Studies on the $B \to \kappa \bar \kappa$ decays in the perturbative QCD approach

TL;DR

This paper investigates $B o ext{kappa} ar{ ext{kappa}}$ decays using perturbative QCD, predicting decay rates and CP asymmetries, and highlighting potential for future experimental tests and new physics searches.

Contribution

First analysis of $B o ext{kappa} ar{ ext{kappa}}$ decays in perturbative QCD with $k_T$ factorization, providing predictions for decay rates and CP asymmetries.

Findings

Large decay rates for $B_s^0 o ext{kappa}^+ ext{kappa}$ decays (~$10^{-5}$).

Pure annihilation $B_d^0 o ext{kappa}^+ ext{kappa}$ decay rate around $3 imes 10^{-6}$.

Pure penguin modes have zero direct and mixing-induced CP violations in the Standard Model.

Abstract

The $B \to \kappa \bar \kappa$ decays are investigated for the first time in the perturbative QCD formalism based on the $k_T$ factorization theorem, where the light scalar $\kappa$ is assumed as a two-quark state. Our numerical results and phenomenological analyses on the CP-averaged branching ratios and CP-violating asymmetries show that: (a) the $B_s^0 \to \kappa^+ \kappa^-$ and $B_s^0 \to \kappa^0 \bar \kappa^0$ decays have large decay rates around ${\cal O}(10^{-5})$, which could be examined by the upgraded Large Hadron Collider beauty and/or Belle-II experiments in the near future; (b) a large decay rate about $3 \times 10^{-6}$ appears in the pure annihilation $B_d^0 \to \kappa^+ \kappa^-$ channel, which could provide more evidences to help distinguish different QCD-inspired factorization approaches, even understand the annihilation decay mechanism; (c) the pure penguin modes $B_d^0 \to \kappa^0 \bar \kappa^0$ and $B_s^0 \to \kappa^0 \bar \kappa^0$ would provide a promising ground to search for the possible new physics because of their zero direct and mixing-induced CP violations in the standard model. The examinations with good precision from the future experiments will help to further study the perturbative and/or nonperturbative QCD dynamics involved in these considered decay modes.

Tables

Table 1. Table 1: The factorization decay amplitudes(in units of 10 − 3 superscript 10 3 10^{-3}  GeV 3 ) of the nonleptonic B → κ ​ κ ¯ → 𝐵 𝜅 ¯ 𝜅 B\to\kappa\bar{\kappa} decays in the PQCD approach at leading order, where only the central values are quoted for clarifications.

Modes	${ℱ}_{fe}$	${ℳ}_{nfe}$	${ℳ}_{nfa}$	${ℱ}_{fa}$
$B^{+}\to {\kappa }^{+}{\overline{\kappa }}^0$	$0.415-i0.173$	$-0.377-i0.216$	$-0.164+i0.814$	$-0.035+i0.859$
$B_d^0\to {\kappa }^{+}{\kappa }^{-}$	$-$	$-$	$-0.597-i1.983$	$0.0008+i0.0005$
$B_d^0\to {\kappa }^0{\overline{\kappa }}^0$	$0.415-i0.173$	$-0.377-i0.216$	$-1.049-i0.036$	$-0.055+i0.907$
$B_s^0\to {\kappa }^{+}{\kappa }^{-}$	$-3.144+i0.377$	$1.130+i0.394$	$5.236+i1.193$	$1.796-i3.709$
$B_s^0\to {\kappa }^0{\overline{\kappa }}^0$	$-3.360$	$1.564+i1.736$	$5.301+i1.782$	$1.788-i3.695$

Equations

$$H_{\mathrm{eff}}=\frac{G_F}{\sqrt{2}}\left\{V_{ub}^{\ast }{V}_{uQ}\left[C_1(\mu )O_1^u(\mu )+C_2(\mu )O_2^u(\mu )\right]-V_{tb}^{\ast }{V}_{tQ}\sum _{i=3}^{10}{C}_i(\mu )O_i(\mu )\right\},$$

$$\displaystyle{\begin{array}[]{ll}\displaystyle O_{1}^{u}\,=\,(\bar{Q}_{\alpha}u_{\beta})_{V-A}(\bar{u}_{\beta}b_{\alpha})_{V-A}\;,&\displaystyle O_{2}^{u}\,=\,(\bar{Q}_{\alpha}u_{\alpha})_{V-A}(\bar{u}_{\beta}b_{\beta})_{V-A}\;;\end{array}}$$

$$\displaystyle{\begin{array}[]{ll}\displaystyle O_{3}\,=\,(\bar{Q}_{\alpha}b_{\alpha})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\beta})_{V-A}\;,&\displaystyle O_{4}\,=\,(\bar{Q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A}\;,\\ \displaystyle O_{5}\,=\,(\bar{Q}_{\alpha}b_{\alpha})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\beta})_{V+A}\;,&\displaystyle O_{6}\,=\,(\bar{Q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A}\;;\end{array}}$$

$$\displaystyle{\begin{array}[]{ll}\displaystyle O_{7}\,=\,\frac{3}{2}(\bar{Q}_{\alpha}b_{\alpha})_{V-A}\sum_{q^{\prime}}e_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\beta})_{V+A}\;,&\displaystyle O_{8}\,=\,\frac{3}{2}(\bar{Q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}e_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A}\;,\\ \displaystyle O_{9}\,=\,\frac{3}{2}(\bar{Q}_{\alpha}b_{\alpha})_{V-A}\sum_{q^{\prime}}e_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\beta})_{V-A}\;,&\displaystyle O_{10}\,=\,\frac{3}{2}(\bar{Q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}e_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A}\;.\end{array}}$$

$$Br(B^{+}\to {\kappa }^{+}{\bar{\kappa }}^0)$$

$$Br(B_d^0\to {\kappa }^{+}{\kappa }^{-})$$

$$Br(B_d^0\to {\kappa }^0{\bar{\kappa }}^0)$$

$$Br(B_s^0\to {\kappa }^{+}{\kappa }^{-})$$

$$Br(B_s^0\to {\kappa }^0{\bar{\kappa }}^0)$$

$$R_{s/d}^{00}(\kappa \bar{\kappa })$$

$$R_{s/d}^{+-}(\kappa \bar{\kappa })$$

$$R_s^{00/+-}(\kappa \bar{\kappa })$$

$${\varphi }_{\kappa }(x,\mu )$$

$${\mathcal{A}}_{\mathrm{dir}}(B^{+}\to {\kappa }^{+}{\bar{\kappa }}^0)$$

$${\mathcal{A}}_{\mathrm{dir}}(B_d^0\to {\kappa }^{+}{\kappa }^{-})$$

$${\mathcal{A}}_{\mathrm{mix}}(B_d^0\to {\kappa }^{+}{\kappa }^{-})$$

$${\mathcal{A}}_{\mathrm{dir}}(B_d^0\to {\kappa }^0{\bar{\kappa }}^0)$$

$${\mathcal{A}}_{\mathrm{mix}}(B_d^0\to {\kappa }^0{\bar{\kappa }}^0)$$

$${\mathcal{A}}_{\mathrm{dir}}(B_s^0\to {\kappa }^{+}{\kappa }^{-})$$

$${\mathcal{A}}_{\mathrm{mix}}(B_s^0\to {\kappa }^{+}{\kappa }^{-})$$

$${\mathcal{A}}_{\Delta {\Gamma }_s}(B_s^0\to {\kappa }^{+}{\kappa }^{-})$$

$${\mathcal{A}}_{\mathrm{dir}}(B_s^0\to {\kappa }^0{\bar{\kappa }}^0)$$

$${\mathcal{A}}_{\mathrm{mix}}(B_s^0\to {\kappa }^0{\bar{\kappa }}^0)$$

$${\mathcal{A}}_{\Delta {\Gamma }_s}(B_s^0\to {\kappa }^0{\bar{\kappa }}^0)$$

Full text

††preprint: JSNU-HEP-2019-1

In the conventional quark model, a meson is composed of one quark and one antiquark, i.e., $q\bar{q}$, with different coupling of the orbital and spin angular momenta GellMann:1964nj ; Zweig:1981pd ; Zweig:1964jf . To date, the structure of the $S$-wave ground state mesons has almost been determined unambiguously, though the $\eta$ and $\eta^{\prime}$ ones may contain the component of gluonium( or pseudoscalar glueball) with different extent Kou:1999tt ; Cheng:2008ss ; Liu:2012ib . However, the components of the $P$-wave mesons are not easily determined. In particular, the description of the inner structure for the light scalar states such as $a_{0}(980)$, $\kappa$ or $K_{0}^{*}(800)$, $\sigma$ or $f_{0}(500)$, and $f_{0}(980)$ is controversial, e.g., $q\bar{q}$, $\bar{q}\bar{q}qq$, meson-meson bound states, etc., and still not well established currently(for a review, see e.g., Refs. Godfrey:1998pd ; Close:2002zu ; Tanabashi:2018oca ). When the light scalar $f_{0}(980)$ was first observed in the $B\to f_{0}(980)K$ channel, performed by the Belle Abe:2002av and BABAR Aubert:2003mi collaborations in 2002 and 2004, respectively, the investigations on the light scalars in the decay productions of the heavy $B$ mesons were naturally considered as a unique insight to explore their underlying structure. With many channels including light scalars of the heavy $B$ meson decays being opened experimentally Amhis:2016xyh ; Tanabashi:2018oca , $B\to SP,SV$ (Here, $P$ and $V$ denote the pseudoscalar and vector meson, respectively) decays have been studied extensively at the theoretical aspects with different approaches/methods, for instance, see Cheng:2009xz ; Liu:2009xm ; Liu:2010zg ; Liu:2013cvx ; Shen:2006ms ; Wang:2006ria ; Wang:2009azc ; Colangelo:2010bg ; Kim:2009dg . With the great development of the Large Hadron Collider beauty(LHCb) and Belle-II experiments Kou:2018nap , more and more modes involving one and/or two scalar states in the $B$ meson decays are expected to be measured with good precision in the future.

In this work, we will study the charmless hadronic $B\to\kappa\bar{\kappa}$ decays (Here, $B$ denotes the nonstrange $B^{+}$ and $B_{d}^{0}$, and strange $B_{s}^{0}$ mesons.) for the first time by employing the perturbative QCD(PQCD) approach Keum:2000wi ; Lu:2000em ; Lu:2000hj based on the $k_{T}$ factorization theorem, where the light scalar $\kappa$ will be considered as a lowest-lying $q\bar{q}$ state. Theoretically, the most important part of a nonleptonic decay amplitude is the effective calculation of the hadronic matrix element, in which the essential inputs are the wave functions (or light-cone distribution amplitudes) of the initial and final hadron states that describe the nonperturbative QCD dynamics independent on the processes. The PQCD approach, as one of the presently three popular QCD-inspired factorizations (the other two are QCD factorization approach Beneke99:qcdf ; Du02:qcdf and soft-collinear effective theory Bauer04:scet , respectively), has the advantages in computing the Feynman amplitudes by conquering the endpoint singularities that exist in the collinear factorization theorem. By keeping the transverse momentum of the valence quark, associated with the Sudakov factors arising from the $k_{T}$ resummation Botts89:ktfact ; Li92:sudakov and threshold resummation Li02:threshold , the PQCD approach can be well applied to calculate the hadronic matrix element of the nonleptonic $B$ meson decays. Apart from the factorizable emission diagrams, the nonfactorizable emission ones and the annihilation ones can also be perturbatively calculated. Furthermore, even though the origin of the CP violation and the annihilation decay mechanism are currently unknown, the experimental measurements Amhis:2016xyh ; Tanabashi:2018oca performed by the BABAR, Belle, and LHCb collaborations have confirmed the direct CP-violating asymmetry of the $B\to K\pi$ decays Keum:2000wi ; Keum:2000ph and the large decay rates of the pure annihilation $B_{d}^{0}\to K^{+}K^{-}$ and $B_{s}^{0}\to\pi^{+}\pi^{-}$ modes Li:2004ep ; Xiao:2011tx predicted in the PQCD approach. Certainly, the predictions made in the PQCD approach about the branching ratios and CP violations of the $B\to PP,PV/VP,$ and $VV$ decays generally agree with the available data within errors.

At the quark level, the considered $B\to\kappa\bar{\kappa}$ decays are induced by the $\bar{b}\to\bar{d}$ or $\bar{b}\to\bar{s}$ transitions, respectively. The weak effective Hamiltonian $H_{\rm eff}$ for the $B\to\kappa\bar{\kappa}$ decays can be written as Buchalla:1995vs ,

[TABLE]

with the Fermi constant $G_{F}=1.16639\times 10^{-5}{\rm GeV}^{-2}$, the light $Q=d,s$ quark, and Wilson coefficients $C_{i}(\mu)$ at the renormalization scale $\mu$. The local four-quark operators $O_{i}(i=1,\cdots,10)$ are written as

• •

current-current(tree) operators

[TABLE]

• •

QCD penguin operators

[TABLE]

• •

electroweak penguin operators

[TABLE]

with the color indices $\alpha,\ \beta$ and the notations $(\bar{q}^{\prime}q^{\prime})_{V\pm A}=\bar{q}^{\prime}\gamma_{\mu}(1\pm\gamma_{5})q^{\prime}$. The index $q^{\prime}$ in the summation of the above operators runs through $u,\;d,\;s$, $c$, and $b$. It is worth mentioning that since we work in the leading order[${\cal O}(\alpha_{s})$] of the PQCD approach, it is consistent to use the leading order Wilson coefficients. For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, the formulas as given in Refs. Keum:2000wi ; Lu:2000em will be adopted directly.

The Feynman diagrams of the $B\to\kappa\bar{\kappa}$ decays at leading order in the PQCD formalism are illustrated in Fig. 1:

• •

Emission topology: Fig. 1(a) and 1(b) describe the factorizable emission diagrams, while Fig. 1(c) and 1(d) describe the nonfactorizable emission ones;

• •

Annihilation topology: Fig. 1(e) and 1(f) describe the nonfactorizable annihilation diagrams, while Fig. 1(g) and 1(h) describe the factorizable annihilation ones.

In 2013, one of us(X. Liu) with Xiao and Zou ever studied the $B\to K_{0}^{*}(1430)\bar{K}_{0}^{*}(1430)$ decays in the PQCD approach Liu:2013lka , where the analytic expressions for the factorization formulas and the decay amplitudes were presented explicitly. Therefore, we just need to replace the $K_{0}^{*}(1430)$ state in Ref. Liu:2013lka with the light $\kappa$ one to obtain easily the corresponding information of the $B\to\kappa\bar{\kappa}$ decays in the PQCD approach. Hence, for simplicity, we will not collect the aforementioned formulas in this paper. The interested readers can refer to Ref. Liu:2013lka for detail.

Then, we can turn to the numerical calculations of the CP-averaged branching ratios and CP-violating asymmetries of the $B\to\kappa\bar{\kappa}$ decays in the PQCD approach. Before proceeding, some essential comments on the nonperturbative inputs are as follows:

1. (a) For the heavy $B$ mesons, the wave functions and the distribution amplitudes, and the decay constants are same as those utilized in Ref. Liu:2013lka , but with the updated lifetimes $\tau_{B_{d}^{0}}=1.52$ ps and $\tau_{B_{s}^{0}}=1.509$ ps, which can be found clearly in the newest Review of Particle Physics Tanabashi:2018oca .
2. (b) For the light scalar $\kappa$, the decay constants and the Gegenbauer moments in the distribution amplitudes have been derived at the normalization scale $\mu=1$ GeV in the QCD sum rule method Cheng:2005nb : the scalar decay constant $\bar{f}_{\kappa}=0.34\pm 0.02$ GeV, the vector decay constant $f_{\kappa}=\bar{f}_{\kappa}/\mu$ with $\mu=m_{\kappa}/(m_{s}-m_{q})$ ($m_{\kappa}$, $m_{s}$, and $m_{q}$ stand for the masses of the light scalar $\kappa$, the strange quark $s$, and the nonstrange light quark $u$ and $d$, respectively.), and the Gegenbauer moments $B_{1}=-0.92\pm 0.11$ and $B_{3}=0.15\pm 0.09$. Here, the running current quark masses $m_{s}=0.12$ GeV and $m_{q}=0.005$ GeV at $\mu=1$ GeV, which are translated from those in a $\overline{\rm MS}$ scale $\mu\approx 2$ GeV Tanabashi:2018oca , are adopted in the calculations. Note that the isospin symmetry is assumed in this work. For the light scalar $\kappa$ mass $m_{\kappa}$, we adopt the value $m_{\kappa}=0.8$ GeV for rough estimations, because this scalar $\kappa$ has been assumed as the lowest-lying $q\bar{q}$ state 222 Moreover, as inferred from the newest Review of Particle Physics Tanabashi:2018oca , this state is also with a finite but indefinite width, whose effect, in principle, has to be included to make relevant predictions more precise. Generally speaking, the width effect could result in the enhancement/reduction of the numerical results with different extent Cheng:2003xc . However, up to now, to our best knowledge, the essential $S$-wave $K\pi$ distribution amplitudes for resonance $\kappa$ state with the constrained parameters, e.g., Gegenbauer moments, are absent. Therefore, the width effect will be left for future investigations elsewhere. .
3. (c) For the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements, we also adopt the Wolfenstein parametrization at leading order, but with the updated parameters $A=0.836$, $\lambda=0.22453$, $\bar{\rho}=0.122^{+0.018}_{-0.017}$, and $\bar{\eta}=0.355^{+0.012}_{-0.011}$ Tanabashi:2018oca .

Now, we present the numerical results of the $B\to\kappa\bar{\kappa}$ decays in the PQCD formalism. Firstly, the PQCD predictions of the CP-averaged branching ratios can be read as follows:

[TABLE]

and

[TABLE]

and

[TABLE]

From the Eqs. (10)-(14), one can find the following points:

• (a) The considered $B\to\kappa\bar{\kappa}$ decays have evidently different CP-averaged branching ratios in the PQCD approach, namely, varying from $10^{-7}$ to $10^{-5}$. Frankly speaking, these numerical results suffer from large theoretical errors mainly induced by the nonperturbative inputs, such as the shape parameter $\omega_{B}$ in the $B$ meson distribution amplitude, the scalar decay constant $\bar{f}_{\kappa}$, especially the Gegenbauer moments $B_{i}(i=1,3)$ in the leading twist light-cone distribution amplitude of $\kappa$. The uncertainties of the above mentioned parameters need to be constrained by the future precise measurements and/or Lattice QCD or QCD sum rule calculations.
• (b) The pure annihilation decay of $B_{d}^{0}\to\kappa^{+}\kappa^{-}$ has the same quark structure as that of the measured one $B_{d}^{0}\to K^{+}K^{-}$, whose decay rate predicted in the PQCD approach has been confirmed by the LHCb experiments Aaij:2012as ; Aaij:2016elb . Therefore, it is expected that the large branching ratio of the $B_{d}^{0}\to\kappa^{+}\kappa^{-}$ mode given in this work could be examined in the LHCb and/or Belle-II experiments. The confirmation of this PQCD result would provide useful hints to understand the inner structure of the light scalar $\kappa$.
• (c) In light of the large $Br(B_{d}^{0}\to\kappa^{+}\kappa^{-})_{\rm PQCD}$ while the small $Br(B_{d}^{0}\to\kappa^{0}\bar{\kappa}^{0})_{\rm PQCD}$, under the assumption of isospin symmetry, it is postulated that a significant cancellation occurred in the $B_{d}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ decay between the contributions induced by the emission and the annihilation topologies, which, as a matter of fact, can be found clearly from the numerical results for the factorization decay amplitudes presented in Table 1.
• (d) The decay rates of the $B_{s}^{0}\to\kappa^{+}\kappa^{-}$ and $B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ modes indicate a very small contamination induced by the tree annihilation diagrams associated with a CKM-suppressed factor $V_{us}\sim\lambda$ in the $\bar{b}\to\bar{s}$ transition. Meanwhile, relative to $V_{td}\sim A\lambda^{3}(1-\rho-{\it i}\eta)$ in the $\bar{b}\to\bar{d}$ transition, the CKM-enhanced factor $V_{ts}\sim A\lambda^{2}$ involved in these two decays finally resulted in the highly large and close branching ratios around ${\cal O}(10^{-5})$.
• (e) As mentioned above, because of the enhanced factor $r_{\rm CKM}=|V_{ts}/V_{td}|^{2}\sim 23.6$ Tanabashi:2018oca , the pure penguin modes $B_{d}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ and $B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ have significantly different decay rates, namely, the former one with $7.75^{+5.55}_{-3.83}\times 10^{-7}$ while the latter one with $1.55^{+1.28}_{-0.86}\times 10^{-5}$, respectively, where the errors have been added in quadrature. In light of the large theoretical errors, a precise ratio of these two branching ratios would be more interested,

[TABLE]

Similarly, another two interesting ratios $R_{s/d}^{+-}(\kappa\bar{\kappa})$ and $R_{s}^{00/+-}(\kappa\bar{\kappa})$ could be easily obtained,

[TABLE]

It is clearly found that the uncertainties in the above ratios $R_{s/d}^{00}$, $R_{s/d}^{+-}$, and $R_{s}^{00/+-}$ are significantly small because the theoretical errors resulted from the hadronic inputs have been cancelled to a great extent. These values are expected to be examined in the future $B$-physics experiments to help further understand the involved QCD dynamics in depth.

• (f) In order to understand the contributions arising from different topologies better, the numerical values for the factorization decay amplitudes are presented explicitly in Table 1. One can find the large nonfactorizable emission contributions and the much larger annihilation contributions in the considered $B\to\kappa\bar{\kappa}$ decays, especially in the two $B_{s}^{0}$ modes. The underlying reason is that the antisymmetric QCD behavior from the only odd terms in the twist-2 distribution amplitude of the light scalar $\kappa$ Cheng:2005nb ,

[TABLE]

where $f_{\kappa}(\mu)$ and $\bar{f}_{\kappa}(\mu)$, $B_{m}(\mu)$, and $C_{m}^{3/2}(t)$ are the vector and scalar decay constants, Gegenbauer moments, and Gegenbauer polynomials, respectively, make the previously destructive interferences become the presently constructive ones between the valence-quark-radiative and valence-antiquark-radiative diagrams in the nonfactorizable emission and annihilation topologies, as already illustrated in Fig. 1. It is worth mentioning that, as can be seen in Table 1, the annihilation diagrams play a dominant role on both of the CP-averaged decay rates and the CP violations of the considered $B\to\kappa\bar{\kappa}$ decays in this work.

• (g) As for the experimental measurements of the predicted large branching ratios, e.g., $Br(B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0})=1.55^{+1.28}_{-0.86}\times 10^{-5}$ and $Br(B_{s}^{0}\to\kappa^{+}\kappa^{-})=1.15^{+0.99}_{-0.67}\times 10^{-5}$, we expect the LHCb and/or Belle-II experiments might measure these channels through the Dalitz plot analysis of $B_{s}^{0}\to(K\pi)_{\kappa}(K\pi)_{\bar{\kappa}}$. In principle, the LHCb and Belle-II experiments have the abilities to detect the $B$ meson decay rates with large branching ratios above $10^{-6}$. Taking $B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ mode as an example, the decay rate ${\cal B}(\kappa^{0}\to K^{+}\pi^{-})$ is $\frac{2}{3}$ based on the assumption of isospin symmetry in the strong interactions. Therefore, we could obtain a branching ratio ${\rm BR}(B_{s}^{0}\to(K^{+}\pi^{-})_{\kappa^{0}}(K^{-}\pi^{+})_{\bar{\kappa}^{0}})\equiv Br(B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0})\cdot{\cal B}(\kappa^{0}\to K^{+}\pi^{-})\cdot{\cal B}(\bar{\kappa}^{0}\to K^{-}\pi^{+})=6.89^{+5.69}_{-3.82}\times 10^{-6}$. We hope this large value above $10^{-6}$ could be measured by the LHCb and/or Belle-II experiments when the events with high statistics are collected. Certainly, more information of the intermediate state $\kappa$ demand the studies on the four-body $B_{s}^{0}\to K^{+}K^{-}\pi^{+}\pi^{-}$ decay armed with the $S$-wave $K\pi$ distribution amplitudes with well constrained nonperturbative parameters for $\kappa$ from Lattice QCD and/or experimental measurements. Unfortunately, they are absent currently to our best knowledge theoretically and experimentally. Therefore, this issue has to be left for future studies elsewhere.

Then, we will discuss the CP-violating asymmetries of the $B\to\kappa\bar{\kappa}$ decays in the PQCD approach. The direct and the mixing-induced CP asymmetries ${\cal A}_{\rm dir}$ and ${\cal A}_{\rm mix}$ are collected as 333It is worth pointing out that, due to the nonzero ratio $(\Delta\Gamma/\Gamma)_{B_{s}^{0}}$ for the $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing as expected in the standard model, the third CP asymmetry ${\cal A}_{\rm\Delta\Gamma_{s}}$ will appear in the $B_{s}^{0}\to\kappa\bar{\kappa}$ decays Liu:2013lka . Here, the quantity $\Delta\Gamma$ is the decay width difference of the $B_{s}$ meson mass eigenstates Beneke99:Bsmixing ; Fernandez06:Bsmixing . Moreover, the three quantities describing the CP violations in the $B_{s}$ meson decays satisfy the relation: $|{\cal A}_{\rm dir}|^{2}+|{\cal A}_{\rm mix}|^{2}+|{\cal A}_{\rm\Delta\Gamma_{s}}|^{2}=1$.

[TABLE]

and

[TABLE]

and

[TABLE]

in which the definitions of the direct CP violation ${\cal A}_{\rm dir}$, the mixing-induced one ${\cal A}_{\rm mix}$, even the third one ${\cal A}_{\rm\Delta\Gamma_{s}}$ arising from the nonnegligible $(\Delta\Gamma/\Gamma)_{B_{s}^{0}}$ term are same as those in Ref. Liu:2013lka . From these numerical results of the CP violations of the $B\to\kappa\bar{\kappa}$ decays in the PQCD approach, some comments are in order:

• •

Generally speaking, these PQCD predictions are not sensitive to the variation of the scalar decay constant $\bar{f}_{\kappa}$ as shown in the above Equations. This can be deduced from the tiny vector decay constant $f_{\kappa}$ in the leading twist light-cone distribution amplitude of the scalar $\kappa$ meson(See Eq. (18) for detail). Furthermore, both of the twist-3 light-cone distribution amplitudes of $\kappa$ are proportional to the scalar decay constant $\bar{f}_{\kappa}$ because of adopting the asymptotic forms for simplicity Liu:2013lka . Based on the definitions, the CP asymmetry is the ratio of the differences of the related branching ratios between $B\to\kappa\bar{\kappa}$ and $\bar{B}\to\bar{\kappa}\kappa$ modes to their corresponding summations, then the scalar decay constant $\bar{f}_{\kappa}$ will be cancelled naturally.

• •

A large direct CP violation for the $B^{+}\to\kappa^{+}\bar{\kappa}^{0}$ mode can be observed, $-87.1^{+26.5}_{-11.8}\%$, which indicates that the involved penguin contributions are sizable, within large theoretical errors. While, due to the small branching ratio predicted in the PQCD approach, it might not be easily measured in the near future.

• •

Both of the $B_{d}^{0}\to\kappa^{+}\kappa^{-}$ and $B_{s}^{0}\to\kappa^{+}\kappa^{-}$ channels exhibit large CP-violating asymmetries, which are expected to be measured with much more possibilities at the LHCb and/or Belle-II experiments because of their large decay rates, namely, $2.86^{+1.62}_{-1.25}\times 10^{-6}$ and $1.15^{+0.99}_{-0.67}\times 10^{-5}$, where the errors have been added in quadrature too. The confirmations from the future measurements on these two modes would provide the evidences not only to support the assumption of the two-quark structure of the light scalar $\kappa$ in the present work, but also to help distinguish different factorization approaches on clarifying the origin of the strong phase in the heavy meson decays Arnesen08:anni-scet ; Chay08:complexanni .

• •

It is interesting to note that the direct and mixing-induced CP violations are naturally zero in both of the pure penguin $B_{d}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ and $B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ decays due to lack of the interferences from the tree contributions in the standard model. Of course, these two channels, especially the latter one with a large branching ratio as $1.55^{+1.28}_{-0.86}\times 10^{-5}$, could provide a promising platform to test the possible new physics beyond the standard model.

In summary, we have studied the CP-averaged branching ratios and the CP-violating asymmetries of the $B\to\kappa\bar{\kappa}$ decays in the PQCD approach based on the $k_{T}$ factorization theorem. The underlying structure of the light scalars are not determined unambiguously yet. Therefore, the light scalar $\kappa$ was assumed as a lowest-lying $q\bar{q}$ meson in the present work. It is expected that the productions of the light scalars in the heavy $B$ meson decays could provide many useful information at another different aspect. The predictions in the PQCD approach showed that: (1) The large decay rates above $10^{-6}$ could be found in the $B_{d}^{0}\to\kappa^{+}\kappa^{-}$, $B_{s}^{0}\to\kappa^{+}\kappa^{-}$, and $B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ channels, which are expected to be measured at the LHCb and/or Belle-II experiments in the near future; (2) The large direct and mixing-induced CP violations could be found in the $B^{+}\to\kappa^{+}\bar{\kappa}^{0}$, $B_{d}^{0}\to\kappa^{+}\kappa^{-}$, and $B_{s}^{0}\to\kappa^{+}\kappa^{-}$ modes, however, the small branching ratio $Br(B^{+}\to\kappa^{+}\bar{\kappa}^{0})$ might limit its future measurements; (3) The zero direct and mixing-induced CP-violating asymmetries in the standard model of the pure penguin $B_{d}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ and $B_{s}^{0}\to\kappa^{0}\bar{\kappa}^{0}$ decays would provide a promising platform to search for the possible new physics beyond the standard model once the nonzero CP violations could be detected evidently in these two modes; (4) The QCD dynamics of the light scalar $\kappa$ is different from that of the $S$-wave pseudoscalar $K$ and vector $K^{*}(892)$ mesons, which turned the previously destructive effects into the presently constructive ones in the nonfactorizable emission and annihilation diagrams, consequently led to the large branching ratios.