Search for the G-parity irregular term in weak nucleon currents extracted from mirror beta decays in the mass 8 system

a Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan b RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan c National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA d Department of Physics, Saitama University, 255 Shimo-okubo, Sakura-ku, Saitama 338-8570, Japan e Fukui University of Technology, 3-6-1 Gakuen, Fukui 910-8505, Japan


Introduction
Symmetry between proton and neutron in the charge space is associated with the symmetric properties of strong interactions, which are charge symmetric and charge conjugation invariant. Consequently, the process on strong interactions is invariant under the G transformation defined as the product of the charge symmetry and the charge conjugation. In the weak interaction, the G-operation invariance claims an important fundamental symmetry in the framework of the standard model, considering the effect of strong interactions on the weak processes [1]. The weak nucleon currents have not only the main terms, which are responsible for the Fermi and Gamow-Teller matrix elements, but also have additional induced terms because of the strong interactions. The induced terms are expected to hold the G symmetry, that is, the decays of a proton and a neutron in a nucleus should be symmetric. A proton and a neutron are, however, a composite particle of a different set of three quarks, (uud) and (udd), respectively, confined by gluons in a nucleon. It is well known that the axial-vector coupling constant is modified from 1 for decay of a free quark to 1.27 for a nucleon [2]. Thus the G-parity violating term may be induced from a small asymmetry caused by such renormalization and also the mass difference between up and down quarks.
Many β-ray correlation-type experiments of nuclei [3] and neutron [4], ν p quasielastic scattering experiment [5] and a measurement of semileptonic-decay branching ratio of τ lepton [6] have been performed to test G-parity violation. The most precise limit has been imposed on the G-parity-violating induced tensor term g II from the β-ray correlation with the nuclear spin alignment of a parent nucleus in the mass A = 12 system by Minamisono et al. [7] as 2M n f T / f A (= g II /g A ) = −0.15 ± 0.12 ± 0.05 (theory) at a 90% confidence level (CL), where g A is the main coupling constant of the axial-vector current and M n is the nucleon mass. So far, there was no reliable confirmation of the result of the A = 12 system in other mass systems.
In the A = 8 system, the β-delayed α angular correlation terms of the mirror pair 8 Li and 8 B have been measured by several groups [8,9]. The induced tenser term was determined as g II /g A = +0.5 ± 0.2 ± 0.3 from the correlation terms by McKeown et al. [9] combined with the M1/E2 transition strength of the analog γ decay [10]. The second error reflects the error from the analogγ -decay measurement. The second-forbidden term f /Ac used in their analysis, which was determined from the E2 strength, however, disagrees with another measurement [11]. f is the second- forbidden matrix element of the vector current, c is the Gamow-Teller matrix element and A is the mass number. If we adopt f /Ac of Ref. [11], g II /g A shifts by about −1.1. The disagreement of f /Ac introduces an additional large systematic uncertainty to the final result. In the A = 20 system, a theoretical prediction of a secondforbidden term j 2 /A 2 c, where j 2 is the second-forbidden matrix element of the axial-vector current, was used to extract g II /g A [12], therefore the result has a large theoretical uncertainty. However, all the highly uncertain terms contributing to the extraction of g II in the β-α angular correlation terms can be experimentally determined combining with the alignment correlation terms in the β-ray angular distribution as discussed later. In the present study, the alignment correlation terms were measured to determine both g II and the other terms in the A = 8 system.

β-ray angular correlation
The β-ray angular distribution W (E, θ Iβ ) from a purely spin aligned nucleus has a correlation term with an alignment A as Here, p, E, E 0 and θ Iβ are the β-ray momentum, energy, end-point energy and ejection angle with respect to the spin-orientation axis, respectively. The 8 Li and 8 B nuclei decay to the broad first excited state of 8 Be and thus the end-point energy E 0 is given by with the population a m of a magnetic substate m, which is normalized to unity as a m = 1. The alignment correlation term, 0), and the β-α angular correlation term, 1), are given [13] by the same equation except for the sign of f /Ac and j 2 /A 2 c terms as where b is the weak magnetism, d I is the time component of the axial-vector current and j 3 is the second-forbidden matrix element of the axial-vector current. The upper and lower signs refer to β decays of 8 Li and 8 B, respectively. The difference, δ − (s) The weak magnetism b/ Ac can be determined from the analog γdecay measurement [10] and the β-delayed α energy spectra [14] based on the Conserved Vector Current (CVC) hypothesis. Thus, the g II /g A and f /Ac are separately determined from the sum of and the difference between δ − (0) and δ − (1), respectively.

Experiment
The experimental procedure consisted of three steps, the production of the polarized nuclei, the spin manipulation from polarization to alignment and the β-ray detection. The present experimental setup was an extension of β-NMR apparatus for measurement of a β-ray-energy dependence of the angular distribution.
A hole was made in a dipole magnet at center axis of iron core coil. A β-ray energy was measured by a set of plastic scintillation counters placed just outside the dipole magnet. The detailed setup was given in Ref. [7].

Production of polarized 8 Li and 8 B
The experimental procedures of 8 Li and 8 B were the same, but several conditions were different. First the experiment of 8 Li is described. The Van de Graaff accelerator at Osaka University was used to provide the pulsed beam of deuteron at 3.5 MeV to bombard a Li 2 O target. The 8 Li nuclei were produced through the nuclear reaction 7 Li(d, p) 8 Li. The recoil angle of the reaction products was selected to 14 • -40 • to optimize the nuclear spin polarization. The polarization of +7.18 ± 0.10% was obtained. The direction of polarization is defined with respect to p beam × p prod , where p beam and p prod are the momentum vectors of beam and reaction product, respectively. The method of polarization measurement is described below. The polarized 8 Li nuclei were implanted into Zn single crystals. The crystals were placed in a static magnetic field B 0 of 60 mT, applied parallel to the polarization direction in order to maintain the polarization and to manipulate the spin orientation with the β-NMR technique. The c axis of the single crystals was set parallel to B 0 .
Here the conditions of 8 B are summarized. The 8 B were produced through the nuclear reaction 6 Li( 3 He, n) 8  To determine the polarization, asymmetries of β-ray angular distribution between θ Iβ = 0 • and 180 • were measured. The β ray was detected by two sets of plastic-scintillation-counter telescopes. The measured asymmetry included the geometrical asymmetry which was caused by the geometrical misalignment between two telescopes. The initial polarization, the geometrical asymmetry and the inversion efficiency α of polarization were deduced using the same test sequence program as Ref. [7]. α was −85.5 ± 0.3% and −94.8 ± 0.9% for 8 Li and 8 B, respectively. These parameters were measured every 31.2 s.

Spin manipulation and spin alignment
The Larmor frequency in a static magnetic field splits into four resonance frequencies due to the hyperfine interaction between the electric quadrupole moment Q of the implanted nucleus and the electric field gradient q at the implantation site in the crystal. The 8 Li nuclei implanted in Zn are known to be located at a single site, while the 8 B nuclei in TiO 2 are located at two different sites.
The quadrupole coupling constant eq Q /h has been determined for all the implantation sites [15]. The relative populations of major and minor sites of boron atoms in TiO 2 are 9 : 1 [16]. The nuclear spin of 8 B implanted only in major site was manipulated [17]. The effect of the unmanipulated 8 B in minor site was negligibly small, that is, a shift of 10 −7 for the alignment correlation terms.
The procedure of the alignment production for spin (I = 2) was newly developed. Fig. 1 shows the schematic procedure of the alignment production for 8 Li. The nuclear spin was manipulated by applying rf oscillating magnetic fields in β-NMR technique. Two methods of rf application were used, i.e., the adiabatic fast passage (AFP) and depolarization methods. The populations between the neighboring two magnetic substates can be interchanged by the AFP method and equalized by the depolarization method. The initial polarization was converted into both positive and negative alignments with ideally zero polarization by applying two depolarizations and four sequential AFPs. After measuring β-ray spectra from the aligned nuclei, the alignment was converted back into polarization to check the consistency of the spin manipulation and to measure the relaxation time of the alignment. Both positive and negative alignments were produced sequentially in each beam cycle to remove a possible systematic uncertainty due to a fluc- tuation of a beam current [7]. In one of the beam cycle, after the positive alignment A + 1 production, the negative alignment A − The is the parameter of an incompleteness of depolarization, which yielded the small residual polarization in the pure alignment section. To determine these population parameters, the βray asymmetries were measured for three orientation patterns shown in Fig. 1, which were at the pure alignment section and at two intermediate steps of the alignment-production procedure.

Beta-ray detection
The β-ray angular distribution was detected by two sets of plastic-scintillation-counter telescopes placed at θ Iβ = 0 • and 180 • .
The typical counting rate of each telescope was 4 kcps for 8 Li and 1.5 kcps for 8 B. Each telescope consists of two thin E counters with 0.5 mm and 1 mm thickness, one energy counter (E counter) with 160 mmφ × 120 mm and one veto counter [7]. The veto counter eliminated the incoming β rays scattered by the magnet surface. The energy was calibrated by determining the β-ray endpoint energies for several β emitters, which were 8 Li itself, 28 Al (E 0 = 3.37 MeV), 20 [18], where the distribution of the β emitters on the catcher was taken into account by considering the kinematics of the nuclear reaction.

Extraction of alignment correlation term
The alignment correlation term was obtained from the ratio of counts at the positive and negative alignment sections, for the up (θ Iβ = 0 • ) and down (180 • ) counters. A and dP in R(E) are the alignment and the residual polarization at the alignment section. The signs given by the superscript in A ± and dP ± are the sign of alignment. The alignment correlation term was extracted from the well approximated formula without the influence of the beam-current fluctuation [7] as where the upper and lower signs are for up and down counters, respectively, The subscript 1 and 2 for R, P and A shows the first and second alignment sections after the beam-off, respectively. A 1+2 was +13.1 ± 0.4% for 8 Li and +17.7 ± 0.8% for 8 B. The alignment correlation terms extracted from the up and down counters were averaged, so that the effect of the residual polarization was canceled.

Corrections
The alignment correlation terms were applied the corrections for the alignment, the angular distribution and the energy spectrum. The alignment of parent nucleus is independent of the β ray, thus the correction for alignment is independent of the β-ray energy. The others depend on the β-ray energy and are shown in Fig. 2.
The alignment was determined from the polarization change, thus the correction for alignment was related to the polarization.
The (p/E) term and the polarization correlation term were neglected in the polarization calculation. The correction for these terms to the alignment correlation terms was evaluated as described in Ref. [7]. The correction factor for (p/E) was 0.997 for both nuclides. The correction factor for polarization correlation term was evaluated to be 0.956 ± 0.030 for 8 Li and 1.004 ± 0.033 The β-ray angular distribution has the cos θ Iβ and P 2 (cos θ Iβ ) terms for polarization and alignment correlated terms, respectively. The influence of the finite solid angle was corrected as a function of energy by using the Monte Carlo simulation. The correction factor C solid for solid angle was shown in Fig. 2. The probability of the large angle scatter of the low energy β ray is higher than the high energy β ray, therefore the correction at low energy becomes large.
The observed alignment correlation term at a certain energy included the contribution of lower and higher energy regions because of the distribution of the counter response function [7]. The correction factor C res for response function was evaluated selfconsistently by using the result of the alignment correlation terms. The correction for 8 B was nearly 1, because the alignment correlation term of 8 B was almost constant above 5 MeV as shown in Fig. 3.
The β-ray angular distribution proportional to the alignment has (p/E) 2 term as (p/E) 2 AB 2 (E)/B 0 (E)P 2 (cos θ Iβ ). The correction factor C (p/E) 2 for (p/E) 2 term was evaluated for each energy. The correction factor C BG for the background in the β-ray energy spectra was negligible in the energy region higher than 5 MeV. The total correction factors C total are shown in Fig. 2.

Systematic uncertainties
The response function could be changed due to uncertainties of the position and thickness of the catcher and the position of the beam spot on the target, because the distribution of 8 Li or 8 B on the catcher could be changed. The thickness of two crystals used as the catcher was 360 ± 40 μm and 250 ± 30 μm for 8 Li and 100 ± 10 μm for 8 B. The influence of these uncertainties was evaluated by using the Monte Carlo simulation. The reliability of the low-energy tail in the simulated response function of monoenergetic β ray was studied experimentally [19]. The 12 B and 12 N were produced as a β emitter. The β-ray energy was selected by a dipole magnet. The shape and amount of tail were confirmed within the 20% statistical error. In the present experimental setup, the tail was caused mainly by the energy loss straggling in the catcher, thus this uncertainty was evaluated by changing the catcher thickness by 20%. The simple sum of the uncertainties for the simulated response function and for the catcher thickness itself was applied. The uncertainties of counter resolution and energy calibration were propagated to the systematic uncertainty of the alignment correlation term. The systematic uncertainty due to gain fluctuation of the E counters, background and pile-up were evaluated. The statistical error in the self-consistent evaluation of C res and C B 1 /B 0 was propagated to the systematic uncertainty. In the alignment determination, it was assumed that the incompleteness parameter of depolarization for two kinds of frequencies was same. The uncertainty due to different was evaluated by assuming the ratio of two 's was 10. The 3rd order orientation of nuclear spin was neglected in the polarization determination. The influence of the correlation term due to the 3rd order orientation, which was formulated [13] with the matrix elements determined in the present study, was evaluated self-consistently.
Each systematic uncertainty described above was less than 0.05% for 8 Li and 0.04% for 8 B in absolute value of the alignment correlation term at 9 MeV.
The statistical error of alignment is included in the systematic uncertainty of the alignment correlation term, because the align- The solid lines are the best fit curves. ment changes the alignment correlation term as a whole such as other systematic uncertainties. The error at 9 MeV was 0.08% for 8 Li and 0.13% for 8 B in absolute value of the alignment correlation term. However, the statistical error of the alignment was included in the statistical error of the final results such as g II /g A .
The total systematic uncertainty at 9 MeV was 5% relative to the alignment correlation term for both 8 Li and 8 B.

Results and discussions
The corrected alignment correlation terms are shown in Fig. 3. To avoid the large correction factor and systematic uncertainties, the data from 6 to 13 MeV for 8 Li and from 5 to 13 MeV for 8 B were used for the extraction of g II /g A . The alignment correlation terms are compared with the β-α angular correlation terms by McKeown et al. [9] in Fig. 3. The obtained difference δ − (s) defined in Eq. (2) is shown in Fig. 4. The deviation of δ − (s) of the alignment correlation terms and the one of the β-α angular correlation terms indicates that the f /Ac term contributes measurably to δ − (s).

Weak magnetism
The reliable evaluation of the weak magnetism b/ Ac is essential to extract g II /g A . The dependence of b(E x ) and c(E x ) on the excitation energy E x of 8 Be were determined from most precise measurements of the analog-γ -transition strength from 8 Be by De Braeckeleer et al. [10] and of the β-delayed-α energy spectra from 8 Li and 8 B by Bhattacharya et al. [14], respectively. The E x dependence can be formulated using the R-matrix theory with four final states [20]. For c(E x )'s of 8  The E x dependence of b(E x ) was redetermined from the analogγ -transition strength [10] using the same R-matrix parameters as c(E x ). The matrix elements of M γ 1 and R γ in b(E x ), which were defined in Ref. [10], were rescaled so as to reproduce the data of the γ -ray-energy distribution to be M γ 1 = −8.71 ± 0.28 and R γ = 1.5 ± 1.4. The b/ Ac as a function of a β-ray energy was de- Free parameters are g II /g A , d I /Ac, f /Ac, j 2 /A 2 c and j 3 /A 2 c. It is assumed that the E x dependence of all the terms except for b(E x ) is same as c(E x ). The induced tensor term was obtained as g II /g A = −0.28 ± 0.28 (stat.) ± 0.15 (syst.), which is consistent with the G-parity conservation and the result in the A = 12 system [7].
The statistical error consists of 0.16 from both the alignment correlation terms and the β-α angular correlation terms, and 0.23 from the transition strength of the isovector M1 decay in determining the weak magnetism b/ Ac. The systematic uncertainty consists of 0.10 from the alignment correlation terms, 0.06 from the β-α angular correlation terms, and 0.09 from the uncertainty of the E x dependence of b(E x ) and c(E x ). The E x dependence of the other terms may differ from that of c(E x ). The uncertainty estimated by assuming the same E x dependence as b(E x ) instead of c(E x ) was less than 0.01 in g II /g A . The other terms were obtained as d I /Ac = 5.5 ± 2.3, f /Ac = 1.0 ± 0.3, j 2 /A 2 c = −490 ± 70 and j 3 /A 2 c = −980 ± 390. The present f /Ac was the middle of the two previous CVC predictions [10,11]. At a 90% CL, we obtained g II /g A = −0.28 ± 0.46 (stat.) ± 0.19 (syst.), where systematic uncertainties evaluated analytically using statistical 1σ errors were multiplied by 1.64, while the others were already evaluated in 90% CL. In the A = 12 system, a possible charge asymmetry of the matrix elements in the mirror transitions was taken into account [7], which yields a shift of 0.10 ± 0.05 in g II /g A . The charge asymmetry in the A = 8 system was not taken into account because the effect was small compared with the error of the present data.
To obtain the weighted mean in the A = 8, 12 and 20 systems, the results of Ref. [7] in the A = 12 system and Ref. [12] in the A = 20 system were used. In the A = 20 system, the theoretical prediction of j 2 was used to extract g II /g A . The value, g II /g A = −0.4 ± 1.1, including 100% uncertainty in j 2 , was used.
The weighted mean of the induced tensor term was obtained to be g II /g A = −0.17 ± 0.16 at a 90% CL and was slightly finite and negative. Shiomi, however, predicted a very small and positive value based on the QCD sum rule as g II /g A = +0.0152 ± 0.0053, which is proportional to the mass difference between up and down current quarks [21]. The experiment was performed for β decay not of free nucleon but of nucleus, therefore the slight difference be-tween the data and the prediction may indicate a renormalization in medium.

Medium effects
To incorporate medium effects such as the off-shell effect and/or the G-parity violating ω meson decay, a model was introduced by Kubodera-Delorme-Rho (KDR) [22]. In the KDR model, the G-parity violating signal is given by κ = ζ + λL instead of g II /2M n , where ζ is the 1-body contribution including the off-shell effect and λ is the 2-body contribution. Since meson exchange current between two nucleons depends on a nuclear structure, the λ contribution in κ is proportional to a matrix element L. Using several mass systems with different L, the contributions of ζ and λ can be separated. The L values without the short range correlation are −0.252, 0.086 and −0.433 in A = 8, 12 and 20 systems, respectively [23]. Since the data of the A = 8 and 12 are almost orthogonal in ζ -λ plane, the result of the A = 8 was crucial in determining the two KDR parameters even if the error of the g II /g A itself was larger than the A = 12 system. From the A = 8, 12 and 20 data, we derived the two KDR parameters to be ζ = −(0.13 ± 0.13) × 10 −3 MeV −1 , λ = +(0.27 ± 0.97) × 10 −3 at a 1σ level. It is shown again that G-parity violating signals are small.

Summary
The G-parity violating induced tensor term, g II /g A , was extracted from the mirror β decay of 8 Li and 8 B, and consistent with the G-parity conservation. The results of three mirror β decays in the A = 8, 12 and 20 systems indicated no G-parity violating signals caused by medium effects. However, in order to clarify whether there is a finite G-parity violation by medium effects at more accurate level, systematic studies in other mass systems are desired. The L of A = 13 system is very small such as 0.024 [22], therefore the 1-body contribution ζ will be clearly detected in the A = 13 system. Systematic studies in the A = 13 and 20 systems [24] are in progress, where no prediction of unknown matrix elements requires to extract g II /g A .