Critical Region of the QCD Phase Transition

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1 Critical Region of the QCD Phase Transition Mean field vs. Renormalization group B.-J. Schaefer 1 and J. Wambach 1,2 1 Institut für Kernphysik TU Darmstadt 2 GSI Darmstadt 18th August 25 Uni. Graz B.-J. Schaefer (TU Darmstadt) 1 / 38

2 Outline 1 QCD phase diagram 2 Mean-Field results 3 Renormalization Group method Introduction to Wilsonian RG Results with Proper-time RG B.-J. Schaefer (TU Darmstadt) 2 / 38

3 Outline 1 QCD phase diagram 2 Mean-Field results 3 Renormalization Group method Introduction to Wilsonian RG Results with Proper-time RG B.-J. Schaefer (TU Darmstadt) 3 / 38

4 QCD Phase Diagram (N f = 3) T, GeV QGP.1 E critical point nuclear vacuum matter quark matter 1 µ B CFL, GeV lattice at µ = : lattice at µ : crossover sign problem eff. models at T = : 1st-order order parameter: qq critical end point E (cf. as in water) Where is the critical point E? What is the size of crit. region around E? B.-J. Schaefer (TU Darmstadt) 4 / 38

5 Lattice Phase Diagram (N f = 3) µ B = T N f =2 χ 175 MeV 2nd order O(4)? N f = 2 1st order Pure Gauge T d 27 MeV T N f =3 χ 155 MeV m tric s m s?? phys. point 1st order crossover 2nd order Z(2) m, m u d N f = 3 N f = 1 B.-J. Schaefer (TU Darmstadt) 5 / 38

6 QCD Phase Diagram (N f = 2) 3D-view (T, µ B, m q) of N f = 2 QCD phase diagram: m q = : O(4)-symmetry 4 modes critical σ, π m q : no symmetry remains only one critical mode σ (Ising) ( π massive) T m q µβ critical line, m q = O(4) universality tricritical point, m q = 3d-Ising universality surface of 1st order transitions triple line, m q = line of end points, m q = / B.-J. Schaefer (TU Darmstadt) 6 / 38

7 QCD Phase Diagram (N f = 2) 3D-view (T, µ B, m q) of N f = 2 QCD phase diagram: m q = : O(4)-symmetry 4 modes critical σ, π m q : no symmetry remains only one critical mode σ (Ising) ( π massive) T m q µβ critical line, m q = O(4) universality tricritical point, m q = 3d-Ising universality surface of 1st order transitions triple line, m q = line of end points, m q = / Landau-Ginzburg potential: order parameter φ = (σ, π) Ω(T, µ; φ) a(t, µ) φ 2 + b(t, µ) φ 4 + c φ 6 +mσ 2nd order line: a(t c, µ c) = O(4) universality ; tricritical point: b(t c, µ c) = B.-J. Schaefer (TU Darmstadt) 6 / 38

8 QCD Phase Diagram (N f = 2) 3D-view (T, µ B, m q) of N f = 2 QCD phase diagram: m q = : O(4)-symmetry 4 modes critical σ, π m q : no symmetry remains only one critical mode σ (Ising) ( π massive) T m q µβ critical line, m q = O(4) universality tricritical point, m q = 3d-Ising universality surface of 1st order transitions triple line, m q = line of end points, m q = / Landau-Ginzburg potential: order parameter φ = (σ, π) Ω(T, µ; φ) a(t, µ) φ 2 + b(t, µ) φ 4 + c φ 6 +mσ 2nd order line: a(t c, µ c) = O(4) universality ; tricritical point: b(t c, µ c) = What are the sizes of the critical regions? B.-J. Schaefer (TU Darmstadt) 6 / 38

9 Ginzburg criterion Ginzburg criterion: size of crit. region break down of mean-field theory Landau-Ginzburg potential for 2nd order phase transition Ω(T, µ; φ) d( φ) 2 + a tφ 2 + bφ 4 ; t = (T T c)/t c Ginzburg-Levanyuk temperature τ GL t T c 2 a d 3 b2 τ GL m 4/5 q B.-J. Schaefer (TU Darmstadt) 7 / 38

10 Ginzburg criterion Ginzburg criterion: size of crit. region break down of mean-field theory Landau-Ginzburg potential for 2nd order phase transition Ω(T, µ; φ) d( φ) 2 + a tφ 2 + bφ 4 ; t = (T T c)/t c Ginzburg-Levanyuk temperature τ GL t T c 2 a d 3 b2 τ GL m 4/5 q size depends on microscopic dynamics universality not applicable He 4 λ-transition: τ GL 1 15 O(2) spin model:τ GL.3 Size of crit. region shrinks as m q (τ GL b 2 ) ; tricritical b B.-J. Schaefer (TU Darmstadt) 7 / 38

11 Outline 1 QCD phase diagram 2 Mean-Field results 3 Renormalization Group method Introduction to Wilsonian RG Results with Proper-time RG B.-J. Schaefer (TU Darmstadt) 8 / 38

12 Mean-field approximation N f = 2 Quark-Meson model L = q (i / g(σ + iγ 5 τ π)) q ( µσ µ σ + µ π µ π) λ 4 (σ2 + π 2 v 2 ) 2 cσ B.-J. Schaefer (TU Darmstadt) 9 / 38

13 Mean-field approximation N f = 2 Quark-Meson model L = q (i / g(σ + iγ 5 τ π)) q ( µσ µ σ + µ π µ π) λ 4 (σ2 + π 2 v 2 ) 2 cσ partition function: Z(T, µ) = 8 Z >< Z D qdqdσd π exp >: i 1/T 9 >= dtd 3 x (L + µ qγ q) >;. B.-J. Schaefer (TU Darmstadt) 9 / 38

14 Mean-field approximation N f = 2 Quark-Meson model L = q (i / g(σ + iγ 5 τ π)) q ( µσ µ σ + µ π µ π) λ 4 (σ2 + π 2 v 2 ) 2 cσ partition function: Z(T, µ) = 8 Z >< Z D qdqdσd π exp >: i 1/T 9 >= dtd 3 x (L + µ qγ q) >;. Mean field approx.: σ σ φ, π π =, integrate quark/antiquarks with Grand canonical potential Ω(T, µ) = T ln Z V = λ 4 ( σ 2 v 2 ) 2 c σ + Ω qq(t, µ) Z d 3 k n o Ω qq(t, µ) = 2N cn f T ln(1 + e (Eq µ)/t ) + ln(1 + e (Eq+µ)/T ) (2π) 3 B.-J. Schaefer (TU Darmstadt) 9 / 38

15 Phase Diagram in MF Approximation st order crossover CEP m π =138 MeV T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 1 / 38

16 Phase Diagram in MF Approximation st order crossover CEP 1 m π =1 MeV m π =138 MeV T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 1 / 38

17 Phase Diagram in MF Approximation m π =69 MeV 1st order crossover CEP 1 m π =1 MeV m π =138 MeV T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 1 / 38

18 Meson Masses (m π = 138 MeV) CEP location: T c 91 MeV, µ c 223 MeV T -dependence mass [MeV] 8 m 7 σ m π µ= T [MeV] B.-J. Schaefer (TU Darmstadt) 11 / 38

19 Meson Masses (m π = 138 MeV) CEP location: T c 91 MeV, µ c 223 MeV T -dependence mass [MeV] 8 m 7 σ m π µ= 1 µ=223 MeV T [MeV] B.-J. Schaefer (TU Darmstadt) 11 / 38

20 Meson Masses (m π = 138 MeV) CEP location: T c 91 MeV, µ c 223 MeV T -dependence mass [MeV] 8 m 7 σ m π µ=3 MeV 2 µ= 1 µ=223 MeV T [MeV] B.-J. Schaefer (TU Darmstadt) 11 / 38

21 Meson Masses (m π = 138 MeV) CEP location: T c 91 MeV, µ c 223 MeV T -dependence µ-dependence mass [MeV] 8 8 m σ m 7 7 σ m π m π µ=3 MeV 3 T= 2 µ= 2 1 µ=223 MeV T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 11 / 38

22 Meson Masses (m π = 138 MeV) CEP location: T c 91 MeV, µ c 223 MeV T -dependence µ-dependence mass [MeV] m σ m 7 σ m π m π µ=3 MeV 3 T= 2 µ= 2 1 µ=223 MeV 1 T=91 MeV T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 11 / 38

23 Meson Masses (m π = 138 MeV) CEP location: T c 91 MeV, µ c 223 MeV T -dependence µ-dependence mass [MeV] 8 m 7 σ m π m 7 σ m π T=15 MeV 3 µ=3 MeV 3 T= 2 µ= 2 1 µ=223 MeV 1 T=91 MeV T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 11 / 38

24 Quark-number density n q (T, µ) (MF) st order crossover CEP m π =138 MeV T [MeV] µ [MeV] Ω(T, µ) n q(t, µ) = µ χ q(t, µ) = 2 Ω(T, µ) ( µ) 2 B.-J. Schaefer (TU Darmstadt) 12 / 38

25 Quark-number density n q (T, µ) (MF) n q [fm -3 ] CEP: T c 91 MeV T = T c ± 5 MeV T>T c µ [MeV] T=T c T<T c T [MeV] st order crossover CEP m π =138 MeV µ [MeV] Ω(T, µ) n q(t, µ) = µ χ q(t, µ) = 2 Ω(T, µ) ( µ) 2 B.-J. Schaefer (TU Darmstadt) 12 / 38

26 Quark-number susceptibility χ q (MF) 3 25 T=T c χ q [fm -2 ] T>T c T<T c µ [MeV] B.-J. Schaefer (TU Darmstadt) 13 / 38

27 Quark-number susceptibility χ q (MF) (isothermal) compressibility κ T V 1 V = P T χq,n nq 2 if χ q large easy to compress interaction attractive (or weakly repulsive) (similar conclusion in Walecka model if m σ ) 3 25 T=T c χ q [fm -2 ] T>T c T<T c µ [MeV] B.-J. Schaefer (TU Darmstadt) 13 / 38

28 Quark-number susceptibility χ q (MF) χ q [fm -2 ] χ q g g c ɛ ; g = T, µ (isothermal) compressibility κ T V 1 V = P T χq,n nq T>T c µ [MeV] T=T c T<T c if χ q large easy to compress interaction attractive (or weakly repulsive) (similar conclusion in Walecka model if m σ ) crit. exp. ɛ = 2/3 (mean field) log (χ q /[fm -2 ]) ε= log ( µ-µ c /[MeV]) B.-J. Schaefer (TU Darmstadt) 13 / 38

29 Critical Region (MF) R q = χ q(t, µ)/χ free q (T, µ) χ free µ 2 q (T, µ) = N cn f π 2 + T «2 3 T [MeV] st order R q = 3 R q = 5 CEP µ [MeV] B.-J. Schaefer (TU Darmstadt) 14 / 38

30 Critical Region w/ scalar susceptibility χ σ (MF) χ σ = 1/m 2 σ R s = m 2 σ(, )/m 2 σ(t, µ) T [MeV] µ [MeV] R s =1 R s =15 R s =2 R s =25 1st order B.-J. Schaefer (TU Darmstadt) 15 / 38

31 Outline 1 QCD phase diagram 2 Mean-Field results 3 Renormalization Group method Introduction to Wilsonian RG Results with Proper-time RG B.-J. Schaefer (TU Darmstadt) 16 / 38

32 Why non-perturbative Renormalization Group? Second-order phase transition long-wavelength fluctuations (ξ ) in systems with many d.o.f large-scale (macroscopic) physical predictions should not depend on microscopic length scale ( Universality ) critical behavior (phase boundaries) best described by renormalization group methods R for quantum fields: Z[J] = 1 N Dφe S[φ,J] ; N = Z[] ill-defined path integral functional diff. equation (FDE) FDE well-defined since original divergences are relegated to the boundary values of its solution (Wilsonian) RG s describe very efficiently universal and non-universal aspects of phase transitions (1st/2nd) B.-J. Schaefer (TU Darmstadt) 17 / 38

33 Wilsonian Renormalization Group split field: Φ(p) = Φ low (p) + Φ high (p) p <k integrate high modes: Z[J] = R DΦ e S[Φ,J] k p Λ = R DΦ low DΦ high e S[Φ low,φ high,j] {z } = R DΦ low e S eff [Φ low,k] high low p Λ k k- k Z k,low = e S eff [Φ low,k] coarse graining (block spin transformation) integrate from micro macro lim k Z k,low = Z direction unique due to relativistic causality of theory B.-J. Schaefer (TU Darmstadt) 18 / 38

34 Flow of effective action S eff for flow of S eff decrease boundary cutoff k Φ S P y Λ k δk change of eff. action S eff when k k δk e S eff [Φ low, k δk] = Z DΦ S e S eff [Φ low + Φ S, k] = e S eff [Φ low, k] Z DΦ S e Z p P x 1 2 Φ 2 S eff S Φ Φ Φ S + O(δk 2 ) {z } 1 «e 2 Tr 2 S eff ln Φ Φ only first order δk needs to be kept to find flow: S eff [Φ low, k] k = 1 «2 Tr 2 S eff ln Φ Φ B.-J. Schaefer (TU Darmstadt) 19 / 38

35 Proper-time Renormalization Group Wilsonian effective action Γ k via Legendre tramsform: Z Γ k [φ] = ln Z k [J] d 4 x J(x)φ(x) ; φ(x) = δ ln Z k δj(x) Γ k [φ] = S eff 1 Tr ln S(2) eff ; 2 Schwinger proper-time representation: (one-loop structure) tγ k [φ] = 1 2 Z dτ τ h i tf k (Λ 2 τ) Tr exp τs (2) eff important ingredient: blocking function f k Γ(d/2, τk 2 ) RG improvement: tγ k [φ] = 1 2 Z dτ τ h i tf k (Λ 2 τ) Tr exp S (2) eff = δ2 S eff δφδφ τγ (2) k ; t = ln «k Λ B.-J. Schaefer (TU Darmstadt) 2 / 38

36 Renormalization Group approaches different realizations: 1 exact RG ERG (averaged action) "!# tγ k [φ] = 1 2 tr 1 tr k + R k Γ (2) k ; Γ (2) k = δ2 Γ k δφδφ 2 proper-time RG PTRG 3 many other tγ k [φ] = 1 2 Z dτ τ h i tf k (Λ 2 τ) tr exp τγ (2) k B.-J. Schaefer (TU Darmstadt) 21 / 38

37 RG Flow Equations N f = 2 Quark-Meson model Z Γ k=λ = d 4 x j q[ /+g(σ+i τ πγ 5)]q ( µσ) ( µ π)2 + V (σ 2 + π 2 ) ff B.-J. Schaefer (TU Darmstadt) 22 / 38

38 RG Flow Equations N f = 2 Quark-Meson model Z Γ k=λ = d 4 x j q[ /+g(σ+i τ πγ 5)]q ( µσ) ( µ π)2 + V (σ 2 + π 2 ) ff flow for grand canonical potential tω k (T, µ) =» k 4 3 coth 12π 2 E π 2NcN f E q «+ 1 coth 2T E σ Eq µ Eπ j tanh 2T «Eσ 2T «+ tanh «ff Eq + µ 2T E 2 π = 1 + 2Ω k/k 2, E 2 σ = 1 + 2Ω k/k 2 + 4φ 2 Ω k /k 2, E 2 q = 1 + g 2 φ 2 /k 2 φ qq quark fluctuations: chiral symmetry breaking meson fluctuations: chiral symmetry restoration B.-J. Schaefer (TU Darmstadt) 22 / 38

39 Solving Flow Equations Two possibilities 1.) Solve coupled flow eqs on φ 2 grid: Ω(φ 2 ) t Ω(φ 2 i+1 ) 2.) Taylor expansion around φ 2 : NX Ω(φ 2 ) = a n(φ 2 φ 2 ) n n= φ 2 φ 2 i φ 2 i+1 Initial condition at e.g. Λ = 5 MeV tree-level parameterization V Λ = 1 4 λ Λ(φ 2 ) 2 symmetric potential Fixed UV parameterization (λ Λ ) to reproduce φ f π 93 MeV in the IR (m q = gf π) B.-J. Schaefer (TU Darmstadt) 23 / 38

40 Scale Evolution of the Potential V UV φ B.-J. Schaefer (TU Darmstadt) 24 / 38

41 Scale Evolution of the Potential V UV scale k φ IR B.-J. Schaefer (TU Darmstadt) 24 / 38

42 Scale Evolution of the Potential V UV scale k φ IR B.-J. Schaefer (TU Darmstadt) 24 / 38

43 Scale Evolution of the Potential V UV scale k φ IR B.-J. Schaefer (TU Darmstadt) 24 / 38

44 Scale Evolution of the Potential V UV scale k φ IR B.-J. Schaefer (TU Darmstadt) 24 / 38

45 V_k [a.u.] Scale Evolution of the Potential V UV scale k φ φ [MeV] IR B.-J. Schaefer (TU Darmstadt) 24 / 38

46 V_k [a.u.] Scale Evolution of the Potential V UV scale k φ V_k [a.u.] φ [MeV] φ [MeV] IR B.-J. Schaefer (TU Darmstadt) 24 / 38

47 The Chiral Phase Transition Finite T, µ = V [MeV/fm 3 ] Evolved potential for different temperatures: Φ = Φ = f π T>T c T=T c T= Φ [MeV] f π [MeV] 1 compare to chiral perturbation theory expansion: scaling ~ 1 (T-T c ) β T [MeV] chiral PT O(T 8 ) RG qq T = 1 T 2 T 4 T 6 ln Λq qq 8fπ 2 384fπ 4 288fπ 6 T + O(T 8 ) Λ q = (47 ± 11) MeV B.-J. Schaefer (TU Darmstadt) 25 / 38

48 Critical exponents parameterize singular behavior of Ω near the phase transition depend only on internal symmetries and dimension of the system φ (k ) Tc T β ; ξ = (T T c) ν ;... altogether 6 exponents but 4 scaling relations T c η δ ν β lattice.254(38) 4.851(22).7479(9).3836(46) N = 4 ɛ.3(1) 4.82(6).73(2).38(1) RG N = 1 RG large-n B.-J. Schaefer (TU Darmstadt) 26 / 38

49 Critical exponents exponent ν for arbitrary N (here for O(N)-model) comparison with 1/N-expansion in N 2 LO «« π2 8 1 ν = O(1/N 3 ) 3π 2 N 6 3π 2 N 1.8 LPA PTRG lattice O(1/N) O(1/N 2 ).6 ν /N B.-J. Schaefer (TU Darmstadt) 27 / 38

50 Chiral Phase Diagram N f = 2: m q 28 MeV nd order 1st order 12 1 O(4) universality class T c [MeV] tricritical point µ [MeV] B.-J. Schaefer (TU Darmstadt) 28 / 38

51 Chiral Phase Diagram N f = 2: m q 28 MeV nd order 1st order 12 1 T c [MeV] Φ Φ = Φ µ [MeV] B.-J. Schaefer (TU Darmstadt) 28 / 38

52 A Second Phase Transition here e.g. V_k [a.u.] T = 6 MeV, µ q = MeV k convex potential φ [MeV] first-order phase transition second-order phase transition V_k [a.u.] V_k [a.u.] φ [MeV] φ [MeV] B.-J. Schaefer (TU Darmstadt) 29 / 38

53 A Second Phase Transition φ [MeV] µ [MeV] T = 1 MeV T = 8 MeV T = 15 MeV tricritical point: T t 52 MeV, µ t 251 MeV splitting point: T 17 MeV µ q 263 MeV gap larger if m q,vac larger V µ = 251 MeV ( µ=1) T=6 MeV IR evolved potential here e.g. T = 6 MeV fixed, µ q 251 MeV φ [MeV] B.-J. Schaefer (TU Darmstadt) 3 / 38

54 Phase diagram: m q 37 MeV (RG) TCP: T c 8.2 MeV TCP : T c 8 MeV TCP 2nd TCP T [MeV] 1 5 [BJS,J.Wambach NPA (25)] µ [MeV] B.-J. Schaefer (TU Darmstadt) 31 / 38

55 Phase diagram: m q 37 MeV (RG) TCP: T c 8.2 MeV 2. TCP : T c 8 MeV CEP: T c 61.5 MeV 2 15 TCP 2nd TCP CEP T [MeV] 1 5 [BJS,J.Wambach NPA (25)] µ [MeV] B.-J. Schaefer (TU Darmstadt) 31 / 38

56 Quark-number density n q (T, µ) (RG) TCP: T c 8.2 MeV T = T c ± 5 MeV.4.35 n q [fm -3 ] T>T c T=T c T<T c µ [MeV] B.-J. Schaefer (TU Darmstadt) 32 / 38

57 Quark-number density n q (T, µ) (RG) TCP: T c 8.2 MeV T = T c ± 5 MeV CEP: T c 61.5 MeV T = T c ± 5 MeV n q [fm -3 ].4.35 T>T c.3.25 T=T c.2 T<T c µ [MeV] T>T c.2.15 T=T c T<T c µ [MeV] B.-J. Schaefer (TU Darmstadt) 32 / 38

58 Quark-number susceptibility χ q (RG) TCP: T c 8.2 MeV T = T c ± 5 MeV CEP: T c 61.5 MeV T = T c ± 5 MeV χ q [fm -2 ] 4 35 T=T c T>T c T<T c µ [MeV] B.-J. Schaefer (TU Darmstadt) 33 / 38

59 Quark-number susceptibility χ q (RG) TCP: T c 8.2 MeV T = T c ± 5 MeV CEP: T c 61.5 MeV T = T c ± 5 MeV χ q [fm -2 ] 4 35 T=T c T>T c T<T c µ [MeV] 4 35 T>T c T=T c T<T c µ [MeV] B.-J. Schaefer (TU Darmstadt) 33 / 38

60 Critical Region (RG) Size of crit. region shrinks as m q (τ GL b 2 ) tricritical b 1 9 R q = 3 R q = 5 T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 34 / 38

61 Critical Region (RG) Size of crit. region shrinks as m q (τ GL b 2 ) tricritical b 1 9 R q = 3 R q = 5 T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 34 / 38

62 Critical exponents χ q µ µ c γ TCP: γ =.5 (Gaussian) log (χ q /fm -2 ) γ= TCP log ( µ-µ c /MeV) B.-J. Schaefer (TU Darmstadt) 35 / 38

63 Critical exponents χ q µ µ c γ TCP: γ =.5 (Gaussian) CEP: MF: ɛ = 2/3 3D Ising: ɛ =.78 log (χ q /fm -2 ) γ= TCP log ( µ-µ c /MeV) CEP ε=.74 ε= log ( µ-µ c /MeV) MF: tricritical exponents different from bicritical exponents B.-J. Schaefer (TU Darmstadt) 35 / 38

64 Meson masses (RG) 8 m 7 σ m π 6 5 µ= MeV T [MeV] B.-J. Schaefer (TU Darmstadt) 36 / 38

65 Meson masses (RG) CEP: T c 61.5 MeV 8 m 7 σ m π 6 5 µ= MeV 4 3 µ=313 MeV T [MeV] T [MeV] µ [MeV] B.-J. Schaefer (TU Darmstadt) 36 / 38

66 Meson masses (RG) CEP: T c 61.5 MeV 8 m 7 σ m π 6 5 µ= MeV 4 3 µ=313 MeV T [MeV] 8 m 7 σ m π 6 T=1 MeV µ [MeV] B.-J. Schaefer (TU Darmstadt) 36 / 38

67 Meson masses (RG) CEP: T c 61.5 MeV 8 m 7 σ m π 6 5 µ= MeV 4 3 µ=313 MeV T [MeV] 8 m 7 σ m π 6 T=1 MeV T=61 MeV µ [MeV] B.-J. Schaefer (TU Darmstadt) 36 / 38

68 Meson masses (RG) CEP: T c 61.5 MeV 8 m 7 σ m π 6 5 µ= MeV 4 3 µ=313 MeV T [MeV] 8 m 7 σ m π 6 T=1 MeV T=61 MeV T=15 MeV µ [MeV] B.-J. Schaefer (TU Darmstadt) 36 / 38

69 Scalar susceptibility χ σ (MF) (RG) R s = m 2 σ(, )/m 2 σ(t, µ) RG R s =1 R s =15 R s =2 R s =25 1st order µ [MeV] B.-J. Schaefer (TU Darmstadt) 37 / 38

70 Scalar susceptibility χ σ (MF) (RG) R s = m 2 σ(, )/m 2 σ(t, µ) RG R s =1 R s =15 R s =2 R s =25 1st order (µ-µ c )/µ c B.-J. Schaefer (TU Darmstadt) 37 / 38

71 Scalar susceptibility χ σ (MF) (RG) MF R s = m 2 σ(, )/m 2 σ(t, µ) RG (T-T c )/T c R s =1 R s =15 R s =2 R s =25 1st order (µ-µ c )/µ c R s =1 R s =15 R s =2 R s =25 1st order (µ-µ c )/µ c B.-J. Schaefer (TU Darmstadt) 37 / 38

72 Scalar susceptibility χ σ (MF) (RG) MF R s = m 2 σ(, )/m 2 σ(t, µ) RG (T-T c )/T c R s =1 R s =15 R s =2 R s =25 1st order (µ-µ c )/µ c R s =1 R s =15 R s =2 R s =25 1st order (µ-µ c )/µ c B.-J. Schaefer (TU Darmstadt) 37 / 38

73 Summary nd order 1st order 12 T c [MeV] Φ Φ = Φ µ [MeV] possibility of 2nd TCP (in χ limit) physical interpretation? B.-J. Schaefer (TU Darmstadt) 38 / 38

74 Summary T [MeV] µ [MeV] R s =1 R s =15 R s =2 R s =25 1st order µ [MeV] R s =1 R s =15 R s =2 R s =25 1st order Comparison: MF vs. RG analysis Size of critical region: χ q(t, µ) and χ σ(t, µ) compressed w/ fluctuations Crit. exponents consistent w/ 3d Ising universality class B.-J. Schaefer (TU Darmstadt) 38 / 38

Non-perturbative Study of Chiral Phase Transition

Non-perturbative Study of Chiral Phase Transition Ana Juričić Advisor: Bernd-Jochen Schaefer University of Graz Graz, January 9, 2013 Table of Contents Chiral Phase Transition in Low Energy QCD Renormalization