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Emergence of long wavelength pion oscillations

arXiv:hep-ph/9308355v1 28 Aug 1993August, 1993Emergence of long wavelength pion oscillationsfollowing a rapid QCD phase transitionKrishna Rajagopal⋆Department of PhysicsJoseph Henry LaboratoriesPrinceton UniversityPrinceton, N.J. 08544ABSTRACTTo model the dynamics of the chiral order parameter in a far from equilibriumQCD phase transition, we consider quenching in the O(4) linear sigma model.We summarize arguments and numerical evidence which show that in the periodimmediately following the quench arbitrarily long wavelength modes of the pionfield are amplified. This results in large regions of coherent pion oscillations, andcould lead to dramatic phenomenological consequences in ultra-relativistic heavyion collisions.My talk was a description of work done with Frank Wilczek [1].In theseproceedings, I sketch our central results, emphasizing several points that wereraised in discussions at the conference.The interested reader should, however,consult Ref.

[1] and our earlier work [2] for a more detailed exposition.⋆Address after Sept. 15: Department of Physics, Harvard University, Cambridge, MA 02138.Talk given at the Quark Matter ’93 conference in Borl¨ange, Sweden, in June, 1993.Research supported by a Charlotte Elizabeth Procter Fellowship.

1. Misalignment of the chiral condensateAmong the most interesting speculations regarding ultra-high energy hadronicor heavy nucleus collisions is the idea that regions of misaligned vacuum mightoccur [3].

Misaligned regions are places where the four-component field φα ≡(σ,⃗π),that in the ground state takes the value (v, 0) is instead partially aligned in the⃗π directions. Because of the explicit chiral symmetry breaking (i.e because thepion is not massless), in such a region φ would oscillate about the σ direction.If they were produced, misaligned vacuum regions would relax by coherent pionemission — they would produce clusters of pions bunched in rapidity with highlynon-Gaussian charge distributions.

In each such cluster, the ratioR ≡nπ0nπ0 + nπ+ + nπ−(1.1)is fixed. Among different clusters, R varies and is distributed according toP(R) = 12R−1/2 .

(1.2)As an example of (1.2), we note that the probability that the neutral pion fraction Ris less than .01 is .1! This is a graphic illustration of how different (1.2) is from whatone would expect if individual pions were emitted with no isospin correlations manypions.

We have proposed [1] a concrete mechanism by which such phenomena mayarise in heavy ion collisions for which the plasma is far from thermal equilibrium.2

2. Emergence of long wavelength pionoscillations following a quenchIn studying the behaviour of the plasma in the central rapidity region of a heavyion collision at RHIC energies or higher, it seems reasonable to assume that aftera time of order 1 fm a hot plasma is formed in which the chiral order parameter isdisordered and in which the baryon number density is low enough that it can beneglected.

Our goal is to study the behaviour of the long wavelength modes of thechiral order parameter as this plasma loses energy and σ develops an expectationvalue.In previous work (Ref. [2], references therein, and Wilczek’s talk at this con-ference) we considered the equilibrium phase structure of QCD.

We argued thatQCD with two massless quark flavours probably undergoes a second order tran-sition. At first sight, this might seem ideal for the development of large regionsof misaligned vacuum, since the long wavelength critical fluctuations characteris-tic of a second order transition are such regions.

Unfortunately, the effect of thelight quark masses spoil this possibility [2]. While in lattice simulations it is inprinciple possible to reduce the light quark masses below their physical values andget arbitrarily close to the second order critical point, in heavy ion experiments wemust live with a pion which has a mass comparable to the transition temperature.Near Tc, the correlation length in the pion channel is shorter than T −1c[2], and asa result the misaligned regions almost certainly do not contain sufficient energy toradiate large numbers of pions.Here, we consider an idealization which 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 witha high temperature heat bath and subsequently evolve mechanically according tozero temperature equations of motion.

In a real heavy ion collision, the phasetransition proceeds by a process in between an equilibrium phase transition inwhich the temperature decreases arbitrarily slowly and a quench in which thermal3

fluctuation ceases instanteously. Our goal is not to quantitatively model a realisticheavy ion collision as a quench.Rather, in studying the dynamics of the longwavelength modes of the chiral order parameter in a quench, our hope is that thequalitative behaviour in this model is representative of the physics which occursin real processes in which the QCD plasma cools rapidly and is far from thermalequilibrium.We use the linear sigma model to describe the low energy interactions of pions:L =Zd4x12∂µφα∂µφα −λ4(φαφα −v2)2 + Hσ,(2.1)where λ, v, and H ∝mq are to be thought of as parameters in the low energyeffective theory obtained after integrating out heavy degrees of freedom.

We treat(2.1) as it stands as a classical field theory, since the phenomenon we are attemptingto describe is basically classical and because as a practical matter it would beprohibitively difficult to do better.Our numerical simulations of quenching in the linear sigma model are describedin more detail in [1]. As initial conditions, we choose φ and ˙φ randomly indepen-dently on each site of a cubic lattice.

Therefore, the lattice spacing a representsthe correlation length in the disordered initial state. In [1] we made a crude at-tempt to choose initial distributions for φ and ˙φ appropriate for a quench from aninitial temperature of T = 1.2Tc.

With initial conditions chosen, we then modelthe T = 0 evolution of the system after the quench by evolving the initial config-uration using a standard finite difference, staggered leapfrog scheme according tothe equations of motion obtained by varying (2.1). After each two time steps, wecompute the spatial fourier transform of the configuration and from that obtainthe angular averaged power spectrum.The central result of our simulations is that the power in the long wavelengthmodes of the pion field grows dramatically.

While the initial power spectrum iswhite and while at late times the system is approaching a configuration in which theenergy is partitioned equally among modes, at intermediate times of order several4

times m−1πthe low momentum pion modes are oscillating with large amplitudes.When we used gaussian initial distributions for φ and ˙φ with width v/2 and vrespectively, the power in modes with k = 0.2a−1 ≃0.3mπ is more than 1000times that in the initial white power spectrum. For initial conditions chosen tomodel an initial temperature of 1.2Tc, the amplification is less, but is still of order100.The amplification of low momentum modes which we observe in the numericalsimulations can be qualitatively understood by approximating φαφα(⃗x, t) in theequations of motion by its spatial average.

After doing the spatial fourier transform,the equation of motion for the pion field becomesd2dt2⃗π(⃗k, t) = −{−λv2 + λ⟨φ2⟩(t) + k2}⃗π(⃗k, t)(2.2)where ⟨φ2⟩(t) means simply the spatial average of φ2. At late times, ⟨φ2⟩becomestime independent and takes its vacuum value, and the quantity in brace bracketsin (2.2) becomes simply m2π + k2.

Immediately after the quench, however, ⟨φ2⟩varies with time, and there are periods when ⟨φ2⟩< v2. A wave vector k mode ofthe pion field is unstable and grows exponentially whenever ⟨φ2⟩< v2 −k2/λ.

As⟨φ2⟩varies, longer wavelength modes are unstable for more and for longer intervalsof time, and, in agreement with the numerical simulations, are amplified relativeto shorter wavelength modes.3. Charge separation does not occurThe striking prediction (1.2) for the probability distribution of the neutral pionfraction R naturally leads to the question of whether there are similarly unusualfluctuations in the electric charge itself, i.e.in the ratio of π+ to π−mesons.Formulae similar to (1.2) hold for the real fields π1 =1√2(π+ + π−) and π2 =1i√2(π+ −π−), but not for π+ and π−.

While the total electric charge must beconserved, there is no conservation law prohibiting the separation of charge into5

regions of net positive and negative charge.We must determine whether longwavelength oscillations of the electric charge density grow. The charge operatorj0 = π1 ∂∂tπ2 −π2 ∂∂tπ1 measures rotary motion in the π1 −π2 plane.

However,the amplification mechanism which operates following a quench kicks ⃗π radiallyoutward, and does not induce rotary motion. This heuristic argument is borne outin the simulations.

Long wavelength oscillations of the electric charge density arenot amplified.Notice that the question of whether charge separation occurs is a dynamicalone, and has a straightforward dynamical answer for the quench mechanism ofgenerating regions of misaligned condensate. In earlier work, Kowalski and Taylor[3] imposed isospin symmetry by hand in order to avoid the possibility of chargeseparation, which they consider physically implausible.

The coherent states wereach are not isospin singlets, and we see no reason to impose that condition;nevertheless the intuition of Kowalski and Taylor is vindicated and there is nocharge separation.4. How large are the regions of coherent pion oscillations?This question is of crucial phenomenological interest.

In order to be observable,these regions must evolve into sufficiently many pions. We must ask, therefore,what are the longest wavelength modes of the pion field which get amplified?In our simulations, the answer is unequivocal — the wavelength of modes whichare amplified is limited only by the size of the lattice on which the simulation isrun.

Alas, it is much harder to determine what will happen in a real heavy ioncollision. The one thing that can be said with certainty is that the effect is notcut offby the inverse pion mass.Modes with k < mπ are amplified.This isin marked contrast to the situation obtained in thermal equilibrium, and is whythe phenomenon will only be detected if the plasma in a heavy ion collision is farenough from thermal equilibrium that quenching is an appropriate idealization.If m−1πis not the long wavelength cut-off, what is?The most optimistic (and6

perhaps implausible) possibility, which is suggested by a literal interpretation ofour simulations, is that coherent oscillations of the pion field in regions as largeas the transverse extent of the plasma are possible. (The rapid expansion in thelongitudinal direction will damp the growth of modes with ⃗k parallel to the beamrelative to those with ⃗k transverse.) In a real collision, the size of modes whichgrow could perhaps be limited by the time available before the pions no longerinteract and therefore can no longer be described by oscillations of a classical field,or perhaps by the size of regions of the plasma in which the energy density isreasonably homogeneous.

At this point, the most that can be said is that longwavelength pion oscillations are amplified after a quench, and that their size is notlimited by any microphysical length like m−1πbut is limited dynamically, perhapsonly by the system size.5. OutlookAlthough we have made many idealizations and approximations, it seems pos-sible that the essential qualitative feature of the phenomenon we have elucidated— long wavelength pion modes experiencing periods of negative mass2 and conse-quent growth following a quench — could occur in real heavy ion collisions.

Giventhe explicit symmetry breaking which gives mass to the pions, one might haveexpected the dynamics following a quench to be featureless. The mechanism herediscussed provides a robust counterexample.

We have not come close to modellinga real heavy ion collision. While our treatment can surely be improved, it seemsdoubtful that quantitative theoretical predictions for heavy ion collisions will bepossible.

At the end of the day, the question of whether or not long wavelengthpion oscillations occur will be answered experimentally. If a heavy ion collision isenergetic enough that there is a central rapidity region of high energy density andlow baryon number, and if such a region cools rapidly enough that the process canbe modelled as a quench, this will be detected by observing clusters of pions ofsimilar rapidity in which the neutral pion fraction R is fixed.

This ratio will be7

different in different clusters and will follow a distribution like (1.2). Were sucha signature to be observed experimentally, it would be clear evidence for an outof equilibrium transition from a QCD plasma in which the chiral order parameterwas initially disordered.REFERENCES1.

K. Rajagopal and F. Wilczek, to appear in Nucl. Phys.

B, hep-ph/9303281,Princeton preprint PUPT-1389, IASSNS-HEP-93/16, 1993.2. K. Rajagopal and F. Wilczek, Nucl.

Phys. B399 (1993) 395.3.

A. Anselm and M. Ryskin, Phys. Letters B226 (1991) 482; J.-P. Blaizotand A. Krzywicki, Phys.

Rev. D46 (1992) 246; J. D. Bjorken, Int.

J. Mod.Phys. A7 (1992) 4189; J. D. Bjorken, Acta Phys.

Pol. B23 (1992) 561; K.L.

Kowalski and C. C. Taylor, preprint hepph/9211282, 1992; J. D. Bjorken,K. L. Kowalski, and C. C. Taylor, SLAC preprint SLAC-PUB-6109, 1993.8

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