CP violation

Possibilities for observing CP violation and measuring Gronau and Wyler[3] have proposed a method of measuring using only measured branching fractions in charged decay[34]. The method is best illustrated by considering the three related decay modes, and the corresponding modes for , , , and , where indicates that the decays into a CP eigenstate. The weak decay diagrams[35] for the first two modes are shown in Fig. . The first decay is a Cabibbo-suppressed version of , while the second is a color suppressed transition where the virtual transforms itself into a pair.

Interference is possible between the two decay modes if the decays into a non-strangeness specific mode. Examples of such final states include , , and . To simplify the discussion, we only use such states which are in specific angular momentum configurations so that their CP is defined as +1 or -1; these states are usually denoted as and . We have that

To further simply the discussion we will consider only 's. The amplitudes for the relevant decay modes are related by

The branching fractions for the decays and need not be equal and a difference between them would be a manifest demonstration of violation in charged meson decay. Two amplitudes are equal in magnitude,

Then denoting the relative hadronic phase between the amplitudes and by [37] we have

To determine we need to measure the branching fractions for the modes in Eqs. and . Two triangles are constructed from the square roots of the branching fractions as illustrated in Fig. . The angles and are then determined.

This method leads to a four-fold ambiguity in the value of . Half the ambiguity is removed if one uses the measurement of in decay which leads to the boundary condition The other ambiguity arises between the determination of and . This ambiguity can be removed by measuring other related decay modes, such as , which are likely to have a different strong phase shift but must have the same CP violating phase. It is also important to realize that this method can find a value for even if is zero. In this case the two triangles for and are congruent.

Observation of CP asymmetry is expedited by having comparable amplitudes for the two interfering decays. At the CKM level (ignoring color suppression), the magnitude of the amplitude is , while the magnitude of the amplitude is where and are the parameters defined by Wolfenstein[39][38][4]. Therefore at the CKM level the ratio of the amplitudes is . However, occurs through a color-allowed diagram while the diagram for is color-suppressed. However, there is a simple way to predict the branching fraction corresponding to the amplitude . Consider the measured decay mode, , which is very similar to the mode . The decay diagrams are shown in Fig. . The difference between the two diagrams is that the decay occurs when there is a transition, while the occurs when there is a transition. We choose to discuss here rather than since is a vector meson as is the . Correcting for phase space effects we have

where is a phase space factor equal to 1.3, and the 's are form-factors. These form-factors are not known; theoretical predictions give %[45]. Using [41], the prediction is that . The error comes from the uncertainty on the branching ratio and the uncertainty in . A similar prediction can be made for the final state and is listed in Table .

The final states with a scalar rather than a vector are more difficult to predict because the analogous mode, , has not been observed, nor are there any upper limits. There are two theoretical predictions that inclusive modes are half the modes[40]. The estimates given in Table use the assumption that is also true for the exclusive modes. For the sum of branching ratios we use a value of for subsequent estimates.

To estimate the branching fraction with amplitude of type , we take the new CLEO measurement[41] %and multiply it by the Cabibbo suppression factor and the enhancement factor [42]. The result is .

In a 20 fb data sample, the expected number of events is given by the product of several factors, including the integrated luminosity, the cross section (1 nb), the detection efficiency (0.09), the efficiency for finding a single charged pion (0.9), and other cuts used to enhance signal to background. The detection efficiency is calculated using the three modes , , and . The cuts used to enhance the signal to background include a cut on the momentum direction of the candidate compared with the thrust axis of the rest of the event. These have an efficiency of 80%. This gives a total of 400 such events summed over and decays[43]. We can add other decay modes, although the analysis must be carried out for them individually because of the ambiguity between and in each mode. Using more than one mode provides a way of resolving this ambiguity. The other obvious final states are , , and . Summing all of these modes leads to approximately 1600 events. The number of type decays, based on the summed branching ratio of , is 20. These rate estimates may be conservative in that they use the current CLEO II efficiencies.

The CP eigenstate decay modes are more prolific. Table shows that the branching ratio times efficiency for these decays is approximately 10%of that of the decays used above.

In order to evaluate how accurate a measurement of sin can be made with 20 fb, we use the expression given by Gronau and Wyler[3] for determining from , , and . The numbers of events going through the channels and have been taken to be 800 and 20. (The use of 800 rather than 1600 events is based on lower measurements of type than are currently available; this change is not expected to significantly affect the accuracy of the measurement.) Choosing values for and defines the number of events in the and channels. These latter values are calculated applying the law of cosines to the triangles in Fig. . For example for , the number of events in the and channels is 53 and 29, respectively, after allowing for the 10%relative detection efficiency of to .

The procedure used is to perform many ``experiments'' starting with initial central values of 800 and 20 events for the and channel for specific values of and . Then we allow all measured event numbers to fluctuate statistically around their central values. The distribution in sin for several choices of and is shown in Fig. . Taking r.m.s. values of as necessary for the observation of CP violation, the potential ``discovery'' region is shown in Fig. .

To be able to discover CP violation using this method requires a detector which has similar (but hopefully better) photon and charged particle reconstruction capabilities as CLEO II, excellent charged particle identification, and excellent vertex detection. The former two are important to increase the discovery potential or guard against smaller branching ratios. The latter two are necessary to reject backgrounds in these rather small signal samples.

It is not clear that particle identification and vertex detection will be sufficient to reduce the background to the required level. Examination of current CLEO data reveals that the equivalent branching ratio of the background in the and signal regions is 10; this is large compared to the branching ratio we are attempting to measure in the channel. However, the particle identification of the current CLEO II detector is minimal for these final states and there is no vertex information on the or the . On the other hand, low background levels are possible, being almost non-existent in the final state. If the background from fake and non- combinations could be eliminated, we would still be faced with the problem of the background from real combinations in continuum events.

Rate asymmetries Any final state which can be produced with two different amplitudes can show CP violating effects. For example, it is possible for a ``penguin'' amplitude to interfere with a ``tree'' level amplitude. The measurement consists of finding a different number of events of versus events. For example, one could search for an asymmetry, , in the conjugate decay modes and where In decays the final states could be , , etc..

The sample size for any particular mode is , where is the branching ratio and is the detection efficiency. The total number of events required to get an standard deviation effect in one mode is . Provided that there is at least one mode with and , an integrated luminosity of 20 fb would be sufficient to establish violation to 4 standard deviations.

Angular correlations in vector-vector decay modes When there is a decay into two vector mesons, and if two amplitudes contribute, then it is possible for CP violation to be observed in differences between the angular distributions of the and . One way of looking for these asymmetries has been suggested by Valencia[44]. Defining the decay in terms of the momentum and polarization vectors as , one can look directly at several asymmetries for both the and the . For example is defined for the decay and a similar quantity, defined for the decay. A difference between and would be evidence of CP violation.

Predictions of angular asymmetries in many channels such as have been given by Kramer and Palmer[46]. The predictions are quite sensitive to hadronic phase shifts. Therefore, it is difficult to make quantitative predictions, but such effects should be explored.

CP-odd energy asymmetries It has been suggested by Burdman and Donoghue that CP violation might show up in three body decays modes, accessible from both and decays, in Dalitz plot asymmetries[47]. One possible mode is . A distribution of interest is the shape of the summed energy spectrum of the and . The average energy distribution can be written as . Finding that is a manifest observation of CP violation. Again, there are no quantitative predictions.