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 statistical anisotropies leave distinct signatures, while isotropic tensor bispectra leave either vanishing or highly suppressed signatures. We discuss how the angular dependence of squeezed bispectra in terms of the short and long momenta determine the non-zero cross-correlations. We also discuss how these non-vanishing configurations are affected by the way in which primordial bispectra transform under parity. By employing the so-called BipoSH formalism to capture the observational effects of statistical anisotropies, we make Fisher-forecasts to assess the detection prospects from $\mu T$, $\mu E$ and $\mu B$ cross-correlations. Observing statistical anisotropies in squeezed $\langle \gamma \gamma \gamma\rangle$ and $\langle \gamma \gamma \zeta\rangle$ 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, statistical anisotropies in squeezed $\langle \zeta \zeta \gamma\rangle$ bispectrum are the most relevant sources of $\mu T$, $\mu E$ and $\mu B$ cross-correlations. In particular, we point out that in anisotropic inflationary models where $\langle \zeta \zeta \zeta \rangle$ leaves potentially observable signatures in $\mu T$ and $\mu E$, the detection prospects of $\langle \zeta \zeta \gamma\rangle$ from $\mu B$ are enhanced.