In this section we make several predictions using the 
's from
table 1 in Ref. [2].
It is clear that new and more accurate data on
will allow for a better determination of 
and 
,
and may correspondingly modify our predictions. However, unless a
dramatic change in the values of these constants occurs, the
modified predictions will be within the errors that we give.
Whereas the slope 
was assumed to coincide
with the slope 
in the final analysis of the data in Ref.
[53], these two quantities may differ in the chiral representation.
Furthermore, our amplitudes allow us to evaluate partial and total decay
rates. In this section, we consider the slope 
and the total rates.
Table 5.3:
Approximations used to evaluate the total rates
in table 5.4. Use of
,
reproduces the one-loop results in table 5.4 to about 
.
Table 5.4:
Total decay rates in sec
. To evaluate the rates at one-loop accuracy,
we have used the 
's from table 1 in Ref. [2].
The final predictions
are evaluated with the amplitudes shown in table 5.3, using
. For the evaluation of the
uncertainties in the
rates see text.
The slope 
We consider the form factor 
introduced in (5.78) and
determine its slope
from the one-loop expression for G. The result is
. As the slope is a one-loop effect,
higher-order corrections may affect its value substantially. For this reason,
we have evaluated 
also from the modified form factor obtained by using the complete
resonance propagators (and the corresponding 
's), see Ref. [62].
The change is 
. We believe this to be a generous error
estimate and obtain in this manner
The central value indeed agrees with the slope 
in (5.37).
Total rates
Once the leading partial waves 
and 
are known from e.g.
decays, the chiral representation allows
one to predict the remaining rates within rather small uncertainties. We
illustrate the procedure for 
.
According to Eq. (5.24),
the relevant amplitude is determined by
and 
. The contribution from H is kinematically
strongly suppressed and completely negligible in all total
rates, whereas the contribution from R is
negligible in the electron modes. Using the chiral representation
of the amplitudes 
and 
, we find that the rate is
reproduced to about 1%,
if one neglects 
altogether and uses only the leading partial wave
in
the remaining amplitude, 
. From the measured [53]
form factor 
we then find 
sec
. Finally, we estimate the error from
where 
.
The final result for the rate is shown in the row ``final prediction" in
table 5.4, where we have also listed the tree and the
one-loop result, together
with the experimental data. The evaluation of the remaining rates is done in a
similar manner -- see table 5.3 for the simplifications used
and table 5.4 for the
corresponding predictions.
We have assessed an uncertainty due to contributions
from
in the following manner.
i) We have checked that the results barely change by using the tree level
expression for 
instead of its one-loop representation. We
conclude
from this that the uncertainties in 
do not matter. ii) The
uncertainty from 
is taken into account by adding to 
in quadrature the
change obtained
by evaluating 
at 
. iii) In 
decays,
we have also added in quadrature the difference generated by evaluating the
rate with 
MeV.
The decay 
has recently been
measured [60] with considerably higher statistics than before [3].
We display the result for the rate in the first column of table
5.4b.
The quoted errors correspond to the errors in the branching
ratio [60] and do not include the uncertainty in the total
decay rate quoted by the PDG [3].
Notice that the value for 
determined in [60] should be multiplied
with -1 [76].
