Time dilation in relativistic quantum decay laws of moving unstable particles

TL;DR

This paper investigates how relativistic effects influence the decay laws of moving unstable particles, revealing a scaling law for survival probability and invariance of decay rate over long times.

Contribution

It introduces a general analysis of relativistic decay laws for particles with power-law mass distributions, deriving a scaling law and showing decay rate invariance at long times.

Findings

Survival probability relates to the rest frame via a scaling law.

Effective mass asymptotically equals the lower bound of the mass spectrum.

Decay rate becomes approximately independent of momentum at long times.

Abstract

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.