Hydrodynamic Approaches in Relativistic Heavy Ion Reactions

We review several facets of the hydrodynamic description of the relativistic heavy ion collisions, starting from the historical motivation to the present understandings of the observed collective aspects of experimental data, especially those of the most recent RHIC and LHC results. In this report, we particularly focus on the conceptual questions and the physical foundations of the validity of the hydrodynamic approach itself. We also discuss recent efforts to clarify some of the points in this direction, such as the various forms of derivations of relativistic hydrodynamics together with the limitations intrinsic to the traditional approaches, variational approaches, known analytic solutions for special cases, and several new theoretical developments. Throughout this review, we stress the role of course-graining procedure in the hydrodynamic description and discuss its relation to the physical observables through the analysis of a hydrodynamic mapping of a microscopic transport model. Several questions to be answered to clarify the physics of collective phenomena in the relativistic heavy ion collisions are pointed out.

Unified Description of Classical and Quantum Behaviours in a Variational Principle

We give a pedagogical introduction of the stochastic variational method and show that this generalized variational principle describes classical and quantum mechanics in a unified way.

Elastic Amplitudes and Observables in pp Scattering

Using a unified analytic representation for the elastic scattering amplitudes of pp scattering valid for all high energy region, the behavior of observables in the LHC collisions in the range $sqrt{s}$ = 2.76 - 14 TeV is discussed. Similarly to the case of 7 TeV data, the proposed amplitudes give excellent description of the preliminary 8 TeV data. We discuss the expected energy dependence of the observable quantities, and present predictions for the experiments at 2.76, 13 and 14 TeV.

pp Interaction in Extended Air Showers

Applying the recently constructed analytic representation for the pp scattering amplitudes, we present a study of p-air cross sections, with comparison to the data from Extensive Air Shower (EAS) measurements. The amplitudes describe with precision all available accelerator data at ISR, SPS and LHC energies, and its theoretical basis, together with the very smooth energy dependence of parameters controlled by unitarity and dispersion relations, permit reliable extrapolation to higher energies and to asymptotic ranges. The comparison with cosmic ray data is very satisfactory in the whole pp energy interval from 1 to 100 TeV. High energy asymptotic behaviour of cross sections is investigated in view of the geometric scaling property of the amplitudes. The amplitudes predict that the proton does not behave as a black disk even at asymptotically high enegies, and we discuss possible non-trivial consequences of this fact for pA collision cross sections at higher energies.

pp Elastic Scattering at LHC Energies

Using a unified analytic representation for the elastic scattering amplitudes of pp scattering valid for all energy region, the behavior of observables in the LHC collisions in the range $sqrt{s}$= 2.76 - 14 TeV is discussed. Similarly to the case of 7 TeV data, the proposed amplitudes give excellent description of the preliminary 8 TeV data. We discuss the expected energy dependence of the observable quantities, and present predictions for the experiments at 2.76, 13 and 14 TeV.

pp Interaction at Very High Energies in Cosmic Ray Experiments

An analysis of p-air cross section data from Extensive Air Shower (EAS) measurements is presented, based on an analytical representation of the pp scattering amplitudes that describes with high precision all available accelerator data at ISR, SPS and LHC energies. The theoretical basis of the representation, together with the very smooth energy dependence of parameters controlled by unitarity and dispersion relations, permits reliable extrapolation to high energy cosmic ray and asymptotic energy ranges. Calculations of the p-air production cross section based on Glauber formalism are made using the input values of the pp forward scattering parameters at high energies, with attention given to the independence of the real and imaginary slope parameters. The influence of contributions of diffractive intermediate states, according to Good-Walker formalism, is examined. The comparison with cosmic ray data is very satisfactory in the whole pp energy interval from 1 to 100 TeV. High energy asymptotic behavior of p-air cross sections is investigated in view of the geometric scaling property of the pp amplitudes. The observed energy dependence of the ratio between p-air and pp cross sections in the data is shown to be related to the nature of the pp cross section at high energies, that does not agree with the black disk image.

Effects of initial state fluctuations in the final state elliptic flow measurements using the NeXSPheRIO model

We present a systematic study of the effects due to initial condition fluctuations in systems formed by heavy-ion collisions using the hydrodynamical simulation code NeXSPheRIO. The study was based on a sample of events generated simulating Au+Au collisions at center of mass energy of 200 GeV per nucleon pair with impact parameter ranging from most central to peripheral collisions. The capability of the NeXSPheRIO code to control and save the initial condition (IC) as well as the final state particles after the 3D hydrodynamical evolution allows for the investigation of the sensitivity of the experimental observables to the characteristics of the early IC. Comparisons of results from simulated events generated using fluctuating initial conditions and smooth initial condition are presented for the experimental observable elliptic flow parameter ($v_2$) as a function of the transverse momentum, $p_t$, and centrality. We compare $v_2$ values estimated using different methods, and how each method responds to effects of fluctuations in the initial condition. Finally, we quantify the flow fluctuations and compare to the fluctuations of the initial eccentricity of the energy density distribution in the transverse plane.

Bulk Viscosity and Relaxation Time of Causal Dissipative Relativistic Fluid Dynamics

The microscopic formulae of the bulk viscosity $zeta $ and the corresponding relaxation time $tau_{Pi}$ in causal dissipative relativistic fluid dynamics are derived by using the projection operator method. In applying these formulae to the pionic fluid, we find that the renormalizable energy-momentum tensor should be employed to obtain consistent results. In the leading order approximation in the chiral perturbation theory, the relaxation time is enhanced near the QCD phase transition and $tau_{Pi}$ and $zeta $ are related as $tau_{Pi}=zeta /[beta {(1/3-c_{s}^{2})(epsilon +P)-2(epsilon -3P)/9}]$, where $epsilon $, $P$ and $c_{s}$ are the energy density, pressure and velocity of sound, respectively. The predicted $zeta $ and $% tau_{Pi}$ should satisfy the so-called causality condition. We compare our result with the results of the kinetic calculation by Israel and Stewart and the string theory, and confirm that all the three approaches are consistent with the causality condition.