A simple exact covariant model in which a scalar particle $\Psi$ 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. [Phys. Rev. D 8, 287 (1973)] by using time-ordered perturbation theory in the infinite-momentum frame. The meaning of the transverse density $\rho (b)$ 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 size $\langle b^2\rangle =\int d^{2}b\hspace{0.5em}{b}^2\rho (b)=-4\frac{\mathit{dF}}{{\mathit{dQ}}^2}(Q^2=0)$. The quantity $\langle b^2\rangle$ 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 nonrelativistic limit is defined precisely, and the ratio of the binding energy $B$ to the mass $\mathcal{M}$ of the lightest constituent is shown to govern the influence of relativistic effects. It is shown 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 $B/\mathcal{M}$, but this only occurs for values of $B/\mathcal{M}$ less than about $0.001$. For masses that mimic the quark-diquark 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 of the order of $15\%\text{−}20\%$.

• Received 18 August 2009

DOI:https://doi.org/10.1103/PhysRevC.80.045210