The theoretical prediction of 
form factors has a long history,
starting
in the sixties with the current algebra evaluation of 
. For
an
early review of the subject and for references to work prior to CHPT
evaluations
of 
we refer the reader to [34] (see also Ref.[35]).
Here we concentrate on the
evaluation of the form factors in the framework of CHPT. We restrict our
consideration to the isospin symmetry limit 
, as a result of which
one has
In Ref. [24], the vector current matrix elements 
have been calculated up to and
including terms of order 
and of order 
and 
in
the invariant form factors. For reasons which will become evident below, we
consider here, in addition to the 
form factors, also
the electromagnetic form factor of the pion
The low-energy representation for 
[24,36] and
[24] reads
The quantity 
is a loop function
displayed in appendix B. It contains the low-energy constant 
.
The indices attached to 
denote the masses running in the loop.
Since 
is the only unknown occurring in 
and
in 
, we need experimental information on the slope of one of these two
form factors to obtain a parameter-free low-energy representation of the
other.
The analogous low-energy representation of the scalar form factor is
The function 
is listed in appendix B, and
and 
stand for
The measured value [26] 
may be used to
obtain a parameter-free prediction of the scalar form factor 
.
In the spacelike interval 
MeV the low-energy representation
(3.27) for the electromagnetic form factor 
is very well
approximated by the first two terms in the Taylor series expansion
around t=0,
Likewise, the linear approximation
reproduces the low-energy representation (3.27) very well, see Fig. 3.1.
Figure 3.1: The vector and scalar form factors 
and 
.
This is in agreement with the observed Dalitz plot distribution, which is consistent with a form factor linear in t. The charge radii are
where
To evaluate these relations numerically, we use the measured charge radius of the pion:
as input and obtain the prediction
in agreement with
the experimental results (3.22), (3.23)
.
From this (and from the considerably more detailed discussion in Ref.
[24]), one concludes, in agreement with other
theoretical investigations [38], that the measured charge radii
and 
are consistent with the low-energy
prediction.
In the physical region of 
decay the low-energy representation
(3.28) for the scalar form factor is approximated by the linear
formula
to within an accuracy of 1 %. (See Fig. 3.1). The curvature generated
by higher-order
terms is also expected to be negligible in the physical region of the
decay [24]. For the slope 
one obtains
where
This (parameter-free) prediction is a modified version of the
Dashen-Weinstein
relation [39], which results if the nonanalytic contribution
is dropped. Dashen, Li, Pagels and Weinstein [40] were the
first to point out that the low-energy singularities generated by the Goldstone
bosons affect this relation. The modified relation is formulated as a
prediction for the slope of 
at the unphysical point 
. Their expression for this slope however has two shortcomings: (i) it
does not account for all corrections of order 
; (ii) The slope at
differs substantially from the slope in the physical region of the decay
[24,41], see Fig. 3.2.
Figure 3.2: The normalized slopes of the vector and the scalar form factors.
Curve 1: the normalized slope 
. Curve 2: the
normalized
slope 
. Near the 
threshold 
,
the vector form factor behaves as 
, whereas
. The slope of the scalar form
factor is therefore singular at 
.
Algebraically, the correction 
is of the same order in the low-energy
expansion as the term involving 
. Numerically, the correction is
however small: 
reduces the prediction by 11 %. With 
the low-energy theorem (3.37) implies
where the error is an estimate of the uncertainties due to higher-order contributions. The prediction (3.39) is in agreement with the high-statistics experiment [28] quoted in (3.23) but in flat disagreement with some of the more recent data listed in (3.24).
In the formulation of Dashen and Weinstein [39], the Callan-Treiman
relation [42] states
that the scalar form factor evaluated at 
differs from
only by terms of order 
: the quantity
is of order 
. Indeed, the low-energy
representation (3.28) leads to
Numerically, 
. The
Callan-Treiman relation should therefore hold to a very high degree of
accuracy. If the form factor is linear from t=0 to 
then
the slope must be very close to
in agreement with (3.39) and with the experimental result of Ref.
[28], but in disagreement with, e.g., the value found in Ref.
[30]. We see no way to reconcile the value 
with
chiral symmetry.
