Symmetry of the order parameter in non-centrosymmetric superconductors: Implementation for CePt$_3$Si and Cd$_2$Re$_2$O$_7$

In noncentrosymmetric metals, the spin degeneracy of the electronic bands is lifted by spin-orbit coupling. We consider general symmetry properties of the pairing function $\Delta(k)$ in noncentrosymmetric superconductors with strong spin-orbit coupling (NSC). We find that $\Delta(k) = \chi(k) t(k)$, where $\chi(k)$ is an even function which transforms according to the irreducible representations of the crystallographic point group and t(k) is a model dependent phase factor. We consider tunnelling between a NSC and a conventional superconductor. It is found that, in terms of thermodynamical properties as well as the Josephson effect, the state of NSC resembles a singlet superconducting state with gap function $\chi(k)$. We propose the gap functions which may account for the experimental properties of the heavy fermion compound CePt$_3$Si and the distorted pyrochlore Cd$_2$Re$_2$O$_7$.