Structure of parton quasi-distributions and their moments

We discuss the structure of the parton quasi-distributions (quasi-PDFs) Q(y,P3) outside the “canonical” −1≤y≤1 support region of the usual parton distribution functions (PDFs). Writing the yn moments of Q(y,P3) in terms of the combined xn−2lk⊥2l -moments of the transverse momentum distribution (TMD) F(x,k⊥2), we establish a connection between the large- |y| behavior of Q(y,P3) and large- k⊥2 behavior of F(x,k⊥2). In particular, we show that the 1/k⊥2 hard tail of TMDs in QCD results in a slowly decreasing ∼1/|y| behavior of quasi-PDFs for large |y| that produces infinite yn moments of Q(y,P3). We also relate the ∼1/|y| terms with the ln⁡z32 -singularities of the Ioffe-time pseudo-distributions M(ν,z32). Converting the operator product expansion for M(ν,z32) into a matching relation between the quasi-PDF Q(y,P3) and the light-cone PDF f(x,μ2), we demonstrate that there is no contradiction between the infinite values of the yn moments of Q(y,P3) and finite values of the xn moments of f(x,μ2).