Validity of Special Relativity at Small Distances and the Velocity Dependence of the Charged-Pion Lifetime

Abstract

The velocity dependence of the lifetime of the pion is investigated under the assumption that the Hamiltonian contains a spatial form factor which in the lab frame (the frame at rest with respect to the neighboring macroscopic bodies) vanishes for distances larger than some length $\alpha$. In this model, there is a violation of the principles of special relativity at small distances. In particular, space-time is anisotropic at distances smaller than $\alpha$. The lifetime of the pion is calculated to second order in $\alpha$, and it is shown that there will be about 1% deviation from the usual formula $\tau \mathrm{}\left(v\right)={(1-\frac{v^2}{c^2})}^{\frac{-1\mathrm{}}{2}}\tau \mathrm{}\left(0\right)\mathrm{}$, (which holds if special relativity is valid at arbitrarily small distances) if, e.g., the pion energy $E_{\pi }={10}^4$ MeV and $\alpha \sim 5\times {10}^{-16\mathrm{}}$ cm. The measurement of the velocity dependence of the pion lifetime at high energies could thus serve as a possible check on the validity of special relativity at small distances. The deviation is of the same order of magnitude as that previously obtained for the muon decay.