Detailed analysis of the PNJL phase diagramtheory.gsi.de/~ebratkov/Conferences/NeD-TORIC-2011/talks...0 Detailed analysis of the PNJL phase diagram Critical properties within an e

0

Detailed analysis of the PNJL phase diagramCritical properties within an effective model of QCD

Hubert Hansen

Institut de Physique Nucléaire de

Lyon, CNRS/IN2P3 and université

Claude Bernard de Lyon

Collaborators: A. Gatti, G.

Chanfray, G. Goessens, P. Costa,

M. Ruivo, C. Da Sousa

TORIC 2011

1

1

Part I: Phases of QCD, symmetries and effective models

Lagrangian of QCD: LQCD = q̄ ( iγµDµ − m̂ ) q −1

4Faµν F

µνa

q = (u, d, s, c, b, t) ; Nc = 3 ; m̂ = diagf(mu,md, . . . ) ; Dµ = ∂µ − igtaAaµ with Aaµ

(a = 1, 2, ..., 8) the gauge field and F µν the gluon field strength tensor.

⇒ Symmetric under SU(3) gauge transformations in color space

Low energy: Relevant to the study of hadronic properties.Non-perturbative structure: existence of quark condensates 〈q̄q〉, appearance of light pseudoscalarparticles ( (quasi) Goldstone bosons). Confinement of quarks.

Group of symmetries for Nf = 3 in the chiral limit mu = md = ms = 0:

→ SUV (3) and UV (1): conservation of isospin and baryon number ;→ SUA(3) and UA(1) (chiral and axial symmetry): alter the parity.

U(3)L ⊗ U(3)R=SUV (3)⊗ SUA(3)⊗ UV (1)⊗ UA(1).

Symmetry Transformation Current Name Manifestation in nature

SUV (3) q → exp(−iλaαa2 )q Vaµ = q̄γµ

λa2 q Isospin Approximately conserved

UV (1) q → exp(−iαV )q Vµ = q̄γµq Baryonic ConservedSUA(3) q → exp(−i

γ5λaθa2 )q A

aµ = q̄γµγ5

λa2 q Chiral Spontaneously broken

UA(1) q → exp(−iγ5αA)q Aµ = q̄γµγ5q Axial “ UA(1) problem”

TORIC 2011 QCD critical properties

2

2

Hadronic spectrum: chiral and axial symmetry breaking

Wigner realization of chiral and axial symmetries:existence of multiplet of particles with same mass and opposite parity for each multiplet of isospin

(chiral partners)

UA(1)⇒ partner with opposite parity to each hadronFalse experimentally in the hadronic spectrum (low energy)

Spontaneous chiral symmetry breaking: Goldstone phaseSUA(3) symmetry: mechanism for the spontaneous breaking of chiral symmetry, related to the

existence of non-zero quark condensates, 〈q̄q〉 (not invariant under SUA(3) ) ; they act as as orderparameters for the spontaneously broken chiral symmetry phase.

Goldstone theorem: existence of eight degenerate Goldstone bosons e.g. pions (Mπ/MN = 0.15).

Lifting of the degeneracy in the pseudoscalar mesons spectrum: explicit symmetry breaking due to

current quark masses mu, md and ms.

Case of the η − η′Classically, massless QCD possesses the UL(3)×UR(3) symmetry but η′ not of the Goldstone type⇒Adler–Jackiw–Bell UA(1) anomaly (whose origin is instantons) breaks this symmetry to SUL(3)×SUR(3) (large mass for the η

′)

⇒ Finally: QCD as (almost) SUL(3)× SUR(3) chiral symmetry ; at low energy, the pertinent degreesof freedom are mesons (QCD experiences a confined/deconfined phase transition) ; mesons give insight

on the non-perturbative vacuum.


3

3

QCD Phase Diagram

Characteristics of the transition well established for µB = 0:Rapid increase of energy density (and other thermodynamic quantities)⇒ transition from hadronicresonance gas to a matter of deconfined quarks and gluons (the rise is interpreted as the liberation of

many degrees of freedom) with the Polyakov loop as order parameter.

Finite temperature and chemical potential: in most three-flavor phase diagram it existsa CEP (critical end point) that ends a first–order chiral phase transition (separating hadronic and quark

phases) that starts at µ 6= 0 and T = 0 (becomes a tricritical point – TCP – in the chiral limit).

From the experimental point of view:

0 100 200 300 400 500 600 700 800 900020406080100120140160180200

E/N = 1.08 GeV s/T3 = 7

SIS

AGS

SPS

T (M

eV)

B (MeV)

RHIC

Freeze-out points for the different beam energies

are showed, Cleymans, Oeschler, Redlich and

Wheaton, Phys. Rev. C73, 034905 (2006).


4

4

UA(1) symmetry breaking and effective t’Hooft interaction

If UA(1) symmetry spontaneously broken: pseudoscalar Goldstone boson with mass√

3Mπ (Weinberg).

Candidates with the correct quantum numbers: η(549) identified as belonging to the octet of

pseudoscalar mesons and η′(985) 'MNucleon (mass too high: “η′ problem”)⇒ If a particle with the characteristics pointed out by Weinberg does not exist, then where isthe ninth Goldstone boson ?⇒ If this Goldstone boson does not exist there is no spontaneous breaking of UA(1) symmetry.1976 (G. ’t Hooft): UA(1) symmetry does not exist at the quantum level (explicitly broken by the axial

anomaly described at the semiclassical level by instantons). Instantons “transform” left handed fermions

into right handed ones and conversely with non zero axial charge variation: ∆Q5 = ±2Nf“’t Hooft determinant” mimics this interaction in a purely fermionic effective theory (“absorbs” Nf left

helicity fermions and convert them):

Linst = gDeiθinst detflavor

(q̄R(x)qL(x)) + h.c.

⇒ Explicit breaking of UA(1) symmetry: η′ about 1 GeV (the mass of η′ has a different origin thanthe other masses of the pseudoscalar mesons ; not the missing Goldstone boson)

UA(1) anomaly responsible for the flavor mixing effect that removes the degeneracy among several

mesons.

⇒ Finally, an interesting question also related to these symmetries is whether both chiralSU(Nf)×SU(Nf) and axial UA(1) symmetries are restored in the high temperature/densityphase and which observables could carry information about these restorations.


5

5

Nambu-Jona-Lasinio effective models of QCD

Motivated by chiral symmetry (Ginzburg-Landau theory) ; used as reference when one add ZNc.

Extended NJL Lagrangian: LNJL = L0 + L4 + L6 with

L0 = q̄(iγµ∂µ − m̂0)q

L4 = G1[(q̄λaq)

2+ (q̄iγ5λaq)

2]

L6 = gD( detflavor

q̄PLq + detflavor

q̄PRq)

q̄ = (ū, d̄, s̄), m̂ = diag(mu,md,ms), PL,R =1∓γ5

2 ; λa,a∈[0,8]: Gell–Mann matrices (flavor).

• Invariant under global color symmetry SU(Nc)• 4-quarks interaction: L4 has UL(3) × UR(3) chiral symmetry (in QCD 4-quarks interactions exist

via exchange of gluons ; G1 mimicks a frozen gluon propagator leading to a contact interaction)

• L6 ⇒ UA(1) anomaly breaks UL(3)× UR(3) to UV (1)× SUL(3)× SUR(3)• m̂0 ⇒ explicit chiral symmetry breaking term: SUL(3)×SUR(3)⇒ SUV (3) if mu = md = ms.

When m0 6= 0 the chiral symmetry is broken. But small breaking (m0

6

6

The Polyakov Loop and the Z3 Symmetry Breaking: Pure Gauge

Polyakov loop in imaginary time and Polyakov gauge Aµ = δµ4A4

L (~x) = P exp[i

∫ β0dτ A4 (~x, τ)

]⇒ effective field Φ = 1Nc TrC L

A4 = iA0 : temporal component of the Euclidean gauge field ( ~A,A4), β = 1/T , P : path ordering.

L transports the field Aµ from the point in space-time (~x, 0) to (~x, β)

⇒ Φ = 0 means confinement ; Φ = 1 means free propagation (deconfinement)Effective potential U(Φ, Φ̄; T ): T0 = 270 MeV

U(Φ, Φ̄;T

)T 4

=

−a(T )2 Φ̄Φ + b(T )ln[1− 6Φ̄Φ + 4(Φ̄3 + Φ3)− 3(Φ̄Φ)2]

where a (T ) = a0 + a1

(T0T

)+ a2

(T0T

)2and b(T ) = b3

(T0T

)3a0 a1 a2 b3

3.51 -2.47 15.2 -1.75

Table 1: Parameters for the effective potential in the pure

gauge sector.

C. Ratti, M. Thaler, W.Weise, hep-ph/0604025 :

lattice: O. Kaczmarek, F. Karsch, P. Petreczky,

F. Zantow, Phys. Lett. B 543, 41 (2002).


7

7

Polyakov – Nambu – Jona-Lasinio effective model

LPNJL = q̄(iγµDµ − m̂)q

+1

2gS

8∑a=0

[ ( q̄ λa q )2 + ( q̄ i γ5 λa q )2 ]

+ gD{det[q̄(1 + γ5)q] + det[q̄(1− γ5)q]}

− U(Φ[A], Φ̄[A]; T

)+µq̄γ0q

Here q = (qu, qd, qs) are the quark fields, m̂ = diag(mu,md,ms), λa are

the flavor SUf(3) Gell–Mann matrices (a = 0, 1, . . . , 8), with λ0 =

√23 I

and Dµ = ∂µ−iAµU(Φ[A], Φ̄[A];T

)acts as a (gluonic) pressure in which quarks

propagate.


8

8

Order parameters

Spontaneous chiral symmetry breaking: Mean field point of view with Nf = 2if G1(q̄q)

2 ' 2G1〈q̄q〉 q̄q then the lagrangian reduces to a free Lagrangian q̄(iγµ∂µ −m)q withmass m = m0 + 2G1〈q̄q〉 ' ΛQCD〈q̄q〉 6= 0 ⇒ spontaneous chiral symmetry breaking〈q̄q〉 ⇒ order parameter (similar to magnetisation): tells you in which phase you are.→ < q̄uqu >=< q̄dqd >: order parameter for the light sector (SUR(2)× SUL(2) chiral symmetry)→ < q̄sqs >: order parameter for the strange sector (topological susceptibility also gives aninformation)

If no explicit breaking, the phase transition is second order.

Confinement / Z3 Symmetry Breaking:→ Φ = 0: no transport of color charges⇒ confinement→ Φ = 1: free “propagation” (parallel transport) of color charges⇒ deconfinement.If no explicit breaking (pure gauge i.e. m→∞), the phase transition is second order.

In QCD there is an explicit breaking of Z3 symmetry due to the kinetic term. Hence, for large mass(chiral symmetry breaking case) the kinetic term is negligible (mψ̄ψ � ψ̄iγµDµψ)⇒ steep crossoverfrom the restored symmetry zone to the broken symmetry zone.

At the contrary when the mass is small (chiral restoration) the crossover is less steep.

The same effect is observed with high chemical potential: the kinetic term is important due to Fermi

motion: mψ̄ψ < ψ̄i/∂ψ + µψ†ψ ' µψ†ψ


9

9

Mean Field Approximation

Chemical equilibrium condition µu = µd = µs = µ. This choice allows for isospin symmetry,

mu = md and approximates the physical conditions at RHIC.

Quark propagator:The quark propagator in the constant background field A4 is then:

Si(p) = −(p/−Mi + γ0(µ− iA4))−1

In the above, p0 = iωn and ωn = (2n+ 1)πT is the Matsubara frequency for a fermion.

Gap equations: constituent quark masses (dynamical masses)

Mi = mi − 2gS 〈q̄iqi〉 − 2gD 〈q̄jqj〉〈q̄kqk〉

where the quark condensates 〈q̄iqi〉, with i, j, k = u, d, s(to be fixed in cyclic order), have to be determined in a self-consistent way with:

〈q̄iqi〉 = − 2Nc∫

d3p

(2π)3

Mi

Ei[θ(Λ

2 − ~p2)− f (+)Φ (Ei)− f(−)Φ (Ei)]

where Ei is the quasi-particle energy for the quark i: Ei =√

p2 +M2i .


10

10

Grand Potential in the Mean Field Approximation

PNJL grand canonical potential density:Ω(Φ, Φ̄,Mi;T, µ) = U

(Φ, Φ̄, T

)+ g

S

∑{i=u,d,s}

〈q̄iqi〉2 + 4gD 〈q̄uqu〉〈q̄dqd〉〈q̄sqs〉 − 2Nc∑

{i=u,d,s}

∫Λ

d3p

(2π)3Ei

− 2T∑

{i=u,d,s}

∫d3p

(2π)3

{Trc ln

[1 + Le

−(Ei−µ)/T]

+ Trc ln[1 + L

†e−(Ei+µ)/T

]}where Ei is the quasi-particle (Hartree) energy for the quark i: Ei =

√p2 +M2i

The propagation of the quarks into the medium filled with (background) gluon fields with pressure Uleads to statiscal suppression of 1- and 2-quarks propagation (statiscal confinement) :

Trc ln[1 + Le

−(Ep−µ)/T]

= ln[1 + 3Φe

−β(Ep−µ) + 3Φ̄e−2β(Ep−µ) + e

−3β(Ep−µ)]

limΦ,Φ̄→0

(Z3 restored) = ln[1 + e

−3β(Ep−µ)]

limΦ,Φ̄→1

(Z3 broken) = Nc ln[1 + e

−β(Ep−µ)]


11

11

Part II: Detailed study of the SU(2) case

(Yves Schutz)


12

12

Chiral condensate and Polyakov loop: crossover vs. 1st order

0

0.2

0.4

0.6

0.8

1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.021

0.0212

0.0214

0.0216

0.0218

0.022

0.0222

0.0224

0.0226

0.0228

0.023

σ/σ

0, Φ

(G

eV

), Φ

bar

(GeV

)

P/T

4

µ (GeV)

T = 0.18 GeV

σ/σ0Φ

ΦbarP/T

4

0

0.2

0.4

0.6

0.8

1

0.3 0.32 0.34 0.36 0.38 0.4 0.42

512

513

514

515

516

σ/σ

0, Φ

(G

eV

), Φ

bar

(GeV

)

P/T

4

µ (GeV)

T = 0.08 GeV

µ1 µ2

σ/σ0Φ

ΦbarP/T

4

Left: Condensate, Polyakov loop and scaled pressure in the chiral cross over zone; the cross-over value is

fixed studying the minimum of −dTdm at µ set.

Right: Condensate, Polyakov loop and scaled pressure in the first order chiral phase transition zone. The

lines µ1, µ2 locate the metastable zone, and the intersection point of the grand potential relating to the

inferior and to the superior branch of the condensate identify the chemical potential that correspond at

the first order transition.


13

13

Chiral phase diagram (PNJL): chemical potential and density

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

T (

GeV

)

µ (GeV)

CEP

crossover region

first order region

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3

T (

GeV

)

ρ/ρ0

CEP

crossover region

unstable region

metastable region

Left: (T, µ) phase diagram studying the transition in the condensate. CEP: (µ = 0.302 GeV,

T = 0.161GeV)

Right: (T, ρ) phase diagram. CEP: (ρ = 2.00ρ0, T = 0.161 GeV).

Density: ρ = 2NcNf

∫Λ

d3p

(2π)3(f

+Φ (Ep)− f

−Φ (Ep))

Third law of thermodynamics satisfied (parameter: Buballa)


14

14

Chiral and deconfinement crossover entanglement

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2

4

6

8

10

12

14

σ/σ

0, Φ

, Φ

bar

χΦ

(G

eV

-1)

T (GeV)

µ = 0.05 GeV

σ/σ0Φ

ΦbarχΦ

χΦ (mfix)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3-2

0

2

4

6

8

10

12

14

16

18

σ/σ

0, Φ

, Φ

bar

χΦ

(G

eV

-1)

T (GeV)

µ = 0.256 GeV

σ/σ0Φ

ΦbarχΦ

χΦ (mfix)

Left: Crossover zone ; red: deconfinement susceptibility ; green: susceptibility with mfix for seeing the

deconfinement transition entanglement.

Right: Crossover zone, near the critical end point ; red: deconfinement susceptibility ; green:

susceptibility with mfix.

The small influence of the mass on Φ allows to desentangle chiral and deconfinement transition.


15

15

High chemical potential region

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3-2

0

2

4

6

8

10

12

14

σ/σ

0, Φ

, Φ

bar

χΦ

(G

eV

-1)

T (GeV)

µ = 0.35 GeV

σ/σ0Φ

ΦbarχΦ

χΦ (mfix)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.05 0.1 0.15 0.2 0.25-2

0

2

4

6

8

10

12

14

σ/σ

0, Φ

, Φ

bar

χΦ

(G

eV

-1)

T (GeV)

µ = 1.4 GeV

σ/σ0Φ

ΦbarχΦ

χΦ (mfix)

Left: Metastable zone ; red: deconfinement susceptibility ; green: susceptibility with mfix.

Right: Zone where the chiral symmetry is restored ; there is no more a minimum of the deconfinement

susceptibility.


16

16

Evolution of the deconfinement susceptibility

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

χΦ

(G

eV

-1)

T (GeV)

χΦ(µ = 0.05)χΦ(µ = 0.3)

χΦ(µ = 0.41)χΦ(µ = 0.6)

χΦ(µ = 0.75)χΦ(µ = 0.9)χΦ(µ = 1.1)χΦ(µ = 1.4)

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

χΦ

bar

(GeV

-1)

T (GeV)

χΦbar(µ = 0.05)χΦbar(µ = 0.3)

χΦbar(µ = 0.41)χΦbar(µ = 0.6)

χΦbar(µ = 0.75)χΦbar(µ = 0.9)χΦbar(µ = 1.1)χΦbar(µ = 1.4)

Left: Φ-susceptibility for different values of µ. The solid lines represent χΦ calculated with the Hartree

mass, the dashed lines χΦ calculated with the constant mass (mean value between the mass at

(T = 0, µ) and m0). From the green line, increasing µ, we cannot see the dashed lines because χΦ is

the same if we take a constant mass or the Hartree mass; actually in this region there is no more a chiral

transition.

Right: Φ̄-susceptibility for different values of µ.


17

17

Phase diagram

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

T (

GeV

)

µ (GeV)

CEP

crossover region

first order region

confinement/deconfinement transition

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6

T (

GeV

)

ρ/ρ0

CEP

crossover region

unstable region

metastable region


Phase diagram for Φ: the green line is the crossover point for Φ (i.e. Max(dΦ/dT )).


18

18

Extended phase diagram

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

T (

GeV

)

µ (GeV)

CEP


µ0

Φ = 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

T (

GeV

)

µ (GeV)

CEP


µ0

Φbar = 0.5

Left: Phase diagram for Φ: the green solid line is the crossover point for Φ (Max(dΦ/dT )) ; the

green dashed lines are the limits of the crossover zone (Max and Min of d2Φ/dT 2). For high

densities we always are in the deconfined phase, except for T = 0, so the Φ-susceptibility approaches no

more to zero, and so the blue line represents the minimum of the Φ-susceptibility that is produced from

this fact. From the blue point it is no more possible to distinguish the maximum of the Φ-susceptibility

because we are almost totally in the deconfinement phase, except for T = 0. The line at µ0 is the point

beyond which χΦ doesn’t approach zero. The violet line is the point where Φ = 0.5.

Right: Phase diagram for Φ̄


19

19

Part III: the SU(3) phase diagram and the role of strangeness

Order parameters evolution

0 50 100 150 200 250 300 350 4000

100

200

300

400

500

600

0.0

0.2

0.4

0.6

0.8

1.0

Mq (

MeV

)

T (MeV)

Ms

Mu

Teff

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

0.0

0.2

0.4

0.6

0.8

1.0

(MeV

3)

T (MeV)

0 50 100 150 200 250 3000

1

2

3

4

5

0

1

2

3

4

5

6

¶ /¶ T (fm-2) ¶ /¶ T (fm-2)

¶

/¶

T, ¶

<ss

>/¶

T

T (MeV)

¶ /¶ T (fm)

¶

/¶ T

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

1/4 /

1/4 (

T=0)

T (MeV)The topological susceptibility, χ in the PNJL model is compared to corresponding lattice results taken

from Alles et al. (Nucl. Phys. B494, 281, 1997) ; Inflexion points of the strange and non-strange quark

condensates are close (slightly separated from the one of Φ).


20

20

Thermodynamic Quantities (p,s and ε) at µ = 0

0 50 100 150 200 250 3000

1

2

3

4

5

p / T

4

T (MeV)

pSB/T4

0 50 100 150 200 250 3000

2

4

6

8

10

12

14

16

/ T 4

T (MeV)

SB/T4

0 50 100 150 200 250 3000246810121416182022

s / T

3

T (MeV)

sSB/T4 Asymptotically, the QCD pressure for Nf massless

quarks and (N2c − 1) massless gluons is given(µB = 0) by:pSB

T 4= (N

2c − 1)

π2

45+ NcNf

7π2

180,

where the first term denotes the gluonic

contribution and the second term the fermionic

one.

Data: Cheng et al. (Phys. Rev. D81, 054504, 2009). The pressure reaches 66% of the strength of the

Stefan-Boltzmann value at T = 300 MeV, a value which attains 85% at T = 400 MeV.


21

21

Phase Transition at Zero Temperature

First order phase transition µcr = 361.7 MeV.

300 320 340 360 380 4000

50

100

150

200

250

300

350

400

4240

4260

4280

4300

stable solutions metastable solutions unstable solutions

B1

B2

Mq (

MeV

)

B (MeV)

MuB

cr

(MeV

fm3 )

300 320 340 360 380 400

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

B2

B1

B /

0

B (MeV)

stable solutions metastable solutions unstable solutions

Bcr

0.0 0.5 1.0 1.5 2.0 2.5-20

-10

0

10

20

30

40

A

CEP

T = 0 (MeV) T = 50 T = 100 T = 111 T = 130 T = 150 T = T CEP T = 170

Pres

s (M

eV fm

3 )

B / 0

B

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1100

1200

1300

1400

1500

1600

1700

1800

T = 130

T = 111T = 100

T = 0T = 50

T (MeV)T = TCEP

T = 150

CEP Gibbs criteria the minimum zero/minimum

of pressure

E/A

(MeV

)

B / 0


22

22

Phase Diagram and the Location of the Critical End Point

Location of the CEP: TCEP = 155.80 MeV and ρCEPB = 1.87ρ0 (µCEP = 290.67 MeV).

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

1st order region

T (M

eV)

B (MeV)

CEP

Chiral Limit

= 0.5

crossover region

= 0.5

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

unstable

Gibbs criteria zero/minimum of pressure

mixed phase

T (M

eV)

B

CEPmetastable

Phase diagram in the SU(3) PNJL model. The left (right) part corresponds to the T − µB (T − ρB)plane. Solid (dashed) line shows the location of the first order (crossover) transition. The dashed lines

shows the location of the spinodal boundaries of the two phase transitions (shown by shading in the

right plot).


23

23

Chiral and deconfinement transition

Opening of a phase with chiral symmetry and statiscally confined at theCEP


24

24

Nernst Principle and Isentropic Trajectories

100 150 200 250 300 350 4000

50

100

150

200

250

T (M

eV)

B (MeV)

Isentropic trajectories in the (T, µB) plane. The following values of the entropy per baryon number

have been considered: s/ρB = 1, 2, 3, 4, 5, 6, 8, 10, 15, 20 (anticlockwise direction).


25

25

Effects of Strangeness and Anomaly on the Critical End Point

Location of the CEP with physical mass: mu = md = 5.5 MeV, ms = 140.7 MeVTCEP = 155.80 MeV and µCEP = 290.67 MeV (ρCEPB = 1.87ρ0) ;

Chiral limit:→ mu = md = ms = 0⇒ no TCP: chiral symmetry is restored via a first order transition for allbaryonic chemical potentials and temperatures

→ in the light sector: mu = md, ms 6= 0⇒ TCP⇒ Both situations expected from universality argument (second order for Nf = 2 and first order forNf ≥ 3, Pisarski, 1983).

0 50 100 150 200 250 300 350 4000

50

100

150

200

250m

s= 140.7 MeV

TCP

T (M

eV)

B (MeV)

CEP

Chiral Limit

crossover

second order

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

CEP

ms (MeV)

ms= 200

T (M

eV)

B (MeV)

ms= 140.7 MeV

ms= 500

ms= 100ms= 50ms= 20ms= 9

Chiral Limit


26

26

Role of the Anomaly Strength in the Location of the CEP

250 260 270 280 290 300 310 320020406080100120140160180200

gD= 1.4 gD0gD= 1.2 gD0

gD= 0.49 gD0

gD= 0.5 gD0

gD= 0.6 gD0

gD= 0.7 gD0

gD= 0.8 gD0

gD= 0.9 gD0

T (M

eV)

B (MeV)

gD= gD0

Dependence of the location of the CEP on the strength of the ’t Hooft coupling constant gD.

⇒ Importance of the strangeness and anomaly for the CEP properties


27

27

Susceptibilities and Critical Behavior in the Vicinity of the CEP

0 50 100 150 200 250 300 3500

50

100

150

200

250

TCP CEP

B/

Bfree = 1.0

B/

Bfree = 2.0

T (M

eV)

B (MeV)

Chiral Limit

Phase diagram: the size of the critical region is

plotted for χB/χfreeB = 1(2) ;

TCP - CEP entanglement: critical exponent goes

from 1/2 to 2/3 when one approaches the CEP.

Left panel: Baryon number susceptibility as

functions of µB for different temperatures around

the CEP: TCEP = 155.80 MeV and T =

TCEP ± 10 MeV. Right panel: Specific heatas a function of T for different values of µBaround the CEP: µCEPB = 290.67 MeV and

µB = µCEPB ± 10 MeV.

260 270 280 290 300 310 3200

5

10

15

20

B (f

m -2

)

B (MeV)

T > T CEP

T < T CEP

T = T CEP

130 140 150 160 170 1800

50

100

150

200

C (f

m -3

)

T (MeV)

B > BCEP

B < BCEP

B = B

CEP


28

28

Conclusions

• The PNJL reproduces well at the mean field level the lattice calculations.• PNJL calculations can be directly deduced from NJL ones (not only for one loop calculation, but to

all orders) by a redefinition of the usual Fermi – Dirac distribution function.

• PNJL results does not destroy the important features like chiral restauration, Goldstone character ofthe pion, etc.

• The introduction of gluons in NJL via a background temporal gauge field embedded in the Polyakovloop add some statistical confinement to the NJL model. No mechanism for true confinement in

PNJL ; yet The results are improved in the right direction (e.g. quarks more “bounded” in meson

below Tc and less bound above Tc) compared to NJL.

• PNJL is a pertinent model to discuss confinement / deconfinement properties.

• Elongation of the critical region in the PNJL model.• Influence of the strange sector on the critical behavior essential (existence/position of the CEP).• Effects of the TCP on the CEP are seen.• The sets of parameters used is compatible with the formation of stable droplets at zero temperature,

insuring the Nernst principle.

• The regularization procedure by allowing high momentum quark states, is essential to obtain therequired increase of extensive thermodynamic quantities, insuring the convergence to the Stefan–

Boltzmann (SB) limit of QCD. In this context the gluonic degrees of freedom also play a special

role.


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