Eur. Phys. J. C (2016) 76:370 DOI 10.1140/epjc/s10052-016-4207-5

Regular Article - Theoretical Physics

μ → eγ and matching at mW

Sacha Davidson1,2,3,a 1 IPNL, CNRS/IN2P3, 4 rue E. Fermi, 69622 Villeurbanne Cedex, France 2 Université Lyon 1, Villeurbanne, France 3 Université de Lyon, 69622 Lyon, France

Received: 18 March 2016 / Accepted: 14 June 2016 / Published online: 4 July 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Several experiments search for μ ↔ e flavour attempt to reconstruct the fundamental Lagrangian of New change, for instance in μ → e conversion, μ → eγ and Physics from the operator coefficients. This is probably not μ → eee¯ . This paper studies how to translate these experi- feasible, but could give interesting perspectives. A first step mental constraints from low energy to a New Physics scale in this “bottom-up” approach, explored in this paper, is to M  mW . A basis of QCD × QED-invariant operators (as use Effective Field Theory (EFT) [31] to translate the exper- appropriate below mW ) is reviewed, then run to mW with imental bounds to the coefficients of effective operators at one-loop Renormalisation Group Equations (RGEs) of QCD the New Physics scale M > mW . and QED. At mW , these operators are matched onto SU(2)- The goal would be to start from experimental constraints invariant dimension-six operators, which can continue to run on μ–e flavour change, and obtain at M the best bound up with electroweak RGEs. As an example, the μ → eγ on each coefficient from each observable. These constraints bound is translated to the scale M, where it constrains two should be of the correct order of magnitude, but not precise sums of operators. The constraints differ from those obtained beyond one significant figure. This preliminary study restricts in previous EFT analyses of μ → eγ , but they reproduce the the experimental input to the bound on BR(μ → eγ), and expected bounds on flavour-changing interactions of the Z makes several simplifications in the translation up to the New and the Higgs, because the matching at mW is pragmatically Physics scale M. Firstly, the EFT has three scales: a low scale performed to the loop order required to get the “leading” mμ ∼ mb, the intermediate weak scale mW , and the high contribution. scale M. Secondly, at a given scale, the EFT contains lighter Standard Model particles and dimension-six, gauge-invariant operators (one dimension-seven operator is listed; however, 1 Introduction dimension-eight operators are neglected). The final simplifi- cation might have been to match at tree level, and run with Neutrino masses and mixing angles imply that “New” one-loop Renormalisation Group Equations (RGEs). How- Physics from beyond the Standard Model(SM) must be ever, a bottom-up EFT should reproduce the results of top- present in the lepton sector, and must induce charged Lepton down model calculations, and it is straightforward to check Flavour Violation (LFV; for a review, see [1]). However, nei- that one- and two-loop matching is required at mW to obtain ther LFV nor the origin of neutrino masses has yet been dis- the correct bounds from μ → eγ on LFV interactions of covered. This study assumes that the required new particles the Z and Higgs. So the matching at mW is performed to the are heavy, with masses at or beyond M > mW . In addition, order required to get the known bounds. between mW and M, there should be no other new particles The paper is organised in two parts: the Sects. 2–4 con- or interactions which affect the LFV sector. One approach struct some of the framework required to obtain experimental to identifying this New LFV Physics, is to construct a moti- constraints on SU(2)-invariant operator coefficients at mW , vated model, and identify its signature in observables [2–30]. then Sect. 5 focusses on using, checking and improving this A more pragmatic approach, which requires optimism but formalism to obtain bounds from μ → eγ on operator coef- no model-building skills, is to parametrise the New Physics ficients at M. The formalism can be organised in four steps: at low energy with non-renormalisable operators, map the matching at mμ, running to mW , matching at mW , then run- experimental constraints onto the operator coefficients, and ning up to the New Physics scale M. Section 2 reviews the basis of QCD×QED-invariant operators, as appropriate a e-mail: [email protected] 123 370 Page 2 of 19 Eur. Phys. J. C (2016) 76 :370 below mW . These operators, of dimension five, six and seven, contribution of the LFV Higgs operator to LFV Z couplings describe three- and four-point functions involving a μ,an was beautifully studied in [45]. There are also many closely e and any other combination of flavour-diagonal light par- related works in the quark sector, reviewed in [46,47]. For ticles. To complete the first step, the experimental bounds instance, the QED anomalous dimension matrix for various should be matched onto these operator coefficients; how- vector four-quark operators is given in [48], and matching ever, this is delayed till Sect. 5, where only the bound on at mW of flavour-changing quark operators is discussed in μ → eγ is imposed on the dipole coefficients (the bounds [49]. However, colour makes the quarks different, so it is not from μ → e conversion and μ → eee¯ are neglected for always immediate to translate the quark results to leptons. simplicity; the strong interaction subtleties of matching to μ → e conversion are discussed in [33–36]). Section 3 dis- cusses the second step, which is to run the coefficients up to 2 A basis of μ–e interactions at low energy mW with the RGEs of QED and QCD. Appendix B gives the anomalous dimension matrix mixing the scalar and tensor 2.1 Interactions probed in muon experiments operators to the dipole (which is responsable for μ → eγ ). The anomalous dimension matrix for vector operators is Experiments searching for lepton-flavour change from μ to neglected for two reasons: although vectors contribute at tree e, probe three- and four-point functions involving a muon, level to μ → e conversion and μ → eee¯ , these experimen- an electron and one or two other SM particles. I focus here tal bounds are not included, and the leading order mixing on interactions that can be probed in μ → eγ , μ → eee¯ of vectors to the dipole is at two-loop in QED, whereas the and μ → e conversion, meaning that the interactions are running here is only performed at one-loop. The next step is otherwise flavour diagonal, and there is only one muon (so to match these operators at mW onto the Buchmuller–Wyler K →¯μe and other meson decays is not considered). [37] basis of SU(2)-invariant operators as pruned in [38], These “new physics” interactions can be written as non- which is referred to as the BWP basis. The tree-level match- renormalisable operators involving a single μ, and some ing for all operators is given in Sect. 4; if this is the leading combination of e,γ,g, u, d, or s. The operators should be contribution to the coefficients, then imposing SU(2) invari- QED and QCD invariant (because we are intested in LFV, ance above mW predicts some ratios of coefficients below not departures from the SM-gauge symmetries), and can be mW , as discussed in Sect. 4.2. Section 5 uses the formalism of any dimension (because the aim is to list the three-point of the previous sections to translates the experimental bound and four-point interactions that the data constrains). The list, on BR(μ → eγ)to sums of SU(2)-invariant operator coeffi- which can be found in [1,33–35] but with different names, cients at mW . Then a few finite-loop contributions are added, is: and the coefficients are run up to M, using a simplified ver- eμ αβ dipole O = mμ(eσ P μ)Fαβ sion of the one-loop QCD and electroweak RGEs [39,40,43]. D,Y Y Finally, Sect. 6 discusses various questions arising from this Oeμee = 1( γ α μ)( γ α ), 4 lepton YY e PY e PY e study, such as the loop order required in matching at mW , 2 μ 1 α α whether the non-SU(2)-invariant basis is required below mW , Oe ee = ( γ μ)( γ ), YX e PY e PX e and the importance of QED running below mW . 2 Oeμee = ( μ)( ) Many parts or this analysis can be found in previous lit- S,YY ePY ePY e erature. Czarnecki and Jankowski [41] emphasized the one- eμuu 1 α α loop QED running of the dipole operator (neglected in the 2 lepton 2 quark O = (eγ PY μ)(uγ PY u), YY 2 estimates here), which shrinks the coefficient at low energy. eμuu 1 α α Degrassi and Giudice [42] give the leading order QED mixing O = (eγ PY μ)(uγ PX u), YX 2 of vector operators to the dipole, which is also neglected here, eμuu O = (eP μ)(uP u), because it arises at two loop. In an early top-down analysis, S,YY Y Y Oeμuu = ( μ)( ), Brignole and Rossi [15] calculated a wide variety of LFV pro- S,YX ePY uPX u cesses as a function of operator coefficients above m , with- Oeμuu = ( σ μ)( σ ) W T,YY e PY u PY u out explicit Renormalisation Group running and a slightly eμdd 1 α α redundant basis. Pruna and Signer [43] studied μ → eγ in O = (eγ PY μ)(dγ PY d) YY 2 EFT, focussing on the electroweak running above mW , which eμdd 1 α α they perform in more detail than is done here. However, they O = (eγ PY μ)(dγ PX d), YX 2 do not obtain the bounds on the LFV couplings of the Z and eμdd O , = (ePY μ)(dPY d) Higgs that arise here in matching at mW . Various one-loop S YY μ → eγ Oeμdd = ( μ)( ), contributions to were calculated in [44], without S,YX ePY dPX d organising them into running and matching parts. Finally, the 123 Eur. Phys. J. C (2016) 76 :370 Page 3 of 19 370

eμdd O , = (eσ PY μ)(dσ PY d), eμll 1 α α T YY 4 lepton O = (eγ PY μ)(lγ PY l), YY 2 Oeμss = 1( γ α μ)( γ α ) YY e PY s PY s eμll 1 α α 2 O = (eγ PY μ)(lγ PX l), YX 2 Oeμss = 1( γ α μ)( γ α ), YX e PY s PX s eμll 2 O , = (ePY μ)(lPY l), eμss S YY O = (eP μ)(sP s) eμττ S,YY Y Y O = (eP μ)(τ P τ), μ S,YX Y X Oe ss = (eP μ)(sP s), S,YX Y X eμττ μ O = (eσ P μ)(τσP τ), Oe ss = ( σ μ)( σ ) T,YY Y Y T,YY e PY s PY s Oeμqq = 1( γ α μ)( γ α ), eμ 1 αβ 2 lepton 2 quark YY e PY q PY q 2 lepton 2 boson O = (eP μ)Gαβ G , 2 GG,Y M Y eμqq 1 α α μ O = (eγ P μ)(qγ P q), Oe = 1 ( μ) αβ YX Y X ˜ , ePY Gαβ G (1) 2 GG Y M Oeμqq = ( μ)( ), S,YY ePY qPY q eμqq O , = (ePY μ)(qPX q), where X, Y ∈{L, R}, and X = Y . These operators are cho- S YX Oeμqq = ( σ μ)( σ ), sen, using Fierz and other spinor identities, to always have the T,YY e PY q PY q (2) lepton flavour-change inside a spinor contraction. Notice also where l ∈{μ, τ}, q ∈{c, b}, X, Y ∈{L, R}, and X = Y . that, following Kuno and Okada [1], the dipole is normalised Including these operators introduces a second “low” scale with a muon Yukawa coupling. The four-fermion operators into the EFT, which in principle changes the running and are labelled with the fermion flavours in superscript, and in requires matching at this second low scale mτ . The running the subscript is the type of Lorentz contraction (scalar, tensor is discussed in the next section. Since the matching is at or vector – except the vector case is implicit), followed by tree level, the operators present below mτ have the same the chiralities of the two fermion contractions in subscript. coefficient just above mτ . Were the dipole to be matched The Lorentz contraction – dipole, scalar, tensor or vector – at one loop, then at mτ , one should compute the finite part will be used throughout this paper to categorise operators. of the diagrams [44] obtained by closing the heavy fermion μ μ μττ The operator coefficients have the same index structure, so loop of the tensors Oe bb , Oe cc and Oe , and attaching ijkl Oijkl T,YY T,YY T,YY CXX is the coefficient of XX, which is a vector contraction a photon (and also there could be similar finite contributions of fermions of chirality X. from four-fermion operators at mμ). Also, scalar operators All the operators appear in the Lagrangian with a coef- involving b, c quarks would match at one loop onto OGG,Y − / 2 ficient C M , and the operator normalisation is cho- [50], as outlined in [51]. sen to ensure that the Feynman rule is −iC/M2.This implies a judicious distribution of 1 s, which is discussed in 2 3 Running up to mW Appendix B. Obtaining constraints from data on the operator coeffi- The operators of Eqs. (1) and (2) can evolve with scale due to μ → cients is reviewed in [1], and e conversion is discussed QED and QCD interactions. QCD effects can be significant, μ → γ in [33–35]. Searches for e probe the dipole operator, and should be resummed, but fortunately they only change μ → ¯ eee probes the four-lepton operators and the (off-shell) the magnitude of operator coefficients, without mixing one μ → dipole, and e conversion probes the two-quark–two- operator into another. This will be taken into account by mul- lepton, diboson and dipole operators. It is interesting that tiplying two-lepton–two-quark operators by an appropriate these three processes are sensitive to almost all the three-and factor (following Cirigliano et al. [35]). The effects of QED four-point functions that involve one muon, any of the lighter running are usually small, of order αem/π, but interesting fermions, or photons or gluons. The only three- or four-point because they give operator mixing. Therefore the QED renor- interactions not probed at tree level are the two-photon inter- malisation of individual operator coefficients is neglected, actions, and only the mixing is included. The scale at which the operators of Eqs. (1) and (2)start eμ αβ eμ αβ O = (eP μ)Fαβ F , O = (eP μ)Fαβ F . running is variable. The lepton operators of Eq. (1) will FF,Y Y F F˜,Y Y start their QED running at mμ, whereas those of Eq. (2) start at mτ . The the two-lepton-two-b operators start run- 2.2 Including heavy fermions ning up at mb. For simplicity, the remaining two-quark–two- lepton operators are taken to start running up at mτ ; that At a slightly higher scale, operators containing c, b μ and τ is, the experimental bounds are assumed to apply at a scale bilinears should be included. These additional operators are: ∼ mτ . 123 370 Page 4 of 19 Eur. Phys. J. C (2016) 76 :370

3.1 Defining the anomalous dimension matrix The vector two-quark–two-lepton operators do not renor- malise under QCD, because the quark vector currents are After including one-loop corrections in MS, the operator conserved: that is, diagram 4 of Fig. 1, with f2 a quark and coefficients will run with scale μ according to the photon replaced by a gluon, cancels against the wave- ∂ α α function renormalisation. However, the same diagram, plus μ C = s Cs + em C ∂μ π π (3) wave-function renormalisation, causes the scalar operators 4 4 s to run like masses in QCD (γ = 6CF ): where the coefficients of all the operators listed in the pre-   4  β vious section have been organised into a row vector C, and eμqq eμqq αs(mW ) 0 α ( ) = ( ) em  CS,XY mW CS,XY mq 4π is the anomalous dimension matrix, which is calculated αs(mq ) 1 as discussed in [47]. μ m (m ) = e qq( ) q W Equation (3) can be approximately solved, by neglecting CS,XY mq (12) mq (mq ) the scale dependence of αem and defining the eigenvalues of s {γ s } for q ∈{u, d, s, c, b} and X, Y ∈{L, R}, so I follow [35] the diagonal to be A ,as γ s in normalising the coefficients with running quark masses as  − A  α ( ) 2β α after the last equality. However, for the light quarks (u, d, s), ( ) s mW 0 δ − em mW C A mW AB [ ]AB log μ τ α ( ) α (mτ ) 4π mτ the QCD running is stopped at m , that is, s mq is s  α ( ) α2 replaced by s mτ . Finally, diagram 4 vanishes for the ten- em 2 mW + [ ] log +··· = C (mτ ) (4) sor four-fermion operators, but the wave-function diagrams π 2 AB B 32 mτ cause the tensor operators to run as β = − / β where 0 11 2N f 3fromtheQCD -function, and   −4/3 m β W . eμqq eμqq αs(mW ) 0 log mτ 3 85. ( ) = ( ) . CT,XY mW CT,XY mτ (13) It is convenient to separate the vector of coefficients below αs(mτ )  mW , C(< mW ), into subvectors:  In QED running, the vector operators mix among them- μ (< ) = qi , u , d , τ ,  , e , selves, but they have no mixing into or from the scalars and C mW C CV CV CV C CV V V tensors. The scalars renormalise themselves and mix to ten- qi u d τ μ C , C , C , C , C μ, sors and in some cases to the dipoles, and the tensors renor- S S S S S  e , qi , u , d , τ ,  ,  , malise themselves and mix to scalars and dipoles (which are CSe C CT CT CT CD CGG (5)  T  dimension five, so they do not mix to other operators). So the  f = eμff, eμff, eμff, eμff , anomalous dimension matrix can be written CV CLL CRR CLR CRL (6)       f eμff eμff eμff eμff  = V 0 C = C , , C , , C , , C , (14) S S LL S RR S LR S RL 0 ST D for f ∈{q , u, d,τ}, (7)  i  with μ μ ⎡ q ,q q ,q ⎤ l e ll e ll γ i j 00000γ i j 00000 C = C , , C , for l ∈{e,μ}, (8) S,S S,T Sl S LL S RR ⎢ γ u,u γ u,u ⎥   ⎢ 0 S,S 0000 0 S,T 0000⎥ μ μ ⎢ d,d d,d ⎥  f e ff e ff ⎢ 00γ , 000 0 0γ , 000⎥ C = C , , C , (9) ⎢ S S τ,τ S T τ,τ ⎥ T T LL T RR ⎢ 000γ 00 0 00γ 00⎥   ⎢ S,S S,T ⎥ ⎢ γ μ,μ γ μ, ⎥  eμ eμ ⎢ 0000S,S 0 0 000S,D 0 ⎥ C = C , C , ⎢ e,e e, ⎥ D D,L D,R (10) ⎢ 00000γ , 0 000γ , 0 ⎥  = , S S , S D,   ST D ⎢ qi qi qi qi qi ⎥ ⎢ γ , 00000γ , 000γ , 0 ⎥ eμ eμ ⎢ T S , T T , T ,D ⎥  = , . ⎢ 0 γ u u 0000 0γ u u 00γ u 0 ⎥ CGG CGG,L CGG,R (11) ⎢ T,S T,T T,D ⎥ ⎢ γ d,d γ d,d γ d, ⎥ ⎢ 00T,S 000 0 0T,T 0 T,D 0 ⎥ ⎢ γ τ,τ γ τ,τ γ τ, ⎥ QCD running concerns the two-lepton–two-quark oper- ⎢ 000T,S 00 0 00T,T T,D 0 ⎥ ⎣ γ ⎦ ators, and the two-lepton-two-gluon operators. The gluon 0 00000 0 000D,D 0 0 00000 0 00000 operators are neglected here, because they do not contribute (15) at LO to μ → eγ , which is the example considered in Sect. 5, and because one-loop matching (not performed here) seems where the first super- and subscript on the γ submatrices indicated in order to correctly account for these operators. belongs to the coefficient labelling the row, and the second indices identify the column. Section 5 runs up the dipole 1 Generically, the one-loop corrections to an operator Q will gener- coefficient, for which the matrix V is not required, so it ate divergent coefficients for other operators {B}. If one computes the one-loop corrections to the amputed Green’s function for the operator will be given in a subsequent publication. For QED mixing of four-fermion operators among them- Q,withn external legs, and Feynman rule ifQ Q, these can be writ- α 1 [] =− [ + n δ ] ten as ifQ 4π B bQBB.Then QB 2 bQB 2 a QB where selves and to the dipole, the relevant diagrams are in − α = μ ∂ ∈ 4π a Z ∂μ Z,andZ renormalises the wave function. Fig. 1, where the gauge boson is the photon, and f2 123 Eur. Phys. J. C (2016) 76 :370 Page 5 of 19 370

µe µe µe µe

f1 f1

f2 f2 f2 f2 f2 f2 f2 f2

µe µe µe µe

f2 f2 f2 f2 f2 f2 f2 f2

Fig. 1 Examples of one-loop gauge vertex corrections to four-fermion the Lorentz or gauge structure of the operators, not the flavour struc- operators. The first two diagrams are the penguins. The last six dia- ture. Missing are the wave-function renormalisation diagrams; for V ±A grams contribute to operator mixing and running, but can only change Lorentz structure in the grey blob, this cancels diagrams 3 and 4

{u, d, s, c, b, e,μ,τ}. These diagrams allow one to com- 1. In some cases, the dimension-six and -eight contributions pute the γ -submatrices of Eq. (14). The results are given arise at the same loop order, but the dimension-six part is in Appendix B. For the second diagram of Fig. 1, f1 = e,μ, from matching, whereas the dimension-eight term arises because Fierz transformations were used to obtain a basis in running and is log2-enhanced. The ratio of dimen- where the μ–e flavour change is within a spinor contraction. sion six to eight is then ∝ z ln2 z, which is ∼0.2for M ∼ 10v. 2. The couplings of the New Physics are unknown, and could have steep hierarchies. In the 2HDM, the heavy 4Atm W Higgs couplings to light fermions can be O(1), rather than of order the fermions’ SM Yukawa coupling. This Above the “intermediate”, weak scale of the EFT, m W increase is parametrised in the 2HDM by tan β, which in m m , two things differ with respect to the low energy h t some cases enhances the dimension-eight operators with theory: the theory and non-renormalisable operators should respect to dimension six. In some 2HDMs, tan β  50, now respect the weak SU(2) symmetry, and the particle con- which I take as a reasonable estimate of the possible tent is extended to include the weak gauge bosons, the higgs, hierarchy of couplings between dimension-six and either and the top. The additional requirement of SU(2) invari- operators. ance will reduce the number of possible four-fermion oper- ators, whereas adding new degrees of freedom (h, W, Z, t) allows more flavour-changing operators involving only two In a generic model, these two enhancements could com- fermions. bine, and other sources of enhancement could perhaps arise.  ∼ v 4.0.1 Neglecting dimension-eight operators So I impose the requirement that M 20 TeV ( 100 ), in the hope that this suppresses dimension-eight operators in The EFT above mW is an expansion in the inverse New many models. Physics scale 1/M, where the lowest order operators that are lepton flavour-changing, but number-conserving, appear at dimension six; they are listed in Appendix D. It is convenient 4.1 Tree matching onto SU(2)-invariant operators to neglect the next order operators, which would appear at dimension eight, because they are numerous, and their RGEs The coefficients of the four-fermion operators from below are unknown. So it is interesting to explore how small must be mW ,giveninEqs.(1) and (2), should be matched at mW the ratio v/M, to justify a parametrisation using dimension- onto the coefficients of the SU(2)-invariant BWP basis, which six operators. are listed in Appendix D. The coefficients on the left of the This question was studied for μ → eγ in a 2 Higgs Dou- equalities are from below mW , the coefficients on the right are blet Model(2HDM) with LFV [64]. Naively, the dimension- SU(2) invariant. Both sets of coefficients should be evaluated 2 2 eight operators are suppressed by v /M ≡ z. However, two at mW , and the fermion masses which appear in the matching enhancements arise: conditions should also be evaluated at mW . 123 370 Page 6 of 19 Eur. Phys. J. C (2016) 76 :370   4.1.1 Dipoles eμττ = eμττ + eττμ − eμ + eμ e , C C C C , C , gL (21) LL LL LL HL 3  HL 1 eμll = eμll − eμ + eμ e , Above mW , there is a dimension-six dipole operator for CLL CLL 2 CHL,3 CHL,1 gL (22) hypercharge, and another one for SU(2). They are given in eμ v eμ m CEH Eqs. (100) and (101). The coefficient of the photon dipole C , =− , (23) μ μ S RR 2 Oe Oe mh operators D,R, D,L from below mW are the linear combi- α α α μe∗ nations (the photon is A = c B +s W , and the negative μττ τμ τ mτ C v W W 3 Ce =−2C e − EH , (24) sign is from τ ): S,LR LE 2 3 mh μ eμ eμ eμ eμ ∗μe ∗μe e = − , = − . eμττ eττμ mτ C v CD,R cW CeB sW CeW CD,L cW CeB sW CeW =− − EH , CS,RL 2CLE 2 (25) mh However, rather than using the Hypercharge and SU(2) μe∗ eμ eμ m C v dipoles O and O , I follow [43], and use the photon eμ =− EH , eB eW CS,LL 2 (26) and Z dipoles above mW , merely changing the names of the mh photon dipole coefficient eμττ = , CT,RR 0 (27) μττ ij = ij = ij − ij , Ce = 0, (28) CD,R Ceγ cW CeB sW CeW T,LL ij =− ij − ij ∈{ ,μ,τ} ∈{ ,μ} = θ CeZ sW CeB cW CeW (16) where e , l e , sW sin W , and the Feyn- g e e man rule for Z couplings to leptons is i (g PL + g PR), where ij ∈{eμ, μe}. (Notice that since the operators are 2cW L R with Oeμ Oμe +. . . Oij D,R and D,R,the h c gives the D,L ). e =− 2 , e = − 2 . gR 2sW gL 1 2sW (29) 4.1.2 Four-lepton operators In the case of vector operators involving three muons or The BWP basis contains only the “vector” four-lepton oper- three electrons of the same chirality, there can be two Z- ators given in Eqs. (94), (95) and (96). There are also new exchange diagrams (u and t channel), which can give a 2 with (¯μ)(ττ)¯ dimension-six interactions of the W, Z and h, described by respect to operators involving e .FromEq.(27) and the operators of Eqs. (102), (103), (104) and (99), which will (28), the tensor coefficients vanish at tree matching. Nonethe- less, these operators are important below mW , because as contribute to four-lepton operators below mW in matching out the Z and h. seen in the previous section, scalars mix to tensors, and ten- There are a few curiosities related to the flavour index sors to the dipole. structure below and above mW . First, since the basis below mW was defined with the e–μ indices inside a spinor contrac- 4.1.3 Two-lepton–two-quark operators tion, there is a scalar operator from below mW which must be Fierzed as given in Eq. (98). Also, there are more dis- Two issues about the CKM matrix V arise in matching opera- tinct flavour structures for operators constructed with SU(2) tors involving quarks at mW : does V appear in the coefficients Oeμff above or below m , and are the quark doublets in the u or doublets, than singlets: the SU(2)-invariant operators LL W μ μ Oef f Oe ff d mass basis? I put V in the coefficients above mW , because and LL , both match onto the below-mW operator LL . However, the two SU(2) operators are distinct2 for f = τ, the experimental constraints are being matched “bottom-up” but not for f = e,μ. onto operator coefficients. So one coefficient from below mW will match onto a sum of coefficients above m , weighted by The coefficients of operators from below mW (on the left W of the equalities) can be matched as follows onto the SU(2)- CKM matrix elements. Secondly, the quark doublets above , , invariant coefficients to the right: mW are taken in the u c t mass eigenstate basis, because it is convenient for translating up in scale the bound on μ → eγ , eμll = eμll − eμ e , CRR CEE 2CHEgR (17) as will be done in Sect. 5. This is because tensor operators mix eμττ = eμττ − eμ e , to the dipoles, and only for u-type quarks are there SU(2)- CRR CEE CHEgR (18)   invariant dimension-six tensors operators. eμ = eμ − eμ + eμ e , CLR CLE CHL,3 CHL,1 gR (19) The BWP basis of two-lepton–two-quark operators has μ μ μ Ce = C e − Ce ge (20) seven vector operators given in Eqs. (81), (82), (83), (84), RL LE HE L (85), (86), (87) and two scalars and a tensor given in Eqs. (88), (89), (90), (91), (92) and (93). For the first two generations 2 The first contracts a flavour-changing neutral current to a flavour- and the b quark, the coefficents from below m (left side conserving neutral current. The second contracts two flavour-changing W neutral currents, or can be fiertzed to make one current flavour- of equality) can be matched to the coefficients above mW as conserving but then both currents are charge-changing (see Eq. 94). follows: 123 Eur. Phys. J. C (2016) 76 :370 Page 7 of 19 370   μ μ μ μ μ e unun = e nn − e nn − u e + e , fewer for the scalar–tensor operators. This means that SU(2)- CLL CLQ(1) CLQ(3) gL CHL(1) CHL(3) invariance should predict some correlations in the scalar– (30)    μ μ μ tensor coefficients below mW . Whereas, if one was trying to e dndn = ∗ e jk + e jk CLL VjnVkn CLQ(1) CLQ(3) reconstruct the coefficients of the SU(2)-invariant operators jk   from data, some additional input (e.g. from Z physics, neu- − d eμ + eμ , trino interactions [52], or loop matching) would be required gL CHL(1) CHL(3) (31) μ μ μ for the vector operators, beyond the coefficients of the oper- e unun = e nn − u e , CRR CEU gRCHE (32) ators of Eqs. (1) and (2). μ μ μ Ce dndn = Ce nn − gd Ce , (33) RR ED R HE  μ μ μ μ e unun = e nn − u e + e , 4.2.1 The vector operators C C gR C ( ) C ( ) (34) LR LU  HL 1 HL 3  μ μ μ μ e dndn = e nn − d e + e , Consider first the vector operators, including the “penguin” CLR CLD gR CHL(1) CHL(3) (35) μ μ μ operators of Eqs. (102), (103) and (104) as well as the four- Ce unun = Ce nn − gu Ce , (36) RL EQ L HE fermion operators. μ μ μ e dndn = ∗ e jk − d e , CRL VjnVknCEQ gL CHE (37) jk • In the case of four-lepton operators with flavour indices μ ∗μ m v μ ∗ e unun = enn − un e , eμee or eμμμ, there are the same number of indepen- CS,LL CLEQU 2 CEH (38) mh dent coefficients above and below. There is one extra four- μ m v μ ∗ μττ e dndn =− dn e , lepton operator above mW for flavour indices e , as can CS,LL 2 CEH (39) mh be seen from Eq. (21). v • eμunun eμnn mun eμ There are fewer two-lepton–two-quark operator coeffi- C , = C − C , (40) S RR LEQU m2 EH cients above mW than below. It is clear that the operators h eμqq eμqq v O , O from below mW with q ∈{u, d, s, c, b} eμdndn mdn eμ LR RR =− , eμu u eμd d eμu u eμd d CS,RR 2 CEH (41) O n n O n n O n n O n n m are equivalent to the LU , LD , EU , ED h eμqq μ m v μ ∗ operators from above. Also it is clear that the O from Ce unun =− un C e , (42) LL S,LR 2 EH below mW with q ∈{u, d, c, s, b} have the same num- m μ μ h ber of independent coefficients as Oe nn and Oe nn . eμd d ∗μenn md v μe∗ LQ(1) LQ(3) n n = − n , eμqq CS,LR CLEDQ CEH (43) O m2 The restriction occurs between RL from below mW , h ∈ μ m v μ where there are five coefficients corresponding to q Ce unun =− un Ce , (44) { , , , , } Oeμnn S,RL 2 EH u d s c b , and EQ above mW , which has a coef- mh v ficient per generation. Neglecting CKM sums, this sug- eμdndn eμnn mdn eμ eμuu eμdd C , = C − C , (45) gests that SU(2) predicts the C − C = 0 and S RL LEDQ 2 EH μ μ RL RL mh e cc − e ss = μ ∗μ CRL CRL 0; however, there is a penguin operator e unun = enn , CT,LL CT,LEQU (46) which contribute to both differences, so only the difference μ e dndn = , of differences is an SU(2) prediction (possibly blurred by CT,LL 0 (47) μ μ CKM). Ce unun = Ce nn , (48) T,RR T,LEQU • The “penguin” operators from above mW (see Eqs. (102), eμd d ¯γ μ C n n = 0, (49) (103), (104)) give the Z a vertex with e PY , which T,RR ¯ matches onto (e¯γ PY μ)( f γ PX f ) operators for all the SM ∈{ , } ∈{ , , } where un u c , dn d s b , and fermions below mW , in ratios fixed by the SM Z cou- 4 4 plings. This contribution adds to the four-fermion oper- gu = 1 − s2 , gu =− s2 , L 3 W R 3 W ator induced at the scale M in the EFT, as given in the 2 2 matching conditions Eqs. (17)–(30). So the coefficient of gd =−1 + s2 , gd = s2 . (50) ¯ / μ L 3 W R 3 W the eZPR penguin operator of Eq. (104) could be deter- eμuu− eμdd mined from CRL CRL , as discussed in the item above. 4.2 Comments on tree matching The coefficients of the two remaining penguin operators are “extra”: in naive coefficient-counting, there are two One observes that the consequences of matching at mW ,at more vector coefficients above mW than below. However, tree level, are different for vector vs. scalar–tensor-dipole they are not completely “free”, because they would match operators. In the vector case, there are more operator coef- at one loop onto the photon dipole operator at mW . These ficients in the SU(2)-invariant theory above mW than in extra penguins are related to the common wisdom, that it is the QCD×QED-invariant theory below, whereas there are interesting for ATLAS and CMS to look for Z → τ ±μ∓ 123 370 Page 8 of 19 Eur. Phys. J. C (2016) 76 :370

and Z → τ ±e∓ decays, but that they are unlikely to see 4.2.3 Matching at “Leading” order Z → μ±e∓ [53]. The point [54] is that an interaction τ¯ /μZ would contribute at tree level to τ → μll¯ , and The aim of a bottom-up EFT analysis is to translate the at one loop to τ → μγ . To be within the sensitivity of bounds from several observables to combinations of oper- the LHC, the coefficient of this coupling needs to exceed ator coefficients at the high scale. So one must compute the the naive bound from τ → μll¯ . However, BR(τ → 3l) numerically largest contribution of each operator to several μτll μτll μ → γ μ → μ → ¯ [55] is controlled by coefficients CXY , CYY , analogous observables ( e , e conversion and eee, to the coefficients on the left of Eqs. (17)–(22), which in the case of μ–e flavour change). It is interesting to have are the sum of SU(2)-invariant four-fermion and penguin constraints from different observables, rather than just the coefficients. So the penguin coefficient could exceed the best bound, because there are more operators than observ- expected bound from τ → 3l, provided that it is tuned ables, so a weaker constraint on a different combination of against the four-fermion coefficient.3 This same argu- coefficients can reduce degeneracies. However, in this paper, ment could apply to a eZ¯ /μcoupling and the bound from only the experimental bound from μ → eγ is considered, μ → eee¯ , although more tuning would be required, since so the aim is to obtain the best bound it sets on all operator the bound on μ → eee¯ [57] is more restrictive. However, coefficients. the Z penguins also contribute at one loop to μ → eγ In the next section, we will see that tree matching and and τ → μγ . And whereas the experimental constraint one-loop running, as performed so far, do not reproduce the on τ → μγ [58,59] is consistent with Z → τ ±μ∓ being correct constraints from μ → eγ on the operators which detectable at the LHC, the bound from μ → eγ implies parametrise LFV interactions of the Higgs and Z; that is, that a eZ¯ /μinteraction, with coefficient of a magnitude that the numerically dominant contributions of these operators to the LHC could detect, would overcontribute to μ → eγ μ → eγ are not included. In addition, two-loop QED run- by several orders of magnitude [54]. ning [42] is required below mW to obtain bounds on vector operators. So it is clear that the simplistic formalism given 4.2.2 The scalar, tensor and dipole operators here, of tree matching and one-loop running, does not work for μ → eγ . It would be interesting to construct a system-

• Above mW , there are two dipoles, given in Eq. (16). At atic formalism, gauge-invariant and renormalisation scheme tree level, the Z-dipole does not match onto any operator independent, that allows one to obtain the best bound on each below mW . operator from each observable. I suppose that such a formal- • There are no dimension-six, SU(2)-invariant four-fermion ism corresponds to “leading order”. Notice that leading order Oeμff is only defined “top-down”, because it describes the contri- operators to match onto the tensor operators T,YY for f ∈{τ,d, s, b}. Furthermore, in tree level matching, the bution of an operator to an observable. So to construct a LO tensors are not generated by any heavy particle exchange. formalism for bottom-up EFT, it seems that one must work They are presumeably generated in one-loop matching by top-down, finding the numerically dominant contribution of the same diagrams that give the mixing below mW ,but each operator to each observable, then ensuring that the com- this should be subdominant because the log is lacking. bination of the contributions from all the operators is scheme • There are no dimension-six, SU(2)-invariant four-fermion independent. μ Oe unun As previously stated, the LO two-loop running is neglected operators to match onto the scalar operators S,YX μ in this paper. However, some attempt is made to perform LO and Oe ff for f ∈{e,μ,τ,d, s, b}, u ∈{u, c} and S,YY n matching at m , where the “LO contribution” of a coefficient X = Y . However, SM Higgs exchange, combined with W above the matching scale to a coefficient below, is pragmati- the H † H LHE¯ operator, will generate these operators in cally defined as the numerically dominant term (and not the tree matching, weighted by m v/m2 or m v/m2.Soit f h un h lowest order in the loop expansion, because this may not be is a tree-level SU(2) prediction that these coefficients are the numerically dominant contribution in presence of hierar- small, as noted by [60]. Since the coefficients of scalar chical Yukawas). operators involving quarks are normalised by a running eμff( ) So, in summary, the “leading order” matching performed quark mass, see Eq. (12), one obtains CS,... mτ μ → γ eμ for e in the next section will consist of the tree equiv- −C (m )m (mτ )m /m2. EH W f t h alences given in this section, augmented by some one- and two-loop contributions of operators that do not mix to the 3 Of course, since the penguin contributes to all four-fermion operators dipole. These loop contributions are obtained by listing all (μγ¯ τ)( ¯γ ) f f , the coefficients of many other operators might need to the operators which do not mix into the dipole above mW , be tuned against the penguin too. An apparently less contrived way to estimating their matching contribution at m , and including engineer this, is to use the equations of motion to replace the penguin W αβ operator by a derivative operator ∂α Z μγ¯ β τ [56], which is suppressed it if it gives an interesting constraint. at low energy by the Z four-momentum. 123 Eur. Phys. J. C (2016) 76 :370 Page 9 of 19 370

5 Translating the μ → eγ bound to M > mW It is also interesting to estimate the loop order probed by the current MEG bound. Counting 1/(16π 2) for a loop (as if In this section, the aim is to use the machinery developed in couplings×logarithm 1), and assuming that M  10 TeV the previous sections to translate the experimental bound on (beyond the reach of the LHC), then Eq. (55) suggests that BR(μ → eγ)to a constraint on operator coefficients at the three-loop effects could be probed. In Sect. 5.4, estimated New Physics scale M. bounds are given on all the operators which MEG can con- strain. Four-fermion operators are defined to be “constrain- 5.1 Parametrising μ → eγ able” if their coefficients C can be bounded C  1 at a scale M ∼ 100mt . It turns that all these operators are within two A flavour-changing dipole operator (in the notation of Kuno SM loops of the dipole. and Okada [1])   5.2 Running up to m 4G F αβ αβ W Lμ→eγ =− √ mμ ARμRσ eL Fαβ + AL μL σ eR Fαβ 2 Between m and mτ , various operators mix into the dipole, (51) W so at mW , the exptal bound (53) applies to the linear combi- can be added to the SM Lagrangian at a low scale ∼ mμ, and nation of the coefficients given on the left-hand-side of Eq. gives a branching ratio (4), when the dummy index B is taken to be a dipole: ⎧ 2 2 2 −13 ⎨ BR(μ → eγ)= 384π (|AR| +|AL | )<5.7 × 10 α       l C (mτ ) = C (m ) − C (m ) γ (52) D D W π ⎩ Sl W SD 4 = ,μ  l e where the constraint is from the MEG experiment [61]. If  α m | |=| | | | < . × −9 + C x (m ) − W AR AL , then AX 8 6 10 , whereas conser- T W π log = , , ,τ 8 mτ vatively only allowing for one coefficient gives the bound x qi u d ⎫ | | < . × −8.  AX 1 2 10 Translated to the coefficients of the    ⎬ ×  x ( ) γ x γ x mW dipole operators of Eq. (1), which are defined including a CS mW ST TD log (56) ⎭ mτ muon Yukawa, this conservative limit gives 2 2 eμee μ A M M ∈{ , , }. ( ) e X −8 where qi s c b The contribution of CS,LL mW will be C , = < 1.2 × 10 . (53) D X m2 m2 neglected, because it is constrained by μ → eee¯ . A linear t t μ μ μ combination of Ce dd(m ) Ce ss (m ), and Ce uu(m ) It is interesting to estimate the scale M to which exper- S,LL W S,LL W S,LL W contributes to μ → e conversion, so possibly an independent iments currently probe. One can consider three possible constraint from μ → eγ on a different combination could guesses for the form of the coefficient of the operator be interesting. However, I neglect these coefficients too, to μσ¯ · FPX e: avoid strong interaction issues and because in tree matching mμ v ev 4 c , c , c , (54) at mW , the first two are Yukawa suppressed to irrelevance. M2 M2 16π 2 M2 eμ In the following, I focus on the “left-handed” dipole CD,L . where c  1 is a dimensionless combination of numerical eμ The evolution of CD,R is similar, so for the “right-handed factors and couplings constants. The first guess is the Kuno– dipole”, only final results and a few non-trivial differences Okada normalisation of (51), corresponding to the Higgs leg are given (which arise due to Higgs loops above mW , where attached to the muon line, but a tree diagram, and suggests Ye  Yμ is neglected). One obtains that the current data probes scales up to ∼ 106 GeV. The  second guess gives the maximum possible scale of ∼ 108 eμ ( ) eμ ( ) + e eμμμ( ) CD,L mτ CD,L mW CS,LL mW GeV – however, it supposes the dipole operator is generated 4π 2 at tree level, with all couplings ∼1. The final guess takes 8Qu Ncmu eμuu 8md eμdd − C , (mW ) + C , (mW ) into account that the dipole operator is generated at one loop mμ T LL mμ T LL  × with a photon leg, and gives a maximum scale of M 3 8ms eμss 6 + C , (mW ) 10 GeV. Notice that this guess is very similar to the Kuno– mμ T LL Okada normalisation used to define the dipoles in this paper: 8Q N m μ 8m μ 2 u c c e cc b e bb e/(16π ) ∼ 3yμ. The maximum scale is relevant, because − C , (mW ) + C , (mW ) mμ T LL mμ T LL it determines how large can be the logarithm from the RGEs above m . I take the third guess with W 4 Equations (26)and(39) show that these operators arise at mW by O  × 6 ⇒ M  . matching out EH, which gives a larger contribution to the dipole via M 3 10 GeV ln 10 (55) top and W loops, as given in Eq. (60). mW 123 370 Page 10 of 19 Eur. Phys. J. C (2016) 76 :370  8mτ eμττ + C , (mW ) T,LL mμ T LL ⎛ 2

α 10^8 C e ⎝ 2mτ eμττ + + C , (mW ) π 3 μ S LL m 1 ⎞  2 2Nc Qq mq eμqq ⎠ + C , (mW ) (57) 0 mμ S LL q=s,c,b where the first parenthesis is first order in , the second -1 parenthesis is the second order scalar→tensor→dipole mix- mW ∼ ing, Qq is the electric charge, and the log mτ was taken 4. -2 The light quark (u, d, s) tensor contributions only include the mixing between mW and mτ ; the (non-perturbative) mixing -3 between mτ and mμ is difficult to calculate, so neglected. -3 -2 -1 0 1 2 3 10^8 C Due to this uncertainty, the light quark tensors are neglected D,L after Eq. (58). With quark masses evaluated at mW , this gives Fig. 2 Between the vertical black lines is the allowed range for the eμ eμ eμuu Oeμ C (mτ ) C (m ) − .0016C (m ) coefficients of the dipole operator D,X (horizontal axis)andc-tensor D,L D,L W T,LL W μ operator Oe cc (vertical axis), evaluated at low energy. At m ,the +. eμdd( ) + . eμss ( ) T,XX W 0017CT,LL mW 0 35CT,LL mW allowed region is between the diagonal blues lines;seeEq.58.This μ μττ μ illustrates that the allowed region changes with scale, in this case due −1.0Ce cc (m ) + 1.0Ce (m ) + 1.8Ce bb (m ) T!,LL W T,LL W T,LL W to operator mixing + −3 . eμμμ( ) + . eμττ ( ) 10 7 6CS,LL mW 4 6CS,LL mW " eμ eμcc eμbb eμcc values of C , (mW ) and C , (mW ) are allowed, provided +1.4C , (mW ) + 1.5C , (mW ) (58) D L T LL S LL S LL they are correlated. Including experimental constraints from μ → ¯ μ → where one notices that the scalar→tensor→dipole mixing eee and e conversion would give other constraints of the “heavy” fermion ( f ∈{τ,c, b}) operators is of on different linear combinations of coefficients, but the prob- the same magnitude as the scalar→dipole mixing of the lem of having more operators than experimental constraints μ operator, because the anomalous dimension mixing ten- would remain. sors to dipoles is large and enhanced by m f /mμ.Thismix- ing is the EFT implementation of the two-loop “Barr–Zee” 5.3 Matching at mW diagrams (see Fig. 3)oftheτ, c and b: contracting the scalar propagator of the Barr–Zee diagram to a point gives The tree-level matching conditions of Sect. 4 allow one to a scalar four-fermion operator, then the photon exchanged translate, at mW , the coefficients of QCD×QED-invariant between the muon and heavy fermion makes a tensor oper- operators to SU(2)-invariant coefficients. With these rules, ator, then the heavy fermion lines are closed to give the the low-scale dipole coefficient can be written dipole. eμ ( ) μe∗( ) − μecc∗ ( )2eQu Ncmc At the weak scale, the experimental bound constrains a C , mτ Ceγ mW C ( ) mW D L LEQU 3 mμπ 2 linear combination of several different operators. It is com- α 2 mon to quote the resulting constraints “one at a time”, that μecc∗ 2e Qu Ncmc +C (mW ) LEQU mμπ 3 is, retaining only one coefficient in the sum of Eq. (58), and # setting the remainder to zero, in order to obtain a bound. v α 2v − μe∗( ) mμ + 2e mτ I will do this later, in listing bounds at the scale M.How- CEH mW 4π 2m2 π 3 mμm2 ever, it is important to remember that the MEG experi- $ h %& h eμ ( ) α v 2 2 + 2 2 ment only ever gives two constraints (on CD,R mτ and 2e Nc Q m Qumc μ + d b , (59) e ( ) π 3 CD,L mτ ) in the multi-dimensional space of operator coef- mh mμmh ficients, and additions or cancellations are possible among eμ ( ) the many contributing operators at mW . This is illustrated in with a similar equation for CD,R mτ . Only four SU(2)- Fig. 2, where the black lines give the experimental bound invariant coefficients are required, because for the leptons and eμ ( ) at low energy on CD,L mτ . The diagonal blue lines are down-type quarks, there are no SU(2)-invariant, dimension- the bound at mW , in a model where only the coefficients six tensor operators, nor scalar operators with the required eμ ( ) eμcc ( ) CD,L mW and CT,LL mW are non-zero: arbitrarily large LL chiral structure. The tensor operators are not generated 123 Eur. Phys. J. C (2016) 76 :370 Page 11 of 19 370

γ γ a flavour-changing Z-penguin vertex, gives contributions to the dipole coefficients: t W μe∗ e e eμ C γ (mW ) g C (mW ) e 16π 2 L HE γ γ h   h  eμ( ) e e eμ ( ) + eμ ( ) , Ceγ mW gR C ( ) mW C ( ) mW e e 16π 2 HL 1 HL 3 μ μ 2 2 (61) Cµe∗ v Cµe∗ v EH M 2 EH M 2 e , e where gL gR are given in Eq. (29), no muon Yukawa appears Fig. 3 The two-loop “Barr–Zee” diagrams which gives the largest con- in the matching coefficient because it is implicit in the dipole H † H LHE¯ m grey tribution of the operator to the dipole below W .The operator definition, and the electron Yukawa was neglected disk is the dimension-six interaction, with two Higgs legs connecting to the vev. The Higgs line approaching the top loop indicates a mass (which is why different penguins mix into the above two μe∗ insertion somewhere on the top loop dipoles). The contribution Ceγ is to be added to the right eμ side of Eq. (59), and Ceγ should be added to the modifica- , , eμ in matching out the W Z h and t at tree level, so their coef- tion of Eq. (59) appropriate to CD,R. ficients can be set to zero as given in Eqs. (27), (28), (47) and (49). (They could arise in one-loop matching, via diagrams 5.4 Running up to M similar to those giving running below mW , so the tensor coef- ficients were retained in the discussion of the Sect. 5.2.) The eμ ( ) At mW , CD,L mτ can be written as a linear combination of scalar operators are generated in matching out the Higgs, see μe∗( ) μe∗( ) μecc∗ ( ) μecc∗ ( ) Ceγ mW , CeZ mW , CLEQU(1) mW , CLEQU(3) mW , Sect. 4.1, which gives the square bracket above. eμ ( ) μe∗( ) CHE mW , and CEH mW . The RGEs to evolve these coef- However, it is well known that this estimate has missed ficients up to M are given in [39,40,43], and generate more Oμe∗ the largest contribution from EH to the dipole opera- intricate and extensive operator mixing than was present tor below mW , which are “Barr–Zee” diagrams with the below mW . The aim here is to present manageable analytic W SM Higgs and a top or loop, as illustrated in Fig. 3. formulae that approximate the “leading” (= numerically most Despite being suppressed by two loops, these diagrams are important) constraints on all the constrainable coefficients at m2/m2 m2 /m2 enhanced by t μ or W μ. In SU(2)-invariant nota- the scale M. Recall that an operator coefficient was defined tion, these diagrams generate a “dimension-eight” dipole here to be constrainable if the current MEG bound, as given † ¯ σ · H H LH FE. However, SU(2) is irrelevant below mW , < eμ in Eq. (53), implies C 1atM 100mt . O( /M2) O μe∗ so this is an 1 matching contribution to D,X .For Consider first C . Neglecting its self-renormalisation lack of good ideas on how to do a well-defined perturbation EH between M and mW , because the anomalous dimension theory in many small parameters (in particular, loops and 2 × ln M/mW < 16π , the “one-operator-at-a-time” con- hierarchical Yukawas), I retrieve from the results of Chang μe∗  . straint at M 100mt is CEH 0 01. So there could be et al. [62], the evaluation of the Barr–Zee diagrams with a Oμe∗ a bound on operators that mix into EH in running between SM Higgs and a top or W loop (which have opposite sign): γ    M and mW . These include the Z and dipoles, which can α be neglected here because they have more direct contribu-  eμ ( ) − μe∗( ) e 2 2 − 7 C , mW C mW Qt NcYt μ → γ D L EH 16π 3Yμ 2 tions to e .ThereisalsoaYμ-suppressed mixing   from the “penguin” operators, which is neglected because μe∗ eα C (mW ) (60) the penguins match at one loop onto the dipole at mW .SoI EH 8π 3Yμ approximate and substitute the square brackets of Eq. (60) for those in eqn μ ∗ μ ∗ C e (m ) = C e (M). (62) (59). EH W EH Having started cherry-picking the “leading” contributions Consider next the penguin operators of Eqs. (102)–(104), from higher order, it is interesting also to include the one- which match at one loop to the dipole. The bound on the loop matching contribution of the “penguin” operators of eμ  . coefficient at M 100mt is CHE 0 1, so I neglect mixing Eqs. (102), (103) and (104). These give a lepton-flavour- into these operators, and approximate changing vertex to the Z, which contributes to Z → μ±e∓ eμ ( ) = eμ ( ), and at one loop to μ → eγ . As discussed in Sect. 4.2,in CHE mW CHE M eμ ( ) = eμ ( ), the context of LHC searches for flavour-changing Z decays, CHL(1) mW CHL(1) M (63) μ → eee¯ give a restrictive bound on a combination of the eμ eμ C ( )(mW ) = C ( )(M). penguins plus four fermion operators. So even if weaker, an HL 3 HL 3 independent constraint from μ → eγ , on a different combi- In running from M → mW , the RGEs given in [39,40,43] nation of operators, is interesting. The one-loop diagram with show that gauge interactions will renormalise the photon 123 370 Page 12 of 19 Eur. Phys. J. C (2016) 76 :370

μe∗ dipole coefficient Ceγ , and cause it to receive contributions Allowing the index B of Eq. (66) to run over the coeffi- μ ∗ μ ∗ μ ∗ μ ∗ μ ∗ e ecc ecc ett ett cients present on the right side of Eq. (59), the anomalous from CeZ , CLEQU(1), CLEQU(3), CLEQU(1), and CLEQU(3). This gauge mixing of scalars to tensors to dipoles is analo- dimension matrix of Eq. (64) and the bound (53)give gous to the QED mixing below mW . In addition, as given 2 M μ ∗ in [39,40], Higgs loops will mix vector four-fermion oper- . × −8  μe∗( ) − . e ( ) 1 2 10 2 Ceγ M 0 016CEH M ators into scalars and tensors. In the following, the third mt order vector→scalar→tensor→dipole mixing is neglected, + . eμ ( ) − . μe∗( ) M 0 001CHE M 0 0043CeZ M ln and only the vector→tensor→dipole is retained for vector mW and tensor operators with a top bilinear. Defining a coefficient μett∗ M − 59C (M) ln vector LEQU(3) m   W   = μett∗, μett∗, μett∗ , μecc∗ , μett∗ , − μecc∗ ( ) . M + . C CEU CEQ CLEQU(1) CLEQU(1) CLEQU(3) CLEQU(3) M 0 43 ln 1 5  mW μecc∗ μ ∗ μe∗ , e , , μ ∗ M CLEQU(3) Ceγ CeZ + . ett ( ) 2 0 039CLEQU(1) M ln mW then, from [39,40,43], the electroweak anomalous dimension α γ μ∂ /∂μ = em γ matrix γ t such that C 4π C is approximately ⎡ ⎤ 00 0 0 − Yt Yμ 000 ⎢ 2e2 ⎥ ⎢ ⎥ ⎢ 00 0 0 − Yt Yμ 000⎥ ⎢ 2e2 ⎥ ⎢ 2 ⎥ ⎢ 00−5 + 15Yt 0 7 000⎥ ⎢ 2e2 3 ⎥ ⎢ 2 ⎥ 00 0 −5 + 15Yc 0 7 00 ⎢ 2e2 3 ⎥ γ ∼⎢ 2 ⎥ γ t ⎢ . + 3Yt 16Yt √8Yt ⎥ ⎢ 0 0 112 0 8 5 2 0 ⎥ ⎢ 2e eYμ 3eYμ ⎥ ⎢ 2 ⎥ ⎢ 0 0 0 112 0 8.5 + 3Yc 16Yc √8Yc ⎥ ⎢ 2e2 eYμ 3eYμ ⎥ ⎢ 2 ⎥ ⎢ 00 0 0 7Yt Yμ 7YcYμ 7 + 3Yt − √24 ⎥ ⎣ e e e2 3 ⎦ 2 00 0 0 22Yt Yμ 22YcYμ √12 − 8 + 3Yt 6e 6e 3 3 e2 (64) where small Yukawa couplings and fractions were neglected,   M μ ∗ sin2 θ = 1/4, and renormalisation and mixing to the vec- + 0.002 1 + ln C ecc (M) W m LEQU(1) tors was neglected because they only affect the dipole at W   eμtt M μ ∗ μ ∗ O(α2 log2). The RGE for the tensor coefficient C , − . × −5 2 ett ( ) + ett ( ) LEQU(3) 4 8 10 ln CEQ M CEU M (67) eμ mW which mixes to the “right-handed” dipole Ceγ would instead √ include the vector contribution: (where mt is written instead of the Higgs vev, to avoid 2 ∂ eμtt αem Yt Yμ issues). This constraint, as well as the equivalent bound on μ C ( ) = ···− eμ ∂μ LEQU 3 π 2 C γ (mτ ):  4 2e  e μ μ μ × C ett + C ett − 3C ett , (65) 2 LU LQ(1) LQ(3) . × −8 M  eμ( ) − . eμ ( ) 1 2 10 2 Ceγ M 0 016CEH M rather than the first two rows of Eq.(64). The approximate m t  solution of these RGEs, if the running of gauge and Yukawa + . eμ ( ) + eμ ( ) 0 001 CHL(1) M CHL(3) M couplings is neglected,5 is  eμ M   − 0.0043C (M) ln αem M eZ m ( ) ( ) δ , − γγ W CB mW C A M A B t A,B ln 4π mW μ M  − 59Ce tt (M) ln α2   LEQU(3) em 2 M mW + γγ γγ ln +··· . (66)   π 2 t t A,B 32 mW − eμcc ( ) . M + . CLEQU(3) M 0 43 ln 1 5 mW 5 α Including s , so the quark operators no longer run as a power of + . eμtt ( ) 2 M α (μ). 0 039CLEQU(1) M ln s mW 123 Eur. Phys. J. C (2016) 76 :370 Page 13 of 19 370

Table 1 Approximate μe∗ eμ −8 Ceγ Ceγ 1.2 × 10 “one-operator-at-a-time” μ ∗ μ C e ln M Ce ln M 3.0 × 10−6 constraints on operator eZ mW eZ mW μ ∗ μ coefficients evaluated at the C ett ln M Ce tt ln M 2.0 × 10−10 LEQU(3) mW LEQU(3) mW scale M, from the MEG bound μ ∗ μ (μ → γ) C ecc (ln M + 3.5) Ce cc (ln M + 3.5) 2.8 × 10−8 [61]onBR e ,asgiven LEQU(3) mW LEQU(3) mW μ ∗ μ in Eqs. (67), (68). For a given C ett ln2 M Ce tt ln2 M 3.1 × 10−7 LEQU(1) mW LEQU(1) mW choice of scale M, the quantity μ ∗ μ C ecc (ln M + 1) Ce cc (ln M + 1) 6.0 × 10−6 in either left column should be LEQU(1) mW LEQU(1) mW μ ∗ μ less than the number in the right C e Ce 7.5 × 10−7 colomn multiplied by M2/m2. EH EH t eμ eμ , eμ . × −5 The operators are labelled in the CHE CHL(1) CHL(3) 1 2 10 μ ∗ μ same way as the coefficients, C ett ln2 M Ce tt ln2 M 2.5 × 10−4 EQ mW LU mW and given in Appendix D μ ∗ μ μ C ett ln2 M Ce tt ln2 M ,3Ce tt ln2 M 2.5 × 10−4 EU mW LQ(1) mW LQ(3) mW

  M μ + . + e cc ( ) First consider operator dimensions above and below mW . 0 002 1 ln CLEQU(1) M mW There is a rule of thumb in EFT [47]: that one matches at a − M loop order lower than one runs, where the loops are counted − 4.8 × 10 5 ln2 in the interaction giving the running. This makes sense if  mW  μ μ μ the loop expansion is in one coupling, or if the same dia- × Ce tt(M) + Ce tt (M) − 3Ce tt (M) (68) LU LQ(1) LQ(3) gram gives the running and one-loop matching, because the running contribution is relatively enhanced by the log. For gives the “one-operator-at-a-time” bounds listed in Table instance, an electroweak box diagram at mW generates a four- 1. These bounds are obtained by assuming that one oper- fermion operator “at tree level” in QCD, which can run down μ → γ ator dominates the e amplitude, so neglect interfer- with one-loop QCD RGEs. One could hope that a similar ences between the various coefficients. If both the left-handed μe∗ eμ argument might apply above mW : a diagram giving one-loop dipole Ceγ and the right-handed Ceγ are generated, then √ matching could contribute to running above mW , so the sub- the right column could be divided by 2. The bounds of the dominant matching could be neglected. However, this is not first six rows agree to within a factor 2 with the constraints the case at mW , because SU(2)-invariant dimension-six oper- given in [43], who do not constrain the coefficients given ators from above mW can match onto operators that would-be in the last four rows. The vector operators, given in the last dimension eight if one imposed SU(2), but that are O(1/M2) two rows, barely pass the “constrainable” threshhold defined and dimension six in the QED×QCD-invariant theory below above (C < 1atM = 100 mt ). This retroactively justifies mW . For example, the LFV Z penguin operators given in Eqs. that the mixing of vectors into scalars was neglected, because (102)–(104) match at one loop onto the “dimension-eight” it would be suppressed by an additional loop. † dipole yμ H H(Le Hσ · FEμ). Similarly, the LFV Higgs † interaction H H(Le HEμ) matches at two loops to the same “would-be dimension-eight” dipole. So the expectation that 6 Discussion of the machinery and its application to running dominates matching can fail at mW . μ → eγ The expectation that one loop is larger than two loops can fail when perturbing in a hierarchy of Yukawa couplings. The The MEG experiment [61] sets a stringent bound on the dipole’s affinity to Yukawas arises because the lepton chiral- dipole operator coefficients at low energy (see Eq. (53)). In ity changes, and the operator has a Higgs leg. The dipole translating this constraint to a scale M > mW , the analysis operator here is defined to include a muon Yukawa coupling here aimed to include the “Leading Order” contribution of Yμ (see Eq. (51)), because in many models, the Higgs leg all “constrainable” operators, where LO was taken to mean attaches to a Standard Model fermion, and/or the lepton chi- numerically largest, and an operator was deemed constrain- rality flips due to a Higgs coupling. And while it is difficult able if a bound C < 1 could be obtained at M ≥ 100mt . to avoid the Yμ in one-loop contributions to the dipole (see However, two-loop running, which gives the leading order the discussion in [63]), there are more possibilities at two mixing of vectors to the dipole, is not included here, so many loops. In particular, it is “well known” [65] that the leading constraints on vector operators are missing. As a result, the contribution to μ → eγ of a flavour-changing Higgs inter- one-operator-at-a-time limits given in Table 1 are obtained action, is via the two-loop top and W diagrams included in from a combination of tree, one- and two-loop matching, the matching contribution of Eq. (60). with RGEs at one loop. Why do these multi-loop matching Its unclear to the author what to do about either of these contributions arise? problems. Perhaps only the LFV operators with at least two 123 370 Page 14 of 19 Eur. Phys. J. C (2016) 76 :370

Higgs legs give their leading contributions in matching rather mW , are generated with small coefficients which give a neg- than running.6 Maybe performing the matching and running ligeable contribution to μ → eγ . The point is that the scalars at two loops would include the leading contributions in loops, and tensors involving leptons and d-type quarks are gener- logs, and Yukawa hierarchies. However, a complete two-loop ated by the Higgs LFV operator, whose leading contribution analysis would take some effort – perhaps it would be sim- to μ → eγ arises in two-loop matching. pler to list all the possible operators at the scale M, locate There are many improvements that could be made to their “Leading Order” contributions, and include them. these estimates. Including the experimental constraints from As discussed above, it is important to match with care at μ → eee¯ and μ → e conversion would directly constrain mW . A slightly different question is whether it is important the vector operators, and give independent constraints on to match onto the extended (non-SU(2)-invariant) operator some of the operators that contribute to μ → eγ . There basis at mW ? The answer probably depends on the low energy are more operators than constraints, so this could allow one observables of interest. In the analysis here of μ → eγ ,the to identify linear combinations of operators that are not con- four-fermion operators that were added below mW (such as strained. One-loop matching is motivated by the restrictive Oeμbb Oeμττ Oeμμμ the scalar four-fermion operators S,YY, S,YY and S,YY experimental bounds, which allow one to probe multi-loop given in Eq. (2)), are numerically irrelevant provided that effects. In addition, there are operators which require one- the matching is performed at two loops. This is because they loop matching, such as the two-gluon operators relevant to were generated in tree matching by the Higgs LFV operator μ → e conversion. Two-loop running is required to get the H † H(LHE), suppressed by the b,τ or μ Yukawa coupling; leading order contribution of vector operators to μ → eγ , see Eqs. (23), (26), (39) and (41). Then, in QED running, they and could be interesting above mW if there are diagrams mix to the dipole (possibly via the tensor), which brings in that dominate the one-loop running due to the presence of another factor of the light fermion mass. With tree matching, large Yukawas, or if quark flavour-off-diagonal operators are this is the best constraint on the Higgs LFV operator, so included, which may contribute to μ → eγ at two loops [64]. is interesting to include. However, it is irrelevant compared It is also motivated by the experimental sensitivity. Finally, with the two-loop diagrams involving a top and W loop, dimension-eight operators can be relevant if the New Physics which match the Higgs LFV operator directly onto the dipole. scale is not to high [64]. This two-loop matching contribution is relatively enhanced by a factor ∼100 as can be seen by comparing the square brackets of Eqs. (59) and (60). So in the case of μ → eγ ,it 7 Summary seems that one would get the correct constraints on operator coefficients at M by using an SU(2)-invariant four-fermion This paper assumes that there is new lepton-flavour violat- operator basis all the way between mμ and M, provided the ing (LFV) physics at a scale M  mW , and no relevant matching at mW is performed to whatever loop order retains other new physics below. So at scales below M, LFV can the “leading” contributions. be described in an Effective Field Theory constructed with The QED mixing between mμ and mW modifies signifi- Standard Model fields and dimension-six operators. The aim cantly the combination of operators that are constrained by was to translate experimental constraints on selected μ ↔ e μ → eγ . This is illustrated in Fig. 2, which shows that flavour changing processes, from the low energy scale of the the constraint has rotated in operator space, to constrain the experiments to operator coefficients at the scale M.Asafirst linear combination of coefficients given in Eq. (58). Coeffi- step, this paper reviews and compiles some of the formal- cients of tensor operators that were of a similiar magnitude ism required to get from low energy to the weak scale: a to the dipole coefficient could give significant enhancement QED×QCD-invariant operator basis is given in Sect. 2,the or cancellations. So the QED running is important. In addi- one-loop RGEs to run the coefficients to mW are discussed tion, the MEG constraint on BR(μ → eγ) is restrictive – in Sect. 3, the anomalous dimensions mixing scalars, tensors as discussed in Sect. 5.1, it could constrain New Physics and dipoles are given in Appendix A, and tree matching onto 7 which contributes at one loop up to a scale M ∼ 10 GeV. SU(2)-invariant operators at mW is presented in Sect. 4. So it would be sensitive to two-loop contributions from LFV As a simple application of the formalism, the experimental 5 operators at a scale of 10 GeV. However, in matching at mW bounds on μ → eγ were translated to the scale M in Sect. 5. onto SU(2)-invariant dimension-six operators, many of the The process μ → eγ was chosen because it is an electromag- tensor and scalar operators which mix with the dipole below netic decay, and it constrains only the coefficients of the two dipole operators. The resulting constraints at M on two lin- ear combinations of operators are given in Eqs. (67) and (68). 6 In tree-level matching, the Z penguins do give their leading contri- These limits are approximative, due to the many simplifica- bution to four-fermion operators; it is only the “leading contribution to μ → eγ ” which arises in one-loop matching. See the discussion in tions discussed in the paper, valid at best to one significant Sect. 4.2.3. figure. Bounds on individual operators can be obtained by 123 Eur. Phys. J. C (2016) 76 :370 Page 15 of 19 370

† assuming one operator dominates the sum; the resulting con- In the case of the dipole, [OD,R] =[OD,L ], so if one writes straints are listed in Table 1. At a scale M ∼ 100mt , μ → eγ eμ μe eμ  7 CD,R eμ CD,R μe CD,L eμ is sensitive to over a dozen operators, whereas, if M 10 − O , − O , − O , μ → γ M2 D R M2 D R M2 D L GeV, then e is sensitive to only a few. μe The formalism of the first sections did not work well for CD,L μe − O , + h.c. (69) μ → eγ . Tree matching and one-loop running missed the M2 D L largest contributions of some operators, as discussed in Sect. Then the +h.c. is double-counting, it just adds all the same 6. This curious problem could benefit from more study, in eμ∗ = μe operators a second time (which implies CD,L CD,R, order to identify a practical and systematic solution. eμ∗ = μe L CD,R CD,L ). So I include in the first and third oper- aters of Eq. (69), and the +h.c.. Acknowledgments I am very grateful to Junji Hisano for interesting questions and discussions, and thank Peter Richardson, Gavin Salam and Aneesh Manohar for useful comments. Appendix B: Anomalous dimension matrix in QED

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm In this appendix are given the various submatrices of the ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, anomalous dimension matrix ST D of Eq. (15). The relevant and reproduction in any medium, provided you give appropriate credit diagrams are given in Fig. 1. to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1. For scalar operators, the penguin diagrams (first and sec- Funded by SCOAP3. ond) do not contribute to one-loop mixing among four- fermion operators, because the photon couples to the vec- Appendix A: Operator normalisation tor current. However, the second penguin diagram, with Oeμll on-shell photon (no fermions) mixes the S,YY operators ∈{ ,μ} All the operators introduced Sect. 2 appear in the Lagrangian for l e , to the dipole. This gives the matrix − / 2 eμ eμ with a coefficient C M , and the operator normalisation is CD,L CD,R 2 μ chosen to ensure that the Feynman rule is −iC/M .This γ l, = Ce ll − ml 0 S,D S,LL emμ (70) implies a judicious distribution of 1 s, which is the subject of μ 2 Ce ll 0 − ml this Appendix. S,RR emμ The Oeμ are flavour-changing, so can be imagined as off- Diagrams 3 and 4 are the same as the mass renormalisa- diagonal elements of the matrix O in lepton-flavour space. tion diagrams (γ = 6 in QED), so combined with the They annihilate a μ, and create an e, so the hermitian m wave-function diagrams, they renormalise scalar opera- conjugate of the operator should appear in the Lagrangian tors, giving a diagonal matrix: too. However, the Lagrangian is a flavour-scalar, so in the 1 [ O] eμff eμff eμff eμff Lagrangian is - 2 Tr C , where the coefficients C are M CS,LL CS,RR CS,LR CS,RL eμff also a matrix in flavour space. (For instance, to obtain only C 6(1 + Q2 ) 000 eμ eμ S,LL f O in the Lagrangian, one takes only C = 0.) Adding f , f eμff γ 1 2 = C 06(1 + Q2 ) 00 1 † † † S,S S,RR f +h.c. [O C ] O = O μ means adding - M2 Tr .If ,asinthe e ff ( + 2 ) CS,LR 0061 Q f 0 case of vector operators, then there are two possibilities for eμff C 0006(1 + Q2 ) the matrix-in-flavour-space C: either take C† = C (so if S,RL f Ceμ = 0, then Cμe = Ceμ∗), so Tr[O†C†]=Tr[CO]. (71) Then in the Lagrangian appears Tr[O†C†]+ Tr[CO],sothe ( + 2 ) operator should be normalised with 1/2 to compensate for where the 1 Q f arises from the photon exchange this double-counting, and thereby ensure that the F-rule is across either current. −iCeμ/M2. Alternatively, one does not impose C† = C, and The last four diagrams mix the YY scalars to the tensors eμ only puts the desired C = 0 coupling in the Lagrangian, (the YX tensor vanishes) with γ = 2Q f : where the +h.c. generates the anti-particle amplitude, and eμff eμff C , C , eμ 2 T LL T RR the Feynman rule is again −iC /M , without the factor of eμff CS,LL 2Q f 0 1/2 in the operator definition. Scalars and tensor operators , γ f f = eμff S,T CS,RR 02Q f (72) are not hermitian, e.g.: μ     Ce ff 00 μ μ S,LR [ ]= ePY e ePY , [ ]† = ePX e ePX , = , eμff S S X Y CS,RL 00 μPY e μPY μ μPX e μPX μ so a scalar or tensor operator Oeμ will induce two distinct 2. The tensors mix to the dipoles, via the first diagram with μ → e flavour-changing interactions of different chirality. the f2 line removed. This gives 123 370 Page 16 of 19 Eur. Phys. J. C (2016) 76 :370

D,L D,R C f C f (78) , , Q N m γ f = C T LL 8 f c f 0 νμ 1 νμ T,D f mμe (73) (aσ PX b)(cσνμPX d) = (aσ PX d)(cσνμPX b) , Q N m 2 C T RR 08f c f f mμe −6(aPX d)(cPX b) (79) σ = i ε σ αβ γ The third and fourth diagrams do not renormalise the where the relation μν 2 μναβ 5 was used to replace α σ σ ( σ αβ γ μ)(ψσ γ χ) = tensors because γ σγα = 0, but the wave-function dia- with PX . It implies that e 5 αβ 5 μν grams do: (eσμνμ)(ψσ χ),so

αβ 1 αβ eμff eμff (eσ PY μ)(ψσαβ PY χ) = (eσ μ)(ψσαβ χ), CT,LL CT,RR 2 γ f, f = eμff − ( + 2 ) ( σ αβ μ)(ψσ χ) = ( = ). T,T CT,LL 2 1 Q f 0 (74) e PY αβ PX 0 X Y (80) eμff − ( + 2 ) CT,RR 0 2 1 Q f

and finally, the last four diagrams mix the tensors to Appendix D: SU(2)-invariant dimension-six operators scalars, giving This appendix lists dimension-six, SM-gauge-invariant oper- eμff eμff ators involving e–μ flavour change. The operators are in the CS,LL CS,RR f, f μ Buchmuller–Wyler basis, as pruned in Grzadkowski et al. γ = Ce ff −96Q 0 (75) T,S T,LL f [38], and this list is referred to as the BWP basis. The opera- eμff − CT,RR 0 96Q f tors are assumed to be added to the Lagrangian +h.c.; when this gives the μ¯ e operator, it is not listed. The τ a are the Pauli → These tensor scalar mixing elements of the QED matrices, with anomalous dimension matrix are large, suggesting that   − one could redefine the operator basis to use a linear τ 2 = 0 i . combination of scalar and tensor operators with smaller i 0 off-diagonal elements. However, QCD does not mix the The four-fermion operators involving e–μ flavour change scalars and tensors, which favours them as basis opera- and two quarks are tors. In addition, the tensor→scalar mixing does not enter ( ) μ 1 the μ → eγ example of Sect. 5, where the scalar–tensor O 1 e nm = ( γ α )( γ α ) LQ Le Lμ Qn Qm operator basis gives the correct behaviour, as verified by 2 μ → γ 1 α α comparing EFT and exact calculations of e in the = [(eeγ PL μ) + (νeγ PL νμ)] 2HDM [64]. 2 ×[( γ α ) + ( γ α )], The dipole also renormalies itself [19], although this un PL um dn PL dm (81) effect is not included here: O(3)eμnm = 1( γ ατ a )( γ ατ a ) LQ Le Lμ Qn Qm   2 = (ν γ α μ)( γ α ) γ = 16 0 . e PL dn PL um D,D α α 016 +(eeγ PL νμ)(unγ PL dm) 1 α α O , O + [(νeγ PL νμ) − (eeγ PL μ)] 3. The diboson operators GG,Y FF,Y are of dimension 2 seven, so the four-fermion operators and dipole operators α α ×[(unγ PL um) − (dnγ PL dm)], (82) do not mix into them. eμnm 1 α α O = (Eeγ Eμ)(Q γ Qm), (83) EQ 2 n eμnm 1 α α Appendix C: Spinor stuff O = (Leγ Lμ)(U nγ Um), (84) LU 2 eμnm 1 α α The Fierz identities can be written for chiral fermions as O = (Leγ Lμ)(Dnγ Dm), (85) LD 2 1 μ μ ( )( ) =− ( γ )( γμ ), e nm 1 α α aPL b cPRd a PRd c PL b (76) O = (Eeγ Eμ)(U nγ Um), (86) 2 EU 2 ( γ μ )( γ ) = ( γ μ )( γ ), a PL,Rb c μ PL,Rd a PL,Rd c μ PL,Rb (77) eμnm 1 α α O = (E γ Eμ)(D γ D ), (87) 1 ED  e   n m (aP b)(cP d) =− (aP d)(cP b) 2 X X X X Oeμnm = A B 2 LEQU Le Eμ AB Qn Um 1 νμ − (aσ PX d)(cσνμPX b), =−(ν P μ)(d P u ) + (e P μ)(u P u ), 8 e R n R m e R n R m 123 Eur. Phys. J. C (2016) 76 :370 Page 17 of 19 370

μe αβ (88) O = yμ(Lμ Hσ E )Bαβ , (101)     eB e μ A B ↔ O enm = L E Q U , O(1)eμ = ( γ α )( † ), LEQU μ e AB n m (89) HL i Le Lμ H Dα H (102) eμnm ( ) μ ↔ O = (L Eμ)(D Q ) O 3 e = ( γ ατ )( † τ ), LEDQ e n m HL i Le Lμ H Dα H (103) ↔ = (νe PRμ)(dn PL um) + (ee PRμ)(dn PL dm), (90) Oeμ = ( γ α )( † ), HE i Ee Eμ H Dα H (104)

μenm ↔ O = (Lμ E )(D Q ), LEDQ e n m (91) ( † α ) ≡ ( † α )− ( α )† α =     where i H D H i H D H i D H H, and D μ A B g a a g Oe nm = σ μν σ , ∂α +i Wα τ +i YBα. (The sign in the covariant derivative T,LEQU Le Eμ AB Qn μνUm (92) 2 2     fixes the sign of the penguin operator and the SM Z vertex.) Oμenm = Aσ μν Bσ , T,LEQU Lμ Ee AB Qn μνUm (93) These signs cancel in matching at mW , so the results of Sect. 4 should be convention-independent. , , where L Q are doublets and E U are singlets (lower case are This covariant derivative leads to Dα = ∂α + ieQAα , Dirac spinors, SU(2) components selected with PL,R), n m after electroweak symmetry breaking, giving a Feynman rule , μ are possibly equal quark family indices, and A B are SU(2) for the photon-electron-electron vertex ieγ . This choice , , indices. The doublet quarks are in the d s b mass eigenstate (opposite to Peskin-Schroeder but agrees with Buras [47]), φ → basis. The operator names are as in [38] with H;the controls the sign of the QED anomalous dimensions mixing flavour indices are in superscript. four-fermion operators to the dipole. The operators involving e–μ flavour change, and leptons, are

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