Measurements of isospin asymmetry and difference of direct $CP$ asymmetries in inclusive $B \to X_s \gamma$ decays

We report measurements of isospin asymmetry $\Delta_{0-}$ and difference of direct $CP$ asymmetries $\Delta A_{CP}$ between charged and neutral $B \to X_s \gamma$ decays. This analysis is based on the data sample containing $772 \times 10^6 B\bar{B}$ pairs that was collected with the Belle detector at the KEKB energy-asymmetric $e^+ e^-$ collider. Using a sum-of-exclusive technique with invariant $X_s$ mass up to 2.8~GeV/$c^2$, we obtain $\Delta_{0-} = \bigl[-0.48 \pm 1.49 {\rm (stat.)} \pm 0.97 {\rm (syst.)} \pm 1.15 {(f_{+-}/f_{00})}\bigr]$\% and $\Delta A_{CP} = \bigl[+3.69 \pm 2.65 {\rm (stat.)} \pm 0.76{\rm (syst.)}\bigr]$\%, where the last uncertainty for $\Delta_{0-}$ is due to the uncertainty on the production ratio of $B^+B^-$ to $B^0\bar{B}^0$ in $\Upsilon(4S)$ decays. The measured value of $\Delta_{0-}$ is consistent with zero, allowing us to constrain the resolved photon contribution in the $B \to X_s \gamma$, and improve the branching fraction prediction. The result for $\Delta A_{CP}$ is consistent with the prediction of the SM. We also measure the direct $CP$ asymmetries for charged and neutral $B \to X_s \gamma$ decays. All the measurements are the most precise to date.


I. INTRODUCTION
The radiative b → sγ decay proceeds predominantly via one-loop electromagnetic penguin diagrams at the lowest order in the standard model (SM). This decay is sensitive to new physics (NP), which can alter the branching fraction, or direct CP asymmetry defined as where Γ denotes the partial width. Precision measurements of B → X s γ branching fraction BðB → X s γÞ [1][2][3][4][5][6] are in good agreement with the SM prediction [7] and set a strong constraint on NP models [8]. The theoretical uncertainty in the prediction of BðB → X s γÞ is about 7% which is comparable with the experimental uncertainty of the current world average [9]. The Belle II experiment is expected to measure the branching fraction with a precision of about 3% [10]. Thus, the reduction of the theoretical uncertainty is crucial to further constrain NP models. The largest uncertainty in the theoretical prediction is due to nonperturbative effects, one of which is the resolved photon contributions [11]. Since the resolved photon contribution from a hard gluon and a light quark scattering to the B → X s γ branching fraction (B 78 RP ) depends on the charge of the light quark and can be hence related to the isospin asymmetry in B → X s γ (Δ 0− ) as [11][12][13] B 78 where the uncertainty of AE0.3 in the right-hand side is associated with SUð3Þ flavor-symmetry breaking. The isospin asymmetry is defined as where N is the number of produced signal events including charge-conjugate decays, τ B − =τB0 ¼ τ B þ =τ B 0 is the lifetime ratio of B þ to B 0 mesons, f þ− and f 00 are the production ratio of B þ B − to B 0B0 in ϒð4SÞ decays, respectively. If the measured value of Δ 0− is consistent with zero, the resolved photon contribution is small and reducing in the theoretical uncertainty on BðB → X s γÞ. Recently, evidence for isospin violation in exclusive B → K Ã ð892Þγ (Δ 0þ ) has been reported [14] where the measured value, Δ 0þ ¼ ðþ6.2 AE 1.5 AE 0.5 AE 1.2Þ%, is consistent with SM predictions [15][16][17][18][19][20]. If the isospin asymmetry for the inclusive decays is consistent with this value, the resolved photon contribution to B → X s γ decays could be sizable. The direct CP asymmetry in B → X s γ is also a sensitive probe for NP [18,[21][22][23][24][25][26][27][28][29][30]. Belle [31] and BABAR [32] measured this quantity, and the current world average ðþ1.5 AE 2.0Þ% [9] is in agreement and of comparable precision, with the SM prediction, −0.6% < A SM CP < þ2.8% [33]. The dominant theoretical uncertainty is due to the limited knowledge of the resolved photon contributions. A newly proposed observable is the difference of the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP 3 . direct CP asymmetries between the charged and neutral B mesons defined as where, By taking the difference, terms with large weak phase in the SM cancel out, and only the spectator-quark-flavor dependent term representing interference between electromagnetic and chromomagnetic dipole operators survives [33]: where α s is the strong coupling constant,Λ 78 is the hadronic parameter denoting the interference between electromagnetic and chromomagnetic dipole diagrams, m b is the bottom quark mass, and C 7 and C 8 are the Wilson coefficients for electromagnetic and chromomagnetic dipole operators, respectively [34]. In the SM, C 7 and C 8 are both real, therefore ΔA CP is zero, but in several NP models ΔA CP can reach the level of 10% in magnitude [33,35,36]. BABAR measured Δ 0− and ΔA CP using data samples of 81.9 fb −1 and 429 fb −1 , respectively, as Δ 0− ¼ ð−0.6 AE 5.8 AE 0.9 AE 2.4Þ% [37] and ΔA CP ¼ ðþ5.0 AE 3.9 AE 1.5Þ% [32], where the first uncertainty is statistical, the second is systematic, and the last one for Δ 0− is due to the uncertainty on the fraction of B þ B − to B 0B0 production in ϒð4SÞ decays. The precisions are limited by statistical uncertainties. Improving these measurements is highly desirable to reduce the theoretical uncertainty of BðB → X s γÞ in the SM as well as to search for NP.
In this article, we report first measurements of Δ 0− and ΔA CP in inclusive B → X s γ at Belle assuming that the two observables have no dependence on decay modes. In addition, we present measurements of individual A CP for the charged and neutral decay and their average with A CP ¼ ðA CP ðB − → X s γÞ þ A CP ðB 0 → X s γÞÞ=2. All measurements are based on the full data sample of 711 fb −1 , containing 772 × 10 6 BB pairs, recorded at the ϒð4SÞ resonance (on-resonance data) with the Belle detector [38] at the KEKB e þ e − collider [39]. In addition, the data sample of 89 fb −1 accumulated 60 MeV below the ϒð4SÞ peak (off-resonance data), which is below the BB production threshold, is used to provide a background description. The result for A CP ðB → X s γÞ supersedes our previous measurement [31].

II. BELLE DETECTOR
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrellike arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL). All the sub-detectors are located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return placed outside of the coil is instrumented to detect K 0 L mesons and muons. The z axis is aligned with the direction opposite the e þ beam. The detector is described in detail elsewhere [38].

III. MC SIMULATION
The selection is optimized with Monte Carlo (MC) simulation samples. The MC simulation events are generated with EvtGen [40] and the detector simulation is done with GEANT3 [41]. We generate two types of signal MC simulation samples, according to the X s mass (M X s ) region: in the region M X s < 1.15 GeV=c 2 , the X s system solely consists of K Ã ð892Þ while in the region M X s > 1.15 GeV=c 2 , X s system is simulated inclusively without any specific resonances, except for K Ã 2 ð1430Þ. In the inclusive signal MC simulation sample, various resonances and final states are simulated. The photon energy spectrum in this sample is produced following the Kagan-Neubert model [42]. The model has two parameters: the b quark mass (m b ) and the Fermi-motion parameter of the b quark inside the B meson (μ 2 π ). The nominal values of these parameters are determined from a fit to the Belle inclusive photon energy spectrum [5]: m b ¼ 4.440 GeV=c 2 and μ 2 π ¼ 0.750 GeV 2 . Further, the generated light quark pair is fragmented into final-state hadrons using PYTHIA [43].
Since the B → K Ã 2 ð1430Þγ decay has a relatively large branching fraction, dedicated MC simulation samples are generated. The decay is generated with the measured branching fraction and then added to the inclusive signal MC simulation sample. To match the photon spectrum with the theoretical one, the M X s distribution for Kπ and K2π modes in the inclusive sample is rescaled. The signal reconstruction efficiency depends on the particle content in the final state; thus, the hadronization of X s is studied using data. We set the branching fraction of B → X s γ to the current world average [9] in order to optimize the background rejection.

IV. EVENT SELECTION
We reconstruct B → X s γ decays with 38 exclusive X s final states listed in Table I. As shown in Table II, we group the final states into ten categories for the purpose of specific selections and fragmentation model calibrations. The reconstructed decay modes cover 59% of the total X s rate, according to the MC simulations. Assuming the K 0 meson to decay equally into K 0 L and K 0 S , the proportion of our measured final states is 77% of the total X s rate. For neutral B decays, all flavor-specific final states are used for the measurements of both A CP and Δ 0− , and 11 flavornonspecific final states, denoted as B fns , are only used for the measurement of Δ 0− . High-energy prompt photons are selected as isolated clusters in the ECL that are not matched to any charged tracks reconstructed by the SVD and the CDC. The cluster energy in the center of mass (CM) system is required to be between 1.5 and 3.4 GeV. The polar angle of the photon direction must be within the barrel ECL, 33°< θ < 132°. We also require the cluster shape to be consistent with an electromagnetic shower, E 9 =E 25 > 0.95, where E 9 =E 25 is the ratio of energy deposits in the 3 × 3 array of CsI(Tl) crystals to that in the 5 × 5 array centered on the crystal with maximum energy. In order to reduce contaminations from asymmetric η → γγ or π 0 → γγ decays, the photon candidate is paired with all other photons in the event with energy greater than 40 MeV. We reject the pairs based on likelihoods (L π 0 and L η ), constructed from their invariant mass, and the energy and polar angle of the additional photon in the CM system [45]. The photon candidate which has L π 0 > 0.05 or L η > 0.10 is discarded.
Charged particles, except pions from K 0 S decays, are required to have a distance of closest approach to the interaction point (IP) within AE5.0 cm along the z axis and AE0.5 cm in the transverse x-y plane, and a laboratory momentum above 100 MeV=c. Charged kaons and pions are identified based on a likelihood ratio constructed from the specific ionization measurements in the CDC, time-offlight information from the TOF, and response from the ACC [46].
Neutral kaon (K 0 S ) candidates are reconstructed from pairs of oppositely charged tracks, treated as pions, and identified by a multivariate analysis [47] based on two sets of input variables [48]. The first set that separates K 0 S candidates from the combinatorial background are: (1) the K 0 S momentum in the laboratory frame, (2) the distance along the z axis between the two track helices at their closest approach, (3) the flight length in the x-y plane, (4) the angle between the K 0 S momentum and the vector joining its decay vertex to the nominal IP, (5) the angle between the π momentum and the laboratory-frame direction of the K 0 S in its rest frame, (6) the distances of closest approach in the x − y plane between the IP and the pion helices, (7) the numbers of hits for axial and stereo wires in the CDC for each pion, and (8) the presence or absence of associated hits in the SVD for each pion. The second set of variables, which identifies Λ → pπ − background that has a similar long-lived topology, are: (1) particle identification information, momentum, and polar angles of the two daughter tracks in the laboratory frame, and (2) the invariant mass calculated with the proton-and pion-mass hypotheses for the two tracks. In total, the first and second sets comprise 13 and 7 input variables, respectively. The selected K 0 S candidates are required to have an invariant mass within AE10 MeV=c 2 of the nominal value [9], corresponding to a AE3σ interval in mass resolution, where σ represents the standard deviation.  Kπ with π 0 3,4 3 K2π without π 0 5,6 4 K2π with π 0 7,8 5 K3π without π 0 9,10 6 K3π with π 0 11,12 7 K4π 13-16 8 K2π 0 17-22 9 Kη 23-32 10 3K

33-38
We reconstruct π 0 candidates from two photons each with energy greater than 50 MeV. We require a minimum momentum of 100 MeV=c in the CM frame and the invariant mass to be within AE10 MeV=c 2 of the nominal π 0 mass, corresponding to about 1.5σ in resolution. To reduce the large combinatorial background, we require the cosine of the angle between two photons in the CM frame to be greater than 0.5.
The η candidates are formed from two photons, each with energy greater than 100 MeV. The photon pairs with invariant mass satisfying 515 MeV=c 2 < M γγ < 570 MeV=c 2 , which corresponds to about 2σ in resolution, are retained. We require a CM momentum to be greater than 500 MeV=c and an absolute value of the cosine of the helicity angle, which is the angle between momentum of one of the photons and direction of laboratory system in the η rest frame, to be less than 0.8.
The 38 X s final states are reconstructed from the selected π þ , π 0 , K þ , K 0 S , and η candidates. In order to reduce the large combinatorial background from events with high multiplicity, we require M X s < 2.8 GeV=c 2 , which corresponds to photon energy threshold of about 1.9 GeV. The K4π and K2π 0 mode categories, listed in Table II, have substantial background. Therefore, the momentum of the first and second leading pions (neutral pions) in K4π (K2π 0 ) category is required to be above 400 and 250 MeV=c, respectively.
B meson candidates are reconstructed by combining an X s with a prompt photon candidate. We form two kinematic variables to select B mesons: the energy difference ΔE ≡ E CM B − E CM beam and the beam-energy constrained mass B are the beam energy, energy and momentum of the B candidate in the CM system, respectively. The B momentum vector p CM B is calculated without using the magnitude of the photon momentum according to p CM as the X s momentum (p CM X s ) and the beam energy are determined with a substantially better precision than that of the photon candidate. We define the signal region in ΔE and M bc as −0.15 GeV < ΔE < 0.08 GeV and 5.27 GeV=c 2 < M bc < 5.29 GeV=c 2 . The ΔE selection is tightened to −0.10 GeV < ΔE < 0.05 GeV for the final states with 2π 0 and ηπ 0 (mode IDs 17-22, 27, 28, 31 and 32) due to larger combinatorial backgrounds. To determine the signal yield and extract physics observables, we fit to the M bc distribution in the wider range of 5.20 GeV=c 2 < M bc < 5.29 GeV=c 2 .

V. BACKGROUND REJECTION
After reconstructing the B meson candidates, two dominant backgrounds still remain: events with D meson decays and continuum e þ e − → qq (q ¼ u, d, s, c) events.
The events with D meson decays, especially the decay chain B → D ðÃÞ ρ þ followed by ρ þ → π þ π 0 with a high energy photon from the π 0 , give rise to a peak in the signal region of M bc . In order to suppress this background, a D veto is applied for candidates with M X s > 2.0 GeV=c 2 . D meson candidates of the major 19 hadronic decay modes are reconstructed with combinations of particles used in the X s reconstruction. The event is rejected if any of the D meson candidates falls in a veto window around the D meson mass. We set the central value and the width of the veto window depending on the charge of the D candidate and whether or not the D candidate is reconstructed in a mode with a π 0 or η meson: the windows are 1835 < M D 0 < 1895 MeV=c 2 and 1840 < M D þ < 1900 MeV=c 2 for the modes without π 0 or η, and 1800 < M D 0 < 1905 MeV=c 2 and 1805 < M D þ < 1910 MeV=c 2 for the modes with π 0 or η.
The continuum background is suppressed using a multivariate analysis with a neural network [47], mostly relying on the difference in topology of continuum (jet-like) and BB (spherical) events. We use the following variables calculated in the CM frame as input parameters to the neural network: (1) the cosine of the angle between the B meson candidate momentum and the z axis, (2) the likelihood ratio of modified Fox-Wolfram moments [49,50], the first sphericity axes of the B candidate and the particles in the ROE, (7) the cosine of the angle between the second sphericity axes of the B candidate and the particles in the ROE, (8) the cosine of the angle between the third sphericity axes of the B candidate and the particles in the ROE, (9) the cosine of the angle between the first sphericity axis of the particles in the event and the z axis, and (10) the signal probability density for the ΔE value. The neural network is trained with signal and qq-background MC simulation events with 2.2 GeV=c 2 < M X s < 2.8 GeV=c 2 .
We obtain a neural network output (O NB ) between −1 and þ1, which can discriminate the continuum background from signal events. The distribution of O NB for simulated samples is shown in Fig. 1. The O NB value is required to be greater than 0.87 in order to maximize the signal significance in the range 2.2 GeV=c 2 < M X s < 2.8 GeV=c 2 , in which the continuum background is the most substantial. This selection suppresses about 98.5% of the qq background while keeping about 51% of the signal events in the MC simulation study.

VI. BEST CANDIDATES SELECTION
After the background suppression, the average number of B candidates per event is about 1.5 for the signal MC simulation sample. In those events we select the candidate with the largest O NB . This selection keeps about 89% of the correctly reconstructed signal events while reducing the number of cross-feed events by 45%, based on the MC simulation study. The ratio of the number of correctly reconstructed signal events to the number of cross-feed events plus correctly reconstructed signal events is improved from 0.67 to 0.77. Since the average number of B candidates per event for BB background is larger, the best candidate selection suppresses this background to 14%.

VII. SIGNAL YIELD EXTRACTION
To extract the signal yield and physics observables, we perform a simultaneous fit with an extended unbinned maximum likelihood method to eight M bc distributions; five for B − , B þ ,B 0 , B 0 , and B fns in the on-resonance data, and three for charged B (B − and B þ ), flavor-specific neutral B (B 0 and B 0 ), and B fns in the off-resonance data. Since the off-resonance data only contain continuum background, this is useful to constrain the continuum background shape.
To take into account the run by run difference of beam energy, the M bc value is shifted with E nom beam − E run beam , where E nom beam is the nominal beam energy set to 5.289 GeV and E run beam is the beam energy for a specific run. By this calibration, the endpoint of the M bc distribution for any run is 5.289 GeV.
The likelihood function consists of probability density functions (PDFs) for signal, cross-feed, peaking and nonpeaking background from BB events, and qq background. All signal and background PDFs are considered for the on-resonance data, while only the qq background PDF is used to fit to the off-resonance data. The signal is modeled with a Crystal Ball function [52]: where m and σ CB are the peak position and width, respectively, and the parameters α and n characterize the non-Gaussian tail. The peak position is determined with a large-statistics B → Dπ data sample. The width is obtained from large simulation samples of B → X s γ decays corrected for the difference between data and MC simulation decays, which is again obtained using the B → Dπ samples. The α and n are fixed to the values obtained from signal MC simulation samples. For the cross-feed background, we construct five histogram PDFs originating from B − , B þ ,B 0 , B 0 , and B fns with the signal MC simulation sample. The fraction of each cross-feed to the corresponding signal is fixed to the MC simulation value.
A Gaussian function is used to model the peaking background. We consider two types of such backgrounds, originating from π 0 decays and others dominated by η decays. All parameters for the π 0 peaking background are fixed using the events in the sideband defined as L π 0 > 0.5 and L η < 0.2. The parameters for other peaking background are fixed with the BB background MC simulation samples.
The nonpeaking background from BB events is modeled with an ARGUS function [53]: where E CM beam is fixed to 5.289 GeV and other shape parameters are determined from the MC simulation. The yields for charged B and neutral B are constrained from the on-resonance and scaled off-resonance events in the sideband defined as M bc < 5.27 GeV, separately.
For the qq background PDF, we use a modified ARGUS function: where a power parameter, p, is introduced to account for the steep slope at low M bc . The c mod parameter is common for on-and off-resonance data and floated in the fit. The p parameter is fixed from the qq MC simulation samples calibrated with the off-resonance data. There are in total 16 free parameters in the simultaneous fit: five signal yields for the five B categories, eight qq yields (five for on-resonance and three for off-resonance), and three c mod shape parameters for qq. Finally, the physics parameters to be extracted can be written in terms of the efficiency corrected signal yields (N i ) as: where A N CP , A C CP and A tot CP are direct CP violation parameters for neutral, charged and combined B → X s γ decays, τ B þ =τ B 0 and f þ− =f 00 are fixed to the PDG values [9,54]. The fitting procedure is validated using the full MC simulation samples and an ensemble test based on toy MC simulation samples.

VIII. CALIBRATION OF X s FRAGMENTATION MODEL
Since the signal efficiency depends on specific decay modes, the fragmentation model in the inclusive MC simulation is calibrated to that of the data to reduce the systematic uncertainties associated to modeling. The final states are divided into ten categories, defined in Table II to calibrate the fragmentation model. We adopt the same calibration method as described in Ref. [6].

IX. SYSTEMATIC UNCERTAINTIES
We evaluate the systematic uncertainties associated with particle detection efficiencies, charge asymmetries in particle detection, selections, physics parameters, X s fragmentation model, background A CP and Δ 0− , fixed parameters in the fit, fitter bias, and MC simulation statistics. We list the systematic uncertainties in Table III.
We estimate the tracking efficiency uncertainty using partially reconstructed D Ãþ → D 0 π þ , D 0 → K 0 S π þ π − events. The uncertainties due to kaon and pion identifications are evaluated with clean kaon and pion samples in D Ãþ → D 0 π þ , followed by D 0 → K − π þ . We determine the uncertainty due to π 0 reconstruction by taking the ratio of the efficiencies of η → 3π 0 to η → π þ π − π 0 or η → γγ. The uncertainty due to K 0 S reconstruction is evaluated by checking the efficiency of K 0 S → π þ π − as functions of flight length, transverse momentum of K 0 S , and polar angle of K 0 S . We measure the charged-pion detection asymmetry using reconstructed B → X s γ candidates (with charged pion in the final state) in the sideband defined as O NB < 0. The charged kaon detection asymmetry is measured using a large clean kaon sample from D 0 → K − π þ decay, where the pion detection asymmetry in the decay is subtracted with pions from D þ s → ϕπ þ decays [56]. The uncertainty due to possible mismodeling of the ΔE distribution we estimate by inflating the ΔE width and shifting the mean value.
We evaluate the uncertainties due to f þ− =f 00 and lifetime ratio by changing these values by AE1σ from the nominal PDG values [9].  The uncertainty due to the fragmentation model we determine by varying the decay channel proportions by their respective uncertainties. The exceptions are the proportions for K4π and K2π 0 in 2.0 GeV=c 2 < M X s < 2.4 GeV=c 2 and all the modes in 2.4 GeV=c 2 < M X s < 2.8 GeV=c 2 , where we use the proportions in MC simulation and a variation of AE50% as uncertainty. The fragmentation uncertainties for each M X s bin are obtained by summing in quadrature the changes for each of the ten mode categories.
Since the threshold between K Ã and the inclusive X s used in the MC simulation is fixed at 1.15 GeV=c 2 , we change this boundary to 1.10 GeV=c 2 and 1.20 GeV=c 2 to evaluate the uncertainty due to the threshold.
The proportion of missing final states that are not included in our reconstructed modes affects the reconstruction efficiency. We evaluate the uncertainty on the relative proportion of each of the 38 measured final states by varying the parameters of the fragmentation model used in the calibration of the MC simulation within their allowed ranges as determined from data. We take the difference from the nominal value as the systematic uncertainty on the missing fraction.
We evaluate the uncertainties due to A CP and Δ 0− in the background decays by changing the A CP and Δ 0− values by AE1σ from the nominal PDG values [9]; if neither A CP nor Δ 0− are measured, we assign AE100% uncertainties.
We evaluate the uncertainties due to tail parameters, α and n, in the signal PDF by floating in turn each of the fixed shape parameters in the fit while fixing the other shape parameters to their nominal values. Then the two uncertainties are added in quadrature. Since the α and n are anticorrelated, this procedure conservatively estimates the uncertainties. The uncertainties due to the other fixed parameters in the signal PDF are evaluated by varying them by AE1σ from the nominal values. The uncertainty due to cross-feed is caused by two sources; one is multiplicity of hadrons in the other B meson decays, the other is fragmentation model for signal. Both change the shape and yield of cross-feed. The former is evaluated with MC simulation by changing the multiplicities of π AE , π 0 , K AE , K 0 and η in the other B meson decays MC simulation from the nominal values to PDG values taking into account their uncertainties [9]. The latter is determined with MC simulation by varying the decay channel proportions by their respective uncertainties. We estimate the uncertainty due to the p parameter in qq background PDF by changing the parameter by AE1σ as obtained from the fit to the offresonance data. To evaluate the uncertainty due to the peaking background from π 0 decays, we vary the parameter values by AE1σ as determined from the sideband data. We evaluate the systematic uncertainties of other peaking backgrounds, which are subleading to the π 0 backgrounds, by changing the normalizations by AE20% which is about twice larger than the uncertainties of the corresponding branching fractions.
We check for possible bias in the fit by performing a large number of pseudo-experiments. In the study, we observe small biases which we add to the systematic uncertainty.
We also take into account the statistical uncertainty of the efficiency estimated with MC simulation samples as systematic uncertainty.
The systematic uncertainties due to efficiencies and background are only relevant for Δ 0− and A tot CP since these cancel out by taking the CP asymmetry in the other observables. The systematic uncertainties due to physics parameters to convert the signal yields to decay widths are only relevant for Δ 0− . The systematic uncertainties due to charged particle detection asymmetry and background A CP are only relevant for A CP as they cancel out for the CP-averaged observable Δ 0− . The largest and dominant systematic uncertainty for Δ 0− is due to f þ− =f 00 . The dominant systematic sources for ΔA CP and A CP are due to peaking background from π 0 decays and charge asymmetries in particle detection.

X. RESULTS
We perform a simultaneous fit to eight M bc distributions shown in Figure 2, with the PDFs as described above, to extract the following results where the first uncertainty is statistical, the second is systematic, and the third for Δ 0− is due to f þ− =f 00 . The fit results for the signal yields are summarized in Table IV. The correlation matrix of the six observables is given in Table V. The χ 2 and number of degrees of freedom in the simultaneous fit calculated from the data points and fit curves in Fig. 2 are 728 and 784, respectively. The measured Δ 0− is consistent with zero and the precision is better than that of BABAR by a factor of three [37]. Thus, this measurement can be used to constrain the resolved photon contribution in B → X s γ as the multiplicative uncertainties for the central value and its uncertainty as fourth and fifth uncertainties, respectively. This result improves the prediction of the branching fraction. The result for ΔA CP is consistent with zero, as predicted in the SM, thus the measurement can be used to constrain NP models, for example, in the supersymmetry model described in Ref. [36], this excludes the parameter space for squark mass below 5.0 TeV=c 2 .
We checked M X s dependences of the observables and find no dependences except Δ 0− in K Ã mass region is larger than the measurement and is consistent with the world average [9].
From our measurement of ΔA CP , we set confidence intervals on ImðC 8 =C 7 Þ based on Eq. (5). The hadronic parameterΛ 78 has a large uncertainty and the range is estimated as 17 MeV <Λ 78 < 190 MeV with a vacuum insertion approximation [33]. Since an uncertainty ofΛ 78 is hard to estimate, we set the 1σ and 2σ confidence level intervals for ImðC 8 =C 7 Þ as a function ofΛ 78 within the range described above as shown in Figure 3. Our result constrains ImðC 8 =C 7 Þ in the positive region better than the only previously available measurement from BABAR [32], and gives a strong constraint on NP models [36]. If we take the average value ofΛ 78 ¼ 89 MeV as a benchmark [35], the 2σ confidence intervals is −0.17 < ImðC 8 =C 7 Þ < 0.86.

XI. CONCLUSION
In summary, we have measured the isospin asymmetry and the difference of the direct CP asymmetries between charged and neutral B → X s γ decays with a sum-ofexclusive technique based on a sample of 772 × 10 6 BB pairs with an assumption that the observables have no dependence on the specific decay modes. The measurement of Δ 0− is consistent with zero and can constrain the resolved photon contribution in B → X s γ, which will improve the prediction of the branching fraction. The result of ΔA CP is consistent with zero as predicted in the SM, enabling constraints on NP models. Our measurements of the CP asymmetries are consistent with zero, and also with the SM predictions. All the results are the most precise to date and will be useful for constraining the parameter space in NP models. Current A CP and ΔA CP measurements are dominated by the statistical uncertainty; thus, the upcoming Belle II experiment will further reduce the uncertainty. To improve the isospin asymmetry at Belle II, reduction of the dominant uncertainty due to f þ− =f 00 is essential, and can be performed at both Belle and Belle II.