Study of $\Upsilon(1S)$ Radiative Decays to $\gamma \pi^+ \pi^-$ and $\gamma K^+ K^-$

We study the $\Upsilon(1S)$ radiative decays to $\gamma \pi^+ \pi^-$ and $\gamma K^+ K^-$ using data recorded with the BaBar detector operating at the SLAC PEP-II asymmetric-energy $e^+ e^-$ collider at center-of-mass energies at the $\Upsilon(2S)$ and $\Upsilon(3S)$ resonances. The $\Upsilon(1S)$ resonance is reconstructed from the decay $\Upsilon(nS)\to \pi^+ \pi^- \Upsilon(1S)$, $n=2,3$. Branching fraction measurements and spin-parity analyses of $\Upsilon(1S)$ radiative decays are reported for the I=0 $S$-wave and $f_2(1270)$ resonances in the $\pi^+ \pi^-$ mass spectrum, the $f_2'(1525)$ and $f_0(1500)$ in the $ K^+ K^-$ mass spectrum, and the $f_0(1710)$ 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 P C = 0 ++ and 2 ++ and to be in the mass region below 2.5 GeV/c 2 [1]. In particular, the J P C = 0 ++ 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 (500), f 0 (1370) [4], f 0 (1500) [5,6], and f 0 (1710) [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 Υ (1S), 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 V (1 −− )→γG, where G indicates the scalar glueball and V (1 −− ) indicates charmonium or bottomonium vector mesons such as J/ψ, ψ(2S), or Υ (1S) [10][11][12][13].
J/ψ decays have been extensively studied in the past [14] and are currently analyzed in e + e − interactions by BES experiments [15,16]. The experimental observation of radiative Υ (1S) 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 Υ (1S) decays to a pair of hadrons have been studied by the CLEO collaboration [17,18] with limited statistics and large backgrounds from e + e − →γ (vector meson). In this work we observe Υ (1S) decays through the decay chain Υ (2S)/Υ (3S)→π + π − Υ (1S). This allows us to study Υ (1S) radiative decays to π + π − and K + 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 + π − and K + 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 DATASET
The results presented here are based on data collected by the BABAR detector with the PEP-II asymmetricenergy e + e − collider located at SLAC, at the Υ (2S) and Υ (3S) 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 fivelayer, 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). Charged-particle 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 Υ (2S) and Υ (3S) 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 well-reconstructed 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 + e − interaction region and have a χ 2 fit probability greater than 0.001. We select muons, electrons, kaons, and pions by applying high-efficiency 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 ∆p i are the missing laboratory three-momenta components and ∆p 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 + i and p e − i indicate the three-momenta of the incident beams. Figure 1 shows the χ 2 distributions for reactions (1) 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 Υ (2S) data, and remove events consistent with being entirely due to background. We note a higher background in the Υ (3S) 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 Υ (2S)/Υ (3S) decays have a soft laboratory momentum distribution (< 600 MeV/c), partially overlapping with the hard momentum distributions for the hadrons originating from the Υ (1S) decay. We therefore require energy balance, following a combinatorial approach.
For each combination of π + 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 (π + s π − s ) is expected to peak at the squared Υ (1S) mass for signal events. Figure 2 shows the combinatorial recoiling mass M rec for Υ (2S) and Υ (3S) data, where narrow peaks at the Υ (1S) 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 = 2.3 MeV/c 2 and σ = 3.5 MeV/c 2 values for the Υ (2S) and Υ (3S) data, respectively. We select signal event candidates by requiring where m(Υ (1S)) f indicates the fitted Υ (1S) mass value. We obtain, in the above mass window, values of signalto-background ratios of 517/40 and 276/150 for Υ (2S) and Υ (3S) data, respectively.
To reconstruct Υ (1S)→γπ + π − decays, we require a loose identification of both pions from the Υ (1S) decay and obtain the distributions of m(γπ + π − ) shown in 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.
At this stage no more than one candidate per event is present.
We reconstruct the final state where Υ (1S)→γK + K − in a similar manner, by applying a loose identification of both kaons in the final state and requiring the m(K + K − γ) mass, shown in Fig. 4, to be in the range 9.1 GeV/c 2 < m(K + K − γ) < 9.6 GeV/c 2 .
IV. STUDY OF THE π + π − AND K + K − MASS SPECTRA The π + π − mass spectrum, for m(π + π − ) < 3.2 GeV/c 2 and summed over the Υ (2S) and Υ (3S) datasets with 507 and 277 events, respectively, is shown in Fig. 5. The spectrum shows I = 0, J P = even ++ 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 (980), and a strong f 2 (1270) signal. The data also suggest the presence of weaker resonant contributions. We also note a small enhancement at a mass of ≈ 750 MeV/c 2 , which we attribute to the presence of ρ(770) 0 background originating from the Υ (3S). This is confirmed by the π + π − mass spectrum for events in the M rec sidebands, satisfying the selection given by Eq. (7), which show an accumulation of events in the ρ(770) 0 mass region. The π + π − mass spectrum from inclusive Υ (3S) decays also shows a strong ρ(770) 0 contribution.
We search for background originating from a possible hadronic Υ (1S)→π + 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 Υ (2S) 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 + π − π 0 effective mass distribution. No Υ (1S) signal is observed, which indicates that the branching fraction for this possible Υ (1S) decay mode is very small and therefore that no contamination is expected in the study of the Υ (1S)→γπ + π − decay mode.
The resulting K + K − mass spectrum, summed over the Υ (2S) and Υ (3S) datasets with 164 and 63 events, respectively, is shown in Fig. 6. The spectrum also shows resonant production, with low background. Signals at the positions of f ′ 2 (1525) and f 0 (1710) can be observed. We make use of a phenomenological model to extract the different Υ (1S)→γR branching fractions, where R is an intermediate resonance. A. Fit to the π + π − mass spectrum We perform a simultaneous binned fit to the π + π − mass spectra from the Υ (2S) and Υ (3S) datasets using the following model.
• We describe the low-mass region (around the f 0 (500)) using a relativistic S-wave Breit-Wigner lineshape having free parameters. We test the Swave hypothesis in Sec.VI and Sec.VIII. We obtain its parameters from the Υ (2S) data only, and we fix them in the description of the Υ (3S) data.
• We describe the f 0 (980) using the Flatté [23] formalism. For the π + π − channel the Breit-Wigner lineshape has the form and in the K + K − channel the Breit-Wigner function has the form where Γ i is absorbed into the intensity of the resonance. Γ π (m) and Γ K (m) describe the partial widths of the resonance to decay to ππ and KK and are given by I: Resonances yields and statistical significances from the fits to the π + π − and K + K − mass spectra for the Υ (2S) and Υ (3S) datasets. The symbol fJ (1500) 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.
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 (980) parameters and couplings are taken from Ref. [24]: m 0 = 0.979 ± 0.004 GeV/c 2 , g π = 0.28 ± 0.04 and g K = 0.56 ± 0.18.
• The total S-wave is described by a coherent sum of f 0 (500) and f 0 (980) as where c and φ are free parameters for the relative intensity and phase of the two interfering contributions.
• In the high π + 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 (2100) having a width fixed to the PDG value (224 ± 22) and unconstrained mass.
• The background is parameterized with a quadratic dependence where p(m) is the π center-of-mass momentum in the π + π − rest frame, which goes to zero at π + π − threshold.
• For the Υ (3S) data we also include ρ(770) 0 background with parameters fixed to the PDG values.
The fit is shown in Fig. 5. It has 16 free parameters and χ 2 = 182 for ndf=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 = ∆χ 2 , where ∆χ 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 ±σ, 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 are added in quadrature.
B. Study of the K + K − mass spectrum.
Due to the limited statistics we do not separate the data into the Υ (2S) and Υ (3S) datasets. We perform a binned fit to the combined K + K − mass spectrum using the following model: • The background is parameterized with a linear dependence starting with zero at threshold.
• The f 0 (980) is parameterized according to the Flatté formalism described by Eq. (10) for the K + K − projection.
• We include an f 0 (2200) contribution having parameters fixed to the PDG values.
The fit shown in Fig. 6. It has six free parameters and χ 2 = 35 for ndf=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 + π − mass spectrum. In the 1500 MeV/c 2 mass region both f ′ 2 (1525) and f 0 (1500) can contribute, therefore we first fit the mass spectrum assuming the presence of f ′ 2 (1525) only and then replace in the fit the f ′ 2 (1525) with the f 0 (1500) resonance. In Table I we label this contribution as f J (1500). 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 ′ 2 (1525) and f 0 (1500) contributions is discussed in Secs. VI and VII.

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.
We define the helicity angle θ H as the angle formed by the h + (where h = π, K), in the h + h − rest frame, and the γ in the h + h − γ rest frame. We also define θ γ as the angle formed by the radiative photon in the h + h − γ rest frame with respect to the Υ (1S) direction in the Υ (2S)/Υ (3S) rest frame.
We compute the efficiency in two different ways.
• We label with ǫ(m, cos θ H ) the efficiency computed as a function of the h + h − effective mass and the helicity angle cos θ H . This is used only to obtain efficiency-corrected mass spectra.
• We label with ǫ(cos θ H , cos θ γ ) 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 ǫ(m, cos θ H ), for Υ (1S)→γπ + π − , we divide the π + 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 m(π + π − ) is then computed using a linear interpolation between adjacent bins. Figure 7 shows the efficiency distributions in the (m(π + π − ), cos θ H ) plane for the Υ (2S) and Υ (3S) datasets.
We observe an almost uniform behavior with some loss at cos θ H close to ±1.
The distributions of the efficiencies as functions of cos θ H and cos θ γ are shown in Fig. 9 for the f 2 (1270)→π + π − and f ′ 2 (1525)→K + K − mass regions, for the Υ (2S) datasets. To smoothen statistical fluctuations, the efficiency projections are fitted using 7-th and 4-th order polynomials, respectively. Similar behaviour is observed for the other resonances and for the Υ (3S) datasets.

B. Efficiency correction
To obtain the efficiency correction weight w R for the resonance R we divide each event by the efficiency ǫ(cos θ H , cos θ γ ) 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 Υ (2S) and Υ (3S) yields for resonances decaying to π + π − while, due to the limited statistics, for resonances decaying to K + K − the two datasets 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 ± 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 + π − and K + K − systems in Υ (1S)→γh + h − we study the dependence of the m(h + h − ) mass on the helicity angle θ H . Figure 10 shows the scatter plot cos θ H vs m(π + π − ) for the combined Υ (2S) and Υ (3S) datasets. We observe the spin 2 structure of the f 2 (1270).
A better way to observe angular effects is to plot the π + 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]: where φ SD is the relative phase. Therefore we expect to observe spin 2 resonances in Y 0 4 and S/D interference in Y 0 2 . The results are shown in Fig. 11. We clearly observe the f 2 (1270) resonance in Y 0 4 and a sharp drop in Y 0 2 at the f 2 (1270) mass, indicating the interference effect. The distribution of Y 0 0 is just the scaled π + 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 (1270) mass. Higher moments are all consistent with zero.
Similarly, we plot in Fig. 12 the K + K − mass spectrum weighted by the Legendre polynomial moments, corrected for efficiency. We observe signals of the f ′ 2 (1525)  Resonance angular distributions in radiative Υ (1S) decays from Υ (2S)/Υ (3S) 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). Figure 13 and Fig. 14 show the resulting S-wave and D-wave contributions to the π + π − and K + K − mass spectra, respectively. Due to the presence of background in the threshold region, the π + π − analysis is performed only on the Υ (2S) 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 + π − mass spectrum we obtain a good separation between S and Dwaves, with the presence of an f 0 (980) resonance on top of a broad f 0 (500) resonance in the S-wave and a clean f 2 (1270) 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 N (S−wave) = 629 ± 128. (17) in agreement with the results from the fit to the π + π − mass spectrum (see Table II). We also compute the fraction of S-wave contribution in the f 2 (1270) mass region defined in Table III and obtain f S (π + π − ) = 0.16 ± 0.02. In the case of the K + K − PWA the structure peaking around 1500 MeV/c 2 appears in both S and D-waves suggesting the presence of f 0 (1500) and f ′ 2 (1525). In the f 0 (1710) mass region statistics is not sufficient to discriminate between the two different spin assignments. This pattern is similar to that observed in the Dalitz plot anal-  region in the range given in Table III, we obtain a fraction of S-wave contribution f S (K + K − ) = 0.53 ± 0.10.

VII. SPIN-PARITY ANALYSIS
We compute the helicity angle θ π defined as the angle formed by the π + s , in the π + s π − s rest frame, with respect to the direction of the π + s π − s system in the Υ (1S)π + s π − s rest frame. This distribution is shown in Fig. 15 for the Υ (2S) data and Υ (1S)→γπ + π − , and is expected to be uniform if π + s π − s is an S-wave system. The distribution is consistent with this hypothesis with a p-value of 65%.
The Υ (nS) angular distributions are expressed in terms of θ γ and θ H . Due to the decay chain used to isolate the Υ (1S) radiative decays (see Eq. (1) and Eq. (2)), the Υ (1S) can be produced with helicity 0 or 1 and the corresponding amplitudes are labeled as A 00 and A 01 , respectively. A spin 2 resonance, on the other hand, can have three helicity states, described by amplitudes C 10 , C 11 , and C 12 . We make use of the helicity formalism [32,33] to derive the angular distribution for a spin 2 resonance: Ignoring the normalization factor |E 00 | 2 , there are two amplitudes describing the Υ (1S) helicity states, which can be reduced to one free parameter by taking the ratio |A 01 | 2 /|A 00 | 2 . Similarly, the three amplitudes describing the spin 2 helicity states, can be reduced to two free parameters by taking the ratios |C 11 | 2 /|C 10 | 2 and |C 12 | 2 /|C 10 | 2 . We therefore have a total of three free parameters.
The expected angular distribution for a spin 0 resonance is given by Ignoring the normalization factors |C 10 | 2 and |E 00 | 2 , the distribution has only one free parameter, |A 01 | 2 /|A 00 | 2 . We perform a 2D 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, ǫ(cos θ H , cos θ γ ) 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 + π − and K + K − mass spectra is negligible in the lowmass regions, we include only the tails of nearby adjacent resonances. In the description of the π + π − data in the threshold region we make use only of the Υ (2S) data because of the presence of a sizeable ρ(770) 0 background in the Υ (3S) sample.
We first fit the f 2 (1270) angular distributions and allow a background contribution of 16% (see Sect. VII) from the S-wave having fixed parameters. Therefore an iterative procedure of fitting the S-wave and f 2 (1270) regions is performed. Figure 16 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 = χ 2 (cos θ H ), χ γ = χ 2 (cos θ γ ) and their sum χ 2 t = (χ H + χ γ )/ndf computed as the χ 2 values obtained from the cos θ H and cos θ γ projections, respectively. We use ndf = N cells − N par , where N par is the number of free parameters in the fit and N cells is the sum of 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 We fit the S-wave region in the π + π − mass spectrum from the Υ (2S) decay including as background the spin 2 contribution due to the tail of the f 2 (1270). The latter is estimated to contribute with a fraction of 9%, with parameters fixed to those obtained from the f 2 (1270) spin analysis described above. Figure 17 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 + K − data in the f J (1500) mass region, where many resonances can contribute: f ′ 2 (1525), f 0 (1500) [31], and f 0 (1710). We fit the data using a superposition of S and D waves, having helicity contributions as free parameters, and free S-wave contribution. We obtain an S-wave contribution of f S (K + K − ) = 0.52 ± 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. 18, giving an adequate description of the data. We assign the spin-2 contribution to the f ′ 2 (1525) and the spin-0 contribution to the f 0 (1500) resonance. We also fit the data assuming the presence of the spin-2 f ′ 2 (1525) only hypothesis. We obtain a likelihood variation of ∆(−2 log L) = 1.3 for the difference of two parameters between the two fits. Due the low statistics we cannot statistically distinguish between the two hypotheses.  We determine the branching fraction B(R) for the decay of Υ (1S) 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 Υ (nS)→π + π − Υ (1S)(→µ + µ − ), 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 B(Υ (1S)→µ + µ − ) = 2.48 ± 0.05% [25]. We determine the reference channel corrected yield using the method of "B-counting", also used to obtain the number of produced Υ (2S) and Υ (3S) [22]. Taking into account the known branching fractions of Υ (2S)/Υ (3S)→π + s π − s Υ (1S) we obtain N (Υ (2S)→π + s π − s Υ (1S)(→µ + µ − )) = (4.35±0.12 sys )×10 5 (22) and N (Υ (3S)→π + s π − s Υ (1S)(→µ + µ − )) = (1.32±0.04 sys )×10 5 (23) events.
As a cross-check we reconstruct Υ (nS)→π + π − Υ (1S)(→µ + µ − ) 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 (1270) yield is corrected first for the π 0 π 0 (33.3%) and then for the ππ (84.2 +2.9 −0.9 %) branching fractions. We also correct the ππ S-wave and f 0 (1710) branching fractions for the π 0 π 0 decay mode. In the case of f J (1500)→K + K − the spin analysis reported in Sec.VI and Sec.VII gives indications of the presence of overlapping f ′ 2 (1525) and f 0 (1500) contributions. We give the branching fraction for f J (1500)→K + K − and, separately, for the f ′ 2 (1525) and f 0 (1500), where we make use of the S-wave contribution f S (K + K − ) = 0.52 ± 0.14, obtained in Sec.VII. The f ′ 2 (1525) branching fraction is corrected for the KK decay mode ((88.7 ± 2.2)%). For all the resonances decaying to KK, the branching fractions are corrected for the unseen K 0K 0 decay mode (50%).
For the f 2 (1270) and f 0 (1710) resonances decaying to π + π − , the relative branching ratios are computed separately for the Υ (2S) and Υ (3S) datasets, obtaining good agreement. The values reported in Table IV are determined using the weighted mean of the two measurements. Since the reference channel has the same number of tracks as the final state, systematic uncertainties related to tracking are negligible with respect to the errors due to other sources. The systematic uncertainty related to the "B-counting" estimate of the event yields in the denominator of Eq. 21 is propagated into the total systematic uncertainty on the branching fractions given in Table IV. Comparing with CLEO results, we note that our results on the S-wave contribution include the f 0 (980) and f 0 (500) contributions, while the CLEO analysis determines the branching fraction for the peaking structure at the f 0 (980) mass. In the same way a direct comparison for the f ′ 2 (1525) branching fraction is not possible due to the f 0 (1500) contribution included in the present analysis. The branching fraction for the f 2 (1270) is in good agreement.

IX. SUMMARY
We have studied the Υ (1S) radiative decays to γπ + π − and γK + K − using data recorded with the BABAR detector operating at the SLAC PEP-II asymmetricenergy e + e − collider at center-of-mass energies at the Υ (2S) and Υ (3S) resonances, using integrated luminosities of 13.6 fb −1 and 28.0 fb −1 , respectively. The Υ (1S) resonance is reconstructed from the decay chains Υ (nS)→π + π − Υ (1S), n = 2, 3. Spin-parity analyses and branching fraction measurements are reported for the resonances observed in the π + π − and K + K − mass spectra. In particular, we report the observation of broad S-wave, f 0 (980), and f 2 (1270) resonances in the π + π − mass spectrum. We observe a signal in the 1500 MeV/c 2 mass region of the K + K − mass spectrum for which the spin analysis indicates contributions from both f ′ 2 (1525) and f 0 (1500) resonances. We also report observation of f 0 (1710) in both π + π − and K + K − mass spectra with combined significance of 5.7σ, and measure the relative branching fraction. For this state, in the gluonium hypothesis, Ref. [10] computes a branching fraction of B(Υ (1S)→γf 0 (1710) = 0.96 +0.55 −0.23 10 −4 . Taking into account the presence of additional, not well measured, f 0 (1710) decay modes, our result is consistent with this predicted branching fraction as well as for the dominance of an ss decay mode. These results may contribute to the long-standing issue of the identification of a scalar glueball.

X. ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the