QCD at ﬁnite temperature and density on the lattice

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QCD at ﬁnite temperature and density on the lattice

Maria Paola Lombardo

INFN, Laboratori Nazionali di Frascati, Italy

Abstract. In the first lecture we briefly summarize the basics of field theory thermodynamics and critical phenomena. We then introduce the lattice gauge field theory approach to QCD at finite temperature and density, which is a non-perturbative scheme allowing first principle calculations using the QCD Lagrangian as a sole input. Some of the general concepts and idea introduced at the beginning are demonstrated by use of simple effective models of QCD. The second lecture is devoted to applications. We emphasize that current methods suffice to study the main phenomena at RHIC and LHC energies, and we discuss the ongoing theoretical efforts devoted to the solution of the sign problem which hampers the simulations of cold and dense matter. We conclude with short overview of the status of the field as of Summer 2008.

References

1. The Proceedings of Quark Matter, the International Conference on Ultrarelativistic Heavy Ion Collisions, usually include reviews on recent lattice results; the pro-ceedings of The International Symposium on Lattice Field Theoryusually include review on more speciﬁc aaspects of lattice thermodynamics.

2. M. P. Lombardo, Finite temperature ﬁeld theory and phase transitions,arXiv:hep-ph/0103141. Lectures given at ICTP Summer School on Astroparticle Physics and Cosmology, Miramare, Trieste, Italy, 12 Jun - 7 Jul 2000. Published in *Trieste 2000, Astroparticle and Cos-mology*

3. F. Karsch,Lattice QCD at high temperature and den-sity, Lect. Notes Phys. 583, 209 (2002) [arXiv:hep-lat/0106019].

4. J. B. Kogut and M. A. Stephanov, The phases of quantum chromodynamics: From conﬁnement to extreme Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 21 (2004) 1.

EPJ Web of Conferences 7, 01004 (2010) 7

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.

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Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)

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(5)

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0.0 0.5 1.0

T/0

0.0 0.5 1.0

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d-dimensional space Imaginary time

and

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0.4 0.8 1.2

0.8

0.4 0 0.5 1.0 T

m

μ

0.2 0.4 0.6

r=0.7 T=133MeV r=0.53

T=186MeV

r=0.43 T=216MeV 1.5

1.0

0.5

as

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0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5

T/T

c

/T_c

q=0

q0

1st 2nd TCP w/o baryons

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2 pi / 12 2 pi/24 T_c

Chiral transition

mu^2 T_E

Roberge Weiss critical line

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0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

T/TC

10 8 6 4 2 0 2 4 6

d

2M/d

2

S

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0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

0 50 100 150 200 250 300

frequency

<> for upd=0.0

0.10 0.20 0.30 0.40 0.50 0.60 0

50 100 150 200 250 300

frequency

<> for upd=0.625

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Glasgow

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Pbp

Mu

Real mu (m.f.) Im mu (m.f.) Im mu (simulation)

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Pbp

Mu

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0.00 0.10 0.20 0.30

0.35 0.40 0.45 0.50

<L>

Imaginary data Imaginary data fit Analytic continuation Uncertain analytic continuation Real Data

Lattice:8x8x8x4

=2.0 m=0.07

0.00 0.10 0.20 0.30 0.40

0.060 0.070 0.080 0.090 0.100

<

>

Imaginary data Imaginary data fit Analytic continuation Uncertain analytic continuation Lattice:8x8x8x4

=2.0 m=0.07

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0.2 0.3 0.4 0.5 0.6

0 0.2 0.4 0.6 0.8 1

T/S0

mu/S0

0 200 400 600 800

_B/MeV

145 150 155 160 165 170 175 180 185

T/MeV

N_f=2, [3]

N

f=2+1, [1]

N_f=2, [2]

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Helmholtz International Summer School

Dense Matter in Heavy Ion Collisions and Astrophysics

JINR Dubna Russia July 14–26 2008

QCD at nite temperature

and density

on the lattice

M.P. Lombardo, INFN Laboratori Nazionali di Frascati

HISS Dubna July 2008 MpL-QCD at nite T andμ

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LATTICE FIELD THEORY : FIRST PRINCIPLE APPROACH TO THE QCD PHASE DIAGRAM

T

μ_B

Hadron Gas

CSC

QGP

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PLAN

T μ μ B I Hadron Gas CSC ? Superfluid

1

2

3

4

QGP

1. Critical line and Critical Point

2. EoS and Critical Behaviour

4. Cold and Dense Matter : QCD−like models 3. Towards FAIR

T μ μ B I Hadron Gas CSC ? Superfluid 1 2 3 4 4 QGP

1. T

c

, THE CRITICAL LINE AND THE CRITICAL POINT

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T

c

:

RBC-B

IELEFELD

100 150 200 250

140 160 180 200 220 240

T [MeV]

0.75*p4fat3

asqtad

T

c

:

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T

c

AT

μ

B

= 0

:

STATUS AS OF

QM2008

RBC-B

IELEFELD

:

T

c

= 190(5)

MeV

W

UPPERTAL

-J

UELICH

:

T

c

= 175(5)

MeV (Glue)

T

c

= 151(6)

MeV (Fermions)

T

HE

CRITICAL

LINE

IN

THE

μ

B

, T

PLANE AT

SMALL

μ

B

T

=

T

c

−

Kμ

2B

K

∝

1 /N

f

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Results for Nf=2+1 from gluonic and fermionic susceptibilities

Fodor, Guse, Katz, K álm án K. Szab ó, QM2008

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C

HALLENGING THE

C

RITICAL

E

NDPOINT OF

QCD

Forcrand-Philipsen : strategy

Scenario I or Scenario II ? To decide, measure slopeKin

m

c

(

μ

)

m

c

(0)

= 1 +

K

_μ

T

2

+

...

K >0: Scenario I , critical endpoint at smallμB

K <0: Scenario II, NO critical endpoint at smallμB: favored at QM2006

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Kogut-Sinclair 2007 : QCD at nite isospin density

NB :μIμBatT TcToublan, Kogut, Sinclair 2004

m

c

(

μ

)

m

c

(0)

= 1

−

3 .

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_μ

_I

T

2 +

...

Current Results:

• Conrm unusual scenario forNt= 4

• Suggest NO critical endpoint forμB <600M eV • See talk by Philippe de Forcrand

SU(3)L×SU(3)RNJL MODEL

60 80

100 120 140_{160 180} 100 200 300 400 500 0 100 200 300 400 500 600 700 800 900 1000

B [MeV]

m [MeV]

mK [MeV] B,CEP physical point diagonal B [MeV]

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2. EOS AND CRITICAL BEHAVIOUR

LATTICE OBSERVABLES FOR THERMODYNAMICS

Number Density : accessible atimaginary chemical potential.

nu,d(T , μu, μd, mu, md) =∂p(T , μu, μd)

∂μu,d ;p(T , μu, μd) = T

VlnZ(T , μu, μd)

Susceptibilities: accessible atatμ= 0

χju,jd(T) =∂

(ju+jd)_p₍_{T , μu, μd}₎

∂μjuu ∂μjd_d

μu=μd=0

.

Test for uctuations.

Taylor coefcients of the excess pressure:

Δp(T , μu, μd)≡p(T , μu, μd)−p(T , μu= 0, μd= 0) =_ju,jdχju,jd(T)μ

ju u ju!

μjd d jd!,

containing information about baryon density effects in the EoS.

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Temperature

Real 2

QGP Hadron Gas Endpoint RW and Chiral

Chiral Transition Roberge Weiss Transition = 0

• In the critical region:

p(T , μ)

=b(T)|t+a(T)(μ2−μ2_c)|(2−α)

implying

n(T , μ) =

A(T)μ(μc2₋_μ2₎(2−α)

• High Temperature, away from critical line

Approach to Free Field

n(T , μ)→nSB(T , μ)

HADRON GAS; CRITICAL BEHAVIOUR; SB from SUSCEPTIBILITIES

0 0.5 1 1.5 2

150 200 250 300 350 T [MeV]

SB nf=2+1, m=220 MeV

nf=2, m=770 MeV

Resonance gas

1.5

2 <Q4>-3<Q2>2 <Q2_>

nf=2+1, m=220 MeV

nf=2, m=770 MeV

Resonance gas 0 0.5 1 1.5 2 2.5 3

50 100 150 200 250 300 350 400 450 T [MeV]

<S4_>-3<S2_>2 <S2>

SB nf=2+1, m=220 MeV

Resonance gas

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CRITICAL BEHAVIOR AND THERMODYNAMICS AT THE ENDPOINT OF THE RW TRANSITION

Critical behavior at imaginaryμ n(μI) =A(T)μI(μc2−μ2I)(2−

α) -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01

0 0.05 0.1 0.15 0.2 0.25 0.3

n

I

Data All data, open critical point >0.15, constrained critical point

Continued to realμ..

n(μ) =A(T)μ(μc2₊_μ2₎(2−α)

nSB(μ) =Aμ+Bμ3)→α= 1

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1

n(

)

Analytic Continuation

Stephan Boltzmann

D’Elia, Di Renzo, Lombardo, 2007, QM2008

CRITICAL BEHAVIOR, THERMODYNAMICS, QUASIPARTICLE MODELS K ¨ampfer, Bluhm Proposal

amenable to an easy comparison with lattice data.

I. Quasiparticlemodel vs Imaginary Chemical Potential Lattice Data, and analytic continuation to Real Chemical Potential

0 0.5 1 1.5

/ T 0 0.5 1 1.5 2 2.5

n / T

i

3

c

0 0.5 1 1.5

/ T 0 0.5 1 1.5 2 2.5 3 3.5 4

n / T

3

c

K ¨ampfer, Bluhm , 2007, QM2008

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0.8 1 1.2 1.4 1.6 1.8 2 T/T

0 0.05 0.1

c₄Q

c

0.8 1 1.2 1.4 1.6 1.8 2 2.2 T/T 0 0.2 0.4 c Crucial ingredients:

• Explicit dependence of the self-energy parts on_μi=μu,dandT

• Implicit dependence via the effective couplingG2(_{T , μu, μd}).

ω2_i=k2 +m2_i+ Πi, Πi=1₃

⎛ ⎝T2 +μ

2

i π2

⎞

⎠G2(_{T , μu, μd}).

K ¨ampfer-Bluhm : see Poster

APPROACH TO SB : ANALYTIC RESULTS VS. LATTICE DATA Based on A. Vuorinen, 2004

0.00 0.02 0.04 0.06 0.08 0.10

Μ Tc 0.88 0.90 0.92 0.94 0.96 N NSB

0.00 0.02 0.04 0.06 0.08 0.10

Μ Tc 0.90 0.92 0.94 0.96 0.98 N NSB

T

= 1

.

5 T

c

—————————-

T

= 3

.

5 T

c

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T

μ

B

I

Hadron Gas

CSC

? Superfluid

1

2

3

4

QGP

3. TOWARDS FAIR

STRONG COUPLING

New Results on The Mass Spectrum

Kawamoto, Miura 2007

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T

phys/Tc= 6T/Tc

( phys

0.5 1/3 0.6 0.4 2 3 0.7 0.8 0.9 5.75 5.62 5.50 5.55 5.70 5.67 5.65 5.80 5.85 1.0 0.95 0.9 0.85 0.8 0.75 0.7 = 0.4

( 2.4 ) ( 3.0 )

0.5 0.6

( 3.6 )

( 4.2 )

( 4.5 )

( 4.8 )

( 5.1 )

( 5.4 )

( 5.7 )

( 6.0 )

5.72

5.60

A B

———–

μ

B

Results forNf = 3:

• Identied phase transition • Indentied Ridge in the T,μplane • Studies of diquark in progress Di Pietro, Feo, Seiler, Stamatescu 2008

CANONICAL FORMALISM

ZC(T , N) =

3

2π

2πT

0 d(μI/T)e−N iμI/TZGC(T , iμI) Forcrand and Kratovchila, 2006,2007 Canonical partition a la Hasenfratz-Toussaint:

ZC(B,β)

ZGC(β₀₌β,μ=iμI₀)

=1

2π

π

−π d

_μI

T

e−i3B μI

T det(U;iμI)

det(U;iμI₀)β₀_,iμI

0

≡ _det(ZCˆ _U(U_;;B)

iμI₀)β₀_,iμI

0

Results forNf= 4:

• First order line-coexistence region • Criticalμand Critical densities→

0.6 0.8 1 1.2 1.4

0 2 4 6 8 10 12 14 16 0 1 2 3 4 5

/T

Baryon number /T3

1 2

T/Tc = 0.92 /T=1.06(2) Weakly interacting massless gas

0 5 10 15 20 25 30

0 0.5 1 1.5 20 1 2 3 4 5 6 7 8

Baryon number B

/T

3

/T = (F(B)-F(B-1))/3T T/Tc = 0.92

Saddle point approximation

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CANONICAL APPROACH II : HISTOGRAMS Proposed by Ejiri,2007

Alternative search for a critical point inNf= 2QCD, large masses

Alexandru, Li, Liu, 2007 :

0 5 10 15

2 3 4 5 6 7 ke Μ T 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10 1.31.41.51.61.7 0 2 4 6 8 10

Β 5.22

Β 5.20

CANONICAL APPROACH III :

Grand Partition Function via a Taylor Expansion. Ejiri, 2007,2008

Z_GC(T ,μq)

Z_GC(T ,0) ≡

e

N_fNtV∞n=1Dn

_μq

T

n

(T ,μq=0),

Z_C(T ,ρV¯ )

Z_GC(T ,0)at large V≈

expVN_fNtn∞=1Dnzn0−ρz¯0

e−iα/2

1

V|D(z₀₎|

(T ,μ=0)

Results forNf= 2:

• Qualitative change atT /Tc0.8 • Indication of First Order Line Ejiri 2008

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Density of states –ρ– constrained partition function:

ρ(x) = DU g(U)δ(φ−x).

Results forNf = 4

• Signal of two phase transition lines

• Indication for a triple point

Fodor, Katz, Schmidt 2007 90

100 110 120 130 140 150 160 170

0 50 100 150 200 250 300 350 400 q [MeV]

[MeV] multiparameter reweighting

DOS method, am=0.05

DOS method, am=0.03

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TWO COLOR QCD

• Symmetry :qq←→qq¯

• Symmetry :μB←→μI • At T=0 μc_B=μc_I =mπ/2 • Diquark Condensate forμB> μcB • Pion Condensate forμB> μcI • No sign problem

QCD atμI= 0and TWO COLOR QCD atμB= 0 No sign problem, many studies!

• Low Temperature, Fermionic Sector

– Strong coupling analysis [Dagotto, Karsch, Moreo,Wollf, 1986]

– Laboratory for diquark condensation [Alford,Rajagopal and Wilczek; Rapp, Sch ¨afer, Shuryak and Velkovsky, 1998]

– Numerical studies of symmetries and spectrum [Hands, Kogut, Morrison, MpL 1998]

– Lattice studies of diquark condensation [Hands, Kogut, Morrison, Sinclair;1998; Aloisio, Azcoiti, di Carlo, Galante, 1999]

– RMT analysis [Akemann, Splittorff, Toublan, Verbaarschot,2001] – Studies of the Dirac spectrum [Bittner, Markum, Pullirsch, MpL, 2001]

– μc(T= 0) =mπ/2: amenable toχPT analysis.

[Kogut, Stephanov, Toublan, Verbaarschot and Zhitnitsky, 2001; Kogut, Toublan, Sinclair 2002]

– EoS and Gluon Condensate fromχPT [Metlitsky,Toublan,Zhitnisky, 2005,2006]

– Model studies of the vector sector [Lenaghan, Sannino, Splittorff 2002]

– Vector spectroscopy on the lattice [Alles, D’Elia, Pepe, MpL, 2001; Muroya,Nakamura,Nonaka, 2003]

– Test Bed for Imaginaryμ[Cea, Cosmai, Giudice, D’Elia, Papa, 2006, 2007]

• High T , Gluon Sector niteμtransition from hadron gas to QGP, similar to the T=0 transition

– Critical Line [Kogut, Sinclair, 2003-2008]

– Topological Susceptibility [ Alles, D’Elia, Pepe, MpL 2002] – Polyakov Loop [Muroya,Nakamura, 2004; Alles, D’Elia, MpL 2006] – Order Parameter for Conement [D’Elia Conradi 2007] – Critical Line [Forcrand, Stephanov, Wenger, 2007]

• Low T : Chiral Symmetry , Connement, BEC Phase

– Connement/Deconnenement,Screening [Hands, Kim, Skullerud 2006–2008] – Glueballs spectrum, screening [Lombardo, Paciello, Petrarca, Taglienti, 2007] – BEC Phase, interface BEC/QGP [Forcrand, Stephanov, Wenger, 2007]

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4.4 4.5 4.6

0 1 2 3 4

I/T

1 1.5 2 2.5

0 1 2 3 4 5

I/T 0.7 0.6 0.5 T/T c BEC

m/T

Binder cumulant

Forcrand, Stephanov, Wenger, 2007

GLUONIC OBSERVABLES IN THE BEC PHASE of QC2D

0++Glueball : lighter in the BEC phase

Susceptibility:χ=< P2>−< P >2peaks atμc

Normal Phase

mπ/mρ m++₀ /mρ

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BEC AND CONFINEMENT: deconnement deep in the BEC phase

0.2 0.4 0.6 0.8 1

0 0.2 0.4

Hands, Kim, Skullerud 2006,2007

DECONFINEMENT IN THE BEC PHASE – BEC/BCS crossover

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.5 1 1.5 2 2.5 3

n_q/n_SB

pII/p_SB

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 5 10 15 20

q/SB nq/nSB p/pSB

Hands, Kim, Skullerud 2008; preliminary, courtesy S. Hands

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From SmallNc= 2to

Large

N

c

: Proposed Phase Diagram

McLerran, Pisarski, 2007

QCD AT FINITE TEMPERATURE AND DENSITY : SUMMARY

Tcknown within 20 % .

Fate of the critical endpoint of QCD???

Smallμ: Mature calculations. Critical line towards continuum Mass spectrum depends onμB

Critical line at lower temperature, largerμ’bends down’ First evidences of a triple point in the T,μBplane

Cold Phases studied atμI= 0/ QC2D atμB = 0

Examples of interesting patterns connement/chiral symmetry.

Critical region needs exact dynamics, models do not sufce