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QFT derivation of the decay law of an unstable particle with nonzero momentum (1804.02728v1)

Published 8 Apr 2018 in hep-ph, hep-th, and quant-ph

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.

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