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Built on a realistic multiband tight-binding model, mirror symmetry is usedto map a calcium-intercalated bilayer graphene Hamiltonian into two independentsingle layer graphene-like Hamiltonians with renormalized hopping. Thequasiparticles exhibit two types of chirality. Here a quasi-particle consistsof two electrons from opposing layers where possess an additional quantumnumber called "cone index" which can be regarded as the eigenvalue of mirrorsymmetry operations. To obtain tight-binding parameters, both effectivemonolayer Schrodinger equations are solved analytically and fitted tofirst-principles band structure results. Two quasi-particles (four electrons)can team up to build a Cooper pair with even or odd chirality. Treatment of thepairing Hamiltonian leads to two decoupled gap equations. The pairing ofquasi-particles with different cone indexes is forbidden. The decoupled gapequations are solved analytically to obtain all the possible superconductingphases. Two nearly "flat bands" crossing the Fermi energy, each related to thegraphene-like structures, are responsible for two distinct superconductivitygaps that emerge.  Depending on how much these bands are affected by the intercalant and whichis closer to the Fermi energy, distorted s-wave or d-wave superconductivity maybecome dominant. Numerical calculations reveal that d-wave superconductivity isdominant in both sectors. For these two dominant phases, within the range of0-6 K which superconductivity has been observed, numerically the transitionfrom single-gap to dual-gap superconductivity is possible. Adopting the two-gapviewpoint of superconductivity in C$_6$CaC$_6$, the dominant $d$-wave statesshould have the same critical temperature. Around $T_c=2K$ these two relationsintersect, otherwise, superconductivity has been realized just in one of thesetwo sectors and disappears in the other one.

Superconducting Symmetry Phases and Dominant bands in (Ca-) Intercalated AA- Bilayer Graphene