Resolving a Discrete Ambiguity in the Ckm Angle Β Through

It is well known that sin(2β), where β is one of the angles of the unitarity triangle of the CKM matrix, can be determined in a theoretically clean way by measuring mixing-induced CP violation in the decay Bd → J/ψKS. Another clean extraction of this CKM angle is provided by the time-dependent angular distribution for the decay products of Bd → J/ψ(→ l +l−)K∗0(→ πKS), where we have more observables at our disposal than in the case of Bd → J/ψKS, so that in addition to sin(2β) also cos(2β) can be probed in a direct way. Unfortunately a sign ambiguity remains in cos(2β). If it could be resolved, a discrete ambiguity in the extraction of the CKM angle β could be resolved as well, which would allow a more incisive test of the CKM model of CP violation. This note shows that detailed time-dependent studies of Bu,d → J/ψK ∗ and Bs → J/ψ φ decay processes can determine the sign of cos(2β), thereby removing the corresponding ambiguity in the extraction of the CKM angle β. CERN-TH/98-123 April 1998 The conventional methods for determining the angles α, β and γ of the usual unitarity triangle [1] of the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix) [2] leave several discrete ambiguities [3]. This is also the case for the “gold-plated” mode Bd → J/ψKS. The mixing-induced CP asymmetry arising in this channel allows only a theoretically clean determination of sin(2β), so that a discrete four-fold ambiguity for the extracted value of β ∈ [0, 360] remains. In the recent literature, several strategies were proposed to resolve ambiguities of this kind [4]. Another clean probe of the CKM angle β is provided by the observables of the angular distributions for the decay products of Bd → J/ψ(→ l l)K(→ πKS) modes [5]–[7]. Such observables can in general be expressed in terms of decay amplitudes as |Af(t)| , Re [A f̃ (t)Af(t) ], Im [A ∗ f̃ (t)Af (t) ], (1) where f and f̃ are labels for specific final-state configurations. The full three-angle distributions for tagged Bd(t) → J/ψK ∗0 (→ πKS) decays are given in [6]–[8]. Throughout this note, by tagging we mean making the distinction of initially, i.e. at t = 0, present unmixed B d,s and B 0 d,s mesons. Weighting functions have been derived to extract the corresponding observables in an efficient way from experimental data [6, 7]. The time evolution of the interference terms in (1), i.e. of the real and imaginary parts of bilinear combinations of certain decay amplitudes, allows the determination [7] of sin(δ1,2) and cos(δ1 − δ2), where δ1 and δ2 are CP-conserving strong phases, and of sin(2β) and cos(δ1,2) cos(2β). (2) The CP-conserving observables |Af(t = 0)|, sin(δ1,2) and cos(δ1 − δ2) can be determined to a higher accuracy from the much larger data samples arising for B → J/ψK transitions, and untagged Bd decays into J/ψK (→ Kπ) and J/ψK∗0(→ Kπ) states [6, 7]. At first sight, one may think that sin(δ1,2) and cos(δ1 − δ2) extracted this way from the Bu,d → J/ψK ∗ angular distributions will allow the determination of cos(2β) with the help of the terms given in (2). A closer look shows, however, that this is unfortunately not the case, since we do not have sufficient information to fix the signs of cos(δ1,2), thereby leaving a sign ambiguity for cos(2β). The purpose of this letter is to point out that this ambiguity can be resolved with the help of tagged, time-dependent studies of Bs → J/ψ φ decays. The angular distributions are given in [7, 8], and weighting functions to extract the observables from experimental data can be found in [7]. An important feature of these observables is that they allow the determination of a CP-violating weak phase φ [7, 9], which takes a very small value, of O(0.03), within the Standard Model, and represents a sensitive probe for new-physics contributions to B s–B 0 s mixing. Provided there is a sizeable mass difference between the mass eigenstates B s and B L s , this phase can even be extracted from untagged Bs data samples [10], where the rapid ∆mst oscillations cancel [11]. Another important feature is the fact that the tagged, time-dependent (−) Bs (t) → J/ψ(→ ll)φ(→ KK) observables corresponding to the “Im” terms in (1) provide