[Paper Review] Proton structure, Partons, QCD, DGLAP and beyond

Alan Martín|arXiv (Cornell University)|Feb 1, 2008

Particle physics theoretical and experimental studies9 references4 citations

TL;DR

This paper provides a comprehensive introduction to deep inelastic scattering (DIS) as a probe of proton substructure, emphasizing quantum chromodynamics (QCD) and the DGLAP evolution equations for parton distribution functions (PDFs). It demonstrates that global global fits to high-precision data at NNLO level describe data down to $ Q^2 = 2~\text{GeV}^2 $ with remarkable accuracy, even without including beyond-DGLAP effects, and reveals a surprising valence-like small-$ x $ behavior of the gluon distribution at low input scales, challenging conventional Regge expectations.

ABSTRACT

We present an introductory discussion of deep-inelastic lepton-proton scattering as a means to probe the substructure of the proton. A resume of QCD is given, emphasizing the running of the coupling constant and the DGLAP evolution equations for the parton densities. The determination of parton distributions is discussed and their importance for predictions of processes at the LHC is emphasized. Going beyond the pure DGLAP regime, we briefly discuss the behaviour of parton densities at low x, and the evidence for non-linear absorptive contributions.

Motivation & Objective

• To provide a pedagogical introduction to deep inelastic scattering as a tool for probing the internal structure of the proton.
• To review the theoretical framework of QCD, focusing on the running coupling constant and the DGLAP evolution equations for parton densities.
• To assess the role of parton distribution functions in predicting cross-sections at the LHC, especially in the context of high-precision global fits.
• To examine the limitations of the DGLAP framework at low $ x $, and to evaluate evidence for non-linear, absorptive effects such as those related to gluon saturation.
• To explore the potential of LHC processes like Drell-Yan and $ J/\psi $ production to probe the small-$ x $ gluon distribution below $ x \sim 10^{-4} $.

Proposed method

• Uses deep inelastic scattering (DIS) kinematics to define the Bjorken scaling variable $ x = Q^2 / 2p \cdot q $, linking it to momentum fractions carried by partons.
• Applies the DGLAP evolution equations to describe the $ Q^2 $-dependence of parton distribution functions, incorporating the running of the strong coupling $ \alpha_s $.
• Analyzes global fits of structure functions from HERA and related experiments to extract parton densities at next-to-next-to-leading order (NNLO).
• Evaluates the impact of non-DGLAP effects such as BFKL dynamics and saturation via the BK equation, particularly at low $ x $.
• Proposes LHC-based processes—Drell-Yan, $ \chi_c $, $ J/\psi $, and prompt photon production—as probes of small-$ x $ parton distributions.
• Uses the diffractive structure function $ F_2^D $ to estimate absorptive corrections via $ \Delta F_2^{\text{abs}} \sim -F_2^D $, assessing saturation signals in data.

Experimental results

Research questions

• RQ1 To what extent can the DGLAP framework describe deep inelastic scattering data down to $ Q^2 = 2~\text{GeV}^2 $, and does including beyond-DGLAP effects improve the fit?
• RQ2 What is the nature of the gluon distribution at small $ x $, and why does it exhibit a valence-like behavior at low input scales despite being a sea parton?
• RQ3 Is there conclusive experimental evidence for gluon saturation or absorptive effects at low $ x $ and low $ Q^2 $ in HERA data?
• RQ4 Can LHC processes such as Drell-Yan and $ J/\psi $ production at high rapidity access the small-$ x $ regime below $ x \sim 10^{-4} $?
• RQ5 How do non-linear effects such as those described by the BK equation influence the behavior of parton densities at low $ x $, and can they be reliably quantified from current data?

Key findings

• Global global fits to high-precision DIS data at NNLO describe the data accurately down to $ Q^2 = 2~\text{GeV}^2 $, even without including beyond-DGLAP effects.
• The inclusion of non-DGLAP contributions such as BFKL or saturation effects does not improve the fit quality, suggesting that DGLAP remains sufficient within the perturbative regime.
• A surprising result from global fits is that the gluon distribution exhibits a valence-like small-$ x $ behavior at low input scales, contrary to expectations from Regge theory where gluons and sea quarks should share the same power-law behavior.
• There is no compelling evidence for gluon saturation or absorptive effects in HERA data at $ Q^2 \gtrsim 1~\text{GeV}^2 $, despite claims based on geometric scaling or the diffractive-to-inclusive ratio.
• LHC processes such as Drell-Yan production at high rapidity can access partons at $ x \sim 1.4 \times 10^{-5} $, offering a viable path to probe the small-$ x $ gluon distribution.
• Prompt $ J/\psi $ production and $ \chi_c $ production via $ gg \to J/\psi $ processes are sensitive to the small-$ x $ gluon distribution and may provide critical constraints if combined with improved theoretical modeling.