Examination of Flavor Symmetry in $B, B_s \to K \pi$ Decays

We study a relation between the weak phase $\gamma$ and the rates and CP asymmetries of several $K \pi$ decays of $B^+$, $B^0$, and $B_s$, emphasizing the impact of the latter measurements. Current data indicate large SU(3) breaking in the strong phases or failure of factorization (including its application to penguin amplitudes) in $K \pi$ modes of $B^0$ and $B_s$. SU(3) and factorization only remain approximately valid if the branching ratio for $B_s \to K^- \pi^+$ exceeds its current value of $(5.27 \pm 1.17) \times 10^{-6}$ by at least 42%, or if a parameter $\xi$ describing ratios of form factors and decay constants is shifted from its nominal value by more than twice its estimated error.

0.010(stat.) ± 0.007(sys.), where f s /f d is the ratio of production fractions of B s and B 0 . Given the world averages [15] f s = (10.4±1.4)%, f d = (39.8±1.0)%, and B(B 0 → K + π − ) = (19.4±0.6)×10 −6 , this implies B(B s → K − π + ) = (5.27±0.74±0.90)×10 −6 . We include this result along with direct CP asymmetries in B 0 → K + π − and B s → K − π + to solve for γ, the strong phases, and the ratio between tree and penguin amplitudes. We find in general a two-fold ambiguity in the solutions for weak and strong phases. Moreover, we find a large SU(3)-breaking effect either between the strong phases or between the magnitudes of strangeness-conserving and strangeness-changing amplitudes, given the present experimental situation [14].
We review the method proposed in Ref. [9]. Employing U-spin symmetry, the decay amplitudes of the relevant modes are where the explicit t-quark dependence is removed using CKM unitarity. Here T and P denote "tree" and "penguin" amplitudes, proportional to the CKM factors V * us V ub and V * cs V cb , respectively. The parameterλ ≡ |V us /V ud | ≃ 0.2317 using λ = 0.2257 [13] and V ud = √ 1 − λ 2 . We also include an overall SU(3)-breaking factor according to the factorization assumption for the amplitudes. 4 Its value is 0.97 +0.09 −0.11 [13,16], corresponding to almost exact SU (3). 5 This should be compared with global fits done within flavor SU(3) [17,18], which associated the breaking factor f K /f π ≃ 1.2 with tree-type amplitudes only. In that case, the predicted branching ratios of the B s → K − π + and K + K − modes [18] agreed with the later experimental measurements. The relative strong phases between T and P are denoted by δ d and δ s for B 0 → K + π − and B s → K − π + , respectively. For consistency, terms of O(λ 2 ) have been ignored in these amplitudes. Since interactions directly involving the spectator quark are expected to be dynamically suppressed, we also ignore their contributions. Consider the charge-averaged ratios [9] Observable Exp. Value Ref.
have been determined to about 5%, their current central values are not likely to vary much in the future. In contrast, B and A CP of B s → K − π + have only been measured by the CDF Collaboration for the first time. The quoted value of B(B s → K − π + ) [14] depends on the fragmentation fractions f s and f d [15] (see also Ref. [19]), whose ratio carries a 14% error. (The total systematic error on B(B s → K − π + ), including this contribution, is 17%.) In the following, we discuss the dependence of solutions on the central value of B(B s → K − π + ). As δ s has been fixed to be the same as δ d , we omit A CP (B s → K − π + ) from the fit and predict its value from the fit parameters. Errors and other measurements are kept at their current values. Fig. 1 shows the dependence of r on B(B s → K − π + ); the χ 2 for the fit to R d , R s , and A CP (B 0 → K + π − ); and the predicted A CP (B s → K − π + ). For B(B s → K − π + ) ∼ > 7.5 × 10 −6 , a solution with strong phases satisfying exact SU(3) can be obtained, as indicated by the vanishing χ 2 min value. This can be attributed to an overall SU(3) breaking factor of at least ξ ≃ 1.2, 2.5 σ from the central value 0.97 given by factorization and echoing the observation in Ref. [18] mentioned above. For B(B s → K − π + ) < 7.5 × 10 −6 , one cannot obtain a satisfactory solution if δ s = δ d . In that case, the value |r| = 0.068 ± 0.034 from Eq. (14) is too small to account for R d and R s . The value of |r| is increased to 0.100 (or larger) if we increase R s by a factor 1.4 (or larger). With such values of r one may obtain the central value of R d and a suitable value of R s using Eqs. (9) and (10). This is the essence of the need for either a larger B(B s → K − π + ) or a larger ξ. Indeed, Fleischer [10] obtained a solution with δ s − δ d ≃ 10 • by increasing R d and R s by 1σ.
Current data thus call for SU(3) breaking in amplitudes at the level of 20% or very different strong phases. As shown in Fig. 1, both r and A CP (B s → K − π + ) decrease with increasing B(B s → K − π + ). These conclusions are qualitatively unchanged if we allow δ d and δ s to differ by ∼ < 10 • for small SU(3) breaking.
We show the dependence of γ and δ on B(B s → K − π + ) in Fig. 2. Their values coincide with each other for small values of B(B s → K − π + ), and start to split into three curves when it is greater than 7.5 × 10 −6 . This occurs when χ 2 min becomes zero for the upper (solid) and lower (dashed) branches. For the dash-dotted branch in the middle, γ and δ still coincide with each other and continue to decrease with B(B s → K − π + ). The χ 2 min values along this branch are small but non-vanishing, corresponding to a "saddle" region in parameter space. The upper and lower branches can represent either γ or δ due to the γ ↔ δ symmetry. However, the weak phase given by the solid curve is more consistent with other analyses. In that case, the corresponding strong phase is given by the dashed curve. As shown in the plot, γ (δ) grows (decreases) monotonically with B(B s → K − π + ) above the fork point.
We now let δ d = δ s , permitting a test of the SU(3) symmetry assumption. With four observables R d , R s , A CP (B 0 → K + π − ), and A CP (B s → K − π + ), one can solve for all four parameters r, γ, δ d and δ s in the decay amplitudes.
As shown in Fig. 3, there are two sets of possible solutions (left and right) as a function of B(B s → K − π + ). For the solution on the left, even though γ falls within the expected range, δ d and δ s differ significantly from each other. For the solution on the right, the strong phases are also quite different and γ is too small when B(B s → K − π + ) ∼ < 6.5 × 10 −6 . However, when B(B s → K − π + ) ≥ 7.5 × 10 −6 , γ becomes reasonable, δ d is between 20 • and 30 • , and δ s approaches 50 • . As the current measurement of the CP asymmetry of B s → K − π + has an error over 40%, we expect it to have a weaker constraint on the parameters, δ s . For the current data, two solutions are found, corresponding to the parameters: In the former, γ is more consistent with results using other methods (for example, adding information based on B 0 → π + π − [21]), and a small strong phase δ d as expected in perturbative QCD [22,23]. However, the strong phase δ s in both solutions is unexpectedly large. The 1σ ranges around the former are The result for |r| here is larger than that from Eq. (14) with δ d = δ s . Even though we no longer have the symmetries between the weak and strong phases mentioned before because of the introduction of an additional strong phase δ s , we still obtain two possible solutions roughly corresponding to γ ↔ δ d . Within this set of observables, it is impossible to resolve the two-fold ambiguity without resorting to some other methods or observables.
For the solutions in Eq. (16), δ s is very different from δ d , contrary to the SU(3) symmetry assumption. More likely possibilities are a B s branching ratio larger than the current value or a value of ξ larger than the factorization estimate given above. These alternatives are impossible to distinguish from one another as the parameters ξ and R s always appear in the combination ξ 2 R s [even in ξ 2 A s = ξ 2 R s A CP (B s → K − π + )]. A larger left-hand side of Eq. (10) would entail cos δ s > 0 rather than the current situation, permitting δ s to be closer to δ d . With ξ = 1.2, one would obtain a solution r = −0.11, γ = 56 • , δ d = 28 • , and δ s = 51 • . The reason that δ s − δ d is still as large as 23 • is because of the pull from A CP (B s → K − π + ). As shown in Fig. 1, a smaller asymmetry is preferred if one hopes to have δ s ≃ δ d .
Even though one often assumes the same SU(3) breaking factor for the tree and penguin amplitudes, they can a priori scale differently. Denote the scaling factors associated with T and P by ξ T and ξ P , respectively. By fixing ξ T = ξ and allowing ξ P to vary around 1, we find that for ξ P ≥ 1.2 the strong phase δ s can lie in the first quadrant, but is still too large (> 70 • ). The weak phase γ also falls below 50 • in this case. However, if we fix ξ P = ξ instead and vary ξ T , the solution improves with increasing ξ T . Taking ξ T = 1.5 as an example, we find r = −0.129, γ = 60 • , δ d = 23 • , and δ s = 41 • . This shows that the scaling behavior of T plays a more dominant role.
Next, we allow both ξ T and ξ P to vary by including γ = (67.6 ± 4.5) • [24] obtained from other methods as another observable constraint. We find that if δ s − δ d ∼ > 20 • , it is possible to obtain a perfect fit to the data. In these cases, the preferred values of r, γ, and δ d become fixed at −0.182, 67.6 • , and 15 • .
The preferred values of ξ T and ξ P as a function of δ s − δ d are shown in Fig. 4. When δ s − δ d ∼ < 20 • , no perfect solution exists. But the most favored ξ T increases linearly with the strong phase difference, while ξ P stays almost as a constant. If the equality between the two strong phases is imposed, we find χ 2 min = 0.82 with r = −0.114, γ = 67 • , δ d = δ s = 26 • , ξ T = 1.59, and ξ P = 0.75. When δ s − δ d ∼ > 20 • , ξ T drops while ξ P increases.  In Table II we compare pairs of solutions for B(B s → K − π + ) = 7.5 × 10 −6 and 10 −5 with those for the current value B(B s → K − π + ) = (5.27 ± 1.17) × 10 −6 , keeping the same 22% error. As B(B s → K − π + ) increases, the values of γ and those of δ d in the two solutions become closer to each other. However, the values of δ s remain significantly different from δ d .
To summarize, the U-spin relation between B 0 → K + π − and B s → K − π + [9] has been utilized to obtain a range of values of the CKM phase γ, thanks to new data on the decay B s → K − π + obtained by the CDF Collaboration [14]. Values of γ consistent with other determinations and strong phases δ d and δ s not differing substantially from one another may be obtained only if the branching ratio B(B s → K − π + ) is at least 42% larger than its currently quoted value of (5.27 ± 1.17) × 10 −6 , or if the parameter ξ [Eq. (4)] describing the ratio of decay constants and form factors is more than about 1.2 (vs. its nominal value of 0.97 +0.09 −0.11 ).
For the nominal values of B(B s → K − π + ) and ξ, one obtains a solution with a twofold ambiguity, whose value of γ in the solution closer to other determinations (using such processes as B 0 → π + π − [21]) is ≃ 60 • . In this solution, however, the strong phases are δ d ≃ 23 • and δ s ≃ 155 • . The latter is inconsistent with perturbative QCD calculations and its large difference from δ d would signal significant SU(3) breaking or failure of factorization. Solutions with smaller SU(3) breaking, such as those which would result if B(B s → K − π + ) were at least 42% larger than its nominal value, would be suggested if recent evaluations of b quark fragmentation [15,19] had overestimated the fraction of b quarks ending up as B s . Alternatively, the SU(3) breaking factor ξ could be larger than estimated, or could differ in tree and penguin amplitudes. Further studies of the B s → K − π + decay and b fragmentation at the Fermilab Tevatron and at LHCb may help to illuminate this question.