Bottomonium spectroscopy and radiative transitions involving the ${\chi }_{\mathrm{b}\mathrm{J}}(1P,2P)$ states at BaBar

We use $(121\pm 1)$ million $\mathrm{{\rm Y}}(3S)$ and $(98\pm 1)$ million $\mathrm{{\rm Y}}(2S)$ mesons recorded by the BABAR detector at the PEP-II $e^{+}{e}^{-}$ collider at SLAC to perform a study of radiative transitions involving the ${\chi }_{b\mathrm{J}}(1P,2P)$ states in exclusive decays with ${\mu }^{+}{\mu }^{-}\gamma \gamma$ final states. We reconstruct twelve channels in four cascades using two complementary methods. In the first we identify both signal photon candidates in the electromagnetic calorimeter (EMC), employ a calorimeter timing-based technique to reduce backgrounds, and determine branching-ratio products and fine mass splittings. These results include the best observational significance yet for the ${\chi }_{b0}(2P)\to \gamma \mathrm{{\rm Y}}(2S)$ and ${\chi }_{b0}(1P)\to \gamma \mathrm{{\rm Y}}(1S)$ transitions. In the second method, we identify one photon candidate in the EMC and one which has converted into an $e^{+}{e}^{-}$ pair due to interaction with detector material, and we measure absolute product branching fractions. This method is particularly useful for measuring $\mathrm{{\rm Y}}(3S)\to \gamma {\chi }_{b1,2}(1P)$ decays. Additionally, we provide the most up-to-date derived branching fractions, matrix elements and mass splittings for ${\chi }_b$ transitions in the bottomonium system. Using a new technique, we also measure the two lowest-order spin-dependent coefficients in the nonrelativistic QCD Hamiltonian.

Figure 1

Schematic representation of the twelve $E1$ channels in the four radiative cascades measured in this analysis: $2S\to 1P\to 1S$, $3S\to 2P\to 2S$, $3S\to 1P\to 1S$, and $3S\to 2P\to 1S$. Each cascade terminates in the annihilation $\mathrm{{\rm Y}}(nS)\to {\mu }^{+}{\mu }^{-}$, not shown. The numbers give the masses [8] of the relevant bottomonium states in $\mathrm{GeV}/c^2$. The first and second transitions in each cascade (except the $3S\to 1P\to 1S$) are referred to in the text as “soft” and “hard” transitions, respectively. Splittings in the photon spectra for these cascades are due to the mass splittings in the intermediate states ${\chi }_{bJ}$ with $J=0$, 1 or 2.

Figure 2

Scatter plot of reconstructed $2S\to 1P\to 1S$ events in the two selection variables $S_{\text{soft}\text{−}\text{hard}}$ and cascade kinematic fit probability for the calorimeter-based analysis. The cluster of in-time and high-probability events in the lower right corner is from the signal process, with a residue at lower probabilities due to tail events. The lack of structure in the scatter plot confirms the complementarity of these two selections. Events with $S_{\text{soft}\text{−}\text{hard}}>2.0$ or fit probability below ${10}^{-5}$ are excluded, as shown by the white dashed lines.

Figure 3

(a) Fit to the soft photon energy $E_{2S\to 1P}$ in the $2S\to 1P\to 1S$ cascade with individual signal (dot-dashed) and background (dash) components for the calorimeter-based analysis. The targeted ${\chi }_{b0}(1P)$ signal corresponds to the small bump on the right; the integral ratio and mean offset of the fit to this peak compared to the $J=1$ peak (center), are defined as $f_0$ and ${\delta }_0$, respectively. Similarly, $f_2$ and ${\delta }_2$ are the integral ratio and mean offset for the fit to the $J=2$ peak (left), also compared to the $J=1$ peak. (b) A zoomed-in view of the ${\chi }_{b0}(1P)$ peak on the same energy scale.

Figure 4

(a) Fit to the soft photon energy $E_{3S\to 2P}$ in the $3S\to 2P\to 2S$ cascade with individual signal (dot-dashed) and background (dash) components for the calorimeter-based analysis. (b) A zoomed-in view of the ${\chi }_{b0}(2P)$ peak on the same energy scale. Discussion of the significance of the $J=0$ peak is contained in Sec. 4c.

Figure 5

(a) Fit to the soft photon energy $E_{3S\to 2P}$ in the $3S\to 2P\to 1S$ cascade with individual signal (signal corresponds to the small bump) and background (dash) components for the calorimeter-based analysis. (b) A zoomed-in view of the ${\chi }_{b0}(2P)$ peak on the same energy scale. Discussion of the significance of the $J=0$ peak is contained in Sec. 4c.

Figure 6

Fit to the photon energy $E_{1P\to 1S}$ in the $2S\to 1P\to 1S$ cascade for the conversions-based analysis. The data are represented by points, the total fit by a solid curve, the background component with a dashed curve, and the individual signal components of the fit by peaking dot-dashed curves in progressively darker shades for $J=0,1,2$. Clear evidence is seen for the $J=2$ and $J=1$ signals.

Figure 7

Fit to the photon energy $E_{3S\to 1S}$ in the $3S\to 1P\to 1S$ cascade for the conversions-based analysis. The data are represented by points, the total fit by a solid curve, the background component with a dotted curve, and the individual signal components of the fit by peaking dot-dashed curves for $J=2$ (darker shade) and $J=1$ (lighter shade). There is clear evidence for the $J=2$ transition, but the $J=1$ signal is not statistically significant.

Figure 8

Fit to the photon energy $E_{2P\to 2S}$ in the $3S\to 2P\to 2S$ cascade for the conversions-based analysis. The data are represented by points, the total fit by a solid curve, the background component with a dotted curve, and the individual signal components of the fit by peaking dot-dashed curves in progressively darker shades for $J=0,1,2$. Clear evidence for the $J=2$ and $J=1$ signals is seen.

Figure 9

Fit to the photon energy $E_{2P\to 1S}$ in the $3S\to 2P\to 1S$ cascade for the conversions-based analysis. The data are represented by points, the total fit by a solid curve, the background component with a dotted curve, and the individual signal components of the fit by peaking dot-dashed curves in progressively darker shades for $J=0,1,2$. Clear signals for the $J=2$ and $J=1$ signals are seen.