Behaviour of observables for neutral meson decaying to two vectors in the presence of $T$, $CP$ and $CPT$ violation in mixing only

When a neutral meson $(P^0 \text{ or } \bar P^0)$ decays to two vector particles, a large number of observables can be constructed from differential decay rate based on the polarization of final state. But, theoretically, all of them are not independent to each other and hence, some relations among observables emerge. These relations have been well studied in the scenario with no $T$ and $CPT$ violation in neutral meson mixing and no direct $CP$ violation as well. In this paper, we have studied the relations among observables in the presence of $T$, $CP$ and $CPT$ violating effects in mixing only. We find that except four of them, all the other old relations get violated and new relations appear if $T$ and $CPT$ violations in mixing are present. Invalidity of these relation will signify the presence of direct violation of $T$, $CP$ and $CPT$ (i.e. violation in the decay itself).


Introduction:
CP T invariance is believed to be a sacred principle of any locally Lorentz invariant quantum field theory. In any axiomatic quantum field theory, this discrete symmetry emerges to be exact up to any order. It has a direct connection with the preservation of Lorentz symmetry [1,2]. Due to its great theoretical importance, it is necessary to test the validity of this principle experimentally. CP T invariance predicts the masses or lifetimes of any particle and its anti-particle to be the same, which has been tested for lots of particles through direct experiments [3]. But one can argue that these quantities are usually dominated by strong or electromagnetic interactions and hence there exits a possibility for tiny CP T violating effects, mediated by weak interactions, to be undetectable in those direct experiments. In this regard, mixing of neutral pseudoscalar meson (K 0 , D 0 , B 0 d , B 0 s ) with its own antiparticle is a promising area [4] to search for CP T violating effects as this phenomenon is a second order electroweak process. However, since the most general mixing matrix includes T and CP violating parameters as well, we have to study the effects of CP , T and CP T violation together.
Searches for CP , T and CP T violation using leptonic and semi-leptonic channels as well as the modes where neutral pseudoscalar meson decays to two other pseudoscalars or one vector and one pseudoscalar have been performed extensively [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. However, effects of CP T violation on the modes where neutral pseudoscalar meson decays to two vectors (P 0 orP 0 → V 1 V 2 ) are not very well studied. Though the Refs. [22,23,24] discuss about these modes involving two vectors, they only consider the SM scenario (i.e. only CP violation in mixing) and its extension to a model with CP T conserving generic new physics effects. However, Ref. [25] has taken CP T violation into account for describing the mode B 0 s → J/ψ φ and Ref. [26] has discussed about triple products and angular observables for B → V 1 V 2 decays in light of CP T violation. In this paper, we have revisited the prospect of searching CP T violation in mixing through P 0 → V 1 V 2 decays using helicity-based analysis for time-dependent differential decay rate. We would also like to emphasize that we have taken a model-independent approach in a sense that we do not specify any definite model that might lead to CP T violation.
The usual technique to deal with the oscillations of neutral pseudoscalar mesons is to consider a final state f to which both P 0 andP 0 can decay. If f consists of two vectors, a large number of observables can be constructed from time-dependent differential decay rate depending on the polarization or orbital angular momentum of the final state. But, all of these observables will not be independent to each other and hence there emerge various relations among them. In Ref. [23,24], these relations have been discussed in the context of SM scenario only for the modes B 0 d orB 0 d decaying to two vectors. In this paper, we study these relations in presence of T , CP and CP T violations in mixing only. We have confined our analysis to the case where CP T violation is small compared to the SM amplitude, which is justified based on the data from several experiments [7,8,12,18,21]. Since independent theoretical parameters for this case are more in number than SM scenario, it is expected to obtain lesser number of relations among observables. We find that except four, all the other old relations in SM get violated and new relations appear if T and CP T violations in mixing are present. These new relations will hold true even if the T , CP and CP T violations become zero; however, they will not form the complete set of relations in that case as they are less in number. These new relations will break down only if T , CP and CP T violating effects are present in decay too (i.e. direct violation).
The paper is organized as follows. In the next section, we briefly describe the theoretical formalism for CP T violation in P 0 −P 0 mixing and express the time dependent differential decay rate of P 0 andP 0 in terms of the mixing parameters. In Sec. 3, we construct helicity-dependent observables from the differential decay rates and express them in terms of T , CP and CP T violating parameters assuming T and CP T violations in mixing to be very small. We also solve for all the unknown theoretical parameters as functions of the observables. In. Sec. 4, we establish the independent relations among these observables in SM case and the scenario with the presence of T and CP T violations in mixing separately. We also discuss how these relations can help us in distinguishing three different scenarios a) SM case, b) T , CP and CP T violation in mixing and c) direct violation of T , CP and CP T . Finally, we summarize and conclude in Sec. 5.

Theoretical Formalism:
We begin by reviewing the most general formalism for P 0 −P 0 mixing, in which CP T and T violation are incorporated. This formalism has already been discussed in Ref. [19]; however, for the sake of completeness we present it in this section. In the (P 0 −P 0 ) basis, the generic mixing Hamiltonian can be expressed in terms of two 2 × 2 Hermitian matrices M and Γ, respectively the mass and decay matrices, as M − (i/2)Γ. It should be noticed that the mixing matrix M − (i/2)Γ is non-Hermitian and it is justified as the probability of finding P 0 andP 0 decreases with time due to presence of the non-null decay matrix Γ. Now, since any 2 × 2 matrix can be expanded in terms of three Pauli matrices σ j and identity matrix I with complex coefficients, we can write: where, E, θ, φ and D are complex entities in general. Comparing both sides of this equation, we obtain: where M ij and Γ ij are (i, j)-th elements of M and Γ matrices respectively. The eigenvectors of the mixing Hamiltonian M − (i/2)Γ are the mass eigenstates (|P L and |P H ) and they can be expressed as linear combinations of the flavour eigenstates (|P 0 and |P 0 ) as follows: where p 1 = N 1 cos θ 2 , q 1 = N 1 e iφ sin θ 2 , p 2 = N 2 sin θ 2 , q 2 = N 2 e iφ cos θ 2 with N 1 , N 2 being two normalization factors and the L,H tags indicate light and heavy physical states, respectively. Since, the physical states, as given by Eq. (3), depend only on the parameters θ and φ, they are called the mixing parameters for P 0 −P 0 system. It should be noticed that the physical states are not orthogonal in general since the mixing matrix is non-Hermitian.
The time evolution of flavour states (|B 0 ≡ |B 0 (t = 0) and |B 0 ≡ |B 0 (t = 0) ) is given by: where, Here Let us now consider a final state f to which both P 0 andP 0 can decay. Using Eq. (4), the time dependent decay amplitudes for the neutral mesons are given by: where dΓ dt where the helicity index λ takes the value {0, , ⊥} and ζ λ takes the value {1, 1, i} for these three helicities respectively. The factor g λ are the coefficients of helicity amplitudes (A λ orĀ λ ) in linear polarization basis and only depend on kinematic angles [27]. In absence of direct violation for CP , T and CP T , these helicity amplitudes can be expressed as: where, a λ and δ λ are two real quantities indicating the magnitudes and phases for different helicity amplitudes. Now, using Eq. (7)-(10), the time-dependent decay rates for P 0 → V 1 V 2 and P 0 → V 1 V 2 modes can be written as [22,23,24,25,26]: where both λ and σ take the value {0, , ⊥}. From Eq. (11) we see that for each of the helicity combination, there are four observables (Λ λσ , η λσ , Σ λσ , ρ λσ ) and six such helicity combinations are possible. Hence, we get total 24 observables for P 0 → V 1 V 2 mode. Similarly, there will be 24 different observables (Λ λσ ,η λσ ,Σ λσ ,ρ λσ ) forP 0 → V 1 V 2 mode too. These observables can be measured by performing a time dependent angular analysis of P 0 (t) → V 1 V 2 and P 0 (t) → V 1 V 2 [22,23,24]. The procedure described in Ref. [26] can be helpful in this regard. On the other hand, probing polarizations of the final state particles may also aid in measurement of these observables. One important point to notice here is that Ref. [22,23,24] did not consider sinh ∆Γt 2 terms in the decays of B 0 d andB 0 d since ∆Γ is consistent with zero [3] for these modes. In that case, η λσ andη λσ remain undetermined and one should work with remaining (18 + 18) = 36 observables for a mode and its conjugate mode. However, since we are considering a general scenario here, we keep all the terms and proceed.

Solutions:
As can be seen from the expansion of observables, given by Eq. (18)- (21), there are total 9 unknown parameters (i.e. 3 of a λ , 3 of ǫ j , 2 of ∆ i and β). In SM case, there are six unknown parameters (3 of a λ , 2 of ∆ i and β), as stated in Ref. [23,24]; however, for our scenario, we have three extra parameters emerging due to T and CP T violation in mixing namely ǫ 1,2,3 , thus resulting in nine theoretical parameters. It should be noted that Ref. [23,24] originally deal with SM scenario plus CP violation in decay, not T and CP T violation in mixing; hence, in addition to six unknown SM parameters, they have three more amplitudes (b λ ), three more strong phases (δ b λ ) and one extra weak phase related to the CP violating part of the decay amplitudes (A λ orĀ λ ). Now, we go back to our scenario and solve the nine theoretical parameters in terms of observables as follows: where, with λ ∈ {0, , ⊥} and i ∈ {0, }. In principle, we should present only 9 equations as the solutions for 9 unknown parameters. But, we have listed more than 9 relations from Eq. (22) to Eq. (30) because the observables involve several angular parameters. Actually, to specify any angular variable without any ambiguity, one must quantify both sin and cos of that angle. However, as can be seen in section 4.2, the extra equations will result in some relations among observables by applying various trigonometric identities.

SM relations:
In SM scenario, all of the three ǫ j become zero and there remain only 6 unknown parameters (3 of a λ , 2 of ∆ i and β) in the theory. But the number of observables for P 0 → V 1 V 2 mode is 24. Hence, 18 independent relations among observables must emerge and they are the following: However, for vanishing ∆Γ, only 18 observables will be accessible to us (as discussed in the section 3.1) and hence, in that case, we should obtain 12 independent relations among observables. Those 12 relations are given by Eq. (32) − Eq. (35), as discussed in Ref. [23,24].
One important point to state is that one can use the solutions, given by Eq. (22)-(29), in the SM scenario also. But, X i , given by Eq. (31), takes the form 0 0 in this case and it causes problem in finding cos ∆ i from Eq. (30). Still, one can express cos ∆ i (i ∈ {0, }) in this scenario as following: which can easily be verified by substituting vanishing ǫ j into the Eq. (18) − Eq. (21). Hence, using Eq. (30), Eq. (32) and Eq. (39), one can write X i (i ∈ {0, }) in the limit ǫ j → 0 (j ∈ {1, 2, 3}) as: Nevertheless, we shall see in the next section that most of these 18 relations from Eq. (32) − Eq. (38) will get violated if T and CP T violations in mixing are also present. On the other hand, if there exists direct violation of T , CP or CP T instead of T and CP T violating effects in mixing, then also most of these relations get violated. Hence, it is impossible to infer from this set of relations whether CP T violation (if it exists at all) is present in mixing or in decay.

T and CP T violation:
In addition to the CP violating weak phase if there exist T and CP T violation in mixing, we have 9 unknown theoretical parameters (3 of ǫ j , 3 of a λ , 2 of ∆ i and β). But the number of observables is still 24. So, there should appear (24 − 9) = 15 number of relations among observables. In order to find them, we substitute the solutions of unknown parameters, given by Eq. (22)−(30), back to the expansion of observables, given by Eq. (18)- (21). Thus we get 11 independent relations, which are given below: There are four more such independent relations among observables which emerges due to the following trigonometric identities: cos(∆ 0 − ∆ ) = cos ∆ 0 cos ∆ + sin ∆ 0 sin ∆ .
Substituting expressions for different angular variables from Eq. (26)−Eq. (30) into the above trigonometric identities, given by Eq. (45) and Eq. (46), we get the remaining four relations as: with i ∈ {0, }. The Eq. (47) contains two relations (for two different i). However, it should be noticed that though sin 2β and cos 2β can be expressed in two ways using the helicities 0 and separately (as shown in Eq. (26) and Eq. (27)), we obtain only one relation among observables from the trigonometric identity: sin 2 2β + cos 2 2β = 1. It happens because of the fact that Eq. (41) ensures: (ρ 00 /Λ 00 ) = (ρ /Λ ) and (η 00 /Λ 00 ) = (η /Λ ). However, one should keep in mind that the relations in Eq. (41)-(46) will not hold true for all orders in ǫ j as we are computing the observables perturbatively up to the first order in ǫ j . The corrections to these relations are quadratic or of higher order in ǫ j and hence can be neglected for sufficiently small values of ǫ j . Now, if one wants to check the validity of the 18 relations of last section (given by Eq. (32) − Eq. (38)) in this scenario, he/she would find ǫ j order correction terms in 14 of them. The 4 relations, which remain intact in both the scenarios are: (ρ ii /Λ ii ) = (ρ 0 /Λ 0 ) and (η ii /Λ ii ) = (η 0 /Λ 0 ) which can easily be observed from Eq.  (21)) and then substituting those expressions for observables into these 15 relations. But it does not mean that we have 15 more independent relations in SM case. Because one can easily check that the 18 relations in last section automatically satisfy the 15 relations of this section. In other words, the 18 relations of previous section is embedded in a complicated form inside the 15 relations of present section. However, as discussed in last section, one has to be careful in dealing with X i while verifying since it takes 0 0 form in SM scenario. Now, if direct violations of T , CP and CP T are present in the decay mode, most of these 15 relations will not hold true and that can be used as a smoking gun signal of confirming those effects. In that case, the 18 relations of SM scenario will be disobeyed too. On the other hand, if these 15 relations are satisfied, then one becomes sure that there is no direct violations of T , CP and CP T , but it cannot be confirmed whether T and CP T violations in mixing are present or not since those 15 relations are satisfied on both the occasions. In this circumstance, the validity of the 18 relations in last section should be examined. If those 18 relations hold true, it would signify the absence of T and CP T violation in mixing and if they get violated, the presence of them will be confirmed.
There is another way to confirm the existence of T , CP and CP T violation in decay. In this analysis, we have used the observables of P 0 → V 1 V 2 mode only for solving all of the 9 unknown parameters, as shown in Eq. (22) − (30). Similarly, it is also possible to solve them by using the observables ofP 0 → V 1 V 2 mode, as given in the Appendix A. These two sets of solutions should match numerically in the absence of NP effects in decay. Hence, significant deviations in the numerical values of the 9 unknown parameters from these two sets of solutions will definitely indicate sizeable contributions of T , CP and CP T violations in decay.

Conclusion:
In conclusion, we have studied the behaviour of observables for neutral meson decaying to two vectors in the presence of T , CP and CP T violation in mixing. Polarizations of final state with two vectors provide us a large number of observables in these modes. We choose the final state in such a way that both P 0 andP 0 can decay to it. We establish the complete set of 15 relations among observables which must be obeyed if there do not exist any direct violations of T , CP and CP T and these relation can be used as the smoking gun signal to confirm their presence or absence. In addition to that we also listed the full set of 18 relations among observables which should be satisfied if there is no violation of T and CP T in mixing of P 0 −P 0 and these relations can be used to probe their existence unambiguously.