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We calculate the gauge terms of the one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory (SM EFT). Combining these results with our previous results for the and Yukawa coupling terms completes the calculation of the one-loop anomalous dimension matrix for the dimensionsix operators. There are 1350 CP-even and 1149 CP-odd parameters in the dimensionsix Lagrangian for 3 generations, and our results give the entire 2499 2499 anomalous dimension matrix. We discuss how the renormalization of the dimension-six operators, and the additional renormalization of the dimension d 4 terms of the SM Lagrangian due to dimension-six operators, lays the groundwork for future precision studies of the SM EFT aimed at constraining the effects of new physics through precision measurements at the electroweak scale. As some sample applications, we discuss some aspects of the full RGE improved result for essential processes such as gg h, h and h Z, for Higgs couplings to fermions, for the precision electroweak parameters S and T, and for the operators that modify important processes in precision electroweak phenomenology, such as the three-body Higgs boson decay h Z + and triple gauge boson couplings. We discuss how the renormalization group improved results can be used to study the flavor problem in the SM EFT, and to test the minimal flavor violation (MFV) hypothesis. We briefly discuss the renormalization effects on the dipole coefficient Ce which contributes to e and to the muon and electron magnetic and electric dipole moments. © The Authors.

We calculate the complete order y 2 and y 4 terms of the 59 × 59 one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory, where y is a generic Yukawa coupling. These terms, together with the terms of order λ, λ 2 and λy 2 depending on the Standard Model Higgs self-coupling λ which were calculated in a previous work, yield the complete one-loop anomalous dimension matrix in the limit of vanishing gauge couplings. The Yukawa contributions result in non-trivial flavor mixing in the various operator sectors of the Standard Model effective theory. © 2014 SISSA.

We calculate the complete order y 2 and y 4 terms of the 59 × 59 one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory, where y is a generic Yukawa coupling. These terms, together with the terms of order λ, λ 2 and λy 2 depending on the Standard Model Higgs self-coupling λ which were calculated in a previous work, yield the complete one-loop anomalous dimension matrix in the limit of vanishing gauge couplings. The Yukawa contributions result in non-trivial flavor mixing in the various operator sectors of the Standard Model effective theory. © 2014 SISSA.

We calculate the gauge terms of the one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory (SM EFT). Combining these results with our previous results for the and Yukawa coupling terms completes the calculation of the one-loop anomalous dimension matrix for the dimensionsix operators. There are 1350 CP-even and 1149 CP-odd parameters in the dimensionsix Lagrangian for 3 generations, and our results give the entire 2499 2499 anomalous dimension matrix. We discuss how the renormalization of the dimension-six operators, and the additional renormalization of the dimension d 4 terms of the SM Lagrangian due to dimension-six operators, lays the groundwork for future precision studies of the SM EFT aimed at constraining the effects of new physics through precision measurements at the electroweak scale. As some sample applications, we discuss some aspects of the full RGE improved result for essential processes such as gg h, h and h Z, for Higgs couplings to fermions, for the precision electroweak parameters S and T, and for the operators that modify important processes in precision electroweak phenomenology, such as the three-body Higgs boson decay h Z + and triple gauge boson couplings. We discuss how the renormalization group improved results can be used to study the flavor problem in the SM EFT, and to test the minimal flavor violation (MFV) hypothesis. We briefly discuss the renormalization effects on the dipole coefficient Ce which contributes to e and to the muon and electron magnetic and electric dipole moments. © The Authors.

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N= L+. δ, with a shift δ that depends on the NDA-weight of operator insertions. The NDA weight of an operator is defined in this Letter, and the general NDA formula for perturbative order N is derived. The formula is used to explain why the one-loop anomalous dimension matrix for dimension-six operators in the Standard Model effective field theory has entries with perturbative order ranging from 0 to 4. The results in this Letter are valid for an arbitrary effective field theory, and they constrain the coupling constant dependence of anomalous dimensions and scattering amplitudes in a general effective field theory. © 2013 Elsevier B.V.

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N= L+. δ, with a shift δ that depends on the NDA-weight of operator insertions. The NDA weight of an operator is defined in this Letter, and the general NDA formula for perturbative order N is derived. The formula is used to explain why the one-loop anomalous dimension matrix for dimension-six operators in the Standard Model effective field theory has entries with perturbative order ranging from 0 to 4. The results in this Letter are valid for an arbitrary effective field theory, and they constrain the coupling constant dependence of anomalous dimensions and scattering amplitudes in a general effective field theory. © 2013 Elsevier B.V.

We calculate the order λ, λ2 and λy2 terms of the 59×59 one-loop anomalous dimension matrix of dimension-six operators, where λ and y are the Standard Model Higgs self-coupling and a generic Yukawa coupling, respectively. The dimension-six operators modify the running of the Standard Model parameters themselves, and we compute the complete one-loop result for this. We discuss how there is mixing between operators for which no direct one-particle-irreducible diagram exists, due to operator replacements by the equations of motion. © SISSA 2013.

The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others. It also has been extensively used to analyze beyond-the-Standard-Model theories. We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact. We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories. Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed. © 2013 SISSA.

The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others. It also has been extensively used to analyze beyond-the-Standard-Model theories. We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact. We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories. Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed. © 2013 SISSA.

We calculate the order λ, λ2 and λy2 terms of the 59×59 one-loop anomalous dimension matrix of dimension-six operators, where λ and y are the Standard Model Higgs self-coupling and a generic Yukawa coupling, respectively. The dimension-six operators modify the running of the Standard Model parameters themselves, and we compute the complete one-loop result for this. We discuss how there is mixing between operators for which no direct one-particle-irreducible diagram exists, due to operator replacements by the equations of motion. © SISSA 2013.