Physical meaning of the form factors $F_1(q^2)$ and $F_2(q^2)$ for $q^2\neq 0$

Question

The charge form factor $F_1(q^2)$ and the anomalous magnetic moment form factor $F_2(q^2)$ have clear interpretations at the value $q^2=0$ i.e. $F_1(0)$ is equal to the charge of the electron and $F_2(0)$ is equal to $(g_e-2)/2$. Is there a physical interpretation for $F_1(q^2)$ and $F_2(q^2)$ when $q^2\neq 0$?

Answer 1

Very crudely, form factors away from the origin represent spatial distributions of charge or anomalous magnetic moment in the Rosenbluth formula, in Fourier space, so the leading (small nonzero) $q^2$ behavior specifies the long-distance extent of these distributions, the next moment shorter distance details, etc... You might extend the intuition to virtual distribution clouds of anything induced by renormalization phenomena.

At the primitive and original level, the idea is clear already in large-angle high-energy Mott scattering: The 1/r point charge potential of the nucleus is modified by a charge distribution $\rho({\mathbf x})$ for the Born approximation, $$ -\frac{Ze}{4\pi |{\mathbf x}|}\longrightarrow -\frac{1}{4\pi} \int \!d^3 {\mathbf x}'\frac{\rho({\mathbf x'})}{ |{\mathbf x}- {\mathbf x}'|} ~, $$ so the factor multiplying the energy conservation $\delta$ is $-Ze F({\mathbf q})/|{\mathbf q}^2|$, where ${\mathbf q}={\mathbf p}'-{\mathbf p}$ : $$ F({\mathbf q})=\frac{1}{-Ze}\int d^3 {\mathbf x} ~\rho({\mathbf x}) e^{-i\mathbf {x\cdot q}} \\ =\frac{1}{-Ze}\int d^3 {\mathbf x} ~\rho({\mathbf x}) (1-i\mathbf {x\cdot q}- (\mathbf {x\cdot q})^2/2+...) $$ as an expansion in small $\mathbf {q}$.

So, for instance, for a spherically symmetric charge distribution, the above integral reduces to $$ 1-\frac{\langle r^2\rangle}{6} |\mathbf { q}|^2+ ..., \qquad \hbox{where}\\ \langle r^2\rangle = {\int d^3 {\mathbf x} ~r^2\rho(r^2) \over \int d^3 {\mathbf x} ~\rho( r)} , $$ is the radius-squared of the proton charge distribution that Hofstadter measured to glorious effect (NP 1961).

Higher moments will reveal more detailed internal features, moving within, and of course lower ones, if not vanishing , as here, dipole distribution asymmetries. It is a Fourier shape-from-moments reconstruction problem. I understand nuclear physics thrives on such maps, if I recall the Jefferson Lab talks copious in the community.