adapted to other cases. For definiteness, we adopt this set of reference threshold values: $ M_2=10^{12}$ GeV, $M_3=10^{15}$ GeV, $\Lambda=2~10^{18}$ GeV. Since, as demanded by perturbativity, $(\2  \3)_{ii} \le O(1)$, O ( 0.02 )     . \[ae\_ss\] This upper estimate is easily adapted to any given model once the relation between $a_0$ and $m_R$ is made explicit. To make the dependence more manifest, let us introduce the following two mSUGRA situations: a) $a_0^2 = 2 m_0^2$; b) $a_0^2 =\ti  M_{1/2}^2$, where $\ti M_{1/2}$ is the universal gaugino mass at the gauge coupling unification scale. In the first case $(a_0/m_R)^2 \approx 2 (1-0.9 r^2)$, with $r \equiv \ti M_1/m_R$. The ratio $a_0/m_R$ thus displays a mild excursion as it decreases from $\sqrt{2}$ down to $0.45$ when moving from the vertical left axes, where $r \ll 1$, to the joining line of the dark grey region, where $r =1$. For case b), the situation is opposite and not so mild: $(a_0/m_R) \approx 2.5~ r$. In the more natural cases in between, the ratio $a_0/m_R$ should thus be rather stable.

Let us now compare the upper estimate (\[ae\_ss\]) with the experimental bound. As appears from eq. (\[FC\]), the latter improves linearly with the experimental 