Singular Three-Body Amplitudes in the Theory of the Third Virial Coefficient

We have shown earlier how the virial coefficients can be expressed in terms of traces taken over products of multiparticle (on-energy-shell) scattering matrices. The traces are to be taken in the angular-momentum representation instead of the momentum representation to avoid the infinite forward N-particle scattering amplitude for N > 2 caused by certain singular diagrams. Here, we analyze and sum the contribution of the singular diagrams to the third virial coefficient. It is also shown that one can get the same result if off-shell amplitudes in the momentum representation are utilized.