Search for CPT and Lorentz Violation in B0-B0bar Oscillations with Dilepton Events

We report results of a search for CPT and Lorentz violation in B0-B0bar oscillations using inclusive dilepton events from 232 million Y(4S) -->BBbar decays recorded by the BABAR detector at the PEP-II B Factory at SLAC. We find 2.8sigma significance, compatible with no signal, for variations in the complex CPT violation parameter z at the Earth's sidereal frequency and extract values for the quantities \Delta(a_\mu) in the general Lorentz-violating standard-model extension. The spectral powers for variations in z over the frequency range 0.26/year to 2.1/day are also compatible with no signal.

quantities ∆aµ in the general Lorentz-violating standard-model extension. The spectral powers for variations in z over the frequency range 0.26 year −1 to 2.1 day −1 are also compatible with no signal.
PACS numbers: 13.25.Hw,12.15.Hh,11.30.Er It was shown recently [1] that an interacting quantum field theory need not be local for CP T violation to imply violation of Lorentz invariance. In the general Lorentzviolating standard-model extension (SME) [2], the parameter for CP T violation in neutral meson oscillations depends on the 4-velocity of the meson [3].
We report a search for this effect using Υ (4S) → BB decays recorded by the BABAR detector at the PEP-II asymmetric-energy e + e − collider. Any observed CP T violation should vary with a period of one sidereal day (≃ 0.99727 solar days) as the Υ (4S) boost direction follows the Earth's rotation with respect to the distant stars [4].
The physical states of the B 0 -B 0 system are where L (H) labels the "light" ("heavy") eigenstate of the effective Hamiltonian. The complex parameter z vanishes if CP T is conserved; T invariance implies |q/p| = 1.
In the SME, CP T -and Lorentz-violating coupling coefficients a qi µ for the two valence quarks in the B 0 meson are contained in quantities ∆a µ = r q1 a q1 µ − r q2 a q2 µ , where the r qi are due to quark-binding and normalization effects. The CP T parameter z depends on the meson 4-velocity β µ = γ (1, β) in each experiment's observer frame as [3] z where β µ ∆a µ is real and varies with sidereal time due to the rotation of β relative to the constant vector ∆ a. The magnitude of the decay rate difference ∆Γ ≡ Γ H − Γ L is known to be small compared to the B 0 -B 0 oscillation frequency ∆m ≡ m H − m L ; hence Eq. 2 constrains Limits on analogous flavor-dependent ∆a µ specific to K 0 K 0 oscillations [5] and to D 0 D 0 oscillations [6] have been reported by the KTeV and FOCUS collaborations, respectively. KTeV has also reported a limit on sidereal variation of the phase φ +− of the CP -violating amplitude ratio η +− = A(K L → π + π − )/A(K S → π + π − ) [7].
The asymmetry between the decay rates at ∆t > 0 and ∆t < 0 compares the probabilities P (B 0 → B 0 ) and P (B 0 → B 0 ). Omitting second-order terms in z gives A CP T (∆t) ≃ −Re z ∆Γ∆t + 2 Im z sin(∆m∆t) cosh(∆Γ∆t/2) + cos(∆m∆t) . (6) The BABAR detector is described elsewhere [12]. We use about 232 million Υ (4S) → BB decays and 16 fb −1 of offresonance data, from 40 MeV below the Υ (4S) resonance, collected during 1999-2004 to search for variations in z with sidereal time of the form For long data-taking periods, any day/night variations in detector response tend to cancel over sidereal time.
We have previously measured [10] time-integrated values of Im z and Re z ∆Γ from the ∆t distribution of the same events. Here, we measure Im z 0 , Re z 0 ∆Γ, Im z 1 , and Re z 1 ∆Γ by extending the likelihood fit to include the event sidereal timet, and extract values for the SME quantities ∆a µ . In a complementary approach, we also measure the spectral power of periodic variations in z over a wide frequency band using the periodogram method [11] developed to study variable stars.
The event selection is the same as in Ref. [10]. Briefly, we suppress non-BB background by event-shape and event-topology requirements, and select events having at least two well-identified lepton candidates with momenta 0. 8 -2.3 GeV/c in the Υ (4S) rest frame that are not part of reconstructed J/ψ , ψ(2S) → e + e − , µ + µ − decays or photon conversions. Lepton candidates must have at least one z-coordinate measurement in the silicon vertex tracker to allow ∆t to be well-measured. We reject events in which either of the two highest-momentum lepton candidates (the dilepton) is classified as a cascade lepton from a b → (c, τ ) → ℓ transition by a neuralnetwork algorithm that uses as input variables the momenta and opening angle of the two leptons together with the event's visible energy and missing momentum. The selected dilepton sample comprises 1.18 million oppositesign events and 0.22 million same-sign events.
We estimate the Υ (4S) decay point in the transverse plane with a χ 2 -fit using the transverse distances to the two lepton tracks and the beam-spot. To measure ∆t, we assume each lepton originates from a direct B meson decay at the point on the lepton track with the least transverse distance to the Υ (4S). The component ∆z, along the Lorentz boost, of the distance between these two points yields ∆t = ∆z/ βγ c. For opposite-sign events ∆z = z + − z − ; for same-sign events we use |∆z|.
We model the ∆t-distribution of the dilepton sample with the probability density functions (PDFs) used in Ref. [10] to represent contributions from B 0 B 0 and B + B − decays and non-BB events. The latter are estimated, using off-resonance data, to be 3.1% of the sample. The fit to data determines that 59% of the BB events are B + B − decays. With minor BB background contributions fixed to values from Monte Carlo (MC) simulation, the fit to data also determines the fractions of B 0 B 0 and B + B − decays that are signal events (≃ 80%) with two direct leptons, and the fractions (≃ 10%) that are events with one direct lepton and a b → c → ℓ cascade decay of the other B meson. Same-sign dilepton events are retained primarily to improve the determination of these fractions.
Each PDF is a convolution of a decay rate in ∆t with a resolution function that is a sum of Gaussians or, for events with a cascade lepton, its convolution with one or two double-sided exponentials accounting for the lifetimes of intermediate τ or D (s) meson states, respectively. We use a sum of three Gaussians for signal events. The fit to data determines their fractions and also their widths except that of the widest, which is fixed to 8 ps. For leptons from different B mesons, our B 0 B 0 decay rate contains z to first-order (cf. Eq. 5) for opposite-sign events and is ∝ e −|∆t|/τ B 0 {cosh(∆Γ∆t/2) − cos(∆m∆t)} for same-sign events; for B + B − decays, it is ∝ e −|∆t|/τ B ± . For leptons from the same B meson, the decay rates are exponentials with effective lifetimes determined from MC simulation. Dilution factors are included to account for wrong flavor tags in cascade decays.
Each event's timestamp yields the time elapsed since the Unix epoch. We use this time, folded over one sidereal day and shifted in phase by 14.0 sidereal-hours, fort.
We extract z from a two-dimensional maximum likelihood fit to the opposite-sign and same-sign data events binned separately in ∆t andt. The likelihood function in ∆t for each of the 24 sidereal-time slices contains a common sum of the PDFs, and z varies witht as in Eq. 7. The likelihood fit corresponds to A CP T in Eq. 6. We obtain the values for z and φ reported in Table I ( upper left). The statistical correlation between Im z 0 and Re z 0 ∆Γ is 76%; between Im z 1 and Re z 1 ∆Γ it is 79%. Table I shows the sources of systematic uncertainties in the asymmetry parameters. Separate contributions are added in quadrature in the totals. We vary separately τ B 0 , τ B ± , and ∆m by 1σ from their known values [13], and vary |∆Γ| over the range 0 -0.1 ps −1 to allow 3σ deviations from the value reported in Ref. [9]. Fixed parameters in the PDF resolution functions for non-signal events are varied separately by 10%, motivated by a comparison of resolution parameters fitted to signal events in data and MC simulation. The fractions of the D (s) meson components in background cascade decays are also varied by 10%. The effects of possible internal misalignments of the silicon vertex tracker (SVT) and uncertainty in the absolute z-scale are evaluated in B 0 B 0 MC samples. The clock that sets the event timestamps is governed by the PEP-II master oscillator, which is stable to within 0.001% of its set frequency. Resynchronization of the clock with U.S. time standards at intervals of less than four months limits relative sidereal phase errors to less than 0.2%. Another small uncertainty in sidereal phase arises in calculating the Υ (4S) boost's right ascension. We use e + e − → µ + µ − (γ) data events, with true ∆z = 0, to check for sidereal variations in measured ∆z that could mimic a Lorentz-violation signal. The measured amplitude (0.022 ± 0.025) µm and mean (0.030 ± 0.018) µm are sources of negligible uncertainties. At the solar-day frequency, the amplitude is (0.028 ± 0.025) µm.
In Fig. 1 we plot the sidereal-time dependence of the measured asymmetry A meas CP T for the opposite-sign dilepton events with |∆t| > 3 ps, thereby omitting highlypopulated bins where any asymmetry is predicted to be small. Figure 2 shows confidence level contours for Im z 1 and Re z 1 ∆Γ. The significance for sidereal variations in z, characteristic of CP T and Lorentz violation, is 2.8σ.
We obtain the results reported in Table I (right). The statistical correlation between Im z 1 and φ is 48%. The significance for sidereal variations in z is again 2.8σ. We  I: Asymmetry parameter values, with statistical errors, for ACP T in Eq. 6 (upper left) and with SME constraint in Eq. 8 (upper right). Equation 7 implies z1 → −z1 for φ → φ + π. Systematic uncertainties are shown in lower part of Table. Without SME constraint With SME constraint  obtain consistent results for Im z 0 , Im z 1 , and φ when second-order terms (Eq. 5) of form |z| 2 = ρ 2 cos 2 (Ωt+φ), motivated by finding |Im z 1 | > |Im z 0 |, are included in the likelihood fit to data with ρ 2 as a free parameter. We use Eqs. 3, 4, and 7 to extract the SME quantities GeV. We now use the periodogram method [11] to compare the spectral power for variations in z at the sidereal frequency with those in a wide band of surrounding frequencies. The spectral power at a test frequency ν is where the data, comprising N measurements w j made at times T j , have variance σ 2 w . Here, T j is the time elapsed since the Unix epoch for opposite-sign dilepton event j, and the weights w j = ∆m∆t j −sin(∆m∆t j ) are suited to the study of periodic variations in z according to Eq. 8.
In the absence of an oscillatory signal, the probability that P (ν) exceeds a value S at a given frequency is exp(−S); if M independent frequencies are tested, the largest P (ν) value exceeds S with probability We use 20994 test frequencies from 0.26 year −1 to 2.1 solar-day −1 , spaced by 10 −4 solar-day −1 . This oversamples the frequency range by a factor of about 2.2 and avoids underestimating the spectral power of a signal. The number of independent frequencies is about 9500. Figure 3 shows the periodogram we obtain. The largest spectral power is P max (ν) = 8.78, for the test frequency ν = 0.46312 solar-day −1 . With no signal, the probability of finding a larger spectral power in our periodogram is 76%. Interpolation to the sidereal frequency (≃ 1.00274 solar-day −1 ) yields P (ν) = 5.28, a value that is exceeded at 78 test frequencies. At the solar-day frequency, where any effects due to day/night variations in detector response should appear, P (ν) = 1.47.
In conclusion, we report results of a search for sidereal variations in the CP T violation parameter z that are complementary to our previous time-integrated measurements [10] using the same events. Neither the likelihood fits nor the periodogram method detect asymmetries large enough to provide evidence for CP T and Lorentz violation. We have constrained the quantities ∆a µ of the Lorentz-violating standard-model extension that parameterize CP T violation in B 0 -B 0 oscillations.
The authors are indebted to Alain Milsztajn (deceased) for his help with the periodogram analysis. We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), MEC (Spain), and STFC (United Kingdom). Individuals have received support from the Marie Curie EIF (European Union) and the A. P. Sloan Foundation.