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  1. This Pati-Salam model doesn't predict gauge mediated proton decay, according to Wikipedia.

  2. However the Georgi-Glashow model predict gauge mediated proton decay, according to Wikipedia.

  • I thought the proton decay in the Georgi Glashow model is due to the $SU(5)$ representation mixes also the $SU(3)$ and $SU(2)$ representations. So the $SU(5)$ gauge boson can interact with the baryons with color and leptons?

  • But the Pati Salam model contains a $SU(4)$ which mix with the color $SU(3)$ and the lepton sector, why the Pati Salam model doesn't allow proton decay? The $SU(4)$ representation mixes also the $SU(3)$ and $U(1)$ representations. So the $SU(4)$ gauge boson can interact with the baryons with color and some leptons? Should it trigger the proton or other hadron decays?

What are the differences? (After posting this, I find a related article Why do gauge bosons/leptoquarks not mediate proton decay in the Pati-Salam model? but I ask a comparison between the two models which is somehow more.)

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  • $\begingroup$ Just want to point out that the argument "$SU(4)$ contains $U(1)_{B−L}$ and preserves it ($B-L$)" in the linked answer is wrong. For example, $SU(2)$ has three generators $L_x, L_y, L_z$, which does NOT imply that the $L_z$ is conserved given that $SU(2)$ contains $U(1)$ of $L_z$. $\endgroup$
    – MadMax
    Jan 19, 2021 at 19:24
  • $\begingroup$ maybe you can inform the original post OP - thanks! $\endgroup$ Jan 20, 2021 at 4:13

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In gauge interaction of Pati-Salam model, the quantum number B+L is accidentally conserved. Two conserved quantities B+L and B-L ensure the conservation of both B and L separatedly. Baryon number non-violation means that gauge bosons of PS group do not mediate proton decay.

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Both the answers to this post and the related post invokes the conservation of $B-L$.

However, argument for conservation of $B-L$ is flawed (or at least insufficient). It goes like this: since Pati Salam's SU(4) contains $$ SU(3)_{color} \times U(1)_{B−L} $$ therefore $B−L$ is conserved and protect by the symmetry $U(1)_{B−L}$.

The above argument is flawed, or at least insufficient. Why? Let's take for example the electroweak theory's $SU(2)$ which contains $$ U(1)_{N_{neutrino}-N_{electron}} $$ generated by $$ \tau^z $$ where $\tau^z$ along with $\tau^x$ and $\tau^y$ constitute the algebra of $SU(2)$.

Obviously, the particle number of electron/neutrino individually is NOT conserved in electroweak theory, since the $W^{\pm}$ gauge bosons change $N_{neutrino}-N_{electron}$. In the other words, even if $U(1)_{N_{neutrino}-N_{electron}}$ is part of $SU(2)$, the $W^{\pm}$ gauge bosons (NOT commuting with $U(1)_{N_{neutrino}-N_{electron}}$) can spoil the conservation of $N_{neutrino}-N_{electron}$.

Similarly, the fact that Pati Salam's $SU(4)$ contains $U(1)_{B−L}$ does NOT automatically imply that $B−L$ is conserved, since there are gauge bosons (leptoquarks in the coset of $su(4)/(su(3)_{color} \times u(1)_{B−L})$) in $SU(4)$ that does NOT commute with $U(1)_{B−L}$. To prove that there is no proton decay in Pati Salam model, one has to provide more robust arguments, if any.

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