Explaining electron and muon $g-2$ anomaly in SUSY without lepton-flavor mixings

We propose a SUSY scenario to explain the current electron and muon $g-2$ discrepancies without introducing lepton flavor mixings. Threshold corrections to the Yukawa couplings can enhance the electron $g-2$ and flip the sign of the SUSY contributions. The mechanism predicts a flavor-dependent slepton mass spectrum. We show that it is compatible with the Higgs mediation scenario.


Introduction
The discrepancy of the lepton anomalous magnetic moment (g − 2) is one of the leading candidates that indicate new physics beyond the standard model (SM). Both in the electron and muon sectors, the anomaly has been reported as ∆a e = a EXP e − a SM e = (−8.7 ± 3.6) × 10 −13 , where a SM µ is the SM prediction of the muon g − 2 [1,2], and a EXP µ is its experimental result [3,4]. Recently, a new discrepancy, ∆a e , was reported in the electron sector, due to the new measurement of the fine structure constant. See Refs. [5,6] for the experimental value of the electron g − 2, Ref. [7] for its theoretical prediction, and Ref. [8] for the new result of the fine structure constant.
It is challenging to explain both anomalies theoretically. In a wide class of new physics models, contributions to the lepton g − 2 are scaled by the lepton mass squared. Suppose the muon g − 2 anomaly is a sign of new physics, the electron g − 2 is expected to receive a contribution, This is too small to explain the result (1). Thus, it seems to require very light new particles, which easily conflict with experimental constraints, e.g., from the LHC.
In addition, the sign of Eq. (1) is opposite to Eq. (2). Extra mechanisms may flip the sign. For instance, flavor violations in the lepton sector can solve the problems, though they are constrained tightly.
New physics models have been studied to explain both anomalies [9][10][11][12][13]. Within the context of the supersymmetry (SUSY), lepton flavor violations are examined [12]. SUSY contributions to the electron g − 2 are enhanced by the tau Yukawa coupling via the mixings of the selectrons with the staus, instead of introducing very light SUSY particles. Further, the sign is chosen appropriately by the mixings. However, it was argued that the lepton-flavor violating τ → eγ restricts the system.
In this letter, we propose a new mechanism to explain both anomalies within the minimal supersymmetric standard model (MSSM). We assume the minimal flavor violation (MFV) for the lepton sector, and thus, the model is free from the lepton flavor violations. The key observation is threshold corrections to the lepton Yukawa couplings. They are non-linear in SUSY particle masses so that even if the SUSY particle masses follow the MFV hypothesis, the relation (3) can be changed drastically. In particular, the SUSY electron Yukawa coupling can be enhanced, and its sign can be opposite to the muon one. The scenario predicts flavor-dependent slepton masses. We will discuss the Higgs mediation scenario as an explicit model [14].
2 Muon and electron g − 2 The SUSY Yukawa couplings of leptons, y i , are matched with the SM ones, m i /v, non-trivially because of radiative corrections ∆ i and a ratio of the Higgs vacuum expectation values tan β ≡ H 0 u / H 0 d as [15][16][17] with i = e, µ, and its superpartnerĩ. Here, m X is a mass of X, g Y is the gauge coupling of U (1) Y , and M 1 is the Bino mass. The loop function is defined as which satisfies I(x, x, x) = 1/2x 2 .
By taking m˜i When the Higgsino mass parameter is much larger than masses of the sleptons and  −500 TeV (right). The red and blue lines denote ∆ µ and ∆ e , respectively. It is found that ∆ i can be around or smaller than −1. In the discontinuity region of the red line, an eigenstate of the smuon becomes tachyonic.
The leading tan β-enhanced radiative corrections are taken into account in Eq. (4), and |∆ i | can be large [16] (cf. Ref. [15]). 3 They include a resummation of the radiative corrections in the form of (g 2 Y µ tan β/M SUSY ) n to all orders, where M SUSY is a typical scale of SUSY particle masses in loops, while other corrections are suppressed.
When |µ| tan β is large, the SUSY contributions to the lepton g − 2 of the i-th generation, (a i ) SUSY , are dominated by the Bino-slepton diagrams. In the mass- insertion approximation, they are represented as [20] where f N (x, y) is the loop function defined in Ref. [20] and satisfies f N (1, 1) = 1/6.
QED corrections beyond the leading order are taken into account by δ QED [27].
The radiative correction ∆ i appears, because (a i ) SUSY is proportional to the SUSY Yukawa coupling of the lepton.
The SUSY contributions (8) are scaled by the lepton mass squared m 2 i . It is noticed that (a i ) SUSY can be affected drastically by ∆ i , when µM 1 is large negative.
For µM 1 < 0, ∆ i is negative. Since f N (x, y) is positive, (a i ) SUSY becomes positive (negative) for 1 + ∆ i < 0 (> 0). In addition, (8) is enhanced significantly around the cancellation point, Thus, (a i ) SUSY /m 2 i can have different size and sign for different flavors, depending on 1 + ∆ i . It is noticed that lepton-flavor mixings are not necessary, and thus, there are no constraints from the lepton flavor violations.
In Fig. 2 we show (a i ) SUSY /m 2 i for the electron (blue line) and the muon (red line). In the horizontal blue band, the observed discrepancy for the electron g − 2 (see Eq. (1)) is explained at the 1σ level, and that for the muon (see Eq. (2)) is shown by the red band. We find that the electron g − 2 discrepancy is explained around the cancellation point ∆ e = −1, corresponding to mẽ L,R 2.3 TeV (6.0 TeV) in the left (right) panel. The selectrons are relatively heavy and satisfy the collider constraints easily. On the other hand, the muon g − 2 anomaly is explained by lighter smuons, because (a µ ) SUSY is required to be positive. Here, all the SUSY particles are set to satisfy the current collider/experimental bounds. 5 Too large |µ| tan β spoils the stability of the electroweak vacuum. In the analysis, we used the formula provided in Ref. [20] to derive the vacuum stability condition. 6 The trilinear coupling associated with µ tan β is proportional to the SUSY Yukawa coupling of the lepton. In the muon case, the vacuum is stable in the pink vertical band. There is a lower bound on the smuon masses, because the potential is stabilized when the smuons become heavy. In addition, an upper bound is obtained when the pink band appears to the left of the mass discontinuity region (see the right panel of Fig. 2). This is because, as the smuon masses increase, |1 + ∆ µ | decreases according to Fig. 1, and thus, the SUSY Yukawa coupling, i.e., the trilinear coupling of the smuons, is enhanced. In Fig. 2, it is found that the smuons are required to be heavier than 4.2 TeV for µ = −100 TeV, while they are limited in 600 GeV mμ 1 TeV for µ = −500 TeV by the vacuum stability condition. In contrast, the vacuum stability constraint for the electron is highly alleviated and does not affect our scenario, because its Yukawa coupling is tiny.
Two sample points are given in Table. 1. In both cases, the electron g −2 discrepancy is explained. For the muon, in order to satisfy the vacuum stability constraint, the smuon should be either heavier (left panel of Fig. 2) or lighter (right panel) than the selectron. In the latter case, the muon g − 2 anomaly is explained with satisfying 5 Light smuons can satisfy the LHC bounds, e.g., by setting the LSP appropriately. 6 The formula fits the result of CosmoTransitions 1.0.2 [28] at the tree level. It may suffer from a large scale uncertainty [29]. In particular, an energy gap exists between the scales of the charge-color breaking vacuum, 10 8 GeV, and the electroweak vacuum, ∼ 100 GeV. Since the potential can be lifted in a large renormalization scale, the constraint might be alleviated.  The scenario may be tested by indirect searches. Since ∆ i is close to −1 for the electron or large negative for the muon, the branching fractions of the (semi-) leptonic B meson decays are affected when the heavy Higgs bosons are relatively light [30,31]. 8 The decays can proceed by exchanging the heavy Higgs bosons, whose couplings to Here, there is no significant hierarchy in the Yukawa couplings. This is in contrast with general SUSY models, where the electron and muon Yukawa couplings involve a tuning to realize their mass relation. 9 8 The quark sector also receives threshold corrections similarly. The SUSY Yukawa couplings of the down-type quarks can be enhanced with certain squark and gluino/Bino masses, and large |µ| tan β. Such effects may be observed in the quark flavor physics. 9 In our scenario, the electron-muon mass relation is explained by a slepton mass spectrum.
Such slepton masses might be chosen by an anthropic selection on the electron mass [33]. Electrons much heavier than MeV scale decouple from the thermal bath before the Big-Bang nucleosynthesis starts, and hence, spoil the success of the standard cosmology.

Higgs mediation scenario
In order to explain the current discrepancies of the electron and muon g − 2, the smuons are required to be lighter than the selectrons. In this section, we provide UV models to realize such a slepton spectrum. Let us assume the MFV for the slepton soft-breaking masses [34][35][36][37][38], 10 where higher order terms of y i are omitted. The first terms, d L and d R , in the right-hand side are flavor-blind contributions, e.g., by SUSY-breaking mediations via gauge interactions. The second terms, c L and c R , depend on the lepton Yukawa couplings, i.e., depend on lepton flavors. Such contributions are yielded by SUSYbreaking mediations via the Higgs sector, as we will discuss below. Here, the lepton Yukawa matrix is diagonalized without loss of generality, and hence, the slepton soft mass matrices are aligned to the Yukawa matrix. Then, there are no lepton-flavor violations. 11 In this Letter, we do not assume anything special for the squarks and the gluino. Their masses depend on details of the UV models.
Let us discuss the Higgs mediation scenario [14,[44][45][46] to realize the flavordependent mass spectrum in Eq. (11). The slepton masses get positive contributions from the Higgs loop diagrams with the Yukawa interactions. By taking m 2 Hu m 2 H d < 0, the slepton masses are generated through RG running as (cf., Ref. [46]), 10 The smuons can also be embedded in N = 2 SUSY multiplets [39,40]. Here, SUSY breaking effects are suppressed due to the N = 2 non-renormalization theorem. The smuons tend to be lighter than other sleptons. 11 This is not the case beyond the MSSM, e.g. when strongly-coupled right-handed neutrinos are introduced to explain the neutrino masses [41]. This may be supported by the thermal leptogenesis [42]. Even in this case, one can introduce a flavor symmetry in the lepton Yukawa couplings,    The selectron, smuon, and stau masses are shown in the middle.
at the leading logarithmic approximation. Here, M GUT is the GUT scale, and Tachyonic mass spectrum is avoided for the pseudo-Higgs boson by assuming [14,44] |µ| ∼ −m Hu , tan β 50.
This setup is favored to achieve large |∆ i |. On the other hand, d L and d R depend on flavor-blind mediation mechanisms. Here, we do not assume them and leave the parameters arbitrary.
There are two types of mass spectra for smuons and selectrons which are consistent with the Higgs mediation. According to the previous section, when mμ mẽ, one obtains |∆ µ | 1 and |y µ | |y e |. On the other hand, when mμ mẽ, |∆ µ | can be so large that |y µ | becomes smaller than |y e |. These two spectra can be realized by the Higgs mediation. In fact, whenm 2 > 0 they satisfy the following relation, Let us provide three data points of the Higgs mediation scenario in Table 2. Points I and III explain the electron g −2 discrepancy, while the muon g −2 anomaly is not.
On the other hand, both are explained at Point II. In all cases, the vacuum stability condition is satisfied for the stau as well as the smuon, because the staus become heavy in the scenario. It is noticed that Points II and III have the same dimensionful input parameters, despite that the results are different. This is because multiple sets of the smuon SUSY-breaking masses satisfy Eq. (4). Then, the dimensionless parameters, particularly y µ , become different due to large threshold corrections.

Conclusions
We proposed an MSSM scenario to explain both the electron and muon g − 2 discrepancies without introducing lepton flavor mixings. The discrepancies are different in scale and sign. The electron g − 2 requires larger SUSY contributions than the muon g − 2 with an opposite sign. In our scenario, this is realized by the threshold corrections to the SUSY Yukawa interactions with the flavor-dependent slepton mass spectrum. The electron Yukawa coupling becomes enhanced by them, and its sign can be opposed to the muon one. In order to explain both anomalies, the smuons are required to be (much) lighter than the selectrons. We discussed that such a mass spectrum is consistent with the Higgs mediation scenario.