Implications for CP asymmetries of improved data on $B \to K^0 \pi^0$

The decay $B^0 \to K^0 \pi^0$, dominated by a $b \to s$ penguin amplitude, holds the potential for exhibiting new physics in this amplitude. In the pure QCD penguin limit one expects $\ckp = 0$ and $\skp = \sin 2 \beta$ for the coefficients of $\cos \Delta m t$ and $\sin \Delta m t$ in the time-dependent CP asymmetry. Small non-penguin contributions lead to corrections to these expressions which are calculated in terms of isospin-related $B\to K\pi$ rates and asymmetries, using information about strong phases from experiment. We study the prospects for incisive tests of the Standard Model through examination of these corrections. We update a prediction $\ckp=0.15\pm 0.04$, pointing out the sensitivity of a prediction $\skp\approx 1$ to the measured branching ratio for $B^0\to K^0\pi^0$ and to other observables.

The decay B 0 → K 0 π 0 , dominated by a b → s penguin amplitude, holds the potential for exhibiting new physics in this amplitude. In the pure QCD penguin limit one expects C Kπ = 0 and S Kπ = sin 2β for the coefficients of cos ∆mt and sin ∆mt in the time-dependent CP asymmetry. Small nonpenguin contributions lead to corrections to these expressions which are calculated in terms of isospin-related B → Kπ rates and asymmetries, using information about strong phases from experiment. We study the prospects for incisive tests of the Standard Model through examination of these corrections. We update a prediction C Kπ = 0.15±0.04, pointing out the sensitivity of a prediction S Kπ ≈ 1 to the measured branching ratio for B 0 → K 0 π 0 and to other observables. One of the most challenging CP asymmetry measurements in B meson decays has involved the coefficients C Kπ and S Kπ in the time-dependent asymmetry measured in B 0 → K S π 0 [1] Γ(B 0 (t) → K 0 π 0 ) + Γ(B 0 (t) → K 0 π 0 ) = −C Kπ cos(∆mt)+S Kπ sin(∆mt) . (1) The parameter C Kπ is related to the direct CP asymmetry by C Kπ ≡ −A CP (B 0 → K 0 π 0 ). The decay B 0 → K 0 π 0 is expected to be dominated by the b → s penguin amplitude and thus is a good place to look for any new physics that may arise in this amplitude [2][3][4]. In the pure QCD penguin limit one expects C Kπ = 0 and S Kπ = sin 2β, respectively, where β = (21.5 ± 1.0) • [5] is an angle in the unitarity triangle. Accounting for small non-penguin contributions leads to corrections to these expressions, which are calculable in terms of isospin-related B → Kπ decay rates and asymmetries. In this Letter we study the prospects for incisive tests of the Standard Model through examination of these corrections. We update a prediction C Kπ = 0.15 ± 0.04 and point out the sensitivity of a recent theoretical prediction S Kπ ≈ 1 [6] to the branching ratio for B 0 → K 0 π 0 and to other observables. Ref.
The current status of measurements of C Kπ and S Kπ is summarized in Table I. The value of C Kπ is consistent with the pure-penguin value of zero, while that of S Kπ is 1.6σ below the pure-penguin value of sin 2β = 0.681 ± 0.025.
A sum rule for direct CP asymmetries in B → Kπ decays has been derived purely on the basis of the ∆I = 0 property of the dominant penguin amplitude, using an isospin quadrangle relation among the four B → Kπ decay amplitudes which depend also on two ∆I = 1 amplitudes [9,10]: In its most precise form the sum rule relates the four CP rate differences [11], where one defines This sum rule includes interference terms of the dominant penguin amplitude with all small non-penguin contributions. A few very small quadratic terms representing interference of tree and electroweak penguin amplitudes vanish in the SU(3) and heavy quark limits [11]. Using the decay branching ratios and CP asymmetries summarized in Table II [5] and the known lifetime ratio τ (B + )/τ (B 0 ) = 1.071 ± 0.009 [5], one can use this relation to solve for the least-well-known quantity ∆(K 0 π 0 ), implying The error on the right-hand-side is dominated by the current experimental errors in A CP (K 0 π + ) and A CP (K + π 0 ). The prediction (5) following from (3) involves a smaller theoretical uncertainty at a percent level from quadratic terms describing the interference of small non-penguin amplitudes. Verification of this prediction would provide evidence that non-penguin amplitudes behave as expected in the Standard Model. [If one uses the corresponding sum rule for CP asymmetries, one predicts A CP (K 0 π 0 ) = −0.138 ± 0.037. Using this relation with A CP (K 0 π + ) = 0, as expected since B + → K 0 π + should be dominated by a penguin amplitude with only a very small annihilation contribution [12], one predicts A CP (K 0 π 0 ) = −0.147 ± 0.028.] Non-penguin amplitudes are generally agreed to increase S Kπ from its pure-penguin value of sin 2β = 0.681 ± 0.025 by a modest amount, generally to 0.8 or below [13][14][15][16]. Model-independent bounds using flavor SU(3) [17,18] also favor at most a deviation of 0.2 from the pure-penguin value. An exception is noted in the treatments of Refs. [19] and [20], and most recently in Ref. [6], where a relation between C Kπ and S Kπ was studied implying a value S Kπ = 0.99 for the central value measured for C Kπ . A geometrical construction is performed which illustrates the way in which such a large value arises.
An aspect of the prediction of S Kπ ≃ 0.99 which has not been sufficiently stressed is its extreme sensitivity to the branching ratio (B 0 → K 0 π 0 ). In the present Letter we analyze the sensitivity of S Kπ to this and other observables within the Standard Model, and highlight those measurements which would shed light on the presence of new physics. In order to restrict the range allowed for S Kπ in the Standard Model one needs certain information about strong phases. Theoretical calculations of strong phases in B → Kπ based on 1/m b expansions are known to fail, most likely because of long distance charming penguin contributions [21,22]. We propose to obtain the necessary information about strong phases directly from experiments. Somewhat different but not completely independent arguments were presented in Ref. [6].
The B → Kπ amplitudes may be decomposed into contributions from various amplitudes as follows [23,24]: The terms T, C and A represent color-favored and color-suppressed tree amplitudes and a small annihilation term, while P stands for a gluonic penguin amplitude. Color-favored and color-suppressed electroweak penguin amplitudes are represented by P EW and P C EW . The sums of the first two and last two amplitudes in Eq. (7) are equal [see Eq. (2)] and both correspond to an amplitude A 3/2 for a Kπ state with isospin I Kπ = 3/2 [9,10]: The contribution −(T + C) to A 3/2 has a magnitude which can be obtained from the decay B + → π + π 0 via flavor SU(3) [25], SU (3) breaking in this amplitude is often assumed to be given by the factor f K /f π = 1.193 ± 0.006 [26]. Here we introduce a parameter ξ T +C = 1.0 ± 0.2 which represents an uncertainty in this factor. The weak phase of T + C is Arg(V * ub V us ) = γ, where γ = (65 ± 10) • [27]. We take its strong phase to be zero by convention. All other strong phases will be taken in the range (−π, π). The penguin amplitude P dominating B → Kπ decays carries the weak phase Arg(V * tb V ts ) = π. Its strong phase relative to that of T + C will be denoted −δ c [28]. Thus The electroweak penguin contribution P C EW + P EW was shown in Refs. [29] and [30] to have the same strong phase as T + C in the SU(3) symmetry limit. In this limit the ratio of these two amplitudes is given numerically in terms of ratios of CKM factors and Wilson coefficients, The parameter ξ EW includes an uncertainty from SU(3) breaking, which we will take as ξ EW = 1.0 ± 0.2, and a smaller uncertainty from CKM factors. We neglect a potential small strong phase of ξ EW which has a negligible effect on our analysis below. Thus we have an amplitude triangle relation, and a similar relation for the CP-conjugate amplitudes in which the sign of γ is reversed. In order to visualize the geometric construction of the triangle (12) and its CPconjugate, as proposed in Ref. [6] but with realistic quantities including the restricted range (5) for A CP (K 0 π 0 ), we express all branching ratios in units of 10 −6 , and take amplitudes as their square roots. (We first divide B + branching ratios by the lifetime ratio τ (B + )/τ (B 0 ) = 1.071 ± 0.009 [5] to compare them with B 0 branching ratios.) The central values of |T + C| for ξ T +C = 1 and the squares |A ij | 2 and |Ā ij | 2 , based on central values of the rates and CP asymmetries in Table II Solutions for the amplitude triangle (12) and its CP-conjugate may be obtained analytically by solving simple quadratic equations for the central values of the parameters which fix A 3/2 in (12), ξ EW = 1, γ = 65 • . The quadratic equation for each triangle has Figure 1: Triangles relating amplitudes for B 0 → K 0 π 0 and B 0 → K + π − to the amplitude A 3/2 , and triangles for the corresponding charge-conjugate processes.
two solutions, which can be visualized by flipping the triangle around the side A 3/2 or A 3/2 which is kept fixed. One thus obtains a total of 2 × 2 = 4 solutions, of which two are illustrated in Fig. 1. The other two solutions correspond to flipping one triangle but not the other.
We have chosen to express the triangles with A 00 orĀ 00 emanating from the origin, in order to illustrate the relative phase of A 00 andĀ 00 which will be important in the evaluation of S Kπ . This relative phase vanishes in the limit of pure penguin dominance and is expected to be smaller than π/2 when including small color-suppressed tree and electroweak penguin contributions in A 00 . This feature holds true for the two illustrated solutions but excludes the two solutions with one triangle flipped, for which the relative phase between A 00 andĀ 00 is larger than π/2.
The expected value of S Kπ is related to the magnitudes and phases of A 00 andĀ 00 in the following manner: The correction φ 00 ≡ Arg(A 00Ā * 00 ) to 2β is found to be positive for both of the displayed solutions. It is quite large, φ 00 = 42.6 • corresponding to S Kπ = 0.99, for the solution (1) with negative real values of the amplitudes A 00 andĀ 00 and smaller, φ 00 = 16.1 • corresponding to S Kπ = 0.85, for the solution (2) with positive real values. Since A 00 is dominated by the penguin amplitude, P = −|P | exp(−iδ c ), solution (1) corresponds to cos δ c > 0 (|δ c | < π/2) while solution (2) involves cos δ c < 0 (|δ c | > π/2).
In order to exclude solution (2) one would have to show unambiguously that cos δ c > 0 or |δ c | < π/2, where δ c is the strong phase difference between T + C and P . A most direct proof for cos δ c > 0 would need an observation of destructive interference between P and T + C in the CP-averaged decay rate of B + → K + π 0 normalized by that of B + → K 0 π + . However, this interference is cancelled by constructive interference of P and P EW +P C EW [31]. Arguments for small strong phase differences including δ c have been presented in studies of B → Kπ and B → ππ based on a heavy quark expansion [32]. These arguments failed, however, when predicting a very small phase Arg(C/T ). This would imply A CP (K + π 0 ) < A CP (K + π − ), contrary to the two asymmetries quoted in Table II, which show that this phase is not very small and must be negative (see argument below [31].) A small value of δ c (|δ c | < 30 • ) was obtained in global flavor SU(3) fits to decay rates and CP asymmetries measured for B → Kπ and B → ππ [13,33]. Within these fits it is difficult to pinpoint a small subset of B → Kπ measurements forcing a small value for δ c . The purpose of the subsequent discussion is to prove cos δ c > 0 using a series of arguments based on specific measurements, stressing the minimal use of untested assumptions about flavor SU(3).
A strong phase which is more directly accessible to experiment than δ c is δ, the strong phase of T relative to that of P . This phase occurs in the amplitude for B 0 → K + π − . Its cosine term appears in the ratio R of CP-averaged decay rates for this process and B + → K 0 π + [34,35]. Neglecting P C EW and A terms in these amplitudes, one would expect R to be smaller than one for cos δ > 0 and larger than one for cos δ < 0. The current value R = 0.899 ± 0.048, obtained from branching ratios in Table II and the above-mentioned ratio of B + and B 0 lifetimes, favors cos δ > 0 over cos δ < 0. This evidence is statistically limited and may suffer from P C EW corrections in B 0 → K + π − . The negative asymmetry A CP (K + π − ) = −0.097 ± 0.012 proves unambiguously that δ is positive.
An argument proving |δ| < π/2 unambiguously is based on the time-dependent CP asymmetry parameter S π + π − in B 0 → π + π − . Assuming flavor SU(3), the ratio of penguin and tree amplitudes and their relative phase are equal in this process to those in B 0 → K + π − , up to CKM factors defining the ratios of amplitudes. Neglecting small W -exchange and penguin annihilation contributions (the resulting systematic uncertainty introduced by this approximation is taken as part of an uncertainty due to SU(3) breaking mentioned below), one has [36] S π + π − = sin 2α + 2r cos δ sin(β − α) − r 2 sin 2β 1 − 2r cos δ cos(β + α) where α = π − β − γ and r is the ratio of penguin and tree amplitudes in B 0 → π + π − . In the absence of a penguin amplitude one has S π + π − = sin 2α, and to first order in the ratio r one finds [37] S π + π − = sin 2α + 2r cos δ sin(β + α) cos 2α .
In order to constrain δ c (the strong phase difference between T + C and P ), using the above range for δ (the strong phase difference between T and P ), one needs information about the strong phase of the ratio C/T . The observation A CP (K + π 0 ) > A CP (K + π − ) implies that Arg(C/T ) is negative and larger in magnitude than δ [31]. A simple proof of this behavior, for terms in the two asymmetries which are linear in |T + C|/|P | and |T |/|P |, respectively, follows from the geometrical identity |T + C| sin δ c = |T | sin δ + |C| sin[δ + Arg(C/T )] (17) illustrated in Fig. 2. The amplitudes T + C interfere constructively in B + → π + π 0 . This follows from the observation that 2(B + → π + π 0 ) > (B 0 → π + π − ) [5], and the above-mentioned constructive interference of T and P in B 0 → π + π − . Thus −π/2 < Arg(C/T ) < −δ < 0 which implies geometrically −π/2 < δ c < δ < π/2, without making any assumption about the magnitude |C/T |. This concludes the proof of cos δ c > 0 which excludes solution (2) in Fig. 1. It is the large value of φ 00 ≡ Arg(A 00Ā * 00 ) in solution (1) in Fig. 1 which is thus responsible for boosting the expected value of S Kπ from its penguin-dominated value of sin 2β ≃ 0.68 to a value very close to 1. We now explore the sensitivity of this effect to small changes in experimental inputs.
We find the greatest sensitivity of S Kπ is to variations of the branching ratio B(K 0 π 0 ) ≡ (B 0 → K 0 π 0 ). In Fig. 3(a) we plot φ 00 and S Kπ versus B(K 0 π 0 ) for nominal values of the parameters noted in the text. We note that S Kπ drops from a value of 0.99 at the central value of B(K 0 π 0 ) to 0.91 and 0.72 at −1σ and −2σ below the central value.   Fig. 3(b)]. The experimental values become considerably more compatible with the Standard Model predictions, and even more so if (B 0 → K + π − ) is increased by 1σ to 20 × 10 −6 [ Fig. 3(c)]. In Figs. 3 the quantity φ 00 is more sensitive than S Kπ to variations in (B 0 → K 0 π 0 ), γ, and (B 0 → K + π − ). For the central value of φ 00 , S Kπ is very close to its maximum value, so it is only for considerably lower values of φ 00 that S Kπ becomes sensitive to these parameters.
For nominal values of the parameters, one has φ 00 = 42.6 • and S Kπ = 0.987. Table III indicates the greatest sensitivity of φ 00 to (B 0 → K 0 π 0 ), followed by γ and ξ EW . There is relatively little sensitivity to ξ T +C .
Other variations are found to have a negligible effect on S Kπ . This includes the asymmetry A CP (B 0 → K + π − ), which involves a very small experimental error, and A CP (B 0 → K 0 π 0 ) ≡ −C Kπ , which is predicted in (5) with a small uncertainty. A large variation in this asymmetry would in any case have little effect on S Kπ , as a geometric construction similar to that in Fig. 1 illustrates. The phases of A 00 andĀ 00 are found to shift nearly together, so that the correction to sin 2β in Eq. (14) changes very little. This insensitivity to C Kπ is displayed for the favored S Kπ solution in Ref. [6], where C Kπ is left unconstrained disregarding the sum rule (3).
Thus the possibility that the above calculation of S Kπ in the Standard Model differs both from its penguin-dominated value of sin 2β ≃ 0.68 and from the data remains intriguing. However, for it to become a robust conclusion about the presence of new physics, accuracies of measurements of the B 0 branching ratios to K 0 π 0 and K + π − and of the CKM angle γ need to be improved. We look forward to such advances in future data, and to more precise measurements of the two asymmetries C Kπ and S Kπ in B 0 → K 0 π 0 . M.G. would like to thank the Enrico Fermi Institute at the University of Chicago for its kind and generous hospitality. We thank Masashi Hazumi for useful discussions, and Dan Pirjol and Jure Zupan for helpful communications. This work was supported in part by the United States Department of Energy through Grant No. DE FG02 90ER40560.
Note added: The measurements of C Kπ and S Kπ given in Table I have been updated very recently by the BaBar and Belle collaborations. New results and their averages are summarized in Table IV. The averaged value of C Kπ agrees with the prediction