CERN-TH/98-319

TUM-HEP-330/98

hep-ph/9810260

A General Analysis of Determinations from Decays

Andrzej J. Buras and Robert Fleischer

[0.5cm] Theory Division, CERN, CH-1211 Geneva 23, Switzerland

[0.3cm] Technische Universität München, Physik Department

D-85748 Garching, Germany

[0.6cm]

We present a general parametrization of , and , decays, taking into account both electroweak penguin and rescattering effects. This formalism allows – among other things – an improved implementation of the strategies that were recently proposed by Neubert and Rosner to probe the CKM angle with the help of , decays. In particular, it allows us to investigate the sensitivity of the extracted value of to the basic assumptions of their approach. We find that certain -breaking effects may have an important impact and emphasize that additional hadronic uncertainties are due to rescattering processes. The latter can be controlled by using flavour symmetry arguments and additional experimental information provided by modes. We propose a new strategy to probe the angle with the help of the neutral decays , , which is theoretically cleaner than the , approach. Here rescattering processes can be taken into account by just measuring the CP-violating observables of the decay . Finally, we point out that an experimental analysis of modes would also be very useful to probe the CKM angle , as well as electroweak penguins, and we critically compare the virtues and weaknesses of the various approaches discussed in this paper. As a by-product, we point out a strategy to include the electroweak penguins in the determination of the CKM angle from decays.

CERN-TH/98-319

October 1998

## 1 Introduction

Last year, the CLEO collaboration reported the observation of several exclusive -meson decays into two light pseudoscalar mesons [1], which led to great excitement in the -physics community. In particular, the decays , and their charge conjugates received a lot of attention [2], since their observables may provide useful information on the angle of the usual non-squashed unitarity triangle of the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix) [3, 4]. So far, only results for the combined branching ratios

(1) | |||||

(2) |

have been published, with values at the level and large experimental uncertainties. As was pointed out in [5], already these combined branching ratios may lead to highly non-trivial constraints on , which become effective if the ratio

(3) |

is found to be smaller than 1. If we use the isospin symmetry of strong interactions and neglect certain rescattering and electroweak penguin effects (for more sophisticated strategies, taking into account also these effects, see [6, 7]), we obtain the following allowed range for [5]:

(4) |

where is given by

(5) |

Unfortunately, the present data do not yet provide a definite answer to the question of whether . The results reported by the CLEO collaboration last year gave [1], whereas a recent, preliminary update yields [8]. A detailed study of the implications of (4) for the determination of the unitarity triangle was performed in [9].

This summer, the CLEO collaboration announced the first observation of another decay, which is the mode [8]. Consequently, it is natural to ask whether we could also obtain interesting information on the angle with the help of this decay. In fact, several years ago, Gronau, Rosner and London (GRL) proposed an interesting strategy to determine , with the help of the decays , , and their charge conjugates, by using the flavour symmetry of strong interactions [10] (see also [11]). However, as was pointed out by Deshpande and He [12], this elegant approach is unfortunately spoiled by electroweak penguins, which play an important role in several non-leptonic -meson decays because of the large top-quark mass [13, 14]. In the case of the mode , electroweak penguins contribute both in “colour-allowed” and in “colour-suppressed” form, whereas only electroweak penguin topologies of the latter kind contribute to the decays and . Performing model calculations within the framework of the “factorization” hypothesis, one finds that “colour-suppressed” electroweak penguins play a negligible role [15]. These crude estimates may, however, underestimate the role of these topologies [4, 16], which therefore represent an important limitation of the theoretical accuracy of the strategies to probe the CKM angle with the help of and decays [2].

In [3, 17], we proposed methods to obtain experimental insights into electroweak penguins with the help of amplitude relations between the decays listed above. Since it is possible to derive a transparent expression for the relevant electroweak penguin amplitude by performing appropriate Fierz transformations of the electroweak penguin operators and using the flavour symmetry [3] (see also [14]), the experimental determination of this amplitude would allow an interesting test of the Standard Model. In two recent papers [18, 19], Neubert and Rosner used a more elaborate, but similar theoretical input to calculate the electroweak penguin amplitude affecting the GRL approach. Provided the electroweak penguin amplitude calculated this way is theoretically reliable, the combined and branching ratios may imply interesting bounds on the CKM angle [18], and the original GRL strategy, requiring the measurement of a CP-violating asymmetry in , is resurrected [19].

In this paper, we point out that the general formulae to probe the CKM angle , with the help of the decays and that were derived in [6], apply also to the combination , of charged decays, as well as to the combination , of neutral decays, if straightforward replacements of variables are performed. In this manner, the virtues and weaknesses of the strategies proposed in [6, 18, 19], and of a new one proposed here, can be systematically investigated and compared with one another. Following these lines, we are in a position to derive the bounds on arising in the , case [18] in a general and transparent way. In contrast to the expressions given in [18], our formalism is valid exactly and does not rely on any expansion in a small parameter. Moreover, it allows us to investigate the sensitivity of the value of to the basic assumptions made in [18], and to take into account certain rescattering processes by using the strategies proposed in [6, 7]. These final-state interaction effects were neglected by Neubert and Rosner in [18, 19], but may in principle play an important role [16, 20–24]. We find that they lead to uncertainties similar to those affecting the , strategies [2], and that furthermore certain -breaking effects may have an important impact. Concerning rescattering effects, the neutral decays , offer theoretically cleaner strategies to probe the CKM angle than the charged modes , . The point is that the decay provides an additional observable, which originates from mixing-induced CP violation. If we use in addition the CP asymmetry arising in the mode to fix the – mixing phase, the rescattering processes can be included completely. We also point out that an experimental analysis of the decay would offer – in combination with the data provided by , and – several simple strategies both to probe the CKM angle and to obtain insights into electroweak penguins. Therefore, an accurate measurement of the decay , which should be feasible at future hadron machines, would be an important goal.

The outline of this paper is as follows: in Section 2, we present a general parametrization of the decay amplitudes and observables, taking into account both electroweak penguin and rescattering effects. In Section 3, we recapitulate the , strategies to constrain and determine the CKM angle in the light of the most recent CLEO data, and point out some interesting features that were not emphasized in previous work. In Section 4, we focus on strategies to probe with the help of the charged decays , , while we turn to a new approach, using the neutral modes , , in Section 5. Several strategies to combine the observables of the modes with those of the decay to determine the CKM angle and to probe electroweak penguins are proposed in Section 6. Finally, the conclusions are summarized in Section 7, where we also critically compare the virtues and weaknesses of the various approaches discussed in this paper. In an appendix, we present a by-product of our considerations, allowing us to include electroweak penguin topologies in the determination of the CKM angle from decays.

## 2 Decay Amplitudes and Observables

In this section, we will closely follow Ref. [6] to parametrize the decay amplitudes and observables of and arising within the framework of the Standard Model. Before turning to these modes, it will be instructive to recall certain features of the decays and , which were already discussed in detail in [6].

### 2.1 The Decays and

In order to probe the CKM angle through these decays, the central role is played by the following amplitude relation:

(6) |

which can be derived by using the isospin symmetry of strong interactions [25]. Here the amplitude , which is usually referred to as a “tree” amplitude, takes the form

(7) |

Owing to a subtlety in the implementation of the isospin symmetry, the amplitude does not only receive contributions from colour-allowed tree-diagram-like topologies, but also from penguin and annihilation topologies [6, 25]. On the other hand, the quantity is due to electroweak penguin contributions, which do not carry the phase , and can be expressed as

(8) |

Note that the remaining electroweak penguin contributions have been absorbed in the amplitude . The label “C” reminds us that only “colour-suppressed” electroweak penguin topologies contribute to . In (7) and (8), and denote CP-conserving strong phases. Explicit formulae for and are given in [6].

The decay amplitude entering (6) can be expressed as follows [6]:

(9) |

where and denote contributions from QCD and electroweak penguin topologies with internal quarks , respectively; describes annihilation topologies, and are the usual CKM factors. If we make use of the unitarity of the CKM matrix and apply the Wolfenstein parametrization [26], generalized to include non-leading terms in [27], we obtain [6]

(10) |

where

(11) |

and

(12) |

In these expressions, and denote CP-conserving strong phases, and is defined in analogy to (11). The quantity is a measure of the strength of certain rescattering effects, and the relevant CKM factors are given by (for a recent update of , see [28]):

(13) |

In the parametrization of the and observables, it turns out to be useful to introduce the quantities

(14) |

with

(15) |

as well as the CP-conserving strong phase differences

(16) |

The CP-conjugate amplitude is obtained from (10) by simply reversing the sign of the weak phase . A similar comment applies also to all other CP-conjugate decay amplitudes appearing in this paper. In addition to the ratio of combined branching ratios defined by (3), also the “pseudo-asymmetry”

(17) |

plays an important role to probe the CKM angle . Explicit expressions for and in terms of the parameters specified above are given in [6].

As we already noted, the electroweak penguins are “colour-suppressed” in the case of the decays and . Calculations performed at the perturbative quark level, where the relevant hadronic matrix elements are treated within the “factorization” approach, typically give [15]. These crude estimates may, however, underestimate the role of these topologies [4, 16]. An improved theoretical description of the electroweak penguins is possible, using the general expressions for the corresponding four-quark operators, appropriate Fierz transformations and the isospin symmetry. Following these lines [6] (see also [3, 14]), we arrive at

(18) |

with

(19) |

and

(20) |

Here are the Wilson coefficients of the current–current operators

(21) |

and the coefficients are those of the electroweak penguin operators

(22) |

As usual, and are colour indices, and denotes the quark charges. It should be kept in mind that two electroweak penguin operators, and , with tiny Wilson coefficients, and electroweak penguins with internal charm- and up-quark exchanges were neglected in the derivation of (18). In our numerical estimates given below, it will suffice to use the leading-order values [29]

(23) |

with . It is possible to rewrite (18) as follows [6]:

(24) | |||||

where we will neglect the first, strongly suppressed term

(25) |

in the following considerations:

(26) |

The combination of Wilson coefficients in this expression is essentially renormalization-scale-independent and changes only by when evolving from down to . Employing and the Wilson coefficients given in (23) yields [6]

(27) |

The quantity is given by

(28) |

where and correspond to a generalization of the usual phenomenological “colour” factors and , describing the intrinsic strength of “colour-suppressed” and “colour-allowed” decay processes, respectively [6]. Note that the “factorization” approach gives , where is the “factorization scale”. Comparing experimental data on and , as well as on and decays gives . Here and are – in contrast to and – real quantities, and their relative sign is found to be positive. Experimental studies of decays favour also . If we assume that the strength of “colour suppression” in decays is of the same order of magnitude, i.e. , we obtain a value of that is larger by a factor of 3 than the “factorized” result

(29) |

corresponding to and in (18). The expression (26) shows nicely that the usual terminology of “colour-suppressed” electroweak penguins in (6) is justified, since is proportional to the generalized “colour” factor . Moreover, it implies a correlation between and , which is given by

(30) |

with

(31) |

The ratio defined by (3) can be expressed as follows [6]:

(32) |

where

(33) | |||||

(34) |

and

(35) | |||||

(36) |

The pseudo-asymmetry (see (17)) takes the form

(37) |

where

(38) |

measures direct CP violation in the decay . Note that tiny phase-space effects have been neglected in (32) and (37) (for a more detailed discussion, see [5]).

### 2.2 The Decays and

Let us now turn to the decays , and their charge conjugates. The isospin symmetry implies the following amplitude relation [30, 31]:

(39) |

where is due to electroweak penguins and refers to a isospin configuration with . Note that there is no component present in (39). Since we have

(40) |

and

(41) |

the phase structure of the amplitude relation (39) is completely analogous to the one given in (6). We just have to perform the replacements

(42) |

The notation of reminds us that this amplitude receives contributions both from “colour-allowed” and from “colour-suppressed” tree-diagram-like topologies [10]. A similar comment applies to the electroweak penguin amplitude , receiving also contributions both from “colour-allowed” and from “colour-suppressed” electroweak penguin topologies [31]. If we neglect electroweak penguin topologies with internal charm and up quarks, as well as the electroweak penguin operators and , which have tiny Wilson coefficients, perform appropriate Fierz transformations of the remaining electroweak penguin operators and and, moreover, apply the isospin symmetry, we arrive at

(43) |

with

(44) | |||||

which is completely analogous to (18) and (19). Since the flavour symmetry of strong interactions implies

(45) |

it is useful to rewrite (43) as follows:

(46) |

where

(47) | |||||

describes -breaking corrections. In the strict limit, we have , and obtain [18]

(48) | |||||

which is related to through

(49) |

Here we have again neglected the strongly suppressed term (25). Within the “factorization” approximation, we have very small -breaking corrections at the level of a few per cent [18] (see also [3]). Unfortunately, we have no insights into non-factorizable breaking at present. Taking into account both the factorizable corrections, which shift from 0.66 to 0.63, and the present experimental uncertainty of (see (13)), Neubert and Rosner give the range of [18].

If we compare (39) with (6), we find that the observables of the charged -meson decays , corresponding to and have to be defined as follows:

(50) | |||||

(51) |

Concerning strategies to probe the CKM angle , the ratio is more convenient than the quantity , which was considered by Neubert and Rosner in [18]. The preliminary results on the CP-averaged branching ratios

(52) | |||||

(53) |

which, very recently, were reported by the CLEO collaboration [8], give

(54) |

Here we have added the errors in quadrature. This result differs significantly from the present value of , although the uncertainties are too large to say anything definite.

In the case of the neutral modes , , we have

(55) | |||||

(56) |

While the CLEO collaboration recently reported the preliminary result [8]

(57) |

there is at present only an upper limit available for the decay , which is given by BR [1].

The parametrization of the observables , and , is completely analogous to (32) and (37) and can be obtained straightforwardly from these expressions by performing appropriate replacements. The most obvious one is the following:

(58) |

Moreover, we have to substitute

(59) |

in the case of the observables and . The parameter , which is defined through the decay amplitude, remains unchanged. This is in contrast to the case of the neutral modes , . Here the decay takes the role of the mode . In analogy to (10), its decay amplitude can be expressed as

(60) |

where takes the form

(61) |

Here and correspond to differences of penguin topologies with internal top and charm and up and charm quarks, respectively (see (11)). In contrast to the