The signal of ill-defined CPT weakening entanglement in the $B_d$ system

In the presence of quantum gravity fluctuations (space-time foam), the CPT operator may be ill-defined. Its perturbative treatment leads to a modification of the Einstein-Podolsky-Rosen correlation of the neutral meson system by adding an Entanglement-weakening term of the wrong exchange symmetry, the $\omega$-effect. In the current paper we identify how to probe the complex $\omega$ in the entangled $B_d$-system using Flavour(f)-CP(g) eigenstate decay channels: the connection between the Intensities for the two time-ordered decays (f, g) and (g, f) is lost. Appropriate observables are constructed allowing independent experimental determinations of Re($\omega$) and Im($\omega$), disentangled from CPT violation in the evolution Hamiltonian Re($\theta$) and Im($\theta$). 2-$\sigma$ tensions for both Re($\theta$) and Im($\omega$) are shown to be uncorrelated.

violating space-time (or otherwise) backgrounds. In the latter case the quantum mechanical operator that implements CPT symmetry is well defined but simply does not commute with the hamiltonian. The ω-effect, if observed, points to an observation of a phenomenon that is exclusively linked to ill-defined nature of the CPT operator, which to date is theoretically linked only to fundamental decoherence [4], independently of any violation of CPT in the hamiltonian.
Recently, a study for separate direct evidence of T, CP, CPT symmetry violation was accomplished [8]. It was based on the precise identification of genuine asymmetry parameters in the time evolution of intensities between the two decays in a B-Factory of entangled neutral B d -meson states. Their values were obtained from the BaBar measurements [2] of the different Flavour-CP eigenstate decay channels. The concept, put forward in [9,10], uses the entangled character of the initial state as the crucial ingredient to (i) connect experimental double decay rates with specific meson transitions probabilities and (ii) identify the transformed transition to that taken as a reference [11,12]. Possible fake effects [8] were demonstrated to be well under control by measurements in the same experiment. The methodology, discussed in [9,10], appears to be [13] crucially dependent on the assumed maximal entanglement between B 0 d andB 0 d , or between two orthogonal superpositions of them, as given by the Einstein-Podolsky-Rosen (EPR) correlation [14] imposed by their decay from the Υ(4S)-state with C = -. The corresponding antisymmetric state of the system has two important implications: (i) the program of using Entanglement and the decays as filtering measurements to prepare and detect the meson states can be implemented at any time for the first decay, even in presence of mixing during the previous entangled evolution; (ii) the coefficients of the different time-dependent terms in the double decay rate intensities for the time-ordered decays to (g, f ) are related to those for the time-ordered decays to (f, g). The antisymmetry of the entangled state is kept for any two independent states of the neutral mesons, so its evolution leads to a trivial time dependence with definite symmetry under the combined exchange (f, t 0 ; g, t 0 + t) → (g, t 0 − t; f, t 0 ). As a consequence, the double decay rate intensity (see eq. (2.10) below) satisfies for the coefficients of its time dependence with ω = 0, where the time-ordered decays (f, g) and (g, f ) are, in general, not connected by any symmetry transformation. At this level, they can be considered as two different experimental ways of measuring the same quantity when ω = 0.
In the application to definite Flavour or CP eigenstates decay products, the preparation by maximal entanglement of the initial state of a single neutral meson is usually referred to as "flavour tagging" B 0 d ,B 0 d , or "CP tagging" B + , B − . The underlying assumption considers B 0 d ,B 0 d as two states of the same field, in order to impose Bose statistics with charge conjugation C and permutation P with CP = +, and it may be invalidated if the CPT operator cannot be intrinsically well defined, as mentioned above. This latter circumstance may occur, for example, in the context of an extended class of quantum gravity models, where the structure of quantum space time at Planckian scales (10 −35 m) may actually be fuzzy, characterised by a "foamy" nature (space-time foam) [5,15,16]. Let us emphasize once more that this kind of CPT breaking is different from an explicit CPT violation in the hamiltonian dynamics such that [CPT, H] = 0, as conventionally introduced, in the context of the Weisskopf-Wigner approach [17][18][19] for the neutral meson system, in the mass matrix. This last CPT violation does not invalidate the analysis followed in [8] and, in fact, genuine observables for CPT violation were found with their values obtained from experiment. However, the CPT breaking associated to "ill-defined" particleantiparticle states modifies the EPR correlation, producing the aforementioned ωeffect [5,20,21]. Treating it in perturbation theory, in such a way that we still talk the language of B 0 d ,B 0 d , the perturbed two-particle state will contain a component of the "wrong" symmetry at the instant of their production by the decay of Υ(4S): where ω = |ω|e iΩ is a complex CPT-breaking parameter [5,20], associated with the non-identical particle nature of the neutral meson and antimeson states. The presence of an ω-effect weakens the entanglement of the initial state (1.2), as follows from the fact that when ω = ±1 the state simply reduces to a product state, whilst when ω = 0 the state is fully entangled. We emphasize that the modification in eq. (1.2) is due to the loss of indistinguishability of B 0 d andB 0 d and not due to violation of symmetries in the production process. Evidently, the probabilities for the two states connected by a permutation are different due to the presence of ω. This modification of the initial state vector has far-reaching consequences for the concept of meson tagging and for the relation of the time dependent intensities between the decays to time-ordered (f, g) and (g, f ) channel 1 .
1 Another important aspect of the ω effect is its dynamical generation during a decoherence evolution, in which the particle interacts with its gravitational environment, for instance. As discussed in [22], a time-dependent contribution to the ω parameter may be generated in specific models of quantum decoherence, which could be present even if the initial state has an ω=0. The relative magnitude of Re (ω(t)) and Im (ω(t)) in this case depends crucially on the decoherence space-time foam model used, but their generic form involves oscillatory dependences on time. Specifically, if one ignores conventional CPT violating effects, then the analysis of [22] has shown that the evolution of an entangled two-particle state contains in certain quantum space-time-foam models time-dependent ω(t) parts, which to leading order in appropriate small quantities assume the form: that is purely generated by the evolution with no ω effect in the initial state t = 0. In the above formula, the superscripts (i), i = 1, 2 refer to individual particles, k is the momentum of the particle (assuming the decaying initial state to be at rest, for brevity), ∆E is the energy difference between In what follows we will study the non-trivial time evolution of eq. (1.2), in the simplified but physically relevant case of a time independent ω, in order to (i) establish the appearance of terms of the (previously forbidden) type |B 0 d |B 0 d and |B 0 d |B 0 d , and (ii) introduce a set of observables, which actually serve as a direct way for measuring ω, based on the violation of the relations eq. (1.1), i.e. using as observables for ω = 0: and checking experimentally the robustness of the correlation between the two states assumed during the tagging. This paper demonstrates that the comparison between the double decay rate Intensities for time-ordered (f = Flavour, g = CP) eigenstate decay products and (g, f ) is sensitive to both Re (ω) and Im (ω).
2 Time evolution

Double decay rates, time dependent intensities
The eigenstates of the effective hamiltonian H are 2 In terms of them where θ is a CP and CPT violating complex parameter given by θ = H 22 −H 11 µ H −µ L . The time evolution of two-meson flavour states is 3) the appropriate single particle states, and the arrows denote the corresponding quantum numbers of a generic two state system, while λ 0 are the energy eigenvalues. The parameter ω 0 in (1.3) is in general complex. In some concrete models of space-time quantum foam it could be purely imaginary [22]. In the present work we shall consider only constant ω in the initial state (1.2). We reserve details for the phenomenology of a time-dependent ω-effect, generated during the evolution, for a future publication. 2 As is commonplace, subindices "H" and "L" correspond to the heavy and light B d states. where In eqs. (2.6)-(2.8), q p is the usual meson mixing quantity given by q 2 Before addressing actual observables, it is worth noting that, attending to eq. (2.3), it is clear that the presence of the symmetric state |S in eq. (1.2) induces the appearance of |B 0 d |B 0 d and |B 0 d |B 0 d states. The transition amplitude for the decay of the first state into |f at time t 0 , and then the second state into |g at time t + t 0 is f, t 0 ; g, t + t 0 |T |Ψ 0 . Squaring and integrating over t 0 , the double decay rate I(f, g; t) is obtained: Expanding to first order in ω, θ and taking ∆Γ = 0, I(f, g; t) has the following form for generic f and g decay channels 3 : where, in terms of the decays amplitudes f |T |B 0 d ≡Ā f and f |T |B 0 d ≡ A f , the following parameters are used 4 : , In addition, x = ∆M Γ 0.77 and δ = 1−|q/p| 2 1+|q/p| 2 1 − 2 × 10 −3 . It is worth reminding that for flavour-specific decay channels X + ± (" ± " for short in the following), we have C ± = ±1, R ± = S ± = 0.

Sensitivity to ω
Coming back to the transition amplitude f, t 0 ; g, t + t 0 |T |Ψ 0 , it has the following structure: (2.16) The prefactor e (−iM −Γ/2)(2t 0 +t) gives a global e −2Γt 0 e −Γt dependence in | f, t 0 ; g, t + t 0 |T |Ψ 0 | 2 . One can readily observe that the ω-dependent terms, even for θ = 0 (i.e. already for the leading ω contribution), do introduce an additional non-trivial t 0 dependence. Ignoring that e (−iM −Γ/2)(2t 0 +t) prefactor, it is clear that combining the transformations t → −t and f g, the first contribution, the standard ω = 0 one, just receives a (−) sign. This implies that, in the absence of ω, in the t-dependence of I(f, g; t), we necessarily have [8]: In the presence of ω = 0 the situation changes drastically. From the remaining contributions in eq. (2.16), the ones induced by the evolution of the ω-dependent term in eq. (1.2), the situation is more involved: the first one, proportional to ωθ and t 0 -independent, is clearly invariant under the combination of f g and t → −t. The last two terms are separately invariant under f g, but have no well defined transformation under t → −t; moreover, contrary to the previous contributions, they depend on t 0 , the time elapsed between production of the BB pair and the first decay 5 . Out of those properties, the simple assignment of symmetry/antisymmetry under f g to the t-even/t-odd terms in e Γ t I(f, g; t), possible when ω = 0, does not apply when ω = 0. This simple remark provides the first understanding of the potential sensitivity to the presence of ω = 0: while in the absence of ω, the measurement of intensities for decays into f and g with the two different orderings (i) first f then g and (ii) first g then f , provides two experimentally independent measurements of the same theoretical quantities, in the presence of ω the situation has changed. Deviations from the standard f g symmetry properties are a gateway to probe for ω.
The BaBar collaboration performed separate analyses [2] for the two different time orderings of the two B meson decays. Previous studies, like [21], exploited the use of two flavour specific decay channels to obtain bounds on Re (ω) through the appearance of |B 0 d |B 0 d and |B 0 d |B 0 d states for t = 0. Equation (2.10) shows that, using flavour specific channels alone, there is no sensitivity to Im (ω): since R ± = 0, the terms in Im (ω) would be absent 6 . Fortunately enough, besides addressing the two different time orderings, in [2], one decay is flavour specific (labelled ± ), while the other is CP specific (decays into J/ΨK S,L , labelled K S,L for short): sensitivity to both Re (ω) and Im (ω) is thus expected.

Experimental observables
In order to reduce experimental uncertainties in the different channels, the BaBar collaboration, in reference [2], fixed the constant term and measured the coefficients C[f, g] and S[f, g] of the decay intensity using for the f and g states one flavour specific channel, X + ν or X −ν , and one CP eigenstate, J/ΨK S or J/ΨK L . Obviously we should have where one should remember that in the coefficients C[f, g] and S[f, g], the ordering of f and g means that f corresponds to the first (in time) decay product of the entangled state evolved in time, and g corresponds to the second (in time) decay product. In the case under consideration, the flavour specific decays simplify significantly the expressions, which are, at linear order in θ, ω, In the presence of ω, the time ordering definite symmetry is not valid anymore and therefore it is relevant to write the completely different coefficients As anticipated, C[ ± , g] − C[g, ± ] and S[ ± , g] + S[g, ± ] are linear in ω, and thus the fact that the BaBar collaboration distinguished the different decay time orderings in [2], now reveals crucial to disentangle the ω effect: These combinations are linearly sensitive not only to Re (ω) but also to Im (ω) when R g = 0. The sensitivity to Im (ω) depends critically on the use of a CP eigenstate channel with large R g , as is the case with J/ΨK S and J/ΨK L .

Results
We are now ready to present the results obtained from a global fit to available BaBar experimental data, following the same statistical treatment as in reference [8]. We use the sixteen experimental observables measured by BaBar in [2]: C[ ± , K S,L ], C[K S,L , ± ], S[ ± , K S,L ] and S[K S,L , ± ]. Taking into account full covariance information on statistical and systematic uncertainties, we perform a fit in terms of the set of parameters {Re (θ), Im (θ), Re (ω), Im (ω), Therefore we generalize the corresponding fit presented in reference [8] to the actual situation where deviations from EPR entanglement are present due to the ω-effect [5]. A more restricted fit is also done in the case where no wrong sign flavour decays are allowed in the B d → J/ΨK decays, that is with λ K S + λ K L = 0.
In table 1(I) we present the general result of the fit, whose most salient features are the following: • Experimental data -more precisely the BaBar measurements in [2] -are sensitive for the first time to Im (ω), revealing a tantalizing 2.4σ deviation from Im (ω) = 0. These observables are also sensitive to Re (ω), but they do not show any significant deviation from Re (ω) = 0, and the previous determination Re (ω) = (0.8 ± 4.6) × 10 −3 [21] -using semileptonic channels -is still better than the present one.
• The results of the fit for the CPT violating parameter θ -in the evolution hamiltonian -are compatible with the previous determination in [8] and the one performed by the BaBar collaboration in reference [23]. An exciting 2σ effect in Re (θ) is still present.
• The parameters that measure the presence of wrong flavour decays in B d → J/ΨK, i.e. C K S − C K L , S K S + S K L and R K S + R K L , do not show any significant deviation from zero and the results are consistent with [8].
• In the case of S K S and R K S we observe that they differ by more than 1σ with respect to the determination in [8] without including the ω effect. Should this persist in the future, it could affect the precise determination of the unitarity triangle angle β.
In table 1(II) we present the results of the same fit with the additional requirement of not having wrong flavour decays, λ K S + λ K L = 0. No significant differences were noticed with respect to the conclusions discussed above for the general case. For completeness we show, when relevant, both analyses together in the same plots without further comments.
(I) Parameters -General analysis Re (θ) ±(6.11 ± 3.45)10 −2 Im (θ) (0.99 ± 1.98)10 −2 Re (ω) (1.09 ± 1.60)10 −2 Im (ω) ±(6.40 ± 2.80)10 −2 S K S −0.624 ± 0.030 In figure 1 is shown the result for the new parameters not previously considered in the analyses where EPR entangled initial states where assumed. A deviation of the complex number ω from zero is found at 95% confidence level. This deviation comes essentially from Im (ω) and it represents a measurement of this parameter for the first time; the measurement of Re (ω) does not improve on the value obtained previously [21] from flavour specific decays. The stability of the fitted value of the complex CPT violating parameter θ is shown in figures 2(a) and 2(b), where it is clear that the results for Re (θ) and Im (θ) do not change from the constrained case ω = 0 to the general case with arbitrary ω. Cross correlations among the different components of θ and ω are shown in figure 3. For example, figure 3(c) shows the independence of Im (ω) and Re (θ): furthermore one can see in that figure that the point (0, 0) in this projection is at more than 2.5σ from the best fit values (or even at 3σ in the λ K S + λ K L = 0 constrained analysis). Finally, in figure 4, one can see the near linear correlation among Im (ω) and R K S . This explains why the presence of ω affects both R K S and S K S .

Conclusions
In the present article we have discussed the possibility of probing the entanglementweakening CPT Violating parameter ω, that potentially signifies the breakdown of CPT operation as a result of quantum decoherence of matter in some models of quantum gravity, by means of identifying appropriate asymmetry parameters in the time evolution of intensities (2.10) between the two decays in a B factory, based on observables that have already been used in previous studies [8] probing independently T, CP and CPT symmetries in the absence of ω. In the current analysis we have included, simultaneously with the ω, also the conventional CPT parameter θ, already considered in [8], which parameterises CPT violation in the case of a well-defined CPT operator which however does not commute with the hamiltonian of the system, indicating a violation of CPT parameterised within the framework of effective field theories (e.g. due to Lorentz symmetry violation by a space-time background), in contrast to the parameter ω that goes beyond that framework.
As we have demonstrated in the present article the set of observables of the B system (2.22), (2.23), (2.24) and (2.25) allow for a simultaneous determination (bounds) of the CPT violating parameters ω and θ, which can thus be disentangled. The results obtained from the experimental data from the BaBar measurements [2] (see table 1(I)) are sensitive for the first time to Im (ω), pointing towards a 2.4σ deviation from Im (ω) = 0, which we interpret as an upper bound. The observables (2.24),(2.25) are also sensitive to Re (ω), but they do not show any significant deviation from Re (ω) = 0, and in this sense they are inferior to the previous analyses [21] using equal sign semileptonic decay asymmetries of the B system, which yield Re (ω) = (0.8 ± 4.6) × 10 −3 . The results (2.22) and (2.23) also allow a fit to the CPT violating parameter θ, and are compatible with the previous determination in [8] and the one performed by the BaBar collaboration in [23], pointing towards a 2σ effect in Re (θ), also interpreted as an upper bound for the corresponding parameter.
Moreover, the parameters that measure the presence of wrong flavour decays in B d → J/ΨK, i.e. C K S −C K L , S K S +S K L and R K S +R K L , do not show any significant deviation from zero and the results are consistent with [8]. In the case of S K S and R K S we observe that they differ by more than 1σ with respect to the determination in [8] without including the ω effect. Should this persist in the future, it could affect the precise determination of the unitarity triangle angle β.
Before closing we stress once more that a quantum-gravity-decoherence-induced CPT violating and entanglement-weakening parameter ω may not only characterise the initial state of an entangled (neutral) meson system, but may also be generated as a result of a decoherening time evolution that goes beyond the local effective field theory framework [22]. A full analysis of that case will appear in a forthcoming publication.