High-precision half-life measurement for the isospin T = 1/2 mirror (cid:12) + decay of 21 Na

The half-life of the isospin T = 1/2 mirror β + decay of 21 Na was measured at the GANIL-SPIRAL radioactive ion beam facility to be T 1 / 2 = 22.422 (10) s, a result that is more than a factor of 5 times more precise than the previous world average, with a resulting ft value that is now two times more precise. The precision of this new result implies that the half-life is no longer the dominant source of uncertainty in the F t value used in the calculation of V ud from mirror decays to test the conserved vector current hypothesis of the Standard Model. The value of V ud deduced from mirror decays using the new half-life result is now in better agreement with the V ud derived from T = 1 superallowed 0 + to 0 + transitions.


I. INTRODUCTION
The unitarity test of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix is crucial for constraining possible extensions to the Standard Model description of electroweak interactions. In particular, the up-down element of the matrix, V ud , which plays by far the dominant role, is presently obtained from measurements of superallowed 0 + to 0 + nuclear β decays [1]. Given the present uncertainty in the neutron lifetime [2,3], the second most precise value of V ud is obtained from isospin T = 1/2 mirror β-decay transitions (mixed Fermi and Gamow-Teller) and thus provides a complementary method of obtaining V ud [4].
The Ft values for mirror transitions of 19 Ne, 21 Na, 29 P, 35 Ar and 37 K are included in the average of V ud since for these cases, both the experimental f t values and the correlation parameters are known with sufficient precision [4]. For T = 1/2 mirror decays, Ft mirror and V ud are related by [5] where K/(hc) 6 = 2π 3 ln2h/(m e c 2 ) 5 = 8120.271(12) × 10 −10 GeV −4 s, G F /(hc) 3 =1.1663787 (6) × 10 −5 GeV −2 is the Fermi constant, C V = 1 is the vector coupling constant, |M 0 F | 2 is the Fermi matrix element and takes the value of 1 for T = 1/2 transitions, ∆ V R is a transition independent radiative correction, f A and f V are statistical rate functions for axial-vector and vector currents, respectively, and ρ is the Gamow-Teller to Fermi mixing ratio. The value of ρ must be deduced experimentally from measurements of angular or asymmetry correlation parameters.
The primary quantity required to calculate Ft mirror is the experimental f t value that is derived from three experimental quantities: the decay Q-value, the branching ratio and the half-life. The experimental f t value then has to be modified by several theoretical corrections for Coulomb and radiative effects that can be written as [5] Ft where f is the statistical rate function, t is the partial half-life, δ R and δ V N S are nucleusdependent radiative corrections and δ V C is the isospin-symmetry breaking correction. Unlike the set of the 14 most precisely determined Ft values for T = 1 superallowed β emitters [1], the uncertainties in the Ft mirror values are presently dominated by experimental sources [4] thereby prompting a need for new precise measurements for the f t values of T = 1/2 mirror β decays. This is particularly important for the five nuclei, 19 Ne, 21 Na, 29 P, 35 Ar and 37 K, that are presently included in the determination of V ud for mirror transitions.
Of these five mirror nuclei, 19 Ne, 21 Na and 35 Ar play a dominant role as the other transitions are not known with comparable precision. Several recent measurements of 19 Ne [6][7][8] have improved the half-life precision to 0.01% making it the most precisely known half-life for any T = 1/2 mirror decay. A recent half-life measurement of 37 K [9] has increased the precision of its Ft mirror value four-fold, however the poorly known value of ρ for this nucleus still limits its contribution in the calculation of V ud .
There has been renewed interest in the study of the β decay of 21 Na: both branching ratios and β-ν correlation parameters have been measured recently [10][11][12], however, its experimental f t value has the lowest precision of the three mirror nuclei that dominate the determination of V ud . In particular, the Ft mirror value for 21 Na is largely limited by the uncertainty in its half-life. There are three previous measurements [13][14][15] that yield an average half-life of 22.487 (54) s after applying a scaling factor to take into account the relatively large reduced χ 2 value of 3.6. The most recent value of 22.47 (3) s was obtained 40 years ago [15].
In this work, a high-precision half-life measurement of 21 Na was performed and a precise value of Ft mirror was extracted, which causes a shift in the value of V ud derived from mirror transitions and leads to an even better agreement with the value of V ud derived from superallowed 0 + to 0 + decays. Our measurement of the 21 Na half-life that is more than 5 times more precise, and the resulting precision of Ft mirror now paves the way for improving V ud via correlation parameter measurements to deduce the Gamow-Teller to Fermi mixing ratio ρ.

II. EXPERIMENT
The experiment was performed at the SPIRAL (Système de Production d'Ions Radioactifs Accélérés en Ligne) rare-isotope beam facility at GANIL (Grand Accélérateur National d'Ions Lourds) in Caen, France. Radioactive ion beams were produced from the fragmentation of a 95 MeV/u primary 36 Ar 18+ beam on a graphite target and were ionized in a VADIS-type (Versatile Arc Discharge Ion Source [16]) FEBIAD (Forced Electron Beam Ion Arc Discharge) ion source operating in its first online production run. Ions were extracted from the source at 10 kV and were separated based on their charge-to-mass ratio with a resolution of m/∆m ≈ 300. Radioactive beams of 21 Na were delivered as 21 Na 1+ at A/q=21 with an average intensity of 4 × 10 7 ions/s to the SPIRAL low-energy identification and tape transport station [17]. Ions were implanted into an aluminized-mylar tape in the collection chamber. An HPGe detector, located at 0 • with respect to the beam axis, was used for identification and to quantify and monitor the beam purity and intensity. Based on gamma-ray spectra, there was no evidence for A=21 isobars such as 21  photomultiplier signals were discriminated and combined in a logical AND to reduce noise effects and the signals were sent to a multi-channel scaler module. These scaler data were time-stamped using a 10 kHz precision clock. The scaler module is free running and is operated independently of the trigger used for the data acquisition. This module has been tested to function without losing events with input frequencies of up to 100 MHz, although in practice the count rate in experiments is limited to significantly less (≈10 kHz). With this method, the raw data are directly used for analysis since no dead time or pile up corrections are required. This method was used recently at GANIL to deduce the half-life of 19 Ne to high precision (0.03%) [6].
Beam intensities were controlled using a series of attenuators in the beam line located 10 m upstream from the experimental station. The reduction factors were chosen so that the initial count rate in the scintillator did not exceed ≈10 kHz. A full description of the SPIRAL identification station, associated electronics and further details about the SPIRAL low-energy facility can be found in Ref. [17].
Data were collected in cycles. A typical cycle consisted of a background collection time of 1 s, followed by a beam-on period between 2 and 14 s to implant ions on the tape, 3 s of tape movement were then required to move the sample from the implantation site to the measurement site in front of the scintillator, a 450 s decay counting period (approximately 20 half-lives of 21 Na), and 5 s of tape movement to remove long-lived contaminant activity away from the scintillator and into a shielded tape storage box before the start of the next cycle. The beam-on time was varied between 2 and 14 s in order to test for any possible sources of rate dependent effects. The total dataset consisted of 44 cycles collected over 6 experimental runs.

III. HALF-LIFE DETERMINATION
Half-life results for 21 Na were obtained by performing fits to the decay activity curves using a modified Levenberg-Marquardt χ 2 minimization method described in previous publications [18,19].
Of the 44 cycles that were collected during the experiment, one cycle was rejected as the run was stopped early and the decay curve was incomplete (less than 5 half-lives of 21 Na).
A total of 43 cycles from 6 runs were included in the analysis.
The half-life of 21 Na was deduced from the sum of the 43 independent decay curves obtained from the time-stamped scaler data. As described above, dead time losses are negligible at standard scaler operating conditions and thus no dead time corrections were applied to the raw data. The summed decay curve was fit (hereafter referred to as the "global fit" method) using a single exponential decay with a constant background for a total of three free parameters (assuming the 42 K activity is constant, see discussion below). The decay period of 450 s was chosen to be sufficiently long so that the constant background level could be well constrained in the fitting procedure. The start of the fit was chosen to be 1 s (1 channel) after the tape-drive motor was stopped (1 s from the start of the decay period) to ensure that the tape was fully stopped in front of the scintillator. The half-life of 21 Na was determined via the global fit method to be T 1/2 = 22.4218 (92) s with a reduced

A. Systematic uncertainties
Several tests were performed to search for potential systematic effects that may bias the high-precision half-life result. As a first step, the fit to the summed decay curve (Fig. 1) was repeated by removing a number of leading channels (the high-rate data) from the analysis in a step-wise manner. The summed decay curve was fit in steps of 3 channels, and up to a total of 45 channels were removed (2 half-lives of 21 Na). The resulting half-life as a function of the number of leading channels removed is plotted in Fig. 2. Since these data are not independent but highly correlated due to the fact that leading channels are removed and the same data are re-fit, the data points are not expected to be scattered around a mean value. Nevertheless, this analysis is sensitive to rate-dependent sources of systematic uncertainty such as PMT gain shifts and detector afterpulsing [7], and to the presence of  Table I. The results of the cycle-by-cycle, the run-by-run, and counting rate variation analysis are all consistent with the global fit result.
The above analysis rules out the presence of potential sources of significant systematic uncertainties such as rate-dependent effects (for example PMT gain shifts and afterpulsing) and the presence of short-lived isobaric contaminants. As mentioned previously, the only known contaminant that was identified based on gamma-ray spectra collected at the implantation site is 42 K, delivered as a 2+ ion at A=21. In the above analysis, decay curves were fit using a single exponential decay for 21 Na, and a single constant background parameter that incorporates both the counter background and the 42 K (T 1/2 = 12.360 h) long-lived component that was assumed to be constant on the 450 s timescale of the 21 Na measurement. The background rate of ≈1 count/s in the scintillator was deduced from dedicated background runs performed before, during, and after the present experiment. The background of 0.97(1) counts/s obtained from the free fit parameter of the global fit (Fig.1)   Group [20] in which the statistical uncertainty of ±0.0092 s (global fit method) is increased by the square root of the largest χ 2 /ν of 1.19 obtained from the cycle-by-cycle analysis. This method yields a total uncertainty of ±0.010 s that is presently dominated by statistical uncertainties. A summary of all uncertainties is presented in Table II. The final result, precise to 0.04%, can be written as where the first uncertainty is statistical and the second is systematic. This result is in agreement with, but is more than a factor of 5 times more precise than the previous world average of 22.487 (54) s [5]. Using the averaging procedure adopted in Ref. [5], the new world average of 21 Na half-life is T 1/2 = 22.428 (14) s. A summary of all 21 Na half-life measurements performed to date is shown in Fig. 6.

IV. Ft mirror VALUE
With the new high-precision 21 Na half-life result, the Ft mirror value for 21 Na can now be calculated using Eq. (2), and the values of δ R , δ V N S and δ V C from Ref. [5]. The result Ar58 [13], Al74 [14], and Az75 [15]. The weighted average of all results is 22.428 (14) s with a χ 2 /ν of 1.92 for 2 degrees of freedom, following the averaging procedure described in Ref. [5].
is two times more precise than the previous value from Ref. [4], and 21 Na now represents the second (after 19 Ne) most precisely determined Ft mirror value for any of the T = 1/2 mirror cases at 0.15% precision.
The improvement in the half-life precision resulting from the present work implies that the leading source of uncertainty in Ft mirror is now no longer the half-life, but the Q ECvalue. The present status of the uncertainties on the quantities that contribute to the error budget of the Ft mirror values is shown in Fig. 7. The contributions to the error budget from 19 Ne, 35 Ar and 37 K, the other most precisely known T = 1/2 mirror nuclei, are also shown for comparison.
In order to evaluate the impact of the new 21 Na half-life on the determination of V ud , the corrected Ft mirror , denoted Ft 0 , must be determined. This corrected value removes the Gamow-Teller to Fermi mixing components to obtain the pure vector contribution and therefore is expected to be constant for all transitions. Ft 0 is defined by [5] Ft The value of Ft mirror is taken from the present work (Eq. 4), and the f A /f V ratio and ρ values are adopted from Refs. [4,5]. The resulting value of Ft 0 for 21 Na is determined to be can be compared to the previous value of 0.9697 (38) for 21 Na determined using the previous average half-life from Ref. [5]. The uncertainty on this new value is not significantly improved due to the fact that the uncertainty in V ud is dominated by the experimentally determined correlation parameters from which the value of ρ is deduced. Figure 8 shows the values for all T = 1/2 mirror nuclei presently considered in the evaluation of V ud [4], all of which have been updated using the most recent measurements [6][7][8][9]. The value of V ud from the present work is in excellent agreement with that of the most precisely known value from 19 Ne and is now better aligned with the other mirror nuclei. The weighted average of all V ud values, determined to be 0.9724 (17), is now in better agreement with the V ud of 0.97417 (21) [1] from superallowed 0 + to 0 + nuclear β decays.

V. CONCLUSION
A high-precision half-life measurement was performed for the T = 1/2 mirror nucleus 21 Na with the aim of improving the precision of the Ft mirror value, that is in turn used to derive adopted from Ref. [5], except the T 1/2 of 19 Ne and 37 K for which the most recent results [6][7][8][9] are included to calculate new world averages of 17.2579 (38) s and 1.23633 (94) s, respectively, following the method from Ref. [5].
V ud . The new half-life was determined to be T 1/2 = 22.422 (10) s, which represents a result that is precise to 0.04% and is an improvement by more than a factor of 5 over the previous world average [5], allowing the precision of the Ft mirror for the 21 Na decay to be reduced to 0.15%, to what is now the second most precise Ft mirror value. As a result, the value of Ft 0 and V ud for 21 Na decay are now in better agreement with the values derived from 19 Ne, however the uncertainty is still dominated by the precision of the correlation parameters.
New measurements of these parameters will be possible at the GANIL-SPIRAL facility using the LPCTrap device [21] and will improve the precision with which V ud can be determined 15 19 Ne and 37 K are updated with the most recent measurements [6][7][8][9]. The previous value of V ud for 21 Na is deduced using Refs. [4,5]. The present V ud value of 21 Na is in excellent agreement with that of the most precisely known value of 19 Ne, and shifts the value of V ud from mirror transitions into better agreement with the value obtained from superallowed 0 + to 0 + transitions.

VI. ACKNOWLEDGEMENTS
We would like to thank the GANIL accelerator and technical staff for their hard work and support. We are especially grateful to B. Blank and E. Liénard for helpful discussions.