9.3 Solutions
511
It may be noted that the beams of sigma hyperons are available only in
recent times. This has been possible because of time dilation for high momenta
particles. Even beams of omega minus have been used.
9.33
dω
dE e
=
2G
2
F
m μ c
2
2 E
2
e
(2π) 3 (c) 6
1 −
4E e
3m μ c 2
This is the differential energy distribution of electrons from the decay of muon.
The distribution has a maximum, which is easily found out by differentiating
the right hand side of the above expression with respect to E e and setting it
equal to zero. We easily find the maximum of distribution to occur at E e =
m μ c
2
/2.
Now, if we regard the electron mass to be negligible then the maximum
value of the electron energy will occur when it is emitted in a direction opposite to the two neutrinos, its maximum energy being also m μ c
2
/2. A rough
plot of the electron energy spectrum is shown in Fig. 9.4.
Fig. 9.4
The orientation of momenta and helicities of the particles in the decay are
shown in Fig. 9.5.
Fig. 9.5 Pion rest-frame
sketch indicating sense of
spin polarization in pion