μ→eγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow e \gamma $$\end{document} in the 2HDM: an exercise in EFT

The 2 Higgs Doublet Model of type III has renormalisable Lepton Flavour-Violating couplings, and its one- and two-loop (“Barr–Zee”) contributions to μ→eγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow e \gamma $$\end{document} are known. In the decoupling limit, where the mass scale M of the second doublet is much greater than the electroweak scale, the model can be parametrised with an Effective Field Theory (EFT) containing dimension-six operators. The 1/M2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/M^2$$\end{document} terms of the exact calculation are reproduced in the EFT, provided that the four-fermion operator basis below the weak scale is enlarged with respect to the SU(2)-invariant Buchmuller–Wyler list. It is found that the dominant two-loop “Barr–Zee” contributions arise mostly in two-loop matching and running, and that dimension-eight operators might be numerically relevant.


Introduction
This exercise was born from a puzzle: experiments that search for μ ↔ e flavour change constrain a long list of QCD×QED-invariant four-fermion operators, some of which turn out to be of dimension eight when SU(2) invariance is imposed. But it is common, when describing New Physics from above m W with Effective Field Theory(EFT) [1,2], to use the SU(2)-invariant basis of dimension 6 operators given by Buchmuller and Wyler [3] and pruned in [4]. To explore when it is justified to neglect the additional fourfermion operators below m W , we were looking for a model where they might give relevant contributions. The first model we tried was the 2 Higgs Doublet Model (2HDM). It turns out that the additional four-fermion operators (not in the Buchmuller-Wyler list) must be included below m W to correctly reproduce the O(1/M 2 ) terms in the μ → eγ amplitude of the 2HDM.
The exercise takes place in a Type III 2HDM in the decoupling limit, where the one-loop and two-loop "Barr-Zee" a e-mail: s.davidson@ipnl.in2p3.fr contributions to the μ → eγ amplitude are known [5]. The aim is to extract the numerically dominant contributions and to identify where they arise in an EFT description.
Section 2 reviews the 2HDM of Type III in the decoupling limit and the calculation in this model of the μ → eγ amplitude by Chang et al. [5] (CHK). Type III 2HDMs include charged-lepton flavour-changing couplings; for simplicity, only a μ ↔ e flavour-changing interaction is allowed. The decoupling limit is taken by requiring the mass scale M of the second doublet to be 10 v, to ensure that EFT can give a reasonable approximation to the μ → eγ amplitude in this model. In the final subsections, the O(1/M 2 ) and O(1/M 4 ) parts of the μ → eγ amplitude are compared for the various classes of diagrams. Section 3 sets up an EFT formalism, based on dimensionsix operators and one-loop RGEs, which correctly reproduces all the O([α log] n /M 2 ) terms of the CHK calculation in the full model. How to obtain the remaining terms of the CHK result is outlined in Appendix B.
Section 4 summarises the less trivial aspects of the calculation and the location in the EFT of the Barr-Zee diagrams. It should be readable independently of the more technical sections, Sects. 2 and 3. There are two-and-a-half issues with the results obtained with dimension-six operators and one-loop RGEs: the four-fermion operator basis below m W only needs to be invariant under QCD and QED, so it should be enlarged with respect to the list [4] of SU(2)-invariant dimension-six operators. Second, two-loop effects are important, because the one-loop contribution is suppressed by the square of the muon Yukawa coupling. Finally, dimension-eight terms can be enhanced, both by logs and by unknown couplings in the 2HDM (encapsulated in tan β).
This exercise overlaps with several studies. Pruna and Signer [6] studied μ → eγ in an EFT consisting of the SM extended by a complete set of SU(2)-invariant operators, but they focused on the electroweak running above m W , rather than the matching at m W , so they did not extend the oper-ator basis below m W . In the context of B physics, Alonso et al. [7] and Aebischer et al. [8] calculated the coefficients of the enlarged operator basis below m W , given a selection of SU(2)-invariant operators above m W . This exercise only agrees approximatively with [7], as discussed in Sect. 4.

μ → eγ in the Type III 2HDM
2.1 Review of the 2HDM Type III in the decoupling limit The 2HDM is a minimal extension of the Standard Model, including one extra Higgs doublet with new unknown interactions to the Standard Model fermions and Higgs-for a review, see [9,10]. In the 2HDM considered here, the extra doublet is taken to be heavy-this is the decoupling limit, and it should be describable with EFT. To allow for LFV, consider a "Type III" model, where there is no discrete symmetry that distinguishes the Higgs, so there is no "symmetry basis" in which to write the Lagrangian (so also no unambiguous definition of tan β). For simplicity, the Higgs potential is taken to be CP invariant.
The Lagrangian can be written in the "Higgs basis", defined such that H 1 = 0, and H 2 = 0, so the doublets are written 1 where the Gs are Goldstones. This shows that the mass eigenstates A and H ± are the CP-odd and charged components of the vev-less H 2 , but there can be some mixing between H 1 and H 2 in the CP-even scalars h, H . The convention In this "Higgs basis", in the notation of [11], the potential parameters are written in upper case, so the potential is ] (in Higgs basis) The angle β − α can be defined from the Higgs potential of a Type III model, unlike β and α. It rotates between the 1 Neglecting the phase ambiguity χ discussed at eqn A.11 of [11]. mass basis of h, H and the Higgs basis: so that what is here called β −α is independent of the angle β, which will later be defined from the Yukawas. If the potential is CP invariant, there is a simple relation for c β−α [11]: The masses of the scalars are related to the potential parameters in the Higgs basis as [11] and the couplings to In the decoupling limit [12], where i v 2 M 2 , the exact relations (6) can be expanded in v 2 /M 2 to obtain and then approximating s β−α 1 in Eq. (5) gives which confirms that, in the decoupling limit, h is mostly H 1 and H is mostly H 2 . The Yukawa couplings in the Higgs basis for the Higgses, and the mass eigenstate basis for the (10) where Q, L are SU(2) doublets, E, U, D are singlets, SU (2) indices are implicit (L H 1 =ν H + 1 +ēH 0 1 ), H i = iσ 2 H * i , the generation indices are explicit, and K is the CKM matrix. The Y matrices are flavour diagonal and equal to the SM Yukawas: as a result of being in the fermion mass eigenstate bases. The ρ P matrices can be flavour-changing. To obtain μ → eγ without other flavour-changing processes, I neglect all off-diagonal elements except ρ μe ρ eμ = 0. To obtain a predictive model and make contact with other 2HDM literature, the diagonal elements follow the pattern of one of the types of 2HDM which have a discrete symmetry that ensures flavour conservation: This requires a definition of tan β which is common to all the fermions. It can for instance be defined from the τ Yukawa: in the mass eigenstate basis for τ R , τ L , define H τ to be the linear combination of Higgses to which couples the τ , and β as the angle in Higgs doublet space between H 1 (the vev) and H τ : Since tan β is defined from the lepton Yukawas, ρ E ∝ tan β in all Types of 2HDM listed above. This is unconventional, but should include the same predictions provided that tan β is allowed to range from 1/50 → 50. From Eq. (10) the couplings to fermions are This gives Feynman rules of the form −i F φ,X i j P X , where where the possible forms for ρ P are given in Eq. 12.

μ → eγ in the 2HDM
The decay μ → eγ can be parametrised by adding the dipole operator to the Standard Model Lagrangian. In the notation of Kuno and Okada [13] which gives where the upper bound is from the MEG experiment [14].
The decay μ → eγ has been extensively studied in the 2HDM [3,[15][16][17][18][19][20][21][22][23][24][25][26][27], particularily in connection [29][30][31][32][33][34][35][36][37] with the recent LHC [28,38] excess in h → τ ± μ ∓ . Chang, Hou and Keung [5] (CHK) calculate the contributions to μ → eγ of neutral Higgs bosons with flavour-changing couplings. Their calculation can be divided into four classes of diagrams, illustrated in Fig. 1: the one-loop diagrams, then three classes of two-loop diagrams, that is, those with a t-loop, with a b-loop and with a W -loop. I use their results, and later, in matching onto SU(2)-invariant operators, assume that the charged Higgs contribution ensures the SU(2) invariance of the results. The results of CHK are referred to as "full-model" results in the following. In the appendix is given a translation dictionary between the notation of CHK and here. The amplitude given by CHK [5] 2 for the one-loop diagram of Fig. 1, with an internal μ, is Notice that these amplitudes are suppressed by two leptonic Yukawas and an additional muon mass insertion to flip the chirality. In the decoupling limit, with φ ∈ {H, A} and μ in the loop, Eq. (18) for A L gives where m H m A M was used in the logarithm, and terms suppressed by more than M −4 were dropped. The result for i j in the above. The one-loop diagram with the light Higgs h and a μ (also in the decoupling limit) gives As noted long ago by Bjorken and Weinberg [39], there are two-loop diagrams which can be relevant for μ → eγ . Some examples are illustrated to the right in Fig. 1; a more complete set of diagrams with broken electroweak symmetry can be found in [40]. The resulting two-loop contributions to μ → eγ can be numerically larger than the one-loop contributions, because the Higgs attaches only once to the lepton line, via the flavour-changing coupling, and otherwise couples to a W, b or t loop. The result for the top loop (neglecting the diagrams with internal Z exchange, see Fig. 1 where the log 2 terms of the approximate equality are the sum of the heavy H and A contributions, and the third term is due to the light h. The approximate equality is in the decoupling limit, so one uses m H m A M in the log, neglects terms suppressed by v 2 /M 2 , and uses the following, which holds for small z [5]: (The function h will appear in the W -loop contribution.) The functions f, g and h are given in CHK, are slowly varying near z ∼ 1 and at as expected because a top mass insertion is required both to have an even number of γ -matrices in the loop, and to provide the Higgs leg of the dipole operator.
A b-quark loop should be described by the same formula as the top loop (again neglecting the Z -exchange diagrams), but the A-exchange contribution will subtract from H -exchange, because of the sign difference in the couplings of A to bs and ts (see Eq. (15)): where the approximate equality is in the decoupling limit, and uses the approximations of Eq. (22). The last term is the light Higgs contribution. Notice that the O(1/M 2 ) contributions of H and A cancelled against each other in the decoupling limit. The formula for A R is obtained by replacing Finally, there is a two-loop contribution involving a Wloop; the third diagram of Fig. 1 is one of the many that contribute. In the CP-conserving 2HDM considered here, A does not couple to the W or the Goldstones, so it cannot appear in these diagrams. CHK compute separately the diagrams with either photon or Z exchange between the lepton and W (see Fig. 1). The Z -mediated amplitude is proportional to 1 − 4 sin 2 θ W and estimated by CHK to be about 10 % of the photon-mediated amplitude, so only the γ -amplitude is considered here. The H and h contributions individually are where z φ = m 2 W /m 2 φ , and -is for φ = H , +is for φ = h. The sum of the contributions, in the decoupling limit, is where the limiting forms of Eq. (22) were used. The first line of the approximate equality is the contribution of H , and the last line is the contribution of h where z h = m 2 W /m 2 h , and the parentheses term evaluates to ∼ 7. The heavy H exchanged between the lepton line and a Goldstone loop generates an O(log /M 2 ) term, which is carefully discussed by CHK, because the 1/M 2 suppression arises from the decoupling limit of c β−α , given in Eq. (9). CHK take the mixing angle c β−α as a free parameter, so one refers to this term as a "non-decoupling" contribution.

The relative importance of the O(1/M 2 )
and For the one-loop diagrams, Eqs. (19,20) give the ratio of Recall that, in this paper, tan β is defined referring to the leptons, see Eq. (13), so So the dimension-eight contribution can be numerically relevant in some areas of parameter space. For where the estimate neglects cancellations and shows that the O(1/M 4 ) contribution is only mildly suppressed, because z ln 2 z does not decrease rapidly.
where v 2 /m 2 where f (m 2 t /m 2 h ) 1 was used. Since α The two-loop top and b contributions contain log 2 terms, which should arise at second order in the one-loop RGEs of dimension-six operators. However, there are significant terms of the top and W contributions without a log, which presumably arise in two-loop matching. The log term of the W amplitude should be generated by two-loop RGEs. So we see that a two-loop analysis would be required to reproduce the dominant O(1/M 2 ) terms.

The EFT version
The aim of this section is to obtain, in EFT, the "leading" O([α log] n /M 2 ) parts of the μ → eγ amplitude. Appendix B discusses where some other parts of the fullmodel calculation would arise. EFT is transparently reviewed in [1,2], and the EFT construction here attempts to follow the recipe given there. The EFT studied here is at "lowest order" in the loop expansion: tree-level matching of operator coefficients, and one-loop RGEs. This should reproduce the O(α n log n ) terms 4 in the amplitude. I do not calculate the two-loop matching, or the two-loop RGEs, which would allow one to reproduce the dimension-six contributions of the top and W .

Setting up the EFT calculation
The aim of this top-down EFT calculation is to reproduce the amplitude for μ → eγ in the 2HDM. So the first step is to match out the heavy Higgses H and A at the "New Physics scale" M. The H ± are neglected because the full-model calculation (to which the EFT is compared) includes only neutral Higgses. Presumably the H ± contribution ensures SU(2) invariance. The 2HDM from above M is matched onto the (unbroken) Standard Model, with its full particle content, and a selection of SU (3)×SU (2)×U (1)-invariant dimension-six operators. A list of possible operators was given by Buchmuller and Wyler [3], and slightly reduced in [4]. Here, only those operators which are required to reproduce the O(1/M 2 ) part of the μ → eγ amplitude are selected. In matching out, the operator coefficients of the effective theory are assigned so as to reproduce the tree-level Green's functions of the full theory, at zero external momentum.
The second step is to run the operator coefficients down to the scale m W . This running should be performed with electroweak RGEs, which are given in [6,41]. However, since CHK separate photon and Z diagrams, and only the photon contributions were retained in the previous section, the running from M → m W is performed with the RGEs of QED. The operator coefficients evolve [42] with scale μ as where the operator coefficients have been organised into a row vector C, and α em 4π is the anomalous dimension matrix. The algorithm to calculate is given, for instance, in [43]. For a square , this equation can be perturbatively solved to give the components of the vector C(μ) at a lower scale μ: At m W , the W, Z , h and t should be matched out. The theory below m W should be Q E D and QC D for all the SM fermions except the top, augmented by a complete set of Q E D × QC D-invariant dimension-six operators. For simplicity, I consider only QED for the b, μ and e, plus those dimension-six operators required in matching onto the treelevel Green's functions of the SU(2)-invariant EFT from above m W .
Finally, the operator coefficients are run down to m μ , where the dipole coefficient can be used to calculate the μ → eγ amplitude. In principle, the RGEs of QED and QCD should be used. However, QCD is neglected because it is not included in the full-model calculation of CHK.

Matching at M and one-loop running to m W
The operator for μ → eγ is given in Eq. (16). It is convenient to rescale the coefficient as where σ is the anti-symmetric tensor i 2 [γ, γ ], and provides an SU(2) contraction. These operators appear in the Lagrangian as so the coefficients C are dimensionless, and the four-fermion operators are normalised such that the Feynman rule is −iC/M 2 . The last pair of operators, Eq. (42) will give the flavourchanging light Higgs interaction required for the one-loop, b-loop and W -loop amplitudes. As can be seen from the right diagram of Fig. 2, matching at the scale M of the tree-level Green's functions of the full theory onto those of the SM + dimension-six operators gives coefficients C eμ eH The running of these coefficients between M and m W is neglected, because it is not required to reproduce the CHK results.
The first four operators of the list above will generate the top loop contribution. Matching at M via the diagram given on the left in Fig. 2 gives coefficients for the scalar L E QU operators, where the negative sign is from the scalar propagator.
where in the brackets after the first equality is the product of the scalar→tensor and tensor→dipole elements of the anomalous dimension matrix . This agrees with the O(1/M 2 ) part of Eq. (21) that is generated by heavy Higgs exchange.
If SU (2) were imposed, these operators would be of dimension eight (for instance, the first operator could be written as (8) H D b )). However, they are of dimension six in the QCD×QED-invariant EFT below m W , and they are required in the 2HDM to correctly reproduce the O(α n log n /M 2 ) terms that dimension-six, one-loop EFT should obtain. The operators of Eq. (47) are not included in the EFT analysis of μ → eγ performed by Pruna and Signer [6], who make the restriction to dimension-six SU(2)-invariant operators. From the diagrams illustrated in Fig. 3, one obtains , which mixes to the dipole, exactly as in the previously discussed case of tops. So with the anomalous dimensions as in Eq. (46), the dipole coefficient . However, this operator is not useful for generating μ → eγ via one-loop RGEs, because there is no tensor operator for it to mix to, on the way to the dipole. The reason that there is no tensor is that σ and σ γ 5 are related: σ μν = i 2 ε μναβ σ αβ γ 5 , which implies that (eσ αβ γ 5 μ)(bσ αβ γ 5 b) = (eσ μν μ)(bσ μν b) or (eσ αβ P L μ)(bσ αβ P R b) = 0. The scalar operator with three muons, O eμμμ S , mixes directly to the dipole via a penguin diagram, so the coef- Fig. 3 The right diagram generates the QCD×QED-invariant, dimension-six operator (eP R μ)(μP R μ), by matching out the light Higgs h (dotted line), which has a SU(2)-invariant dimension-six LFV vertex represented by the grey circle. The free Higgs legs attach to the vev. The left diagram generates a similar operator

Discussion
The CP-conserving 2HDM studied here is a minimal extension of the Standard Model-it has only one LFV coupling , the magnitude of the other new flavoured couplings is controlled by the single parameter tan β, defined in Eq. (13), and the only new mass scale is the heavy doublet mass M, taken 10v. The one-and two-loop contributions to μ → eγ of the neutral Higgses were calculated by Chang et al. [5], and their result, in the decoupling limit, is given in Eqs. (20), (19), (21), (23) and (25).
Section 3 tries to reproduce the amplitude for μ → eγ , in a simple EFT with three scales: M, m W , m μ . At the scale M, the heavy Higgses were "matched out" onto dimension-six, SU(2)-invariant operators. Between M and m W , the operator coefficients should run according to electroweak RGEs, but I used those of QED (because I compare only to the photon diagrams of [5]). At m W , the W, Z , h and t are matched out, so below m W is an EFT containing all Standard Model fermions but the top, interacting via QCD, QED, and various four-fermion operators. Finally the operator coefficients run to m μ according to the RGEs of QED.
There were three issues in reproducing the amplitude for μ → eγ in EFT: 1. In order to obtain the O(1/M 2 ) terms of the full-model calculation, the operator basis below the weak scale needed to include QED×QCD-invariant four-fermion operators which are dimension six, but would have been dimension eight in the SU (2) , as illustrated on the right in Fig. 3. Then, in QED running down to m μ , this operator mixes to the dipole by a penguin diagram obtained by closing a muon loop, inserting a mass and attaching a photon. So the con- However, although the coefficients of (eP R μ)(μP R μ) and (eP R μ)(bP R b) are formally O(1/M 2 ), they are also small, because proportional to light fermion Yukawa couplings. As discussed in Sect. 2.2.2, they do not give a numerically relevant contribution to μ → eγ in the decoupling limit of the 2HDM. So although these extra operators are required below m W to obtain all the O(1/M 2 ) terms, they are not necessary for getting a reasonable approximation to the answer. 6 2. There are two-loop contributions involving a top or Wloop, initially discussed by Bjorken and Weinberg [3], and illustrated in Fig. 1. As discussed in Sect. 2.2.2, these always dominate the one-loop contribution to μ → eγ in the decoupling limit. This disorder in the loop expansion occurs because the one-loop contribution is suppressed by the square of the muon Yukawa coupling, whereas the two-loop diagrams are proportional to the square of the gauge coupling or of the top Yukawa. In the EFT, twoloop matching and running would be required to reproduce the W -loop, and part of the top loop. This "disorder" is partly a feature of the dipole operator, and partly of the 2HDM. The dipole operator has a Higgs leg, and at one loop, that Higgs can attach to the fermion line, or to the boson line, if the boson of the loop has a dimension-three coupling to the Higgs (such as the μ * H * u L E interaction in supersymmetry). However, at two loops, there are many more possibilities for attaching the external Higgs leg with an O(1) coupling. In some models, it might be possible to have O(1) coupling of the external Higgs leg to the fermion or boson of the one-loop diagram. However, in the case of the 2HDM, there are no new fermions so the fermion of the one-loop diagram is at best a τ . The boson is a Higgs without dimension-three interactions, so the one-loop diagram is suppressed by small Yukawa couplings. This issue could be addressed by running and matching at two-loop, 7 as is done for b → sγ [43]. There is also the inevitable ignorance, in EFT, as regards the magnitude of operator coefficients in the full theory. This uncertainty is parametrised by tan β in the 2HDM. To justify neglecting the dimension-eight operators, the restrictions cot β, tan β < v 2 /M 2 had to be imposed. However, since flavour physics is about the hierarchy of couplings, it may not be sensible to assume that all New Physics couplings are O(1) at the New Physics scale.
This exercise located the "Barr-Zee" diagrams (see the right two diagrams of Fig. 1) in an EFT description. The light Higgs contribution, which appears at second order in the one-loop RGEs for dimension-six operators below m W , is suppressed by the square of the SM Yukawa coupling of the b. The heavy Higgs contribution, which could be tan β enhanced, however, only contributes at dimension eight, via second order terms in the one-loop RGEs. This is because the tensor operator (ψ 1 σ P L ψ 2 )(ψ 3 σ P R ψ 4 ) vanishes, so the SU(2)invariant dimension-six scalar operator that can be constructed with leptons and down quarks, (Q D)(E L), cannot mix via RGEs to a tensor and then the dipole.

Summary
This paper used Effective Field Theory (EFT) to calculate the amplitude for μ → eγ in the decoupling limit of a The 2HDM also illustrates that higher dimensional operators may be numerically relevant, because they can be enhanced by unknown large couplings of the high-scale model (e.g. tan β), or by logarithms.
To present the results of CHK, a translation dictionary between their notation and the notation here is useful. CHK give the Lagrangian as where we CHK give their two-loop amplitudes A L ,R in a different normalisation from Eq. (17); the relation is where the negative sign is because Kuno-Okada subtract their dipole operator from the SM Lagrangian, and the hermitian conjugate is because Kuno-Okada write an operator that mediates μ + → e + γ , and CHK compute amplitudes for μ − → e − γ .

B.1. The full-model one-loop contribution
All the terms in Eqs. (20) and (19) Fig. 5 will contribute to the coefficient of the dimension-eight operator: The operator O eμμμ S can then mix via a penguin to the dipole operator.
Then there is also a light Higgs contribution to the same operator at O(1/M 4 ), which can nonetheless by relatively enhanced by tan β with respect to the O(1/M 2 ) term. The corresponding diagram contracts two of the diagrams illustrated on the right in Fig. 2, so it has four external H 1 lines.

B.2. The top loop
The two-loop contribution of H and A is log 2 enhanced, so it arises in the one-loop RGEs. In addition to the previously discussed dimension-six term, that can be suppressed by cot β, the third diagram of Fig. 5  and such a term appears in the full-model calculation of Eq. (21). The top-loop contribution of the light h, which can contribute a significant part of the μ → eγ amplitude, is a twoloop matching contribution at the weak scale.

B.3. The b-loop
The b-loop differs from the top loop, in that the heavy Higgs only contribute at O(1/M 4 ), because there is no dimensionsix scalar operator for bs that can mix to a tensor. Matching out the heavy Higgs onto the dimension-eight operator (L μ H 1 E e )(Q 3 H 1 D b ) would give (8)  As in the one-loop contribution, there is an O(1/M 4 ) term in the light Higgs exchange amplitude, which can be tanβ enhanced, and arises due to two appearances of the dimension-six O eH , with indices eμ and bb.

B.4. The W loop
Despite the fact that the W -loop can give the dominant contribution to μ → eγ in the 2HDM, none of it was obtained in a one-loop EFT calculation using dimension-six operators.
The heavy Higgs part has terms of O(log/M 2 ), O(log 2 / M 4 ), and O(log/M 4 ). Consider here the first two: CHK refer to the dimension-six part as a "non-decoupling" contribution, because they take the mixing angle c β−α as a free parameter, rather than using the decoupling limit dependence given in Eq. (9). They say this contribution arises from Goldstone loops, so in EFT it could be generated by matching out the heavy doublet onto O