Dynamic response functions of graphene with short-range interactions

We consider a short-range interaction between electrons in graphene and determine their dynamic response functions, such as a longitudinal and a transverse electric conductivity and a polarization function. The calculations are performed to all orders in the interaction by taking into account the self-energy renormalization of the electron velocity and using a ladder approximation to account for the vertex corrections, ensuring that the Ward identity (charge conservation law) is satisfied. We discuss whether a collective mode can exist in this system corresponding to the propagation of electric polarization of correlated electron-hole pairs.