Phase transition and axial anomaly in the 3-flavor chiral meson - - PowerPoint PPT Presentation

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Phase transition and axial anomaly in the 3-flavor chiral meson - - PowerPoint PPT Presentation

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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

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Phase transition and axial anomaly in the 3-flavor chiral meson model

Gergely Fej˝

  • s

Research Center for Nuclear Physics Osaka University

Workshop on J-PARC hadron physics 2016 a 4th March, 2016

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral meson

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Outline

aaa Motivation Functional renormalization group Chiral (linear) sigma model and axial anomaly Numerical results Summary

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Motivation

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Motivation

QCD Lagrangian with quarks and gluons: L = −1 4G a

µνG µνa + ¯

ψi(iγµDµ − m)ijψj Approximate chiral symmetry for Nf = 2, 3 flavors: ψL → eiT aθa

LψL,

ψR → eiT aθa

RψR

[vector: θL + θR, axialvector: θL − θR]

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Motivation

QCD Lagrangian with quarks and gluons: L = −1 4G a

µνG µνa + ¯

ψi(iγµDµ − m)ijψj Approximate chiral symmetry for Nf = 2, 3 flavors: ψL → eiT aθa

LψL,

ψR → eiT aθa

RψR

[vector: θL + θR, axialvector: θL − θR] Chiral symmetry is spontaneuously broken in the ground state: < ¯ ψiψi > = < ¯ ψi,Rψi,L > + < ¯ ψi,Lψi,R > = SSB pattern: SUL(Nf ) × SUR(Nf ) − → SUV (Nf ) Anomaly: UA(1) is broken by instantons

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Motivation

QCD Lagrangian with quarks and gluons: L = −1 4G a

µνG µνa + ¯

ψi(iγµDµ − m)ijψj Approximate chiral symmetry for Nf = 2, 3 flavors: ψL → eiT aθa

LψL,

ψR → eiT aθa

RψR

[vector: θL + θR, axialvector: θL − θR] Chiral symmetry is spontaneuously broken in the ground state: < ¯ ψiψi > = < ¯ ψi,Rψi,L > + < ¯ ψi,Lψi,R > = SSB pattern: SUL(Nf ) × SUR(Nf ) − → SUV (Nf ) Anomaly: UA(1) is broken by instantons Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point?

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Motivation

Effective field theory of chiral symmetry breaking: − → three-flavor linear sigma model − → M = T a(sa + iπa) [pseudoscalar mesons: π, K, η, η′, scalar mesons: a0, κ, f0, σ] L = ∂µM∂µM† − µ2 Tr (MM†) − g1 9 [ Tr (MM†)]2 − g2 3 Tr (MM†)2 − Tr [H(M + M†)] − a(det M + det M†)

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Motivation

Effective field theory of chiral symmetry breaking: − → three-flavor linear sigma model − → M = T a(sa + iπa) [pseudoscalar mesons: π, K, η, η′, scalar mesons: a0, κ, f0, σ] L = ∂µM∂µM† − µ2 Tr (MM†) − g1 9 [ Tr (MM†)]2 − g2 3 Tr (MM†)2 − Tr [H(M + M†)] − a(det M + det M†) two independent quartic couplings: g1, g2 explicit symmetry breaking: H = h0T 0 + h8T 8 ’t Hooft determinant breaks only the UA(1) symmetry Goal: take into account quantum- and thermal fluctuations to see the finite temperature behavior of the system

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Functional renormalization group

FRG: follows the idea of Wilsonian renormalization group Zk[J] =

  • Dφe−
  • S[φ]+
  • Jφ+

1

2 φRkφ

  • Rk: IR regulator function

Requirements: 1., scale separation (suppress modes q k) 2., Rk − → ∞ if k − → ∞ 3., Rk − → 0 if k − → 0

k2 k

q Rk(q)

Gradually moving k from Λ to 0, fluctuations are getting integrated out

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Functional renormalization group

scale-dependent effective action: Γk[¯ φ] = − log Zk[J] −

  • J ¯

φ − 1 2

  • ¯

φRk ¯ φ − → k ≈ Λ: no fluctuations ⇒ Γk=Λ[¯ φ] = S[¯ φ] − → k = 0: all fluctuations ⇒ Γk=0[¯ φ] = Γ1PI[¯ φ] the scale-dependent effective action interpolates between classical- and quantum effective actions

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Chiral sigma model

Flow of the effective action is described by the Wetterich equation: ∂kΓk = 1 2 Tr (T)

q,p

∂kRk(q, p)(Γ(2)

k

+ Rk)−1(p, q) − → exact relation, functional integro-differential equation − → approximation(s) needed

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Chiral sigma model

Flow of the effective action is described by the Wetterich equation: ∂kΓk = 1 2 Tr (T)

q,p

∂kRk(q, p)(Γ(2)

k

+ Rk)−1(p, q) − → exact relation, functional integro-differential equation − → approximation(s) needed Derivative expansion of Γk: Γk[M] =

  • x
  • − Vk(M) + Zk(M)∂µM†∂µM + O(∂4)
  • Assumptions:

Vk(M) = Uk(M) + Tr (H(M + M†)) + Ak(M)(det M + det M†) Zk(M) ≈ 1 − → Uk: chiral symmetric part − → Ak: UA(1) breaking term, field dependent!

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Chiral sigma model

The model has 6 free parameters, which have to be fixed using experimental input − → chiral symmetric parameters: µ2, g1, g2 − → explicit breaking terms: h0, h8 − → chiral anomaly parameter: a

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Chiral sigma model

The model has 6 free parameters, which have to be fixed using experimental input − → chiral symmetric parameters: µ2, g1, g2 − → explicit breaking terms: h0, h8 − → chiral anomaly parameter: a First one fixes the explicit symmetry breaking using the PCAC relations: m2

afaˆ

πa = ∂µJ5µ

a

= − ∂ ∂θa

A

Tr (H(M + M†)) (a = 1, ...8) − → h0 ≈ (286MeV )3, h8 ≈ −(311MeV )3

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Chiral sigma model

The model has 6 free parameters, which have to be fixed using experimental input − → chiral symmetric parameters: µ2, g1, g2 − → explicit breaking terms: h0, h8 − → chiral anomaly parameter: a First one fixes the explicit symmetry breaking using the PCAC relations: m2

afaˆ

πa = ∂µJ5µ

a

= − ∂ ∂θa

A

Tr (H(M + M†)) (a = 1, ...8) − → h0 ≈ (286MeV )3, h8 ≈ −(311MeV )3 µ2, g1, g2 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˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Chiral sigma model

Functions need to be determined: − → effective potential Vk(M) − → anomaly function Ak(M) Step I.: solve the renormalization group equations for Vk and Ak at T = 0 to determine the model parameters (µ2, g1, g2, 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 UA(1) anomaly

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Numerical results

200 400 600 800 1000 50 100 150 200 250 300 350 400 π,η σ K η' a0 κ f0 masses [MeV] T [MeV] without anomaly

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Numerical results

200 400 600 800 1000 100 200 300 400 500 π η σ K η' a0 κ f0 masses [MeV] T [MeV] with anomaly

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Numerical results

2 3 4 5 6 7 8 100 200 300 |A| [GeV] √<M+M> [MeV] T = 1.2 Tc T = 0.7 Tc T = 0

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Numerical results

4 4.5 5 5.5 6 6.5 7 7.5 8 100 200 300 400 500 anomaly [GeV] T [MeV] |A[0]| |A[vmin]|

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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Numerical results

0.2 0.4 0.6 0.8 1 100 200 300 400 500 strange non strange vs/ns(T)/vs/ns(0) T [MeV]

dashed: without anomaly, solid: with anomaly

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...

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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 (vns: yes, vs: no) − → temperature dependence of the UA(1) anomaly factor Most important findings: − → mass spectrum might not be useful for UA(1) − → meson fluctuations strengthen the anomaly

Gergely Fej˝

  • s

Phase transition and axial anomaly in the 3-flavor chiral...