Impact-parameter dependent Color Glass Condensate dipole model and new combined HERA data

TL;DR

This work introduces the impact-parameter dependent b-CGC dipole model, built on BK evolution, and fits it to the high-precision HERA combined data to extract its parameters. By comparing with the IP-Sat model, it shows that both approaches reproduce inclusive and exclusive diffractive DIS within HERA kinematics, but predict different small-$x$, high-$Q^2$ behavior and high-$|t|$ diffraction beyond current data due to distinct $x$-power laws and $b$-profiles of the saturation scale. The analysis yields a nucleon saturation scale $Q_S$ of order $<1$ GeV and finds the typical impact-parameter probed in $\gamma^*p$ scattering to be $b\approx2$–$3$ GeV$^{-1}$, underscoring the importance of impact-parameter dynamics. The results provide benchmarks for future small-$x$ studies at RHIC/LHC and upcoming facilities like EIC and LHeC, and point to the need for NLO and non-perturbative refinements to fully map saturation phenomena.

Abstract

The Impact-Parameter dependent Color Glass Condensate (b-CGC) dipole model is based on the Balitsky-Kovchegov non-linear evolution equation and improves the Iancu-Itakura-Munier dipole model by incorporating the impact-parameter dependence of the saturation scale. Here we confront the model to the recently released high precision combined HERA data and obtain its parameters. The b-CGC results are then compared to data at small-x for the structure function, the longitudinal structure function, the charm structure function, exclusive vector meson (J/ψ, φ, ρ) production and Deeply Virtual Compton Scattering (DVCS). We also compare our results with the Impact-Parameter dependent Saturation model (IP-Sat). We show that most features of inclusive DIS and exclusive diffractive data, including the Q^2, W, |t| and x dependence are correctly reproduced in both models. Nevertheless, the b-CGC and the IP-Sat models give different predictions beyond the current HERA kinematics, namely for the structure functions at very low x and high virtualities Q^2, and for the exclusive diffractive vector meson and DVCS production at high t. This can be traced back to the different power-law behavior of the saturation scale in x, and to a different impact-parameter b dependence of the saturation scale in these models. Nevertheless, both models give approximately similar saturation scale Q_S<1 GeV for the proton in HERA kinematics, and also both models lead to the same conclusion that the typical impact-parameter probed in the total γ^{*}p cross-section is about b\approx 2÷3 GeV^{-1}. Our results provide a benchmark for further investigation of QCD at small-x in heavy ion collisions at RHIC and the LHC and also at future experiments such as an Electron-Ion Collider and the LHeC.

Figures (14)

• Figure 1: The exclusive diffractive processes (with $p\ne p^{\prime}$ or $t\ne 0$, and $x<<x^{\prime}<<1$) in the b-CGC dipole model (left) and the IP-Sat dipole model (right) in the rest frame of the target.
• Figure 2: The saturation scale extracted from the b-CGC model, with the parameter sets given in table \ref{['t-2']} (with $m_c=1.27$ GeV) and the old parameterization set from Ref. watt-bcgc as a function of $1/x$, at various impact-parameter $b$.
• Figure 3: Left: The saturation scale in the b-CGC and the IP-Sat dipole models, as a function of $1/x$, at various impact-parameters $b$. For comparison we also show the impact-parameter independent saturation scale obtained from the CGC and the rcBK dipole model. For all models, the saturation scale was obtained from Eq. (\ref{['qs-d']}) with model parameters extracted from the combined HERA data. Right: The saturation scale in the b-CGC and the IP-Sat models as function of the impact-parameter $b$, for various fixed values of $x$, obtained with the parameter set that includes charm mass $m_c=1.27$ GeV.
• Figure 4: Top: The impact-parameter $b$ dependence of the total $\gamma^* p$ cross-section $\sigma^{\gamma^{*}p}$ , at fixed $x$ and various $Q^2$, in the b-CGC and the IP-Sat dipole models. Lower: the dipole-size $r$ dependence of the total $\gamma^* p$ cross-section, for fixed $x$ and $b$, but for various $Q^2$ (left); and for a fixed $Q^2$ and $b$, but various $x$ (right), in the b-CGC dipole model.
• Figure 5: Results for the structure function $F_2(x,Q^2)$ as function of $x$, for various values of $Q^2$, in the b-CGC (solid line) and the IP-Sat (dashed line) dipole models. In order to separate data for each $Q^2$ from the others, the data and model results represented by the lines are multiplied by a factor $2^n$, with $n$ given on the right side of the plot. We used the parameter set of the b-CGC (in table \ref{['t-2']}) and the IP-Sat models with $m_c=1.27$ GeV. The experimental data are from H1 and ZEUS collaborations Aaron:2009aa.