Time dilation in the oscillating decay laws of moving two-mass unstable quantum states

The decay of a moving system is studied in case the system is initially prepared in a two-mass unstable quantum state. The survival probability $\mathcal{P}_p(t)$ is evaluated over short and long times in the reference frame where the unstable system moves with constant linear momentum $p$. The mass distribution densities of the two mass states are tailored as power laws with powers $\alpha_1$ and $\alpha_2$ near the non-vanishing lower bounds $\mu_{0,1}$ and $\mu_{0,2}$ of the mass spectra, respectively. If the powers $\alpha_1$ and $\alpha_2$ differ, the long-time survival probability $\mathcal{P}_p(t)$ exhibits a dominant inverse-power-law decay and is approximately related to the survival probability at rest $\mathcal{P}_0(t)$ by a time dilation. The corresponding scaling factor $\chi_{p,k}$ reads $\sqrt{1+p^2/\mu_{0,k}^2}$, the power $\alpha_k$ being the lower of the powers $\alpha_1$ and $\alpha_2$. If the two powers coincide and the lower bounds $\mu_{0,1}$ and $\mu_{0,2}$ differ, the scaling relation is lost and damped oscillations of the survival probability $\mathcal{P}_p(t)$ appear over long times. By changing reference frame, the period $T_0$ of the oscillations at rest transforms in the longer period $T_p$ according to a factor which is the weighted mean of the scaling factors of each mass, with non-normalized weights $\mu_{0,1}$ and $\mu_{0,2}$.