Time dilation in relativistic quantum decay laws of moving unstable particles

The relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (non-vanishing) lower bound $\mu_0$ of the mass spectrum. The survival probability $\mathcal{P}_p(t)$, the instantaneous mass $M_p(t)$ and the instantaneous decay rate $\Gamma_p(t)$ of the moving unstable particle are evaluated over short and long times for an arbitrary value $p$ of the (constant) linear momentum. The ultrarelativistic and non-relativistic limits are studied. Over long times, the survival probability $\mathcal{P}_p(t)$ is approximately related to the survival probability at rest $\mathcal{P}_0(t)$ by a scaling law. The scaling law can be interpreted as the effect of the relativistic time dilation if the asymptotic value $M_p\left(\infty\right)$ of the instantaneous mass is considered as the effective mass of the unstable particle over long times. The effective mass has magnitude $\mu_0$ at rest and moves with linear momentum $p$ or, equivalently, with constant velocity $1\Big/\sqrt{1+\mu_0^2\big/p^2}$. The instantaneous decay rate $\Gamma_p(t)$ is approximately independent of the linear momentum $p$, over long times, and, consequently, is approximately invariant by changing reference frame.