[Paper Review] Geometric Scaling of $F_2$ and $F_2^c$ in data and QCD Parametrisations

Guillaume Beuf, C. Royon+1 more|ArXiv.org|Oct 28, 2008

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper investigates geometric scaling in the proton structure function $F_2$ and its charm component $F_2^c$ using the Quality Factor (QF) method to compare data and QCD parametrizations. It finds that $F_2^c$ data favor the same scaling behavior as $F_2$ data, but most global PDF fits (MRST, GRV) fail to reproduce this, except CTEQ6.6C4, which includes an intrinsic charm component and best matches the data's scaling properties at low $Q^2$ and low $x$. The results suggest a missing theoretical ingredient in standard PDF fits, possibly related to gluon saturation or low-$x$ resummations.

ABSTRACT

The scaling properties at low $x$ of the proton DIS cross section and its charm component are analyzed with the help of the quality factor method. Scaling properties are tested both in the deep inelastic scattering data and in the structure functions reconstructed from CTEQ, MRST and GRV parametrisations of parton density functions. The results for DIS cross sections are fully compatible between data and parametrisations. Even with larger error bars, the charm component data favors the same geometric scaling properties as the ones of inclusive DIS. This is not the case for all parametrisations of the charm component.

Motivation & Objective

• To test whether the charm component of the proton structure function $F_2^c$ exhibits geometric scaling similar to the inclusive $F_2$ at low $x$.
• To compare the scaling properties of $F_2^c$ in experimental data with those in modern QCD global fits (CTEQ, MRST, GRV).
• To assess whether current PDF parametrizations correctly reproduce the scaling behavior observed in HERA data for both $F_2$ and $F_2^c$.
• To identify which theoretical features—such as intrinsic charm or saturation effects—are necessary to explain the observed scaling in $F_2^c$.
• To evaluate the role of heavy quark mass schemes (FFNS, VFNS, GM-VFNS) in determining scaling behavior in $F_2^c$.

Proposed method

• The Quality Factor (QF) method is used to test scaling properties without assuming a functional form for the scaling function.
• Scaling is tested using the variable $\tau = \log(Q^2 / Q_s^2(Y))$, where $Y = \log(1/x)$, and the scaling parameter $\lambda$ is varied to maximize the QF.
• The QF is computed point-by-point for $F_2/Q^2$ and $F_2^c/Q^2$ in data and PDF parametrizations, comparing correlation between data points.
• The analysis uses data from HERA and PDF sets from CTEQ, MRST, and GRV, with $Q^2$ cuts at 3 and 10 GeV$^2$ to assess low-$Q^2$ behavior.
• The study compares scaling behavior across different parametrizations, including CTEQ6.6C4 with a strong intrinsic charm component.
• The QF peak position identifies the optimal scaling parameter $\lambda$, and its consistency across $F_2$, $F_2^c$, and DVCS data is evaluated.

Experimental results

Research questions

• RQ1 Do the $F_2^c$ data exhibit geometric scaling at low $x$, and if so, does the optimal scaling parameter $\lambda$ match that of $F_2$?
• RQ2 How well do current global PDF fits (CTEQ, MRST, GRV) reproduce the geometric scaling behavior observed in $F_2^c$ data?
• RQ3 Is the scaling behavior of $F_2^c$ in data consistent with the scaling of $F_2$, and what does this imply about the nature of the charm quark in the proton?
• RQ4 Which theoretical features—such as intrinsic charm or gluon saturation—best explain the observed scaling in $F_2^c$?
• RQ5 Why do some PDF parametrizations (e.g., GRV, MRST) fail to reproduce the scaling at low $Q^2$, while CTEQ6.6C4 succeeds?

Key findings

• The $F_2^c$ data favor the same geometric scaling parameter $\lambda$ as $F_2$ data and DVCS data, indicating a consistent scaling behavior across different observables.
• The CTEQ6.6C4 parametrization, which includes a strong sea-like intrinsic charm component, shows the closest agreement with $F_2^c$ data in terms of scaling behavior and $\lambda$ value.
• The MRST and GRV parametrizations do not reproduce the scaling of $F_2^c$ data, especially at $Q^2 < 10$ GeV$^2$, and peak at higher $\lambda$ values than the data.
• Only CTEQ6.6C4 exhibits good geometric scaling for $F_2^c/Q^2$ down to $Q^2 = 3$ GeV$^2$, while other parametrizations fail in this low-$Q^2$ regime.
• The similarity in $\lambda$ values across $F_2$, $F_2^c$, and DVCS suggests a common underlying mechanism, possibly related to gluon saturation or low-$x$ resummations.
• The study indicates that standard PDF fits miss a key theoretical ingredient—potentially related to gluon saturation or low-$x$ dynamics—necessary to describe $F_2^c$ scaling, especially at low $Q^2$.