QCD phase diagram for nonzero isospin-asymmetry SEWM, Barcelona, June 25 th 2018 Sebastian Schmalzbauer with Bastian Brandt & Gergely Endr˝ odi
Outline • QCD with isospin-asymmetry • introduction • motivation • forecast • pion condensation on the lattice • symmetry breaking • simulation details • extrapolating physical results • results for the isospin phase diagram • chiral crossover • pion condensation phase boundary • deconfinement transition • summary 1 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
QCD phase diagram (taken from NICA) 2 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Motivation • two quark flavors u , d 3 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Motivation • two quark flavors u , d • isospin density n I = n u − n d � = 0 : (taken from thescienceexplorer) • systems with charged pions • neutron stars ( udd ) stable 10 14 yr • heavy-ion collisions (N > Z) 160 10 12 yr 10 10 yr 140 10 8 yr 10 6 yr 120 10 4 yr 100 yr 100 1 yr 10 6 s Z = N 80 10 4 s 100 s 60 1 s 40 10 −2 s 10 −4 s 20 10 −6 s 10 −8 s N no data Z 20 40 60 80 100 (taken from wikipedia) 3 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Motivation • two quark flavors u , d • isospin density n I = n u − n d � = 0 : (taken from thescienceexplorer) • systems with charged pions • neutron stars ( udd ) stable 10 14 yr • heavy-ion collisions (N > Z) 160 10 12 yr 10 10 yr 140 10 8 yr • no sign problem ⇒ lattice simulations 10 6 yr 120 10 4 yr 100 yr 100 1 yr 10 6 s Z = N 80 10 4 s 100 s 60 1 s 40 10 −2 s 10 −4 s 20 10 −6 s 10 −8 s N no data Z 20 40 60 80 100 (taken from wikipedia) 3 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Motivation • two quark flavors u , d • isospin density n I = n u − n d � = 0 : (taken from thescienceexplorer) • systems with charged pions • neutron stars ( udd ) stable 10 14 yr • heavy-ion collisions (N > Z) 160 10 12 yr 10 10 yr 140 10 8 yr • no sign problem ⇒ lattice simulations 10 6 yr 120 10 4 yr • analogies to baryon density 100 yr 100 1 yr • Silver Blaze 10 6 s Z = N 80 10 4 s • color neutral compostite particles 100 s 60 • saturation 1 s 40 10 −2 s • technical similarities (small eigenvalues) 10 −4 s 20 10 −6 s 10 −8 s N no data Z 20 40 60 80 100 (taken from wikipedia) 3 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Conjectured QCD isospin phase diagram • baryon chemical potential µ B = 0 • isospin chemical potential µ I = ( µ u − µ d ) / 2 • rich phase structure: • vacuum (white) • quark-gluon plasma • pion condensate • color superconductor 4 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Setup
Pion condensation: symmetry breaking • QCD with light quarks M = / D + m ud 1 • chiral symmetry breaking pattern SU (2) V 5 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Pion condensation: symmetry breaking • QCD with light quarks M = / D + m ud 1 + µ I γ 0 τ 3 • chiral symmetry breaking pattern SU (2) V → U (1) τ 3 • problem: cannot directly observe the spontaneous symmetry breaking • pion condensate � ¯ ψγ 5 τ 1 , 2 ψ � = 0 (finite volume) • accumulation of zero eigenvalues (Goldstone mode) 5 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Pion condensation: symmetry breaking • QCD with light quarks M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 • chiral symmetry breaking pattern SU (2) V → U (1) τ 3 → ∅ • problem: cannot directly observe the spontaneous symmetry breaking • pion condensate � ¯ ψγ 5 τ 1 , 2 ψ � = 0 (finite volume) • accumulation of zero eigenvalues (Goldstone mode) • solution: add explicit unphysical breaking (pionic source) • can observe spontaneous symmetry breaking • no zero eigenvalues 5 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Pion condensation: symmetry breaking • QCD with light quarks M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 • chiral symmetry breaking pattern SU (2) V → U (1) τ 3 → ∅ • problem: cannot directly observe the spontaneous symmetry breaking • pion condensate � ¯ ψγ 5 τ 1 , 2 ψ � = 0 (finite volume) • accumulation of zero eigenvalues (Goldstone mode) • solution: add explicit unphysical breaking (pionic source) • can observe spontaneous symmetry breaking • no zero eigenvalues • need to extrapolate λ → 0 for physical results 5 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
❘ Simulation Details • QCD partition function for N f = 2 + 1 rooted staggered quarks � D [ U ] (det M ud det M s ) 1 / 4 e − S G Z = 6 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
❘ Simulation Details • QCD partition function for N f = 2 + 1 rooted staggered quarks � D [ U ] (det M ud det M s ) 1 / 4 e − S G Z = • quark matrices with η 5 = ( − 1) n t + n x + n y + n z � � / D µ I + m ud λη 5 M s = / M ud = , D 0 + m s / − λη 5 D − µ I + m ud 6 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Simulation Details • QCD partition function for N f = 2 + 1 rooted staggered quarks � D [ U ] (det M ud det M s ) 1 / 4 e − S G Z = • quark matrices with η 5 = ( − 1) n t + n x + n y + n z � � / D µ I + m ud λη 5 M s = / M ud = , D 0 + m s / − λη 5 D − µ I + m ud • no sign problem due to η 5 τ 1 M τ 1 η 5 = M † : � M † M + λ 2 � M = / det M ud = det ∈ ❘ > 0 D µ I + m ud 6 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Simulation Details • QCD partition function for N f = 2 + 1 rooted staggered quarks � D [ U ] (det M ud det M s ) 1 / 4 e − S G Z = • quark matrices with η 5 = ( − 1) n t + n x + n y + n z � � / D µ I + m ud λη 5 M s = / M ud = , D 0 + m s / − λη 5 D − µ I + m ud • no sign problem due to η 5 τ 1 M τ 1 η 5 = M † : � M † M + λ 2 � M = / det M ud = det ∈ ❘ > 0 D µ I + m ud • first studies [Kogut, Sinclair ’02] [de Forcrand, Stephanov, Wenger ’07] • in this work: stout-smeared quarks, physical pion masses, tree-level Symanzik improved gluons [Brandt, Endr˝ odi, Schmalzbauer ’18] 6 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Observables • chiral and pion condensate � π ± � = T = T ∂ ln Z ∂ ln Z � ¯ � ψψ , V ∂ m ud V ∂λ • Polyakov loop N t − 1 � � 1 � � P = tr U t ( n ) V n x , n y , n z n t =0 • need to be renormalized Σ ¯ ψψ , Σ π , P r 7 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
λ -extrapolation: concepts • naive limit � D [ U ] e − S ( λ ) O ( λ ) � O � = lim λ → 0 8 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
λ -extrapolation: concepts • naive limit � D [ U ] e − S ( λ ) O ( λ ) � O � = lim λ → 0 • operator improvement � D [ U ] e − S ( λ ) O (0) � O � = lim λ → 0 8 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
λ -extrapolation: concepts • naive limit � D [ U ] e − S ( λ ) O ( λ ) � O � = lim λ → 0 • operator improvement � D [ U ] e − S ( λ ) O (0) � O � = lim λ → 0 • reweighting of configurations � O � = � OW λ � λ � W λ � λ with full (expensive) or leading order (cheap) reweighting factor ln W LO = − λ V ln W λ = S ( λ ) − S (0) = ln W LO + O ( λ 4 ) , 2 T π ± 8 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Operator improvement: pion condensate • cannot measure pion condensation without pionic source π ± = T λ M † M + λ 2 � − 1 � 2 V tr 9 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
Operator improvement: pion condensate • cannot measure pion condensation without pionic source π ± = T λ M † M + λ 2 � − 1 � 2 V tr • Banks-Casher type relation [Kanazawa, Wettig, Yamamoto ’11] → λ → π �� � d ξ ρ ( ξ )( ξ 2 + λ 2 ) − 1 V →∞ λ → 0 � π ± � − − − − − − − 4 � ρ (0) � 2 with density of singular values ρ ( ξ ) 9 / 16 S. Schmalzbauer (Goethe Universit¨ at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry
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