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

Restoring units in the oscillation phase gives ${\Delta }_{ij}=\Delta m_{ij}^{2}L/(4E\hslash c)$. With $\hslash c=1.97327\times 10^{-7}\mathrm{eV}\mathrm{m}$, one kilometre is $5.06773\times 10^9{\mathrm{eV}}^{-1}$. ThusHere $k_{\mathrm{practical}}$ denotes the coefficient in $\Delta =k_{\mathrm{practical}}\Delta m^2[{\mathrm{eV}}^2]L[\mathrm{km}]/E[\mathrm{MeV}]$. The familiar $1.267$ applies when $E$ is in GeV, not MeV. Strictly, the dimensionless definition $\Delta =2k({\varphi }_i-{\varphi }_j)$ keeps $k=1/4$ once the ${\varphi }_j$ already include the unit conversion. The intended numerical question concerns the practical coefficient multiplying the numerical values of $\Delta m^{2}L/E$.