Abstract
In the present paper, we carry out a systematic study of the flavor invariants and their renormalization-group equations (RGEs) in the leptonic sector with three generations of charged leptons and massive Majorana neutrinos. First, following the approach of the Hilbert series from the invariant theory, we show that there are 34 basic flavor invariants in the generating set, among which 19 invariants are CP-even and the others are CP-odd. Any flavor invariants can be expressed as the polynomials of those 34 basic invariants in the generating set. Second, we explicitly construct all the basic invariants and derive their RGEs, which form a closed system of differential equations as they should. The numerical solutions to the RGEs of the basic flavor invariants have also been found. Furthermore, we demonstrate how to extract physical observables from the basic invariants. Our study is helpful for understanding the algebraic structure of flavor invariants in the leptonic sector, and also provides a novel way to explore leptonic flavor structures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z.-z. Xing, Flavor structures of charged fermions and massive neutrinos, Phys. Rept. 854 (2020) 1 [arXiv:1909.09610] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2020 (2020) 083C01 [INSPIRE].
E. Majorana, Teoria simmetrica dell’elettrone e del positrone, Nuovo Cim. 14 (1937) 171.
G. Racah, On the symmetry of particle and antiparticle, Nuovo Cim. 14 (1937) 322 [INSPIRE].
B. Pontecorvo, Mesonium and anti-mesonium, Sov. Phys. JETP 6 (1957) 429 [Zh. Eksp. Teor. Fiz. 33 (1957) 549] [INSPIRE].
Z. Maki, M. Nakagawa and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28 (1962) 870 [INSPIRE].
C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP-violation, 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].
J. Bernabeu, G.C. Branco and M. Gronau, CP Restrictions on Quark Mass Matrices, Phys. Lett. B 169 (1986) 243 [INSPIRE].
G.C. Branco, L. Lavoura and M.N. Rebelo, Majorana Neutrinos and CP Violation in the Leptonic Sector, Phys. Lett. B 180 (1986) 264 [INSPIRE].
B. Yu and S. Zhou, The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?, Phys. Lett. B 800 (2020) 135085 [arXiv:1908.09306] [INSPIRE].
B. Yu and S. Zhou, Weak-basis invariants and CP conservation in the leptonic sector with Majorana neutrinos, PoS ICHEP2020 (2021) 193 [arXiv:2010.08758] [INSPIRE].
B. Yu and S. Zhou, Sufficient and Necessary Conditions for CP Conservation in the Case of Degenerate Majorana Neutrino Masses, Phys. Rev. D 103 (2021) 035017 [arXiv:2009.12347] [INSPIRE].
A. Pilaftsis, CP violation and baryogenesis due to heavy Majorana neutrinos, Phys. Rev. D 56 (1997) 5431 [hep-ph/9707235] [INSPIRE].
G.C. Branco, T. Morozumi, B.M. Nobre and M.N. Rebelo, A Bridge between CP-violation at low-energies and leptogenesis, Nucl. Phys. B 617 (2001) 475 [hep-ph/0107164] [INSPIRE].
V. Cirigliano, G. Isidori and V. Porretti, CP violation and Leptogenesis in models with Minimal Lepton Flavour Violation, Nucl. Phys. B 763 (2007) 228 [hep-ph/0607068] [INSPIRE].
T. Feldmann, T. Mannel and S. Schwertfeger, Renormalization Group Evolution of Flavour Invariants, JHEP 10 (2015) 007 [arXiv:1507.00328] [INSPIRE].
J. Talbert and M. Trott, Dirac Masses and Mixings in the (geo)SM(EFT) and Beyond, arXiv:2107.03951 [INSPIRE].
B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Wien Austria (2008).
H. Derksen, G. Kemper, V.L. Popov and N.A’ Campo, Computational invariant theory, Springer-Verlag, Berlin Heidelberg (2015).
C. Processi, The Invariant Theory of n × n Matrices, Adv. Math. 19 (1976) 306.
E. Formanek, Invariants and the Ring of Generic Matrices, J. Algebra 89 (1984) 178.
P.H. Chankowski and Z. Pluciennik, Renormalization group equations for seesaw neutrino masses, Phys. Lett. B 316 (1993) 312 [hep-ph/9306333] [INSPIRE].
K.S. Babu, C.N. Leung and J.T. Pantaleone, Renormalization of the neutrino mass operator, Phys. Lett. B 319 (1993) 191 [hep-ph/9309223] [INSPIRE].
S. Antusch, M. Drees, J. Kersten, M. Lindner and M. Ratz, Neutrino mass operator renormalization revisited, Phys. Lett. B 519 (2001) 238 [hep-ph/0108005] [INSPIRE].
Z. z. Xing and S. Zhou, Neutrinos in particle physics, astronomy and cosmology, Springer-Verlag, Heidelberg Germany (2011).
T. Ohlsson and S. Zhou, Renormalization group running of neutrino parameters, Nature Commun. 5 (2014) 5153 [arXiv:1311.3846] [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. Garsia, N. Wallach, G. Xin and M. Zabrocki, Hilbert series of invariants, constant terms and Kostka-Foulkes polynomials, Discrete Math. 309 (2009) 5206.
I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178 [arXiv:2007.14792] [INSPIRE].
G.-y. Huang and S. Zhou, Precise Values of Running Quark and Lepton Masses in the Standard Model, Phys. Rev. D 103 (2021) 016010 [arXiv:2009.04851] [INSPIRE].
J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori, A. Riotto and A. Strumia, Higgs mass implications on the stability of the electroweak vacuum, Phys. Lett. B 709 (2012) 222 [arXiv:1112.3022] [INSPIRE].
Z.-z. Xing, H. Zhang and S. Zhou, Impacts of the Higgs mass on vacuum stability, running fermion masses and two-body Higgs decays, Phys. Rev. D 86 (2012) 013013 [arXiv:1112.3112] [INSPIRE].
G. Degrassi et al., Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP 08 (2012) 098 [arXiv:1205.6497] [INSPIRE].
A. Trautner, Systematic construction of basis invariants in the 2HDM, JHEP 05 (2019) 208 [arXiv:1812.02614] [INSPIRE].
A. Trautner, On the systematic construction of basis invariants, J. Phys. Conf. Ser. 1586 (2020) 012005 [arXiv:2002.12244] [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].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [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].
F.A. Dolan, Counting BPS operators in N = 4 SYM, Nucl. Phys. B 790 (2008) 432 [arXiv:0704.1038] [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].
A. Hanany, Counting BPS operators in the chiral ring: The plethystic story, AIP Conf. Proc. 939 (2007) 165 [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 and N. Mekareeya, Counting Gauge Invariant Operators in SQCD with Classical Gauge Groups, JHEP 10 (2008) 012 [arXiv:0805.3728] [INSPIRE].
A. Hanany, N. Mekareeya and G. Torri, The Hilbert Series of Adjoint SQCD, Nucl. Phys. B 825 (2010) 52 [arXiv:0812.2315] [INSPIRE].
D. Forcella, Master Space and Hilbert Series for N = 1 Field Theories, Ph.D. thesis, CNRS and Ecole Normale Superieure, Paris, 2008. arXiv:0902.2109 [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. Hanany and R. Kalveks, Highest Weight Generating Functions for Hilbert Series, JHEP 10 (2014) 152 [arXiv:1408.4690] [INSPIRE].
A. Bourget and A. Pini, Non-Connected Gauge Groups and the Plethystic Program, JHEP 10 (2017) 033 [arXiv:1706.03781] [INSPIRE].
Y. Xiao, Y.-H. He and C. Matti, Standard Model Plethystics, Phys. Rev. D 100 (2019) 076001 [arXiv:1902.10550] [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].
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].
A. Kobach and S. Pal, Reparameterization Invariant Operator Basis for NRQED and HQET, JHEP 11 (2019) 012 [arXiv:1810.02356] [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].
H.-L. Li, Z. Ren, J. Shu, M.-L. Xiao, J.-H. Yu and Y.-H. Zheng, Complete set of dimension-eight operators in the standard model effective field theory, Phys. Rev. D 104 (2021) 015026 [arXiv:2005.00008] [INSPIRE].
Y. Liao and X.-D. Ma, An explicit construction of the dimension-9 operator basis in the standard model effective field theory, JHEP 11 (2020) 152 [arXiv:2007.08125] [INSPIRE].
L. Graf, B. Henning, X. Lu, T. Melia and H. Murayama, 2, 12, 117, 1959, 45171, 1170086, . . .: a Hilbert series for the QCD chiral Lagrangian, JHEP 01 (2021) 142 [arXiv:2009.01239] [INSPIRE].
G.D. Kribs, X. Lu, A. Martin and T. Tong, Custodial Symmetry (Violation) in SMEFT, arXiv:2009.10725 [INSPIRE].
T. Melia and S. Pal, EFT Asymptotics: the Growth of Operator Degeneracy, SciPost Phys. 10 (2021) 104 [arXiv:2010.08560] [INSPIRE].
C.W. Murphy, Low-Energy Effective Field Theory below the Electroweak Scale: Dimension-8 Operators, JHEP 04 (2021) 101 [arXiv:2012.13291] [INSPIRE].
W. Cao, F. Herzog, T. Melia and J.R. Nepveu, Renormalization and non-renormalization of scalar EFTs at higher orders, arXiv:2105.12742 [INSPIRE].
K. Haddad and A. Helset, Tidal effects in quantum field theory, JHEP 12 (2020) 024 [arXiv:2008.04920] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Tidal effects for spinning particles, JHEP 03 (2021) 097 [arXiv:2012.05256] [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].
A.V. Bednyakov, On three-loop RGE for the Higgs sector of 2HDM, JHEP 11 (2018) 154 [arXiv:1809.04527] [INSPIRE].
M.P. Bento, R. Boto, J.P. Silva and A. Trautner, A fully basis invariant Symmetry Map of the 2HDM, JHEP 21 (2020) 229 [arXiv:2009.01264] [INSPIRE].
M.P. Bento, The invariant space of multi-Higgs doublet models, JHEP 05 (2021) 146 [arXiv:2102.13120] [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].
T. Molien, Über die Invarianten der linearen Substitutionsgruppe, Sitz. König. Preuss. Akad. Wiss. 52 (1897) 1152.
H. Weyl, Zur Darstellungstheorie und Invariantenabzählung der projektiven, der Komplex-und der Drehungsgruppe, Acta Math. 48 (1926) 255.
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: 2107.06274
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
Wang, Y., Yu, B. & Zhou, S. Flavor invariants and renormalization-group equations in the leptonic sector with massive Majorana neutrinos. J. High Energ. Phys. 2021, 53 (2021). https://doi.org/10.1007/JHEP09(2021)053
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2021)053