  We study the degree to which the cosmic microwave background (CMB) can be
used to constrain primordial non-Gaussianity involving one tensor and two
scalar fluctuations, focusing on the correlation of one polarization $B$ mode
with two temperature modes. In the simplest models of inflation, the
tensor-scalar-scalar primordial bispectrum is non-vanishing and is of the same
order in slow-roll parameters as the scalar-scalar-scalar bispectrum. We
calculate the $\langle BTT\rangle$ correlation arising from a primordial
tensor-scalar-scalar bispectrum, and show that constraints from an experiment
like CMB-Stage IV using this observable are more than an order of magnitude
better than those on the same primordial coupling obtained from temperature
measurements alone. We argue that $B$-mode non-Gaussianity opens up an
as-yet-unexplored window into the early Universe, demonstrating that
significant information on primordial physics remains to be harvested from CMB
anisotropies.
