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

Asymptotic freedom means that the gauge running coupling tends to zero as the renormalization scale tends to infinity. In the stated convention this occurs when $b_0>0$, namelyThe high-energy theory then approaches the Gaussian fixed point and perturbation theory becomes increasingly accurate.

The strong-coupling scale $\Lambda$ is defined as the scale at which the one-loop inverse coupling vanishes. Setting $g^{-2}(\Lambda )=0$ givesEquivalently, in terms of the coupling measured at any scale $\mu$,This dimensional transmutation defines ${\Lambda }_{\mathrm{QCD}}$ for Quantum chromodynamics.

For $SU(3{)}_c$, one generation contains two color triplets in the quark doublet and the two right-handed color triplets, so $N_f=4N_g$ and there is no colored scalar. Thuswhich permits at most

For $SU(2{)}_L$, each generation contains three colored copies of the quark doublet and one lepton doublet, again giving $N_f=4N_g$. The one complex Higgs doublet gives $N_s=1$, soHence the maximum is

Let $K=3(4\pi {)}^2$ and $L=\log (M^2/{\mu }^2)$. Running from $M$ down to $\mu$ givesThe assumed unified boundary condition impliesSubtracting the weak equation from the strong and normalized-hypercharge equations therefore givesEliminating $L$ proves

The block-diagonal subgroupis 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 iswhich is traceless and commutes with the $SU(3)$ and $SU(2)$ blocks.

The canonically normalized $SU(5)$ generator must satisfy $\mathrm{Tr}(T_Y^2)=1/2$. Since $\mathrm{Tr}(Y^2)=5/6$,The unified gauge covariant derivative contains $g_5{T}_Y{B}_{\mu }$, whereas the Standard Model convention contains $g_{Y}Y{B}_{\mu }$. 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