Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Heavy-ion collisions: theory review
Andrea Beraudo
CERN, Theory Unit
“QCD at Cosmic Energies”, Paris, 11-15 June 2012
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Outline
The motivation: exploring the QCD phase diagram
Virtual experiment: lattice-QCD simulations
Real experiments: heavy-ion collisions
Soft observables;
Hard probes
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Heavy-ion collisions: exploring the QCD phase-diagram
Critical line (cross-over + C.E.P. + 1st -order)
from lQCD and effective lagrangians (NJL,
linear sigma model..)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
T (MeV)
Heavy-ion collisions: exploring the QCD phase-diagram
quark gluon plasma
Critical line (cross-over + C.E.P. + 1st -order)
from lQCD and effective lagrangians (NJL,
linear sigma model..)
200
E
chemical freeze-out
SIS, AGS
SPS (NA49)
RHIC
100
hadrons
M
0
500
Experimental points from fit of final hadron
multiplicities
color
superconductor
1000
µ (MeV)
B
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Heavy-ion collisions: exploring the QCD phase-diagram
Critical line (cross-over + C.E.P. + 1st -order)
from lQCD and effective lagrangians (NJL,
linear sigma model..)
Experimental points from fit of final hadron
multiplicities
Region explored at LHC: high-T/low-density (early universe, nB /nγ ∼ 10−9 )
From QGP (color deconfinement, chiral symmetry restored)
to hadronic phase (confined, chiral symmetry breaking)
NB hqqi =
6 0 responsible for most of the baryonic mass of the universe: only
∼35 MeV of the proton mass from mu/d 6= 0
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Virtual experiments: lattice-QCD simulations
The best (unique?) tool to study QCD in the
non-perturbative regime
Limited to the study of equilibrium quantities
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
QCD on the lattice
The QCD partition function
Z
Y
Z = [dU] exp [−βSg (U)]
det [M(U, mq )]
q
is evaluated on the lattice through a MC sampling of the field
configurations, where
β = 6/g 2
Sg is the gauge action, weighting the different field configurations;
U ∈ SU(3) is the link variable connecting two lattice sites;
M is the Dirac operator
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
QCD on the lattice: results
From the partition function on gets all the thermodynamical quantities1 :
Pressure: P = (T /V ) ln Z;
1
Wuppertal group, JHEP 1011 (2010) 077
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
QCD on the lattice: results
From the partition function on gets all the thermodynamical quantities1 :
Pressure: P = (T /V ) ln Z;
Trace anomaly:
I ≡ ǫ − 3P = T 5 (∂/∂T )(P/T 4 );
1
Wuppertal group, JHEP 1011 (2010) 077
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
QCD on the lattice: results
From the partition function on gets all the thermodynamical quantities1 :
Pressure: P = (T /V ) ln Z;
Trace anomaly:
I ≡ ǫ − 3P = T 5 (∂/∂T )(P/T 4 );
Energy density: ǫ = I + 3P;
1
Wuppertal group, JHEP 1011 (2010) 077
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
QCD on the lattice: results
From the partition function on gets all the thermodynamical quantities1 :
Pressure: P = (T /V ) ln Z;
Trace anomaly:
I ≡ ǫ − 3P = T 5 (∂/∂T )(P/T 4 );
Energy density: ǫ = I + 3P;
Entropy density:
s = (ǫ + P)/T ;
1
Wuppertal group, JHEP 1011 (2010) 077
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
QCD on the lattice: results
From the partition function on gets all the thermodynamical quantities1 :
Pressure: P = (T /V ) ln Z;
Trace anomaly:
I ≡ ǫ − 3P = T 5 (∂/∂T )(P/T 4 );
Energy density: ǫ = I + 3P;
Entropy density:
s = (ǫ + P)/T ;
Speed of sound: cs2 = dP/dǫ
1
Wuppertal group, JHEP 1011 (2010) 077
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
lattice-QCD results: some comments
One observes a ∼ 20% deviation
from the SB limit even at large T:
how to interpret it?
T µν ≡ diag(ǫ, −P, −P, −P):
the trace anomaly I ≡ ǫ − 3P gives
a measure of the breaking of
conformal invariance (a challenge
for approaches based on AdS/CFT
correspondence?)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Real experiments: heavy-ion collisions
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Heavy-ion collisions: a typical event
Valence quarks of participant nucleons act as sources of strong color
fields giving rise to particle production
Spectator nucleons don’t participate to the collision;
Almost all the energy and baryon number carried away by the remnants
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Heavy-ion collisions: a typical event
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Heavy-ion collisions: a cartoon of space-time evolution
Soft probes (low-pT hadrons): collective behavior of the medium;
Hard probes (high-pT particles, heavy quarks, quarkonia): produced
in hard pQCD processes in the initial stage, allow to perform a
tomography of the medium
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Soft probes and hydrodynamics
Some references...
J.Y. Ollitrault, “Phenomenology of the little bang”,
J.Phys.Conf.Ser. 312 (2011) 012002;
J.Y. Ollitrault, “Relativistic hydrodynamics for heavy-ion collisions”,
Eur.J.Phys. 29 (2008) 275-302
U.W. Heinz, “Hydrodynamic description of ultrarelativistic heavy
ion collisions”,
in *Hwa, R.C. (ed.) et al.: Quark gluon plasma* 634-714
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hydrodynamics and heavy-ion collisions
The success of hydrodynamics in describing particle spectra in heavy-ion
collisions measured at RHIC came as a surprise!
The general setup and its implications
Predictions
Radial flow
Elliptic flow
What can we learn?
Initial conditions
Event-by-event fluctuations and consequences
QCD EOS
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hydrodynamics: the general setup
Hydrodynamics is applicable in a situation in which λmfp ≪ L
In this limit the behavior of the system is entirely governed by the
conservation laws
∂µ T µν = 0 ,
| {z }
four−momentum
where
∂µ jBµ = 0 ,
| {z }
baryon number
T µν = (ǫ + P)u µ u ν − Pg µν
and
jBµ = nB u µ
Information on the medium is entirely encoded into the EOS
P = P(ǫ)
The transition from fluid to particles occurs at the freeze-out
hypersuface Σfo (e.g. at T = Tfo )
Z
E (dN/d~p ) =
p µ dΣµ exp[−(p · u)/T ]
Σfo
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
STAR preliminary
Au+Au central
√sNN = 200 GeV
102
π-
1/(2π) d2N / (mT dmT dy) [c4/GeV2]
1/(2π) d2N / (mT dmT dy) [c4/GeV2]
Hydro predictions: radial flow (I)
STAR preliminary
p+p min. bias
√sNN = 200 GeV
1
π-
10-1
K
10
p
1
0
0.1
0.2
0.3
0.4
0.5
K
-
10-2
0.6
0.7
mT - m0 [GeV/c2]
p
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
mT - m0 [GeV/c2]
√ 2 2
dN
∼ e −mT /Tslope ≡ e − pT +m /Tslope
mT dmT
Tslope (∼ 167 MeV) universal in pp collisions;
Tslope growing with m in AA collisions: spectrum gets harder!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hydro predictions: radial flow (II)
Physical interpretation:
Thermal emission on top of a collective flow
0.5
SPS Pb+Pb: NA49 preliminary
AGS Au+Au: E866
d
mT inverse slope (GeV)
SPS S+S: NA44
1
mhv2⊥ i
2
0.4
φ
Λ
Ξ
−
p
0.3
K
=
±
−
Λ
−
h
π
=⇒
0
E
1 D
2
m (v⊥th + v⊥flow )
2
1
1
mhv2⊥th i + mv2⊥flow
2
2
+
0.2
0.1
=
0.5
1
1.5
1
Tslope = Tfo + mv2⊥flow
2
2
2
Particle mass (GeV/c )
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hydro predictions: elliptic flow
In non-central collisions particle emission
is not azimuthally-symmetric!
y
φ
x
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hydro predictions: elliptic flow
In non-central collisions particle emission
is not azimuthally-symmetric!
0.4
0.35
0.3
0.25
Pb-Pb events at sNN = 2.76 TeV
The effect can be quantified through the
Fourier coefficient v2
Centrality 20-40%
π
K
p
Ξ
Ω
dN N0
=
(1 + 2v2 cos[2(φ − ψRP )] + . . . )
dφ 2π
v2 ≡ hcos[2(φ − ψRP )]i
v2
0.2
0.15
VISH2+1
(MCKLN, η/s=0.20)
π
K
p
Ξ
ALICE
Ω
0.1
0.05
0
0
1
2
3
4
preliminary
5
6
p [GeV/c]
T
Andrea Beraudo
v2 (pT ) ∼ 0.2 gives a modulation 1.4 vs
0.6 for in-plane vs out-of-plane particle
emission!
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Elliptic flow: physical interpretation
y
φ
x
Matter behaves like a fluid whose expansion is driven by pressure
gradients
¤
∂ £
∂P
(ǫ + P)v i = − i ;
∂t
∂x
Spatial anisotropy is converted into momentum anisotropy;
At freeze-out particles are mostly emitted along the reaction-plane.
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Elliptic flow: mass ordering
The mass ordering of v2 is a direct consequence of the hydro expansion
0.4
0.35
0.3
0.25
Particles emitted according to a
thermal distribution
∼ exp[−p·u(x)/Tfo ] in the local
rest-frame of the fluid-cell;
Pb-Pb events at sNN = 2.76 TeV
Centrality 20-40%
π
K
p
Ξ
Ω
Parametrizing the fluid velocity as
u µ ≡ γ⊥ (cosh Y , u⊥ , sinh Y ),
v2
0.2
0.15
VISH2+1
(MCKLN, η/s=0.20)
π
K
p
Ξ
ALICE
Ω
0.1
0.05
0
0
1
2
3
4
one gets (vz ≡ tanh Y )
preliminary
5
p [GeV/c]
T
6
p·u = γ⊥ [m⊥ cosh(y −Y ) − p⊥ ·u⊥ ]
Dependence on mT at the basis of
mass ordering at fixed pT
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: “Bjorken” estimate
It is useful to describe the evolution in term of the variables
p
1 t +z
τ ≡ t 2 − z 2 and ηs ≡ ln
2 t −z
Assuming a boost-invariant purely longitudinal expansion (vz = z/t)
entropy conservation implies:
s τ = s0 τ 0
−→
s0 = (s τ )/τ0
Entropy density is defined in the local fluid rest-frame:
¯
dS ¯¯
1 dS
s≡
=
dx⊥ dz ¯z=0
τ dx⊥ dηs
Entropy is related to the final multiplicity of charged particles
(S ∼ 3.6 N for pions), so that:
s0 =
Andrea Beraudo
1 3.6 dNch 3
τ0 πRA2 dη 2
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
“Bjorken” estimate: results
s0 =
1 3.6 dNch 3
τ0 πRA2 dη 2
From dNch /dη ≈ 1600 measured by
ALICE at LHC and RPb ≈ 6 fm one gets:
s0 ≈ (80 fm−2 )/τ0
τ0 is found to be quite small:
0.1 < τ0 < 1 fm −→ 80 < s0 < 800 fm−3
This should be compared with l-QCD
s(T = 200 MeV) ≈ 10 fm−3
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: Glauber model
Within the Glauber model, given the nuclear thickness function
Z +∞
dz ρA (x, z)
TA (x) ≡
−∞
one can express the initial entropy density in terms of the local
density of
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: Glauber model
Within the Glauber model, given the nuclear thickness function
Z +∞
dz ρA (x, z)
TA (x) ≡
−∞
one can express the initial entropy density in terms of the local
density of
A
B
participants: s(τ0 , x; b) = K (τ0 ) [npart
(x; b) + npart
(x; b)], with
h
¢B i
¡
in
A
TB (x − b/2)/B
npart
(x; b) = TA (x + b/2) 1 − 1 − σpp
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: Glauber model
Within the Glauber model, given the nuclear thickness function
Z +∞
dz ρA (x, z)
TA (x) ≡
−∞
one can express the initial entropy density in terms of the local
density of
A
B
participants: s(τ0 , x; b) = K (τ0 ) [npart
(x; b) + npart
(x; b)], with
h
¢B i
¡
in
A
TB (x − b/2)/B
npart
(x; b) = TA (x + b/2) 1 − 1 − σpp
binary collisions: s(τ0 , x; b) = K ′ (τ0 ) nbin (x; b), where
in
nbin (x; b) = σpp
TA (x + b/2) TB (x − b/2)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: Glauber model
Within the Glauber model, given the nuclear thickness function
Z +∞
dz ρA (x, z)
TA (x) ≡
−∞
one can express the initial entropy density in terms of the local
density of
A
B
participants: s(τ0 , x; b) = K (τ0 ) [npart
(x; b) + npart
(x; b)], with
h
¢B i
¡
in
A
TB (x − b/2)/B
npart
(x; b) = TA (x + b/2) 1 − 1 − σpp
binary collisions: s(τ0 , x; b) = K ′ (τ0 ) nbin (x; b), where
in
nbin (x; b) = σpp
TA (x + b/2) TB (x − b/2)
√
in
( s)
An essential input is the inelastic pp cross section σpp
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
(mb)
140
p̄p (PDG)
ALICE
σtot , σinel , and σel
Glauber model and heavy-ion collisions
120
pp (PDG)
TOTEM
110
Auger + Glauber
100
ATLAS
130
in
≈ 40 − 60 mb at RHIC-LHC
σpp
energies;
σtot (red), σinel (blue) and σel (green)
90
best COMPETE σtot fits
11.4 − 1.52 ln s + 0.130 ln2 s
CMS
80
σtot
70
60
50
σinel
40
30
σel
20
10
0
101
102
103
104
√
s
105
(GeV)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
(mb)
140
p̄p (PDG)
ALICE
σtot , σinel , and σel
Glauber model and heavy-ion collisions
120
pp (PDG)
TOTEM
110
Auger + Glauber
100
ATLAS
130
σtot (red), σinel (blue) and σel (green)
90
best COMPETE σtot fits
11.4 − 1.52 ln s + 0.130 ln2 s
CMS
80
σtot
70
60
50
in
σpp
≈ 40 − 60 mb at RHIC-LHC
energies;
σinel
40
30
σel
20
10
0
101
102
103
104
√
s
105
(GeV)
The Glauber model seems to work
pretty well: nuclear modification factor
RAA (pT ) ≡
(dN/dpT )AA
hNcoll i(dN/dpT )pp
close to 1 for color-neutral probes!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: event-by-event fluctuations
Flow coefficients are defined as vn ≡ hhcos[n(φ − Ψn )]ii.
For hydro simulations with smooth initial conditions
Ψn ≡ ΨRP known exactly;
all odd-harmonics vanish.
Real life is more complicated...
Odd harmonics appear, angles Ψn are not directly measured.
Glauber-MC initial conditions mandatory to study these effects
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Event-by-event fluctuations: experimental consequences
Fluctuating initial conditions giving rise toa :
Non-vanishin v2 in central collisions;
Odd harmonics (v3 and v5 )
a
Andrea Beraudo
ALICE, Phys.Rev.Lett. 107 (2011) 032301
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Initial conditions: Color Glass Condensate
Basic idea:
s0 related to the rapidity density of produced gluons
Spectrum of produced gluons evaluated within kT -factorization:
Z
Z
dp⊥
dNg
∼
s0 ∼
dk⊥ αs φA (x1 , k2⊥ ) φB (x2 , (p⊥ −k⊥ )2 )
dr⊥ dy
p2⊥
where φ(x, k2⊥ ) is an unintegrated gluon distribution
It can be expressed through the dipole scattering amplitude
N (x, r⊥ )
The small-x evolution of the latter is described by the BK-equation
∂N ∼ |{z}
N −
BFKL
N2
|{z}
saturation
A unique setup able to describe data from DIS up to A-A collisions?
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
CGC and particle production
/2
6
MCrcBK 200GeV
MCrcBK 2.76TeV
ALICE 2.76TeV
PHOBOS 200GeV
part
8
dN/dη/N
10
4
2
0
100
200
Npart
300
400
Particle density and its evolution with centrality nicely accomodated2
2
J.L. Albacete, A. Dumitru and Y. Nara, J.Phys.Conf.Ser. 316 (2011)
012011.
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hydro evolution: the role of the Equation of State
In ideal hydro the dependence on the
EOS enters through speed of sound:
∂ ln s
∂v i
1 ∂P
= −cs2
;
=−
i
∂t
ǫ + P ∂x
∂x i
For the transverse expansion one gets:
vx =
cs2 x
t,
σx2
vy =
cs2 y
t
σy2
The larger the speed of sound, the larger
the radial flow!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hard probes
A few experimental results
Jet-quenching
Heavy-flavor
The physical interpretation (with some novel ideas)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Jet-quenching
(in a broad sense: jet-reconstruction in AA possible only recently)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Inclusive hadron spectra: the nuclear modification factor
RAA
PHENIX Au+Au (central collisions):
Direct γ
π0
η
g
GLV parton energy loss (dN /dy = 1100)
10
RAA
1
¡
¢AA
dN h /dpT
≡
pp
hNcoll i (dN h /dpT )
10-1
0
2
4
6
8
10
12
14
16
pT (GeV/ c )
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
RAA
Inclusive hadron spectra: the nuclear modification factor
ALICE, charged particles, Pb-Pb
s NN = 2.76 TeV, | η | < 0.8
1
RAA
¡
¢AA
dN h /dpT
≡
pp
hNcoll i (dN h /dpT )
0-5%
20-40%
0.1
0
ALICE Preliminary
40-80%
10
20
30
40
50
p (GeV/c)
T
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Inclusive hadron spectra: the nuclear modification factor
RAA
Andrea Beraudo
¡
¢AA
dN h /dpT
≡
pp
hNcoll i (dN h /dpT )
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Inclusive hadron spectra: the nuclear modification factor
RAA
¡
¢AA
dN h /dpT
≡
pp
hNcoll i (dN h /dpT )
Hard-photon RAA ≈ 1
supports the Glauber picture (binary-collision scaling);
entails that quenching of inclusive hadron spectra is a final state
effect due to in-medium energy loss.
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Some CAVEAT:
At variance wrt e + e − collisions, in hadronic collisions one starts
with a parton pT -distribution (∼ 1/pTα ) so that inclusive hadron
spectrum simply reflects higher moments of FF
Z
1 X 1
dN h
∼ α
dz z α−1 D f →h (z)
dpT
pT
0
f
carrying limited information on FF (but very sensitive to hard tail!)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Some CAVEAT:
At variance wrt e + e − collisions, in hadronic collisions one starts
with a parton pT -distribution (∼ 1/pTα ) so that inclusive hadron
spectrum simply reflects higher moments of FF
Z
1 X 1
dN h
∼ α
dz z α−1 D f →h (z)
dpT
pT
0
f
carrying limited information on FF (but very sensitive to hard tail!)
Surface bias:
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Some CAVEAT:
At variance wrt e + e − collisions, in hadronic collisions one starts
with a parton pT -distribution (∼ 1/pTα ) so that inclusive hadron
spectrum simply reflects higher moments of FF
Z
1 X 1
dN h
∼ α
dz z α−1 D f →h (z)
dpT
pT
0
f
carrying limited information on FF (but very sensitive to hard tail!)
Surface bias:
leading hadron
hadronization
hard process
QGP
Quenched spectrum does not reflect hLQGP i
crossed by partons distributed in the transverse
plane according to ncoll (x) scaling, but due to
its steeply falling shape is biased by the
enhanced contribution of the ones produced
close to the surface and losing a small amount
of energy!
hadronization
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Di-jet imbalance at LHC: looking at the event display
An important fraction of events display a huge mismatch in ET
between the leading jet and its away-side partner
Possible to observe event-by-event, without any analysis!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Di-jet imbalance at LHC: looking at the event display
An important fraction of events display a huge mismatch in ET
between the leading jet and its away-side partner
Possible to observe event-by-event, without any analysis!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
2
1
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
0.6
10-1
10
evt
1
10-1
10-2
2.5
3
J
evt
0.4
0.6
0.8
10
1
2.5
3
∆φ
Dijet asymmetry Aj ≡
with centrality;
ET1 −ET2
ET1 +ET2
0.2
0.4
0.6
0.8
1
AJ
10
Pb+Pb Data
p+p Data
1
HIJING+PYTHIA
10-1
10-2
2
∆φ
0
0
1
10-1
10-2
2
0.2
AJ
(1/N ) dN/d∆φ
evt
1
0
0
1
1
AJ
(1/N ) dN/d∆φ
evt
10
0.8
sNN=2.76 TeV 0-10%
ATLAS
Pb+Pb
Lint=1.7 µb-1
2
1
AJ
(1/N ) dN/d∆φ
2
1
4
3
evt
0
0
10-20%
3
(1/N ) dN/d∆φ
2
4
(1/N ) dN/dA
J
20-40%
3
evt
evt
4
evt
40-100%
3
(1/N ) dN/dA
J
4
(1/N ) dN/dA
(1/N ) dN/dA
J
Dijet correlations: results
10-2
2
2.5
3
2
∆φ
2.5
3
∆φ
enhanced wrt to p+p and increasing
∆φ distribution unchanged wrt p+p (jet pairs ∼ back-to-back)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Dijet correlations: adding tracking information
Tracks in a ring of radius ∆R ≡
subleading-jet axis:
p
∆φ2 +∆η 2 and width 0.08 around the
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Dijet correlations: adding tracking information
(a)
p
∆φ2 +∆η 2 and width 0.08 around the
(d)
(c)
(b)
2
2
2
p > 120GeV/c
xxx > 8 GeV/c
PYTHIA
0.22 < A J < 0.33
AJ > 0.33
AJ < 0.11
0.11 < AJ < 0.22 xxx
p > 50GeV/c
xxx
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+HYDJET
xxx 4-8 GeV/c
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xxx
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3
xxx 1-4 GeV/c
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0-30%
xxx
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10 xxxxxxxxxxxxxxxxxxxxxx
Leading Jet
Subleading10
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Subleading Jet
Leading Jet
Subleading 10
Jetxxxxxxxxxxxxxxxxxxxxxxx
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0.5
0
0.5
0.5
0
0.5
0.5
0
0.5
102
T,1
T,2
1,2
102
T
Σp per bin (GeV/c)
T
Σp per bin (GeV/c)
Tracks in a ring of radius ∆R ≡
subleading-jet axis:
(e)
2
(f)
2
(g)
2
(h)
CMS
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A J < 0.11
0.11 < AJ < 0.22
0.22 < A J < 0.33
A J > 0.33
PbPb s =2.76 TeV
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∫L dt = 6.7 µb
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10 xxxxxxxxxxxxxxxxxxxxxx
10
10
10
Leading Jet
Subleading Jet
Leading Jet
Subleading Jet
Leading Jet
Subleading Jet
Leading Jet
Subleading Jet
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NN
-1
0.5
∆R
Leading jet
0
∆R
0.5
0.5
Subleading jet
Leading jet
∆R
0
∆R
0.5
Subleading jet
0.5
Leading jet
∆R
0
∆R
0.5
Subleading jet
0.5
∆R
Leading jet
0
∆R
0.5
Subleading jet
Increasing AJ a sizable fraction of energy around subleading jet carried by
soft (pT < 4 GeV) tracks with a broad angular distribution
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Dijet measurements: Fragmentation Fuctions
´
³
ξ ≡ − ln z ≡ − ln pTtrack /pTjet ,
Andrea Beraudo
pTtrack > 4GeV
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Dijet measurements: Fragmentation Fuctions
´
³
ξ ≡ − ln z ≡ − ln pTtrack /pTjet ,
Andrea Beraudo
pTtrack > 4GeV
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Dijet measurements: Fragmentation Fuctions
´
³
ξ ≡ − ln z ≡ − ln pTtrack /pTjet ,
Andrea Beraudo
pTtrack > 4GeV
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Dijet measurements: Fragmentation Fuctions
´
³
ξ ≡ − ln z ≡ − ln pTtrack /pTjet ,
pTtrack > 4GeV
Hard component of jet-FF in AA not strongly modified wrt to pp.
Data (for hard tracks!) compatible with vacuum-like fragmentation of
jets with reduced energy
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Physical interpretation of the data: energy-loss at the parton level!
xE
E (≈ pT )
(1 − x)E
hard process
Interaction of the high-pT parton with the color field of the medium
induces the radiation of (mostly) soft (ω ≪ E ) and collinear
(k⊥ ≪ ω) gluons;
Radiated gluon can further re-scatter in the medium (cumulated q⊥
favor decoherence from the projectile).
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
The basic ingredients
Vacuum-radiation spectrum;
(Gunion-Bertsch) induced spectrum
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Vacuum radiation by off-shell partons
£
¤
A hard parton with pi ≡ p+ , Q 2 /2p+ , 0 loses its virtuality Q through
√
gluon-radiation. In light-cone coordinates, with p± ≡ E ± pz / 2:
#
"
k2
,k
kg ≡ xp+ ,
2xp+
a
~k⊥
"
#
xP +
k2
pf = (1−x)p+ ,
, −k
P+
(1 − x)P +
2(1−x)p+
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Vacuum radiation by off-shell partons
£
¤
A hard parton with pi ≡ p+ , Q 2 /2p+ , 0 loses its virtuality Q through
√
gluon-radiation. In light-cone coordinates, with p± ≡ E ± pz / 2:
#
"
k2
,k
kg ≡ xp+ ,
2xp+
a
~k⊥
"
#
xP +
k2
pf = (1−x)p+ ,
, −k
P+
(1 − x)P +
2(1−x)p+
k⊥ vs virtuality: k2 = x (1−x) Q 2 ;
Radiation spectrum (our benchmark): IR and collinear divergent!
rad
dσvac
= dσ hard
dk + dk
αs
C
R
π2
k + k2
Time-scale (formation time) for gluon radiation:
∆trad ∼ Q −1 (E /Q) ∼ 2ω/k2
Andrea Beraudo
(x ≈ ω/E )
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Medium-induced radiation by on-shell partons
On-shell partons propagating in a color field can radiated gluons.
a
a1
a
a1
a
a1
(a)
(b)
Andrea Beraudo
Heavy-ion collisions: theory review
(c)
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Medium-induced radiation by on-shell partons
On-shell partons propagating in a color field can radiated gluons.
a
a1
a
a1
a
a1
(a)
(b)
(c)
The single-inclusive gluon spectrum: the Gunion-Bertsch result
µ ¶D
µ ¶¿
À
E
dNgGB
αs
q2
L
L
αs
2
[K0 −K1 ] = CR 2
= CR 2
x
dxdk
π
λel
π
λel
k2 (k−q)2
g
g
where CR is the color charge of the hard parton and:
Z
k
k−q
1 dσ el
K0 ≡ 2 ,
K1 ≡
and
h.
.
.
i
≡
dq el
2
(k−q)
σ dq
k
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
The induced spectrum: physical interpretation
αs
dσ ind
= dσ hard CR 2
ω
dωdk
π
µ
L
λel
g
¶¿
¶À
µ
¤
£
sin(ω1 L)
2
2
2
(K0 − K1 ) + K1 − K0 1−
ω1 L
In the above ω1 ≡ (k−q)2 /2ω and two regimes can be distinguished:
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
The induced spectrum: physical interpretation
αs
dσ ind
= dσ hard CR 2
ω
dωdk
π
µ
L
λel
g
¶¿
¶À
µ
¤
£
sin(ω1 L)
2
2
2
(K0 − K1 ) + K1 − K0 1−
ω1 L
In the above ω1 ≡ (k−q)2 /2ω and two regimes can be distinguished:
Coherent regime LPM (ω1 L ≪ 1): dσ ind = 0 −→ dσ rad = dσ vac
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
The induced spectrum: physical interpretation
αs
dσ ind
= dσ hard CR 2
ω
dωdk
π
µ
L
λel
g
¶¿
¶À
µ
¤
£
sin(ω1 L)
2
2
2
(K0 − K1 ) + K1 − K0 1−
ω1 L
In the above ω1 ≡ (k−q)2 /2ω and two regimes can be distinguished:
Coherent regime LPM (ω1 L ≪ 1): dσ ind = 0 −→ dσ rad = dσ vac
®
­
Incoherent regime (ω1 L ≫ 1): dσ ind ∼ (K0 −K1 )2 +K21 −K20
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
The induced spectrum: physical interpretation
αs
dσ ind
= dσ hard CR 2
ω
dωdk
π
µ
L
λel
g
¶¿
¶À
µ
¤
£
sin(ω1 L)
2
2
2
(K0 − K1 ) + K1 − K0 1−
ω1 L
In the above ω1 ≡ (k−q)2 /2ω and two regimes can be distinguished:
Coherent regime LPM (ω1 L ≪ 1): dσ ind = 0 −→ dσ rad = dσ vac
®
­
Incoherent regime (ω1 L ≫ 1): dσ ind ∼ (K0 −K1 )2 +K21 −K20
The full radiation spectrum can be organized as
vac
vac
dσ rad = dσ GB + dσgain
+ dσloss
where
¢­
®
αs ¡
(K0 − K1 )2 (dωdk/ω)
L/λel
g
π2
¢ 2
αs ¡
vac
dσgain
= dσ hard CR 2 L/λel
g hK1 i (dωdk/ω)
π
¢
¡
αs
vac
el
dσloss = 1 − L/λg dσ hard CR 2 K20 (dωdk/ω)
π
dσ GB = dσ hard CR
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Average energy loss
Integrating the lost energy ω over the inclusive gluon spectrum:
h∆E i =
Z
dω
Z
dNgind
CR αs
∼
dk ω
dωdk
4
µ
µ2D
λel
g
¶
L2 ln
E
µD
L2 dependence on the medium-length;
µD : Debye screening mass of color interaction ∼ typical momentum
exchanged in a collision;
µ2D /λel
g often replaced by the transport coefficient q̂, so that
h∆E i ∼ αs q̂L2
2
q̂: average q⊥
acquired per unit length
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Numerical results
Medium-induced radiation: angular distribution
Medium-induced radiation: energy distribution
1.5
6
N=1 opacity
N=1 opacity
5
1
E=10 GeV, L=5 fm, mD=0.46 GeV, λg=1.26 fm, αs=0.3
dN/dθ
dN/dx
4
3
0.5
E=10 GeV, L=5 fm, mD=0.46 GeV, λg=1.26 fm, αs=0.3
2
0
1
0
-0.5
0
0.2
0.4
x
0.6
0.8
1
0
0.5
θ (rad)
1
1.5
At variance with vacuum-radiation, medium induced spectrum
Infrared safe (vanishing as ω → 0);
Collinear safe (vanishing as θ → 0).
Depletion of gluon spectrum at small angles due to their
rescattering in the medium!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Medium-modification of color-flow for
high-pT probes
I will mainly focus on leading-hadron spectra...
...but the effects may be relevant for more differential
observables (e.g. jet-fragmentation pattern)
Essential ideas presented here in a N = 1 opacity calculation3
3
A.B, J.G.Milhano and U.A. Wiedemann, J. Phys. G G38 (2011) 124118
and Phys. Rev. C85 (2012) 031901 + arXiv:1204.4342 [hep-ph]
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Vacuum radiation: color flow (in large-Nc )
i
k
k
high−pT quark
l Nucleus 1
l
l
hard process
i Nucleus 2
i
i
l
Final hadrons from the fragmentation of the Lund string (in red)
First endpoint attached to the final quark fragment;
Radiated gluon – color connected with the other daughter of the
branching – belongs to the same string forming a kink on it;
Second endpoint of the string here attached to the beam-remnant
(very low pT , very far in rapidity)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Vacuum radiation: color flow (in large-Nc )
j k
k
i j
high−pT quark
l Nucleus 1
l
l
hard process
i Nucleus 2
i
i
l
Most of the radiated gluons in a shower remain color-connected
with the projectile fragment;
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Vacuum radiation: color flow (in large-Nc )
j
k
k
ij
high−pT quark
l Nucleus 1
l
l
hard process
i Nucleus 2
i
i
l
Most of the radiated gluons in a shower remain color-connected
with the projectile fragment;
Only g → qq splitting can break the color connection, BUT
·
¸
£ 2
¤
1−z
z
2
Pqg ∼ z + (1 − z)
vs Pqg ∼
+
+ z(1 − z)
z
1−z
less likely: no soft (i.e. z → 1) enhancement!
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hadronization in the presence of
medium-modified color flow
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Hadronization à la PYTHIA
k
j
k
k
Leading string
k
j
j
i
j
i
Medium
high−pT quark
j
Medium
high−pT quark
i
i
l Nucleus 1
l
l
l Nucleus 1
l
j
hard process
i Nucleus 2
i
hard process
i Nucleus 2
i
l
Subleading string
i
i
l
l
“Final State Radiation”
(gluon ∈ leading string)
Gluon contributes to leading hadron
Andrea Beraudo
“Initial State Radiation”
(gluon decohered: lost!)
Gluon contributes to enhanced soft
multiplicity from subleading string
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Fragmentation function
Equark=50 GeV, Eradiated gluon=5 GeV, φgluon=0.1, T=200 MeV
-1
1/Nev(dN/dpT) (GeV )
10
In-medium FSR
In-medium ISR (leading+subleading strings)
In-medium ISR (only leading string)
1
0.1
0.01
0.001
0
10
20
30
40
50
60
pT primary hadrons (GeV)
ISR characterized by:
Depletion of hard tail of FF (gluon decohered!);
Enhanced soft multiplicity from the subleading string
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
FF: higher order moments and hadron spectra
Starting from a steeply falling parton spectrum ∼ 1/pTn at the end
of the shower evolution, single hadron spectrum sensitive to higher
moments of FF:
dN h /dpT ∼ hx n−1 i/pTn
Equark=50 GeV, Eradiated gluon=5 GeV, φgluon=0.1, T=200 MeV
-1
1/Nev(dN/dpT) (GeV )
10
Quenching of hard tail of FF
affects higher moments: e.g.
In-medium FSR
In-medium ISR (leading+subleading strings)
In-medium ISR (only leading string)
FSR: hx 6 i ≈ 0.078;
ISR: hx 6 ilead ≈ 0.052
1
0.1
0.01
0.001
0
10
20
30
40
50
60
pT primary hadrons (GeV)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
FF: higher order moments and hadron spectra
Starting from a steeply falling parton spectrum ∼ 1/pTn at the end
of the shower evolution, single hadron spectrum sensitive to higher
moments of FF:
dN h /dpT ∼ hx n−1 i/pTn
1
Quenching of hard tail of FF
affects higher moments: e.g.
0.8
FSR: hx 6 i ≈ 0.078;
ISR: hx 6 ilead ≈ 0.052
0.6
0.4
6
0
6
<x >ISR/<x >FSR
0.2
0
10
20
30
40
50
60
70
80
90
100
Ratio of the two channels
suggestive of the effect on the
hadron spectrum
pT leading fragment (GeV)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Relevance for info on medium properties
Hadronization schemes developed to reproduce data from
elementary collisions: a situation in which most of the radiated
gluons are still color-connected with leading high-pT fragment;
j k
k
i j
high−pT quark
l Nucleus 1
l
l
hard process
i Nucleus 2
i
i
l
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Relevance for info on medium properties
Hadronization schemes developed to reproduce data from
elementary collisions: a situation in which most of the radiated
gluons are still color-connected with leading high-pT fragment;
In the case of AA collisions a naive convolution
Parton Energy loss ⊗ Vacuum Fragmentation
without accounting for the modified color-flow would result into a
too hard hadron spectrum: fitting the experimental amount of
quenching would require an overestimate of the energy loss at the
partonic level;
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Relevance for info on medium properties
Hadronization schemes developed to reproduce data from
elementary collisions: a situation in which most of the radiated
gluons are still color-connected with leading high-pT fragment;
In the case of AA collisions a naive convolution
Parton Energy loss ⊗ Vacuum Fragmentation
without accounting for the modified color-flow would result into a
too hard hadron spectrum: fitting the experimental amount of
quenching would require an overestimate of the energy loss at the
partonic level;
Color-decoherence of radiated gluon might contribute to reproduce
the observed high-pT suppression with milder values of the medium
transport coefficients (e.g. q̂).
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Heavy-flavor
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Experimental findings
Sizeable suppression of D meson spectra;
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Experimental findings
Sizeable suppression of D meson spectra;
Important suppression also of J/ψ from B decays;
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Experimental findings
Sizeable suppression of D meson spectra;
Important suppression also of J/ψ from B decays;
D mesons seem to follow the collective flow of light hadrons
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Some challenges posed by experimental data
Radiated energy: angular distribution
rad
light quarks (<E> ~1.5 GeV)
rad
2
charm (<E> ~0.9 GeV)
rad
1
rad
d<E> /dθ (GeV/rad)
beauty (<E> ~0.3 GeV)
0
p=5 GeV, L=5 fm, mD=0.46 GeV
λg=1.26 fm, αs=0.3
-1
0
0.5
θ (rad)
1
1.5
Color charge: CF vs CA ;
Mass effect: radiation from b strongly suppressed;
Reconsidering the importance of collisional energy loss?
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
A possible tool to study the
heavy-quark dynamics in the QGP:
the relativistic Langevin equation
Trivial extensions of jet-quenching calculations to the massive
case simply describe the energy-loss of heavy quarks, which
remain external probes crossing the medium;
The Langevin equation allows to follow the relaxation to
thermal equilibrium.4
4
W.M. Alberico et al., EPJC 71, 1666 and J.Phys.G G38 (2011) 124144
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Update of the HQ momentum in the plasma: the recipe
∆p i
= − ηD (p)p i + ξ i (t) ,
|{z}
| {z }
∆t
determ.
stochastic
with the properties of the noise encoded in
hξ i (pt )ξ j (pt ′ )i = b ij (pt )
δtt ′
∆t
b ij (p) ≡ κL (p)p̂ i p̂ j + κT (p)(δ ij −p̂ i p̂ j )
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Update of the HQ momentum in the plasma: the recipe
∆p i
= − ηD (p)p i + ξ i (t) ,
|{z}
| {z }
∆t
determ.
stochastic
with the properties of the noise encoded in
δtt ′
b ij (p) ≡ κL (p)p̂ i p̂ j + κT (p)(δ ij −p̂ i p̂ j )
∆t
Transport coefficients to calculate:
hξ i (pt )ξ j (pt ′ )i = b ij (pt )
Momentum diffusion κT ≡
1 h∆pT2 i
h∆pL2 i
and κL ≡
;
2 ∆t
∆t
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Update of the HQ momentum in the plasma: the recipe
∆p i
= − ηD (p)p i + ξ i (t) ,
|{z}
| {z }
∆t
determ.
stochastic
with the properties of the noise encoded in
δtt ′
b ij (p) ≡ κL (p)p̂ i p̂ j + κT (p)(δ ij −p̂ i p̂ j )
∆t
Transport coefficients to calculate:
hξ i (pt )ξ j (pt ′ )i = b ij (pt )
1 h∆pT2 i
h∆pL2 i
and κL ≡
;
2 ∆t
∆t
Friction term (dependent on the discretization scheme!)
¸
·
κL (p) 1
d − 1 κL (p) − κT (p)
2 ∂κL (p)
Ito
ηD (p) =
−
(1 − v )
+
2TEp Ep2
∂v 2
2
v2
Momentum diffusion κT ≡
fixed in order to insure approach to equilibrium (Einstein relation):
Langevin ⇔ Fokker Planck with steady solution exp(−Ep /T )
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
In a static medium...
0
T=400 MeV
HTL (µ=2πT) t=4 fm
0.04
T=400 MeV
HTL (µ=πT)
t=1 fm
t=1 fm
t=2 fm
0.02
t=2 fm
t=4 fm
2
-1
p f(p) (GeV )
t=8 fm
0
0
1
2
3
0
1
p (GeV)
2
3
0
p (GeV)
For t ≫ 1/ηD one approaches a relativistic Maxwell-Jüttner distribution5
e −Ep /T
fMJ (p) ≡
,
4πM 2 T K2 (M/T )
with
Z
d 3 p fMJ (p) = 1
(Test with a sample of c quarks with p0 = 2 GeV/c)
5
A.B., A. De Pace, W.M. Alberico and A. Molinari, NPA 831, 59 (2009)
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
In an expanding fluid...
The fields u µ (x) and T (x) are taken from the output of two
longitudinally boost-invariant (“Hubble-law” longitudinal expansion
vz = z/t)
p
x µ = (τ cosh η, r⊥ , τ sinh η) with τ ≡ t 2 − z 2
1
u µ = γ̄⊥ (cosh η, v̄⊥ , sinh η) with γ̄ ≡ p
1 − v̄2⊥
hydro codes6 .
u µ (x) used to perform the update each time in the fluid rest-frame;
T (x) allows to fix at each step the value of the transport
coefficients.
6
P.F. Kolb, J. Sollfrank and U. Heinz, Phys. Rev. C 62 (2000) 054909
P. Romatschke and U.Romatschke, Phys. Rev. Lett. 99 (2007) 172301
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
Numerical results: spectra in p-p
Hard production in elementary p-p collisions generated with POWHEG +
PYTHIA PS: nice agreement with FONLL outcome and ALICE results
Andrea Beraudo
Heavy-ion collisions: theory review
Introduction
Virtual experiments: lattice QCD
Real experiments: heavy-ion collisions
Soft probes
Hard probes
RAA
Numerical results: spectra in Pb-Pb
1.2
Pb-Pb, sNN=2.76 TeV
1
Centrality: 0-20%
0.8
D0
D+
ALICE, D mesons
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
16
p (GeV/c)
t
In Pb-Pb collisions c and b quarks are then propagated inside the
medium through the Langevin equation7
7
M. Monteno talk at “Hard Probes 2012”
Andrea Beraudo
Heavy-ion collisions: theory review