  Tractors and twistors can be obtained via gauge reduction of the conformal
Cartan geometry, thanks to the dressing field method. Perhaps surprisingly, it
is possible to reduce the gauge symmetry further still. One can indeed erase
Weyl symmetry thanks to the tractor field, in a way reminiscent of how the
C^2-scalar field allows to erase SU(2) in the electroweak model. This suggests
an alternative to the Weyl or conformal spontaneous symmetry breaking (SSB)
that some authors proposed as improvement for Standard Model physics or
inflationary cosmology. Here, after gauge reduction via dressing, there remains
only the Lorentz gauge symmetry and the twistor field becomes - for all
practical purposes - a Dirac spinor field. In a very simple and natural toy
model, the latter acquires a mass through Lorentz SSB due to the VEV of the
Weyl-invariant tractor field.
