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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

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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].

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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