By P.D.B. Collins, E.J. Squires
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6) 30 Chapter I. Tile S-matrix but that the integrals diverge. 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.
4) c where the contour C is as shown in Fig. 11). 5) where u' + t' + s -----27. The validity of this sort of dispersion relation is a test of our maximal analyticity postulate (v), and for some processes such relations have been checked with experiment, as far as the available data permits. (See, for example, Hamilton, 1964, for review of such comparisons in 7 : - N scattering). In general the behaviour of the amplitude at infinity is not sufficiently convergent for the dispersion relation, as we have written it, to be meaningful.
54 Chapter II. 3) must break down at the t (or u) singularity nearest to the s channel physical region. We can now appreciate better why this is so. 9(2), Squires, 1962~. 1 IPz(z~)] < -7~ exp {llmZ 9 Re~ + ReZ. 11) 0 < Re{zg} < ~ . 4) we see that the partial-wave series will converge if IIm01 ~ ~(z~) = cosh-l(z~) . This defines an ellipse in the complex z~ plane, with foci at z = • 1, and semi-major radius z~ (Lehma~r 1958) ; see Fig. 7). 8. 5) can be used to define A3 (s) for all values of l, even if they are not integers as we have been assuming so far.