Low energy neutrinos from stopped muons in the Earth

We explore the low energy neutrinos from stopped cosmic ray muons in the Earth. Based on the muon intensity at the sea level and the muon energy loss rate, the depth distributions of stopped muons in the rock and sea water can be derived. Then we estimate the $\mu^-$ decay and nuclear capture probabilities in the rock. Finally, we calculate the low energy neutrino fluxes and find that they depend heavily on the detector depth $d$. For $d = 1000$ m, the $\nu_e$, $\bar{\nu}_e$, $\nu_\mu$ and $\bar{\nu}_\mu$ fluxes in the range of 13 MeV $\leq E_\nu \leq$ 53 MeV are averagely $10.8 \%$, $6.3\%$, $3.7 \%$ and $6.2 \%$ of the corresponding atmospheric neutrino fluxes, respectively. The above results will be increased by a factor of 1.4 if the detector depth $d < 30$ m. In addition, we find that most neutrinos come from the region within 200 km and the near horizontal direction, and the $\bar{\nu}_e$ flux depends on the local rock and water distributions.