Quasi-PDFs and pseudo-PDFs

We discuss the physical nature of quasi-PDFs, especially the reasons for the strong nonperturbative evolution pattern which they reveal in actual lattice gauge calculations. We argue that quasi-PDFs may be treated as hybrids of PDFs and the rest-frame momentum distributions of partons. The latter is also responsible for the transverse momentum dependence of TMDs. The resulting convolution structure of quasi-PDFs necessitates using large probing momenta $p_3 \gtrsim 3$ GeV to get reasonably close to the PDF limit. To deconvolute the rest-frame distribution effects, we propose to use a method based directly on the coordinate representation. We treat matrix elements $M(z_3,p_3)$ as distributions ${\cal M} (\nu, z_3^2)$ depending on the Ioffe-time $\nu = p_3 z_3$ and the distance parameter $z_3^2$. The rest-frame spatial distribution is given by ${\cal M} (0, z_3^2)$. Using the reduced Ioffe function ${\mathfrak M} (\nu, z_3^2) \equiv {\cal M} (\nu, z_3^2)/ {\cal M} (0, z_3^2)$ we divide out the rest frame effects,including the notorious link renormalization factors. The $\nu$-dependence remains intact and determines the shape of PDFs in the small $z_3$ region. The residual $z_3^2$ dependence of the ${\mathfrak M} (\nu, z_3^2)$ is governed by perturbative evolution. The Fourier transform of ${\cal M} (\nu, z_3^2)$ produces pseudo-PDFs ${\cal P}(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals. On the basis of these findings we propose a new method for extraction of PDFs from lattice calculations.Comment: 11 pages, Talk at QCD Evolution 2017 Worksho

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" $-1 \leq y \leq 1$ 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) ${\cal F} (x,k_\perp^2)$, we establish a connection between the large-$|y|$ behavior of $Q(y,P_3)$ and large-$k_\perp^2$ behavior of ${\cal F} (x,k_\perp^2)$. In particular, we show that the $1/k_\perp^2$ hard tail of TMDs in QCD results in a slowly decreasing $\sim 1/|y|$ behavior of quasi-PDFs for large $|y|$ that produces infinite $y^n$ moments of $Q(y,P_3)$. We also relate the $\sim 1/|y|$ 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 $f(x, \mu^2)$.Comment: 9 pages, version to appear in Physics Letters

Deeply Virtual Compton Scattering: Facing Nonforward Distributions

Applications of perturbative QCD to deeply virtual Compton scattering process require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements of quark and gluon operators. We discuss two types of functions parametrizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions \cal F_\zeta (X;t) and also their relation to usual parton densities f(x).Comment: 4 pages, to be published in the Proceedings of the International Conference on Deep Inelastic Scattering, Chicago, April 1997, AI

Asymmetric Parton Distributions

Applications of perturbative QCD to hard exclusive electroproduction processes in the Bjorken limit at small invariant momentum transfer t bring in a new type of parton distributions which have hybrid properties, resembling both the parton distribution functions and the distribution amplitudes. Their t-dependence is analogous to that of hadronic form factors. We discuss general properties of these new parton distributions, their relation to usual parton densities and the evolution equations which they satisfy.Comment: 4 pages, to be published in the Proceedings of the International Conference on Deep Inelastic Scattering, Chicago, April 1997, AI

Pion Wave Function from QCD Sum Rules with Nonlocal Condensates

We investigate a model QCD sum rule for the pion wave function $\varphi_{\pi}(x)$ based on the non-diagonal correlator whose perturbative spectral density vanishes and $\Phi(x,M^2)$, the theoretical side of the sum rule, consists of condensate contributions only. We study the dependence of $\Phi(x,M^2)$ on the Borel parameter $M^2$ and observe that $\Phi(x,M^2)$ has a humpy form, with the humps becoming more and more pronounced when $M^2$ increases. We demonstrate that this phenomenon reflects just the oscillatory nature of the higher states wave functions, while the lowest state wave function $\varphi_{\pi}(x)$ extracted from our QCD sum rule analysis,has no humps, is rather narrow and its shape is close to the asymptotic form $\varphi_{\pi}^{as}(x) = 6x(1-x)$.Comment: Talk at the Workshop "Continuous Advances in QCD", 11 pages + appended uuencoded ps file with 8 figures, Latex, CEBAF-TH-94-1

One-loop evolution of parton pseudo-distribution functions on the lattice

We incorporate recent calculations of one-loop corrections for the reduced Ioffe-time pseudo-distribution ${\mathfrak M} (\nu,z_3^2)$ to extend the leading-logarithm analysis of lattice data obtained by Orginos et al. We observe that the one-loop corrections contain a large term reflecting the fact that effective distances involved in the most important diagrams are much smaller than the nominal distance $z_3$. The large correction in this case may be absorbed into the evolution term, and the perturbative expansion used for extraction of parton densities at the $\mu \approx 2$ GeV scale is under control. The extracted parton distribution is rather close to global fits in the $x>0.1$ region, but deviates from them for $x<0.1$.Comment: 10 pages, 11 figures, version to be published in Phys. Rev.

QCD Sum Rules and Compton Scattering

We extend QCD sum rule analysis to moderate energy fixed angle Compton scattering. In this kinematic region there is a strong similarity to the sum rule treatment of electromagnetic form factors, although the four-point amplitude requires a modification of the Borel transform. To illustrate our method, we derive the sum rules for helicity amplitudes in pion Compton scattering and estimate their large-$t$ behavior in the local duality approximation.Comment: 30 pages in Latex, 6 figures not included, available upon request (send email to: [email protected]), ITP-SB-92-70, CEBAF-TH-92-3

Generalized Parton Distributions and Pseudodistributions

We derive one-loop matching relations for the Ioffe-time distributions (ITDs) related to the pion distribution amplitude (DA) and generalized parton distributions (GPDs). They are obtained from a universal expression for the one-loop correction in an operator form, and will be used in the ongoing lattice calculations of the pion DA and GPDs within the parton pseudodistributions approach

Virtuality Distributions in Application to ɣɣ* → π\u3csup\u3e​0\u3c/sup\u3e Transition Form Factor at Handbag Level

We outline basics of a new approach to transverse momentum dependence in hard processes. As an illustration, we consider hard exclusive transition process ɣ*ɣ -\u3e π0 at the handbag level. Our starting point is coordinate representation for matrix elements of operators (in the simplest case, bilocal O(0, z)) describing a hadron with momentum p. Treated as functions of (pz) and z2, they are parametrized through virtuality distribution amplitudes (VDA) Φ(x, σ), with x being Fourier-conjugate to (pz) and σ Laplace-conjugate to z2. For intervals with z+ = 0, we introduce the transverse momentum distribution amplitude (TMDA) Ѱ(x, k┴), and write it in terms of (VDA) Φ(x, σ). The results of covariant calculations, written in terms of Φ(x, σ) are converted into expressions involving Ѱ(x, k┴) Starting with scalar toy models, we extend the analysis onto the case of spin-1/2 quarks and QCD. We propose simple models for soft VDAs/TMDAs, and use them for comparison of handbag results with experimental (BaBar and BELLE) data on the pion transition form factor. We also discuss how one can generate high-k┴ tails from primordial soft distributions