Evolution of Critical Correlations at the QCD Phase Transition

a r X i v :h e p -p h /0610382v 2 8 F e

b 2007Evolution of Critical Correlations at the QCD Phase Transition

N.G.Antoniou,F.K.Diakonos,and E.N.Saridakis ?

Department of Physics,University of Athens,GR-15771Athens,Greece

(Dated:February 2,2008)

We investigate the evolution of the density-density correlations in the isoscalar critical condensate

formed at the QCD critical point.The initial equilibrium state of the system is characterized by a

fractal measure determining the distribution of isoscalar particles (sigmas)in con?guration space.

Non-equilibrium dynamics is induced through a sudden symmetry breaking leading gradually to

the deformation of the initial fractal geometry.After constructing an ensemble of con?gurations

describing the initial state of the isoscalar ?eld we solve the equations of motion and show that

remnants of the critical state and the associated fractal geometry survive for time scales larger than

the time needed for the mass of the isoscalar particles to reach the two-pion threshold.This result

is more transparent in an event-by-event analysis of the phenomenon.Thus,we conclude that the

initial fractal properties can eventually be transferred to the observable pion-sector through the

decay of the sigmas even in the case of a quench.

I.INTRODUCTION Experiments of a new generation,with relativistic nuclei,at RHIC and SPS are currently under consideration with the aim to intensify the search for the existence and location of the QCD critical point in the phase diagram of strongly interacting matter [1,2].Important developments in lattice QCD [3]and studies of hadronic matter at high temperatures [4]suggest that the QCD critical endpoint is likely to be located within reach at SPS energies.It is therefore desirable to explore the range of baryon number chemical potential μB =100-500MeV by studying collisions at relatively low energies,

√δ+1

(δ?5).The crucial question from the observational point

of view is whether in the freeze-out regime,which follows the equilibration stage,the relaxation time-scale τrel of

these?uctuations is long enough compared to the time-scaleτth associated with the development of a massiveσ-?eld beyond the two-pion threshold(mσ≥2mπ).Both time-scales(τrel,τth)are characteristic parameters of the out-of-equilibrium phenomena(σrescattering)which take place during the evolution of the system(towards freeze-out)and the requirementτrel?τth guarantees that critical?uctuations may become observable in theσ-mode(σ→π+π?[6,10]).

In order to quantify these e?ects,we adopt in this work the picture of a rapid expansion(quench)which is a realistic possibility in the framework of heavy-ion collisions.We study the out-of-equilibrium evolution of the initial fractal characteristics of theσ-?eld and we search for time scalesτrel,τth satisfying the above constraint in a particular class of events(event-by-event search).The dynamics of the system is?xed by a two-?eld Lagrangian,L(σ, π),together with appropriate initial conditions.The out-of-equilibrium phenomena are generated by the exchange of energy between theσ-?eld and the environment which consists of massive pions initially in thermal equilibrium.

In section II the formulation of the problem and in particular the equations of motion,the initial conditions and the generation of thermalπ-con?gurations are presented.In section III numerical solutions of the evolution of critical ?uctuations are given and discussed whereas in section IV our?nal results and conclusions are summarized.

II.FORMULATION OFσFIELD DYNAMICS

In our approach we assume an initial critical state of the system in thermal equilibrium,disturbed by a two-?eld potential V(σ, π),in an e?ective description inspired by the chiral theory of strong interactions[11,12].The 3-dimensional Lagrangian density is

L=1

4

(σ2+ π2?v20)2+m2π

2λ2v20,yields 10 λ2 20for400 mσ 600MeV.

The equations of motion resulting from(1)are:

¨σ??2σ+λ2(σ2+ π2?v20)σ=v0m2π

¨ π??2 π+λ2(σ2+ π2?v20) π+m2π π=0,(3) where π2=(π+)2+(π0)2+(π?)2.

Using a constant value v0in eq.2implies a non-vanishing mass(?nite correlation length)for theσ-?eld,mσ=

α2 σn+1i+1,j,k+σn+1i?1,j,k?2σn+1i,j,k + σn+1i,j+1,k+σn+1i,j?1,k?2σn+1i,j,k +

σn+1i,j,k+1+σn+1i,j,k?1?2σn+1i,j,k ?dt2 λ2 σn+1i,j,k 2+ πn+1i,j,k 2?v2 σn+1i,j,k?vm2π ,(5)

and similarly for the other three equations considering theπ-?eld.In eq.(5)αis the lattice spacing while dt is the time step.The upper indices indicate the time instants and the lower indices the lattice sites.As usual we perform an initial fourth order Runge-Kutta step to make our algorithm self-starting,and we impose periodic boundary conditions. We are interested in studying the evolution of the above system using initial?eld con?gurations dictated by the onset of the critical behavior.In this case we expect that theσ-?eld,being the order parameter,will possess critical ?uctuations,and theπ-?elds to be thermal,while the entire system will be in thermodynamical and chemical equilib-rium.Obviously,the subsequent evolution,determined by eqs.(3),will generate strong deviations from equilibrium. Before going on with the detailed study of the dynamics,we?rst describe in the following subsections the generation of an ensemble of?eld con?gurations on a3-D lattice possessing the characteristics of the critical system.This ensemble enters in the subsequent analysis as the initial condition.

A.Generation of initial ensemble of criticalσ-con?gurations

The absolute value of theσ-?eld introduced in the previous subsection is interpreted as local density,and the corresponding critical behavior is described by a fractal measure demonstrated in the dependence of the mean”mass”M( x0,R)on the distance R around a point x0de?ned by:

M( x0,R)= R|σ( x)σ( x0)|d D x ,(6) obeying the power law

M( x0,R)～R D f(7) for every x0.D f is the fractal mass dimension of the system[15,16,17]and the mean value is taken with respect to the ensemble of the initialσ-con?gurations.The production of theσcon?gurations building up the critical ensemble, characterized by the fractal measure given in eqs.(6,7),has been accomplished in[18].In fact the fractal properties of the critical system can be produced as an ensemble average,through the partition function:

Z= δ[σ]e?Γ[σ],(8) withΓ[σ]the scale invariant e?ective action at T=T cr,μ=μcr:

Γ[σ]= R d D x{1

.(10)

δ+1

For the3-D Ising universality class,D=3,the isothermal critical exponent isδ≈5,and the coupling g≈2[20], therefore D f≈5/2.

The power-law behavior of M( x0,R)= R|σ( x)σ( x0)|d3x around a random x0,averaged inside clusters of volume

V,is illustrated in?g.1.The M( x0,R)versus R?gure is drawn as follows:For a given x0of a speci?c con?guration

√

we?nd R of the cluster in which it belongs,taken?3

10

M (R )

R (fm)

FIG.1:M (R )= R R |σ(

x )σ( x 0)|d 3x versus R for the ensemble of σ-?eld con?gurations.The slope Ψ,i.e the fractal mass dimension D f is equal to 5/2within an error of less than 1%.

Calculating d f for each con?guration we obtain a distribution around 5/2with standard deviation ≈0.05.As expected the ensemble average of d f is d f =D f ,within an error of less than 0.5%.

B.Generation of initial thermal π-con?gurations

We generalize the method of [21]in order to produce an ensemble of 3-D π-con?gurations in real space,corresponding to an ideal gas at temperature T 0.The unperturbed Hamiltonian for the classical scalar ?eld theory in three dimensions is

H =

1(2π)3

πk 0e i k x = +∞?∞

d 3k √

(2π)3ξk 0e i k x = +∞?∞

d 3k ωk k 2+m 2π.

Now,choosing an initial classical density distribution [21]

ρ[π,˙π]=Z ?1(β0)exp {?β0H [π,˙π]},

and substituting the Hamiltonian (12)with the free particle solutions (13),we ?nally get ρ[x k ,y k ]=Z ?1(β0)exp ?β0 +∞?∞

d 3k

III.NUMERICAL SOLUTIONS

We study the evolution of the system determined by equations(3)which we solve in3-D25×25×25lattice,using as initial conditions an ensemble of104independentσ-con?gurations on the lattice generated as described above,

i.e possessing fractal characteristics,and104con?gurations for eachπcomponent corresponding to an ideal gas at temperature T0≈140MeV.The initial time derivatives of theσ-?eld,forming the kinetic energy,are assumed to be zero,since this is a strong requirement of the initial equilibrium.The used population is by far satisfactory since the

results converge for ensembles with more than7×103con?gurations(numerically tested)for the considered lattices (sizes from15×15×15to25×25×25sites).In addition,we?nd that the obtained results are independent of the lattice spacingαdespite the discontinuities in the derivatives of the piecewise constant con?gurations,as the corresponding variationδσgoes to zero fast enough so that the limit limα→0δσ

t t0Ψ(t′)dt′.We observe the remarkable phenomenon that the characteristic exponentΨ(t)after reaching the value of the embedding dimension3,it?uctuates and for particular times it becomes

almost equal toΨ(0)=5/2.Thus,after the?rst deformation,the initial critical behavior of the whole ensemble is

<> (M e V

)

t (fm )

FIG.3:Time evolution of the mean ?eld value π for various λ2and τcases.

R (fm )

M (R )FIG.4:M (R )vs R for three successive times,t =0fm,t =0.25fm and t =0.5fm,for typical λ2and τvalues.The solid lines mark the power-law ?t,and the dotted line has slope 2.5and is used as a reference.

partially restored and deformed repeatedly.A detailed explanation of this revival is given in [22].The key point is that the partial restoration takes place when σ(t ) passes through its lower turning point,where the σ-?eld (seen as a system of coupled oscillators)reaches a state similar to the initial one.This phenomenon weakens gradually,and ?nally the dynamics dilutes completely the initial critical behavior.

This phenomenon is also visible in the evolution of the slope ψ(t )of m (R )versus R ,for each con?guration.Indeed,in ?g.6we demonstrate the evolution of the mean ?eld value as well as of ψand of its time average,for λ2=20and τ=1fm,for three independent con?gurations of the ensemble,corresponding to initial value ψ(0)=d f equal to 2.51,2.50and 2.53respectively.We observe an oscillatory behavior of ψ(t )similar to that of Ψ(t )of the whole

(t )FIG.5:Time cases as before.The π.

ensemble In a heavy As the system ?4m 2π

<> (M e V )(t )

FIG.6: σ ,fm.The vertical line of the have ψ(0)=d f ≈P (t )leads to its time average P t τobserve,P (t )is quite as to the peaks of Ψ(t )the initial critical is larger than τth then pions.The time averaged P t the cut 0.5%at τrel words,τrel ≈40fm to ≈1.6fm,=reached after the ?rst it of to the above the ?eld )≈slower quenches τth ,before the decay σin ?g.8for the λ2=fm >1and especially for the ratio τrel dashed

P (t )

FIG.7:Time t .With the dashed line line in ?g.8.

The e?ect of m with decreasing λ2,from the decay for λ2=20a larger number of in order to produce a the 2cases,although the The of the initial pions,despite the It is out of the evolution system using an d f ≈3with a (acquires slope 2.96 P t are always exactly at the freeze out,remains robust for a 2are very likely to The above 9,study

0123

r e l

/

t h

(fm)

FIG.8:The dependence of the ratio τrel /τth on the quench duration τfor λ2=20.The dashed line depicts an exponential ?t.of event-by-event ?uctuations in relativistic nuclear collisions gives rise to a powerful tool in the search for novel,unconventional e?ects associated with QCD phase transitions (critical ?uctuations,disoriented chiral condensates,dissipation at the chiral phase transition).In this work we have examined the evolution of critical ?uctuations neglecting the e?ects of dissipation in this out-of-equilibrium process.Although a detailed study of dissipative aspects of this evolution is beyond the scope of this work,an estimate of their in?uence on the fractal structure of critical ?uctuations is necessary before drawing our ?nal conclusions.For this purpose we have considered a simpli?ed modi?cation of σ?eld dynamics adding a dissipative term Γ˙σ( x ,t )?γν(t )in the ?rst equation (3).ν(t )is the noise term ful?lling the relations ν(t ) =0and ν(t )ν(t ′) =γ2δ(t ?t ′).The coe?cients Γand γare related through the formula γ=

FIG.9: σ ,ψ((Γ?1?0.5 fm),forλ2=

system in an As a out to be a remarkable statistics. In fact,it pattern of intermittent point without

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