November INT-JLab Workshop Amplitude analysis for three- hadron states: Historical perspective Ian Aitchison INT-JLab Workshop, UW Nov

November INT-JLab Workshop Outline The isobar model and 3-h analyses in the 1970s….and more recently But the isobar model doesn’t satisfy unitarity Simplest implementation (“K-matrix –like”) of unitarity relation is wrong Need also analyticity 3-h dynamics Qualitative features of corrections to IM Conclusions (as of mid-1980s)

November INT-JLab Workshop Illinois group (production density matrix) “Partial-wave analysis of the 3 decay of the A2”, G.Ascoli et al., Phys. Rev.Lett. 25 (1970) “Spin-parity analysis of the A3”, G.Ascoli et al., Phys. Rev. D7 (1973) “The reaction at 25 and 40 GeV/c”, Yu. M. Antipov et al., Nucl. Phys. B63 (1973) , and [A1, A2 and A3] P(event) e A2

November INT-JLab Workshop isobar model amplitude factorization

November INT-JLab Workshop SLAC-Berkeley (fit coherent amplitudes) “Generalized isobar model formalism” D.J.Herndon et al., Phys.Rev.D11(1975)3165 “Partial wave analysis of the reaction in the c.m. energy range MeV” D.J. Herndon et al., Phys. Rev. D11(1975) “Amplitude analysis of production at 7 GeV/c” M.Tabak et al., Fourth Int. Conf. Exp. Meson Spectroscopy, 1974 AIP Conf Proc P(event)

November INT-JLab Workshop And more recently…… “ Improved measurement of the CKM angle in decays with a Dalitz plot analysis of decays to and ” B.Aubert et al. (BaBar) Phys. Rev. D78 (2008) The weak phase leads to different and decay rates (direct CPV) and is observable when and decay to common final states. About 0.5M events in the Dalitz plot.

November INT-JLab Workshop Unitarity (1) Two-body elastic unitarity Partial wave amplitude where power series in For example

November INT-JLab Workshop Unitarity(2) Two two-body f.s.i’s Unitarity: (U) Isobar model where are independent of But this does not satisfy (U) Instead, set Then (U) is a linear function of Not factorized

November INT-JLab Workshop Integration in unitarity relation

November INT-JLab Workshop So develops an imaginary part for due to rescattering from the other channel(s) Implementation (1) “ -matrix ” ( eg Ascoli and Wyld, PR D12(1975) ) Set Spurious singularities ( IJRA&Golding, Phys.Lett. 59B(1975)288 )

November INT-JLab Workshop Implementation (2) add analyticity dispersion relation uncorrected isobar model known function two-body data Integral equations for IJRA P. R. 137(1965)B1970, R.Pasquier and J.Y.Pasquier, P.R. 170(1968)1294, IJRA and J J Brehm, P. R. D17(1978)3072 adds up all rescatterings

November INT-JLab Workshop Integration for integral equation

November INT-JLab Workshop is essentially the partial wave projection of the OPE 3 3 process and has logarithmic singularities on the boundary of the Dalitz plot, when all particles in the OPE graph are on-shell. Inside the Dalitz plot, develops an imaginary part.

November INT-JLab Workshop The rescattering corrections to the IM uncorrected IMrescattering corrections where depends on final state interactions independent of production parameters Symbolically, And so Amplitude first rescattering correction provides reasonable approx. for dependence of full solution significant dependence can be generated in full solution

November INT-JLab Workshop Qualitative features of calculations S-waves is complex “scattering length” Effects of “Triangle” singularities All depend on and IJRA & JJBrehm, PR D 20(1979)1131; JJB, PR D21(1980)718, D23(1981) 1194, D25(1982)149

November INT-JLab Workshop Conclusions (as of mid-1980s) The corrections to the IM are unlikely to be larger than of order 20% in magnitude Subenergy corrections can broadly be absorbed into either the two-body parametrizations or the barrier factors, at fixed But (a) there is a - dependence J.J.Brehm, P.R. D25(1982) [ -dependent modulation of ] Study of the heavy-lepton decay “ We can summarize by asserting that rescattering can be a 20% effect relative to the resonance and should be included if the data are refined to that level of accuracy.” (b) the corrections are final-state dependent (eg versus ) Corrections might be needed to reconcile two-body amplitudes derived from different final states if data good enough.