Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuasi-PDFs, momentum distributions and pseudo-PDFs
145
0
0.0
(
0
)
Authors
A. V. Radyushkin
Ask ChatGPT about the research
No Arabic abstract
We show that quasi-PDFs may be treated as hybrids of PDFs and primordial rest-frame momentum distributions of partons. This results in a complicated convolution nature of quasi-PDFs that necessitates using large $p_3 sim 3$ GeV momenta to get reasonably close to the PDF limit. As an alternative approach, we propose to use pseudo-PDFs $P(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals and are related to Ioffe-time distributions $M ( u, z_3^2)$, the functions of the Ioffe time $ u = p_3 z_3$ and the distance parameter $z_3^2$ with respect to which it displays perturbative evolution for small $z_3$. In this form, one may divide out the $z_3^2$ dependence coming from the primordial rest-frame distribution and from the problematic factor due to lattice renormalization of the gauge link. The $ u$-dependence remains intact and determines the shape of PDFs.
rate research
Read More
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 ${cal M} ( u, z_3^2)$ depending on the Ioffe-time $ u = 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 ${mathfrak M} ( u, z_3^2) equiv {cal M} ( u, z_3^2)/ {cal M} (0, z_3^2)$ we divide out the rest frame effects,including the notorious link renormalization factors. The $ u$-dependence remains intact and determines the shape of PDFs in the small $z_3$ region. The residual $z_3^2$ dependence of the ${mathfrak M} ( u, z_3^2)$ is governed by perturbative evolution. The Fourier transform of ${cal M} ( u, z_3^2)$ 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.
The recently proposed large momentum effective theory (LaMET) of Ji has led to a burst of activity among lattice practitioners to perform and control the first pioneering calculations of the quasi-PDFs of the nucleon. These calculations represent approximations to the standard PDFs defined as correlation functions of fields with lightlike separation, being instead correlations along a longitudinal direction of the operator $gamma^z$; as such, they differ from standard PDFs by power-suppressed $1 big/ p^2_z$ corrections, becoming exact in the limit $p_z to infty$. Investigating the systematics of this behavior thus becomes crucial to understanding the validity of LaMET calculations. While this has been done using models for the nucleon, an analogous dedicated study has not been carried out for the $pi$ and $rho$ quark distribution functions. Using a constituent quark model, a systematic calculation is performed to estimate the size and $x$ dependence of the finite-$p_z$ effects in these quasi-PDFs, finding them to be potentially tamer for lighter mesons than for the collinear quasi-PDFs of the nucleon.
We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element ${cal M} (Pz_3, z_3^2)$ as a function of the Ioffe time $ u = Pz_3$ and the distance $z_3$. The next step is to divide ${cal M} (Pz_3, z_3^2)$ by the rest-frame density ${cal M} (0, z_3^2)$. Our lattice calculation shows a linear exponential $z_3$-dependence in the rest-frame function, expected from the $Z(z_3^2)$ factor generated by the gauge link. Still, we observe that the ratio ${cal M} (Pz_3 , z_3^2)/{cal M} (0, z_3^2)$ has a Gaussian-type behavior with respect to $z_3$ for 6 values of $P$ used in the calculation. This means that $Z(z_3^2)$ factor was canceled in the ratio. When plotted as a function of $ u$ and $z_3$, the data are very close to $z_3$-independent functions. This phenomenon corresponds to factorization of the $x$- and $k_perp$-dependence for the TMD ${cal F} (x, k_perp^2)$. For small $z_3 leq 4a$, the residual $z_3$-dependence is explained by perturbative evolution, with $alpha_s/pi =0.1$.
We point out a problem of the phenomenological definition of the valence partons as the difference between the partons and antipartons in the context of the NNLO evolution equations. After demonstrating that the classification of the parton degrees of freedom (PDF) of the parton distribution functions (PDFs) are the same in the QCD path-intergral formulations of the hadronic tensor and the quasi-PDF with LaMET, we resolve the problem by showing that the proper definition of the valence should be in terms of those in the connected insertions only. We also prove that the strange partons appear as the disconnected sea in the nucleon.
The Burkhardt-Cottingham (BC) sum rule connects the twist-3 light-cone parton distribution function (PDF) $g_{T}(x)$ to the twist-2 helicity PDF $g_{1}(x)$. The chiral-odd counterpart of the BC sum rule relates the twist-3 light-cone PDF $h_{L}(x)$ to the twist-2 transversity PDF $h_{1}(x)$. These BC-type sum rules can also be derived for the corresponding quasi-PDFs. We perform a perturbative check of the BC-type sum rules in the quark target model and the Yukawa model, by going beyond the ultra-violet (UV) divergent terms. We employ dimensional regularization (DR) and cut-off schemes to regulate UV divergences, and show that the BC-type sum rules hold for DR, while they are generally violated when using a cut-off. This violation can be traced back to the breaking of rotational invariance. We find corresponding results for the sum rule relating the mass of the target to the twist-3 PDF $e(x)$. Moreover, we supplement our analytical results with numerical calculations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
