Mesonic and nucleon fluctuation effects in nuclear medium

Mesonic and nucleon fluctuation effects in nuclear medium Research Center for Nuclear Physics Osaka University Workshop of Recent Developments in QCD and Quantum Field Theories National Taiwan University, Taipei 11 November, 2017 GF & A. Hosaka, Phys. Rev. D 94, 036005 (2016) aaa GF & A. Hosaka, Phys. Rev. D 95, 116011 (2017)

Outline aaa Motivation Functional Renormalization Group Chiral effective nucleon-meson theory at finite µ B Numerical results Summary

Motivation

Motivation AXIAL ANOMALY OF QCD: U A (1) anomaly: anomalous breaking of the U A (1) subgroup of chiral symmetry vacuum-to-vacuum topological fluctuations (instantons) µ j µa A = g 2 16π 2 ɛµνρσ Tr [T a F µν F ρσ ] 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. B97, 110 (1980). 2 T. Schaefer, Phys. Rev. D57, 3950 (1998).

Motivation η - NUCLEON BOUND STATE: Effective models at finite T and/or density: effective models (NJL 3, linear sigma models 4 ) predict a aaaadrop 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 effective model calculations: they treat model parameters as environment independent constants a v type of terms decrease (a-constant, v-decreases) evolution of a 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. C88, 064906 (2013).

Motivation NEUTRON STARS: Two solar mass neutron star observations: J1614-2230: mass = (1.928 ± 0.017)M s J0348+0432: mass = (2.01 ± 0.04)M s Theoretical challenge: describe stiffness of the equation of state of cold dense matter Effective model calculations stop at mean field level quantum and density fluctuations? Hyperon puzzle hyperons cause an undesired softening of the EoS few times normal nucl. density hyperons should appear

Motivation FLUCTUATION EFFECTS IN FIELD THEORY: 2nd order transitions in statistical field theory diverging correlation length invalidates pert. theory solution: Wilson s momentum space RG explanation of universality, critical exponents, etc. Large logarithms in continuum quantum field theory pert. theory is organized in terms of λ log(e/m) it can fail even for small couplings resummation! solution: Gell-Mann - Low RG Problem in QCD: running coupling grows as energy decreases RG cannot provide a generic solution effective models can help, but are also strongly coupled

Functional Renormalization Group Mathematical implementation: Scale dependent partition function: aa Z k [J] = Dφe (S[φ]+ Jφ) aaaaaaaa e 1 2 φrk φ R k (q) k 2 Scale dependent effective action: 0 0 k q Γ k [ φ] = log Z k [J] J φ 1 2 φr k φ 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]

Motivation Flow equation of the effective action: k Γ k = 1 2 (T ) q,p k R k (q, p)(γ (2) k + R k ) 1 (p, q) = 1 2 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 5 : R k (q) = Z k (k 2 q 2 )Θ(k 2 q 2 ) 5 D. Litim, Phys. Rev. D64, 105007 (2001).

Chiral effective nucleon-meson model Chiral symmetry of QCD: U(3) U(3) (Ψ L/R U L/R Ψ L/R ) Effective model of mesons (M) and the nucleon (N): [M: π, K, η, η and a 0, κ, f 0, σ, N: n, p, ω for short range N N int.] L = Tr [ i M i M] + µ 2 Tr (M M) + g 1 9 [ Tr (M M)] 2 + g 2 3 Tr (M M) 2 + higher order terms in M + a(det M + det M) Tr [H(M + M)] + N( i γ i µ B γ 0 )N g N M 5 N + 1 4 ( iω j + i j) 2 + 1 2 m2 ωω i ω i ig ω Nω i γ i N Model parameters: meson mass parameter (µ 2 ), quartic couplings (g 1, g 2 ), explicit breaking (H = h 0 T 0 + h 8 T 8 ), U A (1) anomaly (a), Yukawa couplings (g,g ω ), vector meson mass (m 2 ω)

Chiral effective nucleon-meson model Local potential approximation: [ Γ k = V k [M] + Tr [ i M i M] Tr [H(M + M )] x V k [M] = V mes k + N( i γ i + µ B γ 0 )N g N M 5 N + 1 4 ( iω j + i j) 2 + 1 ] 2 m2 ωω i ω i ig ω Nω i γ i N [M] + Vk ferm [ M] Chiral invariant expansion: ρ det = det M + det M, aaaaaaaaaaaaaaaaaaaaaaaaρ 4 = Tr [M M Tr [M M]/3] 2 ρ 2 = Tr [M M], V mes k = U k (ρ 2 ) + C k (ρ 2 ) ρ 4 + A k (ρ 2 ) ρ det, V ferm k = Ũ k ( ρ 2 ) Projecting the flow equation onto various operators, one derives individual flow equations for U k, C k, A k, Ũ k

Chiral effective nucleon-meson model Baryon Silver Blaze property: no change in thermodynamics for µ B < m N B µ B,c 6 M. Drews and W. Weise, Prog. Part. Nucl. Phys. 93, 69 (2017).

Chiral effective nucleon-meson model Baryon Silver Blaze property: no change in thermodynamics for µ B < m N B µ B,c At µ B = µ B,c : 6 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 0.8m N v ns,nucl 69.5 MeV ) The first order transition is related to the condensation of the timelike component of the ω vector particle ω couples to v ns that couples to v s jump in all order parameters 6 M. Drews and W. Weise, Prog. Part. Nucl. Phys. 93, 69 (2017).

Chiral effective nucleon-meson model Step I.: solve equations in the vacuum determine model parameters physical masses of π, K, η, η and N are used PCAC relations leads to H (h 0, h 8 ) m 2 αf α π α = µ J 5µ α = θa α Tr (H(M + M )) inputs from the nuclear liquid gas transition: aaaasat. density, Landau mass, binding energy, surface tension Step II.: solve the same equations at finite T and µ B mass spectrum in medium details of symmetry restoration U A (1) anomaly

Numerical results: mass spectrum at finite T 1100 1000 900 800 masses [MeV] 700 600 500 400 f 0κ a 300 0 η η 200 K σ 100 π N 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 T/T C

Numerical results: η η system at finite T 1000 900 η' mass η, η' masses [MeV] 800 700 600 η mass 500 400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 T/T C solid: full solution, dashed: field- and T independent anomaly

Numerical results: anomaly at finite T 9 8 A k 0 [GeV] 7 6 5 T = 1.2 T c T = 0.6 T c T = 0 4 3 0 100 200 300 ρ 2 = (vns 2 + vs 2 )/2 ρ 2 [MeV]

Numerical results: anomaly at finite T

Numerical results: symmetry restoration at finite µ B V eff µ B =915 MeV µ B =922.7 MeV µ B =930 MeV 55 60 65 70 75 80 85 90 95 v ns [MeV]

Numerical results: symmetry restoration at finite µ B 95 90 v s condensate [MeV] 85 80 75 70 v ns 65 910 915 920 925 930 935 940 µ B [MeV] no phase transition beyond µ B,crit 922.7 MeV

Numerical results: anomaly at finite µ B 600 A (µ B ;T=0) [MeV] 500 400 300 200 100 0 T = 0 910 915 920 925 930 935 940 µ B [MeV] vacuum value A k=0 4.7 GeV increases more than 10%

Numerical results: mass spectrum at finite µ B 1100 1000 masses [MeV] 900 800 700 600 500 400 f 0κ a 0 η N ησ 910 915 920 925 930 935 940 µ B [MeV] η mass drop is rather small η N bound state in doubt

Summary Mesonic and nucleon fluctuations effects on chiral symmetry, axial anomaly and mesonic spectrum in nuclear medium Method: Functional Renormalization Group (FRG) approach Findings: mesonic fluctuations make the anomaly coefficient aaaacondensate dependent Nucl. liquid gas transition: 20% drop in (n.s.) chiral cond. (partial) restoration of chiral symmetry seem to aaaaincrease the anomaly ( A 10% relative difference) η mass drop is small η N bound state?