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PITT-97-447; CMU-HEP-97-14; DOE-ER/40682-139;LPTHE-97-

NON-EQUILIBRIUM EVOLUTION OF A `TSUNAMI':

DYNAMICAL SYMMETRY BREAKING

(a) (b) (c)

Daniel Boyanovsky , Hector J. de Vega , Richard Holman ,

S. Prem Kumar , Rob ert D. Pisarski

(a) Department of Physics and Astronomy, University of Pittsburgh, PA. 15260,U.S.A

(b) LPTHE, Universite Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII),

Tour 16, 1 er. etage, 4, Place Jussieu 75252 Paris, Cedex 05, France

(d) Department of Physics, Brookhaven National Laboratory, Upton, NY 11973,

U.S.A.

Abstract

We prop ose to study the non-equilibrium features of heavy-ion collisions by

following the evolution of an initial state with a large numb er of quanta with

a distribution around a momentum jk j corresp onding to a thin spherical shel l

2 2

in momentum space, a `tsunami'. An O (N ) ( ) mo del eld theory in the

large N limit is used as a framework to study the non-p erturbative asp ects of

the non-equilibrium dynamics including a resummation of the e ects of the

medium (the initial particle distribution). In a theory where the symmetry is

sp ontaneously broken in the absence of the medium, when the initial number

of particles p er correlation volume is chosen to b e larger than a critical value

the medium e ects can restore the symmetry of the initial state. We show

that if one b egins with sucha symmetry-restored, non-thermal, initial state,

non-p erturbative e ects automatically induce spino dal instabilities leading to

a dynamical breaking of the symmetry. As a result there is explosive particle

pro duction and a redistribution of the particles towards low momentum due to

the nonlinearity of the dynamics. The asymptotic b ehavior displays the onset

of Bose condensation of pions and the equation of state at long times is that of

an ultrarelativistic gas although the momentum distribution is non-thermal.

11.10.-z,11.30.Qc,1 1.1 5. Tk

Typ eset using REVT X

E 1

I. INTRODUCTION

The Relativistic Heavy Ion Collider (RHIC) at Bro okhaven and the Large Hadron Collider

(LHC) at CERN will provide an unprecedented range of energies and luminosities that will

hop efully prob e the Quark-Gluon Plasma and Chiral Phase transitions. The basic picture of

the ion-ion collisions in the energy ranges prob ed by these accelerators as seen in the center-

of-mass frame (c.m.), is that of two highly Lorentz-contracted `pancakes' colliding and leaving

a `hot' region at mid-rapidity with a high multiplicity of secondaries [1]. At RHIC for Au + Au

26 2

central collisions with typical luminosityof10 =cm :s and c.m. energy 200GeV=n n,a

multiplicity of 500-1500 particles p er unit rapidity in the central rapidity region is exp ected

27 2

[2{4]. At LHC for head on Pb + Pb collisions with luminosity 10 =cm :s at c.m. energy

5TeV=n n, the multiplicityofcharged secondaries will b e in the range 2000 8000 p er

unit rapidity in the central region [4]. At RHIC and LHC typical estimates [1{6] of energy

densities and temp eratures near the central rapidity region are " 1 10 GeV=fm ;T

300 900 MeV .

Since the lattice estimates [4,6] of the transition temp eratures in QCD, b oth for the QGP

and Chiral phase transitions are T 160 200 MeV, after the collision the central region

will b e at a temp erature T >T . In the usual dynamical scenario that one [1] envisages, the

initial state co ols o via hydro dynamic expansion through the phase transition down to a

freeze-out temp erature, estimated to be T 100 MeV [7], at which the mean free-path of

the hadrons is comparable to the size of the expanding system.

The initial state after the collision is strongly out of equilibrium and there are very few

quantitative mo dels to study its subsequent evolution. There are p erturbative and non-

p erturbative phenomena that contribute to the pro cesses of thermalization and hadroniza-

tion. The p erturbative (hard and semihard) asp ects are studied via parton cascade mo dels

which assume that at large energies the nuclei can b e resolved into their partonic constituents

and the dynamical evolution can therefore be tracked by following the parton distribution

functions through the p erturbative parton-parton interactions [8{12]. Parton cascade mo dels

(including screening corrections to the QCD parton-parton cross sections) predict that ther-

malization o ccurs on time scales 0:5fm=c [13]. After thermalization, and provided that the

mean-free path is much shorter than the typical interparticle separation, further evolution

of the plasma can be describ ed with b o ost-invariant relativistic hydro dynamics [1,14]. The

details of the dynamical evolution between the parton cascade through hadronization, and

eventual description via hydro dynamics is far from clear but will require a non-p erturbative

treatment. The non-p erturbative asp ects of particle pro duction and hadronization typically

envisage a ux-tub e of strong color-electric elds, in which the eld energy leads to pro duc-

tion of qq pairs [15,16]. Recently the phenomenon of pair pro duction from strong electric

elds in b o ost-invariant co ordinates was studied via non-p erturbative metho ds that address

the non-equilibrium asp ects and allow a comparison with hydro dynamics [17].

The dynamics near the phase transition is even less understo o d and involves physics be-

yond the realm of p erturbative metho ds. For instance, considerable interest has b een sparked

recently by the p ossibility that disoriented chiral condensates (DCC's) could form during the 2

evolution of the QCD plasma through the chiral phase transition [18]- [23]. Ra jagopal and

Wilczek [24] have argued that if the chiral phase transition o ccurs strongly out of equilibrium,

spino dal instabilities [25] could lead to the formation and relaxation of large pion domains.

This phenomenon could provide a striking exp erimental signature of the chiral phase transi-

tion and could provide an explanation for the Centauro and anti-Centauro (JACEE) cosmic

ray events [26]. An exp erimental program is underway at Fermilab to search for candidate

events [27,28]. Most of the theoretical studies of the dynamics of the chiral phase transition

and the p ossibility of formation of DCC's have fo cused on initial states that are in lo cal

thermo dynamic equilibrium (LTE) [29]- [32].

We prop ose to study the non-equilibrium asp ects of the dynamical evolution of highly

excited initial states by relaxing the assumption of initial LTE (as would be appropriate

for the initial conditions in a heavy-ion collision). Consider, for example, a situation where

the relevant quantum eld theory is prepared in an initial state with a particle distribution

~ ~ ~

sharply p eaked in momentum space around k and k where k is a particular momentum.

0 0 0

This con guration would b e envisaged to describ e two `pancakes' or `walls' of quanta moving

in opp osite directions with momentum jk j. In the target frame this eld con guration

would b e seen as a `wall' of quanta moving towards the target and hence the name `tsunami'

[33]. Such an initial state is out of equilibrium and under time evolution with the prop er

interacting Hamiltonian, non-linear e ects should result in a redistribution of particles, as

well as particle pro duction and relaxation. The evolution of this strongly out of equilibrium

initial state would b e relevant for understanding phenomena such as formation and relaxation

of chiral condensates. Starting from such a state and following the complete evolution of the

system thereon, is clearly a formidable problem even within the framework of an e ective

eld theory such as the linear -mo del.

In this article we consider an even more simplistic initial condition, where the o ccupation

numb er of particles is sharply lo calized in a thin spherical shel l in momentum space around

a momemtum jk j, i.e. a spherical ly symmetric version of the `tsunami' con guration. The

reason for the simpli cation is purely technical since spherical symmetry can be used to

reduce the number of equations. Although this is a simpli cation of the idealized problem,

it will be seen b elow that the features of the dynamics contain the essential ingredients to

help us gain some understanding of more realistic situations.

4 2

We consider a weakly coupled theory ( 10 ) with the elds in the vector

representation of the O (N ) group. Anticipating non-p erturbative physics, we study the

dynamics consistently in the leading order in the 1=N expansion which will allow an analytic

treatment as well as anumerical analysis of the dynamics.

The pion wall scenario describ ed ab ove is realized by considering an initial state describ ed

by a Gaussian wave functional with a large numb er of particles at jk j and a high densityis

achieved by taking the number of particles per correlation volume to be very large. As in

nite temp erature eld theory, a resummation along the lines of the Braaten and Pisarski

[34] program must b e implemented to takeinto account the non-p erturbative asp ects of the

physics in the dense medium. As will b e explicitly shown b elow, the large N limit in the case

under consideration provides a resummation scheme akin to the hard thermal lo op program 3

[34].

The dynamical evolution of this spherically symmetric \tsunami" con guration describ ed

ab ove reveals many remarkable features: i) In a theory where the symmetry is sp ontaneously

broken in the absence of a medium, when the initial state is the O (N ) symmetric, high densty,

\tsunami" con guration, we nd that there exists a critical density of particles dep ending

on the e ective (HTL-resummed) coupling beyond which spino dal instabilities are induced

leading to a dynamical symmetry breaking. ii) When these instabilities o ccur, there is

profuse pro duction of low-momentum pions (Goldstone b osons) accompanied by a dramatic

re-arrangement of the particle distribution towards low momenta. This distribution is non-

thermal and its asymptotic b ehavior signals the onset of Bose condensation of pions. iii)

The nal equation of state of the \pion gas" asymptotically at long times is ultrarelativistic

despite the non-equilibrium distributions.

The pap er is organized as follows: In Section II we intro duce the mo del under consid-

eration and describ e the non-p erturbative framework, namely the large N approximation.

Section III is devoted to the construction of the wave functional and a detailed description

of the initial conditions for the problem. The dynamical asp ects are covered in Section IV.

We rst outline some issues dealing with renormalization and then provide a qualitative

understanding of the time evolution using wave functional arguments. We argue that the

system could undergo dynamical symmetry breakdown and provide analytic estimates for

the onset of instabilities. We present the results of our numerical calculations in Section

V which con rm the robust features of the analytic estimates for a range of parameters.

In Section VI we analyze the details of symmetry breaking and argue that the long time

dynamics can be interpreted as the onset of formation of a Bose condensate even when the

order parameter vanishes.

Finally in Section VI I we present our conclusions and future avenues of study.

II. THE MODEL

As mentioned in the intro duction we consider a theory with O (N ) symmetry in the

large-N limit with the Lagrangian,

1 m

~ ~ ~ ~ ~ ~

L = (@ ):(@ ) ( ) ( ) (2.1)

2 2 8N

1 2 N 1

~ ~

where isanO(N)vector, =(;~) and ~ represents the N 1 pions, ~ =( ; ; :::; ).

We then shift by its exp ectation value in the non-equilibrium state

p p

(~x; t)= N(t)+ (~x; t) ; h (~x; t)i = N(t) : (2.2)

We refer the interested reader to several articles which discuss the implementation of the

large N limit (see for e.g. [35{39,44,45]). The 1=N series may be generated by intro ducing

an auxiliary eld (x) which is an algebraic function of (x), and then p erforming the

functional integral over (x) using the saddle p oint approximation in the large N limit 4

[35{37]. It can b e shown that the leading order terms in the expansion can b e easily obtained

by the following Hartree factorisation of the quantum elds [38,39,44,45],

4 2 2 3 2

! 6h i + constant ; ! 3h i

2 2 2 2 2

(~ ~ ) ! 2h~ i~ h~ i +O(1=N )

2 2 2 2 2 2 2 2

~ ! ~ h i + h~ i ; ~ !h~ i : (2.3)

All exp ectation values are to b e computed in the non-equilibrium state. In the leading order

large N limit we then obtain,

1 1 N

2 2 2 2 0 2 2

L = ~ (@ + M (t))~ (@ + M (t)) V ((t);t)+ h i ; (2.4)

2 2 8

h i

2 2 2 2

M (t)= m + (t)+h i ; (2.5)

h i

2 2 2 2

M (t)= m + 3 (t)+h i ; (2.6)

0 2 2 2

V ((t);t)= N + [ +h i]+m ; (2.7)

2 2

h i = h~ i=N : (2.8)

This approximation allows us to expand ab out eld con gurations that are far from the

p erturbative vacuum. In particular it is an excellent to ol for studying the b ehaviour of

matter in extreme conditions such as high temp erature or high density [17,35{39,44,45].

One way to obtain the non-equilibrium equations of motion is through the Schwinger-

Keldysh Closed Time Path formalism. This is the usual Feynman path integral de ned on a

complex time contour which allows the computation of in-in exp ectation values as opp osed

to in-out S-matrix elements. For details see the references [40]. The Lagrangian density in

this formalism is given by

~ ~

L = L[ ] L[ ] ; (2.9)

neq

with the elds de ned along the forward (+) and backward () time branches. The

non-equilibrium equations of motion are then obtained by requiring the exp ectation value

of the quantum uctuations in the non-equilibrium state to vanish i.e. from the tadp ole

equations [39],

h i = h~ i =0 : (2.10)

In the leading order approximation of the large N limit, the Lagrangian for the eld is

quadratic plus linear and the tadp ole equation for the leads to the equation of motion for

the order parameter or the zero mo de

2 2 2

+ [ (t)+h i(t)](t)+m (t)=0: (2.11)

2 5

In this leading approximation the non-equilibrium action for the N 1 pions is,

Z Z

4 3 + 2 + +2 + +

d x[L L ]= d xdt ( ~ @ ~ M (t)~ ~ ) (+ !) : (2.12)

We have not written the action for the eld uctuations b ecause they decouple from the

dynamics of the pions in the leading order in the large N limit [38,39].

Having intro duced the mo del and the non-p erturbative approximation scheme the next

step is to construct an appropriate non-equilibrium initial state or density matrix.

Although one could continue the analysis of the dynamics using the Schwinger-Keldysh

metho d, we will study the dynamics in the Schrodinger representation in terms of wave-

functionals b ecause this will display the nature of the quantum states more clearly. We nd

it convenient to work with the Fourier-transformed elds de ned as,

ik~x

~ (~x; t)= (t); (2.13) e ~

where we have chosen to quantize in a box of nite volume V that will b e taken to in nity

at the end of our calculations. The Hamiltonian for the pions is given by

0 1

X X

1 1 N

~ ~

@ A

H = + ! (t)~ ~ h~ ~ i ; (2.14)

~ ~ ~ ~ ~ ~

k k k k k k

2 2 8

~ ~

k k

where

2 2 2

! (t)=k +M (t) (2.15)

is the e ective time dep endent frequency and M (t) is given by Eq.(2.5). To leading order

in the large N limit the theory b ecomes Gaussian and the non-linearities are enco ded in a

self-consistency condition, since the frequency (2.15) dep ends on h~ i and this exp ectation

value is in the time dep endent state, as displayed by the set of equations (2.5-2.8).

III. THE INITIAL STATE

As stated in the intro duction, our ultimate goal is to mo del and study the non-equilibrium

asp ects of the evolution of an initial, highly excited state that relaxes following high energy,

heavy-ion collisions. An idealized description of the asso ciated physics would b e to consider

twowave packets made up of very high energy comp onents representing the heavy ions and

moving with a highly relativistic momentum toward each other. The goal would b e to follow

the dynamical evolution of the wavefunctionals corresp onding to this situation, thus clearly

elucidating the non-equilibrium features involved in the phase transition pro cesses following

the interactions of the wave packets. This initial state could be describ ed by a distribution

~ ~

of particles, sharply p eaked around some sp ecial values k and k in momentum space.

0 0

The evolution of this state then follows from the functional Schrodinger equation. 6

Even with the simpli cation of a scalar eld theory such a program is very ambitious

and b eyond the presentnumerical capabilities. One of the ma jor diculties is that selecting

one particular momentum breaks rotational invariance and the evolution equations dep end

on the direction of wave vectors even in the Gaussian approximation. (This statement will

b ecome clear b elow).

In this article however, wecho ose to study a much simpler description of the initial state

which is characterized by a high density particle distribution in a thin spherical `shell' in

momentum space . We prop ose an initial particle distribution that has supp ort concentrated

at jk j. This particular state do es not provide the necessary geometry for a heavy ion collision,

however it do es describ e a situation in which initially there is a large multiplicity of particles

in a small momentum `shell', there is no sp ecial b eam-axis and the pions are distributed

equally in all directions with a sharp spatial momentum. This is a rotation invariant state

that describ es a highly out of equilibrium situation and that will relax during its time

evolution (a spherical \tsunami").

A. The Wave Functional:

Since in the leading order approximation in the large N expansion the theory has b ecome

Gaussian (at the exp ense of a self-consistency condition), we cho ose a Gaussian ansatz for

the wave-functional at t = 0. The reason for this choice is that up on time evolution this

wave functional will remain Gaussian and will b e identi ed with a squeezed state functional

of pions.

" #

A (0)

(t =0)= N (0) exp ~ ~ : (3.1)

~ ~ ~

k k k

This state will then evolve according to the Hamiltonian given in Eq. (2.14) which is es-

sentially a harmonic oscillator Hamiltonian with self-consistent, time-dep endent frequencies.

The functional Schrodinger equation is given by

i = H : (3.2)

The last term in the Hamiltonian (2.14) which is indep endent of the elds (a time dep endent

`vacuum energy term') can be absorb ed in an overall time dep endent phase of the wave

functional. Removing this term by a phase rede nition, the functional Schrodinger equation

b ecomes

" #

i [ ]= + ! (t) ~ ~ [ ] (3.3)

~ ~

k k

2 ~ ~

~ ~

k k

which then leads to a set of di erential equations for the covariance A . The time dep endence

of the normalization factors N is completely determined by that of the A as a consequence

k k

of unitary time evolution. The state for arbitrary time t takes then the form: 7

" #

A (t)

N (t) exp ~ : (3.4) (t)= ~

~ ~ ~

k k k

The evolution equations for the covariance are obtained by taking the functional deriva-

on b oth sides. We obtain the following evolution equations tives and comparing p owers of

[32,38]

2 2

iA (t)=A (t)! (t); (3.5)

k k

0 0

N (t)=N (0) exp A (t )dt ; (3.6)

k k Ik

with A = A (t)+ iA (t). The equal time two-p oint correlation function in the time

k Rk Ik

evolved non-equilibrium state is given by

~ j > < j ~

~ ~

k k

~ i= h~

~ ~

k k

< j >

h i

A (t)

~ )N exp [D ~ ](~ ~ ~

~ ~

~q ~q q q

k k

h i

A (t)

[D ~ ]N exp ~ ~

~q ~q q q

= ; (3.7)

2A (t)

leading to the self-consistency condition

h i(t)= : (3.8)

2A (t)

Formally, one can also represent these two-p oint equal time correlators in terms of func-

tional integrals over the closed time path contour where the initial state is chosen to be

the Gaussian functional describ ed ab ove. However the explicit and rather simple ansatz

for the wave functional enables one to obtain the two-p oint functions directly in a rather

straightforward manner. Moreover, the wave functional approach will p ermit a much clearer

understanding of the physics of the problem. The Ricatti equation (3.5) can be cast in a

simpler form by writing A in terms of new variables as

(t)

A (t)=i ; (3.9)

(t)

leading to the simple equation for the new variables

+ ! (t) =0: (3.10)

k k k

In terms of these mo de functions we nd that the real and imaginary parts of the covariance

A are given by

k 8

_ _

k k

A (t)= ; (3.11)

2 j j

A (t)= ln j (t)j : (3.12)

Ik k

From the di erential equation for the (t) given by Eq. (3.10) it is clear that the

combination that app ears in the numerator of Eq. (3.11) is the Wronskian of the di er-

ential equations and will consequently b e determined from the initial conditions alone. The

2 2

expression for the quantum uctuations h i = h~ i=N is given by,

Z Z

3 2 3

d k j (t)j d k

(t) (t)i = : (3.13) h i(t)= h

k k

3 3

(2 ) (2 ) 2

The mo de functions have a very simple interpretation: they ob ey the Heisenb erg

equations of motion for the pion elds obtained from the Hamiltonian (2.14). Therefore we

can write the Heisenb erg eld op erators as

h i

1 1

~ ~

ik ~x ik ~x

p p

~ (~x; t)= ~a (t) e + ~a (t) e (3.14)

k k

where ~a ; ~a are the time indep endent annihilation and creation op erators with the usual

Bose commutation relations.

B. Initial Conditions:

Within this Gaussian ansatz for the wave functional, the initial conditions are completely

determined by the initial conditions on the mo de functions (t). In order to physically

motivate the initial conditions we now establish the relation between the particle number

distribution and these mo de functions.

Since, in a time dep endent situation there is an ambiguity in the de nition of the particle

numb er, we de ne the particle number with resp ect to the eigenstates of the instantaneous

Hamiltonian (2.14) at the initial time, i.e.

2 2

1 1 ! (0) 1

n^ = + ~ ~

k k

! (0) 2 ~ ~ 2 2

~ ~

k k

" #

1 (0) 1 ! 1

~ ~

= + ~ ~ : (3.15)

~ ~ ~ ~

k k k k

! (0) 2 2 2

Here, ! (0) is the frequency (2.15) evaluated at t = 0, i.e. the curvature of the p otential

term in the functional Schrodinger equation (3.3) at t = 0 and provides a de nition of the

particle number (assuming that ! (0) > 0). The exp ectation value of the number op erator

in the time evolved state is then 9

2 2

[A (t) ! (0)] + A (t)

Rk k

n (t)=< jn^ j >= (3.16)

k k

4 ! (0) A (t)

k Rk

2 2

(t) + (t)

k k

= ; (3.17)

4[1+ (t)]

where and are de ned through the relations,

k k

A (t)=! (0) [1 + (t)] ; A (t)=! (0) (t) : (3.18)

Rk k k Ik k k

In terms of the mo de functions and the exp ectation value of the number op erator is

k k

given by

h i

1 1

2 2 2

n (t)= j (t)j + ! (0)j (t)j : (3.19)

k k k

4 ! (0) 2

k k

The quantity (t) app ears as the phase of the wave function and will b e chosen to be zero

at t = 0 for simplicity,

A (0) = 0 ; (0) = 0: (3.20)

Ik k

Assuming (0) = 0, the initial conditions on the (0) variables can be obtained at once

k k

from Eq. (3.9) and are found to b e,

(0) = i = i! (0) [1 + (0)] ; (0)=1 ; (3.21)

k k k

k k

where is the Wronskian W [ (t) ; (t)]. Hence the wave functional of the system at

k k k

t =0 can b e sp eci ed completely (up to a phase) by the single function (0).

Using Eq. (3.17) one can easily solve for (0) in terms of the initial particle sp ectrum

(0) = 2[n (0) n (0) + n (0)] : (3.22)

k k k k k

Which of the two solutions will giveusinteresting physics is a more subtle question that we

shall address in the next section when we discuss the dynamics of the problem.

Before moving on to the description of the dynamics let us brie y summarize what

we have done. We prop osed a rather simple description of a large multiplicity, high energy

particle collision pro cess by preparing an initial state with an extremely high numb er density

of particles concentrated at momenta given by jk j = k . Consistent with the leading order in

a large N approximation, wechose a Gaussian ansatz for our wave functional, parametrized

by the variables (t), (t) (or alternatively (t) and (t)) and the initial conditions on

k k k k

these variables are determined completely by the choice of the particle distribution function

n (0) at t = 0 (Eq. (3.22)). The next step will be to obtain the renormalized equations of

motion and then to study the dynamics analytically as well as numerically. 10

IV. THE DYNAMICS

From the discussion in the previous sections we see that the following set of equations

for the order parameter (t) and the mo de functions (t) must be solved self-consistently

in order to study the dynamics:

d (t)

2 2 2

+ m + [ (t)+h i ] (t)= 0; (4.1)

dt 2

d (t)

2 2 2 2

+ k + m + [ (t)+h i ] (t)= 0; (4.2)

B k

dt 2

(0) = 1 ; (0) = i ; (4.3)

k k

with the self-consistent condition

2 3

j (t)j d k

: (4.4) h i(t) =

(2 ) 2

The quantities in the ab ove equations must be renormalized. This is achieved by rst

demanding that all the equations of motion b e nite and then absorbing the divergent pieces

into a rede nition of the mass and coupling constant resp ectively,

2 2 2 2 2 2 2

m + [< >(t) + (t)] = m + [< > (t) + (t)] = M (t) : (4.5)

B R

B R R

2 2

A detailed derivation of the renormalization prescriptions requires a WKB analysis of the

mo de functions (t) that reveals their ultraviolet prop erties. Such an analysis has b een

p erformed elsewhere [38,39]. In summary the mass term will absorb quadratic and log-

arithmic divergences while the coupling constant will acquire a logarithmically divergent

renormalization [38,35]. In particular

( )

3 2

d k j (t)j 1 (k )

2 2

< > (t)= M (t) (4.6) +

3 3

(2 ) 2 2k 4k

with an arbitrary renormalization scale.

Intro ducing the e ective mass of the particles at the initial time as

2 2

M = M (t =0); (4.7)

R R

we recognize that this e ective mass has contributions from the non-equilibrium particle

distribution and is the analog of the hard-thermal lo op (HTL) resummed e ective mass

in a scalar eld theory. Recall, however, that the initial distribution is not thermal. In

a scalar theory, the HTL e ective mass is obtained by summing the daisy and sup erdaisy

diagrams [41] which is precisely the resummation implied in the leading order in the large

N approximation. To see this more clearly consider the case in which the order parameter 11

vanishes, i.e. (t) 0; then the e ective mass at the initial time can be written as a gap

equation

k dk

2 2

M = m + (4.8)

R B

4 2 (0)

" #

Z Z

2 2

k dk k dk (0)

= m + +

2 2

4 2! (0) 4 2! (0) 1+ (0)

k k k

where we have used the relation (0) = ! (0)(1 + ). The second term in the ab ove

k k k

expression is the usual contribution obtained at zero temp erature (and zero density) for the

(self-consistent) renormalized mass parameter M i.e the one lo op tadp ole (with ! (0) =

k + M ). The third term contains the non-equilibrium e ects asso ciated with the particle

distributions and vanishes when n ! 0. This term is nite (since n (0) is assumed to be

k k

lo calized within a small range of momenta). For a given distribution n (0), the solution to

the self-consistent gap equation (4.8) gives the e ective mass, dressed by the medium e ects.

This is indeed very similar to the nite temp erature case in which the tadp ole term provides

a contribution / T in the high temp erature limit. We will see later that a term very

similar to this can b e extracted in the limit in which the distribution n (0) is very large.

Since the relevant scale is the quasiparticle mass M , we will cho ose to take M > 0 to

describ e an initial situation in which the O (N ) symmetry is unbroken.

It is convenient for numerical purp oses to intro duce the following dimensionless quanti-

ties:

k (t)

R k R

q = ; g = ; W = ; = M t; ' ( )=

q R

M 2M 8 M

R R

h i

2 2

h i (t)h i (0) : (4.9) g( )=

R R

In terms of these dimensionless quantities the equations of motion (4.1, 4.2) with the initial

conditions (4.3) b ecome

" #

2 2

+1+' () ' (0) + g ( ) '( )=0; (4.10)

" #

2 2 2

+ q +1+' () ' (0) + g ( ) ( )=0 ; (0) = 1 ; (0) = iW ; (4.11)

q q q q

( !)

2 2

j ( )j 1 (q 1) M ()

g ( )=g q dq + 1 ; (4.12)

W 2q M

where we have chosen the renormalization scale = M for simplicity.

In order to make our statements precise in the analysis of the spherically symmetric

\tsunami", we will assume the initial particle distribution to be Gaussian and p eaked at

some value q and width so that

0 12

2 3

" #

N q q

0 0

4 5

n (0) = exp ; (4.13)

where N is the total numb er of particles in a correlation volume M and I is a normalization

factor . The case N >> 1 corresp onds to the high density regime with many particles in

the e ective correlation volume.

A. Preliminary considerations of the dynamics:

Before engaging in a full numerical solution of the evolution equations, we can obtain

a clear, qualitative understanding of the main features of the evolution by lo oking at the

quantum mechanics of the wave functional.

The dynamics is di erent for the di erent solutions for in Eq. (3.22) and can be

understo o d with simple quantum mechanical arguments:

1. Case I

=2[n (0) + n (0) + n (0)]:

q q q

In this case

jq q j

(0) 4 n (0) >> 1 for 1

q q

jq q j

0 for >> 1; (4.14)

and the covariance of the wave functional (3.1) is given by

jq q j

A (0) = (0) = ! (1+ ) >> ! for 1

q q q q q

jq q j

! for >> 1: (4.15)

Now, for each wave vector ~q we have a Gaussian wave-function which is the ground

state of a harmonic oscillator with frequency (0), but whose evolution is determined by

a Hamiltonian for a harmonic oscillator of frequency ! (0) at very early times. For the

mo des q such that >> ! (0), the wave function is very narrow compared to the second

q q

derivative of the p otential and there is a very small probability for sampling large amplitude

eld con gurations (i.e. large ). It is a prop erty of these squeezed states that whereas the

wave-functional is narrowly lo calized in eld space, it is a wide distribution in the canonical

momentum (conjugate to the eld) basis. This wave function will spread out under time

evolution to obtain a width compatible with the frequency ! , i.e. the covariance A ( ) will

q q

diminish in time and the uctuation 13

hj~ j i( ) / (4.16)

A ( )

will increase in time. This will in turn cause the time dep endent frequency in the Hamilto-

nian, ! ( )toincrease with time since the frequency and the uctuations are directly related

via the self-consistency condition. The resulting dynamics is then exp ected to approach an

oscillatory regime in which the width of the wavefunctional and the frequency of the har-

monic oscillator are of the same order. Under these circumstances, there is the p ossibility

that for a particular range of parameters (coupling, central momentum of the distribution

and particle density) parametric ampli cation can o ccur [39,44] that could result in particle

pro duction and redistribution of particles as will be discussed b elow within an early time

analysis. We will see that this case corresp onds to a \tsunami" con guration in a theory

which is symmetric even in the absence of the medium.

2. Case II

=2[n (0) n (0) + n (0)]:

q q q

In this case

1 jq q j

(0) 1+ for 1

jq q j

>> 1; (4.17) 0 for

and the covariance is

! jq q j

q 0

A (0) << ! for 1

q q

4n (0)

jq q j

! for >> 1: (4.18)

Therefore, large amplitude eld con gurations with momenta in the narrow \tsunami" shell,

nowhave a high probability of b eing realized. As b efore, the wave function for each k mo de

corresp onds to the ground state of a harmonic oscillator of frequency

(0) =

4n (0)

which evolves with a Hamiltonian for a harmonic oscillator with frequency ! (0). In this

case the wave function for q q is spread out over eld amplitudes much larger than

! 1=

and it is lo calized in the canonical momentum basis. 14

Under time evolution the wave function will tend to be squeezed i.e. it will be forced

to diminish its width and to b ecome lo calized inside the p otential well. This implies that

the covariance A ( ) will increase under time evolution, while the uctuation (4.16) and the

time dep endent frequency ! ( ) will decrease, i.e. the p otential ` attens out'.

In this case the quantity g ( ) (the renormalized quantum uctuations) in the evolution

equations (4.11) decreases and as will b e seen b elow, under certain conditions, can b ecome

negative.

There is thus a p ossibility of inducing spino dal instabilities in the quantum uctuations.

To see this, consider the case in which ' 0 and the e ective mass squared M ( ) =

1+g( ) in the equation (4.11) b ecomes negative, i.e. when g ( ) < 1. The mo des for

2 2

which q < jM ( )j will see an inverted harmonic oscillator and they will b egin to grow

almost exp onentially, resulting in copious particle pro duction for these mo des as can b e seen

from the expression for the particle number as a function of time (3.19).

This situation, in which the p otential turns into a maximum at the origin dynamical ly,

corresp onds to symmetry breaking, since the minimum will be away from the origin. In

this case the dynamics will result in a re-arrangement of the particle distribution: spino dal

instabilities will arise, long-wavelength mo des will b egin to get p opulated at the exp ense of

the initial non-equilibrium distribution. The spino dal instabilities will in turn result in an

increase in the uctuations that will tend to cancel the negative contribution to g from

the initial non-equilibrium distribution. Eventually a stationary regime should ensue in

which the instabilities are turned-o and the distribution of particles will b e p eaked at low

momenta.

At this p ointwewant to emphasize that the p ossibility for the onset of spino dal instabil-

ities is purely dynamical. In contrast to previous studies of dynamics in spino dally unstable

situations [24,29,25,32] in which an initially symmetric state is evolved with a broken sym-

metry Hamiltonian, in the present case the initial state and the e ective Hamiltonian are

symmetric and the instability is a consequence of the non-equilibrium dynamics.

The ab ove analysis of the dynamics, based on the quantum mechanical analogy will be

shown to be accurate in the next section where we present the details of the numerical

evolution.

B. Early Time Analysis:

A more quantitative understanding of these cases can b e achieved by studying the early

time b ehaviour of the solutions and setting ' 0.

1. Case I

In this case with given by (4.14)and fo cusing on the very early time during which

backreaction e ects can b e ignored, the solution to the mo de equations (Eq. (4.11)) with the 15

initial conditions given by Eq. (4.3) is simply a sup erp osition of plane waves with frequency

! (0):

( ) cos(! (0) ) i(1+ ) sin(! (0) ): (4.19)

q q q

The renormalized quantum uctuations which are dominated by the mo des within the highly

p opulated momentum shell are given by,

sin (! (0) )[1 (1+ ) ]

q q

g ( ) g q dq : (4.20)

! (1+ )

q q

If the initial distribution of particles is suciently sharp, a qualitative understanding of the

early time dynamics can b e obtained by a saddle p oint analysis of the contribution from the

region of large o ccupation numb er.

In this limit g ( ) is approximately given by,

4gN

g ( ) + sin (! ) : (4.21)

The mo de equations now b ecome

# "

2gN 2gN d

0 0

+ q +1+ cos (2! ) ( )=0: (4.22)

q q

d ! !

q q

0 0

This is a Mathieu equation whose solutions are of the Flo quet form [42]. The rst and

broadest instability band is centered at the value of q given by

2gN

2 2

q = q : (4.23)

The width of the unstable band dep ends on the parameter

Q = (4.24)

and can be read o in reference [42]. There is a rather small window of relevant parame-

ters that could allow appreciable parametric ampli cation. Whether the backreaction e ects

allow the unstable band to remain under time evolution resulting in large particle pro duc-

tion and redistribution of particles is a detailed dynamical question that will be studied

numerically b elow.

2. Case II

The dynamics in this case can b e understo o d by the heuristic arguments presented b elow.

Wework with the solution =2[n (0) n (0) + n (0)], which in the limit N >> 1 yields

q q q 0

(4.17) 16

1+ for q q (4.25)

q 0

0 otherwise : (4.26)

leading now to the following approximate form for the uctuation at early times:

4gN

sin (! ) (4.27) g ( )

when the backreaction e ects can be ignored. The rst feature to note is that g and N

app ear together in such away that the e ective coupling is now gN and hence the physics

is intrinsically non-p erturbative when N 1=g . This situation is very similar to that in

high temp erature eld theory wherein the relevant dimensionless quantityisT=m(T )(m(T)

is the temp erature corrected e ective mass) and the e ective coupling constant for long-

wavelength physics is T =m(T ). In this situation the non-p erturbative hard-thermal-lo op

resummation is required.

Secondly the expression for g ( ) is always less than or equal to zero. Notice that unlike

Case I, g ( ) is negative [see Eq.(4.21)]. In particular g (0) = 0 and then g ( ) b ecomes

negative i.e. it b egins to decrease. The fact that the uctuations decrease was exactly what

we had exp ected from the wave functional analysis presented in the previous section.

Furthermore we see that when 4gN =! > 1atvery early times there will b e an unstable

0 q

band of wave-vectors. An estimate of the width of the band can be provided by averaging

the time dep endence of g over one p erio d of oscillation. This estimate yields the band of

wave-vectors

2gN

which will b ecome spino dally unstable. The mo de functions for these wavevectors will grow

exp onentially at early times and their contribution to the uctuation g ( ) (4.9) will grow

{ this is a back-reaction mechanism that will tend to shut-o the instabilities.

This means that if we b egin with a completely O (N ) symmetric state i.e. '(0) = '_ (0) = 0

and if we cho ose N large enough such that 1 + g 1 4gN sin (! )=! < 0, spino dal

0 0 q q

0 0

instabilities will b e triggered and the symmetry will b e sp ontaneouly broken. The condition

for spino dal instabilities to app ear is given by

4gN

> 1 (4.29)

which determines the critical value of the particle number in a correlation volume in terms

of the coupling and the p eak momentum of the distribution.

In the preceding sections we provided an intuitive understanding of the underlying mech-

anism of symmetry breaking in terms of a quantum mechanical analogy. Wenow provide an

alternative argument to clarify the physical mechanism for the dynamical symmetry break-

ing. The argument b egins with the expression for the `dressed' mass in Eq. (4.8) which we

write in terms of dimensionless quantities as 17

# "

q dq

2 2 2

; (4.30) M = m + gM

R R R

! (0) 1+

q q

q dq

2 2

m = m + g : (4.31)

R B

! (0)

The second term is dominated by the p eak in the initial particle distribution. Using eqns.

(4.25,4.26) and using a saddle p oint approximation assuming a sharp distribution, we obtain

the relationship

# "

4gN

2 2

= m : (4.32) M 1

R R

Then cho osing the e ective mass M > 0aswehave done throughout, we see that when

the condition for spino dal instabilities in Eq. (4.29) is ful lled then it must b e that m < 0.

Therefore the renormalized mass squared in the absence of the medium is negative and the

medium e ects, i.e. the non-equilibrium distribution of particles dresses this mass making

the e ective, medium `dressed' mass squared p ositive. Thus in the absence of medium

the p otential was a (sp ontaneous) symmetry breaking p otential. The initial distribution

restores the symmetry at =0 much in the same way as in nite temp erature eld theory

at temp eratures larger than the critical temp erature. However the initial state is strongly

out of equilibrium and its time evolution re-distributes the particles towards low momentum

and the spino dal instabilities result from the squeezing of the quantum state as explained

ab ove.

This situation must be contrasted to that in Case I ab ove. The same argument, now

applied to Case I leads to the result

" #

4gN

2 2

M 1+ = m : (4.33)

R R

We clearly see that with a p ositive e ective mass, Case I corresp onds to the situation in

which the theory was symmetric even without the medium e ects (i.e. m > 0).

Thus we obtain a physical picture of the di erent cases: in Case I the symmetry was un-

broken without a medium and remains unbroken when the large density of particles is added.

By contrast in Case II, the symmetry is sp ontaneously broken in the absence of a medium,

the high density initial state restores the symmetry in a state out of equilibrium. Under

time evolution the dynamics then redistributes the particles pro ducing spino dal instabilities

and breaking the symmetry.

We reiterate that the second case represents a novel situation which is in a sense, contrary

to what happ ens at high temp erature where thermal uctuations suppress the p ossibilityof

long-wavelength instabilities.

The issue of symmetry breaking is a subtle one here. If we b egin with symmetric initial

conditions, '(0) = '_ (0) = 0, the wavefunctional will always be symmetric since the evolu-

tion will maintain this symmetry. In order to test whether the symmetry is sp ontaneously 18

broken or not, one must provide an initial state that is slightly asymmetric, with a very

small initial exp ectation value '(0) 6= 0, and follow the subsequent time evolution. If the

exp ectation value oscillates around zero, then the symmetry is not sp ontaneously broken

since the minimum of the `dynamical e ective p otential' is at the origin in eld space. If the

exp ectation value b egins rolling away from zero and reaches a stationary value away from

zero then one can assert that there is a dynamical minimum away from the origin and the

symmetry is sp ontaneously broken. Thus the test of symmetry breaking requires an initial

condition with a small value of the order parameter.

C. The Late Time Regime

The asymptotic value of the order parameter can be obtained by analyzing the full

dynamics of the theory and dep ends on the initial conditions. This re ects the fact that

there is no static e ective p otential description of the physics. However, some information

ab out the asymptotic state (when '_ (1)='(1) = 0) can b e obtained from the equation of

motion (4.10) by setting '(1) = 0 which yields the sum rule [44,45]

1+' (1)+ g(1)= 0 (4.34)

provided '(1) 6= 0. This sum rule guarantees that the pions are the asymptotic massless

Goldstone b osons since

h i

2 2 2 2

M ( )=m + (1)+h i (1)

(Eq. (2.5)) and the sum rule is a consequence of the Ward identities asso ciated with the

global O (N ) symmetry.

The non-linear evolution of the mo de functions results in a redistribution of particles

within the spino dally unstable band. The distribution b ecomes more p eaked at low momen-

tum and the e ective p otential attens resulting in a non-p erturbatively large distribution

of Goldstone b osons at low momentum.

V. NUMERICAL ANALYSIS

1. Case I

We have investigated the p ossibility of parametric ampli cation in this case in a wide

region of parameters but always in the dense regime N >> 1 and varying the center of

the distribution. We nd numerically that the backreaction e ects shut o the parametric

instabilities rather so on allowing only small particle pro duction and redistribution of parti-

cles. Typically the distribution develops p eaks but remains qualitatively unchanged and the

dynamics is purely oscillatory. 19

2. Case II

The numerical analysis of the problem involves the solution of the coupled set of equations

(4.10), (4.11) and (4.12) app ended with initial conditions. We cho ose the zero mo de initial

conditions to be '(0)=10 ;'_(0)=0 while the mo de functions satisfy (0)=1; (0) =

q q

i! (0)(1 + ). Here ! (0) = q + 1 and

q q q

=2[n (0) n (0) + n (0)] : (5.1)

q q q

We have tested the numerics with a momentum cuto =25 in units of the renormalized

mass M and found that after renormalization the numerical results are insensitive to the

value of the cuto provided it is chosen to b e much larger than the largest wave-vector which

b ecomes spino dally unstable. The initial particle distribution is chosen to b e

(q 5)

n (0) = e ; (5.2)

where the total initial number of particles is taken to be to be N = 2000, the coupling is

2 3

xed at g =10 and the initial value of the order parameter is taken to b e '(0) = 10 .

Results:

Fig.(1) shows g ( ) vs. and the e ective `pion' mass squared M ( ). We see clearly

that spino dal instabilities are pro duced and the quantitative features of the dynamics are

in agreement with the estimates established for the early time dynamics given by Eq.(4.27).

We see that the pions b ecome massless, asymptotically. The distribution function n ( )

multiplied by the coupling g is shown in Fig. (2) at di erent times, clearly demonstrating

how the distribution changes in time. As a consequence of the spino dal instabilities the long-

wavelength mo des grow exp onentially and the ensuing particle pro duction for these mo des

p opulates the band of unstable mo des. In particular the amplitudes of the long wavelength

mo des that b ecome spino dally unstable growtobe non-perturbatively large of order 1=g and

dominate the dynamics completely. At earlier times the initial p eak in the distribution at

q 5 can still be seen, but at later times it is overwhelmed by the distribution at long

wavelengths. Fig. (3) shows a zo om-in of the distribution functions (gn ) vs. q at =30;80

near q = 0 and also near the p eak of the initial distribution, around q 5. We see a remnant

of the original p eak, slighly shifted to the right but much broader than the initial distribution

and with ab out half the original amplitude. After 10 the distributions do not vary much

in this region of momenta, but they do vary dramatically at low momenta. Fig. (4) shows

the total numb er of particles as a function of . We see clearly that initially the total number

of particles diminishes b ecause the uctuations decrease at early times. The long-wavelength

mo des b egin to grow b ecause of spino dal instabilities but their contributions are suppressed

by phase space. Only when their amplitudes b ecome non-p erturbatively large, is the particle

pro duction at long-wavelengths an e ective contribution to the total particle numb er. When

this happ ens, there is an explosive burst of particle pro duction following which the total

number of particles remains fairly constant throughout the evolution. After the spino dal

instabilities are shut-o , which for the values chosen for the numerical evolution corresp ond 20

to 2, the dynamics b ecomes non-linear. Whereas during the initial stages the dynamics

is in the linear regime, after backreaction e ects have shut-o the spino dal instabilities the

further evolution of the distribution functions is a consequence of the non-linearities.

Fig.(5) exhibits one of the clear signals of symmetry breaking. The order parameter

b egins very near the origin, but once the spino dal instabilities kick in, the origin b ecomes

a maximum and the order parameter b egins to roll away from it. Notice that the order

parameter reaches a very large value, which is the dynamical turning p oint of the tra jectory,

b efore settling towards a non-zero value. We nd that the value of the turning p oint and the

nal value of the order parameter dep end on the initial conditions. To illustrate the non-

p erturbative growth of mo des clearly,wehave plotted the quantity g j ( =5)j gj ( =

q q

0)j in Fig. (6) which shows how the amplitude of the long wavelength mo des b ecomes

non-p erturbatively large and of order 1=g .

We have also carried out the numerical evolution with g = 10 ;N = 4000 and g =

10 ;N = 40000 with the same value of q and found the same quantitative b ehavior,

0 0

proving that the relevant combination is gN as revealed by the analytic estimates ab ove. We

have also con rmed that for gN << 1 there are no spino dal instabilities and the dynamics is

purely oscillatory without a redistribution of the particles and with no appreciable particle

pro duction. When the p eak of the initial distribution function is beyond the spino dally

unstable band q > q (see Eq.(4.28)), the original distribution is depleted and broadened

somewhat with irregularities and wiggles but remains qualitatively unchanged (see g. 3).

However, when the p eak of the initial distribution is within the spino dally unstable band there

is a complete re-distribution of particles towards low momentum. The original distribution

disapp ears under time evolution and after the spino dal time only the low momentum mo des

are p opulated.

VI. SYMMETRY BREAKING, ENERGY, PRESSURE AND EQUATION OF

STATE:

A. Onset of Bose Condensation:

We have seen b oth from the numerical evolution and from the argument based on the

sum rule (4.34) which is a result of the Ward identities and Goldstone's theorem, that the

e ective mass term vanishes asymptotically. Therefore the asymptotic equation of motion

for the mo de functions is that of a massless free eld. In particular the asymptotic solution

for the q = 0 mo de is given by

( !1)= A+B (6.1)

where A and B are complex co ecients that can only b e obtained from the full time evolu-

tion. However b ecause the Wronskian

_ _

( ) ( ) ( ) ( )=2iW (6.2)

0 0 0

0 0 21

is constant in time, neither A nor B can vanish [43]. This situation must b e contrasted with

that for the q 6=0 mo des whose asymptotic b ehavior is of the form

iq iq

( !1)= e + e : (6.3)

q q q

This causes the numb er of particles at zero momentum to grow asymptotically as whereas

the number saturates for the q 6=0 mo des. The three dimensional phase space conspires to

cancel the contribution from the q = 0 mo de to the total number of particles, energy and

pressure, which, from the numerical evolution (see Fig.(4)) are seen to remain constant at

long times. This situation is very similar to that in Bose-Einstein condensation where the

excess number of particles at a xed temp erature go es into the condensate, while the total

number of particles outside the condensate is xed by the temp erature and the chemical

p otential. The q = 0 mo de will b ecome macroscopically o ccupied when V where V is

the volume of the system (i.e. the numb er of particles in the zero momentum mo de b ecomes

of the order of the spatial volume). When this happ ens this mo de must be isolated and

studied separately from the q 6= 0 mo des b ecause its contribution to the momentum integral

will b e cancelled by the small phase space at small momentum. Again the situation is very

similar to the case of the usual Bose-Einstein condensation. Notice that this argument is

indep endent of a non-vanishing order parameter ' and leads to the identi cation of the

zero momentum mo de as a Bose condensate that signals sp ontaneous symmetry breaking

even when the order parameter remains zero. Since the e ective mass is zero we identify the

condensing quanta as pions and therefore this mechanism is a novel form of pion condensation

in the absence of direct scattering.

When scattering is included, b eyond the leading order in the large N approximation, the

formation of the Bose condensate will require a detailed understanding of the di erent time

scales. The time scale for the collisionless pro cess describ ed ab ovemust b e compared to the

time scale for collisional pro cesses that would tend to deplete the condensate. If spino dal

instabilities causing non-p erturbative particle pro duction at low momentum o ccur on much

shorter time scales than collisional redistribution then we would exp ect that there will b e a

non-p erturbatively large p opulation at low momenta that could b e interpreted as a coherent

condensate.

The spino dal instabilities seen in this article are similar to those which lead to the for-

mation of Disoriented Chiral Condensates [24,29,25,32]. However we emphasize that unlike

most of the previously studied scenarios for DCC formation in which a `quench' into the

spino dal region was intro duced ad-hoc, in the present situation the spino dal instabilities are

of dynamical origin. We have studied a situation where the vacuum theory has symmetry

breaking minima (with m < 0 in Eq. (4.32)) but the initial state is highly excited with

the particle density larger than a critical value leading to a symmetry restored theory in the

medium. However this initial state is strongly out of equilibrium and its dynamical evolution

automatically induces spino dal instabilities. 22

B. Energy, Pressure and Equation of State:

As mentioned in the intro duction the goal of our study is to understand the dynamical

evolution of strongly out of equilibrium states. In the usual investigations of the dynamics

of the quark gluon plasma one uses a hydro dynamic description in which the energy density,

pressure and all the thermo dynamic variables dep end only on prop er time [1,14]. The hy-

dro dynamic equations are then a consequence of the conservation laws which are app ended

with an equation of state to determine the evolution completely. The hydro dynamic regime

corresp onds to the case when the collisional mean free path is shorter than the wavelength

of the hydro dynamic collective mo des, and therefore the concept of lo cal thermo dynamic

equilibrium is warranted.

Avalid question in the situation that wehave studied in this aricle, is whether and when

an equation of state is a meaningful concept. In the leading order in the large N expansion

there are no collisional pro cesses (these arise at O (1=N )) and therefore the concept of a

hydro dynamic regime in is not applicable in principle. Furthermore since the state considered

is spatially homogeneous the pressure will dep end on time rather than on prop er time. Since

the energy is conserved and the pressure evolves with time, an equation of state will have a

meaning only when the evolution has reached the asymptotic regime.

The energy density is given by

i h

E 1 1 1 k dk

B B

2 2 2 2 2 2 4 2 2

_ _

j ( )j + ! ( )j ( )j = + m + + h i : (6.4)

q q

q B

NV 2 2 8 4 2 8

The last term which arises in a consistent large N expansion, is extremely imp ortantin

that after renormalization it provides a negative contribution which can interpreted as part

of the e ective p otential [44,39]. Using the equation of motion for the order parameter and

the mo de functions it is straightforward to show that the energy is conserved and the last

term is necessary to ensure energy conservation. Since the energy is conserved it can be

renormalized by a subtraction at =0, and therefore is nite in terms of the renormalized

quantities.

The pressure is given by the following expression,

" #

2 2

p + E k 1 k dk

2 2 2

_ _

j ( )j + : (6.5) = + j ( )j

q q

NV 2 2 3

Unlike the energy density, the pressure is not a constant of the motion and needs prop er

subtractions to render it nite. The detailed expressions for b oth the renormalized energy

and pressure can b e found in references [39,45,46]. However, rather than computing the total

energy density and pressure, we will study the contributions from the mo des that are highly

p opulated and whose amplitudes b ecome non-p erturbatively large ( 1=g ). Asymptotically,

when the e ective mass vanishes and the low momentum mo des b ecome highly p opulated

with amplitudes of O (1=g ) the renormalized energy density is given by (see ref. [39,45] for

the explicit expression of the renormalized energy density) 23

h i

E 1 k dk

2 2 2

j ( )j + ! ( )j ( )j + O (g ) (6.6) =

q q

NV 4 2

where k is the largest spino dally unstable wave vector at early times and O (g ) represents

terms that are p erturbatively small. Using the asymptotic solutions for the mo de functions

given by Eq. (6.3) and neglecting the strongly oscillatory phases that average out at long

times we obtain

4 4

h i

E M q dq

2 2

j j + j j + O (g ): (6.7) =

q q

NV 2 2W

Similarly, neglecting the contribution of mo des with small amplitudes, we nd that the

renormalized pressure plus energy density is given by,

4 2

h i

p + E M 4 q dq

2 2

j j + j j ; (6.8) =

q q

NV 3 2 2W

so that in the asymptotic regime

p = ; (6.9)

indep endent of the particle distribution which is non-thermal. This is one of the imp ortant

results of this work. In Fig. (7) we show the trace of the energy momentum tensor E 3P

as a function of time, for the same value of parameters as for Figs. (1-5). Clearly, the trace

vanishes asymptotically.

During the early stages of the dynamics when spino dal instabilities arise and develop with

profuse particle pro duction, an equation of state cannot be de ned. The dynamics cannot

be describ ed in terms of hydro dynamic evolution. Since the pro cesses under consideration

are collisionless, there is no lo cal thermo dynamic equilibrium and an equation of state is

ill-de ned.

VI I. CONCLUSIONS

Wehave studied the evolution of an O (N )-symmetric quantum eld theory, prepared in a

strongly out-of-equilibrium initial state. The initial state was characterized by a particle dis-

tribution lo calized in a thin spherical shell p eaked ab out a non-zero momentum, a spherical

\tsunami". The formulation of this scenario resulted from a simpli cation of the idealized

`colliding-pancake' description of a heavy-ion collision. For a large density of particles in

the initial state, the ensuing dynamics is non-p erturbative and consequently we studied the

O (N ) theory in the leading order in the large N limit which is a systematic non-p erturbative

approximation scheme . When the tree-level theory has vacua that sp ontaneously break the

symmetry and the numb er of particles within a correlation volume at t = 0 is so high that the

symmetry is restored initially , spino dal instabilities are then induced dynamical ly resulting

in profuse particle pro duction for low momenta. 24

This situation is to b e contrasted with the usual studies of DCC's where the initial state

is assumed to satisfy LTE (lo cal thermo dynamic equilibrium) and the spino dal instabilities

are intro duced either via an ad-hoc quench or via co oling due to hydro dynamic expansion

which is also intro duced phenomenologically.

Backreaction of the long wavelength uctuations eventually shuts o these instabilities

and the nonlinearities redistribute particles towards low momenta. Wethus nd asymptoti-

cally in time a novel form of pion condensation at low momentum, out of thermal equilibrium.

Furthermore, a macroscopic condensate of the Bose-Einstein typ e will form at much longer

times (mt V ).

When the spino dal instabilities shut o we nd that the asymptotic `quasiparticles' are

massless pions with a non-thermal, non-p erturbative distribution function p eaked at low

momentum but with an ultrarelativistic equation of state.

We b elieve that these phenomena p oint out to very novel and non-p erturbative mech-

anisms for particle pro duction and relaxation that are collisionless, strongly out of lo cal

thermo dynamic equilibrium and cannot b e describ ed in the early stages via a coarse-grained

hydro dynamic evolution. These are the result of strongly out of equilibrium initial states of

high density that could p otentially b e of imp ortance in the dynamics of heavy ion collisions

at high luminosity accelerators.

A more realistic treatment, mo delling a collision will require an initial state which breaks

the rotational invariance and selects out a b eam-axis along which the colliding pions move

in opp osite directions. However, the analysis of such initial conditions is b eyond the present

numerical capabilities and will b e deferred to a future work.

An upshot of this study of high density, non-equilibrium particle distributions is the fol-

lowing tantalizing theoretical question: can one extract a resummation scheme, or an e ec-

tive theory akin to the Hard Thermal Lo op e ective expansion for arbitrary non-equilibrium,

non-thermal distributions such as the \tsunami" con guration for gauge theories? Possible

answers and consequences of such initial states for gauge theories will be discussed in a

forthcoming article [47].

VIII. ACKNOWLEDGEMENTS:

D. B. thanks the N.S.F for partial supp ort through the grantawards: PHY-9605186 and

LPTHE for warm hospitality. R. H. and S. P. K. were supp orted by DOE grant DE-FG02-

91-ER40682. S. P. K. would like to thank BNL for hospitality during the progress of this

work. H. J. de V. thanks BNL and U. of Pittsburgh for warm hospitality. The work of R.D.P.

is supp orted by a DOE grant at Bro okhaven National Lab oratory, DE-AC02-76CH00016.

The authors acknowledge partial supp ort by NATO. 25

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tion. 28 FIGURES

3 3

1 Σ(τ) 1 Σ(τ) g g -1 -1

-3 -3

-5 -5

-7 -7

-9 -9

-11 -11

-13 -13

-15 -15

0 102030405060708090100110120 01234567891011

τ τ 29 3 (τ) 2 1 M

-1

-3

-5

-7

-9

-11

-13

-15

0 102030405060708090100110120

FIG. 1. g ( ) and M ( ) vs. resp ectively, with the initial distribution Eq.(5.2),

2 3

N = 2000 ; g =10 ; '(0) = 10 ; q =5

0 0

0.5

(τ= 0) 22 q

n 0.4

(τ = 2) 20 q n 18

0.3 16

12 0.2 10

0.1 6

2 0.0 012345678910 0 q 012345678910

q 30 70 220 (τ = 5) q (τ = 10) 200 q

n 60 n 180

50 160

140 40 120

30 100

80 20 60

40 10 20

0 0 012345678910 012345678910

q q

FIG. 2. Distribution function gn vs. q for =0;2;5;10 resp ectively with the same parameters

as in Fig. (1)

800 1.4

(τ = 30) 700 (τ = 30) q 1.2 q n n 600 1.0

500 0.8

400 0.6 300

0.4 200

0.2 100

0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 012345678

q q 31 0.8

1100 0.7 (τ = 80) 1000 q n (τ = 80) q

n 900 0.6

800 0.5 700

600 0.4

500 0.3 400

300 0.2

200 0.1 100

0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2345678

q q

FIG. 3. gn ( =30;80) vs. q . with the same parameters as in Fig. (1), zo omed in the region

near q 0 and the p eak of the initial distribution q =5.

0 32 24 (τ) 23 N

0 10 20 30 40 50 60 70 80 90 100 110 120

FIG. 4. Total numb er of particles N ( ) vs. . with the same parameters as in Fig. (1) 33 0.6 ϕ(τ)

0.5

0.4

0.3

0.2

0.1

0.0 0 102030405060708090100110120

FIG. 5. '( ) vs. . with the same parameters as in Fig. (1) 34 39 (τ = 5) q ∆

-1

012345678910

2 2

FIG. 6. ( =5)=gj ( =5)j gj ( =0)j vs. q for the same parameters as in Fig. (1)

q q q 35 500

400 E - 3P

300

200

100

-100

0 10 20 30 40 50 60 70 80 90 100 110 120

FIG. 7. E 3P vs. for the same parameters as in Fig. (1) 36