An explicit construction of the dimension-9 operator basis in the standard model effective field theory

Abstract

We investigate systematically dimension-9 operators in the standard model effective field theory which contains only standard model fields and respects its gauge symmetry. With the help of the Hilbert series approach to classifying operators according to their lepton and baryon numbers and their field contents, we construct the basis of operators explicitly. We remove redundant operators by employing various kinematic and algebraic relations including integration by parts, equations of motion, Schouten identities, Dirac matrix and Fierz identities, and Bianchi identities. We confirm counting of independent operators by analyzing their flavor symmetry relations. All operators violate lepton or baryon number or both, and are thus non-Hermitian. Including Hermitian conjugated operators there are $384{|}_{\Delta B=0}^{\Delta L=\pm 2}+10{|}_{\Delta B=\pm 2}^{\Delta L=0}+4{|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+236{|}_{\Delta B=\pm 1}^{\Delta L=\mp 1}$ operators without referring to fermion generations, and $44874{|}_{\Delta B=0}^{\Delta L=\pm 2}+2862{|}_{\Delta B=\pm 2}^{\Delta L=0}+486{|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+42234{|}_{\Delta B=\mp 1}^{\Delta L=\pm 1}$ operators when three generations of fermions are referred to, where $\Delta L,\Delta B$ denote the net lepton and baryon numbers of the operators. Our result provides a starting point for consistent phenomenological studies associated with dimension-9 operators.

arXiv comment31 pages, 10 tables; Section 3 is expanded with discussion on relevant phenomenologies; Typos are corrected; The version to be appear in JHEP

References in corpus (31)

• Dimension-Six Terms in the Standard Model Lagrangian
• Dimension-8 Operators in the Standard Model Effective Field Theory
• Extending the Standard Model Effective Field Theory with the Complete Set of Dimension-7 Operators
• Classification of Effective Neutrino Mass Operators
• Heavy Majorana Neutrinos in the Effective Lagrangian Description: Application to Hadron Colliders
• Operators up to Dimension Seven in Standard Model Effective Field Theory Extended with Sterile Neutrinos
• Operator bases, $S$-matrices, and their partition functions
• A Survey of Lepton Number Violation Via Effective Operators
• A neutrinoless double beta decay master formula from effective field theory
• Low-derivative operators of the Standard Model effective field theory via Hilbert series methods
• Neutrinoless double-beta decay and effective field theory
• Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox
• Right-handed neutrino magnetic moments
• A superformula for neutrinoless double beta decay II: The short range part
• Baryon Number, Lepton Number, and Operator Dimension in the Standard Model
• Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation
• BasisGen: automatic generation of operator bases
• The Mu2e Experiment
• Hilbert Series for Flavor Invariants of the Standard Model
• Inclusive Nucleon Decay Searches as a Frontier of Baryon Number Violation
• Enumerating the operators of an effective field theory
• Proton decay into charged leptons
• DEFT: A program for operators in EFT
• An electroweak basis for neutrinoless double $\beta$ decay
• Neutron-antineutron oscillations from lattice QCD
• Unique Neutrino Mass Operator at any Mass Dimension
• Characters and Group Invariant Polynomials of (Super)fields: Road to "Lagrangian"
• Effective field theory approach to lepton number violating decays $K^{\pm }\to {\pi }^{\mp }{l}^{\pm }{l}^{\pm }$: short-distance contribution
• ${\mathbf{\mu }}^{-}$- ${\mathbf{e}}^{+}$ Conversion from Short-Range Operators
• Lepton number violating operators with standard model gauge fields: A survey of neutrino masses from 3-loops and their link to dark matter
• Perturbative Power Counting, Lowest-Index Operators and Their Renormalization in Standard Model Effective Field Theory

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