Weak lensing corrections to tSZ-lensing cross correlation
  The cross correlation between the thermal Sunyaev-Zeldovich (tSZ) effect and
gravitational lensing in wide field has recently been measured. It can be used
to probe the distribution of the diffuse gas in large scale structure, as well
as inform us about the missing baryons. As for any lensing-based quantity,
higher order lensing effects can potentially affect the signal. Here, we extend
previous higher order lensing calculations to the case of tSZ-lensing cross
correlations. We derive terms analogous to corrections due to the Born
approximation, lens-lens coupling, and reduced shear up to fourth order in the
Newtonian potential. Redshift distortions and vector modes are shown to be
negligible at this order. We find that the dominant correction due to the
reduced shear exceeds percent-level only at multipoles of $\ell \gtrsim 3000$.
