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The ’t Hooft-Loop in Coulomb Gauge

Dominik Epple

[email protected]

Eurograd Workshop Todtmoos 2007

The ’t Hooft Loop in Coulomb Gauge Eurograd Workshop Todtmoos 2007

Contents

� Coulomb Gauge Yang-Mills Theory

� The ’t Hooft Loop as confinement criterion

� Dependency of the ’t Hooft Loop behaviour on the Yang-Mills

vacuum

� Results for the ’t Hooft Loop

� Summary

arXiv:0706.0175 [hep-th]; Accepted for publication in Phys.Rev. D

Dominik Epple Page 1


Coulomb Gauge Yang-Mills Theory

� Hamiltonian formulation of Yang-Mills Theory

� Weyl gauge: A0 = 0, D.O.F are cartesian coordinates Aai (x),

canonically conjugated momenta Πai (x) = δ

iδAai (x),

ETCRs[Aa

i (x),Πbj(y)

]= iδabδijδ(x − y),

Hamiltonian H =∫

dxΠ2(x) + B2(x)

� Gauss’ Law DiΠi|Ψ〉 = ρm|Ψ〉 guarantees invariance of the

wave functional under small gauge transformations

� chose Coulomb Gauge ∂iAi(x) = 0 to resolve Gauss’ Law



Coulomb Gauge ∂iAi(x) = 0

� eliminates unphysical degrees of freedom,

� change of coordinates from cartesian to curvilinear ones,

� introduces Jacobian (Fadeev-Popov determinant)

J [A] = det(−∂iDi),

� new physical degrees of freedom are A⊥ai (x).



Coulomb Gauge Yang-Mills Hamiltonian

H =12

∫d3x

[J−1ΠaiJΠa

i + Bai Ba

i

]+

g

2

∫d3x

∫d3x′J−1ρa(x)F ab(x,x′)J ρb(x′)

Physically motivated ansatz for the vacuum wave functional

(strongly peaked at the Gribov Horizon J = 0)

Ψ[A] = J−12N exp(−1

2

∫d3xd3x′A⊥a

i ω(x,x′)A⊥ai (x′))



Find solutions for the Yang-Mills Schrodinger equation for the

lowest energy eigenstate form the variational principle

〈Ψ|H|Ψ〉〈Ψ|Ψ〉 → min .

� coupled set of integral equations (Dyson-Schwinger equations),

to be solved numerically1.

1C. Feuchter and H. Reinhardt, Phys. Rev. D70 (2004) 105021



Quantities which are computed self-consistently are

� Gluon form factor ω(q),〈ω|Aa

i (q)Abj(q

′)|ω〉 = 12(2π)3tij(q)δabδ(q − q′)ω−1(q)

� Ghost form factor d(q), Gω(q) = G0(q)d(q)/g,

G0(x,x′) = 〈x|(−∂2)−1|x′〉 = 14π|x−x′|

� Curvature χ(q), represents the ghost-loop part of the gluon

self-energy,

χ = −12〈δ2 lndet(−D∂)

δAδA 〉,J [A] = det(−D∂) = exp(− ∫

AχA)



The following IR behaviour holds to leading order:

ω(q) = χ(q) = A · k−α, d(q) = B · k−β, 2β − α = 1

The existence of two kinds of solutions has been predicted:

� One with α ≈ 0.795 (Found by C. Feuchter 2004)

� One with α = 1

We have been able to obtain also the solution with α = 1. Very

careful and elaborate numerics is required for it.



1

10

100

1000

0.001 0.01 0.1 1 10 100 1000

k [σ1/2]

ω(k) [σ1/2]χ(k) [σ1/2]

1

10

100

1000

10000

0.001 0.01 0.1 1 10 100 1000

k [σ1/2]

d(k)

Figure 1: (left) The Gluon dressing function ω(q) and the curvature χ(q), calculated by

minimizing the vacuum energy functional with a gaussian ansatz for the wave functional.

These curves are from the α = 1 solution. (right) Ghost dressing function d(q) from the

same calculation.

Phys.Rev.D75:045011,2007; hep-th/0612241



-5

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20x/x0

[V(r)-V(1)]/k0Linear fit

Figure 2: The Coulomb Potential (quark-antiquark-Potential), computed with the form factors

from the solution with α = 1. We find a linear rising potential within excellent precision

(numerical error < 1%).



The ’t Hooft Loop as confinement criterion

The Wilson loop W [A(C)] = tr P exp(− ∮dxμAμ)

� Gauge-invariant quantity

� Measures magnetic flux through surface (for abelian theories)

Temporal Wilson Loop: Order parameter for Yang-Mills Theory

〈W [A(C)]〉 ={

exp(−σ · Area(C)), confined phase

exp(−κ · Perimeter(C)), deconfined phase



Spatial ’t Hooft loop V (C): defined via spatial Wilson Loop

V (C1)W (C2) = ZL(C1,C2)W (C2)V (C1)

with Z = −1 for SU(2) and L = Gaussian linking number.

’t Hooft loop behaviour:

〈V [A](C)〉 ={

exp(−κ · Perimeter(C)), confined phase

exp(−σ · Area(C)), deconfined phase

� dual to the Wilson loop’s behaviour.

(G. ’t Hooft, Nucl. Phys. B138 (1978) 1)



’t Hooft loop creates a Center Vortex (A(C) denotes a center

vortex field, W [A(C1)](C2) = ZL(C1,C2)):

V (C)Ψ[A] = Ψ(A + A)

Continuum Representation:

(H. Reinhardt, Phys. Lett. B577 (2003) 317-323)

V (C) = exp[i

∫d3xA(C)(x)Π(x)

]



The ’t Hooft Loop in Coulomb GaugeTo one-loop order, one can calculate

〈V (C)〉 =∫

DA det(−D∂)Ψ∗(A)Ψ(A + A)

= exp(−S(C)), S(C) := S(R)

S(R) = 4πR

∞∫0

dx K(x

R

)︸︷︷︸physics

π/2∫0

dα (1 − 2 sin2 α) f(2x sin α)

︸︷︷︸=:A(x), geometry

where f(z) = j0(z) − j′′0(z), j0(z) = sin z

z .



The geometrical integral A(x)

A(x) =

π/2∫0

dα (1 − 2 sin2 α) f(2x sinα)

=π

4

1∫−1

dz(1 + z2)J2(2xz)

=π

4x

[2xJ0(2x) + πx

(J1(2x)H0(2x) − J0(2x)H1(2x)

) − 2J1(2x)]

− 3π2

16x2

[J2(2x)H1(2x) − J1(2x)H2(2x)

]




0.001

0.01

0.1

1

0.1 1 10 100

A(x

)

x

Numerical value



The geometrical integral A(x)Asymptotical properties:

� x → 0: A(x) = 2π15x2 − π

35x4 + O

(x6

)� x → ∞: A(x) = π

4x +√

π2 cos

(2x + π

4

)1√x3

+ O(

1√x5

)

0.001

0.01

0.1

1

0.1 1 10 100

A(x

)

x

Numerical valueIR expansion

UV seriesTaylor expansion around x=2




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8 9 10

A(x

)

x

Numerical valueIR expansion

UV seriesTaylor expansion around x=2



Physics: K(q)

K(q) = [ω(q) − χ(q)][1 − 1

2[ω(q) − χ(q)]

ω(q)

]

=12ω(q)

[1 −

(χ(q)ω(q)

)2]

Asymptotic behavior:

� IR: K(q) IR= [ω(q) − χ(q)] ≈ c0 + c1 q + . . .

� UV: K(q) UV= 12ω(q) ≈ 1

2(q + a0) + . . .



Discussion of S(R)/R

S(R) = 4πR

∫dxK

(x

R

)A(x)

� Integral for S(R) linearly and logarithmically UV-divergent

� Large-R-behavior of S(R) tests IR-behavior of K(q)

� need to renormalize by subtraction of UV-leading parts in K(q)

Crucial: do not spoil IR-behavior!



Parametrize ω(q) = A/q + a0 + q.

Replace K(q) → K(q) − K0(q) = K(q):

K(q) =12ω(q)

[1 −

(χ(q)ω(q)

)2]

K0(q) =12(q + a0)

[1 −

(χ(q)ω(q)

)2]

K(q) =A

q

[1 −

(χ(q)ω(q)

)2]



This subtraction has the following properties:

� We do not spoil the IR behavior

� Subtract all terms creating UV divergencies in S(R)

� Subtract also terms creating UV-finite contributions to S(R),however, these are contributions are neglegible

� Renormalized ’t Hooft loop exponent S(R).

S(R) = 4πR

∫dx

A

q

[1 −

(χ(q)ω(q)

)2]

A(x), q = x/R



Behaviour of the ’t Hooft Loop for differentsolutions of the YM vacuum

Using the asymptotic behavior of A(x), one can show that:

� A finite c0 leads to a logarithmically diverging S(R)/R

� c0 = 0 leads to S(R → ∞)/R → const., which corresponds to

a perimeter law for the ’t Hooft Loop.

These points can be verified numerically also with the full A(x).



Result for the ’t Hooft Loop

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12 14 16 18 20

Sba

r(R

)/R

[G

eV]

R [fm]

ξ0=0, ξ= 0.15625ξ0=0, ξ= 0ξ0=0, ξ=-0.15625ξ0=1.44, ξ= 0.15625ξ0=1.44, ξ= 0ξ0=1.44, ξ=-0.15625

Figure 3: The S(R)/R function, calculated numerically using the full expression for A(x)

and the renormalized K(q) as described in the text. The YM-Solution has the property

ω(q) − χ(q) → 0 for q → 0. We see a behaviour of S(R → ∞)/R = const., which

means a perimeter law for the ’t Hooft loop, which corresponds to a confined theory.



Summary

� We have been able to obtain solutions for the Yang-Mills

vacuum which show a exact linear Coulomb potential.

� We can show a perimeter-law for the ’t Hooft Loop, i.e.

confinement, for a special solutions of the Yang-Mills vacuum.


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