Expanding Model Independent Approaches for Measuring the CKM angle $\gamma$ at LHCb

Model independent approaches to measuring the CKM angle $\gamma$ in $B\rightarrow DK$ decays at LHCb are explored. In particular, we consider the case where the $D$ meson decays into a final state with four hadrons. Using four-body final states such as $\pi^+ \pi^- \pi^+ \pi^-$, $K^+ \pi^- \pi^+ \pi^-$ and $K^+ K^- \pi^+ \pi^-$ in addition to traditional 2 and 3 body states and has the potential to significantly improve to the overall constraint on $\gamma$. There is a significant systematic uncertainty associated with modelling the complex phase of the $D$ decay amplitude across the five-dimensional phase space of the four body decay. It is therefore important to replace these model-dependent quantities with model-independent parameters as input for the $\gamma$ measurement. These model independent parameters have been measured using quantum-correlated $\psi(3770) \rightarrow D^0 \overline{D^0}$ decays collected by the CLEO-c experiment, and, for $D\rightarrow K^+ \pi^- \pi^+ \pi^-$, with $D^0-\overline{D^0}$ mixing at LHCb. This poster will give a brief overview the status and prospects of $\gamma$ measurements with $B\rightarrow DK$, $D\rightarrow$ 4-body using a combination of LHCb and charm factory data, with focus on the $D$ meson decays to four pions.