Nucleon Form factors

The nucleon (proton and neutron) electromagnetic form factors describe the spatial distributions of electric charge and current inside the nucleon and thus are intimately related to its internal structure; these form factors are among the most basic observables of the nucleon. Nucleons are the building blocks of almost all ordinary matter in the universe. The challenge of understanding the nucleon's structure and dynamics has occupied a central place in nuclear physics.

Quantum chromo dynamics (QCD) is the theory of the strong interaction, which is responsible for binding quarks through gluons to form hadrons (baryons and mesons). The fundamental understanding of the nucleon form factors in terms of QCD is one of the outstanding problems in nuclear physics. Why do quarks forms colourless hadrons with only two stable configurations, proton and neutron? One important step towards answering this question is to characterize the internal structure of the nucleon. High energy electron scattering provides one of the most powerful tools to investigate this structure.

## Contents

* 1 Introduction
* 2 History
* 3 The new technique
* 4 The new paradigm
* 5 Theoretical progress
* 6 Concluding Remarks
* 7 Acknowledgments
* 8 References
* 9 Further reading
* 10 External links
* 11 See also

## [edit] Introduction

The form factors of the nucleon, the proton and the neutron, are measured by elastically scattering electrons or muons on either a liquid hydrogen for the proton, or light targets, like deuterium or \\(^3He\\) for the neutron. In the lowest approximation the electromagnetic interaction between the scattered projectile and the target nucleon is carried by one virtual photon with invariant mass squared \\(q^2=\omega^2-\vec{q}~^2 < 0\\) in the space-like region; where \\(\omega\\) is the energy transfer and \\(\vec{q}\\) is the three-momentum transfer. The related processes \\(e^{+}+e^{-}\leftrightarrow\bar{p}p\\) is in the complementary time-like region with \\(q^2>0\ .\\) In this article only space like form factors will be discussed.

For the nucleon, the \\(ep\\) cross section is the product of the Mott cross section for a spin-\\(\frac{1}{2}\\) point-like particle, and a form factor, \\(F(Q^2)\ ,\\) which contains the information on the nucleon structure at a given four-momentum transfer \\(Q^2=-q^2\ .\\) The form factor describes how different the nucleon is from a point like particle.

Even though the neutron carries no charge, it can be thought of as a negative pion cloud surrounding a positively charge core; a more recent picture will be discussed below. For the neutron the scattering is dominated by the magnetization, characterized by \\(\mu_n\ ,\\) the magnetic moment of the neutron.

In general two form factors are required to describe elastic scattering of leptons by spin \\(\frac{1}{2}\\) nucleons, one corresponding to the case where the spin state of the proton is the same in the final state as in the initial state, the other when the spin flips in the process. These are the Dirac and Pauli form factors, \\(F_{1}\\) and \\(F_{2}\ ,\\) respectively. These form factors, \\(F_{1}\\) and \\(F_{2}\ ,\\) have the following values in the limit \\(Q^2\rightarrow 0\ ,\\) which correspond to the exchange of a low virtuality photon:

\\[\tag{1} F_{1p}(0)=1\ \ \ \mbox{and}\ \ \ F_{2p}(0)=\kappa_p, \ \ \ F_{1n}(0)=0\ \ \ \mbox{and}\ \ \ F_{2n}(0)=\kappa_n \ ,\\]

where \\(\kappa_p=\mu_p-1\\) and \\(\kappa_n=\mu_n\\) are the anomalous magnetic moment of the proton and neutron respectively, with \\(\mu_p=2.7928\\) and \\(\mu_n=-1.9130\\) in units of nuclear magnetons; this reflects the fact that for \\(Q^2\rightarrow~0\\) the interaction is not localized, and only the overall charge and magnetic moment can be observed.

Experimentalists prefer to use two linear combinations of \\(F_{1}\\) and \\(F_{2}\ ,\\) called electric, \\(G_E\\) and magnetic, \\(G_M\\) form factors, valid for both proton and neutron:

\\[\tag{2} G_{E}=F_{1}-\frac{Q^2}{4M^2}F_2\ \ \ \mbox{and}\ \ \ G_{M}=F_{1}+F_{2}, \\]

where \\(M\\) is the mass of the nucleon.

For very low momentum transfer, \\(G_{E}\\) and \\(G_{M}\\) may be thought of as Fourier transforms of the charge and magnetization current densities inside the nucleon. At larger \\(Q^2\\) the nucleon wave function is different in the initial and final states, and Fourier transform does not apply.

## [edit] History

O. Stern (1933) measured the magnetic moment of the proton in 1933, and observed that it was anomalous, being 2.8 times larger than expected for a spin-\\(\frac{1}{2}\ ,\\) point-like particle. Twenty years later elastic \\(ep\\) cross section experiments at the Stanford High Energy Physics Laboratory were interpreted by Hofstadter and collaborators (Hofstadter et al. 1953, 1956) in terms of a form factor:

\\[\tag{3} \sigma(\theta_e)=\sigma_{M}\left|\int \rho(\vec{r})e^{i\vec q.\vec r}d^{3}{\vec{r}} \right|^2=\sigma_{M}|F({\mathrm q})|^2. \\]

where the integral is over the volume of the target nucleon. The underlying assumption is that the \\(ep\\) interaction is mediated by a single virtual photon.

Following the work of Rosenbluth (Rosenbluth 1950), the cross section can be written in terms of \\(G_{Ep}\\) and \\(G_{Mp}\\) as follows:

\\[\tag{4} \frac{d\sigma}{d\Omega} =\left(\frac{d\sigma}{d\Omega}\right)_{M}\times\left[G_{E}^2 +\frac{\tau}{\epsilon}G_{M}^2\right]\frac{1}{(1+\tau)}, \\]

where \\(\tau=\frac{Q^2}{4M^2}\\) and \\(\epsilon\\) is a kinematic factor \\(\epsilon=[{1+2(1+\tau)\tan^2 \frac{\theta_e}{2}}]^{-1}\ ,\\) which is also the polarization of the virtual photon.

The Rosenbluth separation method takes advantage of the linear dependence in \\(\epsilon\\) of the form factors in the reduced cross section based on Eq. (4), which can be written as: \\[\tag{5} \sigma_R = \left(\frac{d\sigma}{d\Omega}\right)_{exp} / \left(\frac{d\sigma}{d\Omega}\right)_{M}\frac{\epsilon\left(1+\tau \right)}{\tau}=\frac{\epsilon}{\tau}G_{E}^2+G_{M}^2, \\]

where \\(\left(\frac{d\sigma}{d\Omega}\right)_{exp}\\) is a measured cross section. As shown in Fig. 1, a fit to several reduced cross section values at the same \\(Q^2\ ,\\) but for a different \\(\epsilon\\) values, gives independently \\(\frac{1}{\tau}G_{E}^2\\) as the slope and \\(G_{M}^2\\) as the intercept, divided by the square of the dipole form factor, \\(G_D=(1+\frac{Q^2}{0.71})^{-2}\ .\\) The data in this figure comes from Andivahis et al. (1994).

Underlying the interpretation of all \\(ep\\) cross section measurements over half a century, has been the concept that elastic electron nucleon scattering can be described in terms of a one photon exchange process (OPEX). However cross section data must first be corrected for radiative effects, which are multiple photon effects. In first order approximation these corrections include bremsstrahlung by the incident and scattered electron and by the recoiling proton, as well as vertex photon exchange, self-energy and vacuum polarization. An explicit two photon exchange contribution, which traditionally has been evaluated only in the phase space region where one of the two photons has very small energy, is also included. Figure 2 shows several internal radiative correction diagrams involving the electron, there are similar diagrams also for the proton.

Figure 1: Rosenbluth plot for data of Andivahis et al. Filled squares, triangles and circles for 1.75, 3.25 and 5 GeV\\(^{2}\\) with radiative correction; empty symbols before radiative correction.

Figure 2: Born term and lowest order radiative correction graphs for the electron in elastic \\(ep\\) scattering.

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The necessary calculational tools for radiative corrections were worked out by Mo and Tsai (1969), Maximon and Tjon (2000), and others. The corrected cross sections obtained at a fixed \\(Q^2\ ,\\) but with different \\(E_b\\) and \\(\theta_e\ ,\\) can then be analyzed as a function of \\(\epsilon\ ,\\) the kinematic factor defined above, to provide values of \\(G_{E}^2(Q^2)\\) and \\(G_{M}^2(Q^2)\ .\\)

Figure 3: Data base for \\(G_{E{p}}\\) obtained by the Rosenbluth cross section method.

Figure 4: Data base for \\(G_{M{p}}\\) obtained by the Rosenbluth cross section method.

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All form factors obtained from cross section measurements for the proton, \\(G_{Ep}\\) and \\(G_{Mp}\\) are shown in Figure 3 and Figure 4; they have been divided by the dipole form factor \\(G_D\ .\\) Evidently the form factors divided by \\(G_D\\) appear to remain close to \\(1\ .\\) This behavior suggested that \\(G_{Ep}\ ,\\) and \\(G_{Mp}\\) have similar spatial distributions, and this was the paradigm, until the double polarization experiments at the Thomas Jefferson National Accelerator Facility in Virginia (for short JLab) showed that it was not the case.

Early measurements of the neutron form factors used elastic \\(ed\\) scattering. In a second phase, cross section measurements were made comparing \\(^2H(e,e'n)x\\) and \\((e,e'p)\ .\\) More recent experiments use the double-polarization technique described below. In general the neutron data obtained before polarization experiments were not of the same quality as for the proton, and suffered from theoretical uncertainties related to the nuclear structure of the targets used. Therefore the neutron data will be shown and discussed together with the polarization data in the next part.

## [edit] The new technique

The paradigm established on the basis of cross section measurements had to be fundamentally revised following measurements of the proton's electromagnetic form factor ratio, \\(G_{Ep}/G_{Mp}\ ,\\) at JLab. Starting in 1998, three experiments have been done, which measured polarization observables in elastic \\(ep\\) scattering, instead of cross sections, each time to a larger \\(Q^2\ .\\) Earlier work of Akhiezer and Rekalo (1969, 1974), had demonstrated that measuring the polarization of the recoil proton (a method to be called here recoil polarization), would be a more sensitive way to measure \\(G_{Ep}\ ,\\) which is multiplied by \\(G_{Mp}\\) in the transverse component of the polarization, \\(P_t\ ,\\) rather than the cross section (as in the Rosenbluth method), which is increasingly dominated by \\(G_M^2\\) at large \\(Q^2\\) (see Eq. (5). Another major advantage of double polarization experiments is that radiative corrections are minimized, because polarization observables are ratios of cross sections, and \\(G_{Ep}/G_{Mp}\\) therefore is a ratio of ratios. These experiments could not have produced data with the small statistical (and systematical) uncertainties which characterize them, without the technological advances of the Continuous Electron Beam Facility (CEBAF) at JLab, which provided high current beams of polarized electrons, up to ~6 GeV. On the instrumentation side it required the construction of proton polarimeters of much larger size, used at much larger proton energies, than had been done previously. Figure 5 and 6 illustrate two technique to measure polarization observables. Fig. 5 shows the polarization transfer from a longitudinally polarized electron to the struck proton in the one-photon-exchange approximation, Fig. 6 shows the beam-target asymmetry process where, \\(\theta^*\\) and \\(\phi^*\ ,\\) are the polar and azimuthal laboratory angles of the target polarization vector with respect to \\(\vec q\ ,\\) relative to the scattering plane.

Figure 5: Polarization transfer from a longitudinally polarized electron to a proton with exchange of a virtual photon.

Figure 6: Polarized electron scattering from polarized target.

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Double-polarization experiments either measure the only two non-zero components of the polarization of the recoiling proton for longitudinally polarized electrons incident on an un-polarized nucleon target, or the asymmetry in the scattering of longitudinally polarized electrons by polarized protons or neutrons. These double polarization experiments have opened a new \\( {\it {vista} }\\) for the field, both for the proton and the neutron.

In OPEX for elastic \\(eN\\) (\\(N = \\) \\( p \mbox{ or } n \\)), the polarization of the recoiling nucleon has two polarization components, both in the scattering plane, one transverse, \\(P_t\ ,\\) and the other parallel, \\(P_{\ell}\ ,\\) to the vector momentum transfer \\(\vec{q}\\) given by the following relations:

\\[\tag{6} I_0P_t = -2\sqrt{\tau \left (1+\tau \right )}G_{E}G_{M}\tan \frac{\theta_e}{2}, \\]

\\[\tag{7} I_0P_{\ell} = \frac{1}{M}\left(E_e+E^{\prime}_e \right)\sqrt{\tau\left (1+\tau \right )}G_{M}^2 \tan^2\frac{\theta _e}{2} \\]

where \\(I_0 \propto G_{E}^2 + \frac{\tau}{\epsilon}G_{M}^2\ .\\) As first discussed by Perdrisat and Punjabi (1989) in their first proposal at JLab, the form factor ratio, \\(G_{E}/G_M\\) can be obtained directly from these two polarization components, measured simultaneously in a proton polarimeter, with drastically smaller systematic uncertainties than possible in cross section measurements:

\\[\tag{8} \frac{G_{E}}{G_{M}}=-\frac{P_t}{P_{\ell}}\frac{(E_e+E'_{e})}{2M_p}\tan(\frac{\theta_e}{2}) \\]

In the alternate process of beam-target asymmetry measurements, as first described by (Dombey 1969), the form factors can be obtained from the beam helicity asymmetry, keeping the \\(e\\) and \\(N\\) detection angles constant, but alternating the helicity of the \\(e\\)-beam. This method has been recently used in several \\(G_{En}\ .\\) Again, in OPEX, the physical asymmetry \\(A\\) for the elastic \\(eN\\) reaction (\\(N=p\\) or \\(n\\))is:

\\[\tag{9} A= -\frac{2\sqrt{\tau(1+\tau)}\tan(\theta_e/2)}{G_E^2+\frac{\tau}{\epsilon}G_M^2} \Big[ \sin\theta^{\ast}\cos\phi^{\ast}G_E G_M+\sqrt{\tau[1+(1+\tau)\tan^2(\theta_e/2)]} \cos\theta^{\ast}G_M^2 \Big]. \\]

It is evident from Eq. (9) that one can obtain \\(G_{E}/G_M\\) directly with the target polarization perpendicular to the momentum transfer vector \\({\vec q}\\) and within the reaction plane, with \\(\theta^{\ast}= \pi/2\\) and \\(\phi^{\ast}= 0^o\\) or \\(180^o\ ,\\) as illustrated in Fig. 6. This method has typically been used for neutron form factor measurements. The discussion above is strictly applicable only to a free nucleon; measurement of the neutron form factors requires the use of nuclear targets, like \\(^2H\\) or \\(^3He\ ,\\) and corrections are required to take into account final state interaction (FSI) and meson exchange currents (MEC); the size of these corrections decreases with increasing \\(Q^2\ .\\) In the case of a \\(^2H\\) target the equation corresponding to Eq. 9 has been derived by (Tomasi 2004).

Yet another experiment would be the measurement of the recoil polarization for a target nucleon (proton or neutron) polarized along the recoil proton direction, with un-polarized electrons; this polarization correlation would also directly give the \\(G_E/G_M\\)-ratio (Dombey 1969), but the experiment has never been performed.

Figure 7: The ratio \\(G_{E{p}}/G_{M{p}}\\) obtained by the recoil polarization technique. The symbols for Jones et al. (2000) and Punjabi et al. (2005) are circles, for Gayou et al. (2002) and Puckett et al. (2012) are squares, and Puckett et al. (2010) are triangles, respectively. Theoretical curves from Lomon (2002), Guidal et al. (2005), de Melo et al. (2009), Gross et al. (2008), and Cloët et al. (2009), shown as solid (red), short-dashed (blue), dash-dot (orange), dash (green), and short dash-dot (magenta), respectively.

Figure 8: \\(Q^2F_{2{p}}/F_{1{p}}\\) for the data \\(G_{Ep}/G_{M{n}}\\) ratios shown in Figure 7. The same symbols and curves are used as in Figure 7

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Recent recoil polarization experiments at JLab [1] have been made with large proton polarimeters in the momentum range 1 to 5 GeV/c. The technique has been developed, tested and used in a number of experiments at several laboratories (Cheung et al. 1995, Azhgirei et al. 2005, Punjabi et al. 2005). These polarimeters measure the entire azimuthal angular distribution after re-scattering of the recoil proton in a thick analyzer block made either of graphite or polyethylene. An analysis of the azimuthal angular distribution gives the two components of the proton polarization perpendicular to the proton momentum.

It is then necessary to transport these polarization components back to the target, to correct for the spin precession in the magnets of the spectrometer. This precession is essential to the technique, as otherwise the longitudinal component could not be measured. The ratio of the two components at the target gives \\(G_E/G_M\\) according to Eq. (8).

## [edit] The new paradigm

An entirely new picture of the structure of the proton has emerged after two experiments in Hall A at JLab showed that the ratio \\(G_{Ep}/G_{Mp}\\) was not constant, but decreased by a factor of about 3.7 over the \\(Q^2\\) range 1 to 5.6 \\(GeV^2\ .\\) These recoil polarization results are shown in Figure 7, together with results of a third experiment extending the range of \\(Q^2\\) to 8.5 \\(GeV^2\\) (Jones et al. 2000, Punjabi et al. 2005, Gayou et al. 2002, Puckett et al. 2010). The data in Figure 7 display a strikingly different \\(Q^2\\) dependence than the Rosenbluth results shown here in green triangle.

One of the long standing QCD based prediction for the proton form factor is that the ratio \\(\frac{F_{2p}}{F_{1p}}\\) should behave like \\(\frac{1}{Q^2}\\) asymptotically, i.e. for very large (but undefined) momentum transfer. That this is not yet the case is shown in Figure 8, where \\(Q^2\frac{F_{2p}}{F_{1p}}\ ,\\) which can be obtained directly from \\(G_{Ep}/G_{Mp}\ ,\\) is shown versus \\(Q^2\ ;\\) whereas the Rosenbluth data had suggested that asymptotia was near, the new data invalidate this conclusion. This difference is real, not an accident, and has been verified by a number of separate measurements at various \\(Q^2\ ,\\) as seen in Figure 7. The different \\(Q^2\\) behavior of the two proton form factors has led to a fundamental revision of most models of the nucleon. It constitute a true change of paradigm.

The electric form factor of the neutron has also recently been measured to \\(Q^2\\)-values well in excess of 1 \\(GeV^2\ .\\) Ten years ago two experiments were undertaken using a polarized \\(^2H\\) target, and with an un-polarized liquid \\(^2H\\) target and a polarimeter. Most recently a beam-target asymmetry experiment with a polarized \\(^3He\\)-gas target was done at JLab, reaching a \\(Q^2\\) of 3.5 \\(GeV^2\ .\\) In such a target optical pumping polarizes \\(^{85,87}Rb\ ,\\) and the polarization is transferred to \\(^3He\\) through collision; the latest improvement, using a second alkali (\\(Na\\)) to boost the spin transfer resulted in steady, in-beam polarizations of \\(\sim 55\%\\) up to 10\\(\mu A\\) electron beam current. The results of all \\(G_{En}\\) measurements are seen in Figure 9; except for the results from the last experiment, shown as red, filled triangles. Note that at \\(Q^2~=~3.5~ GeV^2\ ,\\) \\(G_{En}/G_D\\) reaches nearly the same value as \\(G_{Ep}/G_D\\) at that same \\(Q^2\\) (see Figure 3). Yet the neutron is neutral!

Figure 9: Data base for \\(G_{E{n}}\\) obtained in double-polarization experiments.

Figure 10: Data base for \\(G_{M{n}}\\) obtained by the ratio method and double polarization experiments.

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Most experiments that determine \\(G_{Mn}\\) are cross section measurements and are shown in Figure 10. The large \\(Q^2\\) coverage achieved recently in the CLAS detector resulted from a simultaneous detection of \\(^2H(e,e'p)'\\) and \\(^2H(e,e'n)\\) events (the ratio method) as well as \\(^1H(e,e'p)\\) for efficiency monitoring.

## [edit] Theoretical progress

Solving the QCD equations from first principles for the nucleon is only possible on the lattice, currently the only fundamental approach available. However, the feasible \\(Q^2\\) range is only up to about 3 to 4 \\(GeV^2\\) because of current computational power limitations. The expectation is that lattice QCD will be applicable up to 10 \\(GeV^2\\) or higher in 5 to 10 years. At the present time only phenomenological models which include some, but not all of the fundamental characteristics of QCD are possible. Some of the most successful models include Vector Meson Dominance (VMD), the relativistic Constituent Quark Model (RCQM), Generalized Parton Distributions (GPD), Dyson-Schwinger QCD, light front Cloudy Bag Model and more.

The earliest models explaining the global features of the nucleon form factors, such as its apparent and approximate dipole behavior, were vector meson dominance (VMD) models. In this picture the photon couples to the nucleon through the exchange of vector mesons. VMD models are a special case of the more general dispersion relation approach which allows to relate time-like and space-like form factors.

More recently extended VMD fits providing a good parametrization of all 4 nucleon electromagnetic form factors have been obtained. For example Bijker and Iachello (2004) achieve good fits to proton and neutron form factors by adding a phenomenological contribution attributed to a small (compared to the proton radius) intrinsic \\(qqq\\) structure to the vector-meson exchange terms.

Constituent quark models (CQM) have a long history too. In these models, the nucleon consists of three constituent quarks, which are thought to be valence quarks dressed with gluons and quark-antiquark pairs, with a mass between 300 and 400 MeV. All other degrees of freedom are absorbed into the masses of these quarks. It is necessary to include relativistic effects (rCQM) because the momentum transfers involved are up to 10 times larger than the constituent quark mass (de Melo 2009). The instant point forms of the rCQM come close to the data for all four nucleon form factors upto 4 \\(GeV^2\\) (Melde 2007).

Perturbative QCD is only valid at very large momentum transfers. In this limit, the virtual photon makes a hard collision with a single valence quark, which then shares the transferred momentum with the other two, nearly collinear quarks through two gluon exchanges. The nucleon electromagnetic form factors provide a famous test for perturbative QCD. Brodsky and Farrar (1975) derived scaling rules for the dominant helicity amplitudes with the expectation that \\(Q^2 F_{2p}/F_{1p}\\) should become constant at sufficiently high momentum transfers \\(Q^2\ .\\) The original pQCD prediction has been modified by Belitsky, Ji, and Yuan (2003). These authors questioned the quark co-linearity assumption at the basis of this prediction. They showed that including quark orbital angular momentum resulted in a logarithmic rise of the \\(Q^2 F_2/F_1\\)-ratio at large \\(Q^2\ .\\)

The generalized parton distributions (GPDs) provide a framework to describe the process of emission and re-absorption of a quark in the non-perturbative region by a hadron in exclusive reactions. The GPDs can be interpreted as quark correlation functions and two of them, \\(H\\) and \\(E\ ,\\) have the property that their first moments exactly coincide with the nucleon form factors. It follows that measurements of elastic nucleon form factors provide stringent constraints on the parametrization of the GPDs. Early theoretical developments in GPDs indicated that measurements of the separated elastic form factors of the nucleon to high \\(Q^2\\) may shed light on the problem of nucleon spin. The first moment of the GPDs taken in the forward limit yields, according to the Angular Momentum Sum Rule (Ji 1997), a contribution to the nucleon spin from the quarks and gluons, including both the quark spin and orbital angular momentum.

Solving a Poincaré covariant Faddeev equation describing two dressed quarks, (Cloet et al. 2009) to obtain nucleon form factors in a model in which two quarks are always correlated. Since recent form factor experiments are at momentum transfers well in excess of the nucleon's mass, a Poincaré covariant description of the nucleon is necessary to describe the data. It is important to note that the Poincaré covariance and the vector exchange nature of QCD guarantee the existence of nonzero quark orbital angular momentum in the nucleon's rest frame bound state amplitude. Recent results from theoretical model calculations discussed here are shown in Figure 7 and Figure 8.

The proton and neutron transverse charge distributions, \\(\rho_{\perp}\ ,\\) versus the impact parameter \\(b\\) in femtometers, in the light front frame as calculated in a model independent approach by Miller (2007), are shown in Figure 11. Most noticeable are the negative transverse charge densities at the center and for large distances for the neutron, which contradict the early ideas about a negative pion cloud surrounding a positively charged core.

Figure 11: The proton and neutron transverse charge distributions in the light front frame, \\(\rho_{\perp}\ ,\\) as calculated by Miller (2007); \\(b\\) is the impact parameter in femtometers. Note the negative sign of the charge at the center and at large distances for the neutron

## [edit] Concluding Remarks

This article contains an overview of the experimental data and the theoretical understanding of the nucleon electromagnetic form factors. These form factors encode the information on the structure of a strongly interacting many-body system of quarks and gluons, such as the nucleon.

The field has a long history, and many theoretical attempts have been made to understand the nucleon form factors, reflecting the intrinsic difficulty of direct calculation from the underlying theory, Quantum Chromodynamics. It is complicated by the fact that it requires, in the few GeV momentum transfer region, non-perturbative methods. Hence, in practice it involves approximations which often have a limited range of applicability. Despite their approximations and limitations, some of these non-perturbative methods do provide some insight on the nucleon structure.

Since the discovery of the drastically different behavior of \\(G_{Ep}\\) and \\(G_{Mp}\\) for \\(Q^2\\) in excess of 1 or 2 GeV\\(^2\ ,\\) most traditional models of the nucleon have been revised, and a few entirely new developments have occurred. Example of the latter are modification of the pQCD prediction which made it necessary to include quark angular momentum. Aggressive development of the GPDs, which originated with the description of Deeply Virtual Compton Scattering (DVCS), but were soon revised to include the unique information provided by elastic form factors, in particular \\(F_2\\) as it relates to the anomalous magnetic moment of the nucleon, which cannot be accessed from inclusive deep inelastic scattering experiments. And finally, form factor calculations within the framework of QCD, solving the Faddeev equation for a quark-diquark system; this approach provides a way to generate the nucleon mass from the kinetic energy of the nucleon constituents. We also mention the great progress made in the calculation of form factors from QCD on the lattice.

The use of the double-polarization technique to measure all four electromagnetic nucleon form factors in the last decade at Jefferson Lab, has resulted in a dramatic improvement of the quality of all four form factors. The upgrade of JLab to 12 GeV beam energy, with 85% polarization, offers promises of measurements of all four form factors up to the highest values of \\(Q^2\\) possible.

For a complete review of nucleon electromagnetic form factors for both data and theory, see review by Perdrisat, Punjabi and Vanderhaeghen (2007).

## [edit] Acknowledgments

The authors are supported by grants from the NSF(USA), PHY-0753777 (CFP), and the DOE(USA), DE-FG02-89ER40525 (VP).

## [edit] References

* Akhiezer A.I. and Rekalo M.P., Sov. Phys. Dokl. 13, 572 (1968).
* Akhiezer A.I. and Rekalo M.P., Sov. J. Part. Nucl. 4, 277 (1974).
* Anderson B. et al. [Jefferson Lab E95-001 Collaboration], Phys. Rev. C 75, 034003 (2007).
* Andivahis L. et al., Phys. Rev. D 50, 5491 (1994).
* Anklin H. et al., Phys. Lett. B 336, 313 (1994).
* Anklin H. et al., Phys. Lett. B 428, 248 (1998).
* Azhgirei L.S. et al., Nucl. Instrum. Meth. A 538, 431 (2005).
* Bartel W. et al., Nucl. Phys. B 58, 429 (1973).
* Becker J. et al., Eur. Phys. J. A 6, 329 (1999).
* Belitsky A.V., Ji X. and Yuan F., Phys. Rev. Lett. 91:092003 (2003).
* Berger Ch. et al., Phys. Lett. B 35, 87 (1971).
* Bermuth J. et al., Phys. Lett. B 564, 199 (2003).
* Bijker R. and Iachello F., Phys. Rev. C 69, 068201 (2004).
* Borkowski F., G. G. Simon, V.H. Walther and R. D. Wendling, Nucl. Phys. B 93, 461 (1975).
* Brodsky S. and Farrar G.R., Phys. Rev. D 11, 1309 (1975).
* Bruins E. E. W. et al., Phys. Rev. Lett. 75, 21 (1995).
* Cheung N.E. et al., Nucl. Instrum. Meth. A 363, 561 (1995).
* Christy M. E. et al., Phys. Rev. C 70, 015206 (2004).
* Cloët I.C., Eichman G., El-Bennich B., Klahn T. and Roberts C.D., Few Body Syst. 46, 1 (2009).
* Coward D. H. et al., Phys. Rev. Lett. 20, 292 (1968).
* de Melo J.P.B., Frederico T., Pace E., Pisano S. and Salmé G., Phys. Lett. B 671, 153 (2009).
* Eden T. et al., Phys Rev. C 50, 1749 (1994).
* Dombey N., Rev. Mod. Phys. 41, 236 (1969).
* Dombey N., Phys. Lett. B 29, 588 (1969).
* Gao H.~Y. et al., Phys Rev. C 50, R546 (1994).
* Gayou O. et al. [Jefferson Lab Hall A Collaboration], Phys. Rev. Lett. 88, 092301 (2002).
* Geis E. et al., Phys. Rev. Lett. 101, 042501 (2008).
* Glazier D. et al., Eur. Phys.J. A 24, 101 (2005).
* Golak J. et al., Phys Rev. C 63, 034006 (2001).
* Gross F., Ramalho G. and Pena M.T., Phys. Rev. C 77:015202 (2008).
* Guidal M., Polyakov M.V., Radyushkin A.V. and Vanderhaeghen M., Phys. Rev. D 72, 054013 (2005).
* Hand L. N., D.G. Miller, and R. Wilson, Rev. Mod. Phys. 35, 335 (1963).
* Hanson K. M. et al., Phys. Rev. D 8, 753 (1973).
* Herberg C. et al., Eur. Phys. J. A 5, 131 (1999).
* Hofstadter R., Rev. Mod. Phys. 28, 214 (1956).
* Hofstadter R., Fechter H.R. and McIntyre J.A., Phys. Rev. 91, 422 (1953).
* Janssens T., R. Hofstadter, E.B. Hughes and M.R. Yearian, Phys. Rev. 142, 922 (1966).
* Ji X., Phys. Rev. Lett. 78, 610 (1997).
* Jones M.K. et al., Phys. Rev. Lett. 84, 1398 (2000).
* Kelly J.J., Phys. Rev. C 70, 068202 (2004).
* Kubon G. et al., Phys. Lett. B 524, 26 (2002).
* Lachniet J. et al. [CLAS Collaboration], Phys. Rev. Lett. 102, 192001 (2009).
* Litt J. et al., Phys. Lett. B 31, 40 (1970).
* Lomon E.L., Phys. Rev. C 66, 045501 (2002).
* Lung A. et al., Phys. Rev. Lett. 70, 718 (1993).
* Madey R. et al. [E93-038 Collaboration], Phys. Rev. Lett. 91, 122002 (2003).
* Markowitz P. et al., Phys. Rev. C 48, 5 (1993).
* Maximon L.C. and Tjon J.A., Phys. Rev. C 62, 054320 (2000).
* Melde T. et al., Phys. Rev. D 76, 074020 (2007).
* Meyerhoff M. et al., Phys. Lett. B 327, 201 (1994).
* Miller G.A., Phys. Rev. Lett. 99, 112001 (2007).
* Mo L.W. and Tsai Y.S., Rev. Mod. Phys. 41, 205 (1969).
* Ostrick M. et al., Phys. Rev. Lett. 83, 276 (1999).
* Passchier I. et al., Phys. Rev. Lett. 82, 4988 (1999).
* Perdrisat C.F. and Punjabi V., http://www.jlab.org/exp_prog/proposals/89/PR89-014.pdf
* Perdrisat C.F., Punjabi V. and Vanderhaeghen M., Prog. Part. Nucl. Phys., 59, 694 (2007) arXiv:hep-ph/0612014.
* Price L. E. et al., Phys. Rev. D 4, 45 (1971).
* Puckett A.J.R. et al., Phys. Rev. C 85, 045203 (2012).
* Puckett A.J.R. et al., Phys. Rev. Lett. 104, 242301 (2010).
* Punjabi V., Perdrisat C.F. et al., [Jefferson Lab Hall A Collaboration] Phys. Rev. C 71, 055202 (2005) [Erratum-ibid. C 71, 069902 (2005)].
* Qattan I.~A. et al., Phys. Rev. Lett. 94, 142301 (2005).
* Riordan S. et al., will be submitted to Phys Rev. Lett. (2010).
* Rock S. et al., Phys. Rev D 46, 24 (1992); Rock S. et al., Phys. Rev. Lett. 49, 1139 (1982).
* Rosenbluth M.N., Phys. Rev. 79, 615 (1950).
* Sill A.~F. et al., Phys. Rev. D 48, 29 (1993).
* Simon G. G., Ch. Schmitt, F. Borkowski and V.H. Walther, Nucl. Phys. A 333, 381 (1980).
* Stern O., Nature 132, 103 (1933).
* Tomasi-Gustafsson, G.I. Gakh, A.P. Rekalo and M.P. Rekalo, Phys. Rev. C 70, 025202 (2004).
* Walker R. C. et al., Phys. Rev. D 49, 5671 (1994).
* Warren G. et al. [Jefferson Lab E93-026 Collaboration], Phys. Rev. Lett. 92, 042301 (2004).

Internal references

* James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

* Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

* David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

* Joseph Grames, Douglas W. Higinbotham, Hugh E. Montgomery (2010) The Thomas Jefferson National Accelerator Facility. Scholarpedia, 5(7):10211.

## [edit] Further reading

* Anthony W. Thomas, Wolfram Weise "The Structure of the Nucleon", (pub. Wiley-VCH, 2001).
* John Dirk Walecka, "Electron Scattering for Nuclear and Nucleon Structure", (pub. Cambridge University Press, 2001).
* I.C. Cloet, G. Eichmann, B. El-Bennich, T. Klahn and C.D. Roberts, Few Body Syst. 46, 1 (2009).
* C.F. Perdrisat, V. Punjabi and M. Vanderhaeghen, Prog. Part. Nucl. Phys., 59, 694 (2007).
* J. Arrington, C.D. Roberts, J.M. Zanotti, J. Phys. G 34, S23 (2007).
* C.E. Hyde-Wright and K.~de Jager, Ann. Rev. Nucl. Part. Sci. 54, 217 (2004).
* H.Y.Gao, Int. J. Mod. Phys. E 12, 1 (2003); H.Y.Gao, Int. J. Mod. Phys. E 12, 567 (2003) (erratum).

## [edit] External links

* Thomas Jefferson National Accelerator Facility

## [edit] See also

Quantum chromodynamics, Generalized Parton Distribution