{'fraction_non_alphanumeric': 0.04565493287021575, 'fraction_numerical': 0.012691202992452074, 'mean_word_length': 3.9880059970014994, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 13, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Disentangling violations of CPT from other new-physics effects

We analyze the prospects for observing CPT violation using neutral-meson P0-P0bar systems. Before one can claim a measurement of CPT violation, one must be able to rule out the possibility that its result is due to simpler new-physics effects. In particular, one must be able to separate the CPT-violating quantities from parameters violating the Delta P = \Delta Q rule of semileptonic decays, and from new-physics contributions to the production mechanism of the neutral mesons. One can isolate CPT violation using the semileptonic decays of single, tagged neutral mesons; unfortunately, this situation cannot be implemented at the Upsilon(4S). For P0-P0bar pairs produced in a correlated parity-odd state we show that, by combining the di-lepton with the single-lepton decays, it is in principle possible to extract unambiguously one CPT-violating parameter. Finally, we develop the formalism necessary for describing new-physics effects in the production mechanism; this includes both cascade decays and violations of the rule of associated production.


Introduction
The "CPT theorem" states that any local field theory with hermitian, Lorentzinvariant interactions obeying the spin-statistics connection is necessarily CPTinvariant [1]. Although the assumptions of this theorem-and thus the validity of its conclusion-are generally taken for granted, the question of whether CPT is violated or not should ultimately be settled through accurate, high-precision experiments.
The most elementary consequences of CPT invariance are the equal values of the masses and lifetimes, and the symmetrical values of the magnetic moments, of a particle and its antiparticle. Unfortunately, in these cases the prospective CPT violation is expected to be a small perturbation in quantities which are dominated by much stronger interactions. For instance, the difference between the masses of K + and K − has an experimental bound [2] |m K + − m K − | / (m K + + m K − ) < ∼ 10 −4 ; however, this bound is not very meaningful, since m K + and m K − are dominated by the strong interaction, while one would expect CPT violation to be at best of miliweak or even superweak strength.
In contrast, the mixing between a neutral meson P 0 and its antiparticle P 0 (here P 0 may be either K 0 , D 0 , B 0 d , or B 0 s ) is a second-order electroweak effect. The smallness of the mixing makes it an ideal setting to look for small violations of the symmetries CP, T, and CPT. In fact, CP violation was first established [3] in the K 0 -K 0 system, and has thus far eluded experimental detection in any other system. Recently, the CPLEAR Collaboration has presented the results of its search for T violation [4] and CPT violation [5] in K 0 -K 0 mixing, and the OPAL Collaboration has looked for CPT-violating effects in the B 0 d -B 0 d system [6]. Detailed experiments on the K 0 -K 0 and B 0 d -B 0 d systems are planned at the φ-and Υ(4S)-factories, respectively.
Most of these experiments involve one or both of the following crucial steps: 1) determining the flavor of the initial meson (this procedure is called "tagging"); 2) determining the flavor of the meson at decay time, which is usually done by looking for semileptonic final states. When one is searching for CPT violation one must face the possibility that both steps are affected by new physics; one must make sure that what is assumed to be a measurement of CPT violation is not, in reality, a measurement of a much less revolutionary new-physics effect.
The semileptonic decays of the mesons obey, in the standard model (SM) and to first order in the electroweak interaction, the ∆P = ∆Q rule (Q is the hadrons' charge, and P the flavor quantum number of the heaviest quark in the decaying meson). That is, in the SM, the decays P 0 → X − l + ν l and P 0 → X + l −ν l are allowed, while the decays P 0 → X + l −ν l and P 0 → X − l + ν l (which have ∆P = −∆Q) are only possible at second order in the electroweak interaction. (Here, X ± denotes a pair of arbitrary CP-conjugate hadronic states.) It is usual to define the parameters Their phases are not rephasing-invariant and, therefore, they are physically meaningless. On the other hand, the magnitudes of x l and ofx l are physically meaningful, and they are generally assumed to be small. If x l andx l do not vanish, then they will obscure the identification of CPT violation. The tagging strategies used in most experiments assume that, for any given event, there is a clear signal of the initial flavor of the neutral meson. The basic idea is rooted in the fact that the interactions of the gluon, photon, and Z 0 are flavor-conserving. For example, in the CPLEAR experiment [4,5] the neutral kaons are produced by the strong interaction through the reactions pp → K − π + K 0 and pp → K + π − K 0 ; the sign of the charge of the charged kaon identifies the initial flavor of the associated neutral meson. This is known as the rule of associated production. Another example occurs at the Z 0 pole, when B 0 is produced in association with B − and a set of particles with total charge +1. Detecting the B − , either by reconstruction or through its subsequent semileptonic decay-assuming the ∆B = ∆Q rule-one tags the initial flavor of the neutral meson. This was the strategy followed by the OPAL Collaboration [6]. In these cases, the production of the "wrong" neutral meson would mean the existence of a |∆P | = 2 interaction. However, given that such an interaction would also contribute to P 0 -P 0 mixing, its contribution to the production process is usually assumed to be negligibly small. This is no longer the case when the production process is due to the |∆P | = 1 interaction of the W boson, such as in cascade decays. For example, one may wish to study the K 0 -K 0 system in the decay chain B 0 d → J/ψK 0 → J/ψf . In the SM, the analysis of this decay [7] is based on the fact that the decay B 0 d → J/ψK 0 does not exist at tree level: there are ∆B = −∆S transitions, but no ∆B = ∆S transitions. However, new-physics effects might alter this situation. One must be able to rule out such effects before these processes can be used to look for violations of CPT. Similarly [8], one has access to the D 0 -D 0 system through the decay chain Here the situation is more complicated because both amplitudes B − → K − D 0 and B − → K − D 0 exist, even within the SM [8,9].
Measurements of CPT violation in tagged decays of neutral kaons have been discussed in the literature, sometimes including the possibility that the ∆S = ∆Q rule is violated [10,11,12,13,14]. The subject has resurfaced in recent analyses [15,16] of the claim of an observation of T violation by the CPLEAR Collaboration [4]. The possibility of a measurement of CPT-violating effects in the regeneration of neutral kaons has also been discussed [14,17].
Measurements of CPT violation at the Υ(4S)-factories have been considered either assuming the ∆B = ∆Q rule [18,19,20], or making some simplifying assumptions about the nature of the ∆B = −∆Q amplitudes [21]. Conversely, the experimental search for ∆S = −∆Q amplitudes in neutral-kaon decays [22], and the theoretical discussion of a search for ∆B = −∆Q amplitudes at the Υ(4S)-factories [23], have been made assuming CPT invariance. Xing [24] has recently shown that it is impossible to disentangle the violation of CPT from ∆B = −∆Q transitions by using exclusively di-lepton decays of the Υ(4S); we confirm his result here.
In this article we present a study of the measurements of CPT violation enabled by semileptonic decays, and of their impediment due to new-physics effects. We consider for the first time the impact of mis-taggings in the production process, and we relate our results to Xing's conclusion on the impossibility to disentangle CPT-violating amplitudes from ∆B = −∆Q amplitudes in dilepton events at the Υ(4S). However, we also consider single-lepton events, and we show that that separation is in principle possible through a combination of single-and di-lepton events. We stress that the situation described by tagged decays cannot be implemented at the Υ(4S), if we allow for violations of the ∆P = ∆Q rule.
We define our notation in section 2. In section 3 we discuss decays in which one has tagged the initial flavor of the neutral meson through the rule of associated production. In section 4 we focus on neutral-meson pairs produced in a parity-odd state, such as the K 0 K 0 pairs from the decay of the φ resonance, and the B 0 d B 0 d pairs produced at the Υ(4S). In section 5 we show that the presence of new physics in the production process will impede the extraction of the CPT-violating parameters. We draw our conclusions in section 6.

Mixing formalism
We start by discussing the mixing of P 0 and P 0 in the Wigner-Weisskopf formalism, when CPT violation is allowed for. We do this in order to introduce a convenient parametrization of the violation of T and CPT in the mixing, and to establish our notation. All equations in this section are exact, and we are careful to identify the reparametrization-invariant quantities; only those quantities are physically meaningful.
The time evolution of is given by The eigenvalues of R are denoted µ a and µ b . Their sum is given by the trace of R: The right-eigenvectors of R corresponding to the eigenvalues µ a and µ b are (p a , q a ) T and (p b , −q b ) T , respectively: Therefore, Equations (4) and (6) imply In order to avoid using the three non-independent quantities q a /p a , q b /p b , and θ, it is convenient to introduce Notice that we do not define the quantities q and p separately; we only define the ratio q/p. From Eqs. (7) and (8) it follows that The CPT-violating parameter θ will later be assumed to be small. We shall then make the approximation √ 1 − θ 2 ≈ 1. It follows from Eqs. (2), (3), and (5) that the states We do not have to make any assumption about the normalization of |P a and of |P b . We also do not have to make any assumption either about the relative phase of |P a and |P b , or about the relative phase of |P 0 and |P 0 . Indeed, one is free to change the phase of the kets |P 0 and |P 0 : The invariance of the state vector |ψ(t) under this rephasing implies that Therefore, from Eq. (3), while R 11 and R 22 do not change. The trace and the determinant of R are invariant under the transformation in Eqs. (14). Therefore, µ a and µ b are invariant too. Thus, θ is invariant under a rephasing of |P 0 and |P 0 . Both the real and the imaginary parts of θ are physically meaningful. They violate CP and CPT. On the contrary, the phase of the parameter q/p in Eq. (8) is not invariant under the rephasing in Eqs. (14); as a result, it is physically meaningless. However, the modulus of q/p is physically meaningful; the real parameter violates CP and T. In summary, the P 0 -P 0 mass matrix has two CP-and CPT-violating parameters (Re θ and Im θ), and one CP-and T-violating parameter (δ). In addition, it has four C-, P-and T-invariant quantities: These are sometimes traded for and 6 3 Tagged decays Let us consider a neutral meson which is identified as a P 0 (P 0 ) at time t = 0. Using Eqs. (7)- (11), one finds that this state is given at time t by respectively. Here, We now seek the quantities which can be measured when |P 0 (t) and |P 0 (t) decay into a final state f . We define Equations (19) depend on two independent functions of the decay time, g + (t) and g − (t). Therefore, one will in principle be able to measure the ratio of the coefficients of the two functions, We cannot compare the normalization of the decay rates corresponding to different final states unless simplifying assumptions are made. For example, some authors assume that there are no electromagnetic final-state interactions, or that CPT violation is absent in the decay process, or that there is no T violation in the mixing of the neutral mesons. We would argue that all these effects must be considered when looking for CPT violation, which is in itself dramatically non-standard. We stress that the observables in Eqs. (22) contain the maximal information that may be extracted from the time dependence of the decay rate. It is possible that particular phenomenological or experimental conditions only allow the extraction of part of this information from the actual decay curves. In order to see this, consider the explicit decay rates: Since the functions |g , and Im g * + (t)g − (t) are linearly independent, one can measure the quantities in Eqs. (22) by tracing the time dependence of the decays of single, tagged P 0 or P 0 . However, if for instance the two eigenstates have equal decay widths, i.e., if Γ a = Γ b , then the function Re g * + (t)g − (t) will vanish and less information will be available. Thus, in the ensuing discussions we address the best possible scenario. Actual experiments may be considerably more problematic.

Decays into semileptonic final states
Consider the particular case of the semileptonic decays. The parameters x l andx l were defined in Eqs. (1). We shall not need to assume any relationship between x l andx l ; in particular, we shall not assume CPT invariance of the decay amplitudes. Now, x l andx l are not invariant under the rephasing of |P 0 and |P 0 in Eqs. (12). The rephasing-invariant, physically meaningful quantities are They will be assumed to be small. If one observes the tagged decays to the semileptonic state X − l + ν l , one can in principle measure the corresponding parameters E andĒ, namely, If one observes the tagged decays to X + l −ν l , one measures In both Eqs. (25) and (26) we have made the approximation of neglecting the products of any two small parameters like θ, λ l , orλ l . One sees that, by using the decays of single tagged mesons, one can in principle separate the CPTviolating parameter θ from the ∆P = −∆Q parameters λ l andλ l . Thus, one can measure CPT violation with tagged decays. Unfortunately, as will be shown in section 5.1, this strategy cannot be implemented at the Υ(4S).

About the searches for ∆P = −∆Q amplitudes
The primary aim of this article is to stress the impact that simple new-physics effects may have on experiments seeking to measure violations of CPT. We now want to point out that the converse is also true. In particular, the experiments performed in the 70's in order to measure violations of the ∆S = ∆Q rule in the semileptonic decays of the neutral kaons have disregarded the possibility of CPT violation. Let us take P 0 to be K 0 and P 0 to be K 0 . For the subscripts which label the eigenstates of propagation we use a → L and b → S, referring to the long-lived and to the short-lived neutral kaon, respectively. Denoting one easily finds These expressions should be compared with those used in fitting the experimental data [22], It is seen that the parameter x e to which the decay curves have been fitted becomes ill-defined when one allows for CPT violation; in one case one has x e ≈ θ − λ e , in the other one it is x e ≈ −λ e . Anyway, we may state that the search for ∆S = −∆Q amplitudes has provided a loose, indirect bound on CPT violation in K 0 -K 0 mixing; since one has obtained x e < ∼ 10 −2 , one can also state that θ < ∼ 10 −2 . Indeed, if θ were much larger than this, its effect should be visible in the analysis of the decay curves in Eqs. (28).

Decays from a correlated state
Let us consider the correlated state with P-and C-parity −1 which, at time where k and − k denote the opposite three-momenta of the two mesons. Consider what can be measured using the correlated state in Eq. (30). Let the meson with momentum k decay at time t 1 into a state f , and the meson with momentum − k decay at time t 2 into a state g. Using Eqs. (19) one finds that the decay amplitude may be written in the form where Since the decay amplitude in Eq. (31) depends on two independent time functions, we may in principle extract the ratio of their coefficients: Indeed, by observing the shape of the dependence on t 1 and on t 2 of the decay rate, one can determine F . It turns out that, as a matter of fact, one can measure F even if one integrates the decay rate over t 1 + t 2 , as long as one still follows its dependence on t 1 − t 2 . Indeed, is identical with the right-hand side of Eq. (34) but for the overall factor, which is

Di-lepton decays
In the measurable quantity of Eq. (33), let us suppose that f is the semileptonic state X − l + ν l , while g is another semileptonic state, which has a charged lepton with the same electric charge, X ′− l ′+ ν l ′ . The quantity F in Eq. (33) then reads We have once again made the approximation of neglecting the products of any two small parameters like θ, λ l , and λ l ′ . Let us now suppose that f is the semileptonic state X + l −ν l , while g is the semileptonic state X ′+ l ′−ν l ′ , with a charged lepton with the same electric charge. The quantity F then reads If, instead, the two semileptonic states detected in opposite sides of the detector have charged leptons with opposite electric charge, then the quantity in Eq. (33) is either We thus conclude that, by using the decays of the correlated state φ − , one can-at least in principle-measure the four linear combinations of small parameters This means that it is impossible to disentangle the CPT-violating parameter θ from the ∆P = −∆Q parameters λ l andλ l by using di-lepton events alone. This conclusion was also arrived at by Xing [24].

Single-lepton events
Let us consider again the correlated state in Eq. (30). We shall now study events in which the final state f is observed in one side of the detector, irrespectively of the decay occurring in the opposite side. This corresponds to integrating over t 2 and summing over all final states g. Using the unitarity conditions in Eqs. (64) and (65) of the Appendix, one easily shows [25] that where Γ[P 0 (t 1 ) → f ] is the rate for a single, tagged P 0 to decay into the final state f at time t 1 , while Γ[P 0 (t 1 ) → f ] is the decay rate for a single tagged P 0 . These rates have been given in Eqs. (23). One finds Consider the particular case in which f is the semileptonic state X − l + ν l . The small parameters are δ (which violates T and CP), θ (which violates CPT and CP), and λ l (which violates the ∆P = ∆Q rule). Working out Eq. (42) to subleading order in those parameters, one finds Let us instead take f to be the semileptonic final state X + l −ν l . We find The functions of time |g + (t)| 2 , |g − (t)| 2 , Re g * + (t)g − (t) , and Im g * + (t)g − (t) are independent. Therefore, the three ratios of their coefficients may in principle be extracted from experiment. One sees that, if one neglects the subleading terms, then the coefficients of the time function Im g * + (t)g − (t) yield the CPT-violating parameter Im θ. On the other hand, Re θ cannot, even in this approximation, be separated from Re λ l or Reλ l .
If one does not neglect the subleading terms, then one can determine the Tviolating parameter δ from the coefficients of the function |g − (t)| 2 . On the other hand, even if one measures a non-zero coefficient of Im g * + (t)g − (t) , one cannot ascertain that one has found CPT violation. This is because of the subleading terms δIm λ l or δImλ l , which are CPT-invariant.

Combining single-lepton and di-lepton events
So far, we have separately discussed the impact that CPT violation and wrongcharge semileptonic decays have on correlated decays into two semileptonic and into one semileptonic final state. Schematically, we have found that Combining the first lines of Eqs. (46) and (47) we may extract 2Re θ +λ l − λ l , which is the same information as in the first line of Eq. (45). Thus, Re θ cannot be disentangled from Re λ l − λ l . On the other hand, one can combine the di-lepton and single-lepton events to extract Im θ. In fact, we may combine the second lines of Eqs. (46) and (47) to extract δIm λ l − λ l . This yields Im λ l − λ l , provided one is able to determine δ from the coefficients of |g − (t)| 2 in the single-lepton events. Then, by comparing Im λ l − λ l with the second line of Eq. (45), one can determine Im θ unambiguously.
We conclude that the decays of the correlated state |φ − can, in principle, be used to disentangle violations of CPT from violations of the ∆P = ∆Q rule in semileptonic decays. However, this only happens in the imaginary parts of the parameters; the real parts remain un-separated, even if one takes into account both di-lepton and single-lepton events.
It is important to observe that this determination of Im θ is possible even in the case Γ a = Γ b , which is expected to hold to a good approximation in the B 0 d -B 0 d system. Indeed, in that case the measurement of the first lines of Eqs. (45)-(47) will be impossible, but the measurement of their second lines will still be feasible. Thus, the extraction of Im θ will not be impeded by an approximate equality of the decay widths of the two eigenstates of mixing.

New-physics effects in the production mechanism
We have shown in section 3 that one may in principle separate the CPT-violating parameter θ from the ∆P = −∆Q parameters λ l andλ l by following the time dependence of the semileptonic decays of single tagged mesons. We recall that by "tagged meson" we mean a neutral meson whose flavor has been unequivocally determined at time t = 0. This is normally done by evoking the rule of associated production, which is based on the flavor-conserving nature of the interactions of the gluon, photon, or Z 0 , that are responsible for most production mechanisms. It states that, if an initial state i decays into a neutral meson together with a tagging state n, then the flavour of n is opposite to the one of the neutral meson.
To be specific, let us consider the conditions at CPLEAR. There one starts with i = pp and one looks for the charge of the kaon in the state n = K − π + . Since the strong interaction preserves flavor, this will identify as K 0 the neutral meson produced in association with n. Conversely, if the charged kaon has positive charge, then n = K + π − and the neutral meson is K 0 . In reality, there could be a small |∆S| = 2 interaction enabling the production process pp → K + π − K 0 . This would destroy the rule of associated production, and one would lose the notion of tagged decays. However, that |∆S| = 2 effect would also contribute to K 0 -K 0 mixing and, thus, one would expect it to be negligibly small.
On the contrary, new-physics effects in the production mechanism may be important when the production is due to the interaction of the W boson. Here we should consider two cases, depending on whether the leading-order SM production mechanism allows for only one (say, P 0 ) or both (P 0 and P 0 ) neutral mesons to be produced in the final state.
The decay B 0 d → J/ψK 0 constitutes an example of the first case. To leading order in the SM this decay takes place, while B 0 d → J/ψK 0 does not. (These processes are called "semileptonic-type decays" by Kostelecký and collaborators [21].) In principle one could use the subsequent evolution of the neutral kaon in order to test its properties; in practice, since those properties are rather well tested in direct decays of neutral kaons, these cascade decays should be used instead to probe the properties of the B 0 d -B 0 d system [7]. In this particular case, the initial state can evolve into its CP conjugate. We shall not consider such cases any further, but we stress that even a small new-physics amplitude to the process B 0 d → J/ψK 0 will affect those analyses. There are also cascade decays in which both neutral mesons can arise in the intermediate state, even within the SM. For example, one may want to use the copious production of B − at the Υ(4S), together with a subsequent decay B − → K − D 0 , in order to probe the decays of the D 0 into some final state f . However, in this case there is another possibility. Although suppressed by about an order of magnitude in amplitude, the process B − → K − D 0 is also allowed in the SM. Thus, if both D 0 → f and D 0 → f are allowed, there are two interfering decay paths: This is actually at the root of the Gronau-London-Wyler [26] and Atwood-Dunietz-Soni [27] methods to determine the CP-violating phase γ. In those methods one assumes that there is no D 0 -D 0 mixing; in that case, there is no interference effect for those final states f into which either D 0 or D 0 cannot decay. Recently, Meca and Silva [8] have shown that the presence of D 0 -D 0 mixing gives rise to a new interference effect, between the amplitudes of the decays into the D 0 -D 0 system, on the one hand, and the mixing in that system, on the other hand. One of the consequences of this result is that, even if we look for a final state f into which D 0 cannot decay, there will be two interfering paths: the unmixed decay path B − → K − D 0 → K − f ; and the mixed decay This effect makes it possible to test new sources of CP violation, and might provide a handle on ∆m in the D 0 -D 0 system [8,9].
We shall consider the general situation in which the initial state i can lead both to the state n together with P 0 , and to the state n together with P 0 . That is, we assume that are both non-vanishing. Then, the production process leads to the superposition of P 0 and P 0 given by Using Eqs. (19), we find that this state evolves into at time t. Suppose that one observes experimentally the time dependence of the overall process i → n P 0 , P 0 → nf.
Recalling that g + (t) and g − (t) are independent functions, we conclude that this allows in principle the determination of Clearly, if the state n correctly tags P 0 , i.e., ifc i = 0, thenÊ coincides with E in Eq. (22). Conversely, if n really identifies P 0 , i.e., if c i = 0, thenÊ =Ē. Let us define [8,9] andξ i = 1/ξ i . The parameter ξ i describes the interference between the production process, represented by the two amplitudes c i andc i , and the subsequent P 0 -P 0 mixing, described by q/p. We want to consider cases in which, although n is not a perfect tag for P 0 (or for P 0 ), the mis-tagging is small. Then, we may treat ξ i (orξ i ) as a small parameter.
When f is the semileptonic state X − l + ν l , we may in principle measure, if or, ifξ i is small, If f = X + l −ν l , then we may in principle measure, if ξ i is small, or, ifξ i is small, In all cases we have made the approximation of neglecting products of small parameters. Equations (54)-(57) should be compared with Eqs. (25) and (26). We see that, if both neutral mesons can be produced in association with n, then we may no longer disentangle the CPT-violating parameter θ from the effects of both mis-tagging and ∆P = −∆Q amplitudes, Notice that we have only considered cases in which the initial state i cannot evolve intoī. This includes cascade decays originating in baryons or charged mesons, but not cascade decays which start from a heavier neutral-mesonantimeson system. In the latter cases the analysis is more complicated [7,9] because there are two distinct neutral-meson systems evolving in time.

Di-lepton decays and mis-tagging
It is instructive to view the difficulties with di-lepton decays, discussed in section 4, as a particular case of mis-tagging. Let us review the tagging strategy usually evoked for measurements at the Υ(4S). It is generally assumed that there are no violations of the ∆B = ∆Q rule in semileptonic decays. Then the following reasoning applies: 1) although the B 0 d and the B 0 d produced at the Υ(4S) oscillate, the antisymmetry of the wave function is preserved by the linearity of the evolution; 2) hence, if at some instant the right-moving meson is found (from its semileptonic decay) to be B 0 d , then the left-moving meson at that instant is certainly B 0 d ; 3) that left-moving meson will evolve from that instant onwards as a tagged B 0 d ; 4) therefore, time-dependent experiments starting from the state Υ(4S) and observing at least one semileptonic decay automatically re-

Conclusions
In this article we have discussed ways to uncover a signal of CPT violation using semileptonic decays. We have stressed the fact that such a signal must be disentangled from new-physics effects in the tagging procedure. These can be due to ∆P = −∆Q amplitudes in semileptonic decays; but also to flavorconservation violations in the production mechanism. If one assumes the latter to be absent, then one concludes that: 1. The separation between violations of CPT and violations of the ∆P = ∆Q rule is possible using tagged decays; unfortunately, these studies cannot be implemented at the Υ(4S) and are difficult to implement elsewhere.
2. One cannot disentangle violations of the ∆P = ∆Q rule from violations of CPT using di-lepton decays of the state |φ − .
We now express the matrix elements of R in terms of the physical parameters µ a , µ b , θ, and δ. We thus obtain the unitarity conditions in the presence of violations of CPT: g |A g | 2 = Γ (1 + x Im θ − y Re θ) , In Eqs. (60)-(62) the sums run over all the available decay modes g. Kenny and Sachs [28] have questioned the use of some simpler unitarity conditions when testing CPT invariance, on the grounds that one of the assumptions of the CPT theorem is the hermiticity of the interactions. However, our derivation of the unitarity conditions is directly rooted on the conservation of probability expressed by Eq. (60), and not on the hermiticity of the Hamiltonian. We would agree with Tanner and Dalitz [13], who argue that any theory which violates Eq. (60) should probably be regarded as physically unacceptable.
Equation (41) follows from Eqs. (64) and (65). A result similar to Eq. (41), but with φ − substituted by the state with P-and C-parity +1, also holds [25], as one would expect on the basic grounds of the conservation of probability.