Constraints on the CKM Angle gamma from B ->K*+- pi-+

We present constraints on the CKM parameter gamma=arg(Vub*) formed within the framework of SU(3) symmetry and based on charmless hadronic B decays to K*+- pi-+ and other pseudoscalar-vector final states. For strong phases of O(10 degrees), our analysis weakly favors cos(gamma)<0. We also estimate that a determination of $\gamma$ with an experimental uncertainty of less than $10^\circ$ can be attained with an order-of-magnitude improvement in the precision of the experimental inputs, but SU(3) symmetry breaking could introduce corrections approaching the size of the current experimental uncertainties.

In the Standard Model, the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix [1] gives rise to CP -violating phenomena through its single complex phase. This phase can be probed experimentally by measuring decay rates and CP asymmetries for charmless hadronic B decays that receive contributions from amplitudes with differing weak phases. In the flavor SU(3) decomposition of the amplitudes in pseudoscalar-vector (P V ) final states [2], the b → uūs transition B → K * ± π ∓ is dominated by two amplitudes, color-allowed tree and gluonic penguin, which interfere with a weak phase π − γ, where γ = arg V * ub , and with an unknown strong phase δ. We extract cos γ in two ways, employing Monte Carlo simulation to propagate experimental uncertainties and ratios of meson decay constants to account for SU(3) symmetry breaking. The first uses only B → K * ± π ∓ , and the second adds information from B → φ (K ) . In both cases, the magnitudes of the penguin and tree amplitudes must be known, and we estimate these from CKM unitarity and measured branching fractions for other B → P V decays.
An alternative method of constraining γ [3] makes use of observables pertaining to b → c transitions and to mixing in the neutral B and K systems, with a resultant 95% confidence level (C.L.) allowed interval for γ of [38 • , 80 • ]. In contrast, the analysis presented in this Letter uses rare, charmless b → u, d, s transitions without reference to mixing-induced CP violation. A discrepancy between the constraints on γ from charmless hadronic B decays and those from B and K mixing might arise from new physics contributions to either B and K mixing or the b → s or b → d penguins.
Global analyses of charmless hadronic B decays in the framework of QCD-improved factorization [4,5] find a value for γ of approximately 80 • . However, these fits predict smaller branching fractions for B → K * ± π ∓ and (K ) * 0 π ± than are observed experimentally, and removing these modes from the above analyses improves the fit quality. It has been suggested [6] that the B → K * π modes may receive dynamical enhancements not accounted for in Refs. [4] and [5]. Our analysis focuses on B → K * ± π ∓ with input from a modest number of other B → P V decays, thus providing a complement to the global fits.
Following the notation in Ref. [2] for SU(3) invariant amplitudes, we denote color-allowed tree amplitudes by t and gluonic penguins by p. Amplitudes for |∆S| = 1 transitions are primed, while those for ∆S = 0 transitions are unprimed. A subscript P or V indicates whether the spectator quark hadronizes into the pseudoscalar or vector meson, respectively. Since B → K * ± K ∓ is dominated by penguin annihilation and W -exchange contributions or rescattering effects, and these decays have not been observed experimentally, we neglect such amplitudes in our analysis. The transition amplitude for B → K * ± π ∓ is A(K * ± π ∓ ) = −(p ′ P + t ′ P ). The amplitudes t ′ and p ′ carry the CKM matrix elements V * ub V us and V * tb V ts with weak phases γ and π, respectively. The amplitudes for the two charge states are given by and we can express the CP -averaged amplitude as We identify squared amplitudes, |A| 2 = A * A, with branching fractions, B, and we absorb all numerical factors, like G F , m B , phase space integrals, decay constants, form factors, and CKM matrix elements, into the definitions of the amplitudes. Most of the B branching fraction measurements in the literature are calculated assuming equal production of charged and neutral mesons. We correct these branching fractions by the ratio of B + B − to B 0B0 production rates, f +− /f 00 , as well as by the ratio of charged to neutral lifetimes, τ + /τ 0 . Because the constraints on γ are constructed from ratios of branching fractions, we scale only the neutral B branching fractions by the product F ≡ f +− f 00 · τ + τ 0 , which is measured directly in Refs. [7] and [8]. Thus, cos γ cos δ can be expressed in terms of the CP -averaged branching fraction B(K * ± π ∓ ): For a given value of δ, γ is determined to a twofold ambiguity. The rate difference between B 0 → K * − π + and B 0 → K * + π − , which is proportional to sin γ sin δ, provides an additional observable that allows us to disentangle γ and δ: which leads to These expressions for γ and δ are subject to a fourfold ambiguity: The charge-separated branching fractions B(K * + π − ) and B(K * − π + ) appearing in Eqs. 5-8 can be determined directly from B(K * ± π ∓ ) and the CP asymmetry A CP (K * ± π ∓ ). In addition, SU(3) symmetry relates the rate difference ∆( [9,10]: To attain greater precision on B(K * + π − ) and B(K * − π + ), we combine information on ∆(ρ − π + ) with the measurements of B(K * ± π ∓ ) and A CP (K * ± π ∓ ) listed in Table I. These inputs are given relative weights that minimize the uncertainties on B(K * + π − ) and B(K * − π + ), and we account for the correlation between CLEO's A CP (K * ± π ∓ ) and B(K * ± π ∓ ) measurements, which are made with the same dataset and technique.
The BaBar analysis of B → π + π − π 0 [11], which determines the CP asymmetry and dilution parameters A, C, and ∆C defined in Ref. [11], allows us to evaluate ∆(ρ − π + ) = −(A + C + A∆C) · B(ρ ± π ∓ ). We propagate the uncertainties on these parameters with their correlations [12] to obtain ∆(ρ − π + ) = −(2.9 ± 4.6) · 10 −6 . Thus, taking the form factor ratio in Eq. 9 to be unity, we find with a correlation coefficient of 0.53. The correlation coefficients between ∆(ρ − π + ) and these two branching fractions are 0.45 for B(K * + π − ) and −0.50 for B(K * − π + ). A second method of estimating γ uses B → K * ± π ∓ and B → φK ± . The possibility of constraining γ from these decays was first noticed by Gronau and Rosner [13], and the concrete formulation of this method was subsequently put forth by Gronau [14,15] EW denotes the electroweak penguin contribution. The weak phase of P ′P EW is the same as that of p ′ P , and its strong phase is expected to be the same as in t ′ P because of the similarity of their flavor topologies [14,15]. Thus, the ratio of the CP -averaged branching fractions for B → K * ± π ∓ and B → φK ± provides a measure of γ up to a twofold ambiguity: where Both B → φK ± and B → φ (K ) 0 receive the same SU(3) amplitude contributions [2], so we can improve the statistical precision of Eq. 11 by combining both channels: where σ + and σ 0 refer to the uncertainties on B(φK ± ) and B(φ (K ) 0 )F , respectively. To determine γ with this method, the size of δ must be known. It is believed, based on perturbative [16] and statistical [17] calculations, that 0 • < |δ| < 90 • . In the simulation, we fix |P ′P EW | to be 1 2 |p ′ P |, as given by factorization calculations [13,18,19], and we evaluate the dependence of our results on |P ′P EW /p ′ P | and δ. In both of the above methods of constraining γ (involving Eqs. 4-8 and Eq. 10), numerical values of |t ′ P | and |p ′ P | are given by other B → P V branching fractions [20]. The penguin amplitude is simply The tree amplitude is taken from the ∆S = 0 transition B → ρ ± π ∓ and related to the |∆S| = 1 amplitude through SU(3)-breaking factors: The experimentally measured B(ρ ± π ∓ ) represents a sum over B 0 → ρ ± π ∓ andB 0 → ρ ± π ∓ decays: We isolate |t P + p P | with the BaBar B → π + π − π 0 analysis [11], which provides Based on the experimental inputs in Table I, we find B(ρ ± π ∓ ) P = (13.9 ± 2.7) · 10 −6 and a correlation coefficient between B(ρ ± π ∓ ) P and ∆(ρ − π + ) of 0.05. Extracting |t P | from B(ρ ± π ∓ ) P requires estimates of the magnitude and phase of p P . Its magnitude is obtained from the analogous |∆S| = 1 amplitude: In the SU(3) limit, p P and t P have the same relative strong phase as that between p ′ P and t ′ P . Their relative weak phase, however, is γ + β, where γ is unknown, a priori. Therefore, we must solve for cos γ and |t ′ P | simultaneously. Using CKM unitarity, the parameters |V td /V ts | and β can be eliminated in favor of |V ub /V cb | and γ via the relations By making these substitutions, we remove our dependence on sin 2β measurements involving b → c transitions and B 0 -B 0 mixing, and we remain sensitive to new physics which may affect these processes and charmless b → u, d, s transitions differently. From the above unitarity relations and the CP -averaged branching fraction B(ρ ± π ∓ ) P = |p P | 2 + |t P | 2 + 2 |p P | |t P | cos(γ + β) cos δ, we find the following expression for |t ′ P |: Using Eq. 23 to calculate |t ′ P | from B → K * ± π ∓ , (K ) * 0 π ± , and ρ ± π ∓ depends on a choice of δ as well as knowledge of γ, and an iterative solution is required. The fixed strong phase appearing in Eq. 24 is distinct from the strong phase in the simulated quantities cos γ cos δ (Eq. 4) and cos(γ ± δ) (Eqs. 5 and 6). To distinguish these two strong phases, we denote the one entering Eq. 24 by δ t ′ P . Below, we verify that the simulated values of cos γ cos δ and cos(γ ± δ) are insensitive to the choice of δ t ′ P . In the second method of constraining cos γ, we simulate Eq. 10 with δ t ′ P = δ. Experimental measurements of the following quantities are given as input to the simulation: F , f K * /f ρ , |V us |, |V ud |, |V ub |, |V cb |, A CP (K * ± π ∓ ), the B → ρ ± π ∓ parameters A, C, and ∆C, and the CP -averaged branching fractions for B → ρ ± π ∓ , (K ) * 0 π ± , K * ± π ∓ , φK ± , and φ (K ) 0 . These parameters are simulated with Gaussian or bifurcated Gaussian (different   widths above and below the peak) distributions, and their values are summarized in Table I. The input that contributes the largest uncertainty to γ is the B → K * ± π ∓ branching fraction.
For the five branching fractions, we combine all publicly presented measurements, with statistical and systematic errors added in quadrature. Where possible, the contribution from f +− /f 00 to the systematic error has been removed, since it is included coherently in the simulation. We neglect all other correlations among the systematic errors. Figure 1 shows the simulated distribution of cos γ cos δ from Eq. 4, with δ t ′ P = 0 • . Fitting this distribution to a bifurcated Gaussian yields the measurement cos γ cos δ = −0.68 +0.63 −0.59 , which suggests constructive interference between t ′ P and p ′ P . The 90%, 95%, and 99% C.L. upper limits on cos γ cos δ| δ t ′ P =0 • are 0.16, 0.42, and 0.94, respectively. Based on the smallness of direct CP asymmetries in B → Kπ, one can infer a strong phase between tree and penguin amplitudes in these decays of (8 ± 10) • [33]. If the strong phases in B → P V decays are as small as in two-pseudoscalar (P P ) final states, then our analysis weakly favors cos γ < 0. The variation of cos γ cos δ with cos δ t ′ P is roughly linear, with a slope of d cos γ cos δ d cos δ t ′ P = 0.11.
From Figure 1, we also find cos(γ + δ) and cos(γ − δ) from Eqs. 5 and 6 to be −0.39 +0.69 by less than 2 • between δ t ′ P = 0 • and δ t ′ P = 80 • . Figure 2 shows the dependence of cos γ from Eq. 10 on δ = δ t ′ P , with |P ′P EW /p ′ P | = 0.5. The peak values are plotted with asymmetric error bars representing the widths of the bifurcated Gaussian distributions. By demanding that cos γ peak in the physical region, one can infer that |δ| < 41 • . The variation of cos γ with |P ′P EW /p ′ P | is linear, with a slope d cos γ d|P ′P EW /p ′ P | = 0.28 − 1.51 cos δ t ′ P . Incorporating the B → φ (K ) decays in the measurement of γ results in greater precision than using B → K * ± π ∓ alone, but the theoretical uncertainties incurred are also larger. Using the simulation of Eq. 4, we also determine the ratio r = 0.30 +0.07 −0.05 at δ t ′ P = 0 • with a δ t ′ P dependence given by r = 0.25 + 0.09 cos δ t ′ P − 0.04 cos 2 δ t ′ P . The inverse ratio for ∆S = 0 decays, |p P /t P | = 1 r |V us /V ud | |V td /V ts |, is found to be 0.43 − 0.49 cos δ t ′ P + 0.26 cos 2 δ t ′ P , which takes the value 0.20 +0.03 −0.02 at δ t ′ P = 0 • . The widths of the generated distributions presented above are dominated by experimental uncertainties on the input branching fractions, A CP (K * ± π ∓ ), A, C, and ∆C. We study the improvement in the resolutions of cos γ cos δ, cos(γ ± δ), and cos γ| δ=0 • , collectively denoted byσ cos γ , as these measurement uncertainties are reduced while maintaining the central values at their current positions, with δ t ′ P = 0 • and |P ′P EW /p ′ P | = 0.5. It is found thatσ cos γ scales with the size of the experimental uncertainties until the latter reach 10% of their current values, where the resolution of γ is O(10 • ). At this point,σ cos γ begins to be dominated by the uncertainty on F , and only by lowering σ F canσ cos γ be reduced any further.
We have modeled SU(3) symmetry breaking effects in ratios of ∆S = 0 to |∆S| = 1 amplitudes with the purely real ratio of decay constants f K * /f ρ . Repeating the simulation without SU(3) breaking (i.e., with f K * /f ρ = 1) results in changes to cos γ cos δ, cos(γ ± δ), and cos γ| δ=0 • of 0.05 or smaller. Recent studies based on QCD-improved factorization [10,34] have suggested that SU(3) breaking could be as large as 30% and that the amplitude ratios may possess a small complex phase. To probe the impact of such effects, we reinterpret f K * /f ρ as a phenomenological parameter and scale it by ±30% of the value given in Table I, neglecting any possible complex phases. We find shifts of +0. 21 −0.32 in cos γ cos δ, +0.32 −0.45 in cos(γ + δ), +0.12 −0.18 in cos(γ − δ), and +0.19 −0.30 in cos γ| δ=0 • . Thus, in this conservative estimate, SU(3) breaking effects are roughly 15%-70% of the current experimental uncertainties. To obtain meaningful constraints on γ, future experimental advances must be accompanied by an improved understanding of SU(3) breaking.
In conclusion, we have formed constraints on γ as a function of δ and |P ′P EW /p ′ P | using branching fractions of and CP asymmetries in B → P V decays. At present, experimental uncertainties overwhelm the theoretical uncertainties arising from the model dependence of |V ub | and |V cb |, but they are the same order of magnitude as the uncertainties in SU(3) symmetry breaking. For strong phases of O(10 • ) or smaller, our analysis favors cos γ < 0, which agrees with indications from B → P P decays [33,35,36]. However, the current experimental precision does not yet permit a stringent comparision with fits reliant upon B and K mixing.