Strange Nucleon Form Factors from $ep$ and $\nu p$ Elastic Scattering

TL;DR

This paper analyzes recent experimental data from electron and neutrino scattering to extract the strange quark contributions to nucleon form factors, revealing configurations different from traditional models and proposing future experiments for improved understanding.

Contribution

It provides a novel interpretation of strangeness in the nucleon based on combined scattering data and discusses future experiments to enhance precision and explore nucleon spin contributions.

Findings

Strangeness contributions to nucleon form factors have been extracted from combined $ep$ and $ u p$ data.

Results suggest configurations different from kaon-loop models.

Future experiments aim to measure strangeness effects at lower $Q^2$ with higher precision.

Abstract

The recent parity-violating $ep$ forward-scattering elastic asymmetry data from Jefferson Lab (HAPPEx and G0), when combined with the $\nu p$ elastic cross section data from Brookhaven (E734), permit an extraction of the strangeness contribution to the vector and axial nucleon form factors for momentum transfers in the range $0.45 < Q^2 < 1.0$ GeV$^2$. These results, combined with the recent determination of the strange vector form factors at $Q^2 = 0.1$ GeV$^2$ (SAMPLE, HAPPEx, PVA4, G0) have been interpreted in terms of $uuds\bar{s}$ configurations very different from the kaon-loop configurations usually associated with strangeness in the nucleon. New experiments are being proposed to improve the state of our knowledge of the $\nu p$ elastic cross section -- these new experiments will push the range of $Q^2$ to much lower values, and greatly increase the precision of the $\nu p$ elastic data. One outcome of this can be a measurement of the strangeness contribution to the nucleon spin, $\Delta s$. Nuclear targets (e.g. C or Ar) are to be used in these neutrino experiments, and so a deep understanding of the nuclear physics, particularly in regard to final state effects, is needed before the potential of these precision experiments can be fully realized.

Equations

$$G_E^{\gamma ,p}$$

$$G_E^{\gamma ,n}$$

$$G_E^{Z,p}$$

$$G_A^{Z,p}=\frac{1}{2}\left(-G_A^u+G_A^d+G_A^s\right).$$

$$G_A^{CC}=G_A^u-G_A^d=\frac{g_A}{(1+Q^2/M_A^2{)}^2}$$

Full text

I Introduction

Ever since the discovery of the first “strange” particles in cosmic ray experiments Rochester and Butler (1947) and the subsequent formulation of the 3-quark model of baryons Gell-Mann (1964), nuclear and particle physicists have sought to understand the role the strange quark plays in “non-strange” particles like the proton. Traditionally these investigations have taken place in the context of deep-inelastic scattering. However, a strong effort has been made to measure the strange quark contribution to the elastic form factors of the proton, in particular the vector (electric and magnetic) form factors. These experiments Mueller et al. (1997); Hasty et al. (2000); Spayde et al. (2004); Ito (2004); Aniol et al. (2004); Maas et al. (2004, 2005); Armstrong et al. (2005); Aniol et al. (2006); Acha et al. (2007) exploit an interference between the $\gamma$-exchange and $Z$-exchange amplitudes in order to measure weak elastic form factors $G_{E}^{Z,p}$ and $G_{M}^{Z,p}$ which are the weak-interaction analogs of the more traditional electromagnetic elastic form factors $G_{E}^{\gamma,p}$ and $G_{M}^{\gamma,p}$ for which copious experimental data are available. The interference term is observable as a parity-violating asymmetry in elastic $\vec{e}p$ scattering, with the electron longitudinally polarized. By combining the electromagnetic form factors of the proton and neutron with the weak form factors of the proton, one may separate the up, down, and strange quark contributions; for example, the electric form factors may be written as follows:

[TABLE]

There is an assumption of charge symmetry, and also an assumption that the role played by the strange quarks in the proton and neutron is the same.

Because of the weak-interaction process at the heart of this measurement program, these parity-violating asymmetries also involve the axial form factor of the proton, which in a pure weak-interaction process takes this form:

[TABLE]

The $u-d$ portion of this form factor is well-known from neutron $\beta$-decay and other charged-current ($CC$) weak interaction processes like $\nu_{\mu}+n\rightarrow p+\mu^{-}$:

[TABLE]

where $g_{A}=1.2695\pm 0.0029$ is the axial coupling constant Eidelman et al. (2004) and $M_{A}=1.001\pm 0.020$ is the so-called “axial mass” which is a very successful fitting parameter for the data on this form factor Bodek et al. (2004). The strange quark portion, $G_{A}^{s}$, is essentially unknown, except for some contradictory indications from polarized deep-inelastic scattering, which we discuss in more detail below. In elastic $ep$ scattering, the axial form factor does not appear as a pure weak-interaction process; there are significant radiative corrections which carry non-trivial theoretical uncertainties. The result is that, while the measurement of parity-violating asymmetries in $\vec{e}p$ elastic scattering is well suited to a measurement of $G_{E}^{s}$ and $G_{M}^{s}$, these experiments cannot cleanly extract $G_{A}^{s}$. The strange axial form factor is of great interest, however, because of the role it plays in the understanding of the spin structure of the proton.

II Strange Quarks and the Spin of the Proton

The strange quark contribution to the proton spin has been a subject of investigation ever since the first polarized inclusive deep-inelastic measurements of the spin-dependent structure function $g_{1}(x)$ by EMC Ashman et al. (1989) demonstrated that the Ellis-Jaffe sum rule Ellis and Jaffe (1974a); Ellis and Jaffe (1974b) did not hold true. Subsequent measurements at CERN and SLAC supported the initial EMC measurements, and a global analysis Filippone and Ji (2001) of these data suggested $\Delta s\approx-0.15$. This analysis carries with it an unknown theoretical uncertainty because the deep-inelastic data must be extrapolated to $x=0$ and an assumption of SU(3)-flavor symmetry must be invoked.

In the meantime, the E734 experiment Ahrens et al. (1987) at Brookhaven measured the $\nu p$ and $\bar{\nu}p$ elastic scattering cross sections in the momentum-transfer range $0.45<Q^{2}<1.05$ GeV2. These cross sections are very sensitive to the strange axial form factor of the proton, $G_{A}^{s}(Q^{2})$, which is related to the strange quark contribution to the proton spin: $G_{A}^{s}(Q^{2}=0)=\Delta s$. Assuming the strange axial form factor had the same $Q^{2}$-dependence as the isovector axial form factor, E734 also extracted a negative value for $\Delta s$. However, this determination was hampered by the large systematic uncertainies in the cross section measurement, as well as a lack of knowledge of the strange vector form factors, and no definitive determination of $\Delta s$ was possible — this conclusion was confirmed by subsequent reanalyses of these data Garvey et al. (1993); Alberico et al. (1999).

The HERMES experiment Airapetian et al. (2005) measured the helicity distribution of strange quarks, $\Delta s(x)$, using polarized semi-inclusive deep-inelastic scattering and a leading order “purity” analysis, and found $\Delta s(x)\approx 0$ in the range $0.03<x<0.3$. This seems to disagree with the analysis of the inclusive deep-inelastic data. This disagreement could be due to a failure of one or more of the assumptions made in the analysis of the inclusive and/or the semi-inclusive data, or it could be due to a more exotic physics mechanism such as a “polarized condensate” at $x=0$ not observable in deep-inelastic scattering Bass (2005).

III Combining $ep$ and $\nu p$ Elastic Data

On account of the apparent discrepancy between the analyses of the two differnt kinds of deep-inelastic data, another method is needed to shed light on the strange quark contribution to the proton spin. Recently Pate (2004) it has become possible to determine the strange vector and axial form factors of the proton by combining data from elastic parity-violating $\vec{e}p$ scattering experiments at Jefferson Lab with the $\nu p$ and $\bar{\nu}p$ elastic scattering data from E734. The parity-violating $\vec{e}p$ data place constraints on the strange vector form factors that were not available for previous analyses of E734 data.

Several experiments have now produced data on forward parity-violating $\vec{e}p$ elastic scattering Aniol et al. (2004); Maas et al. (2004, 2005); Armstrong et al. (2005); Aniol et al. (2006); Acha et al. (2007). Of most interest here are measurements that lie in the same $Q^{2}$ range as the BNL E734 experiment, which are the original HAPPEx measurement Aniol et al. (2004) at $Q^{2}=0.477$ GeV2 and four points in the recent $G^{0}$ data Armstrong et al. (2005). These forward scattering data are most sensitive to $G_{E}^{s}$, somewhat less sensitive to $G_{M}^{s}$, and almost completely insensitive to the axial form factors due to supression by both the weak vector electron charge $(1-4\sin^{2}\theta_{W})$ and by a kinematic factor that approaches 0 at forward angles.

The basic technique for combining the $\vec{e}p$, $\nu p$, and $\bar{\nu}p$ data sets has already been described Pate (2004) and the details of the present analysis will be published Pate et al. (to be published). The results are displayed in Figure 1. The uncertainties in all three form factors are dominated by the large uncertainties in the neutrino cross section data. Since those data are somewhat insensitive to $G_{E}^{s}$ and $G_{M}^{s}$ then the uncertainties in those two form factors are generally very large. However the results for the strange axial form factor are of sufficient precision to give a hint of the $Q^{2}$-dependence of this important form factor for the very first time. There is a strong indication from this $Q^{2}$-dependence that $\Delta s<0$, i.e. that the strange quark contribution to the proton spin is negative. However the data are not of sufficient quality to permit an extrapolation to $Q^{2}=0$, so no quantitative evaluation of $\Delta s$ from these data can be made at this time.

IV Comparison to Model Calculations

It is interesting to compare these results with models that can calculate a $Q^{2}$-dependence for these form factors. Silva, Kim, Urbano and Goeke Silva et al. (2005); Goeke et al. (2006); Silva et al. (2006) have used the chiral quark soliton model ($\chi$QSM) to calculate $G^{s}_{E,M,A}(Q^{2})$ in the range $0.0<Q^{2}<1.0$ GeV2. The $\chi$QSM has been very successful in reproducing other properties of light baryons using only a few parameters which are fixed by other data. In Figure 1 their calculation is shown as the solid line; it is seen to be in reasonable agreement with the available data, although the HAPPEx $G^{s}_{E}$ point at $Q^{2}=0.1$ GeV2 disfavors this calculation. Riska, An, and Zou Zou and Riska (2005); An et al. (2006); Riska and Zou (2006) have explored the stangeness content of the proton by writing all possible $uuds\bar{s}$ configurations and considering their contributions to $G^{s}_{E,M,A}(Q^{2})$. They find that a unique $uuds\bar{s}$ configuration, with the $s$ quark in a $P$ state and the $\bar{s}$ in an $S$ state, gives the best fit to the data for these form factors; see the small-dotted curves in Figure 1. Bijker Bijker (2006) uses a two-component model of the nucleon to calculate $G^{s}_{E,M}(Q^{2})$; the two components are an intrinsic three-quark structure and a vector-meson ($\rho$, $\omega$, and $\phi$) cloud; the strange quark content comes from the meson cloud component. The values of $G^{s}_{E,M}(Q^{2})$ are in good agreement with the data, see the big-dotted line in Figure 1. In the near future, the $G^{0}$ experiment will provided additional data on $G^{s}_{E,M}(Q^{2})$ at 0.23 and 0.63 GeV2 which will help to discriminate between the $\chi$QSM and the models of Bijker and of Riska et al.

V Future Experiments

To provide a useful determination of $\Delta s$, better data are needed for both the form factors and the polarized parton distribution functions. Two new experiments have been proposed to provide improved neutrino data for the determination of the strange axial form factor. FINeSSE FIN (2005) proposes to measure the ratio of the neutral-current to the charged-current $\nu N$ and $\bar{\nu}N$ processes. A measurement of $R_{NC/CC}=\sigma(\nu p\rightarrow\nu p)/\sigma(\nu n\rightarrow\mu^{-}p)$ and $\bar{R}_{NC/CC}=\sigma(\bar{\nu}p\rightarrow\bar{\nu}p)/\sigma(\bar{\nu}p\rightarrow\mu^{+}n)$ combined with the world’s data on forward-scattering PV $ep$ data can produce a dense set of data points for $G_{A}^{s}$ in the range $0.25<Q^{2}<0.75$ GeV2 with an uncertainty at each point of about $\pm 0.02$. Another experiment with similar physics goals, called NeuSpin Miyachi (2007), is being proposed for the new JPARC facility in Japan. It is also important to extend the semi-inclusive deep-inelastic data to smaller $x$ and higher $Q^{2}$ so that the determination of the polarized strange quark distribution $\Delta s(x)$ can be improved. A measurement of this type is envisioned Stösslein and Kinney (2001); Deshpande et al. (2005) for the proposed electron-ion collider facility. At the same time, a future electron-ion collider could provide an improved measurement of the integral $\int_{0}^{1}g_{1}^{p}(x)dx$, which can be combined with a $\nu p$ elastic scattering measurement of the proton’s weak axial charge and a recently developed axial charge sum rule Bass et al. (2002); Bass (2005) to provide an additional determination of $\Delta s$ and perhaps evidence for a “polarized condensate” at $x=0$ Bass (2005). It is only with these improved data sets that we will be able to arrive at an understanding of the strange quark contribution to the proton spin.

VI Nuclear Effects in $A(\nu,p)X$

FINeSSE and NeuSpin will make use of nuclear targets, like carbon or argon perhaps, in order to bring NC $\nu N\rightarrow\nu N$ and CC $\nu N\rightarrow\mu N$ count rates up to a level needed for timely completion of the experiments. In order to reach the level of precision in $G_{A}^{s}$ needed for an extraction of $\Delta s$ then a good understanding of initial and final state nuclear effects is needed. A number of theoretical efforts have been made in the last few years in this direction. There is no space here for any detailed discussion of these calculations but some summary remarks are in order. Maieron et al. Maieron et al. (2003, 2005) studied quasi-elastic $\nu_{\mu}N$ scattering in 16O and 12C using a relativistic shell model and treating final state effects in a distorted wave impulse approximation (DWIA). Meucci et al. Meucci et al. (2004, 2006) used a relativistic DWIA model with a relativistic mean-field model of the nucleus to study the same processes in 12C and also studying final state effects. Martinez et al. Martinez et al. (2006) studied these processes in 12C and 56Fe using a relativistic description of the nucleus and the scattering process; significantly, they argue that nuclear transparencies measured in $A(e,e^{\prime}p)$ experiments can be applied to final state effects in $A(\nu,p)$ processes. The work of van der Ventel and Piekarewicz van der Ventel and Piekarewicz (2004, 2006) centers on the calculation of a set of nuclear structure functions (describing the neutrino-nucleus scattering) via a relativistic plane wave impulse approximation. Finally, the work of Leitner et al. Leitner et al. (2006a, b) uses a fully relativistic formalism including the quasielastic and resonance scattering channels, and taking into account a variety of nuclear and final state effects. The results of several of these studies imply that the ratios $R_{NC/CC}$ and $\bar{R}_{NC/CC}$ to be measured in FINeSSE are relativly insentive to many effects of the nucleus and final state interactions.

Acknowledgements.

The author is grateful to S.D. Bass, H.E. Jackson, D.O. Riska, and A.W. Thomas for useful discussions. This work was supported by the US Department of Energy.