Study of (1S) radiative decays to γπ+π- and γk+K-Permalink

We study the ϒ ð 1 S Þ radiative decays to γπ þ π − and γ K þ K − using data recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy e þ e − collider at center-of-mass energies at the ϒ ð 2 S Þ and ϒ ð 3 S Þ resonances. The ϒ ð 1 S Þ resonance is reconstructed from the decay ϒ ð nS Þ → π þ π − ϒ ð 1 S Þ , n ¼ 2 , 3. Branching fraction measurements and spin-parity analyses of ϒ ð 1 S Þ radiative decays are reported for the I ¼ 0 S -wave and f 2 ð 1270 Þ resonances in the π þ π − mass spectrum, the f 0 2 ð 1525 Þ and f 0 ð 1500 Þ in the K þ K − mass spectrum, and the f 0 ð 1710 Þ in both.


I. INTRODUCTION
The existence of gluonium states is still an open issue for quantum chromodynamics (QCD). Lattice QCD calculations predict the lightest gluonium states to have quantum numbers J PC ¼ 0 þþ and 2 þþ and to be in the mass region below 2.5 GeV=c 2 [1]. In particular, the J PC ¼ 0 þþ glueball is predicted to have a mass around 1.7 GeV=c 2 . Searches for these states have been performed using many supposed "gluon rich" reactions. However, despite intense experimental searches, there is no conclusive experimental evidence for their direct observation [2,3]. The identification of the scalar glueball is further complicated by the possible mixing with standard qq states. The broad f 0 ð500Þ, f 0 ð1370Þ [4], f 0 ð1500Þ [5,6], and f 0 ð1710Þ [7] have been suggested as scalar glueball candidates. A feature of the scalar glueball is that its ss decay mode should be favored with respect to uū or dd decay modes [8,9].
Radiative decays of heavy quarkonia, in which a photon replaces one of the three gluons from the strong decay of J=ψ or ϒð1SÞ, can probe color-singlet two-gluon systems that produce gluonic resonances. Recently, detailed calculations have been performed on the production rates of the scalar glueball in the process Vð1 −− Þ → γG, where G indicates the scalar glueball and Vð1 −− Þ indicates charmonium or bottomonium vector mesons such as J=ψ, ψð2SÞ, or ϒð1SÞ [10][11][12][13].
J=ψ decays have been extensively studied in the past [14] and are currently analyzed in e þ e − interactions by BES experiments [15,16]. The experimental observation of radiative ϒð1SÞ decays is challenging because their rate is suppressed by a factor of ≈0.025 compared to J=ψ radiative decays, which are of order 10 −3 [17]. Radiative ϒð1SÞ decays to a pair of hadrons have been studied by the CLEO Collaboration [17,18] with limited statistics and large backgrounds from e þ e − → γ ðvector mesonÞ. In this work, we observe ϒð1SÞ decays through the decay chain ϒð2SÞ=ϒð3SÞ → π þ π − ϒð1SÞ. This allows us to study ϒð1SÞ radiative decays to π þ π − and K þ K − final states with comparable statistics, but lower background. This paper is organized as follows. In Sec. II, we give a brief description of the BABAR detector, and Sec. III is devoted to the description of event reconstruction. In Sec. IV, we study resonance production in π þ π − and K þ K − final states, and Sec. V is devoted to the description of the efficiency correction. We describe in Sec. VI a study of the angular distributions using a Legendre polynomial moments analysis, while Sec. VII gives results on the full angular analysis. The measurement of the branching fractions is described in Sec. VIII, and the results are summarized in Sec. IX.

II. THE BABAR DETECTOR AND DATA SET
The results presented here are based on data collected by the BABAR detector with the PEP-II asymmetric-energy e þ e − collider located at SLAC, at the ϒð2SÞ and ϒð3SÞ resonances with integrated luminosities [19] of 13.6 and 28.0 fb −1 , respectively. The BABAR detector is described in detail elsewhere [20]. The momenta of charged particles are measured by means of a five-layer, double-sided microstrip detector, and a 40-layer drift chamber, both operating in the 1.5 T magnetic field of a superconducting solenoid. Photons are measured and electrons are identified in a CsI(Tl) crystal electromagnetic calorimeter (EMC). Chargedparticle identification is provided by the measurement of specific energy loss in the tracking devices, and by an internally reflecting, ring-imaging Cherenkov detector. Muons and K 0 L mesons are detected in the instrumented flux return of the magnet. Monte Carlo (MC) simulated events [21], with reconstructed sample sizes more than 100 times larger than the corresponding data samples, are used to evaluate the signal efficiency.

III. EVENTS RECONSTRUCTION
We reconstruct the decay chains and where we label with the subscript s the slow pions from the direct ϒð2SÞ and ϒð3SÞ decays. We consider only events containing exactly four well-measured tracks with transverse momentum greater than 0.1 GeV=c and a total net charge equal to zero. We also require exactly one wellreconstructed γ in the EMC having an energy greater than 2.5 GeV. To remove background originating from π 0 mesons we remove events having π 0 candidates formed with photons having an energy greater than 100 MeV. The four tracks are fitted to a common vertex, with the requirements that the fitted vertex be within the e þ e − interaction region and have a χ 2 fit probability greater than 0.001. We select muons, electrons, kaons, and pions by applying highefficiency particle identification criteria [22]. For each track we test the electron and muon identification hypotheses and remove the event if any of the charged tracks satisfies a tight muon or electron identification criterion. We require momentum balance for the four final states, making use of a χ 2 distribution defined as where Δp i are the missing laboratory three-momenta components and hΔp i i and σ i are the mean values and the widths of the missing momentum distributions. These are obtained from signal MC simulations of the four final states through two or three Gaussian function fits to the MC balanced momentum distributions. When multiple Gaussian functions are used, the mean values and σ quoted are average values weighted by the relative fractions. In Eq. (4), p i indicates the three components of the laboratory momenta of the five particles in the final state, while p e þ i and p e − i indicate the three-momenta of the incident beams. Figure 1 shows the χ 2 distributions for reac- s Þðγπ þ π − Þ, respectively compared with signal MC simulations. The accumulations at thresholds represent events satisfying momentum balance. We apply a very loose selection, χ 2 < 60, optimized using the ϒð2SÞ data, and remove events consistent with being entirely due to background. We note a higher background in the ϒð3SÞ data, but keep the same loose selection to achieve a similar efficiency.
Events with balanced momentum are then required to satisfy energy balance requirements. In the above decays, the π s originating from direct ϒð2SÞ=ϒð3SÞ decays have a soft laboratory momentum distribution (< 600 MeV=c), partially overlapping with the hard momentum distributions for the hadrons originating from the ϒð1SÞ decay. We therefore require energy balance, following a combinatorial approach.
For each combination of π þ s π − s candidates, we first require both particles to be identified loosely as pions and compute the recoiling mass, where p is the particle four-momentum. The distribution of M 2 rec ðπ þ s π − s Þ is expected to peak at the squared ϒð1SÞ mass for signal events. Figure 2 shows the combinatorial recoiling mass M rec ðπ þ s π − s Þ for ϒð2SÞ and ϒð3SÞ data, where narrow peaks at the ϒð1SÞ mass can be observed.
We fit each of these distributions using a linear function for the background and the sum of two Gaussian functions for the signal, obtaining average σ ¼ 2.3 MeV=c 2 and σ ¼ 3.5 MeV=c 2 values for the ϒð2SÞ and ϒð3SÞ data, respectively. We select signal event candidates by requiring where mðϒð1SÞÞ f indicates the fitted ϒð1SÞ mass value. We obtain, in the above mass window, values of signal-tobackground ratios of 517=40 and 276=150 for ϒð2SÞ and ϒð3SÞ data, respectively. To reconstruct ϒð1SÞ → γπ þ π − decays, we require a loose identification of both pions from the ϒð1SÞ decay and obtain the distributions of mðγπ þ π − Þ shown in Fig. 3. The distributions show the expected peak at the ϒð1SÞ mass with little background but do not have a Gaussian shape due to the asymmetric energy response of the EMC to a high-energy photon. The full line histograms compare the data with signal MC simulations and show good agreement.
We reconstruct the final state where ϒð1SÞ → γK þ K − in a similar manner, by applying a loose identification of both kaons in the final state and requiring the mðK þ K − γÞ mass, shown in Fig. 4, to be in the range 9.1 GeV=c 2 < mðK þ K − γÞ < 9.6 GeV=c 2 : IV. STUDY OF THE π + π − AND K + K − MASS SPECTRA The π þ π − mass spectrum, for mðπ þ π − Þ < 3.0 GeV=c 2 and summed over the ϒð2SÞ and ϒð3SÞ data sets with 507 and 277 events, respectively, is shown in Fig. 5(a). The resulting K þ K − mass spectrum, summed over the ϒð2SÞ  and ϒð3SÞ data sets with 164 and 63 events, respectively, is shown in Fig. 5(b). For a better comparison the two distributions are plotted using the same bin size and the same mass range.
We study the background for both π þ π − and K þ K − final states using the M rec ðπ þ s π − s Þ sidebands. We select events in the ð4.5σ − 7.0σÞ regions on both sides of the signal region and require the mðπ þ π − γÞ and mðK þ K − γÞ to be in the ranges defined by Eqs. (7) and (8), respectively. The resulting π þ π − and K þ K − mass spectra for these events are superimposed in gray in Figs. 5(a) and 5(b), respectively. We note rather low background levels for all the final states, except for the π þ π − mass spectrum from the ϒð3SÞ data, which shows an enhancement at a mass of ≈750 MeV=c 2 , which we attribute to the presence of ρð770Þ 0 background. The π þ π − mass spectrum from inclusive ϒð3SÞ decays also shows a strong ρð770Þ 0 contribution.
We search for background originating from a possible hadronic ϒð1SÞ → π þ π − π 0 decay, where one of the two γ's from the π 0 decay is lost. For this purpose, we make use of the ϒð2SÞ data and select events having four charged pions and only one π 0 candidate. We then select events satisfying Eq. (6) and plot the π þ π − π 0 effective mass distribution. No ϒð1SÞ signal is observed, which indicates that the branching fraction for this possible ϒð1SÞ decay mode is very small and therefore that no contamination is expected in the study of the ϒð1SÞ → γπ þ π − decay mode.
The π þ π − mass spectrum, in 30 MeV=c 2 bin size is shown in Fig. 6. The spectrum shows I ¼ 0, J P ¼ even þþ resonance production, with low backgrounds above 1 GeV=c 2 . We observe a rapid drop around 1 GeV=c 2 characteristic of the presence of the f 0 ð980Þ, and a strong f 2 ð1270Þ signal. The data also suggest the presence of weaker resonant contributions. The K þ K − mass spectrum is shown in Fig. 7 and also shows resonant production, with low background. Signals at the positions of f 0 2 ð1525Þ and f 0 ð1710Þ can be observed.
We make use of a phenomenological model to extract the different ϒð1SÞ → γR branching fractions, where R is an intermediate resonance.
and ϒð3SÞ data sets. The gray distributions show the expected background obtained from the corresponding M rec ðπ þ s π − s Þ sidebands. The light-gray distributions evidences the background contribution from the ϒð2SÞ data.
A. Fit to the π + π − mass spectrum We perform a simultaneous binned fit to the π þ π − mass spectra from the ϒð2SÞ and ϒð3SÞ data sets using the following model.
(i) We describe the low-mass region (around the f 0 ð500Þ) using a relativistic S-wave Breit-Wigner lineshape having free parameters. We test the S-wave hypothesis in Secs. VI and VIII. We obtain its parameters from the ϒð2SÞ data only, and we fix them in the description of the ϒð3SÞ data. (ii) We describe the f 0 ð980Þ using the Flatté [23] formalism. For the π þ π − channel the Breit-Wigner lineshape has the form and in the K þ K − channel the Breit-Wigner function has the form where Γ i is absorbed into the intensity of the resonance. Γ π ðmÞ and Γ K ðmÞ describe the partial widths of the resonance to decay to ππ and KK and are given by where g π and g K are the squares of the coupling constants of the resonance to the ππ and KK systems. The f 0 ð980Þ parameters and couplings are taken from Ref. [24]: The total S-wave is described by a coherent sum of f 0 ð500Þ and f 0 ð980Þ as where c and ϕ are free parameters for the relative intensity and phase of the two interfering contributions. (iv) The f 2 ð1270Þ and f 0 ð1710Þ resonances are represented by relativistic Breit-Wigner functions with parameters fixed to PDG values [25]. (v) In the high π þ π − mass region, we are unable, with the present statistics, to distinguish the different possible resonant contributions. Therefore we make use of the method used by CLEO [26] and include a single resonance f 0 ð2100Þ having a width fixed to the PDG value (224 AE 22) and unconstrained mass. (vi) The background is parametrized with a quadratic dependence where pðmÞ is the π center-of-mass momentum in the π þ π − rest frame, which goes to zero at π þ π − threshold. (vii) For the ϒð3SÞ data we also include ρð770Þ 0 background with parameters fixed to the PDG values. The fit is shown in Fig. 6. It has 16 free parameters and χ 2 ¼ 182 for ndf ¼ 152, corresponding to a p-value of 5%. The yields and statistical significances are reported in Table I. Significances are computed as follows: for each resonant contribution (with fixed parameters) we set the yield to zero and compute the significance as σ ¼ , where Δχ 2 is the difference in χ 2 between the fit with and without the presence of the resonance.
The table also reports systematic uncertainties on the yields, evaluated as follows: the parameters of each resonance are modified according to AEσ, where σ is the PDG uncertainty and the deviations from the reference fit are added in quadrature. The background has been modified to have a linear shape. The effective range in the Blatt-Weisskopf [27] factors entering in the description of the intensity and the width of the relativistic Breit-Wigner function have been varied between 1 and 5 GeV −1 , and the average deviation is taken as a systematic uncertainty. The different contributions, dominated by the uncertainties on the resonances parameters, are added in quadrature.
We note the observation of a significant S-wave in ϒð1SÞ radiative decays. This observation was not possible in the study of J=ψ radiative decay to π þ π − because of the presence of a strong, irreducible background from J=ψ → π þ π − π 0 [28]. We obtain the following f 0 ð500Þ parameters: and ϕ ¼ 2.41 AE 0.43 rad. The fraction of S-wave events associated with the f 0 ð500Þ is ð27.7 AE 3.1Þ%. We also obtain mðf 0 ð2100ÞÞ ¼ 2.208 AE 0.068 GeV=c 2 .
B. Study of the K + K − mass spectrum Due to the limited statistics we do not separate the data into the ϒð2SÞ and ϒð3SÞ data sets. We perform a binned fit to the combined K þ K − mass spectrum using the following model: (i) The background is parametrized with a linear dependence starting with zero at threshold.  Fig. 7. It has six free parameters and χ 2 ¼ 35 for ndf ¼ 29, corresponding to a p-value of 20%; the yields and significances are reported in Table I. Systematic uncertainties have been evaluated as for the fit to the π þ π − mass spectrum. The parameters of each resonance are modified according to AEσ, where σ is the PDG uncertainty and the deviations from the reference fit are added in quadrature. The background has been modified to have a quadratic shape. The effective range in the Blatt-Weisskopf [27] factors entering in the description of the intensity and the width of the relativistic Breit-Wigner function have been varied between 1 and 5 GeV −1 , and the average deviation is taken as a systematic uncertainty. The different contributions, dominated by the uncertainties on the resonances parameters, are added in quadrature. In the 1500 MeV=c 2 mass region, both f 0 2 ð1525Þ and f 0 ð1500Þ can contribute, therefore we first fit the mass spectrum assuming the presence of f 0 2 ð1525Þ only and then replace in the fit the f 0 2 ð1525Þ with the f 0 ð1500Þ resonance. In Table I, we label this contribution as f J ð1500Þ. The resulting yield variation between the two fits is small and gives a negligible contribution to the total systematic uncertainty. A separation of the f 0 2 ð1525Þ and f 0 ð1500Þ contributions is discussed in Secs. VI and VII. I. Resonances yields and statistical significances from the fits to the π þ π − and K þ K − mass spectra for the ϒð2SÞ and ϒð3SÞ data sets. The symbol f J ð1500Þ indicates the signal in the 1500 MeV=c 2 mass region. When two errors are reported, the first is statistical and the second systematic. Systematic uncertainties are evaluated only for resonances for which we compute branching fractions.

A. Reconstruction efficiency
To compute the efficiency, MC signal events are generated using a detailed detector simulation [21]. These simulated events are reconstructed and analyzed in the same manner as data. The efficiency is computed as the ratio between reconstructed and generated events. The efficiency distributions as functions of mass, for the ϒð2SÞ=ϒð3SÞ data and for the π þ π − γ and K þ K − γ final states, are shown in Fig. 8. We observe an almost uniform behavior for all the final states.
We define the helicity angle θ H as the angle formed by the h þ (where h ¼ π, K), in the h þ h − rest frame, and the γ in the h þ h − γ rest frame. We also define θ γ as the angle formed by the radiative photon in the h þ h − γ rest frame with respect to the ϒð1SÞ direction in the ϒð2SÞ=ϒð3SÞ rest frame.
We compute the efficiency in two different ways.
(i) We label with ϵðm; cos θ H Þ the efficiency computed as a function of the h þ h − effective mass and the helicity angle cos θ H . This is used only to obtain efficiency-corrected mass spectra. (ii) We label with ϵðcos θ H ; cos θ γ Þ the efficiency computed, for each resonance mass window (defined in Table III), as a function of cos θ H and cos θ γ . This is used to obtain the efficiency-corrected angular distributions and branching fractions of the different resonances. To smoothen statistical fluctuations in the evaluation of ϵðm; cos θ H Þ, for ϒð1SÞ → γπ þ π − , we divide the π þ π − mass into nine 300-MeV=c 2 -wide intervals and plot the cos θ H in each interval. The distributions of cos θ H are then fitted using cubic splines [29]. The efficiency at each mðπ þ π − Þ is then computed using a linear interpolation between adjacent bins. Figure 9 shows the efficiency distributions in the (mðπ þ π − Þ, cos θ H ) plane for the ϒð2SÞ and ϒð3SÞ data sets. We observe an almost uniform behavior with some loss at cos θ H close to AE1. The efficiencies integrated over cos θ H are consistent with being constant with mass and have average values of ϵðϒð2SÞ→π þ π − ϒð1SÞð→γπ þ π − ÞÞ¼ 0.237AE0.001 and ϵðϒð3SÞ → π þ π − ϒð1SÞð→ γπ þ π − ÞÞ ¼ 0.261 AE 0.001.
We also compute the efficiency in the ðcos θ H ; cos θ γ Þ plane for each considered resonance decaying to π þ π − and K þ K − . Since there are no correlations between these two variables, we parametrize the efficiency as ϵðcos θ H ; cos θ γ Þ ¼ ϵðcos θ H Þ × ϵðcos θ γ Þ: ð14Þ The distributions of the efficiencies as functions of cos θ H and cos θ γ are shown in Fig. 11 for the f 2 ð1270Þ → π þ π − and f 0 2 ð1525Þ → K þ K − mass regions, for the ϒð2SÞ data sets. To smoothen statistical fluctuations, the efficiency projections are fitted using seventh-and fourth-order polynomials, respectively. Similar behavior is observed for the other resonances and for the ϒð3SÞ data sets.

B. Efficiency correction
To obtain the efficiency correction weight w R for the resonance R, we divide each event by the efficiency ϵðcos θ H ; cos θ γ Þ, where N R is the number of events in the resonance mass range. The resulting efficiency weight for each resonance is reported in Table II. We compute separately the ϒð2SÞ and ϒð3SÞ yields for resonances decaying to π þ π − while, due to the limited statistics, for resonances decaying to K þ K − the two data sets are merged and corrected using the weighted average efficiency. The systematic effect related to the effect of particle identification is assessed by the use of high statistics control samples. We assign systematic uncertainties of 0.2% to the identification of each pion and 1.0% to that of each kaon. We include an efficiency correction of 0.9885 AE 0.0065 to the reconstruction of the high energy photon, obtained from studies on Data/MC detection efficiency. The efficiency correction contribution due to the limited MC statistics is included using the statistical uncertainty on the average efficiency weight as well as the effect of the fitting procedure. The above effects are added in quadrature and are presented in Table II as systematic uncertainties related to the efficiency correction weight. Finally, we propagate the systematic effect on event yields obtained from the fits to the mass spectra. The resulting efficiency corrected yields are reported in Table II.

VI. LEGENDRE POLYNOMIAL MOMENTS ANALYSIS
To obtain information on the angular momentum structure of the π þ π − and K þ K − systems in ϒð1SÞ → γh þ h − , we study the dependence of the mðh þ h − Þ mass on the helicity angle θ H . Figure 12 shows the scatter plot cos θ H vs mðπ þ π − Þ for the combined ϒð2SÞ and ϒð3SÞ data sets. We observe the spin 2 structure of the f 2 ð1270Þ.
A better way to observe angular effects is to plot the π þ π − mass spectrum weighted by the Legendre polynomial moments, corrected for efficiency. In a simplified environment, the moments are related to the spin 0 (S) and spin 2 (D) amplitudes by the equations [30]:  FIG. 11. Efficiency as a function of (a) cos θ H and (b) cos θ γ for ϒð2SÞ → π þ s π − s ϒð1SÞ → γf 2 ð1270Þð→ π þ π − Þ. Efficiency as a function of (c) cos θ H and (d) cos The lines are the result of the polynomial fits.
where ϕ SD is the relative phase. Therefore, we expect to observe spin 2 resonances in hY 0 4 i and S=D interference in hY 0 2 i. The results are shown in Fig. 13. We clearly observe the f 2 ð1270Þ resonance in hY 0 4 i and a sharp drop in hY 0 2 i at the f 2 ð1270Þ mass, indicating the interference effect. The distribution of hY 0 0 i is just the scaled π þ π − mass distribution, corrected for efficiency. Odd L moments are sensitive to the cos θ H forward-backward asymmetry and show weak activity at the position of the f 2 ð1270Þ mass. Higher moments are all consistent with zero.
Similarly, we plot in Fig. 14 the K þ K − mass spectrum weighted by the Legendre polynomial moments, corrected for efficiency. We observe signals of the f 0 2 ð1525Þ and f 0 ð1710Þ in hY 0 4 i and activity due to S=D interference effects in the hY 0 2 i moment. Higher moments are all consistent with zero.
Resonance angular distributions in radiative ϒð1SÞ decays from ϒð2SÞ=ϒð3SÞ decays are rather complex and will be studied in Sec. VIII. In this section, we perform a simplified partial wave analysis (PWA) solving directly the system of Eq. (16). Figures 15 and 16 show the resulting S-wave and D-wave contributions to the π þ π − and K þ K − mass spectra, respectively. Due to the presence of background in the threshold region, the π þ π − analysis is performed only on the ϒð2SÞ data. The relative ϕ SD phase is not plotted because it is affected by very large statistical errors.
We note that in the case of the π þ π − mass spectrum we obtain a good separation between Sand D-waves, with the presence of an f 0 ð980Þ resonance on top of a broad f 0 ð500Þ resonance in the S-wave and a clean f 2 ð1270Þ in the D-wave distribution. Integrating the S-wave amplitude from threshold up to a mass of 1.5 GeV=c 2 , we obtain an integrated, efficiency corrected yield, in agreement with the results from the fit to the π þ π − mass spectrum (see Table II). We also compute the fraction of S-wave contribution in the f 2 ð1270Þ mass region defined in Table III and obtain f S ðπ þ π − Þ ¼ 0.16 AE 0.02.
In the case of the K þ K − PWA, the structure peaking around 1500 MeV=c 2 appears in both Sand D-waves suggesting the presence of f 0 ð1500Þ and f 0 2 ð1525Þ. In the f 0 ð1710Þ mass region, there is not enough data to discriminate between the two different spin assignments. This pattern is similar to that observed in the Dalitz plot analysis of charmless B → 3K decays [31]. Integrating the Sand  Table III, we obtain a fraction of S-wave contribution f S ðK þ K − Þ ¼ 0.53 AE 0.10.

VII. SPIN-PARITY ANALYSIS
We compute the helicity angle θ π defined as the angle formed by the π þ s , in the π þ s π − s rest frame, with respect to the direction of the π þ s π − s system in the ϒð1SÞπ þ s π − s rest frame. This distribution is shown in Fig. 17 for the ϒð2SÞ data and ϒð1SÞ → γπ þ π − , and is expected to be uniform if π þ s π − s is an S-wave system. The distribution is consistent with this hypothesis with a p-value of 65%.
Ignoring the normalization factors jC 10 j 2 and jE 00 j 2 , the distribution has only one free parameter, jA 01 j 2 =jA 00 j 2 .
We perform a two-dimensional unbinned maximum likelihood fit for each resonance region defined in Table III. If N is the number of available events, the likelihood function L is written as where f sig is the signal fraction, ϵðcos θ H ; cos θ γ Þ is the fitted efficiency [Eq. (14)], and W s and W b are the functions describing signal and background contributions, given by Eq. (18) or Eq. (19). Since the background under the π þ π − and K þ K − mass spectra is negligible in the low-mass regions, we include only the tails of nearby adjacent resonances. In the description of the π þ π − data in the threshold region, we make use only of the ϒð2SÞ data because of the presence of a sizeable ρð770Þ 0 background in the ϒð3SÞ sample.
We first fit the f 2 ð1270Þ angular distributions and allow a background contribution of 16% (see Sec. VII) from the S-wave having fixed parameters. Therefore an iterative procedure of fitting the S-wave and f 2 ð1270Þ regions is performed. Figure 18 shows the uncorrected fit projections on cos θ H and cos θ γ . The cos θ γ spectrum is approximately uniform, while cos θ H shows structures well-fitted by the spin 2 hypothesis. Table III summarizes the results from the fits. We use as figures of merit χ H ¼ χ 2 ðcos θ H Þ, χ γ ¼ χ 2 ðcos θ γ Þ and their sum χ 2 t ¼ ðχ H þ χ γ Þ=ndf computed as the χ 2 values obtained from the cos θ H and cos θ γ projections, respectively. We use ndf ¼ N cells − N par , where N par is the number of free parameters in the fit and N cells is the sum of the number of bins along the cos θ H and cos θ γ axes. We note a good description of the cos θ H projection but a poor description of the cos θ γ projection. This may be due to the possible presence of additional scalar components in the f 2 ð1270Þ mass region, not taken into account in the formalism used in this analysis.
We fit the S-wave region in the π þ π − mass spectrum from the ϒð2SÞ decay including as background the spin 2 contribution due to the tail of the f 2 ð1270Þ. The latter is estimated to contribute with a fraction of 9%, with parameters fixed to those obtained from the f 2 ð1270Þ spin analysis described above. Figure 19 shows the fit  projections on the cos θ H and cos θ γ distributions and Table III gives details on the fitted parameters. We obtain a good description of the data consistent with the spin 0 hypothesis. We fit the K þ K − data in the f J ð1500Þ mass region, where many resonances can contribute: f 0 2 ð1525Þ, f 0 ð1500Þ [31], and f 0 ð1710Þ. We fit the data using a superposition of Sand D-waves, having helicity contributions as free parameters, and free S-wave contribution. We obtain an S-wave contribution of f S ðK þ K − Þ ¼ 0.52 AE 0.14, in agreement with the estimate obtained in Sec. VI. The helicity contributions are given in Table III and fit projections are shown in Fig. 20, giving an adequate description of the data. We assign the spin-2 contribution to the f 0 2 ð1525Þ and the spin-0 contribution to the f 0 ð1500Þ resonance. We also fit the data assuming the presence of the spin-2 f 0 2 ð1525Þ only hypothesis. We obtain a likelihood variation of Δð−2 log LÞ ¼ 1.3 for the difference of two parameters between the two fits. Due the low statistics we cannot statistically distinguish between the two hypotheses.

VIII. MEASUREMENT OF BRANCHING FRACTIONS
We determine the branching fraction BðRÞ for the decay of ϒð1SÞ to photon and resonance R using the expression where N R indicates the efficiency-corrected yield for the given resonance. To reduce systematic uncertainties, we first compute the relative branching fraction to the reference channel ϒðnSÞ → π þ π − ϒð1SÞð→ μ þ μ − Þ, which has the same number of charged particles as the final states under study. We then multiply the relative branching fraction by the well-measured branching fraction Bðϒð1SÞ → μ þ μ − Þ ¼ 2.48 AE 0.05% [25]. We determine the reference channel corrected yield using the method of "B-counting," also used to obtain the number of produced ϒð2SÞ and ϒð3SÞ [22]. Taking into account the known branching fractions of ϒð2SÞ=ϒð3SÞ → π þ s π − s ϒð1SÞ, we obtain Nðϒð2SÞ → π þ s π − s ϒð1SÞð→ μ þ μ − ÞÞ ¼ ð4.35 AE 0.12 sys Þ × 10 5 ð22Þ and Nðϒð3SÞ → π þ s π − s ϒð1SÞð→ μ þ μ − ÞÞ ¼ ð1.32 AE 0.04 sys Þ × 10 5 events. As a cross-check, we reconstruct ϒðnSÞ → π þ π − ϒð1SÞð→ μ þ μ − Þ corrected for efficiency and obtain yields in good agreement with those obtained using the method of "B-counting." Table IV gives the measured branching fractions. In all cases, we correct the efficiency corrected yields for isospin and for PDG measured branching fractions [25]. In these measurements, the f 2 ð1270Þ yield is corrected first for the π 0 π 0 (33.3%) and then for the ππ (84.2 þ2.9 −0.9 %) branching fractions. We also correct the ππ S-wave and f 0 ð1710Þ branching fractions for the π 0 π 0 decay mode. In the case of f J ð1500Þ → K þ K − , the spin analysis reported in Secs. VI and VII gives indications of the presence of overlapping f 0 2 ð1525Þ and f 0 ð1500Þ contributions. We give the branching fraction for f J ð1500Þ → K þ K − and, separately, for the f 0 2 ð1525Þ and f 0 ð1500Þ, where we make use of the S-wave contribution f S ðK þ K − Þ ¼ 0.52 AE 0.14, obtained in Sec. VII.
The f 0 2 ð1525Þ branching fraction is corrected for the KK decay mode (ð88.7 AE 2.2Þ%). For all the resonances decaying to KK, the branching fractions are corrected for the unseen K 0K0 decay mode (50%).
For the f 2 ð1270Þ and f 0 ð1710Þ resonances decaying to π þ π − , the relative branching ratios are computed separately for the ϒð2SÞ and Υð3SÞ data sets, obtaining good