Abstract
Abstract We propose a novel mechanism to reproduce the observed mass hierarchy for scalar mesons lighter than 1 GeV (called the inverse hierarchy), regarding them as mesons made of a quark and an anti-quark ($q\bar{q}$ mesons). The source is provided by the SU(3) flavor-symmetry breaking induced by the U(1) axial anomaly. In particular, the anomaly term including the explicit chiral symmetry breaking plays a significant role in the light scalar meson spectrum. To be concrete, we construct a linear sigma model for scalar mesons of $q\bar{q}$ type together with their pseudoscalar chiral partners, including an anomaly-induced explicit chiral symmetry-breaking term. We find that, due to the proposed mechanism, the inverse hierarchy, i.e., $m\left[ a_0 (980) \right] \simeq m\left[ f_0 (980) \right] > m \left[ K_0^\ast (700) \right] > m \left[ f_0(500) \right]$, is indeed realized. Consequently, the quark content of $f_0 (500)$ is dominated by the isoscalar $\bar uu+ \bar dd$ component, and $f_0 (980)$ by the strange quark bilinear one, $s\bar{s}$.
Highlights
The vacuum structure in QCD is supposed to be governed by the nonpreturbative quark condensate, which spontaneously breaks the chiral symmetry
It gives rise to the associated Nambu-Goldstone bosons, arising as pseudoscalar mesons in the meson spectra. The dynamics among those pseudoscalar mesons is governed by the spontaneously broken chiral symmetry, which is well described in the framework of the chiral perturbation theory [1,2,3]
In addition to those pseudoscalar mesons, scalar mesons made of a quark and an antiquark, so-called qqscalar mesons, are expected to exist as their chiral partners, which should include the fluctuation mode of the flavor-singlet chiral condensate, regarded as the signal particle for the chiral symmetry breaking, to the Higgs boson for the electroweak symmetry breaking
Summary
The vacuum structure in QCD is supposed to be governed by the nonpreturbative quark condensate, which spontaneously breaks the (approximate) chiral symmetry. Some analyses have been attempted without getting into such a complexity: Reference [24] studied the masses of the scalar mesons in an extended Nambu-Jona-Lasino model including a six-quark interaction term induced by the U(1) axial anomaly a la KobayashiMaskawa-t’Hooft (KMT)[25,26,27,28,29,30,31]. Thanks to the flavor singlet nature of the KMT interaction, the vertices contain the strange quark condensate, ss , for the a0 meson and the up or down quark condensates, nn , for the K0∗ meson This contribution is against the simple strange quark number counting and works for resolving the inverse hierarchy. Recent works [32, 33] showed that an NJL-like model extended by including a large number of explicit-flavor violating terms with the current quark masses can reproduce the ”inverse hierarchy” for the masses of the scalar mesons.
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