Prior to CHPT, the most detailed calculations of 
amplitudes were performed by Fearing, Fischbach and Smith
[50] using current algebra techniques.
In the framework of CHPT, the amplitudes are given by (4.17) and (4.18) to leading order in the chiral expansion.
There are in general three types of contributions [44]:
anomaly, local contributions due to 
and loop amplitudes.
Figure 4.2: Loop diagrams (without tadpoles) for 
at 
. For
, the photon must be appended on all charged lines and
on all vertices.
The anomaly contributes to the axial amplitudes
The loop diagrams for 
are shown in
Fig. 4.2. We first write the 
matrix element
in terms of three functions 
which will also appear
in the invariant amplitudes 
. Including the contributions
from the low-energy constants 
in 
, the
matrix element 
is given by
is a scale independent coupling constant and we have traded
the tadpole contribution together with 
for 
in
. The sum over I corresponds to the three loop diagrams
of Fig. 4.2 with coefficients 
displayed in
table 4.3.
Table 4.3: Coefficients for the 
loop amplitudes
corresponding to the diagrams I=1,2,3 in Fig. 4.2.
All coefficients 
must be divided by
.
We use the Gell-Mann--Okubo mass formula
throughout to express 
in terms of 
.
The functions 
and 
can be found in App. B.
The standard 
form factors 
as given in the previous subsection [24] are
It remains to calculate the infrared finite tensor amplitude
. The invariant amplitudes 
can be
expressed in terms of the previously defined functions 
and of
additional amplitudes 
. Diagrammatically, the latter
amplitudes arise from those diagrams in Fig. 4.2 where the
photon is not appended on the incoming 
(non-Bremsstrahlung
diagrams). The final expressions are
The amplitudes 
in Eq.(4.22) are given by
The function 
is given in App. B.
All the invariant amplitudes 
are
real in the physical region. Of course, the same is true for the
matrix element 
.
The 
amplitude has a very similar structure. Both the
matrix element 
and the infrared finite
vector amplitude 
can be obtained from the
corresponding quantities 
and
by the following steps:
and 
;
by 
in 
;
for
listed in table 4.4;
and 
by a
factor 
.
Table 4.4: Coefficients for the 
loop amplitudes
corresponding to the diagrams I=1,2,3 in Fig. 4.2. All
coefficients 
must be divided by 
.
In calculating the rates with the complete amplitudes of the previous subsection, we use the same cuts as for the tree level rates in Subsect. 4.2:
The physical values of 
and 
are used in the amplitudes.
is calculated from the Gell-Mann--Okubo mass formula. The
values of the other parameters can be found in Ref. [2]
and in appendix A.
The results for 
and 
are displayed in
tables 4.5 and 4.6, respectively. For comparison, the
tree level branching ratios of table 4.1 and the rates for
the amplitudes without the loop contributions are also shown. The
separation between loop and counterterm contributions is of course
scale dependent. This scale dependence is absorbed in the scale invariant
constants 
defined in Eqs.(4.20),
(4.23). In other words, the entries in tables 4.5,
4.6 for the amplitudes without loops correspond to setting
all coefficients 
in tables 4.3,
4.4 equal to zero.
Table 4.5: Branching ratios and expected number of events at DA
NE
for 
.
Table 4.6: Branching ratios and expected number of events at DA
NE
for 
.
