Determination of the resonant parameters of chi_c0(3915) with global fit

The chi_c0(3915) was observed firstly by the Belle experiment in the omega J/psi invariant mass spectrum in the process B ->K omegaJ/psi, and then confirmed by the Babar experiment. The two experiments reported the resonant parameters of this particle in the both processes gammagamma ->omegaJ/psi and B ->K omega J/psi by assuming chi_c0(3915) as a S-wave Breit-Wigner resonance. We perform a global fit to the distributions of invariant mass of omega J/psi measured by the Belle and Babar experiments, and incorporate the measurement by the LHCb experiment additionally to extract the mass and width of chi_c0(3915). We obtain M = 3921.0 +/- 0.9 MeV/c2 and Gamma = 17.9 +/- 2.4 MeV which are consistent with the values on PDG within one standard deviation but with improved precision.


I. INTRODUCTION
Since the first charmonium-like state X(3872) was discovered by Belle experiment via the process e + e − → γ ISR π + π − J/ψ, a new era of the study of charmonium-like states was triggered [1].A series of mesons comprised of charmed and anticharmed quark pairs, such as the Y(4260), X(3915), and Zc(3900), were subsequently discovered, and most of them were confirmed by different experiments [2][3][4][5][6][7][8][9][10][11].The understanding of the nature of these particles has turned out to be quite a challenge.Many of them have properties that are quite different from the conventional charmonium, e.g. a low open-charm decay rate or the absence of hadronic transitions to other charmonium states.To explain these anomalous features, many models have been proposed by theorists, including charmonium molecule mixed states [12,13], ccg hybrid states [14], and tetra-quarks [15].In addition, the production rates for some of these charmonium-like states are quite low in most experiments compared to that of the conventional charmonium states, which limits the measurement precision of the resonant parameters and the determination of the corresponding quantum numbers.This makes the interpretation of these particles difficult.Particle Data Group (PDG) [16] renamed the X(3915) and X(3872) as χ c0 (3915) and χ c1 (3872), respectively, according to their spin-parities.We just follow the PDG naming convention in the article.
As a member of charmonium-like family of states, the χ c0 (3915) was first observed by the Belle experiment in the process B → KJ/ψω in a data sample containing 275 × 10 6 B B pairs [9].The mass and width were determined to be 3943 ± 11 ± 13 MeV/c 2 and 87 ± 22 ± 26 MeV with the assumption that the χ c0 (3915) is an S-wave Breit-Wigner (BW) resonance.The particle was confirmed by BaBar experiment in the same decay mode with a 383 × 10 6 B B event data sample [10], their reported mass and width are 3914.6+3.8 −3.4 ± 2.0 MeV/c 2 and 34 +12 −8 ± 5 MeV.BaBar experiment subsequently updated their measurements with a lager data sample of 467 × 10 6 B B events and looser M (π + π − π 0 ) requirement that revealed a X(3872) signal as well [11]; the updated mass and width are 3919.1 +3.8 −3.4 ± 2.0 MeV/c 2 and 31 +10 −8 ± 5 MeV.In addition, the χ c0 (3915) was also observed in the two-photon collision process γγ → ωJ/ψ by both the BaBar and Belle experiments [7,8].Their measured masses and widths are listed in Table I.BaBar performed a spin-parity measurement in their analysis, obtaining the quantum number to be J P = 0 + and identifying the χ c0 (3915) as the χ c0 (2P ) charmonium state.However, this assignment was disputed because of the large rate for the χ c0 (3915) → ωJ/ψ decay and the absence of χ c0 (3915) → D D decays [17,18].Moreover, the mass difference between the χ c2 (2P ) and χ c0 (3915) is only about 10 MeV, which is too small for the fine splitting of P -wave charmonia [17].In 2020, LHCb experiment made an amplitude analysis of the B + → D + D − K + decay [19], and reported that a spin-0 resonance is needed to describe the data well.They determined its mass and width to be 2923.8± 1.5 ± 0.4 MeV/c 2 and 17.4 ± 5.1 ± 0.8 MeV, respectively.
In this article, we perform a simultaneous χ 2 fit to the distributions of invariant mass of ωJ/ψ in the processes γγ → ωJ/ψ measured by BaBar [denoted as (a)], γγ → ωJ/ψ by Belle [(b)], B 0 → ωJ/ψK 0 by BaBar [(c)], B + → ωJ/ψK + [(d)] by BaBar, and B → ωJ/ψK by Belle [(e)] to extract the mass and width of χ c0 (3915).The distributions of M (ωJ/ψ) for these processes are shown in Fig. 1.Furthermore, LHCb's results are taken into account as an additional constraint in the χ 2 calculation.Compared to the values on PDG, which also gives the χ c0 (3915) mass and width by combining the measurements from these experiments, we use more detailed information of the ωJ/ψ invariant mass spectrum, which is expected to provide a result with higher precision.

II. SIMULTANEOUS χ 2 FIT
In the simultaneous χ 2 fit, the functions used to fit the M (ωJ/ψ) spectra in Fig. 1 are similar to those applied in their corresponding publications.The fit function is comprised of signal and background components.The χ c0 (3915) signal shape is described by an S-wave BW function convoluted with the detector resolutions which are also from the publications.The BW is , where M is the peak mass, p * is the momentum of J/ψ momentum in the rest frame of a ωJ/ψ system, and p 0 = p * when m = M [20].M and Γ are common parameters for the five measurements that are allowed to float in the fit.The contributions of the χ c1 (3872) are needed for the distributions (c) and (d) as shown in Fig. 1.The signal shape of the χ c1 (3872) is described with a Gaussian function with a fixed deviation of 6.7 MeV/c 2 but with a free mean value.The detector resolution of γγ → ωJ/ψ for Belle is described by a double Gaussian function, one Gaussian has the mean and deviation values of 4.5 MeV and 0 MeV, respectively, with a coefficient of 0.59, and the other one has a mean and deviation of 16 MeV and -4.0 MeV with a cofficient of 0.41 [7].
Conversely, the resolution for the same process by BaBar is described by a single Gaussian with a deviation of 5.7 MeV and a mean value of zero.Similarly, the resolution for the measurement B → KωJ/ψ by BaBar (Belle) is described by a single Gaussian function with a deviation of 6.7 MeV (6 MeV) and a mean value of zero.All these resolution details are from the corresponding publications.The non-resonance background shapes for the measurement (a) and (b) are described as p * (m)exp(−δp * (m)), where δ is a parameter that is allowed to float in the fit, and m = m(J/ψω) [8].The background shapes of the measurement (c) and (d) are described with a Gaussian function with parameters that float in the fit [11].The shape of the background in the measurement (e) is described with the threshold function of the form p * (m), which is the J/ψ momentum in the rest frame of a ωJ/ψ system.The χ 2 is defined as where i takes values from 1 to 5, corresponding to the measurements from (a) to (e), x ij and σ ij are the observed signals and corresponding errors in each bin as shown in Fig. 1, µ ij is the expected value in each bin calcluated with the fitting shape mentioned above, and N i is the number of bins in each measurement.The numbers of events in many bins for the measurements (a), (b), and (e) are very small, so we reset the bin width to ensure that there is at least nine events in each bin to make a meaningful calculation of their contribution to the χ 2 value.The numbers of bins in the measurement , are from the measurements of the LHCb experiment [19], where M h and Γ h are the measured mass and width, and σ M h and σ Γ h the corresponding statistical uncertainties.
By minimizing the χ 2 with minuit [21], we obtain the fit result with M = 3920.9± 0.8 MeV/c 2 and Γ = 18.2 ± 2.2 MeV.The goodness of fit is χ 2 /ndf = 89.1/66where ndf is the number of degrees of freedom in the fit.The fit results are also shown in Fig. 1.
The total systematic uncertainty of the mass and width, σ sum , is obtained with the formula 1 , where the i has a value from 1 to 5 corresponding to each measurement listed in Table I, and σ i is the systematic uncertainty of the i-th measurement.Combining with the statistical uncertainties from our fit, we finally get the results of M = 3920.9± 0.9 MeV/c 2 and Γ = 18.2 ± 2.4 MeV.In summary, we determine the resonant parameters of χ c0 (3915) by simultaneously fitting the measurements provided by the Belle, BaBar, and LHCb experiments.The mass and width are determined to be M = 3920.9±0.9MeV/c 2 and Γ = 18.2 ± 2.4 MeV, respectively.which are consistent with the average values in PDG as listed in Table I within one standard deviation but with improved precision.
(a) and (b) are N 1 =6 and N 2 =8.For the measurements (c) and (d), the bin width in the mass region of 3.8425 to 3.9925 GeV/c 2 is 10 MeV/c 2 , and 50 MeV/c 2 in the region beyond 3.9925 GeV/c 2 as shown in Fig. 1.The numbers of bins N 3 and N 4 are both 31.For the measurement (e), N 5 =9.The last two components of the χ 2 formula, M h −M σM h and Γ h −Γ σΓ h