\mathcal{I}_{\infty} \mid q \, \rangle\, .\end{aligned}$$

For the spherically symmetric solutions under consideration $\omega$ vanishes and this requires the phase of $X$ to be covariantly constant, i.e.

$$\label{eq:noNUT}
\langle\, \mathcal{I} \mid q \, \rangle=
\langle\, \mathcal{I}_{\infty} \mid q \, \rangle=0\, .$$

We will later show that this is equivalent to the requirement that the NUT charge vanishes. Since there is only dependence on $\rho$, the phase of $X$ can simply be gauged away by means of a K" ahler transformations. The phase of $\mathcal{Z}$ is then also constant, whence $\mathcal{Z}/X$ is real, which can be used to write

$$\label{eq:dX-1}
\frac{d |X|^{-1}}{d\rho} =\pm 2  |\mathcal{Z}|\, .$$

The $\pm$ sign is the sign of $\langle\, \mathcal{R} \mid q \, \rangle$ and we can argue that it has to be positive if the mass is going to be positive: if we take Eq. (\[eq:dgrr\]) at $\rho=0$ ($r=\infty$), we find that the mass of the solution is given by the linear combination of charges and moduli

$$\label{eq:linearBPSmass}
M=  \langle\, \mathcal{R}_{\infty} \mid q \, \rangle\, .$$

Observe that there is no [*a priori*]{} guarantee that $M>0$: this is a condition that has to be imposed independently as to avoid singularities. We will do so and 