Physics Letters B 721 (2013) 118–122

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Physics Letters B

www.elsevier.com/locate/physletb

Flavor-nonuniversal dark gauge bosons and the muon g − 2

Christopher D. Carone

High Energy Theory Group, Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 January 2013Received in revised form 4 March 2013Accepted 7 March 2013Available online 15 March 2013Editor: M. Cvetic

The possibility of explaining the current value of the muon anomalous magnetic moment in models withan additional U(1) gauge symmetry that has kinetic mixing with hypercharge are increasingly constrainedby dark photon searches at electron accelerators. Here we present a scenario in which the couplingsof new, light gauge bosons to standard model leptons are naturally weak and flavor nonuniversal.A vector-like state that mixes with standard model leptons serves as a portal between the dark andstandard model sectors. The flavor symmetry of the model assures that the induced couplings of the newgauge sector to leptons of the first generation are very small and lepton-flavor-violating processes areadequately suppressed. The model provides a framework for constructing ultraviolet complete theories inwhich new, light gauge fields couple weakly and dominantly to leptons of the second or third generations.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, the energy, intensity and cosmic frontiers havecome into focus as the three domains in which future discoveriesin elementary particle physics are likely to arise. Among the pos-sibilities that might be revealed at the intensity frontier are “darksectors” [1], with particles much lighter than the electroweak scalethat have remained undetected due to their very weak interactionswith standard model particles. The gravitational effects of darkmatter is indirect evidence that at least one dark particle exists.The possibility that dark matter may also couple to spontaneouslybroken dark gauge forces has been motivated by the observationthat sub-GeV dark gauge boson decays could explain the electronand positron excesses observed in the cosmic ray spectra [2]. Moregenerally, sectors that are only weakly coupled to the standardmodel arise quite generically in string theory [3], so that inves-tigations of other possible dark sectors [4] continue to be wellmotivated.

Given that the particle content and gauge structure of a genericdark sector is not well constrained, the largest number of anal-yses have assumed the simplest construction, a dark U(1) gaugefield that has a small kinetic mixing with hypercharge [5]. Inter-estingly, it has been pointed out that a discrepancy between theobserved value of the muon anomalous magnetic moment and theexpectation of the standard model can be explained in this sce-nario [6]. However, the region of the mass-kinetic mixing plane inwhich this result can be achieved is very rapidly being excludedby bounds from dark photon searches [8], in particular, the most

E-mail address: cdcaro@wm.edu.

0370-2693/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physletb.2013.03.011

recent results from the Mainz Microtron experiment [9]. Dominantkinetic mixing will always assure that a dark photon couples iden-tically to electrons and muons, so there is no way to escape thisbound. The possibility that dark forces might couple differently tomuons than electrons has been discussed in a somewhat differ-ent context, i.e., the proton charge radius problem, in Ref. [10], butfrom an effective field theory perspective. Here we will presenta complete theory in which such a possibility is naturally real-ized.

Although an explanation of the muon anomalous magnetic mo-ment discrepancy is certainly worthwhile, it is arguably more im-portant that the present work shows, in a complete theory, howa dark sector can naturally couple both weakly and preferentiallyto charged leptons of the second or third generation. This mayprovide motivation for a more intensive phenomenological con-sideration of processes involving muons and taus as a means ofdiscerning couplings to new dark sector particles. The basic struc-ture of the model is this: the dark gauge and higgs particles coupledirectly to a vector-like charged lepton that has gauge quantumnumbers that allow mixing with the right-handed electron, muonor tau. However, the lepton sector is also constrained by a discrete,A4 flavor symmetry that is the origin of neutrino tribimaximalmixing. This symmetry leads to preferential mixing between thevector-like state and only one standard model lepton flavor, whilesimultaneously keeping lepton-flavor-violating effects suppressed.The couplings between the standard model and the dark sectorparticles are naturally small since they are proportional to a smallmixing angle that is set by the ratio of the dark to vector-likemass scales. Lepton-flavor-universal couplings due to kinetic mix-ing with hypercharge are loop suppressed, since the dark sectorgauge group in the proposed model is non-Abelian.

C.D. Carone / Physics Letters B 721 (2013) 118–122 119

Our Letter is organized as follows. In the next section, we de-scribe the basic structure of the theory and show that the muonanomalous magnetic moment discrepancy can be resolved. In Sec-tion 3, we explain how the pattern of couplings assumed in Sec-tion 2 can be achieved via the discrete flavor symmetry A4. Webriefly discuss other phenomenological bounds in Section 4, andin Section 5 we conclude.

2. The portal

The gauge group of the model is GSM × SU(2)D , where GSMis the standard model (SM) gauge group and the additional fac-tor represents the gauge group of the dark sector. Since the darkgauge group is non-Abelian, kinetic mixing with hypercharge doesnot appear immediately through a renormalizable term in the La-grangian. We assume the existence of a vector-like doublet

EaL, Ea

R , (2.1)

where a is an SU(2)D index. These fields have the same GSM quan-tum numbers as a right-handed electron, muon or tau. Note thatthe theory is free of anomalies, since the new fermions come in avector-like pair. Since the E fields carry both SM and dark quan-tum numbers, they serve as a portal between the SM and darksectors.

We assume that the dark gauge symmetry is completely brokenby the vacuum expectation value (vev) of a complex scalar dou-blet φa ,

〈φ〉 =(

0v D/

√2

). (2.2)

A renormalizable coupling between φ and the SM Higgs field H ,namely φ†φH† H , can serve as another portal between the SM anddark sectors. However, to simplify the model parameter space inthe present analysis, we will only consider the case in which thiscoupling is negligibly small. This limit is consistent with LHC dataon the recently discovered 125 GeV boson which (perhaps asidefrom the decay rate to two photons) shows no substantial devia-tions from the expectations for a standard model Higgs boson.

Mass mixing between the E and SM lepton fields can occurwhen the SU(2)D gauge symmetry is broken. This mixing willinduce a coupling between the dark gauge boson multiplet andordinary SM leptons that is suppressed by a small mixing angle,one proportional to v D/v , where v = 246 GeV is the electroweakscale. Like most dark photon models, we will typically assume thatthe dark gauge bosons have masses less than 1 GeV. Mass mixingoriginates through the Yukawa couplings

L ⊃ hμRφ∗a Ea

L + h′εabμRφa EbL + h.c., (2.3)

Here we have taken the SM lepton field to be of the second gen-eration. Generically one expects similar couplings involving the eand τ . Our model, however, is not generic. We will show in Sec-tion 3 that the flavor structure of Eq. (2.3) follows accurately fromthe flavor symmetries of the model. For the remainder of this sec-tion, we will simply assume the mixing given in Eq. (2.3) and studythe new contributions to the muon g − 2.

In order to specify the model, one must know the vector-like lepton mass M , the mass and gauge coupling of the darkgauge bosons, mV and gD respectively, and the mass of the darkhiggs, mϕ . (The dark higgs is the physical fluctuation about v D inthe doublet field φ.) In addition, one must know the mixing an-gles that are functions of the couplings h and h′ in Eq. (2.3). Whilea complete exploration of this six-dimensional parameter space isimpractical, the essential physics can be illustrated by adopting afew simplifying assumptions. We take mV = mϕ , so that a single

Fig. 1. New contributions to the muon anomalous magnetic moment.

mass scale characterizes the dark sector. In addition, we set thecoupling h′ = 0, which renders the analysis of mixing between theSM leptons and the vector-like states more transparent. (This lat-ter assumption can be elevated to a definition of the model if oneimposes a discrete symmetry that forbids the h′ coupling. A Z3symmetry in which EL,R → ωEL,R and φa → ωφa , with ω3 = 1, isa suitable example of such a symmetry.)

Given these assumptions, mass mixing between the SM andvector-like states are given by a two-by-two mass matrix,

L ⊃ Ψ L

(mμ 0δm M

)ΨR + h.c., (2.4)

where Ψ = (μ, E(2))T and δm = hvd/√

2. The relationship betweenthe gauge and mass eigenstate bases is given by

Ψ j =(

c j s j−s j c j

)Ψ mass

j (2.5)

where s j = sin θ j , c j = cos θ j , and j is either L or R . With δm M ,the mixing angles are well approximated by θR ≈ δm/M and θL ≈(mμ/M)(δm/M). Due to this mixing, the SU(2)D gauge bosons,which otherwise would couple only to the vector-like states, nowalso couple to standard model leptons. In terms of mass eigenstatefields, these couplings include

L ⊃[−1

2s2

R gD(μR/A0μR) − 1√2

sR gD(μR/A−E(1)

R + h.c.)

+ 1

2sR cR gD

(μR/A0 E(2)

R + h.c.)] + [R → L]. (2.6)

Here we have parameterized the SU(2) gauge multiplet

AaD T a =

(A0/2 A+/

√2

A−/√

2 −A0/2

), (2.7)

where the T a are the Pauli matrices over two. Note that the A0,A+ and A− are electrically neutral, but the E(1) , E(2) and μ fieldshave identical couplings to the photon. Hence there are numberof diagrams that contribute to the muon anomalous magnetic mo-ment, as shown in Fig. 1.

In addition to the gauge boson exchange diagrams, there arealso diagrams involving exchange of the dark Higgs boson ϕ , alsoshown in Fig. 1. These follow from Eq. (2.3), which includes thefollowing couplings of the mass eigenstate fields

120 C.D. Carone / Physics Letters B 721 (2013) 118–122

Table 1Couplings relevant to the anomalous magnetic moment calculation.

F Xμ C V C A

μ A0 − 14 gD (s2

R + s2L ) − 1

4 gD (s2R − s2

L)

E(1) A± − 12√

2gD (sR + sL) − 1

2√

2gD (sR − sL)

E(2) A014 gD (cR sR + cL sL)

14 gD (cR sR − cL sL)

F S C S C P

μ ϕ − 1√2

h(cR sL) 0

E(2) ϕ 12√

2h(cL cR − sL sR ) − 1

2√

2h(cL cR + sL sR )

Fig. 2. Mass mixing angle plane, for the values of gD and h shown. The regionbetween the dashed lines is consistent with the current experimental value of themuon anomalous magnetic moment, at plus or minus two-standard deviations. Thesolid lines show the vector-like fermion masses, which increase for smaller valuesof θR .

L ⊃ h√2ϕ

(−sLcRμμ + cL sR E(2)E(2)

+ [cLcRμR E(2)

L − sL sRμL E(2)R + h.c.

]). (2.8)

The contribution of new fermions and gauge bosons to themuon anomalous magnetic moment have been computed in a gen-eral framework in Ref. [11], assuming couplings of the form

L = μγ μ(C V + C Aγ 5)F Xμ + h.c., (2.9)

where F is a generic fermion and Xμ a generic gauge boson, and

L = μ(C S + C P γ 5)F S + h.c., (2.10)

where S is a generic scalar. These results can be applied to thepresent model given the identifications shown in Table 1. An exactnumerical evaluation of the g − 2 formulae given in Ref. [11] havebeen incorporated in Fig. 2. Here we show the mV –θR plane, fora fixed choice of the mixing parameter h and the dark gauge cou-pling gD , with mϕ = mV . With mV and gD fixed, the dark scale v D

is determined. The mass scale of the vector-like sector, M , is thenfixed by its relation to the right-handed mixing angle

θR =( √

2h)

mV , (2.11)

gD M

as indicated by the solid lines in the figure. The parameters of themodel are now fixed and the (much smaller) left-handed mixingangle can also be computed.

The region between the dashed lines is where the model ac-counts for the discrepancy between the experimental and SM val-ues of the muon (g − 2)/2 [12]

�aμ = aexpμ − aSM

μ = 287 ± 80 × 10−11, (2.12)

allowing two-standard deviation variation about the central valueof �aμ . The solid lines represent vector-like fermion mass con-tours, for M = 100, 300 and 500 GeV. Collider searches forheavy charged leptons require only that M > 100.8 GeV at the95% C.L. [12]. The solid and dashed bands still overlap for h closerto 1, but for values of M closer to its experimental lower bound.

Although it is known that the presence of an axial-vector com-ponent to the dark gauge boson coupling to muons can cause thefirst diagram in Fig. 1 to contribute to aμ with the wrong sign, thenew gauge boson exchange diagrams in the present model that in-volve heavy vector-like fermions contribute with the correct sign.These diagrams dominate (given that they are less suppressed bypowers of the mixing angle θR ) and the desired result is obtainedwithout fine tuning.

3. The flavor symmetry

There is a vast literature on flavor symmetries that can ac-count for the very non-generic pattern of standard model fermionmasses. In this section, we argue that the same flavor symmetrythat might explain standard model lepton masses can also yieldthe non-generic pattern of couplings assumed in the previous sec-tion, with lepton-flavor-violating effects adequately suppressed.

A variety of models based on the discrete flavor group A4 havebeen proposed after it was discovered that A4 symmetry can pro-vide neutrino mass matrix textures that are consistent with tribi-maximal mixing, at lowest order. Given these promising results, itis well motived to consider the consequences of this symmetry inthe present model. The group A4 has a triplet and three singletrepresentations, 3, 10, 1+ , and 1− . Typical A4 models assign theleft-handed SM leptons to the 3 and the right-handed leptons todistinct singlet representations:

LL ∼ 3, eR ∼ 10, μR ∼ 1+, τR ∼ 1−. (3.1)

The standard model Higgs doublet H is a trivial A4 singlet, 10. Fora review of group theory and model building with A4 symmetry,we refer the reader to the review in Ref. [13]. The symmetry isbroken by two flavon fields, ϕS and ϕT , that both transform as 3’s:

〈ϕS〉 =( v S

v S

v S

), 〈ϕT 〉 =

( vT

00

). (3.2)

By imposing additional, model-dependent symmetries [13], it is ar-ranged that the symmetry breaking in the charged lepton sector isa consequence of the ϕT flavon only, at lowest order, while in theneutrino sector the same statement holds for the ϕS flavon. Sym-metry breaking due to ϕT alone leads to a charged lepton massmatrix that is exactly diagonal,

Y L = vT

ΛF

( ye 0 00 yμ 00 0 yτ

), (3.3)

where ΛF represents the scale at which flavor physics is gener-ated. The reason for this result is that 〈ϕT 〉 breaks A4 down toa Z3 under which all off-diagonal elements of Eq. (3.3) rotate by

C.D. Carone / Physics Letters B 721 (2013) 118–122 121

a nontrivial phase; hence, these elements remain vanishing untilthis symmetry is broken.

We now assign A4 quantum numbers to the vector-like states:

EaL ∼ Ea

R ∼ 1+. (3.4)

Since this charge assignment is the same as that for μR , the mix-ing terms in Eq. (2.3) are A4 invariant. However, the other possibleterms are not:

eRφ∗a Ea

L ∼ 1+, τ Rφ∗a Ea

L ∼ 1−. (3.5)

The representations in Eq. (3.5) rotate by a nontrivial phase underthe Z3 symmetry that is left unbroken by 〈ϕT 〉, and are forbiddenuntil the Z3 symmetry is broken. Hence, the same symmetry struc-ture that leads to a diagonal charged lepton mass matrix whenA4 → Z3 assures that the vector-like states mix only with right-handed muons in the present model. For a different choice inEq. (3.4), one could just as easily arrive at a different model inwhich mixing with τR is preferred.

We must now consider to what extent our desirable resultsare spoiled when the Z3 symmetry is broken by the vacuum ex-pectation value of the other flavon field ϕS . Unwanted couplingsbetween E and eR or τR can occur directly, or via Eq (2.3), whichis no longer in the mass eigenstate basis if Y L is non-diagonal. Thevalue of the Z3 symmetry breaking parameter v S/Λ, however, isnot set by the size of a typical charged lepton Yukawa coupling; itis determined by the mass scale of the right-handed neutrinos thatparticipate in the see-saw mechanism. As is standard in A4-basedmodels, we take the right-handed neutrinos (charge conjugated) totransform as a triplet

νcR ∼ 3. (3.6)

We will now show that it is possible to achieve the proper lightneutrino mass scale with v S v T < ΛF , so that deviations fromthe Z3 symmetric results in the charged SM lepton and vector-like sectors of our model can be made adequately small. In thiscase, contributions to lepton-flavor-violating processes due to theexchange of the non-standard-model states in the theory will alsobe highly suppressed. In what follows, we will take the flavor cutoff ΛF to be the reduced Planck mass, M∗ , which provides thesmallest possible values for v S/ΛF .

The Lagrangian terms that lead to a see-saw mechanism in neu-trino sector are

Lν = 1

2mRνc

RνR + 1

2xνc

RϕSνR + [yLL HνR + h.c.], (3.7)

where x and y are dimensionless coupling constants. This leadsto the conventional tribimaximal form for the Dirac and Majorananeutrino mass matrices [13] that are input into the see-saw

mLR =( 1 0 0

0 0 10 1 0

)yv,

mR R =( A + 2B/3 −B/3 −B/3

−B/3 2B/3 A − B/3−B/3 A − B/3 2B/3

)mR , (3.8)

where A = 1, B = xv S/mR , and v ≈ 246 GeV is the electroweakscale. The light neutrino mass matrix then follows from the see-saw formula mν = mT

LRm−1R RmLR , and has the eigenvalues

m1 = y2 v2

mR

1

(A + B), m2 = y2 v2

mR

1

A,

m3 = y2 v2 1. (3.9)

mR B − A

Neutrino oscillation data indicate the neutrino mass squared split-tings �m2

21 = (7.50 ± 0.20) × 10−5 eV2 and �m232 = 2.32+0.12

−0.08 ×10−3 eV2 [12]. These can be accommodated by Eq. (3.9) fory2 v2/mR = 0.97 × 10−11 GeV and B = xv S/mR = 1.198. This so-lution implies

v S

M∗≈ 3.7 × 10−3 y2

x. (3.10)

The key point is this: Eq. (3.10) determines the deviation from theZ3 symmetric limit our theory in which no lepton-flavor-violatingcouplings appear. For O(1) values of x and small values of theneutrino Dirac Yukawa coupling y, Z3 symmetry breaking can bemade extremely small. It is easy to arrange this from a symmetryperspective by applying, for example, a ZN flavor symmetry on theright-handed neutrinos that allows Majorana but not Dirac massterms at lowest order. Since the size of v S is near the right-handedneutrino mass scale, the smallness of the parameter in Eq. (3.10)is related to how low one can take this scale. For example, withmR ∼ v S ∼ 20 TeV, one would have v S/M∗ ≈ 10−14, with y com-parable in size to the electron Yukawa coupling. This is more thanadequate to render the theory safe from lepton-flavor-violating ef-fects from the coupling of the dark and standard model sectors.As an example, loop effects will generate an effective operator thatcontributes to �i → � jγ which is of the form [13]

1

Λ2NP

LL i Hσμν Fμν ZijeR j (3.11)

where the new physics scale here, ΛNP, is related to the vector-like fermion mass scale by ΛNP ≈ 4π M/gD , and Zij is a couplingmatrix. Since Z has the same flavor structure as Y L and we as-sume an additional symmetry prevents ϕS from contributing atlowest order, we expect that Z ∼ O(v T v S/Λ

2F ). From Eq. (3.3),

we see that v T /ΛF can be no smaller than approximately 0.01if the tau Yukawa coupling is to remain perturbative. So, assum-ing ΛF = M∗ and MR ≈ 20 TeV, we can achieve Zij ∼ O(10−16).In contrast, the bound on Zμe from Ref. [13], updated to takeinto account the latest PDG value [12] for the μ → eγ branch-ing fraction, gives Zμe < 4.5 × 10−9[ΛNP/1 TeV]2, which impliesZμe < 7.1 × 10−7 for gD = 0.1 and M = 100 GeV. This bound iseasily satisfied. Similar conclusions can be drawn from other pos-sible lepton-flavor-violating processes, which we will not considerin explicit detail here.

4. Other phenomenological issues

Before concluding, we comment on a few of the other potentialbounds on the model. It has been noted [14] in the context ofother models [10] in which a right-handed muon couples to a newgauge field Vμ that the effective coupling gRμR/V μR is boundedby the absence of missing mass events in the leptonic kaon decayK → μX . In the present model, the magnitude of gR is s2

R gD/2,which ranges in Fig. 2 from 10−8 to 10−4 between approximately5 to 300 MeV. We comment on this mass range since it roughlycorresponds the to the one over which limits on gR as a functionof vector boson mass are given in Fig. 3 of Ref. [14]; the limit isalways larger than 10−3. Hence, the model parameter space is notseverely constrained based on this consideration.

The bounds on the dark sector states from accelerator experi-ments [7] depend on how these states decay, and their are manypossible scenarios. Here we discuss one interesting case that cor-responds to the largest parameter region shown in Fig. 2: if mV <

2mμ , the dark gauge fields will decay only to electrons via loop-suppressed kinetic mixing and we can compare to the bounds from

122 C.D. Carone / Physics Letters B 721 (2013) 118–122

dark photon searches. The one-loop diagrams that lead to mixingbetween the dark gauge bosons and hypercharge lead to effectiveoperators of the form

Leff = c1

M2φ† F μν

D φFY μν + c2

M2φ† F μν

D φFY μν

+[

c3

M2φ† F μν

D φFY μν + h.c.

], (4.1)

where φ = εabφ∗b . The diagrams that generate these operates

involve the vector-like fermions (and hence the suppressionby M2) as well as two vertices from the interaction terms inEq. (2.3). The couplings c1, c2, and c3 are of order h2 gD gY /(16π2),hh′ gD gY /(16π2), and h′2 gD gY /(16π2), respectively. Kinetic mix-ing is conventionally parameterized as χ F μν

i FY μν/2, where Fi isthe field strength tensor corresponding to the dark gauge field Ai .From the first term in Eq. (4.1), for example, we estimate from aone-loop calculation that

χ ∼ 1

4π2gY

h2

gD

(mV

M

)2

. (4.2)

The most constraining beam dump bound [15] for small kineticmixing is from the SLAC E137 experiment. However, the E137bound does not exclude χ below ∼ 4 × 10−8. For the parame-ter values h = 2 and g = 0.1, used in Fig. 2, and M = 200 GeV,we find that χ is smaller than this value for mV � 67 MeV. FormV � 210 MeV, E137 gives no further bounds. Of course, the darksector could include dark matter candidates into which the darkgauge and Higgs fields decay. In this case, dark photon searches ataccelerators would present no further bounds on the model.

Finally, mixing between the vector-like states and the muon canlead to a shift in the Z boson coupling to muons. We find, how-ever, that this effect is well below current experimental bounds.Since the right-handed vector-like states have exactly the sameelectroweak quantum numbers as the right-handed muon (i.e., thecoupling matrix to the Z is proportional to the identity) there isno effect on the ZμRμR vertex from the rotation in Eq. (2.5). Thesame is not true of the ZμLμL coupling, since the Ea

L states areSU(2)W singlets, unlike μL . In this case, however, the shift in thecoupling is determined by the left-handed mixing angle, and is oforder θ2

L ∼ m2μ/M2 · θ2

R . The most precisely measured Z -pole ob-servable involving muons is Rμ , which is measured to one part in10−3 [12]. However, m2

μ/M2 ∼ 10−6 and θ2R is typically also much

smaller than 1. Hence, we obtain no additional constraints on themodel.

5. Conclusions

In this Letter, we have shown how one can construct a darksector that couples preferentially to a single flavor of standardmodel charged leptons, without generating large lepton-flavor-violating effects. The dark gauge and higgs bosons couple to

O(100) GeV vector-like fermions that have the correct gauge quan-tum numbers to mix with either the right-handed electron, muonor tau. However, the lepton sector is also constrained by an A4flavor symmetry that is the origin of neutrino tribimaximal mix-ing. This symmetry assures that the vector-like state mixes withonly one flavor of charged lepton while the charged lepton Yukawamatrix remains very accurately diagonal. Deviations from this fla-vor structure are related to the ratio of the right-handed neutrinomass scale to the cut off of the effective theory, and can be madeadequately small. Moreover, lepton-flavor-universal couplings viakinetic mixing are loop suppressed since the dark sector gaugegroup is non-Abelian.

If the vector-like states mix with leptons of the second genera-tion (which has been the assumption in this Letter) contributionsfrom the non-standard-model particles can account for the cur-rent deviation of the muon anomalous magnetic moment from thestandard model expectation. More importantly, however, the con-struction described in this Letter can be used to create ultravioletcomplete theories (at least up to the high scale at which flavor isgenerated) in which dark sector gauge bosons couple preferentiallyto either muons or taus. This provides motivation for exploringa vast realm of phenomenological processes involving second andthird generation leptons as another possible window on a dark sec-tor.

Acknowledgements

This work was supported by the NSF under Grant PHY-1068008.In addition, the author thanks Joseph J. Plumeri II for his generoussupport.

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