content_image/342090.jpg content_image/342091.jpg content_image/342092.jpg content_image/342098.jpg content_image/342099.jpg content_image/342101.jpg content_image/342100.jpg content_image/342102.jpg
## IV. PATH INTEGRAL OVER REPARAMETRIZATIONS: ELLIPSE 

In this section we evaluate the path integral over reparametrizations for an ellipse and obtain a prediction for the associated L¨ uscher term. 

The necessary formulas are given in Appendix B of [6] and are partially reproduced in Appendix B. We are interested in the case of a very long ellipse when the ratio b/a → 0 and ν → 1 according to a + b πK $_{(}$√$_{1}$ ν $^{2}$) 

Using the asymptote 

we simplify Eq. (43) to 

For ν → 1 the elliptic function simplifies and we have 2 

for − π/ 2 < σ < π/ 2. Inverting Eq. (46), we find 

This is quite similar to Eqs. (32) and (34) for a rectangle with √ µ replaced by (1 − ν ). 

<img src='content_image/342090.jpg'>

<img src='content_image/342091.jpg'>

<img src='content_image/342092.jpg'>

<img src='content_image/342098.jpg'>

<img src='content_image/342099.jpg'>

The calculation of the path integral over reparametrizations at one loop is quite analogous to that for the rectangle described in the previous section. We see that the large factor of (1 − ν ) − 2 emerges in Eq. (12) because σ ∗ ∝ (1 − ν ) from Eq. (47). To evaluate the coefficient, let us consider the domain of small θ$_{1}$ and θ$_{2}$ : 

which contributes to the integral in (12) 

modulo the powers of b/a ∝ ( ln 8 $^{(1}$− ν ) ) − $^{1}$. The same contribution comes also from the domain of both θ$_{1}$ and θ$_{2}$ near π . We thus have 

<img src='content_image/342101.jpg'>

<img src='content_image/342100.jpg'>

<img src='content_image/342102.jpg'>

