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

TL;DR

This paper derives the decay law for an unstable particle with nonzero momentum using quantum field theory, confirming previous results and revealing surprising implications like the inapplicability of standard time-dilatation.

Contribution

It provides a covariant QFT derivation of the decay law for moving unstable particles, confirming previous approaches and highlighting new theoretical insights.

Findings

Confirmed the spectral function form of decay amplitude for moving particles

Showed the usual time-dilatation formula does not apply

Provided a covariant derivation consistent with previous methods

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: $a_{S}^{\mathbf{p}}(t)=\int_{\sqrt{m_{th} ^{2}+\mathbf{p}^{2}}}^{\infty}dEd_{S}^{\mathbf{p}}(E)e^{-iEt}$ ($m_{th}$ 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 $a_{S}^{\mathbf{p}}(t)=\int_{m_{th}}^{\infty}dmd_{S} ^{\mathbf{0}}(m)e^{-i\sqrt{m_{th}^{2}+\mathbf{p}^{2}}t}$ [here, $d_{S} ^{\mathbf{0}}(m)\equiv d_{S}(m)$ 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.