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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].

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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.

?

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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.

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According to the superconformal picture, we expect to find7

?¯K10/6K10/6?

=

CK

(x2)3+γ+ θ terms;

1

(x2)3+ θ terms;

0,

?¯ O10/6O10/6?

=

?¯K10/6O10/6?

= (4)

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

3Tr?egV¯ΦIe−gVΦI?

(5)

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.

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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

[8]-[14].

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

¯D2?egV¯ΦIe−gV?= −g

¯D2K =g

6ǫIJKTr?ΦI[ΦJ,ΦK]?≡ Kclassical

2ǫIJK[ΦJ,ΦK] (6)

to obtain

10/1

. (7)

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 [1] we have known that in the quantum theory this equation must be

corrected by an “anomaly” term:

¯D2K =g

6ǫIJKTr?ΦI[ΦJ,ΦK]?+g2N

32π2Tr(WαWα) ≡ Kone−loop

10/1

(8)

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

B =g

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

Kquantum

10/1

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.

4DαDα,¯D2= −1

4¯D˙ α¯D˙ α.

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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

λi

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

αǫαβλj

β= λj

αǫαβλi

α(x) + .... Thus, instead

FIJ

6

= FJI

6

= CFTr?∇αΦI∇αΦJ?,

∇αΦI= egV?Dα

?e−gVΦIegV??e−gV. (10)

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

BIJ

6 =g

4ǫIKLTr?ΦJ[¯ΦK,¯ΦL]?+ (I ↔ J)

6. Together they can form a mixture which

(11)

is in the same SU(3)×U(1) representation as FIJ

is the counterpart of K10/1(8):

KIJ

10/6= ZBBIJ

6 + ZFFIJ

6 .(12)

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,

ZF =

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

g2N

32π2Tr?∇αΦIWα

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

g2N

32π2. This is a consequence of the fact that K10/1and K10/6belong to the same

10/3=

10/3and KIJ

g

2Tr?ΦJ[¯ΦJ,ΦI]?+

?. 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

?KK? =

1

(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.

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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

OIJ

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)? =

1

12)2+ θ terms.

(x2

(15)

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),

?O20′/8

?

I

J= Tr?egV¯ΦIe−gVΦJ?−1

3δJ

ITr?egV¯ΦKe−gVΦK?, (16)

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:

D2OIJ

20′/6= −1

2(FIJ

6

− 4BIJ

6) ≡ OIJ

10/6,(17)

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,

D2OIJ

20′/6= ZFFIJ

6

+ ZBBIJ

6

≡ OIJ

10/6. (18)

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 [28], and a more elaborate criterium was proposed in [29].

8

Page 10

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? =

1

(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

?¯K10/6K10/6? =

CK

(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 [30]. There one can obtain [32] 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.

9

Page 11

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

as

?¯D2¯ O20′/6K10/6? = 0. (23)

Finally, the analog of (19) becomes

?¯D2¯ O20′/6D2O20′/6? =

1

(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

K10/6and O10/6.

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

KI

6/3= Tr(ΦIΦJ¯ΦJ) + Tr(ΦI¯ΦJΦJ)(25)

and it has a higher component¯D2KI

6/3, which may mix with

BI

45/3

=

g

4ǫJKLTr(ΦIΦJ[ΦK,ΦL]),

Tr(ΦIWαWα).

(26)

FI

45/3

=(27)

Exactly as in the case of the Konishi multiplet, the field equation of the antichiral superfield

implies the existence of an operator relation

ZK¯D2KI

6/3+ ZBBI

45/3+ ZFFI

45/3= [¯D2KI

6/3]R+ a(g2)[BI]R

(28)

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.

10

Page 12

with a finite coefficient function a(g2). The existence of such a linear relation allows us to

eliminate¯D2KI

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-

traction form.

The appearance of the bare operator¯D2KI

such operator in the 10 of SU(3). Although such an effect was observed e.g. in [20] 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

O(IJK)

50/10= Tr(Φ(IΦJΦK)). (29)

It has a higher component D2O50/10which can mix with

B(IJK)

45/10

=

g

2ǫLM(ITr(ΦJΦK)[¯ΦL,¯ΦM]),

Tr(Φ(I∇αΦJ∇αΦK)).

(30)

F(IJK)

45/10

= (31)

The field equation of the chiral superfield suggests the operator relation

O(IJK)

50/10+3

2(F(IJK)

45/10− 2B(IJK)

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) [27], a

scheme which preserves manifest supersymmetry). In dimensional regularization the action

of a (scalar) field takes the form

S =

?

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)

field becomes

?¯φφ? ∼

(x2)1−ǫ.

1

(34)

11

Page 13

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

?KK?g0 ∼

1

(x2)2−2ǫ. (35)

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

operator Omis

?¯ OmOm?n loop

=

?cn−m+1,n−m+1

1

(x2)m(1−ǫ)[g2(x2µ2)ǫ]n−m+1.

ǫn−m+1

+ ··· +cn−m+1,1

ǫ

+ cn−m+1,0+ O(ǫ)

?

×

(38)

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

12

Page 14

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:

2∂

∂g2lnZK+ x2ǫ

∂

∂g2x2ǫlnG = −∂γ

∂g2lnx2+∂C

∂g2+ O(ǫ). (42)

Next, let us take the derivative x2∂/∂x2of (41):

ǫg2x2ǫ

∂

∂g2x2ǫlnG = −γ + O(ǫ) ⇒ x2ǫ

∂

∂g2x2ǫlnG = −γ

ǫg2+ O(1). (43)

With the help of (43) we can rewrite (42) as follows:

2

∂

∂g2lnZK−

γ

ǫg2= −∂γ

∂g2lnx2+∂C

∂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

2ǫ

?g2

0

γ(τ)

τ

dτ

?

= 1 +γ1

2ǫg2+

?γ2

1

8ǫ2+γ2

4ǫ

?

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

13

Page 15

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

Z2

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

lim

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) =

1

2ǫ

?ˆ g2

0

γ(τ)

τ

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

?¯B6B6?

=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?

?¯D2¯ O20′/6D2O20′/6?

=(50)

=

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 [26]. The expansion of ?¯B6B6? and

14

Page 16

?¯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:

?

?¯B6B6?

?¯D2¯ O20′/6B6?

?¯B6D2O20′/6?

?¯D2¯ O20′/6D2O20′/6?

?

≡ (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:

Z(g2,ǫ)G(g2x2ǫ,ǫ)Z†(g2,ǫ) =

?CK(g2)(x2)−γ(g2)

0

10

?

+ 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:

Z =

?ZK

ZO

10

?

,Z†= ZT=

?ZK

0

1ZO

?

, (53)

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

∂Z

∂g2GZ†+ ZG∂Z†

∂g2+ x2ǫZ

∂G

∂g2x2ǫZ†= O(1)(54)

(the details of the right-hand side do not matter for us). Further, differentiating with

x2∂/∂x2and dividing by ǫg2we obtain

x2ǫZ

∂G

∂g2x2ǫZ†= −γ

g2ǫ

?CK(x2)−γ(g2)

0

00

?

+ O(1). (55)

We can now use this to rewrite (54) in the form

∂Z

∂g2Z−1(ZGZ†) + (ZGZ†)(Z†)−1∂Z†

∂g2−

γ

g2ǫ

?CKx2γ

0

00

?

= O(1). (56)

Finally, substituting the initial equation (52) into (56) we find

∂Z

∂g2Z−1

?CKx2γ

0

10

?

+

?CKx2γ

0

10

?

(Z†)−1∂Z†

∂g2+

γ

g2ǫ

?CKx2γ

0

00

?

= O(1). (57)

15

Page 17

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

differential equations:

Z′

KZ−1

K−

γ

2g2ǫ= O(1);

KZ−1

(58)

Z′

O− (Z′

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),

Z′

O− (Z′

KZ−1

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

Z′

K

in (59) we obtain an equation for ZO:

γ

2g2ǫZO= O(1).

KZ−1

Z′

O−

(61)

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

Z′

Os−γ(g2)

2g2ǫZOs=γ(g2)

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)

?g2

0

ZOf(τ)(Z−1

K)′(τ)dτ .(64)

16

Page 18

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+

?

ω1+ω0γ1

4ǫ

?

g4+

?

ω2+ω1γ1

6ǫ

+ω0γ2

6ǫ

+ω0γ2

24ǫ2

1

?

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.

0ZOf(τ)(Z−1

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),

(67)

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

perturbative expansions:

?¯D2¯ O20′/6B6?

?¯D2¯O20′/6F6?

=g2(x2)−3+3ǫ[b0(ǫ) + g2(x2)ǫb1(ǫ) + ...]

(x2)−3+2ǫ[1 + g2(x2)ǫf1(ǫ) + g4(x2)2ǫf2(ǫ) + ...].

(68)

= (69)

17

Page 19

Here the coefficients have a pole structure according to the general pattern (38):

b0(ǫ) = b00+ b0,−1ǫ + O(ǫ2)

b1(ǫ) =b11

ǫ

f1(ǫ) =f11

ǫ

f2(ǫ) =f22

+ b10+ O(ǫ)

+ f10+ O(ǫ)

ǫ2+f21

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

ǫ

+ ϕ10

?

?

+ g4?ϕ22

+ O(g4).

ǫ2+ϕ21

ǫ

+ ϕ20

?

+ O(g6)

?β11

ǫ

+ β10

(71)

Now, let us put all this in the relation (67) and expand in g2. The condition we find at

level g2is

ϕ11

ǫ

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

ϕ22

ǫ2+ϕ21

ǫ

+ ϕ20+

?β11

ǫ

+ β10

?

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

2β11b00. (76)

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)

18

Page 20

We want to study the consequences of this condition up to level g4. The relevant two-

point functions are

?¯B6B6?

?¯B6F6?

=g2(x2)−3+3ǫ[1 + g2(x2)ǫa1(ǫ) + ...]

g2(x2)−3+3ǫ[d0(ǫ) + g2(x2)ǫd1(ǫ) + ...],

(79)

= (80)

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

4ǫ

+ ω1

?

+ 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

ω0γ1

4ǫ

+ ω1+

?γ1+ 2β11

2ǫ

+ β10+

?γ1

2ǫ+ ϕ10

?

d0(ǫ)

?

(x2)ǫ+ [a1(ǫ) + d1(ǫ)](x2)2ǫ= O(ǫ).

(83)

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

β11= 0.(84)

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

the relation

b10= d10+ a10+ ϕ10d00+ β10,

which determines β10. The other finite correction, ϕ20, can be obtained from the ǫ0terms

in (74):

ϕ20+ ϕ10f10+ f20+ β10b00+ b10= 0.

(87)

(88)

19

Page 21

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:

D2O +1

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

regularization.

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

1Π12)(∂1µΠ12),(90)

with the matter propagator

Π12 =

c0

x2

12

,c0= −

1

4π2. (91)

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

12

(92)

(recall that the definition (11) of B includes a factor of g), while

?D2O¯B?g2 = 4g2N(N2− 1)Π3

12. (93)

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

12−1

2?1?1f(1,2;1,2)?. (94)

f(1,2;1,2) = c5

0

?

d4x5d4x6

15x2

x2

16x2

56x2

25x2

26

. (95)

20

Page 22

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

?D2O¯B?g4

?F¯B?g4

?B¯B?g4

= 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

h=c4

0

?

?

d4x5

x4

d4x3,5,6(∂15∂35∂36∂16)

x2

15x4

25

, (97)

J=c7

0

15x2

35x2

36x2

16x2

25x2

23x2

26

. (98)

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

(99)

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

D2O.

4.2

K10by orthogonalization

Let us introduce the shorthand

?D2O¯B?

?D2O¯D2¯ O?

=g2OB1+ g4OB2+ ... ,

g0OO0+ g2OO1+ ... .

(100)

=

We are looking for a combination

K10 = ZKB + ZOD2O

(101)

orthogonal to D2O up to O(g4). The relevant part of the Z-factors is

ZK

= 1 + g2m11

ǫ

(102)

ZO

=g2n10+ g4?n21

ǫ

+ n20

?

,

i.e. we take ZKto have minimal subtraction form. We impose

?K10¯D2¯ O? = O(ǫ). (103)

21

Page 23

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 [31]. We Fourier-

transform back to x-space to find

OO0

OO1

OB1

OB2

=

−16(1 − 2ǫ + ǫ2)(x2)(2ǫ),

−8M(0 + 12ζ(3)ǫ + ...)(x2)(3ǫ),

−4M(x2)(3ǫ),

+3M2(1/ǫ + 3 + (8 + 12ζ(3))ǫ + ...)(x2)(4ǫ),

(104)

=

=

=

where an overall factor

c1 =

(N2− 1)

(4π2)2x6

12

(105)

has been omitted and M = N/(4π2). For the details on our convention we refer to [14].

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:

m11=3M

2

,n10= −M

4,n21= −3M2

16

,n20= +3M2

16

. (106)

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

m11=γ1

2

(107)

and

n21 =1

2m11n10,(108)

so that ZOis definitely singular. This had already been derived in the preceding sections.

Acknowledgments

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

PST.CLG.978785.

22

Page 24

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 [14]. For convenience

we quote a few formulae. The classical action of N = 4 SYM in terms of N = 1 fields is

SN=4

=

?

1

4

d4xd2θd2¯θ Tr?egV¯ΦIe−gVΦI?

?

(109)

+

d4xLd2θ Tr(WαWα) +

?g

3!

?

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)? = −

δI

J

4π2ˆ x2

12

. (110)

The hatting indicates supersymmetrization by the exponential shift

1

ˆ x2

12

= exp?i[θ1σµ¯θ1+ θ2σµ¯θ2− 2θ1σµ¯θ2]∂µ

The Yang-Mills propagator in Feynman gauge is given by

1

? 1

x2

12

≡ e∆121

x2

12

. (111)

?V (1)V (2)? = +θ2

12¯θ2

4π2x2

12

12

. (112)

The SSDR prescription means to send

1

x2

12

→

1

(x2

12)(1−ǫ)

(113)

while we absorb a corresponding change of the normalization of the propagators into the

mass scale (in the terminology of [14] 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 [14].

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.

2G5+1

2G4− G3− G2+ G1

?. (115)

23

Page 25

5

3

12

1

5

3

1

2

2

1

2

3

5

3

8

123

6

5

4

5

6

3

12

5

12

3

5

6

6

12

3

5

6

7

12

3

5

6

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

?

(117)

The supergraphs G4and G5yield the underlying x-space integrals

G4

G5

=J − 2Π12h,

Π12?1f(1,2;1,2)

(118)

=

(the functions J,h,f were defined in Section 4.1).

24

Page 26

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.

25

Page 27

I

D

D

II

D

D

D

III

D

Figure 3. The correlator ?O20/6¯F?g2.

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