Double scaling limit of N=2 chiral correlators with Maldacena-Wilson loop

We consider $\mathcal N=2$ conformal QCD in four dimensions and the one-point correlator of a class of chiral primaries with the circular $\frac{1}{2}$-BPS Maldacena-Wilson loop. We analyze a recently introduced double scaling limit where the gauge coupling is weak while the R-charge of the chiral primary $\Phi$ is large. In particular, we consider the case $\Phi=(\text{tr}\varphi^{2})^{n}$ , where $\varphi$ is the complex scalar in the vector multiplet. The correlator defines a non-trivial scaling function at fixed $\kappa = n\,g_{\rm YM}^{2}$ and large $n$ that may be studied by localization. For any gauge group $SU(N)$ we provide the analytic expression of the first correction $\sim \zeta(3)\,\kappa^{2}$ and prove its universality. In the $SU(2)$ and $SU(3)$ theories we compute the scaling functions at order $\mathcal O(\kappa^{6})$. Remarkably, in the $SU(2)$ case the scaling function is equal to an analogous quantity describing the chiral 2-point functions $\langle\Phi\overline\Phi\rangle$ in the same large R-charge limit. We conjecture that this $SU(2)$ scaling function is computed at all-orders by a $\mathcal N=4$ SYM expectation value of a matrix model object characterizing the one-loop contribution to the 4-sphere partition function. The conjecture provides an explicit series expansion for the scaling function and is checked at order $\mathcal O(\kappa^{10})$ by showing agreement with the available data in the sector of chiral 2-point functions.


Introduction
Recently, a certain interest has been devoted to the large R-charge limit of four dimensional N = 2 superconformal theories [1,2]. Besides, this limit may be conveniently combined with a weak coupling expansion and tuned in order to provide a non-trivial scaling behaviour. In this mixed regime we can neglect instanton contributions while keeping some interesting (scaled) coupling dependence. Typically, one has a vanishing Yang-Mills coupling g YM → 0 while the R-charge grows like 1/g 2 YM . Extremal correlators of chiral primaries in conformal N = 2 QCD have been computed in such a double scaling limit [3]. If Φ n is a chiral primary whose R-charge increases linearly with n, one may consider the normalized ratio between the 2-point functions in the N = 2 theory and in the N = 4 SYM universal parent theory. This defines the scaling function for gauge group SU (N ) (1.1) where κ = n g 2 YM is the fixed coupling at large R-charge. For certain classes of chiral primaries Φ n , it is possible to compute the perturbative expansion of the function F Φ by exploiting the integrable structure of the N = 2 partition function. In the simplest setup, Φ n is the maximal multi-trace tower Φ n = Ω n ≡ (Trϕ 2 ) n where ϕ is the complex scalar field belonging to the N = 2 vector multiplet. The 2-point functions Ω n Ω n are then captured by a Toda equation following from the four dimensional tt * equations [4], i.e. the counterpart of the topological anti-topological fusion equations of 2d SCFTs [5,6]. By exploiting the Toda equation in order to control the R-charge dependence, it is possible to compute the scaling function in (1.1) at rather high perturbative order [7]. This approach, based on decoupled semi-infinite Toda equations, has been proved to admit generalizations to broader classes of primaries and is believed to be a general feature of Lagrangian N = 2 superconformal theories [8].
In this paper, we reconsider the double scaling limit in (1.1), but for a different class of correlators, i.e. the 1-point function Φ n W of large R-charge chiral primaries Φ n in presence of a circular 1 2 -BPS Maldacena-Wilson loop W [9,10]. In conformal SQCD, it is possible to compute such correlators by localization as thoroughly studied in [11]. Other applications of localization to Wilson loops in N = 2 superconformal theories may be found in [12][13][14][15]. For our purposes, we shall again be interested in the maximal multi-trace case, i.e. Φ = Ω n .
As is well known, the localization computation is based on the partition function of a suitable deformation of the N = 2 theory on S 4 [16][17][18]. However, the map from S 4 to flat space requires to disentangle a peculiar operator mixing induced by the conformal anomaly [19]. This is an annoying feature as far as the analysis of the n → ∞ limit is concerned. Indeed, the mixing structure becomes more and more involved with growing n. In particular, one should need results like those in [11], but with a fully parametric dependence on n. Besides, as in the study of the large R-charge limit of chiral 2-point functions, we want to work with a generic SU (N ) gauge group with (finite) fixed N . 1 The drawback is that finite N results may display a deceiving complexity for large R-charge. For a discussion of what simplifications occur in the mixing problem at large N see [20][21][22][23].
To overcome these difficulties, we exploit special features of the Ω n operators. For generic R-charge n and gauge group SU (N ), we provide the solution to the mixing problem in the maximally supersymmetric N = 4 SYM theory. Similarly, we give as well exact expressions for the first genuine N = 2 correction ∼ ζ(3). These findings are a useful guide to study higher transcendentality contributions. Our results allows to consider the n → ∞ limit of the ratio of 1-point functions taken with fixed κ = n g 2 YM in the SU (N ) theory. The quantity in (1.2) is the simplest natural object to be studied in presence of the Wilson loop and corresponding to (1.1). Our analysis shows that the limiting scaling function G(κ; N ) is well defined and non-trivial. In the SU (2) theory, we check the remarkable equality G(κ; N = 2) = F (κ; N = 2) at least at order O(κ 6 ). This identity does not hold for N > 2 as follows from a study of the G function in the SU (3) theory again at order O(κ 6 ).
The case of SU (2) is definitely special. Based on certain universality arguments we formulate a simple conjecture for the all-order expansion of F (κ; 2) in terms of a certain N = 4 expectation value of the one-loop contribution to the N = 2 matrix model partition function. We have checked the conjecture at order O(κ 10 ) by reproducing the results of [7].
The plan of the paper is the following. In Section 2 we summarize recent developments about the large R-charge double scaling limit in the sector of chiral 2-point correlators. In Section 3 we briefly set up the calculation with the Maldacena-Wilson loop and recall the localization algorithm to map flat space correlators to definite matrix model integrals. In Section 4 we exploit some special features of the operators Ω n to solve the associated mixing problem in the N = 4 theory, to compute the correlator with the Wilson loop for a generic R-charge, and to evaluate the first correction ∼ ζ(3) for a general gauge algebra rank. Section 5 is devoted to a focused study of the large R-charge limit. We first discuss in a rigorous way the universality of the ζ(3) correction. Next, simple educated assumptions allow us to compute the scaling function G(κ; N ) at sixth order in κ for the SU (2) and SU (3) theories. In the final Section 6 we present a conjecture for the all-order expression of the scaling function in the SU (2) theory based on the identification of a particular universal and natural object in the N = 2 matrix model which turns out to encode the scaling function itself.
2 Large R-charge double scaling in N = 2 conformal SQCD We work in flat 4d space and consider N = 2 conformal SQCD, i.e. SYM with gauge group SU (N ) and 2N fundamental hypermultiplets. Chiral primary operators are primaries annihilated by half of the supercharges. They have scaling dimension ∆ and quantum numbers (R, r) of the R-symmetry SU (2) R × U (1) r obeying R = 0 and ∆ = r 2 . If ϕ is the complex scalar field in the vector multiplet, a generic scalar chiral primary is labeled by a vector of integers n = (n 1 , . . . , n ) and reads 2 Superconformal symmetry predicts the diagonal 2-point function where G n,m is a function of the gauge coupling g. In the recent papers [3,7] a special role has been played by the operators Ω n ≡ O 2,...,2 n = (Trϕ 2 ) n , (2.3) and the associated 2-point function coefficients G 2n = (Trϕ 2 ) n (Trϕ 2 ) n have been computed. The functions G 2n are captured by a semi-infinite Toda equation [19,3,8] that allows to compute them, after normalization to their N = 4 SYM value, in the large R-charge limit 3 n → ∞, g → 0, κ = n g 2 = fixed. (2.4) One finds the expansion where the coefficients c ( ) s (N ) have been computed at order O(λ 10 ) in [7]. The first cases are (2. 6) It has been conjectured that all terms associated with multiple products of zeta functions do vanish for N = 2. For this value, i.e. for the SU (2) theory, the expansion (2.5) reduces to 1127171217162240 π 20 κ 10 + O(κ 11 ), (2.8) and, starting at order κ 6 , products of zeta functions appear. This seemingly technical or accidental feature will have a role in the following discussion. A natural issue is than to explore the possibility of non-trivial scaling functions in other partially protected sectors. To this aim, we shall consider here chiral correlators of one primary with a Maldacena-Wilson loop.

Chiral 1-point function with 1 2 -BPS Wilson loop
We consider the 1 2 -BPS Maldacena-Wilson loop defined by [9, 10] where g is the gauge coupling, C is a circle of radius R, ϕ is the complex scalar field in the vector multiplet, and the trace is taken in the fundamental representation. Conformal invariance fully constrains the position dependence of the 1-point function of the chiral primary operator O n with W . For a loop placed at the origin, one has [11] where |x| C is a suitable SO(1, 2) × SO(3) distance between x and the loop respecting the conformal subgroup unbroken by the loop. All the remaining information about the 1-point function is encoded in the coupling dependent normalization A n (g) in (3.2).

Localization results
As discussed in [11], the function A n (g) may be computed by the same matrix model that appears in the partition function and encoding the localization solution of the N = 2 theory on S 4 with a specific (finite) Ω-deformation [16][17][18]. Since we are interested in a weak-coupling expansion, we shall drop the instanton contribution. After this simplification, the sphere partition function is associated with a perturbed Gaussian matrix model. Up to a g-independent normalization it reads , and the non-Gaussian interacting action S int (a) is an infinite series where s n (a) are invariant functions of a. The first terms read explicitly and higher powers of g are associated with higher transcendentality terms. As usual, correlation functions are computed by The N = 4 SYM theory is obtained as a special limit where the interacting action is dropped and the instanton contribution -already discarded at weak-coupling -is also removed. In this limit, the partition function is computed by a very simple Gaussian model and correlators are obtained after full Wick contraction where the basic contraction is a a = δ . In the following we shall be interested in the multi-trace expectation values, cf. (2.1), As discussed in [11] it is possible to compute the function A n (g) in (3.2) by the following prescription where : O n (a) : is obtained by Gram-Schmidt orthogonalization with respect to all operators with dimension smaller than |n|. In [11], several interesting results are obtained for the function A n (g) at specific values of the multi-index n, both at finite N as well as in the planar limit N → ∞ with fixed N g 2 . This is achieved by exploiting recursion relations with respect to n [25]. In general, a potential unavoidable problem is that it may be difficult to obtain results parametric in n, whereas this is essential for our purposes. This is a difficulty already showing up in the N = 4 theory due to the complication of the Gram-Schmidt procedure required to construct normal ordering. In the N = 2 theory, complications are worse due to the g-dependence introduced by the interaction term (3.4).
In the next Section we shall consider the special class of operators Ω n in (2.3), show how to solve the above problems, and derive a set of results depending parametrically in n. This will be the starting point to discuss the large R-charge limit n → ∞ in Section 5.
Remark Concerning notation, in the following we shall need to tell between quantities evaluated in the N = 2 or N = 4 theories. In such cases, we shall denote 4 The correlators W Ω n As we explained in the Introduction, we are interested in the functions As we discussed, to some extent it is straighforward to evaluate the perturbative expansion of A N =2 n at fixed n. However, our main concern is the limit (2.4) and the associated asymptotic ratio For this quantity a different approach is required. Notice that in the case of the chiral 2-point function, the existence of a Toda equation -equivalent to the tt * equations -has been crucial in this respect [7]. In this section, we shall present results for A n (g) in the N = 4 theory. This is a piece entering the ratio (4.2). Next, we move to the N = 2 theory and compute the first non-trivial correction to Although we are ultimately interested in the finite N case, we shall check our results by matching general expressions valid in the planar limit that have been conjectured in [11]. : where the constants c are determined by the condition : Ω : A useful remark is that (4.3) is equivalent to the subtraction of all possible (partial) Wick pairings inside Ω [25,11]. This is why the expansion (4.3) is dubbed normal ordering and denoted by the usual double colon notation. To prove this, we notice that by linearity we can reduce the problem to field monomial in the higher order traces Tr(a n ). Let W (Ω) be the operator constructed by subtraction of Wick pairings. The correlator W (Ω) Ω is obtained after full Wick pairing. If dim Ω < dim Ω, some pairing is necessarily inside W (Ω) and we get zero. By uniqueness, : Ω : ≡ W (Ω).
The case of Ω n We now consider the special case of the operators Ω n = (Tra 2 ) n . Their expectation value is as follows easily from (3.3). According the the remarks in the previous paragraph we can write the following specialized version of (4.3) : Ω n : = Ω n + n−1 k=0 c (n) k Ω k . where α has been defined in (4.4). In the following it will be sometimes convenient to adopt an explicit matrix vector notation for (4.6) and write : To prove (4.6), we start from the orthogonality condition : Ω n : Ω m = 0, with m = 0, 1, . . . , n − 1, that we write as Using (4.4), these conditions may be written The solution is unique and reads 4 c (n) Notice that since c Γ(α + n) Γ(α + m) , (4.12) and noting that the r.h.s. is zero for integer m = n and one for m → n.
A useful elementary identity The specific explicit coefficients (4.9) allow to prove the following elementary identity. For any function F (g a) we can write  We compute O m Ω n by iterating the recursion relation proved in [25] O m Ω 1 = This gives Finally, using the mixing solution (4.6), we have (for |m| > 2n otherwise we get zero by construction of : Ω n :) Replacing this in (4.14) gives which is the same as (4.13).
Remark 1 It may be interesting to compare (4.13) with what is obtained by taking derivatives with respect to 1/g 2 , i.e. essentially with respect to Imτ where τ is the complexified gauge coupling. In this case we would have obtained Gaussian averages with various insertions of powers of Ω 1 . Instead, derivatives with respect to g 2 automatically build up the normal ordered operators : Ω n :. In particular, the N dependence of the normal ordering expansion (4.6) is fully taken into account by the N dependence of the basic expectation F .

Remark 2
The relation (4.13) may be written in explicit form by simply rearranging derivatives. This gives F : Ω n : = (−1) n 2 2 n n p=1 A useful application of formula (4.13) is to the case when F is the Wilson loop (3.8) with expectation value in the SU (N ) theory given by [11]  W : Ω 1 : = g 2 ∂ g W , W : Ω 2 : = g 4 ∂ g (−1 + g ∂ g ) W , W : and so on. As a check, the first two lines are in agreement with Eqs. (4.7, 4.11) of [11]. Thanks to the special structure of W in (4.21), it is also simple to determine the closed expression of (4.22) for any given N , a kind of formula that will be useful in the following. The first cases are (4.24) The SU (2) computation is particularly simple and is easily checked by direct evaluation of the partition function and correlators in the Gaussian matrix model. This is briefly reviewed in App. A. The advantage of the relations proved in this section is that they are parametric in N , i.e. in the gauge group rank.

Checks in the planar limit
From the result (4.34) it is easy to work out the planar limit N → ∞ with fixed λ = N g 2 . Indeed we can expand In (4.34), we have at leading order and therefore Rearranging derivatives, we can write (4.40) in the form Sample cases are These may be compared with the general formula conjectured in [11] that reads and, of course, there is agreement after some rearrangement of the Bessel functions.

Large R-charge limit and universality
We now make full use of the results presented in the previous section to discuss the large R-charge limit of the normalized one-point function of the chiral primary Ω n (ϕ) with the 1 2 -BPS Wilson loop in the N = 2 theory. In particular, the result (4.34) controls exactly the contribution ∼ ζ(3). When combined with explicit low N calculations, it provides a guide to understand the behaviour of higher transcendentality contributions, as we are going to illustrate.

Analysis of the ζ(3) correction
We define the asymptotic ratio of one-point functions, cf. (4.2), The very existence of the large n limit in (5.1) is something that deserves investigation. To begin our analysis, let us consider the simplest SU (2) case. Using (4.41) and the first equation in (4.24) we obtain the ratio 6144n(2n + 1)(2n + 3) + 640g 2 (2n + 1)(2n + 3) + 40g 4 (2n + 3) + g 6 ζ(3) + h.z., (5.2) where "h.z." stands for higher transcendentality terms ζ(5), ζ(3) 2 , and so on. The expression (5.2) is exact in the gauge coupling g, i.e. it resums all contributions ∼ ζ(3) g 4+2 k , with k = 0, 1, 2, . . . . For large n and fixed κ = n g 2 the limit (5.1) reads and comes entirely from the ζ(3) g 4 term in (5.2). The same analysis may be repeated for higher N , see (4.24). In all cases one finds the same leading behaviour as in (5.3). In general, one can show that for SU (N ) the ζ(3) correction reads and thus (5.3) is valid for any N . Indeed, our results allow to check that the higher order terms in coupling g are always accompanied by lower powers of n and do not contribute in the n → ∞ limit. Notice however, that the complexity of the detailed dependence on n increases with N . For instance, even for the simplest ζ(3) correction, the O(g 6 ) term in (5.4) reads for SU (N ) with N = 2, 3, 4 We cannot give a general formula parametric in N , but in all cases, the correction term in the square brackets goes like g 2 /n = κ/n 2 and is negligible at large n.
The series (6.6) has a finite convergence radius, being convergent for |κ| < π 2 . A simple representation of its analytic continuation that may be used for any κ > 0 is where J p are standard Bessel functions. For κ 1, one has log F(κ) = − log 2 π 2 κ + O(log κ), see App. C for more information. Of course, this large κ expansion must be taken with some caution because F(κ) does not include the instanton contribution. Instantons are expected not to contribute at finite κ -and in particular in the weak coupling expansion -but may be important when attempting to reach the large κ regime.

A Direct evaluation of N = 4 correlators in the SU (2) theory
The analysis of the low gauge group rank cases may be carried on explicitly by direct evaluation of the partition function, at least in the simple Gaussian case, i.e. for the N = 4 theory. Let us briefly discuss the SU (2) case as an illustration and to write down some expressions that are used in the main text. The partition function (3.3) may be written in terms of a k , k = 1, . . . , N , the eigenvalues of a 10 Z S 4 = C : Ω n : = (−1) n n! L 1/2 n (2 a 2 ) = where L 1/2 n is the generalized Laguerre polynomial L q n with q = 1/2, and H n (z) are Hermite polynomials. As a check one can compute W : Ω n : = 1 2 2n+1 Now, let us assume that (4.13) holds, with the expansion (4.7) in the l.h.s. We find F : Ω n+1 : = g 2(n+1) d dg 2 g −2n F : Ω n : = g 2(n+1) d dg 2 g −2n F