INTRODUCTION

1.1. THE FIELD OF STUDY

The proton, p, is a positively charged spin ½ particle of mass 938.3 MeV/c2 and the neutron, n, is a neutral spin ½ particle of mass 939.6 MeV/c2. They each have, by conventional definition, positive parity, P = +1 (Sakurai, 1964). The positively and negatively charged pions, π±, are spin 0 particles of mass 139.6 and the neutral charge pion, π0, is a spin 0 particle of mass 135.0. The parity of all three pions is found to be negative, P= -1 (Sakurai, 1964; Nishijima, 1963; Källen, 1964). Pions and nucleons interact strongly, and this interaction is responsible for at any rate the longer range part of the nucleon-nucleon force.

It is the investigation and understanding of these interactions, and of the concomitant structure of the pions and nucleons themselves, together with the broader implications for elementary particles as a whole, that is the field of study of this book. The typical experiment is the collision of a pion with a nucleon target, with the detection of the reaction products. These may be a single pion and a nucleon (elastic or charge-exchange scattering) or many pions and a nucleon, or more rarely it may involve strange particles (strangeness exchange reactions) or baryon-anti-baryon pairs. In all cases the experiments carry implications for the interactions in the collision. Generally, the smaller the number of final particles (except in the special case of the measurement of total cross sections), the less unobvious the implications are. For example, the charge exchange reaction π- + p [vector] π0] + n lends itself to an interpretation in terms of particle (Reggeized ρ- meson) exchange between the pions and the nucleons, while the behavior of the diffraction peak in high energy elastic scattering gives indications of particle structure and the forces between particles or their possible component parts. In pion-nucleon collisions at lower energies one can detect the formation of long-lived intermediate states or resonances, whose width is of the order of 100 MeV, and whose spin and parity is determined by partial wave analysis of elastic and charge exchange scattering or, possibly, through other reactions. These resonances are granted the status of elementary particles and at the same time may be regarded in some sense as excited states of the nucleons so that determination of their energy levels, widths, spins, parities, and other quantum numbers may give important indications on the classification and symmetries of elementary particles and even on their structure. As extended to include strange particles, the study of these energy levels is rather naturally known as baryon spectroscopy.

The two nucleons belong to a set of JP = 1+/2 baryons (shown in Table 1-1) classified as an octet of the SU3 group. The three pions belong to a set of nine JP = 0- mesons (shown in Table 1-2) classified as an octet-singlet mixture under the SU3 symmetry. The SU3 symmetry though broken, as evinced by the large mass differences within multiplets shown in Table 1-1, implies certain relationships between the various pseudoscalar meson-baryon interactions, which though not holding by any means exactly, are observed to exist to a certain approximate degree. Also the structures of the various baryons, as revealed for example in their masses or electromagnetic interactions as well as in meson-baryon interactions, are related, approximately, through SU3, as are the structures of the mesons.

Consequently any conclusions about the pion-nucleon interaction will have certain consequences for, and relations with, other pseudoscalar meson-baryon interactions. Where we discuss the classification of resonances we will naturally make the relations very explicit and complete. In other cases, for example high energy scattering, the detailed discussion will be confined to pion-nucleon scattering, and the application of the methods to processes such as kaon-nucleon scattering will be left implicit. The pion-nucleon interactions form a convenient experimental and theoretical subsystem in which many features of strong interactions can be studied, and techniques developed and expounded with a minimum of discursive interruption.

Among the pions and nucleons only the proton is stable. The neutron undergoes β-decay, n [vector] p + e + ve, with a lifetime of (1.01 ± 0.03) x 103 sec; the charged pions decay by π± with a lifetime of (2.55 ± 0.03) x 10-8 sec; the principal decay mode (98.8%) of the π0 is π0

1.2. ISOSPIN

Historically, isotopic spin has its origin in low energy nuclear physics, where it was observed that the neutron-neutron, neutron-proton, and proton-proton forces were approximately equal. In these circumstances it was natural to regard the neutron and proton as two states of the same particle, the nucleon. In many low energy nuclear physics applications the charge state of the nucleon is an irrelevant internal quantum number. This of course is only approximately true since there are perturbations arising from electromagnetic interactions, in particular the neutron-proton mass difference, itself believed to be due to electromagnetic interactions, and the proton-proton electromagnetic interaction, responsible for the tendency to neutron excess in high-mass nuclei.

The isotopic spin formalism for the nucleons was built on a simple analogy with the nonrelativistic theory of a spin ½ particle, which also has two possible internal states. (As we shall see, especially in Chapter 6, the analogy is mathematically exact since both the nucleon and the spin ½ particle are bases of a 2 × 2 representation of the group SU2). On this analogy the proton and neutron are described by the spinors:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII]

Analogously to the Pauli spin matrices we write the isospin matrices:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII]

which act on the spinors in eq. (1-1). The proton state is an eigenstate of τ3 of eigenvalue +1, and the neutron state is an eigenstate of τ3

where [member of]ijk = ±1 if i, j, k are an even or odd permutation respectively of 1, 2, 3 and [member of]ijk = 0 if any two of i, j, k are equal. It is also easily verified that:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-4)

If we have a state of 2...