Abstract
Decay rates in quantum field theory (QFT) are typically calculated assuming the particles are represented by momentum eigenstates (i.e., plane waves). However, strictly speaking, localized free particles should be represented by wave packets. This yields width corrections to the decay rate and to the clock behavior under Lorentz boosts. We calculate the decay rate of a particle of mass $M$ modeled as a Gaussian wave packet of width $a$ and centered at zero momentum. We find the decay rate to be ${\mathrm{\ensuremath{\Gamma}}}_{0}[1\ensuremath{-}\frac{3{a}^{2}}{4{M}^{2}}+\mathcal{O}(\frac{{a}^{4}}{{M}^{4}})]$, where ${\mathrm{\ensuremath{\Gamma}}}_{0}$ is the decay rate of the particle at rest treated as a plane wave. The leading correction is then of order $\frac{{a}^{2}}{{M}^{2}}$. We then perform a Lorentz boost of velocity $v$ on the above Gaussian and find that its decay rate does not decrease exactly by the Lorentz factor $\sqrt{1\ensuremath{-}{v}^{2}}$. There is a correction of order $\frac{{a}^{2}{v}^{2}}{{M}^{2}}$. Therefore, the decaying wave packet does not act exactly like a typical clock under Lorentz boosts, and we refer to it is a ``WP clock'' (wave packet clock). A WP clock does not move with a single velocity relative to an observer but has a spread in velocities (more specifically, a spread in momenta). Nonetheless, it is best viewed as a single clock as the wave packet represents a one-particle state in QFT. WP clocks do not violate Lorentz symmetry and are not based on new physics; they are a consequence of the combined requirements of special relativity, quantum mechanics, and localized free particles.
Highlights
Localized particles in quantum field theory (QFT) are represented by wave packets [1,2], and localized particles are what take part in decay and scattering processes.1 replacing wave packets by momentum eigenstates leads to a well-known issue at a theoretical level; momentum eigenstates cannot be properly normalized in an infinite volume
We found an analytical result for the leading order width correction to the decay rate of a Gaussian wave packet centered at zero momentum
Where a is the width, and is the mass of the decaying particle. We discussed that this wave packet effect has the possibility in showing up in the highly precise measurements of electroweak pseudo-observables (EWPOs) of planned future eþe− colliders [28–31]
Summary
Localized particles in quantum field theory (QFT) are represented by wave packets [1,2], and localized particles are what take part in decay and scattering processes. replacing wave packets by momentum eigenstates (i.e., plane waves) leads to a well-known issue at a theoretical level; momentum eigenstates cannot be properly normalized in an infinite volume. We obtain analytical results for the leading order correction to decay rates due to the width of a Gaussian wave packet. We first determine the leading order width correction to the decay rate of a particle at rest due to the width a of a Gaussian wave packet centered at zero momentum. We first determine the decay rate of a Gaussian wave packet centered at zero momentum and determine the correction to the plane wave result due to the width a of the Gaussian. The Lorentz-boosted wave packet is not a Gaussian, and we calculate separately the decay rate of a Gaussian wave packet centered at velocity v (i.e., momentum Mγv).
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