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Thermal Evolution of the Axial Anomaly 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: < ¯ > = < ¯R L > + < ¯L R > 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 + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [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 + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [vector: θL + θR , axialvector: θL − θR ] Chiral symmetry is spontaneuously broken in the ground state: < ¯ > = < ¯R L > + < ¯L R > 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 + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [vector: θL + θR , axialvector: θL − θR ] Chiral symmetry is spontaneuously broken in the ground state: < ¯ > = < ¯R L > + < ¯L R > 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? 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@µMy − µ2 Tr (MMy) − 1 [ Tr (MMy)]2 µ 9 g − 2 Tr (MMy)2 − Tr [H(M + My)] − a(det M + det My) 3 Gergely Fej}os Thermal evolution of the axial anomaly 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@µMy − µ2 Tr (MMy) − 1 [ Tr (MMy)]2 µ 9 g − 2 Tr (MMy)2 − Tr [H(M + My)] − a(det M + det My) 3 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 −! 1 if k −! 1 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 y 2 y g1;k (M) y 2 Γk = @µM@ M − µk (M) Tr (MM ) − [ Tr (MM )] x 9 g (M) i − 2;k Tr (MMy)2 − A (M)(det M + det My) − h s 3 k i i −! resummation of operators: aaaa∼ Tr n[MMy] × (det M + det My) and aaaa∼ Tr n[MMy] × ( Tr [MMyMMy]) 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 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 y 2 y g1;k (M) y 2 Γk = @µM@ M − µk (M) Tr (MM ) − [ Tr (MM )] x 9 g (M) i − 2;k Tr (MMy)2 − A (M)(det M + det My) − h s 3 k i i −! resummation of operators: aaaa∼ Tr n[MMy] × (det M + det My) and aaaa∼ Tr n[MMy] × ( Tr [MMyMMy]) Gergely Fej}os Thermal evolution of the axial anomaly First one fixes the explicit symmetry breaking using the PCAC relations: 2 5µ @ y mafaπ^a = @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 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 First one fixes the explicit symmetry breaking using the PCAC relations: 2 5µ @ y mafaπ^a = @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 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 First one fixes the explicit symmetry breaking using the PCAC relations: 2 5µ @ y mafaπ^a = @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 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 a [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 8 7 T = 1.2 Tc T = 0.7 Tc 6 T = 0 [GeV] 5 |A| 4 3 2 0 100 200 300 + √<M M> [MeV] Gergely Fej}os Thermal evolution of the axial anomaly Numerical results 8 7.5 7 6.5 [GeV] 6 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 1 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: T (x)= n(x) p(x) g L = @ M@µMy − µ2 Tr (MMy) − 1 [ Tr (MMy)]2 µ 9 g − 2 Tr (MMy)2 − Tr [H(M + My)] − a(det M + det My) 3 + ¯(−@i γi + µB γ0) + g ¯M~5 ~ ~ P ~a 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: T (x)= n(x) p(x) g L = @ M@µMy − µ2 Tr (MMy) − 1 [ Tr (MMy)]2 µ 9 g − 2 Tr (MMy)2 − Tr [H(M
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