Non-linear evolution and parton distributions at LHC and THERA energies

TL;DR

The paper addresses the breakdown of linear DGLAP evolution at very small x by incorporating nonlinear, high-twist effects through a dipole-based evolution equation. It solves this nonlinear equation numerically, examines the impact-parameter dependence, and introduces a small DGLAP-like correcting term ΔN to reconcile short-distance behavior, validating self-consistency through analytical and numerical checks. The results show substantial damping of the gluon density and a rising saturation scale Q_s(x), with implications for THERA and LHC energy regimes. The study highlights the necessity of nonlinear QCD dynamics to correctly extrapolate parton distributions to high-energy colliders.

Abstract

We suggest a new procedure for extrapolating the parton distributions from HERA energies to higher energies at THERA and LHC. The procedure suggested consists of two steps: first, we solve the non-linear evolution equation which includes the higher twists contributions, however this equation is deficient due to the low ($log(1/x)$) accuracy of our calculations. Second, we introduce a correcting function for which we write a DGLAP type linear evolution equation. We show that this correcting function is small in the whole kinematic region and decreases at low $x$. The nonlinear evolution equation is solved numerically and first estimates for the saturation scale, as well as for the value of the gluon density at THERA and LHC energies are made. We show that non-linear effects lead to damping of the gluon density by a factor of $2 ÷3$ at $x \approx 10^{-7}$.

Figures (8)

• Figure 1: The function $\tilde{N}$ is plotted versus distance (in fermi). The six curves show the convergence of the iterations (from below 1,5,10,14,18,and 25 iterations).
• Figure 2: The comparison between the solutions $\tilde{N}$, Glauber-Mueller formula, and GW model. The four curves correspond to two different solutions $\tilde{N}_{\alpha_S=0.25}$ (large dashes), $\tilde{N}_{\alpha_S - running}$(small dashes), $N_{GM}$ (continuous line), and $N_{GW}$ of the GW model (dots).
• Figure 3: The saturation scale $Q_s$ is plotted as a function $lg x=log_{10}(x)$. (a) - the scale obtained from the equation (\ref{['scale']}), the dotted line ($GW$) corresponds to the saturation scale of the GW model; (b) - the equation (\ref{['scale2']}) is used to determine the scale.
• Figure 4: The $b$-dependence of the solution is compared with the model dependence of the equation (\ref{['Nb']}).The graphs are plotted as functions of $b^2$ for two values of $x$: $x=10^{-3}$ and $x=10^{-5}$. The continuous line is the anzatz (\ref{['Nb']}), while the dashed line is $\tilde{N}(lhs(\ref{['EQ']}))$.
• Figure 5: The solution of the eq. (\ref{['EQ']}) for $b^2= 10 \,{\rm GeV^{-2}}$ is compared with the anzatz dependence of the equation (\ref{['Nb']}). The graphs are plotted as functions of distance for two values of $x$: $x=10^{-3}$ and $x=10^{-5}$. The continuous line is the anzatz (\ref{['Nb']}), while the dashed line is the exact numeric solution.