Measurement of the cross section of $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ at center-of-mass energies from 3.872 GeV to 4.700 GeV

Using data samples with an integrated luminosity of 19 fb$^{-1}$ at twenty-eight center-of-mass energies from 3.872 GeV to 4.700 GeV collected with the BESIII detector at the BEPCII electron--positron collider, the process $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ and the intermediate process $e^{+}e^{-}\to\eta\rho^{0}$ are studied for the first time. The Born cross sections are measured. No significant resonance structure is observed in the cross section lineshape.

Abstract: Using data samples with an integrated luminosity of 19 fb −1 at twenty-eight center-of-mass energies from 3.872 GeV to 4.700 GeV collected with the BESIII detector at the BEPCII electron-positron collider, the process e + e − → ηπ + π − and the intermediate process e + e − → ηρ 0 are studied for the first time. The Born cross sections are measured. No significant resonance structure is observed in the cross section lineshape.
The currently known decays of Y (4220) occur only to open or hidden-charm final states. However, some related theories point out that charmonium-like states are also likely to decay to light hadron final states [12], shedding further light on the Y (4220) [13]. Several measurements of the cross sections for e + e − annihilations to light hadrons have been measured by the BESIII Collaboration, such as e + e − → K 0 S K ± π ∓ π 0 [14], K 0 S K ± π ∓ η [15], pnK 0 S K − + c.c. [16], ppπ 0 [17] etc., but no significant structures have been found so far. In order to better understand the composition and properties of the charmonium-like states, further searches for their decays to charmless light-hadron final states are important.
In this paper, we present the measurements of the Born cross section of the process e + e − → ηπ + π − at center-of-mass (c.m.) energies from 3.872 GeV to 4.700 GeV, and search for possible charmonium (ψ) or vector charmonium-like (Y ) states in the corresponding lineshape.

BESIII detector and data sets
The BESIII detector is a magnetic spectrometer [18] located at the Beijing Electron Positron Collider (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 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 4π solid angle. The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the specific ionization energy loss (dE/dx) resolution is 6% for the 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. The end cap TOF system was upgraded in 2015 with multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [19,20].
The twenty-eight data sets taken at √ s = 3.872 ∼ 4.700 GeV are used in this analysis. The nominal energy of each data set is calibrated by the process e + e − → (γ ISR/FSR )µ + µ − [21,22], where the subscript FSR stands for final-state radiation. The integrated luminosity L is determined by large angle Bhabha events [23,24], and the total integrated luminosity is approximately 19 fb −1 .

Monte Carlo (MC) simulation
Simulated data samples produced with geant4-based [25] MC software, 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 ISR in the e + e − annihilations with the generator kkmc [26]. The inclusive MC simulation sample includes the production of open charm processes, the ISR production of vector charmonium(-like) states, and the continuum processes incorporated in kkmc. The known decay modes are modeled with evtgen [27] using branching fractions taken from the Particle Data Group (PDG) [28], and the remaining unknown ψ decays are modeled with lundcharm [29]. The FSR from charged final state particles is incorporated using photos [30].
In the signal MC simulation samples at each c.m. energy point, three exclusive processes are involved, which are the three-body non-resonant process e + e − → ηπ + π − , and the twobody resonant processes, e + e − → a ± 2 (1320)π ∓ and ηρ 0 . The last process is simulated by the HELAMP model [27] following the dynamics of other vector charmonium decays, while the other two processes are simulated by phase space (PHSP) models. In determining the resulting detection efficiencies, the interference between ηρ and the three-body non-resonant processes is included, while the interference between a 2 (1320)π and the three-body nonresonant process is neglected due to the low statistics. The resulting detection efficiency is obtained by mixing the three processes weighted according to the number of observed events (N obs ) and detection efficiency ( ).

Event selection
The charged tracks detected in the MDC are required to be within a polar angle (θ) range of |cosθ| < 0.93, where θ is defined with respect to the z-axis, which is the symmetry axis of the MDC. All the charged tracks are required to originate from the interaction region V xy < 1 cm and |V z | < 10 cm, where V xy and |V z | are the distances of closest approach of the charged track to the interaction point in the xy-plane and z direction, respectively.
Particle identification (PID) for charged tracks combines measurements of dE/dx in the MDC and the flight time in the TOF to form likelihoods L(h) for each hadron h = p, K, π hypothesis. Tracks are identified as pions when the pion hypothesis has the greatest likelihood (L(π) > L(K) and L(π) > L(p)).
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 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 degrees. 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. Candidate events must have two charged tracks with zero net charge, and the number of photons should be 2 or greater. The two charged tracks must be identified as pions.
To improve the momentum and energy resolution and suppress the potential backgrounds, a four-constraint (4C) kinematic fit, which constrains the total four-momentum of the final state particles to that of the initial colliding beams, is applied to the event under the hypothesis of e + e − → γγπ + π − . If more than one candidate (37% of the selected events) exists in an event, that with the smallest χ 2 4C is selected. Potential backgrounds are investigated with six equivalent-luminosity inclusive MC samples generated at c.m. energies from 4.009 GeV to 4.600 GeV, using an event-type analysis tool, TopoAna [31]. It is found that the main background contributions come from e + e − → γπ + π − , µ + µ − , γγe + e − and e + e − → J/ψ + anything, J/ψ → hh (h = p, π, e, µ) processes. In the first background channel, a reconstructed photon e.g. from beam-related background is combined with the real photon to form a fake η signal. This background is suppressed by requiring the ratio R = |Eγ 1 −Eγ 2 | pη < 0.90, where E γ 1 , E γ 2 are the energies of the two photons from the fake η decay and p η is the momentum of the fake η. The second background channel is suppressed by requiring the hit depth of charged track in the µ counter to be less than 40 cm. The third background channel is suppressed by requiring E/cp <0.7. Here, E and p denote the deposited energy in the EMC and the momentum of the charged track, respectively. The background from J/ψ-related events is vetoed by requiring the invariant mass of π + π − not to fall into the J/ψ mass region [3.05, 3.15] GeV/c 2 . Finally, it is found that the dominant remaining background channel is e + e − → µ + µ − due to µ − π misidentification.
With the above selection criteria, there are significant enhancements close to the η and ρ 0 nominal masses in the two dimensional distribution of the invariant mass of γγ (M γγ ) and π + π − (M π + π − ), as can be seen in Figure 1 (left). The η signal region is defined as 0.513 < M γγ < 0.581 GeV/c 2 , and the lower and upper side-band regions are defined as 0.309 < M γγ < 0.445 GeV/c 2 and 0.649 < M γγ < 0.785 GeV/c 2 , respectively.

Signal yields
The η signal yields are obtained with unbinned likelihood fits to the M γγ spectra. The signal function is described as a MC-simulated shape convolved with a Gaussian function to account for the difference of the detector resolutions between data and MC simulation, while the background function is described by a second-order Chebyshev polynomial. Figure 2 (a) shows the fit result for all energy points combined.
The ηρ 0 signal yields are obtained with a simultaneous unbinned likelihood fit to the M π + π − spectra of the events in the η signal region at all energy points. The ρ 0 resonance is parameterized by a Breit-Wigner (BW) propagator using the Gounaris-Sakurai (GS) model [32]. The parameterized propagator function is expressed as: and h (m) is the derivative of h(m), m is the square of the invariant mass of π + π − , M π is the invariant mass of the π meson, and M and Γ are the mass and width of the ρ. The signal function is described by a coherent probability density function (PDF): where ϕ is the relative phase between the ρ and PHSP amplitudes which describe the non-ρ mode parameterized with a polynomial, and the parameter A is the normalization factor. The parameters of the signal function are left free in the fit. The non-η background shape is obtained by the normalized η side-bands summed over all energies. The number of background events is fixed to f · N sb , where f = 0.25 is the scale factor since the nonη background shape is a linear one and the side-band region is two times wider than the signal region. The parameter N sb is the number of side-band background events at each c.m. energy point. The fits at each point share the parameters of the signal function (Eq. (5.4)). Figure 2 (b) shows the fit result for the sum of all energy points.
The a 2 (1320) signal yield is obtained by a binned likelihood fit to the invariant mass of ηπ ± (M ηπ ± ) spectrum summed over all energy points. The signal function is also described by the MC-simulated shape convolved with a Gaussian function, and the background function is described by a third-order Chebyshev polynomial. Figure 2 (c) shows the fit result.
Numerical results for the fits for events summed over all energy points can be found in Table 1. The number of a ± 2 (1320)π ∓ events is obtained by the fit to the M ηπ ± distribution, the number ηρ 0 + interference events is obtained by the fit to the M π + π − distribution, and number of ηπ + π − (total) events is obtained by the fit to the M γγ distribution. The number of events for the 3-body non-resonant process is given by N (ηπ + π − (3-body non-resonant)) = N (ηπ + π − (total)) − N(ηρ 0 + interference) − N(a ± 2 (1320)π ∓ ).

Cross section calculation
The Born cross section at each energy point is calculated as: where N obs is the number of observed signal events, L is the integrated luminosity, is the detection efficiency, and 1 + δ γ and 1 |1−Π| 2 are ISR and vacuum polarization (VP) factors, respectively. To obtain 1 + δ γ and 1 |1−Π| 2 , we use an energy-dependent power function a/s n as the initial input of the Born cross section, and the final one is obtained by iterating several times until the difference of ·(1+δ γ ) between the last two iterations is less than 1%. The relevant numbers related to the Born cross section measurement for e + e − → ηπ + π − and its intermediate process e + e − → ηρ 0 are listed in Tables 2 and 3, respectively. For the intermediate process e + e − → a 2 (1320) ± π ∓ , we do not report the measurement of its Born cross section due to the low statistics at single energy points.

Systematic uncertainty
The uncertainties in the Born cross section measurements include those of the luminosity measurement, tracking and PID efficiency, photon detection efficiency, intermediate state, R ratio, E/cp ratio, decay depth in the µ counter, η mass window, J/ψ veto, fit of M γγ and M π + π − , kinematic fit, ISR and VP correction and detection efficiency.
• Luminosity measurement. The luminosity is measured using Bhabha events with uncertainty of 1% at all energy points [23,24], which is taken as the systematic uncertainty from the luminosity measurement.
• Tracking efficiency. The pion tracking efficiency is determined by using the control sample J/ψ → ppπ + π − . The difference between data and the MC simulation tracking efficiency is 1% per track [33].
• PID efficiency. The uncertainty related to the pion PID efficiency is studied with the sample e + e − → K + K − π + π − , and the average difference of the PID efficiency between data and MC simulation is determined to be 1% for each charged pion, which is taken as the systematic uncertainty [34].
• Photon detection efficiency. The uncertainty caused by photon reconstruction is 1% per photon, which is studied by the control sample J/ψ → γπ 0 π 0 [35].
• R ratio. The uncertainty caused by the R = |Eγ 1 −Eγ 2 | pη requirement is estimated by changing the range by ±0.05. The larger differences with and without changes are taken as the corresponding uncertainties.
• E /cp ratio. The uncertainty caused by the E/cp ratio requirement is estimated from the control sample J/ψ → π 0 π + π − . The difference between data and MC simulation is found to be 3.88%, which is taken as the systematic uncertainty.
• Decay depth in the µ counter . The systematic uncertainty caused by the requirement on the decay depth in the µ counter is also estimated from the control sample J/ψ → π 0 π + π − . The difference between data and MC simulation is found to be 0.34% and is taken as the systematic uncertainty.
• η mass window . The systematic uncertainty associated with the η mass window requirement is estimated by changing the mass window range by ±1σ, where σ is the η mass resolution, the larger difference with and without change is taken as the systematic uncertainty.
• J /ψ veto. The systematic uncertainty from the J/ψ-related background veto is esti- • Fit of M π + π − . The systematic uncertainties associated with the fit of the M π + π − spectrum come from the choice of the signal function, background function and fit range. They are estimated by: fixing the parameters of the BW function to the values from the PDG; changing the order of the Chebychev polynomial function from second to third; The resulting differences with and without change are taken as the systematic uncertainties.
• Kinematic fit. The uncertainty due to the kinematic fit requirements is estimated by correcting the helix parameters of charged tracks according to the method described in Ref. [36]. The difference between detection efficiencies obtained from MC samples with and without this correction is taken as the uncertainty.
• ISR and VP correction. As mentioned in Section 8, we use the energy-dependent power function f ( √ s) = a/s n to fit the line shape. The systematic uncertainty from the ISR and VP correction is estimated by varying the n value by ±1σ, where σ is the statistical uncertainty of the fitted n value. The larger difference of the cross sections caused by the above changes is taken as the systematic uncertainty.
• Detection efficiency. The detection efficiency is obtained by a weighted average for the three different processes. The weight factors are the respective numbers of signal events. We randomly change the number of signal events for each process according to its statistical uncertainty and get new ratios between different processes. We mix the three processes with the new ratios and get new efficiencies. By repeating the above procedure, we obtain a group of detection efficiencies, which is almost a Gaussian distribution. The corresponding standard deviation is taken as the uncertainty caused by the detection efficiency. It is found that it is negligible.
Due to the limited sample size at other c.m. energies, the systematic uncertainties from the event selection, mass window requirement and background veto are taken to be the same as those at √ s = 4.180 GeV. The total uncertainty in the cross section measurement is obtained by summing the individual contributions in quadrature, and the dominate uncertainties come from the tracking efficiency, PID efficiency, photon efficiency and E/cp ratio requirement. All systematic uncertainties are summarized in Table 4 and Table 5 for the processes e + e − → ηπ + π − and e + e − → ηρ, respectively. 14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.  14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.

Fit to the Born cross sections
The least-squares method is used to fit the Born cross sections under different assumptions. In order to describe purely continuum production, we use the energy-dependent function to fit the Born cross section which only considers the contribution from the one photon exchange process without any resonance. The goodness-of-fit (GOF) is χ 2 /n.d.f. = 47.0/27 = 1.7 for the ηπ + π − process and 47.5/27 = 1.8 for the ηρ process. Here, n.d.f. denotes the number of degrees of freedom. The χ 2 function is constructed as Here, σ D i and σ fit D i are the measured and fitted Born cross sections of the ith energy point, respectively, and δ i is the standard deviation of the measured cross section, which includes the statistical uncertainties only. The goodness of the fits indicates that the data can be described by the energy-dependent function. The fit returns n = 3.5 ± 0.1 and 3.8 ± 0.1 for the processes ηπ + π − and ηρ, respectively. The fit results are shown in Figure 3. Potential contributions from the well-established conventional charmonium states ψ or charmoniumlike states Y , i.e. ψ(4160), Y (4230), Y (4360), ψ(4415), and Y (4660), are investigated by using the coherent sum of the continuum (Eq. (8.1)) and an additional charmonium(-like) state amplitude in the fit to the Born cross section. The fit function can be expressed as where the parameters a and n in f 1 ( √ s) are fixed to those obtained from the fit to the line shape using the function f 1 ( √ s) only. The function BW( √ s) = √ 12πΓee BΓtot s−M 2 +iM Γ 2 tot is used to describe charmonium(-like) states, where M , B, Γ ee , and Γ tot are the mass, branching fraction of the resonance decays, partial width to e + e − and total width, respectively, in which Γ ee and B are left free while the other two parameters are fixed to the values from the PDG, and PS( The statistical significances for the added components are estimated by comparing the change of χ 2 /n.d.f. with and without adding the corresponding component. Since there is interference between the resonance and continuum process, there are two solutions for Γ ee B(ηπ + π − /ηρ) with the same minimum value of χ 2 . Table 6 lists the fit results and the significances for the additional charmonia. The low significances indicate that no obvious charmonium or charmonium-like states are required to describe the measured cross section. Table 6: Results of the fits to the Born cross section. "Solution I" represents the constructive solution, and "Solution II" represents the destructive solution. exponential function Eq. (8.1). The significances for possible contributions from ψ(4160), Y (4230), Y (4360), ψ(4415) or Y (4660) resonances are all less than 2σ. This implies that the charmonium and charmonium-like states disfavor decay to ηπ + π − or ηρ.