[Paper Review] Moments of structure functions for $N_f=2$ near the physical point

Gunnar Bali, Sara Collins+8 more|arXiv (Cornell University)|Nov 27, 2013

Particle physics theoretical and experimental studies14 references3 citations

TL;DR

This study presents lattice QCD calculations of nucleon structure functions—$g_A$, $\langle x\rangle_{u-d}$, $g_S$, and $g_T$—for $N_f=2$ dynamical fermions near the physical pion mass, using non-perturbatively improved clover fermions. By employing multiple $t_{\text{sink}}$ values and simultaneous fits to two- and three-point functions, the authors suppress excited state contamination and find that finite volume effects are small, while $\langle x\rangle_{u-d}$ and $g_A$ show significant dependence on lattice spacing and volume, with results consistent with ETMC and Mainz but inconsistent with earlier QCDSF data due to excited state effects.

ABSTRACT

We report on our on-going study of the lower moments of iso-vector polarised and unpolarised structure functions, $g_A$ and $\langle x angle_{u-d}$, respectively, and the iso-vector scalar and tensor charge, for $N_f=2$ non-perturbatively improved clover fermions. With pion masses which go down to about 150 MeV, we investigate finite volume effects and excited state contributions.

Motivation & Objective

• To precisely determine the iso-vector nucleon matrix elements $g_A$, $\langle x\rangle_{u-d}$, $g_S$, and $g_T$ using lattice QCD with $N_f=2$ dynamical fermions.
• To control systematic errors from excited state contamination, finite volume effects, and lattice spacing by using multiple $t_{\text{sink}}$ values and simultaneous fits to two- and three-point functions.
• To investigate the pion mass dependence and volume dependence of these matrix elements down to $m_\pi \sim 150$ MeV.
• To compare results with experimental values and other lattice studies, particularly to resolve discrepancies in $\langle x\rangle_{u-d}$ and $g_A$.
• To assess the impact of $O(a)$ improvement and non-perturbative renormalization on the final results.

Proposed method

• Simulations were performed on $N_f=2$ ensembles with pion masses from 150 to 490 MeV, using non-perturbatively improved clover fermions at two lattice spacings ($a \sim 0.06$ and $0.07$ fm).
• Wuppertal smearing was applied to both sources and sinks to suppress excited state contributions in the nucleon two-point function.
• The connected quark diagram method was used for $g_A$, $\langle x\rangle_{u-d}$, $g_S$, and $g_T$, with $t_{\text{sink}}$ values chosen to minimize excited state contamination (e.g., $t_{\text{sink}}=15a \sim 1$ fm).
• Simultaneous fits to two- and three-point functions were performed using the form $C_{3pt}/C_{2pt} = B_0 + B_1(e^{-\Delta m(t_{\text{sink}}-t_{\text{ins}})} + e^{-\Delta m t_{\text{ins}}}) + \cdots$, including first excited state contributions.
• Non-perturbative $Z_O$ and $b_O$ factors were used to improve the vector, scalar, and tensor currents to $O(a^2)$, reducing discretization errors.
• Statistical errors were reduced via binning and autocorrelation analysis, and results were extrapolated to the physical point using $m_\pi^2$ dependence.

Experimental results

Research questions

• RQ1 How do excited state contributions affect the determination of $g_A$ and $\langle x\rangle_{u-d}$ in $N_f=2$ lattice QCD simulations near the physical pion mass?
• RQ2 To what extent do finite volume effects influence the extracted values of $g_A$, $\langle x\rangle_{u-d}$, $g_S$, and $g_T$ at $m_\pi \sim 150$ MeV?
• RQ3 How do the results for $g_A$ and $\langle x\rangle_{u-d}$ compare with experimental values and other lattice studies, particularly when accounting for $t_{\text{sink}}$ and smearing choices?
• RQ4 What is the impact of lattice spacing and $O(a)$ improvement on the scalar and tensor charges $g_S$ and $g_T$?
• RQ5 Can the observed discrepancies between this work and earlier QCDSF/ETMC results for $\langle x\rangle_{u-d}$ be attributed to excited state contamination?

Key findings

• The value of $g_A$ shows significant dependence on lattice volume and spacing, with results inconsistent with the experimental value at $Lm_\pi = 2.7$ but consistent with ETMC and Mainz at larger volumes.
• The $\langle x\rangle_{u-d}$ matrix element is found to be significantly lower than earlier QCDSF and ETMC results, primarily due to excited state contamination, and is consistent with recent studies that control such effects.
• Finite volume effects are small for $\langle x\rangle_{u-d}$ and $g_A$ at $Lm_\pi \geq 3.4$, with no significant dependence observed across the $m_\pi$ range studied.
• The scalar charge $g_S$ shows large statistical errors but no significant dependence on $m_\pi$, volume, or lattice spacing, consistent with other recent $N_f=2$ determinations.
• The tensor charge $g_T$ has smaller statistical errors and shows mild $m_\pi$ dependence, consistent with other recent lattice results.
• The ratio $C_{3pt}/C_{2pt}$ shows dominant ground state contribution for $\langle x\rangle_{u-d}$ and $g_A$, suggesting cancellation of excited state effects in the iso-vector channel, unlike the iso-scalar case.