On the thermal sunset diagram for scalar field theories

We study the so-called `` sunset diagram'', which is one of two-loop self-energy diagrams, for scalar field theories at finite temperature. For this purpose, we first find the complete expression of the bubble diagram, the one-loop subdiagram of the sunset diagram, for arbitrary momentum. We calculate the temperature independent part and dependent part of the sunset diagram separately. For the former, we obtain the discontinuous part first and the finite continuous part next using a twice-subtracted dispersion relation. For the latter, we express it as a one-dimensional integral in terms of the bubble diagram. We also study the structure of the discontinuous part of the sunset diagram. Physical processes, which are responsible for it, are identified. Processes due to the scattering with particles in the heat bath exist only at finite temperature and generate discontinuity for arbitrary momentum, which is a remarkable feature of the two-loop diagrams at finite temperature. As an application of our result, we study the effect of the diagram on the spectral function of the sigma meson at finite temperature in the linear sigma model, which was obtained at one-loop order previously. At high temperature where the decay $\sigma\to\pi\pi$ is forbidden, sigma acquires a finite width of the order of $10 {\rm MeV}$ while within the one-loop calculation its width vanishes. At low temperature, the spectrum does not deviate much from that at one-loop order. Possible consequences with including other two-loop diagrams are discussed.