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of A [ θ ∗ ( σ ) + β ] in β are suppressed [5] at large √$_{KS}$$_{min}$$_{. The loop expansion goes in the}$ parameter 1 /KS$_{min}$ and only one loop contributes with the given accuracy. 

A comment is needed about the measure for the integration over β ( θ ). As is explained in Appendix A, the measure for the integration over σ ( θ ) involves a factor 

∗ ∗ ∗ ∗ Because β ∼ 1 $_{/}$√$_{KS}$$_{min}$$_{, the second and third terms on the right-hand side are not essential}$ to the given order. Therefore, the measure for the path integration over β ( θ ) in Eq. (14) is the usual one for smooth functions β ( θ ), while this factor will show up to the next order in 1 /KS$_{min}$ . 

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## III. PATH INTEGRAL OVER REPARAMETRIZATIONS: RECTANGLE 

In this section we show how the path integral over reparametrizations captures the L¨ uscher term for a rectangle. 

The conformal map of the upper half plane onto the interior of a rectangle is given by the Schwarz–Christoffel mapping 

where 

is the incomplete elliptic integral of the first kind. The two parameters A and µ are related 

to the coordinates of the vertices of the rectangle by 

where K ( µ ) = F (1 , µ ) is the complete elliptic integral of the first kind. In deriving Eq. (18) we used the important identity 

Equations (18) relates µ to the ration of R/T as 

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