Two-point Correlators in N=2 Gauge Theories

We consider two-point correlators in SU(N) gauge theories on R4 with N=2 supersymmetry and Nf massless hypermultiplets in the fundamental representation. Using localization on S4, we compute the leading perturbative corrections to the two-point functions of chiral/anti-chiral operators made of scalar fields. The results are compared at two and three loops against direct field theory computations for some special operators whose correlators remain finite in perturbation theory at the specific loop order. In the conformal case, the match is shown up to two loops for a generic choice of operators and for arbitrary N.


Introduction
Non-perturbative effects in gauge theories with N = 2 supersymmetry in four dimensions can be computed in many different ways. A method which has lately received a lot of attention is localization that provides exact results for the field theory path-integral by localizing it around a finite set of critical points 1 . In N = 2 gauge theories this localization is typically achieved after giving a vacuum expectation value (vev) to the scalar field in the vector multiplet, and deforming the space-time geometry to break the gauge and Lorentz symmetries of the system. In the physics literature these deformations are usually called the Ω-background [4,5]. Proceeding in this way, one can explicitly evaluate the partition function Z of the gauge theory on R 4 and derive the celebrated Seiberg-Witten prepotential F that is identified as the free energy of the system after the Ω-background is turned off. Both Z and F, which receive non-perturbative contributions from instantons, are holomorphic functions of the complexified gauge coupling τ of the theory and of the scalar vevs.
The localization ideas can be extended also to more general space-times. For example, the case of N = 2 gauge theories defined on a four-sphere S 4 was worked out in great detail in [6]. In this case the path-integral receives contributions from both instantons and anti-instantons, and localizes on configurations with purely imaginary scalar vevs. The corresponding partition function, which we denote as Z S 4 , is given by the modulus square of the one on R 4 integrated over all possible imaginary vevs. In this picture, holomorphic and anti-holomorphic contributions are associated, respectively, to instantons and antiinstantons that localize around the north and south poles of the sphere. This partition function is at the core of this paper, in which we want to test the predictions of localization against the results obtained by computing the Feynman diagrams in standard perturbation theory. To this aim, we focus on the weak coupling regime where instanton and antiinstanton contributions can be discarded.
Perturbative tests of the localization formulae have already been performed for Wilson loops in N = 4 theories for which both one-loop and instanton corrections to the gauge partition function are absent. In this case, the localization formula reduces to a Gaussian matrix model integral counting the number of ladder diagrams, in full agreement with the field theory predictions [7]. Further investigations in this context have concerned the computation of the OPE coefficients between circular Wilson loops and chiral operators [8][9][10], and the computation of half-BPS Wilson loops [11] in superconformal N = 2 super Yang-Mills theories. In all cases a perfect agreement between the results of localization and those of perturbation theory has been found.
More recently, a series of papers appeared in which two-point correlators of chiral/antichiral operators in N = 2 superconformal theories are computed for low [12][13][14][15] and large [16][17][18] values of the rank of the gauge group. In a conformal theory, the two-point function between a chiral operator O i and an anti-chiral operatorŌ j can be written as , (1.1) where x 12 = x 1 − x 2 , g is the gauge coupling, and ∆ 0 and γ(g) are the engineering and anomalous dimensions of the operators. In [12][13][14][15], it was shown that the functions A ij (g) can be obtained by taking suitable derivatives of a deformed partition function on the sphere, or equivalently by computing two-point correlators in the associated matrix model. This correspondence was tested against the explicit result of standard perturbation theory up to two loops for half-BPS operators of low-dimensions in superconformal SU(N ) gauge theories with N = 2, 3, 4.
In this paper we elaborate on this approach and extend it also to non-conformal models. For definiteness, we consider N = 2 theories with gauge group SU(N ) and N f massless hypermultiplets in the fundamental representation, and study their two-point correlators. Although localization produces a finite result for any choice of operators, when the theory is non-conformal, i.e. when N f = 2N , the two-point functions computed on the gauge theory side are in general divergent and have to be renormalized. To understand in detail the relation between the results in perturbative field theory and those obtained using localization is our ultimate goal. As a first step in this direction, here we focus on correlators that are already finite in field theory at a specific loop order. In this case we are able to provide some highly non-trivial tests of the correspondence. More precisely, we consider two-point functions that vanish up to a given loop order L − 1. Under this assumption, we show that the leading contribution at L loops is finite and the correlator takes the form with specific coefficients A ij L . We find a certain number of operators in this class for L = 2, 3 and for all of them we compare the results for A ij L coming from localization against those obtained by standard field theory methods, finding a perfect match. In the superconformal case we prove the correspondence up to two loops for any operator and for any N , thus generalizing the results of [12][13][14][15].
This paper is organized as follows. In Section 2 we derive the two-point correlators using localization methods. In Section 3 we compute the corrections to the two-point functions in perturbation theory at two and three loops. In Section 4 we summarize our results and discuss the correspondence between localization and field theory. Finally, in Section 5 we present our conclusions. Some technical issues are discussed in the appendices.

Two-point correlators in the matrix model
We consider a four-dimensional N = 2 super Yang-Mills theory with gauge group SU(N ) and N f massless hypermultiplets in the fundamental representation. As discussed in the Introduction, we are interested in the two-point functions between chiral operators made out of the complex scalar field ϕ sitting in the vector multiplet, and anti-chiral operators analogously defined in terms of the conjugate fieldφ. When the gauge theory is superconformal, i.e. when N f = 2N , building on the results of [19,20] it has been shown in [12][13][14][15] that these two-point functions can be obtained by placing the theory on S 4 and taking suitable derivatives of a deformed partition function. The latter can be computed using localization in terms of a matrix model [6]. Thus, matrix model correlators for the theory on S 4 encode information on correlators of the gauge theory in R 4 . Here we extract the field theory correlators from matrix model correlators involving the insertion of "normalordered" operators and argue that the correspondence extends, in a suitable sense, to non-conformal models.
We begin by briefly reviewing the interacting matrix model of [6]; then we show how to efficiently compute correlators of matrix operators for arbitrary N . In this way we do not only retrieve the results of [12][13][14][15] for the SU(N ) superconformal theories with N = 2, 3, 4, but we also extend them to generic N . Later we consider a special class of correlators in non-conformal models.

The partition function on S 4
As shown in [6], the partition function of an N = 2 super Yang-Mills theory with SU(N ) gauge group defined on a four-sphere S 4 can be written in terms of a traceless N × N Hermitian matrix a in the following way: Here a u are the eigenvalues of a, the integration is over the real line, ∆ is the Vandermonde determinant where a uv = a u −a v , and Z(ia, τ ) is the gauge theory partition function on R 4 . The latter is computed using the methods of [5], with the assumption that the adjoint scalar ϕ(x) in the gauge multiplet has a purely imaginary vev given by ϕ = ia, and that the Ω-deformation is parameterized by 1 = 2 = 1/R, where R is the radius of S 4 which, for simplicity, we take to be 1. Finally, the δ-function in (2.1) enforces the special unitarity condition, and τ is the complexified gauge coupling: The gauge theory partition function Z can be written as a product of the classical, one-loop and instanton contributions, namely The classical part produces a simple Gaussian term in the matrix model: while the one-loop contributions from the gauge multiplet and N f matter multiplets can be written as where with G(x) being the Barnes G-function. In the weak-coupling limit g 1 that we will consider, instantons can be discarded since they are exponentially suppressed so that we can put At weak-coupling the integral (2.1) is dominated by the region of small a u , and thus we can expand the functions H appearing in (2.6) for small values of their arguments using Here ζ(n) is the Riemann zeta-function and γ is the Euler-Mascheroni constant. In this way the one-loop contribution can be viewed as an interaction term in a free matrix model: with S n (a) being homogeneous polynomials in a of order n. The first few are: + 30 tr a 4 tr a 2 − 20 tr a 3 2 . (2.12) Performing the rescaling the matrix model gets a canonically normalized Gaussian factor and the sphere partition function becomes The overall g-dependent prefactor in (2.14) will be irrelevant when computing correlators, while the terms of order g 2L in S int will account for effects that take place at L loops in the corresponding field theory computation. Therefore we will refer to the g 2 -expansion of S int as a loop expansion. Note that the Vandermonde determinant ∆(a) in (2.14) arises from the Jacobian of the change of variables from a general N × N matrix to its diagonal form in terms of its eigenvalues a u , and thus the integral (2.14) can be alternatively expressed using a flat integration measure da over all matrix entries, namely where c N is a g-independent constant and da is defined by formula (2.22) below.

Matrix model correlators
Given any function f (a) of the matrix a, its vev in the matrix model described above is defined as , (2.17) where in the second step we used (2.16). Since S int (a) can be expressed as a series in g 2 (see (2.15)), it is natural to view exp(−S int (a)) as an interaction term in a Gaussian matrix model and write Here, we have denoted with a subscript 0 the expectation value in the Gaussian matrix model which, for any function f (a), is defined as (2.20) Having perturbatively reduced the computation of vevs in the interacting matrix model to vevs in a Gaussian model, we now give some details on the latter. If we write a = a b T b where T b are the generators of SU(N ) in the fundamental representation, normalized such that tr and if we normalize the flat SU(N ) measure as For our later purposes, it is convenient to introduce the notation t n 1 ,n 2 ,... ≡ tr a n 1 tr a n 2 . . . 0 . while, after using (2.24), we get Higher order correlators can be computed using the fusion/fission identities (2.29) In fact, one can recursively relate any correlator involving a total of n matrices to the combination of correlators with n − 2 matrices obtained after a single Wick contraction. We find (for n > 2) t m,n−m−2,n 1 ,n 2 − 1 N t n−2,n 1 ,n 2 + n 1 2 t n+n 1 −2,n 2 − 1 N t n−1,n 1 −1,n 2 + n 2 2 t n+n 2 −2,n 1 − 1 N t n−1,n 1 ,n 2 −1 , and so on. For n = 0, 1, 2, these expressions acquire a particularly simple form: t 0,n 1 ,n 2 ,... = N t n 1 ,n 2 ,... , t 1,n 1 ,n 2 ,... = 0 , t 2,n 1 ,n 2 ,... = N 2 − 1 + n 1 + n 2 + . . . 2 t n 1 ,n 2 ,... .

(2.31)
These recursive relations, together with the initial conditions (2.26), can be used to derive finite N formulae in a very efficient way. A few examples are given below: (2.32)

Normal-ordered operators
Another important ingredient is normal ordering. Indeed, since we are ultimately interested in establishing a connection between correlators in the matrix model and correlators in the gauge theory, and since the latter involve chiral and anti-chiral operators that do not have self-contractions, we have to introduce matrix model operators that do not have selfcontractions either, i.e., that are normal-ordered. Given an operator O(a), subtracting all its self-contractions is equivalent to make it orthogonal to all the lower dimensional operators 2 . Let ∆ be the dimension of O(a), and {O p } be a basis of operators with dimensions lower than or equal to (∆ − 2). We denote by C pq the inverse of the matrix of their correlators since the sum in the right hand side of (2.34) precisely subtracts all self-contractions of O(a). We would like to stress that the definition (2.34) of normal ordering introduces an explicit g-dependence and the symbol : : g we have used underlines this feature.
As an example, let us consider O(a) = tr a 2 . The only operator of lower dimension is the identity, so using (2.34) one finds : tr a 2 : g = tr a 2 − tr a 2 . (2.36) The second term precisely subtracts the self-contraction contributions inside a correlator. Indeed, : tr a 2 : g : tr a 2 : g = tr a 2 tr a 2 − tr a 2 2 (2.37) If we explicitly compute the right hand side of (2.36), we find : tr a 2 : g = tr a 2 − (2.38) Notice that the g 2 -term is proportional to (2N − N f ) and thus vanishes in the conformal case. Actually, this is true for the g 2 -terms of all normal-ordered operators, since these corrections originate from insertions of S 2 which is proportional to (2N − N f ).
As will be more clear in the following, for the two-point functions studied in this paper the g-dependent terms in the normal-ordered operators will not contribute to the loop orders we will consider, i.e. they will be always subleading in the gauge coupling and thus can be neglected. For this reason, we find convenient to introduce a specific notation for the g → 0 limit of the normal order, namely Applying the definition (2.34) to operators of the form tr a n 1 tr a n 2 · · · , and using (2.32), it is quite straightforward to obtain the explicit expressions of the leading terms of other normal-ordered operators. For operators of dimensions up to six, beside (2.40), one finds the following results: : tr a 2 2 : = tr a 2 2 − (N 2 + 1) tr a 2 + : tr a 6 : = tr a 6 − 3 2 tr : tr a 3 2 : = tr a 3 2 + 9 2N tr a 2 2 − 9 2 tr a 4 + 9( : tr a 4 tr a 2 : = tr a 4 tr a 2 − For each of the expressions considered above, it is easy to check that all subsequent terms in the right hand side are obtained as self-contractions of the first one. For instance, for tr a 3 2 the terms proportional to tr a 2 2 and tr a 4 correspond to a single Wick contraction of tr a 3 2 , the term proportional to tr a 2 is the result of two Wick contractions and the last one arises from three contractions. As we mentioned above, for the two-point functions considered in this paper the gdependent terms in the normal-ordered operators are not really needed at the loop orders we study. To see this, we observe that in the conformal case the one-loop corrections vanish and therefore the first g-dependent terms are of order g 4 . These corrections lead to modifications of the two-point correlators at order g 8 , i.e. at four loops, which is beyond the approximation considered here. In the non-conformal theories, instead, we will focus on correlators that vanish at weak coupling up to a given loop order (specifically twoand three-loops) and restrict ourselves to the leading contributions (of order g 4 and g 6 , respectively). In these cases one can show that the g-dependent terms in the normal order definition yield subleading contributions to the correlator and thus can be neglected. To check this, it will be useful to observe that : tr a 2 n : g = : tr a 2 n : − : tr a 2 n−1 : + O(g 4 ) .
(2.42) Another useful property is that in the correlator between two normal-ordered operators, the normal ordering can be removed from one of them, namely (2.43)

Two-point correlators
A generic operator of dimension n can be written as .. tr a n 1 tr a n 2 · · · , (2.44) where the sum is over the partitions of n = s n s and the index i labels the various combinations that one can make using coefficients c i n 1 ,n 2 ,... . Of course, for SU(N ) we can restrict to n i > 1 since tr a = 0; thus the independent operators of dimension n are as many as the partitions of n in which the number 1 does not appear.
Alternatively, using the adjoint decomposition a = a b T b , we can write (2.44) as where R i is a gauge tensor. Thus, to any operator O i (n) (a) we can associate a completely symmetric n-index tensor R i .
We now define the two-point correlator A ij (n) (g) as the expectation value of two normalordered operators in the interacting matrix model, namely (2.46) The correlator A ij (n) (g) can thus be computed perturbatively in g using (2.15) and performing Wick contractions in the free matrix model. In this way we write (2.47) The first term of this expansion is simply given by Indeed, at tree level we can put S int = 0 and neglect all g-dependent terms in the normalordered operators; moreover, due to normal ordering, the only possibility we have is to in all possible ways using (2.24); in this way we immediately find the above expression.
At higher loops the calculations are less straightforward, and the results depend on the features of the model considered. In the next subsection we give some explicit examples, starting with the superconformal case.

The conformal case N f = 2N
When N f = 2N , the expansion (2.15) simplifies since (2.12) becomes (2.49) The correlators that appear in the right hand sides can be evaluated in the free Gaussian model for any N using the recursion relations (2.30) in a straightforward manner. Let us consider some explicit examples involving the operators of low dimensions, which we list here: Up to three loops, we find the following correlators: (2.52e) One can check that these formulae reproduce the results in [12][13][14][15] for N = 2, 3, 4 and generalize them to any N . In Section 3 we will compare these expressions with the perturbative computation of the corresponding correlators in field theory at two loops and find perfect agreement.

Special correlators
When N f = 2N things are more complicated. For example, the one-loop interaction term S 2 is no longer vanishing and the perturbative expansion has many more terms. We restrict our attention to a particular class of correlators that vanish up to a given loop order L − 1, i.e. that are of the form with the first non-vanishing contribution of order g 2L .
To make the analysis as simple as possible, we fix one of the operators to be This choice greatly simplifies the calculations in the matrix model but, as we will see in Section 3, it also helps in the perturbative computation of the corresponding correlator in field theory which becomes tractable without being trivial 3 . The two-point correlator we study is then To evaluate this correlator one makes use of the expansion of S int given in (2.15), which implies

(2.57)
Notice that if one already knows that all terms of order less than L vanish, then the denominator in the right hand side of (2.55) does not contribute at order g 2L and thus can be neglected. The first requirement is the vanishing of the zero-loop term, i.e.
Using (2.48) and (2.54), we immediately see that this condition is equivalent to i.e. the R i tensor associated to O i (2m) (a) must be totally traceless.
It is easy to show that if the zero-loop condition (2.58) holds, then also the one-loop contribution vanishes. Indeed, we have (2.60) To obtain the last line, one first writes the product : tr a 2 m : tr a 2 as a linear combination of normal-ordered operators : tr a 2 k : with k ≤ (m + 1), and then uses the fact that only the term with k = m in this combination can contribute to the correlator, yielding in the end a vanishing result because of the condition (2.58). The first non-trivial contributions occur then at two or more loops.

Two loops
Assuming the vanishing of the zero-and one-loop terms, we consider the two-loop contribution to the correlator A i (2m) (g), which reads Above we used the fact that multiple insertions of S 2 automatically vanish given the vanishing of the correlator at the previous orders, as we showed in considering the one-loop contribution, and took into account the explicit form of S 4 given in (2.12). Let us now consider some explicit examples. At dimension two (m = 1), there is a single operator, namely tr a 2 , which does not satisfy (2.59). So we cannot construct special correlators when m = 1.
At dimension four (m = 2), we have two independent operators, tr a 4 and tr a 2 2 , none of which satisfies the tracelessness condition (2.59). However their linear combination does. To see this, we first observe that : O 1 (4) (a) : : tr a 2 2 : 0 = tr a 4 + c 1 2,2 tr a 2 2 : tr a 2 2 : 0 = 2 k=0 c k tr a 4 + c 1 2,2 tr a 2 2 tr a 2 k 0 , where in the first step we have removed the normal ordering from O 1 (4) (a) according to (2.43), and in the second step we have written : tr a 2 2 : = 2 k=0 c k tr a 2 k according to (2.41a). Then, using the definitions (2.25), the zero-loop correlator becomes Exploiting the recursion relations (2.30) we see that each term in the above square bracket is proportional to t 4 + c 1 2,2 t 2,2 , which vanishes because of (2.64). Thus, the condition (2.58) is satisfied. Alternatively, one can check that the R tensor associated to O 1 (4) , namely is traceless in any pair of indices for the choice of c 1 2,2 given in (2.64). Using the above findings, we then obtain This expression follows directly from (2.62), and vanishes in the conformal case when N f = 2N . This analysis can be done also for operators of higher dimensions. For example, when m = 3 we find three operators of dimension six that satisfy the condition (2.58). They are where the coefficients, determined with the method described above, are . (2.71) Using these results in (2.61), we find (2.73) We notice again that these correlators vanish in the superconformal theories.
In the next section we will compare the correlators A i (4) (g) and A i (6) (g) with the corresponding ones computed in field theory using standard Feynman diagrams, finding perfect agreement.

Three loops
Our previous analysis shows that it is possible to find two independent linear combinations O i (6) , with i = 1, 2, of the operators (2.70) such that also the two-loop term of the corresponding correlator vanishes. Using (2.61), we see that this amounts to require The two independent operators can be chosen as . (2.76) When a correlator vanishes up to order g 4 , as is the case of the operators O i (6) we have just introduced, the first non-trivial contribution occurs at three loops and takes the form In (2.77) we used again the fact that terms containing S 2 vanish in this situation and, in the last step, inserted the explicit form of S 6 from (2.12) to obtain C i (6) . The explicit evaluation of this expression for the operators (2.75) gives the following result: (2.79) In the next section these findings will be shown to match with the corresponding ones obtained with a perturbative field theory computation at three loops.

Two-point correlators in field theory
In this section we compute two-point correlation functions in N = 2 gauge theories using standard Feynman diagram techniques. In order to be self-contained we begin by briefly describing the set-up and the Feynman rules in the superspace formalism.

Superfield actions and Feynman rules
We use the N = 1 superfield formulation of N = 2 theories in Euclidean space and collect our conventions and notations in Appendix A. In this formulation, the N = 2 vector multiplet consists of a N = 1 vector multiplet V and a chiral multiplet Φ in the adjoint representation of the gauge group. The corresponding gauge theory action is where g is the gauge coupling (for simplicity we have set to zero the Yang-Mills θ angle) and W α is the chiral superfield-strength of V , namely The last term in (3.1) implements the gauge fixing, with ξ = 1 corresponding to the Fermi-Feynman gauge 4 . With our conventions, the action S gauge , as well as the actions we are going to write hereinafter, is negative defined and thus it appears in the path integral as e Sgauge . Writing V = V a T a , Φ = Φ a T a where T a are the generators of SU(N ) in the fundamental representation normalized as in (2.21), and expanding up to second order in V the action (3.1) with ξ = 1 becomes where f abc are the structure constants of SU(N ).
In N = 1 language, a fundamental N = 2 massless hypermultiplet consists of two chiral multiplets Q andQ transforming in the fundamental and anti-fundamental representation, respectively, together with their anti-chiral partners Q † andQ † . The action for N f such hypermultiplets is which, up to second order in V , explicitly reads Here we have understood the summation over u, v = 1, · · · , N . The action of the N = 2 super Yang-Mills theory with N f massless hypermultiplets can therefore be written as S For our purposes it is convenient to view the pure N = 2 theory as a truncation of the maximally supersymmetric N = 4 theory. Indeed, the latter can be obtained by adding to the pure N = 2 theory the contributions of two adjoint chiral multiplets Φ 2 and Φ 3 , which together with their conjugate Φ † 2 and Φ † 3 build an adjoint N = 2 hypermultiplet H. Thus the action of the N = 4 theory is where in the second line we have used the notation Φ 1 = Φ. Up to second order in V , the action for the adjoint hypermultiplet is For later convenience, we now display the Feynman rules for the various superfields that can be obtained from the actions (3.4), (3.6) and (3.10). In configuration space, the superpropagators can be compactly written, using the following notations [21]: as indicated in Fig. 1. The cubic interaction vertices that can be read from the actions (3.4), (3.6) and (3.10) are listed in Fig. 2. There are also higher order interaction vertices, but they are not needed for our calculations and thus we do not write them explicitly here. In this set-up, a correlator in the N = 2 theory can be written in the form where A H corresponds to the contribution of the diagrams in the N = 4 theory in which the adjoint chiral multiplets Φ 2 and Φ 3 that build the hypermultiplet H run in the internal lines, and A Q corresponds to the diagrams in which matter chiral multiplets Q and Q run in the internal lines. We call them H-and Q-diagrams, respectively. Furthermore, if one considers correlators among BPS protected states, one can simplify (3.12) because such correlators do not receive corrections in the N = 4 theory, namely We will extensively make use of this formula to study the two-point functions of half-BPS operators in N = 2 theories and compare them with the matrix model results of the previous sections. In this respect it is important to realize that the H-diagrams that give rise to A H do not really exist in the N = 2 theories under consideration, but they provide a simple book-keeping device which greatly simplifies our analysis.
In the following we are going to consider two-point functions between chiral and antichiral operators. The former are constructed only with the lowest component of the chiral superfield Φ 1 = Φ of the N = 2 vector multiplet, that is the adjoint scalar while the latter are made only with the conjugate scalar Gauge invariant chiral operators of dimension n can be generically written as where R i is a totally symmetric n-index tensor of SU(N ). The superscript i labels the different such tensors and hence operators that can be considered. The anti-chiral operators can be defined in a similar way by replacing ϕ(x) withφ(x). To simplify the formulae in the following, we adopt the notation Examples of these are the multi-trace operators tr ϕ n 1 (x) · · · tr ϕ n (x) and trφ n 1 (x) · · · trφ n k (x) , (3.18) or combinations thereof. The two-point functions we will consider are for various choices of i and j. At tree-level we can easily compute this correlator using the propagator which is obtained from the ΦΦ † superpropagator in Fig. 1 by setting to zero all fermionic superspace coordinates, and fixing I = J = 1. Performing all contractions, we find We can pictorially describe this result as shown in Fig. 3 where each oriented line represents the tree-level propagator (3.20). At higher orders the calculation of the two-point function (3.19) is less straightforward and crucially depends on the type of N = 2 theory one deals with. In the next subsection we start by considering the superconformal theories corresponding to N f = 2N .

The conformal case N f = 2N
When N f = 2N , the N = 2 theory is superconformal invariant. In this case many simplifications occur: for example, due to the vanishing of the β-function, the gauge coupling is not renormalized and the chiral/anti-chiral operators retain their engineering scaling dimensions without anomalous terms. Furthermore, the one-loop corrections to the propagators and to the three-point coupling exactly cancel between the Q-and H-diagrams, so that any diagram containing this difference as a subdiagram vanishes for arbitrary choices of the operators O i (n) and O j (n) . For example, the one-loop diagram represented in Fig. 4 vanishes. This implies that The only diagrams that can contribute at two loops are those containing the irreducible corrections to the propagator represented in Fig. 6, or the two-loop diagrams drawn in Fig. 7.     Figure 7: The two-loop diagrams that can contribute to the two-point function (3.19). The left diagram refers to the contribution of the matter hypermultiplets while the right diagram refers to the adjoint hypermultiplet H.
Their sum yields the two-loop correlator, which we write as Here we adopted the same symbol A ij (n) (g) used for the matrix model correlators in the previous section since, as we will show, the two coincide.
We start by analyzing the contribution of the irreducible two-loop subdiagrams of Fig. 6. Since the propagators of the superfields that make the hypermultiplets Q and H differ only in their gauge index structure, the difference between the Q-and H-subdiagrams can be written in a factorized form as 5 where W 2 (x 12 ) is the result of the integral over the superspace internal variables and the colour factor C ab 2 , following from the Feynman rules of Fig. 2, is given by with Tr adj being the trace in the adjoint representation. Expressing this trace in terms of traces in the fundamental representation and using the fusion/fission rules (2.28), we can rewrite this factor as follows where in the last step we used the superconformal value N f = 2N . The total contribution of these diagrams to the correlator (3.23) is obtained by performing all contractions between O i andŌ j as in the tree-level computation but with one propagator replaced by (3.24). To perform these contractions in an efficient way, we introduce the notation ϕ = ϕ a T a andφ =φ a T a to denote auxiliary scalars with the Wick contraction ϕ aφb = δ ab . (3.27) We also write the operators as O i (n) (ϕ) and O j (n) (φ) by replacing ϕ(x) andφ(x) with ϕ andφ in O i (n) (x) andŌ j (n) (x), respectively. Then, one can easily realize that replacing one propagator with (3.24) corresponds to inserting in the tree-level correlator the following effective vertex Here and below we will always represent effective vertices by normal-ordered products (denoted by : :) of ϕ andφ in which all self-contractions are discarded. The scalars ϕ and ϕ of V 2 will contract with, respectively, oneφ of O j and one ϕ of O i ; so in this way the 5 The overall prefactor is −(2g) 2 (i √ 2g)(−i √ 2g) = −8g 4 and comes from the normalization of the interaction vertices involved in these diagrams.
replacement of one propagator with the expression in (3.24) is correctly implemented. The contribution of the diagrams of Fig. 6 to the two-loop correlator (3.23) is therefore The last ingredient is the superspace integral W 2 (x 12 ) which, as shown in Appendix C.1, is (3.31) Using this, we see that (3.30) becomes Let us now turn to the diagrams in Fig. 7. Focusing on the irreducible subdiagrams, we see that just as before, the Q − H difference can be written in a factorized form as 6 where W 4 (x 12 ) is the result of the integral over the superspace internal variables and the colour factor following from the Feynman rules is (3.34) The last equation follows from rewriting the adjoint trace in terms of the traces in the fundamental representation. The total contribution of these subdiagrams to the correlator is obtained by performing all contractions as in the tree-level computation but with two propagators replaced by the sub-correlator (3.33). This amounts to inserting in the tree-level correlator the following effective vertex −2g 4 W 4 (x 12 ) V 4 (ϕ,φ) , (3.35) 6 The overall numerical factor 2g 4 is explained as follows: we have a factor of (i coming from the insertions of the four three-point vertices, times a factor of 2 coming from the permutations of the fields Φ2 and Φ3 (or Q and Q) inside the loop. Indeed one can make two independent loops with two Φ2 and two Φ3 (or two Q and two Q) in alternating positions as required by the Feynman rules. In general, the symmetry factor of a loop diagram with 2k three-point vertices is given by ( is the number of independent loops that can be made with k Φ2 and k Φ3 (or k Q and k Q) in alternating positions. The net result is therefore 2 k g 2k k . where : ϕ a 1φb1 ϕ a 2φb2 : = (2N − N f ) : tr(ϕφ) 2 : +4 : (tr ϕφ) 2 : + 2 : tr ϕ 2 trφ 2 : .

(3.36)
In the superconformal case N f = 2N the first term of V 4 is absent. Notice again that the normal ordering in V 4 is necessary to correctly implement the replacement of two propagators with (3.33). Using the effective vertex (3.35), the two-loop correlator corresponding to the sub-diagrams of Fig. 7 is The superspace integral W 4 (x 12 ), which is evaluated in Appendix C.2, is Adding (3.32) and (3.39), we find : (tr ϕφ) 2 : + 2 : tr ϕ 2 trφ 2 : +2(N 2 + 1) : tr ϕφ : .
We have evaluated this expression for the various operators listed in (2.51) and, at two loops, found perfect agreement with the matrix model results (2.52).

Special correlators
When N f = 2N , the N = 2 theories are not superconformal invariant and many of the above simplifications no longer occur. More in general we need to implement a renormalization procedure. For example, the gauge coupling has a non-vanishing one-loop β-function and the chiral/anti-chiral operators acquire anomalous dimensions. It is nevertheless possible to find a set-up where the calculations of the two-point functions remain simple and where a contact with the matrix model results of the previous sections can be established in a direct manner. To do so, we choose operators such that their two-point function vanishes up to a given loop order (L − 1). The L-loop contribution is thus finite and does not need to be renormalized. Therefore, we restrict our attention to special correlators that take the following form for L ≥ 1. Then, in order to keep the discussion and the calculations as simple as possible, we fix the anti-chiral operator to be Thus, we will study the following special correlators for L ≥ 1 and various choices of i. Of course, the first condition we have to require is the vanishing of the tree-level correlator Using (3.43), it is immediate to realize that (3.45) is equivalent to the "tracelessness" condition of the R-tensor, namely R i a 1 a 2 ···a 2m δ a 1 a 2 · · · δ a 2m−1 a 2m = 0 . Indeed, the only Q-and H-diagrams that can contribute correspond to the one-loop correction of the chiral propagator (see Fig. 4). When N f = 2N , the Q − H difference is not vanishing but it is still diagonal in the color indices, and thus leads to a vanishing result after the tracelessness condition (3.46) is used. Therefore, the first non-trivial corrections occur at two or higher loops (L ≥ 2). We are going to consider them in detail in the next subsections.

Two loops
At two loops there are several Q-and H-diagrams that a priori can contribute to the correlator but many of them have a vanishing colour factor. For example, it is easy to realize that all diagrams accounting for the two-loop corrections of the chiral propagator can be disregarded since these corrections are diagonal in the color indices and give a vanishing contribution to the correlator (3.48) upon using the tracelessness condition (3.46). In particular, we can discard the two-loop diagrams of Fig. 6. Similarly, we can disregard the Qand H-diagrams involving corrections to the gauge coupling represented in Fig. 5. Indeed, these diagrams have a colour factor of the form and their contribution to the full correlator will lead either to which vanish because of the tracelessness condition (3.46), or to which vanish due to the symmetry/anti-symmetry properties of R and f . The only two-loop diagrams that can contribute to our special correlators are those represented in Fig. 7. Proceeding as above, we can use the effective vertex (3.35) and show that the two-loop correlator becomes with V 4 given in (3.36). Using the explicit expression (3.38) for the superspace integral W 4 (x 12 ), after collecting all terms we find Specifying to the operators (2.63) and (2.70) that correspond respectively to m = 2 and m = 3, one finds perfect agreement between the field theory correlator (3.54) and the matrix model results (2.69) and (2.73).

Three loops
The computation of the three-loop correlators goes along the same lines. In this case we consider operators such that the two-point function vanishes up to two loops. This condition is equivalent to require that R i a 1 a 2 ···a 2m δ a 1 a 2 · · · δ a 2m−1 a 2m = 0 , R i a 1 a 2 ···a 2m C a 1 a 2 a 3 a 4 4 δ a 5 a 6 · · · δ a 2m−1 a 2m = 0 , where the tensor C 4 is defined in (3.34). Under these conditions, the first contribution to the two-point function is represented by the three-loop correlator For m = 3 we have verified that this three-loop correlator receives contributions only from the Q-and H-diagrams represented in Fig. 8. Following the same steps as before, one finds that the three-loop correlator is obtained by performing the same contractions as in the tree-level computation but with three propagators replaced by the following effective vertex 7 where W 6 (x 12 ) comes from the integration of the superspace internal variables and the colour factor is This amounts to insert in the tree-level correlator the expression : ϕ a 1φb1 ϕ a 2φb2 ϕ a 3φb3 : = (2N − N f ) : tr(ϕφ) 3 : +6 tr : ϕ 3φ trφ 2 : +6 tr :φ 3 ϕ tr ϕ 2 : + 6 tr : ϕ 2φ2 tr ϕφ : +12 tr : (ϕφ) 2 tr ϕφ : −2 tr : ϕ 3 trφ 3 : − 18 tr : ϕ 2φ trφ 2 ϕ : . (3.62) The three-loop contribution to the correlator (3.56) for m = 3 can therefore be written as Thus, collecting all factors we obtain Specifying to the operators (2.75), we again find perfect agreement with the matrix model results (2.79).

Correspondence between matrix model and field theory
In the previous sections we have provided explicit evidence for the localization formula where the correlator in the quantum field theory side is assumed to be finite at a specific loop order. As it stands, this relation has been checked in the superconformal case for any half-BPS operators at two loops, while in the non-conformal case it has been proven for certain class of operators at two and three loops. However, we think it should be possible to generalize and suitably extend this relation to any two-point function. We postpone the interesting question of how to deal with divergences and renormalization to a future work.
To set the stage for these developments, let us further comment our results. At tree level, the relation (4.1) is quite straightforward to prove since the matrix model effectively counts the same tree-level contractions of the field theory. However at higher orders, as we have seen, the correspondence is less obvious.

The conformal case N f = 2N
In the superconformal case, for arbitrary choices of the operators O i (n) andŌ j (n) , there are no corrections at one loop, while at two loops we have shown that the field theory correlator (3.40) matches the one coming from the matrix model using the localization formula (2.50). This confirms the results of [12][13][14][15] and extends them to generic values of N and generic operators. Such a match can be made even more compelling if we recall that the two-loop corrections originate in the matrix model from the insertion of − S 4 (a) − S 4 (a) in the tree-level correlator (see (2.50)). By rewriting this in terms of normal-ordered operators as where the ellipses stand for the terms with different number of ϕ andφ which give a vanishing contribution if inserted in the chiral/anti-chiral two-point function since they violate the U(1) charge conservation. Using these expressions and discarding these unbalanced terms, we therefore find in which it is easy to recognize precisely the same combination appearing in (3.41). This relation explains on general grounds the two-loop agreement between the matrix model and the field theory results for the two-point function of arbitrary operators O i (n) andŌ j (n) in the superconformal case.

Special correlators
A very similar treatment can be made at two and three loops for the special correlators in which the anti-chiral operator is chosen to be tr(φ) 2 m . In this case, on the matrix model side the two-loop corrections arise simply from the insertion of −S 4 (a). Then, making the replacement a → ϕ +φ, we have where the vertex V 4 is defined in (3.36) 8 . Here the ellipses stand again for terms with an unbalanced number of ϕ andφ which give a vanishing contribution, or for terms proportional to the structure constants arising from commutators of ϕ andφ, which also give no contribution in the special correlators due to (3.51). This implies that the match with the localization formula is not limited to the operators we considered in Section 3.3.1, but can be extended to any special correlator involving an operator O i (2m) with a traceless R i tensor and arbitrary m.
At three loops, the special correlators are obtained from the insertion of −S 6 (a). Repeating the same steps as above, we have where the vertex V 6 is defined in (3.62) and the dots have the same meaning as before. This relation suggests that the field theory results should match those derived from the localization formula (2.78) for any choice of operators O j (2m) , and not only for the ones with m = 3 we have considered in Section 3.3.2. It would be nice to explicitly verify this expectation.

Concluding remarks
The results presented in this paper provide some explicit evidence of a direct relation between matrix model correlators and field theory correlators. Of course many important issues remain to be addressed and clarified. From a technical point of view it would be interesting to extend our checks to three loops in the superconformal case and to four loops for the special correlators, and also to generalize the dictionary to three-point functions. From a more conceptual point of view it would be important to understand how to incorporate in this correspondence the effects of the renormalization of the gauge coupling constant and of the anomalous dimensions that arise in the non-superconformal theories when one considers generic operators, and also to include non-perturbative effects due to instantons. Moreover, it would be nice to thoroughly explore the connection between the original matrix model with the Hermitian matrix a and the effective matrix model with the complex matrices ϕ andφ, which we have just highlighted in (4.3), (4.5) and (4.6).
Once all these issues are clarified, the extreme simplicity of the matrix model approach and of localization could be fully exploited for field theory calculations. We leave the discussion of some of these important points to future work.
We raise and lower the indices as follows where ε 12 = ε 21 = ε12 = ε21 = +1. From these rules it follows that The matrices (σ µ ) αβ and (σ µ )α β are defined by where τ are the ordinary Pauli matrices. They are such that and satisfy the Clifford algebra Using these matrices and the above rules, we can prove In Euclidean space, chiral and anti-chiral spinors satisfy the pseudoreality conditions According to this, the chiral superspace coordinates y µ = x µ + i θσ µθ are invariant under this conjugation: y µ † = y µ . Similarly, the anti-chiral coordinatesȳ µ = x µ − i θσ µθ are such thatȳ µ † =ȳ µ . These coordinates satisfȳ where D α andDα are defined in (3.3). Finally, the integration over Grassmann variables is defined such that

B. The Φ (L) -functions
In this appendix we review the derivation of some integrals required in the two and three loop computations. We refer the reader to [22] for details. The integrals we are interested in belong to an infinite sequence computing the L-loop contribution of ladder diagrams to the four-point function in φ 3 -theory. They are given by with y ij = y i −y j . These integrals have been computed in [22] and the result can be written in the compact form as follows 9 where u and v are the cross ratios and Φ (L) (u, v) are the so-called Davydychev functions defined as

(B.4)
Here and Li n is the index n polylogarithm We are interested in the limit x 4 → x 1 , which is equivalent to sending v → u and then u → ∞. Using (B.4), one finds lim v→u u→∞ where the dots stand for subleading terms. Then, inserting this in (B.2) one obtains

C. Superdiagram calculations
In this appendix we compute the two-and three-loop integrals contributing to the two-point functions considered in Section 3. We introduce the short-hand notation where ξ ij = i θ i σθ j denotes the chiral superfield propagator in which the colour and flavour indices are suppressed (see Fig. 1) and denotes the vector superfield propagator without colour indices (see Fig. 1).

C.1 The integral W 2
The first integral we consider is W 2 corresponding to the diagrams in Fig. 6. The two diagrams there differ only in the colour and flavour factors and have the same superspace structure. So for the computation of W 2 it is enough to consider just one of them, say the H-diagram, which we redraw in Fig. 9 without colour indices and where the labels indicate the points where the external fields or the interaction vertices are inserted. Using the notation (C.1) and (C.2), the integral W 2 corresponding to Fig. 9 is given by (C.3) 10 Here, for convenience we relabel yi → x3+2i. Although W 2 is a two-loop contribution, it involves a four-fold coordinate space integral. This is a common feature when computing Feynman diagrams in coordinate space and one may ask why we prefer to use the coordinate space approach instead of the more common one in momentum space where one naively would expect only a two-fold integral at two loops. There are two reasons for this. The first one is that this simple counting works only for correlators of fundamental fields. For composite operators, instead, at a fixed order in g 2 , the number of momentum integrations grows linearly with the number of fields in the operator (see [23,24]). The second reason is that even for finite coordinate space functions, the corresponding momentum space expressions may be ill defined (for example, think of the Fourier transform of 1/x 4 ) and require regularization. Moreover the coordinate space approach is easier to generalize to three and higher point functions.
Let us now evaluate the integrals in (C.3). Note that since the external fields at the points x 1 and x 2 are scalars, we can set θ j =θ j = 0 for j = 1, 2. Performing the θ andθ integrations, and combining all terms we get (C.6)

C.2 The integral W 4
The second superspace integral we have to compute is W 4 corresponding to the diagrams in Fig. 7. Again, the two diagrams there differ only in the colour and flavour factors and have the same superspace structure. So for the computation of W 4 it is enough to consider just one of them, say the H-diagram. Furthermore, it is convenient to "unfold" the irreducible two-loop subdiagram and redraw it as shown in Fig. 10. We write the associated superspace integral as W 4 (x 12 ) = lim This corresponds to a point-splitting regularization of the positions of the external legs that will be made to coincide only at the end of the computation. Indeed, although, the total superdiagram is finite, a regularization is necessary, since at intermediate steps one has to manipulate divergent or ambiguous (like 0/0) expressions. Using the notation in (C.1), the superspace integral W 4 can be written as Then one can perform the integrals in x 6 and x 8 by using (C.5) and exploiting the identity This identity follows by differentiating the explicit expression for the integral in terms of the Davydychev Φ (1) function introduced in Appendix B. Using Tr(σ µσν σ ρστ ) = 2 (δ µν δ ρτ − δ µρ δ ντ + δ µτ δ νρ − µνρτ ) , (C.14) and simplifying the scalar products by means of after some algebra we obtain The integral in the right hand side has been computed in [22] (see also (B.9)). It is finite in the limit x 4 → x 1 but it is quadratically divergent in the limit x 3 → x 2 . However, taking into account also the factor of x 2 23 that is present in W 4 we find in the end a finite result for the two point function. Indeed, W 4 (x 12 ) = lim The last superspace integral we compute is W 6 corresponding to the diagrams in Fig. 8. As in the previous cases, also here the only difference between the two diagrams is in the colour and flavour factors, while the superspace structure is the same. Thus, in order to compute W 6 it is enough to consider just one of them, say the H-diagram, whose irreducible three-loop part is represented in its "unfolded" version in Fig. 11. The relevant integral W 6 (x 12 ) is six-fold and, as suggested by the hexagon diagram, it is regularized by point-splitting, namely W 6 (x 12 ) = lim Inserting this expression in (C.20),we can perform three of the integrations with the help of (C.13). We choose them to be those over x 7 , x 8 , x 9 , but equivalently we could have integrated in x 7 , x 8 , x 9 . Evaluating the trace of the sigma matrices and using (C.15) to express everything in terms of squares, we find