P. Carruthers

Crossing matrices for particles of arbitrary isospin

Abstract An elementary and systematic discussion is given of isospin crossing symmetry for “two-body” collisions involving particles of arbitrary isospins. Special attention is given to the question of phase relations among the states in the various crossed channels. Explicit formulas for the various crossing matrices are given in terms of Racah coefficients. Numerical forms are given for reactions of especial interest. In an appendix the behavior of the relevant fields and operators under charge conjugation and G -conjugation is described.

Pion production and the higher resonances in pion-nucleon scattering☆

The reaction π + N → 2π + N has been studied in the vicinity of the higher resonances in the pion-nucleon cross section. The Low equation for the production amplitude is transformed into an integral equation by isolating the true one-meson intermediate states and discarding higher order contributions. The only part kept in the inhomogeneous term corresponds to the collision of a pion in the nucleon cloud with the incident pion in the resonant T = J = 1 state, which is simulated by an unstable vector Boson. Crossed terms are neglected and the 2π-N state is described by the static model. The terms kept in the sum over states describe the rescattering (off the nucleon) of one of the outgoing pions. The required off-the-energy-shell elastic scattering amplitude is approximated by the 3-3 resonance formula of Chew and Low. With these simplifications the Low equation for the production amplitude reduces to an easily soluble linear integral equation. The rescattering amplitude, which dominates the inhomogeneous term in the resonance region, is proportional to the 3-3 scattering amplitude of one of the outgoing pions. Although the result provides some support for the conventional isobar model, it is important to note that the largeness of the rescattering term arises from scattering far off the energy shell, rather than by “real” excitation as in the phenomenological isobar model. Quantitative calculations for the D32 channel leading to a p-wave (J = 32) and an s-wave pion produce a maximum in the cross section near 600 Mev incident pion lab energy. For a π-π resonance energy squared S = 10, agreement with experiment is obtained with a width about one third that suggested by nucleon electromagnetic structure. In our approximation, the well known 600 Mev D32 isospin 12 resonance occurs at the same energy as the 800 Mev D32 isospin 32 resonance. It is assumed, but not proved, that the neglected terms are responsible for the splitting of the resonance energies. When this splitting is taken into account, the predicted charge state ratios near the second resonance agree well with existing data. The “third” resonance occurs for the state having two p-wave pions, according to the present theory, although no numerical calculations were made for this case. This point of view suggests that the F32, P32, and P12 incident channels contribute to the third resonance.

Local Field Theory and Isospin Invariance. I. Free‐Field Theory of Spinless Bosons

The properties of free‐field theories of spinless bosons are investigated. The self‐conjugate boson field theories of isospin ½, 32, 52, ⋯ are shown to be nonlocal: the energy, isospin, and number densities fail to commute for spacelike separations. The difficulty is traced to the necessary occurrence of non‐independent field variables. For the anomalous case the dependent variables are nonlocally related to the independent ones; elimination of the former makes superficially local quantities depend nonlocally on the independent fields. The connection with canonical field theory is investigated. The p and q coordinates have anomalous commutation relations in the nonlocal case. Although the Hamiltonian appears normal in every case, the anomalous theories are characterized by zero (integrated) Lagrangian. Antiparticle conjugation is investigated in detail with attention to phase questions. For anomalous theories two types of conjugation are found, one of which nonlocally relates the field to a superposition ...

Generalized isospin matrices and projection operators for the reactions NI + π → NI′ + π and NI + NI′ → π + π☆

Abstract Generalized isospin matrices are defined and their properties investigated. These matrices provide a simple treatment of pion vertices whereby the isospin variables of the other particles are treated symmetrically. A phase-consistent set of projection operators and crossing matrices is constructed for the s , t , and u channels of the reaction N I + π → N I ′ + π , where N I , N I ′ are particles of any isospin I , I ′. Applications of these results are discussed briefly.