[Paper Review] Unfolding of target mass contributions from inclusive proton structure function data

M. E. Christy, J. Blümlein+1 more|arXiv (Cornell University)|Jan 3, 2012

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents a novel unfolding procedure to isolate target mass contributions (TMCs) from inclusive proton structure function data in charged lepton deep inelastic scattering, using global fits to $F_2^p$ and $F_L^p$ data without relying on perturbative QCD expansions in $\alpha_s$. The method yields excellent fits ($\chi^2$/DoF = 1.01 for $F_2$, 0.93 for $F_L$) and provides parameterized TMCs that enable cleaner studies of higher-twist effects and perturbative QCD at large $x$ and low $Q^2$.

ABSTRACT

We report on the extraction of the target mass contributions to the unpolarized proton structure functions by applying an unfolding procedure to the available world data from charged lepton scattering. The method employed is complementary to recent and future parton distribution function fits including target mass contributions and the results obtained can be utilized to further study perturbative QCD at large Bjorken x and small $Q^2$ . Global fits are performed to both the available $F^2$ and the separated $F_L$ ($F_1$) data which yield excellent descriptions and provides parameterizations of the extracted target mass contributions.

Motivation & Objective

• Address the challenge of disentangling target mass contributions (TMCs) from higher-twist and perturbative QCD effects in inclusive proton structure function data.
• Overcome limitations of standard PDF fits that exclude high-$x$ data due to kinematic cuts, which reduce $Q^2$ leverage and increase uncertainties.
• Provide a parameterization of TMCs in the unpolarized proton structure functions $F_2^p$ and $F_L^p$ using a non-perturbative unfolding approach.
• Enable improved studies of perturbative QCD at large Bjorken $x$ and low $Q^2$, where TMCs are non-negligible.
• Support future analyses of quark-hadron duality and higher-twist effects by cleanly separating TMCs from massless limit structure functions.

Proposed method

• Apply an unfolding procedure to world data from charged lepton scattering experiments to extract target mass contributions (TMCs) in $F_2^p$ and $F_L^p$.
• Use the Georgi-Politzer formalism within the operator product expansion (OPE) to model TMCs in the structure functions, avoiding finite-order $\alpha_s$ expansions.
• Perform global fits to $F_2^p$ and $F_L^p$ data using MINUIT and an optimized FORTRAN routine (DAIND) for numerical integration of the TMC expressions.
• Utilize data from BCDMS, NMC, SLAC, H1, ZEUS, and Jefferson Lab Hall C (E99-118, E94-110) with L/T separations to minimize correlations between $F_2$ and $F_L$ datasets.
• Construct $F_L$ data from $R = F_L / (2x F_1)$ and the ALLM parameterization of $F_2$, with added uncertainty to account for $R$-data and $F_2$ parameterization errors.
• Fix certain fit parameters (e.g., $C_1^L$) where the fit is insensitive to their value, improving stability and convergence.

Experimental results

Research questions

• RQ1 How can target mass contributions be isolated from inclusive proton structure function data without relying on perturbative QCD expansions in $\alpha_s$?
• RQ2 What is the quantitative impact of target mass effects on the $F_2^p$ and $F_L^p$ structure functions across the kinematic range $x > 3 \times 10^{-5}$ and $Q^2 > 0.1~\text{GeV}^2$?
• RQ3 How well can a global unfolding procedure describe the world data for $F_2^p$ and $F_L^p$ while separating TMCs from massless limit contributions?
• RQ4 What is the role of $F_L^p$ data in constraining the gluon distribution, and how can it be used to improve the separation of TMCs from higher-twist effects?
• RQ5 How do the extracted TMCs compare with those from standard PDF fits that include TMCs via perturbative $\alpha_s$ expansions?

Key findings

• The global fit to $F_2^p$ data yields a $\chi^2$ per degree of freedom of 1.01, indicating an excellent description of the data across $x > 3 \times 10^{-5}$ and $Q^2 > 0.1~\text{GeV}^2$.
• The fit to $F_L^p$ data achieves a $\chi^2$ per degree of freedom of 0.93, demonstrating high consistency with the data, though limited by sparse data in the $0.005 < x < 0.3$ range at $Q^2 > 3~\text{GeV}^2$.
• The extracted TMCs are parameterized via a set of fit coefficients (Tables 3 and 4), enabling precise reconstruction of $F_2^{TM}(x,Q^2)$ and $F_L^{TM}(x,Q^2)$ for future QCD studies.
• Comparison of the fit results with data shows good agreement across all $Q^2$ values, including a detailed zoom-in for $x > 0.5$, where data are most sensitive to TMCs.
• The unfolding method successfully separates TMCs from the massless limit structure functions without assuming a finite-order expansion in $\alpha_s$, offering a complementary approach to standard PDF fits.
• An optimized FORTRAN code with full covariance matrices is provided upon request, enabling reproducibility and further use in global analyses and duality studies.