Unitarity and interfering resonances in ππ scattering and in pion production πN→ππN

The additivity of Breit-Wigner phases has been proposed to describe interfering resonances in partial waves in $\ensuremath{\pi}\ensuremath{\pi}$ scattering. This assumption leads to an expression for partial wave amplitudes that involves products of Breit-Wigner amplitudes. We show that this expression is equivalent to a coherent sum of Breit-Wigner amplitudes with specific complex coefficients which depend on the resonance parameters of all contributing resonances. We use the analyticity of $\ensuremath{\pi}\ensuremath{\pi}$ partial wave amplitudes to show that they must have the form of a coherent sum of Breit-Wigner amplitudes with complex coefficients and a complex coherent background. The assumption of the additivity of Breit-Wigner phases is a new constraint on partial wave amplitudes independent of partial wave unitarity. It restricts the partial waves to analytical functions with a very specific form of residues of Breit-Wigner poles. Since there is no physical reason for such a restriction, we argue that the general form provided by the analyticity is more appropriate in fits to data to determine resonance parameters. The partial wave unitarity can be imposed using the modern methods of constrained optimization. We discuss the production amplitudes in $\ensuremath{\pi}\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\pi}\ensuremath{\pi}N$ reactions, and use analyticity in the dipion mass variable to justify the common practice of writing the production amplitudes in production processes as a coherent sum of Breit-Wigner amplitudes with free complex coefficients and a complex coherent background in fits to mass spectra with interfering resonances. The unitarity constraints on $\ensuremath{\pi}\ensuremath{\pi}$ partial wave amplitudes with resonances determined from fits to mass spectra of production amplitudes measured in $\ensuremath{\pi}\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\pi}\ensuremath{\pi}N$ reactions can be satisfied with an appropriate choice of complex residues of contributing Breit-Wigner poles.