The phenomenology of top quark polarisation in the decay of heavy charged Higgs bosons at LHC and beyond

It has recently been shown that through the $\tau\nu$ decay mode the heavy charged Higgs bosons $H^\pm$ can be discovered at LHC over a vast region of the parameter space previously thought unaccessible. If the charged Higgs bosons are discovered in this decay mode and the measurements of its mass and background contaminated $bt$ decay mode by cuts suppressing phase space for the background. In this paper we study the top quark polarisation in the $bt$ decay mode. When $\tan\beta$ is either moderately large ($\tan\beta\gtrsim9$) or small ($\tan\beta\lesssim4), the correlation effects can be used as a signal of $H^\pm$ production. We outline the analysis procedure for various decay signatures. As an example, we demonstrate the application of our procedure to the dominant production process $gb\to tH^-$ at LHC. The effect is clearly observable for $H^\pm$ mass between 200 GeV and 500 GeV in both leptonic and hadronic channels, but at 500 GeV the boost on the top quark starts to prevent the reconstruction of invariant masses, and hence the observation of the correlation effect, in the hadronic decay channel.