Diboson Interference Resurrection

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Diboson

interference

resurrection

Giuliano Panico

a

,

Francesco Riva

b

,

Andrea Wulzer

b

,

c

,

d

,

e

,

∗

a_{IFAE,UniversitatAutònomadeBarcelona,E-08193Bellaterra,Barcelona,Spain} b_{TheoreticalPhysicsDepartment,CERN,Geneva,Switzerland}

c_{InstitutdeThéoriedesPhénomenesPhysiques,EPFL,Lausanne,Switzerland} d_{DipartimentodiFisicaeAstronomia,UniversitádiPadova,Italy} e_{INFN,SezionediPadova,viaMarzolo8,I-35131Padova,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received14September2017

Receivedinrevisedform24November2017 Accepted28November2017

Availableonline2December2017 Editor:G.F.Giudice

High-energydibosonprocessesattheLHCarepotentiallypowerfulindirectprobesofheavynewphysics, whose effects can be encapsulated in higher-dimensional operators or in modified Standard Model couplings. Anobstructionhowevercomesfromthefactthatleading newphysicseffects oftenemerge indibosonhelicityamplitudesthatareanomalouslysmallintheStandardModel.Assuch,theformally leading StandardModel/NewPhysicsinterferencecontributioncancels ininclusivemeasurements.This paperdescribesasolutiontothisproblem.

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

1. Introduction

Precisiontestsarean increasinglyimportanttool tosearch for dynamicsbeyondtheStandard Model(SM). Effectsof heavynew physicsonSM processesarecapturedby aneffectivefield theory (EFT)[1],genericallydominatedbydimension-6operators (collec-tively denoted as “BSM effects” in what follows). Some of these induceenergy-growth inelectro-weak(or evenstrong,see[2–4]) scatteringprocesses, becoming sensitive targets of the LHC large kinematicreach,providedaccurateenoughmeasurementsare pos-siblein the high-energy regime. The power of thisinterplay be-tweenenergyandaccuracy hasbeendemonstratedinref. [5]for the neutral and charged Drell–Yan processes. High-energy dibo-son[6–12]andboson-plus-Higgs[13,14]productionprocessesare promising channels to be explored in this context, andthey are sensitivetoawidervariety ofBSMeffectsthanDrell–Yan. A pos-sibleobstruction to this program, outlined in ref.[15], takesthe formofa“non-interferencetheorem”,formulatedasfollows.Inthe high-energy limit E



mW amplitudescan be well characterized bythehelicitiesoftheexternalbosons.Inthisregime,2

→

2 tree-level amplitudes involving transversely polarized vector-bosons, turnout to exhibit differenthelicity in theSM andBSM. For in-stance,thedibosonprocessesthatweconsiderhere, f f

→

WTVT,

V

=

W

,

Z

,

γ

havefinal-statehelicity

(

±∓)

intheSMand

(

±±)

in BSM. Thisimpliesthat SM andBSM donot interfereininclusive

*

Correspondingauthor.

E-mailaddresses:gpanico@ifae.es(G. Panico),

francesco.riva@cern.ch

(F. Riva),

andrea.wulzer@pd.infn.it(A. Wulzer).

analyses,sothatthefirstdeparturefromtheSMappearsatorder BSM-squared:animportantobstacletoaprecisionprogramaiming atmeasuringsmalleffects.Inthisarticlewediscussastrategyto “resurrect” the SM-BSM interference, based onthe measurement ofthebosonsazimuthaldecayangles.Similarmeasurementswere proposedlongagoinrefs.[16,17].

Note added While this work was in preparation [18], ref. [19]

appeared, that discusses similar ideas in the context of the W Z

process.

2. Interferenceresurrection

We consider the production of two massive vector bosons

V1,2

_{= {}

_W

_,

_Z

_}

_{, followed by fermionic decays V}1(2)

_→

_f1(2)

+ f−1(2).

The final state fermions are labeled by their helicities, with“ f ” denotingirrespectively particles oranti-particlesofanycharge or flavor. Weare mostly interested in2

→

2 quark-initiated produc-tion,howevermostofwhatfollowsholdsforgenericdiboson pro-duction,possiblyinassociationwithQCDjets.

We choose a “special” coordinate system, defined as follows. Starting fromthe lab frame, and a generic configuration forthe externalstatemomenta,wefirstboostbacktothecenterofmass frameofthediboson(or4-fermions)system. Theboostisas cus-tomaryperformedalong thedirectionof motion(call itr)

ˆ

of the diboson system. In the new systemwe have back-to-backboson momentaandthereferenceunitvectorr,

ˆ

whichwe usetodefine thespecialframeasshowninFig. 1.Namely,wetakethez axisof thespecialframe alongthedirectionofmotionofthefirst boson

https://doi.org/10.1016/j.physletb.2017.11.068

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

Fig. 1. Definition of the decay angles for the diboson system.

V1 whilethex axis isintheplane formedby

ˆ

r andthe diboson axis. The x orientation is taken such that r goes

ˆ

in the positive

x direction or,equivalently, such that the y axis (for left-handed orientation of the x– y–z system) isparallel to the cross-product betweenthe V1 directionandr.

ˆ

Fora2

→

2 productionprocess,

ˆ

r coincides with the collision axis, oriented in the direction of thepartonthat carriedthelargerenergyin thelabframe.Inthe specialframe thecollision thus occursina ratherspecial config-uration,wherethe initialstatesmove inthe x–z planewhilethe intermediatebosonshappentobeproducedexactlyparalleltothe

z-axis.

Thereadermightbe confusedby thefact thatthespecial ref-erence system depends on the kinematical configuration of the event,i.e.differentsystemsareemployedforthecalculationofthe amplitudeatdifferentphase-spacepoints.Theamplitudeobtained inthis waydoesnot indeedcoincide withthe one evaluated di-rectlyinthelabframe.Toobtainthelatterout oftheformerone hastoactwiththephase-spacedependentLorentztransformation that connects the specialframe with the lab, introducing in this wayan additionalandcomplicateddependenceon the kinemati-cal variables.However the physicalexternal statesofthe process are the massless helicity eigenstate fermions, and Lorentz trans-formations act asmultiplicative phase factors on massless states helicity amplitudes. Therefore thisadditional dependence on the kinematics dropsfrom theamplitude modulus square andis un-observable. Stated differently,the amplitude for each kinematical configurationcorrespondstooneindividualquantum-mechanically distinguishableprocess.Assuch,eachonecanbesafelycomputed initsownframe.

Inthespecialframetheamplitudereads

A

∝

g1g2



h1,2

A

h1h2e ih1ϕ1_eih2ϕ2_d h1

(θ

1

)

dh2

(θ

2

) ,

(1)

whereg1(2)arethecouplingsresponsiblefortheV1(2)decaysand

A

h1h2 denotestheamplitudefortheproductionofon-shellvector bosons withhelicities h1 andh2, evaluated in the specialframe. Normalizations and

ϕ

1,2-dependentoverall phases,that will drop from the amplitude modulus square, have been absorbed in the proportionality factor. The above equation relies on the narrow-widthapproximation forthe decaying bosons only to the extent towhichitignorespossibleFeynmandiagramswherethefermion pairsdo notoriginate fromthe virtualvector bosons, andbythe fact that the “hard” amplitude

A

h1h2 is computed with exactly on-shell bosons. Its validity does not require the fermion pairs invariantmassesbeingexactlyequaltothe polemassofthe cor-responding bosons, though the amplitude is peaked around this configuration,because of the usual Breit–Wigner factors that we reabsorbedinthenormalizationfactor.

The variables

θ1

(2)

∈ [

0,

π

]

are the polardecayangles ofeach bosoninitsrestframe, orientedinthedirectionthatgoesfromthe

3-momentum of the V1(2) _{bosonto the one ofthe right-handed} fermion f₊1(2) producedinitsdecay.Inthespecialframethey are obtained fromthe rapidities

η

of the final state fermions by the relations cos

θ

1

=

tanh

η

s

₍

_f1 +

)

−

η

s

(

f−1

)

2

,

cos

θ

2

=

tanh

η

s

₍

_f2 −

)

−

η

s

(

f+2

)

2

,

(2)

where the“s” subscript denotes spacialframe quantities.The az-imuthalvariables

ϕ

1(2)

∈ [

0,2

π

]

aredefinedinthecenterofmass frameofthedibosonsystem(seeFig. 1)astheanglesbetweenthe decayplane ofeachbosonandthex–z planeofthe special coor-dinatesystem. The orientation ofthedecayplane istakenin the directionthatgoesfrom V1(2)to f₊1(2).Inthespecialframe,

ϕ

1(2) aresimplytheazimuthalangles

φ

ofthefinalstatefermions.More precisely

ϕ

1

= φ

s

(

f₊1

)

= φ

s

(

f₋1

)

+

π

,

ϕ

2

= −φ

s

(

f₊2

)

=

π

− φ

s

(

f₋2

) ,

(3)

modulo 2

π

. Notice that our seemingly asymmetric definition of thedecayanglesforthetwobosonsisactuallywhatisneededto describe their decay symmetrically in their own restframes. In-deeditproduces1

↔

2 symmetricalangularfactorsineq.(1).

Withthesedefinitions,eq.(1)iseasilyobtainedbydirect calcu-lation orbyapplying theJacob–Wick partialwave decomposition formula [20] to the case of a J

=

1, m

=

h particle decaying to two particles with helicity difference

λ

= λ

1

− λ

2

= +

1.1 Partial wave decomposition determines the

ϕ

1(2)-dependent phase fac-torsineq.(1)(uptothepreviouslymentionedoverallphases)and givesusdh

(θ )

equaltothed

J

m,λWignerfunction,i.e. d_±1

(θ )

=

1

±

cos

θ

2

,

d0

(θ )

=

sin

θ

√

2

.

(4)

Ourazimuthalangles

ϕ

1(2)aresimilartothosedefinedinHiggs to 4 leptonsdecayanalyses[22,23].There ishoweverone impor-tant difference, namelythefact that their orientationshavebeen specified in termsof fermions ofgiven helicities, while fermions aredistinguishedbytheirelectricchargeinthestandarddefinition. Weareobligedtoworkinthisnon-standardformalismbythefact thattheorientation hastomatchtheoneemployedinthepartial wave decompositionformula, wheretheordering ofthefermions determines whether

λ

= λ

1

− λ

2 equals plus or minus one. Con-cretely this makes a difference only if we consider the leptonic decayoftheZ boson,wherethechargeofthefinalstatefermions ismeasurablewhiletheirhelicityofcourseisnot.Thereforewhile thestandardangles(definedwithfermionchargeorientation)can befullydeterminedinthiscase,ouranglesaresubjecttothe dis-creteambiguity

{θ

1(2)

,

ϕ

1(2)

}

↔ {

π

−θ

1(2)

,

ϕ

1(2)

+

π

}

resultingfrom theinabilityto tellright-handedfromleft-handedleptons.Notice that the left–right ambiguitywe just described doesnot arise in the caseofa leptonicallydecaying W boson, wherefermion chi-ralityisknowntheoreticallyintermsoftheelectricchargeofthe chargedlepton. Howeverinthiscasedeterminingthedecayangles

1 _{Theresultdoesdependonconventionsinthedefinitionofthe vectorboson} polarizationvectors:differentdefinitionscanproducephasesinthevectorboson decayamplitudes,thatcompensatefortheextraphasesthatwillemergefromthe dibosonamplitudecalculation.ThestandardHELASconventions[21]areemployed here.

requires the reconstruction of the neutrino momentum, that in-troducesareconstruction ambiguity correspondingapproximately (forboostedW ) to

{θ

1(2)

,

ϕ

1(2)

}

↔ {θ

1(2)

,

π

−

ϕ

1(2)

}

.We will dis-cussthisindetailinsection3.

Taking the modulus square of eq. (1) we obtain interference termsbetweendibosonamplitudesofdifferenthelicities.Denoting ash

= (

h1,h2) thevectorformedbythetwo bosonhelicitiesand by

h

=

h

−

h thehelicity difference betweenthetwo interfer-ingamplitudes,theinterferencetermscanbesyntheticallywritten as

IV1V2 h⊗h

=

T

V1V2

hh

|

A

h

A

h

|

cos

[

h

·

ϕ

+ δ] ,

(5)

where

ϕ

= (

ϕ

1,

ϕ

2)and

δ

isthe relative phase between

A

h and

A

h, i.e.

A

h

A

∗_h

= |

A

h

A

h

|

eiδ. The dependence on

θ1

(2) and the couplingshasbeenencapsulatedin

TV1V2 hh

=

2 g

2

1g22dh1dh₁dh2dh2

.

(6) Theequationreproducesthewell-knownresultaccordingtowhich intermediate particles of different helicities do interfere, a pri-ori, andthat integrationover the two azimuthal decay angles is neededinordertocanceltheinterferencetermandobtaina fac-torized production-times-branching-ratiocross-section.Noticethat instead integrating over the polar angles

θ1

,2 does not cancel the interference because the Wignerd-functions are not orthog-onal. This means that while the interference effects we are in-terestedin are present in the data, it is not hard to “kill” them bymeasuring only quantities thatare inclusiveon the azimuthal angles. Examples of such quantities are the reconstructed mo-mentaof thetwo bosons andthe variables(transverse or invari-ant mass, pT or rapidity) obtained out of them, which are nor-mally employed in experimental analyses of diboson processes. Interference resurrection requires measuring the azimuthal an-gles

ϕ

1(2), or other kinematical variables that are sensitive to those.

Before moving to concrete examples of interference resurrec-tion,itisworthnoticingthateq.(5)canbefurthersimplifiedifwe restrictourselvesto2

→

2 dibosonprocessesattree-level,by ex-ploitinganinterestingconnectionwithC P symmetry.Thepointis thatthecomplexconjugateofatree-levelamplitude thatreceives no contribution fromnearly on-shell intermediate resonance ex-changeisequaltotheamplitudesforthe“reversed”process with

in andout statesinterchanged.Thisresultisaconsequenceofthe OpticalTheorem andapplies toSM amplitudes aswell as ampli-tudesinducedbyEFT operators.Thereversedamplitudeisinturn relatedtotheoriginalamplitudebytime-reversalsymmetry,which actsontheamplitudeslikeC P becauseoftheC P T theorem.With theseelementsone can prove2 _{that the2}

_→

_{2 amplitudes}

evalu-atedinthespecialframeobey

(

A

h

)

∗

=

ρ

C P

A

h

,

(7)

where

ρ

C P

= +

1 or

−

1 forC P -preservingandC P -violating ampli-tudes,respectively. We areinterested intheinterferencebetween

2 _{Thesimplestwayistoemploythepartial-wavedecompositionoftheamplitude,} noticingthattime-reversalactingonpartialwaveamplitudesjustexchangesin and out stateswithoutextraphases[20].Usingtheopticaltheorem,andthefactthat thebasisfunctionsintheamplitudedecompositionbecomerealifthescattering occurs(asitdoesinthespecialframe)onthex–z plane,theresultisimmediately derived.

theSMterm,whichisC P -even,andBSMcontributionstothe am-plitudeoriginatingfromEFToperators thatareeither C P -evenor

C P -odd.Intheformercasetheproductofthetwointerfering am-plitudes is purely real, while it is purely imaginary in the latter one.Eq.(5)thusbecomes

IV1V2 h⊗h

=

T V1V2 hh



A

SM h

A

BSM₊ h

+

A

BSM₊ h

A

SM h



cos

[

h

·

ϕ

] ,

(8) IV1V2 h⊗h

=

iT V1V2 hh



A

SM h

A

BSM− h

−

A

BSM− h

A

SM h



sin

[

h

·

ϕ

] ,

for C P -even andC P -odd BSM physics, respectively. Interestingly enough,measuringiftheinterferenceassumessineorcosineform (or a combination ofthe two, if both effects are present) would allowustodistinguishCP-evenfromCP-oddnewphysicseffects.

Sofarwehavediscussedmassivedibosonproduction;ifoneof the bosons isa photon, the resultissimpler andcan be worked out along similar lines.In thiscase we have onlyone polarand one azimuthal angle

θ

and

ϕ

associated with the decay of the massive boson V . The photon is a realfinal-state particleof he-licity hγ

= ±

1 andno interference is possible between different photon helicityamplitudes.Furthermore,athigh-energy,the only relevantBSMeffectsemergeinamplitudeswhere V isalso trans-verse, since amplitudes with only one longitudinal vector boson are suppressedbymV

/

E. Theinterferenceamong V -helicity con-figurationsh

= ±

1 andh

= ∓

1 readsinthiscase

I_hV_⊗γ_h,hγ

=

2g2sin2

θ



A

SM h hγ

A

BSM+ hhγ

+

A

BSM+ h hγ

A

SM hhγ



cos



h

ϕ

,

(9) I_hV_⊗γ_h,hγ

=

2ig2sin2

θ



A

SM h hγ

A

BSM− hhγ

−

A

BSM− h hγ

A

SM hhγ



sin



h

ϕ

,

where



h

=

h

−

h.

Intherestofthepaperweworkoutconcreteexampleofsuch interferenceeffects,focusing withgreaterdetailsonthe W

γ

pro-duction process withleptonically decaying W . The most difficult part of the analysis will be the reconstruction of the W boson

decay angle, a technique that furthermore can be useful for in-terference resurrectionalso inother channels. We thusdiscuss it extensivelyinthenextsection.

3. LeptonicW reconstruction

Measuring

ϕ

is essential for interference resurrection, as ex-plainedabove.Thiswouldberelativelystraightforwardfor hadron-icallydecaying W boson,3 _{butthedifficultiesofboostedhadronic}

W tagging and QCD background suppression make the leptonic casesimplertostudy(seehoweversection5).Intheleptoniccase, determiningtheW decayangles

θ

and

ϕ

requiresinsteadneutrino momentum reconstruction. This is performed with the standard strategy of identifying the neutrino transverse momentum (



p_⊥ν )

with the missingtransverse energy vector (E



miss

⊥ ) and

determin-ing the neutrino rapidity by imposing that the lepton–neutrino invariantmassequalsthe W polemassmW.Iftheleptonandthe neutrinoemergefromavirtualW withmassexactlyequaltomW, theequationhastwosolutions

η

±_ν

−

η

l

= ±





cosh−1

[

1

+ 

2

]



 = ±

log



1

+ 



2

+ 

2

₊

2



_,



2

=

m 2 W

−

m2⊥ 2p_⊥lp⊥ν

,

(10)

3 _{Weimplicitlyassumeherethatmomentaofalltheotherparticlesproducedin} associationwiththeW canbemeasureddirectly.

Fig. 2. Correlationbetweentherealandreconstructedanglesϕtrue,ϕreco inW+γ (W+→e+ν)processes,withoutdetectoreffects(left)andincludingdetectoreffects, simulatedwith Delphes (right).Eventsareselectediftheypassthefollowingselectioncriteria:p⊥γ>300GeV,p⊥l,Emiss⊥ >80 GeV,R(γ,l)>3 andηl<2.4.Pointswith auniquesolutionϕ+=ϕ−arehighlightedinorange.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

wherem_⊥

<

mW istheW -bosontransversemass.Onlyoneofthe twosolutionsreproducesthetrueneutrinomomentum,andthere isnowaytotellwhichone.Wethusdecidedtopickoneofthese solutions atrandom on an event-by-eventbasis. This introduces, evenbeforedetectoreffectsaretakenintoaccount,anuncertainty inthedeterminationoftheneutrinomomentum.

WefocusinparticularontheboostedW regime





1,which isthe relevant one forour high-energy analysis. Neutrino recon-struction becomes, from a purely theoretical viewpoint, increas-ingly accurate in this limit because the two solutions for

η

ν in

eq.(10)tendtocoincide,

η

±_ν

=

η

l

±

√

2

+

O

(

3

)

→

η

l.However an interestingsubtletyemerges, relatedwiththefact thatnot all thequantitiescomputedonthetwoneutrinomomentumsolutions coincideinthelimit, butonlythe fourcomponentsofthe recon-structed W momentum andthe polardecayangle

θ.

Forthe W

momentum,whichisjustthesumoftheleptonandofthe recon-structedneutrinomomenta,thisisratherobviouslythe case.The factthat the solutionsgive coincident

θ

can be seenby recalling the standard kinematics ofnearly massless parton splitting, that gives1

+

cos

θ



2x,wherex

=

El

/

EW



p⊥l

/(

p⊥l

+

p⊥ν

)

istheW

energyfractioncarriedawaybythechargedleptoninthesplitting. Since

θ

, in the limit, becomes function of observed components (p_⊥l and p⊥ν ) only, itmust bethe samewhen evaluated onthe

two solutions. The situation is instead very different for

ϕ

, that doesnotconvergetoauniquevalue. Inthelarge-boostexpansion

m2_W

/

p_⊥lp⊥ν



1 wefind4 cot

ϕ

=

1 sin

[φ

ν

− φ

l

]



sinh

[

η

l

−

η

ν

] +

O



m2_W p_⊥lp⊥ν

,

(11)

where

φ

l and

φ

ν are the lepton and neutrino azimuthal

coor-dinates in the lab frame. When evaluated on the two solutions

η

ν

=

η

ν (see± eq. (10)), thefirstterminthe squarebrackettends

tozeroas



∼

mW

/

p⊥ inthe boostedlimit,anddominatesover thesecondone.Thistermhasoppositesignonthetwosolutions. Intheboostedlimit,theleptonandtheneutrinobecomecloseto eachotheralsointhetransverseplane,therefore

φ

ν

− φl

tendsto

zeroanditispossibletoshowthat itscaleslike

.

Eq.(11)thus goestoa constantinthelimit,producingtwo oppositevaluesfor cot

ϕ

± whencomputed inthe twosolutions. So, onlyone ofthe tworeconstructedvaluesof

ϕ

willbe closetothetruedecay an-gle,theotheronewillbe

O(

1)different,andrelatedtotheformer by a discrete operation underwhich the cotangent changes sign.

4 _{Thisimplicitlyassumesa2→2 process.}

Sinceitispossibletoshowthatthesineof

ϕ

approachesaunique limit,therelationamong

ϕ

+and

ϕ

− is

ϕ

+

=

π

−

ϕ

− mod 2

π

,

(12)

andproducesthereconstructionambiguitymentionedinsection2. In reality, thevirtual W mass is not exactly mW, butis very close to that because the W is narrow. The “right” solution will thus provide a goodapproximation of thetrue kinematics. How-ever, experimental errors in the measurement ofthe lepton mo-mentum or of E



miss

⊥ , or the fact that the virtual W mass was

truly slightlyabove mW,canleadtoeventswithm⊥

>

mW.Then eq.(10)hasnorealsolutionandtheneutrinoisreconstructedby requiring that the lepton–neutrino invariant mass is as close as possibleto mW.Thisselectsauniqueconfiguration

η

ν

=

η

l.Inthis situation, the first term in the square bracket of eq. (11) is ex-actlyzero,

φ

ν

− φl

scales like



intheboostedlimitaspreviously

mentioned,while thesecondterminthesquare bracketvanishes as



2_{.The reconstructed}

_ϕ

_{thus approachesaconfigurationwith} cot

ϕ

=

0,correspondingto

ϕ

=

π

/

2 or

ϕ

= −

π

/

2

.

(13)

Theother variables,namelythe W momentumand

θ,

areinstead correctlyreproducedinthelimit.

The peculiarbehavior of the reconstructed

ϕ

, summarized in eqs. (12), (13), is illustrated in Fig. 2, where the true

ϕ

true is compared with the reconstructed one

ϕ

reco in the example of

W

γ

finalstates(thatis relevantfortheanalysisofthenext sec-tion). We haveselected events withphoton transverse momenta

p_⊥γ

>

300 GeV,whilep⊥l

,

Emiss_⊥

>

80 GeVinordertoavoid patho-logicalcaseswhereoneofthefinalstateleptonsisextremelysoft. Generation-level (MadGraph[24],[25]) eventsare shownon the left panel while Delphes[26] detectoreffectsare includedinthe rightone(Pythia 8[27]isusedforshoweringandhadronization). If m_⊥l

<

mW (blue points) we take one of the two solutions at randomaspreviouslydiscussed,howeverweverifiedthatthe fig-ure (and the rest of the analysis) would not change if we had taken systematically the

+

or the

−

solution. The events where

m_⊥l

>

mW, marked in orange, mostly give a reconstructed angle of

±

π

/2,

oftenalso ineventswherethetrueanglewas farfrom

±

π

/2.

Detector resolution has a considerableimpact on the de-terminationof

ϕ

,asitwastobeexpectedbecauseintheboosted regimetheleptonandtheneutrinogetclosetoeachotherandthe determinationofthe scatteringplane becomes increasingly sensi-tivetouncertaintiesinE



miss

⊥ andintheleptonmomentum.Notice

also that detectoreffectspopulate them_⊥l

>

mW region, making indeed moreorange points appear inthe figure. Thisinduces an anomalousconcentrationofpointsat

ϕ

reco

∼ ±

π

/2.

Fig. 3. ReconstructedazimuthalangulardistributionintheSM(blacklines)andBSM(bluearea,withC3W=0.2 TeV−2),normalizedtounitywith Delphes detectorstudy (right)andwithout(left).Sameselectioncutsas

Fig. 2

.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

4. AnomalousgaugecouplingsinW

γ

The only d

=

6 EFT operators that give unsuppressed high-energycontributionstotheW

γ

channelare(withtheconventions ofref.[1])

O3W

=

i jk_Wiν μW jρ ν Wkρμ

,

O

3W

=

i jk



WμiνW jρ ν Wρkμ

,

that are respectively C P -even and C P -odd, and correspond to modificationsof thetrilineargauge couplingsof ref.[6],as

λ

γ

=

6C3Wm2_W

/

g (and similarly for CP-odd quantities), where Ci are thecoefficients, withenergy dimension

−

2,appearing inthe La-grangianas

LBSM

=



Ci

O

i.At highenergythey give a quadrati-callyenhancedcontributiononlytosame-helicityW

γ

finalstates, namely

A

BSM+ + +

=

A

− −BSM+

≈

C3W6e

√

2M2_Wγsin

 ,

A

BSM− + +

= −

A

− −BSM−

≈

iC3W2e

√

2M2_Wγsin

 ,

(14)

where



isthe dibosonscatteringangle and MWγ the invariant

mass of the W

γ

system; e is the electric charge. Their contri-bution is instead not enhanced in the opposite-helicity channel, whichontheother handistheonlysizableoneintheSM,where

A

SM

± ±

∼

m2W

/

M2Wγ .Thisfactistheessenceofthenon-interference

problem[15]mentionedintheintroduction.Byeq.(9),after sum-mingoverthephotonpolarizations(whicharenotobservable),we obtain I_−⊗+Wγ

=

2g2sin2

θ

A

₊₊BSM+



A

SM −+

+

A

+−SM



cos 2

ϕ

,

I_−⊗+Wγ

=

2ig2sin2

θ

A

₊₊BSM−



A

SM −+

−

A

+−SM



sin 2

ϕ

.

(15)

Bylookingattheseequationsonemightworryaboutpossible can-cellations, occurringin one ofthe two interferenceterms, inthe presenceofexactorapproximaterelationsbetweenthe

(

−+)

and

(

+−)

SMamplitudes.Howevernosuchrelationsexistandthetwo interference terms are of comparable magnitudeonce integrated overthedibosonscatteringangledcos.

Following ourdiscussion insection 3, we should average our interferenceformula (15)over thetwo ambiguousconfigurations ineq.(12),obtaining thefollowinginteresting result.Interference withC P -odd newphysics

O3

_W, isopposite in the two ambigu-ous configurations, therefore it cancels in the average giving us no chance to detect it in the W

γ

final state. Interference with

C P -evennewphysics

O3W

,isinsteadinvariantunder

ϕ

→

π

−

ϕ

, henceitisunaffectedbytheaverageandperfectlyvisibleinspite oftheambiguity.ThisisverifiedinFig. 3,whereweshowthe re-constructed

ϕ

distribution withthe same cutsof Fig. 2. The SM

is nearly flat, as expected,5 _{while BSM (taking C}

3W

=

0.2 TeV−2 forillustration) introduces a cos 2

ϕ

behavior. Thelittle bumpsat

±

π

/2 are

duetom_⊥

>

mW configurations.Aside fromthose, the effectofthe Delphes smearingonthedistributionismild.

The rest of the analysis is straightforward. We simulate lep-tonic decays of W+

γ

where, in addition to the cuts for Fig. 2, weconsider p_⊥γ bins of

{

150,210,300,420,600,850,1200

}

GeV,

increasing linearly in size to accommodate experimental resolu-tion on p_⊥γ , but asfine as possibleto maximize the sensitivity

toBSMeffects.Inaddition,weconsider10azimuthalangularbins

∈ [−

π

,

π

]

,wherewefitthenumberofeventstoaquadratic func-tionofC3W.Werepeatthesimulationwithandwithout Delphes detectorsimulation,toquantifytheimpactoftheseeffects.Notice that when quoting generator-level results, we take into account anoverallreconstructionefficiency

∼

0.6 extractedfromthe com-parison with Delphes. Reducible backgrounds are not taken into accountinthesimulation,inspiteofthefactthatjetsfaking pho-tonsgivenearly50% oftheSM W

γ

contributioninexistingrun-1 studies of the W

γ

final state [28].However ref. [28] focuses on lower photonmomenta(p_⊥γ



200 GeV)thanthosethat are

rel-evant for our analysis. We thus expect the jet background to be lessrelevant inour casebecause the photon mistagrate forjets decreaseswithp_⊥γ [29]andbecausetheW j crosssectionshould

decrease fasterthan W

γ

dueto the steeply falling gluon parton distributionfunction.Still,weexpectthisbackgroundtobe signif-icant.

TheresultsareshowninFig. 4,intermsoftheprojected sen-sitivity atthe endofthe High-Luminosity LHC program(3 ab−1, left panel) and at an earlier stage (100 fb−1, right panel). The left vertical axis shows the reach in terms of anomalous cou-plings

λ

γ while the right axisis expressed in terms of C3W.As inref.[5],weshow howthereachdeteriorateswhenhigh-energy (high-p_⊥γ )bins are ignored inthe fit, withtheaim ofoutlining

which kinematicalregime (p_⊥γ



1 TeV,inthis case)is relevant

forthelimit. Accurateexperimental measurementsare neededin this regime, together with a trustable EFT prediction,i.e. an EFT cutoff



>

1 TeV.6Thefullsimulation,witha10%systematic rela-tiveuncertainty,summedinquadraturewiththestatisticalone,is portrayedinblackinthefigure,whiletheanalogousanalysis, but without binning inthe azimuthal angle

ϕ

, isshowndashed. The comparisonofthesetwolinesshowstheaddedvalueofour anal-ysis.Detectoreffectscanbequantifiedinsteadbycomparingwith

5 _{Infact,evenintheSM,interferencebetweenthe±∓andthelongitudinal0∓} amplitudes–whicharesuppressedbyonlyonepoweroftheenergyintheboosted regime–inducesamild∼cosϕ behavior,thatis howeverinvisibleduetothe reconstructionambiguityofeq.(12).

Fig. 4. Projected95%C.L.sensitivityonλγ andC3W forthe14TeVLHCwithanintegratedluminosityof3 ab−1(left)or100 fb−1(right),indifferentcases.Greendotted anddashedcurvescorrespondtoC3W=g/M2andC3W=g3/(16π2M2)(M≈2p⊥γ)respectively.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredto thewebversionofthisarticle.)

thegrayline,whiletheimpactofsystematicerrorsiscapturedby comparisonwiththeblueline.

For reference, we also show in green (dotted, dashed) theo-retical curvescorresponding todifferentpower countings,C3W

=

g

/

2 andC3W

=

g3

/(16

π

2



2

),

reflectingdifferentBSM hypothe-ses,see ref. [33]. Here we approximate





2p_⊥γ to arguethat,

for models that reflect the first power counting (dotted curve), the bounds we obtain are well within the EFT validity, in all

transverse-momentum bins. For weakly coupled models, where

theseeffectsariseatloop-level(dashedcurve),theprojected sensi-tivityisinsteadnot enough.Apopularheuristic methodtoassess the validity of the EFT expansion is to present results with and without the BSM-squared contributions in the cross-section. We havechecked that, withthisprocedure, boundswithout interfer-ence resurrection deteriorate by one order of magnitude, while interference-resurrectionboundsaremuchmorestable.

Ouranalysiscouldbeimprovedbyconsideringadditional vari-ables, such as the polar angle

θ.

A central cut in

θ

would in-deed enhance the interference term (9) compared to the non-interference onesthat are proportional to d2_± andare thus pref-erentiallyforwardorbackward. Wecould alsoexploitthe depen-denceon

,

whichwe could readilyget fromeq.(15).We leave thisforfuturework.

5. Otherchannels

Interferenceresurrectioncouldbeusefulinallchannelswhere BSMeffects hidein vector bosonfinal states that arerare inthe SM.Inthissectionwementionsomeotherinterestingapplications.

HadronicW decaysandCP-oddeffects C P -oddnewphysicscannot bedetectedintheleptonicW

γ

channelbecauseofthe reconstruc-tionambiguity.Inhadronicchannelstheambiguityinthe determi-nationof

ϕ

comesinsteadfromtheinabilitytomeasurefinalstate quarksflavor.Consideringforinstanceacharge-plusW ,wecannot tellwhichoneofthejets(orsubjets,sincetheW isboosted)isthe downanti-quark(whichhasnecessarily

+

1/2 helicity)andwhich oneis theupquark. Ifone ofthetwois pickedatrandomto be thed,

¯

weshouldaveragethecross-sectionovertheleft–right am-biguity

{θ

1(2)

,

ϕ

1(2)

}

↔ {

π

− θ

1(2)

,

ϕ

1(2)

+

π

}

describedinsection2 inthecontext ofZ -decays. Theseoperationsleavesboth C P -even

andC P -oddinterferences(9)invariant,henceitdoesnotprevent their observability.Clearly in the hadronic case we do not even

knowthe W charge,howeversummingovercharges cannot pro-duce a cancellationbecause the total cross-sectionsfor W+ and

W−aredifferentatorderone.Thisremainsachallengingchannel, becauseoftheneedofboostedhadronicW reconstructionand be-causeoftheimportantQCDbackgrounds.

FullyleptonicW Z Despitethesmallerrate,duetothesmall lep-tonic Z branchingratio,herewe canintegrate overthe W decay

angles, which we studied already inleptonic W

γ

, and focus on the Z decayangles

θ

Zand

ϕ

Z.Asdiscussedinsection2itis con-venienttothinkintermsofthe“standard”angles

θ

c_Zand

ϕ

c

Z,with orientations definedin termofthe charge-plusfinal state lepton ratherthanoftheonewith

+

1/2 helicity.Standardanglesarefully measurable,buttheyarerelatedwith“ourangles”

θ

Zand

ϕ

Zina waythatdependsonthechiralityoftheZ bosondecay.Namely,if the Z decaystoleft-handedspinors,thestandardangles coincide with ours, otherwise they are related to the formerby theleft– right ambiguity (section 2)

{θ

1(2)

,

ϕ

1(2)

}

↔ {

π

− θ

1(2)

,

ϕ

1(2)

+

π

}

. When computing the differentialcross-section in the

θ

c_Z and

ϕ

c

Z

angles we should thus sum over the two ambiguous

configura-tions, takingofcourseinto accountthat theleft-handed Z decay

coupling,gL,isdifferentfromtheright-handedone gR.Theresult isreadilyobtainedfromeq.(8)andreads

IW Z_−⊗+

=

2[g2_L

+

g_R2

]

sin2

θ

c_Z

A

₊₊BSM+



A

SM −+

+

A

+−SM



cos 2

ϕ

c_Z

,

IW Z_−⊗+

=

2i

[

g2_L

+

g2_R

]

sin2

θ

c_Z

A

₊₊BSM−



A

SM −+

−

A

+−SM



sin 2

ϕ

c_Z

.

(16)

Since it isinvariant under the left–right ambiguity, hZ

= +

1 in-terference with hZ

= −

1 does not cancel in the sum, neither in theC P -evennorintheC P -oddBSMcase.Thisinprinciplewould allow us to detect C P -odd interference. Notice that in the W Z

channel onecouldalsotake studytheinterferenceofBSMeffects with the longitudinal–longitudinal SM amplitude

A

SM

00 , which is non-vanishing inthehigh-energylimit.Theseeffectscancelifwe integrate over the W boson azimuthal angle, and do not carry radicallynewinformationonnewphysics.Hencecanbesafely ig-nored,atleastatafirststage.

Newphysicsinthelongitudinalpolarizations Asamatteroffact, lon-gitudinal polarizations, though survivinginthehigh-energy limit,

areaccidentally suppressedin theSM, withrespectto the trans-verseones[34].ItwouldthereforebeinterestingtoenhanceBSM effectsinthe longitudinalchannel,7 _{by exploitingitsinterference}

withthetransverse one.The interference termswiththe leading SMamplitudes,assumingafullyleptonicfinalstate,reads I₍W Z_{00)⊗(±∓)}

=

2g2

A

₀₀BSM+sin

ϕ

reco_W sin

ϕ

c_Zd0

(θ

W

)

d0

(θ

cZ

)

×

×



g2_L

[

A

₊₋SMd₊1

(θ

W

)

d−1

(θ

cZ

)

+

A

−+SMd−1

(θ

W

)

d+1

(θ

cZ

)

]

−

g2_R

[

A

₊₋SMd₊1

(θ

W

)

d+1

(θ

cZ

)

+

A

−+SMd−1

(θ

W

)

d−1

(θ

cZ

)

]



,

havingsummed over the W and Z ambiguities. Differentlyfrom the interference terms between the

(

±±)

and

(

±∓)

transverse channels,theaboveformulacancelsifintegratingovereither

ϕ

reco W or

ϕ

c

Z, and interference effects can be observed only if the az-imuthaldecayanglesofbothgaugebosonsaremeasured.Afurther subtletyis connected to the polar decayangles. Integrating over theZ polardecayangle

θ

c_Z,leadsto

I₍W Z_{00)⊗(±∓)}

=

π

2

√

2g 2

_[

_g2 L

−

g2R

]

A

BSM+ 00 sin

ϕ

reco W sin

ϕ

cZ

×

×

d0

(θ

W

)



A

SM +−d+1

(θ

W

)

+

A

₋₊SMg2Ld−1

(θ

W

)



,

(17)

which is suppressed by the small value of g2

L

−

g2R due to the almostexclusively axial couplings ofthe Z bosonto thecharged leptons(gL

 −

gR). Integratingover

θ

W leads insteadtono sup-pression

I₍W Z_{00)⊗(±∓)}

=

π

2

√

2g

2

_A

BSM+

00 sin

ϕ

recoW sin

ϕ

cZd0

(θ

cZ

)



A

SM +−

×

[

g2_Ld₋1

(θ

cZ

)

−

gR2d+1

(θ

cZ

)

] +

A

−+SM

[

g2Ld+1

(θ

cZ

)

−

g2Rd−1

(θ

cZ

)

]



.

Exploitinginterferenceresurrectionfornewphysicsinthe longitu-dinalW Z channelis thusparticularlychallenging inthe leptonic

Z finalstate, sinceit requiresthedetermination ofatleastthree decayangles,namely

θ

c_Z,

ϕ

c

Zand

ϕ

recoW . 6. Conclusionsandoutlook

Manyprocesses involvingelectro-weak bosons havedominant SMandBSM(dimension-6EFT operators)amplitudeswith differ-enthelicities,hencesuppressedinterferenceininclusive measure-ments.ThiscanbeanimportantobstacleintheLHCprecision pro-gram.Fortransversepolarizationsthiscanbeunderstoodthrough simplehelicity selectionrules that holdin thehigh-energy limit, whileforthe longitudinals itis duetoan accidental suppression ofthelongitudinalSMamplitude.

Wehavedescribedamethod,basedonexclusivemeasurements ofazimuthal angular distributions,that provides enhanced sensi-tivity to the interference between SM and BSM effects in dibo-son processes. At the practical level, ambiguities stemming from the W -reconstruction procedure,andthe impossibility of access-ing experimentally the fermion-helicity, singles out a number of processes,andeffects,that suitour proposedanalysis. In particu-lar,wehaveestimatedtheLHCreachforCP-evenmodificationsof trilineargaugecouplings,usingleptonicW

γ

finalstates,seeFig. 4. We have verified the robustness of our results with a detailed simulationincluding detectoreffects, andassessed the impact of luminosity andsystematics. We confirmedthat accessingthe in-terferencesubstantiallyimprovestheBSMreach.Thesameanalysis

7 _{TheseBSMeffectsarepurelyCPeven.}

canbeappliedtohadronicW

γ

orW Z finalstates(seealso[19]) andaccessbothCP-evenandCP-oddeffects.

An alternative strategy to resurrect the interference [2] relies on the emission of one extra parton, that turns on same-sign SMandopposite-signBSMtransversehelicityamplitudes.The ap-proachdescribedinthispaperismoreuniversallyapplicablethan the latterone,andit doesnot relyon next-to-leadingorder par-ton emission, which is potentially suppressed. Other domains of applicabilityofourmethodincludethestudyoflongitudinally po-larizedvectorbosonsscattering,thatappearsinthiscontextasone ofthemostinteresting casesbecauseofthe verysevere acciden-tal suppressionofthe longitudinalwithrespect tothetransverse and because of the BSM relevance of longitudinal vector boson scattering. However it does not fall in the diboson category we considered inthis paperandits studyisleft to future work.We alsoleavetofutureworkacompleteclassificationofdiboson pro-cesses, which one could study with leptonically or hadronically decaying bosons, aswell as a fully-differential studyusing polar, aswellasazimuthaldistributions.

Acknowledgements

We thank Roberto Franceschini, Thomas Gehrmann, Alex Po-marol and Riccardo Rattazzi for useful discussions. The work of G. P. is partly supported by MINECO under Grant CICYT-FEDER-FPA2014-55613-P,FPA2015-64041-C2-1-P,bytheSeveroOchoa Ex-cellenceProgramofMINECOunderthegrantSO-2012-0234andby theGeneralitatdeCatalunyagrant2014-SGR-1450.

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