QCD phase transitions with functional methods...Christian S. Fischer (JLU Giessen) QCD phase transitions with functional methods Feb. 2011 5 / 34. Why we need Functional Methods like

QCD phase transitions with functional methods

Christian S. Fischer

JLU Giessen

Feb. 2011

C.F., A. Maas and J. A. Mueller, EPJ C68 (2010) 165-181.

J. A. Mueller, C.F. and D. Nickel, EPJ C70 (2010) 1037

J. Luecker, C.F., J. A. Mueller, in preparation

Christian S. Fischer (JLU Giessen) QCD phase transitions with functional methods Feb. 2011 1 / 34

Content

1 Introduction

2 Properties of SU(N) Yang-Mills theoryT = 0T 6= 0

3 Chiral and deconfinement transitions in QCD

4 Quark spectral functions


Content

1 Introduction





FAIR: CBM and PANDA


QCD phase transitions

Existence and locationof CEP

Propagation of gluons inQGP

Properties of quarks inQGP

Chiral limit (Mweak → 0): order parameter chiral condensate

〈ψ̄ψ〉 = Z2NcTrD

∫d4p(2π)4 S(p)

Static quarks (Mweak → ∞): order parameter Polyakov-loop

Φ ∼ e−Fq/T


Why we need Functional Methods like the RG

Quark-meson model:

B.-J. Schaefer, J. Wambach PRD 75 (2007) 085015

determine size of critical region from quark number susceptibility

χq(T , µ) = ∂n(T , µ)/∂µ

RG: dramatic decrease in size !→ relevant for CBM


QCD in covariant gauge

quarks, gluons and ghosts:

ZQCD =

∫D[Ψ,A, c] exp

{−

∫d4x

(Ψ(iD/ − m)Ψ

−14

(F aµν

)2+

(∂A)2

2ξ+ c̄(−∂D)c

)}

SQCD =

∫d4x

(

−1+ +

−1+ +

−1+ +

)


QCD in covariant gauge

ZQCD =

∫D[Ψ,A, c] exp

{−

∫ 1/T

0dt∫

d3x(Ψ̄ (iD/ − m)Ψ

−14

(F aµν

)2+

(∂A)2

2ξ+ c̄(−∂D)c

)}

Landau gauge (ξ = 0) propagators in momentum space, q = (~q, ωq):

DGluonµν (q) =

ZT (q)q2 PT

µν(q) +ZL(q)

q2 PLµν(q)

SQuark(q) =1

−i ~γ~q A(q)− i γ4ωn C(q) + B(q)

The Goal:Gauge invariant information from gauge fixed functional approach


Green’s functions

QCD Green’s functionsare connected to confinement:

Gribov-Zwanziger and Kugo-Ojima scenariosRunning CouplingPositivityPolyakov Loop

encode DχSB

are ingredients for hadron phenomenologyBound state equations:Bethe–Salpeter equation / Faddeev equationForm factors, decays etc.

The Goal:Gauge invariant information from gauge fixed functional approach

The Tool:Dyson-Schwinger and Bethe-Salpeter-equations (DSE/BSE)


Lattice QCD vs. DSE/FRG: Complementary!

Lattice simulations

◮ Ab initio

◮ Gauge invariant

Functional approaches:Dyson-Schwinger equations (DSE)Functional renormalisation group (FRG)

◮ Analytic solutions at small momenta

◮ Space-Time-Continuum

◮ Chiral symmetry: light quarks and mesons

◮ Multi-scale problems feasible: e.g. (g-2)µT. Goecke, C.F., R. Williams, arXiv:1012.3886 [hep-ph]

◮ Chemical potential: no sign problem


Content

1 Introduction





Dyson-Schwinger equations (DSEs)

−1

=

−1

-1

2

-1

2-

1

6

-1

2+

−1

=

−1

-


DSEs vs Lattice (T = 0)

0 1 2 3 4 5p [GeV]

0

0.5

1

1.5

ZT=Z

L

Bowman (2004)Sternbeck (2006)DSEFRG

Deep infrared: Interesting and subtle questions

Physics: positivity violation, glueball states

L. von Smekal, R. Alkofer, A. Hauck, PRL 79 (1997) 3591-3594.C.F., A. Maas and J. M. Pawlowski, Annals Phys. 324 (2009) 2408-2437.


Glue at finite temperature T 6= 0

T -dependent gluon propagator from lattice simulations:

0 5 10 15

p2 [Gev

2]

0

0.5

1

1.5

2

2.5

ZT(p

2 )

T=0T=0.6 T

cT=0.99 T

cT=2.2 T

c

0 5 10 15

p2 [Gev

2]

0

0.5

1

1.5

2

2.5

ZL(p

2 )

T=0T=0.6 T

cT=0.99 T

cT=2.2 T

c

Difference between electric and magnetic gluon

Maximum of electric gluon at Tc

Cucchieri, Maas, Mendes, PRD 75 (2007)

C.F., Maas and Mueller, EPJC 68 (2010)


Gluon screening mass

0 0.5 1 1.5 2 2.5T/T

c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4m

el (lattice)

SU(2) - ’new’ data

0 0.5 1 1.5 2 2.5T/T

c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 mel

(lattice)Fits

SU(3) - ’new’ data

C.F., Maas, Mueller, EPJC 68 (2010)

ZT ,L(q,T ) q2Λ2

(q2+Λ2)2

{

(

cq2+Λ2aT ,L(T )

)bT ,L(T )

+ q2

Λ2

(

β0α(µ) ln[q2/Λ2+1]4π

)γ}

SU(2): 2nd order phase transition not yet visible

Fits more reliable at large T than lattice: m ∝ T


Content

1 Introduction





The ordinary chiral condensate�1 = �1 +gluon propagator from DSE/lattice

T = 0 : quark-gluon vertex studied via DSEsAlkofer, C.F., Llanes-Estrada, Schwenzer, Annals Phys.324:106-172,2009.

C.F, R. Williams, PRL 103 (2009) 122001

T 6= 0 : temperature and mass dependent ansatz

Order parameter for chiral symmetry breaking:

〈ψ̄ψ〉 = Z2NcT∑

np

∫d3p(2π)3 TrD S(~p, ωp)


T = 0: Explicit vs. dynamical chiral symmetry breaking

�1 = �1 +

10-2

10-1

100

101

102

103

p2 [GeV

2]

10-4

10-3

10-2

10-1

100

101

Qua

rk M

ass

Fun

ctio

n: M

(p2 )

Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit

C.F. J.Phys.G G32 (2006) R253-R291

M(p2) = B(p2)/A(p2):momentum dependent!

Dynamical massesMstrong(0) ≈ 350 MeV

Flavour dependencebecause of Mweak

〈ψ̄ψ〉 ≈ (250MeV)3


T 6= 0: Chiral symmetry restauration

SQuark(q) =1

−i ~γ~q A(q)− i γ4ωn C(q) + B(q)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

p2

[GeV2]

10-3

10-2

10-1

100

B(i

ω0,p

) [

GeV

]

T= 130 MeVT= 190 MeV

m(µ)=1.85 MeV

10-4

10-2

100

102

104

p2

[GeV2]

1

1.2

1.4

1.6

1.8

A(i

ω0,p

) ,

C(i

ω0,p

) T= 130 MeVT= 190 MeV

m(µ)=1.85 MeV

A(iω0,p)

C(iω0,p)

dynamical effects below Tc ↔ ’HTL-ish’ above Tc


The Polyakov Loop

Φ =

⟨1

NcTrDP exp

{i∫ 1/T

0A4 dt

}⟩∼ e−Fq/T

Lattice:

x

t

Order parameter for centersymmetry breaking:Φ = 0 : confinedΦ 6= 0 : deconfined


The dual condensate I

Consider general U(1)-valued boundary conditions in temporaldirection for quark fields ψ:

ψ(~x ,1/T ) = eiϕψ(~x ,0)

Matsubara frequencies: ωp(nt) = (2πT )(nt + ϕ/2π)

Lattice:ϕ

e i

〈ψψ〉ϕ ∼∑ exp[i ϕn]

(am)l Closed Loops

E. Bilgici, F. Bruckmann, C. Gattringer and C. Hagen, PRD 77 (2008) 094007.F. Synatschke, A. Wipf and C. Wozar, PRD 75, 114003 (2007).


The dual condensate II

Then define dual condensate Σn:

Σn = −

∫ 2π

0

dϕ2π

e−iϕn 〈ψψ〉ϕ

n = 1 projects out loops with n(l) = 1: dressed Polyakov loop

transforms under center transformation exactly like ordinaryPolyakov loop: order parameter for center symmetry breaking

Σ1 is accessible with functional methodsC.F., PRL 103 (2009) 052003

C. Gattringer, PRL 97, 032003 (2006)F. Synatschke, A. Wipf and C. Wozar, PRD 75, 114003 (2007).E. Bilgici, F. Bruckmann, C. Gattringer and C. Hagen, PRD 77 094007 (2008).F. Synatschke, A. Wipf and K. Langfeld, PRD 77, 114018 (2008).J. Braun, L. Haas, F. Marhauser, J. M. Pawlowski, PRL 106 (2011)


Condensate: angular dependence in chiral limit

Σ1 = −

∫ 2π

0

dϕ2π

e−iϕ 〈ψψ〉ϕ

Width of plateau is T -dependent, 〈ψψ〉ϕ(ϕ = 0) ∼ T 2

C.F. and Jens Mueller, PRD 80 (2009) 074029.

Jens Mueller, PhD-thesis, TU Darmstadt, 2010Christian S. Fischer (JLU Giessen) QCD phase transitions with functional methods Feb. 2011 23 / 34

Transition temperatures, quenched

quenched, DSE

0 0.5 1 1.5 2 2.5T/T

c

0.00

0.05

0.10Polyakov loopCondensate

C.F., Maas, Mueller, EPJC 68 (2010).Mueller, PhD-thesis.

SU(2): Tc ≈ 305 MeVSU(3): Tc ≈ 270 MeV

increasing condensate due toelectric part of gluon

quenched, FRG

J. Braun, H. Gies, J. M. Pawlowski, PLB 684 (2010) 262-267.

Polyakov loop potential viagluon/ghost propagators

cf. Buividovich, Luschevskaya, Polikarpov, PRD 78 (2008) 074505.


Transition temperatures, Nf = 2DSE

0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19T [GeV]

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Polyakov LoopCondensate

Luecker, C.F., Mueller, in preparationMueller, PhD-thesis TU Darmstadt 2010.

reduction of Tc: Tc ≈ 165 MeV

’usual’ behaviour ofcondensate

FRG

0

0.2

0.4

0.6

0.8

1

150 160 170 180 190 200 210 220 230

T [MeV]

fπ(T)/fπ(0)

Dual density

Polyakov Loop

160 180 200

χ L,d

ual

J. Braun, L. Haas, F. Marhauser, J. M. Pawlowski, PRL 106(2011) 022002

chiral limit

Tχ ≃ Tconf ≃ 180 MeV


µ 6= 0 : Chiral and deconfinement transition

0 100 200 300µ [MeV]

0

50

100

150

200

T [

MeV

]

Polyakov Loopkonjugate P.-Loopchiral crossoverchiral 1st orderCEP

approximation: backcoupling of quarks onto glue via HTL

CEP at (T , µ) ≃ (85,275) MeV

Small µ: difference between Φ and Φ̄ with Φ < Φ̄C.F., Luecker, Mueller, in preparationJ. Mueller, PhD-thesis, TU Darmstadt, 2010


Content

1 Introduction





Framework

Vital input into transport approaches (HSD,...)

Computation of quark-loop contribution of dilepton productionBraaten, Pisarski, Yuan, PRL 64, 1990

Idea: Fit spectral representation to quark propagatorF. Karsch and M. Kitazawa, PRD 80, 056001 (2009).F. Karsch and M. Kitazawa, PLB 658, 45 (2007).

S(ωp, ~p) =∫

dω′ρ(ω′, ~p)ωp − ω′

Use ansatz for spectral function:

ρ(ω) = Z1 δ(ω − E1) + Z2 δ(ω + E2)

Pseudoparticles: Quark and Plasmino (idealized!)


Results I: Deconfinement and quark analytic structure

0 0.2 0.4 0.6 0.8 1τT

0.04

0.2

1

S(τ)

T≈1.03Tc

T≈0.92Tc

T≈0.73Tc

Mueller, C.F., Nickel, EPJ C70 (2010) 1037

T > Tc : Two pole ansatz works: quark and plasmino

T < Tc : No fit with only real poles; positivity violations

Alternative criterion for deconfinement


Results II: Mass dependence of quark and plasmino

Lattice DSE

1

2

3

4

5

E1/T

, E

2/T

0 0.25 0.5 0.75 1mµ/T

0

0.1

0.2

0.3

0.4

0.5

Z2/(

Z+

Z2) T/T

c= 2.2

T/Tc= 1.5

T/Tc= 1.25

Karsch and Kitazawa, PRD 80, 056001 (2009). Mueller, C.F., Nickel, EPJ C70 (2010) 1037

Qualitative agreement with lattice results

Large quark masses: plasmino disappears


Results III: Dispersion Relation

Lattice DSE

1

2

3

4

5

E1/T

, E

2/T

T/Tc= 2.2

T/Tc= 1.5

T/Tc= 1.25

0 1 2 3 4 5p/T

0

0.1

0.2

0.3

0.4

0.5

Z2/(

Z1+

Z2)

E2/T

(plasmino)

E1/T

(normal)

Karsch and Kitazawa, PRD 80, 056001 (2009). Mueller, C.F., Nickel, EPJ C70 (2010) 1037

Min for Plasmino at p 6= 0

Plasmino enters spacelike region → fit function incomplete!


Results IV: Dispersion Relation - Details

0.6

0.8

1

1.2

1.4

1.6

E1/T

, E

2/T

T/Tc=2.2

T/Tc=1.5

0 0.2 0.4 0.6 0.8 1p/T

0

0.1

0.2

0.3

0.4

0.5

Z2/Z

tot

E2/T

E1/T

0.75

0.8

0.85

0.9

0.95

E1/T

, E

2/T

T/Tc=2.2

T/Tc=1.5

0 0.05 0.1 0.15 0.2p/T

0.42

0.44

0.46

0.48

0.5

Z2/(

Z1+

Z2)

E2/T

E1/T

Include continuum part (Landau damping) into fits:

ρP±(ω, |p|) = 2π

[

Z1δ(ω ∓ E1) + Z2δ(ω ± E2)]

+π

|p|m2

T (1 ∓ω

|p|)Θ(1 − (

ω

|p|)2)

[

(

|p| (1 ∓ω

|p|) ±

m2T

2 |p|

[

(1 ∓ω

|p|) ln

∣

∣

∣

∣

∣

∣

ω

|p| + 1

ω

|p| − 1

∣

∣

∣

∣

∣

∣

± 2])2

+π2m4

T

4|p|2(1 ∓

ω

|p|)2]−1

Dashed lines: HTL-result of slope at p → 0.Mueller, C.F., Nickel, EPJ C70 (2010) 1037


Summary: QCD phase transitions

Summary:

Temperature dependent gluon propagator:characteristic behavior of electric screening mass at Tc

Similar Tc from dressed Polyakov-loop calculated from DSEs

Similar chiral Tc from ordinary quark condensate

Finite chemical potential beyond mean field

Quark spectral functions: quarks and plasminos

Outlook:

Quark spectral functions at finite chemical potential

Thermodynamic observables


Thank you for your attention!

Happy Birthday and Happy Retirement, Ulrich!

Helmholtz Young Investigator Group ”Nonperturbative Phenomena in QCD”


Gluon and Quark-Gluon-Vertex

Ansatz for Quark-Gluon-Vertex:

Γν(q, k ,p) = Z̃3

(δ4νγ4

C(k) + C(p)2

+ δjνγjA(k) + A(p)

2

)×

×

(d1

d2 + q2 +q2

Λ2 + q2

(β0α(µ) ln[q2/Λ2 + 1]

4π

)2δ).


Ghost, Glue and Coupling

10-4

10-3

10-2

10-1

100

101

102

p2 [GeV

2]

10-4

10-3

10-2

10-1

100

101

102

G(p2)

Z(p2)

’decoupling’’decoupling’

10-4

10-3

10-2

10-1

100

101

102

103

104

p2 [GeV

2]

10-1

100

101

α(p2 )

running coupling

dynamically generated scalefixed point of coupling α(p2) = g2/(4π)Z (p2)G(p2) ≈ 9/Nc

deep infrared (p < 50 MeV): scaling vs. decoupling

CF and Alkofer, PLB 536 (2002) 177. C. Lerche and L. von Smekal, PRD 65, 125006 (2002)

C.F., A. Maas and J. M. Pawlowski, Annals Phys. 324 (2009) 2408-2437.


Documents

QCD phase transitions with functional methods...Christian S. Fischer (JLU Giessen) QCD phase transitions with functional methods Feb. 2011 5 / 34. Why we need Functional Methods like