Fast computation of observed cross section for ψ′ → PP decays

It has been conjectured that the relative phase between strong and electromagnetic amplitudes is universally -90° in charmonium decays. ψ′ decaying into a pseudoscalar pair provides a possibility to test this conjecture. However, the experimentally observed cross section for such a process is depicted by the two-fold integral, which takes into account the initial state radiative （ISR） correction and energy spread effect. Using the generalized linear regression approach, a complex energy-dependent factor is approximated by a linear function of energy. Taking advantage of this simplification, the integration of ISR correction can be performed and an analytical expression with accuracy at the level of 1% is obtained. Then, the original two-fold integral is simplified into a one-fold integral, which reduces the total computing time by two orders of magnitude. Such a simplified expression for the observed cross section usually plays an indispensable role in the optimization of scan data taking, the determination of systematic uncertainty, and the analysis of data correlation.