Double resonance production and the equal-phase hypothesis

If all the helicity amplitudes for the reaction

$\text{0}{}^{\mathrm{?}}\text{+}\text{1}\text{2}{}^{\mathrm{+}}\text{→ J}{}_1\text{+ J}{}_2\text{, J}{}_2\text{?}\text{3}\text{2}$

have the same phase, it follows that certain bilinear forms in the statistical tensors t_L1L2^M₁M₂, with L₁ and L₂ even, vanish. An intuitive discussion is given of how these tests of the equal-phase hypothesis arise, their relation to positivity constraints on the joint-density-matrix is studied, and their behavior in the limit of forward production is examined. In order to illustrate the sensitivity of these tests, calculations of helicity amplitudes and the corresponding bilinear forms were made, using reggeized absorption-model calculations for πp → ?Δ. Some of the available data on πp → ?Δ, πp → ωΔ and $\text{K}{}^{\mathrm{+}}\text{p}\text{→}\text{K}{}^1\text{Δ}$ are studied using our tests. Some recent high statistics data on ?Δ indicate that the equal-phase hypothesis is incorrect for this reaction.A by-product of the investigation is the interesting result that if all the helicity amplitudes are in phase, and the absence of a polarized target, one arbitrarily chosen helicity amplitude can be taken to be zero at an arbitrary value of the kinematic variables, without affecting any observable results.