= One-loop normalized hypercharge unification
{title2=$\alpha_Y^{\mathrm{GUT}}=(5/3)\alpha_Y$}
The normalization of an Abelian <gauge coupling> matters in testing <gauge coupling unification>. If $\mu\,dg_i/d\mu=b_ig_i^3$, 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.
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