  It is now well known that mode-coupling effects associated with certain
secondary effects generate higher order correlations in cosmic microwave
background (CMB) temperature anisotropies, beyond the two-point function. In
order to extract the non-Gaussian signal at the three-point level, we suggest a
two-point statistic in the form of an angular power spectrum involving
correlations between squared temperature and temperature anisotropies. This
power spectrum contains compressed information from the bispectrum and can be
easily measured in data with the same techniques that have been considered for
the measurement of the usual temperature-temperature anisotropy power spectrum.
We study the proposed power spectrum resulting from the non-Gaussian signal
generated by correlations involved with gravitational lensing angular
deflections in CMB and the Sunyave-Zel'dovich (SZ) effect due to large scale
pressure fluctuations. Using the Planck frequency cleaned CMB and SZ maps, the
CMB^2-SZ power spectrum provides a direct estimate of the cross-power between
lensing angular deflections and the SZ effect. Through an optimal filter
applied to the squared CMB map, the proposed statistic allows one to obtain all
information from the lensing-SZ bispectrum. The observational measurement of
the lensing-SZ cross-correlation is useful to understand the relation between
large scale structure pressure and dark matter fluctuations.
