Another solution to the problem

is

where are the 2x2 Pauli Matrices. The spinor can be written as a 3 component spinor –

it can be shown that the components are spin ½ wavefunctions.

The equation

for and choosing spherical co-ordinates, the above equation is

thus The wavefunction is effectively a doublet of a fermion and anti-fermion.

The 2 simultaneous equations are

Assume that the spinors are related by a phase transformation –

Where f is real. The following equation arises

It follows that

where Z is the wavelength of the Z boson

It follows that

with the contraints

The masses of the fermion are

gives

For m=91GeV one finds

Thus is follows that the Z can decay into a b quark antiquark pair.