[Paper Review] Pion and Kaon Distribution Amplitudes from Lattice QCD

Jun Hua, Min-Huan Chu+11 more|arXiv (Cornell University)|Jan 23, 2022

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This lattice QCD study computes pion and kaon light-cone distribution amplitudes (DAs) using large-momentum effective theory (LaMET) with non-perturbative renormalization in a hybrid scheme featuring self-renormalization. It achieves continuum and infinite-momentum limit extrapolations at physical pion and kaon masses, revealing significant deviations from the asymptotic DA form and large SU(3) flavor breaking in the kaon DA, with precise Gegenbauer moment determinations including $a_2^\pi = 0.258(70)(52)$ and $a_1^K = -0.108(14)(51)$. The results resolve long-standing ambiguities in DA shapes and provide critical inputs for hard exclusive processes in B decays and form factors.

ABSTRACT

We present the state-of-the-art lattice QCD calculation of the pion and kaon light-cone distribution amplitudes (DAs) using large-momentum effective theory. The calculation is done at three lattice spacings $a\approx\{0.06,0.09,0.12\}$ fm and physical pion and kaon masses, with the meson momenta $P_z = \{1.29,1.72,2.15\}$ GeV. The result is non-perturbatively renormalized in a recently proposed hybrid scheme with self renormalization, and extrapolated to the continuum as well as the infinite momentum limit. We find a significant deviation of the pion and kaon DAs from the asymptotic form, and a large $SU(3)$ flavor breaking effect in the kaon DA.

Motivation & Objective

• To compute the full $x$-dependence of pion and kaon distribution amplitudes (DAs) from first-principles lattice QCD, overcoming limitations of moment-based approaches.
• To address the challenge of non-perturbative renormalization in LaMET, particularly residual linear divergences in quasi-DA matrix elements.
• To achieve continuum and infinite-momentum limit extrapolations at physical pion and kaon masses, ensuring reliable physical predictions.
• To quantify deviations from the asymptotic DA form and assess SU(3) flavor symmetry breaking effects in the kaon DA.
• To provide high-precision inputs for hard exclusive processes such as B-meson decays, pion form factors, and transition form factors.

Proposed method

• Uses large-momentum effective theory (LaMET) to access the full $x$-dependence of meson distribution amplitudes from lattice matrix elements of quasi-parton distributions.
• Employs a recently proposed hybrid renormalization scheme with self-renormalization to eliminate residual linear divergences in the quasi-DA matrix elements.
• Performs non-perturbative renormalization using the ratio scheme, with matching kernels derived in coordinate space to ensure robustness against endpoint artifacts.
• Applies inverse matching in coordinate space to reconstruct the light-cone DA from the renormalized quasi-DA, avoiding oscillations common in momentum-space matching.
• Conducts continuum and infinite-momentum limit extrapolations using three lattice spacings ($a \approx \{0.06, 0.09, 0.12\}$ fm) and three momenta ($P_z = \{1.29, 1.72, 2.15\}$ GeV).
• Uses Gegenbauer polynomial expansion to extract moments, with systematic errors from renormalization scale, large-$\lambda$ extrapolation, and continuum/infinite-momentum limits.

Experimental results

Research questions

• RQ1 How do the pion and kaon distribution amplitudes deviate from the asymptotic form $\phi(x) = 6x(1-x)$ at physical masses and in the continuum limit?
• RQ2 What is the magnitude and nature of SU(3) flavor symmetry breaking in the kaon distribution amplitude compared to the pion?
• RQ3 Can the self-renormalization scheme in LaMET successfully eliminate residual linear divergences and enable reliable continuum extrapolation of the quasi-DA?
• RQ4 How do the Gegenbauer moments of the pion and kaon DAs compare with previous lattice and phenomenological estimates?
• RQ5 What is the impact of coordinate-space matching versus momentum-space matching on the reconstructed DA shape and endpoint behavior?

Key findings

• The pion distribution amplitude shows a significant deviation from the asymptotic form, with $a_2^\pi = 0.258(70)(52)$, indicating a flatter shape than asymptotic.
• The kaon distribution amplitude exhibits a large SU(3) flavor breaking effect, with $a_1^K = -0.108(14)(51)$, suggesting a strongly asymmetric momentum distribution among valence quarks.
• The second moment of the pion DA, $\xi_2^\pi = 0.300(41)$, agrees better with the OPE calculation of Braun et al. (2006) than with the recent lattice result of Bali et al. (2019).
• The $a_4$ moment of the pion DA is $0.122(46)(31)$, indicating non-trivial higher-twist structure beyond the asymptotic form.
• The kaon DA is found to be asymmetric, with $a_1^K = -0.108(14)(51)$, and $a_3^K = -0.043(6)(22)$, suggesting a non-monotonic shape with a dip near $x=0.5$, consistent with flavor-breaking effects.
• The coordinate-space matching implementation shows better stability and reduced endpoint artifacts compared to momentum-space matching, with results agreeing within errors.