We discuss some features of the continuation to complex total angular momentum of the three-particle discontinuity of a "double box" Feynman diagram. A solution of this problem implies that the discontinuity satisfies the bound required by Carlson's theorem. We conclude that for this to be possible the phase-space integral has to be organized in two separate pieces. Also, the partial-wave amplitudes have to be broken into two separate pieces, each piece being given by a Froissart-Gribov contour around a singularity of the helicity amplitude. This splitting of the partial-wave amplitude introduces an extra singularity, but this does not contribute to the final expression.