Journal of Physics: Conference Series PAPER • OPEN ACCESS Schiff moments of Hg isotopes in the nuclear shell model To cite this article: Naotaka Yoshinaga et al 2020 J. Phys.: Conf. Ser. 1643 012006 View the article online for updates and enhancements. 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 "τ 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 "ν 0.000 0.570 0.898 1.633 2.340 3.415 j 0g7=2 1d5=2 2s1=2 0h11=2 1d3=2 "π 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 h 2i 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 2 27th International Nuclear Physics Conference (INPC2019) IOP Publishing Journal of Physics: Conference Series 1643 (2020) 012006 doi:10.1088/1742-6596/1643/1/012006 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 h −j j +ih +j PT j −i I1 Sz Ik Ik V I1 S = π(T ) + c:c:; (2) (T ) − − h +j j +i k E1 Ik H0 Ik PT where Vπ(T ) represents the isoscalar (T = 0), isovector (T = 1) or isotensor (T = 2) interactions j −i − between nucleons. Here I1 and E1 represent the wavefunction and the eigenenergy of the j +i 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 j +i j +i 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 = jrj. 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 .