Abstract
An earlier calculation in a generalized linear sigma model showed that the well-known current algebra formula for low energy pion-pion scattering held even though the massless Nambu Goldstone pion contained a small admixture of a two-quark two-antiquark field. Here we turn on the pion mass and note that the current algebra formula no longer holds exactly. We discuss this small deviation and also study the effects of a SU(3) symmetric quark mass type term on the masses and mixings of the eight SU(3) multiplets in the model. We calculate the s-wave scattering lengths, including the beyond current algebra theorem corrections due to the scalar mesons, and observe that the effect of the scalar mesons is to improve the agreement with experiment. In the process, we uncover the way in which linear sigma models give controlled corrections (due to the presence of scalar mesons) to the current algebra scattering formula. Such a feature is commonly thought to exist only in the nonlinear sigma model approach.
Highlights
A linear sigma model with both quark-antiquark type fields and fields containing two quarks and two antiquarks, seems useful for understanding the light scalar spectrum of QCD
We verified in detail that, as long as the potential of the model satisfied SU(3)L × SU(3)R invariance, the massless version of the famous current algebra theorem [2] on low energy pion pion scattering was correct
We have seen that the intuitive explanation for the existence of a very light scalar meson as well as light scalar mesons with large “four quark” content given in [1] still is good when the physical pion mass is used
Summary
A linear sigma model with both quark-antiquark type fields and fields containing (in an unspecified configuration) two quarks and two antiquarks, seems useful for understanding the light scalar spectrum of QCD. We verified in detail that, as long as the potential of the model satisfied SU(3)L × SU(3)R invariance, the massless version of the famous current algebra theorem [2] on low energy pion pion scattering was correct. We reexamine the pion pion scattering amplitude to try to see if the low energy theorem continues to hold. It turns out that the partially conserved axial vector current, which is required for the theorem, does not hold, unlike for the massless case. The axial vector current has a single particle contribution from the “heavy pion” in this chiral model as well as from the ordinary pion. The ordinary pion does not completely saturate the axial current This is a small effect but is of conceptual interest and may be of more importance for the kaon scattering case. The Appendix explains the method of parameter determination from experiment and a listing of typical values for all the parameters
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