Unitarity and interfering resonances inππscattering and in pion productionπN→ππN

Abstract

The additivity of Breit-Wigner phases has been proposed to describe interfering resonances in partial waves in 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 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 N→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 partial wave amplitudes with resonances determined from fits to mass spectra of production amplitudes measured in N→N reactions can be satisfied with an appropriate choice of complex residues of contributing Breit-Wigner poles.

Representation of spectral functions and thermodynamics

“…We have seen in the previous subsection that two Breit-Wigner resonances in general are combined in a non-linear way. This phenomenon is in accordance with the scattering theory predictions [15], where the S-matrix contributions of the resonances are non-independent, but they satisfy in general a complicated unitarity constraint equation. We should go further and study, what happens when the spectral function can be represented by several resonances.…”

“…Heuristically, the width of the peaks can be treated as a "resolution power". If, according to this resolution, the separation of the peaks is large, then earlier studies [15,17] tell us that these behave as distinct objects: the independent particle approximation works well. If the separation of the peaks starts to be comparable with their width, then by their own resolution they are more and more like one object, and we arrive at the case discussed above.…”

Nuclear and quark matter at high temperature

“…This assumption, however, fails when we consider a system where the quasiparticle peaks occur densely, or if a multiparticle background is present. In such cases the quantum mechanical treatment of the quasiparticle peak contributions to the S matrix have complex coefficients [87][88][89]. The unitarity of the S matrix poses constraints among these coefficients: in this way the pole contributions are no more independent.…”

Hadron melting and QCD thermodynamics

“…In the S-matrix approach this is discussed in Refs. [24,25]. To understand the problem qualitatively, we recall that a quasiparticle is a collective mode, i.e.…”