The block-diagonal subgroup
$$
S(U(3)\times U(2))
=\{\operatorname{diag}(U_3,U_2):\det U_3\det U_2=1\}
\subset SU(5)
$$
is locally $SU(3)\times SU(2)\times U(1)$, with only a finite central quotient distinguishing the global groups. The hypercharge direction in the fundamental representation is
$$
Y=\operatorname{diag}\!\left(-\frac13,-\frac13,-\frac13,
\frac12,\frac12\right),
$$
which is traceless and commutes with the $SU(3)$ and $SU(2)$ blocks.

The canonically normalized $SU(5)$ generator must satisfy $\operatorname{Tr}(T_Y^2)=1/2$. Since $\operatorname{Tr}(Y^2)=5/6$,
$$
T_Y=\sqrt{\frac35}\,Y.
$$
The unified <gauge covariant derivative> contains $g_5T_YB_\mu$, whereas the Standard Model convention contains $g_YYB_\mu$. Hence $g_Y=\sqrt{3/5}\,g_5$. The non-Abelian generators already have the canonical normalization, so <gauge coupling unification> in the <SU(5) grand unified theory> predicts
$$
\boxed{g_s^2=g_w^2=g_5^2=\frac53g_Y^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 2 / v / Solution

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Gauge coupling unification is the convergence of several low-energy gauge couplings to one coupling at a high scale because the gauge factors embed into one simple group.


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