  We show how to include the Jahn-Teller coupling of \mo phonons to the
electrons in the continuum model formalism which describes small-angle twisted
bilayer graphene. These phonons, which strongly couple to the valley degree of
freedom, are able to open gaps at most integer fillings of the four flat bands
around the charge neutrality point. Moreover, we derive the full quantum
mechanical expression of the electron-phonon Hamiltonian, which may allow
accessing phenomena such as the phonon-mediated superconductivity and the
dynamical Jahn-Teller effect.
