On the role of fluctuations in (2+1)-flavor QCD Bernd-Jochen - - PowerPoint PPT Presentation

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On the role of fluctuations in (2+1)-flavor QCD Bernd-Jochen - - PowerPoint PPT Presentation

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On the role of fluctuations in (2+1)-flavor QCD Bernd-Jochen Schaefer Austria Germany Germany September 21 st , 2017 September 20-22, 2017 Zalakaros Conjectured QC 3 D phase diagram expected experimental CEP: existence/location/number

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SLIDE 1

Bernd-Jochen Schaefer

Germany

On the role of fluctuations in (2+1)-flavor QCD

Austria Germany September 21st, 2017

September 20-22, 2017 Zalakaros

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21.09.2017 | B.-J. Schaefer | Giessen University |

Temperature µ

early universe neutron star cores

LHC RHIC SIS AGS

quark−gluon plasma hadronic fluid nuclear matter vacuum

FAIR/JINR SPS

n = 0 n > 0 <ψψ> ∼ 0 <ψψ> = 0 / <ψψ> = 0 /

phases ? quark matter

crossover

CFL

B B

superfluid/superconducting

2SC

crossover

expected experimental trajectory

Conjectured QC3D phase diagram

2

Experiment:

  • CEP: existence/location/number
  • relation between chiral & deconfinement?

chiral ⇔ !deconfinement CEP? [Braun, Janot, Herbst 12/14]

  • Quarkyonic phase: coincidence of both transitions

at μ=0 & μ>0?

  • inhomogeneous phases? ➜ more favored?

[Carignano, BJS, Buballa 14]

  • axial anomaly restoration around chiral transition?

[Mitter, BJS 14]

  • finite volume effects? ➜ lattice comparison/

influence boundary conditions

  • role of fluctuations? so far mostly Mean-Field results

➜ effects of fluctuations important [Rennecke, BJS 16]

examples: size of crit reg. around CEP

  • What are good experimental signatures?

➜ higher moments more sensitive to criticality

deviation from HRG model

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21.09.2017 | B.-J. Schaefer | Giessen University |

Temperature µ

early universe neutron star cores

LHC RHIC SIS AGS

quark−gluon plasma hadronic fluid nuclear matter vacuum

FAIR/JINR SPS

n = 0 n > 0 <ψψ> ∼ 0 <ψψ> = 0 / <ψψ> = 0 /

phases ? quark matter

crossover

CFL

B B

superfluid/superconducting

2SC

crossover

expected experimental trajectory

Conjectured QC3D phase diagram

3

➜ Lattice:

but simulations restricted to small μ

Theory: ➜ Functional QFT methods: FRG,DSE, nPI

parameter dependency

➜ Models: effective theories Experiment: Experiment: (non-equilibrium? ➜ most likely thermal equilibrium) ➜ in a finite box (HBT radii: freeze-out vol. ~ 2000-3000 fm3) (UrQMD ( ): system vol. ~ 50 - 250 fm3) Theoretical aim: deeper understanding & more realistic HIC description ➜ existence of critical end point(s)?

√ s

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21.09.2017 | B.-J. Schaefer | Giessen University | 4

Agenda

  • Motivation: QCD phase diagram
  • Role of Fluctuations:

from mean-field approximations to FRG

  • Instabilities of quark-meson model

truncations

complementary to ➜ long term project

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27.05.2015 | B.-J. Schaefer | Giessen University |

Chiral transition

5

→ ∞ freeze-out close to chiral crossover line How can we probe a transition? ∂np(X) ∂Xn X = T, µ, . . . cn ≡ ∂np(T, µ) ∂(µ/T)n . . .

[HotQCD, QM 2012]

p(T, µ) T 4 =

∞

  • n=0

cn(T) µ T n

Taylor expansion: with

cn(T) = 1 n! ∂np(T, µ)/T 4 ∂(µ/T)n µ T n

  • µ=0
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21.09.2017 | B.-J. Schaefer | Giessen University |

Functional Renormalization Group

6

Γ(2)

k

= δ2Γk δφδφ Γk[φ] scale dependent effective action t = ln(k/Λ) Rk FRG (average effective action)

[Wetterich 1993]

Γk Γk =

  • d4x¯

q[iµ⌃µ − g(⇤+i⌥ ⌅⌥ ⇥5)]q + 1 2(⌃µ⇤)2 + 1 2(⌃µ⌥ ⇥)2 + Vk(⇧2) Vk=Λ(⌅2) = 4 (⇤2+⇧ ⇥2−v2)2 − c⇤

Example: Leading order derivative expansion

  • arbitrary potential

Regulator

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27.05.2015 | B.-J. Schaefer | Giessen University |

Solution techniques

7

Taylor expansion around some point

Vk(φ) =

Nmax

X

n=1

ak,2n n! (φ2 − φ2

0)n

input: initial condition at high UV cutoff (usually a symmetric potential) here: fix parametrization at high scales (UV) such to reproduce vacuum physics (IR)

φk→0 ≡ fπ ∼ 93 MeV

Discretize potential on 1dim-grid

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27.05.2015 | B.-J. Schaefer | Giessen University |

Example: grid technique

8

IR UV evolution

10 20 30 40 50 V_k [a.u.] φ [MeV] 10 20 30 40 50 60 70 80 90 V_k [a.u.] φ [MeV]

Scale evolution

  • f meson potential:

phase transition: First order phase transition: Second Order alternative solution techniques:

  • polynomial (static and co-moving)
  • bilocal expansion technique
  • pseudo-spectral
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27.05.2015 | B.-J. Schaefer | Giessen University |

FRG and QCD

9

pure Yang Mills flow + matter back-coupling full dynamical QCD FRG flow: fluctuations of gluon, ghost, quark and (via hadronization) meson in presence of dynamical quarks: gluon propagator is modified

[Pawlowski et al. 2009/12]

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27.05.2015 | B.-J. Schaefer | Giessen University |

FRG: quark-meson truncation

10

flow for quark-meson model truncation: neglect YM contributions and bosonic fluctuations without bosonic fluctuations: MFA first step:

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27.05.2015 | B.-J. Schaefer | Giessen University |

Phase diagram Nf=2 QM

11 [BJS, J Wambach 2005]

chiral limit mπ = 138 MeV

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27.05.2015 | B.-J. Schaefer | Giessen University |

FRG and QCD

12

Polyakov-loop improved quark-meson flow: fluctuations of Polyakov-loop, quark and meson Yang-Mills flow is replaced by effective Polyakov-loop potential fitted to lattice Yang-Mills thermodynamics

[Herbst, Pawlowski, BJS 2007 2013]

→ UPol(Φ) → UPol(Φ)

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27.05.2015 | B.-J. Schaefer | Giessen University | 13 [Herbst, Pawlowski, BJS 2010,2013]

FRG: Quark-Meson with Polyakov

50 100 150 200 50 100 150 200 250 300 350 T [MeV] µ [MeV] mπ=138 MeV χ crossover Φ crossover

—

Φ crossover χ 1st order σ(T=0)/2 50 100 150 200 50 100 150 200 250 300 350 T [MeV] µ [MeV] mπ=138 MeV χ crossover Φ crossover

—

Φ crossover χ 1st order σ(T=0)/2

without back reaction with back reaction

(T0(µ) = const) (T0(µ))

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21.09.2017 | B.-J. Schaefer | Giessen University | 14

Critical Endpoint

so far: we can exclude CEP for small densities: μB/Τ<2

Higher densities: dynamical baryons needed!

[Fischer, Luecker, Welzbacher 2014]

50 100 150 200 µq [MeV] 50 100 150 200 T [MeV] Lattice: curvature range κ=0.0066-0.0180 DSE: chiral crossover DSE: critical end point DSE: chiral first order DSE: deconfinement crossover µB/T=2 µB/T=3

Exact location of CEP not (yet) accessible with lattice, FRG & DSE

µB/T = 4, 5

[Herbst, Pawlowski, BJS 2011]

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27.05.2015 | B.-J. Schaefer | Giessen University |

  • effective action for scales below

[F. Rennecke, BJS 2017]

(2+1)-flavor effective action

15

k . 2πTc ≈ 1 GeV

  • relevant current quarks: u,d,s

mu ≈ md < ms ⇒ Nf = 2 + 1 Σ = T a(σa+iπa) = 1 √ 2 B @

1 √ 2

  • σl + a0

0 + iηl + iπ0

a−

0 + iπ−

κ− + iK− a+

0 + iπ+ 1 √ 2

  • σl − a0

0 + iηl − iπ0

κ0 + iK0 κ+ + iK+ ¯ κ0 + i ¯ K0

1 √ 2 (σs + iηs)

1 C A

  • (Pseudo)scalar meson nonet via bosonization
  • dominant 4-quark channel with chiral U(N)xU(N) symmetry

λS−P h ¯ q T aq 2 +

  • ¯

q iγ5T aq 2i

  • (Pseudo)scalar mixing angels

mass eigenstates flavor eigenstates

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27.05.2015 | B.-J. Schaefer | Giessen University |

  • gluons fully integrated out: effective action

(2+1)-flavor quark-meson model

16

  • effective potential

Γk = Z

x

n ¯ q Zq,k

  • γµ∂µ + γ0µ
  • q + ¯

qhk·Σ5q + tr

  • ZΣ,k ∂µΣ·∂µΣ†

+ ˜ Uk(Σ)

  • Σ5 = Ta(σa + iγ5πa)

qT = (l, l, s)

˜ Uk = Uk(ρ1, ˜ ρ2) − jlσl − jsσs − ck ξ

explicit chiral symmetry breaking: finite light & strange current quark masses anomalous U(1)A breaking via ’t Hooft determinant

ξ = det(Σ + Σ†)

U(3) x U(3) sym. potential (two chiral invariants)

ρi = tr(Σ·Σ†)i

  • wave function renormalizations
  • Yukawa couplings

hk =   hl,k hl,k hls,k hl,k hl,k hls,k hsl,k hsl,k hs,k  

Zq,k =   Zl,k Zl,k Zs,k   ZΣ,k =   Zφ,k Zφ,k Zφ,k  

p2 + m2 → Zk p2 + m2

[F. Rennecke, BJS 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University |

Chiral Phase Diagram

17

  • different truncations:

[F. Rennecke, BJS 2017]

Γk = Z

x

n ¯ q Zq,k

  • γµ∂µ + γ0µ
  • q + ¯

qhk·Σ5q + tr

  • ZΣ,k ∂µΣ·∂µΣ†

+ ˜ Uk(Σ)

  • truncation

running couplings (TCEP, µCEP) [MeV] LPA0+Y ¯ ˜ Uk, ¯ hl,k, ¯ hs,k, Zl,k, Zs,k, Zφ,k (61,235) LPA+Y ¯ ˜ Uk, ¯ hl,k, ¯ hs,k (46,255) LPA ¯ ˜ Uk (44,265)

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21.09.2017 | B.-J. Schaefer | Giessen University |

Chiral Phase Diagram

18

truncation running couplings (TCEP, µCEP) [MeV] LPA0+Y ¯ ˜ Uk, ¯ hl,k, ¯ hs,k, Zl,k, Zs,k, Zφ,k (61,235) LPA+Y ¯ ˜ Uk, ¯ hl,k, ¯ hs,k (46,255) LPA ¯ ˜ Uk (44,265)

  • Critical Endpoint for different truncations:

[F. Rennecke, BJS 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University |

Masses

19

Ml,0(T) Ms,0(T)

50 100 150 200 100 200 300 400 500 T @MeVD Mq @MeVD

LPA'+Y LPA+Y LPA m = 0 MeV m = 200 MeV

  • quark masses
  • meson masses

mesons decouple more rapidly beyond LPA

driven by the meson wave-function renormalization [F. Rennecke, BJS 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University |

Mixing angles

20

  • mixing angles determine light and strange quark content of σ, f0, η, η’ mesons

50 100 150 200 20 40 60 80 T @MeVD mixing angles

jp HLPA'+YL js HLPA'+YL jp HLPAL js HLPAL

✓ f0 σ ◆ = ✓ cos ϕs − sin ϕs sin ϕs cos ϕs ◆ ✓ σl σs ◆ ✓ η η0 ◆ = ✓ cos ϕp − sin ϕp sin ϕp cos ϕp ◆ ✓ ηl ηs ◆

significant effects on pseudoscalar mixing beyond LPA! consequence: chiral partners of η and η’ change!

140 160 180 200 220 600 700 800 900 1000 1100 1200 T @MeVD M @MeVD

h h' a 0

LPA'+Y LPA

[F. Rennecke, BJS 1610.08748] [F. Rennecke, BJS 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University |

Wave function renormalization

21 50 100 150 200 0.4 0.6 0.8 1.0 1.2 T @MeVD wave function renormalization

Zf,0HTLêZf,0H0L Zl,0HTLêZl,0H0L Zs,0HTLêZs,0H0L m = 0 MeV m = 200 MeV

50 100 150 200 5 6 7 8 9 10 11 T @MeVD Yukawa coupling

h l,0, m = 0 MeV h s,0, m = 0 MeV hl,0, m = 200 MeV hs,0, m = 200 MeV

rapid drop of meson wave function renormalization triggers fast decoupling of mesons above Tc Mφ,k = mφ,k Z1/2

φ,k

[F. Rennecke, BJS 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University | 22

  • Motivation: QCD phase diagram
  • Role of Fluctuations:

from mean-field approximations to FRG

  • Instabilities of quark-meson model

truncations

Agenda

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21.09.2017 | B.-J. Schaefer | Giessen University |

Low Temperatures

23 μ T μ T MF FRG

  • Clausius-Clapeyron relation:

dTc dµc = −∆n ∆s

  • Phase diagram quark-meson model ( )

Nf = 2

  • Entropy density: s/T 3
  • rder-disorder chiral transition: ∆s > 0 if ∆n > 0

gas-liquid transition: ∆s 7 0 depends on size of ∆n FRG: gas-liquid transition to self-bound quark matter where density is not large enough

[R-A Tripolt, BJS, L von Smekal, J Wambach 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University | 24

New phases at low temperatures?

  • possible explanations:

truncation artifact (necessary to go beyond LPA) Note: observation is scheme independent

  • color superconductivity (CS)

as in NJL model studies @low T ➜ pairing transition to CS state pairing interaction in QM model ➜ by σ- and π-exchange between quarks both channels are attractive ➜ Cooper instability of Fermi surface First (real-time in-medium) results: corresponding gap-equations yield gaps of several MeV Within FRG: fermion propagator in Nambu-Gorkov formalism including anomalous off-diagonal terms (see e.g. quark-meson-diquark model [Strodthoff, BJS, L von Smekal 2013] [R-A Tripolt, BJS, L von Smekal, J Wambach 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University | 25

New phases at low temperatures?

example instabilities in pion-direction (p-wave pion condensation):

  • inhomogeneous phases

all FRG studies so far assume spacial homogeneity NJL & QM model studies in MFA ➜ crystalline phases with inhomogeneous order parameter favored [S. Carignano, M. Buballa, BJS 2014] Instabilities ➜ pole(s) of real-time propagators at finite spacial momenta

D−1

π (ω = 0, |⃗

pc|) = 0

100 200 300 400 500 600 700

  • 1.0
  • 0.5

0.0 |p | [MeV] Re Γπ

(2) [GeV2]

k = Λ k = 330 MeV k = 500 MeV k = 0

new phase @ finite k but flow restricted to homogeneity

[R-A Tripolt, BJS, L von Smekal, J Wambach 2017]

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21.09.2017 | B.-J. Schaefer | Giessen University |

Summary & Conclusions

26

  • effects of quantum and thermal fluctuations on QCD phase diagram

full FRG investigation with different truncations LPA, LPA’, LPA’+Y ➜ fluctuations are important

  • thermodynamical instabilities for large chemical potentials

➜ negative entropy density beyond chiral transition