Field theory for zero sound and ion acoustic wave in astrophysical matter

We set up a field theory model to describe the longitudinal low-energy modes in high density matter present in white dwarf stars. At the relevant scales, ions - the nuclei of oxygen, carbon, and helium - are treated as heavy pointlike spin-0 charged particles in an effective field theory approach, while the electron dynamics is described by the Dirac Lagrangian at the one-loop level. We show that there always exists a longitudinal gapless mode in the system irrespective of whether the ions are in a plasma, crystal, or quantum liquid state. For certain values of the parameters, the gapless mode can be interpreted as a zero sound mode and, for other values, as an ion acoustic wave; we show that the zero sound and ion acoustic wave are complementary to each other. We discuss possible physical consequences of these modes for properties of white dwarfs.