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Parton propagation through Strongly interacting Systems– ECT?, Trento, September 2005 –
Collisional energy loss in the sQGP
André Peshier
Institute for Theoretical Physics, Giessen University, Germany
1 Bjorken’s estimate2 Strongly coupled (Q)GP3 = 1 + 2
Why collisional energy-loss?
*
TcQGP
SBCSC
AIP Bulletin, April 20 2005:
Now, for the first time since starting
nuclear collisions at RHIC in the year
2000 and with plenty of data in hand,
all four detector groups operating at
the lab [BNL] . . . believe that the
fireball is a liquid of strongly
interacting quarks and gluons rather
than a gas of weakly interacting
quarks and gluons.
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 2
Why collisional energy-loss?
paradigm: radiative energy-loss [BDMPS] dominates collisional energy-loss
How realistic are commonlyused (pQCD based) inputparameters for e-loss estimates?
‘observed’ at RHIC:
strongly coupled QGP (sQGP)
collective phenomena, in linewith hydrodynamicsfast equilibration, low viscositylarge Xsections, large coupling
⇓
dense liquid ⇔ collisional e-loss
Fokker-Planck eqn. with dragand diffusion parameters relatedto dEcoll/dx [Mustafa, Thoma]
⇒ compatible quenching factors
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 3
Framework
things are involved ⇒ simplify
consider energy-loss of hard jetin static infinite thermalizedmedium (QGP)
consider mostly quenched QCD;
pQCD expectation quarks andgluons differ by group factors(coupling Casimirs, d.o.f.) seenalso for large coupling (lattice)
rescale:
T quenchc = 260MeV → 170 MeV
‘Universality’ in QCD
0
1
2
3
4
5
100 200 300 400 500 600
T [MeV]
p/T4 pSB/T4
3 flavour2+1 flavour
2 flavourpure gauge
0.0
0.2
0.4
0.6
0.8
1.0
1.0 1.5 2.0 2.5 3.0 3.5 4.0
T/Tc
p/pSB
3 flavour2 flavour
2+1 flavourpure gauge
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 4
Bjorken’s formula
[Bjorken ’82] considers energy-loss per length due to elastic collisions
dE
dx=
∫k3
ρ(k) Φ︸ ︷︷ ︸flux
∫dt
dσ
dt∆E
E
k
θ
E′= E − ∆E
small t dominate:dσ
dt= 2πCgg
α2
t2t
E , E ′ � k ∼ T : t = −2(1− cos θ)k∆E
Φ = 1− cos θ
divergences – cut-offs:
∫ t2t1
dtdσ
dt∆E =
94πα
2
k(1− cos θ)ln
t1t2
screening: t2 = −µ2
∆E < ∆Emax : t1 = −2(1− cos θ)k∆Emaxjet persist: ∆Emax ≈ 0.5E
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 5
Bjorken’s formula
dE
dx=
9π
4α2
∫k3
ρ(k)
kln
(1− cos θ)kEµ2
pragmatically: (1− cos θ) → 2∫dk kρ(k) ln k → ln〈k〉
∫dk kρ(k)
〈k〉 → 2T
with ρ(k) = 16nb:dEBdx
= 3πα2T 2 ln4TE
µ2µ2 → m2D = 4παT 2
shortcomings: 1 sloppy k-integral ↔ dE/dx < 0??2 phenomenological IR cut-off
3 relevant scale for coupling?
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 6
Bjorken’s estimate
collisional energy-loss of hard gluons(+1) and quarks (−1)
dEBdx
=
(3
2
)±1 (1 + 16 nf
)2πα2T 2 ln
4TE
µ2
0 5 10 15 20
E [GeV]
0.2
0.4
0.6
0.8
1.0
dE/d
x[G
eV/fm
]
q in QGP
g in GPM = mD
T = 300MeVα = 0.2µ = 0.5...1 GeV
∼ mD =√
4παT
compare to
‘cold’ e-loss ∼ 1 GeV/fmradiative e-loss ∆Erad ∼ Eβ,β = 0, 12 , 1
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 7
Bjorken’s formula – improvements
orderly t and k integrals
0−µ2
t1 = −2(1− cos θ)k∆Emax
t1!< −µ2: ∆E > 0
constrains∫k3
limit ET � µ2:
dẼBdx
= 3πα2T 2 ln0.64TE
µ2
screening systematically withinHTL perturbation theory
dE ?
dx= 3πα2T 2 ln
1.27TE
m2D
[Braaten, Thoma]
µ2? ≈ 0.5m2D
NB: conform with general form of collisional energy loss, in leading-logapproximation, related to cut dressed 1-loop diagrams [Thoma]
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 8
Bjorken’s formula – improvements
running coupling
αpert(Q) =4π
11 ln(Q2/Λ2)
often: Q ∼ 2πT , Λ = Tc/1.14
1 2 3 4
T / Tc
0.5
1.0
1.5
2.0
mD
[GeV
]
(Q), Q = [ ..4 ]T= 0.2
T
NB: even conservative α(T ):
mD > T
(pQCD: mD = gT � T )
1 2 3 4
T/ Tc
2
4
6
dE/d
x[G
eV/fm
] E = 10 GeV
what is relevant scale for α?how reliable is extrapolated pQCD?
A. Peshier, Collisional energy loss Bjorken’s estimate – p. 9
Quasiparticle perspective of s(Q)GP
2PI formalism: thermodynamic potential in terms of full propagator
Ω = 12 Tr(ln(−∆−1) + Π∆
)− Φ, Φ = + + + . . .
Π = 2δΦ
δ∆= + + + . . .
truncation ⇒ thermodyn. consistent resummed approximations
entropy functional of dressed (=quasiparticle) propagator [Riedel, ...]
s[∆] = −∑
i=T ,L
di
∫p4
∂nb∂T
(Im ln(−∆−1i ) + ImΠi Re∆i
)(in Fermi liquid theory: dynamical quasiparticle entropy)
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 10
Quasiparticle perspective of s(Q)GP
dressed propagator for momenta p ∼ T (dg = 16 transverse d.o.f.)
Ansatz: Lorentzian spectral functioncorresponds to self-energy
Π = m2 − 2iγω
( , )
collisionalbroadening
massshift
γ → transport properties
quasiparticle mass and (collisional) width parameterized in form ofpQCD results (gauge invariant, momentum-independent)
m2 = 2παT 2 , γ =3
2πα T ln
c
α
to extrapolate to T ∼ Tc use effective coupling
α(T ) =4π
11 ln(λ(T − Ts)/Tc)2
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 11
Quasiparticle perspective of s(Q)GP
QP model [AP] vs. lattice data [Okamoto et al.]
1 2 3 4
T/Tc
0.0
0.2
0.4
0.6
0.8
1.0
(e-3
p)/3
p 0,s
/s0
(e-3p)/3p0s/s0
1 2 3 4
T/Tc
0
2
4
/Tc
,m/T
c
m
non-perturbative ‘quasiparticles’
m ∼ T heavy excitationsγ ∼ T short mean free path – except very near Tc (crit. slow down)
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 12
s(Q)GP: coupling α(T )
pQCD
αpert(Q) =4π
11 ln(Q2/Λ2)
with Q ∼ 2πT , Λ ∼ Tc
analyze lattice data1 entropy within QP model
2 static qq̄ free energy
F (r ,T ) → Crα
rexp(−mDr)
[Kaczmarek et al.]
for T ∼ O(Tc)IR enhancement of α
1 2 3 4
T / Tc
0.5
1.0
1.5
2.0
pQCD: Q = [ ..4 ]Tlattice (s) & QPlattice (F)
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 13
s(Q)GP: cross section
pQCD & cut-off m2D = 4παT2
σpert =
∫dt
92πα
2
t2=
9
8
α
T 2
→ 98
α(2πT )
T 2∼ O(1 mb)
phenomenology [Molnar, Gyulassy]
σRHIC ∼ O(10 mb)1 2 3 4
T/Tc
0.1
1
10
[mb]
QP: eff
pQCD
QP model, from 2 → m interaction rate d4N/dx4 [AP, Cassing]
Trp1,p2
[2√
λ
2ω12ω2nb(ω1)nb(ω2) σ Θ(P
21 )Θ(P
22 )
]= Trp
[1
2ωγ nb(ω)Θ(P
2)
]
average σ and γ ⇒ σeff = γN+I2∼ O(10 mb)
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 14
Interlude: quasiparticle model – implications
QP-model indicates an almost ideal liquid(?) [AP, Cassing]
large plasma parameter
Γ = 2Ncα
N−1/31
〈Ekin〉
1 2 3 4
T/Tc
10-1
1
10
gas
liquid
(?) 6= ideal gas!!!
large percolation measure
κ2 = σeff N2/3
1 2 3 4
T/Tc
0.5
1.0
1.5
2.0
2.5
dilute regime
very low shear viscosityη
s≈ 0.2 near Tc
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 15
Interlude: quasiparticle model – implications
quantities relevant for radiative energy-loss
mean free path λ = γ−1
1 2 3 4
T / Tc
0.1
0.2
0.5
1
2
5
[fm
]
Tc = 170MeV
g
q
transport coefficient q̂ = m2D/λ
1 2 3 4
T / Tc
0.5
1.0
1.5
2.0
2.5
3.0
[GeV
2 /fm
]
Tc = 170MeV
q̂
NB: γ−1(p ∼ T ) as a lower estimate for mean free path of hard jet
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 16
Non-perturbative parameterization: α(T )
Can pQCD, by using appropriate scales, be extrapolated to ‘near’ Tc?
αpert(Q) =4π
11 ln(Q2/Λ2)
pQCD:Q
Λ∼ 2πT
Tc
non-pert. parameterization(aim at T ∼> 1.3Tc):
Q
Λ→ 1.7 T
Tc
1 2 3 4
T / Tc
0.5
1.0
1.5
2.0
pQCD: Q = [ ..4 ]Tlattice (s) & QPNP: Q/ = 1.7 T/Tclattice (F)
lQCD data [Kaczmarek et al.]
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 17
Non-perturbative parameterization: Debye mass
Can pQCD be consistently extrapolated to ‘near’ Tc?
m2D = 4παT2
pQCD:
α → αpert(Q ∼ 2πT )
non-pert. parameterization:
α → αNP(T )
observation within QP model:
mD ≈ 2.7γ
1 2 3 4
T / Tc
1
2
3
4
mD/
T
NPpQCDQP: 2.7
lattice (F)lattice ( )
lQCD data [Nakamura et al.],[Kaczmarek et al.]
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 18
Non-perturbative parameterization: cross section
Can pQCD be consistently extrapolated to ‘near’ Tc?
σ =
∫ −µ2dt
92πα
2
t2
pQCD: σ = 98α|Q∼2πT
T 2
running α(t) = A/ ln(−t/Λ2):
σ = 9π2
∫ −µ2dt
[A
t ln(−t/Λ2)
]2= 9π2
A2
Λ2
[Ei(− ln(µ2/Λ2)) + Λ
2/µ2
ln(µ2/Λ2)
]→ 9π2
α2(µ)
µ2[1 + . . .]
1 2 3 4
T/Tc
0.1
1
10
[mb]
NPQP: eff
pQCD
µNP = 0.6mD
ΛNP = 420MeV
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 19
Non-perturbative parameterization
Indeed, pQCD can consistently be extrapolated to ‘near’ Tc .
lattice results for α(T ), mD(T ), for T/Tc ∈ [1.3, 4], consistent with
αNP(T ) =4π
11 ln(1.7T/Tc)2
cross section (×10) enhancement consistent with
pert. Xsection dσ/dt ∼ α2/t2
running coupling α(t) =4π
11 ln(−t/Λ2NP), ΛNP = 420MeV
cut-off µ = 0.6mD (compare to µ? = 0.7mD)
relation between αNP(T ) and α(t), assuming√|t̄| = κT ,
κ = 1.7ΛNPTc
≈ 2.74 (compare to 〈k〉 =∫k3 kρ(k)∫k3 ρ(k)
≈ 2.70T )
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 20
Interlude: Cut-off and running coupling
−→ + + + ...α0P2
−→ αP2 − Π
(resummation, renormalization)
vacuum:
Π = Πren ∼ αP2 ln(−P2/µ2)⇓
α(P2) =4π
11 ln(−P2/Λ2)
medium:
Π = Πren + Πmat
Πmat ∼ αT 2 ∼ m2D
IR cut-off: P ∼> mD
running important when mD ∼ gT ∼ Λ (non-pert. regime)
A. Peshier, Collisional energy loss Strongly coupled (Q)GP – p. 21
Collisional e-loss with running coupling
Bjorken:dE
dx=
∫k3
ρ(k) Φ
∫dt
dσ
dt∆E
t-integral, with α(t) = A/ ln(−t/Λ2)
Φ
∫ t2t1
dtdσ
dt∆E = −
94πA
2
k
∫ t2t1
dt
t
1
ln2(−t/Λ2)
=
94πA
k
[α(µ2)− α((1− cos θ)kE )
]constraint (1− cos θ)k ≥ µ2/E
θ-integration leads to logarithmic integrals, li(x) = P∫ x0 dt/ ln(t)
dE
dx=
9A2
8π
∫ ∞k̄
dkkρ(k)
[1− µ2/(2Ek)
ln(µ2/Λ2)+
Λ2
2Ek
(li
µ2
Λ2− li2Ek
Λ2
)], k̄ =
µ2
2E
= T 2 F (µ2
2TE,µ2
Λ2) → dEB
dx
∣∣∣∣α(µ)
[1 +O(α)]
A. Peshier, Collisional energy loss = 1 + 2 – p. 22
Collisional e-loss in s(Q)GP – numerical results
non-perturbative parameterization (T ∼> 1.3Tc)
300 400 500 600
T [MeV]
2
5
1
2
5
10
2
dE/d
x[G
eV/fm
]
E = 10GeVE = 50GeV
= mDBjorken
dE
dx= T 2F (
µ2
2TE,µ2
Λ2)
parameters
Λ → ΛNPµ2 = 0.6m2D
m2D → 4παNP(T )T 2
A. Peshier, Collisional energy loss = 1 + 2 – p. 23
Collisional e-loss in s(Q)GP – numerical results
lQCD/QP parameterization of Debye mass: mD ≈ 2.7γ
small mD(Tc) (∼ phase transition) → increased energy loss
300 400 500 600
T [MeV]
2
5
1
2
5
10
2
dE/d
x[G
eV/fm
]
E = 10GeVE = 50GeV
NPdE
dx= T 2F (
µ2
2TE,µ2
Λ2)
parameters
Λ → ΛNPµ2 = 0.6m2D
m2D → m2D,eff
A. Peshier, Collisional energy loss = 1 + 2 – p. 24
Collisional e-loss in s(Q)GP – numerical results
unquenching: energy-loss of a quark in the sQGP
200 300 400 500
T [MeV]
2
5
1
2
5
10
2
dE/d
x[G
eV/fm
]
E = 10GeVE = 50GeV
quenched
dE
dx= T 2F (
µ2
2TE,µ2
Λ2)
parameters
Tc → 170 MeV
Cgg → Cq(1 + nf /6)
A. Peshier, Collisional energy loss = 1 + 2 – p. 25
Collisional e-loss in s(Q)GP – numerical results
Is dE/dx ∼ 1 GeV/fm enough?
assume
constant dE/dx
Bjorken dynamics
quenching factor
dN
d2pT= Q(pT )
dN0d2pT
=1
2πR2
∫ 2π0
dφ
∫ R0
dr2dN(pT + ∆pT )
d2pT
5 10 15
pT [GeV]
0.1
0.2
0.3
Q(p
T)
dE/dx = 0.8GeV/fmdE/dx = 1.0GeV/fmdE/dx = 1.2GeV/fmMueller (BDMPS)
comparable to [Müller]: BDMPS + transv. profile + Bjorken dynamics
A. Peshier, Collisional energy loss = 1 + 2 – p. 26
Resumé
realistic parameters for sQGP
⇒ enhanced dEcolldx
(quasi) critical screening
⇒ Tc quenching
does sQGP(?) quench toomuch?
far/near side jets vs.geometry+delay(talk Cassing)retardation (talk Gossiaux)
1 2 3 4
T / Tc
1
2
3
4
mD/
T
NPpQCDQP: 2.7
lattice (F)lattice ( )
200 300 400 500
T [MeV]
2
5
1
2
5
10
2
dE/d
x[G
eV/fm
]
E = 10GeVE = 50GeV
quenched
(?)strongly Quenching (Q)GP
A. Peshier, Collisional energy loss = 1 + 2 – p. 27
Bjorken's estimateStrongly coupled (Q)GP=1+2