Search for violations of Lorentz invariance and $ CPT$ symmetry in $ B^0_{(s)}$ mixing

Violations of $ CPT$ symmetry and Lorentz invariance are searched for by studying interference effects in $ B^0$ mixing and in $ B^0_s$ mixing. Samples of $ B^0\to J/\psi K^0_{\mathrm{S}}$ and $ B^0_s\to J/\psi K^+ K^-$ decays are recorded by the LHCb detector in proton--proton collisions at centre-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb$^{-1}$. No periodic variations of the particle-antiparticle mass differences are found, consistent with Lorentz invariance and $ CPT$ symmetry. Results are expressed in terms of the Standard Model Extension parameter $\Delta a_{\mu}$ with precisions of $ \mathcal{O}(10^{-15})$ and $ \mathcal{O}(10^{-14})$ GeV for the $ B^0$ and $ B^0_s$ systems, respectively. With no assumption on Lorentz (non-)invariance, the $ CPT$-violating parameter $z$ in the $ B^0_s$ system is measured for the first time and found to be $ \mathcal{R}e(z) = -0.022 \pm 0.033 \pm 0.005$ and $ \mathcal{I}m(z) = 0.004 \pm 0.011\pm 0.002$, where the first uncertainties are statistical and the second systematic.

Lorentz invariance and the combination of charge conjugation, spatial inversion and time reversal (CP T ) are exact symmetries in the Standard Model (SM) of particle physics, and are deeply connected in any quantum field theory [1]. Quantum theories that aim to describe Planck-scale physics, such as string theory, might break these fundamental symmetries [2]. Present-day experiments are many orders of magnitude away from the Planck energy scale of ∼ 10 19 GeV, however, small effects at low energy might still be observable. Interference effects in the mixing of neutral mesons are sensitive to violations of CP T symmetry, and therefore may provide a window to the quantum gravity scale [3]. Such effects can be quantified in a low-energy, effective field theory, as done in the Standard Model Extension (SME) [4,5]. In this framework, terms that explicitly break Lorentz and CP T symmetry are added to the SM Lagrangian to describe the couplings between particles and (hypothetical) uniform tensor fields. These fields would acquire non-zero vacuum expectation values when these symmetries are spontaneously broken in the underlying theory. The SME couplings are expected to be suppressed by powers of the Planck scale [6]. In the SME, the CP T -violating parameters that can be measured in neutral meson systems also break Lorentz symmetry. The amount of CP T violation depends on the direction of motion and on the boost of the particle. The SME parameters for the B 0 and B 0 s systems can be best measured with a time-dependent analysis of the decay channels B 0 → J/ψ K 0 S and B 0 s → J/ψ K + K − , using the four-velocity of the B mesons [7]. The notation B refers to either B 0 or B 0 s and the inclusion of charge-conjugate processes is implied throughout this Letter. These parameters have been measured previously, albeit with less sensitive decay modes [7], by the BaBar collaboration for the B 0 system [8], and by the D0 collaboration for the B 0 s system [9]. The LHCb detector is a single-arm forward spectrometer described in detail in Refs. [10,11]. Simulated events are produced using the software described in Refs. [12][13][14][15][16]. The data used in this analysis correspond to an integrated luminosity of 3 fb −1 , taken at the LHC at proton-proton centre-of-mass energies of 7 and 8 TeV. The selection of both decay channels is the same as used in Refs. [17] and [18]. The J/ψ meson is reconstructed in the dimuon channel and the K 0 S meson in the π + π − final state. Interference effects from CP T violation can be incorporated generically in the time evolution of a neutral B meson system, described by the Schrödinger equation i∂ t Ψ =ĤΨ. The effective 2 × 2 Hamiltonian is written asĤ =M − iΓ/2 [19]. Diagonalisation gives a heavy-mass eigenstate |B H and a light-mass eigenstate |B L with masses m H,L and decay widths Γ H,L . The differences between the eigenvalues are defined as ∆m ≡ m H − m L and ∆Γ ≡ Γ L − Γ H . The differences between the diagonal matrix elements of the effective Hamiltonian are defined as δm ≡ M 11 − M 22 and δΓ ≡ Γ 11 − Γ 22 . Any difference between the mass or lifetime of particles and antiparticles (i.e., a non-zero δm or δΓ) would be a sign of CP T violation, and is characterised by and the mass eigenstates are given by |B H, Owing to the smallness of the B mixing parameters ∆m and ∆Γ in the denominator, z is highly sensitive to CP T -violating effects.
Considering only contributions to first order in z, the decay rate to a CP eigenstate f as a function of the B proper decay time t becomes where Γ ≡ (Γ 11 + Γ 22 )/2, ζ = +1(−1) for an initial |B (|B ) state and the following definitions are introduced: with A f and A f the direct decay amplitudes of a |B or |B state to the eigenstate f . For the decay B 0 → J/ψ K 0 S , the final state is CP odd, corresponding to CP eigenvalue . Furthermore, in the B 0 system, the approximation ∆Γ d = 0 is made, as supported by experimental data [20].
The decay B 0 s → J/ψ K + K − is similar to B 0 → J/ψ K 0 S , but the decay width difference ∆Γ s cannot be ignored [20]. Another important difference is that the K + K − system mostly originates from the φ(1020) resonance, giving the K + K − pair an orbital angular momentum L = 1 (P wave). Since the J/ψ φ final state consists of two vector mesons, its orbital angular momentum can be L ∈ {0, 1, 2} for the polarisation states f ∈ {0, ⊥, }, respectively, with corresponding CP eigenvalues η f = (−1) L . The K + K − system has a small S-wave contribution [18], which results in another L = 1 component for the J/ψ K + K − final state. These four polarisation states can be separated statistically in the helicity formalism [21], using the three decay angles between the final-state particles. The corresponding weak phases, arg(λ J/ψ K + K − ) = Lπ − φ s , can, in the SM, be expressed in terms of CKM matrix elements, The decay rate has to be modified compared to Eq. 2 to include the angular dependence. It becomes a sum over all ten combinations of the four helicity amplitudes, where f k ( Ω) are angular functions, given in Ref.
[21], and h k (t) are products of the  Table 2: Definition of the coefficients in Eq. 5. The following definitions are used: amplitudes as listed in Table 1. The time dependence of h k (t) is given by with the coefficients listed in Table 2.
In the SME, the dimensionless parameter z is not a constant. It depends on the four-velocity β µ = (γ, γ β) of the neutral meson as [22, 23] thereby breaking Lorentz invariance. The SME parameter ∆a µ describes the difference between the couplings of the valence quarks, within the neutral meson, with the Lorentzviolating fields [22]. Therefore, B 0 and B 0 s mesons can have different values of ∆a µ . Since ∆a µ is real [24], it follows that Re(z)∆Γ = −2Im(z)∆m. For B mesons, ∆m ∆Γ, and so Im(z) is two orders of magnitude smaller than Re(z), and can be ignored in the measurements of ∆a µ . The average boost of B mesons in the acceptance of LHCb is γβ ≈ 20. It follows from Eq. 6 that this large boost results in a high sensitivity to ∆a µ [7].
To measure ∆a µ , the meson direction needs to be determined in an absolute reference frame. Such a frame can be defined with respect to fixed stars [24]. In this frame, the Z-axis points north along the Earth's rotation axis, the X-axis points from the Sun to the vernal equinox on 1 January 2000 (J2000 epoch) and the Y -axis completes the right-handed coordinate system. The latitude of the LHCb interaction point is 46.2414 • N, the longitude is 6.0963 • E, and the angle of the beam east of north is 236.296 • . The beam axis is inclined with respect to the geodetic plane by 3.601 mrad, pointing slightly upwards. The timekeeping is obtained from the LHC machine with a time stamp, t LHC , in UTC microseconds since January 1 st , 1970, 00:00:00 UTC. The time, spatial coordinates and angles have negligible uncertainties and are used to define the rotation from the coordinate system of LHCb to the absolute reference frame. For mesons travelling along the beam axis, Re(z) can be expressed as, where | β| is set to unity, ∆a X,Y,Z = −∆a X,Y,Z , and χ = 112.4 • is the angle between the beam axis and the rotational axis of the Earth. The time dependence results from the Earth's rotation, giving a periodicity with sidereal frequency Ω. The sidereal phase at t LHC = 0 is found to bet = (2.8126 ± 0.0014) hr. The B mesons are emitted at an average angle of about 5 • from the beam axis. This means that the LHCb detector is mostly sensitive to the linear combination ∆a ≡ ∆a 0 + cos(χ)∆a Z = ∆a 0 − 0.38∆a Z , while there is a much weaker sensitivity to the orthogonal parameter, ∆a ⊥ = 0.38∆a 0 + ∆a Z , coming from the smaller transverse component of the B velocity. Both ∆a and ∆a ⊥ are measured and the correlation between them is negligible. Unbinned likelihood fits are applied to the decay-time distributions using Eqs. 2 and 4. To obtain the SME parameters, the sidereal variation of Re(z) is taken into account by including in the fits the LHC time and the three-momentum of the reconstructed B candidate. For the B 0 s sample, the fits are performed to the full angular distribution. In the invariant mass distributions of the B candidates, the background is mostly combinatorial. For both decay channels, this background is statistically subtracted using the sPlot technique [25], which allows to project out the signal component by weighting each event depending on the mass of the B candidate. The mass models are the same as in Refs. [17,18]. The correlation between the shape of the invariant mass distribution and the B momentum or, for the B 0 s sample, the decay angles, leads to a small systematic bias for both samples. This effect is included in the systematic uncertainty. In the B 0 s →J/ψ K + K − sample, there is a small contribution coming from misidentified B 0 →J/ψ K + π − and Λ 0 b → J/ψ pK − decays. This background contribution is statistically removed by adding simulated decays with negative weights. A systematic uncertainty is assigned to account for the uncertainty on the size and shape of this background.
The description of the detection efficiency as a function of the decay time, the decaytime resolution model and the flavour tagging (to distinguish initial B and B mesons) are the same as in Refs. [17] and [18] for the B 0 → J/ψ K 0 S and B 0 s → J/ψ K + K − samples, respectively. This description includes the dilution of the asymmetry due to wrong decisions of the flavour tagging method. The decay-time resolution model and tagging calibration do not lead to a systematic bias in the final result. A possible wrong assignment of the primary interaction vertex (PV) to the B candidate gives a small bias in the ∆a B 0 ⊥ parameter, which is included in the systematic uncertainty. The inefficiency at high decay times, caused by the reconstruction algorithms, is described by an exponential function. For the B 0 sample, this function is obtained from simulation and does not lead to a systematic bias in the result. For the B 0 s sample, the exponential function is obtained from a data-driven method. The change in the final result when using the correction procedure from Ref. [18] is taken as a systematic uncertainty.
The production asymmetry between B 0 and B 0 mesons is included in the modelling of the decay rates, and is taken from Refs. [26,27]. The corresponding uncertainties are included in the statistical uncertainty, while a possible momentum dependence of the production asymmetry is considered as a systematic uncertainty. The B 0 s production asymmetry does not affect the fit to the B 0 s sample, since the fast B 0 s oscillations wash out this effect and since the decay rates for B 0 s and B 0 s tags are normalised separately. In the fit to the B 0 sample, the correlation between Re(z B 0 ) and C J/ψ K 0 S is too large to allow determination of Re(z B 0 ) without making assumptions about the value of C J/ψ K 0 S [7]. On the other hand, to determine ∆a B 0 µ , the averages C J/ψ K 0 S = 0.005 ± 0.020 and S J/ψ K 0 S = 0.676 ± 0.021 [19] as measured by the BaBar and Belle collaborations can be used in the fit. Since the boost of the B 0 mesons is about 40 times lower in these experiments, these values are hardly affected by possible Lorentz violation in the SME. The value of D J/ψ K 0 S is by definition . The uncertainties on these external input values are propagated as systematic uncertainties on ∆a B 0 µ . The mass difference, ∆m d = 0.510 ± 0.003 ps −1 [19], is allowed to vary in the fit within its uncertainty using a Gaussian constraint. Setting ∆Γ d = 0.007 ps −1 , which corresponds to the experimental uncertainty [20], leads to a small change in ∆a B 0 , which is included in the systematic uncertainty. The B 0 lifetime is allowed to vary freely in the fit.
In the fit to the B 0 s sample, the correlation between Re(z B 0 s ) and C J/ψ K + K − is small owing to the additional interference terms from the helicity amplitudes, the non-zero ∆Γ s and the faster B 0 s -B 0 s oscillations. For this reason, the same parameters as in Ref. [18] are varied freely in the fit, in addition to either ∆a µ or z. The detection efficiency is also described as a function of the decay angles. The shape of this angular acceptance is obtained from simulation. The simulated events are weighted to match the kinematic distributions in data. The uncertainty due to the limited number of simulated events and the full effect of correcting for the kinematic distributions in data are added to the systematic uncertainty. Systematic effects due to the decay-angle resolution are negligibly small. The fit to the B 0 s sample is performed simultaneously in bins of the K + K − invariant mass [18]. Each bin has a different interference between the P-and S-wave amplitudes. This effect is included in the fit and no systematic biases are observed.
An overview of the systematic uncertainties is given in Table 3. For the B 0 mixing, the largest contribution comes from the uncertainty on the external parameters C f and S f . A small systematic bias is observed in ∆a B 0 ⊥ due to the momentum dependence of the cross-sections of neutral kaons in the detector material. For the B 0 s mixing, the largest contribution comes from the description of the decay-time acceptance. Effects from the correlation between the mass and decay time and from the accuracy of the length scale and momentum scale of the detector are found to be negligible.
The components of the SME parameter ∆a µ for B 0 mixing, obtained from the fit to the sample of selected B 0 →J/ψ K 0 S candidates, are ∆a B 0 = (−0.10 ± 0.82 (stat) ± 0.54 (syst)) × 10 −15 GeV ,  Since the SME fits consider only one specific frequency, i.e. the sidereal frequency, a wide range of frequencies is scanned by means of the periodogram method. A periodogram gives the spectral power P (ν) of a frequency ν in a signal sampled at discrete, not necessarily equidistant, times. In this analysis, the Lomb-Scargle periodogram [28] is used, as in the BaBar measurement of ∆a B 0 µ [8]. The periodogram is determined for the term in the decay rates proportional to e −Γt Re(z). Since negative weights cannot be used in the periodogram, the B mass windows are narrowed to 5260 < m J/ψ K 0 S < 5300 MeV/c 2 and 5350 < m J/ψ K + K − < 5390 MeV/c 2 compared to those used in the fits [17,18]. In total about 5200 frequencies are scanned in a wide range around the sidereal frequency, from 0.03 solar-day −1 to 2.10 solar-day −1 . The number of frequencies oversamples the number of independent frequencies by roughly a factor of two, thereby avoiding any undersampling [29]. As the data are unevenly sampled, the false-alarm probability is determined from simulation [29], where the time stamps are taken from data.
The two periodograms are shown in Fig. 2. No significant peaks are found. For the B 0 periodogram, the highest peak P (ν max ) = 8.09 is found at a frequency of 1.5507 solar-day −1 and has a false-alarm probability of 0.57. There are 2707 (1559) sampled frequencies with a larger spectral power than the peak at the sidereal (solar) frequency. For the B 0 s periodogram the highest peak P (ν max ) = 10.85 is found at a frequency of 1.3301 solar-day −1 and has a false-alarm probability of 0.06. There are 3386 (2356) frequencies with a larger spectral power than the sidereal (solar) peak. The absence of any signal in the SME fits is confirmed by the absence of significant peaks at the sidereal frequency.
The results presented here are consistent with CP T symmetry and Lorentz invariance. The measurement of ∆a B 0 µ is an improvement in precision of about three orders of magnitude compared to the one from the BaBar collaboration [8] when the SM prediction ∆Γ d = −0.0027 ps −1 [30] is used to scale their result. The measurement of ∆a B 0 s µ is an order of magnitude more precise than the one from the D0 collaboration [9] (note the different definition, ∆a ⊥ ≡ ∆a X 2 + ∆a Y 2 , in Ref. [9]). The measurement of z B 0 s is the first direct measurement of this quantity.