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arXiv:1011.3462v1 [hep-th] 15 Nov 2010

Hamiltonian Dyson-Schwinger and FRG Flow Equations

of Yang-Mills Theory in Coulomb Gauge

Hugo Reinhardt∗, Markus Leder∗, Jan M. Pawlowski†and Axel Weber∗∗

∗Universität Tübingen, Institut für Theoretische Physik, Auf der Morgenstelle 14, 72076 Tübingen, Germany

†Universität Heidelberg, Institut für Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg, Germany

ExtreMe Matter Institute EMMI, GSI, Planckstr. 1, 64291 Darmstadt, Germany

∗∗Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,

Edificio C-3, Ciudad Universitaria, 58040 Morelia, Michoacán, Mexico

Abstract. A new functional renormalization group equation for Hamiltonian Yang-Mills theory in Coulomb gauge is

presented and solved for the static gluon and ghost propagators under the assumption of ghost dominance. The results are

compared to those obtained in the variational approach.

Keywords: Yang-Mills, Hamiltonian, Coulomb gauge, functional renormalization group, Dyson-Schwinger equations

PACS: 12.38.Aw, 05.10.Cc, 11.10.Ef, 11.15.Tk

INTRODUCTION

My talk is devoted to the application of functional renor-

malization group(FRG) flows to the Hamiltonianformu-

lation ofYang-Millstheoryin Coulombgaugedeveloped

in our group [1].

The advantage of the Hamiltonian formulation is its

close connection to physics. In the variational approach

onemakesanansatzfortheunknownvacuumwavefunc-

tional which encodes all the physics [2, 1]. This ansatz

can be systematically improved towards the full theory.

The price to pay is the apparent loss of renormalization

group invariance.

Renormalization group invariance is naturally built-in

in the functional renormalization group approach to the

Hamilton formulation of Yang-Mills theory put forward

in [3]. Such an approach has the advantage of combin-

ing renormalization group invariance with the physical

Hamiltonian picture.

HAMILTONIAN FLOW

In the FRG approach the quantum theory of a field ϕ is

infrared regulated by adding the regulator term

∆Sk[ϕ] =1

2ϕ ·Rk·ϕ ≡1

2

?

ddp

(2π)dϕ(p)Rk(p)ϕ(−p)

(1)

to the classical action. The regulator functionRk(p) is an

effective momentum dependent mass with the properties

lim

p/k→0Rk(p) > 0 , lim

k/p→0Rk(p) = 0 ,

(2)

which ensures that Rk(p) suppresses propagation of

modes with p ? k while those with p ? k are unaffected

and the full theory at hand is recovered as the cut-off

scale k is pushed to zero. Wetterich’s flow equation for

the effective action Γk[φ] of a field φ is given by

∂tΓk[φ] =1

2Tr

Γ(2)

where

Γ(n)

1

k[φ]+Rk

˙Rk,

(3)

k,1...n[φ] =

δnΓk[φ]

δφ1...δφn

(4)

are the one-particle irreducible n-point functions (proper

vertices), for reviews on gauge theories see [4]. The

generic structure of the flow equation (3) is independent

of the details of the underlying theory, but is a mere

consequence of the form of the regulator term (1), i.e.,

that it is quadratic in the field. By taking functional

derivatives of Eq. (3) one obtains the flow equations for

the (inverse) propagators and proper vertices. For the

two-point function this equation reads

∂tΓ(2)

k,12=1

2Tr˙Rk

1

Γ(2)

k+Rk

?

−Γ(4)

k,12

+

?

Γ(3)

k,1

1

Γ(2)

k+Rk

Γ(3)

k,2+(1 ↔ 2)

??

1

Γ(2)

k+Rk

,

(5)

where all cyclic indices (summed over in the trace) have

been suppressed.

In the Hamiltonian approach to Yang-Mills theory

in Coulomb gauge the generating functional of static

correlation functions reads

?

Z[J] =

DADet(−D∂)|ψ[A]|2exp(J·A) ,

(6)

Page 2

k∂k

−1=

−−

−−1

2

FIGURE 1.

ral and dotted lines with black circles denote the regularized

gluon and ghost propagators at cutoff momentum k, respec-

tively. White circles stand for proper vertices at cutoff k, a

regulator insertion˙Rkis represented by a square with a cross.

Flow equation of the gluon propagator. The spi-

where the integration is over transversal gauge fields A

and the Coulomb gauge condition has been implemented

by the usual Faddeev-Popov method. Representing the

Faddev-Popov determinant in the standard fashion by

ghost fields, c, ¯ c,

Det(−D∂) =

?

D ¯ cDce−?¯ c(−D∂)c

(7)

the underlying action reads

S[A, ¯ c,c] = −ln|ψ[A]|2+

?

¯ c(−D∂)c .

(8)

The general flow equation (5) still holds provided that

φ is interpreted as the superfield φ = (A,c, ¯ c). The FRG

flow equations for the gluon and ghost propagators are

diagrammatically given in Figs. 1, 2.

APPROXIMATION SCHEMES AND

NUMERICAL SOLUTION

The FRG flow equations embody an infinite tower of

coupled equationsfor the flow of the propagatorsand the

proper vertices. These equations have to be truncated to

get a closed system. We shall use the following trunca-

tion: we only keep the gluon and ghost propagators, to

wit

Γ(2)

k,AA= 2ωk(p),

Γ(2)

k,¯ cc=

p2

dk(p),

(9)

In addition,we keep the ghost-gluonvertexΓ(3)

we approximate by the bare vertex, i.e., we do not solve

its FRG flow equation. The latter approximation is justi-

fied by Taylor’s non-renormalization theorem extended

to Coulomb gauge. The above truncation removes the

tadpole diagrams from Figs. 1, 2. Moreover, we shall as-

sume infrared ghost dominance and discard gluon loops.

Then the flow equations of the ghost and gluon propaga-

tor reduce to the ones shown in Figs. 3, 4.

k,A¯ cc, which

k∂k

−1=

+

−1

2

−

FIGURE 2.

Flow equation of the ghost propagator.

k∂k

−1= −

−

FIGURE 3.

The bare vertices at k = Λ are symbolized by small dots.

Truncated flow equation of thegluon propagator.

These flow equations are solved numerically using the

regulators

RA,k(p) = 2prk(p) , Rc,k(p) = p2rk(p) ,

?k2

and the perturbative initial conditions at the large mo-

mentum scale k = Λ,

rk(p) = exp

p2−p2

k2

?

(10)

dΛ(p) = dΛ= const. ,

With these initial conditions, the flow equations for the

ghost and gluon propagators are solved under the con-

straint of infrared scaling for the ghost form factor. The

resultingfullflow oftheghostdressingfunctionis shown

in Fig. 5. As the IR cut-offmomentumk is decreased,the

ghost form factor dk(p) (constant at k = Λ) builds up in-

frared strength and the final solution at k = kminis shown

in Fig. 7 together with the one for the gluon energy

ωkmin(p) in Fig. 6. It is seen that the IR exponents, i.e.,

the slopes of the curves dkmin(p),ωkmin(p) do not change

as the minimal cut-off kminis lowered. Let us stress that

we have assumed infrared scaling of the ghost form fac-

tor but not the horizon condition d−1

latter was obtainedfrom the integration of the flow equa-

tion but not put in by hand (the same is also true for

the infrared analysis of the Dyson-Schwinger equations

(DSEs) following from the variational Hamiltonian ap-

proach,i.e., assumingscaling the DSEs yield the horizon

condition).

In Coulomb gauge the inverse ghost form factor

d−1(p) has been shown to represent the dielectric func-

tion of the Yang-Mills vacuum [5], ε(p) = d−1(p). Then

the so-called horizon condition d−1(0) = 0 implies that

the Yang-Mills vacuum is a perfect dual color supercon-

ductor. In the variational approach one can show that the

infrared exponents of the ghost and gluon propagators,

ω(p → 0) ∼ 1/pα,

are relatedbya sum ruleunderthe assumptionofa trivial

scaling of the ghost-gluon vertex [6],

α = 2β −1 .

ωΛ(p) = p+a .

(11)

k=0(p = 0) = 0. The

d(p → 0) ∼ 1/pβ,

(12)

(13)

Page 3

k∂k

−1=

+

FIGURE 4.

Truncated flow equation of the ghost propagator.

1e-04

1e-02

1e+00

1e+02

1e+04

1e-05

1e-02

p

1e+01

1e+04

1e+00

1e+01

1e+02

1e+03

1e+04

k

FIGURE 5.

Flow dk(p) of the ghost form factor.

The infrared exponents extracted from the numerical

solutions of the flow equations are

α = 0.28,

β = 0.64 .

(14)

They satisfy the sum rule found in [6] but are smaller

thantheonesoftheDSE.Moreover,thepresentapproach

allows to prove the uniqueness of the sum rule (13) [3],

analogously to the proof in Landau gauge [7].

Replacing the propagators with running cut-off mo-

mentumscale k underthe loopintegralsof the flow equa-

tion by the propagators of the full theory,

dk(p) → dk=0(p) , ωk(p) → ωk=0(p) ,

(15)

FIGURE 6.

cutoff values kmin.

Inverse gluon propagator ω at three minimal

FIGURE 7.

cutoff values kmin.

Inverse ghost form factor d at three minimal

amounts to taking into account the tadpole diagrams [3].

Then the flow equations can be analytically integrated

and turn precisely into the DSEs obtained in the varia-

tional approach to the Hamiltonian formulation of Yang-

Mills theory [1], with explicit UV regularization by sub-

traction. This establishes the connection between these

two approaches and highlights the inclusion of a con-

sistent UV renormalization procedure in the present ap-

proach.

The above results encourage further studies, which

include the flow of the potential between static color

sources as well as dynamic quarks.

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