Perturbative unitarity bounds for effective composite models

University of Groningen

Perturbative unitarity bounds for effective composite models

Biondini, S.; Leonardi, R.; Panella, O.; Presilla, M.

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Physics Letters B

DOI:

10.1016/j.physletb.2019.06.042

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Biondini, S., Leonardi, R., Panella, O., & Presilla, M. (2019). Perturbative unitarity bounds for effective

composite models. Physics Letters B, 795, 644-649. https://doi.org/10.1016/j.physletb.2019.06.042

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Perturbative

unitarity

bounds

for

effective

composite

models

S. Biondini

a

,

∗

,

R. Leonardi

b

,

O. Panella

b

,

M. Presilla

c

,

d

a_{VanSwinderenInstitute,UniversityofGroningen,Nijenborgh4,NL-9747AGGroningen,Netherlands} b_{IstitutoNazionalediFisicaNucleare,SezionediPerugia,ViaA.Pascoli,I-06123Perugia,Italy}

c_{DipartimentodiFisicaeAstronomia“GalileoGalielei”,UniversitàdegliStudidiPadova,ViaMarzolo,I-35131,Padova,Italy} d_{IstitutoNazionalediFisicaNucleare,SezionediPadova,ViaMarzolo,I-35131,Padova,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received5April2019

Receivedinrevisedform7June2019 Accepted18June2019

Availableonline20June2019 Editor:G.F.Giudice Keywords: Perturbativeunitarity Compositemodels Compositefermions LHCRun2

High-LuminosityandHigh-EnergyLHC

Inthispaperwepresentthepartial waveunitarityboundintheparameterspaceofdimension-5and dimension-6 effective operators that arise in a compositeness scenario. These are routinely used in experimentalsearchesattheLHCtoconstraintcontactandgaugeinteractionsbetweenordinaryStandard Model fermionsand excited(composite) statesofmass M.Afterdeducingtheunitarityboundforthe productionprocessofacompositeneutrino,weimplementsuchboundandcompareitwiththerecent experimental exclusion curvesfor Run 2, the High-Luminosityand High-Energy configurations of the LHC.Ourresultsalsoappliestothesearcheswhereagenericsingleexcitedstateisproducedviacontact interactions. We findthat the unitarity bound, sofar overlooked,is quite compelling and significant portionsoftheparameterspace(M,)becomeexcludedinadditiontothestandardrequestM≥ .

©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Itiswellknownthatpartialwaveunitarityisapowerfultoolto estimatetheperturbativevalidityofeffectivefield theories(EFTs). It has been used in the past to provide useful insights both in strong and electroweak interactions [1] as well as in quantum gravity [2]. Perhapsthebest knownexampleistheboundonthe Higgs mass derived from an analysis of W W

→

W W scattering

within the Standard Model (SM) [1,3]. On the other end, unitar-ity hasalsobeen appliedto a numberofapproachesbeyondthe StandardModel(BSM).ForinstanceincompositeHiggsmodels [4], insearches ofscalar di-bosonresonances [5,6], searchesfor dark mattereffectiveinteractions [7] andongenericdimension-6 oper-ators [8].

OnepossibleBSMalternative,widelydiscussesinliteratureand routinely pursued in high-energy experiments, is a composite-fermions scenario which offers a possible solution to the hierar-chypatternoffermionmasses [9–15].Inthiscontext [14–20],SM quarks“q”andleptons“



”areassumedtobeboundstatesofsome asyetnotobserved fundamentalconstituentsgenerically referred aspreons.Ifquarksandleptonshaveaninternalsubstructure,they areexpectedto beaccompanied byheavy excited states



∗

,

q∗ of

*

Correspondingauthor.

E-mailaddress:s.biondini@rug.nl(S. Biondini).

massesM thatshouldmanifestthemselvesatanunknownenergy scale,thecompositenessscale



.

AscustomaryinanEFTapproach,theeffectsofthehigh-energy physics scale, here



, are capturedin higherdimensional opera-torsthatdescribeprocesseswithinalowerenergydomain,where the fundamental building blocks of the theory cannot show up. Hence, the heavy excited states may interact with the SM ordi-nary fermionsvia dimension-5gaugeinteractions oftheSU(2)L

⊗

U(1)Y SMgaugegroupofthemagnetic-momenttype(sothatthe electromagnetic current conservation is not spoiled by e.g.



∗



γ

processes [18]).In addition,the exchange ofpreonsand/or bind-ingquantaoftheunknowninteractionsbetweenordinaryfermions ( f ) and/or the excited states( f∗) resultsin effective contact in-teractions (CI) that couple the SM fermions and heavy excited states [19–22]. Inthelattercase, thedominanteffectisexpected to be given by the dimension-6 four-fermioninteractions scaling withtheinversesquareofthecompositenessscale



:

L

6

₌

g∗2



2 1 2j μ_j μ

,

(1a) jμ

=

η

Lf

¯

L

γ

μfL

+

η

Lf

¯

_L∗

γ

μfL∗

+

η

Lf

¯

_L∗

γ

μfL

+

h

.

c

.

+ (

L

→

R

) ,

(1b)

where g2_∗

=

4

π

andthe

η

’s factorsareusually setequaltounity. Inthiswork theright-handedcurrentswillbe neglectedfor

sim-https://doi.org/10.1016/j.physletb.2019.06.042

0370-2693/©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

plicity(thisisalsothesettingadoptedbytheexperimental collab-orations).

Asfarasgaugeinteractions(GI)areconcerned, letusconsider thefirst lepton familyandassume that the excited neutrinoand the excited electron are groupedinto left handed singlets and a right-handedSU(2)doublet:e∗_L

,

ν

L∗

,

L∗R

=



ν

∗_R

,

e∗_R



T,sothata mag-netictype couplingbetweenthe left-handedSMdoublet andthe right-handedexcited doublet via the SU(2)L

⊗

U(1)Y gauge fields canbewrittendown [18,23]:

L

5

₌

1 2



L

¯

∗ R

σ

μν



g f

τ

2

·

Wμν

+

g _f_{Y B} μν



LL

+

h

.

c

. .

(2) Here, LT

= (

ν

L

,



L

)

is the ordinary lepton doublet, g and g are theSU(2)L andU(1)Y gaugecouplingsandW μν ,Bμν arethefield strength tensorof the corresponding gauge fieldsrespectively;

τ

arethePaulimatricesandY isthehypercharge, f and fare di-mensionless couplings andare expected (and assumed) to be of orderunity.

ExcitedstatesinteractingwiththeSMsectorthroughthemodel Lagrangians(1a)-(1b) and (2) have beenextensively searched for at high-energy collider facilities. The current strongest bounds are dueto the recentLHC experiments.Charged leptons (e∗

,

μ

∗) havebeensearchedforin thechannel pp

→ 

∗

→ 

γ

[24–30], i.e. produced via CI and then decay via GI, and in the channel

pp

→ 

∗

→ 

qq

¯

 [31] wherebothproductionanddecayproceed throughCI.Neutralexcitedleptonshavebeenalsodiscussedinthe literatureandthecorresponding phenomenologyatLHChasbeen discussedindetailinthecaseofaheavycompositeMajorana neu-trinoN∗ [32].Adedicatedexperimentalanalysishasbeencarried outbytheCMScollaboration[33] onLHCdatacollectedfor

√

s =

13TeVandlookingfortheprocess

pp

→ 

N_∗

→ 

qq

¯

 (3)

withdilepton (dielectrons or dimuons) plus diquark final states. TheexistenceofN∗_ isexcludedformassesup to4

.

60

(

4

.

70

)

TeV at95% confidencelevel,assumingM

= 

.Moreover,thecomposite Majorananeutrinosofthismodelcanberesponsiblefor baryogen-esisvialeptogenesis [34,35].Thephenomenologyofother excited states has also been discussed in a series of recent papers [32,

36–42].

Weemphasizethatinallphenomenologicalstudiesreferenced aboveaswellasall experimentalanalyses thathavesearchedfor excitedstatesatcolliders,itiscustomarytoimposetheconstraint

M

≤ 

onthe parameter spaceofthe model.To thebest ofour knowledgeunitarityhasneverbeentakenintoaccountand/or dis-cussedinconnectionwiththeeffectiveinteractionsofthesocalled excitedstates.Themaingoalofthisworkistoreportinsteadthat theunitarity bounds, asextractedfromEq. (1a)-(1b) and (2), are quitecompellingandshouldbeincludedinfuturestudiesofsuch effectivecompositemodelsbecausetheyconstraint ratherstrongly theparameter space. While we present an explicitcalculation of theunitarityboundforheavycompositeneutrinosearches,we ex-pectthat similarbounds(i.e.equallycompelling)wouldapplyfor excitedelectrons(e∗),muons(

μ

∗)andquarks(q∗).Indeed,the ef-fective operators that describe the latter excited states have the verysamestructureofthosereferredtothecompositeneutrinos.

2. Unitarityinsingle-excited-fermionproduction

Forthederivationoftheunitaritybound,weadoptastandard methodthat makesuseoftheoptical theoremandtheexpansion ofthescatteringamplitudeinpartialwaves.Inordertospecifythe CIandGILagrangiansforadefinitesituation,weconsiderthe pro-ductionoftheexcitedMajorananeutrinoattheLHC.However,we

Fig. 1. Feynmandiagrams depicting themechanisms responsiblefor the process

qq¯→N∗,wherestandsforboth±.Thedarkgreyblob(diagramontheleft) de-scribestheproductionofanon-shellheavyMajorananeutrinoN inproton-proton collisionsatLHC.Theproductionispossiblebothwithgaugeinteractions(first di-agramontheright-handside)andwithfour-fermioncontactinteractions(second diagramontheright-handside).

shallhighlightwhentheresultsapplytoothercompositefermion statesinthefollowing.

The central object that we shall derive fromthe operators in theeffectiveLagrangians(7) and(8) is theinteractingpartofthe

S matrix,indicatedwithT inthisletter.Itentersthepartialwave decompositionofthescatteringamplitudeasfollows

M

i→f

(θ )

=

8

π



j

(

2 j

+

1

)

T_ij_→fd_λj

fλi

(θ ) ,

(4)

where j istheeigenvalueofthetotalangularmomentum J ofthe incoming(outgoing)pair, d_λj

fλi

(θ )

isthe Wigenerd-function and

λ

i (

λ

f) isthetotalhelicityoftheinitial(final)state pair.Without loss of generality, we consider azimuthally symmetric processes andfix

φ

=

0 accordingly. From theoptical theorem andthe de-composition in Eq. (4), one can find the perturbative unitarity conditionofaninelasticprocessforeach j tobe



f =i

β

i

β

f

|

Tij→f

|

2

_≤

₁

_,

₍₅₎

where

β

i (

β

f) is the factor obtained from the two-body phase spaceandreadsfortwogenericparticleswithmassesm1 andm2

β

=



ˆ

s

− (

m1

−

m2)2



ˆ

s

− (

m1

+

m2)2

ˆ

s

.

(6)

Itcorrespondstotheparticlevelocitywhenm1

=

m2.Itis

impor-tanttonoticethattheunitaritybound isimposed onthe subpro-cess involving the proton valence quarks as initial state, namely

qq

¯



→ 

N_∗asshowninFig.1.Then,fortheprocessofinterest,the relevantinteraction(s)areasfollows:

L

CI

=

g2_∗

η



2q

¯



γ

μ_P LqN

¯

γ

μPL



+

h

.

c

. ,

(7)

L

GI

=

g f

√

2



N

¯

σ

μν

_(∂

μWν+

)

PL



+

h

.

c

. .

(8)

Accordingly,inEq. (6),

ˆ

s denotesthecenter-of-massenergyineach collision anditisobtainedfromthe nominalcolliderenergyand thepartonmomentum fractionsas

ˆ

s

=

x1x2s. Asfar asthe

kine-matic is concerned,

β

i

=

1 can be used since the valence quark masses are negligible with respect to the center-of-mass energy. Instead, one finds

β

f

=

1

−

M2

/ˆ

s for the final state, where the compositeneutrinomasshastobekept.

The coreofthemethod reliesonthe derivation ofthe ampli-tudefortheprocessofinterestinducedbythecontactand gauge-mediatedeffectiveLagrangians(7) and(8).Then,onematchesthe so-obtainedresultfor

M

i→f withther.h.sofEq. (4) andextracts the corresponding T_ij_→f for each definite eigenvalue of the total angularmomentum ( j).Thelatterare insertedinto Eq. (5) in or-der to derive the unitarity condition that the model parameters (

,

M

,

g_∗

,

g) andthecenter-of-massenergyhavetoobey.Tothis

end,theamplitude

M

i→f isdecomposedintermsofdefinite he-licitystatesand,therefore,wehavetoexpresstheinitialandfinal stateparticlesspinorsaccordingly[43].Thehelicityofeachparticle intheinitialorfinalstateis

λ

= ±

1

/

2,beingalltheinvolved parti-clesfermions(alsothecompositeneutrinosarespin-1/2fermion). We shall simply use

±

to label the initial and final state helic-itycombinations,

(

+,

+)

,

(

+,

−)

,

(

−,

+)

and

(

−,

−)

. Sinceinthe center-of-mass frame the incoming andoutgoing particles travel inopposite directions,thehelicitiesintheWignerd-functionsare definedas

λ

i

= λ

q

− λ

q¯ and

λ

f

= λ

N∗

− λ

.Onecanadopttwo

dif-ferentbases forexpressing the spinors andthegamma matrices, theDiracandchiral bases(see e.g.appendix inref. [7]).Weused boththeoptionstoderive T_ij_→f andcheckedthatourfindingsare indeedinvariantuponthechoiceofthebasis.

We give the result for the CI Lagrangian in Eq. (7) first. The non-vanishinghelicityamplitudesread

T₍j_{−,+)→(−,+)}=1

= −

ˆ

s g 2 ∗ 12

π



2

1

−

M 2

ˆ

s



1 2

,

(9) T₍j_{−,+)→(+,+)}=1

=

√

ˆ

s M g_∗2 12

√

2

π



2

1

−

M 2

ˆ

s



1 2

.

(10)

Onlytheamplitudewith j

=

1 isnon-zero,duetotheinitial helic-itystate.Thesameoccurswiththevectorandaxial-vector opera-torsstudiedin[7] fordarkmatterpairproductionatcolliders.We noticethatafinitecompositeneutrinomassallowsforthehelicity flip inthe final state originatingtheterm inEq. (10). Weobtain thesame resultifwe workwithright-handed particles intheCI operator,howeverthehelicitiesinEqs. (9) and(10) flipas

+

↔ −

. Using Eq. (4) and summing over the non-vanishing final helicity states,weobtain g4_∗s

ˆ

(

2s

ˆ

+

M2

)

288

π

2

_

4

1

−

M 2

ˆ

s



2

≤

1

.

(11)

As far as the GI process is concerned, we proceed the same way.Adimension-5operatoris involvedand,inthiscase, theW

bosonmediatesthescatteringbetweentheinitialandfinal states. Wekeepthe W bosonmassinourexpression,evenifitismuch smallerthanthetypical

ˆ

s valuesofthepp collisions.TheSM elec-troweakcurrententersbesidestheonefromthecompositemodel andthehelicityamplitudesarefoundtobe

T₍j_{−,+)→(−,+)}=1

= −

ig 2 24

π



ˆ

s3/2

ˆ

s

−

m2 W

1

−

M 2 s



1 2

,

(12) T₍j_{−,+)→(+,+)}=1

=

ig 2 24

√

2

π



ˆ

s M

ˆ

s

−

m2_W

1

−

M 2

ˆ

s



1 2

,

(13)

andthecorrespondingresultfortheunitarityboundis

g4 1152

π

2

_

2

ˆ

s2

(

2

ˆ

s

+

M2

)

(

ˆ

s

−

m2_W

)

2

1

−

M 2

ˆ

s



2

≤

1

.

(14)

A comment is in order. The unitarity bound in Eq. (11) is valid for the more generic production process qq

¯



→

f∗f , i.e. excited chargedor neutralleptons andexcited quarksaccompanied by a SMfermion.Thisstatementtracesbacktotheparticle-blindchoice adoptedintheCIsframework,wherethe

η

’saresettounityinall the cases.More carehas to be takenabouta wider applicability ofthe resultfor GIs inEq. (14). Here,different factorscan enter accordingtothe gaugecouplings andgaugebosons that describe theprocessesinvolvingexcitedchargedleptonsandquarksinstead ofcompositeneutrinos.

3. Implementingthebound

The production of heavy composite Majorana neutrinos has beenstudiedbytheCMSCollaborationbymeasuringthefinalstate withtwoleptonsandatleastonelarge-radiusjet,withdatafrom

pp collisionsat

√

s

=

13 TeVandwithanintegratedluminosityof 2

.

3 fb−1 _{[33].Goodagreement betweenthedata andtheSM}

ex-pectations was observed in the search, butthe whole dataset of the Run 2 of the LHC still needs to be analysed. Therefore, the issueoftheunitarityconditionontheaccessibleparameterspace (M

,



)urgestobeassessed.AsusualinBSMsearches,theabsence ofasignalexcessovertheSMbackgroundistranslatedintoan ex-perimental bound on theparameter space

(

M

,

)

. Moreover, the sensitivity ofthis search was investigated fortwo futurecollider scenarios: the High-Luminosity LHC (HL-LHC), with a centre-of-mass energy of 14 TeV andan integrated luminosity of 3 ab−1, andtheHigh-EnergyLHC(HE-LHC),withacentre-of-massenergy of27 TeVandanintegratedluminosity of15ab−1 [44].The pro-jectionstudies,includedintherecentYellowReportCERN publica-tion [45,46],haveshownthepotentialofsuchfacilitiesinreaching muchhigherneutrinomasses.

In this section, the perturbative unitarity bounds are applied to these searches in the dilepton and a large-radius jet chan-nel with the CMS detector for the three different collider sce-narios. As alreadyclearfromthe ratherdifferent couplingvalues entering the Lagrangians (7) and (8), namely g

/

√

2

≈

1 versus

g2

∗

=

4

π

,theproductionmechanismofaheavycompositeneutrino and other excited states is dominated by the contact interaction mechanism [40]. In particular, it was shown that cross sections incontact-mediatedproductionareusuallymorethantwo orders of magnitudelarger than thegauge mediated ones forall values of the



and M relevant in the analyses. This means that it is a reasonable approximationto consideronly theboundsgivenin Eq. (11) toconstrainttheunitarityviolationofthesignalsamples. In order to estimate the effect of the unitarity condition on LHC searches, we need to implementthe boundsin the case of hadron collisions. Then, the square of the centre-of-mass energy of the collidingpartons system,

ˆ

s

=

x1x2s does not have a

defi-nite value, where x1 and x2 are the partonmomentum fractions

and

√

s isnominalenergyofthecollidingprotons.Tothisaim,we haveestimated

ˆ

s ineacheventgeneratedintheMonteCarlo(MC) samples,andwe haveplugged theresultintoEq. (11) inorderto obtainlevelcurvesontheparameterspaceforwhichtheunitarity boundissatisfiedtosomeextent.Indeed,oneshouldnotinterpret the constraintinEq. (11) toostrictly.A violationofsuch bounds would signal the breakdown of the EFT expansion and call for higherorder operatorsin

√

ˆ

s

/

and/or M

/

to helpinrestoring the unitarity of the process. We implement a theoretical uncer-tainty by allowing up to50% of theevents toviolate the bound, that corresponds to assign a relative correction to the cross sec-tion

δ

σ

/

σ

LO

≤

0

.

5 fromhigherorderterms.Thisprocedureresults inthebandsbetweenthesolid-thickandsolid-thin(violet)linesin Figs.2-5forwhich100%and50%oftheeventssatisfytheunitarity boundrespectively.TheMCsamplesforthesignalaregeneratedat Leading Order(LO) withCalcHEP (v3.6) [47] for

√

s

=

13

,

14 and 27 TeVproton-protoncollisions,usingtheNNPDF3.0LOparton dis-tributionfunctionswiththefour-flavorscheme [48],spanningover the

(,

M

)

regioncoveredby theexperimental searches [44].The informationonthepartonmomentaisthenretrievedfromtheLes HouchesEvent(LHE)filesofeachsignalprocessthrough MadAnal-ysis [49].

We have explicitlychecked that, in the mass range explored, the

ˆ

s-distributions in our MC simulations are peaked at values around M2 almost irrespective of the nominal collider energy

√

Fig. 2. Theunitarityboundinthe(M,)planecomparedwiththeRun2 exclu-sionat95%CLfrom[33],dashedline(blue),fortheeeqq¯finalstatesignature.The solid-thickandsolid-thin(violet)linesrepresenttheunitaritybound,whereasthe dot-dashed(gray)linestandsforthe M≥ condition.Hereandinthefollowing figuresbothandM startat100GeV.

Fig. 3. Theunitarityboundintheplane(M,)comparedwiththeexclusionfrom theHighLuminosityprojectionsstudyin[45] forLHCat√s=14 TeVat3ab−1_of

integratedluminosity.

since in the generated signal eventsthe available energy,

√

ˆ

s, is mostly used to produce a heavy excited state (of mass M). Of course,thelargerthecolliderenergythemoreprominentthe dis-tributiontailsathigh

ˆ

s values.Theseexpectationsarecorroborated by analytical expressions for the

ˆ

s-distributions that can be re-trieved from ref. [32], involving only the product of the parton luminosity functions and the hard production process cross sec-tion.

TheresultsarepresentedinFigs.2,3and4fortheRun2, HL-LHCandHE-LHCscenariorespectively.Thevioletshadedareas,and thecorrespondinguncertaintybandsinadarkerfilling,definethe regionswhere themodelshould notbe trustedbecauseunitarity isviolatedforsuch

(

M

,

)

values.

4. Discussionandresults

Letus elaborate on our findings and explain their impact on theexperimental analysescarried outatthe LHC.First ofall,the experimentaloutcomesaresummarizedwithexclusionregionsin the

(

M

,

)

plane,whichareinturnsetwiththe95%C.L.observed (Run2) [33] and expectedlimit (HL/HE-LHC) [45,46], namelythe dashedbluelinesinFig.2,3and4respectively.Abovetheselines themodelisstillviable,whereasbelowitisexcluded.The experi-mentalcollaborationsquoteroutinelythelargestexcluded excited-statemass byintersecting the 95%C.L. exclusioncurveswiththe

Fig. 4. Theunitarityboundintheplane(M,)comparedwiththeexclusioncurve fromtheHE-LHCprojectionstudiesin[45] for√s=27 TeVat15ab−1_ofintegrated

luminosity.

Fig. 5. Theunitarityboundintheplane(M,)comparedwiththeexclusionfrom theRun2forchargedleptonssearcheswithtwodifferentfinalstates [30,31].

M

≥ 

constraint(dot-dashedgraylineandgrayshadedregionin Fig.2,3and4).Thisisthewidely adoptedcondition imposedon themodelvalidityanditoriginatesfromaskingtheheavyexcited statestobeatmostasheavyasthenewphysicsscale



.Despiteit isareasonableconstraint,itdoesnottakeintoaccountthetypical energyscale that entersthe productionprocess, i.e.

ˆ

s. The corre-spondingmass valuesarereportedinthe firstrawofTable 1for thethreecollidersettings.

TheunitarityconditioninEq. (11) isrepresentedwiththesolid violetlinein Fig.2,3and4,anditdefinestheregion ofthe pa-rameterspace

(

M

,

)

whereunitarityissatisfied(abovethesolid violetline)orviolated(violetshadedarea).Itisclearthatthe uni-tarity boundismuch morerestrictive than M

≤ 

anditshrinks theavailableparameterspacequiteconsiderably.Ifoneappliesthe unitarityboundtotheexperimentalresultsbyfollowingthesame prescriptionasoutline beforefor M

≤ 

,then themaximal neu-trino massvaluesare those collected inTable 1,second raw. For example, for LHC Run 2, we find M

≤

2

.

7 TeV for



=

8

.

6 TeV insteadofM

≤

4

.

6 TeV(



=

4

.

6 TeV).Asanticipated,thenew con-straintsetbytheunitarityboundisquitestrikinganditoffersan alternative theoretical input for ongoing and futureexperimental analyses on this effectivecomposite model. We provide the uni-taritybounddownto M

=

100 GeV.Smallervaluesclashwiththe original modelsettingthat assumessome newphysics abovethe electroweakscaletriggeringfermionexcitations [18,19].

Table 1

InthefirstlinewequotetheboundsreportedintheCMSanalysisoflikesigndileptons anddiquark[33] and subsequentprojectionsstudiesatHL-LHCandHE-LHC[44–46].Insecondline,wequoteinsteadthestrongest massboundobtainedfromFigs.2,3,4whenthelineoftheperturbativeunitaritybound(satisfiedby50%of theevents)crossesthe95%C.L.exclusioncurvefromtheexperimentaland/orprojectionsstudies.

LHC Run 2 HL-LHC HE-LHC

2.3 fb−1_,√_{s=13 TeV 3 ab}−1_,√_{s=14 TeV 15 ab}−1_,√_{s=27 TeV} M=  M≤4.6 TeV [33] M≤7.8 TeV [44–46] M≤12 TeV [44–46] Unitarity 50% M≤2.7 TeV (=8.6 TeV) M≤5.1 TeV (=19.7 TeV) M≤7.2 TeV (=29.5 TeV)

While in this study we have concentrated on the impact of unitarity boundson the heavy composite neutrino production at theLHC,HL-LHCandHE-LHC,we expectthatsimilar boundswill affectthe searchesforchargedexcited leptons. Leaving more de-tailed studies for future work, we apply the unitarity bound in Eq. (11) for the recent experimental results reported in [30,31], wherechargedexcitedleptons areproducedvia CIintheprocess

pp

→ 

∗



,with



∗

=

e∗

,

μ

∗.BeingtheeffectiveLagrangianforthe productionmechanismtheverysameasforexcitedMajorana neu-trinos (if one insists on

η

=

1 for the different states), we can usethe unitarity bound as extractedfor

√

s

=

13 TeV andapply itfortheprocessinvolvingchargedexcited leptons.The compari-sonwiththeexperimentalexclusionlimitat95%C.L.isshownin Fig.5.

Inconclusion, we studied the perturbative unitarity bound ex-tracted from the effective gauge and contact Lagrangians for a composite-fermion model. On general grounds, an effective the-oryisvaliduptoenergy/momentumscalessmallerthanthelarge energyscalethatsetstheoperatorexpansion.Sincecollider exper-imentsare involving moreandmore energetic particlecollisions, theusageandtheapplicabilityofeffectiveoperatorscanbe ques-tioned. Inorder toaddress thisissueandto be onthe safeside, one canimpose theunitarity condition both onthe EFT parame-ters

(

M

,

,

g∗

,

g

)

andthe energyinvolvedin a givenprocess. To thebestof ourknowledge,such aconstraintwas not derived for themodelLagrangiansinEq. (7) and(8),andwehaveobtainedthe corresponding unitarity bounds, namelyEqs. (11) and(14). Thus, theapplicability ofthe effectiveoperators describingthe produc-tion of composite neutrinos (and other excited states) has to be restricted accordingly. We have considered an estimation of the theoreticalerrorontheunitaritybound,whichisderivedat lead-ingorderintheEFTexpansion,byallowingupto50%oftheevents toevadetheconstraint.ThisoriginatesthebandsinFigs.2-5.

Itisthe authors’opinionthat thefindings herediscussedwill have a significant impact on ongoing and future experimental searchesfor excited statescouplingto the SM fermionswiththe interactions given in (7) and (8). At the very least, the unitarity bounds play the role of a collider-driven theoretical tool for in-terpreting the experimental results of the considered composite models,morerigorousthanthesimplerelationM

≤ 

.

Acknowledgements

The authors thankP. Azzi and F. Romeo forusefulcomments anddiscussiononthemanuscript.

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