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5-3-4 Forward Dispersion Relations
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We are now in a position to examine the elastic forward amplitude when ql = q2 = q, Pl = P2 = p, as a function of incoming energy v = q. p/mb = (8 - m;; - mW2mb. We wish to show that it is analytic in a cut plane, with the cuts running along parts of the real axis. This is a nontrivial result in view of the remark at the end of Sec. 5-3-2. Of course, we require that q2 = m;;, the physical mass-shell condition.
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This is obtained by intersecting the surface P, from the previous construction, by the plane z = m~, leading to a hyperboloid. We have to study the corresponding intersection of re. For a complex q lying on the mass shell, the real line L(m~, q) does not intersect this hyperboloid with which it has already the two complex points (m~, q) and (m;, q*) in common. We have noticed that for fixed P
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Figure 5-9 The region i (shaded area), convex hull of the coincidence region in five-dimensional space.
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Lorentz invariance implies that :T is only a function of the component of q along p. Taking the latter along the time axis means that we can restrict our attention to vectors q with only two nonvanishing components, qo and q!, say. The problem is in effect two dimensional with a third component z added for convenience. The complex point (rn~, q) on the hyperbola q6 - qr = rn~ will have (Im q)2 < 0 and will be a point of analyticity provided L(rn~, q) intersects the convex hull ij in the (qo, ql) plane. The condition for this is that C(f meets both branches of the mass-shell hyperbola. Now C(f is given by (p qf < M;, or, equivalently, rnl 2p' q + z < M;'. The condition that C(f meets the upper branch is rnl + 2rnarnb + rn; ::; M~ or (rna + rnb)2 ::; M~. Similarly, the condition that it meets the lower branch is (rna + rnbf ::; M:'. These are therefore the criteria that eliminate the occurrence of complex singularities in the complex v plane. On the other hand, we know that Ml ::; (rna + rnb)2, corresponding to intermediate states A + B or A + B in the sums of Eq. (5-173). From this we conclude that forward dispersion relations (without complex singularities) can at best be proved marginally using the above method. It may happen that M + or M _, or both, are strictly smaller than the elastic threshold. One case frequently encountered can be disposed of without too much trouble. This is when an intermediate isolated state [mass M6 < (rna + rnb)2] occurs below the threshold. Its contribution is then a pole in the energy variable. Multiplying the amplitude by (p q)2 - M6 does not modify the analytic properties but eliminates the singularity. In pion-nucleon scattering, for instance, the nucleon intermediate state contributes such a pole.
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Let us assume that apart from such poles the condition Ml ;::: (rna + rnb)2 is satisfied. We then conclude that in the case of forward scattering the following properties hold:
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1. The amplitude is analytic in the complex v plane except for cuts extending
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along the real axis from rna to + 00 and from - 00 to - rna and possible poles in between. 2. The amplitude is real in between the cuts so that its discontinuity across the cuts is purely imaginary. This follows from Eq. (5-173) applied to forward scattering. 3. The amplitude is bounded by a polynomial for large v. This stems from the tempered character assumed for the fields and holds a fortiori for complex v. 4. Finally, if A == A crossing symmetry requires that :T( - v*) = :T(v)*. U sing this information and Cauchy's formula, a simple analytic representation is obtained. For simplicity assume that, as in property 4, A == A and that :T
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