  While both X-ray emission and Sunyaev-Zel'dovich (SZ) temperature
fluctuations are generated by the warm-hot gas in dark matter halos, the two
observables have different dependence on the underlying physical properties,
including the gas distribution. A cross-correlation between the soft X-ray
background (SXRB) and the SZ sky may allow an additional probe on the
distribution of warm-hot gas at intermediate angular scales and redshifts
complementing studies involving clustering within SXRB and SZ separately. Using
a halo approach, we investigate this cross-correlation analytically. The two
contributions are correlated mildly with a correlation coefficient of
$\sim0.3$, and this relatively low correlation presents a significant challenge
for its detection. The correlation, at small angular scales, is affected by the
presence of radiative cooling or preheating and provides a probe on the thermal
history of the hot gas in dark halos. While the correlation remains
undetectable with CMB data from the WMAP satellite and X-ray background data
from existing catalogs, upcoming observations with CMB missions such as Planck,
for the SZ side, and an improved X-ray map of the large scale structure, such
as the one planned with DUET mission, may provide a first opportunity for a
reliable detection of this cross-correlation.
