Small Bjorken-x Research Articles

Helicity evolution at small x: quark to gluon and gluon to quark transition operators

We include the quark to gluon and gluon to quark shock-wave transition operators into the small Bjorken-x evolution equations for helicity in the flavor-singlet channel derived earlier in [1–3]. While such transitions do not affect the large-Nc version of the evolution equations for helicity, the large-Nc & Nf equations are affected. (Nc and Nf are the numbers of quark colors and flavors, respectively.) We derive the corresponding corrected large-Nc & Nf equations for the polarized dipole amplitudes contributing to the flavor-singlet quark and gluon helicity distributions in the double-logarithmic approximation (DLA), resumming powers of αs ln2(1/x) with αs the strong coupling constant. We solve these equations iteratively and extract the polarized splitting functions up to four loops. We show that our splitting functions agree with the fixed-order perturbative calculations up to and including the existing three-loops results [4–7]. Similar to the large-Nc helicity evolution in the shock-wave approach [8], our large-Nc & Nf small-x splitting functions agree with those obtained in the infrared evolution equations framework from [9, 10] up to three loops, but appear to slightly disagree at four loops.

Unified description of DGLAP, CSS, and BFKL evolution: TMD factorization bridging large and small x

This paper introduces a transverse-momentum dependent (TMD) factorization scheme designed to unify both large and small Bjorken-x regimes. We compute the next-to-leading order quantum chromodynamics (QCD) corrections to the gluon TMD operator for an unpolarized hadron within this proposed scheme. This leads to the emergence of a new TMD evolution, incorporating those in transverse momentum, rapidity, and Bjorken-x. When matched to the collinear factorization scheme, our factorization scheme faithfully reproduces the well-established Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and Collins-Soper-Sterman (CSS) evolutions. Conversely, matching with high-energy factorization not only yields the Balitsky-Fadin-Kuraev-Lipatov evolution but also reveals distinctive signatures of CSS logarithms. The development of this novel TMD factorization scheme, capable of seamlessly reconciling disparate Bjorken-x regimes and faithfully reproducing established QCD evolution equations, has the potential to significantly advance our comprehension of high-energy processes and three-dimensional parton structures of hadrons. Published by the American Physical Society 2024

Orbital angular momentum at small x revisited

We revisit the problem of the small Bjorken-x asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised small-x helicity evolution derived recently in [1]. We relate the quark and gluon OAM distributions at small x to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the OAM distributions, we derive novel small-x evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of αs ln2(1/x) with αs the strong coupling constant). We solve these evolution equations numerically and extract the leading large-Nc, small-x asymptotics of the quark and gluon OAM distributions, which we determine to beLq+q¯xQ2~LGxQ2~∆ΣxQ2~∆GxQ2~123.66αsNc2π,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {L}_{q+\\overline{q}}\\left(x,{Q}^2\\right)\\sim {L}_G\\left(x,{Q}^2\\right)\\sim \\Delta \\Sigma \\left(x,{Q}^2\\right)\\sim \\Delta G\\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{2}\\right)}^{3.66\\sqrt{\\frac{\\alpha_s{N}_c}{2\\uppi}}}, $$\\end{document}in agreement with [2] within the precision of our numerical evaluation. (Here Nc is the number of quark colors.) We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small-x region.

Probing Small Bjorken-x Nuclear Gluonic Structure via Coherent J/ψ Photoproduction in Ultraperipheral Pb-Pb Collisions at sqrt[s_{NN}]=5.02 TeV.

Quasireal photons exchanged in relativistic heavy ion interactions are powerful probes of the gluonic structure of nuclei. The coherent J/ψ photoproduction cross section in ultraperipheral lead-lead collisions is measured as a function of photon-nucleus center-of-mass energies per nucleon (W_{γN}^{Pb}) over a wide range of 40<W_{γN}^{Pb}<400 GeV. Results are obtained using data at the nucleon-nucleon center-of-mass energy of 5.02TeV collected by the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of 1.52 nb^{-1}. The cross section is observed to rise rapidly at low W_{γN}^{Pb}, and plateau above W_{γN}^{Pb}≈40 GeV, up to 400 GeV, entering a new regime of small Bjorken-x (≈6×10^{-5}) gluons being probed in a heavy nucleus. The observed energy dependence is not predicted by current quantum chromodynamic models.

Solutions to the Balitsky-Kovchegov equation including the dipole orientation

Solutions of the target-rapidity Balitsky-Kovchegov (BK) equation are studied considering, for the first time, the complete impact-parameter dependence, including the orientation of the dipole with respect to the impact-parameter vector. In our previous work [1] it has been demonstrated that the spurious Coulomb tails could be tamed using the collinearly-improved kernel and an appropriate initial condition in the projectile-rapidity BK equation. Introducing a different interpretation of the evolution variable, the target-rapidity formulation of the BK equation brings non-locality in rapidity and a kernel modification, removing the term that previously helped to suppress the Coulomb tails. To address this newly emerged non-locality, three different prescriptions are explored here to take into account the rapidities preceding the initial condition. Two of these approaches induce mild Coulomb tails, while the other is free from this effect within the studied rapidity range. The range is chosen to correspond to that of interest for existing and future experiments. To demonstrate that this set up can be used for phenomenological studies, the obtained solutions are used to compute the F2 structure function of the proton and the diffractive photo- and electro-production of J/ψ off protons. The predictions agree well with HERA data, confirming that the target-rapidity Balitsky-Kovchegov equation with the full impact-parameter dependence is a viable tool to study the small Bjorken-x limit of perturbative QCD at current facilities like RHIC and LHC as well as in future colliders like the EIC.

Measurement of the Z boson production cross-section in proton-lead collisions at sqrt{s_{textrm{NN}}} = 8.16 TeV

This article presents the first measurement of the differential Z-boson production cross-section in the forward region using proton-lead collisions with the LHCb detector. The dataset was collected at a nucleon-nucleon centre-of-mass energy of sqrt{s_{textrm{NN}}} = 8.16 TeV in 2016, corresponding to an integrated luminosity of 30.8 nb−1. The forward-backward ratio and the nuclear modification factors are measured together with the differential cross-section as functions of the Z boson rapidity in the centre-of-mass frame, the transverse momentum of the Z boson and a geometric variable ϕ*. The results are in good agreement with the predictions from nuclear parton distribution functions, providing strong constraining power at small Bjorken-x.

Vector meson photoproduction in UPCs with FoCal

We discuss the physics prospects of photon-induced measurements using the high-granularity FoCal detector to be installed at the ALICE experiment, covering the pseudorapidity interval 3.4 ≤ η ≤ 5.8. This new detector, scheduled to be in operation from Run 4, will explore the small Bjorken-x physics region in an unprecedented way. In this region the gluon ,saturation phenomenon is expected to be dominant. Combined with the rest of the ALICE subdetectors, including the zero degree calorimenters, FoCal will serve to reconstruct in a model-independent way the measured photoproduction cross sections for vectors mesons in a wide range of photon-target energies, down to x values of about 7 × 10−6 and 2 × 10−6 in ultra-peripheral photon–proton and photon–lead collisions, respectively.

Quark and gluon helicity evolution at small x: revised and updated

We revisit the problem of small Bjorken-x evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave (s-channel) formalism. Earlier works on the subject in the same framework [1–3] resulted in an evolution equation for the gluon field-strength F12 and quark “axial current” ψ¯γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\psi}\\gamma $$\\end{document}+γ5ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of αs ln2(1/x) with αs the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, D←i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {}_D{}^{\\leftarrow i} $$\\end{document}Di, also sandwiched between the light-cone Wilson lines (with the repeated transverse index i = 1, 2 summed over), was missing in the previous works [1–3]. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, Di − D←i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {}_D{}^{\\leftarrow i} $$\\end{document} , which is related to the Jaffe-Manohar polarized gluon distribution [4]. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (Di − D←i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {}_D{}^{\\leftarrow i} $$\\end{document}, F12, and ψ¯γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\psi}\\gamma $$\\end{document}+γ5ψ), generalizing the results of [1–3]. We also construct closed double-logarithmic evolution equations in the large-Nc and large-Nc&Nf limits, with Nc and Nf the numbers of quark colors and flavors, respectively. Solving the large-Nc equations numerically we obtain the following small-x asymptotics of the quark and gluon helicity distributions ∆Σ and ∆G, along with the g1 structure function,∆ΣxQ2∼∆GxQ2∼g1xQ2∼1x3.66αsNc2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta \\Sigma \\left(x,{Q}^2\\right)\\sim \\Delta G\\left(x,{Q}^2\\right)\\sim {g}_1\\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{x}\\right)}^{3.66\\sqrt{\\frac{\\alpha_s{N}_c}{2\\pi }}} $$\\end{document}in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].

First measurement of the |t|-dependence of coherent J/ψ photonuclear production

The first measurement of the cross section for coherent J/ψ photoproduction as a function of |t|, the square of the momentum transferred between the incoming and outgoing target nucleus, is presented. The data were measured with the ALICE detector in ultra-peripheral Pb–Pb collisions at a centre-of-mass energy per nucleon pair sNN=5.02TeV with the J/ψ produced in the central rapidity region |y|<0.8, which corresponds to the small Bjorken-x range (0.3−1.4)×10−3.The measured |t|-dependence is not described by computations based only on the Pb nuclear form factor, while the photonuclear cross section is better reproduced by models including shadowing according to the leading-twist approximation, or gluon-saturation effects from the impact-parameter dependent Balitsky–Kovchegov equation. These new results are therefore a valid tool to constrain the relevant model parameters and to investigate the transverse gluonic structure at very low Bjorken-x.

Coherent J/ψ photoproduction in ultra-peripheral Pb–Pb collisions with ALICE at the LHC

There are several predictions for the behaviour of the gluon distribution in nuclei at small Bjorken-x and experimental data are needed to choose among them. This is achieved by measuring the cross section of processes specially sensitive to this parton distribution.The high flux of photons from lead ions at the LHC allows us to study photon-induced reactions in ultra-peripheral collisions of Pb–Pb nuclei, in particular of those producing a J/ψ meson exclusively. The study of these collisions, where projectiles do not overlap, provides information about the initial state of nuclei.The newest ALICE results on vector meson photoproduction are presented. The increased statistics and higher collision energy of of Run 2 allow us to put new constraints on available models.

Direct photon production at LHCb

At small Bjorken-x, the large gluon number density in the nucleon leads to gluon recombination competing with gluon splitting, which could result in saturation of the gluon PDF. This gluon saturation has yet to be conclusively observed. Direct photon production provides sensitivity to gluon densities in protons and nuclei, and the forward acceptance of LHCb detector allows for measurements of this process at low Bjorken-x, providing an ideal probe of saturation effects. Progress towards the measurement of forward direct photon production using the LHCb detector is presented.

Resummation of double collinear logs in BK evolution versus HERA data

We present a global fit to HERA data on the reduced cross section measured in electron–proton collisions in the region of small Bjorken-x: x≤x0=10−2 and moderate to high values of the virtuality Q2<Qmax2=650 GeV2. The main dynamical ingredients in the fits are two recently proposed improved BK equations for the description of the small-x evolution of the dipole scattering amplitude [1,2]. These two new equations provide an all-order resummation of double collinear logarithms that arise beyond leading logarithmic accuracy. We show that a very good description of data is possible in both cases, provided the parent dipole or smallest dipole prescriptions are employed for the running of the coupling.

Analysis of combined HERA data in the impact-parameter dependent saturation model

The Impact-Parameter dependent Saturation Model (IP-Sat) is a simple dipole model that incorporates key features of the physics of gluon saturation and matches smoothly to the perturbative QCD dipole expression at large Q^2 for a given x. It was previously shown that the model gives a good description of HERA data suggesting evidence for gluon saturation effects at small x. The model has also been applied to proton-proton and proton-nucleus collisions and provides the basis for the IP-Glasma model of initial conditions in heavy ion collisions. Here we present a reanalysis of available data in electron-proton collisions at small Bjorken-x, including the recently released combined data from the ZEUS and H1 collaborations. We first confront the model to the high precision combined data for the reduced cross-section and obtain its parameters. With these parameters fixed, we compare model results to data for the structure function F_2, the longitudinal structure function F_L, the charm structure function F_2^{c\bar{c}}, exclusive vector meson (J/\psi, \phi and \rho) production and Deeply Virtual Compton Scattering (DVCS). Excellent agreement is obtained for the processes considered at small x in a wide range of Q^2. Our results strongly hint at universality of the IP-Sat dipole amplitude and the extracted impact-parameter distribution of the proton. They also provide a benchmark for further refinements in studies of QCD saturation at colliders.

Charged particle multiplicities inpAinteractions at the LHC from the color glass condensate

The forthcoming LHC measurement of hadron multiplicity in proton-nucleus collisions is a crucial test of the k_t factorization and gluon saturation based models. Here, we provide quantitative predictions for the pseudorapidity distribution of charged particles produced in minimum bias proton-nucleus collisions at the LHC based on the idea of gluon saturation in the color-glass condensate framework. Our formulation gives good descriptions of the LHC and RHIC data for the charged-hadron multiplicities in both proton-proton and nucleus-nucleus collisions, and also the deep inelastic scattering at HERA at small Bjorken-x.

Charmed Meson Production in Deep Inelastic e + A Scattering at Small-x

Based on the nuclear effects considered in the Glauber—Gribov approach, charmed meson production in high energy deep inelastic e+A scattering is investigated in the color dipole formalism. Using the Peterson fragmentation function and the KLR-AdS/CFT color dipole model, which reasonably well describes the HERA data for the inclusive structure functions at small Bjorken-x, we present the predictive results for the D-meson transverse spectra at EIC and LHeC energies. The theoretical results indicate that the x-independence phenomena at very small-x predicted by the KLR-AdS/CFT model are not seen in our calculation.

A predictive phenomenological tool at small Bjorken-x

We present the results from global fits of inclusive DIS experimental data using the Balitsky-Kovchegov equation with running coupling.

Ultrahigh energy neutrino scattering: An update

We update our estimates of charged and neutral current neutrino total cross sections on isoscalar nucleons at ultrahigh energies using a global (x, Q^2) fit, motivated by the Froissart bound, to the F_2 (electron-proton) structure function utilizing the most recent analysis of the complete ZEUS and H1 data sets from HERA I. Using the large Q^2, small Bjorken-x limits of the "wee" parton model, we connect the ultrahigh energy neutrino cross sections directly to the large Q^2, small-x extrapolation of our new fit, which we assume saturates the Froissart bound. We compare both to our previous work, which utilized only the smaller ZEUS data set, as well as to recent results of a calculation using the ZEUS-S based global perturbative QCD parton distributions using the combined HERA I results as input. Our new results substantiate our previous conclusions, again predicting significantly smaller cross sections than those predicted by extrapolating pQCD calculations to neutrino energies above 10^9 GeV.

Comparing AdS/CFT dipole model to HERA F2 data

We apply an AdS/CFT-inspired color-dipole model which contains only three free parameters to describe the HERA data for the inclusive structure function F2 at small Bjorken-x and virtuality. We found that the saturation scale in our AdS/CFT-based parameterization varies in the range of 1 ÷ 3 GeV becoming independent of energy/Bjorken-x at very small x. This leads to the prediction of x-independence of the structure functions at very small x. With the fitted parameters in our model, the predictions for F2, longitudinal structure function, charm structure function and total photo-production cross-sections in the kinematic regions of future experiments can be given.

Hadron multiplicity inppandAAcollisions at LHC from the color glass condensate

We provide quantitative predictions for the rapidity, centrality and energy dependencies of inclusive charged-hadron productions for the forthcoming LHC measurements in nucleus-nucleus collisions based on the idea of gluon saturation in the color-glass condensate framework. Our formulation gives very good descriptions of the first data from the LHC for the inclusive charged-hadron production in proton-proton collisions, the deep inelastic scattering at HERA at small Bjorken-x, and the hadron multiplicities in nucleus-nucleus collisions at RHIC.

Diffractive quarkonium production in association with a photon at the LHC

The rates for diffractive quarkonium plus prompt-photon associated production at the LHC are estimated. The theoretical framework considered in the J/Ψ (or ϒ) production associated with a direct photon at the hadron collider is the non-relativistic QCD (NRQCD) factorization formalism. The corresponding single diffractive cross section is computed based on the hard diffractive scattering factorization supplemented by absorptive corrections. Such processes are sensitive to the gluon content of the Pomeron at small Bjorken-x and they may also be a good place to test the different available mechanisms for quarkonium production at hadron colliders.