Modeling CMB Lensing Cross Correlations with {\sc CLEFT}
  A new generation of surveys will soon map large fractions of sky to ever
greater depths and their science goals can be enhanced by exploiting cross
correlations between them. In this paper we study cross correlations between
the lensing of the CMB and biased tracers of large-scale structure at high $z$.
We motivate the need for more sophisticated bias models for modeling
increasingly biased tracers at these redshifts and propose the use of
perturbation theories, specifically Convolution Lagrangian Effective Field
Theory ({\sc CLEFT}). Since such signals reside at large scales and redshifts,
they can be well described by perturbative approaches. We compare our model
with the current approach of using scale independent bias coupled with fitting
functions for non-linear matter power spectra, showing that the latter will not
be sufficient for upcoming surveys. We illustrate our ideas by estimating
$\sigma_8$ from the auto- and cross-spectra of mock surveys, finding that {\sc
CLEFT} returns accurate and unbiased results at high $z$. We discuss
uncertainties due to the redshift distribution of the tracers, and several
avenues for future development.
