QFT derivation of the decay law of an unstable particle with nonzero momentum

Abstract

We present a quantum field theoretical derivation of the nondecay probability of an unstable particle with nonzero three-momentum $\mathbf{p}$. To this end, we use the (fully resummed) propagator of the unstable particle, denoted as $S,$ to obtain the energy probability distribution, called $d_S^{\mathbf{p}}(E)$, as the imaginary part of the propagator. The nondecay probability amplitude of the particle $S$ with momentum $\mathbf{p}$ turns out to be, as usual, its Fourier transform: $a_S^{\mathbf{p}}(t)={\int }_{\sqrt{m_{th}^2+{\mathbf{p}}^2}}^{\infty }dE{d}_S^{\mathbf{p}}(E)e^{-iEt}$ ($m_{th}$ is the lowest energy threshold in the energy frame, corresponding to the sum of masses of the decay products). Upon a variable transformation, one can rewrite it as $a_S^{\mathbf{p}}(t)={\int }_{m_{th}}^{\infty }dm{d}_S^0(m)e^{-i\sqrt{m_{th}^2+{\mathbf{p}}^2}t}$ [here, $d_S^0(m)\equiv d_S(m)$ is the usual spectral function (or mass distribution) in the rest frame]. Hence, the latter expression, previously obtained by different approaches, is here confirmed in an independent and, most importantly, covariant QFT-based approach. Its consequences are not yet fully explored but appear to be quite surprising (such as the fact that usual time-dilatation formula does not apply), thus its firm understanding and investigation can be a fruitful subject of future research.

Comments: 10 pages