We calculate the mass and decay constant of $I=1/2$ scalar meson based on QCD sum rule. The $I=1/2$ scalar mesons are composed of $s\bar{q}$ or $q\bar{s}$ ($q=u,d$)in quark model, the quantum numbers of spin and orbital angular momentum are S=1, L=1. We obtain the mass of the ground sate in this channel is $1.409\pm 0.026$GeV, which is consistent with the mass of $K_{0}^{\ast}(1430)$. This implies that $K_{0}^{\ast}(1430)$ is the ground state of $s\bar{q}$ or $q\bar{s}$ with quantum number $J^P=0^+$. We also predict the first higher resonance of $K_{0}^{\ast}(1430)$ is above 2.09 GeV.

Light scalar mesons are found to fit rather well a diquark-antidiquark description. The resulting nonet obeys mass formulae which respect, to a good extent, the OZI rule. OZI allowed strong decays are reasonably reproduced by a single amplitude describing the switch of a qbar-q pair, which transforms the state into two colourless pseudoscalar mesons. Predicted heavy states with one or more quarks replaced by charm or beauty are briefly described; they should give rise to narrow states with exotic quantum numbers.

In this article, we suppose the scalar mesons $f_0(980)$ and $a_0(980)$ have both strange and non-strange quark-antiquark components and evaluate the strong coupling constants $g_{f_0 K^+ K^-}$ and $g_{a_0 K^+ K^-}$ with two interpolating scalar currents respectively within the framework of the light-cone QCD sum rules approach. Our observation about the large strong scalar-$KK$ couplings $g_{f_0 K^+ K^-}$ and $g_{a_0 K^+ K^-}$ will support the hadronic dressing model, furthermore, in spite of the constituent structure differences between the $f_0(980)$ and $a_0(980)$ mesons, the strange components have a larger strong coupling constants with the $K^+K^-$ mesons state than the non-strange ones, $g_{f_0 K^+ K^-}^{\bar{s}s}\approx \sqrt{2}g_{f_0 K^+ K^-}^{\bar{n}n}$ and $g_{a_0 K^+ K^-}^{\bar{s}s}\approx \sqrt{2} g_{a_0 K^+ K^-}^{\bar{n}n}$.

We study scalar quark-anti-quark and two-quark-two-anti-quark correlation functions in the instanton liquid model. We show that the instanton liquid supports a light scalar-isoscalar (sigma) meson, and that this state is strongly coupled to both $(\bar{q}q)$ and $(\bar{q}q)^2$. The scalar-isovector $a_0$ meson, on the other hand, is heavy. We also show that these properties are specific to QCD with three colors. In the large $N_c$ limit the scalar-isoscalar meson is not light, and it is mainly coupled to $(\bar{q}q)$.

The spectrum and the mixing angle of the light scalar nonet mesons are studied using the extended Nambu-Jona-Lasinio (NJL) model as well as the improved ladder approximation (ILA) of QCD with U_A(1) symmetry breaking interaction. The U_A(1) breaking is represented by the Kobayashi-Maskawa-'t Hooft (KMT) interaction. The strength of the KMT interaction in the NJL model is determined so as to reproduce the electromagnetic decays of the \eta meson. That in the ILA approach is determined so as to reproduce the pseudoscalar meson spectrum. In the extended NJL model, we study the qualitative features of the scalar meson spectrum. In the scalar nonet spectrum, the KMT interaction is found to give the right ordering of \sigma - a_0 masses and a few hundred MeV mass difference between the \sigma and a_0 mesons. We also find that the strangeness content in the \sigma meson is about 15%. In the ILA approach, we confirm the qualitative features of the results from the extended NJL. We obtain the mass spectra of the light pseudoscalar nonet mesons and the \sigma, a_0 mesons which are consistent with current experimental data. On the other hand, we show that the obtained f_0 and K_0^* state can not be identified directly with the experimental data. It may suggest that the f_0(980) and \kappa(700-900) states may not be explained only by the quark-anti quark state. We also find that the strangeness content in the \sigma meson is about 5%. We obtain the result that the U_A(1) breaking interaction reproducing the mass spectrum of the pseudoscalar meson gives large effects on the scalar meson mass spectrum.

We investigate the implications of large N_c and chiral symmetry for the mass spectra of meson resonances. Unlike for most other mesons, the mass matrix of the light scalars deviates strongly from its large-N_c limit. We discuss the possible assignments for the lightest scalar nonet that survives in the large-N_c limit.

Effects of the U_A(1) breaking interaction on the low-lying nonet scalar mesons are studied using the extended Nambu-Jona-Lasinio model. The strength of the U_A(1) breaking interaction is determined by the electromagnetic decays of the eta meson. We find that the U_A(1) breaking interaction gives rise to about 150 MeV mass difference between the sigma and a_0 mesons. We also find that the strangeness content in the \sigma meson is about 15%. The calculated mass of the I=1/2 state is about 200 MeV heavier than that of the I=1 state. The order of these masses is not likely to change within this model.

Based on the (approximate) chiral symmetry of QCD Lagrangian and bound state assumption of effective meson fields due to strong interactions of gluons, a nonlinearly realized effective chiral Lagrangian for meson fields is resulted from integrating over the quark fields by using the new finite regularization method. As the new method preserves the symmetry principles of the original theory and meanwhile keeps the finite quadratic term given by a physically meaningful characteristic energy scale $M_c$, it then leads the effective chiral field theory to have a dynamically spontaneous symmetry breaking mechanism. The gap equations are yielded as the minimal conditions of the effective potential in the effective theory. The instanton effects are included via the induced interactions discovered by 't Hooft and found to play an important role in obtaining the physical solutions for the gap equations. The lightest nonet scalar mesons ($\sigma$, $f_0$, $a_0$ and $\kappa$) appearing as the chiral partners of the nonet pseudoscalar mesons are found to be as the composite Higgs bosons with masses below the characteristic energy scale $M_c \sim 1.2$ GeV. In particular, the mass of the singlet scalar (or the $\sigma$) is found to be $m_{\sigma} \simeq 688$ MeV.

We evaluate systematically some contributions of the QCD scalar mesons, including radiative decay-productions, not considered with a better attention until now in the evaluation of the hadronic contributions to the muon anomaly. The sum of the scalar contributions to be added to the existing Standard Model predictions a_mu^SM are estimated to be a^S_mu= 11.75(8.25)x10^-10, where the errors are dominated by the ones from the experimental widths of these scalar mesons. This result suggests that the value of a_mu^SM and its errors might have been underestimated in previous works. The inclusion of these new effects leads to a perfect agreement (0.1 sigma) of the measured value a^exp_mu and a_mu^SM from tau-decay and implies a 1.7 sigma discrepancy between a^exp_mu and a_mu^SM from e^+e^- --> hadrons data. More refined unbiased theoretical estimates require improved measurements of the scalar meson masses and widths. The impact of our results to a_mu^SM is summarized in the conclusions.

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 1.49--1.64 GeV.

We summarize the different features of the scalar mesons from QCD spectral sum rule analyses of the two- and three-point functions. The results do not favour the u-bar u + d-bar d interpretation of the broad and low mass sigma (0.6), and the u-bar s resonance nature of the eventually observed kappa(0.9) meson. We also discuss some OZI-violating and classic semileptonic and radiative decay processes which can reveal in a model-independent way the eventual gluon component sigma_B of the sigma. In a meson-gluonium mixing scenario, one also expects an observation of the K K-bar final states from the sigma_B which may compete (if phase space allowed) with the one from a low mass s-bar s state assumed in the literature to be the SU(3) partner of the $\sigma (0.6)$ if this latter is a u-bar u + d-bar d state

I discuss how an extra light scalar meson multiplet could be understood as an effective Higgs nonet of a hidden local U(3) symmetry. There is growing evidence that low energy data requires in addition to a conventional (q bar q) nonet near 1.4 GeV, another light scalar nonet-like structure below 1 GeV, (sigma(600), a_0(980), f_0(980), kappa), which could be interpreted as such a Higgs nonet.

We show that two nonets and a glueball provide a consistent description of data on scalar mesons below 1.7 GeV. Above 1 GeV the states form a conventional (q bar q) nonet mixed with the glueball of lattice QCD. Below 1 GeV the states also form a nonet, as implied by the attractive forces of QCD, but of more complicated nature. Near the center they are 4 quark states of the Jaffe type in S-wave, with some (q bar q) in P-wave, but further out they rearrange in colour to two colourless (q bar q) pairs and finally as meson-meson states. A simple effective chiral model for such a system with two scalar nonets can be made involving two coupled linear sigma models. One of these could be looked upon as the Higgs sector of nonpertubative QCD.\

Complex poles of the unitarized pi-pi scattering amplitude in nuclear matter are studied. Partial restoration of chiral symmetry is modeled by the decrease of in-medium pion decay constant f*_{pi}. For large chiral restoration (f*_{pi}/f_{pi} << 1), 2nd sheet poles in the scalar (sigma) and the vector (rho) mesons are both dictated by the Lambert W function and show universal softening as f*_{pi} decreases. In-medium pi-pi cross section receives substantial contribution from the soft mode and exhibits a large enhancement in low-energy region. Fate of this universality for small chiral restoration (f*_{pi}/f_{pi} ~ 1) is also discussed.

We discuss scalar mesons properties on the light of chiral dynamics. Considering them as the chiral partners of pseudo-scalar mesons we propose an explanation to their unusual properties based on non-trivial vacuum effects coming from the interplay between spontaneous breaking of chiral symmetry and the violation of $U_A(1)$ symmetry by instantons. Including vector mesons as external sources we work out predictions for radiative decays of vector mesons and compare some of them with recent experimental results from high luminosity $\Phi$ factories.

It is shown that most of the unusual properties of the lowest lying scalar (and pseudoscalar) mesons can be understood, at the qualitative and quantitative level, on the basis of the breakdown of the $U_A(1)$ symmetry coupled to the vacuum expectation values of scalars by the spontaneous breaking of chiral symmetry.

The quark-level linear sigma model is revisited, in particular concerning the identification of the f_0(400-1200) (or \sigma(600)) scalar meson as the chiral partner of the pion. We demonstrate the predictive power of the linear sigma model through the pi-pi and pi-N s-wave scattering lengths, as well as several electromagnetic, weak, and strong decays of pseudoscalar and vector mesons. The ease with which the data for these observables are reproduced in the linear sigma model lends credit to the necessity to include the sigma as a fundamental q\bar{q} degree of freedom, to be contrasted with approaches like chiral perturbation theory or the confining NJL model of Shakin and Wang.

Starting with just one bare seed for each member of a scalar nonet, we investigate when it is possible to generate more than one hadronic state for each set of quantum numbers. In the framework of a simple model, we find that in the I=1 sector it is possible to generate two physical states with the right features to be identified with the a_0(980) and the a_0(1450). In the I=1/2 sector, we can generate a number of physical states with masses higher than 1 GeV, including one with the right features to be associated with the K_0^*(1470), but none which can be identified with the light kappa scalar meson. The I=0 sector is the most complicated and elusive: since all outcomes are very strongly model dependent, we cannot draw any robust conclusion. Nevertheless, we find that in that case too, depending on the coupling scheme adopted, the occurrence of numerous states can be achieved. This shows that dynamical generation of physical states is a possible solution to the problem of accounting for more scalar mesons than can fit in a single nonet, as experiments clearly deliver.

We apply QCD Finite Energy Sum Rules to the scalar-isoscalar current to determine the lightest $u \bar{u} + d \bar{d}$ meson in this channel. We use `pinch-weights' to improve the reliability of the QCD predictions and reduce the sensitivity to the cut-off $s_0$. A decaying exponential is included in the weight function to allow us to focus on the contribution from low mass states to the phenomenological integral. On the theoretical side we include OPE contributions up to dimension six and a contribution due to instantons taken from the Instanton Liquid Model. Phenomenologically, we incorporate experimental data by using a coupling scheme for the scalar current which links the vacuum polarisation to the $\pi \pi$ scattering amplitude via the scalar form factor. We find that the sum rules are well saturated for certain instanton parameters. We conclude that the $f_0(400-1200)$ definitely contains a large $u \bar{u} + d \bar{d}$ component, whereas the $f_0(980)$ most likely does not. We are able to estimate the average light quark mass and find $m_q(1 \GeV) = 5.2 \pm 0.6$ MeV.

A model-independent expression resembling Breit-Wigner formulae is derived for the description of resonances which appear in meson-meson scattering. Starting point is a unitarised meson model, but reduced to a simpler form and freed from the specific assumption about the confining force. The parameters of the resulting ``Resonance-Spectrum Expansion'' are directly related to the confinement spectrum and the mechanism of $^3P_0$ valence-quark-pair creation for OZI-allowed hadronic decay, and not to the central positions and widths of resonances. The method also provides a straightforward explanation for the origin of the light scalar mesons without requiring extra degrees of freedom.

We review the analysis of the quark and gluon substructures of the scalar mesons from QCD spectral sum rules and some low-energy theorems applied to the scalar QCD anomaly current. The present data favour equal components of \bar uu+ \bar dd and of gg in the wave functions of the low-mass (below 1 GeV) scalar mesons, which make the wide \sigma and the narrow f_0(980) as \eta'-like particles, which can have strong couplings to meson pairs through OZI violations. A coherent picture of the other I=0 scalar mesons spectra within this mixing scheme is shortly discussed. We also expect the a_0(980) to be the lowest isovector \bar ud state, and the K^*_0(1430) its \bar ds partner.

The relativistic four-quark equations are found in the framework of the dispersion relation technique. The calculations of non-qq meson amplitudes estimate the contributions of three subamplitudes: four-quark amplitude, glueball amplitude and hadronic molekule amplitudes.

Within the increasing experimental evidence of the existence of the I=0 scalar low mass and wide \sigma meson, we review the {\it first} analysis of its quark and gluon substructure from QCD spectral sum rules and some low-energy theorems. The present data favour equal components of \bar uu+ \bar dd and of gg in its wave function, which make the wide sigma and the narrow f_0(980) as \eta'-like particles. A coherent picture of the other I=0 scalar mesons spectra within this mixing scheme is shortly discussed. We also expect the a_0(980) to be the lowest isovector \bar ud state, and the K^*_0(1430) its \bar ds partner.

Within the increasing experimental evidence of the existence of the I=0 scalar low mass and wide \sigma meson, we review the {\it first} analysis of its quark and gluon substructure from QCD spectral sum rules and some low-energy theorems. The present data favour equal components of \bar uu+ \bar dd and of gg in its wave function, which make the wide sigma and the narrow f_0(980) as \eta'-like particles. A coherent picture of the other I=0 scalar mesons spectra within this mixing scheme is shortly discussed. We also expect the a_0(980) to be the lowest isovector \bar ud state, and the K^*_0(1430) its \bar ds partner.

The structure of the light scalar mesons is elucidated by the investigation of the S-matrix poles and the q-qbar spectral density in a coupled channel model that includes pi-pi, K-Kbar and q-qbar channels. It is shown that the dynamical origin of the sigma and f0(980) mesons is consistent with the observed spectrum of the scalar states. The K-Kbar molecular picture of the f0(980) is in good agreement with recent experimental data on the decay phi - gamma pi pi from Novosibirsk.

We calculate the masses, widths and coupling constants of \sigma and color \bar{3} scalar diquark states at finite temperature (T) and chemical potential (\mu) using the effective instanton induced `t Hooft interaction Lagrangian and the N-quantum approach to quantum field theory. These results are then used to construct a non-perturbative solution to the `t Hooft model beyond the mean field approximation valid for any T and \mu both in the Nambu-Goldstone and in the Breit-Wigner phases of the model. We find two branches of the diquark state for non-zero \mu due to the unequal treatment of particles and anti-particles at finite density.

The spectra of some 0++ four-quark states, which are composed of \bar qq pairs, are calculated in QCD. The light four-quark states are calculated using the traditional sum rules while four-quark states containing one heavy quark are computed in HQET. For constructing the interpolating currents, different couplings of the color and spin inside the \bar qq pair are taken into account. It is found that the spin and color combination has little effect on the mass of the four-quark states.

We suggest a mechanism of current quark mass generation which is easily understandable by calculating the corresponding functional integral in the tadpole approximation within the instanton liquid model. Quark condensate excitations influenced by the phononlike modes are analyzed and the Gell-Mann-Oakes-Renner relation with realistic pion mass is obtained. The picture of $\sigma$-meson as being mixed with the soft scalar glueball-like excitation is discussed.

In a U(3)x U(3) quark chiral model of the Nambu-Jona-Lasinio type with the 't Hooft interaction, the ground scalar isoscalar mesons and a scalar glueball are described. The glueball (dilaton) is introduced into the effective meson Lagrangian written in a chirally symmetric form on the base of scale invariance. The singlet-octet mixing of scalar isoscalar mesons and their mixing with the glueball are taken into account. Mass spectra of the scalar mesons and glueball and their strong decays are described.

In the framework of the dispersion relation N/D-method we restore the low-energy pi-pi (IJ^{PC}=00^{++})-wave amplitude sewing it with the previously obtained K-matrix solution for the region 450--1900 MeV [V.V. Anisovich, Yu.D. Prokoshkin and A.V. Sarantsev, Phys. Lett. B389, 388 (1996)]. The restored N/D-amplitude has a pole on the second sheet of the complex-s plane, near the pi-pi threshold. We discuss the hypothesis that this low-mass pole, or the low-mass sigma-meson, corresponds to the dynamical state related to the confinement forces, that is the eyewitness of confinement.

We review how a certain effective chiral Lagrangian approach to pi pi scattering, pi K scattering and eta-prime decay to eta pi pi provides evidence for the existence of light scalars sigma(550) and kappa(900) as well as describing the f0(980) and the a0(980). An attempt to fit these into a nonet suggests that their structure is closer to a dual quark-dual antiquark than to a quark-antiquark. A possible mechanism to explain the next higher mass scalar nonet is also proposed.

The lightest scalar and pseudoscalar nonets are discussed within the [A framework of the broken old UxU3 linear sigma model, and it is shown that already at the tree level this model works remarkably well predicting scalar masses and couplings not far from present experimental values, when all parameters are fixed from the pseudoscalar masses and decay constants. The linear sigma model is the simplest way to implement chiral symmetry together with the broken SU3 of the quark model, and this, not well known, success in understanding experiment is comparable to that of the naive quark model for the heavier multiplets. It is argued that this strongly suggest that the light and very broad sigma resonance exists near 500 MeV.

A Hoelder inequality analysis of the QCD Laplace sum-rule which probes the non-strange (n\bar n) components of the I={0,1} (light-quark) scalar mesons supports the methodological consistency of an effective continuum contribution from instanton effects. This revised formulation enhances the magnitude of the instanton contributions which split the degeneracy between the I=0 and I=1 channels. Despite this enhanced isospin splitting effect, analysis of the Laplace and finite-energy sum-rules seems to preclude identification of a_0(980) and a light broad sigma-resonance state as the lightest isovector and isoscalar spin-zero $n\bar n$ mesons. This apparent decoupling of sigma [\equiv f_0(400-1200)] and a_0(980) from the quark n\bar n scalar currents suggests either a non-q \bar q or a dominantly s\bar s interpretation of these resonances, and further suggests the possible identification of the f_0(980) and a_0(1450) as the lightest I={0,1} scalar mesons containing a substantial n\bar n component.

QCD Laplace sum-rules for light-quark I=0,1 scalar currents are used to investigate candidates for the lightest $q\bar q$ scalar mesons. The theoretical predictions for the sum-rules include instanton contributions which split the degeneracy between the I=0 and I=1 channels. The self-consistency of the theoretical predictions is verified through a H\"older inequality analysis, confirming the existence of an effective instanton contribution to the continuum. The sum-rule analysis indicates that the f_0(980) and a_0(1450) should be interpreted as the lightest $q\bar q$ scalar mesons. This apparent decoupling of the f_0(400-1200) (or $\sigma$) and a_0(980) from the quark scalar currents suggests a non-$q\bar q$ interpretation of these resonances.