CHAPTER 1
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 nucleons, we may denote the isospin operator for the first nucleon by [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] and the isospin operator for the second nucleon by [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] so that the total isospin operator for the system is as follows:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-5)
Two nucleons, which each have isospin ½, can form eigenstates of total isospin of eigenvalue either 0 or 1. We denote the first and second nucleon by superscripts (1) and (2) respectively, and then the normalized eigenstates are:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-6)
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-7)
We have remarked that the interaction of two nucleons is charge-independent; rotations in isospin space are the transformations which bring about changes in the charge states, and charge-independence may be restated as the hypothesis that nuclear forces are invariant under rotations in isospin space. It is this isospin invariance that makes isospin an essential physical concept. The Hamiltonian that describes a 2-nucleon system contains isospin operators; the general isospin invariant form is:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-8)
where H' and H" do not involve [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.]. It is immediately evident, on using the fact (shown in Chapter 6) that the [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] sandwich between isospinors corresponding to nucleon i transforms as a vector under isospin rotations, that eq. (1-8) is invariant under rotations in isospin space and, from eqs. (1-3) and (1-4), it is the only such form.
Historically also, isospin of the pions originated in low energy nuclear physics. Yukawa (1935) predicted the existence of a boson whose exchange between nucleons would give rise to the internucleon force and from the relation between the range of the force and the mass of the exchanged boson, the latter was predicted to be in the region of 100-200 MeV/c2. With the discovery (Anderson and Neddermeyer, 1937) of the muon in cosmic rays of mass ~100 MeV/c2, Yukawa's particle was thought to have been discovered, since the fact that the muon spin is ½ was not at that time known. Since the muons are charged, the problem was to find a meson-nucleon interaction involving charged mesons that yet gave rise to charge-independent nuclear forces. The answer (Kemmer, 1938) was to postulate a triplet of mesons forming a vector in isospin space. Anticipating the discovery of the pion in 1947 (Lattes et al, 1947) these are the π-, π0, π+ having charges -1, 0, +1 respectively, corresponding to third component of isospin being -1, 0, +1 respectively. The internucleon forces were postulated to arise from the virtual emission of a pion at one nucleon and its absorption at the other nucleon. The emission or absorption of pions from nucleons is the basic process, and it is clear that if this basic πNN interaction is invariant under rotations in isospin space, so will be the internucleon force. The form of this interaction in isospin space can be deduced as follows.
A pion is represented by a vector in isospin space, so that the nature of the pion, that is, whether it is positive, negative, or neutral, is denoted by a vector [??] or equivalently three numbers: [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII]. Now we consider the basic πNN interaction, N [right arrow] N + π, and let the initial nucleon have isospinor N1, the final nucleon have isospinor N2 and the pion have isovector [??]. Then the form:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-9)
(† = Hermitian conjugate) is invariant under rotations in isospin space. Expression (1-9) is the unique isospin invariant form of the πNN interaction.
Since for example a proton can only transform to a proton with emission of a π, it is easily seen that the conventions of eqs. (1-1) and (1-2) (together with the further convention that the pion triplet transforms like the angular momentum triplet Y1m (θφ)) imply:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-10)
with corresponding isospin operators for the pion (see also Chapter 6) being:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-11)
In a pion-nucleon scattering experiment the initial state consists of one pion with isospin 1 and one nucleon with isospin ½, so that any initial state can be expressed as a superposition of a state of total isospin ½ and total isospin 3/2. These states are as follows:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-12)
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-13)
There is good evidence that all the strong interactions are invariant under isospin rotat10ns so that for example initial states which have total isospin 3/2 only make transitions to final states with total isospin 3/2, except for possible small pertu rbat10ns, usually unim portant in the study of strong interactions, caused by electromagnetic and weaker interact10ns. Consequently eigenstates of total isospin such as that shown in eqs (1-12) or (1-13) are the appropriate ob1ects of study in pion-nucleon interactions.
As an explicit test of isospin invariance we mention the reaction
d + d [right arrow] He4 + π0 (1-14)
a forbidden reaction since the deuteron and He4 have isospin zero and the pion has isospin 1. Akimov et al. (1960) and Poirier and Pripstein (1963) have shown that the cross sect10n for this react10n is less than 1/100 of the expected rate if isospin were not conserved.
Charge Conjugation and G-parity
The operator of charge conJugation, C, transforms particle into anti-particle so that
C | p > = | [bar.p] > C | p > = | [bar.n] > (1-15)
Both strong and electromagnetic interactions are invariant under charge conjugation (Sakurai, 1964) and the pion properties are
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-16)
We will show that eq. (1-16) follows from the invariance of strong interactions under P, C and isospin transformations, by noting that the pion can transform, through the fundamental πNN transformation into a nucleon-anti-nucleon system whose properties under C can be deduced
[I3, I1 ± i I2] = ± (I1 ± i I2)
so that (I1 ± i I2) applied to a state of an isospin multiplet with eigenvalue of I3 equal to I3' give a state of the same isospin multiplet with eigenvalue of I3 equal to I3' ± 1. Applying these to the proton-neutron multiplet:
| p > = (I1) + iI2) | n >, | n > = (I1) + iI2) | p > (1-17a)
To maintain the linear relation:
Q/e = I3 + 1/2 B (1-18)
between charge Q isospin I3 and baryon number B of nonstrange particles, the third component of isospin of [bar.n] must be + ½ and of [bar.p] must be - ½. Consequently, with a certain phase convention:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-17b)
C evidently anti-commutes with I3, from eq. (1-18), and from eqs. (1-17) and (1-15) it anti-commutes with I1 and commutes with I2:
CI1 = -I1 C, CI2 = I2 C, CI3 = -I3 C. (1-19)
With our phase conventions (-[bar.n], [bar.p]) transforms as (p, n) under isospin transformation, and the nucleon-anti-nucleon isospin wave function corresponding to π0 is thus [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.] Also by parity and angular momentum conservation it must be in a singlet spin state and orbital S-state. Consequently its wave function, anti-symmetric from the Pauli principle, is as follows:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-20)
where α, β are the two possible spin states and R(1, 2) is a symmetrical spatial wave function. Obviously equation (1-20) is an eigenstate of C of eigenvalue +1, which gives the π0 result of eq. (1-16). The π± results can be obtained similarly.
Since C is not a good quantum number for charged particles (in general terms it anti-commutes with I3), it is useful to define the G-parity, which commutes with [??]:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-21)
as a product of charge conjugation and a rotation of 180° about the 2-axis in isospin-space. Such a rotation converts the pion isovectors as follows:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII]
Combined with eq. (1-16) these give:
[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (1-22)
so that the pion has G-parity -1, and obviously any odd number of pions has G-parity -1 and any even number G-parity +1. G-parity is conserved in strong interactions, but not in electromagnetic interactions, and since it commutes with I, it has the same value for any member of the same isospin multiplet.
