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PUPT-1389, IASSNS-HEP-93/16

arXiv:hep-ph/9303281v1 18 Mar 1993PUPT-1389, IASSNS-HEP-93/16March 1993Emergence of Coherent Long WavelengthOscillations After a Quench: Application to QCDKrishna Rajagopal⋆Department of PhysicsJoseph Henry LaboratoriesPrinceton UniversityPrinceton, N.J. 08544Frank Wilczek†School of Natural SciencesInstitute for Advanced StudyOlden LanePrinceton, N.J. 08540⋆ResearchsupportedinpartbyaNaturalSciencesandEngineeringResearchCouncil of Canada 1967 Fellowship and in part by a Charlotte Elizabeth ProcterFellowship.RAJAGOPAL@PUPGG.PRINCETON.EDU† Research supported in part by DOE grant DE-FG02-90ER40542. WILCZEK@IASSNS.BITNET

ABSTRACTTo model the dynamics of the chiral order parameter in a far from equilibriumphase transition, we consider quenching in the O(4) linear sigma model. We ar-gue, and present numerical evidence, that in the period immediately following thequench long wavelength modes of the pion field are amplified.

This results in largeregions of coherent pion oscillations, and could lead to dramatic phenomenologicalconsequences in heavy ion collisions.2

Among the most interesting speculations regarding ultra-high energy hadronicor heavy nucleus collisions is the idea that regions of misaligned vacuum mightoccur [1,2,3,4,5]. In such misaligned regions, which are analogous to misaligneddomains in a ferromagnet, the chiral condensate points in a different directionfrom that favored in the ground state.

If we parametrize the condensate using theusual variables of the sigma model, misaligned vacuum regions are places wherethe four-component (σ,⃗π) field, that in the ground state takes the value (v, 0), ispartially aligned in the π directions. If they were produced, misaligned vacuumregions plausibly would behave as “pion lasers”, relaxing to the ground state bycoherent pion emission.

They would produce clusters of pions bunched in rapiditywith highly non-Gaussian charge distributions. Thus a misaligned vacuum regionstarting with the field in the π1 −π2 plane would emit only charged pions (equallypositive and negative, since the fields are real), while a misaligned vacuum regionstarting with the field pointing in the π3 direction would emit only neutral pions.More generally if we defineR ≡nπ0nπ0 + nπ+π−(1)then assuming that all initial values on the 3-sphere are equally probable, theprobability distribution P(R) is given byP(R) = 12R−1/2 .

(2)As an example of (2), we note that the probability that the neutral pion frac-tion R is less than .01 is 0.1! This is a graphic illustration of how different (2) isfrom what one would expect if individual pions were independently emitted withno isospin correlations.

According to (2) there is a substantial probability that,say, a cluster of 70 pions would all be charged – something that could essentiallynever occur for incoherent emission. There may be some hint of such behavior inthe Centauro (overwhelmingly charged) and anti-Centauro (overwhelmingly neu-tral) events reported in the cosmic ray literature.

[6,7] Here we propose a concrete3

mechanism by which such phenomena might arise in heavy ion collisions, for whichthe plasma is far from thermal equilibrium.In previous work [5,8] we considered the equilibrium phase structure of QCD.We argued that QCD with two massless quark flavors probably undergoes a second-order chiral phase transition. For many purposes it is a good approximation totreat the u and d quarks as approximately massless, and we expect that real QCDhas a smooth but perhaps rapid transition as a function of decreasing temper-ature from small intrinsic to large spontaneous chiral symmetry breaking.Atfirst sight it might appear that a second-order phase transition is especially fa-vorable for the development of large regions of misaligned vacuum.

Indeed thelong-lived, long-wavelength critical fluctuations which provide the classic signatureof a second-order transition are such regions. Unfortunately the effect of the lightquark masses, even though they are formally much smaller than intrinsic QCDscales, spoil this possibility [5].

The pion masses, or more precisely the inversecorrelation length in the pion channel, are almost certainly not small compared tothe transition temperature. As a result the misaligned regions are modest affairs atbest even near the critical temperature.

They extend at most over a few thermalwavelengths, and almost certainly do not contain sufficient energy to radiate largenumbers of correlated pions.Here we will consider an idealization that is in some ways opposite to that ofthermal equilibrium, that is the occurence of a sudden quench from high to lowtemperatures, in which the (σ,⃗π) fields are suddenly removed from contact with ahigh temperature heat bath and subsequently evolve mechanically. We shall showthat long wavelength fluctuations of the light fields (the pions in QCD, which wouldbe massless if not for the quark masses) can develop following a quench from sometemperature T> Tc to T= 0.

The long wavelength modes of the pion fieldsare unstable and grow relative to the short wavelength modes.Before we enter into the details of our model and simulations it seems appropri-ate to discuss the qualitative reason for the result just mentioned, which we suspect4

may be of wider interest. Whenever one has spontaneous breaking of a continuousglobal symmetry, massless Nambu-Goldstone bosons occur.

The masslessness ofthese modes occurs through a cancellation following the schemam2 = −µ2 + λφ2 ,(3)where the second term arises from interaction with a condensate whose expectationvalue ⟨φ⟩2 satisfies ⟨φ⟩2 = µ2/λ ≡v2 in the ground state. However following aquench the condensate starts with its average at 0 rather than v, and generally os-cillates before settling to its final value.

Whenever ⟨φ⟩2 < v2, m2 will be negative,and sufficiently long wavelengths fluctuations in the Nambu-Goldstone boson fieldwill grow exponentially. The same mechanism will work, though less efficiently, ifone has only an approximate symmetry and approximate Nambu-Goldstone modes,or for that matter other modes whose effective m2 is ‘accidentally’ pumped negativeby their interactions with the condensate.Returning to QCD, we shall use the classic Gell-Mann–Levy Lagrangian [9] todescribe the low energy interactions of pions:L =Zd4xn12 ∂iφα∂iφα −λ4φαφα −v22 + Hσo,(4)where φ is a four-component vector in internal space with entries (σ, π).

Here, λ,v, and H ∝mq are to be thought of as parameters in the low energy effectivetheory obtained after integrating out heavy degrees of freedom. We shall treat (4)as it stands as a classical field theory, since the phenomenon we are attemptingto address is basically classical and because as a practical matter it would beprohibitively difficult to do better.

We shall be dealing with temperatures of order200 MeV or less, so that neglect of heavier fields seems reasonable.To model a quench, we begin at a temperature well above Tc. The typical con-figurations have short correlation lengths and ⟨φ⟩∼0.

(⟨φ⟩̸= 0 because H ̸= 0. )One then takes the temperature instantaneously to zero.

The equilibrium config-uration is an ordered state with the φ field aligned in the σ direction throughout5

space, but this is not the configuration in which the system finds itself. The actual,disordered configuration then evolves according to the zero temperature equationsof motion obtained by varying (4).

Quenching in magnet models has been muchstudied in condensed matter physics [10]. However in that context it is usuallyappropriate to use diffusive equations of motion, because the magnet is alwaysin significant contact with other light modes (e.g.

phonons). For this reason thecondensed matter literature we are aware of does not directly apply to our problem.Let us now consider how the physics of quenching may be applied in relativis-tic heavy ion collisions.

In Bjorken’s [11] picture of such a collision, the incidentnuclei as seen in the center of mass frame are both Lorentz contracted into pancakeshapes. They pass through each other, and leave behind a region of hot vacuum.The baryon number of the incident nuclei ends up in that part of the plasma head-ing approximately down the beam pipe, and the central rapidity region which weconsider in this paper consists of plasma with approximately zero baryon num-ber which expands and cools through T = Tc and eventually hadronizes into thedetected pions.

If the collision is energetic enough to create a region of plasmawell above T = Tc, the φ field will indeed be fluctuating among an ensemble ofdisordered configurations. The plasma cools rapidly as it expands, and cannot beexactly in thermal equilibrium.

If it cools fast enough, the configuration of the φfield will “lag,” and as in a quench the system will find itself more disordered thanthe equilibrium configuration appropriate to the current temperature.We have done numerical simulations of quenching to zero temperature in thelinear sigma model with an explicit symmetry breaking term Hσ which makesthe pions massive. Turok and Spergel [12] have considered this scenario with noexplicit symmetry breaking term as a cosmological model for large scale structureformation in the early universe.

They find a scaling solution in which the size ofcorrelated domains grows at the speed of light as larger and larger regions comeinto causal contact and align.When the O(N) symmetry is explicitly broken,however, we do not expect a scaling solution. The Hσ term tilts the potential, thevacuum manifold is not degenerate, and in a time of order m−1πthe scalar field in6

all regions (whether in causal contact or not) will be oscillating about the sigmadirection.In numerically simulating a quench, we choose φ and ˙φ randomly independentlyon each site of a cubic lattice. This means that the lattice spacing a representsthe correlation length in the initial conditions.

If the initial conditions are cho-sen to model a thermal ensemble at some temperature T > Tc, then the O(N)symmetry is not spontaneously broken, and the lattice spacing a represents the πand σ correlation lengths which are approximately degenerate. Because the initialconditions must not distinguish between the σ and π directions in internal space,we must use the linear sigma model (4) instead of integrating out the σ which isheavy at zero temperature to obtain the nonlinear sigma model.

In the simulationwhose results are shown in Figure 1, we chose φ and ˙φ randomly from gaussiandistributions centered around φ = ˙φ = 0 and with root mean square variance v/2and v respectively. The three parameters v, H, and λ in (4) determine mπ, mσ,and fπ = ⟨0|σ|0⟩according toλ⟨0|σ|0⟩⟨0|σ|0⟩2 −v2−H = 0 ,(5)m2π =H⟨0|σ|0⟩,andm2σ = 3λ⟨0|σ|0⟩2 −λv2 .

(6)Note that fπ = ⟨0|σ|0⟩> v for H ̸= 0.In interpreting our results we mustremember that while in the code we are free to choose the energy scale by settingthe lattice spacing a = 1, a actually represents the initial correlation length. Inchoosing the parameters for Figure 1 we assumed that a = (200 MeV)−1 and thenchose v = 87.4 MeV = 0.4372 a−1, H = (119 MeV)3 = 0.2107 a−3, and λ = 20.0so that fπ = 92.5 MeV, mπ = 135 MeV, and mσ = 600 MeV.With parameters and initial conditions chosen, we evolve the initial config-uration according to the equations of motion using a standard finite difference,staggered leapfrog scheme.

(For details of the numerical method see [13,14] and inparticular equations 32-35 of [14].) We used a time step dt = min( a10,2π10mσ ).

We7

verified that our results do not change if the time step is reduced. After each twotime steps, we computed the spatial fourier transform of the configuration, andfrom that obtained the angular averaged power spectrum.

In Figure 1, we plot thepower in modes of the pion and sigma fields with spatial wave vectors of severaldifferent magnitudes k as a function of time. All the curves start at approximatelythe same value at t = 0 because the initial power spectrum is white since we choseφ independently at each lattice site.

The behaviour of the low momentum pionmodes is striking. While the initial power spectrum is white and, as ergodicityarguments would predict, the system at late times is approaching an equilibriumconfiguration in which the equipartition theorem holds, at intermediate times oforder several times m−1πthe low momentum pion modes are oscillating coherentlyin phase with large amplitudes.The simulation was done in a 643 box.

We verified that finite size effects arenot important even for the longest wavelength (ka = 0.20) mode shown in Figure 1by checking that the behaviour of the ka = 0.31 mode is the same in a 323 anda 643 box. The behaviour of only one component of the pion field is shown inFigure 1a.

The other two look qualitatively the same. It should be noted that theexact height of the peaks in the curves depend on the specific initial conditions.If the simulation is run with the same initial distribution for φ and ˙φ but with adifferent seed for the random number generator, the heights of the peaks change,and the relative sizes of the peaks in the three different pion directions change.The qualitative features of Figure 1 — the growth of long wavelength modes of thepion field — do not depend on the specific realization of the initial conditions.

Wediscuss below how the results change as a function of parameters in the potentialand the initial distributions.Let us compare the growth of long wavelength modes found numerically withexpectations from the mechanism previously discussed. Suppose the potential Vin (4) were simply V (φ) = (m2/2)φαφα.

Then, the equations of motion wouldbe linear, and modes with different spatial wave vector ⃗k would be uncoupled.Each curve in Figure 1 would be a sinusoid with period π/pm2 + ⃗k2 and constant8

amplitude. (The power spectrum, being quadratic in the fields, oscillates with onehalf the period of the fields.) The period of the oscillations in Figure 1a is indeedgiven by π/qm2π + ⃗k2, but the amplitudes are far from constant.

This behaviourcan be qualitatively understood by approximating φ2 in the nonlinear term in theequation of motion by its spatial average:φαφα(⃗x, t) ∼⟨φαφα⟩(t) . (7)This approximation is exact in the large N limit [12].

In our problem, N = 4.Using (7) and doing the spatial fourier transform, the equation of motion for thepion field becomesd2dt2⃗π(⃗k, t) = −m2eff(k, t)⃗π(⃗k, t)(8)with the time dependent “mass” given bym2eff(k, t) ≡−λv2 + k2 + λ⟨φ2⟩(t)(9)where k = |⃗k|.Figure 2 shows the time evolution of ⟨φ2⟩in the same simulation whose resultsare shown in Figure 1. In the initial conditions, φ is gaussian distributed with⟨φ2⟩< v2.

Therefore, for a range of wavevectors with k less than some criticalvalue, m2eff < 0 and the long wavelength modes of the pion field start growingexponentially. ⟨φ2⟩grows and then executes damped oscillations about its groundstate value ⟨0|σ|0⟩2.

A wave vector k mode of the pion field is unstable and growsexponentially whenever ⟨φ2⟩< v2 −k2/λ. Modes with k2 > λv2 can never beunstable.

The k = 0 mode is unstable during the periods of time when the ⟨φ2⟩curve in Figure 2 is below v2. Since ⟨φ2⟩is oscillating about ⟨0|σ|0⟩2 > v2, aftersome time the oscillations have damped enough that ⟨φ2⟩never drops below v2,and from that time on all modes are always stable and oscillatory.

In general,longer wavelength modes are unstable for more and for longer intervals of time9

than shorter wavelength modes as ⟨φ2⟩oscillates. Also, m2eff is more negativeand more growth occurs for modes with smaller k. Making the approximation (7)and thus using (8) cannot be expected to completely reproduce the effects of thenonlinear term in (4) which is local in position space.

Nevertheless, it predictsthat the long wavelength modes of the pion field go through alternating periodsof oscillatory behaviour and exponential growth spurts and therefore gives us agood understanding of the behaviour of these modes in Figure 1a. Because thetiming of the growth spurts for different (low) k modes are all determined by⟨φ2⟩, different modes have their growth spurts at the same times and all the longwavelength modes in Figure 1a oscillate in phase.

The longer wavelength modeshave more, stronger, and longer growth spurts, and are therefore amplified moreas in Figure 1a. In an equilibrium phase transition, explicit symmetry breakingkeeps the correlation length at Tc finite and, in the case of QCD, too short to be ofinterest.

[5] After a quench, on the other hand, arbitrarily long wavelength modesof the pion field are amplified even though the pion mass is non-zero.What about the σ modes? For σ2 < v2/3, the effective mass2 for the k = 0mode is negative.

This condition is far less likely to be satisfied than ⟨φ2⟩< v2.Hence, while some growth of the low momentum σ modes is possible, as Figure 1confirms, they do not grow as much as the low momentum pion modes do. Thebehaviour of the sigma modes is also complicated by the fact that oscillations withtwo periods — 2π/mπ and 2π/mσ — contribute.

Roughly, this occurs becauseoscillations ostensibly along the pion directions are oscillations in a curved valleyand are therefore seen in sigma also.At late times ⟨φ2⟩oscillates with small enough amplitude that it never goesbelow v2 and consequently no modes are ever unstable. Hence, if we make the ap-proximation (7) we would expect that at late times each mode in Figure 1a wouldcontinue to oscillate with approximately constant amplitude, with the longer wave-length modes maintaining the large amplitudes acquired during their exponentialgrowth spurts.

That this is not what is seen in Figure 1a reflects the effects weneglected in making the approximation (7). Because the modes are in fact cou-10

pled and the equations of motion are actually nonlinear, ergodicity argumentssuggest that eventually equipartition should apply. If the energy is equally dividedamong modes, then the (amplitude)2 in a mode should be inversely proportionalto (m2π + k2).

Also, the amplitude of the sigma modes at late times should be lessthan that of pion modes of the same k because mσ > mπ. The results shown inFigure 1 are consistent with the assumption that at late times equipartition ap-plies, although at t = 80a the longest wavelength pion modes are still decreasing inamplitude.

(We have verified that at even later times they do reach equipartition. )It is reasonable that longer wavelength modes take longer to decrease in ampli-tude than shorter wavelength modes.

As the system heads toward equipartition,power has to flow from low momentum modes to higher momentum modes. Thisplausibly happens more quickly at higher momentum because higher momentummodes are coupled to a larger number of modes, or, in the continuum, a largervolume of momentum space.

Hence, we end up with the striking phenomena ofFigure 1a: while the system begins and ends with small ⃗π(⃗k, t), in between thelong wavelength pion modes are dramatically amplified.In modelling a heavy ion collision, we should include the effects of the expansionof the plasma after a quench. In an equilibrium phase transition, the expansioncauses a decrease in the temperature.

After a quench, however, T = 0 and theexpansion has the effect of reducing the energy per unit volume in the system. Thesimplest way to model this is to introduce a time dependent lattice spacing a(t).This introduces a ˙φ˙a/a term in the equations of motion which, for a(t) increasingwith time, leads to damping.

Doing this, we found that each of the modes (nowdefined by spatially fourier transforming from comoving ⃗x to comoving ⃗k) redshifts with time. However, as long as the ˙φ term does not dominate over the ¨φterm low momentum modes still become unstable and grow with respect to highmomentum modes.In a real heavy ion collision, the expansion is anisotropic.The plasma expands much more rapidly along the beam direction than in thetransverse directions [11].

While we have not attempted to model this explicitly,it probably means that correlated volumes do not grow large in the longitudinal11

direction.The large expansion rate will rapidly damp modes of the pion fieldwith wave vectors in the longitudinal direction. However, it seems possible thatfor a particular longitudinal position (or, equivalently [11], a particular rapidity)coherent fluctuations on length scales all the way up to the transverse extent ofthe plasma could develop.

Another effect of the expansion is that after some time(perhaps of order 10 fm−1 ∼7m−1π[11]) the energy density is low enough that thedescription in terms of classical fields no longer makes sense. After this time, onehas individual pions flying offtowards the detector.

From Figure 1a, it is plausiblethat at this time there are large long wavelength oscillations of the pion field.If the initial conditions or parameters are such that the initial value of themean energy per unit volume E is much more than λ4v4 no striking growth of lowmomentum modes occurs. Indeed when the total energy is large, the field will beable to climb far up the potential to |φ| ≫v, and therefore ⟨φ2⟩(t) will always begreater than v2, and no modes will become unstable.

Requiring the energy not tobe too large compared to the height of the potential at φ = 0 can be seen either asa constraint on the parameters of the potential (V (φ = 0) ∼λ4v4) for given initialconditions or on the initial conditions for a given potential. In the simulation ofFigure 1, the total energy E ∼2λ4v4.

As parameters (or initial conditions) arechanged so that the total energy is increased, the growth of low momentum modesbecomes less and less prominent, and when E >∼10λ4v4, dramatic growth does notoccur.As long as φ is centered around φ = 0 initially, and as long as E is not toolarge, the choice of initial distributions has little qualitative effect. We have trieddistributions ranging from φ ≡0 with ˙φ chosen from a gaussian distribution to˙φ ≡0 with φ chosen from a uniform distribution with φ2 ≤v2, and in all casesobtained results similar to those of Figure 1.Let us attempt to estimate the conditions that might govern a realistic quench.We will make a crude estimate of the lattice spacing a and initial distributions forφ and ˙φ using the one loop temperature dependent effective potential with H = 0.12

At large temperatures, this is given by [15,16]V (φ, T) = m2(T)φ2 + λ4φ4(10)where the temperature dependent mass is given bym2(T) = λ(−v2 + T 2/4) . (11)While this is a crude approximation, it gives Tc = 2v = 175 MeV which is coinci-dentally quite close to the lattice gauge theory value [17].

If we quench from aninitial temperature Ti, the lattice spacing will be given by the correlation lengthat Tia = m−1(Ti)(12)For Ti = 1.2 Tc = 2.4 v, this gives a−1 = 2.97 v = 260 MeV.We will model initial conditions with correlation length a by choosing φ (and˙φ) independently at each lattice site. φ should therefore be distributed accordingto the probability distributionexp −a3Ti12m2(Ti)φ2 + λ4φ4.

(13)Because it is cut offby the quartic term, this distribution has much less weight atlarge |φ| than a gaussian. We approximate it by the truncated gaussian distributionexp −12m2(Ti)φ2for |φ| < φmax ;0 for |φ| > φmax(14)where φmax is chosen so that φ2 has the same expectation value in the distributions(13) and (14).

To choose initial conditions for ˙φ, we note that ⟨˙φ2⟩= ⟨φ2⟩inlattice units. (This is true for a quadratic potential, and is also true in general ifequipartition applies.) Hence, we use the same distribution for ˙φ as for φ.

We nowhave a crude recipe for choosing initial conditions appropriate to model a quenchfrom an initial temperature Ti to T = 0.13

What does our crude recipe predict? From (13), we expect that in lattice unitsφmax ∼(λ/2)1/4 = 0.56.

In fact, for Ti = 1.2 Tc, we obtain φmax = 0.63 a−1 =1.87 v.The value of φmax is crucial, because it determines over how much ofthe potential φ is distributed. We see that choosing φ gaussian distributed withvariance v/2 as we did in Figure 1 was overly optimistic.

For Ti = 1.2 Tc, it is morereasonable to assume that φ is distributed according to (14) with φmax = 1.87 v.With these initial conditions, the energy per unit volume turns out to be E ∼8λ4v2and the pion field behaves as shown in Figure 3. Low momentum modes do indeedgrow, but less dramatically than in Figure 1.

Also, the system finds its way to afinal state described by equipartition more quickly than in Figure 1. Nevertheless,there is a period of time after the quench when low momentum modes have muchmore power than higher momentum modes.If Ti is taken to be higher, φmax/v and 1/av increase, E/(λ4v4) increases, andthe phenomenon of interest is washed out.

It is not unreasonable, however, toconsider quenching from temperatures larger than but comparable to Tc. If theinitial temperature were much higher, the system could plausibly stay in thermalequilibrium until close to Tc.Hence, choosing Ti = 1.2 Tc and thus obtainingparameters and initial distributions like those of Figure 3 may be a reasonable ifcrude approximation to a real heavy ion collision.We have made many idealizations and approximations.

Nevertheless, it seemspossible to us that the essential qualitative feature of the phenomenon we have elu-cidated — long wavelength pion modes experiencing periods of negative mass2 andconsequent growth following a quench — could occur in real heavy ion collisions.Given the explicit symmetry breaking, one might have expected the dynamics fol-lowing a quench from a “generic” initial state to be featureless. The mechanismhere discussed provides a robust counterexample that should be applicable in othercontexts besides the QCD phase transition.

Examples could include the reheatingof some inflationary universe models and the quenching of spin systems which or-der at low temperatures, for which the interactions of the spins with phonons areless important than their interactions among themselves. If a heavy ion collision14

is energetic enough that there is a central rapidity region of high energy densityand low baryon number, and if such a region cools rapidly enough that it can bemodelled as a quench, this will be detected by observing clusters of pions of sim-ilar rapidity in which the the fraction of neutral pions is fixed. This ratio will bedifferent in different clusters and will follow a distribution like (2).

We can thinkof no process besides a QCD phase transition with the chiral order parameter farout of equilibrium that could produce such a signature.Acknowledgements: We are grateful to David Spergel and Neil Turok for giving usthe code created by the authors of [12, 13, and 14], and to both of them and Ue-LiPen for many very helpful conversations.15

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FIGURE CAPTIONS1) Time evolution of the power spectrum of the pion field and the sigma fieldat several spatial wavelengths after a quench at t = 0. φ and ˙φ were chosenat time t = 0 independently at each lattice site from gaussian distributionscentered at the origin as described in the text. The parameters of the poten-tial were chosen so that for a lattice spacing a = (200 MeV)−1, the physicalvalues mπ = 135 MeV, mσ = 600 MeV, and fπ = 92.5 MeV are obtained.The simulation was performed on a 643 lattice and the time step was a/10.After every two time steps, the spatial fourier transform of φ was computed.For each component of φ, the power in all the modes with k ≡|⃗k| in binsof width 0.057a−1 were averaged.

Figure 1a shows the time evolution of onecomponent of the pion field. The curves plotted are (top to bottom in thefigure) the average power in the modes in the momentum bins centered atka = 0.20, 0.26, 0.31, 0.37, 0.48, 0.60, 0.71, 0.94, 1.16, 1.39, and 1.84.

Theinitial power spectrum is white and all the curves start at t = 0 at approx-imately 0.01 in lattice units. Hence, the longest wavelength pion modes areamplified by a factor of order 1000 relative to the shortest wavelength modeswhich are not amplified at all.

Figure 1b shows the time evolution of thesigma field. Only four modes are plotted — ka = 0.20, 0.48, 0.94, and 1.84.The vertical scale is different than in Figure 1a — the sigma modes do notgrow nearly as much as the pion modes.2) Time evolution of the spatially averaged ⟨φ2⟩for the same simulation whoseresults are shown in Figure 1.

The horizontal line is at ⟨φ2⟩= v2.3) Same as for Figure 1a, except that in this figure we have used a−1 = 260 MeVand have chosen initial conditions from the distribution (14) following thecrude recipe outlined in the text to model a quench from Ti = 1.2 Tc to T = 0.Note that the vertical scale is different than in Figure 1a. All the curvesstart at about 0.03 at t = 0.

With these more realistic initial conditions,long wavelength modes of the pion field grow less than in Figure 1a, but are18

still significantly amplified relative to shorter wavelength modes.19

0102030405060708005101520FIGURE 1at/aPower

010203040506070800.2.4.6.81FIGURE 1bt/aPower

010203040506070800.1.2.3.4FIGURE 2t/a

01020304050607080012345FIGURE 3t/aPower

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