[Paper Review] Moments and power corrections of longitudinal and transverse proton structure functions from lattice QCD

M. Batelaan, Kadir Utku Can+9 more|arXiv (Cornell University)|Jan 1, 2022

Particle physics theoretical and experimental studies1 citations

TL;DR

This paper presents the first lattice QCD calculation of the lowest moments of the proton's longitudinal (FL) and transverse (F2) structure functions, using the second-order Feynman-Hellmann theorem applied to the forward Compton amplitude. It quantifies Q²-dependent moments of F2 and finds significant power corrections, with results in good agreement with experimental data across a range of Q² values.

ABSTRACT

We present a simultaneous extraction of the moments of $F_2$ and $F_L$ structure functions of the proton at a range of photon virtuality, $Q^2$. This is achieved by computing the forward Compton amplitude via an application of the second-order Feynman-Hellmann method. We find the moments of $F_{2,L}$ in good agreement with experimental values. By studying the $Q^2$ dependence of $F_2$ moments, we estimate the power corrections.

Motivation & Objective

• To provide first-principles lattice QCD predictions for the low moments of the proton's longitudinal (FL) and transverse (F2) structure functions.
• To quantify the Q² dependence of the lowest moment of F2, which had not been previously computed in lattice QCD.
• To assess the role of power corrections in structure functions, particularly in the intermediate Q² regime where non-perturbative effects are significant.
• To circumvent operator mixing issues in higher-twist calculations by using the full Compton amplitude instead of matrix elements of local operators.
• To improve theoretical constraints for global PDF analyses by providing ab initio results on higher-twist contributions.

Proposed method

• Computing the forward Compton amplitude on the lattice using the second-order Feynman-Hellmann theorem to extract matrix elements of the electromagnetic current.
• Utilizing the Compton amplitude to reconstruct the structure functions F1, F2, and FL via dispersion relations and analyticity properties.
• Expressing the Compton amplitude as an expansion in Mellin moments of F1, F2, and FL, enabling direct extraction of moments from lattice data.
• Applying a polynomial fit in ω to the Compton amplitude ratios to isolate moments, with stability checks across fit orders.
• Using Bayesian inference to extract posterior distributions of moments, including credible intervals, from lattice data on two ensembles with different lattice spacings and volumes.
• Separating contributions from uu, dd, and ud quark pairs to study flavor dependence of moments.

Experimental results

Research questions

• RQ1 Can lattice QCD accurately compute the lowest moments of the proton's longitudinal structure function FL, particularly at intermediate Q²?
• RQ2 What is the Q² dependence of the lowest moment of F2, and how does it compare to experimental data?
• RQ3 How significant are power corrections in the moments of F2 and FL, and can they be quantified from first principles in lattice QCD?
• RQ4 Does the lattice QCD calculation of the full Compton amplitude avoid the operator mixing problems inherent in traditional OPE-based approaches?
• RQ5 To what extent do lattice results on moments agree with experimental measurements and global PDF fits?

Key findings

• The lowest moment of F2, M²₂(Q²), is successfully quantified for the first time in lattice QCD, with results showing a clear Q² dependence.
• The moments of F2 and FL computed on the lattice are in good agreement with experimental data from HERA and Jefferson Lab across the range of Q² studied.
• Power corrections are found to be significant, particularly in the FL structure function, indicating non-negligible higher-twist contributions.
• The extracted moments of F1 and FL show consistent behavior across different lattice ensembles, with stable results across fit orders and credible intervals well-constrained.
• The uu, dd, and ud quark contributions to the moments exhibit similar trends, with the M(L)₀ moment directly proportional to the lowest F2 moment.
• Density plots of posterior distributions confirm the robustness of the extracted moments, with 68% highest posterior density regions clearly defined and consistent with experimental constraints.