Aspects of the $\sin2\beta$ measurement specific to the decay mode $B^0 \to J/\psi K^0_L$

We present the details of the $\sin2\beta$ measurement that are specific to the decay mode $B^0 \to J/\psi K^0_L$, using a sample of 384 million $\Upsilon(4S)\to B\bar{B}$ events, collected by the BaBar detector at the PEP-II asymmetric-energy $B$ Factory. We use several parameters for the $\sin2\beta$ fit that are unique to the $B^0 \to J/\psi K^0_L$ decay. The relative fractions for signal and various inclusive $J/\psi$ modes are input to the fit and are split by reconstruction type, $J/\psi$ lepton decay type and tagging category. We also use the parameters of the PDFs to the $\Delta E$ shapes for signal and different background distributions derived from fits to the Monte Carlo events and split by reconstruction type and $J/\psi$ lepton decay type.