$\langle x\rangle$ and $\langle x^2\rangle$ of the pion PDF from Lattice QCD with $N_f=2+1+1$ dynamical quark flavours

TL;DR

This study computes the second and third moments of the pion parton distribution function using advanced lattice QCD techniques with multiple quark flavors, pion masses, and lattice spacings to ensure accurate, renormalized results.

Contribution

It provides the first lattice QCD determination of the pion PDF moments with Nf=2+1+1 dynamical quarks, including chiral and continuum extrapolations and finite volume analysis.

Findings

<x>=0.2075(106) at 2 GeV in MS-bar scheme

<x^2>=0.163(33) at 2 GeV in MS-bar scheme

Systematic uncertainties included in final results

Abstract

Using Nf=2+1+1 lattice QCD, we determine the fermionic connected contributions to the second and third moment of the pion PDF. Based on gauge configurations from the European Twisted Mass Collaboration, chiral and continuum extrapolations are performed using pion masses in the range of 230 to 500 MeV and three values of the lattice spacing. Finite volume effects are investigated using different volumes. In order to avoid mixing under renormalisation for the third moment, we use an operator with two non-zero spatial components of momentum. Momenta are injected using twisted boundary conditions. Our final values read $\langle x\rangle=0.2075(106)$ and $\langle x^2\rangle=0.163(33)$, determined at 2 GeV in the $\overline{MS}$-scheme and with systematic and statistical uncertainties summend in quadrature.