Search for the Decay $B_{s}^{0}\rightarrow\eta^{\prime}\eta$

In the Standard Model (SM) charmless hadronic decays $B_{s}^{0}\rightarrow\eta^{\prime}\eta$ proceed via tree-level $b\to u$ and penguin $b\to s$ transitions as shown in Fig. I. Penguin transitions are sensitive to Beyond-the-Standard-Model (BSM) physics scenarios and could affect the branching fractions and CP asymmetries in such decaysbelleiiphysicsbook . Once branching fractions for two-body decays $B_{s}\to\eta\eta,\eta\eta^{\prime},\eta^{\prime}\eta^{\prime}$ are measured, and the theoretical uncertainties are reduced, it would be possible to extract CP violating parameters from the data using the formalism based on SU(3)/U(3) symmetry bf1 . To achieve this goal, at least four of these six branching fractions need to be measured. Only the branching fraction for $B_{s}^{0}\to\eta^{\prime}\eta^{\prime}$ has been measured so far bsepep .

In this paper we report the results of the first search for the decay $B_{s}^{0}\rightarrow\eta^{\prime}\eta$ using the full Belle data sample of $121.4\textrm{fb}^{-1}$ collected at the $\Upsilon(5S)$ resonance. The Belle detector Belle was a large-solid-angle magnetic spectrometer that operated at the KEKB asymmetric-energy $e^{+}e^{-}$ collider KEKB . The detector components relevant to our study include a tracking system comprising a silicon vertex detector (SVD) and a central drift chamber (CDC), a particle identification (PID) system that consists of a barrel-like arrangement of time-of-flight scintillation counters (TOF) and an array of aerogel threshold Cherenkov counters (ACC), and a CsI(Tl) crystal-based electromagnetic calorimeter (ECL). All these components are located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.

The $\Upsilon(5S)$ decays into $B_{s}^{*0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{*0}$, $B_{s}^{*0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{0}$ or $B_{s}^{0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{*0}$, and $B_{s}^{0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{0}$ pairs with relative fractions $f_{B_{s}^{*0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{*0}}=(87.0\pm 1.7)\%$ and $f_{B_{s}^{*0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{0}}=(7.3\pm 1.4)\%$ frac . The data sample contains $(6.53\pm 0.66)\times 10^{6}$ $B_{s}^{(*)0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{(*)0}$ pairs nbsbsb . The excited vector state $B_{s}^{*0}$ decays to $B_{s}^{0}$ by emitting a photon. The daughter $\eta^{\prime}$ meson is reconstructed in the decay mode $\pi^{+}\pi^{-}\eta$, each of the two $\eta$ mesons is reconstructed via its two photon decay. The expected branching fraction for the $B_{s}$ decay of interest spans a wide range: $(2-4)\times 10^{-5}$ bf1 ; bf2 ; bf3 ; bf4 ; bf5 , where the main source of theoretical uncertainty is due to non-perturbative hadronic physics.

To maximize analysis discovery potential and to validate the signal extraction procedure we use a background Monte Carlo (MC) sample equivalent to six times the data statistics. We use a high-statistics signal MC sample to estimate the overall reconstruction efficiency. Both samples are used to develop a model implemented in the unbinned extended maximum likelihood (ML) fit to data. The MC-based model is calibrated using a control data sample of 711 ${\rm fb^{-1}}$ collected at the $\Upsilon(4S)$.

We reconstruct $\eta$ candidates using pairs of electromagnetic showers not matched to the projections of charged tracks to the calorimeter. We require that the reconstructed energy of these showers exceed 50 (100) MeV in the barrel (end-cap) region of the ECL. The larger end-cap ECL energy threshold is due to the larger beam-related background in this region. The ECL energy thresholds have practically no impact on the analysis discussed in this paper. To reject hadronic showers mimicking photons, the ratio of the energy deposited by a photon candidate in the $(3\times 3)$ and $(5\times 5)$ ECL crystal array centered on the crystal with the largest reconstructed energy is required to exceed 0.75. The invariant mass of the $\eta$ candidate is required to be in the range $515\leq M(\gamma\gamma)\leq 580$ ${\rm MeV/c}^{2}$, which corresponds, approximately, to $\pm 3\sigma$ when approximated by a Gaussian resolution function. To suppress misreconstructed $\eta$ candidates, the absolute value of cosine of helicity angle (defined as the angle between the photon momentum in presumed parent’s rest frame and the momentum of the parent in the laboratory frame) is required to be less than 0.97.

Candidates for the decay $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ are reconstructed using pairs of oppositely-charged pions and $\eta$ candidates. We require the reconstructed $\eta^{\prime}$ invariant mass to be in the range $920\leq M(\pi^{+}\pi^{-}\eta)\leq 980$ ${\rm MeV/c}^{2}$, which corresponds, approximately, to the range $[-10,+6]\sigma$ of the Gaussian approximation for the resolution function, after performing a kinematic fit constraining the reconstructed invariant mass of the daughter $\eta$ candidate to the nominal $\eta$ mass PDG . To identify charged pion candidates, the ratios of PID likelihoods, $R_{i/\pi}=L_{\pi}/(L_{\pi}+L_{i})$, are used, where $L_{\pi}$ is the likelihood for the track according to pion hypothesis, while $L_{i}$ is the likelihood according to kaon ($i=K$) or electron ($i=e$) hypotheses. We require $R_{K/\pi}\leq 0.6$ and $R_{e/\pi}\leq 0.95$ for pion candidates. According to MC studies, these requirements reject 28% of background signal candidates (which are primarily due to charged kaons and electrons), while the resulting efficiency loss is below 3%. Charged pion tracks are required to originate from near the interaction point (IP) by restricting their distance of closest approach along and perpendicular to the beam collision axis to be less than 4.0 cm and 0.3 cm, respectively. These selection criteria suppress beam-related backgrounds and reject poorly-reconstructed tracks. To reduce systematic uncertainties associated with track reconstruction efficiency, the transverse momenta of charged pions are required to be greater than 100 MeV/c.

To identify $B_{s}^{0}\rightarrow\eta^{\prime}\eta$ candidates we use beam-energy constrained $B_{s}^{0}$ mass, $M_{\rm bc}=\sqrt{E_{\rm beam}^{2}-p_{B_{s}}^{2}}$, the energy difference, $\Delta E=E_{B_{s}}-E_{\rm beam}$, and the reconstructed invariant mass of the $\eta^{\prime}$, where $E_{\rm beam}$, $p_{B_{s}}$ and $E_{B_{s}}$ are the beam energy, the momentum magnitude and the reconstructed energy of $B_{s}^{0}$ candidate, respectively. All these quantities are evaluated in the $e^{+}e^{-}$ center-of-mass frame. To improve the $\Delta E$ resolution (by approximately 10%), each $\eta$ candidate is kinematically constrained to the nominal invariant mass of $\eta$, the $\eta^{\prime}$ candidates are further constrained to the nominal invariant mass of $\eta^{\prime}$. Signal candidates are required to satisfy selection criteria $M_{\rm bc}>5.3$ ${\rm GeV/c}^{2}$ and $-0.4\leq\Delta E\leq 0.3$ GeV. In Gaussian approximation, the $\Delta E$ resolution is, approximately, 40 MeV. The beam-energy-constrained $B_{s}^{0}$ mass resolution is 4 ${\rm Mev/c^{2}}$. To improve the significance of the signal in case the data indicate its presence, we include the reconstructed invariant mass $M(\pi^{+}\pi^{-}\eta)$ in the 3D ML fit used to statistically separate the signal from background.

Hadronic continuum, i.e. production of light quark pairs in the $e^{+}e^{-}$ annihilation [$e^{+}e^{-}\to q\bar{q}$ ($q=u,d,c,s$)], is the primary source of background in studies of charmless hadronic decays. Because of large initial momenta of the light quarks, continuum events exhibit a “jet-like” event shape, while $B_{s}^{(*)0}\accentset{\rule{4.91673pt}{0.6pt}}{B}_{s}^{(*)0}$ events are distributed isotropically. We use modified Fox-Wolfram moments ksfw , used to describe the topology of the event, to discriminate between signal events and continuum background. A likelihood ratio ($\mathcal{LR}$) is calculated using Fisher discriminant coefficients obtained in an optimization based on these moments. We suppress the background using a discovery-optimized cut on $\mathcal{LR}$ obtained using Punzi’s figure-of-merit punzi :

The background contains real $\eta^{\prime}$ mesons. Such events exhibit a peak in the $M(\pi^{+}\pi^{-}\eta)$ distribution, however, they are distributed uniformly in $M_{\rm bc}$ and $\Delta E$. The fraction of this peaking background is a free parameter in our ML fits.

About 14% of fully-reconstructed signal MC events contain multiple candidates which are primarily (in 75% of such events) due to misreconstructed $\eta$ mesons. In such events we use only the best candidate with the smallest value of $\sum{\chi^{2}_{\eta}}+\chi^{2}_{\pi^{+}\pi^{-}}$, where the values of $\chi^{2}_{\eta}$ are from the mass-constrained fit for the $\eta$ candidates and $\chi^{2}_{\pi^{+}\pi^{-}}$ is from a vertex fit for the charged pion pair. The overall reconstruction efficiency is estimated to be 10% including a 50% relative efficiency loss due to the discovery-optimized background suppression.

To extract the signal yield, we perform an unbinned extended maximum likelihood fit to the three-dimensional (3D) distribution of $M_{\rm bc}$, $\Delta E$, and $M(\pi^{+}\pi^{-}\eta)$. The likelihood function is

We use $B^{0}\to\eta^{\prime}K_{S}^{0}$ data recorded at the $\Upsilon(4S)$ resonance to adjust the PDF shape parameters used to describe the signal. We reconstruct $K_{S}^{0}$ candidates via secondary vertices associated with pairs of oppositely-charged pions ks_reco using a neural network (NN) technique NN . The following information is used in the NN: the momentum of $K_{S}^{0}$ candidate in the laboratory frame; the distance along the $z$ axis between the two track helices at the point of their closest approach; the flight length in the $x-y$ plane; the angle between the $K_{S}^{0}$ momentum and the vector joining the $K_{S}^{0}$ decay vertex to the IP; the angle between the pion momentum and the laboratory-frame $K_{S}^{0}$ momentum in the $K_{S}^{0}$ rest frame; the distance-of-closest-approach in the $x-y$ plane between the IP and the two pion helices; and the pion hit information in the SVD and CDC. The selection efficiency is 87% over the momentum range of interest. We also require that the $\pi^{+}\pi^{-}$ invariant mass be within 12 ${\rm MeV/c^{2}}$ (about 3.5$\sigma$ in resolution) of the nominal $K_{S}^{0}$ mass PDG . We require $5.2\leq M_{\rm bc}\leq 5.3$ ${\rm GeV/c^{2}}$ for $B^{0}$ candidates. All other selection criteria applied to the $B^{0}$ candidates are the same as those used to select $B_{s}^{0}$ candidates.

The presence of four photons in our final state gives rise to a sizable misreconstruction probability for the signal. We study partially misreconstructed signal events, denoted Self Cross Feed (SCF) events, using signal MC sample. A large correlation between $M_{\rm bc}$ and $\Delta E$ for such signal MC events (the Pearson correlation coefficient of 27% for the region of largest same-sign correlations) is taken into account by describing the well-reconstructed part of the signal and SCF separately. SCF events comprise approximately 19% of the reconstructed signal MC sample and are excluded from signal fit model and the efficiency estimate. No sizable correlations among fit variables have been identified for well-reconstructed signal MC events nor for background events.

A sum of a Gaussian and a Crystal Ball xbal function is used to model the signal in each of the three fit variables. For $M_{\rm bc}$ and $M(\pi^{+}\pi^{-}\eta)$ we use a sum with the same mean but different widths, while for $\Delta E$ both mean and width are different. A different approach for the $\Delta E$ parametrization is necessary to provide a better description of its PDF which has a long asymmetric tail due to the additional particles used to evaluate this variable. We use a Crystal Ball function to describe the tails arising from energy leakage expected for photons in the calorimeter. A Bukin function bukin and an asymmetric Gaussian are used to model the SCF contribution in $M_{\rm bc}$ and $\Delta E$, respectively. For $M(\pi^{+}\pi^{-}\eta)$, we use a sum of a Gaussian and a first order Chebyshev polynomial. The signal PDF shape parameters for $M_{\rm bc}$ and $\Delta E$ have been adjusted using the results obtained from the $\Upsilon(4S)$ data.

An ARGUS argus function is used to describe the background distribution in $M_{\rm bc}$, another first-order Chebyshev polynomial is used for $\Delta{E}$. To model the peaking part in $M(\pi^{+}\pi^{-}\eta)$ we use the signal PDF, because the peak is due to real $\eta^{\prime}$ mesons, while an additional first-order Chebyshev polynomial is used for non-peaking contribution.

To test and validate our fitting model, ensemble tests are performed by generating MC pseudoexperiments. In these experiments we use PDFs obtained from simulation and the $B^{0}\rightarrow\eta^{\prime}K_{S}^{0}$ data. The number of signal events is varied between 0 and 50 events, and 1000 pseudoexperiments are performed for each assumed number of signal events. An ML fit is performed for each sample generated in these experiments. For all values of assumed number of signal events the fit signal yield distribution peaks at the expected value, therefore exhibiting good linearity. We use the results of pseudoexperiments to construct classical confidence intervals (without ordering) using a procedure due to Neyman frequentist_approach . For each ensemble of pseudoexperiments the lower and upper ends of respective confidence interval represent the values of fit signal yields for which 10% of the results lie below and above these values, respectively. These confidence intervals are then used to prepare a classical 80% confidence belt belt_method shown in Fig. IX. We use this confidence belt to make a statistical interpretation of the results obtained from ML fit to data.

We fit the 3D fit model described above to the data and obtain $2.7\pm 2.5$ signal and $57.3\pm 7.8$ background events. We show the signal-region projections of the fit to data in Fig. IX. We observe no signal and estimate the 90% confidence-level (CL) upper limit on the branching fraction for the decay $B_{s}^{0}\rightarrow\eta^{\prime}\eta$ using the frequentist approach frequentist_approach and the following formula:

The relative systematic uncertainties for the quantities used in the upper limit estimate are summarized in Table 1. The statistical uncertainty on the reconstruction efficiency can be estimated as $\sqrt{\varepsilon\times(1-\varepsilon)/N}$, where $N$ is the total number of generated signal MC events and $\varepsilon$ is the reconstruction efficiency. This uncertainty is estimated to be 0.1%. We assign a 2.1% systematic uncertainty per $\eta$ candidate eta_syst . Since we have two $\eta$ candidates in our decay, we assign a 4.2% uncertainty for $\eta$ reconstruction. The systematic uncertainty associated with track reconstruction is 0.35% per track track_syst . We therefore assign an uncertainty of 0.7% for two tracks. We assign a 15.3% systematic uncertainty due to the discovery-optimized $\mathcal{LR}$ cut. This uncertainty reflects the relative change in the efficiency when the cut is varied by 0.02 about nominal value of 0.95. Combining all the sources of uncertainties, the total relative systematic uncertainty is 19%.

In summary, we have used the full data sample recorded by the Belle experiment at the $\Upsilon(5S)$ resonance to search for the rare decay $B_{s}^{0}\rightarrow\eta^{\prime}\eta$. We observe no statistically significant signal and set a 90% CL upper limit of $7.1\times 10^{-5}$ on its branching fraction. Our result is 2 times larger than the most optimistic SM-based and QCD-enhanced theoretical prediction and, to date, is the only experimental information on $B_{s}^{0}\rightarrow\eta^{\prime}\eta$. This decay will be probed further at the next-generation Belle II experiment belle2 at the SuperKEKB collider in Japan.

We thank the KEKB group for the excellent operation of the accelerator, the KEK cryogenics group for the efficient operation of the solenoid, and the KEK computer group and the National Institute of Informatics for valuable computing and SINET3 network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Department of Education, Science and Training; the National Natural Science Foundation of China under contract No. 10575109 and 10775142; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea, the CHEP SRC program and Basic Research program (grant No. R01-2005-000-10089-0) of the Korea Science and Engineering Foundation, and the Pure Basic Research Group program of the Korea Research Foundation; the Polish State Committee for Scientific Research; the Ministry of Education and Science of the Russian Federation and the Russian Federal Agency for Atomic Energy; the Slovenian Research Agency; the Swiss National Science Foundation; the National Science Council and the Ministry of Education of Taiwan; and the U.S. Department of Energy.