Mechanical Properties of the Proton from a Deformed AdS Holographic Model

We study the gravitational form factors of the proton and some of its mechanical properties. We use a holographic model based on the AdS/CFT correspondence, in which a deformation in the anti-de Sitter background geometry is considered. By describing the proton as a Dirac field in this background, we numerically evaluate the gloun contribution of its gravitational form factors $A$ and $C$ from its energy-momentum tensor. A comparison of our numerical results with respect to some lattice QCD results and previous results in holography is made. In general, a good agreement is found. We also evaluate the term $D$ and make use of it to compute the pressure and shear distributions in the system, which result in a stable composed particle interpretation consistent with the von Laue stability condition. The energy distribution in the system is also obtained. Internal forces are investigated to support this picture. We are also able to compute the radii associated with these distributions in the proton.

💡 Research Summary

This paper investigates the mechanical properties of the proton by calculating its gravitational form factors (GFFs) within a deformed anti‑de Sitter (AdS) holographic model. The authors begin by motivating the study of the proton’s energy‑momentum tensor (EMT), whose matrix elements encode the mass, spin, pressure, and shear distributions of the nucleon. Direct experimental access to these GFFs is extremely challenging, so indirect extractions from deep‑virtual Compton scattering and forthcoming Electron‑Ion Collider measurements are discussed, alongside recent lattice QCD results for the gluon contributions to the GFFs.

The theoretical framework employed is a bottom‑up AdS/QCD model in which the five‑dimensional background metric is “deformed” by a warp factor
(A(z)= -\ln(z/R) + \frac{k}{2}z^{2}).
The quadratic term breaks conformal invariance and mimics confinement, while the logarithmic term restores pure AdS near the UV boundary. A Dirac field (\Psi) representing the proton is introduced in this background with a bulk mass (m_{5}). By separating left‑ and right‑handed components, performing a Kaluza‑Klein expansion, and redefining the radial wavefunctions, the authors obtain Schrödinger‑like equations for the chiral modes with potentials (V_{L,R}(z)=m_{5}^{2}e^{2A(z)}\mp e^{A(z)}m_{5}A’(z)).

To reproduce the physical proton mass ((M_{p}=0.938) GeV) the model parameters are tuned: the deformation strength (k=(0.343)^{2},\text{GeV}^{2}), an anomalous dimension (\gamma=1.745) (which modifies the relation between (m_{5}) and the canonical scaling dimension), and (m_{5}=1.245) GeV. With these values the ground‑state eigenvalue of the Schrödinger problem matches the proton mass, and the corresponding left‑ and right‑handed wavefunctions are shown to be normalizable and confined.

The next step is to couple the bulk fermion to metric fluctuations (h_{mn}(x,z)) that encode the graviton. Linearizing Einstein’s equations in the deformed background yields a second‑order differential equation for the graviton profile (f(q^{2},z)):
(f’’+3A’ f’ - q^{2}f =0),
subject to boundary conditions (f(q^{2},0)=1) (normalization at the UV) and (f(q^{2},\infty)=0) (regularity in the IR). Numerical solutions are obtained for momentum transfers up to (q^{2}=2) GeV(^2), covering the range of recent lattice calculations.

The gravitational form factors are extracted by varying the total action with respect to the metric. Because the deformed AdS background eliminates the spin‑connection coupling, the form factor (B(q^{2})) vanishes identically in this model. The remaining form factors are given by overlap integrals of the bulk fermion wavefunctions with the graviton profile:
(A(q^{2}) = \frac{1}{2}\int dz, e^{-5A(z)}\bigl