Response functions in multicomponent Luttinger liquids

We derive an analytic expression for the zero temperature Fourier transform of the density-density correlation function of a multicomponent Luttinger liquid with different velocities. By employing a Schwinger identity and a generalized Feynman identity exact integral expressions are derived,and approximate analytical forms are given for frequencies close to each component singularity. We find power-law singularities and compute the corresponding exponents. Numerical results are shown for N - 3 components and implications for experiments on cold atoms are discussed.