Proton Electromagnetic-Form-Factor Ratios at LowQ2

Abstract

We study the ratio R ≡ µG E (Q 2 )/G M (Q 2 ) of the proton at very small values of Q 2 . Radii commonly associated with these form factors are not moments of charge or magnetization densities. We show that the form factor F 2 is correctly interpretable as the two-dimensional Fourier transformation of a magnetization density. A relationship between the measurable ratio and moments of true charge and magnetization densities is derived. We find that existing measurements show that the magnetization density extends further than the charge density, in contrast with expectations based on the measured reduction of R as Q 2 increases.PACS numbers: 13.40.-f,13.40.Em,13.40.Gp,13.60.-r,14.20.Dh Keywords: form factors, charge densities Electromagnetic form factors of the proton and neutron (nucleon) are probability amplitudes that the nucleon can absorb a given amount of momentum and remain in the ground state, and therefore should determine the nucleon charge and magnetization densities. Much experimental technique, effort and ingenuity has been used recently to measure these quantities [1,2].The text-book interpretation of electromagnetic form factors, G E , G M , explained in [2], is that their Fourier transforms are measurements of charge and magnetization densities, and conventional wisdom relates the charge and magnetization mean square radii to the slopes of G E,M at Q 2 = 0. However, this interpretation is not correct because the wave functions of the initial and final nucleons have different momenta and therefore differ, invalidating a probability or density interpretation. A proper charge density is related to the matrix element of an absolute square of a field operator.Here we show that the proper magnetization density is the two-dimensional Fourier transform of the F 2 form factor. We use this and the result that the charge density is the two-dimensional Fourier transform of the F 1 form factor [3,4,5,6] to show that the magnetization density of the proton extends significantly further than its charge density. This is surprising because the observed rapid decrease of the ratio of electric to magnetic form factors with increasing values of Q 2 [1, 2] might lead one to conclude that the charge radius is larger than the magnetization radius.Form factors are matrix elements of the electromagnetic current operator J µ (x ν ) in units of the proton charge:

Realistic transverse images of the proton charge and magnetization densities

“…The transverse densities are known very well indeed. The spatial extent of ρ 2 is broader than that of ρ ch as previously observed [21]. Note that the realistic transverse densities differ substantially from the dipole result of Eq.…”

“…(31), defined by taking the matrix element of 3 rb × j in a transversely polarized state, [10,21]. This Fourier transform involves J 1 (Qb) and therefore the FRA corresponds to that of Eq.…”

Overview of nucleon structure

“…[33] found that the mean square magnetization transverse radius b 2 M is greater than the mean square transverse charge radius b 2 Ch :…”

“…The question of the location of the anomalous magnetization density of the proton was examined in [33]. The magnetization density was constructed by computing the matrix element of µ · B in a nucleon polarized in the xdirection.…”

Model independent results for nucleon structure

“…We recently showed [13], that the two-dimensional Fourier transform of the Pauli form factor F 2 plays the role of the magnetization density.…”