Unitary Phenomenological Description of Three - Particle Systems ?

A general unitary phenomenological description of strongly interacting three-particle systems is developed in terms of two-particle phase shifts and binding energies, two-particle wave functions inside the range of forces, and the three-body wave function in the region where all three force ranges overlap. The two-particle external and internal parameters are unambiguously separated from each other and the same parameters can be determined from many different experiments, while the three-body parameters refer to a specific system at a specific energy, (Submitted to Phys, Rev. Letters.) t Work performed under the auspices of the U. S, Atomic Energy Commission. In order to calculate the wave function, or even the on-shell scattering matrix, for three-particle states, it is necessary to know not only the binding energies of all two-particle subsystems and their phase shifts at all energies but also the interior wave functions connected to these asymptotic parameterizations. 1 Due to virt,ual pion emission and absorption, any system containing strongly interacting two-particle subsystems will experience three-particle forces in any region whose perimeter is less than 3fi/mnc;2 complete knowledge of the twoparticle subsystems does not determine the dynamics of any strongly interacting three-particle system. It has been shown3 that if the wave function is known inside this three-body-force region, the exterior three-particle wave function can be calculated by solving convergent one-variable integral equations. In this letter we convert this demonstration into a practical method for analyzing threeparticle systems by proving that it is possible to parameterize this interior wave function in such a way that only three outgoing particles are present (i.e., that the on-shell three-body T-matrix obtained by solving this equation is unitary) D These interior parameters can be fitted to experiment at a single energy of the three-particle system; they represent an arbitrary (energy-dependent) parameterization of the consequences of unknown twoand three-body forces in the interior region. Additional parameters will have to be included to specify twoparticle interior dynamical quantities e These can be independently determined either by making a complete analysis at additional energies of the three-particle system, or by using different third particles as probes. Hence this interiorexterior separation mades possible the construction of the wave functions of two strongly interacting particles inside the range of forces from quantum mechanical observables D To the extent that this nonrelativistic quantum mechanical prescription makes sense, this description should be the same whatever the third particle in the system. -2The Wick argument2 tells us that whenever the outgoing waves in the two other Faddeev channels (which are the physical mechanism that probe the interior two-particle wave functions in the exterior three-particle region) 3,18 contain momenta exceeding pion production threshold within the system probed, this three-particle description breaks down, But even in this situation, the hadron “soup” which is being probed may have average properties independent of the mode of excitation. To the extent that this is true, our analysis will still give correctly the probability of finding that part of the “soup” which is connected to two, and only two, specified hadrons in the asymptotic region,, The a priori limit of validity of our description is for distances averaged over regions of order h/m,c, or momenta less than mnc; only the study of specific systems will reveal whether it holds down to shorter distances and for higher momenta, We assume4 that the two-particle half off-shell t-matrices are bounded by t$p.kp2) = T;(P) O3 2 JW W2) P 2(p2 -I4k P2 (p2-k2)2 + 2p2(p2-k2) + p4