The energy of the muon in this 2-body decay is given by
equation (
) with 
. Neglecting the neutrino
mass,
the muon kinetic energy in the rest frame of the pion is:
The matrix element of the decay is:
In this case, the vector contribution to the hadronic matrix element must vanish because of parity [Geo-84]. The axial vector contribution is parametrized as:
One notes in passing that if the pion were massless, the axial current would be conserved since:
in which case the pions could be identified as the Goldstone bosons associated
with the spontaneous breaking of chiral symmetry. Using Dirac equation and
zero mass for the neutrino, equation () becomes:
Summing over final spin states and using some trace theorems:
The above expression together with the phase space factor gives, after integrating over fermion energies, the decay rate as:
In the case of a massless muon
(relativistic limit for instance), this decay would be forbidden: in
the relativistic limit, leptons are left-handed and antileptons are right-handed
due to the V-A structure of the interaction and in analogy to the case of
the neutrino discussed in chapter one. By considering the conservation of
helicity and angular momentum, this decay is forbidden.
