Dualities in dense baryonic (quark) matter with chiral and isospin - PowerPoint PPT Presentation

Dualities in dense baryonic (quark) matter with chiral and isospin imbalance Konstantin G. Klimenko, Tamaz G. Khunjua, Roman N. Zhokhov IHEP, MSU, IZMIRAN arXiv:1808.05162 Int. J. Mod. Phys. Conf. Ser. 47 (2018) arXiv:1804.01014 Phys. Rev. D 98, 054030 (2018) arXiv:1710.09706 Phys. Rev. D 97, 054036 (2018) arXiv:1704.01477 Phys. Rev. D 95, 105010 (2017) Seminar of sector of Hadron Matter Physics BLTP JINR October 31, 2018

Broad Group broad group V. Ch. Zhukovsky, N. V. Gubina Moscow state University and D. Ebert, Humboldt University of Berlin

QCD at finite temperature and nonzero chemical potential QCD at nonzero temperature and baryon chemical potential plays a fundamental role in many different physical systems. (QCD at extreme conditions) neutron stars heavy ion collision experiments Early Universe

Methods of dealing with QCD Methods of dealing with QCD First principle calculation – lattice Monte Carlo simulations, LQCD Effective models Nambu–Jona-Lasinio model NJL

lattice QCD at non-zero baryon chemical potential µ B Lattice QCD non-zero baryon chemical potential µ B sign problem — complex determinant ( Det ( D ( µ ))) † = Det ( D ( − µ † ))

Methods of dealing with QCD Methods of dealing with QCD First principle calcultion – lattice Monte Carlo simulations, LQCD Effective models Nambu–Jona-Lasinio model NJL

NJL model NJL model can be considered as effective field theory for QCD . the model is nonrenormalizable Valid up to E < Λ ≈ 1 GeV Parameters G , Λ , m 0

NJL model NJL model can be considered as effective field theory for QCD . the model is nonrenormalizable Valid up to E < Λ ≈ 1 GeV Parameters G , Λ , m 0 dof– quarks no gluons only four-fermion interaction attractive feature — dynamical CSB the main drawback – lack of confinement (PNJL) Relative simplicity allow to consider hot and dense QCD in the framework of NJL model and explore the QCD phase structure (diagram).

chiral symmetry breaking the QCD vacuum has non-trivial structure due to non-perturbative interactions among quarks and gluons lattice simulations ⇒ condensation of quark and anti-quark pairs uu � = � ¯ dd � ≈ ( − 250 MeV ) 3 � ¯ qq � � = 0 , � ¯

Nambu–Jona-Lasinio model Nambu–Jona-Lasinio model � q i γ 5 q ) 2 � q γ ν i ∂ ν q + G qq ) 2 + (¯ L = ¯ (¯ N c q → e i γ 5 α q continuous symmetry � � � σ 2 + π 2 � q − N c � γ ρ i ∂ ρ − σ − i γ 5 π L = ¯ q . 4 G Chiral symmetry breaking 1 / N c expansion, leading order � ¯ qq � � = 0 � � � γ ρ i ∂ ρ − � σ � � σ � � = 0 − → L = ¯ q q

Different types of chemical potentials: dense matter with isotopic imbalance Baryon chemical potential µ B Allow to consider systems with non-zero baryon densities. µ B q γ 0 q = µ ¯ q γ 0 q , 3 ¯

Different types of chemical potentials: dense matter with isotopic imbalance Baryon chemical potential µ B Allow to consider systems with non-zero baryon densities. µ B q γ 0 q = µ ¯ q γ 0 q , 3 ¯ Isotopic chemical potential µ I Allow to consider systems with isotopic imbalance. n I = n u − n d ← → µ I = µ u − µ d The corresponding term in the Lagrangian is µ I q γ 0 τ 3 q 2 ¯

QCD phase diagram with isotopic imbalance neutron stars, heavy ion collisions have isotopic imbalance

Different types of chemical potentials: chiral imbalance chiral (axial) chemical potential Allow to consider systems with chiral imbalance (difference between between densities of left-handed and right-handed quarks). n 5 = n R − n L ← → µ 5 = µ R − µ L The corresponding term in the Lagrangian is q γ 0 γ 5 q µ 5 ¯

Different types of chemical potentials: chiral imbalance chiral (axial) isotopic chemical potential Allow to consider systems with chiral isospin imbalance µ I 5 = µ u 5 − µ d 5 so the corresponding density is n I 5 = n u 5 − n d 5 ← → µ I 5 n I 5 µ I 5 q τ 3 γ 0 γ 5 q Term in the Lagrangian — 2 ¯ If one has all four chemical potential, one can consider different densities n uL , n dL , n uR and n dR

Chiral magnetic effect c = e 2 J = c µ 5 � � B , 2 π 2 A. Vilenkin, PhysRevD.22.3080, K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 78 (2008) 074033 [arXiv:0808.3382 [hep-ph]].

Generation of chiral imbalance in compact stars Due to high baryon densities, magnetic fields and vorticity - Chiral separation effect CSE - Chiral Vortical effect CVE

Chiral separation effect Chiral magnetic (CME) effect has the form J = c µ 5 � � H There is a dual effect so-called chiral sepration effect (CSE) (Son and Zhitnitsky 2004, Metlitski and Zhitnitsky 2005) J 5 = c µ � � J µ 5 = � ¯ ψγ µ γ 5 ψ � H , Then the phenomena looks very similar and dual. J V = c µ A � � J A = c µ V � � H , H

Chiral separation effect in a two flavoured system Let us consider the system with u and d quark flavours J 5 u = N c q u � 2 π 2 µ u � H and for d quark sector the axial current is J 5 d = N c q d � 2 π 2 µ d � H Now let us calculate the chiral current d = N c J 5 = � � u + � 2 π 2 ( q u µ u + q d µ d ) � J 5 J 5 H Now let us express it in terms of µ and ν , taking into account that µ u = µ + ν and µ d = µ − ν one has J 5 = N c � 2 π 2 [( q u + q d ) µ + ( q u − q d ) ν )] � H

Chiral separation effect in a two flavoured system Chiral isospin current and charge J 5 d = N c J I 5 = � � J 5 u − � 2 π 2 ( q u µ u − q d µ d ) � H Expressing it in terms of µ and ν J I 5 = N c � 2 π 2 [( q u − q d ) µ + ( q u + q d ) ν )] � H

Chiral separation effect in a two flavoured system The chiral charge: � d 3 x � ¯ ψγ 0 γ 5 ψ � ⇐ Q 5 = ⇒ µ 5 The chiral isospin charge � d 3 x � ¯ ψγ 0 γ 5 τ 3 ψ � ⇐ Q I 5 = ⇒ µ I 5

Chiral separation effect in a two flavoured system It is quite obvious that the ratio of charges is equal to the ratio of the currents n I 5 n 5 = Q I 5 Q 5 = J I 5 z J 5 z Q I 5 Q 5 = 3 + δ 1 + 3 δ where δ = ν µ For example if ν = 0 then Q I 5 Q 5 = 3

Chiral separation effect: real case The full formula for CSE in the case of finite temperature and non-zero quark mass can be found by Zhitnitsky Metlitski V = e J 5 2 π n m ( T , µ )Φ � � where J 5 d 2 xJ 5 d 2 xB V = 3 and Φ = And the coefficient in front of the magnetic flux is � � � dp 3 1 1 e β ( √ e β ( √ n m ( T , µ ) = − 2 π 3 + m 2 − µ ) + 1 3 + m 2 + µ ) + 1 p 2 p 2 it is a baryon number density of one-dimensional fermions.

Chiral Vortical Effect (CVE) Vorticity ω = 1 � � ∇ × � v 2 Chiral Vortical Effect (CVE) quantifies the generation of a vector current J along the vorticity direction: J = 1 � π 2 µµ 5 � ω Axial current can be generated by the rotation as well � 1 � 1 6 T 2 + 2 π 2 ( µ 2 + µ 2 � J 5 = 5 ) � ω

Chiral imbalance generation due to CVE � 1 � 1 3 T 2 + 2 π 2 ( µ 2 + ν 2 ) J 5 = � � 5 + � J u J d 5 = � ω � 2 � J I 5 = � � 5 − � J u J d 5 = π 2 µν ω �

Model and its Lagrangian We consider a NJL model, which describes dense quark matter with two massless quark flavors ( u and d quarks). � 2 τ 3 γ 0 γ 5 + µ 5 γ 0 γ 5 � γ ν i ∂ ν + µ B 3 γ 0 + µ I 2 τ 3 γ 0 + µ I 5 L = ¯ q + q � τ q ) 2 � G qq ) 2 + (¯ q i γ 5 � (¯ N c q is the flavor doublet, q = ( q u , q d ) T , where q u and q d are four-component Dirac spinors as well as color N c -plets; τ k ( k = 1 , 2 , 3) are Pauli matrices.

Equivalent Lagrangian To find the thermodynamic potential we use a semi-bosonized version of the Lagrangian � � � � q − N c � γ ρ i ∂ ρ + µγ 0 + ντ 3 γ 0 + ν 5 τ 3 γ 1 − σ − i γ 5 π a τ a L = ¯ σσ + π a π a . q 4 G σ ( x ) = − 2 G π a ( x ) = − 2 G q i γ 5 τ a q ) . (¯ qq ); (¯ N c N c Condansates ansatz � σ ( x ) � and � π a ( x ) � do not depend on spacetime coordinates x , � σ ( x ) � = M , � π 1 ( x ) � = ∆ , � π 2 ( x ) � = 0 , � π 3 ( x ) � = 0 . (1) where M and ∆ are already constant quantities.

thermodynamic potential the thermodynamic potential can be obtained in the large N c limit Ω( M , ∆) Projections of the TDP on the M and ∆ axes No mixed phase ( M � = 0 , ∆ � = 0 ) it is enough to study the projections of the TDP on the M and ∆ projection of the TDP on the M axis F 1 ( M ) ≡ Ω( M , ∆ = 0 ) projection of the TDP on the ∆ axis F 2 (∆) ≡ Ω( M = 0 , ∆)

Dualities The TDP (phase daigram) is invariant Interchange of condensates matter content Ω( C 1 , C 2 , µ 1 , µ 2 ) Ω( C 1 , C 2 , µ 1 , µ 2 ) = Ω( C 2 , C 1 , µ 2 , µ 1 )

Dualities Figure: Dualities

Dualities in different approaches Large N c orbifold equivalences connect gauge theories with different gauge groups and matter content in the large N c limit. M. Hanada and N. Yamamoto, JHEP 1202 (2012) 138, arXiv:1103.5480 [hep-ph], PoS LATTICE 2011 (2011), arXiv:1111.3391 [hep-lat]

Dualities in large N c orbifold equivalences two gauge theories with gauge groups G 1 and G 2 with µ 1 and µ 2 Duality G 1 ← → G 2 , µ 1 ← → µ 2 G 2 is sign problem free G 1 has sign problem, can not be studied on lattice