xxxi bienal de la rsef, granada, españa, 10-14 septiembre 2007
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 PresentationTRANSCRIPT
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XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007.
Angel Gómez Nicola
Universidad Complutense Madrid
COEFICIENTES DE TRANSPORTEEN UN GAS DE PIONES
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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
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Hadronic matter at Hadronic matter at high Temperature and high Temperature and
DensityDensity
Heavy Ion Collisions:
QCD phase diagram
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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)
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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
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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.
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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
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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.
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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.
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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
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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
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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
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Tp
0 and resonances in scattering important near the origin ! W.Liu, R.Rapp nucl-th/0604031
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Sound attenuation length (related to elliptic flow):
Results for shear and bulk viscositiesResults for shear and bulk viscosities
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Sound attenuation length (related to elliptic flow):
Results for shear and bulk viscositiesResults for shear and bulk viscosities
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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.