arXiv:hep-th/0501077v2 3 Feb 2005
Operator mixing in N = 4 SYM:
The Konishi anomaly revisited
B. Eden∗, C. Jarczak∗∗, E. Sokatchev∗∗and Ya. S. Stanev∗∗∗
∗Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut,
Am M¨ uhlenberg 1, D-14476 Golm, Germany
∗∗Laboratoire d’Annecy-le-Vieux de Physique Th´ eorique LAPTH,
B.P. 110, F-74941 Annecy-le-Vieux, France1
∗∗∗Dipartimento di Fisica, Universit‘a di Roma “Tor Vergata”
I.N.F.N. - Sezione di Roma “Tor Vergata”
Via della Ricerca Scientifica, 00133 Roma, Italy
In the context of the superconformal N = 4 SYM theory the Konishi anomaly can
be viewed as the descendant K10 of the Konishi multiplet in the 10 of SU(4), carrying
the anomalous dimension of the multiplet. Another descendant O10with the same quan-
tum numbers, but this time without anomalous dimension, is obtained from the protected
half-BPS operator O20′ (the stress-tensor multiplet). Both K10and O10are renormalized
mixtures of the same two bare operators, one trilinear (coming from the superpotential),
the other bilinear (the so-called “quantum Konishi anomaly”). Only the operator K10is
allowed to appear in the right-hand side of the Konishi anomaly equation, the protected
one O10does not match the conformal properties of the left-hand side. Thus, in a super-
conformal renormalization scheme the separation into “classical” and “quantum” anomaly
terms is not possible, and the question whether the Konishi anomaly is one-loop exact is
out of context. The same treatment applies to the operators of the BMN family, for which
no analogy with the traditional axial anomaly exists. We illustrate our abstract analysis
of this mixing problem by an explicit calculation of the mixing matrix at level g4(“two
loops”) in the supersymmetric dimensional reduction scheme.
1UMR 5108 associ´ ee ` a l’Universit´ e de Savoie
Many years ago it has been realized  that the kinetic term of the N = 1 chiral matter
superfields Φ, viewed as a gauge invariant composite operator (usually called the “Konishi
operator” K), satisfies an “anomalous” conservation condition,
where W is the chiral superpotential, V is the N = 1 gauge superfield and Wαis its field
strength. The first term B in the right-hand side of (1) is obtained by applying the field
equations, so it is of classical origin. The second term F is of purely quantum origin and
is referred to as the “quantum Konishi anomaly”. Its coefficient has been obtained by a
one-loop perturbative calculation.
In the free theory (no potential, no coupling to the gauge field) eq.(1) defines a linear
N = 1 multiplet,¯D2K = 0. In particular, this implies that the axial vector component
K = ... +¯θσµθkµ(x) + ... is conserved, ∂µkµ = 0. This vector is sometimes called the
“Konishi current”. It should be pointed out that the conservation of the free vector does
not reflect any symmetry of the interacting theory with a non-vanishing superpotential, as
indicated by the classical term in (1).1Further, the quantum term is often interpreted as
an analog of the standard Adler-Bell-Jackiw axial anomaly. This analogy has been pushed
even further in [2, 3], where it is claimed that the Konishi anomaly satisfies an Adler-
Bardeen theorem, i.e. its coefficient does not receive any quantum corrections beyond one
loop. This claim is substantiated by explicit two-loop perturbative calculations in [2, 3],
but in the rather special context of N = 1 supersymmetric quantum electrodynamics (no
matter self-interaction). The renormalization properties of the Konishi current have also
been discussed in [4, 5], but still without matter self-interaction. Later on, a more general
statement about the one-loop exactness of the Konishi anomaly, this time for non-Abelian
theories, appeared in . More recently, the same subject was discussed in  in relation
to the chiral ring in supersymmetric gauge theories.
The question about the Konishi anomaly becomes particularly interesting in the con-
text of the maximally supersymmetric N = 4 super-Yang-Mills theory (SYM). In this the-
ory the triplet of N = 1 matter superfields ΦI(I = 1,2,3 is an SU(3) index) are in
the adjoint representations of an SU(N) gauge group and have the special superpoten-
tial W = (g/3)ǫIJKTr(ΦIΦJΦK). The N = 4 SYM theory is known to be finite (i.e., its β
function vanishes). Consequently, this is a superconformal theory in four dimensions. In this
context the operator K can be viewed as a gauge invariant composite operator which gives
rise to an entire “long” N = 4 superconformal multiplet, the so-called Konishi multiplet. It
is the simplest example of an operator in the N = 4 SYM theory having anomalous dimen-
sion.2It should be stressed that the Konishi operator is just the first member of the infinite
1Note, however, that without the superpotential the kinetic term of the matter Lagrangian has an extra
U(1) symmetry and kµ can be viewed as the corresponding axial current.
2The anomalous dimension of the Konishi multiplet has been computed at one (level g2) and two (level
g4) loops through OPE analysis of the four-point function of stress-tensor multiplets -; recently, its
three-loop value has been first predicted  and then obtained by direct calculations [13, 14].
family of the so-called “BMN operators” . For instance, the primary state of the dimen-
sion three BMN operator is in the 6 of SU(4); in the N = 1 formulation with residual R
symmetry SU(3)×U(1) it is given by the superfield KI
In the free theory we find¯D2KI
6/3= 0, just like in the Konishi case. In the interacting
6/3obeys an “anomalous” equation similar to (1). The same is true for all the
higher-dimensional BMN operators. However, the superspace condition¯D2K = 0 implies
the conservation of an axial current component only for the bilinear Konishi operator of
dimension two. For this reason the traditional approach to the Konishi anomaly based
on the analogy with the axial anomaly cannot be generalized to the higher BMN opera-
tors. Another special property of the Konishi operator which is sometimes exploited in
the literature is the fact that the B term in (1) coincides with the superpotential of the
N = 4 theory4and the F term with the N = 1 SYM Lagrangian. Again, this does not
generalize to the higher BMN operators. The development of a universal approach to the
Konishi anomaly and to its BMN counterparts, exploiting the superconformal properties of
the N = 4 theory, is one of the main motivations for the present work. Another reason for
it is to clarify some of the ideas of the method for calculation of the anomalous dimensions
of BMN operators proposed in  and further elaborated in .
Before describing our approach, we should recall some basic but important facts about
the renormalization of composite operators (see, e.g. ). In the quantum theory the oper-
ator equation (1) should be understood as a linear relation among renormalized operators,
[¯D2K]R= a(g)[B]R+ b(g)[F]R. (2)
Here [¯D2K]R= ZK¯D2K is the derivative of the renormalized Konishi operator. The latter
is the only scalar singlet gauge invariant operator of dimension two in the SYM theory,
therefore it undergoes multiplicative renormalization [K]R = ZKK with some divergent
factor ZK; the derivative¯D2K in (1) inherits the same renormalization factor. The operators
in the right-hand side of eq. (2) are the properly renormalized versions of the two terms
in the right-hand side of eq. (1). They are in general mixtures of the bare ones, [B]R=
ZBBB+ZBFF +ZBK¯D2K (and similarly for [F]R), where the Zs form a matrix of a priori
divergent renormalization factors. Finally, a(g) and b(g) are finite coefficients whose value
depends on the normalization of the operators, i.e. on the subtraction scheme. The standard
quantum field theory prescription is that the form of the renormalized operators and the
coefficients in (2) should be determined through insertions of the composite operators into
Green’s functions of elementary fields. In practice, already at two loops this procedure
involves rather heavy calculations. To the best of our knowledge, such explicit calculations
have been carried out in a simplified version of the model (without the superpotential term
B) in Refs. [2, 3, 4, 5] using different regularization schemes. The results can be summarized
as follows: if a(g) = 0 (no superpotential), then b(g) is equal to its one-loop value. The
latter statement is the analog of the Adler-Bardeen theorem for this case. In the past it has
been pointed out that the Adler-Bardeen theorem can be viewed as a statement about the
3Here and in what follows the notation 6/3 indicates the SU(4) representation and its SU(3) projection.
4This point may be a source of confusion. It is known that the chiral superpotential term in the action
d4xd2θ W(Φ(x,θ)) is subject to a non-renormalization theorem. This by no means implies that the chiral
operator W(Φ(x,θ)) (i.e. the term in the Lagrangian) is protected. Indeed, a simple one-loop calculation
of its two-point function shows that it is logarithmically divergent.
existence of a scheme in which the anomaly coefficient is one-loop exact (see, e.g., [18, 19]
and especially  where a detailed treatment of the axial anomaly up to two loops in the
dimensional regularization scheme is given). Although the full renormalization procedure in
the presence of the superpotential has not been explicitly worked out beyond one loop, it is
generally assumed that there are no major differences and that the F term in (2) can always
be interpreted as the analog of the axial anomaly subject to the Adler-Bardeen theorem.
The main point we want to make in this paper is that the picture radically changes in
the very special case of the N = 4 SYM theory. Superconformal invariance imposes addi-
tional restrictions on the operator relation (2). Indeed, in the left-hand side we have an
operator with well-defined conformal properties, in particular, with the anomalous dimen-
sion of the Konishi multiplet. So, the renormalized operators appearing in the right-hand
side of eq.(2) must match these conformal properties. We show that there exists only one
such operator, and it is the renormalized version [B]Rof the “classical anomaly” term B.
It can be identified with the SU(3) singlet projection K10/1of a particular superconformal
descendant K10of the Konishi multiplet in the 10 of the R symmetry group SU(4). It
has naive dimension three but as a quantum operator it acquires the anomalous dimension
of the Konishi multiplet. On the contrary, the renormalized version [F]Rof the “quantum
anomaly” term F turns out to be the singlet projection O10/1= F − 4B of the descendant
O10of the so-called stress-tensor multiplet O20′ which has “protected” (canonical) dimen-
sion. Our conclusion is that in the N = 4 case the “anomaly” equation (2) is truncated,
≡ K10/1. (3)
In it there simply is no room for the “quantum anomaly” term [F]R, due to the mismatch
of the conformal properties. To put it differently, the bare F term has been absorbed into
the definition of the renormalized operator mixture K10/1with a coefficient ZF which is
in fact a divergent renormalization factor beyond one loop (such factors are related to the
so-called “matrix elements” of the operator mixture). Exactly the same picture applies to
the BMN operator of dimension three K6.
The idea to interpret the Konishi anomaly as a superconformal descendant of the Konishi
multiplet in the framework of the N = 4 SYM theory was proposed in [21, 22] and was used
for a practical calculation of anomalous dimensions in [16, 14]. The starting point there
is the protected (also called “short”, or half-BPS, or CPO) N = 4 SYM stress-tensor
supermultiplet. Its lowest (primary) component O20′ is a scalar of dimension two in the
20′of the R symmetry group SU(4), and the top-spin descendant of the multiplet is the
conserved stress tensor. Unlike the “long” Konishi multiplet, the short multiplet O20′ is
protected from quantum corrections, and hence has no anomalous dimension.5As shown in
, applying two (non-linear on-shell) N = 4 supersymmetrygenerators to the ground state
O20′, one can construct another member (a superconformal “descendant”) of this protected
multiplet, O10. It is a scalar of dimension three in the 10 of the R symmetry group SU(4).
This descendant is realized as a linear combination of two composite operators, a trilinear
(B10) and a bilinear (F10) ones. In the N = 1 formulation of the theory, restricting B10, F10
5The “protectedness” of the supermultiplet O20′ can be explained by the presence of the conserved stress
tensor among its components. However, the absence of quantum corrections to the two- and three-point
functions of a whole class of BPS operators is a more general phenomenon not related to any conservation
law (for reviews see [23, 24, 25]). The absence of renormalization of the two-point functions of half-BPS
operators was confirmed by explicit perturbative calculations at levels g2and g4in .
to the SU(3) singlet projection, one obtains the terms B and F which appear in the right-
hand side of “anomaly” equation (1), but now they form a different linear combination
O10/1= −1/2(F − 4B). This combination is expected to be protected, i.e. to keep its
canonical dimension. Further, one could attempt to generate a similar scalar descendant K10
of the Konishi multiplet by applying two on-shell N = 4 supersymmetry transformations to
the operator K. In N = 1 superfield language, the SU(3) singlet projection K10/1should
be precisely the right-hand side of eq.(1). However, this naive attempt fails – one only sees
the “classical anomaly” term B in (1), but completely misses the “quantum anomaly” F.
The argument of  goes on to say that the correct form (1) of K10/1can be determined by
requiring orthogonality with O10/1, in the sense that the two-point function ?¯K10/1O10/1?
must vanish. Indeed, the crucial difference between the two descendants K10and O10is that
in the quantum theory the former acquires an anomalous dimension (that of the Konishi
multiplet), while the latter is protected, hence the two operators must be orthogonal. In
other words, we are dealing with a typical operator mixing problem with the additional
requirements that the diagonalized operators must be eigenstates of the superconformal
In this way one can indeed discover the “missing” F term in (1), at least at one loop.
The question we want to address in this paper is what happens beyond one loop. We explic-
itly resolve the operator mixing described above up to level g4(“two loops”) in perturbation
theory. We first give a general description of the expected form of the “pure” supercon-
formal states. Then we verify it by an explicit graph calculation, using manifestly N = 1
supersymmetric Feynman rules and working in the supersymmetric dimensional reduction
scheme (SSDR) . We find that the protected combination O10is not renormalized at
all, it remains in its classical form. However, the Konishi descendant K10changes its form
at every loop level. At one loop we rediscover the correct coefficient of the F term in (1),
but already at two loops this coefficient becomes a divergent renormalization factor. The
treatment of the BMN operator K6follows exactly the same lines.
On the technical side, we can profit from the superconformal properties of the N = 4
theory to further simplify the problem. Generically, [B]Rand [F]Rare mixtures of three
bare operators with the same quantum numbers, B, F and¯D2K. However, remembering
that eq.(1) is the singlet SU(3) projection of a 10 of SU(4), we can switch over to its
projection in the 6 of SU(3). The advantage is that in this channel we can obtain the
protected descendant O10/6directly from a suitable projection of the primary operator
through superspace differentiation, O10/6= D2O20′/6. Then we can use the classical field
equation to find D2O20′/6= −1/2(F6− 4B6). Here we argue that this naive operator
relation remains non-renormalized, unlike that for the long Konishi multiplet. This allows
us to eliminate one of the operators, e.g. F6. Then the Konishi descendant in this channel
is obtained as a mixture of the remaining two, K10/6= ZKB6+ ZOD2O20′/6.
instead of determining the renormalization factors through insertions into Green’s functions
of elementary fields, we do so by diagonalizing the two-point functions of the descendants.
6Note that in the N = 1 formulation of the N = 4 theory the projection K10/6 cannot be obtained
directly from the singlet Konishi operator K through N = 1 superspace differentiation (or, equivalently,
through N = 1 supersymmetry transformations). Similarly, the projection O10/1cannot be obtained from
the primary O20′ since it does not have a singlet SU(3) projection.
According to the superconformal picture, we expect to find7
(x2)3+γ+ θ terms;
(x2)3+ θ terms;
where γ(g) is the anomalous dimension of the Konishi multiplet. This gives us a set of
equations for determining the renormalization factors. Knowing that the primaries K and
O20′ (and hence their superspace derivatives) are orthogonal allows us to carry out the
diagonalization in the most efficient way. Once we have found the form of the renormalized
operator mixtures K10/6and O10/6, we perform an SU(4) rotation back to the singlet SU(3)
channel. This gives us the correct form of K10/1and O10/1. In particular, we see that the
bare operator¯D2K does not mix with B and F.
The paper is organized as follows. In Section 2 we give the detailed definitions of the
descendants K10and O10(and of their analogs for the BMN operator K6), restricting them
to their most convenient SU(3) × U(1) projections. We also formulate the diagonalization
problem. In Section 3 we discuss the renormalization of conformal operators in dimen-
sional regularization. We explain how the anomalous dimension determines the poles in
the renormalization factors, first in the case of a single operator (no mixing). When two
operators are allowed to mix, the structure of their renormalization factors is considerably
more complicated, as we show on the example of K10and O10. Section 4 is devoted to the
graph calculation at level g4.
2 The Konishi operator K, the BMN operator K6, the half-
BPS operators O20′ and O50and their descendants
In this paper we study the two simplest operators from the BMN family in N = 4 SYM
theory, the (naive) dimension two Konishi operator K and the dimension three operator
K6and their superconformal descendants K10and K45, respectively. Descendants with the
same quantum numbers (apart from the anomalous dimension) can also be obtained from
the half-BPS operator O20′ and O50. In this section we give their description in the N = 1
superspace formulation of the N = 4 theory.
2.1The Konishi operator K
The Konishi operator is just the kinetic term of the N = 1 (anti)chiral scalar matter su-
perfields ΦI(¯ΦI), I = 1,2,3 (see the Appendix for the complete N = 4 SYM action):
K = −1
It is a singlet of the R symmetry group SU(4) of the N = 4 SYM theory, and consequently
of the residual SU(3) × U(1) ∈ SU(4), where SU(3) rotates the indices I and U(1) gives
7Notice that these equations become exact only when the dimensional regulator is set to zero, i.e. when
conformal invariance is restored.
the matter superfields R charge, 2/3 for Φ and −2/3 for¯Φ in units in which the R charge
of θ is 1. In the quantum theory this operator is known to develop anomalous dimension
The operator K (5) is an N = 1 superfield, and as such it has a number of components.
For instance, acting on it with two spinor derivatives¯D2, we obtain a scalar of canonical
dimension 3 and of R charge 2. Now, we can use the N = 1 matter classical field equation8
Here the subscript 10/1 means that the operator is an SU(3) singlet projection of a 10 of
SU(4) (see below).
Since the work of  we have known that in the quantum theory this equation must be
corrected by an “anomaly” term:
32π2Tr(WαWα) ≡ Kone−loop
where Wαis the (chiral) N = 1 SYM field strength. The coefficient of the new term has
been computed at one loop (order g2).
As mentioned in the Introduction, an alternative way to view eq.(8) is as an operator
mixing problem. The right-hand side of (8) contains two operators made out of bosonic
(fermionic) superfields, hence the notation B (F),9
6ǫIJKTr?ΦI[ΦJ,ΦK]?,F = Tr(WαWα) ,(9)
having the same quantum numbers (spin 0, dimension 3, R charge 2, SU(3) singlets). In
the quantum theory such operators start mixing. We recall that N = 4 SYM is a conformal
theory, so the way to resolve this mixing is to find “pure states” of the dilatation operator, i.e.
mixtures which have a well-defined conformal anomalous dimension. Thus, the combination
which one should put in the right-hand side of (8) must be such that its anomalous
dimension is that of the Konishi operator K. It is customary to call the operator K10/1a
(superconformal) descendants of the Konishi operator (or simply a member of the Konishi
superconformal multiplet). So, equation (8) identifies two objects, the superfield component
¯D2K with the descendant K10/1. We insist on this difference between component and
descendant: the former is obtained by simple differentiation of the superfield, the later
through use of the field equations. It is well known that in quantum field theory the naive
use of the classical field equations may lead to incorrect results, and the Konishi “anomaly”
is a good example for this. Therefore, the question what is the operator realization of a
particular descendant of a superconformal multiplet must be answered through quantum
calculations, by resolving the corresponding mixing problem.
8The squared derivatives denote D2= −1
9We find it both natural and convenient to include the factor g accompanying the non-Abelian commu-
tator [ΦJ,ΦK] into the definition of the operator B. In particular, this allows us to have a perturbative
expansion in even powers of g.
4¯D˙ α¯D˙ α.
The full N = 4 Konishi multiplet has a scalar descendant K10of dimension 3 in the 10
of SU(4). The N = 1 descendant K10/1that we have derived so far is the SU(3) × U(1)
projection (1,2) in the decomposition 10 → (1,2) + (3,2/3) + (6,−2/3). The easiest
way to see this10is to imagine the generalization of the term F in (9). The N = 1 SYM
multiplet Wα= λα(x) + ... includes one of the four gluinos λi
N = 4 multiplet. The SU(4) covariant counterpart of (the first component of) F then is
β, which indeed forms a 10 of SU(4). In the N = 1 formulation the
other three gluinos are contained in the matter sector, DαΦI= λI
of F (9), we can study a different projection of the 10, for example,
α(i = 1,2,3,4) of the full
α(x) + .... Thus, instead
This is an operator in the (6,−2/3) of SU(3) × U(1). Similarly, the operator B in (9) is
part of a 10 of SU(4); its counterpart
4ǫIKLTr?ΦJ[¯ΦK,¯ΦL]?+ (I ↔ J)
6. Together they can form a mixture which
is in the same SU(3)×U(1) representation as FIJ
is the counterpart of K10/1(8):
6 + ZFFIJ
We have denoted the mixing coefficients in (12) ZB,ZF in anticipation of their nature of
renormalization factors (see Section 3). In Section 4.1 we will show that at the lowest
order (g2) of perturbation theory they have the same values as in the mixture (8), ZB= 1,
SU(4) multiplet K10, and in the N = 4 theory the SU(4) symmetry should be exact at the
quantum level. The quantum calculations at order g2are very simple, no matter whether
we study the projections K10/1or K10/6of K10, but at the next level g4it is much more
convenient to work with K10/6.
We remark that the complete N = 4 descendant K10 of the Konishi multiplet in-
volves a third projection (3,2/3), i.e. the scalar operator KI
directly through the (anomalous) use of the N = 1 field equations.
Finally, we recall that in the quantum theory the Konishi multiplet acquires an anoma-
lous dimension.11This means that its renormalized two-point function has the form
32π2. This is a consequence of the fact that K10/1and K10/6belong to the same
?. Among the three projections K10/1, KI
10/6of K10, only
K10/1is a descendant of K in the restricted N = 1 sense, i.e. only it can be obtained
(x2)2+γ+ θ terms,(13)
where γ(g2) = γ1g2+ γ2g4+ ... is given by a perturbative expansion. The descendants
K10/1and K10/6are supposed to belong to the same superconformal multiplet, so they
must have the same anomalous dimension γ.
10A manifestly SU(4) covariant description is given in [21, 22, 14].
11The terms “anomalous dimension” and “Konishi anomaly” should not be confused. The former is a
general property of all long (unprotected) superconformal multiplets in the N = 4 theory. The latter, as we
argue here, evokes an analogy with the standard axial anomaly which is somewhat misleading.
2.2The half-BPS operator O20′
The two operators F6 (10) and B6 (11) can form another mixture orthogonal to K10/6.
This is a descendant of the other N = 4 scalar multiplet of dimension two, the half-BPS
operator O20′. The latter has its primary state (lowest component) in the 20′of SU(4)
whose decomposition under SU(3)×U(1) is 20′→ (6,4/3) +(¯6,−4/3) +(8,0). In terms
of N = 1 superfields the projection (6,4/3) is realized by the chiral superfield
20′/6= COTr(ΦIΦJ), (14)
where COis a normalization constant. The operator O20′ has the remarkable property of
being “protected” from quantum corrections, i.e., its two-point function keeps its tree-level
form (the SU(4) indices are suppressed):
?¯ O20′(1)O20′(2)? =
12)2+ θ terms.
In particular, this means that it has no anomalous dimension. The fact that the projection
O20′/6(14) of the operator O20′ is chiral and also primary (in the sense that it cannot be
obtained from any other operator of lower dimension through use of the field equations)
has given rise to the popular term “chiral primary operator” (CPO).12We prefer to use the
more relevant term “half-BPS operator”. Indeed, the half-BPS multiplet O20′ also has the
SU(3) × U(1) projection (8,0),
which is not chiral but is nevertheless protected.
The chiral superfield (14) is short, the top component in its θ expansion, D2O20′/6(θ =
0), is a scalar of dimension three in the (6,−2/3) of SU(3)×U(1). Just as we did with the
Konishi operator K, we can obtain a descendant of this protected multiplet by using the
classical field equations:
6) ≡ OIJ
where F,B have been defined in (10), (11). Note the important difference between K10/6
and O10/6: The former is a descendant of the Konishi multiplet only in the N = 4 sense,
while the latter is a descendant of the half-BPS multiplet also in the N = 1 sense.
In principle, in the quantum theory the coefficients in (17) may get renormalized,
One of the aims of this paper is to show that in the renormalization scheme we use eq.(17)
remains exact; in Section 4.1 we verify this up to g2in correlation functions with respect
12A superconformal primary operator satisfying a BPS shortening condition must have a fixed, “quantized”
dimension (for reviews see [23, 24]). Whether a given composite operator is primary or not is a subtle question
which can only be fully answered in the quantum theory. Some indications how to recognize “Chiral Primary
Operators” were given in , and a more elaborate criterium was proposed in .
to D2O and F, and up g4with respect to B. In a sense, eq.(17) is natural for an operator
mixture which is protected, i.e. whose two-point function keeps its tree-level form,
?¯ O10/6O10/6? =
(x2)3+ θ terms. (19)
In other words, we would guess that in this case the naive use of the classical field equations is
justified. However, we are not aware of any general field theory criterium which would allow
us to tell when the classical field equations do receive corrections (the Konishi operator and
its descendants are an example) and when they do not.13Therefore we find it necessary
to carry out an explicit quantum calculation in a particular scheme, which confirms our
conjecture that the mixture (17) is not renormalized while the other one, (12), receives
quantum corrections. The fact that eq.(17) remains exact at the quantum level is very
useful, it allows us to eliminate one of the three operators, e.g. F6(see Section 2.3).
2.3Diagonalization of the operators B and F
Let us summarize the discussion so far. In the N = 1 formulation of the N = 4 SYM theory
there exist two gauge invariant scalar composite operators in the (6,−2/3) of SU(3)×U(1),
F6(10) and B6(11). From them we can prepare two independent mixtures. The first is the
descendant K10/6of the Konishi multiplet and as such it must have the same anomalous
dimension γ. In other words, its two-point function should have the form
(x2)3+γ+ θ terms. (20)
This follows from the fact that K10/6and K belong to a superconformal multiplet of oper-
ators which carry different canonical dimensions, but the same anomalous dimension. This
operator is a projection of the 10 of SU(4) which corresponds to the complete N = 4
superdescendant K10of of the Konishi multiplet. Note that within the N = 1 framework
this particular representative of K10cannot be obtained directly from K through the field
equations, unlike the SU(3) singlet K10/1(8). The relationship between K10/6and K10/1
is indirect, it evokes the full N = 4 supersymmetry of the theory which is not manifest
in the N = 1 formulation. The situation changes in the harmonic superspace formulation
with manifest N = 2 supersymmetry . There one can obtain  a direct descendant
of K which involves two of the four gluinos of the N = 4 theory, and thus mixes together
the fermion operators F (9) and F6(10). The same applies to the operators B and B6.
The existence of such a formulation with a larger manifest supersymmetry is an additional
justification of the path we have chosen to follow here. We want to resolve the mixing
problem of the operators F6and B6and then to use the same mixing matrix for F and B.
The other mixture is the superdescendant O10/6of the protected half-BPS operator
O20′/6. Its two-point function is given in (19), i.e. it has no anomalous dimension. Since
both operators K10/6and O10/6are supposed to be pure conformal states of the same
canonical, but different anomalous dimension, they must be orthogonal to each other,
?¯ O10/6K10/6? = 0.(21)
13This may be a scheme-dependent property.
The superdescendant O10/6is related to the supercomponent D2O20′/6via the (quantum)
dynamical equation (18). This fact can be used to make a change of basis in the operator
mixture K10/6.14For example, we can eliminate the operator F6in favor of D2O20′/6and
replace (12) by
K10/6= ZKB6+ ZOD2O20′/6. (22)
The mixing (renormalization) factors ZK, ZO are of course different from those in (12).
The notation ZKfor the factor of B6suggests that it will turn out identical (up to a finite
overall normalization) with the renormalization factor of the primary Konishi operator K
(see Section 3.3). Similarly, the orthogonality condition (21) can be equivalently rewritten
?¯D2¯ O20′/6K10/6? = 0. (23)
Finally, the analog of (19) becomes
?¯D2¯ O20′/6D2O20′/6? =
(x2)3+ θ terms. (24)
It should be stressed that eq.(24) is an obvious consequence (just a derivative) of equation
(15) stating that the operator O20′/6is a member of a protected multiplet. On the other
hand, eq.(19) is a non-trivial condition on the operator mixture O10/6. The difference
comes from the dynamical nature of the relation (18).
In the rest of the paper we shall see that the abstract analysis of the mixing problem,
as well as the actual graph calculations are considerably simplified if we use the form (22)
instead of (12). The reason for this is that while studying the mixture K10/6we can treat
D2O20′/6as a pure state, thus avoiding the simultaneous determination of the two mixtures
2.4The BMN operator K6, the short operator O50and their descendants
In N = 1 notation the primary operator in the long BMN multiplet of dimension 3 is
6/3= Tr(ΦIΦJ¯ΦJ) + Tr(ΦI¯ΦJΦJ)(25)
and it has a higher component¯D2KI
6/3, which may mix with
Exactly as in the case of the Konishi multiplet, the field equation of the antichiral superfield
implies the existence of an operator relation
14Before doing this it is important to make sure that the Z factors in (18) are not divergent. The reason
is that equations like, e.g. the orthogonality condition (21) only hold up to O(ǫ) terms where ǫ is the
dimensional regulator (they only become exact in the limit ǫ → 0). So, dividing by singular Z factors may
create uncontrollable finite contributions (see Section 3 for details). In the case of eq.(18) not only the Z
factors are finite, but they are given by the classical expression (17), as we argue in this paper.
with a finite coefficient function a(g2). The existence of such a linear relation allows us to
6/3from the mixing problem in the 3 of SU(3). It is quite cumbersome to
obtain the finite constant of proportionality a(g2) in a direct calculation. As before, we
sidestep the problem by appealing to SU(4): we assert that the renormalization factors ZB
and ZFare identical in all components of the 45 of SU(4) and fix them by orthogonalization
in the 10 of SU(3). Orthogonalization does not permit to fix the finite constant of pro-
portionality a(g2), which we leave undetermined. Hence we do not work out the anomaly
equation itself. On the other hand, we gain the freedom of choosing ZB in minimal sub-
The appearance of the bare operator¯D2KI
such operator in the 10 of SU(3). Although such an effect was observed e.g. in  it
cannot occur in this situation because of the underlying SU(4) symmetry.
The protected multiplet with a descendant in the 45 of SU(4) is
6/3in [B]Ris not expected, since there is no
50/10= Tr(Φ(IΦJΦK)). (29)
It has a higher component D2O50/10which can mix with
The field equation of the chiral superfield suggests the operator relation
45/10) = 0, (32)
which we have checked by the same means (and to the same order) as for O20′/6, i.e. by
doing the D-algebra for the supergraphs and exploiting partial integration to reduce to a set
of basic x-space integrals. The calculation is nearly the same as the one in Section 4.1; some
new diagrams can be drawn, but they vanish. The superposition of the other supergraphs
can be simplified by exactly the same manipulations as for O20′/6. Once again, to the given
order eq.(32) is true without the need of introducing renormalization factors.
3 Dimensional regularization and anomalous dimension
3.1 General structure of the two-point functions
The quantum calculation we plan to carry out will be done in the scheme of dimensional
regularization (or, more precisely, supersymmetric dimensional reduction (SSDR) , a
scheme which preserves manifest supersymmetry). In dimensional regularization the action
of a (scalar) field takes the form
d4−2ǫx L(φ(x), ˆ g),ˆ g = gµǫ.(33)
Here µ is a mass parameter which allows the coupling g to remain dimensionless. On the
contrary, the fields φ change their dimension. For example, the propagator for a (scalar)
We can say that even the free field acquires a small “anomalous dimension” −ǫ. Of course,
in the limit ǫ → 0 this anomalous dimension disappears.
dimensional regularization scheme.
We are interested in the two-point functions of composite operators. At each level of
perturbation theory they have a general structure which is explained below. Take, for
instance, the two-point function of the bilinear Konishi operator K ∼¯φφ at tree level (g0).
It is described by a one-loop (in the standard, momentum space counting) graph without
interaction vertices. Its x-space expression simply is the square of (34):
This is a peculiarity of the
At the first non-trivial level g2the graphs have two loops and two interaction vertices, etc.
In general, an n-loop two-point function graph for K has the dimensionful factor
?KK?n loop ⇒ (g2µ2ǫ)n−1.(36)
Similarly, for an operator made out of m fields (for K m = 2), whose free (order g0or,
abusing the term, “tree-level”) two-point function has m − 1 loops, we find
?¯ OmOm?n loop ⇒ (g2µ2ǫ)n−m+1. (37)
The important point here is that the perturbative expansion goes in powers of the single
variable ˆ g2= g2µ2ǫ.
Thus, the general structure of an n-loop two-point function graph for the m-linear
?¯ OmOm?n loop
+ ··· +cn−m+1,1
+ cn−m+1,0+ O(ǫ)
Here the dependence on x is determined by the already known dependence on µ and by the
requirement that the two-point function as a whole must keep the “engineering” dimension
m(1 − ǫ) of the operator Om. The poles in the regulator ǫ come from the expansion of the
divergent n-loop integrals of the corresponding graphs. The fact that the leading singularity
in (38) has the same order as the power of g2has to do with the renormalizability of the
operator, see the next subsection.
The conclusion from the above discussion is that the perturbative two-point function of
the naked operator Omcan be viewed as a function of two variables,
(x2)m?¯ OmOm? = (x2)mǫG(g2(x2µ2)ǫ,ǫ). (39)
The dimensionful parameter µ in (39) is not essential for our subsequent analysis and can
be suppressed. If needed, it can easily be restored by simple dimension counting.
3.2Renormalization of a single operator
The N = 4 SYM theory is supposed to be conformal, i.e. it has vanishing β function. This
means that the coupling g is not renormalized. At the same time, composite operators
have inherent divergences which are responsible for their anomalous dimension. To be more
specific, let us consider the Konishi operator K. Its advantage is its low dimension, so it
is a “pure” state, i.e. cannot mix with any other operator in the N = 4 theory. Thus, we
expect that after the multiplicative renormalization of K its two-point function (39) will
take the following form:
(x2)2?[K]R[K]R? = Z2
K(g2,ǫ)(x2)2ǫG(g2x2ǫ,ǫ) = C(g2)(x2)−γ(g2)+ O(ǫ) (40)
(we have dropped µ). Here γ(g2) = γ1g2+ γ2g4+ ... is the anomalous dimension of the
renormalized operator [K]R= ZKK and ZK(g2,ǫ) is a constant renormalization factor. The
rˆ ole of ZKis to remove all the singularities from G, so that the left-hand side of (40) becomes
finite. After that we can take the limit ǫ → 0, and only in this limit we expect to find the
conformal power behaviour (x2)−γ(g2). It is important to realize that the regulator ǫ (or,
equivalently, the presence of the mass parameter µ) breaks conformal invariance, so the
O(ǫ) terms in the right-hand side of (40) form a complicated function of x. It is clear that
the factor ZKis determined up to an overall finite renormalization factor which modifies
the normalization C(g2) of the two-point function (40).
Now, let us take the log of eq.(40):
2lnZK+ lnG = −γ lnx2+ lnC + O(ǫ). (41)
Further, let us differentiate (41) with ∂/∂g2:
∂g2x2ǫlnG = −∂γ
∂g2+ O(ǫ). (42)
Next, let us take the derivative x2∂/∂x2of (41):
∂g2x2ǫlnG = −γ + O(ǫ) ⇒ x2ǫ
∂g2x2ǫlnG = −γ
ǫg2+ O(1). (43)
With the help of (43) we can rewrite (42) as follows:
∂g2+ O(1). (44)
Note that the left-hand side of (44) does not depend on x, so the O(1) terms in the right-
hand side must compensate the lnx2term.
It is clear that the differential equation (44) only determines the pole structure of ZK,
its O(1) part is kept arbitrary. We can use this freedom in order to choose ZKsuch that
ZK(0,ǫ) = 1 and that it contains only singular terms.
subtraction” (MS) renormalization scheme. Then the solution to eq.(44) is
This is the so-called “minimal
ZK(g2,ǫ) = exp
= 1 +γ1
g4+ O(g6) . (45)
Looking at the result (45), we can easily explain the general structure of the naked two-
point function G (38). Indeed, the leading singularity at the level g2kin the expansion of
(45) is ∼ g2kǫ−kγk
1. This term can cancel the analogous pole in the expansion of G provided
that the leading pole at the level g2kin (38) is also of order ǫ−k. So, the form (38) ensures
the renormalizability of the composite operators O.
Now we can restore the µ dependence. In the presence of µ eq.(40) reads
K(g2,ǫ)G(g2(x2µ2)ǫ,ǫ) = C(g2)(x2µ2)−γ(g2)+ O(ǫ).(46)
The conformal renormalized operator [K]Rshould have the two-point function
ǫ→0(x2)2?[K]R[K]R? = C(g2)(x2)−γ(g2), (47)
where γ is identified with the anomalous dimension. This is achieved by absorbing the µ
dependence into the renormalization factor:
ˆZK(g2,ǫ;µ) = µγ(g2)ZK(g2,ǫ). (48)
Finally, let us make the connection with the renormalization group (or Callan-Symanzik)
equation. Remembering that the mass parameter µ can be associated the coupling constant,
ˆ g = gµǫ, we can rewrite eq.(45) as follows:
lnZK(ˆ g2,ǫ) =
dτ ⇒ µ∂
∂µlnZK= γ(ˆ g2). (49)
We can say that this is the renormalization group equation in a conformal theory (i.e., with
vanishing β function). A peculiarity of the dimensional regularization scheme is that the Z
factors depend on the dimensionful “coupling” ˆ g. This dependence can be factored out as
shown in (48), after which the renormalization group equation takes the form (44), where
the derivatives are taken with respect to g2rather than µ.
3.3 Renormalization and mixing in the case of K10/6and D2O20′/6
The above procedure can be adapted to the case of several operators which mix among
themselves. Here we do this in the simplest case of two operators, one of which has anoma-
lous dimension but the other is already a pure superconformal state and is protected, i.e.,
it has vanishing anomalous dimension. The former is the Konishi descendant K10/6in the
form (22), the latter is the component D2O20′/6of the half-BPS operator O20′/6. In Section
2.3 we explained that these operators should satisfy two conditions, (20) and (23) (condition
(24) is a trivial consequence of the fact that the operator O20′/6is protected). Let us now
see how all this works in the quantum theory. We need to know the two-point functions of
the two bare operators B6and D2O20′/6in the mixture (22). From the Feynman rules and
from the discussion in Section 3.1 we can derive the following general structure
=g2(x2)−3+3ǫ[1 + g2(x2)ǫa1(ǫ) + g4(x2)2ǫa2(ǫ) + ...]
g2(x2)−3+3ǫ[b0(ǫ) + g2(x2)ǫb1(ǫ) + ...]
(x2)−3+2ǫ[1 + g2(x2)ǫc1(ǫ) + g4(x2)2ǫc2(ǫ) + ...].
?¯B6D2O20′/6? = ?¯D2¯ O20′/6B6?
Here the coefficients ai(ǫ) and bi(ǫ) involve poles following the general pattern (38). The
coefficients ci(ǫ) ∼ O(ǫ) are such that in the limit ǫ → 0 they give rise to contact terms
in the two-point function of the protected operator . The expansion of ?¯B6B6? and
?¯D2¯O20′/6D2O20′/6? start with unity, which amounts to tree-level normalization. The over-
all factor g2in ?¯B6D2O20′/6? is due to the fact that the first non-trivial graph involves one
chiral matter coupling.
We can organize the above two-point functions into a 2 × 2 matrix:
≡ (x2)−3+2ǫG(g2x2ǫ,ǫ), (51)
where G is the matrix analog of the function in (39) (we drop the dimensionful constant µ).
This time renormalization means to bring this matrix into diagonal form where we could
read off (in the limit ǫ → 0) the anomalous dimension γ(g2) of the Konishi multiplet (we
recall that K10/6is a descendant of K and so must have the same anomalous dimension) and
the vanishing anomalous dimension of the protected operator O10/6. This can be achieved
with the help of a constant singular renormalization (or mixing) matrix:
+ O(ǫ). (52)
As in the case of a single operator, the conformal behavior indicated in the right-hand side
of (52) only becomes exact in the limit ǫ → 0. Indeed, in the presence of the regulator even
the protected operator D2O20′/6has an “anomalous dimension” −2ǫ corresponding to the
regularized form of the tree graph.
Since we already know that O20′/6(and consequently D2O20′/6) is a pure state normal-
ized at unity, we can choose the mixing matrix in the following triangular form:
where we have taken into account the fact that the mixing coefficients are real. We can
derive differential equations for ZKand ZOby repeating the steps which lead to eq.(44).
Since we are now dealing with matrices, instead of taking the log of eq.(52) we directly
differentiate it. The derivative ∂/∂g2gives
(the details of the right-hand side do not matter for us). Further, differentiating with
x2∂/∂x2and dividing by ǫg2we obtain
+ O(1). (55)
We can now use this to rewrite (54) in the form
∂g2Z−1(ZGZ†) + (ZGZ†)(Z†)−1∂Z†
= O(1). (56)
Finally, substituting the initial equation (52) into (56) we find
= O(1). (57)
Solving this matrix differential equation is greatly simplified due to the triangular struc-
ture (53). Inserting it into (57) and dividing by the finite factor CKx2γ, we obtain two
K)ZO= O(1), (59)
where Z′denotes the derivative with respect to g2. Here we need to make the following
comment. The form of the matrices appearing in (57) is not exact. For example, the zeros
in them are in fact O(ǫ) terms. Since the equation involves the inverse of the singular matrix
Z, we should be careful not to make a mistake because of superposition of poles and O(ǫ)
terms. A closer look shows that this does not happen. For instance, the O(ǫ) terms in the
upper right corners of these matrices induce a shift of ZOin (59),
K)(ZO+ O(ǫ)) = O(1). (60)
Our aim here is to determine just the divergent and the finite parts of ZO, so such a shift
does not matter for us.
Equation (58) is identical with (44) and has the same solution (45), provided we fix the
finite normalization by the same requirement that ZKstart with 1 and that it contain only
poles (minimal subtraction); in this case the right-hand side of eq.(58) vanishes. Replacing
in (59) we obtain an equation for ZO:
At first sight it seems that this equation is the same as (58), so it should have the same
solution. In fact, this is not true because of the presence of finite terms in ZO. Indeed, we
already know that ZOdoes not start with 1 like ZK, but with g2. This follows form the form
of the one-loop “Konishi anomaly” (8), after switching from K10/1to K10/6. For g = 0 both
the B and the F terms in (8) are absent, and the Konishi multiplet becomes “semishort”,
¯D2K = 0. The numerical factor N/32π2in (8) is the first term in the expansion of ZO.
Thus, if the boundary condition for ZKwas ZK(g = 0) = 1, for ZOit is ZO(g = 0) = 0.
Further, when solving for ZKwe declared that its finite part was 1, which can always be
achieved by a suitable overall finite renormalization of K10/6. Once this has been done, we
do not have any freedom left to fix the finite part of ZO, so we must keep it arbitrary in
the right-hand side of (61).
So, let us write down
ZO(g2,ǫ) = ZOs(g2,ǫ) + ZOf(g2) + O(ǫ) (62)
where ZOsis the singular in ǫ part, while ZOfis the finite part. Then eq.(61) becomes
2g2ǫZOf+ O(1) (63)
with the boundary condition ZOs(0,ǫ) = 0. It is easy to check that the solution of (63) has
the following form:
ZOs(g2,ǫ) = −ZK(g2)
In this equation ZKis the renormalization factor of the Konishi operator (45), and the finite
part ZOf(g2) = 0 + ω0g2+ ω1g4+ ω2g6+ ... should be determined by an explicit graph
calculation. Let us also give the first few terms in the perturbative expansion of ZO:
ZO(g2,ǫ) = ω0g2+
g6+ O(g8). (65)
In conclusion we can say that we have resolved the mixing of the operators B6 and
D2O20′/6. We have found two orthogonal pure states, K10/6and D2O20′/6. The entire
information about the mixing coefficients is encoded in two quantities, the anomalous di-
mension of the Konishi multiplet γ and the finite part of ZO. The former has a well-defined
meaning in a conformal theory and is guaranteed to be scheme independent. The latter,
however, does not have any scheme-independent meaning at all. We might pull out the
common renormalization factor ZKand then identify the so-called “Konishi anomaly” with
the remaining factor −?g2
order, this factor becomes singular. Attempting to identify the “Konishi anomaly” with the
finite part of ZOdoes not make much more sense, since it obviously is scheme dependent.
Thus, we are forced to conclude that, in the context of the superconformal N = 4 theory,
the so-called “Konishi anomaly” is nothing but a (divergent) mixing coefficient which may
receive corrections at every loop order.
K)′(τ) in front of D2O20′/6. At the lowest order g2it
reproduces the “one-loop Konishi anomaly”, i.e., the numerical factor in (8). Beyond this
3.4 The superdescendant O10/6
While discussing the mixing problem above, we only dealt with the superdescendant K10/6
of the Konishi multiplet and kept that of the half-BPS operator O20′ implicit, in the form
of the supercomponent D2O20′/6rather than the explicit mixture
D2O20′/6= ZFF6+ ZBB6= O10/6. (66)
This is the generalization of the naive (classical) expression (17), taking into account the
possibility of quantum corrections. In this section we analyze the two-point functions of
O10/6with D2O20′/6and K10/6up to order g4and find some restrictions on the renormal-
ization factors ZF,ZB.
Let us start with the correlator
?¯D2¯O20′/6O10/6? = ZF?¯D2¯O20′/6F6?+ZB?¯D2¯ O20′/6B6? = ?¯D2¯ O20′/6D2O20′/6? = 1+O(ǫ),
which is a component/descendant of the protected correlator ?¯ O20′/6O20′/6? = 1 + O(ǫ).
The two-point functions ?¯D2¯ O20′/6B6? (recall (50)) and ?¯D2¯ O20′/6F6? have the following
=g2(x2)−3+3ǫ[b0(ǫ) + g2(x2)ǫb1(ǫ) + ...]
(x2)−3+2ǫ[1 + g2(x2)ǫf1(ǫ) + g4(x2)2ǫf2(ǫ) + ...].
Here the coefficients have a pole structure according to the general pattern (38):
b0(ǫ) = b00+ b0,−1ǫ + O(ǫ2)
+ b10+ O(ǫ)
+ f10+ O(ǫ)
+ f20+ O(ǫ) (70)
which follows from analyzing the corresponding graphs. In order to compensate these poles
and to assure the standard normalization at unity in (67), we introduce the renormalization
factors ZF,ZBof the form
ZF= 1 + g2?ϕ11
ZB= 1 + g2
Now, let us put all this in the relation (67) and expand in g2. The condition we find at
The expansion in ǫ gives three types of terms, ǫ−1, ǫ0lnx2and ǫ0, which are not present in
the right-hand side of (72) and so must vanish. This implies
+ ϕ10+ [f1(ǫ) + b0(ǫ)](x2)ǫ= O(ǫ). (72)
ϕ11= f11= 0,ϕ10+ f10+ b00= 0.(73)
At level g4, using (73), we find
b0(ǫ)(x2)ǫ+ [f2(ǫ) + b1(ǫ)](x2)2ǫ= O(ǫ). (74)
From the vanishing of the terms ǫ−2and ǫ−1lnx2we obtain
ϕ22= f22= 0. (75)
Similarly, the terms ǫ−1and ǫ0lnx2yield
ϕ21= f21+ b11= −1
The conclusion from the analysis of the protected two-point function (67) is that the
leading poles in ZF must be absent, whereas the subleading pole at level g4is related to
the level g2pole in ZB. The latter can be determined form the orthogonality condition
?¯K10/6O10/6? = O(ǫ).(77)
Substituting the definitions (22) and (66) into (77) and using (67), we can write down
ZKZB?¯B6B6? + ZKZF?¯B6F6? + ZO(1 + O(ǫ)) = O(ǫ). (78)
We want to study the consequences of this condition up to level g4. The relevant two-
point functions are
=g2(x2)−3+3ǫ[1 + g2(x2)ǫa1(ǫ) + ...]
g2(x2)−3+3ǫ[d0(ǫ) + g2(x2)ǫd1(ǫ) + ...],
where a1(ǫ) = a11ǫ−1+a10+O(ǫ), d0(ǫ) = d00+d0,−1ǫ+O(ǫ2) and d1(ǫ) = d11ǫ−1+d10+O(ǫ).
Further, the renormalization factors ZK,ZOwere discussed in Section 3.3, where we found
the following (recall (45) and (65)):
ZK= 1 + g2γ1
2ǫ+ O(g4),ZO= g2ω0+ g4?ω0γ1
+ O(g6). (81)
Let us now substitute all this into the orthogonality condition (78) and expand up to
level g4. We note that the O(ǫ) term multiplying ZO is of order g2, and so is the non-
singular part of ZO, so this term has no effect at the levels we are interested in. Thus, at
level g2we find
ω0+ [1 + d0(ǫ)](x2)ǫ= O(ǫ) ⇒ ω0+ d00+ 1 = 0. (82)
The condition at level g4is
(x2)ǫ+ [a1(ǫ) + d1(ǫ)](x2)2ǫ= O(ǫ).
Expanding in ǫ, we find two conditions following from the vanishing of the terms ǫ−1and
ǫ0lnx2. The first of them is a11+ d11=ω0γ1
4, while the other determines the coefficient
This result, together with (76) and (75), imply the vanishing of all the singular terms in
both renormalization factors ZF,ZB.
Thus, the conclusion from our investigation is that the renormalization factors in (66)
can only contain finite terms:
ZF= 1 + g2ϕ10+ g4ϕ20+ O(g6),
ZB= 1 + g2β10+ O(g4). (85)
These (possible) finite corrections to the naive descendant O10/6(17) obtained through use
of the classical field equations can be determined from the ǫ0parts of the above relations.
One way to find β10is to examine the ǫ0terms in (83), but it is more efficient to go back
to the starting point, the expected operator relation (66) and apply it on the two-point
functions with the operator B6:
?¯D2¯ O20′/6B6? = ZF?¯F6B6? + ZB?¯B6B6?. (86)
Substituting in it the expansions (68), (79) and (80) and working out the ǫ0terms, we find
b10= d10+ a10+ ϕ10d00+ β10,
which determines β10. The other finite correction, ϕ20, can be obtained from the ǫ0terms
ϕ20+ ϕ10f10+ f20+ β10b00+ b10= 0.
4 Explicit calculations up to order g4
4.1 The operator identity between D2O,F,B
For simplicity the SU(4) and SU(3) labels are omitted throughout this section; we discuss
the mixing problem in the 6 of SU(3). Recall that the naive application of the equation of
motion of the chiral superfield implies the operator identity:
2(F − 4B) = 0.(89)
We argue that this equation holds for the bare operators, i.e. that there is no need to
introduce renormalization factors. The analysis of the last section excluded infinite renor-
malization factors, here we show that in the SSDR scheme even finite renormalization effects
are absent. Further, in this scheme the right-hand side of equation (89) is exactly zero (i.e.,
not only in the limit ǫ → 0).
In the Appendix we comment on the calculation of the correlation functions employed
here. We remark that the considerations in this subsection are based on the use of N = 1
superpropagators and partial integration, while the actual regularization is of relevance
only with respect to one detail, namely eq.(123) which is certainly true in dimensional
In order to establish the operator relation (89), we have to verify the vanishing of the
two-point functions of the linear combination on the left-hand side of eq.(89) with each of
the operators F, D2O and B. To lowest order we find
?F¯F?g0 = −2?D2O¯F?g0 = 4?D2O¯D2¯ O?g0
= −16(N2− 1)(∂µ
with the matter propagator
Hence equation (89) is satisfied.
At order g2we first consider the two-point function with¯B. We have
?B¯B?g2 = 2g2N(N2− 1)Π3
(recall that the definition (11) of B includes a factor of g), while
?D2O¯B?g2 = 4g2N(N2− 1)Π3
The latter correlator involves only one diagram which is rational and finite. Since F and B
do not couple at this order, the coefficient of B in (89) is confirmed.
Next, we test eq.(89) with respect to F at order g2:
?F¯F?g2 = −2?D2O¯F?g2 = −32g2N(N2− 1)?Π3
Here we have introduced the double-box integral
f(1,2;1,2) = c5
It is convergent and of dimension two, hence it is proportional to 1/x2
makes it into a contact term. The same contact part occurs in 4?D2O¯D2¯ O?g2, which,
however, has no finite part. Equation (89) remains true also in a two-point function with
respect to D2O because there is a second finite contribution from ?B¯D2¯ O?g2 (see (93)).
Let us finally consider a test of (89) with respect to B at order g4. We find
12. A box operator
= 12g4N2(N2− 1)?J − 2Π12h + Π12?1f?,
6g4N2(N2− 1)?−2Π12h + Π12?1f?,
= 24g4N2(N2− 1)(−J), (96)
where f = f(1,2;1,2) and
Once again, up to O(g4) equation (89) is seen to be identically satisfied. In showing this
we did not rely on the explicit evaluation of any divergent x-space integral.
We conjecture that the operator relation is true to all orders in perturbation theory
without receiving any modification of its coefficients. In conclusion,
−2D2O = F − 4B ≡ [F]R
i.e. we view the combination F − 4B as the correctly renormalised operator F. In the
following we choose to drop the bare operator F from our mixing problem in favour of
Let us introduce the shorthand
=g2OB1+ g4OB2+ ... ,
g0OO0+ g2OO1+ ... .
We are looking for a combination
K10 = ZKB + ZOD2O
orthogonal to D2O up to O(g4). The relevant part of the Z-factors is
= 1 + g2m11
i.e. we take ZKto have minimal subtraction form. We impose
?K10¯D2¯ O? = O(ǫ). (103)
At order g2this yields one equation from the elimination of the finite part, whereas at order
g4we generate three equations relating to the elimination of the simple pole, the simple
logarithm and the finite part. The system is non-singular so that the four constants in the
Z-factors are uniquely determined.
The integrals J,h,f can be calculated in p-space by the Mincer package . We Fourier-
transform back to x-space to find
−16(1 − 2ǫ + ǫ2)(x2)(2ǫ),
−8M(0 + 12ζ(3)ǫ + ...)(x2)(3ǫ),
+3M2(1/ǫ + 3 + (8 + 12ζ(3))ǫ + ...)(x2)(4ǫ),
where an overall factor
has been omitted and M = N/(4π2). For the details on our convention we refer to .
The expressions above are valid in the improved MS scheme, in which fractional powers of
π, the Euler constant γ and ζ(2) are absorbed into the mass scale.
The result of the orthogonalization is:
The explicit calculation is needed, of course, to furnish the principle pieces of information,
i.e. the anomalous dimensions and the finite mixing coefficients ωi. Nevertheless, in the
context of this paper we rather view it as a confirmation of the abstract analysis presented
in the earlier sections, and thus as a demonstration of superconformal invariance. The
consistency conditions on the coefficients in the various correlators are in fact fulfilled in
calculations in SSDR, notably
so that ZOis definitely singular. This had already been derived in the preceding sections.
We profited a lot from enlightening discussions with D. Anselmi, G. Arutyunov, M. Grisaru,
J. Iliopoulos, H. Osborn, G.C. Rossi, K. Stelle, R. Stora, A. Vainshtein. ES is grateful to
the theory group of the Dipartimento di Fisica, Universit` a di Roma “Tor Vergata” and
to the Albert-Einstein Institut, Potsdam for the warm hospitality extended to him. The
work of E.S. was supported in part by the INTAS contract 00-00254 and by the MIUR-
COFIN contract 2003-023852. The work of Ya.S.S. was supported in part by the INFN,
by the MIUR-COFIN contract 2003-023852, by the EU contracts MRTN-CT-2004-503369
and MRTN-CT-2004-512194, by the INTAS contract 03-51-6346 and by the NATO grant
Appendix: Some comments on the graphs
A complete set of conventions with respect to spinor algebra, Fourier transform and regular-
ization by supersymmetric dimensional reduction (SSDR) is given in . For convenience
we quote a few formulae. The classical action of N = 4 SYM in terms of N = 1 fields is
d4xLd2θ Tr(WαWα) +
d4xLd2θ ǫIJKTr(ΦI[ΦJ,ΦK]) + c.c.
where all superfields are in the adjoint of SU(Nc) and the generators are normalised such
that Tr(TaTb) = δab. The matter propagator is
?ΦI(1)¯ΦJ(2)? = −
The hatting indicates supersymmetrization by the exponential shift
= exp?i[θ1σµ¯θ1+ θ2σµ¯θ2− 2θ1σµ¯θ2]∂µ
The Yang-Mills propagator in Feynman gauge is given by
?V (1)V (2)? = +θ2
The SSDR prescription means to send
while we absorb a corresponding change of the normalization of the propagators into the
mass scale (in the terminology of  we use ˜ µ in x-space instead of µ itself). At the same
time, the Grassmann variables are treated as two-component spinors, as in four dimensions.
The most efficient way of reducing supergraphs to ordinary integrals is to expand in the
θ’s the exponential shifts from the matter propagators so as to saturate the Grassmann inte-
grations. After this we are left with some differential operator acting under the x-integrals.
For some simple illustrations of the method see once again .
In the following we calculate with the operators in the 6 of SU(3), as defined in Section
2. Most conveniently one restricts to the 11 projection
B = gTr?Φ1[¯Φ2,¯Φ3]?,F = Tr?∇αΦ1∇αΦ1?.(114)
Let us start with ?D2O¯B?g4, defined by the graphs in Figure 1. The combinatorics produces
8g4N2(N2− 1)?−G8+ G7+ G6+1
Here the extra minus sign from the YM propagator has been included.
2G4− G3− G2+ G1
Figure 1. Graphs G1...8for the correlator ?O20/6¯B?g4.
By means of partial integration in x-space and shrinking of some lines onto which box
operators act, one may show
G8 = G7+ G6− G5− G4− G3− G2, (116)
whereas graph G1vanishes. We conclude:
?D2O¯B?g4 = 12g4N2(N2− 1)?G4+ G5
The supergraphs G4and G5yield the underlying x-space integrals
=J − 2Π12h,
(the functions J,h,f were defined in Section 4.1).
The calculation of ?F¯B?g4 is very similar. The part of this correlator arising from the
two-fermion term in F can be read off from the calculation of ?O¯B?g4: we simply drop
those parts of the graphs in which both derivatives at point 1 act on the same line. In the
YM sector only graphs G8and G7survive; the cancellation of diagram G7against a part
of diagram G8is not affected. We find
−G8+ G7 = 8g4N2(N2− 1)?−2J + 2Π12h?.
Next, there are six graphs in which a connection line emanates from F. Five of these vanish
by θ-counting, the remaining diagram is displayed in Figure 2. It contributes
G9 = 8g4N2(N2− 1)(−2Π12h), (120)
which exactly compensates the h term in (119). Finally, the matter sector graph G4comes
without the h-pieces that it had before. On adding up we obtain:
?F¯B?g4 = 24g4N2(N2− 1)(−J). (121)
Figure 2. The graph G9in the correlator ?F¯B?g4.
The graphs contributing to ?D2O¯F?g2 are given in Figure 3. Graph GIIis zero, while
the other two graphs can rather straightforwardly be summed into
?D2O¯F?g2 = 16g2N(N2− 1)?Π3
To show this it is enough to pull the two box operators in the last equation through the
integrations in f and to distribute the derivatives on the various propagators, although we
rely on the fact that
?15?25f(1,2;1,2) = 0(123)
in dimensional regularization. The last equation can be checked in p-space from topology
T1 in Mincer with numerator p2
in correlators with respect to D2O, F or g4with respect to B) in all regularization schemes
that are compatible with partial integration and in which eq.(123) holds.
2. Our operator equation (89) is actually valid (up to g2
The evaluation of ?F¯F?g2 is not essentially different, while the O(g2) two-point function
of D2O with itself is most conveniently done by considering ?O¯O?g2 and applying the
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