CERN-TH/97-185

EFI 97-34

July 1997

hep-ph/9707521

DECAYS INVOLVING AND

IN THE LIGHT OF THE PROCESS

Amol S. Dighe

Enrico Fermi Institute and Department of Physics, University of Chicago

5640 S. Ellis Avenue, Chicago IL 60615

and

Michael Gronau^{1}^{1}1Permanent Address: Physics
Department,
Technion – Israel Institute of Technology, 32000 Haifa, Israel

Theoretical Physics Division, CERN

CH-1211 Geneva 23, Switzerland

and

Jonathan L. Rosner

Enrico Fermi Institute and Department of Physics, University of Chicago

5640 S. Ellis Avenue, Chicago IL 60615

ABSTRACT

The observation by the CLEO Collaboration of the decays is shown to imply a significant but still uncertain contribution from the flavor-SU(3)-singlet component of the . By comparing the rate for these decays with others for decays of mesons to two pseudoscalar mesons, it is shown that the prospects for observing CP-violating asymmetries in certain modes such as and are quite bright.

PACS numbers: 13.25.Hw, 14.40.Nd, 11.30.Er, 12.15.Ji

CERN-TH/97-185

July 1997

The CLEO Collaboration [1] has recently reported the decays and with branching ratios of and , respectively. In the present note we show that these results, when combined with other information on decays of mesons to pairs of light pseudoscalar mesons, indicate that the decays receive a significant contribution from the flavor-SU(3)-singlet component of the . We use present information to predict the rates for charged and neutral ’s to decay to ( or or ). By searching for processes in which contributions of different weak decay amplitudes are comparable to one another, we show that there is a high likelihood for observable CP-violating asymmetries in the decays and . A similar conclusion was reached earlier by Barshay, Rein, and Sehgal [2] on the basis of a different analysis. Others have emphasized previously the potential for CP-violating rate asymmetries to be exhibited in decays of mesons to pairs of charmless mesons [3].

The contribution to from a new penguin amplitude, occurring only in decays involving a flavor SU(3) singlet component in the final pseudoscalar meson state, was noted in Refs. [4, 5, 6]. While one possibility for this contribution [2, 7, 8] is an intrinsic component in the , more conventional mechanisms [8, 9] (e.g., involving gluons) also seem adequate to explain the observed rate. Some enhancement of conventional mechanisms may be needed to explain the large rate for the inclusive process . We shall not be concerned here with the inclusive process.

We list the relevant decay amplitudes associated with a flavor-SU(3) decomposition [5, 6, 10, 11, 12, 13, 14, 15] in Tables I and II. Unprimed amplitudes denote decays; primed amplitudes denote decays. An amplitude describes a tree-graph contribution, describes a color-suppressed process, a penguin graph contribution coupling to a pair of quark-antiquark mesons, and an additional penguin contribution coupling specifically to the flavor-SU(3)-singlet component of the or . All these amplitudes are defined in such a way [15] as to include contributions from electroweak penguin terms [16].

Decay | Amplitudes | Denom. | rate | |||

factor | rate | rate | (a) | (b) | ||

4.1 | 0 | 0 | 0 | |||

1 | 0 | 0.8 | 0 | 0 | ||

2.8 | 1.0 | 0.06 | 0.24 | |||

1.4 | 0.5 | 0.4 | 1.9 | |||

8.3 | 0.8 | 0 | 0 | |||

0 | 0.4 | 0 | 0 | |||

1 | 0 | 0.8 | 0 | 0 | ||

0 | 0.5 | 0.03 | 0.12 | |||

0 | 0.26 | 0.2 | 0.9 |

(a): Constructive interference between and amplitudes assumed in .

(b): No interference between and amplitudes assumed in .

Decay | Amplitudes | Denom. | rate | |||
---|---|---|---|---|---|---|

factor | rate | rate | (a) | (b) | ||

1 | 0 | 16 | 0 | 0 | ||

0.20 | 8 | 0 | 0 | |||

0.13 | 1.2 | 4.9 | ||||

0.07 | 24 | 9 | 39 | |||

0.4 | 16 | 0 | 0 | |||

0 | 8 | 0 | 0 | |||

0 | 1.2 | 4.9 | ||||

0 | 24 | 9 | 39 |

(a): Constructive interference between and amplitudes assumed in .

(b): No interference between and amplitudes assumed in .

We assume the and are mixed so that and , corresponding to an octet-singlet mixing angle of . The contribution to vanishes for this mixing [14, 15, 17]. More details justifying this assumption are discussed, for example, in Refs. [4], [5], and [12]. Other phase conventions for pseudoscalar mesons may be found in Ref. [13]. We have neglected all annihilation- and exchange-type amplitudes, which are expected to be highly suppressed in comparison with those shown.

In Tables I and II we have also calculated expected squares of contributions of individual amplitudes to decays. We ignore for present purposes any interference between tree ( or ) and other amplitudes. We consider two possibilities for the relative phase of the two predominant amplitudes, and , in the decay . The cases (a) and (b) listed in the Tables correspond to constructive interference and no interference between these amplitudes. [Destructive interference would imply a singlet amplitude so large that the predicted value of would exceed the current 90% confidence level (c.l.) bound [1] .]

Interference between amplitudes becomes important when they are not too different in magnitude, which occurs in several cases which we shall identify presently. We do not quote contributions of color-suppressed amplitudes, neglecting them in the ensuing discussion. We determine amplitudes in the following manner.

(1) The magnitude of the amplitude is estimated by averaging the observed branching ratios [18]

(1) |

and

(2) |

to obtain the estimate , where all squares of amplitudes in the Tables are quoted in branching ratio units of . In we have neglected the small contribution, an assumption which will be seen to be justified. If the rates for (1) and (2) are found to be unequal, the neglect of the amplitude (or of some other contribution) may not be valid. In that case the possibility of a CP asymmetry in (say) may be significantly enhanced.

(2) The magnitude of the amplitude is estimated to be , where and are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. With an uncertainty of about a factor of two, .

(3) The contribution to the decay is estimated to be about 8, whereas [19]

(3) |

Thus there is room for a significant signal. While 90% c.l. upper limits of and are quoted in Refs. [1], Ref. [19] also quotes a signal of

(4) |

and a signal of

(5) |

Taking (4) as an estimate of (neglecting the color-suppressed amplitude in ), and (5) as an estimate of (neglecting the penguin amplitude in ), we find .

(4) The value of is estimated without accounting for SU(3)-breaking to lead to . It could be slightly higher if one applied a correction [14] of a factor of .

(5) The contribution to the branching ratio (in units of ) is ; it cannot account for the observed value of (our average for charged and neutral modes, where and contributions are assumed to be negligible). Assuming constructive interference between and in we find the contribution to the rate to be about , with an additional contribution of 30 from the – interference term. [The enhancement of the rate by a modest amplitude interfering constructively with was noted by Lipkin, last of Refs. [17].] If the interference term is absent (i.e., if the relative phase of the amplitudes is ) then one needs an contribution of to the rate. Henceforth we shall work only with central values of amplitudes for illustrative purposes; the uncertainty in due to the uncertainty in its phase relative to generally exceeds that due to experimental error. (If one allows the branching ratio to be at its value, can even be considerably smaller, with , when and interfere constructively in this decay.)

(6) Since we expect if both and are dominated by the top quark, we choose . (If in fact as a result of charmed quark dominance of this type of penguin contribution, the result is the same.)

The results in the Tables may be interpreted in the following manner.

(i) Any contribution of order 10 or greater (corresponding to a branching ratio of ) has been observed.

(ii) A contribution greater or equal than 1 should be observable in the next generation of CLEO experiments, with improved sensitivity and particle identification. Thus the decays , , , , and should all make their appearance, while and should be resolved from one another. For example, one expects and , where the current upper bounds [1] are and , respectively. The first limit is already quite close to our prediction. The above branching ratios are about a factor of 2 larger than those predicted in Ref. [2].

(iii) The amplitudes for , , and satisfy

(6) |

Aside from small contributions, the terms in this triangle relation are dominated by and contributions. Since and are expected to have the same weak phase, the shape of the triangle will tell us about the relative strong phase of these amplitudes. Neglecting contributions as well, one can write

(7) |

which is easier to measure. The main uncertainty lies in the value of the branching ratio for . If involves strong rescattering from charm-anticharm states [2, 3, 8, 9], its strong phase could differ from that of (and hence also possibly ).

(iv) Processes with two contributions both of which exceed 1 are prime candidates for observable direct CP violation if both strong and weak phases of the two amplitudes differ from one another. The weak phases of the (tree) and (penguin) contributions in are expected to be and [20], respectively, while the relative strong phases are unknown. In the case of , if its contribution is dominated by the charmed quark penguin, a significant strong phase shift could arise from the real intermediate state. One could thus have a large strong phase shift difference between the and amplitudes in . The weak phases of these two amplitudes are also different: the charm penguin is approximately real, while the amplitude has a weak phase .

(v) Our focus has been on the observability of direct CP violation in decays such as and . These processes may not be the first to exhibit CP asymmmetries; asymmetric factories will search for mixing-induced asymmetries, in which the time dependence of the decays must be studied. A time-dependent asymmetry measurement in would provide a clean determination of the weak phase . Our result, , implies a rather large “penguin pollution” in the analysis of the time-dependent decay asymmetry in , compatible with previous estimates [14, 21]. In order to resolve such effects using isospin symmetry [22], one would have to measure , for which the contribution to the branching ratio is only . (The contribution of the color-suppressed amplitude is highly uncertain but unlikely to be much larger.) An alternative way to resolve the penguin pollution question in is to rely on flavor SU(3) to link this decay with various modes [21, 23], all of which have large rates.

To summarize, we have used existing data on and other two-body modes involving pairs of light pseudoscalar mesons to anticipate observable CP-violating effects in the decays and . Since experimental errors are still quite large, the same procedure, based only on flavor SU(3), can and should be applied to better data when they become available. In that case one will be able to test for effects of interference among various amplitudes which have been ignored here (applying, for example, amplitude relations noted in Refs. [4, 5, 6]). One welcome improvement in data will be a better estimate of and , which, under the assumption of top quark dominance, we have taken to be , with an uncertainty of a factor of 2.

A rule of thumb for observable CP-violating effects is that one must at least be able to observe the square of the lesser of two interfering amplitudes at the level in order to observe an asymmetry at this level [24]. Our results indicate that this sensitivity threshold is passed for decays of the form and when branching ratios of order become detectable in experiments sensitive to both charged and neutral final-state particles.

We are grateful to J. Alexander, D. Atwood, K. Berkelman, P. Drell, H. Fritzsch and L. Sehgal for helpful discussions. This work was performed in part at the Aspen Center for Physics, and supported in part by the United States Department of Energy under Grant No. DE FG02 90ER40560 and by the United States – Israel Binational Science Foundation under Research Grant Agreement 94-00253/2.

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