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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)