{J \gamma r_0^{2\eta-1}  }   \int \mathrm{d} r  \cos{ \sqrt{8 \pi \eta } \Theta} \nonumber\\
&&=\alpha {Jr_0 \gamma_r  }   \int \mathrm{d} r  \cos{ \sqrt{8 \pi \eta } \Theta}=-2\mu  \int \mathrm{d} r  \cos{ \sqrt{8 \pi \eta } \Theta},\end{aligned}$$ where $  \sum_r(S^{x}_{r}S^{x}_{r+1}-S^{y}_{r}S^{y}_{r+1} )r_0/r_0^{2\eta}\to \alpha \int \mathrm{d}r\cos{ \sqrt{8 \pi \eta } \Theta}$.

With the help of Eqs. (\[MsG\]) and (\[mG\]) we express the sine-Gordon mass as $$M_{sG}= \left( \frac{ -\alpha \gamma_r Jr_0}{ 2\kappa(\eta)}  \right)^{\frac{1}{2(1-\eta)}}.$$

Equating $M_{XYZ}=M_{sG}$ gives us the following equation $$\label{mu1}
4^{2(1-\eta)} \frac{\gamma_r}{4\sin^2(\pi \eta)}= \frac{ -\alpha \gamma_r Ja }{ 2\kappa(\eta)}.$$

Using the following property of $\Gamma$ functions, $$\Gamma(\eta)\Gamma(1-\eta)= \frac{\pi}{\sin{(\pi\eta)}}$$ we obtain

$$\begin{aligned}
\alpha&=&-\frac{4^{2(1-\eta)} \Gamma^2(\eta) }{4\pi^2(1-\eta)}\left(  \frac{\sqrt{\pi}  \Gamma(\frac{1}{2(1-\eta)}) }{2\Gamma(\frac{\eta}{2(1-\eta)}) }  \right)^{2(1-\eta)}\nonumber\\
&=&-\frac{\Gamma^2(\eta) }{4\pi^{1+\eta} 2^{2(\eta-1)}  (1-\eta)}  \left(  \frac{  \Gamma(\frac{1}{2(1-\eta)}) }{\Gamma(\frac{\eta}{2(1-\eta)}) }  \right)^{2(1-\eta)} .\end{aligned}$$

Note that due to the $U(1)$ symmetry at $\gamma=0$, $$\sum_r S^{x}_{r}S^{x}_{r+1} = -\sum_r S^{y}_{r}S^{y}_{r+1} \to  \frac{\alpha}{2r_0^{1-2\eta}} \int \mathrm{d} r  \cos{ \sqrt{8 \pi \eta } \Theta}.$$

Also note that at $\gamma=0$ the effective theory enjoys conformal invariance and hence a unique normalization of correlation function is carried to all distances Eq. (\[SDN\]). Finally we obtain for $\gamma=0$, $$G^{x,x}_u(r) =G^{y,y}_u(r) = \frac{B_2}{r^{4\eta}},$$ where

$$\begin{aligned}
B_2&=& \frac{\alpha^2 }{2^3}= \frac{1}{2^7}\frac{\Gamma^4(\eta) }{\pi^{2+2\eta} 2^{4(\eta-1)}  (1-\eta)^2}  \left(  \frac{  \Gamma(\frac{1}{2(1-\eta)}) }{\Gamma(\frac{\eta}{2(1-\eta)}) }  \right)^{4(1-\eta)}\nonumber\\
&=& \frac{\Gamma^4(\eta) }{\pi^{2+2\eta} 2^{3+4\eta}  (1-\eta)^2}  \left(  \frac{  