  Built on a realistic multiband tight-binding model, mirror symmetry is used
to map a calcium-intercalated bilayer graphene Hamiltonian into two independent
single layer graphene-like Hamiltonians with renormalized hopping. The
quasiparticles exhibit two types of chirality. Here a quasi-particle consists
of two electrons from opposing layers where possess an additional quantum
number called "cone index" which can be regarded as the eigenvalue of mirror
symmetry operations. To obtain tight-binding parameters, both effective
monolayer Schrodinger equations are solved analytically and fitted to
first-principles band structure results. Two quasi-particles (four electrons)
can team up to build a Cooper pair with even or odd chirality. Treatment of the
pairing Hamiltonian leads to two decoupled gap equations. The pairing of
quasi-particles with different cone indexes is forbidden. The decoupled gap
equations are solved analytically to obtain all the possible superconducting
phases. Two nearly "flat bands" crossing the Fermi energy, each related to the
graphene-like structures, are responsible for two distinct superconductivity
gaps that emerge.
  Depending on how much these bands are affected by the intercalant and which
is closer to the Fermi energy, distorted s-wave or d-wave superconductivity may
become dominant. Numerical calculations reveal that d-wave superconductivity is
dominant in both sectors. For these two dominant phases, within the range of
0-6 K which superconductivity has been observed, numerically the transition
from single-gap to dual-gap superconductivity is possible. Adopting the two-gap
viewpoint of superconductivity in C$_6$CaC$_6$, the dominant $d$-wave states
should have the same critical temperature. Around $T_c=2K$ these two relations
intersect, otherwise, superconductivity has been realized just in one of these
two sectors and disappears in the other one.
