Beyond HTL: The Classical Kinetic Theory of Landau Damping for Selfinteracting Scalar Fields in the Broken Phase

The effective theory of low frequency fluctuations of selfinteracting scalar fields is constructed in the broken symmetry phase. The theory resulting from integrating fluctuations with frequencies much above the spontanously generated mass scale $(p_0>>M)$ is found to be local. Non-local dynamics, especially Landau damping emerges under the effect of fluctuations in the $p_0 \sim M$ region. A kinetic theory of relativistic scalar gas particles interacting via their locally variable mass with the low frequency scalar field is shown to be equivalent to this effective field theory for scales below the characteristic mass, that is beyond the accuracy of the Hard Thermal Loop (HTL) approximation.