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Unitarity and unstable-particle scattering amplitudes
Unitarity is the fundamental property of the S-matrix while its usage for a
scattering of unstable particles has been subtle as unstable particles do not
appear in the asymptotic states. Defining unstable-particle amplitudes as
residues of a higher-point amplitude at an appropriate complex pole, we find
unitarity equations for the 2-to-2 unstable-particle amplitudes from unitarity
and analyticity of stable-particle scattering amplitudes. The unstable-particle
unitarity equations take a form analogous to those of the stable-particle
amplitudes when the in and out states are chosen to be complex-conjugate
positions. In particular, as in the optical theorem, we find a positivity
constraint on a discontinuity of the amplitudes in a positive region of the
momentum transfer variable.Comment: 11 + 5 pages, 7 figures, published versio
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