Scalar, axial-vector and tensor resonances from the $ρD^*$, $ω D^*$ interaction in the hidden gauge formalism

0.06 MB·English

by  R. Molina

r w
Scalar, axial-vector and tensor resonances from the D , D
∗ ∗
interaction in the hidden gauge formalism
R. Molina , H. Nagahiro†, A. Hosaka and E. Oset
∗ ∗∗ ∗
DepartamentodeFísicaTeóricaandIFIC,CentroMixtoUniversidaddeValencia-CSIC,Institutosde
∗
0 InvestigacióndePaterna,Aptdo.22085,46071Valencia,Spain
1 †DepartmentofPhysics,NaraWomen’sUniversity,Nara630-8506,Japan
0
ResearchCenterforNuclearPhysics(RCNP),OsakaUniversity,Ibaraki,Osaka567-0047,Japan
2 ∗∗

n
Abstract. Weapplyaunitaryapproachtogetherwithasetofhidden-gaugeLagrangianstostudythevector-vectorinteraction.
a Westudythecaseofther (w )D interactionwithinthemodel.InI=1/2wegetstrongenoughattractiontobindthesystem.
J ∗
Concretely,wegetoneresonanceforeachspinJ=0,1,2.ForJ=1and2theseresonancescanbeeasilyidentifiedwiththe
8 D (2460)andD (2640).WhereasthestateforJ=0withmassM 2600MeVisapredictionofthemodel.InI=3/2we
∗ ∗
1 getarepulsiveinteractionandhencenoexoticdynamicallygenerate∼dstates.

Keywords: D,mesons,charm,vectors
]
h PACS: 13.75.Lb,12.40.Vv,12.40.Yx,14.40.Cs
p
-
p
INTRODUCTION
e
h
[ In this talk, we report several works that combine coupled channel unitarity and hidden-gaugeLagrangiansfor the

interaction of vector meson among themselves. These Lagrangianswere introduced by Bando-Kugo-Yamawaki[1]
1
v and combined with unitary techniques related with the Bethe-Salpeter equation, provide a useful tool to generate
8 resonancesorboundstatesiftheinteractionisstrongenough.Thesetechniqueswerefirstlyappliedtostudytherr
0 interaction and the interaction was strong enough to bind the rr system [2]. Thus, two bound states appeared as
0 polesin the second Riemannsheet that could be identified with the f (1270)and the f (1370).The decay of these
2 0
3 resonancesto two or fourpionswasprovidedby a boxor crossed boxdiagramfromtwo r mesonsgoingto pions.
.
1 TheseboxdiagramsprovidedawidthcomparablewiththedataquotedatthePDG[3],leadingtosatisfactoryresults
0 onthemass,widthandquantumnumbers,aswellascouplings,andthus,thesedynamicallygeneratedresonancescan
0 be seen as hadronicmolecules.Later worksthatwe quotein the nextSection haveextendedthe modelto study the
1 vector-vectorinteractioninSU(3)andther (w )D system.
: ∗
v Inthismanuscriptwe wanttosummarizebrieflytheformalismofthevector-vectorinteraction.Then,wewantto
i showthesimplecaseofther (w )D interaction.
X ∗
r
a
FORMALISM:THEVV INTERACTION
Within the theorical framework, there are two main ingredients: first, we take the Lagrangians for the interaction
of vector mesons among themselves, that come from the hidden gauge formalism of Bando-Kugo-Yamawaki[1].
Second,weintroducethepotentialV obtainedfromtheseLagrangians(projectedins-wave,spinandisospin)inthe
BetheSalpeterequation:
T =(1ˆ VG) 1V , (1)
−
−
whereGistheloopfunction.Therefore,wearesummingallthediagramscontainingzero,one,two...loopsimplicit
in the Bethe Salpeter equation.Finally,we look for polesof the unitaryT matrix in the second Riemannsheet. All
thisprocedureiswellexplainedin[2,4].Intheseworks,twomainverticesaretakenintoaccountforthecomputation
ofthepotentialV:thefour-vector-contacttermandthethree-vectorvertex,whichareprovidedrespectivelyfromthe
Lagrangians:
g2
L(c)= Vm Vn Vm Vn Vn Vm Vm Vn , (2)
III 2 h − i
L(3V)=ig (¶ m Vn ¶ n Vm )Vm Vn . (3)
III h − i
BymeansofEq.(3),thevectorexchangediagramsinFig.1arecalculated.ThediagraminFig.1d)leadstoarepulsive
p-waveinteractionforequalmassesofthevectors[2]andonlytoaminorcomponentofs-waveinthecaseofdifferent
masses[4].Thus,thefour-vector-contacttermin Fig.1a)andthet(u)-channelvectorexchangediagramsinFig.1c)
areresponsibleforthegenerationofresonancesorboundstatesiftheinteractionisstrongenough.
V V V V
−→ +
V V
V
a) b) c) d)
FIGURE1. TermsoftheLIII Lagrangian: a)fourvector contactterm,Eq.(2);b)three-vectorinteraction,Eq.(3);c)t andu
channelsfromvectorexchange;d)schannelforvectorexchange.
In order to consider the pseudoscalar-pseudoscalar decay mode, and thus compare properly with the available
experimental data for the width, a box diagram is also included, which we show in Fig. 2 for the particular case
of the rr system. Crossed box diagrams and box diagramsinvolving anomalouscouplingswere also calculated in
[2],buttheyweremuchsmaller,speciallyinthecaseoftheanomalouscoupling,thanthecontributionscomingfrom
the box diagramsin Fig. 2 and theywere not consideredin later works[4, 5, 6]. As we are interested in the region
closetothetwovectormesonthreshold,thethreemomentaoftheexternalparticlescanbeneglectedcomparedwith
the massofthe vectormeson,andthus,we can make ~q/M 0 forexternalparticles, whichconsiderablysimplify
| | ∼
the calculation. In Table 1, the eleven states found with this procedure in the work of [4], where the formalism is
applied in SU(3), are shown. As one can see in this table, five of them can be identified with data in the PDG: the
f (1370), f (1710), f (1270), f (1525)andK (1430).Inthiswork,theintegraloftwo-meson-vectorloopfunction
0 0 2 2′ 2∗
is calculated by means of the dimensional regularization method. This requires the introduction of one parameter,
whichisthesubtractionconstant,thatistunedtoreproducethemassofthetensorstates,whereastheotherstatesare
predicted.Inordertocalculatetheboxdiagrams,onehastointroducetwoparameters,thecut-off,L ,fortheintegral,
andoneparameter,L ,involvedintheformfactorused.Theseparameterswereconsideredtobearound1GeVand
b
1.4 GeV respectively in [2] to get reasonable valuesof the f (1370)and f (1270)widths. In the later work of [4],
0 2
thesevaluesofL andL alsoprovideagooddescriptionofthewidthsfortheotherstates,asshowninTable1.
b
As one can see, the modelin SU(3) providesa gooddescriptionof the propertiesof manyphysicalstates. In the
nextsections,wegeneralizethemodeltoSU(4),whereinterestingnewstateswillappear.
ρ+ ρ+ ρ+ ρ− ρ+ ρ+
π+ π0 π0
π0 π0 + π+ π+ + π+ π+
π− π0 π0
ρ− ρ− ρ− ρ+ ρ− ρ−
FIGURE2. Boxdiagramsforthecaseoftherr interaction.
THE r (w ) D SYSTEM
∗
InthissectionwedescribethesimplecaseofthechannelswithC=1andS=0withintheformalismofthevector-
vectorinteractionintroducedbyBando-Kugo-Yamawaki[1].Wehaveasetofthreechannels:r D ,w D andD K¯ .In
∗ ∗ s ∗
fact,wewanttofocustheattentiontotheenergyregionaround2500 2600MeV,wherethereareseveralinteresting
−
resonancesthatcanbe foundinthe PDG.Inviewofthepreviousworksof[2, 4],we hopetofinda bindingenergy
of 200 300MeVandthethresholdoftheD K¯ channelisfaraway,concretely,itis200MeVabovether D and
s ∗ ∗
w D∼thres−holds.Thus,wesimplifyevenmorethestudytothesetwochannels:r D andw D .
∗ ∗ ∗
TABLE1. Propertiesofthe11dynamicallygeneratedstates:polepositions,
massesandwidthsintherealaxesfortwodifferentL (parameterrelatedtothe
b
formfactorusedinthecalculationoftheboxdiagram[4])comparedwiththe
experiment.AllthequantitiesareinunitsofMeV.
IG(JPC) Theory PDGdata
(Mass,Width) Name Mass Width
0+(0++) (1520,257 396) f0(1370) 1200 1500 200 500
0+(0++) (1720,133−151) f0(1710) 17∼24 7 13∼7 8
0−(1+−) (180−2,49) h1 ± ±
0+(2++) (1275,97−111) f2(1270) 1275.1±1.2 185.0+22..49
0+(2++) (1525,45 51) f (1525) 1525 5 7−3+6
− 2′ ± 5
−
1−(0++) (1777,148 172) a0
1+(1+−) (1703−,188) b1
1−(2++) (1567,47 51) a2(1700)??
−
1/2(0+) (1639,139 162) K
1/2(1+) (1743−,126) K1(1650)0?∗
1/2(2+) (1431,56 63) K (1430) 1429 1.4 104 4
− 2∗ ± ±
Inordertofollowtheprocedureoftheworksof[2,4],theV matrixisstraigthforwardextendedtoSU(4),asitwas
doneintheworkof[7]:
r 0 + w r + K + D¯ 0
 √2 √2 ∗ ∗ 
Vm = r − −√r 02+√w2 K∗0 D∗−  , (4)
 K K¯ 0 f D 
 ∗− ∗ ∗s− 
 D 0 D + D + J/y 
∗ ∗ ∗s m
wheretheidealmixinghasbeentakenforw ,f andJ/y .Thedetailsofthefullprocedurethatwetrytosummarizein
thissectionarewellexplainedin[5].Besidesthefour-vectorcontactdiagrams,wehavetwokindsofvector-exchange
diagrams,oneistheexchangeofonelightvectormeson,r orw ,andtheotheroneistheexchangeofaheavyvector
meson, D . Of course, the last terms are proportional to k = m2r 0.15, and this gives rise to corrections of the
∗ m2 ∼
order of 10% of the r -exchange terms. Also, the rrw and theDww∗w vertices violate G-parity, whereas the rww
vertexviolatesisospin,therefore,wedonothavetheexchangeofonelightvectormesonintheV(r D w D )and
∗ ∗
V(w D w D ) potentials. For this reason, these two terms, where only the four-vectorcontact term→plus the D -
∗ ∗ ∗
exchang→etermcontributes,aresmallercomparedwiththeV(r D r D ),termthatmainlyprovidestheinteraction.
∗ ∗
→
AscanbeseeninTable2,thistermisoftheorderof 300MeVinI=1/2.Thismagnitudeisresponsibletobindthe
r D systemandcomesmainlyfromther -exchangete−rmofV(r D r D ).Thisminorrelevanceofthew D channel
∗ ∗ ∗ ∗
→
appearswhenonecalculatethecouplingsg tooneparticularchannelibymeansoftheresiduesoftheamplitudes[5]
i
that we show in Table 3. In I =3/2, it is interesting to see that we get a repulsive interactionwithout any possible
generatedexoticstates.
We getthreestates with I=1/2andJ =0,1 and2 respectively.We havefixedthe valueof m as 1500MeV and
we have fine-tuned the subtraction constant a around its natural value of 2 [8, 9], in order to get the position of
the D (2460) state at the PDG. Thus, we have chosen the value of a = −1.74. Because of the strong interaction
we ge∗2tthree boundstates with practicallyno width exceptbythe small wi−dthprovidedbythe convolutionof the r
propagatortotakeintoaccountitsdecayintwopions.But,thelaterinclusionofthep Dboxdiagramallowsustoget
some more width comparablewith the data in the PDG as can be seen in Table 4. In orderto calculate the p D-box
diagram,twodifferentformfactorsforthevector-two-pseudoscalarvertexareconsidered.Oneisusedintheworksof
[2,4],whichisinspiredbytheempiricalformfactorsusedinthestudyofvectormesondecays[10].Theotherone,
is an exponentialparametrizationfor an off-shellpion evaluated using QCD sum rules [11] using at the same time
the experimentalvalue of the D Dp coupling,gexp [5]. In bothcases we getreasonablevaluesof the width, with
∗ D Dp
thepreference,ofcourse,fortheuseofgexp .In∗Table4,thevaluesobtainedforthelastcaseareshown,wherethe
D Dp
parameterof the exponentialform factor is∗ taken as L =1 GeV. The interesting thing in the calculation of the box
diagram,isthatthisboxdiagramonlyhasJ=0and2,therefore,thestatefoundwithJ =1cannotdecaytop Dby
meansofthismechanism,whichisthereasonwhywegetasmallwidthof4MeV,comparedwiththeotherstatesfor
TABLE2. V(r D r D )forthedifferentspin-isospinchannelsincludingtheexchangeofoneheavy
∗ ∗
vectormeson.Theap→proximateTotalisobtainedatthethresholdofr D .
∗
I J Contact r -exchange D -exchange Total[I(JP)]
∗
∼
1/2 0 +5g2 −2Mg2r2(k1+k3)·(k2+k4) −21kMgr22(k1+k4)·(k2+k3) −16g2[1/2(0+)]
1/2 1 +92g2 −2Mg2r2(k1+k3)·(k2+k4) +21kMgr22(k1+k4)·(k2+k3) −14.5g2[1/2(1+)]
1/2 2 −52g2 −2Mg2r2(k1+k3)·(k2+k4) −21kMgr22(k1+k4)·(k2+k3) −23.5g2[1/2(2+)]
3/2 0 −4g2 +Mg2r2(k1+k3)·(k2+k4) +kMgr22(k1+k4)·(k2+k3) +8g2[3/2(0+)]
3/2 1 0 +Mg2r2(k1+k3)·(k2+k4) −kMgr22(k1+k4)·(k2+k3) +8g2[3/2(1+)]
3/2 2 +2g2 +Mg2r2(k1+k3)·(k2+k4) +kMgr22(k1+k4)·(k2+k3) +14g2[3/2(2+)]
TABLE3. Modulesofthecouplingsg inunitsof
i
GeVforthepolesintheJ=0,1,2; I=1/2sector
withthechannelr D andw D .
∗ ∗
Channel D (2600) D (2640) D (2460)
∗0 ∗1 ∗2
r D 14.32 14.04 17.89
∗
w D 0.53 1.40 2.35
∗
J=0and2.Thus,wefindareasonableexplanationforwhytheD (2640)hasasmallwidth,G <15MeV,whenwe
∗
associatetothisstatethequantumnumbersJP=1+.
InTable5weshowtheuncertaintiesrelatedtotheSU(4)breakingduetotheuseoftheparameterg=m /2f in
V
the Lagrangian. In this table we distinguish three different cases in which we use g2, gg or g2 in the amplitudes
D D
obtained,seeTable2,withg=mr /2fp andgD=mD /2fD,andfixingthea parametertogetthepolepositionofthe
∗
tensorstate.Thisisalwaysdoneandthereforethismethodprovidesarealisticwayoflookingattheuncertainties.We
canseeinthistablethatthechangesinthemassoftheotherstatesarearound20MeV,whichisquitesmall,andeven
ifonefixesalphaandlooksfortheuncertaintiesusingg2,gg org2,thenthechangesinthemassarearound70 90
D D −
MeV,whicharetypicalinanyhadronmodel.
CONCLUSIONS
Wehavestudiedther (w )D interactioncombiningcoupledchannelunitarityandthehiddengaugeLagrangiansfor
∗
theinteractionofvectormesons.WefindastronginteractionthatprovidesoneboundstateforeachspinJ=0,1and
2.WecanassignthestateswithJ=1and2totheD (2640)andD (2460)respectively,whereasthestateobtainedfor
∗ ∗2
J=0isapredictionofthemodel:’D (2600)’.Inthisscheme,theD (2460)andtheD (2600)decaytop Dbymeans
∗0 ∗2 ∗0
ofaboxdiagram.However,thisdecayisnotpossibleforJ=1,providinganaturalexplanationonwhy,whereasthis
stateisheavierthantheD (2460),itswidthfoundinthePDGisverysmallcomparedtothat,G <15MeV,andthe
∗2
decaytop Dhasnotbeenobservedeither.TheresultsobtainedhereshouldstimulatethesearchformoreDstatesin
theregionof2600MeV.
TABLE4. Massesandwidths(inunitsofMeV)obtainedinthecase
oftheuseofanexponentialformfactorwithL =1GeVcomparedwith
theexperiment.
IG[JP] Theory PDGdata
(Mass,Width) Name Mass Width
1/2+(0+) (2608,61) "D (2600)"
∗0
1/2+(1+) (2620,4) D (2640) 2637 <15
∗
1/2+(2+) (2465,40) D (2460) 2460 37 43
∗
−
TABLE 5. Pole positions and subtraction
constant obtained for the three different
cases, g2, gg and g2, when one fixes the
D D
massofthepolewithJ=2.
Constant&a J=0 J=1 J=2
g2& 1.74 2592 2611 2450
−
ggD& 1.53 2571 2587 2450
g2 & −1.39 2551 2565 2450
D −
ACKNOWLEDGMENTS
This work is partly supported by DGICYT contract number FIS2006-03438. We acknowledge the support of the
European Community-Research Infrastructure Integrating Activity Study of Strongly Interacting Matter (acronym
HadronPhysics2,GrantAgreementn.227431)undertheSeventhFrameworkProgrammeofEU.A. H.is supported
inpartbytheGrantforScientificResearchContractNo.19540297fromtheMinistryofEducation,Culture,Science
andTechnology,Japan.H.N.issupportedbytheGrantforScientificResearchNo.18-8661fromJSPS.
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