We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $Pto X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/Hto X$ are treated as classical Higgs fields. Its most comprehensive example is metric-affine gauge theory on the category of natural bundles where gauge fields are general linear connections on a manifold $X$, classical Higgs fields are arbitrary pseudo-Riemannian metrics on $X$, and matter fields are spinor fields. In particular, this is the case of gauge gravitation theory.
We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $Pto X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/Hto X$ are treated as classical Higgs fields. In this theory, matter fields with an exact symmetry group $H$ are described by sections of a composite bundle $Yto P/Hto X$. We show that their gauge $G$-invariant Lagrangian necessarily factorizes through a vertical covariant differential on $Y$ defined by a principal connection on an $H$-principal bundle $Pto P/H$.
Higgs fields are attributes of classical gauge theory on a principal bundle $Pto X$ whose structure Lie group $G$ if is reducible to a closed subgroup $H$. They are represented by sections of the quotient bundle $P/Hto X$. A problem lies in description of matter fields with an exact symmetry group $H$. They are represented by sections of a composite bundle which is associated to an $H$-principal bundle $Pto P/H$. It is essential that they admit an action of a gauge group $G$.
