Precise Discussion of Time-Reversal Asymmetries in B-meson decays

BaBar collaboration announced that they observed time reversal (T) asymmetry through $B$ meson system. In the experiment, time dependencies of two distinctive processes, $B_- \rightarrow \bar{B^0}$ and $\bar{B^0}\rightarrow B_-$($-$ expresses CP value) are compared with each other. In our study, we examine event number difference of these two processes. In contrast to the BaBar asymmetry, the asymmetry of events number includes the overall normalization difference for rates. Time dependence of the asymmetry is more general and it includes terms absent in one used by BaBar collaboration. Both of the BaBar asymmetry and ours are naively thought to be T-odd since two processes compared are related with flipping time direction. We investigate the time reversal transformation property of our asymmetry. Using our notation, one can see that the asymmetry is not precisely a T-odd quantity, taking into account indirect CP and CPT violation of K meson systems. The effect of $\epsilon_K$ is extracted and gives rise to $\mathcal{O}(10^{-3})$ contribution. The introduced parameters are invariant under rephasing of quarks so that the coefficients of our asymmetry are expressed as phase convention independent quantities. Some combinations of the asymmetry enable us to extract parameters for wrong sign decays of $B_d$ meson, CPT violation, etc. We also study the reason why the T-even terms are allowed to contribute to the asymmetry, and find that several conditions are needed for the asymmetry to be a T-odd quantity.


Introduction
T-symmetry is fundamental symmetry in particle physics. T-transformation exchanges an initial state and final state, flipping the momentum and spin of particles. If some rate of a process deviates from the rate of time reversed one, it implies T-violation. T-violation is worth pursuit since it reflects the characterized feature of theory.
In EPR correlating B meson system, if one of a pair of B meson is tagged, another side of B meson is determined as orthogonal state with tagged side B meson. In refs. [1]- [3], a method to observe T-violation using B meson system is suggested. Their idea is based on the difference of the time dependencies for two distinctive processes, B − →B 0 and B 0 → B − . In refs. [1]- [3], it is considered to be T-asymmetry since two distinctive processes are related with flipping time direction. Then, BaBar collaboration announced [4] that they measured non-zero asymmetry and this observation is direct demonstration of T-violation. A review for the BaBar asymmetry and the other T and CPT asymmetries is given in ref. [5].
However, their statement includes ambiguity since we cannot exactly identify B meson such asB 0 or B − . In the BaBar experiment, two methods to identifyB 0 and B − are implemented. The first one is referred as flavor tagging which enables one to identifyB 0 , using the semi-leptonic decay mode of B meson. Another one is referred as CP tagging which allows us to identify B − with final state ψK S .
In ref. [6], it is pointed out that there exist subtleties in BaBar measurement. The main point in ref. [6] is that they consider a process and its authentically time reversed process. Note that the authentic time reversed process includes inverse decay such like l − X →B 0 and ψK L → B 0 . Since an authentic time reversed process is experimentally hard to observe, they substitute another process which does not include inverse decays. Therefore, they derive the conditions that BaBar asymmetry is identical with a T-odd quantity, taking into account inverse decays. The derived conditions are (1) the absence of the wrong sign semileptonic B meson decays, (2) the absence of the wrong strangeness decays, (3) the absence of CPT violation of the strangeness changing decays. All the conditions are derived by assuming that ψK S and ψK L are exact CP eigenstates.
In this paper, we conduct model-independent analysis of an event number asymmetry. Our analysis is extension of the work [6], incorporating the difference of overall constants for the rates that form the asymmetry into calculation. We also include the effect of indirect CP and CPT violation of K meson system. Furthermore, the asymmetry is written in terms of the phase convention independent parameters and one can find contribution of ǫ K explicitly. Some combinations of the coefficients enable one to constrain model-independent parameters. We also discuss the T-even parts of the asymmetry. One can find that the asymmetry is a T-odd quantity when several conditions are satisfied.
In section 2, we introduce the asymmetry of entangled B meson system. In section 3, we define parameters which are definitively T-odd or T-even and describe the relation between notation in [6] and ours. It is also shown that the parameters are phase convention independent quantities. The method to extract the effect of indirect CP violation in Kaon system is also considered in section 3. In section 4, we write the event number asymmetry in terms of the parameters defined in section 3 and show that the asymmetry consists of not only the T-odd part but also T-even part, using our notation. In sections 4.1-4.3, rather than discussing T-transformation property of the asymmetry, some methods to extract the parameters from the asymmetry are investigated. In section 5, we derive the conditions that T-even parts of the asymmetry vanish and examine the intuitive reason why these conditions are required. The conditions are categorized as two types. The first condition is in regards to B meson states that appear in the processes. Including the effect of indirect CP violation, we evaluate how the first condition is violated, in comparison with the result in [6]. The second condition is in regards to overall constant which forms the asymmetry. We find that the second condition is needed when one takes account of the difference of overall constant of the two rates. Section 6 is devoted to summary of our study.
since y ≪ 1 for neutral B meson system. Thus, we expand A with respect to yΓt, where we denote properties of p K , q K and z K are the same as p, q and z, respectively. Following [6], we introduce B meson decay amplitudes and inverse decay amplitudes, where f T is the time reversed state of f , i.e., the state with flipped momenta and spins.
Note that A f (Ā f ) and A ID f (Ā ID f ) are exchanged under T-transformation. Throughout this paper, we introduce the notation G f , S f and C f which are written in terms of amplitude ratio λ f .
Using notation (3.1), we can denote following parameters, Note that ψK L and ψK S are not exact CP eigenstates. G ψK S,L , S ψK S,L and C ψK S,L are parameters written in terms of λ ψK S,L . θ ψK S,L , G ψK S,L , S ψK S,L and C ψK S,L explicitly appear in coefficients of the master formula (A.1-A.5) for the processes. In eq. (3.4), λ ψK S,L is written in terms of the decay amplitude that the final state is the mass eigenstate ψK S,L . A . Note that the wrong strangeness decay amplitudes given as, 6) are numerically smaller than the right strangeness decay amplitudes given as, for the standard model. Thus, we ignore terms with higher power of A where θ K expresses CP and CPT violation in right strangeness decays of B meson and it corresponds toθ ψK of [6]. Note that in [6] indirect CPT violation z K is not taken into account. When deriving eq. (3.8), we calculated at linear order approximation with respect to z K , θ K and wrong strangeness decay amplitudes. Then, we can write λ ψK S,L as, where ∆λ wst consists of wrong strangeness decays as, where eq. (3.15) describes CPT violation in wrong strangeness decays. Similar to eq. (3.12), we can define a parameter including wrong sign decay amplitudes as, Since wrong sign semi-leptonic decay amplitudes and CPT violation, θ ψK 0 andθ ψK 0 , are small, we can expand eqs. (3.13, 3.14) as, Eq. (3.10) indicates that λ ψK S,L is composed of the leading part λ and the sub-leading part suppressed by wrong strangeness decay amplitude. Taking the CPT conserving limit in eq. (3.10), one can obtain the relation in ref. [7]. Note that λ has the definitive transformation property of T, CP and CPT, such as λ One introduces G, S and C analogous to eq.(3.2) by replacing λ f with λ. They are transformed under T as, The CP transformation property of G, S, C is the same as is close to 1 since deviation of |q/p|, |p K /q K | and |Ā ψK 0 /A ψK 0 | from 1 is small. Hence, we can find that C is a small parameter. One can also derive the transformation property of eqs. (3.13-3.14) such like λ wst The CP transformation property of the parameter (3.12, 3.16) is the same as eq. (3.21) Note that parameters G ψK S,L , S ψK S,L and C ψK S,L are related with the parameters G, S and C as, (3.24) where we use notation for an arbitrary complex number A, A R ≡ ReA, A I ≡ ImA, throughout this paper. When deriving eqs. (3.22-3.24), we ignored higher order terms of C and ∆λ wst . Eqs. (3.8, 3.22-3.24) lead relations given as, Since we have included the effect of indirect CP violation of Kaon system, we show how the correction due to ǫ K arises. While the expression of G, C and S in eqs. (3.18-3.20) is invariant under the arbitrary large rephasing such as K 0 | → e −iα K K 0 | and K0 | → e iα K K0 | , the parametrization with ǫ K ≪ 1 allows only the small rephasing α K ≪ 1.
Keeping only the terms which are linear to ǫ K , G, S and C are expanded as follows, where G ′ , S ′ and C ′ are obtained by taking the limit p K q K → 1 in G, S and C. Namely, they are defined by replacing λ with λ ′ in the expression for G, S and C. [6] where indirect CP violation ǫ K is neglected. When one changes the phase convention of states as, the phase of λ ′ changes as follows, Assuming the phase α K is small, G ′ , S ′ , and ǫ I K change as, while C ′ and ǫ R K are invariant, Hereafter, we expand C in terms of C ′ and ǫ R K as shown in eq.(3.29) and we do not expand S and G since they are invariant under the rephasing. The numerical significance of ǫ R K will be discussed in the next section. We turn to definition for parameters including semi-leptonic decay amplitudes of B meson. In the following, from eq.(3.34) to eq.(3.38), we adopt the notations of [6]. Right sign semi-leptonic decay amplitudes are denoted as, Wrong sign semi-leptonic decay amplitudes are similarly given as, For the case of the standard model, wrong sign semi-leptonic decay amplitudes are smaller than right sign decay amplitudes. Thus, we ignore higher powers of wrong sign decay amplitudes than linear order. We define parameters including semi-leptonic decay amplitudes as, where θ l ± expresses CPT violation in semi-leptonic decays of B meson. One can find the transformation law such like λ l + . We assume that CPT violating parameter θ l ± is small. At linear order approximation of θ l ± and wrong sign semi-leptonic decay amplitudes, we obtain, where we can see that contribution of θ l ± approximately vanishes in eq. (3.38). Following ref. [6], we also define G l ± , S l ± and C l ± analogous to eq.(3.2) by replacing λ f with λ l ± . The parameters G l ± , S l ± and C l ± explicitly appear in coefficients of master formula (A.1-A.5) for the processes in which the final states are given as l ± X. Eq.(3.38) gives approximate expressions for G l ± , S l ± and C l ± as, Note that eq. (3.39) implies that G l ± and S l ± are small numbers since λ l + and λ −1 l − are suppressed. We can find the relations, whereλ l and ∆λ l are defined as, They transform definitively under CP, T and CPT. One obtains the transformation property of T as,λ The CP transformation property ofλ l and ∆λ l is the same as (3.43). Hence, the CPT transformation property ofλ l and ∆λ l is also determined asλ l In eq. (3.44), R M expresses mixing-induced CP and T violation for B meson system [6] and is a small number. As for newly introduced parameters, ξ l implies CP and T violation in right sign semi-leptonic decays and we also assume ξ l is a small number. The expression of λ l ξ (3.45) includes right sign semi-leptonic decay amplitude ratio and we assume |A l + /Ā l − | ≃ 1. Therefore, C l ξ is a small number compared with O(1). The parameters defined in eq. (3.44) also appear in the asymmetry in eq.(2.3).
Hereafter, we describe some significant points of the parameters defined in this section. Note that the parameters given as, S, C, G, θ K , R M , z, z K ,λ l , ∆λ l , ξ l , C l ξ ,λ wst and ∆λ wst , have definitive transformation properties exhibited in table 2. In the processes which we consider, K S,L is included as a final state, and the effect of mixing induced T and CP violation, p K /q K , appears in the expressions of G, S, C,λ wst and ∆λ wst . Mixing-induced CP and CPT violation in K meson system, z K , reveals in the asymmetry as well. In the next section, the asymmetry is written in terms of parameters (3.46) and it can be explicitly separated as T-odd parts and T-even parts. The parameters defined as, are small numbers, and our calculation is based on linear order approximation with respect to the parameters (3.48) throughout this paper.  Table 3. Transformation property of the parameters devoted to keep the definitive transformation property of the parameters in table2

Time dependent asymmetry including the overall constants
In this section, we apply the event number asymmetry defined in eq.(2.7) to processes for B-meson decays. One should be aware of that the asymmetry considered in this paper includes the effect of different normalization for two rates; non-zero value of ∆N R defined in eq.(2.8). As the BaBar asymmetry investigated in [4], we also assign f 1 , f 2 , f 3 , f 4 with ψK L , l − X, l + X, ψK S , respectively. We call this process as I. We also consider the other three processes which can be obtained by interchanging l − X with l + X and ψK S with ψK L in the process I. Therefore we identify the four processes as, For all the processes which we consider, we can find ∆N R , ∆σ, yΓt, ∆C,Ŝ,Ĉ are small numbers compared with O(1). Expanding eq. (2.7) with respect to the small parameters, one obtains the asymmetry at linear order approximation, where, We ignore ∆σy in eqs. (4.3-4.6).σ and ∆S are O(1) parameters andσy gives rise to small contribution. The model independent parametrization in eq. (4.2) without the last term can be found in [6]. In each process, we compute the asymmetry and the coefficients We label suffix I ∼ IV on the quantities corresponding to each process to distinguish them. Below and in table 4, we show the asymmetry and the coefficients for the process I. For the other processes, we show them in tables 5-7. We first investigate ∆N R in eq. (2.8) for the process I. With eq. (D.4), one obtains, With eq. (4.7) and eqs. (D.5)-(D.9), one can also derive the coefficients in the asymmetry, Note that C ′ and ǫ R K are phase convention independent parameters due to definition of C ′ . Therefore, we state that all of eqs. (4.8-4.13) are expressed as phase convention independent parameters. In eq. (4.9), effect of indirect CP violation in K meson system explicitly appears and gives rise to O(10 −3 ) contribution to C I T . Assuming |q/p|−1, |Ā ψK 0 /A ψK 0 |−1, |1 + θ K | − 1 are small numbers, we can expand C ′ in eq.(3.30) as, (4.14) A theoretical prediction for the imaginary part of Γ d 12 /M d 12 is calculated [8], and it shows if CPT violations and the wrong sign decay in B → lX in eq. (4.9) are also negligible.
If R T , C T , S T , B T and D T were genuine T-odd quantities, they would vanish in the limit of T-symmetry. In other words, if there are non-vanishing contributions in the limit of T-symmetry, R T , C T , S T , B T and D T are not T-odd quantities. From eqs.(4.8-4.13), we find the T-even contributions. Some of them do not vanish in the limit of T-symmetry and they include C l ξ ,λ R wst θ R K , etc. The others are quadratic with respect to T-odd quantities and they vanish in the limit of T-symmetry. They include S∆λ I l , S∆λ R wst , S 2λI l , etc. Now we study condition that the asymmetry becomes a T-odd quantity. The following equations are needed for T-even terms in each coefficient to vanish, When the real part and imaginary part of λ do not vanish, both G and S are non-zero and the conditions that all the eqs. (4.15)-(4.19) are satisfied become, The conditions except C l ξ = 0 agree with ones obtained in [6]. The additional condition is required since we take account of the overall constants in the asymmetry.
In the first column of table 4, we show how each coefficient of the asymmetry in eq.(4.2) depends on T-odd combination of the parameters and in the other columns we show the dependence on T-even combination of the parameters. As for T-even contribution, we identify the sources of T-even contribution to the coefficients. In the second column, the contribution of θ K which is CP and CPT violation in right strangeness decays is shown. In the third column, the contribution of C l ξ which is CP and CPT violation in the right sign semi-leptonic decays is shown. In the fourth and the fifth column, T-even contribution from the wrong strangeness decays and the wrong sign semi-leptonic decays are shown, respectively.
In tables 5-7, we show the coefficients for the processes (II)-(IV). In Table 4. The coefficients of the asymmetry for the process I with the final state (f 1 , f 2 , f 3 , f 4 ) = (ψK L , l − X, l + X, ψK S ) and the sources which give rise to the non-vanishing contribution to the asymmetry. The sources of the first column corresponds to T-odd terms and the other correspond to T-even terms.
T-odd terms appendix E, we show a rule useful for deriving them.
Although the asymmetry in eq.(4.2) is not exactly T-asymmetry, the measurement of the coefficients are useful for constraining S and G as well as various non-standard interactions. Non-standard interactions include wrong sign decay and CPT violation. In the following subsections, we show how one can determine S and G and also show how one can constrain the various non-standard interactions. We first study the case without any assumption and in later subsections, we investigate two physically interesting cases, one corresponding to the case that CPT is a good symmetry and the other is the case without wrong sign decays. Since there are relations among the coefficients for different processes, we first identify the independent coefficients. From tables 4-7, one finds the Table 5. The coefficients of the asymmetry for the process II with the final state (f 1 , f 2 , f 3 , f 4 ) = (ψK S , l − X, l + X, ψK L ) and the sources which give rise to the non-vanishing contribution to the asymmetry. The sources of the first column corresponds to T-odd terms and the other correspond to T-even terms. Table 6. The coefficients of the asymmetry for the process III with the final states (f 1 , f 2 , f 3 , f 4 ) = (ψK L , l + X, l − X, ψK S ) and the sources which give rise to the non-vanishing contribution to the asymmetry. The sources of the first column corresponds to T-odd terms and the other correspond to T-even terms.

T-odd terms
following relations among the coefficients for the different processes.
They imply that there are ten independent coefficients. In table 8, we show how ten independent combination of the coefficients can be written in terms of CPT even , CPT odd, and wrong sign decay parameters. Since there are eighteen parameters, the number of the independent coefficients is not enough to extract the parameters. However, one can Table 7. The coefficients of the asymmetry for the process IV with the final states (f 1 , f 2 , f 3 , f 4 ) = (ψK S , l + X, l − X, ψK L ) and the sources which give rise to the non-vanishing contribution to the asymmetry. The sources of the first column corresponds to T-odd terms and the other correspond to T-even terms.
T-odd terms CPT even parameters CPT violating parameters wrong sign decays still constrain the combination of the parameters. Below we investigate how to extract the parameters for the three cases.

Extracting the parameters from the coefficients: General case
Let us first examine how one can determine the parameters from the measurements of the coefficients shown in table 8. Hereafter, we discuss a method to determine the values for G and S through observing E T . Since E T is multiplied by y in eq. (4.2), one cannot extract the value of E T solely from the measurement of the asymmetry. Therefore, we need to determine the value of y through the other experiment. y defined in eq. (2.2) is regarding to the width difference of B meson mass eigenstate, and a method to measure y cos β ≃ Gy is suggested in refs. [11]- [12]. Combining the measurement of E I T y ≃ GSy, one can determine S. Since S and G in their leading order satisfy S 2 + G 2 ≃ 1 − O(C 2 ), the measurement of E T determines (±G, S) within the two-fold ambiguity. The ambiguity would be removed if we assume that the standard model contribution is dominant for the width difference. (See and determine |S|. The sign ambiguity for S can be removed because in the leading order S is equal to . Excluding the case that the sub-leading contribution changes the sign of the leading term, one can determine the sign for S through observing . Having determined G and S, we consider constraining the other parameters.
We note that the following relation is satisfied, Since the right-hand side is independent of CPT violating parameters, non-vanishing contribution implies the unambiguous evidence of the presence of the wrong sign decay. Furthermore, since S is determined, one can write the CPT violating parameter θ I K as, If the right-hand side is non-zero, it implies CPT violation in the right strangeness decay. However, the real part of θ K cannot be singly extracted, since (4.23) One also notes the relation, , and D I T is non-zero, it implies CPT violation and/or wrong sign decay. However, if the cancellation between CPT violation and wrong sign decay occurs, they can vanish.

Extracting the parameters from the coefficients: CPT conserving limit
Next we consider the case in the limit of CPT symmetry. In the limit of CPT symmetry, all the contribution in the second column vanishes in table 8. Since all the wrong sign decay parameters are CPT even, the third column of table 8 does not vanish. In the limit C, S and R M − ξ R l can be determined as, Moreover T-odd wrong sign semi-leptonic decay ∆λ l can be determine as, For the other five wrong sign decay parametersλ R wst ,λ R,I l ∆λ R,I wst , one obtains the following three constraints.

Extracting the parameters from the coefficients: Case without wrong sign decay
Lastly, we consider the case without wrong sign decays. The relations in eqs. (4.22-4.24) are satisfied in this case. The right-hand side of eq. (4.21) vanishes. In addition to these, CP and CPT violation of the mixing parameters in B meson system is determined by CP and CPT violation in the neutral K meson system is also determined as, . (4.34) The five parameters C l ξ , θ R K , C, R M and ξ R l satisfy the two constraints eqs. (4.23-4.24).

Conditions for authentic time reversed process
In section 4, we showed the expression of the asymmetry that describes event number difference of figure 2 and figure 3. However, rather than figure 3, figure 4 is an authentic time reversed process of figure 2, since the two processes of figure 2 and figure 4 are related with flipping time direction. In discussion given in refs. [1]- [3], one substituted figure 3 for figure 4 because signal sides of figure 2 and figure 3 are deemed to be a time reversed process to each other. Since figure 3 is not an authentic time reversed process, the asymmetry is slightly deviated from T-odd.
In this section, we clarify why T-even parts are included in the coefficients eqs.(4.8-4.12), although it is naively thought to be a T-odd quantity.
One can show that, when the following conditions are simultaneously satisfied, figure 3 plays the role as a time reversed process of figure 2 and Violation of the conditions (5.1-5.2) is originally calculated in ref. [6]. Including overall factors and using our notation, we show the violation of the conditions(5.1-5.2) as follows, where we used the expression for states defined in eqs. (F.1-F.4, F.6-F.9). In eqs. (5.3-5.4), effect of mixing-induced CP violation in K meson system is included in terms of our nota- A ψK 0 in comparison with ref. [6].
B (l − X→)⊥ |B (→l + X)⊥ = 0 and B (l + X→)⊥ |B (→l − X)⊥ = 0 indicate that one cannot exactly conduct the flavor tagging in the presence of wrong sign semi-leptonic decays. Similarly, B (→ψK S )⊥ |B (ψK L →)⊥ = 0 and B (ψK S →)⊥ |B (→ψK L )⊥ = 0 imply that one cannot exactly carry out the CP tagging in the presence of CPT violation in decays and wrong sign strangeness decays. Therefore, eqs. (5.3-5.4) describe that semi-leptonic decays and strangeness changing decays yield tagging ambiguities, and that are expressed in terms of state non-orthogonality. Then, we turn to explanation of the second condition, ∆N e R = 0. We define the following quantities for the expedient sake.
where X = σ, C and S are given in eqs. (A.3-A.5). Consider the case that the equivalence conditions are satisfied to demonstrate that violation of ∆N e R = 0 gives rise to T-even contribution to the asymmetry. For that case, we can find that X o (X e ) defined in eq. (5.5) is T-odd (even) due to expressions given as follows, where for ∆σ andσ, we write down only the leading part since small parts of ∆σ andσ are neglected when multiplied by y in the asymmetry (2.7). For the process (I), ∆X and X defined in eqs. (2.5-2.6) are written as, One finds that the T-even part of ∆N R leads T-even contribution to ∆X in eq. (5.8).
The same applies toX, and it is shown that ∆X (X) deviates from T-odd (even) when ∆N e R has non-zero value. Therefore, we can demonstrate that ∆N e R gives rise to T-even contribution to the asymmetry given in eqs. (4.3-4.6).

Conclusion
In this paper, the precise meaning of the time reversal-like asymmetry is investigated, based on the most general time dependence of the asymmetry in eq.(4.2). In analysis of BaBar [4] and [6], the difference of the overall constants for the rates is eliminated. The ratio of the overall constants for the two decay rates is deviated from unity, and the deviation ∆N R = N R − 1 is taken into account in our analysis. If one takes the limits ∆N R = 0 and y = 0 in our analysis, the asymmetry of BaBar collaboration [4] is obtained. In our analysis, since the final states ψK S,L are not the exact CP eigenstates, one can find the effect of mixing-induced CP violation in K meson system. The effect of ǫ K is extracted and it gives rise to O(10 −3 ) contribution to C T , the coefficient of cos(xΓt). ǫ R K and direct CP violation |Ā ψK 0 /A ψK 0 | are dominant in C T , if the wrong sign semi-leptonic decay and CPT violations are negligible. As well as ǫ K , the contribution from CPT violation in Kaon system z K is estimated.
We introduced the parameters which have the specific property under CP, T and CPT transformations, including the effect of indirect CP violation in K meson system. Taking account of the difference for overall constants, the coefficients of each time-dependent function are written in terms of such parameters, and one can find that the asymmetry consists of not only T-odd terms, but also T-even terms in the most general time dependent function for the asymmetry. Furthermore, the introduced parameters are invariant under rephasing of quarks. We also found that the asymmetry is expressed as phase convention independent quantities.
We obtained the coefficients of the asymmetry for the processes (I-IV) and studied how to extract the parameters. Assuming that the value of y, i.e., the width difference of B d meson mass eigenstates is known, the three cases to constrain the parameters are discussed. For the most general case, combining the coefficients for different processes enables one to determine the parameters, S and G. We also find that non-zero value of some combination of the coefficient signals either CPT violation or the presence of the wrong sign decays. The other two cases correspond to CPT-conserving limit and the absence of wrong sign decays. In the CPT-conserving case, the coefficients constrain the parameters for wrong sign decays. In the absence of wrong sign decays, indirect CPT violation for B meson and K meson is constrained.
Moreover, we discussed T-even parts in the asymmetry. We found that T-even terms in the asymmetry vanish when several conditions are satisfied. These derived conditions are categorized as two parts. The first one is referred as equivalence conditions, regarding to B meson states for a time reversal-like process and an authentic time reversed process. As suggested in [6], B mesons for the two processes are not equivalent to each other, and we also showed the violation of the equivalence conditions, including the effects of mixing in K meson system. Since non-zero ∆N R is taken into account in our study, ∆N R can be the origin of T-even contribution to the asymmetry. We investigated that the second condition, which requires that T-even parts of ∆N R are zero, is needed for the asymmetry to become a T-odd quantity.

A Coefficients of master formula
We record coefficients of the master formula for the time-dependent decay rate of ref. [6].
where A i andĀ i in eq.(A.1) are defined in eq.(3.1). i and j represent the final state of tagging side (f i ) and signal side(f j ) for a pair of B meson decaying respectively.

B Incoming mass eigenstates and outgoing mass eigenstates in B meson and K meson system
Throughout this paper, the time reversal process of B meson decay often appears. To describe the inverse decay amplitudes, as out-states of B mesons, the reciprocal base must be used for Non-Hermitian Hamiltonian system. This is formulated in several literatures, refs. [13]- [17]. In this appendix, we show the definition of incoming states and outgoing states which are used in this paper.
The incoming mass eigenstates of effective Hamiltonian in B meson system are where p, q are mixing parameters in indirect CP violation and z is a mixing parameter in indirect CP, CPT violation. In terms of matrix elements of effective Hamiltonian, these parameters are written as These expressions help us understand the transformation property of λ, S, G, C, etc. Then, outgoing mass states are determined to fulfill the following conditions, Thus, outgoing mass eigenstates are For K meson system, similarly, Since these mass eigenstates in K meson system are shown in flavor states, we specifically can calculate the amplitudes of transition from incoming states B meson to outgoing states ψK L,S C List of coefficients of time dependent decay rates for process (I) In this appendix, we show the coefficients of the time dependent decay rates in eq. (2.1) which are needed for calculation of the asymmetry of process (I).
where we keep only the leading term for σ κ , since it will be multiplied by a small number y in the formulae of the decay rate eq. (2.1).
D Expressions for N R , ∆S, ∆C, ∆σ,σ,Ŝ andĈ The quantity N R , defined in eq.(2.4), denotes the ratio of a normalization for rates. Since we compute the asymmetry including the effect of N R , its expression should be clarified. In this appendix section, we calculate N R , and obtain the expressions of parameters as ∆S, ∆C, ∆σ,σ,Ŝ andĈ for the process (I). In deriving formulae, we use eqs. (C.1)-(C.14). Expanding N R with respect to small parameters, we obtain the general structure of N R at first order approximation.
We calculate only the leading part of ∆σ I andσ I , since the sub-leading part of ∆σ I and σ I is suppressed when multiplied with yΓt.
∆σ Il = 0,σ Il = 2G. (D.7) We write down the expressions forŜ I andĈ I as follows, E The relation among coefficients of the asymmetries for processes (I)-(IV).
In this appendix, we show the relation among the coefficients for different processes (I-IV). First, we note the coefficients of the process II(IV) are obtained by changing the sign of the mixing parameter q K and z K of I(III). The change of the sign of q K leads to the change of the sign for S, G and λ wst . Next, we show a simple rule which enables one to obtain the coefficients for table 5, with the coefficients of table 7. For this purpose we do not substitute ±1 for C l ± respectively and write the coefficients of asymmetry for process IV, When l + and l − in eq. (D.3) are exchanged, the sign of C l ξ and ξ R l is reversed. According to eqs. (3.40-3.41), the sign ofλ I l and ∆λ R l also changes. Additionally, one needs to interchange C l + and C l − in eqs. (E.1-E.7) and one can obtain the coefficients of asymmetry for process II.

F Calculation of equivalence conditions
In this appendix, we give the derivation of eqs. (5.3-5.4). The expression of the final state of signal side in figure (2) is, since the state is orthogonal to ψK S |. The state orthogonal to |ψK L is Similarly, one can write down, Calculating the inner product of eqs. (F.1) and (F.2), we obtain where we used eqs. (B.12-B.19). The inner product in eq.(F.5) was previously obtained in [6]. In eq.(F.5), we compute it with our notation including the normalization constant and have ignored the second order of small parameters z K , θ ψK 0 ,θ ψK 0 ,λ wst and ∆λ wst . Next, we show the derivation of the first line of eq. (5.3) and the second line of eq. (5.4). The states are given as, Their inner product is, The proportionality to the wrong sign decay amplitude λ l + is derived in [6].