Scalar Nonet Research Articles

Analysis of the scalar nonet mesons with QCD sum rules

In this article, we assume that the nonet scalar mesons below $1\,\rm{ GeV}$ are the two-quark-tetraquark mixed states and study their masses and pole residues using the QCD sum rules. In calculation, we take into account the vacuum condensates up to dimension 10 and the $\mathcal{O}(\alpha_s)$ corrections to the perturbative terms in the operator product expansion. We determine the mixing angles, which indicate the two-quark components are much larger than $50\%$, then obtain the masses and pole residues of the nonet scalar mesons.

Tetraquarks from the Bethe--Salpeter Equation

We present a numerical solution of the four-quark Bethe-Salpeter equation for a scalar tetraquark. We find that the four-body equation dynamically generates pseudoscalar poles in the Bethe-Salpeter amplitude. The sensitivity to the pion poles leads to a light isoscalar tetraquark mass M ~ 400 MeV, which is comparable to that of the sigma/f0(500). The masses of its multiplet partners kappa and a0/f0 follow a similar pattern, thereby providing support for the tetraquark interpretation of the light scalar nonet.

Baryon octet and decuplet phenomenology in a three-flavor extended linear sigma model

We present an effective model, which is an extension of the usual linear sigma model, that contains a low energy multiplet for every hadronic particle type. These multiplets are a scalar nonet, a pseudoscalar nonet, a vector nonet, an axialvector nonet, a baryon octet and a baryon decuplet. Tree level baryon masses and possible two body decuplet decays are calculated. The baryon masses are generated through spontaneous symmetry breaking. The calculated quantities are used to determine the model parameters through a multiparametric minimalization process, which compares the calculated physical quantities with their experimental values. We found that the calculated quantities are in good agreement with the experimental data.

Light quark masses in multi-quark interactions

We suggest and discuss in detail a multi-quark three flavor Lagrangian of the Nambu -- Jona-Lasinio type, which includes a set of effective interactions proportional to the current quark masses. It is shown that within the dynamical chiral symmetry breaking regime, the masses of the pseudo Goldstone bosons and their chiral partners, members of the low lying scalar nonet, are in perfect agreement with current phenomenological expectations. The role of the new interactions is analyzed.

Chiral nonet mixing inππscattering

Pion-pion scattering is studied in a generalized linear sigma model which contains two scalar nonets (one of quark-antiquark type and the other of diquark-antidiquark type) and two corresponding pseudoscalar nonets. An interesting feature concerns the mixing of the four isosinglet scalar mesons which yield poles in the scattering amplitude. Some realism is introduced by enforcing exact unitarity via the K-matrix method. It is shown that a reasonable agreement with experimental data is obtained up to about 1 GeV. The poles in the unitarized scattering amplitude are studied in some detail. The lowest pole clearly represents the sigma meson (or ${f}_{0}(600)$) with a mass and decay width around 500 MeV. The second pole invites comparison with the ${f}_{0}(980)$ which has a mass around 1 GeV and decay width around 100 MeV. The third and fourth poles, resemble some of the isosinglet state in the complicated 1--2 GeV region. Some comparison is made to the situation in the usual SU(3) linear sigma model with a single scalar nonet.

Sigma meson in pole-dominated QCD sum rules

The properties of the $\ensuremath{\sigma}$ meson are studied using the QCD sum rules for tetraquark operators. In the SU(3) chiral limit, we separately investigate SU(3) singlet and octet tetraquark states as constituents of the $\ensuremath{\sigma}$ meson and discuss their roles for the classification of the light scalar nonets $\ensuremath{\sigma}$, ${f}_{0}$, ${a}_{0}$, and $\ensuremath{\kappa}$ as candidates of tetraquark states. All of our analyses are performed in the suitable Borel window which is indispensable to avoid the pseudopeak artifacts outside of the Borel window. We can set up acceptably wide Borel windows after preparing favorable linear combinations of operators and including the dimension 12 terms in the operator-product expansion. Taking into account the possible large widths, we evaluate masses for singlet and octet states as 700--850 and 600--750 MeV, respectively, although the octet operators have a smaller overlap with the tetraquark states than the singlet case, which requires careful interpretations. The splitting of the singlet and octet states emerges from the number of the $\overline{q}q$ annihilation diagrams, which include the color singlet annihilation processes $qq\overline{q}\overline{q}\ensuremath{\rightarrow}(q\overline{q}{)}_{1}$ and the color octet annihilation processes $qq\overline{q}\overline{q}\ensuremath{\rightarrow}G(q\overline{q}{)}_{8}$. The mass of the $\ensuremath{\sigma}$ meson is evaluated as 600--800 MeV, which is much closer to the experimental value $\ensuremath{\sim}500\text{ }\text{ }\mathrm{MeV}$ than the mass evaluated by 2-quark correlator analyses, $\ensuremath{\sim}1.0\text{ }\text{ }\mathrm{GeV}$. This indicates that the tetraquark state shares a larger fraction in the $\ensuremath{\sigma}$ meson than ordinary two quark meson states.

SCALAR NONETS IN POLE-DOMINATED QCD SUM RULES

The light scalar nonets are studied using the QCD sum rules for the tetraquark operators. The operator product expansion for the correlators is calculated up to dimension 12 and this enables us to perform analyses retaining sufficient pole-dominance. To classify the light scalar nonets, we investigate the dependence on current quark mass and flavor dynamics. Especially, to examine the latter, we study separately SU(3) singlet and octet states, and show that the number of annihilation diagrams is largely responsible for their differences, which is also the case even after the inclusion of the finite quark mass. Our results support the tetraquark picture for isosinglets, while that for octets is not conclusive yet.

$\bar KN$ SCATTERING IN CHIRAL CONSTITUENT QUARK MODELS

The chiral SU(3) quark model and the extended chiral SU(3) quark model have proven to be quite successful in reproducing the binding energy of the deuteron, the nucleon–nucleon (NN) scattering phase shifts and the hyperon–nucleon (YN) cross-sections. In the chiral SU(3) quark model, the quark–quark interaction contains one-gluon exchange (OGE), confinement potential and boson exchanges stemming from scalar and pseudoscalar nonets. In the extended chiral SU(3) quark model, the OGE is nearly replaced by vector meson exchange. It was shown in our previous work that these two models give quite similar descriptions for the S, P, D, F wave KN phase shifts, which means that in the KN system the contribution of OGE can be replaced by that of vector meson exchange. In this paper, we use the same models, the same parameters and the same methods to study the [Formula: see text] system with the purpose to test the effects of OGE and vector meson exchange. The cross-sections for K-p scattering into K-p, K0n, π+Σ-, π-Σ+, π0Σ0 and π0Λ channels are dynamically calculated by solving the resonating group method equation. While some channels are well described in one or the other model, a good agreement between the theoretical cross-sections and the experimental data for all channels is not successfully obtained. The present work indicates that both OGE and vector meson exchange are necessary to be included in the quark–quark interactions if one tries to simultaneously describe the KN phase shifts and [Formula: see text] cross-sections using one set of parameters in a constituent quark model.

Regarding the scalar mesons

Based on the main assumption that the ${D}_{sJ}(2860)$ belongs to the ${2}^{3}{P}_{0}$ $q\overline{q}$ multiplet, the masses of the scalar meson nonet are estimated in the framework of the relativistic independent quark model, Regge phenomenology, and meson-meson mixing. We suggest that the ${a}_{0}(1005)$, $K_{0}{}^{*}(1062)$, ${f}_{0}(1103)$, and ${f}_{0}(564)$ constitute the ground scalar meson nonet; it is supposed that these states would likely correspond to the observed states ${a}_{0}(980)$, $\ensuremath{\kappa}(900)$, ${f}_{0}(980)$, and ${f}_{0}(600)/\ensuremath{\sigma}$, respectively. Also ${a}_{0}(1516)$, $K_{0}{}^{*}(1669)$, ${f}_{0}(1788)$, and ${f}_{0}(1284)$ constitute the first radial scalar meson nonet, it is supposed that these states would likely correspond to the observed states ${a}_{0}(1450)$, $K_{0}{}^{*}(1430)$, ${f}_{0}(1710)$, and ${f}_{0}(1370)$, respectively. The scalar state ${f}_{0}(1500)$ may be a good candidate for the ground scalar glueball. The agreement between the present findings and those given by other different approaches is satisfactory.

Lattice study of low-lying nonet scalar mesons in quenched approximation

Using lattice QCD simulation in the quenched approximation, we study the κ meson, which is P03 in the quark model, and compare experimental and other lattice data. The κ is the lowest scalar meson with strangeness and constitutes the scalar nonet. The obtained mass is much higher than the recent experimental value, and therefore the κ(800) is difficult to consider as a simple two-body constituent-quark structure, and may have another unconventional structure.

On the scalar nonet in the extended Nambu–Jona-Lasinio model

We discuss the lightest scalar resonances, f 0 ( 600 ) , κ ( 800 ) , a 0 ( 980 ) and f 0 ( 980 ) in the extended Nambu–Jona-Lasinio model. We find that the model parameters can be tuned, but unnaturally, to accommodate those scalars except the f 0 ( 980 ) . We also discuss problems encountered in the K-matrix unitarization approximation by using N c counting technique.

Multichannel calculation of the very narrow Ds0 *(2317) and the very broad D0 *(2300-2400)

The narrow D s0 * (2317) and broad D 0 * (2300–2400) charmed scalar mesons and their radial excitations are described in a coupled-channel quark model that also reproduces the properties of the light scalar nonet. All two-meson channels containing ground-state pseudoscalars and vectors are included. The parameters are chosen fixed at published values, except for the overall coupling constant λ, which is fine-tuned to reproduce the D s0 * (2317) mass, and a damping constant a for subthreshold contributions. Variations of λ and D 0 * (2300–2400) pole postions are studied for different α values. Calculated cross-sections for S-wave DK and Dπ scattering, as well as resonance pole positions, are given for the value of α that fits the light scalars. The thus predicted radially excited state D s0 * ′(2850), with a width of about 50 MeV, seems to have been observed already.

Two or Four: A Hint from Scalar Mesons in Radiative φ Decays?

In this write-up, we summarize our recent analysis of radiative decays involving light scalar mesons. Our analysis using the vector meson dominance model at tree level indicates that it may be difficult to distinguish qq¯ ¯ q picture and q ¯ q picture for the light scalar nonet. Our result on the process of φ → π 0 ηγ shows that the derivative-type f0K ¯ K interaction reproduces experimental data below 950 MeV well, but gives a poor fit above 950 MeV, i.e., in the energy region around the mass of a0(980), but that the discrepancy can be compensated by the effect of the K loop.

A QCD sum rule study of the light scalar mesons

We examine the interpretation of the light scalar meson nonet as bound states of the scalar diquark and the scalar antidiquark using the QCD sum rule approach. Our results are obtained by means of the operator product expansion (OPE) including operators up to dimension 8. They show no evidence of the coupling of the tetraquark states to the light scalar meson nonet.

Chiral approach to phi radiative decays

The radiative decays of the phi meson are known to be a good source of information about the ${a}_{0}(980)$ and ${f}_{0}(980)$ scalar mesons. We discuss these decays starting from a nonlinear model Lagrangian which maintains the (broken) chiral symmetry for the pseudoscalar (P), scalar (S) and vector (V) nonets involved. The characteristic feature is derivative coupling for the SPP interaction. In an initial approximation which models all the scalar nonet radiative processes together with the help of a pointlike vertex, it is noted that the derivative coupling prevents the ${a}_{0}$ and ${f}_{0}$ resonance peaks from getting washed out (by falling phase space). However, the shapes of the invariant two final PP mass distributions do not agree well with the experimental ones. For improving the situation we verify that inclusion of the charged $K$ meson loop diagrams in the model does reproduce the experimental spectrum shapes in the resonance region. The derivative coupling introduces quadratic as well as logarithmic divergences in this calculation. Using dimensional regularization we show in detail that these divergences actually cancel out among the four diagrams, as expected from gauge invariance. We point out the features which are expected to be important for further work on this model and for learning more about the puzzling scalar mesons.

Implications of a generalized heat kernel expansion for an effective QCD chiral Lagrangian with SU(3) and UA(1) breaking

This work is a follow up of recent investigations where the implications of a generalized heat kernel expansion is studied, constructed to incorporate non-perturbatively the effects of a non-commutative quark mass matrix in a fully covariant way at each order of the expansion. As underlying Lagrangian we use the Nambu – Jona-Lasinio (NJL) model of QCD, with SUf(3) and UA(1) breaking, the latter generated by the 't Hooft flavor determinant interaction. The associated bosonized Lagrangian is derived in leading stationary phase approximation (SPA) and up to second order in the generalized heat kernel expansion. Its symmetry breaking pattern is shown to have a complex structure, involving all powers of the mesonic fields allowed by symmetry. The considered Lagrangian yields a reliable playground for the study of the implications of symmetry and vacuum structure on the mesonic spectra, which we evaluate for the scalar and pseudoscalar meson nonets and compare with other approaches and experiment.

Gaussian Source One-Boson-Exchange Potential between Two Octet Baryons

The nonstatic baryon-baryon potential arising from the Gaussian meson source function, G i exp (- 2 i r 2 /4), is constructed by exchanging the scalar, pseudoscalar and vector meson nonets, where G i is the strength and A i the size parameter of source i (i = 1,2). This potential is referred to as GSOBEP. It has no singularity at the origin. In this potential, the radial function behaves like the Yukawa function in the outer region (r > 1 fm), and like the Yukawa function multiplied by the error function in the inner region (r < 1 fm). The Gaussian source is assumed to originate from the statistical distribution of valence quarks and many sea quarks. The potential parameters consist of the SU(3) parameters, such as the singlet and octet coupling constants, g 1 and g 8 , the coupling ratio α = g F /(g D + g F ), the singlet-octet mixing angle 0 for each meson, and the effective source size parameter, expressed as Λ = Λ 1 Λ 2 /√Λ 2 1 + Λ 2 2 . The potential parameters are determined by the X 2 -fitting to the nucleon-nucleon, hyperon-nucleon and hyperon-hyperon data. The fit is quite good on the whole.

B decays into light scalar particles and glueball

The recent observations of f_0(980) in charmless B-decays motivate further studies of scalar particle and glueball production in these processes. Amplitudes for charmless 2-body B decays involving the members of the scalar nonet are presented based on the symmetries of the dominant penguin contribution. Different scenarios for the lightest scalar nonet are investigated in view of the presently available data. We describe the evidence from B-decays for f_0(1500) with a flavour octet like mixing and the hints towards the members of the q qbar nonet of lowest mass. There is further support for the hypothesis of a broad 0^{++} glueball acting as coherent background especially in B -> K Kbar K. The estimated B decay rates into gluonic mesons represent a sizable fraction of the theoretically derived decay rate for b -> sg.

A diquark chiral effective theory and exotic baryons

A chiral effective theory of quarks and diquarks is formulated and applied to exotic tetraquarks and pentaquarks. The effective theory is similar to the chiral quark effective theory with the addition of diquark degrees of freedom and couplings to quarks. Chiral symmetry through generalized Goldberger–Treiman relations, fixes the mass splitting between the even and odd parity diquarks to be about 600 MeV. We provide an estimate of the parameters of the effective theory from both the random instanton model and known data on the low-lying scalar nonet. We assess the static properties of the exotic baryons, such as their masses, magnetic moments and decay widths. We show that the small decay widths of the newly reported exotics are largely due to a large tunneling suppression of a quark between a pair of diquarks. We find Γ(Θ+→K+n)≃2.5–7.0 MeV and Γ(Ξ−−→Ξ−π−)≃1.7–4.8 MeV.

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