${\Upsilon(5S)}$: What has been learned and what can be learned

We present recent measurements of $B$ and $B^0_s$ production using data collected on the ${\Upsilon(5S)}$ resonance at CLEO and Belle. We also briefly discuss what can be learned using sufficiently larger data samples in the future.

INTRODUCTION
============

The ${\Upsilon(5S)}$ was discovered about 20 years ago in $e^+e^-$ collisions at CLEO @cleo5s_85 and CUSB @cusb5s_85. Its mass is $(10865\pm8)$ MeV, about 300 MeV higher than the ${\Upsilon(4S)}$ resonance, and its width of $(110\pm13)$ MeV is about five times larger than the ${\Upsilon(4S)}$ resonance. Its production cross-section was measured to be about 0.3 nb, about ten times lower than the ${\Upsilon(4S)}$ cross-section. Owing to its mass, the ${\Upsilon(5S)}$ can decay to a variety of final states containing $B$ and $B^0_s$ mesons. The accessible $B$-meson final states include: $\bbbar$, $\bbstar$, $\bstbst$, $\bbbarpi$, $\bbstarpi$, $\bstbstpi$ and $\bbbarpipi$; final states with $B^0_s$ are: $\bsbsbar$, $\bsbsstar$, and $\bsstbsst$. Further study of the ${\Upsilon(5S)}$ is of great interest because of it being a potentially large source of $B^0_s$ mesons at an $e^+e^-$ machine. If suitably large samples can be collected, their study would provide a complementary probe of CP violation and rare decays to LHCb. Studies involving rare decays, particularly those with soft photons, and untagged time-dependent and time-independent measurements, which provide access to the width difference between the $B^0_s$ mass eigenstates, $\Delta\Gamma_s$, offer complementary information to measurements at hadron machines @nierste. Because of the small Lorentz boost at the ${\Upsilon(5S)}$ and the large value of the mixing frequency, $\Delta m_s$ @cdf, time-dependent CP violation measurements in mixing are not possible.

Until recently, there existed only weak limits on the production rate of $B^0_s$ at the ${\Upsilon(5S)}$. Several models attempt to describe the hadronic cross-section in the 10-11 GeV region. The simplest model, the quark-pair creation model (QPC) @martin_yaouanc, allows for quark pairs to be 'popped' out of the vacuum. This model by itself is insufficient to describe the hadronic cross-section. Ono <span>*et al.*</span> @ono_qhm are able to reproduce the spectrum by introducing a $b\bar{b}g$ hybrid at 10683 MeV (the QHM model). A third model, the unitarized quark model (UQM) @tornqvist_byers includes coupled channel effects between $b\bar{b}$ states via real (two-meson) intermediate states. The QHM predicts $\bbstar$ is dominant at 10865 MeV, with relative rates of $\bstbst$:$\bbstar$:$\bbbar$ $\sim$ 1:3:6. On the other hand, the UQM also reproduces the hadronic cross-section, but predicts that $\bstbst$ is dominant, <span>*e.g.,*</span> $\bstbst$:$\bbstar$:$\bbbar$ $\sim$ 2:1:0.3.

A more thorough experimental study of the ${\Upsilon(5S)}$ has been undertaken by CLEO, followed by Belle, to better understand the hadronic cross-section in this region as well as assess the potential for a future high luminosity $e^+e^-$ machine operating at the ${\Upsilon(5S)}$. In 2003, CLEO collected 0.42 $\ifb$ of data at the ${\Upsilon(5S)}$ (10871 MeV) In 2005, Belle collected about 1.9 $\ifb$ on the ${\Upsilon(5S)}$ resonance, followed by a larger sample of 22 $\ifb$ a year later. Here we report on results from CLEO and the first Belle data set.

HADRONIC RESONANCE CROSS SECTION
================================

The hadronic resonance cross section is measured by counting hadronic events recorded at the ${\Upsilon(5S)}$ using various selection criteria employed to reduce the continuum background. The continuum background is estimated using data taken below the ${\Upsilon(4S)}$, and is scaled by: $({\cal{L}}_{{\Upsilon(5S)}}/{\cal{L}}_{{\Upsilon(4S)}})\times(s_{{\Upsilon(5S)}})/s_{{\Upsilon(4S)}})$, where $ {\cal{L}}_i$ and $s_i$ are the integrated luminosities and center-of-mass energies squared at the ${\Upsilon(4S)}$ and ${\Upsilon(5S)}$. The ${\Upsilon(5S)}$ cross-sections are measured to be $(0.301\pm0.002\pm0.039)$ nb @cleo5s_stone and $(0.302\pm0.002\pm0.015)$ nb @drutskoy1, by CLEO and Belle, respectively.

$B$ AND $B^0_s$ OVERVIEW
========================

Many of the same techniques of exclusive reconstruction that are employed at the ${\Upsilon(4S)}$ are used to reconstruct $B$ and $B^0_s$ mesons at the ${\Upsilon(5S)}$. The kinematic variables $\de=\ebeam-E_B$ and the beam-energy constrained mass, $\mbc=\sqrt{\ebeam^2-p_B^2}$ are used. Here, $\ebeam$ is the beam energy and $E_B$ ($p_B$) is the energy (momentum) of the $B$ candidate. Because of the proximity of the $B^0_s$ mass to the beam energy, $\bsbsbar$, $\bsbsstar$, and $\bsstbsst$ are kinematically well-separated. For $B$ mesons, $\bbbar$, $\bbstar$, $\bstbst$ and $\bbbarpipi$ are also kinematically separated. $\bbbarpi$, $\bbstarpi$ and $\bstbstpi$ occupy a broad, but distinct region in $\de-\mbc$ and overlap with one another, and partially with $\bbbarpipi$. Because $B_{(s)}$ candidates are reconstructed (not $B_{(s)}^*$), only for $\bbbar$ and $\bsbsbar$ does $\de$ peak at zero and $\mbc$ peak at the physical $B$ and $B^0_s$ masses. For the other processes, $\de$ and $\mbc$ are offset by a predictable amount. By measuring the yields in each of these kinematically defined regions, one can determine the $B$ and $B^0_s$ fractions at the ${\Upsilon(5S)}$ as well as the contributing subprocesses.

A second approach for obtaining the $B$ and $B^0_s$ yields at the ${\Upsilon(5S)}$ is to use inclusive particle yields. This technique exploits the large difference in inclusive rates for $B$ and $B^0_s$ to various particles, such as $D_s$, $D^0$, and $\phi$.

EXCLUSIVE $B$ MESON FINAL STATES AT THE ${\Upsilon(5S)}$
========================================================

CLEO reconstructs $B$ mesons via their decays $B\to\jpsi (K^+, K^0_S, K^{*0})$ and $B\to D^{(*)}(\pi,\rho)$ (25 channels in total) @b5s_cleo. First, the inclusive $B$ yield is determined by computing the invariant mass of all $B$ meson candidates with $-200< \de < 450$ MeV and $5272<\mbc<5448$ MeV. Unlike $\mbc$, invariant mass does not depend on the particular ${\Upsilon(5S)}$ decay mode. A fit to the invariant mass spectrum yields a signal of $(53.2\pm9.0)$ events. Using efficiencies ($\epsilon$) from Monte Carlo (MC) simulation and branching fractions (${\cal{B}}$) from the PDG @pdg, it is found $\sum{\epsilon_i {\cal{B}}_i}=7.2\times10^{-4}$, where the sum is over the 25 decay modes. From this it is concluded that $\sigma(e^+e^-\to\bbbar (X))=(0.177\pm0.030\pm0.016)$ nb. Using this cross-section and the measured total hadronic cross-section, the $B^0_s$ fraction in ${\Upsilon(5S)}$ decays, $f_s$, is computed to be $f_s = 1-(\sigma_{\bbbar X}/\sigma_{\rm had})=(0.41\pm0.10\pm0.09)$. To determine the contributing subprocesses, events are selected in the expected signal band in the $\de-\mbc$ plane and projected onto the $\mbc$ axis. The resulting distribution of events is shown in Fig. \[fig:mbc\_b\]. A dominant $\bstbst$ peak is observed, along with a smaller contribution from $\bbstar$, and an insignificant yield from $\bbbar$. The distributions are fit to the sum of 3 Gaussians, with resolutions determined from MC and the mass difference between the $\bstbst$ and $\bbbar$ peaks constrained to the expected value of 47.5 MeV. The fitted yields are $3.7^{+3.1}_{-2.4}$ $\bbbar$, $10.3\pm3.9$ $\bbstar$ and $31.4\pm6.1$ $\bstbst$ events. Insignificant yields in $B^{(*)}\bar{B}^{(*)}\pi$ and $\bbbarpipi$ lead to 90% confidence level upper limits of 13.1 and 6.4 events, respectively. Thus, it is found that $(74\pm15\pm8)\%$ of the $B$ production is through $\bstbst$ and $(24\pm9\pm3)$ is from $\bbstar$. For $\bbbar$, $B^{(*)}\bar{B}^{(*)}\pi$ and $\bbbarpipi$ 90% upper limits of 22%, 32% and 14%, respectively, are obtained. These results strongly disfavor the QHM and are consistent with the UQM.

The $\mbc$ peak position for $\bstbst$ and the corresponding peak position from $\bsstbsst$ @bs5s_ian can be used to make a precise measurement of the $B^*_s$ mass. Noting that $M(B_s^*)-M(B^*)=\mbc(\bsstbsst)-\mbc(\bstbst)+1.6$ MeV, and using the well-measured $B^*$ mass @pdg, gives $M(B_s^*)=(5411.7\pm1.6\pm0.6)$ MeV, which provides a significant improvement over the previous measurement. The M1 mass splitting for $B^0_s$ is thus found to be $(45.7\pm1.7\pm0.7)$ MeV, which is consistent with the value in the $B$ system of $(45.78\pm0.35)$ MeV @pdg, as expected from heavy quark symmetry.

![$\mbc$ for $B$ signal candidates in CLEO. The histogram is the data and the line is a fit as described in the text. The shaded regions indicate the shapes expected from $B^{(*)}\bar{B}^{(*)}\pi$ and $\bbbarpipi$. The inset shows the invariant mass for these particular final states, as described in the text. \[fig:mbc\_b\]](mbc_b.eps){height=".33\textheight"}

EXCLUSIVE $B^0_s$ MESON FINAL STATES AT THE ${\Upsilon(5S)}$
============================================================

Both CLEO and Belle reconstruct $B^0_s\to\jpsi(\phi,\eta,\eta^{\prime})$ @cleo_only and $B^0_s\to D_s^{(*)}(\pi,\rho)$ candidates. In Fig. \[fig:bs-excl\] we show the $\de$ versus $\mbc$ and $\mbc$ distribution for $B^0_s\to D_s^{(*)}(\pi,\rho)$ candidates from the CLEO ${\Upsilon(5S)}$ data sample. The three boxes, from left to right, indicate signal regions for $\bsbsbar$, $\bsbsstar$, and $\bsstbsst$. All 10 signal candidates are $\bsstbsst$. An additional 4 events are observed in the $\jpsi$ modes, all of which are also $\bsstbsst$. Using efficiencies from simulation and branching fractions from the PDG @pdg, yields $f_s^*=\sigma({\Upsilon(5S)}\to\bsstbsst)/\sigma({\Upsilon(5S)})=(37^{+14}_{-12}\pm9)\%$, where $f_s^*$ neglects the evidently small $\bsbsbar$ and $\bsbsstar$ contributions. Belle observes a total of 23 events (21 $\bsstbsst$, 2 $\bsbsstar$, 0 $\bsbsbar$) and determines $94^{+6}_{-9}\%$ of $B^0_s$ are from $\bsstbsst$ @drutskoy2.

![$\de$ versus $\mbc$ (left) and $\mbc$ (right) for $B^0_s\to D_s^{(*)}(\pi,\rho)$ candidates in the CLEO ${\Upsilon(5S)}$ data sample. The superimposed boxes are discussed in the text. \[fig:bs-excl\]](mbc_ds.eps){height=".21\textheight"}

INCLUSIVE ANALYSES
==================

Inclusive analyses are senstive to $f_s$, provided the <span>*probe*</span> particle chosen has sufficiently different production rate for $B$ and $B^0_s$ decays. CLEO has used both $D_s$ and $\phi$ mesons, both of which are expected to have substantially higher rates from $B^0_s$ than $B$ mesons @cleo5s_stone. Belle followed up with a similar pair of analyses that use $D_s$ and $D^0$ mesons @drutskoy2. Measurement of the $B^0_s$ fraction, $f_s$, is obtained using an expression of the form:

$$\begin{split}
{\cal{B}}({\Upsilon(5S)}\to D_s X)=N_{{\Upsilon(5S)}}(2f_s{\cal{B}}(B^0_s\to D_s X)+\\
(1-f_s){\cal{B}}({\Upsilon(4S)}\to D_s X)),
\end{split}
\label{eq:eq1}$$

where $N_{{\Upsilon(5S)}}$ is the number of ${\Upsilon(5S)}$ decays, and we have inserted $D_s$ as the probe particle as an example. Here one measures ${\cal{B}}({\Upsilon(5S)}\to D_s X)$ and ${\cal{B}}({\Upsilon(4S)}\to D_s X)$, and makes a model-dependent estimate of ${\cal{B}}(B^0_s\to D_s X)=(92\pm11)\%$ @cleo5s_stone, leaving only $f_s$ unknown. Using similar arguments, it is estimated that ${\cal{B}}(B^0_s\to D^0 X)=(8\pm7)\%$ @drutskoy2. The first measurement of $f_s$ using this technique was reported by CLEO, where they measure: $f_s = (16.8\pm2.6^{+6.7}_{-3.4})\%$ @cleo5s_stone using $D_s\to\phi\pi^+$ mesons. The $D^0$ and $D_s$ scaled momentum spectra from Belle using ${\Upsilon(5S)}$ data (points) and the below-${\Upsilon(4S)}$ continuum (histogram) are shown in Fig. \[fig:dsd0inc\]. The excess above continuum corresponds to the left-hand side of Eqn. \[eq:eq1\]. Using ${\cal{B}}(B\to D_s X)=(8.7\pm1.2)\%$ and ${\cal{B}}(B\to D^0 X)=(64.0\pm3.0)\%$ @pdg, Belle obtains: $f_s = (17.9\pm1.4\pm4.1)\%$ from $D_s$ and $f_s = (18.1\pm3.6\pm7.5)\%$ from $D^0$, which are consistent with each other and with the previous CLEO result.

![Scaled momentum distributions, $x=p_D/\ebeam$, for (a) $D_s^+\to\phi\pi^+$ and (b) $D^0\to K^-\pi^+$ from Belle. The points are data taken at the ${\Upsilon(5S)}$ resonance and the histogram is the continuum obtained from ${\Upsilon(4S)}$ data, normalized by the ratio of luminosities and $s=E_{\rm cm}^2$. \[fig:dsd0inc\]](Fig1b.eps "fig:"){height=".20\textheight"} ![Scaled momentum distributions, $x=p_D/\ebeam$, for (a) $D_s^+\to\phi\pi^+$ and (b) $D^0\to K^-\pi^+$ from Belle. The points are data taken at the ${\Upsilon(5S)}$ resonance and the histogram is the continuum obtained from ${\Upsilon(4S)}$ data, normalized by the ratio of luminosities and $s=E_{\rm cm}^2$. \[fig:dsd0inc\]](Fig2b.eps "fig:"){height=".20\textheight"}

CLEO has also performed an analysis using inclusive $\phi$ production @cleo5s_stone, where $\phi\to K^+K^-$. The measured yields as a function of scaled momentum are shown in Fig. \[fig:phi-5s\]. The first bin $0<x<0.05$ is obtained from MC simulation, because in this bin the kaons generally have too low momentum to be reconstructed. The uncertainty is taken to be 100% of its value. From the measured $D^0\to\phi X$, $D^+\to\phi X$ and $D_s\to\phi X$ branching fractions @cleo_incd, it is deduced that most $\phi$ mesons in $B_{(s)}$ decay come from the cascade $B_{(s)}\to D\to\phi$ or $B_{(s)}\to D_s\to\phi$, which allows one to make a model-dependent estimate ${\cal{B}}(B^0_s\to\phi X)=(16.1\pm2.4)\%$. One then uses Eqn. \[eq:eq1\] to obtain $f_s = (24.6\pm2.9^{+11.0}_{-5.0})\%$.

The combined CLEO average, using exclusive $B$ mesons, and inclusive $D_s$ and $\phi$ mesons, is $f_s=21^{+6}_{-3}\%$.

![ CLEO continuum-subtracted scaled momentum spectra for ${\Upsilon(5S)}\to\phi+X$ (filled circles) and ${\Upsilon(4S)}\to\phi+X$ (triangles). \[fig:phi-5s\]](phi-5s.eps){height=".31\textheight"}

These studies firmly establish the $B^0_s$ production rate to be about 25%, with roughly 90% of it produced as $B^{0*}_s\bar{B}^{0*}_s$. It is also found that $\bstbst$ is dominant, with $\bbstar$ about (1/3)$\bstbst$. Other processes are less than about 25% of $\bstbst$. These results are consistent with the UQM, and inconsistent with the QHM.

$\Delta\Gamma_s$ AND RARE DECAYS
================================

With a large sample of data collected on the ${\Upsilon(5S)}$, one can begin to search for rare $B^0_s$ decays. While many of the decays can be searched for in $p\bar{p}$ (CDF, D0) and $pp$ (LHCb at CERN, starting 2008), the $e^+e^-$ environment enjoys low particle multiplicity and good signal-to-background for low energy photons. Final states with soft photons are particularly difficult for detectors at hadron machines, and so this is an area where machines operating at the ${\Upsilon(5S)}$ can make significant contributions. Taking $\sigma(e^+e^-\to B^0_s\bar{B}^0_s)=0.075$ nb (25% of the resonance cross-section), a 50 $\ifb$ dataset, which could be collected by Belle in a couple of months, yields about 7.5 million $B^0_s$ decays.

In the Standard Model, the width difference, $\Delta\Gamma^{CP}_s$ between the CP eigenstates, $B_s^{odd}$ and $B_s^{even}$ is given by $\Delta\Gamma^{CP}_s=2|\Gamma_{12}|$, where $\Gamma_{12}$ is the off-diagonal matrix element in the $B^0_s$ decay matrix, which connects $B^0_s$ and $\bar{B}^0_s$ via on-shell intermediate states. These matrix elements are saturated by tree-diagrams, such as $B^0_s\to D_s^{+(*)}D_s^{-(*)}$, and therefore $\Gamma^{CP}_s$ is expected to be insensitive to new physics. Because there are more CP even than CP odd final states in $B^0_s$ decays, one expects $\Delta\Gamma^{CP}_s\simeq(0.12\pm0.06)$@nierste. In the presence of CP violation, one finds a width difference between the mass eigenstates of $\Delta\Gamma_s=\Delta\Gamma^{CP}_s\cos\phi$, where $\phi=arg(-M_{12}/\Gamma_{12})$, is the difference between the $B^0_s$ mixing phase and the phase of $-\Gamma_{12}$. In the Standard Model, $\phi$ is expected to be small, of order 0.01. Therefore, one expects $\Delta\Gamma^{CP}_s\simeq\Delta\Gamma_s$. Since new physics could affect $B^0_s$ mixing, <span>*i.e.,*</span> the phase of $M_{12}$, new physics could drive $\Delta\Gamma_s$ away from $\Delta\Gamma^{CP}_s$. Precise measurements of $\Delta\Gamma^{CP}_s$ can be obtained by measuring the branching fractions and CP content of final states accessible to $B^0_s$ and $\bar{B}^0_s$. Such final states are dominated by $D_s^{+(*)}D_s^{-(*)}$, but also include $\jpsi\phi$, $\jpsi\eta$ $\jpsi\eta^{\prime}$, etc. Under certain assumptions @nierste, it can be shown that $B^0_s\to D_s^{+(*)}D_s^{-(*)}$ is pure CP even, and neglecting the charmonium states, one finds $\Delta\Gamma^{CP}_s/\Gamma_s\simeq 2{\cal{B}}(B^0_s\to D_s^{(*)+}D_s^{(*)-})$. With a 50 $\ifb$ ${\Upsilon(5S)}$ data sample, one would expect $\sim$10 events in each decay mode @drutskoy3, yielding $\sim$25% accuracy on ${\cal{B}}(B^0_s\to D_s^{+(*)}D_s^{-(*)})$. If these assumptions are relaxed, one must measure the CP fraction of the various vector-vector final states, which would require substantially larger statistics.

Various rare loop-induced decay modes have been searched for in the Belle data sample, including $B^0_s\to\gamma\gamma$, $B^0_s\to\phi\gamma$, and $B^0_s\to K^+K^-$ @drutskoy1. The limit ${\cal{B}}(B^0_s\to\gamma\gamma)<0.53\times10^{-4}$, is already more stringent than the current world average.

CONCLUSIONS
===========

In summary, we report on measurements of the $B^0_s$ fraction, $f_s$ at CLEO and Belle, and find about 25% of the ${\Upsilon(5S)}$ decays produce $B^0_s$, predominantly in the form of $\bsstbsst$. We also find a dominance of $\bstbst$ among ordinary $B$ decays. Both of these observations are consistent with the UQM and inconsistent with the QHM. An $e^+e^-$ machine operating at the ${\Upsilon(5S)}$ produces about 15 million $B^0_s$ per 100 $\ifb$. Studies of decay channels that produce soft photons could provide complementary information to $B^0_s$ studies at hadron machines.

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[^1]: We gratefully acknowledge the National Science Foundation for support of this work.

{'source': 'arXiv_out/0705'}