Nuclear charge radius of $^{26m}$Al and its implication for V$_{ud}$ in the quark-mixing matrix

Collinear laser spectroscopy was performed on the isomer of the aluminium isotope $^{26m}$Al. The measured isotope shift to $^{27}$Al in the $3s^{2}3p\;^{2}\!P^\circ_{3/2} \rightarrow 3s^{2}4s\;^{2}\!S_{1/2}$ atomic transition enabled the first experimental determination of the nuclear charge radius of $^{26m}$Al, resulting in $R_c$=\qty{3.130\pm.015}{\femto\meter}. This differs by 4.5 standard deviations from the extrapolated value used to calculate the isospin-symmetry breaking corrections in the superallowed $\beta$ decay of $^{26m}$Al. Its corrected $\mathcal{F}t$ value, important for the estimation of $V_{ud}$ in the CKM matrix, is thus shifted by one standard deviation to \qty{3071.4\pm1.0}{\second}.

Introduction.-The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a central cornerstone in the formulation of the Standard Model of particle physics.It connects the quarks' mass with weak eigenstates and, thus, characterises the strength of quark-flavour mixing through the weak interaction.The first element in the top row of the matrix, V ud , manifests in the β decay of pions, neutrons or radioactive nuclei.While individual entries of the quark mixing matrix cannot be predicted within the Standard Model, the CKM matrix is required to be unitary -a tenet which is the subject of intense experimental scrutiny.
At present, superallowed 0 + → 0 + nuclear β decays remain the most precise way to access V ud [10].For these cases, the experimentally measured f t value, characterising a β decay, can be related to a corrected Ft value: where δ ′ R and δ N S constitute the transition-dependent contributions to the radiative corrections while δ C are the isospin-symmetry breaking (ISB) corrections.According to the conserved vector-current hypotheses, the Ft values should be identical for all superallowed β decays.When averaged over all 15 precision cases, they serve to extract V ud .
While the experimental dataset on f t values of superallowed β decays robustly builds on 222 individual measurements [10], theoretical corrections are under scrutiny.As part of this process, the uncertainties in the nuclearstructure dependent radiative corrections δ N S have recently been inflated by a factor of ≈ 2.6 [10].Moreover, the ISB corrections δ C , which are also nuclear-structure dependent, remain an ongoing focus of research which has stimulated new theoretical calculations [17][18][19][20] as well as experimental benchmarks [21][22][23][24][25][26].
For the determination of V ud , 26m Al is of particular importance.The nuclear-structure dependent corrections, δ N S − δ C , in 26m Al are the smallest in size among all superallowed β emitters [10].The same holds true for the combined experimental and theoretical uncertainties in the Ft value of 26m Al [10].Its extraordinary precision is thus almost on par with all other precision cases combined.In times of tension with CKM unitary and rigorous examination of all involved theoretical corrections, it is, therefore, unsettling that one critical input parameter for the calculation of δ C , i.e. the nuclear charge radius, is in the case of 26m Al, based on an extrapolated but experimentally unknown value [27,28].
In this Letter, we report on isotope-shift measurements obtained via collinear laser spectroscopy (CLS) that puts the nuclear charge radius of 26m Al on solid experimental footings.Implications for its Ft value and, thus, V ud are discussed.
Experiment. -Two independent experiments were performed, one at the COLLAPS beamline [29] at ISOLDE/CERN [30] and the other at the IGISOL CLS beamline [31] in Jyväskylä/Finland.Details of the campaign on aluminium isotopes at COLLAPS are described in Ref. [32].In short, radioactive aluminium atoms were synthesised by bombarding a uranium carbide target with 1.4-GeV protons from CERN's PS booster.Once released from the production target, the Al + ion beam was formed via resonant laser ionisation [33], subsequent electrostatic acceleration to 30 keV, and final mass selection via ISOLDE's magnetic high-resolution separator [34].
At IGISOL [35], the radionuclides of interest were produced in 27 Al(p,d) reactions at 25-MeV proton energy.After their release from a thin foil target and extraction from the He-gas filled gas cell, the Al ions were guided towards the high vacuum region of the mass separator via a sextupole ion guide, accelerated to 30 keV and mass separated by a 55 • dipole magnet.
In both experiments, the ions were stopped, cooled and accumulated in a buffer-gas filled radio-frequencyquadrupole cooler buncher [36,37] before they were delivered in 30-keV ion bunches to the respective CLS beamline.There, the ion beam was spatially super-imposed with the laser beam in collinear (COLLAPS) or anti-collinear (IGISOL) fashion.The ions' velocity was adjusted by a Doppler-tuning voltage applied before the neutralisation in a charge exchange cell filled with sodium vapour.In this manner, the laser frequency experienced in the rest frame of the neutral Al atoms could be scanned via Doppler tuning.Once on resonance with the selected transition, fluorescence was detected using a series of photomultiplier tubes and their associated lens systems which surrounded the laser-atom interaction region [38,39].
In both campaigns, the main spectroscopic transition was from the atomic 3s 2 3p 2 P • 3/2 → 3s 2 4s 2 S 1/2 level at 25 235.695cm −1 .Suitable laser light was provided by a continuous wave Ti:Sa ring laser (Sirah Matisse 2) set to an output wavelength of 792 nm.The resulting laser light was frequency doubled using an external cavity frequency doubler (Wavetrain 2), after which the laser beam with a few milliwatts in power was sent through the experimental beamlines.To compensate for long-term drifts in both experiments, the fundamental laser was locked in wavelength to a HighFinesse WSU-10 wavemeter which was regularly calibrated to a frequency-stabilised HeNe laser.
In addition to the low-lying 26m Al isomer with a halflife of T 1/2 = 6.346 02(54) s [10], the long-lived ground state 26 Al (T 1/2 = 7.17(24) × 10 5 y [40]) was also present in the radioactive ion beams, although in different relative intensities which reflected the distinct production methods at ISOLDE and IGISOL.Examples of the obtained resonance spectra of 26,26m Al are shown in Fig. 1.Due to the dense hyperfine structure of 26 Al (nuclear spin I = 5) and the small isomer shift, the single peak associated with 26m Al (I = 0) could not be resolved from the strongest transition in 26 Al.In order to unambiguously demonstrate the presence of the isomer, the transition 3s 2 3p 2 P • 1/2 → 3s 2 3d 2 D 3/2 at 308 nm was additionally utilised during the campaign at IGISOL.As visible in the inset of Fig. 1a, by exploiting the latter transition the multiplets in the hyperfine spectrum of 26 Al were well separated (green line) and offered unobstructed access to the resonance peak of 26m Al (red).However, due to state-mixing with a second close-lying atomic state (∆E ≈ 0.17 meV), this transition is inadequate for the determination of nuclear charge radii [41].
To confirm the presence of the isomer in the ISOLDE beam, we took advantage of the pulsed time-structure of the proton beam and the long release time of Al from the ISOLDE target.In a set of dedicated measurements, subsequent proton pulses were separated in time by at least 12 s corresponding to approximately two half-lives of 26m Al.The recorded fluorescence data was divided into two sets which were measured up to 6 s and between 6 and 12 s after the proton impact, see Fig. 1b  exp.data weighted avg.

FIG. 1. (a)
Example of a resonance spectrum of the main spectroscopic transition 3s 2 3p 2 P • 3/2 → 3s 2 4s 2 S 1/2 obtained in the CLS measurements of 26,26m Al at IGISOL.The inset demonstrates the isomer's presence (red) due to well separated ground and isomer states in the 3s The spectra of 26,26m Al in the main transition at COLLAPS.Ions have been extracted 0 s (b) and 6 s (c) after the proton impact on the ISOLDE target, demonstrating the isomer's presence due to the decrease in intensity consistent with the isomer's half-life.(d,e) Examples of resonance spectra of the 27 Al references studied using the main transition at IGISOL (d) and COLLAPS (e).f ) Extracted isotope shifts (points) and the resulting weighted average (a horizontal line), including systematic uncertainties.lived ground state (green) changed only slightly between these two data sets, likely because of a small time dependence in the Al release.The much stronger decrease in isomer intensity (red) between the first and second 6 s of data taking was consistent with the isomer's half-life when each is normalised to the corresponding groundstate intensity.
Direct comparison of the spectra shows a higher overall rate and thus better statistics for the COLLAPS data set.This statement holds true for both measurements of 26,26m Al as well as stable 27 Al, see Fig. 1d and 1e, which were interleaved with online data as reference measurements.On the other hand, the data from IGISOL benefits from a more favorable isomer-to-ground state ratio, compare Fig. 1a and 1b.The complementarity of the COLLAPS and IGISOL data sets in terms of high statistics versus better isomer-ground state ratio was further strengthened by their distinct control and evaluation of systematic uncertainties.Most importantly, the determination of the ion-acceleration voltage at COLLAPS was achieved by a high-precision voltage divider.At IGISOL, it was calibrated by CLS measurements of stable magnesium (Mg) ions with respect to their precisely known isotope shifts.
Analysis and Results.-The measured resonance spectra of 26,26m,27 Al were fitted to the theoretical model of the hyperfine spectra using the SATLAS package [42].
To constrain the fit in the present work, the ratio of the hyperfine parameters A(P 3/2 )/A(S 1/2 ) was fixed to the precise value of 4.5701 (14), obtained in previous work on Al isotopes at COLLAPS [32].However, this constraint was not applied to the present 27 Al part of the COL-LAPS analysis as the analysed spectra were a subset of the measurements examined in Ref. [32].
For 26,26m Al, a model of the I = 5 ground state and one of the I = 0 isomeric state were superimposed.Within each experimental campaign, all 26,26m Al resonance spectra were fitted simultaneously with the same, shared hyperfine parameters as long as a parameter was not otherwise constrained, see above.Similarly, the isomer shift between ground and isomeric state in 26 Al was implemented as a shared fit parameter across a campaign's entire data set.The isomer centroid ν 26m 0 itself was freely varied for each individual spectrum.For the determination of ν 26m 0 , the Doppler-tuning voltage was converted into frequency based on the isomer's ionic mass.It was verified in fits of simulated spectra that this approach led to accurate results despite the peak overlap with the resonance spectrum of the ground state.
Voigt profiles were chosen for the lineshapes of individ-ual resonance peaks with no intensity constraints in the ground state.The Lorentzian and Gaussian widths were shared between ground state and isomer peaks within each individual spectrum but not shared overall.Due to inelastic collisions in the charge-exchange cell [38,43,44], four equidistant side peaks were considered in the analysis of the COLLAPS data [32].The energy offset of these sidepeaks was determined empirically and the relative intensities were constrained by Poisson's law.Because of lower statistics, the IGISOL data were found to be insensitive to the inclusion of these sidepeaks, thus, they were not considered in the analysis.Each spectrum of 26,26m Al was measured in sequence with an independent 27 Al reference measurement.The isotope shift δν 27,26m = ν 26m 0 − ν 27 0 of each measurement pair was calculated from the frequency centroid ν 26m 0 of 26m Al with respect to the frequency centroid ν 27 0 of the closest 27 Al reference measurement.The results of all individual δν 27,26m determinations are shown in Fig. 1f.Weighted averages in δν 27,26m are calculated separately for the COLLAPS and IGISOL data sets, see Tab.I.
Systematic uncertainties in CLS for measurements of isotope shifts are well understood [45][46][47][48] and are dominated by the imperfect knowledge of the beam energy.The acceleration voltage from the cooler-buncher at IGISOL was calibrated by matching measured isotope shifts in the D1 and D2 lines for singly-charged ions of stable magnesium isotopes to their precisely known literature values in Ref. [49].The remaining uncertainty in beam energy was 1.8 eV.An additional 1 × 10 −4 relative uncertainty was assigned to the scanning voltage in the Doppler tuning.For the COLLAPS data, a 1.5 × 10 −4 relative uncertainty of the incoming ion beam energy was assigned following the specifications of the employed voltage divider (Ohmlabs KV-30A).This was combined with the uncertainties of the calibrated JRL KV10 voltage divider used to measure the scanning voltage and of the employed voltmeters (Agilent 34661A).
Since the systematic uncertainties at COLLAPS and IGISOL were fully independent, statistical and systematic uncertainties of each measurement campaign were first added in quadrature before the weighted average of both measurement results was calculated, see Tab.I. Our final value for the isotope shift between 26m Al and 27 Al is δν 27,26m =377.5 (34) MHz.
With knowledge of the isotope shift δν 27,26m the difference in mean square nuclear charge radii δ⟨r 2 ⟩ between the two isotopes could be calculated according to [50]: , where m e is the electron mass [51] and m A are the nuclear masses obtained when 13 electrons are subtracted from the atomic masses [52] and an excitation energy of 228.305 keV [53] is added for 26m Al.Precision atomicphysics calculations were performed in a multiconfig-uration Dirac-Hartree-Fock framework to evaluate the field and mass shift factors F and M of the investigated atomic transition [32,54].Combining the adopted values of F =76.2 (22) MHz/fm 2 and M =−243(4) GHz u with the isotope shift δν 27,26m of the present work yields δ⟨r 2 ⟩ 27,26m = 0.429(88) fm 2 , see Tab.I. Finally, the root mean square (rms) nuclear charge radius of 26m Al can be derived: Using the previously evaluated rms charge radius of 27 Al, R c ( 27 Al)=3.061(6)fm [32], a value of R c ( 26m Al)=3.130(15) fm is obtained, see Tab.II.
Discussion. -Nuclear charge radii of superallowed β emitters are essential input parameters for the calculation of the ISB corrections δ C when a nuclear shell-model approach with Woods-Saxon radial wavefunctions is employed [27,28].Currently, these δ C calculations are the only ones considered to be sufficiently reliable to evaluate Ft values and thus V ud [10].In the shell-model approach, the ISB corrections are separated into two components, δ C = δ C1 + δ C2 .The former is associated with the configuration mixing within the restricted shell model space while the latter, known as the radial overlap correction, is derived from a phenomenological Woods-Saxon potential and it depends on the nuclear charge radius R c .
Since R c ( 26m Al) was previously unknown, the calculation of δ C2 used R c =3.040(20) fm [27], an extrapolation based on other, known nuclear charge radii.Our experimental result, R c ( 26m Al)=3.130(15) fm, deviates from this extrapolation by 4.5 standard deviations.This significantly impacts the radial overlap correction which is updated to δ C2 =0.310(14) % [55] compared to the previous 0.280(15) % [10].The impact of this sizable change in δ C2 are summarised in Fig. 2a and in Tab.II.
Despite 26m Al being the most accurately studied superallowed β emitter, the corrected Ft value is shifted by almost one full standard deviation to 3071.4(10) s.Its high precision is maintained but, in terms of R c in the Accounting for 0.57 s, this statistical uncertainty contains all experimental as well as those theoretical errors which scatter 'randomly' from one superallowed transition to another.Previously, a single systematic theoretical uncertainty of 0.36 s due to δ ′ R had to be added affecting all superallowed β emitters alike [56].In these circumstances, the shift in the Ft value caused by the new charge radius of 26m Al would have corresponded to ≈ 40% of its total uncertainty.In the latest survey of superallowed β decays [10], however, a systematic theoretical uncertainty of 1.73 s in δ N S was newly introduced, reflecting uncertainties due to previously unaccounted contributions to the nuclear-structure dependent radiative corrections.This represents an almost three-fold increase of the theoretical error associated with δ N S which now dominates the uncertainty in the Ft value.Considering our new charge radius of 26m Al, one thus obtains an Ft value of 3071.96(185)s.
The present work further implies a ∆ CKM in the unitarity test of the first row of the CKM matrix which is brought by ≈ 1/10 σ closer towards unitarity.Although the magnitude of this change is too small to resolve the tension to CKM unitarity, it illustrates the importance of a comprehensive examination of all relevant ingredients to V ud , especially theoretical corrections which involve nuclear-structure dependencies such as radiative and ISB corrections.In terms of δ C2 , there remain seven superallowed β emitters in which the nuclear charge radius is experimentally undetermined [57,58].Among those, 10 C and 14 O are of specific interest given their sensitivity to the Fierz interference term which relates to scalar contributions in β decays.Moreover, it has recently been  [10,56,[59][60][61][62][63] (black) compared to this work (orange).The vertical line to guide the eye corresponds to the value from 2020 [10].
proposed to constrain models of ISB corrections by new, more precise measurements of charge radii in triplets of the isobaric analog states, e.g. 38Ca -38m K -38 Ar [20].
Summary.-Collinear laser spectroscopy has been performed to determine the nuclear charge radius of 26m Al, the most precisely studied superallowed β emitter.The obtained value differs by 4.5 standard deviations from the extrapolation used in the calculation of the isospin-symmetry-breaking corrections [10,27].This notably impacts the corrected Ft value in 26m Al and, thus, the average of all Ft values used in the extraction of V ud .As demanded by the tension in CKM unitarity, this work contributes to the thorough examination of all nuclear-structure dependent corrections in superallowed β decays.Stimulated by the present results, efforts to measure experimentally undetermined charge radii of other cases, for example 54 Co at IGISOL/Jyväskylä, are currently ongoing.
We would like to express our gratitude to the ISOLDE collaboration and the ISOLDE technical teams, as well as the IGISOL collaboration and IGISOL technical teams for their support in the preparation and successful realisation of the experiments.We are thankful for all input and discussions that we received from Ian S. Towner to support this work.S.M-E. is grateful for fruitful discussions with G. Ball.
We acknowledge funding from the Federal Ministry of Education and Research under Contract No.
photon counts

TABLE I .
[32]ured isotope shift δν27,26mbetween27Al and 26m Al obtained at the IGISOL facility and at COL-LAPS/ISOLDE.The weighted average of the two measurements and the resulting difference in mean square charge radius δ⟨r 2 c ⟩ 27,26m is listed.Combined statistical and systematic uncertainties in parentheses.Uncertainty from atomic physics calculations of mass and field shift from[32]in angle brackets.calculation of δ C , the value now stands on a solid experimental basis.The updated Ft value of 26m Al also affects the Ft value, i.e. the weighted average over all 15 precisely studied superallowed β emitters, which is shifted by one half of its statistical uncertainty, see inset in Fig.2a.To our knowledge, this represents the largest shift in the Ft value since 2009, see Fig.2b. b

TABLE II .
Summary of the rms charge radius Rc, the radial overlap correction δC2 and the Ft value of 26m Al, the weighted average of the 15 superallowed β emitters Ft and the result of the CKM unitarity test.Ft values of the 15 superallowed β emitters used to determine V ud .The values in black, taken from [10], include experimental as well as 'statistical' theoretical errors.The previously determined Ft value for 26m Al [10] (blue) is compared to the one (orange) when considering the experimental nuclear charge radius of the present work.The weighted averages for the 15 superallowed β emitters are shown as horizontal bars in the inset (without considering additional, systematic theoretical uncertainties).(b) Evolution of the Ft value with statistical uncertainties in previous reviews