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Grand unification theories (GUT), such as $SU(5)$/$SO(10)$/SUSY variants, suggest proton decay. The lack of observational evidence for proton decay is supposed to rule out basic GUTs, at least for the basic $SU(5)$ GUT.

But does the lack of evidence for proton decay really rules out the basic grand unification theories?

The GUT's prediction of proton half-life is based on the assumption that the primary channel of proton decay in GUTs is usually "proton to electron/positron and meson", which is actually predicated on the assumption of $(e, \nu_e, u, d)$ being in the same generation.

However, 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. The standard generation assignment $$ (e, \nu_e, u, d)\\ (\mu, \nu_\mu, c, s)\\ (\tau, \nu_\tau, t, b) $$ is essentially arbitrary: the only rational of the above generation assignment is the relative magnitudes of masses.

If we adopt an alternative generation assignment, say $$ (\mu, \nu_\mu,u, d)\\ (\tau, \nu_\tau, c, s)\\ (e, \nu_e, t, b) $$ the standard model is still working fine as usual.

If the alternative generation assignment is true, and assuming that GUT transition processes involving GUT gauge interactions are generation preserving (meaning not changing between different particle families), then proton should decay into antimuon, rather than positron. And this means that the proton half-life is longer than envisioned before, since muons are much heavier than electrons.

If this is the case, does it mean that the lack of observational evidence for proton decay might NOT disprove simple GUTs?

There are two kinds of counter arguments to the reasoning above:

  1. $(e, \nu_e)$ and $(u, d)$ have to be grouped together, since the quantum chiral anomaly cancellation conditions demands so. But the quantum chiral anomaly cancellation conditions require that there are same numbers of quark generations and lepton generations. Does anomaly cancellation really preclude the generation grouping of $(\mu, \nu_\mu,u, d)$?
  2. In simple GUTs both electron and muon decays will happen, therefore generation assignment does not matter. However, in the case usual generation assignment, the electron channel decay is NOT suppressed, while the muon channel decay is suppressed due to the dimension 6 nature of the muon channel. Therefore the lack of evidence for proton decay indeed rules out the basic grand unification theories. Whereas in the case of the above alternative generation assignment, BOTH of the electron and the muon channels are suppressed, therefore the does the lack of evidence for proton decay necessarily rule out the basic grand unification theories?
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    $\begingroup$ What is your definition of basic/simple GUT? $\endgroup$
    – Qmechanic
    Feb 1, 2021 at 18:27
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    $\begingroup$ 1. No, anomaly cancellation does not dictate apportionment of specific species to a generation, as the weak quantum numbers of either alternative are identical. Anomalies are irrelevant here. $\endgroup$ Feb 1, 2021 at 20:22
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    $\begingroup$ If you supplant a $\mu^+$ for the $e^+$ in a proton decay to a $\pi^0$, the suppression of the huge phase-space involved is not that large, and I'm sure there are stringent experimental limits for the muon mode, exotic as it might be. My guess for your title answer is "yes". $\endgroup$ Feb 1, 2021 at 20:43
  • $\begingroup$ @CosmasZachos, could you please expand on "the suppression of the huge phase-space involved is not that large" part and formulate it as an answer? $\endgroup$
    – MadMax
    Feb 1, 2021 at 20:51

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My guess to your title answer is "yes", to the extent that the conventional-case phase space for $p\to \pi^0 ~ e^+$ is 1-0.135- 0.0005 ~ 0.865 GeV which is not dramatically larger than that for a $p\to \pi^0 ~ \mu^+$ decay, 1-0.135- 0.106 ~ 0.759 GeV. So if your alternate theory had your gauge boson X decay to $\mu^+ \bar{d}$ instead of $e^+ \bar{d}$, you'd get a comparable rate excluded, as the $\pi^0\to 2\gamma$ has been excluded to minuscule limits... But I have not fussed those in the PDG compilation... For example, Super-Kamiokande excluded Cherenkov rings, such as enter image description here

which wouldn't be dramatically different than the muon Ersatz.

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  • $\begingroup$ Thanks for the answer! Could you please provide references to how the numbers are calculated for the decay channel to muons? $\endgroup$
    – MadMax
    Feb 1, 2021 at 21:37
  • $\begingroup$ References? Back of the envelope: $m_p-m_\pi-m_\mu$. $\endgroup$ Feb 1, 2021 at 21:50

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