Comparison of the rotationally invariant quantity lambda-tilde

The rotationally invariant quantity λ-tilde can be calculated from

which has values that range from -1 to ∞. There is a one-to-one correspondence between λ-tilde and F, which is defined by

and ranges from 0 to 1. The following figure shows an example of one-sigma confidence intervals for λ_θ and λ_φ for the the ϒ(2S) state in the p_T range 2-4 GeV/c superimposed on lines of equal lambda-tilde (left) or F (right).

The following figure shows the comparison of λ-tilde measured in both reference frames for the three ϒ(nS) states as a function of p_T:

Although the parameters λ_θ, λ_φ and λ_θφ in each reference frame are measured using the same data, the data and the Monte Carlo acceptance populate different regions of the solid angle in separate likelihood fits. For this reason, small differences are not unexpected and the size of deviations observed here are typical of statistical fluctuations observed in toy Monte Carlo studies.