No abstract
Many authors have proposed that the masses and coupling constants of the fundamental particles may be determinable from some form of dispersion relations. It has also been proposed that, eventually, the symmetries of strong-interaction physics may occur as results of the dispersion theory rather than as separate assumptions. 1 The purpose of this note is to present an approximate set of dispersion relations from which an interaction symmetry is predicted. The PS (pseudoscalar) mesons -n, K, and r\ (with isospin 0 and G-parity even) are considered, and their mass differences neglected. The V (vector) mesons p, to, andK*, with mass differences neglected, are considered as resonances or bound states of the various pairs of PS mesons, and the coupling constants y p7r77 , y pK K> 7UJKK> 7K*-nK> and YK*riK are computed from a generalization of the "bootstrap" procedure of Zachariasen and Zemach. 2 The ratios of these computed constants are found to be in exact agreement with the predictions of the octet model of unitary symmetry.We choose as a dimensionless energy variable x = \W 2 /m 2 , where Wis the total energy in the center-of-mass system and m is the PS-meson mass. We define the P-wave amplitude for the production of a state j of two PS mesons from a similar state i in terms of the elements of the unitary S matrix by the equationThe only forces considered are those that arise from the exchanges of the various V mesons, with the energy widths of these mesons neglected. The form of the Born approximation resulting from these forces is well known, 2 and is given bywhere /i is related to the mass M of the vector mesons by /i =\M 2 /m 2 > and the Njj are constants that depend on the coupling constants of the exchanged V mesons and on isospin-dependent factors that occur in the crossing matrices. As in reference 2, the amplitude T is set equal to T(Born)/D, where the only singularity contained in D is the right-hand, "unitarity" branch cut, and only the contributions of P-wave states of two PS mesons are retained in the unitarity condition. A dispersion relation is written for D, with one subtraction chosen so that D = 1 at the edge of the left-hand cut, i. e., at the energy Xf = 1 -jit. The V-meson mass and the various coupling constants are then determined from the consistency requirement that the parameters emerging from the dispersion relations agree with those used to specify the forces. In our model the ot > is regarded as a KK resonance, which physically is too light to decay into the KK state. The p and K* are considered as resonances in the coupled TTTT and KK states, and in the coupled TIK and 7]K states, respectively. 3 Hence, we write two-channel dispersion relations for the p and K* 9 using the method of Bjorken. 4 Since our numerator matrix has the simple form of a constant matrix multiplied by the function of energy Fix), it can be shown from the equations of reference 4 that the approximate scattering matrix obtained from our procedure is symmetric. The amplitudes have the form T n =Fix)D-l ix){N u + ...
The number and general nature of the self-consistency equations that may arise in a universal bootstrap theory of all strongly interacting particles is discussed. There are more equations than there are variables to be determined, so that it may be possible to bypass some of the divergence difficulties of dispersion theory by making use of a sufficient number of the equations. A general method of attacking the difficulties associated with the many-particle aspect of the problem is discussed. A simple first approximation to the method is applied to a model of four multiplets (pseudoscalar meson, vector meson, baryon ground state, and y = f+ baryon excited state), under the assumption that unitary symmetry is approximately valid. It is argued that comparison with experiment of the calculated differences in masses of particles within the same multiplet will provide experimental tests that are meaningful in a low-order approximation to the model. It is shown that if the mass splitting of the baryon octet is assumed to be partly self-generating (i.e., not resulting completely from the mass splitting of the meson multiplets), a nondegenerate solution to the model is most likely if there is a large violation of R in variance.
The possibility of determining the relative parity of iT-hyperon pairs by measurements of reactions of the type x (or K) -\-p -> K(or w) +hyperon is investigated. If the protons are polarized in a direction perpendicular to the incident meson beam, it is shown that measurements of the correlation between the proton spin direction and the direction of the meson from the hyperon decay are capable of determining the parity unambiguously, provided that the K spin is zero and the hyperon spin is known. Several alternate paritydetermining experiments that do not require polarized proton targets are discussed; these experiments are all quite difficult, however.
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