Lepton-flavored dark matter

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Lepton-flavored

dark

matter

Jennifer Kile

a

,

Andrew Kobach

b

,

∗

,

Amarjit Soni

c

a_{InstituteforFundamentalTheory,DepartmentofPhysics,UniversityofFlorida,Gainesville,FL32611,USA} b_{NorthwesternUniversity,DepartmentofPhysics&Astronomy,2145SheridanRoad,Evanston,IL60208,USA} c_{PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12December2014 Receivedinrevisedform8March2015 Accepted4April2015

Availableonline8April2015 Editor:B.Grinstein

In this work, we address two paradoxes. The first is that the measured dark-matter relic density can be satisfied with newphysics at O(100 GeV–1 TeV),whilethe null results fromdirect-detection experiments placelower bounds ofO(10 TeV) onanew-physicsscale. Thesecondpuzzle isthat the severesuppressionoflepton-flavor-violatingprocessesinvolvingelectrons,e.g.

μ

→3e,

τ

→e

μμ

,etc., impliesthatgenericnew-physicscontributionstoleptoninteractionscannotexistbelow

O(

10–100 TeV), whereasthe3.6

σ

deviationofthe muong−2 fromthestandardmodel canbeexplainedbya new-physicsscale

<

O(1 TeV).Here,wesuggestthatitmaynotbeacoincidencethatboththemuong−2 and therelicdensitycanbesatisfiedbyanew-physicsscale 1 TeV.Weconsider thepossibilityofa gaugedlepton-flavorinteractionthatcouplesattreelevelonlyto

μ

- and

τ

-flavoredleptonsandthedark sector.Darkmatterthusinteractsappreciablyonlywithparticlesof

μ

and

τ

flavorattreelevelandhas loop-suppressedcouplingstoquarksandelectrons.Remarkably,ifsuchagaugedflavorinteractionexists atascale _O(100 GeV–1 TeV),itallowsforaconsistentphenomenologicalframework,compatiblewith the muon g−2,therelicdensity,directdetection,indirect detection,charged-leptondecays,neutrino tridentproduction,andresultsfromhadronande+e−colliders.Wesuggestexperimentaltestsforthese ideasatcollidersandforlow-energyobservables.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

In this work, we attempt to address two ongoing puzzles in particlephysics.The firstisthat ifdarkmatter isa thermalrelic, its annihilation cross section requires new physics at the elec-troweakscale, i.e.,in therange

O(

100 GeV–1 TeV). However,the null results from direct-detection experiments constrain that a new-physics scale between dark matter and nucleons must be

>

O(

10 TeV) for a dark-matter mass



10 GeV. This tension be-tween the relic density and direct detectionmay be pointingto thepossibilitythatdarkmatter doesnot coupleto quarksattree level.Rather,darkmattermaycoupleprimarilyviainteractions at theelectroweakscaletootherparticlesinthestandardmodel,e.g., leptons. Ifso, such a dark-matter candidate can satisfy the mea-suredrelicdensityattreelevelandaccommodatethenullresults fromdirect detectionby givingrise tointeractions between dark matterandquarksatthelooplevel.

*

Correspondingauthor.

E-mailaddress:[email protected](A. Kobach).

Ifinteractionsattheelectroweakscaleexistbetweendark mat-ter and leptons, then one generically expects such interactions between leptons themselves. There maybe evidence forsuch an interactiongiventhatthecurrent3.6

σ

deviationofthemuon

g

−

2 from the standard model value could be explained by new in-teractions ata scale

<O(

1 TeV). However, the possible existence of newphysics atthis scale introduces a second puzzle: interac-tions at such a scale do not manifest themselves via other pro-cesses among chargedleptons. For example,flavor-violating pro-cesses suchas

μ

→

3e,

τ

→

e

μμ

,

τ

→

ee

μ

,

τ

→

3e,

μ

→

e

γ

[1]

constrainnew-physicsscalestobe

>

O(

10–100 TeV).Additionally, flavor-conservingprocesses,suchasleptonproductionatLEP, con-strain newphysics toscalesabove severalTeV[2–4].Someofthe strongexperimentalconstraintsoninteractionsamongleptonscan be found inTable 1.Since mostof thesestrong constraintscome fromprocessesthatinvolveelectrons,thismotivatesthepossibility ofanewleptonicinteractionattheelectroweakscaleunderwhich electrons,likequarks,donoteffectivelyparticipate.

Here, we considerthe possibilitythat the occurrenceofthese two paradoxes is not a coincidence. We find that they can be resolved simultaneously if we consider a gauged lepton-flavor

http://dx.doi.org/10.1016/j.physletb.2015.04.005

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

Table 1

Constraintson lepton-flavorviolatingand conservingprocesses. For thelastfour observables,the experimentalnullresults are givenintermsofadimension-6operator,suppressedbytwo or-dersof,whichcanbeinterpretedasthenominalscaleofnew physics. Observable Limit Br(μ→3e) <1.0×10−12_[1] Br(μ→eγ) <5.7×10−13_[1] Br(τ→3e) <2.7×10−8_[1] Br(τ→e−μ+μ−) <2.7×10−8_[1] Br(τ→e+μ−μ−) <1.7×10−8_[1] Br(τ→μ−e+e−) <1.8×10−8_[1] Br(τ→μ+e−e−) <1.5×10−8_[1] Br(τ→3μ) <2.1×10−8_[1] Br(τ→μγ) <4.4×10−8_[1] Br(τ→eγ) <3.3×10−8_[1]

μ–e conversion 103_TeV[5] e+e−→e+e− 5 TeV[3]

e+e−→μ+μ− 5 TeV[3]

e+e−→τ+τ− 4 TeV[3]

interaction attheelectroweakscaleunderwhichbothleptonsand darkmatterare charged.Wecall thisframework“lepton-flavored darkmatter”(LFDM).Here,weconsiderthesimplifiedcasewhere theinteraction onlyinvolves

μ

- and

τ

-flavoredleptons anddark matter;thus,darkmatterinteractsonlywithparticlesof

μ

and

τ

flavorat treelevel.We perform a model-independentanalysisof thisscenarioandfind that itcan lead toa consistent framework wheretherelicdensity,directdetection,indirectdetection,results from hadron and e+e− colliders, and low-energy measurements are compatible with flavor gauge bosons with electroweak-scale masses,i.e.,

O(

100 GeV–1 TeV).

Many have investigated the idea that dark matter does not interactwithquarksat thetree level.Mostofthese analyses as-sume an interaction between dark matter andleptons that does not distinguish between lepton flavors [6–27], although a few have considered more general analyses of gauged flavor interac-tions [28–35]. The LFDM framework allows for a more general analysisofinteractionsthat involveonlydarkmatter andleptons atthetree level; itpermits differentcouplingstrengths between leptonflavors,off-diagonalflavorcouplings,andlepton-flavor vio-lation.Fora reviewofflavoreddarkmatter,seeRef. [36]andthe referencestherein.

Ouranalysisisoutlinedasfollows.InSection2,weintroducea parameterizationofLFDMthat,to agood approximation,can en-capsulatemostmodelsofflavor-conservingLFDMthatinvolveonly

μ

- and

τ

-flavored leptons. In Section 3, we survey the relevant constraints on LFDM fromthe relic density, direct-detection, the muon

g

−

2,indirectdetection,andhigh-energyobservablesatLEP andtheLHC.InSection4,weexploretheimportantconstraintsvia aparameterscan.Wediscussthepossibilityoflepton-flavor viola-tion(LFV)inSection5,andinSection6,weofferconclusionsfrom ouranalysisandprospectsforfutureexperimental andtheoretical investigations.

2. Lepton-flavorinteractions

Inthissection,we layout thegeneralframeworkthat weuse toanalyzeLFDM.Wetakedark-sectorparticlestobecomprisedof Diracfermions,sharingcommongaugedlepton-flavor interactions along with right- and left-handed charged leptons. We assume thatneutrinoshaveDiracmassesandconsequentlyintroducethree right-handedpartners also chargedunder the lepton-flavor

sym-metry.1 _{We take thestrong constraintsonprocesses that include}

electronsasevidencethatparticleswithelectronflavordonot ef-fectivelyparticipateinlepton-flavorinteractionsattheelectroweak scale.

Ifthemuon g

−

2 anomalyisduetoLFDM,thismaybean in-dicationthattheinteractionsarepurelyvector-like.2 Itispossible, however,that dueto differentrotations betweenmassand inter-actioneigenstatesintheright- andleft-handedsectors,differences in left- andright-handed couplings can arise in the mass eigen-statebasis.Thiscanintroduceinteractionsthatdeviatefrompurely vectorinteractions.Sinceflavor-violatingcouplingsamongcharged leptonscanbestronglyconstrained,weassumethattheLFDM in-teractionsdonot mediateLFVamongchargedleptons atthe tree level; we thus take the mass andflavor interaction bases inthe charged-leptonsectortobecloselyalignedandtakethedeviations from purely-vector interactions in the mass basis to be negligi-blysmall.However,becausewehaveessentiallynoconstraintson theflavor structure ofthedarksector, we allowdifferent lepton-flavor couplings between left- and right-handed components of darkmatter.WereturntothetopicofLFVinSection5.

WecreateaphenomenologicalLagrangianwithdimension-four (d

=

4)operators thatcanencapsulate mostmodelsofLFDMthat do not violate lepton flavor at the tree-level, butdo permit off-diagonal lepton-flavor vertices. One can introduce an arbitrary number ofgauge bosons, categorized by whether they coupleto diagonal or off-diagonal lepton-flavor currents. We denote their masseigenstates by X and Y , respectively. Wepresently consider thattherearetwo X bosons andone

Y boson,

whichallowsfora sufficientnumberoftermstoaccountforthephenomenology as-sociatedwithawidevarietyofspecificmodels.3_{Therecanalsobe}

an arbitrarynumberofdark-sectorspeciesthat participatein the lepton-flavorinteractions.Therelicdensityanddirectdetectionare onlysensitivetothelightestdark-sectorstate,whichwecall

χ

.

We parameterizetheselepton-flavor interactions withthe fol-lowingphenomenologicalLagrangian4:

L

⊃



i=1,2 Xiα



kiμμJαμ

+

kiτ τJατ

+

kiL

χ

L

γ

α

χ

L

+

ki R

χ

R

γ

α

χ

R



+

Yα



hμτKα

+

hL

χ

L

γ

α

χ

L

+

hR

χ

R

γ

α

χ

R



+

Yα†



hμτ

(

Kα

)

†

+

hL

χ

L

γ

α

χ

L

+

hR

χ

R

γ

α

χ

R



,

(1) where Jα_μ

=

μ

L

γ

α

μ

L

+

μ

R

γ

α

μ

R

+

ν

Lμ

γ

α

ν

Lμ

+

ν

Rμ

γ

α

ν

Rμ

,

(2) J_τα

=

τ

L

γ

α

τ

L

+

τ

R

γ

α

τ

R

+

ν

Lτ

γ

α

ν

Lτ

+

ν

Rτ

γ

α

ν

Rτ

,

(3) Kα

=

μ

L

γ

α

τ

L

+

μ

R

γ

α

τ

R

+

ν

Lμ

γ

α

ν

Lτ

+

ν

Rμ

γ

α

ν

Rτ

.

(4)

Wetake

ν

Rμ and νRτ to bedefinedbytheirinteractionsunderthe flavorgroup;whiletheinteractionbasisdoesnotcoincidewiththe massbasis in theneutrinosector, ouranalysis isnot sensitiveto

1 _{Wedonotaddressthepossibilitythatintroducingadditionallightdegreesof} freedomcaneffectmeasurementsofnefffromthecosmic-microwavebackground.

2 _{Iftheflavorinteractionswerepurelyleft- orright-handed,theycannotaccount} forthemuong−2 anomaly,sincetheywoulddecreasethevalueofaμ.

3 _{Two X bosons allow us to independently vary the strength of the} flavor-conservingfour-Fermiinteractionsbetween

χ

andthechargedleptons,andamong thechargedleptonsthemselves.IfonlyasingleX bosonwereused,forexample, thefour-μ,four-τ,andμμ¯ τ τ¯ effectiveinteractionswouldhaverelatedcoefficients. AsecondX bosonallowsforthesecoefficientstobeindependent.

4 _{WenotethatachievingaLagrangianlikeinEq.(1)inageneralmodelof} fla-vorisnon-trivial;caremustbetakentonothaveflavorviolationinconflictwith experimentalconstraints.Our motivationinchoosingtheformofEq.(1)isthus phenomenological,andnotderivedfromaparticularmodel.

Fig. 1. The1-loopdiagramthatcontributestodirect detectionbymediatingthe interactionbetweenlepton-flavoreddarkmatterandprotons,asparameterizedin Eq.(1).

thismixing. Thedifferencesbetweenthecouplings

k and k

 (and between

h and h

)accountforboththepossibilitythatthecharged leptons and the dark sector are not in the same representation oftheflavor group andthepossible occurrenceofnon-negligible mixinginthedarksector.5 We neglectatree-levelkinetic mixing operator between X and the hypercharge gauge boson, Xμν Bμν ,

whichcanpermitthelepton-flavorinteractionstocoupletoquarks andelectrons attreelevel.If theflavor gaugesymmetry is nota

U

(1),

thistermisdisallowed;inthecaseofa

U

(1)

symmetry,we expect thischoice to be conservative forthe constraintsrelevant forthisanalysis.

3. Flavor-conservingobservables

3.1. Direct detection and relic density

Abasicrequirementforanydark-mattercandidateisthatithas arelicdensitynotinconflictwithobservation.Ifdarkmatter con-sistsofasinglethermalrelic,itimpliesthatDiracdarkmatterhas athermally-averagedannihilationcrosssection toSM particlesof



σ

v

 ≈

4.4

×

10−26cm3

/s

[37].Thedarkmatterwillannihilateto pairsofchargedleptons orneutrinos.6 _{Ifthemassesofthe}

final-state leptons are negligible,the non-relativisticannihilationcross sectionisthefollowing,accordingtotheparameterizationofLFDM giveninEq.(1),



σ

v =m 2 χ 2

π

⎧

⎪

⎨

⎪

⎩

2h2_μτ

(

h_L

+

h_R

)

2 M4_Y

+

⎡

⎢

⎣

⎛

⎝



i=1,2 kiμμ

(

kiL

+

ki R

)

M2_X i

⎞

⎠

2

+

⎛

⎝



i=1,2 kiτ τ

(

kiL

+

ki R

)

M2 Xi

⎞

⎠

2

⎤

⎥

⎦

⎫

⎪

⎬

⎪

⎭

.

(5) If



σ

v

 =

m2

χ

/

4

∼

4.4

×

10−26cm3

/s,

where



is the effective

scaleoftheinteraction,then



∼ (

130 GeV)



mχ

/GeV.

IfLFDMis allowedtobeonlyasubsetofdarkmatter,thensmallerscalescan beobtained.

5 _{Ingeneral,thedark-mattercouplingh}_{canbecomplexduetothepresenceof} CP-violatingphaseswhenrotatingfromtheflavorbasistothemassbasis.However, sinceouranalysisisnotsensitivetotheeffectsofCPviolation,weassumehisreal forsimplicity.

6 _{WeassumethatM}

X1,MX2,MY>mχ andthusneglectthepossibilityof dark-matterannihilationstotheflavorgaugebosons.Wechecktheself-consistencyof thisassumptioninSection4.

Fig. 2. The1-loopcontributiontothemuong−2 duetothelepton-flavor interac-tions,accordingtotheparameterizationinEq.(1).

LFDMdoesnotinteractwithquarksatthetreelevel;the direct-detection crosssection first occursat 1-loop,as showninFig. 1. Because thephotononlycouplesto chargedparticles with flavor-diagonal couplings,the Y boson doesnot participateinthis pro-cess. We estimate the value of this diagram as the running be-tweentheenergyscalerelevantfordirectdetection,whichwetake to be the massof the lepton in the loop, anda renormalization scale

μ

,whichweassumeisoforderthemassoftheflavorgauge bosons,

μ

∼

MXi (seealsoRefs.[38,39]).Weobtain

σ

(

χ

p

→

χ

p

)

=

α

2

_μ

2 N 36

π

3

⎡

⎣



i=1,2 kiμμ

(

ki R

+

kiL

)

M2_X i ln



μ

2 m2 μ



+

kiτ τ

(

ki R

+

kiL

)

M2_X i ln



μ

2 m2 τ



2

,

(6) where

μ

N

=

mχ mp/(mχ

+

mp

).

The LUXexperimentplaceslimitsonthespin-independent in-teraction cross section betweendark matter and nucleons to be



O(

10−45–10−44cm2

)

formχ

>

10 GeV [40]. In order to inter-pret these results as limits on the interaction cross section be-tweendarkmatterandprotons,onecanscaletheexperimental re-sultsbyafactorof Z2

_/

_A2_,where

_{Z and}

_{A are theatomicnumber} andatomicmassnumber,respectively,ofthedetector’sinteraction material.If

σ

(

χ

p

→

χ

p

)

∼

α

2_m2

p

/(36

π

3



4

),

acrosssectionupper limitof10−45_cm2_{atLUXwouldsuggestthat}

_

_

_{400 GeV,which} isconsistentwithelectroweak-scaleLFDMgaugebosonmasses.

3.2. Low-energy observables 3.2.1. Muon g

−

2

Currently, theonlylow-energymeasurement thatdeviates sig-nificantlyfromtheSMisthemuon

g

−

2,whosemeasuredvalueis 3.6

σ

largerthantheSM expectation,

a

SM

μ ,i.e.,

δa

μ

=

a

exp

μ

−

aSMμ

=

(28.8

±

8.0)

×

10−10_{[1].Thisdiscrepancycanbeaccountedforby} LFDM throughthe diagramsshowninFig. 2. Here,the loop con-tainseitheramuonandan X boson, oratauanda

Y boson.

The valueof

aμ is

givenby

aμ

=

aSMμ

+

h2 μτ 4

π

2 m2 μ M2 Y



mτ mμ

−

2 3



+



i=1,2 k2 iμμ 12

π

2 m2 μ M2 Xi

.

(7)

Ifthetheoretical predictionwere tobebroughtto avaluewithin 95% CL ofthe experimentalvalue,7 i.e., 1.3

×

10−9

< δa

_μ

<

4.4

×

10−9_{, then 1/(270 GeV)}2

_<



ik2iμμ

/

M

2

Xi

<

1/(140 GeV)

2 _{for the} case where only muons run in the loop, and 1/(1.9 TeV)2

<

h2_μτ

/

M2_Y

<

1/(1.0 TeV)2 when only taus run in the loop. These valuesareconsistent withtherequirementsfromthedark-matter relicdensityanddirectdetection.

7 _{Forthe muong−2,andthroughoutthisanalysis,weusea}

_χ

2 _functionto estimatelimitsfordifferentCL’s.

Fig. 3. ThecontributiontoneutrinotridentproductionfromX bosons,accordingto theparameterizationinEq.(1).

3.2.2. Tau decays

Lepton-flavored interactions mediated by Y bosons will con-structivelyinterferewiththeSM leptondecays ofthetaulepton,

τ

−

→

μ

−

ν

τ

ν

μ. Using the parameterizationin Eq.(1), thewidth

forsuchdecaysis

(

τ

−

→

μ

−

ν

τ

ν

μ

)

m5τ

π

3



G2 F 192

+

√

2GFh2μτ 384 M2_Y

 

1

−

8m 2 μ m2 τ



,

(8)

whereGF istheFermiconstant. IntheSM, theratioof

(

τ

−

→

μ

−

ν

τ

ν

μ

)/ (

τ

−

→

e−

ν

τ

ν

e

)

isestimatedtobe0.9726,andthis ra-tiohasbeenmeasuredtobe0.979

±

0.004[1].Thismeasurement suggeststhat the 95% CL band requires h_μτ2

/

M2_Y



1/(2.0 TeV)2. Thisposesamildtensionwiththemuon

g

−

2 ifonlythe

Y boson

participatesinLFDM.

3.3. Neutrino trident production

Interactions of the X flavor gauge bosons can contribute to measurementsofneutrinotridentproduction,asdepictedinFig. 3. AscalculatedbytheauthorsofRefs.[41,42],an X boson can con-structivelyinterfere with the SM, resulting inan increase in the measuredcrosssectioncomparedtothatoftheSMexpectation:

σ

SM+X

σ

SM

=

1

+



1

+

4 sin2

θ

w

+

√

2 GF



i k2_iμμ M2_X i



2 1

+



1

+

4 sin2

θ

w



2

.

(9)

The results of the CHARM-II [43] and CCFR [44] experiments are

σ

data

/

σ

SM

=

1.58

±

0.57 and0.82

±

0.28,respectively.Together, thesedatasuggestthat,at95%CL,



i k2_iμμ M2_X i

<

1

(

490 GeV

)

2

.

(10)

This constraint from neutrino trident production strongly disfa-vors the hypothesis that the value of the muon g

−

2 receives a contribution from X bosons alone. Likewise, as mentioned in Section 3.2.2, aparameterization ofLFDM withonly Y bosons is constrained by tau decays and likewise cannot account for the measuredvalueofthemuon

g

−

2.However,ifonepermitsboth X

and

Y bosons

tocontributetothemuon

g

−

2,asshowninFig. 2, thenone canaccount forthemuon g

−

2.We willexplore these ramificationsfordirectdetectionandtheLHCinSection4.

3.4. Indirect detection

Otherpossibleconstraintson LFDMcomefromindirect detec-tion. Our scenario is not relevant for explaining the excesses in thepositron fractionfromAMS [45], Pamela[46],andFermi-LAT

[47],whichrequireanannihilationcrosssectiontoleptons approx-imatelytwotothreeordersofmagnitudelargerthanthatexpected

for a thermal relic [48–50]. Since we assume that

χ

annihilates half the time toneutrinos, indirect detectionis onlyrelevant for ourscenarioifitconstrainstheannihilationcross-sectionto

μ

+

μ

−

or

τ

+

τ

− tobelessthanhalfofthethermalrelicvalue.Forvalues ofthedarkmattermass

mχ



100 GeV,indirectdetectionbounds arenotsufficientlystringenttoruleoutthethermalcrosssection.

However,forsomewhatlight darkmatter,

mχ

∼

few

×

10 GeV, indirect-detection constraints may be important, but are subject tosignificantuncertainties.Dark-matterannihilationto

μ

+

μ

− can be constrained using the AMS positron fractionto be belowthe thermal relic value for dark-matter masses below 100 GeV [51], butsystematicuncertaintiescanalterthelimitsontheannihilation cross-section by afactor ofa few.(Forresults usingthe positron flux,alsoseeRef.[52].)

Indirect-detection constraints are important, however, for the low-massregionmχ



10 GeV.Theannihilationoflightdark mat-tercanbeconstrainedby theeffectsofitsresultant energy injec-tionontheCMB;thermalrelicswithmassesbelowseveralGeVare disfavored [53–55]. Annihilationsof light darkmatter into

τ

+

τ

−

areconstrainedbyFermi-LATobservationstobeassmallasafew

×

10−27cm3

/s

[56–59]. Fermi-LAT also gives similar constraints,

O(10

−27_–10−26_cm3

_/s),

_{onannihilationsoflightdarkmatter into}

μ

+

μ

−[60].SlightlytighterconstraintsarealsopossiblefromAMS, but are subject to effects of solar modulation for dark-matter masses

mχ

<

O(

15 GeV)[51,52,58].

Wenote, however,thattheseconstraintsallgive limitsonthe annihilation cross section of light dark matter to leptons which aresmallerthanthatofathermalrelicbyafactoroforderunity. While we also include annihilations of dark matter to neutrinos (equal inmagnitudeto darkmatter annihilations tocharged lep-tons), some mild tension remains between boundsfrom indirect detectionandthethermalcrosssection.Interestingly,someworks haveconsideredpossibleevidencefordarkmatterannihilationto leptons fromindirectdetectionwithcrosssectionsslightlybelow those ofa thermalrelic[57,60–66]. Wewill briefly returnto the subjectofindirect-detection constraintswhenwe discussthe pa-rameterscansinSection4.

3.5. High-energy colliders

Asdiscussedabove,therequirementsfromthemuon

g

−

2 and therelicdensityimplythatthepreferredscaleofLFDMisthe elec-troweakscale,i.e.,

O(

100 GeV–1 TeV),whichiswithinthereachof colliderexperiments.Whilethelepton-flavor gaugebosonsdonot effectively coupleto electrons andquarks at treelevel, they can contribute, however, to the processes e+e−

→

μ

+

μ

−

,

τ

+

τ

− and

qq

¯

→

μ

+

μ

−

,

τ

+

τ

−viatheone-loopdiagramsinFig. 4.Potentially relevantmeasurementsincludeconstraintsonresonancesearches, effective operators, and couplings of the Z boson to

μ

+

μ

− and

τ

+

τ

−atLEP,aswellasresonancesearchesatLHC.

3.5.1. LEP

ThediagramsinFig. 4canpotentiallygivenewphysics contri-butions to three measurements from the LEP experiments. First, these diagrams may affect the limits from LEP on effective op-erators of the form



−2

(

ei

γ

α ei)( j

γ

α

j), where

=

μ

,

τ

, and i

,

j

=

R or L, for which



is constrained to be larger than sev-eralTeV[3].Second,lepton-flavorphysicscanproducecorrections

to the Z -lepton couplings, as depicted in the three diagrams in

Fig. 4. (Thediagram in Fig. 4(c)only contributesto the Z -lepton

couplingsifthe

s-channel

SMgauge bosonisa Z .) Lastly,narrow resonancesearchesatLEPcan constrainLFDMviathediagram in

Fig. 4(c). Since all ofthe diagramsin Fig. 4 are loop-suppressed, the limitson the scale ofnewphysics resultingfrom these con-siderations are typically weak, MX



200 GeV. However, we give

Fig. 4. (a)and(b):The1-loopcorrectiontothe Z -decayvertexduetoflavorgauge bosons,X andY .(c):The1-loopdiagramforkineticmixingbetween X andthe hyperchargegaugebosonB athadronandleptoncolliders(where f¯f canbeqq or e+e−),whichcancontributetotheZ -leptoncouplings,resonantX production,and effectivefour-fermionoperators.

the case of light X bosons special consideration, because thisis similar to the scale ofLFDM interactions suggested by g

−

2 for thecasewhereonly X bosons andmuonscontribute.Here,wedo notconsiderthecontributionfromthediagram inFig. 4(b),since, compared to the other diagrams inFig. 4, it hasa negligible ef-fect on colliderobservables because ofthe low-energy resultsin Section3.2.

First, we note that when the diagram in Fig. 4(c) contains an s-channel photon,its contributionto e+e−

→

μ

+

μ

−

,

τ

+

τ

− is non-resonant at LEPII center-of-mass energies between 130 GeV and209 GeV.Weobtainaroughconstraintonthiscontributionby approximatingitasaneffectiveoperator



−2

(

e

γ

α e

)(

γ

α

),

which

shouldbeapproximatelyvalidforvaluesof

M

X



250 GeV.Forthe caseof constructive interference withthe SM, the 95% CL lower limiton



forthisoperatoris5.3 TeV for

=

μ

and4.5 TeV for

=

τ

, andforthe caseofdestructive interference,the limitsare 4.6 TeV and3.9 TeV,respectively[3].Weobtain

−

1

(

4

.

6 TeV

)

2

<

e2 12

π

2



5 3

+

ln



μ

2 q2

 

i kiμμ

(

kiμμ

+

kiτ τ

)

M2_X i

<

1

(

5

.

3 TeV

)

2

,

(11)

−

1

(

3

.

9 TeV

)

2

<

e2 12

π

2



5 3

+

ln



μ

2 q2

 

i kiτ τ

(

kiμμ

+

kiτ τ

)

M2 Xi

<

1

(

4

.

5 TeV

)

2

,

(12) where

q

2 _{isthecenter-of-massenergy.WhiletheresultsinRef.[3]} are derived from all LEPII energies, we take the approximation8 q2

_{= (}

_{189 GeV)}2_{. We note that this expression depends} signifi-cantlyonthe renormalizationscale

μ

,which isundeterminedin ouranalysis.9 Toobtaina roughconstraint, wetake thereference value

μ

=

1 TeV,obtaining

8 _{Here,wechooseq}2_{= (189 GeV)}2_{becauseitisthelowestLEPIIcenter-of-mass} energywithalargeluminosity(>100 pb−1_{).Ourresultsareaffectedonlyslightly} bythisapproximation.

9 _{We notethat our expression for the direct-detection cross section,Eq. (6),} is also dependenton the choice ofrenormalizationconditions. For high-energy processes,however,thesmalllogarithms aremorestronglydependent uponthe renormalizationconditions.Here,wequoteourresultsforaspecificvalueof

μ

.

Ad-−

1

(

300 GeV

)

2

<



i=1,2 kiμμ

(

kiμμ

+

kiτ τ

)

M2 Xi

<

1

(

340 GeV

)

2

,

(13)

−

1

(

250 GeV

)

2

<



i=1,2 kiτ τ

(

kiμμ

+

kiτ τ

)

M2 Xi

<

1

(

290 GeV

)

2

.

(14)

Varying

μ

between500GeVand2TeVonlychangesthescalesin Eqs.(13)and(14)by30–50 GeV.

We alsoconsiderthecorrectionstothe Z couplings to

μ

+

μ

−

and

τ

+

τ

−, asshown by the diagramsin Fig. 4. The diagram in

Fig. 4(a)isfinitewhenaddedtoexternallegcorrections;itrescales

the Z vertex, givinganewcontribution tothe Z partial width.10

Thepossiblenew-physicscontributiontothepartialwidthof Z

→

μ

+

μ

−requires



i k2_iμμ M2_X i

<

1

(

200 GeV

)

2

.

(15)

NosignificantconstraintisobtainedfromthepartialwidthofZ

→

τ

+

τ

−.

Additionally,thediagraminFig. 4(c)cancorrectthevector cou-plingsofthe Z to muonsortaus.Thisdiagramdivergesandleads tokineticmixingbetweenthe Z and X bosons andwilldependon the renormalization conditions chosen. Here, we take

μ

=

1 TeV and consider constraints from the vector coupling of the Z to

taus, gVτ

= −

0.0366

±

0.0010[2].Takingthe95% CLrangeasan

approximatemeasureofthesizeofapossiblenew-physics contri-bution,weobtain

!!

!!

!



i kiτ τ

(

kiμμ

+

kiτ τ

)

M2_X i

!!

!!

!

<

1

(

330 GeV

)

2

.

(16)

Ontheotherhand,wedonotgetanyusefulconstraintfromgVμ.

Afewcommentsmustbemadeontheseresults.First,wemust stress that, toobtain arigorousresult, itis notadequate to treat corrections to the Z vertex andcontributions to effective opera-torsseparately; afullfit,simultaneouslyincludingallnewphysics contributions,shouldbeperformed.Additionally,wenotethatour choice fortherenormalizationscale,

μ

=

1 TeV,although reason-able, isarbitrary. Forthesereasons, ourconstraints derived from LEPresultsshouldbetakenasapproximate.Wewillreturntothis whenwediscusstheparameterscansinSection4.

Lastly, ifthe X were light enough tobe producedat LEP, i.e.,

MX



200 GeV,it could beobserved via resonanceproductionin the

μ

+

μ

−,

τ

+

τ

−,ormissingenergyfinalstatesviathediagramin

Fig. 4(c).Searchesfortheresonantpeakofasneutrinodecayingto

μ

+

μ

−arerelevantfor

M

X intherange100

<

MX

<

200 GeV[4]. However, the cross section for X production is loop-suppressed, andsincethesneutrinosearchisonlyvalidforX widths

<1 GeV,

onlysmall,isolatedregionsinthe

M

Xi

−

kiμμ parameter spaceare

constrained.Similarly,analysesconstraininginvisibly-decaying res-onancesatLEParelargelyinsensitivetoourscenario[39].

3.5.2. LHC

TheATLASandCMSexperimentshavesearcheddirectlyfor res-onancesinthe

μ

+

μ

− finalstate, placinglimitsonthecross sec-tionofinclusiveresonantproductionofanewparticleand subse-quentdecayinto

μ

+

μ

− [67,68].Lepton-flavored X bosons canbe

ditionally,wepointoutthatthe5/3 termappearinginEqs.(11)and(12)isrequired tomaintainconsistencywiththerenormalizationconditionsappliedinEq.(6).

10 _{ThediagraminFig. 4(c)willalsocontributetotheZ partialwidths.However,} aftertheapplicationofthelimitsobtainedinEqs.(13)and(14),wefindthatthe contributionofFig. 4(c)tothepartialwidthissubdominanttothatofFig. 4(a).For simplicity,weignoreithere.

producedonresonanceattheLHC,viathediagraminFig. 4(c).As wasthecasewiththeLEPobservablesabove,thisdiagramdepends onrenormalizationconditions.Forthisreason,westressthata rig-orouspredictionforthe crosssection for X production atLHCis notpossible within ourmodel-independent framework. However, approximatecrosssectionspossiblewiththeLFDMframeworkcan beachieved,andtheresultsfromtheLHCexperimentsdonotyet seemtosignificantlyconstraintheLFDMparameterspace.We re-turntothiswhenwediscusstheparameterscansinSection4.

4. Parameterscans

WeillustratetheavailablephasespaceforLFDMbyperforming parameterscansbyrandomlyselectingthevaluesofthecouplings andthemassesofthe gaugebosons inEq.(1).We randomly se-lectvaluesof

k

iμμ,

k

iτ τ ,

k

iL,

k

i R ,hμτ ,

h

L,and

h

R (where

i

=

1,2) within the somewhat-arbitrary range

−

1 and 1. The masses of thegaugebosons, MX1,MX2,andMY arerandomlysampled uni-formlybetween100 GeVand10 TeV,andthevalueof

mχ is

fixed byrequiring thatthe relicdensitybe theobserved value. As dis-cussedinSection3.4,smalltomoderatevaluesof

mχ ,

i.e.,between several GeV and

O(

few

×

10 GeV), may be disfavored given re-sultsfromindirectdetection.Wedonotimposeanyconstrainton thevalue of

mχ when

performing theparameter scan, sincethis amountstoonlychangingtherangeofthe

x-axis

inFig. 5.

Foreachsetofcouplingsandmasses,werequirethatthe the-oretical and experimental values of the muon g

−

2 are within 95% CL ofeach other, whileavoiding tension withthe measured rates of neutrino trident production and

τ

−

→

μ

−

ν

τ

ν

μ, as

de-scribedin Sections 3.3and3.2.2,respectively. For a point inthe phasespace that passestheserequirements, alight-blue point is drawnforthe value

mχ and

thedirect-detectioncross section.A dark-bluepointisdrawnforthoseregionsofthephasespacethat additionallysatisfy theconstraintsfromLEPinEqs.(13)–(16).We reiteratethat the constraints fromLEP are approximate, andthe dark-bluepointsshould betakenasillustrativeofthe rough con-straintsfromLEP.We comparethesescansoftheavailable LFDM phasespacewiththeresultsfromLUX[40]andSuperCDMS[69], as shown in Fig. 5. The LEP constraints affect only the light X

gaugebosonswithmassesofonlyafewhundredGeVorless.We randomlysampleover109 _{pointsinourphasespace,andwefind} that,forawiderangeofcouplings,thelighterofthetwo X gauge

bosonsis preferredto haveamass ofa few hundredGeV, while thepreferredvalue of

M

Y ismultipleTeV.Histogramsofthe val-uesof

M

X1,

M

X2,and

M

Y areshowninFig. 7.

11

We then investigate the implications of LFDM for resonant searchesattheLHC.Scanningover thesamepoints asabove, we roughlyestimatethe pp

→

X cross sectionsachievableattheLHC, assuming a renormalization scale of

μ

=

1 TeV and

√

s

=

7 TeV. Weuse MadGraph5[70]toperformleading-orderresonant cross-section calculations for production of pp

→

X

→

μ

+

μ

− arising fromthediagraminFig. 4(c).Wedonotdoafull mixing calcula-tion,andweneglectinterferencebetweenFig. 4(c)andtheSM.12

WefindthattheresultingcrosssectionsintheLFDMscenariofor

μ

=

1 TeV andcouplingsclosetounityarewithinafactorofthree ofthecurrentlimitsfromCMS[71]for300 GeV

<

MX

<

500 GeV,

11 _{WehavecheckedandfoundthatM}

X1,MX2,MY<mχforonlyO(1%)of phase-spacepointscoveredinthescans.Thisjustifiesourassumption,madeinSection3.1, thatdark-matterannihilationstoX andY gaugebosonscanbeneglected.

12 _{Wecalculatetheresonantcrosssectionforpp→X→μ}+_μ−_forreference val-uesofthe X -leptoncouplingsandwidthX.Becausethecrosssection,toagood

approximation,scalesastheinverseofX,weareabletorescalethese MadGraph5

resultstoaccountforthevaryingvaluesoftheX -fermioncouplingsinthescans.

Fig. 5. Thedark- andlight-bluepointscorrespondtothescanofthedirect-detection crosssection,asdiscussedinSection4,withandwithouttheLEPconstraints, re-spectively.Theredandpurplelinesarethe90%CLupperlimitsfromtheLUX[40] andSuperCDMS[69]experiments,whichhavebeenrescaledbyafactorofZ2

/A2 forXeandGe,respectively.(Forinterpretationofthereferencestocolorinthis fig-ure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6. ThesamescanstoproduceFig. 5areplottedherefortheresonant cross sectionofpp→X→μ+μ−attheLHC,for√s=7 TeV andarenormalizationscale

μ =1 TeV,comparedwiththe95%CLupperlimitsfromCMS[71].Interferencewith theSMhasbeenneglectedandcansignificantlyaffecttheresults,asdiscussedin Section4.

even when constraints from LEP are included. Our results are showninFig. 6.

However, we note that these resultsare only approximate. In particular, the cross section for X production becomes small as

MX approachestherenormalizationscale

μ

;inthiscase,the ap-proximationofneglecting theeffects ofSMinterference withthe diagram of Fig. 4(c)becomes invalid. We find that the effects of interferencewiththe SM are significantandcannot be neglected for300 GeV

<

MX

<

500 GeV. Additionally, interferencewith the SM can cause the shape of the resonance to deviate from that ofaBreit–Wigner, whichcanaffectthesensitivityofsearchesfor narrow resonances.Due to theseeffects,models withlow renor-malization scales13 _{might require dedicated experimental}

analy-ses. For these reasons, we extend the x-axis of Fig. 6 to only

MX

=

500 GeV.However,astheamplituderesultingfromFig. 4(c) 13 _{Wenotethattherenormalizationscale}

_μ

_{dependsontheultravioletcompletion} ofaparticularmodel,andisundeterminedinouranalysis.

Fig. 7. ThesamescanstoproduceFig. 5areplottedhereforthedistributionofthevaluesof(a)MX1 andMX2 (herecombinedandcalledMX)and(b)MY.Thelighterof

thetwoX gaugebosonsispreferredtohaveamassofafewhundredGeV,whilethemassofheavieroneisrelativelyunconstrainedfromabove.TheY gaugebosoncannot haveamass2 TeV duetoconstraintsfromthemuong−2.They-axesoftheseplotsarenotidenticalandareinarbitraryunits.

growsinsizewithincreasing

μ

,ourresultsdoindicatethat mod-els with somewhat larger renormalization scales would be less affected by interference with the SM and would give cross sec-tionssimilartothoseshowninFig. 6.Therefore,wepointoutthat ifalocalexcessisseenattheLHCinthe

μ

+

μ

− channel,butnot inthe

e

+e−channel,LFDMwouldbeacandidateexplanation.

Toillustrate the interplaybetweenthe observables inthe pa-rameterscans,webriefly mentiontwotoy models.First,we con-sider a U

(1)

flavor modelwherethe

μ

and

τ

flavorshaveequal andoppositecouplingstoasingle

X boson

(thisisthewell-known

U

(1)

Lμ

−

Lτ model, thediscussionofsucha setupcanbefound

inRef.[72]andreferencestherein).Weallowmixingbetween fla-vored andunflavored statesinthedarksector, andthus takethe magnitudeoftheright- andleft-handedcouplingsofthedark

mat-tertothe X equal toorlessthanthatofthemuon.Wefindthat

such amodelcan satisfythe relicdensity,directdetection, g

−

2 and tau-decay constraints; X production at colliders is severely suppressedby cancellationbetween

μ

- and

τ

-loopcontributions. However, this toy model is ruled out due to tension between

g

−

2 andthelimitsfromneutrinotridentproduction.Second,we briefly consider an SU

(2)

model, where

μ

- and

τ

-flavored fields areplacedintoaflavordoublet.(Forprevioususeof

SU

(2)

in fla-vor models, see Ref. [72].) This model contains X and Y bosons

ofequalmass,bothofwhichcontributeto g

−

2.Duetothe ten-sionbetweentaudecays,whichrequirethenewphysicsscale for

Y -mediated processestobe above

∼

2 TeV,andg

−

2,which

fa-vors a scale below 1.9 TeV, this model is marginally ruled out. (While the X boson also contributes to g

−

2, this contribution isinadequatetoresolvethistension.)Althoughitisreasonableto speculatethat thislattermodel couldbe embedded intoa larger model with additional contributions to g

−

2 from X bosons or whichcontainsamasssplittingbetweenthe X and Y , wedonot attemptfurthermodel-buildinghere.

Lastly, we briefly mention that the phase space ruled out by directdetectionhasnomodel-independentramificationonthe dis-coverypotentialattheLHC,i.e.,thepointswithintheLUXlimitsin

Fig. 5inhabittheentirephasespacefortheLHC.Anyspecific map-pingbetweendirectdetectionandhadron collidersmustbe done withaspecificmodel.Additionally, togeta roughidea ofthe ef-fectindirectdetectionconstraintscanhaveonourLHCresults,we repeatthescan inFig. 6butrequiring

mχ

>

50 GeV.Wefindthat ifindirectdetectionrules outlow-massdarkmatter candidates,it

doesnot qualitatively changetheavailable phase spaceforLFDM attheLHC.

5. Lepton-flavorviolation

Up tothispoint,we haveonly consideredthecasewhereLFV among chargedleptons isnegligible.Here, webriefly explore the possibility of allowing a small amount of

μ

–

τ

flavor violation, still assuming that the flavor gauge bosons couple negligibly to electrons. The interactions introduced in thissection, iftaken in isolation, are likely inadequateto account formuon g

−

2. How-ever, wepresentthemasan indication ofthelevel ofLFVwhich maybeallowedinmorecompletemodelsofflavoreddarkmatter. The notation using X and Y bosons in Eq. (1) is no longer adequate if we allowfor LFVamong chargedleptons, so we will discussLFVusingeffectivefour-leptonoperators

Oi jkl

≡

Ci jkl

(

1

+ δ)

2



i



αi j

j

 "

k



αkl

l

#

,

(17)

where

a

isachargedleptonofeithermuon

(

a

=

μ

)

ortau

(

a

=

τ

)

flavor,and

δ

=

1(0)if

i

=

k and j

=

l (i

=

k or j

=

l). Whilewe as-sume vector interactions in the flavor interaction basis, we have generalizedtheLorentz structurefrom

γ

α to



α , sincethe inter-actions maydeviate from purely vector when rotatingthe right-andleft-handedsectorstothemassbasis.

Onlythreeoftheseoperators,

O

τ μμμ,

O

μτ τ τ , and

O

τ μτ μ, vi-olate lepton flavor. The first of these,

O

τ μμμ , will contribute to thedecay

τ

−

→

μ

−

μ

−

μ

+,whichisconstrainedtohavea branch-ing fractionof less than 2.1

×

10−8 at 90% CL[1]. The operator

O

μτ τ τ can be constrained by assuming that it must be

consis-tentwiththebranchingfractionof

τ

−

→

μ

−

γ

,whichhasavalue lessthan4.4

×

10−8 at90% CL[1].Lastly,becauseofisospin sym-metry, one generically expects that

O

τ μτ μ can give rise to the process

τ

−

→

μ

−

ν

μ

ν

¯

τ . Becausethisprocessisexperimentally

in-distinguishable fromtheSM process

τ

−

→

μ

−

ν

τ

ν

μ, aconstraint

on thebranchingfractionforthisprocess canbe obtainedby as-suming that it must be consistent with the measured ratio of

(

τ

−

→

μ

−

ν

τ

ν

¯

μ

)/ (

τ

−

→

e−

ν

τ

ν

¯

e) at90%CL [1].The limitson

|

Ci jkl

|/

2 are shown inTable 2. Forsimplicity, we only consider the cases



_{i j}α

= 

_klα

=

γ

α

,

γ

α

₍₁

_±

_γ

5

)/2;

in general,



αi j need not equal



_klα.

AsshowninTable 2,theoperator

O

τ μμμ is morestrongly con-strainedthantheothertwoflavor-violatingoperators.Thisimplies,

Table 2

Limitson|Ci jkl|/2,forthecasesi jα= klα=γα,γα(1±γ5)/2 at90%CL.

|Ci jkl|/2 αi j= klα=γα  α i j= αkl=γα(1±γ5)/2 |Cτ μμμ|/2 1/(15 TeV)2 1/(11 TeV)2 |Cμτ τ τ|/2 1/(2.5 TeV)2 1/(1.5 TeV)2 |Cτ μτ μ|/2 1/(0.7 TeV)2 1/(0.5 TeV)2

forexample,thataflavorgaugebosonwhichcouplestoboth

μμ

¯

and

τ μ

¯

musthaveamass

>

O(

10 TeV)oratleastonesomewhat smallcoupling.However, theseconstraintsdoleaveopen another interestingpossibility.Wecanintroducea

U

(1)

LFVflavorgauge bo-son

V which

interactswithasinglegenerationofleptons.Wetake thatgeneration to be onlyquasi-aligned withthe chargedlepton masseigenstates.Specifically,wetakethe

V boson

tointeractwith anadmixtureof

μ

and

τ

,

L

LFV

⊃

k3Vα

(

L3

γ

αL3

+

R3

γ

α

R3

+

ν

R3

γ

α

ν

R3

),

(18) where

L3

=

Lτcos

θ

L

+

Lμsin

θ

L

,

(19)

R3

=

τ

Rcos

θ

R

+

μ

Rsin

θ

R

,

(20)

andthe

τ

componentdominates,i.e.,cos

θL

∼

cos

θR

∼

1.For sim-plicity,weignoreneutrinomixingandneglectapossiblecomplex phaseinEqs.(19)and(20)andconsiderthecasesin

θL

∼

sin

θR

≡

sin

θ,

although thisdoesnot hold generally. Inthis case,

O

τ μμμ

wouldbesuppressed bythree powers ofsin

θ,

while

O

τ μτ μ and

contributionsto g

−

2 would be suppressed bytwo such factors. Additionally,afour-

τ

operator,

O

τ τ τ τ , suffersnosuchsuppression, andisexperimentallyunconstrained.

We note that these interactions which violate charged-lepton flavor,takenalone,arenotusefultoaccountforthediscrepancyin

g

−

2.Presumablytheywouldneedtobeincorporatedintoamore complete model of LFDM. A V boson with an electroweak-scale mass,i.e.,

∼

1 TeV, andamixingangleaslargeassin

θ

∼

O(

0.1) isphenomenologicallyallowed.Thus,it ispossibletohave signif-icantTeV-scale

μ

–

τ

flavor violation, aslongasthecouplings are chosencarefully.However,wedonotattempttoincorporateflavor violationintoalargermodelhere.

6. Summaryandconclusions

Motivated by the apparent tension between the dark-matter relicdensityandnullresultsfromdirectdetection,aswellasthe 3.6

σ

discrepancy between the measured and predicted value of the muon g

−

2, we posit that dark matter and the mystery of lepton flavor may be related. In our scenario, dark matter does not merely couple primarily to leptons, but it shares a common gaugedflavorinteractionwiththeleptonsintheSM,underwhich onlythe muon and taufamilies are charged.We investigatethis scenario by creating a Lagrangian composed of d

=

4 operators, allowing the SM leptons to have both flavor-diagonal and flavor-changingcouplingstoflavor gaugebosons. Consideringthemuon

g

−

2 andthelackofobservedLFVamongchargedleptons,wetake suchinteractionstobepurelyvector inthechargedleptonsector. However,flavormixingmaybelarge inthedarksector, so differ-entcouplingsforleft- andright-handeddarkmatterarepermitted. Wethenconstrainourscenariousingthedark-matterrelicdensity anddirect detection, low-energy measurements, constraints from LEP,andneutrinotridentproduction,whiledemandingconsistency withthemeasuredvalue ofthemuon g

−

2.Futureprospectsfor LHCanddirectdetectionareinvestigated.

In the LFDM scenario, both the dark-matter relic density

and the muon g

−

2 suggest that such interactions between

dark matter and leptons can exist at the electroweak scale, i.e.,

O(

100 GeV)–

O(

1 TeV). While direct detection,

τ

−

→

μ

−

ν

τ

ν

μ,

and

μ

+

μ

− and

τ

+

τ

− production at LEP do constrain the cou-plingsandmassesoftheflavorgaugebosons,significantparameter spaceremains.ThisisillustratedinFigs. 5 and 6,whereweshow the possible valuesof thecross sections fordirect detectionand resonant production at the LHC, allowed by present constraints. We findthatthereare largeranges offlavorgauge bosonmasses and couplings which give cross sections below but near current sensitivity for both direct detection experiments and resonance searchesattheLHC.

Futuremeasurementsbringsignificanthopeforfurther investi-gationsofLFDM.TheparticularparameterizationofLFDMexplored inthisworkcanaccountforallexperimentalobservations, includ-ing themuon g

−

2,ifthereare bothflavor-diagonal interactions atascaleofafewhundredGeVandoff-diagonalinteractionsata fewTeV.Improvementsinthesensitivityofdirect-detection exper-iments,theprecision ofthemuon g

−

2 measurement,the preci-sionofthebranchingfractionof

τ

−

→

μ

−

νν

,theprecisionofthe rateofneutrinotridentproduction,andresonancesearchesatthe LHCcanfurtherconstrain,orpotentiallydiscover,LFDM. Addition-ally, flavor-violating observables, such as

τ

−

→

μ

−

μ

−

μ

+, could be usefulin constraining orconfirming specific modelsof LFDM, whileprocessesthatinvolveelectrons,e.g.,

μ

→

3e,

μ

→

e

γ

,and

μ

–e conversion,areexpectedtobehighlysuppressed.

The particular parameterization of LFDM considered in this work permitstwo contributions tothe value ofthe muon g

−

2: one contribution froman X boson andanotherfrom a Y boson.

Taken separately, these two interactions are nominally ruled out vianeutrinotridentproductionandtaudecays,respectively.If con-tributions from X and Y bosons are considered simultaneously, then the constraints fromthe muon g

−

2, neutrinotrident pro-duction, andtau decays can each be satisfied at 95% CL, though thereremainssomeoveralltensionbetweenthisparameterization ofLFDMandthedata.Wenote thatthetheory andexperimental valuesofthemuon g

−

2 arepoisedtochangeinthenearfuture, andthe LFDMframework can accommodatea rangeof contribu-tionsto

g

−

2,particularlythosesomewhatsmallerthanthevalue currentlysuggestedbyexperiment.If,forexample,themuon

g

−

2 agreeswiththeSMprediction,higher-mass

X and Y bosons

would easily account for the data. As such, g

−

2 is not necessary for LFDM, though we suggest that given our presentframework, we mightexpectfuturemeasurements andfuturetheorycalculations of

g

−

2 tocomeintobetteragreement.

Wealsopointoutafewpossibleextensionstothiswork.While we haveassumednegligiblecouplingsof theflavor gaugebosons toelectrons, theadditionofsuch smallcouplingscouldbe inves-tigated. Additionally, the possibility of charged-lepton flavor vio-lation, only touchedupon here, could be studied in more detail. And,ofcourse,theconstructionofmoreconcretemodelsofLFDM wouldalsobeaworthyendeavor.

Inconclusion,theLFDMframeworkcanaddressthedark-matter relic density andconstraints from LEP, while explainingthe lack of an observed signal in direct-detection experiments and the muon g

−

2.Lastly, itholds significant promise forfuture exper-iments.

Acknowledgements

We thankAndré deGouvêa forproviding usefulfeedbackand commentsandJulian Heeckforpointingout theconstraintsfrom neutrinotridentproduction.TheworkofA.K. issponsoredinpart bytheDOEgrantNo. DE-FG02-91ER40684andbytheDepartment of Energy Office of Science Graduate Fellowship Program (DOE SCGF),madepossibleinpartbytheAmericanRecoveryand

Rein-vestmentActof2009,administeredbyORISE-ORAUundercontract No. DE-AC05-06OR23100. J.K. is supported in part by the DOE grant No. DE-FG02-97ER41029. The work of A.S. is supported in partbytheDOEcontractNo. DE-AC02-98CH10886(BNL).

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