Gauge coupling unification

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.

The normalization of an Abelian gauge coupling matters in testing gauge coupling unification. If $\mu d{g}_i/d\mu =b_i{g}_i^3$, then ${\alpha }_i^{-1}(\mu )={\alpha }_i^{-1}({\mu }_0)-8\pi b_i\log (\mu /{\mu }_0)$. For normalized hypercharge ${\alpha }_Y^{\mathrm{GUT}}=(5/3){\alpha }_Y$, the inverse slope is $(3/5)b_Y$. Subtracting pairs of inverse couplings at the common scale and eliminating its logarithm gives a low-energy consistency relation. Division by the difference of two slopes requires that difference to be nonzero; a scale above ${\mu }_0$ additionally requires a positive inferred logarithm.

An SU(5) grand unified theory embeds $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$ into $SU(5)$. Canonical normalization of its hypercharge generator predicts $g_s^2=g_w^2=(5/3)g_Y^2$ at the unification scale.

For simply connected space $SU(5)$ broken to the connected Standard Model subgroup, the exact homotopy sequence gives ${\pi }_2(G/H)\cong {\pi }_1(H)\cong \mathbb{Z}$. Stable magnetic monopoles form. Their gauge and scalar cores have scales $(g\eta {)}^{-1}$ and $(\sqrt{\lambda }\eta {)}^{-1}$; their masses are $M_M\sim 4\pi \eta /g$ times a dimensionless profile factor. The physical subgroup is $(SU(3)\times SU(2)\times U(1))/{\mathbb{Z}}_6$, which retains an infinite-cyclic fundamental group.

// Target: cosmology.bigb

Normalize the Abelian coupling by $g_1^2=(5/3)g^{\prime 2}$ and match $g_1=g_2=g_3$ in magnitude at scale $M_X$. With common-flavor coefficients $b_3=(33-2n_f)/3$, $b_2=(22-2n_f)/3$ and $b_1=-2n_f/3$, the one-loop running of the strong coupling extends to $1/{\alpha }_i(\mu )=1/{\alpha }_5-b_i\log (M_X^2/{\mu }^2)/(4\pi )$. The differences obey ${\alpha }_1^{-1}-{\alpha }_2^{-1}=2({\alpha }_2^{-1}-{\alpha }_s^{-1})$. Using ${\alpha }_2^{-1}={\sin }^2{\theta }_W/\alpha$ and ${\alpha }_1^{-1}=3{\cos }^2{\theta }_W/(5\alpha )$ gives the displayed prediction and $\log (M_X/\mu )=\frac{\pi }{11}({\alpha }^{-1}-8/(3{\alpha }_s))$. These simplified coefficients omit the full matter and scalar counting of a realistic unified model; threshold changes require piecewise running with the actual coefficients.