Interpreting the time-dependent CP asymmetry in $B^0 \to \pi^0 K_S$

Flavor SU(3) is used for studying the time-dependent CP asymmetry in $B^0 \to \pi^0 K_S$ by relating this process to $B^0\to \pi^0\pi^0$ and $B^0 \to K^+K^-$. We calculate correlated bounds on $S_{\pi K} - \sin 2\beta$ and $C_{\pi K}$, with maximal magnitudes of 0.2 and 0.3, where $S_{\pi K}$ and $C_{\pi K}$ are coefficients of $\sin\Delta mt$ and $\cos\Delta mt$ in the asymmetry. Stronger upper limits on $B^0 \to K^+K^-$ are expected to reduce these bounds and to imply nonzero lower limits on these observables. The asymmetry is studied as a function of a strong phase and the weak phase $\gamma$.

or U-spin [6,7,8,9,10]. These bounds may be used to indicate when a deviation from the Standard Model is observed in asymmetry measurements [11].
In the present Letter we interpret the results for the two asymmetries S πK and C πK in terms of the two amplitudes contributing to this process and their relative strong and weak phases. The relative weak phase between the two interfering amplitudes is the CKM phase γ ≡ Arg(V * ub ). Using flavor SU(3), we find a relation between deviations from S πK = sin 2β and C πK = 0 and decay rates for B 0 → π 0 π 0 and B 0 → K + K − . The major purpose of this study is to provide, within the CKM framework, both upper and lower bounds on these deviations in terms of measured rates. It will also be shown how to obtain information about γ if such deviations are measured within the range allowed in the Standard Model.
We decompose the amplitude for B 0 → π 0 K 0 into two terms involving CKM factors V * cb V cs and V * ub V us , which we denote by p ′ / √ 2 and −c ′ / √ 2, respectively, This parameterization is true in general within the Standard Model. The two terms, a penguin amplitude p ′ with strong phase δ and a color-suppressed tree amplitude c ′ with weak phase γ, are graphical representations of SU(3) amplitudes [14] of which we make use below. The amplitude p ′ contains color-allowed and color-suppressed contributions from electroweak penguin operators, p ′ ≡ P ′ − P ′ EW − P ′c EW /3 [15]. Expressions for S πK and C πK in terms of p ′ and c ′ can be obtained from definitions, taking into account the negative CP eigenvalue of π 0 K S in B 0 decays: where Using Eq. (4), the asymmetries S πK and C πK are then written in terms of |c ′ /p ′ |, δ, γ, and α ≡ π − β − γ, as The amplitudes p ′ and c ′ are expected to obey a hierarchy, |c ′ | ≪ |p ′ | [14,15], which will be justified later on using experimental data. In the limit of neglecting c ′ , one has the well-known result S πK = sin 2β, C πK = 0. Keeping only linear terms in |c ′ /p ′ |, one has [16] ∆S πK ≡ S πK − sin 2β ≈ −2|c ′ /p ′ | cos 2β cos δ sin γ , Precise knowledge of the ratio |c ′ /p ′ | would permit a determination of sin 2 γ from the two measurements of S πK and C πK [7], Our goal is to obtain information about |c ′ /p ′ | from other B decays using flavor SU (3). For this purpose, we write expressions within flavor SU(3) for the amplitudes of two strangeness conserving B 0 decays [14,15], The amplitudes p and c in ∆S = 0 decays, defined in analogy with p ′ and c ′ in ∆S = 1 decays, involve CKM factors V * cb V cd and V * ub V ud , respectively. The exchange (e) and penguin annihilation (pa) amplitudes occurring in the second process are expected to be negligible, unless enhanced by rescattering [17]. Current branching ratio measurements, averaged over B 0 andB 0 , are [13] B(B 0 → π 0 π 0 ) = (1.89 ± 0.46) × 10 −6 , (14) B(B 0 → K + K − ) < 0.6 × 10 −6 (90% confidence level) .
These values already indicate some suppression of e + pa relative to p − c. Using the 90% confidence level upper bound onB(B 0 → K + K − ) and the central value ofB(B 0 → π 0 π 0 ) we obtain the 90% confidence level bound Although this suppression is not strong enough to allow neglect of the terms e + pa in B 0 → π 0 π 0 , we will make this approximation in the majority of our discussion, anticipating that the bound (15) will be improved in future measurements of B 0 → K + K − . For completeness, we will also discuss the effect of including the amplitude for The other two terms in A(B 0 → π 0 π 0 ), p and c, which are often assumed to dominate this process, are related by SU(3) to the amplitudes p ′ and c ′ in A(B 0 → π 0 K 0 ) through ratios of corresponding CKM factors, where [3] Eqs. (12), (13) and (17) imply This relation between (4), which involves the same hadronic amplitudes p ′ and c ′ with different CKM coefficients, is the basis of our study. We emphasize that it follows purely from SU (3), as can be read form the tables in [7,14]. We start by neglecting the B 0 → K + K − amplitude. Under this approximation, using Eqs. (4) and (19), we calculate the ratio of rates for decays into π 0 π 0 and π 0 K 0 , averaged over B 0 andB 0 and multiplied byλ 2 , The current experimental value of the ratio R π/K obtained from (3) and (14) is For a given value of R π/K in this range, |c ′ /p ′ | is a monotonically decreasing function of cos δ cos γ, Eq. (20) can be used to set bounds on |c ′ /p ′ |. Noting that −1 ≤ cos δ cos γ ≤ 1, one has With R π/K = 0.091 ± 0.012, one finds This implies the following bounds at 95% confidence level: The lower and upper bounds correspond to cos δ cos γ = 1 and cos δ cos γ = −1, respectively. Slightly stronger bounds on |c ′ /p ′ | may be obtained by using current constraints on CKM parameters [18] implying γ > 38 • , or −0.79 ≤ cos δ cos γ ≤ 0.79, at 95% confidence level.
We now turn to ∆S πK and C πK for which we wish to calculate bounds. We proceed in two ways. First, we use the approximate expressions (10) and derive analytically separate bounds on these two measurables. Then we use the exact expressions (7)- (9) in order to draw a graphical plot for correlated bounds.
Eqs. (10) and (22) may be used to calculate maxima for the magnitudes of ∆S πK and C πK when varying δ and γ for fixed values of β and R π/K . Since |c ′ /p ′ | decreases monotonically with cos δ cos γ, the maximum of ∆S πK which is proportional to cos δ is obtained for δ = π and is positive. As for γ, the maximum is obtained for a value given approximately by The current data imply a value γ ≈ 56 • , which lies in the allowed range [18] The most negative value of this measurable in the allowed region of γ is obtained for Since C πK (−δ) = −C πK (δ), one may consider only its magnitude. The maximum of |C πK | is obtained at δ = γ = π/2, for which one finds The value of |C πK | max is essentially the same at γ = 80 • . We will comment on this maximal value below, where we relate it to the CP asymmetry in B 0 → π 0 π 0 . The exact expressions (7)-(9) imply correlated constraints in the S πK -|C πK | plane associated with fixed values of R π/K . We take values of δ with a 15 • step, values of γ satisfying [18] 38 • ≤ γ ≤ 80 • , and values of R π/K between the ±1σ limits of Eq. (21). A scatter plot of the results is shown in Fig. 1. We find The bounds of the allowed region differ only slightly from (27)-(29), for which approximate expressions were used and a central value was chosen for R π/K . An important point demonstrated by the plot is that the measurement of B 0 → π 0 π 0 is seen to imply a minimum deviation from the point (S πK , C πK ) = (sin 2β, 0), which requires a non-zero value for |c ′ /p ′ |. SU(3) breaking in the ratios p ′ /p and c ′ /c is expected to introduce corrections at a level of 20-30 % in these ratios. These effects may be studied using QCD calculations [19,20]. Corresponding effects in ∆S πK and |C πK | are likely to be smaller, since these two quantities involve the ratio of amplitudes |c ′ /p ′ | in which some SU(3) breaking corrections are expected to cancel. We conclude that |∆S πK | and |C πK | are at most as large as 0.2. Larger values would signal physics beyond the Standard Model in B 0 → π 0 K 0 . The possible role of new physics in B → πK decays was studied in [21].
Note that the maximal values of |∆S πK | and |C πK | are obtained for different values of δ. Measuring nonzero values for ∆S πK and C πK , within the above bounds permitted by the Standard Model, could be used to obtain information about tan δ and |c ′ /p ′ | sin γ through rather simple expressions obtained in the linear approximation (10), Since |c ′ /p ′ | in (22) depends on cos δ cos γ, this can in principle be used to determine γ up to discrete ambiguities.
In the above calculation we neglected the contribution of A(B 0 → K + K − ) to the lefthand-side of Eq. (19), anticipating that the upper bound on the corresponding branching ratio (15) will be improved in the future. Including this contribution introduces several unknowns related to magnitudes and strong phases of the terms e and pa, but nevertheless permits a similar analysis of correlated bounds on the asymmetries ∆S πK and C πK in terms of the strong phase δ between p ′ and c ′ and the weak phase γ. That is, one may compute the maximal allowed values of |c ′ /p ′ |, |∆S πK | and |C πK | as functions of δ and γ under the current bound (15).
Starting from Eq. (19), one forms a ratio where A ππ,KK,πK ≡ A(B 0 → π 0 π 0 , K + K − , π 0 K 0 ) andĀ ππ,KK,πK are the amplitudes of the charge-conjugate processes. This ratio is given by the right-hand-side of Eq. (20) in terms of |c ′ /p ′ |, δ and γ. The maximal and minimal allowed values of |c ′ /p ′ | are attained for the largest and smallest possible values of R ′ π/K , respectively, and are calculated from expressions similar to Eq. (24), in which values of R π/K are replaced by corresponding values of R ′ π/K . The maximal values of |∆S πK | and |C πK | correspond to the maximum of R ′ π/K . Although R ′ π/K is not measurable, upper and lower bounds on this quantity follow from the general inequalities The left and right side inequalities become equalities when A KK / √ 2 = ∓rA ππ and A KK / √ 2 = ∓rĀ ππ , where r is defined in Eq. (16). Denoting one then has Thus, we can use the measured limits on R ′ ± to set bounds on ∆S πK and C πK in the same way as before, with B 0 → K + K − now taken into account. We replace the upper bound on R π/K by R ′ + = (1+r max ) 2 R π/K , and the lower bound by R ′ − = (1−r max ) 2 R π/K , where r max = 0.4 from Eq. (16).
Using the central values of the measured rates ofB(B 0 → π 0 π 0 ) andB(B 0 → π 0 K 0 ) and the upper bound onB(B 0 → K + K − ) we get An equation similar to (24), in which R π/K is replaced by R ′ + for an upper bound on |c ′ /p ′ |, and by R ′ − for a lower bound, implies Including errors in R π/K allows a value |c ′ /p ′ | = 0, implying that ∆S πK = C πK = 0 is not forbidden in contrast to the case of neglecting the amplitude for B 0 → K + K − . The above value of R ′ + implies, for δ = π and γ ≃ 61 • given by (26) (in which R π/K is replaced by while for δ = 0 and γ = 80 • we find We also obtain The allowed range of S πK and C πK can be calculated using the exact expressions where extreme values are larger than those in (30) by about 50%. As mentioned, there is now no minimum deviation from the point (S πK , C πK ) = (sin 2β, 0). Such a deviation is expected when improving the upper bound on B 0 → K + K − . We wish to conclude with a few comments: • In the first part of our study we have neglected A(B 0 → K + K − )/ √ 2 relative to A(B 0 → π 0 π 0 ). As we have shown now, including the first amplitude weakens somewhat the upper bounds on |c ′ /p ′ | and on |∆S πK | and |C πK |. We expect that in the next few years the current bound (16) will be improved to imply |e + pa|/|p − c| < 0.2 − 0.3. At this point, the approximation of neglecting these terms will introduce an uncertainty at the same level as SU(3) breaking corrections in p ′ /p and c ′ /c. It would be interesting to study the magnitude of e + pa and SU(3) breaking effects in the above ratios by using QCD calculations [19,20].
• We considered only the direct CP asymmetry −C πK in B 0 → π 0 K 0 . Eventually, one hopes to also measure an asymmetry in B 0 → π 0 π 0 . In the SU(3) approximation and neglecting e + pa, the CP rate differences in these two processes have equal magnitudes and opposite signs [22]. Measuring the two asymmetries may be used to check for SU(3) breaking corrections. Since the charge averaged rate of B 0 → π 0 K 0 is about six times larger than that of B 0 → π 0 π 0 , a small asymmetry C πK implies a six times larger asymmetry in decays to π 0 π 0 . The maximal value calculated for C πK in (29) corresponds to an asymmetry of about 100% in B 0 → π 0 π 0 . Turning things around, an absolute maximal 100% asymmetry in Figure 2: Points in the S πK -|C πK | plane satisfying ±1σ limits on the ratio (1 ± r max ) 2 R π/K , where r max = 0.4, i.e., taking into account upper bound onB(B 0 → K + K − ).
Other notation is the same as in Fig. 1.
In the SU(3) limit, this amplitude is equal to A(B 0 → π 0 π 0 )+A(B 0 → K + K − )/ √ 2 and may replace this sum on the left-hand-side of Eq. (19). In order to obtain bounds on S πK and C πK as above, one would then have to know the ratioB(B s → π 0K 0 )/B(B 0 → π 0 K 0 ). Measuring the charge averaged rate for B s → π 0K 0 in an environment of a hadronic collider may be quite challenging.
• The method for obtaining correlated bounds on ∆S πK and C πK may be applied to CP asymmetries in other processes, such as B 0 → η ′ K S and B 0 → φK S . In [7] upper bounds on quantities analogous to |c ′ /p ′ | were obtained by relating within SU(3) the amplitudes of these processes to the sum of several ∆S = 0 amplitudes. For B 0 → φK S , the bound requires an assumption that a term with weak phase γ is not much larger than in B + → φK + . The SU(3) relations for B 0 → η ′ K S and B + → φK + were shown to follow from U-spin symmetry [9,10]. The bounds on a ratio analogous to |c ′ /p ′ | provided estimates for the maximal values of the asymmetries |S − sin 2β|. In deriving these bounds additive corrections of order (λ 2 ) were neglected in quantities resembling R π/K , and only leading order terms in a |c ′ /p ′ | expansion were kept. Studying the dependence of the asymmetries S and C on c ′ /p ′ , and on strong and weak phases, and avoiding such approximations, one can use the SU(3) relations of [7,9,10] in order to get more precise bounds in the S − |C| plane.