  For the Yang-Mills theory coupled to a single scalar field in the fundamental
representation of the gauge group, we present a gauge-independent description
of the Brout-Englert-Higgs mechanism by which massless gauge bosons acquire
their mass. The new description should be compared with the conventional
gauge-dependent description relying on the spontaneous gauge symmetry breaking
due to a choice of the non-vanishing vacuum expectation value of the scalar
field. In this paper we focus our consideration on the fundamental scalar field
which extends the previous work done for the Yang-Mills theory with an adjoint
scalar field. Moreover, we show that the Yang-Mills theory with a
gauge-invariant mass term is obtained from the corresponding gauge-scalar model
when the radial degree of freedom (length) of the scalar field is fixed. The
result obtained in this paper is regarded as a continuum realization of the
Fradkin-Shenker continuity and Osterwalder-Seiler theorem for the
complementarity between Higgs regime and Confinement regime which was given in
the gauge-invariant framework of the lattice gauge theory. Moreover, we discuss
how confinement is investigated through the gauge-independent
Brout-Englert-Higgs mechanism by starting with the complementary gauge-scalar
model.
