Dealing with T and CPT violations in mixing as well as direct and indirect CP violations for neutral mesons decaying to two vectors

A large number of observables can be constructed from differential decay rate based on the polarization of final state while considering decay of a neutral meson $(P^0 \text{ or } \bar P^0)$ to two vector particles. But all of these observables are not independent to each other since there are only a few independent theoretical parameters controlling the whole dynamics and therefore various relations among observables emerge. In this paper, we have studied the behaviour of observables for neutral meson decaying to two vectors in presence of T and CPT violations in mixing accompanied by both direct and indirect CP violations. We have expressed all of the fourteen unknown theoretical parameters for this scenario in terms observables only and constructed the complete set of thirty four relations among observables whose violation would signify the existence of some new Physics involving direct violation of CPT. In addition, using this formalism we have studied three special cases too: a) SM scenario, b) SM plus direct CP violation c) SM plus T and CPT violation in mixing.

Dealing with T and CPT violations in mixing as well as direct and indirect CP violations for neutral mesons decaying to two vectors Anirban Karan. a a Indian Institute of Technology Hyderabad, Kandi, Sangareddy-502285, Telengana, India E-mail: kanirban@iith.ac.in Abstract: A large number of observables can be constructed from differential decay rate based on the polarization of final state while considering decay of a neutral meson (P 0 orP 0 ) to two vector particles. But all of these observables are not independent to each other since there are only a few independent theoretical parameters controlling the whole dynamics and therefore various relations among observables emerge. In this paper, we have studied the behaviour of observables for neutral meson decaying to two vectors in presence of T and CP T violations in mixing accompanied by both direct and indirect CP violations. We have expressed all of the fourteen unknown theoretical parameters for this scenario in terms observables only and constructed the complete set of thirty four relations among observables whose violation would signify the existence of some new Physics involving direct violation of CP T . In addition, using this formalism we have studied three special cases too: a) SM scenario, b) SM plus direct CP violation c) SM plus T and CP T violation in mixing. effects must be considered together.
In literature, there exist extensive studies on probing T , CP and CP T violation using leptonic, semi-leptonic, two pseudoscalars and one pseudoscalar plus one vector decay modes of neutral pseudoscalar meson [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67]. But the modes with neutral pseudoscalar mesons decaying to two vectors (P 0 orP 0 → V 1 V 2 ) are not very well assessed in light of CP T violation. Refs. [68][69][70] consider the SM scenario (i.e. CP violation in mixing only ) and its extension to models with CP T conserving generic new physics effects only while probing two vectors decay modes of B-mesons. Howbeit, Ref. [71] has taken CP T violation into account for describing the mode B 0 s → J/ψ φ and Ref. [72] has discussed about triple products and angular observables for B → V 1 V 2 decays in context of CP T violation. Furthermore, two vectors decay modes of neutral mesons have been studied in Ref. [73] contemplating T , CP and CP T violation in mixing only. In this paper, we have extended the idea of Ref. [73] to search for T and CP T violation in mixing through P 0 → V 1 V 2 decays using helicity-based analysis in presence of CP violation in decay as well as in mixing. Notwithstanding, the presence of CP violation in decay changes the scenario drastically and complicate all the equations compared to Ref. [73] and consequently recasting the whole approach for this analysis is essential. It should also be noted that we do not consider any specific model that might lead to CP T violation which implies that it's a model-independent approach.
While dealing with oscillations of neutral pseudoscalar mesons (P 0 ,P 0 ), usually a common final state f , to which both P 0 andP 0 can decay, is considered. When f contains two vectors, there emerge three transversity amplitudes for the each of the transitions P 0 → f andP 0 → f depending on the orbital angular momentum of the final state. This fact helps us to construct a large number of observables from timedependent differential decay rates of the two modes. But, all of the observables will not be independent to each other since independent theoretical parameters are lesser in number than the observables. Therefore, various relations among observables appear automatically. These relations have already been addressed in context SM scenario in Refs. [69,70,73], where first two references consider two vector decay modes of B 0 d only with vanishing width deference between the physical states. Moreover, Ref. [73] talks about these relations in the presence of T , CP and CP T violation in mixing. In this paper we advance one step further by exploring these relations in the presence of CP violation in decay in addition to T , CP and CP T violating effects in mixing. These new relations will break down only if any CP T violating effect is present in decay itself. Furthermore, we have used our formalism to study the relations among observables for SM scenario, SM plus direct CP violation case and SM plus indirect violation of T and CP T scenario, which are three special cases of our picture.
The paper is organized as follows. In the next section (Sec. 2), we briefly describe the theoretical formalism for CP T violation in P 0 −P 0 mixing and express the time dependent differential decay rates 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 the "dummy observables" in terms of them as well as small T and CP T violating parameters. Sec. 4 deals with solving unknown theoretical parameters in terms of observables. The relations among observables in present scenario have been established in Sec. 5. In Sec. 6, we use this formalism to find the relations among observables for three special cases: a) SM scenario (CP violation in mixing only), b) SM plus direct CP violation, c) SM plus T and CP T violation in mixing. The phenomenological aspects have been discussed in Sec. 7 and finally, we summarize and conclude in Sec. 8.

Theoretical Framework:
Let us first briefly review the most general formalism incorporating CP T and T violation for P 0 −P 0 mixing, which has already been discussed in Ref. [2,64,73]. In the flavour basis (P 0 ,P 0 ), the mixing Hamiltonian can be expressed in terms of two 2×2 Hermitian matrices, namely mass-matrix M and decay-matrix Γ, as M−(i/2)Γ. Since three Pauli matrices σ j along with identity matrix I constitute a complete set of bases spanning the whole vector-space of 2 × 2 matrices, one 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 mass eigenstates or physical states |P L and |P H are th eigenvectors of the mixing Hamiltonian M − (i/2)Γ 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 indicating light and heavy physical states, respectively. Since, the physical states, as given by Eq. (2.3), depend only on the complex parameters θ and φ, they are called the mixing parameters for P 0 −P 0 system. It should be noted 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: Let us now consider a final state f to which both P 0 andP 0 can decay. Using Eq. (2.4), the time dependent decay amplitudes for the neutral mesons are given by: indicating the Hamiltonian related to the transition from flavour states to f . Therefore, incorporating the mixing, the time dependent decay rates Γ(P 0 (t) → f ) and Γ(P 0 (t) → f ) can be expressed as: (2.7) 3 Observables: 3.1 T and CPT violating parameters: The properties of M and Γ matrices in light of T and CP T symmetry has been discussed in Ref. [75]. First, if CP T invariance holds, then, independently of T symmetry [64,73], Secondly, if T invariance holds, then, independently of CP T symmetry [64,73], Hence, incorporating T , CP and CP T violation in P 0 −P 0 mixing, we parametrize θ and φ as [64,73]: where, β is the CP violating weak mixing phase, ǫ 1 and ǫ 2 are CP T violating parameters and ǫ 3 is T violating parameter other than CP violation. The notation of Belle, BaBar and LHCb collaborations [52][53][54][55] is a bit different from ours; however, the two notations are related to each other by the following transformation [64,73]: or, equivalently: ǫ 1 = Re(z) , ǫ 2 = Im(z) , ǫ 3 = 1 − q p . (3.4)

Decay rates and observables:
Any state consisting of two vectors can have three different values for orbital angular momentum quantum number {0, 1, 2} which correspond to the polarization states {0, ⊥ , }, respectively. Since CP T violation in decay has not been considered, the decay amplitudes for modes and conjugate modes can be expressed in terms of transversity amplitudes as [68-70, 72, 73]: where the factors g λ with λ ∈ {0, , ⊥} are the coefficients of transversity amplitudes (A λ orĀ λ ) in linear polarization basis and depend only on the kinematic angles [74]. Now, using Eq. (2.6)-(3.5), the time-dependent decay rates for P 0 (t) → V 1 V 2 and P 0 (t) → V 1 V 2 modes can be written as [68][69][70][71][72][73]: Λ λσ cosh ∆Γt 2 +η λσ sinh ∆Γt 2 +Σ λσ cos ∆M t +ρ λσ sin ∆M t g λ g σ . where both λ and σ take the value {0, , ⊥}. It is important to note that in the entire paper we have considered a particular ordering for the combination λσ with λ = σ and they are {⊥ 0, ⊥ , 0}. For defining the observables, we have taken the convention of Ref. [73]. On the other hand, Refs. [68][69][70] use a bit different notations involving some additional negative signs. Hence, several inferences of our paper may differ from their results by some signs only; however, all the outcomes of our paper are self-consistent. Now, we see from Eq. (3.6) that for each of the helicity combinations, there are four types of 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 (t) → V 1 V 2 mode. Similarly, there will be 24 different observables (Λ λσ ,η λσ ,Σ λσ ,ρ λσ ) forP 0 ((t) → V 1 V 2 mode too, as shown in Eq. (3.7). These observables can be measured by performing a time dependent angular [68][69][70]. The procedure described in Ref. [72] 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. It should be noticed that Ref. [68][69][70] did not consider sinh ∆Γt 2 terms in the decays of B 0 d andB 0 d since ∆Γ is consistent with zero [44]. In that case, η λσ andη λσ remain undetermined and one should work with remaining (18 + 18) = 36 observables only for a mode and its conjugate mode. However, we have kept all the terms in our analysis since a general scenario has been considered here.

Parametric expansion of observables:
Comparing Eqs. (2.6) and (2.7) to Eqs. (3.6) and (3.7) one can easily infer that all of the observables will be functions of the complex quantities θ and φ. As T and CP T violations are expected to be very small [52][53][54][55]66], we can expand all the observables in terms of ǫ j (j ∈ {1, 2, 3}) keeping up to the linear orders. However, following Refs. [69,70], if we divide all the transversity amplitudes into CP conserving and CP violating parts, it would become very complicated to handle all the unknown parameters. So, we implement a new method to reduce the complexity. After substitution of Eqs. (3.3) and (3.5) into Eqs. (2.6) and (2.7), while expanding the differential decay rates in terms of ǫ j (j ∈ {1, 2, 3}), we find that only twenty four combinations of helicity amplitudes A λ andĀ λ appear as the coefficients of ǫ j . Denoting ξ = e −2iβ , we define these twenty four combinations as follows: where, i ∈ {0, } and λ ∈ {0, , ⊥}. It should be noted that the quantities, mentioned above, which we name as "dummy-observables", are not observables, in general; rather they are some theoretical tools for our convenience. Now, using Eqs. (3.6) and (3.7) one can express the actual observables in terms of the dummy-observables as well as the T and CP T violating parameters ǫ j . For our purpose, we invert those equations and express dummy-observables as functions of the original ones keeping only the linear orders in ǫ j as follows: 9) where, i ∈ {0, }. It is evident from above relations that the dummy-observables become original observable only when there is no T and CP T violation in mixing. Now, we use the same trick for the observables of conjugate mode too and taking i ∈ {0, } rewrite the dummy-observables in terms of them as follows: 4 Solutions of the theoretical parameters: In this section, we discuss how to solve for the unknown theoretical quantities in terms of observables. These theoretical parameters are six helicity amplitudes (A λ ,Ā λ with λ ∈ {0, , ⊥}), which are complex entities and three parameters ǫ j (j ∈ {1, 2, 3}) related to T and CP T violation in mixing. It should be noted that the CP violating weak phase β cannot be probed directly in the presence of direct CP violation; it can only be measured if there is no CP violation in decay itself. Now, it is impossible to measure the absolute phases for all the helicity amplitudes; rather relative phases can be estimated. Hence, we define the following quantities that indicate the relative phases of five transversity amplitudes with respect to A ⊥ : where, 'Arg' implies argument of a complex number, i ∈ {0, } and λ ∈ {0, , ⊥}. Thus we have to solve for fourteen unknown parameters (three of |A λ |, three of |Ā λ |, three of ǫ j , two of Ω i and three ofΩ λ ). For convenience, we define nine angular quantities as follows: where (λ, σ) ∈ {0, , ⊥}. As mentioned earlier, we will consider the combination λσ to be one of {⊥ 0, ⊥ , 0} only; one should not be bothered about reverse ordering. Now, instead of the five relative phases of the transversity amplitudes we use five of the abovedefined angular entities (three of ϕ meas λ and two of ω ⊥i ) as our unknown parameters to solve for. The rest four angular quantities in Eq. (4.2) will be used later in order to find relations among various observables. The relative phases of helicity amplitudes can easily be expressed in terms of the five angular entities mentioned above in the following way: where, i ∈ {0, }. Thus the fourteen theoretical parameters that we are going to solve are three of |A λ |, three of |Ā λ |, three of ǫ j , two of ω ⊥i and three of ϕ meas λ . The modulus of helicity amplitudes are given by: The value of sin Φ meas λ can be found by solving the following cubic equation: in the above expression. The quantities with superscript 'r' and C λ are defined as: where, (Y ≡ ρ,ρ, Σ,Σ) with λ ∈ {0, , ⊥} . Knowing Φ meas i form the above equations, the T and CP T violating parameters in mixing can be obtained from the following equations: (4.7) Thus we solve for twelve of the fourteen unknown parameters. To obtain the solutions we have inverted the expressions for Λ ′ λλ , Σ ′ λλ and ρ ′ λλ in Eqs. (3.9) and (3.10) and then use the definitions of those dummy-observables from Eq. (3.8). It should be noticed that each of ǫ j (j ∈ {1, 2, 3}) can obtained in three ways since λ ∈ {0, , ⊥}. This fact will be used later to find some relations among observables. Now, the remaining two angular quantities ω ⊥i (i ∈ {0, }) are calculated in the following way: . (4.8)

Relations among observables:
In this section we are going to derive complete set of relations among observables for the scenario with T and CP T violation in mixing along with CP violation in both mixing and decay. As discussed before, we have forty eight observables combining mode and conjugate mode, but the number of unknown theoretical parameters are fourteen. Therefore, we must have forty eight minus fourteen equals to thirty four relations among observables. If we simply substitute the solutions of unknown parameters into Eqs. (3.8) -(3.10), we would overcount the number of independent relations among observables. Firstly, it is evident from Eq. (4.7) that each of the ǫ j (j ∈ {1, 2, 3}) can express in three ways depending on the value of λ. Hence, one will give the solution for ǫ j while the rest two can be recast as relations among observables and this happens for each ǫ j . Thus we have two times three equals to six relations among observables which are the following: where, i ∈ {0, }. The above six relations can be interpreted from a different perspective too. Looking at Eqs. (4.4) -(4.7), it can be realized that eighteen observables (three for each of Λ λλ , Σ λλ , ρ λλ ,Λ λλ ,Σ λλ andρ λλ ) have been used to solve for twelve different quantities (three for each |A λ |, |Ā λ |, ǫ j and sin ϕ meas λ ). Hence, eliminating the unknown quantities, one should get eighteen minus twelve equals to six among observables which are given by Eqs. (5.1) -(5.3).
Lastly, the expressions for η ′ λσ ∀ (λ, σ) ∈ {0, , ⊥} have not been utilized so far since η andη become non-measurable in the systems with vanishing ∆Γ (like B 0 d ). That is why we have tried to eliminate them from most of our solutions and relations (although Eq. (5.8) contains them). Nonetheless, one can overcome the problem for systems with vanishing ∆Γ as well as find the rest of the relations in case of general P 0 −P 0 systems. We have to express η ′ λσ ∀ (λ, σ) ∈ {0, , ⊥} in terms of observables and the measured angles (ϕ meas λ and ω λσ ) first (like ρ ′ in the last paragraph) as follows: = (Λ + Σ )(Λ 00 +Σ 00 ) cos (ϕ meas 0 − ω 0 ) + (Λ 00 + Σ 00 )(Λ +Σ ) cos (ϕ meas + ω 0 ) , (5.14) where i ∈ {0, }. Now, substituting the above relations into to twelve equations involving η ′ λσ ∀ (λ, σ) ∈ {0, , ⊥} in Eqs. (3.9) and (3.10), the remaining twelve relations among observables can be established. For vanishing ∆Γ, those twelve relations can be used for theoretical estimation of η λσ andη λσ which can used in Eq. (5.8) to verify those observable relations. Thus excluding the last twelve relations, we have total twenty two observable relations in vanishing ∆Γ scenario, whereas in general cases we have total thirty four relations among observables. Some of these relations will get violated only if there exists direct violation of CP T (i.e. violation in the decay itself.)

Special cases:
In this section, we will study following three special cases using our formalism.

SM scenario:
In SM scenario, there is no violation of T (apart from CP violating effects) and CP T in mixing. Hence, ǫ j = 0 ∀ j ∈ {1, 2, 3} which readily infer from Eqs. (3.9) and (3.10) that Λ λσ =Λ λσ , η λσ =η λσ , Σ λσ = −Σ λσ , ρ λσ =ρ λσ , (6.1) where (λ, σ) ∈ {0, , ⊥}. It should be kept in mind that the forty eight equations in Eqs. (3.9) and (3.10) have been recast as solutions of fourteen theoretical parameter, as given in Sec. 4, and thirty four relations among observables, as described in Sec. 5. Therefore, the twenty four relations in Eq. (6.1) are also embedded in the solutions or relations among observables. But it would take a bit more algebraic complexity to dig them out from there and so we simply derive them from Eqs. (3.9) and (3.10). The other constrain in SM is that each of helicity amplitudes for mode and the conjugate mode is equal to each other (i.e. A λ =Ā λ ). Equating the modulus of helicity amplitude one gets the following three relations from Eqs. (4.4) and (6.1): Here, we have used the Eq. (4.8) and Eqs. (5.4) -(5.6) for the expressions of ω λσ and ω λσ . The expressions for ω ⊥i and ω 0 in this scenario turn out to be following which will be used later: From Eq. (4.2), we also get that ϕ meas λ = −2β. Combining this information with Eqs. (4.5), (4.6), (6.1) and (6.2) results in following two relations: along with the expression of sin 2β as: Thus first part of Eq. (5.7) (i.e.ω λσ = ϕ meas λ −ϕ meas σ +ω λσ ) gets satisfied automatically. After a couple of discussions we will come back to the second part of the equation. Now, substituting the Eqs. (5.9) and (5.10) into Eq. (3.9) and using the expressions of angular quantities ω ⊥i , ω 0 and ϕ meas λ from Eqs. (6.5) and (6.7) along with Eqs. (6.1) and (6.2), one arrive at the following three relations: and, In the same way, using the expressions for η ′ λσ in Eqs. (5.11) -(5.14), the following six relations for i ∈ {0, } can be achieved with the help of a bit of algebraic and trigonometric operations: Finally, we use the last part of Eq. (5.7) (i.e. ω 0 = ω ⊥0 − ω ⊥ ) to reach the last relation: Thus we have six unknown parameters (three of |A λ |, two of ω ⊥i and one β) in this case to solve for which are given by Eqs. (6.3), (6.5) (first part) and (6.7). Therefore, one should get a complete set of forty two independent relations among observables which consists of twenty four in Eq. (6.1), three in each of Eqs. (6.2), (6.4) and (6.10), two in each of Eqs. (6.6), (6.8), (6.11) and one in each of Eqs. (6.9), (6.12) and (6.13) respectively. All the other expressions in Sec. 4 and 5 satisfy automatically. Except the twenty four relations in Eq. (6.1), the other eighteen relation have already been discussed in Ref. [73]. These relations will get violated by the presence of direct CP violation or some CP T non-conserving new Physics effects.
In this case, number of independent relations among observables is forty eight minus eleven equals to thirty seven. Among them twenty four are listed in Eq. (6.14). The remaining thirteen can be found in the following way: 1) four relations can be found from Eq. (5.7), 2) three can be established by substituting Eqs. (5.9) and (5.10) in to Eq. (3.9), 3) the last six can be obtained by replacing Eqs. (5.11) -(5.14) into Eq. (3.9). After a bit of mathematical jugglery, these thirteen relations can be described in the following form: , (6.20) where, (λ, σ) ∈ {0, , ⊥} and λ = σ. It should be noticed that each of the four equations from Eq. (6.16) to Eq. (6.19) contains three relations for different values of λ and σ with λ not being equal to σ, and Eq. (6.20) contains only one. Violation of these relations would definitely imply existence of CP T violating new Physics phenomenon.

SM plus T and CPT violation in mixing:
In this case, one can follow the entire procedure described in Sec. 4 and 5 to get the solutions of theoretical parameters and find the relations among observables. But to reach the expressions in the form of Ref. [73], that already discusses this scenario, one has to encounter various algebraic complexities. As there is no direct CP violation in this case, the helicity amplitudes for mode and conjugate mode will be equal to each other (like SM). All of the three ϕ meas λ become −2β too. Therefore, there will be total nine unknown parameters (three of |A λ |, two of ω ⊥i , one β and three of ǫ j ). It implies that the total number of independent relations among observables is forty eight minus nine equal to thirty nine. It should be noticed from Sec. 6.1 that Σ λλ , Σ 0 , Λ ⊥i , were zero in SM case. However, inverting Eq. (3.9) one can find that they are O(ǫ j ) in the present scenario. Since we are keeping track up to the linear order terms in ǫ j , any quadratic term involving the above expressions will be neglected. The same rule applies for the observables of conjugate mode too.
At first, using the equality of helicity amplitudes for mode and conjugate mode, one achieve the following six relations from Eqs.
where, λ ∈ {0, , ⊥} and i ∈ {0, }. Along with the above six relations we also get the expressions for five unknown parameters as: .
Σ 00 +Σ 00 Λ 00 + Σ 00 − ρ 00 Λ 00 + Σ 00 + ρ ⊥⊥ Λ ⊥⊥ + Σ ⊥⊥ sin 2β = 0 (6.27) With the help of the above expressions the values of ǫ 2 and ǫ 3 in Eq. (4.7) can be written as: Thirdly, substituting the expressions for η ′ λλ for λ ∈ {0, , ⊥} from Eqs. (5.11) and (5.12) into Eqs. (3.9) and (3.10) one would end up with following six relations: It should be noticed that some of the above relations could be further simplified as: Λ λλ = η 2 λλ + ρ 2 λλ andΛ λλ = η 2 λλ +ρ 2 λλ . Nonetheless, to reproduce the relations in Ref. [73], we stick to the former ones only. Using the above relations, the expression for ǫ 1 from Eq. (4.7) can be interpreted as following: With the help of above relations involving η λλ , one can also abandon Eq. (6.27) and recast it as: Fourthly, we use the Eq. (5.8) to arrive at the expressions for cos ω λσ and cosω λσ that can be rewritten as the following six relations: where, with i ∈ {0, }. It is very important to note that to get a correct X i orX i up to O(ǫ j ) one should keep the quadratic terms of ǫ j in the numerator and denominator separately while defining X i orX i , since the leading order terms in both of the numerator and denominator are O(ǫ j ). Notwithstanding, the relations in Eqs. (6.35) and (6.36) can be untangle a bit by using cos ω ⊥i = 1 2 X i √ Λ ⊥⊥ Λ ii and cosω ⊥i = 1 2X i Λ ⊥⊥Λii . Fifthly, by replacing the expressions for ρ ′ λσ with λ = σ from Eqs. (5.9) and (5.10) into Eqs. (3.9) and (3.10), as described in Sec. 5, one would end up with following six relations: where, i ∈ {0, }. Similarly, substituting the expressions for η ′ λσ with λ = σ from Eqs. (5.13) and (5.14) into Eqs. (3.9) and (3.10), would lead to the following six relations: Finally, second part of Eq. (5.7) (ω 0 = ω ⊥0 − ω ⊥ ) indicates the last independent relation as: Thus, the expressions for nine theoretical parameters (three of |A λ |, two of ω ⊥i , one β and three of ǫ j ) in this scenario are given by five equations in Eq. (6.22), two expressions in Eq. (6.28) and one in each of Eq. (6.23) and (6.33). On the other hand, the thirty nine relation among observables are presented as: a) six relations in Eq. (6.21), b) seven equations from Eq. (6.24) -(6.26) (we have not counted Eq. (6.27) since it has been recast as Eq. (6.34)), c) six expressions from Eq. (6.29) -(6.32) and d) twenty relations from Eq. (6.34) -(6.44). If some of these relations do not hold true, that will indicate the presence of CP T or CP violation in decay itself.

Phenomenology:
Various experiments have been performed so far in order to probe CP T violation in neutral meson mixing. Although these experiments measure tiny non-zero values for CP T violating parameters, they become consistent to zero within 2σ due to the presence of experimental error bars with comparable size. In case of kaon system, CP T asymmetry is measured from the semileptonic (π + l −ν l , π − l + ν l ) decay modes of K 0 andK 0 to estimate the CP T -violating complex parameter δ whose real and imaginary parts are directly proportional to ǫ 1 and ǫ 2 respectively in our notation. From the data of KTeV collaboration [76], the real and imaginary parts of this parameter are estimated to be: Re(δ) = (2.51±2.25)×10 −4 and Im(δ) = (−1.5±1.6)×10 −5 which agree with CP T conservation. In case of D 0 −D 0 system, CP T asymmetry, which is constructed by comparing the time dependent decay probabilities of the modes D 0 → K − π + andD 0 → K + π − , has been measured by FOCUS collaboration [77]. This measurement leads to the estimation of CP T -violating complex parameter ξ, whose real and imaginary parts are proportional to ǫ 1 and ǫ 2 respectively, to be: Re(ξ) y−Im(ξ) x = 0.0083±0.0065±0.0041 where, x = ∆M Γ and y = ∆Γ 2Γ . The first measurement of CP T violation in B 0 −B 0 system was performed by BaBar collaboration [52,53]. The last update on it has been carried out by Belle collaboration [54]. For this purpose, they have fitted the time dependent decay rate of the chain: Υ(4S) → B 0 dB 0 d → f rec f tag , where one of the B-meson decays to reconstructed final state f rec at time t rec and the other one decays at time t tag to a final state f tag , that distinguishes between B 0 andB 0 . Several hadronic and semileptonic decay modes of B 0 d (J/ψK S , J/ψK L , D − π + , D * − π + , D * − ρ + and D * − l + ν l ) have been used in this case to find the experimental value for the CP T -violating complex parameter z as: Re(z) = (1.9 ± 3.7 ± 3.3) × 10 −2 and Im(z) = (−5.7 ± 3.3 ± 3.3) × 10 −3 which are consistent with zero. Similarly, using the time dependent decay rate to a CP eigenstate for the mode B 0 s → J/ψK + K − , the CP T violation in B 0 s system has been measured by LHCb collaboration [55] as: Re(z) = −0.022 ± 0.033 ± 0.005 and Im(z) = 0.004 ± 0.011 ± 0.002 that also agrees with conservation of CP T . The dependence of the parameter z on ǫ 1 and ǫ 2 is already given by Eq. (3.4). On the other hand, direct and indirect CP violations as well as T violation in neutral meson mixing 8 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 as well as CP violation in decay. Polarizations of final state with two vectors provide us a large number of observables in these modes. The final state should be chosen in such a way that both P 0 andP 0 can decay to it. We extract all of the fourteen unknown theoretical parameters in terms of the observables and then discuss the procedure to establish the complete set of independent relations among observables containing thirty four equations. These relations can be used as the smoking gun signal to prove the existence of direct violation of CP T (if any) since those effects only can lead to nonobedience of them. Additionally, we explore three special cases e.g. SM case, SM plus direct CP violation scenario and SM plus T and CP T violation in mixing case. Using our new formalism, we derive the expressions for unknown theoretical parameters and construct the complete set of independent relations among observables too in each special case. Experimental verification for each of the sets will signify the existence of some particular type of Physics. For example, the set of relations in SM plus T and CP T violation in mixing scenario can be applied to probe direct violation of CP T or CP , the set of relations in SM plus CP case can be implemented to confirm the existence of any CP T violating new Physics (direct or indirect), whereas the set of observable relations in SM scenario should be used to detect direct CP violation or CP T non-conserving new Physics.