Abstract
We compare the Hilbert series approach with explicit constructions of higher-order Lagrangians for the O(N) nonlinear sigma model. We use the Hilbert series to find the number and type of operators up to mass dimension 16, for spacetime dimension D up to 12 and N up to 12, and further classify the operators into spacetime parity and parity of the internal symmetry group O(N). The explicit construction of operators is done up to mass dimension 12 for both parities even and dimension 10 for the other three cases. The results of the two methods are in full agreement. This provides evidence for the Hilbert series conjecture regarding co-closed but not co-exact k-forms, which takes into account the integration-by-parts relations.
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References
S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev. 166 (1968) 1568 [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
S. Weinberg, Baryon and Lepton Nonconserving Processes, Phys. Rev. Lett. 43 (1979) 1566 [INSPIRE].
W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621 [INSPIRE].
B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].
S. Scherer and H.W. Fearing, Field transformations and the classical equation of motion in chiral perturbation theory, Phys. Rev. D 52 (1995) 6445 [hep-ph/9408298] [INSPIRE].
J. Bijnens, G. Colangelo and G. Ecker, The Mesonic chiral Lagrangian of order p6, JHEP 02 (1999) 020 [hep-ph/9902437] [INSPIRE].
C. Grosse-Knetter, Effective Lagrangians with higher derivatives and equations of motion, Phys. Rev. D 49 (1994) 6709 [hep-ph/9306321] [INSPIRE].
H.W. Fearing and S. Scherer, Extension of the chiral perturbation theory meson Lagrangian to order p6, Phys. Rev. D 53 (1996) 315 [hep-ph/9408346] [INSPIRE].
J. Bijnens, N. Hermansson-Truedsson and S. Wang, The order p8 mesonic chiral Lagrangian, JHEP 01 (2019) 102 [arXiv:1810.06834] [INSPIRE].
J. Bijnens, L. Girlanda and P. Talavera, The Anomalous chiral Lagrangian of order p6, Eur. Phys. J. C 23 (2002) 539 [hep-ph/0110400] [INSPIRE].
T. Ebertshauser, H.W. Fearing and S. Scherer, The Anomalous chiral perturbation theory meson Lagrangian to order p6 revisited, Phys. Rev. D 65 (2002) 054033 [hep-ph/0110261] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: The Plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
J. Gray et al., SQCD: A Geometric Apercu, JHEP 05 (2008) 099 [arXiv:0803.4257] [INSPIRE].
E.E. Jenkins and A.V. Manohar, Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation, JHEP 10 (2009) 094 [arXiv:0907.4763] [INSPIRE].
A. Hanany, E.E. Jenkins, A.V. Manohar and G. Torri, Hilbert Series for Flavor Invariants of the Standard Model, JHEP 03 (2011) 096 [arXiv:1010.3161] [INSPIRE].
L. Lehman and A. Martin, Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox, Phys. Rev. D 91 (2015) 105014 [arXiv:1503.07537] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, Hilbert series and operator bases with derivatives in effective field theories, Commun. Math. Phys. 347 (2016) 363 [arXiv:1507.07240] [INSPIRE].
L. Lehman and A. Martin, Low-derivative operators of the Standard Model effective field theory via Hilbert series methods, JHEP 02 (2016) 081 [arXiv:1510.00372] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485, . . .: Higher dimension operators in the SM EFT, JHEP 08 (2017) 016 [Erratum ibid. 09 (2019) 019] [arXiv:1512.03433] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, Operator bases, S-matrices, and their partition functions, JHEP 10 (2017) 199 [arXiv:1706.08520] [INSPIRE].
M. Ruhdorfer, J. Serra and A. Weiler, Effective Field Theory of Gravity to All Orders, JHEP 05 (2020) 083 [arXiv:1908.08050] [INSPIRE].
J. Dujava, Counting operators in Effective Field Theories, M.Sc. thesis, Charles University, Prague, Czechia (2022) [arXiv:2211.05759] [INSPIRE].
A. Zee, Quantum field theory in a nutshell, Princeton University Press (2003) [ISBN: 9780691010199] [INSPIRE].
A.D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [INSPIRE].
L. Graf et al., 2, 12, 117, 1959, 45171, 1170086, . . .: a Hilbert series for the QCD chiral Lagrangian, JHEP 01 (2021) 142 [arXiv:2009.01239] [INSPIRE].
K. Kampf, J. Novotny and J. Trnka, Tree-level Amplitudes in the Nonlinear Sigma Model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].
C. Cheung, K. Kampf, J. Novotny and J. Trnka, Effective Field Theories from Soft Limits of Scattering Amplitudes, Phys. Rev. Lett. 114 (2015) 221602 [arXiv:1412.4095] [INSPIRE].
J. Bijnens, K. Kampf and M. Sjö, Higher-order tree-level amplitudes in the nonlinear sigma model, JHEP 11 (2019) 074 [Erratum ibid. 03 (2021) 066] [arXiv:1909.13684] [INSPIRE].
R.M. Fonseca, Enumerating the operators of an effective field theory, Phys. Rev. D 101 (2020) 035040 [arXiv:1907.12584] [INSPIRE].
L. Dai, I. Low, T. Mehen and A. Mohapatra, Operator Counting and Soft Blocks in Chiral Perturbation Theory, Phys. Rev. D 102 (2020) 116011 [arXiv:2009.01819] [INSPIRE].
K. Kampf, The ChPT: top-down and bottom-up, JHEP 12 (2021) 140 [arXiv:2109.11574] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. I, Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. II, Phys. Rev. 177 (1969) 2247 [INSPIRE].
S.B. Gudnason and M. Nitta, Reducing the O(3) model as an effective field theory, JHEP 03 (2022) 030 [arXiv:2110.15038] [INSPIRE].
J. Bijnens and L. Carloni, Leading Logarithms in the Massive O(N) Nonlinear Sigma Model, Nucl. Phys. B 827 (2010) 237 [arXiv:0909.5086] [INSPIRE].
J. Bijnens and L. Carloni, The Massive O(N) Non-linear Sigma Model at High Orders, Nucl. Phys. B 843 (2011) 55 [arXiv:1008.3499] [INSPIRE].
J. Bijnens and T. Husek, Six-pion amplitude, Phys. Rev. D 104 (2021) 054046 [arXiv:2107.06291] [INSPIRE].
N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press (2004) [https://doi.org/10.1017/CBO9780511617034] [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
T. Granlund, GNU Multiple Precision Arithmetic Library 6.1.2, (2016).
Acknowledgments
S. B. G. thanks Guilherme Sadovski and Baiyang Zhang for discussions. The work of J. B. is supported in part by the Swedish Research Council grants contract numbers 2016-05996 and 2019-03779. S. B. G. thanks the Outstanding Talent Program of Henan University and the Ministry of Education of Henan Province for partial support. The work of S. B. G. is supported by the National Natural Science Foundation of China (Grants No. 11675223 and No. 12071111) and by the Ministry of Science and Technology of China (Grant No. G2022026021L).
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Bijnens, J., Gudnason, S.B., Yu, J. et al. Hilbert series and higher-order Lagrangians for the O(N) model. J. High Energ. Phys. 2023, 61 (2023). https://doi.org/10.1007/JHEP05(2023)061
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DOI: https://doi.org/10.1007/JHEP05(2023)061