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We build rigorously the attractive five-dimensional model where bulk fermionspropagate along the $ \mathcal{S}^1 / \mathbb{Z}_2$ orbifold and interact witha Higgs boson localised at a fixed point of the extra dimension. The analyticalcalculation of the fermion mass spectrum and effective Yukawa couplings isshown to require the introduction of either Essential Boundary Conditions (EBC)imposed by the model definition or certain Bilinear Brane Terms (BBT) in theaction, instead of the usual brane-Higgs regularisations. The obtained fermionprofiles along the extra dimension turn out to undergo some discontinuities, inparticular at the Higgs brane, which can be mathematically consistent if theaction is well written with improper integrals. We also show that the$\mathbb{Z}_2$ parity transformations in the bulk do not affect the fermionchiralities, masses and couplings, in contrast with the EBC and BBT, but whenextended to the fixed points, they can generate the chiral nature of the theoryand even select the Standard Model chirality set-up while fixing as well thefermion masses and couplings. Thanks to the strict analysis developed, theduality with the interval model is scrutinised.

Rigorous treatment of the $\mathcal{S}^1 / \mathbb{Z}_2$ orbifold model with brane-Higgs couplings