We investigate the superconducting properties of inversion-symmetric twisted
trilayer graphene by considering different parent states, including
spin-singlet, triplet, and SO(4) degenerate states, with or without nodal
points. By placing transition metal dichalcogenide layers above and below
twisted trilayer graphene, spin-orbit coupling is induced in TTLG and, due to
inversion symmetry, the spin-orbit coupling does not spin-split the bands. The
application of a displacement field (D0β) breaks the inversion symmetry and
creates spin-splitting. We analyze the evolution of the superconducting order
parameters in response to the combined spin-orbit coupling and D0β-induced
spin-splitting. Utilizing symmetry analysis combined with both a direct
numerical evaluation and a complementary analytical study of the gap equation,
we provide a comprehensive understanding of the influence of spin-orbit
coupling and D0β on superconductivity. These results contribute to a better
understanding of the superconducting order in twisted trilayer graphene.Comment: 13 pages, 9 figure