Lattice corrections to the quark quasidistribution at one loop

We calculate radiative corrections to the quark quasidistribution in lattice perturbation theory at one loop to leading orders in the lattice spacing. We also consider one-loop corrections in continuum Euclidean space. We find that the infrared behavior of the corrections in Euclidean and Minkowski space are different. We explore features of momentum loop integrals and demonstrate why loop corrections from the lattice perturbation theory and Euclidean continuum do not correspond with their Minkowski brethren, and comment on a recent suggestion for transcending the differences in the results. Further, we examine the role of the lattice spacing $a$ and of the $r$ parameter in the Wilson action in these radiative corrections.

Figure 1

The one-loop perturbative corrections to the quark distributions or quasidistributions. Working in light-front (for the distributions) or axial gauge (for the quasidistributions), diagrams (c) and (d) do not contribute. $p=(p_0,0,0,p_z)$ is the quark momentum.

Figure 2

Poles in the complex $k^0$ plane. Crosses represent poles from the quark propagator, and open circles represent poles from the gluon propagator.

Figure 3

Integration lines and poles of the integrand shown for the complex $k_4$ plane. A gluon propagator pole lies between the two integration paths unless $k_{\perp }$ is large enough.

Figure 4

The vertex correction to the quark quasidistribution, normalized by ${\alpha }_s{C}_F/(2\pi )$, calculated in Minkowski space (blue dashed line) versus the same calculated via LPT in Euclidean space (orange solid line) to leading order in $m^2/p_z^2$ for the momentum fraction $x$ between 0 and 1. Here $p_z$ is 2 GeV, and the quark mass $m$ is 0.02 GeV. $\mathrm{\Lambda }\leftrightarrow n/a$ is 2 GeV. When $\mathrm{\Lambda }<p_z$, the sign of the $x\to 1$ pole flips for the LPT (orange solid line) result, while the Minkowski result remains qualitatively unaffected. The red bar denotes the region where $x=m/p_z$. To the left of this line, the expansion in $m^2/p_z^2$ is not valid.