Measurements of branching fraction and direct $C\!P$ asymmetry in $B^{\pm}\to K^{0}_{S}K^{0}_{S}K^{\pm}$ and a search for $B^{\pm}\to K^{0}_{S}K^{0}_{S}\pi^{\pm}$

TL;DR

This paper measures the branching fractions and direct CP asymmetry in specific charmless B meson decays using a large data sample from the Belle detector, providing precise results and setting upper limits where signals are not significant.

Contribution

First precise measurement of branching fractions and CP asymmetry in $B^{ m ext{±}} o K^{0}_{S}K^{0}_{S}K^{ m ext{±}}$ decays, and the first upper limit for $B^{ m ext{±}} o K^{0}_{S}K^{0}_{S}\pi^{ m ext{±}}$.

Findings

Branching fraction for $B^{ m ext{±}} o K^{0}_{S}K^{0}_{S}K^{ m ext{±}}$ is $(10.42 extpm0.43 extpm0.22) imes10^{-6}$.

CP asymmetry in $B^{ m ext{±}} o K^{0}_{S}K^{0}_{S}K^{ m ext{±}}$ is consistent with zero.

Upper limit on $B^{ m ext{±}} o K^{0}_{S}K^{0}_{S}\pi^{ m ext{±}}$ is $8.7 imes10^{-7}$ at 90% CL.

Abstract

We study charmless hadronic decays of charged $B$ mesons to the final states $K^{0}_{S}K^{0}_{S}K^{\pm}$ and $K^{0}_{S}K^{0}_{S}\pi^{\pm}$ using a $711 fb^{-1}$ data sample that contains $772\times 10^6$ $B\bar{B}$ pairs, and was collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. For $B^{\pm}\to K^{0}_{S}K^{0}_{S}K^{\pm}$, the measured branching fraction and direct $CP$ asymmetry are $[10.42\pm0.43(stat)\pm 0.22(syst)]\times10^{-6}$ and [$+1.6\pm3.9(stat)\pm 0.9(syst)$]%, respectively. In the absence of a statistically significant signal for $B^{\pm}\to K^{0}_{S}K^{0}_{S}\pi^{\pm}$, we obtain a 90% confidence-level upper limit on its branching fraction as $8.7 \times10^{-7}$.

Tables

Table 1. Table 1: List of PDFs used to model the Δ ​ E Δ 𝐸 \Delta E and C N ​ B ′ subscript superscript 𝐶 ′ N B C^{\prime}_{\mathrm{{N\!B}}} distributions for various event categories for B + → K S 0 ​ K S 0 ​ K + → superscript 𝐵 subscript superscript 𝐾 0 𝑆 subscript superscript 𝐾 0 𝑆 superscript 𝐾 B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+} . G, AG, and Poly1 denote Gaussian, asymmetric Gaussian, and first-order polynomial, respectively.

Event category		$\Delta E$		$C_{NB}^{\prime }$
Signal		3 G		G+AG
Continuum $q\overline{q}$		Poly1		2 G
Combinatorial $B\overline{B}$		Poly1		2 G
Feed-across		G+Poly1		G

Table 2. Table 2: Efficiency, differential branching fraction, and 𝒜 C ​ P subscript 𝒜 𝐶 𝑃 {\mathcal{A}_{C\!P}} in each M K S 0 ​ K S 0 subscript 𝑀 superscript subscript 𝐾 𝑆 0 superscript subscript 𝐾 𝑆 0 M_{K_{S}^{0}K_{S}^{0}} bin for B + → K S 0 ​ K S 0 ​ K + → superscript 𝐵 subscript superscript 𝐾 0 𝑆 subscript superscript 𝐾 0 𝑆 superscript 𝐾 B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+} .

$M_{K_S^0{K}_S^0}$ (GeV/$c^2$)	Efficiency (%)		$dℬ/dM\times {10}^{-6}$ ($c^2$/GeV)		${𝒜}_{CP}$ (%)
$1.0-1.1$	$24.0\pm 0.4$		$10.40\pm 1.24\pm 0.38$		$-3.9\pm 10.9\pm 0.9$
$1.1-1.3$	$23.4\pm 0.2$		$8.60\pm 0.85\pm 0.32$		$-0.1\pm 9.3\pm 0.9$
$1.3-1.6$	$22.9\pm 0.1$		$10.23\pm 0.73\pm 0.38$		$+6.6\pm 6.9\pm 0.9$
$1.6-2.0$	$21.8\pm 0.1$		$3.93\pm 0.43\pm 0.15$		$+16.1\pm 10.3\pm 0.9$
$2.0-2.3$	$24.1\pm 0.1$		$3.90\pm 0.47\pm 0.15$		$-3.3\pm 11.3\pm 0.9$
$2.3-2.7$	$25.2\pm 0.1$		$2.45\pm 0.33\pm 0.09$		$-5.7\pm 12.2\pm 1.0$
$2.7-5.0$	$26.3\pm 0.0$		$0.35\pm 0.07\pm 0.01$		$-31.9\pm 19.7\pm 1.2$

Table 3. Table 3: Systematic uncertainties in the branching fraction of B + → K S 0 ​ K S 0 ​ π + → superscript 𝐵 subscript superscript 𝐾 0 𝑆 subscript superscript 𝐾 0 𝑆 superscript 𝜋 B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+} .

Source	Relative uncertainty in $ℬ$ ($\%$)
Tracking	$0.35$
Particle identification	$0.80$
Number of $B\overline{B}$ pairs	$1.37$
Continuum suppression	$0.34$
Requirement on $M_{\mathrm{bc}}$	$0.03$
$K_S^0$ reconstruction	$3.22$
Fit bias	$1.86$
Signal PDF	$1.30$
Combinatorial $B\overline{B}$ PDF	$+1.31,-1.98$
Feed-across PDF	$+3.57,-4.10$
Fixed background yield	$+2.63,-2.27$
Fixed background ${𝒜}_{CP}$	$0.50$
Total	$+6.30$, $-6.67$

Table 4. Table 4: Systematic uncertainties in the differential branching fraction and 𝒜 C ​ P subscript 𝒜 𝐶 𝑃 {\mathcal{A}_{C\!P}} in M K S 0 ​ K S 0 subscript 𝑀 superscript subscript 𝐾 𝑆 0 superscript subscript 𝐾 𝑆 0 M_{K_{S}^{0}K_{S}^{0}} bins for B + → K S 0 ​ K S 0 ​ K + → superscript 𝐵 subscript superscript 𝐾 0 𝑆 subscript superscript 𝐾 0 𝑆 superscript 𝐾 B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+} . “ † † \dagger ” indicates the uncertainty is independent of M K S 0 ​ K S 0 subscript 𝑀 superscript subscript 𝐾 𝑆 0 superscript subscript 𝐾 𝑆 0 M_{K_{S}^{0}K_{S}^{0}} with the listed value being applicable for all the bins. An ellipsis indicates a value below 0.05 % percent 0.05 0.05\% in d ​ ℬ / d ​ M 𝑑 ℬ 𝑑 𝑀 d{\cal B}/dM and 0.001 in  𝒜 C ​ P subscript 𝒜 𝐶 𝑃 {\mathcal{A}_{C\!P}} .

$M_{K_S^0{K}_S^0}$ (GeV/$c^2$)	$1.0-1.1$	$1.1-1.3$	$1.3-1.6$	$1.6-2.0$	$2.0-2.3$	$2.3-2.7$	$2.7-5.0$
Source	Relative uncertainty in $dℬ/dM$ ($\%$)
Tracking†	$0.35$
Particle identification†	$0.80$
Number of $B\overline{B}$ pairs†	$1.37$
Continuum suppression†	$0.34$
Requirement on $M_{\mathrm{bc}}^{†}$	$0.03$
$K_S^0$ reconstruction†	$3.22$
Fit bias†	$0.53$
Signal PDF	${}_{-0.27}{}^{}{}_{}{}^{+0.33}$	${}_{-0.48}{}^{}{}_{}{}^{+0.63}$	${}_{-0.44}{}^{}{}_{}{}^{+0.46}$	${}_{-0.63}{}^{}{}_{}{}^{+0.22}$	${}_{-0.38}{}^{}{}_{}{}^{+0.52}$	$0.67$	$1.10$
Combinatorial $B\overline{B}$ PDF	$0.09$	${}_{-0.13}{}^{}{}_{}{}^{+0.08}$	$0.12$	${}_{-0.21}{}^{}{}_{}{}^{+0.17}$	${}_{-0.34}{}^{}{}_{}{}^{+0.26}$	$0.40$	$0.40$
Feed-across PDF	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
Fixed background yield	$\cdots$	$0.10$	$0.10$	$0.23$	$\cdots$	$0.11$	$0.60$
Fixed background ${𝒜}_{CP}$	$\cdots$	$\cdots$	$\cdots$	$0.20$	$0.10$	$\cdots$	$0.13$
Total	$\pm 3.68$	$\pm 3.72$	$\pm 3.69$	$\pm 3.73$	$\pm 3.72$	$\pm 3.75$	$\pm 3.89$
$M_{K_S^0{K}_S^0}$(GeV/$c^2$)	$1.0-1.1$	$1.1-1.3$	$1.3-1.6$	$1.6-2.0$	$2.0-2.3$	$2.3-2.7$	$2.7-5.0$
Source	Absolute uncertainty in ${𝒜}_{CP}$
Signal PDF	$0.001$	$0.002$	$0.001$	$0.002$	$0.001$	$0.001$	$0.004$
Combinatorial $B\overline{B}$ PDF	$0.001$	$0.001$	$0.001$	$\cdots$	$0.001$	$0.002$	$0.001$
Feed-across PDF	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
Fixed background yield	$\cdots$	$\cdots$	$0.001$	$0.001$	$0.001$	$0.001$	$0.004$
Fixed background ${𝒜}_{CP}$	$\cdots$	$\cdots$	$0.001$	$0.001$	$0.001$	$0.002$	$0.006$
Detector bias†	$0.009$
Total	$\pm 0.009$	$\pm 0.009$	$\pm 0.009$	$\pm 0.009$	$\pm 0.009$	$\pm 0.010$	$\pm 0.012$

Equations

$${\mathcal{A}}_{CP}=\frac{N(B^{-}\to K_S^0{K}_S^0{h}^{-})-N(B^{+}\to K_S^0{K}_S^0{h}^{+})}{N(B^{-}\to K_S^0{K}_S^0{h}^{-})+N(B^{+}\to K_S^0{K}_S^0{h}^{+})},$$

$$C_{\mathrm{N}\mathrm{B}}^{\prime }=\ln \left[\frac{C_{\mathrm{N}\mathrm{B}}-C_{\mathrm{N}\mathrm{B},\min }}{C_{\mathrm{N}\mathrm{B},\max }-C_{\mathrm{N}\mathrm{B}}}\right],$$

$${\cal L}=\dfrac{\mathrm{e}^{-\sum_{j}n_{j}}}{N!}\prod_{i}\Big{[}\sum_{j}n_{j}\mathcal{P}_{j}^{i}\Big{]},$$

$${\cal P}_{j}^{i}\equiv\dfrac{1}{2}(1-q^{i}{\mathcal{A}_{C\!P,j}})\times{\cal P}_{j}(\mbox{$\Delta E$}^{\,i})\times{\cal P}_{j}(C_{\mathrm{{N\!B}}}^{\prime\,i}).$$

$$\mathcal{B}(B^{+}\to K_S^0{K}_S^0{h}^{+})=\frac{n_{\mathrm{sig}}}{ϵ\times N_{B\overline{B}}\times [\mathcal{B}(K_S^0\to {\pi }^{+}{\pi }^{-}){]}^2}\text{mbox},$$

$$\mathcal{B}(B^{+}\to K_S^0{K}_S^0{\pi }^{+})=(6.5\pm 2.6\pm 0.4)\times 10^{-7},$$

$$\mathcal{B}(B^{+}\to K_S^0{K}_S^0{K}^{+})=(10.42\pm 0.43\pm 0.22)\times 10^{-6}\text{mbox},$$

$${\mathcal{A}}_{CP}(B^{+}\to K_S^0{K}_S^0{K}^{+})=(+1.6\pm 3.9\pm 0.9)\%\text{mbox}.$$

Full text

††preprint:

Belle Preprint 2018-26

KEK Preprint 2018-83

Charged $B$-meson decays to the three-body charmless hadronic final states $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{\pm}$ and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{\pm}$ mainly proceed via $b\rightarrow s$ and $b\rightarrow d$ loop transitions, respectively. Figure 1 shows Feynman diagrams of the dominant amplitudes that contribute to these decays. These flavor changing neutral current transitions, being suppressed in the standard model (SM), are interesting as they could be sensitive to possible non-SM contributions PAP:ref .

Further motivation, especially to study the contributions of various quasi-two-body resonances to inclusive $C\!P$ asymmetry, comes from the recent results on $B^{\pm}\rightarrow K^{+}K^{-}K^{\pm}$, $K^{+}K^{-}\pi^{\pm}$ and other such three-body decays LHCb:paper1 ; LHCb:paper2 ; Chialing:paper . LHCb has found large asymmetries localized in phase space in $B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}$ decays LHCb:paper2 . Recently, Belle has also reported strong evidence for large $C\!P$ asymmetry at the low $K^{+}K^{-}$ invariant mass region of $B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}$ Chialing:paper . The fact that the $K\overline{K}$ system of $B^{\pm}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}h^{\pm}(h=K,\pi)$, in contrast to that of $B^{\pm}\rightarrow K^{+}K^{-}h^{\pm}$, cannot form a vector resonance (Bose symmetry) may shed light on the source of large $C\!P$ violation in the latter decays.

The three-body decay $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ charge:ref1 was observed by Belle Belle:paper1 and subsequently studied by BaBar BaBar:paper1 . Belle measured the decay branching fraction as $(13.4\pm 1.9\pm 1.5)\times 10^{-6}$ based on a data sample of $70\mbox{\,fb}^{-1}$ Belle:paper1 , and BaBar reported a branching fraction of $(10.6\pm 0.5\pm 0.3)\times 10^{-6}$ and a $C\!P$ asymmetry of $(+4_{-5}^{+4}\pm 2)\%$ using $426\mbox{\,fb}^{-1}$ of data BaBar:paper1 . The quoted uncertainties are statistical and systematic, respectively.

The decay $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ is suppressed by the squared ratio of CKM matrix CKM:ckm elements $|V_{td}/V_{ts}|^{2}(=0.046)$ with respect to $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$, and has not yet been observed. The most restrictive limit at $90\%$ confidence level on its branching fraction, ${\cal B}(B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+})<5.1\times 10^{-7}$, comes from BaBar BaBar:paper2 .

We present an improved measurement of the branching fraction and direct $C\!P$ asymmetry of the decay $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ as well as a search for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ using a data sample of $711\mbox{\,fb}^{-1}$, which contains $772\times 10^{6}$ $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs and was recorded near the $\mathchar 28935\relax{(4S)}$ resonance with the Belle detector Belle at the KEKB $e^{+}e^{-}$ collider KEKB . The direct $C\!P$ asymmetry is defined as

[TABLE]

where $N$ is the obtained signal yield for the corresponding mode. The detector components relevant for our study are a silicon vertex detector (SVD), a $50$-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), and a barrel-like arrangement of time-of-flight scintillation counters (TOF); all located inside a $1.5$ T solenoidal magnetic field.

To reconstruct $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}h^{+}$ candidates, we begin by identifying charged kaons and pions. A kaon or pion candidate track must have a minimum transverse momentum of $100{\mathrm{\,Me\kern-1.00006ptV\!/}c}$ in the lab frame, and a distance of closest approach with respect to the interaction point (IP) of less than $0.2{\rm\,cm}$ in the transverse $r$–$\phi$ plane and less than $5.0{\rm\,cm}$ along the $z$ axis. Here, the $z$ axis is defined opposite the $e^{+}$ beam. Charged tracks are identified as kaons or pions based on a likelihood ratio ${\cal R}_{K/\pi}={{\cal L}_{K}}/({\cal L}_{K}+{\cal L}_{\pi})$, where ${\cal L}_{K}$ and ${\cal L}_{\pi}$ are the individual likelihoods for kaons and pions, respectively, calculated with information from the CDC, ACC and TOF. Tracks with ${\cal R}_{K/\pi}>0.6$ are identified as kaons while those with ${\cal R}_{K/\pi}<0.4$ are identified as pions. The efficiency for kaon (pion) identification is $86\%$ ($91\%$) with a pion (kaon) misidentification rate of $9\%$ ($14\%$).

The $K^{0}_{\scriptscriptstyle S}$ candidates are reconstructed from pairs of oppositely charged tracks, both assumed to be pions, and are further subject to a selection nisKs based on a neural network neurobayes . The network uses the following input variables: the $K^{0}_{\scriptscriptstyle S}$ momentum in the lab frame; the distance along the $z$ axis between the two track helices at their closest approach; the $K^{0}_{\scriptscriptstyle S}$ flight length in the $r$–$\phi$ plane; the angle between the $K^{0}_{\scriptscriptstyle S}$ momentum and the vector joining the IP to the $K^{0}_{\scriptscriptstyle S}$ decay vertex; the angle between the pion momentum and the lab frame direction in the $K^{0}_{\scriptscriptstyle S}$ rest frame; the distances of closest approach in the $r$–$\phi$ plane between the IP and the two pion helices; the number of hits in the CDC for each pion track; and the presence/absence of hits in the SVD for each pion track. We require that the reconstructed invariant mass be between $491$ and $505{\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$, corresponding to $\pm 3\sigma$ around the nominal $K^{0}_{\scriptscriptstyle S}$ mass PDG with $\sigma$ denoting the experimental resolution.

We identify $B$ meson candidates using two kinematic variables: the beam-energy constrained mass, \mbox{M_{\rm bc}}=\sqrt{E^{2}_{\rm beam}/c^{4}-\left|\sum_{i}\vec{p}_{i}/c\right|^{2}}, and the energy difference, \mbox{\Delta E}=\sum_{i}E_{i}-E_{\rm beam}, where $E_{\rm beam}$ is the beam energy, and $\vec{p}_{i}$ and $E_{i}$ are the momentum and energy of the $i$-th daughter of the reconstructed $B$ candidate; all calculated in the center-of-mass (CM) frame. For each $B$ candidate, we perform a fit constraining its daughters to come from a common vertex, whose position is consistent with the IP profile. Events with 5.271{\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}<\mbox{M_{\rm bc}}<5.287{\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}} and -0.10\mathrm{\,Ge\kern-1.00006ptV}<\mbox{\Delta E}<0.15\mathrm{\,Ge\kern-1.00006ptV} are retained for further analysis. The $M_{\rm bc}$ requirement corresponds approximately to a $\pm 3\sigma$ window around the nominal $B^{+}$ mass PDG . We apply a looser ($-6\sigma$, $+9\sigma$) requirement on $\Delta E$ as it is later used to extract the signal yield.

The average number of $B$ candidates per event is $1.1$ ($1.5$) for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ ($K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$). In case of multiple candidates, we choose the one with the minimum $\chi^{2}$ value for the aforementioned vertex fit. This criterion selects the correct $B$-meson candidate in 75% and 63% of Monte Carlo (MC) events having more than one candidate in $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, respectively.

The dominant background arises from the $e^{+}e^{-}\rightarrow q\overline{q}$ ($q=u,d,s,c$) continuum process. We use observables based on event topology to suppress it. The event shape in the CM frame is expected to be spherical for $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ events, whereas continuum events are jetlike. We employ a neural network to separate signal from background using the following six input variables: a Fisher discriminant formed from $16$ modified Fox-Wolfram moments KSFW ; the cosine of the angle between the $B$ momentum and the $z$ axis; the cosine of the angle between the $B$ thrust and the $z$ axis; the cosine of the angle between the thrust axis of the $B$ candidate and that of the rest of the event; the ratio of the second to the zeroth order Fox-Wolfram moments; and the vertex separation along the $z$ axis between the $B$ candidate and the remaining tracks. The first five quantities are calculated in the CM frame. The neural network training is performed with simulated signal and $q\overline{q}$ events. Signal and background samples are generated with the EvtGen program evtgen ; for signal we assume a uniform decay in phase space. A GEANT-based Geant simulation is used to model the detector response.

We require the neural network output ($C_{\mathrm{{N\!B}}}$) to be greater than $-0.2$ to substantially reduce the continuum background. For both decays, the relative signal efficiency due to this requirement is approximately $91\%$ and the achieved continuum suppression is close to $84\%$. The remainder of the $C_{\mathrm{{N\!B}}}$ distribution strongly peaks near $1.0$ for signal, making it challenging to model it analytically. However, its transformed variable

[TABLE]

where $C_{\mathrm{N\!B},{\rm min}}=-0.2$ and $C_{\mathrm{N\!B},{\rm max}}\simeq 1.0$, can be parametrized by one or more Gaussian functions. We use $C^{\prime}_{\mathrm{{N\!B}}}$ as a fit variable along with $\Delta E$.

The background due to charmed $B$ decays, mediated via the dominant $b\rightarrow c$ transition, is studied with an MC sample. The resulting $\Delta E$ and $M_{\rm bc}$ distributions are found to peak in the signal region for both $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ decays. For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$, the peaking background predominantly stems from $B^{+}\rightarrow D^{0}K^{+}$ with $D^{0}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}$ and $B^{+}\rightarrow\chi_{c0}({\rm{1P}})K^{+}$ with $\chi_{c0}({\rm{1P}})\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}$. To suppress these backgrounds, we exclude candidates for which $M_{K_{S}^{0}K_{S}^{0}}$ lies in the range $[1.85,1.88]{\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ or $[3.38,3.45]{\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, corresponding to a $\pm 3\sigma$ window around the nominal $D^{0}$ or $\chi_{c0}(\rm{1P})$ mass PDG , respectively. In case of $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, the peaking background largely arises from $B^{+}\rightarrow D^{0}\pi^{+}$ with $D^{0}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}$. To suppress it, we exclude candidates for which $M_{K_{S}^{0}K_{S}^{0}}$ lies in the aforementioned $D^{0}$ mass window.

A few background modes contribute in the $M_{\rm bc}$ signal region, but having their $\Delta E$ peak shifted from zero to the positive side for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ or to the negative side for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$. To identify these so-called “feed-across” backgrounds, mostly arising due to $K$–$\pi$ misidentification, we use a $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ MC sample in which one of the $B$ mesons decays via $b\rightarrow u,d,s$ transitions, along with the charmed $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ sample. For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, the feed-across background includes contributions from $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ as well as $B^{+}\rightarrow D^{0}K^{+}$ and $B^{+}\rightarrow\chi_{c0}({\rm{1P}})K^{+}$ that survive the $D^{0}$ and $\chi_{c0}(\rm{1P})$ vetoes. For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$, it comes entirely from $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$. All other events coming from neither the signal, continuum, nor the feed-across components form the so-called “combinatorial” $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ background.

After all selection requirements, the efficiencies for correctly reconstructed signal events are 24% for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and 26% for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$. The fractions of misreconstructed signal events for which one of the daughter particles comes from the other $B$-meson decay are 0.5% for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and 1.1% for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$. We consider these events as part of the signal.

The signal yield and ${\mathcal{A}_{C\!P}}$ are obtained with an unbinned extended maximum likelihood fit to the two-dimensional distribution of $\Delta E$ and $C^{\prime}_{\mathrm{{N\!B}}}$. The extended likelihood function is

[TABLE]

where

[TABLE]

Here, $N$ is the total number of events, $i$ is the event index, and $n_{j}$ is the yield of the event category $j$ ($j\equiv$ signal, $q\overline{q}$, combinatorial, and feed-across). ${\cal P}_{j}$ and ${\mathcal{A}_{C\!P,j}}$ are the probability density function (PDF) and direct $C\!P$ asymmetry corresponding to the category $j$, and $q^{i}$ is the electric charge of the $B$ candidate in event $i$. As the correlation between $\Delta E$ and $C^{\prime}_{\mathrm{{N\!B}}}$ is small (the linear correlation coefficient ranges from 0.5% to 7.0%), the product of two individual PDFs is a good approximation for the total PDF. We apply a tight requirement on $M_{\rm bc}$ instead of including it as a fit variable since it exhibits a large correlation with $\Delta E$ for the signal and feed-across background. We choose $\Delta E$ over $M_{\rm bc}$ in the fit because the former is a better variable to distinguish signal from feed-across background. To account for crossfeed between the two channels, they are fitted simultaneously, with the $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ branching fraction in the correctly reconstructed sample determining the normalization of the crossfeed in the $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ fit region, and vice versa.

Table 1 lists the PDFs used to model the $\Delta E$ and $C^{\prime}_{\mathrm{{N\!B}}}$ distributions for various event categories for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$. For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, we use the same PDF shapes except for the feed-across background component, where we add an asymmetric Gaussian function to the PDFs in Table 1 to accurately describe $\Delta E$ and $C^{\prime}_{\mathrm{{N\!B}}}$ distributions. The free parameters in the fit are the continuum background yields and the branching fractions of $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, and the signal ${\mathcal{A}_{C\!P}}$ for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$. In addition, the following PDF shape parameters of the continuum background are floated in the fit for both $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$: the slope of the first-order polynomial used for $\Delta E$ and the mean and width of the dominant Gaussian component used to model $C^{\prime}_{\mathrm{{N\!B}}}$. The combinatorial $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ yields are fixed to the MC values due to their correlation with the continuum yields. This is because $C^{\prime}_{\mathrm{{N\!B}}}$ is the only variable that offers some discrimination between the two background categories. To improve the overall fit stability, $\mathcal{A}_{C\!P}$ for all components but for the $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ signal are fixed to zero. The other PDF shape parameters for signal and background components are fixed to the corresponding MC expectations for both decays. We correct the signal $\Delta E$ and $C^{\prime}_{\mathrm{{N\!B}}}$ PDF shapes for possible data-MC differences, according to the values obtained with a control sample of $B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$ with $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{+}\pi^{-}$. The same correction factors are also applied for the feed-across background component of $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$.

We determine the branching fraction as

[TABLE]

where $n_{\mathrm{sig}}$, $\epsilon$, and $N_{B\kern 1.25995pt\overline{\kern-1.25995ptB}{}}$ are the total signal yield, average detection efficiency, and number of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs, respectively. Figure 2 shows signal enhanced $\Delta E$ and $C^{\prime}_{\mathrm{{N\!B}}}$ projections of the separate fit to $B^{+}$ and $B^{-}$ samples for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and of the charge-combined fit for $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$. For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, we fit a total of 5103 candidate events to obtain a branching fraction of

[TABLE]

where the first uncertainty is statistical and the second is systematic (described below). Its signal significance is estimated as $\sqrt{-2\ln({\cal L}_{0}/{\cal L}_{\rm max})}$, where ${\cal L}_{0}$ and ${\cal L}_{\rm max}$ are the likelihood values for the fit with the branching fraction fixed to zero and for the best-fit case, respectively. Including systematic uncertainties by convolving the likelihood with a Gaussian function of width equal to the systematic uncertainty, we determine the significance to be $2.5$ standard deviations. In view of the significance being less than 3 standard deviations, we set an upper limit on the branching fraction of $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$. We integrate the convolved likelihood over the branching fraction to obtain the upper limit of $8.7\times 10^{-7}$ at 90% confidence level. This limit is similar to that of BaBar BaBar:paper2 .

For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$, we perform the fit for 2709 candidate events in seven unequal bins of $M_{K_{S}^{0}K_{S}^{0}}$ to decipher contributions from possible quasi-two-body resonances. The efficiency, differential branching fraction, and ${\mathcal{A}_{C\!P}}$ thus obtained are listed in Table 2. Figure 3 shows the differential branching fraction and ${\mathcal{A}_{C\!P}}$ plotted as a function of $M_{K_{S}^{0}K_{S}^{0}}$. We observe an excess of events around $1.5{\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ beyond the expectation of a phase space MC sample. No significant evidence for $C\!P$ asymmetry is found in any of the bins. Upon inspection, no peaking structure beyond kinematic reflection is seen in the $M_{K_{S}^{0}K^{+}}$ distribution. We calculate the branching fraction by integrating the differential branching fraction over the entire $M_{K_{S}^{0}K_{S}^{0}}$ range:

[TABLE]

where the first uncertainty is statistical and the second is systematic. The ${\mathcal{A}_{C\!P}}$ over the full $M_{K_{S}^{0}K_{S}^{0}}$ range is

[TABLE]

This is obtained by weighting the ${\mathcal{A}_{C\!P}}$ value in each bin with the obtained branching fraction in that bin. As the statistical uncertainties are bin-independent, their total contribution is a quadratic sum. For the systematic uncertainties, the contributions from the bin-correlated sources are linearly added, and those from the bin-uncorrelated sources are added in quadrature. The results agree with BaBar BaBar:paper1 , which reported an ${\mathcal{A}_{C\!P}}$ consistent with zero as well as the presence of quasi-two-body resonances $f_{0}(980)$, $f_{0}(1500)$, and $f^{\prime}_{2}(1525)$ in the low $M_{K_{S}^{0}K_{S}^{0}}$ region.

Major sources of systematic uncertainty in the branching fractions are similar for both $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ decays. These are listed along with their contributions in Tables 3 and 4. We use partially reconstructed $D^{*+}\rightarrow D^{0}\pi^{+}$ with $D^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays to assign the systematic uncertainty due to charged-track reconstruction ($0.35\%$ per track). The $D^{*+}\rightarrow D^{0}\pi^{+}$ with $D^{0}\rightarrow K^{-}\pi^{+}$ sample is used to determine the systematic uncertainty due to particle identification. The uncertainty due to the number of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs is 1.37%. The uncertainties due to continuum suppression and $M_{\mathrm{bc}}$ requirements are estimated with the control sample of $B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$ with $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{-}\pi^{+}$. The uncertainty arising due to $K^{0}_{\scriptscriptstyle S}$ reconstruction is estimated from $D^{0}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}$ decays Dash:paper . A potential fit bias is checked by performing an ensemble test comprising $1000$ pseudoexperiments in which signal events are drawn from the corresponding MC sample and background events are generated according to their PDF shapes. The uncertainties due to signal PDF shape are estimated by varying the correction factors by $\pm 1\sigma$ of their statistical uncertainty. Similarly, the uncertainties due to background PDF shape are calculated by varying all fixed parameters by $\pm 1\sigma$. We evaluate the uncertainty due to fixed background yields by varying them up and down by $20\%$ of their MC values. The uncertainty due to fixed background ${\mathcal{A}_{C\!P}}$ is estimated by varying the ${\mathcal{A}_{C\!P}}$ values up and down by one unit of their statistical uncertainties. As for a possible systematics due to efficiency variation across the Dalitz plot in the $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ channel, we find its impact to be negligible.

Systematic uncertainties in ${\mathcal{A}_{C\!P}}$ are listed in Table  4. The systematic uncertainties due to the PDF modeling, fixed background yields and ${\mathcal{A}_{C\!P}}$ are estimated with the same procedure as for the branching fraction. Uncertainties due to the intrinsic detector bias on charged particle detection are evaluated with the samples of $D^{+}\rightarrow\phi\pi^{+}$ and $D_{s}^{+}\rightarrow\phi\pi^{+}$ in conjunction with $D^{0}\rightarrow K^{-}\pi^{+}$ Dphipi:paper . The total systematic uncertainty is calculated by summing all individual contributions in quadrature.

In summary, we have reported measurements of the charmless three-body decays $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ and $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$ using the full $\mathchar 28935\relax{(4S)}$ data sample collected with the Belle detector. We perform a two-dimensional simultaneous fit to extract the signal yields of both decays. For $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}$, a 90% confidence-level upper limit is set on the branching fraction at 8.7$\times$ $10^{-7}$. We measure the branching fraction and ${\mathcal{A}_{C\!P}}$ of $B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+}$ to be ${\cal B}(B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+})=(10.42\pm 0.43\pm 0.22)\times 10^{-6}$ and ${\mathcal{A}_{C\!P}}(B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}K^{+})=(+1.6\pm 3.9\pm 0.9)\%$. These results supersede Belle’s earlier measurements Belle:paper1 and are consistent with those of BaBar BaBar:paper1 ; BaBar:paper2 .

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 Pacific Northwest National Laboratory (PNNL) Environmental Molecular Sciences Laboratory (EMSL) computing group for strong computing support; and the National Institute of Informatics, and Science Information NETwork 5 (SINET5) for valuable network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council including grants DP180102629, DP170102389, DP170102204, DP150103061, FT130100303; Austrian Science Fund under Grant No. P 26794-N20; the National Natural Science Foundation of China under Contracts No. 11435013, No. 11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics (CCEPP); the Shanghai Pujiang Program under Grant No. 18PJ1401000; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. LTT17020; the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Universe, and the VolkswagenStiftung; the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation (NRF) of Korea Grants No. 2015H1A2A1033649, No. 2016R1D1A1B01010135, No. 2016K1A3A7A09005 603, No. 2016R1D1A1B02012900, No. 2018R1A2B3003 643, No. 2018R1A6A1A06024970, No. 2018R1D1 A1B07047294; Radiation Science Research Institute, Foreign Large-size Research Facility Application Supporting project, the Global Science Experimental Data Hub Center of the Korea Institute of Science and Technology Information and KREONET/GLORIAD; the Polish Ministry of Science and Higher Education and the National Science Center; the Grant of the Russian Federation Government, Agreement No. 14.W03.31.0026; the Slovenian Research Agency; Ikerbasque, Basque Foundation for Science, Spain; the Swiss National Science Foundation; the Ministry of Education and the Ministry of Science and Technology of Taiwan; and the United States Department of Energy and the National Science Foundation.