Pion condensate versus chiral density wave at zero temperature - PowerPoint PPT Presentation

Pion condensate versus chiral density wave at zero temperature Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) June 25, 2018 Strong and ElectroWeak Matter, Barcelona 2018 J. O. Andersen, P. Kneschke : “Chiral density wave versus pion condensation at finite density and zero temperature”, Phys. Rev. D 97, 076005

QCD at finite temperature and baryon chemical potential • hadronic matter at low energies ( T , µ B ) • quark-gluon plasma at high energies • exotic phases at large chemical potential − → inhomogeneous phases predicted around µ B = 1 GeV possible Fukushima et al. Rept.Prog.Phys. 74 (2011) 014001 The task is to find phase boundaries and critical points! Perturbative QCD breaks down at low energies Lattice QCD can be used to find the phase transition at µ = 0 , but problematic at non zero chemical potential ⇒ sign problem (complex fermion determinant) Effective theories of QCD, that model chiral symmetry and/or confinement are used to map out the entire T − µ B phase diagram (Nambu-Jona-Lasinio model, quark meson model, chiral perturbation theory) Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 2

QCD at finite temperature and isospin chemical potential The isospin chemical potential µ I introduces an imbalance between up- and down quarks. µ u = µ + µ I , µ d = µ − µ I At large enough densities bose condensation of charged pions is possible. 200 150 T � MeV � • only chiral condensate for small µ I 100 T Χ I = m π • pion condensate above µ c 2 T Ρ 50 T BCS • BEC-BCS crossover at large µ I 0 0 50 100 150 Μ I � MeV � No sign problem for isospin chemical potential ⇒ Lattice QCD works well and can be used for comparison with model calculations Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 3

Pion condensation and inhomogeneous phase Inhomogeneous chiral condensate: q ⇒ complex order parameter ∆ e i � q · � x Cooper pair of quarks with non-zero net momentum � For simplicity we consider a one-dimensional modulation ∆ e iqz At finite temperature the order is destroyed by thermal fluctuations of phonons ⇒ map out the µ − µ I phase diagram at zero temperature previous study in 1+1 dimensional NJL model showed pion and inhomogeneous condensate don’t coexist → We extend this to a 3+1 dimensional QM model P. Adhikari, J.O. Andersen, Phys.Rev. D95 (2017), 054020 Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 4

Quark Meson Model � π ± = � 1 Lagrangian of the QM model in Minkowski space is: 2 ( π 1 ± i π 2 ) √ L = 1 ( ∂ µ σ ) 2 + ( ∂ µ π 3 ) 2 � ∂ µ − 2 i µ I δ µ, 0 � + ( ∂ µ + 2 i µ I δ µ, 0 ) π − � π + � 2 + 1 3 + 2 π + π − ) + λ 2 m 2 ( σ 2 + π 2 24 ( σ 2 + π 2 3 + 2 π + π − ) 2 − h σ ∂ + γ 0 ( µ + µ I τ 3 ) − g ( σ + i γ 5 � � i / � + ψ f τ · � π ) ψ f Replace the meson fields with constant background fields in mean field approximation. Chiral symmetry is broken by chiral condensate, here in the form of a chiral density wave (CDW): < σ > = φ 0 cos( qz ) , < π 3 > = φ 0 sin( qz ) Allow for charged pion condensate: < π 1 > = π 0 , < π 2 > = 0 Introducing the new variables ∆ = g φ 0 and ρ = g π 0 we get the meson potential: m 2 − 4 µ 2 2 q 2 ∆ 2 m 2 V 0 = 1 g 2 + 1 g 2 ∆ 2 + 1 ρ 2 + 1 g 4 (∆ 2 + ρ 2 ) 2 − h λ I g ∆ δ q , 0 2 2 g 2 24 Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 5

Quark Meson Model In the large N c limit we consider only quark corrections to the potential: V 1 = − 1 � det β / � β V log D 2 q τ 3 γ 5 γ 3 − i ργ 5 τ 1 ∂ + γ 0 ( µ + µ I τ 3 ) − ∆ + 1 D = i / / Dirac operator: � 1 � + 1 � �� 1 + e − β ( E ± 1 + e − β ( E ± � − µ ) � + µ ) � 2 E ± 2 E ± V 1 = − N c + T log + T log ¯ , f f f ¯ f p f = u , d with the quark energies: E ± E ± E ± E ± u = E ( ± q , − µ I ) , d = E ( ± q , µ I ) , u = E ( ± q , µ I ) , d = E ( ± q , − µ I ) ¯ ¯ 1   � 2 �� 2 � + ∆ 2 + q � 2 �� p 2 p 2 + ρ 2 E ( q , µ I ) = ⊥ + + µ I   2 Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 6

On-shell Renormalization • The divergences of V 1 are isolated using dimensional regularization d = 3 − 2 ǫ : � 1 � Λ 2 � � + 3 � � p 2 − M 2 ∼ M 4 ǫ + log M 2 2 p The divergence is absorbed into the bare model parameters: m 2 → m 2 ren + δ m 2 etc. 0 = m 2 • We fix the model parameters to physical observables m σ , m π , m q , f π : m 2 σ − m 2 � � tree-level: m 2 = − 1 , λ = 3 π f π , g = m q π m 2 σ − 3 m 2 , h = m 2 � � π f 2 2 f π π � δ m 2 σ − δ m 2 � − λ δ f 2 , δλ = 3 one-loop: δ m 2 = − 1 δ m 2 σ − 3 δ m 2 π π � � π 2 f 2 f 2 π π , − δ g 2 g 2 = δ f 2 δ h = δ m π f π + h δ f π π = δ Z π , δ m q = 0 f 2 f π π • OS renormalization constant from OS conditions: p 2 = m 2 − δ m 2 m 2 σ,π, ren = m 2 � � Σ σ,π σ,π = 0 ⇒ σ,π is pole of the propagator σ,π ∂ ˆ Σ σ,π p 2 = m 2 � � + δ Z σ,π = 0 ⇒ residue at the pole is one σ,π ∂ p 2 Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 7

Parameter Fixing one-loop � � MS = m 2 + 4 g 2 N c m 2 log Λ 2 q − 1 σ ) + 3 m 2 − 2 m 2 m 2 σ − 4 m 2 F ( m 2 2 m 2 π F ( m 2 � � π ) , q (4 π ) 2 m 2 2 q � � � λ MS = λ + 12 g 2 N c log Λ 2 m σ − 4 m 2 + F ( m 2 � � σ ) q (4 π ) 2 f 2 m 2 π q � � log Λ 2 + m 2 + F ( m 2 π ) + m 2 π F ′ ( m 2 π ) σ m 2 q � �� 2 log Λ 2 − m 2 + 2 F ( m 2 π ) + m 2 π F ′ ( m 2 π ) , π m 2 q � � �� 1 + 4 g 2 N c log Λ 2 g 2 MS = g 2 + F ( m 2 π ) + m 2 π F ′ ( m 2 π ) , (4 π ) 2 m 2 q � � �� 1 + 2 g 2 N c log Λ 2 + F ( m 2 π ) − m 2 π F ′ ( m 2 h MS = h π ) (4 π ) 2 m 2 q � ρ 2 ∆ 2 + ρ 2 � 2 2 q 2 ∆ 2 ∆ 2 � + λ MS V = 1 + 1 + 1 ∆ � 2 m 2 m 2 MS − 4 µ 2 − h MS δ 0 , q + V fin I g 2 MS g 2 g 2 g 4 2 24 g MS MS MS MS MS Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 8

Homogeneous Phase Diagram We set q = 0 and choose m π = 140 MeV , m σ = 600 MeV , m q = 300 MeV , f π = 93 MeV 200 �� m q 300 Χ SB 2nd order Ρ 150 250 1st order Ρ � , Ρ � MeV � 200 Μ I � MeV � 150 100 100 50 50 0 0 50 100 150 200 Μ I � MeV � 0 0 100 200 300 400 Μ � MeV � 300 300 250 250 200 � � MeV � � , Ρ � MeV � 200 150 150 100 100 50 50 0 0 100 200 300 400 0 0 50 100 150 200 Μ � MeV � Μ I � MeV � Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 9

Onset of Pion Condensation We determine the position of the phase transition with a Ginzburg-Landau Analysis: V = α 0 + α 2 ρ 2 + α 4 ρ 4 ; �� � ρ 2 ρ 4 p 2 + ∆ 2 ± µ I + 1 − 1 � V 1 ≈ − 2 N c p 2 + ∆ 2 ± µ I p 2 + ∆ 2 ± µ I ) 3 2 � 8 � ( p Subtract expansion in µ I to extract the finite part, divergent part already renormalized: � � � p 2 + ∆ 2 µ 2 � 1 V fin , 2 = − 2 N c ρ 2 I p 2 + ∆ 2 − µ I 2 − p 2 + ∆ 2 − 3 � ( p 2 + ∆ 2 ) p 2 � 1 = − 8 N c ρ 2 µ 2 = − 16 N c � �� (4 π ) 2 ρ 2 µ 2 I F (4 µ 2 1 − s arctan I ) , I (4 π ) 2 s   V fin , 4 = 1 � 1 2 4 N c ρ 4 � 3 −   3  ( p 2 + ∆ 2 )  �� p 2 + ∆ 2 ± µ I p 2 � 1 � � � �� 3∆ 2 = 2 N c s 2 + 1 1 (4 π ) 2 ρ 4 − 2 arctan s 3 µ 2 s I Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 10