In Section 15.7, we speculated that a grand unified group $G$ exists such that
$$
G \supset S U(3) \times S U(2) \times U(1) \text {. }
$$
Georgi and Glashow have shown that the smallest such group of gauge transformations is the group $S U(5)$. Of course, different GUT's with larger groups than $S U(5)$ can be constructed.
Once a group is chosen for investigation, we have to assign the quarks and leptons to multiplets (i.e., irreducible representations) of the group. In the earlier chapters, we presented empirical evidence for distinct families (or generations) of fermions, (u,d; $\left.\nu_e, \mathrm{e}\right),\left(\mathrm{c}, \mathrm{s} ; \nu_\mu, \mu\right), \ldots$, where in the first family, for instance, we have
$$
\begin{aligned}
& \left.\left(\begin{array}{l}
\mathrm{u} \\
\mathrm{d}
\end{array}\right)_L, \mathrm{u}_R, \mathrm{~d}_R\right\} \quad \text { each with three colors, } \\
& \left(\begin{array}{c}
\nu_e \\
\mathrm{e}^{-}
\end{array}\right)_L, \mathrm{e}_R^{-},
\end{aligned}
$$
together with their antiparticles. This grouping ensures that we can construct gauge theories free of anomalies, see Section 12.12 . In a family, there are thus 15 left-handed states, for example, those in (15.61) together with $\overline{\mathrm{u}}_L, \overline{\mathrm{d}}_L$, and $\mathrm{e}_L^{+}$. For the $S U(5)$ model, these can be accommodated in a fundamental $\overline{5}$ - and a 10-representation (the 10 is the antisymmetric part of the product of two fundamental 5-representations, compare (2.58)). Explicitly, we have for the lefthanded states,
$$
\begin{aligned}
\overline{5} & =(1,2)+(\overline{3}, 1)=\left(\nu_e, \mathrm{e}^{-}\right)_L+\overline{\mathrm{d}}_L, \\
10 & =(1,1)+(\overline{3}, 1)+(3,2)=\mathrm{e}_L^{+}+\overline{\mathrm{u}}_L+(\mathrm{u}, \mathrm{d})_L,
\end{aligned}
$$
where we have shown the $\left(S U(3)_{\text {color }}, S U(2)_L\right)$ decomposition of the multiplets.
What are the gauge bosons of $S U(5)$ ? An $S U(N)$ gauge theory has $N^2-1$ gauge bosons. For the $S U(5)$ model, these are
$$
24=(8,1)+\underbrace{(1,3)+(1,1)}_{\text {gluons }}+\underbrace{(3,2)+(\overline{3}, 2)}_{\mathrm{X}, \mathrm{Y}, \mathrm{Y} \text { bosens }}
$$
So, we have a new pair of superheavy gauge bosons, $\mathrm{X}$ and $\mathrm{Y}$. They form a weak doublet and are colored. They mediate interactions which turn quarks into leptons:
$$
(\mathrm{u}, \mathrm{d})_L \rightarrow \mathrm{e}_L^{+}+(\overline{\mathrm{Y}}, \overline{\mathrm{X}}),
$$
or, in $S U(5)$ parlance,
$$
(3,2) \rightarrow(1,1) \otimes(3,2) .
$$
Is the appearance of such transitions really surprising? First, recall that at energies above $M_W$, the distinction between weak and electromagnetic interactions disappears. Similarly, at the GUT-scale $M_{X, \gamma}$, which we identify with (15.58), the strong color force merges with the electroweak force, and the sharp separation of particles into colored quarks and colorless leptons, which interact only through the electroweak force, disappears. This leads to lepton/baryon number-violating interactions such as (15.65).