Nuclear Beta Decay

Abstract

Up to now we have developed the theory of the weak nuclear interaction at the level of the quarks, that is to say, of the constituents of the hadrons. In order to describe the weak interactions of the hadrons themselves we must in addition know how the hadrons are made up of quarks, that is, we must know the wave functions of the quarks within the hadrons. This problem has not yet been completely solved, but there are a number of models which reflect some properties of the hadrons quite well.¹ One of the best known of these models is the so-called MIT bag model,² which we shall now use to calculate the ratio g _A /g _V for the nucleon. In the case of the MIT bag model, one assumes that the quarks can move freely within a sphere of radius R (Fig. 7.1). Since the nucleons represent the lowest baryon states, the quarks should have wave functions without angular or radial nodes. The most general solution of the Dirac equation,

$$Phi \; = \;N\left( {_{\frac{{\sigma .\hat p}} {{E + m}}\Phi }^{\quad \Phi }} \right)\;with\;\Phi \; = \left( {_{{\Phi _2}}^{{\Phi _1}}} \right)$$

((7.1))

is given in this case by

$${\Phi _ \pm }\left( r \right)\; = N\;\left( {_{\frac{{iK}} {{\left( {E + m} \right)}}ji\left( {kr} \right){\sigma _r}X \pm }^{jo\left( {kr} \right)X \pm }} \right),$$

((7.2))

where σ _r = σ · r/r, E² = k² + m² and χ_± are the unit two-component spinors (see Exercise 7.1). The spherical Bessel functions are explicitly

$$jo\left( x \right) = \frac{{\sin \;x}}{x},\;j1\left( x \right) = \frac{{\sin \;x}}{{{x^2}}} - \frac{{\cos \;x}}{x}.$$

((7.3))