Abstract
In this work, we explore an extension of Hilbert series techniques to count operators that include derivatives. For sufficiently low-derivative operators, we conjecture an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts. Specifically, the conjectured technique can be applied whenever there is only one Lorentz invariant for a given partitioning of derivatives among the fields. At higher numbers of derivatives, equation of motion redundancies can be removed, but the increased number of Lorentz contractions spoils the subtraction of integration by parts redundancies. While restricted, this technique is sufficient to automatically recreate the complete set of invariant operators of the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary numbers of flavors). At dimension 8, the algorithm does not automatically generate the complete operator set; however, it suffices for all but five classes of operators. For these remaining classes, there is a well defined procedure to manually determine the number of invariants. Assuming our method is correct, we derive a set of 535 dimension-8 N f = 1 operators.
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References
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, arXiv:1507.07240 [INSPIRE].
P. Pouliot, Molien function for duality, JHEP 01 (1999) 021 [hep-th/9812015] [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].
F.A. Dolan, Counting BPS operators in N = 4 SYM, Nucl. Phys. B 790 (2008) 432 [arXiv:0704.1038] [INSPIRE].
J. Gray, A. Hanany, Y.-H. He, V. Jejjala and N. Mekareeya, SQCD: A Geometric Apercu, JHEP 05 (2008) 099 [arXiv:0803.4257] [INSPIRE].
A. Hanany, N. Mekareeya and G. Torri, The Hilbert Series of Adjoint SQCD, Nucl. Phys. B 825 (2010) 52 [arXiv:0812.2315] [INSPIRE].
Y. Chen and N. Mekareeya, The Hilbert series of U/SU SQCD and Toeplitz Determinants, Nucl. Phys. B 850 (2011) 553 [arXiv:1104.2045] [INSPIRE].
A. Butti, D. Forcella, A. Hanany, D. Vegh and A. Zaffaroni, Counting Chiral Operators in Quiver Gauge Theories, JHEP 11 (2007) 092 [arXiv:0705.2771] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: The plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
D. Forcella, A. Hanany and A. Zaffaroni, Baryonic Generating Functions, JHEP 12 (2007) 022 [hep-th/0701236] [INSPIRE].
S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert Series of the One Instanton Moduli Space, JHEP 06 (2010) 100 [arXiv:1005.3026] [INSPIRE].
A. Hanany, N. Mekareeya and S.S. Razamat, Hilbert Series for Moduli Spaces of Two Instantons, JHEP 01 (2013) 070 [arXiv:1205.4741] [INSPIRE].
D. Rodríguez-Gómez and G. Zafrir, On the 5d instanton index as a Hilbert series, Nucl. Phys. B 878 (2014) 1 [arXiv:1305.5684] [INSPIRE].
A. Dey, A. Hanany, N. Mekareeya, D. Rodríguez-Gómez and R.-K. Seong, Hilbert Series for Moduli Spaces of Instantons on \( {\mathbb{C}}^2/{\mathbb{Z}}_n \), JHEP 01 (2014) 182 [arXiv:1309.0812] [INSPIRE].
A. Hanany and R.-K. Seong, Hilbert series and moduli spaces of k U(N ) vortices, JHEP 02 (2015) 012 [arXiv:1403.4950] [INSPIRE].
L. Begin, C. Cummins and P. Mathieu, Generating functions for tensor products, hep-th/9811113 [INSPIRE].
A. Hanany and R. Kalveks, Highest Weight Generating Functions for Hilbert Series, JHEP 10 (2014) 152 [arXiv:1408.4690] [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].
A. Merle and R. Zwicky, Explicit and spontaneous breaking of SU(3) into its finite subgroups, JHEP 02 (2012) 128 [arXiv:1110.4891] [INSPIRE].
C. Grojean, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Scaling of Higgs Operators and Γ(h → γγ), JHEP 04 (2013) 016 [arXiv:1301.2588] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP 10 (2013) 087 [arXiv:1308.2627] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence, JHEP 01 (2014) 035 [arXiv:1310.4838] [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].
J. Elias-Miró, J.R. Espinosa, E. Masso and A. Pomarol, Renormalization of dimension-six operators relevant for the Higgs decays h → γγ, γZ, JHEP 08 (2013) 033 [arXiv:1302.5661] [INSPIRE].
J. Elias-Miro, J.R. Espinosa, E. Masso and A. Pomarol, Higgs windows to new physics through D = 6 operators: constraints and one-loop anomalous dimensions, JHEP 11 (2013) 066 [arXiv:1308.1879] [INSPIRE].
R. Alonso, H.-M. Chang, E.E. Jenkins, A.V. Manohar and B. Shotwell, Renormalization group evolution of dimension-six baryon number violating operators, Phys. Lett. B 734 (2014) 302 [arXiv:1405.0486] [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, Holomorphy without Supersymmetry in the Standard Model Effective Field Theory, Phys. Lett. B 739 (2014) 95 [arXiv:1409.0868] [INSPIRE].
M. Trott, On the consistent use of Constructed Observables, JHEP 02 (2015) 046 [arXiv:1409.7605] [INSPIRE].
B. Henning, X. Lu and H. Murayama, How to use the Standard Model effective field theory, JHEP 01 (2016) 023 [arXiv:1412.1837] [INSPIRE].
S. Willenbrock and C. Zhang, Effective Field Theory Beyond the Standard Model, Ann. Rev. Nucl. Part. Sci. 64 (2014) 83 [arXiv:1401.0470] [INSPIRE].
J. Elias-Miro, J.R. Espinosa and A. Pomarol, One-loop non-renormalization results in EFTs, Phys. Lett. B 747 (2015) 272 [arXiv:1412.7151] [INSPIRE].
R.S. Gupta, A. Pomarol and F. Riva, BSM Primary Effects, Phys. Rev. D 91 (2015) 035001 [arXiv:1405.0181] [INSPIRE].
G.F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, The Strongly-Interacting Light Higgs, JHEP 06 (2007) 045 [hep-ph/0703164] [INSPIRE].
M. Duehrssen-Debling et al., Higgs Basis, Proposal for an EFT basis choice for LHC HXSWG, LHC Higgs Cross section Working Group 2, LHCHXSWG-INT-2015-001.
C. Cheung and C.-H. Shen, Nonrenormalization Theorems without Supersymmetry, Phys. Rev. Lett. 115 (2015) 071601 [arXiv:1505.01844] [INSPIRE].
L. Berthier and M. Trott, Consistent constraints on the Standard Model Effective Field Theory, arXiv:1508.05060 [INSPIRE].
C.-W. Chiang and R. Huo, Standard Model Effective Field Theory: Integrating out a Generic Scalar, JHEP 09 (2015) 152 [arXiv:1505.06334] [INSPIRE].
R. Huo, Standard Model Effective Field Theory: Integrating out Vector-Like Fermions, JHEP 09 (2015) 037 [arXiv:1506.00840] [INSPIRE].
R. Huo, Effective Field Theory of Integrating out Sfermions in the MSSM: Complete One-Loop Analysis, arXiv:1509.05942 [INSPIRE].
A. Drozd, J. Ellis, J. Quevillon and T. You, Comparing EFT and Exact One-Loop Analyses of Non-Degenerate Stops, JHEP 06 (2015) 028 [arXiv:1504.02409] [INSPIRE].
S. Weinberg, Baryon and Lepton Nonconserving Processes, Phys. Rev. Lett. 43 (1979) 1566 [INSPIRE].
W. Buchmüller 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].
L.F. Abbott and M.B. Wise, The Effective Hamiltonian for Nucleon Decay, Phys. Rev. D 22 (1980) 2208 [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].
G.J. Gounaris, J. Layssac and F.M. Renard, Addendum to off-shell structure of the anomalous Z and gamma selfcouplings, Phys. Rev. D 65 (2002) 017302 [INSPIRE].
C. Degrande, A basis of dimension-eight operators for anomalous neutral triple gauge boson interactions, JHEP 02 (2014) 101 [arXiv:1308.6323] [INSPIRE].
H.D. Politzer, Power Corrections at Short Distances, Nucl. Phys. B 172 (1980) 349 [INSPIRE].
H. Georgi, On-shell effective field theory, Nucl. Phys. B 361 (1991) 339 [INSPIRE].
C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230] [INSPIRE].
C. Grosse-Knetter, Effective Lagrangians with higher derivatives and equations of motion, Phys. Rev. D 49 (1994) 6709 [hep-ph/9306321] [INSPIRE].
H. Simma, Equations of motion for effective Lagrangians and penguins in rare B decays, Z. Phys. C 61 (1994) 67 [hep-ph/9307274] [INSPIRE].
K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Low-energy effects of new interactions in the electroweak boson sector, Phys. Rev. D 48 (1993) 2182 [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,arXiv:1512.03433 [INSPIRE].
J.L. Gross, J. Yellen and P. Zhang, Handbook of Graph Theory, second edition, Chapman & Hall/CRC (2013).
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Lehman, L., Martin, A. Low-derivative operators of the Standard Model effective field theory via Hilbert series methods. J. High Energ. Phys. 2016, 81 (2016). https://doi.org/10.1007/JHEP02(2016)081
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DOI: https://doi.org/10.1007/JHEP02(2016)081