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Analytical electromagnetic form factors of the nucleon in a soft-wall AdS/QCD model

This paper derives analytical expressions for the nucleon's electromagnetic form factors within a soft-wall AdS/QCD framework, demonstrating that the resulting Sachs form factors exhibit correct scaling behavior and normalization while providing a predictive description of nucleon structure and its dependence on the holographic scale.

Original authors: Ibrahim I. Atayev, Konul K. Omarova

Published 2026-10-14

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CC BY 4.0

Original authors: Ibrahim I. Atayev, Konul K. Omarova

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep inside every atom lies the nucleus, a tiny, dense core that holds the universe together. At the heart of that nucleus are protons and neutrons, the building blocks of all visible matter. For decades, physicists have tried to map the invisible landscape within these particles. They know that protons and neutrons are not solid, indivisible spheres, but rather complex clouds of smaller particles called quarks and gluons, bound together by a powerful force. To understand how this internal machinery works, scientists shoot high-energy electrons at protons and neutrons and watch how they bounce off. The way these particles scatter reveals a hidden blueprint known as the electromagnetic form factor. Think of this blueprint as a detailed map showing exactly where the electric charge and magnetic strength are located inside the particle. While we can measure these maps in the lab, predicting them from first principles is incredibly difficult because the forces involved are too strong for standard mathematical tools to handle.

This challenge has led researchers to explore a clever workaround inspired by a theory that connects gravity with the quantum world. By treating the three-dimensional interior of a proton as if it were a projection of a simpler, five-dimensional reality, scientists can use a mathematical framework called the soft-wall AdS/QCD model to calculate these internal structures. In this approach, the messy, tangled behavior of quarks is translated into a smoother, more manageable geometry where the math becomes solvable. It is a way of looking at a complex knot from a different angle to see how the threads are actually arranged.

In a recent study, Ibrahim I. Atayev and Konul K. Omarova used this five-dimensional perspective to derive precise, closed-form equations for the electromagnetic form factors of the proton and neutron. Rather than relying on rough approximations or computer simulations that require heavy processing, they worked through the underlying equations to find exact analytical solutions. They started with the fundamental laws governing the interaction between the particles and the electromagnetic field within this five-dimensional space. By carefully integrating these interactions, they produced clear formulas that describe how the proton's and neutron's charge and magnetism change as the momentum of the collision increases. Their work provides a complete, self-contained description of the nucleon's structure, showing exactly how the internal distribution of charge responds to external pressure.

The results of their calculations match what we expect from nature in several key ways. At low energies, the formulas correctly predict the total electric charge of the proton and the magnetic strength of the neutron. As the energy of the collision rises, the calculated values drop off at a rate that aligns with the known laws of physics for high-speed interactions. The researchers then used these formulas to calculate the size of the proton and neutron, specifically their electric and magnetic radii. Their numbers came out to 0.932 square femtometers for the proton's electric radius and 0.86 square femtometers for its magnetic radius. For the neutron, they found an electric radius of -0.147 square femtometers and a magnetic radius of 0.88 square femtometers. These figures are remarkably close to the values measured in real-world experiments, which are approximately 0.77, 0.73, -0.116, and 0.76 square femtometers respectively. The slight differences suggest that while the model captures the essential physics, there is still room for refinement in how the internal parameters are tuned.

What makes this work particularly valuable is not just the numbers it produces, but the clarity of the path taken to get there. The authors laid out every step of the derivation, showing exactly how the five-dimensional geometry translates into the four-dimensional properties we observe. This transparency turns a black-box calculation into a transparent window, allowing other scientists to see exactly how the internal structure of the nucleon emerges from the theory. By providing these exact expressions, the study offers a solid foundation for future investigations into more complex aspects of nuclear physics, such as how the spin of the proton is distributed among its constituents. The work demonstrates that even in the most chaotic and tightly bound systems in nature, there is an underlying order that can be revealed through the right mathematical lens.

Technical Summary: Analytical Electromagnetic Form Factors of the Nucleon in a Soft-Wall AdS/QCD Model

Problem Statement
Understanding the internal structure of the nucleon remains a central challenge in hadronic physics due to the nonperturbative nature of Quantum Chromodynamics (QCD) at low energies. While experimental data from elastic electron–nucleon scattering reveals rich internal dynamics, including nontrivial scaling behavior and flavor-dependent effects, theoretical frameworks capable of consistently describing both the low-energy regime (dominated by strong coupling) and the high-energy regime (governed by perturbative power-law scaling) remain an open problem. Specifically, deriving analytical expressions for nucleon electromagnetic form factors that reproduce correct normalizations and static properties without relying solely on numerical lattice QCD or phenomenological fits is a significant objective.

Methodology
The authors employ the soft-wall AdS/QCD framework, a holographic model inspired by the gauge/gravity correspondence. In this approach, baryons are modeled as bulk Dirac fields propagating in a five-dimensional Anti-de Sitter (AdS) space, modified by a quadratic dilaton background ( $\phi(z) = \kappa^2 z^2$ ) to introduce confinement and generate linear Regge trajectories.

The methodology proceeds through the following steps:

Model Setup: The authors define the action for bulk fermion fields interacting with a bulk vector field (dual to the electromagnetic current). The metric is the standard AdS metric, and the action includes a mass term and a coupling to the dilaton field.
Bulk Solutions: The normalizable profile functions for the left- and right-handed nucleon components ( $\psi_L$ and $\psi_R$ ) are derived as solutions to the bulk equations of motion. For the ground state ( $n=0$ ), these profiles are expressed in terms of powers of the holographic coordinate $z$ and Laguerre polynomials.
Vector Propagator: The bulk-to-boundary propagator for the vector field is solved using an integral representation involving a parameter $a = Q^2/4\kappa^2$ . This integral form is selected specifically to facilitate analytical computation of matrix elements.
Form Factor Derivation: The electromagnetic interaction is described by the coupling of the bulk fermion to the vector field. The Dirac ( $F_1$ ) and Pauli ( $F_2$ ) form factors are obtained by evaluating bulk matrix elements involving the product of the fermion profiles and the vector propagator.
Analytical Integration: The authors utilize the integral representation of the propagator to transform the form factor expressions into integrals over a variable $x \in [0,1]$ . By evaluating the Gaussian-type integrals over the holographic coordinate $z$ and subsequently integrating over $x$ using Beta function identities, the authors derive closed-form analytical expressions for the form factors.

Key Contributions and Results
The primary contribution of this work is the derivation of explicit, closed-form analytical expressions for the nucleon electromagnetic form factors within the soft-wall AdS/QCD model.

Analytical Expressions: The Dirac ( $F_1$ ) and Pauli ( $F_2$ ) form factors for both protons and neutrons are expressed as rational functions of the variable $a = Q^2/4\kappa^2$ . These expressions are derived directly from the 5D action and bulk profiles, avoiding numerical approximations for the form factors themselves.
Sachs Form Factors: The electric ( $G_E$ ) and magnetic ( $G_M$ ) Sachs form factors are constructed from the Dirac and Pauli components. The resulting formulas exhibit the expected large- $Q^2$ scaling behavior, asymptotically falling as $F(Q^2) \sim 1/Q^4$ .
Normalization and Static Properties: The derived form factors satisfy the correct normalization conditions at zero momentum transfer ( $F_1^p(0)=1, F_1^n(0)=0$ ). The authors compute the electric and magnetic radii by taking the slopes of the form factors at $Q^2=0$ .
Numerical Comparison: The calculated radii are compared with experimental data. The model yields values of $\langle r_E^2 \rangle_p \approx 0.932 \, \text{fm}^2$ , $\langle r_E^2 \rangle_n \approx -0.147 \, \text{fm}^2$ , $\langle r_M^2 \rangle_p \approx 0.86 \, \text{fm}^2$ , and $\langle r_M^2 \rangle_n \approx 0.88 \, \text{fm}^2$ . These are noted to compare reasonably well with experimental values (e.g., $\sim 0.77 \, \text{fm}^2$ for the proton charge radius), though the model slightly overestimates the magnitudes.

Significance and Claims
The paper positions its contribution primarily as methodological rather than the introduction of a novel phenomenological framework. The authors emphasize that while similar calculations have been performed in holographic QCD previously, this work provides a distinct algebraic realization and an explicit, systematic derivation of the results.

Key claims regarding the significance of the work include:

Transparency: The intermediate analytical steps, from the five-dimensional action to the final form factors, are presented explicitly. This contrasts with other works where only final results are stated.
Pedagogical Value: The authors state that the work serves as a pedagogical reference for researchers and students learning AdS/QCD techniques, allowing them to follow the transparent construction of hadronic observables.
Predictive Utility: The compact analytical nature of the results provides a basis for future investigations into generalized parton distributions, transition form factors, and spin-dependent nucleon structures within the holographic framework.

The paper concludes that the soft-wall AdS/QCD model offers a compact and predictive description of nucleon structure, successfully reproducing expected scaling behaviors and normalization conditions through a fully analytical approach.

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