###### Abstract

We try a global fit of the experimental branching ratios and CP-asymmetries of the charmless decays according to QCD factorisation. We find it impossible to reach a satisfactory agreement, the confidence level (CL) of the best fit is smaller than .1 %. The main reason for this failure is the difficulty to accomodate several large experimental branching ratios of the strange channels. Furthermore, experiment was not able to exclude a large direct CP asymmetry in , which is predicted very small by QCD factorisation. Trying a fit with QCD factorisation complemented by a charming-penguin inspired model we reach a best fit which is not excluded by experiment (CL of about 8 %) but is not fully convincing. These negative results must be tempered by the remark that some of the experimental data used are recent and might still evolve significantly.

DAPNIA-02-380 |

LPT Orsay 02-122 |

LMU-02-19 |

PCCF-RI-0218 |

hep-ph/0301165 |

Testing QCD factorisation and charming penguins in charmless

R. Aleksan^{*}^{*}*e-mail :,
P.-F. Giraud^{†}^{†}†e-mail :,
V. Morénas^{‡}^{‡}‡e-mail :,
O. Pène^{§}^{§}§e-mail : and
A. S. Safir^{¶}^{¶}¶e-mail :.

CEA Saclay, DAPNIA/SPP (Bât. 141) F-91191 Gif-sur-Yvette CEDEX, France.

LPC, Université Blaise Pascal - CNRS/IN2P3 F-63000 Aubière Cedex, France.

LPT (Bât.210), Université de Paris XI, Centre d’Orsay, 91405 Orsay-Cedex, France.

LMU München, Sektion Physik,Theresienstraße 37, D-80333 München, Germany.

February 9, 2021

## I Introduction

It is an important theoretical challenge to master the non-leptonic decay amplitudes and particularly non-leptonic decay. It is not only important per se, in view of the many experimental branching ratios which have been measured recently with increasing accuracy by BaBar[?–?], Belle[?–?] and CLEO[?–?], but it is also necessary in order to get control over the measurement of CP violating parameters and particularly the so-called angle of the unitarity triangle. It is well known that extracting from measured indirect CP asymmetries needs a sufficient control of the relative size of the so-called tree () and penguins () amplitudes.

However the theory of non-leptonic weak decays is a difficult issue. Lattice QCD gives predictions for semi-leptonic or purely leptonic decays but not directly for non-leptonic ones. Since long, one has used what is now called “naive factorization” which replaces the matrix element of a four-fermion operator in a heavy-quark decay by the product of the matrix elements of two currents, one semi-leptonic matrix element and one purely leptonic. For long it was noticed that naive factorization did provide reasonable results although it was impossible to derive it rigorously from QCD except in the limit. It was also well-known that the matrix elements computed via naive factorization have a wrong anomalous dimension.

Recently an important theoretical progress has been performed [22, 23] which is commonly called “QCD factorisation”. It is based on the fact that the quark is heavy compared to the intrinsic scale of strong interactions. This allows to deduce that non-leptonic decay amplitudes in the heavy-quark limit have a simple structure. It implies that corrections termed “non-factorizable”, which were thought to be intractable, can be calculated rigorously. The anomalous dimension of the matrix elements is now correct to the order at which the calculation is performed. Unluckily the subleading contributions cannot in general be computed rigorously because of infrared singularities, and some of these which are chirally enhanced are not small, of order , which shows that the inverse power is compensated by . In the seminal papers [22, 23], these contributions are simply bounded according to a qualitative argument which could as well justify a significanlty larger bound with the risk of seeing these unpredictable terms become dominant. It is then of utmost importance to check experimentally QCD factorisation.

Since a few years it has been applied to (two charmless pseudoscalar mesons) decays. The general feature is that the decay to non-strange final states is predicted slightly larger than experiment while the decay to strange final states is significantly underestimated. In [23] it is claimed that this can be cured by a value of the unitarity-triangle angle larger than generally expected, larger maybe than 90 degrees. Taking also into account various uncertainties the authors conclude positively as for the agreement of QCD factorisation with the data. In [24, 25] it was objected that the large branching ratios for strange channels argued in favor of the presence of a specific non perturbative contribution called “charming penguins” [?–?]. We will return to this approach later.

The (charmless pseudoscalar + vector mesons) channels are more numerous and allow a more extensive check. In ref. [31] it was shown that naive factorisation implied a rather small ratio, for decay channel, to be compared to the larger one for the . This prediction is still valid in QCD factorisation where the ratio is of about 3 % (8 %) for the () channel against about 20 % for the one. If this prediction was reliable it would put the channel in a good position to measure the CKM angle via indirect CP violation. This remark triggered the present work: we wanted to check QCD factorisation in the sector to estimate the chances for a relatively easy determination of the angle .

The non-charmed amplitudes have been computed in naive factorisation [32], in some extension of naive factorisation including strong phases [33], in QCD factorisation [?–?] and some of them in the so-called perturbative QCD [38, 39]. In [41], a global fit to was investigated using QCDF in the heavy quark limit and it has been found a plausible set of soft QCD parameters that apart from three pseudoscalar vector channels, fit well the experimental branching ratios. Recently [36] it was claimed from a global fit to that the predictions of QCD factorization are in good agreement with experiment when one excludes some channels from the global fit. When this paper appeared we had been for some time considering this question and our feeling was significantly less optimistic. This difference shows that the matter is far from trivial mainly because experimental uncertainties can still be open to some discussion. We would like in this paper to understand better the origin of the difference between our unpublished conclusion and the one presented in [36] and try to settle the present status of the comparison of QCD factorisation with experiment.

One general remark about QCD factorisation is that it yields predictions which do not differ so much from naive factorisation ones. This is expected since QCD factorisation makes a perturbative expansion the zeroth order of which being naive factorisation. As a consequence, QCD factorisation predicts very small direct CP violation in the non-strange channels. Naive factorisation predicts vanishing direct CP violation. Indeed, direct CP violation needs the occurence of two distinct strong contributions with a strong phase between them. It vanishes when the subdominant strong contribution vanishes and also when the relative strong phase does as is the case in naive factorisation. In the case of non-strange decays, the penguin () and tree () contributions being at the same order in Cabibbo angle, the penguin is strongly suppressed because the Wilson coefficients are suppressed by at least one power of the strong coupling constant , and the strong phase in QCD factorisation is generated by a corrections. Having both and the strong phase small, the direct CP asymetries are doubly suppressed. Therefore a sizable experimental direct CP asymetry in which is not excluded by experiment [9] would be at odds with QCD factorisation. We will discuss this later on. Notice that this argument is independent of the value of the unitarity angle , contrarily to arguments based on the value of some branching ratios which depend on [23].

The Perturbative QCD (PQCD) predicts larger direct CP asymmetries than QCDF due to the fact that penguin contributions to anihilation diagrams, claimed to be calculable in PQCD, contribute to a larger amount to the amplitude and have a large strong phase. In fact, in PQCD, this penguin anihilation diagram is claimed to be of the same order, , than the dominant naive factorisation diagram while in QCDF it is also but smaller than the dominant naive factorisation which is . Hence, in PQCD, this large penguin contribution with a large strong phase yields a large CP asymmetry [40, 42, 43].

If QCD factorisation is concluded to be unable to describe the present data satisfactorily, while there is to our knowledge no theoretical argument against it, we have to incriminate non-perturbative contributions which are larger than expected. One could simply enlarge the allowed bound for those contributions which are formally subleading but might be large. However a simple factor two on these bounds makes these unpredictable contributions comparable in size with the predictable ones, if not larger. This spoils the predictivity of the whole program.

A second line is to make some model about the non-perturbative contribution. The “charming penguin” approach [27, 30] starts from noticing the underestimate of strange-channels branching ratios by the factorisation approaches. This will be shown to apply to the channels as well as to the ones. This has triggered us to try a charming-penguin inspired approach. It is assumed that some hadronic contribution to the penguin loop is non-perturbative. In other words that weak interactions create a charm-anticharm intermediate state which turns into non-charmed final states by strong rescattering. In order to make the model as predictive as possible we will use not more than two unkown complex number and use flavor symmetry in strong rescattering.

In section II we will recall the weak-interaction effective Hamiltonian. In section III we will recall QCD factorisation. In section IV we will compare QCD factorisation with experimental branching ratios and direct CP asymmetries. In section V we will propose a model for non-perturbative contribution and compare it to experiment. We will then conclude.

## Ii The effective Hamiltonian

The effective weak Hamiltonian for charmless hadronic decays consists of a sum of local operators multiplied by short-distance coefficients given in table 1, and products of elements of the quark mixing matrix, or . Below we will focus on decays; where and hold for pseudoscalar and vector mesons respectively. Using the unitarity relation , we write

(1) |

where are the left-handed current–current operators arising from -boson exchange, and are QCD and electroweak penguin operators, and and are the electromagnetic and chromomagnetic dipole operators. They are given by

(2) |

where , are colour indices, are the electric charges of the quarks in units of , and a summation over is implied. The definition of the dipole operators and corresponds to the sign convention for the gauge-covariant derivative. The Wilson coefficients are calculated at a high scale and evolved down to a characteristic scale using next-to-leading order renormalization-group equations. The essential problem obstructing the calculation of non-leptonic decay amplitudes resides in the evaluation of the hadronic matrix elements of the local operators contained in the effective Hamiltonian.

NLO | ||||||
---|---|---|---|---|---|---|

1.137 | 0.021 | 0.010 | ||||

1.081 | 0.014 | 0.009 | ||||

1.045 | 0.009 | 0.007 | ||||

0.096 | 0.331 | — | — | |||

0.060 | 0.223 | — | — | |||

0.039 | 0.144 | — | — | |||

LO | ||||||

1.185 | 0.018 | 0.010 | ||||

1.117 | 0.012 | 0.008 | ||||

1.074 | 0.008 | 0.006 | ||||

0.045 | 0.418 | |||||

0.029 | 0.288 | |||||

0.019 | 0.193 |

## Iii QCD factorization in decays

When the QCD factorization (QCDF) method is applied to the decays , the hadronic matrix elements of the local effective operators can be written as

(3) | |||||

where are leading-twist light-cone distribution amplitudes, and the -products imply an integration over the light-cone momentum fractions of the constituent quarks inside the mesons. A graphical representation of this result is shown in Figure 1.

Here and denote the form factors for and transitions, respectively. , , and are the light-cone distribution amplitudes (LCDA) of valence quark Fock states for , vector, and pseudoscalar mesons, respectively. denote the hard-scattering kernels, which are dominated by hard gluon exchange when the power suppressed terms are neglected. So they are calculable order by order in perturbation theory. The leading terms of come from the tree level and correspond to the naive factorization (NF) approximation. The order of terms of can be depicted by vertex-correction diagrams Fig.2 (a-d) and penguin-correction diagrams Fig.2 (e-f). describes the hard interactions between the spectator quark and the emitted meson when the gluon virtuality is large. Its lowest order terms are and can be depicted by hard spectator scattering diagrams Fig.2 (g-h). One of the most interesting results of the QCDF approach is that, in the heavy quark limit, the strong phases arise naturally from the hard-scattering kernels at the order of . As for the nonperturbative part, they are, as already mentioned, taken into account by the form factors and the LCDA of mesons up to corrections which are power suppressed in .

With the above discussions on the effective Hamiltonian of decays Eq.(1) and the QCDF expressions of hadronic matrix elements Eq.(3), the decay amplitudes for in the heavy quark limit can be written as

(4) |

The above are the factorized hadronic matrix elements, which have the same definitions as those in the NF approach. The “nonfactorizable” effects are included in the coefficients which are process dependent. The coefficients are collected in Sec. III.1, and the explicit expressions for the decay amplitudes of can be found in the appendix A.

According to the arguments in [22], the contributions of weak annihilation to the decay amplitudes are power suppressed, and they do not appear in the QCDF formula Eq.(3). But, as emphasized in [40, 42, 43], the contributions from weak annihilation could give large strong phases with QCD corrections, and hence large CP violation could be expected, so their effects cannot simply be neglected. However, in the QCDF method, the annihilation topologies (see Fig.3) violate factorization because of the endpoint divergence. There is similar endpoint divergence when considering the chirally enhanced hard spectator scattering. One possible way is to treat the endpoint divergence from different sources as different phenomenological parameters [23]. The corresponding price is the introduction of model dependence and extra numerical uncertainties in the decay amplitudes. In this work, we will follow the treatment of Ref. [23] and express the weak annihilation topological decay amplitudes as

(5) |

where the parameters are collected in Sec. III.2, and the expressions for the weak annihilation decay amplitudes of are listed in the appendix B.

### iii.1 The QCD coefficients

We express the QCD coefficients (see Eq.(4) ) in two parts, i.e., . The first term contains the naive factorisation and the vertex corrections which are described by Fig.2 (a-f), while the second part corresponds to the hard spectator scattering diagrams Fig.2 (g-h).

There are two different cases according to the final states. Case I is that the recoiled meson is a vector meson, and the emitted meson corresponds to a pseudoscalar meson, and vice versa for case II. For case I, we sum up the results for as follows:

(6) |

where , and . The vertex parameters and result from Fig.2 (a-d); the QCD penguin parameters and the electroweak penguin parameters result from Fig.2 (e-f).

The vertex corrections are given by:

(7) | |||||

where is the dilogarithm function, whereas the constants and are specific to the NDR scheme.

The penguin contributions are:

(8) | |||||

and the electroweak penguin parameters :

(9) | |||||

where , and where in the expressions for and runs over all the active quarks at the scale , i.e., . The functions and are given respectively by:

(10) | |||||

(11) | |||||

(12) | |||||

The parameters and in , which originate from hard gluon exchanges between the spectator quark and the emitted meson , are written as:

(13) |

For case II (vector meson emitted) except for the parameters of and , the expressions for are similar to those in case I. However we would like to point out that, because , the contributions of the effective operators to the hadronic matrix elements vanish, i.e., the terms that are related to disappear from the decay amplitudes for case II. As to the parameters and in , they are defined as

(14) |

The parameter where are the current quark masses of the meson constituents, is proportional the the chiral quark condensate.

### iii.2 The annihilation parameters

The parameters of in Eq.(5) correspond to weak annihilation contributions. Now we give their expressions, which are analogous to those in [23]:

(15) |

Here the current-current annihilation parameters arise from the hadronic matrix elements of the effective operators , the QCD penguin annihilation parameters from , and the electroweak penguin annihilation parameters from . The parameters of are closely related to the final states; they can also be divided into two different cases according to the final states. Case I is that is a vector meson, and is a pseudoscalar meson (here and are tagged in Fig. 3). Case II is that corresponds to a pseudoscalar meson, and corresponds to a vector meson. For case I, the definitions of in Eq.(15) are

(16) |

For case-II,