Status and prospects for $CPT$ and Lorentz invariance violation searches in neutral meson mixing

An overview of current experimental bounds on $CPT$ violation in neutral meson mixing is given. New values for the $CPT$ asymmetry in the $B^0$ and $B_s^0$ systems are deduced from published BaBar, Belle and LHCb results. With dedicated analyses, LHCb will be able to further improve the bounds on $CPT$ violation in the $D^0$, $B^0$ and $B_s^0$ systems. Since $CPT$ violation implies violation of Lorentz invariance in an interacting local quantum field theory, the observed $CPT$ asymmetry will exhibit sidereal- and boost-dependent variations. Such $CPT$-violating and Lorentz-violating effects are accommodated in the framework of the Standard-Model Extension (SME). The large boost of the neutral mesons produced at LHCb results in a high sensitivity to the corresponding SME coefficients. For the $B^0$ and $B_s^0$ systems, using existing LHCb results, we determine with high precision the SME coefficients that are not varying with sidereal time. With a full sidereal analysis, LHCb will be able to improve the existing SME bounds in the $D^0$, $B^0$ and $B_s^0$ systems by up to two orders of magnitude.


Introduction
In the weak interaction of the Standard Model, the symmetries under transformations of charge conjugation (C), parity (P ), and time reversal (T ) are broken. Nevertheless, the combined CP T transformation is observed to be an exact fundamental symmetry of nature. From a theoretical perspective, CP T symmetry is required by any Lorentz-invariant, local quantum field theory. Many experimental searches for CP T violation have been performed over the last decades. Interferometry in the particle-antiparticle mixing of neutral mesons is a natural and very sensitive method to search to deviations from CP T invariance. Since most CP T tests have been performed with neutral kaons, progress can still be made in the D 0 , B 0 and B 0 s systems. As CP T violation implies Lorentz violation [1], any CP T -violating observable must also break Lorentz invariance. In the framework of the Standard Model Extension [2,3] (SME), spontaneous CP T violation and Lorentz invariance violation appear in a low-energy effective field theory. In this sense, small CP T -violating effects at low energies provide a window to the quantum gravity scale [4]. Such effects are expected to be suppressed by m 2 /M Pl , with M Pl ≈ 10 19 GeV the Planck mass and m the relevant low-energy mass, which depends on the underlying unified theory and possibly ranges from the mass of the neutral meson to the electroweak mass [5]. The Lorentz violation introduces a boost-and direction-dependent variation in the CP T -violating parameters. From an experimental point of view, the direction dependence results in a modulation with the sidereal phase. Such modulations would provide an unambiguous signature of CP T violation.
We will show that a high sensitivity to these effects can be obtained by exploiting the large sample of heavy flavour decays obtained at the LHCb experiment, in particular taking advantage of the forward boost of the neutral mesons. Using published LHCb results, corresponding to a luminosity of 1 fb −1 , we can already deduce improved constraints on the SME parameters. Based on naive extrapolations, further improvements are possible with dedicated analyses on the existing 3 fb −1 data set.

Formalism
The time evolution of a neutral meson system, P 0 -P 0 , is governed by an effective 2 × 2 Hamiltonian H = M − iΓ/2. Following the notation in Ref. [6], we write the light and heavy mass eigenstates, with eigenvalues m L,H − iΓ L,H /2, as where p and q spawn the eigenvectors under CP T symmetry and z is the complex, CP T -violating parameter. The mixing parameters are defined as ∆m ≡ m H − m L and ∆Γ ≡ Γ H − Γ L , and the average mass and decay rate as m ≡ (M 11 + M 22 )/2 and Γ ≡ (Γ 11 + Γ 22 )/2. This definition implies that ∆Γ < 0 for the K 0 , B 0 and B 0 s systems, and ∆Γ > 0 for the D 0 system in the Standard Model. The CP T -violating parameter in P 0 -P 0 mixing can be written independent of phase convention as [7] where δm ≡ (M 11 − M 22 ) and δΓ ≡ (Γ 11 − Γ 22 ) are the differences of the diagonal mass and decay rate matrix elements of H. This equation makes clear that z is sensitive to small values of δm or δΓ due to the smallness of ∆m and ∆Γ in neutral meson systems. By measuring the time-dependent decay rates of an initial P 0 or P 0 state to a final state f or f , information on z can be obtained. For simplicity, we only consider CP T violation in P 0 -P 0 mixing. Direct CP T violation is experimentally difficult to separate from direct CP -violating effects. In both cases, it causes a difference in the instantaneous decay amplitudes, i.e., is the direct decay amplitude of a P 0 (P 0 ) meson to a final state f or f . In the following, any direct CP -violating term implicitly includes possible direct CP T violation. For a complete expression of the decay rates we refer to Ref. [6]. Although those equations apply to the more general case of coherent production of B 0 -B 0 pairs, we will ignore this additional complication here and assume incoherent production by setting the amplitude of the tagging particle to either zero or one. It is instructive to construct an observable CP T asymmetry where P f (P f ) is the time-dependent decay probability of an initial P 0 (P 0 ) meson to a final state f (f ). For decays to pure flavour-specific final states (i.e., A f = A f = 0), this asymmetry can be written as where the direct CP asymmetry is assumed to be small. On the other hand, the CP asymmetry, defined as and the CP T asymmetry become equivalent for decays to CP eigenstates f = f , and their effects become automatically connected. The CP T or CP asymmetry can be written as where The parameter λ f = (q/p)(A f /A f ) is introduced for convenience, and A mix = (1 − |q/p| 4 )/(1 + |q/p| 4 ) describes CP violation in mixing only. In the absence of CP violation in mixing (i.e., |q/p| = 1), C f is equivalent to A dir . Only leading-order terms in λ f and z are retained in Eq. 6. Comparing Eqs. 4 and 6, it becomes apparent that flavour-specific final states and CP eigenstates have different, complementary sensitivities to Re(z) and Im(z). We will come back to this point later.
Up to now we have assumed that z is a constant of nature for each of the four neutral meson systems. We will refer to this assumption as the classical approach. In the SME Lagrangian, CP T -violating and Lorentz-violating terms are introduced for the fermions with coupling coefficients a µ [8]. The observable effect is determined by the contributions from the two valence quarks, q 1 and q 2 , in a meson as ∆a µ a q 1 µ − a q 2 µ , hereby ignoring small effects from binding and normalization. In the SME approach, the equations above remain valid, but now z depends on the four-velocity β µ = γ(1, β) of the neutral meson as An overview of experimental bounds on ∆a µ and other SME parameters is given in Ref. [9].
In the SME, ∆a µ is required to be real [10], which implies δΓ = 0. The real and imaginary parts of z then become connected through As we will see later, this constraint has implications for CP T violation searches within the SME framework.
In such a search, the four-velocity of the neutral mesons at any time needs to be determined with respect to fixed stars. A useful reference frame is the Sun-centred frame defined in Ref. [10]. In this frame, the Z-axis is directed North, following the rotation axis of Earth, the X-axis points away from Sun towards the vernal equinox and the Y -axis complements the right-handed coordinate system. For an experiment where the neutral mesons are produced in a horizontal direction, fixed with respect to the Earth's coordinate system, the dependence on the four-velocity can be written as where Ω is the sidereal frequency and cos χ = cos θ cos λ with θ the azimuth of the neutral mesons and λ the latitude. The time coordinatet is chosen such that the boost direction aligns with the X-axis att = 0. We have used the same convention as in Ref. [10], where the spatial coordinates of the ∆a µ field are defined such that ∆a X,Y,Z = −∆a X,Y,Z . Equation 9 makes clear that z not only depends on the size of the boost, but also that it has a constant component, independent of the sidereal phase, and a component that exhibits a sidereal modulation. The sidereal variation is largest when the experiment is oriented east-west or when it is close to the North Pole. For the LHCb experiment, we determine the latitude λ = 46.24 • N and azimuth θ = 236.3 • east of north, which gives cos χ = −0.38 and sin χ = 0.92. This means that the constant component scales with (∆a 0 − 0.38∆a Z ) and that the sidereal variation at LHCb is close to maximal.

Experimental results and potential measurements
In the following, we present an overview of experimental searches for CP T violation in the four neutral meson systems. We interpret published results that are sensitive to CP T violation. These new values are summarized in Table 1 and discussed in the following. We also include prospects for analyses that can be conducted with current data from the LHCb experiment. The expected sensitivities on the CP T -violating parameters with the existing 3 fb −1 data set are given in Table 2.

Neutral kaons
In the neutral kaon system, there are many experimental searches for CP T violation. Most of them have been performed within the classical framework, i.e., assuming z to be constant.
A search for sidereal variations in the SME framework has been performed at the KLOE  System Parameter Current best value LHCb 3 fb −1 Decay mode [19]. The kaons are produced from the φ resonance, which has a small boost of βγ 0.015, and detected in the π + π − final state. Limits on all four SME parameters are reported with uncertainties on ∆a µ of about 2 × 10 −18 GeV. 1 Another search for sidereal variations has been performed using KTeV data in Ref. [20]. Due to the high boost of the uncorrelated kaons (βγ ≈ 100), strong limits on the sidereal-phase-dependent SME parameters have been set to ∆a X,Y < 9.2 × 10 −22 GeV at 90% confidence level (CL).
Kaons produced at the E773 experiment are also highly boosted (βγ 100). Using E773 results, a bound on the constant SME parameters has been determined in Refs. [8,16] to |∆a 0 − 0.6∆a Z | 5 × 10 −21 GeV. Even though cross sections for kaon and φ production are high at the LHC, it will be difficult for LHCb to compete with the dedicated kaon experiments due to the limited decay time acceptance (roughly up to one K 0 S lifetime), lower boost and larger backgrounds.

Neutral charm
Only the FOCUS collaboration has reported limits on CP T violation in D 0 mixing [17]. About 35k Cabibbo-favoured D 0 → K − π + decays 2 have been analysed, both in the classical and SME approach. This final state is not a pure flavour-specific eigenstate, since there is also a small contribution from doubly Cabibbo-suppressed D 0 → K + π − decays. Due to the small mixing in the D 0 system [12], the CP T asymmetry can be approximated to first order as where x ≡ ∆m/Γ, y ≡ ∆Γ/2Γ, R D = (0.349±0.004)% [12] is the decay rate ratio of doubly Cabibbo-suppressed over Cabibbo-favoured decays and φ and δ are the corresponding weak and strong phases. The second term, the contribution from CP violation, is maximally of O(10 −4 ) [12] and is neglected in the FOCUS analysis. In their classical analysis, a value of Re(z)y − Im(z)x = (0.83 ± 0.77)% is reported. Assuming x ≈ y ≈ 0.5%, this measurement provides only a weak bound on Re(z) − Im(z) of O(1). At LHCb, many more D 0 → K − π + decays are available. In the current 3 fb −1 data sample, more than 50M Cabibbo-favoured decays have been observed [21], which means a possible improvement of the FOCUS measurement by a factor of about 40 and a precision on Re(z)y − Im(z)x of 0.02%. At this precision, the CP -violating term cannot be ignored anymore and needs to be taken into account in the analysis. In addition, the observed CP T asymmetry will include effects from production and detection asymmetries. Fortunately, these effects are expected to be independent of the D 0 decay time, adding to the constant contribution from direct CP violation, A dir .
The same FOCUS paper [17] also presents a full sidereal analysis in the SME framework. The average boost of the D 0 mesons is βγ ≈ 39. Due to the SME constraint, the CP Tviolating term in Eq. 10 is zero and a further expansion in x and y is required, which 1 Natural units are used with c = 1. 2 The inclusion of charge-conjugated decay modes is implicit.
reduces the sensitivity to Re(z). The expansion to second and third order in decay time gives where A dir and all CP -violating terms are omitted. Assuming again x ≈ y ≈ 0.5%, the uncertainties on the ∆a µ parameters are found to be about 3 × 10 −13 GeV. At LHCb, with their large sample of D 0 → K − π + decays and assuming a comparable boost factor, it should be possible to improve the FOCUS bounds by a factor 40. Note, however, that it will not be possible to constrain Re(z) to be smaller than one, since Re(z) is suppressed in Eq. 11 by O(10 −6 ). Nevertheless, no assumptions on the smallness of |z| have been made so far. Extrapolating to the statistically larger sample, LHCb should be able to reach a sensitivity on the ∆a µ parameters of about 1 × 10 −14 GeV.

B 0 mesons
Due to the small value of ∆Γ d in the B 0 system, decays to flavour-specific final states are sensitive to Im(z), while decays to CP eigenstates are sensitive to Re(z) (cf. Eqs. 4 and 6). This is a key point that is used below for B 0 decays, but it is also valid for B 0 s decays. Using only dilepton (i.e., flavour-specific) final states, the BaBar collaboration published Im(z) = (−1.39 ± 0.80)% [22]. Similarly, the Belle collaboration reported Im(z) = (−0.57 ± 0.47)%, mainly using flavour-specific final states [11]. The average value of both results is Im(z) = (−0.8±0.4)% [18]. Using the same dilepton final states, the BaBar collaboration also reported a measurement of Re(z)∆Γ d = (−7.1 ± 4.4) × 10 −3 ps −1 [22]. When inserting the theoretical expectation of ∆Γ d ≈ −(6.8 ± 1.6) × 10 −3 ps −1 [23], this measurement gives only a weak constraint on Re(z) of O(1). Since |z| 2 terms have been ignored in this analysis, this means that a higher sensitivity to Re(z) could have been achieved when including |z| 2 terms in the fits to the decay rates. Due to the relatively low tagging performance in a hadron-collider environment, the untagged asymmetry for flavour-specific decays, defined as gives a higher sensitivity to Im(z) than the tagged asymmetry as defined in Eq. 3. Including experimental effects from a possible detection asymmetry A D and from a production asymmetry A P , the observed asymmetry becomes whereby |z| 2 terms have been ignored and ∆Γ d is approximated to be zero. Compared to Eq. 4, the sensitivity to Im(z) is only reduced by a factor 2, rather than a reduction by a factor 20 − 30, which is the typical loss due to the flavour tagging in a hadronic environment. In Eq. 13, A mix is the flavour-specific CP asymmetry in B 0 mixing. At LHCb, using inclusive B 0 → D ( * )− µ + ν µ decays, a high-precision measurement of Im(z) is possible, since the dilution of the amplitude of the oscillation due to the partial reconstruction is small [24]. We estimate about 3 million inclusive B 0 → D ( * )− µ + ν µ decays in the 3 fb −1 data set, using the observed yields in B 0 s → D − s µ + ν µ decays [25] and the production ratio of B 0 and B 0 s mesons [26]. Hence, a statistical precision on Im(z) of 0.1% is in reach. In B 0 decays to CP final states, Re(z) appears in the cosine term of the time-dependent CP asymmetry. Neglecting CP violation in mixing (i.e., A mix = 0), and setting Γ d = 0 and Im(z) = 0, the time-dependent (tagged) asymmetry from Eq. 6 becomes Effects from Im(z) are expected to be negligible and this assumption can be tested with experimental input from flavour-specific decay modes as described above. Similarly, A mix is also negligible at the current experimental precision [12]. Direct CP violation (C f ) and Re(z) both contribute to the cosine term. In principle, the time-independent offset is also sensitive to Re(z), however, this offset is additionally affected by production, detection and tagging asymmetries. Hence, in practice most information on Re(z) will come from the oscillating term. For B 0 decays to the CP final state J/ψ K 0 S , we can identify C f = 0, D f = cos 2β and S f = sin 2β, where β is the usual CKM parameter. We ignored for simplicity small effects coming from CP violation in kaon and B 0 mixing and direct CP violation due to the penguin contributions. The contribution from direct CP violation gives the dominant uncertainty on C f . Theoretically, it is estimated to be at most a few times 10 −3 [27]. Experimentally, the direct CP asymmetry in B + → J/ψ K + decays is (0.1 ± 0.7)% [18], which is expected to be largely equal to that in B 0 → J/ψ K 0 S decays using isospin symmetry [28]. Another experimental constraint comes from the B 0 → J/ψ π 0 decay, which can be used to determine the direct CP violation in B 0 → J/ψ K 0 S to be (1 ± 1)% [29]. The Belle collaboration has measured Re(z) = (1.9 ± 5.0)%, where the sensitivity mainly comes from B 0 → J/ψ K 0 S,L decays [11]. Similarly, the BaBar collaboration has measured with a small fraction of the data Re(z)Re(λ)/|λ| = (1.4 ± 4.9)% [6]. With Re(λ)/|λ| ≈ 0.72, this measurement translates to Re(z) = (1.9 ± 6.8)%. This result was left unnoticed in the PDG world average of Re(z) [18]. Averaging here both numbers, we find Re(z) = (1.9 ± 4.0)%. Both results neglect the possible contribution from direct CP violation. A more recent and accurate value on Re(z) can actually be obtained using the world average on the cosine coefficient of (0.5 ± 1.7)% [12]. With Re(λ)/|λ| ≈ 0.72 and setting C f = 0, this results in Re(z) = (0.7 ± 2.4)%.
Finally, we briefly mention the BaBar analysis [30] where the CP , CP T and T asymmetries are tested separately. For instance, CP T asymmetries for B 0 mixing are constructed by simultaneously interchanging the time ordering of initial B 0 and B 0 decays and substituting K 0 L and K 0 S states. Although this method is statistically not competitive, it does allow to cleanly separate effects from CP , T and CP T violation. Unfortunately, such tests are only possible at the Υ (4S) experiments, and not at hadron collider experiments where the B 0 mesons are produced incoherently and the reconstruction of K 0 L mesons is much more challenging.
In the SME framework, due to the constraint Re(z)∆Γ d ≈ 2Im(z)∆m, the real part of z is about 150 times larger than the imaginary part. Without loss of generality, we used here the theoretical expectation value of ∆Γ d from Ref. [23]; experimentally 2∆m/∆Γ is already bounded to be larger than 30 at 95% CL [18]. Therefore, B 0 decays to CP eigenmodes are more sensitive to the ∆a µ variables than B 0 decays to flavour-specific modes. The BaBar collaboration published a paper [13] where the ∆a µ parameters are determined in a full sidereal analysis. The boost of the B 0 mesons is βγ = 0.55. They used only dilepton events, rather than CP eigenmodes. Using the expected ∆Γ d value [23], the uncertainties on the ∆a µ parameters are ∼ (3 − 13) × 10 −13 GeV, corresponding to uncertainties on Re(z) of about one. Just as in their classical analysis [22], this means that a higher sensitivity to ∆a µ would have been possible in case |z| 2 terms are not ignored.
When using the location and orientation of the BaBar and Belle experiments and their measurements of Re(z), stronger constraints on the constant ∆a µ term can be set, however, at this point we focus on LHCb where an even higher precision can be reached due to the larger boost of the B 0 mesons. The average momentum of b hadrons at LHCb is p ≈ 80 GeV [31], corresponding to a relativistic boost of βγ ≈ 15. The LHCb collaboration reported a value of C f = (3 ± 9)% using B 0 → J/ψ K 0 S decays in the 1 fb −1 data set [14]. This corresponds to Re(z) = (4 ± 12)%. Using the LHCb beam direction, a measurement of the constant combination of SME parameters of (∆a 0 − 0.38∆a Z ) = (0.9 ± 2.8) × 10 −15 GeV is obtained. This result improves the current best value [13] by two orders of magnitude. By making use of the B 0 momentum in each event and with a full sidereal analysis on the 3 fb −1 data set, it should be possible to further decrease the uncertainties on ∆a µ to about 1 × 10 15 GeV.

B 0 s mesons
The discussion for the B 0 s system is very similar to that for the B 0 system. In this system ∆Γ s is not anymore negligible, but still small enough such that flavour-specific final states primarily give access to Im(z), while CP eigenmodes give access to Re(z). No dedicated CP T measurements have been done with B 0 s mesons to date. In the classical approach, LHCb would be able to measure Im(z) using the flavour-specific B 0 s → D − s π + decays. In the 3 fb −1 data set, N = 100k untagged signal decays can be expected [32]. Following Eq. 13, this corresponds to an uncertainty on Im(z) of 2/N = 0.4% (see Table 2). Alternatively, the more abundant inclusive B 0 s → D − s µ + ν µ decays can be used to measure Im(z). Due to the partial reconstruction, however, the worse time resolution washes out the oscillations already after a 1 ps (see Ref. [24]), reducing the sensitivity.
Constraints on Re(z) can be made using B 0 s decays to the CP eigenstate J/ψ φ. This decay mode is the B 0 s equivalent of B 0 → J/ψ K 0 S . Equation 6 gives the observable asymmetry. The phase arg(λ f ) = φ s is expected [33] and experimentally measured [12] to be small, leading to D f ≈ 1 and S f ≈ 0. Any effect from A mix can be ignored at the current level of precision [12]. The LHCb collaboration has published a value of |λ f | = 0.94 ± 0.04 using the 1 fb −1 data set [15]. Ignoring again direct CP violation, a first evaluation of Re(z) ≈ (1 − |λ f | 2 )/2 = (6 ± 4)% can be made in the B 0 s system.
In the B 0 s system, the SME constraint Re(z)∆Γ s ≈ 2∆m s Im(z) leads to a Re(z) that is a factor 450 larger than Im(z). Even more than for the B 0 system, this means that one should focus on decays to CP final states, such as B 0 s → J/ψ φ. An interesting relation between the K 0 , B 0 and B 0 s systems is pointed out in Ref. [16]. As the expectation value of ∆a µ is dominated by the valence quarks, a sum rule relating these three neutral meson systems can be written as Since the constraints on ∆a K µ are most strong and compatible with zero, this sum rule implies that possible CP T -violating effects in the B 0 and B 0 s system should be of the same order. In that sense, the B 0 system is most interesting, since the production rate of B 0 mesons is higher and the mass difference ∆m d is smaller (c.f. Eq. 7). Ideally, the mass difference should be such that one could just measure one period of oscillation, which is the case for B 0 oscillations. Nevertheless, it remains important to measure possible CP T violation in all possible systems to verify this sum rule.
Using the like-sign dimuon asymmetry measured in the D0 data, a value for ∆a µ has been derived in Ref. [16]. Assuming that the only source of CP T violation comes from B 0 s decays (like-sign dimuons originate from both B 0 and B 0 s mixing) and using the average boost of βγ = 4.1, the constant ∆a µ term becomes (3.7 ± 3.8) × 10 −12 GeV. This corresponds to Re(z) = 1.0±0.8. Stronger limits on Re(z) can be set with the CP eigenmode decay B 0 s → J/ψ φ. Using again Re(z) = (6 ± 4)%, derived from LHCb results [15], and taking as average boost βγ ≈ 15, we find (∆a 0 − 0.38∆a Z ) = (5 ± 3) × 10 −14 GeV, which is an improvement by two orders of magnitude. With the existing LHCb data set and a dedicated sidereal analysis, it should be possible to reach a sensitivity of about 1 × 10 −14 GeV or below.

Conclusion
We have presented new results on CP T violation in B 0 and B 0 s mixing in both the classical and SME approach, derived from published BaBar, Belle and LHCb results. In both approaches there is a significant improvement over previous results (see Table 1). LHCb should be able to further improve these numbers in the B 0 and B 0 s systems, as well as in the D 0 system, with dedicated analyses on the existing 3 fb −1 data set (see Table 2). In most cases these possible LHCb measurements would improve the current best values by orders of magnitude and the corresponding precision on ∆a µ is approaching the interesting region of m 2 /M Pl . Even further improvements can be expected with the LHCb data from run II and in particular after the upgrade. thank Marcel Merk for his feedback on the paper draft, Robert Fleischer for his comments on the possible contribution from direct CP violation, and Patrick Koppenburg and Guy Wilkinson for useful suggestions on the text. This work is supported by the Netherlands Organisation for Scientific Research (NWO) and the Foundation for Fundamental Research on Matter (FOM).