(1.1)
where k=mc/h .
The potential is a solution of the time independent Klein-Gordon equation:
(1.2)
Yukawa proposed that, like the photons that were responsible for carrying the electromagnetic force, a particle existed which carried the nuclear force. This particle was exchanged between nucleons and was responsible for binding a nucleus together.
The relation between the mass of the force-carrier particle and that force's range can be derived using the uncertainty relationship between energy and time [Delta]E[Delta] t~ h which leads to
The maximum distance a particle can travel in time [Delta] t is given by
(1.3)
where the range R would be estimated by the range of the potential of interest. This relation is consistent with the zero mass, infinite range of photons, which mediate the electromagnetic interaction.
The relation (1.3) can also be derived using the assumption that k-1 (k being from (1.1)) should be on the order of the range of the nuclear force (~1 fm),
(1.4)
Yukawa used the relation (1.4) to estimate the mass of a particle mediating the nucleon-nucleon interaction to be about 200 MeV/c2. He named these particles "mesotrons" with "meso" meaning "middle" since the mass of these was between the masses of electrons and nucleons.
By the early 1940's, experiments using cosmic rays indicated the existence of a particle whose mass was ~100 MeV/c2. This was, at first, assumed to be Yukawa's meson. However, the unusually long paths these particles traveled through matter were inconsistent with the predicted, strongly interacting particles. In 1947, C.F. Powell et al. gave experimental evidence of a heavier particle (~150 MeV/c2) which decayed into a lighter one (~100 MeV/c2) [Pow 59]. The heavier particle was then correctly identified as the pion while the lighter particle (which had come to be known as the mu meson) was (eventually) identified as a second-generation member of the lepton family of particles. The term meson now has a more general meaning and is given to any particle constructed of a quark and an anti-quark.
Pions exist in three varieties most easily identified by their electric charge. These are p + , p - , and p 0 with charges +1, -1, and 0, respectively, in units of the fundamental electric charge e. The pions represent an SU(2) triplet and singlet through combinations of up and down quarks. They are all spin zero pseudoscalars. The p + is the antiparticle of the p - and vice versa, while the neutral pion is its own antiparticle. The charged pions have a mass of 139.56995 ± 0.00035 MeV/c2 while the neutral pion has a mass of 134.9764 ± 0.0006 MeV/c2. [PDG 96]
The p + has a number of decay channels as seen in Table 1.1. The p - decay channels are just the charge conjugate of those of the p + . The neutral p 0 has several decay modes with the strongest (~98.8%) being into two photons. The pion beta decay project (pibeta) at the Paul Scherrer Institute (PSI) will attempt to make a precise measurement with an uncertainty of less than 0.5% of the p + -> p 0 e+ n decay rate relative to the p + -> e+ n decay rate.
Table 1.1: The decay modes of the p +. (The decay modes of the p - are the charge conjugates of those for the p +.) [PDG 96]
| Decay
mode
|
Fraction
(
 )
|
Confidence
|
|
(99.98770
± 0.00004)%
|
|
|
(1.24 ± 0.25 ) x 10-4
|
|
|
(1.230 ± 0.004 ) x 10-4
|
|
|
(1.61 ± 0.23 ) x 10-7
|
|
|
(1.025 ± 0.034 ) x 10-8
|
|
|
(3.2 ± 0.5 ) x 10-9
|
|
|
<5 x 10-6
|
90%
|
| Lepton
Family
|
(LF)
or Lepton number (L)
|
violating
modes
|
|
<1.5 x 10-3
|
90%
|
|
<8.0 x 10-3
|
90%
|
|
<1.6 x 10-6
|
90%
|