OPE between the EnergyMomentum Tensor and the Wilson Loop in SuperYangMills theory
Abstract
We analyze the operator product expansion in , 4dimensional SuperYangMills (SYM) theory with the gauge group. We show that a closed Wilson loop does not possess an anomalous dimension and that only the shape of the loop is changed by the conformal transformation.
June 11, 2001
1 Introduction
Conformal field theory (CFT) is important in modern particle physics in various contexts. The most powerful tool in the 2dimensional CFT is the operator product expansion (OPE). The OPE between the energymomentum tensor and an operator extracts its conformal weight.
We find it interesting to consider a similar situation regarding the Wilson loop in a conformally invariant YangMills theory. In this paper, we investigate the OPE between the energymomentum tensor and the Wilson loop in , 4dimensional SuperYangMills (SYM) theory with the gauge group employing dimensional analysis and the properties of the energymomentum tensor. We find that the Wilson loop in SYM theory does not possess an anomalous dimension and that only the deformation of the loop occurs under the conformal transformation.
Another interesting related topic is the AdS/CFT correspondence. [1] The AdS/CFT correspondence enables one to evaluate such physical quantities as the multipoint function [4] and the expectation value of the Wilson loop [2][3] in the strong coupling region. There have been ambitious attempts to compute the expectation value of the Wilson loop in the strong coupling region by means of quantum field theory [6][7] for a direct test of the AdS/CFT correspondence. In particular, Gross and Drukker [7] pointed out that the expectation value of a circular Wilson loop in , 4dimensional SYM theory is determined by an anomaly in the conformal transformation that relates a circular loop to a straight line and computed the expectation value of the circular Wilson loop to all orders in the expansion. However, their analysis is based on the Feynman gauge, and the generalization to the general gauge is nontrivial. We attempt to understand the conformal anomaly through the OPE between the energymomentum tensor and the closed Wilson loop, taking advantage of its gauge invariance.
This paper is organized as follows. Section 2 is devoted to the study of the OPE between and in the gauge theory by means of dimensional analysis and the properties of the energymomentum tensor. Section 3 presents the computation for the gauge theory as a simple example of the general form investigated in the previous section. Section 4 contains the concluding remarks and the outlook for our research. The appendices contain the proofs of the formulae we derive in full detail.
2 General form of the OPE in the SYM theory
In this section, we develop the OPE between the Wilson loop and the energy momentum tensor in the SYM theory. The bosonic part of the Lagrangian and the Wilson loop are as follows:
(1)  
(2)  
(3) 
Throughout this paper, we use the following indices: and . Our analysis is carried out in Euclidean space with the metric . The indices of the scalar fields are contracted by . denotes the coupling constant. is the Wilson loop in , 4dimensional SYM theory, whose derivation is given in detail in a paper of Drukker, Gross and Ooguri. [5] represents the coordinates of the Wilson loop. The parameter of the Wilson loop is an arc length parameter, and it satisfies . is chosen such that .
denotes the energymomentum tensor of , 4dimensional SYM theory, defined by
(4) 
The following two fundamental properties of the energymomentum tensor play a crucial role in our analysis.

tracelessness: . This implies the scale invariance of the action.

divergencelessness: . This implies the conservation of the energy and momentum.
[8]r6.6cm
. Therefore, is the order of the weakest singularity that contributes to the conformal Ward identity. The OPE is expressed by
Before entering the analysis, let us review the wellknown OPE in the 2dimensional CFT. In considering the conformal Ward identity, we perform a contour integral around the operator(lowerdimensional operators)  (5)  
An example of the lowerdimensional operators is the term of the central charge being the energymomentum tensor. The coefficients of and represent the translation and the conformal weight of the operator, respectively. An important special case is a primary field, on which the OPE reduces to , with
(6) 
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and in 4dimensional Euclidean space. We perform the integral over the region wrapping the Wilson loop, and this is translated into an integral over the surface of the manifold. Let be the point on the Wilson loop nearest to the point . In other words, we take the point so that the vector is perpendicular to the tangent vector of the Wilson loop :
Let us next consider the OPE between(7) 
The dependence of the point on the coordinate is given by
(8) 
This can be derived by noting that the vector is also perpendicular to the tangent vector , where is the variation of the parameter accompanying an infinitesimal variation of the coordinate .
Let be the boundary of the 3dimensional ball of a fixed radius that is perpendicular to the tangent vector and has its center at . We wrap the Wilson loop with the surface enveloping these spheres , with running over the whole Wilson loop. We define the region inside this enveloping surface as . Its surface is, of course, the enveloping surface of the spheres . Utilizing Gauss’s theorem, the conformal Ward identity for the Wilson loop is
The meanings of the quantities appearing in the above formula are as follows.

The spacetime integral is performed over the manifold , in which the Wilson loop is included.

is a conformal Killing vector. Its explicit form is as follows:
Translation: (10) Dilatation: (11) Special Conformal Transformation (SCT): (12) 
denotes the spherical integral over , and is the normal vector on the surface .

The measure results from the difference between the inside track and the outside track. When the radius of curvature is and the radius of the sphere is , the ratio of the length of the inside track and the outside track is . Therefore, the measure must be a quantity corresponding to the value . Now, the point resides on the sphere , and the vector corresponds to . The quantity corresponding to is the curvature vector . Therefore, the measure is
(13) The minus sign results from the fact that the vector is directed at the center of curvature.
Since we perform the integral over the sphere whose surface area is , the weakest singularity in the OPE that contributes to the conformal Ward identity is .
We now investigate the OPE . In the following analysis, we separate the OPE into three parts for convenience.
(14) 
We hypothesize that the terms corresponding to the lowerdimensional operators do not emerge in the OPE . In this equation, denotes the terms containing itself without any insertion of the fields into it, which corresponds to the term in the 2dimensional CFT. The other two terms include the insertion of the fields or into . As we find later by means of dimensional analysis, the vector fields and the scalar fields are not inserted simultaneously, and we separate the terms into the contribution of the vector field and that of the scalar field. and denote the contributions with the insertion of the vector and the scalar fields, respectively.
lowerdimensional operators  

2.1 Contribution of itself without field insertion
We first investigate the contribution of itself . We express the OPE as a power series expansion in , where is the nearest point on the Wilson loop to the point . We first list the possible ingredients of this contribution:
(15) 
Here we choose the theta parameter to satisfy , and it immediately follows that

The singularity that contributes to the conformal Ward identity is at least . Therefore, .

The coefficient must have dimensions of , and this gives the condition .

We hypothesize that , , , and have nonnegative powers. Hence , , , and must each be 0 or a positive integer.

Since the coefficient must be a tensor of rank two, the total number of the indices must be even, so that is an even number.

The result should be invariant under the exchange , so that must be an even number.
The second and third conditions lead to the relation . Since , , and are restricted to be 0 or positive integers, the possible singularity in the OPE is thus
(16) 
For convenience we classify the contribution of itself according to the order of the singularity:
(17) 
Here , and denote the contributions with the singularities of , , and respectively.
2.1.1 Terms with singularities of
We first consider the terms with singularities of . Since , the powers of the other ingredients are
(18) 
Thus, we find that the possible form of the OPE is
(19)  
The coefficients , and are determined by the tracelessness and divergencelessness conditions. The former condition is simple, and gives
(20) 
The divergencelessness is less simple due to the dependence of the point on the coordinates, as computed in (8). The divergence is given by
We require that only the strongest singularity of vanish, because the weaker singularities may be canceled by the contribution of the terms in the OPE with weaker singularities. With this assumption, the divergencelessness gives the condition
(21) 
2.1.2 Terms with singularities of
These terms are evaluated in a similar fashion. Since we are now treating the terms of , the powers must satisfy , so that
(22) 
The possible form of the OPE is thus determined to be
These coefficients are again determined by the tracelessness and divergencelessness condition. The former condition is trivial, and yields
(24) 
The latter condition again is less simple, and we require only that the strongest singularity vanish, together with the results of the previous analysis. This gives
(25)  
This gives the condition
(26) 
The conditions (20), (21), (24) and (26) determine the coefficients up to two free parameters:
(27) 
The remaining contribution to the divergence is cancelled together with the terms in the OPE with weaker singularities. But we do not pursue their explicit form, because they do not affect the conformal Ward identities.
2.1.3 Terms with singularities of
We next consider the singularities of . However, it can be shown that these singularities do not contribute to the conformal Ward identity using dimensional analysis. In performing the spherical integral in the conformal Ward identity (LABEL:AZ23cwi2), we utilize the following formulae:
(28)  
Here, is the radius of the sphere. The proof of these formulae is given in full detail in Appendix A. The formula (28) indicates that all we have to do is to verify that the power is an even number. Since is the order of the weakest singularity contributing to the conformal Ward identity, neither the correction of the measure nor the positive power of in the conformal Killing vector contributes any longer. The powers of the ingredients of the OPE must satisfy , so that
(31) 
is thus restricted to be an even number, and we found above that and must be even numbers. It immediately follows that is also an even number, and hence that the terms with singularities of do not contribute to the conformal Ward identity.
2.1.4 The absence of an anomalous dimension in the Wilson loop
We have hitherto derived the contribution of itself to , with two parameters and . We have
(33)  
We now evaluate the conformal Ward identity (LABEL:AZ23cwi2) for the contribution of itself with respect to the translation, dilatation and the special conformal transformation. We utilize the formulae (28) (LABEL:AZ23int3+) in the spherical integral over . Note the following three points in the computation.

Even powers of do not affect the result, as seen from the formula (28).

The quantity . vanishes, because

The positive power of does not contribute, because we set the radius of to be a small value.
First, we compute the conformal Ward identity for the translation, and verify that the translation does not have an anomaly:
(34)  