Damping of zero sound in Luttinger liquids

We calculate the damping gamma(q) of collective density oscillations (zero sound) in a one-dimensional Fermi gas with dimensionless forward scattering interaction F and quadratic energy dispersion k(2)/2m at zero temperature. Using standard many-body perturbation theory, we obtain gamma(q) from the expansion of the inverse irreducible polarization to first order in the effective screened (RPA) interaction. For wave-vectors vertical bar q vertical bar/k(F) << F (where k(F) = mv(F) is the Fermi wave-vector) we find to leading order gamma(q) proportional to vertical bar q vertical bar(3)/(v(F)m(2)). On the other hand, for F << vertical bar q vertical bar/k(F) most of the spectral weight is carried by the particle-hole continuum, which is distributed over a frequency interval of the order of q(2)/m. We also show that zero sound damping leads to a finite maximum proportional to vertical bar k - k(F)vertical bar(-2+2 eta) of the charge peak in the single-particle spectral function, where eta is the anomalous dimension. Our prediction agrees with photoemission data for the blue bronze K0.3MoO3. We comment on other recent calculations of gamma(q).