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BARYON NUMBER DIFFUSION AND INSTABILITIES

arXiv:hep-ph/9209229v1 11 Sep 1992BARYON NUMBER DIFFUSION AND INSTABILITIESIN THE QUARK/HADRON PHASE TRANSITIONFred C. Adams,1,2 Katherine Freese,1,2 and J. S. Langer21Physics Department, University of MichiganAnn Arbor, MI 481092Institute for Theoretical PhysicsUniversity of California, Santa Barbara, CA 93106submitted to Physical Review Letters26 August 1992AbstractHadron bubbles that nucleate with radius Rnuc in a quark sea (if the phase transitionis first order) are shown to be unstable to the growth of nonspherical structure when thebubble radii exceed a critical size of 20−103 Rnuc. This instability is driven by a very thinlayer of slowly diffusing excess baryon number that forms on the surface of the bubblewall and limits its growth.

This instability occurs on a shorter length scale than thosestudied previously and these effects can thus be important for both cosmology and heavyion collisions.PACS Numbers: 12.38.Mh, 27.75.+r, 64.60.–i, 68.70.+w, and 98.80.Dr,Cq,Ft.1

If a first order phase transition occurs when quarks are confined into hadrons at atemperature TC = 100 – 200 MeV, many interesting physical consequences can arise [1].Previous investigations [1, 2] have shown that the baryon number density can be higherin the quark phase than in the hadron phase. In the case of cosmology, this asymmetrycan lead to inhomogeneous distributions of baryons after the phase transition [1, 3].

Thispossibility has led to numerous studies of big bang nucleosynthesis in the presence ofbaryon inhomogeneities [3 – 7]. Another possible relic of such a phase transition is stablemacroscopic quark matter – quark nuggets – which may survive to the present epochand contribute to the dark matter of the universe [1, 8].

The effects of a first orderquark/hadron (Q/H) phase transition are also, in principle, detectable in relativisticheavy ion collisions.In this Letter, we assume that the Q/H phase transition is first order and study anew instability in the growth of hadron bubbles. When the background plasma, eitherin the early universe or in a heavy ion collision, cools to a temperature below the criticaltemperature TC, bubbles of the hadronic phase nucleate and begin growing spherically.Witten [1] has argued that baryon number prefers to be in the quark phase.

As a result,the bubble rejects baryon number as it grows.Because baryon number diffuses veryslowly, it is confined and concentrated into a thin layer on the surface of the bubble.We argue that this boundary layer must limit the growth of the bubble and cause it tobecome morphologically unstable.In this analysis we assume that the latent heat of the phase transition is transportedradiatively, mostly by neutrinos, since they have the longest mean free path. The rela-tive importance of radiation and hydrodynamic flow in removing the latent heat remainsunsettled [1, 4, 9, 10].

Previous work has studied the (stable) evolution of hadron bub-bles when hydrodynamic flow dominates the removal of latent heat [11, 12]. However,radiative processes are present and perhaps dominant from the moment of bubble nucle-ation onward.

In addition, we expect that the new instabilities (due to baryon numberdiffusion) presented here can arise regardless of the type of heat flow.In previous work [9], we showed that when radiative effects dominate, hadron bubblesare unstable for bubble radii greater than the mean free path of the neutrino (∼10 cm)because the heat flow becomes diffusive at that size scale. Hydrodynamic instabilitiesduring hadron bubble growth [13], if present, occur on a smaller scale of order 10−6cm.

The instability proposed here occurs on yet smaller scales, at most 105 fm, andthus greatly alters the manner in which the phase transition proceeds. The resultinginhomogeneities in baryon number are too small (in length scale) to affect big bangnucleosynthesis.In fact, this size scale is small enough that this new instability canoccur in heavy ion collisions, whereas the previously studied instabilities cannot.

Weemphasize, however, that significant uncertainties exist in both the QCD physics and ourknowledge of nonequilibrium pattern formation.We start by considering a locally flat section of a bubble wall at which the hadronicphase is advancing at speed v into the quark sea. Previous work [1, 2] has shown that,when these phases are in equilibrium with each other, the baryon number in the quarkphase nq can be much higher than that in the hadron phase nh; the ratio nh/nq istypically 0.003 – 0.2.

For simplicity, we assume that this ratio vanishes and henceforth2

consider only quantities in the quark phase.In a frame of reference moving with the bubble wall, the diffusion equation for thebaryon number density n in the quark phase is∂n∂t −v ∂n∂z = D∇2n,(1)where the direction of motion is along the z-axis and where D is the diffusion constant.We will assume that v ≈constant; we justify this assumption below. Since quarks carrythe baryon number in this phase, the diffusion constant is given by D = ⟨vq⟩ℓq/3, where⟨vq⟩is the mean speed of the quarks (⟨vq⟩is fairly close to the speed of light c) and ℓq isthe mean free path of a quark (about 1 fm for TC ∼200 MeV).Deep in the quark phase, very far from the advancing hadronic front, the baryonnumber density has a value appropriate for thermal equilibrium at temperature T∞, i.e.,n∞= g (µ/T)∞T 3∞, where the constant g depends on the number of degrees of freedomin the quark phase and where µ is the chemical potential.

For our universe, standard bigbang nucleosynthesis [14, 15] constrains µ to be very small, i.e., (µ/T)∞∼10−9 −10−8.For heavy ion collisions, (µ/T)∞∼10−4 −10−2.The diffusion equation (1) must be supplemented by boundary conditions at theQ/H interface. We assume local thermodynamic equilibrium, so that the baryon num-ber density nI and temperature TI at the interface are related by a Gibbs-Thompsoncondition [16, 17] which, in this case, takes the formTI = TC(1 −d0κ) −βT∆n2I.

(2)Here, κ is the interfacial curvature; d0 ≡σTCCp/L2 is a capillary length and is propor-tional to the surface tension σ; the specific heat is Cp = O(T 3C); and the latent heat isL = O(T 4C). Lattice gauge theory calculations [18] have estimated the surface tension tobe σ ≈0.1 T 3C.

The third term in Eq. (2), βT∆n2I, is the decrease in coexistence temper-ature due to the presence of baryon number (solute).

Notice that this term is quadraticrather than linear in the solute concentration nI. We have defined a convenient energyscale T∆≡L/Cp, which we expect to be O(TC).

To estimate the value of β, we usea bag model for the equation of state (and we take nh/nq = 0). We find that β ≈0.6(TC/T∆)T −6C≈0.6 T −6C , where we have used the results of Ref.

[11] in obtaining thenumerical coefficient.Continuity of baryon number at the interface requiresnIvI = −Dhˆν · ∇niz=zI,(3)where zI is the interface position, ˆν is the unit normal to the interface, and vI is its speedin the ˆν-direction. The physical meaning of Eq.

(3) is that the rate at which excessbaryon number is rejected from the interior of the hadronic phase is balanced by the rateat which baryon number diffuses ahead of the two-phase interface. This effect producesthe destabilizing boundary layer of excess baryon number on the surface of the growingbubble.3

To complete the statement of our model, we must impose a condition of heat continu-ity at the moving interface. We can consider a full treatment of the heat transport eitherthrough a diffusion equation for temperature or through a radiative transfer equation [9].For the parameter regime of interest, however, neutrinos carry away most of the energyand the mean free path of the neutrino is ∼10 cm, much greater than the size scale ofthe bubble.

Hence, the condition of thermal continuity reduces to the simple form [9]v0 = cTI −T∞T∆. (4)Our assumption here is that all of the latent heat produced in the phase transition isefficiently carried out of the system by radiation and that the interface moves at the fastestspeed for which this condition holds.

[We have implicitly assumed that the bubbles arefar enough apart that the latent heat released during the evolution of a given bubbledoes not affect other bubbles. If, however, the mean bubble separation is less than theneutrino mean free path, mutual heating effects should be considered.

]We now consider the stability of the moving bubble wall. As we will see, the situ-ations of interest to us are those for which nI ≫n∞.

In this case, the baryon numberdensity in the boundary layer ahead of the unperturbed wall at zI = 0 is accurately givenby Eqs. (1) and (3) to be nI exp(−2z/ℓ), where the layer thickness ℓ= 2D/v is verysmall for small D. To study the stability of this wall against periodic deformations ofwave-vector k, we consider linear perturbations of the formδn(z, x, t) ∼ˆnk expik · x −qkz + ωkt,δzI(x, t) ∼ˆzk expik · x + ωkt,(5)δTI(x, t) ∼ˆTk expik · x + ωkt,where x denotes the position in the plane of the wall.After performing a standardstability analysis [19], we find that the amplification rate ωk isωk =v (qkℓ−2)/ℓ1 + D(qkℓ−2)/2βn2Icℓ"1 −edℓk24βn2I#,(6)where ed ≡(TC/T∆)d0 and qkℓ≡1 +(1 + k2ℓ2 + ωkℓ2/D)1/2.

The crucial result is thatthe wall is unstable, i.e., ωk > 0, for wave-vectors k < kC, wherekC ≡4βn2I/ edℓ1/2 . (7)We now propose two separate estimates of the characteristic length scales for patternsproduced by this instability.

First, consider a growing hadronic bubble with radius Rand v = ˙R. If we assume that all the baryon number that was swept out by the growingbubble remains in the boundary layer at its surface, then 2πR2ℓnI = 4πR3n∞/3.

With4

this assumption, Eqs. (2) and (4) can be combined to yield a differential equation forR(t):vc =˙Rc = ∆−2 edR −βn2∞R ˙R3D2,(8)where ∆≡(TC −T∞)/T∆is a dimensionless measure of the undercooling.

The condition˙R = 0 determines the nucleation radius Rnuc = 2 ed/∆. For TC = 200 MeV and σ = 0.1T 3C,Rnuc ≈(0.2 fm)/∆.

The relevant value of ∆is unknown, although cosmological studiesusually assume small values, e.g., ∆∼10−2 – 10−4; for heavy ion collisions, we assumea larger value ∆∼1. The bubble becomes unstable for R ≫Rnuc.

When the instabilityoccurs, however, the third term in Eq. (8) is still very small (the results presented belowverify this claim); thus, our implicit assumption of a nearly constant growth speed v isvalid.Our stability analysis implies that the bubble becomes morphologically unstablewhen its radius is appreciably larger than RC = 2π/kC.

(This result can easily beconfirmed by a more systematic analysis in spherical coordinates.) Using our formula forkC and the conservation of baryon number assumption to eliminate nI, we findRCRnuc≈"Dc ed#3/4 "∆βn2∞#1/4.

(9)Notice that D/c ed is of order unity. For cosmology, βn2∞∼10−17.

If we take an un-dercooling of ∆= 10−3, we find RC/Rnuc ∼103, or RC ∼105 fm. [Notice that ourapproximation nI/n∞≫1 is valid; for this case, nI/n∞= 2R/3ℓ≈103.] For heavy ioncollisions, we obtain RC/Rnuc ∼20, or RC ∼4 fm.The problem with the above argument is that it says little about the final con-figuration of the bubble.

Indeed, linear stability analysis is notoriously unreliable forpredicting quantitatively the final relevant length scales for such highly nonlinear pro-cesses. Unfortunately, the present situation is not, as far as we know, directly analogousto pattern-forming systems that have been studied experimentally.

The closest analogythat comes to mind is directional solidification of a dilute alloy where a planar front be-comes unstable and breaks up into elongated cellular structures. The excess solute (here,baryon number) that accumulates ahead of the front diffuses along the boundary layerand ultimately is trapped, in concentrated form, in the narrow, sometimes filamentary,interstices between the cells.

The filaments themselves are usually unstable and break upinto chains of droplets, which are made up of concentrated solute and trail behind themoving front. Although this mechanism is well known in metallurgy, no reliable theoryfor predicting the cellular spacing is available.As our second estimate of characteristic length scales produced by the instability,we propose the following.

Suppose that the Q/H interface leaves behind it baryon-richfilaments with characteristic spacing W. Clearly, W cannot be larger than the stabilitylength 2π/kC; we therefore assume marginal stability and set W ∼k−1C . Let us furtherassume that the radii of the filamentary interstices, where they join the moving interface,are roughly equal to the diffusion length ℓor, equivalently, Rnuc, both of which are5

about the same size and seem to be the only relevant length scales. (We expect that thisassumption is the weakest part of our argument.

In any case, further work should bedone to check its validity.) We can now use global conservation of baryon number, i.e.,n∞W 2 ≈nIℓ2.

From this result and our formula for kC, we findWRnuc∼"∆βn2∞#1/6,(10)which implies cosmological values of W about one order of magnitude smaller than thosepredicted by Eq. (9).We can take this argument one step further and estimate the size of the filaments farbehind the advancing front.

The filaments will contract to a radius ̟ where the baryonnumber density nf is sufficiently large that T = T∞and v = 0; the Gibbs Thompsonrelation applied on a tubular surface is∆+ ed/̟ = βn2f,(11)where the curvature has changed sign (from Eq. [2]) because the hadronic phase is nowon the outside.

Conservation of baryon number requires that nIℓ2 ≈nf̟2, and we thusobtain̟Rnuc∼"βn2∞∆#1/9,(12)which implies that ̟ is of order 1 fm.Thus, the size scale of any baryon numberinhomogeneities produced in the phase transition will be much smaller than that required(> 100 cm) to affect big bang nucleosynthesis.In summary, we have shown that the growth of hadron bubbles during the Q/Hphase transition can be unstable due to a new instability mechanism. The instabilityfirst sets in when the bubble grows to a radius of ∼20−103 times the original nucleationradius.

These instabilities can greatly alter the long term evolution of the bubbles. Wehave presented here a marginal stability hypothesis in which the moving interface leavesbehind tubular regions which are rich in baryon number and which ultimately break upinto droplets of size ̟ ∼1 fm.The most important of our assumptions are as follows: (A) We have assumed thatthe Q/H phase transition is in fact first order.We stress that the evidence on thisissue remains divided.

(B) We have assumed local thermodynamic equilibrium and havewritten the thermodynamic condition at the bubble interface as the Gibbs Thompsonrelation given in Eq. (3).The true form of this relation depends on the details ofQCD physics which are not yet well understood.

(C) We have assumed that hadronbubble growth is limited by the diffusion of baryon number away from the interface.This assumption is expected to be valid for sufficiently large baryon number densities(or, equivalently, large µ/T) and/or sufficiently large bubble radii. We note that the firstof these conditions is more likely to be met in heavy ion collisions, whereas the second is6

more likely to be met in the early universe. (D) We have assumed that all of the latent heatproduced by the phase trasition is carried away by radiation and hence no hydrodynamicflow occurs.

We note that even when this assumption is violated, the requirement ofbaryon number diffusion can still lead to instabilities which are qualitatively similar tothose presented in this paper. (E) We have invoked a marginal stability hypothesisto describe the long term evolution of the system.This type of behavior is not wellunderstood even in laboratory systems, much less in the early universe.AcknowledgementsWe would like to thank D. Kessler, L. Sander, and T. Matsui for helpful conversationsand the ITP at U.C.

Santa Barbara, where the inspiration for this work took place, forhospitality. FCA is supported by NASA Grant No.

NAGW–2802. KF is supported byNSF Grant No.

NSF-PHY-92-96020, a Sloan Foundation fellowship, and a PresidentialYoung Investigator award. JSL is supported in part by NSF Grant No.

NSF-PHY-89-04035 and DOE Grant No. DE-FG03-84ER45108.7

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