Observation of an isoscalar resonance with exotic $J^{PC}=1^{-+}$ quantum numbers in $J/\psi\rightarrow\gamma\eta\eta'$

Using a sample of (10.09$\pm$0.04)$\times$10$^{9}$ $J/\psi$ events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay $J/\psi \rightarrow \gamma\eta\eta'$ is performed. The first observation of an isoscalar state with exotic quantum numbers $J^{PC}=1^{-+}$, denoted as $\eta_1(1855)$, is reported in the process $J/\psi \rightarrow \gamma\eta_1(1855)$ with $\eta_1(1855)\rightarrow\eta\eta'$. Its mass and width are measured to be (1855$\pm$9$_{-1}^{+6}$)MeV/$c^{2}$ and (188$\pm$18$_{-8}^{+3}$)MeV, respectively, where the first uncertainties are statistical and the second are systematic, and its statistical significance is estimated to be larger than 19$\sigma$.

Using a sample of (10.09±0.04)×10 9 J/ψ events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay J/ψ → γηη ′ is performed. The first observation of an isoscalar state with exotic quantum numbers J P C = 1 −+ , denoted as η1 (1855), is reported in the process J/ψ → γη1(1855) with η1(1855) → ηη ′ . Its mass and width are measured to be (1855±9 +6 −1 ) MeV/c 2 and (188±18 +3 −8 ) MeV, respectively, where the first uncertainties are statistical and the second are systematic, and its statistical significance is estimated to be larger than 19σ. The quark model describes a conventional meson as a bound state of a quark and an antiquark. However, due to the non-Abelian nature of QCD, bound states with gluonic degrees of freedom, such as glueballs and hybrids, are also expected. The clear identification of these QCD exotics would validate and advance our quantitative understanding of QCD. Radiative decays of the J/ψ meson provide a gluon-rich environment and are therefore regarded as one of the most promising hunting grounds for gluonic exciations [1][2][3][4].
In this Letter, a partial wave analysis (PWA) of the process J/ψ → γηη ′ is performed. The first observation of an isoscalar state with exotic quantum numbers J P C = 1 −+ , denoted as η 1 (1855), is reported with high statistical significance in the decay chain J/ψ → γη 1 (1855) → γηη ′ . In addition, a large J/ψ → γf 0 (1500) → γηη ′ component is observed, while J/ψ → γf 0 (1710) → γηη ′ is found to be insignificant. More details are presented in a companion paper [27]. The analysis is based on (10.09±0.04)×10 9 J/ψ events accumulated with the BESIII detector [28] operating at the BEPCII storage ring. A detailed description of the BESIII detector can be found in Ref. [29].
Using the GPUPWA framework [30], a PWA is performed for the selected candidate events from the process J/ψ → γηη ′ with η ′ → γπ + π − and η ′ → ηπ + π − . Quasi-twobody decay amplitudes in the sequential decay processes J/ψ → γX, X → ηη ′ and J/ψ → ηX, X → γη ′ and J/ψ → η ′ X, X → γη are constructed using the covariant tensor amplitudes described in Ref. [31]. The resonance X is parametrized by a relativistic Breit-Wigner (BW) propagator with constant width. The complex coefficients of the amplitudes (relative magnitudes and phases) and resonance parameters (masses and widths) are determined by an unbinned maximum likelihood fit to the data. The joint probability for observing the N events in the data sample is where ǫ(ξ i ) is the detection efficiency, Φ(ξ i ) is the standard element of phase space, and M (ξ i ) = X A X (ξ i ) is the matrix element describing the decay processes from the J/ψ to the final state γηη ′ . A X (ξ i ) is the amplitude corresponding to intermediate resonance X. Details of the likelihood function construction can be found in Ref. [27]. The free parameters are optimized using MINUIT [32]. To account for background, the background contribution to the likelihood function is estimated using η ′ sideband events and is subtracted from the total log-likelihood value [33]. The two decay channels J/ψ → γηη ′ , η ′ → ηπ + π − and J/ψ → γηη ′ , η ′ → γπ + π − are combined by adding their log-likelihood values; they share the same set of masses, widths, relative magnitudes, and phases.
The set of two-body amplitudes used in the PWA is determined in three steps. First, a "PDG-optimized" set of amplitudes is determined. To describe the ηη ′ spectrum, all kinematically allowed resonances with J P C = 0 ++ , 2 ++ , and 4 ++ listed in the PDG [34], Ref. [35], and Ref. [36] are considered. Similarly, to describe the γη (′) spectrum, all resonances listed in the PDG with J P C = 1 +− and 1 −− are considered. All possible combinations of these resonances are evaluated. The statistical significance for each resonance is determined by examining the probability of the change in log-likelihood values when including and excluding this resonance in the fits, where the probability is calculated under the χ 2 distribution hypothesis taking into account the change in the number of degrees of freedom. The masses and widths of the resonances near ηη ′ threshold [f 0 (1500), f 2 (1525), f 2 (1565), and f 2 (1640)] as well as those with small fit fractions (<3%) are always fixed to the PDG [34] values. The mass and width of the f 0 (2330), which corresponds to a clear structure around 2.3 GeVc 2 in the ηη ′ mass spectrum, are free parameters. All other masses and widths are also free parameters in the fit. The final PDG-optimized set of amplitudes is the combination where each included resonance has a statistical significance larger than 5σ.
In the second step, a search is performed for additional resonances with by individually adding each possibility to the PDG-optimized solution and scanning over its mass and width. The significance of each additional resonance at each mass and width is evaluated. The result indicates that a significant 1 −+ contribution (>7σ) is needed around 1.9 GeV in the ηη ′ system. The significances for all other additional contributions are less than 5σ. Therefore, an η 1 state is included in the PWA.
In the third step, a baseline set of amplitudes is determined that includes the η 1 state with its mass and width as free parameters. The statistical significances of all resonances in the PDG-optimized set are reevaluated in the presence of the η 1 state. Resonances with significance less than 5σ are removed. The resulting baseline set of amplitudes contains a significant contribution from an isoscalar state with exotic quantum numbers J P C = 1 −+ , denoted as η 1 (1855). Its statistical significance is 21.4σ, and its mass and width are (1855±9 stat ) MeV/c 2 and (188±18 stat ) MeV, respectively. In addition, the baseline set of amplitudes includes four 0 ++ res- , a nonresonant contribution modeled by a 0 ++ ηη ′ system uniformly distributed in phase space (PHSP), and two 1 +− resonances [h 1 (1415), h 1 (1595)] in the γη system. In addition, a 4 ++ resonance f 4 (2050) with statistical significance 4.6σ is included.
The results of the PWA with the baseline set of amplitudes, including the masses and widths of the resonances, the product branching fractions J/ψ → γX → γηη ′ or J/ψ → η (′) X → γηη ′ , and the statistical significances, are summarized in Table I. The measured masses and widths of the f 0 (2020) and f 2 (2010) are consistent with the PDG [34] average values. The measured mass of the f 0 (2330), which is unestablished in the PDG [34], is consistent with the results of Ref. [35], but our measured width is 79 MeV smaller (3.4σ). Figure 1 shows the invariant mass distributions of M (ηη ′ ), M (γη), and M (γη ′ ) for the data (with background subtracted) and the PWA fit projections. Figure 1 also shows the cosθ η distribution, where θ η is the angle of the η momentum in the ηη ′ (Jocob and Wick) helicity frame [37]. This angle carries information about the spin of the particle decaying to ηη ′ . Figure 2 shows the Dalitz plots for the PWA fit projection, the selected data, and the background estimated from the η ′ sideband.   Various checks are performed to validate the existence of the η 1 (1855). The fits are carried out by assigning all other possible J P C (J ≤ 4) to the η 1 (1855), and the log-likelihoods are worse by at least 235 units (>30σ). To probe the significance of the BW phase motion, the BW parametrization of the η 1 (1855) in the baseline PWA is replaced with an amplitude whose magnitude matches that of a BW function but with constant phase (independent of s). This alternative fit has a log-likelihood 43 units (9.2σ) worse than the baseline fit.
To visualize the agreement between the PWA fit results and data, angular moments as a function of M (ηη ′ ) can be calculated for data (with background subtracted) and the PWA model. For events within a given region of M (ηη ′ ), the cosθ η distribution can be expressed as an expansion in terms of Legendre polynomials. The coefficients, which are called the unnormalized moments of the expansion, characterize the spin of the contributing ηη ′ resonances. The moment for the For data, N k is the number of observed events in the kth bin of M (ηη ′ ) and W i is a weight used to implement background subtraction. For the PWA model, N k is the number of events in a PHSP MC sample and W i is the intensity for each event calculated in the PWA model. Neglecting ηη ′ amplitudes with spin greater than 2, and ignoring the effects of symmetrization and the presence of resonance contributions in the γη and γη ′ subsystems, the moments are related to the spin-0 (S), spin-1 (P ) and spin-2 (D) amplitudes by [38,39] √ 4π Y 0 0 = S 2 0 + P 2 0 + P 2 where φ P and φ D are the phases of the P wave and D wave relative to the S wave. Figure 3 shows the moments computed for the data and the PWA model, using Eq. 2, where good data and PWA consistency can be seen. The need for the η 1 (1855) P wave component is apparent in the Y 0 1 moment [ Fig. 3(b)].
Uncertainty associated with the PWA affects both the branching fraction measurements and the resonance parameters. The sources of uncertainty include the background estimation, the resonance description, the resonance parameters, and additional resonances. The statistical significance of the η 1 (1855) is recalculated in each fit variation.
To estimate the uncertainty due to the background estimation, alternative fits are performed using different background normalization factors and different η ′ sideband regions. The statistical significance of the η 1 (1855) is always above 21.1σ. The changes in the branching fractions and resonance parameters are assigned as systematic uncertainties.
Uncertainty arising from the BW parametrization is estimated by replacing the constant width Γ 0 of the BW for the threshold state f 0 (1500) with a mass-dependent width as described in Ref. [27]. The significance of the η 1 (1855) in this case is 21.8σ.
In the baseline fit, the resonance parameters of the f 0 (1500), f 0 (1810), f 2 (1565), f 4 (2050), h 1 (1415)(γη), and h 1 (1595)(γη) are fixed to PDG [34] average values. An alternative fit is performed where resonance parameters are allowed to vary within 1 standard deviation of the PDG values [34], and the changes in the results are taken as systematic uncertainties. The statistical significance of the η 1 (1855) in this case is 20.6σ.
Uncertainties arising from possible additional resonances are estimated by adding the f 0 (1710), f 2 (2220), f 4 (2300), h 1 (1595)(γη ′ ), and ρ(1900)(γη ′ ), which are the most significant additional resonances for each possible J P C , into the baseline fit individually. The resulting changes in the mea-surements are assigned as systematic uncertainties. In all cases, the significance of the η 1 (1855) remains larger than 19.0σ.
Assuming all of these sources are independent, the total systematic uncertainties are +6 −1 MeV/c 2 and +3 −8 MeV for the mass and width of the η 1 (1855), respectively. For the branching fraction of the η 1 (1855), the total relative systematic uncertainty is determined to be +5.9 −13.1 %. Tables VII and VIII of Ref. [27] summarize the systematic uncertainties.