Research Article  Open Access
Branching Fractions and Asymmetries of and Decays in the Family Nonuniversal Model
Abstract
Within the QCD factorization framework, we investigate the branching fractions and the direct asymmetries of decays and under two different scenarios in the standard model and the family nonuniversal model. We find that the annihilation terms play crucial roles in these decays and lead to the major uncertainties. For decays , the new boson could change the branching fractions remarkably. However, for other decays, its contribution might be clouded by large uncertainties from annihilations. Unfortunately, neither the standard model nor the model can reproduce all experimental data simultaneously under one certain scenario. We also noted that the direct asymmetries of could be used to identify the meson and search for the contribution of new boson.
1. Introduction
The study of meson rare decays is a crucial tool in testing the fundamental interactions among elementary particles, exploring the origin of violation, and searching for possible new physics (NP) beyond the standard model (SM). Theoretically and experimentally, such kind of research has been conducted in great detail, especially in the weak interactions of meson. In particular, processes induced by flavorchanging neutral currents (FCNC) attract our attentions, because FCNC only occur at the loop level in SM, and they are always regarded as sensitive probes of NP. FCNC processes have been explored in the  mixing and the semileptonic weak decays, which permit a clean theoretical description. Charmless hadronic meson decays, such as , and , induced by FCNC have been studied extensively. In the past few years, the effects of NP in these decays have been studied widely. These include supersymmetry models, twoHiggs doublet models, models, fourth generation models, and extra dimension models (see review in [1] and references therein).
In the past twenty years, several novel methods have also been proposed to deal with the nonleptonic charmless meson decays, such as the naive factorization [2, 3], the QCD factorization (QCDF) [4–7], the perturbative QCD (PQCD) [8, 9], and the soft collinear effective theory [10–12]. In order to search for effect of NP in the hadronic decays, the most theoretical studies are focused on , , or . However, the studies of the decay modes involving a scalar meson are relatively few, because the underlying structure of the scalar mesons has not been well established theoretically. To describe the component of the scalar mesons, there are usually two possible scenarios (S1 and S2) according to the QCD sum rule method [13]: (i) in S1, we treat scalars above 1 GeV as the first excited states, while the scalars under 1 GeV are regarded as the lowlying states; (ii) in S2, the scalars above 1 GeV are viewed as the ground states, and light scalars are fourquark bound states or hybrid states. Under these two scenarios, many special decays have been examined within the QCDF [14, 15] and PQCD [16–25]. However, because of large uncertainties in SM, the NP effects in these decays are rarely studied.
Recently, the BaBar collaboration has reported their first branching fraction measurements for the decays that are induced by FCNC [26] and also updated their results of [27, 28] in [29]. Both experimental data and previous theoretical predictions are collected in Table 1 for comparison. For , the data are inconsistent with the PQCD predictions in most cases. Moreover, these results are somewhat much lower (or larger) than the QCDF predictions for central values but are consistent with QCDF within rather large uncertainties. For , all predicted central values deviate from the experimental data, though they can be also accommodated within very large theoretical errors. We note that in the following is denoted by in some places for convenience.

The predictions of SM cannot agree with the data well, which permits us to search for possible new physics beyond SM in these decays. Our purpose of this work is to show that a new physics effect of similar size can be obtained from some models with an extra spin1 boson, which are known to naturally exist in some wellmotivated extensions of SM [30]. Interesting phenomena arise when the couplings to physical fermion eigenstates are nondiagonal, which could be realized in the models [31–35], string models [36], and some grand unified theories [37, 38]. For example, in the superstring model advocated by Chaudhuri et al. [36], it is possible to have family nonuniversal couplings, because of different constructions of the different families. It also should be noted that, in such a model, called the family nonuniversal model, the nonuniversal couplings could lead to FCNCs at the tree level as well as introduce new weak phases, which could explain the asymmetries in the current high energy experiments. In fact, the effects of boson have been studied extensively in the low energy flavor physics phenomena, such as meson mixing and decays [39–55], single top production [56], and lepton decays [57].
In this current work, we will adopt the QCDF approach [4–7] to evaluate the relevant hadronic matrix elements of decays, since it is a systematic framework to calculate these matrix elements from QCD theory and holds in the heavy quark limit and the heavy quark symmetry. In such calculations, one requires the additional knowledge about form factors of meson to the scalar or the vector transitions. This problem, being a part of the nonperturbative sector of QCD, lacks a precise solution. To the best of our knowledge, a number of different approaches had been used to calculate the form factors of decays, such as the QCD sum rule [58–60], lightcone QCD sum rule [61, 62], PQCD approach [63], and covariant lightfront quark model (cLFQM) [64]. Among them, the form factors of the cLFQM are first calculated in the spacelike region and their momentum dependence is fitted to a 3parameter form. This parameterization is then analytically continued to the timelike region to determine the physical form factors at . Thus, we will use the results of cLFQM [64] in the following calculations.
We organize this paper as follows. In Section 2, we will reinvestigate and in SM for comparison. In Section 3, we will review the family nonuniversal model briefly and show the effect of to decay modes we are considering. In Section 4, the numerical results and discussions are given. At last, this work will be summarized in Section 5.
2. Revisiting and Decays within the QCDF Framework
2.1. Input Parameters
Because most of parameters have been discussed and presented in [14, 15], we will just list the main inputs as follows.
2.1.1. Decay Constants
To proceed, we discuss the decay constants of the scalar meson. Unlike pseudoscalar meson, each scalar meson has two decay constants, the vector decay constant and the scalar decay constant , namely, which are defined as and they are related by the equation of motion where and are the running current quark masses. Therefore, the vector decay constant is much smaller than the scalar one.
As for the vector meson, there are two kinds of decay constants, longitudinal decay constants and transverse decay constants, which are defined [65] as
2.1.2. Distribution Amplitudes
In practice, the lightcone distribution amplitudes (LCDAs) of light mesons are required, which are nonperturbative and universal. In , , and decays [4–7], the twist3 LCDAs are proven to be important and take about contribution. Thus, we here use the LCDAs up to twist3, and the discussions of higher twists can be found in [66]. The twist2 and twist3 LCDAs of scalar mesons and respect the normalization conditions and . The twist2 LCDA can be expanded in the Gegenbauer polynomials where are Gegenbauer moments and are the Gegenbauer polynomials. The decay constants and the Gegenbauer moments of the twist2 LCDA in two different scenarios have been studied explicitly in [14]. As for the explicit form of the Gegenbauer moments for the twist3 LCDAs, there exist some uncertainties theoretically [67]; thus we adopt the asymptotic form for simplicity: For the vector mesons, the normalization for the twist2 LCDA and the twist3 one is given by where the definitions of can be found in [4–7]. The general expressions of these LCDAs read as where are the Legendre polynomials. The Gegenbauer moments and have been studied using the QCD sum rule method.
2.1.3. Form Factor
Another important nonperturbative parameters in our calculation are form factors of , transitions, which are defined by [2, 3] with , . and denote the vector and axialvector currents. As stated earlier, various form factors for , transitions have been evaluated in cLFQM [64], where form factors are first calculated in the spacelike region and their momentum dependence is fitted to a 3parameter form The parameters , , and relevant for our purposes are summarized in Table 2.

2.2. Amplitudes in QCD Factorization
We will use the QCD factorization approach to study the shortdistance contribution of and decays. In SM, the effective weak Hamiltonian for transitions is given by [68] where is the Fermi coupling constant, () are the products of CabibboKobayashiMaskawa (CKM) matrix elements, are the relevant fourquark operators whose explicit forms could be found, for example, in [68], and are the corresponding Wilson coefficients.
In the QCDF framework, the contribution of the matrix elements is dominated by the form factors and the nonfactorizable impact is controlled by hard gluon exchange. And the total elements can be written as where describes contributions from naive factorization, vertex corrections, penguin contractions, and spectator scattering expressed in terms of the flavor operators , as shown in Figure 1. contains annihilation topology amplitudes characterized by the annihilation operators , as shown in Figure 2. In practice, the flavor operators are basically the Wilson coefficients in conjunction with shortdistance nonfactorizable corrections such as vertex corrections and hard spectator interactions. Combining the shortdistance nonfactorizable corrections, the effective Wilson coefficients have the expressions where , the upper (lower) signs apply when is odd (even), are the Wilson coefficients, and with as the number of colors. The quantities account for vertex corrections, for hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the meson and for penguin contractions. Weak annihilations are described by terms and . For the explicit expressions of above functions, we refer the reader to [15] for details.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
(b)
(c)
(d)
Equipped with these necessary preliminaries, the decay amplitudes could be expressed as where the ratios and are defined as The order of the arguments of the and coefficients is dictated by the subscript , where shares the same spectator quark with the meson and is the emitted meson. For the annihilation part, is referred to the one containing an antiquark from the weak vertex, and contains a quark from the weak vertex. And is the c.m. momentum of the final mesons.
In QCDF, the endpoint singularities appear in calculating the twist3 spectator and annihilation amplitudes. Since the treatment of endpoint divergences is model dependent, subleading power corrections generally can be studied only in a phenomenological way. As the most popular way, the endpoint divergent integrals are treated as signs of infrared sensitive contributions and parameterized by [4–7] with the unknown real parameters and . More discussion about them will be in Section 4.
3. The Family Nonuniversal Model
In this section, we will review the main part of the family nonuniversal model briefly. For simplicity, we only focus on the models in which the interactions between the boson and fermions are flavor nonuniversal for lefthanded couplings and flavor diagonal for righthanded cases. Of course, the analysis can be straightly extended to general cases in which the righthanded couplings are also nonuniversal across generations. The basic formulas of the model with family nonuniversal and/or nondiagonal couplings have been presented in [30, 39], to which we refer readers for detail.
In the gauge basis, the neutral current Lagrangian induced by the boson can be written as where is the gauge coupling associated with the additional group at the scale. Neglecting the renormalization group (RG) running effect between and and the mixing between and boson of SM, we present the chiral current as where the sum extends over the flavors of fermions, the chirality projection operators are , the superscript stands for the weak interaction eigenstates, and () denotes the lefthanded (righthanded) chiral coupling. and are required to be hermitian so as to arrive to a real Lagrangian. Accordingly, the mass eigenstates of the chiral fields can be defined by , and the usual CKM matrix is given by . Then, the chiral coupling matrices in the physical basis of uptype and downtype quarks are, respectively, If the matrices are not proportional to the identity, the matrices will have nonzero offdiagonal elements, which induce FCNC interactions at the tree level directly. In this work, we assume that the righthanded couplings are diagonal for simplicity. Thereby, the effective Hamiltonian of the transitions mediated by the is where and is the mass of the new gauge boson. We note that the above operators of the forms and already exist in SM, so that we represent the effect as a modification to the Wilson coefficients of the corresponding operators. Hence, we rewrite (20) as where the additional contributions to the SM Wilson coefficients at the scale in terms of parameters are given by Thus, we can have a contribution to the QCD penguins as well as the EW penguins , in the light of the results found by Buchalla et al. [68].
In order to simplify the calculations, many additional assumptions are often adopted, such as and . In this paper, we adopt the latter one, because we hope the new physics is primarily manifest in the EW penguins and violates the isospin symmetries. In the literatures, this assumption has been used widely [42, 43, 46–50, 56]. As a result, the contributions to the Wilson coefficients at the weak scale are where Because of the hermiticity of the effective Hamiltonian, the diagonal elements of the effective coupling matrix must be real. However, the offdiagonal elements, such as , generally may contain new weak phases. Moreover, the relation follows from the assumptions of universality for the first two families, as required by and decay constraints [39].
It should be emphasized that the other SM Wilson coefficients may also receive contributions from the boson through RG evolution. With our assumption that there is no significant RG running effect between and scales, the RG evolution of the modified Wilson coefficients is exactly the same as the ones in SM [68]. The numerical results of Wilson coefficients in the naive dimensional regularization (NDR) scheme at the scale ( ) are listed in Table 3 for convenience.

In summary, we list here our simplifications to a general model: we assume (i) no righthanded flavorchanging couplings ( for ), (ii) no significant RG running effect between and scales, (iii) negligible effect on the QCD penguin () so that the new physics is manifestly isospin violating. With these simplifications, we have only three parameters left in the model, , , and . So, this approach provides a minimal way to introduce the effect in the concerned decay modes. Of course, more general models are possible.
Now, the only task left is to constraint the parameters within the existing experimental data. Generally, is expected, if both the gauge groups have the same origin from some grand unified theories. We also hope so that TeV scale neutral boson could be detected at LHC. Theoretically, one can fit the left three parameters , , and new weak phase with the accurate data from factories and other experiments such as Tavatron and LHC. For example, and could be extracted from  mixing as well as decays. To accommodate the mass difference between and , is required [42–44, 48–50]. In [48–50], the authors got that the is about by fitting data of  mixing and decays. Subsequently, with and arrived and experimental data of , and , and could be extracted analogously. Specifically, the asymmetries of and polarizations of constrain , which indicates . However, we have one remark here: in dealing with the nonleptonic decays, because different groups used different factorization approaches, the fitted results are different, but all results have a same order. Note that the detailed constraint of these parameters is beyond the scope of current work and can be found in many references [46–50]. Summing up above analysis, we thereby assume that and so as to probe the new physics effect for a maximum range.
4. Numerical Results and Discussion
We will discuss the decay constants of scalars, firstly. Since it is not clear whether the scalar meson belongs to the first orbital excited states (S1) or the lowlying resonances (S2), we will calculate the processes under both scenarios. In the calculation, the decay constants and Gegenbauer moments obtained within the QCD sum rules method under different scenarios are presented as follows [14]: