Symmetric Phase:

In this case, if there is no external source, then we have $v_{0}=0$ and therefore from Equation (6) the renormalized mass parameter is given by

$$\alpha_{\mathfrak{N}} = \alpha + \lambda \left[ (\mat... ...-1) \sigma_{0}^{2} + 3 \sigma_{\mathfrak{N}}^{2} \right],$$

which is a positive quantity in this phase. It is interesting to observe that, since in the continuum limit $\sigma_{0}$ and $\sigma_{\mathfrak{N}}$ become identical, for the purposes of that limit this equation is identical to the corresponding result for $\alpha_{0}$, shown in Equation (10), thus exhibiting the symmetry of the model in this phase. On the other hand, if there is an external source, then there is also some value of $v_{0}$ associated to it, and therefore according to Equation (6) the renormalized mass parameter changes to

$$\alpha_{\mathfrak{N}} = 3\lambda v_{0}^{2} + \alpha +... ...-1) \sigma_{0}^{2} + 3 \sigma_{\mathfrak{N}}^{2} \right].$$

This means that, given a point $(\alpha,\lambda)$ in the parameter space of the model, the renormalized mass increases with $v_{0}$ and thus with the external source. This is now different from $\alpha_{0}$, since it increases three times as fast with $v_{0}^{2}$. Note that once again $\alpha_{\mathfrak{N}}$ does not depend directly on the external source, but on $v_{0}$ instead.