научная статья по теме QUARK–GLUON PLASMA (SELECTED TOPICS) Физика

Текст научной статьи на тему «QUARK–GLUON PLASMA (SELECTED TOPICS)»

ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 9, с. 1212-1239

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

QUARK-GLUON PLASMA (SELECTED TOPICS)

©2012 V. I. Zakharov*

Institute for Theoretical and Experimental Physics, Moscow, Russia Received November 15, 2011

Introductory lectures to the theory of (strongly interacting) quark—gluon plasma given at the Winter School of Physics of ITEP (Moscow, February 2010). We emphasize theoretical issues highlighted by the discovery of the low viscosity of the plasma. The topics include relativistic hydrodynamics, manifestations of chiral anomaly in hydrodynamics, superfluidity, relativistic superfluid hydrodynamics, effective stringy scalars, holographic models of Yang—Mills theories.

1. INTRODUCTION

1.1. Discovery of "Strongly Interacting Quark—Gluon Plasma "

The discovery of what we call the quark— gluon plasma (for a review see [1]) was announced in the Year 2005 by the American Institute of Physics. The discovery was made in experiments performed at RHIC (Relativistic Heavy Ion Collider) at Brookhaven. At this collider, ions of Au are accelerated to the energy of about 100 GeV per nucleon in the center-of-mass system.

The discovery undoubtedly is the most important observation in the field of elementary particles made in the last decade. What is unusual about the discovery, it was made in terms of differential equations, which are pure classical, while most of us got used to the idea that elementary particles belong to the realm of quantum physics. The differential equations in point are relativistic generalizations of the Navier— Stokes equations of hydrodynamics. The formula of the discovery is that the liquid with the lowest value of the viscosity known in Nature has been observed at RHIC (with no explicit mention of quarks or gluons).

The change of the language implies revisiting of our education and self-education programs. To put it simply, most of us have forgotten what the hydrodynamics means, at least on the professional level. Moreover, in view of the renewed interest the same relativistic hydrodynamics has been developing fast. In particular, the relation of the relativistic hydrodynamics to field theory has been reanalyzed, see in particular [2] and references therein.

Nowadays theory of the quark—gluon plasma is a vast field. These lectures are of introductory nature. But even defining the lectures this way does not fix their content yet. Thus, we choose to concentrate

on topics related to low viscosity of the plasma. Actually, we try to focus on key points by reducing the properties of plasma to a few basic features, for the sake of theoretical discussion. Later we lecture on the theoretical background for each of the "basic points" which we select.

1.2. Basic Properties of Plasma

It might be useful (for the purpose of model building) to reduce the plasma properties to three points, namely, equation of state, viscosity and the role of quantum effects [3].

A. Existence of the plasma was conjectured long time ago. Moreover, the equation of state of the plasma has been known also since long since it was established via numerical experiments within the lattice formulation of QCD, for references see [1]. It turns out that the equation of state is close to that of the ideal gas of quarks and gluons:

V'(T)]plasma ~ [((T)]idealgas (1 - S)

(i)

where the correction 5 & 0.15, e(T) is the energy density as function of temperature and [e(T)]ideaigas is the energy density for noninteracting quarks and gluons.

Thus, the equation of state indicates that the plasma is close to an ideal gas.

B. The observation (1) produces illusion of simplicity of the properties of the plasma. However, analysis of the data obtained at RHIC led to the conclusion that the plasma possesses the lowest viscosity n1 among all the substances known so far:

S / plasma 4n '

(2)

E-mail: vzakharov@itep.ru

''The viscosity n is defined in terms of the force one needs to apply to move through a liquid a thin layer of area A in the direction x: Fx = nAdvx/dy, where the coordinate y is orthogonal to x.

where s is the entropy density (introduced to measure the viscosity in dimensionless units). The value of 1/4n is somewhat symbolical and is quoted for the purpose of memorizing the data. The actual value of n might be larger, say n/s ~ 0.4 [1] or even lower, see [4]. The value n = 1/4n represents the conjectured lower limit [5].

Anyhow, the viscosity observed for the plasma is the lowest one among all the known liquids [1]. Thus, measurements of the viscosity indicate that the plasma is close to an ideal liquid (which is defined as having n = 0). Note that for the ideal gas the viscosity tends to infinity

Ü)

S J ideal gas

oo.

(3)

More precisely, this ratio is inverse proportional to the coupling constant squared n/s ~ l/a2.

C. As a kind of variation of the point B, one argues [5] that such a low value of viscosity implies that the quantum effects are crucial and the liquid cannot be, rigorously speaking, treated classically. Indeed, basing on the estimates common to kinetics

one readily finds that n

- ~ Relaxation (e/^),

where kB is the Boltzmann constant, t is the relaxation time, e is the energy density, and n is the density of particles. From the uncertainty principle, the product of energy of a particle e/n times its free time t cannot be smaller that the Planck constant. Thus:

V Relaxation

- CNJ - t

S Rjuantum

(4)

viscosity? Let us imagine that we are dealing with a two-component substance. One of the components occupies larger phase space c\, and is responsible for the equation of state. The other one has smaller phase space c2, but very small viscosity. Then the total viscosity can be small since, at least naively, to evaluate the total viscosity one adds inverse power of the partial viscosities:

1 _ ci | C2

»Ttot m m ''

(5)

where are normalized by c\ + c2 = 1. Indeed, the meaning of the viscosity n is similar to that of resistance and, if we have two independent motions, then we would apply the rule (5)2).

Thus, the two-component model accommodates naturally points A, B above. Assuming one of the components be superfluid explains, as a bonus, the point C as well.

Another point is worth emphasizing. In the non-relativistic case the superfluid component evaporates at finite temperature Tc. The physics behind is readily understood. Indeed, at T = 0 the superfluid component is related to the condensate of particles with momentum p = 0. At nonvanishing temperature particles are excited by the temperature. Because of the conservation of the number of particles, valid in the nonrelativistic case, the whole of the superfluid component is evaporated at a finite temperature.

In the relativistic case, that is in the absence of conservation of the particles, the theoretical constraints on the phase space occupied by the superfluid component are weaker. Indeed, even at T ^^ the nonperturbative component in case of Yang—Mills theories vanishes only logarithmically:

where the "quantum time" Tquantum ~ h/kB T. Then the observation (2) implies quantum nature of the quark—gluon plasma.

It is a challenge to theory to explain all the three observations (1), (2), (4) which are apparently showing in the opposite directions. Indeed, one starts with the idea that the plasma is an ideal gas and ends up with a kind of a proof that the plasma is a quantum liquid.

1.3. Two-Component Model of Plasma

It is amusing therefore that it is quite straightforward to suggest a model which allows — at least on a qualitative level — to unify all the basic features of the plasma [3]. We have in mind the two-component model of superfluidity as formulated by L.D. Landau.

Indeed, what is "special" about the viscosity? How is it possible to have the equation of state close to that of the ideal gas and, still, nearly vanishing

T

lim C2(T) -g6s(T)

1

(ln T )3

(6)

where g2s(T ) is the coupling of the original 4dtheory3).

1.4. Outline of the Lectures

The lectures address the same issues as mentioned in points A, B, C above but in more detail. According to the standard textbooks of theoretical physics these

2)Equation (5) can be found in old books on classical solutions [6]. In more modern terms, the example of the superfluidity itself might serve as the best illustration to (5). Indeed, the superfluid fraction can be small while the whole liquid is superfluid. On more detailed level, some care should be exercized since one has to distinguish between viscosity with respect to a capillar motion and with respect to rotations, for a recent exposition see [7].

3)Equation (6) does not imply of course that the superfluid

component is necessarily there. Equation (6) refers actually

to any nonperturbative contribution.

r^j

issues belong to different volumes and are not exposed in a relatively compact course. This is the main aim of the lectures: to provide background and introduction to (at least to some of) Sections of theoretical physics which are off the standard list of topics of interest of the reader. The price is that for an expert in any particular field the presentation might be too trivial and incomplete.

In Section 2 we consider relativistic hydrodynamics. In Section 3 we review relation between the hydrodynamics used to describe the plasma and the parton model. In Section 4 we turn to one of the most advanced topics in hydrodynamics, that is chiral effects in plasma. Section 5 is about nonrelativistic superfluidity. In Section 6 we discuss relation between superfluidity and field theory, with emphasize of the role of physics of scalar fields. Section 7, 8 are devoted to physics of effective scalar fields within Yang—Mills theories. An old example of the thermal scalar is the central point of Section 7. Holographic models are introduced in Section 8.

A touch of originality, we hope, is given by our emphasize of the two-component liquid model of the plasma. This model is far from being proven and we use it mo

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