F.S.N. and D.A. Fogaça

IFUSP / BRAZIL

Gluon condensates and the equation of state of cold quark gluon plasmas

arXiv:1012.5266

Introduction

QCD phase diagram

Hot QGP

Cold QGP

Ideal gas of weakly interacting quarks and gluons

Equation of state from the MIT bag model

(perturbative QCD)

Hot QGP

Lattice QCD: significant non-perturbative effects

RHIC: strongly interacting fluid

Non vanishing gluon condensate above deconfinement

David Miller, Phys. Rep. (2007) hep-ph/0608234

Borsanyi et al.,arXiv:1011.4229

Equation of state MIT bag model

quarks vacuum

BBB

3/43/2

3/7

2

3)(

Bp BB

3/43/2

3/7

2

3

3

1)(

Cold QGP

MIT Bag

“Big Bag”

Derive a simple EOS for the cold QGP

Assume that gluon condensates survive in cold QGP

Naively: they go asymptotically to zero ...

Non-trivial behavior: Metlitski, Zhitnitsky, Nucl. Phys. B (2005)

Estimate the effects of the gluon condensates A2 and A4

Our goal

QGP at large densities and zero temperature :

Separation of the gluon fields in soft and hard modes:

aaa AG

a mean field approximation

(“Walecka”)

Hard gluons are generated by intense quark sources

They have large ocupation numbers and become classical

aa00

aA

aa AA

Soft gluons generate the condensates in the plasma

0 aA

0 cba AAA

baba AAAA

Infinite matter: soft and hard fields are uniform !

0 aA 00 a

Soft gluons

Hard gluons

Quarks

The effective Lagrangian :

)()()()( 00000000

2

eeddccbbedacbaaa AAAAffgFF

cbcbaaaa GGfgGGF

jjiaa

jijiiaa

QCD mGTgiFFL

)(4

1

cbcbaa GGfgF Uniform fields:

aaa AG Field decomposition :

002 edcbedacba AAAffg

edcbedacba AAAAffg 2

000000

2 edcbedacba Affg

edcbedacba AAAffg 002

edcbedacba AAAffg 00

2 edcbedacba AAAffg 00

2

00002 edcbedacba AAffg 00

002

edcbedacba AAffg

edcbedacba AAffg 0000

2 0000

2 edcbedacba AAffg

edcbedacba AAffg 00

002

edcbedacba AAffg 00002

000000

2 edcbedacba Affg

000000

2 edcbedacba Affg

edcbedacba Affg 000000

2

00000000

2 edcbedacba ffg

A^4

A^3

A^2

A^1

A^0

mass term for the hard gluons

Expectation values of the soft gluons in the “vacuum” :

dynamical gluon mass

2220

2 AAAgg aa

dimension 2 gluon condensate is a parameter !

22

2

2

240

40

2

)34(4

9F

gFF

gbg aas

dimension 4 gluon condensate is a parameter !

20

22

32

9 gmG

2032

1 gAA baba

0 cba AAA 0 aA

dbcadcbadcba ggggAAAA

[

)34)(32(

40

]cbdagg

jjiaa

jijiiaaG

QCD mTgim

bL )(

2 000

0

240

soft gluons

hard gluons

quarks + hard gluons

Equations of motion

0)( 00 jji

aajiji mTgi

aaG gm 02

jajii

a T 0

Energy - momentum tensor

hard couplingg is a parameter !

The effective Lagrangian

00T

iiTp3

1LgL

T ii

)()(

3/4

FkQ

G

mkkkdm

gb

0

2222

22

240 2

3

2

27)(

FkQ

G mk

kkd

m

gbp

022

4

22

2

240 22

27)(

326 F

Q k

MIT BagModel

From B we can infer thevalue of the condensate in the QGP !

3/43/23/7

2

3)(

B

3/43/23/7

2

3

3

1)(

Bp

The equation of state

When the two EOS coincide0g

Parameters

20 % of the vacuum value

224 AAA20

20

15 % of the vacuum value

GeVm 02.0

3.0s

01.0s

43 )200(/200 MeVfmMeVB

240 444

1FFFFFbB

s

aas

s

aa

42 0006.0 GeVF

22 3.0 GeVA

MeVmG 290

quark mass :

hard coupling :

Numerical results

Pressure and sound velocity

Pressure versus energy density

F. Samarruca, arXiV:1009.1172 [nucl-th]

Comparison with the MIT bag model

200B

100B

0s

Comparison with the MIT bag model

More pressure

More energy

Harder EOS

Hard gluons!

Comparison with the MIT bag model

Conclusion

Simple approach to dense and cold QGP

Weak gluon condensates in QGPBag constant

Massive gluons

Gluon field decomposition = soft + hard

Mean field approximation

Richer version of the MIT bag model with classical hard gluons

Condensates make the EOS softer

2F

2A

Back ups

][)34)(32(

40 cbdadbcadcbadcba ggggggAAAA

BV

V

m

gV 2

202

Compute the Lagrangian, energy-momentum tensor and obtain the EOS :

But we can estimate the Laplacian :

Gluon condensate in a hot QGP :

Gluon condensate in dense and cold QGP ?

David Miller, Phys. Rep. (2007) hep-ph/0608234

Naively: goes asymptotically to zero !

Non-trivial behavior: Metlitski, Zhitnitsky, Nucl. Phys. B (2005)

Introduction

RHIC (2003) : evidence of the strongly interacting QGP (sQGP)

How to include non-perturbative effects in the equation of state ?

Finite temperature: lattice QCD

Finite density: models !

Our model: effects of the gluon condensates in the QGP !

non-perturbative effects !

Borsanyi et al.,arXiv:10114229

The equation of state

MIT BagModel

3/4

From B we can infer thevalue of the condensate in the QGP !

BBB

3/43/2

3/7

2

3)(

Bp BB

3/43/2

3/7

2

3

3

1)(

Finite temperature:

Finite density ?

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