Spin triplet nematic pairing symmetry and superconducting double transition in U$_{1-x}$Th$_{x}$Be$_{13}$.

Motivated by a recent experiment on U$_{1-x}$Th$_{x}$Be$_{13}$ with $x=3\%$, we develop a theory to narrow down the possible pair symmetry to consistently describe the double transition utilizing various theoretical tools; group theory and Ginzburg-Landau theory. It is explained in terms of the two dimensional representation E$_{\rm u}$ with spin triplet. A symmetry breaking causes the degenerate $T_{\rm c}$ to split into the two. The low temperature phase is identified as the cyclic $p$ wave: $\vec {d}({\bf k})={\hat x}k_x+\varepsilon{\hat y}k_y+\varepsilon^2{\hat z}k_z$ with $\varepsilon^3=1$ while the biaxial nematic phase: ${\vec d}({\bf k})={\sqrt 3}({\hat x}k_x-{\hat y}k_y$) is the high temperature one. This allows us to simultaneously identify the uniaxial nematic phase: ${\vec d}({\bf k})=2{\hat z}k_z-{\hat x}k_x-{\hat y}k_y$ for UBe$_{13}$, which breaks spontaneously cubic symmetry of the system. Those pair functions are fully consistent with the above and existing data. We comment on the accidental scenario in addition to this degeneracy scenario and the intriguing topological nature hidden in this long-known material.