Implications of $\textit{SU}(2)_L$ gauge invariance for constraints on Lorentz violation

Lorentz invariance may only be broken far above the electroweak scale, since violations are experimentally stringently constrained. Therefore, the Standard-Model Extension parameterizing Lorentz violation (LV) via (higher-dimensional) field theory operators is manifestly SU(2)$_{L}$ gauge-invariant. As a consequence, LV in neutrinos implies LV in charged leptons and vice versa. This allows us to obtain estimated sensitivities for flavour-changing operators in the charged-lepton sector from neutrino oscillations as well as sensitivities for flavour-diagonal neutrino effects from high-precision electron experiments. We also apply this method to an analysis of time-of-flight data for neutrinos (detected by IceCube) and photons from gamma ray bursts where discrepancies have been observed. Our conclusion is that an explanation of the arrival time difference between neutrino and photon events by dim-5 operators in the neutrino sector would lead to unacceptably large LV effects in the charged-lepton sector.Lorentz invariance is one of the basic ingredients of quantum field theories and violations of it are stringently constrained experimentally. Therefore, the possibility of Lorentz violation (LV) is usually realized at very high energy scales, resulting in a strong suppression of it (by the new scale) in experiments. The Standard-Model Extension (SME) parameterizes LV in a model-independent way, respecting $SU(2)_L$ gauge invariance. This means, e.g., that the neutrino and charged-lepton sectors are linked to each other. Hence, on the one hand, any modification of neutrino properties simultaneously gives rise to effects for charged leptons, which is why the tight limits on flavour-off-diagonal LV for neutrinos imply new bounds on modifications of charged leptons. On the other hand, LV for left-handed charged leptons implies LV for neutrinos. Since LV modifications of the charged-lepton sector are, in general, even more constraining than effects in the flavour-diagonal neutrino sector, we obtain novel tight bounds on LV in the latter. Subsequently, we apply the same approach to an analysis of time-of-flight data for neutrinos (detected by IceCube) and photons from gamma ray bursts where discrepancies have been observed. Our finding is that an explanation of the arrival time difference between neutrino and photon events by dim-5 operators in the neutrino sector would lead to unacceptably large LV effects in the charged-lepton sector