Branching Fractions and 𝐶𝑃 Asymmetries of 𝐵→𝐾^∗₀(1430)𝜌(𝜔) and 𝐵→𝐾^∗₀(1430)𝜙 Decays in the Family Nonuniversal 𝑍^′ Model

Abstract

Within the QCD factorization framework, we investigate the branching fractions and the direct 𝐶𝑃 asymmetries of decays 𝐵→𝐾^∗₀(1430)𝜌(𝜔) and 𝐵→𝐾^∗₀(1430)𝜙 under two different scenarios in the standard model and the family nonuniversal 𝑍^′ model. We find that the annihilation terms play crucial roles in these decays and lead to the major uncertainties. For decays 𝐵⁻→𝐾₀^∗−(1430)𝜌⁰(𝜔), the new 𝑍^′ 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 𝑍^′ model can reproduce all experimental data simultaneously under one certain scenario. We also noted that the direct 𝐶𝑃 asymmetries of 𝐵⁻→𝐾₀^∗−(1430)𝜌⁰(𝜔) could be used to identify the 𝐾^∗₀(1430) meson and search for the contribution of new 𝑍^′ boson.

1. Introduction

The study of 𝐵 meson rare decays is a crucial tool in testing the fundamental interactions among elementary particles, exploring the origin of 𝐶𝑃 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 𝐵 meson. In particular, processes induced by flavor-changing 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 𝐵⁰_𝑞-𝐵⁰_𝑞 mixing and the semileptonic weak decays, which permit a clean theoretical description. Charmless hadronic 𝐵 meson decays, such as 𝐵→𝐾𝜋,𝐾^(∗)𝜙, and 𝐾^(∗)𝜂^(′), 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, two-Higgs doublet models, 𝑍^′ 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 𝐵 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 𝐵 decays, the most theoretical studies are focused on 𝐵→𝑃𝑃, 𝑃𝑉, or 𝑉𝑉. 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 low-lying states; (ii) in S2, the scalars above 1 GeV are viewed as the ground states, and light scalars are four-quark 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 𝐵→𝐾^∗₀(1430)𝜌 that are induced by FCNC [26] and also updated their results of 𝐵→𝐾^∗₀(1430)𝜙 [27, 28] in [29]. Both experimental data and previous theoretical predictions are collected in Table 1 for comparison. For 𝐵→𝐾^∗₀(1430)𝜌, 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 𝐵→𝐾^∗₀(1430)𝜙, 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 𝐾^∗₀(1430) is denoted by 𝐾^∗₀ in some places for convenience.

Table 1 

Branching ratios (in units of 10⁻⁶) of 𝐵→𝐾^∗₀(1430)𝜌 and 𝐵→𝐾^∗₀(1430)𝜙 in QCDF [15], PQCD [19, 23], and experimental data [26, 29].

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 spin-1 𝑍^′ boson, which are known to naturally exist in some well-motivated extensions of SM [30]. Interesting phenomena arise when the 𝑍^′ couplings to physical fermion eigenstates are nondiagonal, which could be realized in the 𝐸₆ 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 𝑍^′ couplings, because of different constructions of the different families. It also should be noted that, in such a model, called the family nonuniversal 𝑍^′ model, the nonuniversal couplings could lead to FCNCs at the tree level as well as introduce new weak phases, which could explain the 𝐶𝑃 asymmetries in the current high energy experiments. In fact, the effects of 𝑍^′ boson have been studied extensively in the low energy flavor physics phenomena, such as 𝐵 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 𝐵 decays, since it is a systematic framework to calculate these matrix elements from QCD theory and holds in the heavy quark limit 𝑚_𝑏→∞ and the heavy quark symmetry. In such calculations, one requires the additional knowledge about form factors of 𝐵 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 𝐵→𝑆 decays, such as the QCD sum rule [58–60], light-cone QCD sum rule [61, 62], PQCD approach [63], and covariant light-front quark model (cLFQM) [64]. Among them, the form factors of the cLFQM are first calculated in the space-like region and their momentum dependence is fitted to a 3-parameter form. This parameterization is then analytically continued to the time-like region to determine the physical form factors at 𝑞²≥0. 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 𝐵→𝐾^∗₀𝜌 and 𝐾^∗₀𝜙 in SM for comparison. In Section 3, we will review the family nonuniversal 𝑍^′ model briefly and show the effect of 𝑍^′ 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 𝐵→𝐾^∗₀(1430)𝜌 and 𝐵→𝐾^∗₀(1430)𝜙 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 𝑓_𝑆 and the scalar decay constant 𝑓_𝑆, namely, which are defined as 𝑆𝑝𝑞₂𝛾_𝜇𝑞₁0=𝑓_𝑆𝑝_𝜇,𝑆𝑝𝑞₂𝑞₁0=𝑚_𝑆𝑓_𝑆, (1) and they are related by the equation of motion 𝑓_𝑆=𝑚₂𝜇−𝑚₁𝜇𝑚_𝑆𝑓_𝑆, (2) where 𝑚₂ and 𝑚₁ 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 𝑉𝑝𝑞₂𝛾_𝜇𝑞₁0=𝑓_𝑉𝑚_𝑉𝜀^∗_𝜇,𝑉𝑝,𝜀^∗𝑞𝜎_𝜇𝜈𝑞^′0=𝑓^⟂_𝑉𝑝_𝜇𝜀^∗_𝜈−𝑝_𝜈𝜀^∗_𝜇. (3)

2.1.2. Distribution Amplitudes

In practice, the light-cone distribution amplitudes (LCDAs) of light mesons are required, which are nonperturbative and universal. In 𝐵→𝑃𝑃, 𝑃𝑉, and 𝑉𝑉 decays [4–7], the twist-3 LCDAs are proven to be important and take about 30% contribution. Thus, we here use the LCDAs up to twist-3, and the discussions of higher twists can be found in [66]. The twist-2 𝜙_𝑆(𝑥) and twist-3 LCDAs of scalar mesons 𝜙^𝑠_𝑆(𝑥) and 𝜙^𝜎_𝑆(𝑥) respect the normalization conditions ¹₀𝑑𝑥𝜙_𝑆(𝑥)=𝑓_𝑆2√6₁₀,𝑑𝑥𝜙^𝑠_𝑆(𝑥)=¹₀𝑑𝑥𝜙^𝜎_𝑆(𝑥)=𝑓_𝑆2√6, (4) and 𝜙^𝑇_𝑆(𝑥)=(1/6)(𝑑/𝑑𝑥)𝜙^𝜎_𝑆(𝑥). The twist-2 LCDA can be expanded in the Gegenbauer polynomials 𝜙_𝑆=1𝑥,𝜇√6𝑓_𝑆𝜇()×6𝑥1−𝑥_∞𝑚=1𝐵_𝑚𝜇𝐶_𝑚^3/2(2𝑥−1), (5) where 𝐵_𝑚 are Gegenbauer moments and 𝐶_𝑚^3/2 are the Gegenbauer polynomials. The decay constants and the Gegenbauer moments of the twist-2 LCDA in two different scenarios have been studied explicitly in [14]. As for the explicit form of the Gegenbauer moments for the twist-3 LCDAs, there exist some uncertainties theoretically [67]; thus we adopt the asymptotic form for simplicity: 𝜙^𝑠_𝑆=1√6𝑓_𝑆,𝜙^𝑇_𝑆=1√6𝑓_𝑆(1−2𝑥). (6) For the vector mesons, the normalization for the twist-2 LCDA Φ_𝑉 and the twist-3 one Φ_𝑣 is given by ¹₀𝑑𝑥Φ_𝑉(𝑥)=𝑓_𝑉,¹₀𝑑𝑥Φ_𝑣(𝑥)=0, (7) where the definitions of Φ_𝑣(𝑥) can be found in [4–7]. The general expressions of these LCDAs read asΦ_𝑉()𝑓𝑥,𝜇=6𝑥1−𝑥_𝑉1+^∞_𝑛=1𝛼^𝑉_𝑛𝜇𝐶_𝑛^3/2(,Φ2𝑥−1)_𝑣𝑥,𝜇=3𝑓^⟂_𝑉2𝑥−1+^∞_𝑛=1𝛼^𝑉_𝑛,⟂𝜇𝑃_𝑛+1(2𝑥−1), (8) where 𝑃_𝑛(𝑥) are the Legendre polynomials. The Gegenbauer moments 𝛼^𝑉_𝑛 and 𝛼^𝑉_𝑛,⟂ 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 𝐵→𝑆, 𝑉 transitions, which are defined by [2, 3] 𝑉𝑝^′𝑉_𝜇𝐵𝑝1=−𝑚_𝐵+𝑚_𝑉×𝜀_{𝜇𝜈𝛼𝛽}𝜀^∗𝜈𝑃^𝛼𝑞^𝛽𝑉^𝐵𝑉𝑞²,𝑉𝑝^′𝐴_𝜇𝐵𝑝=𝑖𝑚_𝐵+𝑚_𝑉𝜀^∗_𝜇𝐴₁^𝐵𝑉𝑞²−𝜀^∗⋅𝑃𝑚_𝐵+𝑚_𝑉𝑃_𝜇𝐴₂^𝐵𝑉𝑞²−2𝑚_𝑉𝜀^∗⋅𝑃𝑞²𝑞_𝜇𝐴₃^𝐵𝑉𝑞²−𝐴₀^𝐵𝑉𝑞²,𝑆𝑝^′𝐴_𝜇𝐵𝑝=−𝑖𝑃_𝜇−𝑚²_𝐵−𝑚²_𝑆𝑞²𝑞_𝜇𝐹₁^𝐵𝑆𝑞²+𝑚²_𝐵−𝑚²_𝑆𝑞²𝑞_𝜇𝐹₀^𝐵𝑆𝑞², (9) with 𝑃_𝜇=(𝑝+𝑝^′)_𝜇, 𝑞_𝜇=(𝑝−𝑝^′)_𝜇. 𝑉 and 𝐴 denote the vector and axial-vector currents. As stated earlier, various form factors for 𝐵→𝑆, 𝑉 transitions have been evaluated in cLFQM [64], where form factors are first calculated in the space-like region and their momentum dependence is fitted to a 3-parameter form 𝐹𝑞²=𝐹(0)1−𝑎𝑞²/𝑚²_𝐵+𝑏𝑞²/𝑚²_𝐵². (10) The parameters 𝑎, 𝑏, and 𝐹(0) relevant for our purposes are summarized in Table 2.

Table 2 

Form factors of 𝐵→𝜌,𝐾^∗₀(1430) transitions obtained in the covariant light-front model [64].

2.2. Amplitudes in QCD Factorization

We will use the QCD factorization approach to study the short-distance contribution of 𝐵→𝐾^∗₀𝜌 and 𝐵→𝐾^∗₀𝜙 decays. In SM, the effective weak Hamiltonian for 𝑏→𝑠 transitions is given by [68] ℋ_eff=𝐺_𝐹√2𝑉_𝑢𝑏𝑉^∗_𝑢𝑠𝐶₁𝑂^𝑢₁+𝐶₂𝑂^𝑢₂+𝑉_𝑐𝑏𝑉^∗_𝑐𝑠𝐶₁𝑂^𝑐₁+𝐶₂𝑂^𝑐₂−𝑉_𝑡𝑏𝑉^∗_𝑡𝑠×¹⁰_𝑖=3𝐶_𝑖𝑂_𝑖+𝐶_7𝛾𝑂_7𝛾+𝐶_8𝑔𝑂_8𝑔+h.c, (11) where 𝐺_𝐹 is the Fermi coupling constant, 𝑉_𝑞𝑏𝑉^∗_𝑞𝑠 (𝑞=𝑢,𝑐,𝑡) are the products of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, 𝑂_𝑖 are the relevant four-quark operators whose explicit forms could be found, for example, in [68], and 𝐶_𝑖 are the corresponding Wilson coefficients.

In the QCDF framework, the contribution of the matrix elements ⟨𝑀₁𝑀₂|𝑂_𝑖|𝐵⟩ is dominated by the form factors and the nonfactorizable impact is controlled by hard gluon exchange. And the total elements can be written as 𝑀₁𝑀₂ℋ_eff𝐵=𝐺_𝐹√2_{𝑝=𝑢,𝑐}𝑉_𝑝𝑏𝑉^∗_𝑝𝑠𝑀₁𝑀₂𝑇^𝑝_𝐴+𝑇^𝑝_𝐵𝐵, (12) where 𝑇^𝑝_𝐴 describes contributions from naive factorization, vertex corrections, penguin contractions, and spectator scattering expressed in terms of the flavor operators 𝑎^𝑝_𝑖, as shown in Figure 1. 𝑇^𝑝_𝐵 contains annihilation topology amplitudes characterized by the annihilation operators 𝑏^𝑝_𝑗, as shown in Figure 2. In practice, the flavor operators 𝑎^𝑝_𝑖 are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. Combining the short-distance nonfactorizable corrections, the effective Wilson coefficients 𝑎^𝑝_𝑖 have the expressions 𝑎^𝑝_𝑖𝑀₁𝑀₂=𝐶_𝑖+𝐶_𝑖±1𝑁_𝑐𝑁_𝑖𝑀₂+𝐶_𝑖±1𝑁_𝑐𝐶_𝐹𝛼_𝑠×𝑉4𝜋_𝑖𝑀₂+4𝜋²𝑁_𝑐𝐻_𝑖𝑀₁𝑀₂+𝑃^𝑝_𝑖𝑀₂, (13) where 𝑖=1,…,10, the upper (lower) signs apply when 𝑖 is odd (even), 𝐶_𝑖 are the Wilson coefficients, and 𝐶_𝐹=(𝑁²_𝑐−1)/(2𝑁_𝑐) with 𝑁_𝑐 as the number of colors. The quantities 𝑉_𝑖(𝑀₂) account for vertex corrections, 𝐻_𝑖(𝑀₁𝑀₂) for hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the 𝐵 meson and 𝑃_𝑖(𝑀₂) for penguin contractions. Weak annihilations are described by terms 𝑏_𝑖 and 𝑏_𝑖,EW. For the explicit expressions of above functions, we refer the reader to [15] for details.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1 

Order of 𝛼𝑠 corrections to the hard scattering kernels. The upward quark lines represent the emission meson from the b quark decay vertex. These diagrams are commonly called vertex corrections, penguin corrections, and hard spectator scattering diagrams for (a)–(d), (e) and (f), and (g) and (h), respectively.

(a)

(b)

(c)

(d)

Figure 2 

Order of 𝛼𝑠 corrections to the weak annihilations.

Equipped with these necessary preliminaries, the decay amplitudes could be expressed as 𝐴𝐵⁻⟶𝐾₀^∗−𝜙𝐺=𝑖^𝐹√2_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)𝑎₃+𝑎^𝑝₄+𝑎₅−𝑟^𝜙_𝜒𝑎^𝑝₆−12𝑎^𝑝₈−12𝑎₇+𝑎₉+𝑎^𝑝_{10𝐾∗0𝜙}×2𝑓_𝜙𝐹^𝐵𝐾∗0₁𝑚²_𝜙𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜙𝑓_𝐾∗0𝑏₂𝛿^𝑝_𝑢+𝑏₃+𝑏_{3,EW𝐾∗0𝜙},𝐴𝐵⁰⟶𝐾₀^∗0𝜙𝐺=𝑖^𝐹√2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)𝑎₃+𝑎^𝑝₄+𝑎₅−𝑟^𝜙_𝜒𝑎^𝑝₆−12𝑎^𝑝₈−12𝑎₇+𝑎₉+𝑎^𝑝_{10𝐾∗0𝜙}×2𝑓_𝜙𝐹^𝐵𝐾∗0₁𝑚²_𝜙𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜙𝑓_𝐾∗0𝑏₃−12𝑏_{3,EW𝐾∗0𝜙},𝐴𝐵⁻⟶𝐾₀^∗0𝜌⁻𝐺=𝑖^𝐹√2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)−𝑎^𝑝₄+𝑟^𝐾∗0_𝜒𝑎^𝑝₆−12𝑎^𝑝₈−12𝑎^𝑝_{10𝜌𝐾∗0}×2𝑓_𝐾∗0𝐴₀^𝐵𝜌𝑚²_𝐾^∗₀𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜌𝑓_𝐾∗0𝑏₂𝛿^𝑝_𝑢+𝑏₃+𝑏_{3,EW𝜌𝐾∗0},𝐴𝐵⁻⟶𝐾₀^∗−𝜌⁰𝐺=𝑖^𝐹2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)−𝑎₁𝛿^𝑝_𝑢+𝑎^𝑝₄+𝑟^𝐾∗0_𝜒(𝑎^𝑝₆+𝑎^𝑝₈)+𝑎^𝑝_{10𝜌𝐾∗0}×2𝑓_𝐾∗0𝐴₀^𝐵𝜌𝑚²_𝐾^∗₀𝑚_𝐵𝑝_𝑐+𝑎₂𝛿^𝑝_𝑢+32𝑎₉+𝑎_{7𝐾∗0𝜌}×2𝑓_𝜌𝐹^𝐵𝐾∗0₁𝑚²_𝜌𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜌𝑓_𝐾∗0𝑏₂𝛿^𝑝_𝑢+𝑏₃+𝑏_{3,EW𝜌𝐾∗0},𝐴𝐵⁰⟶𝐾₀^∗−𝜌⁺𝐺=𝑖^𝐹√2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)−𝑎₁𝛿^𝑝_𝑢+𝑎^𝑝₄+𝑟^𝐾∗0_𝜒𝑎^𝑝₆+𝑎^𝑝₁₀+𝑟^𝐾∗0_𝜒𝑎^𝑝_{8𝜌𝐾∗0}×2𝑓_𝐾∗0𝐴₀^𝐵𝜌𝑚²_𝐾^∗₀𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜌𝑓_𝐾∗0𝑏₃−12𝑏_{3,EW𝜌𝐾∗0},𝐴𝐵⁰⟶𝐾₀^∗0𝜌⁰𝐺=𝑖^𝐹2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)−−𝑎^𝑝₄−𝑟^𝐾∗0_𝜒𝑎^𝑝₆−12𝑎^𝑝₈+12𝑎^𝑝_{10𝜌𝐾∗0}×2𝑓_𝐾∗0𝐴₀^𝐵𝜌𝑚²_𝐾^∗₀𝑚_𝐵𝑝_𝑐+𝑎₂𝛿^𝑝_𝑢+32𝑎₉+𝑎_{7𝐾∗0𝜌}×2𝑓_𝜌𝐹^𝐵𝐾∗0₁𝑚²_𝜌𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜌𝑓_𝐾∗0−𝑏₃+12𝑏_{3,EW𝜌𝐾∗0},𝐴𝐵⁻⟶𝐾₀^∗−𝜔𝐺=𝑖^𝐹2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)𝑎₂𝛿^𝑝_𝑢+2𝑎₃+𝑎₅+12𝑎₉+𝑎_{7𝐾∗0𝜔}×2𝑓_𝜔𝐹^𝐵𝐾∗0₁𝑚²_𝜔𝑚_𝐵𝑝_𝑐−𝑎₁𝛿^𝑝_𝑢+𝑎^𝑝₄+𝑟^𝐾∗0_𝜒𝑎^𝑝₆+𝑎^𝑝₈+𝑎^𝑝_{10𝜔𝐾∗0}×2𝑓_𝐾∗0𝐴₀^𝐵𝜔𝑚²_𝐾^∗₀𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜔𝑓_𝐾∗0𝑏₂𝛿^𝑝_𝑢+𝑏₃+𝑏_{3,EW𝜔𝐾∗0},𝐴𝐵⁰⟶𝐾₀^∗0𝜔𝐺=𝑖^𝐹2×_{𝑝=𝑢,𝑐}𝜆_𝑝^(𝑠)𝑎₂𝛿^𝑝_𝑢+2𝑎₃+𝑎₅+12𝑎₉+𝑎_{7𝐾∗0𝜔}×2𝑓_𝜔𝐹^𝐵𝐾∗0₁𝑚²_𝜔𝑚_𝐵𝑝_𝑐−𝑎^𝑝₄+𝑟^𝐾∗0_𝜒𝑎^𝑝₆−12𝑎^𝑝₈−12𝑎^𝑝_{10𝜔𝐾∗0}×2𝑓_𝐾∗0𝐴₀^𝐵𝜌𝑚²_𝐾^∗₀𝑚_𝐵𝑝_𝑐−𝑓_𝐵𝑓_𝜔𝑓_𝐾∗0𝑏₃−12𝑏_{3,EW𝜔𝐾∗0}, (14) where the ratios 𝑟^𝑉_𝜒 and 𝑟^𝑆_𝜒 are defined as 𝑟^𝑉_𝜒𝜇=2𝑚_𝑉𝑚_𝑏𝜇𝑓^⟂_𝑉𝜇𝑓_𝑉,𝑟^𝑆_𝜒𝜇=2𝑚²_𝑆𝑚_𝑏𝜇𝑚₂𝜇−𝑚₁𝜇. (15) The order of the arguments of the 𝑎^𝑝_𝑖(𝑀₁𝑀₂) and 𝑏_𝑖(𝑀₁𝑀₂) coefficients is dictated by the subscript 𝑀₁𝑀₂, where 𝑀₁ shares the same spectator quark with the 𝐵 meson and 𝑀₂ is the emitted meson. For the annihilation part, 𝑀₁ is referred to the one containing an antiquark from the weak vertex, and 𝑀₂ contains a quark from the weak vertex. And 𝑝_𝑐 is the c.m. momentum of the final mesons.

In QCDF, the endpoint singularities appear in calculating the twist-3 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] ¹₀𝑑𝑦𝑦⟶𝑋_𝐴=1+𝜌_𝐴𝑒^𝑖𝜙𝐴𝑚ln^𝐵Λ_ℎ, (16) with the unknown real parameters 𝜌_𝐴 and 𝜙_𝐴. More discussion about them will be in Section 4.

3. The Family Nonuniversal 𝑍^′ Model

In this section, we will review the main part of the family nonuniversal 𝑍^′ model briefly. For simplicity, we only focus on the models in which the interactions between the 𝑍^′ boson and fermions are flavor nonuniversal for left-handed couplings and flavor diagonal for right-handed cases. Of course, the analysis can be straightly extended to general cases in which the right-handed couplings are also nonuniversal across generations. The basic formulas of the 𝑍^′ 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 𝑍^′ boson can be written as ℒ^𝑍′=−𝑔₂𝐽^′_𝜇𝑍^′𝜇, (17) where 𝑔₂ is the gauge coupling associated with the additional 𝑈^′(1) group at the 𝑀_𝑊 scale. Neglecting the renormalization group (RG) running effect between 𝑀_𝑊 and 𝑀_𝑍′ and the mixing between 𝑍^′ and 𝑍 boson of SM, we present the chiral current as 𝐽^′_𝜇=_𝑖,𝑗𝜓^𝐼_𝑖𝛾_𝜇𝜀_{𝜓𝐿𝑖𝑗}𝑃_𝐿+𝜀_{𝜓𝑅𝑖𝑗}𝑃_𝑅𝜓^𝐼_𝑗, (18) where the sum extends over the flavors of fermions, the chirality projection operators are 𝑃_𝐿,𝑅≡(1∓𝛾₅)/2, the superscript 𝐼 stands for the weak interaction eigenstates, and 𝜀_𝜓𝐿(𝜀_𝜓𝑅) denotes the left-handed (right-handed) chiral coupling. 𝜀_𝜓𝐿 and 𝜀_𝜓𝑅 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 𝜓_𝐿,𝑅=𝑉_{𝜓𝐿,𝑅}𝜓^𝐼_𝐿,𝑅, and the usual CKM matrix is given by 𝑉_CKM=𝑉_𝑢𝐿𝑉^†_𝑑𝐿. Then, the chiral 𝑍^′ coupling matrices in the physical basis of up-type and down-type quarks are, respectively, 𝐵^𝑋_𝑢≡𝑉_𝑢𝑋𝜀_𝑢𝑋𝑉^†_𝑢𝑋,𝐵^𝑋_𝑑≡𝑉_𝑑𝑋𝜀_𝑑𝑋𝑉^†_𝑑𝑋(𝑋=𝐿,𝑅). (19) If the 𝜀 matrices are not proportional to the identity, the 𝐵 matrices will have nonzero off-diagonal elements, which induce FCNC interactions at the tree level directly. In this work, we assume that the right-handed couplings are diagonal for simplicity. Thereby, the effective Hamiltonian of the 𝑏→𝑠𝑞𝑞(𝑞=𝑢,𝑑) transitions mediated by the 𝑍^′ is ℋ^𝑍′_eff=2𝐺_𝐹√2𝑔₂𝑀_𝑍𝑔₁𝑀_𝑍′²𝐵^𝐿∗_𝑠𝑏𝑏𝑠_𝑉−𝐴×_𝑞𝐵^𝐿_𝑞𝑞𝑞𝑞_𝑉−𝐴+𝐵^𝑅_𝑞𝑞𝑞𝑞_𝑉+𝐴+h.c., (20) where 𝑔₁=𝑒/(sin𝜃_𝑊cos𝜃_𝑊) and 𝑀_𝑍′ is the mass of the new gauge boson. We note that the above operators of the forms (𝑏𝑠)_𝑉−𝐴(𝑞𝑞)_𝑉−𝐴 and (𝑏𝑠)_𝑉−𝐴(𝑞𝑞)_𝑉+𝐴 already exist in SM, so that we represent the 𝑍^′ effect as a modification to the Wilson coefficients of the corresponding operators. Hence, we rewrite (20) as ℋ^𝑍′_eff𝐺=−^𝐹√2𝑉^∗_𝑡𝑏𝑉_𝑡𝑠×_𝑞Δ𝐶₃𝑂₃^(𝑞)+Δ𝐶₅𝑂₅^(𝑞)+Δ𝐶₇𝑂₇^(𝑞)+Δ𝐶₉𝑂₉^(𝑞)+h.c., (21) where the additional contributions to the SM Wilson coefficients at the 𝑀_𝑊 scale in terms of 𝑍^′ parameters are given by Δ𝐶₃₍₅₎2=−3𝑉^∗_𝑡𝑏𝑉_𝑡𝑠𝑔₂𝑀_𝑍𝑔₁𝑀_𝑍′²𝐵^𝐿∗_𝑠𝑏𝐵^𝐿(𝑅)_𝑢𝑢+2𝐵^𝐿(𝑅)_𝑑𝑑,Δ𝐶₉₍₇₎4=−3𝑉^∗_𝑡𝑏𝑉_𝑡𝑠𝑔₂𝑀_𝑍𝑔₁𝑀_𝑍′²𝐵^𝐿∗_𝑠𝑏𝐵^𝐿(𝑅)_𝑢𝑢−𝐵^𝐿(𝑅)_𝑑𝑑. (22) Thus, we can have a 𝑍^′ contribution to the QCD penguins Δ𝐶₃₍₅₎ as well as the EW penguins Δ𝐶₉₍₇₎, 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 𝐵^𝐿,𝑅_𝑢𝑢=𝐵^𝐿,𝑅_𝑑𝑑 and 𝐵^𝐿,𝑅_𝑢𝑢=−2𝐵^𝐿,𝑅_𝑑𝑑. 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 𝑍^′ contributions to the Wilson coefficients at the weak scale are Δ𝐶₃₍₅₎=0,Δ𝐶₉₍₇₎𝑉=4^∗𝑡𝑏𝑉_𝑡𝑠𝑉^∗_𝑡𝑏𝑉_𝑡𝑠𝜉^{𝐿𝐿(𝑅)}𝑒^{−𝑖𝜙𝐿}, (23) where 𝜉^𝐿𝑋≡𝑔₂𝑀_𝑍𝑔₁𝑀_𝑍′²𝐵^𝐿∗_𝑠𝑏𝐵^𝑋_𝑑𝑑𝑉^∗_𝑡𝑏𝑉_𝑡𝑠),𝜙(𝑋=𝐿,𝑅_𝐿≡Arg𝐵^𝐿_𝑠𝑏. (24) Because of the hermiticity of the effective Hamiltonian, the diagonal elements of the effective coupling matrix must be real. However, the off-diagonal elements, such as 𝐵^𝐿_𝑠𝑏, generally may contain new weak phases. Moreover, the relation 𝐵^𝐿(𝑅)_𝑠𝑠≃𝐵^𝐿(𝑅)_𝑑𝑑 follows from the assumptions of universality for the first two families, as required by 𝐾 and 𝜇 decay constraints [39].

It should be emphasized that the other SM Wilson coefficients may also receive contributions from the 𝑍^′ boson through RG evolution. With our assumption that there is no significant RG running effect between 𝑀^′_𝑍 and 𝑀_𝑊 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 𝜇=2.1 GeV (𝜇_ℎ=1 GeV) are listed in Table 3 for convenience.

Table 3 

The Wilson coefficients 𝐶_𝑖 within SM and with the contribution from 𝑍^′ boson included in NDR scheme at the scale 𝜇=2.1 GeV and 𝜇_ℎ=1.0 GeV.

In summary, we list here our simplifications to a general 𝑍^′ model: we assume (i) no right-handed flavor-changing couplings (𝐵^𝑅_𝑖𝑗=0 for 𝑖≠𝑗), (ii) no significant RG running effect between 𝑀_𝑍′ and 𝑀_𝑊 scales, (iii) negligible 𝑍^′ effect on the QCD penguin (Δ𝐶_3,5=0) so that the new physics is manifestly isospin violating. With these simplifications, we have only three parameters left in the model, |𝜉^𝐿_𝑞𝑞|, |𝜉^𝑅_𝑞𝑞|, and 𝜙_𝐿. So, this approach provides a minimal way to introduce the 𝑍^′ effect in the concerned decay modes. Of course, more general 𝑍^′ models are possible.

Now, the only task left is to constraint the parameters within the existing experimental data. Generally, 𝑔₂/𝑔₁∼1 is expected, if both the 𝑈(1) gauge groups have the same origin from some grand unified theories. We also hope 𝑀_𝑍/𝑀_𝑍′∼0.1 so that TeV scale neutral 𝑍^′ boson could be detected at LHC. Theoretically, one can fit the left three parameters |𝐵^𝐿_𝑠𝑏|, |𝐵^𝑋_𝑑𝑑|, and new weak phase 𝜙_𝐿 with the accurate data from 𝐵 factories and other experiments such as Tavatron and LHC. For example, 𝐵^𝐿_𝑠𝑏 and 𝜙_𝐿 could be extracted from 𝐵_𝑠-𝐵_𝑠 mixing as well as 𝐵→𝐾^(∗)ℓ⁺ℓ⁻ decays. To accommodate the mass difference between 𝐵_𝑠 and 𝐵_𝑠, |𝐵^𝐿_𝑠𝑏|∼|𝑉_𝑡𝑏𝑉^∗_𝑡𝑠| is required [42–44, 48–50]. In [48–50], the authors got that the 𝜙_𝐿 is about −80^∘ by fitting data of 𝐵_𝑠-𝐵_𝑠 mixing and 𝐵→𝐾^(∗)𝑙⁺𝑙⁻ decays. Subsequently, with 𝐵^𝐿_𝑠𝑏 and 𝜙_𝐿 arrived and experimental data of 𝐵→𝜋𝜋,𝐾𝜋,𝐾𝜌, and 𝐾^(∗)𝜙, 𝐵^𝐿_𝑞𝑞 and 𝐵^𝑅_𝑞𝑞 could be extracted analogously. Specifically, the 𝐶𝑃 asymmetries of 𝐵→𝐾𝜋 and polarizations of 𝐵→𝐾^∗𝜙 constrain |𝐵^𝐿_𝑠𝑏𝐵^𝐿,𝑅_𝑠𝑠|∼|𝑉_𝑡𝑏𝑉^∗_𝑡𝑠|, which indicates |𝐵^𝐿,𝑅_𝑞𝑞|∼1. However, we have one remark here: in dealing with the nonleptonic 𝐵 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 𝜉^{𝐿𝐿(𝑅)}∈(10⁻³,10⁻²) and 𝜙_𝐿∈(−60^∘,−90^∘) 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 𝐾^∗₀(1430) belongs to the first orbital excited states (S1) or the low-lying 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]: S1:𝑓_𝐾∗0(1.0GeV)=−(300±30)MeV;𝑓_𝐾∗0(2.1GeV)=−(370±35)MeV;𝐵₁(1.0GeV)𝐵=0.58±0.07;₁(2.1GeV)𝐵=0.39±0.05;₃(1.0GeV)𝐵=−1.20±0.08;₃(2.1GeV)=−0.70±0.05;S2:𝑓_𝐾∗0(1.0GeV)=(445±50)MeV;𝑓_𝐾∗0(2.1GeV)=(550±60)MeV;𝐵₁(1.0GeV)𝐵=−0.57±0.13;₁(2.1GeV)𝐵=−0.39±0.09;₃(1.0GeV)𝐵=−0.42±0.22;₃(2.1GeV)=−0.25±0.13. (25)

The longitudinal and transverse decay constants of vector mesons are listed as 𝑓_𝜌=216MeV,𝑓_𝜔=187MeV,𝑓_𝜙=215MeV,𝑓^⟂_𝜌=165MeV,𝑓^⟂_𝜔=151MeV,𝑓^⟂_𝜙=186MeV, (26) where the values are taken from [69]. In the LCADs of vectors, the Gegenbauer moments 𝛼^𝑉_𝑛 and 𝛼^𝑉_𝑛,⟂ have been studied, and we will employ the most recent updated values from QCD sum rules [70]: 𝛼₂^𝜌,𝜔=0.15,𝛼^𝜌,𝜔_2,⟂𝛼=0.14,_𝜙2=0.18,𝛼^𝜙_2,⟂=0.14, (27) and 𝛼^𝑉₁=0, 𝛼^𝑉_1,⟂=0.

In [14, 15, 25], it was found that in decay modes with scalars the main theoretical uncertainties are due to the annihilations, especially for the penguin dominated ones. In 𝐵→𝑃𝑃, 𝑃𝑉 decays, the annihilation amplitudes are helicity suppressed because the helicity of one of the final states cannot match with that of its quarks. However, this helicity suppression can be alleviated in the decay modes with scalar because of nonvanishing orbital angular momentum. Thus, annihilation contribution to 𝐵→𝑆𝑃(𝑉) is much larger than the 𝐵→𝑃𝑃(𝑉) case. However, as stated before, the end-point singularity appears in calculating the annihilation contribution, and then two free parameters, 𝜌_𝐴 and 𝜙_𝐴, are introduced phenomenally. In [15], it is found that the behavior of 𝑆𝑉 is similar to the longitudinal part of 𝑉𝑉. Fortunately, with experimental data, it presents the moderate value of nonuniversal annihilation phase 𝜙_𝐴=−40^∘ for 𝐵→𝑉𝑉 decay modes [4–7]. Therefore, for 𝐵→𝑆𝑉, we conservatively take 𝜙_𝐴=(−40±20)^∘ with 𝜌_𝐴=0.6±0.2, which also assures that the hadronic uncertainties are considerably reduced. Furthermore, the endpoint divergence 𝑋_𝐻 in the hard spectator contributions can also be parameterized in the same manner.

Within above parameters and formulas, we calculate the branching fractions of these decays in SM and the family nonuniversal 𝑍^′ model under two different scenarios. Together with partial experimental results, the results are exhibited in Table 4, respectively. For the center values, we adopt 𝜉^{𝐿𝐿,𝐿𝑅}=0.005 and 𝜙_𝐿=−80^∘. For all theoretical predictions, the first errors involved the uncertainties of the Gegenbauer moments 𝐵_1,3, the scalar meson decay constants, and form factors. The second errors arise from the power corrections of annihilation and hard spectator interactions characterized by the parameters 𝑋_𝐴,𝐻. To obtain the third errors of the 𝑍^′ model results, we scan randomly the points in their own possible parameter spaces.

Table 4 

Branching fractions (in units of 10⁻⁶) under the different scenarios.

Compared with predictions in [15] (considering the typos), our results agree with theirs well within uncertainties. However, there still exist few differences, and some reasons are list as follows: (1) in [15], for the parameterizations of singularities, the center values correspond to 𝜌_𝐴,𝐻=0 and 𝜙_𝐴,𝐻=0, while we set 𝜌_𝐴,𝐻=0.6 and 𝜙_𝐴,𝐻=−40^∘; (2) the difference of Wilson coefficients, caused by the top quark mass and other part parameters, will change the results slightly; (3) in this work, the different form factors of 𝐵→𝐾^∗₀(1430) are used under different scenarios, but they adopted same values in [15].

In Table 4, for 𝐾^∗₀(1430)𝜙 channels, though the central values of the prediction under S1 are smaller than the experimental data, they are accommodated with the large uncertainties. However, in S2, the theoretical results are much larger than the data and cannot agree with data even with uncertainties. These theoretical results also agree with the predictions of PQCD [19]. For 𝐵⁰→𝐾₀^∗0𝜌⁰ channels, contrary to 𝐾^∗₀(1430)𝜙, the result of S2 agrees with data well and the prediction of S1 is much smaller than the data. Since for 𝐵⁰→𝐾₀^∗−𝜌⁺ there is large uncertainty in the experimental data, the theoretical results under both S1 and S2 can accommodate the data with large uncertainties theoretically. That is to say, it is impossible to explain all data under one settled scenario simultaneously.

When adding the contribution of the 𝑍^′ gauge boson, as shown in the table, the 𝑍^′ gauge boson changes the branching fractions under both two different scenarios. For 𝐵→𝐾^∗₀𝜙 channels dominated by the annihilation, the 𝑍^′ will enhance the branching fractions in S1, while in S2 the branching ratios are decreased. The reason is that the annihilation is proportional to the decay constant 𝑓_𝐾∗0, which has different sign in different scenarios. For 𝐵⁻→𝐾₀^∗0𝜌⁻ and 𝐵⁰→𝐾₀^∗−𝜌⁺, because the scalar particle is the emitted particle, the whole amplitudes are proportional to the decay constant 𝑓_𝐾∗0; thus the new physics contribution has same trends in different scenarios. For channels with 𝜌⁰ or 𝜔, the spectator quarks enter not only the scalars but also the vectors, the amplitudes become much complicated, and we cannot describe the relations between new physics and branching fractions easily.

Compared to the experimental data, the 𝑍^′ boson affects the branching fractions remarkably and alleviates the disparities. However, we cannot achieve a definite conclusion yet whether 𝐾^∗₀ belongs to the ground states or the first orbital excited states. Moreover, for most modes except 𝐵⁻→𝐾₀^∗−𝜌⁰(𝜔), the new physics contribution might be clouded by the uncertainties taken by the annihilations. Thus, it is also very difficult to search for 𝑍^′ effect in these decays. Specifically, for decays 𝐵⁻→𝐾₀^∗−𝜌⁰(𝜔), 𝑍^′ boson could enhance the branching fractions more than two times; we hope these two channels could be measured in the LHCb experiment or Super-b factories in future so as to probe the 𝑍^′ gauge boson.

To test the isospin symmetry and probe new physics, we define two ratios: 𝑅₁=𝐵𝑟𝐵⁰⟶𝐾₀^∗0𝜌⁰𝐵𝑟𝐵⁰⟶𝐾₀^∗−𝜌⁺=0.96^+1.07_−0.43Exp.,𝑅₂=𝜏𝐵⁰𝜏(𝐵⁻)⋅𝐵𝑟𝐵⁻⟶𝐾₀^∗−𝜙𝐵𝑟𝐵⁰⟶𝐾₀^∗0𝜙=1.68^+0.90_−0.60Exp., (28) where the experimental results are also given, and all uncertainties are added in quadrature. In the isospin limit, 𝑅₁=1/2 and 𝑅₂=1 are expected to hold. Here, we list the theoretical results under different scenarios in different models: 𝑅₁[SM]=0.43^+0.13_−0.10,S10.55^+0.08_−0.08,S𝑅2;₁SM+𝑍^′=0.27^+0.15+0.31_{−0.08−0.17,}S10.62^+0.09+0.20_{−0.09−0.07,}S𝑅2;₂[SM]=1.00^+0.04_−0.04,S10.94^+0.00_−0.01,S𝑅2;₂SM+𝑍^′=0.76^+0.16+0.24_{−0.18−0.18,}S10.73^+0.16+0.19_{−0.27−0.24,}S2. (29) In the above results, the theoretical uncertainties are reduced since they are ratios of branch fractions. Because 𝑅-values are ratios of branching fractions, the uncertainties taken by nonperturbative part are almost zero, and we here ignored them. We see that the symmetries are almost held in SM, while the data shows that the isospin symmetries are violated, which means that the large electroweak penguins or weak annihilations may break the isospin symmetries remarkably. When adding 𝑍^′ contribution, except for 𝑅₁ under S2, the isospin symmetries are broken in opposite directions. However, the family nonuniversal 𝑍^′ model cannot be ruled out due to large uncertainties in both experimental and theoretical sides.

Finally, we will discuss the direct 𝐶𝑃 asymmetries of these decays. Because |𝑉_𝑢𝑏𝑉_𝑢𝑠|(𝜆⁴)≪|𝑉_𝑡𝑏𝑉_𝑡𝑠|(𝜆²) for the charged mode 𝐵⁻→𝐾₀^∗−𝜙 and there is no tree contribution in the neutral mode, thereby the direct 𝐶𝑃 asymmetries of 𝐵→𝐾^∗₀𝜙 are almost zero in both SM and the 𝑍^′ model. For 𝐵→𝐾^∗₀𝜌, although the CKM elements are suppressed, the tree operators with large Wilson coefficients appear in the emission diagrams, so the amplitudes of tree and penguin may have comparable magnitudes. Thus, large 𝐶𝑃 asymmetries in these decays are expected, just like decays 𝐵→𝐾𝜋 and 𝐵→𝐾𝜌. In Table 5, we list the direct 𝐶𝑃 asymmetries of 𝐵→𝐾^∗₀𝜌 under different scenarios. In this table, the uncertainties induced by nonperturbative part are not included, as the direct 𝐶𝑃 asymmetries are also ratios. From the table, we firstly note that 𝐵⁰→𝐾₀^∗0𝜌(𝜔) have large asymmetries, and different scenarios have different signs but with large uncertainties. If we can calculate the annihilation accurately within some effective approach in future, this parameter could be used to distinguish the scenarios. Secondly, for 𝐵⁻→𝐾₀^∗−𝜌⁰(𝜔), the 𝑍^′ could change the signs of the center values, and these two decays can be used in probing new physics effect.

Table 5 

The direct 𝐶𝑃 asymmetry (%) under the different scenarios.

5. Summary

Motivated by recent measurements of decays 𝐵→𝐾^∗₀𝜌 and 𝐾^∗₀𝜙, we studied the branching fractions of these decays both in SM and in the family nonuniversal 𝑍^′ model within the QCDF framework. Because it is not clear whether 𝐾^∗₀ is the lying state or the first orbital excited state, we calculated them under two different scenarios. We noted that, for these decay modes with scalar meson, the annihilations play more important roles than those in 𝐵→𝑃𝑃 and 𝑃𝑉 decays; thus annihilations lead to large uncertainties in these decays. From this point, an effective way that could calculate the annihilations reliably is needed. Compared with the experimental results, we found that different decay modes favor different scenarios. Moreover, in order to account for the large isospin asymmetries in the data, large weak annihilations are also required. Adding the contribution of the family nonuniversal 𝑍^′ boson, the branching fractions and their ratios are both changed remarkably. However, we cannot identify the character of the scalar meson 𝐾^∗₀, either. Furthermore, for most channels, the 𝑍^′ contribution will be buried by large uncertainties, except for decays 𝐵⁻→𝐾₀^∗−𝜌⁰(𝜔). For the isospin asymmetries, the 𝑍^′ boson makes the disparities more worse, but we cannot rule this model out because of large errors in both experimental and theoretical sides.

In this work, we also calculated the direct 𝐶𝑃 asymmetries and found that the 𝐶𝑃 asymmetries of 𝐵→𝐾^∗₀𝜙 are almost zero. In different scenarios, the 𝐶𝑃 asymmetries of 𝐵⁻→𝐾₀^∗−𝜌⁰(𝜔) have different signs; as a result, they can be used to classify the scalar 𝐾^∗₀. If its character could be identified, we accordingly could use these results to probe the new gauge boson 𝑍^′, because it also could changes the signs of the direct 𝐶𝑃 asymmetries. All the above predictions could be tested in the running LHCb or the Super-b factories in the future.

Acknowledgments

This work is supported by the National Science Foundation (nos. 11175151 and 11235005) and the Natural Science Foundation of Shandong Province (ZR2010AM036).