Axial anomaly and hadronic properties in a nuclear medium Gergely - PowerPoint PPT Presentation

Axial anomaly and hadronic properties in a nuclear medium Gergely Fejos Keio University, Department of Physics Topological Science Project Hadron structure and interaction in dense matter KEK Tokai Campus 12 November, 2018 GF & A. Hosaka, Phys. Rev. D 95 , 116011 (2017) aaa GF & A. Hosaka, Phys. Rev. D 98 , 036009 (2018) Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Outline aaa Motivation Functional Renormalization Group Chiral effective nucleon-meson theory at finite µ B Numerical results Summary Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Motivation Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Motivation AXIAL ANOMALY OF QCD: U A (1) anomaly: anomalous breaking of the U A (1) subgroup of U L ( N f ) × U R ( N f ) chiral symmetry − → vacuum-to-vacuum topological fluctuations (instantons) A = − g 2 16 π 2 ǫ µνρσ Tr [ T a F µν F ρσ ] ∂ µ j µ a U A (1) breaking interactions depend on instanton density → suppressed at high T 1 − − → calculations are trustworthy only at high temperature − → is the anomaly present at the phase transition? Very little is known at finite baryochemical potential ( µ B ) 2 − → effective models have not been explored in this direction 1 R. D. Pisarski, and L. G. Yaffe, Phys. Lett. B 97 , 110 (1980). 2 T. Schaefer, Phys. Rev. D 57 , 3950 (1998). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Motivation η ′ - NUCLEON BOUND STATE: Effective models at finite T and/or density: → effective models (NJL 3 , linear sigma models 4 ) predict a − aaaa ∼ 150 MeV drop in m η ′ at finite µ B Effective description of the mass drop: → attractive potential in medium ⇒ η ′ N bound state − → Analogous to Λ(1405) ∼ ¯ − KN bound state 3 P. Costa, M. C. Ruivo & Yu. L. Kalinovsky, Phys. Lett. B 560, 171 (2003). 4 S. Sakai & D. Jido, Phys. Rev. C 88 , 064906 (2013). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Motivation η ′ - NUCLEON BOUND STATE: Effective models at finite T and/or density: → effective models (NJL 3 , linear sigma models 4 ) predict a − aaaa ∼ 150 MeV drop in m η ′ at finite µ B Effective description of the mass drop: → attractive potential in medium ⇒ η ′ N bound state − → Analogous to Λ(1405) ∼ ¯ − KN bound state Problem with mean field calculations: they treat model parameters as environment independent constants − → ” A · v ” type of terms decrease ( A -constant, v -decreases) − → evolution of the ” A ” anomaly at finite T and µ B ? What is the role of fluctuations? 3 P. Costa, M. C. Ruivo & Yu. L. Kalinovsky, Phys. Lett. B 560, 171 (2003). 4 S. Sakai & D. Jido, Phys. Rev. C 88 , 064906 (2013). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Motivation Fluctuation effects in a quantum system is encoded in the effective action Partition function and effective action in field theory: [ S : classical action, φ : dynamical variable, ¯ φ : mean field, J : source field] � � D φ e − ( S [ φ ]+ � J φ ) , Γ[¯ J ¯ Z [ J ] = φ ] = − log Z [ J ] − φ Γ contains the truncated n-point functions − → amplitudes, part. lifetimes, thermodynamics (EoS, etc.) How to calculate the effective action? ⇒ perturbation theory! − → find a small parameter in S and Taylor expand − → fails in QCD & eff. models are not weakly coupled either Non-perturbative methods are necessary: Functional Renormalization Group (FRG) 5 5 C. Wetterich, Phys. Lett. B 301 , 90 (1993) Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Functional Renormalization Group Mathematical implementation of the FRG: k 2 Scale dependent partition function: � D φ e − ( S [ φ ]+ J φ ) � aa Z k [ J ] = R k (q) aaaaaaaa × e − 1 � φ R k φ 2 Scale dependent effective action: 0 k q 0 � ¯ Γ k [¯ J ¯ φ − 1 φ R k ¯ � φ ] = − log Z k [ J ] − φ 2 − → k ≈ Λ: no fluctuations included aaaa ⇒ Γ k [¯ φ ] | k =Λ = S [¯ φ ] − → k = 0: all fluctuations included aaaa ⇒ Γ k [¯ φ ] | k =0 = Γ[¯ φ ] Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Functional Renormalization Group Flow equation of the effective action: � ( T ) ∂ k Γ k = 1 + R k ) − 1 ( p , q ) = 1 ∂ k R k ( q , p )(Γ (2) k 2 2 q , p One-loop structure with dressed and regularized propagators − → RG change in the n -point vertices are aaaadescribed by one-loop diagrams − → exact relation, approximations are necessary 6 D. Litim, Phys. Rev. D 64 , 105007 (2001). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Functional Renormalization Group Flow equation of the effective action: � ( T ) ∂ k Γ k = 1 + R k ) − 1 ( p , q ) = 1 ∂ k R k ( q , p )(Γ (2) k 2 2 q , p One-loop structure with dressed and regularized propagators − → RG change in the n -point vertices are aaaadescribed by one-loop diagrams − → exact relation, approximations are necessary Derivative expansion (local potential approximation): � � � Γ k = Z k ∂ i Φ ∂ i Φ + V k (Φ; x ) x → ”optimized” regulator 6 : R k ( q ) = Z k ( k 2 − q 2 )Θ( k 2 − q 2 ) − 6 D. Litim, Phys. Rev. D 64 , 105007 (2001). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Chiral effective nucleon-meson theory at finite µ B 3 FLAVOR CHIRAL NUCLEON-MESON MODEL: Effective model of chiral symmetry breaking: order par. M [excitations of M : π, K , η, η ′ and a 0 , κ, f 0 , σ ] Tr [ ∂ i M † ∂ i M ] − Tr [ H ( M † + M )] L M = V ch ( M ) + A · (det M † + det M ) + 1 4( ∂ i ω j − ∂ j ω i ) 2 + 1 2 m ω ω 2 i + ¯ L ω + N = N ( ∂ / − µ B γ 0 ) N , N ( g Y ˜ ¯ L Yuk = M 5 − ig ω ω / ) N − → nucleon mass: entirely from Yukawa coupling Goal: calculation of the effective action Γ − → Particular interest: finite µ B − → How does the anomaly behave toward the nuclear aaaaliquid-gas transition? Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Chiral effective nucleon-meson theory at finite µ B Fluctuations are included in the quantum effective action Γ k : � � Tr [ ∂ i M † ∂ i M ] − Tr [ H ( M † + M )] + ¯ Γ k = N ( ∂ / − µ B γ 0 ) N x +1 4( ∂ i ω j − ∂ j ω i ) 2 + 1 � i + ¯ N ( g Y ˜ 2 m 2 ω ω 2 M 5 − ig ω ω / ) N + V k Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Chiral effective nucleon-meson theory at finite µ B Fluctuations are included in the quantum effective action Γ k : � � Tr [ ∂ i M † ∂ i M ] − Tr [ H ( M † + M )] + ¯ Γ k = N ( ∂ / − µ B γ 0 ) N x +1 4( ∂ i ω j − ∂ j ω i ) 2 + 1 � i + ¯ N ( g Y ˜ 2 m 2 ω ω 2 M 5 − ig ω ω / ) N + V k We fluctuation effects in the mesonic potentials: V k = V ch , k ( M ) + A k ( M ) · (det M † + det M ) The chiral potential splits into two parts: k ( ˜ V ch , k ( M ) = V 3 fl k ( M ) + V 2 fl M ) Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Chiral effective nucleon-meson theory at finite µ B Fluctuations are included in the quantum effective action Γ k : � � Tr [ ∂ i M † ∂ i M ] − Tr [ H ( M † + M )] + ¯ Γ k = N ( ∂ / − µ B γ 0 ) N x +1 4( ∂ i ω j − ∂ j ω i ) 2 + 1 � i + ¯ N ( g Y ˜ 2 m 2 ω ω 2 M 5 − ig ω ω / ) N + V k We fluctuation effects in the mesonic potentials: V k = V ch , k ( M ) + A k ( M ) · (det M † + det M ) The chiral potential splits into two parts: k ( ˜ V ch , k ( M ) = V 3 fl k ( M ) + V 2 fl M ) Projecting the flow equation onto chiral invariants lead to k ( ˜ flows of V 3 fl k ( M ), V 2 fl M ) and A k ( M ) � ( T ) ∂ k Γ k = 1 ∂ k R k ( q , p )(Γ (2) + R k ) − 1 ( p , q ) � � Tr k 2 q , p Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Chiral effective nucleon-meson theory at finite µ B Baryon Silver Blaze property: − → no change in the effective action for T = 0 if aaaa µ B < m N − B ≡ µ B , c 7 M. Drews and W. Weise, Prog. Part. Nucl. Phys. 93, 69 (2017). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium

Chiral effective nucleon-meson theory at finite µ B Baryon Silver Blaze property: − → no change in the effective action for T = 0 if aaaa µ B < m N − B ≡ µ B , c At µ B = µ B , c : 7 − → 1st order phase transition from nuclear gas to liquid → nuclear density jumps from zero to n 0 ≈ 0 . 17 fm − 3 − − → non-strange chiral condensate jumps from f π to v ns , nucl aaaa(Landau mass M L ≈ 0 . 8 m N ⇒ v ns , nucl ≈ 69 . 5 MeV ) 7 M. Drews and W. Weise, Prog. Part. Nucl. Phys. 93, 69 (2017). Gergely Fejos Axial anomaly and hadronic properties in a nuclear medium