Light-Front Transverse Nucleon Charge and Magnetisation Densities

TL;DR

Using two RL-based, Poincaré-covariant continuum approaches rooted in Dyson-Schwinger equations, the paper computes the nucleon’s light-front transverse charge and magnetisation densities from the elastic form factors $G_E^N(Q^2)$ and $G_M^N(Q^2)$. The 3-body Faddeev and the $q(qq)$ quark+diquark pictures yield mutually consistent, data-compatible predictions for flavour-separated densities, revealing that valence $u$ and $d$ Dirac radii are nearly equal while the valence $d$ Pauli radius is ~$ imes$0.9 larger than the $u$ one, and that transverse densities in a polarised nucleon break rotational symmetry with charge displaced along the transverse direction. The results support the emergent hadron mass (EHM) picture and suggest that axialvector diquark correlations are essential for a complete description of the proton’s internal magnetisation. The work provides a framework for future lattice-QCD comparisons and extensions to nucleon-to-resonance transitions, advancing our understanding of the three-dimensional structure of hadrons in QCD.

Abstract

Nucleon elastic electromagnetic form factors obtained using both the three-body and quark + fully-interacting-diquark pictures of nucleon structure are employed to calculate an array of light-front transverse densities for the proton and neutron and their dressed valence-quark constituents, viz. flavour separations of the proton and neutron results. These two complementary descriptions of nucleon structure deliver mutually compatible predictions, which match expectations based on modern parametrisations of available data, where such are available. Amongst other things, it is found that transverse-plane valence $u$- and $d$-quark Dirac radii are practically indistinguishable; but regarding kindred Pauli radii, the $d$ quark value is roughly 10% greater than that of the $u$-quark. Moreover, magnetically, the valence $d$ quark is far more active than the valence $u$ quark, probably because it has much greater orbital angular momentum. Both pictures of nucleon structure agree in predicting that, in a polarised nucleon, the transverse-plane charge densities are no longer rotationally invariant. Instead, for a $+\hat x$ polarised nucleon, positive charge is displaced in the $+\hat y$ direction, with the opposite effect for negative charge.

Figures (10)

• Figure 1: With the nucleon's direction of motion defining the longitudinal light-front vector, the light-front transverse plane can be characterised by the usual two cartesian vectors, $(\hat{x},\hat{y})$, as marked by the blue surface in the image. The transverse position vector is indicated by $\vec{b}$ -- orange; and the transverse spin vector by $\vec{S}$ -- red.
• Figure 2: Faddeev equation in RL truncation. Filled circle: Faddeev amplitude, $\Psi$, the matrix-valued solution, which involves 128 independent scalar functions. Spring: dressed-gluon interaction that mediates quark+quark scattering; see Eqs. \ref{['EqRLInteraction']}, \ref{['defcalG']}. Solid line: dressed-quark propagator, $S$, calculated from the rainbow gap equation. Lines not adorned with a shaded circle are amputated. Isospin symmetry is assumed. The sum runs over each of the cases involving quark "$i=1,2,3$" as a spectator to the exchange interaction.
• Figure 3: The nucleon has three valence quarks, so the complete RL nucleon electromagnetic current has three terms: $J_\mu(Q) = \sum_{a=1,2,3} J_\mu^a(Q)$. Using symmetries, one can readily obtain the $a=1,2$ components once the $a=3$ component is known Eichmann:2011pv. $\delta$, $\delta^\prime$ are spinor indices and $\rho$, $\rho^\prime$ are isospin indices. $\Gamma_\mu$ is the dressed-photon+quark vertex, which can be obtained, e.g., following Ref. Xu:2019ilh.
• Figure 4: Proton ($p$) and neutron ($n$) transverse charge densities. Panel A(B) $\rho_{ch}^{p(n)}$. Panel C(D) $b\,\rho_{ch}^{p(n)}$. Legend. Solid purple curve -- density obtained from form factor calculated using three-body Faddeev equation and current Yao:2024uej, with, as therein, like-colour band indicating the associated uncertainty; dot-dashed green -- density obtained using $q(qq)$ form factor Cheng:2025yij; dashed blue -- density from data parametrisation in Ref. Ye:2017gyb; and dotted red -- from data parametrisation in Ref. Kelly:2004hm. (Distributions plotted in units of $\text{fm}^{-2}$ and $b=|\vec{b}|$.))
• Figure 5: $u$- and $d$-quark transverse charge densities. Panel A(B) $\rho_{ch}^{u(d)}$. Panel C(D) $b\,\rho_{ch}^{u(d)}$. Legend. Solid purple curve -- density obtained from form factor calculated using three-body Faddeev equation and current Yao:2024uej, with, as therein, like-colour band indicating the associated uncertainty; dot-dashed green -- density obtained using $q(qq)$ form factor Cheng:2025yij; dashed blue -- density from data parametrisation in Ref. Ye:2017gyb; and dotted red -- from data parametrisation in Ref. Kelly:2004hm. (Distributions plotted in units of $\text{fm}^{-2}$ and $b=|\vec{b}|$.)