heavy-quark langevin dynamics and single-electron spectra...
TRANSCRIPT
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Heavy-quark Langevin dynamics andsingle-electron spectra in AA collisions
Andrea BeraudoUniversity of Torino and CERN - Theory Unit (Centro-Fermi grant)
Les Houches, 20th − 25th February 2011
Work done and in progress in Torino in collaboration with
W.M. Alberico, A. De Pace, M. Nardi, A. Molinari (DFT and INFN),
M. Monteno and F. Prino (INFN and ALICE group)
Nucl. Phys. A 831, 59 (2009) and arXiv:1101.6008 [hep-ph]a
aPresented at “Hot Quarks 2010” (arXiv:1007.4170 [hep-ph]), “Jets in proton-
proton and heavy-ion collisions” (arXiv:1009.2434 [hep-ph]), CERN Th. In-
stitute “The first heavy ion collisions at the LHC” and “Hard Probes 2010”
(arXiv:1011.0400 [hep-ph]).
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Outline
• Heavy quarks as hard probes of the Quark Gluon Plasma
• Theoretical framework:
– the relativistic Langevin equation in an expanding medium
– evaluation of the transport coefficients
• Numerical results (for RHIC and LHC):
from the initial QQ production to the final e-spectra
– Invariant yields E(dN/d3p) (pp vs AA)
– Nuclear modification factor RAA(pT )
– Elliptic flow coefficient v2(pT )
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Space-time cartoon of AA collisions
Hard probes (high-pT partons, heavy quarks and quarkonia):
produced in hard pQCD processes in the very early stages, they cross the
fireball allowing for a tomography of the matter.
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Properties of the QGP @ RHIC (Tmax<∼2Tc)arising from the experimental data :
• It behaves like a fluid (λmfp ≪ L): pT -spectra are well described
by hydrodynamics:
∂µTµν = 0
︸ ︷︷ ︸
four−momentum
, ∂µjµB = 0
︸ ︷︷ ︸
baryon number
and p = p(ǫ)︸ ︷︷ ︸
EOS
Only the conservation laws matter; the properties of the medium
are encoded into the equation of state!
• It is a very opaque medium: sizeable energy loss suffered by
high-pT particles.
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Elliptic flow:in non-central collisions particle emission not azimuthally-symmetric!
dN
d(φ− ψRP )=N0
2π(1 + 2v2 cos[2(φ− ψRP )] + . . . )
xφ
y
.05
.1
V2 /n
q
Matter behaves like a fluid whose expansion is driven by pressure gradients
∂
∂t
h
(ǫ+ P )vii
= − ∂P
∂xi
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Quenching of hadron specra
RAA(pT ) =1
〈Ncoll〉dNAA/dpT
dNpp/dpT∼exp
0.2
(GeV/c)Tp0 1 2 3 4 5 6 7 8 9 10
AA
R
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 charged hadronsneutral pions
Au+Au
d+Au
Sizeable in-medium energy-loss suffered by high-pT partons!
• hard partons from the surface pointing outwards reach the detectors;
• energy of partons crossing the plasma gets dissipated (scattering in the
medium, with radiation of soft gluons).
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Heavy-quark dynamicsin the Quark Gluon Plasma
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Heavy quarks as probes of the QGP
• Relaxation to thermal equilibrium (cumulated effect of many random
collisions):
〈p2〉2M
=3
2T =⇒ pheavy =
√3MT (plight ∼ T )
〈p2heavy〉 ∼
M
T〈p2
light〉 =⇒ τheavy ∼ M
Tτlight
In an expanding fireball with a finite life-time one does not expect the
heavy quarks to reach full equilibrium with the medium.
• One would also expect less quenching of the pT -spectra wrt light
hadrons
– suppression of collinear gluon radiation for θ<∼M/E (dead-cone
effect): collisional energy-loss should play an important role!
– Casimir factor CF entails weaker coupling with the medium wrt
gluons (CA), hence smaller amount of energy-loss.
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The relativistic Langevin equation...
∆pi
∆t= − ηD(p)pi
| {z }
determ.
+ ξi(t)|{z}
stochastic
,
with the properties of the noise encoded in
〈ξi(pt)ξj(pt′)〉=bij(pt)
δtt′
∆tbij(p)≡κ‖(p)p
ipj + κ⊥(p)(δij−pipj)
Transport coefficients to calculate:
• Momentum diffusion κ⊥≡ 1
2
〈∆p2⊥〉
∆tand κ‖≡
〈∆p2‖〉
∆t;
• Friction term (dependent on the discretization scheme!)
ηDIto(p) =
κ‖(p)
2TEp− 1
E2p
»
(1 − v2)∂κ‖(p)
∂v2+d− 1
2
κ‖(p) − κ⊥(p)
v2
–
fixed in order to insure the approach to equilibrium (Einstein relation):
Langevin eq. ⇔ Fokker Planck eq. with steady solution exp(−Ep/T )
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...in a static medium0
0 1 2 3
p (GeV)
0
0.02
0.04
p2 f(p)
(G
eV-1
)
0 1 2 3
p (GeV)
T=400 MeVHTL (µ=πT)
T=400 MeVHTL (µ=2πT)
0
t=1 fm
t=2 fm
t=4 fmt=8 fm
t=1 fm
t=2 fm
t=4 fm
For t≫ 1/ηD one approaches a relativistic Maxwell-Juttner distributiona
fMJ(p) ≡e−Ep/T
4πM2T K2(M/T ), with
Z
d3p fMJ(p) = 1
(Test with a sample of c quarks with p0 =2 GeV/c)
aA.B., A. De Pace, W.M. Alberico and A. Molinari, NPA 831, 59 (2009)
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...in an expanding fluid
The fields uµ(x) and T (x) are taken from the output of two
longitudinally boost-invariant (“Hubble-law” longitudinal expansion
vz = z/t)
xµ = (τ cosh η, r⊥, τ sinh η) with τ ≡√
t2 − z2
uµ = γ⊥(cosh η, v⊥, sinh η) with γ≡ 1√
1 − v2⊥
hydro codesa.
NB Both codes employ the simplifying assumption η≡ηs≡ηL, where
ηS ≡ 1
2lnt+ z
t− zand ηL ≡ 1
2ln
1 + vz
1 − vz,
corresponding to a “Hubble-law” longitudinal expansion vz = z/t.
aP.F. Kolb, J. Sollfrank and U. Heinz, Phys. Rev. C 62 (2000) 054909
P. Romatschke and U.Romatschke, Phys. Rev. Lett. 99 (2007) 172301
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Evaluation κ⊥/‖(p)
Intermediate cutoff |t|∗∼m2D
a separating the contributions of
• soft collisions (|t| < |t|∗): Hard Thermal Loop approximation
• hard collisions (|t| > |t|∗): kinetic pQCD calculation
aSimilar strategy for the evaluation of dE/dx in S. Peigne and A. Peshier,
Phys.Rev.D77:114017 (2008).
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κ⊥/‖(p): hard contribution
P P’
K K’
P
P P
P′
P′
P′
K
K
K
K′ K
′K
′
+ +
(t) (s) (u)
κg/q(hard)⊥ =
1
2
1
2E
Z
k
nB/F (k)
2k
Z
k′
1 ± nB/F (k′)
2k′
Z
p′
1
2E′θ(|t| − |t|∗)×
× (2π)4δ(4)(P +K − P ′ −K′)˛
˛Mg/q(s, t)˛
˛
2q2⊥
κg/q(hard)
‖ =1
2E
Z
k
nB/F (k)
2k
Z
k′
1 ± nB/F (k′)
2k′
Z
p′
1
2E′θ(|t| − |t|∗)×
× (2π)4δ(4)(P +K − P ′ −K′)˛
˛Mg/q(s, t)˛
˛
2q2‖
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κ⊥/‖(p): soft contribution
κsoft⊥ =
1
2
CF g2
4π2v
Z |t|∗
0d|t|
Z v
0dx
|t|3/2
2(1 − x2)5/2ρ(|t|, x)
„
1−x2
v2
«
coth
0
B
@
xq
|t|
1−x2
2T
1
C
A
κsoft‖ =
CF g2
4π2v
Z |t|∗
0d|t|
Z v
0dx
|t|3/2
2(1 − x2)5/2ρ(|t|, x)
x2
v2coth
0
B
@
xq
|t|
1−x2
2T
1
C
A
where
ρ(|t|, x) ≡ ρL(|t|, x) + (v2 − x2)ρT (|t|, x) (|t| ≡ q2−ω2, x ≡ ω/q)
The result is then expressed in terms of the spectral functions of the resummed
gluons exchanged in the collisions with the plasma particles:
ρL/T (ω, q) ≡ 2Im∆L/T (ω + iη, q) where
∆L(z, q) =−1
q2 + ΠL(z, q), ∆T (z, q) =
−1
z2 − q2 − ΠT (z, q)
NB: medium effects embedded in the HTL gluon self-energy!
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The Hard Thermal Loop approximation
It is a one-loop gauge-invariant approximation allowing for the
calculation of thermal corrections to vacuum propagators.
∆L(q0, q) =−1
q2 + ΠL(x), ∆T (q0, q) =
−1
(q0)2 − q2 − ΠT (x)
with x≡q0/q and
ΠL(x) = m2D
(
1 − x
2lnx+ 1
x− 1
)
,
ΠT (x) =m2
D
2
(
x2 + (1 − x2)x
2lnx+ 1
x− 1
)
,
where mD≡gT√
Nc
3+
Nf
6is the Debye mass, responsible for the
screening of electro-static color fields.
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κ⊥/‖(p): numerical results
Combining together the hard and soft contributions...
0 5 10 15 20p (GeV/c)
0
1
2
3
4
5
6
κ (G
eV2 /f
m)
κ T
( |t|*=m
D
2 )
κ T
( |t|*=2m
D
2 )
κ T
( |t|*=4m
D
2 )
κ L
( |t|*=m
D
2 )
κ L
( |t|*=2m
D
2 )
κ L
( |t|*=4m
D
2 )
mc=1.5 GeV, T=400 MeV, µ =1.5π T
0 5 10 15 20p (GeV/c)
0
0.5
1
1.5
2
2.5
3
κ (G
eV2 /f
m)
κ T
( |t|*=m
D
2 )
κ T
( |t|*=2m
D
2 )
κ T
( |t|*=4m
D
2 )
κ L
( |t|*=m
D
2 )
κ L
( |t|*=2m
D
2 )
κ L
( |t|*=4m
D
2 )
mb=4.8 GeV, T=400 MeV, µ =1.5π T
...the dependence on the intermediate cutoff |t|∗ is very mild!
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κ⊥/‖(p): hard and soft contributions
We let the intermediate cutoff |t|∗ vary...
0 2 4 6 8 10p (GeV/c)
0
0.2
0.4
0.6
κ (G
eV2 /f
m)
κ T(soft)
( |t|*=m
D
2 )
κ T(hard)
( |t|*=m
D
2 )
κ T(soft)
( |t|*=4m
D
2 )
κ T(hard)
( |t|*=4m
D
2 )
mc=1.5 GeV, T=400 MeV, µ =1.5π T
0 2 4 6 8 10p (GeV/c)
0
0.5
1
1.5
2
κ (G
eV2 /f
m)
κ L(soft)
( |t|*=m
D
2 )
κ L(hard)
( |t|*=m
D
2 )
κ L(soft)
( |t|*=4m
D
2 )
κ L(hard)
( |t|*=4m
D
2 )
mc=1.5 GeV, T=400 MeV, µ =1.5π T
...but the sum of the contributions remains pretty stable!
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We are ready to perform numerical simulationsfor a realistic case!
• Initial generation of QQ pairs (POWHEG: pQCD@NLO);
• Langevin evolution in the QGP (uµ(x) and T (x) given by hydro);
• At Tc HQs hadronize (fragmentation with PDG branching ratios)
• and decay into electrons (PYTHIA decayer with PDG decay tables).
NB One has first of all to check to be able to reproduce pp results!
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Initial heavy-quark spectra
A B
s−b
b
s
y
x
• Single inclusive HQ spectra E(dN/d3p) generated with POWHEG
(pQCD @ NLO) using the PDF set CTEQ6M (+EPS09 in AA)
• HQ distributed in the transverse plane according to the nuclear
overlap dN/dx⊥∼TAB(x, y)≡TA(x+b/2, y)TB(x−b/2, y), with
TA/B(x⊥) ≡Z +∞
−∞
dz ρA/B(x⊥, z)
• Each quark is given a random k⊥ broadening extracted from a
gaussian distribution, with 〈k2⊥〉 = 〈k2
⊥〉pp + 〈δk2⊥〉AB(~b,~s).
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So far the experimental observables are
single-electron spectra from the decaysof charm
D → Xνe
and bottom hadrons:
B → Dνe
B → Dνe → Xνeνe
B → DY → XνeY
We plot the electrons falling into the PHENIX/ALICE acceptance
(|η|<0.35/0.9)
η ≡ 1
2lnp+ pz
p− pz= − ln tan
θ
2
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pp collisions
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Single-electron spectra: pp results
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
d2 σ/2π
p Tdp
Tdy
(m
b G
eV-2
c2 )
ec
eb
ec+e
b
0 2 4 6 8p
T (GeV/c)
0
0.5
1
1.5
2
ratio
exp
/pow
heg
<kT
2>
NN=0
<kT
2>
NN=1 GeV
2/c
2
(a)
(b)
With default POWHEG values µR =µF =mT , mc=1.5 and mB=4.8 GeV
PHENIX results in pp collisions at√s=200 GeV are nicely reproduced!
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AA collisions:Au-Au @ RHIC and Pb-Pb @ LHC
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Initialization: hydro scenario (0.1 < τ0 < 1 fm)
η/s = 0 η/s = 0.08
τ0 (fm) s0 (fm−3) T0 (MeV) τ0 (fm) s0 (fm−3) T0 (MeV)
0.1 8.4 666
RHIC 0.6 110 357 0.6 140 387
1 84 333
0.1 2438 1000 0.1 1840 854
LHC 0.45 271 482 1 184 420
initial QQ production (from POWHEG)
√sNN σpp
cc (mb) σAAcc (mb) σpp
bb(mb) σAA
bb (mb)
200 GeV 0.254 0.236 1.77×10−3 2.03×10−3
5.5 TeV 3.015 2.288 0.187 0.169
NB huge shadowing effects for cc production in Pb-Pb @ LHC!
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Final spectra: invariant yields
d2N
2πpT dpT dy
∣∣∣∣y=0
nuclear modification factor
RAA(pT ) =1
〈NAAcoll 〉
dN/dpT |AA
dN/dpT |pp
and elliptic flow coefficient
dN
dφ pTdpT=
dN
2π pTdpT
(1 + 2v2(pT ) cos[2(φ− ψRP)] + ...
)
v2≡〈cos[2(φ− ψRP )]〉
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Single-electron invariant yields: RHIC and LHC
0 1 2 3 4 5 6 7 8 9p
T (GeV/c)
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
102
103
d2 N/2
πpT d
p Tdy
(G
eV-2
c2 )
min. bias x 104
0-10% x 102
10-20% x 101
20-40% x 100
40-60% x 10-1
60-92% x 10-2
pp x 10-2
/σNN
Au-Au
pp x Ncoll
(a)
0 2 4 6 8 10 12 14p
T (GeV/c)
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
102
103
d2 N/2
πpT d
p Tdy
(G
eV-2
c2 )
min. bias x 103
0-10% x 101
10-20% x 100
20-40% x 10-1
40-60% x 10-2
60-90% x 10-3
pp x 10-3
Pb-Pb
pp x Ncoll
(b)
• Dashed curves: pp result scaled by 〈Ncoll〉;• Continuous curves: AA result after Langevin (viscous hydro, τ0=1 fm).
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Effects of the medium better dispayed through thenuclear modification factor RAA(pT )
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Heavy-quark RAA: role of the couplingRHIC (left panel) vs LHC (right panel)
(charm: thin lines, bottom: thick lines)
0 2 4 6 8
pT (GeV/c)
0
0.5
1
1.5
RQ A
A
µ=2πTµ=3πT/2µ=πT
0 2 4 6 8 10 12 14
pT (GeV/c)
(a) (b)
Strong dependence on the scale at which the coupling αs(µ) is
evaluated (µ=2πT and µ=πT ): at T =200 MeV αs≈0.34 and 0.63
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Heavy-quark RAA: role of hydrodynamicsRHIC (left panel) vs LHC (right panel)
(charm: thin lines, bottom: thick lines)
0 2 4 6 8
pT (GeV/c)
0
0.5
1
1.5
RQ A
A
τ0=1 fm
τ0=0.1 fm
τ0=0.6 fm
τ0=0.6 fm, ideal
0 2 4 6 8 10 12 14
pT (GeV/c)
viscous, τ0=1 fm
viscous, τ0=0.1 fm
ideal, τ0=0.45 fm
ideal, τ0=0.1 fm
(a) (b)
}viscous
For minimum-bias collisions dependence on the hydrodynamical
scenario is very mild.
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Effects of fragmentation and decays: hc/b and ec/b
RHIC (left panel) vs LHC (right panel)(charm: thin lines, bottom: thick lines)
0 2 4 6 8
pT (GeV/c)
0
0.5
1
1.5
R AA
c/bD/Be
c/e
b
0 2 4 6 8 10 12 14
pT (GeV/c)
(a) (b)
Fragmentation and semileptonic decays (D → Xνe) lead to a
quenching of RAA!
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Single-electron spectra: RHIC (left) vs LHC (right)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
RA
A
datae
ce
be
c+b
0
0.2
0.4
0.6
0.8
1
1.2
1.4
RA
A
0 2 4 6 8
pT (GeV/c)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
RA
A
0 2 4 6 8
pT (GeV/c)
0-10% 10-20%
20-40% 40-60%
60-92%
min. bias
0
0.2
0.4
0.6
0.8
1
RA
A
ec
eb
τ0=1 fm
ec+bec
eb
τ0=0.1 fm
ec+b
0
0.2
0.4
0.6
0.8
RA
A
0 2 4 6 8 10 12 14
pT (GeV/c)
0
0.2
0.4
0.6
0.8
1
RA
A
0 2 4 6 8 10 12 14
pT (GeV/c)
0-10% 10-20%
20-40% 40-60%
60-90%
min. bias
}}
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Some general comments:
• mild dependence on the hydro scenario (ideal/viscous) and the
thermalization time τ0 (in RHIC plots τ0 =1 fm), except for peripheral
collisions;
• high-pT reproduced better with µ∼1.5 ⇒ α=0.32 at T =300 MeV
• intermediate-pT spectra could get increased by coalescence;
• peripheral collisions represent a puzzle (accomodated by a smaller τ0?).
0 1 2 3 4 5 6
pT (GeV/c)
0
0.5
1
1.5
RA
A
viscous, τ0=0.1 fm
viscous, τ0=1 fm
ideal, τ0=0.6 fm
0 2 4 6 8
pT (GeV/c)
60-92% min. bias(a) (b)
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RAA of heavy-flavor electrons vs centrality:RHIC (left panel) vs LHC (right panel)
0 50 100 150 200 250 300
Npart
0
0.2
0.4
0.6
0.8
1
1.2
1.4
RA
A
pT > 0.3 GeV/cpT > 4.0 GeV/c
0 50 100 150 200 250 300 350
Npart
pT > 0.3 GeV/c
pT > 4 GeV/c
(a) (b)
• plots done using the integrated yields;
• parameter set: µ=3πT/2 and viscous hydro with τ0=1 fm;
• similar general trend (medium softens the spectrum conserving N tote )
– pT >0.3 GeV/c: flat RAA∼1 (RAA 6=1 at LHC due to nPDFs!)
– pT >4 GeV/c: suppression increases with centrality.
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The minimum bias case:RHIC (left panel) vs LHC (right panel)
Our analysis for all centrality classes allows a more accurate estimate of
the minimum bias spectrum. Employing in the simulations samples with
the same number of events:
dN
dpT
˛
˛
˛
˛
m.b.
≡
P
i fCi−Ci+1〈Ncoll〉Ci−Ci+1
dNdpT
˛
˛
˛
sample
Ci−Ci+1ˆ
P
i fCi−Ci+1〈Ncoll〉Ci−Ci+1
˜
0 2 4 6 8
pT (GeV/c)
0
0.2
0.4
0.6
0.8
1
RA
A
b=8.44 fmΣ
i(CC)
i
0 2 4 6 8 10 12 14
pT (GeV/c)
b=8.77 fmΣ
i(CC)
i
(a) (b)
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Heavy-quark RAA: AdS/CFT scenario
0 2 4 6 8 10p
T (GeV/c)
0
0.5
1
1.5
2
2.5
3
RA
A
Qc (AdS/CFT λ=10)b (AdS/CFT λ=10)c+b (AdS/CFT λ=10)e
c+e
b (PHENIX min. bias)
PHENIX min. bias
• Plot obtained setting λ=10 in the transport coefficients
κT =√λπT 3γ1/2 κL =
√λπT 3γ5/2
• A more systematic study could be of interest.
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Elliptic flow of heavy-flavor electrons:RHIC (left panel) vs LHC (right panel)
0 1 2 3 4
pT (GeV/c)
0
0.05
0.1
0.15
v 2
PHENIX dataecebec+b
ec+b
, τ0=0.1 fm
0 2 4 6
pT (GeV/c)
ecebec+becebec+b
(a)
(b)
}
}
τ0=1 fm
τ0=0.1 fm
}τ0=1 fm
• v2 with hot-QCD + fragmentation results a bit underestimated;
• Actually very good agreement with τ0 =0.1 fm;
• v2 could be increased by coalescence;
• Much larger flow of ec @ LCH. Milder effect on ec+eb due to role of b.
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Conclusions and perspectives
• The relativistic Langevin equation is a powerful tool to study the
HQ dynamics in the the QGP: it is an effective theory completely
determined by the coefficients κT/L(p) (no matter their
microscopic origin! )
• κT/L(p) have been evaluated considering only 2 → 2 collisions
and distinguishing soft and hard scatterings
• For large pT (pT>∼3 GeV/c) it is possible to accommodate RHIC
data for the single-electron spectra
• Coalescence could further improve the description at lower pT ,
raising RAA and v2
• Preliminary simulations for LHC were attempted. In order to
provide predictions at the current√sNN =2.76 GeV we need a
reliable hydrodynamical scenario....
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Back-up slides
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The easiest algorithm
Going to the fluid rest-frame:
∆pin =−ηD(pn)pi
n∆t+ ξi(tn)∆t ≡ −ηD(pn)pin∆t+ gij(pn)ζi(tn)
√∆t,
∆xn = pn/En∆t
with ∆t=0.02 fm/c (in the fluid rest-frame! ) and
gij(p)≡√
κ‖(p)pipj +
√
κ⊥(p)(δij − pipj) and 〈ζinζ
jn′〉=δijδnn′
Hence one needs simply to:
• extract three independent random numbers ζi from a gaussian
distribution with σ=1;
• update the momentum and position of the heavy quark;
• go back to the Lab-frame: xn+1 and pn+1.
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κ⊥/‖(p): comparison with previous results
0 2 4 6 8 10p (GeV)
0
1
2
κ (G
eV2 /f
m)
κ T
(c) HTL+pQCD ( |t|*=m
D
2 )
κ L
(c) HTL+pQCD ( |t|*=m
D
2 )
κ T
(b) HTL+pQCD ( |t|*=m
D
2 )
κ L
(b) HTL+pQCD ( |t|*=m
D
2 )
κ T
(c) HTLκ
L(c) HTL
κ T
(b) HTLκ
L(b) HTL
T=400 MeV, µ =1.5πT
0 1 2 3 4p (GeV)
0
5
10
15
20
25
κ(p)
/κ(0
)
κT HTL+pQCD (µ =1.5π)
κL HTL+pQCD (µ =1.5π)
κT HTL (µ =1.5π)
κL HTL (µ =1.5π)
κT (Teaney)
κL (Teaney)
κT (Rapp)
κL (Rapp)
κT (Gossiaux)
κL (Gossiaux)
T=200 MeV
• HTL: Hard Thermal Loop approximation applied to any collision;
• Teaney: ∼ shape given by pQCD with κ(0) as a free parameter;
• Rapp: resonant scattering with formation of D-like mesons;
• Gossiaux: ∼ pQCD with mD used as a free parameter.
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Initial heavy-quark spectra: effects of nPDFs
0.0001 0.001 0.01 0.1 1x
0
0.2
0.4
0.6
0.8
1
1.2
1.4Q=1.3 GeVQ=8.6 GeV
gA
(x,Q2)/Ag
proton(x,Q
2)
The dominant process is gg → QQ in which
x1/2 =mQQ√sNN
e±yQQ
→ role of nPDFs different for charm and beauty!
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Glauber and k⊥ broadening
Each HQ is given a k⊥-kick extracted from a gaussian distribution with
〈k2⊥〉AB(~b,~s) = 〈k2
⊥〉pp +agN
2
»
R
dzA ρA(~s, zA)lA(~s, zA)
TA(~s)
+
R
dzB ρB(~s−~b, zB)lB(~s−~b, zB)
TB(~s−~b)
#
due to the length crossed by the incoming partons in nucleus A/B before
the hard event:
lA(~s, zA) ≡Z zA
−∞
dz ρA(~s, z)/ρ0 and lB(~s−~b, zb) ≡Z +∞
zB
dz ρB(~s−~b, z)/ρ0
We choose
agN (GeV2/fm) SPS RHIC LHC
c 0.072 0.092 0.153
b 0.197 0.252 0.420
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Single-electron spectra: procedure
• As an outcome of the initial generation (+ Langevin evolution in AA)
one has two samples (90 · 106) of c and b quarks;
• They are made fragment with Peterson FFs with the branching
fractions given by the PDG;
• Each hadron is sent to the PYTHIA decayer (with updated PDG
decay tables) till producing at least one electron in the final state (in
case of no final electron the event is re-sampled);
• The two sources must be combined with their appropriate weight:
dN
dpT
˛
˛
˛
˛
ec+eb
∼ σcc
X
hc
N inithc
N samplhc
dN
dpT
˛
˛
˛
˛
˛
ec
+ σbb
X
hb
N inithb
N samplhb
dN
dpT
˛
˛
˛
˛
˛
˛
eb
NB We plot the electrons falling into the PHENIX acceptance (|η|<0.35),
where η ≡ 12
ln p+pz
p−pz= − ln tan θ
2
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Centrality classes
fC1−C2 =
R b2b1db b[1 − exp(σNNTAB(b)]
R ∞
0db b[1 − exp(σNNTAB(b)]
,
where TAB(b) =R
dsTA(s + b/2)TB(s − b/2).
Au-Au (√s = 200 GeV) Pb-Pb (
√s = 5.5 TeV)
C1-C2 b (fm) Ncoll C1-C2 b (fm) Ncoll
0-10% 3.27 963 0-10% 3.45 1698
10-20% 5.78 592 10-20% 6.11 1022
20-40% 8.12 282 20-40% 8.58 469
40-60% 10.51 81 40-60% 11.11 123
60-92% 12.80 10 60-90% 13.45 14
0-92% 8.44 247 0-90% 8.77 435
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Which role could be played by coalescence?
Fragm. (left panels) vs Coalescence + Fragm.a (right panels)
0 1 2 3 4 5p
T [GeV]
0
0.5
1
1.5
2
RA
A
c+b resoc+b pQCDc resoPHENIX prel.STAR QM05 (10-40% cent.)
0 1 2 3 4 5p
T [GeV]
0
5
10
15
20
25
-5
v 2 [%
]
c+b resoc+b pQCDc reso
PHENIXSTAR prel.PHENIX QM05
0 1 2 3 4 5p
T [GeV]
0
0.5
1
1.5
2
RA
A
c+b resoc+b pQCDc resoPHENIX prel.STAR QM05 (10-40% cent.)
0 1 2 3 4 5p
T [GeV]
0
5
10
15
20
25
-5
v 2 [%
]
c+b resoc+b pQCDc reso
PHENIXSTAR prel.PHENIX QM05
aH. van Hees, V. Greco and R. Rapp, PRC 73, 034913 (2006)
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Effects of fragmentation
(GeV/c)T
p0 2 4 6 8 10 12 14
AA
R
0
0.2
0.4
0.6
0.8
1
1.2Fragmentation Function:
Delta
=0.040εPeterson
=0.050εPeterson
=0.030εPeterson
=0.020εPeterson
charm
b=8.44 fm
π=2µ
(GeV/c)T
p0 2 4 6 8 10 12 14
AA
R
0.4
0.6
0.8
1
1.2
1.4
Fragmentation Function:
Delta
=0.005εPeterson
=0.008εPeterson
=0.003εPeterson
=0.002εPeterson
bottom
b=8.44 fm
π=2µ
Fragmentation performed with Peterson FF tends to slightly suppress RAA
• Mild dependence on the parameter ǫ
• ǫ=0.04 and 0.005 (for c and b) fixed in order to reproduce HQET FFsa
Fragmentation fractions taken from DESY results and PDG 2009
aE. Braaten, K. Cheung and T.C. Yuan, Phys. Rev. D 48, 5049 (1993)