Measurement of eta_c(1S), eta_c(2S) and non-resonant eta' pi+ pi- production via two-photon collisions

We report the measurement of gamma gamma to eta_c(1S), eta_c(2S) to eta' pi+ pi- with eta' decays to gamma rho and eta pi+ pi- using 941 fb^{-1} of data collected with the Belle detector at the KEKB asymmetric-energy e+e- collider. The eta_c(1S) mass and width are measured to be M = [2984.6\pm0.7 (stat.)\pm2.2 (syst.)] MeV/c^{2} and \Gamma = [30.8^{+2.3}_{-2.2}~(stat.) \pm 2.5~(syst.)] MeV, respectively. First observation of eta_c(2S) to eta' pi+ pi- with a significance of 5.5sigma including systematic error is obtained, and the eta_c(2S) mass is measured to be M = [3635.1\pm3.7~(stat.)\pm2.9~(syst.)] MeV/c^{2}. The products of the two-photon decay width and branching fraction (B) of decays to eta'pi+ pi- are determined to be \Gamma_{gamma gamma}B = [65.4\pm2.6~(stat.)\pm6.9~(syst.)] eV for eta_c(1S) and [5.6^{+1.2}_{-1.1}~(stat.)\pm1.1~(syst.)] eV for eta_c(2S). A new decay mode for the eta_c(1S) to eta'f_0(2080) with f_0(2080) to pi+ pi- is observed with a statistical significance of 20sigma. The f_0(2080) mass and width are determined to be M = [2083^{+63}_{-66}~(stat.)\pm 32~(syst.)] MeV/c^{2} and \Gamma = [178^{+60}_{-178}~(stat.) \pm 55~(syst.)] MeV. The cross sections for gamma gamma to eta' pi+ pi- and eta'f_{2}(1270) are measured for the first time.

Using 637 fb −1 of data, Belle reported the measurement in two-photon fusion of the η c (1S) resonance parameters using its decays to η ′ π + π − with η ′ → ηπ + π − [11]. The above considerations motivate an updated measurement of η c (1S) parameters using the full 941 fb −1 Belle data set, and, addition- * Also at University of Chinese Academy of Sciences.
The cross sections for two-photon production of meson pairs have been calculated in perturbative QCD and measured in experiments in a W region near or above 3 GeV, where W is the invariant mass of the two-photon system. The leading term in the QCD calculation [18][19][20] of the cross section predicts a 1/(W 6 sin 4 θ) dependence for a charged-meson pair, and a 1/W 10 dependence and model-dependent angular distribution for a neutral-meson pair. Here, θ is the scattering angle of a final-state particle in the two-photon CM frame. The handbag model [21] gives the transition amplitude describing energy dependence and predicts a 1/sin 4 θ angular distribution for both charged-and neutral-meson pairs for large W . The Belle results for the cross sections [22] show that the angular distributions for the chargedmeson pairs, γγ → π + π − , K + K − , agree well with the 1/sin 4 θ expectation, while those for the neutralmeson pairs, γγ → π 0 π 0 , K 0 S K 0 S , ηπ 0 and ηη, exhibit more complicated angular behavior. The measured exponent n in the energy dependence 1/W n for both charged-and neutral-meson pairs is found to lie between 7.3 and 11 with a relative error of 7-20%. Further study with improved precision in both experiment and QCD predictions at higher W mass would provide more sensitive comparisons. There is no specific QCD prediction for the two-photon production of either the pseudoscalar-tensor meson pair η ′ f 2 (1270) or the three-body final state η ′ π + π − . Our results for the production of these two-and threebody final states would thus provide new information to validate QCD models.
In this paper, we report the updated measurement of the η c (1S) parameters with the full Belle data sample of 941 fb −1 , the observation of an η c (2S) signal with its decays to η ′ π + π − for the first time, the measurement of the product of the two-photon width of η c (2S) and its branching fraction to η ′ π + π − , and the measurement of non-resonant production of η ′ π + π − with η ′ → ηπ + π − decay via two-photon collisions.

II. DETECTOR AND MONTE CARLO SIMULATION
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, a 50-layer central drift chamber, an array of aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of-flight scintillation counters, and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons. The detector is described in detail elsewhere [23].
We generate the two-photon process γγ → η ′ π + π − using the TREPS code [24], where the η ′ decays generically according to JETSET7.3 [25]. A distribution uniform in phase space is assumed for the η c (1S) and η c (2S) decays to the η ′ π + π − final state. The GEANT3-based [26] simulation package that incorporates the trigger conditions is employed for the propagation of the generated particles through the Belle detector.

III. DATA AND EVENT SELECTION
We use two data samples. The first is collected at the Υ(4S) resonance ( √ s = 10.58 GeV) and 60 MeV below it with integrated luminosity L int,4S = 792 fb −1 , while the other is recorded near the Υ(5S) resonance ( √ s = 10.88 GeV) with L int,5S = 149 fb −1 .
When combining the data in this analysis, a slight dependence of the two-photon cross section on e + e − center-of-mass energy is taken into account, as described in Sec. IV. Two η ′ decay modes, η ′ → ηπ + π − with η → γγ and η ′ → γρ including non-resonant π + π − (denoted as ηππ and γρ, respectively), are included in the reconstruction of the η ′ meson in the η ′ π + π − final state.

A. Selection criteria
At least one neutral cluster and exactly four charged tracks with zero net charge are required in each event. The candidate photons are neutral clusters in the ECL that have an energy deposit greater than 100 MeV and are unmatched with any charged tracks. To suppress background photons from π 0 (π 0 or η) decays for the ηππ (γρ) mode, any photon that, in combination with another photon in the event has an invariant mass within the π 0 (π 0 or η) window |M γγ − m π 0 | < 0.018 GeV/c 2 (|M γγ − m π 0 | < 0.020 GeV/c 2 or |M γγ − m η | < 0.024 GeV/c 2 ) is excluded. Events with an identified kaon (K ± or K 0 S → π + π − ) or proton are vetoed. Charged pion, kaon and proton identification strategies and criteria for the both ηππ and γρ modes, as well as the event selection criteria for the ηππ mode, are the same as those used in Ref. [11] except for the requirement on the transverse momentum |Σp * t | (see Sec. III B). Here, |Σp * t | is the absolute value of the vector sum of the transverse momenta of the η ′ , π + , and π − in the e + e − center-of-mass system. To improve the momentum resolution of the η ′ , two separate fits to the η ′ are applied, one with a constrained vertex and the other with a constrained mass.
For the ηππ mode, the η is reconstructed via its two-photon decay mode, where the two-photon invariant mass is in the window M γγ ∈ [0.524, 0.572] GeV/c 2 (±2σ of the nominal η mass). The η ′ candidate is reconstructed from the η candidate and the π + π − track pair that has an invariant mass within M ηπ + π − ∈ [0.951, 0.963] GeV/c 2 (±2σ of the nominal η ′ mass).
For both the ηππ and γρ modes, we reconstruct η ′ π + π − candidates by combining the η ′ with the remaining π + π − pair, which must satisfy a vertexconstrained fit. For multicandidate events, the candidate with the smallest χ 2 from η ′ mass-constrained fit is selected. For η ′ π + π − combinations with an invariant mass of W = 2.98 (3.64) GeV/c 2 , we find that 8.2% (7.3%) of the signal Monte Carlo (MC) events have more than one candidate per event for the ηππ mode and 15% (9.8%) for the γρ mode, from which the correct candidate is selected 94% (98%) for the ηππ mode and 88% (89%) for the γρ mode. The sum of the ECL cluster energies in the laboratory system and the scalar sum of the absolute momenta for all charged and neutral tracks in the laboratory system for the η ′ π + π − system must satisfy E sum < 4.5 GeV and P sum < 5.5 GeV/c.
The prominent feature for the events from an untagged two-photon process in e + e − collisions is that they tend to carry small transverse momentum. Therefore, a |Σp * t | requirement allows significant background reduction. The |Σp * t | distributions for the ηππ and γρ modes in the signal regions of W ∈ [2.90, 3.06] GeV for η c (1S) and W ∈ [3.60, 3.68] GeV for η c (2S) are shown in Fig. 1.
The |Σp * t | requirement for selection of the η ′ π + π − candidates from both the η c (1S) and η c (2S) decays is optimized using signal and background MC samples. The η c signal and the background are described by a relativistic Breit-Wigner function [see Eq. (1) in section IV] and the exponential of a third-order polynomial, respectively. The background shape in the η c signal region is determined from the fit to the sideband data and normalized. The requirement on |Σp * t | is determined by maximizing the value of s/ √ s + b for both ηππ and γρ modes, where s is the η c signal yield and b is background yield in the η c signal region. We find the best |Σp * t | requirements, which are close to each other in the two η c mass regions, to be |Σp * t | < 0.15 GeV/c for the ηππ mode and |Σp * t | < 0.03 GeV/c for the γρ mode. We find that these values are stable in the range of the expected signal yield based on the previous measurement [11] and theoretical expectation for η c (2S) [27].
The invariant mass distributions for the η ′ π + π − candidates for the ηππ and γρ modes are shown in Fig. 2. In addition to the prominent η c (1S) signal, an evident enhancement in the mass region near 3.64 GeV/c 2 is seen in both modes.

IV. FITTING FOR ηc(1S) AND ηc(2S)
The probability density function f s (W ) for the resonance R is a Breit-Wigner function [28,29] f BW (W ) convolved with a mass-resolution function R ICB after corrections for the detection efficiency ǫ i (W ) and the two-photon luminosity function dL γγ /dW : Here, R ICB is an improved Crystal Ball (ICB) function [30]. The efficiency factor ǫ i (W ) includes the branching fractions of η ′ → ηπ + π − with η → γγ for the ηππ mode (i = 1) and η ′ → γρ with ρ → π + π − for the γρ mode (i = 2). Number of the η c (1S) mesons produced via two-photon process is constrained to be equal for both modes in the simultaneous fit. The luminosity function is evaluated in the Equivalent Photon Approximation (EPA) [28,29] using TREPS [24]. The efficiency for each η ′ decay mode is corrected for the dependence on beam energy in the Υ(4S) and Υ(5S) regions [31,32]: where ǫ 4S (ǫ 5S ) and dL γγ,5S /dW (dL γγ,5S /dW ) are the efficiency and two-photon luminosity functions, respectively, at the Υ(4S) [Υ(5S)] energy. The product of the two-photon decay width and the branching fraction for the R → η ′ π + π − decay is determined as where n obs,i is the yield of decay mode i of the resonance R in the simultaneous fit, while L int is the integrated luminosity. Identical W regions of [2.60, 3.4] GeV/c 2 for η c (1S) and [3.3, 3.8] GeV/c 2 for η c (2S) are chosen in the simultaneous fit for the yield and as the integral interval in the calculation of Γ γγ B.

A. Background estimation
The background in the η ′ π + π − mass spectrum for the R measurement is dominated by three components: (1) non-resonant (N R) events produced via two-photon collisions, which have the same |Σp * t | distribution as that of the R signal; (2) the η ′ sideband (η ′ -sdb) arises from wrong combinations of γγπ + π − (γπ + π − ) for the ηππ (γρ) mode that survive the η ′ selection criteria, estimated using the events in the margins of the η ′ signal in the ηππ (γρ) invariantmass distribution; (3) η ′ π + π − + X (b any ) events having additional particles in the event beyond the R candidate. Other nonexclusive events, including those arising from initial-state radiation, are found to be negligible [11].  For the determination of the background components, two data subsamples, one with |Σp * t | < 0.15 GeV/c (0.03 GeV/c), denoted as p t -balanced, and the other with |Σp * t | ∈ [0.17, 0.2] GeV/c ([0.15, 0.2] GeV/c), denoted as p t -unbalanced, for the ηππ (γρ) mode, are selected. (See Ref. [11] for the details.) The R signal and N R component peak in the p t -balanced sample while the η ′ -sdb and b any backgrounds dominate over the signal plus N R in the p t -unbalanced sample. For the ηππ mode, the η ′ -sdb component is well estimated using the η ′ sideband, defined by M ηπ + π − ∈ [0.914, 0.934] GeV/c 2 and ∈ [0.98, 1.00] GeV/c 2 . The b any component is determined using the events in the p t -unbalanced sample with the η ′ -sdb contribution subtracted. Here, the assumption of the same shape in the invariant mass distribution for the b any component in the p t -balanced and p t -unbalanced samples is implied. For the γρ mode, the sum of η ′ -sdb and b any is determined from the events in the p t -unbalanced sam-ple. These two components are hard to distinguish because of peaking background in the γρ 0 invariant mass distribution, caused by the large width of the ρ meson and the η ′ mass-constraint fit.
The yield and shape for the two components, η ′sdb and b any , separated (combined) for the ηππ (γρ) mode, are fixed in the simultaneous fit. The exponential of a second-order polynomial is used to describe the N R component with the yield and shape floating in the fit for both the ηππ and γρ modes. −68 for the γρ mode. Figure 4 shows the result on the fit for the η c (2S) region, which results in a signal with a statistical significance of 5.5σ, and yields of n 1 = 41 +9 −8 for the ηππ mode and n 2 = 65 +14 −13 for the γρ mode. The η c (2S) mass is determined to be M = (3635.1 ± 3.7) MeV/c 2 ; its width is fixed to the world-average value of 11.3 MeV [33] in the fit. The statistical significance for the η c (2S) signal is calculated with the χ 2 distribution −2ln(L 0 /L max ) for N dof degrees of freedom. Here, L max and L 0 are the maximum likelihoods of the fits with the signal yield floating and fixed to zero, respectively, and N dof = 2 is the difference in the number of floating parameters between the nominal fit and the latter fit.

C. Systematic uncertainties
The systematic uncertainties are summarized in Table II. We estimate the uncertainty in the trigger efficiency using signal MC events. The differences between the two efficiencies with and without simulation of the trigger conditions are evaluated to be 0.5% (0.6%) for η c (1S) (η c (2S)) in the γρ mode, and 1.4% for both η c mesons in the ηππ mode. The contribution to the systematic uncertainty arising from pion identification is studied using an inclusive D * sample. The uncertainties of pion identification are found to be 1.8% (2.3%) in the γρ mode and The averaged values of deviations in the yield, mass, and width between the two simultaneous fits, with |Σp * t | requirement changed by ±0.01 GeV/c in the γρ mode and by ±0.02 GeV/c in the ηππ mode, are treated as systematic uncertainties.
Two methods are applied to evaluate the systematic uncertainty related to the uncertainty in the N R background shape: (1) changing the mass window size in the fit; (2) altering the fit function for the background-shape description. The difference between the average values of the two fit yields calculated by changing the mass window width by ±100 MeV/c 2 is regarded as systematic uncertainty: we find 2.3% (9.0%) in the γρ mode and 2.2% (9.5%) in the ηππ mode for η c (1S) (η c (2S)). The contribution to the uncertainty in the fit yield estimated by varying the order of the polynomial function is found to be minor and thus is neglected.
The uncertainty in the determination of the η ′sdb and b any backgrounds is estimated with changes in the η ′ -sdb window size by ±0.01 GeV/c 2 . The resulting difference in yields is evaluated to be 2.5% for η c (1S) and 4.8% for η c (2S) and is treated as the uncertainty.
The uncertainty from the π 0 -veto is estimated as the difference in efficiency with and without the π 0veto. The uncertainties for the η reconstruction efficiency are studied using an inclusive η sample, and its deviation from the MC simulation plus its error in quadrature is 4.9%. The systematic uncertainties related to charged track reconstruction efficiency, luminosity function calculation, and experimentalconditions dependence are studied via charmonium decay to four charged mesons [7,8]. The evolution of the background conditions over time adds an additional uncertainty of 3% in the yield determination. The accuracy of the two-photon luminosity is estimated to be 5% including the uncertainties from radiative corrections (2%), the uncertainty from the form-factor effect (2%), and the error of the integrated luminosity (1.36%).
To examine the systematic uncertainty in the mass measurement for the R → η ′ π + π − decay, an inclusive control sample of the decay D 0 → η ′ K 0 S with K 0 S → π + π − is selected with a tight mass window for η ′ . The D 0 mass resulting from fits to the invariant mass spectra of η ′ K 0 S is biased from its nominal value by 1.26 MeV/c 2 (0.93 MeV/c 2 ) in the ηππ (γρ) mode. The sum of the bias and statistical er-  Γγγ B is the product of the two-photon decay width and the branching fraction for ηc → η ′ π + π − . The first error is statistical and the second is systematic. ror in quadrature, scaled linearly to the η c mass, is taken as the contribution of the uncertainty for the mass scale. The uncertainty in the width determination is estimated by changing the mass resolution by ±1 MeV/c 2 , and is found to be 1.2 MeV/c 2 for the η c (1S). The uncertainties for the resonance mass and width coming from |Σp * t | and background shape are determined with the same method as that for the Γ γγ B measurement.
Taking the yield-weighted mean of squared uncertainty for the γρ and ηπ + π − modes combined in the fits, the total systematic uncertainties in the measurements of Γ γγ B, mass and width for η c (1S) [η c (2S)] are calculated by adding the individual mean uncertainties in quadrature.  We label it the f 0 (2080), with mass and spin to be given in this section. This structure, produced from η c (1S) decay, is observed for the first time.
We divide the M π + π − distribution into 40 bins of 45 MeV/c 2 width from 0.3 to 2.1 GeV/c 2 . The η c (1S) yields n tot and n s,i , i = 1, ... 40 in the signal window of 2.9 to 3.06 GeV/c 2 are extracted by fitting the η ′ π + π − invariant mass spectra for events in the entire range and in each of the 40 bins, respectively. The unbinned extended maximum likelihood fit is applied. The η c (1S) signal is described by the function f s in Eq. (1) with the mass and width floating in the fit for the entire-range yield but fixed to the values determined from that fit when fitting in each of 40 bins. The background is modeled by the exponential of a third-order polynomial whose shape and normalization parameters float in the fit.
To account for the low statistics in each M π + π − bin, the background-component parameters are held in common for each group of four consecutive bins. The statistical uncertainties of both the signal and background parameters are effectively included in the determination of the η c (1S) yield in each bin. For bins 1 to 30, a mass region of 2.6 to 3.4 GeV/c 2 for the M η ′ π + π − distribution is used. For bins 31 to 40, a restricted fitting mass region is used, with an upper bound of 3.4 GeV/c 2 and a lower bound given by the sum of the η ′ nominal mass plus the mass value at the lower end of the bin, to account for the kinematic limit of the η c (1S) decay.
where the Blatt-Weisskopf barrier factor is f L = f l = 1 as the angular momenta are L = 0 for the η c (1S) and l = 0 for the f 0 (2080) [see later in this section]; the width is Γ(m) = Γ 0 (p/p 0 )(m 0 /m). Here, m and m ′ are reconstructed masses of the f 0 (2080) and η c (1S) in the candidate event; m 0 and Γ 0 are the f 0 (2080) mass and width parameters, respectively, to be fitted. In Eq. (4), q = q(m, m ′ ) is the f 0 (2080) momentum in the η c (1S) rest frame, p(m) is the π momentum in the f 0 (2080) rest frame, and p 0 = p(m 0 ). The quantities p(m) and q(m, m ′ ) in the original Blatt BW function play a role in describing the phase-space behavior of a daughter resonance with medium-size width from the decay of its parent resonance. Our study shows that an improved description for the decay of f 0 (2080), due to complications arising from the broad width of the η c (1S), is required. The improved Blatt-Weisskopf Breit-Wigner is written as where the variable q(m, m ′ ) in Eq. (4) is replaced with its probability density function f q (m) derived according to well-known statistics relations [37]. The function A composite χ 2 -fit to the bin-by-bin η c (1S) yields as a function of M π + π − is performed to extract the f 0 (980), f 2 (1270) and f 0 (2080) components. In this fit, the f 0 (980) and f 2 (1270) yields are floating while the respective shape parameters are fixed to the result [M f0 = (934.9 ± 8.6) MeV/c 2 and Γ f0 = (42±17) MeV] of the fit to the M π + π − ,sdb distribution and to PDG [33]; the f 0 (2080) mass, width and yield float. The b 1 component captures the η c decays to threebody η ′ π + π − and other two-body final states, and is described by a second-order threshold function [38] whose area and shape float in the fit.
The fit results for the f 0 (980), f 2 (1270) and f 0 (2080) components are shown in Fig. 5(b). The f 0 (980) and f 2 (1270) yields are n f0 = 49 ± 17 and n f2 = 77 +28 −29 with statistical significances of 3.1σ and 2.6σ, respectively. The f 0 (2080) yield, mass and width are determined to be n f0 = 451 +43 −41 with a statistical significance of 20σ, M = (2083 +63 −66 ) MeV/c 2 and Γ = (178 +60 −178 ) MeV. Under the assumption of an equal rising rate of the two efficiency curves for the η c (1S) decays to η ′ π + π − and η ′ f 0 (2080), the product of the two-photon decay width and branching fraction for the η c (1S) decay to The upper limit of the product for η c (1S) decays to η ′ f 0 (980) is Γ γγ B[η c (1S) → η ′ f 0 (980)] < 5.6 eV at 95% C.L. Figure 6 shows the distribution of cosθ hel for the f 0 (2080) candidate events, which are extracted by fitting the f 0 (2080) signal in each angular bin, together with MC expectations for J P C = 0 ++ and 2 ++ . Here, θ hel is the f 0 (2080) helicity angle, i.e., the angle between the pion momentum and the direction of the γγ c.m. system in the π + π − rest frame. The χ 2 /N dof values from fitting the angular distribution with a shape fixed to that obtained from MC are 1.7 for J P C = 0 ++ and 14.8 for 2 ++ . We utilize the method previously deployed by LHCb and Belle [39,40] to calculate the exclusion level of the J P C = 2 ++ hypothesis for the f 0 (2080) signal. Two sets of pseudoexperiments, for each J P C hypothesis (0 ++ , 2 ++ ), containing 1000 angular distributions, are generated with their shapes determined from the corresponding MC distributions. These angular distributions, each having the same yields as for the fitted f 0 (2080) signal in data, are fitted with the J P C = 2 ++ and the 0 ++ signal-MC shapes; the likelihood values of the fits are denoted as L 2 ++ and L 0 ++ , respectively. The resulting distribution of ∆(−2lnL) = -2(lnL 2 ++ -lnL 0 ++ ) is fitted to an asymmetric Gaussian function. The exclusion level is characterized by the p-value, which is the integral of the fitted asymmetric Gaussian function of J P C = 2 ++ normalized to 1 from the value of ∆(−2lnL) in the fit to positive infinity. The 0 ++ hypothesis is favored over the 2 ++ hypotheses at the exclusion level of 11σ.

B. Systematic uncertainties
We generate MC events for e + e − → e + e − η c (1S) two-photon production with η c (1S) → η ′ f 0 (2080), f 0 (2080) → π + π − decay. The invariant-mass distribution of the f 0 (2080) is sampled with the improved Blatt BW function in Eq. (5), which is then used to describe the M π + π − distribution for the f 0 (2080) signal in the fit. Differences in the f 0 (2080) mass and width between the input and fitted values are calculated to be 7 MeV/c 2 and 23 MeV, respectively, which are treated as uncertainties caused by the fit bias. Uncertainties coming from the fit range and the η c (1S) signal window are determined by changing these ranges. The effect on the f 0 (2080) parameters with and without f 2 (1270) in the fit are determined and used as systematic errors. The uncertainty in the b 1 background shape is estimated by a comparison of the fitted parameters when using third-order threshold function. The systematic uncertainties in the measurement of the f 0 (2080) mass, width and product Γ γγ B are summarized in Table III.

VI. MEASUREMENTS OF THE CROSS SECTIONS
We utilize the data sample selected in the ηππ mode to measure the non-resonant production of η ′ π + π − final states via two-photon collisions. The cross section of e + e − → e + e − h production is expressed as where h denotes one of two hadronic final states: η ′ π + π − or η ′ f 2 (1270). Here, θ * is the angle between the η ′ momentum and the beam direction in the γγ rest frame. The differential cross section in one W and |cosθ * | two-dimensional (2D) bin is estimated with the formula where the yield ∆N is extracted by fitting the |Σp * t | (M (π + π − )) distribution in a data subsample sliced in each 2D bin for the γγ → η ′ π + π − [γγ → η ′ f 2 (1270)] production. The efficiency ǫ(W, cosθ * ) is evaluated using MC events for each 2D bin.
The W -dependent cross sections of γγ → h are obtained by a summation over |cosθ * | bins as A. Cross sections of γγ → η ′ π + π − (including η ′ f2(1270)) We divide the W distribution between 1.40 and 3.80 GeV into 35 bins and the |cosθ * | distribution into 10 and 5 bins for the W regions of 1.40 to 2.66 GeV and 2.66 to 3.80 GeV, respectively. The defined bin size and total number of bins in W and |cosθ * | are listed in the Table IV. Detection efficiencies as a function of W and |cosθ * | are shown in Fig. 7. The yield ∆N in Eq. (8) is extracted by fitting the |Σp * t | distribution in data for each 2D bin. For the fit, the signal shape in MC is fixed, the η ′ -sdb background in data is normalized and fixed, and the b any background is described by a third-order polynomial with its constant term fixed at 0 and the other parameters floating. A background arising from η ′ → γρ decays in the candidate events of the ηππ mode is studied using the MC sample. One photon and four charged-pion tracks in the MC event, produced for the γρ mode, plus a fake photon, is wrongly chosen as an η ′ π + π − combinatorial candidate for the ηππ mode. Here, the fake photon with low momentum is a neutral track composed of background hits or hit clusters split from charged pion tracks in the ECL. This appears as a background component because of the additional fake photon in the event; it is estimated using the pre-measured cross section for γγ → η ′ π + π − in data for the ηππ mode and is found to be small. The measured cross section for γγ → η ′ π + π − for the ηππ mode after subtraction of this small contamination is shown in Fig. 8. GeV and [2.62, 3.80] GeV, respectively. The efficiency ǫ in each 2D bin, evaluated using signal MC events for γγ → η ′ f 2 (1270) with phase-space distribution, is shown in Fig. 9.
The yield ∆N of f 2 (1270) in Eq. (8) is extracted by fitting the invariant mass spectrum of π + π − for the f 2 (1270) signal using the data subsample in each 2D bin. A broad f 2 (1270) signal in the W region from 2.26 to 2.62 GeV near threshold is described by a D-wave Breit-Wigner function where M and Γ are the f 2 (1270) mass and width. The q and p momentum variables are, respectively, of the f 2 (1270) in the γγ rest frame and of the π meson from the f 2 (1270) decay in the f 2 (1270) rest frame.
In the fits, Γ is fixed to the world-average value, and M is fixed to the value extracted from fitting the π + π − invariant mass spectrum for the f 2 (1270) using events in the full range of W (|cosθ * | < 1). The f 2 (1270) signal in the W region above 2.62 GeV is described by a normal Breit-Wigner function with both M and Γ fixed to the world-average values.
We fix the fraction of the η ′ -sdb background in the fits. The combinatorial background, including nonf 2 (1270) and b any events, is described by a fourthorder polynomial with its parameters fixed to the values extracted from f 2 (1270) fit for each W bin. The W -dependent cross section for γγ → η ′ f 2 (1270) in the ηππ mode, calculated with Eq. (8), is shown in Fig. 10   given in Fig. 11.
We assume that the W and θ * dependencies of the differential cross section follow the power law σ ∝ 1/W n · sin α θ * , which is the same as that for pseudoscalar meson pairs in the Belle data and the QCD predictions [22]. In a fit to the measured cross sections for γγ → η ′ f 2 (1270) in the range of W ∈ [2.5, 3.8] GeV, the resulting W power-law exponent is n = 7.7 ± 1.5 (7.5 ± 2.0) for |cosθ * | ∈ [0.0, 0.8] (∈ [0.0, 0.6]). The differential cross sections in |cosθ * | show an ascending trend in all three W ranges, and its rate of increase is greater for events in the larger W ranges. The complicated behavior for the angular dependence of the cross sections is seen in the range of W < 2.50 GeV with markedly lower power for sin θ * of α < 4, while it tends to match with the power law for the ranges of W ∈ [2.50, 2.62] and [2.62, 3.80] GeV. C. Result for the γγ → η ′ π + π − (excluding η ′ f2(1270)) cross sections In the left plot of Fig. 12, the measured Wdependent cross sections of γγ → η ′ f 2 (1270) and γγ → η ′ π + π − [including η ′ f 2 (1270)] production are shown. The former is obtained by fitting the π + π − invariant mass spectrum for the f 2 (1270) signal and the latter is extracted in fitting the |Σp * t | distribution for the η ′ π + π − signal. Taking the difference between the two yields in each 2D bin in data as input, the cross sections of γγ → η ′ π + π − production without the η ′ f 2 (1270) contribution for the ηππ mode are calculated and shown in the right plot of Fig. 12 and summarized in Table VI. Two peaking structures are evident. The one around 1.8 GeV likely arises from the η(1760) and X(1835) decays to η ′ π + π − [11] and the other around 2.15 GeV is possibly due to γγ → X(2100) → η ′ f 0 (980) production. The η c (1S) contribution near 2.98 GeV has been subtracted. A larger data sample is necessary in order to understand these two structures in more detail.

D. Systematic uncertainty
Systematic uncertainties arising from the pion identification, π 0 -veto and η ′ -sdb background in measurements of the cross sections for both γγ → η ′ π + π − and γγ → η ′ f 2 (1270) production are esti-mated in each 2D bin, using a method similar to that in the determination of the product of twophoton width and branching fraction for the final state, Γ γγ B. Uncertainty in the trigger efficiency is calculated to be 1.2-6.7% for the ηππ mode. The uncertainty in the determination of the b any background shape is estimated by changing each parameter by ±1σ in the fit, and the difference in yields with and without this change in each parameter, added in quadrature, is taken as its contribution to the systematic uncertainty. We study the non-η ′ events with the same final state of γγ → γππππ in MC. We see that these non-η ′ events with a wrong combination of γππ, surviving the η ′ ππ selection criteria, have a peaking feature in the |Σp * t | distribution in the η ′ signal window. The contribution from non-η ′ is regarded as a lower systematic uncertainty of the cross section. The systematic uncertainties in the measurements of the cross sections are summarized in Table VII.

VII. SUMMARY AND DISCUSSIONS
The η c (1S), η c (2S), and non-resonant production of the η ′ π + π − final state via two-photon collisions )(including f are measured. The results for the yields, masses, and widths, as well as the product decay widths are summarized in Table I for the η c (1S) and η c (2S). The differential cross sections for the non-resonant states of two-body η ′ f 2 (1270) with f 2 (1270) → π + π − and three-body η ′ π + π − [excluding η ′ f 2 (1270)] in the ηππ mode are shown in Tables V and VI and Figs. 10-13.
Final state Γγγ B for ηc(1S) Γγγ B for ηc(2S) R(ηc(2S)/ηc(1S)) Γγγ (ηc(2S)) Reference (eV) (eV) (×10 −2 ) (keV) K 0 S K + π − --18 ± 5 ± 2 0.92 ± 0.28 [9] CLEO 2004 KKπ 386 ± 8 ± 21 41 ± 4 ± 6 10.6 ± 2.0 0.54 ± 0.11 [6] BaBar 2011 η ′ π + π − 65.4 ± 2.6 ± 6.9 5.6 ± 1.2 ± 1. In fact, the ratio of the two products of two-photon decay width and branching fraction for the η c (1S) and η c (2S), is a quantity directly measured in experiments. The η c (1S) and η c (2S) mesons in the measurements are all produced via two-photon process, and the dominant contributions to the systematic uncertainty in either product alone, such as those for the twophoton luminosity and reconstruction efficiencies of η and charged pion tracks, cancel almost completely in this ratio. As shown in Table VIII, the R values from the two observations-one by BaBar [6] with KKπ and the other by this analysis with η ′ π + π −are measured to be R = (10.6 ± 2.0) × 10 −2 and (8.6 ± 2.6) × 10 −2 , respectively. They are consistent with each other, while a third measurement with large uncertainty by CLEO [9] is compatible with the former. It implies that the assumption of approximate equality of the branching fractions for η c (1S) and η c (2S) to a specific final state, is reasonable within the errors. Here, the systematic uncertainty contributions in the R values [and thus the ratio of branching fractions for η c (1S) and η c (2S) decays in Eq. (13)] are conservatively estimated, since their cancellation effect in determination of the ratio R errors is not subtracted yet. Under the assumption of equal branching fractions for η c (1S) and η c (2S) decay, the twophoton decay width for η c (2S) is determined to be Γ γγ (η c (2S))= (1.3 ± 0.6) keV by CLEO [9], which lies at the lower bound of the QCD predictions [12][13][14][15][16][17]. The resulting Γ γγ (η c (2S)) value, derived from this work, is less than half of CLEO's (see Table VIII). On the other hand, the measured unequal branching fractions for η c (1S) and η c (2S) decays to KKπ, albeit with good precision for the former [33] but large uncertainty for the latter [10], indicates that an improved test of the assumption with experimental data is indeed needed. Precision measurements of the branching fraction for either η c (2S) decays to K 0 S K + π − (ηπ + π − ) or B decays to Kη c (2S) would be able to clarify the discrepancy in the two-photon decay width of η c (2S) between data and QCD predictions.