value of the integral on the right-hand side of Eq. (C20) is then 2.84 to be compared with π as is expected from Eq. (51). To clarify the calculation, let us keep only three terms on the right-hand side of Eq. (C9b), when we get a quadratic equation and analytical formulas are available. We then find and expanding it either in x or in 1 /x we obtain analogs of Eqs. (C23) and (C24). The two expansions now match at x$_{0}$ = √$_{2 = 1}$$_{.}$$_{41 to be compared with}$ x$_{0}$ = 1 . 95 for all nine terms left on the right-hand side of Eq. (C9b). The value of the integral is now √$_{2 log}$ [ $^{$_{3 + 2}$√$_{2}$}$] = 2 . 49 which is smaller than 2 . 84, which may characterize to which extend the approximation of α$_{m}$ by the eight term in (C9b) is better than by the tree terms. In Appendix D we confirm the value of π by another method. A starting point is the formula [7, 19] that relates the L¨ uscher term to the conformal anomaly. Here the metric ρ is runs inside the ellipse with τ and σ being elliptic coordinates as in Eq. (C2). The function (inverse to (B23) from [6]) where conformally maps ellipse onto a unit disk. To calculate (D2) for a/b ≫ 1, we substitute z ( ω ) = √$_{ν}$ sn ( 2 K π ( π 2 − σ + i τ ) ) = √$_{ν}$ sn ( K − 2 K π ( σ − i τ ) ) = √$_{ν}$ cn ( 2 K π ( σ − i τ ) ) dn 2 K π ( σ i τ (D4) cn ( 2 K π i u, ν ) dn 2 K π i u, ν = 1 dn ( 2 K π u, ν ′ ) (D5) ρ = ∣ ∣$^{d}$$^{z}$ d u ∣ ∣$^{2}$ , u = τ + i σ (D2) ω = h cosh ( τ + i σ ) (D3) L¨ uscher term = 1 96 π ∫ d τ d σ ∂$_{a}$ ln ρ ∂$_{a}$ ln ρ (D1) Appendix D: L¨ uscher term from conformal anomaly 20 f ( x ) = x + √ 1 + x $^{2}$/ 2 (C25)