QCD chiral phase boundary from RG flows. Holger Gies. Heidelberg U.

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1 Heidelberg U.

2 From Micro to Macro DoF

3 From Micro to Macro DoF UV PLM PNJL NJL Quark Meson model Quark models Bag models Skyrmions... IR

4 From Micro to Macro DoF UV PLM PNJL NJL Quark Meson model Quark models Bag models Skyrmions... IR

5 From Micro to Macro DoF UV RG flow PLM PNJL NJL Quark Meson model Quark models Bag models Skyrmions... IR

6 Effective Action universal tool: effective action Γ[φ]

7 Functional RG Flow Equation t Γ k k k Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 = (WEGNER&HOUGHTON 73; WETTERICH 93)

8 Functional RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 RG trajectory: Γ k =Λ = S bare = 1 4 F z µνf z µν + ψ (i / + ga/) ψ

9 Functional RG Flow Equation RG trajectory: t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1

10 Functional RG Flow Equation RG trajectory: t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1

11 Functional RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 RG trajectory: Γ k 0 = Γ

12 Functional RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 RG trajectory: Γ k 0 = Γ

13 RG Flow of the Chiral Sector effective action: Γ k = 1 λ σ [ ( ψ a 2 k 2 ψ b ) 2 ( ψ a γ 5 ψ b ) 2] 1 4 RG flow t λ σ = 2λ σ π 2 l(f) 1 2N c λ σ 2

14 RG Flow of the Chiral Sector effective action: Γ k = 1 λ σ [ ( ψ a 2 k 2 ψ b ) 2 ( ψ a γ 5 ψ b ) 2] 1 4 RG flow t λ σ = 2λ σ π 2 l(f) 1 2N c λ σ 2

15 RG Flow of the Chiral Sector effective action: Γ k = ZF 4 F z µνf z µν ψ (iz ψ / + Z 1 ḡa/) ψ λ σ k 2 [ ( ψ a ψ b ) 2 ( ψ a γ 5 ψ b ) 2] RG flow t λ σ = 2λ σ 1 4π 2 l(f) 1 2N c λ σ 2 1 8π 2 l(fb) 3 1,1 3N c N c 128π 2 l(fb) 1,2 2 1 g 2 λ σ 3N c 2 8 N c g 4

16 RG Flow of the Chiral Sector effective action: Γ k = ZF 4 F z µνf z µν ψ (iz ψ / + Z 1 ḡa/) ψ λ σ k 2 [ ( ψ a ψ b ) 2 ( ψ a γ 5 ψ b ) 2] RG flow t λ σ = 2λ σ 1 4π 2 l(f) 1 2N c λ σ 2 1 8π 2 l(fb) 3 1,1 3N c N c 128π 2 l(fb) 1,2 2 1 g 2 λ σ 3N c 2 8 N c g 4

17 RG Flow of the Chiral Sector effective action: Γ k = ZF 4 F z µνf z µν ψ (iz ψ / + Z 1 ḡa/) ψ + 1 λ σ 2 k 2 (S-P) + 1 λ VA [ 2(V-A) adj. 2 k 2 + (1/N c )(V-A) ] + 1 λ + 2 k 2 (V+A) + 1 λ 2 k 2 (V-A) RG flow t λ σ = 2λ σ 1 4π 2 l(f) 1 1 8π 2 l(fb) 1,1 3 [ 128π 2 l(fb) 1,2 {2N c λ σ 2 2λ λ σ 2N f λ σ λ VA 6λ + λ σ } 3 N c 2 1 N c g 2 λ σ 6g 2 λ + 3N c 2 8 N c g 4 (HG,JAECKEL,WETTERICH 04) ]

18 Chiral Criticality = critical gauge coupling α cr : = if α > α cr : λ (χsb) N f = 3 = N c : α cr 0.85 (HG,JAECKEL 05) = bosonization: λ 1 m 2 φ

19 Chiral Criticality at Finite Temperature quark modes: m 2 T = m 2 f + (2πT (n ))2 = T -dependent critical coupling: α cr (T ) α cr 0.85 (BRAUN,HG 05)

20 Chiral Criticality at Finite Temperature quark modes: m 2 T = m 2 f + (2πT (n ))2 = T -dependent critical coupling: α cr (T ) α cr 0.85 (BRAUN,HG 05) implications for low-energy QCD models: λ init = λ init (T, µ,... )

21 RG Flow of Gluodynamics Operator expansion with the background-field method Γ k [A] = d d x W k (F 2 ), F 2 F a µνf a µν (REUTER,WETTERICH 94; FREIRE,LITIM,PAWLOWSKI 00) W k (F 2 ) = Z F 4 F 2 + W 2 16 (F 2 ) 2 + W 3 3!4 3 (F 2 ) 3 + W 4 4!4 4 (F 2 ) spectrally adjusted flow equation: (HG 02) t Z F t W 2 t W 3 t W 4 t W 5... running coupling: g 2 = Z 1 F ḡ 2 (ABBOTT 82) β function: t g 2 β g 2

22 Running Gauge Coupling (HG 02) (BRAUN,HG 05) T = 0: IR fixed point α cf. vertex expansion in the Landau gauge (V.SMEKAL,ALKOFER,HAUCK 97; FISCHER,ALKOFER 02; ZWANZIGER 02)

23 Running Gauge Coupling (HG 02) (BRAUN,HG 05) T /k : strongly interacting 3D theory α k T α 3D, α 3D α 3D, 2.7, η 3D 1 cf. Landau-gauge vertex expansion: (MAAS,WAMBACH,ALKOFER 05)

24 Chiral Phase Transition α(k, T ) vs. α cr (T /k) χsb triggered by α s single input: α s (m τ ) = (BRAUN,HG 06)

25 Chiral Phase Transition α(k, T ) vs. α cr (T /k) implications for low-energy QCD models: Mind the glue! t Γ model = t Γ model (α,... )

26 Chiral Phase Boundary T N f T cr [Mev] N f small N f : fermionic screening, β quark = 2 3 N f critical flavor number: g 4 8π 2 (BRAUN,HG 05, 06) N cr f 12 (CF. APPELQUIST ET AL. 96; MIRANSKI,YAMAWAKI 96; HG,JAECKEL 05)

27 Chiral Phase Boundary T N f T cr [Mev] N f small N f : fermionic screening, β quark = 2 3 N f critical flavor number: N cr f 12 g 4 8π 2 (BRAUN,HG 05, 06) low-energy models: T cr = T cr (N f, m f,... ) (CF. APPELQUIST ET AL. 96; MIRANSKI,YAMAWAKI 96; HG,JAECKEL 05)

28 Chiral Phase Boundary T N f β(g) = ηg N f = 6, T=0 N f = 8, T=0 N f = 10, T=0 T cr [Mev] Θ G=α s /8π N f fixed-point regime: critical exponent Θ β g 2 Θ (g 2 g 2 )

29 Chiral Phase Boundary T N f β(g) = ηg N f = 6, T=0 N f = 8, T=0 N f = 10, T=0 T cr [Mev] Θ G=α s /8π N f fixed-point regime: critical exponent Θ β g 2 Θ (g 2 g 2 ) shape of the phase boundary for N f N cr f : (BRAUN,HG 05, 06) T cr k 0 N f N cr f 1 Θ, Θ 0.71

30 Quark Mass Dependence NJL, quark-meson model: critical initial conditions T cr (m) T cr (0) T cr (0) 0.3 for m 10eV, m π 200MeV

31 Quark Mass Dependence NJL, quark-meson model: QCD RG flow (BRAUN 06) critical initial conditions T cr (m) T cr (0) T cr (0) 0.3 for m 10eV, m π 200MeV gradual approach to criticality T cr (m) T cr (0) T cr (0) 0.0 for m 10eV, m π 200MeV

32 Quark Mass Dependence NJL, quark-meson model: QCD RG flow (BRAUN 06) critical initial conditions T cr (m) T cr (0) T cr (0) 0.3 for m 10eV, m π 200MeV gradual approach to criticality T cr (m) T cr (0) T cr (0) 0.0 for m 10eV, m π 200MeV lattice gauge theory (KARSCH,LAERMANN,PEIKERT 01) T cr (m) T cr (0) T cr (0) 0.04

33 Conclusions low-energy QCD models: check relevant DoFs chiral phase boundary first guide to full QCD functional methods full QCD calculations RG critical phenomena = finite µ = Polyakov loop (BRAUN,HG,PIRNER 05; BRAUN,HG,PAWLOWSKI XX)