Kaluza–Klein tower of states corresponding to the compactification of the 5D theory on a circle. In order to find the lattice spacing of the corresponding 5D theory on a circle with circumference $N a$, we identify as in [@ACG; @Fermi1; @Fermi2] the low lying mass spectra by $$\frac{2\pi k}{Na}= 2 \sqrt{2} gv \frac{k\pi}{N},$$ from which the lattice spacing is found to be $a=1/(\sqrt{2} gv)$.

$
\begin{array}{cc}
\hskip-1cm (a) & \hskip1cm (b) \\
\hskip-1cm \includegraphics*[bb=140 140 455 630,angle=-90,width=8.5cm]{periodique.eps}
&
\includegraphics*[bb=110 120 500 660,angle=-90,width=8.5cm]{orbifold.eps}
\\
\tilde{A}_k (x_\mu,x_5) = e^{i\, 2\pi k x_5 /Na}\, A (x_\mu)
&
\tilde{A}^+_k (x_\mu,x_5) = \cos \left( \frac{2\pi k x_5}{2 Na} \right)
\\ \\
\langle \tilde{A}_k (x_\mu) \left| A_j (x_\mu) \right. \rangle =
{{\textstyle\frac{1}{\sqrt{N}}}}\, e^{i\, 2\pi k (j-1)  /N}
&
\langle \tilde{A}^+_k (x_\mu) \left| A_j (x_\mu) \right. \rangle =
\sqrt{{{\textstyle\frac{2}{2^{\delta_{k0}}\,N}}}}\, \cos \left( \frac{2\pi k (j-1/2)}{2 N} \right)
\end{array}
$

.5cm

### Scalar masses

The scalar fields in the bifundamental chiral multiplets of (\[fieldcontent\]) receive masses from the $D$-term contributions to the action. In particular, the Lagrangian contains a contribution $\mathcal{L} \supset -{{\textstyle\frac{1}{2}}}\,\sum_{a} D^a_i\,D^a_i$ for each $SU(M)_i$ factor in the gauge group, with $$D^a_i = g \left(
Q_i^{*\alpha \beta} \, T^a_{\alpha\gamma} \,Q_{i}^{\gamma\beta}
-
Q_{i-1}^{*\alpha\beta} \, T^a_{\gamma\beta}\,Q_{i-1}^{\alpha\gamma}
\right)
= g\ 