Gluon condensates and the equation of state of cold quark gluon plasmas F.S.N. Fogaa and IFUSP / BRAZIL arXiv:1012.5266 D.A. Introduction QCD phase diagram

Hot QGP Cold QGP Hot QGP Ideal gas of weakly interacting quarks and gluons Equation of state from the MIT bag model RHIC: strongly interacting fluid (perturbative QCD) Lattice QCD: significant non-perturbative

effects Non vanishing gluon condensate above deconfinement David Miller, Phys. Rep. (2007) hep-ph/0608234 Borsanyi et al., arXiv:1011.4229 Cold QGP MIT Bag Equation of state

MIT bag model Big Bag 3 (B ) 2 7/3 1 3 p (B ) 3 2 7/3

2/3 B 4/3 B 2/3 B 4/3 B quarks vacuum Our goal

Assume that gluon condensates survive in cold QGP Naively: they go asymptotically to zero ... Non-trivial behavior: Metlitski, Zhitnitsky, Nucl. Phys. B (2005) Derive a simple EOS for the cold QGP Estimate the effects of the gluon condensates A2 and A4 QGP at large densities and zero temperature : Separation of the gluon fields in soft and hard modes: Ga Aa a Soft gluons generate the condensates in the plasma Aa

Aa 0 Aa Aa Aa Ab Ac 0 Aa Ab Aa Ab Hard gluons are generated by intense quark sources They have large ocupation numbers and become classical a a 0 0a mean field approximation

(Walecka) Infinite matter: soft and hard fields are uniform ! Aa 0 0a 0 Soft gluons Quarks Hard gluons The effective Lagrangian :

LQCD 1 a a F F i ( i i j g Ti aj Ga i j m ) j 4 F a G a G a g f a b c G b G c Uniform fields: F a g f a b c G b G c Field decomposition : Ga Aa a Fa F a g 2 f a b c f a d e ( Ab 0b 0 ) ( Ac 0c 0 ) ( Ad d 0 0 ) ( Ae e 0 0 ) A^4

g 2 f a b c f a d e Ab Ac Ad Ae g 2 f a b c f a d e Ab Ac Ad e 0 0 g 2 f a b c f a d e Ab Ac d 0 0 Ae g 2 f a b c f a d e Ab 0c 0 Ad Ae g 2 f a b c f a d e 0b 0 Ac Ad Ae g 2 f a b c f a d e Ab Ac d 0 0 e 0 0 g 2 f a b c f a d e Ab 0c 0 Ad e 0 0 2 g f

abc f ad e b c 0 A 0 d0

0 A e g 2 f a b c f a d e 0b 0 Ac d 0 0 Ae 2 g f abc f

ad e b 0 c 0 A A d e0

0 A^3 A^2 g 2 f a b c f a d e 0b 0 0c 0 Ad Ae mass term for the hard gluons A^1 g 2 f a b c f a d e Ab 0c 0 d 0 0 e 0 0 g 2 f a b c f a d e 0b 0 Ac d 0 0 e 0 0 g 2 f a b c f a d e 0b 0 0c 0 Ad e 0 0

g 2 f a b c f a d e 0b 0 0c 0 d 0 0 Ae g 2 f a b c f a d e 0b 0 0c 0 d 0 0 e 0 0 A^0 Expectation values of the soft gluons in the vacuum : Aa Ab Ac 0 a b A A A

c A d Aa 0 04 [ g g a b c d g g a c b d (32)(34) g g a d b c ] 9 2 s a a

2 2 4 4 g 0 b 0 2 F F 2 F2 4(34) g g A a mG2

1 a b 2 A g 0 32 b 9 2 2 g 0 32 dynamical gluon mass dimension 4 gluon condensate is a parameter !

g 2 02 g 2 Aa Aa A2 dimension 2 gluon condensate is a parameter ! The effective Lagrangian LQCD 2 m b04 G 0a a 0 i ( i i j g 0 Ti aj 0a i j m ) j 2 soft gluons

hard gluons quarks + hard gluons Equations of motion 2 G a 0 m g

a ( i i j g 0 Ti aj 0a i j m ) j 0 Energy - momentum tensor T L ( i ) g L ( i ) a i 0 Ti aj j hard coupling g is a parameter ! T0 0

1 p Ti i 3 The equation of state 3 Q 27 g 2 2 4 b 0 2 2 2 m 2

G kF 2 Q 2 27 g 4 p ( ) b 0 2 2

2 m 2 G kF ( ) Q 3 kF 2 6

MIT Bag Model ( ) 2 2 2 d k k k m 0

d k 0 When B p ( ) B 3 2 4/3 k4

k 2 m2 g 0 the two EOS coincide 7/3 1 3 3 2 2/3 4/3 7/3 2/3 4/3 From B we can infer the value of the condensate in the QGP !

Parameters 4 0 B b s a a 1 a a F F F F F2

4 4 s 4 s B 200 MeV / fm 3 (200 MeV ) 4 s 0.3 F 2 0.0006 GeV 4 20 % of the vacuum value A4 A2 A2 02 02 A2 0.3 GeV 2 15 % of the vacuum value

mG 290 MeV quark mass : m 0.02 GeV hard coupling : s 0.01 Numerical results Pressure and sound velocity Pressure versus energy density Comparison with the MIT bag model

s 0 B 200 B 100 F. Samarruca, arXiV:1009.1172 [nucl-th] Comparison with the MIT bag model More energy More pressure Harder EOS Hard gluons!

Comparison with the MIT bag model Conclusio n Simple approach to dense and cold QGP Gluon field decomposition = soft + hard Mean field approximation Weak gluon condensates in QGP F2 Bag constant A2 Massive gluons

Richer version of the MIT bag model with classical hard gluons Condensates make the EOS softer a Back ups Ab Ac Ad 04 [ g g a b c d g g a c b d g g a d (32)(34) But we can estimate the Laplacian :

2 V0 gV 2 B 2 mV Compute the Lagrangian, energy-momentum tensor and obtain the EOS : 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) non-perturbative effects ! 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 !

Borsanyi et al., arXiv:10114229 The equation of state 4/3 MIT Bag Model 3 (B ) 2 7/3

1 3 p (B ) 3 2 7/3 2/3 B 4/3 B 2/3 B 4/3

B From B we can infer the value of the condensate in the QGP ! Finite temperature: Finite density ? difcil acreditar...