Physics of the charmonium-like state X ( 3872 )

We construct spectra of decays of the resonance X(3872) with good analytical and unitary properties which allows to define the branching ratio of the X(3872) → D∗0D̄0 + c.c. decay studying only one more decay, for example, the X(3872) → π+π−J/ψ(1S ) decay, and show that our spectra are effective means of selection of models for the resonance X(3872). Then we discuss the scenario where the X(3872) resonance is the cc̄ = χc1(2P) charmonium which "sits on" the D∗0D̄0 threshold. We explain the shift of the mass of the X(3872) resonance with respect to the prediction of a potential model for the mass of the χc1(2P) charmonium by the contribution of the virtual D∗D̄+ c.c. intermediate states into the self energy of the X(3872) resonance. This allows us to estimate the coupling constant of the X(7872) resonance with the D∗0D̄0 channel, the branching ratio of the X(3872) → D∗0D̄0 + c.c. decay, and the branching ratio of the X(3872) decay into all non-D∗0D̄0 + c.c. states. We predict a significant number of unknown decays of X(3872) via two gluons: X(3872) → gluon gluon → hadrons.


Introduction
The X(3872) resonance became the first in discovery of the resonant structures XYZ (X(3872), Y(4260), Z + b (10610), Z + b (10650), Z + c (3900)), the interpretations of which as hadron states assumes existence in them at least pair of heavy and pair of light quarks in this or that form.
Thousands of articles on this subject already were published in spite of the fact that many properties of new resonant structures are not defined yet and not all possible mechanisms of dynamic generation of these structures are studied, in particular, the role of the anomalous Landau thresholds is not studied.
Anyway, this spectroscopy took the central place in physics of hadrons.Below we give reasons that X(3872), I G (J PC ) = 0 + (1 ++ ), is the χ c1 (2P) charmonium and suggest a physically clear program of experimental researches for verification of our assumption.
The mass spectrum D * 0 D0 + c.c. in the X(3872) → D * 0 D0 + c.c. decay [5] looks as the typical resonance threshold enhancement, see Figure 2. If structures in the above channels are manifestation of the same resonance, it is possible to define the branching ratio BR(X(3872) → D * 0 D0 + c.c.) treating data of the two above decay channels only.
We believe that the X(3872) is the axial vector, 1 ++ [6,7].In this case the S wave dominates in the X(3872) → D * 0 D0 + c.c. decay and hence is described by the effective Lagrangian The width of the X → D * 0 D0 + c.c. decay where k is momenta of D * 0 (or D0 ) in the D * 0 D0 center mass system, m is the invariant mass of the D * 0 D0 pair, The second term in the right side of Eq. ( 2) is very small in our energy region and can be neglected.This gives us the opportunity to construct the mass spectra for the X(3872) decays with the good analytical and unitary properties as in the scalar meson case [8,9].
The mass spectrum in the The branching ratio of X( 3872) In others {i} (non-D * 0 D0 ) channels the X(3872) state is seen as a narrow resonance that is why we write the mass spectrum in the i channel in the form where Γ i is the width of the X(3872) → i decay.
The branching ratio of X(3872 where m 0 is the threshold of the i channel. The inverse propagator D X (m) where Γ = ΣΓ i < 1.2 MeV is the total width of the X(3872) decay into all non-(D * 0 D0 +c.c.) channels. When where

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where Our branching ratios satisfy the unitarity Fitting the Belle data [2], we take into account the Belle results [2]: m X = 3871.84MeV = m D * 0 + m D 0 = m + and Γ X(3872) < 1.2 MeV 90%CL, that corresponds to Γ < 1.2 MeV, which controls the width of the X(3872) signal in the π + π − J/ψ(1S ) channel and in every non-(D * 0 D0 + c.c.) channel.The results of our fit are in the Table 1.
Table 1.BR seen is a branching ratio for m ≤ 3891.84MeV, Γ in MeV, g A in GeV.Our approach can serve as the guide in selection of theoretical models for the X(3872) resonance.Indeed, if 3871.68 MeV < M X < 3871.95MeV and Γ X(3872) = Γ < 1.2 MeV then for g 2 A /4π < 0.2 GeV 2 BR(X → D * 0 D0 + c.c. ; m ≤ 3891.84MeV) < 0.3.That is, unknown decays of X(3872) into non-D * 0 D0 states are considerable or dominant.

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The mass of the X(3872) resonance is 50 MeV lower than predictions of the most lucky naive potential models for the mass of the χ c1 (2P) resonance, and the relation between the branching ratios that is interpreted as a strong violation of isotopic symmetry.But the bounding energy is small, B < (1 ÷ 3) MeV.That is, the radius of the molecule is large, r X(3872) > (3 ÷ 5) fm = (3 ÷ 5) • 10 −13 cm.As for the charmonium, its radius is less one fermi, r χ c1 (2P) ≈ 0.5 fm = 0.5 • 10 −13 cm.That is, the molecule volume is 100 ÷ 1000 times as large as the charmonium volume, V X(3872) /V χ c1 (2P) > 100 ÷ 1000.
The enthusiasts of the molecular scenario do not discuss this question with rare exception.Despite the fact that this means a probability of production of a giant molecule in hard processes, at small distances, is suppressed in comparison with a probability of production of heavy a charmonium by a factor ∼ V χ c1 (2P) /V X(3872) .
By the way, Belle Collaboration supports this conclusion [13].Indeed, and As for the problem of the mass shift, Eq. ( 14), the contribution of the D − D * + and D0 D * 0 loops, see Figure 3, into the self energy of the X(3872) resonance, Π X (s), solves it easily.Let us calculate I D * D(s) in Eq. ( 9) with help of a cut-off Λ.
As for BR(X → ωJ/ψ) ∼ BR(X → ρJ/ψ), Eq. ( 15), this could be a result of dynamics.In our scenario the ωJ/ψ state is produced via the three gluons, see Figure 4.As for the ρJ/ψ state, it is produced both via the one photon, see Figure 5  Close to our scenario is an example of the J/ψ → ρη and J/ψ → ωη decays.According to Ref.

Conclusion
We believe that discovery of a significant number of unknown decays of X(3872) into non-D * 0 D0 +c.c.states via two gluons and discovery of the χ b1 (2P) → ρΥ(1S ) decay could decide destiny of X(3872).
Once more, we discuss the scenario where the χ c1 (2P) charmonium sits on the D * 0 D0 threshold but not a mixing of the giant D * D molecule and the compact χ c1 (2P) charmonium.Note that the mixing of such states requests the special justification.That is, it is necessary to show that the transition of the giant molecule into the compact charmonium is considerable at insignificant overlapping of their wave functions.Such a transition ∼ V χ c1 (2P) /V X(3872) and a branching ratio of a decay via such a transition ∼ V χ c1 (2P) /V X(3872) .

Figure 1 .
Figure 1.(a) The Belle data [2] on the invariant π + π − J/ψ(1S ) mass (m) distribution.The solid line is our theoretical one with taking into account the Belle energy resolution.The dotted line is second-order polynomial for the incoherent background.(b) Our undressed theoretical line.

Figure 2 .
Figure 2. The Belle data [5] on the invariant D * 0 D0 + c.c. mass (m) distribution.The solid line is our theoretical one with taking into account the Belle energy resolution.The dotted line is a square root function for the incoherent background.(a) D * 0 → D 0 π 0 .(b) D * 0 → D 0 γ.

Figure 3 .
Figure 3.The contribution of the D0 D * 0 and D − D * + loops into the self energy of the X(3872) resonance.

Figure 4 .
Figure 4.The three-gluon production of the ω and ρ mesons (the ρ meson via the contribution ∼ m u − m d ).All possible permutations of gluons are assumed.
, and via the three gluons (via the contribution ∼ m u − m d ), see Figure4.

Figure 5 .
Figure 5.The one-photon production of the ρ meson.All possible permutations of photon are assumed.
m D .