Status report: FRG study of the chiral phase transition in a - PowerPoint PPT Presentation

Status report: FRG study of the chiral phase transition in a quark-meson model with (axial-)vector mesons J¨ urgen Eser

FRG with fermions Quark-meson model with (axial-)vector mesons Overview 1 FRG with fermions Recap: Bosonic FRG Extension to fermions 2 Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Recap: Bosonic FRG Quark-meson model with (axial-)vector mesons Extension to fermions Functional renormalization group (FRG) Effective action Γ Theory at hand described by the effective action Γ Scale ( k -)dependent analog: effective average action Γ k Bosonic flow equation along renormalization scale k [Phys. Lett. B301, 90-94] Regulating function R k introduces the k -dependence: � − 1 ∂ k R k ∂ k Γ k = 1 �� � Γ ( 2 ) 2 tr + R k , (1) k δ 2 Γ k � Γ ( 2 ) � αα ′ = (2) k δ Φ α δ Φ α ′ J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Recap: Bosonic FRG Quark-meson model with (axial-)vector mesons Extension to fermions Flow in coupling space Figure 1 : Coupling space { c 1 , c 2 , . . . , c n } ; "Theoryspace" by Morozsergej - Own work. Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Theoryspace.png#/media/File:Theoryspace.png [10/05/15] . J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Recap: Bosonic FRG Quark-meson model with (axial-)vector mesons Extension to fermions Extension to fermions Fermionic flow equation [Phys. Rev. B70, 125111] Grassmann-valued fields and sources: � − 1 ∂ k R k �� � Γ ( 2 ) ∂ k Γ k = − tr + R k , (3) k − → ← − δ δ � Γ ( 2 ) � αα ′ = Γ k (4) k δ Φ α δ Φ α ′ Mixed Bose-Fermi system ∂ k Γ k = 1 � − 1 ∂ k R k �� � Γ ( 2 ) 2 str + R k (5) k J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions Extended linear sigma model (eLSM) (1) Spin-0 fields Definitions: Σ = ( σ a + i π a ) t a , Σ 5 = ( σ a + i γ 5 π a ) t a (6) scalars σ a , pseudoscalars π a , U ( N f ) -generators t a Spin-1 fields Definition of right- and left-handed fields: R µ = ( V a ,µ + A a ,µ ) t a , L µ = ( V a ,µ − A a ,µ ) t a , (7) vector fields V a ,µ and axial-vector fields A a ,µ ; Field strength tensors R µν = ∂ µ R ν − ∂ ν R µ , L µν = ∂ µ L ν − ∂ ν L µ J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions eLSM (2) U ( N f ) R × U ( N f ) L -transformations Σ → U † R µ → U † L µ → U † R Σ U L , (8) R R µ U R , L L µ U L Symmetry breaking � det Σ + det Σ † � Axial anomaly: c A Nonzero quark masses: flavor-diagonal matrices H and ∆ ; � � Σ + Σ † �� � � �� L 2 µ + R 2 tr and tr ∆ H µ J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions eLSM (3) Lagrangian: �� 2 ( D µ Σ) † D µ Σ � � � � � � + m 2 Σ † Σ Σ † Σ L = tr 0 tr + λ 1 tr + 1 �� � 2 � ( L µν ) 2 + ( R µν ) 2 � � Σ † Σ + λ 2 tr 4 tr �� m 2 � � �� 1 L 2 µ + R 2 + tr 2 + ∆ µ � � Σ + Σ † �� � det Σ + det Σ † � − tr − c A ; (9) H Covariant derivative D µ Σ = ∂ µ Σ + ig (Σ L µ − R µ Σ) J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions eLSM (4) Field matrices for N f = 2 π ) · � Σ = ( σ + i η ) t 0 + ( � a 0 + i � t , (10) a 1 µ ) · � R µ = ( ω µ + f 1 µ ) t 0 + ( � ρ µ + � (11) t , a 1 µ ) · � L µ = ( ω µ − f 1 µ ) t 0 + ( � ρ µ − � t (12) Symmetry breaking ( N f = 2) � � Σ + Σ † �� = h 0 tr (13) H 0 σ , = c A σ 2 − � 2 − η 2 + � � det Σ + det Σ † � � π 2 � c A a 0 (14) 2 J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions Effective action with Yukawa coupling Yukawa coupling y y ¯ y ¯ ψφψ , ψ i γ 5 φψ (15) Effective average action of the quark-meson model (QMM) φ i = � ϕ i � J , A i ,µ = � A i ,µ � J , and ψ a = � Ψ a � ¯ ηη : � 1 2 ∂ µ φ i ∂ µ φ i + 1 � 4 F i ,µν F i ,µν + U k ( φ i , A i ,µ ) − h 0 Γ k = 0 σ x � + ¯ ψ a γ µ ∂ µ ψ a + y ¯ ψ a Σ 5 ψ a ; (16) Local potential approximation (LPA): ∂ k Γ k ∝ ∂ k U k J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions Why to study the QMM? (1) Figure 2 : Renormalized masses as a function of the RG-scale (1); arXiv:1504.03585 [hep-ph] . J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions Why to study the QMM? (2) Figure 3 : Renormalized masses as a function of the RG-scale (2); arXiv:1504.03585 [hep-ph] . J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions Preliminary results A 150 1200 125 1000 mass [MeV] 100 800 σ 0 [MeV] 75 600 50 400 25 200 0 0 0 100 200 300 400 0 100 200 300 400 T [MeV] T [MeV] C D 150 1200 125 1000 mass [MeV] 100 800 σ 0 [MeV] 75 600 50 400 25 200 a 0 a 1 0 0 η σ π ρ 0 100 200 300 400 500 0 100 200 300 400 500 T [MeV] T [MeV] Figure 4 : Phase transition: QMM ( A , B ), eLSM ( C , D ). J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Effective action with Yukawa coupling Quark-meson model with (axial-)vector mesons Phase transitions Outlook Next steps: Simulations without axial anomaly Further optimization of meson masses and critical temperature Nonzero chemical potential µ : ¯ ψµγ 0 ψ Phase diagram ( T - µ plane) J¨ urgen Eser Quark-meson model within FRG