# Extracting nucleon strange and anapole form factors from world data

###### Abstract

The complete world set of parity violating electron scattering data up to is analysed. We extract the current experimental determination of the strange electric and magnetic form factors of the proton, as well as the weak axial form factors of the proton and neutron, at . Within experimental uncertainties, we find that the strange form factors are consistent with zero, as are the anapole contributions to the axial form factors. Nevertheless, the correlation between the strange and anapole contributions suggest that there is only a small probability that these form factors all vanish simultaneously.

###### pacs:

13.60.-r 11.30.Er 14.20.Dh 25.30.Bf^{†}

^{†}preprint: JLAB-THY-06-479

Parity-violating electron scattering (PVES) is an essential tool in mapping out the flavour composition of the electromagnetic form factors. Exposing the role of the strange quark via these measurements provides direct information on the underlying dynamics of nonperturbative QCD — a considerable achievement both experimentally and theoretically. The most precise separation of the strange electric and magnetic form factors is available at , where experiments by the SAMPLE Ito:2003mr ; Spayde:2003nr , PVA4 Maas:2004dh and HAPPEx Aniol:2005zf ; Aniol:2005zg collaborations have been performed with varying kinematics and targets. At higher , HAPPEx Aniol:2000at ; Aniol:2004hp , PVA4 Maas:2004ta and the forward angle G0 experiment Armstrong:2005hs provide further information over the range –. Here we use systematic expansions of all the unknown form factors to simultaneously analyze the current data set and extract the values at , independent of theoretical input — other than the constraint of charge symmetry. The results provide a critical test of modern theoretical estimates of the anapole moment of the proton and neutron as well as their strange form factors.

The proton-PVES experiments are sensitive to the strange form factors and , and the electroweak axial form factor — which includes the anapole form factor Musolf:1990ts ; Zhu:2000gn . Previously, limited experimental data made it difficult to carry out a simultaneous separation of all three form factors; instead, assumptions were made on the (in)significance of certain contributions based on the kinematic domain and/or the use of theoretical calculations. In combining proton and deuteron data, there are two independent anapole form factors. Together with the two strange form factors, this analysis presents the first extraction of all four form factors from data. No more than two independent terms have been fit simultaneously in any previous analysis. Further, no analysis has attempted to determine the isoscalar anapole term from data. This contribution is quite poorly constrained by experiment and the design of an appropriate measurement to improve this situation is both a theoretical and experimental challenge.

The role of the strange quark is probed by measuring the PV asymmetry in polarised – scattering, for which the dominant contribution arises from interference between the and exchange. The majority of measurements have been performed on hydrogen: SAMPLE Spayde:2003nr ; Beise:2004py , HAPPEx Aniol:2000at ; Aniol:2005zg , PVA4 Maas:2004ta ; Maas:2004dh and G0 Armstrong:2005hs .

As described in Ref. Musolf:1993tb , the PV asymmetry for a proton target is given by (assuming charge symmetry)

(1) | |||||

(2) | |||||

(3) | |||||

(4) |

The kinematic variables are defined by and . The Standard Model parameters , and are taken from the PDG Eidelman:2004wy . The vector radiative correction factors are defined by , and , with and Eidelman:2004wy . The axial radiative and anapole corrections remain implicit in , as this entire contribution is to be fit to data.

Collaboration | CL | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SAMPLE | — | — | — | — | — | ||||||||||

SAMPLE | — | — | — | — | — | ||||||||||

HAPPEx | — | — | — | — | — | ||||||||||

HAPPEx | — | — | — | — | — | ||||||||||

SAMPLE | 63 | ||||||||||||||

PVA4 | 71 | ||||||||||||||

G0 | 76 | ||||||||||||||

G0 | 96 | ||||||||||||||

G0 | 99 | ||||||||||||||

G0 | 91 | ||||||||||||||

G0 | 91 | ||||||||||||||

G0 | 83 | ||||||||||||||

G0 | 36 | ||||||||||||||

G0 | 18 | ||||||||||||||

G0 | 1 | ||||||||||||||

PVA4 | 1 | ||||||||||||||

G0 | 3 | ||||||||||||||

G0 | 18 | ||||||||||||||

G0 | 5 | ||||||||||||||

G0 | 11 | ||||||||||||||

G0 | 44 | ||||||||||||||

HAPPEx | 28 |

Scattering from targets other than the proton provides access to different flavour components of the nucleon form factors. The HAPPEx Collaboration have recently utilised a helium-4 target to directly extract the strange electric form factor Aniol:2005zf , where the theoretical asymmetry can be written as

(5) |

In the SAMPLE experiment, which detected electrons scattered at backward angles, the contribution from is substantially suppressed. These measurements were primarily sensitive to a linear combination of the axial and strange magnetic form factors. In addition to the proton target, the PV-asymmetry has also been measured on the deuteron Ito:2003mr . While providing a different combination of and , this also introduces sensitivity to the neutron axial form factor.

Scattering from the deuteron is dominated by the quasielastic interaction with the nucleon constituents. The analysis of the deuteron results Beise:2004py has also included nuclear corrections, involving a realistic deuteron wave-function, rescattering effects and the small contribution from elastic deuteron scattering Schiavilla . Further parity-violating contributions arising from the deuteron wavefunction and exchange currents, while small Schiavilla , have been included.

A combined analysis of the current world PV data requires a consistent treatment of the vector and axial form factors and radiative corrections. Our theoretical asymmetries have therefore been reconstructed for each measurement. The theoretical asymmetry is

(6) |

where the values of , given in Table 1, include the latest vector form factors Kelly:2004hm and PDG radiative corrections.

It has been observed that the strange form factors are mildly sensitive to the choice of form factor parameterisation, with an uncertainty dominated by the neutron charge form factor. To test the sensitivity to , we explicitly included the experimental data for Hyde-Wright:2004gh in our global fit. Over the low- domain required in this analysis, the form factor can be parameterised by a Taylor expansion up to . This made no significant difference to the final extraction, and hence the central value of the Kelly parameterisation Kelly:2004hm is taken in the following analysis.

In order to extract all three form factors using as much data as possible, we parameterise their dependence. At low momentum transfer, a Taylor series expansion in is sufficient and minimises the model dependence of the determined form factors. The quality of a Taylor series expansion can be estimated phenomenologically. Vector meson dominance would suggest that the evolution of the form factors be no more rapid than a dipole with mass parameter . Similarly, lattice QCD simulations in the vicinity of the strange quark yield behaviour consistent with a dipole of scale Gockeler:2003ay ; Ashley:2003sn . With the aim of fitting data up to , approximating a dipole by a constant over this range would lead to less than 20% uncertainty (less than 10% at the next order in ).

To isolate the individual form factors at higher-, a combination of neutrino and parity-violating electron scattering should provide the tightest constraint, as described in Ref. Pate:2003rk .

We describe the -dependence of the form factors over the range by

(7) | |||

(8) |

The momentum dependence of the radiative corrections is assumed to be mild, and therefore the axial dipole mass is chosen to be that determined from neutrino scattering, Bernard:2001rs .

The best fit for yields, at leading order in , a reduced , with parameters

(9) | |||||

(10) | |||||

(11) | |||||

(12) |

The second error bar displays the sensitivity to the correlated error in the G0 experiment, where the data has been refit using . The extraction of the strange form factors over the low- range is shown in Fig. 1. We display the joint determination of the strange electric and magnetic form factors at in Fig. 2, where we also show the theoretical calculations of Leinweber et al. Leinweber:2004tc ; Leinweber:2006ug . Similar contours in – and – space are shown in Fig. 3.

The stability of the fits to truncation of the data set at a maximum value has been investigated. The resulting fits are displayed in Table 1, where a clear signal for nonzero strangeness is observed in the vicinity of — with caution that the fits are particularly sensitive to truncation up until . To investigate a potential enhancement near , we include the second-order terms of Eq. (8) and fit all data for . This produces a and best-fit parameters , , , , and , where the errors are statistical only. Figure 1 shows the uncorrelated separation of the electric and magnetic form factors at this order. Where the data is best constrained, , there is only a 55% CL in support of nonzero strangeness. This suggests that the strangeness signal in Table 1, obtained by truncating the data at , is consistent with a random fluctuation.

Previous (non-global) attempts to extract the nucleon strange form factors from world data used a theoretical prediction of Zhu:2000gn . In the following, we compare the axial form factors extracted from the data with this prediction. We write the axial charges, Eq. (7), as

(13) |

with for the proton (neutron). The radiative corrections are implied to be single-quark only , and Eidelman:2004wy . The axial charges are relatively well known, where we use , Filippone:2001ux and Adams:1997tq . The second error in and reflects estimates of the SU(3)-flavour symmetry violations of 20% in the determination of from hyperon -decay Zhu:2000zf . The dominant source of uncertainty in Eq. (13) is the anapole contribution, . Converting the result of Zhu et al. Zhu:2000gn to Kumar:2000eq , the anapole terms are estimated to be and . This gives the total theory estimates for the axial charges in PVES, and , where the second term is the anapole contribution. These estimates are consistent with the present determination, as shown in the right panel of Fig. 3.

As we see from Table 1, the current world data is consistent with the strange form factors being zero at a high level of confidence. The anapole contributions, considered alone, are also consistent with zero. On the other hand, if one interrogates the data for the probability that strange and anapole form factors are simultaneously zero, within the current errors, the hypothesis is only supported at 8%. While the present data set cannot distinguish the origin of this effect, there appears to be significant support for a nonzero signal in at least one of the strange or anapole contributions.

In conclusion, our analysis of the world data set for PVES has yielded the best experimental determination, at low , of the strange electric and magnetic form factors of the proton as well as the anapole form factors of the proton and neutron. While both the strangeness and anapole contributions are consistent with zero, we expect that the additional HAPPEx and G0-backward angle experiments at Jefferson Lab and the PVA4-backward angle experiment at Mainz will soon yield data that, when combined with this analysis, could reveal a nontrivial result for at least one of these form factors.

We wish to express our gratitude to M. Pitt, D. Bowman, K. de Jager, W. Melnitchouk, M. Paris, K. Paschke and R. Schiavilla for helpful discussions. This work was supported by DOE contract DE-AC05-84ER40150, under which SURA operates Jefferson Lab.

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