Abstract
For several decades, the \(\sigma /f_0(500)\) and \(\kappa /K_0^*(700)\) resonances have been subject to long-standing debate. Both their existence and properties were controversial until very recently. In this tutorial review, we compare model-independent dispersive and analytic techniques versus unitarized Chiral Perturbation Theory, when applied to the lightest scalar mesons \(\sigma /f_0(500) \) and \(\kappa /K_0^*(700) \). Generically, the former have settled the long-standing controversy about the existence of these states, providing a precise determination of their parameters, whereas unitarization of chiral effective theories allows us to understand their nature, spectroscopic classification and dependence on QCD parameters. Here we review in a pedagogical way their uses, advantages and caveats.
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Notes
Although we have quoted the last RPP edition, all previous ones can be found in the Particle Data Group web-page at https://pdg.lbl.gov/rpp-archive/.
Literally, the authors of that compilation wrote: “We are beginning to think that \(\kappa \) should be classified along with flying saucers, the Loch Ness Monster, and the Abominable Snowman”.
See reference 15 in [41].
C. Hanhart. Private communication.
On top of that even with the same method and data there were several ambiguities leading to different possible solutions, like for instance the so-called up or down solutions [13, 14]. Over the years it has been possible to disentangle those ambiguities with other input or dispersion relations [93]. We omit this discussion here and refer the reader to the review in [36] and references therein.
In the literature these are also written as \(\delta _{I\ell },\delta ^{(I)}_\ell \) and \(\eta _{I\ell },\eta ^{(I)}_\ell \) as we will see in several figures of this review.
The only exception is the \(\eta '\) whose huge mass is due to another well-understood effect, which is the chiral anomaly, beyond the scope of this tutorial review.
While this work was in the referee process a review has appeared, where large-\(N_c\) behavior of tetraquarks is reviewed in great detail [212].
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Acknowledgements
We acknowledge J.A. Oller for encouraging us to write a tutorial review on this topic. We thank I. Danilkin for his corrections on the manuscript. This project has received funding from the Spanish Ministerio de Ciencia e Innovación grant PID2019-106080GB-C21 and the European Union’s Horizon 2020 research and innovation program under grant agreement No 824093 (STRONG2020). AR acknowledges the financial support of the U.S. Department of Energy contract DE-SC0018416 at William & Mary, and contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. JRE acknowledges financial support from the Swiss National Science Foundation under Project No. PZ00P2 174228.
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Peláez, J.R., Rodas, A. & Ruiz de Elvira, J. Precision dispersive approaches versus unitarized chiral perturbation theory for the lightest scalar resonances \(\sigma /f_0(500) \) and \(\kappa /K_0^*(700) \). Eur. Phys. J. Spec. Top. 230, 1539–1574 (2021). https://doi.org/10.1140/epjs/s11734-021-00142-9
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DOI: https://doi.org/10.1140/epjs/s11734-021-00142-9