  The Higgs field is a connection one-form as the other bosonic fields,
provided one describes space no more as a manifold M but as a slightly
non-commutative generalization of it. This is well encoded within the theory of
spectral triples: all the bosonic fields of the standard model - including the
Higgs - are obtained on the same footing, as fluctuations of a generalized
Dirac operator by a matrix-value algebra of functions on M. In the commutative
case, fluctuations of the usual free Dirac operator by the complex-value
algebra A of smooth functions on M vanish, and so do not generate any bosonic
field. We show that imposing a twist in the sense of Connes-Moscovici forces to
double the algebra A, but does not require to modify the space of spinors on
which it acts. This opens the way to twisted fluctuations of the free Dirac
operator, that yield a perturbation of the spin connection. Applied to the
standard model, a similar twist yields in addition the extra scalar field
needed to stabilize the electroweak vacuum, and to make the computation of the
Higgs mass in noncommutative geometry compatible with its experimental value.
