Muon $g-2$ Anomaly in Anomaly Mediation

The long-standing muon $g-2$ anomaly has been confirmed recently at the Fermilab. The combined discrepancy from Fermilab and Brookhaven results shows a difference from the theory at a significance of 4.2 $\sigma$. In addition, the LHC has updated the lower mass bound of a pure wino. In this letter, we study to what extent the $g-2$ can be explained in anomaly mediation scenarios, where the pure wino is the dominant dark matter component. To this end, we derive some model-independent constraints on the particle spectra and $g-2$. We find that the $g-2$ explanation at the 1$\sigma$ level is driven into a corner if the higgsino threshold correction is suppressed. On the contrary, if the threshold correction is sizable, the $g-2$ can be explained. In the whole viable parameter region, the gluino mass is at most $2-4\,$TeV, the bino mass is at most $2\,$TeV, and the wino dark matter mass is at most $1-2\,$TeV. If the muon $g-2$ anomaly is explained in the anomaly mediation scenarios, colliders and indirect search for the dark matter may find further pieces of evidence in the near future. Possible UV models for the large threshold corrections are discussed.


Introduction
Recently Fermilab has confirmed the long-standing discrepancy of the muon anomalous magnetic moment (g − 2) between the measurement at Brookhaven National Lab and the Standard Model (SM) prediction [1][2][3]. The combined discrepancy is found to be where a EXP µ is the experimental value [1][2][3] (See also Refs. [3][4][5][6][7][8][9]). The deviation is at a significance of 4.2σ. If we adopt the R-ratio analysis in [10] the significance rises to 4.5σ level. This is an important message that there is a beyond SM (BSM) particle coupling to muon and whether muon g−2 can be explained will become an important criterion for the BSM model-building. The BSM should have consistent cosmology, There is a simple and cosmologically safe mediation effect called anomaly mediation [11,12]. This loop effect exists generically and is important if the tree-level mass terms for the gauginos are suppressed. In particular, the tree-level gaugino mass is absent if the SUSY breaking field is charged or sequestered [13]. The resulting gaugino masses are purely induced from the anomaly mediation. The masses follow the pattern: with c i ≈ 1 representing the threshold corrections by integrating out scalar particles, m 3/2 (> 0) is the gravitino mass, representing the Higgsino threshold correction, µ the Higgsino mass parameter, tan β the ratio of the vacuum expectation values (VEVs) of the two higgs fields, and m A the MSSM higgs boson mass. In general, by taking m 3/2 and VEV to be reals, µ is a complex parameter. For simplicity and for ease to evade the CPV bounds, we will limit ourselves to the purely real case, but we will come back this point in the last section. when the higgsino threshold correction is negligible, L vanishes. This spectrum is almost UV model-independent. It is only corrected slightly at the renormalization scale below the splitting scale between the sparticle and particle masses.
For instance, the spectrum remains intact even if the model has any multiplets in the intermediate scales. 1 For −3 L/m 3/2 3, the wino is the lightest gaugino and can be the dominant dark matter component if it is lighter than the other sparticles. 2 Since there is no need to introduce a Polonyi field for generating the gaugino mass, there is no Polonyi/Moduli problem. The gravitino problem is also absent since the heavy gravitino has a lifetime shorter than a second. On the other hand, the late-time decay of the gravitino produces the wino LSP. The wino dark matter abundance can be explained for a certain reheating temperature even if the thermal production is not enough [15,16] (see also Sec.3).
In contrast to the gaugino masses, the sfermion and higgsino mass spectra are model-dependent. There are various simple models categorized by the SUSY spectra. The split SUSY has higgsino as light as gaugino while others are much heavier [17][18][19]. The pure gravity mediation or mini-split SUSY has all other fields much heavier than the gauginos [20][21][22][23]. (See also Ref. [24]) By taking account of a Higgs mediation [25], i.e. in the Higgs-anomaly mediation, the sfermions of the first two generations are as light as gauginos while others are heavy [26][27][28][29]. 3 In all the aforementioned models, flavor violation is suppressed, cosmology is consistent, and the predicted SM Higgs boson mass can easily match the measured one. (See also other early SUSY models explaining the g − 2 [31][32][33][34][35][36]) In this letter, we perform a model-independent analysis to study to what extent 1 A change of the mass spectrum can occur if the Higgs boson is a slepton and there are no Higgsino multiplets at the low energy scale [14]. However, the top Yukawa coupling is difficult to be generated due to holomorphy. 2 Out of this range, the bino can be the LSP. It, however, over-closes the universe and is excluded. 3 The setup is easily realized if the fermion multiplets are sequestered from the SUSY breaking but the Higgs multiplets are not. If the sfermions are pseudo-Nambu-Goldstone boson in a SUSY Non-linear sigma model, a similar tree-level condition can be obtained [27]. However, the loop induced gaugino mass spectrum is found to be different due to the Kähler structure [30], and, interestingly, predicts a bino-wino coannihilation. If there are light moduli, the F-term contribution can affect the gaugino mass spectrum. In these cases, we need a solution to the moduli problem.
These models are not belong to the category of this Letter's focus.
the anomaly mediation scenarios can explain the muon g − 2. We derive the upper limit of the g − 2 in the anomaly mediation scenarios. Then we show that the g − 2 is difficult to be explained if the higgsino threshold correction is negligible, i.e. L ∼ 0.
On the other hand, it can be explained if the higgsino threshold correction is sizable.
The upper bounds of the gaugino masses are derived.
2 Effective theory for g − 2 in anomaly mediation scenarios To perform a model-independent analysis, we consider an effective theory with only gauginos, satisfying Eq. (2), and smuons in addition to the SM particle contents.
We do not consider a light higgsino because the enhanced DM-nucleon coupling is strongly disfavored by the direct detection experiments [37][38][39]. 4 The LHC data, then, sets a stringent bound on the wino LSP and thus the lepton mass: which is reported by ATLAS [41] (CMS [42]). We will take the 660 GeV in the following. 5 This is comparable or more stringent than the indirect detection bound (e.g. [43][44][45]). The wino dark matter satisfying this bound may be tested in the future not only by the collider searches but also by the direct detection experiments.
The LHC bounds other than (4) are much weaker in this model. The smuon bound is almost absent since the wino satisfying (4) is the LSP [46,47]. The predicted gluino mass is almost not constrained if (2) and (4) are satisfied with the wino LSP [48,49]. 6 Since the higgsino is heavy, the only important contribution to the g − 2 is from a bino-smuon loop. The relevant effective interacting Lagrangian is given by 4 When higgsino is much lighter than the wino, this is another model-independent setup for the dark matter and g − 2. For further details of this scenario, see a recent study [40]. 5 This bound depends on the chargino-neutralino mass splitting. Although the light smuons with sizable left-right-mixing contribute to the splitting, the splitting is not generated at the one loop level and thus is negligible compared to the electromagnetic contributions. This pure wino bound applies to our effective theory. 6 A tiny parameter range with large |L| and small masses of the bino and wino is excluded. If we introduce more light sparticles like selectron, the LHC bound may become more severe. We do not do this as we can easily find that the resulting upper bound of g − 2 decreases due to the higher mass scale of the sparticles.
The kinetic terms are normalized.
The smuon has a mass mixing of where the mixing parameter is defined as Here, v ≈ 174 GeV is the SM Higgs VEV. ∆ represents the threshold correction to the muon Yukawa coupling, and ∆/(1 + ∆) is the fraction of the muon mass that is radiatively induced. In addition, we define as diagonal elements of the mass squared matrix for the left-handed and right-handed smuons, respectively.
The most important bound is from the vacuum (meta) stability: This bound can be understood since the action for the bounce solution scales as . A more precise fitting formula, which we adopt in the numerical simulation, can be found in Ref. [50] (see also Ref. [51]). For given smuon diagonal mass components, this gives the maximal left-right mixing parameter, M LR . By taking the mass-insertion approximation justified when vM LR max [m 2 µ L , m 2 µ R ], we obtain [50,[52][53][54][55] where m µ is the muon mass; We have not written down the radiative corrections by the integration of the sparticles above the smuon mass scale because it is model-dependent. This uncertainty will be taken account by varying c i . On the other hand, the electromagnetic correction below the smuon mass scale has been included. One can see that given the smuon and bino masses, the vacuum stability bound sets an upper bound for (a µ ) SUSY . We note that Eqs. (9) and (10) only depend on a combination of µ, tan β, and ∆ in M LR . This means that our analysis does not depend on the size of tan β, or on whether the muon mass is radiatively induced.
In Fig.1, we show the maximized g − 2, a max SUSY [red band], evading the vacuum stability bound by varying the lightest smuon mass for L = 0 (left panel) and L = m 3/2 (right panel). We fix the wino mass to be the lowest value of 660 GeV from the current LHC bound. a max SUSY corresponds to mμ L = mμ R with a given lightest smuon, m min smuon . To show this, we also display a light-blue band with smuon mass splittings, mμ L = 2mμ R and mμ L = 0.5mμ R . They almost overlap. Thus, a smuon mass splitting leads to a smaller a max SUSY than the degenerate mass case. As mentioned, c 1 /c 2 is varied within 1 ± 0.05 to take into account the model-dependent loop corrections, which give the uncertainty of the prediction. As a result, when L = 0, i.e. the higgsino threshold correction is neglected, the g − 2 can be explained at the 1σ level in a narrow region where c 2 /c 1 ∼ 0.95 and M LR is close to the vacuum stability bound.
We have to say that the g − 2 explanation with L = 0 is driven into a corner.
In the right panel, on the other hand, a case with a larger higgsino threshold correction with L = m 3/2 is shown. This shows that the g − 2 at the 1σ level can be explained in a wider parameter range if L = O(m 3/2 ). This is because the bino mass slightly decreases with a given wino mass for larger L, and so the g − 2 contribution is enhanced. As we will see soon in this case we can have a peculiar gaugino mass spectrum, and the gluino mass tends to be lighter than the usual prediction of the anomaly mediation with L = 0. In the next section, we also discuss that the winobino coannihilation can take place with L/m 3/2 = 2-3.
From Fig.1, we can see that the g − 2 is maximized with mμ L ≈ mμ R ≈ M LR ≈ M wino when L is given. By using this property, we can derive the upper bound of gaugino masses. Fig.2 represents a scatter plot with maximized gaugino masses by varying L to explain the g−2 at the 1σ level. We take c 3 /c 1 ≈ 1±0.05, c 2 /c 1 ≈ 1±0.05 at random. The gluino, bino, and wino masses are shown by the collection of the red points from top to bottom. They are obtained by solving 10 9 (a µ ) SUSY = 2.51 − 0.59.
The gray data points in triangle are excluded due to the wino mass bound. We also show the case with 10 9 (a µ ) SUSY = 2.51 + 0.59 by the purple points for comparison.
In summary, we can conclude that if the g − 2 is explained in the anomaly The light gluino and wino masses can be tested in the LHC and future colliders [56][57][58][59][60]. The light bino and smuons are also predicted. Although the bino cannot be produced via electroweak process, we can produce it from the muon collision and then search for its decay in a muon collider [61]. (Muon collider can test all muon g − 2 scenarios [61][62][63][64][65].) In this process, we can even identify the SUSY gauge coupling as well as the bino and smuon masses [61]. Wino dark matter may be also searched for in direct detection experiments. The light gauginos as well as the light smuons with the particular mass pattern will be a smoking-gun evidence of our scenario.

Discussion
Wino dark matter abundance We have assumed the wino LSP composes the dominant dark matter component, although the thermal relic abundance of the pure wino in the mass range is smaller than the observed one. In fact there are two simple possibilities to realize this abundance:

• Coannihilation
The wino LSP mass in the range L/m 3/2 = 2−3 can be similar to the bino mass (see Fig. 2). By increasing L, the thermal relic abundance of the LSP due to the coannihilation tends to increase, and it will be too much if L 3m 3/2 since the LSP is bino. Thus there must be a regime of L in which the wino thermal abundance is comparable to the observed one due to the coannihilation with the bino. This is the case if L ∼ (2 − 3)m 3/2 . In this scenario, the reheating temperature should be much smaller than 10 10 GeV, otherwise the non-thermal component from the gravitino decay would be too large (See the following).

• Gravitino decay
In the anomaly mediation scenario, the dark matter can be produced from the gravitino decay. The decay time is predicted to be after the freeze-out period of the wino when the wino mass is of our interest. Thus there is a non-thermal component of the wino abundance [15,16] When the thermal component is not enough, the total wino abundance can be explained by this component given a correct reheating temperature.
Possible UV models So far, we found that a large L ∼ m 3/2 is favored to explain the g − 2.
Let us discuss what kind of model can allow such a large L. One option is the pure-gravity mediation [20][21][22][23], where tan β = O(1). Indeed, one of the interesting predictions of the model is the possible large L. However we need to slightly modify the model since the smuon mass scale is much heavier than the gaugino masses in the original scenario. To this end, we may consider sequestering some of the lepton multiplets, including the muon multiplet, from the SUSY breaking. 7 The sequestered slepton masses are suppressed compared to the masses of other sfermions and the higgsino. Then we can derive |µ| ∼ (1 + ∆) PeV for a smuon ∼ 1 TeV. This is consistent with the Higgs boson mass and electroweak symmetry breaking in pure-gravity mediation if ∆ is not too large. Note that the charm, top, bottom, and tau multiplets may not be sequestered otherwise they are too light and the very large µ-term triggers the electroweak vacuum to decay into a color/charge-breaking vacuum.
One may also, on the other hand, consider a large tan β case with large µ and m A satisfying m 2 A /µ ∼ tan βm 3/2 . In this case, we can have a Higgs mediation [25] (see also studies relevant to Higgs mediation [68][69][70][71]) if m A µ. Then, all the lepton multiplets may be sequestered to explain the g − 2. In this case the stau is heavy due to the Higgs mediation and, as a result, the stau vacuum decay problem is alleviated. We may also sequester the quark multiplets as in the Higgs-anomaly mediation scenarios [26][27][28][29]. In this case, however, due to the too large µ-term, the higgs mediation would induce large and negative mass squares for the first two generation squarks. Therefore the sequestering should be slightly broken to induce positive squark masses.
By introducing the breaking of the sequestering, we may need to care about the LFV, especially the µ → eγ [72]. In general, L's CP phase is not aligned to that of m 3/2 . Thus, we expect a CP violation. With CPV, interestingly a muon EDM can be tested in the J-PARC [73][74][75] (together with further confirmation of the muon g − 2 ). The gaugino masses are slightly modified due to the CP phase in L, which is linked to the muon EDM and the g − 2. This is also a smoking-gun evidence of our scenario. On the other hand, the electron EDM is severely constrained [76]. In this scenario, since the µ tan β is large, the muon (and electron) Yukawa can be easily generated radiatively, ∆ 1. The loop-induced lepton-photon coupling and the mass basis is automatically aligned and thus the LFV is suppressed [77] (electron EDM can be also suppressed [77][78][79]). 8

Conclusions
The anomaly mediation scenarios of SUSY breaking can be easily freed from the moduli and gravitino problems. The gaugino mass relation is a renormalization invariant and thus it is the UV model-independent prediction. In this paper, we studied to what extent the muon g − 2 anomaly can be explained within models with anomaly-induced gaugino masses. We have built an effective theory involving the gauginos and the smuons. By combining the recent LHC bound and the smuon vacuum stability bound, we found that it is hard to explain the g − 2 if the Higgsino threshold correction is suppressed. On the other hand, if the correction is not suppressed one can still obtain a large enough g − 2. In this case the gluino tends to be lighter than the usual case with suppressed threshold correction, and thus be produced with smaller center-of-mass energies in colliders. The peculiar spectrum of gauginos with L = 0 and light smuons will be the smoking-gun signal in collider experiments in the future.