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where **k** has been replaced by **Q-p**.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:

Mon Jan 15 14:57:06 MET 1996