We study $CPT$ and Lorentz violation in the electroweak gauge sector of the Standard Model in the context of the Standard-Model Extension. In particular, we consider the Lorentz-violating and $CPT$-odd Chern-Simons like parameter for the $W$ boson, which is thus far unbounded by experiment. We demonstrate that any nonzero value of this parameter implies that, for sufficiently large energies, one of the polarization modes of the $W$ boson propagates with spacelike four-momentum. In this scenario, an emission of $W$ bosons by ultra-high-energy cosmic rays is possible. We calculate the induced fermion energy-loss rate, and we deduce the first limit on the pertinent Lorentz- and $CPT$-violating parameter that couples to the $W$ boson. Consistency between the quantum description in various reference frames is preserved by using a recently formulated covariant quantization procedure for massive photons and applying it to the $W$ bosons.

• Received 20 April 2017

DOI:https://doi.org/10.1103/PhysRevD.96.035034