Abstract

Motivated by the powerful capability of measurement for the hadron decays at LHC and SuperKEKB/Belle-II, the nonleptonic , , , , and decays are studied. With the amplitudes calculated with factorization approach and the form factors evaluated with the BSW model, branching fractions and polarization fractions are firstly presented. Numerically, the CKM-favored and decays have branching fractions ~10⁻⁸, which should be sought for with priority and firstly observed by LHC and Belle-II. The and decays are dominated by the longitudinal polarization states, while the parallel polarization fractions of decays are comparable with the longitudinal ones; numerically, + 95% and . Some comparisons between and their corresponding decays are performed, and the relation is found. With the implication of flavor symmetry, the ratios and are discussed and suggested to be verified experimentally.

1. Introduction

The physics plays an important role in testing the flavor dynamics of Standard Model (SM), exploring the source of violation, searching the indirect hints of new physics, investigating the underling mechanisms of QCD, and so forth and thus attracts much experimental and theoretical attention. With the successful performance of BABAR, Belle, CDF, and D0 in the past years, many meson decays have been well measured. Thanks to the ongoing LHCb experiment [1] at LHC and forthcoming Belle-II experiment [2] at SuperKEKB, experimental analysis of meson decays is entering a new frontier of precision. By then, besides mesons, the rare decays of some other -flavored hadrons are hopefully to be observed, which may provide much more extensive space for physics.

The excited states with quantum number of and (, , , , and are the quantum numbers of radial, orbital, spin, total angular momenta, and parity, resp.), which will be referred to as in this paper, had been observed by CLEO, Belle, LHCb, and so on [3]. However, except for their masses, there is no more experimental information due to the fact that the production of mesons is mainly through decays at colliders and the integrated luminosity is not high enough for probing the rare decays. Moreover, decays are dominated by the radiative processes , and the other decay modes are too rare to be measured easily. Fortunately, with annual integrated luminosity ~13 [2] and the cross section of production in collisions nb [4], it is expected that about samples could be produced per year at the forthcoming super-B factory SuperKEKB/Belle-II, which implies that the rare decays with branching fractions are possible to be observed. Besides, due to the much larger production cross section of collisions, experiments at LHC [5, 6] also possibly provide some experimental information for decays.

With the rapid development of experiment, accordingly, the theoretical evaluations for weak decays are urgently needed and worthful. Nonleptonic weak decays allow one to overconstrain parameters obtained from meson decay, test various models, and improve our understanding on the strong interactions and the mechanism responsible for heavy meson weak decay. The observation of an anomalous production rate of weak decays would be a hint of possible new physics beyond SM. In addition, the weak decay provides one unique opportunity of observing the weak decay of a vector meson, where polarization effects can be used as tests of the underlying structure and dynamics of hadrons. To our knowledge, few previous theoretical works come close to studying weak decays. Compared with the decays, which are suppressed dynamically by the orbital angular momentum of final states, decays are expected to have much larger branching fractions and hence are generally much easier to be measured. So, in this paper, we will estimate the observables of nonleptonic two-body weak decay to offer a ready reference.

Our paper is organized as follows. In Section 2, after a brief review of the effective Hamiltonian and factorization approach, the explicit amplitudes of decays are calculated. In Section 3, the numerical results and discussions are presented. Finally, we summarize in Section 4.

2. Theoretical Framework

Within SM, the effective Hamiltonian responsible for nonleptonic weak decay is [7]where or , is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and are Wilson coefficients, which describe the short-distance contributions and are calculated perturbatively; the explicit expressions of local four-quark operators arewhere , and are color indices, is the electric charge of the quark in the unit of , and denotes the active quark at the scale ; that is, , , , , and .

To obtain the decay amplitudes, the remaining and also the most intricate work is how to calculate hadronic matrix elements . With the factorization approach [8–11] based on the color transparency mechanism [12, 13], in principle, the hadronic matrix element could be factorized as

Due to the unnecessary complexity of hadronic matrix element and power suppression of annihilation contributions, we only consider one simple scenario where pseudoscalar meson picks up the spectator quark in meson; that is, , , and in (3) for the moment. Two current matrix elements can be further parameterized by decay constants and transition form factors:where and are the polarization vector, is the decay constant of vector meson, and are transition form factors, , and the sign convention . Even though some improved approaches, such as the QCD factorization [14, 15], the perturbative QCD scheme [16, 17], and the soft-collinear effective theory [18–21], are presented to evaluate higher order QCD corrections and reduce the renormalization scale dependence, the naive factorization (NF) approximation is a useful tool of theoretical estimation. Because there is no available experimental measurement for now, the NF approach is good enough to give a preliminary analysis, and so it is adopted in our evaluation.

With the above definitions, the hadronic matrix elements considered here can be decomposed into three scalar invariant amplitudes :where the amplitudes describe the , , and wave contributions, respectively, and are explicitly written as Alternatively, one can choose the helicity amplitudes (),with

Now, with the formulae given above and the effective coefficients defined aswe present the amplitudes of nonleptonic two-body decays as follows:(i)For decays (the spectator , , and ),(ii)For decays (the spectator and and and ),(iii)For decays,(iv)For decays,

In the rest frame of meson, the branching fraction can be written as where the momentum of final states is

The longitudinal, parallel, and perpendicular polarization fractions are defined aswhere and are parallel and perpendicular amplitudes:

3. Numerical Results and Discussion

Firstly, we would like to clarify the input parameters used in our numerical evaluations. For the CKM matrix elements, we adopt the Wolfenstein parameterization [22] and choose the four parameters , , , and as [23]with and .

The decay constants of light vector mesons are [24]For the decay constants of mesons, we will take [25]which agree well with the results of the other QCD sum rules [26, 27] and lattice QCD with [28].

Besides the decay constants, the transition form factors are also essential inputs to estimate branching ratios for nonleptonic decay. In this paper, the Bauer-Stech-Wirbel (BSW) model [10] is employed to evaluate the form factors , , and , which could be written as the overlap integrals of wave functions of mesons [10]:where is the transverse quark momentum, are the Pauli matrix acting on the spin indices of the decaying quark, and represents the mass of nonspectator quark of pseudoscalar meson. With the meson wave function as solution of a relativistic scalar harmonic oscillator potential [10] and = 0.4 GeV which determines the average transverse quark momentum through , we get the numerical results of the transition form factors summarized in Table 1. In our following evaluation, these numbers and of them are used as default inputs and uncertainties, respectively.

To evaluate the branching fractions, the total decay widths (or lifetimes) are necessary. However, there is no available experimental or theoretical information for until now. Because of the fact that the QED radiative processes dominate the decays of mesons, we will take the approximation . The theoretical predictions on have been widely evaluated in various theoretical models, such as relativistic quark model [29, 30], QCD sum rules [31], light cone QCD sum rules [32], light front quark model [33], heavy quark effective theory with vector meson dominance hypothesis [34], or covariant model [35]. In this paper, the most recent results [33, 35]which agree with the other theoretical results, are approximately treated as in our numerical estimate.

With the aforementioned values of input parameters and the theoretical formula, we present theoretical predictions for the observables of , , , , and decays, in which only the (color-suppressed) tree induced decay modes are evaluated due to the fact that the branching fractions of loop induced decays are very small and hard to be measured soon. Our numerical results for the branching fractions and the polarization fractions are summarized in Tables 2 and 3. In Table 2, the first, second, and third theoretical errors are caused by uncertainties of the CKM parameters, hadronic parameters (decay constants and form factors), and total decay widths, respectively. From Tables 2 and 3, the following could be found:(1)The hierarchy of branching fractions is clear. (i) The branching fractions of and decays are much smaller than the ones of , , and decays, which is caused by the fact that the form factors of transition are much larger than those of and transitions. (ii) For , , and decays, the hierarchy is induced by two factors: one is the CKM factor (see the third column of Table 2), and the other is (see (24), (25), and (26)).(2)Besides small form factors, the , decays are either color suppressed or the CKM factors suppressed. So they have very small branching fractions (see Table 2) and are hardly measured soon. Most of the CKM-favored and tree-dominated , , and decays, enhanced by the relatively large transition form factors, have large branching fractions, , and thus could be measured in the near future. In particular, branching ratios for and decays can reach up to and hence should be sought for with priority and firstly observed at the high statistics LHC and Belle-II experiments. The numerical results and above analyses are based on the NF, in which the QCD corrections are not included. Fortunately, for the color-allowed tree amplitude , the NF estimate is stable due to the relatively small QCD corrections [15]. For instance, in and decays, the results [14] and [15] indicate clearly that the correction is only about and thus trivial numerically. For the color-suppressed decay modes listed in Table 2, even though the NF estimates would suffer significant correction (about in decays, e.g., [36]), they still escape the experimental scope due to their small branching factions and thus will not be discussed further. In the following analyses, we will pay our attention only to the color allowed tree-dominated , , and decays.(3)For the and decays, the flavor symmetry implies the relations Further considering the theoretical prediction (see (24) and (25)) and assumption , one may find the ratios which are satisfied in our numerical evaluations. Experimentally, the first relation (28) is hopefully to be tested soon due to the large branching fractions. For the other potentially detectable , , and decay modes, with branching fractions , the U-spin symmetry implies relations As similar to , one also could get the ratio and relation which is also satisfied in our numerical evaluation. So, it is obvious that such ratios, and , are useful for probing and , respectively, and further testing the theoretical predictions of and in various models, such as the results in [29–35].(4)Besides branching fraction, the polarization fractions are also important observables. For the potentially detectable decay modes with branching fractions , our numerical results of are summarized in Table 3. For the helicity amplitudes , the formal hierarchy pattern is naively expected. Hence, decays are generally dominated by the longitudinal polarization state and satisfy [37]. For decays, in the heavy quark limit, the helicity amplitudes given by (9) could be simplified as The transversity amplitudes could be gotten easily through (19). Obviously, due to the helicity suppression factor , the relation of (32) is roughly fulfilled. As a result, the longitudinal polarization fractions of and decays are very large (see Table 3 for numerical results). It should be noted that the above analyses and (33) are based on the case of and thus possibly no longer satisfied by decays because of the unnegligible vector mass . In fact, for the decays, (9) are simplified as in which, due to , the approximation is used. Because the so-called helicity suppression factor is not small, which is different from the case of decays, it could be easily found that the relation of (32) does not follow. Further considering that are dominated by the term of in (35) due to its large coefficient, the relation could be easily gotten. Above analyses and findings are confirmed by our numerical results in Table 3, which will be tested by future experiments.(5) As known, there are many interesting phenomena in meson decays, so it is worthy to explore the possible correlation between and decays. Taking and decays as example, we find that the expressions of their helicity amplitudes (the former one has been given by (33)) are similar to each other except for the replacements and everywhere in (33). As a result, our analyses in item (4) are roughly suitable for decay, and the relation is generally expected. Interestingly, our prediction is consistent with the result [38], which is in a good agreement with the experimental data [39]. The relation equation (36) follows. In addition, the similar correlation as (36) also exists in the other and corresponding decays.