 Solution

The SU(5) grand unified theory embeds the strong and electroweak groups in a single simple gauge group. Its 24 gauge generators contain the eight gluons, three weak generators and one hypercharge generator, plus twelve additional generators connecting color and weak directions. A common gauge coupling applies at the unification scale with the properly normalized hypercharge generator; low-energy couplings differ through their renormalization group evolution.

Using left-chiral fields throughout, one Standard Model generation fits intoThe dimensions are $3+2=5$ and $6+3+1=10$, so no right-handed neutrino is required in these multiplets. A singlet neutrino can be added. A suitable adjoint scalar vacuum breaks SU(5) to the Standard Model subgroup; a fundamental scalar contains an electroweak doublet and a colored triplet. The extra heavy vector bosons can mediate transitions between quarks and leptons, hence proton decay in the minimal model. This is a proposed unification, not an experimentally established identification of new particles.

The hierarchy problem concerns keeping an elementary weak-scale scalar light in the presence of much heavier physical scales. A heavy state coupled to it generically induces a threshold correction of order $\delta m_H^2\sim c{M}^2/(16{\pi }^2)$, with $c$ a product of couplings. Without protection, a weak-scale result then needs cancellation against a much larger parameter. In SU(5), this is sharpened by doublet-triplet splitting: the electroweak doublet must stay light while its colored partner must be sufficiently heavy to avoid rapid proton decay. Choosing a small tree-level doublet mass is not by itself a mechanism protecting it from subsequent heavy-particle corrections. Supersymmetry and other proposals change this protection problem, but the minimal SU(5) embedding alone does not solve it.