We start with the process (5.1).
The full kinematics of this decay requires five variables.
We will use the
ones introduced by Cabibbo and Maksymowicz [51]. It is
convenient to
consider three reference frames, namely the 
rest system
, the
center-of-mass system 
and the
center-of-mass system 
. Then
the variables are
, the effective mass squared of the dipion system,
, the effective mass squared of the dilepton system,
, the angle of the 
in 
with respect to the dipion line of flight in 
,
, the angle of the 
in 
with respect to the dilepton line of flight in
, and
, the angle between the plane formed by the pions
in
and the corresponding plane formed by the dileptons.
The angles 
, 
and 
are displayed in
Fig.
5.1. In order to specify these variables more
Figure 5.1: Kinematic variables for 
decays. The angle 
is
defined in 
in 
and 
in
.
precisely, let
be the three-momentum of the 
in
and
the three-momentum of the 
in
.
Furthermore, let 
be a unit vector along the direction
of flight of the dipion in
, and 
a unit vector along
the projection
of 
perpendicular to 
,
The vectors 
, 
and 
are indicated in Fig. 5.1. Then, one has
The range of the variables is
It is useful to furthermore introduce the following combinations of four vectors
together with the corresponding Lorentz invariant scalar products
with
Below we will also use the variables
These are related to 
and 
by
