Time reversal symmetry violation in entangled pseudoscalar neutral charmed mesons

The direct observation of time reversal symmetry violation (TV) is important for the test of $CPT$ conservation and the Standard Model. In this paper, we study both time-dependent and time-independent genuine TV signals in entangled $D^0-\bar{D}^0$ pairs. A possible $CPT$-violation effect called the $\omega$ effect is also investigated. In the $C=-1$ entangled state, the asymmetries due to TV are calculated to be of the order of $10^{-5}$ to $10^{-4}$ within the Standard Model, but the modification due to the $\omega$ effect in the $C=-1$ states is found to be about $10\%-30\%$ when $|\omega|\sim 10^{-4}$. This result is consistent with our Monte Carlo simulation, which implies that with $10^9$ to $10^{10}$ events, TV signals can be observed in the entangled $D^0-\bar{D}^0$ pairs, and the bound of $\omega \sim 10^{-3}$ can be reached. The time-dependent and the time-independent asymmetries in the $C=-1$ $D^0-\bar{D}^0$ system provides a window to detect new physics such as the $\omega$ effect, although they are not easily observable.


I. INTRODUCTION
Symmetry, symmetry violation and symmetry breaking have been playing important roles in particle physics. The studies of discrete symmetries P , C, T and their combinations have progressed greatly with the help of large experimental data [1]. There are oscillations between neutral mesons and their antiparticles, such as B 0 −B 0 , D 0 −D 0 and K 0 −K 0 .
In the D 0 −D 0 system, both the mass and the decay width differences between the two mass eigenstates are very small in comparison with the mean values [2]. This provides an opportunity to verify CP violation (CPV) sources from both the Standard Model (SM) and new physics (NP) [3] and even the possibility of CP T violation (CPTV) such as the so-called ω effect, as predicted by some theories of quantum gravity [4,5].
However, direct observation of TV without the presumption of CP T conservation is especially important [7][8][9][10]. The TV signal based on a T-odd product of momentum vectors was observed in the decay D 0 → K + K − π + π − [11]. However, such a signal has a chance of being nongenuine because the initial and final states are not interchanged [8]. The TV signal based on the rate difference between the transformation from K 0 toK 0 and vice versa [12] is controversial [8].
Hence an important development is that a genuine TV signal has been observed in B 0 −B 0 decay, by comparing transitions that are related through time reversal but not through CP conjugation [13,14]. The key idea is to make use of quantum entanglement, also called the Einstein-Podolsky-Rosen correlation [7][8][9]. The initial states of each of the two transitions is prepared by tagging the entangled partners in the corresponding way. The connections between CP , T and CP T asymmetries and the experimental asymmetries are investigated for entangled B 0 dB 0 d mesons [15]. Extension to kaons has been made [9,16]. In this paper, we propose using the time-independent signals to study TV by extending the entanglement approach of TV to D 0 −D 0 systems. The C = −1 entangled D 0 −D 0 pairs can be produced through the strong decay of ψ(3770) [17][18][19] or ψ(4140) [18,19]. ψ(3770) has often been used for the study of CPV of D mesons. The C = +1 entangled state of D mesons can also be produced in the strong decay of ψ(4140) [18,19].
First, we calculate the time-dependent and the time-independent asymmetries between T-conjugate processes for the C = −1 entangled states. Within the SM, the asymmetry of the C = −1 system is found to be at most 10 −5 . We also consider the ω effect in the C = −1 state, which mixes the C = +1 state into it. We find that the ω effect modifies the TV signals by as large as 20% when |ω| ∼ 10 −4 . We also calculate the T asymmetries defined for transitions from D 0 to D − and vice versa, by using event numbers in joint decays of entangled pairs. Finally we use a Monte Carlo simulation [20] to study the C = −1 systems based on the current experimental situation, and demonstrate that if the number of events reachea 10 9 TV signals can be observed; furthermore, if the number of events reaches 10 10 , the bound of ω ∼ 10 −3 can be obtained.
We conclude that in the C = −1 D 0 −D 0 entangled state the time-dependent asymmetry due to TV within the SM requires a large number of events and may provide a window to detect the signal of NP such as the ω effect.
The rest of this paper is organized as follows. In Sec. II we briefly review the idea of studying TV using the entangled states. In Sec. III, we study the joint decay rates of such states. In Sec. IV, we discuss the TV signals in the oscillation of the D 0 −D 0 system. Section. V is a discussion on the relation between the joint decay rate and the experimental measurement. In Sec. VI, we present a Monte Carlo simulation on the TV. Section. VII is a summary.

II. ENTANGLED STATES OF NEUTRAL MESONS
As pseudoscalar neutral mesons consisting of quarks, D 0 = cū andD 0 =cu. In the Wigner-Weisskopf approximation, |D 0 and |D 0 are eigenstates of the flavor, which is the charm in this specific case, with eigenvalues ±1. |D 0 and |D 0 comprise a basis, in which the effective mass matrix is written as where H 00 ≡ D 0 |H|D 0 , H 00 ≡ D 0 |H|D 0 , and so on. The eigenstates of H are where ǫ is the indirect CPV parameter. The corresponding eigenvalues are We can neglect the direct CPV, as done in testing T violation in entangled B mesons [8,15,20], and can also be done in entangled D mesons [2,17,21].
The time evolution of the mass eigenstates is where U(t) represents the time evolution under the effective mass matrix. U(t) evolves the flavor basis states as with where the sign of g − (t) is different from that in Ref. [24], and is same as in Refs. [18,22]. The more general expressions of |D 0 (t) and |D 0 (t) , without the assumption of indirect CP T conservation, are given in Refs. [23,26], and reduce to the expressions here when CP T is indirectly conserved. Note that the two mass eigenstates |D H and |D L are not orthogonal because of indirect CPV parameter ǫ = 0; hence, the basis transformation involving them is not unitary.
There is yet another basis often used, namely, the CP basis, with eigenvalue ±1. The time evolution starting with each of them can be written as Now suppose at time t = 0, the C = ±1 entangled states of two mesons a and b is generated, where the subscripts a and b will be omitted below. Under the mass matrix, Specifically It can be seen that the evolution of Ψ − leaves the entanglement unchanged and provides a good opportunity to study the discrete symmetries. Furthermore, one can define which represents that particle a decays at t a while particle b decays at t b , and is widely used in calculating joint decay rate [8,18,27]. |Ψ C (t a , t b ) can also be written in terms of CP eigenstates as Unless explicitly stated, here, t b ≥ t a is assumed without loss of generality. The free choice between Eqs. (15) and (16)  The entangled meson pairs can be used in the so-called single-tag (ST) and doubletag (DT) methods [17,28,29]. In the case of the C = −1 entangled state, the final state of the first decay at t a tags the partner as D 0 orD 0 or D ± ; one can then study the decay of the tagged partner at t b .
Because at time t, |Ψ − (t) ∝ |Ψ − , the C = −1 entangled state can be used to construct T-conjugate processes. For example, if meson a decays into the l − final state at t a , it implies that meson a has been projected to |D 0 , which decays to the l − final state; hence, meson b is prepared to be |D 0 at t a . Then, by measuring the probability that meson b decays into a  Table I [8,20].
To test TV, we need to compare these T-conjugation transitions. There are several ways to relate the transitions to observables, as discussed below.

B. Joint decay rates
For the entangled meson pairs, an important quantity to study is the joint decay rate, which is the joint rate of the processes in which one of the entangled mesons decays into the The rate of each transition listed in Table I can be obtained from the joint decay rate of the corresponding final states, with meson a decaying to its final state such that the entangled partner b is projected to the initial state in the transition listed.
1. Joint decay rates of C = ±1 states For |Ψ C (t a , t b ) , the joint decay amplitude for the joint processes in which meson a decays to f a at t a while meson b decays to where H a is the weak interaction field theoretic Hamiltonian governing the decay of the meson a and ξ C and ζ C are defined as where A f andĀ f are instantaneous decay amplitudes The joint decay rate is thus where x and y are defined as In experiments, we often use the time-integrated joint decay hence Finally, the time-independent joint decay rate is defined as which is obtained as Note that R C (f a , f b ) is independent of the order of the two final states. In experiments, such time-independent quantities are most easily measured.
2. Joint decay rates under the ω effect.
One kind of CPTV is the so-called ω effect, which is a consequence of some forms of quantum gravity [4,5]. The ω effect affects the entangled source, so the C = −1 entangled state is mixed in by the C = +1 entangled state with a factor ω. For simplicity, in this section we assume the CPV parameters are barely affected by the ω effect.
Because of the ω effect, the C = −1 entangled state is modified to be where ω ≡ |ω|e iΩ is a small mixing factor. The joint decay rate is found to be with where The integrated joint decay rate can be written as where (Ch cosh(yΓ∆t) + Sh sinh(yΓ∆t) + Cs cos(xΓ∆t) + Sn sin(xΓ∆t)) , with In this section, we first establish the TV signals and their behavior predicted within the SM. We use those decay channels in which the direct CPV, i.e. that in the decays, can be neglected and only consider indirect CPV, i.e., that in the oscillation. We consider only the cases in which one of the final states is a CP eigenstate while the other is a flavor eigenstate [8,17,20]. In D 0 −D 0 systems, the indirect CPV parameter is known to be very small [2,30]. Within the SM, the corresponding TV is also expected to be very small.
With direct CPV negligible, we have [2,21] When the final state is a CP eigenstate S ± , within the SM, we have [21] where A f andĀ f are defined in Eq. (21).
where n l = ±1 for l ± final states and n s = ±1 for S ± states.
Experimentally, the semileptonic decay modes and the CP eigenstate decay modes of a C = −1 entangled D 0 −D 0 system have been studied by using DT of the two mesons [17], where the semileptonic decay modes include Keν and Kµν, while the CP eigenstate decay modes include K + K − , π + π − , and K 0 S π 0 π 0 for CP = 1 and K 0 S π 0 , K 0 S ω, and K 0 S η for CP = −1.
q/p is often parametrized as which will be used below. Other frequently used parameters include y CP and A Γ which can be defined as [2,17,31] y CP = y and A Γ = 0 indicate indirect CPV. A Γ is known to be very small. We also define [2] q p which is often used in the studies of D decays.

A. TV signals based on joint decay rates
For the C = −1 entangled state, we can construct four TV signals from time-dependent joint decay rates (depending on the difference ∆t = t b −t a of two decay times), corresponding to the final states listed in Table I. In the first example listed in the Table I, the final states of mesons a and b are l − and S − , with direct CPV neglected, i.e. A l − =Ā l + = 0, where we have assumed no wrong-sign decay. If f a = l − , then Considering Similarly, A similar expression can be for each pair of T-conjugated transitions.
Hence, we can define a T asymmetry, denoted as A 1 − (∆t > 0), and there are three other asymmetries corresponding to the equalities in Eq. (45).
We can also define TV signals independent of |A S ± |, denoted as A 2 − (∆t > 0), There are five other signals similar to Eq. (47) that can be constructed, according to Eq. (45).
One can also use the normalized joint decay rates or the probability density function (PDF), defined as where Therefore, one only needs to consider R − (f a , f b , ∆t) when normalization with respect to various ∆t is taken into account.
Hence, one can construct a TV A 3 − (∆t > 0) as which vanishes only if T symmetry is valid. Note that it was A 3 − (∆t) that was measured in Barbar experiments [13,20].
We now consider the time-independent joint decay rate If time reversal symmetry is respected, then both of the following equations are satisfied Hence we can define the time-independent TV signal of C = −1 states denoted asÂ − , WhenÂ − = 0, at least one of the equalities in Eq. (51) is violated. ThereforeÂ − is the TV signal independent of A S ± .
We emphasize that A 2 − (∆t) = 0 or A 3 − (∆t) = 0 or A − = 0 does not guarantee the time reversal symmetry. However, A 2 − (∆t) = 0 or A 3 − (∆t) = 0 or A − = 0 is a sufficient condition of TV. In experiments, one would like to use the TV signal independent of A S ± , that is, Note that, despite the decays, the antisymmetry of the C = −1 entangled state remains. This is crucial in its use in the construction of genuine TV signals [8]. The C = +1 entangled state of D mesons can also be produced in the strong decay of ψ(4140) [18,19], but it is difficult to extract TV signals from it. When he C = +1 entangled state evolves to t = t a , it becomes |Ψ + (t a ) as given in (14). Consequently, when one of the mesons decays into the f a final state at t a , the other meson becomes a superposition of D 0 andD 0 . If we denote Ψ fa as the state of the second meson tagged by the final state of the first meson f a , Ψ fa can be written as where ∝ implies that these four states are not normalized yet. If, for example, we compare the joint decay rate R + (l − , S − , t a , t b ) with R + (S + , l + , t a , t b ), we are comparing the transitions In the following, we concentrate on the TV signals of the C = −1 entangled states.
Substituting Eq. (36) into Eq. (24), we obtain the time-dependent joint decay rates With Λ ≡ −q/p ×Ā S ± /A S ± , and at the limit at which ∆Γ → 0, which is the case of B mesons [20], the integrated joint decay rates become which reproduces the integrated joint decay rates of B mesons in Refs. [32,33].
In the case of B mesons, we can take the limit ∆Γ → 0 and q/p → e 2iβ ; thus, we find A 1 − (∆t) = − sin(2β) sin(xΓ∆t). This corresponds to the CP asymmetry predicted by the SM, as given in Refs. [32,33].
We can expand A i − (∆t) to the leading order and find We use the parameter values in Ref. [25], Notice that the definition of φ in Ref. [25] is arg (q/p), while in this paper, we define φ ≡ arg (q/p) /2, which is the same as in Ref. [18]. For ∆t = τ D ≡ 1/Γ, we find A 1 − (∆t) ∼ 10 −5 . The time-independent joint decay rate does not depend on the decay times, so we are not able to identify the transition. For example, we need to know which of the final states is the outcome of the earlier decay to distinguish D 0 → D − from D + →D 0 . However, one can construct a time-independent signal for TV.

It is found thatÂ
where x 1 and x 4 are defined in Eq. (58). To the leading order, The error of the signal can be estimated to be related to the event number N as δÂ − ∼ 1/ √ N. Hence the magnitude ofÂ − implies that the number of events should be as large as 10 9 to 10 10 , which will be verified in Monte Carlo simulation in Sec. VI. Such an event number can be obtained at the super-tau-charm factory [34]. state. Then the T-conjugation between each pair of processes in the asymmetries studied above is lost. However, the asymmetries for these pairs of processes can still be investigated to determine the value of ω. We find that these asymmetries are enhanced. For example, for the same final states as in A 1 − (∆t) defined in Eq. (46), the corresponding asymmetry of the C = +1 state is Inserting Eq. (36) into Eqs. (24) and (26), in the case of C = +1, we find The difference between A 1 − (∆t) and A 1 + (∆t) is very large, providing an opportunity to detect the ω effect. The numerical results show that A 1 − (∆t ≈ τ D )/A 1 + (∆t ≈ τ D ) ∼ 10 −4 , which implies that a small ω at the order |ω| ∼ 10 −4 may considerably change the TV signals.
Incidently, this is also the order of magnitude considered in Ref. [5]. So we conjecture the experiment to observe the TV signal in the D system may at the same time provide a window to detect the ω effect with a sensibility up to |ω| ∼ 10 −4 .
For simplicity, we only consider how the TV signal A 2 − (∆t) is affected by the ω effect. Using Eqs. (19), (30)-(35), we find where A M is determined by q/p, as defined in Eq. (39).
The CPV parameters are assumed to be barely affected by the ω effect. Using Eq. (60), the dependence of A ω (∆t) on |ω| and Ω when Γ∆t = 1, i.e., ∆t = τ D ≡ 1/Γ, is shown in Figs. 1 and 2. We find that when |ω| ∼ 10 −4 the change of time-integrated T asymmetry, due to the ω effect, can be as large as 20% of that within the SM. The sensitivity could be competitive with the B or B d meson pairs [35]. In the Monte Carlo simulation presented in Sec. VI, we will find that if the event number is of the order of 10 9 the TV signal can possibly be observed. Such an event number can also set a bound on |ω| at 10 −3 at the same time.
We emphasize when the C = −1 state is mixed with the C = +1 state the signal is no longer a TV signal. However, the deviation from the TV signal calculated within the SM reveals the nonzero ω effect.

SUREMENTS
One can relate the normalized time-integrated joint decay rates to event numbers of the decays [20]. In using normalized time-integrated joint decay rates, the T-conjugated transitions differ in the dependence on the time interval rather than on the number of events.
A similar way to investigate the double decay is to use ST and DT signals [17,28,29].
Suppose the final state of meson a at t a is l − , it tagged the meson b as D 0 , which decays to S − at t b = t a + ∆t, the rate of which can be denoted as Γ(D 0 → S − , ∆t). By assuming that there is no mistake in tagging and that the direct CPV can be neglected, the rate where H is the Hamiltonian governing the decay. As a result In experiments, the decay rate can be related to event numbers as where N l − (t a ) is the number of the events in which meson a decays to l − at t a and N l − ,S − (t a , t a + ∆t 0 ) is the number of the joint events in which meson a decays to l − at t a and then meson b decays to which can be rewritten as where with N l − is the total number of events in which meson a decays to l − and is also called the signal yield of ST decays. N l − ,S − is the the total number of the joint events in which meson a decays to l − while meson b decays to S − and is also called the signal yield of DT decays.
Since T symmetry requires | D − |U(∆t)|D 0 | 2 = | D 0 |U(∆t)|D − | 2 for any ∆t > 0, In experiments, the detection efficiencies should also be considered, so we can write the transition rates as where ε's are the detection efficiencies, with the subscripts the same as those of the corresponding event numbers N 's, which are now understood as the experimental ones.
If time reversal symmetry is conserved, . Then according to Eq. (73), we have By using the ratios between the left-hand sides and right-hand sides of the equalities in Eq. (74), we construct the TV signal A 1 T as which can thus be obtained from the numbers of ST and DT events. Here, A 1 T = 0 is a TV signal. Note that A 1 T = 0 does not guarantee T symmetry; however, A 1 T = 0 is a sufficient condition of TV.
Another T-asymmetry can be constructed as Note that the asymmetries defined in Sec. IV are in terms of joint decay rates, while the asymmetries defined here are in terms of single particle decay rates, some of which are then obtained from joint decay events.

VI. SIMULATION
Through a Monte Carlo simulation [20], we can estimate the significance of the expected time-dependent signal based on current experiments. The time-dependent signal in the D 0 −D 0 mixing is difficult to measure [2,36] because the lifetimes of D mesons are too short, thus requiring a very high resolution of the decay length. We have calculated above that the asymmetries in the C = −1 D 0 −D 0 state are very small. In this section, by using Monte Carlo simulation, we analyze whether we are able to observe such signals or how far experimentally we are away from the required resolution.
Following the idea of Ref. [20], we use R − (f a , f b , ∆t) as the PDF to generate experimental events. For simplicity, we only simulate the D 0 → D − and D − → D 0 transitions. We define The PDF is affected by the mistakes in identifying the final states. In the case of B mesons, only the mistakes in the flavor identification were considered [20]. We assume this is also the case in D mesons. The mistakes in identifying a non-CP eigenstate as CP eigenstate cancel each other between S ± terms in the asymmetries. Similarly, the mistakes in distinguishing the semileptonic decays from background also cancel each other between l ± terms. Moreover, the CP violation in the decays of K 0 S mesons [17], which is used in the CP identification, is known to be small; thus, the mistakes in distinguishing the two CP eigenstates can be neglected. So, we only consider the mistakes in distinguishing the two flavor final states l + and l − .
If ψ(3770) is at rest, the proper time interval ∆t of the decays of the two D mesons is related with the momentum as [24] ∆t where r D and rD are decay lengths of D 0 andD 0 mesons and P is the 3-momentum of D 0 .
The uncertainties mainly come from r D and rD. The average is ≈ 290 µm, and one can use the rms of decay length in Belle, which is < 100 µm [24], and then σ τ /∆τ ≈ 100/290 ≈ 34%.
In Ref. [17], the number of double-tag events is about 5000. Hence, we generate 5000 events for both D 0 → D − and D − → D 0 using the PDF in Eq. (86). With generated events, we are able to obtain the number of events N M C (f a , f b , τ 0 ) in an interval 0 ∼ τ 0 .
The numbers of events that we are interested in are N M C (S + , l + , τ 0 ) and N M C (l − , S − , τ 0 ).
We can also obtain the average decay time ∆t ± M C from generated events, where ± in the superscript represents the transition with the l ± final state.

A. Fitting joint decay rates
Since we use the normalized PDF, we are not able to compare the time-independent joint decay rates of the conjugated transitions. So, we concentrate on comparing time-dependent joint decay rates.
Using Eq. (54), we find that the normalized time-dependent joint decay rate of a C = −1 can be approximately expressed as where r − (f a , f b , ∆t) is defined in Eq. (48) and b and n satisfy b ≡ 2 cos(2φ)y ≈ 2y CP , n ≡ n + + n − 2 , wherex andȳ are defined in Eq. (61). The number of events with ∆τ < τ 0 can be obtained as where the subscript SM represents the expected result in the SM. N f is the total number of events. With the definition N + SM (τ 0 ) ≡ N SM (S + , l + , ∆τ 0 ) and N − SM (τ 0 ) ≡ N SM (l − , S − , ∆τ 0 ), we find that, to the leading order, We can use Eq.   Hence, it is difficult to observe the TV in time-dependent T asymmetry in the C = −1 D 0 −D 0 state because ∆b < δb ± , where ∆b = |b − − b + |, δb ± are the standard deviations of b ± .
We can also estimate how far we are from the observation of the signal. In the SM, we where s ≡ 4∆n + 2u, In the SM, we find s = 7.6 × 10 −5 and 2u = −3.1 × 10 −5 ; therefore, Using Eqs. (91) and (94), we find that with 5000 events the fitting values of b ± are very close to the expected values of b ± ; however, the expected difference ∆b is too small to be observed. The accuracy of b ± needs to be at least smaller than 10 −4 . So we can also conclude that, in consistency with Sec. IV, to observe the TV signal the number of events should be at least four orders of magnitude larger than the one in the current experiments, which is about 5000.

B. Average decay times
In the above, we have used ∆τ ∼ 1, such that ∆t ∼ 1/Γ. Here we verify this assumption, and use the difference between the average decay times in the two conjugate processes as the evidence of TV. Each average decay time does not depend on fitting.
In the SM, the average decay time can be obtained as ∆τ − ≡ To observe the T-violating signal, the accuracy of measuring ∆τ should be about 10 −5 .
It should be noted that the number of events is an important factor that greatly affects the accuracy. We have run the simulation on ∆τ described above with different event numbers.
The results are listed in Table II. To estimate the standard deviation, each simulation with the same number of events is run 300 times. We find that the standard deviation is proportional to 1/ √ N , where N is the event number. According to the trend, if the event number is of the order of 10 9 ∼ 10 10 , which can be expected in the super-tau-charm factory [34], the standard deviation reaches 10 −5 , which is the order of the magnitude of the lifetime difference between the T-conjugate processes, as predicted by the SM and ω effect, Therefore, if the event number is of the order of 10 9 ∼ 10 10 , which can be expected in the super-tau-charm factory, then the TV signal can be observed, and the result can also set a bound on |ω| at about 10 −3 . That is to say, |ω| > 10 −3 can be excluded if not observed.

VII. SUMMARY
In this paper, we have studied TV in the C = −1 entangled D 0 −D 0 systems, and various T asymmetries are considered. We have proposed using the time-independent signals to study TV.
We calculated the time-dependent asymmetries of C = −1 system using joint decay rates, which are expected to be at the order of 10 −5 in the SM. Using the joint decay rates, we also obtained the time-independent asymmetries, which are also expected to be of the order of 10 −5 in the SM. We also studied the contribution of the ω effect caused by a kind of CPTV, which changes the asymmetries by as much as 20% when |ω| ∼ 10 −4 .
We also calculated T asymmetries defined for T-conjugate processes, the transitions from D 0 to D − and vice versa, using the transition rates obtained from the event numbers in joint decays of entangled pairs. These time-independent T asymmetries are also of the order of 10 −4 to 10 −5 .
We used the Monte Carlo simulation to estimate the time-dependent signals in the C = −1 entangled system by using the parameters in the current experimental situation. We estimate that if the event number reaches 10 9 to 10 10 TV signals can be observed in the entangled D 0 −D 0 pairs and the bound of ω ∼ 10 −3 can be reached.
In recent years, quantum entanglement has been found to be a resource of quantum information processing. Likewise, as exemplified by the present work, we may say that quantum entanglement is a resource of precision measurement in particle physics.