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arXiv:hep-th/0501077v2 3 Feb 2005
AEI-2004-127
LAPTH-1084/05
ROM2F/2004/34
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
Abstract
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
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1Introduction
Many years ago it has been realized [1] 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,
¯D2K ≡¯D2Tr?¯ΦegVΦ?
= Tr
?
Φ∂W(Φ)
∂Φ
g2
32π2F ,
?
+
g2
32π2Tr(WαWα)
≡
B +
(1)
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 [6]. More recently, the same subject was discussed in [7] 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 [8]-[11]; recently, its
three-loop value has been first predicted [12] and then obtained by direct calculations [13, 14].
1
Page 3
family of the so-called “BMN operators” [15]. 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
theory KI
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 [16] and further elaborated in [14].
Before describing our approach, we should recall some basic but important facts about
the renormalization of composite operators (see, e.g. [17]). In the quantum theory the oper-
ator equation (1) should be understood as a linear relation among renormalized operators,
6/3= Tr(ΦIΦJ¯ΦJ)+Tr(ΦI¯ΦJΦJ).3
[¯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.
?
2
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existence of a scheme in which the anomaly coefficient is one-loop exact (see, e.g., [18, 19]
and especially [20] 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,
[¯D2K]N=4
R
= a(g)[B]N=4
R
≡ 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
[21], 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 [26].
3
Page 5
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 [21] 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
dilatation operator.
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) [27]. 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.
6Further,
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.
4
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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
(119)
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)
D
12
3
D
5
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
12−1
2?1?2f?. (122)
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.
1p2
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
derivatives afterwards.
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I
D
D
II
D
D
D
III
D
Figure 3. The correlator ?O20/6¯F?g2.
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