Study of radiative bottomonium transitions using converted photons

Bottomonium spectroscopy and radiative transitions between $b\overline{b}$ states can be well-described by effective potential models general_theory . To leading order, radiative decays are expected to be dominantly electric (E1) or magnetic (M1) dipole transitions. In the non-relativistic limit, theoretical predictions for these decays are straightforward and well-understood. However, there are a few notable cases where the non-relativistic decay rates are small or zero, e.g. in “hindered” M1 transitions between S-wave bottomonium such as $\mathchar 28935\relax(nS)\rightarrow\gamma\eta_{b}(n^{\prime}S)$ ($n>n^{\prime}$), and as a consequence of small initial- and final-state wavefunction overlap in the case of $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$ decays nomenclature ; higher-order relativistic and model-dependent corrections then play a substantial role. Measurements of these and other E1 transition rates can lead to a better understanding of the relativistic contributions to, and model dependencies of, interquark potentials. Furthermore, because radiative transitions have a distinct photon energy signature associated with the mass difference between the relevant $b\overline{b}$ states, they are useful in spectroscopic studies for mass measurements, and in the search for and identification of undiscovered resonances.

Radiative transitions within the bottomonium system have been studied previously in several experiments, such as Crystal Ball cb1 ; cb2 , ARGUS with converted photons argus_conv , and iterations of CUSB cusb1 ; cusb2 ; cusb3 ; cusb4 ; cusb_chib and CLEO cleo1 ; cleo2 ; cleo3 ; cleo_inclusive ; cleo_new (including an analysis of photon pair conversions in a lead radiator inserted specifically for that purpose cleo_conv ). These analyses have focused mainly on $\chi_{bJ}(nP)$-related measurements, such as the determination of the masses and the E1 transition rates to and from $\mathchar 28935\relax(mS)$ states. More recently, the BABAR experiment finished its operation by collecting large samples of data at the $\mathchar 28935\relax(3S)$ and $\mathchar 28935\relax(2S)$ center-of-mass (CM) energies. These data are useful for studies of bottomonium spectroscopy and decay and have already led to the discovery of the long-sought $\eta_{b}(1S)$ bottomonium ground state babar_etab1 ; babar_etab2 , an observation later confirmed by CLEO cleo_etab .

In this paper, we present a study of radiative transitions in the bottomonium system using the inclusive converted photon energy spectrum from $\mathchar 28935\relax(3S)$ and $\mathchar 28935\relax(2S)$ decays. The rate of photon conversion and the reconstruction of the resulting $e^{+}e^{-}$ pairs has a much lower detection efficiency than that for photons in the BABAR electromagnetic calorimeter, a disadvantage offset by a substantial improvement in the photon energy resolution. This improvement in resolution is well-suited for performing precise transition energy (hence, particle mass, and potentially width) measurements, and to disentangle overlapping photon energy lines in the inclusive photon energy spectrum. This analysis has different techniques, data selection, and systematic uncertainties than the previous studies babar_etab1 ; babar_etab2 ; cleo_etab , and is relatively free from complications due to overlapping transition peaks, and calorimeter energy scale and measurement uncertainties. We report measurements of $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$, $\chi_{bJ}(1P,2P)\rightarrow\gamma\mathchar 28935\relax(1S)$, observation of $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{b0,2}(1P)$, and searches for the $\eta_{b}(1S,2S)$ states.

In Sec. II we describe the BABAR detector and the data samples used in this analysis. Section III describes the photon conversion reconstruction procedure and the event selection criteria. Each of the following sections (Sec. IV - VII) individually describes the analysis of a particular region of interest in the inclusive photon energy spectrum. Section VIII summarizes the results obtained. Appendix provides specific details of some systematic uncertainties related to this analysis.

The BABAR detector is described in detail elsewhere babar_detector ; a brief summary is provided here. Moving outwards from the collision axis, the detector consists of a double-sided five-layer silicon vertex tracker (SVT) for measuring decay vertices close to the interaction point, a 40-layer drift chamber (DCH) for charged-particle tracking and momentum measurement, a ring-imaging Cherenkov detector for particle identification, and a CsI(Tl) crystal electromagnetic calorimeter (EMC) for measuring the energy deposited by electrons and photons. These detector subsystems are contained within a large solenoidal magnet which generates a 1.5-T field. The steel magnetic flux return is instrumented with a muon detection system consisting of resistive plate chambers and limited streamer tubes babar_lst .

The inner tracking region also contains non-instrumented support structure elements. Interior to the SVT, the interaction region is surrounded by a water-cooled, gold-coated beryllium beam pipe. The SVT support structure consists primarily of carbon-fiber and Kevlar^®. The SVT, beam pipe and vacuum chamber, and the near-interaction-point magnetic elements are mounted inside a cylindrical, carbon-fiber support tube. The inner wall of the DCH is a cylindrical tube of beryllium coated with anti-corrosion paint. A photon at normal incidence traverses approximately 0.01 radiation lengths ($X_{0}$) of material before reaching the SVT, and an additional 0.03$X_{0}$ before the DCH. Due to the asymmetric energy of the incoming $e^{+}e^{-}$ beams, the photons in this analysis tend to be boosted in the direction of $e^{-}$ beam, increasing the typical number of radiation lengths up to 0.02$X_{0}$ and 0.08$X_{0}$ to reach the previously noted detector subsystems. While this extra material is usually considered detrimental to detector performance, it is essential for $\gamma\rightarrow e^{+}e^{-}$ conversions in the present analysis.

The BABAR detector collected data samples of $(121\pm 1)$ million $\mathchar 28935\relax(3S)$ and $(98\pm 1)$ million $\mathchar 28935\relax(2S)$ decays semantics produced by the PEP-II asymmetric energy $e^{+}e^{-}$ collider. This corresponds to an integrated luminosity of $27.9\pm 0.2$ fb^-1 ($13.6\pm 0.1$ fb^-1) taken at the $\mathchar 28935\relax(3S)$ ($\mathchar 28935\relax(2S)$) resonance. Approximately $10\%$ of these data (referred to here as the “test sample”) were used for feasibility studies and event selection optimization; they are excluded in the final analysis. The results presented in this analysis are based on data samples of $(111\pm 1)$ million $\mathchar 28935\relax(3S)$ and $(89\pm 1)$ million $\mathchar 28935\relax(2S)$ decays. An additional $2.60\pm 0.02$ ($1.42\pm 0.01$) fb^-1 of data were taken at a CM energy approximately 30 $\mathrm{\,Me\kern-1.00006ptV}$ below the nominal $\mathchar 28935\relax(3S)$ ($\mathchar 28935\relax(2S)$) resonance energy, to be used for efficiency-related studies.

Large Monte Carlo (MC) datasets simulating the signal and expected background decay modes are used for the determination of efficiencies and the parameterization of lineshapes for signal extraction. The particle production and decays are simulated using a combination of EVTGEN evtgen and JETSET jetset . The radiative decays involving $\chi_{bJ}(nP)$ states are assumed to be dominantly E1 radiative transitions, and the MC events are generated with theoretically predicted helicity amplitudes helicity . The interactions of the decay products traversing the detector are modeled by Geant4 geant4 .

Photon conversions are reconstructed with a dedicated fitting algorithm that pairs oppositely charged particle tracks to form secondary vertices away from the interaction point. The algorithm minimizes a $\chi^{2}$ value ($\chi^{2}_{fit}$) based on the difference between the measured helical track parameters and those expected for the hypothesis that the secondary vertex had originated from two nearly parallel tracks emitted from a $\gamma\rightarrow e^{+}e^{-}$ conversion. The $\chi^{2}_{fit}$ value includes a term to account for an observed finite opening angle between the converted tracks. Requiring $\chi^{2}_{fit}<34$ is found to be the optimal value to select a high-purity converted photon sample. The reconstructed converted photons are also required to have an $e^{+}e^{-}$ invariant mass of $m_{e^{+}e^{-}}<30{\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ (though in practice, $m_{e^{+}e^{-}}$ is typically less than 10${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$). To remove internal conversions and Dalitz decays, and to improve signal purity, the conversion vertex radius ($\rho_{\gamma}$) is required to satisfy $1.7<\rho_{\gamma}<27$ cm. This restricts the photon conversions to the beampipe, SVT, support tube, and inner wall of the DCH, as seen in the plot of conversion vertex position for a portion of the “test sample” in Fig. 1. The efficiency for photon conversion and reconstruction versus energy in the CM frame ($E_{\gamma}^{*}$), as determined from a generic $\mathchar 28935\relax(3S)$ MC sample, is shown in Fig. 2.

Figure 3 shows the inclusive distributions of the resulting reconstructed converted photon energy. The data are divided into four energy ranges, as indicated by the shaded regions in Fig. 3. These ranges and the corresponding bottomonium transitions of interest are, in $\mathchar 28935\relax(3S)$ data:

• •

  $180\leq E_{\gamma}^{*}\leq 300$$\mathrm{\,Me\kern-1.00006ptV}$: $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$

• •

  $300\leq E_{\gamma}^{*}\leq 600$$\mathrm{\,Me\kern-1.00006ptV}$: $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$ and $\mathchar 28935\relax(3S)\rightarrow\gamma\eta_{b}(2S)$

• •

  $600\leq E_{\gamma}^{*}\leq 1100$$\mathrm{\,Me\kern-1.00006ptV}$: $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(1S)$ and $\mathchar 28935\relax(3S)\rightarrow\gamma\eta_{b}(1S)$

and in $\mathchar 28935\relax(2S)$ data:

• •

  $300\leq E_{\gamma}^{*}\leq 800$$\mathrm{\,Me\kern-1.00006ptV}$: $\chi_{bJ}(1P)\rightarrow\gamma\mathchar 28935\relax(1S)$ and $\mathchar 28935\relax(2S)\rightarrow\gamma\eta_{b}(1S)$.

Figure 4 summarizes these energy ranges and the radiative transitions of interest in a pictoral form. Peaks related to some of these transitions are already clearly visible in Fig. 3, where the photon energy in the CM frame of the initial particle for the radiative transition from an initial ($i$) to final ($f$) state is given in terms of their respective masses by

Because we analyse the photon energy in the CM frame of the initial $\mathchar 28935\relax(mS)$ system ($E_{\gamma}^{*}$), the photon spectra from subsequent boosted decays (e.g. $\chi_{bJ}(nP)\rightarrow\gamma\mathchar 28935\relax(1S)$) are affected by Doppler broadening due to the motion of the parent state in the CM frame.

To best enhance the number of signal ($S$) to background ($B$) events, the event selection criteria are chosen by optimizing the figure of merit $\mathcal{F}=\frac{S}{\sqrt{S+B}}$. This is done separately for each energy region. The $180\leq E_{\gamma}^{*}\leq 300$$\mathrm{\,Me\kern-1.00006ptV}$ energy region in $\mathchar 28935\relax(3S)$ uses the same criteria as determined for the similarly low energy $300\leq E_{\gamma}^{*}\leq 600$$\mathrm{\,Me\kern-1.00006ptV}$ range. We determine $S$ from MC samples of $\mathchar 28935\relax(mS)\rightarrow\gamma\eta_{b}(1S)$ weighted to match the measured branching fractions pdg , and $\mathchar 28935\relax(3S)\rightarrow\gamma\eta_{b}(2S)$ assuming the same branching fraction as for the decay to $\gamma\eta_{b}(1S)$. Because the generic decay processes of $\mathchar 28935\relax(3S)$ are not well-known (ie: a large percentage of the exclusive branching fractions have not been measured), the “test sample” data are used to estimate $B$. The optimization is performed by varying the selection criteria for the total number of tracks in the event ($nTRK$), the absolute value of the cosine of the angle in the CM frame between the photon momentum and the thrust axis ($|\cos\theta_{T}|$) thrust , and a $\pi^{0}$ veto excluding converted photons producing an invariant mass ($m_{\gamma\gamma}$) consistent with $m_{\pi^{0}}$ when paired with any other photon (converted or calorimeter-detected) above a minimum energy ($E_{\gamma 2}$) in the event. A requirement on the ratio of the second and zeroth Fox-Wolfram moments fox-wolfram of each event, $R_{2}$, is also applied. The reason for using these particular variables (indicated in parentheses) is to preferentially select bottomonium decays to hadronic final states ($nTRK$) and to remove photons from continuum background events ($|\cos\theta_{T}|$ and $R_{2}$) and $\pi^{0}$ decays ($m_{\pi^{0}}$ veto). Table 1 summarizes the values for the optimized selection criteria.

The efficiency for reconstruction and selection of signal events ($\epsilon$) is determined from MC simulation. A dedicated $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\gamma$ sample is used to study our detector model and converted photon efficiency (discussed in Appendix A), and the correspondence between simulation and data is found to be in very good agreement. Once the optimal selection criteria have been applied, $\epsilon\lesssim 1.5\%$ for conversions compared to $\sim 40\%$ for photons in the EMC for the energy range of interest in this analysis. Conversely, a large improvement is gained in photon energy resolution, e.g. from $\sim 25\mathrm{\,Me\kern-1.00006ptV}$ in the calorimeter to $4\mathrm{\,Me\kern-1.00006ptV}$ or better with converted photons. Figure 3 demonstrates both of these features. The sharply-peaking structures correspond to bottomonium transitions, and are narrow and well-resolved in this analysis. Unlike in the photon energy spectrum expected from the EMC babar_etab1 ; babar_etab2 , the distribution for converted photons drops with energy. The efficiency decreases (also seen in Fig. 2) due to the inability to fully reconstruct the conversion pair as at least one of the individual track momenta approaches the limit of detector sensitivity. We are unable to contribute useful new information on transitions expected below $E_{\gamma}^{*}$ =180 (300)$\mathrm{\,Me\kern-1.00006ptV}$ for the $\mathchar 28935\relax(3S)$ ($\mathchar 28935\relax(2S)$) analysis, which is why those energy ranges are not considered here.

The number of signal events for a given bottomonium transition is extracted from the data by performing a $\chi^{2}$ fit to the $E_{\gamma}^{*}$ distribution in 1$\mathrm{\,Me\kern-1.00006ptV}$ bins. The functional form and parameterization for each photon signal is determined from MC samples, as described below. In general, the lineshape is related to the Crystal Ball function crystal_ball , i.e. a Gaussian function with a power-law tail. This functional form is used to account for bremsstrahlung losses of the $e^{+}e^{-}$ pair. Comparisons between simulation and data made on $e^{+}e^{-}\rightarrow e^{+}e^{-}$ events used for the standard luminosity measurement in BABAR  demonstrate that the bremsstrahlung tails of these distributions are found to be well-described. The underlying smooth inclusive photon background is described by a fourth-order polynominal multiplied by an exponential function. This functional form adequately describes the background in each separate energy range.

The main purpose of the fit to the $180\leq E_{\gamma}^{*}\leq 300$$\mathrm{\,Me\kern-1.00006ptV}$ region of the $\mathchar 28935\relax(3S)$ photon energy spectrum, shown in detail in Fig. 5, is to measure the $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ transitions. The only previous measurements of these transitions were made by CUSB cusb_chib and CLEO cleo2 nearly two decades ago. Those analyses examined the low-energy photon spectrum from exclusive $\mathchar 28935\relax(3S)\rightarrow\gamma\gamma\mathchar 28935\relax(2S)(\ell^{+}\ell^{-})$ decays to derive the branching fractions for $\mathcal{B}(\chi_{b1,2}(2P)\rightarrow\gamma\mathchar 28935\relax(2S))$, and in the case of the CUSB result, to obtain evidence for $\chi_{b0}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$. We present the first fit to $E_{\gamma}^{*}$ to measure the photon from $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ directly. Though this analysis is potentially sensitive to all six $\mathchar 28935\relax(1D_{J})\rightarrow\gamma\chi_{bJ}(1P)$ decays, we treat these decays as a small systematic effect to the $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ measurement.

The $\chi_{bJ}(2P)$ transition lineshapes are parameterized by a Gaussian with power law tails on both the high and low side. This is best understood as a “double-sided” Crystal Ball function with different transition points and exponents for the high and low tails, but with a common Gaussian mean and standard deviation in the central region. The effects of Doppler broadening, due to the motion of the $\chi_{bJ}(2P)$ in the CM frame, are small ($\sim 2\mathrm{\,Me\kern-1.00006ptV}$ width) for these transitions. The $\mathchar 28935\relax(1D_{J})$-related lineshapes are individually parameterized in terms of a single Crystal Ball function. Parameterization of these transitions presents a complication because only the mass of the $J=2$ state has been measured reliably cleo_y1d ; babar_y1d , the value $m_{\mathchar 28935\relax(1D)}=(10163.7\pm 1.4)$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ being obtained when the experimental results are averaged. Marginal evidence for the $J=1$ and 3 states was also seen at $\sim 10152$${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ and $\sim 10173$${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$, respectively cleo_y1d ; babar_y1d . These values are consistent with several theoretical predictions y1d_godfrey_rosner , given a shift to bring the theoretical value for $m_{\mathchar 28935\relax(1D_{2})}$ into agreement with experiment. We therefore assume the $m_{\mathchar 28935\relax(1D_{1,3})}$ mass values stated above to compute the expected energy for transitions from those states. The event yields for these transitions are fixed to the branching fractions expected when $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P))$ pdg is combined with the predictions for $\mathcal{B}(\chi_{bJ}(2P)\rightarrow\gamma\gamma\chi_{bJ}(1P))$ via $\mathchar 28935\relax(1D_{J})$ y1d_kwong_rosner . The efficiencies for the $\mathchar 28935\relax(1D)$ transition signals range from approximately $0.17$ to $0.30\%$, monotonically rising with $E_{\gamma}^{*}$.

Figure 5 shows the measured photon spectrum and results of the fit, before and after subtraction of the inclusive background. In this fit, the parameters describing the background and any systematic offset in the $E_{\gamma}^{*}$ scale are free parameters, together with the signal yields for $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ decays. Table 2 summarizes the fit results. Considering both statistical and systematic uncertainties, we find significant $\chi_{b1,2}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ signals ($>12\sigma$ and $>8\sigma$, respectively, where $\sigma$ represents standard deviation), but do not find evidence for $\chi_{b0}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ decay. The overall energy offset, determined predominantly by the position of the $\chi_{b1,2}(2P)$ transition peaks compared to the nominal pdg values, is found to be inconsequential ($-0.3\pm 0.2$$\mathrm{\,Me\kern-1.00006ptV}$).

The systematic uncertainties on these measurements (with their approximate sizes given in parentheses below and throughout) include the uncertainty in the fit parameters fixed from MC, uncertainty in the converted photon efficiency, assumptions related to the $\mathchar 28935\relax(1D_{J})$ contributions, uncertainty on masses used to calculate the expected $E_{\gamma}^{*}$ values, the $\mathchar 28935\relax(mS)$ counting uncertainty, effects of the fit mechanics, and the effect of the choice for the background shape. For each fit component, all of the parameters fixed to MC-determined values are varied individually by $\pm 1\sigma$ of the statistical uncertainty from the MC determination, and the fit repeated. The maximal variation of the fit result for each component is taken as the systematic uncertainty, and summed in quadrature ($\sim 4\%$). The systematic uncertainty on the converted photon efficiency ($4.7\%$) is estimated using an off-peak control sample and varied selection criteria, as described for all energy regions in Appendix A. The fits are repeated with the $\mathchar 28935\relax(1D_{J})$ masses individually varied by their approximate experimental uncertainties ($\pm 1.8$, $\pm 1.4$, and $\pm 1.5$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ for $J=1,2$ and 3, respectively) babar_y1d , and the fixed yields by $\pm 50\%$ of the theoretical values y1d_kwong_rosner . To make a theory-independent determination of the impact due to $\mathchar 28935\relax(1D_{J})$, the fit is also repeated with four of the $\mathchar 28935\relax(1D_{J})\rightarrow\gamma\chi_{bJ}(1P)$ yields free to vary (the $\mathchar 28935\relax(1D_{1})\rightarrow\gamma\chi_{b1}(1P)$ and $\mathchar 28935\relax(1D_{3})\rightarrow\gamma\chi_{b2}(1P)$ yields are fit as a single component because their $E_{\gamma}^{*}$ values are nearly identical, and the $\mathchar 28935\relax(1D_{1})\rightarrow\gamma\chi_{b2}(1P)$ transition is overwhelmed by the main $\chi_{b1,2}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ peaks and remains fixed). Under this scenario, none of the $\mathchar 28935\relax(1D_{J})$-related transitions is found to be significant, and the yields are consistent with the theoretical predictions within statistical uncertainty. The $\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S)$ yields are not significantly affected. The changes in the fit yields for all of these alternative cases are added in quadrature and taken as the systematic uncertainty due to $\mathchar 28935\relax(1D_{J})$ decays ($\sim 2\%$). It is worth reiterating that the excellent resolution obtained by using converted photons separates the $\mathchar 28935\relax(1D_{J})$- and $\chi_{bJ}(2P)$-related components in $E_{\gamma}^{*}$, which is why the impact of the $\mathchar 28935\relax(1D_{J})$ states does not dominate the measurement uncertainty. The fit is repeated with the bottomonium masses (hence, $E_{\gamma}^{*}$ values) varied according to the PDG uncertainties pdg , and the change in the yield added in quadrature ($\sim 2\%$). The number of $\mathchar 28935\relax(mS)$ mesons and its uncertainty ($1.0\%$) were calculated separately, based on visible cross sections computed from dedicated $e^{+}e^{-}\rightarrow e^{+}e^{-}(\gamma)$ and $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$ control samples. Systematic effects due to the fit mechanics were tested by repeating the fit separately with an expanded $E_{\gamma}^{*}$ range and a bin width of $0.5\mathrm{\,Me\kern-1.00006ptV}$, the difference in results defining a small systematic uncertainty ($1.5\%$). As a cross-check, the fit was repeated with the $\chi_{b0}(2P)$ component restricted to a physical range. The effect on the other signal yields was found to be small ($<2\%$). Finally, the background shape was replaced by a fifth-order polynomial and half of the resulting change in the yield ($<1\%$) taken as the symmetric error due to this assumed parameterization.

We find $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P))\times\mathcal{B}(\chi_{bJ}\rightarrow\gamma\mathchar 28935\relax(2S))$ = $(-0.3\pm 0.2^{+0.5}_{-0.4})\%$, $(2.4\pm 0.1\pm 0.2)\%$, and $(1.1\pm 0.1\pm 0.1)\%$ for $J=0$, 1, and 2, respectively. Using $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P))$ from pdg , we derive $\mathcal{B}(\chi_{bJ}(2P)\rightarrow\gamma\mathchar 28935\relax(2S))$ = $(-4.7\pm 2.8^{+0.7}_{-0.8}\pm 0.5)\%$, $(18.9\pm 1.1\pm 1.2\pm 1.8)\%$, and $(8.3\pm 0.8\pm 0.6\pm 1.0)\%$, where the errors are statistical, systematic, and from the uncertainty on $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P))$, respectively. From these values, we calculate a $90\%$ confidence level upper limit of $\mathcal{B}(\chi_{b0}(2P)\rightarrow\gamma\mathchar 28935\relax(2S))<2.8\%$ ul . Past experimental results cusb_chib ; cleo2 averaged by the PDG pdg rely on assumptions for the branching fractions of $\mathchar 28935\relax(2S)\rightarrow\ell^{+}\ell^{-}$ and $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P)$ and their uncertainties that are no longer valid. In Table 2, we have rescaled these previous results using the current values in order to make a useful comparison. We find our results to be in good agreement with the previous results, and to be the most precise values to date for the $J=$1 and 2 decays.

The $300\leq E_{\gamma}^{*}\leq 600$$\mathrm{\,Me\kern-1.00006ptV}$ range in the inclusive $\mathchar 28935\relax(3S)$ photon energy spectrum, shown in Fig. 6, is complicated by many radiative bottomonium transitions. A principal feature is the photon lines from the three direct $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$ decays. Photons from the secondary decays, $\chi_{bJ}(1P)\rightarrow\gamma\mathchar 28935\relax(1S)$, have energies that overlap with these initial transitions. There are several ways to produce $\chi_{bJ}(1P)$ from $\mathchar 28935\relax(3S)$, each with unique Doppler broadening and relative rate. These decays “feed-down” to produce many extraneous $\chi_{bJ}(1P)$ mesons that contribute substantially to the background level through subsequent $\chi_{bJ}(1P)\rightarrow\gamma\mathchar 28935\relax(1S)$ decay. At the lower edge of this energy range, there are potential contributions from $\mathchar 28935\relax(3S)\rightarrow\gamma\eta_{b}(2S)$ and $\mathchar 28935\relax(2S)$ production from initial state radiation (ISR).

The best known of the $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$ branching fractions comes from the CLEO experiment, which was able to isolate the $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{b0}(1P)$ signal cleo_inclusive . A separate analysis of $\chi_{bJ}(1P)$ decays to multihadronic final states further set upper limits on $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P))$ cleo_hadrons . A recent analysis of $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{b1,2}(1P)\rightarrow\gamma\gamma\mathchar 28935\relax(1S)$ transitions with exclusive $\mathchar 28935\relax(1S)\rightarrow\ell^{+}\ell^{-}$ decays has resulted in a measurement of $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{b1,2}(1P)$ branching fractions cleo_new . Our improved $E_{\gamma}^{*}$ resolution with the converted photon sample allows us to disentangle the overlapping photon lines to make a direct measurement of these radiative transitions as well. We also search for a signal for $\mathchar 28935\relax(3S)\rightarrow\gamma\eta_{b}(2S)$.

The direct $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$ lineshapes are parameterized using the double-sided Crystal Ball function described in Sec. IV plus an independent Gaussian to account for broadening from non-linearities in the $E_{\gamma}^{*}$ resolution due to low momentum tracks encountered in this energy range. The $\mathchar 28935\relax(3S)\rightarrow\gamma\eta_{b}(2S)$ lineshape is modeled with the convolution of a relativistic Breit-Wigner function (natural lineshape for the $\eta_{b}(2S)$) and a Crystal Ball function (experimental resolution function), where the Breit-Wigner function has been modified by a transformation of variables to $E_{\gamma}^{*}$ using Eq. (1). The ISR-produced $\mathchar 28935\relax(2S)$ signal is parameterized with a Crystal Ball function, for which the width is dominated by the spread in the $e^{+}e^{-}$ beam energy.

The lineshapes for the decays $\chi_{bJ}(1P)\rightarrow\gamma\mathchar 28935\relax(1S)$ depend on the initial decays that produced the $\chi_{bJ}(1P)$ states. We consider six main production pathways:

• •

  $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$

• •

  $\mathchar 28935\relax(3S)\rightarrow\gamma\gamma\mathchar 28935\relax(2S)\rightarrow\gamma\gamma\gamma\chi_{bJ}(1P)$

• •

  $\mathchar 28935\relax(3S)\rightarrow\gamma\gamma\mathchar 28935\relax(1D_{J})\rightarrow\gamma\gamma\gamma\chi_{bJ}(1P)$

• •

  $\mathchar 28935\relax(3S)\rightarrow\pi\pi\mathchar 28935\relax(2S)\rightarrow\pi\pi\gamma\chi_{bJ}(1P)$

• •

  $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P)\rightarrow\gamma\pi\pi\chi_{bJ}(1P)$

• •

  $e^{+}e^{-}\rightarrow\gamma_{ISR}\mathchar 28935\relax(2S)\rightarrow\gamma_{ISR}\gamma\chi_{bJ}(1P)$.

The feed-down contribution from $\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(1P)$ is determined directly from the fit to the data. The lineshapes for the subsequent $\chi_{bJ}(1P)\rightarrow\gamma\mathchar 28935\relax(1S)$ decays are distorted by Doppler-broadening effects. We parameterize the $\chi_{bJ}(1P)$ transition lineshape with the convolution of a rectangular function and a Crystal Ball function. Because of the large Doppler width ($\sim 20\mathrm{\,Me\kern-1.00006ptV}$), the resulting shape is relatively broad and non-peaking. In the fit, the relative yields of the direct to the secondary transitions are fixed according to the ratios of the expected efficiencies for each mode, and the branching fractions for the $\chi_{bJ}(1P)\rightarrow\gamma\mathchar 28935\relax(1S)$ decays (to be discussed below).

There are two $3\gamma$ pathways from $\mathchar 28935\relax(3S)$ to $\chi_{bJ}(1P)$. Decays via $\mathchar 28935\relax(2S)$ are fairly well understood, and the precision branching fraction results from Sec. IV are used to determine the expected yields and uncertainties. In contrast, the decays via $\mathchar 28935\relax(1D_{J})$ have not been measured in detail. We rely on theoretical predictions y1d_kwong_rosner , found to be consistent with an experimental measurement of the $4\gamma$ cascade to $\mathchar 28935\relax(1S)$ cleo_y1d , to estimate the total feed-down component. We take the uncertainties on $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\gamma\chi_{bJ}(2P))$ pdg and introduce a $30\%$ uncertainty on each theoretically calculated branching fraction in the decay chain. Doppler effects introduce a smooth $\sim 5\mathrm{\,Me\kern-1.00006ptV}$ broadening in these (and other) multi-step decay processes, thus the lineshapes for the individual $3\gamma$ pathways are adequately parameterized using a standard Crystal Ball function.

There are two di-pion decay chains leading to $\chi_{bJ}(1P)$: either via $\mathchar 28935\relax(3S)\rightarrow\pi\pi\mathchar 28935\relax(2S)$ or $\chi_{bJ}(2P)\rightarrow\pi\pi\chi_{bJ}(1P)$. The former has been precisely measured by BABAR in a recent analysis of the recoil against $\pi^{+}\pi^{-}$ to search for the $h_{b}(1P)$ state babar_pipihb . We combine the branching fraction from that analysis with the PDG average pdg to obtain $\mathcal{B}(\mathchar 28935\relax(3S)\rightarrow\pi^{+}\pi^{-}\mathchar 28935\relax(2S)$ = $(2.7\pm 0.2)\%$. For the $\pi^{0}\pi^{0}$ transition, we use the current world average branching fraction value pdg . The relevant MC samples are generated with the experimentally-determined $m_{\pi^{+}\pi^{-}}$ distribution cleo_dipion_shape . Di-pion transitions between $\chi_{bJ}(2P)$ and $\chi_{bJ}(1P)$ for $J=$1 and 2 have been measured experimentally by CLEO cleo_dipion . The above-mentioned BABAR di-pion analysis babar_pipihb also measured these quantities, which are averaged with the CLEO results to derive $\mathcal{B}(\chi_{bJ}(2P)\rightarrow\pi^{+}\pi^{-}\chi_{bJ}(1P))$ equal to $(9.1\pm 1.0)\times 10^{-3}$ and $(5.0\pm 0.6)\times 10^{-3}$ for $J=1$ and 2, respectively. Decays to the $J=0$ state, with different initial and final $J$ values, and via $\pi^{0}\pi^{0}$ have thus far been below the level of experimental sensitivity. To calculate the expected feed-down, we assume isospin conservation such that $\Gamma_{\pi^{0}\pi^{0}}=\frac{1}{2}\Gamma_{\pi^{+}\pi^{-}}$, and estimate $\mathcal{B}(\chi_{b0}(2P)\rightarrow\pi\pi\chi_{b0}(1P))$ to be about one-fifth of that of the other $J$ states theory_dipion . We assume a $30\%$ uncertainty on all theoretically-estimated branching fractions.