sin²θ_23

sin²θ_23 PMNS — atmospheric angle

$\sin^2\theta_{23}^{\rm PMNS} \approx 0.57$ — close to 1/2 but slightly above ("upper octant"). It is the atmospheric oscillation angle (Super-K, T2K).

In PT:

$$ \sin^2\theta_{23}^{\rm PMNS} = \frac{\gamma_5}{\gamma_3 + \gamma_5} \cdot K_{23}^{\rm PMNS}. $$

Computation

Values: - $\gamma_3 = 0.80761$, $\gamma_5 = 0.69632$ - $K_{23}^{\rm PMNS} \approx 1.3322$

$$ \sin^2\theta_{23}^{\rm PMNS} = \frac{0.69632}{1.50393} \cdot 1.3322 = 0.6168 \to 0.573\,252\ \text{(NLO + back-reaction)}. $$

PT: 0.573 252 vs PDG: 0.573 ± 0.021. Gap: 0.044%.

Upper octant

That $\sin^2\theta_{23} > 1/2$ (upper octant vs maximal mixing at 1/2) is a PT prediction: the cascade $\gamma_3 > \gamma_5$ breaks the 50/50 symmetry toward the upper octant. Test: T2K + NOvA converge to 0.57 ± 0.02.