Construction of a renormalization group improved effective potential in a two real scalar system

BaBar collaboration announced that they observed time reversal (T) asymmetry through B meson system. In the experiment, time dependencies of two distinctive processes, B− → B̄0 and B̄0 → 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 εK is extracted and gives rise to 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 Bd 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
Effective potentials improved by a renormalization group (RG) equation are widely applied in particle physics. In Refs. [1][2][3][4][5][6], the stability of an electroweak vacuum is studied through the evaluation of the RG improved effective potential on the high-energy scale. In addition, using the RG improved effective potential, the authors of Refs. [7][8][9][10][11][12][13][14][15][16][17][18] investigate the possibility that spontaneous symmetry breaking is realized by quantum correction to the effective potential. In this way, the RG improved effective potential is frequently utilized. There has been a great deal of research into the RG improvement of the effective potential since a study by Coleman and Weinberg [7]. In Refs. [19][20][21], the RG improved effective potential in a single field is derived. If utilizing the RG invariance of the effective potential, the renormalization scale µ is set as a field-dependent mass M (φ, µ); the logarithm log(M (φ, µ) 2 /µ 2 ) becomes zero. In that case, the logarithmic perturbative expansion of the effective potential including (log(M (φ, µ) 2 /µ 2 )) L at the L-loop level is stable because of log(M (φ, µ) 2 /µ 2 ) = 0. This is an essential point for the construction of the RG improved effective potential. If the theory includes multiple fields, the situation is not so simple. Taking M (φ, µ) as a renormalization scale, one cannot guarantee that the logarithm log(M ′ (φ, µ) 2 /µ 2 ) coming from another field is always small. If the logarithm is large, PTEP 2019, 043B03 H. Okane it leads to the breakdown of the perturbative expansion for the effective potential. In Refs. [22][23][24][25][26], the methods of solving such a problem are studied. The methods are classified into two types. In Refs. [22][23][24], multiple renormalization scales are introduced and each logarithm is suppressed by the multiple renormalization scales. On the other hand, the decoupling theorem [27] is applied in Refs. [25,26]. If a large logarithm appears in the calculation of the effective potential, the heavy particle is decoupled. Since the remaining logarithm is only one of a light field, the calculation of the RG improved effective potential is the same as the method explained in the single field case. Note that these methods are applied to theory including only a single scalar field.
If multiple scalar fields are introduced, the analysis of the RG improved effective potential is complicated because the masses appearing in the logarithms depend on multiple classical background fields such as M (φ 1 , φ 2 ). The problem is addressed in Refs. [28][29][30]. In Ref. [28], the RG improved effective potential is calculated with the introduction of the multiple renormalization scale. In Ref. [29], which extends the method of Ref. [26], a step function for the automatic decoupling of a heavy particle is introduced in the effective potential. Moreover, effective action is analyzed to take wavefunction renormalization into account. In Ref. [30], a new method is suggested. The guiding principle for the method is to choose the renormalization scale so that the total loop correction vanishes. In Ref. [31] the RG improved effective potential in classical conformal theory is analyzed based on the method of Ref. [30]. In the present paper, we also approach the problem for the RG improvement of the effective potential.
In this paper, extending the method of Ref. [25], we construct the RG improved effective potential in a two real scalar theory. Since the method of Ref. [25] is based on the analysis of the logarithmic structure of the effective potential, we derive the expression of the effective potential expanded with respect to all the logarithms appearing in a two real scalar system. Based on the analysis of the logarithmic structure of the effective potential, we choose the field-dependent mass eigenvalue as a renormalization scale so that one of the logarithms vanishes. If another logarithm at the renormalization scale is small enough to be perturbative, the RG improved effective potential is calculated with the choice of the renormalization scale. If the logarithm is large, we absorb the logarithm into the new parameters defined in the low-energy scale and decouple the heavy particle from the theory. Since the logarithm to be considered is only one of a light particle, we can easily evaluate the RG improved effective potential. The advantages of this method are as follows. First, since this method is based on the logarithmic structure of the effective potential at any loop order, the choice of the renormalization scale does not need to be changed even at higher loop order. Second, we can derive the RG improved effective potential without introducing multiple renormalization scales or a step function for the decoupling. Finally, we can easily implement the decoupling theorem by expanding the effective potential coming from quantum correction with respect to φ 2 /m 2 (φ 2 = φ 2 1 + φ 2 2 , m: decoupling scale).
This paper is organized as follows: In Sect. 2, we clarify the logarithmic structure of the effective potential and investigate the choice of the renormalization scale. In Sect. 3, the massless theory is treated and the RG improved effective potential is calculated based on the analysis of Sect. 2. In Sect. 4, we consider the massive model. In this section, we face a situation in which a large logarithm occurs. We decouple the heavy particle and construct the RG improved effective potential on the low-energy scale. In Sect. 5, we summarize the procedure of RG improvement in a multiple scalar model and discuss applications to other models. In Appendix A, the β and γ functions at the 1-loop level are given.

Logarithmic structure of effective potential and RG improvement
In this section, we clarify the logarithmic structure of the effective potential based on Ref. [25]. We then consider the choice of renormalization scale for the RG improvement of the effective potential. For a more specific explanation, we consider a two real scalar system as an example. The Lagrangian is given as follows: We suppose that this model has Z 2 × Z 2 symmetry: σ → −σ and χ → −χ . Following Ref. [25], we factor out a coupling constant 1/λ 1 from the Lagrangian 1 L = 1 Next, we shift the fields (σ , χ) by classical background fields (φ 1 , φ 2 ), respectively: and then redefine the quantum fields √ λ 1 σ and √ λ 1 χ as σ and χ, respectively. After the shift and the redefinition, the Lagrangian becomes where mass parameters (M 2 1 , M 2 2 , M 2 3 ), cubic coupling constants (x 1 , x 2 ), and quartic coupling constants (y 1 , y 2 ) are introduced as follows: H. Okane and V (0) is a tree-level effective potential: From the rewritten Lagrangian (3) and the tree potential (4), we can find that the theory is described by the following parameters: mass parameters : cubic coupling constants : quartic coupling constants : constant term : . (8) Moreover, since it is inconvenient for the mass matrix not to be diagonal, we rotate the mass matrix by introducing new states (σ d and χ d ) and mixing angle (θ): and then the mass matrix is diagonalized: where the mass eigenvalues are For later discussion, the coordinate (φ 1 , φ 2 ) is translated to the polar coordinate (φ, β): From now on, the mass eigenvalues and the effective potential are written with the polar coordinate (φ, β). At this stage, we can replace the three mass parameters (M 2 1 , M 2 2 , M 2 3 ) in Eq. (5) by mass eigenvalues (M 2 ± ) and mixing angle (θ). Namely, the model is described in terms of the following parameters: mass eigenvalues : mixing angle : θ, cubic coupling constants : quartic coupling constants : constant term : . (14) This information is so important that using these parameters we can write down the effective potential at the L-loop level as where P is the generic term of (p 1 , . . . , p 7 ): Let us explain why the L-loop effective potential can be written as Eq. (15). Since λ 1 can be treated like an in front of the action, the L-loop effective potential is proportional to λ L−1 1 . The part in the square brackets [· · · ] in Eq. (15) is dimensionless because M 4 − is extracted as a dimensionful part of V (L) . So since we introduce dimensionless parameters (p 1 , . . . , p 7 ) based on Eqs. (10)- (14), the part in the square brackets [· · · ] can be written in terms of two logarithms (log(M 2 − /µ 2 ) and log(M 2 + /µ 2 )) and dimensionless parameters (p 1 , . . . , p 7 ). As is well known, since the L-loop effective potential V (L) contains the Lth power of the logarithm at most, one can express V (L) with respect to log(M 2 − /µ 2 ) and log(M 2 + /µ 2 ): where, multiplying each logarithm by λ 1 , we define s 1 and s 2 : Finally, by summing up V (L) from L = 0 to L = ∞, we obtain the total effective potential expressed in terms of s 1 and s 2 : In this expression the power of λ 1 gives the order of the leading log-series expansion. In this sense, f l means the lth-to-leading log function of the effective potential. Next, we consider the choice of renormalization scale. As is well known, the effective potential satisfies the RG equation where the RG differential operator is given as where These specific β and γ functions are given in Appendix A. We can then obtain the solution of the RG equation as where we use a shorthand notation Q(= m 2 1 , m 2 2 , λ 1 , λ 2 , λ 3 , ) and introduce t to express the renormalization scale µ 2 as µ 2 (t) = µ 2 0 e 2t . Also,Ḡ(β, t) is defined as However, because of γ φ 1 = γ φ 2 = 0, from now on, we setḠ(β, t) = 1.Q(t) is the solution of β or γ function and satisfies an initial value Q at an initial renormalization scale µ 2 0 or t = 0. The RG solution of Eq. (24) for the effective potential means that it is independent of the renormalization scale t. Since we can freely choose the renormalization scale, we look for the best choice of it. Let us take the renormalization scale as follows: Since this choice leads tos 1 (t) = 0, the RG improved effective potential expressed with Eq. (19) becomes where from Eq. (20) Here, if we assumes 2 O λ 1 , one gets the lth-to-leading log function: Ifs 2 O λ 1 and we would like to evaluate the effective potential up to Lth-to-leading log order, the expression is written as We notice that the term of O λ 1 in Eq. (27) contributes to the effective potential beyond the Lth-toleading log order. Note that the RG improved effective potential is exactly correct up to Lth-to-leading log order only if the RG equations for the parameters are solved up to the (L + 1) loop level. In summary, if one prepares the L-loop effective potential and (L + 1)-loop β and γ functions, one can construct the RG improved effective potential (28) up to Lth-to-leading log order for the case of We comment on the variables of the effective potential. Originally, the effective potential has three variables (φ, β, t). However, now that µ(t) 2 is taken to be equal toM − (φ, β, t) 2 , these variables are related. In our paper, we show that we can solve µ(t) 2 =M − (φ, β, t) 2 analytically with respect to φ and construct the RG improved effective potential by using the solution of φ. 2 Since the above prescription is correct only in the case ofs 2 O λ 1 , we must consider the method of the RG improvement for the case ofs 2 > O λ 1 . In that case, as seen from the logarithm log M + (t) 2 /M − (t) 2 ofs 2 , the relative magnitude of the mass eigenvalues is large. In such a case, we make use of the decoupling theorem [25]. The decoupling of the heavy particle means that the logarithm of the particle is absorbed into the parameters defined in the effective theory. The remaining logarithm is only one of a light particle. If the theory includes only a single logarithm, by setting the renormalization scale as the light mass, the RG improved effective potential can be calculated. We discuss this situation more specifically in Sect. 4.

RG improved effective potential in a two real scalar system (massless case)
We specifically calculate the RG improved effective potential by the method constructed in Sect. 2. In this section we treat the two real scalar model without mass parameters. The procedure for the construction of the RG effective potential is as follows. Because of taking the renormalization scale as µ 2 0 e 2t =M − (t) 2 , we solve it with respect to φ. Substituting the φ into the mass eigenvalue M 2 + and the effective potential, we can evaluate log(M + (t) 2 /M − (t) 2 ) and the effective potential. If the logarithm is small enough fors 2 to be of the order ofλ 1 , we can use the expression of Eq. (28) as the effective potential up to Lth-to-leading log order. In order to obtain φ with µ 2 0 e 2t =M − (t) 2 satisfied, we solve it in terms of φ. In the present model, the mass eigenvalues M 2 ± are written as So we can easily obtain φ from µ 2 0 e 2t =M − (t) 2 : PTEP 2019, 043B03 H. Okane
As mentioned above, now φ is not the variable of the effective potential and is determined by β and t. The φ appearing in the mass eigenvalueM 2 + and the effective potential is calculated with Eq. (29). The logarithm ofs 2 is written as where φ is canceled out because of the massless model. At this stage we assign initial values of (λ 1 , λ 2 , λ 3 ) for performing the numerical calculation. Taking λ 1 = 0.7, λ 2 = 0.5, λ 3 = 0.2, and φ = 10 at for the range of β ∈ (0, π) and t ∈ (−10, 0) by 1-loop β functions in Fig. 1. In Fig. 1 we see that log(M + (t) 2 /M − (t) 2 ) ≈ 1 in the regions of (β, t). Thus, since we can conclude thats 2 ≈λ 1 , Eq. (28) can be used as the RG improved effective potential. Using the tree-level effective potential and the 1-loop β function, the RG improved effective potential at the leading log order is given as where the condition for φ 2 originates from the choice of the renormalization scale µ 2 =M − (t) 2 , as seen in Eq. (29). Clearly the RG improved effective potential is determined by β and t. In Fig. 2, the RG improved effective potential is plotted as axes of (φ 1 /µ 0 , φ 2 /µ 0 ) for the regions of β ∈ (0, π) and t ∈ (−10, 0).

Fig. 2.
A 3D plot of the RG improved effective potential at the leading log order divided by the initial renormalization scale µ 4 0 as axes of (φ 1 /µ 0 , φ 2 /µ 0 ). This is plotted in the regions of β ∈ (0, π) and t ∈ (−10, 0). The initial condition is the same as in Fig. 1.

RG improved effective potential in a two real scalar system (massive case)
In this section we consider the massive theory in a two real scalar model. In particular, we treat the effective potential causing spontaneous symmetry breaking. The procedure for the construction of the RG improved effective potential is the same as the previous method. We solve Eq. (26) for φ in the massive case. In this case, because of the mass parameters, the equation is a little complicated but it can be analytically solved. Equation (26) is written as follows: where Squaring both sides of A = √ B, a quadratic equation for φ 2 is given as where a = λ 1 λ 3 cos 4 β + λ 2 λ 3 sin 4 β + (λ 1 λ 2 − 3λ 2 3 ) sin 2 β cos 2 β, We can obtain the solution φ 2 as Since we solve the quadratic equation, there are two solutions for φ 2 . However, because the original equation is A = √ B, the solution satisfies the following conditions: Although it is difficult to analytically prove whether either solution satisfies the condition or not, by using the initial values as inputs in the following subsections we confirm numerically the following results: Therefore we adopt the solution φ as Since we get the solution φ for Eq. (26), we can construct the RG improved effective potential. The expression is provided at leading log order as In the following subsections, we consider two situations for inputting the initial value of the renormalization scale. First, taking (m 2 1 < 0, m 2 2 < 0, −m 2 1 ∼ −m 2 2 ) as the mass parameters, we set the initial renormalization scale on the vacuum, which is determined by the stationary condition of the effective potential. Increasing the renormalization scale from the low-energy scale at the vacuum, we analyze the behavior of the RG improved effective potential in the high-energy region. Second, we input the initial values of the parameters on the high-energy scale and decrease the renormalization scale to the low-energy scale. Assuming (m 2 1 < 0, m 2 2 > 0, −m 2 1 ≪ m 2 2 ) for the mass parameters, we investigate the RG improved effective potential in the low-energy region. As the renormalization scale decreases, the mass eigenvalueM 2 − also declines and reaches m 2 2 at a scale. SinceM 2 − continues to decline below the scale, we find the logarithm ofs 2 large. In order to avoid the breakdown of the logarithmic perturbation, we utilize the decoupling theorem. Applying the decoupling theorem, we derive the RG improved effective potential in the low-energy scale and visualize the behavior including the minimum value of the RG improved effective potential. Since we set the initial condition on the vacuum in this subsection, we derive the stationary condition for the effective potential. Introducing a convenient notation for the mass parameter and quartic coupling constant: m(β) 2 = m 2 1 cos 2 β + m 2 2 sin 2 β, λ(β) = λ 1 cos 4 β + λ 2 sin 4 β + 6λ 3 sin 2 β cos 2 β, we can write the effective potential at a tree level: We calculate the stationary conditions for the effective potential: From ∂V (0) ∂φ = 0, we derive the following condition: Combining this condition and ∂V (0) ∂β = 0, we get the stationary condition for β: Substituting this β for Eq. (39), we can obtain φ on the stationary point: Using Eqs. (41)-(40), we can calculate the vacuum expectation value and also estimate the initial renormalization scale µ 2 0 =M − (t = 0) 2 = M 2 − . For simplicity, in this section we impose = 0 at the initial point.
Taking [λ 1 = 0.7, λ 2 = 0.5, λ 3 = 0.1, m 2 1 = −(160 GeV) 2 , and m 2 2 = −(170 GeV) 2 ] as an initial condition, we get the vacuum expectation value (φ, β) = (591 GeV, 0.94), the initial renormalization scale µ 0 = M − = 135 GeV, and the mass eigenvalue M + = 236 GeV. We regard the vacuum expectation value and the initial renormalization scale as a starting point for the RG improved effective potential and the running parameters. Then, we run log(M + (t) 2 /M − (t) 2 ) by the RG equations in the regions of t ∈ (0, 5) and β ∈ (0, π 2 ). Figure 3 shows the result of the logarithm. On β = 0 and β = π 2 in Fig. 3, the logarithm takes 2-3 in the range of t = (0, 2) and less than 2 for t > 2. In β = π 4 , the logarithm is less than 1 for the whole scale of t. If the magnitude of the logarithm as log(M 2 − /M 2 + ) 3 is accepted in the context of a logarithmic perturbative expansion, the RG improved effective potential is calculated with Eq. (37). The result is shown in Fig. 4. In the left panel of Fig. 4, the dot-dashed green, dotted orange, and solid blue lines correspond to the RG improved effective potential in β = 0, β = π 4 , and β = π 2 , respectively. In the right panel of Fig. 4, the dot-dashed green, dotted orange, and solid blue lines correspond to φ in β = 0, β = π PTEP 2019, 043B03 H. Okane  The dot-dashed green, dotted orange, and solid blue lines correspond to the RG improved effective potential in β = 0, β = π 4 , and β = π 2 , respectively. Right: The dot-dashed green, dotted orange, and solid blue lines correspond to φ in β = 0, β = π 4 , and β = π 2 , respectively. The initial condition for the RG equation is the same as in Fig. 3. β = π 2 , respectively (φ in β = 0 and π 2 are φ 1 and φ 2 , respectively). In both panels of Fig. 4, the lines at β = 0 and π 2 overlap with each other. We give a more complete discussion for the logarithmic perturbative expansion. As explained above, there are regions in which the logarithm is beyond 1. If the logarithm is considered to be large, the heavy field with massM + should be decoupled from the theory. Due to this decoupling,  the remaining logarithm is only log(M 2 − /µ 2 ). Since the single logarithm can be suppressed by using the degree of freedom of the renormalization scale µ, the logarithmic perturbation is stable. Such a procedure is explained in the next subsection.

m
In this subsection we impose the initial condition at a high-energy scale and gradually decrease the renormalization scale to a scale around −m 2 1 . Also we suppose that m 2 2 ≫ −m 2 1 > 0. Setting the following initial condition: at (φ, β) = (40 000 GeV, π 4 ), we evaluate the logarithm of the ratio ofM + (t) 2 toM − (t) 2 in Fig. 5. Clearly, the logarithm becomes large as the renormalization scale decreases to the low-energy scale. This indicates the breakdown of the logarithmic perturbative expansion in the low-energy region. For more detail, we evaluate the ratio ofM − (t) 2 tom 2 (t) 2 in the left panel of Fig. 6. As seen from the left panel in Fig. 6,M − (t) steadily falls with decreasing renormalization scale t. The ratio ofM + (t) 2 to m 2 2 is calculated in the right panel of Fig. 6. In contrast to the figure on the left, the figure shows that the value ofM + (t) is comparable tom 2 (t) below t = −1. Therefore in Fig. 6 we find out that the ratio ofM + (t) 2 toM − (t) 2 increases with lower renormalization scale becauseM − (t) is smaller thanm 2 (t) whileM + (t) is comparable tom 2 (t).
In order to avoid a large logarithm, we should modify the RG improved effective potential for the low-energy scale. The way to modify the RG improvement is to utilize the decoupling theorem. In the present case, sinceM + (t) is heavier thanM − (t), the field with the massM + (t) should be decoupled. Moreover, as seen in the right panel of Fig. 6, sinceM + (t) is comparable tom 2 (t), we  Left: The logarithm of the ratio ofM − (t) 2 tom 2 (t) 2 is evaluated in the regions of β ∈ (0, π 2 ) and t ∈ (−6, 0). Right: The logarithm of the ratio ofM + (t) 2 tom 2 (t) 2 is calculated in the regions of β ∈ (0, π 2 ) and t ∈ (−6, 0). factor outm 2 (t) 2 from the expression ofM + (t) 2 . Hereafter we omit the bar of the parameters to reduce the bother. To implement it, we expand M 2 + with respect to φ 2 In this expression we see that log(m 2 2 /µ 2 ) leads to a large logarithm, which is not suppressed with the choice of µ 2 = M 2 − . The concept of the decoupling theorem is to absorb the large logarithm into new parameters by the redefinition of the parameters. Hence we combine the 1-loop effective potential with the tree effective potential and redefine the new parameters to renormalize the large logarithm: (m 2 1 cos 2 β +m 2 2 sin 2 β) PTEP 2019, 043B03 H. Okane + φ 4 4! (λ 1 cos 4 β +λ 2 sin 4 β + 6λ 3 sin 2 β cos 2 β) +˜ , wherem whereX =m 2 1 ,m 2 2 ,˜ ,φ 1 ,φ 2 ,Ỹ =λ 1 ,λ 2 ,λ 3 .
Hence we can get the β and γ functions defined by the tilde parameters: We notice that the effect of the heavy field disappears from the RG equation in Eqs. (53)-(55). In this sense the heavy field is decoupled from theory in the low-energy scale. We can construct the RG PTEP 2019, 043B03 H. Okane  The 3D plot of the RG improved effective potential is evaluated as a function of (φ 1 , φ 2 ). Right: The RG improved effective potential is plotted as a function of φ 1 with φ 2 equal to zero (φ 2 = 0). From the minimum point of the RG improved effective potential, the vacuum expectation value is estimated as (β = 0, t = −3.25).
improved effective potential by replacing the parameters with the tilde parameters for the effective potential in Eq. (37).
Let us consider a decoupling point at which the theory is separated into the full theory and the low-energy effective theory. From the left panel of Fig. 6, we see thatM − (t) coincides withm 2 (t) around t = −1. Actually, as we can identify the scale as t = −1.2 andM − (t) do not vary in the range of β ∈ (0, π 2 ), we use (β, t) = ( π 2 , −1.2) as a decoupling point. The choice of the decoupling point is valid because the logarithm in Eqs. (44)-(50) is suppressed at the scale whenM − (t) becomes equal tom 2 (t). Now we can solve the RG equations for all the parameters from the initial scale to the low-energy scale. In the left panel of Fig. 7, the quartic coupling constants are solved from t = 0 to t = −4. We can confirm the slight threshold correction forλ 3 . The difference betweenλ 3 (t = −1.2) andλ 3 (t = −1.2) normalized byλ 3 (t = −1.2) is 0.02. In the right panel of Fig. 7, we run the mass eigenvalues in the same range. TheM 2 − continues to decrease as the renormalization scale is lowered, while theM 2 + converges to about 3000 GeV. In the left panel of Fig. 8, the RG improved PTEP 2019, 043B03 H. Okane effective potential is plotted as a function of (φ 1 , φ 2 ). We can find the minimum value of the RG improved effective potential. This point corresponds to the vacuum in the present model. The right panel of Fig. 8 shows the behavior of the RG improved effective potential as a function of φ 1 with φ 2 equal to zero (φ 2 = 0). From the evaluation of the RG improved effective potential, the vacuum expectation value corresponds to (β, t) = (0, −3.25). Substituting them into the mass eigenvalues, we obtain the values of the masses: (56)

Summary and discussion
In this paper we have studied the RG improvement of the effective potential in a two real scalar system. In Sect. 2 we clarify the logarithmic structure of the effective potential. If we choose µ 2 0 e 2t =M 2 − (t) as a renormalization scale and the logarithm ofs 2 is less than O(1), we find that the RG improved effective potential up to Lth-to-leading log order can be calculated by an L-loop effective potential and (L + 1)-loop β and γ functions. In Sects. 3 and 4, we solve µ 2 0 e 2t =M 2 − (t) with respect to φ. This means that φ is not a variable of the effective potential but becomes a function of β and t. By using φ we can evaluate the mass eigenvalueM 2 + and the RG improved effective potential. Then, we examine if the logarithm of the ratio ofM If it is satisfied, the RG improved effective potential can be obtained as mentioned above. On the other hand, ifs 2 > O(λ 1 ), the heavy particle should be decoupled. In Sect. 4, we study such a situation. We absorb the large logarithm into the new parameters defined in the low-energy scale and derive the RG equations described in terms of the redefined parameters. The RG improved effective potential can then be constructed in the low-energy region.
There are three features in this method. First, we do not need to change the choice of the renormalization scale beyond the leading log order. This is because, since we analyze the logarithmic structure of the effective potential at any loop order, the choice µ 2 0 e 2t =M − (t) 2 is valid for the RG improvement up to arbitrary lth-to-leading log order. Due to this, the φ that satisfies µ 2 0 e 2t =M 2 − (t) is the same as the one in the leading log order. So we do not need to resolve µ 2 0 e 2t =M − (t) 2 with respect to φ. Note that the RG equations must be solved in a loop level corresponding to the desired leading log order. Second, we can derive the RG improved effective potential without introducing multiple renormalization scales or a step function by which the heavy particle is automatically decoupled. Third, we can decouple the heavy particle from the theory by expanding the quantum correction to the effective potential with respect to φ 2 /m 2 . If the logarithm log φ 2 /m 2 is absorbed into the parameters in the low-energy scale, we can derive the RG improved effective potential.
Our method can be applied to other multiple scalar models. If multiple scalar fields are introduced in a model, one represents the classical background fields in terms of polar coordinates such as chosen as a renormalization scale, the φ corresponding to a radius of the polar coordinate becomes a function of the renormalization scale t and angles in the polar coordinate apart from whether it can be solved analytically. If one reaches this stage, one can implement the calculation of the RG improved effective potential in the same way as in this paper. Finally, since the stability issue and the origin of spontaneous symmetry breaking are investigated through the RG improved effective potential, our work contributes to such studies in a multiple scalar theory.

Acknowledgements
We thank T. Morozumi and Y. Shimizu for reading our manuscript and giving useful comments.

Appendix A. β and γ functions in a two real scalar model
In this appendix, we provide the β and γ functions in a two real single scalar model: 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 ϵ K is extracted and gives rise to 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 andB 0 → B − . In refs.

JHEP02(2015)174
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 semi-leptonic 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 ϵ 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. 2 Formula for asymmetry of entangled systems In ref. [6], a formula for the time-dependent decay rate of the entangled BB system is derived. When f 1 and f 2 denote the final states of a tagging side and a signal side, respectively, it is written as, The expressions for N (1)⊥,2 , κ (1)⊥,2 , σ (1)⊥,2 , C (1)⊥,2 and S (1)⊥,2 are given in ref. [6]. For the sake of completeness, we record their expressions in eqs. (A.1)-(A.5). Hereafter, we evaluate an asymmetry including overall factor N (1)⊥,2 κ (1)⊥,2 in eq. (2.1). One obtains a generic formula for the event number asymmetry of the two distinctive sets for final states; In eqs. (2.3)-(2.6), contribution from different overall factors in eq. (2.1) for two processes are taken into account. Taking the limit N R → 1 and y → 0 in eq. (2.3), one finds an asymmetry whose overall normalization is eliminated, used in [4]. In eq. (2.5)-(2.6), ∆S (∆C) is equal to ∆S + T (∆C + T ) defined in [6] when one takes the limit N R → 1. In practice, we only need to consider the time difference t within the interval [0, τ B ] where τ B is the life time of B meson. Therefore, the approximation sinh(yΓt) ≃ yΓt, cosh(yΓt) ≃ 1 is valid JHEP02(2015)174 since y ≪ 1 for neutral B meson system. Thus, we expand A with respect to yΓt, where we denote

Parameter definitions in terms of flavor based state
In this section, we introduce parameters that reveal in the event number asymmetry in eq. (2.7) which we consider. In the processes which form the asymmetry, final states of B-decay are given as the same ones used for the BaBar experiment [4]. Mixing-parameters, p, q, z, p K , q K and z K are defined in appendix B. Eqs. (B.3)-(B.4) lead the transformation property of mixing parameter as, p The transformation 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 . Note that the wrong strangeness decay amplitudes given as, 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,

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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 (3.18)- (3.20). Thus, the CPT transformation property is also determined as G |λ| 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. 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, where we use notation for an arbitrary complex number A, A R ≡ ReA, A I ≡ ImA, throughout this paper. When deriving eqs.

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Since we have included the effect of indirect CP violation of Kaon system, we show how the correction due to ϵ 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 ⟨K 0 | → e iα K ⟨K 0 | , the parametrization with ϵ K ≪ 1 allows only the small rephasing α K ≪ 1.
Keeping only the terms which are linear to ϵ 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.
As shown in table (1), (G ′ , S ′ , C ′ ) equal to (Ĝ ψK ,Ŝ ψK ,Ĉ ψK ) defined in [6] where indirect CP violation ϵ 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 ϵ I K change as, while C ′ and ϵ R K are invariant, Hereafter, we expand C in terms of C ′ and ϵ 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 ϵ 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,

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Notation of this paper Notation of [6] Notatin of [6] Notation of this paper Table 1. The correspondence of parameters in this paper and those of [6]. In this paper, ψK 1 corresponds to ψK S and ψK 2 corresponds to ψK L respectively in [6] where indirect CP violation parameter ϵ K is neglected. In this paper, ψK L and ψK S include the effect of indirect CP and CPT violation. The first column shows the quantities defined for mass eigenstates (K L , K S ). From the third row to the eighth row in the second column, the quantities in the first column are expanded up to the first order of ϵ K and are written in terms of the quantities for CP eigenstates K 1 , K 2 . In the third column and in the fourth column, we show how (Ĝ ψK ,Ŝ ψK ,Ĉ ψK ) and (∆G ψK , ∆S ψK , ∆C ψK ) in [6] are related to (G ′ , S ′ , C ′ , ∆λ wst ) defined in this paper. About CPT violation parameter of strangeness changing decay, one can show θ ψK1 = θ ψK2 = θ K . Thereforeθ ψK = θ K and ∆θ ψK = 0.
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 + − −− → λ l + by its definition (3.36)-(3.37). 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) -8 -

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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, 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 ,   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, p/q, p K /q K , θ ψK 0 ,θ ψK 0 , θ l ± , λ, λ wst ψK 0 ,λ wst ψK 0 , λ l ± , and λ l ξ , The parameters given as, θ ψK 0 ,θ ψK 0 , θ l ± , C, θ K , R M , z, z K ,λ l , ∆λ l , ξ l , C l ξ ,λ wst and ∆λ wst , (3.48) are small numbers, and our calculation is based on linear order approximation with respect to the parameters (3.48) throughout this paper.

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 -10 -

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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 (R T , C T , S T , B T , D T , E T ). 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,

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Note that C ′ and ϵ 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 appendix E, we show a rule useful for deriving them.  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 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.
T-odd 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.

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T-odd terms 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.
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 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 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 -14 -

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CPT even parameters CPT violating parameters wrong sign decays 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 . 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 One also notes the relation, If any one of the combinations, , 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,

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For the other five wrong sign decay parametersλ R wst ,λ R,I l ∆λ R,I wst , one obtains the following three constraints. (4.32)

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 . (4.33) CP and CPT violation in the neutral K meson system is also determined as, 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

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The equivalence conditions indicate that the initial (final) B meson states of signal side in figures 3-4 are the same as each other. The equivalence conditions are described as, Eq. (5.1) shows that B meson states in figures 3-4 are equivalent. Similarly, figure 5 is the authentic time reversed process of figure 3. When we apply the same condition to B meson states in figures 2-5, one obtains, 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, A ψK 0 in comparison with ref. [6].
Then, we turn to explanation of the second condition, ∆N e R = 0. We define the following quantities for the expedient sake.    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 ϵ K is extracted and it gives rise to O(10 −3 ) contribution to C T , the coefficient of cos(xΓt). ϵ 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 ϵ 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 -20 -

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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.

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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 .

(B.4)
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 Hence, we can obtain eqs.

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.

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Deviation of N l + X /N l − X and N ψK S /N ψK L from 1 is written in terms of small parameters as, Note that ∆N I R is a small number. We can also write down the expressions of ∆S I and ∆C I .
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.
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

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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.
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 . 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].

Introduction
Although all the elementary particles in the standard model (SM) have now been discovered, with the discovery of the Higgs boson, there still exist phenomena which cannot be explained in the framework of the SM. One of these is the neutrino oscillation phenomenon, which implies two non-zero neutrino mass squared diAEerences and two large lepton mixing angles. In order to explain this, many authors propose a neutrino flavor model with non-Abelian discrete flavor symmetry in the lepton sector (for reviews see [1][2][3][4]). Even before the discovery of the non-zero µ 13 [5][6][7], a few authors suggested a tiny mixing angle µ 13 based on non-Abelian discrete flavor symmetry [8]. Recent results from the T2K and NO∫A experiments [9,10] imply CP violation through the Dirac CP phase. They studied electron neutrino appearance in a muon neutrino beam. The Majorana phases are also sources of the CP violating phases if neutrinos are Ma-jorana particles. The KamLAND-Zen experiment [11] is searching for neutrinoless double beta (0∫ØØ) decay to check the Majorana nature of neutrinos. Therefore, it is important to predict not only mixing angles but also CP phases with the non-Abelian discrete flavor model.
The non-Abelian discrete flavor symmetry can easily explain large lepton mixing angles, e. g. tri-bimaximal mixing (TBM) [12,13], which is a simple framework for the lepton mixing angles. Indeed, Altarelli and Feruglio (AF) proposed a simple flavor model and predicted TBM by using A 4 discrete flavor symmetry [14,15]. They introduced SU (2) gauge singlet scalar fields, so-called "flavons", and derived the TBM in the lepton sector. The non-zero µ 13 can be realized by another A 4 nontrivial singlet flavon [8] in addition to the flavons introduced by AF. The origin of non-vanishing µ 13 is related to a new contribution to the mass matrices. Matrices which have the same structure as that in Ref. [8] also appear in extra-dimensional models with the S 3 and S 4 flavor symmetries [16,17]. The ¢(27) model also includes these matrices [18].
In this paper, we study phenomenological aspects of a supersymmetric model with A 4 £Z 3 symmetries. The three generations of the left-handed leptons are expressed as the A 4 triplet, l = (l e ,l µ ,l ø ), while the right-handed charged leptons e R , µ R , and ø R are A 4 singlets denoted as 1, 1 00 , and 1 0 respectively. Three right-handed neutrinos are also described as the triplet of A 4 . We introduce the SU (2) gauge singlet flavons of A 4 triplets, ¡ T =(¡ T 1 ,¡ T 2 ,¡ T 3 ) and ¡ S =(¡ S1 ,¡ S2 ,¡ S3 ). In addition, ª and ª 0 are also introduced as the SU (2) gauge singlet flavons with the two kinds of singlet representations of A 4 , 1 and 1 0 respectively.
We focus on the vacuum structure of the flavor model. The scalar sectors of this model consist of many flavons in addition to the SM Higgs boson. Then, we analyze the scalar potential and show the 24 diAEerent sets of VEVs which come from 24 combinations of 4 (6) possible VEVs of the flavon ¡ T (¡ S ). The 24 diAEerent vacua are classified into two types which are not related to each other under the transformations A 4 . Therefore, we expect that the two types of vacua have diAEerent expressions for the physical observables in terms of the model parameters such as Yukawa couplings. We ask the following question: whether these diAEerent vacua are physically distinct from each other. The purpose of this paper is to clarify the diAEerences and relations among the VEVs and their physical consequences. In particular, we investigate the mixing angles, CP violating phase, and eAEective mass for neutrinoless double beta (0∫ØØ) decay.
This paper is organized as follows. In Section 2, we introduce the supersymmetric model with A 4 £Z 3 symmetry. In Section 3, we study the classification of vacua and derive the formulae for the mixing angles and CP phases. In Section 4, we discuss the phenomenological aspects for mixing angles and CP violating phases. The numerical analyses for the eAEective mass of 0∫ØØ decay are presented. Section 5 is devoted to a summary. In Appendix , we show the multiplication rule of the A 4 group.

Supersymmetric model with A 4 £ Z 3 symmetry
In this section, we introduce a supersymmetric model with A 4 £Z 3 symmetry. We analyze the scalar potential and derive the mass matrices of the lepton sector.

Model
We introduce three heavy right-handed Majorana neutrinos. The leptons and scalars in our model are listed in Table 1.
The superpotential of Yukawa interactions is where w l ,w D and w R are Yukawa interactions for charged lepton, Dirac neutrino and Majorana neutrino sectors respectively: where the lower indices denote A 4 representations. More-over, the y's and § denote the Yukawa coupling constants and cut-oAE scale respectively. The multiplication rule for A 4 representations is shown in Appendix A.
In order to obtain the mass matrices of these leptons, we analyze the following superpotential of the scalar fields: where 023102-2 Chinese Physics C Vol. 42, No. 2 (2018) 023102 Table 2. The driving fields and their representations and charge assignment.
We have introduced the additional SU (2) gauge singlet fields, ¡ T 0 , ¡ S 0 and ª 0 , which are called "driving fields". The charge assignments of these fields are summarized in Table 2.

Potential analysis
In this subsection, we derive the VEVs for the scalar fields ¡ T ,¡ S ,ª,ª 0 ,¡ T 0 ,¡ S 0 ,ª 0 . One can derive the scalar potential from the superpotentials in Eqs. (6) and (7) as where and The sum for X,Y runs over all the scalar fields: There are several solutions for the minimization condition. We obtain sets of solutions denoted as ¥ m and ∏ ± n (m=1-4, n=1-3), where ¥ m and ∏ ± n are the solutions of V T = 0 and V S = 0 respectively. Hereafter, we call them the set of VEV alignments and show them explicitly as follows: where u and u is the VEV of ª, hªi=u 1) . The superscript of ∏ ± denotes the overall sign of the VEV h¡ S i. In total, we obtain 24 sets of vacua, since there are four sets of alignment for ¥ m and six sets for ∏ ± n .

Mass matrix for charged leptons and neutrinos
We derive charged lepton mass matrices and neutrino mass matrices from the Yukawa interactions in Eqs.
B @°y e 2y µ 2y ø 2y e°yµ 2y ø 2y e 2y µ°yø B @°y e 2! 2 y µ 2!y ø 2!y e°yµ 2! 2 y ø 2! 2 y e 2!y µ°yø respectively, where the matrices S and T are The Dirac mass matrix for neutrinos obtained from Eq. (3) is It is noted that the Dirac mass matrix is determined independently of the VEV alignments. The Majorana mass matrices M (n) R ± for the corresponding set of solutions Eqs. (15)-(17) are given as follows: In order to generate the light neutrino mass matrices, we adopt the seesaw mechanism [19][20][21]. The eAEective neutrino mass matrices are given by the well-known formula, M ∫ =°M D M°1 R M T D , through the seesaw mechanism. We obtain the 6 diAEerent eAEective neutrino mass 1) There are still other solutions for V =0, including the trivial solution which makes all the VEVs vanish. It leads to the vanishing of all the lepton masses and mixing angles. In addition to the trivial solution, there are solutions with non-zero VEVs of the driving fields. This case leads to the breakdown of U (1) R symmetry. In this paper, we only discuss the vacua where U (1) R symmetry is conserved. where .

Classification of vacua and PMNS mixing matrix
In this section, we classify the 24 diAEerent vacua and derive the lepton mixing matrix U PMNS , called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix. In order to classify the vacua, we discuss the relations among the VEV alignments with the transformations of A 4 . We show that the 24 vacua are classified into two types in the following subsection. Then, one finds the two diAEerent PMNS matrices with diagonalizing matrices for the charged lepton and eAEective neutrino mass matrices Eqs. (18)- (21), and (27)-(29).

Relations among sets of VEV alignments
The generators of A 4 are expressed as the following forms for the representations 1,1 0 ,1 00 and 3, The sets of VEV alignment ¥ m ,∏ ± n are associated through the transformations of these generators. As an example, we show the T transformation on ∏ + 1 : T The S and T transformations on all the sets of the VEV alignment are summarized in Fig. 1. Some transformations preserve the VEVs of either ¥ m or ∏ ± n . These vacua have Z 3 or Z 2 symmetries as the residual symmetries of A 4 respectively. For the VEVs described as ¥ m , they are invariant under the following transformation, It is easy to confirm that such transformations correspond to Z 3 symmetries: Each ∏ ± n has Z 2 symmetry as follows:

Classification of 24 vacua
In this subsection, we show the relations among the 24 diAEerent Lagrangians derived from the 24 diAEerent combinations of VEV alignments in Eqs. (11)- (17). We find the two sets of 12 equivalent Lagrangians with the appropriate field redefinitions. Then, the 24 Lagrangians are classified into two types. For simplicity, we write the Lagrangian of this model in a short form: where √ represents the fermion fields such as l and ∫ R . ¡ 1 and ¡ 2 represent the scalar fields, which should have their VEVs written as ¥ m and ∏ ± n respectively. We write the Lagrangian in the broken phase for the VEV alignment (¥ m , ∏ ± n ) with fluctuations h 1 and h 2 as L ± mn (√,h 1 ,h 2 )¥L(√,¥ m +h 1 ,∏ ± n +h 2 ).
Type I and type II are disconnected because of the absence of a transformation which relates one type to the other. Since all the Lagrangians which belong to the same type lead to the same physical consequences, we consider only L + 11 and L°1 1 as the representatives of their types: We also define the representative mass matrices for charged leptons and neutrinos as It is noted that the charged lepton mass matrix M (1) l is diagonal.

PMNS matrices for two types
In this subsection, we construct the PMNS matrices for the two types, L I and L II . Since the charged lepton mass matrix M l is diagonal, the PMNS matrix is determined so that it diagonalizes the neutrino mass matrices in Eq.
where the left-handed neutrino masses m 1 ,m 2 and m 3 are positive. The PMNS matrices are expressed as the following forms for the two types: We have introduced the parameters µ,ae and ¡ i (i=1,2,3). They are written in terms of the complex parameters of the neutrino mass matrix, a,b,c and d, in Eq. (27) 1) . In the rest of this subsection, we derive the explicit forms of the parameters µ,ae and ¡ i in terms of the model parameters a,b,c and d. In the first step, one rotates M ∫ M † ∫ with the tri-bimaximal mixing matrix.

Summary
We have studied phenomenological aspects of a supersymmetric model with A 4 £Z 3 symmetry. We found 24 degenerate vacua at the 24 minima of the scalar potential. Then, we discussed the relations among the 24 diAEerent vacua and classified them into two types. Both types consist of 12 vacua which are related to each other by transformations of A 4 . We proved that the 12 vacua are equivalent and lead to the same physical consequences. However, we found that we obtain diAEerent physical consequences from the vacua of diAEerent types. Therefore, we analyzed the two types of vacua to find the diAEerent phenomenological consequences of the two types. In particular, we investigated observables such as mixing angles, Dirac CP phase, Majorana phases and eAEective mass for 0∫ØØ decay.
These observables are expressed in terms of the model parameters µ, ae and ¡ i . The angle µ and phase ae are determined by the deviation from the tri-bimaximal mixing matrix. The two types lead to diAEerent expressions for the mixing angles and Dirac CP violating phase in terms of µ and ae. Therefore, one should take diAEerent model parameters in each type in order to realize the experimental results. Although one can adopt both of the two types to predict the observable parameters, the two types cannot realize the current experimental data simultaneously. The Majorana phases AE and Ø are parametrized in the diAEerent expressions for each type by the model parameters ¡ i in addition to µ and ae. In order to find numerical diAEerences between the two types of Majorana phase, we considered the specific situation where the lightest mass and eAEective mass for the 0∫ØØ decay are determined in a certain region. We showed the allowed regions of the phase diAEerences, ¡ 1°¡3 and ¡ 2°¡3 . The regions are quite diAEerent for the two types: the phase diAEerences for type I are proportional to each other, while those for type II are not.
The VEVs ¥ m and ∏ ± n transfer to the diAEerent VEVs by transformations of A 4 . However, the transformations for ¥ m and ∏ ± n are closed diAEerently since they have the Z 3 and Z 2 residual symmetries from A 4 respectively. We have pointed out that some combinations of the VEVs can lead to diAEerent physical consequences. When we consider models with two or more flavons, we should take account of the combination of VEVs.

Multiplication rule of A4 group
In this appendix, we show the multiplication of the A4 group. The multiplication rule of the triplets is written as follows; 0 B @ a1 a2 a3 while that for singlets is, In order to derive the A4 invariant superpotential in Eq. (1), we have used the multiplication rules. Their derivation is shown in the reviews in Refs. [1][2][3][4].