Quasi-Parton Distribution Function in Lattice Perturbation Theory

Large momentum effective field theory provides a new direction for lattice QCD calculations of hadronic structure functions, such as parton distribution functions (PDFs), meson distribution amplitudes, and so on, directly with $x$-dependence. In the framework of Lattice Perturbation Theory (LPT), we compute the one-loop quark-in-quark quasi-PDF with the na\"ive fermion action ($\tilde{q}^{\mathrm{nv}}$) and quasi-PDF with Wilson-Clover action ($\tilde{q}^{\mathrm{WC}}$) and show that $\tilde{q}^{\mathrm{nv}}$ reduces to the continuum quasi-PDF in the continuum limit. We point out, however, that the continuum limit and massless quark limit do not commute. We find that the condition to recover the same collinear divergence that the quasi-PDF has in continuum QCD is $aP_3^2\approx m$ and $m\ll P_3$, while the condition to fully recover the continuum quasi-PDF is $aP_{3}^{2}\ll m\ll P_{3}$, where $P_3$ is the momentum in the direction of the quark's motion (longitudinal direction). These two conditions are based on perturbation calculations and should not be applied to non-perturbative calculations because the non-perturbative effects cure the collinear divergence. The correction to the quasi-PDF using the na\"ive fermion action is due to the Wilson term and can be viewed as an $\mathcal{O}\left(a^{1}\right)$ correction. For nonzero $r$, the $\mathcal{O}\left(a^{1}\right)$ corrections are subsequently mixed with the quasi-PDF using the na\"ive fermion action.