# Why don't $\rm SO(10)/Spin(10)$ or larger GUTs permit electron-pair decay?

$$\rm Spin(10)$$ unifies all left-handed fermions and anti-fermions, and all right-handed fermions and anti-fermions. And its Pati-Salam subgroup unifies quarks with leptons (SU(4)) and complements SU(2)L with SU(2)R.

And $$\rm Spin(10)$$ includes $$\rm SU(5)$$'s $$u_L^α\to \bar{u}_L^{\bar β}+X^{\bar γ}$$ and even introduces $$d_L^α\to \bar{d}_L^{\bar β}+X^{\bar γ}$$. So why doesn't Spin(10) introduce anything like $$e_L\to \bar{e}_L + X_{???}$$?

An intermediate answer is such vertices also change color, which includes lilac. But in that case, why doesn't Spin(10) introduce constant-color ones like $$u_L^α\to \bar{u}_L^{\bar α}+X_{???}$$? Is it because this would require flipping three indices? Do any larger GUTs do so, even non-viable ones?

• Electrons can't decay into positrons, because positrons already have the same mass as electrons. – knzhou Sep 4 '20 at 5:07
• @knzhou Nope, doesn't apply before SSB. Neutrinos can become quarks via Spin(10) bosons, e.g. $ν\to \bar d + Y^-$. – alexchandel Sep 4 '20 at 5:21
• @knzhou Also up & anti-up quarks have the same post-SSB mass too, yet SU(5) and Spin(10) still contain $u_L^α\to \bar{u}_L^{\bar β}+X^{\bar γ}$. – alexchandel Sep 4 '20 at 5:24
• No, you're saying that these vertices are allowed by the gauge symmetry. That doesn't mean the processes are allowed by energy-momentum conservation. – knzhou Sep 4 '20 at 5:26
• @knzhou No, it isn't: one or more particles can be virtual. I asked for elaboration why Spin(10) symmetry doesn't permit this vertex. I didn't ask what "four-momentum conservation" is. – alexchandel Sep 4 '20 at 5:49