Michael J. Moravcsik

General nondynamical formalism for reactions with particles of arbitrary spin. Number of form factors. Parity conservation

Abstract The general nondynamical formalism for reactions involving particles of arbitrary spins, discussed in several previous papers, is further developed. The number of form factors in the M -matrix is determined under parity conservation as well as other conservation laws, and from this it is derived that many conservation laws have no nondynamical tests of validity in reactions involving spin-zero particles only. The factorization of the M -matrix into irreducible constituents is developed for parity conserving reactions and it is shown how the compound observable can be written down in terms of observables and so-called pseudoobservables of the constituent reactions. It is then demonstrated that for parity conserving reactions, an unpolarized and uncorrelated particle is in general equivalent to a fictitious spin-zero particle possessing both positive and negative parity (dual particle). The structure of observables in terms of the type of traces they contain is studied. The number of terms in a compound observable is derived. It is shown that the number of observables is a parity conserving reaction is always larger than the number of independent bilinear combinations of form factors. Thus, there are always linear relations among the observables. The number and character of these relations is explored and it is shown that they also lead to the complete list of parity experiments, i.e., experiments from which we can determine the intrinsic parity of one of the particles in the reaction, knowing the intrinsic parities of the other participating reactions but knowing nothing about the dynamics of the reaction. It is shown that the observables can be divided into subclasses, a given subclass containing observables which are different linear combinations of the same bilinear combinations of form factors. Each such bilinear combination appears in one and only one subclass.

GENERAL NONDYNAMICAL FORMALISM FOR REACTIONS WITH PARTICLES OF ARBITRARY SPIN: ROTATION INVARIANCE.

Abstract The purpose of this paper is to give a detailed and practical prescription for calculating physical observables from form factors (i.e., invariant amplitudes) of the M -matrix for an arbitrary reaction containing an arbitrary number of particles having arbitrary spins. The treatment is fully relativistic. No invariance principles have been assumed (except rotational invariance in three-space) and the discussion is valid whether or not any of the usual invariance principles (parity conservation, time-reversal invariance, etc.) hold. The prescription is based on those properties of such reactions which have already been treated in the literature, as well as on several remarks that have not been discussed before. Among the latter are the factorizability of the M -matrix in spin-space, the treatment of nonsquare spin matrix tensors, and the general evaluation of traces of products of spin tensors. The expansion of the M -matrix in terms of spin-momentum tensors is explained for the general case, and a similar discussion is given for the density matrix. From these the observables are calculated, and the mathematical background of such calculations are treated in appendices. Some of the results of this paper have been used in several past papers of ours, others will appear in papers completed and soon to be published. A table containing the coefficients of the bilinear products of form factors in the various experimental observables for various reactions containing plausible spins will soon be available also as an unpublished report, as it was calculated by a computer program utilizing the procedure outlined in this paper. The program will also be available to calculate, as the need arises, any reactions not contained in the table.

Unitarity corrections and dispersion modifications of one-particle exchange theories

Amplitudes for elementary particle reactions calculated from a one-particle exchange theory can be modified to satisfy unitarity and to conform to the constraints of a dispersion relation. Unitarity corrections are discussed in Section II for uncoupled and coupled phases and expressions are given for the pure unitarity corrections to phases. Dispersion modifications are discussed in Section III and it is concluded that the schemes proposed so far (1–3) for such modifications are either inconsistent with the unsubtracted dispersion relations or are ambiguous in their definition of the pole contributions. The conclusion is reached that it is not possible to determine experimentally from a dispersion relation, the parameters of a one-particle exchange model. A prescription is given for obtaining the modest amount of information that can in fact be gleaned from the use of such dispersion relations. The specific and quantitative summary of conclusions is given in Section IV.

One-particle exchange theory of pion photoproduction: I. General calculation

Abstract A calculation of single pion photoproduction from single nucleons is given using the one-particle exchange model involving the exchange of one pion, one ϱ, one ω, one φ, one nucleon, one N ∗ , and one N ∗∗ . Crossed diagrams, previously neglected, have also been included in the calculation. An imaginary part of the amplitudes is included using a Breit-Wigner type formula and experimentally observed widths. The theory should be a good representation of pion photoproduction up to about 800 MeV incident laboratory photon energy. A numerical evaluation of the theory will be given in a subsequent paper.

An introduction to x-coefficients and a tabulation of their values**

X -coefficients, as defined here, can be used to simplify the calculation of particle polarizations (vector polarizations as well as higher-order, so called, tensor polarizations) and spin correlations. If not more than two particles have nonzero spin in the reaction under study, then all polarizations and correlations can be expressed as linear combinations of X -coefficients, and only the numbers multiplying them depend on the form factors (invariant amplitudes). If three or four particles have nonzero spin, then bilinear combinations of X -coefficients appear, etc. The X -coefficients do not depend on the dynamics of the reaction and can be tabulated once and for all. They do depend on the spins of the participating particles and are given in this table for those spin values which are the most important ones from a practical point of view.

ONE-PION EXCHANGE AND THE OPTICAL MODEL

Abstract Starting with the two-body scattering amplitude, an optical model potential for nucleon-nucleus scattering has been calculated (1) using the one-pion exchange interaction alone, (2) modifying the s-wave part of this by using effective range theory, and (3) using the modified phase shift analysis for p-p scattering supplemented by charge independence and the Gammel-Thaler potential for the n-p scattering information. Part (1) works sufficiently well to suggest that the one-pion exchange force contributes more importantly to the nucleon-nucleus interaction than to the basic two-nucleon scattering process. Part (2) produces an indifferent improvement on part (1). Part (3) represents the optical potential inferred from the most accurate two-nucleon data presently available. This last evaluation of the optical potential is considered the most reliable to date since it includes, in a good approximation, an infinite number of the high angular momentum states of the two-nucleon amplitude, which play a very essential part in the optical potential. A systematic derivation of the optical model equations is included with special emphasis on the needs of the present calculation; in addition, criteria for the validity of the model are discussed. Allowance has been made for the nuclear volume not being strictly proportional to the mass number (nuclear size effect) and for the inequality of the numbers of neutrons and protons in the nucleus (effect of neutron excess). For heavy nuclei, these effects influence the results up to 25%.

SPIN-ZERO PHOTOPRODUCTION PARITY EXPERIMENTS BEYOND THE THRESHOLD REGION

Abstract It is suggested that parity determining experiments can often be greatly simplified if one can assume that only the few lowest angular momentum states are important. The assumptions made here are not as restrictive as those made for the conventional threshold experiments, and hence the applicability of the schemes is much wider. In particular, a parity experiment is suggested in the photoproduction of spin-zero bosons which does not require polarized photons at all, and which is well within present day experimental techniques. It involves two measurements at the same energy and angle, one measuring the recoil polarization of the fermion, and the other the differential cross section from a polarized target.