Nonlinear corrections to the DGLAP equations in view of the HERA data

TL;DR

This work investigates the first nonlinear corrections to the DGLAP evolution, known as GLRMQ terms, using HERA $F_2(x,Q^2)$ data and LO PDF baselines from CTEQ5L/CTEQ6L. By fitting to H1 data and constructing nonlinear initial PDFs at $Q_0^2=1.4$ GeV$^2$, the authors show that GLRMQ corrections slow the $Q^2$ evolution and can reconcile small-$x$, low-$Q^2$ behavior with the data, yielding a power-like growth of the gluon density at small $x$ that is substantially enhanced relative to CTEQ6L at $Q_0^2$ but converges to negligible differences at $Q^2 uildrel>ar{ extstylear}{ elax} 10$ GeV$^2$. Sets with different charm thresholds (Sets 2a/2b) confirm that the nonlinear approach can maintain large-$x$ fits while improving the small-$x$/low-$Q^2 region. Overall, nonlinear GLRMQ corrections provide a plausible mechanism to account for high parton densities at small $x$ and low $Q^2$, influencing the inferred gluon PDFs and offering guidance for PDF analyses in the near-saturation regime.

Abstract

The effects of the first nonlinear corrections to the DGLAP evolution equations are studied by using the recent HERA data for the structure function $F_2(x,Q^2)$ of the free proton and the parton distributions from CTEQ5L and CTEQ6L as a baseline. By requiring a good fit to the H1 data, we determine initial parton distributions at $Q_0^2=1.4$ GeV$^2$ for the nonlinear scale evolution. We show that the nonlinear corrections improve the agreement with the $F_2(x,Q^2)$ data in the region of $x\sim 3\cdot 10^{-5}$ and $Q^2\sim 1.5$ GeV$^2$ without paying the price of obtaining a worse agreement at larger values of $x$ and $Q^2$. For the gluon distribution the nonlinear effects are found to play an increasingly important role at $x\lsim 10^{-3}$ and $Q^2\lsim10$ GeV$^2$, but rapidly vanish at larger values of $x$ and $Q^2$. Consequently, contrary to CTEQ6L, the obtained gluon distribution at $Q^2=1.4$ GeV$^2$ shows a power-like growth at small $x$. Relative to the CTEQ6L gluons, an enhancement up to a factor $\sim6$ at $x=10^{-5}$, $Q_0^2=1.4$ GeV$^2$ reduces to a negligible difference at $Q^2\gsim 10$ GeV$^2$.

Figures (5)

• Figure 1: The scale evolution of the structure function $F_2(x,Q^2)$ of the free proton for fixed values of $x$ (with constants added to separate the curves). The dashed curves show the LO DGLAP result from CTEQ5L cteq5, and the solid curve the result after the DGLAP+GLRMQ evolution when initial conditions taken from CTEQ5L at $Q_0^2=5$ GeV$^2$. The data is from H1 H1 and the error bars are statistical.
• Figure 2: The goodness parameter $\chi^2$ of the fits of the computed $F_2(x,Q^2)$ to the H1 data, divided by the number of data points, as a function of the $x$ of the data. The cumulative number of the data points is increasing to the left as indicated at the top of the plot: $N(x=0.2)=2$ and $N(x=3.2\cdot10^{-5})=133$ (see also Table 1). The curves are the LO DGLAP results from CTEQ5L (long dashed thick line) and CTEQ6L (dotted-dashed thick line), the DGLAP+GLRMQ result with the initial conditions at $Q^2=5$ GeV$^2$ taken from CTEQ5L (densely dotted) and from CTEQ6L (sparsely dotted), and, our set 1 (solid), set 2a (double dashed) and set 2b (short dashed) .
• Figure 3: As Fig. \ref{['F2_vs_cteq5']} but for LO DGLAP result from CTEQ6L (dotted-dashed) and for the DGLAP+GLRMQ results with our set 1 (solid), set 2a (double dashed) and set 2b (short dashed).
• Figure 4: The parton distribution functions at $Q^2=1.4$ GeV$^2$ as obtained in the DGLAP analyses CTEQ5L cteq5 (dashed), CTEQ6L cteq6 (dotted-dashed) and in the present work based on the DGLAP+GLRMQ evolution. In our set 1 (solid), there is a finite charm contribution, while in the sets 2 (double dashed) charm is zero at this scale. Notice the enhancement from the CTEQ6L glue. The gluon distributions of set 1 and sets 2 are identical.
• Figure 5: The $Q^2$ dependence of the gluon distribution function at fixed values of $x$, from CTEQ5L cteq5 (dashed), CTEQ6L cteq6 (dotted-dashed) and set 1 of the present work (solid). Notice the logarithmic scales and the absolute normalization of the curves.