We report about an ongoing lattice field theory project concerned with the investigation of heavy hybrid mesons. In particular we discuss our computation of the structure of hybrid static potential flux tubes in SU(2) lattice Yang-Mills theory, which is based on the squares of the chromoelectric and chromomagnetic field strength components. Our flux tube results for hybrid static potential quantum numbers $\Sigma_g^-$, $\Sigma_u^+$, $\Sigma_u^-$, $\Pi_g$, $\Pi_u$, $\Delta_g$, $\Delta_u$ are significantly different compared to the flux tube of the ordinary static potential.