# Why in $SU(5)$ we do not consider $\bar{\nu}_L$?

In GUT, why in representation $\bar{5}+10$ of $SU(5)$ we do not consider $\bar{\nu}_L$? One says that there are 15 particles-antiparticles per generation but, for me, there are 16 particles-antiparticles.

• why do you think that $\nu_L \in 5$ describes a different physical state than $\nu_L \in \bar{5}$? – jak May 18 '17 at 8:39
• Of course but I meant why in $\bar{5}+10$ (or $5+\bar{10}$) do we have both $e_L^+$ and $e_L^-$ and not both $\bar{\nu}_L$ and $\nu_L$ ? – ketherok May 19 '17 at 22:10

When Georgi and Glashow discovered the first GUT model, they noticed that all standard model particles fit perfectly in the $\bar{5}\oplus 10$ representation of $SU(5)$. That's why they proposed that it is a good idea to study this kind of model with $SU(5)$ as a GUT group.
However, if you like you can consider an $SU(5)$ model with fermions in the $\bar{5}\oplus 10 \oplus 1$ representation, i.e. simply add the right-chiral fermion by hand to the model.
This is, in fact, exactly what you get when you consider $SO(10)$ as a GUT group. The $15$ standard model fermions plus the right-chiral neutrino fit perfectly in the $16$ of $SO(10)$. When you break $SO(10)$ to $SU(5)$ you get
$$16 \to \bar{5}\oplus 10 \oplus 1$$
• @AccidentalFourierTransform I think the singlet makes no difference for the anomaly cancellation. The anomalies within $\bar{5}\oplus 10$ cancel and equally withing $\bar{5}\oplus 10 \oplus 1$. Otherwise we would have a strong argument for or against right-chiral neutrinos – jak Jun 2 '17 at 6:25