# Unitarization Technics in Hadron Physics with Historical Remarks

## Abstract

**:**

## 1. Introduction

- The tree approximation to the scattering amplitudes violate badly unitarity. This could also be said for perturbative unitarity, at least in some partial waves.
- The Lagrangians are nonlinear and nonrenormalizable, which makes difficult to compute higher-order corrections. Nowadays, we would better say that there is a rapid proliferation of counterterms as the order of the calculation increases in ChPT, with the state of the art at the two-loop level in ChPT. It is typically simpler and much more predictive to implement lower-order calculations of ChPT within non-perturbative methods (several examples are given along this review related to meson-meson scattering and spectroscopy, like for example, the $\pi \pi $ phase shifts, scalar and vector pion form factors, impact of the resonances ${f}_{0}\left(500\right)$ and $\rho \left(770\right)$ in the low-energy phenomenology, $\eta \to 3\pi $ decays, etc.).
- Even if such corrections could be computed, the resultant renormalized perturbation series would probably diverge, since the perturbation parameter has the strength characteristic of strong interactions. This is clear from phenomenology because hadronic interactions are characterized by plenty of resonances and a rapid saturation of unitarity in many partial-wave amplitudes (PWAs).

## 2. Unitarity

## 3. ERE, $\mathit{K}$-Matrix, IAM and Padé Approximants

#### 3.1. ERE and K-Matrix Approaches

#### 3.2. ERE and IAM

#### 3.3. IAM and Padé Approximants

## 4. Final-State Interactions

#### 4.1. ERE, the Omnès Solution and Coupled Channels

- (i)
- If the asymptotic high-energy behavior of $F\left(s\right)$ is known to be proportional to ${s}^{\nu}$, then$$\begin{array}{c}\hfill p-q-\frac{\phi (\infty )}{\pi}=\nu \phantom{\rule{3.33333pt}{0ex}}.\end{array}$$
- (ii)
- Under changes of the parameters when modeling strong interactions one should keep Equation (67) unchanged. As $\nu $ is a known constant, then$$\begin{array}{c}\hfill p-q-\frac{\phi (\infty )}{\pi}=\mathrm{fixed}\phantom{\rule{3.33333pt}{0ex}}.\end{array}$$

#### 4.2. The IAM for FSI

#### 4.3. KT Formalism

## 5. The $\mathit{N}/\mathit{D}$ Method

#### 5.1. Scattering

#### 5.2. FSI

#### 5.3. The Exact $N/D$ Method in NR Scattering

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Coleman, S.; Wess, J.; Zumino, B. Structure of Phenomenological Lagrangians. I. Phys. Rev.
**1969**, 177, 2239. [Google Scholar] [CrossRef] - Callan, C.G.; Coleman, S.; Wess, J.; Zumino, B. Structure of Phenomenological Lagrangians. II. Phys. Rev.
**1969**, 177, 2247. [Google Scholar] [CrossRef] - Weinberg, S. Dynamical approach to current algebra. Phys. Rev. Lett.
**1967**, 18, 188. [Google Scholar] [CrossRef] - Schwinger, J. Chiral dynamics. Phys. Lett. B
**1967**, 24, 473. [Google Scholar] [CrossRef] - Wess, J.; Zumino, B. Lagrangian method for chiral symmetries. Phys. Rev.
**1967**, 163, 1722. [Google Scholar] [CrossRef] - Gasiorowicz, S.; Geffen, D. Effective Lagrangians and field algebras with chiral symmetry. Rev. Mod. Phys.
**1969**, 41, 531. [Google Scholar] [CrossRef] - Gasser, J.; Leutwyler, H. Chiral Perturbation Theory to One Loop. Ann. Phys.
**1984**, 158, 142. [Google Scholar] [CrossRef] [Green Version] - Ecker, G. Chiral perturbation theory. Prog. Part. Nucl. Phys.
**1995**, 35, 1. [Google Scholar] [CrossRef] [Green Version] - Pich, A. Chiral perturbation theory. Rept. Prog. Phys.
**1995**, 58, 563. [Google Scholar] [CrossRef] [Green Version] - Bernard, V.; Meißner, U.-G. Chiral perturbation theory. Ann. Rev. Nucl. Part. Sci.
**2007**, 57, 33. [Google Scholar] [CrossRef] [Green Version] - Peskin, M.E.; Schroeder, D.V. An Introduction to Quantum Field Theory; CRC Press: Boca Raton, FL, USA, 1995. [Google Scholar]
- Schnitzer, H.J. Current algebra and unitarity. Phys. Rev. Lett.
**1970**, 24, 1384. [Google Scholar] [CrossRef] - Schnitzer, H.J. Current algebra beyond the tree approximation. Phys. Rev. D
**1970**, 2, 1621. [Google Scholar] [CrossRef] - Brown, L.S.; Goble, R.L. Pion-Pion Scattering, Current Algebra, Unitarity, and the Width of the Rho Meson. Phys. Rev. Lett.
**1968**, 20, 346. [Google Scholar] [CrossRef] - Weinberg, S. Pion scattering lengths. Phys. Rev. Lett.
**1966**, 17, 616. [Google Scholar] [CrossRef] - Chew, G.F.; Mandelstam, S. Theory of the low-energy pion-pion interaction. Phys. Rev.
**1960**, 119, 467. [Google Scholar] [CrossRef] - Lehmann, H. Chiral invariance and effective range expansion for pion pion scattering. Phys. Lett.
**1972**, 41, 529. [Google Scholar] [CrossRef] - Roiesnel, C.; Truong, T.N. Resolution of the η → 3π Problem. Nucl. Phys. B
**1981**, 187, 293. [Google Scholar] [CrossRef] - Truong, T.N. Chiral Perturbation Theory and Final State Theorem. Phys. Rev. Lett.
**1988**, 61, 2526. [Google Scholar] [CrossRef] - Truong, T.N. Remarks on the unitarization methods. Phys. Rev. Lett.
**1991**, 67, 2260. [Google Scholar] [CrossRef] - Khuri, N.N.; Treiman, S.B. Pion-pion scattering and K
^{+/-}→ 3π decay. Phys. Rev.**1960**, 119, 1115. [Google Scholar] [CrossRef] - Oller, J.A. A Brief Introduction to Dispersion Relations. With Modern Applications; Springer Briefs in Physics; Springer: Heidelberg, Germany, 2019. [Google Scholar]
- Muskhelishvili, W.I. Singular Integral Equations; Springer: Amsterdam, The Netherlands, 1958. [Google Scholar]
- Watson, K.M. Some general relations between the photoproduction and scattering of π mesons. Phys. Rev.
**1955**, 95, 228. [Google Scholar] [CrossRef] - Tanabashi, M. Particle Data Group. Phys. Rev. D
**2018**, 98, 030001. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Oset, E. N/D description of two meson amplitudes and chiral symmetry. Phys. Rev. D
**1999**, 60, 074023. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Oset, E. Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the σ, f
_{0}(980), a_{0}(980) scalar mesons. Nucl. Phys. A**1997**, 620, 438. [Google Scholar] [CrossRef] [Green Version] - Roy, S.M. Exact integral equation for pion-pion scattering involving only physical region partial waves. Phys. Lett. B
**1971**, 36, 353. [Google Scholar] [CrossRef] [Green Version] - Ananthanarayan, B.; Colangelo, G.; Gasser, J.; Leutwyler, H. Roy equation analysis of ππ scattering. Phys. Rep.
**2001**, 353, 207. [Google Scholar] [CrossRef] [Green Version] - Colangelo, G.; Gasser, J.; Leutwyler, H. ππ scattering. Nucl. Phys. B
**2001**, 603, 125. [Google Scholar] [CrossRef] - Kaminski, R.; García-Martín, R.; Grynkiewicz, P.; Peláez, J.R.; Ynduráin, F.J. New dispersion relations in the description of pi pi scattering amplitudes. Int. J. Mod. Phys. A
**2009**, 24, 402. [Google Scholar] [CrossRef] [Green Version] - García-Martín, R.; Kaminski, R.; Peláez, J.R.; Ruiz de Elvira, J.; Ynduráin, F.J. The pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV. Phys. Rev. D
**2011**, 83, 074004. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. Nuclear forces from chiral Lagrangians. Phys. Lett. B
**1990**, 251, 288. [Google Scholar] [CrossRef] - Weinberg, S. Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces. Nucl. Phys. B
**1991**, 363, 3. [Google Scholar] [CrossRef] - Oller, J.A.; Entem, D.R. The exact discontinuity of a partial wave along the left-hand cut and the exact N/D method in non-relativistic scattering. Ann. Phys.
**2018**, 411, 167965. [Google Scholar] [CrossRef] [Green Version] - Entem, D.R.; Oller, J.A. The N/D method with non-perturbative left-hand-cut discontinuity and the
^{1}S_{0}NN partial wave. Phys. Lett. B**2017**, 773, 498. [Google Scholar] [CrossRef] - Kaiser, N.; Siegel, P.B.; Weise, W. Chiral dynamics and the low-energy kaon-nucleon interaction. Nucl. Phys. A
**1995**, 594, 325. [Google Scholar] [CrossRef] [Green Version] - Oset, E.; Ramos, A. Nonperturbative chiral approach to s wave anti-K N interactions. Nucl. Phys. A
**1998**, 635, 99. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Meißner, U.-G. Chiral dynamics in the presence of bound states: Kaon nucleon interactions revisited. Phys. Lett. B
**2001**, 500, 263. [Google Scholar] [CrossRef] - Jido, D.; Oller, J.A.; Oset, E.; Ramos, A.; Meißner, U.-G. Chiral dynamics of the two Λ(1405) states. Nucl. Phys. A
**2003**, 725, 181. [Google Scholar] [CrossRef] [Green Version] - Meißner, U.-G.; Oller, J.A.; Wirzba, A. In-medium chiral perturbation theory beyond the mean field approximation. Ann. Phys.
**2002**, 297, 27. [Google Scholar] [CrossRef] [Green Version] - Birse, M.C. Power counting with one-pion exchange. Phys. Rev. C
**2006**, 74, 014003. [Google Scholar] [CrossRef] [Green Version] - Lacour, A.; Oller, J.A.; Meißner, U.-G. Non-perturbative methods for a chiral effective field theory of finite density nuclear systems. Ann. Phys.
**2011**, 326, 241. [Google Scholar] [CrossRef] [Green Version] - Lacour, A.; Oller, J.A.; Meißner, U.-G. The Chiral quark condensate and pion decay constant in nuclear matter at next-to-leading order. J. Phys. G
**2010**, 37, 125002. [Google Scholar] [CrossRef] - Lacour, A.; Oller, J.A.; Meißner, U.-G. Chiral Effective Field Theory for Nuclear Matter with long- and short-range Multi-Nucleon Interactions. J. Phys. G
**2010**, 37, 015106. [Google Scholar] [CrossRef] - Dobado, A.; Llanes-Estrada, F.J.; Oller, J.A. The existence of a two-solar mass neutron star constrains the gravitational constant G_N at strong field. Phys. Rev. C
**2012**, 85, 012801. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A. An in-medium chiral power-counting scheme for nuclear matter and some applications. J. Phys. G
**2019**, 46, 073001. [Google Scholar] [CrossRef] [Green Version] - Kaiser, N. Resummation of fermionic in-medium ladder diagrams to all orders. Nucl. Phys. A
**2012**, 860, 41. [Google Scholar] [CrossRef] [Green Version] - Kaiser, N. Resummation of in-medium ladder diagrams: S-wave effective range and p-wave interaction. Eur. Phys. J. A
**2012**, 48, 148. [Google Scholar] [CrossRef] [Green Version] - Boulet, A.; Lacroix, D. Approximate self-energy for Fermi systems with large S-wave scattering length: A step towards density functional theory. J. Phys. G
**2019**, 46, 105104. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Llanes-Estrada, F.J.; Sanz-Cillero, J.J. Resonant production of Wh and Zh at the LHC. J. High Energy Phys.
**2018**, 3, 159. [Google Scholar] [CrossRef] [Green Version] - Delgado, R.L.; Dobado, A.; Espada, M.; Llanes-Estrada, F.J.; Merino, I.L. Collider production of electroweak resonances from γγ states. J. High Energy Phys.
**2018**, 11, 10. [Google Scholar] [CrossRef] [Green Version] - Delgado, R.L.; Dobado, A.; Llanes-Estrada, F.J. Unitarity, analyticity, dispersion relations, and resonances in strongly interacting W
_{L}W_{L}, Z_{L}Z_{L}, and hh scattering. Phys. Rev. D**2015**, 91, 075017. [Google Scholar] [CrossRef] [Green Version] - Delgado, R.L.; Dobado, A.; Llanes-Estrada, F.J. Possible new resonance from W
_{L}W_{L}-hh interchannel coupling. Phys. Rev. Lett.**2015**, 114, 221803. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Weinberg, S. Phenomenological Lagrangians. Physica A
**1979**, 96, 327. [Google Scholar] [CrossRef] - Burgess, C.P. Quantum gravity in everyday life: General relativity as an effective field theory. Living Rev. Rel.
**2004**, 7, 5. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Donoghue, J.F. General relativity as an effective field theory: The leading quantum corrections. Phys. Rev. D
**1994**, 50, 3874. [Google Scholar] [CrossRef] [Green Version] - Han, T.; Willenbrock, S. Scale of quantum gravity. Phys. Lett. B
**2005**, 616, 215. [Google Scholar] [CrossRef] [Green Version] - Aydemir, U.; Anber, M.M.; Donoghue, J.F. Self-healing of unitarity in effective field theories and the onset of new physics. Phys. Rev. D
**2012**, 86, 014025. [Google Scholar] [CrossRef] [Green Version] - Calmet, X. The Lightest of Black Holes. Mod. Phys. Lett. A
**2014**, 29, 450204. [Google Scholar] [CrossRef] [Green Version] - Calmet, X.; Casadio, R. The horizon of the lightest black hole. Eur. Phys. J. C
**2015**, 75, 445. [Google Scholar] [CrossRef] [Green Version] - Weinberg, W. The Quantum Field Theory of Fields. Volume I. Foundations; Cambridge University Press: New York, NY, USA, 1995. [Google Scholar]
- Haag, R. Quantum field theories with composite particles and asymptotic conditions. Phys. Rev.
**1958**, 112, 669. [Google Scholar] [CrossRef] - Ruelle, D. On the asymptotic condition in quantum field theory. Helv. Phys. Acta
**1962**, 35, 147. [Google Scholar] - Martin, A.D.; Spearman, T.D. Elementary Particle Theory; North-Holland Publishing Company: Amsterdam, The Netherlands, 1970. [Google Scholar]
- Eden, R.J.; Landshoff, P.V.; Olive, D.I.; Polkinghorne, J.C. The Analytic S-Matrix; Cambridge University Press: Cambridge, UK, 1966. [Google Scholar]
- Oller, J.A. Coupled-channel approach in hadron-hadron scattering. Prog. Part. Nucl. Phys.
**2020**, 110, 103728. [Google Scholar] [CrossRef] [Green Version] - Bethe, H.A. Theory of the effective range in nuclear scattering. Phys. Rev.
**1949**, 76, 38. [Google Scholar] [CrossRef] - Adler, S.L. Consistency conditions on the strong interactions implied by a partially conserved axial vector current. Phys. Rev.
**1965**, 137, B1022. [Google Scholar] [CrossRef] - Au, K.L.; Morgan, D.; Pennington, M.R. Meson dynamics beyond the quark model: Study of final-state interactions. Phys. Rev. D
**1987**, 35, 1633. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gounaris, G.J.; Sakurai, J. Finite-width corrections to the vector-meson-dominance prediction for ρ → e
^{+}e^{-}. Phys. Rev. Lett.**1968**, 21, 244. [Google Scholar] [CrossRef] - Brehm, J.J.; Golowich, E.; Prasad, S.C. Hard-pion effective-range formula for the pion form factor. Phys. Rev. Lett.
**1969**, 23, 666. [Google Scholar] [CrossRef] - Anisovich, V.V.; Sarantsev, A.V. K-matrix analysis of the (IJ
^{PC}= 00^{++})-wave in the mass region below 1900 MeV. Eur. Phys. J. A**2003**, 16, 229. [Google Scholar] [CrossRef] - Moir, G.; Peardon, M.; Ryan, S.M.; Thomas, C.E.; Wilson, D.J. Coupled-channel Dπ, Dη and D
_{s}K¯ scattering from Lattice QCD. J. High Energy Phys.**2016**, 16, 011. [Google Scholar] [CrossRef] [Green Version] - Kawarabayashi, K.; Suzuki, M. Partially conserved axial-vector current and the decays of vector mesons. Phys. Rev. Lett.
**1966**, 16, 255. [Google Scholar] [CrossRef] [Green Version] - Riazuddin; Fayyazuddin. Algebra of Current Components and Decay Widths of ρ and K
^{*}mesons. Phys. Rev.**1966**, 147, 1071. [Google Scholar] [CrossRef] - Gell-Mann, M.; Zachariasen, F. Form factors and vector mesons. Phys. Rev.
**1961**, 124, 953. [Google Scholar] [CrossRef] - Sakurai, J.J. Theory of strong interactions. Ann. Phys.
**1960**, 11, 1. [Google Scholar] [CrossRef] - Oller, J.A.; Oset, E.; Peláez, J.R. Meson meson interaction in a non-perturbative chiral approach. Phys. Rev. D
**1999**, 59, 074001, Erratatum in**1999**, 60, 099906; Erratatum in**2007**, 75, 099903. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Herrero, M.J.; Truong, T.N. Unitarized chiral perturbation theory for elastic pion-pion scattering. Phys. Lett. B
**1990**, 235, 134. [Google Scholar] [CrossRef] [Green Version] - Nieves, J.; Paón Valderrama, M.; Ruiz Arriola, E. The Inverse amplitude method in pi pi scattering in chiral perturbation theory to two loops. Phys. Rev. D
**2002**, 65, 036002. [Google Scholar] [CrossRef] [Green Version] - Basdevant, J.L.; Bessis, D.; Zinn-Justin, J. Padé approximants in strong interactions. Two-body pion and kaon systems. Nuovo Cimento A
**1969**, 60, 185. [Google Scholar] [CrossRef] - Basdevant, J.L.; Lee, B.W. Pade approximation in the σ model unitary ππ amplitudes with the current algebra constraints. Nuovo Cimento A
**1969**, 60, 185. [Google Scholar] [CrossRef] - Basdevant, J.L. The Padé approximation and its physical applications. Fortschritte der Physik
**1972**, 20, 283. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Roca, L. Scalar radius of the pion and zeros in the form factor. Phys. Lett. B
**2007**, 651, 139. [Google Scholar] [CrossRef] [Green Version] - Guo, Z.-H.; Oller, J.A.; Ruiz de Elvira, J. Chiral dynamics in form factors, spectral-function sum rules, meson-meson scattering and semi-local duality. Phys. Rev. D
**2012**, 86, 054006. [Google Scholar] [CrossRef] [Green Version] - Pennington, M.R. Sigma coupling to photons: Hidden scalar in γγ → π
^{0}π^{0}. Phys. Rev. Lett.**2006**, 97, 011601. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Roca, L.; Schat, C. Improved dispersion relations for γγ → π
^{0}π^{0}. Phys. Lett. B**2008**, 659, 201. [Google Scholar] [CrossRef] [Green Version] - Jamin, M.; Oller, J.A.; Pich, A. Strangeness changing scalar form-factors. Nucl. Phys. B
**2002**, 622, 279. [Google Scholar] [CrossRef] [Green Version] - Gasser, J.; Meißner, U.-G. Chiral expansion of pion form-factors beyond one loop. Nucl. Phys. B
**1991**, 357, 90. [Google Scholar] [CrossRef] - Protopopescu, S.D.; Alson-Garnjost, M. ππ Partial wave analysis from reactions π
^{+}p → π^{+}π^{-}Δ^{++}and π+p → K^{+}K^{-}Δ^{++}at 7.1-GeV/c. Phys. Rev. D**1973**, 7, 1279. [Google Scholar] [CrossRef] - Estabrooks, P.; Martin, A.D. ππ phase-shift analysis below the KK threshold. Nucl. Phys. B
**1974**, 79, 301. [Google Scholar] [CrossRef] - Barkov, L.M.; Chilingarov, A.G.; Eidelman, S.I.; Khazin, B.I.; Lelchuk, M.Y.; Okhapkin, V.S.; Pakhtusova, E.V.; Redin, S.I.; Ryskulov, N.M.; Shatunov, Y.M.; et al. Electromagnetic Pion Form-Factor in the Timelike Region. Nucl. Phys. B
**1985**, 256, 365. [Google Scholar] [CrossRef] - Oller, J.A.; Oset, E.; Palomar, J.E. Pion and kaon vector form-factors. Phys. Rev. D
**2001**, 63, 114009. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. The U(1) Problem. Phys. Rev. D
**1975**, 11, 3583. [Google Scholar] [CrossRef] - Gasser, J.; Leutwyler, H. η → 3π to one loop. Nucl. Phys. B
**1985**, 250, 539. [Google Scholar] [CrossRef] [Green Version] - Bijnens, J.; Ghorbani, K. η → 3π at two loops in chiral perturbation theory. J. High Energy Phys.
**2007**, 20071, 030. [Google Scholar] [CrossRef] - Beisert, N.; Borasoy, B. Hadronic decays of eta and eta-prime with coupled channels. Nucl. Phys. A
**2003**, 716, 186. [Google Scholar] [CrossRef] [Green Version] - Borasoy, B.; Nissler, R. Hadronic η and η
^{′}decays. Eur. Phys. J. A**2005**, 26, 383. [Google Scholar] [CrossRef] [Green Version] - Kambor, J.; Wiesendanger, C.; Wyler, D. Final-state interactions and Khuri-Treiman equations in η → 3π decays. Nucl. Phys. B
**1996**, 465, 215. [Google Scholar] [CrossRef] [Green Version] - Anisovich, A.V.; Leutwyler, H. Dispersive analysis of the decay η → 3π. Phys. Lett. B
**1996**, 375, 335. [Google Scholar] [CrossRef] [Green Version] - Guo, P.; Danilkin, I.V.; Fernández-Ramírez, C.; Mathieu, V.; Szczepaniak, A.P. Three-body final state interaction in η → 3π updated. Phys. Lett. B
**2017**, 771, 497. [Google Scholar] [CrossRef] - Colangelo, G.; Lanz, S.; Leutwyler, H.; Passemar, E. Dispersive analysis of η → 3π. Eur. Phys. J. C
**2018**, 78, 947. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Moussallam, B. Extended chiral Khuri-Treiman formalism for η → 3π and the role of the a
_{0}(980), f_{0}(980) resonances. Eur. Phys. J. C**2017**, 77, 508. [Google Scholar] [CrossRef] [Green Version] - Descotes-Genon, S.; Moussallam, B. Analyticity of ηπ isospin-violating form factors and the τ → ηπν second-class decay. Eur. Phys. J. C
**2014**, 74, 2946. [Google Scholar] [CrossRef] [Green Version] - Mandelstam, S. Unitarity condition below physical thresholds in the normal and anomalous cases. Phys. Rev. Lett.
**1960**, 4, 84. [Google Scholar] [CrossRef] - Oller, J.A. The Case of a WW dynamical scalar resonance within a chiral effective description of the strongly interacting Higgs sector. Phys. Lett. B
**2000**, 477, 187. [Google Scholar] [CrossRef] [Green Version] - Meißner, U.-G.; Oller, J.A. J/ψ → ϕππ(K$\overline{K}$) decays, chiral dynamics and OZI violation. Nucl. Phys. A
**2001**, 679, 671. [Google Scholar] [CrossRef] [Green Version] - Guo, Z.-H.; Oller, J.A.; Ríos, G. Nucleon-nucleon scattering from the dispersive N/D method: Next-to-leading order study. Phys. Rev. C
**2014**, 89, 014002. [Google Scholar] [CrossRef] [Green Version] - Castillejo, L.; Dalitz, R.H.; Dyson, F.J. Low’s Scattering Equation for the Charged and Neutral Scalar Theories. Phys. Rev.
**1956**, 101, 453. [Google Scholar] [CrossRef] - Dyson, F.J. Meaning of the solutions of Low’s scattering equation. Phys. Rev.
**1957**, 106, 157. [Google Scholar] [CrossRef] - Kang, X.W.; Oller, J.A. Different pole structures in line shapes of the X(3872). Eur. Phys. J. C
**2017**, 77, 399. [Google Scholar] [CrossRef] [Green Version] - Kang, X.W.; Oller, J.A. Nature of X(3872) from the line shape. In Proceedings of the 18th International Conference on Hadron Spectroscopy and Structure, Guilin, China, 16–21 August 2019. [Google Scholar]
- Guo, Z.H.; Liu, L.; Meißner, U.G.; Oller, J.A.; Rusetsky, A. Towards a precise determination of the scattering amplitudes of the charmed and light-flavor pseudoscalar mesons. Eur. Phys. J. C
**2019**, 79, 13. [Google Scholar] [CrossRef] - Guo, Z.H.; Liu, L.; Meißner, U.-G.; Oller, J.A.; Rusetsky, A. Chiral study of the a
_{0}(980) resonance and πη scattering phase shifts in light of a recent lattice simulation. Phys. Rev. D**2017**, 95, 054004. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Fernandez-Soler, P.; Guo, F.K.; Nieves, J. Two-pole structure of the D
_{0}^{*}(2400). Phys. Lett. B**2017**, 767, 465. [Google Scholar] [CrossRef] - Oller, J.A.; Roca, L. Non-perturbative study of the light pseudoscalar masses in chiral dynamics. Eur. Phys. J. A
**2007**, 31, 534. [Google Scholar] [CrossRef] - Oller, J.A. The Mixing angle of the lightest scalar nonet. Nucl. Phys. A
**2003**, 727, 353. [Google Scholar] [CrossRef] [Green Version] - Guo, Z.-H.; Oller, J.A. Resonances from meson-meson scattering in U(3) CHPT. Phys. Rev. D
**2011**, 84, 034005. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Oller, J.A. Identification of a scalar glueball. Phys. Rev. Lett.
**2008**, 101, 252002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jamin, M.; Oller, J.A.; Pich, A. S-wave Kπ scattering in chiral perturbation theory with resonances. Nucl. Phys. B
**2000**, 587, 331. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Oller, J.A. On the size of the σ meson and its nature. Phys. Rev. D
**2012**, 86, 034003. [Google Scholar] [CrossRef] [Green Version] - Guo, Z.-H.; Oller, J.A. Meson-baryon reactions with strangeness −1 within a chiral framework. Phys. Rev. C
**2013**, 87, 035202. [Google Scholar] [CrossRef] [Green Version] - Khemchandani, K.P.; Martinez Torres, A.; Oller, J.A. Hyperon resonances coupled to pseudoscalar- and vector-baryon channels. Phys. Rev. C
**2019**, 100, 015208. [Google Scholar] [CrossRef] [Green Version] - Khemchandani, K.P.; Martinez Torres, A.; Oller, J.A. Hyperon resonances and meson-baryon interactions in isospin 1. In Proceedings of the 18th International Conference on Hadron Spectroscopy and Structure, Guilin, China, 16–21 August 2019. [Google Scholar]
- Kang, X.W.; Oller, J.A. P-wave coupled-channel scattering of B
_{s}π, Bs*π, B$\overline{K}$, B^{*}$\overline{K}$ and the puzzling X(5568). Phys. Rev. D**2016**, 94, 054010. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Oset, E.; Pelaez, J.R. Nonperturbative approach to effective chiral Lagrangians and meson interactions. Phys. Rev. Lett.
**1998**, 80, 3452. [Google Scholar] [CrossRef] [Green Version] - Ecker, G.; Gasser, J.; Pich, A.; de Rafael, E. The Role of Resonances in Chiral Perturbation Theory. Nucl. Phys. B
**1989**, 321, 311. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Oset, E.; Ramos, A. Chiral unitary approach to meson-meson and meson-baryon interactions and nuclear applications. Prog. Part. Nucl. Phys.
**2000**, 45, 157. [Google Scholar] [CrossRef] [Green Version] - Igi, K.; Hikasa, K.-I. Another look at ππ scattering in the scalar channel. Phys. Rev. D
**1999**, 59, 034005. [Google Scholar] [CrossRef] [Green Version] - Gülmez, D.; Meißner, U.-G.; Oller, J.A. A chiral covariant approach to ρρ scattering. Eur. Phys. J. C
**2017**, 77, 460. [Google Scholar] [CrossRef] - Bando, M.; Kugo, T.; Uehara, S.; Yamawaki, K.; Yanagida, T. Is the ρ meson a dynamical gauge boson of hidden local symmetry. Phys. Rev. Lett.
**1985**, 54, 1215. [Google Scholar] [CrossRef] [PubMed] - Bando, M.; Kugo, T.; Yamawaki, K. Nonlinear Realization and Hidden Local Symmetries. Phys. Rep.
**1988**, 164, 217. [Google Scholar] [CrossRef] - Du, M.-L.; Gülmez, D.; Guo, F.-K.; Meißner, U.-G.; Wang, Q. Interactions between vector mesons and dynamically generated resonances. Eur. Phys. J. C.
**2018**, 78, 988. [Google Scholar] [CrossRef] - Molina, R.; Nicmorus, D.; Oset, E. The ρρ interaction in the hidden gauge formalism and the f
_{0}(1370) and f_{2}(1270) resonances. Phys. Rev. D**2008**, 78, 114018. [Google Scholar] [CrossRef] [Green Version] - Geng, L.S.; Oset, E. Vector meson-vector meson interaction in a hidden gauge unitary approach. Phys. Rev. D
**2009**, 79, 074009. [Google Scholar] [CrossRef] [Green Version] - Geng, L.S.; Molina, R.; Oset, E. On the chiral covariant approach to ρρ scattering. Chin. Phys. C
**2017**, 41, 124101. [Google Scholar] [CrossRef] [Green Version] - Molina, R.; Geng, L.S.; Oset, E. Comments on the dispersion relation method to vector–vector interaction. PTEP
**2019**, 2019, 103B05. [Google Scholar] [CrossRef] - Babelon, O.; Basdevant, J.-L.; Caillerie, D.; Mennessier, G. Unitarity and inelastic final-state interactions. Nucl. Phys. B
**1976**, 113, 445. [Google Scholar] [CrossRef] - Oller, J.A.; Oset, E. Theoretical study of the γγ →meson-meson reaction. Nucl. Phys. A
**1998**, 629, 739. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A. Final state interactions in D decays. Phys. Rev. D
**2005**, 71, 054030. [Google Scholar] [CrossRef] [Green Version] - Frank, W.M.; Land, D.J.; Spector, R.M. Singular potentials. Rev. Mod. Phys.
**1971**, 43, 36. [Google Scholar] [CrossRef] - Case, K.M. Singular potentials. Phys. Rev.
**1950**, 80, 797. [Google Scholar] [CrossRef] - Pavón Valderrama, M.; Ruiz Arriola, E. Renormalization of the deuteron with one pion exchange. Phys. Rev. C
**2005**, 72, 054002. [Google Scholar] [CrossRef] [Green Version] - Pavón Valderrama, M.; Ruiz Arriola, E. Renormalization of NN interaction with chiral two pion exchange potential. Central phases and the deuteron. Phys. Rev. C
**2006**, 74, 054001. [Google Scholar] [CrossRef] [Green Version] - Pavón Valderrama, M.; Ruiz Arriola, E. Renormalization of NN interaction with chiral two pion exchange potential: Non-central phases. Phys. Rev. C
**2006**, 74, 064004. [Google Scholar] [CrossRef] - Meißner, U.-G. Two-pole structures in QCD: Facts, not fantasy! arXiv
**2020**, arXiv:2005.06909. [Google Scholar] [CrossRef]

**Figure 1.**The top row concerns the $\pi \pi $ scalar form factor ${F}_{S}\left(s\right)$ and the bottom one the $\pi \pi $ vector form factor ${F}_{V}\left(s\right)$. In each row the left panel refers to the phase and the right one to the modulus squared of the corresponding form factor. The perturbative calculations are indicated by the (magenta) dot-dashed lines in all cases. The non-perturbative result for ${F}_{S}\left(s\right)$ are shown by the (black) solid lines. For ${F}_{V}\left(s\right)$ we show two lines for the IAM resummation, Equation (84), the (black) solid lines and the (red) dashed ones. The former employs $\langle {r}_{V}^{2}\rangle =0.42$ fm${}^{2}$ (used in Reference [19]) and the latter $\langle {r}_{V}^{2}\rangle =0.41$ fm${}^{2}$. The $\rho -\omega $ mixing, clearly visible at the top of $|{F}_{V}{\left(s\right)|}^{2}$, is not discussed here. The experimental points for the $I=J=1$$\pi \pi $ phase shifts are from References [91,92], and those for $|{F}_{V}{\left(s\right)|}^{2}$ were obtained in Reference [93]. For the $I=J=0$ phase shifts we use the subset of points employed in Figure 2 and that appear on the top in the ${f}_{0}\left(500\right)$ region.

**Figure 2.**Results from Reference [27] with only one free parameter (a natural sized cut-off $\Lambda \simeq 1$ GeV) for the S-wave meson-meson scattering with $I=0$ and 1. From top to bottom and left to right, the isoscalar scalar $\pi \pi \to \pi \pi $ and $K\overline{K}\to \pi \pi $ phase shifts, the $\pi \pi $ inelastic cross-section with the same quantum numbers and a ${\pi}^{0}\eta $ event distribution around the isovector scalar ${a}_{0}\left(980\right)$ resonance are plotted. For more details and references of the experimental papers we refer to Reference [27].

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Oller, J.A.
Unitarization Technics in Hadron Physics with Historical Remarks. *Symmetry* **2020**, *12*, 1114.
https://doi.org/10.3390/sym12071114

**AMA Style**

Oller JA.
Unitarization Technics in Hadron Physics with Historical Remarks. *Symmetry*. 2020; 12(7):1114.
https://doi.org/10.3390/sym12071114

**Chicago/Turabian Style**

Oller, José Antonio.
2020. "Unitarization Technics in Hadron Physics with Historical Remarks" *Symmetry* 12, no. 7: 1114.
https://doi.org/10.3390/sym12071114