Primordial tensor bispectra in $mu$-CMB cross-correlations

Cross-correlations between Cosmic Microwave Background (CMB) temperature and polarization anisotropies and $mu$-spectral distortions have been considered to measure (squeezed) primordial scalar bispectra in a range of scales inaccessible to primary CMB bispectra. In this work we address whether it is possible to constrain tensor non-Gaussianities with these cross-correlations. We find that only primordial tensor bispectra with anisotropies leave distinct signatures, while isotropic tensor bispectra leave either vanishing or highly suppressed signatures. We discuss how the kind of anisotropies and the parity state in primordial bispectra determine the non-zero cross-correlations. By employing the so-called BipoSH formalism to capture the observational effects of these anisotropies, we make Fisher-forecasts to assess the detection prospects from $mu T$, $mu E$ and $mu B$ cross-correlations. Observing anisotropies in squeezed $langle gamma gamma gammarangle$ and $langle gamma gamma zetarangle$ bispectra is going to be challenging as the imprint of tensor perturbations on $mu$-distortions is subdominant to scalar perturbations, therefore requiring a large, independent amplification of the effect of tensor perturbations in the $mu$-epoch. In absence of such a mechanism, anisotropies in squeezed $langle zeta zeta gammarangle$ bispectrum are the most relevant sources of $mu T$, $mu E$ and $mu B$ cross-correlations. In particular, we point out that in models where anisotropies in $langle zeta zeta zeta rangle$ leave potentially observable signatures in $mu T$ and $mu E$, the detection prospects of anisotropies in $langle zeta zeta gammarangle$ from $mu B$ are enhanced.