region for $x_g \lesssim 0.53$. This enhancement is again the result of the parton shower approximation overestimating the exact matrix element in this area [@Gieseke:2003rz]. In general, we see that the parton shower in  produces areas of high emission density which do not correspond to physically enhanced areas of phase space and therefore has a tendency to overpopulate hard regions of phase space. On the other hand, the POWHEG emission is distributed according to the exact real-emission matrix element and so correctly populates the physically enhanced regions of phase space with no additional spurious high density regions. Finally, we also see that the POWHEG corrected distribution fills the dead region of phase space that is not populated by the standalone parton shower.

![Dalitz distributions for the decay $t \rightarrow W b$ with (left) and without (right) the POWHEG style correction. The black outline indicates the physically allowed region of phase space. In the conventional parton shower approach, the region above the green dashed line is populated with radiation from the bottom quark and the regions below the blue dotted lines with radiation from the top quark. These boundaries correspond to the limits of the parton shower with symmetric phase space 