Accurate Determination of the Neutron Skin Thickness of ^{208}Pb through Parity-Violation in Electron Scattering
D. Adhikari, H. Albataineh, D. Androic, K. Aniol, D.S. Armstrong, T. Averett, S. Barcus, V. Bellini, R.S. Beminiwattha, J.F. Benesch, H. Bhatt, D. Bhatta Pathak, D. Bhetuwal, B. Blaikie, Q. Campagna, A. Camsonne, G.D. Cates, Y. Chen, C. Clarke, J.C. Cornejo, S. Covrig Dusa, P. Datta, A. Deshpande, D. Dutta, C. Feldman, E. Fuchey, C. Gal, D. Gaskell, T. Gautam, C. Ayerbe Gayoso, M. Gericke, C. Ghosh, I. Halilovic, J.-O. Hansen, F. Hauenstein, W. Henry, C.J. Horowitz, C. Jantzi, S. Jian, S. Johnston, D.C. Jones, B. Karki, S. Katugampola, C. Keppel, P.M. King, D.E. King, M. Knauss, K.S. Kumar, T. Kutz, N. Lashley-Colthirst, G. Leverick, H. Liu, N. Liyange, S. Malace, R. Mammei, J. Mammei, M. McCaughan, D. McNulty, D. Meekins, C. Metts, R. Michaels, M.M. Mondal, J. Napolitano, A. Narayan, D. Nikolaev, M.N.H. Rashad, V. Owen, C. Palatchi, 14 J. Pan, B. Pandey, S. Park, K.D. Paschke, M. Petrusky, M.L. Pitt, S. Premathilake, A.J.R. Puckett, B. Quinn, R. Radloff, S. Rahman, A. Rathnayake, B.T. Reed, P.E. Reimer, R. Richards, S. Riordan, Y. Roblin, S. Seeds, A. Shahinyan, P. Souder, L. Tang, 16 M. Thiel, Y. Tian, G.M. Urciuoli, E.W. Wertz, B. Wojtsekhowski, B. Yale, T. Ye, A. Yoon, A. Zec, W. Zhang, J. Zhang, X. Zheng
AAn Accurate Determination of the Neutron Skin Thickness of
Pb throughParity-Violation in Electron Scattering (The PREX Collaboration)D. Adhikari, H. Albataineh, D. Androic, K. Aniol, D.S. Armstrong, T. Averett, S. Barcus, V. Bellini, R.S. Beminiwattha, J.F. Benesch, H. Bhatt, D. Bhatta Pathak, D. Bhetuwal, B. Blaikie, Q. Campagna, A. Camsonne, Y. Chen, C. Clarke, J.C. Cornejo, S. Covrig Dusa, P. Datta, A. Deshpande,
11, 14
D. Dutta, C. Feldman, E. Fuchey, C. Gal,
11, 15, 14
D. Gaskell, T. Gautam, C. Ayerbe Gayoso, M. Gericke, C. Ghosh,
17, 11
I. Halilovic, J.-O. Hansen, F. Hauenstein, W. Henry, C.J. Horowitz, C. Jantzi, S. Jian, S. Johnston, D.C. Jones, B. Karki, S. Katugampola, C. Keppel, P.M. King, D.E. King, M. Knauss, K.S. Kumar, T. Kutz, N. Lashley-Colthirst, G. Leverick, H. Liu, N. Liyange, S. Malace, R. Mammei, J. Mammei, M. McCaughan, D. McNulty, D. Meekins, C. Metts, R. Michaels, M.M. Mondal,
11, 14
J. Napolitano, A. Narayan, D. Nikolaev, M.N.H. Rashad, V. Owen, C. Palatchi,
15, 14
J. Pan, B. Pandey, S. Park, K.D. Paschke, ∗ M. Petrusky, M.L. Pitt, S. Premathilake, A.J.R. Puckett, B. Quinn, R. Radloff, S. Rahman, A. Rathnayake, B.T. Reed, P.E. Reimer, R. Richards, S. Riordan, Y. Roblin, S. Seeds, A. Shahinyan, P. Souder, L. Tang,
6, 16
M. Thiel, Y. Tian, G.M. Urciuoli, E.W. Wertz, B. Wojtsekhowski, B. Yale, T. Ye, A. Yoon, A. Zec, W. Zhang, J. Zhang,
11, 14, 32 and X. Zheng Idaho State University, Pocatello, ID 83209, USA Texas A & M University - Kingsville, Kingsville, TX 78363, USA University of Zagreb, Faculty of Science California State University, Los Angeles, Los Angeles, California 90032, USA William & Mary, Williamsburg, Virginia 23185, USA Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Istituto Nazionale di Fisica Nucleare, Sezione di Catania, 95123 Catania, Italy Louisiana Tech University, Ruston, LA 71272 USA Mississippi State University, Mississippi State, MS 39762, USA University of Manitoba, Winnipeg, MB R3T2N2 Canada Stony Brook, State University of New York, NY 11794, USA Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA University of Connecticut, Storrs, CT 06269, USA Center for Frontiers in Nuclear Science, NY 11794, USA University of Virginia, Charlottesville, VA 22904, USA Hampton University, Hampton, Virginia 23668, USA University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA Old Dominion University, Norfolk, Virginia 23529, USA Temple University, Philadelphia, PA 19122, USA Indiana University, Bloomington, Indiana 47405, USA Ohio University, Athens, Ohio 45701, USA Syracuse University, Syracuse, New York 13244, USA Duquesne University, 600 Forbes Avenue, Pittsburgh, PA 15282, USA University of Winnipeg, Winnipeg, MB R3B2E9 Canada Veer Kunwar Singh University, India Virginia Tech, Blacksburg, Virginia 24061, USA Physics Division, Argonne National Laboratory, Lemont, Il 60439 A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute), Yerevan 0036, Armenia Institut f¨ur Kernphysik, Johannes Gutenberg-Universit¨at, Mainz 55122, Germany INFN - Sezione di Roma, I-00185, Rome, Italy Christopher Newport University, Newport News, Virginia 23606, USA Shandong University, Qingdao, Shandong 266237, China (Dated: February 23, 2021)We report a precision measurement of the parity-violating asymmetry A PV in the elastic scatter-ing of longitudinally polarized electrons from Pb. We measure A PV = 550 ± ± F W ( Q = 0 . ) =0 . ± . R n − R p = 0 . ± .
071 fm. The result also yields the first significant direct measurement of theinterior weak density of
Pb: ρ W = − . ± . . ) ± . . ) fm − leading to theinterior baryon density ρ b = 0 . ± . . ) ± . . ) fm − . The measurement accu-rately constrains the density dependence of the symmetry energy of nuclear matter near saturationdensity, with implications for the size and composition of neutron stars. a r X i v : . [ nu c l - e x ] F e b The equation of state (EOS) of nuclear matter [1–5]underlies the structure and stability of atomic nuclei, theformation of the elements, whether stars collapse intoneutron stars or black holes, and the structure of neutronstars themselves. It is remarkable that the physics ofsystems that vary in size by eighteen orders of magnitudeare governed by the same EOS.Observed properties of the full range of atomic nuclei,characterized by a nearly constant central density, pro-vides critical input to the EOS which is in turn applied toinfer the properties of neutron stars, first discovered byJocelyn Bell Burnell [6]. The EOS has been used to ruleout the possibility that the recently observed 2.6 solarmass object is a neutron star [7, 8], and could be used toinfer evidence of new forms of nuclear matter, such as thepresence of a significant non-zero strangeness componentin the neutron star interior [9, 10].Additional constraints to the EOS are obtained fromdetailed studies of neutron star properties (such as size,structure, and cooling). For example, the NICER X-raytelescope has determined a pulsar radius to better than10% [11], and gravitational wave data from LIGO from aneutron star merger event has constrained neutron startidal deformability [12–18].The extensive data on atomic nuclei used by EOSmodels do not yet constrain one critical EOS parame-ter, namely L , the density dependence of the symmetryenergy. Recent progress with chiral effective field the-ory has improved theoretical constraints on L [19]. Apromising avenue to obtain experimental constraints uti-lizes the strong correlation between L and the neutronskin thickness in heavy nuclei R n − R p , that is the dif-ference between the rms radii of the neutron and pro-ton distributions. Precise data on R p are available butnumerous experimental methods to determine R n sufferfrom uncontrolled uncertainties due to hadron dynam-ics [5].A more accurately interpretable method is to measurethe neutral weak form factor F W in elastic electron- Pbscattering, exploiting the significantly larger coupling ofthe Z boson to neutrons compared to protons [20, 21]to achieve an accurate R n extraction. Such measure-ments can provide insights into the dependence of thesymmetry energy on three-nucleon interactions [22] andits role in relativistic heavy-ion collisions [23]. Weakform factors of heavy nuclei lead to a more direct ex-traction of the nuclear central density, which is governedby multi-nucleon interactions [24] and may ultimatelybridge to Quantum Chromodynamics [25]. They providecritical input to dark matter searches and neutrino scat-tering [26] and facilitate tests of neutrino-quark neutralcurrent couplings via measurements of coherent elasticneutrino-nuclear scattering [27].A precise F W extraction can be accomplished by mea-suring the parity-violating asymmetry A P V in longitudi- nally polarized elastic electron scattering off
Pb nuclei: A P V = σ R − σ L σ R + σ L ≈ G F Q | Q W | √ παZ F W ( Q ) F ch ( Q ) , (1)where σ L ( σ R ) is the cross section for the scattering ofleft(right) handed electrons from Pb, G F is the Fermicoupling constant, F ch is the charge form factor, and Q W is the weak charge of Pb. The practical application ofthis formula requires the inclusion of Coulomb distor-tions [28] and experimental parameter optimization suchthat a single kinematic point yields a precise R n deter-mination [21]. The first measurement of A meas P V for
Pbwas published in 2012 [29] (PREX-1); here we report anew result with greatly improved precision (PREX-2).The data measuring A meas P V totaled 114 Coulombs ofcharge from a 953 MeV electron beam on a diamond-lead-diamond sandwich target at an average current of70 µ A in experimental Hall A [30] at Thomas JeffersonNational Accelerator Facility (JLab). The average thick-nesses of the diamond and lead foils, each known to bet-ter than 5% accuracy, were 90 mg/cm and 625 mg/cm respectively. The scattered electrons that passed theacceptance-defining collimator at the entrance of eachHigh Resolution Spectrometer (HRS) [30] were momen-tum analysed and focused by three magnetic quadrupolesand a dipole. Both the Left and Right HRS wereequipped with identical detector packages and were po-sitioned at their most forward angle ≈ ◦ . A sep-tum magnet pair extended the reach of the spectrom-eters to the average desired laboratory scattering angleof ≈ ◦ . The spectrometer achieved a momentum resolu-tion of 0.6 MeV, ensuring that the detector interceptedonly elastic events; the closest inelastic state at 2.6 MeVwas ≈ − − + or + − − + − + + − flip sequence or its com-plement, ensuring cancellation of 60 Hz power line noise.A blinding offset was added to each sequence asymmetryduring decoding and maintained throughout the analy-sis. The data set contained a little over 50 million suchsequences.Approximately every 8 hours, a half-wave plate (HWP)in the injector laser setup was toggled IN or OUT, fa-cilitating a complete asymmetry sign reversal with noother change. The data taken between each such rever-sal were combined into “slugs”. Furthermore, spin ma-nipulation in the injector beam line (using the “double-Wien” [31]) was changed twice during the run to adda 180 ◦ precession, thereby flipping the measured asym-metry sign. With approximately equal amounts of dataat each HWP/Wien state combination, these slow rever-sals provided critical additional cancellation of potentialsources of spurious asymmetries.The scattering angle was calibrated using the differ-ence in nuclear recoil between scattering from hydrogenand heavier nuclei in a water target, with tracks mea-sured using the Vertical Drift Chambers in the HRS [30].The rate-averaged scattering angle was determined to be4 . ± . ◦ and 4 . ± . ◦ for the left and right HRS re-spectively, with an average 4-momentum transfer squaredof Q = 0 . ± . .The beam current was monitored with three radiofre-quency (RF) cavity beam current monitors (BCMs). Theintegrated charge asymmetry between positive and neg-ative helicity bunches was determined every 7.5 seconds,and fed back to a control system which used the injectorPC to minimize this quantity. The cumulative chargecorrection was 20 . ± . A meas P V statistical uncertainty. RF beam positionmonitors (BPMs) were used to monitor the beam trajec-tory throughout the accelerator complex. Careful config-uration of the polarized electron source ensured that thehelicity-correlated difference in the electron beam trajec-tory was small: ≈ ≈ × . × . ) in each spec-trometer. With the long side of each tile oriented alongthe dispersive direction, approximately 7 cm was used tosample the elastically scattered electrons. The rest of thetile was a light-guide to the photomultiplier tube (PMT)on the high-energy side of the elastic peak. The largescattered flux ( ≈ A meas P V = 550 ppb fromthe full data set of 96 slugs.The beam asymmetry correction accounts for helicity-correlated fluctuations in the beam trajectory (position
TABLE I. Corrections and systematic uncertainties to extract A meas PV listed on the bottom row with its statistical uncertainty.Correction Absolute [ppb] Relative [%]Beam asymmetry − . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . A meas PV and statistical error 550 ±
16 100 . ± . - - - sequence asymmetry [%] DataGaussian fit = 93 ppm s FIG. 1. Distribution of 30 million asymmetries measuredover 1/30 s sequences formed with 240 Hz helicity flips. Onlydata taken with a beam current near to 70 µ A is included. and angle in two transverse coordinates) and energy. Aset of 6 BPMs measured the transverse coordinates at lo-cations of varying energy dispersion. The correction wascalculated using a regression analysis over all measuredcoordinates, constrained to be consistent with the dedi-cated modulation data, thus optimizing precision whileaccounting for instrumental correlated noise and reso-lution. The corrections were consistent throughout thedata set, and for the grand average, with the alternative(but less precise) methods based on only regression ordirect modulation-calibrated sensitivities.The asymmetry data are free from any unanticipatedbias as can be seen in Fig. 1, which shows the distributionafter beam corrections of the sequence asymmetry fordata collected with 240 Hz flip rate and 70 µ A beam cur-rent ( ≈
62% of the statistics). The remarkably high levelof agreement between the data and the normal distribu-tion fit over five orders of magnitude is achieved withoutthe application of a single helicity-correlated data qualitycut on any measured parameter.The cumulative beam asymmetry correction was − . ± . - - - normalized deviation from average Entries: 5084 0.01 – = 0.99 s /ndf = 93.5/84 c FIG. 2. Distribution of normalized deviations from the av-erage (blue) for ≈ The beam-corrected asymmetry data are dominated bystatistical fluctuations around a single mean, as demon-strated in Fig. 2. This plot shows the deviations from thegrand average asymmetry for all 5084 ≈ ≈ A corr must be furthercorrected for the beam polarization ( P b ), and the back-ground dilutions ( f i ) and asymmetries ( A i ) to obtain A meas P V : A meas P V = 1 P b A corr − P b (cid:80) i A i f i − (cid:80) i f i . (2)The degree of longitudinal polarization P b of the elec-tron beam was maximized at the beginning of data takingusing the injector Mott polarimeter [33]. It was periodi-cally measured just in front of the target using a Møller polarimeter [30] in dedicated low current runs that wereinterspersed throughout the data taking period. The av-erage beam polarization result was (89 . ± . f C = 6 . ± . A P V for C and
Pb are numerically similar. The effect ofa tiny amount of scattering from magnetized pole-tips inthe spectrometer was found to be negligible. A 0.26 ppbsystematic uncertainty accounted for a possible imperfectcancellation from a residual transverse electron beam po-larization component; no correction was applied.The linear response of the integrated detector signalwas demonstrated to be better than 0.5% in a bench testusing a calibration system with multiple light sources.The linearity of the detector response was also monitoredthroughout the data taking period by comparison withBCM measurements of beam current fluctuations. Theresulting systematic uncertainty was 2 . χ for averaging over the slugs in each configurationis shown. TABLE II. A meas PV for different HWP-Wien state combinations.HWP/Wien A corr sign A meas PV [ppb] χ − ± ± ± − ± For a direct comparison of the measurement to theoret-ical predictions one must convolve the predicted asymme-try variation with the acceptance of the spectrometers: (cid:104) A P V (cid:105) = (cid:82) d θ sin θA ( θ ) d σ dΩ (cid:15) ( θ ) (cid:82) d θ sin θ d σ dΩ (cid:15) ( θ ) , (3)where d σ dΩ is the differential cross section and A ( θ ) is themodeled parity violating asymmetry as a function of scat-tering angle. The acceptance function (cid:15) ( θ ) is defined asthe relative probability for an elastically scattered elec-tron to make it to the detector [34]. The systematicuncertainty in (cid:15) ( θ ) was determined using a simulationthat took into account initial and final state radiationand multiple scattering.Our final results for A meas P V and F W with the acceptancedescribed by (cid:15) ( θ ) and (cid:104) Q (cid:105) = 0 . are: A meas P V = 550 ±
16 (stat . ) ± . ) ppb F W ( (cid:104) Q (cid:105) ) = 0 . ± .
013 (exp . ) ± .
001 (theo . ) . where the experimental uncertainty in F W includes bothstatistical and systematic contributions. Big AppleSIIIFSUgold IUFSUNL3 SLY4TAMUaTAMUbTAMUc PREX-2 = 5.503 fm ch charge radius R Pb [ f m ] W w eak r a d i u s R
520 540 560 580 600 620 [ ppb ] PV PV asymmetry A [ f m ] p - R n n e u t r on sk i n R FIG. 3. Extraction of the weak radius (left vertical axis)or neutron skin (right vertical axis) for the
Pb nucleus. R ch [35] is shown for comparison. The correlation between A P V and the
Pb weak ra-dius R W is obtained by plotting the predictions for thesetwo quantities from a sampling of theoretical calcula-tions [8, 36–41], as shown in Fig. 3, along with the greenband highlighting A meas P V and its 1- σ experimental uncer-tainty.Single nucleon weak form factors are folded with pointnucleon radial densities to arrive at the weak density dis-tribution ρ W ( r ), using Q W = − . ± . γ -Z boxcontributions [42–45] as an overall constraint. The cor-relation slope in Fig. 3 is determined by fitting ρ W ( r )as a 2-parameter Fermi function over a large variety ofrelativistic and nonrelativistic density functional models,determining for each model a size consistent with R W and a surface thickness a . This also determines the smallmodel uncertainty, shown in Fig. 3 (dashed red lines),corresponding to the range of a [24, 46, 47].Projecting to the model correlation to determine theweak radius or alternatively the neutron skin (left andright vertical axes respectively), the PREX-2 results are R W = 5 . ± .
082 (exp . ) ± .
013 (theo . ) fm and R n − R p = 0 . ± .
078 (exp . ) ± .
012 (theo . ) fm.The normalization constant in the Fermi-function formof ρ W ( r ) used to extract R W is a measure of the Pbinterior weak density [47]: ρ W = − . ± . . ) ± . . ) fm − . Combined with the well-measured interior charge density,the interior baryon density determined from the PREX-2 data is ρ b = 0 . ± . − (combining experimen-tal and theoretical uncertainties).This result is consistent with the results from thePREX-1 measurement, which found R n − R p = 0 . ± .
18 fm [48]. Table III summarizes nuclear properties of
Pb from the combined PREX-1 and PREX-2 results,including a 4 σ determination of the neutron skin. TABLE III. PREX combined experimental results for
Pb.Uncertainties include both experimental and theoret-ical contributions.
Pb Parameter ValueWeak radius ( R W ) 5 . ± .
075 fmInterior weak density ( ρ W ) − . ± . − Interior baryon density ( ρ b ) 0 . ± . − Neutron skin ( R n − R p ) 0 . ± .
071 fm radius r [ fm ] ] - [ f m r d e n s i t y Weak skin b r Interior Baryon Density W r - Extracted from PREX W R ch r ch R Pb data ch r FIG. 4.
Pb weak and baryon densities from the combinedPREX data sets, with uncertainties shaded. The charge den-sity [35] is also shown.
Exploiting the strong correlation between R n − R p and the density dependence of the symmetry energy L , the PREX result implies a stiff symmetry energy( L = 106 ±
37 MeV [49]), with important implicationsfor critical neutron star observables. Figure 4 showsthe inferred radial dependence of the
Pb charge, weakand total baryon densities together with their uncertaintybands. The precise 2.5% determination of ρ b for Pbwill facilitate a sensitive examination of its close relation-ship to the nuclear saturation density [24].After the
Pb run, data were also collected to mea-sure A meas P V for Ca (CREX) [50]. The improved sys-tematic control of helicity correlated beam asymmetriesand several other PREX experimental innovations willinform the design of future projects MOLLER [51] andSoLID [52] at JLab measuring fundamental electroweakcouplings, as well as a more precise
Pb radius experi-mental proposal at Mainz [5, 53].We thank the entire staff of JLab for their efforts todevelop and maintain the polarized beam and the exper-imental apparatus, and acknowledge the support of theU.S. Department of Energy, the National Science Foun-dation and NSERC (Canada). This material is basedupon the work supported by the U.S. Department of En-ergy, Office of Science, Office of Nuclear Physics contractDE-AC05-06OR23177. ∗ [email protected][1] S. Novario, G. Hagen, G. Jansen, and T. Papenbrock,Charge radii of exotic neon and magnesium isotopes,Phys. Rev. C , 051303 (2020), arXiv:2007.06684[nucl-th].[2] H. Shen, F. Ji, J. Hu, and K. Sumiyoshi, Effects ofsymmetry energy on equation of state for simulations ofcore-collapse supernovae and neutron-star mergers, As-trophys. J. , 148 (2020), arXiv:2001.10143 [nucl-th].[3] C. Horowitz, Neutron rich matter in the laboratoryand in the heavens after GW170817, Annals Phys. ,167992 (2019), arXiv:1911.00411 [astro-ph.HE].[4] J.-B. Wei, J.-J. Lu, G. Burgio, Z. Li, and H.-J.Schulze, Are nuclear matter properties correlated to neu-tron star observables?, Eur. Phys. J. A , 63 (2020),arXiv:1907.08761 [nucl-th].[5] M. Thiel, C. Sfienti, J. Piekarewicz, C. Horowitz, andM. Vanderhaeghen, Neutron skins of atomic nuclei:per aspera ad astra, J. Phys. G , 093003 (2019),arXiv:1904.12269 [nucl-ex].[6] A. Hewish, S. Bell, J. Pilkington, P. Scott, and R. Collins,Observation of a rapidly pulsating radio source, Nature , 709 (1968).[7] R. Abbott et al. (LIGO Scientific, Virgo), GW190814:Gravitational Waves from the Coalescence of a 23 SolarMass Black Hole with a 2.6 Solar Mass Compact Object,Astrophys. J. Lett. , L44 (2020), arXiv:2006.12611[astro-ph.HE].[8] F. Fattoyev, C. Horowitz, J. Piekarewicz, and B. Reed,GW190814: Impact of a 2.6 solar mass neutron star onnucleonic equations of state, Phys. Rev. C , 065805(2020), arXiv:2007.03799 [nucl-th].[9] L. Tolos and L. Fabbietti, Strangeness in Nuclei and Neu-tron Stars, Prog. Part. Nucl. Phys. , 103770 (2020),arXiv:2002.09223 [nucl-ex].[10] M. Fortin, A. R. Raduta, S. Avancini, andC. Providˆencia, Relativistic hypernuclear compactstars with calibrated equations of state, Phys. Rev. D , 034017 (2020), arXiv:2001.08036 [hep-ph].[11] T. E. Riley et al. , A NICER
View of PSR J0030+0451:Millisecond Pulsar Parameter Estimation, Astrophys. J.Lett. , L21 (2019), arXiv:1912.05702 [astro-ph.HE].[12] K. Chatziioannou, Neutron star tidal deformability andequation of state constraints, Gen. Rel. Grav. , 109(2020), arXiv:2006.03168 [gr-qc].[13] Y. Zhang, M. Liu, C.-J. Xia, Z. Li, and S. K. Biswal,Constraints on the symmetry energy and its associatedparameters from nuclei to neutron stars, Phys. Rev. C , 034303 (2020), arXiv:2002.10884 [nucl-th].[14] H. G¨uven, K. Bozkurt, E. Khan, and J. Margueron, Mul- timessenger and multiphysics Bayesian inference for theGW170817 binary neutron star merger, Phys. Rev. C , 015805 (2020), arXiv:2001.10259 [nucl-th].[15] L. Baiotti, Gravitational waves from neutron star merg-ers and their relation to the nuclear equation ofstate, Prog. Part. Nucl. Phys. , 103714 (2019),arXiv:1907.08534 [astro-ph.HE].[16] J. Piekarewicz and F. Fattoyev, Neutron rich matterin heaven and on Earth, Physics Today , 30 (2019),arXiv:1907.02561 [nucl-th].[17] M. Tsang, W. Lynch, P. Danielewicz, and C. Tsang,Symmetry energy constraints from GW170817 and lab-oratory experiments, Phys. Lett. B , 533 (2019),arXiv:1906.02180 [nucl-ex].[18] M. Fasano, T. Abdelsalhin, A. Maselli, and V. Fer-rari, Constraining the Neutron Star Equation of StateUsing Multiband Independent Measurements of Radiiand Tidal Deformabilities, Phys. Rev. Lett. , 141101(2019), arXiv:1902.05078 [astro-ph.HE].[19] C. Drischler, R. J. Furnstahl, J. A. Melendez, and D. R.Phillips, How well do we know the neutron-matter equa-tion of state at the densities inside neutron stars? abayesian approach with correlated uncertainties, Phys.Rev. Lett. , 202702 (2020).[20] T. Donnelly, J. Dubach, and I. Sick, Isospin Dependencesin Parity Violating Electron Scattering, Nucl. Phys. A , 589 (1989).[21] C. Horowitz, S. Pollock, P. Souder, and R. Michaels, Par-ity violating measurements of neutron densities, Phys.Rev. C , 025501 (2001), arXiv:nucl-th/9912038.[22] W. Bentz and I. C. Clo¨et, Symmetry energy of nuclearmatter and isovector three-particle interactions (2020),arXiv:2004.11605 [nucl-th].[23] H. Li, H.-j. Xu, Y. Zhou, X. Wang, J. Zhao, L.-W. Chen,and F. Wang, Probing the neutron skin with ultrarela-tivistic isobaric collisions, Phys. Rev. Lett. , 222301(2020), arXiv:1910.06170 [nucl-th].[24] C. Horowitz, J. Piekarewicz, and B. Reed, Insightsinto nuclear saturation density from parity violatingelectron scattering, Phys. Rev. C , 044321 (2020),arXiv:2007.07117 [nucl-th].[25] C. Drischler, W. Haxton, K. McElvain, E. Mereghetti,A. Nicholson, P. Vranas, and A. Walker-Loud, To-wards grounding nuclear physics in QCD (2019)arXiv:1910.07961 [nucl-th].[26] J. Yang, J. A. Hernandez, and J. Piekarewicz, Elec-troweak probes of ground state densities, Phys. Rev. C , 054301 (2019), arXiv:1908.10939 [nucl-th].[27] D. Akimov et al. (COHERENT), Observation of Co-herent Elastic Neutrino-Nucleus Scattering, Science ,1123 (2017), arXiv:1708.01294 [nucl-ex].[28] C. Horowitz, Parity violating elastic electron scatteringand Coulomb distortions, Phys. Rev. C , 3430 (1998),arXiv:nucl-th/9801011.[29] S. Abrahamyan et al. , Measurement of the NeutronRadius of 208Pb Through Parity-Violation in Elec-tron Scattering, Phys. Rev. Lett. , 112502 (2012),arXiv:1201.2568 [nucl-ex].[30] J. Alcorn et al. , Basic Instrumentation for Hall A at Jef-ferson Lab, Nucl. Instrum. Meth. A , 294 (2004).[31] C. Sinclair, P. Adderley, B. Dunham, J. Hansknecht,P. Hartmann, M. Poelker, J. Price, P. Rutt, W. Schnei-der, and M. Steigerwald, Development of a high averagecurrent polarized electron source with long cathode oper- ational lifetime, Phys. Rev. ST Accel. Beams , 023501(2007).[32] T. Allison et al. , The Qweak Experimental Apparatus,Nucl. Intrum. Meth. A , 105 (2013).[33] J. M. Grames et al. , High precision 5 MeV Mott po-larimeter, Phys. Rev. C , 015501 (2020).[34] See Supplemental Material for this function. (2021).[35] H. De Vries, C. De Jager, and C. De Vries, Nuclearcharge-density-distribution parameters from elastic elec-tron scattering, Atomic Data and Nuclear Data Tables , 495 (1987).[36] M. Beiner, H. Flocard, N. van Giai, and P. Quentin, Nu-clear ground state properties and selfconsistent calcula-tions with the Skyrme interactions: 1. Spherical descrip-tion, Nucl. Phys. A , 29 (1975).[37] B. G. Todd-Rutel and J. Piekarewicz, Neutron-Rich Nu-clei and Neutron Stars: A New Accurately CalibratedInteraction for the Study of Neutron-Rich Matter, Phys.Rev. Lett. , 122501 (2005), arXiv:nucl-th/0504034.[38] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, andG. Shen, Relativistic effective interaction for nuclei, giantresonances, and neutron stars, Phys. Rev. C , 055803(2010), arXiv:1008.3030 [nucl-th].[39] G. A. Lalazissis, J. Konig, and P. Ring, A Newparametrization for the Lagrangian density of relativis-tic mean field theory, Phys. Rev. C , 540 (1997),arXiv:nucl-th/9607039.[40] F. J. Fattoyev and J. Piekarewicz, Has a thick neutronskin in Pb been ruled out?, Phys. Rev. Lett. ,162501 (2013), arXiv:1306.6034 [nucl-th].[41] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, andR. Schaeffer, A Skyrme parametrization from subnuclearto neutron star densities. 2. Nuclei far from stablities,Nucl. Phys. A , 231 (1998), [Erratum: Nucl.Phys.A643, 441–441 (1998)].[42] J. Erler and S. Su, The Weak Neutral Current, Prog.Part. Nucl. Phys. , 119 (2013), arXiv:1303.5522 [hep-ph]. [43] M. Gorchtein and C. J. Horowitz, Dispersion gamma Z-box correction to the weak charge of the proton, Phys.Rev. Lett. , 091806 (2009), arXiv:0811.0614 [hep-ph].[44] M. Gorchtein, C. J. Horowitz, and M. J. Ramsey-Musolf,Model-dependence of the γZ dispersion correction to theparity-violating asymmetry in elastic ep scattering, Phys.Rev. C , 015502 (2011), arXiv:1102.3910 [nucl-th].[45] J. Erler and M. Gorchtein, (private communication) (Oct2020).[46] B. Reed, Z. Jaffe, C. J. Horowitz, and C. Sfienti, Measur-ing the surface thickness of the weak charge density of nu-clei, Phys. Rev. C , 064308 (2020), arXiv:2009.06664[nucl-th].[47] See Supplemental Material for more information. (2021).[48] C. J. Horowitz et al. , Weak charge form factor and radiusof 208Pb through parity violation in electron scattering,Phys. Rev. C , 032501 (2012), arXiv:1202.1468 [nucl-ex].[49] B. T. Reed, F. J. Fattoyev, C. J. Horowitz, andJ. Piekarewicz, Implications of PREX-II on the equationof state of neutron-rich matter (2021), arXiv:2101.03193[nucl-th].[50] S. Riordan et al. (CREX), CREX: Parity Violating Mea-surement of the Weak Charge Distribution of 48Ca to0.02 fm Accuracy , Tech. Rep. JLAB-PR-40-12-004 (TJ-NAF, 2013).[51] J. Benesch et al. (MOLLER), The MOLLER Ex-periment: An Ultra-Precise Measurement of theWeak Mixing Angle Using Møller Scattering (2014),arXiv:1411.4088 [nucl-ex].[52] P. Souder et al. (SoLID),
Precision Measurement ofParity-violation in Deep Inelastic Scattering Over aBroad Kinematic Range , Tech. Rep. JLAB-PR-09-012-pvdis (TJNAF, 2008).[53] D. Becker et al. , The P2 experiment, The European Phys-ical Journal A54