for every mode. Integrating the Gaussian integral for every mode and using the ζ -function regularization, we get (disregarding the logs to this order) a pre-factor of the type where the product runs over 4 sets of modes, which results in a factor of 4 in the exponent. This coincides with the L¨ uscher term (6) for a rectangle of the size R × T in d = 26 dimensions provided 2 In Appendices C and D we confirm this by an explicit calculation of the determinant of the Laplace operator and the conformal anomaly for an outstretched ellipse. The results of two previous sections demonstrate the already mentioned fact that the ansatz (1) has to be modified in order to describe the L¨ uscher term in d = 4 dimensions. A simple modification is based on the form of the path integral for a rectangle with quadratic action A$_{2}$ [ β ( t )] given by Eq. (36). For an arbitrary curve this path integral can be expressed through the determinant of the corresponding operator, that enters A$_{2}$ , which we denote as O : It is now clear that the following modification of the ansatz (1) will provide the correct value of the L¨ uscher term (6) in d dimensions: This modification of Eq. (1) does not effect the classical limit, leading to the area law, and has the meaning of altering a pre-exponential in the semiclassical approximation. 2 It is worth noting that the Wilson loops for a rectangle and ellipse then coincide [13] in d = 4 (the only dimension with a T/R Coulomb term) to the second order of perturbation theory. ∫ D t e − KA [ x ( t )] (det O ) − ( $^{d}$− 26) / $^{48}$. (55) ∫ D β ( t ) e − A 2 [ β ( t )] = (det O ) − 1 / 2 . (54) ∫ D β ( t ) e − A 2 [ β ( t )] $^{T}$≫R ∝ e πT/R (53) V. A GENERALIZATION TO ARBITRARY DIMENSIONS T R = πa 2 b . (52) ∏ modes (1 − ν ) ∝ ( 1 √$_{1}$ − ν ) 4 ∝ exp ( π 2 2 a b ) , (51) 11