Photoabsorption cross section in the low-$x$ and low-$Q^2$ domain, and DGLAP evolution

TL;DR

The paper investigates the gluon distribution in the proton at low $x$ and low $Q^2$ using a color-dipole picture dominated by two-gluon exchange, leading to $\eta$-scaling of the photoabsorption cross section $\sigma_{\gamma^*p}$. It derives a LO gluon distribution from the longitudinal structure function $F_L$ and analyzes evolution via a $R_3(x,Q^2)$ factor, revealing deviations from standard DGLAP at low $Q^2$ and advocating a low-$Q^2$ modification with a starting scale around $Q_0^2\approx 2$ GeV$^2$. The work provides analytic CDP expressions for cross sections and $F_L$, validates the gluon extraction against data, and justifies the energy dependence of the saturation scale with $C_2\approx 0.29$. Together, these results offer a coherent framework for simultaneously describing DIS and photoproduction in the low-$x$, low-$Q^2$ domain and improve the reliability of gluon distribution determinations in this region.

Abstract

The behavior of the gluon distribution of the proton in the low-$x$, low-$Q^2$ domain of deep inelastic electron-proton scattering (DIS) is being investigated. By considering two-gluon exchange as the dominant interaction in the low-$x$, low-$Q^2$ domain, we imply the well-known result of scaling of the photoabsorption cross section in terms of the scaling variable $η(W^2,Q^2)$. From this, we derive a reliable result for the gluon distribution at the leading order of the perturbative QCD improved parton model, based on evolution from a starting scale of $Q_0^2\cong 2$ GeV$^2$. The validity of evolution, when considering its quantitative modification at low-$Q^2$ without any alteration at larger values of $Q^2$, leads to a quantitative improvement in the extraction of the gluon distribution based on evolution from a starting scale of $Q^2$ conventionally chosen as $Q^2= Q_0^2\cong 2$ GeV$^2$.

Figures (4)

• Figure 1: The theoretical results for the photoabsorption cross section $\sigma_{\gamma^*p} (\eta (W^2,Q^2), \xi)$ in the CDP as a function of the low-x scaling variable $\eta (W^2,Q^2) = (Q^2 + m^2_0)/\Lambda^2_{sat} (W^2)$ for different values of the parameter $\xi$ that determines the (squared) mass range $M^2_{q \bar{q}} \le m^2_1 (W^2) = \xi \Lambda^2_{sat} (W^2)$ of the $\gamma^* \to q \bar{q}$ fluctuations that are taken into account. The experimental results for $\sigma_{\gamma^*p}(\eta(W^2,Q^2),\xi)$ (lower panel) lie on the full line corresponding to $\xi = \xi_0 = 130$, compare to ref. Ku-Schi.
• Figure 2: Experimental data Andreev and the CDP prediction of $F_L(x,Q^2)$ are compared.
• Figure 3: The factor $R_3(x,Q^2)$ in (\ref{['4.7']}), multiplying $F_L(x,Q^2)$ in the CDP, as a function of $Q^2$ at fixed values of $x$.
• Figure 4: The photoabsorption cross section (\ref{['1.2']}) $\sigma_{\gamma^*p}(\eta ={{Q^2+m_0^2}\over{\Lambda_{sat}^2}})$ (black curve) describing the DIS experimental results compared to $\sigma_{\gamma^*p}(\eta)R_3(x,Q^2)$ according to (\ref{['4.12']}) (red curve).