Pion condensation in the twoflavor chiral quark model at finite - PowerPoint PPT Presentation

Pion condensation in the two–flavor chiral quark model at finite baryochemical potential P´ eter Kov´ acs KFKI Research Institute of Particle and Nuclear Physics of HAS, Theoretical Department 2009.04.03 • Motivation • The constituent quark model and its renormalization • Equations at one–loop level, the OPT • Diagonalized propagators • Parametrization • Results at lowest order in ρ (Phys.Rev.D78:116008,2008.) • Conclusion Collaborator: T. Herpay

1 Where can pion condensation occur in nature? • Quark matter can exist in neutron stars − → at very large bariochemical potential ( µ B ≈ 1 GeV) If the isospin chemical potential is also different from zero − → possibility of pion condensation • In heavy ion collisions µ I is tunable with different isotopes of an element Neutrino emission from pion condensed quark matter → direct Urca processes: d → u + e − + ¯ ν u + e − → d + ν = ⇒ It might will be investigated experimentaly

2 In 2 flavoured NJL model ( L. He et al Phys. Rev. D74 , 036005 (2004) ): • if µ I < 140 MeV( = m π ) → no pion condensation • if 140 MeV < µ I < 230 MeV → BEC phase • if µ I > 230 MeV → BCS phase Interesting feature of pion condensation found in SU (2) PNJL model: At sufficiently high temperature the condesate evaporates above a certain µ I . ( e.g. Z. Zhang, Y. Liu: hep-ph/0610221v3 ) Up to now: • Investigations in SU (2) NJL and PNJL model (BEC, BCS and CFL phases) • Investigation in O (4) model in the large N limit (leading order) (BEC phase)

3 The model and its renormalization Our starting point is the renormalized SU (2) L × SU (2) R symmetric Lagrangian with explicit symmetry breaking term 1 − λ ψγ µ ∂ µ ψ − g F 4 φ 4 + hφ 0 + i ¯ ¯ ∂ µ φ∂ µ φ − m 2 φ 2 � � L = ψT i φ i ψ 2 2 1 − δλ δZ∂ µ φ∂ µ φ − δm 2 φ 2 � 4 φ 4 � + 2 ψ = ( u, d ) T − → doublet quark fileds φ = ( φ 0 , φ 1 , φ 2 , φ 3 ) ≡ ( σ, π 1 , π 2 , π 3 ) − → sigma and pion scalar fields h − → symmetry breaking external field T i = ( τ 0 , iτ i γ 5 ) − → quark–boson coupling matrix The renormalized (finite) parameters of the Lagrangian: m 2 , λ, g F δz, δm 2 , δλ are the usual (infinite) counterterms (Fermions are treated at tree level → no wavefunction renormalization)

The genarating functional: 4 � − i β � � � � D φ D Π D ¯ d 3 x (Π ˙ φ + i ¯ ψγ 0 ˙ Z = ψ D ψ exp i dt ψ − H + µ B Q B + µ I Q I ) , 0 where the Hamiltonian is 1 + λ ψγ i ∂ i ψ + g F Π 2 + ( ∇ φ ) 2 + m 2 φ 2 � 4 φ 4 − hφ 0 + i ¯ ¯ � H = ψT α φ α ψ 2 2 1 2 δZ Π 2 + 1 2 δZ ( ∇ φ ) 2 + δλ 4 φ 4 + 1 2 δm 2 φ 2 , − ( i = 1 , 2 , 3) and the canonical momenta of the scalar fields Π = δ L = (1 + δZ ) ˙ φ. δ ˙ φ Q B , Q I are the conserved barion and isospin charges � d 3 x 1 3( u † u + d † d ) Q B = � �� � π 2 ) + 1 d 3 x u † u − d † d � Q I = (1 + δZ )( π 2 ˙ π 1 − π 1 ˙ . 2

5 Symmetry breaking: At small T when either h � = 0 or h = 0 and m 2 < 0 ⇒ � φ 0 � ≡ � σ � ≡ v � = 0 At large µ I ⇒ � φ 1 � ≡ � π 1 � ≡ ρ � = 0 and � φ i � ≡ � π i � = 0 for i = 2 , 3 Shifting the corresponding fields R − i β R − i β R − i β ψGf − 1 ψ e − i Z » Z – d 3 x ¯ d 3 x ˜ d 3 x (Π ˙ φ − ˜ R R R D φ D ¯ e − dt dt L I D Πe i dt H B ) Z = ψ D ψ . 0 0 0 where the Π dependent part of Π ˙ φ − ˜ H B : 1 � � Π ˙ φ − ˜ 0 − 2Π 0 (1 + δZ ) ˙ Π 2 H B = − 2(1 − δZ ) φ 0 1 � � 3 − 2Π 3 (1 + δZ ) ˙ Π 2 − 2(1 − δZ ) φ 3 1 � � 1 − 2Π 1 (1 + δZ )( ˙ Π 2 − 2(1 − δZ ) φ 1 − µ I φ 2 ) 1 � � 2 − 2Π 2 (1 + δZ )( ˙ Π 2 − 2(1 − δZ ) φ 2 + µ I ( φ 1 + ρ )) .

6 Making whole squares in the above brakets Then performing the Π integration → produce the inverse bosonic propagators and the tree-level EoS (linear terms). Propagator matirices: gF gF „ ( − i ω n + 1 3 µ B + 1 − 1 = « 2 µ I ) γ 0 − γ i p i − 2 v − i 2 γ 5 ρ i G f , gF gF ( − i ω n + 1 3 µ B − 1 ij − i 2 γ 5 ρ 2 µ I ) γ 0 − γ i p i − 2 v − 1 = ( − i ω n ) 2 − E 2 i G b π 3 , 44 √ ( − i ω n − µ I ) 2 − E 2 π 3 − λρ 2 − λρ 2 0 1 − 2 λvρ √ − 1 = ( − i ω n + µ I ) 2 − E 2 i G b − λρ 2 π 3 − λρ 2 − 2 λvρ B C kl √ √ @ A ( − i ω n ) 2 − E 2 π 3 − 2 λv 2 − 2 λvρ − 2 λvρ Tree-level EoS: v ( m 2 + λ ( v 2 + ρ 2 )) − h = 0 , EoS tree = σ ρ ( m 2 + λ ( v 2 + ρ 2 ) − µ 2 EoS tree = I (1 + δZ )) = 0 . π 1

7 Renormalizaiton conditions → finiteness of the perturbative N –point functions (the propagator and the four point boson vertex) Practically it is easier to obtain the counterterms from the one–loop EoS Note: there is some arbitrariness in choosing the finite parts With cut-off regularization: − 6 λ (Λ 2 − m 2 ln Λ 2 ) + g 2 δm 2 4 π 2 N c Λ 2 , F = l 2 b 12 λ 2 ln Λ 2 − g 2 32 π 2 N c ln Λ 2 F δλ = , l 2 e l 2 b f 16 π 2 ln Λ 2 g 2 F δZ = − N c , e 2 l 2 f l b , l f → bosonic and fermionic renormalization scales

8 Equations at one–loop level, the OPT At finite temperature tree level mass squares can become negative → some sort of resummation is needed Using the optimized perturbation theory (OPT): • a temperature dependent mass term introduced in the Lagrangian • the difference is treated as a higher order counterterm • the new mass parameter determined by the FAC criterion ( m 1–loop = m tree ) → can be transformed to an equation for a resummed particle mass • conserves Ward–identities

9 Equation for the resummed π 3 mass: π 3 ( T, µ ) = m 2 + δm 2 + ( λ + δλ )( v 2 + ρ 2 ) + Σ π 3 ( ω = p = 0 , m 2 m 2 π 3 , T, µ ) One–loop level EoS for σ : „ Z m 2 + δm 2 + ( λ + δλ )( v 2 + ρ 2 ) + λ X Tr { H b G b ( ω n , p , µ I ) } v p « Z X Tr { H f G f ( ω n , p , µ I , µ B ) } + g F = h p comparing the two equations → a Ward–identity is recognized vm 2 π 3 = h One–loop level EoS for π 1 : „ Z m 2 + δm 2 + ( λ + δλ )( v 2 + ρ 2 ) − µ 2 Tr { R b G b ( ω n , p , µ I ) } X ρ I (1 + δZ ) + λ p « Z X Tr { R f G f ( ω n , p , µ I , µ B ) } + g F = 0 , p

10 Calculation of the loop integrals requires the diagonalization of the propagators → Straightforward but the eigenvalues are non–rational functions of ω n → diagonalization for small ρ → Landau–type analysis Up to O ( ρ 3 ) π 3 = m 2 + λv 2 + t (0) ( m 2 m 2 π 3 , v, T, µ I , B ) + ( λ + t (2) ( m 2 π 3 , v, T, µ I , B )) ρ 2 Due to the Ward identity the EoS for σ remains vm 2 π 3 = h � I − m 2 − λv 2 − r (0) ( m 2 π 3 , v, T, µ I , B )) ρ 2 + O ( ρ 4 ) � µ 2 π 3 , v, T, µ I , B ) − ( λ + r (2) ( m 2 ρ = 0 ⇒ Pion condensate non–zero only if the roots are real If λ + r (2) > 0 and µ 2 I − m 2 − λv 2 − r (0) > 0 � I − m 2 − λv 2 − r (0) ( m 2 µ 2 π 3 , v, T, µ I , B ) ρ = λ + r (2) ( m 2 π 3 , v, T, µ I , B ) ⇒ The transition is of second order

11 Diagonalized propagators √ 1 − | a | 2 ρ 2   b (1 − 2 av ) ρ 2 − 2 aρ √ 1 − | a | 2 ρ 2  + O ( ρ 3 ) , b ∗ (1 − 2 a ∗ v ) ρ 2 2 a ∗ ρ O B = −   √ √  1 − 2 | a | 2 ρ 2 2 a ∗ ρ 2 aρ a = a ( ω n , µ ) = λv/ ( µ 2 + 2 λv 2 + 2i ω n µ ) and b = b ( ω n , µ ) = i λ/ (4 µω n ) I + 2 λv 2 − 4i µ I ω n ) λ (2 µ 2 1 i ˜ − ρ 2 I + 2 λv 2 − 2i µ I ω n ) + O ( ρ 4 ) , G π + = ( ω n + i µ I ) 2 + E 2 (( ω n + i µ I ) 2 + E 2 π ) 2 ( µ 2 π I + 2 λv 2 + 4i µ I ω n ) λ (2 µ 2 1 i ˜ − ρ 2 I + 2 λv 2 + 2i µ I ω n ) + O ( ρ 4 ) , G π − = ( ω n − i µ I ) 2 + E 2 (( ω n − i µ I ) 2 + E 2 π ) 2 ( µ 2 π I + 6 λv 2 + 4 µ 2 − ρ 2 λ ( µ 2 I + 2 λv 2 )( µ 2 I ω 2 1 n ) i ˜ n ) + O ( ρ 4 ) , G σ = I + 2 λv 2 ) 2 + 4 µ 2 σ ) 2 (( µ 2 ω 2 n + E 2 ( ω 2 n + E 2 I ω 2 σ while the π 3 propagator is 1 λ − ρ 2 π ) 2 + O ( ρ 4 ) . i G π 3 = ω 2 n + E 2 ( ω 2 n + E 2 π Important to note − → the transformation matrix depends on ρ, ω n , µ

12 g 2   − i g F 0 ρ 2  1 + F 4 k 0 γ 0 γ 5 ρ 32 k 2  , O F = g 2 − i g F 0 ρ 2 4 k 0 γ 0 γ 5 ρ 1 + F 32 k 2 where k 0 = ( − i ω n + 1 3 µ B ) γ 0 and the matrix is hermitian − ρ 2 g 2 1 1 1 i ˜ F G u/d = − γ 0 p u/d − m f / 8 k 0 p u/d − m f / p u/d − m f / where / p u/d = ( − i ω n + µ u/d ) γ 0 − γ i p i and µ u/d = µ B / 3 ± µ I / 2 . Calculation of one–loop contributions (bosonic case): B b ˜ B ˜ G b } = Tr { ˜ Tr { B b G b } = Tr { B b O − 1 B O B G b O − 1 B O B } = Tr { O B B b O − 1 G b } G b = diag( ˜ where ˜ π + , ˜ π − , ˜ G − 1 G − 1 G − 1 σ )