Abstract
We compute the one-loop anomalous dimensions of the low-energy effective Lagrangian below the electroweak scale, up to terms of dimension six. The theory has 70 dimension-five and 3631 dimension-six Hermitian operators that preserve baryon and lepton number, as well as additional operators that violate baryon number and lepton number. The renormalization group equations for the quark and lepton masses and the QCD and QED gauge couplings are modified by dimension-five and dimension-six operator contributions. We compute the renormalization group equations from one insertion of dimension-five and dimension-six operators, as well as two insertions of dimension-five operators, to all terms of dimension less than or equal to six. The use of the equations of motion to eliminate operators can be ambiguous, and we show how to resolve this ambiguity by a careful use of field redefinitions.
Highlights
Current experimental data at the LHC is well-described by the Standard Model (SM), with electroweak gauge symmetry spontaneously broken by a fundamental scalar doublet, and a Higgs boson of mass ∼ 125 GeV
We have presented the complete one-loop renormalization-group equations for the low-energy effective field theory below the electroweak scale up to and including dimension-six operators
Together with the one-loop renormalization group equations (RGEs) for the Standard Model Effective Field Theory (SMEFT) given in refs. [3,4,5] and the tree-level matching equations presented in ref. [20], these RGEs allow one to evolve all the effects of physics beyond the SM consisting of new particles at some heavy scale Λ down to low energies, where precision measurements are performed, to leadinglog accuracy
Summary
Current experimental data at the LHC is well-described by the Standard Model (SM), with electroweak gauge symmetry spontaneously broken by a fundamental scalar doublet, and a Higgs boson of mass ∼ 125 GeV. We do not restrict the analysis to a scenario that derives from SMEFT as the high-energy theory, but consider all LEFT effects up to dimension six These effects include terms quadratic in the dimension-five dipole-operator coefficients. The one-loop anomalous dimensions of the LEFT operators derived in this work can be used to compute low-energy processes using the more general HEFT as the high-energy effective field theory. The dimension-five operators consist of ∆B = ∆L = 0 dipole operators for fermions ψ = u, d, e, which do not violate baryon number or lepton number, and the ∆L = ±2 dipole operators for the left-handed neutrinos. The dimension-six operator sectors that violate baryon and/or lepton number consist of operators with ∆L = ±4, ∆L = ±2, ∆B = ∆L = ±1, and ∆B = −∆L = ±1 appearing in table 2 of appendix A.
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