Thermal Evolution of the Axial Anomaly

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Thermal evolution of the axial anomaly

Gergely Fej˝os

Research Center for Nuclear Physics Osaka University

The 10th APCTP-BLTP/JINR-RCNP-RIKEN Joint Workshop on Nuclear and Hadronic Physics

18th August, 2016

G. Fejos & A. Hosaka, arXiv: 1604.05982 Gergely Fej˝os Thermal evolution of the axial anomaly Outline

aaa Motivation

Functional renormalization group

Chiral (linear) sigma model and axial anomaly

Extension with nucleons

Summary

Gergely Fej˝os Thermal evolution of the axial anomaly Motivation

Gergely Fej˝os Thermal evolution of the axial anomaly Chiral symmetry is spontaneuously broken in the ground state:

< ψψ¯ > = < ψ¯R ψL > + < ψ¯LψR > 6= 0

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?

Motivation

QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ψ¯ (iγ Dµ − m) ψ 4 µν i µ ij j

Approximate chiral symmetry for Nf = 2, 3 ﬂavors:

iT aθa iT aθa ψL → e L ψL,ψ R → e R ψR

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

Gergely Fej˝os Thermal evolution of the axial anomaly Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point?

Motivation

QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ψ¯ (iγ Dµ − m) ψ 4 µν i µ ij j

Approximate chiral symmetry for Nf = 2, 3 ﬂavors:

iT aθa iT aθa ψL → e L ψL,ψ R → e R ψR

[vector: θL + θR , axialvector: θL − θR ] Chiral symmetry is spontaneuously broken in the ground state:

< ψψ¯ > = < ψ¯R ψL > + < ψ¯LψR > 6= 0

SSB pattern: SUL(Nf ) × SUR (Nf ) −→ SUV (Nf ) Anomaly: UA(1) is broken by instantons

Gergely Fej˝os Thermal evolution of the axial anomaly Motivation

QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ψ¯ (iγ Dµ − m) ψ 4 µν i µ ij j

Approximate chiral symmetry for Nf = 2, 3 ﬂavors:

iT aθa iT aθa ψL → e L ψL,ψ R → e R ψR

[vector: θL + θR , axialvector: θL − θR ] Chiral symmetry is spontaneuously broken in the ground state:

< ψψ¯ > = < ψ¯R ψL > + < ψ¯LψR > 6= 0

Gergely Fej˝os Thermal evolution of the axial anomaly two independent quartic couplings: g1, g2 0 8 explicit symmetry breaking: H = h0T + h8T

’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

Motivation

Effective field theory of chiral symmetry breaking: −→ three-flavor linear sigma model −→ M = T a(sa + iπa) 0 [pseudoscalar mesons: π, K, η, η , scalar mesons: a0, κ, f0, σ] g L = ∂ M∂µM† − µ2 Tr (MM†) − 1 [ Tr (MM†)]2 µ 9 g − 2 Tr (MM†)2 − Tr [H(M + M†)] − a(det M + det M†) 3

Gergely Fej˝os Thermal evolution of the axial anomaly Motivation

two independent quartic couplings: g1, g2 0 8 explicit symmetry breaking: H = h0T + h8T

’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˝os Thermal evolution of the axial anomaly Functional renormalization group

FRG: follows the idea of Wilsonian renormalization group

Z R R 1 − S[φ]+ Jφ+ φRk φ Zk [J] = Dφe 2

R : IR regulator function k k 2

Requirements: 1., scale separation

(suppress modes q k) (q) . k R 2., Rk −→ ∞ if k −→ ∞

3., Rk −→ 0 if k −→ 0 0 0 k q Gradually moving k from Λ to 0, ﬂuctuations are getting integrated out

Gergely Fej˝os Thermal evolution of the axial anomaly Functional renormalization group

scale-dependent eﬀective action: Z 1 Z Γ [φ¯] = − log Z [J] − Jφ¯ − φ¯R φ¯ k k 2 k

−→ 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˝os Thermal evolution of the axial anomaly Derivative expansion + Chiral invariant expansion: Z h µ † 2 † g1,k (M) † 2 Γk = ∂µM∂ M − µk (M) Tr (MM ) − [ Tr (MM )] x 9 g (M) i − 2,k Tr (MM†)2 − A (M)(det M + det M†) − h s 3 k i i −→ resummation of operators: aaaa∼ Tr n[MM†] × (det M + det M†) and aaaa∼ Tr n[MM†] × ( Tr [MM†MM†])

Chiral (linear) sigma model

Gergely Fej˝os Thermal evolution of the axial anomaly Chiral (linear) sigma model

Flow of the eﬀective action is described by the Wetterich equation: Z (T ) 1 (2) −1 ∂k Γk = Tr ∂k Rk (q, p)(Γk + Rk ) (p, q) 2 q,p −→ exact relation, functional integro-diﬀerential equation −→ approximation(s) needed Derivative expansion + Chiral invariant expansion: Z h µ † 2 † g1,k (M) † 2 Γk = ∂µM∂ M − µk (M) Tr (MM ) − [ Tr (MM )] x 9 g (M) i − 2,k Tr (MM†)2 − A (M)(det M + det M†) − h s 3 k i i −→ resummation of operators: aaaa∼ Tr n[MM†] × (det M + det M†) and aaaa∼ Tr n[MM†] × ( Tr [MM†MM†])

Gergely Fej˝os Thermal evolution of the axial anomaly First one ﬁxes the explicit symmetry breaking using the PCAC relations:

2 5µ ∂ † mafaπˆa = ∂µJa = − a Tr (H(M + M )) (a = 1, ...8) ∂θA 3 3 −→ h0 ≈ (286MeV ) , h8 ≈ −(311MeV ) 2 µ and g1 are ﬁxed by the light spectra π, K 0 anomaly parameter a and g2 is selected to reproduce η and η masses in the vacuum

Chiral (linear) sigma model

The model has 6 free parameters, which have to be ﬁxed using experimental input 2 −→ chiral symmetric parameters: µ , g1, g2 −→ explicit breaking terms: h0, h8 −→ chiral anomaly parameter: a

Gergely Fej˝os Thermal evolution of the axial anomaly 2 µ and g1 are ﬁxed by the light spectra π, K 0 anomaly parameter a and g2 is selected to reproduce η and η masses in the vacuum

Chiral (linear) sigma model

First one ﬁxes the explicit symmetry breaking using the PCAC relations:

2 5µ ∂ † mafaπˆa = ∂µJa = − a Tr (H(M + M )) (a = 1, ...8) ∂θA 3 3 −→ h0 ≈ (286MeV ) , h8 ≈ −(311MeV )

Gergely Fej˝os Thermal evolution of the axial anomaly Chiral (linear) sigma model

First one ﬁxes the explicit symmetry breaking using the PCAC relations:

Gergely Fej˝os Thermal evolution of the axial anomaly Chiral (linear) sigma model

Functions need to be determined numerically: 2 −→ eﬀective potential pieces: µk (M), g1,k (M), g2,k (M) −→ anomaly function Ak (M) Step I.: solve the renormalization group equations for these functions at T = 0 to determine the model parameters 2 (µ , 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˝os Thermal evolution of the axial anomaly Numerical results

without anomaly

1000 f0

800 κ

[MeV] 0

600 η' 400 K masses σ 200 π,η

0 0 50 100 150 200 250 300 350 400 T [MeV]

Gergely Fej˝os Thermal evolution of the axial anomaly Numerical results

with anomaly

1000 f 0 κ η' 800 a0 [MeV]

600 η K σ masses 400

200 π

0 100 200 300 400 500 T [MeV]

Gergely Fej˝os Thermal evolution of the axial anomaly Numerical results

1000

η' mass 900

800 [MeV]

700 masses

' η

, 600 η η mass

500

400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

T/TC −→ solid: full solution, dashed: ﬁeld- and T independent anomaly

Gergely Fej˝os Thermal evolution of the axial anomaly Numerical results

7 T = 1.2 Tc T = 0.7 Tc 6 T = 0

[GeV] 5

|A| 4

2 0 100 200 300 + √ [MeV]

Gergely Fej˝os Thermal evolution of the axial anomaly Numerical results

7.5

6.5 [GeV]

5.5 anomaly 5

4.5 |A[0]| |A[vmin]| 4 0 100 200 300 400 500 T [MeV]

Gergely Fej˝os Thermal evolution of the axial anomaly Numerical results

0.8 strange (0)

s/ns 0.6

(T)/v 0.4 s/ns

v non strange 0.2

0 0 100 200 300 400 500 T [MeV] dashed: without anomaly, solid: with anomaly

Gergely Fej˝os Thermal evolution of the axial anomaly Introduction of the nucleon ﬁeld: ψT (x)= n(x) p(x)

g L = ∂ M∂µM† − µ2 Tr (MM†) − 1 [ Tr (MM†)]2 µ 9 g − 2 Tr (MM†)2 − Tr [H(M + M†)] − a(det M + det M†) 3 + ψ¯(−∂i γi + µB γ0)ψ + gψ¯M˜5ψ

˜ ˜ P ã a a M5 is a modified meson field: M5 = ns,1,2,3 T (σ + iπ γ5) −→ element of an embedded U(2) algebra in flavor U(3)

1 1 0 1 Pauli T˜ns = , T˜1,2,3 = 2 0 1 2 matrices

Chiral (linear) sigma model with nucleons

So far: ﬁnite T results with zero baryochemical potential Behavior of the anomaly in nuclear medium?

Gergely Fej˝os Thermal evolution of the axial anomaly Chiral (linear) sigma model with nucleons

So far: ﬁnite T results with zero baryochemical potential Behavior of the anomaly in nuclear medium? Introduction of the nucleon ﬁeld: ψT (x)= n(x) p(x)

g L = ∂ M∂µM† − µ2 Tr (MM†) − 1 [ Tr (MM†)]2 µ 9 g − 2 Tr (MM†)2 − Tr [H(M + M†)] − a(det M + det M†) 3 + ψ¯(−∂i γi + µB γ0)ψ + gψ¯M˜5ψ

˜ ˜ P ã a a M5 is a modified meson field: M5 = ns,1,2,3 T (σ + iπ γ5) −→ element of an embedded U(2) algebra in flavor U(3)

1 1 0 1 Pauli T˜ns = , T˜1,2,3 = 2 0 1 2 matrices

Gergely Fej˝os Thermal evolution of the axial anomaly Chiral (linear) sigma model with nucleons

Due to meson-nucleon interactions, the eﬀective action changes: Z h µ † 2 † g1,k (M) † 2 Γk = ∂µM∂ M − µk (M) Tr (MM ) − [ Tr (MM )] x 9 g (M) − 2,k Tr (MM†)2 − A (M)(det M + det M†) − h s 3 k i i +ψ¯(−∂i γi + µB γ0)ψ + gψ¯M5ψ † † † i +f1,k Tr (M˜M˜ ) + f2,k Tr (M˜M˜ M˜M˜ )

˜ P ˜a a a −→ M = a=ns,1,2,3 T (σ + iπ ) 0 −→ only dynamics of π, η, η (p.s.) and a0, f0, σ (s.) change −→ nucleons do not couple to K and κ mesons The nucleon-meson coupling (g) does not renormalize

µB (and T ) is freely tunable

Gergely Fej˝os Thermal evolution of the axial anomaly Numerical results

0.98

0.96

)/A(0)| 0.94 B µ |A( 0.92

0.9 T=300 MeV T=200 MeV T=100 MeV 0.88 0 100 200 300 400 500 600

µB [MeV]

Gergely Fej˝os Thermal evolution of the axial anomaly Summary

Thermal properties of the axial anomaly via the three-flavor linear sigma model, and its extension with nucleons Fluctuations (quantum and thermal) have been included using the functional renormalization group (FRG) approach Results: −→ thermal evolution of the mass spectrum and condensates −→ temperature dependence of the UA(1) anomaly factor Most important findings: −→ meson fluctuations strengthen the anomaly with respect aaaato the temperature ⇒ no recovery at TC −→ putting the system into a nuclear environment also aaaaaffects the anomaly ⇒ it decreases as µB grows

Gergely Fej˝os Thermal evolution of the axial anomaly