Cosmic Microwave Background Delensing Revisited: Residual Biases and a Simple Fix

The delensing procedure is an effective tool for removing lensing-induced $B$-mode polarization in the Cosmic Microwave Background to allow for deep searches of primordial $B$-modes. However, the delensing algorithm existing in the literature breaks down if the target $B$-mode signals overlap significantly with the lensing $B$-mode ($\sim 300<l<2000$) in multipole-$l$ space. In this paper, we identify the cause of the breakdown to be correlations between the input $B$-map and the deflection field estimator ($EB$). The amplitude of this bias is quantified by numerical simulations and compared to the analytically derived functional form. We also propose a revised delensing procedure that circumvents the bias. While the newly identified bias does not affect the search of degree scale $B$-mode generated by tensor perturbations, the modified delensing algorithm makes it possible to perform deep searches of high-$l$ $B$-modes such as those generated by patchy reionization, cosmic strings, or rotation of polarization angles. Finally, we estimate how well future polarization experiments can do in detecting tensor- and cosmic string- generated $B$-mode after delensing and comment on different survey strategies.