Weak Charge Form Factor and Radius of 208Pb Through Parity Violation in Electron Scattering

Weak charge form factor and radius of 208Pb through parity violation in electron scattering

C. J. Horowitz∗ University of Tennessee, Knoxville, TN, and Indiana University, Bloomington, Indiana 47405, USA

Z. Ahmed, C.-M. Jen, A. Rakhman, and P. A. Souder Syracuse University, Syracuse, New York 13244, USA

M. M. Dalton, N. Liyanage, K. D. Paschke, K. Saenboonruang, and R. Silwal University of Virginia, Charlottesville, Virginia 22903, USA

G. B. Franklin, M. Friend, and B. Quinn Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

K. S. Kumar, D. McNulty,† L. Mercado, S. Riordan,‡ and J. Wexler University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA

R. W. Michaels Thomas Jeﬀerson National Accelerator Facility, Newport News, Virginia 23606, USA

G. M. Urciuoli INFN, Sezione di Roma, I-00161 Rome, Italy (Dated: February 7, 2012) We use distorted wave electron scattering calculations to extract the weak charge form factor 208 FW (¯q), the weak charge radius RW , and the point neutron radius Rn, of Pb from the PREX parity violating asymmetry measurement. The form factor is the Fourier transform of the weak charge −1 density at the average momentum transferq ¯ = 0.475 fm . We find FW (¯q) = 0.204 ± 0.028(exp) ± 0.001(model). We use the Helm model to infer the weak radius from FW (¯q). We find RW = 5.826 ± 0.181(exp) ± 0.027(model) fm. Here the exp error includes PREX statistical and systematic errors, while the model error describes the uncertainty in RW from uncertainties in the surface thickness σ of the weak charge density. The weak radius is larger than the charge radius, implying a “weak charge skin” where the surface region is relatively enriched in weak charges compared to (electromagnetic) charges. We extract the point neutron radius Rn = 5.751 ± 0.175 (exp) ± 0.026(model) ± 0.005(strange) fm, from RW . Here there is only a very small error (strange) from possible strange quark contributions. We find Rn to be slightly smaller than RW because of the nucleon’s size. Finally, we find a neutron skin thickness of Rn − Rp = 0.302 ± 0.175 (exp) ± 0.026 (model) ± 0.005 (strange) fm, where Rp is the point proton radius.

PACS numbers: 21.10.Gv, 25.30.Bf, 24.80.+y, 27.80.+w

Parity violating elastic electron scattering provides a accurately calculated [2], if the charge density ρch [3] is model independent probe of neutron densities, because well known. Many details of a practical parity violat- the weak charge of a neutron is much larger than the ing experiment to measure neutron densities, along with weak charge of a proton [1]. In Born approximation, the a number of theoretical corrections, were discussed in a parity violating asymmetry Apv, the fractional difference long paper [4]. in cross sections for positive and negative helicity elec- Recently, the Lead Radius Experiment (PREX) mea- trons, is proportional to the weak form factor FW . This sured Apv for 1.06 GeV electrons, scattered by about is very close to the Fourier transform of the neutron den- 208 five degrees from Pb, and the neutron radius Rn was sity. Therefore the neutron density can be extracted from extracted [5]. To do this, the experimental A was com- an electro-weak measurement [1]. However, one must in- pv pared to a least squares fit of Rn as a function of Apv, pre- clude the effects of Coulomb distortions, which have been dicted by seven mean field models [6] (see also [7]). In the present paper, we provide a second, more detailed, analysis of the measured Apv. This second analysis provides additional information, such as the weak form factor, and ∗Electronic address: [email protected] †now at Idaho State University, Pocatello, Idaho 83209, USA clarifies the (modest) model assumptions necessary to ex- ‡previously at University of Virginia, Charlottesville, Virginia tract Rn. 22903, USA We start with distorted wave calculations of Apv for an 2 electron moving in Coulomb and weak potentials [2]. We use these to extract the weak form factor from the PREX TABLE I: Least squares fits of Wood Saxon (R, a, see Eq. 2) or Helm model (R , σ, see Eq. 8) parameters to theoretical measurement. In Born approximation, one can deter- 0 mean field model weak charge densities. mine the weak form factor directly from the measured Wood Saxon Helm Apv. However, Coulomb distortions may make Apv sen- Mean field force R (fm) a (fm) R (fm) σ (fm) sitive to the weak form factor for a range of momentum 0 transfers q. In addition, the experimental acceptance for Skyrme I [8] 6.655 0.564 6.792 0.943 PREX includes a range of momentum transfers for labo- Skyrme III [9] 6.820 0.613 6.976 1.024 ratory scattering angles from about 3.5 to 8 degrees [5]. Skyrme SLY4 [10] 6.700 0.668 6.888 1.115 Therefore we will need to make very modest assumptions FSUGold [11] 6.800 0.618 6.961 1.028 about the shape of the weak form factor (how it depends NL3 [12] 6.896 0.623 7.057 1.039 on momentum transfer q) in order to determine the value NL3p06 [6] 6.730 0.606 6.886 1.010 of the form factor at the average momentum transferq ¯ NL3m05 [6] 7.082 0.605 7.231 1.012 [5], Average 0.61 ± 0.05 1.02± 0.09 q¯ = hQ2i1/2 = 0.475 ± 0.003 fm−1. (1)

We initally assume the weak charge density of 208Pb, neutron radii Rn. The average value of a for these mod- ρW (r) has a Wood Saxon form, els is 0.61 ± 0.05 fm. Using a central value of a = 0.6 fm we obtain, ρ0 ρW (r) = , (2) 1 + exp[(r − R)/a] FW (¯q) = 0.204 ± 0.028(exp) ± 0.001(mod). (6) with parameters ρ0, R and a. Note, this form is only Here the first experimental error is from adding the sta- used to access the sensitivity to the shape of the form tistical and systematic errors in Eq. 5 in quadrature. The factor and our results will be independent of this assumed second model error is from varying a by ±0.05 fm. This form. The weak density is normalized to the weak charge shows that the extracted form factor is all but indepen- R 3 QW = d rρW (r), see below. dent of the assumed shape of the weak charge density. We define the weak form factor FW (q) as the Fourier Equation 6 is a major result of this paper. This is the transform of ρW (r), form factor of the weak charge density that is implied by Z the PREX measurement. 1 3 sin qr FW (q) = d r ρW (r). (3) We now explore some of the implications of Eq. 6 us- QW qr ing the Helm model [13] for the weak form factor. In the past, the Helm model has proven very useful for analyz- This is normalized FW (q = 0) = 1. Our procedure is ing (unpolarized) electron scattering form factors [14, 15], to calculate Apv(θ), including full Coulomb distortions see also ref. [16] for an application of the Helm model to [2], assuming ρW from Eq. 2. We average Apv(θ) over neutron rich nuclei. The weak charge density is first as- laboratory scattering angle θ using the experimental ac- sumed to be uniform out to a diffraction radius R0. This ceptance (θ) [5], uniform density is then folded with a gaussian of width σ to get the final weak density. The width σ includes con- R dθ sin θ (θ) dσ A dΩ pv tributions from both the surface thickness of the point hAi = R dσ . (4) dθ sin θ (θ) dΩ nucleon densities and the single nucleon form factor. In the Helm model, the weak form factor has a very simple dσ Here the unpolarized elastic cross section is dΩ . We then form, adjust R until the calculated hAi agrees with the PREX result [5] 3 −σ2q2/2 FW (q) = j1(qR0)e , (7) qR0 P b Apv = 0.656 ± 0.060(stat) ± 0.014(syst) ppm. (5) 2 with j1(x) = sin x/x − cos x/x a spherical Bessel func- Here the first error is statistical and the second error tion. The diffraction radius R0 determines the location includes systematic contributions. For a = 0.6 fm, we q0 of the zero in the weak form factor FW (q0) = 0. In obtain a central value of R = 6.982 fm, see below. Fi- coordinate space, the Helm model weak charge density P b nally from the ρW (r) in Eq. 2, that reproduces Apv , we can be written in terms of error functions (erf), calculate FW (¯q) using Eq. 3. This procedure fully includes Coulomb distortions and depends slightly on the 3QW n R0 + r r − R0 assumed surface thickness a in Eq. 2. In Table I we show ρW (r) = 3 erf √ − erf √ 8πR0 2σ 2σ Wood Saxon fits to seven nonrelativistic and relativistic r 2 σ − 1 ( r+R0 )2 − 1 ( r−R0 )2 o mean field model weak charge densities considered in ref. + e 2 σ − e 2 σ . (8) [6]. Note that these models span a very large range of π r 3

The root mean square radius of the weak charge density R (or weak radius) is R2 = R d3r r2 ρ (r)/Q , TABLE II: Helm model weak charge density parameters R0 W W W W and σ that reproduce the following values for the weak form factor F (¯q), see Eqs. 6 and 7. 3 W R2 = R2 + 5σ2. (9) W 5 0 Density R0 (fm) σ (fm) FW (¯q) Central value 7.167 1.02 0.204 We see that Eq. 6 implies via Eq. 7 a relationship be- Exp error bar 7.417 1.02 0.176 tween allowed values of R0 and σ. This relationship then implies via Eq. 9 a range of weak radii. Thus Eq. 6 does Exp error bar 6.926 1.02 0.232 not, by itself, determine the weak radius. In principle Model error bar 7.137 1.11 0.203 the rms radius follows from the derivative of the form Model error bar 7.194 0.93 0.205 factor with respect to Q2 at q = 0. Because the PREX measurement is at finite q, one needs to assume some that the weak charge density, of a heavy nucleus, is more information about the surface thickness σ in order to ex- extended than the electromagnetic charge density. tract RW . Alternatively within the Helm model, if one In Fig. 1 we show a Helm model weak charge density determined the location of the zero of the form factor q0, that is consistent with the PREX measurement. This in addition to Eq. 6, then this would uniquely fix both figure shows an uncertainty range from the experimental R0 and σ and so determine RW . error and a model uncertainty from the assumed ±0.09 In Table I we collect values of σ determined by least fm uncertainty in σ. Parameters for these densities are squares fits of the Helm density, Eq. 8, to seven model presented in Table II. We also show in Fig. 1 the (elec- mean field densities. The average of σ for the seven mean tromagnetic) charge density [3] and a typical mean field field densities is 1.02 fm and individual results deviate by weak charge density based on the FSUGold interaction, no more than 0.09 fm from this average. If one assumes see Eq. 17 below. This theoretical density is within the σ = 1.02 ± 0.09 fm, Eqs. 6, 7 and 9 imply error bars of the Helm model density.

RW = 5.826 ± 0.181(exp) ± 0.027(mod) fm. (10)

Again the larger experimental (exp) error is from adding the statistical and systematic errors in Eq. 5 in quadrature, while the model (mod) error comes from the coher- 0.08 ent sum of the assumed ± 0.09 fm uncertainty in σ and the ±0.001 model error in FW . The model error in Eq. 10 provides an estimate of the uncertainty in RW that 0.06 ρ

) ch arrises because of uncertainties in the surface thickness. -3

Of course, it is not guaranteed that all theoretical models (fm

w 0.04 ρ

will have a surface thickness within the range 1.02 ± 0.09 - fm. Nevertheless, this result suggests that uncertainties in surface thickness are much less important for R than W 0.02 either the present PREX experimental error or even that of an improved measurement where the experimental error is reduced by about a factor of three [17]. This is 0 0 5 10 consistent with earlier results of Furnstahl [18] suggest- r (fm) ing a nearly unique relation between FW (¯q) and the point neutron radius Rn. We emphasize that if uncertainties in the surface thickness are a concern, one should compare FIG. 1: (Color on line) Helm model weak charge density theoretical predictions for the form factor FW (¯q) to Eq. 208 −ρW (r) of Pb that is consistent with the PREX result 6, instead of comparing theoretical predictions for RW to (solid black line). The brown error band shows the incoher- Eq. 10. ent sum of experimental and model errors. The red dashed Comparing Eq. 10 to the experimental charge radius curve is the experimental (electromagnetic) charge density ρch Rch = 5.503 fm [3, 19] implies a “weak charge skin” of and the blue dotted curve shows a sample mean ﬁeld result thickness based on the FSUGold interaction [11].

RW − Rch = 0.323 ± 0.181(exp) ± 0.027(mod) fm. (11) 208 Finally we wish to extract Rn for Pb from RW in Eq. Thus the surface region of 208Pb is relatively enhanced 10. We start by reviewing the relationship between the in weak charges compared to electromagnetic charges. point proton radius Rp and the measured charge radius This weak charge skin is closely related to the expected Rch. Ong et al. have [20] neutron skin, see below. Equation 11, itself, represents N 3 R2 = R2 + hr2i + hr2 i + + hr2i . (12) an experimental milestone. We now have direct evidence ch p p Z n 4M 2 so 4

2 2 R 3 0 02 s 0 Here the charge radius of a single proton is hrpi = 0.769 where hrs i = d r r GE(r ) is the square of the nucleon 2 2 2 fm and that of a neutron is hrni = −0.116 fm . We strangeness radius. This yields calculate that the contribution of spin-orbit currents to 2 2 2 2 Rch is small because of cancelations between protons Rn = 0.9525RW − 1.671hrs i + 0.7450 fm . (20) 2 2 2 1/2 and neutrons hr iso = −0.028 fm . Finally the Darwin The strangeness radius of the nucleon hrs i is con- contribution 3/4M 2 is also small with M the nucleon strained by experimental data [21–24] and their global 208 2 2 2 2 mass. For Pb we have, Rch = Rp + 0.5956 fm , or, for analysis [27, 28]. Using Table V of ref. [28] for Q < 0.11 2 2 s 2 Rch = 5.503 fm [3, 19], GeV , gives hrs i = −6dGE/dQ = 0.02 ± 0.04 ≈ ±0.04 fm2. Rp = 5.449 fm. (13) The neutron radius then follows from Eq. 10, For the weak charge density of a spin zero nucleus, we R = 5.751 ± 0.175 (exp) ± 0.026(mod) ± 0.005(str) fm . neglect meson exchange and spin-orbit currents and write n (21) [4] Here the very small third (str) error is from possible Z strange quark contributions. The neutron radius Rn is 3 0 Z 0 0 Z 0 0 ρW (r) = 4 d r Gn (|r−r |)ρn(r )+Gp (|r−r |)ρp(r ) . slightly smaller than RW because of the nucleon’s size. (14) Finally, the neutron skin thickness is Z Here the density of weak charge in a single proton Gp (r) Z Rn−Rp = 0.302±0.175(exp)±0.026(mod)±0.005(str) fm. or neutron Gn (r) is the Fourier transform of the nucleon Z 2 Z 2 (22) (Electric) Sachs form factors Gp (Q ) and Gn (Q ). These describe the coupling of a Z0 boson to a proton or neu- This result agrees, within the model error, with the result of ref. [5], R − R = 0.33+0.16 fm. The small difference tron [4], n p −0.18 between the present result and ref. [5] arrises because Z p n s of small limitations of the Helm model in representing 4G = qpG + qnG − G , (15) p E E E theoretical mean field densities. For example the Helm model does not have the correct expodential behavior at Z p n s large distances. However, we have clarified how the ex- 4Gn = qnG + qpGE − GE. (16) E traction of the neutron radius depends upon assumptions 0 At tree level, the weak nucleon charges are qn = −1 and on the weak skin thickness σ and we provide an explicit 0 2 qp = 1 − 4 sin ΘW . We include radiative corrections by model error for the uncertainty in Rn − Rp because of using the values qn = −0.9878 and qp = 0.0721 based uncertainties in σ. on the up C1u and down C1d quark weak charges in refs. We now summarize our results. In this paper we use [25, 26]. The Fourier transform of the proton (neutron) distorted wave electron scattering calculations for 208Pb p n electric form factor is GE(r)(GE(r)) and has total charge to extract the weak charge form factor FW (¯q), Eq. 6, the R 3 p R 3 n s d rGE(r) = 1 ( d rGE(r) = 0). Finally GE describes weak radius RW , Eq. 10, and the point neutron radius strange quark contributions to the nucleon’s electric form Rn, Eq. 21, from the PREX parity violating asymme- factor [21–24]. Note that there may be some small un- try measurement. The weak form factor is the Fourier certainty regarding the Q2 dependence of the radiative transform of the weak charge density at the average mo- 2 corrections. This uncertainty could change Rn, see be- mentum transfer of the experiment. This quantity is es- 2 low, by a very small amount of order (1 + qn)hrpi. sentially model independent and is insensitive to assump- Equation 14 can be rewritten by using a similar ex- tions about the surface thickness. pression for ρch The extraction of RW depends on modest assump- Z tions about the surface thickness. We use the Helm 3 0 p n s model to derive an estimate on the uncertainty in R ρW (r) = qp ρch(r) + d r qn(GEρn + GEρp) − GEρb W because of the uncertainty in surface thickness. We find (17) 208 a “weak charge skin” where the surface region is rela- with ρb = ρn + ρp. The weak charge of Pb is tively enriched in weak charges compared to electromag- Z netic charges. This is closely related to the neutron skin 3 QW = d rρW (r) = Nqn + Zqp = −118.55. (18) where Rn is larger than the point proton radius Rp. Fi- nally, we extract Rn, given RW , and find it to be slightly smaller than RW because of the nucleon’s size. From Eq. 17, we relate the point neutron rms radius Rn, to RW , We thank Witek Nazarewicz for very helpful discus- sions. We gratefully acknowledge the hospitality of 2 QW 2 qpZ 2 2 Z 2 Z + N 2 the University of Tennessee and the Physics Division of Rn = RW − Rch − hrpi − hrni + hrs i, qnN qnN N qnN ORNL where this work was started. This work was sup- (19) ported in part by DOE grant DE-FG02-87ER40365. 5

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