The existence and properties of the sigma meson have been controversial for almost six decades, despite playing a central role in the spontaneous chiral symmetry of QCD or in the nucleon-nucleon attraction. This controversy has also been fed by the strong indications that it is not an ordinary quark-antiquark meson. Here we review both the recent and old experimental data and the model independent dispersive formalisms which have provided precise determinations of its mass and width, finally settling the controversy and leading to its new name: $f_0(500)$. We then provide a rather conservative average of the most recent and advanced dispersive determinations of its pole position $\sqrt{s_\sigma}=449^{+22}_{-16}-i(275\pm12)$. 
In addition, after comprehensive introductions, we will review within the modern perspective of effective theories and dispersion theory, its relation to chiral symmetry, unitarization techniques, its quark mass dependence, popular models, as well as the recent strong evidence, obtained from the QCD $1/N_c$ expansion or Regge theory, for its non ordinary nature in terms of quarks and gluons.

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From controversy to precision on the sigma meson: a review on the status of the non-ordinary $f_0(500)$ resonance

From controversy to precision on the sigma meson: A review on the status of the non-ordinary f(0)(500) resonance

The Pade approximation, its mathematical features, and its practical and physical applications are reviewed. We describe the present status of the convergence problem. Among the physical applications, we give particular emphasis to elementary particle theory. We discuss various proofs of convergence, and show why the approximation is particularly well suited for studying strong interaction dynamics. We finally review the various achievements of the approximation in the determination of ππ, πN and NN scattering amplitudes.

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The Padé Approximation and its Physical Applications

We find a parametrization for the most general ππ partial-wave amplitudes consistent with crossing symmetry and isospin invariance. This enables us to write all the constraints on the partial-wave amplitudes which follow from, and which ensure, crossing symmetry. We then discuss how these relations might be useful, and apply them in particular to Brown and Goble's model for ππ scattering

Crossing restrictions on ππ partial waves

The description of strong interaction physics of low-lying resonances is out of the valid range of perturbative QCD. Chiral effective field theories have been developed to tackle the issue. Partial wave dynamics is the systematic tool to decode the underlying physics and reveal the properties of those resonances. It is extremely powerful and helpful for our understanding of the non-perturbative regime, especially when dispersion techniques are utilized simultaneously. Recently, plenty of exotic/ordinary hadrons have been reported by experiment collaborations, e.g. LHCb, Belle, and BESIII, etc.. In this review, we summarize the recent progress on the applications of partial wave dynamics combined with chiral effective field theories and dispersion relations, on related topics, with emphasis on $\pi\pi$, $\pi K$, $\pi N$ and $\bar{K}N$ scatterings.

A Review on Partial-wave Dynamics with Chiral Effective Field Theory and Dispersion Relation.

The amplitude for the elastic scattering of two spinless particles of equal mass \textonehalf{} is expanded in terms of eigenfunctions which form a complete set for a certain class of functions of the Mandelstam variables $s,t,u(s+t+u=1)$ and which display the threshold behavior of the partial-wave amplitudes. The eigenfunctions are generated by a partial differential operator which commutes with the total angular momentum in any of the three channels and which is invariant under $s$, $t$, $u$ permutations. An infinite number of finite-dimensional crossing relations for the partial-wave amplitudes which are necessary and sufficient for the crossing symmetry of the total amplitude are derived, as well as an explicit form for the corresponding crossing matrices. It is shown that the Fourier coefficients of the expansion satisfy a Froissart-Gribov integral representation whose kernel is determined by the imaginary parts of the partial-wave amplitudes.

J. Nuyts

Papers

Simultaneous "Partial-Wave" Expansion in the Mandelstam Variables: Crossing Symmetry for Partial Waves