JHEP10(2019)197 Springer October 9, 2019 August 21, 2019 October 18, 2019 : : : Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP10(2019)197 -odd sectors. The matching equations express the CP [email protected] , -even and . 3 CP 1908.05295 The Authors. c Effective Field Theories, Renormalization Group

We compute the one-loop matching between the Standard Model Effective , [email protected] -violating theta angles. Our results provide an ingredient for a model-independent La Jolla, CA 92093-0319, U.S.A. E-mail: Department of Physics, University of California at San Diego, 9500 Gilman Drive, Open Access Article funded by SCOAP next-to-leading-log analysis. Keywords: ArXiv ePrint: operators in theEffective low-energy Field theory Theory. in UsingCP terms momentum of insertion, we theanalysis also parameters of obtain constraints of on the physics the matchingorder beyond for Standard the calculations Standard the Model at Model. one-loop They can accuracy be and used for represent fixed- a first step towards a systematic the heavy gauge bosons,complete the set Higgs of particle,the matching and power equations the counting is top ofand quark derived both include are theories. including both integrated effects the We out.masses, present up gauge the couplings, to The results as dimension for well as general six the flavor in coefficients structures of dipole, three-gluon, and four-fermion Abstract: Field Theory and the low-energy effective field theory below the electroweak scale, where Wouter Dekens and Peter Stoffer Low-energy effective field theoryelectroweak below scale: the matching at one loop JHEP10(2019)197 34 11 25 6 12 22 – i – 28 17 30 22 14 28 3 32 14 32 29 36 29 15 19 19 19 21 12 4 3 1 9 17 37 38 5.4.1 Gauge-boson5.4.2 mixing Gauge5.4.3 couplings Theta parameters 5.3.1 Field5.3.2 redefinitions and mass matrices Fermion wave-function renormalization and mixing 4.1.1 Levi-Civita4.1.2 tensor Closed fermion loops 5.6 Gluon three-point5.7 function Four-fermion interactions 5.4 Gauge-boson two-point functions 5.5 Fermion three-point functions 5.1 Expanding loops5.2 Power counting 5.3 Fermion two-point functions 4.2 Evanescent operators 4.3 Tadpoles 4.4 Diagrams and4.5 required vertices Computational tools 4.1 Dimensional regularization and scheme definition 2.1 Scalar sector 2.2 Gauge sector 2.3 Background-field method,2.4 gauge fixing, and Fermions ghosts A Conventions B Neutrinos 6 Results 7 Summary and conclusions 5 Matching procedure 4 Loop calculation 3 LEFT Contents 1 Introduction 2 SMEFT in the broken phase JHEP10(2019)197 52 52 53 45 45 46 46 47 47 48 48 48 49 49 50 50 51 51 51 52 41 41 41 41 42 42 42 43 43 43 44 44 44 45 48 – ii – ¯ ddh 39 , and 52 41 Zh 39 uuh , ¯ 40 h , and 2 + − ν + − γh , ¯ ¯ ¯ , ee uu dd Z 40 2 2 2 2 3 2 2 ¯ ¯ ¯ ¯ ¯ eνW udW duW νννν νν νν νν uug ddg g eeh eeZ ν uuZ ddZ νeW uu dd γ g eeγ ννγ uuγ ddγ γ γZ ZZ hh WW g ee ν 40 D.6.1 D.6.2 D.6.3 D.6.4 D.5.5 D.5.6 D.5.7 ¯ D.5.8 ¯ D.5.9 D.5.1 ¯ D.5.2 ¯ D.5.3 D.5.4 ¯ D.4.2 D.4.3 ¯ D.4.4 D.4.5 ¯ D.4.6 D.4.7 D.3.1 D.3.2 D.4.1 ¯ D.2.6 D.2.7 ¯ D.2.8 D.2.9 ¯ D.2.10 D.2.1 D.2.2 D.2.3 D.2.4 D.2.5 D.6 Four-point functions D.5 Three-point functions with heavy bosons D.3 Two-point functions with momentum insertion D.4 Three-point functions with massless bosons C.1 Background fields C.2 Quantum fields D.1 Tadpoles D.2 Two-point functions D Diagrams C Propagator Feynman rules JHEP10(2019)197 ]. Z 53 54 54 55 55 56 57 9 and W ] for a detailed 8 ], see [ 7 – 2 . Working with these EFTs em U(1) 61 ⊗ c ]. Under the assumption that the Higgs boson 1 – 1 – 57 58 58 = 1) = 1) = 1) L L L ∆ ∆ 59  − 60 = = ∆ = B B B (∆ (∆ (∆ ¯ ¯ ¯ ¯ dd uu ¯ dd du ¯ ¯ ee uu dd ¯ uu dd νe νudd eduu eddd ee ee ee uu D.6.11 D.6.12 D.6.13 D.6.14 ¯ D.6.6 ¯ D.6.7 ¯ D.6.8 ¯ D.6.9 ¯ D.6.10 D.6.5 ¯ It is givenare by invariant the under QED thehas and unbroken the subgroup QCD advantage SU(3) Lagrangian thatthrough scales and are renormalization-group all separated, equations higher-dimension which (RGEs),resum operators both provides large simplifies that the logarithms loop means in calculations to and, a systematically leading-log expansion known as RG-improved perturbation dimension operators that are invariant under thereview. SM For gauge processes group at [ energiesin below which the the electroweak scale, heavy another SMgauge EFT particles should bosons, are be integrated as used, out: wellno these as large are the logarithms the Higgs top are scalar. quark, generatedresulting the low-energy by Since EFT performing (LEFT) these the corresponds particles to matching the have at Fermi comparable one theory of masses, common weak scale. interactions [ The weakly coupled light NP, unless oneenergy is of performing the a new dedicated lightand experiment degree at the of the Goldstone freedom resonance bosons [ complex of scalar spontaneous electroweak doublet symmetry ofModel breaking the Effective transform Field weak as Theory gauge a (SMEFT), group, consisting the of the appropriate SM EFT Lagrangian is and the all higher- Standard at the LHC indicatesheavy that particles New with Physics masses (NP)effects well is above of either the NP very electroweak on weaklycan scale. coupled experiments be In or at described the consists energies by second of This below an scenario, effective the description the effective of threshold field heavy of NP theory remains the (EFT) valid new even that in heavy contains the particles presence the of SM additional very particles only. 1 Introduction The fact that so far no particles beyond the Standard Model (SM) have been observed F LEFT operator basis G Conventions for the supplemental material E SMEFT operator basis JHEP10(2019)197 ] – – ]. B 30 10 16 35 – – 25 33 ], completing 15 , we describe the 5 -meson mixing and decay B , we recall the properties of the 2 ]. ] or support an automated calculation 19 22 [ – eγ 20 ]. , we discuss the technical details of the loop → 4 – 2 – 24 , µ ]. One-loop matching at the weak scale has been , we highlight some aspects of the results of the 23 36 6 , we describe the basis choice for the LEFT that we 3 ] calls for an analysis beyond leading logarithms [ 32 , 31 ], the complete LEFT operator basis up to dimension six was 14 ]. In [ 13 ]. 38 , 37 The structure of the paper is as follows. In section The absence of evidence for NP on the one hand as well as the increasing precision EFT analyses of constraints on NP is a rapidly developing field. For example, the ]. Interestingly, the non-perturbative nature of QCD at low energies has an impact even ]. The low-energy operator basis and RGEs relevant for basis, and gauge fixing.use In for section the one-loopcalculation matching. including regularization In and section matching scheme procedure definitions. and In derivecalculation the section generic of matching observables. equations. In We also section comment on the SMEFT have recently been presentedpreviously [ considered in theset context of of operators. specific processes, Thephysics most [ i.e. complete restricted partial to results a so limited far sub- wereSMEFT given in in the the broken context phase, of mainly to set the conventions for the parameters, choice of which is the natural nextat step fixed in one-loop this orderrequired. direction. with EFT The They methods results also for enable provideaccuracy, cases simplified one where which calculations ingredient no in however resumming an willsults of extension for logarithms require to the is the next-to-leading-logarithmic two- and two-loop three-loop anomalous anomalous dimensions dimensions of as the well. three-gluon operator in Re- the or electric dipole moments [ The motivation to go to higherthat loop is orders is already not constrained to to obtainstructure, be a tiny, as small but correction well to to a obtain aspresent coefficient new the to constraints calculation due assess to of the a the richer uncertainties matching mixing in equations at the the theoretical weak calculation. scale to Here, one-loop we accuracy, calculations of bounds on Wilsonof coefficients processes [ at the tree and loop level [ in experiments on the othertions. hand The require constraining an power improved e.g. accuracy of in the searches for theoretical calcula- lepton-flavor-violating processes [ 18 on processes without any externallepton-flavor-violating NP hadrons, in which is the of process importance e.g. when constraining leading-log calculations have been implemented into software tools that provide automated on, NP. These RGEsdown to allow the one hadronic tocalculated scale, where connect by QCD combining energy becomes the scalesof non-perturbative. running the ranging of Observables effective from are the operators, then Wilson thethe thereby coefficients explicitly problem. scale with separating of At the the the NP perturbative high- matrix calculation hadronic and elements or scale, low-energy a the scales non-perturbative matrix in matching to elements chiral in perturbation general theory [ require a direct non- 12 were derived in [ constructed and the tree-level matchingloop equations to running the in SMEFT the werethe provided. LEFT framework The up for one- a to consistent dimension-six leading-logarithm effects analysis was of the calculated effects in of, [ and constraints theory. The one-loop running in the SMEFT up to dimension six was calculated in [ JHEP10(2019)197 (2.5) (2.2) (2.3) (2.4) (2.1) / D ψ ψ i . 3 ) H  . † X c q,u,d,l,e . H = ( ψ H + h C i l ) + e , H 2 ) + µ µ e Y .  i H 2 D † 2, and the U(1) hypercharge µ yB v µν / )( 1 H 1 2 † D e I B † , ig τ + . H i − µν H i µ + = ( q Q E B i ∗ H I D I µ u ) † 2 t 2 1 C π g W H H + ( 1 u Y I i µ i t θ  32 † X 3 matrices in flavor space. We reproduce 2 D µν λ e † H + ig + H × B − H Q + + ( µν ) i SM H I µν A µ q B – 3 – H L C HD are 3 d 1 4 µ G f W C = A + D I − µν d Y T i )( 3 † u,d,e W † ) + HD Y Iµν ig 2 H 2 2 H H Q ].  π g µ † + SMEFT W 2 θ − D 32 µ L H I 39 HD µν , ( , the SU(2) generators ∂ 2 C + A W   = ( = ) ], in the matching to the LEFT one has to integrate out the 2 14 1 4 T + , v µ H H 12  1 2 † − A µν 12 L – D H e G H − 10 Q ( Aµν µν A   H G G † H H 2 C C 2 3 A µν H π g  G 3 = = θ 32 1 4 λ (6) H . The Yukawa matrices + − − y L , while the appendices provide some further comments on conventions, Majorana = 7 SM We start from the SMEFT Lagrangian L get modified by dimension-six operators in SMEFT: 2.1 Scalar sector The scalar potential and kinetic energy terms with the SU(3) generators generator the SMEFT operators up to dimension six in appendix The covariant derivative is given by where the sum runs over all higher-dimension operators and the SM Lagrangian is the SM, in orderpreviously to discussed specify in our [ conventions. The SMEFT in the broken phase has been While the calculationthe of unbroken the phase divergence [ heavy structure particles, of which the impliesthe that SMEFT following, one can we is discuss becorrections required the performed that to transition are work in from within induced the the by unbroken the broken to phase. presence the of In broken higher-dimension phase operators and compared the to provide them in electronicsection form as supplementalfermions, as material. well as We the offer complete list our of conclusions Feynman diagrams2 in that were calculated. SMEFT in the broken phase calculation. Since the explicit expressions for the matching equations are very long, we JHEP10(2019)197 (2.6) (2.7) (2.8) (2.9) (2.10) (2.14) (2.11) (2.12) (2.13) . Hence, , . , µ ) µ ) . B . B 1  y HB g HB . µν 1 G C ¯ g . C B i = ¯ 2 T 2 T v  v µ µν + ) = 2 T . HWB 1 B I B µ v λ 1 2 T 1 4 Q λ iφ g H v (1 + W 2 (1 + , I µ = 2 C − 1  t I µ ] 3 B g 2 HD 2 . 39 ¯ g HWB [ W C φ i − = = Iµν T λ ]: 2 ( C lead to kinetic terms of the 1 4 v g 1 µ 2 + W 12 ¯ g 2 1 + − kin  √ A = ¯ µ I µν 2 H T H, = G I 2 v c µ HB W A λ , , H X W 1 4 8 Q T kin 0 ,B 2 C 3 ! ) g G, 3 , ¯ G g − 1 3 1 + 2 ] i ) HB ,  − C iφ iφ A µ HW + kin 2 T 1 G , c Aµν HW C µ + + − G, 3  2 T c G ∂ 2 C T g 2 4 λv v v 2 T φ φ A µν = = v = ¯ HW 

– 4 – G = 2 µ Q A µ 2 1 4 (1 + 2 1 HD H G = [1 + I yB µ √ , v (1 + 3 − C 1 HW 2 3 2 T g ) 1 4 W g v = C ig 2 1 H ,M − = = + + µ H 2 , φ 2 I  I µ = µ D h have canonically normalized kinetic terms and the products ¯ g T H i HG 2 v H µ )( W C † I H Q B t † + M  2 H 1 2 -even operators of the class h µ H HG ] ig = h ,W − D C , ) , and + ) kin CP I kin = µ = A µ H, HG = ( H, c HG W G C c 2 , C A 2 T mass even A H 2 µ T , v T 2 h, v G gauge 3 ] uses a different convention for the scalar self-coupling: ] (6) X L L ig L = [1 + 39 1 12 (1 + + 4 ]. + (1 + A µ 3 φ µ G g kin ∂ 39 , H, = = = 12 L 3 A µ The Higgs mass term is given by µ ¯ g Note that [ G 1 D of couplings and fieldsthe is covariant unchanged, derivative takes the same form in terms of the rescaled quantities: We define the following rescaled gauge fields and couplings [ where the fields At dimension six, the gauge fields in the broken phase that are not canonically normalized: 2.2 Gauge sector The Higgs and gauge kinetic terms at dimension four are given by ensure canonically normalized kinetictors [ terms in the presence of the dimension-six opera- The rescaling factors where value (vev) [ The Higgs doublet is given by The minimum of the potential at tree level is given by the modified vacuum-expectation JHEP10(2019)197 (2.24) (2.20) (2.21) (2.17) (2.18) (2.19) (2.22) (2.23) (2.15) (2.16) . e B , . µ . H  Z C 2 2 1 2 µ 2 .... T ¯ g g .... f WB v Z + 2 H − + ¯ µν 2 Z 2 1 π e
2 2 2 1 ) + Q B g µ ¯ ¯ g g M µ 16 µν B 2 1 1 2 B
3 f ¯ ¯ WB g g µ 3 µ B µ + +  H 2 2 B 2 W , 1 1 W .... 3 C µ 1 µ − π g θ 2 ¯ g 1 ! ¯ g + 2 + θ W 1 W µ 32 1 + = 3 µ ] g 1 g e + µ  B 1 ¯ 2¯ g B µν 2¯ W + ¯ θ 2 40 H 2 2 B W g − − g ¯ 2¯ Q θ. 3 2 µν µ W µ , e I µν ! B + ¯ 1 ¯ B − c B W ¯ M g H ¯ 2 1 s f  W 2 µ µ f W µ ¯  g 2 C B H B = B I µν 2 + 1 2 := sin 1 C p µ + + g g ¯ s ¯ HWB 2 s T W B ¯ c v 2 1 = − + ¯ f C W + ¯ 2 2 2 2 g ¯ s π g µ 2 ¯ ¯ c H 3 T s I µ 2 2 2 π – 5 – 2  ¯  v 2 2 θ, + ¯ g Q θ 32 W W 16 , − µ I I µ µ + − f 3 W + ¯ ¯  c s H + = W W W 2 2 1 2 I 2 µ 2 2 C

2 2 ¯ g g := cos ¯ g θ g A µν W (¯ ¯ + c = − + ¯ 2 2 contributes to the gauge-boson mass terms: e = 4 T G 2 e T ¯ HWB g G 2 2 8 2 1 16 v ! 2 v ¯ ¯ g g H ¯ A µν Q µ µ θ 1 2 HD  Q G A Z ¯ ¯ g g HD e Q G ) = 2 , 2 3 

2 C HD π g H e HWB G H 3 C − = C µ C θ H 32 , we perform the transformation [ 2 T 1 2 C D induces an additional kinetic mixing of the weak gauge bosons: = v  = 2 T HD )( v 2 † 2 θ 2 Q 2 HWB g odd H L , 1 + H 2 3 π 2 C HWB µ g + ¯  2 HD 2 (6) X T 16 ¯ g Q 2 1 D v 2 T C L 8 ( ¯ g v + := p -odd sector, the theta terms 3 θ  = CP = ¯ c 3 The gauge-boson mass and mixing terms at dimension four are ¯ θ The mass terms become This diagonalizes both the kineticprovided terms that and the mass matrix of the neutral gauge bosons, where Defining Furthermore, the operator The operator Similarly to the gauge couplings, we define rescaled theta angles get modified in the broken phase by contributions from the operators In the JHEP10(2019)197 , ]. 2 , . ¯ g 45 1 – ¯ g (2.25) (2.27) (2.29) (2.30) (2.26) (2.28) f WB  f WB 41 H 2 T 2 H v C , C 2 T ˆ 2 F T + v , v ,
2 µ
2
2
1 2 ¯ 2 g 2 2 2 . 2 g A µν ¯ 1 g g g e ¯ g  A + ¯ 2 ¯ eQ − + ¯ + ¯ i 2 2 π µν 2 1 2 1 2 ¯ 1 g ¯ g ¯ g ¯ + g A 32 HWB µ 2 2  2 C 2 π ¯ Z g π + 2 T ] ¯ e 1 v HD ¯ 32 g 32 Q 2 2 2 C A 2 g ¯ s ¯ ¯ + θ g 2 T 2 HWB 1 v + ¯ − g + C 2 1 2¯ 3 2 T ¯ g t [ µν v HWB 1 + µν e
Z A C  − 2 ¯ g ¯ i 1 + θ ]. 2 T µν 2 T 2 f 2  v 4 W v ¯ g Z ) ¯ c ˆ 44 ] + + ¯ e 2 F, µν , ¯ 2 s ¯ ¯ e − θ = − π t W + and a classical background field Z 1 43 2 = 3 + Z ¯ ¯ θ g − µ 2 32
2 F 1 1 F 2 2 Z π 2 2 ¯ ¯ g θ 2 – 6 – W ¯ ¯ g g g (
ZA 7→ 32 2 ) 2 ¯
θ + + ¯ 2 ,M 2 2 g 2 2 , ¯ θ F 2 ¯ 2 g 2 ¯ + 1 g + t ¯ g ¯ g + ¯ 4 + − 2 − + T µ 2 1 µν v 2 ¯ 1 g 2 1 e HWB W Z g ¯ g , [ (¯ = A µν C 2 2 µν 2 e 2 G 2 ¯ T g ¯ g ¯ 2 g Z √ W f 1 v WB 1 A i µν 2 g 2 H g ], while internal loop propagators are quantum fields. Splitting ¯ g 4¯ G π 2 4¯ Z ¯ c + C 2 ¯ g 42 1 2 remains massless. 2 T − 32 A 2 3 µ π + ,M v ¯ ) g µ
G Z 2 − ) 2 2 µ 2 32 ¯ θ A 1 ¯ A g π ¯ ¯ s θ 2 3 ¯ θ 1 and the effective couplings are given by 2 4 2 T ¯ W
θ 2 + 1 ¯ g i 16 g 2 3 g 2 2 2 2 ¯ g ¯ g g g = i it ∓ + ¯ = ¯ − − + ¯ + ¯ 1 µ 1 2 ¯ e +  2 (6) θ ¯ ¯ θ 2 1 θ 2 1 ¯ θ 1 µ 4 1 ¯ W g ¯ L g 2 2 t ( ¯ ∂ g + g 2 1 = 1 = 2 4(¯ ¯ θ √  µ t = = = D := Compared to conventional gauge fixing, the background-field method has several ad- The covariant derivative can be rewritten in the mass basis as Z A ¯ ¯ θ θ  µ ZA ¯ θ W the fermion fields into backgroundhence and one quantum fields does does not not need change to the distinguish Feynman rules, them [ vantages: gauge invariance for Green’swhile functions in of the background conventional fields method, is gauge invariance retained is explicitly, broken to BRST invariance, resulting where the quantum fieldsFeynman diagrams, are the the external variables legsto of as the background well integration fields as in [ internal the tree-level propagators functional correspond integral. In 2.3 Background-field method,For gauge the one-loop fixing, matching and calculation, we ghosts willAll employ the the fields background-field method are [ split into a quantum field where the theta angles are given by Including dimension-six effects, the theta terms in the broken phase read where and the photon field where JHEP10(2019)197 ]. 43 and a (2.33) (2.35) (2.36) (2.34) (2.31) (2.32) ˆ h fixes the ,  GF  L -matrix elements can S , . + source terms 0 δJ δW G = GF ˜ F L

7→ . ] for its definition. BG GF ˜ ] F gauge for the quantum fields [ 0 L · , 45 2 ) + ξ ]. The background-field method also x B ˆ ˆ 4 F R F,J [ 41 G d x J ,G ]. Finally, the gauge for the background 4 + A Z d  Z G provides a strong check on the loop cal- 44 F i , G ( log Z ξ ξ AB L i 2 43 ˆ g 7→ −  − ] is still invariant with respect to background- 0] + ] – 7 – − x  4 ˆ F, d = ] = ˆ ˆ F,J F,J [ [ Z GF Z i ˆ = Γ[ F,J W ˆ h , G  L [ + W full ] = exp h Γ ˜ F  7→ A B ˆ F, G h δ δα Γ[ and disregard any subtleties with fermion minus signs in functional derivatives  ]. F 45 det F , while due to unitary background-field gauge, the Goldstone bosons are D coupled to the quantum fields only. The gauge-fixing term is h Z J ] generates disconnected Green’s functions, while the generating functional of ] = accounts for the curved field space due to the higher-dimension SMEFT contri- ˆ g F,J [ ˆ ]. The generating functional is F,J [ Z The gauge-fixing term for the SMEFT in the background-field method has been derived We denote all fields by 45 Z 2 be constructed from for simplicity. generating the one-particle-irreducible (1PI) Green’s functions. The connected Green’s functions is given by The Legendre transform thereof is the effective action where ˆ butions and only involvesgauge the of background the fields, see quantumfields fields, [ gauge while transformations.ghost term The given determinant in [ can be replaced by the Faddeev-Popov with sources in [ independence of theculation. gauge-fixing parameter As anquantum example, field the Higgspure field quantum is fields: split into a background Higgs field malize Green’s functions involving quantumresults fields in [ a simplification offields Ward identities can [ be fixedunitary independently gauge of for the theOn gauge background the fields fixing one and for hand, linear the this quantum reduces fields. the number We choose of possible diagrams, on the other hand, the in gauge-variant (but BRST invariant) counterterms. Note that it is not necessary to renor- JHEP10(2019)197 ⊗ c (2.41) (2.42) (2.37) (2.38) (2.39) (2.40) , ) , − ν ) , ) W ν HB . ∂ ) C HB . 2 2 )( T ¯ C , gauge. η , v i + µ C 2 T ξ ! v . ˆ F are given by  full W 3 R B δ Γ 2 Z µ η 1 δ η (1 + η ∂ = GF (¯ ( B ˆ (1 + J ξM ¯ 2 η L ξ 1

B ! 1 ˆ ¯ c η − √ F = − ¯ s  2 · 0  2 2 = = = ˆ ] in order to obtain the more conventional form ) G J + µ B B +  45 x ¯ s ¯ ¯ u η Z ¯ 4 c − µ d ∂ ¯ , ,M ¯ c s , ( – 8 –  Z  2 2 ) ) , u ξ 2 W 1 2 ) 2 − + − iη HW ¯ ¯ c s − ξM C full HW ∓

generates interaction vertices with two quantum fields 2 2 T ) = C 1 v µ = η 2 T  ( GF v A 2 ) generates the following masses for the Goldstone bosons: G ] = Γ 2 ! L µ ˆ 1 J Z A ∂ √ [ M (1 + ), we define rescaled ghost fields to account for the dimension- ( η η i . The kinetic terms from ¯ 2.33 (1 + η propagators for the quantum gauge bosons. The gauge-fixing ξ W i

1 = 2 η − kin ξ 2.13  − R H, η = = L i i = W W ¯ u u kin GF L 3 has to be fixed as well — here again, we are using ]. It inherits the curved field-space metric of the gauge sector. Hence, in analogy to em Beyond the bilinear terms, The Faddeev-Popov ghost term for the SMEFT with background-field method is given The gauge-fixing term ( For the extraction of the vertex Feynman rules, we note that terms linear in the We change the sign of the anti-ghost compared to [ 45 3 and similarly for the anti-ghosts. The charged ghosts are defined as of the ghost propagator. and we rotate the neutral ghost fields in [ the gauge-field rescaling ( six effects, term does not contributehave to propagators the as kinetic in terms standard of unitary the gauge, background see weak appendix gaugeand fields, up which to four background fields. the mixing term from which lead to standard as well. It also generates a mixing between the Goldstone and gauge bosons that exactly cancels gauge-fixing for the quantum fields.background As fields mentioned of above, We the chooseU(1) weak unitary sector. gauge for The the gauge for the unbrokenquantum fields background do SU(3) not contributemore to than the two 1PI quantum Green’s fields functions. only Furthermore, contribute vertices beyond with one loop, hence we disregard them by considering all tree-level diagrams, wherefield the propagators. 1PI vertices are The connected gauge-fixing by background- term for the background fields is independent of the and its Legendre transform, the generating functional for connected Green’s functions JHEP10(2019)197 ]: . 41 A (2.50) (2.43) (2.44) (2.46) (2.47) (2.48) (2.49) (2.45) G ¯ η − . = : background- B G † 3 η A G i H ¯ em η ) 2 ψ Dµ . , G U(1) ∓ Cµ ¯ η . + ⊗ i G c . c − . A c B µ . η Dµ ˆ , = G ˆ  G + h . Z ( †  A + h η 2 ¯ ¯ η C η µ ) i rs Z ABC l dH ˆ . G A ¯ µ η e f − ˆ 2 Q Z G Z , 3 2 η ¯ Z eY ¯ µ g i rs η dH ∂ † XDB M ξM ( − Z,A C ) it is identical to the SM case [  f ¯ g H η ˆ − Z A µ = ξ 1 + ¯ − η 2 ) G + Z 2.13 ) µ i 2 − η ACX − = rs q − uH η ∂ f u Cµ 2 − − 2 3 Q ) = η η G g † Z,A ¯ µ uY rs ¯ i  η ˆ A uH ,M + ¯ + † + ¯ A – 9 – − C µ ˜ 2 µ ¯ + W η H ∂ , Cµ η ∂ + ( ˆ − + G + A C ,G G µ γ ξM ( i ˆ η ˆ η ξ 2 1 G rs + eH (¯ q ) 2 η d = = Q A W 2 −  η ACB † ¯ G W dY ACB rs + eH 2 i ( f η A G = M ¯ f † η µ g C 3 ξ M 1 ∂ − ¯ g H − ←− 2 = h BG GF 3 , η = − g − 3 L ]). − ∓ H + ¯ η = 2 FP 39 = (6) ψ L = L L AB δ †  QCD GF  L h gauge [ A G , η ξ ¯ η R − Z,A η = = ]: QCD FP 46 † Z,A Unitary gauge for the background fields does not fix the SU(3) The gauge fixing in the QCD sector is independent of the electroweak gauge fixing. In L η are modified in the SMEFT by the dimension-six operators of the class Since the background fields only enter at tree level,2.4 background ghost terms are Fermions irrelevant. The fermion mass matrices and Yukawa couplings field gauge. We use a background-field gauge-fixing Lagrangian The following Hermiticity propertiestian of [ the ghosts ensure that the Lagrangian is Hermi- The ghost Lagrangian reads terms of the canonically normalized gluon fields ( In addition to the bilinearleads terms, the to Faddeev-Popov interaction term in verticesconventional the involving background-field up method to six fields (in contrast to the ghost term in where This leads to canonically normalized kinetic and mass terms for the ghosts: JHEP10(2019)197 ]. 47 (2.57) (2.54) (2.56) (2.51) (2.52) (2.53) . sr ∗ ψH u, d, e . . , where C . † = c 2 . 2 T U v ) . √ 3 , 2 T ν − † v , ψ ) + h 5 V rs , m rs ) Ls ]  2 C b ν ψ − sr ..., ∗ Cν ψH .... M , m ..., , m [ s C = T + 1 Lr + ν . + ν 2 T 2 ] (2.55) kin ( rs v , m Rs ] m Rs Ls d Rs H, ν u rs kin T d ν ] e c − v m 5 † M Lr H, rs Lr [ Lr rs c ¯ ¯ Y u d ] U diag( ¯ [ e 2 1 + 2 ∗ ψ 2 / h / 3 T 3 T / = 3 T Y 3 3 U v ., v 3 [ = v 2 2 [1 + c = 2 .  0 Lr = diag( =  rs L 2 ] d ν T 5 ) = ) = sr v u, d, e , + h ) = √ ∗ ψH C ˜ , ν
[ H H C H – 10 – = s s = s T 2 T Ls Ls u d e v v r r d r rs l q q ¯ ] 3 2 ) are † Cν rs ψ := )( )(¯ )(¯ V − T Lr . , ψ M H H H rs ν c 2.54 [ = ] . † † † rs ] 5 ,M ,M H H H ψ Y rs ) 0 Lr ) [ ] Y + h τ t ., d ν ). Although in the SM these matrices correspond to the usual ][ c = ( = ( = ( . Ls M ) and ( , m , m B [ kin ψ c µ rs rs rs eH dH uH 1 2 + h H, Rr c Q Q 2.52 Q , m − , m ¯ ψ Ls u e h ψ = m m rs [1 + ] Rr L , while the neutrino mass matrix is diagonalized by another unitary trans- ¯ ψ  ψ Y V 2 [ rs 1 ] √ − (see appendix ψ = diag( = diag( = = M e u U [ Y M rs − M ] L ψ = Y We choose a basis, where the weak eigenstates coincide with mass eigenstates for the The dimension-five Weinberg operator generates a neutrino mass in the broken phase: [ M L The mass eigenstates are related to the weak eigenstates by For notational simplicity, we will mostly drop the prime on the mass eigenstates. CKM and PMNS matrices,The respectively, mass they matrices receive in corrections ( within the SMEFT [ up-type quarks, charged leptons,states and and mass right-handed eigenstates down-typetransformation of quarks. the left-handed The down-typeformation weak quarks eigen- are related by a unitary The Higgs boson couples to the neutrinos via is proportional to thedimension Majorana-neutrino six mass in matrix the when SMEFT. keeping only operators up to The Yukawa interaction in the broken phase is modified to Therefore, in the spontaneously broken SMEFT, the fermion mass terms are given by The bilinear terms are JHEP10(2019)197 (3.1) (3.2) µν (2.58) e F . µν F F 2 gauge theory 2 π e ..., em 32 ]. We note that in ..., ) + U(1) QED ) 48 ) . Rs θ s s , ⊗ ..., d ) +  ψ c ) ψ s + µ d µ µ ( i γ ψ γ γ ) + µ ) (with the up-type quark O − s + Rr ) γ Aµν t t d ψ + h.c. 3 u r e r ( i t G (¯ µ ¯ ¯ ψ r ψ 2.56 L γ + ( µ Ls ( ¯ A µν ψ r − ψ ( µ + ¯ i µ G ψ W µ ( X rs 2 W ] W µ Z 5 2 π rs ψ g Hud ≥ Z rs rs (3) rs Hψ d (3) X Hψ 32 (3) Hψ M C [ rs C C (1) Hψ 2 . C + 2 2 T 2 2 2 T T Rr C 2 T √ v v v QCD v √ / √ 2 ψ 2 (3) L T 2 θ 4.2 2 Z 2 2 v L ¯ ¯ g g ¯ ¯ g g . These contributions are given by ) and the Majorana-neutrino mass matrix are + Z + g − − u,d,e − − D – 11 – X = 2 µν 3.2 ψ F H ) = ¯ ) = ) =  2 s s in ( µν ψ Rs ψ ψ − F † ψ d µ µ 1 4 µ γ γ M QCD+QED r I γ / ]. We reproduce the operator basis in appendix Dψ − ¯ τ ψ L r Rr 14 ψi )( ¯ u ψ = and L Aµν H )(¯ )( . The extraction of the Feynman rules and the calculation of ψ µ G H H B M µ I µ D A µν ←→ LEFT X u,d,e,ν i G † L = D iD ←→ 1 4 † ψ i H † ˜ ( H − + ( H ( rs (1) Hψ = rs C rs Hud (3) Hψ C C QCD+QED The mass matrices After the rotation to the mass-eigenstate basis, we introduce Majorana spinors for the The interactions of the fermions with the weak gauge bosons get modified by the L general complex matricesthe in SMEFT, flavor they space. aremass set When matrix equal restricted performing to to theperform the the tree-level a first mass matching basis two matrices generations). to change in between Through ( weak field eigenstates redefinitions, and one mass can eigenstates. In the SM, the The additional operators arewell as the operators Majorana-neutrino at dimension massdimension five six terms and was above. at derived The in dimension complete [ three, list of as LEFT operators up to where the QCD and QED Lagrangian is given by Below the weak scale, we areand working contains with an the EFT SM thatLEFT fermions, is Lagrangian an apart SU(3) consists from of the QCDeffective top and operators: quark, QED as at dimension matter four, fields. plus Therefore, a tower the of additional loop calculations with Majorana fermions,a the subtlety appearance that of we evanescent will operators comment leads on to in section 3 LEFT where the ellipses denote terms involving scalar fields. neutrinos, see appendix diagrams involving Majorana neutrinos are performed according to [ dimension-six operators of the class JHEP10(2019)197 5 γ ]. In (4.1) ]. In 55 39 , 14 problem in [ dimensions, 5 boson is non- γ D (3) Hq C in W with gamma ma- 5 5 γ γ ]. In this case, chiral ]. The 53 49 , 52 6= 4 it implies ]. D ] and one has to perform another 50 , 15 ] = 0 5 γ σ γ 4. However, in many cases it is possible to λ γ – 12 – ν → γ dimensions. As usual for chiral gauge theories, one µ D γ  ]. 2 51 Tr[ − -odd traces, because for 5 = 4 . As is well known, the existence of the chiral anomaly implies γ D µνλσ  boson is integrated out, hence in the basis change from weak eigenstates W ]. This can be achieved most elegantly by algebraic renormalization [ dimensions, as in the ’t Hooft-Veltman (HV) scheme [  54 , 53 Alternatively, one can give up the anticommutation properties of The simplest and most popular scheme is to use an anticommuting At one loop, the corrections to the propagators again generate general complex mass olate the Ward or Slavnov-Taylor identities.again These by symmetry-restoring spurious local anomalies counterterms, havesion to order by be [ order removed in thesome perturbative cases, expan- on-shell renormalizationrestoring conditions counterterms, automatically take but care the of problem the is symmetry- especially severe in connection with minimal avoid the calculation ofanomaly divergent can integrals involving be these obtainedthe traces. by calculation of imposing E.g. a gauge the finite invariance ABJ integral beforehand, triangle [ requiringtrices then in only invariance is explicitly broken, which leads to the appearance of spurious anomalies that vi- called naive dimensionaldivergent regularization diagrams (NDR). involving This scheme does not work for certain at odds with the required limit for or the Levi-Civita tensor that there is nosome symmetry-preserving point regulator. in Therefore, the anythe calculation. scheme context faces of For problems a SMEFT at detailed has recently review, been we discussed refer in to [ [ 4.1 Dimensional regularizationIn and order scheme to definition treatregularization both and UV work in andimmediately IR faces divergences the in problem the how to loop handle diagrams, intrinsically we four-dimensional use objects like dimensional simply is a conventionate — off-diagonal when contributions running to todiagonalization. the the mass scale matrices of [ an experiment, the RGEs4 gener- Loop calculation matrices, which have to bewe rediagonalized choose in the a mass LEFTbutions eigenstate basis. from basis In the where our one-loopmatrices the calculation, matching. is mass always matrices An possiblecorrections contain additional and for complex rotation reshuffles three- that off-diagonal the and diagonalizes contri- off-diagonal four-point the contributions functions. mass into The external-leg choice of basis at the matching scale SMEFT the quark-mixing matrixunitary appearing due in to the an interactionthe additional with LEFT, dimension-six the the contribution proportionalto mass to eigenstates the unitary rotationsinto that the diagonalize Wilson the mass coefficients matrices of are the reabsorbed higher-dimension operators. unitary rotation of the left-handed down-type quarks is the CKM matrix, while in the JHEP10(2019)197 MS ] for (4.6) (4.3) (4.4) (4.5) (4.2) 56 4, hence − D . This allows ). The mentioned , the right-handed 5.1 µ L P µ ] and performs a formal γ R 57 P . ], a modified HV scheme leads  E 49 γ . (1). This implies that products of , ct − ) O , ) C µ } π (  5 ˙ with the gamma matrices, C. ren , γ 2 5 µ C π γ 1 ) + Λ γ d { µ 16 dµ + log(4 ( µ −  1 MS scheme (apart from evanescent operators, 2 ren – 13 –  π ) := = C 4 µ ct − = D C 4 and not contributions of higher order in := 16 µ AC( , which are given in terms of 1 2 ˙ − bare C 6 ). Therefore, the appearance of spurious anomalies in the requires special attention in the calculation of the finite − µ C D 5 = γ  Λ 4, but has matrix elements of ] the anticommutator of − ], see section 49 D matrices. 15 5 , . Vector- and axial-vector vertices are given by the sum and difference γ R ]. With the inclusion of symmetry-restoring counterterms, the HV scheme 12 ), hence the need for symmetry-restoring counterterms would make the HV P – 56 µ 10 γ 4.2 L P ) are still matrices of rank For our calculation, we will use the As observed in [ µ derived in [ hence the counterterms are directly given by The divergences of loopthe difference diagrams of thatfor are the nontrivial expanded SMEFT cross-checks in and with the LEFT the light UV SMEFT scales and divergences, LEFT have see renormalization-group to section equations match Although the problem with pieces of the diagrams, thein divergences the can be SMEFT directlycounterterms and compared LEFT. to the The UV bare counterterms parameters are split into renormalized ones and scheme cumbersome. Therefore, wefew use extra the prescriptions NDR andrenormalization exceptions scheme prescription as whenever amounts explained possible, to in with the the only subtraction following a of subsections. divergent The terms of the form vertices of left- and right-handedgamma vertices. matrix In reduce the the HV vertex scheme, to projectors the four-dimensional on one. bothsee section sides of a single in the HV scheme, they cannotto be the discarded. same results As as shown theexpansion in NDR to [ scheme linear if order one in uses AC( HV chiral scheme vertices [ can be tracedchiral back vertices to are the presence definedfermions of in as higher the the orders Lagrangian, of i.e. vertex the AC( rules left-handed that vertices involve follow from consistently using chiral is a matrix of rank AC( is the only schemeit known remains to be cumbersome fully tomany consistent classes use to of in all diagrams practicalidentical orders the results applications. in HV as scheme perturbation NDR, with in However, theory,even particular symmetry-restoring but as number in counterterms of shown the leads case in to of [ open fermion lines and traces with an subtraction [ JHEP10(2019)197 - CP -odd dimen- CP D , which is 0 σ p λ p µνλσ  -odd operator. CP . 6 -odd triple-gauge operators is part of the 4 dimensions. Since after performing the − CP D – 14 – dimensions until only contractions with external is an intrinsically four-dimensional object, although D µνλσ  -odd dimension-six operators involving the dual field-strength CP ). In the HV scheme with chiral vertices, only the gamma matrix 5 γ -odd traces, we make use of the HV scheme. In the matching for 5 γ ]. In the one-loop matching calculation, the Levi-Civita tensor appears ex- 60 – 58 ) terms appear. In practice, the same result is obtained by using the NDR scheme: We also note that the exact form of the µ to the presence of the three-gluon operators, topproblematic loops traces appear appear in in theseon bulb diagrams, terms however and that the are triangle matching trilinear equations diagrams. inTherefore, only the the depend external In HV momenta, principle, scheme which only leads contain in UV this finite case integrals. to the same result as working directly in four no problem occurs:gauge there bosons, are the twomomentum, remaining open hence Lorentz Lorentz they indices indicesthe cannot are for NDR be all the scheme. contracted antisymmetrized polarizationodd with and of In dipole the no the bosonic same operators, ill-defined two transverse external two-point trace Lorentz to functions appears invariance both with only momenta. in momentum allows No insertion the symmetry-breaking terms structure into can appear even though, due 4.1.2 Closed fermion loops The second potentially criticalappear in case the matching involves calculation in diagrams several places. with In closed regular bosonic fermion two-point functions, loops. They Ward identities for the vertex functions with insertion ofscheme the definition: if theSchouten Lagrangian identity, is evanescent transformed operators byat are four-dimensional one introduced relations loop, that like see can the the give discussion finite at the contributions end of section AC( at first we keeping the all necessary Levi-Civita tensor tensormomenta, reductions uncontracted, polarization in perform vectors, the andscheme, loop we the then integrals Levi-Civita consider includ- these tensoralgebra contractions remain. in in four four Similarly dimensions dimensions. and to In perform the order the to HV remaining ensure the correctness of our results, we verify the tached to a triple-gauge-bosonSM or part Higgs-gauge contributes vertex. atgamma In the matrices the considered (plus fermionic order, vertices, i.e.from only the the the fermion propagator line hastensor contains loop a at integrals, component most its in index three can be contracted only with four-dimensional quantities, no needed, hence we dovertices not appear encounter contractions in of two-pointmatching two functions Levi-Civita for with tensors. the momentum insertion, theta The quantity, present required terms. on to both Here, extract sides the tensors the of the further Levi-Civita matching appear symbol equation. in can Vertices fermionic with be three-point dual kept field-strength functions, as where an the external external boson is at- The totally antisymmetric tensor there are schemessions that [ promote contractionsplicitly in of vertices due two to tensors. Levi-Civita tensors At to the order we are considering, only single insertions of these operators are 4.1.1 Levi-Civita tensor JHEP10(2019)197 µν σ (4.9) (4.7) (4.8) , , are kept , L (4) LL L , P , , E P ev L f µ µν (3) (3) + LR LL P γ σ , µ E E R L L γ ]. We define the P P P . Therefore, the ... R + + , P , µν ⊗ ⊗ L R ]. This definition of 64 σ , P ev , 5.2 P L R dimensions generates ⊗ (4) L LR 63 µ a µ P P L P γ γ i E 63 µ D (2) LR , P } µν L R γ ⊗ E µ + R σ P P L 61 γ L L P L R + R ⊗ ⊗ P P ,P P P µν R L L R µν σ ,P − P ⊗ ⊗ P P P L σ R µ µ L µ . L L L ⊗ P P γ γ γ i 2 P P P P penguin. We evaluate it in the HV L L ) ) R R Γ ) ⊗ ⊗   ⊗  P P P P ) R ) ) R ⊗ ev ev L ev    ⊗ d f eeγ P i 1 e P ,P ev 3 ev ev ) Γ L µν i  a b c R − P a 2 − σ P µ − R γ − i ⊗ 8(2 R X ,P L − – 15 – L ,P = P ,P = (4 = 4(1 + = 4(4 = 4(1 + = 32(2 = 16(1 + 8 R 2 part of the ¯ R P µν L L L R R R A P µ σ P P P P P P γ L ⊗ ν λ σ ν λ σ ⊗ (3) lequ L γ γ γ γ P γ γ 1 P ν µ λ R Q ν µ λ A γ γ γ γ ⊗ γ γ ⊗ µ ν µ ν L ,P R L γ γ γ γ L P R P P µ µ R L P P γ γ ⊗ P P ⊗ µν µ L R ⊗ L γ σ L P ⊗ ⊗ P L L L P P L L ν ν P P P ⊗ ⊗ γ P P { γ ], so that evanescent operators do not contribute to one-loop matrix L L µ λ λ µ P γ γ γ = P γ 63 σ ν ν . Furthermore, the penguin diagrams with heavy external gauge bosons σ L L – } MS for the renormalization of four-fermion operators, but rather use the γ γ γ γ P i 2 P λ µ µ λ 61 Γ (3) lequ γ γ γ γ ν ν Q R R ⊗ γ γ P P i 1 µ µ Γ γ γ { L L P P -dimensional counterterms and the four-dimensional operators are evanescent operators. and write the operators with higher numbers of gamma matrices as linear combinations generic. They can beevanescent assigned operators any value, is defining motivated differentset by schemes of [ the physical following operators derivation in [ the chiral basis and analogous definitions for opposite chirality. The coefficients We use a scheme wherecounterterms the [ contribution ofelements. evanescent operators In is order to compensated definedo by the not local scheme, use the pure evanescent operatorsfollowing definition have to of be evanescent specified. operators for We the NDR scheme: In the case of fermioncounterterms four-point that functions, the correspond loop toeither calculation due elements in to of the appearance the ofor higher operator products due basis of gamma to only matrices Fierz in inD both relations four fermion bilinears that dimensions, only hold in four dimensions. The differences between the the external momentum canonly be problematic diagram set is toscheme the zero, and verify as the explained Wardare identity in generated. in section order to ensure that4.2 no symmetry-breaking terms Evanescent operators three-point functions with aon four-fermion one operator external insertion. momentum, Since problematicstructure the traces in only loop the show depends four-fermion up only structure in vertex. is the The case only of SMEFTare a not Dirac four-fermion critical: tensor operator they with only a contribute to the matching of the four-fermion operators, where dimensions. Finally, closed fermion loops also appear in penguin topologies in the fermion JHEP10(2019)197 ) 4.7 ], e.g. (4.12) (4.13) (4.10) (4.11) 63 four-fermion , L 2) P F µ , ( LL γ . E 1) ⊗ F ( (2) LL LR L ] + E E P L µ j P − γ [ b ) + R L ⊗ , ] for Majorana fermions and P P ) ] = 1) 2 µ L 48 ⊗ F γ P A ( LR L R j 2 E 4( P Γ P [ 1 ) − A ⊗ ) imply ) + ) ev j 1 ] L a L ij L c P 4.7 P [ i [ P would involve contributions with an even a µ Tr[Γ ⊗ ⊗ γ ] ] i (3) (2 + 4 RL R = 1. We abstain from adding further terms . R L , X  -dependence into the definition of these Fierz- P − E – 16 – P P  ij ( ev = c = f − i b R and ] = = P ] = 8( ] = ] = 2( i 2 ) by only making use of the anticommutation relations L L Γ R µν (3) LR ev j 2 P P P e σ E Γ µ 4.7 µ R i 1 µν γ γ = P σ Γ R L j 1 L P P ⊗ [ [ ev P [ d L ⊗ ⊗ P Tr[Γ ⊗ ) ) = ) L L µν L P P σ ev P µ µ L c are then determined by computing the traces γ γ P µν i R R ). The calculation of these traces in NDR is possible since all Lorentz σ = a  P P L ( ( ( P ev O ( b that mix even with odd numbers of gamma matrices. Such terms show up if =  ev up to a i We observe that with the appearance of evanescent operators, a subtlety occurs in Further evanescent operators appear in our calculation: they are given by the differ- Operators with an even higher number of Dirac matrices in both bilinears do not a a fermion flow directionchosen on arbitrarily. However, each it fermion turnsto line. out non-trivial that reversing relations The the between direction fermioncrossing evanescent flow of relations structures, direction and the can such dropping fermion lead that evanescentsults. operators flow the at Consider can naive one a be application loop divergent of loop do not diagram lead with to an correct insertion re- of a connection with Majorana neutrinos.fermion-number-violating interactions The has formalism thedetermination of advantage of [ to relative simplify signsgrams Feynman of since rules, Majorana interfering neutrinos diagrams, the can and be treated to similarly reduce to real the scalar number fields, by of introducing dia- indices and we haveWe not choose included not the to minusevanescent introduce sign operators. any from Note explicit anticommuting that the the definitions fermion ( fields. and analogously for opposite chirality. The parentheses and brackets abbreviate Dirac be related to thefor ones the given in gamma ( matrices. ences of structures that are related by the usual Fierz identities in four dimensions: one uses the above recipethe within operators the corresponding parity basis to number instead of of gamma the matrices, chiral multiplied basis by as in [ appear in our calculation and operators with a different ordering of the Lorentz indices can for indices are contracted.with This derivation leads tolinear evanescent in operators as defined in ( and solving the system The coefficients JHEP10(2019)197 . ]. (4.18) (4.16) (4.17) (4.14) (4.15) are re- 69 -matrix, 1234 i E S ]. However, (3) LL E 68 h , 67 . In this scheme, . ) + 4 are trivially related with the number of u 1243 i L 1243 D.1 L i P 1243 (3) LR µ i (3) γ LR E 3 E (3) LR and external legs u − h E )(¯ I ) − h 2 3 . −h ) u v 4 L n R u V P P L µ and µ , 2) P γ γ n µ . 1 4 − γ u 1 v 3 1234 nV n )(¯ i )(¯ u ( −  2 )(¯ n (3) u n LL ev V 2 X X b L ) MS Lagrangian parameters [ E u 3 h + P = L − v legs. This results in µ – 17 – P L I R γ 2 = n µ 1 P γ = µ u − 1 + 2 : γ I )(¯ u L ν  V E γ ) = 4(4 )(¯ ) 4  λ ) ev + 2 γ u c 4 L ev v E c P )(¯ λ 2 γ (3 + u ν (3 + γ L − ]. µ P − γ , while in general one has to be careful keeping track of crossed and vertices λ 65 3 γ 4(4 ev I u ν c − γ )(¯ 2 µ = = 4(4 u γ MS scheme, as they would introduce gauge dependences into the 1 L u P (¯ = 3 + λ γ − ν ev is the number of vertices with = 1 instead [ γ b in relations between observables and µ n ev 2 4 t T γ V ] for a detailed discussion. We follow the most direct approach and simply calculate c v 1 m 2 H c u 66 (¯ = N M ev where Counting propagators, the numberlated of by internal propagators 4.4 Diagrams and requiredFor vertices any diagram, theinternal propagators Euler formula relates the number of loops see [ all tadpole topologies explicitly,the with top-loop diagrams tadpole shown generates∝ in relatively appendix large corrections duethese to enhanced corrections an have enhancement to factor cancel again in relations between observables [ This implies that the evanescent structures to an expression tensor reduction of the loop integral leads to Dirac structures of the form However, we could also assign a reversed flow direction to the second fermion line, leading operator and a vector-boson correction connecting the two incoming fermion lines. The 4.3 Tadpoles Different treatments of tadpoles canpatible be with found the in the literature, some of which are not com- only for evanescent structures. A schemeb preserving Fierz relations at the one-loop level requires JHEP10(2019)197 , + – 18 – + , + + = 4 external legs. Hence, at one loop the one-point function is E = = For the two-point functions we onlyfollowing need topologies: vertices with three and four legs, resulting in the diagrams with up to given by the tadpole topology: For the matching of the SMEFT to the LEFT up to dimension six, we are interested in For the three-point function verticesfour with different three, topologies: four, and five legs are required, leading to organized in terms ofbosons 1PI (dashed lines), subgraphs. weleg find Distinguishing corrections the between are following fermions not classification shown): (solid (crossed diagrams lines) and and external- The only four-point functions wewe only need include to vertices calculate up are todo dimension four-fermion not six, Green’s need the functions. last any three vertices As topologies with do six not legs appear in and we the matching calculation. The diagrams are best In the case of the four-pointversions, function, there we need are vertices 14 with topologies: up to six legs. Omitting crossed JHEP10(2019)197 , . ]. We ]. Sev- , 75 ] for de- [ 74 , 78 – 73 [ 76 ]. We then used + + ). For the analytic 71 , + 70 Package-X [ FeynCalc . D + + + DiracReduce[] + FeynRules command of + + + , – 19 – + FeynRule[] + + + + and neglected in our calculation as we will explain in ) and Dirac algebra ( 2 ν m TID[] + + + + + routines and the ] to generate the complete set of diagrams. The extraction of the actual -matrix elements for the light-particle processes agree. This can be achieved = = = = S 72 [ . 5.2 Mathematica All diagrams that contribute to the matching equations at the considered order (and 5.1 Expanding loops For the matching procedure,tailed we discussions. follow The standard matching condition EFTand at techniques, SMEFT the see electroweak scale e.g. requires that [ by matching the one-light-particle LEFT irreducible (1LPI) amplitudes including wave-function renor- evaluation of scalar loop functions,compared the we performed results of cross-checks with two independent implementations of the complete computation. 5 Matching procedure FeynArts Feynman rules for thecoded SMEFT in background-field gaugeeral was commands performed of using thee.g. manually same for the package tensor were reduction used ( in different steps of the loop calculation, 4.5 Computational tools The diagrammatic matching of thediagrams. SMEFT and LEFT In involves a order largesystems. to number First, of handle we one-loop this implemented, using complexity,that we encodes dummy fields, the made a type use version of of of fields the and several SMEFT possible computer Lagrangian vertices algebra in tors, hence it issection proportional to some diagrams that turn out to vanish) are listed in appendix The fourth (bulb) diagram in the four-point function involves two dimension-five opera- JHEP10(2019)197 , 4 (5.4) (5.5) (5.1) (5.2) (5.3) . . . ) ) . LEFT loop SMEFT loop M LEFT loop, exp. M LEFT ct + M + + − M LEFT ct M SMEFT ct LEFT ct ( M M − SMEFT ct ], the counterterms for the LEFT ( + . 12 M − – + 10 SMEFT loop SMEFT M SMEFT tree, ren. SMEFT loop, exp. M + M SMEFT loop, exp. M = = – 20 – M + + SMEFT ct LEFT LEFT loop M ]. M M SMEFT ct + 15 + M SMEFT tree, ren. + ) the UV divergences of the expanded SMEFT loop integrals M LEFT ct = 5.5 SMEFT tree, ren. M M + SMEFT tree, ren. ) consists of tree-level and one-loop diagrams, where the bare param- = M 5.1 LEFT tree, ren. = M LEFT tree, ren. LEFT tree, ren. M M LEFT tree, ren. M Each side of ( SMEFT loops are cancelled bythe subtracting expanded the loops LEFT gives UV the matching counterterms.the contribution. only The diagrams Note finite that that part have in to of counterterms this be matching can calculated procedure, be are the extracted expandedare from SMEFT obtained the loops. from RGEs The the in SMEFT RGEs [ in [ cancel in the difference.structure After of the the expansion IR-divergences are of alteredway. the both in loop Since the integrands the SMEFT in and expanded LEFT theminus loops LEFT the light in LEFT integrals scales, the UV the vanish, same divergences. thestructure. The altered expansion Therefore, IR in in the divergences ( are light are scales cancelled equal does by to not the affect the SMEFT UV counterterms, whereas the IR divergences of the expanded prescription Note that the SMEFT and LEFTthe loop contributions LEFT can reproduces both contain the IR IR divergences. structure Since of the SMEFT, these divergences are identical and difference is unaffected. Therefore The LEFT integralsintegrals, with which integrands vanish expanded in in dimensional regularization. the low-energy This scales leads are us scaleless to the matching It is analytic inequation, the both low-energy SMEFT scales, andlow i.e. LEFT scales a loop that polynomial. contributions cancelcannot contain in be non-analytic On the expanded pieces difference. the in in right-handanalytic the In the low-energy side pieces principle, scales, of of the because the the integrands thispieces integrals. of alters in However, the the the if structure loop SMEFT we of integrals and expand the the the non- LEFT integrands, loops both non-analytic are altered in the same way and the analytic The part of the LEFTis tree-level amplitude given that by only contains the renormalized parameters or that perform an explicit scheme translation. eters in the tree-levelcancel part the are UV split divergences into of renormalized the parameters loops: plus counterterms that The goal of therenormalized matching SMEFT is to coefficients. expressis This the important relation renormalized that obviously LEFT the is coefficients presentedthat in scheme results employ term dependent, the are of same hence used the scheme it only definitions in as connection the with matching loop calculation, calculations defined in section malization: JHEP10(2019)197 . m (5.7) (5.8) (5.6) and Λ, where M p/ . The SMEFT i Λ or C . The smallness of v/ . In the LEFT, the vev ]. Through the matching SMEFT m  14 × v Λ. An implicit dependence on v / Λ = or as . . In the following, we will work out 2 T W v , due to the LEFT expansion parameter, µ 5 rs LEFT v Λ  SMEFT C /v v − m × v ,  = = – 21 – v rs m ] Λ m ν = denotes the vacuum expectation values (vev) or the M is absorbed into the values of the Wilson coefficients is analytic in the low scales [ stands for a light SM particle mass. By matching the v i v m L LEFT  an external momentum. Similarly, in the LEFT the expansion p , where p/v or . This implies that we in principle take into account double insertions m/v 2 SMEFT  and . Hence, the matching equations depend on two expansion parameters explicitly appears only as polynomials in 1 i 2 LEFT dimension-five SMEFT operator, Hence, this term canthe be neutrino counted masses either shows that as the scale of lepton-number-violating NP is huge and could operators. At the samecouplings time, containing we up will consider todimension-five corrections LEFT two to dipole additional the coefficients, powers LEFT wewhile of will masses work for and light up gauge the scales. to linear dimension-sixspecial order For operator case in corrections the coefficients is to light we the scales, the will Majorana-neutrino set mass the term, light which scales is to induced zero. at tree A level by the of the heavy SMthese scales equations and at theexpansions, one matching i.e. we scale loop consider contributions and to up the LEFT to Lagrangian parameters dimensionof of the the six order dimension-five in SMEFT operator both and the single LEFT insertions of and the SMEFT dimension-six SMEFT while any further dependenceL on in a polynomial way, and in addition contain pieces that are non-analytic in the ratios contains the full non-analyticThe dependence LEFT on reproduces the thesame heavy same non-analytic and IR structure physics light in asof SM terms the the scales, of LEFT SMEFT, the Wilson i.e. lightv coefficients it SM explicitly scales. contains the Therefore, the contribution which separates LEFT andin the SMEFT previous power section, counting.terms the of matching Following SMEFT determines the parameters.through the procedure Wilson In the outlined coefficients the expansion SMEFT, ofNP parameter the the parameters as scale LEFT is a Λ in absorbed of polynomial NP into in only the 1 appears values explicitly of the Wilson coefficients SMEFT and the LEFT at theof weak SMEFT scale, parameters the and low-energy parameters inheritequations, are the expressed the SMEFT in scale power terms Λ couting appears [ in the LEFT in the form The power counting of bothdimensions. the SMEFT The and dimensionless the expansion LEFTΛ is parameter given is of in the the terms SMEFT of heavymass canonical is mass of mass a scale SM particle, of and parameter NP, is 5.2 Power counting JHEP10(2019)197 ). (5.9) 2.56 in the (5.10) (5.11) (5.12) 5 contribu- Q , these are 3 ν , the inverse m . 5.1 , SMEFT  pr pr ) ) ∗ × R v M M R ∼ R P ν P . In the case of Majorana † m − , R T L M − L M M = Σ P L = R : since P − 3 ν ∗ − m M M R flavor indices. For Dirac fermions, the ∼ Σ T M Σ , ¯ ν R ν R , and Σ T † C p, r ) / pP L / pP δm R = + – 22 – + M = Σ L M ν , we discuss the diagonalization of the mass matrices = (Σ Σ Σ L , with L L L pr δ ,Σ / pP / pP p † 5.3.2 R + := Σ m + , we describe the matching equations for the mass matrices M m M m = Σ − = − pr , are loop-generated matrices in flavor space and the tree-level ]. / p 5.3.1 / p L ( ( R m i i 80 M − , − M -matrix elements. , 79 = = S := L 1 1 − pr − pr in the matching of the four-fermion operators, one should go to dimension M ) can be directly identified with Lagrangian terms of the LEFT. Terms M S S 5 , R 5.9 Q ,Σ L in the matching for the neutrino dipole operator, but neglect insertions of 5 If the loops are expanded in all the light scales as discussed in section Q i.e. the inverse Majorana propagator has the form propagator ( implies the relations where Σ mass term ismatrices given fulfill by neutrinos, the additional Majorana condition We assume that the SMEFTThe tree-level inverse mass matrices fermion havefermions been propagator we diagonalized find is as the in calculated structure ( from the two-point function. For Dirac and derive the necessarycomplex field mass redefinitions. matrices. In We section areand working the in relation a to basis with non-diagonal 5.3.1 Field redefinitions and mass matrices 5.3 Fermion two-point functions The fermion two-point functionsThrough provide field-redefinitions, the they also matching contribute conditions toleg higher-dimension corrections. for operators the In as section external- mass matrices. dimension-seven contributions in the SMEFTinsertions of power counting. Ineight order in to include the double LEFTtions to counting. the neutrino-mass Vice corrections,counting. one versa, should In in work the at order dimension contextbeen to seven considered of in in consistently RGEs, the [ include double SMEFT insertions of the dimension-five operator have Therefore, we consider both countingof schemes and only takematching into for account single the insertions dimension-sixing operators. equations, This the implies neutrinoscorrections that can to for be the the neutrino treated four-fermion masses as of match- massless order Weyl fermions. We also neglect be much larger that the generic scale Λ for lepton- and baryon-number-conserving NP. JHEP10(2019)197 L ψ ψ . 2 )  (5.17) (5.15) (5.16) (5.13) / D R i M, (0) formally ) † )( 0 / † R  Dψ i C M ) M † † R R − C M P R , 2 † ) and L − ) are not fixed and  C (0) B ( ( / L D R † L L,R 1 2 i O − ( Σ ψ B 2 R − L ) MC − − ) / D † L / − R i DP i (0) ( MB C † 0 L B ) (5.14)  ], which are always possible. − B − ( L,R M L † R 15 O L + C (0) + C A P 0 R ( + 2 + . . L ) + B 2 ψ ( Σ M R / p ) D R L 1 2 R i R,L ψ R ( ¯ ) A P ψ R † P ( 3 + † / ψ − ) Dψ R + 3 i M / ¯ ) ψ D M M † L R i / (0) D ψ + A L,R † − i 2 (0) 0 ) B L L )( + M , R . / – 23 – † R D + and the combination R R i Σ / R L (0) + ( C Dψ P A MP correspond to higher-derivative operators that are i 1 2 )( L B † ) † † + L,R − Σ L,R − − MB ψ M M L R M C M L,R / ( † † D R L C + , (0) L i ( (0) + / L,R ( C ¯ B 0 † M DP ψ R ¯ i L ψ A M ¯ L,R − − ψ M M are generic matrices in flavor space of − L Σ + A † + L = † R R R P L + and R M B B C B ψ L L L,R † − P 2 1 † ) , L,R , † ψ C L,R 3 ψ M M M − ) M (0) (0) 0 (0) † R / (0) + 0 (0) + 0 − − D 7→ R − i A L R R L (0) † in Σ 0 R † L Σ L )( Σ Σ Σ Σ , and + B A 2 R † L,R L − M − − − p ], the naive application of the EOM can lead to apparently ambiguous L P ψ C + + L,R 3 MP = = = = = ) 15 L L B + / † † − † † L L L D R † R MA , B A i L ( ( ( C A B C / A D C L L R i ( L,R ¯ ¯ ¯ + + + + ( + + ψ ψ ψ  L A ¯ ¯ L L R R R ψ ψ + + − C A C B Consider the most general field redefinition that removes terms with up to three deriva- In order to remove all the terms with multiple derivatives, we choose a field redefini- A = = L L δ The anti-Hermitian parts of represent chiral transformations. The axial part of the chiral transformation induces a shift tion with where the primes denote a derivative with respect to Up to dimension-six effects,that the can one-loop be written fermion as two-point a function Lagrangian will generate terms where keeps track of power counting. The induced change in the Lagrangian is given by ambiguities are simply related by chiral field redefinitions [ tives: not in the canonical LEFTfield operator redefinition, basis. which They is candiscussed be often removed in referred by performing [ to an asresults. explicit using The ambiguity the is equations duethe of to application the motion of fact (EOM). the that As derivatives EOM. can By be properly integrated by writing parts the before field redefinitions, one sees that the proportional to JHEP10(2019)197 (5.23) (5.20) (5.19) (5.21) (5.22) (5.18) M (0) 0  R † . M Σ † . The induced †  . Writing ] † R m MM δR , A MM MM = 5.4.3 † (0) − 0 † (0)  0 R ] M R M R L = 1 + A A  ψ = , .  L ]  , Tr[ + Σ   L R (0) + A †  M . − (0) + Σ A P R M (0) + ] . M R 0 Σ  ] + Tr[ (0) † L † 0 . † R R (0) Σ δL , R † R (0) † L A M (0) M  ψ R A L 0 † R M A † Σ  1 2 ψ − ] + Tr[ + M  M Rψ † R M L 1 + − (0) † − M , = 1 + † A − − M A  M 0 −  (0) 7→ L † 0 1 + Tr[ P (0) + Σ Tr[ R M Tr[ M 0 (0) M = 0 M  (0) MM M  0 †  L † 0 ,L (0) † L M R i (0) Σ + (0) † (0) ) 1 † 0 2 0 Σ C M 0 ξ M , ψ R ( – 24 – M M M L − † = arg det = arg = Im = M Σ M M (0) − M † f ]. The gauge dependence drops out if the mass 0 M −    R  L δ − M R (0) Lψ ) † 3 2 1 2 1 2 83 M C † P 0 L – + MM † M M 7→ = = = = A + Σ 81 M ], the chiral rotation does not contain an axial part and M L † † † L L L † † R (0) (0) † R δM 0 (0) + ψ † M C A B A − 0 A M L (0) +  (1 + + L † M − − − L − Σ − 1 2 ) † M (0)  Σ L m L R R R R L M M C 1 2 A MP B A A M A Σ  − −  ) = − M (0) + ξ L R ( / ¯ ¯ D − ψ ψ (1 + i M 1 2 1 2   ] = Tr[ M + ¯ ψ † L + + 1-loop = A = f M − L arg det δ L 1-loop A + f M L a bi-unitary transformation We specialize again to real diagonal tree-level mass matrices The mass matrix definedcontains in gauge-dependent this parts waymatrices [ is are diagonalized. not We only choose to complexthe perform gauge and only a dependence non-diagonal, partial from diagonalization but that the removes non-diagonal it mass even matrices. In order to do so, we perform The one-loop mass matrix is given by Since Tr[ does not affect the theta terms. The transformed Lagrangian then becomes We choose phase is proportional to in the theta parameters, which however is of two-loop order, see section JHEP10(2019)197 , . ] = 1-loop (5.26) (5.27) (5.24) (5.25) δL 5.4.3 f M . . Close to the s (0) M ss . , M r r Im m i m -matrix elements in the , one could perform the † pr pr S 2 r 2 r f ) + , f M M 2 s δ m δ m    5.3.1 m ) = 0, which implies Tr[ + − + − ( , , 0 0 ∗ L 2 p 2 p ss † ) † pr ) pr ], which is needed to compute ob- p m p m R ( f ( f M M M 87 δ δ – Rr Lr p p ∗ ∗ ∗ ∗ 0 0 u u 84 , cancelling the denominator. Of course, m m 2 r + Re )    r r ) Lp Rp ζ ζ m pr pr  = ) can again be expanded in the light scales. δ ) δ − = = 2 s 2 p − i i − m r r -violating contribution: δM ( 5.24 m – 25 – p p (1 (1 | | R + − Σ − CP parameters at the two-loop level, see section (0) (0) 1 2 m θ pp pp Lp Rp − = -independent parts is ambiguous and the explicit form ψ ψ δR δL | ξ ) | 2 s 0 0 h h -dependent pieces but the full off-diagonal part for the top Im Im m i i ξ , we define the rotations ( 1-loop u ξ L pr pr δ δ M Σ 1 2 = = − pr pr ) to non-axial rotations, arg det( 1 ) are proportional to δL δR  ) not only s 5.24 -dependent and 5.24 m ξ 5.24 are used as flavor indices. The two-point functions are matrices in spinor = -dependent part of the mass matrix is rotated away, the numerators of the ξ is independent of -independent but includes a ξ 1-loop s -violating parts of the mass matrix can always be rotated away by an axial trans- p, r, s, t δM M ]. The diagonalized mass matrix is directly given by the diagonal entries of CP We restrict ( MS scheme. Consider a set of massive chiral fermion fields that annihilate one-particle δR where and flavor space and they have poles at the values of the physical masses SMEFT and LEFT. Wea chiral briefly theory review in the theservables presence in wave-function of the renormalization fermion SMEFT procedure mixingthe and for [ LEFT. We arestates working as with follows: fields that are renormalized in 5.3.2 Fermion wave-function renormalizationAlternatively and to mixing the fieldmatching redefinitions by specified equating in observables, section i.e. the pole masses and The formation, reshuffling them into the i.e. we add in ( quark. In this case, the denominators in ( Tr[ which is block-diagonal form If only the second terms in ( the splitting of of the non-diagonalintegrating mass out matrices the defines top the quark, particular we choice bring of the basis. mass matrix Since of we the are up-type quarks to the where JHEP10(2019)197 , . , , ) ) ) ) 2 2 s s 2 2 s s (5.35) (5.31) (5.32) (5.33) (5.34) (5.28) (5.29) (5.30) M M M M . = = = = ,  2 2 2 2  ) p p p p  R . This agrees , m R L M  ) amp. 2 s M + i , 0 m m − M ) reads ( L }| )( R ss R . L 5.9 M ... , ) M M ! + (regular at + (regular at + (regular at + (regular at M s x R , ∗ m ∗ ( ∗ ∗ / p s s Lr s Rr r s Lr + M Rr ∗ ) + ζ ζ ζ ! − ζ s ∗  Lr ψ ,Σ 2 s s ) Lr ) ζ L R R ) { m R L L ζ m ( R P P P P T m ( s m Lp ) ) | ) ) 2 2 2 s 2 s s L ζ s Rp s s s − L , 0 s s L ss h ) Σ Σ − M M M M M M M 1 M M s ! 1 R M / p ζ − − − ... − m − −  + + (1 + Σ + + M ∗ LR M 2 / x p 2 m 2 1 2 2 s RR − Rr / / p p m · ∗ / / p p p p s p s ( ( p ζ Rr ( ( S + / pS + ζ ) L + + R R ip ): L L ) that we diagonalized by field redefinitions. s Rp L -matrix elements to amputated Green’s func- P P R P P s Lp R m ζ  xe S ζ ) RL 4 LL )( s s s s 2 s Lp Rp Lp Rp 5.31 (Σ 5.26 (Σ d

− M L ζ ζ ζ ζ 2 S – 26 – 2 / pS m p 2 x x x x s p ( · · · · Z m (1 + Σ 

) ip ip ip ip + L ss / p i M − 2 − − − − i ... Σ

e e e e m − = m (1 + Σ ∗ 1 2 i 4 4 4 4 L s 1 2 r ) ) ) ) S p p p p ζ − p − − − 4 4 4 4 π π π π ) , whose two-point function with the interpolating field for 2 2 d d d d ) s r = r R p (2 (2 (2 (2 2 s X M − M 1 m ∼ M Z Z Z Z → − R ] relates as usual ( = i i i i 2 adj( S p R ss i + 88  M = = = = Σ . In matrix form, the inverse propagator ( pr m i i i i ... 2 m s 1 2 S 0 0 0 0 )( M − }| }| }| }| − R |S| ) + Tr 2 1 2 = (0) (0) (0) (0) , they are given by m  ... 2 m 2 s s s p Lr Lr Rr Rr − p ¯ ¯ ¯ ¯ − m ψ ψ M ψ ψ (1 + Σ h 2 ) ) ) ) 2 2 -even part of the mass matrix ( p x x = x x p p =  ( ( ( (  s 2 CP p Lp Lp Rp Rp M ψ ψ ψ ψ { { { { = det( T T T T | | | | 0 = det 0 0 = det 0 0 0 has a pole at h h h h depending only on thewith the diagonal entries ofThe the propagator matrices is Σ given by the inverse of ( with the solution To one-loop accuracy, this gives s which has zero eigenvalues at the physical masses of the fermions: tions: We need to sum over all fields We allow for the casements. where The LSZ the formula propagator [ contains off-diagonal, i.e. flavor-changing ele- In matrix form, the propagator is poles at JHEP10(2019)197 ]. 92 – (5.39) (5.37) (5.40) (5.38) (5.36) 89 . , , . #   87 ) ) ) 2 s s 2 2 s  m m m ) ( ( ( , , 2 s M ps , 1 1 ps L R rs m 2 s − − Σ ( Σ Σ 0 2 s s 2    2 s M ) ) ) ). The square-rooted M ss m = m 2 s L m R ) that diagonalize the 2 Σ p − m 2 s MS schemes [ −

  − ( M 5.29 M ) ) )

) m 1 2 s 5.24 s 2 ) ) ∗ ss 2 s + + − 2 s 2 s − m m m S ( M ( m m ) m ( m ∗ ( ( ( 2 s ( 0 0 ps R L rs M ps m R ss R ss − Σ adj( − ) + Σ ( Σ s 2 s r 2 ) s ) s ) M ss L R m m M m 2 s 2 p 2 r 2 p m p ( s Σ = p ) + Σ ) + Σ m 2 m 1 2 m ss m m m 2 s 2 s ( p m s

− ) − ss − . ) m m − − , − M 1 m − ( ( 2 s (1 + Σ 2 s (1 + Σ 2 s 2 s ) 0 0 )  ) 1 − 1 ) 1 s 2 M 2 s 2 s m 1 2 LL RR m 2 s L ss L ss − m m − S ( ) m ) s S S m m Σ Σ m Im ( ) ) ( ( + ss R L ( s ) m L R – 27 – ps † rs ss  2 s 2 s det( κ ∗ ps m M ) M M 2 2 2 2 s m m M M − M d + M s + s dp Im m s − − ( (1 Im m m m (1 + Σ m (1 + Σ ) ) m i i κ = M ss i 1 1 2 2 s 2 s )( )( − − Σ − S R L m m ) + ) ) + ) − ) ) + ( ( ) ) + 2 s 2 s R L 2 − s 2 s ) 2 s 2 s L ss R ss 2 s m m 1 m ( m M m ( M M Σ Σ  ( ( ( m 0 0 s 2 1 2 1 ps ( † rs − 0 (1 + Σ (1 + Σ + + ps ) correspond to the chiral rotations ( ss ss 2 2 m 2 2 p ss − − M M p p p m m M p r ( M   m M = p 2 M s m m s m = = ( = ( = − M Re Re   s Re s M ) ) " 2 s → s lim ) 2 LL m m RL LR rs ps RR m p m ps δ δ ( S S S S δ / − − 1 + 1 + − 1 +    ∗ ps rs s ps Rp δ δ + (1 + (1 is a free phase parameter for the physical states. The off-diagonal terms with the ζ δ + (1 = = κ = = In the case of Majorana neutrinos, the physical mass at one-loop order is given by s s Lp Rr s Lp ζ ζ ζ and the square-rooted residues are Here, denomiators 1 mass matrices. The denominatorsand appear e.g. as show up usual in in the non-degenerate renormalization perturbation of theory the mixing matrices in leading to In the vicinity of theresidues poles, can the be propagator extracted matrix as has the form ( where JHEP10(2019)197 5.4.2 (5.48) (5.43) (5.44) (5.45) (5.46) (5.47) (5.41) (5.42) ) = 0, 2 (0) = 0. p , ( 2 AA T Z , AA L M  2 Z .
) − 1 M 2 − ab 2 p
− p ( ) , 2 ab L  p ) ( ) = ) = field and with general 2 2 2 L p p p ˆ ( ( ( Z , ) + Σ 2 ab L ) p ZZ ZZ 2 Z Π ) + Σ ( 2 ν ab M p 2 p l ( . ( p µ . l p ν ph ZZ T 2 p 2 c, Σ p µ , = 0 b c p ) + ζ ) = λ µ M
− , t 2 , l a 2 c ) g p ζ p + 2 . The gauge-boson two-point functions 2 − Z ( ( p
2 ( ab T ) ab L M p ) = 2 ab T ) = 0 ) = 0 Π p Π p 2 ( 2 = ( ph 5.4.1  p p 2 c, ( ( ν νλ ab T , ph 2 p , M – 28 – ∼ 1 ) + Σ . p µ 2 Z ZA )Π ZA 2 → − ab t l p p 2 p
( p ( ) ) + Σ − 2 2 ) µν ab p ) = ) = 2 p 5.4.3 t ( ,M 2 2 ( Σ p µν ( T p p g ab ( ( t ab T = 0 det  Π field) AZ AZ i  ph ) + Σ ]. The inverse propagator can be written in terms of transverse ˆ ν − , A 2 2 p 2 A p 95 p µ ( – p t M ) = p 93 − ( [ ˆ for the ab µν , l Z ) = µν , t 2 g 2 γ Π γ 2 ˆ p ˆ ξ ξ p p (  and ab T i ˆ Π A ) = ) = ) denote the one-loop corrections. The Ward identity requires Σ 2 2 2 p p ) = p ( ( p ( ( AA AA µν ab l t ab T,L Σ At one loop, the physical masses are given by The physical masses of the gauge bosons are determined by In the vicinity of the poles, the transverse part of the propagator matrix is given by which implies The propagator matrix is given by satisfying and Σ i.e. higher-order corrections are transverse. The absence of poles then implies Σ where the tree-level contributionsgauge are parameter (in unitary gauge for the 5.4.1 Gauge-boson mixing We consider mixing of vectorground fields, fields as it isand present longitudinal at contributions: one loop between the neutral back- We proceed with the gauge-bosonbetween two-point functions the and neutral we gauge firstare discuss bosons then one-loop in used mixing section to extractand the the matching theta expressions parameters for in the section gauge couplings in section 5.4 Gauge-boson two-point functions JHEP10(2019)197 ) Z – A 5.50 (5.51) (5.52) (5.49) (5.50) -violating . ) = 0, which . CP 2 ) p GF 2 Z ( L . M ZA L ( A µ ]: the G . ZA T ,  (0) + 96 Σ  00 ) = Σ 2 (0) Z 2 (0) 1 0 (0) p GG T 0 and the Higgs scalar. In M ( ZA T Σ (0) = 0, i.e. there is no GG T − (0) Σ GG T A AZ L Z 00 ) Σ 2 AZ T Z Σ ]. 1 2 1 λν 1 2 AA T ) = M 43 2 G − Z − λ )Σ 1 1 M D λν (  (  (0) = Σ F 3 (0) = A λ AZ T g ) 7→ ZA T ∂ Σ AZ T A µ µν = ¯ )( 2 Z Σ G 1 2 µν Z µ M 1 F D ,G M , g − µ – 29 – ( µ ∂ 1 4  = = ( A 1 4 − Z A A Z  (0) 0 ) can be interpreted as terms in the LEFT Lagrangian  − (0) 0 AA T  (0) 5.41 0 Σ AA T (0) 1 2 = 0 implies that Σ 0 Σ GG T 2 − 1 2 , ζ p AA T 1 ) − 2 , ζ Z  1 1 + Σ e M   1 + Σ (0) ( = ¯ 0 0  e 7→ Aµν ZZ T AA T µν µ G Σ Σ F A 1 2 1 2 A µν µν G F − − 1 4 1 4 − − = 1 = 1 = Z A Z A ζ ζ L Note that there is also one-loop mixing between the the Lagrangian are totaldiagrams derivatives and of therefore ordinary usuallyperformed perturbation do with theory. not standard contribute However,parameters in methods in the Feynman the by SMEFT matching using can canacquires be a multiplied a nevertheless trick by vev a be proposed equal non-propagating scalar to in dummy one. field [ that By calculating diagrams with a single external dummy field, By another field redefinition (orare by transformed applying into the four-fermion EOM), operators. the dimension-six terms5.4.3 in ( Theta parameters The matching of the theta parameters is complicated by the fact that the theta terms in Matching the covariant derivativeLEFT then gauge directly couplings: leads to the matching equations for the The kinetic terms can be made canonical by a redefinition of the fields: fermion masses. Expandedinverse gauge-boson in propagator ( all light scales up to dimension-six effects, the one-loop gauge invariance. 5.4.2 Gauge couplings The matching equations for theboson gauge two-point couplings functions can by be directly performing obtained field from redefinitions, the gauge- similarly to the case of the In the background-field method, gaugetogether invariance with requires analyticity at Σ one-loop mixing for on-shellgauge, photons this even is in only minimal enforced by schemes, on-shell while renormalization in [ background-field the gauge, conventional the mixing of the photon with the Higgs scalar vanishes due to and for the square-rooted residues, we find JHEP10(2019)197 (5.53) (5.54) ] it was argued that the sec- 96 -odd SMEFT operators or theta ] by inserting momentum into a is not gauge invariant. Using the 96 µ CP φ ) K µ -odd SMEFT terms to the LEFT theta ] is simplified in the current setting as we K µ 96 CP φ . ∂ µ . The matching can be performed in several ∂ = ( -violating mass term is generated only at one µ φ – 30 – K D.4 CP − µν e F . µν is set to its vev. In [ F φ D.3 ] puts this argument on solid ground: it ensures that gauge 42 , 41 -odd contributions to the mass matrices. We work in a basis, where CP -violating mass terms away by an axial rotation and would expect an CP ] a potential problem of the method was mentioned: since the theta terms are 96 In addition to the direct contributions to the theta terms, additional effects are gener- In [ -violating mass term and calculating the fermion-loop contribution to the gauge-boson extracted from the three-pointrelevant diagrams functions are with shown two inways. fermions appendix One and could a calculateresult photon the with or expanded a gluon. fully setat off-shell of All the operators three-point LEFT and functions, operator perform identify the a basis. field In redefinition this (or case, use the many EOM) contributions to would arrive directly correspond or matching calculations at thetheta two-loop terms level, requires the special effect attention. of axial field5.5 redefinitions on the Fermion three-point functions The matching equations for the electromagnetic and chromomagnetic dipole operators are additional contribution to thesult theta of terms the through anomalous theCP rotation chiral can anomaly. be However,two- obtained the or as re- three-point in function. [ loop, Since its the insertion intoequations the for fermion the loop theta corresponds angles to and here a it two-loop can effect be in disregarded. the We matching remark that in running invariance with respect to the background fields is maintainedated in through the the loop calculation. these contributions are left into the rotate non-diagonal the complex mass matrices. One could choose While the first variantvanishes gives when a contribution the to dummyond the field variant LEFT should theta not terms,background-field appear method the because [ second the variant current or theta parameters. Because ondimension-six operators the but SMEFT also side, intoin the momentum SMEFT this is theta modified inserted terms, not perturbation theydiagrams appear only theory are as into as shown new the they in vertices appendix are no longer totaltotal derivatives. derivatives, the All generated relevant counterterms could either be of the form perturbation theory and the contributionterms of can the be calculated. Theare general method only of interested [ in termsthe linear method in amounts the to higher-dimension insertingterms SMEFT momentum and operators. into matching it In the with practice, the LEFT theta terms, where the momentum is inserted in the which allows one to insert momentum into the diagram, the matching can be performed in JHEP10(2019)197 ) 5.56 (5.57) (5.58) (5.59) (5.55) (5.56) , µν ) ) and ( ) correspond A F ( 6 L p , A ψ 5 5.15 ) 5.51 − γ 0 µ † ψγ p P L † R = + A ) . A ( 5 + µν µν µν , k A L , F F F A µν ) µ ) L R A L k P G k , ( , ψ ψ ψ ( µ † ψγ L R + µ Rr  † ψγ ψγ  L † ψγ ) ) ψ ψ ψ ) ( p L L L A A p ( ( 4 † L † ψγ ψγ ( † µν R r µν T r A L L σ u B B σ 5 u ) † L † µν R R 0 ) γ R 0 µν ¯ µν σ µ ψ B ¯ B ψ k p, p Lp µν / + µν / p, p ( Dσ + ←− ¯ Dσ ←− ( + ψ σ σ i i ) µν R µ pr Aµ pr L µν µν pr µν A ψG ¯ ¯ F ψ Γ Γ ( 3 ψ F F – 31 – F i i L R ) ) R A − 0 0 − ψ − − µ ψ + p p ) k ( ( L R µν µν p p ψγ ψγ µν ψ ψ F + F u u ) affecting the two-point functions, the field redefini- L L F ) L R † L A 7→ 7→ µν Rr ( 2 A σ / / L 5.15 R ψ Dψ Dψ )) = ¯ )) = ¯ A L 0 0 i i 5 ψ ψ + ¯ p p µν L ψ R γ ( ( R σ µ p p B B = γ A ψ ψ Lp † ψγ ψγ ¯ L + ψ ψγ L L → → ) L pr ψγ A ) ) µν ( µν ( 1 k k L σ σ ( ( A µν L R γ µ A σ = ¯ ¯ ) ψ ψ γ g L . In order to match the vertex functions to the LEFT operators, we first p ) 0 L ¯ ( ψ + + p p r ( ) = r ψ = 0 + ( ψ ( p L ) generates additional terms that only show up in three-point (and higher-point) δ M p, p i ( = ), we have employed the tree-level matching to express the coefficients of the dipole M ) i A 5.13 P ( µ The general structure of a photonic or gluonic Dirac fermion three-point function is In addition to the terms ( Γ 5.55 with write down an intermediate sethave already of removed operators fermion up EOM to operators: dimension six, where field redefinitions where the vertex function can be decomposed into the following Lorentz structures: to the external-leg corrections to the three-point function. given by which does not affecttogether with the the two-point shift functions. induced by The the Lagrangian redefinition shifts of the ( gauge bosons ( In ( operators already in termsAnalogous shifts of are LEFT present coefficients forbe the (using eliminated gluonic matrix dipole by operators. the notation EOM in The or derivative flavor operators another space). can field redefinition gets shifted up to dimension-six terms by tion ( functions: the original Lagrangian required by gauge invariance.by the After matching having offor performed the the two-point the functions, off-shell field these three-pointfermions redefinitions operators and functions as only are as keep implied removed well. thethat from gauge generate the boson One four-fermion off-shell, result can operators in then order through to the directly identify gauge-field work the EOM. with field redefinitions on-shell to the same operators found in the matching of the two-point functions, due to relations JHEP10(2019)197 . D.5 ) and (5.62) (5.60) (5.61) 5.62 , .
R pr E 4.1.2 . − A A ) L (0). Through field pr , ) 00
E µν µν . pr ) G GG T † ψγ G r ν A ν L Σ ) m 1 4 D A µν D ( − ( λν − + G Rr G pr Lr p ψγ , , Lr λ ψ = ψ L

m ψ A D A ( ( A E pr pr T † ψγ † ψγ 1 2 T − A T µ µ L L ) γ = = γ µν + − µν σ Rp Lp pr pr G ¯ ] ] ¯ ), i.e. ψ pr pr ψ ψγ ψγ µ 4 6 Rp R ¯ pr L L L pr ψ A A D [ [ ˜ ˜ ( E E ) ) 5.50 pr r r † ψG E , L m m ) +
+ ) + – 32 – + R e + − pr µν G µν p p -odd three-gluon operators from the EOM operator F E F O µν ν ν e m m G F + ∂ ∂ ( L CP ( , the three-point diagrams are listed in appendix Lr L pr + ( + ( Rr ψ Lr + . Second, there are contributions from 1LPI four-fermion E ,

ψ ψ G µν
D.2 R L µ ) pr pr µ σ r γ O γ pr E E D.6 † ψγ G m Rp L Lp Rp + − ¯ L ¯ ¯ ψ ψ ψ − -even and + L R pr pr p = L R pr pr pr † ψγ E E pr L E E ψγ m L CP ( L 1 2 1 2 + + + 1 2 2 2 k k − − = = = = pr pr pr pr ] ] ] ] 1 2 3 5 A A A A [ [ [ [ diagrams with tree-level propagatorsterms of of heavy gauge 1PI bosons. diagramstions are with We collected organize heavy in them gaugeThird, appendix again there bosons: in are the loopgauge contributions diagrams bosons to that for 1LPI generate the matching two- contributions and two-point to three-point four-fermion func- operators: functions with either, these massless 5.7 Four-fermion interactions In order to extract the matchingto contribution collect to several the contributions. LEFT First four-fermiondiagrams, of operators, shown all, we in a have appendix direct contribution arises from 1PI four-point already identified in theredefinitions two-point or function the gauge-field ( The EOM, separation it of is the transformedis into most a easily set obtained of bymatching four-quark writing operators. at down the the Feynman amplitudefermion rules loops level. for in the the Lagrangian Some three-gluon ( amplitude technicalities have concerning been the discussed in calculation section of the function can be identified with a set of operators Due to gauge invariance, the three-gluon part of the EOM operator reappears that we contributions to the four-fermion operators. 5.6 Gluon three-pointAfter function having applied thefunction field and redefinitions render that the are kinetic determined terms with canonical, the the gluon expanded two-point off-shell gluon three-point and similar relations forwe the obtain coefficients the of matching the equations gluonic for structures. the dipole From these operators relations as well as gauge-field EOM The Feynman rules foridentification this Lagrangian and the Gordon identity provide the immediate JHEP10(2019)197 ]. The 14 ] consists only of contact 14 . However, in all these diagrams ) = 0. ) (5.63) 2 2 p D.5.1 p ( ( two-point function, which receives a non- M M µ Z p – 33 – ) = p ( µ = 0, hence M µ M µ p three-point function, as shown in appendix ¯ Finally, at one loop the last possibly relevant Higgs-exchange diagrams consist of a This does not apply directly to the Higgs- The Higgs-photon two-point function vanishes in the background-field method due to Two-point functions with a Higgs boson could in principle appear inside of four-fermion The tree-level matching of the SMEFT to the LEFT [ ddh Hence, only diagrams with internalthe top quarks need tothe be SM considered, vertex which structure contributeloop to either produces contains an an external externalvanish. momentum. (light) down-type External-leg Since quark the corrections mass light do scales or not are the contribute set either to as zero, in all these diagrams diagrams the Higgs coupling to two othertributions light to fermions. the At Higgs dimensiondiagrams one-loop six that three-point in require functions the the are insertion SMEFT,neglect needed of only diagrams and higher-dimensional the that we SMEFT SM do involve operators.to con- not scalar We a consider can couplings light also to scale.a light derivative fermions, As coupling, which the a are Higgs light proportional three-point scale function necessarily requires appears either if a the chirality internal change fermions or are light. vanishing contribution due to athe top-quark amplitude loop. to However, be Lorentzthat proportional invariance is still to set requires an to external zero momentum, in which the brings four-fermion in matching. dimension-six a coupling light to scale the light fermions, the Higgs propagator, and a SM loop-induced and gauge invariance requires dimension-six SMEFT coefficient. Since into the zero four-fermion due operators to all the light LEFT scaleswould power are be counting, set required, two insertions which of gives dimension-six a SMEFT operators dimension-eight contribution ingauge the invariance: SMEFT the counting. amplitude needs to have the form diagrams. The Higgsup two-point to function dimension does six: notto for give the a any Higgs four-fermion propagator, contribution Higgs-exchange the to diagramHiggs same the with coupling argument one-loop matching as to in corrections the the light tree external level fermions matching applies is [ either proportional to a light scale or to a the tree-level matching of the four-fermion operators in termsterms of as external-leg well corrections. as exchangecoupling diagrams is of proportional weak to gauge a lightrequire bosons. scale, two insertions Since no of Higgs-exchange the dimension-six diagrams dimension-four operators. appear,as Higgs This since we holds it discuss true would even in at the the one-loop following. level, propagators of massless gauge bosons,matched or, directly equivalently off and shell. moreand conveniently, In they the three-point can second be functions method, areredefinition, EOM turned as operators discussed into generated in four-fermion the by previous 1LPItermined operators sections. two- with with Finally, the the the fermion field two-point help redefinitions functions that of to we de- a make the field kinetic terms canonical modify 1LPI diagrams can be considered as sub-diagrams in a four-point function with tree-level JHEP10(2019)197 :
 W 4 Z 2 (6.1) Z µ 2 W r M p µ u M u + m  m
4
. W 2 Z 2 ], which we log Z 2 ) M ) π M M 14 

2 Z 4 2 t . All the results  + 2 Z W 2 2 W + 5 M 4 t 2 m π H µ M M G 2 W  4 2 Z 5 W 72 m µ  M c + 2 Z 2 W M M −  N MS mass matrices are 2 4 µ M W log 4 ). We do not reabsorb 3 W  108 − 3 M ! T ¯ g M v 2 2 Z 2 8 + log 2 π 2.56 Z log T pr π M (3) 2 Hq 3 2 v 5 2 M T Z 2 2 3 C W  √ v ¯ g − 3 T M p + M 48 ¯ g v u 2 2 2 8 H +36 Z , see ( 1 4 2 r W π ¯ g m + u π 2 H 8 U M − M 3 M 2 Z 2 ¯ g 6 m T 4 2 M Z v denotes the explicit top-quark flavor
√ M 
2 M e HWB t 2 2 2 4 2 4 t and G H W Z H π 5 π C 144 H 2 m H M M M M 2 Z V c +  ( 4 4 ]. All parameters are understood to be 2 iC M N 2 M Z pr 2  √ − − 2 W 14 δ + + 35 µ √ 72 M 2 6 2 W Z Z p 2 W 4 + 576 u  2 µ W M M M HG – 34 –

m C qu M  (8) 27 pttr 3
− 2  Z ¯ g 4 C Z + log  log 2 + M + pr (1) t M Hq π 1 2 2 p W − c 2 W  C  u m µ 3 N M 2 +14 2 Z
H m ¯ + g 2 W + 35 2 T 2  Z 4 32
W into the definition of the SMEFT Wilson coefficients, but µ M p v M MS) SMEFT parameters, with implicit dependence on the 2 Z 4 2 Z 2 u W M M  − + 2 π log 5 U M m 2 M M 4 π Z 32 2
π T W 2 − Z 4 12 qu Z log 17 2 (1) v 2 pttr . In order to illustrate the form of the results, we display the Z M 2 W 2 M π + M 2 T W C M and − 8 π M 2 v t Z 4 F M 4 2 H W 2 40 2 432 2 M H − 2 4 π H W m V 8 M 864 π − M  M √ 3 − 2 M -dependent unitary matrices 3 2 M 16 M ¯ g Z 2 2 Z 18 2 . 48 3 T √ 4 W W π W v 32 M √ + M W 2 3 µ 32 8 M M ¯ g µ 54 3 3 √ 3 are flavor indices, and the subscript 1 pr ¯ ¯ uG g g 4 ¯ + + 32 g ¯ g ( C r 3 pr 2 pr pr pr HD uW Hu uG uB T ¯ g v √ C C C C C pr and δ + + + + + + p = The results are given for a SMEFT basis, where the renormalized pr uG L the mixing matrices we keep themwith explicitly. a SMEFT It basis isdiagonalized: with also in off-diagonal possible a mass fixed-orderto matrices to calculation, — be use only the restored the the off-diagonal in matching top mass the quark matrices results tree-level needs then matching in to only contribution. connection be have In a next-to-leading-log analysis, index. The firstrepresents term the is the tree-level SMthe SMEFT running one-loop matching (i.e. matching [ renormalized contribution,matching while scale the second term diagonalized by Here, of both theories. Asform the in the explicit supplemental expressions material, areare with very conventions given specified long, in in we terms appendix onlyreproduce of provide the in them LEFT in appendix coefficients digital coefficient of of the the operator LEFT basis up-type derived quark in gluonic [ dipole operator at the matching scale contribute to the matching. 6 Results The main result of thisthe work SMEFT is the with complete the set LEFT, of one-loop including matching effects equations up connecting to dimension six in the power counting boson couples to a light fermion. Therefore, even at one loop no Higgs-exchange diagrams JHEP10(2019)197 ). and 5.24 dγ L . In the whole 4.1 that receives a tree-level V,LL νedu L ) would result in a dimension-7 problem in section 5.24 5 γ – 35 – gauge for the quantum fields, keeping the gauge pa- ]. In this comparison, we used the compilation of the ξ R 15 , 12 – 10 boson and top quark, which is not suppressed by a light scale. ]. Note that the divergences only have to agree up to a change of W 20 and can be neglected (we assume the differences of the light masses that ) to be of the same order as the light masses themselves). We stress that uG L 5.24 ), which can be determined by comparing the divergences of the mass matrices. generic. The gauge-parameter independence of the results provides a very strong ξ 5.24 In addition to the checks mentioned above, we compared our results to the literature We briefly comment on the checks that we performed to test the correctness of the We emphasize again that our results are provided for a LEFT basis where the matching SM matching contribution.the Taking complete into divergence account structure, the confirmingwith basis the the consistency change, SMEFT of and we the LEFT find matching RGEs. agreement calculation of in several cases. Starting with the flavor-changing parts of the dipole operators ing calculation agreefrom with the the respective difference RGEs [ SMEFT of RGEs SMEFT from and [ basis LEFT ( counterterms extracted This basis change includeshigher axial loop field order), redefinitions feeding e.g. (which into affects the the operator theta coefficient terms only at calculation, we worked with linear rameter check, both of theproblem) correctness and of of the the combination of diagrams allwe (again the checked contributions in to that the the the matching context equations. counterterms of Finally, that the cancel regularization the UV and IR divergences in the match- light scales. results. We compared twoas independent an implementations internal of cross-check. the Wefield checked complete gauge the loop invariance, Ward see calculation identities also in the order discussion to of ensure the background- from the diagonalization of thedue mass matrix, to corresponding a to a loopSimilarly, SM for with external-leg the correction a coefficients ofin the the four-fermion tree-level operators matching, thatleg receive corrections the a with diagonalization SM dimension-six of contribution SMEFT the operator insertions mass that matrices are results not suppressed in by external- In the case of up-typecubic quarks, in the the off-diagonal light SMcorrection scales, to contributions to hence the the massappear diagonalization matrices in are ( ( the same does not apply in the case of down-type quarks: here, a contribution appears contribution itself generates off-diagonal andDiagonalizing complex the contributions mass to matrices theefficients reshuffles mass in these matrices. the contributions form intotree-level of the matching starts external-leg LEFT at Wilson corrections. dimension co- matrices six, In therefore has the only to the case be SM ofwith contribution taken real to the diagonal into the mass dipole account. mass matrices operators, can be The the obtained additional from a external-leg field correction redefinition of for the a form ( basis the off-diagonal mass matriceslight everywhere in scales the explicitly one-loop appeardimension matching less only equations. than in Since six, the the itmatrices into should matching off-diagonal be to mass straightforward coefficients matrices. to recombine of the LEFT masses operators and mixing of one should then either perform the full diagonalization at the matching scale or restore JHEP10(2019)197 ] , ] ∝ 108 111 , and , , uγ, uG L 98 ,LR 107 1 , V dd ], as well as L ]. 105 , ]. In addition, 37 98 , 102 97 V,LL dd ]. Note that the lat- – ] and the tree-level L 15 99 108 , , 37 ] for the matching induced 107 , ]. A subtlety appears in the 110 105 ]. Since they differ by an evanescent 106 – 109 couplings [ 103 ] and LEFT [ 12 to the flavor-diagonal dipoles, f W – C 10 f WB , while in parts of the literature an operator H 2 C , and – 36 – f WB H , and C ] provide a consistent framework for EFT analyses at f W , 14 H C Hud , C e B = 2 parts of the four-fermion operators class, we find agreement with [ H 3 F C -violating dipole moments calls for EFT treatments of increased X , e G H CP C For the , are in agreement with the results of [ 4 operator for which the definition that appears in the literature [ eγ was used to compute the matching onto the dipoles [ f W L ρ Q − µ . Finally, for the ∆ W , we checked that the matching is in agreement with the known SM result [ , and + νρ tt uG Our results provide the complete one-loop matching equations at the electroweak scale. The one-loop running within the SMEFT [ , we find that the calculation reproduces the known SM matching [ ,LR Here we use the operator as defined in table W C 8 4 V dd dγ, dG dG µν e They can be used for fixed-order one-loop calculations within theF LEFT/SMEFT in cases ter definition corresponds toterm, the both triple-gauge definitions coupling should defined lead in to [ the same predictions for observables if used consistently. off-diagonal matching corrections to theto mass mass matrices matrices appear. also Off-diagonal arisehas contributions in to the perform RGEs. ain basis When additional change computing external-leg in a corrections order process to to at the diagonalize a coefficients the given of mass scale, effective matrices, one operators. which can result leading-log accuracy. The increasing precision ofviolating experimental processes searches e.g. or in lepton-flavor- accuracy. We have derivedpower the counting one-loop of both matchinge.g. equations the we keep up SMEFT the and to fullare LEFT. dimension flavor of structure No six and dimension further in include approximations six the mass are in corrections applied, to the dipole power operators counting. that The results are provided in a basis, where the matched Wilson coefficientsSM prediction to found the in constraintsnon-vanishing experiments coefficients in running of at the the an SMEFT. higher-dimension energy SMEFT Deviations below operators. the from scale the ofmatching at NP the result electroweak in scale [ calculations below the scaleabove of and NP the insearches LEFT a and below model-independent low-energy the manner. precisionand electroweak experiments expressed Using scale, as can the bounds all be SMEFT model on constraints evaluated can the then in on Wilson be the NP coefficients tested simply same of from by framework effective integrating collider operators. out the Any heavy new particular particles and comparing 7 Summary and conclusions In the absence ofanalyses evidence are for becoming more NP relevant. in Underabove the direct the assumption searches electroweak that at new scale, particles high effective have field masses energies, theories model-independent provide an efficient tool to perform all by a finite part. by L as well as with the contributions due to four-fermion operators derived in [ the terms proportional tothe the contributions of L case of the differs by an evanescent operator from the one used here, leading to a result that differs L JHEP10(2019)197 (A.8) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.1) . c N , . CD C . δ ) 5 T T γ = C A − ABC C d (1 + = 1 2 + CD † , δ C = AB = AB − R δ δ , c 0 = 1 2 1 ν N 0 1 , fulfilling ABD µ ,P 0 . − ) = f σ ] γ ) = 0 5 C ν 2 ν } B γ . 0 − B , γ iγ T ABC − µ A = µνµ γ = ,T = 1  [ T (1 ∗ A i i 2 2 1 2 C C , f T – 37 – − Tr( { = 0123 1 = =  c = L − µν , , 5 N 2 σ c γ 2 C A 2 N λ ,C T µν ,P T µ σ 3 αγ = γ δ γ ABC − 2 A = γ T if ]. We provide the complete matching results in electronic form 1 = F γ 1 0 C 21 − , ] = iγ C αγ B 20 δ µ := ,T = Cγ 5 A γ T A βγ [ T and the chiral projectors are defined as 5 A αβ γ T are the Gell-Mann matrices. The antisymmetric and symmetric structure con- A λ The SU(3) generators are normalized to and the color factors are given by where stants are defined by The charge-conjugation matrix is given by In four space-time dimensions, we have the relation where the fully antisymmetric Levi-Civita tensor is normalized as The matrix Financial support by the DOE (Grant No. DE-SC0009919) isA gratefully acknowledged. Conventions The commutator of Dirac matrices is denoted by Acknowledgments We thank J.A. de V. Manohar, Vries, E.lar, B. Mereghetti, we are S. Grinstein, grateful Pal, to F. and B. M. Grinstein Jegerlehner, and Trott A. for E. V. useful Manohar E. discussions. for Jenkins, comments In on particu- B. the manuscript. A. Kniehl, leading-log analysis, which is desirableA for systematic some phenomenological high-precision observables analysis atwork. of low To the energies. this one-loop end, the matching inclusiontools effects of would is the be matching left useful results for [ inas existing future supplemental SMEFT/LEFT software material, which should facilitate their further use. where the logarithms are not large. They also provide a first step towards a next-to- JHEP10(2019)197 .  (B.5) (B.6) (B.7) (B.8) (B.9) (B.1) (B.2) (B.3) (B.4) T (B.11) (B.10) L ¯ ν , C † ν Mr M ν . L . ) ν i 3 ν R M + ¯ , ν P L 0 ν 0 ) , m ν γ 2 ν T ν Lr † rp ¯ ν CM ¯ Γ C , m 0 T CM M 0 1 γ ν ν T L γ R 1 2 ν m † pr , P  ¯ Γ − 2 1 0 + ) γ C, . M = 0 L − c T . Lp P ν . / ∂ν L ψ . ν = diag( R ∗ pr c , T 0 M 0 ν η / ¯ ∂ν P Ci Γ + h + ¯ ! c ¯ i ν L ∗ . L T ν M M L = = C ν Lr Lr P ∗ c ν ν 2 c ν ν η !  = R 1 2 = ψ L 0 = ¯ pr pr iσ 0 ∗ + ν c ¯ ¯ Γ Γ = c 0 M η U  + 2

. , ν † ν C C ν c
− ∗ c M – 38 – . T ν , ν = η ν T T R Lp Lp M

¯ 0 := ψ ν † ν ν ν = = C = U M CP M c ν ) + h c + ( + ( ν η M T rp L M = ν ν ¯ Lr ν Γ P Lr Lr 1 2 U := ν ν C ν = Cν c L pr pr − ψ L M P T Lp Γ Γ ν T ν M − ( Lp Lp ν ν / ∂ν L pr ] i P ν = ¯ = ¯ ,U M pr ¯ 0 ν L M [ Γ int 1 2 2 1 R ν, Uν P L  = = ν h − L L ¯ L Γ collect expressions involving Dirac structures and other fields. In terms of ν Mp ¯ ν / ∂ν i 1 2 is a complex phase. The charge-conjugate neutrino is then L c ν = η = ¯ int ν ν, L L where Γ and the Majorana fields, this can be rewritten as The SMEFT bilinear interaction terms of the left-handed neutrinos has the form which fulfills the Majorana condition In terms of the Majorana field, the neutrino Lagrangian can be rewritten as The mass matrix can be diagonalized by a unitary rotation in flavor space After diagonalization, we can define a Majorana fermion The neutrino kinetic termof and the (lepton-number-violating) SMEFT mass read term in the broken phase where We define the charge-conjugate field of a four-spinor by We consider a purely left-handed neutrino i.e. in the chiral four-component basis B Neutrinos JHEP10(2019)197 (C.1) (B.12) (B.13) (B.14) . Mr ν i
∗ µν rp ¯ C , R , P  ν , µν µν ]. We do not list the vertex 2 p , σ σ , p µ  R 43 p  ν =  P ) 2 p ν ν g 0 2 p µ 2 W p ] obviously arise only in vertices ˆ Z + ξ p γ u, d, e , p µ µ M ) p M − p † µν µν 39 pr γ = , ¯ ˆ σ ξ C − (1 0 µν − pr L 24 ¯ C − P − µν µν g µν , ψ (1 . g µν , γ σ , µν  ) σ ) ,  i i g i − i L 2 2 µ ν ψ ν ψ + pr 2 2 2  H W Z gauge [
P C = 1 µν M M M M pr µ ξ µ i rp M M M g i 0 ¯ – 39 – γ C i C AB γ − − R 2 + + −  − − − − 0 R p 2 2 + 2 = = i iδ / / p p 2 2 2 2 P ( ( p p
p i p p p − − i µ pr pr ∗ rp γ ¯ Γ Γ , γ ¯ ======C L µ R P γ P − = + µ pr + i i ˆ C ˆ ˆ C ˆ ˆ h pr G ˆ A ν Z ψ ˆ ¯ T µ L C W L P µ Cγ P γ R P + 2 h Mp ¯ ν 1 2 = int ν, L C.1 Background fields In this appendix, wequantum fields, list which the agree with Feynmanrules the rules for expressions for in the the the SMEFT SMDifferences propagators in [ to of background-field the the gauge rules backgroundthat because for depend and conventional of on the the large gauge-fixing number Lagrangian. of vertices. C Propagator Feynman rules we find writing the Dirac structure as and using where transposition acts only in Dirac space and flavor indices are specified explicitly. By JHEP10(2019)197 the . , , ) (C.2) − i i 2 2 Z Z 2 ν ν η M M , ξM ξM i i 0 − + − − 2 2 AB / p p 2 2 ( p iδ i p p = = = = ˆ h G 0 i Z G ν η η G + , ˆ h G ,   2 W 2 Z ν ν , p + ξM Z ξM p µ η Z µ  p − − p ν 2 , 2 2 p p p p µ ) – 40 – ) p  ξ ξ ˆ ˆ ) h h ν g 2 p − ξ − p µ u, d, e , p (1 − (1 ) = ξ − (1 + −  − η W − µν µν g , ψ (1 g , , µν  ) ,  i i g − 2 2 W W 2 ψ ψ 2 2 2  H W Z ˆ ˆ h h µν M M i ξM ξM M M M g i i i i AB − 2 + −  − , − − − − − p 2 iδ i / p 2 2 2 2 2 2 2 ( p i p − p p i p p p p − ======ˆ ˆ h h h t + i + A  h G A Z ψ η η G W lated for the one-loop matchingterms of of 1PI the functions SMEFT and to dofor not the explicitly the LEFT. show heavy We 1LPI organize diagrams background-field with the gauge tree-level diagrams bosons. propagators in D.1 Tadpoles charge flow is in the opposite direction). D Diagrams In this appendix, we provide the complete list of relevant Feynman diagrams that we calcu- In the ghost propagators, the arrow denotes the ghost number flow (in the case of C.2 Quantum fields JHEP10(2019)197 ˆ ˆ Z Z ˆ ˆ ˆ ˆ A A Z Z ˆ h + + + + + h Z G G W W W ˆ ˆ ˆ ˆ A A Z A ˆ ˆ Z A ˆ ˆ ˆ ˆ A Z Z Z ˆ ˆ ˆ A Z Z ˆ h + + + + +     h η η η η G G W W W ˆ ˆ ˆ Z Z A ˆ ˆ ˆ ˆ A A Z A ˆ ˆ ˆ ˆ ˆ A Z Z Z Z ˆ ˆ ˆ A Z Z ˆ h + 0 0 + + + + + + + h h G G G G G G G G G W ˆ ˆ ˆ – 41 – Z A A ˆ ˆ ˆ ˆ ˆ A A Z Z A ˆ ˆ ˆ ˆ A Z Z Z ˆ ˆ ˆ ˆ Z A Z Z + + + + + + 0   h h Z η η G W W W W W W , no diagrams with background Higgs fields contribute to the ˆ ˆ ˆ ˆ A A Z A ˆ ˆ ˆ ˆ A A Z Z 5.7 Zh ˆ ˆ ˆ ˆ A Z Z Z ˆ ˆ ˆ ˆ Z , and A Z Z 0 + +    t t t t t t γh η η η G G G , ˆ ˆ ˆ ˆ A A Z A ˆ ˆ ˆ ˆ 2 A A Z Z hh ZZ γZ γ D.2.4 As discussed in section one-loop matching up to dimension six. D.2.3 D.2.2 D.2.1 D.2 Two-point functions JHEP10(2019)197 + + p ˆ ˆ + + ˆ e p ˆ W W G ˆ e ˆ ˆ W W + + + u Z A + h h h ν η η η G W W + + r ˆ + + G ˆ ˆ ˆ e r W W ˆ e ˆ ˆ W W + + p ˆ ˆ + + ˆ e p ˆ W W G ˆ e ˆ ˆ W W + + 0 0 0 0 + u t h ν G G G G G W W + + r ˆ + + G ˆ ˆ ˆ e r W W ˆ e ˆ ˆ W W + + + p ˆ ˆ ˆ + ˆ e p ˆ W W W G ˆ e ˆ W + 0 + + + u Z − h Z Z e η η G G G G W + r – 42 – ˆ + + + G ˆ ˆ e r W ˆ e ˆ ˆ ˆ W W W + + + p ˆ ˆ ˆ + + ˆ e p ˆ W W W G ˆ e ˆ ˆ ˆ h W W + + 0 + u A  − t A Z e η η η G G W W + + r ˆ + + + G ˆ ˆ ˆ e r W W ˆ e ˆ ˆ ˆ W W W + + + p ˆ G ˆ ˆ ˆ + + ˆ e p p ˆ W W W G ˆ e ˆ e ˆ ˆ ˆ ˆ h h W W + + u u Z + t t t t h A Z e d η η G G + + r r ˆ + + + ˆ G e ˆ ˆ ˆ e r ˆ W W 2 G ˆ e ˆ ˆ ˆ W W W ee g WW D.2.7 ¯ D.2.6 D.2.5 JHEP10(2019)197 ]: (D.2) (D.1) 48 . p p p ˆ 5 ˆ p ˆ p ν d u ˆ ˆ ν u γ µ 0 + + + u u u γ h e u u G G G G r r = , ˆ r ˆ r r ν u ˆ ) 1 ˆ ˆ ν d u p − ( r C u T 1 ) 5 − p p p ˆ γ ˆ p ˆ p ν d u C ˆ ˆ ν u µ ) γ + + 0 p + + u u u ( e u u G G G − W W ( r r ˆ r ˆ r r ν u T rp ˆ ˆ ˆ ν d u ,C Γ µ C γ ) p − ( p p p p ˆ ˆ p = ˆ p ν d u p u ˆ ˆ ν u ˆ u 1 ˆ h + u u u − t t h Z e d u ) = ¯ W C r p r r – 43 – T , µ ( ˆ u ˆ r ˆ r r ν u ˆ p ˆ ) ˆ ν d u v p ) ( r p u − , Cγ ) ( p 5 p p p p p ( ˆ ˆ ν rp γ ˆ p ˆ ˆ p ν d u u ˆ ˆ ν u pr ˆ h )Γ = + 0 + u u u u t p h h G )Γ e ν d d 1 G ( G W p r − r r ( ¯ v ˆ r r ˆ r r ν u p r C ˆ ˆ ˆ ˆ ν d u u ˆ ν u − T 5 = ¯ = D rev p p p p p , Cγ ˆ ˆ ν ˆ p ˆ ˆ p ν d u u D ˆ ˆ ν u 1 + 0 0 u u u u t h A Z Z = ν ν d d G G W 1 r r ˆ r r ˆ r r ν u r ¯ − 2 ˆ dd ˆ ˆ ˆ ν d u u ˆ ν uu ν C T 1 C D.2.10 D.2.9 ¯ where the transposition acts inexplicitly. Dirac We space use and the transposition relations in the flavor indices is given Note that the reversed diagrams can be easily obtained from the original diagrams [ D.2.8 JHEP10(2019)197 A theta e A + Z ˆ A p ˆ ˆ W ˆ G A d + 0 0 h and G G W r ˆ ˆ e ˆ G A d + ˆ A A W A ˆ A p ˆ ˆ ˆ G A d + + 0 + + G G G W W r ˆ ˆ ˆ G A d ˆ A + ˆ W A p ˆ ˆ ˆ G d A + + + t G W W r ˆ – 44 – ˆ ˆ A G d ˆ + A W ˆ ˆ G A p ˆ ˆ ˆ ˆ G A A d ˆ ˆ h h + + t t t t W W r ˆ ˆ ˆ ˆ G A A d ˆ ˆ G A + ˆ ˆ ˆ G A A p p ˆ ˆ W ˆ G ˆ A d d ˆ h + t t t t t h h W r r ): the background-field method ensures that there is no mixing between ˆ ˆ ˆ ˆ G d ) are compensated by the explicit inclusion of the squared coupling in the A d + ˆ ˆ 2 ˆ 2 A G A g γ W 2.28 2.28 , and the external-leg corrections for the photon are compensated by the explicit Z operator. The additional one-loopterm contributions ( from external-leg corrections to the gluon theta and inclusion of the squared coupling in theD.3.2 operator. There are noterms ( contributions from external-leg corrections to the D.3 Two-point functionsD.3.1 with momentum insertion JHEP10(2019)197 p p p p ˆ ˆ ˆ ˆ ν e ν e + + + p p + + ˆ ˆ e ν G G G h W G p p ˆ ˆ ˆ ˆ ˆ ˆ ˆ e ν A A A A A u u u u u u e e e ν e e ˆ ˆ ˆ r r A A A ˆ ˆ e ν + + + + + Z h + G G G r r W r r r r W ˆ ˆ e W ν ˆ ˆ ˆ ˆ e e ν ν p p p p p p ˆ ˆ ˆ ˆ ν e ν e ˆ + ˆ e ν + + + + u G G W tt W A e W p p ˆ ˆ ˆ ˆ ˆ ˆ e ν p A A A A ˆ e u u + u u u e ν e e ν G ˆ ˆ ˆ ˆ ˆ r r A A A A A ˆ ˆ e ν v + h + r r r + e ˆ ˆ ˆ e e ν G W r r r r W ˆ ˆ ˆ ˆ e e ν ν p p p p + p p + ˆ ˆ ˆ ˆ ν e ν e ˆ ˆ e ν + G p p + ˆ u u ˆ e ν W G tt h G e e p ˆ ˆ ˆ ˆ e p A A A ˆ ν u u + u u u e u ν h e e r G ν e ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν r A A A A A A A – 45 – ˆ e v v + r r A e e + + ˆ ˆ e ν G r r r r r r G G ˆ ˆ e ν ˆ ˆ ˆ ˆ e e ν ν p p p p p + p p + ˆ ˆ ˆ ˆ ˆ ν ν e ν e ˆ ˆ e ν 0 + G p p + + ˆ u u ˆ e ν G W W G G e e ˆ ˆ p A A ˆ e u u + 0 u u u u e e u e e ν G r ν e W ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e A A A A A A A A v v + + + r r e e + + ˆ ˆ e ν G G r r r r r r r W G ˆ ˆ e W ν ˆ ˆ ˆ ˆ ˆ e e ν ν ν p p p p p p p p + ˆ ˆ ˆ ˆ ˆ ˆ ν e ν e ν e ˆ ˆ e ν + + p p + ˆ u u ˆ e ν h W G Z W W e e ˆ ˆ ˆ A A A u u + u u u u e e u u Z e e e ν ννγ eeγ e e W ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A A A A A v v + h + + r r 0 e e + ˆ ˆ e ν G G r r W r r r r r r W ˆ ˆ e W ν ˆ ˆ ˆ ˆ ˆ ˆ e e e ν ν ν D.4.2 D.4.1 ¯ D.4 Three-point functions with massless bosons JHEP10(2019)197 p p p p p ˆ ˆ + ˆ ˆ d u d u ˆ u (D.3) p u u ˆ u W tt h h p u d ˆ ˆ ˆ ˆ u A A A u + + u u u d d u G G d ˆ ˆ ˆ ˆ r A A A A , ˆ u v v ) r A A d u + p ˆ u r ( r r r r G ˆ ˆ ˆ u r ˆ ˆ d d u u u 1 − p p p p p + ˆ ˆ ˆ ˆ d u d u C ˆ u 0 G p + + ) u 0 u ˆ u G G G u d p ˆ p A − ˆ u + + u u u u u d u r d p, W W ˆ ˆ ˆ ˆ ˆ ˆ ˆ u A A A A A A − v v + + ( r d u + ˆ u r T rp r r r r W W W ˆ ˆ ˆ u ˆ ˆ d d u u Γ C ) 0 p p p p p p p ˆ ˆ ˆ ˆ ˆ u d u d u ˆ u + + ( p p u u ˆ u h Z W W d u u ˆ ˆ A A u u u u u h h u d u u u ) = ¯ ˆ ˆ ˆ ˆ ˆ 0 A A A A A – 46 – p v v h + + ( r 0 d u p , ˆ u G G G r r r r r r v ) ˆ ˆ ˆ u ˆ ˆ ˆ ) d d u u u p 0 ( p r − u p p p p p ) ˆ ˆ ˆ ˆ ˆ u d u d u p, p + + + , p u u ˆ u 0 G − h p G G d u ( p ˆ ˆ ˆ ˆ u ( A A A 0 0 rp u u u u pr d u u G G u ˆ ˆ ˆ ˆ )Γ r ]: A A A A )Γ p ˆ u v v 0 + + ( + Z d h u p 48 r G G ( G r r r r r r ¯ v p ˆ ˆ ˆ u ˆ ˆ ˆ d d u u u − u = = ¯ p p p p p p ˆ ˆ ˆ ˆ ˆ u d u d u ˆ + u + + + D rev u u G W A p W W d u D ˆ ˆ ˆ ˆ u p A A A ˆ u u u u u d ¯ Z Z d u u ddγ uuγ ˆ ˆ ˆ ˆ ˆ r A A A A A ˆ u v v + + h r r + d u ˆ ˆ u u W W r r r r r W ˆ ˆ ˆ ˆ ˆ d d u u u D.4.4 explicitly. D.4.3 ¯ the original diagrams [ where the transposition acts in Dirac space and transposition in the flavor indices is given Similarly to the two-point functions, the reversed diagrams can be easily obtained from JHEP10(2019)197 p p p p p ˆ + ˆ ˆ ˆ d u ˆ d d u 0 0 p ˆ u u d G W G tt u d p ˆ ˆ d p p A ˆ ˆ u u u d u + + u d u u G G u ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ G G G G G G r A A ˆ d v v r r r r r d u + ˆ ˆ ˆ ˆ ˆ d d d u u r r r G ˆ ˆ d ˆ d u p p p + p p ˆ ˆ ˆ ˆ d u ˆ d d u 0 G p p p ˆ ˆ u u ˆ d d u h G h u d p ˆ u u u d + + u u u d d u r u u d W W ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d G G G G G G A A A v v r r r d u + + + ˆ ˆ ˆ d d u r r r r r G G ˆ ˆ W ˆ ˆ d d u ˆ d u p p p p ˆ ˆ ˆ ˆ d u d d p p p ˆ ˆ u u ˆ d d u h tt tt Z p u d p ˆ ˆ ˆ d u ˆ ˆ G G A u u u u u h h d d d d u ˆ ˆ ˆ ˆ ˆ ˆ G G G G r r A A – 47 – ˆ ˆ d u v v h r 0 0 0 d u ˆ d G G G r r r r r r ˆ ˆ ˆ ˆ ˆ d d u ˆ d d u p p p p p ˆ ˆ ˆ ˆ ˆ d u d u d p + p p ˆ ˆ u u ˆ d d u G G G h d u p ˆ ˆ ˆ d ˆ ˆ G A G A 0 0 u u u u u u d d u G G d d u ˆ ˆ ˆ ˆ ˆ G G G G r A ˆ d v v h h + Z d h h h u G r r r r r r r r ˆ ˆ ˆ ˆ ˆ ˆ d d u ˆ ˆ d d d u u p p p p p p p p ˆ ˆ ˆ ˆ ˆ ˆ ˆ d u d u d ˆ + d d u + + + u u G G G W h h A d u p ˆ ˆ ˆ d p ˆ ˆ G A G A ˆ u u u d u u u d u u ¯ Z Z d d u ddg uug ˆ ˆ ˆ ˆ ˆ ˆ G G G G r A A ˆ d v v h G G r r r r + d u ˆ ˆ ˆ ˆ d d d u r r r r r W ˆ ˆ ˆ ˆ ˆ d d d u u D.4.6 D.4.5 ¯ JHEP10(2019)197 p ˆ e h h + + G p G ˆ ˆ ˆ e p Z Z C ˆ e ˆ u G ν r ˆ ˆ A e r p Z ˆ ˆ ˆ e d G + + + B ˆ G W G r G ˆ e ˆ h t p 0 0 ˆ e + + + G G W W r W p ˆ ˆ d ˆ ˆ e p Z Z C C ˆ e u ˆ ˆ G G ν r ˆ ˆ A A e r p Z ˆ ˆ ˆ ˆ e d G G + + + B B G ˆ ˆ G G G W r ˆ e ˆ h t p + + ˆ e + G G + u G G r ν p C t ˆ ˆ d ˆ e ˆ p G Z C ˆ e + ˆ G r t W ˆ ˆ ˆ A e r p p Z Z A – 48 – ˆ ˆ ˆ ˆ three-point function with internal top quarks: e v d d G ˆ G + ν + B ˆ B G r G W ˆ ˆ G e ¯ ˆ h ddh + t G ˆ h p p C C t t ˆ ˆ e e ˆ + + ˆ G + ¯ G ddh G A W r r Z W p tt A ˆ ˆ ˆ G ˆ d d ˆ ˆ ˆ e ˆ Z Z Z G t u u , the only possibly relevant diagrams with background Higgs t e ν B r p p , and t ˆ ˆ ˆ ˆ e d d h G + 5.7 + + W B r W r W ˆ ˆ e ˆ e uuh G ˆ h + , ¯ t h W ˆ h p p C 2 ˆ t t e ˆ e ˆ + G ν u tt tt , W r r e C p h t A A ˆ ˆ ˆ ˆ d d ˆ ˆ e ˆ ˆ G Z Z G G 3 u t Z e eeZ eeh g ˆ r B Z t ˆ e v ˆ G Z e B r r ˆ ˆ ˆ e e G However, with the light scales set to zeroD.5.2 all these diagrams ¯ vanish. As discussed in section fields are SM loop corrections to the D.5 Three-point functionsThe with remaining heavy three-point bosons functions onlyfour-fermion appear as operators. sub-diagrams in Up the toto matching dimension onto zero. the six, in these diagrams allD.5.1 light scales can ¯ be set D.4.7 JHEP10(2019)197 p p p ˆ ˆ ν u ˆ ν + h 0 + + G G tt p W p p h G ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν ν u p p p p Z Z Z Z ˆ ˆ e ν ˆ ˆ u u u u u e ν d r ˆ ˆ ˆ ˆ ˆ ˆ ˆ u r r r Z Z Z Z Z Z ˆ ˆ ˆ ν ν u + + h Z r r r + ˆ ˆ e ν ˆ u G W W r r r ˆ ˆ ν ν ˆ u p p p 0 ˆ ˆ ˆ ν ν u + + + + h G p p p ˆ ˆ ˆ ν u e Z W p W W W p ˆ ˆ ˆ ˆ ˆ ν u p p Z Z Z ˆ ν ˆ u u u u u u u e d e r r e d ν ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν u r r Z Z Z Z Z Z Z ˆ ˆ ν u + + + h + + + + G G G r r r W r r r ˆ ˆ e W W W ν ˆ u ˆ ν ˆ ν ˆ u p p p p 0 + p ˆ + + ν ˆ ˆ u ν ˆ ˆ u e + G G + + p + ˆ u ν W W G G p W e p G ˆ ˆ ˆ ˆ ν u p p Z Z ˆ ν ˆ u u + u u u u ν d d e r r e W ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν u r r Z Z Z Z Z Z Z Z – 49 – ˆ ˆ ν u v + + r r + + e + ˆ e ˆ u G r W W r r r W ˆ W ν ˆ ˆ ν ν ˆ u p p p p + p ˆ + ν ˆ ˆ ˆ u u ν ˆ + + ν G 0 0 + u u W G W W G p p Z d e p p G ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e ν u u p Z Z Z Z Z Z ˆ ν u + + u u e u e r W W ˆ ˆ ˆ ˆ ˆ ν r r r r Z Z Z Z ˆ ˆ ˆ ˆ e ν u u v v h + h h r + + e d ˆ ν G W r r r r W ˆ ˆ ν ν ˆ u ˆ u p p p p p p ˆ + u ˆ ˆ ˆ ˆ ν ν u ν ˆ ν + + u u W Z G tt Z p p Z W u ν p p h ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e ν u u Z Z Z Z Z Z Z u 2 u u u e e ν Z Z uuZ ν u ˆ ˆ ˆ ˆ r r r r Z Z Z Z ˆ ˆ ˆ ˆ e ν u u v v h + h h Z r + ν u ˆ ν G r r r r r W ˆ ν ˆ ˆ ˆ ˆ ν ν u u D.5.4 ¯ D.5.3 JHEP10(2019)197 p p p + p p ˆ ˆ + ν ˆ ˆ d ν ˆ ν d + + h G + 0 Z W + h G W W p p G ˆ ˆ ˆ ˆ ˆ ν d p p W Z Z ˆ ˆ ν u u d u u u ν u e u ν r r ˆ ˆ ˆ ˆ e d + + + + r r Z Z ˆ ˆ 0 e d ˆ ˆ ˆ ˆ + + r r + + W W W W G ˆ ˆ e d G W r r r W W ˆ ˆ e ˆ d e p p p p 0 ˆ ˆ ˆ ˆ d ν d ν + + + G 0 0 + + h G G W Z Z W p p p p ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν ν d d p p W Z W Z Z Z ˆ ˆ ν d u u u u d e ν u r ˆ ˆ e + + + r r r r Z ˆ ˆ ˆ ˆ e e d d h ˆ ˆ ˆ + + h AW r + + + W W W ˆ d G W W r r r r W W ˆ ˆ ˆ ˆ e e d d p p p p + ˆ ˆ ˆ ν d ˆ ν d + G 0 p + ˆ d + + G Z A tt p p p p h G ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν ν d d p W Z W Z Z ˆ ν + u u u e e d G r t ˆ ˆ ˆ e + + + r r r r Z Z – 50 – ˆ ˆ ˆ ˆ t t e e d d ˆ ˆ ˆ h A Z ZW + + W W W + G r r r r r G W ˆ ˆ ˆ d ˆ ˆ e e d d p p p p p ˆ ˆ ˆ ν ˆ ˆ ν d ν d h + p + + ˆ u u d + + A G Z Z G e u p p p G ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν ν d p W Z W Z ˆ d + + u u u t ν e r u W W ˆ ˆ ˆ ˆ d + + + r r r Z Z Z ˆ ˆ ˆ e e d v v ˆ ˆ ˆ + + Z + + + W W W ν u + G G G G r r r r r r W ˆ W ˆ ˆ d ˆ ˆ e e ˆ d d e p p p p p 0 p ˆ ˆ ˆ ν ˆ ˆ ˆ d ν ν d + d + + G + + u u u G + + Z d W G W ν d p p p h ˆ ˆ ˆ ˆ ˆ ˆ ˆ ν ν d p W Z W Z ˆ t d u u u ¯ e Z Z u ν νeW ddZ r ˆ ˆ ˆ ˆ d + + + r r r Z Z Z ˆ ˆ ˆ e e d v v h t ˆ ˆ ˆ + + r + + e W W W d ˆ d G G W W r r r r r ˆ ˆ ˆ ˆ e e ˆ d d e D.5.6 D.5.5 JHEP10(2019)197 p p p p ˆ ˆ ˆ u u u ˆ u + h Z + h p A Z ˆ ˆ u p p W ˆ ˆ u u u u u u u d u u r ˆ d + + + + + + r ˆ d ˆ ˆ ˆ ˆ ˆ ˆ + + AW r r + W W W W W W ˆ ˆ d d G W r r r W ˆ ˆ ˆ d d d p p p 0 ˆ u ˆ ˆ u u + G p 0 + ˆ u + + + h G p p Z G ˆ ˆ ˆ ˆ ˆ u u p W W W ˆ u u u u d u d r t ˆ d + + + + r r ˆ ˆ d d ˆ ˆ ˆ ˆ + A ZW + + W W W W + G G r r r r G W ˆ ˆ ˆ ˆ d d d d p p p + class by Hermitian conjugation. ˆ ˆ u u ˆ class by Hermitian conjugation. u G p 0 + u ˆ u + + + + + G A p p u h G ˆ ˆ ˆ ˆ ˆ u u p p W W W ˆ ν ˆ u u u u Z d u r u udW νeW ˆ d + + + + + r r – 51 – ˆ ˆ d d v ˆ ˆ ˆ ˆ ˆ + Z r + + W W W W W d + ˆ e G r W r r r W ˆ W ˆ ˆ ˆ d d d d p p p p ˆ ˆ u u ˆ ˆ u u + p p 0 ˆ ˆ ν u + + + + A Z p p W t h G ˆ ˆ ˆ ˆ ˆ ˆ u u W W W W u u u u t G d u e d + + + + r r ˆ ˆ d d v h ˆ ˆ ˆ ˆ + + + + W W W W d Z Z G G r r W r r r r W ˆ ˆ e ˆ ˆ ˆ d ˆ d d d d p p p p p p ˆ + + u ˆ ˆ ˆ u u u ˆ ˆ u ν + − + − u 0 W W + + Z d p p A W t G ˆ ˆ ˆ ˆ u u W W u u u u e t d ¯ A d u duW udW eνW + + + + + + r r ˆ ˆ 0 d d v t ˆ ˆ ˆ ˆ ˆ ˆ + + r r + W W W W W W d G ˆ ˆ e d G G W r r r r ˆ ˆ ˆ ˆ d d d d D.5.9 These diagrams are trivially related to the ¯ These diagrams are trivially related to the D.5.8 ¯ D.5.7 ¯ JHEP10(2019)197 2 2 6 2 2 × × × × × p s p s p u ˆ ˆ e ν ˆ ˆ ˆ . ν ν ν Z e p s x v ˆ ˆ e ν s p s p ν + ˆ e ˆ ν ˆ e ˆ ν s u v × ˆ e A W v u ν ν v u Z ν e + t t r e ν A ˆ ˆ ˆ e ν ν r t r t r t u ˆ ˆ ν e W ˆ ˆ ˆ ˆ e ν e ν ν u Z r t r e Z ˆ ˆ e ν ˆ ν Z 12 2 2 2 2 2 2 × × × × × × , × + s p p s p p ˆ ˆ p ν ν ˆ ˆ ˆ e ν ˆ ν ν ˆ + ν + W p s ˆ v ˆ e ν s p s s W W e u v ˆ e ˆ ˆ ν e ˆ s u + e e ˆ e + v v u u u u t r u v e ν e e ν e W ˆ ˆ ν ν t ν e W ˆ t t e r r t t r t ˆ u e ˆ ˆ ˆ ˆ ˆ u ν e ν e ˆ ˆ + e ν + + e r r + r W W Z W ˆ ν + ˆ ˆ ν ν W W 2 6 2 2 2 2 2 – 52 – × × × × × × × p s p s Z s v Z u ˆ ˆ e ν ˆ ˆ e ν Z ˆ e e ν s p v s ˆ s p s p ˆ p ν u s p e + ˆ ˆ ˆ e ˆ ˆ e ˆ ν ν ν ν ˆ e ˆ ν p u v Z v ˆ u ν + v ν u u W Z v u ν e ν ν Z Z e ν t r r + Z W ˆ ˆ r t u ν ˆ ν r r t t r t ˆ ˆ ν ν u ˆ ˆ ˆ ˆ e ν e ν W ˆ ˆ t ν e r e ˆ e ˆ Z ν u t t u r Z + ˆ ν e ˆ e e ˆ ν W 2 2 × 12 + 2 2 2 2 × × W × × × × + s p + ˆ ˆ p ν s u s v v W u ˆ ˆ e ν ˆ W e e ν u v e e v d s p s p s p s s p p ¯ ν + ¯ ˆ ˆ e ˆ uu ee e ˆ ν ˆ ν ˆ t ν ˆ ˆ ν e e ˆ ˆ r ν ν p ˆ ˆ ν u v ˆ u ν + + + v u Z W e ν e A Z e νν νν νννν r + + Z W W W ˆ r t ν r r t t r t ˆ ˆ u ν ν ˆ ˆ ˆ ˆ ν ν e e W ˆ ˆ t W e ν r ν ˆ e ˆ ν u t t u u + r e ˆ e ˆ e e ν ˆ ν W D.6.3 D.6.2 D.6.1 Crossed versions of the diagrams are not shown explicitly but denoted by D.6 Four-point functions JHEP10(2019)197 2 2 2 4 2 × × × s s × × s ˆ Z ˆ Z d d Z ˆ u p + ˆ e v + s p Z s s p p p G ˆ t ˆ d ˆ ν u ˆ ˆ ˆ ˆ ν d ν ν u p p s p s W v ˆ u ν v ˆ ˆ u ν ˆ e + ˆ e v e u t ν d ν u ν d Z r G v u u r ˆ ν t r r e e ˆ ν ν t ˆ ˆ ˆ ν d ν u t + t r t r t r ˆ u ˆ d ˆ e ν ˆ ˆ e e ˆ ˆ u t ν + G t t ˆ Z ˆ d ˆ d u r W Z ˆ e 2 2 2 × × 2 × + 2 + s s × ˆ W + ˆ d + d G × s + v s W p p s p ˆ W u ˆ u ˆ d ˆ ν ˆ ν ˆ d ν p W Z v s p v ˆ u ν + d ˆ v ˆ ν e u t t s s p p u p e ˆ ˆ r ˆ ˆ e e e e v ˆ u ν W + ˆ ν t r r e d v ˆ v u u u ˆ ˆ ν d ν r t + r W e ν e ν u ˆ ˆ ν d ˆ ν t G u t r t r t + ˆ t r ˆ d d ˆ ˆ ˆ ˆ d e e e e ˆ ˆ u ν t W ˆ u + A 2 W 2 × 2 – 53 – × + 2 2 × s p ˆ d ˆ p ν W s p + × × ˆ ν ˆ s v ˆ p ν d s p p G v ˆ ˆ ν u ν ˆ ˆ ˆ d ν ν s A Z ν ˆ s d v t t ˆ u s s v p p u e Z t u ˆ ˆ ˆ ˆ e e v e e ν d Z ˆ t d t r u ν r t r ˆ v v ˆ u u u u ˆ ν d ˆ ˆ t ν ˆ + r ν d e e e e u ν ˆ ˆ u ν ν r t t r t r + W ˆ ˆ r ν ˆ ˆ ˆ ˆ d e e e e Z ˆ ν G Z . A Z ν 2 2 2 2 2 2 m × × × × × × + 2 2 or p p s p ˆ ˆ ˆ ν ν ˆ + d ν s W p 5 s p × × ˆ ˆ v ˆ u ν v ˆ v d ν s C p s e W d e ˆ ˆ s ˆ d ν d s p A Z ˆ ˆ + u ˆ d ν + u v u u v u s s p p v Z t e e u d e ¯ t ˆ ˆ r ˆ ˆ W e e e e u e W dd ˆ ˆ u ¯ ee ˆ t ν d t t r r t r u ˆ ˆ v v u u u u d ˆ ˆ ˆ d ν ˆ u ν ˆ + d ν e e e e e e d ee νν r t r r t r t r + ˆ W ˆ ν + d ˆ ν + ˆ ˆ ˆ ˆ ˆ e e e e ν W W W Z Z We do not consider diagrams consistingproportional of to lepton-number-violating parts, because they are D.6.5 ¯ D.6.4 JHEP10(2019)197 p s p ˆ e ˆ ˆ + u e A Z s s p p ˆ v v ˆ ˆ ˆ u e d e s s p p s W u ν ˆ ˆ ˆ ˆ e e ˆ d d s u p Z ˆ ˆ e d + u u v v u u u v v Z e e ν d u s t p r d ν d e W ˆ ˆ ˆ e e ˆ t t e d t r r t t r r ˆ ˆ u u d ˆ ˆ ˆ d e u e ˆ v ˆ ˆ ˆ u + e u d e ν u e e t r r W Z Z ˆ ˆ e u + ˆ e Z t r W ˆ e ˆ e + + s s s s s ˆ ˆ ˆ u d ˆ ˆ + u u d Z W G + p v v ˆ e s s s p p p p p W G d d ˆ ˆ ˆ ˆ ˆ e e e ˆ ˆ ˆ d d e e u p p ˆ ˆ e e u + Z v v u u u v t t u Z e ν ν d d d t r r W ˆ ˆ ˆ t e u e r r t t r r r r ˆ ˆ v u u ˆ ˆ ˆ e e ˆ ˆ ˆ d e u e d e ˆ + + e e d d t t G t t + + ˆ W s p Z ˆ ˆ d u ˆ d u ˆ ˆ e e W W + u v s p s e e p ˆ Z ˆ ˆ e d ˆ + d e Z Z W t r + ˆ ˆ e e v s p s s s s p p p p G W t e ˆ e ˆ ˆ ˆ u ˆ ˆ ˆ e e e ˆ ˆ ˆ d d e s u u p ˆ v e ˆ u u Z + u v u v v u e t t u u e e ν ν d u r r G u t ˆ ˆ t e e r t t r r r t t ˆ ˆ – 54 – ˆ ˆ ˆ e u ˆ ˆ ˆ d e e u d e ˆ t + + u r ˆ ˆ 4 e u Z t t r + ˆ W W A A ˆ d ˆ d e × G s ˆ e + + v s s p ˆ W e ˆ d ˆ + d e A A W Z p ˆ e v v s p s s s s p p p p p W u e ˆ Z e ˆ ˆ ˆ u ˆ ˆ ˆ ˆ e e e e ˆ ˆ ˆ d d e s u u p r ˆ v ˆ e ˆ u e + u u u u v u v v v t u ν d Z e e e ν d u u u t e r W ˆ ˆ t t t e u r t t r r r r ˆ ˆ ˆ u u ˆ e ˆ ˆ e ˆ ˆ ˆ ˆ d e e u e d u e t + r e u ˆ ˆ e u t G r t + ˆ A Z Z ˆ ˆ d e d W 4 s × p ˆ Z d ˆ + e Z Z Z A s s p ˆ v e ˆ ˆ u e s p s s s s p p p p p W ˆ ν ˆ e d ˆ ˆ v ˆ ˆ ˆ ˆ e e e e ˆ ˆ ˆ d d e s u u ν v ˆ u + u p u u u ¯ u u v v v v ¯ e u dd uu d v ˆ e e e e e + d d u u u t u e W ˆ t t u r t t r r r r t r r ee ee W ˆ ˆ u ˆ ˆ ˆ e ˆ ˆ ˆ ˆ e d u e d e e u ˆ e ˆ t ˆ + e u r ν ˆ u ˆ d e r t ν W A Z Z Z ˆ ˆ e d t Z ˆ e D.6.7 ¯ D.6.6 ¯ JHEP10(2019)197 p p s s s ˆ ˆ 4 2 2 4 ˆ ˆ d u d u ˆ d Z p ˆ e × × × × s p ˆ s + d ˆ u s G ˆ u u ˆ + u G Z Z v v u v u u W d v d d d u u u t s s p p p W ˆ u r r r d t t t ˆ ˆ ˆ ˆ ˆ u u p u u u r ˆ ˆ ˆ v ˆ ˆ ˆ d u d u d u ˆ ˆ e u e v v u u u Z d u u u u r + A Z ˆ r t r r t u u ˆ ˆ ˆ ˆ ˆ u u u u u W + u t t W A Z ˆ u ˆ u p s s ˆ ˆ d u ˆ d Z Z Z p ˆ e s s s p p p ˆ ˆ ˆ d d d ˆ ˆ ˆ u u u + 2 Z u v v v u v u u u 4 2 2 u W d d d d u u u u t × ˆ r r r r d t t t t r × × × ˆ ˆ ˆ ˆ v ˆ ˆ ˆ ˆ + d u d u d u d u ˆ e ν s ˆ u Z W A + G G Z v s s s p p p d W ˆ ˆ ˆ ˆ ˆ ˆ u u u p u u u ˆ + u v v u u v u d p d u u + u u s r p W ˆ ˆ d u ˆ t r r t r t u ˆ e G G G u ˆ ˆ ˆ ˆ ˆ ˆ u u u u u u v d p s s s p p e ˆ ˆ ˆ t ˆ u d d d ˆ ˆ s u u + A Z ˆ ˆ u A d v v u v u u v t u W d d d u u u d r d ˆ r r r r e t t t t t ˆ ˆ ˆ ˆ – 55 – ˆ ˆ ˆ ˆ d u u d d u d u ˆ d 2 2 2 Z + + Z Z × × × W W Z Z A Z + p s s p s s s p p p p ˆ ˆ d u ˆ ˆ u u ˆ e ˆ ˆ ˆ W Z Z ˆ ˆ ˆ u u u u u u v u v v v v u u u s s s p p p u ν u ˆ ˆ ˆ u u u u u u + d d d ˆ ˆ ˆ s u u u ˆ t r r t r t d v v v v u u u t u W ˆ ˆ ˆ ˆ ˆ ˆ u u u u u u d t d d d u u u r r u ˆ u ˆ u ˆ r r r e t t t t r t G ˆ ˆ ˆ ˆ Z Z ˆ ˆ ˆ d u d d u d u ˆ ˆ u d + + G A A W G 2 2 2 + p s × × × p ˆ ˆ d u ˆ e W A A v Z Z G s p s s s p p p e ˆ ˆ ˆ ˆ ˆ u u + d d d ˆ ˆ ˆ s u u u s s s p p p ˆ ¯ d ¯ ˆ ˆ ˆ ˆ ˆ ˆ dd uu u u u v v v u u u u v u u u W Z d d d d u u u u v v v u u u t u v ˆ r r r r d u u u u u u t t t t u u uu uu ˆ ˆ ˆ ˆ u ˆ ˆ ˆ ˆ u d d u u d u d r r t t r t t r e ˆ ˆ ˆ ˆ ˆ ˆ u u u u u u ˆ ˆ u u r + Z Z Z ˆ e G Z Z W Z D.6.9 ¯ D.6.8 ¯ JHEP10(2019)197 p s s p ˆ s 2 v u ˆ ˆ ˆ d d ˆ d u d p s + 4 2 2 ˆ v ˆ p × d u s p G t u ˆ u ˆ d × × × ˆ s p u + ˆ + d ˆ + s u + u ˆ t u v u d Z + d G Z W r r G d u d W W ˆ ˆ r r u u t r t t r s s s p p p t p W ˆ ˆ ˆ u u d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ + ˆ u d d d d d u d d d d Z Z t G v u u v v t t u u ˆ t ˆ ˆ d + d d d d d d d u u r t r t r r t ˆ W ˆ ˆ ˆ ˆ ˆ ˆ d d d d d d d + p s G t p s s + ˆ ˆ s s A Z v u d ˆ ˆ ˆ d ˆ ˆ ˆ d d u d d d G p + v ˆ p u s p G u ˆ u ˆ d ˆ p u + u ˆ + + u + Z u 2 u t d A d u r t 4 2 2 W W W W ˆ ˆ t × r r u d t r r r ˆ ˆ v u d ˆ ˆ u u d ˆ + ˆ ˆ u × × × u u A Z u u + s t u t t ˆ ˆ W d ˆ ˆ d + d d d Z A W s s s p p p p W ˆ ˆ ˆ ˆ ˆ ˆ ˆ d d d d d d d p s v u u v v u u p s s ˆ p ˆ v u d ˆ ˆ Z d d d d u u u ˆ d d u ˆ + u u r t r t r r t v ˆ s ˆ ˆ ˆ ˆ ˆ ˆ p d s d d d d d d p p W + ˆ u d ˆ ˆ u d ˆ ˆ s u u ˆ t v u d + + + W u u A Z Z ˆ t d d u G Z u u t W W W ˆ t r r r d t r ˆ ˆ – 56 – u d ˆ ˆ ˆ u u d u t + + ˆ r u A u ˆ d ˆ u G r 2 2 2 2 u t t W ˆ ˆ ˆ u d d u × × × × + + Z Z G A p s p s s ˆ p ˆ v u d ˆ ˆ W ˆ s s s s d d u p p p p ˆ + u u ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d d d d d d d d v s p s p p W ˆ v v v u u u d d ˆ ˆ t t u d ˆ ˆ s u u d d d d d d ˆ v u d + + u u u A Z r t r t r t r t d u Z ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d u u d d d d d d d d t W W ˆ t r r r d t r ˆ ˆ u d ˆ ˆ ˆ u u d u + G t + + + ˆ Z Z r u d ˆ d ˆ u G r u t t W W W ˆ ˆ ˆ u d d u 2 2 2 2 s p p ˆ v s v d × × × × ˆ ˆ ˆ u u + d u d + p s ˆ v ˆ p u d Z G G Z s s p s W d ˆ u ˆ ˆ ˆ ¯ d d ˆ d p u dd s s s s p p p p ˆ + + u ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ u u A d d d d d d d d v u G A Z d d ¯ r d u v u v v v u u u dd W W ˆ t t r u t t d d d d d d r u u t r r ˆ ˆ ˆ ˆ u d d ˆ d u d ˆ ˆ + + ˆ ˆ u d u u r r t r t t r t d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d d d d d d d d t u u r ˆ W W d d u ˆ Z u + G Z Z W D.6.10 JHEP10(2019)197 + + s p W ˆ d W ˆ ν s p u ˆ s s ˆ v p d ν s p p Z ˆ u ˆ e ˆ ˆ ˆ d u u d ν p ˆ s ν ˆ ˆ v u d d u v u Z v d u u u e v d d d r d u e Z t r ˆ s e p r r t t r ˆ ˆ ˆ ˆ u e ˆ ˆ t e d d t d ˆ ˆ r ˆ + u u e ˆ ν ˆ ˆ ν v + d u t Z d d W ˆ u Z W t r ˆ ˆ d + d s p ˆ Z ˆ d W d s p v u ˆ s ˆ d ν s s p p p A d d ˆ ˆ ˆ ˆ ˆ u d ν d ν s p ˆ ν ˆ ˆ d v u u u v u v + ν u v e d Z d d t u 4 d ν t r ˆ ν r W r t r ˆ ˆ u u e ˆ ˆ e t d × ˆ t ˆ + u e r d ˆ ν ˆ ˆ e u r s t A ˆ ˆ W d d + ˆ u s p ˆ ˆ v d d d W p + ˆ d s s u v p s Z ˆ W r ˆ ˆ ˆ ν d Z d d u + d ˆ p d Z v t r ˆ p u ν s s p p ˆ ˆ s p d d Z W ˆ u ˆ ˆ ˆ d ˆ d d ν d ν ˆ s p ˆ d ν t ˆ v ˆ d v u u u v ˆ u Z u d u d e Z d ν Z d u ν u r r t ˆ ˆ t r d d ˆ t u r r – 57 – ˆ ˆ u u u e v ˆ ˆ e u t ˆ + e r ν ν e Z ˆ ˆ e u t t 4 4 + t r ˆ ˆ W ν ν ˆ ˆ e u W × × s s ˆ ˆ d d + + s p + s p ˆ p ˆ ˆ G ν d ˆ t + + d ν W s p ˆ = 1) p d p W ˆ ˆ u d v ˆ ˆ p ν d s p s + p W W ˆ ν d t ˆ ˆ L s ˆ ˆ d ν d s ν + r G ˆ v ˆ u u d ˆ d u u r v u v u u ∆ t ˆ u W Z e e d u u r t t ν d + ˆ d ˆ ˆ t t u u r r r  t t r G u t v u ˆ ˆ ˆ ˆ e u ˆ ˆ u e ˆ ν d ˆ ˆ d e e A Z d ν e t r = t r ˆ ν Z ˆ e ˆ ˆ Z e u B 4 4 + p × × s p (∆ Z ˆ ˆ ν ˆ + ν d W Z s s p s ˆ ˆ ˆ ˆ d v u d v s p d s s p p s + W ˆ ν d ˆ d u ˆ ˆ v ˆ ¯ ˆ ˆ d d ν ν d s p du p G ˆ v u u ˆ d ν ˆ u u v d v p u v d u u u ˆ Z e ν e d + u u e u r r νudd νe u t t ˆ ˆ t t e e r r r t t r W u ˆ ˆ t ˆ ˆ u ˆ ˆ ˆ e u e u ˆ d ν ˆ r d u r A ˆ ν u ˆ ˆ u d d Z t + r + u A ˆ + u ˆ e t t ˆ W ˆ d W d W D.6.12 D.6.11 JHEP10(2019)197 + W u s s p ˆ ˆ d d d ˆ u p v u ˆ u v d u u Z r ˆ t t u r ˆ ˆ e e ˆ u p ˆ u v u s ˆ u u d s p ˆ d ˆ u v + Z t d ˆ e W r u ˆ t u u ˆ e r ˆ u s ˆ d Z v p d ˆ u s p ˆ v d ˆ u u Z r r ˆ ˆ u u – 58 – u u e e t t ˆ ˆ e e + p s = 1) W ˆ ˆ d u p ˆ = 1) u s L ˆ v d v d u L ∆ r e u ˆ u r t − u ˆ ˆ e u ν t = ∆ ˆ = e Z B B Z (∆ (∆ p s ˆ ˆ d u s p ˆ d ˆ u v u v u d u ν d eddd eduu r t ˆ t ˆ u e r ˆ e ˆ u + W There is no SMEFT contribution in this class up to dimension six. D.6.14 ¯ D.6.13 JHEP10(2019)197 are r ) ) ) ) e H H . `t H ) ) ) and ) r r ) r c r ) t ) ) r ) ) . r d u r e jt l q ) r r r Cl r t p p p l e q u d µ µ p four-fermion d l ¯ q Ce q µ µ µ µ µ Cl γ γ µ T γks )( I I 4 . γ γ γ γ γ Ce )(¯ )(¯ + h T γs γ q p p p p p τ τ c T γis p u l ¯ ¯ ) ψ 3 H . ) T γs q e H p p H d u q )( t t u † l ¯ † † q u )( )( )(¯ )(¯ u d )( )(¯ )( H )(¯ )( H ) )(¯ ) )( H ) H ) A A βjr ) 2 ) H t t ( t H H t ( ( H t βjr H t βr H H µ e u T d T H µ βr µ µ ψ e u d µ µ I µ D µ µ µ µ µ Cq I µ µ µ µ Cq D ) 2 γ γ γ γ γ Cu D D ←→ D D γ γ γ D ←→ ←→ ←→ ←→ Cu s s s s s D i † s s s D i ←→ = 1 + h i 5 : T i αip i ←→ ¯ ¯ eH † dH uH e ¯ d d i T αp u u † † e H T † αip d i † u q e T αp † H ¯ d † q 2 ( Q Q )(¯ Q L RR )( )( H )(¯ )(¯ )(¯ d ( )( ( H ( )(¯ H H H ]. r r r ( r r i r ( H r ( ( ψ r ( jk H ( l q ij l q q ij )( l q q ( 7  (   µ µ µ µ µ µ A A i` γ γ γ γ αβγ γ γ  = ∆ 7 : T T ¯ p p p p  LL p p
l ¯ l αβγ ¯ µ αβγ l ¯ q µ q q ( (  (  (¯ (¯ (¯ γ γ B H αβγ p p µ  ) q q ∆ 8 : ( (¯ (¯ D + h.c. H (1) Hl (3) Hl He (1) Hq (3) Hq Hd Hu † † Q Q Q Q Q Q Q H H . le ld lu qqql ( qe duql qque Hud (1) qu (8) qu (1) qd (8) qd duue 2 c ∗ Q Q  Q Q . Q Q Q Q Q Q Q Q Q
) D ` are weak-eigenstate indices. 4 H H H µ † + h j H D ) H ) ) † H t ( t t ) d 3 : u d H ) ) ) ) ) A ) kr t ) t . t t ) A t t HH t I I A µν µν A µν t µν µν c u d T I d e u d ) µν d T p, r, s, t . in the table heading. Subscripts u σ µ µ µ µν Cl µν µ µ µ µ µν k k s s k k . s s ) γ γ γ γ γ γ γ q q . T ip q q s s s c e s s s s  l (¯ (¯ HG e HG LL ¯ ¯ . (¯ (¯ ¯ c ¯ d d HW in SMEFT. There is also the Hermitian conjugate HW e u d HW ( d ) u ) e . H HD ) HB ) HB ) HB jk jk ¯ r r jk jk )( )( RR ) r )(¯ )(¯ )( ) r ) ) + h r )( k`   )(¯ 5  – 59 –  Q d u Q r r r r r r r r r ) )  e r d u ) ) + h )( e r r e A A d e u u d I u I u e I r r ij Q µ µ µ µ u u τ µ µ e τ e ¯ τ A  T T LR µν j µν p µν j p γ γ γ γ γ γ A l ¯ T ¯ q p µν σ p p p µν µν σ µν p p σ )( µν µν RR ( (¯ µ ¯ T 3 e p p e p e d u σ σ σ u σ XH σ l ¯ σ j ) p (¯ (¯ (¯ ( q γ q (¯ = 2 ( j (¯ p p p p ( p 2 p q (¯ l l ¯ (¯ ¯ p q q q ¯ q (¯ H ( ( LR (¯ (¯ u 6 ψ (¯ (¯ † L 8 : ( 5 (¯ H H Q ∆ ( 6 : 8 : ( eB dG ee dB uG ed uB dd eu (1) lequ (3) lequ uu eW dW uW (1) ud (8) ud (1) quqd (8) quqd 2 : Q Q Q Q Q Q Q Q Q Q Q Q Q Q H Q Q Q Q Q Q = 2 operator L ) ) t t q q ) ) I ) t I µν µν t t Kµ Kµ Cµ ρ Cµ ρ q τ ρ ρ l q τ Iµν Iµν µν µν , as indicated by + h . B B µ µ Aµν Aµν µ µ µ G G c † 5 W W ) γ γ B B ) . γ γ γ I I W W G G µν µν s s s s s tj Bρ ν Bρ ν Q q l q ¯ Jρ Jρ µν µν q q I I ν ν q µν µν A µν A µν f W 2 ¯ e s G G LL )( )(¯ )(¯ B )(¯ )(¯ e ¯ W W G r r r d f W 3 r r l HW H H )( q q l l Aν µ Aν µ ) + h HB H I I )( µ Iν Iν I I 2 µ µ µ µ HG H † † e X HW H r τ τ G G γ † † τ τ γ γ † † e † † ¯ p f W W LL X µ µ ¯ p p H H j p l ¯ RL l ¯ H H l ¯ q H H ( γ γ H H ( 1 : ( (¯ p p )( ABC ABC l ¯ q 4 : IJK IJK ( (¯ f f 8 : (   ¯ e e LR G B f W f 2 operator WB HG H HB H HW H e ll . Dimension-five ∆ . The dimension-six operators in SMEFT. The operators that conserve baryon and lepton G G f HWB H (1) qq (3) qq (1) lq W (3) lq W ledq − Q Q Q Q Q Q Q have Hermitian conjugates. The subscripts Q Q Q Q Q Q Q Q Q Q 8 : ( Q . c = . L + h Table 2 number are divided intooperators eight classes are according further to divided their into field subclasses content. according The to class-8 their chiral properties. Operators with ∆ weak-eigenstate indices. In this appendix, we reproduce the SMEFT operator basis from [ Table 1 E SMEFT operator basis JHEP10(2019)197 ) ) ) ) ) ) ) Rt Rt Rt Rt Rt Rt Rt ) ) ) d u u ) ) ) ) ) d u ) ) u d ) A A A A Rt Lt Rt Rt Lt µν µν Lt µν Rt Rt Rt Rt Rt T d T u . . d u T u T d u σ σ σ d u e u c c . . Ls Ls Ls Ls Ls Ls Ls Ls Ls Ls Ls Rs Rs Ls Ls Ls Rs Ls ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ d d d d d d u u e d d d d d d u u u )( )( )( )( )( )( )(¯ )(¯ )(¯ )( )( )( )( )( )( )(¯ )(¯ )(¯ Rr Rr ) + h ) + h Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Rr Cµ ρ Cµ ρ e e e e d d d d u u e e e e e u u e G G A A A A Lp µν µν Lp Lp Lp Lp Lp µν Lp Lp Lp Lp Lp RL LR ¯ T T T e T e e ν ν e e d σ u u σ σ u Bρ ν Bρ ν (¯ (¯ (¯ (¯ (¯ (¯ (¯ ( (¯ (¯ )( )( (¯ Lp Lp Lp Lp G G Lp Lp Lp ¯ ]. d u u u e ν e ( (¯ (¯ (¯ (¯ 3 (¯ (¯ LR LR Aν µ Aν µ ( ( e 14 G G X ,RR ,RR ,RR ,RR ,RR ,RR ,RR ,RR 8 1 8 1 8 1 8 1 S,RL S,RL S,RL νedu ed eu S,RR S,RR S,RR S,RR T,RR T,RR T,RR νedu ed eu ee νedu ed eu ABC ABC S S S S S S S S uddu uddu dd dd ud ud uu uu O O O f f O O O O O O O O O O O O O O O e G G O O . c . . . ) ) ) ) c c . . A µν A µν Rt Rt Rt Rt ) ) ) ) + h ) ) ) ) ) d u ) ) ) d u ) µν G µν G µν . A A A A F Rt F Rt Rt Rt F Rt Rt Rt Rt Rt Rt Rt Rt Rt c ) + h ) + h Rr T d T u e . Rr e T d T u u d u e d u e µ µ µ µ µ u d µ µ µ µ µ Rr µ µ µ µ µ µ Rr Rr A Rt Rt γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ e u d A A T u u + h T T µ µ µ Rs Rs Rs Rs Rs Rs Rs Rs Rs Rs µν µν µν Rs Rs Rs Rs Rs Rs ¯ ¯ ¯ ¯ γ γ ¯ ¯ e γ e d d e d d u u e d u u d u u σ σ σ ) µν µν X )(¯ )(¯ )( )( )(¯ )( )( )(¯ )(¯ )(¯ )( )(¯ )(¯ )( )(¯ ) Rs Rs σ σ )(¯ Rs Lp Lp Lp ¯ ¯ ¯ d d d ¯ ¯ e ¯ d Lr u Lr Lr Lr Lr Lr Lr Lr Lr Lr Lr Lr Lr Lr Lr Lr Lp Lp RR )( )( ¯ )( d u e ν d d u u e ν d d u u e ν ¯ d u LR µ µ µ µ µ µ µ µ )( µ µ µ µ A A ( A A Lr Lr Lr γ γ γ γ γ γ γ γ γ γ γ γ T T d d e T T eγ dγ uγ dG uG µ µ µ µ µ µ A Lp Lp Lp Lp LL Lp Lp Lp Lp Lp Lp Lp Lp O O ¯ O ¯ O γ γ γ O γ ( ¯ e γ γ ν d e ν – 60 – u e d ν d u u T (¯ (¯ ( (¯ (¯ (¯ (¯ ( (¯ ( (¯ (¯ µ Lp Lp Lp Lp Lp Lp ¯ γ ¯ ν d d u u u (¯ ( ( (¯ (¯ (¯ Lp µν u F (¯ ) . c . Lr ν ,LR ,LR ,LR ,LR ,LR ,LR ,LR ,LR ,LR ,LR 8 1 8 1 8 1 8 1 8 1 µν V,LR V,LR V,LR V,LR V,LR V,LR V,LR V,LR V,LR νedu de ue ed eu νd νu ee νe + h V V V V V V V V V V uddu uddu dd dd du du ud ud uu uu O O O O O O O O O O O O O O O O O O O Cσ X ) T Lp ν ) . ) νν ( c ( . Rt Lt ) ) ) ) ) d ) ) ) d ) ) ) ) ) ) ) ) νγ A A Rt Rt Lt Rt Rt Lt Lt Rt Lt Lt Lt O Rt Lt Lt Lt Lt T d u d T d u ) + h d u e d d u e ν d u e µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ Lt γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ ) u Rs Rs Rs Ls Ls Ls µ Rs Ls Ls Ls Ls Ls Rs Rs Rs Ls Ls Ls . Lr ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ d d d d γ e d u e d u e ν d d d u c u u ) ) . )( )( )( )( )(¯ )( )(¯ )(¯ )( )(¯ )(¯ )(¯ )( )( )( )(¯ )(¯ )(¯ Ls Cν ¯ d Lr Lr Lr Lr Lr Lr Lr Rr Rr Lr Lr Rr Rr Rr Lr Lr Rr RR LL Rr T Lp + h )( ν e ν u u d e ν e u e ν u u d e u e ν µ µ µ )( )( µ µ µ µ µ µ µ µ µ µ µ ( µ µ A A Lr γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ T e T νν ν µ µ µ LL Lp Lp Lp Lp Lp Lp Lp Lp Lp Lp Rp Rp Rp Rp RR Rp Rp O ¯ ( γ γ e γ ν ¯ e ν ν e d e ν u e ( u e d u u (¯ (¯ (¯ (¯ (¯ (¯ ( (¯ (¯ (¯ (¯ (¯ (¯ ( (¯ (¯ Lp Lp Rp ν u u (¯ (¯ (¯ . LEFT operators of dimension three and five, as well as LEFT operators of dimension six ,LL ,LL ,RR ,RR 8 1 8 1 V,LL V,LL V,LL V,LL V,LL V,LL V,LL V,LL V,LL V,LL dd uu νedu ed eu νd νu νe ee νν V,RR V,RR V,RR V,RR V,RR dd uu ed eu ee V V ud ud V V ud ud O O O O O O O O O O O O O O O O O O O Table 3 that conserve baryon and lepton number. We reproduce the list of LEFT operators up to dimension six from [ F LEFT operator basis JHEP10(2019)197 . , p W µ ) ) ) ) ) ) ) ) could γ Lt γ Rt γ Lt γ Rt γ Rt γ Rt γ Rt . as well d d d d d d u c . Ls Ls Rs Ls Rs Ls Ls F ν e e ν e e ν 2.25 )(¯ )(¯ )(¯ )(¯ )(¯ )(¯ )(¯ β Lr β Lr β Lr β Rr β Rr β Rr β Lr = 1 + h Cd Cd Cd Cd Cd Cd Cd L αT Lp αT Lp αT Lp αT Rp αT Rp αT Rp αT Lp d d d d u d u ∆ ( ( ( ( ( ( ( − αβγ αβγ αβγ αβγ αβγ αβγ αβγ , and examples for the =        6 B ∆ S,LL ddd S,LR S,LR S,LR S,RL udd ddu ddd ddd S,RR S,RR udd ddd O O O O O O O MS parameters at the scale ) ) ) ) ) ) ) ) ) Lt Lt Lt Lt Rt Lt Lt Rt Rt . Ce Ce Ce Ce Cν Cν Ce Ce Cν c ) . , which also specifies the conventions γT Ls γT Ls γT Rs γT Ls γT Ls γT Ls γT Rs γT Rs γT Ls Lt d d u d d u u u u 7 . Most standard variables and functions )( )( )( )( )( )( )( )( )( Cν . 6 c β Rr β Lr β Lr β Lr β Rr T β Rr β Lr β Rr β Rr Ls . ν = 1 + h Cd Cu Cu )( Cu Cd Cu Cu Cu Cu L Lr αT Lp αT Rp αT Lp αT Lp αT Rp αT Rp αT Rp αT Lp αT Rp u d d d d u d u d – 61 – ( ( ( ( ( ( ( ( ( Cν = 4 + h = ∆ T Lp αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ ), the four-fermion operators carry flavor indices L ν file and a subdirectory containing all the results in          ( B ), see table ∆ ri ∆ ti S,LL S,LL duu udd S,RL S,RL S,RL S,RL S,LR S,LR ddu dud duu uud duu uud S,RR S,LL duu νν , file can be used to directly read in and further process O O O O O O O O O O and si , ) ) ) ) pi ) ) Lt Lt Lt Lt ri Rt Lt u d e u ) ) ) ) ) ) ) ) , u u µν µν µν Lt µν Lt Lt Rt µ µ Rt Lt Rt Rt σ σ σ d u e γ γ σ u u d e u pi conventions. Further code conventions for SMEFT variables are . Mathematica Ls Rs Rs Rs Ls Rs Ls Rs Rs Ls Rs Rs Rs Ls c ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ e e e d d d d d d u d d . u u )(¯ )(¯ )(¯ )( )( )( )( )( )( )(¯ )( )( )(¯ )(¯ Lr Lr Lr Lr Lr Lr Rr Lr Lr Lr Lr Lr Lr Rr ν ν e e ν e Mathematica µ µ , conventions for flavor indices are given in table (denoted by Cν Cν Cν Cν Ce Ce Cν Cν µν µν µν µν = 2 + h T T r T T Lp Lp T T 5 T T Lp Lp Lp Lp Lp Lp Cγ Cγ ν ν . LEFT operators of dimension six that violate baryon and/or lepton number. ν ν ν ν Cσ ν ν Cσ Cσ Cσ ( ( ( ( ( ( L ( ( T T Lp Lp T T Lp T T Lp Lp ν Lp ν (denoted by ∆ ( ν ( ν ν ν ( and ( ( ( t p Mathematica Table 4 S,LL S,LL S,LL S,LL T,LL T,LL T,LL T,LL νd νedu νu νe S,LR S,LR S,LR S,LR V,RL νd νedu νu νe νu νd νedu νe νedu V,RR νedu In the results we include the contribution from the tree-level matching. The parameters The results for the LEFT mass matrices and dipole operators carry external flavor O O O O O O O O O O O O O O , and s , for the ordering of the flavor indices of thein coefficients the of tree-level Hermitian expressions conjugate areAn operators. the implicit renormalized dependence running ofresults all are expressed the in parameters terms on of masses, the gauge matching couplings, scale and is the vev, understood. even if The ( indices r follow listed in table labelling of Wilson coefficients are shown in table as the mass matrices,documentation gauge of couplings, the andmaterial conventions theta consists and parameters. of variable a This namestext appendix form. that is we The athe use. short results. The supplemental G Conventions for the supplementalThe complete material results for the one-loopmaterial, matching are containing provided in the digital form coefficients as supplemental of all LEFT operators listed in appendix JHEP10(2019)197 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ]) + ) + 4.7 2.6 A.8 2.57 2.56 2.56 2.25 2.13 2.57 2.17 2.56 2.56 2.10 2.56 pi ^4) + vT ^2])/ [ ^2)) + mZ mZ ])/864 + mu ) + ^2* ^2) + mZ pi vT mH ^2/ ])/ [ vT ^2 W ^2]* ] pi mu ])/8)) + Pi [2]* ^2* [ ^4* ^4) + mH [2]* Mu pi ])/16 - ^2) - ^2 + 35* mZ [2]))* mu mZ mZ [ ^4)* ^2/ mZ Sqrt mZ [\[ ]* W mu ^2))/ ri ^2* Sqrt mZ ] , Sqrt ^2) + mZ ^2* mH Log ^2) + ri Mu t [2]* ^2])/ , ^2]* mZ , mW mZ ^4)/ mZ pi ])/(432* ])/(108* [\[ mH t [2]* [ mW , Sqrt )/(48* ^4))* pi pi ^2])/(24* ^2])/(8* [2]* ^2/ [ [ Log ^2/ ])/(72* pi vT W CHu * W [ mZ Sqrt ) + mW mZ ] mu mu ] ])/(72* ri * -boson mass ( [ ^4 + 14* vT ^2* ]* Sqrt Mu Mu ri Z ^2/ ^2/ ^2)* vT [ color factor ( mu Cqu1 W mW W ^2 + ^4)* g3bar c * [2]) + ] ] ri [\[ [\[ mu ]* ) + ^2))/(48* [2]* mZ , mZ mZ mt Mu g3bar Mu ^2)* ]* – 62 – - and -boson mass ( * Nc ))/(576* pi )/(96* Log *(-17 + (32* /8)* Log -quark mass ( SU(3) number of colors Higgs vevunitary quark-mixing matrixunitary lepton-mixing matrix ( ( ( lepton massneutrino mass ( ( down-type-quark mass ( up-type-quark mass ( t H W matching scale ri mZ Sqrt ^2* [ * mZ I * , [\[ Nc Sqrt [\[ vT ^2 + * ri ^2*(32* mW vT , pi )-( mW mW g3bar pi [ Log ^4* Log CuB g3bar ^2 + * ^2 + 35* ] + ( g3bar [ mH vT ^2)* , . . . rescaled theta parameters ( mt ^2)* *(- ])/(32* mW mZ pi ^4* mZ CHD [ CHq3 ^2])/(4* ^2)* ^2* )) + mZ CHq1 ^2 + mW ^4 - 5* ri ^2* ^3 - 5* mu mW mZ ^2])/(32* mH vT , *(8* * mZ mW g3bar , . . . parameters defining evanescent scheme ( , . . . rescaled gauge couplings ( ]*(( mW mW ^2)* mH [2] + (( t mW mZ ^2)* ^2/ ^4 - 4* ^2* , * Vdag Udag , [Nu][i] ])/16 + ( mZ W ^4* [2]* ^2 - 5* , , \ [Theta]1bar [Mu]W ri ] ^2/ mZ ^2 - 5* t aEvan cf Nc vT V U m me[i] md[i] mu[i] mt mH mW \ g1bar \ , *(8* g3bar pi mZ mH mZ W mW Sqrt , [ Mu ^2 + 5* ] ^2*(4* mW i pi Sqrt ^2] + (3* ^2* pi g3bar } [ mu mW Mu mH ^2)) + ( ])/ [ 3 i i i ^4 - 40* * ^2 + [\[ ^4 + mt mH *(8* ν } } } g3bar ^2 + 5* *(8* mZ τ t b kd mW * )/16)* mW mW [\[ ri , m 2 I mW Log ]*((54* )/(4* ^2/ , Cqu8 ]*(-( , m , m , m ν Z * *(-8* W [2]* * c s µ g3bar Log Nc ] g3bar [2]* pi ) as it is given in the supplemental material: , m mW mW ri ^6 - 4* * g3bar [ mt ri † † , m ,M , m , m 1 = 3 ^2) + *(32* g1bar *(-4* *(4* * * , Mu * * , ν e d u ,... H W ,... ,... t 6.1 * ^2*(2* ^2* ^4* mZ Sqrt c F W ev m m m m T 1 1 *(((3* Sqrt mZ CuG pi pi . SMEFT variables appearing in the code with the one-loop matching results, which is ¯ ¯ a C N v V,V U, U { { { { m M M θ g µ variable code name explanation equation ref. *((3* mZ mH mt [\[ * [ [ g1bar g3bar g3bar g1bar g3bar CHWB vT g3bar g3bar g3bar In order to enable a direct comparison of the code conventions, here we print the text CHGt CHG ( ( (( ( (48* (18* ( ( 36* (3* (-( Log ((27* (144* ( ( CuW ( ( CuG be used to eliminatesimplify the certain expressions. parameters. A mixed formform is of chosen ( with the intention to Table 5 provided as supplemental material. JHEP10(2019)197 Higgs (1979) 1571 (1980) 269 (1979) 1566 43 43 B 173 , talk given at Nucl. Phys. Phys. Rev. Lett. , , Phys. Rev. Lett. , ]. ]. SPIRE IN Possible tests of time reversal invariance in [ SPIRE , April 15–18, Louvain, Belgium (2019). C5DagC5[pi,ri] Cquqd1Dag[pi,ri,si,ti] Cquqd8[pi,ri,si,ti] CuBDag[pi,ri] CHBox CHWtB IN – 63 – [ ∗  ∗ (1986) 621  rpts (1) quqd Effective Lagrangian analysis of new interactions and flavor rp uB C Kronecker delta fixed top-quark flavor index implicit sum over three generations internal flavor indices, always appearing in pairs, explicit sum over three generations (including the top quark) together with an explicit sum over three generations internal flavor index, appearing multiple times, always fixed external flavor indices, no implicit or explicit sum 5 ), which permits any use, distribution and reproduction in . Code conventions for flavor indices. ur C (1957) 517   C B 268 Operator analysis of new physics := 5 † pu Operator analysis of nucleon decay := 106 † C f  WB u pr prst prst (1) (8) † quqd quqd uB H H Table 6 ... P C C C C C coefficient code name CC-BY 4.0 Effective theories for precision Higgs and flavor physics Nucl. Phys. Baryon and lepton nonconserving processes This article is distributed under the terms of the Creative Commons , Phys. Rev. kd[pi,ri] [t] [uu,vv,ww,xx] sum[ui] [ui] [pi,ri,si,ti] , ]. ]. ]. u t uvwx u prst SPIRE SPIRE SPIRE . Examples of SMEFT Wilson coefficients appearing in the code with the one-loop matching pr δ {} {} P {} {} formula code name explanation IN IN IN [ [ conservation Beta decay [ and Effective Field Theory (HEFT 2019) H.A. Weldon and A. Zee, W. Buchm¨ullerand D. Wyler, J.D. Jackson, S.B. Treiman and H.W. Wyld, S. Weinberg, F. Wilczek and A. Zee, A.V. Manohar, [5] [6] [2] [3] [4] [1] any medium, provided the original author(s) and source areReferences credited. Table 7 results, provided as supplemental material. Open Access. Attribution License ( JHEP10(2019)197 , ]. ]. eγ → (2017) ] C 78 SPIRE 12 793 µ ] SPIRE IN 09 158 IN [ ][ ]. JHEP , JHEP , (2013) 087 (1934) 1 (1968) 1568 SPIRE Phys. Rept. Eur. Phys. J. , 10 IN 11 , Annals Phys. ][ 166 ]. , (2014) 035 arXiv:1709.04486 arXiv:1711.05270 [ JHEP [ 01 , SPIRE arXiv:1704.04504 [ IN Dimension-six terms in the ]. ][ Non-perturbative effects in Nuovo Cim. Phys. Rev. JHEP Renormalization group evolution of , ]. , DsixTools: the standard model (2018) 016 (2018) 084 ]. SPIRE dependence 03 arXiv:1008.4884 IN 01 physics beyond the standard model at one [ λ [ (2017) 405 SPIRE IN B SPIRE [ Low-energy effective field theory below the Low-energy effective field theory below the (in Italian), Renormalization group evolution of the standard Renormalization group evolution of the standard IN JHEP ]. ]. Wilson: a Python package for the running and C 77 JHEP , ][ ]. , – 64 – ]. arXiv:1312.2014 (2010) 085 [ SPIRE SPIRE SPIRE IN 10 (1985) 465 IN ]. radiation [ The SMEFTsim package, theory and tools IN SPIRE ][ β ][ IN arXiv:1810.08132 Chiral perturbation theory to one loop Chiral perturbation theory: expansions in the mass of the ]. ]. ][ Eur. Phys. J. SPIRE JHEP , B 250 (2014) 159 , , IN ][ The standard model as an effective field theory 04 SPIRE SPIRE arXiv:1810.05675 [ IN IN ]. ][ ][ JHEP SmeftFR — Feynman rules generator for the Standard Model Effective , Nucl. Phys. arXiv:1904.03204 flavio: a Python package for flavour and precision phenomenology in the SPIRE Nonlinear realizations of chiral symmetry , arXiv:1804.05033 , arXiv:1709.06492 IN [ [ [ (2019) 088 Trends to a theory of arXiv:1706.08945 ]. ]. ]. [ 01 arXiv:1704.06639 [ SPIRE SPIRE SPIRE IN IN IN arXiv:1308.2627 arXiv:1310.4838 Standard Model and beyond (2017) 070 Field Theory effective field theory toolkit matching of Wilson coefficients(2018) above and 1026 below the electroweak scale strange quark JHEP [ [ (1984) 142 electroweak scale: operators and matching [ electroweak scale: anomalous dimensions the standard model dimensionphenomenology six operators III: gauge coupling dependence and loop: complete renormalization group evolution158 below the electroweak scale model dimension six operators I:[ formalism and model dimension six operators II:[ Yukawa dependence standard model lagrangian (2019) 1 I. Brivio, Y. Jiang and M. Trott, A. Dedes et al., J. Aebischer, J. Kumar and D.M. Straub, D.M. Straub, J. Gasser and H. Leutwyler, W. Dekens, E.E. Jenkins, A.V. Manohar and P. Stoffer, A. Celis, J. Fuentes-Martin, A. Vicente and J. Virto, S. Weinberg, J. Gasser and H. Leutwyler, E.E. Jenkins, A.V. Manohar and P. Stoffer, E.E. Jenkins, A.V. Manohar and P. Stoffer, R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, J. Aebischer, M. Fael, C. Greub and J. Virto, E.E. Jenkins, A.V. Manohar and M. Trott, E.E. Jenkins, A.V. Manohar and M. Trott, I. Brivio and M. Trott, E. Fermi, B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, [8] [9] [7] [23] [24] [21] [22] [18] [19] [20] [16] [17] [14] [15] [12] [13] [11] [10] JHEP10(2019)197 , , ]. (2017) ] Nature ]. , ]. 05 -gauges Phys. Lett. ξ (1981) 189 , SPIRE ]. R Phys. Rev. IN SPIRE with the full , ][ SPIRE IN γ JHEP [ International IN , + SPIRE B 185 e st ][ decay IN → γ ][ (2016) 037 + + e ]. µ arXiv:1605.05081 05 [ → + ]. Nucl. Phys. µ SPIRE hep-ph/9410338 , IN [ , KEK-2009-10 (2009). JHEP , ][ SPIRE arXiv:1907.04923 Renormalisation-group improved ]. , (2016) 434 IN arXiv:1810.09413 ]. [ ][ arXiv:1408.3565 , talk given at the 31 Matching of gauge invariant [ Two- and three-loop anomalous dimensions (1995) 95 ]. , August 28–September 1, Vancouver, SPIRE The background field method and the S C 76 IN SPIRE transitions ]. IN c ][ Application of the background field method to the EFT approach to the electron electric dipole ][ SPIRE B 440 decay in a systematic effective field theory approach → (2019) 090 ]. Matching for FCNC effects in the flavour-symmetric IN – 65 – b (2014) 014 [ arXiv:1310.7534 eγ SPIRE 04 [ IN 10 [ → and ]. SPIRE Eur. Phys. J. s µ IN arXiv:1903.00500 , [ [ JHEP ]. A search for muon to electron conversion in muonic gold → Operator analysis for precision electroweak physics , JHEP Nucl. Phys. b SPIRE The (1983) 372 ]. , , IN (2014) 269 arXiv:1303.0754 ]. (2013) C01. ][ [ COMET conceptual design report A search for muon-to-electron conversion at J-PARC: the COMET SPIRE Improved limit on the electric dipole moment of the electron Order of magnitude smaller limit on the electric dipole moment of arXiv:1704.03888 IN SPIRE 343 New constraint on the existence of the Search for the lepton flavour violating decay [ (2006) 337 B 229 ][ (2019) 029 022 IN SPIRE processes in a systematic effective-field-theory approach [ IN 06 e Feynman rules for the standard model effective field theory in arXiv:1112.0242 The Mu2e Experiment at Fermilab collaboration, [ C 47 The background field method beyond one loop → Science PTEP , µ , (2017) 143 (2013) 201801 JHEP ]. ]. collaboration, collaboration, , Nucl. Phys. (1991) 326 06 collaboration, collaboration, , 110 arXiv:1702.03020 (2018) 355 [ SPIRE SPIRE IN arXiv:1512.02830 IN [ matrix electroweak standard model JHEP B 265 dimension-six operators for [ SMEFT 117 moment at the two-loop level of Weinberg’s dimension-six CP-odd gluonic operator 562 with dimension 6 operators analysis of experiment ACME the electron ACME Eur. Phys. J. Conference on Physics inCanada collisions (2011), (PIC 2011) Lett. MEG dataset of the MEG[ experiment SINDRUM II MEG L.F. Abbott, M.T. Grisaru and R.K. Schaefer, A. Denner, G. Weiglein and S. Dittmaier, A. Dedes et al., B. Grinstein and M.B. Wise, L.F. Abbott, J. Aebischer, A. Crivellin, M. Fael and C. Greub, T. Hurth, S. Renner and W. Shepherd, G. Panico, A. Pomarol and M. Riembau, J. de Vries, G. Falcioni, F. Herzog and B. Ruijl, G.M. Pruna and A. Signer, A. Crivellin, S. Davidson, G.M. Pruna and A. Signer, COMET collaboration, COMET collaboration, R.K. Kutschke, [42] [43] [39] [40] [41] [37] [38] [35] [36] [33] [34] [30] [31] [32] [28] [29] [26] [27] [25] JHEP10(2019)197 , Nucl. ] B 256 , Phys. ]. (1996) ] , D 41 Annals , SPIRE C 69 (1995) 3036 IN ]. [ (2019) 172 ]. Phys. Lett. , (2019) 056023 05 D 52 Phys. Rev. SPIRE hep-th/0005255 , Z. Phys. IN ]. [ SPIRE ]. , [ hep-ph/9609422 IN (1990) 66 JHEP D 100 -function in the Standard ][ , β SPIRE scheme SPIRE IN 5 Phys. Rev. IN γ (1979) 1 B 333 , ][ Top quark decay at next-to-leading (2001) 673 ][ Gauge invariance and dimensional 66 (1996) 3645 [ ]. Phys. Rev. ]. , C 18 ]. B 27 Feynman rules for fermion number violating SPIRE Lectures on elementary particles and quantum ]. Nucl. Phys. SPIRE IN Evaluation of the flavor changing vertex arXiv:1803.08001 , IN SPIRE Practical algebraic renormalization [ [ The background field formulation of the , in IN ][ -function of the strong coupling in the SM at Gauge fixing the standard model effective field β ][ SPIRE – 66 – arXiv:1508.02680 arXiv:1508.03624 IN [ Eur. Phys. J. [ Four-loop strong coupling [ , Regularization and renormalization of gauge fields 5 Dimensional renormalization and the action principle γ (1992) 467 ]. ]. On the vanishing of evanescent operators ]. ]. (2018) 251801 Acta Phys. Polon. Prog. Theor. Phys. Suppl. QCD nonleading corrections to weak decays in dimensional , (1977) 11 , (2016) 151 SPIRE The CKM parameters in the SMEFT (2016) 095 SPIRE B 387 hep-ph/9302240 120 IN SPIRE SPIRE IN 52 Local covariant operator formalism of nonabelian gauge theories and [ [ hep-ph/9907426 02 in flavor changing neutral processes IN IN ][ [ 5 ]. ]. ][ ][ γ B 762 Facts of life with Spurious anomalies in dimensional renormalization JHEP , SPIRE SPIRE Nucl. Phys. (1993) 113 Top-Yukawa effects on the (1972) 189 IN IN The renormalization of the axial anomaly in dimensional regularization , Perturbation theory anomalies , S. Deser et al. eds., MIT Press, Cambridge U.S.A. (1970). (2001) 197 [ [ ]. ]. ]. using the Breitenlohner-Maison-’t Hooft-Veltman Phys. Rev. Lett. Phys. Lett. 288 B 44 , , B 303 hep-ph/9504315 sH [ SPIRE SPIRE SPIRE → IN IN arXiv:1907.00997 arXiv:1812.08163 IN four-loop level regularization and ’t Hooft-Veltman schemes (1991) 239 Lett. Model 525 b (1990) 888 regularization with [ Phys. field theory Phys. Commun. Math. Phys. [ order in the Standard[ Model Effective Field Theory [ interactions electroweak standard model [ theory quark confinement problem A.J. Buras and P.H. Weisz, M.J. Dugan and B. Grinstein, S.A. Larin, A.V. Bednyakov and A.F. Pikelner, M.F. Zoller, T.L. Trueman, J.G. Korner, N. Nasrallah and K. Schilcher, R. Ferrari, A. Le Yaouanc, L. Oliver and J.C. Raynal, P.A. Grassi, T. Hurth and M. Steinhauser, S.L. Adler, G. ’t Hooft and M.J.G. Veltman, P. Breitenlohner and D. Maison, F. Jegerlehner, R. Boughezal, C.-Y. Chen, F. Petriello and D. Wiegand, S. Descotes-Genon et al., A. Denner, H. Eck, O. Hahn and J. Kublbeck, A. Helset, M. Paraskevas and M. Trott, T. Kugo and I. Ojima, A. Denner, S. Dittmaier and G. Weiglein, [61] [62] [58] [59] [60] [56] [57] [54] [55] [51] [52] [53] [49] [50] [47] [48] [45] [46] [44] JHEP10(2019)197 , ] ]. , Comput. (1997) SPIRE , ]. Comput. ]. IN ]. , Les Houches ][ ]. D 56 renormalization SPIRE rule in Comput. Phys. SPIRE hep-ph/9606222 IN SPIRE Comput. Phys. , 1 2 [ IN -quark running , IN ][ ]. MS b SPIRE ][ = ][ decay in SMEFT IN ]. I b ¯ ][ b ∆ ]. ]. SPIRE Phys. Rev. → , On the IN ]. ]. 3 SPIRE (1997) 311 [ h (2018) 095014 IN SPIRE SPIRE arXiv:1612.00660 ][ IN α/m [ ]. 479 IN , July 3–28, Les Houches, France ]. SPIRE SPIRE ][ [ D 98 IN IN ]. hep-ph/0406019 , in the proceedings of the ][ ][ arXiv:1503.01469 ]. Gauge-independent (1991) 345 [ SPIRE arXiv:1505.02646 [ -Hamiltonian in the next-to-leading order SPIRE hep-ph/0204085 [ IN [ IN 64 SPIRE ][ (2017) 463 = 2 ][ IN Renormalization of lepton mixing for SPIRE (1982) 419 | New developments in FeynCalc 9.0 ][ S IN Phys. Rev. ∆ , NLO corrections to hep-ph/9412375 ][ | (2005) 269 Flavor mixing, gauge invariance and wave function (2015) 276 Majorana neutrino masses in the renormalization [ B 916 – 67 – (2015) 151 Lect. Notes Phys. B 115 , FEYN CALC: computer algebraic calculation of arXiv:1607.07352 ]. 197 07 [ (2002) 076002 arXiv:1310.1921 B 705 arXiv:1601.01167 [ [ FeynRules — Feynman rules made easy ]. (1995) 39 hep-ph/9604330 SPIRE ]. [ JHEP On one-loop corrections in the standard model effective field D 66 IN arXiv:0806.4194 [ Evanescent operators, scheme dependences and double Nucl. Phys. , The complete hep-ph/0012260 Two-loop calculations in QCD and the [ Phys. Lett. , [ SPIRE (2016) 115 , SPIRE IN B 455 case arXiv:1904.06358 Comput. Phys. Commun. IN 09 ][ ) (2016) 432 (2014) 2250 Nucl. Phys. [ , ][ (1996) 27 , γ γ Phys. Rev. 207 185 FeynRules 2.0 — A complete toolbox for tree-level phenomenology Effective field theories The HQET/NRQCD Lagrangian to order Introduction to effective field theories → , (2009) 1614 (2001) 418 JHEP h , B 476 Package-X: a Mathematica package for the analytic calculation of one-loop Γ( Nucl. Phys. 180 140 (2019) 173 Generating Feynman diagrams and amplitudes with FeynArts 3 , Comput. Phys. Commun. ]. , arXiv:1804.05863 08 hep-ph/9701294 [ SPIRE arXiv:1807.04283 IN group equations for lepton[ flavor violation renormalization summer school: EFT in(2018), Particle Physics and Cosmology Majorana neutrinos integrals [ 230 Feynman amplitudes Phys. Commun. Phys. Commun. Commun. JHEP theory; the Commun. in the 2HDM mass in QCD and the SM insertions non-leptonic weak decays Nucl. Phys. S. Davidson, M. Gorbahn and M. Leak, D. Espriu, J. Manzano and P. Talavera, A.V. Manohar, A. Broncano, M.B. Gavela and E.E. Jenkins, A.V. Manohar, A.V. Manohar, R. Mertig, M. B¨ohmand A. Denner, V. Shtabovenko, R. Mertig and F. Orellana, H.H. Patel, A. Alloul et al., T. Hahn, C. Hartmann and M. Trott, N.D. Christensen and C. Duhr, A. Denner, L. Jenniches, J.-N. Lang and C. Sturm, A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner and O.L. Veretin, J.M. Cullen, B.D. Pecjak and D.J. Scott, N. Tracas and N. Vlachos, S. Herrlich and U. Nierste, S. Herrlich and U. Nierste, [80] [81] [78] [79] [76] [77] [73] [74] [75] [71] [72] [69] [70] [66] [67] [68] [64] [65] [63] JHEP10(2019)197 , D D , ] sγ Nucl. , 11 ]. → B 347 b Phys. Rev. Phys. Rev. , ]. SPIRE JHEP , , ]. IN (1981) 297 ][ hep-th/0612033 ]. SPIRE Nucl. Phys. 65 [ , SPIRE ]. IN ]. extensions of the IN ][ ]. couplings from ][ SPIRE ]. U(1) SPIRE IN SPIRE [ × IN IN SPIRE [ W W γ R ]. ][ IN SPIRE ][ (2006) 116003 IN arXiv:1210.5296 [ ]. [ ]. SU(2) SPIRE (1982) 1 On the formulation of quantized field × Prog. Theor. Phys. (1982) 163 IN D 74 , L arXiv:0901.0114 ][ hep-ph/9310332 0 73 SPIRE [ [ SPIRE ¯ ]. K 140 IN IN Effective Hamiltonian for weak radiative B ]. [ ↔ (2013) 263 mixing in the Weinberg-Salam model: effective ][ SU(2) The renormalization group evolution of the CKM (1981) 2001 Renormalization of mixing angles ]. hep-ph/0402130 0 ]. Z [ – 68 – SPIRE 85 hep-ph/9810350 K SPIRE IN [ Phys. Rev. Strong CP-violation without instantons (2009) 208 IN , Physical renormalization condition for the quark mixing ][ D 23 Gauge theories of the strong and electroweak interaction SPIRE (1994) 5894 [ and SPIRE decay in IN (1988) 138 ¯ Radiative corrections to Higgs decays in the extended ν IN Photon- Annals Phys. ][ hep-ph/9309228 ν sγ , Mixing renormalization in Majorana neutrino theories [ + B 673 D 49 π Constraints on the anomalous → Simple on-shell renormalization framework for the A novel formulation of Cabibbo-Kobayashi-Maskawa matrix (1999) 197 Radiative corrections in precision electroweak physics: a hep-ph/9601390 b (2004) 033002 B 202 → Renormalization of the quark mixing matrix Effects of superheavy quarks and leptons in low-energy weak gauge [ Phys. Rev. (1955) 205 ]. + , 3 C 9 Rev. Mod. Phys. 1 K − , Prog. Theor. Phys. Suppl. ]. , D 70 (1994) 165 , hep-ph/0203210 (1981) 1772] [ = ¯ µ [ SPIRE Phys. Lett. Phys. Rev. µ Electroweak theory. Framework of on-shell renormalization and study of a , , IN 65 [ Phys. Lett. SPIRE (1996) 286 → Gauge and scheme dependence of mixing matrix renormalization , arXiv:1808.03466 IN L B 320 [ [ K Nuovo Cim. ]. Eur. Phys. J. Phys. Rev. , , , B 474 (1981) 469 (2002) 115013 SPIRE Erratum ibid. IN [ Standard Model Phys. Lett. 23 meson decay processes charges and the B.G. Teubner, Stuttgart, Germany (2001). (2018) 104 Weinberg-Salam model matrix matrix historical perspective 65 theories higher order effects (1990) 203 Phys. Cabibbo-Kobayashi-Maskawa matrix [ renormalization X.-G. He and B. McKellar, P.L. Cho and M. Misiak, B. Grinstein, R.P. Springer and M.B. Wise, T. Inami and C.S. Lim, L. Baulieu and R. Coquereaux, M. B¨ohm,A. Denner and H. Joos, H. Georgi, T. Tomaras and A. Pais, A. Denner, S. Dittmaier and J.-N. Lang, J. Fleischer and F. Jegerlehner, A. Denner, E. Kraus and M. Roth, A. Sirlin and A. Ferroglia, A. Pilaftsis, H. Lehmann, K. Symanzik and W. Zimmermann, C. Balzereit, T. Mannel and B. Plumper, A. Denner and T. Sack, B.A. Kniehl and A. Pilaftsis, B.A. Kniehl and A. Sirlin, K.I. Aoki et al., B.A. Kniehl and A. Sirlin, [99] [97] [98] [94] [95] [96] [92] [93] [90] [91] [87] [88] [89] [85] [86] [83] [84] [82] [100] JHEP10(2019)197 B , , ]. SPIRE ]. Nucl. Phys. IN , ]. ][ (1991) 2223 SPIRE (2011) 059903] SPIRE IN IN ]. ][ ]. D 43 ][ D 84 SPIRE ]. SPIRE IN Anomalous moments of quarks IN ][ arXiv:1303.3156 ][ [ Phys. Rev. SPIRE , Signets of CP-violation IN ]. ]. ][ arXiv:1309.7822 Weak decays beyond leading logarithms coupling effects in the weak radiative Erratum ibid. -violating dimension-6 electroweak gauge [ [ arXiv:1903.03625 [ SPIRE (2013) 149 CP SPIRE W tb IN IN 05 ][ [ Studying trilinear gauge couplings at next linear arXiv:1301.2597 hep-ph/9612378 – 69 – [ [ The evolution of Weinberg’s gluonic CP violation JHEP (2014) 076004 (2008) 077501 hep-ph/9512380 , Anomalous [ Searches for (2019) 051801 (1990) 1709 ]. D 89 Renormalization group running of dimension-six sources of (2013) 004 (1998) 365 D 78 123 64 ]. 06 C 2 SPIRE arXiv:1703.04751 Probing charged matter through Higgs diphoton decay, gamma ray [ IN (1996) 1125 CP violation in Higgs-gauge interactions: from tabletop experiments to ][ SPIRE JHEP 68 Phys. Rev. IN Phys. Rev. , , Right-handed charged currents in the era of the Large Hadron Collider [ , (2017) 086 Phys. Rev. Lett. Phys. Rev. Lett. ]. Eur. Phys. J. , , , 05 (1991) 311 SPIRE IN arXiv:0802.1413 Rev. Mod. Phys. boson operators collider operator the LHC and leptons from nonstandard[ W W gamma couplings lines and EDMs parity and time-reversal violation B-meson decay [ JHEP 357 G. Buchalla, A.J. Buras and M.E. Lautenbacher, B. Gripaios and D. Sutherland, G.J. Gounaris and C.G. Papadopoulos, E. Braaten, C.-S. Li and T.-C. Yuan, V. Cirigliano et al., F. Boudjema, K. Hagiwara, C. Hamzaoui and K. Numata, J. Fan and M. Reece, W. Dekens and J. de Vries, S. Alioli et al., A. De Rujula, M.B. Gavela, O. Pene and F.J. Vegas, B. Grzadkowski and M. Misiak, [111] [108] [109] [110] [106] [107] [104] [105] [102] [103] [101]