Optimal multifrequency weighting for CMB lensing
  Extragalactic foregrounds in Cosmic Microwave Background (CMB) temperature
maps lead to significant biases in CMB lensing reconstruction if not properly
accounted for. Combinations of multi-frequency data have been used to minimize
the overall map variance (internal linear combination, or ILC), or specifically
null a given foreground, but these are not tailored to CMB lensing. In this
paper, we derive an optimal multi-frequency combination to jointly minimize CMB
lensing noise and bias. We focus on the standard lensing quadratic estimator,
as well as the "shear-only" and source-hardened estimators, whose responses to
foregrounds differ. We show that an optimal multi-frequency combination is a
compromise between the ILC and joint deprojection, which nulls the thermal
Sunyaev-Zel'dovich (tSZ) and Cosmic Infrared Background (CIB) contributions. In
particular, for a Simons Observatory-like experiment with
$\ell_{\text{max},T}=3000$, we find that profile hardening alone (with the
standard ILC) reduces the bias to the lensing power amplitude by $40\%$, at a
$20\%$ cost in noise, while the bias to the cross-correlation with a LSST-like
sample is reduced by nearly an order of magnitude at a $10\%$ noise cost,
relative to the standard quadratic estimator. With a small amount of joint
deprojection the bias to the profile hardened estimator can be further reduced
to less than half the statistical uncertainty on the respective amplitudes, at
a $20\%$ and $5\%$ noise cost for the auto- and cross-correlation respectively,
relative to the profile hardened estimator with the standard ILC weights.
Finally, we explore possible improvements with more aggressive masking and
varying $\ell_{\text{max,}T}$.
