Supersymmetric Alignment Models for (g − 2)µ

Yuichiro Nakai1, Matthew Reece2, and Motoo Suzuki1

1Tsung-Dao Lee Institute and School of Physics and Astronomy,Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

2Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA, 02138, U.S.A

Hierarchical masses of quarks and leptons are addressed by imposing horizontal symmetries. Insupersymmetric Standard Models, the same symmetries play a role in suppressing flavor violatingprocesses induced by supersymmetric particles. Combining the idea of spontaneous CP violation tocontrol contributions to electric dipole moments (EDMs), the mass scale of supersymmetric particlescan be lowered. We present supersymmetric models with U(1) horizontal symmetries and discussCP and flavor constraints. Models with two U(1) symmetries are found to give a viable solution tothe muon g − 2 anomaly. Interestingly, the parameter space to explain the anomaly will be probedby future electron EDM experiments.

I. INTRODUCTION

Precise comparison between theory and experiment forthe anomalous magnetic moment of the electron led usto the foundation of quantum field theory to describethe nature of elementary particles. In 1947, Kusch andFoley discovered a tiny deviation from g = 2 in the gy-romagnetic ratio of the electron [1], which was shown tooriginate from the one-loop effect of QED with the propertreatment of divergences by Schwinger [2]. Likewise, thereported discrepancy in the anomalous magnetic momentof the muon (g−2)µ may lead us to a deeper understand-ing of nature. Combining Brookhaven and Fermilab datato measure (g−2)µ, the discrepancy between theory andexperiment is ∆aobs

µ = aexpµ −atheory

µ = (25.1±5.9)×10−10

with aµ ≡ (g−2)/2 [3–5] (for a review of the status of theStandard Model (SM) calculation, see ref. [6]1). Since thediscrepancy is of the order of the SM electroweak contri-bution, aµ(EW) = (15.4±0.1)×10−10, it may indicate anew contribution to the muon g− 2 from physics beyondthe SM which lies at a scale . O(1) TeV.

Low-energy supersymmetry (SUSY) has been an at-tractive candidate of physics beyond the SM to explainthe muon g−2 anomaly as well as to address the natural-ness problem of the electroweak scale. Loops of sleptonsand electroweakinos can generate a new contribution tothe muon g − 2 with the correct size [13], an idea thatwas explored after the first Brookhaven result [14–18] andhas continued to draw recent attention (e.g., [19–23]).However, the required light sleptons and electroweakinosgenerally lead to dangerously large lepton flavor viola-tion (LFV) in processes such as µ → eγ and µ → econversion. To make matters worse, an arbitrary CPviolation (CPV) in SUSY breaking parameters inducesa large electric dipole moment (EDM) of the electron,

1 For an alternative take, see the recent lattice calculation ofref. [7]. Intriguingly, if this calculation is correct, the discrep-ancy with the data-driven calculation of the hadronic vacuumpolarization could point to different tensions with the StandardModel [8–12].

which is severely constrained by the ACME measurement[24] (whose implications for SUSY have been exploredin refs. [25, 26]). Even with minimal flavor violation,new physics explaining the muon g − 2 anomaly with anO(1) CPV phase would predict the electron EDM to befive orders of magnitude larger than observations allow.Elaborate mechanisms of SUSY breaking such as gaugemediation (see refs. [27, 28] for reviews) and gaugino me-diation [29, 30] are able to address the issue.

In this paper, we assume the simplest and the mostnaive version of SUSY breaking without elaborate mech-anisms but introduce U(1) horizontal symmetries, whichcan explain the hierarchical masses of quarks and lep-tons [31]. A virtue of such supersymmetric SMs, whichwe call SUSY alignment models, is that horizontal sym-metries also control the structure of sfermion masses andsuppress flavor violating processes [32–35]. Generic CPviolation is still dangerous, but the idea of spontaneousCP violation [36] makes it possible to suppress contri-butions to EDMs and at the same time accommodatethe correct Cabibbo-Kobayashi-Maskawa (CKM) phasein the quark sector. Then, the application of U(1) hori-zontal symmetries and spontaneous CP violation to thelepton sector can suppress SUSY contributions to LFVand CPV processes [37, 38], which opens up a possibilityto explain the muon g − 2 anomaly without contradict-ing CP and flavor constraints. Our starting point is amodel with a single U(1) horizontal symmetry. We inves-tigate contributions to LFV and CPV processes and findthat the model can achieve supersymmetric particles ataround 10 TeV for a large tanβ. Then, we consider mod-els with two U(1) symmetries. These models can furtherrelax CP and flavor constraints and provide a viable so-lution to the muon g − 2 anomaly. We discuss a relationbetween the SUSY contribution to the muon g − 2 andthat of the electron EDM and find that the model pa-rameter space to explain the muon g−2 anomaly will beprobed by near-future electron EDM experiments.

The rest of the paper is organized as follows. In sec-tion II, we review the SM flavor structure which will beexplained by U(1) horizontal symmetries. Section IIIpresents a SUSY alignment model with a single U(1).

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We introduce three different flavon fields which break theU(1) symmetry and CP spontaneously and discuss LFVand CPV constraints on the model. In section IV, weconsider models with two U(1) symmetries to pursue thepossibility of explaining the muon g − 2 anomaly. LFVand CPV constraints on these models are investigatedin section V. Section VI is devoted to conclusions anddiscussions. Some details on the presented models andanalyses are described in appendices.

II. THE SM FLAVOR STRUCTURE

The quark mass ratios and mixings in the CKM matrixcan be expressed in terms of the Wolfenstein parameteri-zation [39]. The mass ratios are parameterized by powersof λ ∼ 0.2 as

mc/mt ∼ λ3 , mu/mt ∼ λ6 − λ7,

mb/mt ∼ λ2 , ms/mb ∼ λ2 , md/mb ∼ λ4 .(1)

Here, the subscripts of u, c, t, d, s, b denote up, charm,top, down, strange and bottom quarks, respectively. Theorders of magnitude of different CKM entries are deter-mined by λ as well,

|V CKM12 | ∼ λ , |V CKM

23 | ∼ λ2 , |V CKM13 | ∼ λ3 , (2)

where |V CKMij | is the absolute value of the (i, j) compo-

nent of the CKM matrix. The CPV effect in the quarkmixings is parameterized by a phase δCKM ' 1.2 in thestandard parameterization [40].

Similarly, the mass ratios and mixing angles of the lep-ton sector can be expressed in powers of λ. The chargedlepton mass ratios are written as

mµ/mτ ∼ λ2 , me/mτ ∼ λ5 , mτ/mt ∼ λ3 , (3)

where subscripts e, µ, τ denote electron, muon and tauleptons, respectively. In the present paper, we focus onthe normal ordering of neutrino masses with ν3 (ν1) beingthe heaviest (lightest) neutrino and for simplicity assumethe neutrino mass ratios,

mν1/mν3 . λ , mν2/mν3 ∼ 1 . (4)

Note that a very light or even massless ν1 is consistentwith the neutrino mass-squared difference measurementsand the cosmological constraint on the sum of neutrinomasses [41]. The absolute value of the (i, j) component ofthe Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrixcan be expressed in terms of λ as

|V PMNS12 | ∼ λ , |V PMNS

13 | ∼ λ , |V PMNS23 | ∼ 1 . (5)

A CPV effect in the lepton mixings parameterized by aphase δCP can be measured by the neutrino oscillation,and δCP & π is favored in the T2K experiment [42]. TheCP-preserving scenario, i.e., δCP = π, is still consistent

within the 3σ confidence level (and the NOνA experi-ment has results in mild tension with those of T2K [43]),but future measurements at the Hyper-Kamiokande [44]and DUNE [45] experiments will provide a more accuratemeasurement of the lepton sector CPV.

III. THE MODEL WITH A SINGLE U(1)

We here assume our supersymmetric SM respects a sin-gle U(1)H horizontal symmetry under which quark andlepton supermultiplets have nontrivial charges. Let usfirst describe a simple example with the U(1)H symme-try to motivate a model with three flavon fields. In thesimple model, the fermion mass ratios and mixing ma-trices are explained by a flavon chiral superfield S1 withcharge −1 under the U(1)H symmetry. The flavon isregarded as a spurious field whose vacuum expectationvalue (VEV) is given by 〈S1〉 = λΛ. Here, Λ denotessome UV mass scale, and we use Λ = 1 units in thefollowing discussion. The U(1)H charges of quarks andleptons are non-negative integers, and with appropriatecharge assignments, the fermion mass ratios and mixingmatrices presented in the previous section are realized.In particular, the CKM and PMNS mixing matrices areapproximately given by

|V CKMij | ∼ λ|H(Qi)−H(Qj)| ,

|V PMNSij | ∼ λ|H(Li)−H(Lj)| ,

(6)

where H(X) denotes the horizontal U(1)H charge for achiral superfield X, and Qi and Li (i = 1, 2, 3) are theleft-handed doublet quarks and leptons of the i-th gen-eration, respectively. The charges H(Q1), H(Q2), H(Q3)are chosen to realize the CKM entries (2), and this chargeassignment also determines the scaling of the soft scalarmass-squared matrix in terms of λ,

M2Q∼ m2

q

1 λ λ3

λ 1 λ2

λ3 λ2 1

, (7)

where mq denotes a typical mass scale of squarks. Thesizable off-diagonal components in the above matrix leadto dangerously large flavor changing neutral currents(FCNCs). In particular, the (1, 2) element is less sup-pressed compared to the other off-diagonal elements, andthe stringent neutral Kaon mixing constraint requiresmq � 10 TeV. A similar argument can be applied to thelepton sector. A reasonable charge assignment for theleft-handed doublet leptons to explain the PMNS mixingmatrix leads to sizable off-diagonal elements in the left-handed slepton soft mass-squared matrix, and then theconstraint from the Br(µ→ e+γ) measurement requiresm` � 10 TeV for a large tanβ where m` is a typical massscale of sleptons. However, such stringent constraintson the soft mass scales are not a generic consequence ofSUSY alignment models. We will next see that weakerbounds can be obtained by extending the model.

3

A. Three flavons

Let us now introduce three flavon fields S2, S2, S3 withhorizontal charges −2, 2,−3, respectively. We supposethese flavons obtain VEVs, 〈S2〉 ∼ λ2, 〈S2〉 ∼ λ2, 〈S3〉 ∼λ3. As in the case of the simple model above, appropriateU(1)H charge assignments will lead to the correct fermionmass ratios and mixing matrices. However, in the presentmodel, the (1, 2) element of M2

Qis given by, e.g.,

(M2Q

)12 ∼ 〈S2〉〈S3〉m2 ∼ λ5m2, (8)

which is significantly suppressed compared to that of thesimple model. The same idea is also applied to the leptonsector.

We consider spontaneous CPV to control complexphases in SUSY breaking parameters but still generate(at least) the CKM phase. A CP invariant superpoten-tial of the flavon fields is given by

WS = ξX(S2S2 − λ4) + Y (c1S43 + c2S

23S

32 + c3S

62) , (9)

where ξ and c1,2,3 are real coefficients and X,Y are SMgauge singlet chiral superfields but Y has the horizon-tal charge +12. The above superpotential respects theU(1)R symmetry under which X,Y have the charge +2and the three flavons are neutral. The F -term conditionfor X leads to the VEVs of 〈S2〉 = 〈S2〉 = λ2, while thecondition for Y gives

c1〈S3〉4 + c2〈S3〉2〈S2〉3 + c3〈S2〉6 = 0 , (10)

which is solved as

〈S3〉2 =−c2 ±

√c22 − 4c1c3

2c1〈S2〉3 . (11)

We can see that the VEV 〈S3〉 obtains a complex phasewhen

c22 − 4c1c3 < 0 , (12)

and the CP symmetry is broken spontaneously.The superpotential for quarks and leptons in the gauge

eigenbasis is given by

WYukawa = Yu ijQiujHu + Yd ijQidjHd + Ye ijLiejHd

+ Yν ij(HuLi)(HuLj)

MN, (13)

where ui, di, ei (i = 1, 2, 3) and Hu,d denote theright-handed up-type quarks, the right-handed down-type quarks, the right-handed charged leptons and thetwo doublet Higgs fields, respectively. Yu,d,e,ν are theirYukawa couplings and MN is some UV mass scale. Theorigin of the Majorana neutrino mass terms in the secondline through the seesaw mechanism [46–48] is discussed in

Appendix B. To realize the fermion mass ratios and mix-ing matrices, we take the following charge assignments for(s)quarks and (s)leptons under the horizontal symmetry,

Q1

(3)Q2

(2)Q3

(0)u1

(4)u2

(1)u3

(0)d1

(−7)d2

(−4)d3

(0)

L1

(2)L2

(1)L3

(1)e1

(3)e2

(−3)e3

(−1)

(14)

We also assume H(Hu) = H(Hd) = 0. The above chargeassignments work well for a large tanβ ∼ 50 because,e.g., the bottom and tau Yukawa couplings are not sup-pressed by the horizontal symmetry.

The Yukawa matrices are given by couplings with thethree flavons, and their orders of magnitude can be pa-rameterized by λ,

Yu ∼

λ7 λ4 λ3

λ6 λ3 λ2

λ4 λ5 1

, Yd ∼

λ4 λ7 λ3

λ11 λ2 λ2

λ13 λ4 1

,

Ye ∼

λ5 λ7 λ5

λ4 λ2 1λ4 λ2 1

, Yν ∼

λ4 λ3 λ3

λ3 λ2 λ2

λ3 λ2 λ2

,

(15)

where an O(1) coefficient and a possible phase of eachmatrix element has been omitted for notational simplic-ity. The CKM matrix is written in terms of the diago-nalization matrices of the left-handed up-type quarks Vuand the left-handed down-type quarks Vd, i.e., V CKM =V †uVd. The up-type quark Yukawa matrix Yu provides adiagonalization matrix close to the CKM matrix V †u ∼VCKM, while Yd does not provide a sizable mixing be-tween Q1 and Q2 to explain |V CKM

12 | ∼ λ because the cor-responding off-diagonal entries of Yd are very small. Thesmall (1, 2) mixing of Vd makes it possible to suppressSUSY contributions to, e.g., the neutral Kaon mixingcompared to the simple model. A similar discussion alsoapplies to the lepton sector, where the sizable |V PMNS

12 | ismainly provided via the neutrino Yukawa matrix Yν . CPphases are introduced through couplings with S3. Forthe CKM phase, for example, the (1, 3) entry of Yd hasa phase via the coupling with S3 and the phase can pro-vide δCKM ∼ 1 in a similar manner to that discussed inref. [36].

The structure of the soft scalar mass-squared matricesare constrained as

M2Q∼ m2

q

1 λ5 λ3

λ5 1 λ2

λ3 λ2 1

, M2˜u ∼ m2q

1 λ3 λ4

λ3 1 λ5

λ4 λ5 1

,

M2˜d ∼ m2q

1 λ3 λ7

λ3 1 λ4

λ7 λ4 1

,

M2L∼ m2

`

1 λ5 λ5

λ5 1 1λ5 1 1

, M2˜e ∼ m2`

1 λ6 λ4

λ6 1 λ2

λ4 λ2 1

.

(16)

4

The trilinear soft SUSY breaking terms have a similarstructure to the Yukawa matrices due to the horizontalsymmetry, but they are not exactly proportional to theYukawa matrices in general due to their O(1) undeter-mined coefficients.

So far, we have implicitly assumed the kinetic termsof quarks and leptons are canonical, but in general, theirKahler potential can be

Kmatter = X†i ZijXj , (17)

where Xi = Qi, ui, di, Li, νi or ei, and Zij is a Hermitianmatrix. The off-diagonal elements of Zij are suppressedby some powers of λ, following their horizontal charges.To obtain the canonical kinetic terms, we diagonalize Zijby rotating the field as Xi = V Xij X

′j with a unitary ma-

trix V X and further redefine X ′j to remove the remainingO(1) numbers in the diagonal parts. This process canchange the orders of magnitude of some entries of theYukawa and soft mass-squared matrices in the basis ofthe canonically normalized kinetic terms. (Because wedid not take all U(1)H charges to have the same sign,the argument of Appendix B of ref. [34] does not apply.)We take account of this effect in our numerical analyses.

The U(1)H horizontal symmetry is anomalous with thecharge assignments presented above. The anomalies arisefrom U(1)2

Y −U(1)H , U(1)2H −U(1)Y , U(1)3

H , U(1)H −(Gravity)2, U(1)H−SU(3)2

C and U(1)H−SU(2)2L. While

the anomaly cancellation may be realized by the 4dGreen-Schwarz mechanism [49, 50], instead the anoma-lies could be canceled by adding new fields charged underthe horizontal symmetry. Another possibility is to find ananomaly-free discrete group ZN ⊂ U(1)H . If N is largeenough, like N ≥ 20, the powers of λ for the Yukawa andsoft mass-squared matrices are not affected.

B. R-symmetry and the µ problem

Throughout this paper, we follow [38] in assuming aZ4 R-symmetry [51], under which the SM matter fieldshave charge 1 and the Higgs fields, the flavons, and theSUSY-breaking spurions have charge 0. Lagrange mul-tiplier fields like X and Y in (9) carry charge 2. Thissymmetry has the effect of forbidding a superpotentialµ-term, but allowing the Giudice-Masiero mechanism togenerate effective µ and bµ terms from Kahler potentialterms [52]. We assume that the SUSY-breaking spurionsdo not violate flavor or CP symmetries. As a result, theCP phase of the µ term arises only from higher order,flavon-suppressed terms. For example, in the model dis-cussed above, one expects to generate a µ term of order

µ ∼ m3/2

(1 +O(S3

2S23))∼ m3/2

(1 + iO(λ12)

). (18)

Here S32S

23 is the leading flavor-invariant but CP-

violating term that can be constructed from the flavonfields. This model predicts that EDM contributions sen-sitive to argµ are extremely suppressed, by a factorλ12 ≈ 4× 10−9 relative to naive expectations.

C. CP and flavor bounds

Let us now discuss CP and flavor constraints on themodel presented in the previous subsection. The mostrelevant observables are summarized in Tab. I. We esti-mate SUSY contributions to these observables by usingthe mass insertion approximation. As a demonstration,we take mq = M3 = 20 TeV, m` = M1,2 = µ = 10 TeVand tanβ = 50 where M1,2,3 are three gaugino masses.The typical mass scales of the trilinear soft SUSY break-ing terms are taken as mq, m` for squarks and sleptons,respectively. We have checked that the estimation is con-sistent with that of the public code, susy flavor v2.5 [53–55].

The CP-violating parameter εK measured from theneutral Kaon oscillation is generally sensitive to super-symmetric particles whose masses are much higher thanthe TeV scale (see, e.g., ref. [56]). The current experi-mental value is |εK | = 2.228(11) × 10−3 [57]. The theo-retical uncertainty of the SM prediction is, however, morethan one order of magnitude larger than the experimen-tal value due to the large uncertainty of |Vcb| [58]. In themodel with three flavons, the SUSY contribution to εKis estimated as [59]

|εSUSYK | ' 102

(20 TeV

mq

)2

Im[(δd12)LL(δd21)RR] . (19)

Here, (δXij )HH is defined through the soft mass-squaredmatrix for the scalar partner of a SM fermion X in theYukawa diagonal basis, (M2

X)ij ≡ m2(δij + (δXij )HH),

where H denotes the helicity of the fermion X. To obtainthe soft mass-squared matrices or the trilinear soft SUSYbreaking parameters in the Yukawa diagonal basis, we usea method similar to the one discussed in ref. [38]. We in-troduce a random parameter for each entry of the flavormatrices and find parameter sets which satisfy criteria tofit with the observed fermion masses, mixing angles andCP phases. Finding 1000 data sets to satisfy the criteria,we compute, e.g., averaged soft mass-squared matrices.See appendix A for more details. By using this method,we obtain (δd12)LL(δd21)RR ≈ λ6.7, for the averaged values,which lead to |εSUSY

K | ≈ 10−3 with an O(1) phase. Thiscontribution is comparable to the observed value.

For the D− D mixing, the current experimental valueof the mass difference is given by |∆MD| = 0.63+0.27

−0.29 ×10−14 GeV [57]. The uncertainty of the SM predictionis also expected to be large due to long distance effects(see, e.g., ref. [60] and references therein). The SUSYcontribution to ∆MD is given by [61, 62]

|∆MSUSYD | ' 10−12 GeV ×

(20 TeV

mq

)2

× Re [(δu12)LL(δu21)RR] ,

(20)

where we have found (δu12)LL(δu21)RR ≈ λ3.5 for the av-eraged values by using the method described above and

5

obtain |∆MSUSYD | ≈ 10−15 GeV comparable to the ob-

served value.The neutron EDM is also used to put a constraint on

flavor mixings and CP phases in the (s)quark sector. Thecurrent limit is given by |dn| < 1.8×10−26 e cm [63]. TheSUSY contribution to the neutron EDM is [64]

|dSUSYn | ' 10−23 e cm×

(20 TeV

mq

)3(M3

20 TeV

)× Im

[(Au11)LRmq

],

(21)

where (Auij)LR denotes the (i, j) entry of the trilinear softSUSY breaking term of the up-type squarks. Using theaveraged value of (Au11)LR ≈ λ6.2 with an O(1) phase, wefind |dSUSY

n | ≈ 10−28 e cm.Among LFV processes, Br(µ → e + γ) gives the most

stringent bound on the SUSY parameter space. The cur-rent upper bound on this process is Br(µ → e + γ) <4.2×10−13 [65]. The SUSY contribution to Br(µ→ e+γ)is estimated in the mass insertion approximation as [66]

Br(µ→ e+ γ) ' 5× 10−9

(tanβ

50

)2(10 TeV

m`

)4

×(µM1

m2`

)2 ∣∣(δ`23)LL(δ`31)RR∣∣2 .

(22)

Using the averaged value |(δ`23)LL(δ`31)RR| ≈ λ3.3, we ob-tain Br(µ → e + γ) ≈ 10−13, comparable to the currentexperimental bound.

The electron EDM is also an important probe for flavormixings and CP phases in the (s)lepton sector. The cur-rent upper limit is given by |de| < 1.1× 10−29 e cm [24].The SUSY contribution to the electron EDM is

|dSUSYe | ' 5× 10−25 e cm×

(tanβ

50

)(10 TeV

m`

)2

×(µM1

m2`

)Im[(δ`13)LL(δ`31)RR] .

(23)

(There are additional contributions that are independentof flavor violation but proportional to arg µ, but theseare more suppressed because the phase of µ is very smallin this model, as discussed above in section III B.) Us-ing the averaged value |(δ`13)LL(δ`31)RR| ≈ λ6.8 with anO(1) phase, we obtain |dSUSY

e | ≈ 5 × 10−30 e cm, whichis comparable to the current limit.

IV. MODELS WITH TWO U(1) SYMMETRIES

The model presented in the previous section can ac-commodate sleptons whose typical mass scale is requiredto be around 10 TeV by CP and flavor constraints. Onthe other hand, the muon g−2 anomaly can be addressedby light sleptons and electroweakinos whose masses are

less than about 1 TeV. Here, we explore SUSY modelswith two U(1) symmetries, U(1)H1

× U(1)H2, to pursue

the possibility of explaining the muon g−2 anomaly with-out violating CP and flavor constraints. The symmetriesare spontaneously broken by flavon chiral superfields, S1

and S2, which are singlets under the SM gauge symmetry.Their charges under U(1)H1 × U(1)H2 are

S1

(−1, 0)

S2

(0,−1)(24)

and their VEVs are assumed to be 〈S1〉 ∼ λ and 〈S2〉 ∼λ2.

Quarks and leptons carry charges (H1, H2) under theU(1) symmetries. Defining H = H1 + 2H2, the SMfermion mass ratios in Eq. (1) and Eq. (3) impose thefollowing constraints on charge assignments for (s)quarksand (s)leptons under the horizontal symmetries,

H(Q3) +H(u3) = 0, H(Q2) +H(u2) = 3,

H(Q1) +H(u1) = 7,

H(Q3) +H(d3) = 0, H(Q2) +H(d2) = 2,

H(Q1) +H(d1) = 4,

H(L3) +H(e3) = 0, H(L2) +H(e2) = 2,

H(L1) +H(e1) = 5.

(25)

We have assumed H(Hu) = H(Hd) = 0, and the con-straints work well for a large tanβ ∼ 50. Using Eq. (6),the constraints from the CKM and PMNS mixing matri-ces in Eq. (2) and Eq. (5) are expressed as

H(Q1)−H(Q2) = 1, H(Q1)−H(Q3) = 3,

H(Q2)−H(Q3) = 2,

H(L1)−H(L2) = H(L1)−H(L3) = 1,

H(L2)−H(L3) = 0.

(26)

Many models can satisfy the conditions of Eq. (25) andEq. (26). Here, we present a working example of possiblecharge assignments for (s)quarks and (s)leptons,

Q1

(3, 0)

Q2

(0, 1)

Q3

(0, 0)

u1

(−2, 3)

u2

(1, 0)

u3

(0, 0)

d1

(−3, 2)

d2

(2,−1)

d3

(0, 0)

L1

(5, 0)

L2

(0, 2)

L3

(0, 2)

e1

(−4, 2)

e2

(2,−2)

e3

(0,−2)

(27)

As in the case of the model with a single U(1)H , theYukawa matrices are given by couplings with the flavonsS1, S2 and their orders of magnitude can be parameter-

6

Observable Experimental bound Model with S3 Model with S4

|εK | 2.228(11)× 10−3 [57] ∼ 10−3 ∼ 10−7

|∆MD| 0.63+0.27−0.29 × 10−14 GeV [57] ∼ 5× 10−15 GeV ∼ 5× 10−15 GeV

nEDM 6 10−26 e cm [63] ∼ 10−28 e cm ∼ 10−28 e cm

Br(µ→ e+ γ) 6 4.2× 10−13 [65] ∼ 10−16 ∼ 10−16

eEDM 6 1.1× 10−29 e cm [24] ∼ 5× 10−29 e cm ∼ 10−30 e cm

TABLE I. CP and flavor observables and their current experimental bounds. The estimation of the SUSY contribution to eachobservable in the models with U(1)H1 × U(1)H2 is also shown. We take mq = M3 = 5 TeV, m` = M1,2 = µ = 500 GeV andtanβ = 50. The typical mass scales of the trilinear soft SUSY breaking terms are taken as mq, m` for squarks and sleptons,respectively.

ized by λ,

Yu ∼

λ7 λ4 λ3

0 λ3 λ2

0 λ 1

, Yd ∼

λ4 0 λ3

0 λ2 λ2

0 0 1

,

Ye ∼

λ5 0 0

0 λ2 1

0 λ2 1

, Yν ∼

λ10 λ9 λ9

λ9 λ8 λ8

λ9 λ8 λ8

.

(28)

One important difference from the case of a single U(1)His that some entries are zero because of the holomorphicnature of superpotential terms [34, 67]. For example, ifH1(Qi)+H1(uj) < 0 or H2(Qi)+H2(uj) < 0, then (Yu)ijvanishes. In the previous model with a single U(1)H ,the flavon field S2 with a positive charge was introduced,while the current model does not have such a flavon field.

Spontaneous CPV is not realized with only the twoflavons S1, S2 because phases of their VEVs are rotatedaway by the U(1)H1

× U(1)H2. Then, we introduce an

additional singlet chiral superfield SN where N ≥ 3 is apositive integer. Its horizontal charges are (−N, 0) andthe VEV, 〈SN 〉 ∼ λN , is complex. Such a VEV is easilyobtained by considering superpotential terms,

WS = Z(aS2N + bSNS

N1 + cS2N

1 ) , (29)

where a singlet chiral superfield Z has horizontal charges(2N, 0). The F -term conditions FS1 = FSN

= 0 aresolved by 〈Z〉 = 0, and FZ = 0 leads to

a〈SN 〉2 + b〈SN 〉〈S1〉N + c〈S1〉2N = 0 . (30)

This equation is solved as

〈SN 〉〈S1〉N

=−b±

√b2 − 4ac

2a. (31)

We take b2 − 4ac < 0 to get a complex 〈SN 〉. Throughsuperpotential couplings with SN , CP phases are intro-duced to the Yukawa matrices. For N = 3, for exam-ple, (Yd)13 receives a new contribution with |〈S3〉| ∼ λ3,which leads to an O(1) phase of the CKM matrix. ForN = 4, only the (Yu)12 obtains a phase, which also leadsto an O(1) CKM phase. For N ≥ 5, the quark Yukawa

matrices do not receive O(1) CKM phases,2 so we focuson the cases of N = 3, 4 in the following discussion. Forthe lepton sector, an O(1) phase is also introduced toYν . Notice that the flavor-invariant, CP-violating com-bination that can contribute to argµ (or other flavor-invariant terms originating in the Kahler potential) is

SN1 S†N , of order λ2N . Thus, the arg µ contribution to

EDMs can be significantly larger than that discussed insection III B.

With the charge assignments of Eq. (27), the structureof the soft scalar mass-squared matrices for squarks andsleptons is constrained as

M2Q∼ m2

1 λ5 λ3

λ5 1 λ2

λ3 λ2 1

, M2˜u ∼ m2

1 λ9 λ8

λ9 1 λ

λ8 λ 1

,

M2˜d ∼ m2

1 λ11 λ7

λ11 1 λ4

λ7 λ4 1

,

M2L∼ m2

1 λ9 λ9

λ9 1 1

λ9 1 1

, M2˜e ∼ m2

1 λ14 λ12

λ14 1 λ2

λ12 λ2 1

.

(32)

The trilinear soft SUSY breaking terms have the samestructure as the Yukawa matrices, up to O(1) undeter-mined coefficients.

As in the case of the model with a single U(1)H , wetake account of the effect of the non-canonically normal-ized kinetic terms for (s)quarks and (s)leptons in the fol-lowing analysis. The U(1)H1

× U(1)H2horizontal sym-

metries are also anomalous. The same comment can beapplied to the present model. We have not provideda concrete mechanism to generate VEVs 〈S1〉 ∼ λ and〈S2〉 ∼ λ2. In the Green-Schwarz mechanism, the Fayet-Iliopoulos (FI) term is generated for an anomalous U(1)via the gravitational anomaly [68, 69], which leads to a

2 Contributions from the non-canonical kinetic terms to the CKMphase are negligible.

7

nonzero flavon VEV [70]. This mechanism may be ableto be applied to the present model, which is left for afuture study.

.

V. FLAVOR CONSTRAINTS & (g − 2)µ

We now discuss CP and flavor constraints on the modelwith two U(1) horizontal symmetries presented in theprevious section and see if the model can successfully ad-dress the muon g− 2 anomaly. Here, for reference valuesof the model parameters, we take mq = M3 = 5 TeV,m` = M1,2 = µ = 500 GeV and tanβ = 50. The typi-cal mass scales of the trilinear soft SUSY breaking termsare also taken as mq, m` for squarks and sleptons, re-spectively. The observed Higgs mass can be explainedwith the stop trilinear soft SUSY breaking parameter fora squark mass around 5 TeV [77]. We have checked theconsistency of the estimates with those obtained by thepublic code susy flavor v2.5. The SUSY contributions tothe quark and lepton sector observables are summarizedin Tab. I and generally suppressed compared to those ofthe model with a single U(1) due to the further suppres-sion of the off-diagonal entries of the Yukawa and softmass-squared matrices.

Let us first discuss the quark sector observables. Hereagain, 1000 good trials to realize the observed fermionmasses, mixing angles and CP phases are found, and thesoft mass-squared matrices are averaged over the trialsto compute the observables. In the present model, themost dominant contribution to ∆MD is given by [61, 62]

|∆MSUSYD | ' 10−11 GeV ×

(5 TeV

mq

)2

× |Re [(δu12)LL(δu21)RR]| .(33)

Here, the averaged values of (δu12)LL(δu21)RR ≈ λ4.8 leadto |∆MSUSY

D | ≈ 5 × 10−15 GeV, comparable to the ex-perimental value. The SUSY contribution to εK givenby Eq. (19) is quite different between the models withS3 and S4. For the former case, by using the aver-aged values of (δd12)LL(δd21)RR ≈ λ8 with O(1) phases, weobtain |εSUSY

K | ≈ 10−3, which is comparable to the ob-served value. On the other hand, for the latter case, theSUSY contribution is quite suppressed as |εSUSY

K | ≈ 10−7

with Im[(δd12)LL(δd21)RR] ≈ λ14. This is because theflavon S4 has a larger U(1)H1 charge compared to thatof S3 and CP phases are provided for fewer entries ofthe mass matrices of, in particular, the down (s)quarksector. The SUSY contribution to the neutron EDMis presented in Eq. (21). Using the averaged value ofIm[(Au11)LR/mq] ≈ λ8, we obtain |dSUSY

n | ≈ 10−28 e cm,smaller than the current limit.

For the lepton sector observables, the dominant con-tribution to Br(µ→ e+ γ) is now estimated in the mass

insertion approximation as [66]

Br(µ→ e+ γ) ' 5× 10−4

(tanβ

50

)2

×(

500 GeV

m`

)4(µM1

m2`

)2

×(∣∣0.5(δ`21)LL

∣∣2 +∣∣(δ`23)RR(δ`31)LL

∣∣2) .(34)

Here, the averaged values are given by (δ`21)LL ≈ λ8.7,(δ`23)RR(δ`31)LL ≈ λ9.7, which lead to Br(µ → e + γ) ≈10−16. The contribution is much smaller than the exper-imental upper bound.

The SUSY contribution to the electron EDM fromthe off-diagonal entries of the SUSY breaking parame-ters is negligible, and the dominant contribution is ob-tained from the flavor diagonal entries with the phase ofµ. Then, the contribution to the electron EDM is relatedto that of the muon g − 2,

|dSUSYe | ∼ eme

mµ

aµ2mµ

|arg(µ)|

' 10−24

(aµ

2× 10−9

)|arg(µ)| e cm .

(35)

This relation shows that a tiny phase of µ is requiredto explain the muon g − 2 anomaly and to be consistentwith the current upper bound on the electron EDM, i.e.,|arg(µ)| . 10−5 for aµ ≈ 2× 10−9. The electron EDM isthen a powerful probe for SUSY models addressing themuon g − 2 anomaly. In the model with S3, we esti-mate arg(µ) ∼ λ6 and obtain |dSUSY

e | ' 5 × 10−29 e cmfor aµ ' 2 × 10−9, which requires an O(10)% fine-tuning to be consistent with the current electron EDMbound. The model with S4 gives |arg(µ)| ∼ λ8 leadingto |dSUSY

e | ' 10−30 e cm for aµ ' 10−9. Future electronEDM experiments (see, e.g., refs. [78–83]) will search forthe favored parameter space to explain the muon g − 2anomaly.

Figure 1 shows the parameter space where the muong−2 anomaly can be addressed by the SUSY contributionin the U(1)H1

×U(1)H2models with S3 and S4. Here, we

assume only the left-handed sleptons and electroweakinos(bino, wino and higgsinos) are light and the right-handedsleptons, all squarks, gluino and heavy Higgs bosons aredecoupled. To estimate the SUSY contribution to themuon g − 2, the public code susy flavor v2.5 [53–55] isused. The result is consistent with the one-loop calcu-lation presented in ref. [13]. We also take account ofsearches for slepton pair production at the LHC [71–74],as discussed in refs. [75, 76]. The black solid lines inthe left panel correspond to the current electron EDMconstraint |de| = 1.1× 10−29 e cm with different amountsof tuning in the nonzero CP phase contribution to theµ parameter. The black dashed lines in the right paneldenote the future reach of electron EDM measurements

8

FIG. 1. The muon g − 2 anomaly can be addressed by the SUSY contribution at the 1σ (2σ) level in the orange (yellow)shaded region. The left and right panels correspond to the U(1)H1 × U(1)H2 models with S3 and S4, respectively. In bothpanels, we assume only the left-handed sleptons and electroweakinos (bino, wino and higgsinos) are light and the right-handedsleptons, all squarks, the gluino, and the heavy Higgs bosons are decoupled. We take tanβ = 50 and µ = M2 = 2M1. Toestimate the SUSY contribution to the muon g−2, the public code susy flavor v2.5 [53–55] is used. The result is consistent withthe one-loop calculation presented in ref. [13]. The purple shaded region is excluded by searches for slepton pair production atthe Large Hadron Collider (LHC) [71–74], as discussed in refs. [75, 76]. The black solid lines in the left panel correspond tothe current electron EDM bound, |de| = 1.1× 10−29 e cm, by using µ = |µ|(1 + i κλ6) where λ = 0.2 and κ = 0.1, 0.2, 0.3 fromthe left to the right, respectively. The black dashed lines in the right panel correspond to the future reach of electron EDMmeasurements, |de| = 10−30 e cm, by using µ = |µ|(1 + i κ λ8) where λ = 0.2 and κ = 0.2, 0.4, 0.6, 1 from the left to the right,respectively.

|de| = 10−30 e cm. As we discussed above, the favored pa-rameter space to explain the muon g−2 anomaly withouttuning is all covered by the future measurements.

VI. CONCLUSIONS AND DISCUSSIONS

SUSY provides an attractive possibility to explain thereported muon g−2 anomaly, but SUSY contributions toLFV and CPV processes must be sufficiently suppressed.In this paper, we have considered U(1) horizontal sym-metries to address hierarchical masses of quarks and lep-tons. Such SUSY alignment models with spontaneousCP violation can also control the structure of sfermionmasses and suppress CP and flavor violating processes.The correct CKM phase in the quark sector is realizedat the same time. We started with a model with a sin-gle U(1) horizontal symmetry and investigated CP andflavor constraints. The model can viably achieve super-symmetric particles at around 10 TeV for a large tanβ.Then, we considered a model with two U(1) horizontalsymmetries. The model can further relax CP and flavorconstraints and realize sleptons and electroweakinos ata scale . O(1) TeV to provide a viable solution to themuon g − 2 anomaly. We found that the favored param-eter space to address the muon g − 2 anomaly will beextensively investigated by future electron EDM experi-ments.

The lepton sector CPV is hinted by the neutrino oscil-lation, although δCP = π is still consistent. Our modelsgenerically predict a sizable CP phase in the lepton sec-

tor, and hence its discovery would support the models.Unlike gauge mediation or gaugino mediation of SUSYbreaking, the selectron and smuon masses are generallynot degenerate in SUSY alignment models. If the LHCobserves the selectron and smuon with different masses,it may indicate the existence of horizontal symmetries.

ACKNOWLEDGMENTS

We would like to thank Daniel Aloni and Pouya Asadifor discussions. MR is supported in part by the DOEGrant DE-SC0013607 and the Alfred P. Sloan Founda-tion Grant No. G-2019-12504.

Appendix A: Good trials

To estimate SUSY contributions to CP and flavor ob-servables, we need to find the soft mass-squared matri-ces of squarks and sleptons in the Yukawa diagonal basiswhere Yu, Yd and Ye are diagonal. Following the proce-dure of ref. [38], we first introduce an O(1) random num-ber for each entry of the Yukawa and soft mass-squaredmatrices. Then, going to the Yukawa diagonal basis, ifthe generated Yukawa matrices satisfy the following crite-ria to realize the observed pattern of the quark and leptonmasses, the mixing angles and the CKM phase, we call ita good trial and compute the effective soft mass-squaredmatrices. In a good trial, the ratio of a quark mass mq

and the observed value mobsq is within a factor of λ = 0.2,

9

that is,

λ ≤ mq

mobsq

≤ 1

λ. (A1)

The absolute value of each entry of the CKM matrix iswithin the range, 0.8 0.11 0.002

0.11 0.8 0.02

0.004 0.02 0.8

≤ |VCKM| ≤

1 0.44 0.008

0.44 1 0.08

0.016 0.08 1

,

(A2)

and the absolute value of the CKM phase defined in thestandard notation satisfies

0.5 ≤ | sin(δCKM)| ≤ 1 . (A3)

The criteria for the lepton sector is the same as that ofref. [38]. We get 1000 good trials among which the ab-solute values of the real and imaginary parts of each softmass-squared matrix element are averaged respectively.Finally, we divide the matrix by the average of the ab-solute eigenvalues to make the diagonal entries close toone.

Appendix B: The right-handed neutrinos

One way to UV complete the Majorana neutrino massterms in Eq. (13) is to introduce three right-handed neu-

trinos Ni (i = 1, 2, 3) and consider the superpotential,

Wν = YLiYNj

LiHuNj − YNkYNl

MNNkNl , (B1)

where YLi, YNi

are given by couplings to flavons and MN

may be regarded as the order parameter for the spon-taneously broken U(1)B−L symmetry. Here, the sameindices i, j, k, l are summed over. Then, the F-term con-dition of Ni leads to

YNjNj =

YLiLiHu

2MN

. (B2)

Integrating out the heavy right-handed neutrinos Ni byusing this relation, the effective superpotential is ob-tained as

Wν,eff =(YLi

LiHu)(YLjLjHu)

4MN

. (B3)

Defining Yν ij/MN ≡ YLiYLj

/(4MN ), we reproduce theMajorana neutrino mass terms in Eq. (13). Note that thesuperpotential (B3) does not depend on YNi

and hencehorizontal charges of the right-handed neutrinos Ni arenot relevant in the discussion.

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