replace $\breve{\gamma}_i(\ell=0) = \gamma_i(\ell=0)/v=\gamma_i/v$. Setting $\breve{\gamma}_i(\xi_i)= C$, an $O(1)$ constant, we obtain $$\label{eq:33}
  \xi_i = \left(\frac{C v}{\gamma_i}\right)^{\frac{1}{2-\Delta_i}}.$$ We expect that operator 1 (2) is dominant if $\xi_1$ ($\xi_2$) is the shorter length. Hence the boundary between the two regimes, in which one or the other operator dominates, occurs when $\xi_1=\xi_2$, or $$\label{eq:34}
  \gamma_1^{2-\Delta_2}= (C v)^{\Delta_1-\Delta_2} \gamma_2^{2-\Delta_1}.$$ Note that, although the scaling of this boundary is determined by this argument, the precise location is not, due to the ambiguity of $C$. We can understand this conclusion also from the scaling of the ground state energy density, $\mathcal{E}_0$, which obeys $$\label{eq:35}
  \mathcal{E}_0(\gamma_1,\gamma_2) = b^{-2} \mathcal{E}_0\left(\frac{\gamma_1}{v}
  b^{2-\Delta_1},\frac{\gamma_2}{v} b^{2-\Delta_2}\right).$$ Choosing $b = (\frac{\gamma_1}{v})^{-1/(2-\Delta_1)}$, we obtain $$\label{eq:36}
    \mathcal{E}_0(\gamma_1,\gamma_2) = \left(\frac{\gamma_1}{v}\right)^{\frac{2}{2-\Delta_1}} f( \gamma_2/ (\gamma_1/v)^{(2-\Delta_2)/(2-\Delta_1)}),$$ where $f({{\cal X}})$ is a universal scaling function. If there is a phase transition as $\gamma_1$ and $\gamma_2$ are varied, there must therefore be a non-analyticity at ${{\cal X}}={{\cal X}}^*$ in $f({{\cal X}})$, for some ${{\cal X}}^*$. This gives an equivalent condition to Eq. (\[eq:34\]). To precisely determine the phase boundary, we need to know ${{\cal X}}^*$ (or $C$), which means we need knowledge of $f({{\cal X}})$. Such scaling functions are generally determined by the full RG flows out of the scale-invariant 