Inhomogeneous condensation in effective models for QCD using the - PowerPoint PPT Presentation

Inhomogeneous condensation in effective models for QCD using the finite mode approach Achim Heinz Chiral group, April 27 th 2015

The phase diagram of Quantum Chromodynamics q − 1 4 G a µν G a µν � ı / � L QCD = ¯ D − m q D = γ µ � ∂ µ − ı g QCD A a µ t a � G a µν = ∂ µ A a ν − ∂ ν A a µ − g QCD f abc A b µ A c / , ν

Parity doublet model Chiral limit m q → 0 (bare masses) QCD Lagrangian is then invariant under global chiral U ( N f ) L × U ( N f ) R transformations → chiral condensate φ order parameter of QCD. Parity doublet model for nucleons Besides the nucleon also a chiral partner of the nucleon is introduced. Chiral symmetry is preserved at all densities, this is in contrast to other models e.g. Walecka model.

The chiral density wave Inhomogeneous condensation: < σ > = φ ( x ) Recent studies of quark models show the relevance of inhomogeneous condensation at moderate densities. Are spatial modulations relevant for a state-of-the-art hadron model? Ansatz for chiral density wave: � σ � ∼ φ cos(2 f x ) , � π � ∼ φ sin(2 f x )

Dynamical quantities as a function of µ Φ , Χ � MeV � Φ , Χ � MeV � Φ , Χ � MeV � Φ , Χ � MeV � Φ , Χ � MeV � Φ , Χ � MeV � Φ , Χ � MeV � f � MeV � 150 150 150 150 150 150 150 400 300 100 100 100 100 100 100 100 homogeneous homogeneous homogeneous homogeneous 200 phase phase phase phase CDW CDW CDW 50 50 50 50 50 50 50 100 1100 Μ � MeV � 1100 Μ � MeV � 1100 Μ � MeV � 1100 Μ � MeV � 1100 Μ � MeV � 1100 Μ � MeV � 1100 Μ � MeV � 1100 Μ � MeV � 0 0 0 0 0 0 0 0 900 950 1000 1050 900 900 900 900 900 900 900 950 950 950 950 950 950 950 1000 1000 1000 1000 1000 1000 1000 1050 1050 1050 1050 1050 1050 1050 f is of the size of 1 . 5 fm solid line: φ , dashed line: ¯ χ µ = 923 MeV transition to nuclear matter, µ = 975 MeV transition to the CDW. Homogeneous nuclear matter at ρ 0 , mixed phase between 2 . 49 ρ 0 to 10 . 75 ρ 0 , purely inhomogeneous phase above 10 . 75 ρ 0 .

Finite-mode approach “Wish list” for the calculation of arbitrary modulations arbitrary modulations: σ ( x ) is not an input several inhomogeneous fields: σ ( x ), π ( x ), ω ( x ), ... no limitations on the dimension of the model: 1 + 1, 1 + 2, ... higher dimensional modulations: σ ( x , y ), σ ( x , y , z ), ... New numerical tool: so-called “finite-mode approach” Lattice inspired method, which allows to access all points of the “wish list”. Not so fast, there are a few limitations: numerically complex finite size corrections optimization and tests

Gross-Neveu model in the finite-mode approach � N c � 2 N c ψ i ( γ µ ∂ µ + γ 0 µ ) ψ i + 1 � ¯ � ¯ 2 g 2 L GN = ψ i ψ i i =1 i =1 in 1 + 1 dimensions renormalizable asymptotic freedom discrete chiral symmetry in the large N c limit chiral symmetry is broken

GN model in the finite-mode approach Effective action of the GN model: � 1 � � d 2 x σ ( x ) 2 − ln (det Q ) S = N c , Q = γ µ ∂ µ + γ 0 µ + σ ( x ) 2 λ finite spacetime volume ˆ L 0 ˆ L 1 fermion fields as superposition of plane waves: e − ı (ˆ x 0 +ˆ k 0 ˆ k 1 ˆ x 1 ) ˆ � ψ j (ˆ x 0 , ˆ x 1 ) = η j , n 0 , n 1 � ˆ L 0 ˆ L 1 n 0 , n 1 with the momenta: k 0 = 2 π k 1 = 2 π ˆ ˆ ( n 0 − 1 / 2) , n 1 , n 0 , n 1 ∈ N ˆ ˆ L 0 L 1 finite number temporal modes N 0 and space modes N 1

GN model in the finite-mode approach ˆ Discretized effective action for a homogeneous condensate σ ( x ) ≡ ˆ σ : N 0 N 1 ��� 2 π � 1 � 2 π � 1 σ 2 � 2 �� 2 � 2 �� 2 � S ˆ 2 π � = ˆ L 1 ˆ � � σ 2 − ˆ µ 2 L 0 ln − n 0 + n 1 + ˆ + 2ˆ µ + n 0 − ˆ ˆ ˆ N c 2 λ L 0 2 L 1 L 0 2 n 0=1 n 1= − N 1 finite number of modes N 0 and N 1 introduces momentum cutoffs: = 2 π = 2 π ˆ ˆ k cut k cut , ( N 1 + 1 / 2) N 0 0 1 ˆ ˆ L 0 L 1 Effective action as a function of: λ , N 0 , N 1 , ˆ and ˆ k cut k cut 0 1 2 π 2 N 0 ( N 1 + 1 / 2)ˆ σ 2 S = λ ˆ k cut ˆ k cut N c 0 1 + N 0 N 1 � 2 ��� � 2 � n 0 − 1 / 2 � 2 � 2 n 0 − 1 / 2 � n 1 � k cut k cut σ 2 − ˆ µ 2 k cut � � ˆ ˆ µ ˆ ln + + ˆ + 2ˆ − 0 1 0 N 1 + 1 / 2 N 0 N 0 n 0=1 n 1= − N 1

Choosing and determining suitable parameters The temperature ˆ T is proportional to 1 / N 0 : low temperature ˆ σ ( ˆ T ≈ 0, where ˆ T ) ≈ ˆ σ 0 : N 0 → N 00 critical temperature ˆ T = ˆ σ ( ˆ T c , where ˆ T ) = 0: N 0 → N 0 c ≪ N 00 T c = ˆ ˆ T c ( λ, N 00 , N 0 c , N 1 , ˆ k cut , ˆ k cut ) 0 1 N 00 and N 1 are limited by computer resources λ and ˆ k cut d N for ˆ S T ≈ 0 and ˆ T = ˆ follow from the gap equation 0 = T c 0 d ˆ σ N 0 c and ˆ ∂ N 0 c ˆ ∂ ∂ ˆ k cut are fixed due to T c = 0 and T c = 0 1 ∂ ˆ k cut 1 Only N 00 and N 1 are free parameters

T c as a function of of ˆ k cut and N 1 1 � � T T c c 0.570 0.570 0.568 0.568 0.566 0.566 0.564 0.564 0.562 0.562 � cut k 1 cut � 0.560 0.560 300 k 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 N 1 � 0.5 0 50 100 150 200 250 for small ˆ for large ˆ k cut k cut no improvement with 1 / ( N 1 + 0 . 5) on 1 increasing N 1 improvement with increasing N 1 N 1 = 64 (green curve) N 1 = 128 (red curve) N 1 = 256 (blue curve) optimal choice for ˆ k cut ∂ ˆ at T c = 0 1 ∂ ˆ k cut 1 N 00 = 256 and N 0 c = 36, λ and ˆ k cut are fixed, black lines indicate the infinite volume 0 T c = e C /π continuum result ˆ

Performance of the finite mode approach � T c 0.56710 N 00 � 48 0.56705 N 00 � 72 0.56700 N 00 � 96 N 00 � 120 0.56695 � � � � � � � � � � � � � � � � � � � N 00 � 480 0.56690 0.56685 0.56680 N 1 100 200 300 400 ˆ T c as a function of N 1 and N 00 , the black line the infinite volume continuum result T c = e C /π . Fast convergence for symmetric choices N 00 = N 1 (filled black circles) ˆ

GN model phase diagram � T 0.6 � � � � � 0.5 Σ � 1.0 � � � 0.5 �� � � 0.4 � x � 0.2 0.4 0.6 0.8 1.0 � � � 0.5 � � � � � II � 1.0 � � � � 0.3 � � � � � The shape of the condensate � � � � � I � � � � � � � � � � � � � σ ( x ) close to I / III phase ˆ 0.2 � � � � � � � � � � � � � � ��������� �� � � � � � � � ��������������������� � � � � � � � � � � � � boundary. � � � � � � � III 0.1 � � � � � � � � � � � � � 0.0 Μ 0.0 0.5 1.0 1.5 + M � c m e − i ˆ p ˆ x 1 I: homogeneous broken phase ˆ σ (ˆ x 1 ) = m = − M II: restored phase p = 2 π ˆ m , c + m = ( c − m ) ∗ III: inhomogeneous phase ˆ L 1 ˆ ˆ σ := σ/σ 0 , µ := µ/σ 0 , ˆ t := T /σ 0

χ Gross-Neveu model in the finite-mode approach � N � N � 2 � 2   N ψ i ( γ µ ∂ µ ) ψ i + g 2 � ¯ � ¯ � ¯ L χ GN = ψ i ψ i + ψ i ıγ 5 ψ i   2 i =1 i =1 i =1 discrete chiral symmetry → continuous chiral symmetry N 10 � ¯ � � c n e ı nx � σ ( x ) = ˆ ψ i ψ i = � t n = − 10 i =1 0.7 restored phase 10 N � ¯ 0.6 � � d n e ı nx � η ( x ) = ˆ ψ i ıγ 5 ψ i = 0.5 i =1 n = − 10 0.4 CDW only two phases are present: 0.3 restored phase with ˆ σ ( x ) = ˆ η ( x ) = 0 0.2 � 0.1 chiral density wave (CDW) with Μ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 σ ( x ) = ˆ ˆ σ cos(2 f x ) , and η ( x ) = ˆ ˆ σ sin(2 f x )

χ Gross-Neveu model in the finite-mode approach � , Η � Σ 1.0 CDW is not an input! It follows 0.5 � from a dynamical minimization of x 0.2 0.4 0.6 0.8 1.0 � 0.5 the σ - and the η -condensate. � 1.0 µ = 0 . 295 at ˆ ˆ t = 0 . 0945, first coefficient � S eff � , Η � Σ � Μ 1.0 0.2 0.4 0.6 0.8 1.0 0.5 � 1 � x 0.2 0.4 0.6 0.8 1.0 � 0.5 � 2 � 1.0 µ = 0 . 585 at ˆ ˆ t = 0 . 0945, second coefficient � 3 � , Η � Σ 1.0 � 4 0.5 effective action at ˆ t = 0 . 189. The different � x 0.2 0.4 0.6 0.8 1.0 � 0.5 colours correspond to different local � 1.0 minima µ = 0 . 875 at ˆ ˆ t = 0 . 0945, third coefficient

The NJL model in the finite-mode approach Lagrangian of the NJL model � N c � N c  � 2 � 2  N c ∂ψ i − G � ¯ � ¯ � ¯ ψ i / L NJL = ψ i ψ i + ψ i ıγ 5 ψ i   N c i =0 i =0 i =0 � λ � S � d 4 x m ∗ 2 ( x ) − ln (det Q ) Q = γ µ ∂ µ + γ 0 µ + m ∗ 2 ( x ) = , N c 2 Looks similar but: 1 + 3 dimensional model not renormalizable continuous chiral symmetry regularization scheme effects phase diagram computational effort increases strongly