  In this paper we put forward a systematic and unifying approach to construct
gauge invariant composite fields out of connections. It relies on the existence
in the theory of a group valued field with a prescribed gauge transformation.
As an illustration, we detail some examples. Two of them are based on known
results: the first one provides a reinterpretation of the symmetry breaking
mechanism of the electroweak part of the Standard Model of particle physics;
the second one is an application to Einstein's theory of gravity described as a
gauge theory in terms of Cartan connections. The last example depicts a new
situation: starting with a gauge field theory on Atiyah Lie algebroids, the
gauge invariant composite fields describe massive vector fields. Some
mathematical and physical discussions illustrate and highlight the relevance
and the generality of this approach.
