Neutron charge radius and the Dirac equation

Abstract

We consider the Dirac equation for a finite-size neutron in an external electric field. We explicitly incorporate Dirac-Pauli form factors into the Dirac equation. After a non-relativistic reduction, the Darwin-Foldy term is cancelled by a contribution from the Dirac form factor, so that the only coefficient of the external field charge density is $e/6 r^2_{En}$, i. e. the root mean square radius associated with the electric Sachs form factor . Our result is similar to a recent result of Isgur, and reconciles two apparently conflicting viewpoints about the use of the Dirac equation for the description of nucleons.Comment: 7 pages, no figures, to appear in Physical Review

Effective theories of scattering with an attractive inverse-square potential and the three-body problem

“…Using the fact that our potential acts at R =R, we can combine Eqs. (28,29) to obtain an equation relating v 3 and the phase shift:…”

“…Because of its relevance to three-body EFTs, the attractive ISP has been the subject of several recent papers [29,30,31]. It will also play a central role in this paper, since the DWRG analysis for scattering in the presence of this potential will also provide the basic power counting for the three-body system.…”

“…An alternative to introducing a boundary condition to fix the self-adjoint extension of the ISP is to regulate the singularity of the potential at short distances and introduce a counterterm [29,37]. In essence this is what the momentum-space regularisation of the STM equation does for the three-body problem [19,21,22].…”

“…Other authors have shown explicitly how a short-range force can be used to select an extension [29,31,37], but have not developed the power counting for all possible terms in that force. Furthermore, the three-body forces obtained in this manner turn out to be multi-valued, with no obvious procedure to resolve this [29,31].…”

“…An interesting feature of the approaches based on regularisation of the singular interaction is that they exhibit a cyclic behaviour as the scale of the cut-off is varied [19,21,29,37]. In fact, as pointed out by Wilson [38], they form a novel kind of limit cycle of the RG.…”