of neutron radii on heavier nuclei. A tree-level approach was used to extract the 27 Al weak radius R w ¼ 3.00 AE 0.15 fm, and the weak skin thickness R wk -R ch ¼ -0.04 AE 0.15 fm. The weak form factor at this Q 2 is F wk ¼ 0.39 AE 0.04. DOI: 10.1103/PhysRevLett.128.132501

As beam properties and experimental techniques have improved over the last two decades, so has the precision of parity-violating (PV) asymmetry measurements in elastic electron scattering. These experiments initially focused on carbon [1], then hydrogen and helium targets to study strange quark form factors [2]. The improving precision of these experiments has led to standard model tests [3,4], and even more recently neutron radius determinations in heavy nuclei [5,6] which impact our understanding of the structure and composition of neutron stars [7].

The proton's weak charge was determined in the Q weak experiment [4,8] by measuring the PV asymmetry in ⃗ ep elastic scattering with high precision at low Q 2 . By far the largest background in that experiment (≈24%) came from the aluminum alloy cell that contained the hydrogen. To accurately account for that background, precise additional asymmetry measurements were made on aluminum interspersed between data taking on hydrogen.

Those same aluminum asymmetry results that served to account for background in the Q weak experiment have been further analyzed in this Letter to isolate the 27 Al asymmetry A PV for elastic electron scattering at Q 2 ¼ 0.02357 GeV 2 . A successful comparison with theory [9] would provide additional confidence in the empirical background subtraction used in the Q weak experiment [4].

However, the most important aspect of the first 27 Al A PV measurement presented here is the test case it provides for the electroweak (EW) technique [10] used to determine the neutron radius R n of a complex nucleus in ⃗ eA scattering. In conjunction with the more easily determined proton radius R p , this also delivers the neutron skin R n -R p .

For a light complex nucleus like 27 Al with a neutron excess of only 1, we expect the neutron skin to be very thin. If this naïve expectation is confirmed by our measurement, it would serve as a benchmark for the application of the EW technique to heavier nuclei like 208 Pb, where the resulting neutron skin can be related to neutron star physics [7]. The EW technique has recently been applied to 208 Pb [5], and the resulting neutron skin was found to be in some tension with earlier non-EW results [11,12] which favor a thinner skin. The benchmark of the EW technique which our result can provide is especially important in light of this observed tension.

Beyond providing the 27 Al asymmetry A PV , neutron radius R n , and neutron skin thickness R n -R p , we also report the 27 Al weak form factor F wk at our Q 2 , the 27 Al weak radius R wk and weak skin thickness R wk -R ch , where R ch is the charge radius. R wk should closely track the neutron radius because the weak charge comes primarily from the neutrons-the proton's weak charge is much smaller [4].

A PV asymmetry is a nonzero difference between differential cross sections σ AE ðθÞ measured with a beam polarized parallel (þ) or antiparallel (-) to its incident momentum. In the Born approximation the elastic ⃗ e -27 Al asymmetry can be expressed [9] as

where G F is the Fermi constant, α is the fine structure constant, -Q 2 is the four-momentum transfer squared, Q W ¼ -12.92 AE 0.01 is the predicted [13] weak charge of 27 Al including all radiative corrections, and Z is the atomic number of 27 Al. F W ðQ 2 Þ and F EM ðQ 2 Þ are the weak and electromagnetic (EM) form factors for 27 Al, normalized to unity at Q 2 ¼ 0. This measurement was conducted in Hall C of Jefferson Lab using the Q weak experimental apparatus [14] and the polarized electron beam of the CEBAF accelerator. The helicity of the polarized electron beam was selected at a rate of 960 Hz, allowing the beam to be produced in a sequence of "helicity quartets," either ðþ --þÞ or ð-þþ-Þ, with the pattern chosen pseudorandomly at 240 Hz. In addition, every 8 h an insertable half-wave plate (IHWP) was placed in or out of the source laser's path to reverse the polarization direction. A "double Wien" spin rotator was also used to reverse the electron spin direction twice during the 27 Al data-taking.

A 60 μA longitudinally polarized 1. 16 GeV electron beam was incident on a 3.68 mm thick by a 2.54 cm square 7075-T651 aluminum alloy target. This target was machined from the same lot of material used for the LH 2 target window components of the weak charge measurement, so it could also be used to account for the background aluminum asymmetry that contaminated the measured hydrogen asymmetry [3,4]. Other elements in this alloy, as determined during a postexperiment assay, include Zn (5.87 wt%), Mg (2.63 wt%), Cu (1.81 wt%), and other (0.47 wt%).

Electrons scattered from the target were first selected by a series of three collimators and were then focused by a toroidal magnetic field onto an azimuthally symmetric array of eight synthetic-quartz Cherenkov detectors, each with a 2-cm-thick lead preradiator. The polar-angle (θ) acceptance was 5.8°to 11.6°, the azimuthal-angle acceptance was PHYSICAL REVIEW LETTERS 128, 132501 (2022) 132501-2 49% of 2π, and the energy acceptance was large: ≈150 MeV. Cherenkov light generated in the quartz from the passing electrons was collected by photomultiplier tubes (PMTs) attached to each end of each detector in the array. The current from the PMTs was integrated over each helicity state, normalized to the beam current, and then averaged together to form the raw asymmetry A raw , as shown in Fig. 1 and Table I. Several small systematic corrections were applied to A raw to derive a measured asymmetry A msr :

where A BCM is a beam current monitor (BCM) normalization uncertainty, A reg is a helicity-correlated beam motion correction, A BB is a beam line background correction, A L is a nonlinearity correction, A T is a residual transverseasymmetry correction, and A bias is a rescattering bias correction. Each of these corrections is discussed below.

The raw asymmetry charge normalization adopted the same technique and BCMs as used in the weak charge measurement [4], leading to a correction of A BCM ¼ 0.0 AE 2.1 ppb, dominated by the BCM accuracy.

Helicity-correlated variations in the beam position and energy also required a correction, A reg ¼ 0.4 AE 1.4 ppb. This was determined with a linear regression method [3,15], to correct the effects of natural beam motion using helicity-correlated differences measured with different beam position monitors.

Electrons in the beam halo interacted with beam line components causing a false asymmetry. Auxiliary detectors placed close to the beam line were used to form a correlation with the main detectors to correct for this false asymmetry. The overall correction was A BB ¼ -4.7 AE 6.6 ppb.

Nonlinearity effects in the main detector PMTs and BCMs used for asymmetry normalization were quantified in bench-top tests. The correction for this effect was A L ¼ -2.0 AE 7.0 ppb [15,16].

Any residual transverse components to the beam polarization will cause a parity-conserving azimuthal variation in the asymmetry, which coupled with imperfections in the azimuthal symmetry of the detectors may lead to a false asymmetry. This was measured using a transversely polarized beam [17] and scaled to the measured azimuthal variation in the present data, leading to a correction, A T ¼ -3.4 AE 8.8 ppb [15].

As described in earlier publications [4,8], lead preradiators placed in front of the main detectors were needed to reduce low-energy backgrounds. However, scattered electrons with spins precessing from longitudinal to transverse in the spectrometer magnetic field acquired an analyzing power from Mott scattering in the lead, which led to a correction of A bias ¼ 4.3 AE 3.0 ppb.

Determination of a purely elastic asymmetry A PV required additional corrections for beam polarization, background asymmetries, and a combination of radiative and acceptance corrections:

where R tot ¼ 0.9855 AE 0.0087, determined primarily by simulation [4], accounts for the radiative and finite acceptance effects, f i is the signal fraction of a particular background asymmetry, and A i is its corresponding asymmetry. These can be found in Table II.

The beam polarization was monitored continuously using a Compton polarimeter [18] and periodically with dedicated measurements using a Møller polarimeter [19]. Both were found to agree [20] during the experiment and yielded a combined polarization of P ¼ 88.80 AE 0.55%.

Nonelastically scattered electrons entering the large acceptance of the apparatus contaminated the measured asymmetry with backgrounds which had to be estimated FIG. 1. Raw asymmetries (statistical errors only) plotted against 8-h IHWP IN or OUT "data subsets" (lower axis), and monthly L or R Wien spin rotator orientation (upper axis). The configuration consistent with Eq. ( 1) is given by Wien Left and IHWP IN, i.e., IN L , which is equivalent to OUT R . The opposite sign asymmetry arises when either the Wien or the IHWP is flipped, but not both. During Wien A, there was an additional (g -2) spin flip which arose from running the JLab recirculating linac with two passes at half the gradient instead of one pass with full gradient. The green lines (bands) denote weighted averages (uncertainties) of the positive and negative asymmetries. and subtracted in Eq. ( 3). Nonelastic processes considered in this analysis include quasielastic, single-particle, and collective excitations, and inelastic scattering with a Δ in the final state. Correction for each of these backgrounds required knowledge of the fraction of events that fell into the acceptance, f i , derived from the cross section of each process at the kinematics of the experiment, and A i , the asymmetry for each process. Both of these were determined using models and/or experimental data from previous measurements. The relevant dilutions for each of these background processes were reported in Ref. [17]. The quasielastic asymmetry A QE was estimated for 27

Al from a relativistic Fermi gas model [21], with a conservative 50% relative uncertainty.

The inelastic asymmetry A inel was determined by dropping the spectrometer magnetic field to about 75% of its nominal value to move the inelastic events onto the detectors. The corresponding polarization-corrected 27 Al asymmetry

was briefly measured, with f 75 inel estimated from simulation to be ð20 AE 5Þ% on top of the elastic tail, and A 75 el scaled down from its value at full field by 1.181, the ratio of the corresponding Q 2 at each field. A value for A inel ¼ -0.58 AE 5.83 ppm at full field was obtained by solving Eq. ( 4) for A 75 inel and then scaling up by the Q 2 ratio.

Following the work of Ref. [9], the asymmetry for the giant dipole resonance was estimated using the Born approximation for an N ¼ Z nucleus, with a negative sign A GDR ¼ -2.2 AE 1.1 ppm appropriate for this isovector transition, and a conservative 50% relative uncertainty.

Asymmetries A nucl ≈ 2.5 ppm for the 11 strongest excited states of 27 Al up to 7.477 MeV were also obtained using the Born approximation for elastic scattering, with small corrections made for the acceptance-averaged Q 2 . States with large E2 transition rates or which were strongly populated by T ¼ 0 probes were assumed to be isoscalar and assigned 50% uncertainties. The remaining states were assumed to be isovector. Since the sign of the asymmetry depends on whether those isovector states were proton or neutron excitations, a 200% uncertainty was used to encompass both possibilities.

For the asymmetries A alloy associated with the contaminant elements in the alloy used for the target, the Born approximation calculation was again used as described in Ref. [9] for each of the dominant six elements. These calculations include Coulomb distortions, but assume spherically symmetric proton and neutron distributions, so only include the leading multipole term. As before, 50% uncertainties were used.

Background contributions from pions, neutrals, and the beam line were negligible, and are discussed in Ref. [15].

After all corrections, the elastic 27 Al asymmetry is

at Q 2 ¼ 0.02357 AE 0.00010 GeV 2 , which corresponds to hθ lab i¼7.61°AE 0.02°. This result, the first on 27 Al, agrees well with previously published distorted wave Born calculations [9] as shown in Fig. 2.

The neutron distribution radius R n was determined using a many-models correlation method first employed by the PREX Collaboration [22]. A selection of relativistic mean-field models [23][24][25][26][27][28][29] was chosen based on their ability to reasonably predict several nuclear structure observables: nucleon binding energies, charge radii, and strengths of isoscalar and isovector giant resonances in selected nuclei. The relationship between R n and A PV was found to be R n ¼ð-0.6007 AE 0.0002Þ

A PV ppm þð4.1817 AE 0.0011Þ fm ð6Þ with a correlation coefficient 0.997. Using this relation our final asymmetry yielded R n ¼ 2.89 AE 0.12 fm; see Fig. 3.

To determine the neutron skin R n -R p , we use the proton distribution radius R p following Ref. [30] for spherical nuclei,

where m N is the nucleon mass, and N denotes the number of neutrons. Here and below we use an 27 Al charge radius R ch ¼ 3.035 AE 0.002 fm [31], and correct for the proton charge radius hr p i¼0.8751 AE 0.0061 fm [32], the neutron charge radius hr 2 n i¼-0.1161 AE 0.0022 fm 2 [13], and a spin-orbit nuclear charge correction hr 2 so i¼-0.017 fm 2 following Ref. [30]. For consistency these parameters must be the same as those used to extract R n using Eq. ( 6).

The neutron skin is R n -R p ¼ -0.04 AE 0.12 fm, confirming the naive expectation for a light nucleus such as 27 Al where N ≈ Z that the neutron skin should be close to zero within our uncertainty. To illustrate the sensitivity of R p to its input parameters, using other recent values for hr p i [13] and R ch [33] would only raise R p by 1%, which is small compared with our 4.2% precision for R n .

In order to proceed to estimates of EW observables to which this experiment is sensitive (see Table III), we follow the Born approximation (tree-level) formulation presented in Ref. [34]. Although this leads only to approximate EW results, the 9.1% precision of our asymmetry is large enough to blunt the need for a more precise treatment. In addition, Fig. 2 shows that the Born approximation accurately predicts our asymmetry. Moreover, the relatively low Z of 27 Al reduces the corrections from Coulomb distortions (∝ Z) relative to a heavier nucleus like Pb.

Following Ref. [34],weintroduceatermΔ which accounts for hadronic and nuclear structure effects at Q 2 > 0:

where

Inserting our A PV result [Eq. ( 5) into either Eq. (1) or Eq. ( 8)], and using an F EM ¼ 0.384 AE 0.012 calculated following the prescription outlined in Ref. [35], we obtain a weak form factor F wk ðQ 2 ¼ 0.0236 GeV 2 Þ¼0.393 AE 0.038.T h eF EM calculation (corrected for small Coulomb distortions) is good to about 3% [35], which we verified by comparing with differential cross section data [36].

With our A PV result, Δ ¼ 0.025 AE 0.094. To lowest order in Q [34], from which we obtain R wskin ¼ -0.04 AE 0.15 fm, consistent as expected with our small neutron skin result. Employing the R ch introduced earlier, R wk ¼ 3.00 AE 0.15 fm. The relative difference between the weak and charge radii λ ≡ ðR wk -R ch Þ=R ch ¼ -1.3% AE 5.0%.

In conclusion, the agreement between predictions [9] and this first measurement of the elastic asymmetry on 27 Al supports the background procedures used in the Q weak experiment [4] on hydrogen. The tree-level EW results obtained above for R wk and R wskin are consistent with broad FIG. 2. Parity-violating asymmetry vs laboratory scattering angle. The measured value is shown with statistical (inner error bar) and total (outer error bar) uncertainties. The theoretical prediction [9] at our beam energy is shown for spherically symmetric neutron and proton densities in Born approximation (blue dots), for a distorted wave calculation with spherical densities (dashed green line) and the full calculation with nonspherical proton density (red solid line). The red shaded band indicates nuclear structure and Coulomb distortion uncertainties. FIG. 3. Models (symbols indicated in the legend) used to establish the correlation (Eq. ( 6), and solid black line) between the 27 Al A PV and its neutron radius R n . The dashed black lines indicate where on the many-models correlation plot the central value of our asymmetry determines R n . The shaded bands indicate the total uncertainty associated with our result. Similarly, our 27 Al neutron skin is close to zero, as expected, providing some validation and a benchmark for the application of the many-models approach and EW technique [10] to the measurement of heavier nuclei [5,6,22]. This is especially interesting in light of the tension which exists [11,[37][38][39] between the recent EW neutron skin determination R n -R p ¼ 0.283 AE 0.071 fm for 208 Pb [5], and the 2012 average of several disparate but self-consistent non-EW determinations R n -R p ¼ 0.184 AE 0.027 fm [12]. The older non-EW determinations have come under additional scrutiny and even some criticism recently [40]. However, we note that they appear to be more consistent with the latest constraints on neutron star properties from LIGO and Virgo (especially for the tidal deformability) [41], from NICER [7], and astrophysical models in general.