A pr 2 01 9 Shell model results for nuclear β − – decay properties of sd shell nuclei

We evaluate the allowed β−decay properties of nuclei with Z = 8 − 15 systematically under the framework of the nuclear shell model with the use of the valence space Hamiltonians derived from modern ab initio methods, such as in-medium similarity renormalization group and coupledcluster theory. For comparison we also show results obtained with fitted interaction derived from the chiral effective field theory and phenomenological USDB interaction. In this work, we have tested predictive power of ab initio effective interactions by comparing calculated results with the experimental data for β− – decay properties of sd shell nuclei. We have performed calculations for O → F, F → Ne, Ne → Na, Na → Mg, Mg → Al, Al → Si, Si → P and P → S transitions. Theoretical results of B(GT ), logft values and half-lives, are discussed and compared with the available experimental data. Ab initio calculations of β-decay properties are very limited for the sd shell nuclei, thus present comprehensive study will add more information to it.


I. INTRODUCTION
The study of β − -decay properties of unstable-nuclei have been extensively investigated in the past [1]. There are several experimental data available for half-lives, logf t values, Gamow-Teller (GT ) strengths, Q values and branching fractions [2][3][4]. On the other hand with the recent development in ab initio approaches it is possible to predict these properties. The β-decay half-lives are very important for understanding r-process nucleosynthesis. The GT strength is one of the important tools to study the structure of atomic nuclei. The experimental GT strengths can be obtained from β-decay studies and charge-exchange (CE) reactions. The reliable estimates of GT strength distributions in neutron-rich nuclei can be of great interest for description of β-decay properties.
Study of β-decays based on ab initio methods is very limited, and used to be available only for few-body systems and several light nuclei. Study of GT transitions with the inclusion of the effects of three-nucleon forces and two-body currents from chiral effective field theory are reported for 14 N and 22,24 O in Ref. [5]. The origin of anomalous long life time for the β-decay of 14 C to 14 N was investigated using ab initio no-core shell model with the Hamiltonian from the chiral effective field theory including three-nucleon force terms [6]. The no-core shell model results for B(GT ) strengths for p-shell nuclei with A = 10 − 13, using two-nucleon (N N ) and three-nucleon (3N ) interactions derived from chiral effective field theory, were reported in Ref. [7]. The chiral low-energy constants c D and c E are constrained by means of accurate ab initio calculations of the binding energies of the A = 3 system and half-life of triton [8]. The study of the tritium β-decay with the chiral effective field theory was reported in Ref. [9]. The uncertainties in constraining low-energy constants from 3 H β decay were reported in Ref. [10].
In very early years, features of the beta-decay of neutron-rich sd shell nuclei with five or more excess neutrons were predicted by Wildenthal et al [11]. The comprehensive study of β-decay properties of sd-shell nuclei for A = 17-39 were reported by Brown and Wildenthal in Ref. [1]. The Gamow-Teller beta-decay rates for A≤18 nuclei were reported in Ref. [12]. For the f p-shell nuclei, the calculated Gamow-Teller matrix elements of 64 decays of nuclei in the mass range A=41-50 were reported in Ref. [13]. In these studies, a substantial quenching of the axial-vector coupling in the GT strength was found in p-shell and sd-shell nuclei about by 20% and also in f p-shell nuclei by 25-26%. Theoretical studies on the quenching of the GT strengths were carried out for nuclei with closed-core ± 1 nucleon [14,15], and the effects of coupling to 2p-2h configurations and roles of two-body meson-exchange currents were found to be important.
In later years, the shell model calculations for β − -decay properties of neutron-rich Z = 9 − 13 nuclei with N ≥ 18 were reported by Li and Ren in Ref. [16]. The nuclear β − -decay half-lives for f p and f pg shell nuclei were reported in Ref. [17]. The shell model description of GT strengths in pf -shell nuclei were available in Ref. [18]. The systematic shell-model study of β-decay properties and GT strength distributions in A ≈ 40 neutron-rich nuclei were reported in Ref. [19]. Theoretical calculations for half-lives of medium-mass and heavy mass neutronrich nuclei from QRPA based on the Hartree-Fock Bogoliubov theory or other global models were available in the literature [20][21][22][23][24].
More recently, results of the study on GT and doublebeta decays of heavy nuclei within a framework of an effective theory were presented in Ref. [25]. The ab initio calculations of GT strengths in sd shell nuclei for 13 different nuclear transitions including electron-capture reac-tion rates for 23 Na(e − , ν) 23 Ne and 25 Mg(e − , ν) 25 Na were reported in Ref. [26]. Here, a need for the quenching of the GT strength was reported also for the ab initio shellmodel interactions.
In the present work we have reported β − -decay properties of Z = 8−15 nuclei using nuclear shell model based on ab initio interactions and newly fitted interaction derived from the chiral effective field theory. The purpose of the present work is to study how well the recent ab initio and newly developed shell-model interactions based on chiral interactions can describe the β-decay properties in sd-shell, and also to find how much quenching is necessary for these interactions by comparing with many more experimental data than in Ref. [26]. This work will add more information to earlier works [1,11,16], where shell model results with phenomenological effective interactions were reported. This paper is organized as follows. In Sec. II, we present details of ab initio interactions. The formalism of the calculations for β − -decay properties are presented in Sec. III. In Sec. IV, we present theoretical results along with the experimental data. Finally, a summary and conclusions are drawn in Sec. V.

II. AB INITIO HAMILTONIANS
To calculate GT , logf t values and half-lives for the sd shell nuclei, we have performed shell-model calculations using two ab initio interactions : IM-SRG [27,28] and CCEI [29,30]. Also we have performed calculations with newly fitted interaction derived from the chiral effective field theory [31]. For comparison, we have also performed calculations with the phenomenological USDB effective interaction [32] in addition to the above three interactions. For the diagonalization of matrices we used J-scheme shell-model code NuShellX [33].
The USDB starts from single-particle energies and twobody matrix elements, where the effects of three-nucleon interactions are considered to be included implicitly. The ab initio interaction, on the other hand, starts from chiral two-nucleon and three-nucleon interactions, and onebody and two-body terms outside a core are constructed. The effects of the three-nucleon forces are thus more properly treated in the ab initio approach compared with the phenomenological one.
Glazek and Wilson [34] and Wegner [35] developed techniques to diagonalize many-body Hamiltonians in free space known as the similarity renormalization group (SRG). The SRG consists of a continuous unitary transformation, parametrized by the flow parameter s, and splits the Hamiltonian H(s) into diagonal and offdiagonal parts where H(s = 0) is the initial Hamiltonian. Taking the derivative of the Hamiltonian with respect to s, one gets where is the anti-Hermitian generator of the unitary transformation. For an appropriate value of η(s), the off-diagonal part of the Hamiltonian, H od (s), becomes zero as s → ∞. Instead of the free space evolution, in-medium SRG (IM-SRG) has an attractive feature that one can involve 3,...,A-body operators using only two-body mechanism. The starting Hamiltonian H with respect to a finite-density reference state |Φ 0 is given as Here the normal-ordered strings of creation and annihilation operators obey Φ|{a † i · · ·a j }|Φ =0, and the E 0 , f ij , Γ ijkl , and W ijklmn are the normal-ordered zero-, one-, two-, and three-body terms, respectively (see Ref. [36][37][38][39] for full details). In case of IM-SRG, targeted normal ordering with respect to the nearest closed shell rather than 16 O is adopted to take into account the threenucleon interaction among the valence nucleons.
We use another ab initio approach to study β − -decay properties of nuclei in the sd shell region, named as coupled-cluster effective interactions (CCEI). For this effective interaction, the intrinsic A-dependent Hamiltonian is given as (for IM-SRG interaction also): . (5) The N N and 3N parts are taken from a next-to-nextto-next-to leading order (N3LO) chiral nucleon-nucleon interaction, and a next-to-next-to leading order (N2LO) chiral three-body interaction, respectively. For both IM-SRG and CCEI, we use Λ NN = 500 MeV for chiral N3LO N N interaction [40,41], and Λ 3N = 400 MeV for chiral N2LO 3N interaction [42], respectively.
In the CCEI to achieve faster model-space convergence, the similarity renormalization group transformation has been used to evolve two-body and three-body forces to the lower momentum scale λ SRG = 2.0 fm −1 (see Ref. [43] for further details). Also, for the coupled-cluster calculations, a Hartree-Fock basis built from thirteen major harmonic-oscillator orbitals with frequency Ω = 20 MeV have been used.
We can expand the Hamiltonian for the suitable modelspace using the valence-cluster expansion [44] given as Here A is the mass of the nucleus for which we are doing calculations, H AC 0 is the core Hamiltonian, H AC +1 1 is the valence one-body Hamiltonian, and H AC +2 2 is the twobody Hamiltonian. The two-body term is derived from Eq. (6) by using the Okubo-Lee-Suzuki (OLS) similarity transformation [45,46]. After using this unitary transformation the effective Hamiltonian become non-Hermitian.
For changing the non-Hermitian to Hermitian effective Hamiltonian the matric operator [S † S] = P 2 (1 + ω † ω)P 2 is used, where S is a matrix that diagonalize the Hamiltonians (see Ref. [47] for further details ). After using the matric operator the Hermitian shell-model Hamiltonian is then obtained as [S † S] 1/2Ĥ A CCEI [S † S] −1/2 . Using IM-SRG targeted for a particular nucleus [48] and CCEI interactions, the shell model results for spectroscopic factors and electromagnetic properties are reported in Refs. [49,50]. In case of CCEI, the core is fixed to be 16 O and no target normal ordering is carried out.
Recently, L. Huth et al. [31] derived a shell-model interaction from chiral effective field theory. The valencespace Hamiltonian for sd shell is constructed as a general operators having two low energy constants (LECs) at leading order (LO) and seven new LECs at next-toleading order (NLO) and fitted the LECs of CEFT operators directly to 441 ground-and excited-state energies. For the chiral EFT interaction they have taken the expansion in terms of power of (Q/Λ b ) ν based on Weinberg power counting [51], where Q is a low-momentum scale or pion mass m π and Λ b ∼ 500 MeV is the chiral-symmetrybreaking scale.

III. FORMALISM
In the beta decay, the f t value corresponding to GT transition from the initial state i of the parent nucleus to the final state f in the daughter nucleus is expressed as [52] where B(GT ) is the Gamow-Teller transition strength, and f A is the axial vector phase space factor that contains the lepton kinematics. In this work, we have calculated the phase space factor f A with parameters given by Wilkinson and Macefield [53] together with the correction factors given in Refs. [54,55]. The f t values are very large, so they are defined in term of "logf t" values. The logf t is expressed as logf t= log(f A t i→f ). The total half-life T 1/2 is related to the partial half-life as where f runs over all the possible daughter states that are populated through GT transitions.
The partial half-life is related to the total half-life T 1/2 of the allowed β − -decay as where, b r is called the branching ratio for the transition with partial half-life t i→f . The Gamow-Teller strength B(GT ) is calculated using the following expression: where g A (=-1.260) is the axial-vector coupling constant of the weak interaction, and |i and |f are the initial and final state shell-model wave functions, respectively, here the τ ± refers to isospin operator for the β ± decay, for the β − -decay we use the convention τ − |n = |p , J i is the initial-state angular momentum, and q is the quenching factor. Following Refs. [1,12,13], we define which is independent of the direction of the transitions. R(GT ) values are defined as where the total strength W is defined by In the β − -decay the endpoint energy of the electron E 0 (in units of MeV) is an essential quantity to calculate the phase space factor f A . E 0 is given by the expression [1] where the Q is the β-decay Q value, and E i and E f are excitation energies of the initial and final states. Here, we have taken Q values from the experimental data [4].

IV. RESULTS AND DISCUSSIONS
In Table I we compare calculated and experimental values of the matrix elements M (GT ). Calculated values of M (GT ) presented here are those with q=1. The β-decay energies (E (decay)), branching ratios (I β ) and logf t values as well as the values of W are given in Table I as the root-mean-square (RMS) deviations for the effective interactions considered here are given in Table II. The quenching factors are slightly different for different effective interactions. Their values are in the range of q =0.62-0.77. The value for USDB, q=0.77±0.02, is consistent with the one, q =0.764, in Ref. [56]. The RMS deviations for the ab initio and CEFT interactions are enhanced compared with the USDB by 25-32% and 52%, respectively.
We have plotted the experimental R(GT ) values with respect to the theoretical R(GT ) values for the sd shell nuclei in Fig.1. For further calculations of observables, we take the quenching factors from the Table II. The different sources of renormalization [57,58] affecting the values of g A depends on (i) missing configuratins outside the sd-shell, (ii) non-nucleonic degree's of freedom such as ∆ 33 resonance, and (iii) many-body operators induced by unitary transformations in the ab initio method.
In case of IM-SRG and CCEI, the intrinsic two-body current operator is determined from 3N interaction by current conservation, and induced effective operator also arises from unitary transformations. Some part of the quenching can be attributed to these contributios, and the rest to the configurations outside the sd shell. It is natural that the r.m.s. deviation of the quenching factor is the least for the USDB as free fitting procedure is carried out for the two-body terms. On the other hand, the connection of the two-body current operator to 3N interaction is lost leading to non-uniquee choice of the two-body operator. The quenching factor of IM-SRG is larger than that of CCEI, which can be due to the use of targeted normal ordering in the IM-SRG. In case of CCEI, missing contributions from 3N interaction among valence nucleons can become large in higher-mass sd-shell nuclei. The CEFT is constructed within two-body terms up to NLO in contrast to N3LO (NN) and N2LO (3N) for the IM-SRG and CCEI. Moreover, the fitting 441 energy data was done while 608 energy data were used for the USDB. These points naturally lead to larger r.m.s. deviation for the quenching factor. The comparison between calculated and experimental excitation energies, logf t values, and branching fractions of the β − -decays for the sd shell nuclei are shown in Table III. The first and second columns present the parent and daughter nuclei with spin and parity, respectively. In column 3, 4, and 5 the experimental excitation energies, logf t values, and the branching fractions are presented, respectively. For the different interactions, calculated excitation energies, logf t values, and the branching fractions in the framework of the shell model are listed in the columns 6-17. In general, results for IM-SRG and CCEI can deviate more from experiment for larger-mass nuclei as over-binding is seen near Ca region for these methods. Owing to the targeted normal ordering in IM-SRG, the deviation can be reduced for the IM-SRG compared with the CCEI. These tendencies are seen for M(GT) in Table  I. Large deviations from experiment are also seen more often for CEFT that is consistent with the largest r.m.s. deviation for the quenching factor.
In Fig. 2 we show the distribution of calculated logf t values with the experimental data for some β − -decays nuclei for which experimental logf t values are available for excited states also. In case of 21 F, although results of the ab initio interactions for excitation energy for the excited 3/2 + , 5/2 + and 7/2 + states slightly differ from the experimental data, all the interactions give calculated logf t values close to the experimental data. For 28 Ne, the calculated logf t values with the CCEI are better in comparison to other interactions. The calculated value for excitation energy for 2 + 1 state is in good agreement for all the interactions for 28 Na. For this nucleus all the four interactions give reasonable results for logf t values.
In Table IV, we compare the theoretical and the experimental β-decay half-lives of sd shell nuclei. The first and second columns denote the parent and daughter nuclei, respectively. In the column 3 we list the experimental Q values, which are taken from [4]. These experimental Q values are evaluated by subtracting the mass of the daughter nuclei from the mass of the parent nuclei. The experimental half-lives are presented in column 4. The half-lives calculated from the USDB interaction are presented in column 5. The half-lives calculated with IM-SRG, CCEI, and CEFT are reported in this Table in columns 6-8, respectively. The total half-life is calculated from the partial half-lives of all the transitions. For many isotopes, the β−decay half-lives decrease rapidly from order of s or h to order of ms with increasing neutron numbers. For 21 F(5/2 + ) → 21 Ne, 28 Ne(0 + ) → 28 Na and 28 Na(1 + ) → 28 Mg transitions, the difference in the half-lives is almost determined by the difference of the B(GT) value of the transition to the state with the largest branching ratio, and also difference in phase space factors. For 21 F(5/2 + ) → 21 Ne, the transition to the g.s.
(3/2 + ) gives some contributions. For 32 Mg(0 + ) → 32 Al transition, the agreement between calculations and experiment is not so good for logf t and half-lives. This is due to large values of the calculated B(GT) strength compared with the experimental one. For 28 Mg(0 + ) → 28 Al(1 + ) transition, the phase space factor, which is estimated to be roughly proportional to (decay energy) 5 , depends very much on the interactions as the Q value for this transition is as small as 1.832 MeV. The excitation energies for the 1 + 1 state of 28 Al obtained for the interactions are smaller than the experimental one, E x =1.373 MeV, by 0.176, 0.571, 0.251 and 0.018 MeV for USDB, IM-SRG, CCEI and CEFT, respectively, which leads to an enhancement of the phase space factor by nearly 10 times for IM-SRG. Though the difference of the B(GT) values is within a range of a factor of about 3, large difference in the phase space factors leads to larger difference in the half-lives.
Here, we make some general comments on the halflives. (1) (2) The discrepancy between calculated and experimental half-life becomes large (a) when the discrepancy between the calculated and experimental B(GT ) is large, or (b) when the transition with the dominant branching ratio is different between the calculation and the experiment, or (c) when the Q value for the transition is small and the difference between the calculated and experimental excitation energies is large enough to lead to a substantial change of the phase space factor for the transition. In case of 22 O with IM-SRG, a large discrepancy comes from combined effects of (a) and (b). Nuclei in the island of inversion such as 32 Mg can not be well described for both the ab initio and phenomenological interactions due to the reason (a). 28 Mg discussed above corresponds to the case (c). (3) For isotopes with Z =10-13 (Z=14-15), there are one or two cases (or more cases) for each Z in which the calculated half-lives differ from the experimental ones by a factor more than 3 due to the reasons (a), (b) or (c) in case of IM-SRG and CCEI.
The β−decay half-lives from the CEFT and USDB interactions for 31,33,34 Si give resonable aggrement with the experimental results, but for the ab initio approaches, these results are not in good argreement. For 32 Si, the half-lifves for all the interactions are very far from the experimental data. The calculated half-lives from the ab initio and CEFT interactions for P isotopes do not show good agreement with the experimental data, while for 33,34 P the half-lives from the USDB interaction are in fairly good agreement with the experimental data. In the higher mass region of the sd shell nuclei, the ab initio interactions do not work well, since we have done the calculation without including intruder orbitals from the pf shell.  [2]. I β are the branching ratios. All other quantities are explained in the text.

V. SUMMARY AND CONCLUSIONS
In the present work we have performed shell model calculations using ab initio approaches along with interaction based on chiral effective field theory and phenomenological USDB interaction, and evaluated B(GT ), logf t values and half-lives for the sd shell nuclei. We also obtained quenching factors corresponding to different interactions for the calculation of the B(GT ) strengths.
All the ab initio interactions as well as the fitted interaction based on chiral effective theory considered here need certain quenching of the GT strengths, as large as by 44-62%, as for the phenomenological USDB interaction. The quenching factor can be attributed to (i) configurations outside the sd shell, (ii) induced effective Gamow-Teller operator due to the unitary transformation in the ab initio approach, and (iii) the intrinsic twobody Gamow-Teller operator connected with 3N interaction. In case of ab initio approaches, IM-SRG and CCEI, both the intrinsic and induced two-body operators can be constructed in principle, and the contributions from configurations outside the sd-shell can be reliably con-   strained. It would be an interesting problem to study how much of them can be explained by the contributions from two-body currents as well as effective Gamow-Teller operator produced by ab initio techniques. The effects of the two-body currents have been studied for GT transitions in tritium [8,9] as well as in 14 C and 22,24 O [5]. The effects have been studied also for electromagnetic moments and transitions in few-body and light nuclei with A ≤9 [59,60]. The two-body currents have been taken into account in the study of electromagnetic moments and transitions in p-, sd-and pf -shell nuclei [61] though they are limited to the induced part.