v$.

### Gauge boson masses

The analysis of the gauge boson mass matrix follows exactly that of the non-supersymmetric models analyzed in [@ACG; @Fermi1; @Fermi2], which we repeat only for completeness. The mass matrix is obtained by expanding the kinetic term $\sum_i (D_\mu Q_i)^{\dagger} D^\mu Q_i$ of the scalar components of the bifundamentals $Q_i$, which gives a contribution to the Lagrangian of the form [@ACG; @Fermi1; @Fermi2], $${\cal L}\supset
{g^2 v^2}\sum_i (A^{a\, \mu}_i-A^{a\, \mu}_{i-1})^2,$$ where we have used the normalization ${\rm tr}\, T^a T^b =\delta^{ab}$ for the generators of the $SU(M)$ gauge groups (this normalization will ensure a canonically normalized kinetic term for gauginos, see later), and $g$ is the gauge coupling of the $SU(M)$ groups. This gives the following mass term: $${{\textstyle\frac{1}{2}}}\, A_{i\, \mu}^a \  \mathcal{M}^2_{ij \, ab} \ A^{b\, \mu}_j$$ where the mass matrix is a direct product of the identity in the gauge index space times a more involved matrix in the lattice index space $$\label{gb-eq}
\mathcal{M}^2_{ij \, ab} = {2 g^2 v^2} \delta_{ab} \Omega_{ij}
\ \ \ \mbox{with} \ \ \
\Omega =
\left( \begin{array}{rrrrr}
2 & -1 & & &-1\\
-1& 2 & -1 & & \\
& \ddots & \ddots& \ddots & \\
& 