Interactions between different Kaluza-Klein modes in brane world

In brane-world theory, through Kaluza-Klein (KK) reduction, a higher-dimensional U(1) gauge field manifests on the brane as a series of vector and scalar KK modes, while a bulk fermion field manifests as left- and right-handed components. However, these conclusions rely on the common assumption that there is no interaction between different levels of KK modes. Recent experimental phenomena, such as flavor mixing in particles, suggest that such interactions should be taken into account. To address this, we propose an \emph{Orthonormal Completeness Hypothesis} (OCH) for the basis functions used to expand the higher-dimensional field. By applying the OCH, we demonstrate that the effective action of a free bulk U(1) gauge field is intrinsically gauge-invariant in brane models with codimension-\(d\) (\(d \geq 1\)). This effective action suggests the existence of interactions between different levels of KK modes, which can only be eliminated by choosing specific basis functions, provided the warp factors satisfy special commutation relations. In general, such interactions are universally present. We show the numerical calculations for these coupling coefficients in an interesting 6D brane world. This method can be extended to fermion fields, and it is shown that interactions between different levels of left- and right-handed KK modes exist, providing new insights into phenomena such as flavor mixing.