Pion Distribution Amplitude from Lattice QCD

TL;DR

This work addresses how to obtain the full $x$-dependence of the pion lightcone distribution amplitude $\phi_\pi(x)$ from lattice QCD. It uses the LaMET framework to relate Euclidean quasi-DAs $\tilde{\phi}(x, P_z)$ to $\phi_\pi(x)$ via a one-loop matching kernel $Z_\phi$ and all-order mass corrections controlled by $c=m_\pi^2/(4P_z^2)$. A nonperturbative Wilson-line renormalization is implemented through a mass counterterm $\delta m$, removing the linear divergences that plagued the bare quasi-DA. Using $N_f=2+1+1$ HISQ lattices with $m_\pi \approx 310$ MeV and boosted momenta up to $P_z$ corresponding to $2$ and $3$ units, the extracted DA is broad and single-humped, in agreement with Dyson-Schwinger and Belle-based parametrizations within uncertainties. The work demonstrates feasibility of direct lattice extraction of $\phi_\pi(x)$ via LaMET and outlines concrete steps to reach physical pion mass and higher precision.

Abstract

We present the first lattice-QCD calculation of the pion distribution amplitude using the large- momentum effective field theory (LaMET) approach, which allows us to extract lightcone parton observables from a Euclidean lattice. The mass corrections needed to extract the pion distribution amplitude from this approach are calculated to all orders in $m^2_π /P_z^2$. We also implement the Wilson- line renormalization which is crucial to remove the power divergences in this approach, and find that it reduces the oscillation at the end points of the distribution amplitude. Our exploratory result at 310-MeV pion mass favors a single-hump form broader than the asymptotic form of the pion distribution amplitude.

Figures (4)

• Figure 1: The pion quasi-distribution amplitude (at $\mu=2$ GeV) after one-loop and mass correction for $P_z=2$ (blue) and $3$ (green) (in units of $2\pi/L$). The extrapolation to infinite momentum to remove the remaining higher-twist effects is shown in red. The Wilson-line renormalization that removes the power divergent contribution is not included in this plot, and will be implemented later in the results of improved pion quasi-DA. The purple dashed line is the asymptotic form $6x(1-x)$.
• Figure 2: The energy of the static-quark pairs fit to the functional form of Eq. \ref{['V']}. The point at $r=1$ is excluded from the fit to reduce discretization error. If we further exclude the $r=2$ point, then $c_2$ is increased by 15%, still in the range of Eq. \ref{['dm']}.
• Figure 3: The improved pion distribution amplitude at $\mu=2$ GeV using $\delta m=0.38\delta m_{\text{1-loop}}$ in Eq. \ref{['impDA']} for $P_z=2$ (blue) and $3$ (green) (in units of $2\pi/L$) and extrapolation to infinite-momentum limit (red), along with the asymptotic form $6x(1-x)$ (dashed line).
• Figure 4: The improved pion distribution amplitude at $\mu=2$ GeV with $\delta m=(0.38 \pm 0.28)\delta m_{\text{1-loop}}$ (red band with the central value denoted by red dot-dashed) obtained in this work (labeled as "LaMET"), along with that obtained from the Dyson-Schwinger equation (labeled "DSE") analysis of the pion (blue), a fit to the Belle data (labeled "Belle", cyan), parametrized fits to the lattice moments (labeled "Param 1" and "Param 2", respectively, gray and green) and the asymptotic form (labeled "Asymp", purple).