content_image/341712.jpg content_image/341713.jpg content_image/341714.jpg content_image/341715.jpg
To make the structure of the operator O more explicit, it is convenient to use the ex- pansion (9) of the direct function θ ( σ ) rather than that (10) of the inverse function σ ( θ ) as above. Using the identity 

<img src='content_image/341712.jpg'>

which stems from the definitions (9) and (10), we then obtain for the real-axis parametriza- tion: 

which determines the “momentum” (with respect to s ) space operator 

<img src='content_image/341713.jpg'>

<img src='content_image/341714.jpg'>

with 

We can finally substitute t ∗ ( s ) by t ( s ) in this formula without changing the semiclassical approximation. 

<img src='content_image/341715.jpg'>

It is worth noting that in contrast to the Laplace operator of Ref. [14], where the L¨ uscher term was obtained for the Polyakov string, the present operator O lives in the boundary, which makes the construction nontrivial. 

## VI. CONCLUSIONS 

The conclusion is that the reparametrization of the boundary curve involved in Eq. (1) carries information on the transverse fluctuations in 26 dimensions. As is shown in the previous section, it is possible to generalize this to any dimensions. 

Our motivation for the present paper is our previous work on the QCD/string scattering amplitudes [15], where we used that the amplitude can be expressed in terms of a Wilson loop through Feynman path integration. There we only considered the leading area behavior. However, having developed a path integral expression for the next term, we hope that the x µ integrals can be performed, thereby providing a momentum space analogue of the L¨ uscher term. We hope this may help to answer a very interesting question as to how the intercept of the Regge trajectory changes under such a modification of the ansatz for the Wilson loops. 

