Bled Workshops in Physics Vol. 8, No. 1 p. 57
Parity violating electron scattering on 4He
M. Viviani
INFN, Sezione di Pisa, Pisa, Italy
Abstract. In order to isolate the contribution of the nucleon strange electric form factor to the parity-violating asymmetry measured in 4He(e, e')4He experiments, it is crucial to have a reliable estimate of the magnitude of isospin-symmetry-breaking (ISB) corrections in both the nucleon and 4He. Isospin admixtures in the nucleon are determined in chiral perturbation theory, while those in 4He are derived from nuclear interactions, including explicit ISB terms. At the low momentum transfers of interest in recent measurements reported by the HAPPEX collaboration at Jefferson Lab, it results that both contributions are of comparable magnitude to those associated with strangeness components in the nucleon electric form factor.
One of the challenges of modern hadronic physics is to determine, at a quantitative level, the role that quark-antiquark pairs, and in particular ss pairs, play in the structure of the nucleon. Parity-violating (PV) electron scattering from nucleons and nuclei offers the opportunity to investigate this issue experimentally. The PV asymmetry (APV) arises from interference between the amplitudes due to exchange of photons and Z-bosons, which couple respectively to the electromagnetic (EM) and weak neutral (NC) currents. These currents involve different combinations of quark flavors, and therefore measurements of APV, in combination with electromagnetic form factor data for the nucleon, allow one to isolate, in principle, the electric and magnetic form factors G| and GM, associated with the strange-quark content of the nucleon.
Experimental determinations of these form factors have been reported recently by the Jefferson Lab HAPPEX [1] and G0 [2] Collaborations, Mainz A4 Collaboration [3], and MIT-Bates SAMPLE Collaboration [4]. These experiments have scattered polarized electrons from either unpolarized protons at forward angles [1-3] or unpolarized protons and deuterons at backward angles [4]. The resulting PV asymmetries are sensitive to different linear combinations of G| and GM as well as the nucleon axial-vector form factor G^. However, no robust evidence has emerged so far for the presence of strange-quark effects in the nucleon.
Last year, the HAPPEX Collaboration [5,6] at Jefferson Lab reported on measurements of the PV asymmetry in elastic electron scattering from 4 He at four-momentum transfers of 0.091 (GeV/c)2 and 0.077 (GeV/c)2. Because of the Jn=0+ spin-parity assignments of this nucleus, transitions induced by magnetic and axial-vector currents are forbidden, and therefore these measurements can lead to a direct determination of the strangeness electric form factor G| [7,8], provided that isospin symmetry breaking (ISB) effects in both the nucleon and 4 He, and
relativistic and meson-exchange (collectively denoted with MEC) contributions to the nuclear EM and weak vector charge operators, are negligible. A realistic calculation of these latter contributions [8] found that they are in fact tiny at low momentum transfers.
Recently, we have completed the first realistic calculations of the ISB and here we will discuss their effect on the PV asymmetry (see Ref. [9] for details). The PV asymmetry measured in (e, e') elastic scattering from 4He is given by

47Tct\/2
,2fl ,F")(q) 2G£-Gf
4 sin2 0w - 2
F(0)(q) (GE + Gn)/2
(1)
where Gp is the Fermi constant as determined from muon decays, and 0W is the Weinberg mixing angle. The term G|(Q2) is the strange electric form factor of the nucleon, while the terms gE and F(1) (q)/F(0) (q) are the contributions to APV, associated with the violation of isospin symmetry at the nucleon and nuclear level, respectively.
The most accurate measurement of APV has been recently reported at four-momentum transfer of Q2=0.077 (GeV/c)2 [6]:
Apv = [+6.40 ± 0.23 (stat) ± 0.12 (syst)] ppm ,	(2)
from which one obtains
F(1 ) (q) 2 G1 — Gs
r^2Fi4l-(GftG#2^-010±0-038	(3)
at Q2 = 0.077 (GeV/c)2. This result is consistent with zero. In the following, we discuss the estimates for the ISB corrections first in the nucleon and then in 4He, respectively gE(Q2) and F(1)(q), at Q2=0.077 (GeV/c)2 (corresponding to q=1.4 fm-1).
For GE(Q2) we use the estimate obtained in Ref. [10], combining a leadingorder calculation in chiral perturbation theory with estimates for low-energy constants using resonance saturation. At the specific kinematical point of interest Q2 = 0.077 (GeV/c)2, it results that gE(Q2) = -0.0017 ± 0.0006, and with
[11], we obtain
GE(Q2) = 0.799 and G£(Q2) = 0.027
2 G1
E - 0.008 ± 0.003	(4)
(GE + Gn)/2
at Q2 = 0.077 (GeV/c)2.
We now turn to F(0) (q) and F(1' (q), the isoscalar and isovector form factors of4He, respectively. The form factor F( 1' (q) is very small because 4He is predominantly an isospin T = 0 state, but it contains also tiny T = 1 and 2 components. We have computed such components using a variety of Hamiltonian models, in order to have an estimate of the model dependence. We have considered: i) the AV18 NN potential [12]; ii) the AV18 plus Urbana-IX 3N potential [13] (AV18/UIX); iii)
the CD Bonn [14] NN plus Urbana-IXb 3N potentials (CDBonn/UIXb); and iv) the chiral N3LO [15] NN potential (N3LO). The Urbana UlXb 3N potential is a slightly modified version of the Urbana UIX (in the UlXb, the parameter U0 of the central repulsive term has been reduced by the factor 0.812), designed to reproduce, when used in combination with the CD Bonn potential, the experimental binding energy of 3H.
The form factors F(0) (q) and F(1) (q) calculated with the AV18/UIX Hamil-tonian model, are displayed in Fig. 1, where the effect of inclusion of mesonexchange contributions (MEC) is also shown. The dashed curves are the calculations including only the one-body operators, while the solid curves have been obtained including the relativistic one-body and MEC [8] in the electromagnetic charge operators. The experimental data are from Refs. [16]. From the figure it is evident that for q < 1.5 fm-1, the effect of MEC in both F(0)(q) and F(1)(q) is negligible.
In the inset of Fig. 1, we also show the the model dependence of the ratio |F(1' (q)/F(0) (q)| (all calculations include MEC). The various Hamiltonian models give predictions quite close to each other; the remaining differences reflect the different percentages of the T=1 component in the 4He wave function.
q (fm-1)
Fig. 1. The F(0) (q) and F(1) (q) form factors for the AV18/UIX Hamiltonian model. The F(0) (q) is compared with the experimental4He charge form factor [16]. The dashed curves are the calculations including only one-body operators. The solid curves include also MEC. The ratio |F(1' (q)/F(0) (q)| (all calculations include MEC) is shown in the inset for the four Hamiltonian models considered in this paper.
The calculated ratios F(1) (q)/F(0) (q) at Q2=0.077 (GeV/c)2 are of the order of -0.002. The value corresponding to the N3LO is somewhat larger than for the other models, as can be seen in Fig. 1, reflecting the larger percentage of T=1 admixtures predicted by the N3LO potential. The inclusion of 3N potentials tends
to decrease the magnitude of F(1 VF(0), and relativistic and MEC are, at this value of Q2, negligible.
Therefore, at Q2=0.077 (GeV/c)2, both contributions F(1)/F(0) and Gf are found of the same order of magnitude as the central value of r in Eq. (3). Using in this equation the value F(1)/F(0) « -0.00157 obtained with the Hamiltonian models including 3N potentials, and the chiral result for G EE = -0.0017 ± 0.0006, one would obtain Gf [Q2 = 0.077 (GeV/c)2] = -0.001 ± 0.016 thus suggesting that the value of r is almost entirely due to isospin admixtures. Of course, the experimental error on r is still too large to allow us to draw a more definite conclusion. A recent estimate of Gf using lattice QCD input obtains [17] Gf (0.1 (GeV/c)2) = +0.001 ± 0.004 ± 0.003. An increase of the experimental accuracy of one order of magnitude would be necessary in order to be sensitive to Gf at low values of Q2. Indeed, if the lattice QCD prediction above is confirmed, the present data would suggest that the leading correction to the PV asymmetry is from isospin admixtures in the nucleon and/or 4He.
References
1.	K.A. Aniol et al., Phys. Rev. Lett. 82, 1096 (1999); Phys. Lett. B 509, 211 (2001); Phys. Rev. C 69, 065501 (2004); Phys. Lett. B 635, 275 (2006).
2.	D.S. Amstrong et al., Phys. Rev. Lett. 95, 092001 (2005).
3.	F.E. Maas et al., Phys. Rev. Lett. 93, 022002 (2004); Phys. Rev. Lett. 94,152001 (2005).
4.	D.T. Spayde et al, Phys. Lett. B 583, 79 (2004); T.M. Ito et al., Phys. Rev. Lett. 92,102003 (2004).
5.	K.A. Aniol et al, Phys. Rev. Lett. 96, 022003 (2006).
6.	A. Acha et al., Phys. Rev. Lett. 98, 032301 (2007).
7.	M.J. Musolf et al, Phys. Rep. 239,1 (1994).
8.	M.J. Musolf, R. Schiavilla, and T.W. Donnelly, Phys. Rev. C 50, 2173 (1994).
9.	M. Viviani et al, Phys. Rev. Lett. 99,112002 (2007).
10.	B. Kubis and R. Lewis, Phys. Rev. C 74, 015204 (2006).
11.	M. A. Belushkin, H. W. Hammer and U.-G. Meißner, hep-ph/0608337; M. A. Be-lushkin, private communication.
12.	R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995).
13.	B.S. Pudliner et al, Phys. Rev. C 56,1720 (1997).
14.	R. Machleidt, Phys. Rev. C 63, 024001 (2001).
15.	D.R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003).
16.	R.F. Frosch et al, Phys. Rev. 160, 874 (1968); R.G. Arnold et al, Phys. Rev. Lett. 40,1429 (1978).
17.	D.B. Leinweber et al, Phys. Rev. Lett. 97, 022001 (2006).