Constraints on superheavy dark matter decaying into $h\nu$, $Z\nu$ and $W\ell$ -- Benchmark example within an extended seesaw framework

Dark matter particles could be superheavy (mass $M_X\gtrsim10^9~$GeV) provided that their lifetime $\tau_X$ is extremely long, i.e. greater than $\simeq 10^{22}~$yr. Such stringent constraints on $\tau_X$ are generally obtained by limiting the prompt emission of ultrahigh energy ($\gtrsim10^9~$GeV) gamma rays and neutrinos from the decay processes to below the corresponding flux upper bounds. In this paper, we show that even more severe bounds can be obtained for $M_X\gtrsim10^{13}~$GeV from the synchrotron radiation of electron decay byproducts in the Galaxy. We illustrate the power of these constraints using generic Higgs-induced $h\nu$ and gauge-induced $Z\nu/W\ell$ decay channels, motivated by particle-physics setups invoking right-handed neutrinos. As a concrete benchmark, we consider a superheavy dark-matter candidate within an extended type-I seesaw framework and show that the lower bounds on lifetime can be translated into upper bounds on a mass-mixing parameter $\delta M$, which must satisfy approximately $\delta M\lesssim 2\times 10^{-17}/[M_X/(10^9~\mathrm{GeV})]^{0.5}$~GeV for $M_X\gtrsim 10^9$~GeV. Some implications in the context of inflationary cosmologies are discussed.