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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 ($χ$) in the Earth's core. We consider a scenario of long-lived ($τ_χ\sim 10^{28}$ s), super heavy ($m_χ=10^7-10^{10}$ GeV) dark matter that decays via $χ\to ν_τ\barν_τ$ or $χ\to ν_μ\barν_μ$. 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 $ε_ρ$ combined with the partial decay width $B_{χ\to ν_τ\barν_τ}Γ_χ$ in the range of $(ε_ρ/10^{-10}) B_{χ\to ν_τ}Γ_χ\lesssim 1.5\times 10^{-29}-1.5\times 10^{-28}$ s$^{-1}$. For $χ\to ν_μ\barν_μ$, $m_χ= 10^8-10^{10}$ GeV and $E_μ>10^7$ GeV, the range of constraints is $(ε_ρ/10^{-10}) B_{χ\to ν_μ}Γ_χ\lesssim 3\times 10^{-29}-7\times 10^{-28}$ s$^{-1}$.