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This review of the quark-level linear \sigma model is based upon the dynamical realization of the pseudoscalar and scalar mesons as a linear representation of SU(2) x SU(2) chiral symmetry, with the symmetry weakly broken by current quark masses. In its simplest SU(2) incarnation, with two non-strange quark flavors and three colors, this nonperturbative theory, which can be selfconsistently bootstrapped in loop order, is shown to accurately reproduce a host of low-energy observables with only one parameter, namely the pion decay constant f_\pi. Extending the scheme to SU(3) by including the strange quark, equally good results are obtained for many strong, electromagnetic, and weak processes just with two extra constants, viz. f_K and $\langle\pi|H_{\mbox{\scriptsize weak}}|K\rangle$. Links are made with the vector-meson-dominance model, the BCS theory of superconductivity, and chiral-symmetry restoration at high temperature. Finally, these ideas are cautiously generalized to the electroweak sector, including the W, Z, and Higgs bosons, and also to CP violation., Comment: Review, 34 pages, 14 figures with 24 graphs, 1 table, version accepted for publication in Fortschritte der Physik; v2: typos and 3 references corrected, Ref. [175] updated

A status report on the topic Compton scattering and polarizabilities is presented with emphasis on the scalar t-channel as entering into dispersion theory. Precise values for the polarizabilities are obtained leading to $\alpha_p = 12.0\pm 0.6$ $(12.0)$, $\beta_p=1.9\mp 0.6$ $(1.9)$, $\alpha_n= 12.5\pm 1.7$ $(13.4)$, $\beta_n= 2.7 \mp 1.8$ $(1.8)$ in units of $10^{-4}$ fm$^3$ and $\gamma^{(p)}_\pi = -36.4 \pm 1.5$ $(-36.6)$, $\gamma^{(n)}_\pi = 58.6 \pm 4.0$ $(58.3)$, $(\gamma^{(p)}_0= -0.58\pm 0.20)$, $(\gamma^{(n)}_0 = +0.38\pm 0.22)$ in units of $10^{-4}$ fm$^4$, for the proton (p) and neutron (n), respectively. The data given with an error are {\it recommended} experimental values with updates compared to [1] where necessary, the data in parentheses are predicted values. These predicted values are not contained in [1], but are the result of a newly developed analysis which is the main topic of the present paper. The most important recent discovery is that the largest part of the electric polarizability and the total diamagnetic polarizability of the nucleon are properties of the $\sigma$ meson as part of the constituent-quark structure, as expected from the mechanism of chiral symmetry breaking. This view is supported by an experiment on Compton scattering by the proton carried out in the second resonance region, where a large contribution from the $\sigma$ meson enters into the scattering amplitudes. This experiment led to a determination of the mass of the $\sigma$ meson of $m_\sigma = 600 \pm 70$ MeV. From the experimental $\alpha_p$ and predicted differences $(\alpha_n - \alpha_p)$ neutron polarizabilities in the range $\alpha_n= 12.0 - 13.4$ are predicted, where the uncertainties are related to the $f_0(980)$ and $a_0(980)$ scalar mesons.

We comment on a recent paper by Giacosa, Gutsche, and Lyobovitskij, in which it is argued that a quarkonium interpretation of the $\sigma$ meson should give rise to a much smaller two-photon decay width than commonly assumed. The reason for this claimed discrepancy is a term in the transition amplitude, necessary for gauge invariance, which allegedly is often omitted in the literature, including the work of the present authors. Here we show their claims to be incorrect by demonstrating, in the context of the Quark-Level Linear $\sigma$ Model, that the recently extracted experimental value $\Gamma_{\sigma\to2\gamma}=(4.1\pm0.3)$ keV is compatible with a $q\bar{q}$ assignment for the $\sigma$, provided that meson loops are taken into account as well., Comment: RevTeX, 4 pages, Comment on Phys. Rev. D 77 (2008) 034007 [0710.3403 [hep-ph]]

Light constituent quark masses and the corresponding dynamical quark masses are determined by data, the quark-level linear sigma model, and infrared QCD. This allows to define effective nonstrange and strange current quark masses, which reproduce the experimental pion and kaon masses very accurately, by simple additivity. In contrast, the usual nonstrange and strange current quarks employed by the Particle Data Group and Chiral Perturbation Theory do not allow a straightforward quantitative explanation of the pion and kaon masses., Comment: 4 pages, AIP style, contribution to conference ``Quark Confinement and the Hadron Spectrum VII'', Azores, Portugal, 2-7 September 2006

The contribution of strange sea quarks to the proton mass and spin, as well as the related pion-nucleon sigma term, are briefly revisited, in the light of new experimental and lattice results. Also the predictions of chiral perturbation theory for these quantities are discussed., Comment: 5 pages, plain LaTeX

Chiral symmetry breaking (ChSB) is reviewed to some extent within the quark-level-linear-sigma-model (QLL$\sigma$M) theory and standard chiral perturbation theory (ChPT). It is shown, on the basis of several examples related to the pion, as a well-known Goldstone boson of chiral symmetry breaking, that even the non-unitarized QLL$\sigma$M approach accounts, to a good approximation, for a rather simple, self-consistent, linear, and very predictive description of Nature. On the other hand, ChPT -- even when unitarized -- provides a highly distorted, nonlinear, hardly predictive picture of Nature, which fits experiment only at the price of a lot of parameters, and requires a great deal of unnecessary theoretical effort. As the origin of this distortion, we identify the fact that ChPT, reflecting only direct ChSB by nonvanishing, current-quark-mass values, does not -- contrary to Quantum Chromodynamics (QCD) and the QLL$\sigma$M -- contain any mechanism for the spontaneous generation of the dynamical component of the constituent quark mass. This leads to a very peculiar picture of Nature, since the strange current quark mass has to compensate for the absence of nonstrange dynamical quark masses. We thus conclude that standard ChPT -- contrary to common wisdom -- is unlikely to be the low-energy limit of QCD. On the contrary, a chiral perturbation theory derived from the QLL$\sigma$M, presumably being the true low-energy limit of QCD, is expected instead to provide a distortion-free description of Nature, which is based on the heavy standard-model Higgs boson as well as light scalar mesons, as the source of spontaneous generation of current and dynamical quark masses, respectively., Comment: 16 pages, 1 EPS figure

After describing the SU(2) linear sigma model (L$\sigma$M), we dynamically generate it using the B.W. Lee null tadpole sum (characterizing the true vacuum) together with the dimensional regularization lemma. Next we generate the chiral-limiting (CL) nonstrange and strange constituent quark masses ${\hat m}=325.7$ MeV; $m_s=486$ MeV away from the CL. Finally, we study vector meson dominance (VMD) and the pion, kaon charge radii and the loop-order $\rho\to\pi\gamma$, $\pi^0\to\gamma\gamma$ amplitudes in the quark model. Lastly, we verify this procedure using tree-order VMD graphs., Comment: 8 pages, 2 figures

We start by studying the Goldberger-Treiman discrepancy (GTd) $\Delta = (2.259\pm 0.591)%$. Then we look at the $\pi N$ $\sigma$ term, with the dimensionless ratio $\sigma_N/2m_N=3.35%$. Finally we return to predicting (via the quark model) the $\pi N$ coupling constant, with GTd $\Delta\to 0$ as $m_q\to m_N/3$., Comment: 5 pages, no figures, LaTeX; v2: corrected typo in eq. (6); v3: a discussion concerning chiral perturbation theory, and a co-author added

Data indicate that $\Delta I=1/2$ transitions account for 4.5-4.7% of both CP conserving and CP violating $K \to 2\pi$ decays, as well as CP conserving radiative $K \to \pi\pi\gamma$ processes. Observed $K \to \pi\pi\gamma/\pi\pi$ branching ratios are shown to scale near $\alpha/\pi$ or $\alpha/2\pi$. The $K_L$-$K_S$ mixing angle $\phi$ and the semileptonic weak-rate asymmetry $\delta$ are reviewed, and theory is shown to be consistent with data. Also, $K \to 2\pi$ $\Delta I=1/2$ dominance is studied in the context of the chiral constituent quark model, displaying again excellent agreement with data. Finally, indirect and direct kaon CP violation are successfully described in the framework of photon-mediated loop graphs. This suggests that kaon CPV can be understood via second-order weak transitions, radiatively corrected., Comment: Plain LaTeX, 12 pages, 4 EPS figures. PACS numbers: 11.30.Er, 12.15.Lk, 13.20.Eb, 13.40.Ks. Version 2: 14 pages, improved presentation, introductory and concluding remarks as well as references added; accepted for publication in Mod. Phys. Lett. A (2004)

By comparing SU(3)-breaking scales of linear mass formulae, it is shown that the lowest vector and scalar mesons all have a $\bar{q}q$ configuration, while the ground-state octet and decuplet baryons are $qqq$. Also, the quark-level linear $\sigma$ model is employed to predict similar $\bar{q}q$ and $qqq$ states. Furthermore, the approximate mass degeneracy of the scalar $a_0$(985) and $f_0$(980) mesons is demonstrated to be accidental. Finally, it is shown that various strong, electromagnetic, and weak mesonic decay rates are successfully explained within the framework of the quark-level linear $\sigma$ model., Comment: 11 pages, REVTeX4, no figures. PACS nos.: 14.40.-n, 12.40.-y, 11.30.Rd, 13.25.Jx, 13.40.Hq, 13.25.Es, 13.25.Ft

The quark-level linear sigma model is employed to compute a variety of electromagnetic and weak observables of light mesons, including pion and kaon form factors and charge radii, charged-pion polarizabilities, semileptonic weak $K_{\ell3}$ decay, semileptonic weak radiative pion and kaon form factors, radiative decays of vector mesons, and nonleptonic weak $K_{2\pi}$ decay. The agreement of all these predicted observables with experiment is striking. In passing, the tight link between the linear sigma model and vector-meson dominance is shown. Some conclusions are drawn on the linear sigma model in connection with lattice and renormalization-group approaches to QCD., Comment: plain LaTeX, 21 pages, 8 EPS figures, 1 table. PACS numbers: 12.40.-y, 11.30.Rd, 13.40.Gp, 13.40.Hq, 13.20.Cz, 13.20.Eb, 13.25.Es, 12.40.Vv

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., Comment: 15 pages, plain LaTeX, 3 EPS figures

The assignments of the isoscalar scalar mesons f0(980), f0(1370), and f0(1500) in terms of their qqbar substructure is still a matter of heated dispute. Here we employ the weak and electromagnetic decays D(s)(+) to f0+pi(+) and f0 two-photon decays, respectively, to identify the f0(980) and f0(1500) as mostly ssbar, and the f0(1370) as dominantly nonstrange, in agreement with previous work. The two-photon decays can be satisfactorily described with quark as well as with meson loops, though the latter ones provide a less model-dependent and more quantitative description., Comment: v1, 15 pages, plain LaTeX, 1 eps figure. v2, 18 pages, plain LaTeX (figure included). More discussion, especially on the f0(1370) and its empirical two-photon width

The scalar-meson assignments of Shakin and Wang in a generalized Nambu--Jona-Lasinio model are contradicted by recent experimental information. Also the strict distinction made by these authors between ``intrinsic'' and ``dynamically generated'' states is contested, as well as a number of other statements., Comment: 7 pages, (v2 cosmetics, v3 corrections in one reference), to be published in Phys.Rev.D

The SU(3) linear sigma model is dynamically generated in loop-order using the nonstrange-strange basis. Only self-consistent logarithmic divergent graphs are needed, with quadratic divergent graphs replaced by SU(3) mass-shell equal splitting laws. The latter lead to an eta'-eta mixing angle of 41.84 deg which is consistent with phenomenology. Finally this above SU(3) linear sigma model in turn predicts strong decay rates which are all compatible with data., Comment: 14 pages, 1 figure, elsart; version to appear in J. Phys. G, appendix added, some typos corrected

The bound-state Schwinger-Dyson and Bethe-Salpeter (SD-BS) approach is chirally well-behaved and provides a reliable treatment of the eta-eta' complex although a ladder approximation is employed. Allowing for the effects of the SU(3) flavor symmetry breaking in the quark-antiquark annihilation, leads to the improved eta-eta' mass matrix., Comment: 11 pages, 1 figure, revtex. Invited talk presented by D. Klabucar at the workshop "Quark Matter in Astro- and Particle Physics" (27.-29. November 2000, Rostock, Germany), to appear in the proceedings of this workshop

We exploit the W-emission process to study the measured weak decay of the D(s,+)(1.9686) meson into f(0)(980) and a positively charged pion. We conclude that the scalar f(0)(980) meson contains mostly strange-antistrange flavors, which is supported by different model studies., Comment: 5 pages, plain LaTeX, minor changes, acknowledgments