Charged charmonium-like Z+(4430) from rescattering in conventional B decays

In a previous paper we suggested an explanation for the peak designated as $Z(4430)^+$ in the $\psi^\prime\pi^+$ mass spectrum, observed by Belle in ${\bar{\!B}} \to {\psi^\prime} \pi^+ K$ decays, as an effect of $\bar{\!D}{}^{\,*0}D^{\,+} \to \psi^\prime\pi^+$ rescattering in the decays ${\bar{\!B}} \to {D_{s}^{\,\prime -}} {D}$, where the $D_{s}^{\,\prime -}$ is an as-yet unobserved radial excitation of the pseudoscalar ground state $D_{s}^{\,-}$-meson. In this paper, we demonstrate that this hypothesis provides an explanation of the double $Z^+$-like peaking structures, which were studied by LHCb with much higher statistics. While according to our hypothesis, the origin of the peaking structures is purely kinematical, reflecting the presence of a conventional resonance in the hidden intermediate state, the amplitude of the $Z(4430)^+$ peak carries a Breit-Wigner-like complex phase, arising from the intermediate $D_{s}^{\,\prime -}$ resonance. Thus, our hypothesis is entirely consistent with the recent LHCb measurement of the resonant-like amplitude behaviour of the $Z(4430)^+$. We perform a toy fit to the LHCb data, which illustrates that our approach is also consistent with all the observed structure in the LHCb $M({{\psi^\prime}\pi^+})$ spectrum. We suggest a critical test of our hypothesis that can be performed experimentally.

Many XYZ states above the open charm threshold, and decaying into charmonium and light hadron(s) have been observed in the past decade. Their conventional interpretations as charmonium states remain too controversial as their properties, especially their large decay rates into final states without open charm, do not easily match the levels of heretofore unobserved charmonia. Various exotic explanations, such as tetraquarks, molecular states, charmonium hybrids and hadrocharmonium are also not fully embraced by the physics community, as they can not describe the variety of observed states, and all their measured properties, within a single selfconsistent approach.
The first charmonium-like state, the Z(4430) + , which is fully inconsistent with the charmonium spectrum, was observed by Belle [1,2] in 2007 as a peak in the ψ ′ π + mass near M ∼ 4430 MeV in B decays. Since this peak in being interpreted as a real resonance is charged and also contains a cc pair, its minimal quark content, udcc, is necessarily exotic. The existence of the Z(4430) + was cast into doubt by BaBar [3], but the recent Z(4430) + observation by LHCb [4] unambiguously (with the sig-nificance of ∼ 14σ) supports Belle's claim.
Among the exotic explanations of the Z(4430) + nature the most popular are the tetraquark [5], hadrocharmonium [6] and DD * * molecules [7]. There are also non-resonant interpretations such as the "cusp effect" [8] and the initial single pion emission mechanism [9]. However, the latter two seem to be excluded by the recent LHCb analysis of a complex phase rotation in the Z(4430) + region: the extracted Argand diagram for the Z(4430) + amplitude is found to be consistent with resonance behavior. In our previous paper [10] we have suggested another possible explanation of the Z(4430) + peak resulting fromD * 0 D + → ψ ′ π + rescattering in the decaysB → D ′− s D + . If our ad hoc hypothesis is correct, the origin of the peaking structure is purely kinematical, reflecting the presence of a conventional resonance (the D ′− s meson) in the hidden intermediate state. However, our explanation also implies a new interesting underlying phenomena: namely, a nonvanishing rescattering amplitude over a wide range of M(D * 0 D + ). In this Letter, we demonstrate that our approach is fully consistent with all the experimental data, including the recent Z(4430) + phase study by LHCb, as the Z(4430) + phase would then arise from the Breit-Wigner D ′− s amplitude. We show that other structures that are evident in the LHCb ψ ′ π + spectrum can be attributed to similar effects. We also suggest here a critical test of our hypothesis that can be performed by Belle, BaBar and LHCb.
First, we note that in our previous paper [10] we have predicted the quantum numbers of the Z(4430) + to be J P = 1 + based on the simple argument that thē D * 0 D + → ψ ′ π + rescattering should be dominated by S -waves in both the collidingD * 0 D + and also the produced ψ ′ π + systems. This prediction was confirmed by subsequent Belle [11] and LHCb [4] measurements. We also predicted the presence of other structures in the ψ ′ π + spectrum, in particular near M ∼ 4200 MeV, that arises from anotherB → D * ′− s D decay chain. Such a broad peak at M = 4239 MeV is, indeed, observed in the LHCb data, which has been interpreted as another Z + resonance.
We recall that our explanation of the peaking structures in ψ ′ π + spectrum proposed in [10] is based on the effect of rescattering into ψ ′ π + of aD * 0 D + pair produced in the decay chains or of a D * +D0 pair from chains where the D ( * )′− s are radial D ( * ) s meson excitations. We introduce a common notation, (DD) * + , to refer to both D * 0 D + and D * +D0 systems in the reactions (1) and (2), respectively. In theseB decays, two charmed mesons are produced spatially at the same point and relatively slowly fly apart with v/c ≈ 0.3 − 0.5. The amplitude for reconfiguration of the charm quark from one charmed meson and the charm antiquark from another one into ψ ′ , with the simultaneous merging of a light quark-antiquark pair into π + , is determined by the integral of overlapping of two products of wave functions with the same quark content, taking into account color suppression. We assume this amplitude is small but not vanishing, and does not change dramatically within the range of interest (M D + M D * < M (DD) * + 4.8 GeV).
The inclusiveB →D * 0 D + K − andB → D * +D0 K − branching fractions are large: both of order 0.5% [14]. It is natural to assume that they should be saturated by two-body modes with radial D − s and D * − s excitations, because the contribution of orbital D − s excitations to these final states is small [14]. A similar mode, [12]. Other channels and even the D ′− s have not, thus far, been searched for experimentally.
We use the notation Z + to refer to a pseudoparticle with J P = 1 + formed by (DD) * + combination before its decay to ψ ′ π + . As in our previous paper we calculate the amplitude of interest in the on-shell approximation of the triangle diagram ( Fig. 1, Ref. [10]), taking into account the D ( * )′− s Breit-Wigner amplitude. It also includes the D * spin rotation amplitudes, which provide the proper D * helicity in the Z + system, corresponding to S -wave formation of the Z + . Unlike the previous paper, here we have taken into account that different D ( * )′− s Breit-Wigner regions, carrying different complex phases, can result in the same (DD) * + mass (with different D ′− s decay angle). Thus, the total amplitude should be calculated as an integral of all contributions with different phases. This procedure is more rigorous, and the Z + shape is also slightly changed, relative to our previous calculations. The full decay amplitude has the following form in the helicity formalism: where J is the D ( * )′− s spin; θ dec is the decay angle of the D ( * )′− s (the angle between theB andD ( * )0 in the D ( * )′− s rest frame); θ rot is the rotation angle of theD * 0 spin from the D ( * )′− s frame for the reaction (1) or theB frame for the reaction (2) to the Z + frame; θ form is the formation angle of Z + , i.e. the angle between theB and D * in the Z + rest frame. The first Wigner D-function is responsible for the proper angular distribution of the D ( * )′− s decay (in the (2) case only the zero helicity projection is operative). The second function, D 1 0,λ (θ rot ), describes the D * spin rotation from the frame, where it is produced, to the frame where it is absorbed. Finally, the D 1 0,0 (θ form ) corresponds to the proper formation of the spin-1 Z + pseudostate from the vector (D * ) and the pseudoscalar (D). Two variables, M D ( * )′− s and θ dec , fully describe the three-body kinematics, thus M (DD) * + , θ rot and θ form are functions of these two variables.
We have performed calculation of equation (3) [14]); the D ′− s parameters are fixed to M = 2610 MeV and Γ = 100 MeV as in our previous paper [10] (the expected 2S 1 − 2S 3 splitting is (60−100) MeV [13]). The resulting Z + shape for all four chains are shown in Fig. 1; we plot separately the contributions of different D * helicities in the case of decay into the D * ′− s resonance. In particular, the decay (2b) (Pseudo-scalar to Vector Vector) is described by three independent amplitudes.
The phase of the Z + amplitude, arg(A Z + ), from the reaction (1a), which is responsible for the most prominent peak of the Z(4430) + is presented in Fig. 2. In the region around the Z(4430) + the phase turns out to have an inverse behavior compared to the conventional Breit-Wigner definition: it tends to rotate counterclockwise in the Argand diagram. However, experimentally the direction of amplitude rotation can not be determined as there is a two-fold ambiguity (A ↔Ā) in the extraction of the Z + amplitude from the measured |A Z + + A non−Z + | 2 . Thus, our hypothesis is fully consistent with the LHCb Argand diagram.
To illustrate that our hypothesis is plausible we use the LHCb ψ ′ π + mass spectrum with vetoed K * and K * 2 (1430) resonances ( Fig. 4 from [4]) and perform a toy fit to this spectrum ignoring interference between major B → ψ ′ K * ( * ) and rescattering contributions. This is not a fully correct procedure, we thus use it for illustration only. We first estimate the remaining contributions of K − π + < 1.8 GeV 2 borrowed from [4] (black points), orange, green and magenta histograms are contributions from K * (890), K * 2 (1430) and S -wave three body phase space, respectively, expected by LHCb fit. b) Distribution of M 2 ψ ′ π + after incoherent subtraction of contributions from K * (890), K * 2 (1430) and non-resonance three body decays. The black curve represents our fit to the data points. Red, blue and cyan curves represent contributuion of (1a) process, and (2b) with λ = 1 and λ = 0, respectivly. K * (890) K * 2 (1430) and S -wave three body phase space, after selection in the 1.0 < M 2 K − π + < 1.8 GeV 2 interval, using Figs. 3 a) and b) from [4]. The LHCb data points with these three contributions superimposed (the histogram colors correspond to the LHCb notations) are shown in Fig. 3 a). The spectrum in Fig. 3 b) is obtained after a bin-by-bin subtraction of K * ( * ) and nonresonance three body decays. This remaining spectrum we attribute to the rescattering contribution and perform the fit to this spectrum with a sum of contributions from the reactions (1a) and (2b) only, thus with five free parameters. We note that all intermediate B decay channels with various D ( * )′− s states contribute to Z + production coherently with the same universal amplitude of rescattering. The fit results are plotted in Fig. 3 b) with the black solid line, and nicely describe all the features observed in data.
A real test of our hypothesis can be achieved with a 4D-fit performed by Belle, BaBar and LHCb for B → ψ ′ π + K − decays using amplitudes (3) instead of resonance-like Z + 's. Obviously the fitting model with rescattering comprises too many free parameters: at least 7 complex amplitudes to describe all possible contributions as well as the yet-undetermined parameters of the D ′− s resonance. It is important to fix these amplitudes using a study of B →D * 0 D + K − and B → D 0 D * + K − , which is possible at B-factories or LHCb. However, there is an easier way to check our hypothesis experimentally. The Z + -like structures should appear in the distributions of M (D * ⊥D ) + ×cos 2 (θ form ) in either B →D * 0 D + K − or B →D 0 D * + K − decays, or in both. The M (D * ⊥D ) + × cos 2 (θ form ) is the (D * ⊥D ) + combination mass spectrum corrected in each bin for the fraction of the D * transverse component in the (DD) * + rest frame, and also the 1 + formation factor D 2 (θ form ) = cos 2 (θ form ).
In summary, we show thatD * 0 D + → ψ ′ π + rescattering in the decay chainB → D ′− s D + , D ′− s →D * 0 K − can explain the appearance of a peak in the ψ ′ π + mass spectrum inB → ψ ′ π + K − decays around M ∼ 4430 MeV and also correctly describes the quantum numbers and amplitude resonance-like behavior. This approach allows also to describe another peak at M ∼ 4.2 GeV that is observed in LHCb data and has been interpreted as another exotic resonance, as well as a high mass structure at the upper bound of the mass spectrum, which remains still undersaturated by the LHCb fit (with many K * * and two Z(4430) + 's included).
The authors thank Yu. Kalashnikova for useful comments and D. Besson for the paper English correction.