Probing the nature of Z_c states via the eta_c rho decay

The nature of the so-called XYZ states is a long-standing problem. It has been suggested that such particles may be described as compact four-quark states or loosely bound meson molecules. In the present work we analyze the Z_c(') ->eta_c rho decay using both approaches. Such channel might provide useful insights on the nature of the Z_c('), helping discriminating between the two different models.


Introduction
Since the first observation of the X(3872), made by Belle in 2003 [1], a considerable number of "exotic" particles has been discovered in the heavy quark sector (for a review see [2]). In particular, the finding of charged charmoniumlike resonances is a compelling evidence of the fact that these resonances cannot be fitted into the known frameworks. However, their nature still lacks of a comprehensive theoretical understanding. The most accepted phenomenological models interpret them as compact tetraquark states [3], as loosely bound meson molecules [4,5], as a quarkonium state interacting with light matter via residual strong forces [6], or as hybrid gluonic states [7].
These two states happen to be close to the DD * and D * D * thresholds, respectively, which might suggest a molecular interpretation [11]. However, the mechanism for which these would-be-molecules are pushed slightly above the threshold (by tens of MeV) is still unclear. On the other hand, the constituent diquark-antidiquark model predicted the existence of a 1 +− resonance around 3882 MeV [3,12], and of its radial excitation around 4470 MeV. The confirmation by the LHCb collaboration of a charged Z(4430) state decaying into ψ(2S ) π + [13] and far from open charm thresholds strengthens the whole picture [14]. The Z c (4020) lacked an interpretation in the original model, but this is fixed by a recent "type II" tetraquark paradigm [15].
The Belle collaboration recently started an analysis with the aim of searching for such exotic resonances decaying into the η c charmonium [16]. A possible interesting channel could be Z ( ) c → η c ρ, as it can provide a good way to discriminate between the interpretations mentioned before. In Section 2 we will discuss this decay channel according to the main tetraquark models. In Section 3 we will evaluate the branching fractions according to the molecular hypothesis, by means of a non-relativistic effective theory. Our conclusions are in Section 4.

The compact tetraquark
In the constituent diquark-antidiquark models, the ground state tetraquarks are the eigenstates of the color-spin Hamiltonian: In the type I model, the κ i j coefficents are extracted from ordinary meson and baryon masses, and the X(3872) mass is used as input to fit the diquark mass. The spin-1 eigenstates X and Z in table 1 are identified as the X(3872) and the Z(3900) predicting a mass of 3882 MeV for the latter, with θ −47 • . The Z (1 +− ) state would have a mass of 3755 MeV, but no resonances have been found with this mass and quantum numbers. In this framework, the decay of tetraquarks into charmonia is due to the overlap of the diquark-antidiquark wave function with the charmonium-light meson one. In other words, it is possible to re-arrange the cc quarks into a color singlet pair with a Fierz transformation (see table 1). It is natural to expect the effective coupling of the tetraquark to a charmonium to be proportional to the cc component with the appropriate spin content, which takes into account heavy quark spin conservation. Note that the 0 cc component of Z c (3900) is cot 2 • 30 times larger than the 1 cc one, thus predicting a peak in η c ρ much more intense than the one in the discovery channel J/ψ π.
Recently, it was proposed to neglect the spin-spin interaction outside the diquarks (type II) [15], getting to the identification of the X, Z, Z resonances with the states in table 1, setting θ = 0 • with the predicted masses. With this choice, both Z c s are expected to have similar coupling to J/ψ π and η c ρ.
The kinematics can be taken into account parametrizing the matrix elements as an effective coupling times the most general Lorentz-invariant combination of polarization vectors and momenta with appropriate behaviour under parity and charge conjugation [12], i.e.: In the original models, there is no recipe to take into account orbital or radial excitations of the cc pair: the coupling to each heavy quark spin configuration is somewhat universal, and the differences are only of kinematical nature. Hence, the P-wave decays into h c π are highly suppressed by phase space. Recently, Brodsky et al. [17] proposed that a pair diquark-antidiquark tend to convert all its kinetic energy into potential energy of the color flux tube, and can be considered at rest at a fixed relative distance which satisfies V(r Z ) = M − 2m cq , where m cq = 1.86 GeV is the constituent diquark mass, and V(r Z ) is the spinless Cornell potential. Thus, the larger a charmonium wave function is at r Z , the more favored the decay into such state will be 1 . In other words, we can assume the effective coupling to be proportional to |ψ cc (r Z )| 2 . In table 2 we report the predicted branching fractions to η c ρ. Given the rawness of these 1 Light mesons are not expected to play a role in this mechanism, having a much broader wave function. The argument is someway similar to the hadrocharmonium theory [6].
Kinematics only Dynamics included type I type II type I type II 199 199 4.05 4.05 Table 2: Predicted ratios of branching ratios for Z ( ) c states according to the main tetraquark models. The results in the first column are computed considering the couplings as universal. In the second column, instead, we estimated them using the method of [17]. models, the values reported have to be considered as order-of-magnitude estimates. Anyway, it turns out that at least one of the Z ( ) c is expected to decay into η c ρ with a branching ratio larger than the respective discovery mode.

The molecular picture
In the meson molecule model, the Z ( ) c is interpreted as a D * D( * ) C=−1 resonant state. We will evaluate the branching fraction Z ( ) c → η c ρ by means of the Non-Relativistic Effective Field Theory (NREFT) [5], a framework based on HQET and NRQCD. The interaction vertices between the molecular Z ( ) c , the η c and the D ( * ) mesons have been discussed in [5], and are given by: where · · · stands for a trace over Dirac indices. The details on the definitions of the HQET superfields and our conventions are reported in Appendix A. This kind of HQET Lagrangian could describe the decays of Z c s regardless of the internal structure of such states, and provide model-independent results in terms of unknown effective couplings. In these effective theories the molecular nature is taken into account by forcing these states to couple to their own constituents only, and forbidding all other tree level vertices. The transition to charmonia would thus be allowed by heavy meson loops. The interaction between heavy mesons and charmonia is given by [18,5]: where Ψ is the superfield containing the S -wave charmonia, and χ the superfield of the P-wave charmonia. The coupling g 2 is related to the electronic width of J/ψ, while g 1 can be estimated via QCD sum rules. An estimate gives g 1 −2.09 GeV −1/2 and g 2 1.16 GeV −3/2 [18].
One key ingredient of our analysis is the interaction between heavy mesons and light mesons. The relevant HQET Lagrangian can be found in [19]. We report here the terms involving two heavy mesons and one ρ: where the imposition of Vector Meson Dominance implies β = 0.9, and QCD sum rules give λ = 0.56 GeV −1 [20].

Power counting
The relevant one-loop diagrams are reported in Fig. 1. As shown in [5], the importance of a certain diagram in NREFT can be estimated using a power counting procedure. The heavy meson velocity relevant for the decay/production of some particle X can be evaluated as v X ∼ √ |M X − 2M D | /M D , which in our case gives v Z 0.12 and v η 0.68. Every meson loop counts as v 5 X /(4π) 2 , while the heavy meson propagator scales as 1/v 2 X . In case there is more than one heavy quark external line joined to a loop, we will use an average velocityv. Moreover, depending on the possible presence of derivatives in the interaction vertex, the diagram may also scale either as a power of the external momentum of the ρ meson, q 426.7 MeV, or as an additional power of v X . According to these rules, the combination of the diagrams in Fig. 1 scale as: Powers of M D have been introduced to make everything dimensionless. We now need to evaluate the contribution from higher number of loops since it may happen for them to be as relevant as the one loop contributions [5]. Among the possible processes, a typical contribution is given by the one reported in Fig. 2, where a pion is exchanged inside the loop. Each pion vertex counts as gp π /F π , where g 0.5 is the axial coupling, p π is the pion momentum and F π is the pion decay constant. The power counting of the internal pion propagator cancels with pion momenta of the vertices. The leading contribution to the two loop amplitude is of order: that is an order of magnitude smaller than the one loop diagrams. We have few diagrams of this kind, to be conservative we consider our predictions on the branching ratios as order-of-magnitude estimates.
The case of Z c is very close to the one presented here and we will thus omit it. However, it is worth noting that, since its constituents are D * D * , there is one diagram less at the lowest order (Fig. 3).

Branching fractions
We now have all the tools to evaluate the ratio between the branching fractions BR(Z c → η c ρ)/BR(Z c → J/ψ π) and BR(Z c → η c ρ)/BR(Z c → h c π). The amplitudes for the Z ( ) c → J/ψ(h c ) π processes are taken from [5]. We obtain: Other interesting ratios can be evaluated. We can assume the total width to be saturated by the decays into D ( * ) D * , η c ρ, h c π, J/ψ π, ψ(2S )π, and use the measured widths [22] to extract the coupling of the molecular Z ( ) c to its constituents: |z| (1.28 ± 0.13) GeV −1/2 and |z | (0.67 ± 0.20) GeV −1/2 , where the uncertainties are only due to the experimental uncertainties of the total widths. Note the relevant amount of spin symmetry violation, much larger than in the bottom sector [5]. The ratios we are looking for are therefore: While this work was written up, a paper discussing the decays Z ( ) c → η c ρ in an Effective Lagrangian approach appeared [23], quite in agreement with our results.

Conclusions
In this letter we discuss the possible decays Z ( ) c → η c ρ within the most accepted phenomenological models. Concerning the tetraquark pictures, we used both the type I [3] and type II [15] paradigms, with and without taking into account a recently developed dynamical picture [17]. The outcome of this analysis is that all the considered models predict at least one of Z c and Z c to decay copiously into η c ρ.
On the other hand, within the molecular model, we used NREFT [5] to evaluate the branching fractions. We found that the Z ( ) c → η c ρ channels are strongly suppressed with respect to both Z c → J/ψ π and Z c → h c π, and hence there is a fair probability for these decays not to be seen. Moreover, we showed that the molecular interpretation predicts the branching fraction into h c π to be slightly larger for Z c (3900) than for Z c (4020), at odds with current experimental results [10]. On the other hand, a slight suppression of Z c → J/ψ π w.r.t. Z c → J/ψ π is expected, in agreement with current data [8,12]. Anyway, a higher statistics should make both Z c s visible in both channels.
According to these results, the analysis of the η c ρ final state could shed some light on the long-standing problem about the interpretation of the Z c and Z c states.
where v µ is the heavy meson velocity and a is a flavor index. The fields V µ (V µ ) and P (P) annihilate a (anti-)vector and a (anti-)pseudoscalar respectively according to V µ |V(q, ) = µ √ M V |0 , P |P(q) = √ M P |0 . In the definition of χ µ , we omit the C = +1 P-wave charmonia. For light vector mesons, we follow the anti-hermitian convention of [19], i.e. ρ µ = ig Vρµ / √ 2, where:ρ and g V 5.8 [19,20]. The vector field V µ contains information on pion pairs, which are not of interest for the present case. Starting from the Lagrangians (5) one can obtain the Feynman rules for different processes, which we report in Table A.3. The non-relativistic limit can be obtained by letting v → (1, 0).

Process
Relativistic rule Non-relativistic rule

Appendix B. Amplitudes and loop integral
According to the Feynman rules found in Tab. A.3 and in [5] the non-relativistic one loop amplitude associated with the processes in Fig. 1 is: where λ and η are the spatial polarizations of the Z c and of the ρ respectively and q ρ is momentum of the outgoing ρ. The overall factor of 2 comes from the charge conjugate diagrams. I i (m 1 , m 2 , m 3 ; q) is the loop integral with three propagators: For the case of the Z c we have, instead: