Proton Structure Functions from Holographic Einstein-Dilaton Models

TL;DR

The paper develops a holographic framework to compute proton DIS structure functions F1 and F2 within a self-consistent Einstein–Dilaton model. By solving the bulk equations for the background geometry from a chosen dilaton potential and introducing probe fermions and a virtual photon in the Polchinski–Strassler formalism, it derives expressions for F1 and F2 that reflect the underlying confining geometry. Using a B1-type potential that reproduces the tensor glueball spectrum and YM thermodynamics, and fixing the proton mass via an anomalous dimension, the authors achieve very good agreement with experimental F2 data at large Bjorken x, demonstrating the viability of ED holography for DIS in the strong-coupling regime. The approach provides a flexible, parameter-efficient route to DIS predictions and can be extended to other x regimes and finite-temperature contexts.

Abstract

We study the proton structure functions $F_1$ and $F_2$ in the context of holography. We develop a general framework that extends previous holographic calculations of $F_1$ and $F_2$ to the case where the bulk geometry stems from bottom-up Einstein-Dilaton models, which are commonly used in the literature to describe some properties of QCD in the strong coupling regime. We focus on a choice of the dilaton potential that leads to a holographic model able to reproduce known lattice QCD results for the glueball masses at zero temperature and pure Yang-Mills thermodynamics above deconfinement. Once the parameters of the background holographic model are fixed, we introduce probe fermionic and gauge fields in the bulk {\it a la} Polchinski and Strassler to determine the corresponding structure functions. This particular realization of the model can successfully describe the proton mass and provide results for $F_2$ at large $x$ in very good agreement with experimental data.

Figures (9)

• Figure 1: Schematic description of a DIS process. A lepton with momentum $\ell$ interacts with a proton with momentum $P$ through the exchange of a virtual photon with momentum $q$. The final hadronic state is represented by multiple particles generically called $X$.
• Figure 2: The Einstein-Dilaton model potential, $V(\phi)$, defined in Eq. \ref{['Pot']}, with $R=1$ and parameters fixed in Table \ref{['table:1']}, corresponds to the solid orange curve. Its UV asymptotic expression is plotted in dashed blue for comparison.
• Figure 3: The $A(\phi)$ (blue curve) and $B(\phi)$ (orange curve) functions, given by Eqs. \ref{['A']} and \ref{['B']}, respectively, which fully determine the geometry of the asymptotic $AdS_5$ spacetime given by Eq. (\ref{['gs']}), after fixing the potential $V(\phi)$ in Eq. \ref{['Pot']} and solving the master equation for $G(\phi)$ given in Eq. \ref{['Geq']}.
• Figure 4: The photon field $f(\phi,q)$, Eq. \ref{['phimunorm']}, obtained from numerical solutions of Eq. \ref{['eqfphiq']}, for the first four values of $q^2$.
• Figure 5: $V_L(\phi)$ and $V_R(\phi)$ represent the potentials of the fermionic probes in the DIS problem, Eq. \ref{['PotSch1']}. Since we choose $\phi$ to represent the holographic coordinate, we have $\phi\thicksim1/E$. We then see that deep in the IR, besides largely overlapping, the potentials show a confinement profile in this region, which corresponds to a very low energy scale.