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.
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