Abstract
Axions and axion-like particles (ALPs) are ubiquitous in popular attempts to solve supercalifragilisticexpialidocious puzzles of Nature. A widespread and vivid experimental programme spanning a vast range of mass scales and decades of couplings strives to find evidence for these elusive but theoretically well-motivated particles. In the absence of clear guiding principle, effective field theories (EFTs) prove to be an efficient tool in this experimental quest. Hilbert series technologies are a privileged instrument of the EFT toolbox to enumerate and classify operators. In this work, we compute explicitly the Hilbert series capturing the interactions of a generic ALP to the Standard Model particles above and below the electroweak symmetry scale, which allow us to build bases of operators up to dimension 8. In particular, we revealed a remarkable structure of the Hilbert series that isolates the shift-symmetry breaking and preserving interactions. In addition, with the Hilbert series method, we enumerate the sources of CP violation in terms of CP-even, CP-odd and CP-violating operators. Furthermore, we provide an supplementary file of the Hilbert series up to dimension 15 to supplement our findings, which can be used for further analysis and exploration.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R.D. Peccei and H.R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].
R.D. Peccei and H.R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D 16 (1977) 1791 [INSPIRE].
S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40 (1978) 223 [INSPIRE].
F. Wilczek, Problem of Strong P and T Invariance in the Presence of Instantons, Phys. Rev. Lett. 40 (1978) 279 [INSPIRE].
J.E. Kim, Weak Interaction Singlet and Strong CP Invariance, Phys. Rev. Lett. 43 (1979) 103 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Can Confinement Ensure Natural CP Invariance of Strong Interactions?, Nucl. Phys. B 166 (1980) 493 [INSPIRE].
M. Dine, W. Fischler and M. Srednicki, A Simple Solution to the Strong CP Problem with a Harmless Axion, Phys. Lett. B 104 (1981) 199 [INSPIRE].
A.R. Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions (in Russian), Sov. J. Nucl. Phys. 31 (1980) 260 [INSPIRE].
J. Preskill, M.B. Wise and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. B 120 (1983) 127 [INSPIRE].
L.F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. B 120 (1983) 133 [INSPIRE].
M. Dine and W. Fischler, The Not So Harmless Axion, Phys. Lett. B 120 (1983) 137 [INSPIRE].
D.J.E. Marsh, Axion Cosmology, Phys. Rept. 643 (2016) 1 [arXiv:1510.07633] [INSPIRE].
L. Di Luzio, M. Giannotti, E. Nardi and L. Visinelli, The landscape of QCD axion models, Phys. Rept. 870 (2020) 1 [arXiv:2003.01100] [INSPIRE].
S.W. Hawking, Quantum Coherence Down the Wormhole, Phys. Lett. B 195 (1987) 337 [INSPIRE].
S.B. Giddings and A. Strominger, Loss of Incoherence and Determination of Coupling Constants in Quantum Gravity, Nucl. Phys. B 307 (1988) 854 [INSPIRE].
M. Kamionkowski and J. March-Russell, Planck scale physics and the Peccei-Quinn mechanism, Phys. Lett. B 282 (1992) 137 [hep-th/9202003] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
Q. Bonnefoy, Heavy fields and the axion quality problem, Phys. Rev. D 108 (2023) 035023 [arXiv:2212.00102] [INSPIRE].
P.W. Graham, D.E. Kaplan and S. Rajendran, Cosmological Relaxation of the Electroweak Scale, Phys. Rev. Lett. 115 (2015) 221801 [arXiv:1504.07551] [INSPIRE].
J.R. Espinosa et al., Cosmological Higgs-Axion Interplay for a Naturally Small Electroweak Scale, Phys. Rev. Lett. 115 (2015) 251803 [arXiv:1506.09217] [INSPIRE].
R. Franceschini et al., Digamma, what next?, JHEP 07 (2016) 150 [arXiv:1604.06446] [INSPIRE].
M. Srednicki, Axion Couplings to Matter. 1. CP Conserving Parts, Nucl. Phys. B 260 (1985) 689 [INSPIRE].
H. Georgi, D.B. Kaplan and L. Randall, Manifesting the Invisible Axion at Low-energies, Phys. Lett. B 169 (1986) 73 [INSPIRE].
Q. Bonnefoy, C. Grojean and J. Kley, Shift-Invariant Orders of an Axionlike Particle, Phys. Rev. Lett. 130 (2023) 111803 [arXiv:2206.04182] [INSPIRE].
P. Draper and D. McKeen, Diphotons from Tetraphotons in the Decay of a 125 GeV Higgs at the LHC, Phys. Rev. D 85 (2012) 115023 [arXiv:1204.1061] [INSPIRE].
M. Bauer, M. Neubert and A. Thamm, Collider Probes of Axion-Like Particles, JHEP 12 (2017) 044 [arXiv:1708.00443] [INSPIRE].
M. Bauer, M. Heiles, M. Neubert and A. Thamm, Axion-Like Particles at Future Colliders, Eur. Phys. J. C 79 (2019) 74 [arXiv:1808.10323] [INSPIRE].
H. Davoudiasl, R. Marcarelli, N. Miesch and E.T. Neil, Searching for flavor-violating ALPs in Higgs boson decays, Phys. Rev. D 104 (2021) 055022 [arXiv:2105.05866] [INSPIRE].
I. Brivio, O.J.P. Éboli and M.C. Gonzalez-Garcia, Unitarity constraints on ALP interactions, Phys. Rev. D 104 (2021) 035027 [arXiv:2106.05977] [INSPIRE].
M. Bauer, M. Neubert and A. Thamm, Analyzing the CP Nature of a New Scalar Particle via S->Zh Decay, Phys. Rev. Lett. 117 (2016) 181801 [arXiv:1610.00009] [INSPIRE].
J. Quevillon, C. Smith and P.N.H. Vuong, Axion effective action, JHEP 08 (2022) 137 [arXiv:2112.00553] [INSPIRE].
F. Björkeroth, E.J. Chun and S.F. King, Flavourful Axion Phenomenology, JHEP 08 (2018) 117 [arXiv:1806.00660] [INSPIRE].
M. Bauer et al., The Low-Energy Effective Theory of Axions and ALPs, JHEP 04 (2021) 063 [arXiv:2012.12272] [INSPIRE].
J. Martin Camalich et al., Quark Flavor Phenomenology of the QCD Axion, Phys. Rev. D 102 (2020) 015023 [arXiv:2002.04623] [INSPIRE].
L. Calibbi, D. Redigolo, R. Ziegler and J. Zupan, Looking forward to lepton-flavor-violating ALPs, JHEP 09 (2021) 173 [arXiv:2006.04795] [INSPIRE].
M. Bauer et al., Flavor probes of axion-like particles, JHEP 09 (2022) 056 [arXiv:2110.10698] [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].
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, Hilbert series and operator bases with derivatives in effective field theories, Commun. Math. Phys. 347 (2016) 363 [arXiv:1507.07240] [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].
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].
M. Ruhdorfer, J. Serra and A. Weiler, Effective Field Theory of Gravity to All Orders, JHEP 05 (2020) 083 [arXiv:1908.08050] [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].
U. Banerjee, J. Chakrabortty, S. Prakash and S.U. Rahaman, Characters and group invariant polynomials of (super)fields: road to “Lagrangian”, Eur. Phys. J. C 80 (2020) 938 [arXiv:2004.12830] [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].
D. Kondo, H. Murayama and R. Okabe, 23, 381, 6242, 103268, 1743183, . . . : Hilbert series for CP-violating operators in SMEFT, JHEP 03 (2023) 107 [arXiv:2212.02413] [INSPIRE].
L. Gráf et al., Hilbert series, the Higgs mechanism, and HEFT, JHEP 02 (2023) 064 [arXiv:2211.06275] [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].
J. Bijnens, S.B. Gudnason, J. Yu and T. Zhang, Hilbert series and higher-order Lagrangians for the O(N) model, JHEP 05 (2023) 061 [arXiv:2212.07901] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: The Plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
C. Grojean, J. Kley and C.-Y. Yao, CHINCHILLA: a Mathematica package for the construction of invariants using the Hilbert series, to appear.
J. Bonilla, I. Brivio, M.B. Gavela and V. Sanz, One-loop corrections to ALP couplings, JHEP 11 (2021) 168 [arXiv:2107.11392] [INSPIRE].
M. Chala, Á. Díaz-Carmona and G. Guedes, A Green’s basis for the bosonic SMEFT to dimension 8, JHEP 05 (2022) 138 [arXiv:2112.12724] [INSPIRE].
S. Weinberg, Baryon and Lepton Nonconserving Processes, Phys. Rev. Lett. 43 (1979) 1566 [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].
L. Lehman, Extending the Standard Model Effective Field Theory with the Complete Set of Dimension-7 Operators, Phys. Rev. D 90 (2014) 125023 [arXiv:1410.4193] [INSPIRE].
C.W. Murphy, Dimension-8 operators in the Standard Model Eective Field Theory, JHEP 10 (2020) 174 [arXiv:2005.00059] [INSPIRE].
I. Brivio et al., ALPs Effective Field Theory and Collider Signatures, Eur. Phys. J. C 77 (2017) 572 [arXiv:1701.05379] [INSPIRE].
T. Melia and S. Pal, EFT Asymptotics: the Growth of Operator Degeneracy, SciPost Phys. 10 (2021) 104 [arXiv:2010.08560] [INSPIRE].
M. Chala, G. Guedes, M. Ramos and J. Santiago, Running in the ALPs, Eur. Phys. J. C 81 (2021) 181 [arXiv:2012.09017] [INSPIRE].
B. Gripaios and D. Sutherland, An operator basis for the Standard Model with an added scalar singlet, JHEP 08 (2016) 103 [arXiv:1604.07365] [INSPIRE].
E. Bertuzzo, C. Grojean and G.M. Salla, ALPs, the on-shell way, to appear.
S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022 [INSPIRE].
S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial-vector current. II, Phys. Rev. 139 (1965) B1638 [INSPIRE].
C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation, Phys. Rev. Lett. 55 (1985) 1039 [INSPIRE].
C. Jarlskog, A Basis Independent Formulation of the Connection Between Quark Mass Matrices, CP Violation and Experiment, Z. Phys. C 29 (1985) 491 [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].
Q. Bonnefoy, E. Gendy, C. Grojean and J.T. Ruderman, Opportunistic CP violation, JHEP 06 (2023) 141 [arXiv:2302.07288] [INSPIRE].
E.E. Jenkins and A.V. Manohar, Rephasing Invariants of Quark and Lepton Mixing Matrices, Nucl. Phys. B 792 (2008) 187 [arXiv:0706.4313] [INSPIRE].
Q. Bonnefoy et al., The anomalous case of axion EFTs and massive chiral gauge fields, JHEP 07 (2021) 189 [arXiv:2011.10025] [INSPIRE].
W. Altmannshofer, J.A. Dror and S. Gori, New Opportunities for Detecting Axion-Lepton Interactions, Phys. Rev. Lett. 130 (2023) 241801 [arXiv:2209.00665] [INSPIRE].
G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, The QCD axion, precisely, JHEP 01 (2016) 034 [arXiv:1511.02867] [INSPIRE].
M. Bauer et al., Consistent Treatment of Axions in the Weak Chiral Lagrangian, Phys. Rev. Lett. 127 (2021) 081803 [arXiv:2102.13112] [INSPIRE].
Y. Liao and J.-Y. Liu, Generalized Fierz Identities and Applications to Spin-3/2 Particles, Eur. Phys. J. Plus 127 (2012) 121 [arXiv:1206.5141] [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].
E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03 (2018) 016 [arXiv:1709.04486] [INSPIRE].
Y. Liao, X.-D. Ma and Q.-Y. Wang, Extending low energy effective field theory with a complete set of dimension-7 operators, JHEP 08 (2020) 162 [arXiv:2005.08013] [INSPIRE].
C.W. Murphy, Low-Energy Effective Field Theory below the Electroweak Scale: Dimension-8 Operators, JHEP 04 (2021) 101 [arXiv:2012.13291] [INSPIRE].
S.-S. Kim, H.M. Lee and K. Yamashita, Positivity bounds on Higgs-Portal dark matter, JHEP 06 (2023) 124 [arXiv:2302.02879] [INSPIRE].
S. Das Bakshi, J. Machado-Rodríguez and M. Ramos, Running beyond ALPs: shift-breaking and CP-violating effects, arXiv:2306.08036 [INSPIRE].
H. Song, H. Sun and J.-H. Yu, Effective Field Theories of Axion, ALP and Dark Photon, arXiv:2305.16770 [INSPIRE].
H. Song, H. Sun and J.-H. Yu, Complete EFT Operator Bases for Dark Matter and Weakly-Interacting Light Particle, arXiv:2306.05999 [INSPIRE].
Acknowledgments
We thank Enrico Bertuzzo and Gabriel Massoni Salla for enlightening discussions on the axion scattering amplitudes that give an alternative way to obtain a basis of the axion interactions to the SM particles. We thank Guilherme Guedes for discussions on operator basis redundancies and Pham Ngoc Hoa Vuong for discussions on shift-breaking effects in ALP EFTs. Furthermore, we thank Emanuele Gendy for collaboration at early stages of this project and Quentin Bonnefoy for numerous discussions on this topic and comments on the manuscript. We thank Katharina Albrecht for cross-checking some of the results in this paper.
This work is supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 “Quantum Universe” — 390833306, as well as by the grant 491245950. This project also has received funding from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie Staff Exchange grant agreement No 101086085 - ASYMMETRY. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation. C.Y.Y. is supported in part by the Grants No. NSFC-11975130, No. NSFC-12035008, No. NSFC-12047533, the National Key Research and Development Program of China under Grant No. 2017YFA0402200, the China Postdoctoral Science Foundation under Grant No. 2018M641621 and the Helmholtz-OCPC International Postdoctoral Exchange Fellowship Program.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2307.08563
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Grojean, C., Kley, J. & Yao, CY. Hilbert series for ALP EFTs. J. High Energ. Phys. 2023, 196 (2023). https://doi.org/10.1007/JHEP11(2023)196
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2023)196