Weak Interactions of Pions and Muons

Abstract

Let me begin with a brief summary of the present status of weak interaction theory. The basic interaction is described in terms of charged currents

$${{H}_{W}}=\frac{G}{\sqrt{2}}{{g}_{\lambda }}{{g}_{\lambda }}^{+}$$

(1)

$${{g}_{\lambda }}={{\overline{v}}_{\mu }}{{\gamma }_{\lambda }}\left( 1+{{\gamma }_{5}} \right)\mu +\overline{{{v}_{e}}}{{\gamma }_{\lambda }}\left( 1+{{\gamma }_{5}} \right)e+{{J}_{\lambda }}$$

(2)

where J is the hadronic current assumed to be of the Cabibbo form

$${{J}_{\lambda }}=\cos \theta \left( {{J}_{\lambda }}^{1}+i{{J}_{\lambda }}^{2} \right)+\sin \theta \left( {{J}_{\lambda }}^{4}+i{{J}_{\lambda }}^{5} \right)$$

(3)

where the superscripts define components of an SU₃ octet. Here we will be interested only in the strangeness-conserving current cosθ(J_λ ¹ + iJ_λ ²) where 1,2 are isospin indices. Beyond the assumed isovector property we desire to specify the hadronic current as com-pletely as possible. This is much harder than in the case of the leptonic current (the first two terms of Eq.(2)) because we do not know what elementary particles to use to describe the current. A standard assumption is that this current has the basic properties described by the quark representation

$$\cos \theta \left( {{J}_{\lambda }}^{1}+i{{J}_{\lambda }}^{2} \right)\equiv {{V}_{\lambda }}+{{A}_{\lambda }}\propto \overline{p}{{\gamma }_{\lambda }}\left( 1+{{\gamma }_{5}} \right)n$$

(4)

where p,n are the field operators of the quarks. Unfortunately, this leads to very few conclusions without a knowledge of the strong interactions of quarks.