\(\mu \rightarrow e \gamma \) and matching at \(\varvec{m_W}\)
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Abstract
Several experiments search for \(\mu \leftrightarrow e\) flavour change, for instance in \(\mu \rightarrow e ~\mathrm{conversion}\), \(\mu \rightarrow e \gamma \) and \(\mu \rightarrow e \bar{e} e\). This paper studies how to translate these experimental constraints from low energy to a New Physics scale \(M \gg m_W\). A basis of QCD \(\times \) QEDinvariant operators (as appropriate below \(m_W\)) is reviewed, then run to \(m_W\) with oneloop Renormalisation Group Equations (RGEs) of QCD and QED. At \(m_W\), these operators are matched onto SU(2)invariant dimensionsix operators, which can continue to run up with electroweak RGEs. As an example, the \(\mu \rightarrow e \gamma \) bound is translated to the scale M, where it constrains two sums of operators. The constraints differ from those obtained in previous EFT analyses of \(\mu \rightarrow e \gamma \), but they reproduce the expected bounds on flavourchanging interactions of the Z and the Higgs, because the matching at \(m_W\) is pragmatically performed to the loop order required to get the “leading” contribution.
1 Introduction
Neutrino masses and mixing angles imply that “New” Physics from beyond the Standard Model(SM) must be present in the lepton sector, and must induce charged Lepton Flavour Violation (LFV; for a review, see [1]). However, neither LFV nor the origin of neutrino masses has yet been discovered. This study assumes that the required new particles are heavy, with masses at or beyond \(M > m_W\). In addition, between \(m_W\) and M, there should be no other new particles or interactions which affect the LFV sector. One approach to identifying this New LFV Physics, is to construct a motivated model, and identify its signature in observables [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. A more pragmatic approach, which requires optimism but no modelbuilding skills, is to parametrise the New Physics at low energy with nonrenormalisable operators, map the experimental constraints onto the operator coefficients, and attempt to reconstruct the fundamental Lagrangian of New Physics from the operator coefficients. This is probably not feasible, but could give interesting perspectives. A first step in this “bottomup” approach, explored in this paper, is to use Effective Field Theory (EFT) [31] to translate the experimental bounds to the coefficients of effective operators at the New Physics scale \(M> m_W\).
The goal would be to start from experimental constraints on \(\mu \)–e flavour change, and obtain at M the best bound on each coefficient from each observable. These constraints should be of the correct order of magnitude, but not precise beyond one significant figure. This preliminary study restricts the experimental input to the bound on \(BR(\mu \rightarrow e \gamma )\), and makes several simplifications in the translation up to the New Physics scale M. Firstly, the EFT has three scales: a low scale \(m_\mu \sim m_b\), the intermediate weak scale \(m_W\), and the high scale M. Secondly, at a given scale, the EFT contains lighter Standard Model particles and dimensionsix, gaugeinvariant operators (one dimensionseven operator is listed; however, dimensioneight operators are neglected). The final simplification might have been to match at tree level, and run with oneloop Renormalisation Group Equations (RGEs). However, a bottomup EFT should reproduce the results of topdown model calculations, and it is straightforward to check that one and twoloop matching is required at \(m_W\) to obtain the correct bounds from \(\mu \rightarrow e \gamma \) on LFV interactions of the Z and Higgs. So the matching at \(m_W\) is performed to the order required to get the known bounds.
The paper is organised in two parts: the Sects. 2–4 construct some of the framework required to obtain experimental constraints on SU(2)invariant operator coefficients at \(m_W\), then Sect. 5 focusses on using, checking and improving this formalism to obtain bounds from \(\mu \rightarrow e \gamma \) on operator coefficients at M. The formalism can be organised in four steps: matching at \(m_\mu \), running to \(m_W\), matching at \(m_W\), then running up to the New Physics scale M. Section 2 reviews the basis of QCD\(\times \)QEDinvariant operators, as appropriate below \(m_W\). These operators, of dimension five, six and seven, describe three and fourpoint functions involving a \(\mu \), an e and any other combination of flavourdiagonal light particles. To complete the first step, the experimental bounds should be matched onto these operator coefficients; however, this is delayed till Sect. 5, where only the bound on \(\mu \rightarrow e \gamma \) is imposed on the dipole coefficients (the bounds from \(\mu \rightarrow e ~\mathrm{conversion}\) and \(\mu \rightarrow e \bar{e} e\) are neglected for simplicity; the strong interaction subtleties of matching to \(\mu \rightarrow e ~\mathrm{conversion}\) are discussed in [33, 34, 35, 36]). Section 3 discusses the second step, which is to run the coefficients up to \(m_W\) with the RGEs of QED and QCD. Appendix B gives the anomalous dimension matrix mixing the scalar and tensor operators to the dipole (which is responsable for \(\mu \rightarrow e \gamma \)). The anomalous dimension matrix for vector operators is neglected for two reasons: although vectors contribute at tree level to \(\mu \rightarrow e ~\mathrm{conversion}\) and \(\mu \rightarrow e \bar{e} e\), these experimental bounds are not included, and the leading order mixing of vectors to the dipole is at twoloop in QED, whereas the running here is only performed at oneloop. The next step is to match these operators at \(m_W\) onto the Buchmuller–Wyler [37] basis of SU(2)invariant operators as pruned in [38], which is referred to as the BWP basis. The treelevel matching for all operators is given in Sect. 4; if this is the leading contribution to the coefficients, then imposing SU(2) invariance above \(m_W\) predicts some ratios of coefficients below \(m_W\), as discussed in Sect. 4.2. Section 5 uses the formalism of the previous sections to translates the experimental bound on \(BR(\mu \rightarrow e \gamma )\) to sums of SU(2)invariant operator coefficients at \(m_W\). Then a few finiteloop contributions are added, and the coefficients are run up to M, using a simplified version of the oneloop QCD and electroweak RGEs [39, 40, 43]. Finally, Sect. 6 discusses various questions arising from this study, such as the loop order required in matching at \(m_W\), whether the nonSU(2)invariant basis is required below \(m_W\), and the importance of QED running below \(m_W\).
Many parts or this analysis can be found in previous literature. Czarnecki and Jankowski [41] emphasized the oneloop QED running of the dipole operator (neglected in the estimates here), which shrinks the coefficient at low energy. Degrassi and Giudice [42] give the leading order QED mixing of vector operators to the dipole, which is also neglected here, because it arises at two loop. In an early topdown analysis, Brignole and Rossi [15] calculated a wide variety of LFV processes as a function of operator coefficients above \(m_W\), without explicit Renormalisation Group running and a slightly redundant basis. Pruna and Signer [43] studied \(\mu \rightarrow e \gamma \) in EFT, focussing on the electroweak running above \(m_W\), which they perform in more detail than is done here. However, they do not obtain the bounds on the LFV couplings of the Z and Higgs that arise here in matching at \(m_W\). Various oneloop contributions to \(\mu \rightarrow e \gamma \) were calculated in [44], without organising them into running and matching parts. Finally, the contribution of the LFV Higgs operator to LFV Z couplings was beautifully studied in [45]. There are also many closely related works in the quark sector, reviewed in [46, 47]. For instance, the QED anomalous dimension matrix for various vector fourquark operators is given in [48], and matching at \(m_W\) of flavourchanging quark operators is discussed in [49]. However, colour makes the quarks different, so it is not always immediate to translate the quark results to leptons.
2 A basis of \(\mu \)–e interactions at low energy
2.1 Interactions probed in muon experiments
Experiments searching for leptonflavour change from \(\mu \) to e, probe three and fourpoint functions involving a muon, an electron and one or two other SM particles. I focus here on interactions that can be probed in \(\mu \rightarrow e \gamma \), \(\mu \rightarrow e \bar{e} e\) and \(\mu \rightarrow e ~\mathrm{conversion}\), meaning that the interactions are otherwise flavour diagonal, and there is only one muon (so \(K\rightarrow \bar{\mu }e\) and other meson decays is not considered).
All the operators appear in the Lagrangian with a coefficient \(C/M^2\), and the operator normalisation is chosen to ensure that the Feynman rule is \(i C/M^2\). This implies a judicious distribution of \(\frac{1}{2}\)s, which is discussed in Appendix B.
2.2 Including heavy fermions
Including these operators introduces a second “low” scale into the EFT, which in principle changes the running and requires matching at this second low scale \(m_\tau \). The running is discussed in the next section. Since the matching is at tree level, the operators present below \(m_\tau \) have the same coefficient just above \(m_\tau \). Were the dipole to be matched at one loop, then at \(m_\tau \), one should compute the finite part of the diagrams [44] obtained by closing the heavy fermion loop of the tensors \(\mathcal{O}^{e\mu bb}_{T,YY}\), \(\mathcal{O}^{e\mu cc}_{T,YY}\) and \(\mathcal{O}^{e\mu \tau \tau }_{T,YY}\), and attaching a photon (and also there could be similar finite contributions from fourfermion operators at \(m_\mu \)). Also, scalar operators involving b, c quarks would match at one loop onto \(\mathcal{O}^{GG,Y}\) [50], as outlined in [51].
3 Running up to \( m_W\)
The operators of Eqs. (1) and (2) can evolve with scale due to QED and QCD interactions. QCD effects can be significant, and should be resummed, but fortunately they only change the magnitude of operator coefficients, without mixing one operator into another. This will be taken into account by multiplying twolepton–twoquark operators by an appropriate factor (following Cirigliano et al. [35]). The effects of QED running are usually small, of order \(\alpha _{em}/\pi \), but interesting because they give operator mixing. Therefore the QED renormalisation of individual operator coefficients is neglected, and only the mixing is included.
The scale at which the operators of Eqs. (1) and (2) start running is variable. The lepton operators of Eq. (1) will start their QED running at \(m_\mu \), whereas those of Eq. (2) start at \(m_\tau \). The the twoleptontwob operators start running up at \(m_b\). For simplicity, the remaining twoquark–twolepton operators are taken to start running up at \(m_\tau \); that is, the experimental bounds are assumed to apply at a scale \(\sim m_\tau \).
3.1 Defining the anomalous dimension matrix
For QED mixing of fourfermion operators among themselves and to the dipole, the relevant diagrams are in Fig. 1, where the gauge boson is the photon, and \(f_2 \in \{ u,d,s,c,b,e,\mu ,\tau \}\). These diagrams allow one to compute the \(\gamma \)submatrices of Eq. (14). The results are given in Appendix B. For the second diagram of Fig. 1, \(f_1 =e,\mu \), because Fierz transformations were used to obtain a basis where the \(\mu \)–e flavour change is within a spinor contraction.
4 At \(m_W\)
Above the “intermediate”, weak scale of the EFT, \(m_W \simeq m_h \simeq m_t\), two things differ with respect to the low energy theory: the theory and nonrenormalisable operators should now respect the weak SU(2) symmetry, and the particle content is extended to include the weak gauge bosons, the higgs, and the top. The additional requirement of SU(2) invariance will reduce the number of possible fourfermion operators, whereas adding new degrees of freedom (h, W, Z, t) allows more flavourchanging operators involving only two fermions.
4.0.1 Neglecting dimensioneight operators
The EFT above \(m_W\) is an expansion in the inverse New Physics scale 1 / M, where the lowest order operators that are lepton flavourchanging, but numberconserving, appear at dimension six; they are listed in Appendix D. It is convenient to neglect the next order operators, which would appear at dimension eight, because they are numerous, and their RGEs are unknown. So it is interesting to explore how small must be the ratio v / M, to justify a parametrisation using dimensionsix operators.
 1.
In some cases, the dimensionsix and eight contributions arise at the same loop order, but the dimensionsix part is from matching, whereas the dimensioneight term arises in running and is log\(^2\)enhanced. The ratio of dimension six to eight is then \(\propto z\ln ^2 z\), which is \({\sim }0.2\) for \(M \sim 10 v\).
 2.
The couplings of the New Physics are unknown, and could have steep hierarchies. In the 2HDM, the heavy Higgs couplings to light fermions can be \(\mathcal{O}(1)\), rather than of order the fermions’ SM Yukawa coupling. This increase is parametrised in the 2HDM by \(\tan \beta \), which in some cases enhances the dimensioneight operators with respect to dimension six. In some 2HDMs, \(\tan \beta \lesssim 50\), which I take as a reasonable estimate of the possible hierarchy of couplings between dimensionsix and either operators.
4.1 Tree matching onto SU(2)invariant operators
The coefficients of the fourfermion operators from below \(m_W\), given in Eqs. (1) and (2), should be matched at \(m_W\) onto the coefficients of the SU(2)invariant BWP basis, which are listed in Appendix D. The coefficients on the left of the equalities are from below \(m_W\), the coefficients on the right are SU(2) invariant. Both sets of coefficients should be evaluated at \(m_W\), and the fermion masses which appear in the matching conditions should also be evaluated at \(m_W\).
4.1.1 Dipoles
4.1.2 Fourlepton operators
The BWP basis contains only the “vector” fourlepton operators given in Eqs. (94), (95) and (96). There are also new dimensionsix interactions of the W, Z and h, described by the operators of Eqs. (102), (103), (104) and (99), which will contribute to fourlepton operators below \(m_W\) in matching out the Z and h.
There are a few curiosities related to the flavour index structure below and above \(m_W\). First, since the basis below \(m_W\) was defined with the e–\(\mu \) indices inside a spinor contraction, there is a scalar operator from below \(m_W\) which must be Fierzed as given in Eq. (98). Also, there are more distinct flavour structures for operators constructed with SU(2) doublets, than singlets: the SU(2)invariant operators \(\mathcal{O}^{e \mu ff}_{L L} \) and \(\mathcal{O}^{e ff \mu }_{L L} \), both match onto the below\(m_W\) operator \(\mathcal{O}^{e \mu ff}_{L L} \). However, the two SU(2) operators are distinct^{2} for \(f =\tau \), but not for \(f = e,\mu \).
4.1.3 Twolepton–twoquark operators
Two issues about the CKM matrix V arise in matching operators involving quarks at \(m_W\): does V appear in the coefficients above or below \(m_W\), and are the quark doublets in the u or d mass basis? I put V in the coefficients above \(m_W\), because the experimental constraints are being matched “bottomup” onto operator coefficients. So one coefficient from below \(m_W\) will match onto a sum of coefficients above \(m_W\), weighted by CKM matrix elements. Secondly, the quark doublets above \(m_W\) are taken in the u, c, t mass eigenstate basis, because it is convenient for translating up in scale the bound on \(\mu \rightarrow e \gamma \), as will be done in Sect. 5. This is because tensor operators mix to the dipoles, and only for utype quarks are there SU(2)invariant dimensionsix tensors operators.
4.2 Comments on tree matching
One observes that the consequences of matching at \(m_W\), at tree level, are different for vector vs. scalar–tensordipole operators. In the vector case, there are more operator coefficients in the SU(2)invariant theory above \(m_W\) than in the QCD\(\times \)QEDinvariant theory below, whereas there are fewer for the scalar–tensor operators. This means that SU(2)invariance should predict some correlations in the scalar–tensor coefficients below \(m_W\). Whereas, if one was trying to reconstruct the coefficients of the SU(2)invariant operators from data, some additional input (e.g. from Z physics, neutrino interactions [52], or loop matching) would be required for the vector operators, beyond the coefficients of the operators of Eqs. (1) and (2).
4.2.1 The vector operators

In the case of fourlepton operators with flavour indices \(e \mu ee\) or \(e\mu \mu \mu \), there are the same number of independent coefficients above and below. There is one extra fourlepton operator above \(m_W\) for flavour indices \(e\mu \tau \tau \), as can be seen from Eq. (21).

There are fewer twolepton–twoquark operator coefficients above \(m_W\) than below. It is clear that the operators \(\mathcal{O}_{LR}^{e \mu qq}\), \(\mathcal{O}_{RR}^{e \mu qq}\) from below \(m_W\) with \(q \in \{u,d,s,c,b\}\) are equivalent to the \(\mathcal{O}_{LU}^{e \mu u_nu_n}\), \(\mathcal{O}_{LD}^{e \mu d_nd_n}\), \(\mathcal{O}_{EU}^{e \mu u_nu_n}\), \(\mathcal{O}_{ED}^{e \mu d_nd_n}\) operators from above. Also it is clear that the \(\mathcal{O}_{LL}^{e \mu qq}\) from below \(m_W\) with \(q \in \{u,d,c,s,b\}\) have the same number of independent coefficients as \(\mathcal{O}_{LQ (1)}^{e \mu nn}\) and \(\mathcal{O}_{LQ (3)}^{e \mu nn}\). The restriction occurs between \(\mathcal{O}_{RL}^{e \mu qq}\) from below \(m_W\), where there are five coefficients corresponding to \(q \in \{u,d,s,c,b\}\), and \(\mathcal{O}_{EQ}^{e \mu nn}\) above \(m_W\), which has a coefficient per generation. Neglecting CKM sums, this suggests that SU(2) predicts the \(C_{RL}^{e \mu uu}  C_{RL}^{e \mu dd}=0\) and \(C_{RL}^{e \mu cc}  C_{RL}^{e \mu ss} = 0\); however, there is a penguin operator which contribute to both differences, so only the difference of differences is an SU(2) prediction (possibly blurred by CKM).

The “penguin” operators from above \(m_W\) (see Eqs. (102), (103), (104)) give the Z a vertex with \(\bar{e} \gamma P_Y \mu \), which matches onto \((\bar{e} \gamma P_Y \mu ) (\bar{f} \gamma P_X f)\) operators for all the SM fermions below \(m_W\), in ratios fixed by the SM Z couplings. This contribution adds to the fourfermion operator induced at the scale M in the EFT, as given in the matching conditions Eqs. (17)–(30). So the coefficient of the \(\bar{e} \, Z \! \! \! \! / ~ P_R \mu \) penguin operator of Eq. (104) could be determined from \(C_{RL}^{e \mu uu}  C_{RL}^{e \mu dd}\), as discussed in the item above. The coefficients of the two remaining penguin operators are “extra”: in naive coefficientcounting, there are two more vector coefficients above \(m_W\) than below. However, they are not completely “free”, because they would match at one loop onto the photon dipole operator at \(m_W\). These extra penguins are related to the common wisdom, that it is interesting for ATLAS and CMS to look for \(Z \rightarrow \tau ^\pm \mu ^\mp \) and \(Z \rightarrow \tau ^\pm e^\mp \) decays, but that they are unlikely to see \(Z \rightarrow \mu ^\pm e^\mp \) [53]. The point [54] is that an interaction \(\bar{\tau } \, Z \! \! \! \! / ~ \mu \) would contribute at tree level to \(\tau \rightarrow \mu \bar{l}l\), and at one loop to \(\tau \rightarrow \mu \gamma \). To be within the sensitivity of the LHC, the coefficient of this coupling needs to exceed the naive bound from \(\tau \rightarrow \mu \bar{l}l\). However, \(BR(\tau \rightarrow 3l )\) [55] is controlled by coefficients \(C_{XY}^{\mu \tau ll}\), \(C_{YY}^{\mu \tau ll}\), analogous to the coefficients on the left of Eqs. (17)–(22), which are the sum of SU(2)invariant fourfermion and penguin coefficients. So the penguin coefficient could exceed the expected bound from \(\tau \rightarrow 3l \), provided that it is tuned against the fourfermion coefficient.^{3} This same argument could apply to a \(\bar{e} \, Z \! \! \! \! / ~ \mu \) coupling and the bound from \(\mu \rightarrow e \bar{e} e\), although more tuning would be required, since the bound on \(\mu \rightarrow e \bar{e} e\) [57] is more restrictive. However, the Z penguins also contribute at one loop to \(\mu \rightarrow e \gamma \) and \(\tau \rightarrow \mu \gamma \). And whereas the experimental constraint on \(\tau \rightarrow \mu \gamma \) [58, 59] is consistent with \(Z \rightarrow \tau ^\pm \mu ^\mp \) being detectable at the LHC, the bound from \(\mu \rightarrow e \gamma \) implies that a \(\bar{e} \, Z \! \! \! \! / ~ \mu \) interaction, with coefficient of a magnitude that the LHC could detect, would overcontribute to \(\mu \rightarrow e \gamma \) by several orders of magnitude [54].
4.2.2 The scalar, tensor and dipole operators

Above \(m_W\), there are two dipoles, given in Eq. (16). At tree level, the Zdipole does not match onto any operator below \(m_W\).

There are no dimensionsix, SU(2)invariant fourfermion operators to match onto the tensor operators \(\mathcal{O}^{e\mu ff}_{T,YY}\) for \(f \in \{ \tau ,d,s,b\}\). Furthermore, in tree level matching, the tensors are not generated by any heavy particle exchange. They are presumeably generated in oneloop matching by the same diagrams that give the mixing below \(m_W\), but this should be subdominant because the log is lacking.

There are no dimensionsix, SU(2)invariant fourfermion operators to match onto the scalar operators \(\mathcal{O}^{e\mu u_nu_n}_{S,YX}\) and \(\mathcal{O}^{e\mu ff}_{S,YY}\) for \(f \in \{ e,\mu ,\tau ,d,s,b\}\), \(u_n \in \{ u,c\}\) and \(X\ne Y\). However, SM Higgs exchange, combined with the \(H^\dagger H \bar{L}HE\) operator, will generate these operators in tree matching, weighted by \(m_f v/m_h^2\) or \(m_{u_n} v/m_h^2\). So it is a treelevel SU(2) prediction that these coefficients are small, as noted by [60]. Since the coefficients of scalar operators involving quarks are normalised by a running quark mass, see Eq. (12), one obtains \(C^{e\mu ff}_{S,\ldots } (m_\tau ) \simeq C^{e\mu }_{EH} (m_W) m_f(m_\tau ) m_t/m_h^2 \).
4.2.3 Matching at “Leading” order
The aim of a bottomup EFT analysis is to translate the bounds from several observables to combinations of operator coefficients at the high scale. So one must compute the numerically largest contribution of each operator to several observables (\(\mu \rightarrow e \gamma \), \(\mu \rightarrow e ~\mathrm{conversion}\) and \(\mu \rightarrow e \bar{e} e\), in the case of \(\mu \)–e flavour change). It is interesting to have constraints from different observables, rather than just the best bound, because there are more operators than observables, so a weaker constraint on a different combination of coefficients can reduce degeneracies. However, in this paper, only the experimental bound from \(\mu \rightarrow e \gamma \) is considered, so the aim is to obtain the best bound it sets on all operator coefficients.
In the next section, we will see that tree matching and oneloop running, as performed so far, do not reproduce the correct constraints from \(\mu \rightarrow e \gamma \) on the operators which parametrise LFV interactions of the Higgs and Z; that is, the numerically dominant contributions of these operators to \(\mu \rightarrow e \gamma \) are not included. In addition, twoloop QED running [42] is required below \(m_W\) to obtain bounds on vector operators. So it is clear that the simplistic formalism given here, of tree matching and oneloop running, does not work for \(\mu \rightarrow e \gamma \). It would be interesting to construct a systematic formalism, gaugeinvariant and renormalisation scheme independent, that allows one to obtain the best bound on each operator from each observable. I suppose that such a formalism corresponds to “leading order”. Notice that leading order is only defined “topdown”, because it describes the contribution of an operator to an observable. So to construct a LO formalism for bottomup EFT, it seems that one must work topdown, finding the numerically dominant contribution of each operator to each observable, then ensuring that the combination of the contributions from all the operators is scheme independent.
As previously stated, the LO twoloop running is neglected in this paper. However, some attempt is made to perform LO matching at \(m_W\), where the “LO contribution” of a coefficient above the matching scale to a coefficient below, is pragmatically defined as the numerically dominant term (and not the lowest order in the loop expansion, because this may not be the numerically dominant contribution in presence of hierarchical Yukawas).
So, in summary, the “leading order” matching performed for \(\mu \rightarrow e \gamma \) in the next section will consist of the tree equivalences given in this section, augmented by some one and twoloop contributions of operators that do not mix to the dipole. These loop contributions are obtained by listing all the operators which do not mix into the dipole above \(m_W\), estimating their matching contribution at \(m_W\), and including it if it gives an interesting constraint.
5 Translating the \(\mu \rightarrow e \gamma \) bound to \(M > m_W\)
In this section, the aim is to use the machinery developed in the previous sections to translate the experimental bound on \(BR(\mu \rightarrow e \gamma )\) to a constraint on operator coefficients at the New Physics scale M.
5.1 Parametrising \(\mu \rightarrow e \gamma \)
5.2 Running up to \(m_W\)
5.3 Matching at \(m_W\)
5.4 Running up to M
At \(m_W\), \(C^{e \mu }_{D,L} (m_\tau )\) can be written as a linear combination of \(C^{\mu e *}_{e\gamma }(m_W)\), \(C^{\mu e *}_{eZ}(m_W)\), \(C^{\mu e cc *}_{LEQU(1)}(m_W)\), \(C^{\mu e cc *}_{LEQU(3)}(m_W)\), \( C^{e \mu }_{HE}(m_W)\), and \(C^{\mu e *}_{EH}(m_W)\). The RGEs to evolve these coefficients up to M are given in [39, 40, 43], and generate more intricate and extensive operator mixing than was present below \(m_W\). The aim here is to present manageable analytic formulae that approximate the “leading” (= numerically most important) constraints on all the constrainable coefficients at the scale M. Recall that an operator coefficient was defined here to be constrainable if the current MEG bound, as given in Eq. (53), implies \(C < 1\) at \(M \simeq 100 m_t\).
Approximate “oneoperatoratatime” constraints on operator coefficients evaluated at the scale M, from the MEG bound [61] on \(BR(\mu \rightarrow e \gamma )\), as given in Eqs. (67), (68). For a given choice of scale M, the quantity in either left column should be less than the number in the right colomn multiplied by \({M^2}/{m_t^2}\). The operators are labelled in the same way as the coefficients, and given in Appendix D
\( C^{\mu e *}_{e \gamma }\)  \( C^{e \mu }_{e \gamma }\)  \(1.2\times 10^{8}\) 
\(C^{\mu e *}_{e Z} \ln \frac{M}{m_W}\)  \( C^{e \mu }_{e Z} \ln \frac{M}{m_W}\)  \(3.0\times 10^{6}\) 
\( C^{\mu e tt*}_{LEQU(3)} \ln \frac{M}{m_W} \)  \( C^{e \mu tt}_{LEQU(3)} \ln \frac{M}{m_W} \)  \(2.0 \times 10^{10}\) 
\( C^{\mu e cc*}_{LEQU(3)} (\ln \frac{M}{m_W} +3.5) \)  \( C^{e \mu cc}_{LEQU(3)} (\ln \frac{M}{m_W} +3.5) \)  \(2.8\times 10^{8}\) 
\( C^{\mu e tt*}_{LEQU(1)} \ln ^2 \frac{M}{m_W} \)  \( C^{e \mu tt}_{LEQU(1)} \ln ^2 \frac{M}{m_W} \)  \(3.1\times 10^{7}\) 
\( C^{\mu e cc*}_{LEQU(1)} ( \ln \frac{M}{m_W}+1) \)  \( C^{e \mu cc}_{LEQU(1)}( \ln \frac{M}{m_W}+1) \)  \(6.0\times 10^{6}\) 
\( C^{\mu e *}_{EH}\)  \( C^{e \mu }_{EH}\)  \(7.5\times 10^{7}\) 
\(C^{e \mu }_{HE}\)  \(C^{e \mu }_{HL(1)}, C^{e \mu }_{HL(3)}\)  \(1.2\times 10^{5}\) 
\( C^{\mu e tt*}_{EQ} \ln ^2 \frac{M}{m_W} \)  \( C^{e \mu tt}_{LU} \ln ^2 \frac{M}{m_W} \)  \(2.5\times 10^{4}\) 
\( C^{\mu e tt*}_{EU} \ln ^2 \frac{M}{m_W} \)  \( C^{e \mu tt}_{LQ(1)} \ln ^2 \frac{M}{m_W} \), \( 3 C^{e \mu tt}_{LQ(3)} \ln ^2 \frac{M}{m_W} \)  \(2.5\times 10^{4}\) 
6 Discussion of the machinery and its application to \(\mu \rightarrow e \gamma \)
The MEG experiment [61] sets a stringent bound on the dipole operator coefficients at low energy (see Eq. (53)). In translating this constraint to a scale \(M> m_W\), the analysis here aimed to include the “Leading Order” contribution of all “constrainable” operators, where LO was taken to mean numerically largest, and an operator was deemed constrainable if a bound \(C<1\) could be obtained at \(M\ge 100 m_t\). However, twoloop running, which gives the leading order mixing of vectors to the dipole, is not included here, so many constraints on vector operators are missing. As a result, the oneoperatoratatime limits given in Table 1 are obtained from a combination of tree, one and twoloop matching, with RGEs at one loop. Why do these multiloop matching contributions arise?
First consider operator dimensions above and below \(m_W\). There is a rule of thumb in EFT [47]: that one matches at a loop order lower than one runs, where the loops are counted in the interaction giving the running. This makes sense if the loop expansion is in one coupling, or if the same diagram gives the running and oneloop matching, because the running contribution is relatively enhanced by the log. For instance, an electroweak box diagram at \(m_W\) generates a fourfermion operator “at tree level” in QCD, which can run down with oneloop QCD RGEs. One could hope that a similar argument might apply above \(m_W\): a diagram giving oneloop matching could contribute to running above \(m_W\), so the subdominant matching could be neglected. However, this is not the case at \(m_W\), because SU(2)invariant dimensionsix operators from above \(m_W\) can match onto operators that wouldbe dimension eight if one imposed SU(2), but that are \(\mathcal{O}(1/M^2)\) and dimension six in the QED\(\times \)QCDinvariant theory below \(m_W\). For example, the LFV Z penguin operators given in Eqs. (102)–(104) match at one loop onto the “dimensioneight” dipole \( y_\mu H^\dagger H(\overline{L}_e H \sigma \cdot F E_\mu )\). Similarly, the LFV Higgs interaction \(H^\dagger H(\overline{L}_e H E_\mu )\) matches at two loops to the same “wouldbe dimensioneight” dipole. So the expectation that running dominates matching can fail at \(m_W\).
The expectation that one loop is larger than two loops can fail when perturbing in a hierarchy of Yukawa couplings. The dipole’s affinity to Yukawas arises because the lepton chirality changes, and the operator has a Higgs leg. The dipole operator here is defined to include a muon Yukawa coupling \(Y_\mu \) (see Eq. (51)), because in many models, the Higgs leg attaches to a Standard Model fermion, and/or the lepton chirality flips due to a Higgs coupling. And while it is difficult to avoid the \(Y_\mu \) in oneloop contributions to the dipole (see the discussion in [63]), there are more possibilities at two loops. In particular, it is “well known” [65] that the leading contribution to \(\mu \rightarrow e \gamma \) of a flavourchanging Higgs interaction, is via the twoloop top and W diagrams included in the matching contribution of Eq. (60).
Its unclear to the author what to do about either of these problems. Perhaps only the LFV operators with at least two Higgs legs give their leading contributions in matching rather than running.^{6} Maybe performing the matching and running at two loops would include the leading contributions in loops, logs, and Yukawa hierarchies. However, a complete twoloop analysis would take some effort – perhaps it would be simpler to list all the possible operators at the scale M, locate their “Leading Order” contributions, and include them.
As discussed above, it is important to match with care at \(m_W\). A slightly different question is whether it is important to match onto the extended (nonSU(2)invariant) operator basis at \(m_W\)? The answer probably depends on the low energy observables of interest. In the analysis here of \(\mu \rightarrow e \gamma \), the fourfermion operators that were added below \(m_W\) (such as the scalar fourfermion operators \(\mathcal{O}^{e\mu bb}_{S,YY}\), \(\mathcal{O}^{e\mu \tau \tau }_{S,YY}\) and \(\mathcal{O}^{e\mu \mu \mu }_{S,YY}\) given in Eq. (2)), are numerically irrelevant provided that the matching is performed at two loops. This is because they were generated in tree matching by the Higgs LFV operator \(H^\dagger H(\overline{L} H E)\), suppressed by the \(b,\tau \) or \(\mu \) Yukawa coupling; see Eqs. (23), (26), (39) and (41). Then, in QED running, they mix to the dipole (possibly via the tensor), which brings in another factor of the light fermion mass. With tree matching, this is the best constraint on the Higgs LFV operator, so is interesting to include. However, it is irrelevant compared with the twoloop diagrams involving a top and W loop, which match the Higgs LFV operator directly onto the dipole. This twoloop matching contribution is relatively enhanced by a factor \({\sim }100\) as can be seen by comparing the square brackets of Eqs. (59) and (60). So in the case of \(\mu \rightarrow e \gamma \), it seems that one would get the correct constraints on operator coefficients at M by using an SU(2)invariant fourfermion operator basis all the way between \(m_\mu \) and M, provided the matching at \(m_W\) is performed to whatever loop order retains the “leading” contributions.
The QED mixing between \(m_\mu \) and \(m_W\) modifies significantly the combination of operators that are constrained by \(\mu \rightarrow e \gamma \). This is illustrated in Fig. 2, which shows that the constraint has rotated in operator space, to constrain the linear combination of coefficients given in Eq. (58). Coefficients of tensor operators that were of a similiar magnitude to the dipole coefficient could give significant enhancement or cancellations. So the QED running is important. In addition, the MEG constraint on \(BR(\mu \rightarrow e \gamma )\) is restrictive – as discussed in Sect. 5.1, it could constrain New Physics which contributes at one loop up to a scale \(M \sim 10^7\) GeV. So it would be sensitive to twoloop contributions from LFV operators at a scale of \(10^5\) GeV. However, in matching at \(m_W\) onto SU(2)invariant dimensionsix operators, many of the tensor and scalar operators which mix with the dipole below \(m_W\), are generated with small coefficients which give a negligeable contribution to \(\mu \rightarrow e \gamma \). The point is that the scalars and tensors involving leptons and dtype quarks are generated by the Higgs LFV operator, whose leading contribution to \(\mu \rightarrow e \gamma \) arises in twoloop matching.
There are many improvements that could be made to these estimates. Including the experimental constraints from \(\mu \rightarrow e \bar{e} e\) and \(\mu \rightarrow e ~\mathrm{conversion}\) would directly constrain the vector operators, and give independent constraints on some of the operators that contribute to \(\mu \rightarrow e \gamma \). There are more operators than constraints, so this could allow one to identify linear combinations of operators that are not constrained. Oneloop matching is motivated by the restrictive experimental bounds, which allow one to probe multiloop effects. In addition, there are operators which require oneloop matching, such as the twogluon operators relevant to \(\mu \rightarrow e ~\mathrm{conversion}\). Twoloop running is required to get the leading order contribution of vector operators to \(\mu \rightarrow e \gamma \), and could be interesting above \(m_W\) if there are diagrams that dominate the oneloop running due to the presence of large Yukawas, or if quark flavouroffdiagonal operators are included, which may contribute to \(\mu \rightarrow e \gamma \) at two loops [64]. It is also motivated by the experimental sensitivity. Finally, dimensioneight operators can be relevant if the New Physics scale is not to high [64].
7 Summary
This paper assumes that there is new leptonflavour violating (LFV) physics at a scale \(M \gg m_W\), and no relevant other new physics below. So at scales below M, LFV can be described in an Effective Field Theory constructed with Standard Model fields and dimensionsix operators. The aim was to translate experimental constraints on selected \(\mu \leftrightarrow e\) flavour changing processes, from the low energy scale of the experiments to operator coefficients at the scale M. As a first step, this paper reviews and compiles some of the formalism required to get from low energy to the weak scale: a QED\(\times \)QCDinvariant operator basis is given in Sect. 2, the oneloop RGEs to run the coefficients to \(m_W\) are discussed in Sect. 3, the anomalous dimensions mixing scalars, tensors and dipoles are given in Appendix A, and tree matching onto SU(2)invariant operators at \(m_W\) is presented in Sect. 4.
As a simple application of the formalism, the experimental bounds on \(\mu \rightarrow e \gamma \) were translated to the scale M in Sect. 5. The process \(\mu \rightarrow e \gamma \) was chosen because it is an electromagnetic decay, and it constrains only the coefficients of the two dipole operators. The resulting constraints at M on two linear combinations of operators are given in Eqs. (67) and (68). These limits are approximative, due to the many simplifications discussed in the paper, valid at best to one significant figure. Bounds on individual operators can be obtained by assuming one operator dominates the sum; the resulting constraints are listed in Table 1. At a scale \(M\sim 100 m_t\), \(\mu \rightarrow e \gamma \) is sensitive to over a dozen operators, whereas, if \(M \gtrsim 10^{7}\) GeV, then \(\mu \rightarrow e \gamma \) is sensitive to only a few.
The formalism of the first sections did not work well for \(\mu \rightarrow e \gamma \). Tree matching and oneloop running missed the largest contributions of some operators, as discussed in Sect. 6. This curious problem could benefit from more study, in order to identify a practical and systematic solution.
Footnotes
 1.
Generically, the oneloop corrections to an operator Q will generate divergent coefficients for other operators \(\{ B\}\). If one computes the oneloop corrections to the amputed Green’s function for the operator Q, with n external legs, and Feynman rule \(i f_Q Q\), these can be written as \( i f_Q \frac{\alpha }{4\pi } \frac{1}{\epsilon } \sum _B b_{QB} B\). Then \([{\varvec{\Gamma }}]_{QB} = 2 [ b_{QB} + \frac{n}{2} a \delta _{QB}] \) where \(\frac{\alpha }{4\pi } a = \frac{\mu }{Z} \frac{\partial }{\partial \mu } Z\), and Z renormalises the wave function.
 2.
The first contracts a flavourchanging neutral current to a flavourconserving neutral current. The second contracts two flavourchanging neutral currents, or can be fiertzed to make one current flavourconserving but then both currents are chargechanging (see Eq. 94).
 3.
Of course, since the penguin contributes to all fourfermion operators \((\bar{\mu } \gamma \tau ) (\bar{f} \gamma f)\), the coefficients of many other operators might need to be tuned against the penguin too. An apparently less contrived way to engineer this, is to use the equations of motion to replace the penguin operator by a derivative operator \(\partial _\alpha Z^{\alpha \beta } \bar{\mu } \gamma _\beta \tau \) [56], which is suppressed at low energy by the Z fourmomentum.
 4.
 5.
Including \(\alpha _s\), so the quark operators no longer run as a power of \(\alpha _s(\mu )\).
 6.
In treelevel matching, the Z penguins do give their leading contribution to fourfermion operators; it is only the “leading contribution to \(\mu \rightarrow e \gamma \)” which arises in oneloop matching. See the discussion in Sect. 4.2.3.
Notes
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.
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