$X(3872)\to J/\psi\pi^+\pi^-$ and $X(3872)\to J/\psi \pi^+\pi^-\pi^0$ decay widths from QCD sum rules

New spectroscopy from the B factories, the advent of CLEO-c and the BES upgrade renewed the interest in charmonia. Among the new measurements, the state X(3872) has received special attention due to its unexpected properties. Its structure has been studied with different theoretical approaches, most of them being able to reproduce the measured mass. A further test for the theoretical descriptions of the X(3872) is to explain its narrow decay width. In this work we address the decays $X\to J/\psi \pi^+\pi^-\pi^0$ and $X\to J/\psi \pi^+ \pi^-$, using QCD sum rules with the hypothesis that $X$ is a four quark state.

This decay suggests an appreciable transition rate to J/ψ ω and establishes strong isospin violating effects. The measured X(3872) mass can be reproduced in several approaches and it is not yet possible to discriminate between the different structures proposed for this state: tetraquark [4,5], cusp [6], hybrid [7], glueball [8] or DD * bound state [9,10,11,12,13]. The theoretical study of the decay width can help in clarifying this situation. In this work we use the method of QCD sum rules (QCDSR) [14,15,16] to study the hadronic decays of X(3872) given in Eq. (1), considering X as a four-quark state. In recent calculations [17,18,19], the QCDSR approach was used to study the light scalar mesons, the D + sJ (2317) meson and the X(3872) meson considered as four-quark states and a good agreement with the experimental masses was obtained. In particular, in ref. [19] we have considered the X(3872) as the J P C = 1 ++ state with the symmetric spin distribution: [cq] S=1 [cq] S=0 + [cq] S=0 [cq] S=1 . The interpolating field for X q is given by: where a, b, c, ... are colour indices, C is the charge conjugation matrix and q represents the quark u or d.
As pointed out in [4], isospin forbidden decays are possible if X is not a pure isospin state. Pure isospin states are: If the physical states are just X u or X d , the mass eigenstates, maximal isospin violations are possible. Deviations from these two ideal situations are described by a mixing angle between X u and X d [4]: In ref. [4], by considering the ratio of branching ratios given in Eq.(1), they arrived at θ ∼ 20 0 and at Γ(X → J/ψππ) ∼ 5 MeV. However, to arrive at such small decay width they had to make a bold guess about the order of magnitude of the XJ/ψV (where V stands for the ρ or ω vector meson) coupling constant: g XψV = 0.475. In this work we evaluate the XJ/ψV coupling constant directly from the QCD sum rules. For the light scalar mesons, considered as diquark-antidiquark states, the study of their vertex functions using the QCDSR approach was done in ref. [17]. The hadronic couplings determined in ref. [17] are consistent with existing experimental data. In the case of the meson D + sJ (2317) considered as a four-quark state, the QCDSR evaluation of the hadronic coupling constant g DsJ Dsπ [20] gives a partial decay width in the range 0.2 keV ≤ Γ(D + sJ (2317) → D + s π 0 ) ≤ 40 keV. The QCDSR calculation for the vertex, X(3872)J/ψV , centers around the three-point function given by where p = p ′ + q and the interpolating fields are given by: with N = I = 1 for V = ρ, N = 1/3, I = 0 for V = ω and where j q α is given in Eq. (2) and (see Eq.(4)).
Using the above definitions in Eq.(5) we arrive at with where S q ab (x − y) = 0|T [q a (x)q b (y)]|0 is the full quark q propagator. To evaluate the phenomenological side of the sum rule we insert, in Eq.(5), intermediate states for X, J/ψ and V . Using the definitions: we obtain the following relation: where the dots stand for the contribution of all possible excited states, and the form factor, g XψV (q 2 ), is defined by the generalization of the on-mass-shell matrix element, J/ψV |X , for an off-shell V meson: which can be extracted from the effective Lagrangian that describes the coupling between two vector mesons and one axial vector meson [4]: From Eq.(13) we see that we have four independent structures in the phenomenological side. For each one of these structures, i, we can write Π q(phen) i In Eq.(16), ρ cont i (p 2 , q 2 , u), gives the continuum contributions, which can be parametrized as ρ cont [20,21,22], with s 0 and u 0 being the continuum thresholds for X and J/ψ respectively. Taking the limit p 2 = p ′ 2 = −P 2 and performing a single Borel transformation to P 2 → M 2 , we get (Q 2 = −q 2 ): where B i and ρ cc i (u, Q 2 ) stand for the pole-continuum transitions and pure continuum contributions. For simplicity, one assumes that the pure continuum contribution to the spectral density, ρ cc i (u, Q 2 ), is given by the result obtained in the OPE side for the structure i. Asymptotic freedom ensures this equivalence for sufficiently large u. Therefore, one uses the Ansatz: (17), B i is a parameter which, together with the form factor, g XψV (Q 2 ), has to be determined from the sum rule. In the OPE side we work at leading order and consider the condensates up to dimension five, as shown in Fig. 1. To keep the charm quark mass finite, we use the momentum-space expression for the charm quark propagator. We calculate the light quark part of the correlation function in the coordinate-space, which is then Fourier transformed to the momentum space in D dimensions. The resulting light-quark part is combined with the charm-quark part before it is dimensionally regularized at D = 4. For each structure i, we can write the Borel transform of the correlation function in the OPE side in terms of a dispersion relation: where the spectral density, ρ q(OP E) i , is given by the imaginary part of the correlation function. The perturbative term (diagram in Fig. 1(a)) contributes only to the structures ǫ αµνσ p ′ σ and ǫ αµσγ p ′ σ q γ q ν , while the quark condensate and mixed condensate (diagrams (b) to (e) in Fig. 1) contribute to the structures ǫ αµνσ q σ and ǫ ανσγ p ′ σ q γ p ′ µ . Therefore, to get more terms contributing in the OPE side we have two options for the structures: ǫ αµνσ q σ and ǫ ανσγ p ′ σ q γ p ′ µ . In order to test the dependence of the results with the chosen structure, we will work with these two structures.
Transferring the pure continuum contribution to the OPE side we get for the structure ǫ ανσγ p ′ σ q γ p ′ µ (which we call structure 1): and for the structure ǫ αµνσ q σ (which we call structure 2) we get: In Eqs. (19) and (20) we have used the relation qgσ.Gq = m 2 0 qq . Making use of Eqs. (9) and (10), and working at the SU(2) limit, i.e., considering the quarks u and d degenerate, we arrive at three sum rules for each structure, that can be written in the general expression: where and Since from Eq. (21) we see that the OPE side of the sum rule determines only one value for C XV for each structure (for a fixed value of Q 2 ), we arrive at the following relations between the form factors:  [19]. We also use three different values for s 0 = (3.872 + ∆s 0 ) 2 GeV 2 : ∆s 0 = 0.4 GeV, ∆s 0 = 0.5 GeV and ∆s 0 = 0.6 GeV. For u 0 we use u 0 = (m ψ + 0.5) 2 GeV 2 . The meson-current coupling, λ q , defined in Eq. (12), can be determined from the two-point sum rule [19]. In Table I we give the results obtained from ref. [19] for three different values of s 0 . We start with the structure 1. In Fig. 2 we show, through the circles, the right-hand side (RHS) of Eq.(21) for Q 2 = 3 GeV 2 , as a function of the Borel mass.
From Eq. (23), we see that all form factors are related with the function A XV (Q 2 ). Since the coupling constant is defined as the value of the form factor at the meson pole: Q 2 = −m 2 V , to determine the coupling constant we have to extrapolate the QCDSR results to a Q 2 region where the sum rules are no longer valid (since the QCDSR results are valid in the deep Euclidian region). To do that we parametrize the QCDSR results through a analytical form. In Fig. 3 we also show that the Q 2 dependence of A XV (Q 2 ) can be well reproduced by the monopole parametrization (solid line):  Doing the same kind of analysis for the other values of the continuum threshold we show, in Table II, the monople parametrizations of the QCDSR results, as well as their values at the off-shell meson pole.  The Q 2 behaviour of A XV (Q 2 ) can also be well represented by a monopole form in the case of structure 2, with a precision similar to the one shown in Fig. 3. In Table III we give the monople parametrizations of the QCDSR results for the structure 2, as well as their values at the off-shell meson pole.

1.28
Comparing the results in Tables II and III we see that, althought the results from the structure 2 are somewhat smaller than the results from the structure 1, they are still compatible with each other. We will use these differences to estimate the uncertainties in our results.
From Eq. (23) we see that, in the case of the meson ω, there is no mixing angle dependence in the relation between A XV and g Xψω . Therefore we can use the results in Tables II and III to directly estimate the XJ/ψω coupling constant. We get which is much bigger than the guess made in ref. [4]: g XψV = 0.475. Having the coupling constant and the relations in Eqs. (23) and (24), we can estimate the decay widths of the processes X → J/ψ π + π − π 0 and X → J/ψ π + π − by supposing that the 2π and 3π decays are dominated by the ρ and ω vector mesons respectively. In the narrow width approximation we have: with n = 2, 3 for V = ρ, ω. In Eq.(27), s is the invariant mass-squared of the pions, Γ V and B V →nπ are, respectively, the total decay width and the branching ratio of the V → nπ decay. The decay momentum p(s) is given by with λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc.
This procedure may appear somewhat unjustified. However, we do believe that there should be a particular choice of the interpolating field, which represents a genuine four-quark state, for which CD diagrams vanish. From our calculation we find out that the interpolating field in Eq.(2) has a component similar to a J/ψ − V molecule.
To summarize: we have presented a QCD sum rule study of the three-point functions of the hadronic decays of X(3872) meson, considered as a diquark antidiquark four quark state. Supposing that the physical state is a mixture between the isospin eigenstates, we find that the QCD sum rules result for the mixing angle is compatible with the result found in [4]. However, we get a partial decay width much bigger than the experimental total decay width. Therefore, we conclude that our particular choice of the interpolating field has a J/ψV molecule component, and is not the most appropriate candidate to explain the very small width of the meson X(3872). Further studies, using different interpolating fields, are necessary for a better understanding of the structure of the meson X(3872).