In the lepton Lagrangian there are no mass terms allowed for the gauge bosons, due to gauge symmetry. To fix the problem of existing gauge boson masses, we introduce another field (Higgs) and say that here we
1) have self-interactions
(Why do we have self-interactions for the Higgs field but not in the lepton case? Is the Higgs "charged" like the self-interacting gluon is?)
2) just flip the sign of the $|\phi|^4$ term
(Why are we allowed do to that?)
3) hence break the gauge symmetry and get gauge boson masses
(Why are we allowed to do break the symmetry here but not for the lepton field?)
→ We have similar initial complex fields (lepton and Higgs) with similar couplings to the other gauge bosons, so why are we allowed to treat both fields in such a different way?