$\text{TM}_1$ neutrino mixing with $\sin \theta_{13}=\frac{1}{\sqrt{3}}\sin \frac{\pi}{12}$

We construct a neutrino model using the flavour group $S_4\times C_4 \times C_3\times C_2$ under the type-1 seesaw mechanism. The vacuum alignments of the flavons in the model lead to $\text{TM}_1$ mixing with $\sin \theta_{13}=\frac{1}{\sqrt{3}}\sin \frac{\pi}{12}$. The mixing also exhibits $\mu\text{-}\tau$ reflection symmetry. By fitting the eigenvalues of the effective seesaw mass matrix with the observed neutrino mass-squared differences, we predict the individual light neutrino masses. The vacuum alignment of the $S_4$ triplet appearing in the Majorana mass term plays a key role in obtaining the aforementioned $\text{TM}_1$ scenario. Since the symmetries of the flavour group are not sufficient to define this alignment, we apply the recently proposed framework of the auxiliary group in our model. Using this framework, the $S_4$ triplet is obtained by coupling together several irreducible multiplets that transform under an expanded flavour group consisting of the original flavour group as well as an auxiliary group. The vacuum alignment of each of these multiplets is uniquely defined in terms of its residual symmetries under the expanded flavour group. As a result, the $S_4$ triplet constructed from these multiplets also becomes uniquely defined.