Electromagnetic form factors and charge densities from hadrons to nuclei

Abstract

A simple exact covariant model in which a scalar particle Ψ is modeled as a bound state of two different particles is used to elucidate relativistic aspects of electromagnetic form factors F (Q 2 ). The model form factor is computed using an exact covariant calculation of the lowest-order triangle diagram. The light-front technique of integrating over the minus-component of the virtual momentum gives the same result and is the same as the one obtained originally by Gunion et al. by using time-ordered perturbation theory in the infinite-momentum-frame. The meaning of the transverse density ρ(b) is explained by providing a general derivation, using three spatial-coordinates, of its relationship with the form factor. This allows us to identify a mean-square transverse sizeThe quantity b 2 is a true measure of hadronic size because of its direct relationship with the transverse density. We show that the rest-frame charge distribution is generally not observable by studying the explicit failure to uphold current conservation. Neutral systems of two charged constituents are shown to obey the conventional lore that the heavier one is generally closer to the transverse origin than the lighter one. It is argued that the negative central charge density of the neutron arises, in pion-cloud models, from pions of high longitudinal momentum that reside at the center. The non-relativistic limit is defined precisely, and the ratio of the binding energy B to the mass M of the lightest constituent is shown to govern the influence of relativistic effects. We show that the exact relativistic formula for F (Q 2 ) is the same as the familiar one of the three-dimensional Fourier transform of a square of a wave function for very small values of B/M, but this only occurs values of B/M less than about 0.001. For masses that mimic the quark-di-quark model of the nucleon we find that there are substantial relativistic corrections to the form factor for any value of Q 2 . A schematic model of the lowest s-states of nuclei is developed. Relativistic effects are found to decrease the form factor for light nuclei but to increase the form factor for heavy nuclei. Furthermore, these lowest s-states are likely to be strongly influenced by relativistic effects that are order 15-20%.

Baryons as relativistic three-quark bound states

“…Unfortunately these relations only hold non-relativistically because relativistic boost corrections enter with Q 2 /m 2 and obscure the interpretation in terms of charge densities. Such effects are negligible as long as the binding energies are tiny, which is still the case for nuclei, but the typical masses of hadrons are small enough to induce corrections at all values of Q 2 and even affect their charge radii [563]. In this sense, the Sachs form factors are not directly related to charge distributions and the definitions (4.42) only reflect a generic measure for the size of a hadron.…”

Realistic transverse images of the proton charge and magnetization densities

“…This interpretation is deeply buried in the thinking of nuclear physicists and continues to guide intuition, as it has since the days of the Nobel prize-winning work of Hofstadter [7]. Nevertheless, the relativistic motion of the constituents of the system causes the text-book interpretation to be incorrect [8]. The difficulty is that in electron-proton scattering the initial and final nucleon states have different momenta and therefore different wave functions.…”

Generalized parton distributions and transverse densities in a light-front quark–diquark model for the nucleons

“…The charge and magnetization densities in the transverse plane are defined as the Fourier transform of the electromagnetic form factors. The form factors involve initial and final states with different momenta and the three dimensional Fourier transforms cannot be interpreted as densities whereas the transverse densities (i.e., Fourier transformed only for the transverse momenta) defined at fixed light-front time are free from this difficulty and have a proper density interpretation [25,26]. We calculate the transverse charge and anomalous magnetization densities for both proton and neutron in the light-front diquark model and compare with the two different global parameterizations proposed by Kelly [27] and Bradford et al [28].…”