Effective field theory interpretation of lepton magnetic and electric dipole moments

We perform a model-independent analysis of the magnetic and electric dipole moments of the muon and electron. We give expressions for the dipole moments in terms of operator coefficients of the low-energy effective field theory (LEFT) and the Standard Model effective field theory (SMEFT). We use one-loop renormalization group improved perturbation theory, including the one-loop matching from SMEFT onto LEFT, and one-loop lepton matrix elements of the effective-theory operators. Semileptonic four-fermion operators involving light quarks give sizable non-perturbative contributions to the dipole moments, which are included in our analysis. We find that only a very limited set of the SMEFT operators is able to generate the current deviation of the magnetic moment of the muon from its Standard Model expectation.


Introduction
The electric and magnetic dipole moments of the electron and the muon are precision low-energy probes of the Standard Model (SM) of particle physics, and provide strong indirect constraints on physics beyond the Standard Model (BSM). The electric dipole moment is CP -odd, whereas the magnetic moment is CP -even. With the current experimental precision, the electron magnetic moment is sensitive not only to QED effects but also to hadronic contributions. Due to the larger muon mass, the muon magnetic moment is more sensitive to heavier mass scales, including hadronic and electroweak contributions, as well as BSM physics (see Ref. [1] for a comprehensive recent review). The current experimental value for the anomalous magnetic moment of the muon a µ [2][3][4][5][6]  is a hint for new physics is not yet clear, but the E989 experiment at Fermilab is aiming to further reduce the experimental uncertainty to 0.14 ppm [29]. On the theory side, the latest determination of hadronic vacuum polarization (HVP) from lattice QCD [30] improves the uncertainties compared to previous lattice-QCD calculations, but it is in a 2.1 σ tension with the value derived from the e + e − → hadrons cross section [11][12][13][14][15][16][31][32][33][34]. Meanwhile, for the hadronic light-by-light contribution, lattice-QCD determinations [27,35] are in agreement with the phenomenological estimate [1].
In the case of the electron anomalous magnetic moment, the agreement between experiment [36], which correspond to −2.4 σ or +1.6 σ discrepancies, respectively. The change in a due to a change in α QED is dominated by the one-loop Schwinger correction α/(2π) [39], which is the same for a e and a µ . However, the experimental uncertainty for a µ is much larger than for a e , and the theoretical prediction for the anomalous magnetic moment of the muon is not affected by the small difference in α QED .
In the examples we consider in this paper, ∆a ∝ m , so it is useful to write the anomalous magnetic moment of the electron discrepancy as ∆a e = m e m µ −18(7) Cs 10(6) Rb × 10 −11 , (1.8) in order to compare BSM contributions to ∆a µ and ∆a e simultaneously. Dipole moments require a chirality change of the lepton, which can introduce an additional factor of the lepton mass, resulting in the "naive scaling" ∆a ∝ m 2 [40]. A chiral enhancement results if this second mass factor is replaced by some heavy internal mass scale. The electric dipole moments (EDMs) of leptons are CP -odd observables that are highly suppressed in the SM. Any measurement of an EDM would be a clear indication of BSM physics, and the current experimental bounds [41,42] |d e | < 1.1 × 10 −29 e-cm @ 90% CL , |d µ | < 1.5 × 10 −19 e-cm @ 90% CL , (1.9) place strong constraints on new physics which is CP -violating. 1 The bound on the muon EDM could be improved by about three orders of magnitude in a proposed experiment at PSI [44].
In this paper, we investigate the possibility that the discrepancies, Eqs. (1.3) and (1.7), are due to BSM physics. While many models explaining the muon (and in some cases also the electron) g − 2 discrepancy have been constructed, we instead analyze possible BSM contributions in a model-independent way using effective-fieldtheory (EFT) techniques. EFT methods have been applied at the one-loop and even two-loop level to extract constraints from the EDM of the electron [45,46], but in the case of magnetic moments, the usual treatment only involves a discussion of the dipole operator itself. Recent studies investigated the connection to muon-collider [47,48] and muon-proton collider observables [49]. Logarithmic one-loop terms were included in Ref. [47].
In this work, we extend the available EFT analyses by exploring the lepton dipole moments at one-loop order including dimension-six operators. We consider two scenarios: (a) BSM physics appears above the electroweak scale and can be analyzed using dimension-six operators in the SM effective field theory (SMEFT), and (b) BSM physics occurs between the electroweak and QCD scales, and can be analyzed using a low-energy effective field theory (LEFT). Both effective field theories contain dipole operators that contribute to lepton dipole moments at tree level, so one trivial possibility is that the g − 2 discrepancies are explained by a BSM contribution to the EFT dipole operator. In addition to this possibility, we take into account other operators that can contribute to the dipole moments at loop level. Recent calculations have determined the one-loop anomalous dimensions in SMEFT [50][51][52][53] and LEFT [54,55], as well as the one-loop matching between the two theories [56]. We determine all operators which can contribute to g − 2 including one-loop matching and perturbative integration of the renormalization group (RG) evolution up to second order in the anomalous dimensions. We present our results in the spirit of Ref. [57][58][59] as master formulae in terms of LEFT, as well as SMEFT, Wilson coefficients. We find that, although there are several operators that could provide an explanation if new physics arises below the electroweak scale, an explanation for the discrepancy based on other EFT operators arising from BSM physics heavier than the electroweak scale is highly constrained. In this case, new physics would have to directly induce the dipole operator itself or certain semileptonic tensor operators, which can give significant contributions to ∆a through charm-or top-quark loops. Thus, while there are a plethora of new-physics models that aim to address ∆a , within the EFT language, all of the models can be described by only a few operators. We end by showing how our results can be applied to specific BSM scenarios, using the case of scalar leptoquarks as a representative example.
As the magnetic and electric dipole moments are related to the real and imaginary parts of the Wilson coefficients in SMEFT and LEFT, we also discuss the suppression of the imaginary parts relative to the real parts that is required to produce the g − 2 anomalies while still being consistent with the electric dipole moment constraints in Eq. (1.9).

Effective field theory and dipole moments
The low-energy effects of BSM physics above the electroweak scale can be described by an EFT that only contains the SM particles and respects the SM gauge group SU (3) c × SU (2) L × U (1) Y , known as SMEFT [60][61][62]. 2 We limit our analysis to dimension-six operators. For operators of even higher dimension to be relevant, the BSM model would have to be tuned so that dimension-six contributions to g − 2 are smaller than power-suppressed higher-dimension contributions. If the BSM theory has light particles below the electroweak scale, but heavier than the muon mass, we can use a low-energy EFT that only contains light SM particles, and is invariant under SU (3) c × U (1) em , known as LEFT.
Our sign convention for the SU with SU (3) c generators T A , SU (2) L generators t I = τ I /2, and the U (1) Y hypercharge generator y. Our sign convention for the SU (3) c × U (1) em covariant derivative is where the QED coupling constant is positive, e = |e|, and the eigenvalues of the electric charge operator Q are q e = −1, q u = 2/3, q d = −1/3 for the charged leptons, up-type quarks, and down-type quarks, respectively. The results for the SMEFT and LEFT running and matching calculations have been implemented in publicly available codes [63][64][65][66], 3 which we partly use for our numerical analysis. As the weak interactions are a chiral gauge theory, the finite parts of one-loop diagrams depend on the specific scheme chosen for γ 5 . For example, the Fierz identityĀ holds in d = 4 dimensions. Away from d = 4 dimensions, the difference of the two sides of Eq. (2.3) is an evanescent operator, which has matrix elements proportional to (d − 4), and gives a non-zero finite contribution in one-loop matrix elements if the loop integral is divergent. We will use the SMEFT operator basis in Ref. [62], the LEFT operator basis in Ref. [54], and follow the conventions for evanescent operators and γ 5 explained in Ref. [56]. The anomalous magnetic moment and electric dipole moment of a lepton are defined through the vertex function γ = (−ieq e )ū(p )Γ µ (p, p )u(p) , (2.4) which is decomposed in terms of Lorentz-invariant form factors as where k = p − p is the incoming photon momentum. The mass m is the physical (pole) lepton mass and the external lepton states are on shell, p 2 = p 2 = m 2 . The vertex function includes external-leg corrections on the photon and fermion lines.
The prefactor e in Eq. (2.4) is fixed by the renormalization condition F E (0) = 1, and corresponds to the static on-shell electric charge e QED , measured, for example, using Coulomb's law at large distances. The two experimental values for α QED ≡ e 2 QED /(4π) are given in Eq. (1.5).
The anomalous magnetic moment is defined as the limit of the magnetic (or Pauli) form factor at zero momentum transfer: 6) and the electric dipole moment is given by the limit of the electric dipole form factor at zero momentum transfer: The anapole moment F A (0) is not an observable [68]; it can be absorbed into fourfermion operators using the equations of motion.
To illustrate the LEFT description of magnetic dipole moments, consider the SM one-loop electroweak contribution to the anomalous magnetic dipole moments [69] The terms in the LEFT Lagrangian that we need for this paper, which will allow us to describe both SM and BSM contributions, are with implicit sums over the flavor (generation) indices p, r, s, t. We will denote the generation indices by 1, 2, 3, or equivalently by e, µ, τ for leptons, and u, c, t for uptype quarks. For down-type quarks, the generation index is the weak-eigenstate index d , s , b and is related to the mass eigenstate index d, s, b by the CKM matrix, as discussed in Ref. [54, §2.2]. We use the convention that the coefficients satisfy the same symmetry relations as the operators, e.g., In constructing the LEFT Lagrangian induced by the SM, there is a tree-level contribution to the four-fermion operator O V,LR ee prst due to Z exchange, and a finite matching contribution to the dipole operator O eγ pr at one-loop, The total one-loop electroweak contribution to the anomalous magnetic moment is given by the sum of the tree-level contribution from the dipole operator Eq. (2.12) and the one-loop contribution of the V, LR operator Eq. (2.11) from the graph shown in Fig. 1; the two contributions sum to Eq. (2.8). While the W contribution only enters as a direct matching to the dipole operator, numerically about 60% of the Z contribution comes from the LEFT loop and 40% from the direct matching to the dipole. For BSM contributions, one generally expects an analogous situation: some effects only appear as a direct matching onto the dipole operators, while other contributions can indirectly feed into g − 2 through EFT loops. We can evaluate the complete RG-improved BSM contribution since one-loop running, matching, and matrix elements are now available.
Lepton dipole moments are predictions of the SM that only depend on the experimental inputs of particle masses and coupling constants. In the EFT, the dipole moments also depend on higher-dimension operator coefficients. The dipole moments get tree-level contributions from the dimension-five dipole operators as well as loop contributions from dimension-six interactions in the LEFT from the graph shown in Fig. 1. The dipole moments therefore depend on Wilson coefficients that are free parameters in the EFT, making it difficult to correlate g − 2 with other observables in general. This feature explains why, for the magnetic moments, the EFT language is mostly employed only up to tree level, i.e., models are matched directly onto the dipole operators. We will show that the EFT approach beyond leading order is nevertheless useful to systematically organize viable mechanisms that can explain the g − 2 discrepancies and to explicitly distinguish logarithmically enhanced contributions.
As is well known, the SM electroweak contribution to the muon g − 2 up to two loops [9,10] is comparable in magnitude to the discrepancy Eq. (1.3), which already implies that heavy BSM physics needs an enhancement mechanism to be a viable explanation [40,43]. In the case of the electron g−2, both the differences Eq. (1.7) and the uncertainties are much larger than the SM electroweak contribution [70], (2.14)

LEFT interpretation
At tree level, the magnetic and electric dipole moments a and d are related to the real and imaginary parts of L eγ and are conventionally given using different units, with a dimensionless and d in e-cm, At one-loop order, a is where e now denotes the renormalized MS coupling, but m is still the physical pole mass of the lepton. The first term is the famous QED one-loop contribution computed by Schwinger [39]. The QED contribution is known to five loops [7]. The second term is the tree-level contribution of the dipole operator O eγ , together with its one-loop correction from the graphs in Fig. 2. The a 4 contribution arises from one-loop insertions of four-lepton operators. In the MS scheme, and using the conventions of Ref. [56] for evanescent operators, it is given by where µ denotes the renormalization scale, and we have used the symmetry of the Wilson coefficients Eq. (2.10). The corresponding expressions for the electric dipole moment are given by the imaginary parts using the relative normalization in Eq. (3.1) between a and d . In the SM, the scalar Wilson coefficients L S,RR ee are proportional to Yukawa couplings and only appear as dimension-eight effects [54]. The same remains true in the SMEFT at one loop [56]. However, these Wilson coefficients could be present in the LEFT if there is BSM physics below the weak scale. On the other hand, L V,LR ee is already present in the SM due to the one-loop electroweak contribution (2.11).
Finally, a 2 2q in Eq. (3.2) results from semileptonic tensor operators given in Eq. (2.11). In perturbation theory and in the scheme of Ref. [56], only logarithmic one-loop contributions are generated, where the sums on p are over heavy active quark flavors at the scale µ. The light-quark matrix elements at low energies have to be evaluated non-perturbatively [71], and are not included in Eq. (3.4). They give a dipole contribution instead of the perturbative contribution Eq. (3.4). Here Λ χ = 4πF π is the scale of chiral symmetry breaking, with F π = 92 MeV the pion decay constant. c T is a non-perturbative parameter, of order unity by naive dimensional analysis [72,73], and depends on the renormalization scale µ of the operators, which is chosen to be 2 GeV. In terms of the chiral low-energy constant Λ 1 of Ref. [74], Lattice-QCD input from Ref. [75] for the pion tensor charge together with vectormeson saturation [76] lead to c T ≈ −1.0(2), where the error does not include model uncertainties, see Refs. [71,77]. The scalar operators L S,RR ed , L S,RR eu mix into the corresponding tensor operators L T,RR ed , L T,RR eu due to electromagnetic corrections, which subsequently contribute to the dipole via penguin graphs, as shown in Fig. 3. This scalar-tensor mixing means that there are also non-perturbative contributions to the dipole from L S,RR eu uu , L S,RR ed dd , L S,RR ed ss similar to Eq. (3.5), but with an additional αq q q e /(4π) electromagnetic suppression due to photon exchange between the quark and lepton lines, from the left-hand graph in Fig. 3. We will use the estimate where c S is a non-perturbative constant of order unity, and depends on µ, which is chosen to be 2 GeV. 4 At a scale of µ = 2 GeV, the charm quark is an active degree of freedom. While 4 In chiral perturbation theory, the terms in Eq. (3.7) arise from structures ∝ Tr(Q 2 ). There also exist structures ∝ Tr(LU † )Tr(Q 2 L )O eγ from diagrams like the one in the right-hand panel of Fig. 3, which give rise to a different q u,d dependence. We neglect them for simplicity, as these contributions, like those in Eq. (3.7), have an electromagnetic suppression and are very small. non-perturbative effects in its contribution may not be negligible, we use perturbation theory, Eq. (3.4), for a rough estimate of the charm contribution: T is expected to be of order unity, and is a reminder that there are nonperturbative contributions to the charm-quark loop. For the scalar charm-quark semileptonic operator, we will use the estimate where the additional αq u q e /(4π) suppression relative to Eq. (3.8) again comes from the photon exchange in the left-hand diagram in Fig. 3. The (LR)(RL) operators L S,RL ed , L S,RL eu also can have non-perturbative contributions from graphs where there is photon exchange between the quark and lepton lines. We will use the estimates Eq. (3.7) and Eq. (3.9) with the non-perturbative constants replaced by c S and c (c) S , respectively. The non-perturbative contribution from semileptonic vector operators requires, in addition to the photon exchange, a helicity flip on the lepton line, with a resultant m chiral suppression. As a result, their contribution to a is suppressed relative to the scalar operators, and can be neglected.
In Eq. (3.2), we have neglected terms quadratic in the dipole-operator coefficients [54,55], which in principle appear at dimension six as well. Second-order contributions from two flavor-changing dipoles are negligible due to constraints on flavor-violating decays of charged leptons. Since the flavor-diagonal dipole operators contribute at tree level to the dipole moments, second-order dipole contributions are much smaller than the first-order tree-level contributions, and can be neglected.
In order to interpret the g − 2 discrepancies in terms of new physics, we subtract from Eq. (3.2) the corresponding expression in the SM: the Schwinger term drops out, and as we only keep terms linear in the Wilson coefficients, all LEFT coefficients can be replaced by their purely BSM contribution.
Let us evaluate Eq. (3.2) numerically in order to estimate the size of the relevant Wilson coefficients needed to explain the anomalies. Using coefficients renormalized at µ = 2 GeV, one finds: (3.10) For definiteness, we take the MS gauge couplings at the scale µ = M Z as input [78] and evolve them to 2 GeV using the one-loop RG. For two generations of up-and down-type quarks and three lepton generations, the LEFT Lagrangian has 1176 CP -even baryon-and lepton-number-preserving terms at dimension six, and 25 at dimension five. Only 10 and 1 of these, respectively, contribute to each ∆a . Without any assumption on the flavor structure of BSM physics, the muon and electron dipole moments are essentially uncorrelated: 9 of the 11 free parameters in each ∆a e and ∆a µ are unique and only the two parameters L S,RR ee eµµe and L V,LR ee eµµe enter both electron and muon dipole moments due to the symmetry relation Eq. (2.10).
Assuming L eγ ∼ Λ −1 , L ee,ed,eu ∼ Λ −2 where Λ is the scale of new physics, the largest impact on a is from the dimension-five operator L eγ . However, in many models L eγ is induced at dimension-six, L eγ ∼ v/Λ 2 , and its contribution scales like the four-fermion operators. The most important four-fermion operators are the leptonic current-current operator L V,LR ee τ τ involving the τ , the semileptonic tensor operators of Eq. (3.5), which give a non-perturbative contribution to the dipole moments, and the semileptonic charm-tensor operator of Eq. (3.8).
Assuming only a single Wilson coefficient to be non-zero, the needed values of the Wilson coefficients to explain ∆a µ or ∆a e individually are collected in the upper panel of Table 1. For ∆a e , we give two sets of numbers using the two values for α QED in Eq. (1.5). Of course, an explicit BSM model can generate more than one non-zero coefficient, as discussed for a leptoquark model in Sec. 6. In the case of L S,RR ee eµµe and L V,LR ee eµµe , the symmetry relation Eq. (2.10) implies that they contribute to both muon and electron anomalies, which require different values. The second panel of Table 1 shows a minimal combined scenario for these two parameters, where both are non-zero, and adjusted to explain the muon anomaly while largely avoiding effects in ∆a e , which leads to a very strong correlation. Five significant digits are shown to illustrate that a simultaneous explanation of the electron anomaly ∆a Cs e or ∆a Rb e requires a fine-tuning of the two coefficients.  . In this case SU (2) L invariance implies that the LEFT operator is related to a charged-current interaction that can be probed by π → µν [79]. Similarly, constraints on four-lepton operators can be derived if one assumes that L V,LR ee µτ τ µ originates from the SMEFT operator C e µτ τ µ . In this case, L V,LR ee µτ τ µ is related to a charged-current interaction that can be constrained by τ → µν τνµ , leading to L V,LR ee µτ τ µ ≤ 12 TeV −2 [78,80,81]. For both operators these bounds are not stringent enough to exclude the values collected in Table 1. In addition, it should be noted that neither of the above constraints hold in scenarios where the SMEFT approach does not apply, as would be the case when BSM physics appears at or below the electroweak scale.
The expressions for the magnetic moments can be used to obtain the EDMs of the leptons, d , by multiplying Eq. (3.10) by q e e QED /(2m ) and replacing Re by Im. Combined with the experimental bounds in Eq. (1.9), this allows one to constrain the imaginary parts of the Wilson coefficients. These bounds can be determined by multiplying the middle of the ranges given in the = µ and = e, Cs columns in Table 1 by the combination 2m d exp /(e QED |∆a |) = {(640.0) µ , (6.5 × 10 −7 ) e } for the muon and electron, respectively, with ∆a and d exp given by Eqs. (1.3), (1.7), and (1.9). From the values of these rescaling factors one sees that d exp µ allows the imaginary parts of the Wilson coefficients to be a factor of ∼ 6 × 10 2 larger than the size of the real parts required to explain ∆a µ . Instead, for the case of the electron, the imaginary parts are constrained to be roughly six orders of magnitude smaller than the size of the real parts needed to explain ∆a e , leading to limits ∼ 10 −10 − 10 −15 GeV −2 on the imaginary parts of the coefficients of the four-fermion operators. 5 Although simple rescaling can be used to obtain limits on most of the CP -odd Wilson coefficients, it is not appropriate in all cases due to the fact that currently the most stringent limit on d e results from measurements on the paramagnetic molecule ThO. CP -odd effects in this molecule are not only induced by d e , but also by electron-nucleon interactions of the formN Nēiγ 5 e [82][83][84], generated by the semileptonic scalar operators in Eq. (2.9) that contain light quarks. Similarly, the stringent limit on the EDM of the diamagnetic atom d Hg [85,86] can be used to probe spin-dependent nucleon-electron interactions that are induced by the semileptonic tensor operators in Eq. (2.9) involving light quarks [87]. Following Refs. [71,88], these effects lead to constraints of the order of ∼ 10 −15 GeV −2 on the semileptonic scalar and ∼ 10 −14 GeV −2 on the semileptonic tensor operators.
The discrepancies ∆a µ and ∆a e can be explained simultaneously within the LEFT by choosing suitable linear combinations of the Wilson coefficients in Table 1. This result is not surprising since LEFT contains a dimension-five dipole operator that contributes to the dipole moments. The values in Table 1 also provide an estimate of the scale at which BSM physics must occur to be able to explain the dipole anomalies, based on the low-energy operator generated in the model. Naive dimensional analysis (NDA) scaling [72,73] of the operators Eq. (2.11) gives L 5 4π/Λ NDA for the dimension-five dipole operators, and L 6 (4π/Λ NDA ) 2 for the dimension-six operators, where Λ NDA is the scale of new physics as given by the derivative expansion, i.e. the p/Λ NDA expansion. These inequalities can be interpreted as bounds on the coefficients if the theory is strongly coupled, i.e. at the unitarity bound. The Wilson coefficients can have additional perturbative suppression factors if some of the interactions are weakly coupled. For example, if the dipole operator has a coupling constant g, the inequality becomes L 5 g/Λ NDA . If the photon is weakly coupled in the BSM theory, one expects a coupling constant g = e in the dipole-operator coefficient. Similarly, if the four-fermion operators have a factor of g 2 , as occurs in the low-energy operators generated by weak gauge-boson exchange, then L 6 g 2 /Λ 2 NDA . The SM electroweak coupling g 2 ∼ 0.6, so in the following discussion, we will use L 5 ∼ 1/Λ and L 6 ∼ 1/Λ 2 to estimate the scale of new physics. One can get the corresponding values of Λ NDA for a strongly coupled theory by multiplying Λ by 4π.
For ∆a Cs e , the values in Table 1 mostly correspond to rather high BSM scales: for the dimension-five dipole operator, the scale is Λ ∼ 1/ |L| ∼ require low scales in the 10 − 50 GeV range. While the scale for the dipole operators is high, in most models, the dipole coefficients are effectively dimension six, L ∼ v/Λ 2 , so that ∆a Cs e requires a dipole scale Λ ∼ 10 3 TeV and ∆a µ a dipole scale Λ ∼ 360 TeV. If BSM models induce dipole operators with a loop suppression of ∼ α/(4π), the scale is lowered even further, to ∼ 30 TeV and ∼ 9 TeV, respectively. If the four-fermion operators are generated in BSM models with small coupling constants, then the scale at which they are generated will be lower than the values given here.
The values presented in this section are for couplings renormalized at the scale µ = 2 GeV. We include RG running effects and operator mixing between Λ and 2 GeV in the next sections.

Running effects below the electroweak scale
In the previous section, we investigated the LEFT contributions to the anomalous magnetic moments using parameters at the low scale µ = 2 GeV. The LEFT is valid up to the scale of BSM physics, or the electroweak scale µ W , whichever is lower. We can run the LEFT parameters from µ = 2 GeV to a higher scale, to investigate the effects of RG evolution, which is due to QCD and QED. To illustrate this running effect and its implications on the anomalous magnetic moment, we will scale Eq. (3.10) from 2 GeV up to µ = 60 GeV in the LEFT. Taking one-loop QCD and QED anomalous dimensions [55,89,90] into account, one finds for the electron or muon magnetic moment, with terms in descending order of size of their coefficients, where we have used the dimensionless constants with d i the dimension of L i and Λ = 60 GeV, and we have dropped terms with coefficients smaller than 10 −9 . If the BSM scale is 60 GeV, the dimensionless constantsL are expected to be at most of order unity, and can be smaller if there are suppressions due to small coupling constants. Note that the contribution of L S,RR ee τ τ is generated by first running into the L S,RR ee τ τ , which subsequently induces the dipole operator, effectively making it a two-loop effect. Similar two-loop mixing effects occur for the semileptonic scalar operators. Apart from the effect of the scalar operators, the RG resummation beyond one loop typically induces corrections of only a few percent, reaching 6.3% in the case of the b-quark tensor operator L T,RR ed bb . The non-perturbative contributions at low energies, which depend on the parameters c T , c S and c S , are important for the light-quark operators.
As the scale of new physics increases, the size of L i decreases, and eventually the contribution from some operators becomes too small to explain the g − 2 discrepancy. At the same time, RG evolution and operator mixing implies that new operators can contribute via mixing into those present in Eq. (3.10). All Wilson coefficients present at µ = 2 GeV still contribute at Λ = 60 GeV, although L V,LR ee ee has been dropped since its coefficient is a bit smaller than 10 −9 . Furthermore, two additional semileptonic b-quark operators, as well as additional leptonic scalar operators, contribute to a , when RG effects are taken into account: . Such high scales motivate an analysis within the SMEFT instead of the LEFT, which we will consider in the next section. 6 With an increasing scale of new physics, more correlations between different observables and additional constraints arise and one might wonder if the dipole operators themselves can be constrained through their mixing effects. Since mixing only is possible with lower-dimensional operators, one could consider the effects on the running of the electromagnetic coupling. Dipole insertions in a vacuum-polarization diagram, as shown in Fig. 5, induce the running [55] 16π 2 µ d dµ e = −16e 2 q e m ReLeγ . with dimensionless dipole-operator coefficients defined as in Eq. (4.2), but for Λ = M Z . Determinations of the QED coupling at the weak scale through global electroweak fits are largely compatible with input on the hadronic running from e + e − data [15,16]. The electroweak fits lead to a precision on ∆α at the weak scale of ∼ 39 × 10 −5 [91], which constrains the rescaled τ -dipole-operator coefficient at the level of 8.4 × 10 −2 . This corresponds to a tree-level contribution to a τ of which is comparable to the constraint on a τ itself [78]. However, this bound only applies if the τ -dipole operator is generated at or above the electroweak scale. If it is loop induced, the suppression factor pushes the naive scale well below the weak scale.
The constraints from the running of α have mainly been studied in the context of γ γ Figure 5: Mixing between dipole operators and the QED charge through the vacuum polarization graph.
the HVP contribution to a µ [31][32][33][34]92]. In Ref. [93], an EFT language was employed, parameterizing changes in the HVP function in terms of a dimension-six operator ∂ λ F µν ∂ λ F µν . In the LEFT basis, this operator is replaced via the equations of motion by a set of current-current operators including O V,LR ee . This might seem puzzling at first sight, as after the basis change the four-fermion operators do not contribute to the photon polarization function, whereas the equation-of-motion operator did. However, to relate the effective running QED coupling to observables, it should not be defined as usual in terms of the basis-dependent photon 1PI function, but rather through four-fermion processes, which include the contact contributions of four-fermion operators. As before, the EFT approach only applies if the scale of the operator lies at or above the electroweak scale. In Ref. [93], changes in the HVP function were related to a scale of a few GeV, so that no model-independent connection between the HVP contribution to a µ and α at the weak scale could be established.

SMEFT interpretation
Above the EW scale µ W , the LEFT is replaced by the SMEFT which consists of the SM plus higher dimension operators. The LEFT coefficients can be computed in terms of those in SMEFT by matching the two EFTs at the electroweak scale. The matching equations have been computed at tree level [54,94] and at one loop [56,95,96], and we include the one-loop effects. The heavy SM particles, i.e., the top quark, W , Z, and Higgs bosons, are integrated out at one common scale µ W , which we set equal to the MS top mass, µ W = m t = 162.5 GeV [78]. As an input in the matching equations, we use the on-shell W , Z boson masses and α QED evolved from the scale µ = M Z to µ = m t at one loop. The exact choice of scheme changes the numerical results by a few percent. For simplicity, we neglect additional dimension-6 corrections that appear when expressing the SM electroweak contribution to a in terms of the Fermi constant determined from muon decay, as the discrepancy ∆a µ is larger than the SM electroweak contribution Eq. (2.13).
The SMEFT results will be given at two scales, Λ = 250 GeV and Λ = 10 TeV, to illustrate the contributions to the dipole moments. Within the SMEFT, we evolve the coefficients from the matching scale µ W to the scale Λ using the oneloop RG evolution [50][51][52]. In terms of SMEFT Wilson coefficients renormalized at Λ = 250 GeV, this results in the following expression for the magnetic moments: where we used the mass basis for the up-type quarks and charged leptons and only contributions up to 10 −9 have been retained. The C W and C W contributions violate holomorphy [97] due to the one-loop matching contribution, which depends on the scheme choice for Q W , see the discussion in Ref. [56] for details. We have used the convention of Eq. (4.2), where couplings with a tilde are couplings scaled by powers of Λ = 250 GeV to make them dimensionless. The results for the dipole operators and C e are comparable to the findings in Ref. [98]. The terms from line five to eight in Eq. (5.1) are shown for completeness but are subject to strong constraints. Using the python code smelli [99] and assuming only one operator is present at a time, these operators are excluded as explanations for the discrepancy: one finds that C HW B , C HB , C HW , C HD , and C W are strongly constrained by the signal strength of h → Zγ from gluon fusion production, µ gg (h → Zγ); C He

22
is excluded by the Z-pole observable give too large contributions to R(K * ) and R(D * ), respectively. Finally, C is constrained by a combination of meson decays.
In the end, very few SMEFT operators can explain the a µ discrepancy; the electroweak dipoles C eB , C eW , and the four-fermion operators C e , C (1) equ . The particle content of possible tree-level models [100] generating the Wilson coefficients in Eq. (5.1) that are not excluded phenomenologically are summarized in Table 2.
As before, the EDMs of the leptons can be obtained from Eq. (5.1) by a simple rescaling and replacing Re by Im. For example, to obtain |d /d exp | the relevant factor Spin Rep. , are constrained to be smaller than 1. The limits due to d exp e are much stronger, leading to nontrivial constraints even for the contributions that are suppressed by an additional factor of m e /m µ . Several of the constraints that can be obtained in this way were recently discussed in Refs. [45,46,101]. In comparison, Ref. [46] is able to constrain additional operators by considering two-loop diagrams that induce next-to-leading-log contributions which were neglected in our analysis. On the other hand, we explicitly include the, sometimes sizable, non-perturbative contributions due to the semileptonic operators, which we parameterized by c S,T and c (c) S,T and which were not considered in Ref. [46]. In a next step, we evolve the SMEFT Wilson coefficients to a higher scale of Λ = 10 TeV, finding for the magnetic moments: If the BSM scale is in the multi-TeV range, the only possible operators that provide viable explanations of the muon g − 2 discrepancy are the electroweak dipole operators C eW and C eB , and the semileptonic tensor operators C equ .

Leptoquarks
The EFT analysis presented in the previous sections can be connected to specific models proposed to explain the muon anomaly. In this section, we review a particular example in the form of scalar leptoquark models, which can provide a NP contribution that is large enough to account for the discrepancy. The leptoquark models together with several other tree-level mediators generating the SMEFT Wilson coefficients are collected in Table 2.
Leptoquark models are prominent candidates to explain the a µ discrepancy (see, for example, Refs. [102][103][104][105][106] and references therein). The possible leptoquark models include SU (2) L singlet leptoquarks S 1 transforming as (3, 1, 1/3) under SU (3) c × SU (2) L × U (1) Y , SU (2) L triplet leptoquarks S 3 transforming as (3, 3, 1/3), or SU (2) L doublet leptoquarks R 2 , transforming as (3, 2, 7/6). Direct collider searches at the LHC constrain leptoquarks to be heavier than 1 − 1.5 TeV [107,108]. For the purposes of this paper, the inputs we need from the BSM theory are the leptoquark masses, and the Yukawa couplings of the leptoquarks to SM particles, where C = iγ 2 γ 0 is the charge conjugation matrix, p, r are generation indices, i, j, I are weak SU (2) L indices, and α is a color index. In these expressions, q and are left-handed SU (2) L quark and lepton doublets, while d, u, and e are right-handed SU (2) L singlet down-type, up-type, and charged lepton fields. The leptoquark Yukawa couplings y L pr , y R pr , y S pr , y (u) pr , and y (q) pr are arbitrary 3 × 3 matrices in flavor space. At energy scales below the leptoquark mass, the leptoquarks can be integrated out, generating four-fermion SMEFT operators with the tree-level coefficients where M 1,2,3 are the masses of S 1 , R 2 and S 3 . The S 1 and S 3 contributions agree with the results in Ref. [109] and the ones for R 2 with Refs. [100,110]. From Eq. (6.2) it can already be seen that S 1 and R 2 give a large contribution to the magnetic moments through the tree-level matching to C if they couple to t quarks and the corresponding leptons. Finally, the QCD corrections to Eq. (6.2) can be shown to be about 10% [111]. The one-loop diagrams shown in Fig. 6 generate contributions to the leptonic dipole operators Y * e rt (y e + y u + 2y q ) + where the results for S 1 and R 2 are consistent with Ref. [88]. 7 The S 1 and S 3 contributions were computed in Ref. [109], and we have some minor differences with their result. Although all three leptoquarks contribute to a l via dipole operators, only the S 1 and R 2 leptoquarks give rise to contributions that are chirally enhanced. These leptoquark contributions are proportional to the up-type Yukawa couplings, Y u , and provide a large contribution from t-quark loops. S 3 only contributes proportional to the lepton Yukawa couplings Y e . Between µ = M 1,2 and the electroweak scale the semi-leptonic tensor operators, with coefficients C (3) equ given in Eq. (6.2), mix into the dipole operators via RG running. At the electroweak scale C eB and C eW match onto L eγ at tree level, while C Including the RG resummation at leading log, in addition to the above fixed-order result, has the effect of decreasing the contributions in Eq. (6.5) by 20 − 30% for masses M 1,2 between 1 − 10 TeV. Eq. (6.5) thus implies that the a µ discrepancy can be explained using M 1 in the few to 10 TeV range and leptoquark Yukawa couplings close to unity. The experimental upper limit Br(µ → eγ) < 4.2 × 10 −13 [116] severely constrains explanations of (g − 2) e,µ that use leptoquarks. In particular, for any leptoquark explanation that induces both (g − 2) e and (g − 2) µ through top-quark loops, the predicted Br(µ → eγ) is too large. Consider the S 1 scenario as an example. Assuming that the S 1 leptoquark only couples the electron and muon to the top quark, the chirally enhanced contributions to ∆a µ and ∆a e are In terms of these quantities, a lower limit on the branching ratio of µ → eγ is given where α is the fine-structure constant, τ µ the muon lifetime, and ξ = y R te y R tµ . The minimum value of the above expression can be obtained by minimizing with respect to ξ 2 . Using the observed values of ∆a e/µ we obtain BR(µ → eγ) min = τ µ αm 3 µ 8 |∆a e ∆a µ | m e m µ = 1.5 × 10 −4 , (6.8) with ∆a e = ∆a Cs e and τ µ = 2.2 µs = 3.3 × 10 18 GeV −1 . Therefore, it is not possible to satisfy the constraints from µ → eγ if one assumes that ∆a e and ∆a µ are both due to loops involving only the top and a single leptoquark [43].
Finally, we note that two-Higgs-doublet models (2HDM) are also popular explanations of the ∆a µ anomaly, where the dominant contribution usually is from two-loop Barr-Zee diagrams shown in Fig. 7. These models often have a light CP -odd scalar below the electroweak scale and could be described by a two-loop matching onto LEFT [118]. A detailed analysis of these models is beyond the scope of the present paper.

Conclusions
We have analyzed the lepton magnetic and electric dipole moments in a modelindependent way in terms of LEFT and SMEFT. Our main results are expressions for ∆a e,µ as a function of the LEFT and SMEFT Wilson coefficients that include one-loop running and matching effects. We also consider the contributions of semileptonic operators involving light quarks, which get modified by the strong interactions at low energies. We have parameterized these effects in terms of non-perturbative parameters of O(1), whose precise determination requires a non-perturbative matching calculation with lattice QCD. Along with the current experimental measurements, our results place strong constraints on the Wilson coefficients in these effective field theories. We find that the current muon anomaly ∆a µ ∼ 2 × 10 −9 can only be generated by a very limited set of higher-dimensional operators, especially if the BSM physics scale is above a TeV. In this case, the only interactions that can produce a large enough ∆a µ are the dipole operators, C eW and C eB , and the semileptonic four-fermion operators C . These operators need to be induced at one loop and tree level, respectively, to produce a large enough ∆a µ . Even if there is BSM physics below the electroweak scale, only a few other semileptonic operators contribute (given in Eq. (4.1)).