Dark gauge bosons: LHC signatures of non-abelian kinetic mixing

Dark gauge bosons: LHC signatures of non-abelian kinetic mixing

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Argüelles, Carlos A., et al. “Dark Gauge Bosons: LHC Signatures of

Non-Abelian Kinetic Mixing.” Physics Letters B, vol. 770, July 2017,

pp. 101–07. © 2017 The Authors

As Published

http://dx.doi.org/10.1016/j.physletb.2017.04.037

Publisher

Elsevier

Version

Final published version

Citable link

http://hdl.handle.net/1721.1/118442

Terms of Use

Creative Commons Attribution 4.0 International License

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Dark

gauge

bosons:

LHC

signatures

of

non-abelian

kinetic

mixing

Carlos

A. Argüelles

a

,

b

,

c

,

Xiao-Gang He

d

,

e

,

f

,

Grigory Ovanesyan

g

,

Tao Peng

a

,

Michael J. Ramsey-Musolf

g

,

h

,

∗

a_{DepartmentofPhysics,UniversityofWisconsin,Madison,WI53706,USA} b_{WisconsinIceCubeParticleAstrophysicsCenter,Madison,WI53703,USA} c_{MassachusettsInstituteofTechnology,Cambridge,MA02139,USA}

d_{INPAC,DepartmentofPhysicsandAstronomy,ShanghaiJiaoTongUniversity,Shanghai200240,China} e_{DepartmentofPhysics,NationalTaiwanUniversity,Taipei10617,Taiwan}

f_{PhysicsDivision,NationalCenterforTheoreticalSciences,Hsinchu30013,Taiwan}

g_{AmherstCenterforFundamentalInteractions,DepartmentofPhysics,UniversityofMassachusettsAmherst,Amherst,MA01003,USA} h_{KelloggRadiationLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received15May2016

Receivedinrevisedform4April2017 Accepted18April2017

Availableonline20April2017 Editor: B.Grinstein

We consider non-abeliankineticmixingbetweenthe Standard ModelSU(2)L and adarksector U(1) gaugegroupassociatedwiththepresenceofascalarSU(2)Ltriplet.Themagnitudeoftheresultingdark photoncoupling



isdeterminedbytheratioofthetripletvacuumexpectationvalue,constrainedtoby 4 GeV byelectroweakprecisiontests,tothescaleoftheeffectivetheory.Thecorrespondingeffective operator Wilson coefficientcan be O(1) whileaccommodatingnull resultsfor dark photon searches, allowingforadistinctiveLHCdarkphotonphenomenology.AfteroutliningthepossibleLHCsignatures, weillustratebyrecastingcurrentATLASdarkphotonresultsintothenon-abelianmixingcontext.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Thesearch forweakly coupledlight vector bosonshas beena subjectofconsiderableinterestinrecentyears.Searcheshavebeen carriedoutinanumberofdifferentcontexts,includinglowenergy colliders,mesondecays,beamdumpexperiments,andhigh-energy colliders(see, e.g., Refs. [1,2] and references therein). Theoretical studies typically assume that interactions of the “dark photon” withthevisiblesectoraremediatedbyabeliankineticmixing be-tween the Standard Model (SM) hypercharge andthe dark U

(

1

)



gaugegroups[3–5].Forthe“darkZ ”,mixingwiththeSM Z -boson

mayalsooccurvia themasstermsintheLagrangian[6,7].Forboth abelianandmass-mixing,theeffectsariseatthelevelof renormal-izableoperators. Theresultingcouplingofthe darkvector bosons totheSMarethenparameterizedbyadimensionlessparameter



thatis constrainedbyexperimentto be



10−3 _{orsmallerwhen}

fordarkboson massesbelow

∼

10 GeV.Thesmallscale of



has noobviousorigininthiscontext,soonemustresorttomodelsto explainwhyitisnot

O(

1

)

.

*

Correspondingauthor.

E-mailaddresses:caad@mit.edu(C.A. Argüelles),hexg@phys.ntu.edu.tw

(X.-G. He),ovanesyan@umass.edu(G. Ovanesyan),tpeng23@wisc.edu(T. Peng),

mjrm@physics.umass.edu(M.J. Ramsey-Musolf).

In thisstudy, we observethat non-abelian kinetic mixing be-tween the U

(

1

)

 and the SM SU

(

2

)

L gauge groups, encoded in

non-renormalizable operators, can provide a simple explanation withoutassumingtinyoperatorcoefficientsintheeffectivetheory. DoingsorequiresaugmentingtheSMfieldcontentwithadditional bosons gaugebosonstransformingnon-triviallyunderSU

(

2

)

L.For

concreteness, we consider the scalar triplet1



∼ (

1

,

3

,

0

,

0

)

and focusonthedimension-fiveoperator

O

(5) W X

= −

β



Tr



Wμν





Xμν (1.1)

where Xμν andW μν aretheU

(

1

)

andSU

(

2

)

L fieldstrength

ten-sors,respectively;



= 

aTa _withTa_beingtheSU

₍

₂

₎

L generators;

and



is the mass scale associated with fields that have been integratedoutingeneratingtheoperator.A non-zerovacuum ex-pectationvalue



0



≡

v willlead tomixingbetweenthe U

(

1

)



boson Xμ and the neutralSU

(

2

)

L gauge boson Wμ.3 The mixing

parameteristhengivenby



= β

sin

θ

W



_v 





,

(1.2)

1 _{We listthe quantumnumbersintheorder SU(3)}

C×SU(2)L×U(1)Y×GD,

whereGDisthedarkgaugegroup. http://dx.doi.org/10.1016/j.physletb.2017.04.037

0370-2693/©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

102 C.A. Argüelles et al. / Physics Letters B 770 (2017) 101–107

where

θ

W istheweakmixingangle.Fornon-vanishingmixing

pa-rameter, Xμ inheritsall couplingsof the photon to SM fermions butrescaled by the universal factor



,whose magnitude is con-trolledbythescaleratiov

/

.Importantly,constraintsfrom

elec-troweak precision tests constrain the triplet vev to be relatively small: v



4 GeV. Thus,



will satisfy the experimental bounds

for



largerthanaboutoneTeVfor

β

∼

O(

1

)

.

The idea of non-abelian kinetic mixing is not original to us. TheauthorsofRef. [8]consideredU

(

1

)

Y

×

SU

(

2

)

,withthelatter

factor beinga dark SU(2)gauge group [9]. Dark SU(2) gauge in-variancerequiresintroductionofanadditionalscalartriplet



D

∼

(

1

,

1

,

0

,

3

)

, allowing for a dimension five mixing operator analo-goustothatofEq.(1.1).Incontrasttothe presentcase, however, thedarktripletvevcanhaveanymagnitude,andforlargevalues, a small



requires a commensurately small operator coefficient. Ina follow-upwork [10] applicationsforastrophysical anomalies andotherconstraintsarestudiedinthisscenario.InRef. [11]this non-abelian kinetic mixing is used to explain the X-ray line at 3.55 keV.

More recently, the authors of Ref. [12] considered SU

(

2

)

L

×

U

(

1

)

kineticmixingvia thedimensionsixoperator

C



2H

†_Ta_{H W}a

μνXμν (1.3)

whereH istheSMHiggsdoublet,leadingto



∼

C

(

v

/)

2. Assum-ing this operator arises at one-loop, one has



∼

4

π

mϕ , where

mϕ is the mass of the mediator

ϕ

in the loop. For





10 TeV (ormϕ



1 TeV),onemaysatisfy theexperimentalconstraintson



for C

∼

O(

1

)

. The authors of this work consider an explicit model witha scalar mediator

ϕ

∼ (

1

,

3

,

0

,

qD

)

and a dark Higgs

hD

∼ (

1

,

1

,

0

,

qD

)

that isresponsible forgeneratingthedark

pho-tonmass.A detailedanalysisofthe collidersignaturesassociated withthedarkbosonsisgiven.

Inwhat follows,weconcentrate onthe collidersignatures as-sociated with the dimension five operator (1.1) rather than on constructionof anexplicit darksectormediator model.In partic-ular, we note that final states containing one ormore X bosons

maybeproducedthroughtwodistinctmechanisms,eachofwhich involves

O

(_{W X}5) directly:(1)Drell–Yanpair productionof



states,

pp

→

V

→ 

, followed by the

O

_{W X}(5) -induced decay



→

X V ,

resulting in a X X V V topology; (2) direct production via

O

(_{W X}5) ,

pp

→

V∗

→

X



,followedbythe decay



→

X V ,generatinga fi-nalstateofthetopology X X V .Forsufficientlylarge

β/

thedirect production mechanism(2) may dominate. In this case, v must

be sufficientlysmall toensure the experimental constraintson



are satisfied. Conversely, forsmaller

β/

(larger v fora given



),productionwilloccurprimarilythroughtheDrell–Yanprocess.2 Forsimilarreasons,the



-decaybranchingratioswillalsocarrya dependenceon

β/

(and,thus,onvforfixed



).Inwhatfollows,

we delineate severalgeneral parameter space regimesassociated withthisinterplayofparameters.

Forconcreteillustration,wethenconsiderthepresentLHC sen-sitivityfor the regions ofparameter space wherethe direct pro-ductionmechanismdominatesandwherethe



→

V X branching

ratio is close to unity. For this parameter space region and for

mX

>

2mμ, one expects displaced vertices associated with X

→

μ

+

μ

− decays,wherethedimuonpairappearsasaleptonjet.The

2 _{Inprinciple,thesamesetofpossibilitiesappliestotheoperator(1.3);in}

prac-tice,theyarelesslikelytoberealized,sincetheminimumvalueofisroughlyten timeslargerthanfortheinteraction(1.1)andsincethedimensionsixoperator car-riesaquadraticdependenceontheinversemassscale.Thus,considerationofthe darksectormediatorsresponsiblefor(1.3)asanalyzedinRef.[12]maybethemost promisingprobeinthelattercase.

ATLAScollaborationhasperformedasearchforeventsofthistype thatinvolvetwoorfourleptonjets[13].Wecarryoutasimple re-castofthecorresponding ATLASboundonlong-liveddarkbosons forourscenario,notingthattheATLASsearch isinclusiveand ac-commodatesadditional,unobserved, final state SM gauge bosons. FordarkbosonmassmX in therange0

.

2 GeV

≤

mX

≤

2 GeV we

find that the presentATLAS exclusioncan extendto

/β

∼

sev-eralhundredGeV,dependingonthevalueofv.Aswediscussin

Section4,thepresentreachmaylieontheborderoftheregionof validity oftheeffectivetheory.Consequently,one shouldconsider ourresultsasindicativeoftheLHC8TeVsensitivitytothe param-eters ofthisscenario ratherthan asquantitativelydefinitive. We, thus,alsodiscussthepossibilities forfutureLHCtestsofthis sce-nariothatwouldprobehighermassscales,includingsearchesthat wouldidentifytheSMfinalstategaugebosons.

Ourdiscussionofthisscenarioandcollideranalysisisorganized asfollows.InSection2wereviewthesetupofthetriplet-assisted non-abeliankineticmixing.InSection3weoutlinedistinctiveLHC signatures for our scenario andin Section 4 we present the re-castofATLAS boundsondarkphotonsforthenon-abeliankinetic mixing.Finally,weconcludeinSection5.

2. Themodel

We add toSM Lagrangian dimension four operators involving dark photon andthereal triplet fields,aswell asdimension five effectiveoperators:

L

=

L

SM

+

L

(d=4)

+

L

(d=5)

+ . . . .

(2.1)

Thedimensionfourandfiveoperatorswetaketobeoftheform:

L

(d=4)

= −

1 4XμνX μν

₊



0 2 cW BμνXμν

+

Tr



Dμ





† Dμ





−

V

(,

H

)

+ ˜

L

(d=4)

,

L

(d=5)

= −

1



Tr



Wμν



 

α

Bμν

+ β

Xμν



≡

O

(W B5)

+

O

(5) W X

.

(2.2)

Here,

L

(d=4)containstheusualabelian( X B)kineticmixingterm andcW isthecosineoftheweakmixingangle.Theterms

break-ing the dark U

(

1

)

 gauge group are not explicitlypresented and arepartof

˜

L

(d=4)_{.Therealtripletfield}

_

_andthescalar

triplet-doubletpotentialaregivenby[14]:



=

1 2





0

√

2



+

√

2



−

−

0

,

Dμ



= ∂

μ



+

ig

₃



a=1 W_μaTa

, 



,

(2.3) V

(

H

, )

= −

μ

2_H†_H

_{+ λ}

0



H†H



2

−

μ

2 G

+

b4G2

+

a1H†



H

+

a2H†H G

,

(2.4) where G

≡

Tr



†



=

 02

2

+ 

+



−.In the notation of Ref. [14], G

=

F

/

2.

GivenaUVcompletetheoryonemayintegrateoutheavystates that haveboth SM anddark charges,asillustrated inFig. 1. We leave the model-dependent details ofthe full theory unspecified, focusing instead on

O

(_{W X}5) and the corresponding collider phe-nomenology. In addition it is possible that similar graphs as in Fig. 1 generate the effectivedimension five operator

O

(_{W B}5).After

Fig. 1. Feynmangraphsthatmaygeneratenon-abelianmixingSU(2)L×U(1).Here,themediatorsintheloopmaybe(a)fermions,(b)scalars,or(c)otherdegreesoffreedom

associatedwithnon-perturbativedynamics.

Fig. 2. FeynmangraphsforLHCproductionanddecayofthe particlesinthe triplet-assistednon-abelianmixingmodel.Diagrams(a,b)indicatescalarpairproduction, followedbyO(5)

W X-mediatedscalardecays.Diagrams(c,d)indicateO (5)

W X-mediatedproductionanddecays.Inallgraphs,theincomingvectorbosonisvirtual.

electroweak symmetry breaking (EWSB), this operator will con-tributetothe S parameter:

α

emS

=

4cWsW

α

v



.

(2.5)

This sets a 90% CL bound

α

v

/



0

.

0008.We will henceforth

set

α

=

0 andconcentrateonthephenomenology associatedwith

O

(5)

W X.

Before proceeding, we comment here that kinetic mixing of gaugebosonscanalsobe realizedfornon-abeliangroups. For ex-ample, for a SU(N)

×

SU(M) gauge theory with gauge fields W

and Y ,one can introduce a scalar field

ab transforming asthe adjointrepresentationunderboththeSU(N)

×

SU(M) groups,with indices “a” and “b” corresponding to SU(N) and SU(M), respec-tively.Inanalogywith

O

_{W X}(5) ,onecanconstructthed

=

5 operator

Wa_{μν Y}b

μν

ab. A non-vanishing vev for

ab will lead to kinetic

mixing between W and Y . One may also construct renormaliz-ablemodelsthatgeneratethisoperatorattheone-looplevel.We deferadetailedconsiderationofthispossibilitytoafuturestudy. 3. Colliderphenomenology

Inthepresenceof

O

(_{W X}5) ,thecolliderphenomenologyassociated withthe realtriplet can differsubstantially from what hasbeen considered previously in Ref. [14]. To illustrate the key features, wewillmakethefollowingassumptions:

(a) The potential parameters are chosen so as to render the doublet-triplet mixing angle – proportional to v – to be

small, butnon-vanishing.Inthiscasetheneutralscalarsector will consist of twostates, H1,2,with H1 beingprimarily the

SM Higgs boson and H2 beingprimarily



0. Inthe charged

scalar sector, doublet-triplet mixing impliesthat thephysical chargedtripletstatesH±arenotpuretripletstates,butrather mixtures of



± andthechargedcomponentsof thedoublet, with the other combination providing the longitudinal com-ponents of themassive weak gaugebosons. Note that inthe absenceofdoublet-tripletmixing,SU

(

2

)

L

×

U

(

1

)

Y gauge

invari-anceprecludes



fromcouplingtotheSMfermions.The pres-enceofa non-vanishing mixinganglethenintroduces a

cou-plingof H±, H2 to theSMfermions throughtheSM Yukawa

interactions.3

(b) For v

=

0, the triplet states havea common mass, give by

m2_

= −

μ

2



+

a2v

2

_/

_{2.Electroweakloopsraisethemassofthe}

chargedcomponentswithrespecttothat oftheneutral com-ponent by

∼

166 MeV,allowing forthe decay H+

→

H2

π

+.

Ourchoice ofthe potential parameters will not substantially alterthissplittingevenforv

=

0.

With thesecomments inmind, we now consider the production anddecaysofthetriplet-likescalars.

3.1. Production

The LHC production and decay mechanisms of interest are shown in Fig. 2. Graphs (a) and(b) indicate Drell–Yan pair pro-duction, pp

→

V∗

→ φφ

, where

φ

denotes any of the physical scalars,withthesubsequentdecays

φ

→

X V ,leadingtothe topol-ogyX X V V .Asdiscussedabove,the

φ

stateswillbepredominantly triplet-like. Graphs (c)and(d) show the

O

(_{W X}5) -mediated produc-tion pp

→

V∗

→ φ

X , witha subsequentdecay

φ

→

X V ,leading tothetopology X X V .(FeynmanrulesfortheverticesinFig. 2are listedintheAppendix A.)

In Fig. 3 we show the LHC productioncross sections for dif-ferent channels at

√

s

=

8 TeV. The left panel corresponds to

mφ

=

130 GeV andtherightonecorrespondstomφ

=

300 GeV.For

bothmassesweobservethatfor

β/



1

/

TeV theDrell–Yanpair production dominates, while for

β/



1

/

TeV

O

(_{W X}5) -mediated productionisthedominantmechanism.For

√

s

=

14 TeV the cor-respondingtransitionbetweenDrell–Yanand

O

(_{W X}5) -mediated pro-ductionoccursforapproximatelythesamevalueof

β/

.

3.2. Triplet-likescalardecaybranchingratios

The triplet-like scalars H± and H2 will decay to W±X and Z

/

γ

X respectively as well asto other final states asconsidered

3 _{Forgenericchoicesofscalarpotentialparameters,themagnitudeoftheneutral}

doublet-tripletmixinganglefallswellbelowtheupperboundimpliedby Higgs-bosonsignalstrengths[15].SeeRef.[14]foradetailedanalysisofthedependence ofthemixingangleonthepotentialparameters.

104 C.A. Argüelles et al. / Physics Letters B 770 (2017) 101–107

Fig. 3. Productioncrosssectionsforpp→V→ φφandpp→V→Xφforassociatedtriplet-likestatesφ=H+,H2andadarkphotonX withmassmX=0.4 GeV.Forthe

finalstatescontainingasinglechargedscalarandoneneutralboson,wehavesummedthecrosssectionsforbothcharges[e.g.σ(H+H2)+σ(H−H2)].Theleftandright

panelscorrespondtomφ=130 GeV andmφ=300 GeV,respectively.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtotheweb

versionofthisarticle.)

Fig. 4. BranchingratiosforH+ decaysasafunctionofβ/(bottomhorizontalaxis)and(upperhorizontalaxis)formX=0.4 GeV.Thetop(bottom)rowcorresponds tov=1 GeV (v=10−3GeV),whiletheleft(right)columncorrespondstomH+=130 GeV (mH+=300 GeV).Thesolidblacklineindicatesthebranchingratiofor H+→W+X .Branchingratiosforotherfinalstatesareasindicatedbythelegendinsert.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.)

inRef.[14].Forillustrativepurposesweshowthedecaywidthfor

H±

→

W±X ,whichissufficientfortheanalysisthatweconsider below.Thetreelevel H±

→

W±X decayrateisgivenby

(

H±

→

W±X

)

(3.1)

=

1

−

2(m 2 X+M2W±) M2 H±

+

(m2X−M2W±)2 M4 H± 16πMH+

×



1 2



M2_H±

−

m2X

−

M2W±



2

+

M2_XM2_W±



β

2



2c 2 ∓

,

where c_∓ is the mixing angle associated with diagonalizing the charged scalarsector. Combined with the other H+ decay

chan-nels [14] we compute the branching ratios shown in Fig. 4. The left andright panels correspond to mH+

=

130 GeV and mH+

=

300 GeV, respectively. The top panelscorrespond to v

=

1 GeV

andthebottomonesto v

=

1 MeV.

From the plots inFig. 4 we seethat for v

=

1 GeV, a value

near the maximum allowed by electroweak precision tests, the branching ratio for H+

→

W+X is essentially 100% when





10−4.Forthesmallervalueofv

=

1 MeV,thebranchingratiois

essentially 100%forall valuesof



.This translatesinto therange

β/



0

.

1

/

TeV for thebranching ratioto beessentially 100% in-dependent on thevalue ofthe vev. Forlower valuesof

β/

any branchingratiofromzerotooneispossible,andtheprecisevalue dependsstronglyonthevaluev.

Fig. 5. Constrainsontriplet-assistednon-abeliankineticmixing,recastfromtheATLASsearchreportedRef.[13].Theleftpanelgivestheexclusioninthe(cτ,σ×BR)plane, wheretheregionabovetheparabolaisexcluded.Thediagonallinesindicatethedependenceofσ×BR oncτfordifferentrepresentativechoicesofv.Therightpanelgives

theexclusionregioninthe(v,/β)planeformX=0.4 GeV (redregion)andmX=1.5 GeV (yellowregion).(Forinterpretationofthereferencestocolorinthisfigure

legend,thereaderisreferredtothewebversionofthisarticle.)

3.3.Variousregimesforcolliderphenomenology

From the foregoing discussion of production and decays, the LHCsignaturesanddetectionstrategieswillvaryaccordingtothe value of

β/

. We delineate three regimes leading to distinctive phenomenologyfor8TeV pp centerofmassenergy:

(1)

β/

∼

1

/

TeV.In thisregimewe see thatDrell–Yanpair pro-duction p

→ φφ

dominatesInadditionthebranchingratiofor

φ

→

X V decayisclosetohundredpercent.

(2)

β/



0

.

1

/

TeV. Inthis regime Drell–Yan pair production re-mains the dominantmechanism. However, BR

(φ

→

X V

)

can rangefromzerotoone,dependingonvalueofv.

(3)

β/



1

/

TeV. In this regime the

O

(_{W X}5) -mediated process

pp

→

X

φ

is dominant and BR

(φ

→

X V

)

is close to one. In thiscase,thepossiblefinalstatesareindicatedinFig. 2(c,d). (Recallthat for

√

s

=

14 TeV, the transitionbetween

O

(_{W X}5) -medi-ated production and Drell–Yan pair production also occurs for

β/

∼

1

/

TeV.) While all three possibilities above are worth ex-plorationinfuture,forillustrativepurposeswe focushereonthe thirdregime.

4. ATLASrecast

Consideringnow regime(3),werecasttheATLAS darkphoton search results [13] into constraints on ourscenario. The analysis ofRef. [13] assumesthe presence ofa SM Higgsboson decaying totwo new statesthat radiate two (orfour) dark photons, lead-ingtodisplacedverticesandleptonjets.Thisistobecomparedto ourproductionscenariomediatedbyanoff-shellvectorbosonV∗, leading toa final state containing two X bosons andan on-shell

V . Note, that the ATLAS study [13] only applied cuts to isolate eventswithlepton jets anddisplaced vertices.No reconstruction oftheHiggs bosoninvariant masswas preformed, nor werecuts on the missing energy applied. Thus, although the ATLAS study wascarriedoutassumingdifferentunderlying X -bosonproduction dynamics,theanalysisissufficientlyinclusivetoaccommodatethe scenarioconsidered here aswell. Looking to the future,we note thatone could likely improvetheLHC sensitivityto

O

(_{W X}5) by in-cludingadditionalcriterianeededtoidentifythefinalstate V .

We then translate the ATLAS bounds on



−

mX parameter

space[13] to the parameter space relevant to our scenario.

Cer-taindistinctionsbetweentheanalysisofRef.[13]andthatforour scenario haveto be accounted forproperly. Specifically, Ref. [13] presents the 95% C.L. exclusionplots for the signal cross section

σ

(

H

)

×

Br

(

H

→

2X

+ · · · )

asa function of the darkphoton life-timec

τ

(seetheleftpanelofFigure 16inthatwork4).Inourcase, the 95%C.L. bound appliesto

σ

(φ

X

)

×

Br

(φ

→

V X

)

.In addition,

σ

(

H

)

andBr

(

H

→

2X

+· · · )

areindependentof



(thedependence on mX is negligible for very light dark bosons). The production

cross section andbranching ratios forour scenario,on the other hand,dependonvariouscombinationsoftheparametersthat gov-ern



,viz,

σ

(

H X

)

∼ (β/)

2

∼

1

/(

τ

v2_

)

,where

τ

isthe X lifetime.

Inmakingthe translationfromRef.[13] wethen usetherelation inEq.(1.2).

Intheleft panelofFig. 5we showtheATLAS 95%CLlimit on

σ

(φ

X

)

×

Br

(φ

→

V X

)

,summingoverall

φ

,formX

=

0

.

4 GeV (solid

black) and lines of constant cross section

σ

(

pp

→ φ

X

)

(again, summed over all

φ

) for threerepresentative values of v: v

=

1 MeV (solidred),v

=

1

.

5 MeV (dashedolive)andv

=

2

.

5 MeV

(dottedmagenta).Ineachcase,Br

(φ

→

V X

)

≈

100%.Foreachline ofconstantvthepointsofintersectionwiththesolidblackcurve

determinetheboundariesoftheregionofexcludedc

τ

.Weobserve that the ATLAS exclusion then applies to v in the MeV range,

well belowthe

ρ

-parameter bound. These results,together with Eq.(1.2),leadtoconstraintsinthe(v,

β/

)plane,showninthe

rightpanelofFig. 5.Forillustrationweconsiderthistranslationfor two valuesofmX:0.4GeV (red)and1.5GeV(gold). We observe

thattheexclusioncanreach

/β

uptoseveralhundredGeV, de-pendingonthevalueofmX andv.Notethatforfixed

/β

(fixed

σ

×

BR),c

τ

(



)increases(decreases)withdecreasingv.Thus,for

agiven

/β

andsufficientlysmall v (equivalently



),

τ

willfall

belowtheATLASexclusioncurveintheleftpanelofFig. 5;hence, the exclusionlimits on

/β

inthe rightpanel weakenwith de-creasingv.

The foregoing illustrative analysis has endeavored to remain as model-independent as possible. Nevertheless, it is interest-ing to consider briefly the possible dynamics that may generate

O

(5)

W X andthecorrespondingimplications fortheinterpretationof

presentandprospectiveLHCresults.Fig. 1 indicatesa fewof the

4 _{Thisboundis obtainedbyexcluding fromthe analysisTYPE2–TYPE2events,}

whichcorrespondtobothdarkphotonsdecayingtojets.Thisleadstoastronger boundduetocorrespondingbackgrounds.

106 C.A. Argüelles et al. / Physics Letters B 770 (2017) 101–107

possibilities:(a)loopsinvolvingnewvector-likefermions;(b) loops involving new scalars; (c) non-perturbative dynamics. We com-ment on the first two possibilities. Considering new vector-like fermions F with mass MF, naïve dimensional analysis suggests

that

/β

∼

16

π

2_M

F

/

y,where y isthe F F

¯



couplingandwhere

we take the gauge couplings to be

O(

1

)

. Since F carries SU(2) charge, it would likely have been observed if sufficiently light. For example, the non-observation of pairs of new charged par-ticles (e.g., vector-like leptons) at LHC [16], may imply a lower boundof200GeV



MF insomecases,5 implying

/β



3

.

2 TeV

for y

∼

O(

1

)

. Significantly larger integrated luminosity and/or a search that exploitsfinal stategauge boson reconstruction would be neededto reach thislevel ofsensitivity. Fornewelectroweak scalars S withmassMS,onehas

/β

∼

16

π

2M2SFS

/

aS,whereaS

istheS S



couplingwithdimensionsofmass,andFS willdepend

inpartontheSU

(

2

)

representationofS.AssumingMS



100 GeV

inordertoevadeLEP IIlimitsandtakingaS

∼

MS,andtaking FS

tobe oforder

O(

1

)

, italsogives

/β

∼

1

.

6 TeV.However, noth-ing precludesaS frombeinga fewtimeslarger than MS, soit is

notunreasonabletoanticipate

/β

beingtotheupperendofthe exclusionregioninFig. 5.

Itispossibleforthechargedtripletstate H+todecaytoapair ofmediatorsattree-level.Inorderforthisdecaytooccur,the me-diatormassmust bylessthan halfmH+.Forthe limitsshownin

Fig. 5anddiscussedabove,wehavetakenmH2

=

mH+

=

130 GeV. Inthiscase,notree-leveldecaytomediatorswithmasses satisfy-ing presentcolliderbounds ispossible. The other caseismH+

=

300 GeV whichwasusedinsomeofourBRplotsinFig. 4.Butwe didnot useit inFig. 5 in deriving ourlimitson

/β

. Therefore, ourassumptionof100%branchingofH+

→

W+X isconsistent.

Finally,itisworth notingthatforinternalparticlemassesnear 100 GeV, one may be near the border of the region of validity ofa pure effective theory treatment ofthe collider phenomenol-ogy.Inprinciple,invokingan explicitmodelforgeneration ofthe operatorcoefficientand/or inclusionofaformfactorwouldlikely provideamorequantitativelyrealisticassessment.Similar consid-erationsapply to the applicationof the Higgseffectivetheory in studies of Higgs boson observables (see, e.g., Ref. [17] for a dis-cussion in the context of SM di-Higgs production in association withan additional,high-pT jet). Inthe presentinstance,the

AT-LAS lepton jet reconstruction efficiency peaks in the vicinity of

pX

T

∼

40 GeV, while the massesof the intermediate H±

/

H2 and

final state W -boson are not so large. Thus, we would expect at mosta modest degradation of the signal strength in a more re-alistic, model-dependent analysis. Nonetheless, we consider our statementsaboutthepresentLHCreachasindicativeofthe8TeV sensitivityratherthanasquantitativelydefinitive.

5. Outlook

MixingbetweenthedarkU

(

1

)

 andSU

(

2

)

L gaugegroups,

me-diatedbytheoperator

O

(_{W X}5) ,leadstoasmallmixingparameter



, whosemagnitudeissetbythescaleratiov

/

withan

O(

1

)

Wil-son coefficient,

β

. The resulting collider phenomenology is quite distinctive, as

O

(_{W X}5) may dominatethe production offinal states containing X bosonswhen

/β



1 TeV atboth

√

s

=

8 TeV and

√

s

=

14 TeV.CurrentATLASbounds, basedonaninclusivesearch forpairsofleptonjetsassociatedwithdisplacedvertices,exclude

/β

uptoabout600GeV,dependingonthevalueofmX andthe

triplet vev v. Looking to thefuture,the collectionofadditional

data during Run II will extendthe reach of the inclusivesearch.

5 _{Wethank S.Martinfor usefuldiscussionofthe assumptionsunderlyingthe}

workofRef.[15].

Intheadventofadiscovery,inclusionofadditionalsearchcriteria associatedwiththefinalstatevectorboson(s)wouldallowoneto distinguishthisscenariofromthoseassociatedwithabeliankinetic mixing.Ananalysisofthispossibility,alongwiththeLHC sensitiv-ity to other regions of the (mX,



) plane, will appear in future

work.

Acknowledgements

We thank Patrick Draper, Jesse Thaler, and Jiang-Hao Yu for useful discussions. This work was supported in part by U.S. Department of Energy contract DE-SC0011095 (G.O., T.P., and M.J.R.-M.). XGH was supported in part by MOE Academic Excel-lent Program (Grant No. 105R891505) and MOST of ROC (Grant No. MOST 104-2112-M-002-015-MY3),andinpartbyNSFCofPRC (Grant No. 11575111).Thiswork wasalsosupported byKey Lab-oratory forParticle Physics,AstrophysicsandCosmology,Ministry ofEducation,andShanghaiKeyLaboratoryforParticlePhysicsand Cosmology (SKLPPC) (Grant No. 11DZ2260700). He also thanks Korea Institute for Advanced Study (KIAS) for their hospitality during the completion of this work. CA was supported in part by the NationalScience Foundation (ANT-0937462,PHY-1306958, PHY-1505855, and PHY-1505858) and by the University of Wis-consin ResearchCommitteewithfunds grantedby the Wisconsin AlumniResearchFoundation.

Appendix A. Feynmanrulesrelevantforcollidersignatures Feynman rules of interactions between the dark bosons, lep-tons,gaugebosons,chargedandneutralHiggsbosonsarelistedin thetablebelow:

Interaction Feynman rule

Xl+l− ie





0

−

βvsW_



W±H∓X i_β



gμν pp

−

pν pμ



c_∓ Z H1X i_β



gμν pp

−

pν pμ



cWs0 Z H2X i_β



gμν pp

−

pν pμ



cWc0 A H1X i_β



gμν pp

−

pν pμ



sWs0 A H2X i_β



gμν pp

−

pν pμ



sWc0 W_μ+

(

p1

)

Wν−

(

p2

)

H1Xα

(

p3

)

iβ_g



pμ₃gνα

−

pν₃gμα



s0 W_μ+

(

p1

)

Wν−

(

p2

)

H2Xα

(

p3

)

iβ_g



pμ₃gνα

−

pν₃gμα



c0 W_μ±

(

p1

)

Zν

(

p2

)

H∓Xα

(

p3

)

∓

iβ_g



pμ₃gνα

−

pν₃gμα



cWc∓ W_μ±

(

p1

)

Aν

(

p2

)

H∓Xα

(

p3

)

∓

iβ_g



pμ₃gνα

−

pν₃gμα



sWc∓ Feynman rulesentering intheverticesofthe graphsinFig. 2. Wherec_∓

≡

cos

θ

_∓andc0

≡

cos

θ

0 areasdefinedinRef.[14].

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