In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs). These functions, referred to as TMDs, are defined as the Fourier transforms of hadronic matrix elements of nonlocal combinations of gluon fields. The nonlocality is bridged by gauge links, which have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. For gluons, the specific correlator with one future and one past pointing gauge link is, in the limit of small x, related to a correlator of a single Wilson loop. We present the parametrization of Wilson loop correlators in terms of Wilson loop TMDs and discuss the relation between these functions and the small-x 'dipole' gluon TMDs. This analysis shows which gluon TMDs are leading or suppressed in the smallx limit. We discuss hadronic targets that are unpolarized, vector polarized (relevant for spin-1/2 and spin-1 hadrons), and tensor polarized (relevant for spin-1 hadrons). The latter are of interest for studies with a future Electron-Ion Collider with polarized deuterons.

We propose to measure the linear polarization of the external electromagnetic fields of a relativistic heavy ion through azimuthal asymmetries in dilepton production in ultraperipheral collisions. The asymmetries estimated with the equivalent photon approximation are shown to be sizable.PACS numbers: I. INTRODUCTIONTransverse momentum dependent(TMD) parton distribution function [1] is one of the most powerful theoretical tools that are utilized to explore the three-dimensional imaging of nuleon/nuclei. Among many TMD parton distributions, the linearly polarized gluon distribution [2] has received growing attentions in recent years. It describes the correlation between gluon transverse momentum and its polarization vector inside an unpolarized nucleon or nucleus. It is of particular interest to study linearly polarized gluon distribution at small x [3, 4], as it is predicted to grow equally rapidly towards small x as compared to the unpolarized gluon distribution in the dilute limit. In the saturation limit, the dipole type linearly polarized gluon distribution and the dipole type unpolarized gluon distribution remain identical, whereas the linearly polarization of Weizsäcker-Williams gluons is suppressed. Though it has been found promising to probe the linearly polarized gluon distribution by measuring cos 2φ azimuthal asymmetry for two particle production in various high energy scattering processes at RHIC, LHC, or a future Electron-Ion Collider(EIC) [3][4][5][6][7][8][9][10][11][12][13], this gluon distribution so far has not yet been studied experimentally.In analogy to the QCD case, one also can define a linearly polarized photon distribution for an unpolarized nucleon or nuclei target, which can be accessed by measuring the azimuthal asymmetries in di-lepton production in hadronhadron collisions [8]. However, it is not very practical to extract the polarized photon distribution in hadronic reactions due to the di-lepton Drell-Yan production background. Instead, the cleaner and more promising way to probe the linearly polarization of photons would be the purely electromagnetic two photon reaction γγ → l + l − in heavy-ion ultra-peripheral collisions(UPCs) where the hadronic background is absent. Though photon-photon collisions in the UPC case has been extensively studied [14][15][16][17][18][19][20][21][22][23][24][25][26][27], to the best of our knowledge, the polarization dependent effects have not yet been addressed so far. Both the unpolarized photon distribution and the polarized one in the UPC case can be determined using the external classical field approximation [14,15]. It is not surprising to find that they are identical to each other in this approximation, just like the relation established between the dipole amplitude and the polarized gluon distributions [3,[28][29][30]. In the present paper, we propose to test this theoretical predication by measuring cos 2φ and cos 4φ asymmetries in di-lepton production induced by the linearly polarized photon distribution.Recently, the STAR collaboration at RH...

We investigate the impact parameter dependence of the cos 4φ azimuthal asymmetry for electromagnetic lepton pair production in heavy ion collisions. The asymmetry induced by linearly polarized coherent photons exhibits strong impact parameter dependence.

The first evidence of spin alignment of vector mesons (K Ã0 and ϕ) in heavy-ion collisions at the Large Hadron Collider (LHC) is reported. The spin density matrix element ρ 00 is measured at midrapidity (jyj < 0.5) in Pb-Pb collisions at a center-of-mass energy (ffiffiffiffiffiffiffi ffi s NN p) of 2.76 TeV with the ALICE detector. ρ 00 values are found to be less than 1=3 (1=3 implies no spin alignment) at low transverse momentum (p T < 2 GeV=c) for K Ã0 and ϕ at a level of 3σ and 2σ, respectively. No significant spin alignment is observed for the K 0 S meson (spin ¼ 0) in Pb-Pb collisions and for the vector mesons in pp collisions. The measured spin alignment is unexpectedly large but qualitatively consistent with the expectation from models which attribute it to a polarization of quarks in the presence of angular momentum in heavy-ion collisions and a subsequent hadronization by the process of recombination.

Abstract:We perform a phenomenological analysis of the cos 2φ azimuthal asymmetry in virtual photon plus jet production induced by the linear polarization of gluons in unpolarized pA collisions. Although the linearly polarized gluon distribution becomes maximal at small x, TMD evolution leads to a Sudakov suppression of the asymmetry with increasing invariant mass of the γ * -jet pair. Employing a small-x model input distribution, the asymmetry is found to be strongly suppressed under TMD evolution, but still remains sufficiently large to be measurable in the typical kinematical region accessible at RHIC or LHC at moderate photon virtuality, whereas it is expected to be negligible in Z/W -jet pair production at LHC. We point out the optimal kinematics for RHIC and LHC studies, in order to expedite the first experimental studies of the linearly polarized gluon distribution through this process. We further argue that this is a particularly clean process to test the k t -resummation formalism in the small-x regime.

We present a detailed study of vector meson photoproduction in ultraperipheral heavy ion collisions (UPCs). Using the dipole model, we develop a framework for the joint impact parameter and transverse momentum dependent cross sections. We compute the unpolarized cross section and cos 2ϕ azimuthal angular correlation for ρ0 photoproduction with ϕ defined as the angle between the ρ0’s transverse momentum and its decay product pion meson’s transverse momentum. Our result on unpolarized coherent differential cross section gives excellent description to the STAR experimental data. A first compari- son between theoretical calculation and experimental measurement on the cos 2ϕ azimuthal asymmetry, which results from the linearly polarized photons, is performed and reasonable agreement is reached. We find out the characteristic diffractive patterns at both RHIC and LHC energies and predict the impact parameter dependent cos 2ϕ azimuthal asymmetries for ρ0 photoproduction by considering UPCs and peripheral collisions. The future experimental measurements at RHIC and LHC relevant to our calculations will provide a tool to rigorously investigate the coherent and incoherent production of vector meson in UPCs, as well as to probe the nuclear structure in heavy ion collisions.

The longitudinal polarization of hyperon in e + e − annihilation at high energies depends on the longitudinal polarization of the quark produced at the e + e − annihilation vertex whereas the spin alignment of vector mesons is independent of it. They exhibit very different energy dependences. We use the longitudinal polarization of Lambda hyperon and the spin alignment of K * as representative examples to present numerical results of energy dependences and demonstrate such distinct differences. We present the results at the leading twist with perturbative QCD evolutions of fragmentation functions at the leading order.

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