Partners of the $X(3872)$ and HQSS breaking

0.07 MB·

by  D. R. Entem

Partners of the X(3872) and HQSS breaking
D.R. Entem1,a), P.G. Ortega2 and F. Ferna´ndez1
6 1GrupodeF´ısicaNuclearandIUFFyM,UniversidaddeSalamanca,E-37008Salamanca,Spain.
1 2CERN(EuropeanOrganizationforNuclearResearch),CH-1211Geneva,Switzerland.
0
2
a)Correspondingauthor:[email protected]

n
a
J Abstract.Sincethediscoveryofthe X(3872)thestudyofheavymesonmoleculeshasbeenthesubject ofmanyinvestigations.
5 Ontheexperimentalsidedifferentexperimentshavelookedforitsspinpartnersandthebottomanalogs.Onthetheoreticalside
1 differentapproacheshavebeenusedtounderstandthisstate.SomeofthemareEFTthatimposeHQSSandsotheymakepredictions
forthepartnersoftheX(3872),suggestingtheexistenceofaJPC =2++partnerinthecharmsectororJPC =1++or2++analogsin

] thebottom.
h Inour work,inorder tounderstand the X(3872), weuseaChiral quarkmodel inwhich,due totheproximitytothe DD
∗
p threshold,weincludecc¯statescoupledtoDD molecularcomponents.Inthiscoupledchannelmodeltherelativepositionofthe
∗
- barecc¯stateswithtwomesonthresholdsareveryimportant.WehavelookedfortheX(3872)partnersandwedon’tfindabound
p
stateintheD D JPC =2++.InthebottomsectorwefindtheoppositesituationwheretheB B withJPC =2++isboundedwhile
e ∗ ∗ ∗ ∗
h theJPC =1++isnotbounded.Theseresultsshowshowthecouplingwithcc¯statescaninduceddifferentresultsthanthoseexpected
[ byHQSS.Thereasonisthatthissymmetryisworseintheopenheavymesonsectorthaninthehiddenheavymesonsector.

1
v
INTRODUCTION
1
0
9 Thetheoreticalstudyofexoticstates,notwellaccommodatedinthenaivequarkmodel,hasmotivatedagreatinterest
3 since the discoveryof the X(3872).This state was discoveredby the Belle collaborationin 2003[1] and verysoon
0
after confirmed by the CDF [2], D0 [3] and BaBar [4] collaborations.The state lies well below where the χ (2P)
. c1
1 state is expected by quark models and is very close to the DD threshold. However it also presents very peculiar
∗
0 decayproperties,beingthemostrelevantthedecaysintoJ/ΨππthroughaρandJ/Ψπππthroughanωwhichimplies
6 somekindofisospinviolation.Thesepropertiesruleoutcompletelya cc¯ interpretationofthestate. Howeveritcan
1
beeasilyunderstoodinthemolecularpictureasa DD molecule.Thentheisospinviolationcanbeobtaineddueto
: ∗
v thedifferenceonthemassofdifferentchargedstatesofthecharmedmesonswithouttheneedofintroducingisospin
i violatinginteractions.Forthesereasonsitisnowacceptedasthebestcandidatetobeanonqq¯state.
X
The first consequence of the molecular picture is the existence of other analog states. In fact, if you assume
r
a heavyflavorsymmetry(HFS),whichisanapproximategoodsymmetryofQCDandimpliesthattheinteractionsare
thesamewhenyouchangeacquark(antiquark)byabquark(antiquark),thenaJPC =1++analoginthebottomonium
sectorshouldexistduetothereductionofthekineticenergybythebiggermassofthebottommesons.Soyoushould
expectastatewithbiggerbindingenergyandprobablysmallerisospinbreakingpropertiesduetothesmalldifference
inthemassesofthebottommesons.
Otherapproximatesymmetriescanbeusedtostudytheexistenceofotheranalogslike,forinstance,heavyquark
spin symmetry (HQSS) which implies the independence of the interactions on the spin of the heavy quark. This
symmetry has been use by the authors of Reference [5] to study the analogs in the charmonium sector with other
quantum numbers. The strongest conclusion of this work is the existence of a JPC = 2++ B B analog which has
∗ ∗
beencalledX(4012)sincetheinteractioninthischannelisthesameasinthechanneloftheX(3872).Inordertoget
conclusionsforotherquantumnumbersfurtherassumptionsareneeded.AssumingthattheX(3915)isa0++analoga
totalofsixmolecularstatesarefoundinthecharmoniumsector.
UsingsimilarideasandHFStheauthorsofReference[6]studiedthebottomoniumsectorwherethe1++ analog
shouldbefoundandextendedthestudytotheisovectorchannelsinbothsectors.
Althoughthepuremolecularpictureisverypopularonecouldexpectthatthemolecularstatescouldcoupleto
nearbycc¯states.WefollowedthisinvestigationintheframeworkoftheChiralQuarkModel(CQM)inReference[7].
InthisworkwefoundthattheJPC =1++ DD channelisunboundedifwedonotincludethecouplingtocc¯statesand
∗
isboundedoncethecouplingtotheχ (2P)isincluded.Inalatterwork[8]wealsoincludeotherchannelslikethe
c1
J/Ψωfindingitirrelevant.Thereasonisthatthecouplingof J/Ψωandcc¯ statesisanOZIforbiddenprocessandso
verysuppressedandthecouplingtoDD statesgoesthrougharearrangementprocesswhichisalsoverysuppressed.
∗
SothemostimportantcomponentsaretheDD andχ (2P).
∗ c1
Theirrelevanceofthe J/Ψωchannelhasbeenconfirmedrecentlyinlattice simulations[9] andalso thestrong
coupling between the DD components and cc¯. In this work only the bound DD state is found and the expected
∗ ∗
χ (2P)stateisnotfound.Forthisreasonithasbeeninterpretedasacc¯largelydressedbytheDD components.This
c1 ∗
differsfromourpicturesinceforustheX(3872)appearsasanadditionalstatetothenaivequarkmodelexpectations.
Ifoneassumesthatthecouplingtocc¯statescanbeimportantoneimportantquestionisifthiscouplingcanmake
resultsdeviatefromexpectationsfromHQSSandHFS.Thisiswhatweanalyzeinthepresentwork.
THECHIRALQUARKMODEL
The present work use the CQM which has been extensivelyused to study baryonand meson phenomenology.The
SU(2)versionwaspublishedinReference[10]andlateronwasextendedtotheSU(3)versionandtotheheavyquark
sectorinReference[11].Hereweonlydescribethemainingredientsofthemodel.
QCD has Chiral Symmetry as an approximate good symmetry at the level of the Lagrangian. However this
symmetryisnotrealizedinthehadronspectrumandisspontaneouslybrokenatthequantumlevel.Thisfactimplies
theexistenceofmasslessGoldstonebosons(thepions)withthequantumnumbersofthebrokensymmetry.However
thesymmetryisnotexactandtheGoldstonebosonsacquireadynamical(small)masswhichisthemostsatisfactory
waytounderstandthesmallnessofthemassofthepionmeson.WhenwebreakthesymmetryintheSU(3)version
thenthekaonsandtheetamesonappears.Thesenewdegreesoffreedomgeneratesinteractionsbetweenquarksasone
Goldstonebosonexchanges.Multibosonexchangesarenotincludedbuttheyaretakenintoaccountbytheexchange
of scalar bosons. However Chiral Symmetry Breaking also generates a dynamicalmass for quarks, the constituent
quarkmass.
Besides the Chiral Symmetry Breaking another crucial non-perturbative effect of QCD is confinement which
makesallhadronstoappearascolorsinglets.We includeconfinementina phenomenologicalwayasa colorlinear
screenedinteractionbetweenquarks.
QCD perturbative effects are included by the interaction induced by one gluon exchange which is of special
relevanceonthedescriptionofheavyquarksystems.
AllthedetailsoftheinteractionsusedcanbefoundinReferences[11]and[12]wheretheparametersusedcan
befound.
THETWO MESON INTERACTION
Oncewehavetheinteractionbetweentheconstituentsofmesonswecanstudytheinteractionbetweenmesonsgen-
eratedbythem.WeobtainthisinteractionsusingtheResonatingGroupMethod(RGM)consideringthatweknowthe
internalwave functionsof the interacting mesons. In our calculation these wave functionsare obtained solving the
twobody(qq¯)system.
Inthesystemsconsideredinthisworknoantisymmetryeffectsbetweenquarksindifferentmesonsarepresent.
SotheinteractionisgivenbythesocalledRGMdirectkernel
RGMV (P~ ,P~) = d~p d~p d~p d~p φ (~p )φ (~p )V (P~ ,P~)φ (~p )φ (~p ) (1)
D ′ Z ξA′ ξB′ ξA ξB ∗A ξA′ ∗B ξB′ ij ′ A ξA B ξB
iXA,j B
∈ ∈
Howeverquarksorantiquarkscanbeexchangedbetweendifferentmesonscouplingdifferentmesonstates,like,for
instance, the DD J/Ψω. This kind of processes are rearrangementdiagramsthat are suppressed by the meson
∗
→
wavefunctionsoverlaps.
COUPLINGMOLECULAR AND QQ¯ COMPONENTS
AsmentionedintheintroductiontheclosestQQ¯ statestothetwomesonthresholdscancoupledwiththesestates.A
measuredofthiscouplingisgivenbythewidthsoftheQQ¯ statesdecayingtoopenheavyquarkmesons,whichisof
theorderortensorhundredsofMeV.
Inorderto studythiseffectwe usethe 3P modelto coupledtwoquarkandfourquarksectors. Itis important
0
to notice thatthe modelonly introducesa couplingthatcan be fitted to anydecay andthen allthe othersare given
bythemesonwavefunctionsandquarksymmetries.Inordertogetanoverallgooddescriptionoftwomesondecays
we performed an analysis of strong decays in differentsectors and we allowed the coupling to logarithmically run
with thescale ofthesystem givenbythe reducedmassof thetwo quarksofthedecayingmeson.Thiswasdonein
Reference[13]wherepredictionsinsectorswherethecouplingwasnotfittedweregivenwithverygoodagreement
withtheexperimentaldata.
Thenwetakethehadronicwavefunctiongivenby
Ψ = c ψ + χ (P)φ φ β (2)
α β M1 M2
| i | i | i
Xα Xβ
where ψ arethehiddenheavymesonsand φ φ β arethetwomesonstateswithβquantumnumbers.
M1 M2
Th|eicouplingwithQQ¯ statesinduceda|neffectiveienergydependentpotentialbetweenthetwomesonsgivenby
h (P)h (P)
Veff(P,P)= β′α ′ αβ (3)
β′β ′ Xα E−Mα
WesolvethecoupledchannelproblemfollowingReference[14].AllthedetailscanbefoundinReference[8].
HEAVYQUARKSPIN SYMMETRY
Since we want to analyze the discrepancies with HQSS expectations induced in our model by the coupling to QQ¯
states is important to check if we don’t have additional HQSS breaking effects. In our model HQSS breaking is
presentsincewetakefiniteheavyquarkmasses,howeveroneshouldexpectsmalleffectsduetothebigvaluesofthe
heavyquarkmasses.
ForS-wavetwomesonstatesitiseasytofindthefollowingrelationsbetweenmatrixelements
2
D D (0++)HDD(0++) = DD(0++)HDD(0++) D D (0++)HD D (0++) (4)
∗ ∗ ∗ ∗ ∗ ∗
√3h | | i h | | i−h | | i
DD (1++)HDD (1++) = D D (2++)HD D (2++) (5)
∗ ∗ ∗ ∗ ∗ ∗
h | | i h | | i
3 1
= DD(0++)HDD(0++) D D (0++)HD D (0++) (6)
∗ ∗ ∗ ∗
2"h | | i− 3h | | i#
2 DD (1+ )HDD (1+ ) = DD(0++)HDD(0++) + D D (0++)HD D (0++) (7)
∗ − ∗ − ∗ ∗ ∗ ∗
h | | i h | | i h | | i
whicharegivenjustbyrecouplingcoefficients.TheserelationsarecheckedinFigures1and2.ForexactHQSSall
the solid lines should be the same, so we see just small HQSS breaking effects in the two meson interaction. The
smallbreakingeffectsareinducedbythesmalldifferenceinthewavefunctionsofthepseudoscalarandvectorheavy
mesonswhichareshowninFigure3andforexactHQSSshouldbethesame.
RESULTS
AsshownintheprevioussectionthemodelpreservesHQSSinagoodapproximationalthoughsmallbreakingeffects
arepresentduetotheuseoffiniteheavyquarkmasses.HoweveritisinterestingtonoticethattheHQSSbreakingin
themassesofopencharmandbottommesonsisbiggerthaninthehiddencharmandhiddenbottomsectors.Forthis
reasontherelationsbetweenthetwomesonthresholdsandthe QQ¯ stateschangewhenwechangethe JPC quantum
numbers.This can be seen in Figure 4 where we present the predictionsof the pure QQ¯ states of the modelin red
comparedwiththetwomesonthresholdsandthestatesoftheParticleDataGroup(PDG)fromReference[15]inblue.
1.0 1.0
0.8 0.8
0.6 0.6
0.4
0.4
-2V) 0.2 -2V) 0.2
Ge Ge 0.0
p) ( 0.0 p) ( -0.2
V(p, -0.2 V(p, -0.4
-0.4
-0.6
-0.6 -0.8
-0.8 -1.0
-1.0 -1.2
0 500 1000 1500 2000 0 500 1000 1500 2000
p (MeV) p (MeV)
FIGURE1.Diagonal matrixelementsofthetwomesoninteractioninmomentumspaceforthe D()D() sector(leftpanel) and
∗ ∗
B()B()sector(rightpanel).ThedashedbluelinegivestheD D (0++)matrixelement,thedashedredlinetheDD(0++),thesolid
∗ ∗ ∗ ∗
bluelinetherighthandsideofEquation4andthesolidredlinelefthandsideofthesameEquation.
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
-2V) 0.2 -2V) 0.2
e e
G G
p) ( 0.0 p) ( 0.0
p, -0.2 p, -0.2
V( V(
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1.0 -1.0
0 500 1000 1500 2000 0 500 1000 1500 2000
p (MeV) p (MeV)
FIGURE2.Diagonal matrixelementsofthetwomesoninteractioninmomentumspaceforthe D()D() sector(leftpanel) and
∗ ∗
B()B()sector(rightpanel).ThedashedbluelinegivestheD D (0++)matrixelement,thedashedredlinetheDD(0++),thesolid
∗ ∗ ∗ ∗
bluelinetherighthandsideofEquation5,thesolidredlinelefthandsideofthesameEquationandthesolidgreenlinetheright
handsideofEquation6.
ForthePDGstatesweonlyincludestateswithwellknownquantumnumberswithsomeexceptions.Inthe1+ sector
−
althoughtheIG quantumnumbershasnotbeenmeasureditisawellacceptedcandidatefortheh (1P)stateasshown
c
bytheagreementwithourquarkmodelresult.FortheX(3940)theputinthe1++sectorsinceithasbeenseeninDD
∗
decaysandsoitisagoodcandidatefortheχ (2P)state.Inthebottomoniumsectorthesameappliestotheh (1P)
c1 b
andh (2P)(includedintheupdatedversion).Fortheχ (3P)onlyC =+hasbeenmeasuredandwehaveincludedas
b b
astatecoveringthethreepossibleassignments.Wealsohaveincludedinlightbluethecandidatesfortheχ (2P)and
b1
χ (2P)measuredrecentlybyLHCb[16].
b2
It is important to consider the following aspects. In the charmonium sector the relevant thresholds, in which
an S-wave two mesonstate is possible, are DD and D D in 1+ , DD and D D in 0++, DD in 1++ and D D in
∗ ∗ ∗ − ∗ ∗ ∗ ∗ ∗
2++.Thesameoccursinthebottomoniumsectorchangingthe D( ) mesonsby B( ) mesons.Then,a QQ¯ stateabove
∗ ∗
thresholdgivesattractionandbelowgivesrepulsion.Thestrengthisinverselyproportionaltothedifferencebetween
thethresholdpositionandthemassofthebareQQ¯ state.
Ourprescriptioninthepresentworkistoincludethecloserstateaboveandbelowtherelevantthresholdandwe
onlyincludetwomesonstateswhereanS-waveispresent.Thepartialwavesincludedfortwomesonstatesare
1. Inthe0++sectorweincludeDD(BB)andD D (B B )1S waves
∗ ∗ ∗ ∗ 0
2. Inthe1++sectorweincludeDD (BB )3S and3D waves
∗ ∗ 1 1
8.0 7.0
7.0 6.0
6.0
5.0
-3/2V) 5.0 -3/2V) 4.0
Ge 4.0 Ge
Ψ(p) ( 3.0 Ψ(p) ( 3.0
2.0
2.0
1.0 1.0
0.0 0.0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
p (GeV) p (GeV)
FIGURE3.Wavefunctionforthepseudoscalar(red)andvector(blue)heavymesons.Theleftpanelcorrespondstocharmmesons
(D())andtherightpaneltobottommesons(B()).
∗ ∗
TABLE1.AdditionalstatestothedressedQQ¯statesgivenbythemodel.
ThestatespredictedusingHFSandHQSSsymmetriesgiven inRefer-
ences[5,6]aregivenforcomparison.
Charmonium 1+ 0++ 1++ 2++
−
Reference[5] 3815/3955 3710/Input Input 4012
Reference[6] Input 4012
Thiswork X(3872)
Bottomonium 1+ 0++ 1++ 2++
−
Reference[6] 10580 10600
Thiswork 10621 ? 10648
3. Inthe1+ sectorweincludeDD (BB )andD D (B B )3S and3D waves
− ∗ ∗ ∗ ∗ ∗ ∗ 1 1
4. Inthe2++sectorweincludeD D (B B )5S ,1D and5D waves
∗ ∗ ∗ ∗ 2 2 2
The resultsare summarizedin Table 1 wherewe onlyquoteadditionalstates to the dressed QQ¯ states thatwe
findasresonances.
Let’sconsiderfirstthe1++ DD and2++ D D channelswhereHQSStellsusthattheinteractionisthesame.In
∗ ∗ ∗
thecharmsectorweseethatinthe1++wegetmoreattractionthanrepulsionwhileinthe2++wegetrepulsion(thestate
abovethresholdis an F-wave cc¯ state thatis weakly coupledto the two meson sector). As shown in Reference [7]
the two meson interaction is not enough to bind the system, however the couplingwith the χ (2P) state gives the
c1
additional attraction and we get a new state that we assigned to the X(3872). In the 2++ sector we don’t get this
additionalattractionandforthisreasonnonewstateisfound.Howeverthedressedcc¯(3F )stategetsdressedanda
2
stateclosetothepredictedX(4012)byHQSSisfoundbutnotpresentasanadditionalstate.Forthisreasonalthough
anstatewithsimilarmassisfoundweexpectdifferentdecaysproperties.
Inthebottomoniumwehavetheoppositesituation.Nowthetwomesoninteractionisenoughtobindthesystem.
Inthe1++weexpectmorerepulsionfromthestatebelowtheBB threshold.InTable1wedon’tgiveananswertothe
∗
existenceornotexistenceofthisstatesincewecanfindaveryshallowboundstateornostatewhenwevarythevalue
ofthestrengthparameterofthe3P modelwithinitsuncertainties.Inthe2++ wegetsimilarrepulsionandattraction
0
fromthestatesbelowandabovethresholdandherewefindanadditionalstate.Thissituationdifferscompletelyfrom
themodelindependentexpectationsofHQSSandHFSwithoutcouplingtoQQ¯ states.
FortheotherquantumnumbersHQSSneedsomeassumptionsandthefinalresultdependsontheseassumptions.
In Table 1 we compareour results in the 0++ and 1+ sectors with the conclusionsfrom References[5, 6] where a
−
betteragreementisfoundwithReference[6].
As a summary, in this work we only include the closest QQ¯ state above and below the relevant thresholds,
howevera more complete description includingall the closest states is necessary to give definite conclusions.This
4.3 10.8
4.2
10.7
4.1
B*B*
4 D*D* 10.6 BB*
BB
GeV) 3.9 DD* GeV) 10.5
M( 3.8 M(
DD
3.7 10.4
3.6
10.3
3.5
3.4 10.2
1+- 0++ 1++ 2++ 1+- 0++ 1++ 2++
FIGURE4.Charm(leftpanel)andbottom(rightpanel)spectrum.BlueboxesshowsthestatesintheParticleDataGroup[15].See
commentsinthetextforthePDGstatesselected.Wealsohaveincludedinlightbluethecandidatesfortheχ (2P)andχ (2P)
b1 b2
measuredrecentlybyLHCb[16].ThestatesinredarethepureQQ¯ statespredictedbythemodel.
willbepresentedinaforthcomingpaper.Howeverthemainconclusionofthisworkisthatthecouplingbetweentwo
mesonstatesandQQ¯ statescanmodifiedHQSSandHFSconclusionsinthepuremolecularpicture,andwethinkthat
thisproblemshouldbeaddressinmodelsathadronlevel.
ACKNOWLEDGMENTS
ThisworkhasbeenpartiallyfundedbyMinisteriodeCienciayTecnolog´ıaunderContractno.FPA2013-47443-C2-
2-P, by the Spanish Excellence Network on Hadronic Physics FIS2014-57026-REDTand by the Spanish Ingenio-
Consolider2010ProgramCPAN(CSD2007-00042).P.G.OrtegaacknowledgesthefinancialsupportfromtheEuro-
peanUnion’sMarieCurieCOFUNDgrant(PCOFUND-GA-2011-291783).
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