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Dilepton production as a measure of QGPthermalization time
Mauricio Martinez1
Work done in collaboration with M. Strickland
1Helmholtz Research School,Frankfurt Institute for Advanced Studies,
Johann Wolfgang Goethe Universitat Frankfurt
Quark Matter 2008Jaipur, India
9th February 2008
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 1 / 16
Pre-equilibrium phase of the QGP
τ0
~ Qs
-1 τ
1
0.1-0.2 fm/c
Expansion rate and isotropization
via interactions balance
<p >T
<p >L
CG
C/G
lasm
a
Bo
ltzm
an
n-
Vla
sov
Tra
nsp
ort
Vis
cou
s H
ydro
System is momentarily isotropicSystem is momentarily isotropicSystem is momentarily isotropic
Expansion rate is much faster
than the interaction time scale
1/τ >> 1/τ int
small plarge p large p
∼ τiso
As a result of the rapid expansion along the beam axis, ananisotropy in the momentum-space is developed.
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 2 / 16
Electromagnetic probes in heavy ion collisions
Electromagnetic signatures giveinformation about initial partondistributions and early timedynamics of the collision.
Photons are more difficult forexperimentalists to measure due tolarge backgrounds.
Dileptons offer a better option fromthe experimental point of view.
Influence of non equilibrium dynamics on dilepton production?
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 3 / 16
Dilepton emission from an anisotropic QGP
Dilepton rate d4R/d4P depends on the direction ofthe anisotropy and the angle of the dilepton pair withrespect to the longitudinal axis.
As an ansatz, we choose an anisotropic phase spacedistribution in momentum-space:
f i (p, x ) = f iiso
(
p2T + (1 + ξ)p2
L
)
ξ measures the strength of the anisotropy and it’s relatedwith the kinematic variables:
ξ =12〈p2
T 〉
〈p2L〉
− 1
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 4 / 16
Model for an anisotropy in momentum-spaceIn a free streaming plasma:
ξFS(τ) =( τ
τ0
)2− 1
limτ≫τ0
E(τ) ⇒ E0
(τ0
τ
)
“T ” = T0
In a hydrodynamical plasma:
ξ(τ) = 0
E(τ) = E0
(τ0
τ
)4/3
T = T0
(τiso
τ
)1/3
Propose a model that interpolates between free streaming andhydrodynamical expansion :
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 5 / 16
Space-time evolution with anisotropies
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 6 / 16
Dilepton production vs. M
⇐ No difference!!!
A K factor of 1.5 was applied to account for NLO corrections.
T0= 845 MeV, τ0= 0.088 fm/c, Tc = 160 MeV.
Cuts: PT > 8 GeV.
M. Martinez and M. Strickland, arXiv:0709.3576 [hep-ph].
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 7 / 16
Dilepton production vs. PT
Large enhancement atintermediate PT fromanisotropy!!!
A K factor of 6 was applied to account for NLO corrections.
T0= 845 MeV, τ0= 0.088 fm/c, Tc = 160 MeV.
Cuts: M > 2 GeV.
M. Martinez and M. Strickland, arXiv:0709.3576 [hep-ph].
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 8 / 16
Conclusions
We construct a model that interpolates between freestreaming and hydrodynamics evolution. The model takesinto account the time dependence of the anisotropy in themomentum-space.
Dilepton production in the kinematic range 3< PT <8 GeVprovides an estimate of the momentum-space anisotropyand a possible measure of the isotropization time, τiso.
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 9 / 16
Backup slides
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 10 / 16
Dilepton rate at leading order
From relativistic kinetic theory, the dilepton rate production forqq → l+l− is:
dNd4xd4p
=dRd4P
=
∫
d3p1
(2π)3
d3p2
(2π)3 fq(p1, T )fq(p2, T )
× υrelσLOqq→l+l−δ4(P − p1 − p2)
The invariant distributions of dileptons as a function of invariantmass and transverse momementum are respectively:
dNdM2dy
= πR2∫
d2PT
∫ τf
τ0
∫
∞
−∞
dRd4P
τdτdη.
dNd2PT dy
= πR2∫
dM2∫ τf
τ0
∫
∞
−∞
dRd4P
τdτdη.
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 11 / 16
Some details about the interpolating model
Smeared stepfunction λγ(τ − τiso)
Time-dependence of E , phard and ξ
E(τ) = EFS(τ) [U(τ)/U(τ0) ]4/3 ,
phard(τ) = T0 [U(τ)/U(τ0) ]1/3 ,
ξ(τ) = a2(1−λ(τ)) − 1 ,
U(τ) =
[
R
(
(τiso
τ
)2− 1)]
3λ(τ)4 (τiso
τ
)λ(τ)
R(ξ(τ)) =
(
11 + ξ(τ)
+arctan(
√
ξ(τ))√
ξ(τ)
)
limτiso≫τ
E(τ) ⇒ E0
(τ0
τ
)
Free streaming limit
limτ≫τiso
E(τ) ⇒ E0
(τiso
τ
)4/3Hydrodynamical limit
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 12 / 16
Dependence on the model parameter γ
A K factor of 6 was applied to account for NLO corrections.
T0= 845 MeV, τ0= 0.088 fm/c, Tc = 160 MeV.
Cuts: M > 2 GeV.
M. Martinez and M. Strickland, arXiv:0709.3576 [hep-ph].
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 13 / 16
Produced fraction of Medium Dileptons in time
T0= 845 MeV, τ0= 0.088 fm/c, Tc = 160 MeV, τiso= 0.5 fm/c.
Cuts: M > 2 GeV.
M. Martinez and M. Strickland, arXiv:0709.3576 [hep-ph]
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 14 / 16
Fixing multiplicities
We could modify the model for the non equilibrium region fixingthe final multiplicities. For doing so, we demand that after agiven τiso, the hard momentum scale is the same.
2 5 10 20 50Τ�Τ0
0.3
0.5
0.7
1
p har
d�
T 0
Τiso =1.6 fm�c
Τiso =0.8 fm�c
Τiso =0.4 fm�c
Τiso =0.1 fm�c
As a consequence, the initial conditions will change and willdepend on τiso.
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 15 / 16
Fixing multiplicities: dilepton spectrum vs. PT
2 4 6 8 10PT @GeVD
-9
-8
-7
-6
-5
-4
-3
log
10H
dN
e+
e
-
�d
yd
pT2@
Ge
V-
2DL
Medium HΤhydro = 2 fm�cL
Medium HΤhydro = 0.088 fm�cL
Jet Conversion
Drell Yan
Dilepton production will be smaller compared with the casewhen the initial conditions are fixed but the effect from ananisotropy remains.A K factor of 6 was applied to account for NLO corrections.
T0= 845 MeV, τ0= 0.088 fm/c, Tc = 160 MeV, τ0 ≤ τiso ≤ 1 fm/c. Cuts: M > 2 GeV.
M. Martinez and M. Strickland, forthcoming
Mauricio Martinez Dilepton production as a measure of QGP thermalization time Quark Matter 2008 16 / 16