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Wilson Loops for a quark anti-quark pair in D3-brane space

We calculate static Wilson loops for a heavy quark anti-quark pair in different positions in the space generated by a large number of coincident D3-branes. Simple results are obtained from limiting cases of the geodesic shape. In particular, quark anti-quark static potentials for flat and AdS spaces are reproduced.


Introduction
It was shown by Maldacena that large N superconformal gauge theories have a dual description in terms of string theory in AdS space [1]. Soon after, this proposal was used to calculate Wilson loops for gauge theories from the corresponding dual geometry [2,3]. This allows a computation of the interaction energy associated with gauge fields leading to a geometric criterion for confinement 1 . For instance, the energy of a quark anti-quark pair in large N superconformal N = 4 Yang-Mills theory is obtained from the Wilson loop of the corresponding string in AdS space. For this space the energy has a non-confining Coulomb like behaviour, as expected for a conformal field theory. This approach was applied to multicentre and rotating branes [4,5] and many other spaces and models [6,7,8,9]. More recently this approach to Wilson loops has been used in the search for confining geometries [10,11,12,13] and their related properties [14,15,16,17,18,19]. Wilson loops have also been applied to discuss scattering amplitudes [20,21] and radiation [22] using AdS/CFT correspondence.
Here we will study Wilson lines in the space generated by a large number of coincident D3-branes. This space is a particular case of the ones considered in [4,5] but here we will consider the quarks located at different positions rather than considering them always at infinity. This is interesting because the curvature of the D3-brane space is not constant. Near the branes this space is asymptotically AdS while far from the branes the metric is asymptotically flat. So the behaviour of the geodesics depends on the position of the quarks. We obtain general relations for the energy of the quark pair in the D3-brane space. Then considering limiting cases we obtain interesting simple results for this energy, including those corresponding to flat and AdS spaces.
The area of the Wilson loop for a stationary configuration is the product of the time interval times the string length. Then the energy of a quark anti-quark pair is proportional to this area. In order to obtain this energy we start describing the string dynamics by the Nambu-Goto action where we are setting the string scale α ′ = 1. The geometry of interest corresponds to a ten-dimensional metric of the form where i = 1, 2, 3 and ds 2 represents five extra transverse directions that will not be relevant in our discussion. We consider an infinitely heavy quark anti-quark pair located at r = r 1 so that their configuration is always stationary. For simplicity we can take them as sitting on one of the x i axis separated by a coordinate distance L. The string connecting these quarks represents the geodesic in this space and reaches a minimum value r = r 0 . This minimum value is determined from the equations of motion in terms of r 1 and L. Following [8] we write a relation between these parameters as Then the corresponding interaction energy of the quark anti-quark pair is given by where m q = m q (r 1 ) is the energy of each non interacting quark.

D3-brane space
The invariant measure for the ten dimensional geometry generated by a large number N of coincident D3-branes can be written as [23,24] Using this metric in eq. (1.3) and (1.5) we obtain where we defined y 1 ≡ r 1 /r 0 and This implies that the energy and the quark coordinate separation are related by The dependence of this energy on L indicates if the theory is confining or not. Note that r 0 depends on L because of equation (2.2) so that this relation may lead to different confining behaviours. To determine these behaviour one must evaluate these elliptic integrals that depend on the quark position. We are going to consider some limiting cases of the parameter y 1 = r 1 /r 0 that represents the shape of the geodesics. These simple particular cases correspond to interesting physical situations.

First case: highly curved geodesics
This situation is defined by the condition y 1 ≡ r 1 /r 0 >> 1. We will find different subcases corresponding to the minimum of the geodesic r 0 close to or far from the branes.
In this case it is convenient to consider the series expansion of the integrals (2.4,2.5): where C 1 and C 2 are constants. In order to calculate these constants we evaluate the integrals in the limit y 1 → ∞ using beta functions. The first integral is divergent in this limit but the difference I 1 − y 1 is finite. So we get Inserting in eqs. (2.2) and (2.3) these polynomial approximations we find where we disregarded terms of order O(y −3 1 ) . Note that eq. (3.4) can be solved for r 0 in terms of the position r 1 of the quarks, their separation L and the constant R. Using again the fact that r 1 >> r 0 this equation reduces to a cubic one Note that r + 0 ≥ r − 0 and there is also one non physical negative solution. Note that only one of these two roots minimizes the energy. One can see from eq. (3.5) which of the solutions will be appropriate, depending on the ratio r 0 /R. For the interesting limiting cases r 0 << R and r 0 >> R the solutions that minimize the energy are r + 0 and r − 0 respectively. Let us now consider these particular subcases.

Geodesic minimum close to the branes
In this case, corresponding to r 0 << R, the minimum of the energy is given by r 0 = r + 0 . Then one could substitute expression (3.7) in the energy (3.5). However we can get a nicer picture if we use the limit r 0 << R directly in eqs. (3.4) and (3.5) finding Identifying r 1 /2π as the energy of each quark and using the relation between L and r 0 we recover the result of [2,3] for the AdS space This result is consistent with the fact that close to the branes the D3-brane space is asymptotically AdS. Since this energy decreases with L it shows a non confining behaviour. Note that the above result is an approximation valid only for r 1 >> r 0 . So this expression becomes exact if r 1 → ∞ . If one wants to obtain the corrections to this energy in terms of powers of r 1 one should include the negative powers appearing in expansion (3.1) and also expand the cosine that shows up in (3.7). Although this approach would be valid for any r 1 > r 0 , it is not convenient in the case r 1 ∼ r 0 . We will consider an alternative approach for r 1 ∼ r 0 in section 4.

Geodesic minimum far from the branes
In this case, corresponding to r 0 >> R, the minimum of the energy is given by r 0 = r − 0 . As in the previous case, it is more convenient to consider the approximation r 0 >> R directly in eqs. (3.4) and (3.5) obtaining This leading order approximation for the energy coincides with the flat space case. This is consistent with the fact that far from the branes the D3-brane space is asymptotically flat. This energy increases with L so that it exhibit a confining behaviour.

Second case: almost straight geodesics
This situation is characterized by the condition r 1 ∼ r 0 , which means y 1 ≡ r 1 /r 0 = 1 + ǫ with ǫ << 1. As in the previous case we will also find subcases depending on the position of the geodesic minimum r 0 far from or close to the branes. In this case equations (2.4) and (2.5) reduce to so that quark coordinate separation (2.2) and interaction energy (2.3) become In order to obtain simple solutions for this system with interesting physical interpretation we consider next the limiting cases of the ratio r 0 /R as in the previous section.

Quarks far from the branes
If we put the quarks far from the branes, that means r 1 >> R the condition r 1 ∼ r 0 implies r 0 >> R , so that eqs. (4.3) and (4.4) take the approximate form This interaction energy between quarks corresponds to the flat space case. This is expected since the space felt by the quarks far from the branes is approximately flat, as can be seen from the metric (2.1) in the limit r >> R.

Quarks near the branes
Putting the quarks close to the branes, that means r 1 << R the condition r 1 ∼ r 0 implies r 0 << R and the expressions (4.3) and (4.4) are approximated by In order to obtain a relation between the energy and the quark separation L one needs to solve equation (4.7) for r 0 . Using the approximations considered in this case we find This implies that the energy (4.8) is approximated by Then the interaction energy between quarks is proportional, to leading order, to the coordinate separation L times the AdS metric factor (r/R) 2 . This can be understood noting that in this case the quark anti-quark pair is very close to the branes. So the geodesic is almost a straight line because the transverse part of the metric in this region as well as its variations against radial direction are very small. This result may be surprising if compared with the result of references [2,3] reproduced in eq. (3.11). Both cases correspond to quark anti-quark energy configurations in AdS space but in section 3.1 the quarks are far from the geodesic minimum while in this section we consider the quarks near the geodesic minimum r 1 ∼ r 0 . One may wonder if the result (4.10) above could be obtained from (3.11) but as we discussed in the end of section 3.1 the approximations considered there are only valid for r 1 >> r 0 . Both cases correspond to particular approximations of equation (2.3) or equivalently (2.6).

Conclusions
We calculated the Wilson lines for a quark anti-quark pair in the space generated by a large number of D3-branes with different quark positions. We have seen that it is possible to recover from the D3-brane space the Wilson line behaviour corresponding to both AdS and flat spaces by choosing conveniently the quark position and the geodesic curvature. It is interesting to note that the D3-brane space exhibits different confining behaviours depending on the quark position and geodesic shape.