Search

We study the small-$x$ asymptotics of the flavor singlet, leading-twist quark Transverse Momentum Dependent parton distribution functions (TMDs). We apply the recently developed Light-Cone Operator Treatment to rewrite the TMD operator definitions at small $x$ in terms of polarized dipole amplitudes, present the small-$x$ evolution equations and their solutions in the large-$N_c$, linearized, Double-Logarithmic Approximation (DLA), where $N_c$ is the number of quark colors. Previously, this approach was used to determine the small-$x$ asymptotics of the unpolarized and helicity TMDs. In this work we obtain the small-$x$ asymptotics of the Sivers function, helicity worm-gear, transversity, pretzelosity, Boer-Mulders and transversity worm-gear TMDs, thus completing the set of the asymptotic small-$x$ behavior of all the leading-twist flavor-singlet quark TMDs., Comment: 8 pages, 1 table

We phenomenologically constrain the small-$x$ and small-$\xi$ gluon generalized parton distributions (GPDs) with the deeply virtual $J/\psi$ production (DV$J/\psi$P) in the framework of GPDs through universal moment parameterization (GUMP). We use a hybrid cross-section formula combining collinear factorization to the next-to-leading order (NLO) accuracy of the strong coupling $\alpha_s$, with corrections from non-relativistic QCD to account for the power corrections due to the heavy $J/\psi$ mass. We reach reasonable fit to the measured differential cross-sections of DV$J/\psi$P by H1 at Hadron-Electron Ring Accelerator (HERA) as well as forward gluon PDFs from JAM22 global analysis. We find that both NLO and non-relativistic corrections are significant for heavy vector meson productions. Of course, the gluon GPD we obtain still contain considerable freedom in need of inputs from other constraints, particularly in the distribution-amplitude-like region., Comment: 29 pages 7 figures; v2,v3: fix some bugs

We study the small-$x$ asymptotics of the leading-twist quark transverse momentum dependent parton distribution functions (TMDs) which encode couplings between the polarization of the quarks and that of their parent hadron, with at least one of the two polarizations in the transverse direction: the two worm-gear TMDs $g_{1T}$ and $h_{1L}^{\perp}$, and the pretzelosity $h_{1T}^{\perp}$. We apply the recently developed Light Cone Operator Treatment, finding that in the flavor non-singlet sector all three TMDs reduce in the small-$x$ limit to previously known polarized dipole amplitudes, and thus the large-$N_c$, linearized, Double Logarithmic Approximation (DLA) asymptotics have all been solved for previously. For the worm-gear TMD $g_{1T}$ we find \begin{align} g_{1T}^{\textrm{NS}} (x \ll 1, k_T^2) \sim \left( \frac{1}{x} \right)^0 , \end{align} while for the worm-gear TMD $h_{1L}^{\perp}$ we find \begin{align} h_{1L}^{\perp \textrm{NS}} (x \ll 1, k_T^2) \sim \left( \frac{1}{x} \right)^{-1}. \end{align} Finally, for the pretzelosity TMD $h_{1T}^{\perp}$ we find \begin{align} h_{1 T}^{\perp \textrm{NS}} (x \ll 1, k_T^2) \sim \left( \frac{1}{x} \right)^{-1 + 2 \sqrt{\frac{\alpha_s N_c}{2\pi}}}. \end{align} We have compiled these asymptotics together with those of the other five flavor non-singlet, leading-twist quark TMDs into a single unified table., Comment: 14 pages, 2 figures, 1 table

We present the first global analysis of generalized parton distributions (GPDs) combing lattice quantum chromodynamics (QCD) calculations and experiment measurements including global parton distribution functions (PDFs), form factors (FFs) and deeply virtual Compton scattering (DVCS) measurements. Following the previous work where we parameterize GPDs in terms of their moments, we extend the framework to allow for the global analysis at non-zero skewness. Together with the constraints at zero skewness, we fit GPDs to global DVCS measurements from both the recent JLab and the earlier Hadron-Electron Ring Accelerator (HERA) experiments with two active quark flavors and leading order QCD evolution. With certain choices of empirical constraints, both sea and valence quark distributions are extracted with the combined inputs, and we present the quark distributions in the proton correspondingly. We also discuss how to extend the framework to accommodate more off-forward constraints beyond the small $\xi$ expansion, especially the lattice calculated GPDs., Comment: 23 pages, 7 figures and 3 tables

We study the small-$x$ asymptotics of the flavor non-singlet T-odd leading-twist quark transverse momentum dependent parton distributions (TMDs), the Sivers and Boer-Mulders functions. While the leading eikonal small-$x$ asymptotics of the quark Sivers function is given by the spin-dependent odderon, we are interested in revisiting the sub-eikonal correction considered by us earlier. We first simplify the expressions for both TMDs at small Bjorken $x$ and then construct small-$x$ evolution equations for the resulting operators in the large-$N_c$ limit, with $N_c$ the number of quark colors. For both TMDs, the evolution equations resum all powers of the double-logarithmic parameter $\alpha_s \, \ln^2 (1/x)$, where $\alpha_s$ is the strong coupling constant, which is assumed to be small. Solving these evolution equations numerically (for the Sivers function) and analytically (for the Boer-Mulders function) we arrive at the following leading small-$x$ asymptotics of these TMDs at large $N_c$: \begin{align} f_{1 \: T}^{\perp \: NS} (x \ll 1 ,k_T^2) & = C_O (x, k_T^2) \, \frac{1}{x} + C_1 (x, k_T^2) \, \left( \frac{1}{x} \right)^{3.4 \, \sqrt{\frac{\alpha_s \, N_c}{4 \pi}}} , \notag \\ h_1^{\perp \, \textrm{NS}} (x \ll 1, k_T^2) & = C (x, k_T^2) \left( \frac{1}{x} \right)^{-1}. \notag \end{align} The functions $C_O (x, k_T^2)$, $C_1 (x, k_T^2)$, and $C (x, k_T^2)$ can be readily obtained in our formalism: they are mildly $x$-dependent and do not strongly affect the power-of-$x$ asymptotics shown above. The function $C_O$, along with the $1/x$ factor, arises from the odderon exchange. For the sub-eikonal contribution to the quark Sivers function (the term with $C_1$), our result shown above supersedes the one obtained in our previous work due to the new contributions identified recently., Comment: 50 pages, 5 figures

We apply the formalism developed earlier for studying transverse momentum dependent parton distribution functions (TMDs) at small Bjorken $x$ to construct the small-$x$ asymptotics of the quark Sivers function. First, we explicitly construct the complete fundamental "polarized Wilson line" operator to sub-sub-eikonal order: this object can be used to study a variety of quark TMDs at small-$x$. We then express the quark Sivers function in terms of dipole scattering amplitudes containing various components of the "polarized Wilson line" and show that the dominant (eikonal) term which contributes to the quark Sivers function at small $x$ is the spin-dependent odderon, confirming the recent results of Dong, Zheng and Zhou. Our conclusion is also similar to the case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou (see also the work by Szymanowski and Zhou). We also analyze the sub-eikonal corrections to the quark Sivers function using the constructed "polarized Wilson line" operator. We derive new small-$x$ evolution equations re-summing double-logarithmic powers of $\alpha_s \, \ln^2 (1/x)$ with $\alpha_s$ the strong coupling constant. We solve the corresponding novel evolution equations in the large-$N_c$ limit, obtaining a sub-eikonal correction to the spin-dependent odderon contribution. We conclude that the quark Sivers function at small $x$ receives contributions from two terms and is given by \begin{align} f_{1 \: T}^{\perp \: q} (x, k_T^2) = C_O (x, k_T^2) \, \frac{1}{x} + C_1 (k_T^2) \, \left( \frac{1}{x} \right)^0 + \ldots \end{align} with the function $C_O (x, k_T^2)$ varying slowly with $x$ and the ellipsis denoting the sub-asymptotic and sub-sub-eikonal (order-$x$) corrections., Comment: 41 pages, 11 figures, 1 table. Added new discussion paragraphs in section III.A and section IV, added a footnote in section IV, corrected typos, updated references. v3: Minor typos corrected, the sub-eikonal term in the asymptotics of the Sivers function is revised in the follow up paper, see footnote on page 23

We present the calculation of the leading contribution to the quark Sivers function at small-Bjorken $x$. This calculation uses the high energy scattering approximation and operator formalism developed by Kovchegov and Sievert to obtain a dominant contribution to the quark Sivers function coming from the spin-dependent odderon, in agreement with the results of Dong, Zhen, and Zhou. We then calculate this dominant contribution in the diquark model of the proton to obtain a small-$x$ estimate for the Sivers function., Comment: 6 pages, 2 figures, contribution to the proceedings of the XXVIII International Workshop on Deep Inelastic Scattering and Related Subjects (DIS2021), Stony Brook, New York, USA, 12-16 April, 2021, for submission to SciPost Physics Proceedings

We calculate the single transverse spin asymmetry (STSA) in polarized proton-proton ($p^{\uparrow}+p$) and polarized proton-nucleus ($p^{\uparrow}+A$) collisions ($A_N$) generated by a partonic lensing mechanism. The polarized proton is considered in the quark-diquark model while its interaction with the unpolarized target is calculated using the small-$x$/saturation approach, which includes multiple rescatterings and small-$x$ evolution. The phase required for the asymmetry is caused by a final-state gluon exchange between the quark and diquark, as is standard in the lensing mechanism of Brodsky, Hwang and Schmidt. Our calculation combines the lensing mechanism with small-$x$ physics in the saturation framework. The expression we obtain for the asymmetry $A_N$ of the produced quarks has the following properties: (i) The asymmetry is generated by the dominant elastic scattering contribution and $1/N_c^2$ suppressed inelastic contribution (with $N_c$ the number of quark colors); (ii) The asymmetry grows or oscillates with the produced quark's transverse momentum $p_T$ until the momentum reaches the saturation scale $Q_s$, and then only falls off as $1/p_T$ for larger momenta; (iii) The asymmetry decreases with increasing atomic number $A$ of the target for $p_T$ below or near $Q_s$, but is independent of $A$ for $p_T$ significantly above $Q_s$. We discuss how these properties may be qualitatively consistent with the data on $A_N$ published by the PHENIX collaboration and with the preliminary data on $A_N$ reported by the STAR collaboration., Comment: 23 pages, 10 figures