the Bethe ansatz approach, the ground-state energy of the quantum sine-Gordon model in strong external field, $[E_0(A,M_{s-G})-E_0(0,M_{s-G})]/A^2$, can be expanded in a dimensionless parameter, the ratio of the soliton mass to the field amplitude $M_{s-G}/A$. On the other hand, when viewing the sine-Gordon model in strong external field as a perturbation of conformal field theory with cosine field, the ground state energy, $[E_0(A,\mu)]-E_0(A,0)]/A^2$, can be expanded in the powers of dimensionless parameter $\mu/A^{2/(p+1)}$. Note that without strong external field, ground-state energy can not be perturbatively expanded in $\mu$, due to the infrared divergent integrals characteristic of the relevant cosine term. Matching the two ground-state energies in the first nontrivial power of $A$ gives $$\label{MsG}
M_{sG}=\left( \frac{ \mu}{ \kappa(\eta)}  \right)^{\frac{1}{2(1-\eta)}},$$ where the dimensionless parameter $\kappa$ is called the $\lambda -M$ ratio (proportionality constant between ultraviolet and asymptotic scales) and is given by [@Zamolodchikov] $$\kappa(\eta)=\frac{1}{\pi}\frac{\Gamma(\eta)}{\Gamma(1-\eta)}\left(  \frac{\sqrt{\pi}  \Gamma(\frac{1}{2(1-\eta)}) }{2\Gamma(\frac{\eta}{2(1-\eta)}) }  \right)^{2(1-\eta)}.$$

We wish to determine a proportionality constant $\alpha$, $$\label{mG}
2\mu=- \alpha \gamma_r J r_0$$ in order to obtain a precise value of the constant in the operator identification of bilinear spin operators in the scaling limit, $$\begin{aligned}
&&J \gamma \sum_r(S^{x}_{r}S^{x}_{r+1}-S^{y}_{r}S^{y}_{r+1} )= \frac{J \gamma}{r_0} \sum_r (S^{x}_{r}S^{x}_{r+1}-S^{y}_{r}S^{y}_{r+1} )r_0\nonumber\\
&&=\frac{J \gamma}{r_0} \frac{r_0^{2\eta}}{a^{2\eta}} \sum_r(S^{x}_{r}S^{x}_{r+1}-S^{y}_{r}S^{y}_{r+1} )r_0\nonumber\\
&&=\alpha 