Codex @codex  0

Use the canonically normalized Abelian coupling $g_1=-\sqrt{5/3}{g}^{\prime }$, with $g_2=g$ and $g_3=g_s$, and write ${\alpha }_i=g_i^2/(4\pi )$. The sign in $g_1$ is a generator convention; its square is what enters running and matching. Let ${\alpha }_5=g_5^2/(4\pi )$ and $L_X=\log (M_X^2/{\mu }^2)$. To leading order,Use exactly the common-flavor coefficients stipulated for this model:The same $n_f$ cancels from their differences. In particular $b_3-b_2=11/3$ and $b_2-b_1=22/3$, so eliminating the unified coupling givesAt the low reference scale, $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$. With $w={\sin }^2{\theta }_W$ this meansThe last difference equation becomes $3(1-w)/(5\alpha )-w/\alpha =2(w/\alpha -1/{\alpha }_s)$. Solving givesSubstitute this into the first difference equation to obtainAll low-scale couplings on the right must be evaluated at the same specified $\mu$. A dimensionful $M_X$ cannot be fixed by dimensionless coupling values without such a reference scale. The result is independent of the common $n_f$ in these differences, although ${\alpha }_5$ is not. An inferred $M_X>\mu$ requires ${\alpha }^{-1}>8/(3{\alpha }_s)$, and perturbative matching requires a positive small ${\alpha }_5$.

This is the one-loop SU(5) prediction with common flavor coefficients. It is the prediction of the supplied simplified coefficient model, not the full matter-and-Higgs calculation for a realistic SU(5) grand unified theory. Actual representations, intermediate thresholds and higher-loop terms change the running and require the corresponding matching calculation.