Hamiltonian Dyson-Schwinger and FRG Flow Equations of Yang-Mills Theory in Coulomb Gauge

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.

Full text

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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)

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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)

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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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