  In this work we establish in a rigorous manner, and a model independent way,
the conditions for bypassing Bekenstein's no-scalar-hair theorem for static,
spherically symmetric, and asymptotically flat black holes, while maintaining
the validity of the energy conditions. Specifically, we argue that a hidden
assumption in the theorem, namely the vanishing of the quantity $\mathcal{G} =
\mathcal{E} + T^\theta\!_\theta$, where $\mathcal{E}$ is the energy density and
$T^\theta\!_\theta$ the corresponding component of the energy-momentum tensor
of the scalar field theory, can be relaxed. Indeed, if $\mathcal{G}$ is
positive, as a consequence of the assumption on the validity of the energy
conditions, then scalar hair is potentially allowed in the black hole's
exterior, consistently with the gravitational equations and the generic
properties of the (non-trivial) energy momentum tensor. As an explicit example,
in which such a behaviour is realised, we discuss the well known model of a
(3+1)-dimensional Schwarzschild black hole coupled to a spontaneously broken
Yang-Mills $SU (2)$ gauge theory interacting with a Higgs scalar. We present a
rather novel approach to obtain analytical black hole solutions for this system
(in contrast to the numerical ones in the existing literature) by applying an
appropriate perturbative treatment whereby the black hole configuration is
derived as a result of backreaction of Higgs and gauge fields onto an initially
fixed flat spacetime. The massive nature of the scalar and gauge fields in this
example requires a special treatment, because of their asymptotic form, which
we discuss in some detail.
