Nuclear Theory’21 ed. V. Nikolaev, Heron Press, Sofia, 2002 The Electric Form Factor of the Neutron D. Day Institute of Nuclear and Particle Physics, Department of Physics, University of Virginia, Charlottesville, VA 22904 Abstract. The elastic form factors provide valuable information about the charge and magnetization currents inside the proton and neutron. Information on the neutron electric form factor, G n E , has proven the most elusive, primarily due to the lack of a free neutron target. The traditional experimental methods used to extract G n E are briefly reviewed before discussing the advantages of spin dependent measurements. Details of Jefferson Lab experiment E93026 which measured G n E through � e, e ′ n ) p , will be presented. D ( � 1 Introduction The magnetic moments measurements by Otto Stern in 1934 were the first evi- dence that the neutron and the proton were composite particles, ones with internal structure. Without compositeness, one would expect the magnetic moment of the proton to be one nuclear magneton and that of the neutron to be zero. The source of the nucleon anomalous magnetic moments is the strong inter- action which gives rise to complex electromagnetic currents of quarks and an- tiquarks in the nucleon. The non-zero value of the neutron’s magnetic moment implies that the neutron must have a charge distribution. Precise knowledge of this charge distribution will give important information about the strong force that binds quarks together in neutrons and protons and other composite particles. The distribution of the charge is contained in an experimentally determined quan- tity, the electric form factor, G n E , a function of momentum transfer. Nucleons are composed of quarks and gluons and information about their in- ternal structure is critical for testing quark models. For example in a symmetric quark model, with all the valence quarks with the same wavefunction, the charge would everywhere be zero and G n E = 0. Any deviation from zero exposes the details of the wavefunctions. G n E is critical for any study of nuclear structure – 1
2 The Electric Form Factor of the Neutron without an accurate description of the nucleon form factors it is almost impossi- ble to obtain information from the few body structure functions, our best testing ground for FSI, MEC, and NN potentials (see, for example, [1]). A precise de- termination of the charge distribution in the neutron has frustrated physicists for more than 40 years, primarily from the lack of a free neutron target and the fact that the electric form factor is so small. The situation is finally improving because of recent advances in beam and target technology. 2 Nucleon Electromagnetic Form Factors The diagram in Figure 1 represents the exchange of a virtual photon, carrying energy ν and three momentum q between the electron and the nucleon target, at rest in the laboratory. The large oval, labeled here as G E,M , represents all the information about the structure of the nucleon. In one photon exchange, the elastic scattering of an relativistic electron from a nucleon is described in terms of the Dirac and Pauli form factors, F 1 and F 2 respectively as in, dσ E ′ ( F 1 ) 2 + τ 2 ( F 1 + F 2 ) 2 tan 2 ( θ e / 2) + ( F 2 ) 2 �� � � d Ω = σ Mott . (1) E F 1 and F 2 are functions of Q 2 and have the following normalization: F p 1 (0) = 1 , F p 2 (0) = 1 . 79 , F n 1 (0) = 0 , and F n 2 (0) = − 1 . 91 . The four momentum transfer Q 2 = q 2 − ν 2 = 4 EE ′ sin 2 ( θ e / 2) and τ = Q 2 / (4 M 2 ) . The interpretation of the nucleon form factors has proven more convenient by taking a linear combination of F 1 and F 2 , resulting in the Sachs electric and E ′ , − → E R , − → k ′ P R γ ✄✁✄✂✄ �✁�✂� ✄✁✄✂✄ �✁�✂� G E,M ✄✁✄✂✄ �✁�✂� ✄✁✄✂✄ �✁�✂� ✄✁✄✂✄ �✁�✂� electron ✄✁✄✂✄ �✁�✂� nucleon �✁�✂� ✄✁✄✂✄ �✁�✂� ✄✁✄✂✄ �✁�✂� ✄✁✄✂✄ E , − → M k Figure 1. Elastic electron scattering in the one-photon approximation.
D. Day 3 magnetic form factors; G E ( Q 2 ) ≡ F 1 ( Q 2 ) − τF 2 ( Q 2 ) , G M ( Q 2 ) ≡ F 1 ( Q 2 ) + F 2 ( Q 2 ) . (2) In the Q 2 = 0 limit they are given by: G E (0) = Q/e and G M (0) = µ/µ N , where Q and µ are the charge and the magnetic moment of the nucleon respec- tively. Specifically, for the proton and neutron: G p G p G n G n E (0) = 1 , M (0) = 2 . 79 , E (0) = 0 , M (0) = − 1 . 91 . Making use of the Sachs form factors in Eq. 1 the electron-nucleon cross section expression becomes dσ d Ω = σ Mott E ′ G 2 E + τ (1 + (1 + τ )2 tan 2 ( θ e / 2)) G 2 � � . (3) M (1 + τ ) E 0 This expression is the Rosenbluth formula [2] and unlike Eq. 1 it contains no interference between the electric and magnetic terms. By making measurements at a fixed momentum transfer but different scattering angles the two form factors can, in principle, be separated (via a “Rosenbluth separation”). In the nonrelativistic limit, Q 2 = q 2 , the form factors G E,M can be identi- fied as the Fourier transforms of the symmetric charge and magnetization densi- ties, e.g. G n E is Fourier transform of the neutron charge distribution ρ ( r ) : 1 � G n q 2 � d 3 rρ ( r ) e ( i q · r ) � = E (2 π ) 3 d 3 r ρ ( r ) − q 2 d 3 r ρ ( r ) r 2 + · · · = 0 − q 2 � � r 2 � � = + · · · (4) ne 6 6 We can then relate the slope at Q 2 = 0 of any the form factors to the mean squared radius of the associated distribution. Specifically, in the case of G n E we � r 2 � can relate to the decomposition ne = − 6 dG n = − 6 dF n E (0) 1 (0) 3 r 2 F n r 2 r 2 � � � � � � + 2 (0) = + . (5) ne 1n Foldy dQ 2 dQ 2 2 M 2 n The second of these terms, the Foldy term, 3 n = ( − 0 . 126) fm 2 , has 2 µ n /M 2 nothing to do with the rest frame charge distribution while the first is the spa- tial charge extension seen in F n � r 2 � 1 . has been measured [3] through thermal ne = − 0 . 113 ± 0 . 003 ± 0 . 004 fm 2 . Conse- � r 2 � neutron–electron scattering: ne � r 2 � = − 0 . 113+0 . 126 ≈ 0 . This result suggests that the spatial charge quently 1 n extension seen in F n 1 is about 0 (or very small) and has left the interpretation of G n E controversial [4, 5]. This issue now appears to have been resolved [6, 7]
4 The Electric Form Factor of the Neutron whereby the Foldy term is exactly canceled by a contribution to F 1 that is not re- lated to the charge distribution: G n E arises from the rest frame charge distribution of the neutron. That the neutron have a negative charge radius (and consequently that E have a positive slope at Q 2 = 0 ) was expected many years ago, given the G n anomalous magnetic moment of neutron and Yukawa theory of mesons. A nega- tive charge radius can be understood in both a hadronic picture in which there ex- ists a pπ − component in the neutron wavefunction that gives rise to a π − cloud at large radii, and in the constituent quark model in which spin-spin forces between the quarks gives rise to a charge segregation. G p E and G P 2.1 M Measurements The electric and magnetic form factors of the proton have been separated via the Rosenbluth technique out to large momentum transfers. The magnetic form fac- tor has been extracted with good precision; however the proton charge form fac- tor data has suffered from the limitations of the Rosenbluth technique at large momentum transfer ∗ . The early form factor data was well described (to the 20% level) by a phenomenological dipole parametrization, where the form factors scaled as � − 2 � G D = G p Q 2 = G n G p M M = , G D = 1 + . E 0 . 71(GeV / c) 2 µ p µ n 1 + Q 2 /k 2 � − 2 is gener- � From Eq. 4 we see that a ’dipole” form factor, G D = ated by an exponential charge distribution: ρ ( r ) ∝ e − kr . In the past, when de- scribing electromagnetic nuclear responses, G n E has either been taken to be zero or taken to follow the Galster parametrization [8] of G n E from elastic e -D scatter- ing: G n E = − τG D µ n / (1 + 5 . 6 τ ) . The proton electric and magnetic form factors are shown in Figure 2. 2.2 Neutron Form Factors The deuteron serves as an approximation of a free neutron target but a firm un- derstanding of the ground and final state wavefunctions is required in order to extract reliable information about the form factors. The lack of a free neutron target and the dominance of G n M over G n E has, (setting aside recent progress that I will address shortly) left the data set on the neutron form factors much less than desired. The traditional techniques ∗∗ (restricted to the use of unpolarized beams ∗ Absolute cross section measurements require precise knowledge of the current and target thick- nesses, the solid angles (difficult for magnetic spectrometers), and deadtime and detector efficiencies (when scattering at both forward and backward angles the rates can vary by more than an order of magnitude). ∗∗ A nearly complete tabulation of the long history of all the form factor measurements can be found in [10].
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