Table Of ContentCharge screening effect on hadron-quark mixed
phase in compact stars
6
0
0
2
n TomokiEndo∗
a
J DepartmentofPhysics,KyotoUniversity,Kyoto606-8502,Japan
1 E-mail: [email protected]
1 ToshikiMaruyama
v
JapanAtomicEnergyResearchInstitute,†Tokai,Ibaraki319-1195,Japan
7
1 E-mail: [email protected]
0
1 SatoshiChiba
0 JapanAtomicEnergyResearchInstitute,†Tokai,Ibaraki319-1195,Japan
6
E-mail: [email protected]
0
/
h ToshitakaTatsumi
p
DepartmentofPhysics,KyotoUniversity,Kyoto606-8502,Japan
-
o E-mail: [email protected]
r
t
s
a Westudythechargescreeningeffectinthehadron-quarkmixedphase. Byincludingthecharge
:
v screeningeffect,rearrangementofchargedparticlesoccursandsomepartbecomeslocallycharge-
i
X neutral. Asaresulttheequationofstateforthemixedphasebecomesclosetothatgivenbythe
r Maxwell construction, which means that the Maxwell construction would effectively gain the
a
physicalmeaningagaineveninthesystemwithtwoormorechemicalpotentials.Wealsodiscuss
theinterplayofthesurfacetensionandtheCoulombinteraction. Botheffectswouldrestrictthe
regionofthemixedphaseinthecoreofhybridstars.
29thJohnsHopkinsWorkshoponcurrentproblemsinparticletheory:strongmatterintheheavens
1-3August
Budapest
∗Speaker.
†Presentname:JapanAtomicEnergyAgency
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
1. Introduction
It is widely believed that quark matter exists in high-temperature and/or high-density situa-
tions like in the relativistic heavy-ion collisions (RHIC) [1] or in the core of neutron stars [2, 3],
and the “deconfinement transition” has been actively searched. Theoretical studies using model
calculations or based on the first principle, lattice QCD [4] have been also carried out by many
authors to find the critical temperature. Although many exciting results have been reported, the
deconfinement transition hasnotbeenclearlyunderstood yet.
We, hereafter, consider the phase transition from hadron phase to three-flavor quark phase in
neutron-star matter in a semi-phenomenological way. Since many theoretical calculations have
suggested that the deconfinement transition should be of first order in low-temperature and high-
density area [5, 6], we assume the first-order phase transition in this paper. If the deconfinement
transitionisoffirstorder,onemayexpectamixedphaseduringthetransition. Actually,thehadron-
quark mixed phase has been considered during the hadronization era in RHIC [7, 8, 9] or in the
coreregionofneutronstars[10,11,12].
Thereisanissue about themixed phase forthe first-order phase transitions withtwoormore
chemical potentials [13]. We often use the Maxwell construction (MC) to derive the equation
of state (EOS) in thermodynamic equilibrium, as in the liquid-vapor phase transition of water,
where bothphases consist ofsingle particle species. However, ifmanyparticle species participate
in the phase transition as in neutron-star matter, MC is no more an appropriate method. Before
Glendenning first pointed out [13], many people have applied MC to get EOS for the first-order
phasetransitions [14,15,16,17]expected inneutron stars,suchaspionorkaoncondensation and
thedeconfinement transition.
Let us consider a deconfinement transition in neutron-star matter. We must introduce many
chemicalpotentialsforparticlespecies,buttheindependentonesinthiscasearereducedtotwo,i.e.
baryon-number chemical potential m and charge chemical potential m , due to chemical equilib-
B Q
riumandtotalchargeneutrality. Theyarenothingbuttheneutronandelectronchemicalpotentials,
m and m ,respectively. Inthemixed-phase thesechemicalpotentials should bespatially constant.
n e
When we naively apply MC to get EOS in thermodynamic equilibrium, we immediately notice
thatm isconstantinthemixed-phase,whilem isdifferentineachphasebecauseofthedifference
B e
of the electron number density in these phases. This is because MC uses EOS of bulk matter in
each phase, which is of locally charge-neutral and uniform matter; many electrons are needed in
hadronmattertocompensatethepositivechargeofprotons,whileinquarkmatterchargeneutrality
isalmostfulfilledwithoutelectrons. Thus
m Q=m H, m Q6=m H, (1.1)
B B e e
in MC, where superscripts “Q” and “H” denote the quark and hadron phases, respectively. Glen-
denningemphasizedthatwemustusetheGibbsconditions(GC)inthiscaseinsteadofMC,which
relaxesthecharge-neutrality conditiontobegloballysatisfiedasawhole,notlocallyineachphase
[13]. GCimposesthefollowingconditions,
m Q =m H, m Q=m H,
B B e e
PQ =PH, TQ=TH. (1.2)
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Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
Hedemonstrated awideregionofthemixedphase,wheretwophaseshaveanetchargebuttotally
charge-neutral: EOS thus obtained, different from that given by MC, never exhibits a constant-
pressure region. Hesimply considered amixed phase consisting oftwobulk matters separated by
a sharp boundary without any surface tension and the Coulomb interaction, which we call “bulk
Gibbs”forconvenience. “BulkGibbs”requiresthateachmattercanhaveanetchargebutthetotal
charge isneutral,
f r Q+(1− f )r H =0, (1.3)
V ch V ch
where f meansthevolumefraction ofquarkmatterinthemixedphaseand“r Q,H”meanscharge
V ch
density in each matter. Figure 1 shows the phase diagram in the m - m plane. We can see that
B e
there is a discontinuous jump in m for the case of MC, while the curve given by “bulk Gibbs”
e
smoothly connects uniform hadron matter and uniform quark matter; the mixed phase can appear
inawidem regionin“bulkGibbs”,incontrastwithMC[18].
B
Hadron
200 MC
]
V
e
M
[ 100 bulk Gibbs
e
m
Quark
0
1000 1100 1200 1300 1400
m [MeV]
B
Figure1:Phasediagraminthem −m plane.Thereappearsnoregionofthemixedphasebythecalculation
B e
withMC,whileawideregionofthemixedphasebythe“bulkGibbs”calculation.
However, this “bulk Gibbs” is too simple to study the mixed phase, since we must consider
matter with non-uniform structures instead of two bulk uniform matters; the mixed phase should
have various geometrical structures where both the number and charge densities are no more uni-
form. Then we have to take into account finite-size effects like the surface and the Coulomb
interactionenergies. Weshowthatthemixedphaseshouldbenarrowinthem spacebythecharge
B
screening effect, andderive EOSforthedeconfinement transition inneutron-star matter. Weshall
seeEOSresultsinbeingsimilartothatgivenbyMC.WealsodiscusstheinterplayoftheCoulomb
interactioneffectandthesurfaceeffectinthecontextofthehadron-quarkmixedphase. Preliminary
results forthedropletcasehasbeenalreadyreported inRef.[19].
2. Brief review ofthe previous works
Heiselberg et al. [20] studied a geometrical structure in the mixed phase by considering the
spherical quark droplets embedded in hadron matter. They introduced the surface tension and
treated its strength as a free parameter because the surface tension at the hadron-quark interface
3
Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
has not been clearly understood. They pointed out that if the surface tension parameter s is large
(s ≥90MeV/fm2),theregion ofthemixedphaseislargely limitedorcannot exist. Subsequently
Figure2: Schematicimageofstructuredmixedphase.
Glendenning and Pei [10] have suggested “crystalline structures of the mixed phase” which have
some geometrical structures, “droplet”, “rod”, “slab”, “tube”, and “bubble” (Fig. 2), assuming a
smalls [10,21]. Thefinite-sizeeffectsareobviousinthesecalculationsbyobservingenergies. We
mayconsideronlyasinglecell,bydividingthewholespaceintoequivalentWigner-Seitzcells: the
cell size is denoted by R and the size of the lump (droplet, rod, slab, tube or bubble) by R (Fig.
W
3). Thenthesurface energydensity isexpressed intermsofthesurfacetension parameters as
f s d
e = V , (2.1)
S
R
where d denotes the dimensionality of each geometrical structure; d =3 for droplet and bubble,
d =2forrodandtube,andd=1forslab. TheCoulombenergydensityreads
2
e = 2p e2 r H−r Q R2F (f ) , (2.2)
C ch ch d V
(cid:16) (cid:17) 1
F (f ) ≡ 2(d−2)−1 1− df1−2/d + f (d+2)−1, (2.3)
d V 2 V V
(cid:20) (cid:18) (cid:19) (cid:21)
where we simply assume an uniform density distribution in each phase. When we minimize the
sum of e and e with respect to the size R for a given volume fraction f , we can get the well-
S C V
knownrelation,
e =2e . (2.4)
S C
This implies that an optimal size of the lump is determined by the balance of these finite-size
effects. Eventually we can express the sum of the surface and Coulomb energy densities with an
optimalcellsize[20]:
2 1/3
ps 2 r H−r Q F (f )
e (d)+e (d)=3f d ch ch d V . (2.5)
C S V (cid:16) 2d (cid:17)
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Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
Thuswecancalculatetheenergyofanygeometricallystructuredmixedphasewith(2.5)bychang-
ingtheparameterd. Manyauthorshavetakenthistreatmentforthemixedphase[10,18,20]. Note
that the energy sum in Eq. (2.5) becomes larger as the surface tension gets stronger, while the
relation Eq.(2.4)isalwayskept.
However, this treatment is not a self-consistent but a perturba-
tive one, since the charge screening effect for the Coulomb potential
or the rearrangement effect of charged-particles in the presence of the
Coulomb interaction is completely discarded. We shall see that the
Coulomb potential is never weak in the mixed phase, and thereby this
treatment overestimates theCoulombenergy. Thecharge screening ef-
fect is included only if we introduce the Coulomb potential and con-
sistently solvethe Poisson equation withother equations ofmotion for
charged particles. Consequently it is a highly non-perturbative effect.
Norsen and Reddy [22]have studied the Debye screening effect in the
context of kaon condensation to see a large change of the charged-
particle densities likekaonsandprotons. Maruyamaetal.havenumer-
ically studied it in the context of liquid-gas phase transition at subnu-
cleardensities[23],wherenuclearpastacanberegardedasgeometrical
structuresinthemixedphase. Subsequently, wehavealsostudiedkaon
condensation at high-densities [24], where kaonic pasta structures ap-
peared in the mixed phase. Through these works we have figured out
Figure 3: Wigner-Seitz
theroleoftheDebyescreening inthemixedphase.
approximation for the
Voskresensky etal.[25]explicitly studiedtheDebyescreening ef-
droplet case. R is the
droplet radius and R fect for afew geometrical structures ofthe hadron-quark mixedphase.
W
thecellradius. Theyhaveshownthattheoptimalvalueofthesizeofthestructurecan-
not be obtained due to the charge screening, if the surface tension is
not so small. They called it mechanical instability. It occurs because
the Coulomb energy density is suppressed for the size larger than the Debye screening length (cf.
Eq.(4.2)). Theyalsosuggestedthatthepropertiesofthemixedphasebecomeverysimilartothose
given by MC,if the charge screening effectively works. The apparent violation of the Gibbs con-
ditionsinMC(Eq.(1.1))canberemediedbyincluding theCoulombpotentialinagauge-invariant
way: the number density of charged particles is given by a gauge-invariant combination of the
chemical potential and the Coulomb potential, and thereby it can be different in each phase for
a constant charge chemical potential if the the Coulomb potential takes different values in both
phases. However,theyusedalinearapproximation tosolvethePoissonequation analytically.
IftheCoulombinteractioneffectissolarge,itwouldbeimportanttostudyitwithoutrecourse
toanyapproximation. Inthispaperwenumerically studythecharge screening effectonthestruc-
tured mixed phase during the deconfinement transition in neutron-star matter in a self-consistent
way. Actuallyweshallseeimportance ofnon-linear effectsincluded inthePoissonequation.
3. Self-consistent calculationwiththe Coulombpotential
Wehave presented our formulation in detail inRef. [26, 27]. Here weonly briefly explain it.
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Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
Weconsider thegeometrically structured mixedphase(SMP)whereonephaseisembedded inthe
other phase with a certain geometrical shape. We divide the whole space into equivalent charge-
neutralWigner-Seitzcells. TheWigner-Seitzcellisarepresentativeofequallydividedspaceunder
approximation of a geometrical symmetry: sphere in three dimensional (3D) case, rod in 2D case
and slab in 1D case with the size R . The cell consists of two phases in equilibrium: one phase
W
withasizeRisembeddedintheotherphaseasillustrated inFig.3.
Quark phase consists of u, d, s quarks and electron. We incorporate the MIT Bag model and
assumethesharpboundaryatthehadron-quark interface: uanddquarksaretreatedtobemassless
and s is massive (m = 150MeV), and they interact with each other by the one-gluon-exchange
s
interaction inside the bag. Hadron phase consists of proton, neutron and electron. The effective
potentialisusedtodescribetheinteractionbetweennucleonsandtoreproducethesaturationprop-
ertiesofnuclearmatter.
Thentotalthermodynamic potential (W )consists ofthehadron, quark andelectron contri-
total
butions andthesurfacecontribution:
W =W +W +W , (3.1)
total H Q S
whereW denotes thecontribution ofhadron(quark) phase. Wehereintroduce thesurface con-
H(Q)
tributionW ,parametrizedbythesurfacetensionparameters ,W =s SwithSbeingtheareaofthe
S S
interface. Note that itmay be closely related with the confining mechanism and unfortunately we
have no definite idea about how to incorporate it. Actually many authors have treated its strength
as a free parameter and seen how the results are changed by its value[10, 18, 20], which we also
followinthisreport.
To write down the thermodynamic potential we use the idea of the density functional theory
(DFT)withinthelocal density approximation (LDA)[28,29]. TheCoulombinteraction energy is
included inW andcanbeexpressed intermsofparticledensities,
H(Q)
E = 1(cid:229) d3rd3r′Qir i(~r)Qjr j(~r′), (3.2)
V 2 |~r−~r′|
i,j ZVW
wherei=u,d,s,p,n,e withQ beingtheparticle charge(Q=−e<0fortheelectron) andV the
i W
volumeofWigner-Seitz cell. Accordingly theCoulombpotential isdefinedas
V(~r)=−(cid:229) d3r′eQir i(~r′)+V , (3.3)
0
|~r−~r′|
i ZVW
whereV isanarbitrary constant representing thegaugedegree offreedom. Wefixthegaugebya
0
conditionV(R )=0 in this paper. Operating a Laplacian (cid:209) 2 on the Coulomb potentialV(~r), we
W
automatically derivethePoissonequation.
There are introduced six chemical potentials in Eq. (3.1) corresponding to particle species
in the cell. Then we have to determine now eight variables, i.e., six chemical potentials, m (i=
i
u,d,s,p,n,e), and the radii Rand R . Fixing Rand R , wehave four conditions due tochemical
W W
6
Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
equilibrium ineachphaseandattheinterface:
m +m =m ,
u e d
m =m ,
d s
m +m =m ≡m ,
p e n B
m =m +2m ,
n u d
m =2m +m . (3.4)
p u d
Therefore,oncetwochemicalpotentials(cid:0)m andm aregi(cid:1)ven,wecandetermineotherfourchemical
B e
potentials, m , m ,m and m . Wedetermine m bytheglobalchargeneutrality condition:
u d s p e
f r Q+(1− f )r H =0, (3.5)
V ch V ch
d
where the volume fraction f = R , and d denotes the dimensionality of each geometrical
V RW
structure. At this point fV is still(cid:16)fixe(cid:17)d. The particle densities r i are given by the equations of
d W
motion, whicharederivedformtheextremumcondition, tot =0,
dr
i
d E
m = kin+str −NV(~r), (3.6)
i dr (~r) i
i
where E stands for the kinetic and strong interaction energy except the Coulomb interaction
kin+str
energy.
Then we find the optimal value of R (R is fixed and thereby f is changed by R) by using
W V
oneofGC;
PQ=PH+Ps , (3.7)
wherethepressure comingfromthesurface tensionisgivenby
dS
Ps =s . (3.8)
dV
Q
ThepressureineachphasePQ(H) isgivenbythethermodynamic relation: PQ(H)=−W /V ,
Q(H) Q(H)
where W is the thermodynamic potential in each phase and given by adding electron and the
Q(H)
CoulombinteractioncontributionstoW inEqs.(3.1). Finally,wedetermineR byminimizing
Q(H) W
thermodynamic potential. Thus, once m is given, all other values m (i=u,d,s,p,e) and R, R
B i W
canbeobtained. Accordingly, theparticledensityprofilesaregivenbyEq.(3.6).
We numerically solve Eqs. (3.6) and the Poisson equation in a self-consistent way. In nu-
merical calculation, every point inside the cell is represented by a grid point (the number of grid
points N ≈100). Equations of motion are solved by a relaxation method for a given baryon-
grid
numberchemicalpotentialunderconstraintsoftheglobalchargeneutrality. NotethatwekeepGC
throughout thenumericalprocedure.
4. Numerical results anddiscussions
First, we present the thermodynamic potential density of each uniform matter and that given
by “bulk Gibbs” in Fig. 4. In uniform matter, hadron phase is thermodynamically favorable for
7
Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
Q
uniform matter
] - 50 (charge neutral)
3
m
MC
f
/
V
e - 100
M bulk Gibbs
[ (No Coulomb nor Surface) H
V
/
- 150
W
1100 1200 1300
m [MeV]
B
Figure4: Thermodynamicpotentialdensity(W /V =w )inthem space.
B
m <1225MeVandquarkphaseaboveit. Thereforeweplotdw ,differenceofthethermodynamic
B
potential density betweenthemixedphaseandeachuniformmatter:
w −w uniform m ≤1225MeV,
dw = total H B (4.1)
(w total−w Quniform m B≥1225MeV,
wherew =W /V ,etc.
total total W
The results are shown in Figs. 5 and 6. In these figures we also depict two results for com-
parison: one is given by“bulk Gibbs”, where the finite-size effects are completely discarded. The
other is the thermodynamic potential given by a perturbative treatment of the Coulomb interac-
tion, which is denoted by “no screening”; discarding the Coulomb potential V(~r), we solve the
equations ofmotiontogetuniformdensityprofiles(norearrangement), thenevaluatetheCoulomb
interaction energy byusingthedensityprofilesthusdetermined. Notethat“noscreening” violates
the gauge invariance, whileour treatment respects it[25]. Since chemical potential itself is gauge
variant,wehavetoincludeV(~r)togetherlikeinEq.(3.6)tosatisfythegaugeinvariance: whenthe
Coulombpotentialisshiftedbyaconstantvalue,V(~r)=⇒V(~r)−V ,thechargechemicalpotential
0
should be also shifted as m =⇒ m +NV . Incidentally, the phase diagram in the m −m plane
i i i 0 B e
(see,e.g.,Fig.1)isnotwell-definedbythisreason,whilemanypeoplehavewrittenit[18,30]. We
can see the screening effects by comparing the results given by “no screening” with those by the
self-consistent calculation denoted by “screening”. dw given by MC appears as a point denoted
by a circle in Figs. 5 and 6 where two conditions, PQ =PH and m Q = m H, are satisfied. On the
B B
other handthemixedphasederived from“bulkGibbs”appears inawideregion of m . Therefore,
B
iftheregionofthemixedphasebecomesnarrower,itsignalsthatthepropertiesofthemixedphase
becomeclosetothosegivenbyMC.Onemayclearlyseethatw becomesclosetothatgivenby
total
MCduetothefinite-sizeeffects, theeffectsofthesurface tensionandtheCoulombinteraction.
The large increase of dw from the “bulk Gibbs” curve comes from the effects of the surface
tension and the Coulomb potential. Since the surface tension parameter is introduced by hand,
wemustcarefully studytheeffectsofthesurface tension andtheCoulombinteraction, separately.
From the difference between the result given by “no screening” and that by “bulk Gibbs”, wecan
roughly say that about 2/3 of the increase comes from the effect of the surface tension and 1/3
8
Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
5 5
s = 40 MeV/fm2 MC screening s = 60 MeV/fm2 MC screening
droplet droplet
]3 0 rod ]3 0 rod
m slab m slab
f tube f tube
V/ bubble V/ bubble
Me - 5 no screening Me - 5 no screening
droplet droplet
rod rod
dw[ - 10 bulk Gibbs slab dw[ - 10 bulk Gibbs slab
tube tube
bubble bubble
1100 1200 1300 1400 1100 1200 1300 1400
m [MeV] m [MeV]
B B
Figure 5: Difference of the thermodynamic po- Figure 6: Same as Fig. 5 for s =60 MeV/fm2.
tential density as a function of baryon-number Thenegativedw regionisnarrowerthanthes =
chemicalpotentialm fors =40MeV/fm2.Ifdw 40MeV/fm2case.
B
is negative, the mixed phase is a thermodynami-
callyfavorablestate. MCdeterminesonepointof
phase transition in uniform matter, denoted as a
circleinthem -dw plane.
B
from the Coulomb interaction (see Eq. (2.4)). Comparing the result of self-consistent calculation
withthatof“noscreening”, wecanseethatthechangeofenergy caused bythescreening effectis
notsolarge,butstillthesameorderofmagnitudeasthatgivenbythesurface effect.
100 r · 100 V(r)
r p· 100 r p· 100
r n· 100 ]V 100 r n· 100
]3 r eu· 100 Me rr e·· 110000
-[fm 50 rr ds·· 110000 -][3m, 50 rr uds·· 110000
f
[
0 0
0 10 20 0 10 20
r [fm] r [fm]
Figure 7: Density profiles in the droplet phase Figure 8: Density profiles and the Coulomb po-
givenby“noscreening”form =1189MeVand tentialgivenbytheself-consistentcalculationfor
B
s = 60 MeV/fm2. They are uniform in each thesameparametersetasFig.5. R=7.7fmand
phase. R=7.2fmandR =12.8fm. R =18.9fm.
W W
If the surface tension is stronger, the relative importance of the charge screening effect be-
comes smaller and theeffect of the surface tension becomes moreprominent, asisseen in Figs. 5
and6.
Although the charge screening has not so large effects on bulk properties of the matter, we
shallseethatitisremarkableforthechargedparticlestochangethepropertiesofthemixedphase.
The screening effect induces the rearrangement of the charged particles. We can see this effect
by comparing Fig. 7 with Fig. 8. The quark phase is negatively charged and the hadron phase
is positively charged. The negatively charged particles in the quark phase such as d, s, e and
the positively charged particle in the hadron phase p are attracted toward the boundary. On the
9
Chargescreeningeffectonhadron-quarkmixedphaseincompactstars TomokiEndo
contrary thepositively charged particle inthequark phase uandnegatively charged particle inthe
hadron phase e are repelled from the boundary. The charge screening effect also reduces the net
charge in each phase. In Fig. 9, we show the local charge densities of the two cases shown in
Figs. 7 and 8. The change of the number of charged particles due to the screening is as follows:
In the quark phase, the numbers of d and s quarks and electrons decrease, while the number of u
quark increases. In the hadron phase, on the other hand, the proton number should decrease and
the electron number should increase. Consequently the local charge decreases in the both phases.
In Fig. 9 we can see that the core region of
the droplet tends to be charge-neutral and near 1
r screening
the boundary of the Wigner-Seitz cell is almost r h screening
r q no screening
charge-neutral. InFigs.10and11wepresentthe r h no screening
q
lumpandcellradiiforeachdensity. Aswehave ]3
-m 0
shown in the previous paper[25], the Coulomb
f
[
energy is suppressed for larger R by the screen-
ing effect. The R dependence of the total ther-
modynamic potential comes from the contribu- - 1
0 10 20
tions of the surface tension and the Coulomb r [fm]
interaction: the optimal radius giving the mini-
mumofthethermodynamic potential isthende- Figure 9: Local-charge densities for the “no
screening”caseandthecaseoftheself-consistent
termined by the balance between two contribu-
with the screening effect. For “no screening”,
tions, since the former gives a decreasing func-
chargedensityof each phase is constantoverthe
tion, while the latter an increasing one. If the
region. The absolute value of the charge density
Coulomb energy issuppressed, theminimum of
islargerthanthatgivenbytheself-consistentcal-
the thermodynamic potential is shifted to larger culation in each phase. In the hadron phase, the
radius. As a result the size of the embedded chargedensitybecomesalmostvanishednearthe
phase (R) and the cell size (RW) become large. cellboundaryr=RW.
InRef.[25]theydemonstratedthattheminimum
disappears for alarge value of the surface tension parameter: the structure becomes mechanically
unstable inthiscase.
Wecannotshowitdirectlyinourframeworkbecausesuchunstablesolutionsareautomatically
excluded during thenumerical procedure, whilewecanseeitstendency inFigs.10and11: Rand
R getlargerbythescreening effect.
W
WealsoseetherelationbetweenthesizeofthegeometricalstructureandtheDebyescreening
length. TheDebyescreeninglengthappearsinthelinearizedPoissonequationandisthengivenas
l q −2=4p (cid:229) Q ¶ hr cfhi , l p −2=4p Q ¶ hr pchi , (l e)−2=4p Q ¶ hr echi , (4.2)
D f ¶m D p ¶m D e ¶m
f f ! p ! (cid:18) e (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
wherehr chistands fortheaveraged density inquark phase, hr chiisproton number averaged den-
f p
sity in the hadron phase and hr chi is the electron charge density averaged inside the cell. It gives
e
aroughmeasureforthescreening effect: Atadistance largerthantheDebyescreening length, the
Coulombinteraction iseffectivelysuppressed.
In Fig. 10 we show sizes of geometrical structure for “no screening” case. If we ignore the
screening effect, the size of the embedded phase is comparable or smaller than the corresponding
10
