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The charmonium-molecule hybrid structure of the X(3872)
Makoto Takizawa (Showa Pharmaceutical Univ.)Sachiko Takeuchi (Japan College of Social Work)Kiyotaka Shimizu (Sophia University)
International conference on the structure of baryons,BARYONS’10, Osaka, Japan. Dec. 8, 2010
Ref: M. Takizawa and S. Takeuchi, EPJ web of conference 3, (2010) 03026; Prog. Theor. Phys. Suppl. 186 (2010) 160.
ContentsProblems of X(3872) as C C-bar stateProblems of X(3872) as D0 D*0-bar moleculeCoupling between C C-bar core state,
D0 D*0-bar state and D+D*- stateInteraction between D and D*Wavefunction and isospin symmetry
breaking Energy spectrumNumerical resultsDiscussionSummary
About X(3872)First observation: 2003, Belle, KEKB
cited more than 500 timesB- → K- π+ π- J/ψ decay
Sharp peak of the invariant mass distribution of π+ π- J/ψ
Mass: (3871.5 ± 0.19) MeV about 0.3 MeV below D0 D*0-bar thresold
Width: less than 3.0 MeV Quantum Number: JPC = 1++ ?Other decay mode:
X(3872) → γ J/ψ 、 γψ(2S) X(3872) → π+ π- π0 J/ψ
International Journal of Theoretical Physics International Journal of Theoretical Physics
B+ → K+ + J/ψ + ππ(π)
11 Sep 2010 jps fall meeting @ 九州工業大学
B+ → X(3872) + K+ → J/ψ + vector meson
→π’s
Problems of X(3872) as C C-bar State1. Estimated energy of 2 3P1 c c-bar state
by the potential model is 3950 MeV, which is about 80 MeV higher than the observed mass of X(3872).
2. If X(3872) is c c-bar state, it is isoscalar.X(3872) → ρ0 J/ψ → π+ π- J/ψ : isovectorThis decay means large isospin breaking.
Is X(3872) isospin mixed state?
Isovector component is smaller than isoscalar component : 10~20%
Estimation of isospin component from this value is an issue of the discussion
D. Gamermann and E. Oset, Phys. Rev. D80:014003,2009.M. Kerliner and H. J. Lipkin, arXiv:1008.0203.K. Terasaki, Prog. Theor. Phys. 122:1205,2010.
X(3872) as D0 D*0-bar Molecule
mD0 + mD*0 = (3871.81 ± 0.36) MeV
mX(3872) = (3871.50 ± 0.19) MeV
X(3872) is a very shallow bound state of D0 D*0-bar: D0 D*0-bar Molecule
Problem of X(3872) as D0 D*0-bar Molecule
1. D0 D*0-bar is 50% isovector and 50% isoscalar: Too big the isovector component
2. Why are there no charged X(3872)?D+ D*0-bar, D0 D*- molecules
3. The production rate of such molecular-like state may be too small.
Coupling between C C-bar core and D0 D*0-bar, D+ D*-
Structure of X(3872): c c-bar core state (charmonium) is coupling to D0 D*0-bar and D+ D*- states
Effect of the isospin symmetry breaking is introduced by the mass differences between neutral and charged D, D* mesons
Coupling between C C-bar core and D0 D*0-bar, D+ D*-
c c-bar core D*0-bar
D0 D+
D*-
+. . . . .
Coupling between C C-bar core, D0 D*0-bar and D+ D*-
cc-bar core state:D0 D*0-bar state :
D+ D*- state : in the center of mass frameq is the conjugate momentum of the relative coordinate
Coupling between C C-bar core, D0 D*0-bar and D+ D*-
Charge conjugation + state is assumed
Interaction: Isospin symmetric
Interaction between D0 and D*0bar, D+ and D*-
cc
u-bar
u-bar
c-bar
c-bar
D0
D*0-bar
ccd-bar
c-bar
c-bar
D+
D*-
u u
d-bar
d d
Like the σ-meson exchangeNo isospin symmetry breaking
Interaction between D0 and D*0bar, D+ and D*-
Interaction:
)()(
11)(*)(*
**
22220000
qDDUqDD
qqqDDUqDD
Diagram
c c-bar core
D*0-bar
D0 D+
D*-
+. . . . .
c c-bar core
Coupling between C C-bar core, D0 D*0-bar and D+ D*-X(3872) is a mixed state:
Isospin base:
Isospin symmetric case: c2 = c3 No isovector component
Coupling between C C-bar core, D0 D*0-bar and D+ D*-
Schroedinger Equation
Energy spectrumWe consider c c-bar core state is
produced in the production process
Transition strength S(E):B
K
E=Energy transfer
X(3872)
Numerical results: Mass Mass of the cc-bar core: 3.95 GeV
from S. Godfrey, N. Isgur, Phys. Rev. D 32 (1985) 189.
Cutoff: 0.3GeV and 0.5 GeV
Lambda = 0.5 GeV, Calculated bound state energy is 3.871 GeV with coupling strength g = 0.0185 GeV3/2
Lambda = 0.3 GeV, Calculated bound state energy is 3.871 GeV with coupling strength g = 0.0094 GeV3/2
Numerical results: WavefunctionLambda = 0.5 GeV
Lambda = 0.3 GeV
Large isospin symmetry breakingCutoff dependence is small
Why so large isospin symmetry breaking?
mD0 + mD*0 = (3871.81 ± 0.36) MeV
mD+ + mD*- = (3879.89 ± 0.37) MeV
mX = 3871.5 MeV
Binding EnergyNeutral D case: 0.81 MeVCharged D case: 8.89 MeV
Large difference
Numerical results: WavefunctionLambda = 0.5 GeV
D0 D*0-bar
D+ D*-
r [fm]
r psi(r) [GeV1/2]
Numerical results: Energy spectrum
Lambda = 0.3 GeV
GeVX(3872) bound state
CC-bar state
Numerical results: Energy spectrum
Lambda = 0.5 GeV
GeVX(3872) bound state
CC-bar state disappears
Numerical results:
Mass of the cc-bar core: 3.95 GeVfrom S. Godfrey, N. Isgur, Phys. Rev. D 32 (1985) 189.
Cutoff: 0.5 GeV & 1.0 GeV
Determination of the interaction strengthsFirst, we set λ=0, then g is fixed so as to reproduce mass of X(3872) to be 3.8715 GeV
Then, we change the value of g from 0.9g, 0.8g, 0.7g, … and determine the value of λ so as to reproduce mass of X(3872) to be 3.8715 GeV
Numerical results: X(3872) componentsΛ=0.5 GeV
0 0.10.20.30.40.50.60.70.80.9 10
0.2
0.4
0.6
0.8
1
D+ D*-
D0 D*0
c c-bar core
g / g (lambda=0)
Numerical results: X(3872) componentsΛ=0.5 GeV
0 0.10.20.30.40.50.60.70.80.9 10
0.10.20.30.40.50.60.70.80.9
1
D D* I =0D D* I=1c c-bar core
g / g (lambda=0)
Numerical results: X(3872) componentsΛ=1.0 GeV
g / g (lambda=0)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
D+ D*-D0 D*0c c-bar core
Numerical results: X(3872) componentsΛ=1.0 GeV
g / g (lambda=0)
0 0.10.20.30.40.50.60.70.80.9 10
0.10.20.30.40.50.60.70.80.9
1
D D* I =0D D* I=1c c-bar core
DiscussionBound state of hadrons
Kinetic energy v.s. Potential energy
Bound state of two hadronsDeuteron
Heavier Hadron -> Smaller kinetic termAbout 1 GeV mass (proton, neutron)-> bound state exists
DiscussionCharm quark hadrons -> mass is
bigger than 1 GeVPossibility of forming the bound
statesBottom quark hadrons -> more
probablePossibility of the exotic hadrons
Discussion -X(3872)-Quarkonium-hadronic molecule
hybrid structure ー> new style of hadrons
Properties of X(3872) can be explained by the charmonium-hadronic molecule structure, naturally
Quark model result of the charmonium ismeaningful as the core state.
Discussion -X(3872)-Large Isospin symmetry breaking
can be explained by the present picture
No observation of isospin multiplet is clearly explained, since the intermediate isoscalar ccbar state causes the attractive force between D and D*
Discussion -X(3872)-Production rate
compact ccbar component
Radiative decay
Charmonium state is 2 3P1
Easy to transit to ψ(2S) than J/ψ(need the calculation to confirm it)
B X (2S) B X J / 3.41.4
SummaryCharmonium-hadronic molecule
hybrid structure can explain the observed properties of the X(3872) naturally.
Backup
Numerical results: WavefunctionLambda = 0.5 GeV
D0 D*0-bar
D+ D*-
r [fm]
psi(r) [GeV3/2]
Numerical results: WavefunctionLambda = 0.3 GeV
D0 D*0-bar
D+ D*-
r [fm]
r psi(r) [GeV1/2]
Numerical results: WavefunctionLambda = 0.3 GeV
D0 D*0-bar
D+ D*-
r [fm]
psi(r) [GeV3/2]
Numerical results: cc-bar core in complex energy planeΛ=0.5 GeV