Cosmological decay of Higgs-like scalars into a fermion channel

We study the decay of a Higgs-like scalar Yukawa coupled to massless fermions in post-inflationary cosmology, combining a non-perturbative method with an adiabatic expansion. The renormalized survival probability \(\mathcal{P}_\Phi(t)\) of a (quasi) particle ``born'' at time \(t_b\) and decaying at rest in the comoving frame, \(\mathcal{P}_\Phi(t) = \Big[\frac{t}{t_b}\Big]^{-\frac{Y^2}{8\pi^2}}~ e^{ \frac{Y^2}{4\pi^2}\,\big(t/t_b\big)^{1/4} } \,e^{-\Gamma_0\,(t-t_b)}~ \mathcal{P}_\Phi(t_b) \), with \(\Gamma_0\) the decay rate at rest in Minkowski space-time. For an ultrarelativistic particle we find \(\mathcal{P}_\Phi(t) = e^{-\frac{2}{3}\Gamma_0\,t_{nr}\,(t/t_{nr})^{3/2}}~ \mathcal{P}_\Phi(t_b)\) before it becomes non-relativistic at a time \(t_{nr}\) as a consequence of the cosmological redshift. For \(t\gg t_{nr}\) we find \(\mathcal{P}_\Phi(t) = \Big[\frac{t}{t_{nr}}\Big]^{-\frac{Y^2}{8\pi^2}}~ e^{ \frac{Y^2}{4\pi^2}\,\big(t/t_{nr}\big)^{1/4} }~\Big[\frac{t}{t_{nr}}\Big]^{\Gamma_0 t_{nr}/2} \,e^{-\Gamma_0\,(t-t_{nr})}~ \mathcal{P}_\Phi(t_{nr})\). The extra power is a consequence of the memory on the past history of the decay process. We compare these results to an S-matrix inspired phenomenological Minkowski-like decay law modified by an instantaneous Lorentz factor to account for cosmological redshift. Such phenomenological description \emph{under estimates the lifetime of the particle}. For very long lived, very weakly coupled particles, we obtain an \emph{upper bound} for the survival probability as a function of redshift \(z\) valid throughout the expansion history \(\mathcal{P}_\Phi(z) \gtrsim e^{-\frac{\Gamma_0}{H_0}\,\Upsilon(z,z_b)}\,\mathcal{P}_\Phi(z_b)\), where \(\Upsilon(z,z_b)\) only depends on cosmological parameters and \(t_{nr}\).