The QCD phase transition probed by fermionic boundary conditions Falk Bruckmann (Univ. Regensburg) Bogoliubov readings Dubna, September 2010 partly with E. Bilgici, C. Gattringer, C. Hagen, Z. Fodor, K. Szabo, B. Zhang Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 0 / 19
QCD need to understand confinement and chiral symmetry breaking Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 19
QCD and its phase diagram need to understand confinement and chiral symmetry breaking but also deconfinement and chiral symmetry restoration at finite temperature and/or density ⇒ new phases of matter 200 175 Quark � gluon plasma 150 Temperature � MeV � Early Universe 125 100 75 2SC Hadron phase 50 25 CFL NQ 250 500 750 1000 1250 1500 1750 2000 Baryon chemical potential � MeV � Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 19
QCD and its phase diagram need to understand confinement and chiral symmetry breaking but also deconfinement and chiral symmetry restoration at finite temperature and/or density ⇒ new phases of matter 200 175 Quark � gluon plasma 150 Temperature � MeV � Early Universe 125 100 75 2SC Hadron phase 50 25 CFL NQ 250 500 750 1000 1250 1500 1750 2000 Baryon chemical potential � MeV � here finite temperature: x 0 ∈ S 1 β . . . Eucl. and compact, β ≡ 1 / T both effects related? Dual quantities generic? Random Matrix Theory Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 19
Theory challenge: Deconfinement � β Polyakov loop: P ( � � 0 dx 0 A 0 ( x 0 ,� � x ) = P exp i x ) ∈ SU ( 3 ) � tr P� � x in complex plane [one point per configuration] 0.3 0.3 T < T c T ≈ T c T > T c 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.2 -0.1 0 0.1 0.2 0.3 order parameter like magnetization, but inverse behavior free energy of infinitely heavy quarks � e −∞ = 0 T < T c � tr P� ∼ e − β F quark = e − # � = 0 T > T c breaks center symmetry Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 2 / 19
Theory challenge: Chiral symmetry restoration spectral density ρ ( λ ) of the Dirac operator: 0.3 0.3 T < T c T ≈ T c T > T c 0.24 0.24 0.18 0.18 0.12 0.12 0.06 0.06 0 0 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 order parameter of chiral symmetry: ρ ( 0 ) ∼ � ¯ ψψ � Banks-Casher i.e. for massless quarks [mass breaks chiral symmetry explicitly] Confinement and chiral symmetry related? Dual quantities Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 3 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes ( → action) how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes ( → action) how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes Polyakov loop U 0 ( 0 ,� x ) U 0 ( a ,� ( → action) x ) . . . how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes Polyakov loop “Polyakov loops U 0 ( 0 ,� x ) U 0 ( a ,� ( → action) x ) . . . with detours” how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes Polyakov loop “Polyakov loops loops U 0 ( 0 ,� x ) U 0 ( a ,� ( → action) x ) . . . with detours” winding twice how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes Polyakov loop “Polyakov loops loops U 0 ( 0 ,� x ) U 0 ( a ,� ( → action) x ) . . . with detours” winding twice how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
Dual quantities: idea and definition lattice: gauge invariant quantities ⇋ link products along closed loops plaquettes Polyakov loop “Polyakov loops loops U 0 ( 0 ,� x ) U 0 ( a ,� ( → action) x ) . . . with detours” winding twice how to distinguish these classes of loops? ⇒ phase factor e i ϕ multiplying U 0 at fixed x 0 -slice & Fourier component Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19
≃ quarks with general boundary conditions Gattringer ’06 ψ ( x 0 + β ) = e i ϕ ψ ( x 0 ) physical quarks are antiperiodic: ϕ = π general quark propagator: cf. Synatschke, Wipf, Wozar, ’07 1 γ µ D µ ϕ + m (physical) chiral condensate: 1 1 � � ρ ( 0 ) = � ¯ ψψ � = lim ≡ Σ ϕ = π m → 0 lim tr γ µ D µ V ϕ = π + m V →∞ dual condensate: Bilgici, FB, Gattringer, Hagen ’08 � 2 π Σ 1 ≡ 1 d ϕ e − i ϕ 1 1 � � ˜ tr γ µ D µ 2 π V ϕ + m 0 Fourier component picks out all contributions that wind once ≡ dressed Polyakov loop: chiral symmetry connected to confinement Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 5 / 19
Dual quantities and center symmetry center: commutes with all group elements for SU ( 3 ) : { 1 , e 2 π i / 3 ≡ z , e − 2 π i / 3 ≡ z ∗ } · 1 3 center transformation: non-periodic gauge transformation, e.g. U 0 → zU 0 in some time slice invariance: action invariant, Polyakov loop: tr P → z tr P center symmetric = confined phase: tr P = 0 at low T center broken = deconfined phase: tr P ≈ { 1 , z , z ∗ } � = 0 at high T [transform into each other] dual quantities like dual condensate ˜ Σ 1 : same behaviour under center: ˜ Σ 1 → z ˜ Σ 1 Synatschke, Wipf, Langfeld ’08 Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 6 / 19
Dual condensate: order parameter I SU ( 3 ) quenched: Bilgici, FB, Gattringer, Hagen ’08 (bare) ˜ Σ 1 with m = 100MeV (bare) Polyakov loop 3 x 4 3 x 4 8 8 0.25 0.25 3 x 4 3 x 4 10 10 3 x 6 3 x 6 10 10 3 ] 0.20 0.20 3 x 4 3 x 4 12 <Tr P>/3V [GeV 12 3 x 6 3 x 6 12 12 3 ] Σ 1 [GeV 3 x 4 3 x 6 14 14 0.15 0.15 3 x 6 14 3 x 8 14 0.10 0.10 0.05 0.05 0.00 0.00 100 200 300 400 500 600 100 200 300 400 500 600 T [MeV] T [MeV] less renormalisation ← detours = dressing Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 7 / 19
Dual condensate: order parameter II SU ( 3 ) with dynamical fermions: FB, Fodor, Gattringer, Szabo, Zhang preliminary N f = 2 + 1 staggered fermions at phys. masses Aoki et al. ’06 ⇒ crossover with T � ¯ ψψ � = 155 ( 2 )( 3 ) MeV and T P c = 170 ( 4 )( 3 ) MeV c (bare) ˜ Σ 1 with m = 60MeV (bare) Polyakov loop 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 75 100 125 150 175 200 225 250 75 100 125 150 175 200 225 250 similar behaviour (center symmetry not an exact symmetry anymore) Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 8 / 19
Dual condensate: mechanism I � 2 π Σ 1 = 1 1 1 d ϕ e − i ϕ · � � ˜ tr γ µ D µ 2 π V ϕ + m 0 Fourier integrand � ... � as a function of ϕ : Bilgici, FB, Gattringer, Hagen ’08 0.45 0.40 T < T c , am = 0.10 T < T c , am = 0.05 0.35 I( ϕ) T > T c , am = 0.10 T > T c , am = 0.05 0.30 0.25 0.20 0 π/ 2 π 3π/ 2 2 π ϕ [for real Polyakov loops, others shift plot by 2 π/ 3] ⇒ depends on ϕ only at high temperatures ⇒ ˜ Σ 1 � = 0 � in particular: chiral condensate survives at high T for periodic bc.s dummy several lattice works Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 9 / 19
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