Falk Bruckmann (Univ. Regensburg)crunch.ikp.physik.tu-darmstadt.de/qghxm/2011/TALKS/... · 2011. 3....

Dual and dressed quantities in QCD

Falk Bruckmann(Univ. Regensburg)

Quarks, Gluons, and Hadronic Matter under Extreme ConditionsSt. Goar, March 2011

Falk Bruckmann Dual and dressed quantities in QCD 0 / 22

Motivationphase transition/crossover at finite temperature

Deconfinement: Polyakov loopprobes center symmetry for infinitely heavy quarksorder parameter: 0→ finite / small→ large

Chiral restoration: Quark condensate ρ(0)

probes chiral symmetry for massless quarksorder parameter: finite→ 0 / large→ small

related? Dual quantities

Dressed Polyakov loop from phase boundary conditionsidea and definition Bilgici, FB, Gattringer, Hagen 08lattice results: quenchedand dynamical Zhang, FB, Fodor, Gattringer, Szabo 1012.2314 + Borsanyi

Dressed Wilson loops from magnetic fields FB, Endrödi in prep.


Dual quantities: idea

lattice: gauge invariant quantities link products along closed loops

plaquettes(→ action)

how to distinguish these classes of loops? Gattringer 06

⇒ phase factor eiϕ multiplying U0 at fixed x0-slice




plaquettes Polyakov loop(→ action) U0(0, ~x)U0(a, ~x) . . .






plaquettes Polyakov loop “Polyakov loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours”






plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice






plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice


⇒ phase factor eiϕ multiplying U0 at fixed x0-slice & Fourier component



phase factor eiϕ multiplying U0 at fixed x0-slice

≡ quarks with general boundary conditions

ψ(x0 + β) = eiϕψ(x0)

physical quarks are antiperiodic: ϕ = π

boundary angle ϕ: tool on the level of observables (valencequarks)

formally an imaginary chemical potential


Dual condensate: definition

various dual quantities possible cf. Synatschke, Wipf, Wozar 07

general quark propagator:1

Dϕ + m

general quark condensate:⟨tr

1Dϕ + m

⟩≡ Σϕ lim

m→0Σπ : chiral condensate

dual condensate:

Σn ≡1

2π

∫ 2π

0dϕ e−inϕΣϕ

Fourier component picks out all contributions that wind n times





Dϕ + m


1Dϕ + m

⟩≡ Σϕ lim


dual condensate:

Σ1 ≡1

2π

∫ 2π

0dϕ e−iϕΣϕ

Fourier component picks out all contributions that wind once





Dϕ + m


1Dϕ + m

⟩≡ Σϕ lim


dual condensate:

Σ1 ≡1

2π

∫ 2π

0dϕ e−iϕΣϕ


≡ dressed Polyakov loop: chiral symmetry connected to confinement


Dual condensate: definitionvarious dual quantities possible cf. Synatschke, Wipf, Wozar 07


Dϕ + m


1Dϕ + m

⟩≡ Σϕ lim


dual condensate:

Σ1 ≡1

2π

∫ 2π

0dϕ e−iϕΣϕ


≡ dressed Polyakov loop: chiral symmetry connected to confinement

dual quantities: same behavior under center symmetry (← special ϕ’s)⇒ order parameter


Dressed Polyakov loops: order parameter I

SU(3) quenched

on lattices with different resolution and volume:

(bare) Σ1 with m = 100MeV

100 200 300 400 500 600T [MeV]

0.00

0.05

0.10

0.15

0.20

0.25

Σ1

[GeV

3 ]

83 x 4

103 x 4

103 x 6

123 x 4

123 x 6

143 x 4

143 x 6

143 x 8

(bare) Polyakov loop

100 200 300 400 500 600T [MeV]

0.00

0.05

0.10

0.15

0.20

0.25

<T

r P

>/3

V [

GeV

3 ]

83 x 4

103 x 4

103 x 6

123 x 4

123 x 6

143 x 6

less UV renormalisation in a ∼ 1Nt← ‘detours’


Dressed Polyakov loops: order parameter II

SU(3) with dynamical quarks

2 + 1 flavors of improved staggered fermions on 243 · 8 Aoki et al. 06

⇒ crossover with T chiralc = 157(4) MeV, T deconf

c = 170(5) MeV

(bare) Σ1 with m = 60MeV

75 100 125 150 175 200 225 2500

0.01

0.02

0.03

0.04

0.05

(bare) Polyakov loop

75 100 125 150 175 200 225 250

0.01

0.02

0.03

0.04

0.05

0.06

similar behaviour (center symmetry not an exact symmetry anymore)mass dependence of Σ1? prelim. plots


Dressed Polyakov loops: order parameter II

SU(3) with dynamical quarks

2 + 1 flavors of improved staggered fermions on 243 · 8 Aoki et al. 06

⇒ crossover with T chiralc = 157(4) MeV, T deconf

c = 170(5) MeV

(bare) ‘free energy’ − log Σ1/β

75 100 125 150 175 200 225 2500.3

0.35

0.4

0.45

0.5

0.55

(bare) free energy − log P/β

75 100 125 150 175 200 225 250

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

similar behaviour (center symmetry not an exact symmetry anymore)mass dependence of Σ1? prelim. plots


Dressed Polyakov loops: mechanism I

Σ1 =1

2π

∫ 2π

0dϕ e−iϕ ·

⟨tr

1Dϕ + m

⟩Fourier integrand 〈...〉 as a function of ϕ and T :

0 π/2 π 3π/2 2πϕ

0.20

0.25

0.30

0.35

0.40

0.45

I(ϕ)

T < Tc, am = 0.10

T < Tc, am = 0.05

T > Tc, am = 0.10

T > Tc, am = 0.05150

250

350

T@MeVD

0Π��2

Π

3 Π��2

2 Π

j

0

0.2

0.4

0.6

0.8

150

250

350

T@MeV

0

0.2

0.4

0.6

0.8

quenched [real Polyakov loops chosen] dynamical, m =1MeV

⇒ depends on ϕ only at high temperatures⇒ Σ1 6= 0 Xin particular: chiral condensate survives at high T for periodic bc.sdummy several lattice works, class. dyons FB 09


Dressed Polyakov loops: mechanism IIspectral representation (tr means sum over all eigenmodes):

Σϕ =⟨

tr1

Dϕ + m

⟩=⟨ ∑λϕ>0

2mλ2ϕ + m2

⟩Σ1 =

∫ 2π

0

dϕ2π

e−iϕ . . .

smallest λϕ . m contribute (denominator): IR dominance

0.000

0.005

0.010

0.015

0.020

0.025

T < Tc

T > Tc

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

T < Tc

T > Tc

0 2000 4000|λ| [MeV]

0.0

0.5

1.0

1.5

T < Tc

T > Tc

0 2000 4000|λ| [MeV]

0.0

0.5

1.0

1.5

T < Tc

T > Tc

Individual contributions Individual contributions

Accumulated contributions Accumulated contributions

m = 100 MeV m = 1 GeV

m = 100 MeV m = 1 GeV

valid for dual condensate at even higher massesadd. effect: only lowest modes respond to bc.s


Dressed Polyakov loops: mechanism II

accumulated condensates as a function of λ

dual condensate vs. physical condensate

10 20 30 40 50 60 70

0.0005

0.001

0.0015

0.002

0.0025

0.003

10 20 30 40 50 60 70

0.05

0.1

0.15

0.2

0.25

⇒ dual condensate has converged at even higher masses, wherethe conventional one has not yet (at our resolution)

additional effect: high modes respond to bc.s only weakly,no contribution after Fourier transform in Σ1


Dressed Polyakov loops: mechanism III

geometric series for lattice D hopping once (like staggered we use):

Σ1 ≡∫ 2π

0

dϕ2π

e−iϕ⟨

tr1

Dϕ + m

⟩; tr

∞∑k=0

(−1)k (Dϕ)k

mk ; tr(U − U†

2am

)k

k = length of loop

⇒ large mass suppresses long loops (many U ’s⇒many 1/m’s)

m→∞: conventional Polyakov loop reproduced since shortest

but less IR dominated FB, Gattringer, Hagen 06

⇒mass gives inv. width of dressed Polyakov loops


Dressed Polyakov loops: other applications

through Schwinger-Dyson equations Fischer, Mueller 09

from Random matrix models FB in prep.

Gaussian integrals replace QCD dynamicsmodified Matsubara frequencies: ω = ϕT

ω = πT for physical bc.sω = 0 for periodic bc.s← ‘no twist’

Σϕ from saddle point method:

0

1

2

3

4

T�T_c0

2

4

6j

0.0

0.5

1.0


Dressed Polyakov loops: other applications

at imag. µ through functional RG Braun, Haas, Marhauser, Pawlowski 09

in Polyakov loop extended NJL modeldummy Kashiwa, Kouno, Yahiro 09, Mukherjee, Chen, Huang 09

incl. magnetic field Gatto, Ruggieri 10

in gauge group G2 (no center) Danzer, Gattringer, Maas 08

in SU(2) with adjoint quarks (Tchiral ' 4Tdeconf)dummy Bilgici, Gattringer, Ilgenfritz, Maas 09

canonical determinants Bilgici et al. 11, Nagata, Nakamura 10


‘Wilson Xloop 800’


Dressed Wilson loops: idea and definitionWilson loops of fixed area (in some plane)⇒ phase factor eifS according to area S

magnetic / electric field: constant and abelian f = da (charge=1)

Uµ → Uµeiaaµ W → W eiH

C a = W eiRR

f = W eifS [Stokes]

again in quark condensate (or other observables):⟨tr

1Df + m

⟩≡ Σf = # · tr1 · e0 + # · trW1×1 · eif + . . .

dual condensate:ΣS ≡

∑f

∫e−iSf Σf

picks out all Wilson loops of area S and any circumference≡ dressed Wilson loopsconstant magn. fields⇒Wilson loops ← beyond lattice!?


Dressed Wilson loops: details

Dirac operator restricted to some 2D plane (cheap)e.g. magn. field in z:⇒ D in (x , y)-plane, for different z and t (many)⇒Wilson loops in (x , y)-plane = spatial (area law, too)

finite volume:magn. flux quantised: f = 2π n

Vol2n ∈ Z (like momentum)

area bounded, i.e. periodic⇒ Fourier series

discrete lattice:flux bounded: f ≤ 2π 1

a2 (like momentum)area quantised⇒ discrete Fourier transform (both variables discrete)

for Wilson loops: need all magn. fields


Dressed Wilson loops: results2 + 1 staggered fermions on a 163 · 4 lattice (T = 146 MeV)

condensate Σf over magn. field f = 2π n256 , n = 0,1, ..,256

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 50 100 150 200 250

m_lightm_strange

(averaged over 64 spatial planes and 5 configurations)

grows with magn. field X shoulders?heavy quarks: effect washed out X

not flatFourier components⇒ dual condensate = dressed Wilson loop


Dressed Wilson loops: results

dressed Wilson loop ΣS (Smax = 256)

1e-05

0.0001

0.001

0.01

0.1

1

5 10 15 20 25 30

m_lightm_strange

Wilson loops

together with conventional Wilson loops

decays with area X

pattern for small sizes? geometry of lattice loops?


Dressed Wilson loops: mechanism I

geometric series as before (on the lattice):

ΣS ≡∑

f

∫e−iSf

⟨tr

1Df + m

⟩; tr

∞∑k=0

(−1)k (Df )k

mk ; tr(U − U†

2am

)k

k = length of loop

large mass suppresses long loops

e.g. loops with area S = 4:

circumference k = 8 circumference k = 10Falk Bruckmann Dual and dressed quantities in QCD 18 / 22

Some geometry of lattice loops

ideal lattice loop≡ max. area at fixed circumference:

rectangles, not circle-like

but entropy = number of loops of different shape!?

no disconnected loops since trDk = D(x , y)D(y , z) . . .D(w , x)

extreme cases for area ΣS=p2 :

ideal loop = square: factor 4(2am)4p

many plaquettes after each other: factor 4p2

(2am)4p2

for large mass square loop favored

for large mass ΣS contains mainly rectangular Wilson loops

how many such Wilson loops ; combinatorical factor(for given area at miminal circumference)


Dressed Wilson loops: results

again: dressed Wilson loop ΣS

now for large mass and incl. combinatorical factor

0.1

1

1 2 3 4 5 6 7 8

1000 m_light (~m_b)5000 m_lightWilson loops

agrees with Wilson loops X← used for scale settingat large area S precision limited: tiny ΣS × huge combin. factor


Dressed Wilson loops: mechanism II

IR dominance? only the lowest modes respond to the magnetic field?NO:

eigenvalue variation at fixed index= standard deviation wrt. all magn. fields

50 100 150 200 250 300 350ev. index

0.02

0.04

0.06

0.08

ev. variation

high modes do respond to magnetic field(in contrast to phase bc.s for dressed Polyakov loops)⇒ dressed Wilson loops converge slowly in spectral representation,

especially at large masses


Summary and Outlookdressed Polyakov loops dressed Wilson loops

center symmetry Σ1 6= 0 string tension ΣS ∼ e−σS

related to quark condensate: chiral symmetry

dummywinding from phase bc.s ϕ area from magn. field f

loops with long circumference suppressed by large mass

IR dominance no IR dominancemass range limited

log Σ1,bare 6∼ 1a + . . .? log ΣS,bare 6∼ c

a + . . .?

mass: growing Tc? what is the meaning oflog ΣS(m <∞)?

(sensitive to lattice geometry?)

beyond lattice! beyond lattice?


dual condensates Σ1 and its ‘free energies’ − log Σ1/β as a function oftemperature for masses m = 1 MeV, 10 MeV, 100 MeV (smeared)

100 150 200 2500

0.005

0.01

0.015

0.02

0.025

100 150 200 2500

0.005

0.01

0.015

0.02

0.025

100 150 200 2500

0.005

0.01

0.015

0.02

0.025

0.03

100 150 200 250

0.3

0.35

0.4

0.45

0.5

100 150 200 250

0.3

0.32

0.34

0.36

0.38

0.4

0.42

100 150 200 250

0.38

0.4

0.42

0.44


Falk Bruckmann (Univ. Regensburg)crunch.ikp.physik.tu-darmstadt.de/qghxm/2011/TALKS/... · 2011. 3....

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Transcript of Falk Bruckmann (Univ. Regensburg)crunch.ikp.physik.tu-darmstadt.de/qghxm/2011/TALKS/... · 2011. 3....