Linear Boltzmann Transport for Jet Propagation in the Quark-Gluon Plasma: Elastic Processes and Medium Recoil
Yayun He, Tan Luo, Xin-Nian Wang, Yan Zhu
备注：
19 pages in RevTeX with 19 figures, version typos are corrected in Eqs. (1), (4) and (9) in the final published version. An Erratum is submitted to PRC
A Linear Boltzmann Transport model within perturbative QCD is developed for the study of parton propagation inside the quark-gluon plasma. Both leading partons and thermal recoil partons are tracked so that one can also study jet-induced medium excitations. In this study, we implement the complete set of elastic parton scattering processes and investigate elastic parton energy loss, transverse momentum broadening and their nontrivial energy and length dependence. We further investigate medium modifications of the jet shape and fragmentation functions of reconstructed jets. Contributions from thermal recoil partons are found to have significant influences on jet shape, fragmentation functions and angular distribution of reconstructed jets.
Jet energy transported to quark-gluon plasma during jet-medium interaction excites the QGP medium and creates energetic thermal partons -- recoil particles or recoils. Modification of the jet structure in heavy ion collisions is studied using \textsc{martini}, in which recoil simulation is enabled. In large systems such as central Pb-Pb collisions, the recoil effect is expected to be critical due to strong jet-medium interaction. We show the results of the jet mass function and jet shape function are improved when the recoil particles are included in the reconstructed jets. We conclude that th…
A Boltzmann equation is given for the early stages of evolution of the gluon system produced in a head-on collision. The collision term is taken from gluon- gluon approximation. < p_\perp > and < p_z^2 > are evaluated as a function of time using initial conditions taken from the McLerran-Venugopalan model.
A Linear Boltzmann Transport (LBT) model has been developed for the study of jet propagation inside a quark-gluon plasma. Both leading and thermal recoiled partons are transported according to the Boltzmann equations to account for jet-induced medium excitations. In this talk, we present our study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate elastic energy loss and transverse shape of reconstructed jets. Contributions from the recoiled …
Recoil Geometry Unmasks Gluon Saturation in Forward $Z^0$ Production
Fudan University（复旦大学）；Peking University（北京大学）；Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；Shandong University (Qingdao)（山东大学（青岛））
We derive the generalized Boltzmann equation (GBE) near equilibrium from the Kadanoff-Baym equation for quark excitation with ultrasoft momentum (~g^2T, g: coupling constant, T: temperature) in quantum chromodynamics (QCD) at extremely high T, and show that the equation is equivalent to the self-consistent equation derived in the resummed perturbation scheme used to analyze the quark propagator. We obtain the expressions of the dispersion relation, the damping rate, and the strength of a quark excitation with ultrasoft momentum by solving the GBE. We also show that the GBE enables us to obtain…
We show how the Landau-Pomeranchuk-Migdal effect on photon production rates in a quark-gluon plasma can be derived via the usual Boltzmann equation. To do this, we first derive the electromagnetic polarization tensor using linear response theory, and then formulate the Boltzmann equation including the collisions mediated by soft gluon exchanges. We then identify the resulting expression for the production rate with that obtained by the field-theoretic formalism recently proposed by Arnold, Moore and Yaffe. To illustrate the LPM effect we solve the Boltzmann equation in the diffusion approximat…
We have developed a numerical framework for a full solution of the relativistic Boltzmann equations for the quark-gluon matter using the multiple Graphics Processing Units (GPUs) on distributed clusters. Including all the $2 \to 2$ scattering processes of 3-flavor quarks and gluons, we compute the time evolution of distribution functions in both coordinate and momentum spaces for the cases of pure gluons, quarks and the mixture of quarks and gluons. By introducing a symmetrical sampling method on GPUs which ensures the particle number conservation, our framework is able to perform the space-ti…
We study the evolution of a gluon system under conditions of density and temperature similar to those explored in the early stage of ultra-relativistic heavy-ion collisions. We first describe the implementation of Relativistic Boltzmann-Nordheim (RBN) transport approach that includes in the collision integral the quantum effects of Bose-Einstein Statistics. Then, we describe the evolution of a spatially uniform gluon system in a box under elastic collisions solving the RBN for various initial conditions. We discuss the critical phase-space density that leads to the onset of a Bose-Einstein con…
The famous formula for the emission angle of Cherenkov radiation should be modified when applied to hadronic reactions because of recoil effects. They impose the upper limit on the energy of the gluon emitted at a given angle. Also, it leads to essential corrections to the nuclear refractive index value as determined from the angular position of Cherenkov rings.
A Linearized Boltzmann Transport (LBT) model coupled with hydrodynamical background is established to describe the evolution of jet shower partons and medium excitations in high energy heavy-ion collisions. We extend the LBT model to include both elastic and inelastic processes for light and heavy partons in the quark-gluon plasma. A hybrid model of fragmentation and coalescence is developed for the hadronization of heavy quarks. Within this framework, we investigate how heavy flavor observables depend on various ingredients, such as different energy loss and hadronization mechanisms, the mome…
We develop a new heavy quark transport model, QLBT, to simulate the dynamical propagation of heavy quarks inside the quark-gluon plasma (QGP) created in relativistic heavy-ion collisions. Our QLBT model is based on the linear Boltzmann transport (LBT) model with the ideal QGP replaced by a collection of quasi-particles to account for the non-perturbative interactions among quarks and gluons of the hot QGP. The thermal masses of quasi-particles are fitted to the equation of state from lattice QCD simulations using the Bayesian statistical analysis method. Combining QLBT with our advanced hybrid…
We consider the meson and baryon production in the quark-gluon plasma as applied to the early Universe and ultrarelativistic heavy ion collisions in collider. The essential feature of our investigation is to take into account a potential barrier for quarks under meson and baryon production. We have used the Boltzmann equation to describe kinetic processes of quark fusion. We have derived analytic expressions for solutions of kinetic equations. This permits to find the dependence of reaction rate on a time, temperature, mass and charge of quarks as well as on a string tension connected with the…
Wakes created by a parton moving through a static and infinitely extended quark-gluon plasma are considered. In contrast to former investigations collisions within the quark-gluon plasma are taken into account using a transport theoretical approach (Boltzmann equation) with a Bhatnagar-Gross-Krook collision term. Within this model it is shown that the wake structure changes significantly compared to the collisionless case.
We develop a framework of coupled transport equations for open heavy flavor and quarkonium states, in order to describe their transport inside the quark-gluon plasma. Our framework is capable of studying simultaneously both open and hidden heavy flavor observables in heavy-ion collision experiments and can account for both, uncorrelated and correlated recombination. Our recombination implementation depends on real-time open heavy quark and antiquark distributions. We carry out consistency tests to show how the interplay among open heavy flavor transport, quarkonium dissociation and recombinati…
We apply the GLV reaction operator solution to the Vitev-Gunion-Bertsch (VGB) boundary conditions to compute the all-order in nuclear opacity non-abelian gluon bremsstrahlung of event-by-event fluctuating beam jets in nuclear collisions. We evaluate analytically azimuthal Fourier moments of single gluon, $v_n^M\{1\}$, and even number $2\ell$ gluon, $v_n^M\{2\ell\}$ inclusive distributions in high energy p+A reactions as a function of harmonic $n$, %independent target recoil cluster number, $M$, and gluon number, $2\ell$, at RHIC and LHC. Multiple resolved clusters of recoiling target beam jets…
The azimuthal decorrelation between a vector boson and a jet is an essential hard probe in high energy proton-proton and heavy-ion collisions. We overcome intrinsic limitations of previous studies by using a recoil-free axis, achieving unprecedented next-to-next-to-leading logarithmic accuracy with small nonperturbative corrections. This choice of axis also makes the observable robust in the presence of a large background. Furthermore, the azimuthal angle distribution is minimally changed when determined using only charged particle tracks, which offer superior angular resolution for precise me…
Jet energy loss and single inclusive jet suppression in high-energy heavy-ion collisions are studied within a pQCD parton model that includes both elastic and radiative interactions between jet shower and medium partons as they propagate through the quark-gluon plasma. The collisional energy loss of jets with a given cone-size is found to be relatively small comparing with the radiative energy loss. However the effect of transverse momentum broadening due to elastic scattering is significant in the calculation of radiative energy loss within the higher-twist formalism. The nuclear modification…
We present a kinetic description of Bose-Einstein condensation for particle systems being out of thermal equilibrium, which may happen for gluons produced in the early stage of ultra-relativistic heavy-ion collisions. The dynamics of bosons towards equilibrium is described by a Boltzmann equation including Bose factors. To solve the Boltzmann equation with the presence of a Bose-Einstein condensate we make further developments of the kinetic transport model BAMPS (Boltzmann Approach of MultiParton Scatterings). In this work we demonstrate the correct numerical implementations by comparing the …
We study the time evolution of a high-momentum gluon or quark propagating through an infinite, thermalized, partonic medium utilizing a Boltzmann equation approach. We calculate the collisional energy loss of the parton, study its temperature and flavor dependence as well as the the momentum broadening incurred through multiple interactions. Our transport calculations agree well with analytic calculations of collisional energy-loss where available, but offer the unique opportunity to address the medium response as well in a consistent fashion.
We investigate how thermalization of gluons depends on the initial conditions assumed in ultrarelativistic heavy ion collisions at RHIC. The study is based on simulations employing the pQCD inspired parton cascade solving the Boltzmann equation for gluons. We consider independently produced minijets with $p_T > p_0=1.3 \sim 2.0$ GeV and a color glass condensate as possible initial conditions for the freed gluons. It turns out that full kinetic equilibrium is achieved slightly sooner in denser system and its timescale tends to saturate. Compared with the kinetic equilibration we find a stronger…
We develop a set of coupled Boltzmann equations to describe the dynamical evolution of heavy quarks and quarkonia inside the quark-gluon plasma. The quarkonium dissociation and recombination terms are calculated from pNRQCD. Their interplay drives the system to a detailed balance. The heavy quark energy loss term is necessary for the system to reach kinematic thermalization. By coupling the transport equations with initial particles' momenta generated by PYTHIA and hydrodynamic medium evolutions, we can describe the $R_{AA}$ of $Υ$ family at both RHIC and LHC energies. The transverse momentum …
We discuss the propagation of heavy quarks (charm and bottom) through the QGP by means of a relativistic Boltzmann transport approach including both collisional and radiative energy loss mechanisms. In particular we investigate the impact of induced gluon radiation by dynamical QCD medium implementing in our transport model a formula for the emitted gluon spectrum calculated in a higher-twist scheme. We notice that in the region of high transverse momentum ($p_T > 10$ GeV) radiative processes play an essential role giving a dominant contribution to the generation of $R_{AA}$ and $v_2$ at momen…
We investigate the electromagnetic response of a viscous quark-gluon plasma in the framework of the relativistic Boltzmann equation with current conserving collision term. Our formalism incorporates dissipative effects at all orders in linear response to the electromagnetic field while accounting for the full space- and time-dependence of the perturbing fields. As an example, we consider the collision of two nuclei in a stationary, homogeneous quark-gluon plasma. We show that for large collision energies the induced magnetic fields are governed by the response of quark-gluon plasma along the l…
We study the production and the evolution of quark-gluon plasma expected to be formed in ultra relativistic heavy-ion collisions, within the color flux-tube model. We introduce the gluonic component in the Boltzmann equation, which we solve in the phase space which is extended to include the SU(3) color degree of freedom. The color degree of freedom is shown to play a decisive role in equilibration, and in fixing the temperature of the plasma. We further find that the soft partons that we study here contribute substantially to the bulk properties. Finally, the model is shown to provide a detai…
The quark gluon plasma produced in ultra-relativistic heavy-ion collisions exhibits remarkable features. It behaves like a nearly perfect liquid with a small shear viscosity to entropy density ratio and leads to the quenching of highly energetic particles. We show that both effects can be understood for the first time within one common framework. Employing the parton cascade Boltzmann Approach to Multi-Parton Scatterings (BAMPS), the microscopic interactions and the space-time evolution of the quark gluon plasma are calculated by solving the relativistic Boltzmann equation. Based on cross sect…
We derive the splitting kernels for partons produced in large $Q^2$ scattering processes that subsequently traverse a region of strongly-interacting matter using a recently-developed effective theory \SCETG. We include all corrections beyond the small-$x$ approximation, consistent with the power counting of \SCETG. We demonstrate how medium recoil, geometry and expansion scenarios, and phase space cuts can be implemented numerically for phenomenological applications. For the simplified case of infinite transverse momentum kinematics and a uniform medium, we provide closed-form analytic results…
The study of transport coefficients of strongly interacting matter got impetus after the discovery of perfect fluid ever created at ultrarelativistic heavy ion collision experiments. In this article, we have calculated one such coefficient viz. electrical conductivity of the quark gluon plasma (QGP) phase which exhibits a momentum anisotropy. Relativistic Boltzmann's kinetic equation has been solved in the relaxation-time approximation to obtain the electrical conductivity. We have used the quasiparticle description to define the basic properties of QGP. We have compared our model results with…
Within the framework of Boltzmann transport equation with a Bhatnagar-Gross-Krook (BGK) collisional kernel, we study the wake potential induced by fast partons traveling through the high-temperature QCD plasma which is anisotropic in momentum-space. We calculate the dielectric response function of a collisional anisotropic quark-gluon plasma (AQGP) for small $ξ$ (anisotropic parameter) limit. Using this, the wake potential for various combinations of the anisotropy parameter ($ξ$) and the collision rate ($ν$) is evaluated both for parallel and perpendicular directions of motion of the fast par…
Emergence of thermal recoil jets in high-energy heavy-ion collisions
Institute of Frontier and Interdisciplinary Science, Shandong University（山东大学前沿交叉科学研究院）；Department of Modern Physics, University of Science and Technology of China（中国科学技术大学近代物理系）；School of Physics, Southeast University（东南大学物理学院）；Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University（华中师范大学粒子物理与夸克轻子物理教育部重点实验室）
It is shown that the recently developed hybrid code VISHNU, which couples a relativistic viscous fluid dynamical description of the quark-gluon plasma (QGP) with a microscopic Boltzmann cascade for the late hadronic rescattering stage, yields an excellent description of charged and identified hadron spectra and elliptic flow measured in 200 A GeV Au+Au collisions at the Relativistic Heavy-Ion Collider (RHIC). Using initial conditions that incorporate event-by-event fluctuations in the initial shape and orientation of the collision fireball and values eta/s for the specific shear viscosity of t…
We estimate here the electrical and Hall conductivity using a quasiparticle approach for quark matter. We use a Boltzmann kinetic approach in presence of external magnetic field. We confront the results of model calculations with Lattice QCD simulations for vanishing magnetic field. In general electrical conductivity decreases with magnetic field. The Hall conductivity on the other hand can show a non monotonic behaviour with magnetic field due to an intricate interplay of behaviour of relaxation time and strength of the magnetic field. We argue for vanishing quark chemical potential Hall cond…
We perform the first systematic study of the nonlinear electromagnetic currents induced by external electromagnetic field in quark-gluon plasma, both in the two cases that the inhomogeneity of electromagnetic field is small (large) so that the collision effect is important (negligible). In the former case, we list and classify possible components of the currents in a systematic way, and make an order estimate of each component by using the Boltzmann equation in the relaxation time approximation. In the latter case, we explicitly calculate the quadratic current by using the Vlasov equation, and…
Photons produced in the pre-equilibrium/pre-hydro stage of the quark-gluon plasma produced in relativistic heavy-ion collisions were computed using parton distribution functions obtained from solutions of the Boltzmann equation. The effect of the initial gluon momentum anisotropy $ξ$ and the dependence on the saturation momentum $Q_s$ was investigated. We see that small $Q_s$ results in a photon yield enhancement, whereas a larger $Q_s$ results in a pre-equilibrium photon suppression, owing to the strict constraint of matching to experimental energy density.
We apply the momentum integrated Boltzmann transport equation to study the time evolution of various quark flavours in the central region of ultra-relativistic heavy ion collisions. The effects of thermal masses for quarks and gluons are incorporated to take into account the in-medium properties of these ingredients of the putative quark gluon plasma. We find that even under very optimistic conditions, complete chemical equilibration in the quark gluon plasma appears unlikely.
By coupling the Boltzmann transport equations of both quarkonium and open heavy quarks, we investigate their dynamical evolution inside the quark-gluon plasma and study quarkonium production in heavy ion collisions. The Boltzmann transport equation of quarkonium is derived from the open quantum system formalism and effective field theory of QCD by assuming quarkonium interacts weakly with the plasma. The dissociation and recombination terms in the Boltzmann equation are calculated in potential nonrelativistic QCD. It is shown that the combination of quarkonium dissociation, recombination, open…
We investigate the relativistic equation of state of hadronic matter and quark-gluon plasma at finite temperature and baryon density in the framework of the nonextensive statistical mechanics, characterized by power-law quantum distributions. We study the phase transition from hadronic matter to quark-gluon plasma by requiring the Gibbs conditions on the global conservation of baryon number and electric charge fraction. We show that nonextensive statistical effects play a crucial role in the equation of state and in the formation of mixed phase also for small deviations from the standard Boltz…
Measurement of medium-induced acoplanarity in central Au-Au and pp collisions at $\sqrt{s_{\rm NN}}=200$ GeV using direct-photon+jet and $π^{0}$+jet correlations
A system of equations of anisotropic hydrodynamics that describes mixture of quark and gluon fluids is studied. The equations are based on the zeroth, first, and second moments of the RTA kinetic equations. Tests of this formulation are performed by comparing the results of anisotropic hydorodynamics with the exact solutions of the Boltzmann equations for a mixture of fluids in the Bjorken flow limit. One finds a very good agreement between the hydrodynamic and kinetic-theory results.
Production, elliptic flow and the nuclear modification factor of charm and bottom quarks are studied in central and non-central heavy-ion collisions at RHIC and LHC using the partonic transport model Boltzmann Approach of MultiParton Scatterings (BAMPS). Employing an initial heavy quark yield obtained with PYTHIA the full space-time evolution of charm and bottom quarks in the quark-gluon plasma (QGP) is carried out with BAMPS, taking also secondary production in the QGP into account. Only elastic collisions of heavy quarks with particles from the medium cannot describe the experimentally obser…
We report on a numerical study of the Boltzmann equation including $2\leftrightarrow 2$ scatterings of gluons and quarks in an overoccupied Glasma undergoing longitudinal expansion. We find that when a cascade of gluon number to the infrared occurs, corresponding to an infrared enhancement analogous to a transient Bose-Einstein condensate, gluon distributions qualitatively reproduce the results of classical-statistical simulations for the expanding Glasma. These include key features of the distributions that are not anticipated in the "bottom-up" thermalization scenario. We also find that quar…
We study the properties of the Lindbladian quantum mechanical evolution of quarkonia with non-Abelian charges (color-singlet and octet) in the quark-gluon plasma. We confirm that heavy quark recoils in the Lindblad equation correctly thermalize quarkonium colorful states within statistical errors from the simulation method. We also demonstrate that the Lindblad equation in the dipole limit can provide an efficient alternative method, which is applicable to a finite time evolution before thermalization and dramatically reduces the numerical cost. Our findings will serve as a foundation for larg…
Evolution of spatially anisotropic perturbation created in the system formed after Relativistic Heavy Ion Collisions has been studied. The microscopic evolution of the fluctuations has been examined within the ambit of Boltzmann Transport Equation (BTE) in a hydrodynamically expanding background. The expansion of the background composed of Quark Gluon Plasma (QGP) is treated within the framework of relativistic hydrodynamics. Spatial anisotropic fluctuations with different geometry have been evolved through Boltzmann equation. It is observed that the trace of such fluctuation survive the evolu…
Gribov quantization is a method to improve the infrared dynamics of Yang-Mills theory. We study the thermodynamics and transport properties of a plasma consisting of gluons whose propagator is improved by the Gribov prescription. We first construct thermodynamics of Gribov plasma using the gauge invariant Gribov dispersion relation for interacting gluons. When the Gribov parameter in the dispersion relation is temperature dependent, one expects a mean field correction to the Boltzmann equation. We formulate covariant kinetic theory for the Gribov plasma and determine the mean-field contributio…
We derive the in-medium gluon radiation spectrum off a quark within the path integral formalism at finite energies, including all next-to-eikonal corrections in the propagators of quarks and gluons. Results are computed for finite formation times, including interference with vacuum amplitudes. By rewriting the medium averages in a convenient manner we present the spectrum in terms of dipole cross sections and a colour decoherence parameter with the same physical origin as that found in previous studies of the antenna radiation. This factorisation allows us to present a simple physical picture …
Among the key features of hot and dense QCD matter produced in ultra-relativistic heavy-ion collisions at RHIC is its very low shear viscosity, indicative of the properties of a near-ideal fluid, and a large opacity demonstrated by jet energy loss measurements. In this work, we utilize a microscopic transport model based on the Boltzmann equation with quark and gluon degrees of freedom and cross sections calculated from perturbative Quantum Chromodynamics to simulate an ideal Quark-Gluon-Plasma in full thermal and chemical equilibrium. We then use the Kubo formalism to calculate the shear visc…
Based on the constituent quasiparticle model of quark-gluon plasma (QGP), the matrix elements of the density operator and the Wigner function in the color phase space are presented in form of color path integrals over Wiener and SU(3) group Haar measures. Monte Carlo calculations of quark and gluon momentum distributions and spatial pair distribution functions have been carried out for the strongly coupled QGP plasma in thermal equilibrium at zero baryon chemical potential. The Debye screening mass and the running coupling constant have been obtained from the spatial pair distribution function…
We calculate the shear viscosity $η$ to entropy density $s$ ratio $η/s$ of a gluon plasma in kinetic theory including the gg->gg and gg<->ggg processes. Due to the suppressed contribution to $η$ in the gg->gg forward scattering, it is known that the gluon bremsstrahlung gg<->ggg process also contributes at the same order $O(α_{s}^{-2})$ in perturbative QCD. Using the Gunion-Bertsch formula for the gg<->ggg matrix element which is valid for the limit of soft bremsstrahlung, we find that the result is sensitive to whether the same limit is taken for the phase space. Using the exact phase space, …
We derive equations for the time evolution of the reduced density matrix of a collection of heavy quarks and antiquarks immersed in a quark gluon plasma. These equations, in their original form, rely on two approximations: the weak coupling between the heavy quarks and the plasma, the fast response of the plasma to the perturbation caused by the heavy quarks. An additional semi-classical approximation is performed. This allows us to recover results previously obtained for the abelian plasma using the influence functional formalism. In the case of QCD, specific features of the color dynamics ma…
Experimental data measured in sqrt(s)=2.76 TeV Pb+Pb collisions at the LHC show a significant enhancement of events with an unbalanced pair of reconstructed jet momenta in comparison with p+p collisions. This enhancement of momentum imbalance is supposed to be caused by the different momentum loss of the initial back-to-back di-partons by scatterings within the created dense medium. For investigating the underlying partonic momentum loss we employ the on-shell transport model BAMPS (Boltzmann Approach for Multi-Parton Scattering) for full heavy-ion collisions, which numerically solves the 3+1D…
We consider the quarkonium diffusion, dissociation and recombination inside quark-gluon plasma. We compute scattering amplitudes in potential nonrelativistic QCD for relevant processes. These processes include the gluon absorption/emission at the order $gr$, inelastic scattering at the order $g^2r$ and elastic scattering with medium constituents at the order $g^2r^2$. We show these amplitudes satisfy the Ward identity. We also consider one-loop corrections. The dipole interaction between the color singlet and octet is not running at the one-loop level. Interference between the tree-level gluon…
We use an expansion in angular mode functions in order to solve the Boltzmann equation for a gluon plasma undergoing longitudinal expansion. By comparing with the exact solution obtained numerically by other means we show that the expansion in mode functions converges rapidly for all cases of practical interest, and represents a substantial gain in numerical effort as compared to more standard methods. We contrast the cases of a non expanding plasma and of longitudinally expanding plasmas, and follow in both cases the evolutions towards thermalization. In the latter case, we observe that, alth…
