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 Quark-meson model with (axial-)vector mesons Recap: Bosonic FRG 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 Γk Bosonic flow equation along renormalization scale k [Phys. Lett. B301, 90-94] Regulating function Rk introduces the k-dependence: ∂kΓk = 1 2 tr

• Γ(2)

k

+ Rk

−1 ∂kRk

• ,

(1)

• Γ(2)

k

• αα′ =

δ2Γk δΦαδΦα′ (2)

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Recap: Bosonic FRG Extension to fermions

Flow in coupling space

Figure 1 : Coupling space {c1, c2, . . . , cn};

"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 Quark-meson model with (axial-)vector mesons Recap: Bosonic FRG Extension to fermions

Extension to fermions

Fermionic flow equation [Phys. Rev. B70, 125111] Grassmann-valued fields and sources: ∂kΓk = − tr

• Γ(2)

k

+ Rk

−1 ∂kRk

• ,

(3)

• Γ(2)

k

• αα′ =

− → δ δΦα Γk ← − δ δΦα′ (4) Mixed Bose-Fermi system ∂kΓk = 1 2 str

• Γ(2)

k

+ Rk

−1 ∂kRk

• (5)

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

Extended linear sigma model (eLSM) (1)

Spin-0 fields Definitions: Σ = (σa + iπa)ta , Σ5 = (σa + iγ5πa)ta (6) scalars σa, pseudoscalars πa, U(Nf )-generators ta Spin-1 fields Definition of right- and left-handed fields: Rµ = (Va,µ + Aa,µ)ta , Lµ = (Va,µ − Aa,µ)ta , (7) vector fields Va,µ and axial-vector fields Aa,µ; Field strength tensors Rµν = ∂µRν − ∂νRµ, Lµν = ∂µLν − ∂νLµ

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

eLSM (2)

U(Nf )R × U(Nf )L-transformations Σ → U†

RΣUL ,

Rµ → U†

RRµUR ,

Lµ → U†

LLµUL

(8) Symmetry breaking Axial anomaly: cA

• det Σ + det Σ†

Nonzero quark masses: flavor-diagonal matrices H and ∆; tr

• H
• Σ + Σ†

and tr

• ∆
• L2

µ + R2 µ

• J¨

urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

eLSM (3)

Lagrangian: L = tr

• (DµΣ)† DµΣ
• + m2

0 tr

• Σ†Σ
• + λ1
• tr
• Σ†Σ

2

+λ2 tr

• Σ†Σ

2

+ 1 4 tr

• (Lµν)2 + (Rµν)2

+ tr

• m2

1

2 + ∆ L2

µ + R2 µ

• − tr
• H
• Σ + Σ†

− cA

• det Σ + det Σ†

; (9) Covariant derivative DµΣ = ∂µΣ + ig(ΣLµ − RµΣ)

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

eLSM (4)

Field matrices for Nf = 2 Σ = (σ + iη)t0 + ( a0 + i π) · t , (10) Rµ = (ωµ + f1µ)t0 + ( ρµ + a1µ) · t , (11) Lµ = (ωµ − f1µ)t0 + ( ρµ − a1µ) · t (12) Symmetry breaking (Nf = 2) tr

• H
• Σ + Σ†

= h0

0σ ,

(13) cA

• det Σ + det Σ†

= cA 2

• σ2 −

a0

2 − η2 +

π2 (14)

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

Effective action with Yukawa coupling

Yukawa coupling y y ¯ ψφψ , y ¯ ψiγ5φψ (15) Effective average action of the quark-meson model (QMM) φi = ϕiJ, Ai,µ = Ai,µJ, and ψa = Ψa¯

ηη:

Γk =

• x

1

2∂µφi∂µφi + 1 4Fi,µνFi,µν + Uk(φi, Ai,µ) − h0

0σ

+ ¯ ψaγµ∂µψa + y ¯ ψaΣ5ψa

• ;

(16) Local potential approximation (LPA): ∂kΓk ∝ ∂kUk

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling 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 Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling 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 Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

Preliminary results

100 200 300 400 25 50 75 100 125 150 T [MeV] σ0 [MeV]

A

100 200 300 400 200 400 600 800 1000 1200 T [MeV] mass [MeV] 100 200 300 400 500 25 50 75 100 125 150 T [MeV] σ0 [MeV]

C

100 200 300 400 500 200 400 600 800 1000 1200 T [MeV] mass [MeV]

D

a0 η σ π a1 ρ

Figure 4 : Phase transition: QMM (A,B), eLSM (C,D).

J¨ urgen Eser Quark-meson model within FRG

FRG with fermions Quark-meson model with (axial-)vector mesons Effective action with Yukawa coupling Phase transitions

Outlook

Next steps: Simulations without axial anomaly Further optimization of meson masses and critical temperature Nonzero chemical potential µ: ¯ ψµγ0ψ Phase diagram (T-µ plane)

J¨ urgen Eser Quark-meson model within FRG