XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007.
18
XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007. Angel Gómez Nicola Universidad Complutense Madrid COEFICIENTES DE TRANSPORTE EN UN GAS DE PIONES
description
COEFICIENTES DE TRANSPORTE EN UN GAS DE PIONES. Angel Gómez Nicola. Universidad Complutense Madrid. XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007. t. (fm/c). T (MeV). 120-200. 300-400. HADRON GAS ( p, K , h, r,...). ¿QGP?. 50. 1. 5-10. PERTURBATIVE QCD - PowerPoint PPT Presentation
Transcript of XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007.
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.
