$1/M$ and diverges as $M\rightarrow 0$. This is illustrated in Figure \[fig:m2M2\]. Since $a=0$ is always semi-stable, the phase transition cannot be of the second order.

The $1/M$ singularity in originates from the fact that the kinetic operator $\left[M^2+(|\vec q|-q_0)^2\right]$ vanishes at $M^2=0$ and $|\vec{q}| = q_0$. If the zero of the kinetic operator were at $\vec{q}=0$, it would not have caused the $1/M$ singularity because the factor $q^{d-1}$ in the phase space volume $q^{d-1}dq$ would have suppressed it. The singularity is generated in our case because of the larger phase space volume $\sim q_0^{d-1}dq$ at $|\vec{q}|=q_0$ in . It ensures that there is a positive $M^2$ solution to and that the origin of the field space is meta-stable.

This feature is shared by the gravity theory considered in this paper. In our classical analysis in the last section, we found that the phase transition happens at $T=T_c$ and the new phase for $T < T_c$ is represented by the non-linear static solutions constructed in section \[sec:MCSSBHAdS\]. Since the phase transition is of the second order, the size of the non-linear solution grows linearly in $\sqrt{T_c - T}$. This means that, at $T=T_c$, there are static solutions to the linearized 