will only consider the positive sign above; Eq. (\[eq:dX-1\]) is then the expression found in the literature.

If we take another derivative of Eq. (\[eq:dgrr\]) and use Eq. (\[eq:dX-1\]), we find

$$\frac{d^{2}(-g_{rr})}{d\rho^{2}} =
2\frac{d |X^{-1}|}{d\rho}|\mathcal{Z}| +2|X|^{-1}\frac{d|\mathcal{Z}|}{d\rho}=
4|\mathcal{Z}|^{2} +2|X|^{-1}\left(\frac{dZ^{i}}{d\rho}\partial_{i}|\mathcal{Z}|
+{\rm c.c.}\right)\, .$$

Now, at $\rho=\rho_{\rm fix}=0$ we have $Z^{i}=Z_{\rm fix}$ and $dZ^{i}/d\rho=0$, and the above equation takes on the form

$$\label{eq:AZfix}
\frac{A}{2\pi} = 4|\mathcal{Z}_{\rm fix}|^{2}\, .$$

Again, there is no [*a priori*]{} guarantee that $|\mathcal{Z}_{\rm fix}|\neq 0$, which therefore is another condition that has to be imposed independently as to avoid singularities. Actually, even though in this expression $A$ is basically an absolute value, the positivity of $A$ is only guaranteed if the scalar fields take admissible values, the mass is positive etc.

These identities allow us to find two interesting expressions for $|\mathcal{Z}_{\rm fix}|$. Expanding the two sides of Eq. (\[eq:dgrr\]) as a power series in $\rho$ we find

$$\frac{A}{2\pi} = 
2 \langle\, \left.\frac{d\mathcal{R}}{d\rho}\right|_{\rho=0} 
\mid q \, \rangle\, .$$

Using the expressions in Appendix \[sec-proofs\] we get [@Ceresole:1995ca; @Ferrara:1996dd]

$$|\mathcal{Z}_{\rm fix}|^{2}
={\textstyle\frac{1}{2}}
\langle\, \left.\frac{d\mathcal{R}}{d\rho}\right|_{\rho=0} 
\mid q \, \rangle  
=-{\textstyle\frac{1}{2}}
q^{T}\mathcal{M}(\mathcal{F}_{\rm fix})q\, ,$$

where

$$\mathcal{M}(\mathcal{F}) \equiv
\left(
\begin{array}{cc}
\Im{\rm m}\mathcal{F} 
+\Re{\rm e}\mathcal{F}\Im{\rm m}\mathcal{F}^{-1}\Re{\rm e}\mathcal{F} & 
-\Im{\rm m}\mathcal{F}^{-1}\Re{\rm e}\mathcal{F} \\
& \\
