Kinetics of hydrodynamic pions in chiral perturbation theory
Juan M. Torres-Rincon, Derek Teaney
备注：
30.5 pages, 7 figures. Generalization to arbitrary number of flavors, by considering pion-pion scattering in SU(N) instead of SU(2). Explanations and extra references added. Accepted for publication in Physical Review D journal
We determine the kinetic coefficients of ultrasoft pions using chiral perturbation theory at finite temperature close to the chiral limit. This is used to compute the axial charge diffusion and damping coefficients in the hydrodynamic effective theory for these pion waves. We show that to provide a leading order answer for these coefficients one needs to explore the dynamics of hard, soft, and ultrasoft pion modes, which are represented microscopically by the appropriate kinetic and hydrodynamic descriptions..
The space-time evolution of the (1+1)-dimensional viscous hydrodynamics with an initial quark-gluon plasma (QGP) produced in ultrarelativistic heavy ion collisions is studied numerically. The particle-emitting sources undergo a crossover transition from the QGP to hadronic gas. We take into account a usual shear viscosity for the strongly coupled QGP as well as the bulk viscosity which increases significantly in the crossover region. The two-pion Hanbury-Brown-Twiss (HBT) interferometry for the viscous hydrodynamic sources is performed. The HBT analyses indicate that the viscosity effect on th…
We investigate the transverse momentum dependence of pion HBT radii on the basis of a hydrodynamical model. Recent experimental data show that $R_{\text{out}}/R_{\text{side}} < 1$, which suggests a strong opaqueness of the source. In addition to the opaqueness naturally caused by transverse flow, we introduce an extrinsic opacity by imposing restrictions on the pion emission angle. Comparing the HBT radii obtained from the normal Cooper-Frye prescription and the opaque emission prescription, we find that $R_{\text{out}}/R_{\text{side}}$ is less than unity only for small values of the transvers…
An analytically calculable model has been proposed recently for the simultaneous description of particle spectra and correlations, which is able to explain in a natural manner the $m_t$ scaling of the radius parameters of the Bose--Einstein correlation functions. We present here preliminary results on fitting this model to data sampled by NA44 collaboration in central S + Pb reactions at CERN SPS, which indicate that both pions and kaons freeze-out at a surprisingly low temperature of T approximately 108 MeV.
Using and comparing kinetic theory and second-order Chapman-Enskog hydrodynamics, we study the non-conformal dynamics of a system undergoing Bjorken expansion. We use the concept of `free-streaming fixed lines' for scaled shear and bulk stresses in non-conformal kinetic theory and hydrodynamics, and show that these `fixed lines' behave as early-time attractors and repellors of the evolution. In the conformal limit, the free-streaming fixed lines reduce to the well-known fixed points of conformal Bjorken dynamics. A new fixed point in the free streaming regime is identified which lies at the in…
The kinetic Freeze Out process of a pion gas through a finite layer with time-like normal is considered. The pion gas is described by a Boltzmann gas with elastic collisions among the pions. Within this model, the impact of the in-medium pion mass modification on the Freeze Out process is studied. A marginal change of the Freeze Out variables temperature and flow velocity and an insignificant modification of the frozen out particle distribution function has been found.
In this paper, we continue our phenomenological studies of heavy-ion collisions using 3+1d anisotropic hydrodynamics (aHydro). In previous works, we compared quasiparticle aHydro (aHydroQP) with ALICE 2.76 TeV Pb-Pb and RHIC 200 GeV Au-Au collision results. At both energies, the agreement was quite good between aHydroQP and the experimental data for many observables. In this work, we present comparisons of the Hanbury Brown--Twiss (HBT) radii and their ratios determined using pi+ pi+ pairs produced in 200 GeV Au-Au collisions. We first present comparisons with STAR results for the HBT radii an…
We present the derivation of second-order relativistic viscous hydrodynamics from an effective Boltzmann equation for a system consisting of quasiparticles of a single species. We consider temperature-dependent masses of the quasiparticles and devise a thermodynamically-consistent framework to formulate second-order evolution equations for shear and bulk viscous pressure corrections. The main advantage of this formulation is that one can consistently implement realistic equation of state of the medium within the framework of kinetic theory. Specializing to the case of one-dimensional purely-lo…
We investigate the effect of early chemical freeze-out on radial flow, elliptic flow and HBT radii by using a fully three dimensional hydrodynamic model. When we take account of the early chemical freeze-out, the space-time evolution of temperature in the hadron phase is considerably different from the conventional model in which chemical equilibrium is always assumed. As a result, we find that radial and elliptic flows are suppressed and that the lifetime and the spatial size of the fluid are reduced. We analyze the p_t spectrum, the differential elliptic flow, and the HBT radii at the RHIC e…
The total angular momentum is conserved in the evolution of the Quark-Gluon Plasma (QGP) created in heavy-ion collision, and consists of two sectors: the orbital angular momentum (OAM) caused by kinetic motion, and the spin, an intrinsic degree of freedom of quarks and gluons. Microscopic scattering processes allow the conversion between these two components, hence the spin density could eventually have non-trivial influence on the QGP evolution. A hydrodynamic theory, with the spin polarization effect properly taken into account, is required to quantitatively study the polarization rate for o…
We present comparisons between 3+1D quasiparticle anisotropic hydrodynamics (aHydroQP) predictions for a large set of bulk observables and experimental data collected in 5.02 TeV Pb-Pb collisions. We make aHydroQP predictions for identified hadron spectra, identified hadron average transverse momentum, charged particle multiplicity as a function of pseudorapidity, the kaon-to-pion ($K/π$) and proton-to-pion ($p/π$) ratios, and integrated elliptic flow. We compare to data collected by the ALICE collaboration in 5.02 TeV Pb-Pb collisions. We find that these bulk observables are quite well descri…
Within hydrodynamic approach, we study the Bose-Einstein correlation of identical pions by taking into account both event-by-event fluctuating initial conditions and continuous pion emission during the whole development of the hot and dense matter formed in high-energy collisions. Considerable deviations occur, compared to the usual hydro calculations with smooth initial conditions and a sudden freeze-out on a well defined hypersurface. Comparison with data at RHIC shows that, despite rather rough approximation we used here, this description can give account of the $m_T$ dependence of $R_L$ an…
The kinetic freeze-out for the hydrodynamical description of relativistic heavy ion collisions is discussed using a background-fluctuation splitting of the hydrodynamical fields. For a single event, the particle spectrum, or its logarithm, can be written as the sum of background part that is symmetric with respect to azimuthal rotations and longitudinal boosts and a part containing the contribution of fluctuations or deviations from the background. Using a complete orthonormal basis to characterize the initial state allows one to write the double differential harmonic flow coefficients determi…
We use effective kinetic theory to study the pre-equilibrium dynamics in heavy-ion collisions. We describe the evolution of linearized energy perturbations on top of out-of-equilibrium background to the energy-momentum tensor at a time when hydrodynamics becomes applicable. We apply this description to IP-Glasma initial conditions and find an overall smooth transition to hydrodynamics. In a phenomenologically favorable range of $η/s$ values, early time dynamics can be accurately described in terms of a few functions of a scaled time variable $τT/(η/s)$. Our framework can be readily applied to …
High-energy nuclear collisions produce a nonequilibrium plasma of quarks and gluons which thermalizes and exhibits hydrodynamic flow. There are currently no practical frameworks to connect the early particle production in classical field simulations to the subsequent hydrodynamic evolution. We build such a framework using nonequilibrium Green's functions, calculated in QCD kinetic theory, to propagate the initial energy-momentum tensor to the hydrodynamic phase. We demonstrate that this approach can be easily incorporated into existing hydrodynamic simulations, leading to stronger constraints …
We compute the gradient expansion for anisotropic hydrodynamics. The results are compared with the corresponding expansion of the underlying kinetic-theory model with the collision term treated in the relaxation time approximation. We find that a recent formulation of anisotropic hydrodynamics based on an anisotropic matching principle yields the first three terms of the gradient expansion in agreement with those obtained for the kinetic theory. This gives further support for this particular hydrodynamic model as a good approximation of the kinetic-theory approach. We further find that the gra…
We develop a set of kinetic equations for hydrodynamic fluctuations which are equivalent to nonlinear hydrodynamics with noise. The hydro-kinetic equations can be coupled to existing second order hydrodynamic codes to incorporate the physics of these fluctuations, which become dominant near the critical point. We use the hydro-kinetic equations to calculate the modifications of energy momentum tensor by thermal fluctuations from the earliest moments and at late times in Bjorken expansion. The solution to the kinetic equations precisely determines the coefficient of the first fractional power (…
Anisotropic transverse flow is studied in Pb+Pb and Au+Au collisions at SPS and RHIC energies. The centrality and transverse momentum dependence at midrapidity of the elliptic flow coefficient v_2 is calculated in the hydrodynamic and low density limits. Hydrodynamics is found to agree well with the RHIC data for semicentral collisions up to transverse momenta of 1-1.5 GeV/c, but it considerably overestimates the measured elliptic flow at SPS energies. The low density limit LDL is inconsistent with the measured magnitude of v_2 at RHIC energies and with the shape of its p_t-dependence at both …
In this work, we start with chiral kinetic theory and construct the spin hydrodynamic framework for a chiral spinor system. Using the 14-moment expansion formalism, we obtain the equations of motion of second-order dissipative relativistic fluid dynamics with non-trivial spin polarization density. In a chiral spinor system, the spin alignment effect could be treated in the same framework as the Chiral Vortical Effect (CVE). However, the quantum corrections due to fluid vorticity induce not only CVE terms in the vector/axial charge currents but also corrections to the stress tensor. In this fra…
We examine the space-time structure of the particle-emitting sources with fluctuating initial conditions in smoothed particle hydrodynamics. The two-pion correlation functions of single events for the sources exhibit event-by-event fluctuations. The large event-by-event fluctuations and wide distributions of the error-inverse-weighted fluctuations between the HBT correlation functions of single and mixed events are important features for the sources with the fluctuating initial conditions. The root-mean-square of the weighted fluctuations is a signal to detect the inhomogeneity of the systems …
We are analyzing the hydrodynamics of 200 A GeV S+S collisions using a new approach which tries to quantify the uncertainties arising from the specific implementation of the hydrodynamical model. Based on a previous phenomenological analysis we use the global hydrodynamics model to show that the amount of initial flow, or initial energy density, cannot be determined from the hadronic momentum spectra. We additionally find that almost always a sizeable transverse flow deve- lops, which causes the system to freeze out, thereby limiting the flow velocity in itself. This freeze-out dominance in tu…
Hydrodynamics can be formulated in terms of a perturbative series in derivatives of the temperature, chemical potential, and flow velocity around an equilibrium state. Different formulations for this series have been proposed over the years, which consequently led to the development of various hydrodynamic theories. In this work, we discuss the relativistic generalizations of the perturbative expansions put forward by Chapman and Enskog, and Hilbert, using general matching conditions in kinetic theory. This allows us to describe, in a comprehensive way, how different out-of-equilibrium definit…
We use the entropy production variational method to associate a one particle distribution function to the assumed known energy-momentum and entropy currents describing a relativistic conformal fluid. Assuming a simple form for the collision operator we find this one particle distribution function explicitly, and show that this method of linking the hydro and kinetic description is a non trivial generalization of Grad's ansatz. The resulting constitutive relations are the same as in the conformal dissipative type theories discussed in J. Peralta-Ramos and E. Calzetta, Phys. Rev. D {\bfseries 80…
During the expansion of a heavy ion collision, the system passes close to the $O(4)$ critical point of QCD, and thus the fluctuations of the order parameter $(σ, \vecπ)$ are expected to be enhanced. Our goal is to compute how these enhanced fluctuations modify the transport coefficients of QCD near the pseudo-critical point. We also make a phenomenological estimate for how chiral fluctuations could effect the momentum spectrum of soft pions. We first formulate the appropriate stochastic hydrodynamic equations close to the $O(4)$ critical point. Then, working in mean field, we determine the cor…
Identical-particle (pion and kaon) femtoscopy in Pb$-$Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV with Therminator2 modeled with (3+1)D viscous hydrodynamics
Indian Institute of Technology Bombay（印度理工学院孟买分校）；University of Tsukuba（筑波大学）
利用 (3+1)D 粘性流体动力学和 THERMINATOR 2 代码研究 Pb-Pb 碰撞中介子和 K 介子的费米简缩，发现半径随对偶动量增加而减小，表明强集体性。
We use principal component analysis to study the hydrodynamic attractor in Yang-Mills kinetic theory undergoing the Bjorken expansion with Color Glass Condensate initial conditions. The late time hydrodynamic attractor is characterized by a single principal component determining the overall energy scale. How it is reached is governed by the disappearance of single subleading principal component characterizing deviations of the pressure anisotropy, the screening mass and the scattering rate. We find that for wide range of couplings the approach to the hydrodynamic attractor at late times is wel…
Using the Poisson bracket method, we construct the hydrodynamics of nuclear matter in the chiral limit, which describes the dynamics of all low-energy degrees of freedom, including the fluid-dynamical and pionic ones. The hydrodynamic equations contain, beside five Euler equations of relativistic fluid dynamics, Nf^2-1 second order equations describing propagating pions and Nf^2-1 first order equations describing the advection of the baryonic vector isospin charges. We present hydrodynamic arguments showing that the pion velocity vanishes at the second order phase transition at Nf=2.
Anisotropic-hydrodynamics framework is used to describe a mixture of quark and gluon fluids. The effects of quantum statistics, finite quark mass, and finite baryon number density are taken into account. The results of anisotropic hydrodynamics are compared with exact solutions of the coupled kinetic equations for quarks and gluons in the relaxation time approximation. The overall very good agreement between the hydrodynamic and kinetic-theory results is found.
In a relativistic setting, hydrodynamic calculations which include shear viscosity (which is first order in an expansion in gradients of the flow velocity) are unstable and acausal unless they also include terms to second order in gradients. To date such terms have only been computed in supersymmetric N=4 Super-Yang-Mills theory at infinite coupling. Here we compute these second-order hydrodynamic coefficients in weakly coupled QCD, perturbatively to leading order in the QCD coupling, using kinetic theory. We also compute them in QED and scalar lambda phi^4 theory.
We report part of our recent work on viscous hydrodynamics with consistent phase space distribution $f(x,\p)$ for freeze out. We develop the gradient expansion formalism based on kinetic theory, and with the constraints from the comparison between hydrodynamics and kinetic theory, viscous corrections to $f(x,\p)$ can be consistently determined order by order. Then with the obtained $f(x,\p)$, second order viscous hydrodynamical calculations are carried out for elliptic flow $v_2$.
We derive system of equations describing fluidity of the medium consisting of non-relativistic particles with finite mass-widths. For that we use expressions for the kinetic Noether 4-current and the Noether energy-momentum tensor being conserved provided one uses self-consistent approximations to the gradient expanded Kadanoff-Baym equations. Kinetic coefficients entering equations of non-ideal hydrodynamics of resonances are obtained in terms of the real and imaginary parts of the self-energies within a relaxation time approximation.
We analyze the evolution of hydrodynamic fluctuations for QCD matter below $T_c$ in the chiral limit, where the pions (the Goldstone modes) must be treated as additional non-abelian superfluid degrees of freedom, reflecting the broken $SU_L(2) \times SU_R(2)$ symmetry of the theory. In the presence of a finite pion mass $m_π$, the hydrodynamic theory is ordinary hydrodynamics at long distances, and superfluid-like at short distances. The presence of the superfluid degrees of freedom then gives specific contributions to the bulk viscosity, the shear viscosity, and diffusion coefficients of the …
Based on the (3+1)-dimensional hydrodynamic model, the space-time evolution of hot and dense nuclear matter produced in non-central relativistic heavy-ion collisions is discussed. The elliptic flow parameter v_2 is obtained by Fourier analysis of the azimuthal distribution of pions and protons which are emitted from the freeze-out hypersurface. As a function of rapidity, the pion and proton elliptic flow parameters both have a peak at midrapidity.
The evaluation of hydrodynamic transport coefficients in relativistic field theory, and the emergence of an effective kinetic theory description, is examined. Even in a weakly-coupled scalar field theory, interesting subtleties arise at high temperatures where thermal renormalization effects are important. In this domain, a kinetic theory description in terms of the fundamental particles ceases to be valid, but one may derive an effective kinetic theory describing excitations with temperature dependent properties. While the shear viscosity depends on the elastic scattering of typical excitatio…
Kinetic Freeze-Out Properties from Transverse Momentum Spectra of kaon, Pion, and (anti-)proton production in U+U collisions at $\sqrt{s_{NN}}$ = 193 GeV
How physical systems approach hydrodynamic behavior is governed by the decay of nonhydrodynamic modes. Here, we start from a relativistic kinetic theory that encodes relaxation mechanisms governed by different timescales thus sharing essential features of generic weakly coupled nonequilib- rium systems. By analytically solving for the retarded correlation functions, we clarify how branch cuts arise generically from noncollective particle excitations, how they interface with poles arising from collective hydrodynamic excitations, and to what extent the appearance of poles remains at best an amb…
Starting from Boltzmann equation with relaxation time approximation for the collision term and using Chapman-Enskog like expansion for distribution function close to equilibrium, we derive hydrodynamic evolution equations for the dissipative quantities directly from their definition. Although the form of the equations is identical to those obtained in traditional Israel-Stewart approaches employing Grad's 14-moment approximation and second moment of Boltzmann equation, the coefficients obtained are different. In the case of one-dimensional scaling expansion, we demonstrate that our results are…
We study deuteron production using no-coalescence hydrodynamic + transport simulations of central AuAu collisions at $\sqrt{s_{NN}} = 7 - 200$ GeV. Deuterons are sampled thermally at the transition from hydrodynamics to transport, and interact in transport dominantly via $πp n \leftrightarrow πd$ reactions. The measured proton, Lambda, and deuteron transverse momentum spectra and yields are reproduced well for all collision energies considered. We further provide a possible explanation for the measured minimum in the energy dependence of the coalescence parameter, $B_2(\sqrt{s_{NN}})$ as well …
Transverse momentum spectra of negative and positive pions produced at mid-(pseudo)rapidity in inelastic or non-single-diffractive proton-proton collisions over a center-of-mass energy, $\sqrt{s}$, range from a few GeV to above 10 TeV are analyzed by the blast-wave fit with Boltzmann (Tsallis) distribution. The blast-wave fit results are well fitting to the experimental data measured by several collaborations. In a particular superposition with Hagedorn function, both the excitation functions of kinetic freeze-out temperature ($T_0$) of emission source and transverse flow velocity ($β_T$) of p…
We develop further a kinetic theory of strings and derive a transport equation for a network of cosmic strings with Nambu-Goto evolution, interactions and background gravitational effects taken into account. We prove an H-theorem and obtain necessary and sufficient conditions for a thermodynamic equilibrium. At the lowest order the equilibrium is estimated by the von Mises-Fisher distributions parametrized by mean directions and dispersions of the right- and left-moving tangent vectors. Under assumption of a local equilibrium we derive a complete set of hydrodynamic equations that govern the e…
