Dual quark condensate and dressed Polyakov loops

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1 Dual quark condensate and dressed Polyakov loops Falk Bruckmann (Univ. of Regensburg) Lattice 28, William and Mary with Erek Bilgici, Christian Hagen and Christof Gattringer Phys. Rev. D77 (28) 947, [hep-lat], Phys. Lett. B647 (27) 56-61, hep-lat/6122 Falk Bruckmann Quark condensate and Polyakov loops 1 / 13

2 Motivation QCD at finite temperature: confinement and chiral symmetry breaking quenched Yang-Mills theory: same T c Polyakov loop: P( x) = P exp ( i β dx A (x, x) ), β = 1/k B T tr c P in SU(3): T < T c T T c T > T c order parameter for confinement: related to the free energy of a single quark confined phase: tr c P = (F quark ) Falk Bruckmann Quark condensate and Polyakov loops 2 / 13

3 spectral density ρ(λ) of the Dirac operator (in background A µ ): T < T c T T c T > T c order parameter of chiral symmetry: ρ() ψψ... chiral condensate Banks-Casher Falk Bruckmann Quark condensate and Polyakov loops 3 / 13

4 spectral density ρ(λ) of the Dirac operator (in background A µ ): T < T c T T c T > T c order parameter of chiral symmetry: ρ() ψψ... chiral condensate Banks-Casher Is there an underlying mechanism connecting the two? does confinement leave a trace in the Dirac spectrum? quarks should know that they are confined! dressed Polyakov loops as a new order parameter Falk Bruckmann Quark condensate and Polyakov loops 3 / 13

5 The idea work on the lattice (regulator) Polyakov loop: P(x) N τ=1 U (x + τ, x) Falk Bruckmann Quark condensate and Polyakov loops 4 / 13

6 The idea work on the lattice (regulator) Polyakov loop: P(x) N τ=1 U (x + τ, x) Dirac operator, here staggered D(x, y) 1 2a η µ (x) [ U µ (x)δ x+ˆµ,y h.c. ] µ Kogut, Susskind hopping by one link D l (x, x) products of links along closed loops of length l, at x Falk Bruckmann Quark condensate and Polyakov loops 4 / 13

7 The idea work on the lattice (regulator) Polyakov loop: P(x) N τ=1 U (x + τ, x) Dirac operator, here staggered D(x, y) 1 2a η µ (x) [ U µ (x)δ x+ˆµ,y h.c. ] µ Kogut, Susskind hopping by one link D l (x, x) products of links along closed loops of length l, at x how to distinguish Polyakov loops from trivially closed loops? Falk Bruckmann Quark condensate and Polyakov loops 4 / 13

8 The idea work on the lattice (regulator) Polyakov loop: P(x) N τ=1 U (x + τ, x) Dirac operator, here staggered D(x, y) 1 2a η µ (x) [ U µ (x)δ x+ˆµ,y h.c. ] µ Kogut, Susskind hopping by one link D l (x, x) products of links along closed loops of length l, at x how to distinguish Polyakov loops from trivially closed loops? phase twisted boundary conditions, as a tool: Gattringer 6 ψ(x + β, x) = z ψ(x, x), z = e iφ imag. chem. potential realized by U zu at some time slice Polyakov loops: P zp, trivial loops stay the same Falk Bruckmann Quark condensate and Polyakov loops 4 / 13

9 P itself turned out to be not suitable (UV dominated) FB et al. 6 propagator: cf. Synatschke, Wipf, Wozar 7 tr 1 m + D φ = 1 m ( 1) l l= m l tr(d φ ) l... all powers of D φ Falk Bruckmann Quark condensate and Polyakov loops 5 / 13

10 P itself turned out to be not suitable (UV dominated) FB et al. 6 propagator: cf. Synatschke, Wipf, Wozar 7 tr 1 m + D φ = 1 m ( 1) l l= = 1 (±1) m (2am) l loops of length l m l tr(d φ ) l... all powers of D φ tr c U µ (x) e iφq(loop) loop q(loop) Z: how many times the loop winds around [, β] Falk Bruckmann Quark condensate and Polyakov loops 5 / 13

11 P itself turned out to be not suitable (UV dominated) FB et al. 6 propagator: cf. Synatschke, Wipf, Wozar 7 tr 1 m + D φ = 1 m ( 1) l l= = 1 (±1) m (2am) l loops of length l m l tr(d φ ) l... all powers of D φ tr c U µ (x) e iφq(loop) loop q(loop) Z: how many times the loop winds around [, β] project onto particular winding q: 1 2π dφ e iφq 2π let s specify to a single winding q = 1 like the Polyakov loop: Falk Bruckmann Quark condensate and Polyakov loops 5 / 13

12 A new observable dummy FB et al. 8 Σ 1 2π dφ 1 1 2π e iφ tr V m + D φ = 1 mv loops (±1) (2am) l of length l, winding once tr c l U µ (x) dual condensate dressed Polyakov loops Falk Bruckmann Quark condensate and Polyakov loops 6 / 13

13 A new observable dummy FB et al. 8 Σ 1 2π dφ 1 1 2π e iφ tr V m + D φ = 1 mv loops (±1) (2am) l of length l, winding once tr c l U µ (x) dual condensate massless limit: lim lim m V Σ 1 = massive limit: 2π lim Σ 1 tr c P m dφ 2π e iφ ρ() φ dressed Polyakov loops dual chiral condensate ρ() ψψ (integrated over phase bc.s) thin Polyakov loop (shortest) detours suppressed by 2am Falk Bruckmann Quark condensate and Polyakov loops 6 / 13

14 Σ 1 is an order parameter numerical results (quenched): Σ 1 [GeV 3 ] x x x x x x x x T [MeV] Σ 1 as a function of temperature for m = 1MeV Falk Bruckmann Quark condensate and Polyakov loops 7 / 13

15 Spectral representation Σ 1 2π dφ 1 1 2π e iφ tr = V m + D φ 2π dφ 1 2π e iφ V i 1 m + λ (i) φ truncate the sum: IR dominance expected since λ in denominator! confirmed by lattice data (if m not too large): Individual contributions m = 1 MeV T < T c T > T c Individual contributions m = 1 GeV T < T c T > T c Accumulated contributions m = 1 MeV Accumulated contributions m = 1 GeV T < T c T < T c.5 T > T c T > T c. 2 4 λ [MeV] 2 4 λ [MeV]. Falk Bruckmann Quark condensate and Polyakov loops 8 / 13

16 how is a vanishing/finite Polyakov loop built up by the eigenvalues? respond differently to bc.s in confined and deconfined phase.45 I(ϕ) T < T c, am =.1 T < T c, am =.5 T > T c, am =.1 T > T c, am = V i.2 1 m+λ (i) φ π/2 π 3π/2 2π ϕ as a function of φ for real P nonvanishing cos φ-part only in the deconfined phase Σ 1 non-real P: the plot is shifted by ±2π/3 periodicity 2π/3, known from imag. µ Lombardo et al. Falk Bruckmann Quark condensate and Polyakov loops 9 / 13

17 How about the chiral condensate? remember: Σ 1 m,v confined phase: 2π dφ e iφ ρ() φ = 2π dφ e iφ ψψ φ ψψ, but independent of φ vanishing Σ 1 Falk Bruckmann Quark condensate and Polyakov loops 1 / 13

18 How about the chiral condensate? remember: Σ 1 m,v confined phase: 2π dφ e iφ ρ() φ = 2π dφ e iφ ψψ φ ψψ, but independent of φ vanishing Σ 1 deconfined phase: ψψ =, spectral gap: ρ() =!? Falk Bruckmann Quark condensate and Polyakov loops 1 / 13

19 How about the chiral condensate? remember: Σ 1 m,v confined phase: 2π dφ e iφ ρ() φ = 2π dφ e iφ ψψ φ ψψ, but independent of φ vanishing Σ 1 deconfined phase: ψψ =, spectral gap: ρ() =!? no: ρ() periodic for real P Gattringer, Schaefer 3 always one bc. where ρ() ψψ φ δ(φ + φ P ) nonvanishing Σ 1 for all T > T c Falk Bruckmann Quark condensate and Polyakov loops 1 / 13

20 Center symmetry the deconfinement transition of pure gauge theory can be described as spontaneous breaking of the center symmetry: the action is invariant under U z U at some time slice, z center(su(3)) the Polyakov loop changes as tr c P z tr c P Falk Bruckmann Quark condensate and Polyakov loops 11 / 13

21 Center symmetry the deconfinement transition of pure gauge theory can be described as spontaneous breaking of the center symmetry: the action is invariant under U z U at some time slice, z center(su(3)) the Polyakov loop changes as tr c P z tr c P same for the dressed Polyakov loops Σ 1 as they wind once as well therefore order parameter for confinement Falk Bruckmann Quark condensate and Polyakov loops 11 / 13

22 Center symmetry the deconfinement transition of pure gauge theory can be described as spontaneous breaking of the center symmetry: the action is invariant under U z U at some time slice, z center(su(3)) the Polyakov loop changes as tr c P z tr c P same for the dressed Polyakov loops Σ 1 as they wind once as well therefore order parameter for confinement all functions of the form Synatschke, Wipf, Langfeld 8 2π dφ 2π e iφ f (D φ ) transform this way, thus are order parameters for center symm. Falk Bruckmann Quark condensate and Polyakov loops 11 / 13

23 Generalisation: Locally resolved Polyakov loops so far: x P(x) eigenvalues λ(i) φ now: P(x) eigenvalues λ (i) φ and eigenvectors ψ(i) φ Falk Bruckmann Quark condensate and Polyakov loops 12 / 13

24 Generalisation: Locally resolved Polyakov loops so far: x P(x) eigenvalues λ(i) φ now: P(x) eigenvalues λ (i) φ and eigenvectors ψ(i) φ static quark potential V q q ( x y ) ln tr P( x) tr P( y) SU(2): Synatschke, Wipf, Langfeld mode sum N t =6 Polyakov line N t =6 mode sum N t =2 Polyakov line N t =2 V(r) / T r/a string tension preserved by a truncated mode sum mechanism not fully clear Bilgici, Gattringer 8 Falk Bruckmann Quark condensate and Polyakov loops 12 / 13

25 Summary the response of Dirac spectra to different temporal bc.s contains information about confinement the dressed Polyakov loop Σ 1 is a novel deconfinement order param. that relates the dual chiral condensate to the thin Polyakov loop... and is dominated by IR modes many center sensitive functions of D can be defined Falk Bruckmann Quark condensate and Polyakov loops 13 / 13

26 Summary the response of Dirac spectra to different temporal bc.s contains information about confinement the dressed Polyakov loop Σ 1 is a novel deconfinement order param. that relates the dual chiral condensate to the thin Polyakov loop... and is dominated by IR modes many center sensitive functions of D can be defined outlook: random matrix theory description of D φ gauge group G(2): no nontrivial center Bruckmann, Verbaarschot in progr. Gattringer, Maas in progr. full QCD and 4-fermi deformation (Sinclair): T χsb T deconf how in the formalism?! Falk Bruckmann Quark condensate and Polyakov loops 13 / 13