Structure of Parton Quasi-Distributions and Their Moments

We discuss the structure of the parton quasi-distributions (quasi-PDFs) Q (y, P-3) outside the canonical -1 \u3c= y \u3c= 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(perpendicular to)(2l)-moments of the transverse momentum distribution (TMD). F(x,k(perpendicular to)(2)), we establish a connection between the large-vertical bar y vertical bar behavior of Q (y, P-3) and large-k(perpendicular to)(2) behavior of F(x,k(perpendicular to)(2)). In particular, we show that the 1/k(perpendicular to)(2) hard tail of TMDs in QCD results in a slowly decreasing similar to 1/vertical bar y vertical bar behavior of quasi-PDFs for large vertical bar y vertical bar that produces infinite y(n) moments of Q(y, P-3). We also relate the - 1/vertical bar y vertical bar terms with the lnz(3)(2)-singularities of the Ioffe-time pseudo-distributions m(v, z(3)(2)). Converting the operator product expansion for m(v,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)