On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory
On the basis of the general considerations such as R-operation and Slavnov-Taylor identity we show that the effective action, being understood as Legendre transform of the logarithm of the path integral, possesses particular structure in N = 4 supersymmetric Yang-Mills theory for kernels of the effe...
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Інститут математики НАН України
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irk-123456789-1464532019-02-10T01:25:50Z On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory Cvetic, G. Kondrashuk, I. Schmidt, I. On the basis of the general considerations such as R-operation and Slavnov-Taylor identity we show that the effective action, being understood as Legendre transform of the logarithm of the path integral, possesses particular structure in N = 4 supersymmetric Yang-Mills theory for kernels of the effective action expressed in terms of the dressed effective fields. These dressed effective fields have been introduced in our previous papers as actual variables of the effective action. The concept of dressed effective fields naturally appears in the framework of solution to Slavnov-Taylor identity. The particularity of the structure is independence of these kernels on the ultraviolet regularization scale Λ. These kernels are functions of mutual spacetime distances and of the gauge coupling. The fact that β function in this theory vanishes is used significantly. 2006 Article On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory / G. Cvetic, I. Kondrashuk, I. Schmidt // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 28 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81Q30 http://dspace.nbuv.gov.ua/handle/123456789/146453 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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On the basis of the general considerations such as R-operation and Slavnov-Taylor identity we show that the effective action, being understood as Legendre transform of the logarithm of the path integral, possesses particular structure in N = 4 supersymmetric Yang-Mills theory for kernels of the effective action expressed in terms of the dressed effective fields. These dressed effective fields have been introduced in our previous papers as actual variables of the effective action. The concept of dressed effective fields naturally appears in the framework of solution to Slavnov-Taylor identity. The particularity of the structure is independence of these kernels on the ultraviolet regularization scale Λ. These kernels are functions of mutual spacetime distances and of the gauge coupling. The fact that β function in this theory vanishes is used significantly. |
| format |
Article |
| author |
Cvetic, G. Kondrashuk, I. Schmidt, I. |
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Cvetic, G. Kondrashuk, I. Schmidt, I. On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Cvetic, G. Kondrashuk, I. Schmidt, I. |
| author_sort |
Cvetic, G. |
| title |
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory |
| title_short |
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory |
| title_full |
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory |
| title_fullStr |
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory |
| title_full_unstemmed |
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory |
| title_sort |
on the effective action of dressed mean fields for n = 4 super-yang-mills theory |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/146453 |
| citation_txt |
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory / G. Cvetic, I. Kondrashuk, I. Schmidt // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 28 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT cveticg ontheeffectiveactionofdressedmeanfieldsforn4superyangmillstheory AT kondrashuki ontheeffectiveactionofdressedmeanfieldsforn4superyangmillstheory AT schmidti ontheeffectiveactionofdressedmeanfieldsforn4superyangmillstheory |
| first_indexed |
2025-07-11T00:00:37Z |
| last_indexed |
2025-07-11T00:00:37Z |
| _version_ |
1837306528496877568 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 002, 8 pages
On the Effective Action of Dressed Mean Fields
for N = 4 Super-Yang–Mills Theory
Gorazd CVETIC †, Igor KONDRASHUK †‡ and Ivan SCHMIDT †
† Departamento de F́ısica, Universidad Técnica Federico Santa Maŕıa,
Avenida España 1680, Casilla 110-V, Valparaiso, Chile
E-mail: gorazd.cvetic@usm.cl, igor.kondrashuk@usm.cl, ivan.schmidt@fis.utfsm.cl
‡ Departamento de Ciencias Basicas, Universidad del Bio-Bio,
Campus Fernando May, Casilla 447, Avenida Andreas Bello, Chillan, Chile
Received October 30, 2005, in final form January 01, 2006; Published online January 09, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper002/
Abstract. On the basis of the general considerations such as R-operation and Slavnov–
Taylor identity we show that the effective action, being understood as Legendre transform of
the logarithm of the path integral, possesses particular structure in N = 4 supersymmetric
Yang–Mills theory for kernels of the effective action expressed in terms of the dressed effective
fields. These dressed effective fields have been introduced in our previous papers as actual
variables of the effective action. The concept of dressed effective fields naturally appears
in the framework of solution to Slavnov–Taylor identity. The particularity of the structure
is independence of these kernels on the ultraviolet regularization scale Λ. These kernels are
functions of mutual spacetime distances and of the gauge coupling. The fact that β function
in this theory vanishes is used significantly.
Key words: R-operation; gauge symmetry; N = 4 supersymmetry; Slavnov–Taylor identity
2000 Mathematics Subject Classification: 81Q30
Slavnov–Taylor (ST) identity [1] is an important tool in quantum field theory. It is a con-
sequence of BRST symmetry [2] of the tree level action of gauge theories, and it consists in an
equation written for a functional that is called effective action [3]. An approach to solving the
ST identity in gauge theories has been proposed recently [4, 5]. In this Letter we re-consider
our analysis for a particular case of N = 4 supersymmetric theory. Our analysis will be based
on five theoretical tools: R-operation [6], gauge symmetry, N = 4 supersymmetry, the ST iden-
tity itself, and absorbing two point Green’s functions into a re-definition of the effective fields.
Effective fields are variables of the effective action [3].
N = 4 super-Yang–Mills theory is widely considered from the point of view AdS/CFT cor-
respondence [7]. Anomalous dimensions of gauge invariant operators are related to energies of
string states [8]. In this Letter we consider N = 4 super-Yang–Mills theory from a different
point. We analyze one particle irreducible (proper) correlators of this theory, which are kernels
of the effective action. For example, a kernel can be proper vertex of several gluons. We hope
this analysis can have application to calculation of maximal helicity violating amplitudes of pro-
cesses with n gluons [9, 10]. N = 4 super-Yang–Mills theory is useful theoretical playground to
understand better the problems that stand in QCD. This model has special particle contents. In
addition to one gluon and four Majorana fermions it contains six scalar fields. All the particles
are in the adjoint representation of SU(N) gauge group.
It has been known for a long time that for N = 4 Yang–Mills supersymmetric theory beta
function of gauge coupling vanishes [11, 12, 13, 14]. We extensively use this fact in our analysis.
mailto:gorazd.cvetic@usm.cl
mailto:igor.kondrashuk@usm.cl
mailto:ivan.schmidt@fis.utfsm.cl
http://www.emis.de/journals/SIGMA/2006/Paper002/
2 G. Cvetic, I. Kondrashuk and I. Schmidt
The basic notation of this paper coincides with notation of [4]. The main ST identity is [3]
Tr
[∫
dx
δΓ
δAm(x)
δΓ
δKm(x)
+
∫
dx
δΓ
δc(x)
δΓ
δL(x)
−
∫
dx
δΓ
δb(x)
(
1
α
∂mAm(x)
)]
+
∫
dx
δΓ
δφ(x)
δΓ
δk(x)
+
∫
dx
δΓ
δk̄(x)
δΓ
δφ̄(x)
= 0. (1)
The effective action is a functional of all the effective fields and external sources participating
in this equation, Γ ≡ Γ[Am, b, c, φ, φ̄,Km, L, k, k̄] [24, 3]. The external sources Km, L, k, k̄ are
coupled in the exponential of the path integral to the BRST transformations of fields from the
measure of the path integral [3], that is, to the BRST transformations of fields Am, c, φ, φ̄ of
the tree level action, respectively. The effective fields Am, b, c, φ, φ̄ are traditionally designated
by the same letters as are the fields Am, b, c, φ, φ̄, which are variables of the path integral. The
effective fields are defined as variational derivatives of the logarithm of the path integral with
respect to the corresponding external sources coupled to these variables of integration in the
path integral [3]. The matter effective field φ stands for spinors as well as for scalars. We assume
summations over all indices of the representation of matter fields. The traditional Lorentz gauge
fixing is taken and the corresponding Faddeev–Popov ghost action introduced according to the
line of Ref. [13]. These terms break supersymmetry of the tree level action. The β function is
zero but anomalous dimensions of propagators are non-zero [13].
Consider the vertex Lcc. Here we do not specify arguments of the effective fields. It is the
only vertex, which is invariant with respect to the ST identity at the classical level. At the
quantum level it transforms to the form
〈Lcc〉 × 〈Lcc〉+ 〈LccA〉 × 〈Km∂mc〉 = 0. (2)
This is a direct consequence of the main ST identity (1) and is a schematic form of the ST identity
relating the Lcc and LccA field monomials. The precise form of this relation can be obtained by
differentiating the identity (1) with respect to L and three times with respect to c and then by
setting all the variables of the effective action equal to zero. The brackets in (2) mean that we
have taken functional derivatives with respect to fields in the corresponding brackets at different
arguments and then have put all the effective fields equal to zero.
We know from the theory of R-operation [6] that in Yang–Mills theory the divergences can be
removed by re-defining the fields and the gauge coupling. Thus, there are four renormalization
constants that multiply the ghost, gluon, spinor, and scalar fields [3]. The gauge coupling
also must be renormalized but this is not the case in the theory under consideration. In this
paper we concentrate on two regularizations: regularization by higher derivatives described
in [3], Λ is the regularization scale, and regularization by dimensional reduction [15]. The
regularization by higher derivatives has been constructed for supersymmetric theories in [16, 17].
Having used this regularization, new scheme has been proposed in [18, 19, 20]. We assume here
that the component analog of that scheme can be constructed. The regularization by higher
derivatives provides strong suppression of ultraviolet divergences by introducing additional terms
with higher degrees of covariant derivatives acting on Yang–Mills tensor into the classical action,
which are suppressed by appropriate degrees of the regularization scale Λ. In addition to this,
it is necessary to introduce a modification of the Pauli–Villars regularization to guarantee the
convergence of the one-loop diagrams [3]. This scheme does not break gauge invariance beyond
one loop level. Moreover, it has been suggested in [3] that such modification by Pauli–Villars
terms to remove one-loop infinities is gauge invariant by construction. To regularize the fermion
cycles, the usual Pauli–Villars regularization can be used. However, this approach, when applied
to explicit examples, is known to yield incorrect results in Landau gauge [21]. A number of
suggestions have been put forward to treat this problem [22, 23]. As was shown in [23], the
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang–Mills Theory 3
contradiction, noticed in [21] is related to the singular character of Landau gauge. In all other
covariant gauges the method works, and to include also the Landau gauge one has to add
one more Pauli–Villars field to get the correct result [22, 23]. We analyze the theory in the
regularization by the dimensional reduction in a parallel way [15].
At one-loop level the part associated with the divergence of Lcc term must be invariant itself
under the ST identity since the second term in identity (2) is finite in the limit of removing
regularization Λ → ∞ [4]. According to Ref. [4], this results in the following integral equation
for the part of the correlator Lcc corresponding to the superficial divergence ∼ ln p2
Λ2 :∫
dxΓΛ(y′, x, z′)ΓΛ(x, y, z) =
∫
dx ΓΛ(y′, y, x)ΓΛ(x, z, z′)
=
∫
dx ΓΛ(y′, x, z)ΓΛ(x, z′, y), (3)
where ΓΛ(x, y, z) is this scale(Λ−) dependent part of the most general parametrization Γ(x, y, z)
of the correlator Lcc,
Γ ∼
∫
dx dy dz Γ(x, y, z)fabcLa(x)cb(y)cc(z). (4)
Here fabc is the group structure constant. The only solution to the integral equation (3) is [4]
ΓΛ(x, y, z) =
∫
dx′ Gc(x′ − x) G−1
c (x′ − y)G−1
c (x′ − z). (5)
The subscript Λ means scale-dependent part of the correlator. As can be seen, all scale-
dependence of this correlator is concentrated in the dressing function. The complete correlator
Lcc at one loop level can be then written as∫
dx dy dz Γ(x, y, z)
i
2
f bcaLa(x)cb(y)cc(z)
=
∫
dx′ dy′ dz′ dx dy dz Γ̃(x′, y′, z′)Gc(x′ − x)G−1
c (y′ − y)
×G−1
c (z′ − z)
i
2
f bcaLa(x)cb(y)cc(z).
Here Γ̃(x′, y′, z′) is scale-independent kernel of Lcc correlator1.
We absorb this dressing function Gc into the corresponding re-definition of the fields L and
c, and then divide the ghost propagator into two parts, one of which is related to the dressing
function of the ghost field Gc, and we call another the dressing function of the gluon field GA.
The effective field K and the antighost field b get opposite re-definition by integrating with the
dressing function G−1
A [4]. The important point here is covariance of the part of the ST identity
without gauge fixing term with respect to such redefinitions [4]. In terms of the dressed effective
fields, we have a useful relation which is a consequence of the main ST identity (1) and can be
obtained by differentiating the main ST identity two times with respect to c̃ and one time with
respect to b̃. This resulting identity is
〈Ãmb̃c̃〉 × 〈K̃mc̃〉+ 〈L̃c̃c̃〉 × 〈b̃c̃〉 = 0. (6)
Again, this identity is written in a schematic way. However, a new important point appears
here. Namely, since the two-point proper functions in terms of dressed effective fields are trivial
1In [4] we conjectured, on the basis of equations (3) and (4), that the complete Γ(x, y, z) has the structure (5);
this would correspond to Γ̃(x′, y′, z′) ∝ δ(x′ − y′)δ(x′ − z′).
4 G. Cvetic, I. Kondrashuk and I. Schmidt
tree level two-point proper functions, the divergences of 〈Ãmb̃c̃〉 and 〈L̃c̃c̃〉 coincide. Since
〈L̃c̃c̃〉 is scale-(or Λ-) independent, the 〈Ãmb̃c̃〉 is scale-independent also. Concerning the gluon
propagator, one part of the divergence is in the dressing function GA, and the rest of divergence
would be absorbed in the re-definition of the gauge coupling constant. This last divergence is
absent in N = 4 theories. All the other correlators are solved by the ST identity in terms of
the dressed effective fields and their kernels are finite (do not possess divergence in the limit of
removing regularization) and scale-independent.
Infinite parts of the dressing functions will be one loop counterterms corresponding to the
re-definition of the fields. Then we can repeat this procedure at two loop level and so on, up
to any order in loop number. Indeed, re-definition by multiplication of the fields of the tree
level action results in re-defining external legs of proper correlators in comparison with the
unrenormalized theory. This property was used by Bogoliubov and Shirkov [6] in the derivation
of renormalization group equations. Re-defining the effective fields by dressing functions does
not bring new aspects in this sense. Indeed, re-defining variables of the path integral by dressing
will result in the dressing of the external legs of the proper correlators. By proper correlators we
mean kernels of Γ, that is, these kernels are one particle irreducible diagrams. We have already
used this re-definition in the dressing functions in the L̃c̃c̃ correlator. Thus, new superficial
divergences will have to satisfy the integral equation (3) at two loop order too in terms of
effective fields L̃c̃c̃, which are effective fields dressed by one loop dressing function Gc. We then
repeat this re-definition in the same procedure in each order of the perturbation theory as we
did in the previous paragraph at one loop level.
Until now the pure gauge sector has been considered. Fermions are necessary for providing
supersymmetry. Consider vertex kcφ at one loop level. The superficial divergence of this vertex
is cancelled by the divergence of the vertex Lcc. This means that in the part of the correlator
kcφ corresponding to superficial divergence ∼ ln p2
Λ2 this divergence can be absorbed into the
dressing functions in the following way:∫
dx dx1 dy1 dy2 Gφ(x− x1) G−1
c (x− y1)G−1
φ (x− y2)k(x1)c(y1)φ(y2). (7)
Note that the dressing functions for the fermions and scalars G−1
φ (x) are not fixed yet. We set
them equal to halves of the two point matter functions. The remaining vertices are restored in
the unique way because the ST identity works.
This theory has intrinsic on-shell infrared divergences, like those canceled by bremsstrahlung
of soft gluons (such a cancellation happens on shell). To regularize these divergences we can
introduce mass parameter µ [25]. Such a trick breaks the ST identity by terms dependent on µ.
At the end the dependence on µ will disappear in physical matrix elements. We mean by the
physical matrix element a connected diagram on-shell contribution to amplitudes of particles. In
comparision, the dimensional reduction regulates both the ultraviolet and the infrared divegen-
ces. On the infrared side, this is better than to introduce small mass parameter µ, the reason
being that in a non-Abelian gauge theory the limit for infrared regulator going to zero is in
general singular. Infrared divergences cancel in the physical matrix elements in dimensional
reduction too. However, one can think that in the effective action (off shell) they could be
present since cancellation of the infrared divergences happens (on shell) between proper and one
particle reducible graphs [25]. Below we will indicate that off shell these infrared divergences do
not exist at all in the position space2.
In principle, infrared divergences in the effective action represent an outstanding problem
that is not treated in the present work in necessary details. We show in this paragraph that
in the regularization by the dimensional reduction this problem does not appear. It is enough
2We treat the theory in the position space. Infrared divergences are absent off shell also in the momentum
space by the same reasons.
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang–Mills Theory 5
L
c c
Figure 1. One-loop contribution to the Lcc vertex. The wavy lines represent the gluons, the straight
line are for the ghosts.
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� �
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�
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�
�
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�
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�
�
�
�
�
L
L (d)
(a)
(c)
(b)
ccc
c c c c
c
L
L
Figure 2. Two-loop diagrams for the Lcc vertex. The wavy lines represent the gluons, the straight
lines the ghosts. The black disc in (c) is for one-loop contribution in the renormalization of the vector
propagator from scalar, spinor and ghost fields.
to show this for the Lcc correlator, because other correlators can be expressed in terms of that
one by ST identity, if we work in terms of dressed effective fields. The one-loop contribution
shown in Fig. 1 (the only diagram that can be drawn) is apparently convergent in the Landau
gauge. The point is that the derivatives can be integrated out of the graph due to the property
of transversality of the gluon propagator in the Landau gauge that immediately makes this
graph convergent in the ultraviolet region, but it is safe in the infrared. Note that in other
gauges this correlator remains divergent in the ultraviolet, and its scale dependence is contained
in the dressing function Gc. The two-loop diagrams (planar) are drawn below in Fig. 2. The
first two diagrams are apparently convergent in ultraviolet in Landau gauge since all subgraphs
are convergent. This is due to the property of transversality that again allows to integrate
out the derivatives. The scale dependence in the diagram (c) cancels the scale dependence of
diagram (d) because of N = 4 supersymmetry (absence of the gauge coupling renormalization).
Infrared region is also not dangerous since even in Landau gauge the gluon propagator is safe in
the infrared region.
However, no invariant regularization scheme is known for supersymmetric theories, due to
the well-known γ5-problem: on general grounds finite renormalizations have to be recursively
performed order by order in the loop expansion in order to preserve the relevant functional iden-
6 G. Cvetic, I. Kondrashuk and I. Schmidt
tities [26]. In this paragraph we discuss how one can treat this problem by using the technique
described in this paper. The point is that the vertex Lcc is always convergent superficially in
the Landau gauge. In N = 4 super-Yang–Mills theory ultraviolet divergences in the subgraphs
of the Lcc vertex should cancel each other at the end. The insertion of the operator of the
conformal anomaly into vacuum expectation values of operators of gluonic fields at different
points in spacetime is proportional to the beta function of the gauge coupling [27]. Due to
the algebra of the four-dimensional supersymmetry the beta function should be zero [14]. Al-
gebra of the supersymmetry operators in the Hilbert space created by dressed fields can be
considered as four-dimensional in Lorentz indices as well as in spinor indices since the limit
ε → 0 is non-singular at one-loop order, two-loop order and higher orders as we have seen in the
previous paragraphs. Thus, we can consider each correlator as pure four-dimensional, solving
order-by-order the problem of dimensional discrepancy of convolutions in Lorentz and spinor
indices.
The behavior of the theory in the IR region is not spoiled by the higher derivative regu-
larization too. This is clear from the structure of the gluon propagator (in Landau gauge, for
example) [3]:
Dab
µν = δab
[
−
(
gµν −
kµkν
k2
)
1
k2 + k6/Λ4
]
.
Sixth degree of momentum in the denominator improves significantly ultraviolet behavior but
in the infrared it is negligible in comparison with second degree of momentum. The infrared
divergence appears when we en force put the on-shell condition p2
i = 0, where pi are external
momenta. In general, infrared region is not dangerous off shell in component formulation in
Wess–Zumino gauge when we regularize the theory by higher derivatives.
In such way we come to our main conclusion in this paper. Namely, N = 4 supersymmetric
theory has scale-independent effective action in terms of the dressed effective fields. All the
dependence on the dimensionful parameter of ultraviolet regularization remains in the dressing
functions only. This is in correspondence with direct calculation of anomalous dimensions and
beta function in N = 4 theory [13]. At one-loop level, kernels of this scale-independent theory
are in general dilogarithms in momentum space. These dilogarithms are Fourier transforms of
the kernels in position space as given below. For example, the correlator of the dressed effective
fields L̃, c̃, and c̃ at one loop level in any SU(N) gauge theory has, among others, the following
contribution:
〈L̃a(x)c̃b(y)c̃c(z)〉 ∼ g2N
1
((z − y)2)2(x− y)2(z − x)2
fabc, (8)
where the dressed effective fields are made of undressed effective fields convoluted to the dressing
functions. The latter are unspecified but they are parts of the two point proper Green functions.
The terms of the type (8) can be obtained, for example, by calculating the one-loop Lcc Green
function in the Landau gauge (where Gc(x) = δ(4)(x)) and then using repeatedly the identity
1
(2π)4
∫
d4k
e−ikx
k2 + iε
= lim
η→+0
i
4π2
1
[(|x0| − iη)2 − x2]
.
By the ST identity the correlators (8) are related to the vertex KAc which in its turn is related
to the tree gluon vertex AAA. The relation of these vertices is dictated by the ST identity and
can be explicitly verified. Thus, the contribution similar to (8) can be found in the proper
correlator of the three dressed gluons at one-loop level 〈Ãa
µ(x)Ãb
ν(y)Ãc
λ(z)〉. At the same time,
in N = 4 supersymmetry we do not need to make additional renormalization in two point gluon
function to absorb the rest of infinities from it into renormalization of the gauge coupling, since
the β function is zero.
On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang–Mills Theory 7
We found that the dressed effective fields are the actual variables of the effective action.
The effective action is to be written in terms of these dressed effective fields. In general, in
non-supersymmetric gauge theory like QCD the dependence on UV regularization scale will be
present inside the correlators of the dressed effective fields because it is necessary to remove
the dependence on this scale by renormalization of the gauge coupling constant. In N = 4
supersymmetric theory such a renormalization does not take place. Thus, the kernels for the
dressed fields do not depend on scale. This might make possible an analysis of these kernels by
tools of conformal field theory in all orders of perturbation theory.
We have shown in this paper that such a scale independent structure of correlators is a direct
consequence of the Slavnov–Taylor identity and it is encoded in the Lcc correlator of the dressed
effective fields. In general, by solving step-by-step the ST identity it is possible to reproduce
structure of all n-gluonic proper correlators in terms of dressed effective fields. Here the question
how to define the concept of scattering may arise. The knowledge of correlators is not sufficient
to define a scattering matrix. Indeed, it follows by the above construction that these correlators
of dressed mean fields in N = 4 supersymmetry do not have dependence on any mass parameter,
or, stated otherwise, the theory in terms of dressed mean fields is conformally invariant. It is
known that S-matrix in conformal field theory cannot be constructed, since we do not have any
dimensional parameter like mass or scale to define scattering concepts like typical scattering
length, or size of a meson and so on. The only observables in this theory are correlators of the
gauge invariant operators and their anomalous dimensions. However, on shell, when we go to the
amplitudes, the scale appears due to infrared on shell divergences, so that scattering concepts
can be introduced in a traditional way [28].
To conclude, in N = 4 supersymmetric Yang–Mills theory there are two types of scale-
independent correlators. They are, first, correlators of BPS gauge invariant operators which can
be found from AdS/CFT correspondence and, second, correlators of the dressed mean fields, for
example, dressed gluons in Landau gauge, that have been considered in this paper. Correlators
of the gauge invariant operators are surely gauge independent. Non-BPS operators are scale-
dependent and their anomalous dimensions are related to the infrared singularities of gluon
amplitudes that are on-shell correlators in the momentum space [28]. As to correlators of the
dressed mean fields, their gauge dependence is an open question. We make a conjecture that
all gauge dependence is contained in the dressing functions of the dressed mean fields, and that
the kernels of the dressed mean fields are gauge invariant.
Acknowledgements
The work of I.K. was supported by Ministry of Education (Chile) under grant Mecesup FSM9901
and by DGIP UTFSM, by Fondecyt (Chile) grant #1040368, and by Departamento de Inves-
tigación de Universidadad del Bio-Bio, Chillan. The work of G.C. and I.S. was supported by
Fondecyt (Chile) grants #1010094 and #1030355, respectively. I.K. is grateful to Boris Kope-
liovich for the discussions of infrared divergences.
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