Vector meson dominance in photon structure functions at small $x$ from holography

TL;DR

The paper addresses real photon structure functions at small $x$ where hadronic (vector-meson) contributions are significant. It uses a holographic QCD framework with the BPST Pomeron kernel in AdS$_5$ to compute $F_i^{\gamma}(x,Q^2)$ as a convolution of photon wave functions with the Pomeron kernel, and implements vector meson dominance via the $\rho$-meson gravitational form factor. Two channels are analyzed—$\gamma\gamma$ scattering and $\gamma\rho$ scattering—yielding predictions for $F_2^{\gamma}(x,Q^2)$, $F_L^{\gamma}(x,Q^2)$, and the longitudinal-to-transverse ratio $R_{L/T}$, with good agreement to OPAL data and consistency with the Gluck-Reya-Schienbein photon PDFs. The results indicate that vector meson dominance emerges naturally in this holographic setting, with a small number of fit parameters and predictive power for small-$x$ photon structure, providing a framework for future collider tests.

Abstract

We investigate the photon structure functions via the photon-photon and photon-vector meson scattering within the framework of holographic QCD, focusing on the small Bjorken $x$ region and assuming that the Pomeron exchange dominates. The quasi-real photon structure functions are formulated as the convolution of the known U(1) vector field wave function with the Brower-Polchinski-Strassler-Tan (BPST) Pomeron exchange kernel in the five-dimensional AdS space. Assuming the vector meson dominance, the photon structure functions can be calculated in a different way with the BPST kernel and vector meson gravitational form factor, which can be obtained in a bottom-up AdS/QCD model, for the Pomeron-vector meson coupling. It is shown that the obtained $F_2$ structure functions in the both ways agree with the experimental data, which implies the realization of the vector meson dominance within the present model setup. Calculations for the longitudinal structure function and the longitudinal-to-transverse ratio are also presented.

Figures (5)

• Figure 1: The overlap functions $zP_{24}(z)$ in the integrand of Eq. (\ref{['F_i']}). The photon is shown by the red solid curve, and one for the $\rho$ meson by the green dashed curve.
• Figure 2: The structure function $F_2^\gamma(x, Q^2)$ as a function of $x$ for $Q^2 = 1$ and $10$$\text{GeV}^2$. The parameter value $z_0 = 6.0$$\text{GeV}^{-1}$ is used to generate the curves with the modified BPST kernel. The uncertainties for $g_0^2$ are shown in the figure.
• Figure 3: The structure function $F_2^\gamma(x, Q^2)$ as a function of $x$ for various values of $Q^2$, compared with the experimental data from OPAL Collaboration OPAL:2000nfx and those calculated from the GRS PDF set Gluck:1999ub. In each panel, the red solid and green dashed curves represent our calculations, while the blue dotted curves indicate the GRS predictions. The uncertainties for $g_0^2$ are shown in the figure.
• Figure 4: The longitudinal structure function $F_L^\gamma(x, Q^2)$ as a function of $x$ for $Q^2 = 1$ and $10$$\text{GeV}^2$. The parameter value $z_0 = 6.0$$\text{GeV}^{-1}$ is used to obtain the curves with the modified kernel. The uncertainties for $g_0^2$ are shown in the figure.
• Figure 5: The longitudinal-to-transverse ratio $R_{L/T} = F_{\gamma}^L(x, Q^2)/F_{\gamma}^T(x, Q^2)$ as a function of $x$ for $Q^2 = 1$ and $10$$\text{GeV}^2$. The parameter value $z_0 = 6.0$$\text{GeV}^{-1}$ is used to obtain the curves with the modified kernel.