Phase transition and axial anomaly in the 3-flavor chiral meson model Gergely Fej˝ os Research Center for Nuclear Physics Osaka University Workshop on J-PARC hadron physics 2016 a 4th March, 2016 Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral meson
Outline aaa Motivation Functional renormalization group Chiral (linear) sigma model and axial anomaly Numerical results Summary Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Motivation Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Motivation QCD Lagrangian with quarks and gluons: L = − 1 µν G µν a + ¯ ψ i ( i γ µ D µ − m ) ij ψ j 4 G a Approximate chiral symmetry for N f = 2 , 3 flavors: ψ L → e iT a θ a ψ R → e iT a θ a L ψ L , R ψ R [vector: θ L + θ R , axialvector: θ L − θ R ] Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Motivation QCD Lagrangian with quarks and gluons: L = − 1 µν G µν a + ¯ ψ i ( i γ µ D µ − m ) ij ψ j 4 G a Approximate chiral symmetry for N f = 2 , 3 flavors: ψ L → e iT a θ a ψ R → e iT a θ a L ψ L , R ψ R [vector: θ L + θ R , axialvector: θ L − θ R ] Chiral symmetry is spontaneuously broken in the ground state: < ¯ ψ i ψ i > = < ¯ ψ i , R ψ i , L > + < ¯ ψ i , L ψ i , R > � = 0 SSB pattern: SU L ( N f ) × SU R ( N f ) − → SU V ( N f ) Anomaly: U A (1) is broken by instantons Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Motivation QCD Lagrangian with quarks and gluons: L = − 1 µν G µν a + ¯ ψ i ( i γ µ D µ − m ) ij ψ j 4 G a Approximate chiral symmetry for N f = 2 , 3 flavors: ψ L → e iT a θ a ψ R → e iT a θ a L ψ L , R ψ R [vector: θ L + θ R , axialvector: θ L − θ R ] Chiral symmetry is spontaneuously broken in the ground state: < ¯ ψ i ψ i > = < ¯ ψ i , R ψ i , L > + < ¯ ψ i , L ψ i , R > � = 0 SSB pattern: SU L ( N f ) × SU R ( N f ) − → SU V ( N f ) Anomaly: U A (1) is broken by instantons Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point? Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Motivation Effective field theory of chiral symmetry breaking: − → three-flavor linear sigma model → M = T a ( s a + i π a ) − [pseudoscalar mesons: π, K , η, η ′ , scalar mesons: a 0 , κ, f 0 , σ ] ∂ µ M ∂ µ M † − µ 2 Tr ( MM † ) − g 1 9 [ Tr ( MM † )] 2 L = g 2 3 Tr ( MM † ) 2 − Tr [ H ( M + M † )] − a (det M + det M † ) − Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Motivation Effective field theory of chiral symmetry breaking: − → three-flavor linear sigma model → M = T a ( s a + i π a ) − [pseudoscalar mesons: π, K , η, η ′ , scalar mesons: a 0 , κ, f 0 , σ ] ∂ µ M ∂ µ M † − µ 2 Tr ( MM † ) − g 1 9 [ Tr ( MM † )] 2 L = g 2 3 Tr ( MM † ) 2 − Tr [ H ( M + M † )] − a (det M + det M † ) − two independent quartic couplings: g 1 , g 2 explicit symmetry breaking: H = h 0 T 0 + h 8 T 8 ’t Hooft determinant breaks only the U A (1) symmetry Goal: take into account quantum- and thermal fluctuations to see the finite temperature behavior of the system Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Functional renormalization group FRG: follows the idea of Wilsonian renormalization group � 1 � � � S [ φ ]+ � J φ + D φ e − 2 φ R k φ Z k [ J ] = R k : IR regulator function k 2 Requirements: 1., scale separation R k (q) (suppress modes q � k ) 2., R k − → ∞ if k − → ∞ 3., R k − → 0 if k − → 0 0 q 0 k Gradually moving k from Λ to 0, fluctuations are getting integrated out Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Functional renormalization group scale-dependent effective action: � φ − 1 � Γ k [¯ J ¯ φ R k ¯ ¯ φ ] = − log Z k [ J ] − φ 2 ⇒ Γ k =Λ [¯ φ ] = S [¯ − → k ≈ Λ: no fluctuations φ ] ⇒ Γ k =0 [¯ φ ] = Γ 1 PI [¯ → k = 0: all fluctuations φ ] − the scale-dependent effective action interpolates between classical- and quantum effective actions Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Chiral sigma model Flow of the effective action is described by the Wetterich equation: � ( T ) ∂ k Γ k = 1 ∂ k R k ( q , p )(Γ (2) + R k ) − 1 ( p , q ) 2 Tr k q , p → exact relation, functional integro-differential equation − − → approximation(s) needed Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Chiral sigma model Flow of the effective action is described by the Wetterich equation: � ( T ) ∂ k Γ k = 1 ∂ k R k ( q , p )(Γ (2) + R k ) − 1 ( p , q ) 2 Tr k q , p → exact relation, functional integro-differential equation − − → approximation(s) needed Derivative expansion of Γ k : � � � − V k ( M ) + Z k ( M ) ∂ µ M † ∂ µ M + O ( ∂ 4 ) Γ k [ M ] = x Assumptions: U k ( M ) + Tr ( H ( M + M † )) + A k ( M )(det M + det M † ) V k ( M ) = Z k ( M ) 1 ≈ → U k : chiral symmetric part − − → A k : U A (1) breaking term, field dependent! Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Chiral sigma model The model has 6 free parameters, which have to be fixed using experimental input → chiral symmetric parameters: µ 2 , g 1 , g 2 − − → explicit breaking terms: h 0 , h 8 → chiral anomaly parameter: a − Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Chiral sigma model The model has 6 free parameters, which have to be fixed using experimental input → chiral symmetric parameters: µ 2 , g 1 , g 2 − − → explicit breaking terms: h 0 , h 8 → chiral anomaly parameter: a − First one fixes the explicit symmetry breaking using the PCAC relations: = − ∂ m 2 π a = ∂ µ J 5 µ Tr ( H ( M + M † )) a f a ˆ ( a = 1 , ... 8) a ∂θ a A → h 0 ≈ (286 MeV ) 3 , h 8 ≈ − (311 MeV ) 3 − Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Chiral sigma model The model has 6 free parameters, which have to be fixed using experimental input → chiral symmetric parameters: µ 2 , g 1 , g 2 − − → explicit breaking terms: h 0 , h 8 → chiral anomaly parameter: a − First one fixes the explicit symmetry breaking using the PCAC relations: = − ∂ m 2 π a = ∂ µ J 5 µ Tr ( H ( M + M † )) a f a ˆ ( a = 1 , ... 8) a ∂θ a A → h 0 ≈ (286 MeV ) 3 , h 8 ≈ − (311 MeV ) 3 − µ 2 , g 1 , g 2 are fixed by the light spectra (i.e. π , K , and σ ) anomaly parameter a is tuned so that η and η ′ masses get close to their experimental values at T = 0 Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Chiral sigma model Functions need to be determined: → effective potential V k ( M ) − − → anomaly function A k ( M ) Step I.: solve the renormalization group equations for V k and A k at T = 0 to determine the model parameters ( µ 2 , g 1 , g 2 , a ) Step II.: solve the same equations at T > 0 to get: − → thermal effects on the mesonic spectrum − → details of symmetry restoration → finite temperature behavior of the U A (1) anomaly − Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Numerical results without anomaly f 0 1000 κ 800 masses [MeV] a 0 600 η ' 400 K σ 200 π , η 0 0 50 100 150 200 250 300 350 400 T [MeV] Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Numerical results with anomaly f 0 1000 κ η ' masses [MeV] a 0 800 600 η K σ 400 200 π 0 100 200 300 400 500 T [MeV] Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Numerical results 8 7 T = 1.2 T c T = 0.7 T c T = 0 6 |A| [GeV] 5 4 3 2 0 100 200 300 √ <M + M> [MeV] Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Numerical results 8 7.5 7 anomaly [GeV] 6.5 6 5.5 5 4.5 |A[0]| |A[v min ]| 4 0 100 200 300 400 500 T [MeV] Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Numerical results 1 0.8 strange v s/ns (T)/v s/ns (0) 0.6 0.4 non strange 0.2 0 0 100 200 300 400 500 T [MeV] dashed: without anomaly, solid: with anomaly Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
Summary Thermal properties of strongly interacting matter via the three-flavor linear sigma model Fluctuations (quantum and thermal) have been included using the functional renormalization group (FRG) approach − → neglecting the change in the wavefunction renormalization − → derivative expansion up to next-to-leading order Results: → thermal evolution of the mass spectrum − − → evaporation of the condensate ( v ns : yes, v s : no) → temperature dependence of the U A (1) anomaly factor − Most important findings: → mass spectrum might not be useful for U A (1) − − → meson fluctuations strengthen the anomaly Gergely Fej˝ os Phase transition and axial anomaly in the 3-flavor chiral...
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