The matrix element for 
has the
structure
with
Here, 
denotes the polarization vector of the photon with
, whereas A,
V stand for two Lorentz
invariant amplitudes which occur in the general decomposition of the
tensors
The form factor A 
is related to the matrix element of
the axial (vector)
current in (1.4).
In appendix C we display the general
decomposition of 
,
for 
and provide also the link with the notation
used by the PDG [3] and in [4,5].
The term proportional to 
in (1.2) does not contain
unknown quantities -- it
is determined by the amplitude of the nonradiative decay 
. This part of the amplitude is usually referred to as ``inner
Bremsstrahlung (IB) contribution", whereas the term proportional to 
is called ``structure dependent (SD) part" .
The form factors are analytic functions in the complex 
-plane cut
along the positive real axis. The cut starts at 
for
A (at 
for V). In our phase convention, A and
V are real in the physical region of 
decays,
The kinematics of (spin averaged) 
decays needs two
variables, for which we choose the conventional quantities
In the K rest frame, the variable x (y) is proportional to the photon (charged lepton) energy,
and the angle 
between the photon and the charged lepton is
related to x and y by
In terms of these quantities, one has
We write the physical region for x and y as
or, equivalently, as
where
