Dynamics of Wilson Loops

• Wilson Loop Distributions and Diffusion

• Casimir Scaling and Screening in Confining Phase

• Wilson Loop Effective Action

• Approximate Center Symmetry

• Correlation functions of large Wilson loops

• Summary

A.Brzoska - D. Steinbacher - J. Negele - M.Thies

Wilson Loop Distributions and Diffusion

Wilson-loops

W =1

2tr P exp{ig

∮CdxµAµ(x)}

Vacuum expectation values of Wilson loops

〈0|W |0〉 =

∫d[A]e−S[A]W [A] , S = −1

4

∫d 4xFa

µνFaµν

of sufficiently large size satisfy in the confining phasean area law

〈0|W |0〉 ∼ e−σA

with string tension σ.

For rectangular loops A = RT

x

t

R

T

and for sufficiently large times T � R Wilson loopsdetermine the interaction energy V (R) of static quarks

g2 1

R, R → 0

〈0|W |0〉 ∼ e−T V (R) , V (R) ∼σR , R → ∞

Distribution of Wilson loops

p(ω) = 〈0|δ (ω − W ) |0〉

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

2 40−1

−0.5

0

0.5

1

SU(2) lattice gauge theory

Wilson loop distribution for rect-

angular loops of different areas

Values of Wilson loops in different

planes centered at the origin as function

of the area (fm2) for a single configuration

Diffusion of Wilson Loops

When increasing size of loop in 4-dimensional Euclideanspace as function of parameter t - ”time ”, value ofthe (untraced) Wilson loop carries out Brownian mo-tion on the group manifold, S3 (SU (2)). The motionof an ensemble of these degrees of freedom is in turndescribed by a diffusion equation. Due to gauge invari-ance of Wilson-loops diffusion is independent of 2-ndpolar and of azimuthal angle on S3. With

ω = cos ϑ

Laplacian given by

∆ =1

sin2 ϑ

d

dϑsin2 ϑ

d

dϑ

Diffusion on the sphere

ϑ

NLaplacian

∆ = 1sin ϑ

∂∂ϑ

sin ϑ ∂∂ϑ

+ 1sin2 ϑ

∂2

∂ϕ2

= −L2

0 ϑ π

Uniform Distribution

on the sphere

1

Diffusion equation on S3

(d

dt− ∆) G(ϑ, t) =

1

sin2 ϑδ(ϑ)δ(t)

Eigenfunctions of Laplace operator

ψn =

√2

π

sin nϑ

sin θ, −∆ ψn = (n2 − 1)ψn

Solution of diffusion equation

G(ϑ, t) =2

πsin ϑθ(t)

∞∑n=1

n sin nϑ e−(n2−1)t

t−3/2 exp−ϑ2

4t, t → 0

G(ϑ, t) ∼2

π, t → ∞

QED-behavior for small t, uniform distribution on S3

asymptotically.Relation between ”time” t and Wilson-loop size R

not determined. ”Time” t obtained from fit of expecta-tion value of Wilson loop, 〈0|W (R)|0〉 = exp(−3t)

−1 −0.5 0 0.5 10

1e+06

2e+06

1 x 12 x 13 x 12 x 24 x 15 x 13 x 24 x 25 x 2

Wilson loop distribution in SU(2) lattice gauge theory

Casimir Scaling and Screening in Confining PhaseHigher representation Wilson loops

Expectation value of an observable O(ϑ) in a diffusionprocess on the S3

〈O(ϑ)〉 =2

π

∫ π

0

sin ϑ d ϑ O (ϑ)∞∑

n=1

n sin n ϑ e−(n2−1)t

Wilson loops in the 2 j + 1 representation of SU(2)

Wj(ϑ) =1

2j + 1tr exp 2iϑ

−j

. . .j

=1

2j + 1

sin(2j + 1)ϑ

sin ϑ∼ ψ2j+1(ϑ)

〈Wj〉 = e−4j(j+1)·t

Laplace-Beltrami = Quadratic Casimir

Approximate Casimir Scaling SU(3) Lattice GaugeTheories

Potentials and their ratios in different representations of SU(3), G.S. Bali

Casimir Scaling ⇔ Wilson Loop Diffusion

Two regimes in confining phase

• Regime of Casimir ScalingConfinement of fundamental and adjoint chargesWilson loop distribution result of diffusion

• Screening RegimeConfinement of half-integer (fundamental) chargesand screening of integer (adjoint) charges

Strong coupling implies screening

C

C

Planar tiling of 3-d tiling offundamental loop adjoint loop

3-d tiling suppressed in regime of Casimir scaling

Screening

Diffusion of Wilson loops in the presence of forces de-scribed by potential V (ϑ)

-1 -0,5 0 0,5 1-1

-0,5

0

0,5

1

Haar Measure and eigenfunctions

of Laplacian ψ2j+1(ϑ) as a func-

tion of cos ϑ for j = 1/2, 1, 3/2, 2

Vanishing of fundamental Wilson loop for size → ∞guaranteed if V (ϑ) symmetric around equator of S3

ϑ = π2

V (π/2 − ϑ) = V (ϑ − π/2)

Symmetric potentials admix j = 1/2 to half-integerstates, j = 0 to integer Laplace - eigenfunctions

Confinement → Symmetric V

Casimir scaling → Vanishing V

Wilson Loop Effective ActionConnecting to QCD

Effective action of Wilson loops from diffusion (=Eu-clidean Quantum-Mechanics of point particle)

S ∼∫

dtϑ̇2

In the confinement regime t ∼ ρ2

Seff[ϑ] =3

8πσ

∫dρ

ρ

(dϑ

dρ

)2

Yang-Mills action in gauge

Aaϕ = δa,3

1

gπρ

(π

2− a(t, ρ, z)

)

where (circular) Wilson loops become elementary de-

grees of freedom

W (t, ρ, z) = sin(a(t, ρ, z))

SYM =

∫ρ dρ dϕ dz dt

×{ 1

2π2g2 ρ2[(∂ρa)2 + (∂za)2 + (∂ta)2] + L′} ,

Assume

• coupling of Wilson loops at different t, z to eachother to be negligible beyond a range t2 + z2 ≤ λ2

• within this range no dependence of the Wilson loopson t, z

• negligible coupling to other degrees of freedom

SYM ≈ λ2

g2

∫dρ

ρ(∂ρa)2

From comparison with effective action

λ2 =3g2

8πσApproximate Center Symmetry

As suggested by the diffusion model Lagrangian is sym-metric under reflections of the Wilson loops at the equa-tor of S3

Z̃ : a(t, ρ, z) → −a(t, ρ, z)

accompanied by changes in the other fields. Residualgauge transformation with twist - in analogy with cen-ter symmetry if a space time dimension is compact.

Unless for vanishing ρ, ρAϕ → 0, a singular mag-netic field ∼ δ(ρ2) of infinite action is generated andtherefore

ρ → 0 : a(t, ρ, z) → π

2Boundary condition at ρ = 0 prevents symmetry Z̃

to be realized and gives rise to distributions of Wilson-loops for small ρ concentrated around a = π/2. Withincreasing loop size, the effect of the ”boundary condi-tion” at ρ = 0 decreases and the symmetry is approxi-matively restored as in diffusion of Wilson loops. Thevacuum expectation value of the Wilson loop is a mea-sure of the restoration of the symmetry

〈0|W (t, ρ, z)|0〉 ∼ exp(−σπρ2)

If exact the symmetry (Z̃) would require 〈0|W |0〉 = 0.

Correlation function of large Wilson loops

x

y

Approximate center symmetry makes thedynamics of Polyakov and Wilson loopsof large size similar. Symmetries protectgauge strings from decaying.In particular from

〈0|P (d)P (0)|0〉 ∼ exp(−σβd)

one expects

〈0|W (d, ρ)W (0, ρ)|0〉 ∼ exp(−σ2πρd)

Accounting for exp(−πρ2) suppressed contributionsfrom the violation of the Z̃ symmetry

〈0|W (d, ρ) W (0, ρ)|0〉 ≈ e−2σπρ2(c0 + c1e

−mgbd + ..)

+ d1e−σ2πρd

• The first class of contribution arises from the inter-mediate ground-state and glueballs. Not present forPolyakov loops - forbidden by exact center symme-try Z

• Second class of contributions - common to Wilsonand Polyakov loops - arises from intermediate stateswhich are odd under Z and Z̃ respectively

-1 -0,5 0 0,5

-10

-5

0

5

10R=2R=3R=4R=5R=6R=7R=8R=9R=10R=11R=12

Logarithm of correlation func-- tion of quadratic Wilson-loops- (R×R) as a function of- ξ = 2Rd − R2

For sufficiently large Wilson loops we expect[〈0|W (d,R) W (0, R)|0〉 − 〈0|W (0, R)|0〉2] /〈0|W (0, R)|0〉2

≈ c1e−mgb

ξ+R2

2R + d1e−2σξ + ...

Geometrical interpretation

d

R

∼ exp{−2πσRd} ∼ exp{−2πσR2−mgbd}For large loops, up to the glueball correction, the sur-

face of minimal area dominates. Change in configura-tion around d ≈ R - Gross-Ooguri transition.

Wilson loop operators applied to the vacuum generateapproximative stationary states of energy

E[W (R)] ≈ σ2πR

Single (gauge) string states are approximative eigen-states of the Hamiltonian.

Summary

• Wilson loop distribution - result of diffusion process

• Casimir scaling: a new regime in the confinementphase, different from strong coupling limit

• Approximate center symmetry associated with con-finement

• Sufficiently large Wilson loops generate approxima-tive stationary single string states