Hadron Molecules

0.18 MB·

by  Thomas Gutsche

CPC(HEP&NP),2009,33(X):1—6 Chinese Physics C Vol.33,No.X, Xxx,2009
Hadron Molecules*
Thomas Gutsche 1) Tanja Branz Amand Faessler Ian Woo Lee Valery E. Lyubovitskij
Institutfu¨rTheoretischePhysik,Universit¨atTu¨bingen,KeplerCenterforAstroandParticlePhysics,
AufderMorgenstelle14,D–72076Tu¨bingen,Germany
0
1
0 Abstract WediscussapossibleinterpretationoftheopencharmmesonsD∗ (2317),D (2460)andthehidden
s0 s1
2 charm mesons X(3872), Y(3940) and Y(4140) as hadron molecules. Using a phenomenological Lagrangian
n approach we review the strong and radiative decays of the Ds∗0(2317) and Ds1(2460) states. The X(3872)
a is assumed to consist dominantly of molecular hadronic components with an additional small admixture of a
J
charmoniumconfiguration. Determingtheradiative(γJ/ψ andγψ(2s))andstrong(J/ψ2π andJ/ψ3π)decay

2
modes we show that present experimental observation is consistent with the molecular structure assumption
1
of the X(3872). Finally we give evidencefor molecular interpretations of the Y(3940) and Y(4140) related to

] theobserved strong decay modes J/ψ+ω or J/ψ+φ , respectively.
h
p Key words charm mesons, hadronic molecule, strong and radiative decay
-
p PACS 12.38.Lg, 12.39.Fe, 13.25.Jx, 14.40.Gx, 36.10.Gv
e
h
[ 1 Introduction a scalar DK and a axial D∗K molecule because of a

1 relatively small binding energy of 50 MeV. These
v ∼
stateswerediscoveredandconfirmedjustafew years
0
The complexity of hadronic mass spectra allows ago by the Collaborations BABAR [4], CLEO [5] and
7
8 for the possibility that existing and newly observed Belle [6]. The simplest interpretation of these states
1 hadrons can possibly be interpreted as hadronic is that they are the missing j = 1/2 (the angular
. s
1 molecules. Such an interpretation is possible when momentum of the s-quark) members of the cs¯L=1
0 themassoftheobservedmesonliesclosetoorslightly multiplet. However, this standard quark model sce-
0
1 belowthethresholdofacorrespondinghadronicpair. nario is in disagreement with experimental observa-
: In the following we focus on some of the candidates tion since the D∗ (2317) and D (2460) states are
v s0 s1
i in the heavy meson sector which are discussed and narrower and their masses are lower when compared
X
considered as such hadronic bound states. to theoretical expectations (see e.g. discussion in
ar TheX(3872)isoneofthepeculiarnewmesonres- Ref. [7]).
onanceswhichwasdiscoveredduringthelastyears[1] The CDF Collaboration recently reported evi-
and where its properties cannot be simply explained dence for a narrow near-threshold structure, termed
inthecontextofconventionalconstituentquarkmod- the Y(4140) meson, in the J/ψφ mass spectrum
els. Inthemolecularapproaches,theearliestgivenin in exclusive B+ J/ψφK+ decays with the mass
→
Refs.[2,3],itisarguedthattheX(3872)canbeiden- m =4143.0 2.9(stat) 1.2(syst)MeVandnat-
Y(4140) ± ±
tified with a weakly–bound hadronic molecule whose uralwidthΓ =11.7+8.3(stat) 3.7(syst)MeV[8].
Y(4140) −5.0 ±
constituents are D and D∗ mesons. This natural in- As alreadystressedin [8], the new structure Y(4140),
terpretation is due to the fact that its mass is very whichdecaystoJ/ψφjustabovetheJ/ψφthreshold,
close to the D0D¯∗0 threshold andhence is in analogy is similar to the previously discovered Y(3940) [9,10],
to the deuteron — a weakly–bound state of proton which decaysto J/ψω near this respective threshold.
and neutron. Both observedstates, Y(4140)and Y(3940),are well
In the open charm sector the scalar D∗ (2317) above the threshold for open charm decays. A con-
s0
and axial D (2460) mesons could be candidates for
s1
Received24December 2009
∗Supported by Deutsche Forschungsgemeinschaft Contract No. FA67/31-2 and No. GRK683. This research is also part of
theEuropeanCommunity-ResearchInfrastructureIntegratingActivity”StudyofStronglyInteractingMatter”(acronymHadron-
Physics2, Grant Agreement No. 227431), Russian President grant “Scientific Schools” No. 3400.2010.2, Russian Science and
Innovations FederalAgencycontractNo. 02.740.11.0238.
1)E-mail:[email protected]
(cid:13)c2009ChinesePhysicalSocietyandtheInstituteofHighEnergyPhysicsoftheChineseAcademyofSciencesandtheInstitute
No.X T.Gutscheetal: HadronMolecules 2
ventionalcc¯charmoniuminterpretationisdisfavored, (HHCHPT)). Further details of this procedure can
sinceopencharmdecaymodeswoulddominate,while be found in Refs. [12, 21].
the J/ψφ orJ/ψω decayratesareessentially negligi-
ble [8,11]. This could be a signal for nonconventional 3 Strong and radiative decays of
structure of these Y states. D∗ (2317) and D (2460)
s0 s1
In the following we give a compact overview of
our recent calculations related to the strong and
For the D∗ (2317) we consider a possible inter-
electromagnetic decay properties of the D∗ (2317), s0
s0 pretation as a hadronic molecule - a bound state of
D (2460), X(3872), Y(3940) and Y(4140) mesons
s1 D and K mesons. Using an effective Lagrangian ap-
interpretedashadronmolecules. Detailscanbefound
proach we calculated the strong D∗ D π0 and ra-
in the related publications [12–21] on this topic. s0→ s
diative D∗ D∗γ decays [14]. A new impact related
s0→ s
to the DK molecular structure of the D∗ (2317) me-
2 The method s0
sonisthatthepresenceofu(d)quarksintheDandK
mesons gives rise to a direct strong isospin-violating
We first briefly discuss the formalism behind the transition D∗ D π0 (see Fig.1) in addition to the
s0→ s
study of hadronic molecules. As an example we con- decaymechanisminducedbyη π0 mixingconsidered
sider the Ds∗0±(2317) meson as a bound state of a previously (see Fig.2). −
D and a K meson. We use the current results for
the quantum numbers of isospin, spin and parity: π0 π0
I(JP) = 0(0+) and mass m = 2.3173 GeV [1].
Ds∗0 D+ D∗+ D0 D∗0
Our framework is based on an effective interac- D∗s0+ D+s D∗s0+ D+s
tion Lagrangian describing the coupling between the
D∗ (2317) meson and their constituents - D and K K0 K+
s0 (a) (b)
mesons:
π0 π0
(x) = g D∗−(x) dyΦ (y2)
LDs∗0 Ds∗0 s0 Z Ds∗0 · D∗s0+ K+ K∗+ D+s D∗s0+ K0 K∗0 D+s
DT(x+w y)K(x w y)+H.c.(1)
K − D
D0 D+
where D and K are the meson doublets and w = (c) (d)
ij
mi is a kinematic variable. The correlation func- Fig. 1. Direct isospin-violating transition for
mi+mj
tionΦDs∗0 characterizesthefinitesizeoftheDs∗0(2317) Ds∗0→Dsπ0
mesonasa(DK)boundstateanddependsontherel-
ative Jacobi coordinate y with x being the center of Weshow[14] thatthedirecttransitiondominatesover
mass (CM) coordinate. The Fourier transformof the theη π0 mixingtransitionintheD∗ D π0 decay.
. − s0→ s
correlation function is Φ˜ (p2) = exp( p2/Λ2 ),
Ds∗0 E − E Ds∗0 π0 π0
where p is the Euclidean Jacobi momentum. Here
E
Λ is a size parameter,whichparametrizesthe dis- η η
Ds∗0 D+ D∗+ D0 D∗0
tributionofDandK mesonsinsidetheD∗ molecule. D∗s0+ D+s D∗s0+ D+s
s0
The D∗ DK coupling constant g is determined
s0 Ds∗0 K0 K+
by the compositeness condition [22,23], which implies
(a) (b)
thattherenormalizationconstantofthehadronwave π0 π0
functionissetequaltozero: Z =1 Σ′ (m2 )=
Ds∗0 − Ds∗0 Ds∗0 η η
0,whereΣ′ isthederivativeoftheD∗ mesonmass K+ K∗+ K0 K∗0
operator. DTs∗h0e effective Lagrangian iss0the basis for D∗s0+ D+s D∗s0+ D+s
the study of the decay properties of the hadronic D0 D+
molecules. It defines the coupling of the molecule (c) (d)
to its constituents but also supplies the hadron loop
Fig. 2. η π0 mixing transition for D∗ D π0
connectedtothe finalstateparticlesconsidered. The − s0→ s
interaction of the constituents with external parti- TheradiativetransitionD∗+ D∗+γ isgeneratedby
s0 → s
cles is further described by effective Lagrangians ei- theloopdiagramsofFig.3,wherethelastgraphshave
ther adjusted by data or by theory (as for example to be included to guarantee full gauge invariance for
by heavy hadron chiral perturbation theory [24,25] the case of a non-local vertex.
No.X T.Gutscheetal: HadronMolecules 3
γ γ while for the radiative modes we have
Γ(D∗ D∗γ) = 0.47 0.63 keV,
D+ D+ K+ K+ s0→ s −
D∗s0+ D∗s+ D∗s0+ D∗s+ Γ(D D γ) = 2.7 3.7 keV. (3)
s1→ s −
K0 D0 Thevariationinresultsreflectstheuncertaintyinthe
(a) (b)
vertex function. The corresponding ratios of rates
γ γ with
D+ K+
D∗s0+ D∗s+ D∗s0+ D∗s+ RDs∗0 = Γ(Ds∗0→Ds∗γ)/Γ(Ds∗0→Dsπ)=0.01
R = Γ(D D γ)/Γ(D D∗π)=0.05(4)
K0 D0 Ds1 s1→ s s1→ s
(c) (d)
satisfy qualitatively the present experimental results
γ γ ofRDs∗0≤0.059andRDs1=0.44±0.09[1]. Hencefrom
D+ K+ thepresentobservationoftheradiativeandstrongde-
D∗s0+ D∗s+ D∗s0+ D∗s+ caysoftheD∗ andD aninterpretationasDK and
s0 s1
D∗K hadron molecules seems feasible.
K0 D0
In Ref. [15] we further investigated the weak de-
(e) (f)
caysB D(∗)+D∗ (D ),wherefullconsistencywith
→ s0 s1
γ γ present data is achieved. This further strengthens
D+ K+ thepresentedmolecularpicture. Wealsogivepredic-
D∗s0+ D∗s+ D∗s+ D∗s0+ D∗s+ D∗s+ tions[21] fortheweakdecaysinvolvingf (980),which
0
serves as a further indicator for this structure inter-
K0 D0
pretation.
(g) (h)
Fig. 3. Radiative transition D∗+ D∗+γ 4 Decay analysis of X(3872)
s0 → s
The X(3872) with quantum numbers JPC =1++
The other strong coupling vertices are fixed by
is considered as a composite state containing both
HHCHPT or by data, for full details see Ref. [14].
molecular hadronic and a cc¯ component. We re-
Similar graphs apply for the strong and radiative
centlyshowed[18] thatslightbinding ofthe DD∗ sys-
transitionsoftheD (2460)interpretedinthepresent
s1
framework as a D∗K hadronic molecule [13]. Results tem canbe achievedin a full meson-exchangemodel.
Following Refs. [17, 26] we consider the X(3872)
for the strong decays are
state as a superposition of the dominant molecular
D0D∗0 component, of other hadronic configurations–
Γ(Ds∗0→Dsπ0) = 47−112 keV, D±D∗∓,J/ψω,J/ψρ,aswellasofthecc¯charmonium
Γ(D D∗π0) = 50 79 keV, (2) configuration as
s1→ s −
Z1/2 Z1/2
X(3872) = cosθ D0D∗0(D0D¯∗0 + D∗0D¯0 )+ D±D∗∓(D+D∗− + D−D∗+ )+Z1/2 J ω +Z1/2 J ρ
| i " √2 | i | i √2 | i | i Jψω| ψ i Jψρ| ψ i#
+ sinθcc¯. (5)
Here θ is the mixing angle between the hadronic tential model. For example, for a binding energy
and the charmonium components: cos2θ and sin2θ ǫ m +m m =0.3 MeV the estimate of
D0D∗0 D0 D∗0− X
represent the probabilities to find a hadronic and Ref. [26] provides
charmonium configuration, respectively, for the nor-
malization ZD0D∗0 = 0.92, ZD±D∗∓=0.033,
Z = 0.041, Z =0.006. (7)
ZD0D∗0+ZD±D∗∓+ZJψω+ZJψρ=1. (6) Jψω Jψρ
For comparison with data we employ values for The coupling of the X to its constituents, once the
Z , Z , Z and Z as derived in a po- amplitudes of Eq. (7) are chosen, are determined
D0D∗0 D±D∗∓ Jψω Jψρ
No.X T.Gutscheetal: HadronMolecules 4
again by the compositeness condition [16,17,19]. 60keVandismuchlargerthantheonefortheψ(2S)γ
A new measurement by the BABAR Collabora- channel (about 0.3 keV). Inclusion of a charmonium
tion gives clear evidence for a strong radiative decay component in the X(3872) configuration leads to the
modeinvolvingtheψ(2S)[27]. Theyindicatethemea- resultsofFig.5indicatingtheradiativedecaywidths
sured ratio of in dependence on the cc¯admixture.
(X(3872) ψ(2S)γ)
B → =3.5 1.4. (8)
(X(3872) J/ψγ) ± 103
B →
Asknownfrompreviouscalculationsinthemolecular 102
approachtheradiativedecaywidthoftheX(3872)
→ V)101
ψ(2S)γ is always smaller than the one involving e
k
J/ψγ. ItisthereforeexpectedthattheratioofEq.(8) Γ(100
givessomeconstraintonapossiblecharmoniumcom-
10-1
ponent in the X(3872).
In Fig.4 we display the diagrams relevant for the 10-20.1 0.15 -s0i.n2θ 0.25 0.3
radiative decays X J/ψγ and X ψ(2S)γ. The
→ → Fig.5. RadiativedecaywidthsoftheX(3872).
lastdiagramofFig.4whichisgeneratedbytheJ/ψV
Solidanddottedcurvesaretheresultsforthe
(V =ρ,ω)componentoftheX(3872)onlycontributes transitions X →J/ψγ and X →ψ(2S)γ, re-
to the J/ψγ final state. spectively.
γ γ
D(D∗) D±(D∗±) A small admixture of a cc¯component is essential
X D∗(D) X D±(D∗±) to reproduce the large measured value for R2S [16].
Thisfeatureisdue tothe destructiveinterferencebe-
D∗(D) D∗∓(D∓)
J/ψ,ψ(2S) J/ψ,ψ(2S) tweenthe J/ψγ decayamplitudes arisingfromthe cc¯
γ and the hadronic components. For mixing values of
γ
c about sinθ 0.16 in the case of ǫ = 0.3 MeV the
X X ρ0(ω) ≈ −
c decay width for ψ(2S)γ may exceed the one of J/ψγ
c and a consistent explanation for the observed ratio
J/ψ
J/ψ,ψ(2S) R is obtained.
2S
Fig. 4. Diagrams contributing to the radiative To get estimates [17] for the decay widths of
transitionsX(3872)→J/ψ+γandX(3872)→ X(3872) J/ψ+h with h = π+π−π0,π+π0,π0γ,γ
→
ψ(2S)+γ. we use the results of Ref. [28], which are based on
the assumption that these decays proceed through
Results [16] for the radiative decay modes involving the processes X to J/ψω and J/ψρ. Here we esti-
the hadronic components of the X(3872) are con- mate[17]bothshortandlong-distanceseffectsonlyfor
tained in Table 1. The value of the binding energy theX γJ/ψdecaysusingourpreviousresult,while
→
is set to 0.3 MeV with the probabilities taken from theX J/ψ+hdecaysonlytakeintoaccountshort–
→
Eq. (7). distanceeffects. ResultsfortheJ/ψ decaymodesare
given in Table 2.
Table 1. Radiative decays of hadronic components.
Table 2. Properties of X → J/ψ+h decays.
Hadronicconfig. Γ(X(3872)→γJ/ψ, γψ(2S))keV Numbersinbracketsarevalidforexplicitval-
DD∗ 60-120(J/ψ) 0.3(ψ(2S)) ues for ZJψρ and ZJψω of Eq. (7). Data for
J/ψV 6(J/ψ) 0(ψ(2S)) ratios are from Refs. [27, 29].
DD∗ andJ/ψV 30-65(J/ψ) 0.3(ψ(2S))
Quantity Nonlocalcase
The inclusion of the J/ψV components in the Γ(X→J/ψπ+π−),keV 9.0×103ZJψρ (54.0)
structure of the X(3872) leads to a decrease in the Γ(X→J/ψπ+π−π0),keV 1.38×103ZJψω (56.6)
X J/ψγ rate because ψ(2s)V components are ab- Γ(X→J/ψπ0γ),keV 0.23×103ZJψω (9.4)
sen→t in the X(3872). Note that taking into account ΓΓ(X(X→→JJ/ψ/ψππ++ππ−−π)0) 1.05
the hadronic components only the ratio of Eq. (8) =1.0±0.4±0.3
cannot be explained. If no cc¯ component is present Γ(X→J/ψγ) 0.10
Γ(X→J/ψπ+π−)
in the X(3872) the decay width for J/ψγ is about =0.14±0.05;0.33±0.12
No.X T.Gutscheetal: HadronMolecules 5
Presentdatafortheratioofrates Γ(X→J/ψπ+π−π0) 5 J/ψV(= ω,φ) decays of Y(3940) and
Γ(X→J/ψπ+π−)
indicate a strong isospinviolating J/ψπ+π−π0 decay Y(4140)
mode. Similarly, the measured ratio Γ(X→J/ψγ)
Γ(X→J/ψπ+π−)
points to a large radiative decay channel. Both ra-
ItwassuggestedinRef.[30]thatboththeY(3940)
tios of rates can be fully reproduced in the molec- and Y(4140) are hadronic molecules. These hadron
ular picture. Note that all the decay channels con- bound states can have quantum numbers JPC =
sidered so far are fed by the subleading J/ψω, J/ψρ 0++ or 2++ whose constituents are the vector charm
and cc¯ components. To work out the influence of D∗(D∗) mesons:
s
the leading molecular components we consider the
hadronic X→χcJ+(π0,2π) and X→J/ψ+(2π,3π) Y(3940) = 1 D∗+D∗− + D∗0D∗0 ,
decays[17]. HerethevaluesofJ=0,1,2correspondto | i √2 | i | i
the JP =0+,1+,2+ quantum numbers ofthe charmo- Y(4140) = D∗+(cid:0)D∗− . (cid:1) (9)
| i | s s i
nium states. Becauseof the dominance ofthe D0D∗0
For the observed Y(3940) and Y(4140) states we
component in the transitions of X into charmonium
adopt the convention that the spin and parity quan-
states χ and pions we estimate these decays using
cJ
tum numbers of both states are JPC = 0++. The
only this component. The two-body decays are de-
coupling of the scalar molecular states to their con-
scribedbythe (D0D∗0)loopdiagramshowninFig.6.
stituents is again set up by the compositeness condi-
π0 π0 π0 tion. To determine the strong Y J/ψV and two-
→
D0 D∗0 D∗0 photon Y γγ decays we have to include the cou-
X X X →
D∗0 D∗0 D0 plings of D∗(D∗) mesons to vector mesons (J/ψ, ω,
s
D∗0 χc0(c2) D0 χc1 D0 χc0 φ) and to photons. The couplings of J/ψ, ω, φ to
Fig. 6. Diagrams contributing to the hadronic vector D∗(Ds∗) mesons are taken from the HHChPT
transitions X(3872)→χ +π0. Lagrangian [24,25]. The leading-order process rele-
cJ
vant for the strong decays Y(3940) J/ψω and
→
Y(4140) J/ψφ is the diagram of Fig.7 involving
Inclusion of the charged (D±D∗∓) loops gives a →
the vector mesons D∗ or D∗ in the loop.
further correction to the decay widths. The possible s
couplings of D(D∗) mesons to pions and charmonia
states is taken from HHCHPT. Similar graphs ap- J/ψ
D∗
ply for the three-body decay modes. Results for the
hadronic decays involving the neutral (D0D∗0) com- Y D∗
ponent only and the full results are indicated in Ta-
D∗
V
ble 3.
Fig. 7. Leading order diagram for Y →J/ψV
Table 3. Properties of X → χ +nπ decays.
cJ decay.
Here ZD≡ZD0D∗0.
Quantity D0D∗0 loop D0D∗0+ WealsoconsidertheradiativeY(3940)/Y(4140)
→
+D−D∗+ γγ decay widths [20], which serve as a prediction for
Γ(X→χc0+π0),keV 41.1ZD (37.8) 61.0 possibly identifying the molecular structure of these
Γ(X→χc0+2π0),eV 63.3ZD (58.2) 94.0 Y states.
ΓΓ((XX→→χχcc11++2ππ0)0,),keeVV 1714.31ZZDD((61803..26)) 1160.945.2 D∗ γ D∗ γ
Γ(X→χc2+π0),keV 15ZD (13.8) 22.1 Y D∗ Y γ
Γ(X→χc2+2π0),eV 20.6ZD (19.0) 30.4 D∗ γ D∗
(a) b)
γ γ
Note that explicit numbers refer to a binding en- γ
D∗
ergyof0.3MeVwiththeprobabilitiestakenfromEq.
Y γ Y D∗
(7). The decay pattern of Table 3 involving pions
D∗
and χ states is sensitive to the leading molecular
cJ (c) (d)
(D0D∗0) component. These predictions [17] can serve
Fig. 8. Diagrams relevant for the radiative de-
to possibly identify the full hadronic composition of
cay Y →γγ.
the X(3872) in running and planned experiments.
No.X T.Gutscheetal: HadronMolecules 6
The relevant diagrams which arise from coupling of and Iizuka rule [11]. Further tests of the presented
the charged D∗±(D∗±) mesons to photons and from scenarioconcernthe two-photondecaywidths,which
s
gauging the strong interaction Lagrangian are dis- we predict to be of the order of 1 keV [20]. Results
played in Fig.8. The numerical results for the quan- for the strong J/ψ decays are quite similar for the
tities characterizing the strong J/ψV (V =ω,φ) and JPC=2++ assignment, which cannot be ruled out at
radiative 2γ decays of Y(3940)and Y(4140)are con- this point.
tained in Table 4.
Table 4. Decay properties of Y(3940) and 6 Conclusions
Y(4140).
Quantity Y(3940) Y(4140) The approach to hadron molecules based on the
Γ(Y →J/ψV),MeV 5.47±0.34 3.26±0.21 compositenessconditionconstitutesaconsistentfield
Γ(Y →γγ),keV 0.33±0.01 0.63±0.01 theoretical tool. The determination of the decay
Γ(Y →γγ)
R= ×104 0.61±0.06 1.93±0.16 properties of the open charm mesons D∗ (2317),
Γ(Y →J/ψV) s0
D (2460) and the hidden charm mesons X(3872),
s1
The predictions of Γ(Y(3940) J/ψω) = Y(3940) and Y(4140) in comparison to present data
→
5.47 MeV and Γ(Y(4140) J/ψφ) = 3.26 MeV for isfullyinlinewithamolecularinterpretationofthese
→
the observed decay modes are sizable and fully con- states. Wealsogivefurtherpredictionsofdecayprop-
sistent with the upper limits set by present data on ertieswhichcanbetestedinfutureexperiments. Cur-
the total widths. The result for Γ(Y(3940) J/ψω) rent results are very encouraging in ultimately iden-
→
is also consistent with the lower limit of about 1 tifying hadronic molecules in the meson spectrum.
MeV[11]. ValuesofafewMeVforthesedecaywidths We thank Yubing Dong, Yong-Liang Ma and
naturally arise in the hadronic molecule interpreta- Sergey Kovalenko for extensive collaborations on the
tion of the Y(3940) and Y(4140), whereas in a con- topic of hadron molecules. V.E.L. is on leave from
ventional charmonium interpretation the J/ψV de- the Department of Physics, Tomsk State University,
cays are strongly suppressed by the Okubo, Zweig 634050 Tomsk, Russia.
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