Model independent constraints on four-lepton operators

We obtain constraints on four-lepton interactions in the effective field theory with dimension-6 operators. To this end, we combine the experimental input from Z boson measurements in LEP-1, W boson mass and decays, muon and tau decays, lepton pair production in LEP-2, neutrino scattering on electrons, and parity violating electron scattering. The analysis does not rely on any assumptions about the flavor structure of the dimension-6 operators. Our main results are the confidence intervals for Wilson coefficients of 16 lepton-flavor conserving four-lepton operators, together with the full correlation matrix. Consequences for leptophilic models beyond the Standard Model are discussed.


Introduction
An effective field theory (EFT) provides a model-independent framework to characterize new physics beyond the Standard Model (SM). It gives an adequate description of physical processes in current and past experiments if the new particles are much heavier than the weak scale. Then new physics effects can be represented, without introducing new degrees of freedom, by operators with canonical dimensions D > 4 added to the SM Lagrangian. Assuming lepton number conservation, leading effects are expected to originate from D=6 operators [1]. It is important to understand and describe, with minimal theoretical assumptions, the existing experimental constraints on these operators. While many D=6 operators can be probed at the LHC collider, others can be better constrained by previous experiments at high and low energies.
The program of systematic characterization of experimental constraints on the Wilson coefficients D=6 operators was pioneered in Ref. [2], and later continued e.g. in Refs . Most often, due to a huge number of independent D=6 operators [31,32], additional assumptions about the flavor structure of fermionic operators are made. The exception was Ref. [21] where a completely generally flavor structure was allowed. That work derived electroweak precision constraints on the subset of D=6 operators that yields the so-called vertex corrections to Z and W boson interactions with the SM fermions. Only observables that are not affected at leading order by four-fermion operators were considered, so as to separate the analysis of vertex corrections from that of (more numerous) four-fermion operators. Assuming that D=6 operators give the dominant new physics contributions to observables, and that loop suppressed new physics contributions are sub-dominant, 20 flavor preserving vertex corrections can be simultaneously constrained. A Gaussian likelihood function in the space of the vertex corrections was given, which can be used to constrain any particular model beyond the SM predicting a more restricted pattern of vertex corrections in the low-energy EFT.
In this paper we extend the analysis of Ref. [21] so as to also obtain constraints on some D=6 four-fermion operators. This step is important in order to probe a larger class of theories than what can be achieved using vertex corrections only. For example, if the UV theory contains vector boson coupled to the SM fermions but not mixing with the Z or W boson, then it gives rise only to 4fermion operators in the low-energy EFT. Again, the main goal is to derive experimental constraints without assuming anything about the flavor structure of D=6 operators. We focus on four-lepton operators, leaving four-fermion operators with quarks for future publication. These operators were probed most precisely by lepton pair production in LEP-2 and W mass measurements. However, these observables leave several unconstrained directions in the space of four-lepton operators. Therefore, we also include in our analysis low-energy precision experiments, such as neutrino scattering on electrons, parity violating electron scattering, and leptonic decay of muons and taus. This way we are able to simultaneously constrain 16 linear combinations of 27 Wilson coefficients of lepton-flavor conserving four-lepton operators, in addition to the previous constraints on the vertex corrections. The correlation matrix is given, which allows one to reconstruct the full likelihood function. Given that, our results can be used to constrain large classes of models that give rise to an arbitrary pattern of vertex corrections and 4-lepton operators in the low-energy EFT.
The paper is organized as follows. In Section 2 we lay out our formalism where the effects of D=6 operators are parametrized by vertex corrections and four-fermion operators. In Section 3 we discuss how relevant experimental observables are affected by these EFT parameters. In Section 4 we give the confidence intervals for the leptonic vertex corrections and a subset of lepton-flavor con-serving four-lepton operators, together with the full correlation matrix (relegated to Appendix C). In Section 5 we discuss the consequences of our general analysis for particular models with new leptophilic particles. In Appendix A we discuss the relationship between our formalism and the more familiar oblique parameters, and in Appendix B we review current constraints on lepton-flavor violating operators.

Formalism
We first summarize our conventions. The SU (2) × U (1) gauge couplings of the SM are denoted by g L , g Y . The photon coupling strength is e = g L s θ , where s θ = g Y / g 2 L + g 2 Y is the weak mixing angle. The vacuum expectation value (VEV) of the SM Higgs field H is H † H = v 2 /2. The SM fermions are written using the two-component spinor notation, and we follow the conventions of Ref. [33]. The left-and right-handed charged leptons are denoted by e I = (e, µ, τ ), e c I = (e c , µ c , τ c ), while the neutrinos are denoted by ν I = (ν e , ν µ , ν τ ), where I = 1 . . . 3 is the flavor index. We work in the basis where these fermions fields are mass eigenstates.
We consider an EFT with the Lagrangian Here L SM is the SM Lagrangian, and O D=6,i is a complete basis of SU (3) × SU (2) × U (1) invariant operators of canonical dimension D=6 constructed out of the SM fields. The Wilson coefficients c i are formally O(v 2 /Λ 2 ) where Λ is the scale of new physics that sets the EFT expansion. We assume here that O D=6,i conserve the baryon and lepton number. Below we also assume that individual U (1) e × U (1) µ × U (1) τ lepton numbers are conserved; see Appendix B for the discussion of lepton flavor violating operators. In Eq. (1) we omit higher-dimensional operators with D > 6. While these are always present in an EFT derived as a low-energy description of specific UV models, throughout this paper we will assume that their contribution to the relevant observables is negligible. This assumption is generically true when the scale Λ of new physics is much larger than the electroweak scale v, since the Wilson coefficients of higher dimensional operators are formally O(v 4 /Λ 4 ) or smaller. Consequently, our analysis will be performed at O(Λ −2 ), that is to say, we will take into account the corrections to observables that are linear in c i , and ignore quadratic corrections which are formally O(Λ −4 ). We do not impose any constraints on the Wilson coefficients c i . In particular, all D=6 operators can be simultaneously present in the Lagrangian, and the Wilson coefficients of fermionic operators can be flavor dependent. Rather than parametrizing the theory space by the Wilson coefficients of D=6 operators, we find it more convenient to work directly with parameters describing interactions of mass eigenstates after electroweak symmetry breaking. Our formalism follows that in Refs. [13,34]. Without any loss of generality, L eff can be brought to a form where the kinetic terms of all mass eigenstates are diagonal and canonically normalized. Then the quadratic Lagrangian for the electroweak gauge boson and lepton mass eigenstates is given by Here, δm parametrizes the relative correction to the W boson mass that may arise in the presence of D=6 operators. By construction, there is no correction to the Z boson mass: a possible shift due to D=6 operators has been absorbed into the definition of the electroweak parameters g L , g Y and v. For the sake of our analysis we need to define the interactions of leptons with the SM gauge fields in the effective Lagrangian: Here, the effects of D = 6 operators are parameterized by the vertex corrections δg. All δg's in Eq. (3) are independent parameters, which in general may depend on the lepton flavor. By construction, there is no vertex corrections to photon interactions. The parameters δg can be related by a linear transformation to Wilson coefficients of D=6 operators in any particular basis, see Ref. [34] for a map to popular bases used in the literature. Therefore The main focus of this paper is on the lepton-flavor conserving 4-lepton operators in Eq. (1) summarized in Table 2. Overall, there is 3 × 3 + 3 × 6 = 27 such operators. Three of those, denoted [O e ] IJJI , are complex, in which case the corresponding Wilson coefficient is complex, and the Hermitian conjugate operator is included in Eq. (1). The goal of this paper is to derive simultaneous constraints on the Wilson coefficients of (as many as possible) 4-lepton operators and the leptonic vertex corrections in Eq. (3). In our framework, the remaining parameter introduced above -the W mass correction δm in Eq. (2) -is related to the leptonic vertex corrections and one 4-lepton operators [34]: Again, this relation is a consequence of the linearly realized SM gauge symmetry and the absence of operators with dimensions greater than 6. It also ensures that the Fermi constant G F measured in muon decays is given at tree-level by G F = 1/ √ 2v 2 . This way, the tree-level relations between the electroweak parameters g L , g Y and v and the input observables α em , m Z and G F are the same as in the SM.

Experimental Input
In this section we discuss observables that will allow us to place constraints on 4-lepton operators and leptonic vertex corrections.

Z-and W-pole observables
The leptonic vertex corrections in Eq. (3) can be probed by measurements of leptonic decays of on-shell Z and W bosons. Precise measurements of observables ultimately related to various Z and W partial decay widths were performed in LEP-1 (Z) [35] and LEP-2 (W) [36]. The dependence of these observables on leptonic and quark vertex corrections is correlated. On the other hand, the dependence on four-fermion operators is suppressed by Γ V /m V and can be neglected [2]. Simultaneous constraints on all flavor-preserving vertex corrections were derived recently in Ref. [21], and we use directly these results. Marginalizing the likelihood over the quark vertex corrections, the constraints on the leptonic ones are given by: with the correlation matrix Note that all leptonic vertex corrections are strongly constrained by the data in a model-independent way. In particular, the constraints on charged leptons couplings to Z (dominated by LEP-1) are at a per-mille level, while the constraints on lepton couplings to W (dominated by LEP-2) are at a percent level.
By virtue of Eq. (4), this result constrains a combination of leptonic vertex correction and one four-lepton operator.

Fermion pair production in LEP-2
The LEP-2 experiment measured differential cross sections for the processes e + e − → + − , = e, µ, τ at energies above the Z boson resonance. Away from the Z-pole, these processes probe not only Z couplings to leptons but also 4-lepton operators, and the effect of the latter increases with increasing center-of-mass energy. Let us first focus on the processes e − e + → µ − µ + (e − e + → τ − τ + is analogous). For the experimental input, we will use the total cross-sections and forward-backward asymmetries measured at 12 different center-of-mass energies between √ s ≈ 130 GeV and √ s ≈209 GeV [36]. We are interested in O(Λ −2 ) corrections to these observables from D=6 operators, which translates to linear corrections in the vertex corrections and Wilson coefficients of 4-fermion operators (i.e. the interference term between SM and new physics). At that order, the observables are affected by 5 In the limit of vanishing fermion masses, their effect on the forward (σ F ) and backward (σ B ) e − e + → µ − µ + cross sections is given by where g Ze L,SM = − 1 2 + s 2 θ , g Ze R,SM = s 2 θ are the couplings of the Z to left-and right-handed electrons. The effect of the vertex corrections δg Ze L , δg Ze R , δg Zµ L , and δg Zµ R is also taken into account in the fit, but is not displayed here. The operator [O e ] 1221 does not interfere with the SM due to the different helicity structure; thus it enters only at the quadratic ( O(Λ −4 )) level and is neglected in this analysis.
One observes that measurements of the total cross section and asymmetry in e − e + → µ − µ + in principle can constrain 3 linear combinations of the 5 four-lepton operators that enter in Eq. ( We move to the process e − e + → e − e + (Bhabha scattering). In Ref. [36], LEP-2 quotes the differential cross sections for the scattering angle cos θ in the interval [−0.9, 0.9], and the center-ofmass energies from 189 GeV to 207 GeV. Bhabha scattering is affected by the three four-leptons In the limit of vanishing fermion masses their effect on the differential cross section is given by where t = − s 2 (1 − cos θ) and u = − s 2 (1 + cos θ). Again, the dependence on the vertex corrections δg Ze L , δg Ze R is taken into account in our analysis but not displayed here. In principle, Bhabha scattering at LEP-2 constrains independently all 3 four-electron operators, but again an approximate flat direction along the direction

Low-energy neutrino scattering
Interactions of SM leptons can be probed by neutrino scattering on electrons. We focus on processes with muon neutrinos: ν µ e − → ν µ e − , and muon anti-neutrinos: ν µ e − → ν µ e − , which were studied at center-of-mass energies far below the Z-pole by the CHARM [38], CHARM-II [39], and BNL-734 [40] experiments. The results are usually presented as constraints on the vector (g V ) and axial (g A ) coupling strength of the Z boson to electrons: where the SM predicts g V = −0.0396, g A = −0.5064 [41].
In the presence of D=6 operators, the scattering cross sections measured in these experiments are sensitive not only to the Z boson couplings but also to the four-leptons operators involving the 2nd generation doublet: [O ] 1122 and [O e ] 2211 . Nevertheless, at energies below the Z-pole, the measurements of g V and g A can be easily recast as constraints on the parameters in our framework.
At the linear level, the vector and axial couplings are effectively modified as Notice that the dependence on the four-lepton operators is different than for the LEP-2 observables discussed in the previous subsection. Therefore, low-energy neutrino scattering provides us with complementary information that will allow us to constrain additional directions in the space of D=6 Wilson coefficients. Experimental results on low-energy scattering of electron neutrinos [42] and anti-neutrinos [43] on electrons are also available. These probe the 4-electron operators [O ] 1111 and [O e ] 1111 . However, the current experimental accuracy is worse than for muon neutrinos scattering, and including this additional input would not affect the global fit in an appreciable way.

Parity violating electron scattering
The SLAC E158 experiment made a precise measurement of parity-violating asymmetry in Møller scattering e − e − → e − e − [44]. The asymmetry is defined as is the cross-section for incident left-(right-) handed electrons. The E158 experiment used a polarized electron beam of energy E ≈ 50 GeV against an electron target at rest which corresponds to a center-of-mass energy of √ s ≈ √ 2m e E ≈ 0.2 GeV, far below the Z pole. The results are presented as a measurement of the weak mixing angle at low energies: where the SM predicts s 2 θ (Q 2 = 0.026GeV 2 ) = 0.2381 ± 0.0006 [45]. cancels out in σ R − σ L ). At the linear order in the EFT parameters and leading order in s/m 2 Z , the effect of these operators and the vertex corrections can be effectively represented as a shift of the measured weak mixing angle: Although Møller scattering probes the same 4-electron operators as LEP-2, c.f. Eq. (9), its importance rests in the sensitivity to the combination that is accidentally very weakly constrained by unpolarized electron scattering in LEP-2.

Tau and muon decays
The leptonic tau decays τ − → e − ν τνe , τ − → µ − ν τνµ , and the conjugates provide additional information on 4-lepton operators involving τ . In particular, the provide the only constraint we are aware of on lepton-flavor conserving 4-lepton operators with muons and taus. The decays can be described by the following effective Lagrangian: where f = e, µ. At the linear level, the relative strength of the Fermi constant measured in the tau decays normalized to that measured in the muon decay is affected by the vertex corrections and four-lepton operators as where the W mass corrections δm can be expressed by other EFT parameters, c.f. Eq. (7). The experimental values quoted by the PDG are [46] A e = 1.0029 ± 0.0046, and the SM prediction is A f = 1.
For the muon decay, µ − → e − ν µνe and the conjugate, the total rate defines the SM input parameter v and by itself it does not probe new physics. However, additional information can be extracted from differential distributions in (polarized) muon decay. Customarily, these measurements are presented in the language of Michel parameters [47]. From the EFT perspective the most interesting are the so-called η and β /A parameters, because they are the only ones that may receive contributions at O(1/Λ 2 ) [48,49]: These parameters have been measured in an experiment in the PSI [50]: Analogous limits from tau decays are much weaker.

General Fit
We now do a global fit to all the data discussed above so as to simultaneously constrain D=6 operators in the EFT Lagrangian that give rise to leptonic vertex corrections and 4-lepton interactions. Previously, constraints on 4-lepton (and other 4-fermion) operators were obtained in Refs. [2,3] and recently updated in Ref. [28], assuming the Wilson coefficients are the same for all 3 fermion generations. The novel aspect of our analysis is that we allow for a completely general flavor structure of the D=6 operators. We combine the following experimental inputs discussed in Section 3: • Z boson production and decay in LEP-1 and leptonic W decays in LEP-2, • W mass measurement, • Two-lepton production in LEP-2, • Muon-neutrino scattering on electrons, • Parity violation in low-energy Møller scattering, • G F measurements in τ decays.
We consider the EFT Lagrangian with operators up to D=6, neglecting possible contributions of D=8 operators. 1 Consistently, in our analysis we only include corrections to observables that are linear in Wilson coefficients of D=6 operators. These are formally O(v 2 /Λ 2 ) in the EFT counting, and come from interference between tree-level SM and D=6 contributions to the relevant amplitudes. We also ignore loop-suppressed effects proportional to D=6 Wilson coefficients. We use the experimental results, the SM predictions, and the analytic expression for D=6 contributions discussed in Section 3 to construct a global Gaussian likelihood in the space of the relevant Wilson coefficients. With this procedure, we get the following global constraints: with the correlation matrix written down in Eq. (53).
A few general comments are in order: • In the global fit, the constraints on the leptonic vertex corrections are the same as the ones in Eq. (5) determined from on-shell Z and W data. The additional experimental input considered in this analysis constrains 4-lepton operators without affecting the limits on the vertex corrections in an appreciable way. Nevertheless, the correlations between vertex corrections and 4-lepton operators are non-negligible in some cases, as can be observed in Eq. (53).
• Not all 4-lepton operators can be constrained by the current data. In particular, we are not aware of any experiments probing four-muon or four-tau interactions. On the other hand, most of the Wilson coefficients of 4-lepton operators involving electrons are constrained, in a model-independent way, at a percent level accuracy.
One can also constrain the analogous operators with tau leptons. Translating the constraints on the form factor g S RR in leptonic tau decays [46] one obtains: Re([c e ] 2332 ) = 0.19 ± 0.15, and |[c e ] 1331 | < 0.70, |[c e ] 2332 | < 0.72 at 95% confidence level (CL). Stronger limits may arise via 1-loop contributions of these operators to anomalous electric and magnetic moments [53], however in this case the limits concern in fact for a linear combination of [c e ] IJJI and the Wilson coefficients of D=6 dipole operators in the effective Lagrangian.
• Combining the results from this paper with the ones in Ref. [21] one could also perform a global analysis of leptonic parameters together with quark vertex corrections. However, correlations between 4-lepton operators and quark vertex corrections are small: we find that the correlation coefficients are typically of order 0.01, and the largest is 0.07. All in all, the constraints on the quark vertex corrections and their correlations with the leptonic vertex corrections quoted in [21] are not affected by the combination.
Using the results in Eq. (19) and the correlation matrix ρ one can reconstruct the complete Gaussian likelihood function in the space of leptonic vertex corrections and four-lepton operators: where x is a 14-dimensional vector collecting the δg's and c's as in Eq. (19), and x 0 , ∆ x are the corresponding central values and 1 σ errors. In specific extensions of the SM only a subset of the general EFT parameters will be generated. In such a case, constraints on the model parameters can be obtained by restricting the full likelihood to the smaller subspace, and then minimizing the restricted likelihood. In the next section we perform this procedure for a handful of scenarios beyond the SM that affect only leptonic observables.

One by one
Before we attack specific models, we first discuss a general scenario where only one four-lepton operator and no vertex corrections is generated by new physics. Setting all but one Wilson coefficient to zero in the likelihood in Eq. (21), and then minimizing the resulting function we obtain the constraints summarized in Fig. 1. The strongest constraint, at a per-mille level, is the one on [c ] 1221 . The reason is that the corresponding operator affects the measurement of the Fermi constant G F in muon decays, and this way, unlike other 4-fermion operators, it affects the electroweak precision observables very accurately measured in LEP-1. The constraints on the remaining operators containing electrons are dominated by lepton pair production in LEP-2 and are somewhat weaker. Finally, the muon-tau four-lepton operators are only weakly constrained by tau decays.
We also visualize these constraint in terms of the new physics scale probed by each operator. To this end, we write c i = ±g 2 * /Λ 2 i , where Λ i can be interpreted as the mass scale of new particles and g * their coupling strength. Then we derive the 95% CL lower limit on the ratio Λ i /g * . In general, the limit depends on the sign of the Wilson coefficient, and for our presentation we always choose the lower one of the two possibilities. The results are shown in Fig. 2. Current data allow one to probe new particles with masses up to 5 TeV if they are coupled to the SM with order one strength.

Z prime
We now consider a model with a new neutral vector boson V µ of mass m V coupled to leptons as L ⊃ V µ κ L,I¯ Iσµ I + κ R,I e c I σ µē c I .
We assume the vector does not mix with the Z-boson, and does not couple to quarks. In such a case, it can be constrained neither by Z-pole observables nor by LHC dilepton resonance searches. Four-lepton operators: one by one Figure 1: Summary of the 68% CL intervals for the Wilson coefficients of 4-lepton operators, assuming that only 1 such operator is present at a time and that leptonic vertex corrections are absent.
However, it is constrained by the off-Z-pole observables discussed here. Integrating out the vector we get an effective theory with the Wilson coefficient of 4-lepton operators: Plugging these expressions in the general likelihood in Eq. (21) we obtain the likelihood as a function of κ/m V . In Fig. 3 we show examples of this likelihood for 2 scenarios: one where the vector couples to electrons only, and another one where the vector couples universally to all leptons, κ L/R,I = κ L/R . In both cases, we find that κ/m V 0.1-0.3/TeV, depending on the ratio of the left-and right-handed couplings. We can also observe that the vector-like couplings, κ L ≈ κ R , are more strongly constrained than the axial ones, κ L ≈ −κ R .

Vector Triplet
Consider a model with a new triplet of vector bosons V i µ of mass m V coupled to left-handed leptons as Integrating out the triplet we get the following 4-lepton operators in the EFT: These operators were not introduced previously. The reason is that they are related to other 4-lepton operators in Table 2 via Fierz transformations. Using, the identity one identifies the Wilson coefficients in the low-energy EFT as When the triplet couples to electrons only, κ 1 = 0 and κ 2,3 = 0, only one four-lepton operator [O ] 1111 is generated. Plugging the above expression in the general likelihood in Eq. (21) one obtains the following bound on the ratio of the vector mass and coupling: TeV, @ 95% CL.
The limit is stronger than what might be inferred from the one-by-one limits plotted in Fig. 2, because the model predicts the negative sign of the Wilson coefficient [c ] 1111 , for which the experimental constraints are stronger than for the positive one. On the other hand, when the triplet couples with the same strength to all leptons, κ I = κ, we find a slightly stronger bound:

Inert Higgs
The last example we study is a model with a scalar S of mass m S transforming, much like the SM Higgs, as 2 1/2 under SU (2) L × U (1) Y , and coupled to leptons as We assume that S does not get a VEV. Integrating out the scalar we get the following 4-lepton operators in the EFT: Once again these operators do not appear in Table 2, but using the Fierz transformation,σα α µ σ µ ββ = 2δ α β δα β we can rewrite them as This way, we identify the Wilson coefficients of the 4-lepton operators induced in the EFT by integrating out the scalar S: When the scalar couples to electrons only, Y 1 = 0 and Y 2 = Y 3 = 0, its mass over coupling is constrained as m S |Y 1 | ≥ 2.6 TeV, @ 95% CL.
When the scalar couples to all 3 generations of leptons then the constraints on the Michel parameters discussed around Eq. (20) can be relevant. If the couplings Y I are the same for all 3 generations, Y I = Y , then these constraints have a small effect, and the bound m S /|Y | is the same as in Eq. (34). However, if Y I are proportional to fermion's mass, Y I = Y 3 m e I /m τ , then the constraint on [c e ] 2332 from tau decays is the dominant one, leading to m S /|Y 3 | > 60 GeV at 95% CL.

Conclusions
In this paper we discussed constraints on 4-lepton D=6 operators in the EFT beyond the SM. For the first time, the analysis was performed without any assumptions about the flavor structure of the D=6 operators. We presented our results such that they can readily be recast as constraints on specific models beyond the SM that, after integrating the new heavy particles, lead to leptonic vertex corrections and 4-lepton operators in the low-energy EFT. Our results are particularly relevant in relation to models where lepton flavor universality is not preserved. We find that the typical current experimental sensitivity to the scale suppressing 4-lepton operators is of order a few TeV. In the best case, one can probe 50 TeV particles provided they interact strongly with the SM leptons (g * ∼ 4π), and they generate the best constrained operator [O ] 1221 (e.g, via exchange of an SU (2) L triplet of vector bosons). Our analysis also reveals several blind spots where the current sensitivity is weaker, which would be interesting targets for future experiments. In particular, certain linear combinations of 4-lepton operators involving electrons and taus are very weakly constrained. This can be cured by future e + e − colliders once polarization information about the initial state is available. Moreover, experimental information on 4-lepton operators involving muons and taus is currently very limited. Bounds on two such operators involving left-handed doublets could be improved by more precise measurements of the rate and differential distributions in τ → µνν decays; probing the remaining operators would be one of the strong points of the physics program of a future µ + µ − collider.
Including the available experimental information about e + e − → jets in LEP-2 and pp → leptons and jets at the LHC, as well as about low energy scattering on nuclei one can generalize this analysis so as to also constrain 4-fermion operators involving quarks. This is left for a future publication.

Acknowledgements
AF is supported by the ERC Advanced Grant Higgs@LHC.

A Relation to universal theories
In this paper we used the formalism where new physics effects in the D=6 EFT were represented by vertex corrections and 4-fermion operators. At the same time, the quadratic terms of electroweak gauge bosons in the Lagrangian were assumed to be the same as in the SM, except for a correction to the W boson mass. That can always be achieved without loss of generality, via field redefinitions and integration by parts. On the other hand, in the literature, precision constraints on new physics are often expressed in the language of oblique parameters, such as the Peskin-Takeuchi S, T , U parameters [54]. These are, on the contrary, defined via corrections to kinetic terms of electroweak gauge bosons. In this appendix we discuss the relationship between the two formalisms.
Oblique parameters fully characterize the new physics effects for the so-called universal theories [4,55]. The theory is universal if one can recast it to a form where new physics affects only propagators of the SM gauge bosons. Before introducing the oblique parameters, we first define the momentum expansion of the 2-point functions of electroweak gauge bosons: where p is the 4-momentum of the incoming gauge boson. We are interested in corrections δΠ with respect to the 2-point function in the SM. Two-point functions are not directly measurable, but certain combinations of δΠ V 1 V 2 affect measurable quantities. Up to order p 2 , the physical combinations are the 3 Peskin-Takeuchi oblique parameters: At order p 4 one can define [4] further oblique parameters: BB . (37) Compared to Ref. [4], we rescaled these parameters by α = e 2 /4π. Now we want to relate the oblique parameters defined above to the vertex corrections and 4-fermion operators in the EFT Lagrangian of Eq. (1). In universal theories, by definition, the new physics effects in the EFT Lagrangian can be represented by only bosonic operators with D > 4. These operators may lead to corrections to the gauge boson propagators, and one can relate their Wilson coefficients to the oblique parameters in Eq. (36) and Eq. (37). It turns out that, in an EFT with operators up to D=6, only the parameters S, T , W , Y can be generated at tree level [4]. Using field redefinitions and integration by parts we can get rid of the corrections to the gauge boson propagators, trading them for flavor diagonal vertex corrections and flavor conserving four-fermion operators as in our Eq. (1). Completing this procedure, we obtain the following map: Thus, the oblique parameters S, T , W , Y correspond to a special pattern of vertex corrections and 4-fermion operators [55]. In our language, the theory can is universal if the pattern of vertex corrections and 4-fermion operators can be matched to that in Eq. (38). Note that both leptonic and quark operators are necessarily present in universal theories. Our formalism is more general and applies to a large class of models, as we don't need to make any assumptions about the pattern of vertex corrections or 4-fermion operators. None of the specific models discussed in Section 5 is universal, and cannot be properly described by the oblique parameters. The current constraints on the oblique parameters are These constraints are dominated by the leptonic observables discussed in this paper. To derive Eq. (39), we also used the experimental input from the hadronic observables e + e − → qq, bb, cc in LEP-1 and LEP-2 [35,36] and atomic parity violation [56]. If, instead, we plugged in Eq. (38)  with I = J may arise from D=6 operators in the EFT. These operators do not interfere with the SM and thus, at the leading order, they do not affect the constraints on flavor conserving operators discussed in Section 4. In this appendix, for completeness, we review experimental constraints on some lepton flavor violating vertex corrections and four-lepton operators. See also [53,58,59] for recent reviews.

B.1 From lepton flavor violating Z decays
Lepton flavor violating Z boson vertices can be probed by on-shell Z decays at LEP and the LHC, as recently discussed Ref. [21]. The current experimental limits are: At tree level, this translates to the following constraints on the vertex corrections:

B.2 From lepton flavor violating lepton decay
Searches for lepton flavor violating muon and tau decays have, so far, all given negative results and set tight constraints on lepton flavor violating operators. In what follows, we perform a tree-level computation, neglecting the masses of the daughter leptons.
The 90% CL constraints on the branching ratios given by PDG [46] are:

B.3 From lepton decay parameters
In experiments studying lepton decays → 1 νν such as Ref. [63], the two emitted neutrinos are not detected and are assumed to conserve lepton flavor. A more general analysis of muon decay allowing lepton number violation was presented in Ref. [64]. The authors show that there is a one-to-one correspondence between the form factors g γ µ defined e.g. in Ref. [63] in the lepton flavor conserving case and combinations of parameters in the lepton number violating case.
In our D=6 EFT framework, for the decay I → J νν, this correspondence is: where a JI = 2 and a kl = 1 in all other cases.