  We develop an analytic theory to describe the interaction between electrons
and K-phonons and study its influence on superconductivity in the bare bands of
twisted bilayer graphene (TBG). We find that, due to symmetry and the
two-center approximation, only one optical K-phonon (~ 160meV) of graphene is
responsible for inter-valley electron-phonon interaction. This phonon has
recently been found in angular-resolved photo-emission spectroscopy to be
responsible for replicas of the TBG flat bands. By projecting the interaction
to the TBG flat bands, we perform the full symmetry analysis of phonon-mediated
attractive interaction and pairing channels in the Chern basis, and show that
several channels are guaranteed to have gapless order parameters. From the
linearized gap equations, we find that the highest Tc pairing induced by this
phonon is a singlet gapped s-wave inter-Chern-band order parameter, followed
closely by a gapless nematic d-wave intra-Chern-band order parameter. We
justify these results analytically, using the topological heavy fermion mapping
of TBG which has allowed us to obtain an analytic form of phonon-mediated
attractive interaction and to analytically solve the linearized and T=0 gap
equations. For the intra-Chern-band channel, the nematic state with nodes is
shown to be stabilized in the chiral flat band limit. While the flat band
Coulomb interaction can be screened sufficiently enough - around Van-Hove
singularities - to allow for electron-phonon based superconductivity, it is
unlikely that this effect can be maintained in the lower density of states
excitation bands around the correlated insulator states.
