Large- correlation functions
in superconformal QCD
Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, Gideon Vos
Institute for Theoretical Physics, KU Leuven, 3001 Leuven, Belgium
Department of Mathematical Sciences and Center for Particle Theory
Durham University, Durham, DH1 3LE, UK
Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland
Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
We study extremal correlation functions of chiral primary operators in the large- superconformal QCD theory and present new results based on supersymmetric localization. We discuss extensively the basis-independent data that can be extracted from these correlators using the leading order large- matrix model free energy given by the four-sphere partition function. Special emphasis is given to single-trace 2- and 3-point functions as well as a new class of observables that are scalars on the conformal manifold. These new observables are particular quadratic combinations of the structure constants of the chiral ring. At weak ’t Hooft coupling we present perturbative results that, in principle, can be extended to arbitrarily high order. We obtain closed-form expressions up to the first subleading order. At strong coupling we provide analogous results based on an approximate Wiener-Hopf method.
- 1 Introduction
2 Exact 2- and 3-point functions in the chiral ring
- 2.1 chiral ring
- 2.2 Extremal correlation functions in the chiral ring
- 2.3 2- and 3-point functions from partition functions and matrix models
- 3 Correlation functions at large
- 4 Weak coupling results
- 5 Partial strong coupling results
- A Deformed matrix integrals from supersymmetric localization
- B Proof of ()
- C Large- matrix model at weak coupling
- D Large- matrix model at strong coupling
Dedicated to the memory of Ioannis Bakas
References [1, 2] computed the exact (extremal) correlation functions of chiral primary operators in the 4d superconformal gauge theory with gauge group coupled to massless hypermultiplets. These correlation functions are highly nontrivial functions of the complexified coupling constant and include all-order perturbative and instanton corrections. At the moment, they are the only known example of nontrivial, exactly computed 3-point functions in a 4d QFT. The computation of [1, 2] relied on the constraints imposed on the chiral ring correlators by the 4d equations , together with input from supersymmetric localization , and made use of the relation, proposed in , between the sphere partition function and the Zamolodchikov metric on the conformal manifold. The relationship between extremal correlation functions in the chiral ring and the sphere partition function was further clarified and extended in , which paved the way towards concrete computations in general 4d SCFTs with conformal manifolds. More generally, it would be interesting to know if there are also other correlation functions that can be computed in practice by employing similar techniques (see  for a recent analogous computation of correlation functions in 3d superconformal field theories).
In this paper we consider the family of superconformal field theories with gauge group and hypermultiplets. We focus on the large- ’t Hooft-Veneziano limit and explain how correlators of chiral primary operators can be computed as a function of the ’t Hooft coupling . One reason why these correlators are interesting is that they encode information about a putative string theory dual for this family of large- theories111Since we consider a ’t Hooft-Veneziano limit where the ratio is fixed and non-vanishing, this duality would have the peculiar feature where mesonic hypermultiplet bilinears would lead to an number of gauge invariant operators with low conformal dimension. A related discussion of similar limits in two-dimensional theories can be found in . We thank S. Minwalla for comments related to this feature. (see  for an earlier discussion of such duality). Moreover, similar techniques could be applied to closely related theories (e.g. orbifolds of super-Yang-Mills theory (SYM) ) with known AdS/CFT duals, and lessons obtained in this paper could be easily extended there as well. A recent discussion of conformal manifolds in the context of the AdS/CFT correspondence from the supergravity point of view can be found in . More generally, having a solid understanding of a large- correlator as an exact function of in QFT, at leading and subleading orders in the -expansion, could be a useful guide towards a concrete analysis of various formal aspects of the large- expansion in a full-fledged 4d gauge theory.
chiral primary correlators in the ’t Hooft limit of the theory were recently considered in [12, 13], where 2-point functions of single-trace chiral primaries were computed perturbatively in at leading order in . In this paper we substantially extend these results by computing more general correlation functions at large . Specifically, we focus on two main classes of observables.
The first are 3-point functions of chiral primary single-trace operators. 3-point functions of chiral primary operators in SYM theory have of course been widely studied in the context of holography starting with [14, 15]. In the theory, all the non-vanishing 3-point functions are extremal, and are especially sensitive to mixing with multi-trace operators . We point out that there is a well-motivated and unambiguous definition of the basis of chiral primary operators near the weak-coupling point based on parallel transport that is formulated in terms of a natural connection on the space of operators in conformal perturbation theory. This definition works particularly well in our class of theories in the large- limit, where the conformal manifold is essentially one-dimensional. A different basis of chiral primary operators is defined implicitly through the relation with the partition function . We compute 3-point functions of the form in the first few orders in around the weak coupling point in both bases. We check that to leading order our methods reproduce the results of  for the SYM, as expected. Unlike the SYM theory, however, in theories correlators of chiral primaries receive quantum corrections that we can easily compute up to any desired order in .
In order to bypass the subtleties that arise from the mixing between single-trace and multi-trace operators we also consider a new class of observables obtained by certain quadratic combinations of the chiral ring structure constants. These quantities are geometric scalars on the conformal manifold, they are manifestly independent of the choice of basis, and therefore can be meaningfully computed and compared at arbitrary values of the coupling constant. Furthermore, an infinite subset of them obey a very simple recursion relation, coming from the equations, that can be solved explicitly in terms of 2-point function data immediately available at large .
We show how both classes of observables can be computed at leading order in the large- limit from the planar free energy of the theory on deformed by higher chiral primary sources. The latter is also the planar free energy of a corresponding matrix model, which arises from localization, and can be determined from the solution of the saddle-point integral equation
where . The sum on the r.h.s. originates from the higher chiral primary source deformations of the theory. It is a polynomial whose degree is suitably adjusted to the correlator we want to compute. The planar free energy follows directly from the eigenvalue density . Eq. (1.1) was first considered in  and further used in . We have not been able to solve (1.1) analytically for arbitrary values of , so we will limit ourselves to analyzing its solutions in two regimes, at weak and strong coupling . Given a solution of (1.1) (approximate or exact), there is a well-defined procedure [6, 1] to recover correlation functions of the physical theory by combining appropriate derivatives of .
Computations based on the weak coupling expansion of the solutions of (1.1) are pretty straightforward and, technically, they follow closely the logic of [17, 12]. At strong coupling the analysis of eq. (1.1) is considerably harder. As was first pointed out in  approximate solutions can be obtained with the use of the Wiener-Hopf method.222We should point out that these approximations are not parametrically controlled, so we cannot prove conclusively that the large- scalings obtained in this way persist in the exact solution of the saddle-point equations. Using this method we estimate the large- scaling of 2-point functions of single-trace operators in the chiral ring, extending partial results in , and the large- scaling of 3-point functions. The large- scaling of 2-point functions is also discussed from an independent point of view based on the analysis of the density of connected 2-point functions in the matrix model.
Plan of the paper and summary of the main results. In the main text of the paper we focus on properties and results of correlators on . Intermediate results based on localization and the corresponding matrix model are relegated to the appendices, where the reader can find all the pertinent details.
In section 2 we discuss in detail the correlation functions of interest and we set the conventions that are used in the rest of the paper. In addition, we review the relation between extremal correlation functions in the chiral ring, the deformed partition function on and the matrix model that arises from localization.
In section 3 we discuss general properties of correlation functions in the large- limit. We explain what contributions can be extracted from the leading order large- free energy of the partition function and how issues involving the mixing of single-trace and multi-trace operators affect our computations. We also define appropriate quadratic combinations of the structure constants and show that they obey a recursion relation, coming from the equations, that can be solved in closed form.
Results specific to the weak coupling expansion of the theory are presented in section 4. We provide closed form expressions for 2- and 3-point functions both at tree level and at the first nontrivial subleading order in perturbation theory. Along the way, we present a method, specific to the large- limit, that allows us to determine the correlation functions of single-trace operators without going through the full Gram-Schmidt orthogonalization procedure proposed in . In this section we also discuss how the use of parallel transport on the conformal manifold leads to unambiguous perturbative expressions for the single-trace 3-point functions. The basis-independent structure constant squared combinations, defined in section 3, are computed perturbatively in at the end of the section.
Finally, partial results in the strong coupling limit of the theory are discussed in section 5. We emphasize the large- scaling of 2- and 3-point functions and discuss the technical difficulties associated to the current use of the Wiener-Hopf method.
Four appendices at the end of the paper provide the technical background for the computations presented in the main text. Appendix A summarizes the matrix model that arises from localization and the corresponding saddle-point equations in the large- limit. In this appendix the reader can also find the derivation of an integral equation obeyed by the density of connected 2-point functions, as well as an explicit solution of this equation at infinite ’t Hooft coupling. Appendix C describes the perturbative solution of the saddle-point equations at weak coupling and appendix D the approximate solution based on the Wiener- Hopf method. Appendix B provides the proof of a technically efficient general relation between 3-point functions in the gauge theory and derivatives of the matrix model planar free energy in the large- limit.
2 Exact 2- and 3-point functions in the chiral ring
In this paper we focus on extremal correlation functions of chiral primary operators in a specific class of superconformal field theories defined as SYM theory with gauge group coupled to hypermultiplets (in short, superconformal-QCD, or SCQCD). We will mostly follow the conventions of [1, 18], where one can also find a detailed description of generic properties of the chiral primary operators and further useful references to the literature.
We begin with a quick summary of the operators of interest tailored to the specific features of the SCQCD theories and the goals of this paper. Then, we proceed to define the correlation functions that will play a central role in our discussion and to summarize recent developments that allow their exact non-perturbative computation. Along the way, we emphasize the implications of the recent developments on 2- and 3-point functions.
Operator notation. In the course of the paper we will consider the theory either on or . To keep the distinction between these cases explicit at all times, we will refer to the chiral primary operators on as and the corresponding operators on as . is an appropriate multi-index that labels the operator. Moreover, for notational economy we will frequently refer to single-trace generators on as inside correlation functions, double-trace operators as , etc. The operators may acquire a further label, , that refers to specific linear combinations of single/multi-trace operators to be defined.
Correlation function notation. Correlation functions on will be denoted as (or simply as without index), correlation functions on as and correlation functions on the associated matrix model as .
2.1 chiral ring
We begin by considering the SCQCD theory on flat space, . The chiral primary operators are, by definition, local superconformal primary operators annihilated by all four left-chiral Poincaré supercharges , where is an index and a spinor index. In the SCQCD theory these operators have a simple description as generic multi-trace operators of the adjoint complex scalar field in the vector multiplet. Using a multi-index we denote them as
where are arbitrary non-negative integers. The proportionality symbol refers to an overall normalization factor that will be fixed later. Corresponding multi-trace operators built out of the complex conjugate field will be denoted as ; those are anti-chiral primary operators annihilated by all four right-chiral Poincaré supercharges .
The scaling dimension of each of the operators is half their charge
This relation holds non-perturbatively for generic values of the exactly marginal coupling constant of the theory , where as usual is the theta-angle of the theory and the gauge coupling.
It is clear from the definition (2.1) that the full class of chiral primary multi-trace operators can be generated by Operator Product Expansion (OPE) multiplication from a finite set of single-trace operators , . In what follows we will adopt a normalization convention, consistent with the so-called holomorphic gauge , where the leading term in the OPE between two chiral primary operators,
The dots indicate higher-dimension descendant operators. is the multi-trace operator . The absence of a spacetime singularity in the OPE of two chiral primary operators is a characteristic property of chiral primary operators. The convention (2.4), which sets333In this expression is the obvious multi-index Kronecker delta.
allows us to fix the normalization of all multi-trace operators in terms of the normalization of the single-trace generators .
2.2 Extremal correlation functions in the chiral ring
The main interest of the paper lies in the so-called extremal correlation functions, defined as correlation functions of chiral and anti-chiral primary operators with a single anti-chiral insertion
The charge conservation requires the -charge relation
otherwise the correlator vanishes.
is the common scaling dimension of the two insertions in the 2-point function (2.8), and etc. the scaling dimensions of each operator in the 3-point function (2.9). The interesting datum in each of these correlation functions is the position independent, but generally coupling constant dependent, numerator in the 2-point functions and in the 3-point functions. In the rest of the text it will be convenient to refer to these coefficients using the notation
There is a simple well-known relation between the 2- and 3-point function coefficients and the OPE coefficients in the chiral ring
As an explicit illustration of this relation, consider the computation of the 3-point function of single-trace operators
where following the aforementioned convention we denote the single trace operator simply as in a correlation function. Equation (2.12) implies that this is equal to
which is a 2-point function between a double-trace operator and a single-trace operator.
2.3 2- and 3-point functions from partition functions and matrix models
So far we exclusively discussed correlation functions of the SCQCD theory on . In recent developments, however, a concrete relation has been put forward between the 2-point function coefficients of the theory on and the derivatives of a suitably deformed partition function of the theory on the four-sphere [5, 19, 20, 6]. The latter is further related by supersymmetric localization  to the partition function of a corresponding matrix model.
2.3.1 Deformed partition functions on and their localization
The first step of the procedure starts, quite generally, by placing the superconformal field theory on in a manner that preserves the supergroup of a general massive theory, . In addition, we deform the theory by F-term interactions that are upper components of short multiplets containing the chiral primary fields . It is enough for our purposes to consider deformations restricted to the single-trace chiral primary fields . In superspace form the deformations of interest are
where is the chiral density. is the exactly marginal deformation of the SCQCD theory.
Now consider the partition function of this theory
The finite part of this quantity is physical [5, 20] and depends non-trivially on the complex couplings . Interestingly, although this quantity is given by a complicated path integral, it can be reduced by supersymmetric localization to a corresponding matrix integral that can be analyzed with standard methods . The precise form of the matrix integral in the case of the SCQCD theories is presented in appendix A.
2.3.2 Relation between the partition function and 2-point functions on
Recently,  put forward a concrete general prescription that relates the -derivatives of to the flat-space 2-point function coefficients . One way to summarize the prescription is the following.
Assume we want to evaluate the 2-point function coefficient 444At this point we will include an index or in the notation of the correlation functions to denote explicitly whether we refer to correlation function on or . for two operators , of the same scaling dimension in the chiral ring. Consider the same (single or multi-trace) operators on — for the counterpart of — and construct linear combinations where operators at scaling dimension mix with all operators of smaller dimension (including the identity operator when is even)
The sum runs over the set , which is defined to include all the chiral primaries of scaling dimension , mod 2. The coefficients , are clearly dimensionful and therefore proportional to an appropriate power of the sphere radius. They are fully fixed by implementing the Gram-Schmidt orthogonalization procedure,
The key statement of  is the relation555Notice that the conformal mapping between the sphere and the plane introduces an additional factor of in the relation between the sphere and the plane 2-point functions. To avoid clutter in the equations, we absorb this factor in the normalization of the operators . Of course this has no effect on the normalized correlators that we discuss in the rest of the paper.
where the matrix has, by definition, the elements
The final element is the statement that the 2-point function coefficients are simply given by derivatives of the deformed partition function as follows
denotes a correlation function in the matrix model of appendix A.
Having determined the 2-point functions in this manner we have essentially fixed the normalization conventions for all the chiral primary operators. At this point one should wonder if this prescription is consistent with the choice (2.5) for the OPE coefficients. Following the work in , Ref.  demonstrated that the ansatz (2.22) satisfies the full set of equations with (2.5) incorporated. This is a strong explicit check that (2.22) is indeed consistent with (2.5).
2.3.3 Formulae for 3-point functions
All 2-point functions on the r.h.s. of this equation can be expressed via (2.22) in terms of derivatives of the deformed partition function, or alternatively in terms of derivatives of the free energy
of the corresponding matrix model.
As an explicit example consider again the 3-point function of three single-trace operators. The above prescription gives
Obviously, the structure of the sum on the r.h.s becomes increasingly complicated with increasing scaling dimension.
For a more concrete illustration consider a 3-point function that involves the lowest lying single-trace operators, e.g. . In this case the matrix appearing on the r.h.s. of eq. (2.26) is
Explicit evaluation gives the following simple 2- and 3-point function formulae
where the final result is expressed directly in terms of derivatives of the matrix model free energy .
The correlation functions of operators with higher scaling dimensions can be expressed similarly solely in terms of , but the final expression is considerably more complicated. Further simplifications occur, however, in the large- limit, which is the main topic of the following sections.
3 Correlation functions at large
In this section we study extremal correlation functions in the large- limit and their relation to the matrix model. Due to large- factorization, the behavior of correlators involving multi-trace operators is dominated at large by the factorized answer. Therefore, we introduce a notion of “connected” 2-point functions, which involves, as usual, the full 2-point functions minus the factorized pieces. We argue that these correlators can be determined by the leading contribution to the free energy in the large- limit, which in turn can be computed by the saddle-point method.
At a later part of this section we specialize to the two main classes of observables that we are interested in. First, we study in detail the relation between single-trace 3-point functions and the free energy, and discuss some useful simplifications that occur at large . We also discuss in detail issues related to mixing between single- and multi-trace operators, which in principle can affect the results of our computation, and propose one way to get around these difficulties by using the natural connection provided by conformal perturbation theory.
Lastly, we define a new interesting class of observables, which are quadratic combinations of the structure constants that enjoy many useful properties. Most notably, these observables are manifestly free from ambiguities related to the choice of basis of chiral operators, so they are not affected by the subtleties associated to large- mixing between single- and multi-trace operators. In addition, the equations provide a very simple recursion relation for these observables, which can be solved in closed form in terms of simple geometric data on the conformal manifold.
The explicit analysis of these quantities at weak and strong coupling is the subject of subsequent sections.
3.1 Correlation functions and the matrix model free energy at large
We consider the large- limit at fixed ’t Hooft coupling constant . Similarly, we rescale the sources of the higher Casimir operators so that the parameters666In , a different convention for the couplings was used, namely . We find that our choice is more convenient for the purpose of this paper, as it avoids various explicit factors of that would otherwise appear in intermediate formulae. This effectively corresponds to a different overall normalization of the chiral operators compared to , which of course does not have any effect on the normalized correlators.
is the leading large- contribution. It can be evaluated using the saddle-point approximation, details of which we review in appendix A.
In the previous section, we reviewed how generic 2-point functions (of single-trace or multi-trace operators) in the chiral ring on can be expressed in terms of an algebraic functional of derivatives of the free energy . In the large- limit, and after the Gram-Schmidt procedure has been properly applied, the result contains a finite number of derivatives of with respect to the parameters . The leading contribution to this result comes from , and may scale with in different ways depending on the specifics of the operator insertions.
For instance, the 2-point function of two single-trace operators
scales at large as a constant. Similarly, the 2-point function of a multi-trace operator with a single-trace operator scales like
So, for example, the leading order scaling of the 2-point function of a double-trace and a single-trace operator is of order in agreement with the 3-point function scaling and eq. (2.12). 3-point functions of single-trace operators are one of the main quantities we will consider explicitly in the rest of the paper.
The scaling of 2-point functions between general multi-trace operators is more intricate because of large- factorization. For example, the 2-point function of two double-trace operators behaves at leading order as
The leading order behavior is dominated by factorization, unless the single-trace 2-point functions above vanish. Clearly, at this order the 2-point function (3.5) does not contain any new information beyond (3.3). However, the connected version of ,
is far more interesting and scales with a subleading power of , as . The leading contribution to the connected correlator is also determined by suitable combinations of derivatives of the free energy term . Hence, quantities like (3.6) are also accessible within the saddle-point approximation of the matrix model and contain useful information about the large- gauge theory. We will consider observables related to (3.6) in subsection 3.3.
More generally, we can consider the 2-point function of multi-trace operators
This quantity is precisely what we would get if we started from a connected -point function and took the limit where the insertions of all the chiral operators go to infinity and the insertions of the anti-chiral operators go to zero. Again, since these objects are expressed in terms of 2-point functions of chiral primaries, they can be computed in terms of the matrix model free energy . Their leading behavior in is determined by the leading term of the free energy, .
3.2 Single-trace 2- and 3-point functions
Next let us take a closer look at the Gram-Schmidt procedure, , at large . It was argued in  that mixing between single- and multi-trace operators can be ignored for the purpose of computing single-trace 2-point functions in flat space from the sphere correlators. As a consequence, the 2-point functions on the plane can be easily calculated from . Using standard formulae for the Gram-Schmidt diagonalization in terms of matrix determinants, we thus obtain777As explained in , when we map 2-point functions from the sphere to the plane, we get an additional factor coming from the conformal mapping.
where is the matrix given by
The single-trace 3-point functions can also be determined in terms of . More concretely, we are interested in computing
The following (streamlined) procedure leads to the desired result. First, we perform the Gram-Schmidt orthogonalization procedure by diagonalizing the matrix of sphere 2-point functions of single-trace operators only. This leads to the following formal identification
where and the remaining ’s are determined from the condition for . Our claim is that
The proof of this statement is presented in appendix B. This formula is non-trivial because in principle it differs from the prescription presented in subsection 2.3.3. According to that prescription , to compute (3.10) we would have to perform the Gram-Schmidt diagonalization for the operator directly, whose expression in terms of sphere operators differs from the square of (3.11). The reason why we can work directly with the single-trace operators at large is due to large- factorization of correlators, as explained in appendix B.
3.2.1 Mixing with multi-trace operators
The single-trace 3-point functions are affected by the following subtlety: -charge conservation implies that the only non-vanishing 3-point functions are “extremal”, or more specifically in (3.10). This means that the single-trace 3-point functions, unlike the 2-point functions, are sensitive to mixing with multi-trace operators . As a consequence, different choices for the basis of operators away from the weakly coupled point will lead to different answers for these 3-point functions.
This is best illustrated in an example. Let us consider the operator
where is an arbitrary function of the coupling constant . Its 2-point function at large is identical to the one for , so it cannot be used to distinguish the two operators. However, the 3-point function of this operator with two operators reads
Since both terms on the r.h.s. contribute at the same order, , the leading term at large for this correlator depends on the arbitrary function . At tree level, we can explicitly check that the correlators computed from the sphere partition function match the ones computed with Feynman diagrams in the standard trace basis (4.22), (4.23), so . As we move away from the weakly coupled point, however, it is not obvious a priori that the scheme we are employing matches the one of ordinary perturbation theory in flat space.
There are two possibilities to get around this issue. One is to work with quantities that are manifestly free from ambiguities related to the choice of basis. This is the approach that is described in the following subsection. Alternatively, one can fix the basis of operators away from the weakly-coupled point using the following well-motivated procedure. Conformal perturbation theory provides us with a preferred connection on the space of operators . This connection can be used to parallel transport the operators away from the weakly-coupled point. While this procedure depends on the path chosen to connect the points on the conformal manifold, at large a preferred path emerges, since the conformal manifold effectively becomes one-dimensional. We explain how to implement this procedure explicitly, and provide various examples, in section 4.3.
3.3 Basis-independent 3-point functions
So far we have discussed correlation functions in a specific basis of chiral operators, where both 2- and 3-point functions are non-trivial. In general, however, it is customary to work in a different basis, where the 2-point functions are unit normalized, and all the non-trivial information is encoded in the coupling-constant dependence of higher-point correlators. Alternatively, we can work directly with quantities that are manifestly free from ambiguities arising from the choice of basis. In geometric language, we can look at scalar quantities on the conformal manifold. The simplest such quantity that can be constructed solely from the chiral ring data is
This object is closely related to the “properly normalized” 3-point functions defined in .
For example, in the case of gauge group , the chiral ring is generated by and the 2-point functions in the chiral ring are given by , so we have
where are the 3-point functions written in a basis where the 2-point functions are unity. These quantities were computed exactly in .
In order to compute (3.15) at large , we would need to compute the 2-point functions of all the chiral operators (both single- and multi-trace) to the appropriate order in and then take the large- limit. The leading order term in is a combinatoric constant determined by large- factorization, so to get non-trivial results we have to consider the terms of order in (3.15).
It is easy to see that these corrections are captured by the leading order free energy . Indeed, we can work in a basis where , so that (schematically)
is the limit of the connected 4-point function where the (anti-)chiral operators are sent to the same point. In the large- limit, the first two terms in the expression above give the factorized contribution to , while the connected 4-point function behaves as . The free energy is the generating function for the connected correlators, hence we conclude that the correction to can indeed be computed (at least in principle) from the leading term in the free energy .
In fact, in the case where , , we can derive an explicit relation for the correction to in terms of the single-trace 2-point functions (3.8). This is possible because this quantity obeys a very simple recursive relation coming from the equations that can be solved explicitly.
3.3.1 equations and
We recall the general equations for a (complex) 1-dimensional moduli space
If we contract the indices appropriately and define the quantity