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So oo N f A (s, t, u) = F N-1 (s, t) + -~x ~II= 1(t - t~) ~ Dr(s, t') dr" (r -- tl) . . 8) where F x - 1 (s, t) is a function of s multiplied by a polynomial of degree N - 1 in t. Thus if D t (s, t) t~'~ tx-~ at fixed s, e > 0, the integral converges, and the dispersion relation is well defined. ) If Dt (s, t) is bounded by some power of t it will always be possible to do this. 6), A (s, t, u) is no longer completely determined by D t(s, t) [and D,]. N. Thus, requiring that the amplitude satisfy maximal analyticity of the first kind, with all its singularities given by the Landau-Cutkosky rules, is not necessarily sufficient to determine it completely.
In this chapter we shall discuss in some detail the partial-wave decomposition of the scattering amplitude. In the first instance we shall consider only integer values of angular momentum, which are of course the only physically allowed values (for bosons) given the quantization of angular momentum. But we shall subsequently find it interesting to extend our definition of partial-wave amplitudes to include unphysical non-integer, and indeed complex, values. In so doing we shall encounter singularities of the amplitudes in the angular momentum plane, "Regge" poles and cuts, which are connected with the divergences of the Mandelstare representation.
7) co 1f +~- j Dr(s, t') dr" (t'--t:)~_:~'~-5~) (t'--t) + (uterm), to where we have picked up a contribution from the poles at t = t~. So oo N f A (s, t, u) = F N-1 (s, t) + -~x ~II= 1(t - t~) ~ Dr(s, t') dr" (r -- tl) . . 8) where F x - 1 (s, t) is a function of s multiplied by a polynomial of degree N - 1 in t. Thus if D t (s, t) t~'~ tx-~ at fixed s, e > 0, the integral converges, and the dispersion relation is well defined. ) If Dt (s, t) is bounded by some power of t it will always be possible to do this.