Schiff Moments of Hg Isotopes in the Nuclear Shell Model

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PAPER • OPEN ACCESS Schiff moments of Hg isotopes in the nuclear shell model

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This content was downloaded from IP address 170.106.33.22 on 26/09/2021 at 06:57 27th International Nuclear Physics Conference (INPC2019) IOP Publishing Journal of Physics: Conference Series 1643 (2020) 012006 doi:10.1088/1742-6596/1643/1/012006

Schiﬀ moments of Hg isotopes in the nuclear shell model

Naotaka Yoshinaga1, Kota Yanase2 and Koji Higashiyama3 1Department of Physics, Saitama University, Saitama City 338-8570, Japan 2Center for Nuclear Study, University of Tokyo, Hongo, Bunkyo, Tokyo 113-0033, Japan 3Department of Physics, Chiba Institute of Technology, Narashino, Chiba 275-0023, Japan E-mail: [email protected]

Abstract. Existence of the permanent electric dipole moment of a fundamental particle implies violation of time reversal invariance. The electric dipole moment of a diamagnetic neutral atom is mainly induced by the nuclear Schiﬀ moment. In this study the Schiﬀ moment induced by the interaction which violates parity and time reversal invariance is calculated for the 199Hg nucleus in terms of the nuclear shell model.

1. Introduction The electric dipole moment (EDM) is a physical observable which violates time reversal symmetry. Since the simultaneous application of charge (C), parity (P ) and time (T ) reversal operators keeps the total symmetry of a system, violation of T reversal symmetry is equivalent to the violation of CP reversal symmetry. The Standard Model in particle physics violates CP invariance only through a single phase in the Cabibbo-Kobayashi-Maskawa matrix that mixes quark flavors [1, 2]. Thus, it is important to evaluate subtle effects on EDMs from various aspects. The resulting T reversal violation is therefore expected to produce only tiny EDMs. Recently, the effect of the P , CP -odd electron-nucleon interaction on the electric dipole moment of the 199Hg atom is considered by evaluating the nuclear spin matrix elements in terms of the nuclear shell model [3]. At present the upper limit on the neutron EDM is experimentally 2.9×10−26ecm [4]. However, the Standard Model predicts quite a small value, 10−32ecm [5, 6, 7]. Some theories beyond the Standard Model predict larger EDMs [7, 8, 9, 10]. EDMs originating from CP violation in the hadron sector are searched for in neutron and diamagnetic atoms such as 129Xe, 199Hg and 225Ra. Measurements of EDMs for these atoms have been attempted and their upper limits are 4.1 × 10−27ecm for 129Xe [11], 7.4 × 10−30ecm for 199Hg [12], and 5.0 × 10−22ecm for 225Ra [13]. Concerning 129Xe, two groups have reported improved measurements of EDM. One of these measurements gives its value as (0.26 ± 2.33 stat ± 0.72 sys)×10−27ecm (95% C.L.) [14], while the other measurement reports as (−4.7±6.4)×10−28ecm (95% C.L.) [15]. The EDM of a neutral diamagnetic atom arises mainly from the Schiff moment of the nucleus. The nuclear Schiff moment originates mainly from two different sources; one from nucleon intrinsic EDMs, and the other from the two-body nuclear interaction which violates P and T invariance. In the latter case theoretical calculations have been carried out for Hg, Rn, and Ra isotopes using mean field theories [16, 17, 18, 19, 20, 21]. In the previous study [22] the Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 27th International Nuclear Physics Conference (INPC2019) IOP Publishing Journal of Physics: Conference Series 1643 (2020) 012006 doi:10.1088/1742-6596/1643/1/012006

Table 1. Single-particle energies ετ for neutrons (τ = ν) and protons (τ = π) in units of MeV. j 2p1/2 1f5/2 2p3/2 0i13/2 1f7/2 0h9/2 εν 0.000 0.570 0.898 1.633 2.340 3.415 j 0g7/2 1d5/2 2s1/2 0h11/2 1d3/2 επ 2.500 1.683 0.000 1.348 0.351

Table 2. Strengths of two-body interactions between neutrons (ν-ν), protons (π-π), and neutrons and protons (ν-π). G0 and G2 indicate the strengths of the monopole-pairing (MP ) and quadrupole-pairing (QP ) interactions between like nucleons. The κ2’s indicate the strengths of the quadrupole-quadrupole (QQ) interactions between like and unlike nucleons. MP interactions are given in units of MeV, and QP and QQ interactions are given in units of MeV/b4 using the oscillator parameter b. G0 G2 κ2 ν-ν 0.0800 0.0060 0.016 π-π 0.0800 0.0060 0.004 ν-π 0.040

Schiff moments of the lowest 1/2+ states for 135Xe, 133Xe, 131Xe, and 129Xe nuclei are calculated assuming two-body interactions violating P and T invariance. Particularly effects of the particle- hole excitations from the magic core of the nucleus are found to be important. In this work we report the Schiff moment of 199Hg nucleus, whose atomic EDM has the smallest upper limit in experiment.

2. The shell model calculation The nuclear shell model is one of the most successful models to describe various aspects of nuclear structure. In order to calculate the nuclear structure of 199Hg, it is preferable to exploit the full 82-126 neutron space and also 50-82 proton space. It is, however, very hard to perform full shell-model calculations for 199Hg due to the large number of shell-model configurations. To get around this problem, the pair-truncated shell model (PTSM) is utilized [23, 24, 25], where the full shell-model space is truncated within subspaces composed of collective pairs of nucleons. In this work the configuration spaces are five orbitals between magic numbers 50 and 82, (0g7/2, 1d3/2, 1d5/2, 2s1/2, and 0h11/2), for protons, and six orbitals between magic numbers 82 and 126, (0h9/2, 1f7/2, 0i13/2, 2p3/2, 1f5/2, and 2p1/2), for neutrons. As an effective interaction, the pairing plus quadrupole-quadrupole interaction is employed with parameters G0, G2, κ2 for like particles and κ2 between neutrons and protons [26]. Adopted single-particle energies and strengths of two-body interactions are listed in Table 1 and Table 2, respectively. Both neutrons and protons are considered as holes. in comparison with the experimental data. The details will be reported elsewhere, but the overall agreement to the experimental energy levels is well described. However, spin and parity of the experimental ground state (1/2−) is not reproduced. For the reproduction of spin and parity of the experimental ground state, a delicate treatment of nucleon effective interactions would be necessary.

3. Theoretical framework The Schiff moment operator S coming from the asymmetric charge distribution in a nucleus is P A ei 2 − 5 ⟨ 2⟩ expressed as [27] S = i=1 10 ri ri 3 r ch ri , where i represents the ith nucleon. Here A is the mass number of a specific nucleus, and ri indicates ith nucleon position. In this study the Schiff moments are calculated for the lowest states with spin I = 1/2 and negative parity. Here

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ei is the charge for the ith nucleon. ei = e is taken for a proton and ei = 0 is assumed for a 2 neutron. The r ch is the mean squared radius of the nuclear charge distribution [28]. By perturbation theory, the expectation value of the Schiﬀ moment operator is expressed as X S = S(T ), (1) T =0,1,2 X ⟨ −| | +⟩⟨ +| PT | −⟩ I1 Sz Ik Ik V I1 S = π(T ) + c.c., (2) (T ) − − ⟨ +| | +⟩ k E1 Ik H0 Ik PT where Vπ(T ) represents the isoscalar (T = 0), isovector (T = 1) or isotensor (T = 2) interactions | −⟩ − between nucleons. Here I1 and E1 represent the wavefunction and the eigenenergy of the | +⟩ lowest state with spin I and negative parity for the Hamiltonian H0, respectively. Ik represents the kth intermediate state with spin I and positive parity. Note that this expression is valid as | +⟩ | +⟩ long as Ik forms an orthonormal complete system [22] and each state Ik is not necessary to be the eigenstate of the Hamiltonian H0. In the present study, only I = 1/2 states are considered. All these states have their projection (spin third component) +1/2. At the leading order, the CP -odd (i.e., P and T violating) nuclear force is a one-pion exchange process made by combining the CP -even and CP -odd pion nucleon interactions. In this paper PT the P and T violating two-body interactions Vπ(T ) in Eq. (2) are considered as follows, which are explicitly written as [29, 30, 31], PT · − · Vπ(0) = F0(τ 1 τ 2)(σ1 σ2) rf(r), (3) PT − − · Vπ(1) = F1[(τ1z + τ2z)(σ1 σ2) + (τ1z τ2z)(σ1 + σ2)] rf(r), (4) PT − · − · Vπ(2) = F2 (3τ1zτ2z τ 1 τ 2)(σ1 σ2) rf(r), (5) exp(−mπr) 1 where f(r) = 2 1 + with r = r −r , and r = |r|. The coeﬃcients F (T = 0, 1, 2) mπr mπr 1 2 T 2 2 2 1 mπ (0) 1 mπ (1) 1 mπ (2) are expressed as F0 = − g¯ gπNN ,F1 = − g¯ gπNN ,F2 = − g¯ gπNN , 8π MN πNN 16π MN πNN 8π MN πNN where MN is mass of a nucleon, mπ is mass of a pion, and gπNN is the strong πNN coupling (T ) constant, andg ¯πNN is the strong πNN constant which violates P and T invariance with isospin (T ) (T ) T . In the followingg ¯ and EgπNN are denoted asg ¯ and g for short, respectively. πNN Any intermediate state I+ in Eq. (2) is represented as a one-particle and one-hole excited k E

− state (1p1h-state) from the I1 state. Since the Schiﬀ moment operator is a one-body operator working only on protons, it is enough to consider proton excited 1p1h-states. To evaluate the Schiﬀ moment in Eq. (2), kth intermediate 1p1h-state is given as h i (I) | +⟩ | +⟩ (K) † (K) ⊗ | −⟩ Ik = (ij)K; I = Nij [ciπc˜jπ] I1 , (6) † where ciπ (cjπ) represents the proton creation (annihilation) operator in the orbital i (j), with j−m c˜jm = (−1) cj−m. Namely, a 1p1h-state with spin K, in which one proton excites E from orbital

j to orbital i by the Schiﬀ moment operator, is coupled with the lowest state I− to form an E D E 1

+ (K) + + excited state Ik . Nij is a normalization constant determined as Ik Ik = 1. Here K can take 1 or 0 for I = 1/2. By neglectingD the residualE interaction, the energy denominator in − − + | | + ∼ − ≡ − Eq. (2) is approximately treated as E1 Ik H0 Ik ( Eij) where Eij εi εj represents the single particle-hole excitation energies from orbital j to i. Then Eq. (2) is written as X ⟨ −| | +⟩⟨ +| PT | −⟩ I1 Sz (ij)K; I (ij)K; I V I1 S = − π(T ) + c.c. (7) (T ) E Kij ij

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199 −2 3 Table 3. Calculated results of a(T ) for Hg (in units of 10 efm ). T Type I Type II Type III Sum 0 −0.884 1.230 −0.175 0.171 1 −1.337 −0.350 0.078 −1.608 2 3.898 2.760 −0.060 6.599

Figure 1. The schematic illustration of three types of 1p1h-excitations considered in this study.

To calculate Eq. (7), three types of 1p1h-excitations are considered. The first type is a set of excitations from an orbital between 50 and 82 to an orbital over 82. These excitations are called type-I excitations. The second type is a set of excitations from an orbital under 50 to an orbital between 50 and 82. These excitations are called type-II excitations. The third type is a set of excitations from an orbital under 50 to an orbital over 82. These excitations are called type-III excitations. Note that excitations among orbitals between 50 and 82 are vanished since these orbitals are not connected by the Schiff moment operator. The schematic picture representing three types of 1p1h-excitations is shown in Fig. 1. For the type-I excitation, an intermediate state is explicitly written as |I−⟩ = h i k type-I (I) (K) † (K) ⊗ | +⟩ † Nph [apπc˜hπ] I1 . Here apπ represents the proton creation operator in the orbital p, where p indicates an orbital over 82.c ˜hπ represents the proton annihilation operator in the orbital h, where h indicates an orbital betweenh 50 and 82. For thei type-II excitation, an intermediate (I) | −⟩ (K) † ˜ (K) ⊗ | +⟩ † state is written as Ik type-II = Nph [cpπbhπ] I1 . Here cpπ represents the proton ˜ creation operator in an orbital p, where p indicates an orbital between 50 and 82. bhπ represents the proton annihilation operator in the orbital h, where h indicates an orbitalh under 50. Fori the (I) | −⟩ (K) † ˜ (K) ⊗| +⟩ type-III excitation, an intermediate state is written as Ik type-III = Nph [apπbhπ] I1 . In this study, all orbitals under the magic number 50 are considered for core orbitals. However, 0d3/2, 1s1/2, and 0s1/2 orbitals are not connected by the Schiff moment operator. For orbitals over the magic number 82, all orbitals up to the primary quantum number N = 8 ¯hω from the bottom are considered. Orbitals over 8 ¯hω have no contributions to the Schiff moment. This ⟨ 2 ⟩ ⟨ 2⟩⟨ ⟩ is because the Schiff moment operator in Eq. (??) is constructed to the ri ri and ri ri , and the following constraints are imposed; |∆n| ≤ 3, |∆ℓ| = 1, 3, and |∆j| ≤ 1, where n, ℓ, and j indicate the radial oscillator quantum number, the orbital angular momentum, and the total spin angular momentum, respectively. The energy of each single particleD orbitalE is taken 3 2 2 from the Nilsson energy [32] as εnℓj = 2n + ℓ + ¯hω − κ 2ℓ · s + µ ℓ − ℓ ¯hω, with D E 2 N κ = 0.0637 and µ = 0.60, where ℓ2 = 1 N(N + 3) and ¯hω = 41A−1/3 MeV. N 2 The total Schiff moment is the summation of three isospin components. In this study, Schiff

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199 Table 4. Calculated results of a(T ) in comparison with other models for Hg (in units of efm3). The mark (⇔) indicates the range of errors using different nucleon effective interactions. Models a0 a1 a2 Simple SM [34] 0.087 0.087 -0.174 RPA [21] 0.00004 0.055 0.009 QRPA [17] 0.002 ⇔ 0.010 0.057 ⇔ 0.090 0.011 ⇔ 0.025 Mean-field [20] 0.009 ⇔ 0.041 −0.027 ⇔ 0.005 0.009 ⇔ 0.024 Present work 0.0017 −0.016 0.066

moments are evaluated as coeﬃcients in front ofg ¯(T )g,

(0) (1) (2) S = a(0) g¯ g + a(1) g¯ g + a(2) g¯ g. (8)

The present results for 199Hg are summarized in Table. 3. The contributions of type-III excitations are roughly ten times smaller than those from type-I and type-II for most of the isospin components. In Table. 4 the present results for 199Hg are compared with those in other models. It is interesting to note that the mean-field approximations predict large range of errors among them due to different choices of effective interactions, indicating that a proper treatment of the ground state wavefunction is inevitable in the evaluation of Schiff moments.

4. Estimation of the atomic EDM in terms of the nuclear shell model Our results are summarized as S = 0.171¯g(0)g − 1.608¯g(1)g + 6.599¯g(2)g × 10−2efm3. (9)

The relation between the Schiﬀ moment and the atomic EDM of 199Hg is calculated as in Ref. [35], S d 199Hg = −2.8 × 10−17 ecm. (10) efm3 Now let us estimate the atomic EDM within the Standard Model. The strong πNN coupling constantsg ¯(T ) in the Standard Model are recently estimated by Yamanaka et al. [36]; g¯(0) = −1.1 × 10−17, g¯(1) = −1.3 × 10−17, g¯(2) = +3.3 × 10−21. Using these values and the standard value of g = 13.5, the atomic EDM for 199Hg in the Standard Model is estimated as 199 −35 dSM Hg = 7.2 × 10 ecm. (11)

If a larger EDM than that in Eq. (11) is observed in experiment, it becomes obvious evidence for physics beyond the Standard Model in particle physics.

5. SUMMARY In the present study the nuclear Schiﬀ moments induced by the interaction which violates parity and time reversal invariance are calculated for the lowest 1/2− states of the 199Hg isotope. The wavefunctions of the Hg isotopes are calculated in terms of the nuclear shell model approach. Excitations from orbitals between the magic numbers 50 and 82 to orbitals over 82 (type-I excitations), the excitations from orbitals under 50 to orbitals between the magic numbers 50 and 82 (type-II excitations), and the excitations from orbitals under 50 to orbitals over 82 (type- III excitations), are considered for the one-particle and one-hole excitations. It is found that the contributions of type-III excitations are roughly ten times smaller than those from the type-I

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and type-II excitations. In summary, the upper limit for the electric dipole moment of 199Hg neutral atom is estimated using its Schiﬀ moment. In the Standard Model it is obtained as 199 −35 dSM Hg = 7.2 × 10 ecm. If a larger EDM is observed, it would provide evidence for physics beyond the Standard Model.

Acknowledgements This work was supported by Grant-in-Aid for Scientiﬁc Research (C) (No. 16K05341) and (No. 17K05450) from Japan Society for the Promotion of Science (JSPS).

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