  We study the finite temperature, low energy, long wave-length spectrum of the
dynamic structure factor of the spin-$1/2$ antiferromagnetic Heisenberg chain
in the presence of exchange anisotropy and external magnetic fields. Using
imaginary-time quantum Monte-Carlo we extract parameters, relevant to
characterize a {\it renormalized} Luttinger liquid. For small momentum our
results are consistent with a change from propagating spinon density waves to
spin diffusion, described by a finite-frequency spin-current relaxation rate.
Results for this relaxation rate as well as other Luttinger liquid parameters
are presented versus temperature, momentum, magnetic field, and anisotropy,
including finite-size analysis, and checks for anomalous diffusion. Our results
are consistent with exact diagonalization and Bethe Ansatz, where available,
and with corroborate findings of other previous studies using bosonization,
transfer matrix renormalization group, and quantum Monte-Carlo.
