Travis NorsenDepartment of Physics, University of Washington, Seattle, Washington 98195
~Received 14 January 2002; published 10 April 2002!
We study the transition from npe-type nuclear matter~consisting of neutrons, protons, and electrons! tomatter containing strangeness, using a Walecka-type model predicting a first-order kaon-condensate phasetransition. We examine the free energy of droplets ofK matter as the density, temperature, and neutrino fractionare varied. Langer nucleation rate theory is then used to approximate the rate at which critical droplets of thenew phase are produced by thermal fluctuations, thus giving an estimate of the time required for the new~mixed! phase to appear at various densities and various times in the cooling history of the protoneutron star.We also discuss the famous difficulty of ‘‘simultaneous weak interactions’’ which we connect to the literatureon nontopological solitons. Finally, we discuss the implications of our results to several phenomenologicalissues involving neutron star phase transitions.
A recent ‘‘hot’’ topic in nuclear physics has been the atempt to understand the production of strangeness in thepanding plasma thought to be created in relativistic collisioof heavy ions. There are several aspects to this complexdifficult problem, including how~and, indeed, whether! athree-flavor deconfined quark-gluon plasma is producedthe initial collision; how this strange matter equilibrates, epands, and cools; and how it eventually rehadronizes, hofully giving rise to a unique observable signal. The goalthese studies is to better understand the phase structuQCD in the low-density, high-temperature region of tphase diagram.
A different but complementary problem involves thstudy of nuclear matter at high density and low temperatusuch as that existing in neutron star interiors. Here maconsisting of neutrons, protons, and electrons~npe matter! iscrushed during the gravitational collapse of the parent s~Actually, what we call npe matter in this paper may alinclude muons and neutrinos.! As the star collapses weainteractions convert electrons and protons into neutrons, welectron neutrinos produced copiously as a by-product.ditionally, neutrinos of all flavors are created throubrehmstrahlung-type collisions among the warm nucleoAs the neutrinos diffuse outward and radiate away, the ptoneutron star cools from an initial temperature of sevetens of MeV, and eventually settles into the familiar groustate of a neutron star with central density several timnuclear matter density.
It is now understood that at sufficiently high density,phase transition will occur in which neutrons and protonsnpe matter are replaced by deconfined quarks~includingstrange quarks! @1–3#. A related mechanism for the appeaance of strangeness in neutron star matter is kaon condetion, which may become possible at densities somewlower than the density required for the deconfining transitdescribed above. The possibility of kaon condensationfirst pointed out by Kaplan and Nelson, who were motivaby the chiral Lagrangian prediction of an attractive interation betweenK2 and nuclear matter@4#.
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Intuitively, one can understand the phenomenon of kacondensation in the following way. As the density of nmatter is increased, and assuming that all particle speciesin equilibrium with respect to the weak interactions, the eletron chemical potential increases. Simultaneously, the eftive mass of an in-mediumK2 will decrease, due to theattrative interaction mentioned above. Therefore, at socritical density, the electron chemical potential~i.e., the en-ergy of the electrons at the top of the Fermi sea! will becomegreater than the effective mass of theK2. It then becomesenergetically favorable for the high-energy electrons to deaccording to
e2→ne1K2 ~1!
at which point a condensate ofK2 will form.This picture is somewhat oversimplified, as the produ
tion of kaons does not proceed exclusively by the decayenergetic electrons, but may also occur via fully hadroweak couplings, e.g.,n→p1K2. Also, the notion of a uni-form condensate of kaons is probably wrong. As pointedby Glendenning@5#, the presence of two separate conservcharges in neutron star matter~namely, the baryon numbeand the electric charge! gives rise to the possibility of amixed phase ofN matter andK matter existing over a widerange of densities and pressures in the star.~By ‘‘ N-matter’’we mean matter not containing kaons, but without thestriction of charge neutrality implied by ‘‘npe matter.’’! Thismixed phase will have the form of a Coulomb lattice—tnow standard ‘‘pasta’’ structure consisting of droplets of tnew phase immersed in a background ofN matter ~at lowdensities!, with rods and slabs replacing droplets as the dsity is increased, and finally with the role of the two phasinterchanging in the transition toward homogenousK matteras the density is increased still further@6#.
In this paper, we will assume a model which predicts juthis kind of first-order phase transition to kaon-condensmatter. For the description of nuclear matter, we useWalecka-type model in which the interactions of stronginteracting particles are mediated bys, v, and r mesons,
treated in the mean-field approximation. Kaons are incluin the model on the same footing as the nucleons, asscribed in Sec. II.
However, our goal is not to study the structure of tground state predicted by this model~and its subsequent effect on the global neutron star properties such as the mradius relation!, but rather to study thenucleationof kaons,that is, the process by which the initial protoneutron smatter with zero strangeness acquires strangeness thrthe spontaneous appearance ofK-matter droplets. We willbegin this study by using the theory of homogenous nuation from Langer@7#. Here one assumes that at a givtemperature, there are constant fluctuations producing sshort-lived droplets of the new phase. Below a certain critisize, the surface energy cost of these droplets wins outthe volume term and the droplets shrink away. For socritical size, however, the free energy gained from the pduction of a large volume of the new~energetically favored!phase is just large enough to cancel the cost of surfaceergy, and the droplet will spontaneously grow. In the castudied here, because the two phases have nonzero elcharge densities, there is also a Coulomb term in theenergy, which becomes large for large droplets. Hence apercritical droplet will not grow forever, but will reachstable size at which the energy gain between the volumesurface energies just balances the energy cost of the Clomb energy.
Langer showed that the expected time needed forkind of phase transition to proceed depends on the probaity of a critical droplet being produced by thermal fluctutions, and also on the growth rate of such a critical dropalong the unstable direction in configuration space. Assuing a thermal distribution of these fluctuations, the nucleatrate per unit volume can be written
G;I 0e2DF(Rcrit )/T, ~2!
whereDF(Rcrit) is the excess free energy of a critical drolet, T is the temperature, andI 0 ~the ‘‘prefactor’’! is a micro-scopic fluctuation rate related to the growth rate of a supcritical droplet, the thermal conductivity of the medium, aother properties@7–9, 42#. The prefactor is often approximated by dimensional analysis as simplyI 0;T4. We willdiscuss this approximation later. Our goal is to apply tnucleation rate theory to the case of cooling protoneutstar matter in order to better understand exactly when, whand how the transition from npe-type matter to the kacondensed phase occurs.
Our work generally follows previous work on the nuclation of quark matter droplets in neutron star matter, whhas been studied extensively@10–13#. The case of kaon condensation is physically unique, however, since it is the orealistic proposed example of a direct first-order transitfrom npe-type neutron star matter to matter containstrangeness. In the case of the deconfinement transition mtioned above, for example, and even at densities, temptures, and pressures for which three-flavor quark matteenergetically favored, the transition is indirect in the following sense: an intermediate stage of two-flavor quark ma
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will be produced first, with strange quarks then slowly apearing during a smooth crossover or second-order transto the three-flavor ground state. This latter transition is,course, slow due to the weakness of the weak interactionthis scenario, however, the original nucleation events whtake one from npe matter to deconfined quark matter, nnot involve the weak interactions at all.
The case of kaon condensation is radically different, sinhere there is no intermediate zero-strangeness state wmight allow for fast nucleation followed by a slow busmooth growth of the strangeness-containing fields. Instethe thermal fluctuations responsible for nucleation evemust directly involve the weak-interaction processes whproduce kaons. The difficulties posed by the widely varyitime scales involved in these two sorts of processes~thermalfluctuations and weak interactions! will form the theme ofour discussion.
The outline of the rest of the paper is as follows. In Secwe present the details of the nuclear mean field theory ufor the subsequent discussion. In Sec. III we use this moto extract information about the free energy ofK-matterdroplets of different sizes, as the background baryon numdensity and temperature are varied. This allows us to use~2! to estimate the nucleation rate for the kaon-condensphase transition. In Sec. IV we discuss in more detailunderlying mechanism for the fluctuations which producenucleation, and thereby analyze the trustworthiness ofestimates. Here we also make contact with the literature oQballs, and argue that the problem of direct strangeness nuation in neutron star matter may be profitably consideredan instance ofQ-ball nucleation. Finally, in Sec. V we summarize the findings, discuss the relevance of our resultphenomenological issues in neutron star physics, and icate some proposals for future investigation.
II. MEAN-FIELD THEORY DESCRIPTION OF KAONCONDENSATION
In this section we briefly review the model proposedGlendenning and Schaffner@6# which predicts a first-ordertransition from nuclear matter to the kaon-condensed phThe model begins with a relativistic Walecka-type Lagranian describing the neutron and proton fields, as well as thes,v, andr mesons which mediate their interactions:
LN5C̄N~ igm]m2mN* 2gvNgmVm2grNgmtWN•RW m!CN
11
2]ms]ms2
1
2ms
2s22U~s!21
4VmnVmn
11
2mv
2 VmVm21
4RW mn•RW mn1
1
2mr
2RW m•RW m, ~3!
wheremN* 5mN2gsNs is the nucleon effective mass, whicis reduced compared to the free nucleon mass due toscalar fields. The vector fields corresponding to the omeand rho mesons are given byVmn5]mVn2]nVm , andRW mn
5]mRW n2]nRW m, respectively.CN is the nucleon field operator, with tWN the nucleon isospin operator.
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In addition to the usual kinetic, mass, and interactterms for the nucleon and meson fields, the model alsocludes cubic and quartic self-interactions of thes field,
U~s!51
3bmN~gsNs!31
1
4c~gsNs!4, ~4!
whereb and c are dimensionless coupling constants. Thecoupling constants~as well as the three nucleon-meson coplings:gsN , gvN , andgrN! are chosen to reproduce the empirical properties of nuclear matter at saturation dens@14,15#.
Kaons are included in the model in the same fashionthe nucleons, by coupling to thes, v, andr meson fields.There exist in the literature several meson-exchaLagrangians which attempt to describe kaon-nucleon intetions. A detailed discussion of these models and their relato the Chiral Lagrangian description proposed by KaplanNelson @4# can be found in papers by Ponset al. @16# andPrakash and co-workers@17,18#. In the present paper we employ the Lagrangian proposed by Glendenning aSchaffner-Bielich@6#. The kaon Lagrangian is given by
LK5~DmK !†~D mK !2mK*2K†K, ~5!
whereK denotes the isospin doublet kaon field. The covaant derivativeDm5]m1 igvKVm1 igrKtWK•RW m couples thekaon field to the vector mesons, and the kaon effective mterm mK* 5mK2gsKs describes its coupling to the scal
meson.tWK is the kaon isospin operator.The vector coupling constants are determined by isos
and quark counting rules@6# and are given bygvK5gvN/3and grK5grN . The scalar coupling is fixed by fitting to aempirically determined kaon optical potential in nuclear mter. The real part of this quantity has been determined frproperties of kaonic atoms to lie in the somewhat wide ra80 MeV&UK(no)&180 MeV @19,20#. Here we chooseUK(no)5120 MeV to fix gsK . Lower values of the kaonoptical potential reduce the strength of the first-order trantion and, eventually, instead produce a second-order ptransition, while higher values make the first-order transitstronger. While the issues discussed in this paper rely onexistence of a first-order transition, the qualitative concsions are generally independent of the specific value ofcoupling. So long as the transition is first order, the meffect of changingUK(no) will be to change the critical density for the onset of the mixed phase, without severelyfecting our discussion of the nucleation properties nearcritical density.
The model as presented so far is a complicated, stroninteracting field theory which cannot be solved in any resonable way. It is therefore standard to make a mean-fiapproximation, in which the meson field operators areplaced by their expectation values. Because of rotationavariance only the time component of the vector fieldsVm andRW m can have a nonzero expectation value. Likewise, onlyisospin 3 component of the isovector fieldRW m can be non-
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zero. The equations of motion for the meson~mean! fieldscan be simply derived from the above Lagrangians, andgiven by
ms2s52
dU
ds1gsB~nn
(s)1np(s)!1gskmK* f K
2 u2, ~6!
mv2 v5gvN~nn1np!2gvK f K
2 u2~mK1gvKv1grKr !,~7!
mr2r 5grN~np2nn!2grK f K
2 u2~mK1gvKv1grKr !, ~8!
where the meson fieldss,v,r now represent the appropriatmean values. Herenn andnp represent the neutron and proton number densities, respectively, whilenn
(s) andnp(s) are the
corresponding scalar densities. We have substitutedK5(0,K2) andK25(1/A2) f Kue2 imKt , wheref K is the kaondecay constant andu is a dimensionless kaon field strengparameter. We have neglected to write down the additiosmall contributions to the kaon-coupling terms due to finitemperature effects, though these are included in our c~See Ref.@16# for a more explicit presentation of the detaiof this model for finite temperature.! We will be working inthe bulk approximation~in which the meson fields do novary with position!, and so have set to zero the gradieterms which would otherwise appear in the equations of mtion.
The equation of motion for the kaon field~in terms ofu)is
05@~mK* !22~mK* !2#u, ~9!
which indicates that the kaon effective massmK* 5mK
2gsKs and the effective chemical potentialmK* 5mK1Xmust be equal in order for the kaon field to take on a nonzvalue. HereX5gvKv1grKr is the vector field contributionto theK2 energy. We also include a Langrangian describnoninteracting spin-12 particles to account for the presenceelectrons, muons, and neutrinos.
The thermodynamic potential per unit volume for thnucleon sector is
VN
V5
1
2ms
2s21U~s!21
2mv
2 v221
2mr
2r 2
22T (i 5n,p
E d3k
~2p!3ln~11e[E(k)2n i ]/T!, ~10!
where the nucleon energyE(k)5Ak21mN*2. The chemical
potentials are given bymp5np1gvNv1 12 grNr andmn5nn
1gvNv2 12 grNr .
The other thermodynamic quantities can be calculafrom VN in the standard way. The nucleonic contributionthe pressure, for example, isPN52VN /V, while the num-ber densities and entropy densities are given by
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TRAVIS NORSEN PHYSICAL REVIEW C 65 045805
nn,p52]VN /V
]mn,p~11!
52E d3k
~2p!3f F@E~k!2nn,p#, ~12!
sn,p52]VN /V
]T, ~13!
where f F(e)5(e2e/T11)21 is the Fermi-Dirac distributionfunction. The neutron and proton scalar densities which ein the equation of motion for thes field are given by
nn,p(s) 52E d3k
~2p!3
mN*
E~k!f F@E~k!2nn,p#. ~14!
The energy density is determined, through the usual relaTsN5eN1PN2( im ini , to be
eN51
2ms
2s21U~s!21
2mv
2 v221
2mr
2r 2
12 (i 5n,p
E d3k
~2p!3E~k! f F@E~k!2n i #
1 (i 5n,p
ni~m i2n i !. ~15!
The thermodynamic potential for the lepton species pres~electrons, muons, and neutrinos! is given by
VL
V52(
lTglE d3k
~2p!3@ ln~11e2[El (k)2m l ]/T!
1 ln~11e2[El (k)1m l ]/T!#, ~16!
wherem l denotes the chemical potential for lepton speciel,and gl are the spin degeneracies:g52 for electrons andmuons,g51 for neutrinos.b equilibrium requires the fol-lowing constraints on the chemical potentials:
mK5me2mne5mn2mp5mm2mnm
. ~17!
The lepton contributions to the pressure, energy density,entropy are determined from Eq.~16! in the usual way.
Finally, the thermodynamic potential for the kaonsgiven by
VK
V5
1
2f K
2 u2@~mK* !22~mK1X!2#
1TE d3p
~2p!3ln~12e2[v2(p)2mK]/T! ~18!
wherev2(p)5Ap21(mK* )22X is the in-medium energy oa K2 with momentump. Again, the kaon contribution to thother thermodynamic quantities can be found by differention. For a detailed derivation and presentation of the th
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modynamics of this model, see Ref.@16#. For discussions ofsimilar studies involving mixed phases in mean-field theorin several different physical contexts see, e.g., Refs.@21–24#.
Our goal here is to study the rate at whichK matter nucle-ates in a background of npe matter at various temperatand densities. As discussed previously, this involves calating the free energy of droplets of various sizes, in partilar, the free energy of the critical droplet configuration. Tfirst step toward this end is to produce a description odroplet ofK matter of arbitrary radius. This involves solvinthe meson field equations of motion subject to various cstraints.
Generally, one solves simultaneously two versions ofmeson field equations above: one withu50 describing theN-matter component of the mixed phase, the other withuÞ0 @with the specific value ofu determined self-consistentlthrough Eq.~9!# describing theK matter. One requires achemical equilibrium between the two phases, i.e., thatrelevant chemical potentials in the two phases match. Adtionally, the requirement of overall electric charge neutralmeans that the electric charge densities in the two phamust be of opposite sign. One then uses the respective chdensities to calculatea52q(N)/(q(K)2q(N)), the volumefraction of theK-matter phase.~Here q(N) and q(K) are thecharge densities of theK and N matter, respectively.! Be-cause we are concerned with droplets of small radius, ialso crucial to include correct mechanical equilibrium btween the two phases. In Ref.@25# this requirement wasshown to affect rather dramatically the bulk properties oftwo phases.~Also see Ref.@26#.! For a spherical droplet ofKmatter in a background ofN matter, this constraint reads
PK2PN52s/R, ~19!
wheres is the surface tension between the two phases, anRis the radius of the droplet.
Finally, it should be noted that we work at constabaryon number density. That is, in constructing a sequencdroplets of varying radii, we require that the overall barydensitynB5anB
(K)1(12a)nB(N) be held fixed at some spec
fied value. HerenB(N,K) are the baryon number densities
the N andK phases.Once the equations of motion are solved self-consiste
subject to these constraints, it is possible to calculate theenergy of a given droplet. The total bulk energy density ogiven droplet configuration is defined analogously to toverall baryon number density
ebulk5ae (K)1~12a!e (N), ~20!
where the energy density in each phase includes the cobutions from the nucleons, leptons, and kaons, as approate. The total bulk energy is then found by multiplying thenergy density by the total volume of a Wigner-Seitz c~defined as the spherical region including one droplet acontaining zero total electric charge! VWS. This energy isthen supplemented by the surface and Coulomb contrtions,
E~R!5ebulkVWS~R!14pR2s1ECoul , ~21!
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STRANGENESS NUCLEATION IN NEUTRON STAR MATTER PHYSICAL REVIEW C65 045805
FIG. 1. Free-energy cost of kaon droplets~asa function of the radius! for two different valuesof the surface tensions. Each plot is shown neathe critical density for the transition~which isslightly different for the two values ofs), withadditional curves drawn just above and just blow the critical density, showing the developmeof an energetically favored droplet structurethe density is increased past the critical densit
yl.ednaglyta
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whereECoul is found by integrating the electric field energdensityeCoul(r )5 1
2 uE(r )u2 throughout the Wigner-Seitz cel@HereuE(r )u is the magnitude of the electric field determinby Gauss’s law.# The total entropy of a droplet configuratiois found analogously, by a volume-fraction-weighted avering of the individual phase entropy densities, then multiping by the volume of the Wigner-Seitz cell. Thus the tofree energy of a droplet can be calculated as
F~R!5E~R!2TS~R!. ~22!
Only theR dependence has been indicated explicitly, but,course, the energy and entropy both depend strongly onfixed baryon number density, the temperature, the neutfraction, etc.
However, we are interested not merely in the free eneof a given configuration, but, rather, the change in freeergy required to produce various droplets. Hence we asolve for pure, charge neutral npe matter at a given bardensity, and calculate its free energy density,f npe5e2Ts.We are then led to define
DF~R!5F~R!2 f npeVWS, ~23!
which represents the free energy cost of a transition frelectrically neutral npe matter~the initial state of an evolvingprotoneutron star! to a single droplet of kaon-condensed mter of radiusR. Figure 1 showsDF(R) as a function ofRnear the critical density for the transition, for two differevalues of the surface tension,s520 and 30 MeV/fm2.
We have shown the free-energy curves for two valuess in order to illustrate the role this quantity plays. As epected, smaller values ofs reduce both the size and freeenergy cost of a critical droplet. In what follows, we wsimply pick the values530 MeV/fm2, a value suggestedby Glendenning’s study of the boundary between the tphases@27#, as well as by our own earlier work on this mod
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@25#. In Sec. II we will use this type of curve to acquirinformation about the free-energy cost of a critical dropunder various conditions of temperature, density, and ntrino fraction.
III. DROPLET FREE ENERGY AND NUCLEATION RATES
As mentioned in Sec. I, one can understand the prodtion of kaons in neutron star matter as being due to the wdecay of an electron~or, equivalently, the change of a neutron into a proton plus a kaon!. By definition, above thecritical density this transition is energetically favorable, i.exothermic. So we may schematically write
e2ne1K21~Heat!. ~24!
The excess heat generated by the production of kaonseventually be radiated away in the form of photons and ntrinos as the neutron star cools, but that is not our intehere. Rather, our goal is to understand how the free energdroplet configurations depends on the temperature. Quatively, one can guess the correct answer by applying LeCelier’s principle to the equilibrium indicated in Eq.~24!: rais-ing the temperature will tend to push the equilibrium batoward the left. That is, raising the temperature should lowthe energetic favorability of kaon droplets.
This prediction is borne out by our calculations, as illutrated in Fig. 2. As the temperature is increased from zethe free energy of kaon droplet configurations increases rtive to that of neutral npe matter at the same densitytemperature. This increase~which is as large as several hundred MeV for typical droplets! means that, at a density wherthe mixed phase would be the ground state atT50, themixed phase is no longer favored at higher temperatureother words, turning up the temperature increases~slightly!the critical density for the transition.
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TRAVIS NORSEN PHYSICAL REVIEW C 65 045805
FIG. 2. Free energy of droplets~relative toneutral npe matter with the same baryon numbdensity! as a function of radius at temperaturevarying between zero and 30 MeV. As expecteincreasing the temperature decreases the egetic favorability of the kaon phase.
pclhicngra
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However, because the kaon phase droplets must beduced by thermal fluctuations, we expect the thermal nuation rate of the kaon phase to be much larger at higtemperatures. In order to estimate the speed with whdroplets of the kaon phase are produced, we use the Lanucleation rate theory discussed in Sec. I. The nucleationper unit volume is estimated as
G;T4e2DF(Rcrit )/T, ~25!
whereT is the temperature andRcrit is the critical dropletradius~at a given temperature and baryon density!. The free-energy cost of this critical droplet configuration~i.e., theheight of the barrier which must be crossed to produc
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stable droplet of the new phase! can be easily read off ographs like the ones already shown. One may then plugrectly into Eq.~25! to give the nucleation rate per unit voume. In order to convert this rate into an intuitively meaingful quantity, we calculate the expected time for a singnucleation event in a single typical Wigner-Seitz cell of voumeVWS5103 fm3. This time is given by
t51
GVWS5
eDF(Rcrit )/T
VWST4
. ~26!
In Fig. 3 we show this nucleation time as a function of desity for several different temperatures~note the log scale!. As
az
ity,
FIG. 3. Expected time for the nucleation ofsingleK-matter droplet in a typical Wigner-Seitvolume. Shown is log10 of the nucleation time inseconds as a function of baryon number densat several different temperatures.
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STRANGENESS NUCLEATION IN NEUTRON STAR MATTER PHYSICAL REVIEW C65 045805
FIG. 4. Free energy of droplets~relative toneutral npe matter with the same baryon numbdensity! as a function of radius for various smaneutrino densities.
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expected, the nucleation time is a very strong function oftemperature. At a temperature of 0.1 MeV, the expecnucleation time is many, many times longer than the agethe universe across the entire density range of the mphase. AtT51.0 MeV the nucleation time is prohibitivelylong at the lower end of the mixed phase density regime,is less than one second at densities above aboutnB;0.55 fm23, which is something like1
2 no above the criticaldensity. At higher temperatures, the nucleation proceedsmost immediately across the entire density range.
Naively, this leads to the conclusion that the neutron ssettles into its ground state without any delay as it codown from an initial temperature of several tens of MeV.the initial, hot conditions, thermal fluctuations are sufciently fast and sufficiently numerous to seedK-matter drop-lets wherever those droplets are energetically favored.sizes and distances between adjacent kaon structuresundergo tiny changes as the matter cools, and perhapsouter edge of the mixed phase extends outward somewhthe critical density decreases, but generally the neutronincludes the full mixed phase from birth.
This picture is complicated by at least two factors whihave not been discussed explicitly until now. The firstthese is the presence, in the early stages of protoneutroncooling, of a significant neutrino fraction. The second qution is whether or not we should believe the rate estimajust given, since the nucleation rate prefactor~estimatedabove asT4) describes the rate of microscopic fluctuationwhich, in the present case, consist ofweak interaction pro-cesses. These processes are, after all, weak, so onedoubt that the naive estimate based simply on dimensioanalysis is appropriate. This issue will be discussed in a ssequent section; for now, we will turn to the question of teffects of neutrinos on the nucleation ofK matter. ~For arelated discussion, see Ref.@28#.!
Returning to Eq.~24! and again applying LeChetalier’principle, we guess that the presence of a nonzero densielectron neutrinos will~like high temperature! suppress theappearance ofK matter. This guess turns out to be corre
04580
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ightalb-
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,
but for slightly complicated reasons. Turning on a nonzeelectron neutrino fraction
Yne5
nne
nB~27!
forces the chemical potentials for electrons, kaons, andbaryons to adjust according to the constraint of constantnBand b equilibrium @Eq. ~17!#. Surprisingly, doing this actu-ally lowers the free-energy cost of kaon droplets, as indicain Fig. 4.
The reason for this can be understood as follows. Tmain effect of turning on a chemical potential for neutrinois to increase as well the chemical potential for electrosince the constraint of fixed baryon number density doesallow mn andmp much freedom to adjust. But increasing thnumber density of electrons decreases the~positive! electriccharge density of theN matter outside a kaon droplet. Hencmore of this matter is needed to cancel the negative chargthe droplet itself, and the volume fraction ofK matter,a,decreases. But the overall free-energy density depends oathrough Eq.~20!. Finally, since the energy density ofK mat-ter is somewhat larger than that for electrically neutral nmatter~due not only to the presence of kaons, but also tohigher local baryon density! while that of the surroundingNmatter is somewhat lower than neutral npe matter~due to thecorrespondingly lower baryon number density here!, theoverall energy density of the mixed phase is actuallycreased by the decrease ina coming from the nonzero neutrino density.
However, because of the requirement of overall eleccharge neutrality and the fact that theK matter is alwaysnegatively charged, theN-matter component outside of thdroplet must have a positive electric charge density. Ashave just seen, the presence of even a small nonzero chcal potential for electron neutrinos also increases the denof electrons, and thereby reduces the positive charge inregion. As expected, a larger neutrino fraction will decrea
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ta
v-
TRAVIS NORSEN PHYSICAL REVIEW C 65 045805
FIG. 5. Minimum droplet radius consistenwith the overall electric charge neutrality, asfunction of the baryon number density, for seeral different values of the neutrino fractionYn .Here the temperature is fixed atT50.
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the electric charge density here even more, and eventuapoint is reached at which the charge density ceases topositive. Then it is no longer possible to define a WignSeitz cell, i.e., no longer possible to satisfy the requiremof electric charge neutrality.
Actually, this point is reached for very moderate valuesthe neutrino fraction, especially at low densities~near thecritical density for the transition! where, by definition, theelectric charge density in theN-matter phase is positive, busmall. We find that the pressure equilibrium condition E~19! also has a comparable effect on the electric charge dsities via the electron chemical potential.~This issue wasdiscussed in Ref.@25#.! In particular, larger structures atgiven baryon density have lower electron chemical pottials; hence it is possible to elude the effect of nonzeroYn byproducing largerK-matter droplets. Thus, at a givenYn and agiven baryon density, there will be a minimum sizeRminconsistent with charge neutrality, and such structurestherefore not allowed. Electron Debye screening will aplay a role in forbidding structures with large radii, so tregion ofnB2Yn parameter space in which no mixed phacan exist is even somewhat larger than suggested here@25#.
In Fig. 5 we show this minimum radius as a functionbaryon density, for several different values of the electneutrino fraction,Yn . At a given Yn , the minimum radiusallowed by global charge neutrality begins to diverge ascomes down in density. The density at which this quandiverges~or, if we were to consider also the effects of Debscreening, the density at which this quantity exceeds 5fm! acts as an effective critical density for the onset ofmixed phase. Hence, early in the evolution of the proneutron star~PNS!, when the neutrino fraction is~at least! afew percent, no mixed phase will be produced at densiless than 0.60–0.65 fm23. This is to be compared to thcritical density at zero temperature and zero neutrino frtion, 0.49 fm23. Actually, the electron neutrino fraction ithe first seconds of PNS evolution may be closer to 10%which case the effective critical density below which
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-
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stable kaon mixed phases exists will be pushed upwardaround 0.80 fm23, several factors ofn0 above the nominal(T50,Yn50) critical density. The upshot is that, in thneutrino-rich conditions of the early PNS evolution, the crical density for the onset of the kaon mixed phase is increasignificantly compared to the nominal critical density.
This obviously modifies the naive inference from Fig.that the mixed phase will be produced quickly and easduring the early, hot part of the PNS evolution. Becausehigh temperatures occur at a time of high neutrino densthere is a wide range of densities over which no mixed phcan be formed at these early times. Hence, due to thepression ofK matter by neutrinos, a PNS with central densnot too far above the nominal critical density for the onsetthe mixed phase may fail to produce anyK matter during itscooling, until the temperature is so low that nucleaticannot occur in reasonable times. This suggests a scenawhich the neutron star could exist indefinitely in a metastastate. However, further analysis is required to supporthypothesis—in particular, we must address the questionthe reliability of the nucleation rates in Fig. 3 given the prolem of simultaneous weak interactions@2#.
IV. FLUCTUATIONS AND THE PROBLEM OFSIMULTANEOUS WEAK INTERACTIONS
As a first guess, one might suppose that the nucleatioK-matter droplets is driven by thermal fluctuations in tlocal number density of baryons. At a sufficiently higbaryon number density, the phase transition to the kacondensed phase becomes second order, and the kaonmay grow smoothly without having to overcome an enerbarrier. Hence one might suppose that the first-order tration could occur by this same mechanism operating locallythermal fluctuation produces a baryon overdensity in a loregion of radius 1 –2 fm. The kaon field then spontaneou‘‘fills in’’ this region, producing a stable droplet ofK matterwhich could then grow to the stable size. In this picture,
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STRANGENESS NUCLEATION IN NEUTRON STAR MATTER PHYSICAL REVIEW C65 045805
FIG. 6. Number density of thermal kaons inpe matter as a function of the baryon numbdensity. The three curves correspond~very! ap-proximately to three times during the cooling hitory of the protoneutron star. As the star coothe density of thermal kaons drops substantiaand is effectively zero in the context of the probability of seed production considered below.
ue
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nucleation rate would be governed by the frequency of sficiently overdense and sufficiently large baryon numbfluctuations.
However, there is an immediate and fatal problem wthis basic picture. Sufficient fluctuations in baryon densare no doubt plentiful. A typical 1 –2-fm region contains othe order of ten baryons, so, assuming simpleAN fluctua-tions, a factor of 2 overdensity is only three standard devtions away from the mean. But the lifetime of such a flutuation is limited to strong interaction time scales. Indeednuclear matter at these densities, the speed of soundsizable fraction of the speed of lightc. Hence the lifetime ofa baryon number density fluctuation is of ordert;R/c;10223 sec. As is well known from studies of a seconorder kaon condensate phase transition, however, thescales needed for the development of an appreciable valuthe kaon field are 10–15 orders of magnitude longer than@29#. Similar conclusions are found for the appearancestrange quark matter from an initial state of two-flavor quamatter in neutron stars@30,31#. That is, not surprisingly, thetime scale for the development of the kaon field is typicalthe weak interactions.
The production ofK-matter droplets, therefore, cannot bdriven by density fluctuations in the background of baryoThe microscopic kaon-production rate is simply too slowkeep up with the strong interaction time scales that govsuch fluctuations; thus long before any appreciable valuethe kaon field in the fluctuation has developed, the flucttion evaporates.
We conclude that the nucleation rate will be governedthermal fluctuations producing fluctuations in the local kanumber density itself. Moreover, so long as the time scalethe spontaneous growth of a critical droplet is small copared to weak interaction time scales, this rate can deponly on the thermal number density of kaons,
nK(thermal)5E d3p
~2p!3f B@v2~p!2mK#, ~28!
04580
f-r
y
--n
a
eofisf
k
f
.
nor-
ynr-nd
where f B is the Bose-Einstein occupation probability anv2(p) was defined in Sec. II. In Fig. 6 we show this thermkaon number density for several values of the temperaand neutrino fraction in npe-type neutron star matter afunction of density.
It is then simple to estimate the probability of a sufficie‘‘seed’’ of kaons appearing. This calculation is reminiscentthe standard undergraduate statistical mechanics problecalculating the probability that all of the atoms in a boxgas are found to be in a certain small region of the box. Hwe must calculate the probability that a certain numberN ofthe kaons in a kaon gas of densitynK
(thermal) spontaneouslyappear in some small region of space with volumeV.
For definiteness, consider a large box of volumeV0 andtotal kaon numberN0, with N0 /V05nK
(thermal) . If we treatthe kaons as classical particles, then the probability oNkaons being found in a small subregionV of V0 is given bythe binomial theorem:
P~N!5N0!
N! ~N02N!! S V
V0D NS 12
V
V0D (N02N)
. ~29!
Assuming V!V0 and N!N0, we may approximateN0!;(N02N)!N0
N , and neglectN in the exponent (N02N).This gives
P~N!;1
N! S N0V
V0D NS 12
V
V0D N0
. ~30!
But N0V/V05^N&, where ^N&5VnK(thermal) is the average
number of thermal kaons in the volumeV. Using ^N&/V5N0 /V0 we have, finally,
P~N!;^N&N
N! S 12^N&N0
D N0
;^N&N
N!e2^N&. ~31!
~In practice,nK(thermal)!1 fm23, so the exponential in the
last line above can be ignored.!
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TRAVIS NORSEN PHYSICAL REVIEW C 65 045805
FIG. 7. Expected time for a critical droplet oN kaons to form out of a spontaneous densfluctuation of kaons with an average densinK
(thermal) .
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The number of kaons in a critical droplet varies somewwith density. Near the critical density~at zero temperatureand zero neutrino fraction! the radius of a critical droplet is afew fm, for a critical kaon number of order 100. At highdensities, the critical radius drops to only 1 –2 fm, andcorresponding critical kaon number drops to only a few.Fig. 7 we plot the time needed to form such a critical se~for various Ncrit ! as a function of thermal kaon numbedensity. To estimate the seed-production time from the prability considered above, we multiply each probability by tmaximum possible number of discrete seeding locatiwithin a typical Wigner-Seitz cell. This is of orderVWSnB;1000 fm330.5/fm3;500. We then assume that the arangement of thermal kaons is ‘‘reshuffled’’ on the fastpossible time scale, sayt;10223 sec. ~Note that this as-sumption is extremely generous. The actual reshuffling tmust be at least several orders of magnitude greater thanbut here a generous upper limit on the seed-productionwill suffice.! Then the typical time needed for a seed ofNkaons to be produced by this method is given by
t~N!;10223 sec
500P~N!, ~32!
with P(N) given as a function ofnK(thermal) above. Actually,
we should replaceP(N) here with(n>NP(n), but in prac-tice each term in the sum is negligible compared to the pvious sum, so the difference will not affect the result.
We see that the numberNcritical is indeed critical to thedetermination of the rate. If ten~or more! kaons are needeto produce a critical droplet, the probability of a sufficienumber all showing up in the same place at the same timextremely small, and one must wait prohibitively long forseed to ever be produced, especially at lower temperatwhere nK
(thermal) is extremely small. At higher densitiewhereNcrit is lower, of order 3–5, the probability is muc
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higher and the time needed for a seed to be producedrandom reshuffling may be of order seconds or less for rsonable values of the temperature.
Comparing Figs. 6 and 7, we see that the time neededa seeding event during the initial hot protoneutron star cfiguration (T530 MeV, Yn50.03) is on the order of a fewseconds or less ifNcrit<5210. However, as we see fromFig. 5, electric charge neutrality forbids droplets smaller th;2 fm for densities below;0.65 fm23. This is well abovethe (T50, Yn50) critical density, and it is likely that suchhigh density is not reached in the protoneutron star coespecially during the early evolution when the star is still hand relatively rarefied.
After a few tens of seconds, the PNS temperature dropT510 MeV, with the neutrino fraction also dropping tosay,Yn50.01. Looking again at Fig. 6, we see that the nuber density of thermal kaons has dropped dramatically duthe lower temperature. The electric charge neutrality cstraint is relaxed somewhat, with droplets of radius 1 –2now being allowed at densities below aboutnB;0.58 fm23. There is a possibility, considering Fig. 7, thseeding may occur with reasonable speed for densities athis value, whereNcrit is of order 3–5. But again, there isfairly rigid effective critical density below which nucleatiocannot occur. As the star continues to cool, the number dsity of thermal kaons drops dramatically, and the seedtime becomes prohibitively long at all but the very highedensities, 5–7 times nuclear matter density, reasonathought to exist in the neutron star core.
The crude estimates made here for seed productionhomogenous but fluctuating background of thermal kaonssimilar to the estimates in Refs.@32,33# of the growth pro-cess for subcriticalQ balls in a scalar field with a conservecharge. These authors assumed a thermal distribution ofcritical Q balls with relatively largeN, and then consideredthe rate at which additional charge is accreted to~and re-leased by! the Q ball via a random-walk process in orderestimate the rate at which critical droplets are produced
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STRANGENESS NUCLEATION IN NEUTRON STAR MATTER PHYSICAL REVIEW C65 045805
our case, the number of kaons in a critical droplet ismuch greater than unity, so we use the Poisson statisticEq. ~31! rather than a Gaussian distribution withAN fluctua-tions.
Actually, there is an exact analogy between the physickaon nucleation and the problem ofQ-ball nucleation, i.e.,the decay of metastable field configurations in the preseof a conserved charge@34–37#. The effective theory for thekaon field in our model will be precisely the theory ofcomplex scalar with an effective potential supporting nonpological solitons, i.e., stable droplets ofK matter. In thelimit of infinitely slow weak interactions, this effectivetheory will contain a globalU(1) flavor symmetry corre-sponding to the conservation of strangeness. In this limit,seeding ofK matter droplets will occur via a process fomally identical to that described in Ref.@37#, namely, a smalluniform initial charge density~in our case, the background othermal kaons! producing the required ‘‘bounce’’ configuration by flowing toward a seeding center. As discussed in R@37#, the action associated with this configuration will bsignificantly greater than the corresponding case for ascalar which carries no conserved charge. The typical nuation rates in the case of a charged scalar will thus be exnentially slower, due to the need for a global redistributioncharge.
The two estimates of nucleation rates in the present pacan be thought of as limits in the cases of infinitely slow ainfinitely fast weak interactions. In reality, of course, thweak interactions proceed at some intermediate speed. Ineffective theory of the kaon field described above, inclusof strangeness-changing weak reactions will corresponthe addition of a small symmetry-breaking term in the effetive potential, e.g.,
Ue f f~ uKu!→Ue f f~ uKu!1lK, ~33!
where l is a strongly temperature-dependent constantcharacterizes the rate at which kaons can be produceabsorbed by background scattering processes involnucleons and leptons.
The actual rate ofK-matter nucleation will depend oboth types of symmetry breaking, that is,~1! the initial back-ground thermal kaon density and~2! the dynamical produc-tion of new kaons, though it is not obvious which mechaniwill dominate the nucleation for physically relevant tempetures. In order to treat this problem reliably, one must cstruct realistic bounce configurations in the kaon effecttheory as the initial background density and dynamibreaking parameterl are varied with temperature, imposimaginary-time periodicity corresponding to the~inverse!temperature, and calculate the action of the bounce. Quatively, one expects the resulting nucleation times to be inmediate between the fast-weak-interaction and no-weinteraction limits in the current paper. At the very least, oshould expect that the slowness of the weak interactishould modify the nucleation rate prefactorI 0 to a morerealistic dimensional analysis estimate
04580
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-
e
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ale-o-f
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thento-
atorg
--el
ta-r-k-es
I 0→GF2T8;
T8
MW4
, ~34!
whereMW is the mass of the weak gauge bosons. At a teperature of 10 MeV, this estimate would reduce the expecnucleation time by a factor of approximately 10216 from thetimes shown in Fig. 3. As mentioned above, the issuenearly conserved strangeness will also affect the exponefactor in the rate equation, thus suppressing the rate efurther and perhaps bringing the results closer to the emates in Fig. 7. Only the reliable calculation outlined abo~and currently underway by the present author! will answerthis issue with any certainty, however.
V. DISCUSSION AND CONCLUSIONS
We have, then, an intriguing possible scenario in whthe PNS manages to settle into its ground state without foing the kaon-condensate mixed phase which is theground state of the system. At high initial temperatures,nucleation ofK-matter droplets is suppressed by the preseof neutrinos, even though at these high temperaturesseeding of droplets is in principle fast. Over a wide rangedensities kaon droplets are produced copiously by therfluctuations, but they are not yet stable due to the increaseffective critical density. As the star cools, the restricticoming from the presence of neutrinos is relaxed, butintrinsic fluctuation rate drops.
For an initial PNS core density that is not too far abothe nominal (T50) critical density for the formation of aK-matter mixed phase; therefore, it is likely that the star wcool not into the true ground state, but, rather, into a mestable state consisting of electrically neutral npe matter. Tscenario may be relevant to understanding various phenenological issues.
For example, the apparent existence of anomalouheavy neutron stars with massesM;2Msun @38–40# mightbe explained by the anomalously stiff equation of statenpe-type matter relative to matter with a kaon condensGenerally, if the various possible phase transitions thoughoccur in neutron star matter can be avoided by the impobility ~or, equivalently, extreme slowness! of nucleation ofthe new phase, a relatively stiff equation of state may somtimes be maintained over a more extended range of denthan would be naively expected. Hence the existence of sheavy stars may not be sufficient evidence to rule outexistence of kaon condensation at (3 –5)n0, especially ifthese stars were born with smaller masses and only suquently~that is, once cold! acquired larger masses via accrtion from companion stars.
Additionally, metastability of the sort introduced abovmay potentially be useful in understanding the propertiesGRB’s or other poorly understood explosive events. A retively light PNS, as we have argued, may cool and deleptize without theK-matter mixed phase forming, even whethe star’s core density exceeds the critical density fortransition. As is evident from Fig. 6, however, the thermkaon density increases monotonically and steeply with d
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TRAVIS NORSEN PHYSICAL REVIEW C 65 045805
sity, so that, even at the very low temperatures ofT!1 MeV eventually attained in the neutron star, theresome density at which seeding may become possiblereasonable time. At the very least, with increasing densone eventually encounters the second-order point at wthe kaon field may be produced smoothly with no needseeding. Hence if an initially metastable neutron star begto accrete matter from a companion binary~or, additionally,if an initially rotating metastable neutron star gets spun dovia accretion! the central density may increase sufficienfor K matter to begin to appear.
Once the kaon matter appears, however, there will bfeedback effect, due to the softening of the equation of stThe production of a small quantity ofK matter in the corewould allow the star to contract slightly, thus increasing tdensity in the core, and increasing the size of the regionwhich kaonic matter can appear. Further kaon producleads to further collapse, and vice versa. Thus if mass action and/or spin-down results in the critical density for tonset of a second-order kaon-condensate transition breached in the neutron star core~or, equivalently, if onereaches the low-temperature effective critical densitywhich a mixed phase can be nucleated spontaneouslysufficient speed! one would expect an explosive eventwhich the star contracts significantly, resulting in the releof a tremendous amount of energy~of order tenths ofMsun).Cheng and Dai discussed a similar proposal in whaccretion-induced conversion to strange quark matter isgested as a possible explanation forg-ray bursters@41#.
One especially interesting aspect of such a collapse i
ty
tt
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e
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n
ae.
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ng
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e
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its
potentially turbulent nature. The picture is of a pureK-mattercore seeding the mixed phase through several kilometermaterial above it. In effect, theK matter boils off of the outeredge of the second-order core, and floats upward to formmixed phase throughout the entire region of the mixphase’s energetic favorability. This implies an upward adownward transfer of matter that closely resembles turbuconvection, but in which strangeness rather than heat issubstance being convected.
As mentioned at the end Sec. IV, more reliable calcutions need to be performed in order to better understandthe slowness of the weak interactions affect the originucleation rate estimates based on Langer’s formula.have argued that the correct framework for these futureculations involves the formalism of quantum tunneling~orthermal activation! in a theory with a~nearly! conserved glo-bal charge representing strangeness. It is also worth mening that the scenario outlined here is the most realistictential application of theQ-ball nucleation formalismdeveloped in Refs.@36,37#; theorists up to now have relieon supersymmetric models to find possible theories conting charged scalars supporting nontopological solitons.
ACKNOWLEDGMENTS
Sanjay Reddy, Eduardo Fraga, and Guy Moore areknowledged for helpful discussions. This work was suported in part by a National Science Foundation GraduResearch Fellowship and by U.S. Department of EneGrant No. 62-4928.
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