Formulating Weak Lensing from the Boltzmann Equation and Application to
  Lens-lens Couplings
  The Planck mission has conclusively detected lensing of the Cosmic Microwave
Background (CMB) radiation from foreground sources to an overall significance
of greater than $25\sigma$. The high precision of this measurement motivates
the development of a more complete formulation of the calculation of this
effect. While most effects on the CMB anisotropies are widely studied through
direct solutions of the Boltzmann equation, the non-linear effect of CMB
lensing is formulated through the solutions of the geodesic equation. In this
paper, we present a new formalism to the calculation of the lensing effect by
\emph{directly solving the Boltzmann equation}, as we did in the calculation of
the CMB anisotropies at recombination. In particular, we developed a
diagrammatic approach to efficiently keep track of all the interaction terms
and calculate all possible non-trivial correlations to arbitrary high orders.
Using this formalism, we explicitly articulate the approximations required to
recover the usual remapping approach used in current studies of the weak
lensing. In addition, we point out additional unexplored corrections that are
manifest in our formalism to which experiments may be sensitive. As an example,
we calculate the correction to the CMB temperature power spectrum for the
\emph{lens-lens} coupling effects which are neglected in standard calculations.
We find that the correction is $\lesssim 0.1\%$ of the CMB temperature power
spectrum for $\ell$ up to 3000 and thus is comparable to the cosmic variance.
