The differential decay width is:
where the matrix element is
Figure: The opening angle between the two gamma rays
from the 
decays. The directions of the gamma rays deviate from collinearity by at
most 
due to the recoil energy of the 
.
The hadronic matrix element 
can only be a
vector as explained in chapter one. With the similar method used to write
down equation 1.42, the hadronic matrix element is:
with
Equation (
) follows from the fact that since there is no
spin, 
and 
are the only vectors available
to construct the matrix element. The CVC hypothesis requires that
Figure: The energy spectrum of one of the
gamma rays originating from the decay of the 
. The spread of this energy
reflects the spread in the recoil energy of the 
.
which leads to
In the exact isospin symmetry, 
hence
. The square matrix element is:
where
Using some trace theorems [Hal-84], it is a straightforward exercice to show that
In equation (), the terms proportional to 
are
negligible compare to the rest due to the electron mass. In addition,
since the range of 
is small 
,
is replaced by its value at 
. However, 
is
fixed by the CVC hypothesis:
as was demonstrated in chapter one. With all these simplifications, the matrix element becomes:
At this point, the decay width is written as:
where
and
The above equation can be written as:
One may write 
more elegantly as:
and 
need to be isolated and calculated. To that end, one
contracts ()
with 
and with
. The results are two equations which are quickly solve to
get:
and
After the isolation of 
and 
, the integrations can be
performed with the aid of the following properties of the
-function:
and
with
is the step function. The
integral () reduces to:
where k has been replaced by Q-p. One gets:
and
Putting all this together,
Integrating over the 
-function:
To get the limits 
and 
, one realizes that after the
-function integration, () becomes:
Since
equation () sets the range of integration over 
which
is determined as follows: solve () for
and use the
condition () to get:
One arrives at the discriminant
and finds the roots
from which the desired range of 
is obtained:
Finally, the integration () is carried out to get 
:
With similar approach,
The decay width can now be written as:
From 
,
and from the energy-momentum relation
where 
is the pion mass difference:
In addition,
Furthermore,
and
The decay width now appears as:
where small terms of the order 
have been
neglected. It is useful to introduce a variable z and a
parameter 
such that:
and
Finally, the decay width is obtained as:
where
It is straightforward to show that with 
,
The integration () can also be carried out:
Equation () together with the main decay mode of the 
,
that is 
, gives the branching ratio of
the pion
beta process: 
. The other decay channels of 
are
listed
below with their probability of occurrence in parentheses:
