Symmetric Phase:

In this case, if there is no external source, then we have $v_{0}=0$ and therefore from Equation (5) the renormalized mass parameter is given by

$$\alpha_{0} = \alpha + \lambda \left[ (\mathfrak{N}+1) \sigma_{0}^{2} + \sigma_{\mathfrak{N}}^{2} \right],$$ (D.10)

which is a positive quantity in this phase. On the other hand, if there is an external source $j_{0}$, then there is also some value of $v_{0}$ associated to it, and therefore according to Equation (5) the renormalized mass parameter changes to

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

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. Note however that $\alpha_{0}$ does not depend directly on the external source, but on $v_{0}$ instead. This indicates that, in the case of localized external sources, the renormalized mass should depend both on the external source and on the relative position between the external source and the point of measurement of the mass.