Sensitivity of parity-violating A(e→ , e′)A scattering and atomic parity nonconservation to neutron distributions in nuclei

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Nuclear Physics A663&664 (2000) 381c-384c

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Sensitivity of Parity-Violating A( e, e')A Scattering and Atomic Parity Nonconservation to Neutron Distributions in Nuclei S.J. Pollock

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and M.C. Welliver

"University of Colorado, Boulder CB 390, Boulder, CO 80309, USA Parity-violating electron scattering (PYES) could provide a unique means to determine spatial neutron distributions and their moments in heavy nuclei. Knowledge of the neutron distribution is of fundamental interest for nuclear structure models, and the first moment is of special interest for atomic parity experiments. We have examined what could be learned from a hypothetical measurement of the parity-violating asymmetry in elastic electron scattering on barium and lead nuclei (both spin-O and Ni-Z). We find that a single measurement of this quantity could determine the rms neutron radius to within a couple of percent, to be compared with the 5-10% existing uncertainties. We also compute the quantitative connection to atomic parity nonconservation, and the resulting limits on possible low energy Standard Model tests which could be achieved. 1. Introduction and motivation Elastic polarized electron scattering from nuclei can provide a unique tool for probing the electroweak structure of nucleons and nuclei. [1-4] In the case of spin-O nuclei, such asymmetry measurements could provide a clean measurement of the nuclear weak charge distribution, just as unpolarized electron scattering determines the electric charge distribution in nuclei. Because the proton's weak charge is small, the weak charge distribution is closely related to the spatial distribution of neutrons within a nucleus. Experimental knowledge of neutron distributions is presently obtained through the use of strongly interacting probes, where systematic theoretical uncertainties can be quite large and uncontrolled[5]. Current nuclear structure models involve parameter sets where quantities which would help fix neutron distributions are poorly determined. Even a small number of precise measurements directly involving spatial neutron distributions would provide badly needed constraints on such models. In addition, knowledge of neutron distributions may soon be relevant in the analysis of atomic parity nonconservation (PNC). The weak charge of Cs was recently found to be [6] Q~Pt = -72.06(28)expt(34)theory, in mild disagreement at the 2.50" level with the Standard Model prediction of -73.20(13)theory- [7] Cesium has a fairly large fractional neutron excess, and predicted differences between neutron and proton distributions cause a small modification to Qw, on the order 0.1 ± 0.3. [1] While uncertainties associated with differences in neutron and proton distributions are 'This work was supported by a DOE research grant, #DE-FG03-93ER40774 0375-9474/00/$ see front matter © 2000 Elsevier Science RY. All rights reserved. PH S0375-9474(99)00622-3

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8.J Pollock, MC Wellioer/Nuclear Physics A663&664 (2000) 381c-384c

not yet a limiting factor in the interpretation of atomic PNC, improved Standard Model tests using atomic PNC will require improved knowledge of neutron distributions. In this note, we consider the possibility of determining the neutron rms radius in lead, via a plausible PVES experiment, such as has been proposed at Jefferson Lab [8].

2. Formalism and results We consider elastic PVES from 208Pb, a spin-O and Ni-Z nucleus. This process gives rise to a parity-violating asymmetry, given in Born approximation (neglecting isospin violation) by

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q is the momentum transfer, A?r(q) = -G F lq21/(41fay'2). The nucleon form factors are defined by Fp(n) (q2) == J jO(qr)pp(n) (r)d3r, where jo is the zeroth spherical Bessel function and Pn(p) is the spatial distribution of neutrons (protons), normalized to unity. In addition to the asymmetry itself, the limiting statistical uncertainty in Air is

(2) where P is beam polarization, L is luminosity, T is running time, and dn is the solid angle of the detector. We consider a hypothetical experimental setup with a 100% polarized beam running for 500 hours at 400 MeV, a luminosity of 2 x 1036 S-l cm- 2, and 10 msr solid angle for the detector. To make initial estimates of how sensitive PVES is to the rms neutron radius, R,., we arbitrarily constrain the neutron spatial distribution to match the shape of the proton distribution, but not necessarily the radius. We parameterize nuclear densities with a three-parameter Gaussian form [9], p(r) = pO(l + wr 2/c2)/ (1 + exp[(r 2 - C2)/Z2]). To vary the rms radius but constrain the shape, we scale the neutron parameters Cn and Zn keeping wn(= wproton) fixed, and compute oRn/Rn ~ (l/Rn) 6.Ar:at/(fJAlr/fJRn)' The curves in Fig. 1 show the predicted uncertainty in extracted Rn versus momentum transfer for a single such measurement on lead. The solid curve corresponds to a nominal value of the rms radius, R~OM, equal to the proton radius. The dashed curve assumes R~OM = 1.1Rp . There is a wide, flat minimum, whose location is largely independent of R~OM. This suggests a broad kinematic region in which an experiment could optimally constrain Rn at a level significantly below current limits. The optimal q value assuming the neutron distribution is identical to the protons ("constrained calculation") is 0.46 fm>'. At this point, (oR,./R,.)stat=0.46%. If R,. is scaled uniformly to 1.1 Rp, The location in q shifts down negligibly, and the value of oR,./ R,. is unchanged. For a more general, unconstrained investigation, we calculate Air and 6.Ar:at as above, but then search for all possible alternative neutron distributions that give rise to an asymmetry which falls within this statistical uncertainty of our nominal value, yielding a set of allowable parameterizations. We invoke an additional constraint on the parameterizations of the neutron distribution, requiring that the half-density parameter Co and

s.i Pollock, u.c. Wellioer /Nuciear Physics A663&664

(2000) 381c- 384c

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Figure 1. Estimated uncertainty in the extracted rms neutron radius of lead versus momentum transfer for a hypothetical experiment . Solid curve: R.;;0M = R p ; dashed curve: R.;;0M = 1.1 Rp ; in both cases the neutron distribution is constrained so that its shape mat ches the proton distribution. Diamonds: R~OM = R p ; squares: R~OM = 1.1 Rp ; here the shape parameters are allowed to vary by up to 30% from the nominal values.

thickness parameter Zn differ by no more than 30% from their nominal values. This restriction is completely arbitrary, but is more than large enough to accommodate essentially all nuclear struct ure models, making this a model independent calculation. For each neutron parameterization in the set , we calculate R n , yielding a distribution of neutron rms radii which would be experimentally allowed. In Fig. 1, these model independent uncertainty estimates are plotted as discrete points. Allowing the shape of the neutron distribution to vary narrows the optimal kinemati c region and pushes it to lower values of the momentum transfer, around 0.25 fm- I, with (c5Rn/Rn)stat=1.4%. Below the first diffraction minimum, both the optimal value of the momentum transfer and the extracted value of (c5Rn/ Rn)stat appear to be quite insensit ive to the (a priori unknown) value of the rms neutro n radius . A single low-q measurement of the asymmetry could thus allow a nuclear model independent constraint on the neutron rrns radius to better than 2% (ignoring radiative corrections and systematic uncertainties, which we have not yet considered). A second measurement at larger momentum transfer could provide additional information on the shape of the neutron distribution , decreasing the resulting uncertainty in extracted radius . We are presently working on quantify ing this possibility[2). Coulomb distortions will modify the asymmetry, as shown by Horowitz [10], but are known and computable. The effects of radiative corrections are also strongest in the diffraction minima , which are probably not the best kinemati c regions in which to perform

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S.J Pollock, u.c. Welliver/Nuclear Physics A663&664 (2000) 381c-384c

an experiment. In light of the modifications due to radiative corrections, it is probably best to interpret our results as rough guides to the statistical uncertainty in extracting R.n, but we expect the dependence of this uncertainty on momentum transfer, and the optimal kinematic regions, should not be significantly effected by Coulomb distortions.

3. Connections to Atomic Parity Nonconservation In Boulder, experiments on atomic Cs [6] have measured the nuclear weak charge Qw to about 1%. Differences between Pn(r) and pp(r) give rise to a shift in the observed weak charge, which we call LlQ~-P[1,2]. If the uncertainties in atomic PNC experiments (in particular uncertainties in atomic theory) are significantly improved, the sensitivity to new physics may thus soon be limited by existing uncertainties in spatial neutron distributions. Given an allowable range of neutron parameter sets, we have calculated [2] the allowed range OLlQ~P consistent with a single plausible PYES experiment, as described above. The optimal value of the momentum transfer for measuring LlQ~P in this way is slightly higher than what is needed to best measure Rn. For R~OM = Rp, we find qopt = 0.32 frn"", at which point LlQ~P = 0 ±O.13. If R~OM = l.1Rp, qopt is still 0.32 fm", at which point LlQ~P = 2.1 ± 0.14 This optimal q value indicates that shape information can be important in constraining LlQ~P. Below the first diffraction minimum, the optimal value of momentum transfer and resulting uncertainty in LlQ~P are insensitive to the assumed value of R~OM. Therefore, the choice of optimal experimental kinematics should not be sensitively affected by the a priori unknown neutron distribution itself. Even if one chooses to use the momentum transfer which minimizes (oRn!Rn)stat, e.g. q = 0.25 fm- 1 (with R~OM = Rp ) we obtain OLlQ~P = ±O.26, considerably worse than the optimal value, but approaching levels needed for significant tests of the Standard Model. In summary, improved knowledge of neutron distributions obtainable from a single PVES experiment, as described above, could significantly improve our knowledge of R,., perhaps down to 2%, and reduce the nuclear structure uncertainty in LlQ~P to below ±0.15 for 208Pb. Even a single 48 hour PYES experiment on lead could improve our knowledge of R,. to around 3%, and LlQ~P to roughly the ±0.40 level. This may allow for significant improvements in our understanding of nuclear structure, and for Standard Model tests via atomic PNC convincingly independent of nuclear structure effects.

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7. 8. 9. 10.

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