Neutrino constraints on long-lived heavy dark sector particle decays in the Earth

Recent theoretical work has explored dark matter accumulation in the Earth and its drift towards the center of the Earth that, for the current age of the Earth, does not necessarily result in a concentration of dark matter ($\chi$) in the Earth's core. We consider a scenario of long-lived ($\tau_\chi\sim 10^{28}$ s), super heavy ($m_\chi=10^7-10^{10}$ GeV) dark matter that decays via $\chi\to \nu_\tau \bar{\nu}_\tau$ or $\chi\to \nu_\mu \bar{\nu}_\mu$. We show that an IceCube-like detector over 10 years can constrain a dark matter density that mirrors the Earth's density or has a uniform density with density fraction $\epsilon_\rho$ combined with the partial decay width $B_{\chi\to \nu_\tau \bar{\nu}_\tau}\Gamma_\chi$ in the range of $(\epsilon_\rho/10^{-10}) B_{\chi\to \nu_\tau}\Gamma_\chi \lesssim 1.5\times 10^{-29}-1.5\times 10^{-28}$ s$^{-1}$. For $\chi\to \nu_\mu \bar{\nu}_\mu$, $m_\chi = 10^8-10^{10}$ GeV and $E_\mu>10^7$ GeV, the range of constraints is $(\epsilon_\rho/10^{-10}) B_{\chi\to \nu_\mu}\Gamma_\chi \lesssim 3\times 10^{-29}-7\times 10^{-28}$ s$^{-1}$.