Table Of ContentNuclear Chiral Dynamics and Phases of QCD
W. Weise
Physik-Department, Technische Universita¨t Mu¨nchen,
D-85747 Garching, Germany
2 January 5, 2012
1
0
2
n Abstract
a
J This presentation starts with a brief review of our current picture of QCD phases, derived from
4 lattice QCD thermodynamics and from models based on the symmetries and symmetry breaking
patterns of QCD. Typical approaches widely used in this context are the PNJL and chiral quark-
]
h meson models. It is pointed out, however, that the modeling of the phase diagram in terms of
t
- quarks as quasiparticles misses important and well known nuclear physics constraints. In the
l
c hadronic phase of QCD governed by confinement and spontaneously broken chiral symmetry, in-
u
mediumchiraleffectivefieldtheoryistheappropriateframework,withpionsandnucleonsasactive
n
[ degrees of freedom. Nuclear chiral thermodynamics is outlined and the liquid-gas phase transition
1 is described. The density and temperature dependence of the chiral condensate is deduced. As a
v consequence of two- and three-body correlations in the nuclear medium, no tendency towards a
0
first-order chiral phase transition is found at least up to twice the baryon density of normal nuclear
5
9 matter and up to temperatures of about 100 MeV. Isospin-asymmetric nuclear matter and neutron
0 matter are also discussed. An outlook is given on new tightened constraints for the equation-
.
1 of-state of cold and highly compressed matter as implied by a recently observed two-solar-mass
0
neutron star.
2
1
:
v
1 Introduction
i
X
QCD phase diagram: visions and facts
r
a
A theorist’s sketch of the QCD phase diagram, plotted in the plane of temperature T and baryon
chemical potential µ , usually looks like the one shown in Fig.1. The hadronic phase, with quarks and
B
gluons confined in mesons and baryons, is thought to be separated from the deconfined quark-gluon
phase by a crossover transition at low baryon chemical potential µ and at temperatures T of order
B
Λ 0.2 GeV. At a critical point this crossover supposedly ends and turns, at larger values of the
QCD
∼
baryon chemical potential, into a first-order phase transition. When the baryon chemical potential is
further increased, various low-temperature color-superconducting phases are expected to emerge.
At zero chemical potential, lattice simulations of full QCD thermodynamics with three quark flavors
[1, 2, 3, 4] do indeed suggest a crossover scenario for the temperature dependence of the chiral (quark)
condensate, an order parameter spontaneously broken chiral symmetry, with a transition temperature
T around 170 MeV. Lattice QCD extrapolations to non-zero (real) chemical potentials are notoriously
c
difficult, however. Extensions of the phase diagram to baryonic matter and various forms of quark
1
T
[GeV] N =2 (q=u,d)
qufark gluon phase
−
deconfinement
0.2
quark gluon phase
Tc −
e
r criticalpoint
u
t
a
er hhaaddrroonnicpphhasaese
p
m q¯q =0 CCSSCCpphhaassees
e ! "#
t qq =0
nuclear ! "#
matter
1GeV µB
q¯q
baryon chemical potential
condensation
Cooper
Spontaneous
pairing
Chiral Symmetry high density phases:
Breaking Color Super Conductivity
Figure 1: Schematic picture of the QCD phase diagram in terms of temperature T and baryon chemical
potential µ , sketching the deconfined quark-gluon phase at high temperature, the low-temperature and
B
low-density hadronic phase with spontaneously broken chiral symmetry, and the domain of high-density
(color-superconducting - CSC) phases featuring various forms of quark Cooper pairing. The dashed line
illustrates a chiral crossover transition ending at a hypothetical critical point where a first-order phase
transition boundary (solid line) begins. The first-order liquid-gas phase transition of nuclear matter is
also indicated.
matter rely so far entirely on models. Only at asymptotically large quark chemical potentials the
picture becomes simple again. At Fermi momenta of the order of several GeV, the quarks at the Fermi
surface can be treated using perturbative QCD. The gluon exchange quark-quark interaction is weakly
attractive in color-antitriplett pairs. Such a weakly attractive interaction is sufficient to form Cooper-
pair-like diquarks in various combinations so that color superconducting phases emerge [5]. However,
the more quantitative density scale at which such transitions take place cannot be predicted.
Basically all models of the QCD phase diagram (briefly summarized in Section 2) work with dynam-
ical quarks as quasiparticles and do not take into account baryons and their correlations. An apparently
generic result of such models is the fact that the first order transition line at low temperature meets
the chemical potential axis at a quark chemical potential around µ 0.3 GeV, the typical constituent
q
∼
quark mass scale, corresponding to a baryon chemical potential µ close to the nucleon mass. This,
B
however, is the terrain of nuclear physics with its enormously rich and well explored phenomenology.
No sign of a first-order chiral or deconfinement phase transition is visible in that area right in the center
of the QCD phase diagram. The only known phase transition in nuclear matter is the one from a
Fermi liquid to an interacting Fermi gas. This motivates the present essay as a proposal for taking the
well-known nuclear physics constraints more seriously into account (see Section 3). This presentation
is in parts an update of a previous review [6]. It closes with a brief discussion of new constraints from
neutron star observations on the equation of state at high density and low temperature.
2
2 Modeling the QCD phase diagram
Confinement and spontaneous chiral symmetry breaking are governed by two basic symmetry principles
of QCD:
The symmetry associated with the center Z(3) of the local SU(3) color gauge group is exact
c
•
in the limit of pure gauge QCD, realized for infinitely heavy quarks. In the high-temperature,
deconfinement phase of QCD this Z(3) symmetry is spontaneously broken, with the Polyakov
loop acting as the order parameter.
Chiral SU(N ) SU(N ) symmetry is an exact global symmetry of QCD with N massless
f R f L f
• ×
quark flavors. In the low-temperature (hadronic) phase this symmetry is spontaneously broken
down to the flavor group SU(N ) (the isospin group for N = 2 and the “eightfold way” for
f V f
N = 3). As a consequence there exist N2 1 pseudoscalar Nambu-Goldstone bosons and the
f f −
QCD vacuum hosts a strong quark condensate.
These symmetries and symmetry breaking patterns serve as guiding principles for constructing
modelsoftheQCDphases. Thebasicquestionisabouttheintertwiningofspontaneouschiralsymmetry
breaking with confinement.
2.1 Chiral condensate and Polyakov loop
The order parameter of spontaneously broken chiral symmetry is the quark condensate, q¯q . The
(cid:104) (cid:105)
disappearence of this condensate, by its melting above a characteristic transition temperature scale,
signals the restoration of chiral symmetry in its unbroken Wigner-Weyl realization. The transition from
confinement to deconfinement in QCD is likewise controlled by an order parameter, the Polyakov loop.
A non-vanishing Polyakov loop Φ reflects the spontaneously broken Z(3) symmetry characteristic of the
deconfinement phase. The Polyakov loop vanishes in the low-temperature, confinement sector of QCD.
Two limiting cases are of interest in this context. In the pure gauge limit of QCD, corresponding
to infinitely heavy quarks, the deconfinement transition is established as a first order phase transition
with a critical temperature of about 270 MeV. In the limit of massless u and d quarks, on the other
hand, the isolated chiral transition appears as a second order phase transition at a significantly lower
critical temperature. This statement is based on calculations using Nambu - Jona-Lasinio (NJL) type
models [7, 8, 9] which incorporate the correct spontaneous chiral symmetry breaking mechanism but
ignore confinement. The step from first or second order phase transitions to crossovers is understood
as a consequence of explicit symmetry breaking. The Z(3) symmetry is explicitly broken by the mere
presence of quarks with non-infinite masses. Chiral symmetry is explicitly broken by non-zero quark
masses. It is then a challenging question whether and how the chiral and deconfinement transitions
get dynamically entangled in just such a way that they finally occur within overlapping transition
temperature intervals.
Confinement implies spontaneous chiral symmetry breaking, but the reverse is not necessarily true.
There is no a priori reason why the chiral and deconfinement transitions should be intimitely connected.
NonethelessthisappearstobethecaseinlatticeQCDcomputationswithalmostphysicalquarkmasses.
Results from lattice QCD thermodynamics [1] with 2+1 flavors (at zero baryon chemical potential)
gave a chiral crossover transition temperature of about 190 MeV. Recent improved computations [2, 3]
indicate a shift of the chiral transition to somewhat lower temperatures, around 160 MeV, consistent
with earlier lattice simulations [4]. Lattice QCD results for the temperature dependence of the Polyakov
loop have remained remarkably stable over the years. Pure gauge QCD on the lattice, with full gluon
dynamics but without quarks (or equivalently, with infinitely heavy quarks) has established a first-order
deconfinement transition at a critical temperature T 270 MeV. Once light quarks are added, the Z(3)
c
(cid:39)
3
1.0 1.0
0 0.8 0.8
q LQCD
q 0.8 0.8
chiralcondensateqq 0000....0246 !!q¯q¯qq""T0 Polloyoapkov 000...246 Polyakovloop 0000....0246 noPn2N-+loJ1Lc flaalvors sNNNptoΤΤΤuuret846g,,,a18pu24g,1e6,gpauurge000e ...246 Polyakovloop
0.5 1.0 1.5 2.0 0.1 0.2 0.3 0.4 0.5
T Tc T GeV
Figure 4: Solid curves: calculated temperature dependence of the chiral condensate q¯q (left)
Figure 2: Pure gauge and full QCD lattice results for the Polyakov loop as function of temperature [1].
and of the Polyakov loop Φ (right) normalizedto the transition temperature Tc = 205MeV as
The solid line is calculated in an advanced (non-local) PNJL model [15], to be discussed later in the
obtainedinthenonlocalPNJLmodelconsideredhere. Thedashedlinesshowthechiralconden-
sate for the pure fermionic case and the Polyakov loop for the pure gluonic castee,xrte.spectively.
center symmetry is explicitly broken by the presence of these quarks and the first-order deconfinement
1.0
transition turns into a crossover, with a significantly reduced transition temperature around 200 MeV.
0
This is shown in Fig.2 where Polyakov loop results from pure gauge lattice computations are compared
q
q 0.8 to full QCD simulations with N = 2+1 quark flavors.
f
q
q T 270MeV
0
0.6 2.2 The non-local PNJL model
e
at T0208MeV
s
n Insights concerning the issue of a possible intertwining of chiral and deconfinement transitions can
e
d 0.4
n be gained from a model based on a minimal synthesis of the NJL-type spontaneous chiral symmetry
o
c breaking mechanism and confinement implemented through Polyakov loop dynamics. This PNJL model
l
ra 0.2 [10, 11] is specified by the following action:
i
h
c
β=1/T V
= dτ d3x ψ ∂ ψ (ψ,ψ ,φ) (Φ,T) . (1)
0.0 † τ †
0.10 0.15 0.20 0.25 0.30 S (cid:90)0 (cid:90)V (cid:104) −H (cid:105)− T U
T GeV It introduces the Polyakov loop,
Φ = N 1Trexp(iφ/T) , (2)
c−
Figure 5: The T dependence of the chiral condensate. The chiral transition tempera-
0
ture decreases from 205MeV to about 170MeV when reducing T0(Nf = 0)w=ith27a0MhoemVotgoeneous temporal gauge field, φ = φ3λ3 + φ8λ8 ∈ SU(3), coupled to the quarks. The
T (N =2)=208MeV. dynamics of Φ is controlled by a Z(3) symmetric effective potential , designed such that it reproduces
0 f
U
the equation of state of pure gauge lattice QCD with its first order phase transition at a critical
temperature of 270 MeV. The field φ acts as a potential on the quarks represented by the flavor doublet
Finally, the pressure P = Ω is computed after subtracting a divergent v(afocuruNm (=T =2)0o)r triplet (for N = 3) fermion field ψ. The Hamiltonian density in the quark sector is
− f f
term. The result is shown in Fig. 7. This figure displays, in addition, the separate contributions
(cid:126)
= iψ (α(cid:126) +γ mˆ φ)ψ + (ψ,ψ ) , (3)
† 4 †
H − ·∇ − V
20
with the quark mass matrix mˆ and a chiral SU(N ) SU(N ) symmetric interaction .
f L f R
× V
Earliertwo-flavorversionsofthePNJLmodel[10,11,12]havestillusedalocalfour-pointinteraction
of the classic NJL type, requiring a momentum space cutoff to regularize loops. A more recent version
4
[13], using a non-local interaction between quarks, does not need any longer an artificial NJL cutoff. It
generates instead a momentum dependent dynamical quark mass, M(p), along with the non-vanishing
quark condensate. A further extension to three quark flavors [14] includes a U(1) breaking term
A
implementing the axial anomaly of QCD. This term is constructed as a non-local generalization of the
Kobayashi-Maskawa-’tHooft 3 3 determinant interaction.
×
A basic relation derived in this non-local PNJL model is the gap equation generating the momentum
¯
dependent quark mass self-consistently together with the chiral condensate ψψ . Its form (written
(cid:104) (cid:105)
here for the two-flavor case) is reminiscent of the corresponding equation emerging in Dyson-Schwinger
approaches to QCD:
d4q M(q)
M(p) = m +8N G (p q) , (4)
0 c (2π)4C − q2 +M2(q)
(cid:90)
where m = m = m is the current quark mass, G is a coupling strength of dimension (length)2 and
0 u d
(q) is a momentum space distribution, the Fourier transform of which represents the range over which
C
the non-locality of the effective interaction between quarks extends in (Euclidean) 4-dimensional space-
time. This non-local approach permits to establish contacts with the high-momentum limit of QCD
with its well-known behaviour M(p) α (p) ψ¯ψ /p2 at p . At p 1 GeV the distribution (p)
s
∝ − (cid:104) (cid:105) → ∞ ≤ C
is designed to follow lattice QCD and Dyson-Schwinger results, or it may be deduced from an instanton
liquid model (see Ref. [14] for more details). The non-local PNJL model can indeed be “derived” from
QCD as demonstrated in Refs. [16, 17].
Once the input is fixed at zero temperature by well-known properties of the pseudoscalar mesons,
the thermodynamics of the PNJL model can now be investigated with focus on the symmetry breaking
pattern and on the intertwining of chiral dynamics with that of the Polyakov loop. The primary role
of the Polyakov loop and its coupling to the quarks is to suppress the thermal distribution functions of
color non-singlets, i.e. quarks and diquarks, as the transition temperature T is approached from above.
c
Color singlets, on the other hand, are left to survive below T . This is seen by analyzing the relevant
c
piece of the thermodynamic potential Ω = (T/V)ln , the one involving the quark quasiparticles:
− Z
∆Ω d3p
ln 1+3Φe (Ep µ)/T +3Φ e 2(Ep µ)/T +e 3(Ep µ)/T
− − ∗ − − − −
T ∝ (2π)3
(cid:90) (cid:104) (cid:105)
d3p
+ ln 1+3Φ e (Ep+µ)/T +3Φe 2(Ep+µ)/T +e 3(Ep+µ)/T . (5)
∗ − − −
(2π)3
(cid:90) (cid:104) (cid:105)
Thissuppressionofcolornon-singletsinthehadronicphasebelowT should, however, notbeinterpreted
c
as dynamical confinement. The unsuppressed color singlet three-quark degrees of freedom are not
clustered but spread homogeneously over all space. At this stage, nucleons are not treated properly as
localized, stronglycorrelatedthree-quarkcompoundsplusseaquarks. Thusonecannotexpectthatsuch
a model describes properly the low-temperature phase of matter at finite baryon chemical potentials
around 1 GeV, the domain of nuclear many-body systems.
At zero baryon chemical potential, a remarkable dynamical entanglement of the chiral and decon-
finement transitions is nonetheless verified in the PNJL model, as demonstrated in Fig.3. In the absence
of the Polyakov loop the quark condensate (left dashed line), shows the expected chiral crossover tran-
sition, but at a temperature way below and far separated from the 1st order deconfinement transition
controlled by the pure-gauge Polyakov loop effective potential (right dashed line). Once the coupling
U
of the Polyakov loop field to the quark density is turned on, the two transitions move together and
end up at a common transition temperature around 0.2 GeV. The deconfinement transition becomes a
crossover (with Z(3) symmetry explicitly broken by the coupling to the quarks).
It is interesting to compare this PNJL description of the interplay between the chiral and decon-
finement transitions, with a scenario based on strong-coupling lattice QCD with inclusion of Polyakov
5
1.0 1.0
0
q
q 0.8 0.8
q
q p
0.6 0.6 o
e o
t l
a v
s o
n k
de 0.4 0.4 ya
n l
o o
c P
al 0.2 0.2
r
i
h
c
0.0
0.5 1.0 1.5 2.0
T T
c
¯
Figure 3:FSiogulirde c4u: rSvoelsid: ctuermvepse:rcaatlucurelatdeedpteenmdpeenrcaetuoref tdhepeecnhdiernaclecoofntdheencshairtael cq¯oqnde=nsau¯teu q¯+q (dledft)(left) and
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
of the Polaynadkoovf tlhoeopPoΦly(arkiogvhlto)onpoΦrm(raiglihzte)dntoormthaleizterdatnositthieontrtaenmsitpioenratteumrpeeTratu=re2T05=M2e0V5,MaesVcaalsculated in
c c
the nonlocoabltaPinNedJLinmthoedneolnwloictahlNPNJ=Lm3oadnedlcpohnysisdiecraeldqhuearer.kTmhaesdsaessh[e1d5l]i.neTshsheodwatshheedchlirinalecsosnhdoewn-the chiral
f
condensatseatfeorfotrhteheppuurreeffeerrmmiioonniiccccaaseseanadndthethPeolPyaoklyovaklooovplofoorpthfeorputhreegpluuorneicgcluasoen,ircescpaescet,ivreelys.pectively.
loop dynamics [19]. Altho1u.0gh the starting points and frameworks are quite different in those two ap-
proaches, recent studies [19] arrive at similar results concerning the close correspondence between chiral
0
and deconfinement tranqsition temperature ranges.
q 0.8
Pions also have a pronouncedinfluence [14] on the chiral condensate as it approaches the transition
q
region from the hadronqic phase.The chiral crossover becomes siTg0ni2fi7c0aMnteVly smoother close to Tc. Lattice
0.6
QCD results [2] with pehysical pion masses and realistic extrapolations to the continuum limit show a
at T0208MeV
similar tendency. s
n
e
Undoubtedly a primde c0h.4allenge in the physics of strong interactions is the exploration of the QCD
n
phase diagram at non-ozero baryon density, extending from normal nuclear matter all the way up to
c
very large quark chemiralcal0p.2otentials µq at which color superconducting phases are expected to occur.
PNJL calculations at fihinite µ give a pattern of the chiral order parameter in the (T,µ ) plane showing
c q q
a crossover at small µ that ends at a critical point. From there on a first-order transition line extends
q
down to a quark chemical0p.0otential µ 0.3 - 0.4 GeV at T = 0.
0.10 q0∼.15 0.20 0.25 0.30
The typical phase diagram that emerges from the most recent version of the non-local three-flavor
T GeV
PNJL model [15] is shown in Fig.4. The phase to the right of the 1st order transition line, but below the
deconfinement boundary that decouples from the chiral transition beyond the critical point, has been
Figure 5: The T dependence of the chiral condensate. The chiral transition tempera-
0
named “quarkyonic”[21] and is hypothetically assumed to exist until superconducting phases take over
ture decreases from 205MeV to about 170MeV when reducing T (N = 0) = 270MeV to
0 f
at large quark chemical potential. A recent discussion about the possible relationship between such a
T (N =2)=208MeV.
0 f
hypothetical phase and the hadron rates produced in high-energy heavy-ion collisions [22] may not yet
be conclusive but is pursued with great intensity.
One mustFirneaclalyl,l tahgeaipnreasstutrehiPs p=ointΩtihsactomcapluctueldataifotnerssaunbdtraecxttinragpaoldaivtieorngesntbavsaecduuomn(PTN=JL0)or related
−
approachetsersmti.llThhaevreesluimltiitsesdhovwanlidinitFyiga.t7.mTohdiesrfiagtuereanddisphlaigysh, ibnaardydointiodne,ntshietiseespasirnatceectohnetyribduotionnost properly
incorporate baryonic degrees of freedom. They miss the important constraints imposed by the existence
of nucleons and nuclear matter. This becomes clear when converting the PNJL phase diagram, Fig.4,
20
into a plot that translates quark chemical potentials into baryon densities via the derivative of the pres-
sure, ρ = ∂P/∂µ , with respect to baryon chemical potential µ = 3µ . The 1st order transition line
B B B q
6
1.0 deconfinement
hadronic phase
c
T
chiral
T
0.5 PolyakovLoop
1st order
chiralcrossover
chiral1storder
0.0
0.1 0.2 0.3
Μ GeV
q
Figure4: PFhigausreed1i1a:grPahmasfeorditahgeratmhrefoer-fltahveo(r2n+on1l)o-flcaavloPrNnoJnLlomcaoldPeNl[J1L5]m. oTdheleaotramnegaena-finedldblleuveelb.andsshow
The orange band shows the confinement-deconfinement crossover transition as described by
the deconfinement and chiral crossover transitions. The solid black line indicates the chiral first-order
the Polyakov loop in the range 0.1 < Φ < 0.3. The dashed black line corresponds to the chi-
transition.
ral crossover (blue band: 0.3 < ψ¯ψ / ψ¯ψ < 0.7). The solid black line indicates the chiral
0
first-order transition. The temperature scale is set by T =0.2GeV.
c
converts to a broad coexistence region along the density axis that encloses what is well known as nuclear
physics terra(ii)n,Inbculutdinnogwwadvees-cfurnibcteidonw-reitnhormthaeliz“awtiornonegff”ecqtsuaresqipuairretsicalecadreegfurleeres-aosfsefsrsemeednotmo:f cPhNi-JL quarks
rather than interaralclotwin-egnenrugcylethoenosr.emQsu.aPrskesudaorsecanlaortmtheseonremleavsasnestaancdticvoerrqesupaosnidpianrgtidcelceasyactonlostwanttesmperature
at zero temperature have been re-derived. The results clearly show that the formal-
and baryon chemica potentials. Consequently, interpretations based on the schematic phase diagram of
ism incorporates fundamental chiral relations such as the Gell-Mann–Oakes–Renner and
Fig.4 along the chemical potential axis should be exercised with caution.
Goldberger-Treiman relations. In the three-flavor case, the inclusion of the ’t Hooft-
Several further important questions are being raised in this context. A principal one concerns the
Kobayashi-Maskawa interaction leads to the correct mass splitting between the η and the
existenceandloηcamtieosnono.fthecriticalpointinthephasediagram. ExtrapolationsfromlatticeQCD,either
by Taylor expansions[4] around µ = 0 or by analytic continuation from imaginary chemical potential
(ii) ThePNJLthermodynamicshasnowbeendevelopedwithsystematicinclusionofthequark
[23], have so far not reached a consistent conclusion. A second question relates to the sensitivity of
quasiparticle renormalization factor Z(p). The temperature dependence of the chiral con-
the first order transition line in the phase diagram with respect to the axial U(1) anomaly in QCD
densateandofthePolyakovloophasbeencalculated,indicatingchiralanddecoAnfinement
[24]. This issuecrhoasssobveerentraandsditrioensss.edWienhRaveef.c[o2m5p]auresdinoguar rgeesunletrsawliGthinrezcbeuntrgla-Lttaicned-QaCuDancsoamtpzuftoar- the chiral
SU(3) effectivetaiocntsi.onFinwailtlyh, aaxqiuaalrkUc(h1e)maicnaolmpoatleyn.tiaIlthwasasbepeoninintterdodoucuetdtthhaatt,endaebpleesnedxitnegnsoionnsdteotails of the
axial U(1) bretahkeifinngitien-dteenrasictytioreng,ioan soefctohnedQCcrDitpichaalsepdoiiangtrammi.ght appear such that the low-temperature
A
evolution to high density is again just a smooth crossover, or the first order transition might disappear
(iii) The impact of the wave-function renormalization factor Z(p) compared to previous calcu-
altogether andlgaitvioensw[1a1y,1t2o]saetstimngoZot(hp)cro1sissogveenrertahllryoquugihteosumt.allIonveartthherweeh-oflleavreolrevNanJtLmotympeenturemalization of
≡
this approach [r2a6n]g,ei.tThhaisscraencebnetulyndberesetnooddecmonosnidsetrriantgetdhehgoawp etqhueatiinotnesraatctzieorno tbemetpweeraetnurceh:isrianlcecondensate
and diquark deZgr(epe)sdoevfifarteeesdsoigmni,ficbaanstelydforonmthuenigtyenounilnyeinUt(h1e)mobmreeantkuimngrsainxg-eppoint1vGeretVe,x,itscaenf-open up a
A
fect does not contribute much to the relevant integrals because of its suppression by the
smooth transition corridor along the chemical potential axis in which chiral and diquark condensates
integration measure.
may coexist.
Interesti(nivg) dWeivtehloinpcmluseinontsofaZre(pp),rethseencthliyralcaonnddudecctoendfinteomweanrtdcsrofissxoivnegrttrhaensiitniopnustteonfdtthoebnecoonm-leocal PNJL
smoother compared to our previous investigations.
model by direct comparison with lattice QCD thermodynamics at imaginary chemical potential [23, 27,
28] and then(vs)tuTdhyeiflnagvoimr dpelpiceantdieonncse,oTn(tNhe),pohfathsee ddeiacognrfianmemaetntretaelmµp.erature scale is an important
0 f
issue in lowering chiral-transition temperatures in accordance with the tendency recently
2.3 The Polyakov - quark-meson model
26
AvariantofthePNJLapproachisthePolyakov-quark-meson(PQM)model[29]. Itisbasedonalinear
sigma model with quarks coupled to chiral pion and sigma fields and propagating in a Polyakov loop
7
9
200 200
150 150
] ]
V V
e e
M M
100 100
[ [
T ! crossover T ! crossover
" crossover " crossover
— —
50 " crossover 50 " crossover
CEP CEP
! first order ! first order
0 0
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
µ [MeV] µ [MeV]
q
FIG. 6: Chiral and deconfinement phase diagraFmigufroer5:aPchoansestdainagtraTm f=or2th0e8twMoe-flVav(olrePftQpMan(Peol)lyaaknodv -foqruaTrk-(mµe)sown)itmhoγˆde=l [300.]8. 5Th(erisgohlitd and
0 0
¯
dotted lines show the deconfinement crossover transition (Φ and Φ crossover), while the dashed line
panel). The (grey) band corresponds to the width of dΦ/dT at 80% of its peak height. Close to the intersection point of
marks the chiral crossover transition line (χ crossover) ending in the critical point. The chiral first order
the chiral transition and the deconfinement transition at mid chemical potential a double peak structure in the temperature
transition is restricted to a small low-temperature region.
derivative of the Polyakov-loop variables emerges. The (green) dashed line in this region follows the highest peak.
background. Combined with renormalization group techniques, the PQM model has recently improved
0.03 in a version [30] that includes the 0b.0a3ck-reaction of the quarks at finite µ on the Polyakov loop effective
1.2 µ = 0 MeV 1.2 µ = 0q MeV
pµo t=en 1t5ia0l. MWeVith increasing quark chemical potential, this backµ-r e=a c1t5io0n MheasVthe effect of aligning the
1 chµi r=a l2a9n0d MdeeVconfinement tran 1sition borders in the T µ diagraµm = s2u9c0h MtheaVt these transitions tend to
−
go in parallel not only at small chemical potential, but also as µ increases. This important feature is
0 0 q
0.8 apparent in Fig.5. Note tha 0t.8the critical point is now moved to low temperature and the chiral first
0 25 50 0 25 50
B B
S order transition is restrSicted to a very small region around quark chemical potential µ 0.3 GeV in
p 0.6 p 0.6 q ∼
/ the phase diagram. Ho/wever, once again, this region corresponds to the density range of nuclear matter
p p
where the quark-meson picture is not expected to be valid.
0.4 0.4
0.2 2.4 Dyson-Schwing e0r.2approach to QCD thermodynamics
The approach that is, by design and construction, closest to QCD itself is the one based on solutions
0 0
0 50 100 150 200 o 2f5(t0run 3c0at0ed) 3D5y0son-Schwinger e0quat i5o0ns fo 1r0q0uar k15a0nd g2l0u0on c 2o5rr0ela t3io0n0fu n3c5t0ions. Dyson-Schwinger
thermodynamics has now advanced to the point of producing its generic version of the QCD phase
T [MeV] T [MeV]
diagram [31], as displayed in Fig.6. It is again remarkable that the chiral and deconfinement crossovers
go jointly in parallel from T 170 180 MeV at µ = 0 down to a critical point at T 100 MeV and
c q e
(cid:39) − (cid:39)
FIG. 7: Pressure normalized to the Stefan-Boltµzman28n0pMreeVss.uFrreomfotrhearecoonnsdtoawnntwTard=a c2h0ir8alMfiresVt or(dleefrttrpaannsietilo)narnegdiowniitshinµdi-ccaotrerde.ctions
q 0
(cid:39)
(right panel) for three different chemical potentiaWls.haTththeeCDEysPon-iSschlowcinagteerdanadpPpQroMxiamppartoealcyheasthaµve=in2c9o3mmMoenV, h.oTwehveer,inwsiethtsthsehPoNwJLthmeodels
pressure at µ = 290 MeV for small temperaturiess.the absence of nucleon formation in the range of densities characteristic of nuclear (rather than
constituent quark) matter. The range of quark chemical potentials around µ 0.3 GeV and low
q
∼
temperatures is nuclear terrain! The baryon densities
A similar trend is seen in the entropy, Fig. 8, and cide over the whol1e p∂hPase diagram and as a consequence
ρ = , (6)
B
quark number density, Fig. 9, if the µ-corrections are the quark numbe3r(cid:32)d∂eµnq(cid:33)siTty approaches much faster the
taken into account. The entropy density decreases for SB-limit (right panel of Fig. 9).
determined by the derivative of the pressure P with respect to chemical potential, turn out to be in the
small temperatures at µ = 290 MeV since the number
of active degrees of freedom decreases when approach- 8
ing the first-order transition from below. At the transi-
In Fig. 10 the scaled quark number density (left panel)
tion the entropy jumps. The bump around T 90 MeV
∼ and the corresponding scaled quark number susceptibility
(left panel) is a remnant from the smooth chiral crossover
(right panel) for three different temperature slices around
transition. This effect is completely washed out when the
the critical endpoint (TCEP,TCEP 5 MeV) as a function
µ-corrections are included (right panel). Similar to the
±
of the quark chemical potential are collected. In this fig-
findings for the pressure these corrections become more
ure the µ-corrections in T are omitted while in Fig. 11
significant at larger chemical potential. 0
they are taken into account. Due to the chiral critical
This also appears in the quark number density n = endpoint which is a second-order transition the suscepti-
q
∂Ω/∂µ which is plotted in Fig. 9. For comparison the bility diverges with a certain power law [41]. There are
−
corresponding SB-limits (dashed lines) are also shown no strong modifications in the structure of the suscepti-
in this figure. The quark density approaches the SB- bility divergence if the back-reaction of the matter sector
limit always from below. Without the µ-corrections the is taken into account or not. As a consequence it seems
Polyakov loop suppresses the quark densities for chemi- that the size of the critical region around the CEP is not
cal potential larger than the intersection point of the chi- strongly modified by these fluctuations. The only differ-
ral and deconfinement transition in the phase diagram ence is that including the µ-corrections the peak height of
Fig. 6. With the T (µ) corrections both transitions coin- the susceptibility is more pronounced towards the CEP.
0
5
200
[1] P. de Forcrand and O. Philipsen, PoS LAT-
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Figure 6: Phase diagraFmIGd.e4ri:vTedhefrpomhastehediDagyrsaomn-SfochrwchinirgaelrsaypmpmroeatcrhytboreQaCk-D thermodPyhnyasm. Licestt[3.1B].696 (2011) 58-67.
The curves show the cihnirgal(χan)danddecodneficnoenmfinenemt cernotssoofveqrutarraknssi(tΣio1n)s athnadt atunrtni-out to[8a]lGm.osEtncdorinoidcii,deZ. Fodor, S. D. Katz, K. K. Szabo,
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− (cid:39)
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(cid:39) S. Ejiri, et al. Prog. Theor. Phys. Suppl. 153, 118
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71, 114014 (2005);
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range 0.1 0.2 fm, covering typical nuclear matter densities. Therefore models and[a1p0p]rCoa.cDhe.sRdoeableinrtgs, S. M. Schmidt, Prog. Part. Nucl.
− crossover line goes over into a critical point at ap-
with dynamical (constituent) quarks as quasiparticle degrees of freedom in that area cannPohtybs.e4c5orre(c2t0:00) S1-S103.
proximately(T ,µ ) (95,280)MeV, followed
it is crucial to take into account the cEluPsteriEnPg of quarks into nucleons and their[c1o1r]reJl.atBiornasuninatnhde H. Gies, JHEP 0606 (2006) 024;
≈
by the coexistence region of a first order transi-
nuclear medium. J. Braun, H. Gies, J. M. Pawlowski, Phys. Lett.
tion. Thus we find the comparatively large value B684 (2010) 262-267. J. Braun, Eur. Phys. J. C
µ /T 3. We have checked, that these val- 64 (2009) 459.
EP EP
≈
3 Nuclear chuiersaalrethnoetromveorlydysennsaitmiveictso the details of our [12] C.S.Fischer, Phys.Rev.Lett.103 (2009) 052003;
C. S. Fischer, J. A. Mueller, Phys. Rev. D80
truncation: when changing the parameters in the
(2009) 074029.
At this point it is nowvearptperxoparniastaetztowtituhrnintaowraeradssonthaebldeisrcaunsgsieonweofonbuscelrevaer thermodynamics using
[13] C. S. Fischer, A. Maas and J. A. Muller, Eur.
the framework of chiral effective field theory, the low-energy realization of QCD in its sector with
variations of (T ,µ ) of the order of ten per-
EP EP Phys. J. C 68 (2010) 165.
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cent. Thus a firm conclusion of the present ap-
[14] J. Braun, L. M. Haas, F. Marhauser, J. M.
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D 75 (2007) 114003;
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| | (cid:39)
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included in our truncation of the quark-gluon in- F π
(cid:28) ∼
GeV. Short-distance correlations reflect large nucleon-nucleon scattering lengths and sho(u2l0d0i3n)st0e9a4d0b20e.
teraction may play an important role here. Also
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Acknowledgements
et al. [ CP-PACS ], Phys. Rev. D63 (2001)
We thank Daniel Mueller and Jan Pawlowski for
034502; F. Burger, E. -M. Ilgenfritz, M. Kirch-
discussions. This work has been supported by the ner, M. P. Lombardo, M. Muller-Preussker,
Helmholtz Young Investigator Grant VH-NG-332 O. Philipsen, C. Urbach, L. Zeidlewicz,
and the Helmholtz International Center for FAIR [arXiv:1102.4530 [hep-lat]].
within the LOEWE program of the State of Hesse.
NUCLEAR MATTER and QCD PHASES
nuclei T
[GeV] Nf =2 (q=u,d)
quark gluonphase
0.2 −
Tc quark?−gluonphase
e
r criticalpoint
u
t
a
er hadronphase ?
p
em !q¯q"=# 0 sCuCpSeS(rCcCcoolonpprd)hhuacatssoeers
Scales in t nunculceleaarr !qq"=# 0
mamtatteterr
nuclear matter: 0
1GeV µB
baryonchebmariycoanlpchoetemnitciaallpotential
momentum scale: 0 0.15 density [fm−3]
Fermi momentum pkFF 1.4 fm−1 2mπ
! ∼
NN distance: d 1.8 fm 1.3 m 1
NN ! ! −π
energy per nucleon: E/A 16 MeV
!−
compression modulus: K=(260 30) MeV 2m
π
± ∼
Figure 7: Illustration of the position of nuclear matter in the QCD phase diagram, outlining relevant
scales of the nuclear many-body problem.
be fitted to selected empirical data.
Within the last decade such an approach to the nuclear many-body problem has been developed
based on the understanding of the nucleon-nucleon interaction itself in terms of chiral effective field the-
ory [32, 33, 34]. In this approach, chiral one- and two-pion exchange processes in the nuclear medium
are treated explicitly while unresolved short-distance dynamics are encoded in contact interactions.
Three-body forces emerge naturally and play a significant role in this framework. The pion mass m ,
π
the nuclear Fermi momentum p and the mass splitting M M 2m between the ∆(1232) and
F ∆ N π
− (cid:39)
the nucleon are all comparable “small” scales that figure as expansion parameters. The relevant, ac-
tive degrees of freedom at low energy are therefore pions, nucleons and ∆ isobars. Two-pion exchange
interactions produce intermediate-range Van der Waals - like forces involving the large spin-isospin po-
larizablityoftheindividualnucleons. ThePauliprincipleplaysanimportantroleactingonintermediate
nucleons as they propagate in two-pion exchange processes within the nuclear medium.
3.1 In-medium chiral perturbation theory
A key ingredient of chiral perturbation theory applied to nuclear matter is the in-medium nucleon
propagator,
i
S (p) = (γ pµ +M ) 2πδ(p2 M2)θ(p0)θ(p p(cid:126) ) , (7)
N µ N (cid:34)p2 M2 +iε − − N F −| | (cid:35)
− N
that takes into account effects of the filled nuclear Fermi sea (with Fermi momentum p ). These prop-
F
agators enter in the loop diagrams generating the free energy density. Thermodynamics is introduced
using the Matsubara formalism. The free energy density is then computed as a function of temperature
T and baryon density ρ at given Fermi momentum p . It is derived in the form reminiscent of a virial
F
10
