The amplitudes 
and 
have been worked out in the framework of
various approaches, viz., current algebra, PCAC, resonance exchange,
dispersion relations, ... . For a rather detailed review together with an
extensive list of references up to 1976 see [9]. Here, we concentrate
on the predictions of V, A in the framework of CHPT.
The amplitudes A and V have been evaluated [10,11] in the framework of CHPT to one loop. At leading order in the low-energy expansion, one has
As a consequence of this, the rate is entirely given by the IB contribution at leading order. At the one-loop level, one finds
where 
and 
are the renormalized low-energy couplings
evaluated at the scale 
(the combination 
is scale
independent). The vector form factor stems from the Wess-Zumino term
[12] which enters the low-energy expansion at order 
,
see Ref. [2].
Remarks:
-dependence. A nontrivial 
-dependence only occurs at the
next order in the energy expansion (two-loop effect, see the discussion
below). Note that the available analyses of experimental data of 
decays [7,4,5,8] use constant form factors
throughout.
has been pinned down
from other processes, Eq. (1.36) allows one to
evaluate
A,V unambiguously at this order in the low-energy expansion. Using 
and 
, one has
The result for the combination 
agrees with (1.30) within the
errors,
while 
is consistent with (1.32).
We display in table 1.3 the branching ratios 
(1.21)
which follow from
the prediction (1.37). These predictions satisfy of course the
inequalities found from experimental data (see table 1.2).
Table 1.3: Chiral
prediction at order 
for the branching ratios
. The
cut used in the last column is given in Eq. (1.20).
The chiral prediction gives constant form factors at order 
. Terms of
order 
have not yet been calculated. They would, however, generate a
nontrivial 
- dependence both in V and A. In order to estimate the
magnitude of these corrections, we consider one class of 
- contributions:
terms which are generated
by vector and axial-vector resonance exchange with
strangeness [9,13],
where V,A are given in (1.36). We now examine the effect of the
denominators in (1.38) in the region 
which has been explored in 
[4].
We put 
and evaluate the rate
where 
denotes the total number of 
decays
considered, and 
sec.
Figure 1.3: The rate 
in (1.39), evaluated with the form
factors
(1.38) and
.
The solid line corresponds to 
MeV, 
GeV. The dashed line is evaluated with
MeV, 
and the dotted line corresponds to
.
The total number of events is also indicated in each case.
The function 
is displayed in Fig. (1.3) for
three different values of 
and 
, with
.
The total number of events
is also indicated in each case. The difference between the dashed and the
dotted line shows that
the nearby singularity in the anomaly form factor influences the decay rate
substantially at low photon energies.
The effect disappears at 
, where 
. To minimize the effect of resonance exchange, the
large-x region should thus be
considered. The low-x region, on the other hand, may be used to explore the
-dependence of V and of A. For a rather exhaustive discussion
of the relevance of this 
- dependence
for the analysis of 
decays we refer the reader to Ref.
[9].
