Instanton corrections to twist-two operators

We present the calculation of the leading instanton contribution to the scaling dimensions of twist-two operators with arbitrary spin and to their structure constants in the OPE of two half-BPS operators in $\mathcal N=4$ SYM. For spin-two operators we verify that, in agreement with $\mathcal N=4$ superconformal Ward identities, the obtained expressions coincide with those for the Konishi operator. For operators with high spin we find that the leading instanton correction vanishes. This arises as the result of a rather involved calculation and requires a better understanding.


Introduction
In this paper we continue the study of instanton corrections to correlation functions in maximally supersymmetric N = 4 Yang-Mills theory. Although these corrections are exponentially small in the planar limit, they are expected to play an important role in restoring the S−duality of the theory. At weak coupling, the leading instanton contribution can be found in the semiclassical approximation by neglecting quantum fluctuation of fields. In this approximation, the calculation amounts to evaluating the product of operators in the background of instantons and integrating the resulting expression over the collective coordinates. For a review see [1][2][3].
Previous studies revealed [4] that the leading instanton contribution to four-point correlation function of half-BPS operators in N = 4 SYM scales at weak coupling as e −8π 2 /g 2 . An OPE analysis showed, however, that this correction does not affect twist-two operators [5] and, therefore, does not modify the leading asymptotic behaviour of correlation functions in the light-cone limit. This led to the conclusion [6,7] that the leading instanton contribution to the conformal data of twist-two operators (scaling dimensions ∆ S and OPE coefficients C S ) should be suppressed by a power of the coupling constant and scale as g 2n e −8π 2 /g 2 with some n ≥ 1. The calculation of such corrections within the conventional approach is way more complicated as it requires going beyond the semiclassical approximation.
In [8] we argued that, by virtue of N = 4 superconformal symmetry, the above mentioned instanton effects can be determined from the semiclassical computation of two-and threepoint correlation functions for another operator in the same supermultiplet. Following this approach, we computed the leading non-vanishing correction to the scaling dimension of the Konishi operator, ∆ (inst) K = O(g 4 e −8π 2 /g 2 ) and to its structure constant in the OPE of two half-BPS operators, C (inst) K = O(g 2 e −8π 2 /g 2 ) (see [8] for explicit expressions). In this paper we extend the analysis to twist-two operators O S with arbitrary even Lorentz spin S. For spin zero, the operator O S=0 coincides with the half-BPS operator and is protected from quantum corrections. For spin-two, the operator O S=2 belongs to the same supermultiplet as the Konishi operator and, therefore, has the same conformal data. For S ≥ 4, quite surprisingly, our calculation yields a vanishing result for the instanton contribution. This implies that the leading instanton corrections to the conformal data of twist-two operators O S with S ≥ 4 are suppressed at least by a power of g 2 as compared with those for the Konishi operator (1.1) Notice that these two expressions differ by a power of the coupling constant, ∆ (inst) S /C (inst) S = O(g 2 ), whereas the leading perturbative corrections to both quantities have the same scaling in g 2 at weak coupling.
The paper is organized as follows. In Section 2 we define operators of twist two and discuss their relation to light-ray operators. In Section 3 we construct the one-instanton solution to the equations of motion in N = 4 SYM for the SU (2) gauge group. In Section 4 we present the calculation of correlation functions involving half-BPS and twist-two operators in the semiclassical approximation and discuss its generalization to the SU (N ) gauge group. Section 5 contains concluding remarks. Some details of the calculation are summarized in four Appendices.

Twist-two operators
All twist-two operators in N = 4 SYM belong to the same supermultiplet and share the same conformal data. This allows us to restrict our consideration to the simplest twist-two operator, of the form O S (x) = tr Z D S + Z(x) + . . . , (2.1) where Z(x) is a complex scalar field and D + = n µ D µ (with n 2 = 0) is a light-cone component of the covariant derivative D µ = ∂ µ + i[A µ , ]. All fields take values in the SU (N ) algebra, e.g. Z(x) = Z a (x)T a with the generators normalized as tr(T a T b ) = δ ab /2. The dots on the right-hand side of (2.1) denote a linear combination of operators with total derivatives of the form ∂ ℓ + tr ZD S−ℓ + Z(x) with 0 ≤ ℓ ≤ S. The corresponding expansion coefficients are fixed by the condition for O S (x) to be a conformal primary operator and depend, in general, on the coupling constant. To lowest order in the coupling, they are related to those of the Gegenbauer polynomials (see Eq. (2.9) below).
In this paper, we compute the leading instanton corrections to correlation functions of twist-two operators (2.1) and half-BPS scalar operators of the form where the complex scalar fields φ AB = −φ BA (with A, B = 1, . . . , 4) satisfy reality condition φ AB = 1 2 ǫ ABCD φ CD . The auxiliary antisymmetric tensor Y AB is introduced to project the product of two scalar fields onto the representation 20 ′ of the SU (4) R−symmetry group. It satisfies ǫ ABCD Y AB Y CD = 0 and plays the role of the coordinate of the operator in the isotopic SU (4) space. The scalar field Z entering (2.1) is a special component of φ AB where (Y Z ) AB has the same properties as the Y −tensor in (2.2) and has the only nonvanishing components (Y Z ) 14 Here the scaling dimension of twist-two operator ∆ S , the normalization factor N S and threepoint coefficient function C S depend on the coupling constant whereas the scaling dimension of the half-BPS operator is protected from quantum corrections.

Light-ray operators
To compute the correlation functions (2.4), it is convenient to introduce a generating function for the twist-two operators (2.1), the so-called light-ray operator, In distinction with (2.1), it is a nonlocal operator -the two scalar fields are separated along the light-ray direction n µ and two light-like Wilson lines are inserted to restore gauge invariance, with E(z 1 , z 2 )E(z 2 , z 1 ) = 1. The scalar variables z 1 and z 2 define the position of the fields along the null ray. Making use of gauge invariance of (2.5), we can fix the gauge n µ A µ (x) = 0 in which Wilson lines (2.6) reduce to 1. Then, the expansion of (2.5) in powers of z 1 and z 2 takes the form where ∂ + = (n∂). To restore gauge invariance, it suffices to replace ∂ + → D + in this relation. The local operators on the right-hand side of (2.7) are not conformal primaries but, for given S = k + n, they can be expanded over the conformal primary operators O S (0) and their descendants ∂ ℓ + O S−ℓ (0). As mentioned at the beginning of this section, the conformal operators have the following general form with the expansion coefficients c kn depending on the coupling constant. To lowest order in the coupling, these coefficients coincide (up to an overall normalization) with those of ( involving the Gegenbauer polynomial [9,10] Note that the sum in (2.8) vanishes for odd S and the conformal operators are defined for even nonnegative S. Inverting (2.8) we can expand the light-ray operator (2.7) over the conformal twist-two operators and their descendants. In this way, we find that the operators O S (0) appear as the coefficients in the expansion of the light-ray operator (2.5) Here the dots denote the contribution from descendant operators of the form ∂ ℓ + O S (0). This contribution is fixed by conformal symmetry, see [12].

From light-ray to twist-two operators
The rationale for introducing (2.5) is that finding instanton corrections to light-ray operators proves to be simpler as compared to that for twist-two operators. Then, having computed the correlation function 2 we can then apply (2.10) to obtain the three-point correlation function To lowest order in the coupling, we can simplify the calculation by making use of the following relation between the operators O S (0) and O(z 1 , z 2 ) where the integration contour in both integrals encircles the origin. Indeed, replacing O(z 1 , z 2 ) on the right-hand side with (2.7) and computing the residue at z 1 = 0 and z 2 = 0 we obtain (2.8) with c kn given by (2.9). Using the operator identity (2.12) inside corrrelation functions we arrive at We would like to emphasize that relations (2.12) and (2.13) hold to the lowest order in the coupling constant. Beyond this order, we have to take into account O(g 2 ) corrections to (2.9). Relation (2.13) offers an efficient way of computing the correlation functions of twist-two operators. As an example, we show in Appendix C how to use (2.13) to obtain the correlation functions (2.4) in the Born approximation (see Eqs. (C.4) and (C.6)).

Instantons in N = 4 SYM
The general one-instanton solution to the equations of motion in N = 4 SYM with the SU (N ) gauge group depends on 8N fermion collective coordinates, see [1][2][3]. Among them 16 modes are related to N = 4 superconformal symmetry. The remaining 8(N − 2) fermion modes are not related to symmetries in an obvious way and are usually called 'nonexact modes'. This makes the construction of the instanton solution more involved.
For the SU (2) gauge group the general one-instanton solution to the equations of motion of N = 4 SYM can be obtained by applying superconformal transformations to the special solution corresponding to vanishing scalar and gaugino fields and gauge field given by the celebrated BPST instanton [14] where η a µν are the 't Hooft symbols and the SU (2) generators are related to Pauli matrices T a = σ a /2. It depends on the collective coordinates ρ and x 0 defining the size and the position of the instanton, respectively.
In this section, we present explicit expressions for the one-instanton solution in N = 4 SYM for the SU (2) gauge group. Then, in the next section, we explain how to generalize the expressions for the correlation functions (2.4) to the case of the SU (N ) gauge group.
The field configuration (3.1) is annihilated by (the antichiral) half of the N = 4 superconformal generators. Applying to (3.1) the remaining (chiral) N = 4 superconformal transformations (see (A.11) in Appendix A), we obtain a solution to the N = 4 equations of motion that depends on 16 fermionic collective coordinates, ζ A α andηα A . The resulting expressions for gauge and scalar fields can be expanded in powers of fermion modes (14) , denotes the component of the gauge field that is homogenous in ζ A α andηα A of degree n and similar for scalars. Gaugino fields admit similar expansions (see (B.4) in Appendix B) but we will not need them for our purposes. Explicit expressions for various components of (3.2) are given below.
By virtue of superconformal invariance, the action of N = 4 SYM evaluated on the instanton configuration (3.2) does not depend on the fermionic modes and coincides with the one for pure Yang-Mills theory 3 where τ is the complex coupling constant Notice that, due to our definition of the Lagrangian (see (A.9) in Appendix A), the instanton solution (3.1) and (3.2) does not depend on the coupling constant. It is straightforward to work out the leading term of the expansion (3.2) by subsequently applying N = 4 superconformal transformations to (3.1), see [1]. The direct calculation of the subleading terms becomes very involved due to the complicated form of these transformations (see (A.11) in Appendix A). There is, however, a more efficient approach to computing higher components in (3.2) which is presented in Appendix B. It makes use of the known properties of fields with respect to conformal symmetry, R−symmetry and gauge transformations and allows us to work out the expansion (3.2) recursively with little efforts.
To present the resulting expressions for the instanton configuration (3.2) for the SU (2) gauge group it is convenient to switch to spinor notation and use a matrix representation for (3.1) (see Appendix A for our conventions) where the four-dimensional vector of 2 × 2 matrices σ µ = (1, σ) and the SU (2) generators T a = σ a /2 are built from Pauli matrices. This field carries two pairs of indices, Lorentz indices (α,α = 1, 2) and SU (2) indices (i, j = 1, 2). Here we distinguish lower and upper SU (2) indices and define the product of two matrices by All indices are raised and lowered with the help of the antisymmetric tensor, e.g.
where the second relation follows from (3.1) and (3.5). The advantage of (A αα ) ij as compared to (3.5) is that it is symmetric with respect to the SU (2) indices. The instanton (3.1) and (3.7) is a self-dual solution to the equations of motion in pure Yang-Mills theory, F (0) αβ = 0. The corresponding (chiral) strength tensor is given by where ∂ αβ = (σ µ ) αβ ∂ µ and angular brackets denote symmetrization with respect to indices, A (αβ) = A αβ + A βα . Here a shorthand notation was introduced for the instanton profile function It is easy to verify that (F αβ ) ij is symmetric with respect to both pair of indices and satisfies the equations of motion Dα α F αβ = [∂α α + Aα α , F αβ ] = 0. Notice that (3.7) and (3.8) do not depend on the position of the instanton x 0 . To restore this dependence it suffices to apply the shift x → x − x 0 . We can further simplify (3.7) and (3.8) by contracting all Lorentz indices with auxiliary (commuting) spinors |n ≡ λ α and |n] ≡λα depending on their chirality, e.g.
The resulting expressions only have SU (2) indices and are homogenous polynomials in λ and λ. In particular, where the superscript '(0)' indicates that these expressions correspond to the lowest term in the expansion (3.2). An unusual feature of the expressions on the right-hand side of (3.10) is that the chiral Lorentz indices are identified with the SU (2) indices.
To obtain the subleading corrections in the instanton solution (3.2), depending on fermion modes, we apply the method described in Appendix B. Namely, we make use of (B.9) and replace the gauge field by its expression (3.10). Going through the calculation we get where ζ stands for a linear x−dependent combination of fermion modes and a shorthand notation is used for various contractions of Lorentz indices Note that the dependence on the fermion modes enters into n|F (4) ij |n through the linear combination (3.12). As explained in Appendix B, this property can be understood using conformal symmetry.
We recall that, to leading order, the instanton solution (3.10) satisfies the self-duality condition F (0) αβ = 0. Using the obtained expression for A (4) αα , we find that the anti-self-dual component of the gauge field strength tensor [n|F ij |n] ≡λ α λ β Fαβ ,ij receives a nonvanishing correction [n|F (4) ij |n] = − which contains four fermion modes. This relation illustrates that higher components of fields depend on fermion modes in a nontrivial manner.
Repeating the same analysis we can evaluate subleading corrections to all fields in (3.2). In particular, using the relations (B.7) and (B.9) we get the following expressions for the scalar field on the instanton background where brackets in the first relation denote antisymmetrization with respect to the SU (4) indices, the variable ζ is defined in (3.12) and (ζ 2 ) AB = (ζ 2 ) BA = ζ βA ǫ βγ ζ γB . The expressions (3.11) and (3.15) depend on the size of the instanton, ρ, as well as on 16 fermion modes, ξ A α andηα A . To restore the dependence on the position of the instanton, we apply the shift x → x − x 0 .

Correlation functions in the semiclassical approximation
In the semiclassical approximation, the calculation of correlation functions reduces to averaging the classical profile of the operators over the collective coordinates of instantons [1-3] where the gauge invariant operators O i on the right-hand side are evaluated on the instanton background (3.2). For the SU (2) gauge group, the collective coordinates of the one-instanton solution are the size of the instanton ρ, its localtion x µ 0 and 16 fermion modes, ξ A α andη Ȧ α . The corresponding integration measure is [15] dµ phys e −S inst = g 8 2 34 π 10 e 2πiτ d 4 x 0 where the complex coupling constant τ is defined in (3.4). We recall that, due to our normalization of the Lagrangian (A.9), the instanton background (3.2) does not depend on the coupling constant. The same is true for the operators (2.2), (2.5) and (2.8) built from scalar and gauge fields. As a consequence, the instanton correction to the correlation function of these operators scales in the semiclassical approximation as O(g 8 e 2πiτ ) independently on the number of operators n. At the same time, the same correlation functions in the Born approximation scale as O(g 2n ) with one power of g coming from each scalar field (see Eqs. (C.4) and (C.6)). 4 Thus, the ratio of the two contributions scales as O(g 8−2n e 2πiτ ) with n being the number of operators. As we show in a moment, the instanton correction vanishes in the semiclassical approximation for n > 5. Later in the paper we shall compute this correction explicitly for n = 2 and n = 3.
For the correlation function (4.1) to be different from zero upon integration of fermion modes, the product of operators O 1 . . . O n should soak up the product of all fermion modes (ξ) 8 Let us denote by k i the minimal number of fermion modes in O i . Then, by virtue of the SU (4) invariance, the total number of modes in the product O 1 . . . O n is given by k min + 4p with k min = k 1 + · · · + k n and p = 0, 1, . . . . For k min > 16 the product O 1 . . . O n is necessarily proportional to the square of a fermion mode and, therefore, the correlation function (4.1) vanishes. To obtain a nontrivial result for the correlation function, we have to go beyond the semiclassical approximation and take into account quantum fluctuations of fields. For k tot ≤ 16 and k min multiple of 4, the integral over fermion modes in (4.1) does not vanish a priori. This case corresponds to the so-called minimal correlation functions [3,16]. Another interesting feature of these correlation functions is that the results obtained in the semiclassical approximation for the SU (2) gauge group can be extended to the general case of SU (N ) gauge group for the one-instanton solution [2,17] and for multi-instanton solutions at large N [18].
Since the operators (2.2), (2.5) and (2.8) involve two scalar fields, we deduce from (3.2) that each of them has at least four fermion modes, k = 4. Then, for the product of these operators O 1 . . . O n we have k min = 4n and, as a consequence, the correlation function O 1 . . . O n inst is minimal for n ≤ 4.

Instanton profile of twist-two operators
To compute the correlation functions (2.4) in the semiclassical approximation, we have to evaluate the operators O 20 ′ and O S on the instanton background by replacing all fields by their explicit expressions (3.2) and, then, expand their product in powers of 16 fermion modes.
For the half-BPS operators (2.2) we find [8] O (4) where f (x) is the instanton profile function (3.9) and the superscript on the left-hand side indicates the number of fermion modes. Since O 20 ′ (x) is annihilated by half of the N = 4 supercharges, its expansion involves only four fermion modes.
As argued above, in order to study correlators involving twist-two operators it is convenient to introduce light-ray operators (2.5). In order to evaluate those in the instanton background we replace scalar and gauge fields by their expressions Z = Z (2) + Z (6) + . . . and A = A (0) + A (4) + . . . , which leads to where the leading term contains four fermion modes and each subsequent term has four modes more. The last term is proportional to the product of all 16 modes (ξ) 8 (η) 8 . The first two terms on the right-hand side of (4.4) are given by where E (n) denote specific terms in the expansion of the light-like Wilson line (2.6) on the instanton background Explicit expressions for E (0) and E (4) can be found in Appendix D.
The remaining terms O (12) and O (16) are given by a priori lengthy expressions but their evaluation leads to a surprisingly simple result As we show in Appendix D, these terms are necessarily proportional to the square of a fermion mode and, therefore, have to vanish. Let us consider the correlation function (2.11). As was explained at the beginning of the section, this correlation function is minimal and scales as O(g 8 e 2πiτ ) in the semiclassical approximation. Applying (4.1), we have to find the instanton profile of and identify the contribution containing 16 fermion modes. Making use of (4.3) and (4.4) we find (4.8) Later we shall use this relation to find the instanton correction to the three-point correlation In a similar manner, in order to find O S (0)Ō S ′ (x) inst we have to consider the correlation function of two light-ray operators separated by a distance x and conjugated to each other. For this purpose, we have to generalize the definition (2.5) by allowing the light-ray to pass through an arbitrary point x where P µ is the operator of total momentum. The operator (4.9) is given by the same expression (2.5) with scalar and gauge fields shifted by x. Obviously, for . Substituting (4.4) into (4.9), we find that O(z 1 , z 2 |x) has a similar expansion in powers of fermion modes. Then, we take into account (4.7) to get whereŌ(z 3 , z 4 |x) is a conjugated operator. Notice that the lowest O (4) term of the expansion (4.4) does not contribute to (4.8) and (4.10).
We are now ready to compute the correlation functions (4.8) and (4.10). Using the expression for the integration measure (4.2), we first perform the integration over fermion modes.
Replacing O  .5), respectively, and going through a lengthy calculation we find (see Appendix D for details) In a similar manner, we find from (4.10) (see Appendix D for details) Notice that (4.11) and (4.12) vanish quadratically for z 1 → z 2 (as well as for z 3 → z 4 ). This property can be understood as follows. For z 2 → z 1 we can use the definition of the light-ray operator (2.5) to show that The first two terms of the expansion involve the half-BPS operator tr Z 2 , it can be obtained from the general expression (2.2) for Y = Y Z . According to (4.3), the half-BPS operators have exactly four fermion modes and, therefore, the first two terms on the right-hand side of (4.13) do not contribute to To obtain the correlation functions, we have to integrate (4.11) and (4.12) over the bosonic coordinates ρ and x 0 with the measure (4.2). The resulting integrals can be expressed in terms of the so-called D−functions where x i0 = x i − x 0 . Multiplying (4.11) and (4.12) by the additional factors coming from the integration measure (4.2), we finally find the leading instanton correction to the correlation functions (4.16) The same correlation functions in the Born approximation are given by (C.2) and (C.5), respectively, evaluated for N = 2. Dividing (4.15) and (4.16) by the Born level expressions, we find that the instanton corrections to the two correlation functions scale as O(g 2 e 2πiτ ) and O(g 4 e 2πiτ ), respectively, in agreement with the analysis at the beginning of this section.

Instanton corrections to correlation functions
Let us apply (4.15) and (4.16) to derive the correlation functions of twist-two operators. We recall that to the lowest order in the coupling constant, the correlation functions involving light-ray and twist-two operators are related to each other through the relations (2.12) and (2.13). It is not obvious however whether the same relations should work for the instanton corrections in the semiclassical approximation. To show that this is the case, we apply below where I S (x 1 , x 2 ) denotes the following integral with ∆ = 4. In what follows we relax this condition and treat ∆ as an arbitrary parameter.
The reason for this is that the same integral with ∆ = 3 enters the calculation of (4.16).
Replacing the D−function in (4.18) by its integral representation (4.14) and exchanging the order of integration, we find that the integrand has triple poles at z 1 , z 2 = (ρ 2 + x 2 0 )/(2(x 0 n)). Blowing up the integration contour in (4.18) and picking up the residues at these poles, we find that the integral over z 1 and z 2 vanishes for all nonnegative integer S except for S = 2, leading to For ∆ = 4 the calculation of this integral yields (4.21) Surprisingly enough, this expression vanishes for all spins except S = 2. For S = 0 the corresponding twist-two operator O S=0 = tr Z 2 is half-BPS. In this case, the three-point correlation function is protected from quantum corrections and, therefore, the instanton correction (4.21) should vanish for S = 0.
To obtain the two-point correlation function O S (0)O S ′ (x) from (4.16) we apply (2.12) to both operators to get where I SS ′ (x) is given by a folded contour integral over four z−variables. Using (4.18) this integral can be rewritten as where I S is evaluated for ∆ = 3. Replacing I S with its integral representation (4.19), we evaluate the integral by picking up the residue at the poles z 3 , in the same manner as (4.19). In this way, we arrive at where the D−function is defined in (4.14). Similar to (4.21), this expression vanishes for all spins except S = S ′ = 2. For S = S ′ = 0 this property is in agreement with protectiveness of two-point correlation functions of half-BPS operators. A close examination of (4.14) shows that D 66 (0, x) develops a logarithmic divergence. It comes from integration over small size instantons, ρ → 0, located close to one of the operators, x 2 0 → 0 and (x − x 0 ) 2 → 0. This divergence produces a logarithmically enhanced contribution ∼ ln x 2 which modifies the scaling dimensions of twist-two operators. To identify this contribution, we regularize the integral by modifying the integration measure over x 0 Relations (4.21) and (4.26) define one-instanton corrections to the correlation functions. Antiinstanton corrections are given by the complex conjugated expressions. Finally, we combine (4.21) and (4.26) with analogous expressions in the Born approximation (given by (C.4) and (C.6) for N = 2) and add the anti-instanton contribution to get the following expressions for the correlation functions in N = 4 SYM for the SU (2) gauge group where the normalization factor c S is defined in (C.6). We recall that these relations were derived in the one (anti)instanton sector in the semiclassical approximation and are valid up to corrections suppressed by powers of g 2 . Comparing (4.28) with the general expression for a two-point correlation function (2.4), we obtain the leading instanton correction to the scaling dimension of twist-two operators ∆ S = 2 + S + γ S  In a similar manner, matching (4.27) and (2.4) we obtain the following result for the properly normalized OPE coefficient As follows from (4.28), the instanton correction to the normalization factor N S entering the first relation in (2.4) has the same dependence on the coupling constant as (4.29) and does not affect the leading correction to the structure constants in the OPE of two half-BPS operators. As was already mentioned, for S = 0 the correlation functions (4.27) and (4.28) are protected from quantum corrections and, therefore, γ S=0 = 0 and C S=0 = 1 for arbitrary coupling constant. For S > 2, there are no reasons for the same relations to hold beyond the semiclassical approximation. For S = 2, the corresponding conformal operator O S=2 ∼ tr(ZD 2 + Z) − 2 tr(D + ZD + Z) belongs to the same N = 4 supermultiplet as Konishi operator K = tr(φ ABφ AB ). As a consequence, the two operators should have the same anomalous dimension, γ S=2 = γ K , as well as OPE coefficients, C S=2 = C K . As a nontrivial check of our calculation we use the results of [8] to verify that both relations are indeed satisfied in the semiclassical approximation.
As was already mentioned in Section 4, the correlation functions (4.27) and (4.28) belong to the class of minimal correlation functions. Following [2,17,18], we can then generalize the relations (4.27) and (4.28) to the SU (N ) gauge group and, in addition, include the contribution of an arbitrary number of (anti)instantons at large N . As before, for S = 2 the resulting expressions for γ S and C S do not receive corrections in the semiclassical approximation, whereas for S = 2 they coincide with those for the Konishi operator and can be found in [8].

Conclusions
In this work we have presented the explicit calculation of the instanton contribution to twoand three-point correlation functions involving half-BPS and twist-two operators. A somewhat surprising outcome of our analysis is the vanishing of the leading instanton corrections to the scaling dimensions and the OPE coefficients of twist-two operators with spin S > 2. This result comes from a rather involved calculation and requires a better understanding. Note that the situation here is very different from that for twist-four operators. As it was shown in [8], crossing symmetry implies that twist-four operators with arbitrarily high spin acquire instanton corrections already at order O(g 2 e −8π 2 /g 2 ).
According to (1.1), the instanton corrections to twist-two operators with S > 2 are pushed to higher order in g 2 . Their calculation remains a challenge as it requires going beyond the semiclassical approximation. There is however an interesting high spin limit, S ≫ 1, in which we can get additional insight on the instanton effects. In this limit, the scaling dimensions of twist-two operators scale logarithmically with the spin [19,20] where Γ cusp (g 2 ) is the cusp anomalous dimension. The instanton contribution to ∆ S should have the same asymptotic behavior and produce a correction to Γ cusp (g 2 ). According to the first relation in (1.1), it should scale at least as O(g 6 e −8π 2 /g 2 ). We can find however the same correction using the fact that the cusp anomalous dimension governs the leading UV divergences of light-like polygon Wilson loops. The light-like polygon Wilson loop W L is given by the product of gauge links (2.6) defined for L different light-like vectors n i . In the semiclassical approximation, the instanton contribution to W L can be found using (4.1). Since W L does not depend on the coupling constant on the instanton background, the dependence on g 2 only comes from the integration measure (4.2) leading to W L inst = O(g 8 e −8π 2 /g 2 ). At the same time, in the Born approximation, we have W L Born = 1. As a consequence, the leading instanton correction to the cusp anomalous dimension scales at least as Combining the last two relations we conclude that ∆ (inst) S = O(g 8 e −8π 2 /g 2 ln S) at large S. Thus, if the leading O(g 6 e −8π 2 /g 2 ) correction to ∆ (inst) S in (1.1) is different from zero, it should approach a finite value for S → ∞. 6 It would be interesting to compute explicitly the instanton correction (5.2).

A N = 4 SYM in spinor notations
Performing the calculation of instanton corrections in N = 4 SYM it is convenient to employ spinor notations. We use Pauli matrices σ µ = (1, σ) to map an arbitrary four-dimensional Minkowski vector x µ into a 2 × 2 matrix and use the completely antisymmetric tensor to raise and lower its indices with ǫ αβ ǫ αγ = δ γ β , ǫαβǫαγ = δγβ and ǫ 12 = ǫ 12 = 1. Then, For derivatives we have similarly Throughout the paper we use the following conventions for contracting Lorentz indices in the product of 2 × 2 matrices 6 Similar considerations should apply to the structure constants C (inst) S in the large spin limit. Indeed, from the analysis of [21] it follows that the structure constants for S ≫ 1 are given in terms of the cusp anomalous dimension.
Using these definitions we obtain for the gauge field A µ and the stress tensor where the additional factor of i is introduced for convenience. The symmetric matrices F αβ and Fαβ describe (anti)self-dual parts of the strength tensor where angular brackets denote symmetrization with respect to indices and the covariant derivative D αα = D µ (σ µ ) αα is defined as The Lorentz indices are raised and lowered according to (A.2). The Lagrangian of N = 4 super Yang-Mills theory takes the following form in spinor notations where gaugino fields λ A α andλα A , and scalar fields, φ AB andφ AB , carry the SU (4) indices (A, B = 1, . . . , 4) and satisfy the reality conditionφ AB = 1 2 ǫ ABCD φ CD . All fields are in the adjoint representation of the SU (N ) gauge group, e.g. A αα = A a αα T a , with the generators satisfying [T a , T b ] = if abc T c and normalized as tr(T a T b ) = 1 2 δ ab . In the special case of the SU (2) gauge group the generators are expressed in terms of Pauli matrices T a = σ a /2.
As follows from (A.9), gauge fields, gaugino and scalars satisfy equations of motion where brackets in the second relation denote antisymmetrization of the SU (4) indices. These relations describe transformations of fields under combined Q− andS−transformations with the corresponding parameters being ξ A α andη Ȧ α , respectively. All relations in (A.11) except the last one depend on the linear (x−dependent) combination This property plays an important role in our analysis and it can be understood as follows. We recall thatS−transformations can be realized as composition of the inversion and Q−transformations where (ηS) =η Ȧ αSα A and (ξ Q) = ξ Aα Q αA . Let us consider the last relation in (A.11). We first apply inversions and take into account that I changes the chirality of Lorentz indices Then, we obtain from (A.11) and (A.13) . Combining the relations (A.15) together, we find that δλα A = (ξ · Q +η ·S)λα A agrees with the last relation in (A.11). The additional correction to δλ proportional toηα B comes from the inhomogenous term in I(Dβ βφ AB ). In other words, the appearance of O(η) term in the expression for δλ in (A.11) is ultimately related to the fact that Dα βφ AB does not transform covariantly under the inversion, or equivalently, that δλ involves the operator Dα βφAB which is not conformal primary.
The question arises whether the relations (A.11) are consistent with conformal symmetry. Supplementing (A.11) with the relation that follows from (A.13) and (A.14), we verify that Q +S variations of all fields, δF αβ , δFαβ, δφ AB , δλ A α and δλα A , transform under the conformal transformations in the same manner as the fields themselves. The same property should hold for higher order variation of fields, e.g. for δ n φ AB with n = 2, 3, . . . .
The explicit calculation of δ n φ AB from (A.11) is very cumbersome and is not efficient for higher n due to proliferation of terms. Instead, we can use conformal symmetry to simply the task. Namely, conformal symmetry restricts the possible form of δ n φ AB and allows us to write its general expression in terms of a few arbitrary coefficients. The latter can be fixed by requiring the fields to satisfy the N = 4 SYM equations of motion (A.10). where Φ stands for one of the fields (scalar, gaugino and gauge field). Here Φ (0) is given by (B.1) and the notation was introduced for

B Iterative solution to the equations of motion
where the variation of fields δΦ is given by (A.11). It is important to emphasise that applying (B.3) we first perform superconformal transformations (A.11) and, then, replace Φ (0) by their explicit expressions (B.1).
Substituting (B.4) into (A.10) and matching the number of fermion modes on both sides of the relations, we obtain the system of coupled equations for various components of fields. To the leading order, we have from (A.10) where the covariant derivative D αα is given by (A.8) with the gauge field replaced by A αα . To the next-to-leading order we find in a similar manner αA } + [φ (2),BC , D ααφ

Leading order solutions
Applying relations (B.3) together with (A.11) and (B.1) we obtain the leading order corrections to scalar and gaugino fields αβ is nonvanishing self-dual part of the gauge strength tensor and x−dependent variable ζ A α is defined in (A.12). It is straightforward to check that (B.7) satisfy the equations of motion (B.5). We verify that, in agreement with (B.3), the fields (B.7) are related to each other as where expressions on the right-hand side follow from (A.11). As was mentioned above, the form of (B.7) is restricted by conformal symmetry. For instance, the expression forλ involves (ζD βα F ζ) CD = ζ γC D βα F γδ ζ δD which is not a conformal primary operator. As was explained in Section A, this leads to the appearance of the second term inλ (3) αA proportional tō η Ḃ α which is needed to restore correct conformal properties of the gaugino field. The relative coefficient between the two terms in the expression forλ (3) αA is uniquely fixed by the conformal symmetry whereas the overall normalization coefficient is fixed by the equations of motion (B.5).

Next-to-leading order solutions
Direct calculation of subleading corrections to fields (B.4) based on (B.3) and (A.11) is very cumbersome. We describe here another, more efficient approach.
We start with next-to-leading correction to the gauge field and try to construct the general expression for A (4) αα which has correct properties with respect to conformal and R symmetries. By construction, A (4) αα is a homogenous polynomial of degree 4 in fermion modes ξ A α andηα A . To begin with, we look for an expression that depends on their linear combination ζ A α defined in (A.12) and has quantum numbers of the gauge field. Since ζ A α has scaling dimension (−1/2), the product of four ζ's should be accompanied by an operator carrying the scaling dimension 3. It can only be built from the self-dual part of the strength tensor F αβ and covariant derivatives D αα . In virtue of the equations of motion, ǫ αβ D αα F βγ = 0, such operator takes the form D (αα F β)γ . Contracting its Lorentz indices with those of the product of four ζ's we obtain A (4) αα ∼ ǫ ABCD ζ A α ζ βB (ζD βα F ζ) CD . Since this operator is not conformal primary, it should receive correction proportional to ǫ ABCD ζ A αη Ḃ α (ζF ζ) CD , the relative coefficient is fixed by the conformal symmetry. The overall normalization coefficient can be determined by requiring A (4) αα to satisfy the first relation in (B.6). As we will show in a moment, there is much simpler way to fix this coefficient using the second relation in (B.3).
Repeating the same analysis for scalar and gaugino fields we get where the notation was introduced for (ζ 2 ) AB = (ζ 2 ) BA = ζ βA ǫ βγ ζ γB , It is straightforward to verify that the fields (B.9) satisfy the system of coupled equations (B.6). Notice that the last two relations in (B.9) involve a conformal primary operator F 2 αβ of dimension 4 and, as a consequence, the dependence on fermion modes only enters through the linear combination ζ A α . 7 We recall that the subleading corrections to fields have to satisfy (B.3). In application to (B.9) these relations read BC ]ζ C α , defines the correction to self-dual part of the gauge strength tensor. Replacing the fields with their explicit expressions (B.7) and (B.9), we verify that the relations (B.11) are indeed satisfied.
We can now turn the logic around and apply the relations (B.8) and (B.11) to compute subleading corrections to the fields. Indeed, we start with the expression for λ

C Projection onto twist-two operators
In this Appendix we explain how to use the light-ray operators (2.5) to compute the correlation functions involving twist-two operators in the Born approximation.
To begin with, we consider the correlation function (2.11) of the light-ray operator and two half-BPS operators in N = 4 SYM with the SU (N ) gauge group. In the Born approximation, we can neglect gauge links in the definition (2.5) of O(z 1 , z 2 ) and express (2.11) in terms of free propagators of scalar fields φ AB ( where D(x) = 1/(4π 2 x 2 ) and the additional factor of g 2 appears due to our normalization of the Lagrangian (A.9). The generators of the SU (N ) gauge group are normalized as tr(T a T b ) = δ ab /2, so that O 20 ′ (x i ) = Y i,AB Y i,CD φ a,AB φ a,CD . We recall that the light-ray operators (2.5) are built out of the complex scalar field Z = φ 14 . It is convenient to represent this field as Z = Y Z,AB φ AB , with Y Z having the only nonvanishing components Y Z,14 = −Y Z,41 = 1/2. Then, we find where x 12 = x 1 − x 2 and the notation was introduced for ( we can rewrite this expression as To obtain the three-point correlation function of local twist-two operator, we substitute (C.3) into (2.13). Blowing up the integration contour in (2.13) and picking up the residue at This relation coincides with the general expression for the correlation function of twist-two operators (2.4). Notice that (C.4) vanishes for odd S, in agreement with the fact that the twist-two operators carry nonnegative even spin S. The same technique can be used to compute two-point correlation function of twist-two operators O S (0)Ō S ′ (x) . We start with the correlation function of two light-ray operators separated by distance x. In the Born approximation we have with ǫ = 2(nx)/x 2 . Substituting this relation into (2.13) and performing integration over z 1 and z 2 , we can project the light-ray operator tr [Z(nz 1 )Z(nz 2 )] onto the twist-two operator O S (0). Repeating the same procedure with respect to z 3 and z 4 , we obtain the expression for O S (0)Ō S ′ (x) that is different from zero only for even positive S = S ′ and is given by with c S = (2S)! /(S! ) 2 and D(x) defined in (C.1).

D Instanton profile of operators Light-like Wilson line
The calculation of the leading term E (0) (z 1 , z 2 ) of the expansion of the light-like Wilson line (2.6) relies on the following identity where we replaced the instanton field with its explicit expression (3.7) for x = nt − x 0 and took into account that n 2 = 0. Here in the second relation we introduced the following 2 × 2 matrix Σ ij with Σ ± being projectors, Σ 2 ± = Σ ± , Σ + Σ − = 0 and Σ + + Σ − = 1. Notice that Σ + n = n Σ − = n and n Σ + = Σ − n = 0.
Since the Σ−matrix in (D.1) does not depend on the integration variable, the pathordered exponential reduces to the conventional exponential leading to E (0) (z 1 , z 2 ) = exp (ΣI(z 1 , z 2 )) = e I(z 1 ,z 2 ) Σ + + e −I(z 1 ,z 2 ) Σ − , where the 2 × 2 matrices Σ ± are independent on z i and I(z 1 , z 2 ) = −I(z 2 , z 1 ) is given by We recall that the matrix indices of Σ ± are identified with the SU (2) indices of E (0) (z 1 , z 2 ). The first subleading correction to the Wilson line, E (4) (z 1 , z 2 ), comes from the A (4) term in the expansion of the gauge field. It is given by Replacing n|A (4) |n] with its explicit expression (3.11) (evaluated for x = nt − x 0 ) and taking into account (D.3), we obtain where ζ t ≡ ζ(tn − x 0 ) = ξ + (tn − x 0 )η depends on the integration variable. Here we used shorthand notations for contraction of the indices, e.g.

Light-ray operators
The expansion of the light-ray operator on the instanton background in powers of fermion modes takes the form (4.4). We show below that the last two terms of the expansion vanish, Eq. (4.7). The underlying reason for this is that, by virtue of N = 4 superconformal symmetry, the light-ray operator O(z 1 , z 2 ) only depends on 12 fermion modes, ξ A α and [nη A ]. According to its definition (2.5), the operator O(z 1 , z 2 ) depends on scalar and gauge fields, Z(x) and (nA(x)), evaluated on the light-ray x µ = n µ z. Examining the explicit expressions for the lowest components of these fields, Eqs. (3.11)  is necessarily proportional to the square of a fermion mode and, therefore, vanishes. The next-to-top component O (12) (z 1 , z 2 ) contains the product of all 12 fermion modes. R−symmetry fixes its form to be where the first four Θ's carry the SU (4)−charge of two scalar fields Z = φ 14 and the remaining factors are the SU (4) singlets. Here we did not display the lower index of Θ A i . Counting the total number of Grassmann variables in (D.7), we find that it is proportional to Θ 1 Since the lower index can take only three values this product vanishes leading to (4.7).
Derivation of (4.12) According to (4.9), the instanton profile of O(z 1 , z 2 |x) can be obtained from that of O(z 1 , z 2 ) by shifting the coordinates of all fields by x. As follows from (3.1), this transformation is equivalent to shifting the position of the instanton, x 0 → x 0 − x. As before we decompose the light-ray operator as O (8) = O A (z 1 , z 2 ) andŌ (8) A =Ō (8) A (z 3 , z 4 )| x 0 →x 0 −x . Let us consider separately four terms in the first line of (D.23). The instanton profile of O (8) A is given by (D.9). To get an analogous expression forŌ (8) A , we apply the shift x 0 → x 0 −x to (D.9), change the coordinates, z 1 → z 3 and z 2 → z 4 , and replace Y Z with conjugated YZ−variables defined asZ =φ 14 = φ 23 = (YZ ) AB φ AB and satisfying (Y Z YZ) = 1. In this way we get whereĪ (8) A is obtained from (D.10) through transformations described above. Integrating out fermion modes we find d 8 ξd 8η I  where I (8) B is given by (D.21) andĪ (8) B is obtained from I B through the same transformation as before. Integration over fermion modes yields , (D.29) leading to the following relation Combining together (D.27), (D.30) and (D.33), we find that the sum of four terms in the first line of (D.23) is 1/4 × Eq.(4.12). Since it is invariant under the exchange of points, z 1 ↔ z 2 and z 3 ↔ z 4 , the contribution of terms in the second line of (D.23) is three times larger. As a result, the total contribution of (D.23) is given by (4.12).