[Paper Review] Moments of pion distribution amplitude using operator product expansion on the lattice

William Detmold, Issaku Kanamori+3 more|arXiv (Cornell University)|Oct 29, 2018

Particle physics theoretical and experimental studies13 references4 citations

TL;DR

This paper proposes a lattice QCD method to compute higher moments of the pion light-cone distribution amplitude (LCDA) using a valence relativistic heavy quark to avoid power divergences in operator mixing. The approach computes the Euclidean hadronic tensor via current-current correlators, yielding clean signals for the tensor at coarse lattice spacings (0.05–0.075 fm), but lattice artifacts remain significant, necessitating finer spacing data for reliable continuum extrapolation and future extraction of higher Mellin moments.

ABSTRACT

We report an exploratory study of the current-current matrix elements that are relevant to the extraction of moments of the pion light-cone distribution amplitude, employing the method of introducing a valence relativistic heavy quark. The numerical investigation is carried out in the quenched approximation with the physical volume $L\approx 2.4$ fm at two values of lattice spacing (0.05 and 0.075 fm). We obtain clean signals for the relevant Euclidean hadronic tensor with reasonable statistics, but observe that the lattice artefacts are non-negligible in our results. The key conclusion from the analysis hitherto is that although our approach has the potential for making significant contributions to parton physics, data at finer lattice spacings that are currently being produced are needed in order to control the continuum extrapolation.

Motivation & Objective

• To develop a lattice QCD method for computing higher Mellin moments of the pion light-cone distribution amplitude (LCDA), which are otherwise obstructed by power divergences in standard approaches.
• To overcome the challenge of operator mixing and power divergences in lattice calculations of parton distribution functions by introducing a valence relativistic heavy quark.
• To test the feasibility of extracting the pion LCDA moments via the operator product expansion (OPE) in Euclidean space using the hadronic tensor from current-current correlators.
• To assess the impact of lattice artifacts on the extracted hadronic tensor and determine the necessity of finer lattice spacings for reliable continuum extrapolation.
• To explore the effectiveness of momentum smearing techniques in improving signal-to-noise ratios for matrix elements at non-zero pion momentum.

Proposed method

• The method employs a valence relativistic heavy quark to render the current-current matrix elements free of power divergences, enabling direct extraction of LCDA moments via the OPE.
• The hadronic tensor $ U^{[ u au]}_A(q,p) $ is computed from Euclidean current-current correlators involving the axial current of the heavy and light quarks.
• The tensor is analytically continued to Minkowski space by replacing $ q_4 \to i q_0 $, allowing extraction of $ a_n $ moments through the OPE expansion in Eq. (7).
• The OPE expansion relates the hadronic tensor to the LCDA moments $ a_n $ via Wilson coefficients $ \mathcal{C}_W^{(n)} $, Gegenbauer polynomials $ C_n^2(\eta) $, and a scale-dependent $ \tilde{Q}^2 $.
• Momentum smearing is applied to the pion interpolating operator to improve signal-to-noise ratios in correlators at non-zero momentum.
• Lattice data are generated at two coarse spacings (0.05 and 0.075 fm) with physical volume $ L \approx 2.4 $ fm, and one additional point at $ a = 0.06 $ fm for artifact analysis.

Experimental results

Research questions

• RQ1 Can the valence heavy quark method successfully produce clean signals for the Euclidean hadronic tensor relevant to pion LCDA moments in lattice QCD?
• RQ2 To what extent do lattice artifacts affect the hadronic tensor at coarse lattice spacings (0.05–0.075 fm), and can they be controlled?
• RQ3 Is the current approach viable for extracting higher Mellin moments of the pion LCDA, given the need for continuum extrapolation?
• RQ4 How effective is momentum smearing in improving the signal-to-noise ratio for matrix elements at non-zero pion momentum?
• RQ5 What level of lattice spacing is required to ensure reliable continuum extrapolation for this method?

Key findings

• Clean signals are obtained for the relevant Euclidean hadronic tensor $ \text{Im}[U_A^{[12]}(q,p)] $ at $ m_\Psi = 1.3 $ and $ 2 $ GeV with reasonable statistics.
• Lattice artifacts are found to be non-negligible at the current lattice spacings of 0.05–0.075 fm, particularly evident in the $ q_4 $-dependence of the tensor at fixed momentum.
• The method successfully avoids power divergences due to operator mixing, validating its theoretical foundation for higher moment extraction.
• Momentum smearing significantly improves signal quality for non-zero pion momentum correlators, as shown in comparisons between Gaussian and momentum-smeared sources.
• The inclusion of data at $ a = 0.06 $ fm confirms the presence of lattice artifacts, reinforcing the need for finer spacing data.
• The study concludes that while the method is promising, reliable continuum extrapolation requires data at finer lattice spacings than currently available.