Abstract
We present a detailed study of a linear sigma model containing one chiral nonet transforming under $\mathrm{U}(1{)}_{\mathrm{A}}$ as a quark-antiquark composite and another chiral nonet transforming as a diquark-antidiquark composite (or, equivalently from a symmetry point of view, as a two meson molecule). The model provides an intuitive explanation of a current puzzle in low energy QCD: Recent work has suggested the existence of a lighter than 1 GeV nonet of scalar mesons which behave like four quark composites. On the other hand, the validity of a spontaneously broken chiral symmetric description would suggest that these states be chiral partners of the light pseudoscalar mesons, which are two quark composites. The model solves the problem by starting with the two chiral nonets mentioned and allowing them to mix with each other. The input of physical masses in the SU(3) invariant limit for two scalar octets and an excited pion octet results in a mixing pattern wherein the light scalars have a large four quark content while the light pseudoscalars have a large two quark content. One light isosinglet scalar is exceptionally light. In addition, the pion pion scattering is also studied and the current algebra theorem is verified for massless pions which contain some four quark admixture.
Highlights
The topic of anomalously light scalar mesons in QCD has become a subject of increasing interest in the last fifteen years or so [1]-[39]
The difficulty people had previously in accepting the light scalars was largely due to the great success of the simple quark model, in which the lightest scalars are expected to be p-wave quark- antiquark composite states and to be in the 1 to 1.5 GeV range, like the other p-wave states
We have given a detailed treatment of a systematic approach to the study of a linear sigma model containing one chiral nonet transforming under U(1)A as a quark-antiquark composite and another chiral nonet transforming as a diquark-anti diquark composite
Summary
The topic of anomalously light scalar mesons in QCD has become a subject of increasing interest in the last fifteen years or so [1]-[39]. In the zero quark mass limit with just a few explicit chiral invariant terms contained, there was a possibility of a situation in which the lightest pseudoscalars could have zero mass (i.e. be Nambu-Goldstone bosons) and be primarily qqtype while the heaviest mesons could be scalars, primarily of qqqqtype. A treatment [43] of the model with similar chiral invariant terms and several different quark mass terms found that light scalars with relatively large admixtures of qqqqtype states are favored. In [44], the present authors studied the more general version of the model in which all possible chiral invariant, even non renormalizable, terms were included together with the single usual realization of the quark mass term.
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