Abstract
We introduce a systematic method to classify the Standard Model Effective Field Theory (SMEFT) operators based on their CP properties with the Hilbert series techniques. Our method makes it possible to enumerate operators violating CP symmetry in a few seconds. We present the complete classification of dimension eight operators under CP transformation, and the number of CP-odd or CP-violating operators are listed up to dimension 14. We also provide a companion code in Form that allows anybody to reproduce our results.
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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].
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].
Y. Liao and X.-D. Ma, Operators up to Dimension Seven in Standard Model Effective Field Theory Extended with Sterile Neutrinos, Phys. Rev. D 96 (2017) 015012 [arXiv:1612.04527] [INSPIRE].
L. Merlo, S. Saa and M. Sacristán-Barbero, Baryon Non-Invariant Couplings in Higgs Effective Field Theory, Eur. Phys. J. C 77 (2017) 185 [arXiv:1612.04832] [INSPIRE].
A. Kobach and S. Pal, Hilbert Series and Operator Basis for NRQED and NRQCD/HQET, Phys. Lett. B 772 (2017) 225 [arXiv:1704.00008] [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].
Anisha, S. Das Bakshi, J. Chakrabortty and S. Prakash, Hilbert Series and Plethystics: Paving the path towards 2HDM- and MLRSM-EFT, JHEP 09 (2019) 035 [arXiv:1905.11047] [INSPIRE].
C.B. Marinissen, R. Rahn and W.J. Waalewijn, … , 83106786, 114382724, 1509048322, 2343463290, 27410087742, . . . efficient Hilbert series for effective theories, Phys. Lett. B 808 (2020) 135632 [arXiv:2004.09521] [INSPIRE].
L. Gráf et al., 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian, JHEP 01 (2021) 142 [arXiv:2009.01239] [INSPIRE].
B. Yu and S. Zhou, Hilbert series for leptonic flavor invariants in the minimal seesaw model, JHEP 10 (2021) 017 [arXiv:2107.11928] [INSPIRE].
B. Yu and S. Zhou, Spelling out leptonic CP violation in the language of invariant theory, Phys. Rev. D 106 (2022) L051701 [arXiv:2203.00574] [INSPIRE].
B. Yu and S. Zhou, CP violation and flavor invariants in the seesaw effective field theory, JHEP 08 (2022) 017 [arXiv:2203.10121] [INSPIRE].
L. Gráf et al., Hilbert series, the Higgs mechanism, and HEFT, JHEP 02 (2023) 064 [arXiv:2211.06275] [INSPIRE].
R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 04 (2014) 159 [arXiv:1312.2014] [INSPIRE].
G.N. Remmen and N.L. Rodd, Consistency of the Standard Model Effective Field Theory, JHEP 12 (2019) 032 [arXiv:1908.09845] [INSPIRE].
Q. Bonnefoy, E. Gendy, C. Grojean and J.T. Ruderman, Beyond Jarlskog: 699 invariants for CP violation in SMEFT, JHEP 08 (2022) 032 [arXiv:2112.03889] [INSPIRE].
G. Arias-Tamargo, A. Bourget, A. Pini and D. Rodríguez-Gómez, Discrete gauge theories of charge conjugation, Nucl. Phys. B 946 (2019) 114721 [arXiv:1903.06662] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, Outer automorphism anomalies, JHEP 02 (2022) 094 [arXiv:2111.04728] [INSPIRE].
S. Weinberg, Quantum fields and antiparticles, in The Quantum Theory of Fields, Cambridge University Press (1995), p. 191–258 [https://doi.org/10.1017/cbo9781139644167.007].
W. Grimus and M.N. Rebelo, Automorphisms in gauge theories and the definition of CP and P, Phys. Rept. 281 (1997) 239 [hep-ph/9506272] [INSPIRE].
H. Georgi, Lie algebras in particle physics: from isospin to unified theories, second edition, Taylor & Francis (2000) [ISBN: 9786613261465].
I. Bischer, C. Döring and A. Trautner, Telling compositeness at a distance with outer automorphisms and CP, arXiv:2212.07439 [ULB-TH/22-16] [INSPIRE].
H. Sun, Y.-N. Wang and J.-H. Yu, Hilbert Series and Operator Counting on the Higgs Effective Field Theory, arXiv:2211.11598 [INSPIRE].
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ArXiv ePrint: 2212.02413
The title sequence is the total number of independent CP-violating operators, as defined in this paper, in the SMEFT with one fermion generation for mass dimensions 6, 8, 10, 12, 14, ….
Hamamatsu Professor. (Hitoshi Murayama)
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Kondo, D., Murayama, H. & Okabe, R. 23, 381, 6242, 103268, 1743183, … : Hilbert series for CP-violating operators in SMEFT. J. High Energ. Phys. 2023, 107 (2023). https://doi.org/10.1007/JHEP03(2023)107
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DOI: https://doi.org/10.1007/JHEP03(2023)107