Measurement of Time-dependent $CP$ Asymmetries in $B^{0}\to K_S^0 \eta \gamma$ Decays

We report a measurement of time-dependent $CP$ violation parameters in ${B^0 \to K_S^0 \eta \gamma}$ decays. The study is based on a data sample, containing ${772 \times 10^6 B\bar{B}}$ pairs, that was collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$ collider. We obtain the $CP$ violation parameters of ${{\cal S} = -1.32 \pm 0.77 {\rm (stat.)} \pm 0.36{\rm (syst.)}}$ and ${{\cal A} = -0.48 \pm 0.41 {\rm (stat.)} \pm 0.07{\rm (syst.)}}$ for the invariant mass of the $K_S^0 \eta$ system up to 2.1 GeV/$c^2$.


INTRODUCTION
The radiative b → sγ decay proceeds dominantly via one-loop electromagnetic penguin diagrams at lowest order in the standard model (SM). Since heavy unobserved particles might enter in the loop, such decays are sensitive to new physics (NP). Precision measurements of the branching fraction for B → X s γ by CLEO [1], BaBar [2][3][4] and Belle [5,6] are consistent with SM predictions [7,8] and give a strong constraint to NP models [9]. Another important observable that is sensitive to NP signatures in the b → sγ process is the photon polarization. Within the SM, the photon is mostly produced with left-handed polarization; the right-handed contribution is suppressed by m s /m b at leading order, where m s (m b ) is the strange (bottom) quark mass. Various NP models, such as supersymmetry [10][11][12][13][14][15], left-right symmetric models [16] and extra-dimensions [17][18][19][20][21][22], allow right-handed currents in the loops and hence can enhance the right-handed photon contribution [23][24][25][26][27]. Thus, a measurement of the photon polarization in the b → sγ process is an important tool to search for NP.

TIME-DEPENDENT CP VIOLATION
At the KEKB asymmetric-energy collider (3.5 GeV e + on 8.0 GeV e − ), the Υ(4S) is produced with a Lorentz boost of βγ = 0.425 nearly along the z axis, which is antiparallel to the e + beam direction. In the decay chain Υ(4S) → B 0B0 → f rec f tag , one of the B mesons decays at proper time t rec to a final state f rec (our signal mode), and the other (B tag ) decays at proper time t tag to a final state f tag that is used to determine the flavor of the signal B meson. The distribution of the proper time difference ∆t = t rec − t tag is given by where S (A) is the mixing-induced (direct) CP violation parameter, τ B 0 is the B 0 lifetime, ∆m d is the mass difference between the two B 0 mass eigenstates, and q = +1 (−1) is the b-flavor charge when the tagging B meson is a B 0 (B 0 ). Since the B 0 andB 0 mesons are approximately at rest in the Υ(4S) center-of-mass (CM) frame, ∆t can be determined from the displacement in z in the laboratory frame between the f rec and f tag decay vertices: ∆t ≃ (z rec − z tag )/βγc ≡ ∆z/βγc, where z rec and z tag are the decay positions along the z axis of the signal and tag-side B mesons.

BELLE DETECTOR
The Belle detector [54] 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 barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic caloriemeter (ECL) comprised of CsI(Tl) crystals. All these detector components are located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented with resistive plate chambers to detect K 0 L mesons and muons. Two inner detector configurations were used: A 2.0 cm radius beampipe and a 3-layer SVD was used for the first sample of 152 × 10 6 BB pairs (SVD1), while a 1.5 cm radius beampipe, a 4-layer SVD and a small-inner-cell CDC were used to record the remaining 620 × 10 6 BB pairs (SVD2) [56].

EVENT SELECTION
The most energetic isolated cluster in the ECL in the CM frame of an event that is not associated with any charged tracks reconstructed in the SVD and CDC is selected as the prompt photon. Its energy must lie between 1.8 and 3.4 GeV. We require that its shower shape be consistent with an electromagnetic shower by imposing the criterion E 9 /E 25 > 0.95 for the ratio of energy deposits in a 3 × 3 array of CsI(Tl) crystals to that in a 5 × 5 array, both centered on the crystal with the largest energy deposit. To reduce contamination from the decays π 0 → γγ or η → γγ, the prompt photon candidate is paired with all other photons in the event with energy exceeding 40 MeV in the laboratory frame. We reject the event if the pair is consistent with the above decays, based on a likelihood constructed from the invariant mass, the energy and polar angle of the second photon in the laboratory frame [57].
Neutral pion candidates are reconstructed from two photons whose energies exceed 50 MeV in the laboratory frame. We require the invariant mass of the photon pairs to lie between 114 and 147 MeV/c 2 , which corresponds approximately to a ±3σ window in resolution about the nominal π 0 mass [58]. To reduce the combinatorial background, we retain candidates with a momentum greater than 100 MeV/c in the CM frame.
Charged particles, except for pions from K 0 S decays, are required to have a distance of closest approach to the interaction point (IP) within 5.0 cm along the z axis and 0.5 cm in the transverse plane. Charged kaons and pions are identified with a likelihood ratio constructed from specific ionization measurements in the CDC, time-of-flight information from the TOF, and the number of photoelectrons in the ACC.
Neutral kaon (K 0 S ) candidates are reconstructed from pairs of oppositely-charged tracks, treated as pions, and identified by a multivariate analysis [59] based on two sets of input variables [60]. 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 ±10 MeV/c 2 of the nominal value, corresponding to a ±3σ interval in mass resolution.
We reconstruct η candidates from the γγ and π + π − π 0 final states, denoted as η 2γ and η 3π , respectively. For the η 2γ mode, we require that the photon energy in the CM system be greater than 150 MeV. The candidates satisfying the di-photon invariant mass requirement of 510 MeV/c 2 < M γγ < 575 MeV/c 2 are retained. For the η 3π mode, the invariant mass of the three-pion system is required to be in the range 537 MeV/c 2 < M πππ < 556 MeV/c 2 . These requirements correspond to about ±2σ windows in mass resolution.
We reconstruct B candidates by combining a K 0 S with an η and a γ candidate. We form two kinematic variables to select B mesons: are the energy and momentum, respectively, of the B candidate in the CM system. We define the signal region in ∆E and M bc for the measurement of CP violation as −0.15 GeV < ∆E < 0.08 GeV and 5.27 GeV/c 2 < M bc < 5.29 GeV/c 2 .
To determine the signal fraction, a larger fitting region, |∆E| < 0.5 GeV and 5.20 GeV/c 2 < M bc < 5.29 GeV/c 2 , is employed. The average number of B candidates in an event with at least one candidate is 1.47; this is primarily due to multiple η candidates. If there is more than one B candidate in the fitting region, the candidate whose η daughter's mass is closest to the nominal value is selected. If still necessary, the B candidate with the K 0 S daughter's mass closest to the nominal value is retained.

BACKGROUND SUPPRESSION
To suppress the dominant e + e − → qq (q ∈ {u, d, s, c}) continuum background, we use a neural network based on four input variables calculated in the CM frame: (1) the cosine of the angle between the B momentum and the z axis, (2) the likelihood ratio of modified Fox-Wolfram moments [61,62] that gives the strongest separation power, (3) the cosine of the angle between the third sphericity axes [63] calculated from the B candidate and all other particles in the rest of the event (ROE), and (4) the cosine of the angle between the first sphericity axis in the ROE and the z axis. The network is trained with a GEANT3based Monte Carlo (MC) simulation [64]. The output variable, O NB , in the range [−1, 1], is used as one of the variables to determine the signal fraction. To enable a simple analytical modeling, O NB is transformed into where O min NB and O max NB are chosen to be −0.7 and 0.935 (0.915), respectively, for the η 2γ (η 3π ) mode. The events with O NB < O min NB are discarded; this selection keeps 80% (73%) of the signal while removing 92% (95%) qq background for the η 2γ (η 3π ) mode.
The decay modes of the following CP eigenstates constitute peaking backgrounds: , where a X and K X represent a light unflavored resonance and a kaonic resonance, respectively. To suppress these backgrounds, we require 2.0 GeV/c 2 < M γη < 2.9 GeV/c 2 or M γη > 3.2 GeV/c 2 to eliminate J/ψ → ηγ and a X → ηπ 0 , M Kη < 1.82 GeV/c 2 or M Kη > 1.90 GeV/c 2 to removeD 0 → K 0 S η, and M γK > 2.0 GeV/c 2 to suppress K X → K 0 S π 0 andD 0 → K 0 S π 0 , where a soft photon from the π 0 decay is undetected.
One of the decays arising from the b → sγ transition, B 0 → K 0 S π 0 γ, is a major peaking background. This decay is exclusively reconstructed and rejected if the candidate is found to satisfy the following requirements: 0.12 GeV/c 2 < M γγ < 0.15 GeV/c 2 , 1.6 GeV < E CM γ < 3.4 GeV, −0.20 GeV < ∆E < 0.10 GeV, and M bc > 5.27 GeV/c 2 .

HELICITY ANGLE AND MASS DISTRIBUTIONS
As the spin and invariant mass of the Kη system are not well known, we study B + → K + ηγ [65] assuming the isospin symmetry breaking to be small between B 0 → K 0 ηγ and B + → K + ηγ [66]. The selections on B + → K + ηγ are the same as those on B 0 → K 0 S ηγ except for kaon selections. We define the helicity angle (θ hel ) as the angle between the K momentum and the opposite of the B-meson momentum in the Kη rest frame. The signal yields are extracted by fitting to ∆E and M bc in bins of cos θ hel and the K + η invariant mass; later, the efficiency-corrected yield is obtained. We fit to the cos θ hel distribution with spin-1 and spin-2 hypotheses, as a spin-3 resonance in B decays is only found in a B 0 s decay and is highly suppressed compared to the spin-1 states [68]. Figures 1 and 2 show the background-subtracted and efficiency-corrected θ hel and invariant-mass distributions for B + → K + ηγ. We find that the signal is concentrated in the region M Kη < 2.1 GeV/c 2 and has the signature of a spin-1 system. From these studies, we apply two selection criteria,  −0.7 < cos θ hel < 0.9 and M Kη < 2.1 GeV/c 2 , to B 0 → K 0 S ηγ candidates to maximize the signal sensitivity.

SIGNAL EXTRACTION
We extract the signal yield with a three-dimensional extended unbinned maximumlikelihood fit to ∆E, M bc , and O ′ NB . For the signal ∆E-M bc distribution, a two-dimensional histogram is used as the two variables have 40% correlation due to the imperfect energy measurement for the prompt photon. The O ′ NB distribution is modeled with the sum of two bifurcated Gaussian functions sharing a common peak position and right-side width. For the qq background, the ∆E and M bc distributions are parameterized by a second-order Chebyshev polynomial and an ARGUS function [69], respectively. The sum of a bifurcated Gaussian and a Gaussian function reproduces its O ′ NB distribution. For background from B meson decays, the ∆E distribution is described by an exponential function; O ′ NB is modeled with a bifurcated Gaussian function; the M bc distribution is described by the sum of an ARGUS function and a Gaussian function. The fit results projected onto ∆E, M bc and O ′ NB are shown in Fig. 3. We obtain 69.5 +13.4 −12.4 and 22.4 +7.3 −6.4 signal events for the η 2γ and η 3π decay modes, respectively, with purities in the signal region of 28.4% and 22.5%.

FLAVOR TAGGING
The flavor of the B tag meson is determined from inclusive properties of particles in the ROE based on a multi-dimensional likelihood method. The algorithm for flavor tagging is described in detail elsewhere [70]. Two parameters, q defined in Eq. (1) and r, are used to represent the tagging information. The parameter r is an event-by-event MCdetermined flavor tagging quality factor that ranges from 0 for no flavor information to 1 for unambiguously determined flavor. The data are sorted into seven intervals of r in which the fractions of wrongly tagged B flavor (w l , l = 1, ..., 7) as well as the differences between B 0 andB 0 (∆w l ) are determined from self-tagged semileptonic and hadronic b → c decays.

VERTEX RECONSTRUCTION
The vertex positions of signal-side decays of B 0 → K 0 S η 3π γ and B 0 → K 0 S η 2γ γ is determined from the charged tracks. For B 0 → K 0 S η 3π γ decays, we require at least one of the charged pions from η 3π decays, which originate from the B decay position, to have at least one (two) hit in the SVD r-φ (z) layers. To improve the B-vertex resolution, we use an additional constraint from the transverse-plane beam profile at the IP (σ beam x ∼ 100 µm, σ beam y ∼ 5 µm) smeared with the finite flight length of the B 0 meson in the x-y plane. The estimated uncertainty of the reconstructed vertex position in the z direction (σ rec z ) determined with single (two) charged track is required to be less than 500 µm (200 µm) to ensure enough quality for time dependent analysis. For B 0 → K 0 S η 2γ γ decays, the K 0 S trajectory, reconstructed from its pion daughters, is used to determine the vertex position with the aforementioned constraint on the smeared beam profile; this strategy is adopted since the decay vertex of the long-lived K 0 S is displaced from the B decay vertex. To have good resolution of the K 0 S trajectory, both pions daughters must satisfy SVD-hit requirements of at least one (two) hit in the r-φ (z) layers for SVD1, and at least two hits in both r-φ and z layers for SVD2. We apply a selection on the σ rec z to be less than 500 µm. The vertex position of B tag is determined from well-reconstructed charged particles in the ROE [71]. The |∆t| is restricted to be less than 70 ps for further analysis.

EVENT MODEL
We determine S and A by performing an unbinned maximum-likelihood fit to the observed ∆t distribution in the signal region. The probability density function (PDF) expected for the signal distribution, P sig (∆t, q, w l , ∆w l ; S, A), is given by Eq. (1), modified to incorporate the effect of incorrect flavor assignment. Two of the parameters in the PDF expression, τ B 0 and ∆m d , are fixed to their world average [72]. The distribution is convolved with the proper-time resolution function, R sig (∆t), which is a function of the event-by-event ∆t uncertainties. The resolution function R sig (∆t) incorporates the detector resolution, contamination of non-primary tracks in the vertex reconstruction of B tag , and the kinematic energy generated by the Υ(4S) decay. As in Ref. [73], universal R sig parameters are used for the vertex reconstruction for η 3π and the long-lived K 0 S . A detailed description can be found in Ref. [74]. The PDF for BB background events (P BB ) is modeled in the same way as for signal, but with different lifetime and CP violation parameters while using the same resolution function (R BB = R sig ). The effective lifetime of the BB background is obtained from a fit to the MC sample for each η decay mode. The PDF for qq background events, P qq , is modeled as the sum of exponential and prompt components, and is convolved with a double Gaussian representing the resolution function R qq . All parameters in P qq and R qq are determined by a fit to the ∆t distribution of a background-enhanced sample in the ∆E-M bc sideband.
For each event i, the following likelihood function is calculated: where P ol is a broad Gaussian function that represents an outlier component with a small fraction f ol [74]. The signal and background probabilities, f sig and f BB , are calculated on an event-by-event basis from the function obtained by the same ∆E-M bc -O ′ NB fit used to extract the signal yield, and are then multiplied by a factor that depends on the flavor tagging r-bin. The r distributions of the signal and the qq background are estimated by repeating the ∆E-M bc -O ′ NB fit procedure for each r interval with the three background shape parameters fixed to the full-range result. The BB background distribution is estimated from MC samples and found to be small.

RESULTS
The only free parameters in the final fit are S and A, which are determined by maximizing the likelihood function L = Π i P i (∆t i ; S, A), where the product is over all events. We obtain S = −1.32 and A = −0.48, and find that the central values are outside of the physical boundary defined by S 2 + A 2 = 1. We extract the statistical uncertainties from the root-mean-square of the CP violation parameter distributions obtained using an ensemble test with input values of (S true , A true ) = (−0.94, −0.34), which is the closest point on the physical boundary to the fit result [75], as δS = ±0.77 and δA = ±0.41 [76]. The correlation between S and A is found to be 0.15. We define the raw asymmetry in each ∆t interval as (N q=+1 − N q=−1 )/(N q=+1 + N q=−1 ), where N q=±1 is the number of observed candidates with the given q. The ∆t distributions and raw asymmetries for events in the signal-enhanced 0.5 < r ≤ 1.0 region for q = ±1 are shown in Fig. 4.

VALIDATIONS
Various cross-checks are performed to confirm the validity of our procedure. The CP asymmetry fit to MC signal samples shows good linearity. Dedicated lifetime fits to B + → K + ηγ samples yield 2.0±0.3 ps and 2.3±0.4 ps for η 2γ and η 3π , respectively. A lifetime fit to B 0 → J/ψK 0 S using only K 0 S to determine the signal vertex results in 1.528±0.027 ps. A CP asymmetry fit to the B + → K + ηγ control samples yields (S,A)=(0.01±0.35, 0.06±0.29) and (0.2±0.6, 0.2±0.4) for η 2γ and η 3π , respectively. Lastly, a CP asymmetry fit to B 0 → J/ψK 0 S only using K 0 S to determine the signal vertex position yields (S,A)=(0.73±0.05, 0.00±0.03). These results are consistent with either their world-average or expected values [58].

SYSTEMATIC UNCERTAINTIES
We calculate systematic uncertainties in the following categories by fitting the data with each fixed parameter being varied by its uncertainty: values of physics parameters such as ∆m d and τ B 0 , effective lifetime and CP asymmetry of the BB background, imperfect knowledge of the qq background ∆t PDF, the flavor-tagging determination, the signal and background fractions, and the resolution functions. A possible bias in the fit is checked by performing a large number of pseudo-experiments. The fit result is consistent with the input value within the statistical uncertainty. We quote this uncertainty as the possible fit bias. The uncertainty due to the vertex reconstruction is estimated by changing the requirements on the track quality. For the effect of SVD misalignment, we use the value from the latest sin 2φ 1 measurement at Belle [77], which is estimated from MC samples by artificially displacing the SVD sensors in a random manner. Effects of tag-side interference [78] are estimated with a control sample of B → D * ℓν events. A detailed description of the evaluation of the systematic uncertainties is found in Ref. [79]. The dominant systematic contributions for S arise from the uncertainties in the resolution function and vertex reconstruction. The systematic uncertainty in A is dominated by the resolution function. These contributions are added in quadrature and summarized in Table I. CONFIDENCE LEVEL CONTOURS Figure 5 shows confidence intervals calculated using the Feldman-Cousins frequentist approach [80], incorporating a smearing by additional Gaussian functions to represent the systematic uncertainties discussed above. Our result is less than 2σ away from zero, and is consistent with the BaBar result [44] as well as the SM predictions [48][49][50][51][52][53] with the assumption that time-dependent CP asymmetries in B 0 → K * 0 γ and B 0 → K 0 S ηγ are the same.

CONCLUSION
In summary, we have measured CP violation parameters in B 0 → K 0 S ηγ decays using a data sample of 772 × 10 6 BB pairs. The obtained parameters S = −1.32 ± 0.77(stat.) ± 0.36(syst.), A = −0.48 ± 0.41(stat.) ± 0.07(syst.) are consistent with the null-asymmetry hypothesis within 2σ as well as with SM predictions [48][49][50][51][52][53]. Our measurement is dominated by statistical uncertainty. Therefore, with much higher statistics and also higher acceptance and reconstruction efficiencies, the forthcoming Belle II experiment should significantly improve upon the precision of this measurement.