Observational Aspects of Symmetries of the Neutral B Meson System

We revisit various results, which have been obtained by the BABAR and Belle Collaborations over the last twelve years, concerning symmetry properties of the Hamiltonian, which governs the time evolution and the decay of neutral B mesons.We find that those measurements, which established CP violation in B meson decay, 12 years ago, had as well established T (time-reversal) symmetry violation. They also confirmed CPT symmetry in the decay (T_CPT = 0) and symmetry with respect to time-reversal (epsilon? = 0) and to CPT (delta? = 0) in the B0 ?B0bar oscillation.


Introduction
A system of neutral mesons such as B 0 ,B 0 or K 0 ,K 0 is a privileged laboratory for the study of weak-interaction's symmetries. Even though the phenomenological framework is well understood since long time [1][2][3][4], recent discussions in the physics community [5] show that it may be useful to revisit a few points, in order to fully (and correctly) exploit the experimental results. This process is then at the origin of the present note.
We focus on the B 0B0 system, and refer to experimental results [6][7][8][9][10] that have been achieved by measurements of the decay products of B 0B0 pairs created in the entangled antisymmetric state where the first B in this notation moves in direction p and the second in direction − p.
Within the Weisskopf-Wigner approximation [1] the time evolution of a neutral B-meson, and its decay into a state f is described by the amplitude A Bf , where T and Λ are represented by constant, complex 2 × 2 matrices T = (T ij ) = f i |T|B j and Λ = (Λ ij ) = B i |Λ|B j , i, j = 1 (2). We consider experiments with final states f i = J/ψK i or f i = µ i νµ(νµ)X. Here B 1(2) , K 1 (2) and µ 1 (2) stand for the flavour eigenstates B 0 (B 0 ), K 0 (K 0 ) and µ + (µ − ) or e + (e − ), respectively. We recall that a symmetry is a property of the hermitian Hamiltonian (H = H0 + H weak ) of the Schrödinger equation which is defined in a space sufficiently complete to include all the particle states under consideration, also the decay products [1]. Thus the aim of the experiments is to establish properties of the weak interaction Hamiltonian H weak by measuring observable combinations of the elements of Λ and of T, which represent these properties.
As CP violation implies T and/or CP T violation, we specifically consider the classical aim posed by the discoverers of T violation [11] "to express quantitatively the fraction of the observed CP violation due to T violation and CP T violation separately".
In passing, we show that a more recent treatment, which attempts to define T -symmetry violation as "motion-reversalsymmetry violation", without reference to the weak interaction Hamiltonian [12], is a special case within our phenomenology.

Observables of Symmetries
Together with a parametrization of the matrices Λ and T, the equations (1) and (2) are a sufficient basis for the description of the symmetry properties of the experimental results [6][7][8][9][10]. Symmetry properties of the Hamiltonian often manifest themselves in an especially simple and direct way in relations between measured quantities. Here, Table 1 gives a summary, with definitions and derivations as found in [1][2][3][4], and the phase conventions of [2]. Our approach is analogous to [13]. Channels are assumed to have one single amplitude.
In In order to calculate the amplitude A Bf in eq.(2), we need to evaluate the exponential in terms of Λ. We do this by summing up the power series (as explained in [13]). Let U = (U ij ) = e −iΛt and find U 11 = U0(cos(ωt) + i 2δ sin(ωt) ), U 22 = U0(cos(ωt) − i 2δ sin(ωt) ), For the matrix (T ij ) = ( J/ψK i |T|B j ), we assume with complex T 11 , T 22 , corresponding to the "∆b = ∆S rule". From Table 1 , and with the (arbitrary) normalization | T 11 | 2 + | T 22 | 2 = 2 , we deduce the useful identity among the (diagonal) elements of T, Results based on eqs.
(1) to (11) will turn out to be sensitive to all the four symmetry parameters in Table 1.
Throughout this work, we assume that channels have one single amplitude. Two interfering amplitudes may fake nonvanishing values of TCP T or TT , depending on their weak and strong phases, without the presence of the corresponding symmetry violations in the Hamiltonian.

Experiments 3.1 General description
Call A f 1 ,f 2 (t) the amplitude for the decay of an entangled, antisymmetric B 0B0 pair into a final state with the two observed particles f1 (at time t0) and f2 (at later time t > 0). With specific choices of the two final states f1, f2 , we can uniquely represent the complete set of results of the CP -, T -and CP T -symmetry violation studies listed in Table 2 and performed by [6][7][8] through [10], by making use of eq. (12) below [2,16], whose derivation we sketch here. We note with ( [13], section 2.7), that the time evolution acts on the two-particle state |Ψ of eq. (1) solely by a multiplicative factor, which is independent of the symmetry violations under consideration, and which does not influence the decay properties of |Ψ . We may thus, without loss of generality, arbitrarily choose t0 = 0, t > 0, and apply eq. (2) to the single-particle components in |Ψ , to obtain In rewriting (12), we can explicitly derive the formula for the state |S f 1 , which survived the decay to f1, and its (single particle) time evolution and decay to f2 as with The variety of expected frequency distributions | A f 1 ,f 2 (t) | 2 is displayed in Table 2. We find that the parameters of the data analysis are the T and CP T violation parameters of the T matrix, TT and TCP T , concerning the decay, and those, pi, qi, (i = 1, 2, 5, 6), concerning mainly the B 0B0 oscillation matrix Λ. In the limit of CP symmetry of Λ the pi, qi all vanish. Then, TT and TCP T are exactly associated each with its own proper time dependence: TT with ± sin(∆mt), and TCP T with ± cos(∆mt). Table 2 also allows one to read off the relations of the measured distributions to the symmetry violating parameters of Λ and T, as demonstrated below, and also to construct combinations of data which are true signatures for specific violations.

The earlier results
The experiments [6][7][8][9] have measured in 2001/2 all the data sets listed in Table 2, and thereby discovered CP violation in the matrix T. We show now that these data furthermore establish time-reversal symmetry violation in H weak , and are compatible as well with CP T symmetry of the T matrix as with ǫ = 0, δ = 0, i.e. CP symmetry of Λ.
The experiment [9] has set a stringent limit on T -symmetry violation in the Λ matrix of the B 0B0 system with a direct measurement of ǫ. See Table 2 (entries {9} and {10}) and Table 3. The method is analogous to the one of the CPLEAR experiment [17,18] for the K 0K 0 system, where also a signature for T -violation ("Kabir asymmetry") has been directly measured. The experiments make use of the general identity, valid in two dimensions (see [13]), Λ 21 /Λ 12 ≡ (e −iΛt ) 21 /(e −iΛt ) 12 = U 21 /U 12 from which the connection from the data to the T -symmetry violation signal ǫ , follows -without any assumptions on CP T symmetry or on the value of ∆Γ of the Λ matrix. A reanalysis of the results in 2007 of the BABAR and Belle collaborations by [19] has shown that the data contradict motion-reversal symmetry (see [5]) in the B 0B0 system.
In summary, the discovered CP violation in the B 0B0 system is T -symmetry violation in the decay-amplitude matrix Table 3: A selection of expectations for the experiment of Ref. [10].
Due to the presence of T CP T , of the p i and q i , our results contradict the attempt [12,20] to define the differences {2a} to {2d}, each as a signature for T violation. In the lower part, signatures for T -and CP T -symmetry violations are indicated.
Display in [10] Rates compared Expected ∝ a + b cos(∆m t) + c sin(∆m t) a b c Signatures are for T, TT = 0 with TCP T ≈ 0. In the K 0K 0 system, however, the CP -violation is T -symmetry violation in oscillations, ΛT = 0 with ΛCP T ≈ 0 .

Recent results
The analysis by [10] is based on [12] with novel notions of CP T -, CP -, and T -symmetry, which, in contrast to the classical definitions [1], are not related to properties of the weak interaction Hamiltonian, but to comparisons of surviving states |S f 1 with suitably motion-reversal transformed ones of type |S f ′ 1 . The novel definitions are less general than the classical ones as they need the assumption of TCP T = 0. This new analysis then becomes a special case of our present work, and in turn looses the possibility to address the "classical aim", mentioned in our Introduction. (Details below).
To prove that the phenomenology of [12] uses TCP T = 0, it is sufficient to express their eq.(A.5 of [12]) in terms of the elements of the matrix T, T 11 and T 22 , to find αβ * = −1 = − | T 11 | 2 / | T 22 | 2 or TCP T = 0. The work of [12] specifies 3 sets of 4 pairs of measurements, whose comparisons are supposed to indicate the violations of the 3 symmetries mentioned above. (See Tables 1, 2, 3 of [12]). Each of the 24 measurements is completely determined by the products of the first and the second decay of the antisymmetric, entangled B 0B0 pair. Their amplitudes are thus uniquely given by our eq. (12). The corresponding rates are listed in our Table 2, labeled {1} to {8}.
The envisaged T -violating comparisons, labelled {2a} to {2d} in Table 3, depend also on TCP T , and thus contradict the affirmation in [10], that "Any difference in these two rates is evidence for T -symmetry violation", since a T -symmetric, CP T -violating Hamiltonian H weak (TT = 0, TCP T = 0) would just also create such rate differences.
The CP -violating comparisons in Table 2 of [12] also depend on TCP T cos(∆mt) and on TT sin(∆mt). This confirms that T -and/or CP T -violation imply CP -violation. T -violation in the (2 by 2 dimensional) B 0B0 system is thus never independent of CP violation. See also [14].
The CP T -violating comparisons in Table 3 of [12] neither depend on TCP T nor on TT , and are thus, contrary to the authors' intentions, unable to detect CP T symmetry violation in the matrix T.
Nevertheless, the measured frequency distributions {2a} to {2d} show a dominant sin(∆m t) time-dependence, meaning, for this reason, that TCP T ≈ 0, and with the previous knowledge about the vanishing of the qi, that TT = 0, i. e. T -symmetry violation is confirmed. (More combinations are discussed in [5]). In the lower part of Table 3, we indicate rate combinations which are true signatures of T -or CP T -symmetry violations.

Conclusion
The experiments [6] and [7] have discovered CP violation in the B 0B0 system. Our analysis shows that this CP violation is dominantly T violation, with the same statistical significance. Furthermore, their data sets contain the information which allows for the estimation of all symmetry-violating parameters indicated in Table 1. CP symmetry of the matrix Λ, which governs the B 0B 0 oscillation, is confirmed.
The novel definitions of the symmetries (CP, T, CP T ) used by [12,20] are more restrictive than the classical ones [1].