\\
  \mathcal{S}^+_{\pm 2\delta}(x) & = & \pm \frac{i}{2} A_2 e^{i\beta \theta}
  e^{\pm i 2\pi \phi/\beta}, \\
  \label{eq:bosonizespinops+3}
  \mathcal{S}^+_\pi(x) & = & A_3 e^{i\beta\theta}.\end{aligned}$$ The parameter $\beta$ is obtained by solving the integral equations,[@Bogoliubov-Izergin-Korepin; @Qin-Fabrizio; @Cabra-Honecker-Pujol] and the amplitudes $A_1,A_2,A_3$ have been determined numerically in Ref. . We note that the above effective field theory describes the long-distance correlations of the spin chain, and is a good approximation beyond some cut-off length $a_0$. For “generic” values of the magnetization, this is of the order of lattice spacing; however, it diverges near saturation ($|M|\rightarrow 1/2$), where it scales roughly as the distance between flipped spins (antiparallel to the field), $a_0 \sim
(1/2-|M|)^{-1}$.

The parameter $\beta = 2\pi R$ is related to the “compactification radius” $R$. At zero magnetization $M=h=0$, the $SU(2)$ invariant Heisenberg chain has $2\pi R^2 = 1$. In the field $\beta$ and $R$ decrease (see Fig. \[fig:Beta\_curve\] of Appendix \[sec table\]) toward the limit $2\pi R^2=1/2$ as $|M|\rightarrow 1/2$. The scaling dimensions for general $M$ are given in terms of $R$, and in the limits of zero and full polarization, are listed in Table \[tab:dims\].

               Operator                      $\Delta$         $M=0$   $M\rightarrow 1/2$
  ----------------------------------- ---------------------- ------- --------------------
           $\mathcal{S}^z_0$                   $1$             $1$           $1$
   $\mathcal{S}^z_{\pi \pm 