The theoretical predictions of 
form factors have a long
history which started in the sixties with the current algebra
evaluation of F, G, R and H. For an early review of the
subject and for references to work prior to CHPT we refer the
reader to [34] (see also [35]).
Here we concentrate
on the evaluation of the form factors in the framework of CHPT
[54,55,61,62].
The chiral representation of the form factors at leading order was originally given by Weinberg [63],
The next-to-leading order corrections are displayed
below, and the later sections contain an estimate of yet higher-order
contributions. Here we note that the
total decay rates which follow from Eq. (5.49) are typically
a factor of two
(ore more) below the data. As an example, consider the channel 
. Using (5.49), the total decay rate
becomes
1297
sec
,
whereas the experimental value is 3160
140 sec
[3].
The one-loop result for F [54],[55] may be written in the form
The contribution
denotes the unitarity correction generated
by the one-loop graphs which appear at order 
in the low-energy
expansion. Its expression will be given in appendix D.
The contribution 
is a polynomial in
obtained from the tree graphs at order 
. We
find
where
The remaining coefficients 
are zero.
The contributions 
contain
logarithmic terms, independent of 
and u:
The corresponding decomposition of the form factor G is [54],[55]
For the expression of 
see appendix D.
The polynomials
are
The remaining 
vanish. The logarithms contained in
are
The form factor R contains a pole part 
and a
regular piece Q. [Since the axial current acts as an interpolating
field for a kaon, the residue of the pole part is related to
the 
amplitude in the standard manner.] We write
According to (5.49), the Born terms 
are [63]
The one-loop corrections have been worked out in Ref. [62].
The unitarity corrections 
are displayed in the appendix D.
The residues 
and 
are
with
The remaining 
vanish. The logarithms in 
are
For the nonpole part Q we find:
with
The remaining 
vanish. The logarithms in 
are
The first nonvanishing contribution in the chiral expansion of the form factor H is due to the chiral anomaly [12]. The prediction is [64]
in excellent agreement with the experimental value [53] 
. The
next-to-leading order corrections to H have also been calculated
[65].
If the new low-energy parameters are
estimated using the vector mesons only, these corrections are small.
The results for F,G and R must satisfy two nontrivial
constraints: i) Unitarity requires that F,G and R contain, in
the physical region 
, imaginary
parts governed by S- and P-wave 
scattering [these
imaginary parts are contained in the functions
. ii) The scale dependence
of the low-energy constants 
must be compensated by
the scale dependence of 
and 
for all values
of
. [Since we work at order 
in the
chiral expansion, the
meson masses appearing in the above expressions satisfy the
Gell-Mann-Okubo mass formula.] We have checked that these
constraints are satisfied.
Because the one-loop contributions are rather large, one expects still substantial corrections from higher orders. In the following section, we therefore first estimate the effects from higher orders in the chiral expansion, using then this improved representation for the form factors in a comparison with the data.
To investigate the importance of higher-order terms, we employ the method
developed in Ref. [66]. It amounts to writing a dispersive
representation of the partial wave amplitudes, fixing the subtraction
constants using chiral perturbation theory. Here, we estimate the
higher-order terms in the S-wave projection of the amplitude 
,
because
this form factor plays a decisive role in the determination of 
and 
, and it is influenced by S-wave 
scattering which is known
[67] to produce substantial corrections.
Only the crossing-even part
contributes in the projection (5.67). The partial wave f has the following analytic properties:
, it is analytic in the complex 
-plane, cut along the real
axis for Re 
and Re 
.
, it is real.
, its phase coincides with the
isospin zero S-wave phase 
in elastic 
scattering,
follows from the relations
Since 
and 
have cuts at 
,
the claim is proven.
We introduce the Omnès function
where 
will be chosen of the order of 1 GeV below.
According to (5.69), multiplication by 
removes the cut in f for 
.
Consider now
where 
has only the left-hand (right-hand) cut, and introduce
Then v has only a right-hand cut, and we may represent it in a dispersive way,
We expect the contributions from the integral beyond 1 GeV
to be small.
Furthermore, 
is approximately real between 
and
1 GeV
, as a result of which one has
For given 
and 
, the form factor f is finally obtained
from
The behaviour of 
at 
is
governed by the large-|t| behaviour of 
and 
, see (5.70).
Therefore, instead
of using CHPT to model 
, we approximate the left-hand cut by resonance
exchange. To pin down the
subtraction constants 
and
, we require that the threshold expansion of f and 
agree
up to and including terms of order 
. For a specific choice of 
,
this fixes 
in terms of the quantities which occur in the one-loop
representation of 
and 
.
In the case where 
, f has then a particularly simple form at 
,
The details of the evaluation of 
and of 
may be found in Ref.
[62].
In the partial wave f, the effects of the final-state interactions are
substantial, because they are related to the I=0,S-wave 
phase
shift. On the other hand, for the leading partial wave in
, these effects are reduced, because the phase
is small at low energies. We find it more difficult to assess an
estimate for the higher-order corrections in this case -- we come back to this
point in the following sections.
We add a remark concerning the choice of the subtraction point in the Omnès function (5.71). To remove the right-hand cut in f, the modified Omnès function
would do as well, with any value of 
. To illustrate the
meaning of 
, we consider for simplicity the form factor f
in the case where 
. The choice 
then amounts
to the statement that, at threshold, there are no contributions from two loops
and beyond by fiat. We
consider this to be rather unlikely. (For a different opinion see
[69].) On the other hand, we have checked that our results do not
vary significantly if 
is taken of the order of the pion mass
squared, 
.
