Threshold corrections of $\chi_{\rm c}(2P)$ and $\chi_{\rm b}(3P)$ states and $J/\psi \rho$ and $J/\psi \omega$ transitions of the $X(3872)$ in a coupled-channel model

We calculate the masses of $\chi_{\rm c}(2P)$ and $\chi_{\rm b}(3P)$ states with threshold corrections in a coupled-channel model. Here, the meson quarkonium core is augmented by higher Fock components due to pair-creation effects. According to our results, we interpret the resonances characterized by very small threshold corrections, like $\chi_{\rm b}(3P)$'s, as almost pure quarkonia, and those states characterized by non-negligible threshold corrections, like the $X(3872)$, as quarkonium cores plus meson-meson components. We also study the $J/\psi \rho$ and $J/\psi \omega$ hidden-flavor strong decays of the $X(3872)$. The decays are calculated as the dissociation of one of these components ($D^0 \bar D^{0*}$) into a $c \bar c$ state ($J/\psi$) plus a light meson ($\rho$ or $\omega$) in a potential model. In particular, our result for the ratio between the $X(3872) \rightarrow J/\psi \omega$ and $J/\psi \rho$ widths (0.6) is compatible with the present experimental data ($0.8\pm0.3$) within the experimental error.

The quark structure of the X(3872), discovered by Belle in B meson decays [2], is still an open puzzle. This resonance is characterized by 1 ++ quantum numbers, a very narrow width, and a mass 50 − 100 MeV lower than quark model (QM) predictions [1]. This is why the pure charmonium interpretation of the X(3872) as a χ c1 (2 3 P 1 ) state is incompatible with the present experimental data. Nevertheless, the charmonium picture provides precise estimations for other observables, suggesting that the wave function of the X(3872) may contain a non-negligible charmonium component [58]. Because of this, a possible solution to the X(3872) puzzle is to describe the meson as a cc core plus continuum effects, The χ b (3P ) system was discovered by ATLAS in 2012 [60] and later confirmed by D0 [61]. The two collaborations gave an estimate of the multiplet mass barycenter, but they did not provide results for the mass splittings between the members of the multiplet. It is worth noting that in Ref. [61] the authors stated: "Further analysis is underway to determine whether this structure is due to the χ b (3P ) system or some exotic bottom-quark state". Indeed, χ b (3P ) resonances are not very far from the first open-bottom decay thresholds thus, in principle, their wave functions may include non-negligible continuum components [36,39,62]. The previous exotic interpretations are contradicted by the recent CMS results for the masses of the χ b1 (3P ) (10513.42 ± 0.41 ± 0.18 MeV) and χ b2 (3P ) (10524.02 ± 0.57 ± 0.18 MeV) [63].
In this paper, we discuss the interpretation of quarkonium-like exotics as QQ cores plus meson-meson components. The starting point is the spectrum. Here, differently from the case of Refs. [35,36], we do not perform a global fit to the quarkonium spectrum. We use an effective model, where the quarkonium core is augmented by higher Fock components due to virtual particles, and extract the net threshold corrections within a multiplet after a certain zero-mode energy is subtracted. The latter is just the smallest self-energy correction (in terms of absolute value) of a multiplet member. According to the previous procedure, we can interpret the resonances with zero net threshold correction, like χ c0 (2P ) or χ b (3P )'s, as pure quarkonia, while the states with non-null net threshold correction, like X(3872), as quarkonium cores plus meson-meson components.
We also study the hadronic J/Ψρ and J/Ψω transitions of the X(3872). The J/Ψρ and J/Ψω amplitudes are computed as the dissociation of the X(3872)'s D 0D0 * component into a cc state (J/Ψ) plus a light meson (ρ or ω) by means of the diagrammatic approach to mesonmeson scattering of Refs. [64,65]. Our results are compared with the existing experimental data [1].

A. Higher Fock components of quarkonium-like mesons
The effects of qq sea pairs are introduced explicitly into the quark model through a QCD-inspired 3 P 0 paircreation mechanism [30,35,36,58,66,67]. The paircreation mechanism is inserted at the quark level and one-loop diagrams are computed by summing over a complete set of accessible SU(N) f ⊗ SU(2) s ground-state mesons, the intermediate states |BC . To leading order in pair creation, the meson wave function is given by (1) Here, |ψ A is made up of a zeroth order quark-antiquark configuration, |A , plus a sum over all the possible higher Fock components, |BC , due to the creation of 3 P 0 quarkantiquark pairs. The sum is extended over a complete set of intermediate meson-meson states |BC , with energies E b,c = M 2 b,c + q 2 ; q and ℓ are the relative radial momentum and orbital angular momentum of B and C, and J is the total angular momentum, with J = J b + J c + ℓ.

B. Threshold mass shifts
In Refs. [35,36], the physical masses of charmonia and bottomonia were computed via where E a is the bare mass of meson A, calculated within the relativized QM of Ref. [68], and a self-energy correction, where the sum is extended over a complete set of intermediate meson-meson states BC.
To get results, Eq. (2) was fitted to the experimental data [1] in a recursive procedure, obtaining a new set of relativized QM parameters [35,36].
Here, we calculate the mass shifts within a certain meson multiplet using a different method. We hypothesize that only the closest thresholds (e.g. 1S1S, 1S1P or 1S2S) can influence the multiplet structure. The other thresholds are supposed to give some kind of global contribution, which can be then subtracted. We also hypothesize that the presence of a certain threshold does not affect the properties of a single resonance, but it influences those of all the multiplet members. Thus, the net effect of a certain threshold on a quarkonium-like meson multiplet is similar to that of a spin-orbit or hyperfine splitting. The physical masses are now calculated via where E a is the bare mass from Ref. [68], Σ(M a ) the self-energy correction of Eq. (3), where we substitute the bare mass in the denominator with the physical one, and ∆ is the smallest self-energy correction (in terms of absolute value) of the multiplet (see Sec. II C). The substitution E a → M a in the denominator of Eq. (3)  We calculate the threshold mass shifts of the χ c (2P ) multiplet members due to a complete set of ground state 1S1S meson loops, like DD, DD * , and so on.
The values of the bare masses are extracted from Ref. [69], those of the physical masses from the PDG [1] with the exception of h c (2P ). As the latter state is still unobserved, for its physical mass we use the same value as the bare one [69]. The self-energy corrections, Σ(M a ), are computed as discussed in Sec. II B, using the model parameter values of Refs. [35,58]; we get: ). Finally, in Table I we report the values of the calculated physical masses of the χ c (2P ) multiplet members.
It is worth noting that: I) Our theoretical predictions agree with the data within the typical error of a QM calculation, of the order of 30 − 50 MeV; II) According to our theoretical results, the χ c0 (2P ) has no continuum component, as expected. Indeed, the relativized QM prediction [68] for the resonance bare mass agrees with the experimental data from PDG [1] within the experimental error; III) Among the χ c (2P ) multiplet members, the χ c1 (2P ) receives the largest contribution from the continuum. This non-negligible continuum contribution, resulting in a large self-energy correction, is necessary to lower the relativized QM prediction, 3.95 GeV, towards the observed value of the mass, 3871.69 MeV [1]; IV) In the χ c (2P ) case, threshold effects break the usual mass pattern of a χ-type multiplet, namely   [70], where the authors used predicted multiplet mass splittings in combination with the measured χ b1 (3P ) mass.
We calculate the mass shifts within the χ b (3P ) multiplet due to 1S1S thresholds. The values of the physical masses are extracted from Refs. [1,70], those of the bare masses from Ref. [70]. The self-energy corrections are calculated with the model parameter values of Ref. [36]; we get: Table I we report the values of the calculated physical masses of the χ b (3P ) multiplet members and compare them with the experimental data.
According to our results: I) The threshold effects are supposed to be negligible and compatible with zero in the χ b (3P ) case. Because of this, we interpret χ b (3P ) states as pure bottomonia; II) Unlike the χ c (2P ) case, the usual mass pattern within a χ-type multiplet, namely M χ0 < M χ1 ≈ M h < M χ2 , is now respected.
Our predictions are in agreement with those of Refs. [68,70] within the error of a QM calculation. On the contrary, the results of Refs. [36,39] and the phenomenological predictions of Ref. [62] would suggest the presence of non-negligible mixing effects between χ b (3P ) states and continuum or molecular-type components. Recent experimental data from CMS [63], M χ b1 (3P ) = 10513.42 ± 0.41 ± 0.18 MeV and M χ b2 (3P ) = 10524.02 ± 0.57 ± 0.18 MeV, agree with the present results for the masses of the χ b (3P ) multiplet.

III. J/Ψρ AND J/Ψω HADRONIC TRANSITIONS OF THE X(3872) DUE TO THRESHOLD EFFECTS
Here, we show our results for the J/Ψρ and J/Ψω hadronic transitions of the X(3872) due to continuumcoupling (or threshold) effects [35,36,58,66].
The transitions can be seen as two-step processes. At a first stage, the cc χ c1 (2 3 P 1 ) meson is "dressed" with open-charm meson-meson continuum components, like DD, DD * , and so on. See Eq. (1) and [58, Table VIII]. The wave function of the X(3872) is thus made up of a χ c1 (2 3 P 1 ) core [indicated as |A in Eqs. (1) and (9)] plus open-charm DD, DD * , ... higher Fock components [|BC in Eqs. (1) and (9)]. At a second stage, the D 0D0 * continuum component of the X(3872) dissociates into a cc meson, J/Ψ, and a light one, ρ or ω, indicated as |D and |E in Eq. (9). The BC → DE scattering (or dissociation) amplitudes are computed with the nonrelativistic potential model formalism of Refs. [64,65] and Sec. III A where, for simplicity, we consider a single harmonic oscillator parameter, α, for the wave functions of the five mesons, A, B, C, D and E.

A. Diagrammatic approach to meson-meson scattering
In this section, we briefly remind the diagrammatic approach to meson-meson scattering of Refs. [64,65], where the low-energy scattering of qq mesons is described by means of a non relativistic potential model. The Hamiltonian of the model is where αs rij − 3β 4 r ij − 8παs 3mimj S i · S j δ(r ij ) , (5b) λ a i and λ a j are Gell-Mann color matrices and r ij is the relative coordinate between the quarks i and j. The matrix elements of the interacting Hamiltonian are given by where P i = P b + P c and P f = P d + P e . There are four diagrams which contribute to the O(H I ) BC → DE scattering amplitude. See [64,]. The matrix elements h fi of a particular diagram can be written as the product of five contributions h fi(particular diagram) = SI flavor I color I spin I space , where S = −1 is the "signature" phase, I flavor , I color and I spin are flavor, color and spin matrix elements, respectively, and I space the spatial matrix element of the potential (5b) [64]. The matrix elements of Eq. (7) are calculated with harmonic oscillator wave functions for the hadrons B, C, D and E, with a single harmonic oscillator parameter, α = 0.5 GeV, and taking P d along theẑ axis. Finally, the cross section of the scattering process BC → DE is given by:

B. Results
Analogously to what is done in the positronium case [71], the decay amplitude is written as where the term in square brackets is the convolution product between a distribution function, describing the probability to find the |BC = D 0D0 * component in the wave function of |A = |X(3872) [58, Table VIII], the term |v b − v c | is the difference between the velocities of the mesons B and C, and the term Γ 2 a 4 [where Γ a is the experimental total width of the X(3872)] in the denominator is necessary for the convergence of the calculation, the mass of the X(3872) being very close to the D 0D0 * threshold [58]. We also have to multiply by |Ψ BC (0)| 2 , which is the squared harmonic oscillator wave function of the BC meson-meson molecule evaluated in the origin, namely its probability density. The harmonic oscillator parameter of Ψ BC (r), α bc , is determined from the condition where r molecule X(3872) ≈ 10 fm is the dimension of the X(3872) in the D 0D0 * molecular interpretation [23]. To get results, the width of Eq. (9) has to be integrated over the Breit-Wigner mass distribution of the ρ and ω mesons, because M J/Ψ + M ρ/ω > M D 0 + MD0 * . Finally, the results of our calculation are compared with the experimental data [1] in Table II. It is worth noting that our result for the ratio between the X(3872) → J/Ψω and X(3872) → J/Ψρ amplitudes, Γ th is compatible with the present experimental data [1,72] Γ exp  ≈ 2 Ref. [74] 1.0 ± 0.3 Ref. [75] 0.42 Ref. [76] 1.27 − 2.24 Ref. [77] 1.4 Present work 0.6 In Ref. [73], the author estimated the ratio between the X(3872) → J/Ψω and X(3872) → J/Ψρ amplitudes, R ω/ρ ≈ 2, in a semi-quantitative way, in which both J/Ψρ and J/Ψω are produced through D andD *exchange between the DD * pair. In Ref. [74], the ratio between the X(3872) → J/Ψρ and X(3872) → J/Ψω widths was calculated using the re-scattering mechanism and effective Lagrangians, based on the chiral symmetry and heavy quark symmetry. The final result for the ratio is R ρ/ω = 1.0 ± 0.3. In Ref. [75], the description of the X(3872) as a D 0D0 * molecule is analyzed within the framework of both one-pion-exchange and one-bosonexchange models and the result for the ratio between the X(3872) → J/Ψ 3π and X(3872) → J/Ψ 2π widths is 0.42. In Ref. [76], the authors calculated the ratio between the X(3872) → J/Ψ 3π and X(3872) → J/Ψ 2π widths, R ω/ρ = 1.27 − 2.24, in a cc-two-meson hybrid model, where the two-meson state consists of the D 0D0 * , D +D− * , J/Ψρ and J/Ψω channels.

IV. CONCLUSION
We calculated the mass shifts within the χ c (2P ) and χ b (3P ) multiplets by introducing the effects of the qq sea pairs into the quark model (QM) formalism through a QCD-inspired 3 P 0 pair-creation mechanism [35,36,66]. The pair-creation mechanism is inserted at the quark level and the one-loop diagrams are computed by summing over a complete set of accessible SU(N) f ⊗ SU(2) s ground-state mesons. In order to calculate the mass shifts within a certain meson multiplet, we made the following hypotheses: only the closest thresholds influence the multiplet structure, while the other thresholds give some kind of global contribution, which can be subtracted; the presence of a certain threshold does not affect the properties of a single resonance, but it influences the behavior of all the multiplet members.
In our interpretation, χ c0 (2P ) is pure charmonium, while χ c1 (2P ) and χ c2 (2P ) are characterized by a relevant threshold component; χ b (3P ) states are pure bottomonia. The possible importance of coupled-channel effects within a certain quarkonium multiplet depends on the energy distance between the meson-meson thresholds and the masses of the multiplet members. For example, in the χ c (2P ) multiplet, the X(3872) − DD * energy difference is of the order of 1 MeV, while in the χ b (3P ) case the closest open-flavor thresholds are a few tens of MeV away.
We also calculated the hidden-flavor X(3872) → J/Ψρ and X(3872) → J/Ψω hadronic decays. The previous transitions can be seen as two-step processes. At a first stage, the cc core of the X(3872) is "dressed" with opencharm meson-meson components, like DD, DD * , and so on [35,36,58,66]. At a second stage, the D 0D0 * continuum component of the X(3872) dissociates into a cc meson, J/Ψ, and a light one, ρ or ω. The dissociation amplitude is computed with the non-relativistic potential model formalism of Refs. [64,65]. It is worth observing that our results for the X(3872) → J/Ψρ and X(3872) → J/Ψω amplitudes are of the same order of magnitude as the experimental data [1] (see Table  II). These are not precisely measured quantities; indeed, what we know is that the total width of the X(3872) is smaller than 1.2 MeV and that the branching fractions of the two decays are larger than 2.6% and 1.9%, re-spectively. Thus, it is more interesting to compare our theoretical result, Eq. (11a), and the corresponding experimental data, Eq. (11b), for the ratio. One can see that our theoretical result is compatible with the experimental data for the ratio within the experimental error.
In conclusion, the results of the present paper and Ref. [58] suggest the interpretation of the X(3872) as the superposition of a cc core and meson-meson continuum (or molecular-type) components, like DD, DD * , D * D * , and so on [35,58]. On the contrary, the present results are compatible with a pure bottomonium interpretation for χ b (3P ) states.