Intrinsic Electromagnetic Structure of Nucleons

• Intrinsic Electromagnetic Structure of Nucleons is the study of how quark and gluon interactions generate spatially resolved electromagnetic charge and current distributions within protons and neutrons.
• The topic utilizes scattering experiments and theoretical models such as light-front dynamics, chiral constituent quark models, and GPD frameworks to quantify form factors and transverse densities.
• Practical insights from measurements of nucleon deformation, meson-cloud contributions, and heavy quark effects are key to resolving puzzles like the proton radius and understanding medium modifications.

The intrinsic electromagnetic structure of nucleons encapsulates how quark and gluon degrees of freedom inside nucleons (protons and neutrons) generate spatial and dynamical distributions of electromagnetic charge and current. These internal properties are revealed and quantified through electromagnetic form factors, transverse spatial densities, and related observables measured in various scattering experiments. Theoretical interpretation of these observables employs frameworks such as light-front dynamics, chiral constituent quark models, generalized parton distributions, and continuum QCD methods. These approaches are essential for linking empirical data to the underlying quantum field theory of the strong interaction, Quantum Chromodynamics (QCD).

1. Electromagnetic Form Factors and Experimental Probes

The electromagnetic structure of nucleons is probed through processes including elastic electron-nucleon scattering, deep-inelastic scattering (DIS), semi-inclusive DIS (SIDIS), and deeply-virtual Compton scattering (DVCS) (Miller, 2010). In elastic scattering, the nucleon's response is encoded in the Dirac ($F_1$) and Pauli ($F_2$) form factors, related to the spatial distributions of charge and magnetization: $$\Gamma_\mu = \gamma_\mu F_1(Q^2) + \frac{i \sigma_{\mu\nu} q^\nu}{2M} F_2(Q^2)$$ These are combined into Sachs form factors,

$$G_E(Q^2) = F_1(Q^2) - \frac{Q^2}{4M^2}F_2(Q^2), \quad G_M(Q^2) = F_1(Q^2) + F_2(Q^2)$$

which are the natural variables for experimental extractions.

SIDIS and DVCS enable access to partonic (quark and gluon) densities dependent on both momentum fraction and spatial information, expressed through generalized parton distributions (GPDs) and transverse momentum dependent distributions (TMDs) (Miller, 2010, Selyugin, 2013). SIDIS observables such as the TMD $h_{1T}^\perp$ reveal spin-dependent internal structures with direct connection to nucleonic deformation and orbital angular momentum.

Advances in recoil polarization and double-polarization experiments have clarified the $Q^2$-dependence of ratios such as $\mu_p G_E^p/G_M^p$ (Punjabi et al., 2015), while neutron electric form factors, previously challenging due to the neutron's net neutrality, are now accessible via recoil and target polarization techniques.

2. Transverse Charge and Magnetization Densities; Non-spherical Nucleons

Utilizing light-front (infinite momentum frame) dynamics, the two-dimensional transverse densities provide model-independent spatial maps of nucleon charge and current (Miller, 2010). The Dirac form factor's two-dimensional Fourier transform,

$$\rho(b) = \int \frac{d^2q}{(2\pi)^2} F_1(Q^2 = q^2) e^{-i\mathbf{q}\cdot \mathbf{b}},$$

gives the transverse charge density, exhibiting notable features:

• The central (impact parameter $b=0$) charge density of the neutron is negative, reflecting the internal arrangement of $d$ and $u$ quark distributions and potential pionic contributions.
• The mean squared magnetization radius $\langle b^2 \rangle_M$, extracted from Pauli form factor $F_2$, exceeds the transverse charge radius $\langle b^2 \rangle_{\text{Ch}}$. The spatial magnetization distribution extends further than the charge, as

$$\langle b^2 \rangle_M - \langle b^2 \rangle_{\text{Ch}} \approx \frac{\mu}{M^2} > 0$$

where $\mu$ is the nucleon magnetic moment.

Spin-dependent densities, defined by operators of the form $$\hat\rho(\mathbf{r},\hat{n}) = \sum_i [\frac{1}{2}(1+\sigma_i\cdot\hat{n})]\delta(\mathbf{r}-\mathbf{r}_i)$$, reveal deformations tied to internal orbital angular momentum. Distributions display non-round ("cloverleaf") shapes in the transverse plane—distinctly non-spherical even for spin-$1/2$ nucleons—observable via the measurement of transverse polarization observables and TMDs such as $h_{1T}^\perp$ (Miller, 2010).

3. Theoretical Approaches: Light-Front, Chiral Constituent Quark, and GPD Models

The structure is modeled using a diversity of approaches:

• Light-front dynamics grants the most direct interpretation of form factors as Fourier transforms of two-dimensional densities (Miller, 2010). Using light-cone variables $x^+$ (light-front time) and frames with $q^+ = 0$, form factors can be mapped onto spatial distributions without ambiguities from Lorentz boosts.
• Chiral Constituent Quark Models (χCQM) introduce Goldstone boson clouds around valence quarks, naturally accounting for sea quark effects and chiral symmetry breaking. The electromagnetic properties—form factors, radii, and intrinsic quadrupole moments—are derived from multi-quark spin-flavor operator structures and configuration-mixed nucleon wave functions (Dahiya et al., 2010, Dahiya et al., 2010, Dahiya et al., 2017). The quadrupole moment in the proton is found to be proportional to $-\langle r^2 \rangle_n$, implying positive (prolate) deformation if $\langle r^2 \rangle_n$ is negative.
• GPD-based models link the Dirac and Pauli form factors to integrals of the GPDs $H^q(x, t)$ and $E^q(x, t)$ over $x$, with model-dependent $t$-dependence encoding spatial structure (Selyugin, 2013, Selyugin, 2019, Vaziri et al., 12 Nov 2024). Recent advancements include refined ansatzes for the $t$-dependence (e.g., the VS24 model (Vaziri et al., 12 Nov 2024)), employing high-precision PDFs at N$^3$LO accuracy to better capture experimental trends in both form factors and radii.

The comparison of predictions with experimental observables such as electric radii, magnetic moments, and flavor-separated form factors demonstrates the impact of model ansatz and PDF choice, particularly for the neutron where sensitivity is enhanced (Selyugin, 2013, Selyugin, 2019).

4. Meson-Cloud and Medium Effects on Intrinsic Structure

The nontrivial QCD vacuum and low-energy structure is manifested in nucleon-meson couplings, most prominently the pion cloud (Kupelwieser et al., 2013). Confinement-based relativistic constituent-quark models, when extended to include explicit pion-coupling to quark fields, produce electromagnetic and strong vertex form factors for the "bare" nucleon and the pion. These incorporate the dressing effects, systematically improving the match to low-$Q^2$ electromagnetic data, and identify meson-cloud corrections as crucial for achieving accurate predictions for static properties and the $Q^2$-dependence of form factors.

Embedding nucleons in nuclear environments introduces additional modifications:

• Electromagnetic stress from external fields can deform internal structure, with effects scaling as the electromagnetic polarizability $\alpha_E$ over the nucleon volume, $\delta \sim \alpha_E / r_p^3$, and become relevant for understanding EMC-like effects in nuclear matter (Koch, 2019).
• Medium modifications associated with nucleon virtuality or short-range correlations induce small but measurable suppressions of the high-$x$ component of quark distributions (EMC effect) and intrinsic form factors. The change in form factors can be described within light-front holographic QCD (LFHQCD) via two-state nucleon models (mixing "blob-like" and "point-like" configurations). The resulting suppression in the high-$Q^2$ regime is on the order of a few percent, consistent with experimental constraints (Kim et al., 23 Apr 2024).

In the context of relativistic density functional theory for nuclei, inclusion of nucleon intrinsic EM structure—both proton size and neutron spin-orbit terms—is required for accurate reproduction of charge radii evolution and features such as isotopic kinks (e.g., at $N=126$ in Pb isotopes). The neutron intrinsic effects are especially influential for heavy and neutron-rich isotopes (Xie et al., 2023).

5. Advanced QCD, Diquark Correlations, and Heavy Quark Effects

Contemporary continuum QCD approaches employ Poincaré-covariant Faddeev equations to describe nucleon structure as arising from interacting, nonpointlike diquark correlations (isoscalar-scalar $0^+$ and isovector-pseudovector $1^+$) within three-quark systems (Segovia, 2019). The dominance of $S$-wave structure in ground and first excited states (Roper resonance), and the direct calculation of nucleon-to-Roper transition form factors, support this picture. Observable consequences, such as the node structure in transition amplitudes and zero-crossings in Pauli transition form factors, are robust features directly tied to diquark dynamics.

Further, QCD predicts both intrinsic (nonperturbative) and extrinsic (perturbative) heavy quark content in nucleons. Intrinsic contributions from higher Fock states (e.g., $|uud + c\bar c\rangle$) produce small, negative contributions to magnetic moments and radii, consistent with lattice QCD and recently established at a ~1% level for intrinsic charm (Brodsky et al., 2022). The associated analytic light-front wave functions enable direct computation of heavy quark–induced modifications to the nucleon form factors.

6. Interpretation and Future Directions

The intrinsic electromagnetic structure of nucleons is complex, reflecting both the spatial and dynamical distributions of quark and gluon degrees of freedom, as well as intricately correlated sea effects, meson-cloud contributions, and nuclear medium modifications. Modern measurement techniques and theoretical tools are converging toward a three-dimensional, flavor- and spin-resolved mapping of nucleons.

The proton’s magnetization density exceeds its charge distribution in spatial extent, the neutron's transverse charge density at its center is negative, and the nucleon’s shape, as seen in spin-dependent densities, is non-spherical due to orbital angular momentum. Theoretical approaches, from light-front dynamics and chiral quark models to GPD frameworks and continuum QCD, each supply essential insights, enable robust connections to data, and continue to be refined with improved parameterizations, higher-order corrections, and lattice QCD benchmarks.

Prospective advances, such as deeper precision data from Jefferson Lab 12 GeV, global analysis of time- and space-like form factors, and full implementation of multi-channel coupled dynamics, are anticipated to resolve outstanding issues—such as the proton radius puzzle, the detailed nature of nucleon deformation, heavy quark sea content, and the mechanism of medium modifications—thereby completing the quantitative mapping of the intrinsic electromagnetic structure of nucleons across all scales.