suppression of quarkonia (and jets) in the quark gluon...

IntroductionQuenching of jets

Quarkonium suppression

Suppression of quarkonia (and jets) in the QuarkGluon Plasma

Andrea Beraudo

INFN - Sezione di Torino

Lattice-QCD workshop,22-23 December 2014 Torino

1 / 54



Heavy-ion collisions: exploring the QCD phase-diagram

QCD phases identified through the orderparameters

Polyakov loop 〈L〉 ∼ energy cost toadd an isolated color charge

Chiral condensate 〈qq〉 ∼ effectivemass of a “dressed” quark in a hadron

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 breaking1)

NB 〈qq〉 6=0 responsible for most of the baryonic mass of the universe: only

∼35 MeV of the proton mass from mu/d 6=0

1V. Koch, Aspects of chiral symmetry, Int.J.Mod.Phys. E6 (1997)2 / 54



Heavy-ion collisions: a typical event

Valence quarks of participant nucleons act as sources of strong colorfields 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

3 / 54



Heavy-ion collisions: a typical event

4 / 54



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): producedin hard pQCD processes in the initial stage, allow to perform atomography of the medium

5 / 54



Jet quenching

(in a broad sense: jet-reconstruction in AA possible only recently)

6 / 54



Inclusive hadron spectra: the nuclear modification factor

)c (GeV/Tp0 2 4 6 8 10 12 14 16

AA

R

-110

1

10

PHENIX Au+Au (central collisions):γDirect

0πη

/dy = 1100)g

GLV parton energy loss (dN

PHENIX Au+Au (central collisions):γDirect

0πη

/dy = 1100)g

GLV parton energy loss (dN

RAA ≡

(dNh/dpT

)AA

〈Ncoll〉 (dNh/dpT )pp

7 / 54




(GeV/c)T

p0 10 20 30 40 50

AA

R

0.1

1

0-5%

20-40%

40-80%

ALICE, charged particles, Pb-Pb

| < 0.8η = 2.76 TeV, | NNs

ALICE Preliminary

RAA ≡

(dNh/dpT

)AA


7 / 54




RAA ≡

(dNh/dpT

)AA


7 / 54




RAA ≡

(dNh/dpT

)AA


Hard-photon RAA ≈ 1

supports the Glauber picture (binary-collision scaling);

entails that quenching of inclusive hadron spectra is a final stateeffect due to in-medium energy loss.

7 / 54



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!

8 / 54



Dijet correlations: results

JA0 0.2 0.4 0.6 0.8 1

J)

dN/d

Aev

t(1

/N

0

1

2

3

440-100%

JA0 0.2 0.4 0.6 0.8 1

J)

dN/d

Aev

t(1

/N0

1

2

3

420-40%

JA0 0.2 0.4 0.6 0.8 1

J)

dN/d

Aev

t(1

/N

0

1

2

3

410-20%

JA0 0.2 0.4 0.6 0.8 1

J)

dN/d

Aev

t(1

/N

0

1

2

3

40-10%

ATLASPb+Pb

=2.76 TeVNNs

-1bµ=1.7 intL

φ∆2 2.5 3

φ∆)

dN/d

evt

(1/N

-210

-110

1

10

φ∆2 2.5 3

φ∆)

dN/d

evt

(1/N

-210

-110

1

10

φ∆2 2.5 3

φ∆)

dN/d

evt

(1/N

-210

-110

1

10

φ∆2 2.5 3

φ∆)

dN/d

evt

(1/N

-210

-110

1

10Pb+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)

9 / 54



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 mediuminduces 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).

10 / 54



Average energy loss

Integrating the lost energy ω over the inclusive gluon spectrum:

〈∆E 〉 =

∫

dω

∫

dk ωdN ind

g

dωdk∼

CRαs

4

(µ2

D

λelg

)

L2 lnE

µD

L2 dependence on the medium-length;

µD : Debye screening mass of color interaction ∼ typical momentumexchanged in a collision;

µ2D/λel

g often replaced by the transport coefficient q2, so that

〈∆E 〉 ∼ CRαs qL2

q: average q2⊥ acquired per unit length

2For a lattice-QCD determination see M. Panero et al.,Phys.Rev.Lett. 112(2014) 16, 162001

11 / 54




(charmonium and now also bottomonium)

12 / 54



Original idea by Matsui and Satz

Statement3: the J/ψ anomalous suppression in high energy AAcollisions represents an unambiguous signature of deconfinement

Underlying assumptions:

The J/ψ’s are produced in the very early stage of the collisionThe medium resulting from the HIC thermalizes in a timeτtherm ≈ 0.5 − 1fm/c;Crossing a deconfined medium the cc bound states tend tomelt (Debye screening of the Coulomb interaction):

V (r) ∼ −α

r+ σr → −

α

re−mD r

Consequences: one expects a sequential suppression pattern, with

Tψ′

diss < TJ/ψdiss and T

Υ(3s)diss < T

Υ(2s)diss < T

Υ(1s)diss

3T. Matsui and H. Satz, PLB 178 (1986).13 / 54



Experimental evidence

Wonderful results found by CMS through µ+µ− invariant mass spectra

14 / 54



Experimental evidence

Wonderful results found by CMS through µ+µ− invariant mass spectra

Suppression dependent on the binding energy of the state and on the

centrality of the collision

14 / 54



The challenge

Is it possible to make this picture more quantitative through afirst-principle calculation?

A possible answer: take advantage of the results provided bythe lattice-QCD simulations.

15 / 54



Quarkonium in hot-QCD: two independent (?) approaches

Heavy-quark free-energy calculations:evaluate ∆F occurring once a static QQ pair is placed in a thermalbath of gluons and light quarks

e−β∆F

(1)

QQ(x−y,T )+C

=1

3〈W (x)W †(y)〉

Meson Spectral Function reconstruction:look for resonance-peaks4 in the spectral densities extracted fromin-medium quarkonium euclidean propagators

GM(τ) ≡ 〈JM(τ)J†M(0)〉

4S. Datta, F. Karsch, P. Petreczky and I. Wetzorke, PRD 69, 094507 (2004)16 / 54



Polyakov-line correlators

e−β∆FQQ(x−y,T ) ∼ 〈χ(β, y)ψ(β, x)ψ†(0, x)χ†(0, y)〉

They describe a QQ pair propagating from τ =0 to τ =β and can be

measured on the lattice5

-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]

0.76Tc0.81Tc0.90Tc0.96Tc1.00Tc1.02Tc1.07Tc1.23Tc1.50Tc1.98Tc4.01Tc

0

500

1000

1500

0 0.5 1 1.5 2

U1 [MeV]

r [fm]

1.09Tc1.13Tc1.19Tc1.29Tc1.43Tc1.57Tc1.89Tc

Can one exploit this information to get an effective QQ potential?

state J/ψ χc ψ′

Td/Tc (Veff ≡ F1) 1.1 0.74 0.1-0.2Td/Tc (Veff ≡ U1) 1.78-1.92 1.14-1.15 1.11-1.12

5O. Kaczmarek and F. Zantow, PoS LAT2005:192 (2006) 17 / 54



Meson Spectral Functions

One measures the imaginary-time propagator

GM(τ) ≡ 〈JM(τ)J†M(0)〉

of a meson produced by the current

JM(τ) ≡ q(τ)ΓMq(τ)

From GM(τ) the MSF has to be reconstructed:

GM(τ) =

∫ ∞

0dω σM(ω)

︸︷︷︸

MSF

cosh(ω(τ − β/2))

sinh(βω/2)

NB: Typically GM(τ) is known for a quite limited set of points(<∼50) → problems in inverting the above transform.

18 / 54



MSFs on the lattice: charmonium

Attempting the inversion through a MEM procedure6,7...

2 4 6 8 10ω (GeV)

0

10

20

30

ρ(ω

)/ω²

T = 221 MeVT = 294 MeVT = 353 MeVT = 392 MeVT = 441 MeV

J/ψm= 0.092

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

5 10 15 20

σ(ω

)/ω

2ω[GeV]

T=0, Ndata=16

T=1.5Tc

The vector (left) and pseudoscalar (right) MSFs displaywell-defined ground-state peaks up to temperature T ∼2Tc .

6G. Aarts et al., arXiv:0705.2198 [hep-lat]7A. Jakovac et al., Phys.Rev. D75 (2007) 014506.

19 / 54



MSFs on the lattice: bottomonium

Employing NRQCD one gets rid of the contribution from energiesω ≪ 2M and it is possible to simulate also bottomonium on the lattice

20 / 54





NB Vector MSF directly related to the dilepton production rate

dNµ+µ−

dp0d~p

∣∣∣∣~p=0

∼α2

em

p20

nB(p0)σV (p0)

20 / 54





NB Vector MSF directly related to the dilepton production rate

dNµ+µ−

dp0d~p

∣∣∣∣~p=0

∼α2

em

p20

nB(p0)σV (p0)

One is tempted to establish a direct connection with exp data, however...20 / 54



Some open problems: a brief summary

Potential models: which effective potential from the QQfree-energy data?

MSF: in principle would contain the full information on thein-medium quarkonium properties, BUT large uncertaintiesfrom inverting the transform. So far, most of the availableresults reliable just for the existence of a ground-state peak;

Is it possible to establish a link between screened potentialmodels and spectral studies?

21 / 54



The basic objectof our study

G>(t,r1; t,r2|0,r′1; 0,r

′2)≡〈χ(t,r2)ψ(t,r1)

︸︷︷︸

JM(t)

ψ†(0,r′1)χ†(0,r′2)

︸︷︷︸

J†M

(0)

〉

NB: In the M → ∞ limit quarks are frozen to their positions andG>

M=∞(t) reduces to the closed Wilson loop W(t,r)

22 / 54



Real-time static potential in hot-QCD: weak-coupling

Evaluate perturbatively [M. Laine et al., JHEP 0703 (2007) 054]

G>M=∞(t, r) = G (0)>(t, r) + G (2)>(t, r) + . . .

23 / 54





G>M=∞(t, r) = G (0)>(t, r) + G (2)>(t, r) + . . .

Assuming that, for t → ∞, G>M=∞(t, r) is solution of

(i∂t − Veff (r))G>M=∞(t, r) = 0

identify the LO perturbative contribution to the effective potential:

Veff = V(2)eff + . . .

23 / 54





G>M=∞(t, r) = G (0)>(t, r) + G (2)>(t, r) + . . .

Assuming that, for t → ∞, G>M=∞(t, r) is solution of

(i∂t − Veff (r))G>M=∞(t, r) = 0

identify the LO perturbative contribution to the effective potential:

Veff = V(2)eff + . . .

Use it in the finite mass case to solve the equation

(i∂t − T − V(2)eff (r))G>(t) = 0

23 / 54



Some questions to answer

Does G>(t) obey a closed Schrodinger equation? Is the concept ofan effective potential meaningful/necessary?

What’s the link of the effective potential with the QQ free-energy?Does the latter contain all the possible information?

Is it possible to include the effect of collisions in a consistent way?

24 / 54



Our goal

We wish to perform a study resulting

numerically less expensive then lattice calculations (henceallowing a more robust reconstruction of the spectralfunction);

capable to get a deeper physical insight on the processesinvolved.

NB Since we wish to study very general medium effects, notpeculiar of QCD, for the sake of simplicity we will consider the caseof heavy charged particles placed in a QED plasma. This will besufficient to provide and answer to the previous questions

25 / 54



In-medium correlators

G>(t) ≡ 〈O(t)O†(0)〉

O† creates a Q or a QQ pair;

Spectral decomposition

G>(t) = Z−1∑

n

e−βEn∑

m

〈n|O(t)|m〉〈m|O†(0)|n〉

= Z−1∑

n

e−βEn∑

m

e i(En−Em)t |〈m|O†(0)|n〉|2,

G>(t) is an analytic function in the strip −β< Imt< 0 ⇒unified description of real and imaginary-time propagation;HQs: external probe placed in a hot/dense medium of lightparticles =⇒ {|n〉} do not contain heavy quarks.

26 / 54



In-medium spectral functions

In the general case the spectral density of a correlator wouldbe given by

σ(ω) ≡ G>(ω) ∓ G<(ω);

Dealing with the propagation of an external probe one hasG< ≡ 0, so that

σ(ω) = G>(ω) =⇒ G>(t) =

∫ +∞

−∞

dω

2πe−iωtσ(ω);

The standard procedure to get σ(ω) is then, exploiting theanalyticity of G>:

G>(t =−iτ)︸︷︷︸

evaluated

=

∫ +∞

−∞

dω

2πe−ωτ σ(ω)

︸︷︷︸

reconstructed

.

27 / 54



General setup

A heavy quark (magnetic effects negligible!) coupled to a mediumof light particles (for a relativistic hot plasma M >> T >> m):

H = HQ + Hint + Hmed , where

HQ =

∫

d3r ψ†(r)

(

M −∇2

2M

)

ψ(r),

Hint = g

∫

d3 r ψ†(r)ψ(r)A0(r), with

Hmed the hamiltonian describing the medium;A0 the electrostatic potential created by the light particles.

[H, NQ ] = 0 ⇒ NQ ≡

∫

d3r ψ†(r)ψ(r) is conserved

EoM : i∂tψ(t, r) = [ψ(t, r), H] =

(

M −∇2

2M+ gA0(t, r)

)

ψ(t, r)

28 / 54



Path-integral formulation

Let us fix the main ideas with the simple case of a single particle.The HQ propagator for a given configuration of the backgroundgauge-field reads:

G>A (t, r) =

∫ r

0[Dz] exp

[

i

∫ t

0dt ′

(1

2M z2 − gA0(t

′, z(t ′))

)]

After taking a medium-average over the configurations of thebackgroung field one has:

G>(t, r) = Z−1

∫

[DA]

∫ r

0[Dz] exp

[

i

∫ t

0dt ′

1

2M z2

]

× exp

[

−ig

∫ t

0dt ′A0(t

′, z(t ′))

]

e iSeff [A]

29 / 54



Which action to employ to weight the field configurations fora hot gauge plasma?

30 / 54



Scales in a weakly-coupled relativistic plasma

λ hard = 1/g^4T

λ soft = 1/g^2T

d~1/T

λ dB=1/2piT

λ D~1/gT

most of the scattering processes involve the exchange ofsoft momenta Q∼gT .

31 / 54



The HTL effective action

The propagation of soft (long wave-length) gauge-bosons(Q∼gT ) is dressed by the interactions with the lightplasma-particle which are hard (K ∼T )

µ νQ Q

K − Q

K

hard

hard

soft soft

The HTL effective action (for an abelian gauge plasma):

SHTL[A] =1

2

∫

d4x

∫

d4y Aµ(x)(D−1

)HTL

µν(x − y)Aν(y).

32 / 54



A heavy “quark” in a hot gauge plasma

Neglecting possible non-abelian effects we perform Monte Carlosimulations for

G>(−iτ, r1|0, r′1)=

∫ z(τ)=r1

z(0)=r′1

[Dz]exp

[

−

∫ τ

0

dτ ′

(

M +1

2M z2

)]

×

× exp

[g2

2

∫ τ

0

dτ ′

∫ τ

0

dτ ′′∆TL (τ ′ − τ ′′, z(τ ′) − z(τ ′′))

]

where

∆L(τ,q) ≡ ∆vacL (τ,q) + ∆T

L (τ,q)

=−1

q2δ(τ) +

∫ +∞

−∞

dq0

2πe−q0τρL(q0,q)[θ(τ) + N(q0)]

is expressed in terms of the HTL spectral function

ρL(ω > 0, q) ≡ 2π[

ZL(q)δ(ω−ωL(q))︸︷︷︸

plasmon pole

+ θ(q2−ω2)βL(ω, q)︸︷︷︸

Landau damping

]

33 / 54



HTL longitudinal spectral function

ρL(ω) ≡ 2 ImDretL (ω) = 2 Im∆L(ω + iη),

where:

∆L(q0, q) =

−1

q2 + m2D

(

1 − q0

2qln q0+q

q0−q

)

0 0.2 0.4 0.6 0.8 1ω/m

D

0

2

4

6

8

10

ρ L(ω

,q)/

m2 D

q=0.5mD

Pole + Continuum. The width is put by hand! 34 / 54



Our long term goal...

...would be to address the QQ case within the same approach:

G>(−iτ ; r1, r2|0; r′1, r′2) = e−(M1+M2)τ

∫ r1

r′1

[Dz1]

∫ r2

r′2

[Dz2]×

× exp

[

−

∫ τ

0

dτ ′

(1

2M1z1

2−g2

2

∫ τ

0

dτ ′′∆TL (τ ′−τ ′′, z1(τ

′)−z1(τ′′))

)]

×

× exp

[

−

∫ τ

0

dτ ′

(1

2M2z2

2−g2

2

∫ τ

0

dτ ′′∆TL (τ ′−τ ′′, z2(τ

′)−z2(τ′′))

)]

×

× exp

[

−g2

∫ τ

0

dτ ′

∫ τ

0

dτ ′′∆L(τ′−τ ′′, z1(τ

′)−z2(τ′′))

]

The formulation fully accounts for retardation effects,without any need of defining an effective potential

35 / 54



In terms of Feynman diagrams...

36 / 54



The static limit

For M =∞ the HQs are frozen to their positions. The asymptoticbehavior of the real-time QQ propagator allows the to identify thein-medium effective potential:

G (t, r1−r2) ∼t→∞

exp[−iVeff(r1 − r2)t],

with

Veff(r1 − r2)effective potential

≡ g2

∫dq

(2π)3

(

1 − e iq·(r1−r2))

D00(ω=0,q)

= g2

∫dq

(2π)3

(

1 − e iq·(r1−r2)) [ 1

q2 + m2D

︸︷︷︸

screening

−iπm2

DT

|q|(q2 + m2D)2

︸︷︷︸

collisions

]

=−g2

4π

[

mD +e−mD r

r

]

− ig2T

4πφ(mDr)

One gets ∆FQstatic

= −αmD/2.37 / 54



Consistent treatment of screened self-energy andinteraction8

Veff(r) = −αmD −α

re−mD r

∼r→0

−αmD −α

r+αmD = −

α

r

For bound states of very small size medium effects cancel!

An analogous problem in solid-state physics...

Veff(r) turns out to coincide with the Ecker-Weitzel potential usedto study excitons (e-h bound states) in semiconductors.

8See also R. Rapp, D. Blaschke and P. Crochet, arXiv:0807.247038 / 54



Numerical resultsfrom the MC simulations for the path-integral

39 / 54



Spectral function reconstruction

G>(t =−iτ)︸︷︷︸

evaluated

≡ G (τ) =

∫ +∞

−∞

dω

2πe−ωτ σ(ω)

︸︷︷︸

reconstructed

.

Spectral analysis performed with the Maximum EntropyMethod9, requiring the introduction of a default model towhich the outcome reduces in case of very poor data for G (τ)

We require the default model to fullfill the (general!) sumrules

∫dω

2πσdef(ω) = 1,

∫dω

2πω σdef(ω) = M

9M. Jarrell and J.E. Gubernatis, Phys. Repts. 269 133 (1996)40 / 54



In order to interpret the numerical outcomes of thesimulations....

...some physical insight from (weak-coupling)thermal field theory calculations

41 / 54



Dyson equation for HQ propagator

Analytic non-relativistic HQ propagator

G (z) =−1

z − Ep − Σ(z ,p),

where Ep =M+p2/2M and setting z =ω+iη corresponds toretarded boundary conditions;

HQ spectral function:

σ(ω) ≡ 2Im GR(ω) =Γ(ω)

[ω − Ep − Re Σ(ω)]2 + Γ2(ω)/4,

with Γ(ω)≡−2Im ΣR(ω) =⇒ HQ spectral functionnon-vanishing only for energies for which the self-energydevelops an imaginary-part.

42 / 54



Resummed one-loop calculation

p

p0

0

t

t

a

a

The zero-momentum HQ self-energy reads:

Σ(p0) = g2CF

∫dk

(2π)3

∫ +∞

−∞

dk0

2πρL(k

0, k)1 + N(k0)−nF (Ek)

p0 − Ek − k0

Test-particle limit recovered setting nF (Ek)=0, which arisesnaturally in the regime T/M ≪ 1

Σtest(p0)=g2CF

∫dk

(2π)3

∫ +∞

0

dk0

2πρL(k

0, k)

[1 + N(k0)

p0 − Ek − k0+

N(k0)

p0 − Ek + k0

]

43 / 54



HQ spectrum: physical processes

ω ω ω ω

Ek E E Ek k k

kk0

0ω

ωL

L(k)

(k)

(a) (b) (c) (d)

Plasmon-pole contribution (a and b)

Γpole(ω) = g2CF

∫dk

(2π)3(2π)ZL(k)×

×[(1 + N(ωL(k))) δ(ω − Ek − ωL(k)) + N(ωL(k))δ(ω − Ek + ωL(k))]

Continuum contribution (c and d)

Γcont(ω) = g2CF

∫dk

(2π)3

∫ k

0

dk0 βL(k0, k)×

× (2π){[

1 + N(k0)]δ(ω − Ek − k0) + N(k0)δ(ω − Ek + k0)

}

44 / 54



HQ spectral funtion: one-loop result

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ω−M (GeV)

0

10

20

30

40

50

60

σ(ω

) (G

eV-1

)

M=1.5 GeVM=4.5 GeVM=45 GeVM=infinite

Negative shift and broadening (larger as M→∞!) of the main peak;

45 / 54




-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ω−M (GeV)

0

10

20

30

40

50

60

σ(ω

) (G

eV-1

)

M=1.5 GeVM=4.5 GeVM=45 GeVM=infinite


Appearance of secondary peaks at energies corresponding to a largedensity of states for plasmon absorption/emission processes;

45 / 54




-3 -2 -1 0 1 2 3(ω-M)/T

0

5

10

15

20

σ (ω

) T

T/M=0.133 (one-loop)T/M=0.200 (one-loop)T/M=0.333 (one-loop)T/M=0.133 (G

one-loop+MEM)

T/M=0.200 (Gone-loop

+MEM)

T/M=0.333 (Gone-loop

+MEM)


Appearance of secondary peaks at energies corresponding to a largedensity of states for plasmon absorption/emission processes;

MEM applied to Gone−loop(τ) captures just part of the features ofthe spectrum 45 / 54



HQ spectral funtion: path-integral result

-3 -2 -1 0 1(ω -M)/T

0

2

4

6

8

10

12

σ (ω

) T

M=infinite (gaussian prior)T/M=0.133 (gaussian prior)T/M=0.200 (gaussian prior)T/M=0.267 (gaussian prior)

-3 -2 -1 0 1 2 3 4 5(ω -M)/T

0

1

2

3

4

σ (ω

) T

M=infinite (constant prior)T/M=0.100 (constant prior)T/M=0.133 (constant prior)T/M=0.200 (constant prior)T/M=0.267 (constant prior)

Proper dimensionless units account for most of the T dependence;

Broadening and negative shift of the main peak and low-energystrenght (common feature);

Secondary bump (gauss.) and high-ω tail (const.) model-dependent;

Vertical lines at ω−M =−αmD

2 and lines at ω−M =±ωpl.

46 / 54



Some recent developments: the setup

We have shown how a static in-medium QQ potential can beobtained from the large-time behavior of the real-time Wilson loop:

V (r) = limt→∞

i∂tW (t, r)

W (t, r)

47 / 54





V (r) = limt→∞

i∂tW (t, r)

W (t, r)

The spectral function ρ(ω) allows one to establish a link betweenthe Euclidean Wilson loop W (τ) measured on the lattice and W (t)

W (τ) =

∫

dωe−ωτρ(ω) ↔

∫

dωe−iωtρ(ω) = W (t)

47 / 54





V (r) = limt→∞

i∂tW (t, r)

W (t, r)


W (τ) =

∫


∫


The in-medium QQ potential is then given by

V (r) = limt→∞

∫

dω ωe−iωtρ(ω, r)/

∫

dω e−iωtρ(ω, r)

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V (r) = limt→∞

i∂tW (t, r)

W (t, r)


W (τ) =

∫


∫


The in-medium QQ potential is then given by

V (r) = limt→∞

∫

dω ωe−iωtρ(ω, r)/

∫

dω e−iωtρ(ω, r)

The reconstruction of ρ(ω) exploiting the knowledge of W (τi ) at allvalues of τi is crucial!

47 / 54



Some recent developments: results

A new bayesian approach significantly improves the standard MEMreconstruction of the spectral funtion ρ(ω). One gets for the potential10

The real part turns out to coincide with the color-singlet free energy;

The imaginary part looks in qualitative agreement with theweak-coupling HTL result

10Y.Burnier et al., PRL 111, 182003 (2013) and arXiv:1410.2546 48 / 54



Perspectives

The challenge of interest for QGP phenomenology is toextract real-time information (i.e. transport coefficients) fromeuclidean simulations: viscosity (for the hydrodynamicevolution), diffusion coefficient (for HQ thermalization),electric conductivity (evolution of strong B-fields)

I’m waiting for volunteers for the QQ path-integralsimulations...

49 / 54



Back-up slides

50 / 54



Maximum Entropy Method

G (τ) =

∫ ∞

0

dω K (ω, τ)σ(ω)

G (τ): known for 1 ≤ τi/a ≤ Nτ (∼ 20);

σ(ω): for it one wants a very fine scan. ωl = l ·∆ω, with1 ≤ l ≤ Nω(∼ 102 − 103) =⇒ χ2 method not applicable;

H: a priori information on general properties (e.g. sum rules,positivity...) of the spectral function (key ingredient!)

One looks for the most probable spectral function compatible with thedata and the constraints:

δP[σ|G ,H]

δσ= 0,

where, from Bayes’ theorem

P[σ,G ,H] = P[σ|G ,H] × P[G |H] × P[H] = P[G |σ,H] × P[σ|H] × P[H]

51 / 54



Maximum Entropy Method

G (τ) =

∫ ∞

0

dω K (ω, τ)σ(ω)

G (τ): known for 1 ≤ τi/a ≤ Nτ (∼ 20);

σ(ω): for it one wants a very fine scan. ωl = l ·∆ω, with1 ≤ l ≤ Nω(∼ 102 − 103) =⇒ χ2 method not applicable;

H: a priori information on general properties (e.g. sum rules,positivity...) of the spectral function (key ingredient!)

One looks for the most probable spectral function compatible with thedata and the constraints:

δP[σ|G ,H]

δσ= 0,

hence:

P[σ|G ,H] ∼ P[G |σ,H]︸︷︷︸

likelihood function

× P[σ|H]︸︷︷︸

prior probability 52 / 54



Likelihood function: P[G |σ, H] ∼ e−L, with

L =1

2

∑

i ,j

[G (τi )︸︷︷︸

MCdata

−Gσ(τi )]

C−1ij

︸︷︷︸

cov matrix

[G (τj)︸︷︷︸

MCdata

−Gσ(τj)].

Maximizing it would correspond to the standard χ2-fitting.

Prior probability: P[σ|H] ∼ eαS , with

S =

∫ ∞

0

[

σ(ω) − m(ω) − σ(ω) lnσ(ω)

m(ω)

]

dω.

playing the role of an entropy term.

Default model m(ω): contains the a priori information on thespectral density;The entropy is maximum (S = 0) when the spectral functioncoincides with the default model. That’s what happens in theabsence of data!

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(Bryan’s) Maximum Entropy Method

For a given value of α one looks for the maximum ofP[σ|G , H] ∼ e−L+αS ≡ eQ[σ]:

δQ[σ|G , H]

δσ

∣∣∣∣σα(ω)

= 0,

where α controls the relative weight between

L: tends to fit σ to the data;S: tends to fit σ to the default model.

It’s like minimizing the free-energy in statistical mechanics!

One finally integrates over different values of α.

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suppression of quarkonia (and jets) in the quark gluon...

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