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

Abstract 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 $$r=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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 ( $$G_{En}, G_{Ep}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mi>En</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mi>Ep</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ), and proton magnetic $$G_{Mp}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mi>Mp</mml:mi> </mml:mrow> </mml:msub> </mml:math> form factors. The validation is performed through comparison of the results of the approach to the analytically determined Maclaurin expansion - in the vicinity of $$r=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> - of the radial density function. The method is also applied to the relativistic Dirac form factor $${{F_{1p}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> 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.