A generalized linear sigma model for low-energy QCD is employed to study the quark structure of eight low-lying scalar isomultiplets as well as eight low-lying pseudoscalar isomultiplets. The model, building on earlier work, assumes the possible mixing of quark antiquark states with others made of two quarks and two antiquarks. No a priori assumption is made about the quark contents of the states, which emerge as predictions. An amusing and contrasting pattern for the quark structure is found; the lighter conventional pseudoscalars are, as expected, primarily of two-quark type whereas the lighter scalars have very large four-quark admixtures. The new feature of the present paper compared to earlier ones in this series involves the somewhat subtle and complicated effects of SU(3) flavor breaking. They do not alter the general pattern of two-quark vs four-quark mixing obtained in the SU(3)symmetric case but, of course, give a more detailed picture.

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Global aspects of the scalar meson puzzle

We present a two-flavor linear sigma model with global chiral symmetry and (axial-)vector mesons as well as an additional glueball degree of freedom. We study the structure of the well-established scalar resonances ${f}_{0}(1370)$ and ${f}_{0}(1500)$: by a fit to experimentally known decay widths we find that ${f}_{0}(1370)$ is predominantly a $\overline{q}q$ state and ${f}_{0}(1500)$ is predominantly a glueball state. The overall phenomenology of these two resonances can be well described. Other assignments for our mixed quarkonium-glueball states are also tested, but turn out to be in worse agreement with the phenomenology. As a by-product of our analysis, the gluon condensate is determined.

Glueball in a chiral linear sigma model with vector mesons

A collective treatment of the I=0 scalar mesons below 2 GeV [σ(550), f0(980), f0(1370), f0(1500) and f0(1710)] in a nonlinear chiral Lagrangian framework that is constrained by the mass and the partial decay widths of the I=1/2,1 scalars [κ(900), $K_0^*(1430)$, a0(980) and a0(1450)] is presented. The sub-structure of these states in terms of two and four quark components, as well as a glueball component is explored, and its correlation with the mass of f0(1370) is studied. Consistency with the available experimental data suggests that the σ(550) is dominantly a nonstrange four-quark state, whereas the sub-structure of other I=0 states are sensitive to the input mass of f0(1370). This investigation estimates the scalar glueball mass in the range 1.47–1.64 GeV.

ISOSINGLET SCALAR MESONS BELOW 2 GeV AND THE SCALAR GLUEBALL MASS

It is well known that the instanton approach to QCD generates an effective term which looks like a three-flavor determinant of quark bilinears. This has the right behavior to explain the unusual mass and mixing of the $\ensuremath{\eta}(958)$ meson, as is often simply illustrated with the aid of a linear SU(3) sigma model. It is less well known that the instanton analysis generates another term which has the same transformation property but does not have a simple interpretation in terms of this usual linear sigma model. Here we point out that this term has an interpretation in a generalized linear sigma model containing two chiral nonets. The second chiral nonet is taken to correspond to mesons having two quarks and two antiquarks in their makeup. The generalized model seems to be useful for learning about the spectrum of low-lying scalar mesons which have been emerging in the last few years. The physics of the new term is shown to be related to the properties of an excited ${\ensuremath{\eta}}^{\ensuremath{'}}$ state present in the generalized model and for which there are some experimental candidates.

Note on a Sigma Model Connection with Instanton Dynamics

A collective treatment of the I=0 scalar mesons below 2 GeV [sigma (550), f0(980), f0(1370), f0(1500) and f0(1710)] in a non-linear chiral Lagrangian framework that is constrained by the mass and the partial decay widths of the I=1/2,1 scalars [kappa(900), K0*(1430), a0(980) and a0(1450)] is presented. The sub-structure of these states in terms of two and four quark components, as well as a glueball component is explored, and its correlation with the mass of f0(1370) is studied. Consistency with the available experimental data suggests that the sigma(550) is dominantly a non-strange four-quark state, whereas the sub-structure of other I=0 states are sensitive to the input mass of f0(1370). This investigation estimates the scalar glueball mass in the range 1.47--1.64 GeV.

Isosinglet Scalar Mesons Below 2 GeV and the Scalar Glueball Mass

Following the re-establishment of the σ(500) and the κ(900), the light scalar mesons a0(980) and f0(980) together with the σ(500) and the κ(900) are considered as the chiral scalar partner of pseudoscalar nonet in SU(3) chiral symmetry, and the high mass scalar mesons a0(1450), K*0(1430), f0(1370) and f0(1710) turned out to be considered as the L = 1 q scalar mesons. We assume that the high mass of the L = 1 q scalar mesons is caused by the mixing with the light scalar mesons. For the structure of the light scalar mesons, we adopted the qq model in order to explain the 'scalar meson puzzle'. The inter-mixing between the light scalar nonet and the high mass L = 1 q nonet and the intra-mixing among each nonet are analysed by including the glueball into the high mass scalar nonet.

https://iopscience.iop.org/article/10.1088/0954-3899/28/6/318/pdf

Mixing among light scalar mesons and L = 1 qq̄ scalar mesons

Following the re-establishment of the \sigma(600) and the \kappa(900), the light scalar mesons a_0(980) and f_0(980) together with the \sigma(600) and the \kappa(900) are considered as the chiral scalar partner of pseudoscalar nonet in SU(3) chiral symmetry, and the high mass scalar mesons a_0(1450), K^*_0(1430), f_0(1370) and f_0(1710) turned out to be considered as the L=1 q\bar{q} scalar mesons. We assume that the high mass of the L=1 q\bar{q} scalar mesons is caused by the mixing with the light scalar mesons. For the structure of the light scalar mesons, we adopted the qq\bar{q}\bar{q} model in order to explain the "scalar meson puzzle". The inter-mixing between the light scalar nonet and the high mass L=1 q\bar{q} nonet and the intra-mixing among each nonet are analyzed by including the glueball into the high mass scalar nonet.

Mixing among light scalar mesons and L=1 q\bar{q} scalar mesons

We analyse the mass spectroscopy of low mass scalar nonet and high mass scalar nonet+glueball and get the result that the strengths of the inter-mixing between low and high mass scalar mesons are very strong and the strengths of mixing for I = 1, 1/2 scalar mesons and those of I = 0 scalar mesons are almost the same. We perform a χ2 fit based on the decay widths and ratios of these mesons to estimate the effects of inter-mixing and get the result that the inter-mixing has large effects for these decay processes and f0(1710) is preferred as glueball rather than f0(1500). Best-fit values for the coupling constants A, A' and A'' which represent the couplings of low mass scalar meson N to two pseudoscalar mesons PP, high mass scalar meson N' to PP and glueball G to PP, respectively are obtained as (A, A', A'') = (4.0 ± 1.1 GeV−1, 1.8 ± 0.3 GeV−1, − 0.64 ± 0.04 GeV−1).

Effects on scalar meson decays of strong mixing between low and high mass scalar mesons

We analyze the mass spectroscopy of low and high mass scalar mesons and get the result that the coupling strengths of the mixing between low and high mass scalar mesons are very strong and the strengths of mixing for $I=1, 1/2$ scalar mesons and those of I=0 scalar mesons are almost same. Next, we analyze the decay widths and decay ratios of these mesons and get the results that the coupling constants $A'$ for $I=1, 1/2$ which represents the coupling of high mass scalar meson $N'$ -> two pseudoscalar mesons $PP$ are almost same as the coupling $A'$ for the I=0. On the other hand, the coupling constant $A$ for $I=1, I=1/2$ which represents the low mass scalar meson $N$ -> $PP$ are far from the coupling constant $A$ for I=0. We consider a resolution for this discrepancy. Coupling constant $A''$ for glueball $G$ -> $PP$ is smaller than the coupling $A'$. $\theta_P$ is $40^\circ \sim 50^\circ$.

Effects to Scalar Meson Decays of Strong Mixing between Low and High Mass Scalar Mesons

We analyze the mass spectroscopy of low and high mass scalar mesons and get the result that the coupling strengths of the mixing between low and high mass scalar mesons are very strong and the strengths of mixing among I = 1, 1/2 scalar mesons and those among I = 0 scalar mesons are almost same. Next, we analyze the decay widths and ratios of these mesons and get the results that the coupling constants A′ for I = 1, 1/2 which represents the coupling of high mass scalar meson N′ → pseudoscalar mesons PP are almost same as the coupling A′ for the I = 0. On the other hand, the coupling constant A for I = 1, 1/2 which represents the low mass scalar meson N → PP are far from the coupling constant A for I = 0. We consider a resolution of this discrepancy.

N. Morisita

Papers

Mixing among light scalar mesons and L = 1 qq̄ scalar mesons

Mixing among light scalar mesons and L=1 q\bar{q} scalar mesons

Effects on scalar meson decays of strong mixing between low and high mass scalar mesons

Effects to Scalar Meson Decays of Strong Mixing between Low and High Mass Scalar Mesons

Effects to Scalar Meson Decays of Strong Mixing between Low and High Mass Scalar Mesons