how to understand the underlying structures of x 4140 x 4274 x 4500 and x 4700

Blaizot Keywords: Exotic state Tetraquark Rescattering effect Triangle singularity We investigatethe possiblerescattering effects whichmaycontributeto theprocess B+→ J /ψφ K+.. Rescatter

Contents lists available atScienceDirect

www.elsevier.com/locate/physletb

Xiao-Hai Liu

Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan

Article history:

Received 20 July 2016

Received in revised form 22 December 2016

Accepted 7 January 2017

Available online 11 January 2017

Editor: J.-P Blaizot

Keywords:

Exotic state

Tetraquark

Rescattering effect

Triangle singularity

We investigatethe possiblerescattering effects whichmaycontributeto theprocess B+→ J /ψφ K+.

It isshownthatthe ψφrescatteringviathe ψK1loopcansimulatethestructureofX (4700).Thecusp effectduetothe D∗+

s D−

s rescatteringmaypossiblysimulatethe X (4140) structure,butitdependson the cuspmodelparameters.Ifthe quantumnumbersof X (4274) (X (4500) are 1++ (0++), it ishard

toascribetheobservationof X (4274)and X (4500)tothe P -wavethresholdrescatteringeffects,which impliesthat X (4274)and X (4500)couldbegenuineresonances.Wealsosuggestthat X (4274)maybe theconventionalorbitallyexcitedstateχc1 ( 3P )

©2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

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

IntheStandardModel,quarksandgluonsarerelatedto

color-singletmesons andbaryonsby the long-distanceregime ofQCD,

which remains the least understood aspect of the theory.In

ex-periments, there has been a renewal of hadron spectroscopy in

thelastdecade,initiated bythefindingsofnumerousX Y Z states

Mostofthesestatesdonotfitintothepredictionsofthe

conven-tionalquark model,whichhasbeenproved tobe verysuccessful

indescribing the heavy quarkonia belowthe open-flavor

thresh-olds.Someofthe X Y Z states, suchaschargedcharmonium-likeZ c

states,chargedbottomonium-like Z b states,andheavypentaquark

candidates P c(4450) and P c(4380), definitely cannot be the

con-ventionalq q-mesons ¯ or qqq-baryons Thesenewobservations not

onlyenrichourknowledgeaboutthehadronspectroscopybutalso

bring new challenges We refer to Refs [1–5] for both

theoret-ical and experimental reviews about the recent study on exotic

hadrons

Veryrecently, theLHCbcollaborationreportedthe observation

ofseveralresonance-likestructuresin J/ψφ invariantmass

distri-butionsin B+→ J/ψφK+ decays[6,7].Theirmasses, widthsand

favorable quantumnumbersare

M X(4140)=4146.5±4.5+4.6

−2.8MeV,

X(4140)=83±21+21

−14MeV, J P C=1++,

E-mail address:liuxh@th.phys.titech.ac.jp

M X(4274)=4273.3±8.3+17.2

−3.6 MeV,

X(4274)=56±11+8

−11MeV, J P C=1++,

M X(4500)=4506±11+12

−15MeV,

X(4500)=92±21+21

−20MeV, J P C=0++,

M X(4700)=4704±10+14

−24MeV,

X(4700)=120±31+42

−33MeV, J P C=0++, (1)

among which the higher states X(4500) and X(4700) are firstly reported by the LHCb collaboration X(4140) and X(4274) were firstly observed by the CDF collaboration in the J/ψφ invariant massdistributionfromB →K J/ψφ decays[8,9].Thepresenceof

X(4140)in B decays waslaterconfirmedbytheCMSandD0 col-laborations [10–12] Another state X(4350) was reported by the Bellecollaborationfromthetwophotonprocess γ γ →J/ψφ[13]

X(4140) and X(4274)were alsoexpectedto be produced in the twophotonfusionreaction,butneitherofthemwasobserved[13]

Ifthequantumnumbersof X(4140)and X(4274)are1++,as

re-portedbytheLHCbcollaboration,theirnon-observationinthetwo photofusionreactionscanbeunderstoodaccordingtothe Landau– Yang theorem, which forbids the transitions between a massive spin-1 particle andtwo real photons[14,15] But fortwo virtual photonfusionorone realandonevirtual photonfusion,the pro-ductionofmassivespin-1particlesisnotforbidden

http://dx.doi.org/10.1016/j.physletb.2017.01.008

0370-2693/©2017 The Author 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

3

Fig 1 Rescattering processes via the open-charm meson loops.

Theseresonance-like peaksin the J/ψφ invariant mass

spec-trumareveryintriguing,becausetheymaycontainbotha cc pair ¯

and an s ¯s pair, whichimpliesthatthesestatesmaybeexotic

Tak-ingintoaccounttheirmassesanddecaymodes,someresearchers

suggest that X(4140) and X(4274) could be the hadronicbound

statesof D∗+

s D∗−

s and D+

s0 D−

s,respectively [16–26].By means of quarkmodelorQCDsumrules,these“ X ”stateobservedin J/ψφ

distributionsare alsosuggestedtobe some csc¯ ¯s tetraquark states

withproperquantumnumbers[27–32]

Concerning those exotic states, apart from the genuine

reso-nances explanations, such as molecular states, tetraquark states,

or hybrid, some non-resonance explanations were also proposed

in literatures There have been many theoretical attempts to try

to connect the singularities of the rescattering amplitudes with

the resonance-like peaks in experiments, such as the cusp

ef-fect [33–38], or the triangle singularity (TS) mechanism The TS

mechanismwas firstnoticein1960s[39–49].Unfortunately,most

ofthoseproposed processeswerelackofexperimental supportat

thattime.Itwas rediscoveredbypeopleinrecentyearsandused

to interpret some exoticphenomena, such as the largely isospin

violationin η (1405) →3π, theproduction of a1(1420),the

pro-ductionofsome X Y Z particles andsoon[50–64].Itisshownthat

sometimes it is not necessary to introduce a genuine resonance

todescribearesonance-likepeak,becausesome kinematic

singu-laritiesoftherescatteringamplitudescouldbehavethemselvesas

bumps in the corresponding invariant mass distributions, which

maybring ambiguities to ourunderstanding about the natureof

exoticstates.Before claiming thatone resonance-likepeak

corre-spondsto onegenuine particle,it isalsonecessary toexclude or

confirmthesepossibilities

Inthiswork, we investigatesome possiblerescatteringeffects

intheprocess B+→J/ψφK+.Theopen-charmmesons

rescatter-ingandψφrescatteringarestudiedinSection2.Theconclusions

andsomediscussionsaregiveninSection3

2.1 Open-charm meson rescattering effect

In experiments, the rates of B decays into a charmed meson

andacharmed-strangemesonare foundtobe quitelarge among

thehadronicdecaymodes.Sincethevelocitiesoftheopen-charm

mesonswillnotbelarge,theyhaveadequatetimetogetinvolved

inthefinalstateinteractions.Ithasbeensuggestedthatthe

rescat-teringeffectmayplayanimportantroleinthehadronicdecaysof

bottommeson[65–69].Forhigherexcited charmedmesons,they

mayfurtherdecayintoakaonandacharmed-strangemeson,and

then wemayexpect that therescatteringprocesses illustrated in

Fig 1wouldcontributetothedecaychannel B+→K+J/ψφ.The

lowestcharmedmesonswhichcandecayinto D ( ∗)

s K could bethe first radially excited states D and D∗ . The experimentally

ob-served resonancescorresponding to D and D∗  are D(2550) and

D(2600) respectively[70],whichiswidelyaccepted.Accordingto

thelatestresultsofLHCb[71],theirmassesandwidthsare

Table 1

Thresholds for theD ( s J ∗)+ D ( s ∗)− Threshold [MeV] D+s D∗+s D+s0 (2317) D+s1 (2460) D+s1 (2536)

D−s 3936.6 4080.4 4286.0 4427.8 4503.4

D∗−s 4080.4 4224.2 4429.8 4571.6 4647.2

M D=2579.5±3.4±5.5 MeV,

D=177.5±17.8±46.0 MeV,

M D∗ =2649.2±3.5±3.5 MeV,

D∗ =140.2±17.1±18.6 MeV. (2)

Sincethemassof Dissomewhatlowerthanthethresholdof D∗s K

(∼2606 MeV), its contribution to the rescattering amplitude is supposedtobesmallercomparedwith D∗ .Thereexisttheoretical and experimental indications that the ratesof B decays into ex-citedcharmedmesonswouldbesizable,althoughsuchdecaysare supposedtobesuppressedbytheheavyquarksymmetry(HQS)at theleadingorder[72–76].Thisimpliesthattherescatteringeffects inducedbythoseexcitedstatesmaybeimportant

Anotherinteresting propertyofFig 1isthat thethresholdsof some D ( ∗)+

s J D ( ∗)−

s combinationsare rathercloseto the“ X ” states observed in J/ψφ distributions For convenience, we use D s0,

D s1 and D

s1 to represent the P -wave charmed-strange mesons

D s0(2317), D s1(2460) and D s1(2536), respectively From Table 1, one can see that the thresholds of D∗

s D s, D∗

s D∗

s (D s0 D s), and

D

s1 D s (D s1 D s, D s1 D∗

s, Ds0 D∗

s)areclosetothemassesof X(4140),

X(4274)andX(4500),respectively.Duetothesingularitiesaround these thresholds may be present in the rescattering amplitudes One may wonder whether there are some connections between the singularitiesandthe“ X ” states.There aretwo intriguing sin-gularitieswhichmayappearinthetrianglerescatteringdiagrams When two of thethree intermediate states are on-shell,the sin-gularity at threshold is a finite square-root branch point, which corresponds to the cusp effect In some special kinematic con-figurations, all of the three intermediate states can be on-shell simultaneously, which corresponds to the leading Landau singu-larity of the triangle diagram This leading Landau singularity is usually called the TS, which may result in narrow peaks in the correspondingspectrum.Moreover,whentheTSoccurs,the trian-glerescatteringdiagramcanbeinterpretedasaclassicalprocessin space–time,andtheTSwilljustbelocatedonthephysical bound-aryofthescatteringamplitude[45].InRef.[49],itwasarguedthat forthesingle channelrescatteringprocess,when the correspond-ing resonance-productiontreediagramisaddedcoherentlytothe trianglerescatteringdiagram,theeffectofthetrianglediagramis nothingmorethanamultiplicationofthesingularityfromthetree diagrambyaphasefactor.Thereforethesingularitiesoftriangle di-agramcannotproduceobviouspeaksintheDalitzplotprojections Thisistheso-calledSchmidtheorem.Butforthecoupled-channel cases,thesituationwillbequitedifferentfromthesinglechannel casediscussed in Ref [49].For therescatteringdiagrams studied

inthispaper,theintermediateandfinalstatesaredifferent, there-forethesingularitiesinducedbytherescatteringprocessesarestill expected to be visible in the Dalitz plot projections The reader

is referred toRefs [43,77]forsome comments abouttheSchmid theorem,andRefs.[78,79]formorediscussionsaboutthe coupled-channelcase

2.1.1 The model

Inthefactorizationapproach,ifthecontributionsfrompenguin operators are neglected, the decays B →D ( ∗)

s J D¯( ∗) receive

contri-butions only from the external W -emission diagram The weak



D ( ∗)

s J D¯( ∗)|H W|B



canthen befactorizedintothe prod-uctoftwomatrixelements,i.e.,



D ( ∗)

s J D¯( ∗)|H W|B



= G F

√

2V

∗

cb V cs a1



¯

D ( ∗)|(V−A)μ|B



× D ( ∗)

s J|(V−A)μ|0



withtheWilson-coefficientcombination a1=c1+c2/N c

Thematrixelement ¯D ( ∗)|(V−A)μ|B

canbeparametrizedby

aseriesofhadronicformfactors:

 ¯D|V μ|B

= M B M D

h+( ω )(v+v)μ+h−( ω )(v−v)μ

,

 ¯D|A μ|B

=0,

 ¯D∗ |V μ|B

= M B M D∗ 

ih V( ω ) εμνα βν∗v

β v α ,

 ¯D∗ |A μ|B

= M B M D∗ 

h A1( ω )( ω +1) ∗μ− (h A2( ω )v μ

+h A3( ω )vμ)( ∗·v) 

wherev (v)isthevelocityof B (D¯( ∗)), ωistheproductof

veloc-ities v ·v,and  isthe polarizationvector of D¯∗ .Forthedecay

process B →D ( ∗)

s J D¯( ∗),intherestframeofB, both D ( ∗)

s J and D¯( ∗)

nearly stay at rest, which is very close to the zero recoil limit

(ω =1) As an approximation, we will set v=v= (1,0,0,0) in

thefollowing sections,andcalculatethe numericalresults atthe

zerorecoillimit.Itshouldbementionedthatforsomeofthe

sub-diagramsinFig 1,thevelocityoftheD¯( ∗)mesonmaynotbevery

small.Forinstance, inthe process B+→ ¯D∗ D∗+

s thevelocity of

¯

D∗  is around 0.4atthe restframe of B+, whichis actually not

very small.This may bring some theoretical uncertainties to the

strengthofrescatteringamplitudes.Butthe positionsofthe cusp

structures(ortheTS),whichmainlydependonthekinematic

con-figurationsofrescatteringprocesses,willnotbeaffectedbythe

ap-proximationofzerorecoillimit.Fortheprocesses B+→ ¯D∗ D ( )+

s1 , the velocity of D¯∗  is smaller, and the approximation would be

better

InEq.(3),thematrixelement

D ( ∗)

s J|(V−A)μ|0

isrelatedwith thedecayconstantofthecorresponding D( s J ∗),whichisdefinedas



D s|A μ|0

= f D s M D s v μ,



D∗

s|V μ|0

= f D∗

s M D∗

sμ∗,



D s0|V μ|0

= f D s0 M D s0 v μ,



D ( )

s1|A μ|0



= f

D ( )

s1

M

D ( )

At zero recoil, it can be noticed that in Eq (4) only the form

factors h+ and h A1 have the non-vanishing contributions

Cor-respondingly, for the rescattering processes B+→ J/ψφK+ via

the D ( ∗)

s J D¯( ∗)-loops, only the diagrams illustrated in Fig 1 can

survive Explicitly, the sub-diagrams involved in the calculations

are D+

s D¯0[D∗−

s ], D+

s0 D¯0[D∗−

s ], D∗+

s D¯∗ 0[D−

s], D∗+

s D¯∗ 0[D∗−

s ],

D+

s1 D¯∗ 0[D−

s], D+

s1 D¯∗ 0[D∗−

s ], D+

s1 D¯∗ 0[D−

s], and D+

s1 D¯∗ 0[D∗−

s ]

loops, of which the particles in the brackets represent the

ex-changedmesonsbetweenD ( ∗)

s J andD¯( ∗).

Inordertoestimatetherescatteringamplitudes,wealsoneed

to know the relevant strong couplings in Fig 1 To proceed, the

momentum for external and internal particles are denoted as

B+(P) →K+(p

1)J/ψ(p2)φ (p3)and D( s J ∗)+(q3) ¯D ( ∗)0(q

2)[D ( ∗)−

s (q1)], respectively In the framework of heavy hadron chiral

perturba-tion theory [66,80–84], the decay amplitudes for D¯( ∗)→D ( ∗)

s K

aregivenby

A ( ¯D→D∗

s K) = g

f K

M DM D∗

s p1· ∗(D∗

s),

A ( ¯D∗ →D s K) = g

f K



M D∗ M D s p1·  ( ¯D∗ ),

A ( ¯D∗ →D∗

s K) =i g

f K

M D∗ M D∗

s εμνα β p1μ v να( ¯ D∗ ) ∗

β(D∗

s),

(6)

wheretherelativecouplingstrengthbetweendifferentchannelsis determined bytheHQS.Forthe S-wave scattering D ( ∗)+

s D ( ∗)−

J/ψφ,acontactinteractionwhichrespectstheHQSisconstructed

inRef.[58],andtherelevantscatteringamplitudetakestheform

A (D∗+

s D−

s → J/ψφ) =iβSεα β γ δ v αβ(D∗+

s ) ∗γ(φ) δ∗(J/ψ ),

A (D∗+

s D∗−

s → J/ψφ) = βS(−g α β g γ δ+g α δ g β γ+g αγ g βδ)

× α( D∗+

s ) β(D∗−

s ) γ∗(φ) ∗δ(J/ψ ), (7)

whereβS isthecouplingconstantforthecontactinteraction

Itis worthnoticingthat accordingtoEqs.(4),(6)and(7),the rescattering amplitude corresponding to the D∗+

s D¯∗ 0[D∗−

s ]-loop actually vanishes.Because ofparityandangularmomentum con-servations, if D∗+

s and D∗−

s scatter in relative S-wave, the quan-tumnumbersof D∗+

s D∗−

s systemcanonlybe 0++ or2++,which

means the quantum numbers of the produced J/ψφ can only

be 0++ or 2++. Withan anti-symmetric tensorappearing in the

rescattering amplitude, this sub-diagram finally gives a vanish-ing contribution We can understand this conclusion by firstly considering the J/ψφ system as a single scalar particle The

¯

D∗ 0D∗−

s K+vertexinthe D∗+s D¯∗ 0[D∗−

s ]-loopisaVector–Vector– Pseudoscalar type coupling, where an anti-symmetric tensor is requiredinthecorrespondingcovariantamplitude Theloop inte-gralsintherescatteringamplitudewithatensorstructurecanbe reduced to linearcombinations ofLorentz-covariant tensors con-structedfromthemetrictensorandalinearlyindependentsetof the external momentum, accordingto the Passarino–Veltman re-ducedmethod[85].Thesetensorloopintegralswillbecontracted withthemetrictensors,externalmomentum,polarizationsvectors and anti-symmetric tensors to obtain the covariant rescattering amplitude There is no anti-symmetric tensor from the B D¯∗ D∗

s

vertex dueto the zerorecoil limit approximation in Eq.(4) For

B decaying into K and a scalar particle ( J/ψφ system) via the

D∗+

s D¯∗ 0[D∗−

s ]-loop,thereareonlytwo linearlyindependent mo-mentum that will be contracted with the anti-symmetric tensor from the D¯∗ 0D∗−

s K+ vertex, which gives a vanishing

rescatter-ing amplitude.The Lorentz indicesofthe polarizationtensors for

atensorparticleare symmetric.According tothesimilaranalysis,

ifthequantum numbersof J/ψφ systemare setto be 2++,one

willalso obtaina vanishingrescatteringamplitudecorresponding

tothe D∗+s D¯∗ 0[D∗−

s ]-loop

If the quantum numbers of J/ψφ system are J P C =0++ or

J P C=1++,the P -wave and S-wave charmed-strangemesonscan scatterinto J/ψφ viarelative P -wave Tosimplifythediscussion, the quantum numbers of J/ψφ are firstly set to be 0++, then

the P -wave scatteringamplitudeswhichrespecttheHQStake the form

A (D+

s0 D∗−

s → J/ψ φ)



M D s0 M D∗

s

q D s0·  (D∗

s) ∗(φ) · ∗(J/ψ ),

A (D ( )+

s1 D−

s → J/ψ φ)

M

D ( ) M D s q D s·  (D ( )

s1) ∗(φ) · ∗(J/ψ ),

A (D ( )+

s1 D∗−

s → J/ψ φ)

= iβP

M

D ( )

s1

M D∗

s

εμνα β q α D∗

s q β

D ( )

s1

μ(D∗

s) ν(D ( ) s1) ∗(φ) · ∗(J/ψ ),

(8)

whereβP isthe dimensionlesscouplingconstant forthe P -wave

scattering Other assignments of thequantum numbers of J/ψφ

systemwillbediscussedlater

Bymeans ofthe above scatteringamplitudes,the rescattering

amplitudeof B+→K+J/ψφ via theopen-charmmesonloopsin

Fig 1isgivenby

T[D ( ∗)

s J D¯( ∗)]

B+→K+J/ψ φ=1

i

 d4q 1

(2π )4

×A( B→D ( s J ∗) D¯( ∗ ) ) A( ¯ D ( ∗)→D ( ∗)

s K) A( D ( ∗)

s J D∗

s→J/ψ φ) (q2−M2

D∗s )(q2−M2

¯

D ( ∗)+iM D¯( ∗)  D¯( ∗) )(q2−M2

D ( ∗)

s J ) ,

(9)

wherethesumoverpolarizationsofintermediatestatesisimplicit

TheTScorrespondstoalogarithmicsingularityoftheloopintegral

Aslong asthekinematicconditions oftheTSare fulfilled,it

im-plies that one of the intermediate state in the triangle diagram

must be unstable, and we have to take into account the width

effect.Inourmodel,weuseaBreit–Wignertypepropagatorto

re-placethenormalpropagatorintheloopintegral,asdidinEq.(9)

The complex masswill remove the singularity from thephysical

boundary by a smalldistance, andmakes the physical scattering

amplitudefinite.However,thismethodisjustanapproximationof

some more complete theories InRef [44], it isargued that this

approximation is appropriate forcalculating enhancement effects

duetotheTS

2.1.2 Numerical results

Ignoringthecommoncouplingconstantsandformfactors,the

relativestrengthbetweendifferentrescatteringamplitudesmainly

depend on the decay constants of D ( ∗)

s J and the form factors h+

and hA1.Forthedecayconstantof D s,weadopttheexperimental

value,i.e., f D s=257.5MeV[70],andmake f D∗s= f D s.Intheheavy

quarklimit,wehavethefollowingrelations[86–89]

f D s0= f D s1, f D

But theserelations are not consistent with the experimental

ob-servations very well In the numerical calculation, we adoptthe

valuescalculatedinthecovariantlight-frontmodel[90,91],which

gives

f D s0=71 MeV, f D s1=121 MeV, f D

s1=38 MeV. (11)

Fortheformfactors h+and hA1,weadoptthevaluescalculatedin

theframeworkofrelativisticquarkmodel[72],whichgives

h+(1) 0.012,h A1(1) 0.098. (12)

Since hA1(1) ismuch largerthan h+(1), correspondinglyone can

expectthatthecontributionofFig 1(b)wouldbemuchlargerthan

thatofFig 1(a).Thenumericalresultsof J/ψφinvariantmass

dis-tributionsviatherescatteringprocessesaredisplayedinFig 2.The

experimental data from Ref [6] is also displayed in Fig 2 to be

comparedwiththenumericalresult.The resultinFig 2(a)is

ob-tainedbysettingthevaluesof MD¯( ∗) andD¯( ∗) asthoseinEq.(2)

Toinvestigatethedependenceonthemassofintermediatestates,

wealsocalculatetherescatteringamplitudesbysettingthevalues

of M¯( ∗) and¯( ∗) asthose ofthesecondradially excited states,

Fig 2 Thesolid line represents the invariant mass distributions of J/ψφ via the rescattering processes of Fig 1 The dotted line represents the distributions expected from the phase space The mass (width) ofD ( ∗)istakentobethatof(a)thefirst

and (b) the second radially excited state ofD ( ∗),respectively.Theverticaldashed

lines indicate the positions ofX(4140),X(4274)and X(4500)respectively The ex-perimental data points with error bars are from Ref [6]

and the result is displayed in Fig 2(b) There is no experimen-tal measurement concerning thesecond radially excited charmed mesons D ( ∗)(3S),andthefollowingresultscalculatedinthequark

modelareadoptedincalculations[92]:

M D( 3S )=3068 MeV, D( 3S )=106 MeV,

M D∗( 3S )=3110 MeV, D∗( 3S )=103 MeV. (13)

In Figs 2(a) and (b), one may notice that there are several cusps whichstay around the thresholds of D s1 D s, D

s1 D s, D s1 D∗

s

and D

s1 D∗

s,respectively.Asdiscussed previously,thesub-diagram

D∗+

s D¯∗ 0[D∗−

s ]-loop gives vanishing contribution, therefore there

is nocusp appearing aroundthe D∗+

s D∗−

s threshold Correspond-ingly, for the rescattering processes studied in this paper, there

is no cusp that can simulate the structure of X(4274) Because the thresholdof J/ψφ islarger thanthat of D∗+

s D−

s,there isno cusp corresponding to the D∗+

s D−

s threshold either In the the-oretical analysis of the LHCb collaboration [6,7], a cusp model proposed by Swanson was introduced to fit the structures of

X(4140)[37].Inthiscuspmodel,theintroductionofan exponen-tial form factor, with a momentum scale (β0) characterizing the hadron size, makes the cusp slightly above the sum of masses

of the rescattering mesons In Refs [6,7], the value of β0 ob-tained by the fit to the data is 297±20 MeV The fittingresult indicates that the below- J/ψφ-threshold cusp can simulate the

X(4140) structure [6,7] Our model in a sense is similar with the cusp modelproposed bySwanson In therescattering ampli-tudes corresponding to the sub-diagrams D+

s D¯0[D∗−

s ]-loop and

D∗+

s D¯∗ 0[D−

s]-loop of Fig 1, there is also a D±

s D∗∓

s -cut corre-sponding to the D±

s D∗∓

s -threshold cusp One of the differences between our model and the cusp model is that the

exponen-tial form factor is not introduced in our model As a result, in

our model the cusp of the rescattering amplitude just stays at

the D±

s D∗∓

s -threshold Since the D±

s D∗∓

s -threshold is below the

J/ψφ-threshold,onecannotexpectthatthe D±s D∗∓

s -cuspappears

inthe J/ψφ invariant massdistributions To show the influence

ofthe D±

s D∗∓

s -cut ofthe rescatteringamplitude, the distribution

curvesexpectedfromthephasespacearedisplayedinFig 2.Itcan

beseenthatinFigs 2(a)and(b)thresholdenhancementsappear

inthe J/ψφ distribution,but there is no narrow peak structure

appearingaround4140MeV.InRef.[56],theauthorsarguedthat

thekinematicthresholdcuspcannotproduceanarrowpeakinthe

invariantmassdistributionintheelasticchannelincontrastwitha

genuineS-matrixpole.Weagreewiththisargument.Italsoshould

bementionedthatforthecuspmodelintroducedinRef.[37],the

lineshapeofthecuspstructure isverysensitive totheparameter

β0 in theexponentialformfactor,whichimpliesthat the

conclu-siondrawnfromthismodelisquitemodel-dependent.Wereferto

Ref.[56] formorediscussions aboutthecuspmodelandits

limi-tations.However,takingintoaccountthatthe X(4140)structureis

actuallynotanarrowpeakbutratherabroadthreshold

enhance-mentintheexperimental observations,therescatteringeffect(or

cuspeffect)cannotbesimplyignoredinunderstandingthenature

ofX(4140)

Allof thecusps around X(4500) are toobroad andtoosmall

tosimulatethestructureof X(4500).Therearetwomainreasons

forthisresult Oneisthat the kinematicconditionsforthe

pres-enceofTSarenotsatisfiedinthoserescatteringprocessesofFig 1

[57].Thereforetherescatteringamplitudecannotgetenhancement

induced by the TS Another reason is that, if the J/ψ and φ

stay in relative S-wave, to preserve the parity, only via P -wave

can D ( )

s1 D ( ∗)

s (or D s0 D ( ∗)

s ) scatter into J/ψφ Usually the near-thresholdP -wave scatteringswillbehighlysuppressedduetothe

smallmomentumofscatteringparticles.Becauseoftheunsatisfied

kinematicconditionsofTSandthecharacteristicofnear-threshold

P -wave scatterings,nomatterwhat thequantumnumbersofthe

J/ψφ system are (0++, 1++, or 2++), there will be no obvious

narrowpeaksappearedintheinvariantmassdistributionsaround

4500 MeV

It should be mentioned that in Figs 2(a) and (b), the

distri-butioncurvesarerescaled tobeapproximatelymatchedwiththe

datapoints.Becausesomeofthecouplingsintherescattering

pro-cesses are not well determined, we only focus on the lineshape

behavior,butdonotaimtofitthedata

According to the above analysis, it seems hard to ascribe the

observationof X(4274) and X(4500) in B decays to the

rescat-tering effects This implies that X(4274) and X(4500) may

cor-respondtosome genuineresonances,such astetraquark statesor

somehigherexcitedcharmoniumstates.Thecuspeffectduetothe

D∗+

s D−

s rescatteringmaypossiblysimulatethe X(4140)structure,

butitmaydependsontheparametersofthecuspmodel

2.2 ψφrescattering effect

2.2.1 The model

Inthenaivefactorizationapproach,theamplitudeforthecolor

suppresseddecays B →M c¯c K ( ∗∗),with Mc c¯ and K( ∗∗)representing

acharmoniaandakaon(excited kaon)mesonrespectively,isgiven

by



M c¯c K ( ∗∗)+|H W|B+

= G F

√

2V

∗

cb V cs a2



K ( ∗∗)+|(V−A)μ|B+

× M c c¯|(V−A)μ|0

with a2=c2+c1/N c.Thematrixelement

M c¯c|(V−A)μ|0

is pro-portional to the decay constant of M¯ It can be expected that

Fig 3 Rescattering process via theψK1 loop.

the ratio of the amplitude |A(B → J/ψK∗∗) | to the amplitude

|A(B → ψK∗∗)| willbe close to f J/ψ/f ψ For J/ψ andψ, the

discrepancybetweenthedecayconstants f J/ψ (416±5MeV)and

f ψ (294±5MeV)isnotverylarge.Forthedecay B+→K+J/ψφ,

fromLHCbresultsitisshownthatthereisarichspectrumof ex-citedkaonresonances,whichhassignificantcontributionsvia the sequentialdecaysB+→J/ψK∗∗+→J/ψφK+ [6,7].Althoughthe

reflectionofexcitedkaonscannotresultinobviousresonance-like structuresin J/ψφ distributions,they maycontribute tothe pro-ductionofthose“ X ”statesinanotherway.Asdiscussedabove,it can be expected that the decay rateof B → ψK∗∗ would be at

thesameorderofmagnitudewiththatof B →J/ψK∗∗.Forsome

higherexcitedkaons,theiron-shellproductionmaybe prohibited duetothephasespace, buttakingintoaccounttheirbroaddecay widths,theycanstillcontributetotheprocess B+→ ψφK+.

Interestingly,itisfoundthatthethresholdofψφ(∼4706MeV)

is verycloseto the massof X(4700)(∼4704MeV).Therefore it

is possible that the rescatteringprocess illustrated inFig 3 may resultinsomeresonance-likepeaksin J/ψφ distributionsaround theψφthreshold,whichmaysimulatethe X (4700)signal.Among theexcited kaons,thedominantcontributionscomefromthe ax-ialvector states,asshowninRefs.[6,7].Inthefollowinganalysis,

we willfocusontherescatteringeffects inducedby K1,ofwhich thequantumnumbersare J P=1+.

The general invariant amplitude for B → ψK1 can be

writ-ten as:

A (B→ ψK1) =a∗(ψ) · ∗(K1)

(M B+M K1)2p B· ∗(ψ)p

B· ∗(K1)

(M B+M K1)2iεμνα β p μ B p ν K

1∗α(ψ) ∗β(K

1).

(15)

For B decaying into ψ andthehigher excited state K

1, both ψ

and K1 willnearlystayatrestintherestframeofB Thereforein theaboveequation,onlythefirsttermontherighthandsidewill contributesignificantly.Asanapproximation,weonlykeepthefist term inthe calculation, and setthe form factor a as a constant Theaxial-vectormeson K1 candecayintoφK in S-wave, andthe amplitudetakestheform

where gK 1isthecouplingconstant.ψφcanscatterinto J/ψφvia

exchangingsoftgluons.Tosimplifythemodel,weonlyconstructa contactinteractionforthisscattering,andtheamplitudereads

A (ψφ → J/ψφ) =g C T  (ψ) ·  (φ) ∗(J/ψ) · ∗(φ), (17)

where the quantum numbers of J/ψφ system are 0++. For the

rescattering process ψK1→K J/ψφ via strong interactions, the totalangularmomentum(orspin)ofψK1 systemshouldbe

con-served.IfweonlytakeintoaccountthefirsttermofEq.(15)inthe calculation,itimpliesthattheangularmomentumofψK1system

is0.Furthermore,since K decaysintoφK in S-wave, itmeansthe

Fig 4 Invariantmass distributions ofJ /ψφvia the rescattering processes of Fig 3 ,

where the mass/width of the intermediate stateK1 is taken to be 1650/150 MeV

(solid line), 1793/365 MeV (dotted line), and 1968/396 MeV (dashed line)

sepa-rately The vertical dashed line indicates the position ofX(4700) The experimental

data points with error bars are from Ref [6]

angular momentum of the intermediate ψφ systemis 0,which

further meansthe angularmomentum of thefinal J/ψφ system

isalso0.As aresult, forthe S-wave scatteringψφ →J/ψφ,

al-thoughthequantumnumbersof J/ψφ (ψφ)systemcanbe0++,

1++or2++,onlythe0++partcansurviveintherescattering

am-plitude.Thisisconsistentwiththeexperimentalfittingresultsthat

thequantumnumbersof X(4700)are0++.

The rescattering amplitude of B+→K+J/ψφ via the ψK1

-loopinFig 3isgivenby

T [ψK1]

B+→K+J/ψ φ=1

i

d4q1

(2π )4

A (B→ ψK

1) A (K1→ φK) A (ψφ → J/ψφ) (q21−M2φ)(q22−M2K

1+iM K1K1)(q23−M2ψ) ,

(18)

wherethesumoverpolarizationsofintermediatestatesisimplicit

2.2.2 Numerical results

Forthemomentwejustfocusonthelineshapebehaviorofthe

J/ψφ distributions via therescattering processes,butignore the

explicitvaluesoftherelevantcouplingconstantsandformfactors

Thenumericalresultsaredisplayed inFig 4.Accordingtothe

fit-tingresultsofLHCb [6,7],the distributionsin Fig 4are obtained

by setting the mass (width) of K1 to be 1650 MeV (150 MeV),

1793MeV (365MeV)and1968MeV (396MeV)separately.From

Fig 4, one can see that there is a clear peak appearing in the

vicinityof4.7 GeV,whichcorresponds tothe ψφthreshold.The

S-wave near threshold scattering makes this peak very obvious

Furthermore,althoughthekinematicconfigurationofthe

rescatter-ingprocessinFig 3doesnotmeettheconditionsofTSverywell,

itisalreadyveryclosetothekinematicregionwherethetriangle

singularitycanbepresent[57].Thereforethephysicalrescattering

amplitude will be influenced by the triangle singularity to some

extent.From Fig 4,it can be seen that thepeak induced by the

rescatteringeffect issimilarwiththestructureof X(4700)

The Argandplot corresponds to therescattering amplitude in

Eq (18) is displayed in Fig 5 It can be seen that the phase of

theamplitudeshowsabehaviorofrapidcounter-clockwisechange,

whichissimilarwithagenuineresonance

In conclusion, it is possible that the ψφ rescattering via the

ψK loopcouldsimulatethe X (4700)structurein J/ψφ

distribu-Fig 5 Realand imaginary parts of the rescattering amplitude in Eq (18) Motion with the increasing invariant massM J /ψ φis counter-clockwise.

tions Thebelow- J/ψφ-thresholdcuspduetothe D∗+

s D−

s rescat-teringmaypossiblysimulatethe X(4140)structure,butitdepends

onthecuspmodelparameters.However,ifthequantumnumbers

of X(4274)( X(4500)) are1++(0++),duetotheconservationsof

parity and angular momentum, and the suppression of the near threshold P -wave scattering, it ishard to describe thestructures

of X(4274) and X(4500) withrescatteringeffects, whichimplies that X (4274)and X (4500)couldbegenuineresonances

Concerning X(4274), although itis observed inthe J/ψφ in-variant mass distributions, it does not necessarily mean that it contains s¯s as the valence quarks We suggest that it may be the conventional orbitally excited state χc1(3P) This suggestion

isbasedonthefollowingarguments:

•First,thepredictedmassof χc1(3P)intheframeworkofquark modelsisabout4271MeV or4317MeV[93,94],whichisvery closetothemassof X (4274).Thepredictedwidth(∼39 MeV)

is also closeto the observed width of X(4274) (∼56 MeV) Althoughforthehighercharmoniumstates,thepredictionof conventional quark modelsisnot veryreliable, itcan stillbe takenasaguidance

•Second,ifthe X(4274)is χc1(3P),apartfrom J/ψφ,onemay expectthatitcanalsoeasilydecayinto J/ψ ω.FromtheFig 2

inRef [95],it canbenoticed thatapart from X(3872),there arealsosomebumpsaround4.3 GeV appearinginthe J/ψ ω

distributions Although the current statics for B → J/ψ ωK

maynotbe largeenoughto confirmthispoint,itcanstill be takenasanevidence

•Third,therescatteringeffectscannotdescribetheobservation

of X(4274),as discussedin thispaperandRefs [6,7],which

to some extentexcludes thepossibilityof non-resonance ex-planations

•Fourth,underthefactorizationansatz,since χc0, χc2andtheir radiallyexcitedstatescannotbeproducedviathe V−A

cur-rent,itcanbeexpectedthatthedecayrateof B →Kχc1(3P)

will be largerthan thatof B →Kχc0(3P)or B →Kχc2(3P) This may lead to that in the J/ψφ distributions, only the

χc1(3P)signalissignificant

Ifthe X (4274)isreally χc1(3P),onemaysearchforitinthe radia-tive decays ofψ(4415) (ψ(4S)),takinginto account thatthis E1

theframeworkofquarkmodel[93,94]

Acknowledgements

HelpfuldiscussionswithFeng-KunGuo,GangLiandQiangZhao

are acknowledged This work is supported in part by the Japan

Society for the Promotion of Science under Contract No.P14324,

andtheJSPSKAKENHI(GrantNo 25247036)

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of< i >X< /i>(4140< h3>)

Allof thecusps around X< /i>(4500< h3>) are toobroad andtoosmall

tosimulatethestructureof X< /i>(4500< h3>).Therearetwomainreasons... willnearlystayatrestintherestframeofB Thereforein theaboveequation,onlythefirsttermontherighthandsidewill contributesignificantly.Asanapproximation,weonlykeepthefist term inthe calculation, and setthe form... the lineshape

behavior,butdonotaimtofitthedata

According to the above analysis, it seems hard to ascribe the

observationof X< /i>(4274< h3>) and