How would our world change?
What would be good?
What would be bad?
What wouldn't change?
How would our world change if protons have no mass?
atoms would no longer exist, which means nothing would exist, just a sea of electrons and positrons that went on forever. You decide if that is good or bad.
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How would our world change if protons have no mass?
If protons had no mass, it would mean that they wouldn't have all that strong binding energy, which probably means they couldn't exist at all with their current 3 quark structure. So the question is kind of nonsense.
Setting that aside. Suppose the proton were just a massless fundamental particle, subject to electromagnetic forces (because it's charged). It would also be subject to gravity (because it carries energy of motion even without mass), but that isn't especially significant. A massless proton wouldn't be bound into atoms even. They would just zip around at the speed of light. This means the electrons would be unbound, and the universe would be a big glowing plasma, not unlike it was shortly after the big bang. There wouldn't be a world.
Antiprotons would be massless, too, so p+ and p- could be electromagnetically pair-produced in abundance. Depending on if these hypothetical massless particles interacted weakly, the positrons and electrons would decay into protons/antiprotons. Given enough time, even the neutrinos would interact weakly and collide with one another and scatter into proton/anti-proton pairs.
Assuming the universe started with the same amount of energy in the first place, there would have to be a lot more particles running around now. Most of the energy (at least most of the small percent that we can see) in the universe is bound up in protons. If protons were massless, there would be more of them and more of everything else (although there wouldn't be any other stable massive particles I think--they'd all go back into the massless protons. Structure formation on a cosmic scale would be completely different.
