Universal properties of superconformal OPEs for 1 / 2 BPS operators in 3 ≤ D ≤ 6

. We give a general analysis of OPEs of 1 / 2 BPS superﬁeld operators for the D = 3 , 4 , 5 , 6 superconformal algebras OSp (8 / 4 , R ) , P SU (2 , 2 / 4) , F 4 and OSp (8 ∗ / 4) which underlie maximal AdS supergravity in 4 ≤ D + 1 ≤ 7 . The corresponding three-point functions can be formally factorized in a way similar to the decomposition of a generic superconformal UIR into a product of supersingletons. This allows for a simple derivation of branching rules for primary superﬁelds. The operators of protected conformal dimension which may appear in the OPE are classiﬁed and are shown to be either 1 / 2 or 1 / 4 BPS, or semishort. As an application, we discuss the ‘non-renormalization’ of extremal n -point correlators.


Introduction
Maximal AdS supergravities in four, five, six and seven dimensions are dual to superconformal field theories on the world volume of M 2 , D 3 , D 4 /D 8 and M 5 branes, respectively [1]. Although only the D 3 brane dynamics has a perturbative description in the superconformal regime, some general properties of abstract superconformal field theories can be obtained by using the BPS nature of a certain class of superconformal primary operators and the model-independent nature of OPEs (for reviews see, e.g., [2]- [4]).
In the classification of UIRs of superconformal algebras an important role is played by the representations with 'quantized' conformal dimension, since in the quantum field theory 2.3 so-called 'supersingleton' (or massless supermultiplet) [36]- [39]. This description of the threepoint functions results in a unified treatment of all theories where the supersingleton constituents are identified with the microscopic 'degrees of freedom' of the brane world volume dynamics †.
We show that the selection rules or, to put it differently, the 'protection mechanism' for certain channels in the OPE of two 1/2 BPS operators has a very simple origin, which can be illustrated by the following example from ordinary conformal field theory [40,41]. Consider the three-point function of two scalar fields with a rank s symmetric tensor field: (1.1) These fields have conformal dimensions A , B and , respectively. Now, suppose that the scalars are massless ('singletons'), These equations are conformally invariant only if the scalars have the canonical dimension A = B = (D −2)/2. However, this is not all: a direct calculation shows that the condition (1.2) also fixes the dimension of the tensor j {µ 1 ···µs} at its canonical value = s+D −2 and, moreover, forces this tensor to be conserved. Thus, imposing a condition on the operators at points 1 and 2 of the three-point function can have the effect of 'protecting' the operator at point 3.
A similar phenomenon takes place with the three-point functions involving two 1/2 BPS short multiplets. We write them down formally in a factorized form in which the R symmetry quantum numbers at the third point are associated with a BPS factor whereas the spin and the (possibly anomalous) dimension are carried by a singlet factor. Depending on the choice of the R symmetry irrep at point 3, this singlet factor turns out to be either trivial, or of the type (1.1), (1.2) above, or unconstrained. Correspondingly, we find the following selection rules for the third operator: it is either BPS short, or semishort, or unconstrained. Thus, in the first two cases the operator at point 3 is 'protected' while in the third case it is 'unprotected'.
The paper is organized as follows. In section 2 we describe the supersingleton degrees of freedom (brane supercoordinates) for all these theories and their relation to supermultiplet shortening. We also explain how various superconformal UIRs, including different kinds of BPS short and semishort multiplet, can be obtained as composite operators made out of supersingletons. In section 3 we give a unified treatment of the three-point functions involving two 1/2 BPS operators by writing them down in the factorized form described above and deriving the selection rules for the third operator. The results are interpreted in section 4, where particular attention is paid to the occurrence and the role of the semishort protected multiplets. In section 5 we give an application of these results in the form of a general non-renormalization theorem for 'extremal' n-point correlators of 1/2 BPS multiplets.

Standard description of the supersingletons
Supersingletons are massless representations of the D-dimensional superconformal algebra or, equivalently, superfields satisfying conformally covariant massless field equations. It is well 2.4 known [42]- [44] that there exist only a finite number of them in D = 3, 5 while there are infinite sets in D = 4, 6. Here we restrict ourselves to the so-called 1/2 BPS supersingletons, which have been identified with the basic brane degrees of freedom in the context of the AdS/CFT correspondence. These supersingletons are 'ultrashort' supermultiplets with maximal spin s max = 1/2 in D = 3, 5 and s max = 1 in D = 4, 6. Such supermultiplets can be characterized by the quantum numbers of their lowest component: vanishing Lorentz spin, canonical conformal dimension of a massless scalar (D − 2)/2 and R symmetry irrep with Dynkin labels (DL) according to the following list:  (8) triality. Here we restrict ourselves to the OPE of two supersingletons of the same type 8 s . These supersingletons can be described in terms of scalar superfields carrying external R symmetry indices according to (2.1) and satisfying the following on-shell constraints [45]- [49]: Here the indices i, j, k belong to the fundamental representation (or its complex conjugate) of the R symmetry groups, except for the case D = 3, where i is an 8 v index and a, b are 8 s indices; Γ i denote the gamma matrices of SO (8). The symbols (), [] and {} mean symmetrization, antisymmetrization and traceless part, respectively. These constraints eliminate most of the components of the superfields and put the remaining ones on the massless shell.

Supersingletons as 1/2 BPS short superfields
Harmonic superspace allows us to write down all these supersingletons as

2.5
In the case D = 3 the four sets of U (1) charges are denoted by ±(±) [±]{±}; in the other cases it is more convenient to use the individual projections of the indices of the fundamental representation (i = 1, 2 for SU (2) and i = 1, 2, 3, 4 for SU (4) and USp(4)) to label the different states of an R symmetry irrep. These superfields are in general functions of the harmonic variables having infinite expansions on the harmonic coset R/H. The condition which cuts these expansions down to polynomials in the harmonics u is the condition of R symmetry irreducibility. It takes the familiar form of the definition of a HWS: Here D ↑ u denotes the set of raising (step-up or creation) operators of the group R realized in the form of covariant harmonic derivatives on the coset R/H. If one uses a complex parametrization of the coset, conditions (2.4) become covariant Cauchy-Riemann conditions (harmonic analyticity [50]). The combination of Grassmann analyticity (2.3) with irreducibility under the R symmetry group (2.4) is equivalent to the original formulation (2.2) of the supersingletons.
Supersingletons are 'ultrashort' superfields in the sense that their θ expansion contains just a few massless fields. Here we show only the bosonic content of the G-and H-analytic superfields (2.3): (2.5) The massless scalar fields φ belong to the R symmetry irreps listed in (2.1); the on-shell two-form and three-form field strengths F are singlets. Concluding this section we mention that (2.3) is not the only possible realization of the supersingletons as Grassmann analytic superfields. Instead of projecting the external R symmetry indices onto the HWS, we could take any other state and accordingly choose the half of the θs that the superfield depends on, for example (there is no need to do this in the case D = 5). Unlike the superfields (2.3), the new ones are not harmonic analytic (since they are not HWS), i.e. they are not annihilated by all of the raising operators (harmonic derivatives). Instead, they are related to the HWS (2.3) by the action of some of the raising operators: The use of this alternative realization is explained in the next section.

BPS short multiplets as products of supersingletons
An important advantage of the description of the supersingletons as Grassmann and harmonic analytic superfields is the possibility of obtaining new BPS short objects by simply multiplying the basic supersingletons [8]. The reason is that analytic objects form a ring structure, i.e. a set closed under multiplication. Thus, any power [W ] k is automatically Grassmann analytic, i.e. depends on the same half of the Grassmann variables; recall (2.3). Further, since the supersingleton W is the HWS of the R symmetry irrep with DL listed in (2.1), the power [W ] k satisfies the constraint which defines it as the HWS of one of the following irreps: of USp(4).
where in the left column we have indicated the R symmetry DL instead of the U (1) charges. These superfields satisfy the same conditions of BPS shortness (i.e. of Grassmann and harmonic analyticity), but their component content strongly depends on the value of k. In the case of the supersingleton (k = 1) we have seen that the combination of the two conditions puts the superfield on the massless shell. An even stronger constraint is obtained for k = 0: a singlet analytic object can only be a constant, as follows from the obvious property (W ) 0 = 1. However, for k ≥ 2 the constraints become much weaker. In particular, the first component of the superfield, a scalar of dimension k(D − 2)/2, satisfies no constraint whatsoever †. Indeed, if we realize the 1/2 BPS superfield as a composite operator (W ) k for k ≥ 2, we see that the first component is a generic scalar composite made out of the massless scalars φ from (2.5). This crucial distinction among the cases k = 0, 1 and k ≥ 2 is at the origin of the selection rules for the three-point functions which are derived in section 3. Another possibility of obtaining BPS short composite operators is to multiply together two different realizations of the basic supersingleton. For instance, the product of Grassmann analytic superfields of the types (2.3) and (2.6), or of any of their powers, is a superfield which does not depend on 1/4 of the full set of θs. According to the AdS terminology, such operators are called '1/4 BPS short': (2.10) † It should be mentioned that for k = 2 in D = 4, 5, 6 and for k = 2, 3 in D = 3 some of the higher components of this composite superfield are conserved vectors or tensors. The best-known example is the N = 4, D = 4 SYM stress-tensor multiplet which is described by the supersingleton bilinear W 12 W 12 .

2.7
This time, since the factor of the type (2.6) is not harmonic analytic, the product does not automatically define a HWS of a new R symmetry irrep. This can be achieved by imposing further irreducibility conditions. For example, in the case D = 3 the harmonic condition (recall (2.7)) defines the HWS of the irrep [0jp0] of SO (8). Similarly, in the case D = 4 (or D = 6) the condition (4)).
We remark that equation (2.10) does not exhaust the list of composite BPS objects obtained by multiplying various realizations of the basic supersingleton [8]. For example, in D = 3 one could have 1/8 and 3/8, in D = 4 1/8 BPS multiplets, etc. We do not consider them here because they are associated with R symmetry irreps different from those appearing in the OPE of two 1/2 BPS operators (see equation (3.2)).

Semishort multiplets
In what follows the so-called 'semishort' (or 'current-like') multiplets will play an important role. In this section we give a brief summary of the origin of such multiplets as limiting cases or as isolated points in the series of UIRs of the superconformal algebras. We also give their realization as composite operators made out of supersingletons which satisfy some 'current-like' superspace constraints †.
The semishort multiplets are to some extent the analogues of the 'conserved' tensor representations of the ordinary conformal group SO(D, 2). It is well known that the rank s symmetric traceless tensor field j {µ 1 ···µs} (x) of the so-called 'canonical' dimension = s+D −2 forms a reducible but indecomposable representation of the conformal group SO(D, 2) [41]. This means that its divergence ∂ µ 1 j {µ 1 ···µs} transforms covariantly and can be set to zero. The resulting 'transverse' tensor is already irreducible ‡.
The conserved tensors can be viewed as a limiting case of the generic series of UIRs of SO(D, 2). A generic conformal operator carrying conformal dimension and Lorentz spin s can be written down in the following composite form: (2.13) Here j {µ 1 ···µs} is a conserved tensor and φ is a massless scalar ('singleton') field: Note that j {µ 1 ···µs} can itself be represented as a composite operator made out of singletons, for example for s = 1 15) † Note that if the supersingletons carry a colour index N c , under which the composite is a singlet, there are in principle different operators with the same spin and R symmetry quantum numbers [14,21,25,51].
‡ The representation is indecomposable because the 'longitudinal' part cannot be projected out by a local conformal operator.

2.8
where φ is another copy of the singleton. The parameter δ in equation (2.13) can take non-integer values, δ ≥ 0 (for s > 0) or δ ≥ 1 (for s = 0). This accounts for the possible 'anomalous' dimension of the operator O s subject to the unitarity bound ≥ s + D − 2 (for s > 0) or ≥ (D − 2)/2 (for s = 0). From the 'composite' form (2.13) it is clear that the unitarity bound is saturated if δ = 0, s > 0 (no massless scalar appears) or if δ = 1 and s = 0. Thus, the conserved tensor is at the threshold of the continuous series of UIRs represented by the composite operators (2.13).
A similar phenomenon takes place in the classification of the superconformal UIRs. Let us first recall some of the known series of UIRs [52]- [54]. We restrict ourselves to those which can possibly form a three-point function with two 1/2 BPS UIRs. They must carry Lorentz indices corresponding to a symmetric traceless tensor of rank s and R symmetry quantum numbers which are listed in (3.2).

OSp(8/4) (D = 3)
. The Lorentz quantum number (spin) is an integer J = s and we are dealing with SO (8) representations of the type [0a 2 a 3 0]. There exist two series of UIRs: (2.16) The discrete series (B) contains the BPS multiplets.

P SU (2,2/4) (D = 4).
We consider Lorentz spins J 1 = J 2 = s/2 and SU(4) representations of the type [a 1 a 2 a 1 ]. Two of the three existing series of UIRs are relevant in this case: (2.17) The discrete series (C) contains the BPS multiplets.

Series (B) is an 'isolated' series and series (C) contains the BPS multiplets.
OSp(8 * /4) (D = 6). We consider Lorentz spins J 1 = 0, J 2 = s, J 3 = 0 and USp (4) representations of the type [a 1 a 2 ] (with even a 1 ). There exist four series of UIRs: . conformal dimension , spin s and R symmetry DL a i as a formal product of three factors [8]: (2.20) The first factor accounts for the Lorentz spin, the second for the conformal dimension and the third for the R symmetry labels of the composite operator O s . Each of these factors can in turn be viewed as a 'fake composite' operator obtained from the basic supersingletons. Thus, the spin factor has the form of a bilinear composite of dimension s + D − 2, for example for s = 1 and similarly for s > 1 (see, e.g., [46]). Note that these composites satisfy superspace 'conservation conditions' following from the massless superfield equations (2.2). Using spinor notation, they can be written down as follows:  The defining property of the semishort superfields is that they satisfy some superspace constraints obtained as the intersection of the supercurrent constraints (2.22) (or of their analogues for the scalar supercurrent Φ) and the Grassmann analyticity constraints on the BPS factor.  These constraints are significantly weaker in the sense that the corresponding 'current-like' superfield does not contain any conserved tensor components. Without going into the details of the θ expansion, this is quite clear from the factorized form of the 'current-like' operators, which is at least trilinear in the basic supersingletons. The role of the constraints now is to simply eliminate some components from the θ expansion (but not entire projections of θs, as in the BPS case). Thus, the 'current-like' multiplets do not reach the maximal spin of the generic superfield of the same type, and for this reason we call them 'semishort'.

Selection rules for three-point functions involving two 1/2 BPS operators
In this section we address the main subject of the present paper. The OPE of two 1/2 BPS operators is determined by the three-point functions of the following type: Here BPS 1/2 denote two 1/2 BPS short operators described in section 2.3. The third operator in equation (3.1) is characterized by the quantum numbers of its lowest (θ 3 = 0) component ('superconformal primary field'). These are conformal dimension (a priori arbitrary), Lorentz spin s (meaning that the component is a symmetric traceless rank s tensor) and an R symmetry irrep labelled by a pair of integers (jk). The latter appears in the tensor product (m) ⊗ (n) of two of the irreps listed in (2.8) (we assume that m ≥ n):  In what follows we show that the few rather elementary facts about supersingletons and their products we have presented in section 2 are sufficient to explain the selection rules on the operator O (jk) s in (3.1). Although we will be discussing three-point functions in superspace,

2.11
we will hardly need to go into any details of their θ dependence. The examination of the lowest (θ 1 = θ 2 = θ 3 = 0) component will give us all the necessary information. The reason for this is the remarkable property of the three-point functions (3.1) that they are uniquely fixed by conformal supersymmetry. Indeed, the superfunction (3.1) depends on half of the Grassmann variables at points 1 and 2 and on a full set of such variables at point 3. Thus, the total number of odd variables exactly matches the number of supersymmetries (Poincaré Q plus special conformal S). Therefore there exist no nilpotent superconformal invariants made out of the θs and the complete θ 1,2,3 expansion of (3.1) is determined by its lowest component. The latter is the three-point function of two scalars and one tensor field, and is fixed by conformal invariance up to an overall factor.
Before proceeding, we would like to compare the method we follow here with earlier approaches [20,24,25,32,31]. There the origin of the selection rules was related to the requirement of absence of harmonic singularities (harmonic analyticity) at the higher levels of the θ expansion of the three-point function. This is certainly an equivalent explanation; however, here we insist upon the fact that harmonic analyticity is nothing but the coordinate expression of R symmetry irreducibility. Thus, by just analysing the occurrence of the different R symmetry irreps in conjunction with our knowledge of the supermultiplet structure, we are able to derive the same selection rules without inspecting the actual harmonic or Grassmann coordinate dependence.

Factorization of the three-point functions
The crucial observation is that the lowest component of the three-point function (3.1) can be factorized as follows: where the new 'fake operator' O ds is an R symmetry singlet, but it carries spin s and dimension The two factors in the rhs of (3.3) have the following structure. The factor carrying the spin at point 3 is made out of the three-point conformal vector and of the supersingleton two-point function W (1)W (2) θ 1,2 =0 . The latter is completely determined by just R symmetry, translation and dilatation invariance and is given by (3.6) Here (12) symbolizes the irreducible harmonic structure which carries the quantum numbers of a HWS of the R symmetry group corresponding to the basic supersingleton: Thus, the complete first factor in the rhs of (3.3) has the form The second factor in the rhs of (3.3) accounts for the R symmetry quantum numbers at point 3.
It is entirely made out of supersingleton two-point functions: In order to reproduce the structure of the R symmetry representations at point 3 listed in equation (3.2) we have to use both types of supersingleton, according to equation (2.10). Thus, while the harmonic factors (13), (23) originate from two-point functions involving only HWS (recall (3.6)), the factor (23 − ) (and similarly (13 − )) comes from a two-point function of the mixed type, i.e. where the supersingleton at point 3 is not the HWS of the corresponding representation:

BPS shortness and selection rules
The form of the lowest component of the three-point function that we found in the preceding subsection satisfies the basic requirements of conformal covariance and R symmetry irreducibility. The construction we presented clearly shows that this form is unique (up to an overall constant factor). Next, we have to extend this lowest component to a full superspace three-point function. According to the counting argument from the beginning of this section, this superextension is also unique (if it exists). One way to proceed would be to use various techniques to construct the superconformal three-point covariant starting from its first component [31,55]- [58]. This results in rather complicated expressions which are not so easy to analyse. Here we present a different approach based on the factorized form (3.3) which directly leads to the conditions for existence of such a superextension.
The origin of possible constraints on the three-point functions is in the fact that the operators at points 1 and 2 are 1/2 BPS short. The second factor (3.9) from equation (3.3) can immediately be extended to a superfunction having the required properties at points 1 and 2. Indeed, each factor in equation (3.9) is the lowest component of a two-point function of supersingletons, so the obvious superextension of equation (3.9) is Here W − (3) denotes the alterative realization (2.6) of the supersingleton. As explained in section 2, BPS shortness is equivalent to analyticity, i.e. it is a multiplicative property. The product (3.12) of BPS objects automatically has the required properties at points 1 and 2. So, we need to concentrate on the first factor (3.8). We can formally treat this factor as the lowest component of another, 'fake' three-point function involving two 1/2 BPS short objects of identical labels (k) at points 1 and 2 and an R symmetry singlet, a priori long multiplet at  (this does not affect the differential operator 1 ) and using translations and conformal boosts to set x 3 = 0, x 2 = ∞, we obtain (denoting It is easy to see that this equation admits two solutions. The first solution is d = −s or, equivalently, In order for this solution to not violate the unitarity bounds on the superconformal representation at point 3 of section 2.4, we must set s = 0, which results in The second solution to equation (3.15) is d = s + D − 2, which gives rise to for arbitrary s. We stress the fact that although the constraints above have been derived as necessary conditions, they are also sufficient for the existence of the corresponding three-point functions. The argument is the same as stating that given a massless scalar in the appropriate R symmetry representation, conformal supersymmetry enables us to reconstruct the entire supersingleton multiplet whose lowest component this scalar is. Similarly, when k ≥ 2 we are dealing with the situation in which conformal supersymmetry restores the full 1/2 BPS composite operator [W ] k starting from an unconstrained scalar lowest component. Hence, for k ≥ 2 we do not find any restrictions on the quantum numbers , s at point 3, apart from the unitarity bounds from section (2.4). This is true even in the particular case k = 2 when the bilinear [W ] 2 is a supercurrent multiplet: the required conserved vector (tensor) components will be automatically created as conformal supersymmetry descendents of the lowest, unconstrained scalar component.

Interpretation of the results. Protected operators
In this section we show that the three cases (3.13), (3.17) and (3.18) correspond to protected operators at point 3, namely, to 1/2 or 1/4 BPS short operators in cases (i) and (ii.a) and to semishort operators in case (ii.b).
Case (i). The simplest situation occurs when k = 0. Then the factor (3.8) in equation (3.3) becomes trivial and the entire three-point function is reduced to the second factor (3.9), which is just a product of two-point functions of supersingletons (see equation (3.12)): (4.1) Note the absence of a two-point function connecting points 1 and 2. Now we can identify the operator at point 3 with a composite BPS operator (recall equation (2.10)): Case (ii.a). In this case the three-point function still factorizes into two-point functions of supersingletons: and the operator at point 3 is s has the right quantum numbers is not yet sufficient to claim that it is indeed semishort. A simple counterexample illustrating this point is the threepoint function of two scalars of different dimensions 1,2 and a vector of canonical dimension j = D − 1: Thus, we can interpret (3.8) as the lowest component of a 'fake' three-point function, which proves (4.5).
It is important to realize that the above factorization is only formal, it just helps us investigate the supermultiplet structure. In fact, the singlet factor O ds which looks like a 'supercurrent' is not the true operator at point 3. The full operator O (j1) s is only semishort and not a supercurrent (and, consequently, does not contain conserved tensor components). The reason is that in the case k = 1 the BPS factor BPS (j1) is always present. Indeed, from (3.2) it follows that in order not to have a BPS factor the operator O (j1) s must be an R symmetry singlet. Thus, we should set j = 0 and m + n = 2, which implies m = n = 1. However, this corresponds to putting just one supersingleton at points 1 and 2, which is not a gauge-invariant object and thus is not of physical relevance.
The same argument shows that if k ≥ 2 one can have a situation where the BPS factor is absent and the operator O (jk) s is a true 'supercurrent'. Going back to the generic factorized form (2.20), we see that this may happen if j = 0, m + n = 2k ≥ 4 and if we choose to set s > 0, δ = 0 or s = 0, δ = 1. A well-known example is the Konishi multiplet in N = 4, D = 4 SYM which appears in the OPE of two stress-tensor multiplets and corresponds to m = n = k = 2, j = s = 0. In the free field theory it is known to satisfy a superspace constraint and to contain conserved tensor components. However, in the presence of interactions the Konishi multiplet acquires an anomalous dimension [59] and thus ceases to be a 'supercurrent'. Further examples of operators which have anomalous dimension are the higher-spin and R symmetry singlet multiplets (m = n = k = 2, j = 0 and s > 0) considered in [60]. These operators again reduce to 'supercurrents' in the free field theory.
The above discussion clearly shows the key difference between the cases k = 0, 1 and k ≥ 2. In the cases k = 0, 1 the conformal dimension at point 3 is fixed by the branching rules and thus O necessarily becomes BPS or semishort. In the case k ≥ 2 there is no reason to maintain the conformal dimension at one of these fixed values, so O may be a BPS, a semishort 2.17 or a generic long multiplet. It follows that for k = 0, 1 any operator O (jk) s appearing in the OPE of two 1/2 BPS operators is protected by the superconformal kinematics whereas for k ≥ 2 it is unprotected, i.e. its conformal dimension is determined by the dynamics of the theory.

Extremal correlators
One of the possible applications of the branching rules that we have found is the proof that a certain class of n-point correlation functions of 1/2 BPS operators W m ≡ [W ] m , W m 1 (1)W m 2 (2) · · · W mn (n) , Using AdS supergravity considerations, in [29,30] it was shown that the extremal correlators are not renormalized and factorize into products of two-point functions. Here we give a simple explanation of this fact from the CFT point of view based on our results on the three-point functions and the related OPEs of 1/2 BPS operators. The same argument has already been presented in [32] for the case † D = 6. For simplicity we restrict ourselves to four-point extremal correlators. They can be represented as the convolution of two OPEs: To find out their spectrum, we first examine the R symmetry quantum numbers. From (3.2) we see that they are given by a pair of integers: where we have assumed m 3 ≥ m 4 . Since in the extremal case m 1 = m 2 +m 3 +m 4 (recall (5.2)), the intersection is given by the following conditions: (5.7) † A different proof in the case D = 4, based on a direct analysis of the n-point superconformal covariants, was given in [61]. See also [31] for a recent argument.

2.18
Finally, in this particular case the three-point functions in (5.3) degenerate into products of two two-point functions (recall (4.1)), so ( This clearly shows that the extremal four-point correlator factorizes into a product of two-point functions. In other words, it always takes its free (Born approximation) form, so it remains non-renormalized. The generalization of the above result to an arbitrary number of points is straightforward and it follows the D = 6 pattern exhibited in [32]. Further, the argument concerning the non-renormalization of 'next-to-extremal' [62,63] D = 6 correlators (i.e. those for which m 1 = n i=2 m i − 2) presented in [32] applies to the cases D = 3, 4, 5 as well. Notice that three-point functions of protected operators in D = 4, other than 1/2 BPS, have recently been proved not to suffer from renormalization [31,64].