gomez_nicola

EM PROPERTIES AND

TRANSPORT COEFFICIENTS

OF A PION GAS





Angel Gómez Nicola

Universidad Complutense

Madrid



Strong and Electroweak Matter 2006. BNL, 10-14 May.

Linear Response to soft EM fields





Different limits of the retarded EM current-current correlator







Im(  ) ( , q )

R i 

m (T )     lim  (, q )

2 R

 (T )  lim lim i

D lim 00

 0  q 0  3 q 0  0





DC Electrical Conductivity Debye screening mass





In a HIC enviroment





 q2  0 Real, Photon Spectra

R 

Im(  )  (, q )

q 2  M 2 Virtual, Dilepton Spectra

Chiral Perturbation Theory:



Relevant for low and moderate temperatures below Chiral SSB



Most general derivative and mass expansion of NGB mesons compatible

with the SU L ( N f )  SU R ( N f )  SUV ( N f ) SSB pattern of QCD:



f2

2  UU T  m f2U 0

2

NLSM

2

  2  4  ... U  (U 0 ,U )  SU (2), U



, U 0  1 | U |2

f



U 0,3   U 0,3 , U 1,2   U 1,2  e ijk AU 2,1



Weinberg’s chiral power counting: p  E , m, T , e     4f  1 GeV

D

 p 

  D  2( N L  1)   (d  2) N d

 

   d

Pion EM Form Factors at T>0:

AGN, F.J.LLanes-Estrada, J.R.Peláez, PLB606, 351 (2005)





Enter the dilepton rate from annihilation of thermal pions:



e+

 g



 e-







D=4 ChPT diagrams:









4

The pion EM charge distribution at T>0



V0 ( p1 , p2 )  0 J 0EM (0)   ( p1 )  ( p2 )  k0 Ft ( S0 , S )

k  p1  p2 , S  p1  p2





 6 dFt ( S0  0, S )

QT  Ft ( S0  0, S  0 ) , r 2

 2

T QT dS

S 0





Q0  1 , r 2  0.45 fm 2

0







+

Net pion charge is screened:

Gale&Kapusta ‘91

1  q 2  Eq

2 2

mD (T ) Kapusta ‘92

QT  1 

2 f

2 2 

0

dq

Eq

nB ( Eq )  1  2 2

2e f Eq  q2  m

2

Model independent



Confirms Dominguez et al ’94 (QCD sum rules)



Provides a rough estimate of deconfinement:



3

d k 4 2 3/ 2

n (T )  3 n ( Ek ) r n (Tc )  1 Tc  200 MeV

(2 ) 3 B

3 Tc





Tc  265 MeV using r 2 (Kapusta)

0

Unitarization: The Inverse Amplitude Method



ChPT does not reproduce resonances (r,,...)

due to the lack of exact unitarity.



 

In the c.om. frame ( p1   p2 , back to back dileptons): (S0  2m )



2

Im t ( S0 ; T )   T ( S0 ) t ( S )

( 4)

IJ

(2)

IJ

2

0 Thermal Perturbative



Im F ( 4 ) ( S0 ; T )   T ( S0 )t11 ) ( S0 ) F ( 2 ) ( S0 )

(2 2

  Unitarity



AGN, F.J.LLanes-Estrada, J.R.Peláez, PLB550, 55 (2002)



2

 T ( S0 )  1  2 1  2nB ( S0 / 2)

4m

Two-pion thermal phase space

S0

(1  nB )  nB

2 2





Enhancement Absorption

Exact unitarity at T>0 *+ ChPT matching at low energies



Im F IAM   T t11 F IAM

IAM

  F IAM  1  F (1)  

Im t IAM

IJ  T t IAM 2

IJ (Im t IAM 1

IJ   T ) t IAM  t ( 0 )  t (1)  



t ( S0 ; T )  ( 2 ) 2

IAM t 2

( S0 )( 2)

 2





t ( S0 )  t ( 4 ) ( S0 ; T )



1  Re F ( 2 ) ( S0 ; T )

F IAM

( S0 ; T )  ( 2 ) 2 IAM

t11 ( S0 ; T )

t11 ( S0 )  Re t11 ) ( S0 ; T )

(4









Thermal  and r poles

A.Dobado, AGN, F.J.LLanes-Estrada, J.R.Peláez, PRC66:055201 (2002)









* To O(nB) (only two-pion states, dilute gas).

The unitarized form factor









Significant peak reduction and broadening around Mr (slight mass shift)

compatible with dilepton spectrum (NA45,NA60) and VMD calculations.

Electrical Conductivity of a pion gas:

D.Fernández-Fraile, AGN, PRD73:045025 (2006)







Nonperturbative corrections needed to account for the inverse

width dependence of transport coefficients (“pinching poles”):







 '2  E p 

 

Im  G R (' , p )G A (' , p )  2



(  0 ,0) 2 E p p

0  p  E p



e 2 N ch

as expected from Kinetic Theory   

m

In ChPT:

1 d 3 k  s ( s  4m )

2



p  O( p )  E p

5

 DG

   tot ( s ) exp  Ek / T 

2 (2 ) 3

p

2 Ek E p

32  2



 tot ( s )   (2 J  1)(2 I  1) t

2

IJ

3s J 0 I 0

Chiral counting revisited for transport coefficients:



“Double lines” with equal momentum carry ≠0 propagators and

do not count for chiral loops: assign a “nonperturbative” factor Y



 ( 0)  e2m Y

Identify diagrams with larger powers of Y:



Ladders O( p 2 kY k 1 )





Bubbles O(Y k 1 )







  ( k )  O ( 2 ) ( 0) explicitly for Γ  0

k 1

For T  m : Nonrelativ istic KT :

 m T  m 2 

Relevant p  m T  m 

  n v tot   

m T 

3



 m

 4 

 f 

    



m m

 (T )  2.7e m

( 0) 2

Y Potentially large

T T





Detailed analysis of the effective ladder vertex

Loop spectral function ~ pCM / m  T / m  Y 1



m

 (k )

 O( p 2 k ) for T  m

T

T dependence PERTURBATIVE

factorizes IN CHPT

For T  m :



Unitarization effects relevant for

Relevant p  T partial waves in the thermal width





2

 m 

YNU   

 T  Ladder diagrams perturbative within ChPT

YU  const





Those with constant (  m / f2 ) vertices much more suppressed than

2





those with derivative vertices (  T 2 / f2 ).

Unitarization modifies

 (T ) / m

( 0)

decreasing behaviour

( ~ T expected in QGP phase)









 (1) /  ( 0) Ladder vertex integral equation

(Boltzmann-like) important

where ChPT starts breaking up

Estimates for photon spectrum near zero energy:



 3   3 nB ( ) Im  ( 2  q )

dRg 1 R  2

In equilibrium:

d q 4

 

q R



   

 0   R 0 (  0 , q  0)  0

0









dRg  3T (T )

timelike !   (  0 , q  0)   2  103 GeV - 2 fm - 4

d 3q 4 3



 unit (T  150 MeV)

( 0)







Compatible with values near the origin of recent hadronic gas analysis

(S.Turbide, R.Rapp, C.Gale PRC69, 014903 (2004), W.Liu, R.Rapp nucl-th/0604031)









  0 and resonances in  scattering

important near the origin !

In fact, assuming a Bjorken expansion:

dNg 3RAnuc  f

2





4 2  i

 3 ( pT  0 )  

d T ( ) (T ( ))  5.6  102 GeV -2

d p

RA  7.7 fm, nuc  10 (WA98 208

Pb collisions )

T ( )  Ti ( i /  )1 / 3

 i  3 fm/c, Ti  170 MeV

 f  13 fm/c, T f  104 MeV

  pT cosh( y   )





Consistent with a linear

extrapolation from the origin









M.M.Aggarwal et al (WA98)

PRL93, 022301 (2004)

ChPT provides model-independent predictions for EM properties of a

pion gas at low temperatures.



Unitarized ChPT describes a thermal r in the pion EM form factor

consistent with dilepton data: large r  (T) and small Mr  (T).



Transport coefficients in ChPT require a modification of the standard chiral

power counting to account for O(1/) collision effects (“pinching poles”).



The DC electrical conductivity is dominated by the LO at low T and

unitarization raises (T) for T>100 MeV. Results are consistent with

photon yield data near zero energy, highlighting   0 importance.



As TTc ladder-type diagrams have to be resummed.



Viscosities, kaons, qT>0 photons, ,K ,…