hepth/0303107
BMN operators with vector impurities,
symmetry and ppwaves
ChongSun Chu, Valentin V. Khoze and Gabriele Travaglini

Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300, R.O.C.

Centre for Particle Theory, Department of Mathematical Sciences,
University of Durham, Durham, DH1 3LE, UK

Centre for Particle Theory, Department of Physics and IPPP,
University of Durham, Durham, DH1 3LE, UK
Email: chongsun.chu, valya.khoze,
Abstract
We calculate the coefficients of threepoint functions of BMN operators with two vector impurities. We find that these coefficients can be obtained from those of the threepoint functions of scalar BMN operators by interchanging the coefficient for the symmetrictraceless representation with the coefficient for the singlet. We conclude that the symmetry of the ppwave string theory is not manifest at the level of field theory threepoint correlators.
1 Introduction
The ppwave/SYM correspondence of Berenstein, Maldacena and Nastase (BMN) [1] represents all massive modes of type IIB superstring theory in a plane wave background in terms of composite BMN operators in Super YangMills in four dimensions. Until now, most of the calculations on the gauge theory side of the correspondence were restricted to the BMN operators with scalar impurities.
The goal of the present paper is to extend the study of correlation functions of scalar BMN operators [2, 3, 4, 5, 6, 7, 8] to correlators of vector BMN operators. In particular we will address the relevance of a symmetry of the ppwave string theory for the threepoint functions of vector BMN operators in the gauge theory. Twopoint correlators of BMN operators with vector impurities have already been considered in [9, 10, 11]. We will compute threepoint functions of BMN operators with two vector impurities. These threepoint functions are essential for the vertex–correlator ppwave duality [2, 3, 7]. Our goal is to compare the coefficients of the threepoint functions of vector BMN operators with those for scalar BMN operators. One would expect that the symmetry of string theory in the ppwave background (explained below) requires the equality of these two coefficients. The main result of this paper is that this two coefficients are different.^{1}^{1}1The earlier version of this paper reported an agreement between the vector and scalar coefficients. This was due to an incorrect handling of the compensating terms in the vector BMN operators, last term in (16). Our result is that the vector threepoint function (71) is related to the scalar threepoint function (52) by exchanging the contribution for the symmetrictraceless operator with that of the singlet. This conclusion can also be derived from an earlier work of Beisert [10]. From these results it appears that the symmetry of the ppwave string theory is not respected at the level of threepoint functions of BMN operators with definite scaling dimensions in interacting field theory.
On the string theory side, the ppwave background has a bosonic symmetry of , where the exchanges the action of the two groups. This symmetry acts quite trivially at the free string level [12, 13]. However, its realisation in the dual field theory is not manifest and, therefore, highly nontrivial. In the ppwave/SYM duality, the rotation groups in the lightconegauge string theory are mapped to the product of the Lorentz (Euclidean) symmetry and the Rsymmetry, , in the field theory. Thus, on the field theory side, the factor swaps the action of with . A symmetry between spacetime and the internal (R)space is novel, and might possibly be expected only in the large double scaling limit. The understanding of the symmetry, both in interacting string theory [12, 13] and in field theory, is one of the most challenging and exciting topics in the ppwave/SYM duality.
In field theory, the BMN operators that are dual to string excitations in the first four directions, i.e. related to the factor , carry impurities of the form (vector impurities). Twopoint functions and anomalous dimensions of conformal primary vector BMN operators have been considered and determined in [9, 11]. The minimal form of the BMN correspondence is based on the mass–dimension type duality relation which maps the masses of string states to the anomalous dimensions of the corresponding BMN operators in the gauge theory:
(1) 
This relation has been verified for scalar BMN operators in the planar limit of SYM perturbation theory in [1, 14, 15]. Calculations in the BMN sector of gauge theory at the nonplanar level were performed in [16, 2, 4, 5] also taking into account mixing effects of planar BMN operators. The relation was extended in [18, 17, 19, 20, 21] to all orders in the effective genus expansion parameter . In [9, 11] anomalous dimensions of vector BMN operators were found to be equal to those of scalar BMN operators. This verifies the consistency of the symmetry with the relation (1).
However, no further statement has been made so far about the symmetry beyond the massdimension duality (1). As we mentioned earlier, we find that the symmetry of the ppwave string theory is not respected at the level of threepoint functions of BMN operators with definite scaling dimensions in interacting field theory.
We will first need to carry out a field theory analysis of the threepoint function involving BMN operators with vector impurities. This part of the analysis is new and contains some of the main results of this paper.
Let us recall that in a conformal theory, two and threepoint functions of conformal primary operators are completely determined by conformal invariance. One can always choose a basis of scalar conformal primary operators such that the twopoint functions take the canonical form:
(2) 
and all the nontrivial information of the threepoint function is contained in the independent coefficient :
(3) 
where . Since the form of the dependence of conformal threepoint functions is universal, it is natural to expect that the spacetime independent coefficient is related to the interaction of the corresponding three string states in the ppwave background. Note that, in order to be able to use the coefficients , it is essential to work on the SYM side with BMN operators. These operators are defined in such a way that they do not mix with each other (i.e. have definite scaling dimensions ) and are conformal primary operators. Conformal invariance of the theory then implies that the twopoint correlators of scalar BMN operators are canonically normalized, and the threepoint functions take the simple form (3). Defined in this way, the basis of BMN operators is unique and distinct from other BMN bases considered in the literature. For two scalar impurities, this BMN basis was constructed in [4].
However, due to their nontrivial transformation properties under the conformal group, conformal primary vector BMN operators have in general more complicated two and threepoint functions. Thus, a priori, it is not clear whether it is possible (and how) to extract in the vector case a spacetime independent coefficient, similar to the of the scalar correlators, that can then be compared with the ppwave string interaction. In our opinion, this is one of the main obstacles in the understanding of how the ppwave/SYM duality works for vector impurities and of the rôle of the symmetry beyond the level of the twopoint functions in the ppwave/SYM correspondence. In this paper, we make the observation that in a certain large distance limit, the two and threepoint correlation functions for vector BMN operators reduce to the same form as that for the scalar case. This allows one to make a direct comparison with the corresponding scalar threepoint functions.
The paper is organised as follows. In Section 2, we present the BMN operators with vector impurities and with positive Rcharge. To obtain nonvanishing correlators, one also needs to know the conjugate BMN operators, i.e. the BMN operators with negative Rcharge. We construct these operators by employing a new conjugation operation which is a product of the usual hermitian conjugation with the inversion operation. We explain why this construction is the most natural one in the present context. An important advantage of our construction is that the vector BMN operators are orthonormal with respect to the inner product defined using this conjugation. In Section 3 we compute the threepoint functions involving vectorBMN operators with definite scaling dimensions in interacting field theory.
******
Note on notation and conventions
We write the bosonic part of the Lagrangian as
(4) 
where , are the six real scalar fields transforming under an Rsymmetry group . The covariant derivative is , where , and . If we define the complex combinations
(5) 
the Lagrangian can be reexpressed as
(6) 
where
(7)  
are the Fterm and Dterm of the scalar potential respectively. In the last equalities we write only the terms which will be relevant for our analysis. Our generators are normalised as
(8) 
so that, for example,
(9) 
The ppwave/SYM duality is supposed to hold in the BMN large double scaling limit,
(10) 
In this limit there remain two free finite dimensionless parameters [1, 16, 2]: the effective coupling constant of the BMN sector of gauge theory,
(11) 
and the effective genus counting parameter
(12) 
of Feynman diagrams. The right hand sides of (11), (12) express and in terms of the ppwave string theory parameters.
2 Conformal primary vector BMN operators
Here we will study the BMN operators with vector impurities^{2}^{2}2CSC and VVK acknowledge an early collaboration with Michela Petrini, Rodolfo Russo and Alessandro Tanzini on the radial quantisation method and its applications to vector BMN operators discussed in section 2 of this paper. . We will be concerned with the operators
(13) 
and, for ,
(14) 
where we defined
(15) 
The normalisation of the operator is such that its twopoint function takes the canonical form (2) in the planar limit. As for the vector BMN operator , it is normalized in such a way that Eq. (38) below holds. We note that this choice of normalisation constant is different from that^{3}^{3}3In particular we have the same normalisation constant for both cases and . This is related to our prescription for the operator conjugation and the definition of the inner product. We will explain how this prescription is dictated by the ppwave/SYM correspondence. adopted in [11].
The first operator, , is a chiral (halfBPS) primary operator, and corresponds to the vacuum state of ppwave string theory. For , the second operator, is a nonchiral vector conformal primary BMN operator, and corresponds to a string state . Here and are indices of bosonic excitations of the first in the lightcone ppwave string theory.^{4}^{4}4We adopt the convention that BMN operators with vector (resp. scalar) impurities correspond to bosonic excitations of the first (resp. second) in the lightcone ppwave string theory. The operator has a definite scaling dimension, , which implies that the singletrace expression on the right hand side of (14) must be accompanied with multitrace corrections (and other mixing effects) at higher orders in [22, 4]. The dots on the right hand side of (14) indicate these corrections. These mixing terms are important in general, but in this paper we will show how to calculate correlation functions involving operators (14) without the need of knowing the precise analytical expressions for these mixing terms.
To be more precise, we should distinguish between symmetrictraceless, antisymmetric and singlet representations:
(16)  
(17)  
(18) 
where
(19)  
(20) 
Notice that the compensating term is present only in the definition of the symmetrictraceless operator in (16) and not in the singlet (18). The precise form of the operators (16)–(18) is determined by acting with supersymmetry transformations on the scalar BMN operators in (42), and it was first obtained in [10] (Eqs. (B.10), (B.11) and (B.12)), which are valid also at finite . Our operators (16), (17) and (18) follow in the large limit from those in [10]. Supersymmetry dictates that the singletrace bosonic operators in (17) and (18) must be accompanied by fermionic bilinears and scalar bilinears – see Appendix B of [10] for the precise form of these terms. All these corrections, as well as the multitrace corrections, will not be relevant for the calculation of threepoint functions presented in the following sections, hence we will include them in the dots in (16)–(18) and discard them.
For , the operator is a supergravity translational descendant of the vacuum:
(21)  
(22) 
This operator is protected, hence its conformal dimension is given by the engineering dimension.
We now note that the operators are not orthogonal with respect to the scalar product , and therefore cannot correspond to the (orthonormal) basis of string states (at least not directly). For example, one has [11] for the translational descendant defined in (21),
(23) 
which is nonzero for . We also note that, in order to keep the right hand side of (23) finite as , an additional factor of would be required in the definition (21) of [11].
The right hand side of (23) has nothing to do with an orthonormality of the string states. We therefore introduce a different notion of conjugation, which will allow a direct correspondence to string (and supergravity) states defined as hermitian conjugation followed by an inversion:^{5}^{5}5We illustrate the following procedure for the symmetrictraceless operators (16). The extension to the antisymmetric and singlet representations is straightforward.
(i) We define the barredoperator as
(24)  
where is the usual inversion tensor, in terms of which the Jacobian of the inversion is expressed .
(ii) We introduce the inner product
(25) 
and,
(iii) propose the correspondence between field theory and string theory inner products:
(26) 
where is the string state that is in correspondence with the field theory operator .
We remark that the introduction of the barredoperator is completely natural in the context of the radial quantisation of field theory [23], where hermitian conjugation is always accompanied by an inversion. Indeed, under inversion a scalar field of conformal dimension transforms as [24, 25]
(27) 
Differentiating both sides of (27) with respect to we obtain
(28) 
Combining the action of hermitian conjugation with an inversion, we get
(29) 
from which it follows^{6}^{6}6 A note on conventions: a bar applied to a composite operator will always mean hermitian conjugation times an inversion as in (24). For ordinary fields we continue to use . that
(30)  
which is the freetheory expression for (24).
We note that the expression for the string operator (14) can be more compactly written as [9, 11]
(31) 
The corresponding expression for the free barredoperator is given then by
(32) 
We now apply (24), or, equivalently (30), to the protected supergravity operator in (21)
(33) 
and (23) is now replaced by the inner product
(34)  
(35) 
Unlike (23), this expression is consistent with an operator–supergravitystate correspondence. This is the first consistency check of our proposal (24) and (26).
We now move on to consider string states, and compute in the free theory the twopoint function in the limit . To this end, it is convenient to observe that the only terms which survive in this overlap are the ones where one derivative operator originating from the barred operator and one from the unbarred operator act on the same propagator, . For these terms
(36) 
where we have used that , and . Keeping this in mind, one easily computes in the limit ,
(37)  
(38) 
This result is again consistent with our operatorstring state correspondence (26). This is the second, nontrivial consistency check of our proposal (24) and (26). The normalisation chosen in (14) was designed to lead, on the right hand side of (38), to the product of Kronecker deltas with coefficient equal to 1 .
A few general remarks are in order:
1. In distinction with Eqs. (29a)–(29d) of [11], in our case (38), the overlap between supergravity and string states vanishes.
2. On general grounds, conformal invariance requires that the twopoint function of vector conformal primary operators of scaling dimension should have the form [24, 25]:
(39) 
In our approach, we amputate the coordinate dependence on the right hand side of (39), and contract vector indices with (appropriate tensor products of) the inversion tensor , thus directly computing
(40) 
see our result (38). We take the limit because of the barredoperator is the inversion of and, in the radial quantisation formalism, states are obtained from operators at the point . The corresponding state in radial quantisation would be
(41) 
which is precisely our definition. The twopoint functions of vector operators are now correctly normalised, and take the canonical form. As a result, they are suited for a correspondence with the (orthonormal) string theory basis of states.
3. For the BMN operators with scalar impurities,
(42) 
one can follow the same procedure as above, and define the barredoperators as . Obviously, whether or not we introduce an inversion for the scalar fields is rather irrelevant: all the previous results for scalar Green functions are modified in a straightforward manner and the relation (26) is verified. However, as we have shown, this leads to important differences for vector operators.
4. It has been argued already in [9, 10] that the vector conformal primary BMN operators, i.e. BMN operators with various numbers of vector impurities are bosonic supersymmetry descendants of the scalar conformal primary BMN operators. This construction has been systematically carried out in [10]. Supersymmetry is important as it ensures that BMN operators with one vector and one scalar impurity [9] or two vector impurities [11] have exactly the same anomalous dimension as BMN operators with two scalar impurities, [9, 10], in agreement with string theory expectations.
3 Threepoint functions of vector conformal primary BMN operators
Conformal invariance constrains the expression of threepoint functions of conformal primary operators. For the particular class of threepoint functions , involving vector conformal primary operators with , one has
(43) 
where , ’s are the scaling dimensions of , and respectively; and is a dimensionless function of . In the quantum theory, , , , where , are the anomalous dimensions of , . Therefore
(44)  
(45)  
(46) 
Notice that the anomalous dimensions for vector conformal primary operators with one vector and one scalar impurity [9] or with two vector impurities [11] are the same as for the original BMN operators with two scalar impurities [1].
Conformal invariance requires to depend on the vector indices , , , through appropriate tensorial products of the inversion tensor, , thus it contains dependence^{7}^{7}7See, e.g. , section III.2 of [26]. and cannot be compared directly to the coefficient of the scalar threepoint function (3), nor with a threestring interaction vertex. As in the previous section, we propose to consider instead the threepoint functions involving the barredoperators and, moreover, to work in the limit^{8}^{8}8As before, the limit is a consequence of the formalism of radial quantisation. We also note that translational invariance, broken by radial quantisation, is restored in this limit. . Using our definition (30) for the barredoperator, we will therefore compute
(48)  
(49) 
for (and , finite), where and are given by (14) and (24). This is one of the key observation of this paper. Now can be compared directly to the scalar threepoint function coefficient , defined below.
The threepoint functions of BMN operators with scalar impurities (42) have the form
(50) 
or, introducing the barredoperators and working in the limit (and , finite),
(51) 
The expression for the coefficient of the threepoint function for BMN operators with two scalar impurities is
(52) 
where is the Rcharge ratio, and the symmetric traceless and antisymmetric traceless combinations of two Kronecker deltas are defined as
(53) 
These results were first obtained in the simple case in [3]. The general expression (52) was derived in [4].
We now explain how the computation of the vector threepoint functions proceeds. In subsection 3.1 we will describe the freetheory computation, and devote 3.2 to the planar corrections at oneloop. In order to efficiently organise our analysis, we will make a stepbystep comparison with the known computation for the case of scalar impurities. More precisely, our strategy will consist in identifying the “building blocks” which lead to the expression (52) for the coefficient of the threepoint function of scalar BMN operators, and comparing them to the corresponding building blocks for the case of BMN operators with vector impurities.
3.1 The calculation in free theory
Let us briefly review the free theory computation for the threepoint function with (complex) scalar impurities,^{9}^{9}9For the considerations in free theory presented in this section, we can set all anomalous dimensions equal to zero. say and [2]. For calculations with scalars we use the complex basis (5), but continue calling the BMN operators as and .
Obviously, to get a nonzero result an impurity in the barred operator must be contracted with an impurity in and the result boils down to the evaluation of the Feynman diagram in Figure 1, which gives
(54) 
The factor comes from carefully summing the BMN phase factors over all the position of and impurities in the operators. Its explicit form is given in Appendix B, and will not be needed here. When , (54) is the only contribution to the threepoint function at the free level. When the mixing with multitrace operators must be taken into account [4, 22] and will modify even free theory results at leading order in . These mixing effects being added to the contributions of Figure 1 lead to the result of (52) [4].
We now consider the vector impurity case. First, notice that, in the free theory, covariant derivatives can be replaced with simple derivatives. The second key observation is that, in the limit we are considering ( and , finite), the only nonvanishing contractions are those where an impurity in the operator is connected to an impurity in . The result of such contractions has been analysed in (36). This observation leads to the immediate conclusion that there is only one Feynman diagram contributing to the free vector impurity case (Figure 2). The associated phase factor is the same as for the scalar impurity case of Figure 1. Therefore the free theory result for the vector threepoint function is given by
(55) 
In writing (55) we have taken into account that a factor of from two free contractions of the vector impurities (see the right hand side of (36)) is precisely cancelled by a factor of from the normalisation of the vector BMN operators.^{10}^{10}10 Notice that the normalisation constant for vector BMN operators is half the normalisation of the scalars, . Therefore the free result (55) for vector BMN operators leads to the same result as for the scalars (see (54)).
As in the scalar case, there are mixing effects of the barred singletrace operator with barred doubletrace operators. These mixing effects will affect the freetheory contribution of Figure 2. However, as we argued above, in the region , the vector impurities inside a BMN operator are orthonormal to each other with respect to the inner product (25), and hence behave in the same way as scalar impurities inside a BMN operator. As a result, it is easy to convince oneself that the modifications due to mixing effects to the freetheory threepoint function coefficient are the same for both the scalar and the vector case. Hence, the free threepoint function with vector impurities reproduces precisely its counterpart for the case of scalar impurities.
Before concluding this section, we would like to discuss further the issue of mixing. The mixing of singletrace BMN operators with doubletrace operators is crucial in order to obtain conformal expressions such as (43) (or(48)). However, here we are not concerned with deriving the conformal expression on the right hand side of (43), which must be correct anyway, as far as the mixing effects are such that we are dealing with vector conformal primary operators. Our goal is rather to compute the coefficient of the threepoint function with two nonchiral operators, . At leading order in , the only mixing effect which contributes to the right hand side of (43) (or (48)) is the mixing of the barred operator with doubletrace operators^{11}^{11}11To see it immediately, note that the doubletrace corrections to the singletrace expression for a BMN operator is of , i.e. suppressed with . This can be compensated by factorising the threepoint function into a product of two twopoint functions. This is possible only for the doubletrace mixing in the operator . [4]. These mixing effects will affect not only the freetheory contribution to , but also the logarithmic terms and due to interactions of the doubletrace corrections in with the BMN operators sitting at and . However, it is important to note that these mixing effects cannot affect the remaining logarithm, [8]. Hence the coefficient of this logarithm can be computed in planar perturbation theory at order without taking into account mixing altogether.
Our programme will therefore consist in assuming the conformal form (rather than deriving it), and evaluating the terms proportional to , thus determining the full coefficient of the vector threepoint function. In doing so we are allowed to neglect the doubletrace corrections, and work directly with the original singletrace BMN expressions.
3.2 The calculation in the interacting theory
The observations made at the end of the last section allow us to limit ourselves to the Feynman diagrams which can generate a term. Notice that selfenergy corrections cannot generate such a dependence, and will thus be completely irrelevant for our purposes.