Abstract
First it is shown that the tree amplitude for pion–pion scattering in the minimal linear sigma model has an exact expression which is proportional to a geometric series in the quantity [Formula: see text], where mB is the sigma mass which appears in the Lagrangian and is the only a priori unknown parameter in the model. This induces an infinite series for every predicted scattering length in which each term corresponds to a given order in the chiral perturbation theory counting. It is noted that, perhaps surprisingly, the pattern, though not the exact values, of chiral perturbation theory predictions for both the isotopic spin 0 and isotopic spin 2 s-wave pion–pion scattering lengths to orders p2, p4 and p6 seems to agree with this induced pattern. The values of the p8 terms are also given for comparison with a possible future chiral perturbation theory calculation. Further aspects of this approach and future directions are briefly discussed.
Highlights
The chiral perturbation theory approach [1]-[4] provides a systematic method for improving the ”current algebra” or tree level “non-linear chiral Lagrangian” results for low energy QCD in powers of a characteristic squared momentum, p2
The main new feature in the present approach seems to be the realization that the use of the simplest linear sigma model at tree level does not give just one number but gives an infinite series of numbers which can be conveniently compared with the series resulting from chiral perturbation theory
Another amusing feature is that this approach provides a specific model for the expansion parameter of this series; namely m2π/(m2B − m2π)
Summary
The chiral perturbation theory approach [1]-[4] provides a systematic method for improving the ”current algebra” or tree level “non-linear chiral Lagrangian” results for low energy QCD in powers of a characteristic squared momentum, p2 (or number of derivatives). For example vector meson dominance is known to be good at low energies; a typical well known immediate prediction gives the squared charge radius of the pion as rπ2 = 6/m2ρ This kind of approach may be theoretically justified to some extent by invoking the 1/N expansion of QCD [5], [6] which yields tree level dominance. A simple sigma dominance approximation is not viable because it would not guarantee the nearly spontaneous breakdown of chiral symmetry mechanism which is crucial for QCD Such a mechanism is guaranteed by the use of a linear sigma model of some type. The actual low energy scattering is known to result from an enormous cancellation between the sigma pole and the contact contributions This unpleasant feature is mitigated in the non-linear sigma model (which forms the basis of the chiral perturbation scheme). In order to compare it with something, it is natural to compare it with another power series in squared momentum- chiral perturbation theory
Sign-in/Register to view highlights & summary 
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call