XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007.

Angel Gómez Nicola

Universidad Complutense Madrid

COEFICIENTES DE TRANSPORTEEN UN GAS DE PIONES

Hadronic matter at Hadronic matter at high Temperature and high Temperature and

DensityDensity

Heavy Ion Collisions:

(fm/c)

T (MeV)

5-10

120-200

300-400

50

¿QGP?¿QGP? HADRON GASHADRON GAS

((K

PERTURBATIVE QCDquarks and and gluons

EFFECTIVE LOW-ENERGY MODELSLight hadrons

Hadronic matter at Hadronic matter at high Temperature and high Temperature and

DensityDensity

Heavy Ion Collisions:

QCD phase diagram

RELEVANT D.O.F AT LOW ENERGIES E<< 1 GeV

“NGB” are the lightest pseudoscalars ( mq 0 mBG0)

Nf = 2:’s (3)

Nf = 3:’s (3), K’s (4),

qq LR 2

1 5,

QCD with Nf massless quarks is chiral invariant:

RRLL RqqLqq ,)(, fNSURL

Chiral Symmetry of QCD:

Spontaneously Broken at T=0 (no chiral degeneracy of SU(Nf ) multiplets):

)()()( fRLfRfL NSUNSUNSU (ISOSPIN FOR Nf =2)

T

, qqq m B Msimilarly to a ferromagnet with

Chiral Symmetry is restored at Chiral Symmetry is restored at TTc 150-200 MeV

The order parameter is the quark condensate:

free energy density

qT m

zqq

Chiral Perturbation Theory:

Relevant for low and moderate temperatures below Chiral SSB

Weinberg’s chiral power counting:

)below ( GeV 14 ,,, cTTfeTmEp

Dp /

NLSM

...42

0222

2 2UfmUU

f T

200 ||1 , ),2(),( UUf

USUUUU

1,22,12,13,03,0 , UAeUUUU ijk

Most general derivative and mass expansion of NGB mesons compatible with the SSB pattern of QCD model-independent low-energy predictions.

Transport coefficients of a pion gas in ChPT:

D.Fernández-Fraile,AGN: PRD73:045025,2006, NPA785:166-169,2007, EPJA31:848,2007

)(xJTC measure the response of the system to restore equilibrium after the action of an external force coupled to a conserved current .

Diagramatically, nonperturbative corrections needed to account for the inverse collisional width dependence (“pinching poles”):

)0,0(

22'

2),'(),'( p

pp

AR EE

pGpG

pp E0

as expected from Kinetic Theory

m

Ne chel

2

Im

222mpEp

Linear Response (Kubo formula) in terms of retarded correlators:

,)]0(),([)(Imlimlim0000

JxJtx

qC xq

q

i4 ediq

)3/1( ty conductivi electrical DC thegives instance,For el CJ iel

In equilibrium at temperature T

1)4()2(11

]Re[Re , )()(Re

1)(

tttsist

st IJIJ

IJ

In the dilute gas regime:

Data up to 1 GeV.

(770), f0(600)

Exactly Unitary.

1)4()2(11

]Re[Re , )()(Re

1)(

tttsist

st IJIJ

IJ

In the dilute gas regime:

Data up to 1 GeV.

(770), f0(600)

Exactly Unitary.

Transport coefficients: revising chiral counting

Identify diagrams with larger powers of Y:

Ladders

Bubbles

)( 12 kkYpO

)( 1kYO

0for explicitly )( )0(2

1

)( ΓOk

k

Ymeel 2)0(

“Double lines” with equal momentum carry ≠0 propagators and carry a “nonperturbative” factor Y for chiral loops:

WPC)by )( of (instead 22 pOme

T

mmeTel

2)0( 7.2)(

:For mT

4

2

: KTisticNonrelativ

f

m

m

Tmv

N tot mTmp Relevant

T

mY Potentially large

Detailed analysis of the effective ladder vertex

mTpO

T

m kkel for )( 2)(

Loop spectral function ~ 1// YmTmpCM

T dependence factorizes

PERTURBATIVE IN CHPT

constY

T

mY

U

NU

2

:For mT

Tp Relevant Unitarization is important !

Ladders still formally ChPT perturbative.

Unitarity gives larger T-dependence.

mTel /)()0(

up. breaks ChPT henrelevant w

bemay like) (Boltzmann equation integralladder

important become )/( verticesderivativeBut 22

fT

)0()1( / elel

Tp

0 and resonances in scattering important near the origin ! W.Liu, R.Rapp nucl-th/0604031

Sound attenuation length (related to elliptic flow):

Results for shear and bulk viscositiesResults for shear and bulk viscosities

Sound attenuation length (related to elliptic flow):

Results for shear and bulk viscositiesResults for shear and bulk viscosities

The leading order gives rigourously the T<<m behaviour, consistently with nonrelativistic Kinetic Theory.

Transport coefficients in ChPT require a modification of the standard

chiral power counting to account for O(1/) collision effects.

Unitarity is crucial to ensure the correct high-T behaviour.

As TTc ladder-type diagrams may have to be sumed, although

results suggest strong cancellations.

Kaons, chemical potentials …

At higher temperatures, the LO agrees reasonably with KT analysis and phenomenological/theoretical results.