Estimate of SU(3) flavour symmetry breaking in vs. decay
Abstract
PITHA 03/04, hepph/0308040, 27 June 2003
I estimate the SU(3) flavour symmetry breaking in the
ratio of penguintotree ratios of the decays
and , given
an assumption on the flavour symmetry breaking of the
hadronic input parameters. The decay amplitudes are calculated
in QCD factorization. Implications for the determination of
are discussed.
Workshop on the CKM Unitarity Triangle, IPPP Durham, April 2003
1 Introduction
In this note I perform a study of SU(3) breaking for the decays and . This system is interesting, because it allows for a determination of the angle from mixinginduced and direct CP asymmetries [1, 2] provided the SU(3) symmetry breaking corrections to a certain double ratio of amplitudes are known. The analysis of SU(3) breaking is done in the theoretical framework of QCD factorization [3, 4, 5], which expresses the hadronic decay amplitudes in terms of fundamental constants, decay constants, form factors etc. in the heavy quark limit. The following is a preliminary version of work in progress in collaboration with M. Neubert.
We write the decay amplitudes as
(1) 
where . The “tree” and “penguin” amplitudes are defined as the coefficients of the two terms with different CKM matrix elements. They also contain subleading penguin, electroweak penguin and weak annihilation amplitudes. In the SU(3) symmetry limit and . We will be interested in the deviations of the ratios
(2) 
from unity.
The ratio of tree amplitudes is the product of a factorizable and a nonfactorizable term, . The factorizable term is given by
(3)  
where we have included the light meson masses and phase space effects. The form factor is taken from [6], but the assumed value for the form factor is only an “educated guess”. In the context of QCD sum rules this form factor is expected to depend sensitively on the poorly known first Gegenbauer moment of the kaon lightcone distribution amplitude, since the form factor is dominated by the soft spectatorquark overlap term.
2 Estimate of SU(3) breaking
The factorizable SU(3) breaking correction cancels in the double ratio [2]. This ratio and deviate from unity due to nonfactorizable effects that can be computed in QCD factorization assuming that . The computation is a straightforward extension of the results of [4]. I refer to this paper and [5] for all details concerning the method and the values of the hadronic parameters. The parameters relevant to SU(3) breaking are:

vs. in the normalization of scalar penguin terms;

the ratio vs. in the normalization of the hardscattering and weak annihilation terms;

the Gegenbauer moments , vs. , , and the first inverse moments of the meson distribution amplitudes, vs. ;

the parameters for soft power corrections from hard scattering and weak annihilation, and .
Since these parameters are often not wellknown, we will make assumptions on the SU(3) breaking and exhibit the effect of these assumptions on the ratios and . This should be distinguished from more ambitious approaches as discussed in [7], where the SU(3) breaking in the input parameters is also computed.
For a given pair of related parameters denoted , for instance , we set all other parameters and to their standard values. We then write and vary , which controls the amount of SU(3) breaking, between and 0.3. For the complex parameters and we vary the magnitude and phase simultaneously and independently by this amount. We make two exceptions to this treatment. For the light meson Gegenbauer moments, which are already small corrections to the SU(3) symmetric asymptotic distribution amplitudes, we take , , and . Second, we use
(4) 
with taken from [8].
The resulting variations of the magnitudes and phases of and about their default values
(5) 
are shown in the upper part of Table 1. Adding up the various uncertainties we obtain nonfactorizable SU(3) breaking effects of order for the ratio of tree amplitudes and of order for the double ratio of penguin and tree amplitudes. Since the strong phases of the ratios are very small, the SU(3) breaking effect on the phases is also small in absolute magnitude.
These estimates have to be regarded with caution, since they rely on the default estimate of weak annihilation (), in which the annihilation amplitude does not have a strong phase. To study the impact of weak annihilation in more detail we pick the four values corresponding to with phase 0, 90, 180, 270 degrees for the pion mode and allow the corresponding parameters to vary by for the kaon mode. All the values of the annihilation parameter chosen by this procedure lie (almost) within our standard error assumption . The lower part of Table 1 demonstrates that the SU(3) breaking effect on the ratio of penguin amplitudes can be large, up to in magnitude and on the phase of . We conclude that unless a better understanding of SU(3) breaking effects in the weak annihilation amplitude is found the assumption should not be made.
Parameter  

Sum  
3 Determination of
To estimate the theoretical uncertainty in the determination of the angle due to the SU(3) breaking effects, we assume that the mixing phase is , corresponding to , and define the timedependent CP asymmetry through
(6)  
We assume that and have been measured. We can then determine
(7) 
as a function of ,^{1}^{1}1The definition of differs from [2] by a sign, so that is near zero in QCD factorization. and predict , the direct CP asymmetry in decay, for a given assumed value of the SU(3) breaking ratio . A measurement of then results in a determination of with a theoretical error due to the uncertainty of .
We dismiss solutions with as unphysical, since a large penguin amplitude is or will be excluded by branching fraction measurements. We generically find two solutions for . The second solution satisfies only in a small range around and will not be discussed further. The other solution already constrains from alone. The question is how a measurement of narrows this range further. In Figure 1 we show as a function of for five values of : the central value according to (5), and for the most pessimistic error assumptions corresponding to , , and . The figure also shows and .
The CP asymmetry exhibits a resonancelike behaviour after has gone through its minimal value, since the denominator of the expression for goes through a minimum when becomes small and positive. We find that the largest error is introduced by the uncertainty in the phase of , which in our analysis is entirely due to weak annihilation. For instance, if is found, the strategy would determine two values of in the absence of theoretical (and experimental) errors. Including the SU(3) breaking error, we obtain and . Since is expected to be positive, the second range is theoretically favored. In general, the theoretical error can become larger or smaller depending on as seen from the figure. It will also depend on what the mixinginduced and direct CP asymmetries in will eventually turn out to be, but our result should be generic.
We therefore conclude that the strategy to determine from the to system may suffer from considerable theoretical uncertainties, unless additional information is available. This can come from two sources: (1) Excluding the possibility of a weak annihilation amplitude with a large strong rescattering phase would already eliminate the largest uncertainty that we could identify in the context of QCD factorization. This could be achieved experimentally by excluding a large direct CP asymmetry in and neglecting SU(3) breaking effects in the estimate of the weak annihilation phase. (2) If the mixing phase is known, the mixinginduced CP asymmetry in decay provides a fourth observable, which could be used to determine (or eliminate) , leaving only the SU(3) breaking error on . To make this work in practice, a significant experimental effort is required.
References
 [1] I. Dunietz, Extracting CKM parameters from B decays, FERMILABCONF93090T, presented at the Summer Workshop on B Physics at Hadron Accelerators, Snowmass, Colorado, June 21 – July 2, 1993.

[2]
R. Fleischer,
Phys. Lett. B 459 (1999) 306;
R. Fleischer and J. Matias, Phys. Rev. D 66 (2002) 054009.  [3] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B 591, 313 (2000).
 [4] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B 606 (2001) 245.
 [5] M. Beneke and M. Neubert, [hepph/0308039].
 [6] P. Ball, talk at this Workshop [hepph/0306251].
 [7] A. Khodjamirian, talk at this Workshop.
 [8] H. Leutwyler, Phys. Lett. B 378 (1996) 313.