Compton Scattering Sum Rules for Massive Vector Bosons

1.3 S-Matrix Formalism and Dispersion Theory
It immediately follows that F (ν) is analytic. Due to the Schwarz reflection principle,
F (ν ∗ ) = F ∗ (ν), the amplitude is also analytic in the lower half plane.
1.3.2 Dispersion Relations
For a massive particle, the amplitude F (ν) will have branch cuts along the real axis,
see fig. 1.1. Due to analyticity we can apply Cauchy’s theorem,
F (ν) = 1 ∮
2πi
C
dν ′
ν ′ − ν F (ν′ ). (1.41)
The integration path is shown in fig. 1.1. Since we are usually interested in physical
Im F (ν)
Re F (ν)
−ν 0
ν 0
Figure 1.1: Contour of integration in the complex energy plane used to derive the
dispersion relation. The half-circles are blown up to infinity.
energies, we choose to evaluate F (ν) at ν = x + iε for real x. We can split the integral
into curve integrals along the semicircles of radius R and parts along the real axis. If we
let R → ∞, the parts along the contour vanish—assuming a sufficiently well-behaving
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