Using time-dependent indirect $CP$ asymmetries to measure $T$ and $CPT$ violation in $B^0$-${\bar B}^0$ mixing

Quantum field theory, which is the basis for all of particle physics, requires that all processes respect $CPT$ invariance. It is therefore of paramount importance to test the validity of $CPT$ conservation. In this Letter, we show that the time-dependent, indirect $CP$ asymmetries involving $B$ decays to a $CP$ eigenstate contain enough information to measure $T$ and $CPT$ violation in $B^0$-${\bar B}^0$ mixing, in addition to the standard $CP$-violating weak phases. Entangled $B^0{\bar B}^0$ states are not required (so that this analysis can be carried out at LHCb, as well as at the $B$ factories), penguin pollution need not be neglected, and the measurements can be made even if the $B^0$-${\bar B}^0$ width difference vanishes.

We begin by reviewing the most general formalism for B 0 -B 0 mixing, in which CP T and T violation are incorporated. The 2 × 2 hermitian matrices M and Γ, respectively the mass and decay matrices, are defined in the (B 0 ,B 0 ) basis. When M − (i/2)Γ is diagonalized, its eigenstates are the physical light (L) and heavy (H) states B L and B H . Now, any 2 × 2 matrix can be expanded in terms of the three Pauli matrices σ i and the unit matrix with complex coefficients: Comparing both sides of this equation, we obtain We can define complex numbers E, θ and φ as follows: The eigenvalues of M − (i/2)Γ are E − iD and −E − iD, with eigenstates where p 1 = cos θ 2 , q 1 = e iφ sin θ 2 , p 2 = sin θ 2 and q 2 = e iφ cos θ 2 . In Ref. [17] (pgs. 349-358), T.D. Lee discusses the CP T and T properties of M and Γ. First, if CP T invariance holds, then, independently of T symmetry, In addition, if T invariance holds, then, independently of CP T symmetry, From Eqs. (2) and (3), we have Thus, if T is a good symmetry, the left-hand quantity is a pure phase, and the modulus of the square root is one. Note that it is usually said that the absence of CP violation implies |e iφ | = 1. However, strictly speaking, this is due to the absence of T violation. The two reasons are equivalent only if CP T is conserved.
Using Eq. (4), we have Now, if CP T is a good symmetry, then θ = π 2 [Eq. It will be useful to define the complex θ and φ in terms of real parameters as θ = θ 1 + iθ 2 and φ = φ 1 + iφ 2 . In the absence of both T and CP T violation in B 0 -B 0 mixing, the parameters take the following values: where β mix is the weak phase describing B 0 -B 0 mixing. In the standard model, β mix = β for the B 0 d meson. Now, if T and CP T violation are present in the mixing, the parameters θ 1 , θ 2 and φ 2 will deviate from these values. We define ǫ 1,2,3 via ǫ 1 and ǫ 2 are CP T -violating parameters, whereas ǫ 3 indicates T violation.
The values for ǫ 1 , ǫ 2 and ǫ 3 have been reported by the BaBar and Belle Collaborations [18,19]. Their notation is related to ours as follows: so that ǫ 1 and ǫ 2 are expected to be very small, as they are CP T -violating parameters. As for ǫ 3 , note that |q/p| has been measured at the Υ(4S) using the same-sign dilepton asymmetry, assuming CP T conservation [20]: Thus, ǫ 3 is also very small. Above, we have called ǫ 1 and ǫ 2 the CP T -violating parameters. But one must be careful about such names. ǫ 1 and ǫ 2 do not contribute only to observables measuring CP T violation. They also lead to CP -and T -violating effects. Similarly, the Tviolating parameter ǫ 3 also contributes to CP -violating observables. And the reverse is true: recall that, in Ref. [14], the BaBar Collaboration measured a large true Tviolating asymmetry. This does not suggest that ǫ 3 is large, as there are also large contributions to the asymmetry coming from CP -violating effects (assuming CP T invariance). The point is that ǫ 1 , ǫ 2 and ǫ 3 are also sources of CP violation, and it is this fact that allows their measurement in the time-dependent indirect CP asymmetry, as we will see below.
In the presence of T and CP T violation in B 0 -B 0 mixing, the time evolution of the flavor eigenstates (|B 0 ≡ |B 0 (t = 0) and |B 0 ≡ |B 0 (t = 0)) is given by Here We consider a final state f to which both B 0 andB 0 can decay. Using Eq. (15), the time-dependent decay amplitudes for uncorrelated or tagged neutral mesons are given by If we set θ = π 2 and Im φ = 0 in the above expressions, we recover expressions for the differential decay rates that are commonly found elsewhere in the literature.
We now write θ = θ 1 + iθ 2 and φ = φ 1 + iφ 2 , with [see Eqs. (10) and (11)] where the ǫ 1,2,3 are very small. In order to probe T and CP T violation in B 0 -B 0 mixing, one must measure the parameters ǫ 1,2,3 . Below, we illustrate how this can be done in the B 0 d system. For B 0 s mesons, the procedure is similar, though a bit more complicated.
First, as regards Γ d , the value of y d = ∆Γ d /2Γ d has been measured to be small: y d = −0.003±0.015 with the B 0 d lifetime of 1.520±0.004 ps [21]. This means that we can approximate sinh(∆Γt/2) ≃ ∆Γt/2 = y d Γ d t and cosh(∆Γt/2) ≃ 1. In principle, for large enough times, this approximation will break down. However, even at time scales of O(10) ps, the approximation holds to ∼ 10 −4 , and by this time most of the B 0 d s will have decayed. The observable we will use to extract the T -and CP T -violating parameters ǫ 1,2,3 is the time-dependent indirect CP asymmetry A f CP (t) involving B-meson decays to a CP eigenstate. It is defined as In the limit of CP T conservation, T conservation in the mixing, and ∆Γ = 0, one has the familiar expression where Here, C is the direct CP asymmetry and ϕ is the measured weak phase, which differs from the mixing phase If there is no penguin pollution, then ϕ cleanly measures a weak phase and C = 0. But if there is penguin pollution, then neither of these holds.
In the presence of T and CP T violation in the mixing, we use Eqs. (18) and (19) to obtain the time-dependent CP asymmetry. We first expand the various functions in the two equations, keeping only terms at most linear in the small quantities ǫ 1,2,3 and ∆Γ d : The denominator [Eq. (27)] has the form A(1 + x), with x small, so we can appriximate 1/A(1 + x) ≃ (1 − x)/A. Combining all the pieces, and again keeping only terms at most linear in ǫ 1,2,3 and y d , we obtain 6 where the coefficients are given by The seven pieces have different time dependences so that, by fitting A f CP/CP T (t) to the seven time-dependent functions, all coefficients can be extracted.
The five observables c 0 , c 1 , c 2 , s 1 and s 2 can be used to solve for the five unknown parameters C, ϕ and ǫ 1,2,3 . In practice, a fit will probably be used, but there is an analytical solution. The parameter C is simply given by The solution for sin ϕ is obtained by solving the following quartic equation: Of course, there are four solutions, but, since the ǫ i are small, the correct solution is the one that is roughly s 1 / √ 1 − C 2 . Finally, ǫ 1 , ǫ 2 , ǫ 3 are given by This shows that it is possible to measure the parameters describing T and CP T violation in B 0 d -B 0 d mixing using the time-dependent indirect CP asymmetry, and this can be carried out at LHCb.
The parameters c ′ 1 and s ′ 1 depend only on ϕ and y d . Thus, given knowledge of ϕ, the value of y d can be found from measurements of these parameters. Note that, even if the width difference ∆Γ d between the two B-meson eigenstates vanishes, the T -violating parameter ǫ 3 can still be extracted. This is contrary to the claim of Refs. [6] and [9].
Above, the method was described for the B 0 d system. In the case of B 0 s mesons, ∆Γ s is not that small, so the functions sinh (∆Γ s t/2) and cosh (∆Γ s t/2) must be kept throughout. This modifies the forms of Eqs. (26), (27) and (28), but the idea does not change. A f CP/CP T (t) still depends on seven different time-dependent functions, a fit can be performed to extract their coefficients, and C, ϕ, ǫ 1,2,3 and ∆Γ S can be found using the measurements of these coefficients.
The values of c 1 and s 1 have been measured for several B 0 d decays to CP eigenstates [20], and the value of ǫ 3 is independent of the decay mode. Using these values, we can estimate c 0 , c 2 and s 2 from Eq. (33), which assumes that CP T is conserved. As an example, for the final state J/ψK S , we find Should the measurements of c 0 , c 2 and s 2 deviate significantly from the above values, this would indicate the presence of CP T violation in B 0 d -B 0 d mixing. To sum up, we have shown that the time-dependent, indirect CP asymmetries involving B 0 ,B 0 → f CP contain enough information to extract not only the CPviolating weak phases, but also the parameters describing T and CP T violation in B 0 -B 0 mixing. These measurements can be made at the Υ(4S) (e.g., BaBar, Belle) or at high energies (e.g., LHCb). There is no need to neglect penguin pollution in the decay, and the method can be applied to B 0 d -or B 0 s -meson decays.