Study of charmoniumlike states by amplitude analyses at Belle

A review of charmoniumlike state studies at Belle which use amplitude analyses is presented, including the Zc(4430)+ , Zc(4200)+ and χc0(3860).


Measurement of the Z c (4430) + quantum numbers
The Z c (4430) + quantum numbers were measured by performing an amplitude analysis of the decaysB 0 → ψ(2S )π + K − [2] (inclusion of charge-conjugate modes is implied hereinafter). The Belle Collaboraton previously observed the Z c (4430) + in a one-dimensional analysis of the same decay [5] and later using a Dalitz analysis [6]. However, the previous Belle analyses either ignored the interference with the contribution of the K * resonances completely or reduced the sensitivity by integrating over the angular variables; consequently, they were not sensitive to the Z c (4430) + quantum numbers. Also, before the Z c (4430) + quantum-number determination, a model-independent analysis had been performed by the BaBar Collaboration [7]; the Z c (4430) + existence had not been confirmed, but the result had not contradicted the Belle observation because of a smaller data sample.
The default model included six K * resonances (K * 0 (800), K * (892), K * (1410), K * 0 (1430), K * 2 (1430), and K * (1680)) and the Z c (4430) + . All possible quantum numbers of the Z c (4430) + with J ≤ 2 were considered: 0 − , 1 + , 1 − , 2 + , 2 − . The preferred Z c (4430) + spin-parity hypothesis was found to be 1 + . Projections of the fit results onto M 2 ψ(2S )π + in the models without the Z c (4430) + and with the Z c (4430) + (J P = 1 + ) are shown in figure 1. The Z c (4430) + parameters were measured to be M = 4485 +22 The points with error bars are the data, the hatched histogram is the background, the blue solid line is the fit with the Z c (4430) + , and the red dashed line is the fit without its contribution. The narrow K * veto is applied: Comparison of the quantum-number hypotheses was performed using Monte-Carlo (MC) pseudoexperiments generated in accordance with the fit results. It was performed for various amplitude models to take the systematic uncertainty into account. The resulting exclusion levels of the 0 − , 1 − , 2 − , and 2 + hypotheses were found to be 3.4σ, 3.7σ, 4.7σ, and 5.1σ, respectively.
The observation of the Z c (4430) + and the results of the measurement its quantum numbers were later confirmed by the LHCb Collaboration [8]. The LHCb Collaboration additionally observed the resonant character of the Z c (4430) + by measuring the dependence of its amplitude on mass. The Z c (4430) + existence was also confirmed by LHCb in a model-independent way [9].

Observation of the Z c (4200) +
The Z c (4200) + was observed in the decaysB 0 → J/ψπ + K − [3]. The default model included 10 K * resonances: . The Z c (4430) + was also included to the default model. Optionally, an additional Z + c resonance was added. All possible quantum numbers of the new Z + c state with J ≤ 2 were considered. Its significance was calculated globally with the expected null-hypothesis distribution of ∆(−2 ln L) determined from MC pseudoexperiments. A new charmoniumlike state Z c (4200) + with J P = 1 + was found with a significance of 6.2σ. Its parameters were measured to be M = 4196 +31  J/ψπ + . The points with error bars are the data, the hatched histogram is the background, the blue solid line is the fit with the Z c (4200) + , and the red dashed line is the fit without its contribution. The region with 1.2 < M 2 K − π + < 2.05 GeV/c 2 is selected. Figure from reference [3].
The comparison of the quantum numbers was performed by using ∆(−2 ln L); it was checked using the pseudoexperiment method. The 0 − , 1 − , 2 − , and 2 + hypothesis were found to be excluded at the levels of 6.1σ, 7.4σ, 4.4σ, and 7.0σ, respectively.
In addition, the mass dependence of the Z c (4200) + amplitude was measured by performing an alternative fit with the Z c (4200) + contribution represented by complex constants in six mass bins. The resulting Argand plot for the H 1 amplitude shows a resonant dependence; it is presented in figure 3. The H 0 amplitudes have much larger relative errors and it is not possible to determine whether they also change resonantly.
The significance of the Z c (4430) + was measured to be 4.0σ. Thus, evidence for Z c (4430) + → J/ψπ + was found. The Z c (4430) + interferes destructively with the K * resonances.
The presence of exotic J/ψπ + contribution in the decaysB 0 → J/ψπ + K − was recently confirmed by the LHCb Collaboration [10]. The observed distribution of the J/ψπ + invariant mass suggests the existence of two resonances with masses of two resonances with M J/ψπ + ≈ 4200 MeV/c 2 and M J/ψπ + ≈ 4600 MeV/c 2 . The analysis performed by LHCb is modelindependent and it consequently does not measure the masses, widths, and quantum numbers of the J/ψπ + resonances. In addition, the Z c (4200) + state may be the same as the Z c (4240) + (or R c0 (4240) + ) state found by LHCb in the decaysB 0 → ψ(2S )π + K − [8] if the Z c (4240) + quantum numbers are J P = 1 + (the preferred hypothesis is 0 − , but it is preferred over 1 + at 1σ level only).
Since the signal yield in the processes e + e − → J/ψD ( * )D( * ) is known to be rather low even with a partial reconstruction, which includes only the J/ψ and one of the D ( * ) mesons, the event selection was optimized to improve the sensitivity. A multivariate analysis was performed for each D decay channel. After that, a global optimization of the selection requirements has been performed, which included the channel-specific definitions of the signal region and the neural-network output requirements.
The default model of the process e + e − → J/ψDD included the nonresonant amplitude (with three different parameterizations) and an additional resonance X * decaying to DD. The X * produced in this process has even spin and positive parity; the C-parity should also be positive. Thus, two quantum-number hypotheses were considered: J PC = 0 ++ and 2 ++ . A new charmoniumlike state X * (3860) was observed with a significance of 6.5σ. The preferred quantum-number hypothesis is J PC = 0 ++ (thus, the new state can be denoted alternatively as the χ c0 (3860) [16]), however, the 2 ++ hypothesis was not excluded: the 0 ++ hypothesis is preferred at 2.5σ level. The χ c0 (3860) parameters were measured to be M = 3862 +26 The new χ c0 (3860) resonance seems to be a better candidate for the χ c0 (2P) charmonium state than the X(3915), because its properties are well matched to expectations for the χ c0 (2P) resonance. Such properties include the mass, the mass difference with the χ c2 (2P), the largewidth DD observation decay mode, the production that is consistent with pure S -wave as is the case for the process e + e − → J/ψχ c0 (1P). In addition, the χ c0 (3860) mass and width agree with the χ c0 (2P) parameters determined from an alternative fit to the Belle and BaBar γγ → DD data [11].