Why Neutrino Masses Cannot Arise from Nonzero VEV of Charged Higgs Field in the Only Higgs Doublet

A scheme of neutrino mass generation is proposed by [arXiv:2003.12069], in which the small nonzero neutrino masses come from nonzero vacuum expectation value (VEV) of the charged Higgs field in the Higgs doublet in the Standard Model of particle physics (SM). However, the introduction of nonzero VEV for the charged Higgs field implies broken $\mathrm{U}(1)$ symmetry in the electroweak theory, which leads to non-conservation of electric charge and nonzero mass of photon. By comparing the predicted ratio of VEVs of charged Higgs field and neutral Higgs field in [1] with the ratio constrained by the present experimental upper bound of photon mass, we show that this scheme does not work.