Why should proton decay into positron rather than antimuon?

Question

Grand unification theories, such as $SU(5)$/$SO(10)$/SUSY variants, suggest proton decay. The lack of observational evidence for proton decay rules out simple GUTs.

But wait a minute! The GUT's prediction of proton half-life is based on the assumption that proton decays into positron (along with a neutral pion). According to Cosmas Zachos's comment on another PSE post, there is NO established generational linkage between up/down quarks (making up the proton) and electron/position. We might as well assume up/down quarks and muon/antimuon are in the same generation (as suggested here, whereas electrons and, say, top/bottom quarks are in the same generation). Hence proton should decay into antimuon, rather than positron (assuming GUT vertices involving X bosons are generation preserving).

If this is the case, the proton half-life is much longer than envisioned before, since muons are heavier than electrons. Therefore, the lack of observational evidence for proton decay does NOT disprove simple GUTs at all.

Answer 1

I don't think that fixes any problems. It doesn't matter how you set up the generations; in simple GUTs both of the decays you mention will happen.

In more detail: we already know that lepton number is violated, which means that if the decay $p \to \mu^+ \pi_0$ is allowed, so is $p \to e^+ \pi_0$. From an effective field theory point of view, the fact that one of these process is allowed "directly" by an exchange of a GUT X or Y-boson, while the other requires "extra" violations, doesn't really matter. Both are described by dimension 6 operators consistent with the symmetries, so both must happen at comparable rates, suppressed by $1/\Lambda_{\text{GUT}}^2$.