Search for $X(3872)$ and $X(3915)$ decay into $\chi_{c1} \pi^0$ in $B$ decays at Belle

TL;DR

This paper searches for specific decay modes of the $X(3872)$ and $X(3915)$ particles in B meson decays, setting upper limits on their branching fractions and ratios, based on a large dataset from the Belle experiment.

Contribution

It provides the first upper limits on the decay of $X(3872)$ and $X(3915)$ into $ ext{chi}_{c1} ext{pi}^0$ in B decays, improving understanding of these exotic states.

Findings

Upper limit on $B^+ o X(3872) K^+$ with $X(3872) o ext{chi}_{c1} ext{pi}^0$

Upper limit on $B^+ o X(3915) K^+$ with $X(3915) o ext{chi}_{c1} ext{pi}^0$

Ratio $ ext{Br}(X(3872) o ext{chi}_{c1} ext{pi}^0)/ ext{Br}(X(3872) o J/\psi ext{pi}^+ ext{pi}^-)$ constrained to be less than 0.97

Abstract

We report a search for $X(3872)$ and $X(3915)$ in $B^+ \to \chi_{c1} \pi^0 K^+$ decays. We set an upper limit of $\mathcal{B}(B^+ \to X(3872) K^+) \times \mathcal{B}(X(3872) \to \chi_{c1} \pi^0)$ $ < 8.1 \times 10^{-6}$ and $\mathcal{B}(B^+ \to X(3915) K^+) \times \mathcal{B}(X(3915) \to \chi_{c1} \pi^0)$ $ < 3.8 \times 10^{-5}$ at 90\% confidence level. We also measure $\mathcal{B}(X(3872) \to \chi_{c1} \pi^0)/\mathcal{B}(X(3872) \to J/\psi \pi^+ \pi^-) < 0.97$ at 90\% confidence level. The results reported here are obtained from $772 \times 10^{6}$ $B\overline{B}$ events collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider.

Tables

Table 1. Table 1: Summary of the systematics uncertainties for the ℬ ​ ( B + → X ​ K + ) × ℬ ​ ( X → χ c ​ 1 ​ π 0 ) ℬ → superscript 𝐵 𝑋 superscript 𝐾 ℬ → 𝑋 subscript 𝜒 𝑐 1 superscript 𝜋 0 \mathcal{B}(B^{+}\to XK^{+})\times\mathcal{B}(X\to\chi_{c1}\pi^{0}) and R χ c ​ 1 / ψ X subscript superscript 𝑅 𝑋 subscript 𝜒 𝑐 1 𝜓 R^{X}_{\chi_{c1}/\psi} .

Source	$ℬ$ (%)		$R_{{\chi }_{c1}/\psi }^X$ (%)
	$X\left(3915\right)$	$X\left(3872\right)$
Lepton identification	2.3	2.2	-
Kaon identification	1.0	1.0	-
Efficiency	0.5	0.5	2.2
$B\overline{B}$ pairs	1.4	1.4	-
$B$ production	1.2	1.2	-
Tracking	1.1	1.1	0.7
$\gamma$ identification	2.0	2.0	2.0
${\pi }^0$ veto	1.2	1.2	1.2
${\pi }^0$ reconstruction	2.2	2.2	2.2
Signal extraction	${}_{-19.5}{}^{}{}_{}{}^{+16.1}$	${}_{-44.4}{}^{}{}_{}{}^{+37.0}$	${}_{-44.5}{}^{}{}_{}{}^{+37.1}$
Secondary $ℬ$	3.0	3.0	2.9
Total	${}_{-20.2}{}^{}{}_{}{}^{+17.0}$	${}_{-44.7}{}^{}{}_{}{}^{+37.4}$	${}_{-44.8}{}^{}{}_{}{}^{+37.4}$

Equations

$${\chi }^2={\chi }_V^2+{\chi }_{{\pi }^0}^2+(\frac{M_{{\chi }_{cJ}}-m_{{\chi }_{cJ}}}{{\sigma }_{{\chi }_{cJ}}}{)}^2+(\frac{M_{\mathrm{bc}}-m_B}{{\sigma }_{M_{\mathrm{bc}}}}{)}^2,$$

$$\mathcal{B}(B^{+}\to X(3872)K^{+})\times \mathcal{B}(X(3872)\to {\chi }_{c1}{\pi }^0)<8.1\times 10^{-6}$$

$$\mathcal{B}(B^{+}\to X(3915)K^{+})\times \mathcal{B}(X(3915)\to {\chi }_{c1}{\pi }^0)<3.8\times 10^{-5}.$$

Full text

The $X(3872)$ state was observed for the first time by the Belle collaboration in 2003 via its decay to $J/\psi\pi^{+}\pi^{-}$ in the $B^{+}\to J/\psi\pi^{+}\pi^{-}K^{+}$ decays Choi:2003ue . Its mass ($3871.69\pm 0.17$) MeV/$c^{2}$, narrow width ($\Gamma<1.2$ MeV) pdg:2018 , and other properties suggest it to be a non-conventional $c\bar{c}$ state. The $X(3872)$ has also been seen in other decay modes: $D^{0}\bar{D}^{*0}$, $J/\psi\gamma$, $\psi(2S)\gamma$, and $J/\psi\pi^{+}\pi^{-}\pi^{0}$  Belle:DDstr ; BaBar:Radiative ; Belle:Radiative ; LHCb:Radiative ; BaBar:Jpsiomega . Very recently, a new decay mode, $\chi_{c1}\pi^{0}$, was reported by BESIII BESIII_X in $e^{+}e^{-}\to\chi_{c1}\pi^{0}\gamma$. According to their measurement, $R^{X}_{\chi_{c1}/\psi}\equiv\mathcal{B}(X(3872)\to\chi_{c1}\pi^{0})/\mathcal{B}(X(3872)\to J/\psi\pi^{+}\pi^{-})$ $=$ $0.88^{+0.33}_{-0.27}\pm 0.10$, where the first uncertainty is statistical and the second is systematic. In comparison with conventional charmonium, this ratio seems to be large; e.g., $\mathcal{B}(\psi(2S)\to J/\psi\pi^{0})/\mathcal{B}(\psi(2S)\to J/\psi\pi^{+}\pi^{-})=3.66\times 10^{-3}$.

If the $X(3872)$ structure is dominated by a charmonium $\chi_{c1}(2P)$ component, we expect the branching fraction for the pionic transition, $X(3872)\to\chi_{c1}\pi^{0}$, to be very small due to isospin breaking by the light quark masses Volshin_0709.4474 , significantly suppressed compared to that for $X(3872)\to\chi_{c1}\pi^{+}\pi^{-}$ ($R\approx$ 4.0%). The BESIII result disfavors the $\chi_{c1}(2P)$ interpretation of the $X(3872)$ and suggests instead a tetraquark or molecular state with a significant isovector part in its wave function, which results in an enhanced single-pion transition Volshin_0709.4474 .

In the search for $X(3872)\to\chi_{c1}\pi^{+}\pi^{-}$ Belle:inclusivechi , the Belle Collaboration determined the branching fraction $\mathcal{B}(B^{+}\to X(3872)K^{+})\times\mathcal{B}(X(3872)\to\chi_{c1}\pi^{+}\pi^{-})$ to be less than $1.5\times 10^{-6}$ at 90% confidence level (C.L.). In addition, the Belle Collaboration observed $B^{+}\to\chi_{c1}\pi^{0}K^{+}$ and published the background-subtracted ${}_{s}\mathcal{P}lot$ pivk distribution for $M_{\chi_{c1}\pi^{0}}$, which showed no structure at the $X(3872)$ mass. We use a similar technique to provide a limit on $R^{X}_{\chi_{c1}/\psi}$.

The $X(3915)$ was first observed, via its decay to $J/\psi\omega$, by the Belle Collaboration in $B\to J/\psi\omega K$ decay Belle:Choi:2005 . The quantum numbers of $X(3915)$ were identified to be $J^{PC}$ = $0^{++}$ Babar:X3915 , suggesting it may be $\chi_{c0}(2P)$. If $X(3915)$ is $\chi_{c0}(2P)$, its width should be larger Guo . However, the measured width ($20\pm 5$ MeV$/c^{2}$) pdg:2018 is significantly narrower than theoretical expectations ($>$ 100 MeV${/c^{2}}$). The $J/\psi\omega$ is also expected to be suppressed by the Okubo-Zweig-Iizuka (OZI) rule in the $\chi_{c0}(2P)$ scenario Olsen . A $J^{PC}$ = $2^{++}$ assignment is also consistent with our observation Zhou_PRL . If $X(3915)$ is a non-conventional $c\bar{c}$ state, then one may expect the single pion transition to be enhanced in $X(3915)$ decays as compared to charmonium, where it is suppressed due to isotopic symmetry breaking.

In the study reported here, we reproduce the previous result for $B^{+}\to\chi_{c1}\pi^{0}K^{+}$ Belle:inclusivechi ; charge_conjugate , search for the intermediate states $X$ ($X$ denotes $X(3872)$ and $X(3915)$), and measure the product branching fraction $\mathcal{B}(B^{+}\to XK^{+})\times\mathcal{B}(X\to\chi_{c1}\pi^{0})$.

We use a sample of $772\times 10^{6}$ $B\bar{B}$ events collected with the Belle detector abashian at the KEKB asymmetric-energy $e^{+}e^{-}$ collider, operating at the $\Upsilon(4S)$ resonance kurokawa . The Belle detector is a large-solid-angle spectrometer, which includes a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) comprised of 8736 CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux return yoke located outside the coil is instrumented to detect $K^{0}_{L}$ mesons and identify muons. The detector is described in detail elsewhere abashian . Two inner detector configurations were used. A first sample of $152\times 10^{6}$ $B\bar{B}$ events was collected with a 2.0-cm-radius beam pipe and a 3-layer SVD, and the remaining $620\times 10^{6}$ $B\bar{B}$ pairs were collected with a 1.5-cm-radius beam pipe, a 4-layer SVD and a modified CDC SVD2_NIMA_560_1_2006 .

We use EVTGEN EvtGen with QED final-state radiation by PHOTOS PHOTOS for the generation of Monte Carlo (MC) simulation events. GEANT3-based GEANT MC simulation is used to model the response of the detector and determine the efficiency of the signal reconstruction. Signal MC is used to estimate the efficiency and selection criteria for reconstructing $B^{+}\to X(\to\chi_{c1}\pi^{0})K^{+}$ decay.

We reconstruct the $B^{+}\to\chi_{c1}\pi^{0}K^{+}$ decay mode with the same selection criteria as those used in the previous analysis Belle:inclusivechi . To suppress continuum background, we require the ratio of the second to the zeroth Fox-Wolfram moment FoxWolf to be less than 0.5. Charged tracks are required to originate from the vicinity of the interaction point (IP): the distance of closest approach to the IP is required to be within 3.5 cm along the beam direction and within 1.0 cm in the plane transverse to the beam direction. An ECL cluster is treated as a photon candidate if it is isolated from the extrapolated charged tracks, and its energy in the lab frame is greater than 100 MeV. We reject a photon candidate if the ratio of energy deposited in the central 3$\times$3 square of cells to that deposited in the enclosing 5$\times$5 square of cells in its ECL cluster is less than 0.85. This helps to reduce photon candidates originating from neutral hadrons.

The $J/\psi$ meson is reconstructed via its decay to $\ell^{+}\ell^{-}$ ($\ell$ = $e$ or $\mu$) and selected by the invariant mass of the $\ell^{+}\ell^{-}$ pair ($M_{\ell\ell}$). For the dimuon mode, $M_{\ell\ell}$ is the invariant mass $M_{\mu^{+}\mu^{-}}$; for the dielectron mode, the four-momenta of all photons within 50 mrad cone of the original $e^{+}$ or $e^{-}$ direction are absorbed into the $M_{\ell\ell}\equiv M_{e^{+}e^{-}(\gamma)}$ to reduce the radiative tail. The reconstructed invariant mass of the $J/\psi$ candidates is required to satisfy 2.95 GeV$/c^{2}<M_{e^{+}e^{-}(\gamma)}<3.13$ GeV$/c^{2}$ or 3.03 GeV$/c^{2}<M_{\mu^{+}\mu^{-}}<3.13$ GeV$/c^{2}$. For the selected $J/\psi$ candidates, a vertex-constrained fit is applied to the charged tracks and then a mass-constrained fit is performed to improve the momentum resolution. The $\chi_{c1}$ candidates are reconstructed by combining a $J/\psi$ candidate with a photon. To reduce background from $\pi^{0}\to\gamma\gamma$, a likelihood function is employed to distinguish isolated photons from $\pi^{0}$ daughters using the invariant mass of the photon pair, photon energy in the laboratory frame and the polar angle with respect to the beam direction in the laboratory frame koppenburg . We combine the candidate photon with any other photon and then reject both photons of a pair whose $\pi^{0}$ likelihood is larger than 0.8. For further analysis, we keep the $\chi_{c1}$ candidates with a reconstructed invariant mass satisfying 3.467 GeV$/c^{2}<M_{J/\psi\gamma}<$ 3.535 GeV$/c^{2}$, which corresponds to $[-4.5\sigma,+2.8\sigma]$ about the nominal mass of the $\chi_{c1}$ pdg:2018 , where $\sigma$ is the $\chi_{c1}$ mass resolution from the fit to the MC simulated $J/\psi\gamma$ mass distribution. To improve the momentum resolution a mass-constrained fit is applied to the selected $\chi_{c1}$ candidates.

Particle identification is performed using specific ionization information from the CDC, time measurements from the TOF, and the light yield measured in the ACC. Charged kaons and pions are identified using the $K$ likelihood ratio, $R_{K}=\mathcal{L}_{K}/(\mathcal{L}_{K}+\mathcal{L}_{\pi})$, where $\mathcal{L}_{K}$ and $\mathcal{L}_{\pi}$ are likelihood values for the kaon and pion hypotheses pid . Kaon tracks are correctly identified with an efficiency of $89.4\%$, whereas the probability of misidentifying a pion as a kaon is $10.1\%$ for $B^{+}\to X(3872)(\to\chi_{c1}\pi^{0})K^{+}$.

Photon pairs are kept as $\pi^{0}$ candidates whose invariant mass lies in the range 120 MeV$/c^{2}<M_{\gamma\gamma}<$ 150 MeV$/c^{2}$ ($\pm 3\sigma$ about the nominal mass of $\pi^{0}$). To reduce combinatorial background, the $\pi^{0}\to\gamma\gamma$ candidates are also required to have an energy balance parameter $|E_{1}-E_{2}|/(E_{1}+E_{2})$ smaller than 0.8, where $E_{1}$ ($E_{2}$) is the energy of the first (second) daughter photon in the laboratory frame. For each selected $\pi^{0}$ candidate, a mass-constrained fit is performed to improve its momentum resolution.

To identify the $B$ meson, two kinematic variables are used: the beam-energy-constrained mass $M_{\rm bc}$ and the energy difference $\Delta E$. The former is defined as $\sqrt{E_{\rm beam}^{2}/c^{2}-(\sum_{i}\vec{p}_{i})^{2}}/c$ and the latter as $\sum_{i}E_{i}-E_{\rm beam}$, 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 particle in the center-of-mass (CM) frame; the summation is over all final-state particles used to reconstruct the $B$ candidate. We reject candidates having $M_{\rm bc}$ less than 5.27 GeV$/c^{2}$ or $|\Delta E|~{}>$ 120 MeV. After the reconstruction, an average of 1.24 $B$ candidates per event is found. When there are multiple $B$ candidates in one event, we retain only the candidate with the the lowest $\chi^{2}$ value defined as:

[TABLE]

where $\chi^{2}_{V}$ is the reduced $\chi^{2}$ returned by the vertex fit of all charged tracks, $\chi^{2}_{\pi^{0}}$ is the reduced $\chi^{2}$ for the $\pi^{0}$ mass-constrained fit, $M_{\chi_{cJ}}$ is the reconstructed mass of $\chi_{cJ}$, and $m_{\chi_{cJ}}$ and $m_{B}$ are the nominal masses of the $\chi_{cJ}$ and $B$ mesons, respectively. This method has 95% efficiency for selecting the true candidate.

We extract the signal yield from an unbinned extended maximum likelihood (UML) fit to the $\Delta E$ distribution. The signal probability density function (PDF) is modeled by a sum of a Gaussian function and a logarithmic Gaussian function lg . The mean and width of the core Gaussian with larger fraction are floated and the remaining parameters of tail distribution are fixed from studies of MC simulation.

To study the background from events with a $J/\psi$, we use MC-simulated $B\to J/\psi X$ sample corresponding to 100 times the integrated luminosity of the data sample. Possible peaking backgrounds from the feed-across of $B^{+}\to\chi_{c2}\pi^{0}K^{+}$ are found in the $\Delta E$ distribution around $-$50 MeV, which are due to the mass-constrained fit to $\chi_{c1}\to J/\psi\gamma$ candidates; we estimate that only five such events are expected in real data. Thus, we fix this peaking background contribution in the fit. The PDF for the peaking background is modeled by an asymmetric Gaussian distribution for which the parameters are fixed according to MC simulation after MC/data correction (using the signal events whose mean and sigma of the core Gaussian are floated).

The rest of the background is combinatorial and modeled by using a first-order Chebyshev polynomial. The fit to the $\Delta E$ distribution for $B^{+}\to\chi_{cJ}\pi^{0}K^{+}$ is shown in Fig. 1(a). We obtain $806\pm 69$ signal events for the $B^{+}\to\chi_{c1}\pi^{0}K^{+}$ decay mode, which is consistent with our previous study Belle:inclusivechi . In order to improve the resolution on the invariant mass of the combined $\chi_{c1}$ and $\pi^{0}$ candidates ($M_{\chi_{c1}\pi^{0}}$), we scale the energy and momentum of the $\pi^{0}$, such that $\Delta E$ (defined below) is equal to zero while the $M_{\pi^{0}}$ is kept constant to its already mass-constrained value. This corrects for the incomplete energy measurement of the $\pi^{0}$ detection. The corrected four-momentum of the $\pi^{0}$ is then used to improve the invariant mass $M_{\chi_{c1}\pi^{0}}$ and $M_{K^{+}\pi^{0}}$.

To search for the $X$, we examined the background-subtracted $M_{\chi_{c1}\pi^{0}}$ distribution produced with the ${}_{S}\mathcal{P}$lot technique splot for the range (3.75 GeV$/c^{2}~{}<~{}M_{\chi_{c1}\pi^{0}}~{}<$ 4.05 GeV$/c^{2}$) as shown in Fig. 1(b). Figure 1(c) shows the $M_{K\pi^{0}}$ ${}_{S}\mathcal{P}$lot distribution in the range of interest (3.75 GeV$/c^{2}~{}<~{}M_{\chi_{c1}\pi^{0}}~{}<$ 4.05 GeV$/c^{2}$), where most events come from the $K^{*}$ decays.

In order to extract the $X$ signal yield, we use the $M_{\chi_{c1}\pi^{0}}$ distribution within the signal-enhanced window of $-30$ MeV $<\Delta E<$ 20 MeV for $B^{+}\to(\chi_{c1}\pi^{0})K^{+}$ candidates. We veto events from $B^{+}\to\chi_{c1}K^{*+}$ decay by rejecting events with 791.8 MeV/$c^{2}~{}<~{}M(K^{+}\pi^{0})~{}<$ 991.8 MeV$/c^{2}$. This requirement reduces the background by 32% with a signal efficiency of 84%. We extract the signal by performing a 1D UML fit to the $M_{\chi_{c1}\pi^{0}}$ distribution. The signal PDFs for both $X(3872)$ and $X(3915)$ are modeled by the sum of two Gaussians. All the PDF parameters are fixed from the MC simulation after a MC/data correction estimated from the $B^{+}\to\psi(2S)(\to\chi_{c1}\gamma)K^{+}$ sample is applied Belle:chic1gamma (the mean and sigma of the core Gaussian were fixed after scaling, while the tail parameters were fixed from signal MC).

The efficiency ($\epsilon$) is estimated to be $5.35\%$ and $5.37\%$ for $B^{+}\to X(3872)(\to\chi_{c1}\pi^{0})K^{+}$ and $B^{+}\to X(3915)(\to\chi_{c1}\pi^{0})K^{+}$ using the MC simulations, respectively. This efficiency has been calibrated by the difference between MC simulation and data, as described later. A fit to the data shown in Fig. 2 results in a signal yield of $2.7\pm 5.5$ ($42\pm 14$) events having significance of 0.3 $\sigma$ (2.3 $\sigma$) for the $B^{+}\to X(3872)(\to\chi_{c1}\pi^{0})K^{+}$ ($B^{+}\to X(3915)(\to\chi_{c1}\pi^{0})K^{+}$) decay mode. The systematic uncertainty (explained later) has been included in the significance calculation.

With the absence of any significant signal, we estimate an upper limit (U.L.) at 90% C.L. We apply a frequentist method that uses ensembles of pseudoexperiments. For a given signal yield, sets of signal and background events are generated according to their PDFs and fits are performed. The C.L. is determined from the fraction of samples that give a yield larger than that of data. We estimate the branching fraction according to the formula $\mathcal{B}=Y^{U.L.}/(\epsilon\times\mathcal{B}_{s}\times N_{B\bar{B}})$; here $Y^{U.L.}$ is the estimated U.L. yield at $90\%$ C.L., $\epsilon$ is the reconstruction efficiency, $\mathcal{B}_{s}$ is the product of secondary branching fraction taken from Ref. pdg:2018 , and $N_{B\bar{B}}$ is the number of $B\bar{B}$ mesons in the data sample. Equal production of neutral and charged $B$ meson pairs in the $\Upsilon(4S)$ decay is assumed. For this assumption, an uncertainty of 1.2% is added to the total systematics.

We estimate the U.L. on the product of branching fractions $\mathcal{B}(B^{+}\to X(3872)K^{+})\times\mathcal{B}(X(3872)\to\chi_{c1}\pi^{0})$ directly from the above MC pseudoexperiment samples. The limit includes the systematic uncertainties from efficiency, particle identification, and signal extraction method into the yield obtained by smearing the assumed values by their uncertainties. Along with that we also smear the $N_{B\bar{B}}$ and secondary branching fraction by adding their systematic uncertainties as a fluctuation of the value used to calculate the branching fraction. Using the MC pseudoexperiment samples we estimate the U.L. (90% C.L.) on the product branching fraction as:

[TABLE]

To measure the $R^{X}_{\chi_{c1}/\psi}$, we use the previous Belle measurement of $\mathcal{B}(B^{+}\to X(3872)K^{+})\times\mathcal{B}(X(3872)\to J/\psi\pi^{+}\pi^{-})$ = $(8.63\pm 0.82({\rm stat.})\pm 0.52({\rm syst.}))\times 10^{-6}$ Belle:Choi:2011 . Some of the systematic uncertainties cancel, such as lepton identification, $\mathcal{B}(J/\psi\to\ell\ell)$, some tracking systematics, and kaon identification. The U.L. on $R^{X}_{\chi_{c1}/\psi}$ is estimated in the same manner as that on $\mathcal{B}(B^{+}\to X(3872)K^{+})\times\mathcal{B}(X(3872)\to\chi_{c1}\pi^{0})$. We remove the cancelled systematic uncertainties and smear the pseudoexperiments with the remaining ones. We further smear $\mathcal{B}(B^{+}\to X(3872)K^{+})\times\mathcal{B}(X(3872)\to J/\psi\pi^{+}\pi^{-})$ by its statistical uncertainty and uncancelled systematic uncertainties. For each toy sample, $R^{X}_{\chi_{c1}/\psi}$ is estimated for the generated $R^{X}_{\chi_{c1}/\psi}$. The C.L. value is then determined from the fraction of samples of pseudoexperiments having $R^{X}_{\chi_{c1}/\psi}$ larger than the central value of data. We estimate the U.L. to be $R^{X}_{\chi_{c1}/\psi}$ $<$ 0.97 at $90\%$ C.L.

Table 1 summarizes systematic uncertainties for the measured product branching fraction $\mathcal{B}(B^{+}\to XK^{+})\times\mathcal{B}(X\to\chi_{c1}\pi^{0})$ and the ratio $R^{X}_{\chi_{c1}/\psi}$. A correction for the small difference in the signal detection efficiency between MC and data is applied for the lepton identification requirements, which are determined from $e^{+}e^{-}\to e^{+}e^{-}\ell^{+}\ell^{-}$ and $J/\psi\to\ell^{+}\ell^{-}$ ($\ell=e$ or $\mu$) samples. Dedicated $D^{*+}\to D^{0}(K^{-}\pi^{+})\pi^{+}$ samples are used to estimate the kaon (pion) identification efficiency correction. The uncertainty on the efficiency due to limited MC statistics is 0.5%, and the uncertainty on the number of $B\bar{B}$ pairs is 1.4%. The uncertainty on the track finding efficiency is found to be 0.35% per track by comparing data and MC for $D^{*}\to D^{0}\pi$ decay, where $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $K_{S}^{0}\to\pi^{+}\pi^{-}$. The uncertainty on the photon identification is estimated to be 2.0% from a sample of radiative Bhabha events. The systematic uncertainty associated with the difference of the $\pi^{0}$ veto between data and MC is estimated to be 1.2% from a study of the $B^{+}\to\chi_{c1}(\to J/\psi\gamma)K^{+}$ sample. For $\pi^{0}$ reconstruction, the efficiency correction and systematic uncertainty are estimated from a sample of $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decays. The errors on the PDF shapes are obtained by varying all fixed parameters by $\pm 1\sigma$ and taking the change in the yield as the systematic uncertainty. The largest uncertainty in the PDF parameterization for $X(3872)$ ($X(3915)$) is $30\%$ (${}^{+15}_{-17}\%$) from fixing the mass (width) of the $X(3872)$ ($X(3915)$) to the value reported in Ref. pdg:2018 . In order to estimate the uncertainty coming from the background shape, we used a third-order polynomial and took the difference as the uncertainty. Further, we also used large fitting range and added the difference in quadrature to the uncertainty coming from signal extraction procedure. The uncertainties due to the secondary branching fractions are also taken into account. Assuming all the sources are independent we add them in quadrature to obtain the total systematic uncertainties.

To summarize, in our searches for $X(3872)$ and $X(3915)$ decaying to $\chi_{c1}\pi^{0}$, we did not find a significant signal. We obtained $2.7\pm 5.5$ ($42\pm 14$) events, with a signal significance of 0.3 $\sigma$ (2.3 $\sigma$) for the $B^{+}\to X(3872)(\to\chi_{c1}\pi^{0})K^{+}$ ($B^{+}\to X(3915)(\to\chi_{c1}\pi^{0})K^{+}$) decay mode. We determine an U.L. on the product branching fractions $\mathcal{B}(B^{+}\to X(3872)K^{+})\times\mathcal{B}(X(3872)\to\chi_{c1}\pi^{0})$ $<$ $8.1\times 10^{-6}$ and $\mathcal{B}(B^{+}\to X(3915)K^{+})\times\mathcal{B}(X(3915)\to\chi_{c1}\pi^{0})$ $<$ $3.8\times 10^{-5}$ at 90% C.L. The null result for our search is compatible with the interpretation of $X(3872)$ as an admixture state of a $D^{0}\bar{D}^{*0}$ molecule and a $\chi_{c1}(2P)$ charmonium state Volshin_0709.4474 . One can further estimate $R^{X}_{\chi_{c1}/\psi}$ $<$ 0.97 at 90% C.L. Our U.L. does not contradict the BESIII result BESIII_X . This information can be used to constrain the tetraquark/molecular component of the $X$ states.

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 (FWF); 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.