We derive an analytical expression for the two-gluon production in the pA (light-heavy) collisions, and focus specifically on the rapidity dependent part. We approximate the gauge field from the heavy target as the Color Glass Condensate which interacts with the light projectile whose source density allows for a perturbative expansion. We discuss the longitudinal correlations of produced particles. Our calculation goes in part beyond the eikonal limit for the emitted gluons so that we can retain the exponential terms with respect to the rapidity difference. Our expression can thus describe the short-range correlations as well as the long-range ones for which our formula is reduced to the known expression. In a special case of two high-pt gluons in the back-to-back kinematics we find that dependence on the rapidity separation is only moderate even in the diagrammatically connected part.
We investigate hydrodynamic evolution of the quark gluon plasma for the colour glass condensate type initial conditions. We solve full second-order viscous hydrodynamic equations in the longitudinal direction to find that non-boost invariant expansion leads to visible deformation on the initial rapidity distribution. The results indicate that hydrodynamic evolution with viscosity plays an important role in determining parameters for the initial distributions.
The initial distribution of gluons at the very early times after a high energy heavy ion collision is described by the bulk scale $Q_s$ of gluon saturation in the nuclear wavefunction. The subsequent evolution of the system towards kinetic equilibrium is described by a non-linear Landau equation for the single particle distributions \cite{Mueller1,Mueller2}. In this paper, we solve this equation numerically for the idealized initial conditions proposed by Mueller, and study the evolution of the system to equilibrium. We discuss the sensitivity of our results on the dynamical screening of colli…
We prove that the inclusive single-gluon production cross section for a hadron colliding with a high-density target factorizes into the gluon distribution function of the projectile, defined as usual within the DGLAP collinear approximation, times the cross section for scattering of a single gluon on the strong classical color field of the target. We then derive the gluon-proton (nucleus) inelastic cross section and show that it is (up to logarithms) infrared safe and that it grows slowly with center of mass energy. Furthermore, we discuss jet transverse momentum broadening for the case of nuc…
We review the Color Glass Condensate effective theory, that describes the gluon content of a high energy hadron or nucleus, in the saturation regime. The emphasis is put on applications to high energy heavy ion collisions. After describing initial state factorization, we discuss the Glasma phase, that precedes the formation of an equilibrated quark-gluon plasma. We end this review with a presentation of recent developments in the study of the isotropization and thermalization of the quark-gluon plasma.
We study exclusive dijet production in electron-proton deep inelastic scattering at a future Electron Ion Collider. We predict the elliptic modulation of the cross section as a function of the angle between the dijet transverse momentum and the recoil momentum, and show that this modulation is due to non-trivial angular correlations between the transverse coordinate and transverse momentum in the Wigner (or Husimi) distribution. The small-$x$ evolution is shown to decrease the elliptic modulation in the EIC kinematics, because of the growth of the proton with decreasing $x$.
We apply the effective theory of color glass condensate to the analysis of gluon states in dense quark matter, in which the saturation region of gluons is also present. We find that in the region two point function of gluons shows algebraic long range order. The order is completely the same as the one gluons show in the dense quark matter, which form quantum Hall states. The order leads to the vanishing of massless gluon pole. We also find that the saturation region of gluons extends from small $x$ to even large $x\lesssim 1$ in much dense quark matter. We point out a universality that the col…
We extend the McLerran-Venugopalan model for the gluon distribution functions of very large nuclei to larger values of the longitudinal momentum fraction xF. Because gluons with larger values of xF begin to resolve the longitudinal structure of the nucleus, we find that it is necessary to set up a fully three-dimensional formalism for performing the calculation. We obtain a relatively compact expression for the gluon number density provided that the nucleus is sufficiently large and consists of color-neutral nucleons. Our expressions for the gluon number density saturate at small transverse mo…
The study of the perturbative Odderon at high gluon densities in the color glass condensate requires to take into account states with more than three gluons. We formulate evolution equations for these states in which the number of gluons is not fixed during the evolution. We further determine the coupling of these Odderon states to the gamma to eta_c impact factor for arbitrary numbers of gluons. We find an exact solution of the evolution equation for the four-gluon Odderon state in terms of the three-gluon Odderon state. It is shown that a similar solution exists also for a different class of…
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…
We study the medium-induced gluon emission from a hard quark jet traversing the dense nuclear matter within the framework of deep inelastic scattering off a large nucleus. We extend the previous work and compute the single gluon emission spectrum including both transverse and longitudinal momentum exchanges between the hard jet parton and the medium constituents. On the other hand, with only transverse scattering and using static scattering centers for the traversed medium, our induced gluon emission spectrum in the soft gluon limit reduces to the Gyulassy-Levai-Vitev one-rescattering-one-emis…
In a two-flavor color superconductor, the SU(3)_c gauge symmetry is spontaneously broken by diquark condensation. The Nambu-Goldstone excitations of the diquark condensate mix with the gluons associated with the broken generators of the original gauge group. It is shown how one can decouple these modes with a particular choice of 't Hooft gauge. We then explicitly compute the spectral density for transverse and longitudinal gluons of adjoint color 8. The Nambu-Goldstone excitations give rise to a singularity in the real part of the longitudinal gluon self-energy. This leads to a vanishing gluo…
The Color Glass Condensate is a universal state of matter which can manifest itself in hadronic processes involving small-x partons, like DIS and pp, pA and AA collisions at high energy. Observables are given in terms of multi-gluon correlators, whose ensemble evolves according to a RG equation, the JIMWLK equation. We focus on recent progress towards its solution which lead to quasi-exact, analytic expressions for the multi-gluon correlators at high energy.
The distribution of longitudinal gluons in a nucleon, introduced earlier by Gorsky and Ioffe, is estimated from below by making use of the positivity of density matrix and the analogue of the Wandzura-Wilczek relation for the light-cone distributions of polarized gluons in a transversely polarized nucleon.
Gluonic nucleon energy correlators and fracture functions for Color Glass Condensate
University of Jyväskylä（于韦斯屈莱大学）；Helsinki Institute of Physics（赫尔辛基物理研究所）；Shandong University（山东大学）；École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）；Chulalongkorn University（朱拉隆功大学）
The purpose of these lectures is to provide an introduction to the physics issues which are being studied in the RHIC heavy ion program. These center around the production of new states of matter. The Quark Gluon Plasma is thermal matter which once existed in the big bang which may be made at RHIC. The Color Glass Condensate is a universal form of matter which controls the high energy limit of strong interactions. Both such forms of matter might be produced and probed at RHIC.
We make a critical, next-to-leading order, study of the relationship between the longitudinal structure function FL and the gluon distribution proposed in Cooper- Sarkar et al. (Z. Phys. C 39:281, 1988; Acta Phys. Pol. B 34:2911 2003), which is frequently used to extract the gluon distribution from the proton longitudinal structure function at small x. The gluon density is obtained by expanding at particular choices of the point of expansion and compared with the hard Pomeron behavior for the gluon density. Comparisons with H1 data are made and predictions for the proposed best approach are al…
We obtain an approximate analytical form of the gluon distribution using the GLAP equation with a factorization ansatz,and test its validity by comparing it with that of Gluck,Reya and Vogt at low $x$ regime. We also present calculations of the longitudinal structure functions.
Non-Abelian plasma instabilities play an important role in the non-equilibrium dynamics of a weakly coupled quark-gluon plasma. Using the discretized hard loop framework we calculate the time evolution of soft gluonic fields in a longitudinally free streaming background. Extrapolating our results to energies probed in relativistic heavy-ion collisions we find a pressure anisotropy that persists for a few fm/c. However, the chromofields quickly develop a Boltzmann longitudi- nal energy spectrum, suggesting fast longitudinal thermalization of the quark gluon plasma even though it remains momentu…
In the framework of the QCD effective action the vertices of gluon emission in interaction of reggeons are studied in the limit of small longitudinal momenta of the emitted gluon. It is found that the vertices drastically simplify in this limit so that the gluon becomes emitted from a single reggeon coupled to the projectile and target via multireggeon vertices. Contribution from this kinematical region is studied for double and 2x2 elementary collisions inside the composite projectile and target.
We study gluon production in high energy proton-nucleus collisions in the semi-classical framework of the Color Glass Condensate. We develop a general formalism to compute gluon fields in covariant gauge to lowest order in the classical field of the proton and to all orders in the classical field of the nucleus. The use of the covariant gauge makes the diagrammatic interpretation of the solution more transparnt. k_t-factorization holds to this order for gluon production -- Our results for the gluon distribution are equivalent to the prior diagrammatic analysis of Kovchegov and Mueller. We also…
In recent years, Fractal Inspired Models of quark and gluon densities at small x have been proposed. In this paper, we investigate longitudinal structure function F-L (x, Q2) within this approach. We make predictions using the QCD based approximate relation between the longitudinal structure function and the gluon density. As the Altarelli-Martinelli equation for the longitudinal structure function cannot be applied to Model I due to the presence of a singularity in the Bjorken x-space we consider Model II only. The qualitative feature of the prediction of Model II is found to be compatible wi…
We discuss application of formalism of small-$x$ effective action for reggeized gluons, \cite{Gribov,LipatovEff,BFKL}, for the calculation of classical gluon field of relativistic color charge, similarly to that done in CGC approach of \cite{Venug,Kovner}. The equations of motion with the reggeon fields are solved in LO and NLO approximations and new solutions are found. The results are compared to the calculations performed in the CGC framework and it is demonstrated that the LO CGC results for the classical field are reproduced in our calculations. Possible applications of the NLO solution i…
We present results for the structure function F_L for a gluon target having nonzero transverse momentum square at order α_s. The results of double convolution (with respect to Bjorken variable x and the transverse momentum) of the perturbative part and the unintegrated gluon densities are compared with recent experimental data for F_L at low x values and with predictions of other approaches.
Calculations are presented of the longitudinal structure function $F_L(x, Q^2)$. We use next-to-leading order expressions in QCD $({\cal{O}}(α_s^2))$ plus parton densities determined previously from global fits to data on deep inelastic lepton scattering, prompt photon production, and bottom quark production. Anticipated data from the DESY ep collider HERA should provide discriminating information on the gluon density, particularly for values of $x < 0.05$.
We report on a first NLO computation of photon production in p+A collisions at collider energies within the Color Glass Condensate framework, significantly extending previous LO results. At central rapidites, our result is the dominant contribution and probes multi-gluon correlators in nuclei. At high photon momenta, the result is directly sensitive to the nuclear gluon distribution. The NLO result contains two processes, the annihilation process and the process with $q\bar{q}$ pair and a photon in the final state. We provide a numerical evaluation of the photon spectrum from the annihilation …
We complete the construction of the renormalization group equation (RGE) for the Color Glass Condenstate begun in Paper I. This is the equation which governs the evolution with rapidity of the statistical weight function for the color glass field. The coefficients in this equation --- one-loop real and virtual contributions --- are computed explicitly, to all orders in the color glass field. The resulting RGE can be interpreted as the imaginary-time evolution equation, with rapidity as the ``imaginary time'', for a quantum field theory in two spatial dimensions. In the weak field limit it redu…
We calculate $γ_l(0)$, the damping rate for longitudinal on-shell gluons with zero momentum in hot QCD using the hard-thermal-loop (htl) scheme. We find it to be divergent in the infrared, which means that in this scheme $γ_l(0)$ is different from $γ_t(0)$, the corresponding damping rate for transverse gluons which is known to be finite. This result suggests that the htl scheme is infrared sensitive and thus may need to be improved upon in this sector. We discuss this issue after we present our calculation.
We extend previous work on high energy nuclear collisions in the Color Glass Condensate model to study collisions of finite ultrarelativistic nuclei. The changes implemented include a) imposition of color neutrality at the nucleon level and b) realistic nuclear matter distributions of finite nuclei. The saturation scale characterizing the fields of color charge is explicitly position dependent, $Λ_s=Λ_s(x_T)$. We compute gluon distributions both before and after the collisions. The gluon distribution in the nuclear wavefunction before the collision is significantly suppressed below the saturat…
We study the coupling of two longitudinal Z-bosons to two gluons via technicolor interactions. Noticing a similarity of this process to $2 γ\rightarrow 2 π^{0}$, we calculate the amplitude in chiral perturbation theory in one-generation technicolor model. At the invariant mass of $ Z_{L}Z_{L}$ above the colored pseudo-Goldstone boson threshold, we find the signal in proton collisions at $\sqrt{s} = 14\, TeV$ is stronger than the standard model background by an order of magnitude, and is large enough to be easily observable at LHC. The cross section for $W_{L}^{+}W_{L}^{-}$ pair production is a…
A Slavnov-Taylor identity is derived for the gluon polarization tensor in hot QCD. We evaluate its implications for damping of gluonic modes in the plasma. Applying the identity to next to the leading order in hard-thermal-loop resummed perturbation theory, we derive the expected equality of damping rates for static transverse and longitudinal (soft) gluons. This is of interest also in view of deviating recent reports of $γ_t(p=0)\neqγ_l(p=0)$ based on a direct calculation of $γ_l(p=0)$.
Thermalization of a longitudinally expanding color glass condensate with Bjorken boost invariant geometry is investigated within microscopical parton cascade BAMPS. Our main focus lies on the detailed comparison of thermalization, observed in BAMPS with that suggested in the \BUp scenario. We demonstrate that the tremendous production of soft gluons via $gg \to ggg$, which is shown in the \BUp picture as the dominant process during the early preequilibration, will not occur in heavy ion collisions at RHIC and LHC energies, because the back reaction $ggg\to gg$ hinders the absolute particle mul…
We investigate the dependence of the deep inelastic proton structure function $F_L(x, Q^2)$ on different forms of the transverse momentum dependent (TMD, or unintegrated) gluon distribution. We present a comparison of theoretical results with the latest experimental data taken by the H1 and ZEUS Collaborations at HERA. We demonstrate that these data, despite of having quite large uncertainties, can test different kinds of the gluon TMD in a proton. Moreover, different phenomenological models at low $x$ could be tested by future experiments on deep inelastic scattering.
The longitudinally polarized gluon density is probed sensitively in hard collisions of polarized protons under the condition that the dominant dynamics are perturbative and of leading twist origin. First data have recently been presented by Phenix on the double-spin asymmetry A_{LL}^{pi} for pi^0 production at moderate transverse momentum p_\perp \simeq 1 - 4 GeV and central rapidity. By means of a systematic investigation of the relevant degrees of freedom we show that the perturbative QCD framework at leading power in p_\perp produces an asymmetry that is basically positive definite in this …
Probing quark transverse momentum distributions in the Color Glass Condensate: quark-gluon dijets in Deep Inelastic Scattering at next-to-eikonal accuracy
We perform a detailed comparison of long range rapidity correlations in the Color Glass Condensate framework to high multiplicity di-hadron data in proton-proton and proton-lead collisions from the CMS, ALICE and ATLAS experiments at the LHC. The overall good agreement thus far of the non-trivial systematics of theory with data is strongly suggestive of gluon saturation and the presence of subtle quantum interference effects between rapidity separated gluons. In particular, the yield of pairs collimated in their relative azimuthal angle $Δϕ\sim 0$, is sensitive to the shape of unintegrated glu…
We show that the numerical solution of the classical SU(3) Yang-Mills equations of motion in the McLerran-Venugopalan model for gluon production in central heavy ion collisions leads to a suppresion at low $p_t$ and an enhancement at the intermediate $p_t$ region as compared to peripheral heavy ion and pp collisions at the same energy. Our results are compared to previous, Color Glass Condensate inspired calculations of gluon production in heavy ion collisions. We revisit the predictions of the Color Glass Condensate model for $pA$ ($dA$) collisions in Leading Order and show that quantum evolu…
We consider a nonlinear evolution equation recently proposed to describe the small-$x$ hadronic physics in the regime of very high gluon density. This is a functional Fokker-Planck equation in terms of a classical random color source, which represents the color charge density of the partons with large $x$. In the saturation regime of interest, the coefficients of this equation must be known to all orders in the source strength. In this first paper of a series of two, we carefully derive the evolution equation, via a matching between classical and quantum correlations, and set up the framework …
We study numerically the dependence of the difference between the longitudinal and transverse gluon propagators, $Δ=D_L-D_T$, on the momentum and temperature at $T\gtrsim T_c$ both in SU(2) and SU(3) gluodynamics. It is found that the integral of $Δ$ with respect to the 3-momentum is sensitive only to infrared dynamics and shows a substantial correlation with the Polyakov loop. At $T=T_p\sim 1.2 T_c$ it changes sign giving some evidence that $T_p$ can serve as a boundary of the postconfinement domain.
In [arXiv:0804.2630], we have analyzed the leading logarithms of energy that appear in the inclusive spectrum of gluons produced in heavy ion collisions, calculated in the Color Glass Condensate framework. The main result of this paper was that these logarithms are intrinsic properties of the colliding projectiles, and that they can be resummed by letting the distributions of color sources in the nuclei evolve according to the JIMWLK equation. An essential step in the proof of this factorization result is the calculation of the gluon spectrum at Next-to-Leading order, and in particular a funct…
We consider the implications of the Color Glass Condensate for the central region of p+A collisions. We compute the k_t distribution of radiated gluons and their rapidity distribution dN/dy analytically, both in the perturbative regime and in the region between the two saturation momenta. We find an analytic expression for the number of produced gluons which is valid when the saturation momentum of the proton is much less than that of the nucleus. We discuss the scaling of the produced multiplicity with A. We show that the slope of the rapidity density dN/dy provides an experimental measure fo…
We consider an initially at rest colored particle which is struck by an ultra-relativistic nucleus. The particle is treated classically both with respect to its motion and its color charge. The nucleus is treated as a sheet of colored glass within the context of the Color Glass Condensate framework. We compute both the momentum and coordinates of the struck classical particle and the emitted radiation. Our computations generalize the classic electrodynamics computation of the radiation of an accelerated charged particle to include the radiation induced by the charged gluon field. This latter c…
We study the phase space distributions of gluons inside a nucleon/nucleus in the small-$x$ regime including the gluon saturation effect. This can be done by using the relation between the gluon Wigner distribution and the dipole S-matrix at small-$x$, the latter satisfies the Balitsky-Kovchegov (BK) equation. By efficiently solving the BK equation with impact parameter dependence, we compute the Wigner, Husimi and generalized TMD (GTMD) distributions in the saturation regime. We also investigate the elliptic angular dependence of these distributions which has been recently shown to be measurab…
We discuss proton-nucleus collisions in the framework of the color glass condensate. By assuming that the proton can be described as a low density color source, we solve exactly the Yang-Mills equations corresponding to this type of collision, and then use this solution in order to calculate inclusive gluon production or quark-antiquark production. Our result shows that k_T-factorization, while valid for gluon production, is violated for quark pair production in proton-nucleus collisions.
The high energy limit of QCD is controlled by very high energy density gluonic matter, the Color Glass Condensate. In the first instants of the collisions of two sheets of Colored Glass Condensate, a Glasma is formed with longitudinal flux tubes of color electric and magnetic fields. These flux tubes decay and might form a turbulent liquid that eventually thermalizes into a Quark Gluon Plasma
In these lectures, we develop the theory of the Colour Glass Condensate. This is the matter made of gluons in the high density environment characteristic of deep inelastic scattering or hadron-hadron collisions at very high energy. The lectures are self contained and comprehensive. They start with a phenomenological introduction, develop the theory of classical gluon fields appropriate for the Colour Glass, and end with a derivation and discussion of the renormalization group equations which determine this effective theory.
We consider a version of the McLerran-Venugopalan model by Lam and Mahlon where confinement is implemented via colored noise in the infrared. This model does not assume an infinite momentum frame, hence the boosted nuclei are not infinitely thin. Instead, the nuclei have a finite extension in the longitudinal direction and therefore depend on the longitudinal coordinate. In this fully three dimensional framework an x dependence of the gluon distribution function emerges naturally. In order to fix the parameters of the model, we calculate the gluon distribution function and compare it with the …
We study, for the first time, the spatial extension of the "source" that produces quark gluon plasma (QGP) in ultra relativistic heavy ion collisions. The longitudinal dimension is studied as a function of time as the system evolves. The source size is found to exhibit a novel non-classical feature.
We consider two-gluon production in dilute-dense collisions within the Color Glass Condensate framework, applicable to both proton-nucleus and heavy-light ion collisions. We go beyond the glasma graph approximation which is valid in the dilute-dilute limit and show the correspondence between the glasma graphs and the $k_T$-factorized approach that we use in our calculation. We then identify the classical uncorrelated, and the Hanbury-Brown-Twiss (HBT) and Bose enhancement correlated contributions, with the Bose enhancement contribution being suppressed by the number of degrees of freedom with …
In a model of relativistic heavy ion collisions wherein the unconfined quark-gluon plasma is condensed into glass, we derive the Vogel-Fulcher-Tammann cooling law. This law is well known to hold true in condensed matter glasses. The high energy plasma is initially created in a very hot negative temperature state and cools down to the Hagedorn glass temperature at an ever decreasing rate. The cooling rate is largely determined by the QCD string tension derived from hadronic Regge trajectories. The ultimately slow relaxation time is a defining characteristic of a color glass condensate.
In the usual real-time finite-temperature gauge theory both the physical and the unphysical degrees of freedom are thermalised. We discuss the alternative approach where only the physical transverse components of the gauge field have bare thermal propagators, whereas the unphysical degrees of freedom are not heated. We show how pinch singularities are avoided: sometimes this requires resummation. If only the hard thermal loop is included in the resummation, the spatially-longitudinal component of the gauge field, which contains an extra collective plasmon mode, becomes fully thermalised, thoug…
