47 pages, 2 figures. A new appendix and extra discussions are added, some typos are corrected, new references are added, acknowledgments are modified. Results and conclusions are unchanged
We derive Next-to-Eikonal (NEik) corrections to the background quark propagator, which stem from (i) considering a finite longitudinal width target instead of an infinitely thin shockwave and (ii) including the interaction of the quark with the transverse components of the background field. These two different corrections to the eikonal approximation combine together and provides a gauge covariant expression for the quark propagator at NEik accuracy. We then apply our results to quark (or antiquark) scattering on a nucleus at NEik accuracy, considering both unpolarized cross section and helicity asymmetry.
We extend the study of corrections to the eikonal approximation that was initiated in Ref. \cite{Altinoluk:2014oxa} to higher orders. These corrections associated with the finite width of the target are investigated and the gluon propagator in background field is calculated at next-to-next-to-eikonal accuracy. The result is then applied to the single inclusive gluon production cross section at central rapidities and the light-front helicity asymmetry, in pA collisions, in order to analyse these observables beyond the eikonal limit. The next-to-next-to-eikonal corrections to the unpolarized cro…
We present a new method to systematically include corrections to the eikonal approximation in the background field formalism. Specifically, we calculate the subleading, power-suppressed corrections due to the finite width of the target or the finite energy of the projectile. Such power-suppressed corrections involve Wilson lines decorated by gradients of the background field - thus related to the density - of the target. The method is of generic applicability. As a first example, we study single inclusive gluon production in pA collisions, and various related spin asymmetries, beyond the eikon…
We revisit the calculation of quark and scalar propagators in a gluon background field at Next-to-Eikonal accuracy, which will provide the first power correction to scattering processes in the high-energy limit, including gluon saturation effects. The two main results of our study are as follows. First, we derive a very general Next-to-Eikonal expansion of the quark propagator in terms of the scalar propagator, which allows us to isolate spinor-specific Next-to-Eikonal corrections from the universal ones. Second, we present a new method allowing, for the first time, a systematic calculation of…
In the framework of our model of soft interactions at high energy based on CGC/saturation approach,we show that Bose-Einstein correlations of identical gluons lead to large values of $v_n$. We demonstrate how three dimensional scales of high energy interactions: hadron radius, typical size of the wave function in diffractive production of small masses (size of the constituent quark), and the saturation momentum, influence the values of BE correlations, and in particular, the values of $v_n$. Our calculation shows that the structure of the `dressed' Pomeron leads to values of $v_n$ which are cl…
Multiple initial state parton interactions in p(d)+A collisions are calculated in a Glauber-Eikonal formalism, which incorporates the competing pattern of low-pT suppression due geometrical shadowing, and a moderate-pT Cronin enhancement of hadron spectra. Dynamical shadowing effects, which are not included in the computation, may be extracted by comparing experimental data to the baseline provided by the Glauber-Eikonal model. Data for neutral pion production at midrapidity show absence of dynamical shadowing in the RHIC energy range, sqrt(s)~20-200 GeV. Recent preliminary data at forward rap…
Within the Color Glass Condensate effective theory, we reconsider the next-to-leading order (NLO) calculation of the single inclusive particle production at forward rapidities in proton-nucleus collisions at high energy. Focusing on quark production for definiteness, we establish a new factorization scheme, perturbatively correct through NLO, in which there is no `rapidity subtraction'. That is, the NLO correction to the impact factor is not explicitly separated from the high-energy evolution. Our construction exploits the skeleton structure of the (NLO) Balitsky-Kovchegov equation, in which t…
Color Glass Condensate (CGC) provides a classical description of dense gluon matter at high energies. Using the McLerran-Venugopalan (MV) model we calculate the initial energy density ε(τ) in the early stage of the relativistic nucleus-nucleus collision. Our analytical formula reproduces the quantitative results from lattice discretized simulations and leads to an estimate ε(τ=0.1fm)=40-50GeV/fm^3 in the Au-Au collision at RHIC energy. We then formulate instability with respect to soft fluctuations that violate boost invariance inherent to hard CGC backgrounds. We find unstable modes arising a…
Multi-particle production in QCD is dominated by higher twist contributions. The operator product expansion is not very effective here because the number of relevant operators grow rapidly with increasing twist. The Color Glass Condensate (CGC) provides a framework in QCD to systematically discuss "classical" (multiple scattering) and "quantum" evolution (shadowing) effects in multi-particle production. The apparently insuperable problem of nucleus-nucleus scattering in QCD simplifies greatly in the CGC. A few examples are discussed with emphasis on open problems.
Nucleus-nucleus scattering is studied in the local Reggeon Field Theory in the quasi-classical approximation with non-eikonal boundary conditions corresponding to the Glauber picture at low energies. Comparison with the commonly used eikonal boundary conditions shows that the new conditions make both the action and nucleus-nucleus total cross-sections lower by $3÷5$ %. They also substantially change the behaviour of the solutions of the equations of motion at low energies. Using expressions for the double inclusive cross-sections derived earlier in the Reggeon Field Theory \cite{CM} long-range…
We study the production of heavy quarkonium states in high energy proton-nucleus collisions. Following earlier work of Blaizot, Fujii, Gelis, and Venugopalan, we systematically include both small $x$ evolution and multiple scattering effects on heavy quark pair production within the Color Glass Condensate (CGC) framework. We obtain for the first time expressions in the Non-Relativistic QCD (NRQCD) factorization formalism for heavy quarkonium differential cross sections as a function of transverse momentum and rapidity. We observe that the production of color singlet heavy quark pairs is sensit…
Monte-Carlo implementations of kT -factorization formula with both KLN and running-coupling BK unintegrated gluon distributions for nucleus-nucleus collisions are used to analyze recent experimental data on the particle multiplicities from RHIC(Au+Au@200GeV) and LHC(Pb+Pb@2.76TeV). We also compare the predicted transverse energy at midrapidity to new data from ALICE.
We analyze the structure of running coupling corrections to the gluon production cross section in the projectile-nucleus collisions calculated in the Color Glass Condensate (CGC) framework. We argue that for the gluon production cross section (and for gluon transverse momentum spectra and multiplicity) the inclusion of running coupling corrections brings in collinear singularities due to final state splittings completely unaffected by CGC resummations. Hence, despite the saturation/CGC dynamics, the gluon production cross section is not infrared-safe. As usual, regularizing the singularities r…
We discuss the photon production in the $pA$ collision in a framework of the color glass condensate (CGC) with expansion in terms of the proton color source $ρ_p$. We work in a regime where the color density $ρ_A$ of the nucleus is large enough to justify the CGC treatment, while soft gluons in the proton could be dominant over quark components but do not yet belong to the CGC regime, so that we can still expand the amplitude in powers of $ρ_p$. The zeroth-order contribution to the photon production is known to appear from the Bremsstrahlung process and the first-order corrections consist of t…
We continue our studies of possible generalization of the Color Glass Condensate (CGC) effective theory of high energy QCD to include the high $p_t$ (or equivalently large $x$) QCD dynamics as proposed in [JJM-elastic]. Here we consider scattering of a quark from both the small and large $x$ gluon degrees of freedom in a proton or nucleus target and derive the full scattering amplitude by including the interactions between the small and large $x$ gluons of the target. We thus generalize the standard eikonal approximation for parton scattering which can now be deflected by a large angle (and th…
Heavy flavor measurements in high multiplicity proton-proton and proton-nucleus collisions at collider energies enable unique insights into their production and hadronization mechanism because experimental and theoretical uncertainties cancel in ratios of their cross-sections relative to minimum bias events. We explore such event engineering using the Color Glass Condensate (CGC) effective field theory to compute short distance charmonium cross-sections. The CGC is combined with heavy-quark fragmentation functions to compute $D$-meson cross-sections; for the $J/ψ$, hadronization is described e…
Virtual radiative corrections due to the long range Coulomb forces of heavy nuclei with charge Z may lead to sizeable corrections to the Born cross section usually used for lepton-nucleus scattering processes. An introduction and presentation of the most important issues of the eikonal approximation is given. We present calculations for forward electroproduction production of rho mesons in a framework suggested by the VDM (vector dominance model), using the eikonal approximation. It turns out that Coulomb corrections may become relatively large. Some minor errors in the literature are correcte…
We examine the origins of azimuthal correlations observed in high energy proton-nucleus collisions by considering the simple example of the scattering of uncorrelated partons off color fields in a large nucleus. We demonstrate how the physics of fluctuating color fields in the color glass condensate (CGC) effective theory generates these azimuthal multiparticle correlations and compute the corresponding Fourier coefficients v_n within different CGC approximation schemes. We discuss in detail the qualitative and quantitative differences between the different schemes. We will show how a recently…
Proton-nucleus collisions (p+A) at LHC energies provide a rigorous test of the Color Glass Condensate (CGC), a model proposed to describe the high energy limit of Quantum Chromodynamics. In the CGC the average multiplicity of charged particles at midrapidity in p+A collisions depends logarithmically on the number of participants, $N_{part}$. In contrast, the Wounded Nucleon Model (WNM) of independent nucleon-nucleon scatterings, verified at RHIC energies, predicts that multiplicity in p+A depends linearly on $N_{part}$. We argue that for the first time, the dependence of mean multiplicity on $…
We consider forward inclusive production of two quarks in the high energy p-A collisions in the CGC formalism. We demonstrate that the production cross-section is determined by the convolution of the proton generalized double transverse momentum-dependent distribution (2GTMD) functions with two independent eikonal scattering amplitudes: the product of two dipoles and a quadrupole. We explicitly demonstrate that the quadrupole amplitude term accounts for all the (initial and final state) effects of quantum statistics for identical fermions, and the correlations due to these effects. We also dem…
We study the effects on multigluon production at mid-rapidity in the Color Glass Condensate of the non-eikonal corrections that stem from relaxing the shockwave approximation and giving the target a finite size. We extend previous works performed in the dilute-dilute approximation suitable for proton-proton collisions, to the dilute-dense one applicable to proton-nucleus. We employ the McLerran-Venugopalan model for the projectile averages. For the target averages, we use the Golec-Biernat--Wüsthoff model and restrict to the leading contributions in overlap area that allow a factorization of e…
Non-eikonal corrections to dijet production in DIS
Instituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago de Compostela（加利西亚高能物理研究所，圣地亚哥-德孔波斯特拉大学）；CPHT, CNRS, École Polytechnique, Institut Polytechnique de Paris（法国国家科学研究中心，巴黎综合理工学院，巴黎理工学院）
We investigate the scattering of a quark on a heavy nucleus at high energies using the time-dependent basis light-front quantization (tBLFQ) formalism, which is the first application of the tBLFQ formalism in QCD. We present the real-time evolution of the quark wave function in a strong classical color field of the relativistic nucleus, described as the Color Glass Condensate. The quark and the nucleus color field are simulated in the QCD SU(3) color space. We calculate the total and the differential cross sections, and the quark distribution in coordinate and color spaces using the tBLFQ appr…
The BFKL fan diagram equation for the scattering on the nucleus is solved numerically with the eikonalized initial condition and for a realistic nuclear density. The gluon density has a soliton-like form in the $\log q - y$ space. Inclusive cross-sections for jet production in hA and AB collisions are calculated.
We study single inclusive hadron production in proton-proton and proton-nucleus collisions in the CGC framework. The parameters in the calculation are determined solely by standard nuclear geometry and by electron-proton deep inelastic scattering data, which is fit using the running coupling BK equation. We show that it is possible to obtain a good fit of the HERA inclusive cross section also without an anomalous dimension in the initial condition. We argue that one must consistently use the proton transverse area as measured by a high virtuality probe in DIS also for the single inclusive cros…
In the framework of the effective field theory for interacting BFKL pomerons, applied to nucleus-nucleus scattering, boundary conditions for the classical field equations are discussed. Correspondence with the QCD diagrams at the boundary rapidities requires pomeron interaction with the participating nuclei to be exponential and non-local. Commonly used 'eikonal' boundary conditions, local and linear in fields, follow in the limit of small QCD pomeron-nucleon coupling. Numerical solution of the classical field equations, which sum all tree diagrams for central gold-gold scattering, demonstrate…
We compute the cross section for photons emitted from sea quarks in proton-nucleus collisions at collider energies. The computation is performed within the dilute-dense kinematics of the Color Glass Condensate (CGC) effective field theory. Albeit the result obtained is formally at next-to-leading order in the CGC power counting, it provides the dominant contribution for central rapidities. We observe that the inclusive photon cross section is proportional to all-twist Wilson line correlators in the nucleus. These correlators also appear in quark-pair production; unlike the latter, photon produ…
We use the Color Glass Condensate (CGC) framework to study the production of forward heavy quark-antiquark pairs in unpolarized proton-nucleus or proton-proton collisions in the small-x regime. In the limit of nearly back-to-back jets, the CGC result simplifies into the transverse-momentum dependent (TMD) factorization approach. For massless quarks, the TMD factorization formula involves three unpolarized gluon TMDs: the Weizsäcker-Williams gluon distribution, the adjoint-dipole gluon distribution, and an additional one. When quark masses are kept non-zero, three new gluon TMDs appear -- each …
In this thesis the Color Glass Condensate (CGC) framework, which describes quantum chromodynamics (QCD) at high energy, is applied to various scattering processes. Higher order corrections to the CGC evolution equations, known as the BK and JIMWLK equations, are also considered. It is shown that the leading order CGC calculations describe the experimental data from electron-proton deep inelastic scattering (DIS), proton-proton and proton-nucleus collisions. The initial condition for the BK evolution equation is obtained by performing a fit to deep inelastic scattering data. The fit result is u…
We present an analysis of data on single inclusive pion production measured by the LHCf collaboration in high-energy proton-proton and proton-nucleus at ultra-forward rapidities, $8.8 \leq\! y\leq \!10.8$. We also analyse forward RHIC data for calibration purposes. Our analysis relies on the use of a Monte Carlo event generator that combines a perturbative description of the elementary scattering process at partonic level based on the hybrid formalism of the Color Glass Condensate with an implementation of hadronization in the framework of the Lund string fragmentation model. This procedure al…
Simple power counting arguments in the dilute-dense framework of the Color Glass Condensate (CGC) Effective Field Theory predict that even and odd azimuthal anisotropy harmonics of two-particle correlations in proton-nucleus collisions at the LHC will respectively satisfy v^2_{2n} ~ N_{ch}^0 and v^2_{2n+1} ~ N_{ch}, where N_{ch} denotes the number of charged particles. We show that these expectations are borne out qualitatively, and even quantitatively, within systematic uncertainties, for v_2 and v_4 in comparisons with data from the ATLAS collaboration. We also observe that ATLAS data for th…
We re\,examine the theory of hadronic photon-nucleon interactions at the quark-gluon level. The possibility of multiple parton collisions in a single photon-nucleon collision requires an eikonal treatment of the high-energy scattering process. We give a general formulation of the theory in which the $γp$ cross section is expressed as a sum over properly eikonalized cross sections for the interaction of the virtual hadronic components of the photon with the proton, with each cross section weighted by the probability with which that component appears in the photon, and then develop a detailed mo…
We consider the next-to-leading order (NLO) calculation of single inclusive particle production at forward rapidities in proton-nucleus collisions and in the framework of the Color Glass Condensate (CGC). We focus on the quark channel and the corrections associated with the impact factor. In the first step of the evolution the kinematics of the emitted gluon is kept exactly (and not in the eikonal approximation), but such a treatment which includes NLO corrections is not explicitly separated from the high energy evolution. Thus, in this newly established "factorization scheme", there is no "ra…
In this series of three lectures, we discuss several aspects of high energy scattering among hadrons in Quantum Chromodynamics. The first lecture is devoted to a description of the parton model, Bjorken scaling and the scaling violations due to the evolution of parton distributions with the transverse resolution scale. The second lecture describes parton evolution at small momentum fraction x, the phenomenon of gluon saturation and the Color Glass Condensate (CGC). In the third lecture, we present the application of the CGC to the study of high energy hadronic collisions, with emphasis on nucl…
We investigate the low transverse momentum heavy quarkonium ($J/ψ$, $Υ$) productions in the forward rapidity region of $pp$ and $pA$ collisions at the LHC as an important probe to the transverse momentum tomography of the gluons in hadrons in the Color Glass Condensate (CGC) framework. By implementing the Sudakov resummation consistently in the CGC formalism, we achieve an excellent agreement between the improved CGC calculations and the LHC data. We show that both the small-$x$ and the Sudakov effects are essential for a complete description of heavy quarkonium productions in the low transver…
For studying small-$x$ gluon saturation in forward dijet production in high-energy dilute-dense collisions, the improved TMD (ITMD) factorization formula was recently proposed. In the Color Glass Condensate (CGC) framework, it represents the leading term of an expansion in inverse powers of the hard scale. It contains the leading-twist TMD factorization formula relevant for small gluon's transverse momentum $k_t$, but also incorporates an all-order resummation of kinematical twists, resulting in a proper matching to high-energy factorization at large $k_t$. In this paper, we evaluate the accur…
With the advent of very powerful particle accelerators, such as RHIC and the LHC, it becomes possible to study QCD in high energy collisions, in which the gluon content of the proton or nucleus is probed and its density becomes often large enough for nonlinear effects to play a role. This small-x regime of QCD is well described by an effective theory known as the Color Glass Condensate (CGC). In this thesis, we introduce the CGC and apply it to two different problems. First, we use the CGC to study forward heavy-quark production in pA collisions. When the quarks are nearly back-to-back, the CG…
We compute the differential cross-section for inclusive prompt photon production in deeply inelastic scattering (DIS) of electrons on nuclei at small $x$ in the framework of the Color Glass Condensate (CGC) effective theory. The leading order (LO) computation in this framework resums leading logarithms in $x$ as well as power corrections to all orders in $Q_{s,A}^2/Q^2$, where $Q_{s,A}(x)$ is the nuclear saturation scale. This LO result is proportional to universal dipole and quadrupole Wilson line correlators in the nucleus. In the soft photon limit, the Low-Burnett-Kroll theorem allows us to…
We study the azimuthal asymmetries for quark pair production in proton-nucleus collisions using a hybrid approach in which the nucleus is treated in the Color Glass Condensate (CGC) framework while the Lipatov approximation is applied on the proton side. Our treatment goes beyond the large N_c limit. We particularly focus on the so-called correlation limit where the imbalance of the transverse momentum of the quark pair is much smaller than the out-going individual quark transverse momenta. In this kinematic region, a matching between the hybrid approach and a factorization in terms of transve…
We propose a new parton model and demonstrate that the model describes the relevant experimental data at high energies. The model is based on Pomeron calculus in 1+1 space-time dimensions, as suggested in Ref. [18] and on simple assumptions regarding the hadron structure, related to the impact parameter dependence of the scattering amplitude. This parton model evolves from QCD, assuming that the unknown non-perturbative corrections lead to fixing size of the interacting dipoles. The advantage of this approach is that it satisfies both t-channel and s-channel unitarity, and can be used for summ…
CGC Fit to HERA DIS and Forward Hadron Production in p+p Collisions
Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）；Doctoral School of Exact and Natural Sciences, Jagiellonian University（雅盖隆大学数理博士学院）；Department of Physics, Temple University（天普大学物理系）；RIKEN-BNL Research Center, Brookhaven National Laboratory（布鲁克海文国家实验室理研-布鲁克海文研究中心）；Physics Department, Brookhaven National Laboratory（布鲁克海文国家实验室物理部）
We find the DIS structure functions at strong coupling by calculating R-current correlators on a finite-size shock wave using AdS/CFT correspondence. We improve on the existing results in the literature by going beyond the eikonal approximation for the two lowest orders in graviton exchanges. We argue that since the eikonal approximation at strong coupling resums integer powers of 1/x (with x the Bjorken-x variable), the non-eikonal corrections bringing in positive integer powers of x can not be neglected in the small-x limit, as the non-eikonal order-x correction to the (n+1)st term in the ei…
At very high energies or small values of Bjorken x, the density of partons, per unit transverse area, in hadronic wavefunctions becomes very large leading to a saturation of partonic distributions. When the scale corresponding to the density per unit transverse area, the saturation scale Q_s, becomes large (Q_s\gg Λ_{QCD}), the coupling constant becomes weak (α_S(Q_s)\ll 1) which suggests that the high energy limit of QCD may be studied using weak coupling techniques. This simple idea can be formalized in an effective theory, the Color Glass Condensate (CGC), which describes the behavior of th…
We investigate the scattering of a quark jet on a high-energy heavy nucleus using the time-dependent light-front Hamiltonian approach. We simulate a real-time evolution of the quark in a strong classical color field of the relativistic nucleus, described as the Color Glass Condensate. We study the sub-eikonal effect by letting the quark jet carry realistic finite longitudinal momenta, and we find sizeable changes on the transverse coordinate distribution of the quark. We also observe the energy loss of the quark through gluon emissions in the $\ket{q}+\ket{qg}$ Fock space. This approach provid…
We calculate isolated photon production at forward rapidities in proton-nucleus collisions in the Color Glass Condensate framework. Our calculation uses dipole cross sections solved from the running coupling Balitsky-Kovchegov equation with an initial condition fit to deep inelastic scattering data and extended to nuclei with an optical Glauber procedure that introduces no additional parameters beyond the basic nuclear geometry. We present predictions for future forward RHIC and LHC measurements. The predictions are also compared to updated results for the nuclear modification factors for pion…
In this paper we calculate the contribution to rapidity and angular correlations of the first Pomeron loop diagram in the dense partonic environment. This diagram is expected to give the largest contribution to the density variation mechanism of the angular correlations. We show that this diagrams leads to sizable contributions to the rapidity correlation functions of the order of $σ_{in} /\Lb πR^2\Rb$ where $σ_{in}$ is the inelastic cross section and $R$ is the size of the typical dipole inside the proton saturation scale. Therefore, the correlations do not depend on the saturation scale. We …
Inclusive production of $J/ψ$ mesons, especially at forward rapidities, is an important probe of small-x gluons in protons and nuclei. In this paper we re-evaluate the production cross sections in the Color Glass Condensate framework, where the process is described by a large x gluon from the probe splitting into a quark pair and eikonally interacting with the target proton or nucleus. Using a standard collinear gluon distribution for the probe and an up to date dipole cross section fitted to HERA data to describe the target we achieve a rather good description of the cross section in proton-p…
Forward trijet production in proton-nucleus collisions: gluon initiated channel
SUBATECH UMR 6457 (IMT Atlantique, Universite de Nantes, IN2P3/CNRS)（SUBATECH UMR 6457 (IMT大西洋学院，南特大学，IN2P3/CNRS)）；Department of Physics, Temple University（天普大学物理系）；RIKEN-BNL Research Center, Brookhaven National Laboratory（理研-布鲁克海文国家实验室研究中心）；Physics Department, Brookhaven National Laboratory（布鲁克海文国家实验室物理系）
