Testing new physics with the electron g-2

We argue that the anomalous magnetic moment of the electron (a_e) can be used to probe new physics. We show that the present bound on new-physics contributions to a_e is 8*10^-13, but the sensitivity can be improved by about an order of magnitude with new measurements of a_e and more refined determinations of alpha in atomic-physics experiments. Tests on new-physics effects in a_e can play a crucial role in the interpretation of the observed discrepancy in the anomalous magnetic moment of the muon (a_mu). In a large class of models, new contributions to magnetic moments scale with the square of lepton masses and thus the anomaly in a_mu suggests a new-physics effect in a_e of (0.7 +- 0.2)*10^-13. We also present examples of new-physics theories in which this scaling is violated and larger effects in a_e are expected. In such models the value of a_e is correlated with specific predictions for processes with violation of lepton number or lepton universality, and with the electric dipole moment of the electron.


Introduction
The anomalous magnetic moment of the muon a µ = (g − 2) µ /2 is one of the most celebrated tests of the Standard Model (SM). Indeed, the high precision of its theoretical and experimental determinations makes a µ a powerful test on new physics. The situation has become especially intriguing with the ∼ 3.5σ reported discrepancy between the SM prediction and the experimental value [1][2][3][4] ∆a µ = a EXP µ − a SM µ = 2.90(90) × 10 −9 .
On the other hand, the anomalous magnetic moment of the electron a e has never played a role in testing ideas beyond the SM. In fact, it is believed that new-physics contaminations of a e are too small to be relevant and, with this assumption, the measurement of a e is employed to determine the value of the fine-structure constant α.
The aim of this paper is to emphasize that the situation has now changed, thanks to advancements both on the theoretical and experimental sides. Indeed, the theoretical prediction of a e has been refined to an unprecedented accuracy and its experimental value is now known with smaller errors. At the same time, good determinations of the fine-structure constant have been obtained from atomic physics experiments, providing a value of α that is completely independent of the measurements of a e . As a result, a e can now be viewed as a very useful probe of physics beyond the SM and the situation is going to become even more promising soon, as efforts are underway to reduce significantly both theoretical and experimental errors. The most exciting aspect of the story is that a e will soon provide us with a crucial consistency check of new-physics interpretations of the alleged discrepancy in a µ . Moreover, in certain classes of models, if ∆a µ is caused by new physics, then it is possible to correlate the value of a e with various other rare processes violating lepton universality or individual lepton number.
The paper is organized as follows. In sect. 2 we carefully review the present status of the SM prediction of a e and confront it with the experimental measurement. In sect. 3 we discuss future improvements in the theoretical and experimental results and show their impact for probing new physics. We use the prototype examples of supersymmetry (sect. 4), of a light pseudoscalar (sect. 5), and of vector-like fermions (sect. 6) to illustrate generic features of theories beyond the SM. Our results are summarized in sect. 7.  [5,6]. More recently, a new determination of the electron g−2 has been performed by Gabrielse and his collaborators at Harvard University, with the result [7] a EXP e = 115 965 218 07.3 (2.8) × 10 −13 . ( The uncertainty of this result, δa EXP e = 2.8 × 10 −13 , i.e. 0.24 parts in a billion (ppb), is 15 times smaller than that reported back in 1987. The two measurements differ by 1.8 standard deviations.

The Standard Model prediction
The SM prediction a SM e is usually split into three parts: QED, electroweak (EW) and hadronic. Here we provide a summary of the present status of these contributions.

QED contribution
The QED term a QED e arises from the subset of SM diagrams containing only leptons and photons. This dimensionless quantity can be cast in the general form [8] where m e , m µ and m τ are the electron, muon and tau lepton masses. The term A 1 , arising from diagrams containing only photons and electrons, is mass and flavor independent. In contrast, the terms A 2 and A 3 , generated by graphs containing also muons and taus, are functions of the indicated mass ratios. The muon contribution to a QED e , although suppressed by m 2 e /m 2 µ ∼ 2.34 × 10 −5 , is about ten times larger than the experimental uncertainty in eq. (2) (the tau contribution, suppressed by m 2 e /m 2 τ , is of order 10 −14 ). The functions A i (i = 1, 2, 3) can be expanded as power series in α/π and computed order-by-order: A i = A (2) i (α/π) + A (4) i (α/π) 2 + A (6) i (α/π) 3 + · · · . Only one diagram is involved in the evaluation of the one-loop (first-order in α, second-order in the electric charge) contribution; it provides the famous result by Schwinger [9] Seven two-loop diagrams contribute to the fourth-order coefficient A 1 , one to A 2 (m e /m µ ) and one to A (4) 2 (m e /m τ ), while A (4) 3 (m e /m µ , m e /m τ ) = 0. The exact mass-independent coefficient has been known for more than fifty years, A (4) 1 = 197/144 + π 2 /12 + 3ζ(3)/4 − (π 2 /2) ln 2 = −0.328 478 965 579 193 78... [10], where ζ(s) is the Riemann zeta function. The numerical evaluation of the exact expression for the mass-dependent coefficient A (4) 2 [11,12] with the latest CO 2 (m e /m τ ) = 1.837 98 (33)×10 −9 [13]. The tiny errors are due to the uncertainties of the mass ratios. The sum of the above values provides the two-loop QED coefficient 2 (m e /m τ ) = −0.328 478 444 002 55 (33).
The standard error δC 2 = 3.3 × 10 −13 leads to a totally negligible O(10 −18 ) uncertainty in a QED e . More than one hundred diagrams are involved in the evaluation of the three-loop (sixthorder) QED contribution. The exact result for the coefficient A 1 , mainly due to Remiddi and his collaborators [14], yields the numerical value A 2 (r) for arbitrary values of the mass ratio r was completed in 1993 by Laporta and Remiddi [15]. The exact formula contains hundreds of polylogarithmic functions, including harmonic polylogarithms [16]. Numerically evaluating these analytic expressions with the latest CODATA mass ratios given above, we obtain A 2 (m e /m τ ) = −6.5830 (11) × 10 −8 . The same values can be obtained with the simple series expansions of [17], thus avoiding the complexities of these numerical evaluations. The contribution to a QED e of these three-loop mass-dependent coefficients is −0.9 × 10 −13 , i.e. about a third of the present experimental uncertainty δa EXP e , see eq. (2). The contribution of the three-loop diagrams with both muon and tau loop insertions in the photon propagator can be calculated numerically from the integral expressions of [18]. We get A The error δC 3 = 1.1 × 10 −11 leads to a totally negligible O(10 −19 ) uncertainty in a QED e . Almost one thousand four-loop diagrams contribute to the mass-independent coefficient A (8) 1 , and only few of them are known analytically. However, in a formidable effort that has its origins in the 1960s, Kinoshita and his collaborators calculated A (8) 1 numerically [19]. Their latest result is A (20), from Ref. [20]. In the same, very recent, article they also computed the tiny mass-dependent four-loop coefficients A The error δC 4 = 0.0020, caused by the numerical procedure, leads to an uncertainty of 5.8×10 −14 in a QED e . Independent work on the computation of the four-loop coefficient is in progress [21]. Very recently, Kinoshita and his collaborators completed the heroic calculation of the 12672 five-loop diagrams contributing to the tenth-order coefficient A (10) 1 [20,22]. Their result is A (10) 1 = 9.16 (58). They also computed the tiny mass-dependent term A (10) 2 (m e /m µ ) = −0.003 82 (39). As the tenth-order contribution of the τ lepton loops can be safely neglected, the sum of these two numbers provides the five-loop coefficient The error δC 5 = 0.58 leads to an uncertainty of 3.9 × 10 −14 in a QED e . The complete QED contribution is given by As (α/π) 6 = 1.6 × 10 −16 , terms of order i ≥ 6 are assumed to be negligible at present.

Standard Model prediction of a e and value of α
The sum of the QED contribution a QED e (α) plus the hadronic and weak terms discussed above yields the SM prediction of the electron g−2: (the dependence on α of any contribution other than a QED e is negligible). To compare it with experiment, we need the value of the fine-structure constant α. The latest determination of α by CODATA [13], α (CODATA) = 1/137.035 999 074 (44) cannot be employed for our purpose, as it is mainly driven by the value obtained equating the theoretical SM prediction of the electron g−2 with its measured value, (thus assuming the absence of any significant new-physics contribution). 1 Indeed, solving eq. (14) with the experimental value of eq. (2) we obtain α (g−2) = 1/137.035 999 174 (34) in agreement with Refs. [20]. This is the most precise value of α available today. The difference between this number and α(CODATA) in eq. (13) is mainly due to the very recent QED five-loop result of Ref. [20], which is included in our derivation, but not in the CODATA one. Clearly, in order to compute a SM e (α) and compare it with a EXP e we must use a determination of α independent of the electron g−2 . At present, the two most accurate ones are They differ by less than one standard deviation. The first value was obtained from the ratio h/M Cs (h is Planck's constant and M Cs is the mass of the 133 Cs atom), which was determined by measuring the atomic recoil frequency shift of photons absorbed or emitted by 133 Cs atoms using atom interferometry [28]. The second was deduced from the measurement of the ratio h/M Rb (M Rb is the mass of the 87 Rb atom) with an experimental scheme that combines atom interferometry with Bloch oscillation [29][30][31]. The values of α in eqs. (16,17) were inferred from the ratios h/M Cs,Rb combining them with the very precisely known Rydberg constant and the mass ratios M Cs,Rb /m e [13]. Given the higher precision of α( 87 Rb) vs. α( 133 Cs) (by more than one order of magnitude), the former is the value of α we employ to compute a SM e (α). We note that α( 87 Rb) agrees with α(g−2) in eq. (15) (the difference is 1.3 standard deviations), and its uncertainty δα( 87 Rb) is larger than δα(g−2) just by a factor of 2.7.
The first (second) error is determined by the uncertainty of the four(five)-loop QED coefficient, the third one is δa HAD e , and the last (7.60 × 10 −13 ) is caused by the error δα( 87 Rb). The uncertainties of the EW and two/three-loop QED contributions are totally negligible. When combined in quadrature, all these uncertainties yield δa SM e = 7.64 × 10 −13 . Note that the present precision of the SM prediction, which is about three times worse than the experimental one, is limited by the uncertainty of the fine-structure constant α( 87 Rb).

Standard Model vs. measurement
The SM value in eq. (18) is in good agreement with the experimental one in eq. (2). They differ by ∆a e = a EXP e − a SM e = −10.6 (8.1) × 10 −13 , i.e. 1.3 standard deviations, thus providing a beautiful test of QED at four-loop level! (The fourloop contribution to a QED e is −5.56 × 10 −11 .) Once again, the uncertainty δ∆a e = 8.1 × 10 −13 is dominated by that of the SM prediction, through the error caused by δα( 87 Rb).

New Physics tests with a e
New physics effects in the electron g−2 are usually expected to be comparable with the EW contribution, a EW e = 0.3854 (42) × 10 −13 , see eq. (10), and therefore much smaller than the uncertainty δ∆a e = 8.1 × 10 −13 reported above. Indeed, as we mentioned earlier, the commonly used CODATA (and PDG) value of α is mainly derived from a EXP e under this assumption. However, as we will discuss in the next section, the uncertainty in ∆a e is expected to be reduced. Then, the anomalous magnetic moment of the electron will provide us with important information on new physics effects.
The first one, 0.6 × 10 −13 , is caused by the numerical procedure used to determine the four-loop QED coefficient. It is already rather small and can be further reduced to 0.1 × 10 −13 with a large scale numerical recalculation [32]. Also the second error, induced by the five-loop QED term recently computed by Kinoshita and his collaborators, is already small, and may soon drop to less than 0.1×10 −13 [32]. As the tiny -but hard to reduce -hadronic uncertainty is 0.16×10 −13 and those of the EW and two/three-loop QED contributions are totally negligible, the overall purely theoretical error of ∆a e (i.e. the value of δ∆a e obtained setting to zero δa EXP e and the the error induced by δα) is, at present, 0.7 × 10 −13 . This is likely to decrease even further, by a factor of two or three, in a relatively near future.
Thanks to these recent theoretical improvements, the precision on ∆a e is now limited only by the experimental uncertainties δa EXP e and δα. At present they affect δ∆a e by 2.8 × 10 −13 (δa EXP e ) and 7.6 × 10 −13 (δα). It seems reasonable to expect a reduction of the former error to a part in 10 −13 (or better) in ongoing efforts to improve the measurement of the electron (and positron) anomalous magnetic moment [33,34]. Work is also in progress for a significant reduction of the latter error [31,35].
In conclusion, a determination of ∆a e at the level of 10 −13 (or below) is a goal that can be achieved not too far in the future with ongoing experimental work. As we will discuss in the next section, this will bring a e to play a pivotal role in probing new physics in the leptonic sector.

General structure of new-physics contributions
The one-loop SM electroweak contribution to the anomalous magnetic moment of the lepton ( = e, µ, τ ) is This can be viewed as a benchmark for contributions from new physics at the electroweak scale. For the muon, this is about the same size as the observed discrepancy, see eq. (1). New physics effects for the leptonic g−2 can be accounted for by means of the effective which describes dipole transitions ( → γ) in the leptonic sector. Starting from eq. (22), we can evaluate ∆a as ∆a = 2m 2 Re(A ) .
Let us consider now some new particles with typical mass Λ NP and couplings g L and g R to left-and right-handed leptons , respectively. The one-loop new-physics contribution to the amplitude A is then of the form and therefore ∆a reads now With f 1,2 we indicate loop functions which depend on ratios (x k ) of unknown masses of the new particles contributing to the amplitude → γ, and k is a lepton flavor index. In the term proportional to f 1 , the chiral flip required by the dipole transition occurs through a mass insertion in the external lepton line. In the term proportional to f 2 , the mass insertion is in the internal line of some new particle, thus explaining the parametric factor v/m . Although f 2 must be proportional to the lepton Yukawa coupling, as a consequence of chiral symmetry, in practice this term can become very sizeable whenever a new large coupling leads to a chiral enhancement.
In a broad class of theories beyond the SM, g L,R and f 1 are flavor universal (i.e. are the same for any ) and f 2 vanishes, such that We will refer to this case as "naive scaling" (NS). NS applies, for instance, if the new particles have an underlying SU(3) flavor symmetry in their mass spectrum and in their couplings to leptons (which is the case for gauge interactions). An interesting consequence of NS is that an explanation of the muon g−2 anomaly makes definite predictions for new effects in the anomalous magnetic moments of electron and τ , It is very exciting that the sensitivity in ∆a e is not far from what is required to test whether the discrepancy in the muon g−2 also manifests itself in the electron g−2 under the NS hypothesis. Thus, determining ∆a e with a precision below 10 −13 is an important goal with rich physics consequences. The measurement of a τ may play a similar role if a precision below 10 −6 will be attained (see sect. 3.3).
Although the NS case is especially simple and common to a large class of new-physics interactions, it is by no means the only possibility. Many theories make predictions beyond NS, either because of the chirally-enhanced term proportional to f 2 , or because of lepton nonuniversality in the couplings (g L,R ) or the mass spectrum (f 1 ). For instance, in a multi-Higgs doublet model, the couplings g L,R are related to Yukawa couplings and therefore one would find the scaling ∆a i /∆a j = m 4 i /m 4 j . A different example is offered by supersymmetry with non-degenerate sleptons, in which ∆a loses its correlation with lepton masses.
The case of non-naive scaling is interesting from the theoretical point of view because it allows for exploration of the structure of lepton symmetries. It is also interesting experimentally because it can lead to effects in ∆a e testable already with present sensitivity. Moreover, as we will show with various examples in the next sections, any lepton non-universality in ∆a can be related to other experimental observables, offering the possibility of cross-checking new physics effects.
The underlying → γ transition described by the effective Lagrangian of eq. (22) can generate, in addition to the anomalous magnetic moments ∆a , also lepton flavor violating (LFV) processes ( = ), such as µ → eγ, and CP violating effects, such as the leptonic electric dipole moments (EDMs, d ), Using the general parametrization for A of eq. (24) we find On general grounds, one would expect that, in concrete NP scenarios, ∆a , d and BR( → γ), are correlated. In practice, their correlations depend on the unknown flavor and CP structure of the couplings g L and g R , and thus we cannot draw any firm conclusion. In the following, we will point out the general conditions that have to be fulfilled by any NP theory in order to account for large effects in ∆a while satisfying the constraints from d and BR( → γ).
Regarding the leptonic EDMs, we find the following model-independent relations d e ∆a e 7 × 10 −14 10 −24 tan φ e e cm , where we have defined φ = arg(A ). We have normalized ∆a µ to the central value of the current anomaly and ∆a e,τ to their values in naive scaling, ∆a /∆a µ = m 2 /m 2 µ . From eq. (31) we learn that an explanation of the ∆a µ anomaly implies, for a natural CPV phase φ µ ∼ O(1), a model-independent upper bound on d µ 3 × 10 −22 e cm which is still far from the current bound d µ 10 −18 e cm, but well within the expected future sensitivity d µ < 10 −24 e cm [36]. Therefore, any experimental effort to improve the resolution on d µ would be valuable.
On the other hand, the electron EDM imposes a bound on the corresponding CPV phase φ e at the level of 10 −3 , if NS is at work. Such a condition could be realized for instance if φ e is generated radiatively while φ µ arises already at the tree level. Going beyond NS, one could also envisage scenarios where the electronic dipoles are suppressed compared to the muonic dipoles because of hierarchical couplings g L,R e g L,R µ , as it happens for instance in a multi-Higgs doublet model where g L,R are related to Yukawa couplings. In general, as shown by eq. (30), the EDMs (but not ∆a µ ) vanish if g L = g R as it could arise in a left-right symmetric theory. Now we discuss the potential constraints on ∆a arising from LFV processes. An inspection of eq. (30) shows that BR( → γ) and ∆a are generally correlated as BR( → γ) ∼ (∆a µ ) 2 |θ | 2 where θ stands for the relevant flavor mixing angle, which is approximately LFV contributions to ∆a can be particularly important as they are typically chirally enhanced by m τ /m , as we will discuss in the case of supersymmetry. Such a chiral enhancement is not effective in τ LFV processes and therefore, in this case, it might be possible to keep BR(τ → γ) under control, while generating large effects especially for ∆a e . Before turning our attention to some prototype theories beyond the SM that predict nonnaive scaling in ∆a , we will now briefly summarize the status of the anomalous magnetic moment of the τ .

Status of the τ anomalous magnetic moment
The present experimental sensitivity of the τ lepton g−2 is only 10 −2 . In fact, while the SM prediction for a τ is precisely known [37], the very short lifetime of this lepton (2.9 × 10 −13 s) makes it very difficult to determine its anomalous magnetic moment by measuring its spin precession in a magnetic field, like in the electron and muon g−2 experiments. Instead, experiments focused on high-precision measurements of τ pair production in various high-energy processes and comparison of the measured cross sections with the SM predictions. The present PDG limit on the τ g−2 was derived by the DELPHI collaboration from e + e − → e + e − τ + τ − total cross section measurements at LEP2: −0.052 < a EXP τ < 0.013 at 95% confidence level [38]. This reference also quotes the result in the form: (17).
Comparing eqs. (33) and (34) (their difference is roughly one standard deviation), it is clear that the sensitivity of the best existing measurements is still more than an order of magnitude worse than needed. In [39], the reanalysis of various measurements of the cross section of the process e + e − → τ + τ − , the transverse τ polarization and asymmetry at LEP and SLD, as well as of the decay width Γ(W → τ ν τ ) at LEP and Tevatron allowed the authors to set a stronger model-independent limit on new physics contributions: −0.007 < a NP τ < 0.005. The possibility to improve these limits is certainly not excluded. Future high-luminosity B factories, such as Belle-II [40] or SuperB [41], offer new opportunities to improve the determination of the τ magnetic properties. The authors of [42] proposed to measure the τ anomalous magnetic moment form factor at the Υ resonance with sensitivities down to 10 −5 or 10 −6 , and work is in progress for the determination of a τ via the measurement of radiative leptonic τ decays [43].

Supersymmetry and a e
The supersymmetric contribution to a comes from loops with exchange of chargino/sneutrino or neutralino/charged lepton. Therefore, violations of "naive scaling" can arise through sources of non-universalities in the slepton mass matrices. Such non-universalities can be realized in two ways.
1. Lepton flavor conserving (LFC) case. The charged slepton mass matrix violates the global non-abelian flavor symmetry, but preserves U(1) 3 . This case is characterized by nondegenerate sleptons (mẽ = mμ = mτ ) but vanishing mixing angles because of an exact alignment, which ensures that Yukawa couplings and the slepton mass matrix can be simultaneously diagonalized in the same basis. We will discuss these two cases separately.

Lepton flavor conserving case
In the LFC case, we assume non-degenerate slepton masses for different families (mẽ = mμ = mτ ) but flavor alignment between lepton and slepton mass matrices. This is reminiscent of the well-known alignment mechanism [44], which was proposed to solve the supersymmetric flavor problem in the quark sector by aligning the down quark/squark mass matrices and which might arise naturally in the context of abelian flavor models [45].
In this case the supersymmetric contribution to ∆a LFC is given by the following approximate expression The loop functions f n , f c and g n (with x 1(2) = M 2 1(2) /m 2 , and x µ = µ 2 /m 2 ) are given in the appendix, and we use standard notation for the supersymmetric parameters. In our numerical analysis we use the exact expressions for ∆a LFC in the mass eigenstate basis [46].
In the illustrative case of a single mass scale (M 1,2 = µ = m˜ ) and if arg(µM 1,2 ) = arg(A M 1 ) = 0, the result simplifies to 2 This shows that the discrepancy in the muon g−2 can be explained by supersymmetric particles with masses around 100 (300) GeV, for tan β = 3 (20). On the other hand, the contribution proportional to the term A , which could induce in principle a violation of the NS if A e /A µ = m e /m µ , is too small to explain the muon g−2 anomaly after imposing the vacuum stability bound |A |/m˜ 3.
For values of ∆a µ explaining the muon g−2, non-degenerate sleptons at the level mμ ≈ 3 mẽ lead to ∆a e ≈ 10 −12 , which is at the limit of present experimental sensitivity. This naive expectation is confirmed by a complete numerical analysis, as illustrated in the left panel of fig. 1, which shows ∆a e as a function of the normalized selectron/smuon mass splitting X eµ = (m 2 e −m 2 µ )/(m 2 e +m 2 µ ). The plot has been obtained by scanning the supersymmetric parameters that account for the (g − 2) µ anomaly at the level of 1 ≤ ∆a µ × 10 9 ≤ 5 (black points), or 2 ≤ ∆a µ × 10 9 ≤ 4 (red points in the inner region). We have imposed all current bounds on sparticle masses arising from direct searches and required M 1 ≤ 100 GeV, M 2 = 2M 1 , µ ≤ 200 GeV, 3 ≤ tan β ≤ 30, mẽ ,μ ≤ 500 GeV. As can be clearly seen, values of ∆a e almost up to 10 −12 are reachable for highly non-degenerate selectron and smuon masses such that |X eµ | ∼ 1.
In the right panel of fig. 1 we show ∆a τ as a function of X µτ = (m 2 µ − m 2 τ )/(m 2 µ + m 2 τ ). The plot has been obtained following the same procedure described for the left panel. We observe that ∆a τ can reach the level of 10 −5 if mτ mμ, therefore violating NS, whose prediction is shown by the horizontal line in fig. 1. The condition mτ mμ is justified in the so-called split family models [47,48], where the first two generations of squarks and leptons are substantially heavier than the third generation.
Before concluding this section, it is worth to comment about the potential constraints arising from the leptonic EDMs, on the same line done in the model-independent analysis of sect. 3.2. From eq. (31) we learn that ∆a e ≈ 10 −13 (12) is allowed only if arg(µM 1,2 ) 10 −3 (4) . Moreover, since we are dealing now with flavor-blind CPV phases, that is φ = φ e = φ µ = φ τ = arg(µM 1,2 ), the electron EDM limit d e

Correlation between a e and violation of lepton universality in LFC
In supersymmetric theories, NS violations for the leptonic (g − 2) can arise through sources of non-universalities in the slepton masses. In turn, these non-universalities will induce violations of lepton flavor universality in low-and high-energy processes such as P → ν, τ → P ν (where P = π, K), i → jν ν, Z → and W → ν through loop effects. Lepton universality has been probed at the few per-mill level so far, see table 1. It is interesting to study the correlation between such violations of lepton universality and departures from NS for ∆a . Taking for example the process P → ν, we can define the quantity where the term v 2 /min(m 2 e,μ ) stems from SU(2) breaking effects. Within supersymmetry, such SU(2) breaking sources arise from left-right soft breaking terms, from mixing terms in the chargino/neutralino mass matrices, or from D-terms. For highly non degenerate sleptons and if min(m 2 e,μ ) ∼ v 2 , ∆r e/µ P can reach the few per-mill level.
In fig. 2, on the left, which has been obtained through the same scanning procedure described for fig. 1, we show the quantity ∆r e/µ K,π , defined in eq. (38), as a function of ∆a e . We learn that large NS violations typically imply also breaking effects of lepton universality at the few per-mill level, which are expected to be within the future experimental reach.
Similarly, in fig. 2, on the right, we show breaking effects of lepton universality in the µ/τ sector, accounted for by the quantity ∆r µ/τ K,π which can be constructed for instance combining processes like τ → P ν and P → µν (where P = π, K). Large NS violations, bringing ∆a τ to the level of 10 −5 , are typically correlated with violations of lepton universality in the µ/τ sector at the per-mill level. These effects will be tested experimentally at the SuperB, which aims at LFU tests in τ decays well below the per-mill level [58].

Lepton flavor violating case
We now consider LFV contributions to a . The importance of this case lies in the fact that LFV can generate contributions to a e and a µ that are chirally enhanced, being proportional to m τ . Using the mass-insertion approximation, which is valid for near degenerate sleptons with mass m˜ , we find We defined δ ij AB = (m 2 AB ) ij /m 2 (with A, B = L, R), where m 2 LL and m 2 RR stand for the left-and right-handed slepton mass square matrices. The function l n is given in the appendix. In the illustrative case where m˜ = M 1 , we find

Correlation between a e and τ → eγ in LFV
The case of LFV has to be confronted with the strong constraints from processes like i → j γ, see table 2. The decay rate of i → j γ is given by  measures violations of lepton universality in Γ(P → µν)/Γ(τ → P ν) with P = K, π. Black points satisfy the condition 1 ≤ ∆a µ ×10 9 ≤ 5, while red points correspond to 2 ≤ ∆a µ ×10 9 ≤ 4. BR(µ − → e − γ) < 2.4 × 10 −12 [59] BR(τ − → µ − γ) < 4.4 × 10 −8 [60,61] BR(τ − → e − γ) < 3.3 × 10 −8 [60,61]  The supersymmetric contributions to µ → eγ and τ → γ ( = e, µ) amplitudes arise at one loop level through the exchange of charginos/sneutrinos and neutralinos/sleptons. Although our numerical results are based on the exact expressions for A ij L,R in the mass eigenstate basis [62], in the following we provide their expressions in the mass-insertion approximation [63] A ij In the region of parameter space that maximizes ∆a e , corresponding to moderate/large tan β values and a large µ term, BR(τ → γ) can be approximated as Combining eq. (45) with eq. (41), one can set an upper bound on ∆a as a function of BR(τ → γ). We have studied this correlation numerically and we show in fig. 3 our results for ∆a e vs. BR(τ → eγ). The plot has been obtained by means of a scan with M 1 ≤ 2 TeV, M 2 = 2M 1 , µ ≤ 5 TeV, 3 ≤ tan β ≤ 30, m˜ ≤ 2 TeV, |δ τ e RR | < 1, |δ τ e LL | < 1. As illustrated in fig. 3, the experimental limit BR(τ → eγ) < 3.3 × 10 −8 (90% CL) curiously corresponds to a limit on ∆a e close to 10 −12 , which is just at the edge of present experimental sensitivity. Therefore, in the case of LFV, any future positive indication for ∆a e will necessarily imply that τ → eγ is just beyond the present bound. Moreover, even if τ → eγ will not be detected at the upcoming SuperB and Belle II facilities, which are expected to reach the sensitivity of BR(τ → eγ) 3 × 10 −9 , we can still expect values for ∆a e up to the level of ∆a e 2 × 10 −13 , which is roughly a factor of three above the expectations of the NS scenario.
However, in contrast to the LFC scenario, LFV is not able to account for the ∆a µ anomaly, with departure from NS. This is because by combining eq. (45) with eq. (41) we find that the experimental limit BR(τ → µγ) < 4.4 × 10 −8 (90% CL) sets a very strong upper bound on  ∆a LFV µ . Moreover, large effects in ∆a LFV e require slepton masses roughly at the TeV, in order to keep BR(τ → eγ) under control. In such a regime, we cannot employ the contribution to (g − 2) µ from LFC because ∆a LFC µ is not large enough in order to account for the anomaly, see eq. (36). This can be clearly seen in fig. 3 on the right where we show ∆a µ vs. |∆a e | in the LFV scenario, imposing the bound BR(τ → eγ) < 3.3 × 10 −8 . The black line shows the correlation between ∆a µ and ∆a e in the case of NS where LFV contributions are absent. In the interesting region where 10 −13 ∆a e 10 −12 , it turns out that ∆a µ is below the level of 10 −9 .
Finally, there is a formidable constraint from µ → eγ, a process that receives contributions from the combination of flavor mixing angles δ µe ≈ δ µτ δ τ e . Therefore, in order to fulfill the BR(µ → eγ) bound while generating sizable effects for ∆a e ∝ δ eτ δ τ e , we need a strong hierarchy such that δ µτ AB δ eτ AB . In conclusion, a large contribution to ∆a e in the LFV case is possible if the only sizable mixing is between the first and third generation of sleptons.
Also in the LFV case a potential constraint can arise from the leptonic EDMs. In particular, in order to obtain ∆a e ≈ 10 −13 (12) we need a suppression for the relevant CPV phase arg(µM 1 δ τ RR δ τ LL ) 10 −3 (4) . However, in contrast with the LFC case, we have now flavordependent CPV phases and therefore the electron EDM does not constrain directly d µ,τ . Still, whenever d µ,τ are induced by flavor-dependent phases (coming from LFV sources), powerful bounds are obtained by the LFV processes i → j γ. In particular, we find that the current bound BR(τ → µγ) < 4.4 × 10 −8 sets the upper bounds d µ 3 × 10 −23 e cm and d τ 1.5 × 10 −24 e cm.

Correlation between a e and violation of lepton universality in LFV
Violations of lepton universality can arise also in the LFV scenario. The quantity that is determined experimentally and accounts for deviations from µ-e universality is R e/µ P The sums extend over all neutrino (or antineutrino) flavors since they cannot be distinguished experimentally. This is important for our purposes because in the LFV case one expects new contributions to P → eν τ , a process that is detected as breaking of lepton universality, rather than violation of lepton flavor. One would naively expect that the LFV channels P → i ν k (i = k) are suppressed compared with the LFC ones (i = k). An interesting exception is provided by charged Higgs mediated LFV contributions, which can be sizable for large tan β and which are chirally-enhanced by the factor m τ /m [64]. Indeed, the dominant contribution to ∆r e/µ P in the case of mixing between the first and third generation sleptons is The coefficients ∆ 11 RL and ∆ 31 RR measure the effective couplings H +ν eL e R and H +ν τ L e R respectively, which are induced at one-loop level by exchange of Bino or Bino-Higgsino and sleptons.
Since these effective Yukawa interactions are of dimension four, the quantities ∆ 11 RL and ∆ 31 RR are not sensitive to the overall soft scale, hence avoiding supersymmetric decoupling. In the region of parameter space where ∆a e receives large effects, i.e. for µ m˜ ∼ M 1 , it turns out that 3 ∆r e/µ P ≈ 8 × 10 −3 0.5 TeV An inspection of eqs. (40,48) reveals that LFV effects contributing to ∆a LFV and ∆r e/µ K are correlated, as shown in fig. 4. The plot has been obtained by means of a scan with 10 ≤ tan β ≤ 50, M H + ≤ 1 TeV, M 1 ≤ 2 TeV, M 2 = 2M 1 , µ ≤ 5 TeV, m˜ ≤ 2 TeV, and |δ τ e AB | < 1. It is interesting that ∆a LFV and ∆r e/µ K can simultaneously reach experimentally testable values even for supersymmetric masses beyond the LHC reach.
Turning to pion physics, one finds that ∆r e/µ π ∼ (m 2 π /m 2 k ) × ∆r e/µ K . Therefore ∆r e/µ π susy is negligible after the constraints from ∆r e/µ K susy are imposed.

Disoriented A-terms
So far, we have investigated various supersymmetric scenarios and their capability to account for the muon g−2 anomaly. In particular, we have analyzed the possibility of breaking the NS among different leptonic g−2. As discussed in the previous sections, both model-independently as well as in supersymmetric frameworks, the major challenge we have to deal with, when generating large effects for the leptonic g−2, is to keep under control other dipole transitions like the electron EDM and µ → eγ. Yet, the correlation among ∆a , d and BR( → γ) depends on the unknown flavor and CP structure of new physics. Therefore, from a phenomenological perspective, d and BR( → γ) do not provide a direct bound on new contributions to ∆a .
However, it would be desirable to have a concrete scenario where the above conditions are naturally fulfilled. In the following, we point out that this happens in the case of the socalled "disoriented A-terms", invoked in ref. [66] to account for the recently observed direct CP violation in charm decays D → KK, ππ.
The assumption of disoriented A-terms is that flavor violation is restricted to the trilinear terms where θ f ij are generic mixing angles. This pattern can be obtained when the trilinear terms have the same hierarchical pattern as the corresponding Yukawa matrices but they do not respect exact proportionality. A natural realization of this ansatz arises in scenarios with partial compositeness, as recently pointed out in ref. [67]. Interestingly, the structure of eq. (49) allows us to naturally satisfy the very stringent flavor bounds of the down-sector thanks to the smallness 3 In particular, starting from the exact expressions for ∆ 31 RR and ∆ 11 RL in the mass eigenstate basis [65], which we use in our numerical analysis, one can find that ∆ 31 Notice that Im(δ eτ LL δ τ e RR ) is strongly constrained by the electron EDM. However, sizable contributions to ∆r e/µ P can still be induced by Re(δ eτ LL δ τ e RR ).
of down-type quark masses. Similarly, also the bounds from the lepton sector can be satisfied under the (natural) assumption that the unknown leptonic flavor mixing angles are of the form θ ij ∼ m i /m j [67].
In the disoriented A-term scenario, the dominant amplitude for µ → eγ is where x 1 = M 2 1 /m 2 . Assuming that the only possible sources of CP violation arise from A terms, the electron EDM d e is generated by the one-loop exchange of Bino and charged sleptons. One can find the following approximate expression On the other hand, the g−2 does not require any source of CP violation and therefore it always receives effects also from SU (2) interactions. In particular, the leading effects is where we have considered the illustrative case where M 2 = µ, so that x 2 = M 2 2 /m 2 = µ 2 /m 2 . The loop function is such that f (1) = 1/8 and f (0) = 1/2. Therefore, the leading SUSY contribution for g−2 is parametrically enhanced relatively to the amplitude generating d e and µ → eγ by a factor of ∼ tan β/ tan 2 θ W ≈ 100×(tan β/30). This naturally raises the question of whether it is possible to account for the (g − 2) µ anomaly, while satisfying in a natural way the constraints from d e and µ → eγ. Indeed, the latter constraints generally require slepton masses around the TeV scale, if we assume A ∼m with O(1) phases and θ ij ∼ m i /m j . Even with such large slepton masses, it is still possible to account for the (g − 2) µ anomaly if gauginos are significantly lighter than sleptons, which is also welcome from naturalness arguments. This can be seen observing that if x 2 1, that is if M 2 , µ m, the relevant loop function for ∆a µ is enhanced by a factor of four compared to the case where M 2 = µ =m. At the same time, both BR(µ → eγ) and d e tend to decrease when M 1 m, see eqs. (50,51). Clearly, one could equivalently consider a lighter supersymmetric spectrum to explain the muon g−2 and A terms somewhat smaller thanm.
In particular, settingm = |A e | = 1 TeV, sin φ Ae =1, M 2 = µ = 2M 1 = 0.2 TeV, and tan β = 30, we find that  It is interesting that disoriented A-terms can account for (g−2) µ , satisfy the bounds on µ → eγ and d e , while giving predictions within experimental reach. However, we expect that the electron g−2 follows NS. A potential source of NS breaking comes from the trilinear terms (provided A e /A µ = m e /m µ ). In practice, as already discussed in previous sections, their effects are very small after the vacuum stability bound |A |/m 3 are imposed and therefore the NS relations are preserved.

Light (pseudo)scalars and a e
In this section we will investigate scenarios where the muon g−2 anomaly is accounted for by contributions arising from light (pseudo)scalar particles. Irrespectively of the underlying theoretical motivations, such framework is interesting because it typically predicts large and very special NS violations.
We parametrize the Yukawa interactions between the light scalar field φ and pseudoscalar A with leptons with the following effective Lagrangian where C φ and C A are arbitrary constants. Although we will not discuss specific models, the field A could arise as a pseudo-Goldstone boson of an extended Higgs sector and the field φ could be a light gauge singlet coupled through a dimension-five effective interaction to the ordinary Yukawa terms. Very light scalar or pseudoscalar particles with Yukawa-like couplings are generally subject to stringent constraints both from low-energy data, such as meson decays, as well as reactor and beam dump experiments (for a review see ref. [68]). Most of these bounds disappear for M A > ∼ 10 GeV, which is the mass regime of interest for us in order to account for the (g−2) µ anomaly, as we will show below. For simplicity, we can also consider the case in which φ and A are coupled to leptons, but not to quarks. Because of the smallness of the electron Yukawa coupling, this case is even less constrained.
where the loop functions are with asymptotic limits L τ φ = 6 ln r+13 9 r r 1 ln 2 r + 2 ln r + 4 + π 3 As shown by eq. (58), two loop effects for ∆a exhibit NS and can be positive or negative depending on the sign of Re(C C τ * ). Moreover, the enhancement factor m 2 τ /m 2 e,µ of two loop effects relative to one-loop effects, which is particularly important for the case of the electron g−2, can easily compensate the additional loop suppression, as we will see later. On the other hand, in the case of the τ , one-loop effects are typically dominant compared to two-loop effects.
Hereafter, we will focus on the case of pseudoscalar particles, which is especially interesting. Specializing to the mass regime m M A where the ∆a µ anomaly can find an explanation, we have the following situation: 1) ∆a e is always dominated by two-loop effects, 2) ∆a µ receives comparable one-and two-loop contributions, and 3) ∆a τ is always dominated by one-loop effects. As a result, we expect significant NS violations that we are going now to study numerically.
In the left plot of fig. 6, we show the anatomy of the contributions to ∆a µ setting C A = 50: the red line corresponds to the magnitude of the negative ("−") one-loop effects, the green line refers to the positive ("+") two-loop effects and the black line stands for the magnitude of the total contribution. The most prominent feature emerging from this plot is the different decoupling properties of the one-and two-loop effects as one can check directly from eq. (60). In particular, two-loop effects have a much milder decoupling with M A compared to one loop-effects. This implies, in turn, the existence of a value for M A where the two effects have comparable (and opposite) size. The exact value of M A where this happens depends on the masses and couplings of the particles circulating in the second loop of the Barr-Zee diagram. In our case, an almost exact cancellation occurs for M A ≈ 6 GeV and therefore the total contribution to ∆a µ is positive for M A 6 GeV and negative for M A 6 GeV.
In fig. 7 we show the values attained by ∆a induced by pseudoscalar effects monitoring, in particular, whether NS is at work or not.
In the left plot of fig. 7, we show ∆a µ vs. ∆a e for different pseudoscalar masses M A and varying C A . As we can see, NS (black line) is systematically violated by a large amount and, in particular, ∆a e always lies above its naive expectation. The actual amount of NS violations depends on the degree of cancellation between one-and two-loop effects entering ∆a µ . Overall, in the regions where the ∆a µ anomaly is accommodated, ∆a e typically exceeds the 10 −13 level, thereby providing a splendid opportunity to test the g−2 anomaly via the electron one.
In fig. 7, on the right, we show ∆a µ vs. ∆a τ for different pseudoscalar masses M A and varying C A . Similarly to the case of the electron g−2, NS (black line) is largely violated and ∆a τ can reach values up to the level of 10 −3 , while explaining the ∆a µ anomaly. Such values should be well within the experimental resolutions expected at a SuperB factory.  Let us discuss now the predictions for the leptonic EDMs induced by light pseudoscalars. At one-loop level, the effective Lagrangian of eq. (54) leads to a real dipole amplitude and therefore the EDMs are vanishing at this order. Two loop effects are generally complex and provide very important contributions, as already discussed for the case of the leptonic g−2. Since two-loop effects follow NS, it is expected that the electron EDM is the most sensitive probe of this scenario among the leptonic EDMs. In particular, from the model-independent expectations of eq. (31), we deduce that, in order to accommodate the (g −2) µ anomaly while satisfying the electron EDM bounds, we need the condition Im(C C f * ) 10 −3 , where f stands here for the heavy fermion running in the second loop. This condition could be naturally satisfied either if Im(C ) = Im(C f ) or if a dynamical mechanism suppresses both Im(C ) and Im(C f ), for instance by a loop factor.

Vector-like fermions and a e
In the following, we will consider the impact of heavy vector-like fermions on the leptonic (g − 2) (see also [70]). The introduction of heavy vector-like fermions, mixing with the SM fermions, can be motivated by the explanation of flavor hierarchies in the SM, as we will discuss shortly.
Hereafter, we will focus on the leptonic sector only and therefore we introduce SU(2) vectorlike doublets (L L ⊕ L R ) and singlets (E L ⊕ E R ) governed by the Lagrangian The fields L L (L R ) and E R (E L ) have the same (opposite) quantum numbers as the SM fields L and e R , respectively. Note that the SM Yukawas are assumed to be vanishing and they will be generated dynamically after electroweak symmetry breaking through the mixing between light and heavy fermions, once heavy fermions are integrated out. The above Lagrangian is reminiscent of the fermionic sector of composite Higgs models which are dual of warped 5d models. In such a scheme, the chiral fermions correspond to weakly-coupled elementary fields while the vector-like fermions are composite fields belonging to a strongly interacting sector. The Higgs boson also belongs to the strong sector and does not couple to elementary fields.
The mass eigenstates, before electroweak symmetry breaking, are obtained diagonalizing the mass mixing in L through the following 2 × 2 unitary matrices where Hereafter we use the notation s L(R) = sin θ L(R) and c L(R) = cos θ L(R) . After performing the rotations of eq. (62), the Lagrangian of eq. (61) becomes where and therefore we can define the following mass mixing matrix As a result, the fermionic spectrum consists of two heavy fermions with masses approximatively given by M E and M L and light SM fermions with masses given by where i = 1, 2, 3 is a lepton flavor index. The leptonic g−2 are generated at the loop-level by the exchange of the Higgs, W , Z bosons and heavy fermions. The dominant contribution arises from the diagram with an underlying mixing of SU(2) doublets and singlets (since it is chirally enhanced by a factor of v/m ). Using the mass relation of eq. (67), neglecting terms of order O(v 4 /M 4 L,E ) and performing an explicit loop calculation one can find that the dominant contribution is given by [70] ∆a c 16π 2 where c ∼ O(1). From eq. (68) we learn that if the heavy leptons have masses around the EW scale M L , M E ∼ v and if Re(λ LEλLE ) ∼ 1 then the muon g−2 anomaly can be solved. We stress that, in general, the parameters λ LE ,λ LE , M L,E , and c L,R are not flavor universal and therefore naive scaling might be violated. The Yukawa couplings λ LE andλ LE could also violate lepton flavor and CP. As a reference framework, we consider the so-called anarchic scenario, where all the entries in λ LE andλ LE are assumed to be of order one. For the branching ratio of µ → eγ we obtain BR(µ → eγ) ≈ c 2 × 10 −6 |λ LEλLE and therefore, in order to satisfy the experimental constraint BR(µ → eγ) 2 × 10 −12 , we need |s e L(R) /s µ L(R) | 2 2 × 10 −6 me mµ for c ∼ 1. Such a suppression for the flavor mixing angles could be achieved by introducing a flavor symmetry determining the structures of λ LE andλ LE .
For the electron EDM we find d e e ≈ 5c × 10 −25 Im(λ LEλLE ) 300GeV and the experimental bound is satisfied for |Im(λ LEλLE )| 2 × 10 −3 for c ∼ 1. Thus, we conclude that both the flavor and CP structures of λ LE andλ LE have to be highly non-generic to satisfy the constraints from i → i γ and the electron EDM.
Finally, we mention that interactions with heavy leptons generate also corrections to the fermion couplings of the Z and Higgs bosons. Indeed, after integrating out the heavy states using their equations of motion we obtain dimension-six operators leading to corrections of the fermion couplings of the form v 2 /M 2 L,E .

Conclusions
The aim of this paper was to show that the anomalous magnetic moment of the electron a e can be viewed today as a new player among the low-energy processes that are able to probe newphysics effects. This novel status of a e stems from recent improvements on both the experimental and theoretical fronts. One important ingredient is the measurement of α from atomic-physics experiments, which are becoming competitive with a e in the determination of the fine-structure constant. The second ingredient is the ongoing effort to measure a e with better experimental accuracy. The third element is a more precise theoretical determination of a e in the SM. At present, the experimental measurement of a e is in good agreement with the SM and the uncertainty in the quantity ∆a e = a EXP e − a SM e is about 8 × 10 −13 . As discussed in this paper, future progress can reduce this error by about one order of magnitude.
From the theoretical point of view, the great interest in testing new-physics effects in a e comes from the well-known discrepancy between the experimental measurement and the SM prediction of a µ . Observing or excluding an anomaly in a e could become the most convincing way to establish the origin of the a µ discrepancy. In a large class of models, new-physics contributions to a (for = e, µ, τ ) are proportional to m 2 , a situation that we call "naive scaling". In the case of naive scaling, the present value of the (g−2) µ anomaly, see eq. (1), corresponds to ∆a e = (0.7 ± 0.2) × 10 −13 . While theoretical predictions have recently achieved an impressive precision O(10 −13 ) via formidable calculations, confirmation of the a µ discrepancy through a e still requires experimental improvements at the utmost level.
In many well-motivated cases, new physics effects do not respect naive scaling, as we have shown with several examples of such theories. In the context of supersymmetry, this can happen with lepton flavor conservation (for non-degenerate but aligned sleptons) or with lepton flavor violation (with relative misalignment between lepton and sleptons). In the first case, once we normalize ∆a e in such a way to reproduce ∆a µ ≈ 3 × 10 −9 , we obtain specific predictions on violation of lepton universality in P → µν/P → eν and P → µν/τ → P ν, for P = π, K. The case of lepton flavor violation cannot explain the anomaly in a µ , but can lead to new effects in a e that are correlated with the prediction of τ → eγ and violation of e/µ lepton universality. In a fairly model-independent way, it is also possible to correlate new effects in ∆a e with the corresponding electric dipole moment.
We have also considered a class of theories with a light (pseudo)scalar interacting with matter proportionally to ordinary Yukawa couplings. In this case we have found an interesting pattern of violations of naive scaling coming from the interplay between one-loop contributions with ∆a ∝ m 4 and two-loop contributions with ∆a ∝ m 2 . For the parameters capable of explaining the (g−2) µ anomaly, we found that ∆a e is larger than its naive-scaling value and can be close to its present experimental bound.
We believe that a e offers a special opportunity to test new-physics effects and to shed new light on the current discrepancy in the muon anomalous magnetic moment. A robust and ambitious experimental program is necessary to exploit its full potential.