Observation and study of the decay $J/\psi\rightarrow\phi\eta\eta'$

TL;DR

This paper reports the observation of the decay $J/ar{psi} ightarrowar{\phi}ar{\eta}ar{\eta'}$ and evidence for a new structure, $X$, in the $ar{\phi}ar{\eta'}$ mass spectrum, with detailed measurements of its properties.

Contribution

First observation of the decay $J/ar{psi} ightarrowar{\phi}ar{\eta}ar{\eta'}$ and evidence for a new resonance $X$ in the $ar{\phi}ar{\eta'}$ spectrum with measured mass, width, and quantum number hypotheses.

Findings

Branching fraction of $J/ar{psi} ightarrowar{\phi}ar{\eta}ar{\eta'}$ is $(2.32\pm0.06\pm0.16) imes10^{-4}$.

Evidence for a structure $X$ with mass around 2.0-2.1 GeV/$c^2$ and width about 130-177 MeV.

Significance of the $X$ structure is 4.4$\sigma$ for $J^P=1^-$ and 3.8$\sigma$ for $J^P=1^+$.

Abstract

We report the observation and study of the decay $J/\psi\rightarrow\phi\eta\eta'$ using $1.3\times{10^9}$ $J/\psi$ events collected with the BESIII detector. Its branching fraction, including all possible intermediate states, is measured to be $(2.32\pm0.06\pm0.16)\times{10^{-4}}$. We also report evidence for a structure, denoted as $X$, in the $\phi\eta'$ mass spectrum in the $2.0-2.1$ GeV/$c^2$ region. Using two decay modes of the $\eta'$ meson ($\gamma\pi^+\pi^-$ and $\eta\pi^+\pi^-$), a simultaneous fit to the $\phi\eta'$ mass spectra is performed. Assuming the quantum numbers of the $X$ to be $J^P = 1^-$, its significance is found to be 4.4$\sigma$, with a mass and width of $(2002.1 \pm 27.5 \pm 21.4)$ MeV/$c^2$ and $(129 \pm 17 \pm 9)$ MeV, respectively, and a product branching fraction $\mathcal{B}(J/\psi\rightarrow\eta{}X)\times{}\mathcal{B}(X\rightarrow\phi\eta')=(9.8 \pm 1.2 \pm 1.7)\times10^{-5}$. Alternatively, assuming $J^P = 1^+$, the significance is 3.8$\sigma$, with a mass and width of $(2062.8 \pm 13.1 \pm 7.2)$ MeV/$c^2$ and $(177 \pm 36 \pm 35)$ MeV, respectively, and a product branching fraction $\mathcal{B}(J/\psi\rightarrow\eta{}X)\times{}\mathcal{B}(X\rightarrow\phi\eta')=(9.6 \pm 1.4 \pm 2.0)\times10^{-5}$. The angular distribution of $J/\psi\rightarrow\eta{}X$ is studied and the two $J^P$ assumptions of the $X$ cannot be clearly distinguished due to the limited statistics. In all measurements the first uncertainties are statistical and the second systematic.

Tables

Table 1. Table 1: Systematic uncertainties in ℬ ​ ( J / ψ → ϕ ​ η ​ η ′ ) ℬ → 𝐽 𝜓 italic-ϕ 𝜂 superscript 𝜂 ′ \mathcal{B}(J/\psi\rightarrow\phi\eta\eta^{\prime}) . The correlated sources between the two η ′ superscript 𝜂 ′ \eta^{\prime} decay modes are denoted with asterisks.

Sources	Mode I ($\%$)	Mode II ($\%$)
MDC tracking*	4.0	4.0
PID*	4.0	4.0
Photon detection*	1.8	2.4
Kinematic fit	2.5	1.1
Sideband regions	0.1	0.3
Mass window for $\eta$	0.5	0.7
Mass window for $\varphi$	0.9	1.0
Mass window for ${\eta }^{\prime }$	0.7	0.6
MC statistics	0.6	0.9
Branching fractions*	2.1	2.1
Number of $J/\psi$*	0.6	0.6
2D binning	3.9	2.2
Total	8.0	7.2

Table 2. Table 2: Measured N sig subscript 𝑁 sig N_{\rm sig} and ℬ ​ ( J / ψ → η ​ X ) × ℬ ​ ( X → ϕ ​ η ′ ) ℬ → 𝐽 𝜓 𝜂 𝑋 ℬ → 𝑋 italic-ϕ superscript 𝜂 ′ \mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime}) values for the 1 − superscript 1 1^{-} and 1 + superscript 1 1^{+} assumptions.

$J^P$	$N_{\mathrm{sig}}$	$ℬ\left(J/\psi \to \eta X\right)\times ℬ\left(X\to \varphi {\eta }^{\prime }\right)$
$1^{-}$	$658\pm 77$	$\left(9.8\pm 1.2\right)\times {10}^{-5}$
$1^{+}$	$642\pm 88$	$\left(9.6\pm 1.4\right)\times {10}^{-5}$

Table 3. Table 3: Systematic uncertainties in the mass and width of the structure, as well as ℬ ​ ( J / ψ → η ​ X ) × ℬ ​ ( X → ϕ ​ η ′ ) ℬ → 𝐽 𝜓 𝜂 𝑋 ℬ → 𝑋 italic-ϕ superscript 𝜂 ′ \mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime}) (denoted as ℬ X subscript ℬ 𝑋 \mathcal{B}_{X} in this table) for the 1 − superscript 1 1^{-} assumption.

	Mass	Width
Sources	(MeV/$c^2$)	($\mathrm{MeV}$)	${ℬ}_X$ ($\%$)
Signal parametrization	9.1	2	2.9
$f_0\left(1500\right)$	9.5	5	11.6
PHSP assumption	15.2	5	9.8
Fitting range	6.3	3	3.1
$M\left({\pi }^{+}{\pi }^{-}\right)$ requirement	1.8	2	0
Extra structures	2.5	0	1.1
Momentum calibration	0.7	-	-
Sideband regions	0.9	2	0.4
MDC tracking	-	-	4.0
PID	-	-	4.0
Photon detection	-	-	2.4
Kinematic fit	-	-	3.0
Mass window for $\eta$	-	-	0.7
Mass window for $\varphi$	-	-	1.0
Mass window for ${\eta }^{\prime }$	-	-	0.7
MC statistics	-	-	0.9
Branching fractions	-	-	2.1
Number of $J/\psi$	-	-	0.6
Total	21.4	9	17.5

Table 4. Table 4: Systematic uncertainties in the mass and width of the structure, as well as ℬ ​ ( J / ψ → η ​ X ) × ℬ ​ ( X → ϕ ​ η ′ ) ℬ → 𝐽 𝜓 𝜂 𝑋 ℬ → 𝑋 italic-ϕ superscript 𝜂 ′ \mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime}) (denoted as ℬ X subscript ℬ 𝑋 \mathcal{B}_{X} in this table) for the 1 + superscript 1 1^{+} assumption.

	Mass	Width
Sources	(MeV/$c^2$)	($\mathrm{MeV}$)	${ℬ}_X$ ($\%$)
Signal parametrization	2.4	0	0.4
$f_0\left(1500\right)$	2.6	19	13.4
PHSP assumption	5.9	28	12.4
Fitting range	1.1	6	3.2
$M\left({\pi }^{+}{\pi }^{-}\right)$ requirement	1.3	1	0
Extra structures	0.7	1	1.9
Momentum calibration	0.7	-	-
Sideband regions	0.7	1	0.4
MDC tracking	-	-	4.0
PID	-	-	4.0
Photon detection	-	-	2.4
Kinematic fit	-	-	2.3
Mass window for $\eta$	-	-	0.7
Mass window for $\varphi$	-	-	1.0
Mass window for ${\eta }^{\prime }$	-	-	0.7
MC statistics	-	-	0.9
Branching fractions	-	-	2.1
Number of $J/\psi$	-	-	0.6
Total	7.2	35	20.0

Equations

$$\begin{array}{r}\mathcal{B}=\frac{N_{\mathrm{corr}}}{N_{J/\psi }\mathcal{B}(\eta \to 2\gamma )\mathcal{B}(\varphi \to K^{+}{K}^{-}){\mathcal{B}}_{{\eta }^{\prime }}},\end{array}$$

$$\begin{array}{r}(|\frac{1}{m^2-M^2+iM\Gamma /c^2}{|}^2\times (pq{)}^3\times ϵ)\otimes R,\end{array}$$

$$\begin{array}{r}\mathcal{B}(J/\psi \to \eta X)\times \mathcal{B}(X\to \varphi {\eta }^{\prime })\\ =\frac{N_{\mathrm{sig}}}{N_{J/\psi }\mathcal{B}(\eta \to 2\gamma )\mathcal{B}(\varphi \to K^{+}{K}^{-})\overline{ϵ}},\end{array}$$

Full text

I INTRODUCTION

Exotic hadrons, e.g., glueballs, hybrid states and multiquark states, are allowed in the framework of Quantum Chromodynamics (QCD), but no conclusive evidence for them has yet been found in the light hadron sector. The decay $J/\psi\rightarrow{VPP}$ (where $V$ denotes vector and $P$ denotes pseudoscalar) is an ideal probe to study light hadron spectroscopy and to search for new hadrons. There have been theoretical c_introduction_the_1 ; c_introduction_the_2 ; c_introduction_the_3 ; c_introduction_the_4 and experimental c_introduction_exp_MARKIII ; c_introduction_exp_DM2 ; c_introduction_exp_omega_pipi ; c_introduction_exp_omega_kk ; c_introduction_exp_phi_pipi_kk ; c_introduction_exp_kstr_k_pi studies performed, which have mainly been focused on the $V$ recoil system to search for exotic hadrons. The $P$ recoil system, on the other hand, could also be utilized to do a similar study. For example, the $Y(2175)$, denoted as $\phi(2170)$ by the Particle Data Group (PDG) c_introduction_pdg , was confirmed in the process $J/\psi\rightarrow\eta{Y(2175)},Y(2175)\rightarrow\phi{f_{0}(980)}$ by BESII c_introduction_bes and BESIII c_introduction_besiii . Searching for its decay to the $\phi\eta^{\prime}$ state provides valuable input for understanding its nature c_introduction_ssg . The decay $J/\psi\rightarrow\phi\eta\eta^{\prime}$ has not been studied before, and could aid in our understanding of $J/\psi$ decay mechanisms and offers an opportunity to study possible intermediate states.

In this article, we report the observation and study of the decay $J/\psi\rightarrow\phi\eta\eta^{\prime}$ using $(1310.6\pm 7.0)\times{10^{6}}$ $J/\psi$ events c_data_samples collected with the BESIII detector. Its branching fraction, including all possible intermediate states, is measured. We also report evidence for a structure denoted as $X$ in the $\phi\eta^{\prime}$ mass spectrum in the $2.0-2.1$ GeV/$c^{2}$ region. The mass and width of this structure, as well as the product branching fraction $\mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime})$, are measured. The $\phi$ meson is reconstructed through its $K^{+}K^{-}$ decay mode, $\eta$ through $\gamma\gamma$, and $\eta^{\prime}$ through both $\gamma\pi^{+}\pi^{-}$ and $\eta\pi^{+}\pi^{-}$ (with the $\eta\rightarrow\gamma\gamma$), denoted as mode I and mode II, respectively.

II BESIII EXPERIMENT AND MONTE CARLO SIMULATION

The BESIII detector is a magnetic spectrometer c_besiii_detector located at the Beijing Electron Position Collider (BEPCII) c_bepcii . The cylindrical core of the BESIII detector 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 (0.9 T in 2012) magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93$\%$ over 4$\pi$ solid angle. 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 of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps.

Simulated data samples produced with a geant4-based c_MC_GEANT4 Monte Carlo (MC) package, including the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate the backgrounds. The simulation of the $e^{+}e^{-}$ collisions includes the beam energy spread and initial state radiation (ISR) and is modeled using the generator kkmc c_MC_KKMC . The inclusive MC sample consists of the production of the $J/\psi$ resonance and the continuum processes incorporated in kkmc c_MC_KKMC . The known decay modes are modeled with evtgen c_MC_BesEvtGen using branching fractions taken from the PDG c_introduction_pdg , and the remaining unknown decays from the charmonium states with lundcharm c_MC_LUNDCHARM . Final state radiation (FSR) from charged final state particles is incorporated with the photos package c_MC_PHOTOS .

III Event selection and data analysis

Charged tracks are reconstructed from hits in the MDC. We select four charged tracks with net charge zero in the polar angle range $|\cos\theta|<0.93$, and require their points of closest approach to the $e^{+}e^{-}$ interaction point to be within $\pm 10$ cm in the beam direction and $1$ cm in the plane perpendicular to the beam direction. The $dE/dx$ and TOF measurements are combined to form particle identification (PID) confidence levels for the $\pi$, $K$ and $p$ hypotheses. We require that one $K^{+}K^{-}$ pair and one $\pi^{+}\pi^{-}$ pair are identified. A vertex fit that assumes the $\pi^{+}\pi^{-}K^{+}K^{-}$ tracks all come from a common vertex is applied.

Photons are reconstructed from electromagnetic showers in the EMC. At least three photons are required for mode I and four for mode II. The minimum energy for showers to be identified as photons in the barrel region ($|\cos\theta|<0.8$) is 25 MeV, and in the end caps ($0.86<|\cos\theta|<0.92$) is 50 MeV. Showers out of the above regions are poorly reconstructed and not used in this analysis. To suppress showers from charged particles, a photon must be separated by at least 10 degrees from the nearest charged track. EMC cluster timing requirements suppress electronic noise and energy deposits unrelated to this event.

Four-constraint (4C) kinematic fits are applied to all combinations of photons, and only the combination with the smallest $\chi^{2}_{\rm 4C}$ is kept. We only keep those events with $\chi^{2}_{\rm 4C}\leq{40}$ for mode I and $\chi^{2}_{\rm 4C}\leq{80}$ for mode II. To suppress background events containing $\pi^{0}$’s, those events with the invariant mass of any photon pair within a $\pi^{0}$ mass window [$0.12\leq{}M(\gamma\gamma)\leq{}0.15$ GeV/$c^{2}$] are rejected. For mode I, the combination with the smallest value of $\delta^{2}_{1}=[M(\gamma_{1}\gamma_{2})-m_{\eta}]^{2}/\sigma_{\eta}^{2}+[M(\gamma_{3}\pi^{+}\pi^{-})-m_{\eta^{\prime}}]^{2}/\sigma_{\eta^{\prime}}^{2}$ is used to assign photons to the $\eta$ and $\eta^{\prime}$. Here $m_{\eta}$ and $m_{\eta^{\prime}}$ are the nominal $\eta$ and $\eta^{\prime}$ masses c_introduction_pdg , respectively; $\sigma_{\eta}$ and $\sigma_{\eta^{\prime}}$ are the mass resolutions determined from signal MC simulation. Mass windows for the $\eta$, $\phi$ and $\eta^{\prime}$ mesons are (in GeV/$c^{2}$) $0.522\leq{}M(\gamma\gamma)\leq{}0.573$, $1.010\leq{}M(K^{+}K^{-})\leq{}1.030$ and $0.936\leq{}M(\gamma\pi^{+}\pi^{-})\leq{}0.979$. $M(\pi^{+}\pi^{-})$ is required to be less than 0.87 GeV/$c^{2}$ to suppress the background from the $J/\psi\rightarrow\eta\phi f_{0}(980)$ process as shown in Fig. 1. For mode II, we use the combination with the smallest $\delta^{2}_{2}=[M(\gamma_{1}\gamma_{2})-m_{\eta}]^{2}/\sigma_{\eta}^{2}+[M(\gamma_{3}\gamma_{4})-m_{\eta}]^{2}/\sigma_{\eta}^{2}$ for the best $\eta$ meson combination; the $\eta$ for which $M(\pi^{+}\pi^{-}\eta)$ is closest to $m_{\eta^{\prime}}$ is attributed to the candidate decaying from the $\eta^{\prime}$. Mass windows for the $\eta$, $\phi$ and $\eta^{\prime}$ mesons are (in GeV/$c^{2}$) $0.509\leq{}M(\gamma\gamma)\leq{}0.586$, $1.010\leq{}M(K^{+}K^{-})\leq{}1.030$ and $0.920\leq{}M(\eta\pi^{+}\pi^{-})\leq{}0.995$.

Figure 2 shows the distributions of $M(\gamma\pi^{+}\pi^{-})$ versus $M(K^{+}K^{-})$ for mode I and $M(\eta\pi^{+}\pi^{-})$ versus $M(K^{+}K^{-})$ for mode II. The background inferred from the $\eta$ sidebands is negligible according to both the study of the data and the corresponding inclusive MC samples for $J/\psi$ decays. The non-$\phi$ and/or non-$\eta^{\prime}$ backgrounds are determined by the weighted sums of the horizontal and vertical sidebands with the entries in the diagonal sidebands subtracted to compensate for the double counting of background components. The different sidebands are illustrated in Fig. 2 and the weighting factors are obtained from the 2-dimensional (2D) fits to the mass spectra of $M(\gamma\pi^{+}\pi^{-})$ versus $M(K^{+}K^{-})$ and $M(\eta\pi^{+}\pi^{-})$ versus $M(K^{+}K^{-})$. The $\phi$ and $\eta^{\prime}$ meson signals are seen clearly in both modes. The three body decay $J/\psi\rightarrow\phi\eta\eta^{\prime}$ is thus established, which is the first observation of this decay.

IV Measurement of $\mathcal{B}(J/\psi\rightarrow\phi\eta\eta^{\prime})$

The branching fraction for $J/\psi\rightarrow\phi\eta\eta^{\prime}$, including all possible intermediate states, is measured. Following the procedure in Ref. c_3body_br_efficiency , the regions of $M^{2}(\phi\eta^{\prime})$ versus $M^{2}(\phi\eta)$ are divided into $40\times{40}$ areas (each area is tagged by $i$ and $j$) and the numbers of events ($n_{\rm data}^{ij}$), non-$\phi$ and/or non-$\eta^{\prime}$ background ($n_{\rm bkg}^{ij}$) and efficiency ($\epsilon_{ij}$) are obtained individually in each area. Then $\mathcal{B}(J/\psi\rightarrow\phi\eta\eta^{\prime})$ is determined by

[TABLE]

where $N_{\rm corr}$ is the efficiency-corrected number of signal events and is determined from $N_{\rm corr}=\Sigma_{ij}[(n_{\rm data}^{ij}-n_{\rm bkg}^{ij})/\epsilon_{ij}$]; $N_{J/\psi}$ is the total number of $J/\psi$ events c_data_samples ; $\mathcal{B}$ is the PDG branching fraction c_introduction_pdg ; $\mathcal{B}_{\eta^{\prime}}$ is $\mathcal{B}(\eta^{\prime}\rightarrow\gamma\pi^{+}\pi^{-})$ for mode I and $\mathcal{B}(\eta^{\prime}\rightarrow\eta\pi^{+}\pi^{-})\times{}\mathcal{B}(\eta\rightarrow\gamma\gamma)$ for mode II. The total signal yield after background subtraction is $1684\pm 48$ for mode I and $510\pm 25$ for mode II; $\mathcal{B}(J/\psi\rightarrow\phi\eta\eta^{\prime})$ is determined to be $(2.31\pm 0.07)\times 10^{-4}$ for mode I and $(2.34\pm 0.12)\times 10^{-4}$ for mode II. The uncertainties are statistical only. The weighted average c_combine_3body of the results for the two $\eta^{\prime}$ decay modes is $(2.32\pm 0.06\pm 0.16)\times 10^{-4}$, after taking into account the correlations between uncertainties from the two modes, as denoted with asterisks in Table 1.

The systematic uncertainties in $\mathcal{B}(J/\psi\rightarrow\phi\eta\eta^{\prime})$ measurements are shown in Table 1. The uncertainties from MDC tracking and PID efficiencies are established to be 1.0$\%$ per pion/kaon in Refs. c_uncertainty_track ; c_uncertainty_PID . The uncertainty related to photon detection is determined to be 0.6$\%$ per photon in Ref. c_uncertainty_photon . The uncertainties associated with the 4C kinematic fit are studied with the track parameter correction method c_uncertainty_kinematic and the differences between the efficiencies with and without corrections are regarded as uncertainties; the influence of the $\chi^{2}_{\rm 4C}$ requirement is also considered in the uncertainty determination. The sideband regions of the $\phi$ and $\eta^{\prime}$ mesons are shifted by 1$\sigma$ (the nominal width of signal region corresponds to 3$\sigma$), and the effects on the results are assigned as uncertainties. The uncertainties from mass windows are determined by smearing the mass spectra from MC simulation to compensate for the differences between the resolutions from data and MC; the differences between efficiencies before and after smearing are taken as uncertainties. The influences of finite MC statistics are taken into account. The uncertainties due to quoted branching fractions and number of $J/\psi$ events are from the PDG c_introduction_pdg and Ref. c_data_samples , respectively. The uncertainties from the 2D binning method are obtained by changing the numbers of areas in the $\mathcal{B}(J/\psi\rightarrow\phi\eta\eta^{\prime})$ determination. The total systematic uncertainties are obtained by summing all contributions in quadrature, assuming they are independent.

V Study of an intermediate state in the $\phi\eta^{\prime}$ mass spectrum

Figure 3 shows Dalitz plots for modes I and II. Both have concentrations of events with $M^{2}(\phi\eta^{\prime})$ values near 4.5 $(\rm GeV/\it{c}^{\rm 2})^{\rm 2}$. There are also diagonal bands in both modes corresponding to the process $J/\psi\rightarrow\phi{}f_{0}(1500),f_{0}(1500)\rightarrow\eta\eta^{\prime}$ according to studies of the MC samples. Apart from these, no other structures are evident.

V.1 Simultaneous fit

With the assumption that there is an $X$ structure in the $\phi\eta^{\prime}$ mass spectrum in the $2.0-2.1$ GeV/$c^{2}$ region, corresponding to the clusters near 4.5 $(\rm GeV/\it{c}^{\rm 2})^{\rm 2}$ visible in Fig. 3, a simultaneous fit is performed on the $\phi\eta^{\prime}$ mass spectra for modes I and II. Since the spin-parity value ($J^{P}$) of the structure could affect the relative orbital angular momenta between the decay products of $J/\psi\rightarrow\eta X$ and $X\rightarrow\phi\eta^{\prime}$, the fits with two different assumptions on the $J^{P}$ value are both performed. However, due to the limited statistics, they cannot clearly be distinguished. In the simultaneous fits, the interference between the structure and the direct decay $J/\psi\rightarrow\phi\eta\eta^{\prime}$ is not considered.

Assuming the $J^{P}$ value of the structure to be $1^{-}$, the signal component is parameterized by

[TABLE]

where $m$ is the reconstructed mass of the $\phi\eta^{\prime}$ system; $M$ and $\Gamma$ are the mass and width of the structure in the constant-width relativistic Breit-Wigner (BW) function; the P-wave phase space (PHSP) factor $(pq)^{3}$ is considered in the partial width, where $p$ is the $\phi$ momentum in the $\phi\eta^{\prime}$ rest frame, and $q$ is the $\eta$ momentum in the $J/\psi$ rest frame; $\epsilon$ denotes the efficiency and $R$ is the double-Gaussian resolution function, both of which are determined from a signal MC simulation. The mass and width of the BW function are allowed to float but are constrained to be the same for both modes; the signal ratio of the two modes is fixed based on PDG $\eta^{\prime}$ branching fractions c_introduction_pdg and MC-determined efficiencies. The total signal yield for the two modes is allowed to float in the fit. The background components consist of non-resonant $\phi\eta\eta^{\prime}$, $J/\psi\rightarrow\phi{}f_{0}(1500),f_{0}(1500)\rightarrow\eta\eta^{\prime}$ and non-$\phi$ and/or non-$\eta^{\prime}$ processes. For the non-resonant $\phi\eta\eta^{\prime}$ process, the line shapes are derived from the MC simulation of $J/\psi\rightarrow\phi\eta\eta^{\prime}$ process generated according to PHSP, and the ratio of background numbers for the two modes is fixed, similar to the signal case. For $J/\psi\rightarrow\phi{}f_{0}(1500),f_{0}(1500)\rightarrow\eta\eta^{\prime}$ background, whose influence on the structure is small, the shapes are from MC simulation; $\mathcal{B}(J/\psi\rightarrow\phi{}f_{0}(1500))\times{\mathcal{B}(f_{0}(1500)\rightarrow\pi\pi)}$ and $\mathcal{B}(J/\psi\rightarrow\phi{}f_{0}(1500))\times{\mathcal{B}(f_{0}(1500)\rightarrow{}K\bar{K})}$ from BESII c_introduction_exp_phi_pipi_kk , together with $\mathcal{B}(f_{0}(1500)\rightarrow\pi\pi)$, $\mathcal{B}(f_{0}(1500)\rightarrow{}K\bar{K})$ and $\mathcal{B}(f_{0}(1500)\rightarrow\eta\eta^{\prime})$ from the PDG c_introduction_pdg , are used to obtain the expected number of $f_{0}(1500)$, and the background number is fixed to the expected value. The non-$\phi$ and/or non-$\eta^{\prime}$ backgrounds are determined from the 2D sidebands of the $\phi$ and $\eta^{\prime}$ mesons as shown in Fig. 2.

Figure 4 shows the results of the simultaneous fit, where the mass and width of the structure are determined to be $(2002.1\pm 27.5)$ MeV/$c^{2}$ and $(129\pm 17)$ MeV, respectively. The log-likelihood value is 15591.8, with a goodness-of-fit value of $\chi^{2}/\rm{d.o.f.}$ of $20.98/26=0.81$ for mode I and $25.97/26=1.00$ for mode II. The statistical significance of the new structure is calculated to be larger than 10$\sigma$, determined from the change of the log-likelihood values and the numbers of free parameters in the fits with and without the inclusion of the structure. After smearing the likelihood curve with the Gaussian-distributed systematic uncertainties (Table 3), the significance is evaluated to be 4.4$\sigma$. Many checks have been done to make sure that none of the possible background contributions could produce peaking backgrounds in the $2.0-2.1$ GeV/$c^{2}$ region in the $\phi\eta^{\prime}$ mass spectrum. A comparison between data and MC also indicates no significant structures in the $\phi\eta$ mass spectrum.

Assuming the $J^{P}$ value of the structure to be $1^{+}$, the simultaneous fit with the S-wave PHSP factor $pq$ in the partial width is performed with results shown in Fig. 5. The mass and width of the structure are determined to be $(2062.8\pm 13.1)$ MeV/$c^{2}$ and $(177\pm 36)$ MeV, respectively. The log-likelihood value is 15595.9, with a goodness-of-fit value of $\chi^{2}/\rm{d.o.f.}$ of $16.68/26=0.64$ for mode I and $24.36/26=0.94$ for mode II. The significance of the structure after considering the systematic uncertainties (Table 4) is evaluated to be 3.8$\sigma$.

V.2 Angular distribution

The $J^{P}$ assignment for the structure is investigated by examining the distribution of $|\rm cos\theta|$, where $\theta$ is the $\eta$ polar angle in the $J/\psi$ rest frame. If $J^{P}=1^{-}$, the decay $J/\psi\rightarrow\eta X$ takes place through a P wave, neglecting the higher orbital angular momenta due to the closeness of the threshold, and the $|\rm cos\theta|$ is expected to follow a $1+\rm cos^{2}\theta$ distribution. If $J^{P}=1^{+}$, the above decay takes place through an S wave, where the $|\rm cos\theta|$ distribution is expected to be flat.

The events are divided into four intervals of $|\rm cos\theta|$, and the total signal yield in each interval is obtained with the same simultaneous fit method with a $1^{+}$ assumption, as described above. After efficiency correction and normalization, the $|\rm cos\theta|$ distribution of data is shown in Fig. 6, together with the fitting results with the $1^{-}$ and $1^{+}$ assumptions. The $1^{-}$ assumption has $\chi^{2}/$d.o.f. value being $10.55/3=3.52$ while for the $1^{+}$ assumption it is $4.41/3=1.47$. Although the $\chi^{2}/$d.o.f. value favors the $1^{+}$ assumption, these two assumptions cannot clearly be distinguished due to the limited statistics. The $0^{+}$ assumption is ruled out because it violates $J^{P}$ conservation, and the $0^{-}$ assumption is rejected at 99.5$\%$ confidence level from the Pearson $\chi^{2}$ test. The results of simultaneous fit with $1^{-}$ assumption are consistent with those from $1^{+}$.

V.3 Measurement of the product branching fraction

The product branching fraction to the $\eta\phi\eta^{\prime}$ final state via $X$ is

[TABLE]

where $N_{\rm sig}$ is the total signal yield from the two modes in the simultaneous fit; $\bar{\epsilon}$ is $\mathcal{B}(\eta^{\prime}\rightarrow\gamma\pi^{+}\pi^{-})\epsilon_{\rm I}+\mathcal{B}(\eta^{\prime}\rightarrow\eta\pi^{+}\pi^{-})\mathcal{B}(\eta\rightarrow{}2\gamma)\epsilon_{\rm II}$, where $\epsilon_{\rm I}$ and $\epsilon_{\rm II}$ are the detection efficiencies determined from signal MC simulation after considering the $J^{P}$ value of the structure and the angular distributions of the $\eta$, $\phi$ and $\eta^{\prime}$; the other variables have been defined before. The measured $N_{\rm sig}$ and $\mathcal{B}(J/\psi\rightarrow\eta X)\times{\mathcal{B}(X\rightarrow\phi\eta^{\prime})}$ values for the $1^{-}$ and $1^{+}$ assumptions are summarized in Table 2, where the uncertainties are statistical only.

V.4 Systematic uncertainties

Tables 3 and 4 summarise the systematic uncertainties in the measurements of mass and width of the structure, as well as $\mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime})$ for the $1^{-}$ and $1^{+}$ assumptions, respectively. In case there are differences between the uncertainties from the two modes, the more conservative values are used.

The signal parametrization is changed from a constant-width BW function to a BW with mass-dependent width. The impact on the signal yield is taken as the uncertainty of $\mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime})$. The pole mass ($m_{\rm pole}$) and pole width ($\Gamma_{\rm pole}$) are obtained by solving for the complex equation $P=m_{\rm pole}-i\Gamma_{\rm pole}/2$ for which the BW denominator is zero, and the differences between the mass and width from the nominal fit and $m_{\rm pole}$ and $\Gamma_{\rm pole}$ are considered as the uncertainties of mass and width, respectively. To obtain the uncertainties associated with the $f_{0}(1500)$ component of the data, the background levels in the simultaneous fit are varied by $\pm 1\sigma$ c_introduction_pdg ; c_introduction_exp_phi_pipi_kk , where $\sigma$ denotes the uncertainty on the determined number of the $f_{0}(1500)$, and the maximum changes in the fit results are regarded as uncertainties. We also vary the non-resonant $\phi\eta\eta^{\prime}$ background levels by $\pm 1\sigma$, and take the largest influences on the fit results as the uncertainties due to the PHSP assumption. We vary the range of the simultaneous fit by 5$\%$ and take the largest deviations of the fitting results as uncertainties. To obtain the uncertainties due to the $M(\pi^{+}\pi^{-})$ requirement for mode I, it is relaxed from 0.87 to 0.90 GeV/$c^{2}$ and the effects on the fitting results are considered as uncertainties. The two possible extra structures around 2.3 GeV/$c^{2}$ in Figs. 4 (b) and 5 (b) are considered. Following the procedure in Ref. c_introduction_besiii , we use BW functions convolved with a resolution function to describe them and the corresponding significances are determined to be less than 1.1$\sigma$, and they are not considered in the nominal result. However, their impacts on the fitting results are taken as systematic uncertainties. The difference between the fitted $\eta$ mass and that from the PDG c_introduction_pdg is taken as the uncertainty due to momentum calibration. The descriptions of other items are included in Table 1. The total systematic uncertainties are the quadrature sums of the individual contributions, assuming they are independent.

VI Summary and discussion

In summary, using $(1310.6\pm 7.0)\times{10^{6}}$ $J/\psi$ events collected with the BESIII detector, we report the observation and study of the process $J/\psi\rightarrow\phi\eta\eta^{\prime}$. Its branching fraction, including all possible intermediate states, is determined to be $(2.32\pm 0.06\pm 0.16)\times 10^{-4}$. Evidence for a structure denoted as $X$ in the $\phi\eta^{\prime}$ mass spectra in two dominant $\eta^{\prime}$ decay modes is reported, and a simultaneous fit is performed. Assuming the $J^{P}$ value of the structure to be $1^{-}$, the significance of the structure is evaluated to be 4.4$\sigma$; the mass and width are determined to be $(2002.1\pm 27.5\pm 21.4)$ MeV/$c^{2}$ and $(129\pm 17\pm 9)$ MeV, respectively; the product branching fraction $\mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime})$ is measured to be $(9.8\pm 1.2\pm 1.7)\times 10^{-5}$. The mass of the structure is over 5$\sigma$ away from that of the $Y(2175)$ in the PDG c_introduction_pdg , suggesting the structure might not be the $Y(2175)$. For a $1^{+}$ assumption, the significance is evaluated to be 3.8$\sigma$; the mass and width are determined to be $(2062.8\pm 13.1\pm 7.2)$ MeV/$c^{2}$ and $(177\pm 36\pm 35)$ MeV, respectively; the product branching fraction $\mathcal{B}(J/\psi\rightarrow\eta X)\times{}\mathcal{B}(X\rightarrow\phi\eta^{\prime})$ is measured to be $(9.6\pm 1.4\pm 2.0)\times 10^{-5}$. The angular distribution is studied and the $1^{-}$ and $1^{+}$ assumptions cannot clearly be distinguished due to the limited statistics. No meson candidate in the PDG has mass, width and $J^{P}$ values that are compatible with the structure. More studies with a larger $J/\psi$ data sample in the future might help to better understand the structure, including a $J^{P}$ determination and precise measurements of the mass, width, and product branching fraction.

Acknowledgements.

This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11335008, 11425524, 11625523, 11635010, 11735014; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; the Institute of Nuclear and Particle Physics and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0010504, DE-SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI) Darmstadt; Institute for Basic Science (Korea) under project code IBS-R016-D1.