Negative moments as the signature of the radial density at small distances

The present paper proposes a robust evaluation of any radial density at small distances using negative-order radial moments evaluated in momentum space. This evaluation provides a valuable insight into the behavior of a given radial density in the vicinity of \(r=0\), and puts strong emphasis on the importance of measuring form factors at large squared four-momentum transfer, a domain essential for the determination of negative order moments. A specific attention is paid to the regularization scheme directly affecting the numerical determination of the radial density’s parametrization. The proposed method is applied to non-relativistic study cases of the nucleon electric (\(G_{En}, G_{Ep}\)), and proton magnetic \(G_{Mp}\) form factors. The validation is performed through comparison of the results of the approach to the analytically determined Maclaurin expansion - in the vicinity of \(r=0\) - of the radial density function. The method is also applied to the relativistic Dirac form factor \({{F_{1p}}}\) of the proton. In such a non-trivial case, the Maclaurin development might not exist for the radial density, rendering the determination from the proposed method extremely important.