5.10 Improvements at DANE

The chiral analysis of decays has been used so far for three purposes:

1. The data from Ref. [53] allows one to make predictions for the slope of the G form factor, for the total rates in all the channels and for the scattering lengths. These are given in Eq. (5.85), in table 5.4 and in table 5.2, respectively.

2. The same data allow one to test the large- prediction, see Eqs. (5.82) and (5.83).

3. The full set of and scattering data allows the best determination of the coefficients and in the chiral Lagrangian, see (5.80).

In the next generation of decay experiments, there is the opportunity to improve the phenomenology of (see table 5.1):

1. A very useful innovation would be to analyze the experimental data with a modified chiral representation. In the latter, the full S- and P- wave parts of and could be inserted, using the chiral representation solely to describe the small background effects due to higher partial waves . To be more precise, one would take for R and H the one-loop chiral representation, whereas for G one writes

   

   and similarly for F. The unknown amplitudes and the phases would then be determined from the data. We have checked that, if the errors in the form factors determined in this manner can be reduced by e.g. a factor 3 with respect to the ones shown in (5.37), one could pin down particular combinations of and to considerably better precision than was shown above. This is true independently of an eventual improvement in the theoretical determination of the higher-order corrections in the form factor G -- which is a theoretical challenge in any case.

2. The present experimental uncertainty on G is still too large to provide a precise value for the large- parameter . decays are mainly sensitive to which in turn can be used to pin down . is mainly sensitive to which contains and .)

3. The observation of all reactions with high statistics could provide a cleaner separation of the various isospin amplitudes.

4. Finally, we come to a most important point. As we mentioned already, has been used [59] to determine the isoscalar S-wave scattering length with the result . This value must be compared with the SU(2)SU(2) prediction [77,78] . Low-energy scattering is one of the few places where chiral symmetry allows one to make a precise prediction within the framework of QCD. In their article, Rosselet et al. comment about the discrepancy between and the leading-order result [79] in the following manner: ``... it appears that this prediction can be revised without any fundamental change in current algebra or in the partial conservation of axial-vector current [80,81]." Today, we know that this is not the case: The standard picture of the vacuum structure in QCD [82] would have to be revised, should the central value be confirmed with a substantially smaller error. For recent work which supports this scenario see the contribution of Knecht and Stern in this Handbook [83].

   decays are -- at present [75] -- the only available source of clean information on S-wave scattering near threshold. We refer the reader to Ref. [84] for a detailed analysis of the issue.