Re-examining sin(2beta) and Delta m(d) from evolution of B(d) mesons with decoherence

In the time evolution of neutral meson systems, a perfect quantum coherence is usually assumed. The important quantities of the B(d) system, such as sin (2beta) and Delta m(d), are determined under this assumption. However, the meson system interacts with its environment. This interaction can lead to decoherence in the mesons even before they decay. In our formalism this decoherence is modelled by a single parameter lambda. It is desirable to re-examine the procedures of determination of sin(2beta) and Delta m(d) in meson systems with decoherence. We find that the present values of these two quantities are modulated by lambda. Re-analysis of B(d) data from B-factories and LHCb can lead to a clean determination of lambda, sin(2beta) and Delta m(d).

In the time evolution of neutral meson systems, a perfect quantum coherence is usually assumed. The important quantities of the B 0 d system, such as sin 2β and ∆m d , are determined under this assumption. However, the meson system interacts with its environment. This interaction can lead to decoherence in the entangled mesons even before they decay. In our formalism this decoherence is modelled by a single parameter λ. It is desirable to re-examine the procedures of determination of sin 2β and ∆m d in meson systems with decoherence. We find that the present values of these two quantities are modulated by λ. Re-analysis of B 0 d data from B-factories and LHCb can lead to a clean determination of λ, sin 2β and ∆m d .
Introduction.-In neutral meson systems, quantum coherence plays a crucial role in the determination of many observables. However, any real system interacts with its environment and this interaction can lead to a loss of quantum coherence. The environmental effects may arise at fundamental level, such as the fluctuations in a quantum gravity space-time background [1,2]. They may also arise due to the detector environment itself. Irrespective of the origin of the environment, its effect on the neutral meson systems can be taken into account by using the ideas of open quantum systems [3][4][5]. This formalism enables the inclusion of effects such as decoherence and dissipation in a systematic manner [6].
The time evolution of neutral mesons, which are coherently produced in meson factories, are used to measure a number of parameters of the standard model of particle physics and also to search for physics beyond the standard model. With the inclusion of the decoherence effects, the measured values of some of these parameters can change. In this letter, we study the effect of decoherence on the important observables in the B 0 d meson system, such as the CP violating parameter sin 2β and the B 0 d −B 0 d mixing parameter ∆m d . We also suggest methods which will enable clean determination of such observables.
The evolution of the B 0 d system is built up from first principles. The effect of the environment forces the evolution to be a semi-group rather than a unitary one [6][7][8]. We use the density matrix formalism to represent the time evolution of the B 0 d system. This ensures the complete positivity of the state of the system and hence its physical validity. In this formalism, the decoherence is modelled by a single parameter λ. By construction, the density matrices are trace preserving.
The work presented here, we hope, would lead to the inclusion of the effects of decoherence in the analysis of data from the B 0 d systems. It may be worthwhile to reanalyze the data from the B factories and LHCb to verify if a singature of decoherence is already inherent in it. Given the wealth of data expected from the KEK Super B factory, it is conceivable that a signal for the decoher-ence may well be found. Thus a detailed study of B 0 d observables can lead to tests of physics at scales much higher than those typical of flavour physics.
We first study the parameter sin 2β, whose measurement is the first singal for CP violation outside the neutral kaon system. The precision measurement of its value is the corner stone in establishing the CKM mechanism for CP violation. With the inclusion of the decoherence effects, it turns out that the experimentally measured CP asymmetry depends both on the decoherence parameter λ and the angle β of the unitarity triangle. Next we study ∆m d , which denotes the mixing in the B 0 d system and is an important input in extracting sin 2β from the measured time dependent CP asymmetry. We find that the measured value of ∆m d is also affected by the decoherence effects. Finally, we suggest a method of analysis by which the three quantities, (a) λ, (b) ∆m d and (c) sin 2β can all be measured.
Determination of sin 2β.-In the following, we develop the formalism which applicable to B 0 d as well as B 0 s mesons. We are interested in the decays of B 0 and B 0 mesons as well as B 0 ↔B 0 oscillations. To describe the time evolution of all these transitions, we need a basis of three states: B 0 , B 0 and |0 , where |0 reprents a state with no B meson and is required for describing the decays. In this basis, we can define ρ B 0 (B 0 ) (0), the initial density matrix for the state which starts out as B 0 (B 0 ). The time evolution of these matrices is governed by the [9]. The Kraus operators are constructed taking into account the decoherence in the system which occurs due to the evolution under the influence of the environment [10,11]. The time dependent density matrices are for B 0 andB 0 respectively. In the above equation, where Γ L and Γ H are the respective decay widths of the decay eigenstates B 0 L and B 0 H . Also λ is the decoherence parameter, representing interaction between one-particle system and its environment. Here we assume no CP violation in mixing which is a valid approximation in B systems.
We define the decay amplitudes The probability, Let us now consider B 0 d → J/ψK S decay. One can define a CP violating observable Calculating the probabilities using Eqs. (1) and (2) we get With this approximation, the above expression simplifies to (5) Putting λ = 0 in the above equation, we get the usual expression for CP asymmetry in the interference of mixing and decay. Thus the presence of decoherence modifies the expression for CP asymmetry in the interference of mixing and decay. For . Assuming |λ f | = 1, i.e., no direct CP asymmetry, we get A J/ψKS (t) = sin 2β e −λt sin (∆m d t) .
Therefore we see that the coefficient of sin (∆m d t) in the CP asymmetry is sin 2β e −λt and not sin 2β! The measurement of sin 2β is masked by the presence of decoherence. Thus in order to have a clean determination of sin 2β, an understanding of λ is imperative.
In [12][13][14], it was shown that there can be some discrepancy between the experimental value and the standard model (SM) theoretical prediction of sin 2β. This discrepancy may hint towards possible physics beyond the SM. However, if λ is not small enough to be neglected, this discrepancy can also be attributed to the ignorance of decoherence in the analysis.
Determination of ∆m d .-It is obvious that in order to determine sin 2β, we need to know ∆m d and λ. If ∆m d is measured using observables which are independent of λ, then we only need to determine λ for the clean extraction of sin 2β. If the determination of ∆m d is also masked by the presence of decoherence then we need to have a clean determination of ∆m d . The present world average of ∆m d quoted in PDG is (0.510±0.003) s −1 which is an average of measurements of ∆m d from OPAL [15], ALEPH [16], DELPHI [17], L3 [18], CDF [19], BaBar [20], Belle [21], D0 [22] and LHCb [23] experiments. There are several ways in which ∆m d can be determined experimentally. LHCb, CDF and D0 experiments determine ∆m d by measuring rates that a state that is pure B 0 d at time t = 0, decays as either as B 0 d orB 0 d as function of proper decay time. In the presence of decoherence, the survival (oscillation) probability of initial B 0 d meson to decay as B 0 d (B 0 d ) at a proper decay time t is given by where we have neglected the width difference ∆Γ d . The positive sign applies when the B 0 d meson decays with the same flavor as its production and the negative sign when the particle decays with opposite flavor to its production. We see that the survival (oscillation) probability of B 0 d is λ dependent! The time dependent mixing asymmetry, used to determine ∆m d , is then given by A mix (t, λ) = P + (t, λ) − P − (t, λ) P + (t, λ) + P − (t, λ) = e −λt cos(∆m d t) .  7), except that the proper time t is replaced by the proper decaytime difference ∆t between the decays of the two neutral B d mesons. Therefore, we see that the determination of ∆m d at LHCb, CDF, D0, Belle and BaBar experiments is also masked by the presence of λ. The true value of ∆m d can be determined by a two parameter (∆m d , λ) fit to the time dependent mixing asymmetry A mix (t, λ) defined in Eq. (8). This in turn will enable a determination of true value of sin 2β using Eq. (6).
Determination of ∆m d in the LEP experiments is mainly based on time independent measurement, i.e., from the ratio of the total same-sign to opposite-sign semileptonic rates (R) or the total B 0 d −B 0 d mixing probability (χ). We shall now see that these observables are also λ dependent. Therefore all the methods used to determine ∆m d depend upon λ.
Correlated B 0 d meson semi-leptonic decays.-The entangled B 0 d −B 0 d mesons, produced in the decay of the Υ(4S) resonance, can both decay semi-leptonically. The effects of decoherence on the resulting dilepton signal was studied in [24]. Here we calculate these effects using the formalism described in the previous section. The entangled B 0 d −B 0 d state can be written as The time evolution of the above state is described by the following density matrix: where and are given by Here the parameters are as in Eq. (1). The double decay rate, G(f, t 1 ; g, t 2 ), that the left-moving meson decays at proper time t 1 into a final state f , while the right-moving meson decays at proper time t 2 into the final state g, is then given by Tr ]. From this a very useful quantity called the single time distribution, Γ(f, g; t), can be defined as Γ(f, g; t) = ∞ 0 dτ G(f, τ + t; g, τ ), where t = t 1 − t 2 is taken to be positive.
We now consider the decays of B 0 d mesons into semileptonic states h l ν, where h stands for any allowed charged hadronic state. Under the assumption of no CP violation in mixing, CPT conservation and no violation of ∆B = ∆Q rule, the amplitudes for B 0 d /B 0 d into h − l + ν can be written as There are two important observables which can be affected by interaction with the environment. One is the ratio of the total same-sign to opposites-sign semileptonic rates and the other is the total B 0 d −B 0 d mixing probability .
(15) Time independent probabilities, Γ(f, g), can be obtained by integrating the distribution Γ(f, g; t) over time.
The expressions for R and χ are obtained to be where we have neglected ∆Γ d . Thus we see that along with ∆m d , these observables also depend upon the decoherence parameter λ.
For the observable R, the last experimental update was given about two decades ago [25]. This value was used in ref. [24] to estimate the value of λ to be (−0.72±1.18)×10 11 s −1 . In order to determine λ from R (or χ), we need ∆m d as an input. As shown above, the present method of determining ∆m d is masked by the presence of λ. Therefore it is important to reanalyze the data on the time dependent mixing asymmetry in terms of the two parameter (λ, ∆m d ) expression given in Eq. (8). It is important to find the value of R from the data of BaBar and Belle and the value of χ from CDF, DO and LHCb. Then the expressions in Eqs. (16) and (17) can be verified using the values obtained from the fit to the time dependent asymmetry. This will provide an additional consistency check on assumptions made regarding decoherence. Finally, the values of λ and ∆m d from the A mix (t, λ) fit can be used in Eq. (6) to obtain a clean measurement of sin 2β.
The present analysis can easily be extended to the B 0 s system as well. Running on similar lines, one can derive an expression for the time dependent CP asymmetry in the mode B s → ψφ, similar to Eq. (4). This expression is a function of four parameters: λ, sin 2β s , ∆m s and ∆Γ s . An expression for the time dependent mixing asymmetry, with the condition ∆Γ s = 0 has to be derived, which will be a function of λ, ∆m s and ∆Γ s . These two time-dependent asymmetries should be re-analysed using a four parameter fit for a clean determination of sin 2β s , ∆m s and ∆Γ s .
Conclusions.-In this work, we have studied the effect of decoherence on two important observables sin 2β and ∆m d in a neutral meson system. We find that the asymmetries which determine these quantities are also functions of the decoherence parameter λ. Hence it is imperative to measure λ for a clean determination of these quantities. We suggest a re-analysis of the data on the above asymmetries for an accurate measurement of all the three quantities λ, sin 2β and ∆m d . The present analysis can easily be extended to the B 0 s system as well.