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Next: 5.4 Isospin decomposition Up: 5 decays Previous: 5.2 Matrix elements

5.3 Decay rates

The partial decay rate for (5.1) is given by

 

The quantity is a Lorentz invariant quadratic form in F,G,R and H. All scalar products can be expressed in the 5 independent variables and , such that

 

Carrying out the integrations over the remaining variables in (5.13) gives [51]

 

where

 

The form factors F,G,R and H are independent of and . It is therefore possible to carry out two more integrations in (5.15) with the result

 

The explicit form of is

 

For data analysis it is useful to represent this result in a still different form which displays the and dependence more clearly [52]:

 

One obtains

 

where

 

The definition of in (5.21) corresponds to the combinations used by Pais and Treiman [52] (the different sign in the terms is due to our use of the metric ). The form factors agree with the expressions given in [52]. We conclude that our convention for the relative phase in the definition of the form factors in Eq. (5.12) agrees with the one used by Pais and Treiman. The comparison of (5.18) with [53, table II,] shows furthermore that it also agrees with this reference.

The quantity can now easily be obtained from (5.19) by integrating over and ,

 



Carlos E.Piedrafita