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The importance of $S$-matrix unitarity in realistic meson spectroscopy is reviewed, both its historical development and more recent applications. First the effects of imposing $S$-matrix unitarity on meson resonances is demonstrated in both the elastic and the inelastic case. Then, the static quark model is revisited and its theoretical as well as phenomenological shortcomings are highlighted. A detailed account is presented of the mesons in the tables of the Particle Data Group that cannot be explained at all or only poorly in models describing mesons as pure quark-antiquark bound states. Next the earliest unitarised and coupled-channel models are revisited, followed by several examples of puzzling meson resonances and their understanding in a modern unitarised framework. Also, recent and fully unquenched lattice descriptions of such mesons are summarised. Finally, attention is paid to production processes, which require an unconventional yet related unitary approach. Proposals for further improvement are discussed., Comment: Invited review paper, 51 pages, 16 figures, 4 tables; to be published in "Progress in Particle and Nuclear Physics" 2021. v2: several typos corrected, in particular in Eqs. (15), (17), (36), and (37) in Section 2; some accompanying remarks rephrased for clarity; Eq. (41) improved. NOTE: v3 is identical to v2; published version (in print and online) largely corresponds to v1; v3 (=v2) is correct

An unambiguous definition of meson resonance masses requires a description of the associated phase shifts in terms of a manifestly unitary $S$-matrix and its complex poles. However, the commonly used Breit-Wigner (BW) parametrisations can lead to appreciable deviations. We demonstrate this for a simple elastic resonance, viz. $\rho(770)$, whose pole and BW masses turn out to differ by almost 5 MeV. In the case of the very broad $f_0(500)$ and $K_0^\star(700)$ scalar mesons, the discrepancies are shown to become much larger, while also putting question marks at the listed PDG BW masses and widths. Furthermore, some results are reviewed of a manifestly unitary model for meson spectroscopy, which highlight the potentially huge deviations from static model predictions. Finally, a related unitary model for production amplitudes is shown to explain several meson enhancements as non-resonant threshold effects, with profound implications for spectroscopy., Comment: 6 pages, Acta Physica Polonica B style; Talk by G. Rupp at Workshop Excited QCD 2019, Schladming, Austria, Jan. 30 - Feb. 3, 2019. Version 2: some rephrasing on Breit-Wigner parametrisations; version accepted for publication in Acta Phys. Pol. B Proc. Suppl

Besides the unitarity and symmetry requirements for a multi-resonance scattering amplitude, several other natural conditions can easily exclude unrealistic proposals. In particular, the behaviour of singularities under the variation of model parameters yields important information. We discuss how resonance poles should move in the complex-energy plane when coupling constants and masses are varied, how resonances above threshold can turn into bound states below threshold and how the light-quark spectrum can be turned into the spectrum of heavy quarks, with one and the same analytic expression for the scattering amplitude. Moreover, it is shown that perturbative approximations usually do not satisfy those natural conditions., Presented at Excited QCD 2017, Sintra (Portugal), 8-12 May 2017, 6 pages, plain LaTeX; minor changes to the text, one reference corrected and one reference added

Masses and widths of the four light scalar mesons σ , κ , a 0 ( 980 ) and f 0 ( 980 ) may be reproduced in a model where mesons scatter via a q q ¯ loop. A transition potential is used to couple mesons to q q ¯ at a radius of ∼ 0.57 fm . Inside this radius, there is an infinite bare spectrum of confined q q ¯ states, for which a harmonic oscillator is chosen here. The coupled-channel system approximately reproduces the features of both light and heavy meson spectroscopy. The generation of σ , κ , a 0 ( 980 ) and f 0 ( 980 ) is a balance between attraction due to the q q ¯ loop and suppression of the amplitudes at the Adler zeros. Phase shifts increase more rapidly as the coupling constant to the mesons increases. This leads to resonant widths which decrease with increasing coupling constant—a characteristically non-perturbative effect.

The very recently observed D * sJ (2463) meson is described as a $J^{P}\! = \!1^{ + }$ $c\bar{s}$ bound state in a unitarised meson model, owing its existence to the strong OZI-allowed $^{3}\!P_{0}$ coupling to the nearby S-wave $D^{*}\!K$ threshold. By the same non-perturbative mechanism, the narrow axial-vector D s1(2536) resonance shows up as a quasi-bound-state partner embedded in the $D^{*}\!K$ continuum. With the same model and parameters, it is also shown that the preliminary broad 1 + D 1(2400) resonance and the established narrow 1 + D 1(2420) may be similar $c\bar{n}$ partners, as a result of the strong OZI-allowed $^{3}\!P_{0}$ coupling to the nearby S-wave $D^{*}\!\pi$ threshold. The continuum bound states D 1(2420) and D s1(2536) are found to be mixtures of 33% $^{3\!}P_{1}$ and 67% $^{1\!}P_{1}$ , whereas their partners D 1(2400) and D * sJ (2463) have more or less the opposite $^{2S + 1}\!P_1$ -state content, but additionally with some $D^{*}\!\pi$ or $D^{*}\!K$ admixture, respectively. The employed mechanism also reproduces the ratio of the K L -K S mass difference and the K S width, by describing K L as a bound state embedded in the $\pi\pi$ continuum. The model’s results for $J^{P}\! = \!1^{ + }$ states containing one b quark are also discussed.

It is shown that the incomplete scalar meson nonets found by Tornqvist [1] and Tornqvist & Roos [2], employing a new version of the Helsinki unitarised quark model, should in fact be complete, including an as yet unconfirmed light $K_0^*$ below 1 GeV (old $\kappa$ ) and the established $f_0$ (1500). A detailed comparison is presented with the predictions of the Nijmegen unitarised meson model, in which two complete scalar nonets show up below 1.5 GeV [3]. The reason for the flavour-nonet breaking found in [1,2] we argue to originate in the use of coupling constants for the three-meson vertex which are not independent of flavour. Also some statements made in [1] are critically reviewed.

A measurement of the branching ratio for K s → e + e − decay was performed with the CPLEAR detector at LEAR. Full event reconstruction together with calorimeter e/π separation allowed for powerful background rejection and high signal acceptance. Analysis of data taken up to 1994 yields the result: Br(K s → e + e − ) −7 (90 % CL).

Antiproton annihilation at rest in gaseous hydrogen (16 bar pressure) to the final states ρ(770)X andf2(1270)X, whereX stands for one or more neutral mesons, has been studied at LEAR using the CPLEAR detector. Relative branching ratios are given forX =π0,2π0, 3π0 ω in the ρ case, and forX =π0, η in thef2 case. The annihilation channel\(\bar pp \to \rho 3\pi ^0 \) has not been measured before. The fraction of P-wave annihilation is deduced from the ratiof2π0/ρπ0 to be 0.38±0.07, in good agreement with the result previously obtained by CPLEAR from the ratioK+K-/π+π-.

Radiative transitions in charmonium and bottomonium are calculated in a unitarized meson model, and an analysis of the results and the validity of common approximations is given. The results for charmonium indicate the need for relativistic corrections in our model; those for bottomonium agree very well with experiment (as far as they are available).

The ratio of the branching fractions for p p →K+K− and p p →π+π− was determined with the CPLEAR detector, by stopping antiprotons in a gaseous hydrogen target at 15 bar pressure. It was found to be BR(K+K−)/BR(π+π−)=0.205± 0.016. The fraction of P-wave annihilation at rest at this target density was deduced to be (38±9)%.

The resonances in meson meson scattering and the bound states of quarks andantiquarks are described within the same formalism. Comparison with experiment shows that this strategy is very promising for a detailed study of strong interactions and confinement.

A unitarized nonrelativistic meson model which is successful for the description of the heavy and light vector and pseudoscalar mesons yields, in its extension to the scalar mesons but for the same model parameters, a complete nonet below 1 GeV. In the unitarization scheme, real and virtual meson-meson decay channels are coupled to the quark-antiquark confinement channels. The flavor-dependent harmonic-oscillator confining potential itself has bound states epsilon(1.3 GeV), S(1.5 GeV), delta(1.3 GeV), kappa(1.4 GeV), similar to the results of other bound-state qqbar models. However, the full coupled-channel equations show poles at epsilon(0.5 GeV), S(0.99 GeV), delta(0.97 GeV), kappa(0.73 GeV). Not only can these pole positions be calculated in our model, but also cross sections and phase shifts in the meson-scattering channels, which are in reasonable agreement with the available data for pion-pion, eta-pion and Kaon-pion in S-wave scattering., Comment: A slightly revised version of Zeitschrift fuer Physik C30, 615 (1986)

In this talk we review the limitations of including meson loops as perturbative corrections in a solvable quark model. We first discuss meson-meson scattering within a formalism which treats confined quark pairs and mesons on an equal footing. The interaction between the mesons proceeds through $s$-channel meson-exchange diagrams. Next, we develop a perturbative expansion of the model, and show that the resonance poles found in such a treatment, even by accounting for contributions up to fourth order, do not coincide with those obtained with the full model. We conclude that the resonance predictions based on perturbative approximations in quark models are not reliable, especially in those cases where the coupling to the scattering channels is large., Conference proceedings for EEF70, Workshop on Unquenched Hadron Spectroscopy:Non-Perturbative Models and Methods of QCD vs. Experiment

Low-energy meson-meson scattering data are a powerful testing ground for quark models. Here, we describe the behaviour at threshold of S-wave scattering-matrix singularities. The majority of the full scattering-matrix mesonic poles stem from an underlying confinement spectrum. However, the light scalar mesons K0*(830), a0(980), f0(400-1200), and f0(980) do not, but instead originate in 3P0-barrier semi-bound states. We show that the behaviour of the corresponding poles is identical at threshold. In passing, the light-meson sector is given a firm basis., Comment: Plain LaTeX, 16 pages Contribution to the Second International Workshop on Hadron Physics, Effective Theories of Low Energy QCD, 25-29 September, 2002 (Coimbra, Portugal)

We study meson (π,Κ) form factors in general, and specialize at a later stage to a specific scheme, namely the quark‐level linear σ model (LσM). We 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 Κl3 decay, semileptonic weak radiative pion and kaon form factors, radiative decays of vector mesons, and nonleptonic weak Κ2π decay. We find good to very good agreement with experiment of all these predicted observables.

By comparing SU(3)-breaking scales of linear mass formulae, it is shown that the lowest vector, axial-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. Finally, the approximate mass degeneracy of the scalar $a_0$(980) and $f_0$(980) mesons is demonstrated to be accidental., Comment: 8 Pages, AIP Proc. Style; Contribution to the High-Energy Physics Workshop "Scalar Mesons: an Interesting Puzzle for QCD", May 16-18, 2003, SUNY Institute of Technology, Utica (NY)

Because of the controversial X(3872) meson's very close proximity to the $D^0\bar{D}^{*0}$ threshold, this charmonium-like resonance is often considered a meson-meson molecule. However, a molecular wave function must be essentially of a meson-meson type, viz. $D^0\bar{D}^{*0}$ in this case, with no other significant components. We address this issue by employing a simple two-channel Schrodinger model, in which the $J^{PC}=1^{++}$ $c\bar{c}$ and $D^0\bar{D}^{*0}$ channels can communicate via the $^3P_0$ mechanism, mimicked by string breaking at a sharp distance $a$. Thus, wave functions and their probabilities are computed, for different bound-state pole positions approaching the $D^0\bar{D}^{*0}$ threshold from below. We conclude that at the PDG X(3872) mass and for reasonable values of $a$, viz. 2.0 to 3.0 GeV$^{-1}$, the $c\bar{c}$ component remains quite substantial and certainly not negligible, despite accounting for only about 6 to 10% of the total wave-function probability, owing to the naturally long tail of the $D^0\bar{D}^{*0}$ component.

A potential model for mesons is presented, which combines quark confinement and strong decay in a realistic approach. The multichannel Schr\"odinger formalism is employed to describe a system of one or more permanently closed quark-antiquark channels in interaction with several two-meson channels. For the potential in the $q\overline{q}$ channels a harmonic oscillator with constant frequency is taken. As for the meson channels only Okubo-Zweig-Iizuka-rule-allowed decays into two mesons of the pseudoscalar or vector type are considered. Final-state interactions between these mesons are not yet taken into account. The communication between confined and free channels is supposed to take place via the $^{3}P_{0}$ mechanism, for which a locally approximated transition potential is derived. In order to obtain an analytic solution for the $S$ matrix, the transition potential is treated by using a multi-$\ensuremath{\delta}$-shell method. Kinematically relativistic corrections and color splitting allow a fairly successful treatment of pseudoscalar as well as vector mesons for all quark flavors. The results are confronted with the data and discussed.

In a multichannel treatment of the\(q\bar q\)-system it is shown that the impact of decay on the spectra and wavefunctions of hadrons is substantial. Some consequences for potential models are worked out in detail. We conclude that an “equivalent” confining\(q\bar q\)-potential to be used in an effective single-channel treatment might deviate significantly from the\(q\bar q\)-potential in a realistic multichannel treatment.

The dependence on quark flavour of hyperfine mass separations is studied within the quark flavour number framework for the case of pseudoscalar and vector meson ground states. It is shown that for mesonic partners with at least one beauty valency the flavour dependence with respect to the flavour of the remaining valency differs from the corresponding dependence for partners with at least one nonstrange or at least one strange valency. For partner mesons with at least one charmed valency the hyperfine splittings are almost independent of the other flavour.

We present flavour-symmetric results for the couplings of quark-antiquark systems to meson-meson channels in the harmonic-oscillator expansion. We tabulate their values for all possible open and closed decay channels of pseudo-scalar, vector and scalar mesons. We compare the predictions of a model that employs these flavour-symmetric couplings, both with the results of a model which uses explicitly flavour-dependent couplings, and with experiment., 29 pages, plain LaTeX, rewritten version