Abstract
We perform a one-loop study of the small-z32 behavior of the Ioffe-time distribution (ITD) M(ν,z32), the basic function that may be converted into parton pseudo- and quasi-distributions. We calculate the corrections on the operator level, so that our results may be also used for pseudo-distribution amplitudes and generalized parton pseudodistributions. We separate two sources of the z32-dependence at small z32. One is related to the ultraviolet (UV) singularities generated by the gauge link. Our calculation shows that, for a finite UV cut-off, the UV-singular terms vanish when z32=0. The UV divergences are absent in the reduced ITD given by the ratio M(ν,z32)/M(0,z32). Still, it has a non-trivial short-distance behavior due to lnz32Λ2 terms generating perturbative evolution of parton densities. We give an explicit expression, up to constant terms, for the reduced ITD at one loop. It may be used in extraction of PDFs from lattice QCD simulations. We also use our results to get new insights concerning the structure of parton quasi-distributions at one-loop level.
Highlights
The usual parton distribution functions (PDFs) f (x) [1] are often mentioned as “light-cone PDFs”, since they are related to matrix elements M(z, p) of the p|φ(0)φ(z)|p type taken on the light cone z2 = 0
We present an expression for the reduced Ioffe-time distribution (ITD), and we show how a simple ln(z32m2) evolution logarithm proliferates into a rather complicated structure for quasi-PDF
Our goal is to describe this on the operator level, as a modification of the original soft bilocal operator by gluon corrections
Summary
The usual parton distribution functions (PDFs) f (x) [1] are often mentioned as “light-cone PDFs”, since they are related to matrix elements M(z, p) of the p|φ(0)φ(z)|p type taken on the light cone z2 = 0. The parton pseudo-distributions P(x, −z2) [2,3] generalize PDFs for a situation when z is off the light cone, in particular, when z is spacelike z2 < 0. One should treat M(z, p) as a function M(ν, −z2) of the Ioffe time (pz) ≡ −ν [4] and z2. Ji [5], using purely spacelike separations z = (0, 0, 0, z3) (or, for brevity, z = z3) allows one to study matrix elements M(z3, p)
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