Pion Distribution Amplitude and Quasi-Distributions

TL;DR

This work extends the quasi-distribution program to the pion distribution amplitude by employing the virtuality distribution amplitude (VDA) and transverse-momentum dependent amplitude (TMDA) formalisms. It derives a direct relation between the pion quasi-distribution amplitude $Q_π(y,p_3)$ and the TMDA, and analyzes how nonperturbative $p_3$-evolution—driven by the $k_ot$-dependence of soft TMDAs—resembles, competes with, and eventually yields the light-cone DA $_pi(y)$ in the large-$p_3$ limit. The paper presents explicit soft-model constructions (Gaussian and slow-tail TMDA) to study $p_3$-evolution for different baseline DAs (CZ, flat, asymptotic), and discusses the onset of perturbative hard tails at higher $p_3$ via one-gluon exchange. The results provide guidance for lattice calculations of the pion DA using quasi-distributions and illuminate how 3D hadron structure encoded in TMDA affects the extraction of light-cone information. Overall, the work clarifies the interplay between nonperturbative and perturbative evolution in the quasi-distribution framework for the pion, with practical implications for future lattice studies at a few GeV scales.

Abstract

We extend our analysis of quasi-distributions onto the pion distribution amplitude. Using the formalism of parton virtuality distribution amplitudes (VDAs), we establish a connection between the pion transverse momentum dependent distribution amplitude (TMDA) $Ψ(x, k_\perp^2)$ and the pion quasi-distribution amplitude (QDA) $Q_π(y,p_3)$. We build models for the QDAs from the VDA-based models for soft TMDAs, and analyze the $p_3$ dependence of the resulting QDAs. As there are many models claimed to describe the primordial shape of the pion DA, we present the $p_3$-evolution patterns for models producing some popular proposals: Chernyak-Zhitnitsky, flat and asymptotic DAs. Our results may be used as a guide for future studies of the pion distribution amplitude on the lattice using the quasi-distribution approach.

Figures (3)

• Figure 1: Quasi-distribution amplitude $Q^{\rm CZ}_\pi (y,P)$ for $P/\Lambda =1,3,5,10$ in the Gaussian (left) and "slow" models (right).
• Figure 2: Quasi-distribution amplitudes $Q_\pi (y,P)$ in the "slow" model for $P=\Lambda$ (left) and $P= 3\Lambda$ (right) evolving to CZ, flat and asymptotic DAs.
• Figure 3: Quasi-distribution amplitudes $Q_\pi (y,P)$ in the "slow" model for $P= 5 \Lambda$ (left) and $P= 10 \Lambda$ (right) evolving to CZ, flat and asymptotic DAs.