The Proton Gluon Distribution from the Color Dipole Picture

TL;DR

The paper demonstrates that the proton gluon distribution at small $x$ can be extracted from DIS data within the color-dipole picture (CDP) by relating $F_L$ to $G(x,Q^2)$ through a two-gluon exchange mechanism and a saturation-scale $\,\Lambda^2_{sat}(W^2)$. At large $Q^2$ the CDP-derived gluon distribution evolves consistently with pQCD and approaches an asymptotic form set by the saturation scale with exponent $C_2\approx0.29$, while at low $Q^2$ the CDP predicts substantial deviations from conventional evolution, indicating that using a fixed low-$Q^2$ starting scale in global fits is questionable. A Froissart-bounded representation of $F_2$ provides a cross-check that yields similar high-$Q^2$ behavior, yet highlights model dependence in the low-$Q^2$ region. Overall, the work argues for a CDP-consistent treatment of DIS data across $Q^2$, with implications for the interpretation of gluon distributions and the validity of standard evolution at low $Q^2$.

Abstract

Employing the representation of the experimental data on deep inelastic electron-proton scattering (DIS) in the color-dipole picture (CDP), we determine the gluon distribution of the proton at small Bjorken $x$. At sufficiently large momentum transfer, $Q^2$, the extracted gluon distribution fulfills the standard evolution equation for the proton structure function. For low values of $Q^2$, e.g. for $Q^2 = 1.9 {\rm GeV}^2$, the evolution equation for the proton structure function is violated. The standard procedure of adopting a low-$Q^2$ starting scale for the extraction of the gluon density is questionable and requires further investigations.

Figures (18)

• 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 $\sigma_{\gamma^*p}(\eta(W^2,Q^2),\xi)$ lie on the full line corresponding to $\xi = \xi_0 = 130$, compare refs. Ku-SchiKuroda.
• Figure 2: In Fig.2a we show the experimental data for $F_2(x \cong Q^2/W^2, Q^2)$ as a function of $1/W^2$, and in Fig.2b, for comparison, as a function of $x$. The theoretical prediction based on (\ref{['eq:3.19']}) with (\ref{['eq:3.20']}) is also shown in Fig.2a.
• Figure 3: The gluon distribution $\alpha_s (Q^2) x g(x,Q^2) \equiv \alpha_s (Q^2) G (x,Q^2)$ of the CDP, compare (\ref{['eq:5.1']}), as a function of $W^2$ for various values of $Q^2$. The solid line shows the asymptotic limit (\ref{['eq:5.9']}) that is reached at $Q^2 \mathrel{\hbox{$>$$\sim$}} 30 {\rm GeV}^2$.
• Figure 4: As Fig. 3, but based on (\ref{['eq:5.10']}), employing $R = const = 1/2 \rho = 3/8$. Compare text for details.
• Figure 5: The gluon-distribution function $\alpha_s (Q^2) xg (x,Q^2)$ of the CDP as a function of $x \cong Q^2/W^2$ at the (low) value of $Q^2 = 1.9~ {\rm GeV}^2$, the scale frequently used as input scale Pelicer. The solid curve is due to $C_{1}=0.31$ and $C_{2}=0.29$ and the uncertainties are due to $C_{1}=0.34{\pm}0.05$ and $C_{2}=0.27{\pm}0.01$Schild (G. Cvetic, D. Schildknecht, B. Surrow, M. Tentyukov, Eur. Phys. J. C 20, 77 (2001)).