In this section we would like to present results
for some extensions of the minimal parametrization we have used
so far: (1) we include a p-wave term in F, and simultaneously
adopt a notation consistent with the partial wave expansion of
Ref. [1]; (2) we consider an 
dependence in f, g, etc.;
(3) we consider two parametrizations of 
in terms
of s-wave scattering length 
to
give a first indication on what errors can be expected on this
parameter.
F thus becomes
where 
, 
and X are defined in Ref. [1].
Our previous parametrization is thus equivalent to this one
for 
.
For the 
dependence, chiral perturbation theory to one loop
predicts [9]
approximately the same slope for s_ 
and s_l 
, or, more precisely,
To be more model-independent, and to have a separate determination
of the s_ 
and s_l 
slopes, we choose the parametrization
We then find, for 300000 events,
to be compared with table 1. We have displayed the errors for
the old parameters as well as the added ones, as some of them
have increased slightly. We have taken central values
and 
(which maintains the normalization
unchanged); if other parameters are preferred it should be remembered
that the absolute errors of 
and the
slope remain essentially constant.
For 
we have first considered the parametrization
used [10]
by the previous experiment, to compare our estimated errors
with the ones they determined. They use
where 
, the difference between the s-wave slope
and the p-wave scattering length. There is moreover
a possible relation between b and 
For 30000 events (to compare with Ref. [5]) we find
, to be compared with 0.05 in Ref. [5],
if we use both eqs. 21 and 22.
If we use only eq. 21 we find 
and
, to be compared with 0.11 and 0.16 in
Ref. [5]. We have used the central values 
in
the first case, and 
, 
in the second case,
as found in Ref. [5]. So, in one year running at DA
NE at
a luminosity of 
cm
s
, we
expect a factor of five improvement in the error on the
scattering length, meaning that DA
NE should be able
to determine if the existing discrepancy
between measurements and predictions for 
is statistically
significant.
We have also considered a more recent parametrization, due to Schenk. [11] He gives
where 
s_ 
and
Here I denotes isospin, l angular momentum, and 
is the
Kronecker delta. We need to calculate 
and 
; 
and 
(the values at which the phase shifts should pass through 
)
are the squares of the 
and 
meson masses.
There are far too many parameters here to be able to
determine them
by K_e4 
measurements alone; to begin with, we fix the higher order
coefficients 
and 
to zero.
The remaining set 
, 
, 
and 
is still too large: if we plot 
versus s_ 
, we find that
while changing 
from its central value of 0.037 to a new
value of 0.087 changes 
essentially uniformly by 15
percent, changing instead 
from its central value of 0.24
to 0.19 has the same effect, to within about 1 percent.
Moreover, with appropriate variations in 
and 
,
one can quite well mock up a variation in 
(or 
,
naturally).
Thus, one can only hope to determine two independent parameters
without recourse to other experiments or other theoretical constraints.
The MLM quantifies these conclusions. Assuming Schenk's central
values (
, 
, and above), 
or 
can be determined to 0.03 (for Rosselet statistics of 30,000 events),
or 
or 
to 0.04, if only one parameter is considered, and
the rest held fixed. If one determines 
and 
simultaneously,
holding the rest of the parameters fixed,
the correlation is 0.71 (defined as the covariance of the two parameters
divided by the 
's
for each, 1 for a maximally correlated pair of
variables), and the errors are 
and 
respectively.
If one determines instead 
and 
simultaneously,
the correlation is 
and the errors are 
and 
;
for 
and 
the correlation is 
and the
errors are 
and 
. To determine more than two parameters
simultaneously is not possible.
