Direct CP Violation in Untagged B-Meson Decays

Direct CP violation can exist in untagged, neutral B-meson decays to certain self-conjugate, hadronic final states. It can occur if the resonances which appear therein permit the identification of distinct, CP-conjugate states -- in analogy to stereochemistry, we term such states ``CP-enantiomers.'' These states permit the construction of a CP-odd amplitude combination in the untagged decay rate, which is non-zero if direct CP violation is present. The decay B\to \pi^+\pi^-\pi^0, containing the distinct CP-conjugate states \rho^+ \pi^- and \rho^- \pi^+, provides one such example of a CP-enantiomeric pair. We illustrate the possibilities in various multi-particle final states.

The measurement of a non-zero value of Re(ǫ ′ /ǫ) in K → ππ decays establishes the existence of direct CP violation in nature [1], and provides an important first check of the mechanism of CP violation in the Standard Model (SM). Numerically, however, Re(ǫ ′ /ǫ) is very small. In the SM, this results, in part, from the weakness of inter-generational mixing [2]; the associated CP-violating parameter δ KM in the Cabibbo-Kobayashi-Maskawa (CKM) matrix need not be small [3]. Indeed, the measurement of a large CP-asymmetry in B 0 (B 0 ) → J/ψK s decay and related modes [4], induced through the interference of B 0 −B 0 mixing and direct decay, suggests that δ KM ∼ O(1) [5]. Nevertheless, the observation of direct CP violation in the B-meson system is needed to clarify the mechanism of CP violation, to confirm that the Kobayashi-Maskawa (KM) phase drives the CP-violating effects seen. In the SM, direct CP violation is anticipated to be much larger in B-meson decays than in K-meson decays [6]. The observation of direct CP violation in B-meson decays would falsify models in which the CP-violating interactions are "essentially" superweak [7,8]. In this paper, we discuss how the presence of direct CP violation can be elucidated in untagged B-meson decays -the practical advantage of this strategy is the far larger statistical sample of events available.
The rich resonance structure of the multiparticle (n ≥ 2) final states accessible in heavy meson decays provides the possibility of observing direct CP violation without tagging the flavor of the decaying, neutral meson. The familar condition for the presence of direct CP violation, |Āf /A f | = 1, can be met by a non-zero value of the partial rate asymmetry, so that, seemingly, one would want to distinguish empirically a decay with amplitude A f from that of its CP-conjugate mode with amplitudeĀf . However, in neutral B, D-meson decays to self-conjugate final states [9,10,11], direct CP violation in untagged decays may nevertheless occur. It can occur if we can separate the self-conjugate final state, via the resonances which appear, into distinct, CP-conjugate states. This condition finds it analogue in stereochemistry: we refer to molecules which are non-superimposable, mirror images of each other as enantiomers [12]. Accordingly, we refer to non-superimposable, CPconjugate states as CP enantiomers. In B → π + π − π 0 decay, e.g., the intermediate states ρ + π − and ρ − π + form CP enantiomers, as they are distinct, CP-conjugate states. As a result, the untagged decay rate contains a CP-odd amplitude combination. The empirical presence of this CP-odd interference term in the untagged decay rate would be realized in the Dalitz plot as a population asymmetry, reflective of direct CP violation.
We shall use B → π + π − π 0 decay as a paradigm of how direct CP violation can occur in untagged B-meson decays. In what follows, we shall largely follow the notation and conventions of Quinn and Silva [13]. Consider the amplitudes for B 0 (B 0 ) → π + π − π 0 decay: where the two-body decay amplitudes are given by a +− = A(B 0 → ρ + π − ), a −+ = A(B 0 → ρ − π + ), and a 00 = A(B 0 → ρ 0 π 0 ) and f i is the form factor describing ρ i → ππ. We have used s = (p − + p 0 ) 2 , t = (p + + p − ) 2 , and u = (p + + p 0 ) 2 1 . For clarity, note that . Since ρ + π − and ρ − π + are distinct, CP-conjugate states, the amplitudes a g = a +− + a −+ and a u = a +− − a −+ have distinct properties under CP. That is, if we defineā g =ā +− +ā −+ andā u =ā +− −ā −+ , we see, under an appropriate choice of phase conventions, that the CP conjugate of a g isā g , whereas the CP conjugate of a u is −ā u . With a n = 2a 00 we have Neglecting the width difference of the B-meson mass eigenstates, as ∆Γ ≡ Γ H − Γ L and |∆Γ| ≪ Γ ≡ (Γ H + Γ L )/2, the decay rate into π + π − π 0 for a B 0 meson at time t = 0 is given by [15] (3) whereas the analogous decay rate for aB 0 meson at time t = 0 is given by (4) Note that λ 3π ≡ qĀ 3π /pA 3π and ∆m ≡ M H − M L . We neglect ∆Γ, so that we set |q/p| = 1. Untagged observables, for which the identity of the B meson at t = 0 is unimportant, where i, j ∈ g, u, n, noting that i, j labels are not repeated in the sum labelled "i < j".
The different products f i f * j are distinguishable through the Dalitz plot of this decay, so that the coefficients of these functions are empirically distinct [13]. For our purposes the crucial point is that these observables, as first noted by Quinn and Silva [13], can be of CP-odd character. In particular, the presence of a g a * u +ā gā * u and/or a n a * u +ā nā * u is reflective of direct CP violation. Physically these observables correspond to a population asymmetry under the exchange of u and s (or of p + and p − ) across the Dalitz plot. To make the geometric sense of this construction clear, consider a Dalitz plot in u versus s, that is, in the invariant masses of the π + π 0 and π − π 0 pairs, respectively -such a plot is shown in Fig. 1 of Ref. [16]. The presence of the CP-odd amplitude a g a * u +ā gā * u , e.g., engenders a population asymmetry about the u = s "mirror line;" specifically, the number , or a more sophisticated function, consistent with the theoretical constraints of analyticity, time-reversal-invariance, and unitarity, see Ref. [14] for all details. Note that, e.g., of charged ρ events in the u > s region differs from that in the s < u region. Note that the functional form of f + (u) and f + (s) restrict the product f g f u to the ρ ± bands in the Dalitz plot. The asymmetry is largest in the regions where the ρ i bands overlap, though the restricted number of events in the overlap region make it more efficacious to compare the entire population of the charged ρ bands in the u > s and u < s regions [17]. The second amplitude combination of Eq. (6) is determined by the population asymmetry across the u = s line in the regions in which the ρ ± and ρ 0 bands overlap. A population asymmetry in B,B → π + π − π 0 decay about the u = s line is also a signature of direct CP violation. However, non-zero values of the amplitude combinations of Eq. (6) do not guarantee its existence as cancellations, though likely incomplete, can occur. The direct CP-violating observables of Eq. (6) can persist even if the strong phases of the a j amplitudes were zero. To illustrate, we parametrize a j = T j exp(−iα) + P j and P j /T j = r j exp(iδ j ), where r j > 0 and δ j is the strong phase of interest 2 . Thus The real and imaginary parts of this relation are each observable, as they correspond to distinct f i -dependent terms in Eq. (5). The combination T g T * u can be complex, though we assume it to be real for crispness of discussion. In the imaginary part, we see that direct CP violation can exist if the strong phases of a j vanish, i.e., if δ u = δ g = 0; merely the difference of r g and r u must be non-zero to realize direct CP violation were sin α = 0. If δ j = 0 the strong phase is provided by the resonance width, Im(f i f * j ) = 0. Theoretical estimates suggest that r g and r u are both non-zero and unequal [18]. In constrast, a partial rate asymmetry can be written as |a g | 2 − |ā g | 2 = −4|T g | 2 r g sin δ g sin α , yielding the familiar result that both r g and δ g must be non-zero to yield direct CP violation were sin α = 0. Such conditions are realized in the real part of Eq. (7) as well, so that the direct CP-violating observables we propose can be manifest irrespective of the strong phases of a j , as they can be non-zero were δ j either zero or 90 degrees. This greater flexibility arises as the combination P g /T g − P * u /T * u appears in Eq. (7), whereas P g /T g − P * g /T * g , e.g., appears in the partial rate asymmetry.
Interestingly, similar considerations arise in the angular analysis of B → V 1 V 2 decays: there, too, a CP-odd interference term can beget direct CP violation in untagged decays [19,20]. There are three helicity amplitudes, labelled by the helicity λ ∈ (0, ±1) of either vector meson in B → V 1 V 2 decay. Working in a transversity basis [21], we can define the amplitudes [22]. The full angular distribution of the summed amplitudes for B 0 andB 0 decay permits the extraction of the imaginary part of the amplitude combinations of Eq. (6), under the identification a g → A , a u → A ⊥ , and a n → A 0 . Moreover, these untagged contributions are insensitive to the strong phase [23].
The conditions which permit the realization of direct CP violation in untagged modes are quite general. We need only consider self-conjugate final states whose resonances encode enantiomeric pair correlations. Self-conjugate final states can be realized not only through the b → dqq decays of B d mesons but also through the b → sqq decays of B s mesons, where q ∈ u, d, s, c quarks. The KM picture of CP violation suggests that direct CPviolating effects ought be suppressed by a factor of O(λ 2 ) ∼ 1/20 in B s meson decay to charmed, self-conjugate states. Thus the goals of direct CP violation searches in B d and B s meson decays can be distinct. The appearance of direct CP violation in B d -meson decays would substantiate the KM picture of CP violation, whereas its appearance in any significant measure in B s decays to charmed final states would signal the presence of new physics. Physics with B s mesons is important to the future B-physics programs at the Tevatron [24] and at the LHC [25]. The effective tagging efficiency ǫ eff is significantly smaller in a hadronic environment, cf. ǫ eff ∼ 7% [26] with ǫ eff ∼ 27% [27,28] at the B-factories, so that the untagged studies we propose significantly enable direct CP violation searches at these facilities 3 .
Let us enumerate three-, four-, and five-particle final states in B d decay which could yield direct CP violation in the KM picture. We thus focus on b → duū and b → dcc decays, and some possibilities are given in Table I -we do not attempt to be exhaustive. The CP-enantiomers are useful in the sense we have illustrated in B → ρπ decay: they permit the formation of manifestly CP-odd amplitude combinations which can be probed through asymmetries in the population of events in the regions where the resonances of the CP-enantiomeric pair occur. We expect the CP-violating effects to be larger for broad resonances such as the ρ and K * (892). Note that the final states K + K − π 0 and K + K − π + π − , with the CP enantiomers indicated, also lend themselves to direct CP violation searches in B s decay. Multiparticle final states can support more than one CP-enantiomeric pair, as illustrated in B d → π + π − π + π − π 0 decay. In the case of CP enantiomers which have more than one spin one particle, as in (a 1 (1260) + ρ − , a 1 (1260) − ρ + ), or which are not realized by a quasi-two-body decay, as in (ρ + π − π + π − , ρ − π + π + π − ), a caution is in order. For example, the presence of two spin-one particles in the final state implies that partial waves with L = 0, 1, or 2 can occur; the factor (−1) L impacts the CP of the state. The sum and difference of the amplitudes associated with B 0 → a 1 (1260) + ρ − andB 0 → a 1 (1260) − ρ + decay still yield combinations with definite CP properties for any particular L, but for L = 0 or 2 the sum of amplitudes, with a suitable choice of phase conventions, does not change sign under CP, whereas for L = 1 the sum of amplitudes do change sign under CP. In either event, for fixed L, the CP-odd amplitude combination of Eq. (7) appears and drives a population asymmetry under the exchange of the momentum of a π + emerging from the a 1 (1260) + and that of the π − from the ρ − in the region of the Dalitz plot where the resonances of the CP-enantiomeric pair occur. States of fixed L can be realized through a helicity analysis; the formation of the A ⊥ amplitude, e.g., selects the L = 1 state [21]. In the absence of a helicity analysis, both CP-even and CP-odd contributions are subsumed in "g × u" term of Eq. (7), so that a population asymmetry in this case can exist without direct CP violation. Thus for pairs with two spin one particles, a helicity analysis is required; similar considerations apply to pairs for which the decays are not quasi-two-body in nature -an ancillary angular analysis is necessary.
The observation of direct CP violation in B-meson decays in itself is crucial to establishing the mechanism of CP violation. Nevertheless, we would also like to interpret such results in terms of the parameters of the CKM matrix. An assumption of isospin symmetry can codify TABLE I: B d decays to certain three-, four-, and five-particle, self-conjugate final-states and some of the CP-enantiomers they contain.

5-particles CP-enantiomers
a A helicity and/or angular analysis is required; see text.
and potentially determine the hadronic parameters needed to interpret the mixing-induced CP-asymmetry in b → dqq transitions to charmless final states. Relevant to the modes we discuss are the isospin-based analyses which yield sin(2α) in B → ρπ [13,29,30] and B → a 1 π [31] decays. These analyses, however, do not determine the parameters necessary to interpret direct CP violation; the terms containing sin α and cos α are multiplied by unknown hadronic parameters. Nevertheless, were sin(2α) determined and direct CP violation observed, the SM value of sin α could be inferred, modulo discrete ambiguities. Interpreting direct CP-violating observables directly in terms of the underlying weak parameters may not prove possible. Theoretical progress has been made, however, in the computation of partial-rate asymetries in some two-body decays, see, e.g., Refs. [32,33]. Alternatively, more phenomenological treatments indicate that the presence of resonances in certain channels can enhance the associated partial rate asymmetry [34,35] and aid in the extraction of weak phase information [36]. We have discussed the conditions under which the rich resonance structure of hadronic B decays can be exploited to search for direct CP violation in untagged decays. Our method is sufficiently general to enable direct CP violation searches in B s and D meson decays as well. In some channels the untagged search we propose complements tagged, time-dependent analyses in B → ρπ and B → a 1 π decays. Nevertheless, the gain in statistical power realized in untagged versus tagged searches, i.e., roughly a factor of 2 at the B-factories and of 4 in a hadronic environment such as at CDF, argues for a more comprehensive program.