What isn't Lorenz invariant about adding a rest-mass term to the dispersion relation for your particle propagators? That seems to be what everyone does in the intro chapters/lectures in QFT courses. The Klein-Gordon equation and Dirac equation with a nonzero rest-mass term transforms into itself...
I've seen the same popular news articles everyone else has about the discovery of the Higgs boson. I have always had some questions about it though that I haven't seen addressed, and I'm still beating my head against introductory QFT, so I'm not to the level where I'd pick up the answers from...