Unitarity in the Isobar Model: Guide to the Literature

There has been a lot of work on unitary corrections to the isobar model. Here we mainly follow the work of Aitchison and Brehm. Their series of papers describes in great detail a method how to encorporate unitarity (and analyticity) into the isobar model. We plan to write a note summarizing these methods and applying them to COMPASS analysis cases. Especially the implementation of the methods described here into a useful software has to be worked out yet.

• "Three body unitarity" by Aitchison and Pasquier: Phys. Rev. 152, 1274 (1966). Shows that requiring unitarity in the 2-body isobar channel is sufficient to generate amplitudes that satisfy three-body discontinuity equations. Can this approach be generalized to n-body final states?

• The Pasquier-Inversion I/II: Phys Rev 170, 1294 (1968) and Phys. Rev. 177, 2482(1969). This paper gives the basic method how to deal with the integral equations that result from the application of unitarity via dispersion relations. The reduction to single-variable integral equations is derived. The second part also discusses how to separate kinematic singularities that stem from spin effects. The papers discuss 3-body processes, it remains to be worked out how to apply the formalism to general N-particle final states.

• "Analysis of Three-Hadron Final States" by Aaron, Amando: Phys Rev Lett 31, 1157 (1973). This paper is usually quoted as the one that first explicitly spelled out the violation of unitarity by the isobar model.

• "Relativistic three-pion dynamics" by Aitchison: J. Phys. G: Nucl. Phys., Vol.3, No.2, 121 (1077). Application of the dispersion relation approach with Pasquier-inversion to the 3pi system. Probably the best starting point to follow the topic.

• "Unitarity and the Isobar Model" by Brehm: Ann. Phys. Vol. 108 (1977) 454. Brehm's first paper on the topic where he starts to develop the formalism of 2-body discontinuities for the KN->KpiN process.

• "Unitary analytic isobar model .." by Aitchison and Brehm: Phys. Rev. D 17, 3072 (1978). Aitichson and Brehm use Brehm's 2-body discontinuities to set up the dispersion relations. Only s-wave isobars are used. A system of integral equations for the isobar amplitudes is given.

• "Medium-energy Npipi dynamics I/II"" Phys. Rev. D 20, 1119 (1979) and Phys. Rev. D 20, 1131 (1979). The formalism is spelled out in detail for the Npipi system. The second part of this monumental work contains also a detailed discussion in human language.

• "Are there Important Unitarity Corrections to the Isobar Model?": Phys. Lett. 84B (1979) 349. The answer is: no big ones. But who knows what happens at high statistics!?

• "Final-state interactions in a system of three pions" by Brehm: Phys. Rev. D23 (1981) 1194. Brehm returns to the (relatively simple) 3pion system. Using the rho-pi and eps-pi isobar channels he develops a useful method for solving the integral equations in terms of resolvents.

• "Simplified parameterization of 3pi final-state interactions" by Brehm: Phys. Rev. D25 (1982) 3069. Brehm discusses the results of his methods in a simplified parameterization. I am not sure how useful this is.

-- SebastianNeubert - 04 May 2010