]>PLB34432S03702693(19)30094210.1016/j.physletb.2018.12.070The AuthorsPhenomenologyFig. 1The neutralino (bino–bino) diagram to generate electron g − 2. The 1–3 flavor violation in left and righthanded slepton masses can break the lepton mass scaling of g − 2 in Eq. (3).Fig. 1Fig. 2The plot with δLR mass insertion. δLL,RR13 and δLR13,31 are generated randomly. The mass parameters are fixed as mL = mR = 450 GeV, M1 = 310 GeV, M2 = 350 GeV, μ = 450 GeV for tanβ=40 to realize the central value of muon g − 2.Fig. 2Fig. 3Contour plot of log10Br(τ→eγ) on the M2–μ plane. We fix the parameters as M1 = −220 GeV, mL = mR = 300 GeV, tanβ=40, Aτ = −3 TeV, and δLL,RR13=0.1.Fig. 3Table 1The mass spectrum for gaugino masses, M1, M2, Higgsino mass μ, SUSY breaking left and righthanded slepton masses, mL, mR, and ratio of the Higgs vacuum expectation values, tanβ=vu/vd, in the scenarios given in the text. The electron g − 2 is adjusted to the central values by assuming δLL13=δRR13, and τ → eγ is canceled by the freedom of δLR13,31. The selectron and sneutrino masses are split due to these offdiagonal elements. We note that the mass eigenstates of the selectron (stau) contain stau (selectron) of the current basis at O(10)%. We choose Aτ = 0.Table 1Scenario 1Scenario 2Scenario 3Scenario 4
M1310420220150
M2350260800300
μ450250230620
mL = mR450530300540
tanβ40304045

mχ˜10301197192149
mχ˜20332257235293
mχ˜30455312254625
mχ˜40482428809632

mχ˜1+327202227293
mχ˜2+480320809634

me˜1,2429, 461510, 543288, 312517, 548
mμ˜1,2450, 454531, 533302, 305539, 544
mτ˜1,2406, 507493, 578266, 343487, 608
mν˜417, 445, 472499, 526, 552271, 293, 314506, 536, 564

Δaμ × 1092.82.72.92.4
Electron g − 2 with flavor violation in MSSMBhaskarDuttaaYukihiroMimurab⁎mimura@riko.shimaneu.ac.jpaDepartment of Physics, Texas A&M University, College Station, TX 778434242, USADepartment of PhysicsTexas A&M UniversityCollege StationTX778434242USAbInstitute of Science and Engineering, Shimane University, Matsue 6908504, JapanInstitute of Science and EngineeringShimane UniversityMatsue6908504Japan⁎Corresponding author.Editor: J. HisanoAbstractThe muon g−2 anomaly is a longstanding discrepancy from its standard model prediction. The recent improved measurement of the fine structure constant makes the electron g−2 anomaly very interesting for both sign and magnitude in comparison to the muon g−2 anomaly. In order to explain both muon and electron g−2 anomalies, we introduce flavor violation in the minimal supersymmetric standard model (MSSM) as a low energy theory. The lepton flavor violating process τ→eγ is one of the major constraints to explain both g−2 anomalies simultaneously emerging from flavor violating terms. We show various mass spectra of sleptons, neutralinos, and charginos, consistent with the LHC results, to explain both anomalies after satisfying the flavor violation constraint.KeywordsAnomalous magnetic momentSupersymmetryLepton flavor violation1IntroductionThe anomalous magnetic moment (g−2) of muon is one of the longstanding deviations from its standard model (SM) prediction. The discrepancy between the experiment [1,2] and the SM prediction [3,4] of aμ=(g−2)μ/2 is more than 3.5σ:(1)Δaμ=(2.74±0.73)×10−9. The SM prediction is smaller than the experimental measurement. There will be new measurements of aμ at Fermilab very soon and at JPARC. The theory prediction is expected to have a better accuracy.On the other hand, the electron g−2 has been consistent [5,6] with the measurement. However, a recent precise measurement of the fine structure constant [7] has changed the situation, which leads to a 2.4σ discrepancy in the electron g−2 [8](2)Δae=(−8.7±3.6)×10−13. The SM prediction is larger than the experimental measurement in this case, and the sign is opposite. Without any flavor violation in the lepton sector, the anomalous magnetic moments of electron and muon obey the lepton mass scaling as(3)Δae/Δaμ=me2/mμ2≃2.4×10−5, even if there is an effect from the physics beyond SM. In that sense, both sign and magnitude have discrepancies. The theoretical implication has been studied on this issue [8–10].The minimal supersymmetric (SUSY) standard model (MSSM) is one of the promising candidates of the models beyond SM. However the SUSY particles have not yet been observed at the LHC. Based on the recent results, the colored SUSY particles, e.g., squarks (q˜), gluinos (g˜) are heavier than 1.6–2 TeV [11,12]. However the constraints on the noncolored sparticles, e.g., sleptons (ℓ˜), charginos (χ˜+), neutralinos (χ˜0) etc. are not very good [13,14] and a lot parameter space is available in the mass range 100 GeV to 1 TeV for these particles. This mass range is very important for these particles to contribute to the g−2 calculations.The muon g−2 anomaly has been studied in the context of MSSM [15–23]. In MSSM, there can be a loop diagram in which a slepton and a chargino (neutralino) propagate, and it can explain the muon g−2 anomaly provided the sleptons and chargino (neutralino) are adequately light, say less than a TeV. The LHC constraints are discussed in [20] and it was shown that a large region of parameter space is still available and a sizable parameter space will still be available even after the LHC acquires 3000 fb−1 of luminosity.However, even if the central value of muon g−2 can be explained by the sparticle loop diagrams, the electron g−2 contribution only provides Δae≃6.5×10−14 if there is no flavor violation. In this paper, we consider the MSSM with flavor violation as a weak scale theory. We show that the major constraint to reproduce the central value of the electron g−2 is τ→eγ, and we study if g−2 anomalies of muon and electron can both be accommodated after satisfying the τ→eγ constraint. The mass spectrum to reproduce the electron g−2 depends on the choice of flavor violation, and the slepton masses need to be adequately light. In this work we will use the masses of sleptons, charginos, and neutralinos allowed by the LHC results to explain the electron and muon g−2 anomalies.This paper is organized as follows. In section 2, we discuss the possible explanations of the electron g−2 anomaly in the context of the MSSM and associated flavor violations. In section 3, we describe our numerical fit of both electron and muon g−2 anomalies satisfying the LHC and the flavor violation constraints and section 4 contains our conclusion.2Electron g−2 and flavor violationIn the MSSM, the chargino loop diagram with Higgsino–wino propagation (H˜–W˜) gives the dominant contribution to the anomalous magnetic moment, g−2, using a simple mass spectrum with gaugino mass unification. In addition to the Higgsino–wino propagator, there is also a contribution from the neutralino diagram with bino–bino propagator (B˜–B˜). For the Higgsino–wino (and Higgsino–bino) contribution, the Higgsino vertex contains the Yukawa coupling of muon/electron, while for the bino–bino contribution, the left–right smuon/selectron mixing contains the muon/electron mass. As a result, if there is no flavor violation, the amplitude is proportional to the muon/electron mass in any diagrams and the lepton mass scaling in Eq. (3) is observed for g−2.We introduce the 1–3 flavor violation in the slepton mass matrices to break the lepton mass scaling. The lepton flavor violating decay μ→eγ process has a strict experimental bound, and we do not introduce any 1–2 flavor violation. The coexistence of the 1–3 and 2–3 flavor violations can induce μ→eγ, and thus we introduce only 1–3 flavor violation. Under this assumption, the muon g−2 is generated from the diagonal elements in the slepton masses (without any flavor change) by chargino and neutralino loop diagrams. On the other hand, if there are both left and righthanded 1–3 flavor violation, the neutralino loop diagram for electron g−2 can contain the τ mass instead of the electron mass as shown in Fig. 1, and thus the lepton mass scaling can be violated.Using the conventional mass insertion parametrization,(4)δLL13=(mℓ˜2)LL13mℓ˜2,δRR13=(mℓ˜2)RR13mℓ˜2,δLR13=(mℓ˜2)LR13mℓ˜2, where mℓ˜2 is an average slepton mass, one can express the contribution to the electron g−2 via the B˜–B˜ diagram as(5)(δLL13δLR31+δLR13δRR31)+δLL13δLR33δRR31, where δLR33=(Aτ−μtanβ)mτ/mℓ˜2. We note that δRL31=(δLR13)⁎, δLL31=(δLL13)⁎, etc. are satisfied due to the hermiticity of the slepton mass matrix.11Since the experimental bound of the electron electric dipole moment (eEDM) is quite severe, we assume that all the elements in the slepton mass matrix are real (as we will comment later). One can find that the lepton mass scaling can be violated with and without δLR13,31 flavor violation (induced by the scalar trilinear coupling A which is not proportional to the charged lepton Yukawa matrix). By choosing the signs of the offdiagonal elements, one can obtain the opposite sign of Δae compared to Δaμ. The magnitude of the electron g−2 discrepancy, Δae, is about 10 times larger compared to the one which is expected from Δae≃me2/mμ2Δaμ by the lepton mass scaling. If the bino–bino contribution saturates the muon g−2, one estimates that δLL13δRR31∼10me/mτ∼0.052 can realize the magnitude of electron g−2. Usually, the bino–bino contribution is subdominant to muon g−2, and we need larger flavor violation to obtain the central value of electron g−2, but one can expect that δLL,RR13∼O(0.1) can reproduce the electron g−2.The Br(τ→eγ) [24] provides a constraint to achieve the central value of the electron g−2 by the flavor violation:(6)Br(τ→eγ)<3.3×10−8. In other words, τ→eγ may be observed soon (but τ→μγ will not) if this realization of the electron g−2 is true.The τ→eγ amplitudes can be expressed by the mass insertion method as follows:1.τL→eRγ(7)(δLR31+δLR33δRR31)AB˜−B˜+δRR31AH˜−B˜L.2.τR→eLγ(8)(δRL31+δRL33δLL31)AB˜−B˜+δLL31AH˜−W˜(B˜)R. If the LR flavor violation is turned on, the δLR31,13 can tune the amplitudes to satisfy the experimental bound of τ→eγ. When the muon g−2 anomaly is satisfied (for gaugino and Higgsino masses: M1,M2,μ>0), one can obtain negative Δae by choosing the signs of the offdiagonal elements as(9)δLL13:±,δLR13:∓,δRR31:±,δLR31:∓.Even without any LR flavor violation, negative Δae can be generated by δLL13δRR31≠0. In this case, however, the contributions to the τ→eγ amplitude have to be canceled between bino–bino diagram and Higgsino–wino (bino) diagrams, and thus the SUSY mass spectrum is constrained. The bino–bino contribution to the amplitudes for both τL→eRγ and τR→eLγ behaves as(10)AB˜−B˜∝αYM1μmℓ˜L2mℓ˜R2fN(mℓ˜L,mℓ˜R,M1), where fN stands for a loop correction for neutralino diagram. The Higgsino–bino contribution to τL→eRγ amplitudes behaves as(11)AH˜−B˜L∝−αYM1μfN(M1,μ,mℓ˜R), and the behavior of the Higgsino–wino (bino) contribution to τR→eRγ amplitudes can be roughly written as(12)AH˜−W˜(B˜)R∝α2M2μfC(M2,μ,mν˜)−α22M2μfN(M2,μ,mℓ˜L)+αY2M1μfN(M1,μ,mℓ˜L), where fC stands for a loop function for chargino diagram. We find that M1/M2<0 is needed (where M1 and M2 are bino and wino masses) and sleptons need to be enough light to satisfy the experimental bound of τ→eγ and to obtain the large magnitudes of Δae. The opposite signs of Δae compared to Δaμ can be obtained by δLL13δRR31>0.3Numerical worksIn this section, we show our numerical calculations of g−2 of muon and electron. In the previous section, in order to illustrate the qualitative feature, we have used the mass insertion approximation, but here, we calculate the ℓ¯iσμνℓjFμν operator without using the mass insertion approximation.First, we show the case where LR flavor violation is turned on. As we have explained, the τ→eγ amplitudes can be canceled by choosing the LR offdiagonal elements in this case. As long as the muon g−2 anomaly is satisfied, the mass spectrum does not get further constrained in order to obtain the electron g−2 anomaly, since the electron g−2 can be adjusted by choosing the 1–3 offdiagonal elements of the slepton mass matrices. In this case, therefore, if the mass spectrum, which can reproduce the muon g−2, can satisfy the collider bounds, the electron g−2 can be also reproduced without contradicting the experimental bounds in principle. Since the main contribution to the electron g−2 emerges from the bino–bino diagram, a heavier Higgsino mass μ and lighter sleptons and bino can reproduce the electron g−2 by smaller flavor violation. Fig. 2 shows the plots created by randomly generated δLR13,31 and δLL,RR13 for a fixed mass spectrum which can satisfy the muon g−2, and one finds that the central value of the electron g−2 can be obtained satisfying the experimental bound of τ→eγ.Since we need to satisfy the g−2 of muon using the sparticles allowed by the LHC constraint, let us discuss the LHC constraints on the noncolored sparticle masses [13,14] and the parameter space which is still allowed. We assume that the lightest neutralino is the lightest SUSY particle.1.We first discuss the slepton masses. The selectron and smuon masses do not have any constraint if the mass difference between the lightest neutralino and the selectron mass is ≤60 GeV. Also, if the lightest neutralino mass is above 300 GeV then there is no constraint on the selectron and smuon masses. The stau masses do not have any constraint from the LHC yet.2.It is also interesting to note that if the lightest neutralino mass is above 300 GeV then there is no constraint on the nexttolightest neutralino and chargino masses provided the selectron and smuon masses are heavier than these particle masses.3.If the lightest and nexttolightest neutralinos and chargino are primary Higgsino then the maximum constraint on this mass scale is 150–200 GeV.4.In addition to the above straightforward scenarios, many other scenarios could be available by investigating the branching ratios (BR) of the heavy neutralinos and charginos into various final states, e.g., W,Z,h,τ,e,μ,ν plus χ˜01. The constraints shown by CMS are ATLAS mostly use BR=1 for each of these final states.Following these prescriptions, we are showing 4 points which are not ruled out by the LHC data in Table 1. The lightest neutralino are chosen to be Higgsino, wino–Higgsino, bino–Higgsino or bino types to show different possibilities.•Scenario 1:All the heavier neutralinos or charginos dominantly decay via W, Z, τ+χ˜10. LHC constraints are satisfied since the lightest neutralino mass, mχ˜10, is 300 GeV. There exists no constraint on the selectron, smuon masses since mχ˜10 is 300 GeV and all the LHC constraints are satisfied.•Scenario 2:The lightest neutralino and the lightest chargino are within 10 GeV and they are around 200 GeV. The lightest chargino and the neutralino masses are required to be above 160 GeV in such a degenerate case [25,26]. Two other neutralinos and the heaviest charginos are within 120 GeV of the lightest neutralino and they decay dominantly via W, Z,h+χ˜10. In such final states the mass difference between the heavier neutralino/chargino and the lightest neutralino needs to be at least 200 GeV for mχ˜10∼200 GeV in order to have any constraint from the LHC. The lightest neutralino is wino–Higgsino type. The heaviest neutralino is more than 95% bino, which would make it hard to be produced at the LHC. There is no constraint on the selectron and smuon masses above 500 GeV for mχ˜10∼200 GeV. All the LHC constraints are satisfied for this scenario.•Scenario 3:Three lighter neutralinos and the lightest chargino are within 60 GeV and the lightest neutralino is bino–Higgsino type. These heavier particles decay into the lightest neutralino via W⁎,Z⁎ and there exists no constraint on these particles since for a lightest neutralino around 200 GeV, the mass difference is needed to be at least 200 GeV for W,Z final sates to have constraints from the LHC. The heaviest neutralino and the chargino are winotype with mass around 800 GeV. The heaviest neutralino decays into νLν˜L (36%), ττ˜ (12%), lLl˜L (24%) where l=e,μ and ν˜L decays mostly into W⁎+χ˜10+τ and χ˜10+ν and the heavy chargino decays into lLνL+χ˜10 (35%), τν+χ˜10 (10%) and others. One can find that the lllν final state crosssection is around 0.02 fb which is quite small compared to the crosssection (∼ 1 fb) that can be constrained for this final state for a 800 GeV χ˜40,χ˜2± with ml˜=0.05mχ˜10+0.95mχ˜2± and a 200 GeV χ20 [27]. The χ˜40,χ˜2± masses are too large to be constrained by any other final state. The selectron and smuon masses do not have constraint for masses up to 330 GeV for mχ˜10∼200 GeV. The scenario 3, therefore, cannot be constrained by the LHC results so far.•Scenario 4:The selectron and smuon masses are heavier than the two lighter neutralino and the lightest chargino masses. The final states of the χ˜20/χ˜1± will be dominated by W, Z and there exists no constraint on these masses up to 300 GeV for mχ˜10∼150 GeV. The heavier neutralinos and charginos are primarily Higgsinos in such a scenario and they decay via W,Z,h. Their masses are heavier than 600 GeV which leads to no constraint from the LHC. There is no constraint on the selectron and smuon masses above 500 GeV for mχ˜10∼200 GeV. The LHC constraints do not apply to this scenarioLet us now present the case of LR mixing is 0. In the case where LR flavor violation is absent, the mass spectrum is constrained to satisfy the τ→eγ bound. Fig. 3 shows the contour plot of Br(τ→eγ) fixing bino mass M1 and slepton masses, mL and mR. As we have mentioned, M1/M2 has to be negative to cancel the amplitudes of τ→eγ. Since the chargino contribution dominates the muon g−2 in the region where τ→eγ is suppressed, the smaller M2 and μ can generate larger muon g−2. On the other hand, the magnitude of the electron g−2 is dominated by the bino–bino diagram, and it can be larger for larger μ and larger flavor violation. When we choose the flavor violation to reach the central value of electron g−2 and select the mass spectrum to cancel τ→eγ amplitudes, we find that it is not easy to reach the central value of muon g−2 (as far as the smuon mass is the same as the average of selectron and stau masses), but one can find a solution to achieve the 1σ range of the muon g−2. From the cancellation of τL→eRγ amplitude between Eqs. (10) and (11), we need M1μ∼mℓ˜2 naively. If Aterm coefficient is large, the Higgsino mass can be lowered to cancel the τL→eRγ amplitude and enlarge muon g−2. (A large Aterm coefficient can lead the existence of charge breaking global minimum, but it can be avoided if the CP odd Higgs mass is adequately heavy (roughly more than Aτ/3) in this case.) In the example given in Fig. 3, we obtain Δaμ=2.3×10−9 at M2=800 GeV and μ=230 GeV, and the electron g−2 can reach the central value, Δae=−8.7×10−13, satisfying the τ→eγ bound.We show one typical benchmark point for the chargino, neutralino, and slepton spectrum which can satisfy the τ→eγ bound and reach the 1σ range of g−2:(13)M1=−220GeV,M2=800GeV,μ=230GeV,mL=mR=300GeV,tanβ=40, and Aτ=−3 TeV, in the convention that the LR component of slepton mass matrix is (Aτ−μtanβ)mτ. The mass spectrum is(14)mχ˜1,2,3,40=197,226,258,809GeV,mχ˜1,2+=227,809GeV,(15)me˜1,2=289,317GeV,mμ˜1,2=301,305GeV,mτ˜1,2=250,349GeV,(16)mν˜=265,293,319GeV. This scenario is similar to scenario 3 described above and therefore is not constrained by any LHC data.We note that the electron mass is modified by finite loop correction at O(10)% when the central value of the electron g−2 is reproduced.Before concluding this section, we note on the bounds of electron EDM [28]:(17)de<1.1×10−29e⋅cm. Both g−2 and EDM of electron is generated by the e¯LσμνFμνeR operator, and thus, if the SUSY contribution saturates the deviation from the SM prediction of g−2, we obtain(18)deSUSY=e2meaeSUSYtanϕ≃2×10−11Δaetanϕe⋅cm, where ϕ is a phase of the amplitude. We usually suppose that the gaugino, Higgsino mass and the (diagonal elements of) scalar trilinear coupling matrix are real to satisfy the electron and neutron EDM bounds. The phases of the offdiagonal elements also need to be almost real, and the bound of the phases is(19)argδLL,RR,LR,RL13<O(10−6). If the phases are aligned as(20)argδLL13=argδRR13=argδLR13=argδRL13, the phases are unphysical since they can be removed by field redefinition, and the electron EDM vanishes.4ConclusionIn conclusion, we have investigated the recently reported more than 2σ electron g−2 anomaly. The discrepancy between the SM prediction and the experimental measurement for the electron case has opposite sign compared to the muon g−2 case where the anomaly is more than 3σ. Further, the ratio of the measured muon and electron anomalies is about 10 times less than that predicted by the lepton mass scaling mμ2/me2. One requires flavor violation in the leptonic sector to induce such a breakdown of the scaling.In this work we showed that it is possible to explain electron and muon g−2 anomalies simultaneously in the MSSM using the parameter space which is allowed by the LHC data. The satisfaction of the g−2 anomalies require noncolored particles, such as selectrons, smuons, staus, neutralinos and charginos, and the LHC constraints on these particles allow sufficient parameter space for masses between 100 GeV–1 TeV. 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