QCD phase diagram at high temperature and density

0.58 MB·English

by  Mei Huang

QCD phase diagram at high temperature and density
Mei Huang1,2 ∗
1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China
2 Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing, China
This article reviews recent progress of QCD phase structure, including color superconductor at
high baryon densityand strongly interacting quark-gluonplasma (sQGP) at high temperature cre-
ated through relativistic heavyion collision. A brief overview is given on thediscovery of sQGP at
RHIC. The possibility of locating the critical end point (CEP) from the property of bulk viscosity
overentropy density is discussed. For the phase structureat high baryon density, thestatus of the
unconventionalcolorsuperconductingphasewithmismatchedpairingisreviewed. Thechromomag-
neticinstability, Sarmainstability and Higgs instability in thegapless color superconductingphase
are clarified.
0
1
PACSnumbers:
0
2
n I. INTRODUCTION
a
J
Quantum Chromodynamics (QCD) is an asymptoti-
9
cally free theory [1] and regarded as the fundamental
1
theory of quarks and gluons. At very high energies, in-

teraction forces become weak, thus perturbation calcu-
]
h lations can be used. The perturbative QCD predictions
p
have been extensively confirmed by experiments, while
-
p QCD in the non-perturbative regime is still a challenge
e totheorists. ThefundamentalquarksandgluonsofQCD
h have not been seen as free particles, but are always con-
[
fined within hadrons. It is still difficult to construct the

1 hadrons in terms of nearly massless quarks and gluons. FIG. 1: QCD phase diagram at finite temperature and
v The observed baryon spectrum indicates that the (ap- baryon density.
6 proximate) chiral symmetry is spontaneously broken in
1 the vacuum. As a result, the eight pseudoscalar mesons
2
π, K and η are light pseudo-Nambu-Goldstone bosons, that the “little Bang” can be produced at RHIC and
3
andtheconstituentquarkobtainsdynamicalmass,which LHC. Recently, it was shown that the new state of mat-
.
1 contributes to the baryon mass. Besides conventional terproducedatRHICisfarawayfromtheasymptotically
0 mesonsandbaryons,QCDitselfdoesnotexcludetheex- hotQGP,but ina stronglycoupledregime. This stateis
0
istence of the non-conventional states such as glueballs, called strongly coupled quark-gluon plasma (sQGP)[8].
1
hybrid mesons and multi-quark states [2]. FormostrecentreviewsaboutQGP,e.g.,seeRef.[9–12].
:
v
Since1970s,peoplehavebeeninterestedinQCDatex- Studying QCD at finite baryon density is the tra-
i
X treme conditions. It is expected that the chiral symme- ditional subject of nuclear physics. The behaviour of
try can be restored, and quarks and gluons will become QCD at finite baryon density and low temperature is
r
a deconfined at high temperatures and/or densities [3–6]. central for astrophysics to understand the structure of
Fig. 1 is the typical QCD phase diagram, which shows compact stars, and conditions near the core of collaps-
the system is in deconfined quark-gluonplasma phase at ing stars (supernovae, hypernovae). Cold nuclear mat-
hightemperature,andincolorsuperconductingphaseat ter, such as in the interior of a Pb nucleus, is at T = 0
high baryon density. and µ m = 940MeV. Emerging from this point,
B N
≃
Resultsfromlatticeshowthatthequark-gluonplasma there is a first-order nuclear liquid-gas phase transition,
(QGP) does exist. For the system with zero net baryon which terminates in a criticalendpoint at a temperature
density, the deconfinement and chiral symmetry restora- 10MeV [13]. If one squeezes matter further and fur-
∼
tion phase transitions happen at the same critical tem- ther, nuleons will overlap. Quarks and gluons in one
perature [7]. At asymptotically high temperatures, e.g., nucleon can feel quarks and gluons in other nucleons.
during the first microseconds of the “Big Bang”, the Eventually, deconfinement phase transition will happen.
many-bodysystemcomposedofquarksandgluonscanbe Unfortunately, at the moment, lattice QCD is facing the
regardedas an ideal Fermi and Boson gas. It is believed “sign problem” at nonzero net baryon densities. Our
understanding at finite baryon densities has to rely on
effective QCD models. Phenomenological models indi-
cated that, at nonzero baryon density, the QGP phase
∗[email protected] and the hadron gas are separated by a critical line of
2
roughly a constant energy density ǫ 1GeV/fm3 [14].
cr
≃
sysItnemtheiscaasceoolofrassyumpeprtcootnicdaulclyorh.igThhbisawryaosnpdreonpsoitsye,dtbhye )(pvT200.1.28 b » 6.8 Fm (16-24% Central)
Frautschi [15] and Barrois [16]. Based on the Bardeen,
Cooper, and Schrieffer (BCS) theory [17], because there 0.16 STAR Data
is a weak attractive interaction in the color antitriplet 0.14
channel, the system is unstable with respect to the for- 0.12 G s/t o = 0
mationofparticle-particleCooper-paircondensateinthe
0.1
momentum space. Detailed numerical calculations of G /t = 0.1
s o
color superconducting gaps were firstly carried out by 0.08
BailinandLove[18]. Theyconcludedthatthe one-gluon 0.06
exchange induces gaps on the order of 1 MeV at several G /t = 0.2
0.04 s o
timesofnuclearmatterdensity. Thissmallgaphaslittle
effect on cold dense quark matter, thus the investigation 0.02
of cold quark matter lay dormant for several decades. 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
It was only revived recently when it was found that the p(GeV)
T
colorsuperconductinggapcanbeoftheorderof100MeV
[19], which is two orders larger than early perturbative
FIG.2: Ellipticflowv2 asafunctionofpT fordifferentvalues
estimates in Ref. [18]. For this reason, the topic of color
of Γs/τ0. The figuretaken from Ref.[23].
superconductivitystirredalotofinterestinrecentyears.
Forreviewarticlesonthecolorsuperconductivity,seefor
example, Refs. [20]. with uµ the flow velocity, ǫ,p the energy density and
In this article, I will focus on recent progressof sQGP
pressure density, respectively.
createdatRHIC and colorsuperconductingphase struc-
In the Navier-Stokes hydrodynamics, the energy mo-
ture at intermediate baryon density regime. The outline
mentum tensor decomposes into ideal and dissipative
of this article is as follows: I will give a brief overview
parts as
on the discovery of sQGP at RHIC in Sec. II. Then
introduce the status of the color superconducting phase Tµν =Tµν +τµν, (3)
NS ideal
especiallythegaplesscolorsuperconductingphaseinSec.
III. At last, I will give a brief outlook in Sec. IV. with
2
τµν =η( µuν+ νuµ µν uα+ζ µν uα. (4)
α α
II. STRONGLY INTERACTING ∇ ∇ −3△ ∇ △ ∇
QUARK-GLUON PLASMA(SQGP)
Where µν = gµν uµuν, µ = µν∂ , η,ζ are the
ν
△ − ∇ △
shear viscosity and bulk viscosity, respectively.
A. Discovery of sQGP at RHIC
It was expected that deconfined quark matter formed
at high temperature should behave like a gas of weakly
Studying Quantum chromodynamics (QCD) phase
interacting quark-gluon plasma (wQGP). The perturba-
transition and properties of hot quark matter at high
tive QCD calculation gives a large shear viscosity in the
temperature has been the main target of heavy ion col-
wQGP with η/s 0.8 for α = 0.3 [21]. Therefore,
s
lision experiments at the Relativistic Heavy Ion collider ≃
it turned out as a surprise that the RHIC data of el-
(RHIC), the forthcoming Large Hadron Collider (LHC)
liptic flow v can be described very well by requiring a
2
and FAIR at GSI.
very small shear viscosity over entropy density ratio η/s
The deconfined quark-gluon plasma, if it can be cre-
[22, 23]. Lattice QCD calculationconfirmedthat η/s for
ated through heavy-ion collisions, is an intermediate
thepurelygluonicplasmaisrathersmallandintherange
state and cannot be measured directly. In experiment,
of 0.1 0.2 [24].
thedetectorcanonlymeasurethefreeze-outhadrons. In −
It is now believed that the system created at RHIC is
order to extract the property of the intermediate state,
a strongly coupled quark-gluon plasma (sQGP) and be-
hydrodynamicsis oftenused to simulate the evolutionof
haveslikeanearly”perfect”fluid[25,26]. TheAdS/CFT
the fluid.
duality gives a lower bound η/s = 1/4π [27]. Therefore,
The hydrodynamicalequationsofmotionarethe local
it is conjectured that the sQGP created at RHIC might
conservation laws of energy-momentum and net charge
be the most perfect fluid observed in nature.
∂ Tµν =0, ∂ Nµ =0. (1) However, a perfect fluid should have both vanishing
µ µ c shear and bulk viscosities.
In ideal hydrodynamics, the energy-momentum tensor The perturbative QCD calculation gives ζ/s=0.02α2s
takes the form of for 0.06 < αs < 0.3 [28]. In the hydrodynamic simula-
tionusedto describe the evolutionof the fireballcreated
Tµν =(ǫ+p)uµuν pgµν, (2) at RHIC, the bulk viscosity ζ has often been neglected.
ideal −
3
Ζ(cid:144)s
0.8 0.8
0.6
0.6
0.4
0.4 0.2
0.0
0.2 1.02 1.04 1.06 1.08 1.10 1.12 1.14
0.0 T(cid:144)Tc
1 2 3 4 5
FIG. 3: The bulk viscosity over entropy density ratio as a
functionofscaledtemperatureT/Tc. Thefigureistakenfrom
Ref.[29].
The zero bulk viscosity is for a conformal equation of
FIG. 4: Searching for thecritical end point at RHIC.
stateandalsoareasonableapproximationfortheweakly
interacting gas of quarks and gluons. However, recent
latticeQCDresultsshowthatthe bulkviscosityoveren-
shown from lattice QCD [44] and effective QCD mod-
tropydensityratioζ/srisesdramaticallyuptotheorder
els [45] that this phase transition is of second order and
of 1.0 near the critical temperature T [29–31]. (There
c
belongs to the universality class of O(4) spin model in
are still some subtle issues to determine the bulk viscos-
three dimensions [46]. For real QCD with two quarks of
ity of QCD through calculating the correlations of the
small mass,the secondorder phase transition becomes a
energy-momentum tensor on the lattice, see more de-
smooth crossoverat finite temperature. At finite baryon
tailed discussion in Ref. [32].) The sharp peak of bulk
chemicalpotential,there arestillnoreliableresults from
viscosityatT hasalsobeenobservedinthelinearsigma
c
lattice QCD due to the severe fermion sign problem.
model[33]andintherealscalarmodel[34]. Theincreas-
HoweverQCD effective models [45] suggestthat the chi-
ing tendency of ζ/s has been shown in a massless pion
ral phase transition at finite µ is of first order. It is ex-
gas [35] and in the NJL model below T [36]. The large
c
pectedthatthereexistsacriticalendpoint(CEP)inthe
bulkviscositynearphasetransitionisrelatedtothenon-
T µ QCD phase diagram. The CEP is defined as the
conformal equation of state [37, 38], and the correlation
−
endpointofthe firstorderphase transition,andbelongs
between the bulk viscosity and the conformal anomaly
to the Z(2) Ising universality class [47]. The signature
has been investigated in Ref. [39].
of CEP has been suggested in Refs. [48]. The precise
The sharp rise of the bulk viscosity will lead to the
locationofthe CEP is still unknown. In the future plan,
breakdown of the hydrodynamic approximation around
RHIC is going to lower the energy and trying to locate
the critical temperature. The effect of large bulk vis-
the CEP as shown in Fig. 4.
cosityonhadronizationandfreeze-outprocessesofQGP
createdatheavyioncollisionshasbeendiscussedinRefs. Recently, the authors of Ref. [49, 50] suggested us-
[40–43]. The authors of Ref. [40] pointed out the possi- ing the shear viscosity over entropy density ratio η/s to
bility that a sharprise of bulk viscosity near phase tran- locate the CEP by observing the ratio of η/s behaves
sition induces an instability in the hydrodynamic flow of differently in systems of water, helium and nitrogen in
the plasma,andthismodewillblowupandtearthesys- first-, second-order phase transitions, see the system of
temintodroplets. AnotherscenarioispointedoutinRef. water for example in Fig. 5. The ratio of η/s shows a
[29,42]thatthelargebulkviscositynearphasetransition cusp at Tc for second order phase transition, and a shal-
mightinduce“softstatisticalhadronization”,i.e. theex- lowvalleynearTc forcross-over,andshowsajumpatTc
pansion of QCD matter close to the phase transition is for first-order phase transition.
accompanied by the production of many soft partons, Due to the complexity of QCDin the regime ofstrong
whichmaybe manifestedthroughboth adecreaseofthe coupling, results on hot quark matter from lattice cal-
average transverse momentum of the resulting particles culation and hydrodynamic simulation are still lack of
and an increase in the total particle multiplicity. analytic understanding. In recent years, the anti-de Sit-
ter/conformal field theory (AdS/CFT) correspondence
has generatedenormousinterest in using thermal =4
B. Searching for the critical end point super-Yang-Mills theory (SYM) to understand sNQGP.
The shear viscosity to entropy density ratio η/s is as
At small baryon chemical potential µ, for QCD with small as 1/4π in the strongly coupled SYM plasma [27].
two massless quarks, the spontaneously broken chiral However, a conspicuous shortcoming of this approach is
symmetry is restored at finite temperature, and it is theconformalityofSYM:thesquareofthespeedofsound
4
symmetry breaking, φ¯ = 0 in the vacuum), we shift the
field as φ φ¯+φˆ. In terms of the shifted field, the
→
Lagrangianis given by
1 1 10
= (φ¯)+ (∂ φˆ)2 m2φˆ2 (bφ¯+ cφ¯3)φˆ3
L L0 2 µ − 2 0 − 3
b 5 1
( + cφ¯2)φˆ4 cφ¯φˆ5 cφˆ6, (6)
− 4 2 − − 6
with
a b c
(φ¯)= φ¯2+ φ¯4+ φ¯6 Hφ¯. (7)
0
L 2 4 6 −
Itisnoticedthatthenewfieldφˆobtainsatree-levelmass
of m2 = a + 3b φ¯2 + 5c φ¯4. The induced interaction
0
termsincluding the cubic interactiontermwith coupling
FIG. 5: The shear viscosity overentropydensity ratio η/s in strength bφ¯+10/3cφ¯3, the quartic term with coupling
thewater system. The figure is taken from Ref.[50]. strength b/4 + 5/2cφ¯2, the quintic term with coupling
strengthcφ¯,andthe six-pointinteractiontermwithcou-
c2 always equals to 1/3 and the bulk viscosity is always pling strength 1/6c.
s
zeroatalltemperaturesinthistheory. Thoughζ/satT Assuming translation invariance, we consider effective
c
is non-zero for a class of black hole solutions resembling potentialΩ insteadofeffective actionΓ, these twoquan-
the equationofstate ofQCD,the magnitude is lessthan tities are related via:
0.1 [51], which is too small comparing with lattice QCD
V
results. Γ= Ω, (8)
−T
An alternative nonperturbative approach to study
QCD phase transition is by using effective models. In where V is the 3-volume of the system. The effective
the following, we investigate the thermodynamical and potential in the CJT formalism reads
transport properties in two toy models, one is the sim-
plestrealscalarmodel[34,52],theotherismorerelativis- Ω[φ¯,G¯] = Ω0(φ¯) + Ω2[φ¯,G¯]
tic QCD effective model, i.e, the Polyakov-linear-sigma 1
+ lnG¯−1(K)+G¯−1(K)G¯(K) 1 (,9)
model(PLSM)[53],whichcandescribechiralphasetran- 2 0 −
ZK
sition as well as deconfinement phase transition success- (cid:2) (cid:3)
fully. where Ω0(φ¯) = 0(φ¯) is the tree-level potential, and
G¯(G¯ ) is the full(Ltree-level) propagator:
0
Real scalar model
G¯−1(K,φ¯)= K2+M2(φ¯),G¯−1(K,φ¯)= K2+m2(φ¯).
Weintroducetherealscalartheoryincludingthesextet − 0 − 0
(10)
interaction which is described by the Lagrangian
In the Hartree approximation, the momentum depen-
= 1(∂ φ)2 1aφ2 1bφ4 1cφ6+Hφ. (5) dent contributions are neglected, Ω2 denotes the contri-
µ bution from two-particleirreducible diagrams,and takes
L 2 − 2 − 4 − 6
the form of
WhenH =0,this theoryisinvariantunderφ φand
→−
hwahsicahZd2etseyrmmimneetrtyh.e hvearceuuam,b,pcraorpeermtioesd.el Tpahreamsyestteerms, Ω2[φ¯,G¯]= 3b+ 15cφ¯2 G¯(K) 2+15c G¯(K) 3.
4 2 6
at finite temperature will be evaluated in the Cornwall- (cid:18) (cid:19)(cid:18)ZK (cid:19) (cid:18)ZK (cid:19)
(11)
Jackiw-Tomboulis(CJT)formalism[54]. We willdiscuss
The self-consistent one- and two-point Green’s func-
the following four cases: 1) c = 0,b > 0,a > 0,H = 0,
tions satisfy
the system is always in the symmetric phase. 2) c =
0,b > 0,a < 0,H = 0, the vacuum at T = 0 breaks δΩ δΩ
0, 0 . (12)
theZ2 symmetryspontaneously,andthesymmetryisre- δφ¯ ≡ δG¯ ≡
stored at higher T with a second-order phase transition. (cid:12)φ¯=φ,G¯=G (cid:12)φ¯=φ,G¯=G
(cid:12) (cid:12)
3) c = 0,b > 0,a < 0,H = 0, the Z(2) symmetry is ex- Allther(cid:12)(cid:12)modynamicalinform(cid:12)(cid:12)ationofthesystemiscon-
6
plicitlybroken,andthesystemwillexperienceacrossover tained in the grand canonical potential Ω, evaluated at
at high temperature. 4) c > 0,b < 0,a > 0,H = 0, the themeanfieldlevel. Theentropydensitysisdetermined
brokensymmetry is restoredat high T with a first-order bytakingthederivativeofeffectivepotentialwithrespect
phase transition. to the temperature, i.e,
If symmetry is spontaneously broken in the vacuum,
φ has a vacuum expectation value φ¯, (in the case of no s= ∂Ω(φ)/∂T. (13)
−
5
As the standard treatment in lattice calculation, we in-
troduce the normalized pressure density p which is nor-
26000
malized to vanish atT =µ=0 andthe energydensity ε
c=0, b=0.1, a=0, H=0
as c=0, b=0.1, a=100, H=0
25000 c=0, b=0.1, a=-100, H=0
c=0, b=0.1, a=-100, H=103
p= Ω, ε= p+Ts. (14)
− −
24000
The equation of state p(ε) is an important input into
hydrodynamics. The square of the speed of sound C2 is
s
related to p/ε and has the form of /s23000
h
dp s s
C2 = = = , (15) 22000
s dε Tds/dT C
v
where 21000
C =∂ε/∂T, (16)
v 20000
25 50 75 100 125 150 175 200 225 250
T
is the specific heat. At the critical temperature, the
entropy density as well as the energy density change
most quickly with temperature, thus one expect that C2 FIG.6: Theshearviscosityoverentropydensityη/sasafunc-
s
should have a minimum at T . tionofthetemperatureT,forcaseswithasecond-orderphase
c
The shear viscosity η is calculated by using the Boltz- transition (solid curve), a crossover (dash-dotted curve),and
mann equation [55]. The two-particle elastic scattering with no phase transition for massive field (dashed curve)
and massless field (dotted curve). The figure is taken from
amplitude, whichgovernsparticlecollisionsinthe Boltz-
Ref.[52].
mann equation, is
1 1 1
i =λ +λ2 + + , (17)
T 4 3 s m2 t m2 u m2
(cid:20) − − − (cid:21)
107
where s,t and u are Mandelstam variables, and λ =
3
6φ (b+10cφ2+10c G¯(K,φ¯)) andλ =12(b+5cφ2+
0 3 0 K 4 2 0
5c K G¯(K,φ¯)) are eRffective couplings. 106
The shear viscosity over entropy density ratio η/s in
a=100,b=-0.12,c=0.000025
R
therealscalarmodelinshowninFig.6and7fordifferent
ordersofphasetransitions. Thereis clearlyaqualitative
difference in the η/s behavior between cases with and /s 105
h
without a phase transition. It is seen that η/s shows a
cusp at T for the case of 2nd-order phase transition, a
c 104
shallow valley near T for crossover, and shows a jump
c
at T for the case of 1st-order phase transition. This
c
behavior is qualitatively the same as that in the classic
103
systemssuchastheinH OsystemasshowninFig. 5. If
2
thereis nophasetransition. η/s isalwaysmonotonically
0 10 20 30 40 50 60 70
decreasing. T
Thebulkviscosityisrelatedtothecorrelationfunction
of the trace of the energy-momentum tensor θµ:
µ FIG. 7: The shear viscosity over entropy density η/s as a
functionofthetemperatureT,forthecaseof1storderphase
1 1 ∞
ζ = lim dt d3reiωt [θµ(x),θµ(0)] . (18) transition.
9ω→0ω Z0 Z h µ µ i
Accordingtotheresultderivedfromlowenergytheorem, with the negative vacuum energy density ε = Ω =
v v
in the low frequency region, the bulk viscosity takes the Ω(φ) , and the parameter ω = ω (T) is a scale at
T=0 0 0
form of [29, 30] which| the perturbation theory becomes valid. From the
above formula, we can see that the bulk viscosity is pro-
ζ = 1 T5 ∂ (ε−3p) +16ε , portional to the specific heat Cv near phase transition,
9ω0 (cid:26) ∂T T4 | v|(cid:27) thus ζ/s behavesas 1/Cs2 nearTc in this approximation.
1 The bulk viscosity over entropy density ratio ζ/s as a
= 16ε+9TS+TC +16ε . (19)
9ω {− v | v|} functionofT isshowninFigs.8,9,10,and11. Itisshown
0
6
0.05 0.8
a=100,b=-0.125,c=0.000025
0.04 a=10000,b=-1.2,c=0.000025
b=0.1 0.6 a=40000,b=-4.0,c=0.000069
b=10
0.03 b=30
/s b=60 /s 0.4
0.02
0.2
0.01
0.00 0.0
0.0 100 200 300 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10
T(MeV) T/Tc
FIG.8: Thebulkviscosityoverentropydensityζ/sasafunc- FIG. 11: The bulk viscosity over entropy density ζ/s as a
tion of T for the case without phase transition in the real functionofT fora1st-orderphasetransitionintherealscalar
scalar model. The figure is taken from Ref.[34]. model. The figureis taken from Ref.[34].
that in the case without symmetry breaking, the bulk
viscosity over entropy density ζ/s decreases monotoni-
callywiththeincreaseofthe temperature. Inthe caseof
1.0
2nd-order phase transition, ζ/s decreases with T at low
0.8 b=0.1 temperatureregion,thenrisesupatthecriticaltempera-
b=10
b=30 tureTc andshowsanupwardcusp,anddecreasesfurther
/s 0.6 b=60 in the temperature T >Tc. In the case ofcrossover,it is
observed the cusp behavior of ζ/s is washed out. In the
0.4 case of 1st-order phase transition, ζ/s shows divergent
behavior at T .
c
0.2 From Refs.[50, 52], we know that η/s shows a shal-
low valley in the case of crossover and a jump at T for
c
0.0
0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 first-orderphase transition. But it is hard to distinguish
T/Tc whetherthe systemexperiencesa crossoverorfirst-order
phasetransitionjustfromthevalueofη/sextractedfrom
FIG.9: Thebulkviscosityoverentropydensityζ/sasafunc- the elliptic flow v .
2
tion of T for a 2nd-order phase transition in the real scalar
From our results in the real scalar model, it is found
model in the real scalar model. The figure is taken from
thattheratioofζ/sshowsaverysharppeakatT inthe
Ref.[34]. c
case of first order phase transition, and there is no obvi-
ous change of ζ/s for crossover. As pointed out in Ref.
[40] that a sharp rise of bulk viscosity near phase tran-
sition induces an instability in the hydrodynamic flow
of the plasma, and this mode will blow up and tear
0.20 the system into droplets. Therefore, one can distinguish
0.18 whether the system experiences a first order phase tran-
0.16 a=-10000,b=10,H=0.03 sition or a crossover from observables at RHIC exper-
a=-10000,b=10,H=40
0.14 iments. This result supports the idea of using ζ/s to
0.12
/s 0.10 locate the CEP as suggested in Ref. [30].
0.08 The Polyakov-linear-sigma model
0.06
0.04 We have shown the behavior of shear viscosity over
0.02
entropydensity η/s andbulk viscosityoverentropyden-
0.00
50 100 150 200 sity ζ/s near T for different orders of phase transitions
c
T(MeV) in a toy model, i.e, the real scalar model. In the follow-
ing,weuse amorerealisticQCDeffective model,i.e, the
FIG. 10: The bulk viscosity over entropy density ζ/s as a Polyakov-linear-sigmamodel(PLSM),whichisdescribed
function of T for the case of crossover (the solid line). The
by the Lagrangian[56]
figure is taken from Ref.[34].
= (φ,φ∗,T) (20)
chiral
L L −U
7
where we have separated the contribution of chiral de- Lagrangian in Eq.(20), and in the chiral limit, this La-
grees of freedom and the Polyakovloop. The chiral part grangian is invariant under the chiral flavor group, just
of the Lagrangian, = + consists of the like the original QCD Lagrangian. The trace of the
chiral q m
fermionic part L L L Polyakov-loop, φ and its conjugate φ∗ can be treated as
classical field variables in this work.
= ψ (iγµD gT (σ +iγ π ))ψ (21) The temperature dependent effective potential
Lq f µ− a a 5 a f (φ,φ∗,T) is used to reproduce the thermodynamical
f
X U
behavior of the Polyakov loop for the pure gauge
and the purely mesonic contribution case in accordance with lattice QCD data, and it has
the Z(3) center symmetry like the pure gauge QCD
m = Tr(∂µΦ†∂µΦ m2Φ†Φ) λ1[Tr(Φ†Φ)]2 Lagrangian. In the absence of quarks, we have φ = φ∗
L − −
λ Tr(Φ†Φ)2+c[Det(Φ)+Det(Φ†)] and the Polyakov loop is taken as an order parameter
2
+−Tr[H(Φ+Φ†)], (22) for deconfinement. For low temperatures, U has a
single minimum at φ = 0, while at high temperatures
it develops a second one which turns into the absolute
the sum is over the three flavors (f=1,2,3 for u, d, s).
minimum above a critical temperature T , and the
In the above equation we have introduced a flavor-blind 0
Z(3) center symmetry is spontaneously broken. In this
Yukawa coupling g of the quarks to the mesons and the
paper, we will use the potential (φ,φ∗,T) proposed in
couplingofthe quarksto a backgroundgaugefieldA =
µ Ref.[58], which has a polynomialUexpansion in φ and φ∗:
δ A via the covariant derivative D = ∂ iA . The
µ0 0 µ µ µ
−
tΦheisscaalcaormσpleaxnd3×pse3umdoastcraixlaranπd imsedseofinnneodnientst,erms of U(φ,φ∗,T) = b2(T) φ2 b3(φ3+φ∗3)+ b4(φ2)2,
a a T4 − 2 | | − 6 4 | |
(28)
Φ=T (σ +iπ ). (23)
a a a
with
The 3 3 matrix H breaks the symmetry explicitly and
×
is chosen as T T 2 T 3
0 0 0
b (T)=a +a +a +a . (29)
2 0 1 2 3
T T T
H =T h , (24) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
a a
A precision fit of the constants a ,b is performed to re-
i i
where h are nine external fields. The T = λ /2 are
a a a produce the lattice data for pure gauge theory thermo-
the generators of the U(3) symmetry, λ are the Gell-
a dynamics and the behavior of the Polyakov loop as a
Mann matrices with λ = 21. The T are normalized function of temperature. The corresponding parameters
0 3 a
are
to Tr(TaTb) = δab/2 andqobey the U(3) algebra with
[T ,T ] = if T and T ,T = d T respectively,
a b abc c { a b} abc c a0 =6.75, a1 = 1.95, a2 =2.625,
heref andd fora,b,c=1,...,8arethestandardan- −
abc abc
a = 7.44, b =0.75, b =7.5. (30)
tisymmetricandsymmetricstructureconstantsofSU(3) 3 3 4
−
group and
ThecriticaltemperatureT fordeconfinementinthepure
0
gauge sector is fixed at 270 MeV, in agreement with the
2
f 0, d = δ . (25) lattice results.
ab0 ab0 ab
≡ r3 we obtain the thermodynamical potential density as
The quantity (φ,φ∗,T) is the Polyakov-loop effec- Tln
tive potential expUressed by the dynamics of the traced Ω(T,µf) = − V Z =U(σx,σy)+U(φ,φ∗,T)+Ωψ¯ψ,
Polyakovloop
(31)
φ=(TrcL)/Nc, φ∗ =(TrcL†)/Nc. (26) with the quarks and antiquarks contribution
The Polyakov loop L is a matrix in color space and ex- d3~p
plicitly given by Ωψ¯ψ =−2TNq (2π)3{
Z
ln[1+3(φ+φ∗e−(Eq−µ)/T)e−(Eq−µ)/T +e−3(Eq−µ)/T]
β
L(~x)= exp i dτA4(~x,τ) , (27) +ln[1+3(φ∗+φe−(Eq+µ)/T)e−(Eq+µ)/T +e−3(Eq+µ)/T]
P " Z0 # d3~p }
2TN
with β = 1/T being the inverse of temperature and − s (2π)3{
Z
A4 =iA0. InthePolyakovgauge,thePolyakov-loopma- ln[1+3(φ+φ∗e−(Es−µ)/T)e−(Es−µ)/T +e−3(Es−µ)/T]
trix can be given as a diagonal representation [57]. The
coupling between Polyakov loop and quarks is uniquely +ln[1+3(φ∗+φe−(Es+µ)/T)e−(Es+µ)/T +e−3(Es+µ)/T] .
}
determined by the covariant derivative D in the PLSM (32)
µ
8
Here, N = 2, N = 1, and E = p~2+m2 is the va-
q s q q 0.35
lence quarkandantiquarkenergyfoqru anddquarks,for
0.30
strange quark s, it is E = p~2+m2, and m , m is
s s q s
the constituent quark mass for u, d and s. The purely 0.25
p
mesonic potential is
p/ 0.20 =0 MeV
m2 c w=
U(σx,σy) = 2 (σx2+σy2)−hxσx−hyσy − 2√2σx2σy 0.15 PLSLSMM
0.10 N=6
λ 1 1
+ 1σ2σ2+ (2λ +λ )σ4+ (λ +λ )σ4.
2 x y 8 1 2 x 4 1 2 y 0.05
(33)
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Minimizing the thermodynamical potential in Eq.(31) T/Tc
with respective to σ , σ , φ and φ∗, we obtain a set of
x y
FIG.12: Theequation-of-stateparameterw(T)=p(T)/ε(T)
equations of motion
for µ=0 MeV. The Polyakov linear sigma model prediction
∂Ω ∂Ω ∂Ω ∂Ω (solid line) and thelinearsigma model prediction (dash line)
∂σ =0, ∂σ =0, ∂φ =0, ∂φ∗ =0.(34) are compared with Nf =2+1 lattice QCD data for Nτ =6.
x y Lattice data taken from Ref.[38]. The figure is taken from
The set of equations can be solved for the fields as func- Ref.[53].
tions of temperature T and chemical potential µ, and
the solutions of these coupled equations determine the
behavior of the chiral order parameter σ , σ and the 0.30
x y
Polyakov loop expectation values φ, φ∗ as a function of
0.25
T and µ.
Fig.12 shows the pressure density over energy density
0.20
p/ε, which is represented in terms of equation-of-state =0 MeV
(EOS) parameter, at zero density and finite density, re- /s 0.15
spectively. We observethatthepressuredensityoveren-
PLSM
ergydensityincreaseswithtemperatureandsaturatesat 0.10 LSM
=1 GeV
high temperature. Both the linear sigma model and the
0.05
Polyakov linear sigma model give very similar results at
high temperature, the pressure density over energy den-
0.00
sity p/ε saturates at a value smaller than 1/3. Another 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
common feature of the p/ε in the linear sigma model T/Tc
and the Polyakov linear sigma model is that there is a FIG. 13: The bulk viscosity over entropy density ratio ζ/s
bumpappearingatlowtemperatureregion,whichisalso as a function of the temperature for µ = 0 MeV. The solid
observed in the lattice result. Around the critical tem- line denotes the Polyakov linear sigma model prediction and
peratureT ,thepressuredensityoverenergydensityp/ε the dashed line denotes the linear sigma model prediction.
c
shows a downward cusp. However, the minimum value Lattice data taken from Ref.[30]. The figure is taken from
of the p/ε around T is 0.2 in the linear sigma model, Ref.[53].
c
which is much larger than the result from the Polyakov
linear sigma model and the lattice QCD data. For the
Polyakovlinearsigmamodel,theminimumofp/εaround InRef. [34],wehaveinvestigatedtheequationofstate
T is0.075,whichisconsistentwiththelatticeQCDdata and bulk viscosity in the real scalar model and O(4)
c
[38]. modelinthecaseof2ndorderphasetransition,crossover
In Fig.13, we plot the bulk viscosity overentropy den- and 1st order phase transition, and we have found that
sity ratio ζ/s as a function of the temperature for zero the thermodynamic properties and transport properties
chemicalpotential. Itisshownthat,atzerochemicalpo- inthesesimplemodelsnearthecriticaltemperatureT at
c
tentialµ=0,thebulk viscosityoverentropydensityζ/s strongcoupling aresimilartothose ofthe complexQCD
decreasesmonotonicallywiththe increaseofthetemper- system. In a more realistic QCD effective model, i.e, the
ature in both the Polyakov linear sigma model and lin- Polyakovlinearsigmamodel[53],wehavesystematically
ear sigma model, and at high temperature, ζ/s reaches investigatedthethermodynamicpropertiesandbulkvis-
its conformal value 0. In [30], the bulk viscosity over cosityandfoundthesepropertiesmatchwithlatticedata
entropy density of the three flavor system is extracted very well in the case of zero chemical potential. We fur-
from lattice result, which is shown by the square. It is ther evaluate the chiral phase transitions of u,d and s
observed that ζ/s in PLSM near phase transition is in quarksanddeconfinementphasetransitionatfinite tem-
verygoodagreementwiththe lattice resultin [30],i.e, it perature and finite density, and show the T µ phase
−
rises sharply near phase transition. structure of the Polyakovlinear sigma model in Fig.14.
9

280 1.0
LSM CEP
240 PLSM CEP
0.8
200
=0 MeV
T (MeV) 160 /s 0.6 ===811036 90 . 5 MMMeeeVVV
120 0.4
crossover
80
1st order
y’(T) 2nd peak 0.2
40 ’ (T) 2nd peak
’(T) 2nd peak
0 0.0
0 50 100 150 200 250 300 350 400 450 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
(MeV) T/Tc
FIG. 14: The T −µ phase diagram in the Polyakov linear FIG. 15: The bulk viscosity over entropy density ratio ζ/s
sigma model. The figure is taken from Ref.[53]. in the PLSM for different chemical potentials as functions of
T/Tc. The figureis taken from Ref.[53].
For the Polyakov linear sigma model, the result
divergent at the critical temperature.
shows that the critical end point is around (T ,µ ) =
E E Asdiscussedearlier,thesharpriseofthebulkviscosity
(188 MeV,139.5 MeV), which is close to the lattice re-
will lead to the breakdownof the hydrodynamic approx-
sult (T ,µ ) = (162 2 MeV,µ = 120 13 MeV)
E E ± E ± imation around the critical temperature, and will affect
[59]. For the linear sigma model without the Polyakov
the hadronization and freeze-out processes of QGP cre-
loop, the critical end point is located at (T ,µ )
E E ≃ ated at heavy ion collisions Refs. [40–43]. For example,
(92.5MeV,216MeV). The criticalchemicalpotentialµ
E inRef. [40],itispointedoutthatasharpriseofbulkvis-
in PLSM is much lower than that in the PNJL model
cosity near phase transition might induce an instability
with three quark flavorswhere the predicted critical end
in the hydrodynamic flow of the plasma, and this mode
point is µ >300 MeV[60, 61].
E will blow up and tear the system into droplets. Another
The chiral phase transition for the strange quark and
scenarioispointedoutinRef. [29,42]thatthelargebulk
thedeconfinementphasetransitionintheT µplaneare
viscositynear phase transitionmight induce “soft statis-
−
shown by the dash-dotted line and dotted line, respec-
tical hadronization”, i.e. the expansion of QCD mat-
tively. It is found that with the increase of chemical po-
ter close to the phase transition is accompanied by the
tential, the critical temperature for strange quark to re-
production of many soft partons, which may be mani-
storechiralsymmetrydecreases. However,forthedecon-
fested through both a decrease of the averagetransverse
finement phase transition, with the increase of chemical
momentum of the resulting particles and an increase in
potential, the deconfinement critical temperature keeps
the total particle multiplicity. Therefore the critical end
almost a constant around 220 MeV. It can be seen that
point might be located through the observables which
in the Polyakov linear sigma model, there exists two-
are sensitive to the ratio of bulk viscosity over entropy
flavor quarkyonic phase [62] at low density, where the
density.
u,d quarks restore chiral symmetry but still in confine-
However,itisnoticedthatthe resultsofbulkviscosity
ment, and three-flavorquarkyonic phase at high density,
inthis paperarebasedonEq. (19),wheretheansatzfor
where the u,d,s quarksrestorechiralsymmetry but still
the spectral function
in confinement.
Because the Polyakov-loop in the PLSM is not intro- ρ(ω,~0) 9ζ ω2
= 0 (35)
duced dynamically, it is difficult to calculate the trans- ω π π(ω2+ω2)
port properties from the Boltzmann equation. However,
we can use Eq.(19) to calculate the bulk viscosity. In has been used in the small frequency, and ω is a scale
0
Fig.15, we plot the bulk viscosity over entropy density at which the perturbation theory becomes valid. In our
ratio ζ/s as a function of the temperature for different calculation, ω =1 GeV,its magnitude atT is inagree-
0 c
chemicalpotentials. Itshowsζ/sasfunctionofthescaled ment with that obtained in ChPT for massive pion gas
temperature T/T for different chemical potentials with system in Ref. [39]. Qualitatively, the bulk viscosity
c
µ = 0,80,139.5,160 MeV. We can see that when the corresponds to nonconformality, thus it is reasonable to
chemical potential increases up to µ = 80 MeV, there is observe a sharp rising of bulk viscosity near phase tran-
anupwardcuspappearinginζ/srightatthecriticaltem- sition. Ref. [39]hasinvestigatedthe correlationbetween
peratureT . Withtheincreaseofthechemicalpotential, the bulk viscosity and conformalbreaking, and supports
c
the upward cusp becomes sharper, and the height of the the results in Ref.[29, 30]. The sharp rising of bulk vis-
cusp increases. At the critical end point µ and when cosity has also been observed by another lattice result
E
µ>µ for the first order phase transition, ζ/s becomes [31] and in the linear sigma model [33]. However, till
E
10
now, no full calculation has been done for the bulk vis- Cooper pairing of the electrons breaks the electromag-
cosity. The frequencydependence ofthe spectraldensity netic gauge symmetry, and the photon obtains an effec-
has been analyzed in Refs. [63] and [32] and the limita- tive mass. This indicates the Meissner effect [66], i.e., a
tion of the ansatz Eq.(35) has been discussed. From Eq. superconductor expels the magnetic fields.
(19),weseethatthebulkviscosityisdominatedbyCv at In QCD case at asymptotically high baryon density,
Tc. IfCv divergesatTc,thebulkviscosityshouldalsobe the dominant interaction between two quarks is due to
divergent at the critical point and behave as t−α. How- the one-gluon exchange. This naturally provides an at-
ever,the detailedanalysisin the Ising modelin Ref. [64] tractive interaction between two quarks. The scattering
shows a very different divergent behavior ζ t−zν+α, amplitude forsingle-gluonexchangeinanSU(N )gauge
∼ c
with z 3 the dynamic critical exponent and ν 0.630 theory is proportional to
≃ ≃
the critical exponent in the Ising system. More careful
calculation on the bulk viscosity is needed in the future. N +1
c
(T ) (T ) = (δ δ δ δ ) (37)
a ki a lj jk il ik jl
− 4N −
c
N 1
c
+ − (δ δ +δ δ ).
III. COLOR SUPERCONDUCTING PHASES 4N jk il ik jl
c
It is knownthat the deconfined colddense quark mat- WhereTaisthegeneratorofthegaugegroup,andi,jand
ter is in color superconducting phase. k,l are the fundamental colors of the two quarks in the
Let us start with the system of free fermion gas. incomingandoutgoingchannels,respectively. Underthe
FermionsobeythePauliexclusionprinciple,whichmeans exchangeofthecolorindicesofeithertheincomingorthe
no two identical fermions can occupy the same quantum outgoing quarks, the first term is antisymmetric, while
state. The energy distribution for fermions (with mass the secondterm is symmetric. For Nc =3, Eq. (38) rep-
m) has the form of resentsthatthetensorproductoftwofundamentalcolors
decomposesintoan(antisymmetric)colorantitripletand
1 a (symmetric) color sextet,
f(E )= , β =1/T, (36)
p eβ(Ep−µ)+1
[3]c [3]c =[3¯]c [6]c. (38)
⊗ a⊕ s
here E = p2+m2, µ is the chemical potential and
p
T is the temperature. At zero temperature, f(E ) = In Eq. (38), the minus sign in front of the antisymmet-
p
p
θ(µ E ). The ground state of the free fermion gas is ric contribution indicates that the interactionin this an-
p
−
a filled Fermi sea, i.e., all states with the momenta less titripletchannelisattractive,whiletheinteractioninthe
than the Fermi momentum p = µ2 m2 are occu- symmetric sextet channel is repulsive.
F
−
pied, and the states with the momenta greater than the For cold dense quark matter, the attractive interac-
p
Fermi momentum p are empty. Adding or removing a tion in the color antitriplet channel induces the conden-
F
single fermion costs no free energy at the Fermi surface. sate of the quark-quark Cooper pairs, and the ground
ForthedegenerateFermigas,theonlyrelevantfermion state is called the “color superconductivity”. Since the
degreesoffreedomarethoseneartheFermisurface. Con- diquark cannot be color singlet, the diquark conden-
sideringtwofermionsneartheFermisurface,ifthereisa sate breaks the local color SU(3) symmetry, and the
c
net attraction between them, it turns out that they can gauge bosons connected with the broken generators ob-
form a bound state, i.e., Cooper pair [65]. The binding tain masses. Comparing with the Higgs mechanism of
energy of the Cooper pair ∆(K) (K the total momen- dynamical gauge symmetry breaking in the Standard
tum of the pair), is very sensitive to K, being a maxium Model, here the diquark Cooper pair can be regardedas
where K = 0. There is an infinite degeneracy among acompositeHiggsparticle. Thecalculationoftheenergy
pairs of fermions with equal and opposite momenta at gapandthe criticaltemperaturefromthefirstprinciples
the Fermi surface. Because Cooper pairs are composite has been derived systematically in Refs. [67–74].
bosons,theywilloccupythesamelowestenergyquantum In reality, we are more interested in cold dense quark
state at zero temperature and produce a Bose-Einstein matter at moderate baryon density regime, i.e., µ
q
∼
condensation. Thus the ground state of the Fermi sys- 500MeV,whichmayexistintheinteriorofneutronstars.
tem with a weak attractive interaction is a complicated Itislikelythatcolddensequarkdropletmightbecreated
coherent state of particle-particle Cooper pairs near the in the laboratory through heavy ion collisions in GSI-
Fermi surface [17]. Exciting a quasiparticle and a hole SPSenergyscale. Atthese densities, anextrapolationof
which interact with the condensate requires at least the theasymptoticargumentsbecomesunreliable,wehaveto
energy of 2∆. relyoneffectivemodels. Calculationsintheframeworkof
In QED case in condensed matter, the interaction be- pointlikefour-fermioninteractionsbasedontheinstanton
tween two electrons by exchanging a photon is repul- vertex[19,75–77],aswellasinthe Nambu–Jona-Lasinio
sive. The attractiveinteractiontoformelectron-electron (NJL) model [78–82] show that color superconductivity
Cooper pairs is by exchanging a phonon, which is a col- does occur at moderate densities, and the magnitude of
lective excitation of the positive ion background. The diquark gap is around 100 MeV.