mode. The fermions modes would be denoted as $(f,f^c)_{3,1}^{n}$, where the subscript labels the $SU(3)$ triplet or singlet component and the superscript is $0$ for the zero mode and $KK$ otherwise. The low energy fermions are always the zero mode triplets $f,f^c=(f,f^c)^0_3$ so that their sub-indices and super-indices can be suppressed without confusion.

The Higgs doublet is fixed on a brane, which only has $SU(3)\times U(1)$ flavour symmetry, and is chosen to be an $SU(3)$ flavour singlet with a $U(1)$ charge of $-2$. There is no ordinary Yukawa term for the zero modes, so that all of them are higher order. However the Higgs does have a renormalizable coupling to $SU(3)$ singlet fermions.

There are two effective flavon triplets for each extra dimension $\phi_i,\bar\phi_i$. After integrating the KK modes we can obtain the effective Yukawa terms $$\mathcal{L}_Y\sim Tr\Big[f f^c H  \left(\bar{\phi}\bar{\phi}+\bar\phi\phi^\dagger+\phi^\dagger\bar{\phi}+\phi^\dagger\phi^\dagger\right)\Big],
\label{effyuk}$$ where all the $f, f^c$ are the $SU(3)$ triplet zero modes and the corresponding subscripts and superscripts are suppressed. If there is more than one extra dimension, there would be these terms for each flavon as well as all possible mixings.

The terms in Eq. \[effyuk\] come from diagrams as in figure \[fig:kk\]. The flavon insertions can be 