Instanton effects in correlation functions on the light-cone

Instanton effects in correlation functions on the light-cone

Abstract

We study instanton corrections to four-point correlation correlation function of half-BPS operators in SYM in the light-cone limit when operators become null separated in a sequential manner. We exploit the relation between the correlation function in this limit and light-like rectangular Wilson loop to determine the leading instanton contribution to the former from the semiclassical result for the latter. We verify that the light-like rectangular Wilson loop satisfies anomalous conformal Ward identities nonperturbatively, in the presence of instantons. We then use these identities to compute the leading instanton contribution to the light-like cusp anomalous dimension and to anomalous dimension of twist-two operators with large spin.

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IPhT-T17/053

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Institut de Physique Théorique1, Université Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette

1 Introduction

Maximally supersymmetric Yang-Mills theory ( SYM) possesses a remarkable electric-magnetic duality, also known as duality [1, 2, 3]. It establishes the equivalence between certain correlation functions computed at weak and at strong coupling. Testing the duality proves to be a complicated task as it requires understanding these functions at strong coupling [4, 5].

In the case of SYM with the gauge group, the duality predicts that the correlation functions of half-BPS operators should be invariant under modular transformations acting on the complexified coupling constant . Two- and three-point correlation functions of half-BPS operators are protected from quantum corrections and trivially verify the duality. For higher number of points, the functions receive quantum corrections and have a nontrivial dependence on the coupling constant. In order to test the duality, we have to find perturbative contribution to for finite and supplement it with instanton corrections. Although these corrections are exponentially small at large , they play a crucial role in restoring the duality.

In this paper we study instanton effects in four-point correlation function of half-BPS operators. Quantum corrections to are described by a single function of two cross ratios and (with ). It has the following general form at weak coupling

(1)

where the first term is a perturbative correction and the second one is a nonperturbative correction due to (anti) instantons. The function describes the contribution of quantum fluctuations of instantons and runs in powers of . 2

At present, the instanton corrections to (1) are known to the lowest order in . The corresponding function can be found in the semiclassical approximation following the standard approach (for a review, see [6, 7, 8]). In this approximation the quantum fluctuations are frozen and the correlation function is given by a finite-dimensional integral over the collective coordinates of instantons. An explicit expression for the function is known in one-instanton sector () as well as for an arbitrary number of instantons in the large limit. To go beyond the semiclassical approximation, we have to include quantum fluctuations of instantons. Their contribution to scales as where integer positive counts the number of instanton loops. It is much more difficult to compute such corrections and a little progress has been made over the last decade.

Perturbative corrections to (1) are known to have some additional structure [9, 10] which allows us to construct integral representation for the function to any order in without going through a Feynman diagram calculation. Moreover, SYM is believed to be integrable in the planar limit [11]. In application to the correlation function (1), this opens up the possibility to determine perturbative contribution for arbitrary ’t Hooft coupling at large [12, 13]. A natural question is whether some of these remarkable properties survive in (1) in the instanton sector.

As a first example, we examine behaviour of the correlation function at short distances , or equivalently for and . As follows from the OPE, the leading contribution to (1) in this limit, , comes from Konishi operator, unprotected operator with smallest scalling dimension . The anomalous dimension of this operator, , and its structure constant in the OPE of two half-BPS operators, , have expansion at weak coupling similar to (1). The leading instanton correction to and can be found by examining the asymptotic behaviour of the second term on the right-hand side of (1). In this way one obtains, using the known results for the function in the semiclassical approximation, that and do not receive instanton correction [14, 15, 16, 17]. Thus, the leading corrections to and can only come from quantum instanton corrections to (1).

The same quantities can be also extracted from the two- and three-point correlation functions, and , respectively. Computation of these correlation functions in the semiclassical approximation yields the following result for the leading instanton corrections and (explicit expressions can be found in [18]). Notice that both expressions have additional factors of as compared with the semiclassical contribution to (1). To get the same expressions for and from the four-point correlation function , one would have to take into account one- and two-loop instanton corrections to (1), respectively.

This example illustrates a hidden simplicity of instanton effects – finding the leading quantum instanton contribution to the four-point correlation function at short distances, , can be mapped into a semiclassical calculation of two- and three-point correlation functions of the Konishi operator [18].

We show in this paper that analogous phenomenon also happens for in the light-like limit (with ) when four half-BPS operators become light-like separated in a sequential manner. In this limit, the correlation function is expected to have the following form [19]

(2)

where the product of four scalar propagators defines the leading asymptotic behavior and the function is given by (1) for . At weak coupling, perturbative corrections to (1) are enhanced by powers of logarithms of and . Such corrections can be summed to all orders in leading to [19, 20]

(3)

Here dots denote subleading corrections and is the light-like cusp anomalous dimension in the adjoint representation of the .

The same anomalous dimension controls divergences of cusped light-like Wilson loops and its appearance in (3) is not accidental. As was shown in [19], the leading asymptotics of the function is described by (an appropriately regularized) rectangular light-like Wilson loop

(4)

evaluated along light-like rectangle with vertices at points . The subscript indicates that is defined in the adjoint representation of the . The subleading (logarithmically enhanced) corrections to (3) come from the so-called jet factor . Its form is fixed by the crossing symmetry of the four-point correlation function [20].

In this paper, we compute the leading instanton correction to the four-point correlation function (1) in the light-cone limit . Notice that this limit is Minkowskian in nature whereas instantons are defined in Euclidean signature. To find instanton corrections to (1), we shall determine the function in Euclidean domain of and and, then, analytically continue it to .

As in the previous example, we start with the semiclassical approximation to (1). As was shown in [21], the instanton corrections to scale in this approximation as and, therefore, they do not modify the asymptotic behaviour (3). To go beyond the semiclassical approximation, we analyze the light-cone asymptotics of and argue that the relation between and light-like rectangular Wilson loop mentioned above also holds in the presence of instantons. This relation allows us to establish the correspondence between the leading (quantum) instanton correction to and the semiclassical result for .

We show that the resulting expression for takes the same form as in perturbation theory (3) with the important difference that the light-like cusp anomalous dimension in (3) is modified by the instanton correction. In the simplest case of the gauge group, this correction in one-(anti)instanton sector is given by

(5)

Following [22, 23, 7], this result can be generalized to the gauge group and to the case of multi-instantons at large . To obtain the same result (5) from the direct calculation of the four-point correlation function, one would have to compute quantum instanton corrections to (1) at order .

As a byproduct of our analysis, we verify that the light-like rectangular Wilson loop satisfies the anomalous conformal Ward identities [24] nonperturbatively, in the presence of instantons. We also determine the leading instanton correction to anomalous dimension of twist-two operators with large spin . This anomalous dimension scales logarithmically with the spin (see Eq. (9) below) and its behaviour is controllled by the cusp anomalous dimension [25]. Making use of (5) we find that the leading instanton contribution scales as . This agrees with the finding of [26] that does not receive correction for any spin .

The paper is organized as follows. In Section 2 we analyze asymptotic behaviour of the four-point correlation function in the light-cone limit in the presence of instantons and discuss its relation with the light-like Wilson loop. In Section 3 we compute instanton contribution to the light-like rectangular Wilson loop in the semiclassical approximation. We then use it in Section 4 to determine the leading instanton correction to the cusp anomalous dimension. Section 5 contains concluding remarks. Details of the calculation are presented in four appendices.

2 Correlation functions in the light-cone limit

In this section we examine instanton corrections to a four-point correlation function of scalar half-BPS operators

(6)

Here auxiliary tensors satisfy and serve to project the operator onto representation of the . In virtue of superconformal symmetry, the dependence of the four-point correlation function

(7)

on variables can be factored into a universal kinematical factor independent on the coupling constant [27]. In what follows we do not display this factor and concentrate on the dynamical part that depends on the cross ratios only.

In the Born approximation, reduces to the sum of terms each given by the product of free scalar propagators. In the light-like limit, , the leading contribution to comes from only one term of the form (2) with 3 Going beyond this approximation, we apply the OPE to each pair of neighbouring operators in (7), e.g.

(8)

where the sum runs over local operators with Lorentz spin , dimension and twist . For the dominant contribution to (8) comes from twist-two operators with arbitrary spin and scaling dimension 4 It scales as and yields the expected asymptotic behaviour (2). In the similar manner, the remaining factors in (2) come from the twist-two operators propagating in other OPE channels.

A detailed analysis shows [19, 20], that the leading asymptotic behaviour of the function for is governed by twist-two operators with large spin or depending on the OPE channel. The anomalous dimension of such operators grows logarithmically with the spin,

(9)

and generates corrections to enhanced by powers of and . As we see in a moment, this observation simplifies the calculation of instanton corrections to (2).

Instantons are classical configurations of fields (scalar, gaugino and gauge fields) satisfying equations of motion in Euclidean SYM [28]. To compute their contribution to the correlation function (7) at weak coupling, we have to go through few steps. First, we decompose all fields (that we denote generically by ) into classical, instanton part and quantum fluctuations,

(10)

Here we introduced the factor of to emphasize that does not depend on the coupling constant. Then, we substitute (10) into (7) and integrate over quantum fluctuations and over collective coordinates of instantons. Finally, we match the resulting expression for into (2), identify the function and analytically continue it to small and .

In the semiclassical approximation, we neglect quantum fluctuations in (10) and obtain the following expression for

(11)

where the half-BPS operators (6) are replaced by their expressions in the instanton background and are integrated over the collective coordinates of the instantons. The relation (11) can be represented diagrammatically as shown in Figure 1(a).

For the one-instanton configuration in SYM with the gauge group the integration measure is given by [29]

(12)

Here bosonic collective coordinates and define the position of the instanton and its size, respectively. Fermionic coordinates and (with and ) reflect the invariance of SYM under superconformal transformations. For the correlation function (11) to be different from zero, the product of four half-BPS operators in (11) should soak all fermion modes.

For the gauge group the instanton depends on the additional bosonic and fermion modes. In what follows we shall concentrate on the case and discuss generalization to the later in Section 4.4.

In SYM with the gauge group, the one-instanton solution can be obtained [30] by applying superconformal transformations to the special field configuration consisting of vanishing scalar and gaugino fields and gauge field given by the celebrated BPST instanton. 5 For gauge field this leads to

(13)

where is the BPST instanton [28] and denotes component containing fermion modes and . In virtue of the symmetry, each subsequent term of the expansion has four modes more. Expressions for the scalar and gaugino fields have a form similar to (13) with the only difference that the lowest term of the expansion has a nonzero number of fermion modes whose value is dictated by the charge of the fields. Notice that the expansion (13) is shorter than one might expect as the symmetry allows for the presence of terms with up to fermion modes. It turns out, however, that all field components with the number of fermion modes exceeding vanish due to superconformal symmetry [26]. The explicit expressions for various components in (13) can be found in [6, 26].

\psfrag

x1[cc][cc]\psfragx2[cc][cc] \psfragx3[cc][cc]\psfragx4[cc][cc] \psfragI[cc][cc] \psfrag(a)[cc][cc](a)\psfrag(b)[cc][cc](b)\psfrag(c)[cc][cc](c)\psfrag(d)[cc][cc](d)

Figure 1: Instanton corrections to the four-point correlation function: (a) contribution in the semiclassical approximation; (b) leading contribution in the light-cone limit; (c) example of subleading contribution. Solid, wavy and dashes lines denote scalars, gauge fields and gauginos, respectively. Lines attached to the central blob represent the instanton background.

We recall that relations (13) define the classical part of (10). Replacing the scalar field in the definition of the half-BPS operator (6) with we find that scales in the instanton background as and contains fermion modes. Then, we can apply (11) and (12) to arrive at the following result for the four-point correlation function in the semiclassical approximation [29]

(14)

where function is defined in Appendix C. The contribution of anti-instanton is given by a complex conjugated expression. The relation (14) holds for arbitrary . For we find from (2) that the instanton contribution vanishes as for . Thus, the one-instanton correction (14) does not modify asymptotic behaviour of in the light-cone limit [21].

The same result can be obtained using the OPE. In the semiclassical approximation, the product of the operators on the left-hand side of (8) reduces to the product of two functions describing the classical profile of half-BPS operators. It is obviously regular for and, therefore, cannot produce singularity that is needed to get a finite result for in (2). For such singularity to arise, we have to go beyond the semiclassical approximation in (10) and exchange quantum fluctuations between the two operators in (8). To lowest order in the coupling we have

(15)

where . Notice that the quantum fluctuation produces singularity but its contribution is suppressed by the factor of as compared with the semiclassical result.

For the correlation function to have the expected form (2) with nonvanishing , at least one quantum fluctuation has to be exchanged between each pair of neighboring operators in (7). As follows from the above analysis, the corresponding contribution to has the following dependence on the coupling constant

(16)

where each fluctuation brings in the factor of . Comparing this relation with (1) we find that the first three terms in the expansion of the instanton induced function in powers of should vanish in the light-cone limit , in the one-instanton sector at least.

To the leading order in , the dominant contribution to (16) comes from Feynman diagrams shown in Figure 1(b). They contain four scalar propagators connecting the points and . In the first-quantized picture, these diagrams describe a scalar particle propagating between the points in an external instanton gauge field. Notice that the particle can also interact with instanton fields of gaugino and scalar but this leads to a subleading contribution. To show this, consider the diagram shown in Figure 1(c). It contains two Yukawa vertices and its contribution to has the same dependence on the coupling constant as (16). However, in distinction from the diagram shown in Figure 1(b), it does not produce singularity. Indeed, as follows from (8), the leading behaviour is controlled by the twist of exchanged operators. For the diagram shown in Figure 1(c) such operators are built from two scalar and two gaugino fields and their twist satisfies . For the diagram shown in Figure 1(b), the leading operators have twist two and are of a schematic form , where is a light-cone component of the covariant derivative.

Thus, the leading contribution to for only comes from diagrams shown in Figure 1(b). Denoting the scalar propagator in the instanton background as , we obtain the following result for the correlation function in the light-like limit

(17)

where denotes integration over the collective coordinates of instantons with the measure (12). This result is rather general and it holds for multi-instanton contribution to in SYM with an arbitrary gauge group.

We can argue following [19] that the relation (17) leads to the same factorized expression (3) for the function as in perturbation theory. The propagator depends on two momentum scales, and , which define a proper energy of the scalar particle and its interaction energy with the instanton background, respectively. For , or equivalently , the instanton carries small energy and its interaction with the scalar particle can be treated semiclassically. In this limit, reduces to a free scalar propagator multiplied by the eikonal phase given by the Wilson line evaluated along the light-cone segment . Taking the product of four Wilson lines corresponding to four propagators in (17), we obtain that the contribution of instanton with is described by the rectangular light-like Wilson loop defined in (4). The gauge fields in (4) are now replaced by the instanton solution (13) and integration over its moduli is performed with the measure (12). For the eikonal approximation is not applicable. The contribution from this region, denoted by in (3), can be determined from the crossing symmetry of in the same way as it was done in perturbation theory [20].

3 Light-like Wilson loop in the instanton background

We demonstrated in the previous section that the leading light-cone asymptotics of the correlation function is described by light-like rectangular Wilson loop . Due to the presence of cusps on the integration contour, develops specific ultraviolet divergences. In the expression for the correlation function (3), these divergences cancel against those of the function in such a way that the UV cut-off of the Wilson loop is effectively replaced with .

3.1 Conformal Ward identities

Let us start with summarizing the properties of . As was shown in [24], the conformal symmetry restricts the dependence of on kinematical invariants

(18)

where and are the divergent and finite parts, respectively. The dependence of on the UV cut-off is described by the evolution equation

(19)

where the light-like cusp anomalous dimension depends on the representation of the gauge group in which the Wilson loop is defined. The general solution to this equation depends on the so-called collinear anomalous dimension . It appears as a coefficient in front of in the expression for and depends on the choice of the regularization. The finite part of (18) is uniquely fixed by the conformal symmetry 6

(20)

Combining together (18) and (20) we obtain the following relation for

(21)

where the dependence on the UV cut-off disappears since the second derivative annihilates the divergent part of . This relation allows us to find from directly, without introducing a regularization, by computing its second derivative (21).

We would like to emphasize that relations (19) – (21) follow from the conformal Ward identities and should hold in the presence of instantons. We shall verify this property below.

We recall that the instanton correction to should match the leading correction to the correlation function for . Taking into account (16), we expect that the instanton correction to the cusp anomalous dimension should scale as .

3.2 Semiclassical approximation

To compute instanton corrections to the light-like Wilson loop, we have to define in Euclidean signature. This can be achieved by allowing the cusp points to take complex values, such that . Having determined as a function of and , we shall continue it to Minkowski signature.

To find in the semiclassical approximation, we have to evaluate the Wilson loop (4) in the instanton background and, then, integrate it over the collective coordinates with the measure (12)

(22)

We recall that is defined in the adjoint representation of the . It proves convenient to generalize (4) and define the Wilson loop in an arbitrary representation

(23)

where is a light-line Wilson line stretched between the points and

(24)

Here the gauge field is integrated along the light-cone segment and are the generators in the representation . Notice that for zero value of the coupling constant, the Wilson loop (23) is equal to the dimension of the representation .

In special cases of the fundamental and adjoint representations of the , the generators are related to Pauli matrices, , and completely antisymmetric tensor, , respectively. The corresponding Wilson loops, and , satisfy the fusion relation

(25)

where is complex conjugated to .

Applying (22), we have to replace the gauge field in (23) with its expression in the instanton background, (see Eqs. (10) and (13)). As follows from (23) and (24), the resulting expression for does not depend on the coupling constant. It depends however on fermion modes of the instanton, and . This dependence has the following general form

(26)

where denotes a homogenous invariant polynomial in and of degree . The Wilson loop in the adjoint representation has similar form. The relation (25) allows us to express in terms of .

In order to compute the Wilson loop (22) we only need the top component containing fermion modes. The remaining components give vanishing contribution upon integration over fermion modes in (22). For the Wilson loop in the fundamental representation, the top component is given by . For the Wilson loop in the adjoint representation we get from (25) and (26)

(27)

Since contains fermion modes, it has the following form

(28)

where and similar for . The scalar function depends on four points (with ) and on the bosonic collective coordinates and . It also depends on the representation of the gauge group.

Substituting (28) into (22) and taking into account (12), we obtain the following expression for the instanton correction to in the semiclassical approximation 7

(29)

The dependence of this expression on the coupling constant matches (16). We expect that the instanton effects should modify the light-like cusp anomalous dimension. For this to happen, the integral in (29) has to develop UV divergences. Indeed, as we show below, instantons of small size () located in the vicinity of the cusp points () provide a divergent contribution to (29).

3.3 Cusp anomalous dimension

The light-like Wilson loop is invariant under conformal transformations at the classical level. At the quantum level, its conformal symmetry is broken by cusp singularities. In application to (29) this implies that if the integral in (29) were well-defined, should be conformally invariant. The conformal transformations act nontrivially on the bosonic moduli and leaving the integration measure in (29) invariant. Therefore, invariance of under these transformations translates into conformal invariance of the function . To regularize cusp singularities of (29) we can modify the integration measure as

(30)

leaving the function intact. 8

The conformal symmetry dictates that the function can depend on the bosonic moduli, and , and four points through conformal invariants only. The latter have the following form 9

(31)

We recall that the points define the vertices of light-like rectangle and satisfy . As a consequence, and we are left with only two nonvanishing invariants, .

Since is obtained from the Wilson loop (28) evaluated in background of instanton field, it is an intrinsically classical quantity. We therefore expect it to be a rational function of and . In addition, should vanish for or . The reason for this is that for (or ) the rectangular contour in (23) collapses into a closed backtracking path. The Wilson lines in (23) cancel against each other for such path leading to or equivalently . These properties suggest to look for in the form

(32)

where expansion coefficients are symmetric due to the cyclic symmetry of (23). Moreover, as we show in Appendix D, is actually a polynomial in both variables, so that the sum in (3.3) contains a finite number of terms.

Replacing the function in (29) with its general expression (3.3), we find that the integrals over and can be expressed in terms of functions defined in Appendix C. These functions are finite for generic but develop logarithmic divergences for . A close examination shows that divergences arise from integration over and and have a clear UV origin. These are the cusp divergences that were mentioned at the end of the previous subsection. Regularizing divergences according to (30), we obtain the following expression for the instanton correction to the light-like Wilson loop (29)

(33)

Here we introduced notation for the regularized integral

(34)

evaluated for . This integral is well-defined for and the cusp divergences appear as poles in . The details of calculation can be found in Appendix C.

Substituting the resulting expression for (see (C) in Appendix C) into (33) we find that takes a remarkable simple form

(35)

where is the dimension of the representation in which the Wilson loop is defined. Here we denoted the residue at the double pole as anticipating that the same quantity defines the instanton correction to the cusp anomalous dimension. It is given by the following expression

(36)

We recall that are coefficients of the expansion of the top component of the Wilson loop (28) in powers of the conformal invariants (3.3). Notice that for the function in (36) develops a pole. For the sum in (36) to be finite the corresponding coefficient has to vanish.

The residue at the simple pole and the constant term in (3.3), and , respectively, are given by expressions similar to (36). However, in distinction from they depend on the choice of the regularization in (34). That is why we do not present their expressions.

Let us now compare (3.3) with the expected properties of light-like Wilson loop. We combine (3.3) with the Born level contribution to the Wilson loop, , and require that has to satisfy (19) and (21). This leads to the following relations

(37)

It is easy to check that (3.3) verifies these relations. In this way, we find that (36) defines indeed the leading instanton correction to the cusp anomalous dimension.

In addition to (3.3), the light-like Wilson loop also receives perturbative corrections that run in powers of . To lowest order in , these corrections have exactly the same form (3.3) although expressions for the anomalous dimensions are different [31]. The reason for such universality can be understood as follows. As explained in Section 3.1, the conformal symmetry fixes the dependence of the light-like rectangular Wilson loop on kinematical invariants. In particular, it allows us to determine the finite part of the Wilson loop in terms of the cusp anomalous dimension, Eq. (20). The fact that the instanton corrections (3.3) verify (19) and (21) implies that the conformal Ward identities found in [24] hold nonperturbatively, in the presence of instanton effects.

In the next section, we apply (36) to compute the leading instanton contribution to the cusp anomalous dimension in the fundamental and adjoint representations of the .

4 Instanton contribution to the cusp anomalous dimension

According to (33), the instanton corrections to the light-like Wilson loop are determined by the coefficients . To find them, we have to identify the top component of the Wilson loop (28) containing fermion modes and, then, expand the corresponding function in powers of the conformal invariants (3.3).

The top component of the Wilson loop depends on the choice of the representation . Making use of the relation (27) (and its generalization for higher spin representations of the ) we can express in terms of various components of the Wilson loop in the fundamental representation (26).

4.1 Wilson loop in the fundamental representation

To compute the Wilson loop in the fundamental representation of the , we have to replace the gauge field in (23) and (24) with its expression (13) in the instanton background, .

The resulting expression for the Wilson line (24) depends on fermion modes and admits an expansion similar to (26)

(38)

In distinction from (26), each term on the right-hand side is gauge dependent. The explicit expressions for the first three terms on the right-hand side of (38) are

(39)

where parameterizes the light-like segment and . Expressions for the remaining components of (38) are more involved. Going through their calculation we find (see Appendix B) that they are proportional to the square of a fermion mode and, therefore, have to vanish

(40)

Thus, the light-like Wilson line (38) has at most fermion modes.

Substituting (38) into (23) we obtain that the Wilson loop (26) is given by a linear combination of terms of the form

(41)

where integers