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Next: Decay Up: Pion Beta Decay Previous: Kinematics

Pion Beta Decay Rate

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:




Equation (gif) 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:




Using some trace theorems [Hal-84], it is a straightforward exercice to show that


In equation (gif), 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:






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 (gif) with and with . The results are two equations which are quickly solve to get:




After the isolation of and , the integrations can be performed with the aid of the following properties of the -function:






is the step function. The integral (gif) reduces to:


where k has been replaced by Q-p. One gets:





Putting all this together,


Integrating over the -function:


To get the limits and , one realizes that after the -function integration, (gif) becomes:




equation (gif) sets the range of integration over which is determined as follows: solve (gif) for and use the condition (gif) to get:


One arrives at the discriminant


and finds the roots


from which the desired range of is obtained:


Finally, the integration (gif) 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,




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:




Finally, the decay width is obtained as:




It is straightforward to show that with ,


The integration (gif) can also be carried out:


Equation (gif) 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:


next up previous contents
Next: Decay Up: Pion Beta Decay Previous: Kinematics

Bernward Krause
Mon Jan 15 14:57:06 MET 1996