Leggett modes and the Anderson-Higgs mechanism in superconductors without inversion symmetry

We develop a microscopic and gauge-invariant theory for collective modes resulting from the phase of the superconducting order parameter in non-centrosymmetric superconductors. Considering various crystal symmetries we derive the corresponding gauge mode $\omega_{\rm G}({\bf q})$ and find, in particular, new Leggett modes $\omega_{\rm L}({\bf q})$ with characteristic properties that are unique to non-centrosymmetric superconductors. We calculate their mass and dispersion that reflect the underlying spin-orbit coupling and thus the balance between triplet and singlet superconductivity occurring simultaneously. Finally, we demonstrate the role of the Anderson-Higgs mechanism: while the long-range Coulomb interaction shifts $\omega_{\rm G}({\bf q})$ to the condensate plasma mode $\omega_{\rm P}({\bf q})$, it leaves the mass $\Lambda_0$ of the new Leggett mode unaffected and only slightly modifies its dispersion.