Optical theorems and steinmann relations

Formulas that express in terms of physical scattering functions the discontinuity of any 3-to-3 scattering function across any basic normal threshold cut are derived from field theory. These basic cuts are the cuts in channel energies that start at lowest normal thresholds and extend to plus infinity. The discontinuity across such a cut generally depends on whether it is evaluated above or below each of the remaining basic cuts. Formulas are obtained for all cases. Generalized Steinmann relations are found to hold: the 2282 boundary values from which the discontinuities across basic cuts are formed have a unique extension to a set of 2¹⁶ = 65,536 functions, one for each combination of sides of the 16 basic cuts, such that for any pair of overlapping channels the corresponding double discontinuity vanishes. The ordinary Steinmann relations require this property to hold only for the double discontinuities formed from the original 2282 boundary values. The results are derived from the field-theoretic formalism of Bros, Epstein, and Glaser, which is slightly developed and cast into a form suited for calculations of the kind needed here.