Analytic continuation of functional renormalization group equations

Functional renormalization group equations are analytically continued from imaginary Matsubara frequencies to the real frequency axis. On the example of a scalar field with O(N) symmetry we discuss the analytic structure of the flowing action and show how it is possible to derive and solve flow equations for real-time properties such as propagator residues and particle decay widths. The formalism conserves space-time symmetries such as Lorentz or Galilei invariance and allows for improved, self-consistent approximations in terms of derivative expansions in Minkowski space.Comment: 32 pages, 5 figures, published versio

Few-body hierarchy in non-relativistic functional renormalization group equations and a decoupling theorem

For non-relativistic quantum field theory in the few-body limit with instantaneous interactions it is shown within the functional renormalization group formalism that propagators are not renormalized and that the renormalization group equations of one-particle irreducible vertex functions are governed by a hierarchical structure. This hierarchy allows to solve the equations in the n-body sector without knowledge or assumptions about the m-body sectors where m>n.Comment: 9 pages, 8 figures, published versio

Mode-by-mode hydrodynamics: ideas and concepts

The main ideas, technical concepts and perspectives for a mode resolved description of the hydrodynamical regime of relativistic heavy ion collisions are discussed. A background-fluctuation splitting and a Bessel-Fourier expansion for the fluctuating part of the hydrodynamical fields allows for a complete characterization of initial conditions, the fluid dynamical propagation of single modes, the study of interaction effects between modes, the determination of the associated particle spectra and the generalization of the whole program to event-by-event correlations and probability distributions.Comment: 6 pages, 1 figure, Plenary contribution to the International Conference on the Initial Stages of High-Energy Nuclear Collisions 2013 (IS2013

Variational principle for theories with dissipation from analytic continuation

The analytic continuation from the Euclidean domain to real space of the one-particle irreducible quantum effective action is discussed in the context of generalized local equilibrium states. Discontinuous terms associated with dissipative behavior are parametrized in terms of a conveniently defined sign operator. A generalized variational principle is then formulated, which allows to obtain causal and real dissipative equations of motion from the analytically continued quantum effective action. Differential equations derived from the implications of general covariance determine the space-time evolution of the temperature and fluid velocity fields and allow for a discussion of entropy production including a local form of the second law of thermodynamics.Comment: 34 pages, improved discussion of dissipative terms in energy momentum tensor, references adde

Isotropization from Color Field Condensate in heavy ion collisions

The expanding fireball shortly after a heavy ion collision may be qualitatively described by a condensate of color fields or gluons which is analogous to Bose-Einstein-condensation for massive bosonic particles. This condensate is a transient non-equilibrium phenomenon and breaks Lorentz-boost symmetry. The dynamics of color field condensates involves collective excitations and is rather different from the perturbative scattering of gluons. In particular, it provides for an efficient mechanism to render the local pressure approximately isotropic after a short time of 0.2 fm/c. We suggest that an isotropic color field condensate may play a central role for a simple description of prethermalization and isotropization in the early stages of the collision.Comment: 30 pages, 11 figures, published versio

Fluid dynamic propagation of initial baryon number perturbations on a Bjorken flow background

Baryon number density perturbations offer a possible route to experimentally measure baryon number susceptibilities and heat conductivity of the quark gluon plasma. We study the fluid dynamical evolution of local and event-by-event fluctuations of baryon number density, flow velocity and energy density on top of a (generalized) Bjorken expansion. To that end we use a background-fluctuation splitting and a Bessel-Fourier decomposition for the fluctuating part of the fluid dynamical fields with respect to the azimuthal angle, the radius in the transverse plane and rapidity. We examine how the time evolution of linear perturbations depends on the equation of state as well as on shear viscosity, bulk viscosity and heat conductivity for modes with different azimuthal, radial and rapidity wave numbers. Finally we discuss how this information is accessible to experiments in terms of the transverse and rapidity dependence of correlation functions for baryonic particles in high energy nuclear collisions.Comment: 36 pages, 11 figures; v3: change in the format of the latex file. Minor changes in the text. Typos corrected and updated references. Tweak in Fig. 1-11. Accepted for publication in Phys. Rev.

Causality of fluid dynamics for high-energy nuclear collisions

Dissipative relativistic fluid dynamics is not always causal and can favor superluminal signal propagation under certain circumstances. On the other hand, high-energy nuclear collisions have a microscopic description in terms of QCD and are expected to follow the causality principle of special relativity. We discuss under which conditions the fluid evolutions for a radial expansion are hyperbolic and how the properties of the solutions are encoded in the associated characteristic curves. The expansion dynamics is causal in relativistic sense if the characteristic velocities are smaller than the speed of light. We obtain a concrete inequality from this constraint and discuss how it can be violated for certain initial conditions. We argue that causality poses a bound to the applicability of relativistic fluid dynamics. }Comment: 23 pages, 13 figures; Added references, corrected typos, added discussion as section 2, results unchange

Characterization of initial fluctuations for the hydrodynamical description of heavy ion collisions

Event-by-event fluctuations in the initial conditions for a hydrodynamical description of heavy-ion collisions are characterized. We propose a Bessel-Fourier decomposition with respect to the azimuthal angle, the radius in the transverse plane and rapidity. This allows for a complete characterization of fluctuations in all hydrodynamical fields including energy density, pressure, fluid velocity, shear stress and bulk viscous pressure. It has the advantage that fluctuations can be ordered with respect to their wave length and that they can be propagated mode-by-mode within the hydrodynamical formalism. Event ensembles can then be characterized in terms of a functional probability distribution. For the event ensemble of a Monte Carlo Glauber model, we provide evidence that the latter is close to Gaussian form, thus allowing for a particularly simple characterization of the event distribution.Comment: 40 pages, 16 figure

Statistics of initial density perturbations in heavy ion collisions and their fluid dynamic response

An interesting opportunity to determine thermodynamic and transport properties in more detail is to identify generic statistical properties of initial density perturbations. Here we study event-by-event fluctuations in terms of correlation functions for two models that can be solved analytically. The first assumes Gaussian fluctuations around a distribution that is fixed by the collision geometry but leads to non-Gaussian features after averaging over the reaction plane orientation at non-zero impact parameter. In this context, we derive a three-parameter extension of the commonly used Bessel-Gaussian event-by-event distribution of harmonic flow coefficients. Secondly, we study a model of N independent point sources for which connected n-point correlation functions of initial perturbations scale like 1/N^(n-1). This scaling is violated for non-central collisions in a way that can be characterized by its impact parameter dependence. We discuss to what extent these are generic properties that can be expected to hold for any model of initial conditions, and how this can improve the fluid dynamical analysis of heavy ion collisions.Comment: 35 pages, 5 figures, published versio