Question

Briefly discuss Yangs-Mills SU(2) gauge theory of the Strong nuclear force, the nature of the proton and neutron and the type and reason for the number of exchange particles of the theory. Describe how and why a free neutron will decay and what particles are involved in the transition.

Answer 1

SU(2) is the special unitary group of degree 2. In SU(2) gauge theory, the fields transform under the SU(2) gauge

The Lagrangian of the theory remains invariant under the above transformation. U is an element of SU(2) group and can be written as

The are the generators of the group.

According to the notion of isospin (or isobaric spin) invariance the proton and the neutron can be regarded as an isospin doublet.

Conservation of isospin is just the requirement of invariance under isospin rotations.

For SU(2) the number of generators is 3.

(Note: The number of generators of SU(N) group is N² - 1. For SU(2) it is 2² - 1= 3 and for SU(3) it is 3² - 1 = 8.)

Because of the gauge degrees of freedom, we can add gauge fields to the theory. The number of gauge fields in the theory is equal to the number of generators of the group. For SU(2) gauge, the number of gauge fields is 3.

The of exchange particles of the SU(2) theory are the gauge particles and hence its number is 3.

since the free neutron has a mass slightly greater than proton, it decays to proton and a W boson. The later then decays to an electron and an antineutrino to conserve the charge and lepton number