Ambiguities in the definition of local spatial densities in light hadrons

The relationship between the matrix element of a local operator and the Fourier transform of the associated form factor fails for systems such as the nucleon where its intrinsic size is of order its Compton wavelength. Although one can conceive of an intrinsic charge density distribution in the proton, there does not seem to be an unambiguous way to define, compute, or measure it precisely.

Figure 1

Radial charge density distributions, $4\pi r^2\rho (r)$, in a Gaussian model for a hydrogen atom. (a) The difference between the charge density distribution for a hydrogen atom localized within 10 fm and the naive charge density distribution obtained by Fourier transform of its charge form factor. (b) The radial charge distribution of a Gaussian hydrogen atom localized within $0.25a_0$ compared to the naive charge distribution.

Figure 2

The difference between the radial charge density distribution for a carbon nucleus localized within 0.2 fm and the naive radial charge density distribution obtained by Fourier transform of its charge form factor.

Figure 3

(a) Radial charge density distributions, $4\pi r^2\rho (r)$, in a Gaussian model for the proton. The solid curve is the naive radial charge density distribution of Eq. (21). The dotted curves are the radial charge density distributions for a proton localized with $R=0.05$, 0.2 and 0.5 fm. (b) The fractional difference between the naive and localized charge density distributions for $R=0.2\mathrm{fm}$ and $R=0.5\mathrm{fm}$.

Figure 4

(a) Radial charge density distributions, $4\pi r^2\rho (r)$, in a Gaussian model for the deuteron. The solid curve is the naive radial charge density distribution of Eq. (21). The dotted curves are the radial charge density distributions for a deuteron localized with $R=0.2$, 0.5 and 1.0 fm. (b) The fractional difference between the naive and localized charge density distributions for $R=0.3\mathrm{fm}$ and $R=0.5\mathrm{fm}$ and $R=0.7\mathrm{fm}$.