Color neutrality effects in the phase Color neutrality effects in the phase diagram of the PNJL modeldiagram of the PNJL model

A. Gabriela GrunfeldTandar Lab. – Buenos Aires - Argentina

In collaboration with

D. BlaschkeD. Gomez Dumm N. N. Scoccola

Motivation

Understanding of the behavior of strongly interacting matter at finite T and/or density is of fundamental interest and has important applications in cosmology, in the astrophysics of neutron stars and in the physics of URHIC.

CBM@FAIR

From RHIC

(from Jürgen Schaffner-Bielich)

For a long time, QCD phase diagramrestricted to 2 phases

HADRONIC PHASE: “our world”color neutral hadrons, SB

QGP: S is restored

In recent years phase diagram richer and more complex structure T

Hadrons

QG Plasma

2SC CFL

E

T

Hadrons

QG Plasma

2SC CFL

E

Rajagopal

The treatment of QCD at finite densities and temperatures is a problem of very high complexity for which rigorous approaches are not yet available

NJL model is the most simple and widely used model of this type.

• Development of effective models for interacting quark matter that obey the symmetry requirements of the QCD Lagrangian

• Inclusion of simplified quark interactions in a systematic way

local interactions

Effective theories Lattice results at μ -> 0

Reproduce ?

extrapolate at high μ

Nambu Jona-Lasinio model + Polyakov loop dynamiccs

( , )U

Lattice simulations of P in a pure gauge theory

Chiral symmetry breaking

confinement

It reasonable to ask what happens with color neutrality in presence of PL• important in URHIC• could be extended to compact stars imposing electric charge neutrality + β decay

Higher Tc than NJL

The model

NJL SU(2) flavor + quarks with a background color field related to the Polyakov loop Φ:

mc (current q mass), G and H parameters of the model

1

2 0t

C i

5 2 2Ci

In our case: SU(2) flavor + diquarks + color neutrality

H/G = ¾ from Fierz tr. OGE

diquarks

*S. Rößner, C. Ratti and W. Weise, PRD75, 034007 (2007)

3 3 8 8

Polyakov loop:

quarks with a background color field aa tΑgiAiA

004

LTrN C

C

1

then

0

4exp AdiL

TiC eTr /

31

Polyakov gauge => diag representation

order parameter for confinement

conf

deconf

0

0

We considered the polynomial form for the effective potential *:

2

0 00 1 2

3

03

( )

( )

T Ta T a a a

T T

Tb T b

T

with

2*33*** )(3)(461ln)()(21

),( tbTaTU

T0 = 270 MeV from lattice crit temp for deconf.

δSE (Φ,T) -> (V/T) U(Φ,T)

gluon dynamics,

effective potential, confinement-deconf. transition

over Dirac, flavor and color indices

1 0 0 5 2 2*

5 2 2 0 0

. ( )( , )

. ( )n

n

n

i p iS i p

i p i

Matsubara frequencies ωn=(2n+1) π T

Then, we obtain the Euclidean effective action

where Matsubara frequencies ωn=(2n+1) π T

1

22 3

(4) 3( , ) ln ( ; )22 2 (2 )

MFAMFAE

nn jE

TU T Tr S i p

S d pV G H

• MFA -> drop the meson fluctuations• (+ Usual 2SC ansatz Δ5 = Δ7= 0 and Δ2 = Δ)

Then, the thermodynamic potential per volume reads:

2 2

2ln 2 ln 1 exp( / )n

n

ET E T E T

T

22 3

(4) 3

1( , ) ln 1 exp ( / )

22

2 2 (2 )

MFAMFAE

f j jjE

U T E T ES d p

NV G H

1,2

2 23,4 3

2 25,6 3

2 2

( )

( )

b

r

E

E i

E i

where

p

8

8

3

23

r

b

Thermodynamic equilibrium -> minimum of thermodynamic potential. The mean fields and are obtained from the coupled gap equations

together with

We impose color charge neutrality

We consider * 8 0 311 2cos

3 T

To Ω be real

0),(

;0),(

fc

MFAfc

MFA TT

0),(

3

fcMFA T

0),(

8

fcMFA T

=> μ3 = 0

NUMERICAL RESULTS

we use the set of parameters from PRD75, 034007 (2007) • G = 10.1 GeV-2

• Λ = 0.65 GeV effective theory, fluctuations, at T = μ = 0• H = ¾ G, 0.8G• mc = 5.5• a0 = 3.51• a1 = -2.47• a2 = 15.2 from lattice• b3 = -1.75• T0 = 270.

Phase diagram: Low μ -> XSB and XSB + 2SCHigh μ -> 2SC

Phase diagrams

Low temperature expansion

11210

)()(3

20

2)(

2)(2

0

0

22

0

22

8

0

2

2

pdpH

SgSgpdp

SgSgEp

dpM

GmM

rrbb

rrbbc

)()()()(3

20

0

22

8rrbb SgSgSgSgpdp

3/

3/2

3200

00

80

MifM

MifM

M

T = 0

• for μ ≠ 0 (Δ still 0)

Trivially satisfied for a wide range of μ8

• for μ = 0, Δ = μ8= μr = μb = 0, Mo = 324.11 Mev

Step beyond: μ8 from fin T and then T -> 0 For μ < M0/3

2ln238 T

For μ > M0 (before 1st order ph.tr)

2SC ->

HG

pdpG

0

22

112

G

f )(

• T = 0

If H/G > 0.783 f(Δ) ≠ 0

in region μr = cte f(Δ) ≠ 0

• T ≠ 0

in region μr = cte f(Δ) ≠ 0 untilT = 20 MeV, 2nd order

Summary and outlookSummary and outlook

• we have studied a chiral quark model at finite T and µ NJL + diquarks + Polyakov loop + color neutrality• ansatz PRD75, 034007 (2007) ϕ8 = 0 => μ8 ≠ 0, then μ3 = 0• to enforce color neutr color neutrality => μ8 ≠ 0 • without PL, symmetric case, with PL non symmetric densities in color space• different quark matter phases can occur at low T and intermediate µ• coexisting phase XSB + 2SC region• Next step: starting with ϕ3 ϕ8 ≠ 0, => μ3 μ8 ≠ 0 more general…

فرامرز

Some ReferencesS. Rößner, C. Ratti and W. Weise, PRD75, 034007 (2007) F. Karsch and E. Laermann, Phys. Rev. D 50, 6954 (1994) [arXiv:hep-lat/9406008].C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73, 014019 (2006) [arXiv:hep-ph/0506234].M. Buballa, Phys. Rept. 407, 205 (2005) [arXiv:hep-ph/0402234].K. Fukushima Physics Letters B 591 (2004) 277–284S. Rößner, T. Hell, C. Ratti and W. Weise, arXiv:0712.3152v1 hep-ph

THANKS!