(0)=0,~\nu(0)=0,~\varphi(0)=\varphi_c,
  ~\psi(0)=0,~\tilde{p}(0)=\tilde{p}_c,$$ and Eqs. (\[eq:beq1\])-(\[eq:beq6\]) are integrated numerically up to the stellar surface at which $\tilde{p}=0$. Thereafter the solution is matched with the static and spherically symmetric “vacuum” solution, where the term “vacuum” means only the absence of matter, i.e., $\tilde{T}_{\mu\nu}=0$. This solution is given in [@de1992]. The solutions are parametrized by three parameters $b$, $d$, and $\varphi_0$. From the matching conditions at the surface, we can obtain $\nu(r)$ including the constant term. In order to set $A(\varphi_0)$ to unity, we rescale the raw quantities to the renormalized ones as follows: $$\begin{aligned}
  r_{ren}&=&A_0^2 r,~~ \mu_{ren}= A_0^2 \mu,~~ \nu_{ren}=\nu,
  ~~\varphi_{ren}=\varphi,~~ \psi_{ren}=A_0^{-2}\psi, \nonumber \\
  \varphi_{0 ren} &=& \varphi_0,~~ a_{ren}=A_0^2 a,
  ~~b_{ren}=A_0^2 b, ~~d_{ren}=A_0^2 d, ~~N_{ren}=A_0^3 N.\end{aligned}$$ Then, from the asymptotic properties at spatial infinity for a static and isolated system $$\begin{aligned}
  g_{*\mu\nu}&=&\eta_{\mu\nu}+\frac{2G_*m}{r_{ren}}\delta_{\mu\nu}
  +O\left(\frac{G_*^2}{r_{ren}^2}\right), \\
  \varphi&=&\varphi_0+\frac{G_*\omega}{r_{ren}}
  +O\left(\frac{G_*^2}{r_{ren}^2}
    \right), \end{aligned}$$ where $\eta_{\mu\nu}$ is the Minkowskian metric. We call $m$ the Arnowitt-Deser-Misner (ADM) energy and $\omega$ the scalar charge [@de1992]. $b_{ren}$ and $d_{ren}$ are related to $m$ and $\omega$ as $$\begin{aligned}
  G_* m=\frac{b_{ren}}{2}, \\
  G_*\omega= -d_{ren}.\end{aligned}$$ Hereafter the subscripts “[*ren*]{}” are omitted for simplicity. We use units in which $G_*=1$.

stability criteria
==================

In scalar-tensor theory, control parameters of the static, spherically symmetric, and isolated 