Pion and Kaon PDFs from Lattice QCD with Complementary Approaches

TL;DR

This work delivers a first-principles lattice QCD determination of unpolarized twist-2 PDFs for the pion and kaon by computing forward matrix elements of non-local vector operators with Wilson lines on a $N_f=2+1+1$ ensemble and analyzing them with both LaMET and short-distance factorization. Through renormalization, Fourier transforms, and one-loop matching, the authors extract light-cone PDFs $q(x,\mu)$ at $\mu=2$ GeV, including valence and flavor-separated combinations, and perform infinite-momentum extrapolations to assess convergence. The dual methodology enables cross-checks of systematic effects and provides insights into SU(3) flavor breaking, showing that the kaon’s strange quark can carry a larger momentum fraction than the up quark, with comparable pion-kaon shapes in certain regimes. The results offer valuable first-principles constraints for meson structure and furnish a framework for refining lattice techniques in extracting $x$-dependent hadron parton distributions, including potential extensions to higher-twist and generalized parton distributions.

Abstract

In this work, we present a first-principles lattice-QCD calculation of the unpolarized quark PDF for the pion and the kaon. The lattice data rely on matrix elements calculated for boosted mesons coupled to non-local operators containing a Wilson line. The calculations on this lattice ensemble correspond to two degenerate light, a strange, and a charm quark ($N_f=2+1+1$), using maximally twisted mass fermions with a clover term. The lattice volume is $32^3\times 64$, with a lattice spacing of 0.0934 fm, and a pion mass of 260 MeV. Matrix elements are calculated for hadron boosts of $|P_3| = 0,~0.41,~0.83,~1.25,~1.66,$ and 2.07 GeV. To match lattice QCD results to their light-cone counterparts, we employ two complementary frameworks: the large-momentum effective theory (LaMET) and the short-distance factorization (SDF). Using these approaches in parallel, we also test the lattice data to identify methodology-driven systematics. Results are presented for the standard quark PDFs, as well as the valence sector. Beyond obtaining the PDFs, we also explore the possibility of extracting information on SU(3) flavor-symmetry-breaking effects. For LaMET, we also parametrize the momentum dependence to obtain the infinite-momentum PDFs.

Figures (43)

• Figure 1: The ground state energies for the pion (left) and kaon (right) at the various values of the momentum boost, $a p$. The red curve corresponds to the dispersion relation, and the blue points correspond to the energy obtained from a plateau fit on the actual lattice data.
• Figure 2: The ratio of Eq. \ref{['eq:Ratio']} for $R_\pi^u$ (left), $R_K^u$ (center) and $R_K^s$ (right) for $z=0$ (no imaginary part) and $P_3=1.25$ GeV.
• Figure 3: The ratio of Eq. \ref{['eq:Ratio']} for $R_\pi^u$ (left), $R_K^u$ (center) and $R_K^s$ (right) for $z=5$ and $P_3=1.25$ GeV. The top (bottom) panel corresponds to the real (imaginary) part.
• Figure 4: Pion bare matrix element, $F_\pi^u$ for momentum boost $|P_3|=0,~0.41,~0.83,~1.25,~1.66,~2.07$ GeV. The real and imaginary components are shown in the left and right panels, respectively.
• Figure 5: Bare matrix elements for the kaon up flavor for momentum boost $|P_3|=0,~0.41,~0.83,~1.25,~1.66,~2.07$ GeV. The real and imaginary parts are shown in the left and and plots, respectively.