To illustrate the usefulness of forthcoming more accurate 
data, we
determine here the low-energy constants 
and 
from data on 
decays and on 
threshold parameters, using the
improved S-wave amplitude f set up above.
For a comparison with earlier work [55] we refer the reader
to Ref. [62].
We perform fits
to 
as given in (5.37) and to
the 
threshold
parameters listed in table 5.2. We introduce for this purpose the
quantities
where the factor 
appears because G is expanded
in derivatives of Legendre polynomials.
Below, we identify [
, 
] with [
,
], which depend on 
. Furthermore, we compare
the slope 
with
which depends on both 
and 
. We use these dependences to
estimate systematic uncertainties in the determination of the low-energy
couplings. [In future high-statistics experiments, the 
-dependence of the
form factors will presumably be resolved. It will be easy to adapt the
procedure to this case.]
We have used MINUIT [70] to perform the fits. The results for the
choice 
are given in table 5.2.
Table 5.2:
Results of fits with one-loop and unitarized form factors, respectively. The
errors quoted for the 
's are statistical only. The 
are given in
units of 
at the scale 
, the scattering lengths 
and the slopes 
in appropriate powers of 
.
In the columns denoted by ``one-loop", we have evaluated 
and
from the one-loop representation given above
.
In the fit with the unitarized form
factor (columns 3 and 5), we have evaluated 
from
Eqs. (5.76), inserting in the Omnès function the
parametrization of the 
S-wave phase shift proposed by Schenk
[72, solution B,]. For the form factor G,
we have again used the one-loop representation.
The statistical errors
quoted for the 
's are the ones generated by the procedure MINOS in MINUIT
and correspond to an increase of 
by one unit.
A few remarks are in order at this place.
scattering data is better using
the unitarized form factors, in particular so for
the D-wave scattering lengths.
's in
column 4 and 5 are apparently consistent with each other within one error bar,
the 
in column 5 increases from 4.9 to 30.7 if
the 
's from column 4 are used in the evaluation of 
in
column 5. (For a discussion about the interpretation of the errors see
[70]).
which occur in 
analyses may be evaluated from a given set of 
and 
[44]. Their value changes in a significant way by
using the unitarized amplitude instead of the one-loop formulae: the
values for 
in column 4 and 5 are
and 
, respectively.
and 
are related to 
phase shifts through sum rules [73,74]. In principle, one
could
take these constraints into account as well. We do not consider them here, because we find it
very difficult to assess a reliable error for the integrals over the
total 
cross sections which occur in those relations.
The statistical error in the data is only one source of the uncertainty in the low-energy constants, which are in addition affected by the ambiguities in the estimate of the higher-order corrections. These systematic uncertainties have several sources:
have not been taken into account.
and 
depend on 
, and 
is a
function of both 
and 
.
phase shift and on the cutoff 
used.
We have considered carefully these effects [62], and found that the best
determination of 
, and 
which takes them into
account is
These values are the ones quoted in table 1 in Ref. [2].
For 
analyses it is useful to know the corresponding
values for the constants 
and 
,
The value and uncertainties in these couplings play a decisive role in a
planned experiment [75] to measure the lifetime of 
atoms,
which will provide a completely independent measurement of the 
scattering lengths 
.
One motivation for the analysis in [54,55] was to test the
large-
prediction 
. The above result shows that a small non-zero
value is preferred.
To obtain a clean error analysis, we have repeated the fitting
procedure using the variables
We
performed a fit to 
and 
data, including the theoretical error
in G as discussed above, and found
The result is that the large-
prediction works remarkably well.
