Locating QCD’s critical end point (with functional...

Christian S. Fischer

Justus Liebig Universität Gießen

Christian S. Fischer (University of Gießen) Locating QCD’s critical end point / 28

Eichmann, CF, Welzbacher, PRD in press, arXiv:1509.02082

20th of January 2016

Locating QCD’s critical end point(with functional methods)

1


+

dq

+

dq

N

−1=

−1+

NB

1.Introduction

2.Gluons, quarks and the CEP

3.Baryon effects on the CEP

Overview

2


Texttext

Text2text

Text3

QCD phase transitions

3


Texttext

Text2text

Text3


3

Is this happening ??

chemical potential


Texttext

Text2text

Text3


Lattice-QCDpresent: extrapolationfuture: exact methods ?

DSE/FRGnot exact, but allow for ’10%-physics’

3

Is this happening ??

chemical potential


Taylor expansion (Nf=2):

Reweighting (Nf=2+1):

Search for the CEP

4

Fodor, Katz, JHEP 0404 (2004) 050

Datta, Gavai and Gupta, NPA 904-905 (2013) 883c!Gavai, Gupta, PRD 71 (2005) 114014

Analytic continuation (Nf=3):de Forcrand, Philipsen, JHEP 0811 (2008) 012; ! NPB 642 (2002) 290


Chiral transition line from analytic continuation

5

μ

10

Lattice method:

Results:

Larger curvature than previous results (but: different definitions and error budget)

Bellwied, Borsanyi, Fodor, Günther, Katz, Ratti and Szabo, PLB B 751 (2015) 559

iµ

Calc. boundary at imaginary μ and extrapolate to real μ Control systematics


QCD in covariant gauge

6

MweakImaginary time formulation:

ZQCD =

ZD[�, A] exp

(�Z 1/T

0dt

Zd3x

⇣� (iD/ + �4µ�m)�

� 1

4

�F aµ�

�2+ gauge fixing

⌘�

Landau gauge propagators in momentum space, p = (⇥p,�p) :

The Goal: gauge invariant information in a gauge fixed approach.

DGluon

µ⌫ =ZT (p)

p2PTµ⌫(p) +

ZL(p)

p2PLµ⌫(p)

SQuark(p) = [i~�~pA(p) + i �4!̃n C(p) +B(p)]�1


dressed Polyakov loop

Polyakov loop potential

QCD order parameters from propagators

7

h�̄�i = Z2NcTrD1

T

X

�

Zd3p

(2�)3S(⇤p,⇥)

Chiral order parameter:

Deconfinement:

� = �Z 2�

0

d⇥

2�e�i⇥ h⇥̄⇥i⇥

Synatschke, Wipf, Wozar, PRD 75, 114003 (2007)Bilgici, Bruckmann, Gattringer, Hagen, PRD 77 094007 (2008)CF, PRL 103 052003 (2009)

L =1

NcTr eig�A0

� +�1

6�� (�� S)

�A0=

1

2Braun, Gies, Pawlowski, PLB 684, 262 (2010)Braun, Haas, Marhauser, Pawlowski, PRL 106 (2011)Fister, Pawlowski, PRD 88 045010 (2013)CF, Fister, Luecker, Pawlowski, PLB 732 (2014) 273

−1=

−1−


+

dq

+

dq

N

−1=

−1+

NB

1.Introduction



Overview

8


The DSE for the quark propagator

9

• dressed Gluon propagator

• dressed Quark-Gluon-Vertex

Input:

Two strategies: I. use model for gluon and vertex!

→ ok for first insights → not good enough for systematic study!II. determine gluon and vertex explicitly

[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)

Qin, Chang, Chen, Liu and Roberts, PRL 106 (2011) 172301

−1=

−1−


Nonperturbative gluon is massive Crucial difference between magnetic and electric gluonMaximum of electric gluon near Tc

Glue at finite temperature (T≠0)

10

T-dependent gluon propagator from quenched lattice simulations:

0 2 4 6 8 10p2 [Gev2]

0

0.5

1

1.5

2

2.5

Z L(p2 )

T=0T=0.6 TcT=0.99 TcT=2.2 Tc

0 2 4 6 8 10p2 [Gev2]

0

0.5

1

1.5

2

2.5

Z T(p)

T=0T=0.6 TcT=0.99 TcT=2.2 Tc

Cucchieri, Maas, Mendes, PRD 75 (2007)CF, Maas, Mueller, EPJC 68 (2010)Maas, Pawlowski, von Smekal and Spielmann, PRD 85 (2012) 034037Aouane, Bornyakov, Ilgenfritz, Mitrjushkin, Muller-Preussker and Sternbeck, PRD 85 (2012) 034501

FRG: Fister, Pawlowski, arXiv:1112.5440


under study at T=0

!

T≠0: ansatz, T,m,μ dependent

DSEs of QCD

11

quenched, T-dependent lattice propagator

quark gluon vertex

−1=

−1− 1

2

− 12 − 1

6

+ − 12

+

−1=

−1−

Skullerud, Kizilersu, JHEP 0209 (2002) 013CF, Williams PRL 103 (2009) 122001Mitter, Pawlowski and Strodthoff, PRD 91 (2015) 054035Williams, Fischer, Heupel, PRD in press, arXiv:1512.00455Sternbeck et al. in preparation


Approximation for Quark-Gluon interaction

12

Abelian WTI

perturbation theory

Infrared ansatz: • d2 fixed to match gluon input• d1 fixed via quark condensate (see later)• correct UV (quant.) and IR-behavior (qual.)

�⇥(q, k, p) = eZ3

✓⌅4⇥⇤4

C(k) + C(p)

2+ ⌅j⇥⇤j

A(k) +A(p)

2

◆⇥

⇥

d1d2 + q2

+q2

⇥2 + q2

✓⇥0�(µ) ln[q2/⇥2 + 1]

4⇧

◆2�!

T,μ,m-dependent vertex:

CF, Pawlowski, PRD 80 (2009) 025023 Mitter, Pawlowski and Strodthoff, PRD 91 (2015) 054035Williams, Fischer, Heupel, PRD in press, arXiv:1512.00455


Expect: Transitions controlled by deconfinementSU(2) second order, SU(3) first order

QCD phase transition: heavy quark limit/quenched

13

Quark mass dependence:−1

=−1

−1

2

−1

2−

1

6

+ −1

2

+ +

−1=

−1−

−1=

−1−


Deconfinement transition in agreement with lattice QCDCorrect tricritical scaling

Critical line/surface for heavy quarks

14

0

0.2

0.4

0.6

0.8

1

200 220 240 260 280

Φ

T [MeV]

m = 320 MeVm = 363 MeVm = 400 MeV

Mean-field scaling

0.2

0.4

0.6

0 0.015 0.03

(T-Tc)/Tc

Polyakov Loop:

Fromm, Langelage, Lottini, Philipsen, JHEP 1201 (2012) 042

CF, Luecker, Pawlowski, PRD 91 (2015) 1


Deconfinement transition in agreement with lattice QCDCorrect tricritical scaling

Critical line/surface for heavy quarks

14

0

0.2

0.4

0.6

0.8

1

200 220 240 260 280

Φ

T [MeV]

m = 320 MeVm = 363 MeVm = 400 MeV

Mean-field scaling

0.2

0.4

0.6

0 0.015 0.03

(T-Tc)/Tc

Polyakov Loop:

Fromm, Langelage, Lottini, Philipsen, JHEP 1201 (2012) 042

CF, Luecker, Pawlowski, PRD 91 (2015) 1


Physical up/down and strange quark masses Transition controlled by chiral dynamics at μ=0: compare to available lattice results

QCD phase transitions: Nf=2+1

15

−1=

−1−

1

2

−1

2−

1

6

+ −1

2

+ +

−1=

−1−

−1=

−1−


Nf=2+1, zero chemical potential

16

Lattice: Borsanyi et al. [Wuppertal-Budapest Collaboration], JHEP 1009(2010) 073DSE: CF, Luecker, PLB 718 (2013) 1036, CF, Luecker, Welzbacher, PRD 90 (2014) 034022

Crossover

bare quark masses

100 150 200 250T [MeV]

0

0,2

0,4

0,6

0,8

1

∆l,s

(T)/∆

l,s(0

)

Lattice QCDQuark Condensatedressed Polyakov Loop

zero chemical potential


quantitative agreement

Nf=2+1, zero chemical potential

16

Lattice: Borsanyi et al. [Wuppertal-Budapest Collaboration], JHEP 1009(2010) 073DSE: CF, Luecker, PLB 718 (2013) 1036, CF, Luecker, Welzbacher, PRD 90 (2014) 034022

Crossover

bare quark masses

100 150 200 250T [MeV]

0

0.2

0.4

0.6

0.8

1

∆l,s

(T)/∆

l,s(0

)

Lattice QCDQuark Condensatedressed Polyakov Loop

zero chemical potential


quantitative agreement: DSE prediction verified by lattice

Unquenched Gluon DSE vs Lattice

17

Aouane, Burger, Ilgenfritz, Muller-Preussker and Sternbeck, PRD 87 (2013) 11 [arXiv:1212.1102]

CF, Luecker, PLB 718 (2013) 1036 [arXiv:1206.5191]DSE:Lattice:

0 1 2 3p [GeV]

0

1

2

3

4

ZL

T = 187 MeV Quenched



T = 187 MeV LatticeT = 215 MeV LatticeT = 235 MeV LatticeT = 187 MeV DSET = 215 MeV DSET = 235 MeV DSE


Nf=2+1: Condensate

18

Quark condensate


evaluated from Polyakov-Loop potentialimportant input for P-models: PQM, PNJL !

Nf=2+1: Polyakov loop potential at finite μ

19

� +�1

6�� (�� S)

�A0=

1

2

L =1

Nctr eig

RA0

Polyakov-Loop

CF, Fister, Luecker, Pawlowski, PLB 732 (2014) 273

Herbst, Mitter, Pawlowski, Schaefer, Stiele, PLB 731 (2014) 248


Nf=2+1: Polyakov loop and phase diagram

20

0 50 100 150 200µq [MeV]

0

50

100

150

200

T [M

eV]

DSE: chiral crossoverDSE: critical end pointDSE: chiral first orderDSE: deconfinement crossover



combined evidence of FRG and DSE: no CEP at μB/T<2

20

CF, Luecker, PLB 718 (2013) 1036,CF, Fister, Luecker, Pawlowski, PLB 732 (2014) 273CF, Luecker, Welzbacher, PRD 90 (2014) 034022

CEP at large μ0 50 100 150 200

µq [MeV]0

50

100

150

200

T [M

eV]


µB/T=2µB/T=3




20


CEP at large μ0 50 100 150 200

µq [MeV]0

50

100

150

200

T [M

eV]


µB/T=2µB/T=3

Extrapolated curvature from latticeKaczmarek at al. PRD 83 (2011) 014504,Endrodi, Fodor, Katz, Szabo, JHEP 1104 (2011) 001Cea, Cosmai, Papa, PRD 89 (2014) 074512

0 50 100 150 200µ

q [MeV]

0

50

100

150

200

T [

MeV

]

Lattice: curvature range κ=0.0066-0.0180

DSE: chiral crossoverDSE: critical end point

DSE: chiral first orderDSE: deconfinement crossover

µB/T=2

µB/T=3




20

Nc=2: Brauner, Fukushima and Hidaka, PRD 80 (2009) 74035 Strodthoff, Schaefer and Smekal, PRD 85 (2012) 074007

Caveat: baryon effects missing...


CEP at large μ0 50 100 150 200

µq [MeV]0

50

100

150

200

T [M

eV]


µB/T=2µB/T=3

Extrapolated curvature from latticeKaczmarek at al. PRD 83 (2011) 014504,Endrodi, Fodor, Katz, Szabo, JHEP 1104 (2011) 001Cea, Cosmai, Papa, PRD 89 (2014) 074512

0 50 100 150 200µ

q [MeV]

0

50

100

150

200

T [

MeV

]

Lattice: curvature range κ=0.0066-0.0180

DSE: chiral crossoverDSE: critical end point

DSE: chiral first orderDSE: deconfinement crossover

µB/T=2

µB/T=3


Nf=2+1+1-QCD with DSEs

21

−1=

−1+ + +

−1=

−1−

−1=

−1−

−1=

−1−

up/down strange

charm

Physical up/down, strange and charm quark masses

Transition controlled by chiral dynamics

no lattice or model results available yet


Nf=2+1+1-QCD with DSEs

21

−1=

−1+ + +

−1=

−1−

−1=

−1−

−1=

−1−

up/down strange

charm

Physical up/down, strange and charm quark masses

Transition controlled by chiral dynamics

no lattice or model results available yet

CF, Luecker, Welzbacher, PRD 90 (2014) 034022

0 50 100 150 200µu/d [MeV]

0

50

100

150

200

T [M

eV]

Nf=2+1Nf=2+1+1


+

dq

+

dq

N

−1=

−1+

NB

1.Introduction



Overview

22


QCD phase transitions I

23

Fukushima, Hatsuda, Rept. Prog. Phys. 74 (2011) 014001

Low temperatures, large chemical potential: baryons are important degrees of freedom

How do baryons affect the quark condensate ??


‘Off-shell baryons’ do affect quark condensate…

Baryon effects onto quark I

24

−1=

−1−


Dependence on T and μ via -propagators -wave functionsExploratory calculation: use wave functions from T=μ=0

Baryon effects onto quark II

25

−1=

−1+

NB

+

B

+

dq

+

dq

N

−1=

−1+

NB


DSE/Faddeev landscape (T=μ=0)

26

Eichmann, AlkoferNicmorus, Krassnigg

Eichmann, N*-Workshop, Trento 2015

Oettel, AlkoferRoberts, BlochSegovia et al.

Roberts et al Eichmann, AlkoferSanchis-Alepuz, CF

Sanchis-Alepuz, CFWilliams



26








26








26







Zero chemical potential: no effects after rescaling

Baryon effects - results (Nf=2)

27

+

dq

+

dq

N

−1=

−1+

NB





27

CEP: almost no effects

0 50 100 150 200µ

q [MeV]

100

150

200

250

T [

MeV

]

gluon-dressing loop only+ baryon loop + diquark loop (rescaled strength)+ baryon loop + diquark loop (rescaled strength, prefactor)

gluon dressing only+ baryon + diquark (rescaled strength)+ baryon + diquark (rescaled strength, prefactor)

+

dq

+

dq

N

−1=

−1+

NB





27

CEP: almost no effects

0 50 100 150 200µ

q [MeV]

100

150

200

250

T [

MeV

]

gluon-dressing loop only+ baryon loop + diquark loop (rescaled strength)+ baryon loop + diquark loop (rescaled strength, prefactor)

gluon dressing only+ baryon + diquark (rescaled strength)+ baryon + diquark (rescaled strength, prefactor)

But: strong μ-dependence of baryon wave function may change things…

+

dq

+

dq

N

−1=

−1+

NB



Temperature dependent gluon propagator

characteristic behaviour of electric gluon

‘melting’ of magnetic gluon with temperature

Deconf. Tpc from dressed Polyakov-loop/Polyakov-loop potential

QCD with finite chemical potential (beyond mean field)

back-reaction of quarks onto gluons important

Nf=2+1 and Nf=2+1+1: CEP at μc/Tc > 3

charm quark does not influence CEP

Baryon effects may or may not be significant for CEP…

Summary

28

Work in progress: - mesons and baryons at finite T and μ - quark-gluon vertex at finite T and μ


Back-up

29

Back-up slides


spacelike momenta: good agreement with lattice

fully dressed gluon is not massless !

recent improvement: 3g-vertex

Landau gauge gluon propagator

30

−1=

−1− 1

2

− 12 − 1

6

+ − 12

−1=

−1−

CF, Maas, Pawlowski, Annals Phys. 324 (2009) 2408.

Dµ⌫(p) =

✓�µ⌫ � pµp⌫

p2

◆Z(p2)

p2

0 1 2 3 4 5p [GeV]

0

0.5

1

1.5

Z(p2 )

Sternbeck et. al. (2006)DSE

Eichmann, Williams, Alkofer, Vujinovic PRD 89, (2014) 10


spectral function: positivity violations

Landau gauge gluon propagator

31

spectral function

Strauss, CF, Kellermann, Phys. Rev. Lett. 109, (2012) 252001

Gluon cannot appear in detector!

Cornwall, Papavassiliou,...


Quark mass: flavor dependence

32

M(p2): momentum dependent!

Dynamical mass: Mstrong≈350 MeV

Flavour dependence because of mweak

Chiral condensate:

10-2 10-1 100 101 102 103

p2 [GeV2]

10-4

10-3

10-2

10-1

100

101

Qua

rk M

ass F

unct

ion:

M(p

2 ) [G

eV]

Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit

Typical solution:

h�̄�i ⇡ (250MeV)30.0 1.0 2.0 3.0 4.0p [GeV]

0.0

0.1

0.2

0.3

0.4

M(p

) [G

eV]

Lattice: quenched Lattice: unquenched (N

f=2+1)

DSE: quenchedDSE: unquenched (N

f=2)

CF, Nickel, Williams, EPJ C 60 (2009) 47

[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)


Quark mass: flavor dependence

32

M(p2): momentum dependent!

Dynamical mass: Mstrong≈350 MeV

Flavour dependence because of mweak

Chiral condensate:

10-2 10-1 100 101 102 103

p2 [GeV2]

10-4

10-3

10-2

10-1

100

101

Qua

rk M

ass F

unct

ion:

M(p

2 ) [G

eV]

Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit

Typical solution:

h�̄�i ⇡ (250MeV)3

Not directly measu

rable !

0.0 1.0 2.0 3.0 4.0p [GeV]

0.0

0.1

0.2

0.3

0.4

M(p

) [G

eV]

Lattice: quenched Lattice: unquenched (N

f=2+1)

DSE: quenchedDSE: unquenched (N

f=2)

CF, Nickel, Williams, EPJ C 60 (2009) 47

[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)

Christian Fischer (University of Gießen) Glueballs, tetraquarks and excited baryons from DSEs / 38

irreducible three-body forcestwo-body interactions:

non-perturbative gluon exchangemeson exchange two-body forces beyond one-particle exchange

numerically expensive but manageable !

Faddeev - equation

33

++= +

Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010)

Sanchis-Alepuz, CF, Kubrak, PLB 733 (2014)

Sanchis-Alepuz, Williams, work in progress…

Sanchis-Alepuz, Williams, PLB 749 (2015) 592

http://file://localhost/users/chfi/downloads/baryonbse3body.svg





Faddeev - equation

33

++= +










Faddeev - equation

33

++= +







large temperatures: behaviour as expected from HTLfirst order transition at large chemical potential

Nf=2+1: thermal electric gluon mass

34

CF, Luecker, PLB 718 (2013) 1036

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