DESY 02–040 UAB–FT–523 hep–ph/0204101 April 2002

Exploring CP Violation through Correlations in

, , Observable Space

Robert Fleischer^{*}^{*}*E-mail: Robert.F

Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany

Joaquim Matias^{†}^{†}†E-mail:

IFAE, Universitat Autònoma de Barcelona, E–08193 Barcelona, Spain

Abstract

[10pt] We investigate allowed regions in observable space of , and decays, characterizing these modes in the Standard Model. After a discussion of a new kind of contour plots for the system, we focus on the mixing-induced and direct CP asymmetries of the decays and . Using experimental information on the CP-averaged and branching ratios, the relevant hadronic penguin parameters can be constrained, implying certain allowed regions in observable space. In the case of , an interesting situation arises now in view of the recent -factory measurements of CP violation in this channel, allowing us to obtain new constraints on the CKM angle as a function of the – mixing phase , which is fixed through up to a twofold ambiguity. If we assume that is positive, as indicated by recent Belle data, and that is in agreement with the “indirect” fits of the unitarity triangle, also the corresponding values for around can be accommodated. On the other hand, for the second solution of , we obtain a gap around . The allowed region in the space of and is very constrained in the Standard Model, thereby providing a narrow target range for run II of the Tevatron and the experiments of the LHC era.

## 1 Introduction

One of the most exciting aspects of present particle physics is the exploration of CP violation through -meson decays, allowing us to overconstrain both the sides and the three angles , and of the usual non-squashed unitarity triangle of the Cabibbo–Kobayashi–Maskawa (CKM) matrix [1]. Besides the “gold-plated” mode [2], which has recently led to the observation of CP violation in the system [3, 4], there are many different avenues we may follow to achieve this goal.

In this paper, we first consider modes [5]–[14], and then focus on the , system [15], providing promising strategies to determine . In a previous paper [16], we pointed out that these non-leptonic decays can be characterized efficiently within the Standard Model through allowed regions in the space of their observables. If future measurements should result in values for these quantities lying significantly outside of these regions, we would have an immediate indication for the presence of new physics. On the other hand, a measurement of observables lying inside these regions would allow us to extract values for the angle , which may then show discrepancies with other determinations, thereby also indicating new physics. Since penguin processes play a key rôle in , and decays, these transitions actually represent sensitive probes for physics beyond the Standard Model [17].

Besides an update and extended discussion of the allowed regions in observable space of appropriate combinations of decays, following [16], the main point of the present paper is a detailed analysis of the , system in the light of recent experimental data. These neutral -meson decays into final CP eigenstates provide a time-dependent CP asymmetry of the following form:

(1) | |||||

where we have separated, as usual, the “direct” from the “mixing-induced” CP-violating contributions. The time-dependent rates refer to initially, i.e. at time , present - or -mesons, denotes the mass difference of the mass eigenstates, and is their decay width difference, which is negligibly small in the system, but may be as large as in the system [18]. The three observables in (1) are not independent from one another, but satisfy the following relation:

(2) |

If we employ the -spin flavour symmetry of strong interactions, relating down and strange quarks to each other, the CP-violating observables provided by and allow a determination both of and of the – mixing phase , which is given by in the Standard Model [15]. Moreover, interesting hadronic penguin parameters can be extracted as well, consisting of a CP-conserving strong phase, and a ratio of strong amplitudes, measuring – sloppily speaking – the ratio of penguin- to tree-diagram-like contributions to . The use of -spin arguments in this approach can be minimized, if we use as an input. As is well known, this phase can be determined from mixing-induced CP violation in ,

(3) |

up to a twofold ambiguity. Using the present world average

(4) |

which takes into account the most recent results by BaBar [19] and Belle [20], as well as previous results by CDF [21] and ALEPH [22], we obtain

(5) |

On the other hand, the – mixing phase , which enters , is negligibly small in the Standard Model. It should be noted that we have assumed in (3) that new-physics contributions to the decay amplitudes are negligible. This assumption can be checked through the observable set introduced in [23].

Whereas is already accessible at the -factories operating at the resonance, BaBar, Belle and CLEO, the mode can be studied nicely at hadron machines, i.e. at run II of the Tevatron and at the experiments of the LHC era, where the strategy sketched above may lead to experimental accuracies for of [24] and [25], respectively. Unfortunately, experimental data on are not yet available. However, since is related to through an interchange of spectator quarks, flavour-symmetry arguments and plausible dynamical assumptions allow us to replace approximately by , which can already be explored at the -factories. A key element of our analysis is the ratio of the CP-averaged and branching ratios, which can be expressed in terms of and hadronic penguin parameters. As pointed out in [26], constraints on the latter quantities can be obtained from this observable, allowing an interesting comparison with theoretical predictions.

In our analysis, we shall follow these lines to explore also allowed regions in the space of the CP asymmetries of the , system, and constraints on . To this end, we first use (3) to fix the – mixing phase , yielding the twofold solution (5). For a given value of the mixing-induced CP asymmetry , the ratio of the CP-averaged and branching ratios allows us then to determine the direct CP asymmetry as a function of . Consequently, measuring these observables, we may extract this angle. Moreover, the corresponding hadronic penguin parameters can be determined as well. On the other hand, if we assume that lies within a certain given range, bounds on and can be obtained, depending on the choice of . In particular, we may assume that the mixing-induced CP asymmetry is positive or negative, leading to very different situations.

Since experimental data for the direct and mixing-induced CP
asymmetries of are already available from the
factories, we may now start to fill these strategies with
life:^{1}^{1}1The connection between our notation and those
employed in [27, 28] is as follows:
and
.

(6) |

(7) |

yielding the naïve averages

(8) |

Unfortunately, the BaBar results, which are an update of the values given in [29], and those of the first Belle measurement are not fully consistent with one another. In contrast to BaBar, Belle signals large direct and mixing-induced CP violation in , and points towards a positive value of . As we shall point out in this paper, the following picture arises now: for a positive observable , as indicated by Belle, the solution of being in agreement with the “indirect” fits of the unitarity triangle [30], yielding , allows us to accommodate also the corresponding values for around , whereas a gap around arises for the second solution of . On the other hand, varying within its whole negative range, remains rather unconstrained in the physically most interesting region. Using the experimental averages given in (8), we obtain and . Interestingly, there are some indications that may actually be larger than , which may then point towards the unconventional solution of . The negative sign of implies that a certain CP-conserving strong phase has to lie within the range . In the future, improved experimental data will allow us to extract and the relevant hadronic parameters in a much more stringent way [15, 26].

Following a different avenue, implications of the measurements of the CP asymmetries of were also investigated by Gronau and Rosner in [31]. The main differences to our analysis are as follows: in [31], the observables are expressed in terms of and , the “tree” amplitude is estimated using factorization and data on , and the “penguin” amplitude is fixed through the CP-averaged branching ratio with the help of flavour-symmetry and plausible dynamical assumptions. In contrast, we express the observables in terms of and the general – mixing phase , which is equal to in the Standard Model, and use the ratio of the CP-averaged and branching ratios as an additional observable to deal with the penguin contributions, requiring also flavour-symmetry and plausible dynamical assumptions. We prefer to follow these lines, since we have then not to make a separation between tree and penguin amplitudes, which is complicated by long-distance contributions, and have not to use factorization to estimate the overall magnitude of the tree-diagram-dominated amplitude ; factorization is only used in our approach to take into account -breaking effects. As far as the weak phases are concerned, we prefer to use and , since the results for the former quantity can then be compared directly with constraints from other processes, whereas the latter can anyway be determined straighforwardly from mixing-induced CP violation in up to a twofold ambiguity, also if there should be CP-violating new-physics contributions to – mixing. This way, we obtain an interesting link between the two solutions for and the allowed ranges for , as we have noted above.

It should be emphasized that the parametrization of the CP-violating observables in terms of and is actually more direct than the one in terms of and , as the appearance of is due to the elimination of with the help of the unitarity relation . If there were negligible penguin contributions to , mixing-induced CP violation in this channel would allow us to determine the combination , which is equal to in the Standard Model. On the other hand, in the presence of significant penguin contributions, as indicated by experimental data, it is actually more advantageous to keep and in the parametrization of the observables. Moreover, we may then also investigate straightforwardly the impact of possible CP-violating new-physics contributions to – mixing, which may yield the unconventional value of . These features will become obvious when we turn to the details of our approach.

Another important aspect of our study is an analysis of the decay , which is particularly promising for hadronic experiments. Using the experimental results for the ratio of the CP-averaged and branching ratios, we obtain a very constrained allowed region in the – plane within the Standard Model. If future measurements should actually fall into this very restricted target range in observable space, the combination of with through the -spin flavour symmetry of strong interactions allows a determination of , as we have noted above. On the other hand, if the experimental results should show a significant deviation from the Standard-Model range in observable space, a very exciting situation would arise immediately, pointing towards new physics.

The outline of this paper is as follows: in Section 2, we first turn to the allowed regions in observable space of decays, and give a new kind of contour plots, allowing us to read off directly the preferred ranges for and strong phases from the experimental data. In Section 3, we then discuss the general formalism to deal with the , system, and show how constraints on the relevant penguin parameters can be obtained from data on . The implications for the allowed regions in observable space for the decays and will be explored in Sections 4 and 5, respectively. In our analysis, we shall also discuss the impact of theoretical uncertainties, and comment on certain simplifications, which could be made by using a rather moderate input from factorization. Finally, we summarize our conclusions and give a brief outlook in Section 6.

## 2 Allowed Regions in Observable Space

### 2.1 Amplitude Parametrizations and Observables

The starting point of analyses of the system is the isospin flavour symmetry of strong interactions, which implies the following amplitude relations:

(9) |

Here and denote the strong amplitudes describing colour-allowed and colour-suppressed tree-diagram-like topologies, respectively, is due to colour-allowed and colour-suppressed EW penguins, is a CP-conserving strong phase, and denotes the ratio of EW to tree-diagram-like topologies. A relation with an analogous phase structure holds also for the “mixed” , system. Because of these relations, the following combinations of decays were considered in the literature to probe :

Interestingly, already CP-averaged branching ratios may lead to non-trivial constraints on [8, 11]. In order to go beyond these bounds and to determine , also CP-violating rate differences have to be measured. To this end, it is convenient to introduce the following sets of observables [13]:

(10) |

(11) |

(12) |

where the are ratios of CP-averaged branching ratios and the represent CP-violating observables. In Tables 1 and 2, we have summarized the present status of these quantities implied by the -factory data. The averages given in these tables were calculated by simply adding the errors in quadrature.

Observable | CLEO [32] | BaBar [33] | Belle [34] | Average |
---|---|---|---|---|

Observable | CLEO [35] | BaBar [27, 33] | Belle [28, 36] | Average |
---|---|---|---|---|

The purpose of the following considerations is not the extraction of , which has been discussed at length in [7]–[14], but an analysis of the allowed regions in the – planes arising within the Standard Model. Here we go beyond our previous paper [16] in two respects: first, we consider not only the mixed and charged systems, but also the neutral one, as advocated in [13, 14]. Second, we include contours in the allowed regions that correspond to given values of and , thereby allowing us to read off directly the preferred ranges for these parameters from the experimental data. The “indirect” fits of the unitarity triangle favour the range

(13) |

which corresponds to the Standard-Model expectation for this angle
[30]. Since the CP-violating parameter , describing
indirect CP violation in the neutral kaon system, implies a positive value
of the Wolfenstein parameter [37],^{2}^{2}2For a negative
bag parameter , which appears unlikely to us, negative
would be implied [38]. we shall restrict to
.

To simplify our analysis, we assume that certain rescattering effects [39] play a minor rôle. Employing the formalism discussed in [13] (for an alternative description, see [12]), it would be possible to take into account also these effects if they should turn out to be important. However, both the presently available experimental upper bounds on branching ratios and the recent theoretical progress due to the development of the QCD factorization approach [40, 41] are not in favour of large rescattering effects.

Following these lines, we obtain for the charged and neutral systems

(14) | |||||

(15) |

where denotes a CP-conserving strong phase difference between tree-diagram-like and penguin topologies, measures the ratio of tree-diagram-like to penguin topologies, corresponds to the electroweak penguin parameter appearing in (9), and

(16) |

A detailed discussion of these parametrizations can be found in [13]. Using the flavour symmetry to fix through [5], we arrive at

(17) |

(18) |

where the ratio of the kaon and pion decay constants takes into account factorizable -breaking corrections. In [41], also non-factorizable effects were investigated and found to play a minor rôle. In Table 3, we collect the present experimental results for and following from (17) and (18), respectively. The electroweak penguin parameter can be fixed through the flavour symmtery [11] (see also [7]), yielding

(19) |

with

(20) |

Taking into account factorizable breaking, the central value of is shifted to . For a detailed analysis within the QCD factorization approach, we refer the reader to [41].

We may now use (14) to eliminate in (15):

(21) |

allowing us to calculate for given as a function of ; if we vary between and , we obtain an allowed region in the – plane. This range can also be obtained by varying and directly in (14) and (15), with and .

A similar exercise can also be performed for the mixed system. To this end, we have just to make appropriate replacements of variables in (14) and (15). Since electroweak penguins contribute only in colour-suppressed form to the corresponding decays, we may use in this case to a good approximation. Moreover, we have , where the determination of requires the use of arguments related to factorization [7, 9] to fix the colour-allowed amplitude , or the measurement of [42], which is related to through the -spin flavour symmetry of strong interactions. The presently most refined theoretical study of can be found in [41], using the QCD factorization approach. In our analysis, we shall consider the range . Since we have to make use of dynamical arguments to fix and in the case of the mixed system, it is not as clean as the charged and neutral systems.

### 2.2 Numerical Analysis

In Figs. 1 and 2, we show the allowed regions in observable space of the charged and neutral systems, respectively. The crosses correspond to the averages of the experimental results given in Tables 1 and 2, and the elliptical regions arise, if we restrict to the Standard-Model range specified in (13). The labels of the contours in (c) refer to the values of for , and those of (d) to the values of for . Looking at these figures, we observe that the experimental data fall pretty well into the regions, which are implied by the Standard-Model expressions (14) and (15). However, the data points do not favour the restricted region, which arises if we constrain to its Standard-Model range (13). To be more specific, let us consider the contours shown in (c) and (d), allowing us to read off the preferred values for and directly from the measured observables. In the charged system, the -factory data point towards values for larger than , and smaller than . In the case of the neutral system, the data are also in favour of , but prefer to be larger than . These features were also pointed out in [14]; in Figs. 1 and 2, we can see them directly from the data points. If future measurements should stabilize at such a picture, we would have a very exciting situation, since values for larger than would be in conflict with the Standard-Model range (13), and the strong phases and are expected to be of the same order of magnitude; factorization would correspond to values around . A possible explanation for such discrepancies would be given by large new-physics contributions to the electroweak penguin sector [14]. However, it should be kept in mind that we may also have “anomalously” large flavour-symmetry breaking effects. A detailed recent analysis of the allowed regions in parameter space of and that are implied by the present data can be found in [43], where also very restricted ranges for were obtained by contraining to its Standard-Model expectation. Another study was recently performed in [44], where the were calculated for given values of as functions of , and were compared with the present -factory data.

In Fig. 3, we show the allowed region in observable space of the mixed system. Here the crosses represent again the averages of the experimental -factory results. Since the expressions for and are symmetric with respect to an interchange of and for , the contours for fixed values of and are identical in this limit. Moreover, we obtain the same contours for . The experimental data fall well into the allowed region, but do not yet allow us to draw any further conclusions. In the charged and neutral systems, the situation appears to be much more exciting.

Let us now turn to the main aspect of our analysis, the , system. In our original paper [16], we have addressed these modes only briefly, giving in particular a three-dimensional allowed region in the space of the CP asymmetries , and . Here we follow [26], and use the CP-averaged branching ratio as an additional input to explore separately the allowed regions in the space of the CP-violating and observables, as well as contraints on . The experimental situation has improved significantly since [16] and [26] were written, pointing now to an interesting picture, although the uncertainties are still too large to draw definite conclusions. However, these uncertainties will be reduced considerably in the future due to the continuing efforts at the factories. Once the mode is accessible at hadronic experiments, more refined studies will be possible. In the LHC era, the physics potential of the , system can then be fully exploited. In this paper, we point out that the Standard-Model range in observable space is very constrainted, thereby providing a narrow target range for these experiments.

## 3 Basic Features of the , System and the Connection with

### 3.1 Amplitude Parametrizations and Observables

The decay originates from quark-level transitions. Within the Standard Model, it can be parametrized as follows [45]:

(22) |

where is due to “current–current” contributions, the amplitudes describe “penguin” topologies with internal quarks (, and the

(23) |

are the usual CKM factors. Employing the unitarity of the CKM matrix and the Wolfenstein parametrization [37], generalized to include non-leading terms in [46], we arrive at [15]

(24) |

where

(25) |

with , and

(26) |

The quantity is defined in analogy to , , and was already introduced in (20). The “penguin parameter” measures – sloppily speaking – the ratio of the “penguin” to “tree” contributions.

Using the Standard-Model parametrization (24), we obtain [15]

(27) | |||||

(28) |

where can be determined with the help of (3), yielding the twofold solution given in (5). Strictly speaking, mixing-induced CP violation in probes , where is related to the weak – mixing phase and is negligibly small in the Standard Model. However, due to the small value of the CP-violating parameter of the neutral kaon system, can only be affected by very contrived models of new physics [47].

In the case of , we have [15]

(29) |

where

(30) |

and

(31) |

correspond to (25) and (26), respectively. The primes remind us that we are dealing with a transition. Introducing

(32) |

we obtain [15]

(33) | |||||

(34) |

where the – mixing phase

(35) |

is negligibly small in the Standard Model. Using the range for the Wolfenstein parameter following from the fits of the unitarity triangle [30] yields . Experimentally, this phase can be probed nicely through , which allows an extraction of also if this phase should be sizeable due to new-physics contributions to – mixing [47]–[49].

It should be emphasized that (27), (28) and (33), (34) are completely general parametrizations of the CP-violating and observables, respectively, relying only on the unitarity of the CKM matrix. If we assume that is negligibly small, as in the Standard Model, these four observables depend on the four hadronic parameters , , and , as well as on the two weak phases and . Consequently, we have not sufficient information to determine these quantities. However, since is related to through an interchange of all down and strange quarks, the -spin flavour symmetry of strong interactions implies

(36) |

Making use of this relation, the parameters , , and can be determined from the CP-violating , observables [15]. If we fix through (3), the use of the -spin symmetry in the extraction of can be minimized. Since and are defined through ratios of strong amplitudes, the -spin relation (36) is not affected by -spin-breaking corrections in the factorization approximation [15], which gives us confidence in using this relation.

### 3.2 Constraints on Penguin Parameters

In order to constrain the hadronic penguin parameters through the CP-averaged and branching ratios, it is useful to introduce the following quantity [26]:

(37) |

where

(38) |

denotes the usual two-body phase-space function. The branching ratio BR can be extracted from the “untagged” rate [15], where no rapid oscillatory