= One-loop SU(5) prediction with common flavor coefficients
{title2=$\sin^2\theta_W=\frac16+\frac{5\alpha}{9\alpha_s}$}
Normalize the Abelian coupling by $g_1^2=(5/3)g'^2$ 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.
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