Three-pion scattering in chiral perturbation theory


  • Tomas Husek Lund University



Chiral perturbation theory, pions, hadron-hadron interactions, scattering amplitudes


We present the results on the relativistic six-pion scattering amplitude at low energy, calculated at O(p 4 ) within the framework of the massive O(N) nonlinear sigma model extended to the next-to-leading order in the chiral counting. For N = 3, this approach corresponds to the two(- quark)-flavor Chiral Perturbation Theory. We also present the expressions for the pion mass, pion decay constant and the four-pion amplitude in the case of N (meson) flavors at O(p 4 ).


S. Weinberg, “Phenomenological Lagrangians,” Physica A 96 (1979) 327,

J. Gasser and H. Leutwyler, “Chiral Perturbation Theory to One Loop,” Annals Phys. 158 (1984) 142,

H. Osborn, “Implications of Adler zeros for multipion processes,” Lett. Nuovo Cim. 2 (1969) 717,

I. Low and Z. Yin, “Soft Bootstrap and Effective Field Theories,” JHEP 11 (2019) 078,

J. Bijnens, K. Kampf, and M. Sjö, “Higher-order tree-level amplitudes in the nonlinear sigma model,” JHEP 11 (2019) 074, [Erratum: JHEP 03 (2021) 066].

M. Mai and M. Doring, “Finite-Volume Spectrum of π +π + and π +π +π + Systems,” Phys. Rev. Lett. 122 (2019) 062503,

T. D. Blanton, F. Romero-López, and S. R. Sharpe, “ I = 3 Three-Pion Scattering Amplitude from Lattice QCD,” Phys. Rev. Lett. 124 (2020) 032001,

M. Mai, M. Döring, C. Culver, and A. Alexandru, “Three-body unitarity versus finite-volume π +π +π + spectrum from lattice QCD,” Phys. Rev. D 101 (2020) 054510,

C. Culver, M. Mai, R. Brett, A. Alexandru, and M. Döring, “Three pion spectrum in the I = 3 channel from lattice QCD,” Phys. Rev. D 101 (2020) 114507,

M. Fischer, B. Kostrzewa, L. Liu, F. Romero-López, M. Ueding, and C. Urbach, “Scattering of two and three physical pions at maximal isospin from lattice QCD,” Eur. Phys. J. C 81 (2021) 436,

Hadron Spectrum, Collaboration, M. T. Hansen, R. A. Briceño, R. G. Edwards, C. E. Thomas, and D. J. Wilson, “EnergyDependent π +π +π + Scattering Amplitude from QCD,” Phys. Rev. Lett. 126 (2021) 012001,

R. Brett, C. Culver, M. Mai, A. Alexandru, M. Döring, and F. X. Lee, “Three-body interactions from the finite-volume QCD spectrum,” Phys. Rev. D 104 (2021) 014501,

T. D. Blanton, A. D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López, and S. R. Sharpe, “Interactions of two and three mesons including higher partial waves from lattice QCD,” JHEP 10 (2021) 023,

J. Bijnens and T. Husek, “Six-pion amplitude,” Phys. Rev. D 104 (2021) 054046,

A. Dobado and J. Morales, “Pion mass effects in the large N limit of chiral perturbation theory,” Phys. Rev. D 52 (1995) 2878,

J. Bijnens and L. Carloni, “Leading Logarithms in the Massive O(N) Nonlinear Sigma Model,” Nucl. Phys. B 827 (2010) 237,

J. Bijnens and L. Carloni, “The Massive O(N) Non-linear Sigma Model at High Orders,” Nucl. Phys. B 843 (2011) 55,

J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, and M. E. Sainio, “Elastic ππ scattering to two loops,” Phys. Lett. B 374 (1996) 210,

J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, and M. E. Sainio, “Pion-pion scattering at low energy,” Nucl. Phys. B 508 (1997) 263, [Erratum: Nucl. Phys. B 517 (1998) 639].

J. Bijnens and G. Ecker, “Mesonic low-energy constants,” Ann. Rev. Nucl. Part. Sci. 64 (2014) 149,

G. Colangelo, J. Gasser, and H. Leutwyler, “ππ scattering,” Nucl. Phys. B 603 (2001) 125, .

S. Aoki et al., “Review of lattice results concerning low-energy particle physics,” Eur. Phys. J. C 77 (2017) 112,




How to Cite

Husek T. Three-pion scattering in chiral perturbation theory. Supl. Rev. Mex. Fis. [Internet]. 2022 Jun. 13 [cited 2024 May 20];3(3):0308073 1-5. Available from: