  The class of higher-dimensional scenarios, based on a brane-localised Higgs
boson coupled to bulk fermions, can address both the flavour puzzle and gauge
hierarchy problem. A key question arises due to the possibility of fermion
profile discontinuities at the Higgs boundary: how to calculate rigorously the
fermion mass spectrum and effective four-dimensional (4D) Yukawa couplings? We
show that the proper treatment, leading to physically consistent solutions,
does not rely on any Higgs peak regularisation but requires the presence of
certain Bilinear Brane Terms (BBT). In particular, no profile jump should
appear and the Higgs regularisations turn out to suffer from mathematical
discrepancies reflected in two non-commutativities of calculation steps debated
in the literature. The introduction of BBT can by replaced by vanishing
conditions for probability currents at the considered flat interval boundaries.
Both contribute to the definition of the field geometrical configuration of the
model, even in the free case. The BBT could allow to elaborate an ultra-violet
origin of the chiral nature of the Standard Model and of its chirality
distribution among quarks/leptons. The current conditions are implemented via
essential boundary conditions to be contrasted with the natural boundary
conditions derived from the action variation. All these theoretical conclusions
are confirmed in particular by the converging exact results of the 4D versus 5D
approaches. The analysis is completed by a description of the appropriate
energy cut-off procedure. The new calculation methods presented, implying the
independence of excited fermion masses and 4D Yukawa couplings on the
'wrong-chirality' Yukawa terms, have impacts on phenomenological results like
the relaxing of previously obtained strong bounds on Kaluza-Klein masses
induced by flavour changing reactions generated through exchanges of the Higgs
field.
