lightest Dirac neutrino mass. We will show is that unless the lightest Dirac neutrino is sufficiently light, the form of the scalar potential for $\sigma$ is not consistent with the 3D version of equation (\[ours\]), for some value of $R$ the scalar interaction becomes weaker than gravitation.

We want to check if the effective potential of the SM compactified on a circle verifies Eq. (\[ours\]). For practical reasons it is useful to define: $$\frac{\tilde{\chi}}{M^2_\text{p}} \equiv2\left(\frac{V^{\prime\prime\prime}}{V^{\prime\prime}}\right)^{2}-\frac{V^{\prime\prime\prime\prime}}{V^{\prime\prime}},$$ since the plots become easier to read. On the other hand, the intuition on what could change if a perturbation to the potential is included is lost, since we are taking ratios. In terms of this new variable Eq. (\[ours\]) is $\tilde{\chi} \geq 1$. In computing $\tilde{\chi}$ all derivatives are taken with respect to the canonical field $\phi$. However, in Fig (\[neutrinos\]) we plot it with respect to $R$, for simplicity. The derivatives can be computed analytically using standard formulas involving the $K_{n}$ functions. We find that for normal neutrino hierarchy the Strong SWGC is violated unless the lightest neutrino is lighter than $1.5\times 10^{-3}$ eV, see fig.(\[neutrinos\]).

![ Bound on neutrino mass for normal hierarchy[]{data-label="neutrinos"}](neutrinos.pdf)

Interestingly, in the case of inverted hierarchy we obtain 