Neutrino-antineutrino Asymmetry of C$\nu$B on the Surface of the Round Earth

It has been claimed that the coherent scattering of relic neutrinos with the Earth will result in a neutrino-antineutrino asymmetry of $\mathcal{O}(10^{-4})$ on the Earth surface, which is five orders of magnitude larger than the naive model expectation. In this work we show that this overdensity was overestimated for the perfectly round Earth by solving the exact solution with partial waves. The maximal asymmetry after summing over all the angular modes is only around $10^{-8}$ above the ground. To achieve the proposed asymmetry of $\mathcal{O}(10^{-4})$, a special geography may be needed as the experimental site.