recent studies on nuclear spectroscopy have shown that low-energy spectra are reproduced by the LSSM calculations involving these realistic interactions in a specific area of the nuclear chart [29, 30, 31, 32, 33, 34].

The effective one-body operator of the nuclear magnetic moment is expressed as

where $\bm{l}_{a}$ and $\bm{s}_{a}$ denote the orbital and spin angular momenta of the nucleon, respectively. We adopt a quenching factor of $0.7$ for the spin $g$ factor, which is attributed to the two-body current correction and core polarization. This value is fixed to reproduce the magnetic moments of ^{129,131,133,135}Xe, which is consistent with other studies [30, 35]. The LSSM calculations of ¹²⁹Xe employ the standard model space between the magic numbers 50 and 82 and then lack the particle-hole configurations, which have large one-body matrix elements of the magnetic moment, such as $(0g_{9/2}^{-1},0g_{7/2})$ and $(0h_{11/2}^{-1},0h_{9/2})$. The spin dependence of the second-order NSM operator in Eq. (4) indicates that the one-body matrix elements between the spin-orbit partners are also significant. We then apply the quasiparticle vacua shell model (QVSM) with the model space extended to include the $0g_{9/2}$ and $0h_{9/2}$ orbitals. The QVSM is a variational approach to describe nuclear wave functions as linear combinations of the proton and neutron numbers, parity, and angular momentum projected basis states. For odd-mass nuclei, the basis states are given by one-quasiparticle states characterized by variational parameters, which are determined to minimize the energy of the projected and superposed wave functions. The framework of the QVSM calculation is detailed in Ref. [36].

We adopt phenomenological extensions of the SNV effective interaction with the full $sdgh$ shell for the QVSM calculation. The SNV effective interaction is employed for the standard model space, and the extended part is given by the VMU interaction [27]. This interaction is referred to as SNV-$sdgh$. Another set, SNJKV interaction, consists of the SNV, JUN45, and Kuo-Herling interactions, and the remaining proton-neutron part is given by the VMU interaction. To compensate for the enhancement in the pairing correlation by extending the model space, we reduce the pairing interaction strengths between like nucleons. The reduction factors are determined by low-energy spectra of ¹³⁴Sn and ¹³⁴Te as 0.72 for proton and 0.76 for neutron in the SNV-$sdgh$ interaction, and 0.74 for proton and 0.76 for neutron in the SNJKV interaction.

Table 1 shows the excitation energies of the $2^{+}$ and $4^{+}$ states obtained by the QVSM calculations. While the $2^{+}$ states are systematically reproduced, the $4^{+}$ states are higher than the experimental values by 0.2 MeV on average. Here, benchmark calculations, based on the energy variance extrapolation method, show slower convergence of high-spin states [36]. This implies that the excitation energies of the $4^{+}$ states are overestimated in the present QVSM calculations. A natural progression of this study is to perform a global calculation using such an extrapolation procedure to further improve the effective interaction.

Figure 1 shows the convergence of the magnetic moment and second-order NSM coefficient $s_{n}$ as the number of QVSM basis states increases. The standard deviation of the last 10 points is $5\%$, at most. In fact, this level of accuracy does not qualitatively change our conclusion. Comparing the magnetic moment and NSM coefficient in the region of a small number of basis states, we find that they fluctuate in opposite directions. This feature stems from the spin dependence of the operators in Eq. (4) and (8). A similar behavior is found between the NSM coefficients of the ground state and the second-lowest $1/2^{+}$ state. As discussed below, these relations help reducing the uncertainty of the NSM coefficients.

The second-order NSM coefficients $s_{N}$ of ¹²⁹Xe were investigated by using the pair-truncated shell model (PTSM), where many-body configurations are made of building blocks composed of nucleon pairs with specific angular momenta [25, 37, 38]. This reduces the dimension of the many-body Hamiltonian matrix to be diagonalized by orders of magnitude in comparison with the conventional $J$-scheme dimension. The PTSM calculation of $s_{N}$ was originally performed in Ref. [25]. The authors employed a phenomenological monopole and quadrupole pairing plus quadrupole-quadrupole interaction as an effective interaction. The interaction strengths were tuned to reproduce the low-energy spectra of Xe, Ba, Ce, and Nd isotopes close to the neutron magic number. Here, we refer to this effective interaction as YH04. Another parameter set of the same form of the effective interaction (YH11) was suggested in Ref. [39] and applied to compute the NSM coefficients [37]. This revision was introduced for the better reproduction of odd-mass nuclei.

In this study, we revisit the PTSM calculations with certain modifications. We adopt the standard formula $\hbar\omega=41A^{-1/3}$ (MeV) for the harmonic oscillator frequency to evaluate the one-body matrix elements as well as in the LSSM and QVSM calculations. This choice corresponds to a root-mean-square charge radius of $4.6$ fm for ¹²⁹Xe, which is consistent with the experimental value of $4.8$ fm [40]. In contrast, an incorrect value of $\hbar\omega=41$ MeV was adopted in Ref. [39], and the root-mean-square charge radius was severely underestimated. Moreover, the original study in Ref. [37] adopted an extended model space for protons, whereas we found that the influence to be negligible. Therefore, we employ the standard model space for consistency with the LSSM calculations.

Table 2 presents the results of the shell model calculations for the second-order NSM coefficients. We adopt the mean values of the last 10 points in Fig. 1 as results of the QVSM calculations. It is confirmed that the NSM of the neutron-odd nucleus, ¹²⁹Xe, is more sensitive to the neutron EDM, whereas the sensitivity to the proton EDM is weak and strongly dependent on the effective interaction. However, the PTSM calculations predict the opposite sign of $s_{n}$ to the LSSM and QVSM calculations. This infers a serious discrepancy, which will be investigated hereafter.

The second-order contribution is decomposed with respect to the one-body transition density as follows:

where $i,j$ indicate the single-particle orbitals. Figure 2 (a) shows each term obtained by the QVSM calculation using the SNJKV interaction. The odd neutron occupies the $2s_{1/2}$ orbital in the IPM, and the diagonal element of this orbital is still dominant in the QVSM calculation. The spin-flipping contributions reduce $s_{n}$ by half with respect to the IPM estimate, while preserving the sign. To investigate the significant decrease in the PTSM calculations, we performed the same analysis for the second-lowest $1/2^{+}$ state. As shown in Fig. 2 (b), the single-particle nature of the $2s_{1/2}$ orbital disappears, and the negative contributions of the $d$ orbitals become dominant.

Figures 2 (c) and (d) show the same decomposition of the magnetic moment. One can find similarity with the NSM coefficient, which recalls the synchronized behavior shown in Fig. 1. Considering that the second-order NSM operator (4) is proportional to the spin operator, associating $s_{n}$ with the magnetic moment is reasonable. In fact, the magnetic moment is dominated by the neutron spin part, because the odd neutron in the $2s_{1/2}$ orbital has zero orbital angular momentum.

The possible correlation between the second-order NSM coefficient $s_{n}$ and the magnetic moment mentioned above is demonstrated more explicitly in Fig. 3. This correlation is understood by considering a two-level model that describes the ground state as

where $\big{|}\psi_{\text{g.s.}}\big{\rangle}$ and $\big{|}\psi_{1/2^{+}_{2}}\big{\rangle}$ denote the ideal ground state that reproduces the experimental value of the magnetic moment, and the second lowest $1/2^{+}$ state, respectively. If a nuclear model calculation represents the ground state as $\alpha\simeq 1$, the result should be plotted on the upper left in Fig. 3 corresponding to the experimental value of the magnetic moment. However, the result approaches the predicted values of the second-lowest $1/2^{+}$ state when $\alpha$ approaches zero. This model can explain the strong quenching predicted by the PTSM calculations, which is corroborated by the isotopic dependence. As shown in Fig. 2 in Ref. [25], the NSM coefficient $s_{n}$ of ¹³⁵Xe is close to the IPM estimate in the PTSM calculations with the YH04 interaction. Because the ¹³⁵Xe nucleus consists of one neutron hole in the $50\leq N\leq 82$ model space, the ground-state wave function is similar to the IPM wave function in shell model calculations with any truncation schemes and effective interactions. However, the NSM coefficient $s_{n}$ is drastically reduced as the neutron number decreases and many-body correlations come into play. Considering the inconsistency with the experimental value of the magnetic moment, the PTSM calculations would fail to describe the relatively complicated wave function of ¹²⁹Xe, which consists of seven neutron holes in the conventional model space. Consequently, the contamination of the ideal second-lowest $1/2^{+}$ state causes a strong quenching of the NSM coefficient and magnetic moment.

Here, we mention the effect of extending the model space in the QVSM calculations. As explained above, the $0h_{11/2}$ orbital is strongly connected with its spin-orbit partner, the $0h_{9/2}$ orbital, by the second-order NSM and magnetic moment operators. The $0h_{9/2}-0h_{11/2}$ contributions placed at the bottom right corners of Figs. 2 (a) and (c) are opposite to the major components of the $2s_{1/2}$ orbital. However, the magnetic moment is larger than the results of the LSSM calculations without the $0h_{9/2}$ orbital, possibly because the magnetic moment of ¹²⁹Xe is sensitive to configuration mixing inside the conventional $50\leq Z,N\leq 82$ model space [41].

Using the correlation with the magnetic moment, the theoretical uncertainty of the NSM coefficient $s_{n}$ can be considerably reduced. Although the PTSM results are counted with equal weights, we obtain $s_{n}=0.29\pm 0.10\,\text{fm}^{2}$, which supports the results of the LSSM and QVSM calculations. Note that the third-order contribution $a_{n}$ does not disturb this correlation. In fact, the LSSM calculation with the SNV effective interaction yields $a_{n}=0.053\,\text{fm}^{2}$ and $a_{p}=-0.001\,\text{fm}^{2}$ for ¹²⁹Xe.

No apparent correlation with magnetic moment was found for the ¹⁹⁹Hg case. The predicted value of the magnetic moment in the second-lowest $1/2^{-}$ state is very close to the value of the ground state. This implies that the magnetic moment is useless for discussing the contamination in the ground-state wave function of ¹⁹⁹Hg.

Table 3 summarizes the NSM coefficients for ¹⁹⁹Hg. The IPM estimate is attributed to the last neutron in the $2p_{1/2}$ orbital, whereas the lower neutron orbitals are fully occupied. The LSSM calculation is performed by employing the Kuo-Herling effective interaction. We use the same empirical formula for the harmonic oscillator frequency in the case of ¹²⁹Xe. The resulting root-mean-square charge radius $5.3\,\text{fm}$ is consistent with the experimental value of $5.4\,\text{fm}$.

The LSSM calculation shows that the NSM coefficient $s_{n}$ is moderately reduced from the IPM estimate. This result corresponds to a higher sensitivity to the neutron EDM, whereas the current experimental limit of ¹⁹⁹Hg obtained using the RPA results is as tight as the direct measurement. The second term in Eq. (4) yields the main contribution $2.98~{}\text{fm}^{2}$ in the ground state, whereas the contributions of the first and second terms are comparable in the second-lowest $1/2^{-}$ state. This is qualitatively consistent with the RPA results. However, the HF and HFB calculations in Ref. [22] yielded significantly smaller second-order NSM coefficients. Their results, which correspond to the long-range contribution from pion exchange, can be interpreted as the summation of $s_{p}$ and $s_{n}$. If the proton EDM contribution $s_{p}$ is negligible, as expected from the LSSM and RPA calculations, one can find possible correlations between $s_{n}$, $a_{0}$, and $a_{2}$. As discussed in Ref. [19], these correlations might be attributed to the contamination of excited states, as in the case of ¹²⁹Xe. The ground state of ¹⁹⁹Hg can be identified by the spectroscopic factor and collective nature of some excited states instead of the magnetic moment.