The Higss mechanism was cooked up in the context of the unification of the weak and electromagnetic unification. A strange Mexican hat ootential was introduced, having a non-zero value when the wavefunctions, or fields (though strictly spoken these are operator valued distributions) are zero! When performing local gauge (phase) transformations, the Lagrangian must stay the same. Together with the field shifting to the brim of the hat (symmetry breaking) this results in a shiffing of 4 pre SB field into a massless one (the photon), the weak intermediate massive bosons, and a Higgs, giving the mass to the latter (it is said also that they have eaten a ghost Goldstone boson...I ask you!).

Now, all of this is based on mathematics and it fits. But it's very artificial. I say the Higgs particle clearly exists. But the Higgs mechanism not. A mere mathematical ghost mechanism. An imaginary one, a virtual one (though in technical language of QFT, tbis is exactly what it is, a virtual process). What say you?