Along the same lines with previous answers suggesting "mass cannot be negative," I'd like to add an insight for why that might probably be the case. If Higgs field and particles' varying degrees of interaction with the field is what gives rise to what we call mass, then the theory suggests that protons don't have mass (and constitute the velocity limit through space) because they don't interact with the field at all. I don't think the framework allows for negative interaction with the field or an "anti-Higgs" field.

Along the same lines with previous answers suggesting "mass cannot be negative," I'd like to add an insight for why that might probably be the case. If Higgs field and particles' varying degrees of interaction with the field is what gives rise to what we call mass, then the theory suggests that photons don't have mass (and constitute the velocity limit through space) because they don't interact with the field at all. I don't think the framework allows for negative interaction with the field or an "anti-Higgs" field.