Lifshitz scaling to Lorentz-violating high derivative operator and gamma-ray bursts

In this work we have used a Ho\v{r}ava-Lifshitz scaling to rewrite a Lorentz-violating higher-order derivative electrodynamics controlled by a background four-vector $n_{\mu}$. The photon propagator was obtained and we have analyzed the dispersion relation and the observational results of gamma-ray burst (GRB) experiments were used. The limits of the critical exponent were discussed in the light of the GRB data and the physical implications were compared with the current GRB-Lorentz-invariance-violation literature. We show that the bound for the Lorentz-violating coupling for dimension-six operators, obtained from a Ho\v{r}ava-Lifshitz scaling, is eight orders of magnitude better than the result found without considering a Ho\v{r}ava-Lifshitz scaling, also this bound is nearby one, which is expected to be relevant phenomenologically.