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Heavy-ion collisions: theory review Andrea Beraudo CERN, Theory - - PowerPoint PPT Presentation

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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

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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

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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

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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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions

Heavy-ion collisions: exploring the QCD phase-diagram

(MeV)

B

µ 500 1000 T (MeV) 100 200

hadrons quark gluon plasma

chemical freeze-out SIS, AGS SPS (NA49) RHIC

E M

color super- conductor

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

Andrea Beraudo Heavy-ion collisions: theory review

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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 qq=0 responsible for most of the baryonic mass of the universe: only ∼35 MeV of the proton mass from mu/d =0

Andrea Beraudo Heavy-ion collisions: theory review

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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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions

QCD on the lattice

The QCD partition function Z =

  • [dU] exp [−βSg(U)]
  • q

det [M(U, mq)] 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

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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;

1Wuppertal group, JHEP 1011 (2010) 077 Andrea Beraudo Heavy-ion collisions: theory review

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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);

1Wuppertal group, JHEP 1011 (2010) 077 Andrea Beraudo Heavy-ion collisions: theory review

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SLIDE 10

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;

1Wuppertal group, JHEP 1011 (2010) 077 Andrea Beraudo Heavy-ion collisions: theory review

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SLIDE 11

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;

1Wuppertal group, JHEP 1011 (2010) 077 Andrea Beraudo Heavy-ion collisions: theory review

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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: c2

s = dP/dǫ

1Wuppertal group, JHEP 1011 (2010) 077 Andrea Beraudo Heavy-ion collisions: theory review

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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 µ

ν ≡ 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

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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

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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

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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

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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

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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

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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

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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 ∂µT µν = 0

  • four−momentum

, ∂µjµ

B = 0

  • baryon number

, where T µν = (ǫ + P)uµuν − Pg µν and jµ

B = nBuµ

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) E(dN/d p) =

  • Σfo pµdΣµ exp[−(p · u)/T]

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Hydro predictions: radial flow (I)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 10 102 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-2 10-1 1

1/(2π) d2N / (mT dmT dy) [c4/GeV2] mT - m0 [GeV/c2] mT - m0 [GeV/c2] 1/(2π) d2N / (mT dmT dy) [c4/GeV2] STAR preliminary Au+Au central √sNN = 200 GeV STAR preliminary p+p min. bias √sNN = 200 GeV π- π- K- p π- π- K- p

dN mTdmT ∼ e−mT /Tslope ≡ e−√

p2

T +m2/Tslope

Tslope(∼ 167 MeV) universal in pp collisions; Tslope growing with m in AA collisions: spectrum gets harder!

Andrea Beraudo Heavy-ion collisions: theory review

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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

0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 SPS Pb+Pb: NA49 preliminary AGS Au+Au: E866 SPS S+S: NA44 π+ h− K± p φ Λ Λ

−

Ξ− d

Particle mass (GeV/c2) mT inverse slope (GeV)

1 2mv2

⊥

= 1 2m

  • (v⊥th + v⊥flow)2

= 1 2mv2

⊥th + 1

2mv2

⊥flow

= ⇒ Tslope = Tfo + 1 2mv2

⊥flow

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Hydro predictions: elliptic flow

x φ y

In non-central collisions particle emission is not azimuthally-symmetric!

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Hydro predictions: elliptic flow

[GeV/c]

T

p

1 2 3 4 5 6

2

v 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

= 2.76 TeV

NN

s Pb-Pb events at Centrality 20-40% π K p Ξ Ω VISH2+1 /s=0.20) η (MCKLN, π K p Ξ Ω

ALICE preliminary

In non-central collisions particle emission is not azimuthally-symmetric! The effect can be quantified through the Fourier coefficient v2 dN dφ = N0 2π (1 + 2v2 cos[2(φ − ψRP)] + . . . ) v2 ≡ cos[2(φ − ψRP)] v2(pT) ∼ 0.2 gives a modulation 1.4 vs 0.6 for in-plane vs out-of-plane particle emission!

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Elliptic flow: physical interpretation

x φ y Matter behaves like a fluid whose expansion is driven by pressure gradients ∂ ∂t

  • (ǫ + P)v i

= − ∂P ∂xi ; 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

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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

[GeV/c]

T

p

1 2 3 4 5 6

2

v 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

= 2.76 TeV

NN

s Pb-Pb events at Centrality 20-40% π K p Ξ Ω VISH2+1 /s=0.20) η (MCKLN, π K p Ξ Ω

ALICE preliminary

Particles emitted according to a thermal distribution ∼exp[−p·u(x)/Tfo] in the local rest-frame of the fluid-cell; Parametrizing the fluid velocity as uµ ≡ γ⊥(cosh Y , u⊥, sinh Y ),

  • ne gets (vz ≡tanh Y )

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

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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 τ ≡

  • t2 − z2

and ηs ≡ 1 2 ln t + z t − z Assuming a boost-invariant purely longitudinal expansion (vz =z/t) entropy conservation implies: s τ = s0 τ0 − → s0 = (s τ)/τ0 Entropy density is defined in the local fluid rest-frame: s ≡ dS dx⊥dz

  • z=0

= 1 τ dS dx⊥dηs Entropy is related to the final multiplicity of charged particles (S ∼3.6 N for pions), so that: s0 = 1 τ0 3.6 πR2

A

dNch dη 3 2

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

“Bjorken” estimate: results

s0 = 1 τ0 3.6 πR2

A

dNch dη 3 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

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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 TA(x) ≡ +∞

−∞

dz ρA(x, z)

  • ne can express the initial entropy density in terms of the local

density of

Andrea Beraudo Heavy-ion collisions: theory review

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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 TA(x) ≡ +∞

−∞

dz ρA(x, z)

  • ne can express the initial entropy density in terms of the local

density of participants: s(τ0, x; b)=K(τ0) [nA

part(x; b) + nB part(x; b)], with

nA

part(x; b) = TA(x + b/2)

  • 1 −
  • 1 − σin

ppTB(x − b/2)/B

B

Andrea Beraudo Heavy-ion collisions: theory review

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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 TA(x) ≡ +∞

−∞

dz ρA(x, z)

  • ne can express the initial entropy density in terms of the local

density of participants: s(τ0, x; b)=K(τ0) [nA

part(x; b) + nB part(x; b)], with

nA

part(x; b) = TA(x + b/2)

  • 1 −
  • 1 − σin

ppTB(x − b/2)/B

B binary collisions: s(τ0, x; b)=K ′(τ0) nbin(x; b), where nbin(x; b) = σin

pp TA(x + b/2) TB(x − b/2)

Andrea Beraudo Heavy-ion collisions: theory review

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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 TA(x) ≡ +∞

−∞

dz ρA(x, z)

  • ne can express the initial entropy density in terms of the local

density of participants: s(τ0, x; b)=K(τ0) [nA

part(x; b) + nB part(x; b)], with

nA

part(x; b) = TA(x + b/2)

  • 1 −
  • 1 − σin

ppTB(x − b/2)/B

B binary collisions: s(τ0, x; b)=K ′(τ0) nbin(x; b), where nbin(x; b) = σin

pp TA(x + b/2) TB(x − b/2)

An essential input is the inelastic pp cross section σin

pp(√s)

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Glauber model and heavy-ion collisions

10 20 30 40 50 60 70 80 90 100 110 120 130 140 σtot, σinel, and σel (mb) 101 102 103 104 105 √s (GeV) σtot σinel σel ¯ pp (PDG) pp (PDG) Auger + Glauber ATLAS CMS ALICE TOTEM best COMPETE σtot fits 11.4 − 1.52 ln s + 0.130 ln2 s σtot (red), σinel (blue) and σel (green)

σin

pp ≈ 40 − 60 mb at RHIC-LHC

energies;

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Glauber model and heavy-ion collisions

10 20 30 40 50 60 70 80 90 100 110 120 130 140 σtot, σinel, and σel (mb) 101 102 103 104 105 √s (GeV) σtot σinel σel ¯ pp (PDG) pp (PDG) Auger + Glauber ATLAS CMS ALICE TOTEM best COMPETE σtot fits 11.4 − 1.52 ln s + 0.130 ln2 s σtot (red), σinel (blue) and σel (green)

σin

pp ≈ 40 − 60 mb at RHIC-LHC

energies; The Glauber model seems to work pretty well: nuclear modification factor RAA(pT) ≡ (dN/dpT)AA Ncoll(dN/dpT)pp close to 1 for color-neutral probes!

Andrea Beraudo Heavy-ion collisions: theory review

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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 ≡ cos[n(φ − Ψn)]. 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

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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)

aALICE, Phys.Rev.Lett. 107 (2011) 032301 Andrea Beraudo Heavy-ion collisions: theory review

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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: s0 ∼ dNg dr⊥dy ∼ dp⊥ p2

⊥

  • dk⊥ αs φA(x1, k2

⊥) φB(x2, (p⊥−k⊥)2)

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 ∼ N

  • BFKL

− N 2

  • saturation

A unique setup able to describe data from DIS up to A-A collisions?

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

CGC and particle production

part

N 100 200 300 400 /2

part

/N η dN/d 2 4 6 8 10 MCrcBK 200GeV MCrcBK 2.76TeV ALICE 2.76TeV PHOBOS 200GeV

Particle density and its evolution with centrality nicely accomodated2

2J.L. Albacete, A. Dumitru and Y. Nara, J.Phys.Conf.Ser. 316 (2011)

012011.

Andrea Beraudo Heavy-ion collisions: theory review

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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: ∂v i ∂t = − 1 ǫ + P ∂P ∂xi = −c2

s

∂ ln s ∂xi ; For the transverse expansion one gets: vx = c2

s x

σ2

x

t, vy = c2

s y

σ2

y

t The larger the speed of sound, the larger the radial flow!

Andrea Beraudo Heavy-ion collisions: theory review

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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

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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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Inclusive hadron spectra: the nuclear modification factor

) c (GeV/

T

p 2 4 6 8 10 12 14 16

AA

R

  • 1

10 1 10

PHENIX Au+Au (central collisions): γ Direct π η /dy = 1100)

g

GLV parton energy loss (dN PHENIX Au+Au (central collisions): γ Direct π η /dy = 1100)

g

GLV parton energy loss (dN

RAA ≡

  • dNh/dpT

AA Ncoll (dNh/dpT)pp

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Inclusive hadron spectra: the nuclear modification factor

(GeV/c)

T

p 10 20 30 40 50

AA

R 0.1 1

0-5% 20-40% 40-80% ALICE, charged particles, Pb-Pb | < 0.8 η = 2.76 TeV, |

NN

s ALICE Preliminary

RAA ≡

  • dNh/dpT

AA Ncoll (dNh/dpT)pp

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Inclusive hadron spectra: the nuclear modification factor

RAA ≡

  • dNh/dpT

AA Ncoll (dNh/dpT)pp

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Inclusive hadron spectra: the nuclear modification factor

RAA ≡

  • dNh/dpT

AA Ncoll (dNh/dpT)pp 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

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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/pα

T) so that inclusive hadron

spectrum simply reflects higher moments of FF dNh dpT ∼ 1 pα

T

  • f

1 dz zα−1Df →h(z) carrying limited information on FF (but very sensitive to hard tail!)

Andrea Beraudo Heavy-ion collisions: theory review

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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/pα

T) so that inclusive hadron

spectrum simply reflects higher moments of FF dNh dpT ∼ 1 pα

T

  • f

1 dz zα−1Df →h(z) carrying limited information on FF (but very sensitive to hard tail!) Surface bias:

Andrea Beraudo Heavy-ion collisions: theory review

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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/pα

T) so that inclusive hadron

spectrum simply reflects higher moments of FF dNh dpT ∼ 1 pα

T

  • f

1 dz zα−1Df →h(z) carrying limited information on FF (but very sensitive to hard tail!) Surface bias:

hard process leading hadron hadronization hadronization QGP

Quenched spectrum does not reflect LQGP 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

  • f energy!

Andrea Beraudo Heavy-ion collisions: theory review

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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

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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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Dijet correlations: results

J

A

0.2 0.4 0.6 0.8 1 J

) dN/dA

evt

(1/N

1 2 3 4

40-100%

J

A

0.2 0.4 0.6 0.8 1 J

) dN/dA

evt

(1/N

1 2 3 4

20-40%

J

A

0.2 0.4 0.6 0.8 1 J

) dN/dA

evt

(1/N

1 2 3 4

10-20%

J

A

0.2 0.4 0.6 0.8 1 J

) dN/dA

evt

(1/N

1 2 3 4

0-10% ATLAS Pb+Pb =2.76 TeV

NN

s

  • 1

b µ =1.7

int

L φ ∆

2 2.5 3

φ ∆ ) dN/d

evt

(1/N

  • 2

10

  • 1

10 1 10

φ ∆

2 2.5 3

φ ∆ ) dN/d

evt

(1/N

  • 2

10

  • 1

10 1 10

φ ∆

2 2.5 3

φ ∆ ) dN/d

evt

(1/N

  • 2

10

  • 1

10 1 10

φ ∆

2 2.5 3

φ ∆ ) dN/d

evt

(1/N

  • 2

10

  • 1

10 1 10 Pb+Pb Data p+p Data HIJING+PYTHIA

Dijet asymmetry Aj ≡

ET1−ET2 ET1+ET2 enhanced wrt to p+p and increasing

with centrality; ∆φ distribution unchanged wrt p+p (jet pairs ∼ back-to-back)

Andrea Beraudo Heavy-ion collisions: theory review

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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 ≡

  • ∆φ2+∆η2 and width 0.08 around the

subleading-jet axis:

Andrea Beraudo Heavy-ion collisions: theory review

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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 ≡

  • ∆φ2+∆η2 and width 0.08 around the

subleading-jet axis:

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per bin (GeV/c)

T

p Σ 1 10

2

10

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0.5 0.5

< 0.11

J

A

Leading Jet Subleading Jet

PYTHIA +HYDJET 0-30% (a)

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per bin (GeV/c)

T

p Σ 1 10

2

10

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Leading jet

R ∆ 0.5

Subleading jet

R ∆ 0.5

< 0.11

J

A

Leading Jet Subleading Jet

(e) CMS

=2.76 TeV

NN

s PbPb

  • 1

b µ L dt = 6.7

∫

0-30%

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1 10

2

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0.5 0.5

< 0.22

J

0.11 < A

Leading Jet Subleading Jet > 120GeV/c

T,1

p > 50GeV/c

T,2

p π 3 2 >

1,2

φ ∆

(b)

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1 10

2

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Leading jet

R ∆ 0.5

Subleading jet

R ∆ 0.5

< 0.22

J

0.11 < A

Leading Jet Subleading Jet

(f)

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1 10

2

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0.5 0.5

xxx xxx xxx xxx xxx xxx xxx xxx

> 8 GeV/c 4-8 GeV/c 1-4 GeV/c

< 0.33

J

0.22 < A

Leading Jet Subleading Jet

(c)

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1 10

2

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Leading jet

R ∆ 0.5

Subleading jet

R ∆ 0.5

< 0.33

J

0.22 < A

Leading Jet Subleading Jet

(g)

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1 10

2

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> 0.33

J

A

Leading Jet Subleading Jet

(d)

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1 10

2

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Leading jet

R ∆ 0.5

Subleading jet

R ∆ 0.5

> 0.33

J

A

Leading Jet Subleading Jet

(h)

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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Dijet measurements: Fragmentation Fuctions

ξ ≡ − ln z ≡ − ln

  • ptrack

T

/pjet

T

  • ,

ptrack

T

> 4GeV

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Dijet measurements: Fragmentation Fuctions

ξ ≡ − ln z ≡ − ln

  • ptrack

T

/pjet

T

  • ,

ptrack

T

> 4GeV

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Dijet measurements: Fragmentation Fuctions

ξ ≡ − ln z ≡ − ln

  • ptrack

T

/pjet

T

  • ,

ptrack

T

> 4GeV

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Dijet measurements: Fragmentation Fuctions

ξ ≡ − ln z ≡ − ln

  • ptrack

T

/pjet

T

  • ,

ptrack

T

> 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

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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!

E (≈ pT) (1 − x)E xE 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

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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

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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+, Q2/2p+, 0
  • loses its virtuality Q through

gluon-radiation. In light-cone coordinates, with p± ≡E ± pz/ √ 2:

a P + (1 − x)P +

  • k⊥

xP +

kg ≡

  • xp+,

k2 2xp+ , k

  • pf =
  • (1−x)p+,

k2 2(1−x)p+ , −k

  • Andrea Beraudo

Heavy-ion collisions: theory review

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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+, Q2/2p+, 0
  • loses its virtuality Q through

gluon-radiation. In light-cone coordinates, with p± ≡E ± pz/ √ 2:

a P + (1 − x)P +

  • k⊥

xP +

kg ≡

  • xp+,

k2 2xp+ , k

  • pf =
  • (1−x)p+,

k2 2(1−x)p+ , −k

  • k⊥ vs virtuality: k2 = x (1−x) Q2;

Radiation spectrum (our benchmark): IR and collinear divergent! dσrad

vac = dσhard αs

π2 CR dk+ k+ dk k2 Time-scale (formation time) for gluon radiation: ∆trad ∼ Q−1(E/Q) ∼ 2ω/k2 (x ≈ ω/E)

Andrea Beraudo Heavy-ion collisions: theory review

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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 a1 a1 a a (a) (c) (b)

Andrea Beraudo Heavy-ion collisions: theory review

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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 a1 a1 a a (a) (c) (b)

The single-inclusive gluon spectrum: the Gunion-Bertsch result x dNGB

g

dxdk = CR αs π2 L λel

g

[K0−K1]2 = CR αs π2 L λel

g

q2 k2(k−q)2

  • where CR is the color charge of the hard parton and:

K0 ≡ k k2 , K1 ≡ k−q (k−q)2 and . . . ≡

  • dq 1

σel dσel dq

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

The induced spectrum: physical interpretation

ω dσind dωdk = dσhardCR αs π2 L λel

g

(K0 − K1)2 + K2

1 − K2

1− sin(ω1L) ω1L

  • In the above ω1 ≡(k−q)2/2ω and two regimes can be distinguished:

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

The induced spectrum: physical interpretation

ω dσind dωdk = dσhardCR αs π2 L λel

g

(K0 − K1)2 + K2

1 − K2

1− sin(ω1L) ω1L

  • In the above ω1 ≡(k−q)2/2ω and two regimes can be distinguished:

Coherent regime LPM (ω1L≪1): dσind =0 − → dσrad = dσvac

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

The induced spectrum: physical interpretation

ω dσind dωdk = dσhardCR αs π2 L λel

g

(K0 − K1)2 + K2

1 − K2

1− sin(ω1L) ω1L

  • In the above ω1 ≡(k−q)2/2ω and two regimes can be distinguished:

Coherent regime LPM (ω1L≪1): dσind =0 − → dσrad = dσvac Incoherent regime (ω1L≫1): dσind ∼

  • (K0−K1)2+K2

1−K2

  • Andrea Beraudo

Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

The induced spectrum: physical interpretation

ω dσind dωdk = dσhardCR αs π2 L λel

g

(K0 − K1)2 + K2

1 − K2

1− sin(ω1L) ω1L

  • In the above ω1 ≡(k−q)2/2ω and two regimes can be distinguished:

Coherent regime LPM (ω1L≪1): dσind =0 − → dσrad = dσvac Incoherent regime (ω1L≫1): dσind ∼

  • (K0−K1)2+K2

1−K2

  • The full radiation spectrum can be organized as

dσrad = dσGB + dσvac

gain + dσvac loss

where dσGB = dσhardCR αs π2

  • L/λel

g

(K0 − K1)2 (dωdk/ω) dσvac

gain = dσhardCR

αs π2

  • L/λel

g

  • K2

1 (dωdk/ω)

dσvac

loss =

  • 1 − L/λel

g

  • dσhardCR

αs π2 K2

0 (dωdk/ω)

Andrea Beraudo Heavy-ion collisions: theory review

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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: ∆E =

  • dω
  • dk ω dNind

g

dωdk ∼ CRαs 4 µ2

D

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; µ2

D/λel g often replaced by the transport coefficient ˆ

q, so that ∆E ∼ αsˆ qL2 ˆ q: average q2

⊥ acquired per unit length

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Numerical results

0.2 0.4 0.6 0.8 1

x

1 2 3 4 5 6

dN/dx

N=1 opacity

Medium-induced radiation: energy distribution

E=10 GeV, L=5 fm, mD=0.46 GeV, λg=1.26 fm, αs=0.3 0.5 1 1.5

θ (rad)

  • 0.5

0.5 1 1.5

dN/dθ

N=1 opacity

Medium-induced radiation: angular distribution

E=10 GeV, L=5 fm, mD=0.46 GeV, λg=1.26 fm, αs=0.3

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

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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

  • bservables (e.g. jet-fragmentation pattern)

Essential ideas presented here in a N = 1 opacity calculation3

3A.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

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Vacuum radiation: color flow (in large-Nc)

i i high−pT quark Nucleus 1 Nucleus 2 hard process l l l l i k k i

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

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Vacuum radiation: color flow (in large-Nc)

i high−pT quark Nucleus 1 Nucleus 2 hard process l i j j k k i l l i l

Most of the radiated gluons in a shower remain color-connected with the projectile fragment;

Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Vacuum radiation: color flow (in large-Nc)

i i high−pT quark Nucleus 1 Nucleus 2 hard process l l l l i k k i j j

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 Pqg ∼

  • z2 + (1 − z)2

vs Pqg ∼ 1 − z z + z 1 − z + z(1 − z)

  • less likely: no soft (i.e. z → 1) enhancement!

Andrea Beraudo Heavy-ion collisions: theory review

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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

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Hadronization ` a la PYTHIA

i i i i i i Medium high−pT quark Nucleus 1 Nucleus 2 hard process j j k k l l l l

“Final State Radiation” (gluon ∈ leading string) Gluon contributes to leading hadron

i i Medium high−pT quark Nucleus 1 Nucleus 2 hard process l l l l i i j j j j k k Subleading string Leading string

“Initial State Radiation” (gluon decohered: lost!) Gluon contributes to enhanced soft multiplicity from subleading string

Andrea Beraudo Heavy-ion collisions: theory review

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Fragmentation function

10 20 30 40 50 60

pT primary hadrons (GeV)

0.001 0.01 0.1 1 10

1/Nev(dN/dpT) (GeV

  • 1)

In-medium FSR In-medium ISR (leading+subleading strings) In-medium ISR (only leading string)

Equark=50 GeV, Eradiated gluon=5 GeV, φgluon=0.1, T=200 MeV

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

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FF: higher order moments and hadron spectra

Starting from a steeply falling parton spectrum ∼ 1/pn

T at the end

  • f the shower evolution, single hadron spectrum sensitive to higher

moments of FF: dNh/dpT ∼ xn−1/pn

T

10 20 30 40 50 60

pT primary hadrons (GeV)

0.001 0.01 0.1 1 10

1/Nev(dN/dpT) (GeV

  • 1)

In-medium FSR In-medium ISR (leading+subleading strings) In-medium ISR (only leading string)

Equark=50 GeV, Eradiated gluon=5 GeV, φgluon=0.1, T=200 MeV

Quenching of hard tail of FF affects higher moments: e.g. FSR: x6 ≈ 0.078; ISR: x6lead ≈ 0.052

Andrea Beraudo Heavy-ion collisions: theory review

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FF: higher order moments and hadron spectra

Starting from a steeply falling parton spectrum ∼ 1/pn

T at the end

  • f the shower evolution, single hadron spectrum sensitive to higher

moments of FF: dNh/dpT ∼ xn−1/pn

T

10 20 30 40 50 60 70 80 90 100

pT leading fragment (GeV)

0.2 0.4 0.6 0.8 1

<x

6>ISR/<x 6>FSR

Quenching of hard tail of FF affects higher moments: e.g. FSR: x6 ≈ 0.078; ISR: x6lead ≈ 0.052 Ratio of the two channels suggestive of the effect on the hadron spectrum

Andrea Beraudo Heavy-ion collisions: theory review

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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;

i high−pT quark Nucleus 1 Nucleus 2 hard process l i j j k k i l l i l Andrea Beraudo Heavy-ion collisions: theory review

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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

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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. ˆ q).

Andrea Beraudo Heavy-ion collisions: theory review

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Heavy-flavor

Andrea Beraudo Heavy-ion collisions: theory review

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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

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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

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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

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Some challenges posed by experimental data

0.5 1 1.5

θ (rad)

  • 1

1 2

d<E>

rad/dθ (GeV/rad)

light quarks (<E>

rad~1.5 GeV)

charm (<E>

rad~0.9 GeV)

beauty (<E>

rad~0.3 GeV)

Radiated energy: angular distribution

p=5 GeV, L=5 fm, mD=0.46 GeV λg=1.26 fm, αs=0.3

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

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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

4W.M. Alberico et al., EPJC 71, 1666 and J.Phys.G G38 (2011) 124144 Andrea Beraudo Heavy-ion collisions: theory review

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Update of the HQ momentum in the plasma: the recipe ∆pi ∆t = − ηD(p)pi

determ.

+ ξi(t)

  • stochastic

, with the properties of the noise encoded in ξi(pt)ξj(pt′)=bij(pt)δtt′ ∆t bij(p)≡κL(p)ˆ piˆ pj + κT(p)(δij −ˆ piˆ pj)

Andrea Beraudo Heavy-ion collisions: theory review

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Update of the HQ momentum in the plasma: the recipe ∆pi ∆t = − ηD(p)pi

determ.

+ ξi(t)

  • stochastic

, with the properties of the noise encoded in ξi(pt)ξj(pt′)=bij(pt)δtt′ ∆t bij(p)≡κL(p)ˆ piˆ pj + κT(p)(δij −ˆ piˆ pj) Transport coefficients to calculate: Momentum diffusion κT ≡ 1 2 ∆p2

T

∆t and κL ≡ ∆p2

L

∆t ;

Andrea Beraudo Heavy-ion collisions: theory review

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Update of the HQ momentum in the plasma: the recipe ∆pi ∆t = − ηD(p)pi

determ.

+ ξi(t)

  • stochastic

, with the properties of the noise encoded in ξi(pt)ξj(pt′)=bij(pt)δtt′ ∆t bij(p)≡κL(p)ˆ piˆ pj + κT(p)(δij −ˆ piˆ pj) Transport coefficients to calculate: Momentum diffusion κT ≡ 1 2 ∆p2

T

∆t and κL ≡ ∆p2

L

∆t ; Friction term (dependent on the discretization scheme!) ηD

Ito(p) = κL(p)

2TEp − 1 E 2

p

  • (1 − v 2)∂κL(p)

∂v 2 + d − 1 2 κL(p) − κT(p) v 2

  • 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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

In a static medium...

1 2 3

p (GeV)

0.02 0.04

p

2f(p) (GeV

  • 1)

1 2 3

p (GeV)

T=400 MeV HTL (µ=πT) T=400 MeV HTL (µ=2πT)

t=1 fm t=2 fm t=4 fm t=8 fm t=1 fm t=2 fm t=4 fm

For t ≫ 1/ηD one approaches a relativistic Maxwell-J¨ uttner distribution5 fMJ(p) ≡ e−Ep/T 4πM2T K2(M/T), with

  • d3p fMJ(p) = 1

(Test with a sample of c quarks with p0 =2 GeV/c)

5A.B., A. De Pace, W.M. Alberico and A. Molinari, NPA 831, 59 (2009) Andrea Beraudo Heavy-ion collisions: theory review

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

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 codes6. uµ(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.

6P.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

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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

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Introduction Virtual experiments: lattice QCD Real experiments: heavy-ion collisions Soft probes Hard probes

Numerical results: spectra in Pb-Pb

(GeV/c)

t

p

2 4 6 8 10 12 14 16

AA

R

0.2 0.4 0.6 0.8 1 1.2

=2.76 TeV

NN

s Pb-Pb, Centrality: 0-20%

D

+

D ALICE, D mesons

In Pb-Pb collisions c and b quarks are then propagated inside the medium through the Langevin equation7

  • 7M. Monteno talk at “Hard Probes 2012”

Andrea Beraudo Heavy-ion collisions: theory review