Angular analysis of B→J/ψK1: Towards a model independent determination of the photon polarization with B→K1γ

Angular analysis of B→J/ψK1 Towards a model independent determination of the photon polarization with B→K1γ Physics Letters B 763 (2016) 66–71 Contents lists available at ScienceDirect Physics Letters[.]

Contents lists available atScienceDirect

www.elsevier.com/locate/physletb

E Koua, ∗ , A Le Yaouancb, A Tayduganovc

aLaboratoire de l’Accélérateur Linéaire, Univ Paris-Sud, CNRS/IN2P3 (UMR 8607), Université Paris-Saclay, 91898 Orsay Cédex, France

bLaboratoire de Physique Théorique, CNRS/Univ Paris-Sud 11 (UMR 8627), 91405 Orsay, France

cCPPM, Aix-Marseille Université, CNRS/IN2P3 and Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332, 13288, Marseille, France

Article history:

Received 16 May 2016

Received in revised form 3 October 2016

Accepted 12 October 2016

Available online 14 October 2016

Editor: B Grinstein

Weproposeamodelindependentextractionofthehadronicinformationneededtodeterminethephoton polarizationoftheb→sγ processbythemethodutilizingtheB→K1γ→Kπ π γ angulardistribution

We show that exactly the same hadronicinformation can beobtained by using the B→ J /ψ K1→

J /ψ Kπ πchannel,whichleadstoamuchhigherprecision

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3

1 Introduction

The circular polarization of the photon in the b→sγ

pro-cesshasaunique sensitivitytonewphysics,namelytothe

right-handedchargedcurrent(seee.g [1–3]) Whileitisavery

funda-mentalobservable,the experimentaldetermination ofthephoton

polarizationwasnotachievedatahighprecisionintheprevious B

factory experiments.Therefore,thisisa veryimportantchallenge

forLHCb aswell asforthe upgradeof B factory,Belle II

experi-ment.Varioustheoreticalideastomeasurethephotonpolarization

havebeenproposed (pioneeredby[4–8] andfollowedby [9–12])

and many experimental efforts are currently on-going [13,14]

Since the photon polarization measurement determines the

Wil-son coefficient C ( )

7 ,itwill havean importantconsequencetothe

globalfitaswell[15]

Recently the LHCb Collaboration has presented an interesting

result [16] on the so-called up-down asymmetry of the B →

Kπ π γ decay,originallyproposedin[7,8].Theup-down

asymme-try,which isthedifference ofthenumberofeventswithphoton

emittedabove andbelowthe Kπ π decayplane inthe Kπ π

ref-erenceframe, can indeedprovide theinformation onthe photon

polarization The basic idea isto determine the photon

polariza-tion by measuring the K1 polarization, which is correlated with

the photon polarization, through its angular distribution in the

B→Kπ π γ decay

TodeterminethephotonpolarizationfromtheLHCbresult,we

need the detailedprediction of the K1→Kπ π strong decay.In

* Corresponding author.

E-mail address:kou@lal.in2p3.fr (E Kou).

our previous works[9,17],we have obtainedthisinformation by usingtheotherexperimentalresults,mainlytheisobarmodel de-scriptionfrom theACCMOR Collaboration[18],complemented by thetheoretical modelcomputation usingthe3P0 model[19].The

B→K1(1270) γ →Kπ π γ channel, different from the K1(1400)

channel, requiresvarious unconventionaltreatmentsand unfortu-nately, our conclusion is that there are certain uncertainties re-mainingtodescribethischannel.Themaindifficultiesare(see[17]

forthedetaileddiscussions):

•theexistenceoftwointermediateprocesses,K1(1270) →K∗π

and K1(1270) →Kρ, withthe latter beingjust on the edge

ofthe Kρphase spaceandhavinghoweveralargebranching ratio.Quasi-thresholdeffectsmustbetakenintoaccount;

•furthermore,aswe found, thefinal estimationofphoton po-larizationisalsosensitivetothecontributionofthe K1(1270)

decaychannelswithscalarisobars,K1(1270) →K( π π )S−wave

orK1(1270) → (Kπ )S−waveπ,whicharenotwelldetermined, neitherbyexperimentnorbytheory

These problemsmustbe solved inthe futurewithmoredetailed analysisof K1 resonances,whichareproducedfrom B, τ or J/ψ

decays

In this article, we rather propose a model independent ap-proachtocircumventtheproblem.Inallthepreviousworks,onlya partialangulardistributionwasconsidered,i.e.takingintoaccount onlyone θ angle.We showinthisarticlethatwithamore com-pleteangulardescription,theinformationontheK1decayneeded for photon polarization determination can be extracted directly from B→Kπ π + γ decay.Thatis,usingtheanglesinvolvingnot

http://dx.doi.org/10.1016/j.physletb.2016.10.013

0370-2693/©2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) Funded by SCOAP 3

onlythecosθ likedistributionwhichyieldstheup-down

asymme-try,butalsotheazimuthalangleφdependence,wecanobtainthe

fullhadronicinformationwithoutthe isobarmodeldescriptionof

theresonances

Infact,withthelimitedstatisticsavailable forB→Kπ π + γ,

thismethodiscurrentlydifficult.Ontheother hand,itturns out

that we can obtain the samehadronic information fromanother

channelB→Kπ π + J/ψwheretwoordersofmagnitudeshigher

statistics,withrespecttothephotonchannel,isavailable[20].We

showthat thefullangulardistributionmeasurement allowsusto

separate the B decay and K1 decay partsso that we can extract

thesamehadronicinformationfromtheB→Kπ π + J/ψdecay

Forthemoment,forasimplerillustrationoftheapproach,we

considerthecaseofonly one K1 resonance,which maybe

prac-tically supported by the fact that B→K1(1270) γ seems largely

dominantoverB→K1(1400) γ [14,16,21,22]

Therestofthearticleisorganized asfollows:insection2,we

introducethe kinematicalvariables includingthe θ andφ angles

whichare crucial forourwork In section 3,we write down the

decayamplitudesofB→K1J/ψand B→K1γ withK1 decaying

toKπ π.Insection4,wederivetheangulardistributionsforthese

decays.Then,wedemonstrateinsection5thatthehadronic

infor-mationweneedtodeterminethephotonpolarizationinB→K1γ

can be obtained directly from the measurement of angular

co-efficients in B→K1J/ψ and/or B→K1γ, and we conclude in

section6

2 Kinematics ofB+→V K+

1 →V K+π+π− decay(V = J/ψ, γ)

Inthissection,wedescribeall thedefinitionsofthe

kinemati-calvariables(see Fig 1).WeuseB+→V K+

1 →V K+π+π−decay

asanexample butone canobtain thesimilar formulaeforother

charge combinations.Throughoutthis article,we work inthe K1

restframe.Wecan movetotheconventional B restframeorany

other frame simply by a Lorentz transformation First, we assign

thethreemomentaas

π+( p1) , π−( p2) , K+( p3) (1)

Now,we define a standard orthogonal frame, with respect to

thespindirectionofK1,orV =J/ψ, γ.First,theO z isdefinedas



e z= p V

|p V| =

−p B

Wedefinetheaxisperpendiculartothe Kπ π decayplanebyn:



n= p1× p2

Then,the O y is chosenasnormaltothe O z and V = J/ψ, γ

di-rectionby



e y= p V× n

Finally, O x is then chosen as the normal to O y and O z: e x=



e y× e z

Onealsodefinesapolarangleθ,ofn with respecttothee z:

cosθ = e z· n (5)

Letusheresetaconditionforθ as



e x· n=sinθ >0, 0< θ < π (6)

Now we rotateex onto the Kπ π decayplane anddefine the

resultase

x whichcanbewrittenas



e= e y× n (7)

Fig 1 Kinematics of the B→K1(→K π π ) V decay.

Wecanthendefineasecondorthogonalframe,whichisbasedon the K1 decay plane, e,ey,n. Defining φ1,2 to be the azimuthal anglefromthee

x axisinthis(x,y)decayplane,thecomponents

ofthepionsthreemomenta,



p1,2= |p1,2|(cosφ1,2e

x+sinφ1,2e y) , (8)

canbeexpressedintermsofθ,φ1,2 inthestandardframeas:

( p1,2)x= |p1,2|cosθcosφ1,2, ( p1,2)y= |p1,2|sinφ1,2, ( p1,2)z= −|p1,2|sinθcosφ1,2.

(9)

Theadvantageisthattheanglesθ,φ1,2 areconnecteddirectly withthedecayplane.We notethatthelinearcombinationofthe

φ1,2 angles,

isafunctiontheDalitzvariablesdefinedby

s= (p K1)2

s13= (p1+p3)2= (p K1−p2)2,

s23= (p2+p3)2= (p K1−p1)2,

s12= (p1+p2)2= (p K1−p3)2.

(11)

In the K1 rest frame, p K1 =0 and |p1,2,3| can be expressed in termsofs23,s13,s12respectively.Sinceonlytwoofthemare inde-pendent,wechooses23,s13forsymmetry.Thentherelativeangle betweenthethreemomentaofthetwopions

cosδ = p1· p2

|p1||p2| =

|p3|2− |p1|2− |p2|2

2|p1||p2| , (12)

isexpressibleintermsofs, s13,s23.Thesameholdsfortheother relativeanglesbetweenthethreemomenta.1 Thismeansthat the

Kπ π systemisrigidoncethemassesofthetwo Kπ subsystems havebeenchosen.Itisstillallowed torotatehowever:ifthe nor-mal isfixed by a definite θ, there remains a freerotation of the rigid Kπ π systemaround n in the decay plane We choose the angledefiningthisrotationas:

1 We have furthermore 0< δ < π, (sinδ >0), because the anglesφ1 ,φ2 are mea-sured in the plane oriented by the normaln= p × p /|p × p |.

φ ≡ φ1+ φ2

Inthisway,the angleφ in thereference [7]isnow fixed,which

allowstoperformdefinitecalculations Notethat ourdefinitionis

justonepossibleamongmanyotherswhilewehavefoundit

con-venientbecauseitsimplifiesthecalculations

Then,re-expressingφ1,2 as

φ1,2= φ ∓ δ

2,

onecangetthecomponentsofp1,2 inEq.(9),expressedinterms

ofφandtheDalitzvariables

3 The decay amplitudes and rates

Thefourbodydecayratecan bewrittenastheproductofthe

decay ratesof B→K 1s z V s z and K 1s z →Kπ π summed over the

differentV polarizations:

d V( ) ≡d B→K1V→ (Kπ π )V)s (14)

s z

(2π )4

2M B



 MV

s z(B→K 1s z V→ (Kπ π )V)s 2

(2π )3dsd2d3,

wheres z isthepolarizationofV = J/ψ, γ:

s z=0, ±1 (for V= J/ψ ), s z= ±1 (for V = γ ) (15)

WefollowthePDGconvention, i.e.

d2= 1

(2π)5

|p∗

V|

2M B,

ψ d3=

1

32(2π)8

1

in the previous section Here, B can be B±, B0 or B0 Denoting

theamplitude of B→K1( )V asAs z( )andof K1( ) →Kπ π as

μ

K 1szJμ,onecanwrite:

MV

s z(B→K 1s z V→ (Kπ π )V)s= A

V

s z( ) × ( μ

K 1szJμ( 13, 23)s) ( −m2K

1) +im K1 K1( ) .

(16)

Inthefollowing,weconsideronlythedominantK1=K1(1270)for

simplicity,though itcanbereadilyextendedtoincludeK1(1400)

The propagator of the K1, which is parametrized hereas Breit–

Wignerfunction,isintroduced inordertousetheKπ π invariant

mass m K π π ≡ √s as thevarying K1 mass The K1 restframe is

meantastheactual Kπ π system.Thisisnotaconvention,butan

assumption onthe off-shellextrapolationof amplitudes,partially

justified by unitarity Note that this implies that the Dalitz plot

(13,s23)dependsons aswell

InEq.(16),thefullkinematicalvariabledependenceofJ isleft

implicitbutitcan be displayed withhelpoftwo formfactors as

C1,2[9]:

Jμ( 13, 23)s≡ C1( , 13, 23)p1μ− C2( , 13, 23)p2μ. (17)

Theseformfactorscouldbemadeexplicitinaquasi-two-body

ap-proach to the K1 decay [17] Here, on the contrary, we want to

determinetheminamodelindependentwaybyusingthe

experi-mentaldatatoavoidtheambiguitiesdescribedintheintroduction

4 Angular distribution

Now, we define the probability density function (PDF) for a

given value of s. First, the different transverse (s z= ±) andthe

longitudinal(s z=0)polarizationsofV statedonotinterfere,thus

thedecayrateiswrittenas2:

2 ForV=J /ψ, we integrate over the J /ψdecay angle here so that the

interfer-ence term disappears.

d B→K1V→ (Kπ π )V)s

ds13ds23d(cosθ )dφ

= (2π )4

2M B

(2π )3ds 1

(2π )5

|p∗

V|

2M B

32(2π )8s







1

( −m2K

1) +im K1 K1( )





 2

s z

| AV

s z( ) |2 

K 1sz·  JK1(13, 23)s 2

,

(18)

where p∗

V isthethreemomentumof V inthe B referenceframe, while the K1 polarization vector K1 and J K1 are definedin the

K1 reference frame Therefore, the θ and φ dependence is con-tained inthe factor  

K 1sz·  JK1(13,s23)s 2

.Note that inEq.(18),

we have to add all charge combinations, K+

1 →K+π+π− and

1 →K0π+π0 forK+

1 and K10→K+π0π− and K10→K0π+π−

forK0 (andsimilarforthechargeconjugations)

ThePDFWV(13,s23,cosθ, φ)sisobtainedfromEq.(18)andis normalizedas:



ds13



ds23



d(cosθ )



dφ WV( 13, 23,cosθ, φ)s=1. (19)

Thus,thePDFcanbewrittenintermsofthesquareddecay ampli-tudes,whicharethefunctionsofthekinematicalvariablesweare interestedin,withouttheirrelevantpre-factors:

WV( 13, 23,cosθ, φ)s=



s z| AV

s z( ) |2 

K 1sz·  JK1(13, 23)s 2



ds13



ds23



d(cosθ ) 

dφ 

s z| AV

s z( ) |2 

K 1sz ·  JK1( 13, 23)s 2

(20)

NextwemakeexplicittheangulardistributionofWV usingthe definitionofthecoordinatesystemandanglesgiveninsection2:

WV( 13, 23,cosθ, φ)s

≡a V+ (a V1 +a2Vcos 2φ +a3Vsin 2φ)sin2θ +b Vcosθ , (21)

wheretheangularcoefficientsdependontheDalitzvariablesand fixedvalueofs.Theycanbewrittenas:

a V( , 13, 23) =N s Vξa V



|c1|2+ |c2|2−2Re(c1c∗

2)cosδ



, (22)

a1V( , 13, 23) =N s Vξa V i



|c1|2+ |c2|2−2Re(c1c∗

2)cosδ



, (23)

a2V( , 13, 23) =N s Vξa V

i



( |c1|2+ |c2|2)cosδ −2Re(c1c∗

2)



, (24)

a3V( , 13, 23) =N s Vξa V

i



( |c1|2− |c2|2)sinδ



b V( , 13, 23) = −N s Vξb V

2Im(c1c∗

wherethefactorN V s >0 isthenormalizationfactor,whichisequal

totheinverseofthedenominatorofEq.(20) The ξ’s representthe B→K1 V decay,and thus,depend only

ons

ξa V( ) ≡ | AV+( ) |2+ | AV−( ) |2

ξa V

i( ) ≡ −(| AV+( ) |2+ | AV−( ) |2) +2| AV( ) |2

ξb V( ) ≡ | AV+( ) |2− | AV−( ) |2

(27)

Infact,for V = γ,thelongitudinalamplitude vanishes (Aγ

0 =0), which simplifies the above expressions, giving as a result aγ =

−2a γ1

Thecoefficientsc1 ,2 are relatedto theformfactorsinEq.(17)

as:

c1( , 13, 23) = C1( , 13, 23) |p1|,

c2( , 13, 23) = C2( , 13, 23) |p2| ,

wherewewroteexplicitlytheDalitzvariablesdependence.The

an-gleδ(with0< δ < π)isdefinedas

cosδ = p1· p2

|p1||p2| .

Letusalso remindthat all therelevantkinematicalvariables can

beexpressedintermsoftheDalitzvariables:

|p1,2|2=E21,2−m21,2, p1· p2=E1E2−s12−m21−m22

2 ,

E1,2=s−s23,13+m

2

1,2

2√

s .

5 Photon polarization: relating theB→K1γ andB→K1J/ψ

amplitudes

The photon polarization in the B→K1γ process which we

wanttodetermineisdefinedasfollowing:

Pγ ≡ | A

γ

+( ) |2− | Aγ−( ) |2

Strictly speaking, Pγ is differentfrom the “polarization

parame-ter”

λγ ≡ |C+|2− |C−|2

where C± represents only the short-distance b→sγ decay, i.e

C+/C− m s b )/m b ( for B(B)decays,whiletheamplitude Aγ

±( )

iswrittenastheproductofC±andthehadronicformfactorT1(0)

whichcontains the long-distance effect.Now, when we consider

onlyone K1 finalstate,weexpectasingleformfactorforboth±

polarization,i.e.Aγ

±( ) ∝T1(0) Thus,the long-distancepart

theso-calledcharmloopcontributionsdeviatetheformfactorsfor

the±polarization, whichinducesasmalldifference betweenPγ

andλγ Wewillcomebacktothisissuelater-on Notethat Pγ is

±( )forradiative

decays We will also discuss on a possible s-dependence of Pγ

later-on

NowusingEq.(27),onecanfind

Pγ = ξ

γ

b

Weshownowthatthiscanbedeterminedfromthemeasurement

ofangularcoefficients of B→K1γ and B→K1J/ψ, i.e.a V,a V

i,

ourmainfinding,is:

Pγ = ξ

γ

b

ξa γ = ∓b γ( , 13, 23)

a γ( , 13, 23)

1−

a V2( s13,23)

a V

1( s13,23)

2

−

a V

3( s13,23)

a V

1( s13,23)

Letusbrieflyderivethisequation.First,weobtainξa γ via:

ξa γ= a γ( , 13, 23)

N γ s

|c1|2+ |c2|2−2Re(c1c∗

Theterminthesquarebracketsinthedenominatoriscommonfor

V= J/ψ, γ andcanbeobtainedforgivenpointof( ,s13,s23)as

|c1|2+ |c2|2−2Re(c1c∗

2)cosδ =a V( , 13, 23)

N V sξa V( ) =a V1( , 13, 23)

N s Vξa V i( ) .

(33)

Next,wedetermineξb γ fromtheexperimentalmeasurementof

bγ( ,s13,s23):

ξb γ= − b γ( , 13, 23)

N γ s

2 Im(c1c∗

Nowweobtainthedenominatorfactor2Im( 1c∗

2)sinδ.Bywriting

Im(c1c∗

2) = ±  |c1|2|c2|2− [Re(c1c∗

2) ]2,

we findthat we need toobtain independentlythese twofactors,

|c1|2|c2|2 andRe(1c∗

2), fromtheaboveequations.Then, by using Eqs.(23)–(25),wefind

2 Im(c1c∗

N s Vξa V i( )

×  (a V1( , 13, 23))2− (a2V( , 13, 23))2− (a3V( , 13, 23))2

(35)

Finally,thesignambiguityremains, whichcannotberesolved

atthispoint

NowbyinsertingEqs.(32)–(35)intoEq.(30),wecanobtainthe polarizationwhichwewanttodetermineasEq.(31)

ThemainresultinEq.(31)implies:

• The photon polarization in B→K1γ can be obtained from the measurement of the angular coefficients aγ( ,s13,s23),

bγ( ,s13,s23)whichcanbemeasured onlywiththestandard cosθ distribution, together with the coefficientsa V1,2,3( ,s13,

s23)whichrequirestheazimuthalangleφdistribution.The ad-vantageisthatthelattercoefficientscanbemeasuredequally

by using either B→ J/ψK1 or B→K1γ decays Therefore,

we can take advantage of the much higher statistics of the

J/ψprocess

• Thefinalresultsdependonlyontheratiooftheangular coef-ficientssothatthereisnoneedforthenormalization

• The photon polarization Pγ does not depend on s nor any Dalitz variables (sub-dominant effects which could induce

ex-pressioninEq.(31)isconstantatanypointofthe( ,s13,s23)

plane.Whenweusethe J/ψtodeterminethedenominatorof thisterm,wesimplyneedtomappointbypointontheDalitz plane

• Concerningthe sign ambiguity, inpractice, we maymeasure theabsolutevalueofthepolarization parameter| Pγ|.Inthis way,weareleftwiththesignambiguityofoverallsignofPγ

butwecanneglectthesignvariationofbγ/aγ termsincePγ

mustbeconstantinthe( ,s13,s23)plane

Weshouldmakeabriefcommentonthes-dependenceofPγ

Althoughit issub-dominant, acontamination fromthe K1(1400)

resonancecould causethe s-dependence.Also,the largewidthof

theK1(1270)itselfinducingan s-dependencecannotbe

impossi-ble[23].However, forbothcases,thes-dependencewouldappear

onlyat farthe K1 pole.Therefore, instudyingthe amplitudesin

the vicinityof the peak,we expect thefinal s-dependenceto be

verymoderate

Asstressedearlier,thepolarizationPγ differsinprinciplefrom

andisnotincludedthereforeintheC±coefficients.Theevaluation

ofthiseffectisverydifficult.Ithasbeen discussedquantitatively

onlyinthesimplercasesB→K∗γ andB→K∗l+l−whererather

differentevaluationshavebeenproposed:onebeingaparametric

oneinthe1/m b expansion[24],anotherbeingthroughQCD sum

rules [25,26].Inour paper[1], we havetried todiscussthe

con-nectionbetweenthetwoevaluations.Ontheother hand,an

eval-uationofcharm contributionsto B→K1γ hasnot beendoneso

far Since theshort-distance contributions, includingnew physics

effects,shouldbethesameforB→K∗γ andB→K1γ,an

obser-vation ofdifferentphotonpolarizations betweenthesetwo

chan-nelsshouldbeattributedtothelong-distanceeffect,inparticular,

to thecharm contributions Therefore,such an observationcould

provideanimportantkeytounderstandthecharmloop

contribu-tions

Before closingthe section, let us discussthe reliability ofthe

method.Ourargumentbelowisonlyqualitativesincefora

quan-titative discussion, detailed Monte Carlo simulations would be

needed B→ J/ψKπ π has been studied by the Belle

Collab-oration [20] In order to separate B→ J/ψK1 event from the

J/ψKπ π spectrum, a careful resonance study has to be done,

namelyvetoingothercharmoniumchannelssuchasB→ ψ(2S)K1

as well as the exotic resonances which decay into J/ψ π π, i.e

B→X(3872)K or B→Y(4260)K Nearly 2.5×103 events are

identified as B→ J/ψK1 in [20] Approximately 20(100) times

more events are expected at Belle II with 10(50)ab−1 of data,

whichwillalloweasilytoextractdetailedDalitzandangular

distri-butionsof K1 decays.Thereforetheerrorsexpectedinthesecond

part ofEq (31) (those written interms of a V i) would be nearly

negligible

ThemainuncertaintywillcomefromthefirstpartofEq.(31),

i.e.theratiooftheangularcoefficientofB→K1γ,bγ( ,s13,s23)/

aγ( ,s13,s23).IntherecentanalysisofBabar[14],about2.5×103

B+→K+π+π−γ events are reconstructed, among which 60%

are known to come from B+→K+

1(1270) γ Thus, with Belle II with10(50)ab−1 ofdata,weexpect 5(25) ×103 B→K1(1270) γ

events With LHCb run one data (3 fb−1

), 1.4× 104 B+ →

K+π+π−γ eventsarereconstructed,whichextrapolateto∼2.2×

104 events for B+→K+

1(1270) γ at the end of LHCb run II (8 fb−1).Withthissizeofdata,wecaneasilymakeoverahundred

ofbinsontheDalitzplane,whichcanbefurtheroptimizedby

us-ing the known decay property of K1(1270) This naive estimate

tellsthatwecanhaveorderof10MeVresolutionon π π andKπ

invariantmass,whichcanleadtoahighenoughsensitivitytoPγ

6 Conclusions

The angular distribution in the polar angle θ of the B →

Kresγ →Kπ π γ processhasrecentlybeenmeasuredbytheLHCb

Collaboration[16].Amongvarious kaonicresonances Kres,alarge

B→K1(1270) γ contribution has beenidentified, confirming the

previous result [14,21,22] The extraction of the b→sγ photon

polarization from this datarequires a detailedknowledge of the

K1 decays,inparticular, theimaginarypartoftheproductofthe

twoformfactors,Im( 1c∗

2).Theimaginarypartis,ingeneral,very sensitive tothe resonancestructure ofthe decaywhilethere are

manyuncertaintiesintheresonancedecaystructureofK1(1270),

especially due to i) the limited phase space for the main decay

channel K1(1270) → ρK resultinginstrongdistortioneffects,ii) a possible K1(1270) → κπ contributions, neither well determined experimentallynortheoreticallytractable

Inorderto circumventthisproblem, wepropose a determina-tion of the strong interaction factor Im( 1c∗

2) independent of an isobarmodelfortheK1decay.ThismethodrequirestheDalitzplot

oftheangularcoefficientsincludingbothpolarandazimuthal an-gles.Inthisarticle,wehaveshownthatthesameDalitzplot anal-ysis can be also obtained through the B→ J/ψK1→ J/ψKπ π

channel.TheB decaypartofthesetwochannelsareverydifferent whilewe foundthat wehaveenoughobservablestoseparate the

de-tailedMonteCarlostudies,inparticularbyevaluatingthebinning effect

Acknowledgements

We would like to thank François Le Diberder for many dis-cussions, inparticular, on the feasibility of the method We also acknowledge Patrick Roudeau, Akimasa Ishikawa and Yoshimasa Onofordiscussions TheworkofA.T.hasbeencarriedout thanks

the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the “In-vestissements d’Avenir” Frenchgovernmentprogram managed by theANR

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oftheangularcoefficientsincludingbothpolarandazimuthal an-gles.Inthisarticle,wehaveshownthatthesameDalitzplot anal-ysis can be also obtained through the B→... b→sγ photon

polarization from this datarequires a detailedknowledge of the

K1 decays,inparticular, theimaginarypartoftheproductofthe

twoformfactors,Im(...

Thus,thePDFcanbewrittenintermsofthesquareddecay ampli-tudes,whicharethefunctionsofthekinematicalvariablesweare interestedin,withouttheirrelevantpre-factors:

WV(