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$_{min}$ . 1 σ ′ = 1 σ ′ ∗ + β ′ = 1 σ ′ ∗ − β ′ ( σ ′ ∗ ) 2 + ( β ′ ) 2 ( σ ′ ∗ ) 3 + O ( ( β ′ ) $^{3}$) . (15) ω = AF ( z √ µ , µ ) − i AK ( √ 1 − µ 2 ) 2 , (16) AK ( µ ) = R 2 , AK ( √ 1 − µ 2 ) = T, (18) F (1 /µ, µ ) = K ( µ ) + i K ( √ 1 − µ 2 ) . (19) 2 T R = K ( √ 1 − µ 2 ) K ( µ ) . (20) Equations (18) relates µ to the ration of R/T as where K ( µ ) = F (1 , µ ) is the complete elliptic integral of the first kind. In deriving Eq. (18) we used the important identity 5 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 The conformal map of the upper half plane onto the interior of a rectangle is given by the Schwarz–Christoffel mapping In this section we show how the path integral over reparametrizations captures the L¨ uscher term for a rectangle. F ( z √ µ , µ ) = ∫ z 0 d x √ µ − x 2 √ 1 − µx 2 (17) A and µ are related III. PATH INTEGRAL OVER REPARAMETRIZATIONS: RECTANGLE