Study of dipion transitions among ${\rm Y}(3S)$, ${\rm Y}(2S)$, and ${\rm Y}(1S)$ states

We present measurements of decay matrix elements for hadronic transitions of the form ${\rm Y}(nS)\to {\rm Y}(mS)\pi \pi$, where $(n,m)=(3,1),(2,1),(3,2)$. We reconstruct charged and neutral pion modes with the final state Upsilon decaying to either ${\mu }^{+}{\mu }^{-}$ or $e^{+}{e}^{-}$. Dalitz plot distributions for the 12 decay modes are fit individually as well as jointly assuming isospin symmetry, thereby measuring the matrix elements of the decay amplitude. We observe and account for the anomaly previously noted in the dipion invariant mass distribution for the ${\rm Y}(3S)\to {\rm Y}(1S)\pi \pi$ transition and obtain good descriptions of the dynamics of the decay using the most general decay amplitude allowed by partial conservation of the axial-vector current considerations. The fits further indicate that the ${\rm Y}(2S)\to {\rm Y}(1S)\pi \pi$ and ${\rm Y}(3S)\to {\rm Y}(2S)\pi \pi$ transitions also show the presence of terms in the decay amplitude that were previously ignored, although at a relatively suppressed level.

Figure 1

(Left) The decay ${\rm Y}(nS)\to {\rm Y}(mS)\pi \pi$ follows the production of an initial state labeled ${{\rm Y}}^{\prime }$ which decays to an ${\rm Y}\pi \pi$ state. In our analysis, the final state ${\rm Y}$ decays to a lepton pair whose momentum vectors are very nearly back-to-back due to the large energy release. (Center) The decay of the initial state Upsilon is governed by two kinematic variables, the Dalitz masses $M_{{\rm Y}\pi }$ and $M_{\pi \pi }$. (Right) Alternately one can think of the $\pi \pi$ system as a composite, $X$, and study its structure via the pion “decay” angles.

Figure 2

Dilepton invariant mass distributions for lepton pairs; the abscissa is the dilepton invariant mass, showing peaks at the masses of the ${\rm Y}(1S)$ and ${\rm Y}(2S)$ mesons. The hatching indicates the limits to the invariant mass selection windows. Candidates are plotted after the signal selection described in Sec. 2d. At left are the dimuon candidates and at right the dielectron candidates.

Figure 3

Recoil mass, $M_{\mathrm{rec}}$, distributions for all modes. The upper plot is generated from neutral decays, ${\rm Y}(nS)\to {\rm Y}(mS){\pi }^0{\pi }^0$, and the lower from charged decays, ${\rm Y}(nS)\to {\rm Y}(mS){\pi }^{+}{\pi }^{-}$. The final signal selections (track quality, pion quality, dilepton mass, etc.) have been applied. The peaks at the ${\rm Y}(1S)$ and ${\rm Y}(2S)$ masses correspond to decays to these resonances from an ${\rm Y}(3S)$ parent. The peaks at $9.8\mathrm{GeV}/c^2$ are from ${\rm Y}(2S)\to {\rm Y}(1S)\pi \pi$ decays. The hatching shows the bounds on the recoil mass values for the three transitions. See also the window definitions in Table 1. Yields are set to zero in the regions that correspond neither to signal nor to sidebands.

Figure 4

Candidate events that passed all selection criteria, and that have the final state Upsilon decaying to ${\mu }^{+}{\mu }^{-}$. In the middle is the decay ${\rm Y}(3S)\to {\rm Y}(1S){\pi }^{+}{\pi }^{-}$. To the left is its neutral counterpart ${\rm Y}(3S)\to {\rm Y}(1S){\pi }^0{\pi }^0$. To the right is the charged transition ${\rm Y}(2S)\to {\rm Y}(1S){\pi }^{+}{\pi }^{-}$, with the bulk of its distribution at large values of dipion invariant mass. In each plot there are ten degrees of gray scale ranging from white (lowest occupancy per bin) to black (highest occupancy).

Figure 5

Candidate events that have passed all selection criteria, and that have the final state Upsilon decaying to $e^{+}{e}^{-}$. As in the prior plot, three transitions are, left to right, ${\rm Y}(3S)\to {\rm Y}(1S){\pi }^0{\pi }^0$, ${\rm Y}(3S)\to {\rm Y}(1S){\pi }^{+}{\pi }^{-}$, and ${\rm Y}(2S)\to {\rm Y}(1S){\pi }^{+}{\pi }^{-}$.

Figure 6

The efficiency-sculpted phase space in the two-dimensional plane for the transitions ${\rm Y}(3S)\to {\rm Y}(1S){\pi }^{+}{\pi }^{-}$ (left) and ${\rm Y}(3S)\to {\rm Y}(1S){\pi }^0{\pi }^0$ (right). Note that the neutral final state has a more uniform efficiency, especially in the region of moderate dipion mass and large $|\cos ({\theta }_X)|$.

Figure 7

The three functions used in the fit for the ${\rm Y}(3S)$ decay to ${\rm Y}(1S)\pi \pi$. From left to right these are for the pure $\mathcal{A}$ term, the interference term, and the pure $\mathcal{B}$ term.

Figure 8

Plots overlaying projections of the data (points with error bars) and the fit result (histograms) onto the $M_{\pi \pi }$ and $\cos {\theta }_X$ variables. The plots are summed over electrons and muons, but are differentiated by pion charge. The neutral modes (open symbols, dashed lines) show only a positive distribution in $\cos {\theta }_X$ because the two pions are indistinguishable. For the charged modes (solid symbols, solid lines) the angle is that of the ${\pi }^{+}$.

Figure 9

Plots of the bin-by-bin deviations of the data from the fit templates normalized to the expected uncertainty on the bin content for the transitions between ${\rm Y}(3S)$ and ${\rm Y}(1S)$. The left plot is for the charged pion modes while the right plot is for the neutral pion modes. The data are summed over lepton species. No strong concentration of deviations is apparent.

Figure 10

Variation of $\mathcal{B}$ with $\mathcal{C}$ magnitudes. The points indicate the fit and error for $\mathcal{B}$ at fixed values of $\mathcal{C}$. The ellipse indicates the one sigma bound on the free fit, the axis of which agrees well with the point-by-point fits. The bands indicate the one standard deviation error limits on $\mathcal{B}$ when $\mathcal{C}$ is fixed to zero.

Figure 11

The left plot shows the amplitude component functions ${\mathcal{S}}_{\mathcal{A}}$, ${\mathcal{S}}_{\mathcal{B}}$, ${\mathcal{D}}_{\mathcal{A}}$, and ${\mathcal{D}}_{\mathcal{B}}$ as a function of $M_{\pi \pi }\equiv \sqrt{q^2}$. These are summed to obtain the total amplitude. The partial rate to $S$-wave and $D$-wave components is shown in the right plot for the ${\rm Y}(3S)\to {\rm Y}(1S)\pi \pi$ decay as determined from the results of this analysis: $\mathcal{B}/\mathcal{A}=-2.52+1.19i$. Note that the $D$-wave contribution is largest in the low to intermediate range of $q^2$, and is suppressed at both extrema by angular momentum barrier effects. Note further that this is not a resonance phenomenon despite its shape in $M_{\pi \pi }$ and the changing angular structure.

Figure 12

Complex values of matrix element ratio $\mathcal{B}/\mathcal{A}$ from combined fits for the three transitions under the assumption that $\mathcal{C}=0$. Note the twofold ambiguity in the imaginary part.