Partial-Wave Analysis of the Deck Amplitude forπN→πππN

We present a straightforward method for calculating the amplitudes of the various $\mathrm{JMLS}$ states in the final $3\pi$ system of the reaction $\pi N\to \pi \pi \pi N$, assuming a Deck-model form for the production process. This calculation allows us to study relative phases of different partial waves, and the nucleon spin structure of the production amplitudes, in an unambiguous fashion. We find that the magnitudes and phases of $3\pi$ partial waves obtained by this method agree reasonably well with those obtained by applying an experimental fitting program to Monte Carlo events generated using the Deck formula; this may be interpreted as a check on the assumptions of the fitting program. Likewise, the magnitudes and phases agree reasonably well with those obtained from the data, confirming the belief that the $A_1$ and $A_3$ effects are probably produced by the Deck mechanism. In the rest system of the three pions, only one nucleon helicity-amplitude combination is dominant. The method of analysis can be applied to similar models for other production reactions.