Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology
Rodrigo Alonso, Elizabeth E. Jenkins, Aneesh V. Manohar, Michael Trott
aa r X i v : . [ h e p - ph ] J u l Prepared for submission to JHEP
Renormalization Group Evolution of the StandardModel Dimension Six OperatorsIII: Gauge Coupling Dependence and Phenomenology
Rodrigo Alonso, a Elizabeth E. Jenkins, a Aneesh V. Manohar, a Michael Trott b, a Department of Physics, University of California at San Diego, 9500 Gilman Drive,La Jolla, CA 92093-0319, USA b Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We calculate the gauge terms of the one-loop anomalous dimension matrix forthe dimension-six operators of the Standard Model effective field theory (SM EFT). Combin-ing these results with our previous results for the λ and Yukawa coupling terms completesthe calculation of the one-loop anomalous dimension matrix for the dimension-six operators.There are 1350 CP -even and CP -odd parameters in the dimension-six Lagrangian for3 generations, and our results give the entire × anomalous dimension matrix. Wediscuss how the renormalization of the dimension-six operators, and the additional renormal-ization of the dimension d ≤ 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 ef-fects of new physics through precision measurements at the electroweak scale. As some sampleapplications, we discuss some aspects of the full RGE improved result for essential processessuch as gg → h , h → γγ and h → Zγ , for Higgs couplings to fermions, for the precisionelectroweak parameters S and T , and for the operators that modify important processes inprecision electroweak phenomenology, such as the three-body Higgs boson decay h → Z ℓ + ℓ − and triple gauge boson couplings. We discuss how the renormalization group improved resultscan be used to study the flavor problem in the SM EFT, and to test the minimal flavor viola-tion (MFV) hypothesis. We briefly discuss the renormalization effects on the dipole coefficient C eγ which contributes to µ → eγ and to the muon and electron magnetic and electric dipolemoments. Corresponding author. ontents γ ij G F C H , C HD , C H (cid:3) h → f f h → W W and h → ZZ gg → h h → γγ h → γ Z µ → eγ , magnetic moments, and electric dipole moments 27 P i operators to the standard basis 31C Results 31 C.1 X H H D X H ψ H
33– i –.6 ψ XH ψ H D ψ ( LL )( LL ) ( RR )( RR ) ( LL )( RR ) ( LR )( RL ) ( LR )( LR ) The LHC experiments have recently found strong evidence for a scalar particle with mass126 GeV, and properties consistent with the Higgs boson of the Standard Model (SM) [1, 2].The absence of any clear evidence of new particles at energies up to several times the scalarboson mass allows one to parametrize the effects of arbitrary new physics residing at energies Λ ≫ v on physical observables at the electroweak scale in terms of higher dimension operatorsbuilt out of SM fields. Experimental measurements of the properties of the scalar boson andother observables at the electroweak scale can then be used to constrain or determine thecoefficients of the higher dimension operators, and hence the effects of arbitrary beyond-the-standard-model (BSM) theories with characteristic energy scale Λ in a model independentway.In this paper, we adopt the assumption that the scalar boson observed at LHC is the SMHiggs boson, and that the Higgs mechanism generates the mass of the SM gauge fields andfermions. Specifically, we assume that the observed scalar boson h is part of a SU (2) L doublet H with hypercharge y h = , and that the electroweak SU (2) L × U (1) Y gauge symmetry is alinearly realized symmetry in the scalar sector which is spontaneously broken by the vacuumexpectation value of H . These assumptions yield the simplest and most direct interpretation ofthe LHC data, and the related experimental observations from LEP and the Tevatron. . TheSM effective field theory (SM EFT) based on these assumptions consists of the SM Lagrangianplus all possible higher dimension operators.The leading higher dimension operators built out of SM fields that preserve baryon andlepton number are 59 dimension-six operators [8, 9]. It is important to keep in mind that manyof these operators have flavor (generation) indices. For n g = 3 generations, the dimension-six Lagrangian has 1350 CP -even and 1149 CP -odd couplings, for a total of 2499 hermitianoperators and real parameters. The flavor indices obviously cannot be neglected — there isno reason in general, for example, why the new physics contribution to µ → eγ should be There are other alternatives being investigated, such as a nonlinearly realized SU (2) L × U (1) Y gaugesymmetry in the scalar sector with a light scalar h ; see [3–7] and references therein – 1 –he same as the new physics contribution to the muon magnetic moment. Despite the largenumber of operators, it is important to realize that the SM equations of motion (EOM) havebeen used extensively in reducing the operator basis. As a result, the coefficient of a removedoperator is distributed among the remaining operators.In this work, we complete the full calculation of the × one-loop anomalousdimension matrix of the 59 dimension-six operators in the operator basis of Ref. [8, 9], includingflavor indices for an arbitrary number of generations n g . We present the gauge coupling termsin the one-loop anomalous dimension matrix in this paper. Combined with our past results [10–13], this gives the full one-loop renormalization group evolution (RGE) of the dimension-sixoperators of the SM EFT. Having the full one-loop RGE of an independent set of dimension-six operators in the SM EFT has the advantage that all physical effects are included, andthere can be no cancellation of terms between independent operators.To precisely interpret any pattern of deviations of SM processes using higher dimensionaloperators, one has to map the pattern of deviations observed at the electroweak scale backto the scale Λ , where the BSM physics was integrated out of the effective field theory. Dueto operator mixing, the pattern of Wilson coefficients that are observed at the low scale ∼ m H is not identical to the pattern of Wilson coefficients at the matching scale Λ . OurRG calculation determines all of the logarithmically enhanced terms in observables at therenormalization group scale µ = m H due to RG running from the high-energy scale of newphysics µ = Λ .There are also other contributions from the finite parts of one-loop graphs at the low scale µ ∼ m H , which we have not computed. For Λ ∼ TeV, ln(Λ /m H ) ∼ , so there is a modestenhancement of the log terms over the finite terms. As experiments get more precise, and thescale Λ is pushed higher, the log terms become even more important relative to the finite terms.Nevertheless, the calculation of finite terms is important, and these terms will eventually berequired for a precise comparison of data with the SM EFT. The anomalous dimensions canalso be viewed as computing the ln Λ /m H enhanced finite terms. The anomalous dimensioncomputation is easier because it can be done in the unbroken theory, whereas the computationof finite terms needs to be done in the broken theory.An important application of the SM EFT is to test the hypothesis of minimal flavorviolation [14, 15]. The dimension-six operators can have arbitrary flavor structure, and therenormalization group equations derived in Refs. [10–13] and in this paper give non-trivialmixing between different particle sectors. MFV assumes that the only sources of U (3) flavorsymmetry violation are the Yukawa coupling matrices Y e , Y u and Y d . The SM respects MFVby definition. Since MFV is formulated in terms of symmetries, it is preserved by the RGevolution. If the dimension-six Lagrangian respects MFV, then the RG evolution preservesthis property.The general dimension-six Lagrangian does not have to respect MFV, and RG evolutionthen feeds non-minimal flavor violation into different operator sectors. By constraining theparameters of the SM EFT, one can experimentally test the MFV hypothesis taking this RGrunning into account. It is important to test MFV directly in a model-independent way. The– 2 –M EFT provides a model-independent formalism to test the MFV hypothesis.The outline of this paper is as follows. In Section 2, we discuss our notation, and thegauge coupling constant terms reported in this work. Some generalities about the structureof the anomalous dimension matrix are given in Sec. 2.1. Some interesting cancellationsare pointed out in Sec. 2.3. A detailed presentation of the gauge coupling constant termsin the RG equations of the dimension-six operator coefficients is relegated to Appendix C.Section 3 compares the standard operator basis of Refs. [8, 9] with SILH operators [16]. Abrief discussion of MFV and its implications is given in Section 4. Section 5 presents themain applications of the SM EFT to phenomenology. We discuss the SM parameters attree level, and how their values are modified by the SM EFT dimension-six operators. Inparticular, we discuss the modifications to the Higgs mass and couplings, and to the gaugeboson masses. We also discuss the scale dependence of the dimension-six operators, and howthe dimension-six operators contribute to the running of the d ≤ parameters of the SMLagrangian. The complete expressions for the running of the gg → h , h → γγ and h → γZ amplitudes are given in Secs. 5.8, 5.9, and 5.10, respectively. In Secs. 5.11 and 5.12, we discussthe operators corresponding to the electroweak precision data (EWPD) parameters S and T ,and operators modifying critical processes for precision electroweak phenomenology, such astriple gauge boson couplings and the three-body decay h → Z ℓ + ℓ − . In Sec. 5.13, we discussthe dipole coefficients C eγ which contribute to the decay µ → eγ and to the muon and electronmagnetic and electric dipole moments. We present our conclusions in Section 6. The countingof parameters in L (6) is summarized in Appendix A, and the conversion of SILH operators tothe standard basis is given in Appendix B. The complete list of 59 independent dimension-six operators is given in Table 1. The operatorsare divided into eight classes by field content and number of covariant derivatives. The eightoperator classes are X , H , H D , X H , ψ H , ψ XH , ψ H D and ψ , where X = G Aµν , W
Iµν , B µν represents a gauge field strength, H denotes the Higgsdoublet scalar field, ψ is a fermion field ψ = q, u, d, l, e , and D is a covariant derivative. Thedimension-six Lagrangian is L (6) = X i C i Q i (2.1)where the Q i are the dimension-six operators of Table 1 and the operator coefficients C i havedimensions of / Λ . The one-loop anomalous dimension matrix γ ij is defined by the RGequation of the operator coefficients ˙ C i ≡ π µ d C i d µ = γ ij C j . (2.2)The explicit RG equations are given in Appendix C as differential equations, rather than aselements of the matrix γ . We will use γ ij to represent the × block form of the anomalous– 3 –imension matrix, where the subscripts on γ refer to the eight operator classes i, j = 1 , . . . , .For example, γ is the × anomalous dimension submatrix which mixes the 3 independentclass 5 operator coefficients into the 2 independent class 3 operator coefficients (see Table 1).Although there are 59 independent operators, many of them have flavor indices whichtake on n g = 3 values. Table 2 gives the number of CP -even and CP -odd coefficients for eachoperator class. For n g = 3 , there are (107 n g + 2 n g + 213 n g + 30 n g + 72) / CP -evencoefficients and (107 n g + 2 n g + 57 n g − n g + 48) / CP -odd parameters, for a totalof 2499 parameters which need to be constrained by experiment. The counting of parametersis summarized in Appendix A.Such a large number of terms makes the calculation of the complete anomalous dimensionmatrix a formidable task. In Ref. [13], we began by computing the × one-loop anomalousdimension matrix γ for the class-4 Higgs-gauge operators X H , since these operators con-tribute directly to the experimentally interesting Higgs production and decay channels gg → h , h → γγ , and h → γZ , which first occur at one loop in the SM. The × submatrix γ hasbeen subsequently verified by several independent calculations (e.g. Ref. [17]). In Ref. [12], wecalculated the λ -dependent terms of the full anomalous dimension matrix for vanishing gaugecoupling constants, as well as the complete running of the SM d ≤ parameters due to thedimension-six operators. The running of the SM parameters resulting from the dimension-six operators is of order m H / Λ , which is of the same order as the tree-level contribution ofdimension-six operators. The Yukawa-dependent terms of the anomalous dimension matrixfor vanishing gauge couplings were computed in Ref. [10]. In this paper, we complete the fullcalculation of the one-loop anomalous dimension matrix of the dimension-six operators bycomputing the gauge coupling terms.The one-loop anomalous dimension matrix has the usual / (16 π ) suppression of a one-loop calculation. However, there are several anomalous dimensions with large numerical fac-tors. In Ref. [12], for example, we found that π µ dd µ C H = 108 λ C H + . . . . (2.3)Since m H = 2 λv , the anomalous dimension coefficient is λ = 54 m H /v ≃ , independentof the normalization convention for the quartic coupling λ . In the study of the Yukawacoupling terms of Ref. [10], the numerical factors were generally O (1) . These Yukawa termsgive interesting nontrivial flavor mixing between the various operators. The gauge termscalculated in this paper also contain several large coefficients. For example, the mixing of theclass 4 operators X H into the class 2 operator H gives π µ dd µ C H = − (48 g y h + 12 g g y h ) C HB . . . (2.4)The lengthiest contributions to gauge coupling constant terms come from the well-knownpenguin graph Fig. 1. The penguin graph itself is simple to compute. However, there are25 possible ψ operators in the L (6) Lagrangian, and the penguin graph is proportional to– 4 – igure 1 . A penguin diagram. The solid square is a ψ vertex from L (6) , and the dot is a SM gaugecoupling. D µ X µν , which is replaced by a gauge current summed over all fermion and scalar fields. Theresulting four-fermion and fermion-scalar operators then have to be Fierzed to the canonicaloperator basis, resulting in the bulk of the terms given in Appendix C.One finds a substantial amount of operator mixing in the SM EFT, and such mixingaffects observables measured at the electroweak scale in a manner which must be unraveled tounderstand BSM theories. One of the consequences of this mixing is the propagation of CP violation through different sectors of the Lagrangian. For instance, dipole operators receivecontributions from CP violating class 4 operators (that enter, e.g., h → γZ at tree level), thelatter are therefore subject to electric dipole moment constraints, see Sec. 5.13. On the otherhand, it is already known [13] that mixing effects are relevant for studies of h → γγ . γ ij The complication of dealing with a large operator basis naturally leads to the desire to simplifythe calculation, or to look for hidden structure in the anomalous dimension matrix to moreeasily understand the physics of the one-loop RGE flow. In Ref. [12], we showed that thestructure of the anomalous dimension matrix can be understood using Naive DimensionalAnalysis (NDA) [18]. The argument is simplest using rescaled operators b Q i . The rescaledoperators b Q i are given by g X , H , H D , g X H , yψ H , gyψ XH , ψ H D and ψ ,where each gauge field strength X has been rescaled by a gauge coupling g , and the chirality-flip operators ψ H and ψ XH , which change chirality by one unit, have been rescaled byan additional Yukawa coupling y . The dimension-six Lagrangian can be rewritten in terms ofthe rescaled operators and their corresponding coefficients b C i , L (6) = X i C i Q i = X i b C i b Q i . (2.5)The RG equations for the original and rescaled operator coefficients are given by ˙ C i = γ ij C j , ˙ b C i = b γ ij b C j , (2.6)– 5 –here the one-loop anomalous dimension matrices γ ij and b γ ij are related to each other by therescaling factors and their derivatives. In Ref. [12], we showed that the anomalous dimensionmatrix b γ for the rescaled operators has entries proportional to b γ ∝ (cid:18) λ π (cid:19) n λ (cid:18) y π (cid:19) n y (cid:18) g π (cid:19) n g , N = n λ + n y + n g (2.7)where N , the perturbative order of the anomalous dimension, is defined as the sum of thenumber of factors n λ of the Higgs self-coupling λ , the number of factors n y of y , and thenumber of factors n g of g . For the rescaled dimension-six operators, N ranges from 0 to4. In Ref. [11], we derived a general formula for the perturbative order N of the anomalousdimension matrix b γ ij , N = 1 + w i − w j , (2.8)where w i is the NDA weight of the operators b Q i in the i th class [11]. The class 2 operator b Q H has NDA weight w = 2 ; the operators in classes { , , , } have NDA weight ; the operatorsin classes { , } have NDA weight ; and the class 1 operators have NDA weight w = − .Using Eq. (2.8), the possible coupling constant dependences of b γ ij are obtained. Our previouswork calculated all anomalous dimensions with nontrivial n λ and n y with n g = 0 . The presentwork completes the calculation of all terms with n g = 0 .Although the coupling constant dependence of the anomalous dimension matrix is simplestfor the NDA rescaled operators, the RGE in Refs. [10, 12, 13] and in this work are quotedin terms of the original unrescaled operators Q i of Refs. [8, 9]. The possible entries of γ ij were classified in Ref. [12] by studying all possible one-loop diagrams including EOM terms.The classification is a bit subtle. The non-zero entries arise directly from diagrams whichcontribute to a given term, but also indirectly via EOM. For example, the H D − H D entry of the anomalous dimension matrix is computed from graphs with one insertion of a H D operator, Q H (cid:3) or Q HD , with 4 external H lines. These graphs contribute to the γ submatrix for the running of the coefficients C H (cid:3) and C HD . The graphs contributing to γ also require a counterterm proportional to the EOM operator E H (cid:3) of Ref. [12]. This operatorcan be eliminated in favor of other operators such as the ψ H operators in the standardbasis. Thus, the γ graphs also contribute to the γ submatrix via the EOM, even thoughthey do not have any external fermion lines.The NDA weights w i for the NDA rescaled operators b Q i of the eight operator classes,and the coupling constant dependence of the allowed anomalous dimensions b γ ij are shownin Table 3, with the operators ordered according to decreasing NDA weight. Now that theentire matrix has been computed, we can compare with the classification of Ref. [12]. Thecross-hatched entries in the table are anomalous dimension entries which could exist based onthe allowed diagrams, but which vanish by explicit computation. These entries vanish becausethe relevant diagram vanishes, has no infinite part despite being naively divergent, or, in someinteresting cases, by cancellation between different contributions such as a direct contribu-– 6 –ion to γ ij and an indirect contribution obtained by using the EOM. These cancellations arediscussed further in Sec. 2.3.The diagonal blocks in Table 3 have N = 1 since w i = w j . Blocks one below the diagonalhave N = 0 , whereas blocks one above the diagonal have N = 2 , etc. When N is less than , γ vanishes, and we find that this is always the case. However, there are many additionalanomalous dimensions which vanish. Indeed, almost all of the N = 0 entries vanish. Thenotable exception of a N = 0 submatrix which does not vanish is γ which mixes class 8four-fermion operators ψ into the class 6 dipole operators ψ XH in violation of the general“no tree-loop mixing” claim of Refs. [19–21]. Other examples which violate no tree-loop mixingexist [22]. “Tree-loop” classification [23] of terms in an EFT Lagrangian has limited usefulness,and does not apply in general when the UV theory generating the dimension-six operatorsis itself an EFT, or is a strongly interacting theory. Attempts to broaden this classificationscheme in a very general manner relied critically on the assumption of minimal coupling.However, in Ref. [24], we showed that the concept of minimal coupling is ill defined in general. The calculations in this paper are done in background field with gauge fixing parameter ξ ,and cancellation of ξ -dependence provides a check on the results. The gauge dependence onlycancels for gauge-invariant interactions, i.e. if the relations y q = y d + y h , y q = y u − y h , y l = y e + y h , (2.9)are satisfied. Although the expressions for the anomalous dimensions have been written interms of all six hypercharges, y i cannot be thought of as varying independently, but mustsatisfy the constraints Eq. (2.9). A check of the results that follows from custodial SU (2) symmetry is discussed at the end of Sec. 5.11.The SM Yukawa couplings L Yukawa = − (cid:20) H † j d r [ Y d ] rs q js + e H † j u r [ Y u ] rs q js + H † j e r [ Y e ] rs l js + h.c. (cid:21) , (2.10)where r, s are flavor indices and j is an SU (2) index, are only gauge invariant because the of SU (2) is self-conjugate, so that H j and e H j = ǫ jk H † k belong to the same SU (2) representation.The SU (2) group cannot be generalized to a SU ( N ) group. While some of the SU (2) grouptheory factors have been written as Casimirs such as c A, and c F, , the results are only validwhen they take on their SU (2) values c A, = 2 and c F, = 3 / .The SU (3) results are written for an SU ( N c ) theory. Anomaly cancellation does not holdfor the SU ( N c ) × U (1) Y anomaly for arbitrary N c , but the results can still be useful in othercontexts for the SU ( N c ) anomalous dimensions. The SU (3) Fierz identity T Aαβ T Aλσ = 12 δ ασ δ λβ − N c δ αβ δ λσ (2.11)– 7 – igure 2 . Graphs with insertions of the X operator which cancel after using the equations of motion. has been used to rearrange color indices and put operators into standard form. This identityis valid for the fundamental representation of SU ( N c ) , but is not valid for arbitrary represen-tations. Thus, the quadratic Casimir c F, is equivalent to ( N c − / (2 N c ) , and the fermionsmust be in SU ( N c ) fundamental or anti-fundamental representations. The one-loop anomalous dimension matrix does not contain all possible terms that can arisefrom the allowed one-loop graphs and the EOM. In a few cases, the entries vanish becausethe graph has no divergent part. An example from Ref. [12] is the y contribution to γ , or H − ψ H D mixing.There also are a few cases with interesting non-trivial cancellations which arise whendifferent contributions to the same anomalous dimension are added together after using theequations of motion. An example is the contribution of insertions of the CP -even operators X to the anomalous dimension from the graphs shown in Fig. 2. The divergent part of thefirst graph is proportional to A = − c A, g C W D µ W Iµλ D ν W I νλ − c A, g C G D µ G Aµλ D ν G A νλ . (2.12)The divergent part of the sum of the second and third graphs is proportional to A = − ig c A, C W D µ H † τ I D ν HW Iµν . (2.13)There is no gluon term, since gluons do not couple to the Higgs field. The divergent part ofthe fourth graph is proportional to A = g c A, C W D µ W Iµν j I νψ + g c A, C G D µ G Aµν j A νψ , (2.14)where j I µψ = X ψ = q,l ψ γ µ τ I ψ, j A µψ = X ψ = q,u,d ψ γ µ T A ψ, (2.15)– 8 –re the SU (2) and SU (3) fermion currents, respectively. The operator Eq. (2.13) is equalto g c A, C W P HW , where P HW is given in Eq. (3.3). Integrating by parts, and writing thecommutator of two covariant derivatives as a field-strength tensor gives the identity P HW = − ig D µ H † τ I D ν H W
Iµν = g j I νH D µ W Iµ ν − g H † HW Iµν W Iµν − g g y h H † τ I HW Iµν B µν , (2.16)where j I µH = i H † τ I ←→ D µ H ) , (2.17)is the Higgs doublet SU (2) current. The total is A + A + A = − g c A, C W D µ W Iµλ h D ν W I νλ − g j I λψ − g j I λH i − c A, g C G D µ G Aµλ h D ν G A νλ − g j A λψ i − g c A, C W (cid:20) g H † HW Iµν W Iµν + 12 g g y h H † τ I H W
Iµν B µν (cid:21) . (2.18)Using the gauge field equations of motion D µ W I µν = g (cid:16) j I µH + j I µψ (cid:17) , D µ G A µν = g j A µψ , (2.19)only the second line survives, A + A + A = − g c A, C W (cid:20) g H † HW Iµν W Iµν + 12 g g y h H † τ I H W
Iµν B µν (cid:21) . (2.20)The gluon term C G cancels completely and most of the C W term cancels. There is a residualcontribution from Eq. (2.20) to the anomalous dimension of C HW and C HW B , the coefficientsof the X H Higgs-gauge boson operators. The graphs in Fig. 2 contribute to the running of C HW and C HW B even though none of the diagrams have two external gauge bosons and twoexternal Higgs lines, the field content of X H operators. The cancellation of C G and C W terms in various anomalous dimensions is the reason for the absence of several terms in thelast column of Table 3.The C f W and C e G contributions to the anomalous dimension arise from the same graphsas in Fig. 2, with the insertions of the CP -odd operators e XXX . In this case, one obtainsEqs. (2.12) and (2.14) with D µ W Iµν and D µ G Aµν replaced by D µ f W Iµν and D µ e G Aµν , respectively,and Eq (2.13) with W Iµν replaced by f W Iµν . The equations of motion for e X are D µ e X µν = 0 ,rather than Eq. (2.19), so naively there can be a difference between the C f W , e G and C W,G contributions to the anomalous dimension. However, the total sum A + A + A is − g c A, C f W D µ f W Iµλ h D ν W I νλ − g j I λψ − g j I λH i − c A, g C e G D µ e G Aµλ h D ν G A νλ − g j A λψ i − g c A, C f W (cid:20) g H † H f W Iµν W Iµν + 12 g g y h H † τ I H f W Iµν B µν (cid:21) , (2.21)– 9 –nstead of Eq. (2.18). The first lines in both Eq. (2.18) and Eq. (2.21), which would haveproduced a difference in the C f W , e G and C W,G contributions, are proportional to the gauge fieldequations of motion (2.19) and vanish. Thus, the contributions to the anomalous dimensionfrom the CP -odd coefficients C f W , e G are the same as the contributions from the CP -evencoefficients C W,G .Another interesting cancellation occurs in the contribution of the ψ XH dipole operators.The coefficients C eW , etc. of these operators will be denoted generically by C ψX , where ψ = e, u, d . The dipole operators contribute to the running of ψ H coefficients C ψH , such as C eH ,and to the running of ψ H D coefficients C Hψ , such as C He . The anomalous dimension forthe running of C ψH gets multiple contributions from C ψX and C ∗ ψX which arise from graphswith insertions of the ψ XH dipole operators and their hermitian conjugates. As above,the multiple contributions arise from using the EOM to bring all divergences to the canonicalbasis. The total contribution of C ∗ ψX to the running of C ψH cancels after using the hyperchargeconstraints Eq. (2.9), even though individual contributions do not vanish. The contributionof C ψX to the running of C ψH does not cancel. The total contribution of both C ψX and C ∗ ψX to the running of the ψ H D coefficients C Hψ exactly cancels, which is why there is no g y entry in the anomalous dimension γ from ψ H D - ψ XH mixing in Table 3.The contributions of the dipole operators and the gauge operators with X and e X arerelated by factors of i . This simple factor follows from the complex self-duality of σ µν P R .There is no C ∗ ψX contribution to the running ˙ C ψX , or to the runnings ˙ C (1) quqd , ˙ C (8) quqd and ˙ C (3) lequ ,which are the ψ operators to which the dipole operators contribute.The examples above indicate that the RG contribution of the dipole operators respectsholomorphy in C ψX . Several of the gauge coupling terms of the one-loop anomalous dimension matrix have beencalculated before. However, we emphasize that with the results reported in this work, we havedetermined the complete one-loop anomalous dimension matrix for dimension-six operatorsof the SM EFT for the first time.Previous calculations of individual elements of the anomalous dimension matrix includethe following works. The anomalous dimension of Q G and Q ˜ G were determined in Refs. [25–27]. We agree with this result. Ref. [25] computed the anomalous dimension of dimension-fiveand dimension-six operators in QCD. Parts of our calculation in which the Higgs field canbe treated as an external constant field agree with these results. The renormalization offour-fermion operators has been studied for many years in the context of the low-energytheory of weak interactions, and provides a check on the ψ − ψ anomalous dimension. Thecomplete one-loop RGE of the operators in class was calculated for the first time in Ref. [13].Previously, some individual terms in this running result were calculated in Refs. [28–31], and Due to the number of operators renormalized, and the fragmentary literature on the subject, we apologizein advance to authors whose works are overlooked in this discussion. – 10 –hese terms are consistent with our calculation. Ref. [19] calculated the mixing of dipoleoperators Q uG , Q uW and Q uB with the combination of Wilson coefficients Q HW , Q HB and Q HW B that corresponds to h → γ γ , see Section 5.9, which corresponds to a set of entries in γ . We agree with these results. Ref. [32] reports the running of the operators Q uH and Q uG due to the QCD coupling, which corresponds to entries in γ and γ . We agree with theresults of this paper.The papers mentioned in the previous paragraph allow a relatively direct comparisonbetween results computed in the same operator basis. Many other results in the literature arereported in a different basis, making a comparison difficult. Ref. [20] presents a few terms inthe anomalous dimension matrix without flavor indices (i.e. for n g = 1 ), and only includingthe top Yukawa coupling. The exact translation between such partial results and this workrequires that a complete non-redundant operator basis be defined, which often is not the case.Ref. [20] does not define such a mapping to allow us to compare our results to the termsreported, see the next Section for more discussion on this point. Nevertheless, some otherclassic past results in Refs. [33–46] overlap with some of the results presented here, as do somemore recent works [47–54]. A minimal basis of dimension-six operators is obtained by removing all redundant operatorsusing the SM EOM. This paper uses the dimension-six operators Q i of Ref. [9] which hasno redundancies. It is a well-established result in quantum field theory that operators whichvanish by the classical equations of motion do not contribute to S -matrix elements even at thequantum level [55], and so EOM can be used to simplify the effective Lagrangian. Formally,the redundant operators can be eliminated by a change of variables in the functional integral.It is clearly a nuisance to use a redundant operator basis.Including redundant operators introduces extra parameters in the Lagrangian which canbe eliminated by field redefintions, and do not contribute to any measurable quantity [55].This redundancy is not always obvious, since intermediate steps and partial results can dependon the redundant parameters. It is only when the complete S -matrix element is carefullycomputed that one sees that certain combinations of parameters drop out due to the EOM.Redundant operators have led to enormous confusion in the literature over many decades, forexample, this was a source of significant confusion in the early days of heavy quark effectivetheory. For this reason, when choosing a basis, it is advantageous to not introduce redundantparameters.Recently, some authors [20, 56, 57] have advocated using the “SILH-basis.” The definitionof this operator basis varies in the papers, and the original SILH paper [16] does not definea complete basis. We will discuss the version presented in Ref. [56]. The basis of Refs. [8, 9]contains nine CP -even operators made out of only gauge and Higgs fields, Q G , Q W , Q H , Q H (cid:3) , Q HD , Q HG , Q HW , Q HB , Q HW B . (3.1)– 11 –he SILH basis defined in Ref. [56] contains 14 CP -even operators made out of only gaugeand Higgs fields with the operator coefficients ¯ c H , ¯ c T , ¯ c , ¯ c W , ¯ c B , ¯ c HW , ¯ c HB , ¯ c γ , ¯ c g , ¯ c W , ¯ c G , ¯ c W , ¯ c B , ¯ c G . (3.2)The six operators Q G , Q W , Q H , Q H (cid:3) , Q HG , Q HB coincide with the operators corresponding to ¯ c G , ¯ c W , ¯ c , ¯ c H , ¯ c g , ¯ c γ , up to simple rescalings by couplings. In Ref. [56], it is argued that thethree operators corresponding to ¯ c W , ¯ c B and ¯ c G can be removed by the SM EOM in favorof other operators retained in the SILH operator basis. This removal leaves five flavor-singletoperators P HW = − i g ( D µ H ) † τ I ( D ν H ) W Iµ ν , P HB = − i g ( D µ H ) † ( D ν H ) B µ ν , P W = − i g H † τ I ←→ D µ H ) ( D ν W Iµ ν ) , P B = − i g H † ←→ D µ H ) ( D ν B µ ν ) , P T = ( H † ←→ D µ H ) ( H † ←→ D µ H ) , (3.3)in the SILH basis, instead of the three operators Q HW = H † H W
Iµν W µνI , Q HW B = H † τ I H W
Iµν B µν , Q HD = ( H † D µ H ) ⋆ ( H † D µ H ) , (3.4)in the standard basis.Since Eq. (3.3) has five operators, and Eq. (3.4) has only three operators, two additionaloperators from the standard Q i basis can be eliminated if the operators in Eq. (3.3) are usedinstead of those in Eq. (3.4). The five P i operators can be written in terms of the standard basis Q i using the equations of motion, and the conversion is given in Appendix B. The relationsinvolve non-bosonic Q i operators, a fact that is used in Ref. [56] to remove the lepton-Higgsoperators Q (1) Hl and Q (3) Hl together with Q HW , Q HW B and Q HD in favor of the 5 P i operatorsof Eq. (3.3). However, only the flavor-singlet combinations Q (1) Hlpp , Q (3)
Hlpp , (3.5)enter the relations in Eq. (B.1). One can modify the singlet part of the coefficients of Q (1) Hl and Q (3) Hl by the shift C (1 , Hlrs → C (1 , Hlrs + a (1 , δ rs , (3.6)and absorb the change in the P i operator coefficients. The constants a (1 , can be chosen toeliminate the trace Eq. (3.5), or to set the electron operator C (1) Hlee = 0 , etc. However, thecoefficients of the flavor non-singlet parts C (1) Hlrs − n g δ rs C (1) Hlpp , C (3)
Hlrs − n g δ rs C (3) Hlpp (3.7) The SILH basis operators are denoted by P i to avoid confusion with similarly labelled operators Q i in thestandard basis. – 12 – annot be removed, and must be retained. Removal of the flavor-singlet portions of C (1) Hl and C (3) Hl makes the treatment of BSM flavor violation in the SILH basis cumbersome. Furthermore,a careful and consistent treatment of EOM effects is necessary in all calculations using the“SILH-basis,” otherwise the basis remains redundant.The lepton-Higgs operators Q (1) Hl and Q (3) Hl can be removed completely if one assumescompletely unbroken U (3) flavor symmetry of the UV theory, so that the coefficients ofthese operators are unit matrices in flavor space. This assumption was implicitly adoptedin the initial work of Ref. [58] that identified this field redefintion, and it is also adopted inRefs. [20, 56, 57]. This assumption is stronger than assuming MFV, which only says that thecoefficients of the lepton operators is a function of Y † e Y e , not that it is proportional to the unitmatrix. Ref. [20] computes a few of the anomalous dimensions in the case of a U (3) flavor-symmetric BSM sector, in an attempt to circumvent this difficulty. While the assumption offlavor-symmetric BSM physics can be adopted, it limits the applicability of the EFT. One ofthe important features of the SM EFT is that it can be used to test MFV, but this is onlypossible if MFV is not put in by hand. Many SILH basis results cannot be used to test MFVin a straightforward manner, since stronger assumptions than MFV have already been builtinto the formalism.In reducing the SILH operators to the operator basis of Ref. [9], the EOM relationsin Appendix B also include the SM dimension-four operator ( H † H ) , which is the usual λ ( H † H ) Higgs interaction term. This means that the connection of the two bases alsoinvolves the redefinition of SM parameters. Explicitly, the RGE for the SM parameters alsohave contributions from dimension-six operators, as pointed out in Ref. [12]. These effects arenot taken into account in Ref. [20] preventing a comparison of our results with Ref. [20]. Also note that Ref. [20] advocates retaining redundant operators in intermediate stepsof the analysis. Retaining redundant operators in partial results for an anomalous dimensionmatrix introduces spurious gauge and scheme dependence, see the discussion in Ref. [12]. It isnot defined in Ref. [20] how the partial results for the anomalous dimension matrix presentedthere can be converted to the full results valid for any BSM flavour structure. This is anotherreason we cannot compare our results with the partial calculation in Ref. [20].
The SM EFT provides a way to test the hypothesis of MFV in new physics. The SM has a U (3) symmetry in the limit of vanishing Yukawa couplings under which q → U q q, l → U l l, u → U u u, d → U d d, e → U e e. (4.1)The MFV hypothesis [14, 15] is that the only source of flavor violation is the Yukawa matrices,so that the full theory is flavor invariant if the Yukawa matrices transform as Y u → U u Y u U † q , Y d → U d Y d U † q , Y e → U e Y e U † l . (4.2) For an example of this effect, see Sec. 5.5, Eq. 5.34. – 13 –f the new physics respects MFV, then the SM EFT derived from it also does. Thisassumption severely restricts the dimension-six coefficients. The coefficients of the flavorinvariant operators in classes 1–4 can only depend on the flavor invariants Tr f ( Y † e Y e ) , Tr f ( Y † d Y d , Y † u Y u ) , (4.3)In an EFT setup, the dependence on such invariants can be absorbed into an effective coeffi-cient.The ψ H operators have coefficients C dHrs = h f ( Y † d Y d , Y † u Y u ) Y † d i rs , C uHrs = h f ( Y † d Y d , Y † u Y u ) Y † u i rs , C eHrs = h f ( Y † e Y e ) Y † e i rs , (4.4)where it is implicit that the above functions also can depend on the invariants of Eq. (4.3).For example, the quark functions can depend on the lepton invariant Tr f ( Y † e Y e ) and vice-versa. Analogous formulae to Eq. (4.4) hold for the ψ XH dipole operators { C eW , C eB } , { C uG , C uW , C uB } and { C dG , C dW , C dB } , respectively.The ψ H D operators have coefficients C (1 , Hqrs = h f ( Y † d Y d , Y † u Y u ) i rs , C (1 , Hlrs = h f ( Y † e Y e ) i rs ,C Hurs = aδ rs + h Y u f ( Y † d Y d , Y † u Y u ) Y † u i rs , C Hdrs = aδ rs + h Y d f ( Y † d Y d , Y † u Y u ) Y † d i rs ,C Hers = aδ rs + h Y e f ( Y † e Y e ) Y † e i rs , C Hudrs = h Y u f ( Y † d Y d , Y † u Y u ) Y † d i rs . (4.5)Again, dependence of the above functions of the invariants of Eq. (4.3) is implicit.Similar expressions hold for the ψ operators, with coefficients in flavor space which areproducts of the cases considered above. As is well-known, one can make U (3) rotations tobring the Yukawa matrices into the form Y e → diag ( m e , m µ , m τ ) , Y d → diag ( m d , m s , m b ) , Y u → diag ( m u , m c , m t ) K, (4.6)where K is the CKM matrix. At this stage, the masslessness of neutrinos allows for thediagonalization of Y e and the absence of flavor violation in the lepton sector. The introductionof neutrino masses can be accomplished in the model-independent spirit of this paper via the d = 5 Weinberg Operator. This operator is naturally suppressed by a scale higher than Λ since it violates lepton number. Assuming this hierarchy of scales, the RGEs of d = 5 and d = 6 operators are independent and the inclusion of neutrino masses is orthogonal and doesnot affect the results presented here.Since MFV is implemented as a symmetry which is respected by the SM Lagrangian, theRG evolution of L (6) maintains MFV if the coefficients at scale Λ satisfy the MFV hypothesis. In this section, f denotes an arbitrary function, and all the f s do not have to be the same. Some U (1) s areanomalous, and one also can have dependence on certain combinations of det Y u,d,e and the θ angles [59–61]. – 14 –n this case, the flavor structure of L (6) is the same as corresponding amplitudes computedfrom loop graphs in the SM. However, it is important to emphasize that the assumption ofMFV does not imply that the coefficients of ψ H D and ψ operators are proportional tothe unit matrix, which is a stronger assumption that requires that the functions f have aperturbative expansion in Y with small coefficients. In view of Eq. (4.6), this expansion inpowers of Yukawa matrices can be justified for off-diagonal elements inducing flavor violation,as customary, but not for the diagonal entry of the third generation, see Ref. [62] for somediscussion on this point.One of the important applications of the SM EFT is to test the hypothesis of MFV inBSM physics in a model-independent way. Interestingly, the full SM RGE transfers flavorviolation in one set of operators to other operator sectors. Testing the consistency of MFV inlow-energy measurements, taking into account the full SM EFT, is important for increasingour understanding of the flavor structure of new physics. A quick look at the anomalousdimensions in Refs. [10, 12] and Appendix C should convince the reader that any flavor ansatznot based on a symmetry will not be preserved by the RGE. In this section, we outline the generalization of the analysis of observables measured at theelectroweak scale from the SM to the SM EFT, and how the full one-loop RGE for thedimension-six Wilson coefficients measured at a low scale ∼ v can be used to obtain the Wilsoncoefficients at the high scale Λ . An important point we emphasize is that if constraints atthe scale v are to be mapped to a high scale BSM theory, then all corrections of the order v / (16 π Λ ) in the SM EFT have to be included in the analysis. Otherwise, the analysis isinconsistent.Our aim is not to perform a precision analysis, but to simply outline some issues that aprecision Higgs and electroweak phenomenology program should take into account, and howthe one-loop RGE result aids in this program. Some aspects of how the SM EFT modifies SMphenomenology have been discussed previously in Refs. [20, 56, 63, 64] and other works. How-ever, many aspects of how the SM EFT affects precision predictions have not been discussedin detail before, and we outline some of them below.The Lagrangian of the SM EFT is L = L SM + L (6) + . . . (5.1)where the . . . denote operators of dimension greater than six suppressed by additional powersof Λ . The dimension-six terms L (6) can be treated perturbatively, i.e. we only need to includethese to first order, since second-order contributions from L (6) are as important as first-order– 15 –ontributions from L (8) , etc. The SM Lagrangian is L SM = − G Aµν G Aµν − W Iµν W Iµν − B µν B µν + ( D µ H † )( D µ H ) + X ψ = q,u,d,l,e ψ i /D ψ − λ (cid:18) H † H − v (cid:19) − (cid:20) H † j d Y d q j + e H † j u Y u q j + H † j e Y e l j + h.c. (cid:21) , (5.2)and L (6) is defined in Eq. (2.1). We start by discussing the modification of the SM parametersat tree-level due to L (6) . The dimension-six Lagrangian of the SM EFT alters the definition of SM parameters at treelevel in a number of ways. The operator Q H changes the shape of the scalar doublet potentialat order v / Λ to V ( H ) = λ (cid:18) H † H − v (cid:19) − C H (cid:16) H † H (cid:17) , (5.3)yielding the new minimum h H † H i = v (cid:18) C H v λ (cid:19) ≡ v T , (5.4)on expanding the exact solution ( λ − p λ − C H λv ) / (3 C H ) to first order in C H . The shiftin the vacuum expectation value (VEV) is proportional to C H v , which is of order v / Λ .The scalar field can be written in unitary gauge as H = 1 √ c H, kin ] h + v T ! , (5.5)where c H, kin ≡ (cid:18) C H (cid:3) − C HD (cid:19) v , v T ≡ (cid:18) C H v λ (cid:19) v. (5.6)The coefficient of h in Eq. (5.5) is no longer unity, in order for the Higgs boson kinetic termto be properly normalized when the dimension-six operators are included. The kinetic terms L = ( D µ H † )( D µ H ) + C H (cid:3) (cid:16) H † H (cid:17) (cid:3) (cid:16) H † H (cid:17) + C HD (cid:16) H † D µ H (cid:17) ∗ (cid:16) H † D µ H (cid:17) , (5.7)and the potential in Eq. (5.3) yield L = 12 ( ∂ µ h ) − c H, kin v T (cid:2) h ( ∂ µ h ) + 2 vh ( ∂ µ h ) (cid:3) − λv T (cid:18) − C H v λ + 2 c H, kin (cid:19) h − λv T (cid:18) − C H v λ + 3 c H, kin (cid:19) h − λ (cid:18) − C H v λ + 4 c H, kin (cid:19) h + 34 C H vh + 18 C H h , (5.8) One can always replace v by v T in terms that depend on the L (6) coefficients, since the change is order / Λ . – 16 –or the h self-interactions. The Higgs boson mass is m H = 2 λv T (cid:18) − C H v λ + 2 c H, kin (cid:19) . (5.9) The definition of the fermion mass matrices and the Yukawa matrices are modified by thepresence of ψ H operators. The Lagrangian terms in the unbroken theory L = − (cid:20) H † j d r [ Y d ] rs q js + e H † j u r [ Y u ] rs q js + H † j e r [ Y e ] rs l js + h.c. (cid:21) + (cid:20) C ∗ dHsr (cid:16) H † H (cid:17) H † j d r q js + C ∗ uHsr (cid:16) H † H (cid:17) ˜ H † j u r q js + C ∗ eHsr (cid:16) H † H (cid:17) H † j e r l js + h.c. (cid:21) , (5.10)yield the fermion mass matrices [ M ψ ] rs = v T √ (cid:18) [ Y ψ ] rs − v C ∗ ψHsr (cid:19) , ψ = u, d, e (5.11)in the broken theory. The coupling matrices of the h boson to the fermions L = − h u Y q + . . . are [ Y ψ ] rs = 1 √ Y ψ ] rs [1 + c H, kin ] − √ v C ∗ ψHsr = 1 v T [ M ψ ] rs [1 + c H, kin ] − v √ C ∗ ψHsr , ψ = u, d, e (5.12)and are not simply proportional to the fermion mass matrices, as is the case in the SM. Ingeneral, the fermion mass matrices and Yukawa matrices will not be simultaneously diagonal-izable (these parameters have different RGEs), so that the couplings of the Higgs boson tothe fermions will not be diagonal in flavor due to terms of order v / Λ . G F The value of the VEV in the SM is obtained from the measurement of G F in µ decay, µ − → e − + ¯ ν e + ν µ . Define the local effective interaction for muon decay as L G F = − G F √ ν µ γ µ P L µ ) (¯ e γ µ P L ν e ) . (5.13)The parameter G F is fixed by measuring the muon lifetime. In the SM EFT, − G F √ − v T + (cid:18) C llµeeµ + C lleµµe (cid:19) − C (3) Hlee + C (3) Hlµµ ! . (5.14)The C ll terms are from the four-lepton interaction in L (6) , and the C (3) Hl terms are from W exchange, where one W lν vertex is from the Q (3) Hl operator, and the other is the usual SM e and µ are generation indices 1 and 2, and are not summed over. – 17 –ertex. There are contributions to µ decay from C llµers , and C llrsµe with r = e, s = µ , as wellas from ( LL )( RR ) currents, but these do not interfere with the SM amplitude, and theircontributions to the muon lifetime are higher order in / Λ .Similar expressions hold for other weak decay processes, and G F in τ decay, or in quarkdecays, can differ from µ decay due to the C ll and C (3) Hl terms. The definition of the gauge fields and the gauge couplings are affected by the dimension-sixterms. The relevant dimension-six Lagrangian terms are L (6) = C HG H † HG Aµν G Aµν + C HW H † HW Iµν W Iµν + C HB H † HB µν B µν + C HW B H † τ I HW Iµν B µν + C G f ABC G Aνµ G Bρν G Cµρ + C W ǫ IJK W Iνµ W Jρν W Kµρ . (5.15)In the broken theory, the X H operators contribute to the gauge kinetic energies, L SM + L (6) = − W + µν W µν − − W µν W µν − B µν B µν − G Aµν G Aµν + 12 v T C HG G Aµν G Aµν , + 12 v T C HW W Iµν W Iµν + 12 v T C HB B µν B µν − v T C HW B W µν B µν , (5.16)so the gauge fields in the Lagrangian are not canonically normalized, and the last term inEq. (5.16) leads to kinetic mixing between W and B . The mass terms for the gauge bosonsfrom L SM and L (6) are L = 14 g v T W + µ W − µ + 18 v T ( g W µ − g B µ ) + 116 v T C HD ( g W µ − g B µ ) . (5.17)The gauge fields need to be redefined, so that the kinetic terms are properly normalizedand diagonal. The first step is to redefine the gauge fields G Aµ = G Aµ (cid:0) C HG v T (cid:1) , W Iµ = W Iµ (cid:0) C HW v T (cid:1) , B µ = B µ (cid:0) C HB v T (cid:1) . (5.18)The modified coupling constants are g = g (cid:0) C HG v T (cid:1) , g = g (cid:0) C HW v T (cid:1) , g = g (cid:0) C HB v T (cid:1) , (5.19)so that the products g G Aµ = g G Aµ , etc. are unchanged. This takes care of the gluon terms.The electroweak terms are L = − W + µν W µν − − W µν W µν − B µν B µν − (cid:0) v T C HW B (cid:1) W µν B µν + 14 g v T W + µ W − µ + 18 v T ( g W µ − g B µ ) + 116 v T C HD ( g W µ − g B µ ) . (5.20)The mass eigenstate basis is given by [65] " W µ B µ = " − v T C HW B − v T C HW B cos θ sin θ − sin θ cos θ Z µ A µ , (5.21)– 18 –here the rotation angle is tan θ = g g + v T C HW B (cid:20) − g g (cid:21) , (5.22)so that sin θ = g p g + g (cid:20) v T g g g − g g + g C HW B (cid:21) , cos θ = g p g + g (cid:20) − v T g g g − g g + g C HW B (cid:21) . (5.23)The photon is massless, as it must be by gauge invariance, since U (1) Q is unbroken. The W and Z masses are M W = g v T ,M Z = v T g + g ) + 18 v T C HD ( g + g ) + 12 v T g g C HW B . (5.24)The covariant derivative is D µ = ∂ µ + i g √ (cid:2) W + µ T + + W − µ T − (cid:3) + i g Z (cid:2) T − s Q (cid:3) Z µ + i e Q A µ , (5.25)where Q = T + Y , and the effective couplings are given by e = g g p g + g (cid:20) − g g g + g v T C HW B (cid:21) = g sin θ −
12 cos θ g v T C HW B ,g Z = q g + g + g g p g + g v T C HW B = e sin θ cos θ (cid:20) g + g g g v T C HW B (cid:21) ,s = sin θ = g g + g + g g ( g − g )( g + g ) v T C HW B . (5.26)The ρ parameter, defined as the ratio of charged and neutral currents at low energies [66], is ρ ≡ g M Z g Z M W = 1 + 12 v T C HD . (5.27)Measurements of the W and Z masses and couplings, and the photon coupling fix g , g , v T , C HW B and C HD . The couplings of the gauge bosons to fermions are also modified, and arecent discussion can be found in Ref. [67]. C H , C HD , C H (cid:3) The discussion in Sections 5.1–5.4 studied the impact of higher dimensional operators on themeasured SM parameters at tree level. The coefficients C HD , etc. of the higher dimensionoperators that enter the expressions are renormalized at the low scale, and are related to the– 19 –arameters at the high scale Λ by the RGE. As mentioned earlier, the RGE contributionsare the same as the log Λ /m H enhanced contributions from the finite parts of the one-loopdiagrams.The RGE for C H , C HD and C H (cid:3) which enter the Higgs and gauge Lagrangian are ˙ C H = (cid:18) λ + 6 Y ( S ) − g − g (cid:19) C H − g y h (cid:0) g y h + g − λ (cid:1) C HB − g (cid:0) g y h + 3 g − λ (cid:1) C HW − g g y h (cid:0) g y h + g − λ (cid:1) C HW B − (cid:0) (4 y h g + g ) + 8( g − g y h ) λ − λ (cid:1) C HD + 403 ( g λ − λ ) C H (cid:3) + 16 g λ C (3) Hltt + 16 g λC (3) Hqtt + 8 λ ( η + η ) − (cid:18) [ Y e Y † e Y e ] wv C eHvw + 3[ Y d Y † d Y d ] wv C dHvw + 3[ Y u Y † u Y u ] wv C uHvw + h.c. (cid:19) , (5.28) ˙ C H (cid:3) = (cid:18) − y h g − g + 24 λ + 4 Y ( S ) (cid:19) C H (cid:3) + 2 g C (3) Hltt + 2 g N c C (3) Hqtt + 203 g y h C HD + 4 g y h (cid:18) N c y d C Hdtt + y e C Hett + 2 y l C (1) Hltt + 2 N c y q C (1) Hqtt + N c y u C (1) Hutt (cid:19) − η , (5.29) ˙ C HD = (cid:18) − y h g + 92 g + 12 λ + 4 Y ( S ) (cid:19) C HD + 803 g y h C H (cid:3) + 16 g y h (cid:18) N c y d C Hdtt + y e C Hett + 2 y l C (1) Hltt + 2 N c y q C (1) Hqtt + N c y u C (1) Hutt (cid:19) − η , (5.30)where η , , , are defined in our previous paper Ref. [10]. The precision electroweak parameter T is C HD , so these RGE are also used in Sec. 5.11. Note that the dimension-six operatorcoefficients from the operators in parentheses on the second lines of Eqs. (5.29) and (5.30)drop out of the running of the combination ( C H (cid:3) − C HD / appearing in c H, kin . The RGEfor C HW B is given in Sec. 5.11.The RGE in Eqs. (5.28)–(5.30) depend on other coefficients in L (6) . If the scale Λ is afew TeV, the RGE can be integrated perturbatively, so that C ( µ ) ≈ C (Λ) − π γ C ln Λ µ + . . . where ˙ C = γ C , (5.31)and the . . . are part of the leading-log series γ C ln Λ /µ given by exact integration of the RGE.The ln Λ /µ terms in Eq. (5.31) must be the same as the ln µ terms in the finite parts of theone-loop graphs. Thus the anomalous dimensions are another way of computing the ln Λ /µ enhanced terms in the finite parts of the one-loop graphs.– 20 – .6 h → f f The decay of the Higgs boson into fermions is another important test of the symmetry breakingstructure of the SM. Define the effective coupling Y b of the b quark to the Higgs by L Yuk = −Y b h ¯ b b . The decay width is given by Γ( h → ¯ b b ) = Y b m H N c π (cid:18) − m b m H (cid:19) / , (5.32)where all parameters are renormalized at µ ∼ m H .In the SM, the effective coupling of the b quark to the Higgs field can be predictedvery accurately. The b -quark mass can be determined very precisely from global studies of ¯ B → X c ℓ ¯ ν and X s γ [68], and then used to determine the b -quark Yukawa coupling at thescale m H using the SM RGE. The relation Y b = √ m b /v between the Higgs coupling andquark mass is modified in the SM EFT, and is given by Eq. (5.12), with Y b = [ Y d ] bb , and therelation between v and G F is modified as in Eq. (5.14) due to tree level effects from L (6) .The scaling of parameters from m b to m H is also modified. The dimension-six operatorcontribution to the one-loop running of the effective coupling of the SM Higgs to fermions isgiven in Ref. [12]. We repeat the result for the down quarks here for the sake of completeness. The running of the Y d is modified by the terms µ dd µ [ Y d ] rs = m H π (cid:20) C ∗ dHsr − C H (cid:3) [ Y d ] rs + 12 C HD [ Y d ] rs + [ Y d ] rt (cid:18) C (1) Hqts + 3 C (3) Hqts (cid:19) − C Hdrt [ Y d ] ts − [ Y u ] ts C ∗ Hudtr − C (1) ∗ qdsptr + c F, C (8) ∗ qdsptr ! [ Y d ] tp + C ledqptrs [ Y e ] ∗ pt + N c C (1) ∗ quqdptsr [ Y u ] ∗ tp + 12 C (1) ∗ quqdsptr + c F, C (8) ∗ quqdsptr ! [ Y u ] ∗ tp (cid:21) . (5.33)These terms are of order v / Λ , and are just as important as the running of the C ψH and c H, kin contributions in Eq. (5.12), and must be included for a consistent calculation.The net effect of including the RGE in Eq. (5.12) and Eq. (5.33) is to introduce a shift ofthe form Γ( h → ¯ b b ) = ( Y b + ∆ Y b ) m H N c π (cid:18) − m b m H (cid:19) / , (5.34)where the running effects induced by new physics are included in ∆ Y b : ∆ Y b = m H π log (cid:18) m H m b (cid:19) C + m H π log (cid:16) m H Λ (cid:17) C . (5.35) Note that the usual one loop running of the SM parameters summarized in Ref. [69–71] should be addedto this result for the full scale dependence of these effective couplings in the SM EFT. – 21 –he expression for C is obtained by setting r = s = 3 in the expression in square bracketson the r.h.s. of Eq. (5.33). The expression for C is C = 12 √ λ [ Y d ] (cid:18) ˙ C H (cid:3) −
14 ˙ C HD (cid:19) − λ ˙ C ∗ dH (5.36)where the anomalous dimensions ˙ C H (cid:3) , ˙ C HD , and ˙ C dH are given in Eqs. (5.29), (5.30) andSec. C.5 respectively. Note that as we are considering Λ ∼ TeV , the log enhancement ismodest and of about the same size for running from m b to m H and from Λ to m H . The log( m H /m b ) contribution in Eq. (5.35), and analogous terms in other amplitudes, have beenneglected in Ref. [20], and need to be included for a consistent calculation including / Λ RGE effects.The discussion above also applies to Higgs decays into other fermions, such as cc and τ + τ − . Using newly developed charm tagging techniques [72], it may be possible to measuredeviations in Γ( h → ¯ c c ) at the LHC (see the discussion in Ref. [73]).There are also flavor-changing Higgs-fermion couplings from L (6) , which contribute toflavor-changing Higgs decays, such as h → bs . These do not interfere with the SM Higgsamplitude, which is flavor diagonal, so the flavor-changing decay rates are order / Λ . Nev-ertheless, as the running of C eH , C dH and C uH is not the same as the running of the SMYukawa couplings, searches for Higgs flavor violation is well-motivated. For some recent workon this subject, see Refs. [74, 75]. h → W W and h → ZZ The h → W W and h → ZZ amplitudes receive direct contributions from L (6) . The relevant CP -even Lagrangian terms are L = ( D µ H ) † ( D µ H ) − (cid:0) W Iµν W Iµν + B µν B µν (cid:1) , + C HW Q HW + C HB Q HB + C HW B Q HW B + C HD Q HD , (5.37)which lead to the interactions L = 14 g v T h (cid:2) ( W µ ) + ( W µ ) (cid:3) [1 + c H, kin ] + C HW v T h (cid:2) ( W µν ) + ( W µν ) (cid:3) (5.38)for the W , and L = 14 ( g + g ) v T h ( Z µ ) (cid:2) c H, kin + v T C HD (cid:3) + 12 g g v T h ( Z µ ) C HW B + v T h ( Z µν ) (cid:18) g C HW + g C HB + g g C HW B g + g (cid:19) (5.39)for the Z .A ratio of deviations in the SM gauge boson coupling to the Higgs, reported in [76], isdefined as λ W Z ≡ Γ( h → W W )Γ( h → W W ) SM Γ( h → ZZ ) SM Γ( h → ZZ ) (5.40)– 22 –rom Eqs. (5.38,5.39), we see that c H, kin cancels out in λ W Z , but there are corrections from theHiggs-gauge operators C HW , C HB and C HW B . This correction depends on the off-shellnessof the W and Z , since it is proportional to the field-strength tensors, and thus momentum-dependent. In the SM EFT, the ratio λ W Z depends on the L (6) parameters C HW , C HB and which are not custodial SU (2) violating, as well as C HW B and C HD which are custodial SU (2) violating. The couplings of the gauge bosons to fermions are also modified. For arecent discussion on these corrections in this basis see Ref. [67]. gg → h The Higgs-gluon operators Q HG and Q H e G contribute to the Higgs production rate via gluonfusion. The L (6) contribution to gg → h is important because the SM amplitude starts at oneloop order, with no tree-level contribution. A similar enhancement of L (6) corrections occursfor h → γγ and h → γZ discussed in the next two sections.Define C gg and e C gg by rescaling C HG and C H e G by g , C HG = g C gg C H e G = g e C gg . (5.41)The scaling by g simplifies the RGE, and makes contact with the notation of Refs. [13, 77]which uses C gg = − c G e C gg = − ˜ c G (5.42)since a factor of − / (2Λ ) was included in the normalization of the operators. The otheradvantage of the rescaling is that the field and coupling constant renormalizations Eq. (5.18)and (5.19) cancel out.The change in gg → h relative to the SM is given by [77] σ ( gg → h ) σ SM ( gg → h ) ≃ Γ( h → gg )Γ SM ( h → gg ) ≃ (cid:12)(cid:12)(cid:12)(cid:12) π v C gg I g (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π v e C gg I g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.43)where I g ≈ . is the numerical value of a Feynman parameter integral[77, 78]. We haveneglected corrections from c H, kin and the Yukawa couplings Eq. (5.12) which are v / Λ , butnot enhanced by π . If C gg from BSM physics is loop suppressed as in the SM, then theseterms must be included.The complete one loop RGE of C gg and e C gg are relatively simple, ˙ C gg = (cid:18) λ + 2 Y ( S ) − g − g (cid:19) C gg − (cid:18) [ Y d ] wv C dGvw + [ Y u ] wv C uGvw + h.c. (cid:19) ˙ e C gg = (cid:18) λ + 2 Y ( S ) − g − g (cid:19) e C gg + 2 (cid:18) i [ Y d ] wv C dGvw + i [ Y u ] wv C uGvw + h.c. (cid:19) (5.44)where C dGvw = g C dGvw , C uGvw = g C uGvw , (5.45)– 23 –re rescaled coefficients of the color magnetic dipole operators, and Y ( S ) = Tr h N c Y † u Y u + N c Y † d Y d + Y † e Y e i . (5.46)The Higgs-gluon contributions in the first term of Eq. (5.44) were computed in Ref. [13]. Theonly new contribution from the full RGE is the second term from the color dipole operatorswhich was also calculated in Ref. [19] for the special case of no flavor indices, and with onlya non-zero top quark Yukawa coupling. h → γγ A very important process is h → γγ , which played a key role in the discovery of the SM scalar.Again, it is convenient to define C γγ = 1 g C HW + 1 g C HB − g g C HW B , (5.47)in terms of which our previously defined coefficients [13, 77] are C γγ = − c γγ , e C γγ = − ˜ c γγ . (5.48)The h → γγ rate is Γ( h → γγ )Γ SM ( h → γγ ) ≃ (cid:12)(cid:12)(cid:12)(cid:12) π v C γγ I γ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π v e C γγ I γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.49)where I γ ≈ − . is a Feynman parameter integral [77, 78]. Again, as in the gluon case,we are dropping other v / Λ terms that must be included if C γγ from BSM physics is loopsuppressed.The effective amplitude is C γγ e F µν F µν hv (5.50)where g = e cos θ W g = e sin θ W (5.51)are the definitions of e and θ W without a bar. These differ from the coupling constants inEq. (5.19) (with a bar) at order / Λ .The complete one-loop RGE is ˙ C γγ = (cid:18) λ − g − g + 2 Y ( S ) (cid:19) C γγ + (cid:0) λ − g (cid:1) C HW B g g − g C W + (cid:0) C dγrs [ Y d ] sr + 4 C eγrs [ Y e ] sr − C uγrs [ Y u ] sr + h.c. (cid:1) , ˙ e C γγ = (cid:18) λ − g − g + 2 Y ( S ) (cid:19) e C γγ + (cid:0) λ − g (cid:1) C H f W B g g − g C f W + (cid:0) − i C dγrs [ Y d ] sr − i C eγrs [ Y e ] sr + 8 i C uγrs [ Y u ] sr + h.c. (cid:1) . (5.52)– 24 –he first line of each equation is the contribution from the × submatrix of Higgs-gaugeoperators computed in Ref. [13]. The second line gives the additional terms including all 59operators. There are contributions from the triple-gauge operators Q W = ǫ IJK W Iνµ W Jρν W Kµρ , Q f W = ǫ IJK W Iνµ W Jρν , f W Kµρ (5.53)and the dipole operator coefficients defined in Sec. 5.13.This result is the first truly complete one-loop result of the RGE running of C γγ . h → γ Z The measurement of h → γ Z at LHC has not yet reached the sensitivity required to observethe SM rate [79, 80]. Nevertheless, this process is interesting in several BSM scenarios be-cause a suppression of BSM effects in h → γ γ, gg due to a pseudo-Goldstone Higgs does notnecessarily imply a suppression of BSM effects in h → γ Z (for a recent discussion see [81]).We define the effective Wilson coefficient in this case to be C γZ = 1 g g C HW − g g C HB − (cid:18) g − g (cid:19) C HW B (5.54)so that the modification of the decay rate is Γ( h → γZ )Γ SM ( h → γZ ) ≃ (cid:12)(cid:12)(cid:12)(cid:12) π v C γZ I Z (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π v e C γZ I Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.55) I Z ≈ − . [77, 78], again neglecting v / Λ terms due to c H, kin , etc., and our previouslydefined coefficients are C γZ = − c γZ , e C γZ = − ˜ c γZ (5.56)The one loop RGE results for the CP-even term ˙ C γZ = 12 csc θ W sec θ W n (2 cos 2 θ W + 1)[ Y d ] wv C dγvw + (2 cos 2 θ W − Y e ] wv C eγvw − (4 cos 2 θ W − Y u ] wv C uγvw + h.c. o + 2 (cid:18) [ Y d ] wv C dZvw + [ Y e ] wv C eZvw − Y u ] wv C uZvw + h.c. (cid:19) + (cid:18) λ + 2 Y ( S ) − e + 193 e sec θ W − e csc θ W (cid:19) C γZ + e (cid:18)
113 cos 2 θ W − (cid:19) csc θ W sec θ W C γγ + e (cid:18)
32 sec θ W −
332 cot θ W csc θ W (cid:19) C W + (cid:0) e − e csc θ W + 4 λ cos 2 θ W (cid:1) csc θ W sec θ W C HW B g g . (5.57)The RGE for ˙ e C γZ is given by the substitution Y ψ → − iY ψ , C γZ → e C γZ , C γγ → e C γγ , C W → C f W , and C HW B → C H f W B , as for the gg and γγ amplitudes.– 25 – .11 Electroweak precision observables We are assuming Λ is parametrically higher than the EW scale v , so the usual S , T and U parametrization [82–85] of the oblique electroweak precision data (EWPD) can be used. Anoperator based analysis of EWPD was first developed in Ref. [65]. The standard operatorbased approach identifies the S parameter with the operator Q HW B , and the T parameterwith the operator Q HD , S = 16 π v g g C HWB , T = − πv (cid:18) g + 1 g (cid:19) C HD . (5.58a)A shift in the definition of v is order / Λ for this expression, and we neglect this effect. The U parameter corresponds to the dimension-eight operator ( H † W µ ν H )( H † W µ ν H ) , which weneglect. A fit that treats m h = 126 GeV as an input value [86] to EWPD finds S = 0 . ± . and T = 0 . ± . with a correlation coefficient between S and T of . . S and T depend on C HW B and C HD evaluated at the weak scale. The RG evolution of C HD is given in Eq. (5.30), and the RG evolution of C HW B is ˙ C HW B = (cid:18) λ + 2 Y ( S ) + 43 g + 193 g (cid:19) C HW B + 2 g g ( C HW + C HB ) + 3 g g C W + g (cid:18) Y u ] wv C uBvw − Y d ] wv C dBvw − [ Y e ] wv C eBvw + h.c. (cid:19) + g (cid:18) Y u ] wv C uWvw + [ Y d ] wv C dWvw + 3[ Y e ] wv C eWvw + h.c. (cid:19) . (5.59)The T parameter is usually interpreted as a measure of custodial symmetry violation,whereas the S parameter is considered to be sensitive to the difference between the numberof left-handed and right-handed fermions. Interestingly, the SM EFT one loop RGE does notmix the operators C HW B , C HD . However, this does not follow from custodial symmetry. TheSM violates custodial symmetry in g interactions, and through mass splittings of the SU (2) L doublets. If we take the limit Y d → Y u , Y e → and y d → y u , then y h → from Eq. (2.9). Inthis limit, the standard model preserves custodial SU (2) , as does the RGE. This provides anon-trivial check of our results.The consequences of the RGE for precision electroweak parameters was studied in Ref. [13].The RGE allows one to compute the ln Λ /m H contribution to these observables, which wascomputed previously in the broken theory [28–30]. Our computation agrees with their resultsfor the terms they computed, but has additional effects (e.g. due to the top quark Yukawa)which were not in the previous results. Another promising source of information on EW interactions are triple gauge couplings (TGC).For some recent studies on the phenomenology of these measurements see Refs. [57, 63, 87, 88].Some of the scale dependence of the operators involved in this process (in another basis) has– 26 –een determined [89, 90]. In the basis used here, only the operator Q W directly contributes toTGC measurements. (Other contributions come about indirectly due to field redefinitions.)The full RGE of the Wilson coefficient of the operator Q W has the simple form ˙ C W = (24 − b , ) g C W , or µ dd µ (cid:18) C W g (cid:19) = 24 g (cid:18) C W g (cid:19) , (5.60)where b , is the first coefficient in the g β -function. The triple gauge boson operators donot mix with any other dimension-six operators. This multiplicative renormalization can belargely understood using the results of Ref. [11]. Consequently, TGC measurements providea very clean probe of this dimension-six operator.Recently, Refs. [91, 92] have shown that the decay spectra of the three-body decay h → V ℓ + ℓ − are particularly rich sources of information on the possible effects of anoma-lous couplings of the Higgs boson, and BSM contact interactions. The full decomposition ofthe modification of the V ℓ + ℓ − decay spectra in the operator basis used here was given inRef. [67], which shows that the relevant terms depend on the coefficients C W B , C HD , C HW , C HB , C Hl , C Hl , C He , as well as the coefficient c H, kin which only modifies the total decay rate.It has been argued that TGC measurements probe the same physics as h → V ℓ + ℓ − decays [57] in the SILH basis. This claim comes about by arbitrarily setting the operator C W , which is present in the SILH basis, and in the analysis in Ref. [57], to zero. Thisoperator contributes to TGC measurements, but not to h → V ℓ + ℓ − decays at tree level.It is by using this arbitrary choice that Ref. [57] claims a strong relationship between theseexperimentally measurable quantities. This makes the results in Ref. [57] model-dependent,and not general. For example, the exactly solvable model of Ref. [93] produces C W but noHiggs-lepton operators. In the non-redundant basis of Ref. [9], TGC measurements are alsonot related to h → V ℓ + ℓ − decays since the combination of Wilson coefficients that contributeto the two processes is not identical. Measurable results are basis independent, and modelindependent results do not arbitrarily set operators to zero, as was done in Ref. [57]. Wedisagree with the conclusions of Ref. [57] which are stated as broad, model-independent,conclusions. µ → eγ , magnetic moments, and electric dipole moments The lepton dipole operators L = C eWrs l r,a σ µν e s τ Iab H b W Iµν + C eBrs l r,a H a σ µν e s H a B µν + h.c. (5.61)contribute to radiative transitions such as µ → eγ which is a remarkably clean window tophysics BSM. In the broken phase, Eq. (5.61) gives the charged lepton operators L = ev √ C eγrs e r σ µν P R e s F µν + ev √ C eZrs e r σ µν P R e s Z µν + h.c. (5.62)– 27 –here r and s are flavor indices ( { e e , e µ , e τ } ≡ { e , µ , τ } ) and C eγrs = 1 g C eBrs − g C eWrs C eZrs = − g C eBrs − g C eWrs C dγrs = 1 g C dBrs − g C dWrs C dZrs = − g C dBrs − g C dWrs C uγrs = 1 g C uBrs + 1 g C uWrs C uZrs = − g C uBrs + 1 g C uWrs (5.63) C uW has the opposite sign for u -type quarks in Eq. (5.63) because of the opposite sign for T L . The RGE for C eγ is ˙ C eγrs = (cid:26) Y ( s ) + e (cid:18) −
94 csc θ W + 14 sec θ W (cid:19)(cid:27) C eγrs + 2 C eγrv [ Y e Y † e ] vs + (cid:18)
12 + 2 cos θ W (cid:19) [ Y † e Y e ] rw C eγws + e (12 cot 2 θ W ) C eZrs − (2 sin θ W cos θ W ) [ Y † e Y e ] rw C eZws − cot θ W [ Y † e ] rs (cid:0) C HW B + iC H f W B (cid:1) + 4 e [ Y † e ] rs (cid:16) C γγ + i e C γγ (cid:17) + e (cot θ W − θ W ) [ Y † e ] rs (cid:16) C γZ + i e C γZ (cid:17) + 16[ Y u ] wv C (3) leqursvw . (5.64)The current experimental limit [94] on BR ( µ → eγ ) is . × − from the MEG experiment,which implies v √ m e C eγµe . . × − TeV − (5.65)at the low energy scale µ ∼ m µ .The lepton Yukawa couplings are diagonal in the mass eigenstate basis, so the µ → eγ transition amplitude depends on C eγ , C eZ and C (3) lequ . The bound Eq. (5.65) implies m t m e C (3) lequµett . . × − TeV − (5.66)using the estimate ln(Λ /m H ) / (16 π ) ∼ . for the renormalization group evolution, andassuming that this term is the only contribution to C eγµe at low energies.The anomalous magnetic moment of the muon is δa µ = − m µ v √ Re C eγµµ (5.67)which yields the limits | C HW B | . . TeV − , | C γγ | . TeV − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m t m µ Re C (3) lequµµtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . TeV − , (5.68)– 28 –ssuming that each of these is the only contribution to C eγµµ .The bound on the electric dipole moment of the electron translates to the limits (cid:12)(cid:12) C H f W B (cid:12)(cid:12) . × − TeV − , (cid:12)(cid:12)(cid:12) e C γγ (cid:12)(cid:12)(cid:12) . × − TeV − , (cid:12)(cid:12)(cid:12)(cid:12) m t m e Im C lequeett (cid:12)(cid:12)(cid:12)(cid:12) . × − TeV − , (5.69)using the recently measured upper bound [95], d e < . × − e cm from the ACME collab-oration, again assuming each of these terms is the only contribution. This paper completes the full calculation of the one-loop renormalization of the dimension-six Lagrangian of the SM EFT. We present all of the remaining gauge terms in the × anomalous dimension matrix of the baryon number conserving operators. The anomalousdimension matrix of the dimension-six baryon number violating operators have been computedin Ref. [96].Many of the results are lengthy, but a few important cases such as gg → h , h → γγ and h → γZ have simple RG equations which are given explicitly in this paper. We have computedthe modification of the Higgs mass, self-interactions, and couplings to fermions and gaugebosons from L (6) . The dimension-six terms change the relation between the Higgs vacuumexpectation value and G F , and also contribute to the ρ parameter. The RGE improvementof all of these relations is now known, and will be useful for future precision studies of theSM EFT. A complete analysis of the SM EFT is a formidable task, because L (6) has 2499independent parameters.We have also discussed how the SM EFT provides a model-independent way to test theMFV hypothesis, and how the full SM EFT RGE mixes flavor violation between the differentoperator sectors. A few applications of our results have been given in this paper. Acknowledgments
This work was supported in part by DOE grant DE-SC0009919. MT thanks W. Skiba forhelpful conversations. RA and AM thank B. Shotwell, D. Stone, H.-M. Chang and C. Murphyfor useful discussions. We would also like to thank C. Zhang [97], G. Pruna and A. Signer [98],J. Brod, A. Greljo, E. Stamou, and P. Uttayarat [99], and C. Cheung and C.-H. Shen [100]for pointing out typos/errors in previous versions of the manuscript.
A Flavor representations and parameter counting
In this appendix, we briefly discuss the flavor representations of the operators, and the pa-rameter counting of Table 2.Operators in classes 1–4 have no flavor indices, and the counting is trivial.– 29 –lass 5 and 6 operator coefficients are n g × n g complex matrices M rs in flavor space,with n g complex entries. The real matrix elements give the n g CP -even parmeters and theimaginary matrix elements yield n g CP -odd entries.Class 7 operators, other than Q Hud are hermitian, so their coefficients are n g × n g hermitianmatrices H rs in flavor space, which can be written as H rs = S rs + iA rs , where S is real-symmetric and CP -even with n e = n g ( n g + 1) / parameters, and A is real-antisymmetric and CP -odd, with n o = n g ( n g − / parameters. Q Hud , which is not hermitian, is an n g × n g complex matrix with n g CP -even and n g CP -odd parameters.The four-fermion operators in Class 8 are the only non-trivial case. The ( LR )( RL ) and ( LR )( LR ) operators are not hermitian, and each has n g CP -even and n g CP -odd parameters,since the operator has 4 independent flavor indices. The ( LL )( RR ) operators are the productof L and R currents, each of which has n e CP -even and n o CP -odd components, for n e + n o CP -even and n e n o CP -odd terms. The counting for ( LL )( LL ) and ( RR )( RR ) operatorswhen the currents are different, Q (1 , lq , Q eu , Q ed , Q (1 , ud , is the same as for the ( LL )( RR ) operators. The interesting case is for Q ll , Q (1 , qq , Q uu , Q dd where the two currents are identical,so that all four flavor indices transform under the same SU ( n g ) flavor group. The operatorstransform as the + adj + adj + aa + ss where adj is the adjoint representation, aa is therepresentation T [ ij ][ kl ] antisymmetric in the upper and lower indices, and ss is the representation T ( ij )( kl ) symmetric in the upper and lower indices. The aa representation vanishes for n g = 3 .The singlet has one CP -even parameter, the adjoint has ( n g − n g + 2) / CP -even and n g ( n g − / CP -odd parameters, aa has n g ( n g − n g + n g + 2) / CP -even and n g ( n g − n g − n g + 2) / CP -odd parameters, and ss has n g ( n g − n g + 1)( n g + 2) / CP -even and n g ( n g − n g + 3 n g − / CP -odd parameters. The operator Q ee is a special case, becauseof the Fierz identity (¯ e p γ µ e r )(¯ e s γ µ e t ) = (¯ e s γ µ e r )(¯ e p γ µ e t ) , (A.1)which implies that the operator must be symmetric in the two e indices and in the two e indices. This identity does not hold for the other fermions, because they have SU (2) or colorindices. Q ee transforms as adj + ss because of the Fierz identity.Adding up the individual contributions gives Table 2. The relevant group theory results can be found, for example, in Refs. [101, 102]. – 30 –
Conversion of P i operators to the standard basis The equations of motion can be used to express the operators P i in the standard basis. Theidentifications are P B = 12 y h g Q H (cid:3) + 2 g y h Q HD + 12 g (cid:20) y l Q (1) Hltt + y e Q Hett + y q Q (1) Hqtt + y u Q Hutt + y d Q Hdtt (cid:21) , P W = 34 g Q H (cid:3) − g m H ( H † H ) + 2 g λQ H + 14 g (cid:20) Q (3) Hltt + Q (3) Hqtt (cid:21) + 12 g (cid:18) [ Y † u ] rs Q uHrs + [ Y † d ] rs Q dHrs + [ Y † e ] rs Q eHrs + h.c. (cid:19) , P HB = 12 g y h Q H (cid:3) + 2 g y h Q HD − y h g Q HB − g g Q HW B + 12 g (cid:20) y l Q (1) Hltt + y e Q Hett + y q Q (1) Hqtt + y u Q Hutt + y d Q Hdtt (cid:21) , P HW = 34 g Q H (cid:3) − g m H ( H † H ) + 2 g λQ H − g Q HW − y h g g Q HW B + 14 g h Q (3) Hl + Q (3) Hq i + 12 g (cid:18) [ Y † u ] rs Q uHrs + [ Y † d ] rs Q dHrs + [ Y † e ] rs Q eHrs + h.c. (cid:19) ,P T = − Q H (cid:3) − Q HD . (B.1) C Results
The renormalization group equations by operator class are given below. The complete RGequations for the dimension-six operators are given by adding Eqs. (6.1)–(6.4) of Ref. [12],the equations in the appendices of Ref. [10] and the equations given below. Eqs. (4.3)–(4.5)of Ref. [12] give the renormalization group evolution of SM couplings due to dimension-sixoperators.The parameters η − are defined in the appendix of Ref. [10]. Some equations use ξ B ,defined by ξ B = 43 y h ( C H (cid:3) + C HD ) + 83 (cid:20) y l C (1) Hltt + 2 y q N c C (1) Hqtt + y e C Hett + y u N c C Hutt + y d N c C Hdtt (cid:21) (C.1)The other parameters are c A, = 2 , c F, = 3 / , c A, = N c , c F, = ( N c − / (2 N c ) with N c = 3 , b , = − / − n g / , b , = 43 / − n g / and b , = 11 − n g / . C.1 X ˙ C G = (12 c A, − b , ) g C G ˙ C e G = (12 c A, − b , ) g C e G ˙ C W = (12 c A, − b , ) g C W ˙ C f W = (12 c A, − b , ) g C f W – 31 – .2 H ˙ C H = (cid:18) − g − g (cid:19) C H + λ (cid:20) g C H (cid:3) + (cid:0) − g + 24 g y h (cid:1) C HD (cid:21) − (cid:0) y h g + g (cid:1) C HD + 12 λ (cid:0) g C HW + 4 g y h C HB + 2 g g y h C HW B (cid:1) − (cid:0) g g y h + 9 g (cid:1) C HW − (cid:0) g y h + 12 g g y h (cid:1) C HB − (cid:0) g g y h + 6 g g y h (cid:1) C HW B + 163 λg (cid:18) C (3) Hltt + N c C (3) Hqtt (cid:19)
C.3 H D ˙ C H (cid:3) = − (cid:18) g + 163 g y h (cid:19) C H (cid:3) + 203 g y h C HD + 2 g (cid:18) C (3) Hltt + N c C (3) Hqtt (cid:19) + 43 g y h (cid:18) N c y u C Hutt + N c y d C Hdtt + y e C Hett + 2 N c y q C (1) Hqtt + 2 y l C (1) Hltt (cid:19) ˙ C HD = 803 g y h C H (cid:3) + (cid:18) g − g y h (cid:19) C HD + 163 g y h (cid:18) N c y u C Hutt + N c y d C Hdtt + y e C Hett + 2 N c y q C (1) Hqtt + 2 y l C (1) Hltt (cid:19)
C.4 X H ˙ C HG = (cid:0) − y h g − g − b , g (cid:1) C HG ˙ C HB = (cid:0) y h g − g − b , g (cid:1) C HB + 6 g g y h C HW B ˙ C HW = − g C W + (cid:0) − y h g − g − b , g (cid:1) C HW + 2 g g y h C HW B ˙ C HW B = 6 g g y h C W + (cid:0) − y h g + 92 g − b , g − b , g (cid:1) C HW B + 4 g g y h C HB + 4 g g y h C HW ˙ C H e G = (cid:0) − y h g − g − b , g (cid:1) C H e G ˙ C H e B = (cid:0) y h g − g − b , g (cid:1) C H e B + 6 g g y h C H f W B ˙ C H f W = − g C f W + (cid:0) − y h g − g − b , g (cid:1) C H f W + 2 g g y h C H f W B ˙ C H f W B = 6 g g y h C f W + (cid:0) − y h g + 92 g − b , g − b , g (cid:1) C H f W B + 4 g g y h C H e B + 4 g g y h C H f W – 32 – .5 ψ H ˙ C eHrs = [ Y † e ] rs (cid:20) g C H (cid:3) + (cid:18) − g + 6 g y h (cid:19) C HD (cid:21) − (cid:20) y l + 3 y e − y l y e ) g + 274 g (cid:21) C eHrs + 3[ Y † e ] rs (cid:0) g ( C HW + iC H f W )+ 4( y h + 2 y l y e ) g ( C HB + iC H e B ) + 2 g g y l ( C HW B + iC H f W B ) (cid:1) − (cid:0) g y e C eBrt + g C eWrt (cid:1) [ Y e Y † e ] ts − Y † e Y e ] rv (cid:0) g ( y l + y e ) C eBvs − g C eWvs (cid:1) − (cid:0) g y h y e + 4 g y h y l + g g y h (cid:1) C eBrs − (cid:0) g g y h y e + 4 g g y h y l + 3 g (cid:1) C eWrs + (cid:0) g + 12 g y l y h (cid:1) [ Y † e ] rt C Hets + 12 g y e y h C (1) Hlrt [ Y † e ] ts + 12 g y e y h C (3) Hlrt [ Y † e ] ts + 43 g [ Y † e ] rs (cid:18) C (3) Hltt + N c C (3) Hqtt (cid:19) ˙ C uHrs = [ Y † u ] rs (cid:20) g C H (cid:3) + (cid:18) − g + 6 g y h (cid:19) C HD (cid:21) − (cid:20) y q + 3 y u − y q y u ) g + 274 g + 6 c F, g (cid:21) C uHrs + 3[ Y † u ] rs (cid:0) g c F, ( C HG + iC H e G ) + 3 g ( C HW + iC H f W )+ 4( y h + 2 y q y u ) g ( C HB + iC H e B ) − y q g g ( C HW B + iC H f W B ) (cid:1) − Y † d Y d ] rt g C uWts − g C dWrt [ Y d Y † u ] ts − (cid:0) g c F, C uGrt + g C uWrt + (3 y u + y d ) g C uBrt (cid:1) [ Y u Y † u ] ts − Y † u Y u ] rv (cid:0) c F, g C uGvs − g C uWvs + 2( y q + y u ) g C uBvs (cid:1) − (cid:0) g y h y u + 4 g y h y q − g g y h (cid:1) C uBrs + 3 (cid:0) g g y h y u + 4 g g y h y q − g (cid:1) C uWrs − (cid:0) g − g y q y h (cid:1) [ Y † u ] rt C Huts + 3 g [ Y † d ] rt C ∗ Hudst + 12 g y u y h C (1) Hqrt [ Y † u ] ts − g y u y h C (3) Hqrt [ Y † u ] ts + 43 g [ Y † u ] rs (cid:18) C (3) Hltt + N c C (3) Hqtt (cid:19) ˙ C dHrs = [ Y † d ] rs (cid:20) g C H (cid:3) + (cid:18) − g + 6 g y h (cid:19) C HD (cid:21) − (cid:20) y q + 3 y d − y q y d ) g + 274 g + 6 c F, g (cid:21) C dHrs + 3[ Y † d ] rs (cid:0) c F, g ( C HG + iC H e G ) + 3 g ( C HW + iC H f W )+ 4( y h + 2 y q y d ) g ( C HB + iC H e B ) + 2 y q g g ( C HW B + iC H f W B ) (cid:1) − Y † u Y u ] rt g C dWts − g C uWrt [ Y u Y † d ] ts − (cid:0) c F, g C dGrt + g C dWrt + (3 y d + y u ) g C dBrt (cid:1) [ Y d Y † d ] ts − Y † d Y d ] rt (cid:0) c F, g C dGts − g C dWts + 2 ( y q + y d ) g C dBts (cid:1) − (cid:0) g y h y d + 4 g y h y q + g g y h (cid:1) C dBrs − (cid:0) g g y h y d + 4 g g y h y q + 3 g (cid:1) C dWrs + (cid:0) g + 12 g y q y h (cid:1) [ Y † d ] rt C Hdts + 3 g [ Y † u ] rt C Hudts + 12 g y d y h C (1) Hqrt [ Y † d ] ts + 12 g y d y h C (3) Hqrt [ Y † d ] ts + 43 g [ Y † d ] rs (cid:18) C (3) Hltt + N c C (3) Hqtt (cid:19) – 33 – .6 ψ XH ˙ C eWrs = (cid:2) (3 c F, − b , ) g + (cid:0) − y e + 8 y e y l − y l (cid:1) g (cid:3) C eWrs + g g (3 y l − y e ) C eBrs − [ Y † e ] rs (cid:0) g ( C HW + iC H f W ) + g ( y l + y e )( C HW B + iC H f W B ) (cid:1) ˙ C eBrs = (cid:2) − c F, g + (cid:0) y e + 4 y e y l + 3 y l − b , (cid:1) g (cid:3) C eBrs + 4 c F, g g (3 y l − y e ) C eWrs − [ Y † e ] rs (cid:0) g ( y l + y e )( C HB + iC H e B ) + 32 g ( C HW B + iC H f W B ) (cid:1) ˙ C uGrs = (cid:2) (10 c F, − c A, − b , ) g − c F, g + (cid:0) − y u + 8 y u y q − y q (cid:1) g (cid:3) C uGrs + 8 c F, g g C uWrs + 4 g g ( y u + y q ) C uBrs − Y † u ] rs g ( C HG + iC H e G ) − g c A, [ Y † u ] rs (cid:0) C G + iC e G (cid:1) ˙ C uWrs = (cid:2) c F, g + (3 c F, − b , ) g + (cid:0) − y u + 8 y u y q − y q (cid:1) g (cid:3) C uWrs + 2 c F, g g C uGrs + g g (3 y q − y u ) C uBrs − [ Y † u ] rs (cid:0) g ( C HW + iC H f W ) − g ( y q + y u )( C HW B + iC H f W B ) (cid:1) ˙ C uBrs = (cid:2) c F, g − c F, g + (cid:0) y u + 4 y u y q + 3 y q − b , (cid:1) g (cid:3) C uBrs + 4 c F, g g ( y u + y q ) C uGrs + 4 c F, g g (3 y q − y u ) C uWrs − [ Y † u ] rs (cid:0) g ( y q + y u )( C HB + iC H e B ) − g ( C HW B + iC H f W B ) (cid:1) ˙ C dGrs = (cid:2) (10 c F, − c A, − b , ) g − c F, g + (cid:0) − y d + 8 y d y q − y q (cid:1) g (cid:3) C dGrs + 8 c F, g g C dWrs + 4 g g ( y d + y q ) C dBrs − Y † d ] rs g ( C HG + iC H e G ) − g c A, [ Y † d ] rs (cid:0) C G + iC e G (cid:1) ˙ C dWrs = (cid:2) c F, g + (3 c F, − b , ) g + (cid:0) − y d + 8 y d y q − y q (cid:1) g (cid:3) C dWrs + 2 c F, g g C dGrs + g g (3 y q − y d ) C dBrs − [ Y † d ] rs (cid:0) g ( C HW + iC H f W ) + g ( y q + y d )( C HW B + iC H f W B ) (cid:1) ˙ C dBrs = (cid:2) c F, g − c F, g + (cid:0) y d + 4 y d y q + 3 y q − b , (cid:1) g (cid:3) C dBrs + 4 c F, g g ( y d + y q ) C dGrs + 4 c F, g g (3 y q − y d ) C dWrs − [ Y † d ] rs (cid:0) g ( y q + y d )( C HB + iC H e B ) + 32 g ( C HW B + iC H f W B ) (cid:1) C.7 ψ H D ˙ C (1) Hlrs = 12 ξ B g δ rs y l + 43 g y h C (1) Hlrs + 43 g N c y d y h C ldrsww + 43 g y e y h C lersww + 83 g y h y l C llrsww + 43 g y h y l C llrwws + 43 g y h y l C llwsrw + 83 g y h y l C llwwrs + 83 g N c y h y q C (1) lqrsww + 43 g N c y h y u C lursww – 34 – C (3) Hlrs = 16 g C H (cid:3) δ rs + 23 g C (3) Hltt δ rs + 23 g N c C (3) Hqtt δ rs + 13 g C (3) Hlrs + 13 g C llrwws + 13 g C llwsrw + 23 g N c C (3) lqrsww − g C (3) Hlrs ˙ C Hers = 12 ξ B g δ rs y e + 43 g y h C Hers + 43 g N c y d y h C edrsww + 43 g y e y h C eersww + 43 g y e y h C eerwws + 43 g y e y h C eewsrw + 43 g y e y h C eewwrs + 43 g N c y h y u C eursww + 83 g y h y l C lewwrs + 83 g N c y h y q C qewwrs ˙ C (1) Hqrs = 12 ξ B g δ rs y q + 43 g y h C (1) Hqrs + 83 g y h y l C (1) lqwwrs + 43 g N c y d y h C (1) qdrsww + 43 g y e y h C qersww + 83 g N c y h y q C (1) qqrsww + 43 g y h y q C (1) qqrwws + 43 g y h y q C (1) qqwsrw + 83 g N c y h y q C (1) qqwwrs + 4 g y h y q C (3) qqrwws + 4 g y h y q C (3) qqwsrw + 43 g N c y h y u C (1) qursww ˙ C (3) Hqrs = 16 g C H (cid:3) δ rs + 23 g C (3) Hltt δ rs + 23 g N c C (3) Hqtt δ rs + 13 g C (3) Hqrs + 23 g C (3) lqwwrs + 13 g C (1) qqrwws + 13 g C (1) qqwsrw + 23 g N c C (3) qqrsww − g C (3) qqrwws − g C (3) qqwsrw + 23 g N c C (3) qqwwrs − g C (3) Hqrs ˙ C Hurs = 12 ξ B g δ rs y u + 43 g y h C Hurs + 43 g y e y h C euwwrs + 83 g y h y l C luwwrs + 83 g N c y h y q C (1) quwwrs + 43 g N c y d y h C (1) udrsww + 43 g N c y h y u C uursww + 43 g y h y u C uurwws + 43 g y h y u C uuwsrw + 43 g N c y h y u C uuwwrs ˙ C Hdrs = 12 ξ B g δ rs y d + 43 g y h C Hdrs + 43 g N c y d y h C ddrsww + 43 g y d y h C ddrwws + 43 g y d y h C ddwsrw + 43 g N c y d y h C ddwwrs + 43 g y e y h C edwwrs + 83 g y h y l C ldwwrs + 83 g N c y h y q C (1) qdwwrs + 43 g N c y h y u C (1) udwwrs ˙ C Hudrs = − g ( y u − y d ) C Hudrs – 35 – .8 ψ C.8.1 ( LL )( LL )˙ C llprst = 23 g y h y l C (1) Hlst δ pr − g C (3) Hlst δ pr + 13 g C (3) Hlsr δ pt + 13 g C (3) Hlpt δ rs + 23 g y h y l C (1) Hlpr δ st − g C (3) Hlpr δ st + 43 g y l C llprww δ st + 43 g y l C llstww δ pr + 43 g y l C llwwst δ pr + 43 g y l C llwwpr δ st + 23 g y l C llpwwr δ st + 23 g y l C llswwt δ pr + 23 g y l C llwrpw δ st + 23 g y l C llwtsw δ pr − g C llpwwr δ st − g C llswwt δ pr − g C llwrpw δ st − g C llwtsw δ pr + 13 g C llswwr δ pt + 13 g C llpwwt δ rs + 13 g C llwrsw δ pt + 13 g C llwtpw δ rs + 43 g N c y l y q C (1) lqprww δ st + 43 g N c y l y q C (1) lqstww δ pr − g N c C (3) lqprww δ st − g N c C (3) lqstww δ pr + 23 g N c C (3) lqsrww δ pt + 23 g N c C (3) lqptww δ rs + 23 g N c y l y u C luprww δ st + 23 g N c y l y u C lustww δ pr + 23 g N c y d y l C ldprww δ st + 23 g N c y d y l C ldstww δ pr + 23 g y e y l C leprww δ st + 23 g y e y l C lestww δ pr + 6 g C llptsr − (cid:0) g − y l g (cid:1) C llprst ˙ C (1) qqprst = 23 g y h y q C (1) Hqst δ pr + 23 g y h y q C (1) Hqpr δ st + 43 g y l y q C (1) lqwwst δ pr + 43 g y l y q C (1) lqwwpr δ st + 43 g N c y q C (1) qqprww δ st + 43 g N c y q C (1) qqstww δ pr + 43 g N c y q C (1) qqwwst δ pr + 43 g N c y q C (1) qqwwpr δ st + 23 g y q C (1) qqpwwr δ st + 23 g y q C (1) qqswwt δ pr + 23 g y q C (1) qqwrpw δ st + 23 g y q C (1) qqwtsw δ pr + 16 g C (1) qqswwr δ pt + 16 g C (1) qqpwwt δ rs + 16 g C (1) qqwrsw δ pt + 16 g C (1) qqwtpw δ rs − N c g C (1) qqpwwr δ st − N c g C (1) qqswwt δ pr − N c g C (1) qqwrpw δ st − N c g C (1) qqwtsw δ pr + 2 g y q C (3) qqpwwr δ st + 2 g y q C (3) qqswwt δ pr + 2 g y q C (3) qqwrpw δ st + 2 g y q C (3) qqwtsw δ pr + 12 g C (3) qqswwr δ pt + 12 g C (3) qqpwwt δ rs + 12 g C (3) qqwrsw δ pt + 12 g C (3) qqwtpw δ rs − N c g C (3) qqpwwr δ st − N c g C (3) qqswwt δ pr − N c g C (3) qqwrpw δ st − N c g C (3) qqwtsw δ pr + 23 g N c y q y u C (1) quprww δ st + 23 g N c y q y u C (1) qustww δ pr + 23 g N c y d y q C (1) qdprww δ st + 23 g N c y d y q C (1) qdstww δ pr + 112 g C (8) qusrww δ pt + 112 g C (8) quptww δ rs − N c g C (8) quprww δ st − N c g C (8) qustww δ pr + 112 g C (8) qdsrww δ pt + 112 g C (8) qdptww δ rs − N c g C (8) qdprww δ st − N c g C (8) qdstww δ pr + 23 g y e y q C qeprww δ st + 23 g y e y q C qestww δ pr + 3 g C (1) qqptsr + 9 g C (3) qqptsr + 9 g C (3) qqprst − N c (cid:0) g − N c y q g (cid:1) C (1) qqprst – 36 – C (3) qqprst = 16 g C (3) Hqst δ pr + 16 g C (3) Hqpr δ st + 13 g C (3) lqwwst δ pr + 13 g C (3) lqwwpr δ st + 16 g C (1) qqswwt δ pr + 16 g C (1) qqwtsw δ pr + 16 g C (1) qqpwwr δ st + 16 g C (1) qqwrpw δ st + 16 g C (1) qqpwwt δ rs + 16 g C (1) qqswwr δ pt + 16 g C (1) qqwtpw δ rs + 16 g C (1) qqwrsw δ pt + 13 g N c C (3) qqprww δ st + 13 g N c C (3) qqstww δ pr + 13 g N c C (3) qqwwst δ pr + 13 g N c C (3) qqwwpr δ st − g C (3) qqpwwr δ st − g C (3) qqswwt δ pr − g C (3) qqwrpw δ st − g C (3) qqwtsw δ pr + 12 g C (3) qqpwwt δ rs + 12 g C (3) qqswwr δ pt + 12 g C (3) qqwtpw δ rs + 12 g C (3) qqwrsw δ pt + 112 g C (8) quptww δ rs + 112 g C (8) qusrww δ pt + 112 g C (8) qdptww δ rs + 112 g C (8) qdsrww δ pt − g C (3) qqptsr − N c g C (3) qqprst − g C (3) qqprst + 12 y q g C (3) qqprst + 3 g C (1) qqptsr + 3 g C (1) qqprst ˙ C (1) lqprst = 43 g y h y l C (1) Hqst δ pr + 43 g y h y q C (1) Hlpr δ st + 83 g y l y q C llprww δ st + 83 g y l y q C llwwpr δ st + 43 g y l y q C llpwwr δ st + 43 g y l y q C llwrpw δ st + 83 g N c y q C (1) lqprww δ st + 83 g y l C (1) lqwwst δ pr + 83 g N c y l y q C (1) qqstww δ pr + 83 g N c y l y q C (1) qqwwst δ pr + 43 g y l y q C (1) qqswwt δ pr + 43 g y l y q C (1) qqwtsw δ pr + 4 g y l y q C (3) qqswwt δ pr + 4 g y l y q C (3) qqwtsw δ pr + 43 g N c y l y u C (1) qustww δ pr + 43 g N c y d y l C (1) qdstww δ pr + 43 g y e y l C qestww δ pr + 43 g N c y q y u C luprww δ st + 43 g N c y d y q C ldprww δ st + 43 g y e y q C leprww δ st + 12 y l y q g C (1) lqprst + 9 g C (3) lqprst ˙ C (3) lqprst = 13 g C (3) Hqst δ pr + 13 g C (3) Hlpr δ st + 23 g N c C (3) lqprww δ st + 23 g C (3) lqwwst δ pr + 13 g C (1) qqswwt δ pr + 13 g C (1) qqwtsw δ pr + 23 g N c C (3) qqstww δ pr + 23 g N c C (3) qqwwst δ pr − g C (3) qqswwt δ pr − g C (3) qqwtsw δ pr + 13 g C llpwwr δ st + 13 g C llwrpw δ st + 3 g C (1) lqprst − (cid:0) g − y l y q g (cid:1) C (3) lqprst C.8.2 ( RR )( RR )˙ C eeprst = 23 g y e y h C Hest δ pr + 23 g y e y h C Hepr δ st + 43 g y e y l C lewwpr δ st + 43 g y e y l C lewwst δ pr + 43 g N c y e y q C qewwpr δ st + 43 g N c y e y q C qewwst δ pr + 23 g N c y e y u C euprww δ st + 23 g N c y e y u C eustww δ pr + 23 g N c y d y e C edprww δ st + 23 g N c y d y e C edstww δ pr + 23 g y e C eeprww δ st + 23 g y e C eestww δ pr + 23 g y e C eewwpr δ st + 23 g y e C eewwst δ pr + 23 g y e C eepwwr δ st + 23 g y e C eeswwt δ pr + 23 g y e C eewtsw δ pr + 23 g y e C eewrpw δ st + 12 y e g C eeprst – 37 – C uuprst = 23 g y h y u C Hust δ pr + 23 g y h y u C Hupr δ st + 23 g y e y u C euwwst δ pr + 23 g y e y u C euwwpr δ st + 43 g y l y u C luwwpr δ st + 43 g y l y u C luwwst δ pr + 43 g N c y q y u C (1) quwwst δ pr + 43 g N c y q y u C (1) quwwpr δ st + 13 g C (8) quwwpt δ rs + 13 g C (8) quwwsr δ pt − N c g C (8) quwwst δ pr − N c g C (8) quwwpr δ st + 23 g N c y u C uuprww δ st + 23 g N c y u C uustww δ pr + 23 g N c y u C uuwwpr δ st + 23 g N c y u C uuwwst δ pr + 23 g y u C uupwwr δ st + 23 g y u C uuswwt δ pr + 23 g y u C uuwrpw δ st + 23 g y u C uuwtsw δ pr + 13 g C uupwwt δ rs + 13 g C uuswwr δ pt + 13 g C uuwtpw δ rs + 13 g C uuwrsw δ pt − N c g C uupwwr δ st − N c g C uuswwt δ pr − N c g C uuwrpw δ st − N c g C uuwtsw δ pr + 23 g N c y d y u C (1) udprww δ st + 23 g N c y d y u C (1) udstww δ pr + 16 g C (8) udptww δ rs + 16 g C (8) udsrww δ pt − N c g C (8) udprww δ st − N c g C (8) udstww δ pr + 6 g C uuptsr − (cid:18) N c g − y u g (cid:19) C uuprst ˙ C ddprst = 23 g y d y h C Hdst δ pr + 23 g y d y h C Hdpr δ st + 23 g N c y d C ddprww δ st + 23 g N c y d C ddstww δ pr + 23 g N c y d C ddwwpr δ st + 23 g N c y d C ddwwst δ pr + 23 g y d C ddpwwr δ st + 23 g y d C ddswwt δ pr + 23 g y d C ddwtsw δ pr + 23 g y d C ddwrpw δ st + 13 g C ddpwwt δ rs + 13 g C ddswwr δ pt + 13 g C ddwtpw δ rs + 13 g C ddwrsw δ pt − N c g C ddpwwr δ st − N c g C ddswwt δ pr − N c g C ddwtsw δ pr − N c g C ddwrpw δ st + 43 g y d y l C ldwwpr δ st + 43 g y d y l C ldwwst δ pr + 43 g N c y d y q C (1) qdwwpr δ st + 43 g N c y d y q C (1) qdwwst δ pr + 13 g C (8) qdwwsr δ pt + 13 g C (8) qdwwpt δ rs − N c g C (8) qdwwpr δ st − N c g C (8) qdwwst δ pr + 23 g y d y e C edwwpr δ st + 23 g y d y e C edwwst δ pr + 23 g N c y d y u C (1) udwwpr δ st + 23 g N c y d y u C (1) udwwst δ pr + 16 g C (8) udwwpt δ rs + 16 g C (8) udwwsr δ pt − N c g C (8) udwwpr δ st − N c g C (8) udwwst δ pr + 6 g C ddptsr − (cid:18) N c g − y d g (cid:19) C ddprst ˙ C euprst = 43 g y e y h C Hust δ pr + 43 g y h y u C Hepr δ st + 83 g y l y u C lewwpr δ st + 83 g y e y l C luwwst δ pr + 83 g N c y q y u C qewwpr δ st + 83 g N c y e y q C (1) quwwst δ pr + 43 g N c y e y u C uustww δ pr + 43 g N c y e y u C uuwwst δ pr + 43 g y e y u C uuswwt δ pr + 43 g y e y u C uuwtsw δ pr + 43 g y e y u C eeprww δ st + 43 g y e y u C eewwpr δ st + 43 g y e y u C eepwwr δ st + 43 g y e y u C eewrpw δ st + 43 g N c y u C euprww δ st + 43 g y e C euwwst δ pr + 43 g N c y d y e C (1) udstww δ pr + 43 g N c y d y u C edprww δ st + 12 y e y u g C euprst – 38 – C edprst = 43 g y e y h C Hdst δ pr + 43 g y d y h C Hepr δ st + 43 g N c y d y e C ddstww δ pr + 43 g N c y d y e C ddwwst δ pr + 43 g y d y e C ddswwt δ pr + 43 g y d y e C ddwtsw δ pr + 43 g y d y e C eeprww δ st + 43 g y d y e C eewwpr δ st + 43 g y d y e C eepwwr δ st + 43 g y d y e C eewrpw δ st + 83 g y e y l C ldwwst δ pr + 83 g y d y l C lewwpr δ st + 83 g N c y e y q C (1) qdwwst δ pr + 83 g N c y d y q C qewwpr δ st + 43 g N c y d y u C euprww δ st + 43 g N c y e y u C (1) udwwst δ pr + 43 g N c y d C edprww δ st + 43 g y e C edwwst δ pr + 12 y e y d g C edprst ˙ C (1) udprst = 43 g y h y u C Hdst δ pr + 43 g y d y h C Hupr δ st + 43 g N c y d y u C uuprww δ st + 43 g N c y d y u C uuwwpr δ st + 43 g y d y u C uupwwr δ st + 43 g y d y u C uuwrpw δ st + 43 g N c y d y u C ddstww δ pr + 43 g N c y d y u C ddwwst δ pr + 43 g y d y u C ddswwt δ pr + 43 g y d y u C ddwtsw δ pr + 83 g N c y d y q C (1) quwwpr δ st + 83 g N c y q y u C (1) qdwwst δ pr + 83 g y d y l C luwwpr δ st + 83 g y l y u C ldwwst δ pr + 43 g N c y d C (1) udprww δ st + 43 g N c y u C (1) udwwst δ pr + 43 g y d y e C euwwpr δ st + 43 g y e y u C edwwst δ pr + 12 y u y d g C (1) udprst + 3 (cid:18) N c − N c (cid:19) g C (8) udprst ˙ C (8) udprst = 43 g C uupwwr δ st + 43 g C uuwrpw δ st + 43 g C ddswwt δ pr + 43 g C ddwtsw δ pr + 43 g C (8) quwwpr δ st + 43 g C (8) qdwwst δ pr + 23 g C (8) udprww δ st + 23 g C (8) udwwst δ pr + 12 (cid:18) y u y d g − N c g (cid:19) C (8) udprst + 12 g C (1) udprst C.8.3 ( LL )( RR )˙ C leprst = 43 g y h y l C Hest δ pr + 43 g y e y h C (1) Hlpr δ st + 83 g y e y l C llprww δ st + 83 g y e y l C llwwpr δ st + 43 g y e y l C llpwwr δ st + 43 g y e y l C llwrpw δ st + 83 g N c y e y q C (1) lqprww δ st + 83 g N c y l y q C qewwst δ pr + 43 g y e C leprww δ st + 83 g y l C lewwst δ pr + 43 g N c y e y u C luprww δ st + 43 g N c y d y e C ldprww δ st + 43 g N c y l y u C eustww δ pr + 43 g N c y d y l C edstww δ pr + 43 g y e y l C eestww δ pr + 43 g y e y l C eeswwt δ pr + 43 g y e y l C eewtsw δ pr + 43 g y e y l C eewwst δ pr − y l y e g C leprst ˙ C luprst = 43 g y h y l C Hust δ pr + 43 g y h y u C (1) Hlpr δ st + 83 g y l y u C llprww δ st + 83 g y l y u C llwwpr δ st + 43 g y l y u C llpwwr δ st + 43 g y l y u C llwrpw δ st + 83 g N c y q y u C (1) lqprww δ st + 83 g N c y l y q C (1) quwwst δ pr + 43 g N c y u C luprww δ st + 83 g y l C luwwst δ pr + 43 g N c y d y u C ldprww δ st + 43 g y e y u C leprww δ st + 43 g N c y d y l C (1) udstww δ pr + 43 g y e y l C euwwst δ pr + 43 g N c y l y u C uustww δ pr + 43 g N c y l y u C uuwwst δ pr + 43 g y l y u C uuswwt δ pr + 43 g y l y u C uuwtsw δ pr − y l y u g C luprst – 39 – C ldprst = 43 g y h y l C Hdst δ pr + 43 g y d y h C (1) Hlpr δ st + 83 g y d y l C llprww δ st + 83 g y d y l C llwwpr δ st + 43 g y d y l C llpwwr δ st + 43 g y d y l C llwrpw δ st + 83 g N c y d y q C (1) lqprww δ st + 83 g N c y l y q C (1) qdwwst δ pr + 43 g N c y d C ldprww δ st + 83 g y l C ldwwst δ pr + 43 g N c y d y u C luprww δ st + 43 g y d y e C leprww δ st + 43 g N c y l y u C (1) udwwst δ pr + 43 g y e y l C edwwst δ pr + 43 g N c y d y l C ddstww δ pr + 43 g N c y d y l C ddwwst δ pr + 43 g y d y l C ddswwt δ pr + 43 g y d y l C ddwtsw δ pr − y l y d g C ldprst ˙ C qeprst = 43 g y h y q C Hest δ pr + 43 g y e y h C (1) Hqpr δ st + 83 g N c y e y q C (1) qqprww δ st + 83 g N c y e y q C (1) qqwwpr δ st + 43 g y e y q C (1) qqpwwr δ st + 43 g y e y q C (1) qqwrpw δ st + 4 g y e y q C (3) qqpwwr δ st + 4 g y e y q C (3) qqwrpw δ st + 83 g y e y l C (1) lqwwpr δ st + 83 g y l y q C lewwst δ pr + 43 g y e C qeprww δ st + 83 g N c y q C qewwst δ pr + 43 g N c y e y u C (1) quprww δ st + 43 g N c y d y e C (1) qdprww δ st + 43 g N c y q y u C eustww δ pr + 43 g N c y d y q C edstww δ pr + 43 g y e y q C eestww δ pr + 43 g y e y q C eewwst δ pr + 43 g y e y q C eeswwt δ pr + 43 g y e y q C eewtsw δ pr − y q y e g C qeprst ˙ C (1) quprst = 43 g y h y q C Hust δ pr + 43 g y h y u C (1) Hqpr δ st + 83 g N c y q y u C (1) qqprww δ st + 83 g N c y q y u C (1) qqwwpr δ st + 43 g y q y u C (1) qqpwwr δ st + 43 g y q y u C (1) qqwrpw δ st + 4 g y q y u C (3) qqpwwr δ st + 4 g y q y u C (3) qqwrpw δ st + 83 g y l y u C (1) lqwwpr δ st + 43 g y e y u C qeprww δ st + 43 g N c y d y u C (1) qdprww δ st + 43 g N c y u C (1) quprww δ st + 83 g N c y q C (1) quwwst δ pr + 83 g y l y q C luwwst δ pr + 43 g y e y q C euwwst δ pr + 43 g N c y d y q C (1) udstww δ pr + 43 g N c y q y u C uustww δ pr + 43 g N c y q y u C uuwwst δ pr + 43 g y q y u C uuswwt δ pr + 43 g y q y u C uuwtsw δ pr − y q y u g C (1) quprst − (cid:18) N c − N c (cid:19) g C (8) quprst ˙ C (8) quprst = 43 g C (1) qqpwwr δ st + 43 g C (1) qqwrpw δ st + 4 g C (3) qqpwwr δ st + 4 g C (3) qqwrpw δ st + 23 g C (8) quprww δ st + 23 g C (8) qdprww δ st + 43 g C (8) quwwst δ pr + 23 g C (8) udstww δ pr + 43 g C uuswwt δ pr + 43 g C uuwtsw δ pr − (cid:18) y q y u g + 6 (cid:18) N c − N c (cid:19) g (cid:19) C (8) quprst − g C (1) quprst – 40 – C (1) qdprst = 43 g y h y q C Hdst δ pr + 43 g y d y h C (1) Hqpr δ st + 83 g N c y d y q C (1) qqprww δ st + 83 g N c y d y q C (1) qqwwpr δ st + 43 g y d y q C (1) qqpwwr δ st + 43 g y d y q C (1) qqwrpw δ st + 4 g y d y q C (3) qqpwwr δ st + 4 g y d y q C (3) qqwrpw δ st + 83 g y d y l C (1) lqwwpr δ st + 43 g y d y e C qeprww δ st + 43 g N c y d y u C (1) quprww δ st + 43 g N c y d C (1) qdprww δ st + 83 g N c y q C (1) qdwwst δ pr + 83 g y l y q C ldwwst δ pr + 43 g y e y q C edwwst δ pr + 43 g N c y q y u C (1) udwwst δ pr + 43 g N c y d y q C ddstww δ pr + 43 g y d y q C ddswwt δ pr + 43 g y d y q C ddwtsw δ pr + 43 g N c y d y q C ddwwst δ pr − y q y d g C (1) qdprst − (cid:18) N c − N c (cid:19) g C (8) qdprst ˙ C (8) qdprst = 43 g C (1) qqpwwr δ st + 43 g C (1) qqwrpw δ st + 4 g C (3) qqpwwr δ st + 4 g C (3) qqwrpw δ st + 23 g C (8) quprww δ st + 23 g C (8) qdprww δ st + 43 g C (8) qdwwst δ pr + 23 g C (8) udwwst δ pr + 43 g C ddswwt δ pr + 43 g C ddwtsw δ pr − (cid:18) y q y d g + 6 (cid:18) N c − N c (cid:19) g (cid:19) C (8) qdprst − g C (1) qdprst C.8.4 ( LR )( RL )˙ C ledqprst = − (cid:18) y d ( y q − y e ) + y e ( y e + y q )) g + 3 (cid:18) N c − N c (cid:19) g (cid:19) C ledqprst C.8.5 ( LR )( LR )˙ C (1) quqdprst = 4 g ( y q + y u ) C dBst [ Y † u ] pr − g C dWst [ Y † u ] pr − N c g ( y q + y u ) C dBpt [ Y † u ] sr + 12 N c g C dWpt [ Y † u ] sr − N c − N c g C dGpt [ Y † u ] sr + 4 g ( y q + y d ) C uBpr [ Y † d ] st − g C uWpr [ Y † d ] st − N c g ( y q + y d ) C uBsr [ Y † d ] pt + 12 N c g C uWsr [ Y † d ] pt − N c − N c g C uGsr [ Y † d ] pt − (cid:18)(cid:0) y d + 2 y d y u + 3 y u (cid:1) g + 3 g + 12 (cid:18) N c − N c (cid:19) g (cid:19) C (1) quqdprst − N c (cid:18)(cid:0)(cid:0) y d + 10 y d y u + 3 y u (cid:1) g − g (cid:1) + 8 (cid:18) N c − N c (cid:19) g (cid:19) C (1) quqdsrpt − (cid:18) − N c (cid:19) (cid:18)(cid:0) y d + 10 y d y u + 3 y u (cid:1) g − g + 4 (cid:18) N c − N c (cid:19) g (cid:19) C (8) quqdsrpt + 2 (cid:18) − N c (cid:19) g C (8) quqdprst – 41 – C (8) quqdprst = 8 g C dGst [ Y † u ] pr − g ( y q + y u ) C dBpt [ Y † u ] sr + 24 g C dWpt [ Y † u ] sr + 16 N c g C dGpt [ Y † u ] sr + 8 g C uGpr [ Y † d ] st − g ( y q + y d ) C uBsr [ Y † d ] pt + 24 g C uWsr [ Y † d ] pt + 16 N c g C uGsr [ Y † d ] pt + 8 g C (1) quqdprst + (cid:18) − (cid:0) y d + 10 y d y u + 3 y u (cid:1) g + 6 g + 16 1 N c g (cid:19) C (1) quqdsrpt + (cid:18)(cid:18) − y d − y d y u − y u (cid:19) g − g + 2 (cid:18) N c − N c (cid:19) g (cid:19) C (8) quqdprst + 1 N c (cid:18)(cid:0) y d + 10 y d y u + 3 y u (cid:1) g − g + 4 (cid:18) − N c − N c (cid:19) g (cid:19) C (8) quqdsrpt ˙ C (1) lequprst = − (cid:18) (cid:0) y e + y e ( y u − y q ) + y q y u (cid:1) g + 3 (cid:18) N c − N c (cid:19) g (cid:19) C (1) lequprst − (cid:0)
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Aµν G Aµν Q H e G H † H e G Aµν G Aµν Q HW H † H W
Iµν W Iµν Q H f W H † H f W Iµν W Iµν Q HB H † H B µν B µν Q H e B H † H e B µν B µν Q HW B H † τ I H W
Iµν B µν Q H f W B H † τ I H f W Iµν B µν ψ XH + h.c. Q eW (¯ l p σ µν e r ) τ I HW Iµν Q eB (¯ l p σ µν e r ) HB µν Q uG (¯ q p σ µν T A u r ) e H G
Aµν Q uW (¯ q p σ µν u r ) τ I e H W
Iµν Q uB (¯ q p σ µν u r ) e H B µν Q dG (¯ q p σ µν T A d r ) H G
Aµν Q dW (¯ q p σ µν d r ) τ I H W
Iµν Q dB (¯ q p σ µν d r ) H B µν ψ H DQ (1) Hl ( H † i ←→ D µ H )(¯ l p γ µ l r ) Q (3) Hl ( H † i ←→ D Iµ H )(¯ l p τ I γ µ l r ) Q He ( H † i ←→ D µ H )(¯ e p γ µ e r ) Q (1) Hq ( H † i ←→ D µ H )(¯ q p γ µ q r ) Q (3) Hq ( H † i ←→ D Iµ H )(¯ q p τ I γ µ q r ) Q Hu ( H † i ←→ D µ H )(¯ u p γ µ u r ) Q Hd ( H † i ←→ D µ H )( ¯ d p γ µ d r ) Q Hud + h.c. i ( e H † D µ H )(¯ u p γ µ d r )8 : ( ¯ LL )( ¯ LL ) Q ll (¯ l p γ µ l r )(¯ l s γ µ l t ) Q (1) qq (¯ q p γ µ q r )(¯ q s γ µ q t ) Q (3) qq (¯ q p γ µ τ I q r )(¯ q s γ µ τ I q t ) Q (1) lq (¯ l p γ µ l r )(¯ q s γ µ q t ) Q (3) lq (¯ l p γ µ τ I l r )(¯ q s γ µ τ I q t ) 8 : ( ¯ RR )( ¯ RR ) Q ee (¯ e p γ µ e r )(¯ e s γ µ e t ) Q uu (¯ u p γ µ u r )(¯ u s γ µ u t ) Q dd ( ¯ d p γ µ d r )( ¯ d s γ µ d t ) Q eu (¯ e p γ µ e r )(¯ u s γ µ u t ) Q ed (¯ e p γ µ e r )( ¯ d s γ µ d t ) Q (1) ud (¯ u p γ µ u r )( ¯ d s γ µ d t ) Q (8) ud (¯ u p γ µ T A u r )( ¯ d s γ µ T A d t ) 8 : ( ¯ LL )( ¯ RR ) Q le (¯ l p γ µ l r )(¯ e s γ µ e t ) Q lu (¯ l p γ µ l r )(¯ u s γ µ u t ) Q ld (¯ l p γ µ l r )( ¯ d s γ µ d t ) Q qe (¯ q p γ µ q r )(¯ e s γ µ e t ) Q (1) qu (¯ q p γ µ q r )(¯ u s γ µ u t ) Q (8) qu (¯ q p γ µ T A q r )(¯ u s γ µ T A u t ) Q (1) qd (¯ q p γ µ q r )( ¯ d s γ µ d t ) Q (8) qd (¯ q p γ µ T A q r )( ¯ d s γ µ T A d t )8 : ( ¯ LR )( ¯ RL ) + h.c. Q ledq (¯ l jp e r )( ¯ d s q tj ) 8 : ( ¯ LR )( ¯ LR ) + h.c. Q (1) quqd (¯ q jp u r ) ǫ jk (¯ q ks d t ) Q (8) quqd (¯ q jp T A u r ) ǫ jk (¯ q ks T A d t ) Q (1) lequ (¯ l jp e r ) ǫ jk (¯ q ks u t ) Q (3) lequ (¯ l jp σ µν e r ) ǫ jk (¯ q ks σ µν u t ) Table 1 . The 59 independent dimension-six operators built from Standard Model fields which conservebaryon number, as given in Ref. [9]. The operators are divided into eight classes: X , H , etc.Operators with + h.c. in the table heading also have hermitian conjugates, as does the ψ H D operator Q Hud . The subscripts p, r, s, t are flavor indices. – 49 – lass N op CP -even CP -odd n g n g n g n g n g n g n g (9 n g + 7) 8 51 n g (9 n g −
7) 1 308 : ( LL )( LL ) 5 n g (7 n g + 13) 5 171 n g ( n g − n g + 1) 0 1268 : ( RR )( RR ) 7 n g (21 n g + 2 n g + 31 n g + 2) 7 255 n g (21 n g + 2)( n g − n g + 1) 0 1958 : ( LL )( RR ) 8 4 n g ( n g + 1) 8 360 4 n g ( n g − n g + 1) 0 2888 : ( LR )( RL ) 1 n g n g LR )( LR ) 4 4 n g n g All n g (107 n g + 2 n g + 89 n g + 2) 25 1191 n g (107 n g + 2 n g − n g −
2) 5 1014
Total (107 n g + 2 n g + 213 n g + 30 n g + 72) 53 1350 (107 n g + 2 n g + 57 n g − n g + 48) 23 1149 Table 2 . Number of CP -even and CP -odd coefficients in L (6) for n g flavors. The total number ofcoefficients is (107 n g + 2 n g + 135 n g + 60) / , which is 76 for n g = 1 and 2499 for n g = 3 . H H D yψ H ψ H D ψ g X H gyψ XH g X Class
NDA Weight − H λ, y , g λ , λg , g λy , y λy , λg , /// y λg , g //// λg H D λ, y , g /// y y , g /// g ///// y g /// g yψ H λ, y , g λ, y , g λ, y , g λ, y g //// g λ, g , g y /// g ψ H D g , y /// y g , // λ, y g , y /// g ///// g y /// g ψ g , y g , y g y /// g g X H / / λ, y , g y g gyψ XH / / g g , y g g X / g Table 3 . Form of the one-loop anomalous dimension matrix b γ ij for dimension-six operators b Q i rescaled according to naive dimensional analysis. The operators are ordered by NDA weight, ratherthan by operator class. The possible entries allowed by the one-loop Feynman graphs are shown. Thecross-hatched entries vanish.rescaled according to naive dimensional analysis. The operators are ordered by NDA weight, ratherthan by operator class. The possible entries allowed by the one-loop Feynman graphs are shown. Thecross-hatched entries vanish.