QCD with isospin chemical potential: low densities and Taylor - PowerPoint PPT Presentation

QCD with isospin chemical potential: low densities and Taylor expansion QCD with isospin chemical potential: low densities and Taylor expansion Bastian Brandt and Gergely Endr˝ odi Goethe University Frankfurt 28.07.2016

QCD with isospin chemical potential: low densities and Taylor expansion Contents 1. (short) Introduction 2. Simulation setup and λ extrapolation 3. QCD at small isospin chemical potential 4. Comparison to Taylor expansion around µ I = 0

QCD with isospin chemical potential: low densities and Taylor expansion Introduction 1. Introduction

QCD with isospin chemical potential: low densities and Taylor expansion Introduction QCD at finite isospin chemical potential QCD at finite chemical potential ( N f = 2): u quark: µ u d quark: µ d ◮ Can be decomposed in baryon and isospin chemical potentials: µ B = 3( µ u + µ d ) / 2 and µ I = ( µ u − µ d ) / 2 ◮ Non-zero µ I introduces an asymmetry between isospin ± 1 particles Positive µ I : ⇒ More protons than neutrons! ◮ Such situations occur regularly in nature: ◮ Within nuclei with # neutrons > # protons. ◮ Within neutron stars. ◮ . . . ◮ However: Usually µ I ≪ µ B . ◮ Finite µ I breaks SU V (2) explicitly to U τ 3 (1). Here: consider µ B = 0!

QCD with isospin chemical potential: low densities and Taylor expansion Introduction Expected phase diagram Exploring the phase diagram using χ PT at finite µ I : [ Son, Stephanov, PRL86 (2001) ] crossover 2nd O (2)? T 1st pion condensation 0 µ I m π / 2

QCD with isospin chemical potential: low densities and Taylor expansion Introduction Expected phase diagram Exploring the phase diagram using χ PT at finite µ I : [ Son, Stephanov, PRL86 (2001) ] crossover 2nd O (2)? T 1st pion condensation 0 µ I m π / 2 First lattice simulations ( N t = 4, m π > m phys ): π 1st order deconfinement and 2nd order curve join? ⇒ Existence of tri-critical point? [ Kogut, Sinclair, PRD66 (2002); PRD70 (2004) ]

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation 2. Simulation setup and λ extrapolation

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation Lattice action [ G. Endr˝ odi, PRD90 (2014) ] ◮ Gauge action: Symanzik improved ◮ Mass-degenerate u / d quarks: Fermion matrix: [ Kogut, Sinclair, PRD66 (2002); PRD70 (2004) ] � D ( µ ) � λγ 5 M = − λγ 5 D ( − µ ) D ( µ ): staggered Dirac operator with 2 × -stout smeared links λ : small explicit breaking of residual symmetry ◮ Necessary to observe spontaneous symmetry breaking at finite V . ◮ Serves as a regulator in the pion condensation phase. ◮ Strange quark: rooted staggered fermions (no chemical potential) ◮ Quark masses are tuned to their physical values. 6 × 16 3 , 24 3 , 32 3 , 8 × 24 3 , 32 3 , 40 3 , 10 × 28 3 , 40 3 ◮ Lattice sizes: . . .

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation Lattice action [ G. Endr˝ odi, PRD90 (2014) ] ◮ Gauge action: Symanzik improved ◮ Mass-degenerate u / d quarks: Fermion matrix: [ Kogut, Sinclair, PRD66 (2002); PRD70 (2004) ] � D ( µ ) � λγ 5 M = − λγ 5 D ( − µ ) D ( µ ): staggered Dirac operator with 2 × -stout smeared links λ : small explicit breaking of residual symmetry ◮ Necessary to observe spontaneous symmetry breaking at finite V . ◮ Serves as a regulator in the pion condensation phase. ◮ Strange quark: rooted staggered fermions (no chemical potential) ◮ Quark masses are tuned to their physical values. 6 × 16 3 , 24 3 , 32 3 , 8 × 24 3 , 32 3 , 40 3 , 10 × 28 3 , 40 3 ◮ Lattice sizes: . . .

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation λ -extrapolations For physical results: λ needs to be removed! Problem: dependence on λ is not known! (at least for most of the observables) T = 124 MeV, µ I = 68 MeV 2.1 T = 124 MeV, µ I = 85 MeV T = 162 MeV, µ I = 68 MeV 1.8 1.5 � n I � /T 3 1.2 0.9 0.6 0.3 0 0.002 0.004 0.006 0.008 aλ

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation λ -extrapolations For physical results: λ needs to be removed! Problem: dependence on λ is not known! (at least for most of the observables) 5e-05 T = 114 MeV, µ I = 34 MeV 0 T = 124 MeV, µ I = 34 MeV T = 148 MeV, µ I = 34 MeV -5e-05 R � -0.0001 ψψ � ¯ -0.00015 a 4 m R -0.0002 -0.00025 -0.0003 0 0.002 0.004 0.006 0.008 aλ

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation λ -extrapolations For physical results: λ needs to be removed! Problem: dependence on λ is not known! (at least for most of the observables) Best possibility for model independence: ◮ Use a (cubic) spline extrapolation. ◮ Fix one of the external points. ◮ Leave the associated outer deriatives free. (additional free parameters) ◮ To stabilise the extrapolation: Need to assume that last two points lie on a (cubic) curve! Remaining systematic effect: Position of nodepoints influences the result!

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation λ -extrapolations Possible solution: Perform a “spline Monte-Carlo” [ see S. Borsanyi ] ◮ Average “all” splines with a similarly good description of the data. Allow for changes of # of nodes and node positions. ◮ Splines are weighted according to some suitable “action” S . Two possibilities: ◮ Use the Akaike information criterion: S AIC = 2 N P + χ 2 ◮ Use the negative goodness of the fit: S GOD = P ( χ 2 , N dof ) − 1 P ( χ 2 , N dof ) = γ ( χ 2 / 2 , N dof / 2) – cumulative χ 2 distribution function Γ( N dof / 2) ( γ : lower incomplete gamma fct.) ◮ Problem: oscillating solutions ⇒ Include some measure δ for oscillations Full action: S = S AIC / GOD + f × δ (parameter f needs to be tuned)

QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation λ -extrapolations Results with S AIC and f = 10 . 0: T = 124 MeV, µ I = 68 MeV 2.1 T = 124 MeV, µ I = 85 MeV T = 162 MeV, µ I = 68 MeV 1.8 1.5 � n I � /T 3 1.2 0.9 0.6 preliminary 0.3 0 0.002 0.004 0.006 0.008 aλ

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential 3. QCD at small isospin chemical potential

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential Definition of the transition point Investigate the finite temperature transition (crossover) for µ I < µ C I . � ¯ � Transition temperature T C is defined by the behaviour of ψψ : ◮ Standard: Use the inflection point of the condensate. ◮ Easier alternative for µ I < µ C I : Use the point where subtracted condensate reaches a certain value. (that value has to be known from µ = 0 – Silver Blaze)

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential Definition of the transition point Investigate the finite temperature transition (crossover) for µ I < µ C I . � ¯ � Transition temperature T C is defined by the behaviour of ψψ : ◮ Standard: Use the inflection point of the condensate. ◮ Easier alternative for µ I < µ C I : Use the point where subtracted condensate reaches a certain value. (that value has to be known from µ = 0 – Silver Blaze)

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential Definition of the transition point Investigate the finite temperature transition (crossover) for µ I < µ C I . � ¯ � Transition temperature T C is defined by the behaviour of ψψ : ◮ Standard: Use the inflection point of the condensate. ◮ Easier alternative for µ I < µ C I : Use the point where subtracted condensate reaches a certain value. (that value has to be known from µ = 0 – Silver Blaze) Here: Use subtracted u / d condensate renormalised by the quark mass: � ¯ � � ¯ � ¯ � � � � � m R ψψ R = m u / d ψψ − ψψ � T =0 ,µ I =0 � ¯ R = − 7 . 407 10 − 5 GeV 4 � Value at the transition (in continuum): m R ψψ [ BW: Borsanyi et al , JHEP1009 (2010) ]

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential Definition of the transition point µ I = 0 MeV 0 R [GeV 4 ] -3e-05 -6e-05 � ψψ � ¯ -9e-05 m R preliminary -0.00012 120 140 160 180 T [MeV] Curves: Simple spline interpolation.

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential Definition of the transition point µ I = 0 MeV 0 µ I = 17 MeV R [GeV 4 ] -3e-05 -6e-05 � ψψ � ¯ -9e-05 m R preliminary -0.00012 120 140 160 180 T [MeV] Curves: Simple spline interpolation.

QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential Definition of the transition point µ I = 0 MeV 0 µ I = 17 MeV µ I = 34 MeV R [GeV 4 ] -3e-05 -6e-05 � ψψ � ¯ -9e-05 m R preliminary -0.00012 120 140 160 180 T [MeV] Curves: Simple spline interpolation.