Dips in partial wave amplitudes from final state interactions.

We consider the dip-peak structures in the J=0 partial wave amplitudes for processes γγ → WW and γγ, gg → tt taking into account the corresponding Born term process and the rescattering process where the intermediate state is rescattered through the exchange of Higgs resonance state in the direct channel. Near a direct channel resonance, final state interactions can be important even for relatively weakly interacting particles. Consider the production by photons of a pair of W s, γγ → WW. The Born amplitude is of order e and rescattering by the W s naively makes the amplitude of order e. However, if the rescattering process corresponds to the exchange of, such as in this example, the Higgs in the direct channel, near the Higgs resonance the rescattering amplitude goes as e m Γ ∼ e, where Γ is the resonance width, and m is the resonance mass. A characteristic signature of a final state interaction near a resonance is a dip in the scattering amplitude.[1-7] This dip occurs for the part of the amplitude where the rescattering particles are on mass shell and when there is only a single channel involved. In practice,the dip may be washed out by the part of the amplitude with the particles off-shell, other particles produced and rescattered into the desired final state (including other polarizations of the final state particles), and even contributions from other particles than the final state particles to Γ. Our goal in this note is to consider a few simple reactions in the Standard Model and check to what extent a dip will manifest itself in the partial wave amplitude. We choose to work directly with the J=0 partial wave amplitude, since the interference effect should show up most prominently in partial wave amplitudes. As a definite, and uncomplicated example consider again γγ → WW. First take only the Goldstone boson part of the W s, γγ → χχ and work in the ξ = 1 gauge where mχ = MW . The Born J = 0 amplitude is a ++ = e 16π (1− β) β ln 1 + β 1− β (1) where β is the χ velocity, β = (1−M2 W/E) 1 2 , with E the initial photon energy. The amplitude for two photons to produce an off-shell Higgs is ǫμ(k1)ǫν(k2)T μν = eMWπ 2 sin θW (2π)4 ( g − 2k ν 1k μ 2 s ) ǫμ(k1)ǫν(k2) × [ mH M W + mH s I ] (2) See in particular the discussion in the Appendix of Ref. 5.