Study of the h c ( 1 1 P 1 ) meson via ψ (2 S ) → π 0 h c decays at BESIII

Using 448 million ψ (2 S ) events, the spin-singlet P -wave charmonium state h c (1 1 P 1 ) is studied via the ψ (2 S ) → π 0 h c decay followed by the h c → γη c transition. The branching fractions are measured to be B Inc ( ψ (2 S ) → π 0 h c ) Using the center of gravity mass of the three χ cJ (1 3 P J ) mesons ( M (c . o . g . ) ), the 1 P hyperﬁne mass splitting is estimated to be ∆ hyp = M ( h c ) − M (c . o . g . ) = (0 . 03 ± 0 . 06 ± 0 . 15) MeV / c 2 , which is consistent with the expectation that the 1 P hyperﬁne splitting is zero at the lowest-order.

Using the center of gravity mass of the three χcJ (1 3 PJ ) mesons (M (c.o.g.)), the 1P hyperfine mass splitting is estimated to be ∆ hyp = M (hc) − M (c.o.g.) = (0.03 ± 0.06 ± 0.15) MeV/c 2 , which is consistent with the expectation that the 1P hyperfine splitting is zero at the lowest-order.

I. INTRODUCTION
Despite extensive studies of the charmonium system since its discovery in 1974 [1][2][3][4], knowledge of the singlet state h c (1 1 P 1 ) is sparse. Only nine decay modes have been observed, with the h c → γη c predominant, with a branching fraction of (50 ± 9)% [5]. And although several measurements of its mass have been performed, more precision is desirable. This would allow better tests of the hypothesis of zero spin-spin hyperfine mass splitting relative to the center-of-gravity mass of the three χ cJ (1 3 P J ) states, defined as [6] M (c.o.g.) = M (χ c0 ) + 3M (χ c1 ) + 5M (χ c2 ) 9 , where the M (χ cJ ) are the masses of the χ cJ (1 3 P J ) states. Only one measurement of the h c (1 1 P 1 ) width exists, provided by the BESIII experiment based on 106 million ψ(2S) events [7]. The dataset used in the current analysis includes that sample.
A more precise knowledge of the h c (1 1 P 1 ) resonance parameters is also important given the recent discoveries in the XY Z (charmonium-like) sector with the h c resonance as an intermediate state. Indeed, BESIII observed the Z ± c (4020) decaying to π ± h c [8] as well as two resonant structures in the cross section for e + e − → π + π − h c [9]. The signal decays of these exotic states all employ the tagged channel h c → γη c , η c → hadrons. Thus, decreasing the uncertainty on the h c branching fractions and resonance parameters is of the utmost importance.
This article reports an improved determination of the h c (1 1 P 1 ) resonance parameters taking advantage of (448.1 ± 2.9) × 10 6 ψ(2S) collected by the BESIII detector in 2009 and 2012 [10]. The mass and width are extracted by studying the ψ(2S) → π 0 h c → (γγ)(γη c ) process following the same approach as Ref. [11]. In particular, this study uses the π 0 recoil mass distribution to reconstruct the h c (1 1 P 1 ) mass, both inclusively (ψ(2S) → π 0 h c ) and by tagging the decay via the electric dipole (E1) transition photon from h c → γη c . For the rest of this article, the two data samples will be referred as inclusive and tagged (Inc or Tag as a subscript), respectively. The choice of using two channels is motivated by the necessity of measuring the branching fractions B Inc (ψ(2S) → π 0 h c ) and B Tag (h c → γη c ), the uncertainties of which are still large [5].

II. BESIII DETECTOR AND DATASETS
The BESIII detector [12] records symmetric e + e − collisions provided by the BEPCII storage ring [13], which operates in the center-of-mass energy range from 2.0 to 4.9 GeV. BESIII has collected large data samples in this energy region [14]. The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel. The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for electrons from Bhabha scattering. The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end cap) region. The time resolution in the TOF barrel region is 68 ps, while in the end cap is 110 ps.
Simulated data samples produced with a geant4based [15] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector and the detector response, are used to determine detection efficiencies and to estimate backgrounds. The simulation models the beam energy spread and initial state radiation (ISR) in the e + e − annihilations with the generator kkmc [16,17]. The inclusive MC sample consists of the production of the ψ(2S) resonance, the ISR production of the J/ψ, and the continuum processes (e + e − → e + e − , e + e − → hadrons, and e + e − → γγ) incorporated in kkmc. The known decay modes are modelled with evtgen [18,19] using branching fractions taken from the Particle Data Group (PDG) [5], and the remaining unknown charmonium decays are modelled with lundcharm [20,21]. Final state radiation (FSR) from charged final state particles is incorporated using the photos [22] package. In the signal MC samples, the ψ(2S) → π 0 h c decay and the E1 transition h c → γη c are both generated by evtgen.

III. EVENT SELECTION AND ANALYSIS PROCEDURE
Although charged tracks are not directly used in the reconstruction, candidate events must have at least two good tracks to suppress background. Good tracks reconstructed in the MDC must pass the following fiducial and production vertex cuts. Only tracks with momenta less than 2.0 GeV/c are considered and they are required to satisfy |cosθ| < 0.93, where θ is the angle between the momentum and beam axis. The distance of closest approach to the interaction point must be less than 10 cm along the beam axis, and less than 1 cm in the transverse plane. Photon candidates are identified using showers in the EMC. The deposited energy of each shower must be more than 25 MeV in the barrel region (|cosθ| < 0.80) and more than 50 MeV in the end cap region (0.86 < |cosθ| < 0.92). To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within (0, 700) ns. To exclude showers that originate from charged tracks, the angle between the position of each shower in the EMC and the closest extrapolated charged track must be greater than 10 • . Finally, to suppress the background contribution from the continuum processes, such as Bhabha scattering, the total energy deposit in the EMC (E Tot EMC ) is required to satisfy 0.6 < E Tot EMC < 3.2 GeV. Candidate events are required to have at least two (three) photon candidates passing the fiducial and energy cuts for the inclusive (tagged) channel. The π 0 candidate is reconstructed via its decay to γγ, with both photon candidates laying in the barrel (|cosθ| < 0.80) and having an energy ≥ 40 MeV. The di-photon pair is accepted as a π 0 candidate if its invariant mass satisfies 120 < M γγ < 145 MeV/c 2 . Multiple π 0 candidates are allowed in one event. A one-constraint (1C) kinematic fit fixing the π 0 mass to its known value [5] is used to improve the energy resolution. Candidates with a fit χ 2 > 200 are rejected. The dominant background for the inclusive sample comes from combining photons from two different π 0 s;the shape of this background is discussed below. The photon coming from the E1 decay (γ Tag ) is expected to peak around 500 MeV. Taking into consideration that Γ ηc ≈ 30 MeV [5], the E1 photon is required to satisfy 465 < Eγ Tag < 535 MeV and must not form a π 0 with any other photons in the event. In the tagged channel, if more than one π 0 is found in the π 0 recoil mass signal region (3.500-3.550 GeV/c 2 ) the π 0 with the minimum 1C fit χ 2 is kept.
The three main background sources are ψ(2S) → J/ψ π + π − , ψ(2S) → J/ψ π 0 π 0 , and ψ(2S) → γχ c0 . The first two are suppressed by requiring all combinations of the π + π − (π 0 π 0 ) recoil mass to be outside the range M J/ψ ± 4 MeV/c 2 (M J/ψ +38 −8 MeV/c 2 ), where M J/ψ is the nominal J/ψ mass [5]. This veto window is optimized based on the figure of merit S √ S+B , where S and B are the numbers of signal and background events estimated from the MC. No suppression of the γχ c0 decay is needed since the photon energy is not in the tagged photon energy range, and thus this decay contributes to the background with a typical combinatorial shape.
The signal efficiencies, Tag = 12.37% and Inc = 14.25%, are estimated for the tagged and inclusive channels, respectively, based on two signal MC samples of 300000 events each. The signal shape is modeled with a resolution function (shown in Fig. 1) based on MC. It is a sum of a Gaussian function and a Crystal Ball function; both contribute to symmetric smearing while the latter also includes the low-energy tail due to π 0 reconstruction. The parameters of this resolution function are obtained from a signal MC dataset where the h c (1 1 P 1 ) width is set to 0. The convolution of the resolution function and a Breit-Wigner distribution is used to describe the signal in the final dataset to extract N Inc and N Tag (i.e., the number of the inclusive and tagged events, respectively). Defining N (ψ(2S)) as the total number of ψ(2S) events, the various branching fractions, B, are obtained as To assess the shape of the irreducible background, a MC dataset (described in Sec. II) of 400 million inclusive ψ(2S) decays with the signal contributions are removed is studied. This shows that a 5 th -order Chebychev polynomial satisfactorily describes the irreducible background in both the channels.
Fits to the distributions of the π 0 recoil mass are performed minimizing the negative log-likelihood on the inclusive and tagged channels separately. The total probability function is constructed from a 5 th -order Chebychev polynomial, to describe the background, added to a convolution of the resolution function and a Breit-Wigner distribution for the signal. In the tagged channel, the parameters of the Chebychev polynomial are allowed to  Fig. (a) shows the resolution function for the inclusive channel, while the one for the tagged chain is presented in Fig. (b).
float, as well as the mass and the width of the h c (1 1 P 1 ) resonance. But for the inclusive channel, the signal shape parameters are fixed to the values found in the tagged channel, while the parameters of the Chebychev polynomial are left floating. Tests on MC samples are used to validate both the model and the fitting procedure; no bias is found.
The fit results are summarized in Table I. A comparison between this analysis and the PDG [5] shows compatibility between the central values with improved precision. Fig. 2 presents the fits for the π 0 recoil mass spectra, RM(π 0 ), for both the inclusive and tagged channels. Separate fits to the 2009 and 2012 datasets yield parameters compatible with each other within 1.5σ. The systematic uncertainties reported in Table I are described in detail in the next section.

IV. SYSTEMATIC UNCERTAINTIES
Sources of systematic uncertainties on the measurement of the h c (1 1 P 1 ) resonance parameters and branching fractions include the background line-shape, the mass ranges for the veto of J/ψ from ψ(2S), the photon energy calibration and reconstruction efficiency, the π 0 reconstruction efficiency, and the luminosity. The contributions from each source are shown in Table II. For each measurement, the total systematic uncertainty corresponds to a quadrature sum of all individual sources, which are discussed in detail next.
Background line-shape. To determine the systematic uncertainties associated to the fit function, a 4 th -order Chebychev polynomial to describe the background behaviour in both the inclusive and tagged channels is tested. The discrepancy with respect to the nominal fit result is taken as a systematic uncertainty. Assuming that part of the background events might be miscounted as signal, a first-order polynomial is added to the resolution function, maintaining a 4 th order Chebychev polynomial to describe the main background contribution. The discrepancy between this method and the nominal one is taken as a systematic uncertainty and summed in quadrature with the result from the first variation.
In both cases, smaller veto ranges give the largest changes with respect to the nominal vetoes; these changes are taken as systematic uncertainties.
Photon reconstruction efficiency. The systematic uncertainties arising from potential inconsistencies of the photon-energy measurements between data and MC simulation are obtained from Ref. [7].
Photon energy calibration. The uncertainty due to the photon energy distribution is obtained from Ref. [7]. Signal shape. This uncertainty includes contributions from the signal line-shape and the 1-C kinematic fit, and is estimated in Ref. [7].
π 0 reconstruction efficiency. The uncertainty on the branching fractions due to the π 0 reconstruction efficiency is estimated in Ref. [24].
Number of ψ(2S) events. This uncertainty is estimated in Ref. [10], and included in the branching fraction uncertainties.
Other sources of the possible systematic uncertainties are studied, but found to be negligible. These include the bin size, the signal region, the trigger efficiency, the numbers of π 0 and charged tracks, the masses and widths of the ψ(2S) and η c , and the sample sizes of the MC simulations.
The measurements of the h c (1 1 P 1 ) resonance parameters and branching fractions are performed with the world's largest ψ(2S) data sample, collected by the BE-SIII experiment [10]. Table I shows the improved precision of this measurement with respect to the PDG values [5], and compatibility within one standard deviation.
These results on ∆ hyp show that, with the foreseen increase in ψ(2S) statistics by the BESIII experiment [14], significant efforts to reduce systematic uncertainties will be necessary.