Structure of parton quasi-distributions and their moments

We discuss the structure of the parton quasi-distributions (quasi-PDFs) $Q(y, P_3)$outsidethe\mathrm{"}canonical\mathrm{"}$-1 \leq y \leq 1$ support region of the usual parton distribution functions (PDFs). Writing the $y^n$ moments of $Q(y,P_3)$ in terms of the combined $x^{n-2l}{k}_{\perp }^{2l}$-moments of the transverse momentum distribution (TMD) $\mathcal{F}(x,k_{\perp }^2)$, we establish a connection between the large-$\mathrm{\mid }y\mathrm{\mid }$ behavior of $Q(y,P_3)$ and large-$k_{\perp }^2$ behavior of $\mathcal{F}(x,k_{\perp }^2)$. In particular, we show that the $1\mathrm{/}{k}_{\perp }^2$ hard tail of TMDs in QCD results in a slowly decreasing $\sim 1\mathrm{/}\mathrm{\mid }y\mathrm{\mid }$ behavior of quasi-PDFs for large $\mathrm{\mid }y\mathrm{\mid }$ that produces infinite $y^n$ moments of $Q(y,P_3)$. We also relate the $\sim 1\mathrm{/}\mathrm{\mid }y\mathrm{\mid }$ terms with the $\ln z_3^2$-singulariies of the Ioffe-time pseudo-distributions $\mathfrak{M}(\nu ,z_3^2)$. Converting the operator product expansion for $\mathfrak{M}(\nu ,z_3^2)$ into a matching relation between the quasi-PDF $Q(y,P_3)$ and the light-cone PDF $f(x,{\mu }^2)$, we demonstrate that there is no contradiction between the infinite values of the $y^n$ moments of $Q(y,P_3)$ and finite values of the $x^n$ moments of $f(x, \mu^2)$.