Past exam of the particle physics course of the University of Oxford / 2022 / 6 / d / vi

The relevant resolution condition is on the oscillation phase, not solely on which mixing angle is being measured. Since $\Delta \propto L/E$,At the same medium baseline, the fast ${\theta }_{13}$-dependent ripples have phases about $37.5$ times larger than the slow ${\theta }_{12}$ component. Resolving those ripples therefore requires a much better fractional energy resolution. At 47 km, $|{\Delta }_{23}|\simeq 59.5$; avoiding substantial phase smearing requires a resolution appreciably better than $1/59.5\simeq 1.7\%$.

However, a short-baseline ${\theta }_{13}$ experiment near 1.24 km and a ${\theta }_{12}$ experiment near 46.5 km each operate near their relevant first maximum, $\Delta \simeq \pi /2$. Their intrinsic fractional phase-resolution requirements are then comparable. The smaller ${\theta }_{13}$ disappearance amplitude demands careful relative calibration and background control, but it does not by itself impose a universally finer energy resolution. The proposal is conditional on baseline and analysis, rather than a general requirement.