CERNTH/98293
EFI9844
hepph/9809311
Determination of the Weak Phase from
Rate Measurements in Decays
Matthias Neubert
[0.1cm] Theory Division, CERN, CH1211 Geneva 23, Switzerland
[0.3cm] and
[0.3cm] Jonathan L. Rosner
[0.1cm] Enrico Fermi Institute and Department of Physics
University of Chicago, Chicago, IL 60637, USA
A method is described which, under the assumption of SU(3) symmetry, allows one to determine the angle of the unitarity triangle from timeindependent measurements of the branching ratios for the rare twobody decays and , as well as of the CPaveraged branching ratios for the decays and , all of which are of order . The effects of electroweak penguin operators are included in a modelindependent way, and SU(3)breaking corrections are accounted for in the factorization approximation.
(To appear in Physical Review Letters)
September 1998
The study of CP violation in the weak decays of mesons will provide important tests of the flavor sector of the Standard Model, which predicts that all CP violation results from the presence of a single complex phase in the quark mixing matrix. The precise determination of the sides and angles of the unitarity triangle, which is a graphical representation of the unitarity relation , plays a central role in this program [1]. Whereas the angle will be accessible at the firstgeneration factories through the measurement of CP violation in the decay , the angle is harder to determine. The sum can be extracted in a theoretically clean way from measurements of CP violation in the decays (or in the related decays and ), but because of experimental difficulties such as the detection of the mode this will be a longterm objective. A method to determine proposed by Gronau and Wyler uses rate measurements for six decay modes [2], some of which require the reconstruction of the neutral charmmeson CP eigenstate . A variant of this approach using decays has been discussed by Dunietz [3]. Unfortunately, these methods rely either on measurements of some processes with very small branching ratios, posing experimental [4] and theoretical [5] challenges, or on measurements requiring considerable precision (see, e.g., Refs. [6, 7] and references therein).
In view of these difficulties, approximate methods to determine the angle have received a lot of attention. The simplest of these methods was proposed by Gronau, Rosner and London (GRL), who suggested a triangle construction involving the amplitudes for the decays , , and , as well as for the corresponding CPconjugated decays [8]. Besides a plausible dynamical assumption this method relies on SU(3) flavor symmetry in relating with decays. Later, it was argued that the GRL method is spoiled by electroweak penguin contributions, which have an important impact in decays and upset the naive SU(3) triangle constructions [9, 10]. More sophisticated methods based on quadrangle constructions involving other decay modes such as [10] or [11, 12] were invented to circumvent this problem. There have also been proposals for deriving bounds on using CPaveraged rate measurements in decays [13, 14, 15, 16, 17, 18], and for combining these measurements with those of rate asymmetries and other decays like to obtain further information [19, 20].
In the present note we propose a variant of the original GRL method, which based on the findings of our previous work [18] includes the potentially dangerous electroweak penguin contributions in a modelindependent way using Fierz identities and SU(3) symmetry. We thus obtain an approximate method for learning that is conceptually as simple and uses the same experimental input and theoretical assumptions as the GRL method, though the actual triangle constructions are somewhat more complicated. The main advantage of our approach is that it is based on rare twobody decays that are relatively easy to access experimentally, and that have larger branching ratios than the decays needed for all other methods of measuring . Although the accuracy of this extraction may ultimately be limited by theoretical uncertainties, even an approximate value for will be very useful, if only to help eliminating discrete ambiguities inherent in other determinations [21].
The basis of our method is the amplitude relation
(1)  
where is an isospin amplitude parametrizing the transition , is a stronginteraction phase, and is the weak phase associated with the quark decay . The second relation is strictly valid in the SU(3) flavorsymmetry limit; however, the factor accounts for the leading (i.e., factorizable) corrections to that limit. The crucial new ingredient in (1) with respect to the corresponding relation used in Ref. [8] is the presence of the parameter accounting for the contributions of electroweak penguin operators. We have recently shown that in the SU(3) limit this parameter is real (i.e., it does not carry a nontrivial stronginteraction phase) and calculable in terms of Wilson coefficients and electroweak parameters [18]. The result is
(2) 
where is the electromagnetic coupling at the weak scale, is the Wolfenstein parameter, [22], and accounts for factorizable SU(3)breaking corrections. The derivation of this result uses the fact that the relevant electroweak penguin operators are Fierzequivalent to the usual current–current operators and of the effective weak Hamiltonian for decays [15], and that in the SU(3) limit the isospin amplitude receives a contribution only from the combination , but not from the difference .
As in the original GRL method, we must rely on the dynamical assumptions that and . Whereas the first relation follows from the fact that only the current–current operators contribute to decays (electroweak penguin contributions can be neglected in this case [23]), the second one assumes that there are only negligible contributions proportional to the weak phase to the amplitude for the decay , which thus can be taken to have the simple form , where is the weak phase of the leading top and charmpenguin amplitudes, and is a stronginteraction phase. Possible contributions to this amplitude proportional to the weak phase are indeed expected to be very small, because they could only come from upquark penguins or annihilation topologies [24]. However, this intuitive argument could be invalidated if soft finalstate rescattering effects were very important [14, 15, 16, 17, 19, 20]. We stress, therefore, that the assumption is a working hypothesis of our method, which must be tested independently. A necessary condition for the validity of this assumption is the absence of a sizable direct CP asymmetry in the decays . If we write , where measures the strength of possible rescattering contributions and is a stronginteraction phase, then . Since the global analysis of the unitarity triangle prefers values of such that , and since is unlikely to be small because without sizable strong phases there would not be a rescattering contribution in the first place, a small experimental value for the asymmetry would be a strong indication that our working hypothesis is justified.
Let us define the amplitude ratios
(3) 
which under the assumptions stated above can be determined experimentally through timeindependent rate measurements via
(4) 
A future measurement of would signal direct CP violation in the decays . At present, preliminary data reported by the CLEO Collaboration [25] imply and, combined with some theoretical guidance, [18]. Moreover, we define
(5) 
with , so that
(6) 
In terms of these quantities, the triangle relation (1) and its CPconjugate take the form
(7) 
where is an unknown stronginteraction phase difference, while the phases contain both strong and weak contributions. It follows that
(8) 
Combining these results, we find that the allowed solutions for can be obtained from the real zeros of the equation
(9) 
which, taking into account the dependence of and , correspond to the zeros of a fourthorder polynomial in .
A simplified analysis can be performed if the phase difference turns out to be small or close to – a possibility that can be tested for experimentally. To this end, one exploits the following exact relations:
(10) 
The global analysis of the unitarity triangle prefers values of in the range [26], which would imply . Then the second relation can be used to obtain a reasonable estimate and upper limit for . If it turns out that is small, corresponding to a situation where or , one can set in the first relation to obtain
(11) 
which determines up to a possible twofold ambiguity. From (10), it follows that a criterion for the validity of this approximation is that the deviation of from be less than the uncertainty in the product , i.e. . With present uncertainties on the parameters and , which are unlikely to be improved much in the near future, this implies . With the current experimental values for the various parameters, and in the absence of independent experimental results for and , the relations (10) do not yet provide for a useful estimate of ; however, they may become valuable with more precise measurements. It is remarkable that even in the case , i.e., in the absence of direct CP violation in decays, can be determined using relation (11), which becomes exact in that limit.
In practise, the determination of using (9) or (10) is limited by experimental as well as theoretical uncertainties in the extraction of the parameters , , and . Let us illustrate the situation with a realistic example. Assume that the true values of the parameters are (the center of the region preferred by the global analysis), and (the current central values), and that the strong phase difference takes the value . It then follows that and . Let us assume that we can measure the values of these parameters with some errors given by , , and . We do not anticipate that and will soon be known with an accuracy much better than today, because these quantities are affected by theoretical uncertainties such as the estimate of SU(3)breaking effects. We thus assign a 15% error to them [27]. The assumed error on the amplitude ratios corresponds to a measurement of the corresponding ratios of branching ratios with a precision of about 10%. In this example, the approximate value for obtained by setting in (10) is , which is close to the correct value . We have quoted the various sources of errors separately. It is apparent that the precision in the measurements of the ratios is the limiting factor of our method. The approximate solution obtained with is , which is excluded by the global analysis of the unitarity triangle. In Figure 1, we show the distribution of the exact real solutions of equation (9) for 1000 random choices of the Gaussian errors in the various input quantities. The solutions where can again be excluded based on the global analysis of the unitarity triangle. From the central peak, we obtain , implying at one standard deviation .
To conclude, we have shown that the weak phase can be determined using timeindependent measurements of the branching ratios for the decays and , as well as of the CPaveraged branching ratios for the decays and . The new development that makes this method practical is the observation that the strong phases of the electroweak penguin and tree amplitudes are related to one another by Fierz identities and SU(3) flavor symmetry. SU(3)breaking corrections can be accounted for in the factorization approximation. On the other hand, like many earlier proposals our method relies on the dynamical assumption that finalstate rescatterings do not induce a sizable contribution proportional to the weak phase in the amplitude for the process . The validity of this assumption can be tested for experimentally by searching for direct CP violation in this decay.
Acknowledgments: Part of this work was done during the Workshop on Perturbative and NonPerturbative Aspects of the Standard Model at St. John’s College, Santa Fe, July–August 1998. We would like to thank the organizer Rajan Gupta, as well as the participants of the workshop, for providing a stimulating atmosphere and for many useful discussions. We also wish to thank Michael Gronau, Sheldon Stone and Lincoln Wolfenstein for helpful comments. One of us (J.L.R.) was supported in part by the United States Department of Energy through contract No. DE FG02 90ER40560.
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