# Analysis of CKM-Favored Quasi-Two-Body $B\to D(R\to)K\pi$ Decays in PQCD
Approach

Zhi-Tian Zou  Department of Physics, Yantai University, Yantai 264005, China
Wen-Sheng Fang  Department of Physics, Yantai University, Yantai 264005, China
Xin Liu  School of Physics and Electronic Engineering, Jiangsu Normal
University, Xuzhou 221116, China  Ying<EMAIL_ADDRESS>Department of
Physics, Yantai University, Yantai 264005, China

###### Abstract

LHCb Collaboration studied the resonant structure of
$B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ decays using the Dalitz plot analysis
technique, based on a data sample corresponding to an integrated luminosity of
$3.0{\rm fb}^{-1}$ of $pp$ collision. The $K^{-}\pi^{+}$ components have been
analyzed in the amplitude model, where the decay amplitude is modeled to be
the resonant contributions with respect to the intermediate resonances
$K^{*}(892)$, $K_{0}^{*}(1430)$ and $K_{2}^{*}(1430)$. Motivated by the
experimental results, we investigate the color-favored quasi-two-body
$B\to\overline{D}^{0}K\pi$ decays in the framework of the perturbative QCD
(PQCD) approach. We calculate the the branching fractions by introducing the
appropriate wave functions of $K\pi$ pair. Our results are in agreement well
the available data, and others can be tested in LHCb and Belle-II experiments.
Using the narrow-width-approximation, we also extract the branching fractions
of the corresponding two-body $B\to\overline{D}R$ decays, which agree to the
previous theoretical calculations and the experimental data within the errors.
There are no $CP$ asymmetries in these decays in the standard model, because
these decays are all governed by only the tree operators.

##  1 Introduction

Three-body non-leptonic $B$ meson decays constitute a large portion of the
branching fraction and therefore attract a lot of attentions for several
phenomenological applications, such as deeply testing the standard model (SM),
the study of $CP$ violation and the exaction of the CKM angles [1]. The
branching fractions and $CP$ asymmetries of a large number of channels have
been studied extensively by BaBar [2, 3, 4, 5, 6, 7, 2], Belle [8, 9, 10, 11]
and LHCb [12, 13, 14, 15, 16]. Stimulated by the abundant experimental
measurements, the theoretical explorations have been carried out in the recent
years. Many three-body nonleptonic decays of heavy mesons have been studied in
detail in QCD factorization [17, 18, 19, 20, 21, 22, 23], PQCD approach [24,
25, 26, 27, 28, 29, 30, 31, 32, 33] and other theoretical methods [34, 35, 36,
37].

Different from the two-body $B$ decays where the momenta of final states are
fixed, The kinematics of the three-body decay is completely determined by two
of the three kinematic invariants $s_{12}$, $s_{13}$ and $s_{23}$, with the
definition $s_{ij}=(P_{i}+P_{j})^{2}/M_{B}^{2}$. The physical kinematical
region in the plane of two invariants (the Dalitz plot) is given by a
triangle. The Dalitz plot analysis has been proved to be a powerful tool for
studies of multi-body decays, and has been applied widely in the experimental
analysis. The analysis of three-body $B$ decays using this technique enables
one to study the properties of various resonances. Indeed, most of the quasi-
two-body $B$ decays are extracted from the Dalitz-plot analysis of three-body
ones. In addition, the Dalitz plot analysis of three-body $B$ decays provides
a nice methodology for extracting information on the unitarity triangle in SM.
For example, the Dalitz analysis together with the isospin symmetry allows one
to extract the angle $\alpha$ from the $B\to\pi\pi\pi$ decays with the vector
resonance $\rho$ [38].

On the theoretical side, three-body decays of heavy $B$ mesons are rather more
complicated as they receive resonant and nonresonant contributions and involve
3-body matrix elements. The interference between nonresonant and resonant
amplitudes makes it difficult to disentangle these two distinct contributions
clearly. The experimental measurements show that some three-body $B$ decays
are dominated by intermediate vector, scalar and tensor resonances, namely,
they proceed via quasi-two-body decays containing a resonance state and a
bachelor. While for some others, the nonresonant contributions govern the
decay amplitudes [18, 39]. Summarizing the experimental measurements, we find
that there is a large nonresonant fraction in the penguin-dominant modes. For
example, the nonresonant fraction reaches about $90\%$ in $B\to KKK$ decays,
$17-40\%$ in $B\to K\pi\pi$ decays, and only $14\%$ in $B\to\pi\pi\pi$ decays
[40]. However, there is not any reliable theoretical description of the
amplitudes of $B$ meson three-body decays so far, and most studies is are
still model dependent. The isobar model [41] and the K-matrix formalism [42]
are popularly adopted in the experimental analysis, especially the former
[12]. In these models, the resonant amplitudes are modeled and the non-
resonant contributions are often described by an empirical distribution in
order to reproduce the whole phase space [43].

Besides a large number of charmless three-body $B$ decays, a lot of charmed
three-body decays have also been measured in $B$ factories [44] and LHCb [45,
46]. For example, $B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ have been detailed
analyzed in LHCb in ref.[45], where the resonant structures of the
$\overline{D}^{0}K^{-}$ and $K^{-}\pi^{+}$ components have been studied using
the Dalitz plot technique. For the $\overline{D}^{0}K^{-}$ component, a
structure is found at $m_{\overline{D}^{0}K^{-}}\approx 2.86$ GeV, which is
viewed as an admixture of spin-1 and spin-3 resonances. For the $K^{-}\pi^{+}$
component, $K^{*}$ resonances and the corresponding fit fractions have been
well measured and the branching fractions of the corresponding quasi-two-body
decays were also reported, based on the fit fraction and the branching
fraction of the $B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ from previous LHCb
measurement [47]. Motivated by this, we shall investigate the CKM-favored
$B_{(s)}\to\overline{D}K\pi$ decays in PQCD approach, and focus on the $S$,
$P$ and $D$-wave resonant contributions with respect to the $K\pi$ component.

We all know that the resonant three-body decays (quasi-two-body decays)
correspond to the edge of the Dalitz plot, where the two particles move
collinearly with large energy and the bachelor particle recoils back. The
meson-pair and the rest one move fast and back-to-back in the $B$ meson rest
frame so that the interaction between the meson-pair and the bachelor meson is
highly suppressed. If we regard the meson-pair as a whole, the quasi-two-body
decay is very similar to a two-body decay and the factorization formula for
two-body decay can be applied safely. In practice, we also need to introduce a
new wave function to describe the meson-pair, including the angular momenta.
The soft interactions between two mesons are also absorbed in this
nonperturbative wave function.

In PQCD approach, the amplitude of quasi-two-body $B$ decay can be factorized
into different parts according to the “characteristic” scales. As we know, the
physics above the mass of $W$ boson $m_{W}$ is weak interaction and can be
calculated perturbatively. The physics between $m_{W}$ scale and $b$ quark
mass $m_{b}$ scale can be contained in the Wilson coefficient $C(\mu)$, which
can be calculated within the renormalization group equation and the Wilson
coefficient at the scale $m_{W}$. The physics between $m_{b}$ scale and the
factorization scale $\Lambda_{h}$ is dominated by exchanging one hard gluon
and can be calculated perturbatively. This part is also called hard kernel
$H$. The physics below $\Lambda_{h}$ scale is soft and nonperturbative, which
can be described by the universal wave functions. Therefore, the decay
amplitude of $B\to D(R\to)K\pi$ decay in PQCD can be decomposed into the
convolution as [48]

$\displaystyle\mathcal{A}\sim\int
dx_{i}db_{i}\Big{[}C(t)\otimes{H}(x_{i},b_{i},t)\otimes\Phi_{B}(x_{1},b_{1})\otimes\Phi_{D}(x_{2},b_{2})\otimes\Phi_{K\pi}(x_{3},b_{3})\otimes
e^{-S(t)}\Big{]},$ (1)

where $x_{i}$ are the momentum fraction of the quarks in the initial and final
mesons, and $b_{i}$ are the conjugate variables of the intrinsic transverse
momentum $k_{Ti}$ of the quarks. $\Phi_{B}$, $\Phi_{D}$ are the wave functions
of the $B$ meson and the $\overline{D}$ meson, respectively. $\Phi_{K\pi}$ is
the new introduced two-meson wave function. The exponential term $e^{-S(t)}$
is the Sudakov form factor obtained from the resummation of the double
logarithms arising from the retained intrinsic transverse momentum $K_{T}$ of
the inner quarks [49, 50, 51].

The layout of the present paper is as follows. In Sec.2, we firstly show the
Hamiltonian governing those quasi-two-body $B\rightarrow\overline{D}K\pi$
decays considered in this paper. The wave functions of $K-\pi$ pair are also
discussed. In Sec.3 we address the numerical results of the branching
fractions of those quasi-two-body decays. In this section based on the
obtained branching fractions we will also probe the corresponding two-body
$B\to\overline{D}R$ ($R$ denoting the intermediate resonance) decays and
compare them with the previous results and the experimental data. Finally, we
conclude this work in Sec.4

##  2 The Decay Formalism and Wave Function

For the quasi-two-body decays, the Dalitz plot analysis help us to interpret
the decay amplitude in isobar model conveniently, which is popularly used to
describe the complex amplitude of three-body decays by the experimentalists.
In this model the total decay amplitude is represented by a coherent sum of
amplitudes from $N$ individual decay channels with different resonances,

$\displaystyle\mathcal{A}=\sum_{i=1}^{N}a_{i}\mathcal{A}_{i},$ (2)

where $a_{i}$ is the complex coefficient showing the relevant magnitude and
the phase of different decay channels. The phase is a new source of the $CP$
asymmetry, namely, the $CP$ asymmetry caused by the interference among the
different channels. $\mathcal{A}_{i}$ is the amplitude of the quasi-two-body
decay associated with the certain resonance, which can be perturbatively
calculated in PQCD.

The weak Hamiltonian $\mathcal{H}_{eff}$ of $\bar{b}\to\bar{c}{u}\bar{q}$
governing the $B_{(s)}\to\overline{D}K\pi$ decays can be expressed as [52]

$\displaystyle\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}\Big{\\{}V_{uq}^{*}V_{cb}(C_{1}O_{1}+C_{2}O_{2})\Big{\\}}\left|{}_{q=d,s}\right.,$
(3)

where $V_{cb}$ and $V_{uq}$ are the CKM matrix elements. The $C_{1}$ and
$C_{2}$ are the so-called Wilson coefficients corresponding to the four-quark
operators $O_{1}$ and $O_{2}$ respectively, which are all tree operators. The
explicit expressions of the operators are written as

$\displaystyle
O_{1}=(\bar{b}_{\alpha}c_{\beta})_{V-A}(\bar{u}_{\beta}q_{\alpha})_{V-A};\;\;\;O_{2}=(\bar{b}_{\alpha}c_{\alpha})_{V-A}(\bar{u}_{\beta}q_{\beta})_{V-A}.$
(4)

We all know that the direct $CP$ asymmetry is related to the interference
between the tree type contribution and that of the penguin operators.
Therefore, there are no direct local $CP$ asymmetries in decays induced by the
$\bar{b}\to\bar{c}{u}\bar{q}$ transition in SM, because the penguin operators
do not involved in these decays.

It can be found from the factorization formula as shown in Eq.(1) that the
most important inputs are the wave functions of the initial and final mesons.
For the initial $B$ meson and the final $\overline{D}$ meson, their wave
functions have been well determined and adopted widely in the studies of two-
body $B_{(s)}\to PP,PV,VV$ [50, 53, 54, 55] and $B_{(s)}\to DP,DV,DS,DT$ [56,
57, 58, 59, 60, 61, 62] decays and other decay processes [63, 64, 65], and we
will not discuss them any more in this paper. In the quasi-two-body decays,
the new introduced parameters are the two-meson wave functions corresponding
to different resonances. In current work, according to the LHCb analysis [45],
we will consider the resonant contributions from the scalar resonance
$K^{*}_{0}(1430)$, the vector resonances $K^{*}(892)$ and $K^{*}(1410)$, and
the tensor resonance $K^{*}_{2}(1430)$ to the $B_{(s)}\to\overline{D}K\pi$
decays. So, three kinds of wave functions will be discussed in the following
sections. It should be noted that at present the forms of the two-meson wave
functions based on QCD-inspired approach are still absent, and many
phenomenological attempts have been performed to determine these forms and to
constrain the involved parameters based on the available experimental data.

We firstly discuss the $S$-wave $K\pi$ pair wave function $\Phi_{S,K\pi}$,
which can be given as [66],

$\displaystyle\Phi_{S,K\pi}=\frac{1}{2\sqrt{N_{c}}}\left[P\mkern-10.5mu/\phi_{S}(z,\zeta,\omega)+\omega\phi_{S}^{s}(z,\zeta,\omega)+\omega(n\mkern-10.5mu/v\mkern-10.5mu/-1)\phi_{S}^{t}(z,\zeta,\omega)\right],$
(5)

with $P$ and $\omega$ denoting the momentum and the invariant mass of the
$K\pi$ pair respectively, satisfying $P^{2}=\omega^{2}$. The light-like
vectors $n=(1,0,\mathbf{0}_{T})$ and $v=(0,1,\mathbf{0}_{T})$ are the
dimensionless vectors. $\phi_{S}$ and $\phi_{S}^{s,t}$ are the twist-2 and
twist-3 light-cone distribution amplitudes (LCDAs), respectively. The inner
parameter $z$ is the momentum fraction of the spectator quark, and $\xi$ is
the momentum fraction of the $K$ meson in the $K\pi$ pair. Similar to the LCDA
of the meson, the LCDAs of the $K\pi$ pair can also be decomposed into the
Gegenbauer polynomials together with the corresponding Gegenbauer moments,
which are given as

$\displaystyle\phi_{S}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\frac{6}{2\sqrt{2N_{C}}}F_{S}(\omega)z(1-z)\left[\frac{1}{\mu_{S}}+B_{1}C_{1}^{3/2}(1-2z)+B_{3}C_{3}^{3/2}(1-2z)\right],$
(6) $\displaystyle\phi_{S}^{s}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2N_{c}}}F_{S}(\omega),$ (7)
$\displaystyle\phi_{S}^{t}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2N_{C}}}F_{S}(\omega)(1-2z),$ (8)

with

$\displaystyle\mu_{S}=\frac{\omega}{m_{2}-m_{1}},$ (9)

where $m_{1,2}$ are the masses of the running current quarks in the resonance
$K_{0}^{*}(1430)$. $C_{1,3}^{3/2}(t)$ and $B_{1,3}$ are the Gegenbauer
polynomials and corresponding Gegenbauer moments respectively. For the twist-3
LCDAs, the asymptotic forms are adopted for simplicity. However, the values of
the inner parameters $B_{1,3}$ in twist-2 LCDA have not been determined by the
QCD-inspired approach so far. We determined them to be $B_{1}=-0.4\pm 0.2$ and
$B_{3}=0.7\pm 0.4$ by phenomenological method together with the available
experimental measurements of the $B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ decay
[45]. In ref.[29], we had investigated the $B_{s}\to
K^{0}(\overline{K}^{0})K^{\pm}\pi^{\mp}$ and fitted $B_{1}=-0.4\pm 0.2$ and
$B_{3}=-0.8\pm 0.4$. It is obvious that the values of $B_{1}$ are almost same
and the signs of $B_{3}$ are different. We also note that, in previous studies
of the charmless $B_{s}\to K^{0}(\overline{K}^{0})K^{\pm}\pi^{\mp}$ decays the
terms proportional to the fraction $(\frac{m_{i}}{M_{B}})^{2}$ have been
neglected since the mass of the final $K$ or $\pi$ is small enough to be
omitted, in comparison with the mass $M_{B}$ of $B$ meson. In this work, when
we consider the charmed $B$ meson decays with massive $\overline{D}$ meson
involved in final states, the terms proportional to the fraction
$(\frac{M_{D}}{M_{B}})^{2}$ can not be neglected any more, which will change
the behaviour of the propagator of the inner quarks. Compared with $B_{1}$,
$B_{3}$ has a large fluctuation in the fitting procedure. As a result, the new
fitted $B_{3}$ differs from the previous result, which is also confirmed in
ref.[67]. In ref. [68], these parameters were also taken as the same as those
in wave function of the $K_{0}^{*}(1430)$ meson with large uncertainties
considered.

Unlike the distribution amplitudes of meson, the time-like form factor
$F_{S}(\omega)$ is involved. In particular, this form factor is parameterized
by the relativistic Breit-Wigner (RBW) model which is adopted extensively in
experimental analysis, and has been proved to be a valid model for describing
the narrow resonance that can be well separated from the other resonant and
non-resonant contributions. However, RBW model fails to describe the
$F_{S}(\omega)$ associated with the $K_{0}^{*}(1430)$ resonance, because
$K_{0}^{*}(1430)$ resonance interferes strongly with a slowly varying non-
resonant contribution [69]. In order to overcome this deficiency, the so-
called LASS line shape [70, 71] is developed to describe the combined
contributions including the resonant and non-resonant parts, which explicit
expression is given as

$\displaystyle F_{S}(\omega)=\frac{\omega}{\mid
p_{1}\mid(\cot\delta-i)}+e^{2i\delta}\frac{m_{0}\Gamma_{0}\frac{m_{0}}{\mid
p_{0}\mid}}{m_{0}^{2}-\omega^{2}-im_{0}\Gamma_{0}\frac{\mid
p_{1}\mid}{\omega}\frac{m_{0}}{p_{0}}},$ (10)

where the phase $\delta$ is determined by $\cot\delta=\frac{1}{a\mid
p_{1}\mid}+\frac{r\mid p_{1}\mid}{2}$ with the scattering length $a=1.95\pm
0.09~{}{\rm GeV}^{-1}$ and the effective range $r=1.76\pm 0.36~{}{\rm
GeV}^{-1}$ [72]. The $|p_{1}|$ is the magnitude of the momentum of one of
daughter of the resonance $K_{0}^{*}(1430)$ in the center-of-mass frame of the
$K\pi$ pair and the $|p_{0}|$ is the value of the $|p_{1}|$ when
$\omega=m_{0}$. Finally, $m_{0}$ and $\Gamma_{0}$ are the pole mass and width
of the resonance $K_{0}^{*}(1430)$. From Eq.(10), one can easily find that the
first term represents the non-resonant contribution, while the second term is
the resonant one. Usually, a cutoff at $\omega=1.7~{}{\rm GeV}$ is suggested
for the nonresonant contributions, which implies that there is only resonant
contribution when the invariant mass is larger than $1.7~{}\rm GeV$. In this
paper we will also adopt the LASS model to evaluate the $S$-wave contributions
to the $B_{(s)}\to\overline{D}K\pi$ decays, which has also been adopted by
LHCb.

Phenomenologically, the wave function of the $P$-wave $K\pi$ pair can be
modeled from the wave function of the vector meson. Due to the law of
conservation of angular momentum, only the longitudinal polarization component
contributes to the decay amplitude of $B\to\overline{D}(R\to)K\pi$, so we only
keep the longitudinal wave function

$\displaystyle\Phi_{P,K\pi}=\frac{1}{\sqrt{2N_{C}}}\left[P\mkern-10.5mu/\phi_{P}(z,\xi,\omega)+\omega\phi_{P}^{s}(z,\xi,\omega)+\frac{P\mkern-10.5mu/_{1}P\mkern-10.5mu/_{2}-P\mkern-10.5mu/_{2}P\mkern-10.5mu/_{1}}{\omega(2\xi-1)}\phi_{P}^{t}(z,\xi,\omega)\right],$
(11)

where the expressions of twist-2 and twist-3 LCDAs are

$\displaystyle\phi_{P}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\parallel}(\omega)}{\sqrt{2N_{C}}}z(1-z)\left[1+a_{1}C_{1}^{3/2}(t)+a_{2}C_{2}^{3/2}(t)\right](2\xi-1),$
(12) $\displaystyle\phi_{P}^{s}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\perp}(\omega)}{2\sqrt{2N_{C}}}t(2\xi-1),$ (13)
$\displaystyle\phi_{P}^{t}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\perp}(\omega)}{2\sqrt{2N_{C}}}t^{2}(2\xi-1),$
(14)

with $t=1-2z$. For the twist-3 LCDAs $\phi_{P}^{s,t}$, we also take the
asymptotic forms. The Gegenbauer moments $a_{1,2}$ in the twist-2 LCDA are
taken as $a_{1}=-0.4$ and $a_{2}=0.46$ [67].

For the time-like form factor $F_{P}^{\parallel}$ corresponding to the
resonances $K^{*}(892)$ and $K^{*}(1410)$, it can be well modeled by the RBW
model, and be given as [71]

$\displaystyle F_{\rm
RBW}(\omega)=\frac{m_{0}^{2}}{m_{0}^{2}-\omega^{2}-im_{0}\Gamma(\omega)},$
(15)

with the nominal mass $m_{0}$ of the resonance. The mass-dependent width
$\Gamma(\omega)$ corresponding to a resonance with spin $L$ can be expressed
as [3]

$\displaystyle\Gamma(\omega)=\Gamma_{0}\left(\frac{|p_{1}|}{|p_{0}|}\right)^{2L+1}\left(\frac{m_{0}}{\omega}\right)X_{L}^{2}(r|p_{1}|),$
(16)

with the same definitions as the eq.(10) for the $\Gamma_{0}$, $|p_{1}|$ and
$|p_{0}|$. The $X_{L}$ is the Blatt-Weillkopf barrier factor [73], which
depends on the angular momentum of the meson pair, with the expression as

$\displaystyle L=0,X_{0}(a)=1,$ (17) $\displaystyle
L=1,X_{1}(a)=\sqrt{\frac{1+a_{0}^{2}}{1+a^{2}}},$ (18) $\displaystyle
L=2,X_{2}(a)=\sqrt{\frac{a_{0}^{4}+3a_{0}^{2}+9}{a^{4}+3a^{2}+9}},$ (19)

$a_{0}$ being the value of the $a$ at the pole mass of the resonance. The
inner parameter $r$ in eq.(16) reflects the effective barrier of the
resonance, and is usually taken to be $r=4~{}{\rm GeV}^{-1}\approx 0.8$ fm
[74] for each resonance, because it dose not affect the numerical results
remarkably. Currently, $X_{1}(a)$ is used for the $P$-wave longitudinal time-
like form factor, that is

$\displaystyle F_{P}^{\parallel}(\omega)=F_{\rm RBW}(\omega)\mid_{L=1}.$ (20)

For the transverse time-like form factor, $F_{P}^{\perp}(\omega)$ can be
obtained using the relation [25]

$\displaystyle\frac{F_{P}^{\perp}(\omega)}{F_{P}^{\parallel}(\omega)}\approx\frac{f_{V}^{\perp}}{f_{V}},$
(21)

with the decay constants $f_{V}^{(\perp)}$ of the vector resonance.

Now, we turn to discuss the wave function of the $D$-wave $K\pi$ pair. Because
the initial $B$ meson and final state $D$ meson are all pseudoscalars, the
helicity $\lambda=\pm 2$ components vanish due to the conservation of the
angular momentum. Therefore, the behavior of the $D$-wave $K\pi$ pair are very
similar to the $P$-wave one [75]. The only differences are the LCDAs
$\phi_{D}$, $\phi_{D}^{s}$, and $\phi_{D}^{t}$, whose expressions are

$\displaystyle\phi_{D}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\frac{6F_{D}^{\parallel}(\omega)}{2\sqrt{2N_{C}}}z(1-z)\left[3a_{D}(2z-1)\right]P_{2}(2\xi-1),$
(22) $\displaystyle\phi_{D}^{s}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\frac{-9F_{D}^{\perp}(\omega)}{4\sqrt{2N_{C}}}\left[a_{D}(1-6z+6z^{2})\right]P_{2}(2\xi-1),$
(23) $\displaystyle\phi_{D}^{t}(z,\xi,\omega)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\frac{9F_{D}^{\perp}(\omega)}{4\sqrt{2N_{C}}}\left[a_{D}(1-6z+6z^{2})(2z-1)\right]P_{2}(2\xi-1),$
(24)

where the Gegenbauer moment $a_{D}$ is determined to be 0.25 and function
$P_{2}(t)$ is the Legendre polynomial. $F_{D}^{\parallel}(\omega)$ and
$F_{D}^{\perp}(\omega)$ are the $D$-wave longitudinal and transverse time-like
form factors respectively, which can also be well parameterized by RBW model
with the spin-2 resonance $K_{2}^{*}(1430)$.

Figure 1: Typical Feynman diagrams for the quasi-two-body decay
$B^{0}\to\overline{D}^{0}(R\to)K\pi$ in PQCD, where the black squares stand
for the weak vertices, large (purple) spots on the quark lines denote possible
attachments of hard gluons, and the green ellipse represents $K\pi$-pair and
the red one is the light bachelor $D^{0}$ meson.

In light of the effective Hamiltonian eq.(3), we plot the Feynman diagrams
contributing to quasi-two-body $B_{(s)}\to\overline{D}K\pi$ decays at the
leading order in Figure 1. Based on the factorization formula and the wave
functions of initial and final states, we then calculate the total decay
amplitudes with the CKM matrix elements and the wilson coefficients, which are
presented as

$\displaystyle\mathcal{A}(B^{0}\to\overline{D}^{0}(K^{+}\pi-))$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{us}\left[(C_{1}+C_{2}/3)\mathcal{F}_{K\pi}^{LL}+C_{2}\mathcal{M}_{K\pi}^{LL}\right],$
(25) $\displaystyle\mathcal{A}(B^{0}\to D^{-}(K^{+}\pi^{0}))$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{us}\left[(C_{1}/3+C_{2})\mathcal{F}_{D}^{LL}+C_{1}\mathcal{M}_{D}^{LL}\right],$
(26) $\displaystyle\mathcal{A}(B^{0}\to D_{s}^{-}(K^{+}\pi^{0}))$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{ub}\left[(C_{1}+C_{2}/3)\mathcal{A}_{K\pi}^{LL}+C_{2}\mathcal{W}_{K\pi}^{LL}\right],$
(27) $\displaystyle\mathcal{A}(B^{+}\to\overline{D}^{0}(K^{+}\pi^{0}))$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{us}\left[(C_{1}/3+C_{2})\mathcal{F}_{D}^{LL}+C_{1}\mathcal{M}_{D}^{LL}\right.$
(28)
$\displaystyle\left.+(C_{1}+C_{2}/3)\mathcal{F}_{K\pi}^{LL}+C_{2}\mathcal{M}_{K\pi}^{LL}\right],$
$\displaystyle\mathcal{A}(B_{s}\to\overline{D}^{0}(K^{-}\pi^{+}))$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{ud}\left[(C_{1}+C_{2}/3)\mathcal{F}_{K\pi}^{LL}+C_{2}\mathcal{M}_{K\pi}^{LL}\right],$
(29) $\displaystyle\mathcal{A}(B_{s}\to D_{s}^{-}(K^{+}\pi^{0}))$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{us}\left[(C_{1}/3+C_{2})\mathcal{F}_{D}^{LL}+C_{1}\mathcal{M}_{D}^{LL}\right.$
(31)
$\displaystyle\left.+(C_{1}+C_{2}/3)\mathcal{A}_{K\pi}^{LL}+C_{2}\mathcal{W}_{K\pi}^{LL}\right].$

In above decay amplitudes, the symbols $\mathcal{F}_{K\pi(D)}^{LL}$,
$\mathcal{M}_{K\pi(D)}^{LL}$, $\mathcal{A}_{K\pi(D)}^{LL}$ and
$\mathcal{W}_{K\pi(D)}^{LL}$ denote the contributions from the factorizable
emission type diagrams, the hard-rescattering emission diagrams, the $W$ boson
annihilation type diagrams, and the $W$ boson exchange type diagrams,
respectively. The subscript $K\pi(D)$ represents the $K\pi$ pair ($D$ meson)
is recoiled, and the superscript $LL$ stands for the contributions coming from
$(V-A)\bigotimes(V-A)$ current. For the sake of simplicity, the expressions of
the $\mathcal{F}_{K\pi(D)}^{LL}$, $\mathcal{M}_{K\pi(D)}^{LL}$,
$\mathcal{A}_{K\pi(D)}^{LL}$ and $\mathcal{W}_{K\pi(D)}^{LL}$ are not given
any more, which can be obtained by making the following substitutions in the
results of ref. [28],

$\displaystyle\phi_{K}^{a,p}\to\phi_{D},\;\phi_{K}^{t}=0,\;r_{K}\to
r_{D}=m_{d}/m_{B}.$ (32)

We also point out that when considering the resonant contribution from the
$D$-wave resonance $K_{2}^{*}(1430)$, the contribution $\mathcal{F}_{D}^{LL}$
disappears due to the fact that the tensor structure can not be produced
through $(V-A)$ current.

##  3 Numerical Results and Discussions

In this section, we firstly list the parameters we used in the numerical
calculations, including the QCD scale, the masses and the lifetimes of the
$B/B_{s}$ mesons, the mass of the $\overline{D}$ meson, the masses and the
widths of the intermediate resonances, and the CKM matrix elements,

$\displaystyle\Lambda_{QCD}^{f=4}=0.25\pm 0.05~{}{\rm
GeV},\;m_{B}=5.279~{}{\rm GeV},\;m_{B_{s}}=5.366~{}{\rm GeV},$
$\displaystyle\tau_{B^{+}}=1.638~{}{\rm ps},\;\tau_{B^{0}}=1.519~{}{\rm
ps},\;\tau_{B_{s}}=1.520~{}{\rm ps},$ $\displaystyle m_{D^{0}}=1.865~{}{\rm
GeV},\;m_{D^{+}}=1.869~{}{\rm GeV},\;m_{D_{s}}=1.96~{}{\rm GeV},$
$\displaystyle m_{K^{*}(892)}=0.892~{}{\rm GeV},\;m_{K^{*}(1410)}=1.414~{}{\rm
GeV},\;m_{K_{0}^{*}(1430)}=1.425~{}{\rm GeV},$ $\displaystyle
m_{K^{*}_{2}(1430)}=1.427~{}{\rm GeV},\;\Gamma_{K^{*}(892)}=51.4~{}{\rm
MeV},\;\Gamma_{K^{*}(1410)}=232~{}{\rm MeV},$
$\displaystyle\Gamma_{K^{*}_{0}(1430)}=270~{}{\rm
MeV},\;\Gamma_{K_{2}^{*}(1430)}=100~{}{\rm MeV},$ $\displaystyle
V_{cb}=0.04182^{+0.00085}_{-0.00074},\;V_{ud}=0.97435\pm
0.00016,\;V_{us}=0.22500\pm 0.00067.$ (33)

In Table.1, we present the numerical results of quasi-two-body
$B_{(s)}\to\overline{D}K\pi$ decays, together with the experimental data from
LHCb [45]. In our calculations, we take three kinds of errors into accounts to
evaluate the theoretical uncertainties. The first uncertainties arise from the
nonperturbative parameters in the wave functions of the initial and final
states, such as the decay constants $f_{B}=(0.19\pm 0.02)~{}{\rm GeV}$ and
$f_{B_{s}}=(0.23\pm 0.02)~{}{\rm GeV}$ of $B$ and $B_{s}$ meson respectively;
the shape parameter $\omega_{B}=(0.4\pm 0.04)~{}{\rm GeV}$ and
$\omega_{B_{s}}=(0.5\pm 0.05)~{}{\rm GeV}$ for $B$ meson and $B_{s}$ meson;
the Gegenbauer moments in LCDAs of $K\pi$ pair and the $\overline{D}$ meson.
We emphasize that this kind of uncertainties is dominant, but these
uncertainties could be reduced with the improvements of the experimental
measurements and the developments of the nonperturbative approaches. The
second uncertainties origin from the unknown contributions from QCD radiative
corrections [76, 77] and the power corrections [78, 79], which are
characterized by varying the $\Lambda_{\rm QCD}=0.25\pm 0.05~{}{\rm GeV}$ and
the factorization scale $t$ from $0.8t$ to $1.2t$. In recent years, there are
some attempts to study these two kinds of corrections in two-body $B$ decays
[80, 81, 82], but the relate researches on three-body $B$ decays have not been
carried out yet. The last uncertainties are caused by the uncertainties of the
CKM matrix elements $V_{cb}$ and $V_{ud(s)}$, which have marginal effects on
the branching fractions.

Table 1: The flavour-averaged branching ratios (in $10^{-5}$) of $B_{(s)}\to\overline{D}K\pi$ decays with resonances $K^{*}(892)/(1410)$, $K_{0}^{*}(1430)$ and $K_{2}^{*}(1430)$ in PQCD approach, together with the experimental data [45] from LHCb. Decay Modes | PQCD | EXP
---|---|---
$B^{0}\to\overline{D}^{0}(K^{*0}(892)\to)K^{+}\pi^{-}$ | $1.71^{+1.00+0.33+0.02}_{-0.82-0.31-0.08}$ | ….
$B^{0}\to\overline{D}^{0}(K^{*0}(1410)\to)K^{+}\pi^{-}$ | $0.10^{+0.05+0.04+0.01}_{-0.04-0.03-0.01}$ | ….
$B^{0}\to\overline{D}^{0}K^{+}\pi^{-}(LASS)$ | $2.01^{+1.10+0.10+0.20}_{-0.86-0.44-0.02}$ | $.....$
$B^{0}\to\overline{D}^{0}(K_{0}^{*0}(1430)\to)K^{+}\pi^{-}$ | $1.58^{+0.81+0.27+0.15}_{-0.68-0.43-0.03}$ | $....$
$B^{0}\to\overline{D}^{0}K^{+}\pi^{-}(LASSNR)$ | $1.10^{+0.60+0.20+0.12}_{-0.47-0.26-0.00}$ | $.....$
$B^{0}\to\overline{D}^{0}(K_{2}^{*0}(1430)\to)K^{+}\pi^{-}$ | $0.25^{+0.15+0.06+0.03}_{-0.10-0.05-0.00}$ | $.....$
$B^{0}\to D^{-}(K^{*+}(892)\to)K^{+}\pi^{0}$ | $12.4^{+5.5+1.5+0.6}_{-4.0-1.8-0.5}$ | ….
$B^{0}\to D^{-}(K^{*+}(1410)\to)K^{+}\pi^{0}$ | $0.80^{+0.35+0.09+0.03}_{-0.25-1.10-0.03}$ | ….
$B^{0}\to D^{-}K^{+}\pi^{0}(LASS)$ | $0.18^{+0.08+0.00+0.01}_{-0.06-0.01-0.00}$ | $.....$
$B^{0}\to D^{-}(K_{0}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.12^{+0.06+0.02+0.01}_{-0.05-0.02-0.00}$ | $....$
$B^{0}\to D^{-}K^{+}\pi^{0}(LASSNR)$ | $0.10^{+0.05+0.01+0.00}_{-0.04-0.01-0.00}$ | $....$
$B^{0}\to D^{-}(K_{2}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.17^{+0.10+0.09+0.01}_{-0.08-0.08-0.00}$ | $...$
$B^{0}\to D_{s}^{-}(K^{*+}(892)\to)K^{+}\pi^{0}$ | $5.15^{+1.41+0.46+0.45}_{-1.19-0.76-0.15}$ | ….
$B^{0}\to D_{s}^{-}(K^{*+}(1410)\to)K^{+}\pi^{0}$ | $0.36^{+0.10+0.04+0.02}_{-0.07-0.04-0.01}$ | ….
$B^{0}\to D_{s}^{-}K^{+}\pi^{0}(LASS)$ | $0.23^{+0.11+0.03+0.02}_{-0.16-0.13-0.00}$ | $.....$
$B^{0}\to D_{s}^{-}(K_{0}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.21^{+0.10+0.03+0.01}_{-0.13-0.00-0.00}$ | $....$
$B^{0}\to D_{s}^{-}K^{+}\pi^{0}(LASSNR)$ | $0.12^{+0.06+0.01+0.01}_{-0.07-0.01-0.00}$ | $....$
$B^{0}\to D_{s}^{-}(K_{2}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.93^{+0.52+0.09+0.10}_{-0.36-0.06-0.00}$ | $...$
$B^{+}\to\overline{D}^{0}(K^{*+}(892)\to)K^{+}\pi^{0}$ | $16.7^{+7.1+3.2+0.7}_{-5.3-3.4-0.7}$ | ….
$B^{+}\to\overline{D}^{0}(K^{*+}(1410)\to)K^{+}\pi^{0}$ | $1.23^{+0.51+0.24+0.05}_{-0.40-0.25-0.05}$ | ….
$B^{+}\to\overline{D}^{0}K^{+}\pi^{0}(LASS)$ | $2.14^{+1.76+0.06+0.03}_{-1.08-0.27-0.00}$ | $.....$
$B^{+}\to\overline{D}^{0}(K_{0}^{*+}(1430)\to)K^{+}\pi^{0}$ | $1.03^{+0.61+0.19+0.11}_{-0.46-0.18-0.00}$ | $....$
$B^{+}\to\overline{D}^{0}K^{+}\pi^{0}(LASSNR)$ | $1.20^{+0.90+0.16+0.12}_{-0.63-0.20-0.00}$ | $....$
$B^{+}\to\overline{D}^{0}(K_{2}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.25^{+0.16+0.11+0.02}_{-0.10-0.08-0.00}$ | $...$
$B_{s}\to\overline{D}^{0}(\overline{K}^{*0}(892)\to)K^{-}\pi^{+}$ | $28.6^{+16.7+4.3+0.5}_{-13.3-5.6-0.8}$ | $28.6\pm 0.6\pm 0.7\pm 0.9\pm 4.2$
$B_{s}\to\overline{D}^{0}(\overline{K}^{*0}(1410)\to)K^{-}\pi^{+}$ | $1.74^{+0.97+0.54+0.03}_{-0.79-0.50-0.07}$ | $1.7\pm 0.5\pm 0.2\pm 1.4\pm 0.2$
$B_{s}\to\overline{D}^{0}K^{-}\pi^{+}(LASS)$ | $25.7^{+16.1+3.0+2.9}_{-11.1-6.1-0.0}$ | $21.4\pm 1.4\pm 1.0\pm 4.7\pm 3.1$
$B_{s}\to\overline{D}^{0}(\overline{K}_{0}^{*0}(1430)\to)K^{-}\pi^{+}$ | $19.7^{+11.0+3.7+0.22}_{-8.5-6.9-0.00}$ | $20.0\pm 1.6\pm 0.7\pm 3.3\pm 2.9$
$B_{s}\to\overline{D}^{0}K^{-}\pi^{+}(LASSNR)$ | $14.1^{+8.7+2.8+1.7}_{-6.2-3.5-0.0}$ | $13.7\pm 2.5\pm 1.5\pm 4.1\pm 2.0$
$B_{s}\to\overline{D}^{0}(\overline{K}_{2}^{*0}(1430)\to)K^{-}\pi^{+}$ | $3.95^{+2.41+1.13+0.46}_{-1.65-0.96-0.00}$ | $3.7\pm 0.6\pm 0.4\pm 1.1\pm 0.5$
$B_{s}\to D_{s}^{-}(K^{*+}(892)\to)K^{+}\pi^{0}$ | $11.1^{+4.5+2.0+0.5}_{-3.3-2.1-0.4}$ | ….
$B_{s}\to D_{s}^{-}(K^{*+}(1410)\to)K^{+}\pi^{0}$ | $0.66^{+0.27+0.13+0.03}_{-0.20-0.13-0.03}$ | ….
$B_{s}\to D_{s}^{-}K^{+}\pi^{0}(LASS)$ | $0.16^{+0.11+0.02+0.04}_{-0.05-0.03-0.00}$ | $.....$
$B_{s}\to D^{-}(K_{0}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.12^{+0.08+0.03+0.02}_{-0.04-0.02-0.00}$ | $....$
$B_{s}\to D^{-}K^{+}\pi^{0}(LASSNR)$ | $0.09^{+0.05+0.02+0.02}_{-0.03-0.01-0.00}$ | $....$
$B_{s}\to D_{s}^{-}(K_{2}^{*+}(1430)\to)K^{+}\pi^{0}$ | $0.24^{+0.16+0.13+0.02}_{-0.10-0.08-0.00}$ | $...$

From Table.1, it can be seen that for the
$B_{s}\to\overline{D}^{0}(K^{*}\to)K^{-}\pi^{+}$ decays the branching
fractions we calculated are in good agreement with the LHCb experimental
measurements that are based on a data sample corresponding to an integrated
luminosity of 3.0 $fb^{-1}$ of $pp$ collision data with respect to the
$K^{*}(892)$, $K^{*}(1410)$, $K_{0}^{*}(1430)$ and $K^{*}_{2}(1430)$
resonances. Using the fitted wave functions of $K\pi$ pair we also calculate
the branching fractions of other $B_{(s)}\to\overline{D}(K^{*}\to)K\pi$
decays, and they are expected to be tested in experiments in future. Due to
the enhancement by the large CKM elements, most branching fractions are at the
range $10^{-6}-10^{-4}$, which are measurable in LHCb and Belle-II
experiments.

For the quasi-two-body $B_{s}\to\overline{D}^{0}(R\to)K^{-}\pi^{+}$ decays
corresponding to the $S$, $P$ and $D$ wave intermediate resonances $K^{*}$,
the branching fractions are larger than the other corresponding decays,
because the quasi-two-body $B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ decays are
governed by the process $\bar{b}\to\bar{c}u\bar{d}$, which is enhanced by
$\mid\frac{V_{ud}}{V_{us}}\mid^{2}$ compared to the other decays such as
quasi-two-body $B^{0}\to\overline{D}^{0}K^{+}\pi^{-}$ decays governed by
$\bar{b}\to\bar{c}us$. As a result, the $B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$
decays are firstly measured in the LHCb experiment. However, for the quasi-
two-body $B^{0}\to D_{s}^{-}K^{+}\pi^{0}$ decays that are also governed by the
$\bar{b}\to\bar{c}ud$ transition and enhanced by the large CKM elements
$V_{cb}V_{ud}$, the branching fractions are far smaller than those of
$B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ decays, because they are pure
annihilation type decays. In particular, they are viewed as to be power
suppressed in comparison with the emission diagrams. For the
$B_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ and
$B^{0}\to\overline{D}^{0}K^{+}\pi^{-}$ decays with the $\overline{D}$ emitted,
the branching fractions corresponding to the $S$ and $D$ wave resonances are
also sizable or even similar to that of the $P$ wave resonance. However, the
$S$ and $D$ wave resonant branching fractions of the $B^{0}\to
D^{-}K^{+}\pi^{0}$, $B^{+}\to\overline{D}^{0}K^{+}\pi^{0}$ and $B_{s}\to
D_{s}^{-}K^{+}\pi^{0}$ decays with $K\pi$ pair emitted are much smaller than
that of $P$ wave resonant branching fractions, and it is because the
amplitudes with $S$ wave or $D$ wave $K-\pi$ pair emission are highly
suppressed or forbidden.

In light of SU(3) symmetry, we define a ratio between the quasi-two-body
$B_{s}\to\overline{D}^{0}(K^{*}\to)K^{-}\pi^{+}$ and
$B^{0}\to\overline{D}^{0}(K^{*}\to)K^{+}\pi^{-}$ decays, and the naive
estimation is given as

$\displaystyle
R=\frac{Br(B_{s}\to\overline{D}^{0}(K^{*}\to)K^{-}\pi^{+})}{Br(B^{0}\to\overline{D}^{0}(K^{*}\to)K^{+}\pi^{-})}\sim|\frac{V_{ud}}{V_{us}}|^{2}\frac{\tau_{B_{s}}}{\tau_{B^{0}}}\sim
18.$ (34)

Accordingly, we also evaluate these ratios from Table.1 as

$\displaystyle
R=\frac{Br(B_{s}\to\overline{D}^{0}(\bar{K}^{*}(892)\to)K^{-}\pi^{+})}{Br(B^{0}\to\overline{D}^{0}(K^{*}(892)\to)K^{+}\pi^{-})}\sim\frac{28.6^{+17.2}_{-14.4}}{1.71^{+1.05}_{-0.88}}\sim
16.7^{+14.3}_{-12.1},$ (35) $\displaystyle
R=\frac{Br(B_{s}\to\overline{D}^{0}(\bar{K}^{*}(1410)\to)K^{-}\pi^{+})}{Br(B^{0}\to\overline{D}^{0}(K^{*}(1410)\to)K^{+}\pi^{-})}\sim\frac{1.74^{+1.11}_{-0.93}}{0.10^{+0.06}_{-0.05}}\sim
17.4^{+15.2}_{-12.7},$ (36) $\displaystyle
R=\frac{Br(B_{s}\to\overline{D}^{0}(\bar{K}_{0}^{*}(1430)\to)K^{-}\pi^{+})}{Br(B^{0}\to\overline{D}^{0}(K_{0}^{*}(1430)\to)K^{+}\pi^{-})}\sim\frac{19.7_{-10.9}^{+11.6}}{1.58_{-0.80}^{+0.87}}\sim
12.5^{+10.0}_{-9.4},$ (37) $\displaystyle
R=\frac{Br(B_{s}\to\overline{D}^{0}(\bar{K}_{2}^{*}(1430)\to)K^{-}\pi^{+})}{Br(B^{0}\to\overline{D}^{0}(K_{2}^{*}(1430)\to)K^{+}\pi^{-})}\sim\frac{3.95_{-1.90}^{+2.70}}{0.25_{-0.11}^{+0.16}}\sim
16.0^{+14.7}_{-10.3}.$ (38)

It is obvious that there are large uncertainties in the results eqs. (35 \-
38). The acceptable deviation between our calculations and naive estimations
attributes to the complicate wave functions of the $K\pi$ pair. As
aforementioned, although there are some attempts to study the wave functions
of meson pair, the wave functions based on the first principle are still
absent. Therefore, we have to employ the phenomenological models that are
introduced in Sec.2. In this regard, the accuracy of wave functions of heavy
mesons should be further improved in future.

Table 2: The branching ratios (in $10^{-5}$) of $B_{(s)}\to\overline{D}K^{*}$ decays probed from the quasi-two-body $B_{(s)}\to\overline{D}(K^{*}\to)K\pi$ decays based on the narrow-width-approximation(NWA), together with the experimental data [83] and the former PQCD predictions from refs.[57, 59, 62] Decay Modes | NWA | EXP | former PQCD
---|---|---|---
$B^{0}\to\overline{D}^{0}K^{*0}(892)$ | $2.57^{+1.57}_{-1.32}$ | $4.5\pm 0.6$ | $2.6_{-0.9}^{+1.1}$
$B^{0}\to\overline{D}^{0}K^{*0}(1410)$ | $1.51^{+0.96}_{-0.81}$ | $<6.7$ | $...$
$B^{0}\to\overline{D}^{0}K_{0}^{*0}(1430)$ | $2.55^{+1.41}_{-1.33}$ | $0.7\pm 0.7$ | $3.19^{+3.21}_{-2.43}$
$B^{0}\to\overline{D}^{0}K_{2}^{*0}(1430)$ | $0.75^{+0.49}_{-0.33}$ | $2.1\pm 0.9$ | $1.45_{-0.50}^{+0.51}$
$B^{0}\to D^{-}K^{*+}(892)$ | $37.2^{+17.1}_{-13.2}$ | $45\pm 7$ | $38.3^{+13.5}_{-13.9}$
$B^{0}\to D^{-}K^{*+}(1410)$ | $36.3^{+17.8}_{-14.2}$ | … | …
$B^{0}\to D^{-}K_{0}^{*+}(1430)$ | $0.39^{+0.19}_{-0.17}$ | $....$ | $0.79_{-0.48}^{+0.40}$
$B^{0}\to D^{-}K_{2}^{*+}(1430)$ | $1.02^{+0.78}_{-0.66}$ | $...$ | $1.16_{-0.62}^{+0.72}$
$B^{0}\to D_{s}^{-}K^{*+}(892)$ | $15.4^{+4.6}_{-4.2}$ | $3.5\pm 1.0$ | $18.2^{+6.26}_{-6.45}$
$B^{0}\to D_{s}^{-}K^{*+}(1410)$ | $16.3^{+5.9}_{-4.8}$ | … | …
$B^{0}\to D_{s}^{-}K_{0}^{*+}(1430)$ | $0.67^{+0.33}_{-0.42}$ | $...$ | $1.41^{+0.70}_{-0.65}$
$B^{0}\to D_{s}^{-}K_{2}^{*+}(1430)$ | $5.59^{+3.18}_{-2.17}$ | $...$ | $6.06^{+1.81}_{-1.96}$
$B^{+}\to\overline{D}^{0}K^{*+}(892)$ | $50.1^{+23.4}_{-18.9}$ | $53\pm 4$ | $63.7^{+20.5}_{-21.6}$
$B^{+}\to\overline{D}^{0}K^{*+}(1410)$ | $55.9^{+27.4}_{-23.6}$ | … | …
$B^{+}\to\overline{D}^{0}K_{0}^{*+}(1430)$ | $3.32^{+2.14}_{-1.71}$ | $...$ | $4.72^{+2.46}_{-2.06}$
$B^{+}\to\overline{D}^{0}K_{2}^{*+}(1430)$ | $1.50^{+1.14}_{-0.78}$ | $...$ | $3.33^{+1.60}_{-1.38}$
$B_{s}\to\overline{D}^{0}\overline{K}^{*0}(892)$ | $42.9^{+25.8}_{-21.6}$ | $44\pm 6$ | $43.6^{+22.0}_{-17.4}$
$B_{s}\to\overline{D}^{0}\overline{K}^{*0}(1410)$ | $39.5^{+26.4}_{-22.5}$ | $39\pm 35$ | …
$B_{s}\to\overline{D}^{0}\overline{K}_{0}^{*0}(1430)$ | $31.9^{+19.0}_{-17.9}$ | $30\pm 7$ | $53.9^{+27.2}_{-23.8}$
$B_{s}\to\overline{D}^{0}\overline{K}_{2}^{*0}(1430)$ | $11.8^{+8.1}_{-5.7}$ | $11\pm 4$ | $20.3^{+8.7}_{-7.9}$
$B_{s}\to D_{s}^{-}K^{*+}(892)$ | $33.3^{+14.7}_{-11.7}$ | … | $28.1^{+16.7}_{-13.8}$
$B_{s}\to D_{s}^{-}K^{*+}(1410)$ | $30.0^{+14.8}_{-12.4}$ | … | …
$B_{s}\to D_{s}^{-}K_{0}^{*+}(1430)$ | $0.38^{+0.25}_{-0.13}$ | $....$ | $0.39_{-0.16}^{+0.21}$
$B_{s}\to D_{s}^{-}K_{2}^{*+}(1430)$ | $1.44^{+1.26}_{-0.78}$ | $...$ | $1.97_{-0.96}^{+1.09}$

Under the narrow-width-approximation (NWA), the quasi-two-body decay and
corresponding two-body process satisfy the factorization relation

$\displaystyle BF(B\to M_{1}R\to M_{1}M_{2}M_{3})\approx BF(B\to M_{1}R)\times
BF(R\to M_{2}M_{3}),$ (39)

where the $R$ represents the resonance. Therefore, we could extract the
branching fractions of the related two-body decays, by combining our numerical
results and the corresponding branching fractions of the resonances decay to
the $K\pi$. In turn, the comparison with the experimental data of two-body
decays can help us to further verify the reliability of the wave functions of
meson pair. In PDG [83], the branching fractions of the resonances decays to
$K\pi$ are summarized as follows,

$\displaystyle BF(K^{*}(892)\to K\pi)$ $\displaystyle\sim$ $\displaystyle 1,$
(40) $\displaystyle BF(K^{*}(1410)\to K\pi)$ $\displaystyle=$
$\displaystyle(6.6\pm 1.3)\%,$ (41) $\displaystyle BF(K^{*}_{0}(1430)\to
K\pi)$ $\displaystyle=$ $\displaystyle(93\pm 10)\%,$ (42) $\displaystyle
BF(K_{2}^{*}(1430)\to K\pi)$ $\displaystyle=$ $\displaystyle(49.9\pm 1.2)\%.$
(43)

Using above data and the results in Table.1, we roughly extract the branching
fractions of the corresponding two-body $B\to\overline{D}R$ decays and present
the results in Table. 2. It can be seen that most branching fractions
determined from NWA agree well with the available experimental data [83] and
the previous predictions [57, 59, 62] based on PQCD within the uncertainties.
Of course, if we wonder to test PQCD approach with these decays, the
uncertainties in both theoretical and experimental sides should be reduced in
future. We also note that for the pure annihilation $B^{0}\to
D_{s}^{-}K^{*+}(892)$ decay channel, the branching fraction
$(15.4^{+4.6}_{-4.2})\times 10^{-5}$ we obtained is much larger than the
experimental data $(3.5\pm 1.0)\times 10^{-5}$, though it is still in
agreement with the previous study $(18.2^{+6.3}_{-6.4})\times 10^{-5}$ [57].
In fact, the $B^{0}\to D_{s}^{-}K^{*+}(892)$ decay occurs only with the
annihilation diagrams, which are power suppressed. This deviation may indicate
that the contributions from high power corrections become important, and we
left this study to be our future work.

##  4 Summary

In this work we have investigated the resonant contributions of the quasi-two-
body $B\to DK\pi$ decays with the vector resonances $K^{*}(892)/(1410)$, the
scalar resonance $K_{0}^{*}(1430)$ and the tensor resonance $K_{2}^{*}(1430)$.
We first fitted the parameters in the wave functions of the $K\pi$ pair with
the experimental results from LHCb. With the wave functions we then calculated
the other quasi-two-body $B_{(s)}\to\overline{D}(K^{*}\to)K\pi$ decays. Most
branching fractions are at the range of $10^{-6}\sim 10^{-4}$, which is very
promising to be measured experimentally. With the narrow-width-approximation,
we also explored the branching fractions of the corresponding two-body decays,
and most results are in agreement with the available experimental data. We
hope our results to be further tested in experiments, so as to test the wave
functions of $K\pi$ pair and PQCD approach. Because all decays are governed by
only tree operators, there are no direct $CP$ asymmetries in these decays. If
large $CP$ asymmetry is observed in experiments, it would be a signal of new
physics beyond SM.

## Acknowledgment

This work is supported in part by the National Science Foundation of China
under the Grant No. 11705159, and the Natural Science Foundation of Shandong
province under the Grant No. ZR2022MA035 and No. ZR2019JQ04.

## References

  * [1] Y. Grossman, Z. Ligeti, and A. Soffer, Measuring $\gamma$ in $B^{\pm}\to K^{\pm}(KK^{*})_{D}$ decays, Phys. Rev. D 67 (2003) 071301, [hep-ph/0210433].
  * [2] BaBar Collaboration, B. Aubert et al., Dalitz Plot Analysis of $B^{\pm}\to\pi^{\pm}\pi^{\pm}\pi^{\mp}$ Decays, Phys. Rev. D 79 (2009) 072006, [arXiv:0902.2051].
  * [3] BaBar Collaboration, J. P. Lees et al., Study of CP violation in Dalitz-plot analyses of $B^{0}\to K^{+}K^{-}K^{0}_{S}$, $B^{+}\to K^{+}K^{-}K^{+}$, and $B^{+}\to K^{0}_{S}K^{0}_{S}K^{+}$, Phys. Rev. D85 (2012) 112010, [arXiv:1201.5897].
  * [4] BaBar Collaboration, B. Aubert et al., Time-dependent amplitude analysis of $B^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$, Phys. Rev. D80 (2009) 112001, [arXiv:0905.3615].
  * [5] BaBar Collaboration, J. P. Lees et al., Amplitude Analysis of $B^{0}\to K^{+}\pi^{-}\pi^{0}$ and Evidence of Direct CP Violation in $B\to K^{*}\pi$ decays, Phys. Rev. D 83 (2011) 112010, [arXiv:1105.0125].
  * [6] BaBar Collaboration, J. P. Lees et al., Amplitude analysis and measurement of the time-dependent CP asymmetry of $B^{0}\to K_{S}^{0}K_{S}^{0}K_{S}^{0}$ decays, Phys. Rev. D 85 (2012) 054023, [arXiv:1111.3636].
  * [7] BaBar Collaboration, B. Aubert et al., Evidence for Direct CP Violation from Dalitz-plot analysis of $B^{\pm}\to K^{\pm}\pi^{\mp}\pi^{\pm}$, Phys. Rev. D 78 (2008) 012004, [arXiv:0803.4451].
  * [8] Belle Collaboration, Y. Nakahama et al., Measurement of CP violating asymmetries in $B^{0}\to K^{+}K^{-}K^{0}_{S}$ decays with a time-dependent Dalitz approach, Phys. Rev. D 82 (2010) 073011, [arXiv:1007.3848].
  * [9] Belle Collaboration, J. Dalseno et al., Time-dependent Dalitz Plot Measurement of CP Parameters in $B^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ Decays, Phys. Rev. D 79 (2009) 072004, [arXiv:0811.3665].
  * [10] Belle Collaboration, A. Garmash et al., Dalitz Analysis of Three-body Charmless $B^{0}\to K^{0}\pi^{+}\pi^{-}$ Decay, Phys. Rev. D 75 (2007) 012006, [hep-ex/0610081].
  * [11] Belle Collaboration, A. Garmash et al., Evidence for large direct CP violation in $B^{\pm}\to\rho(770)^{0}K^{\pm}$ from analysis of the three-body charmless $B^{\pm}\to K^{\pm}\pi^{\pm}\pi^{\mp}$ decay, Phys. Rev. Lett. 96 (2006) 251803, [hep-ex/0512066].
  * [12] LHCb Collaboration, R. Aaij et al., Amplitude analysis of the $B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decay, Phys. Rev. D 101 (2020), no. 1 012006, [arXiv:1909.05212].
  * [13] LHCb Collaboration, R. Aaij et al., Observation of Several Sources of $CP$ Violation in $B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ Decays, Phys. Rev. Lett. 124 (2020), no. 3 031801, [arXiv:1909.05211].
  * [14] LHCb Collaboration, R. Aaij et al., Resonances and $CP$ violation in $B_{s}^{0}$ and $\overline{B}_{s}^{0}\to J/\psi K^{+}K^{-}$ decays in the mass region above the $\phi(1020)$, JHEP 08 (2017) 037, [arXiv:1704.08217].
  * [15] LHCb Collaboration, R. Aaij et al., Amplitude analysis of $B^{0}_{s}\rightarrow K^{0}_{\textrm{S}}K^{\pm}\pi^{\mp}$ decays, JHEP 06 (2019) 114, [arXiv:1902.07955].
  * [16] LHCb Collaboration, R. Aaij et al., Observation of the decay $B^{0}_{s}\to\phi\pi^{+}\pi^{-}$ and evidence for $B^{0}\to\phi\pi^{+}\pi^{-}$, Phys. Rev. D 95 (2017), no. 1 012006, [arXiv:1610.05187].
  * [17] B. El-Bennich, A. Furman, R. Kaminski, L. Lesniak, B. Loiseau, and B. Moussallam, CP violation and kaon-pion interactions in $B\to K\pi^{+}\pi^{-}$ decays, Phys. Rev. D 79 (2009) 094005, [arXiv:0902.3645]. [Erratum: Phys.Rev.D 83, 039903 (2011)].
  * [18] S. Kränkl, T. Mannel, and J. Virto, Three-body non-leptonic B decays and QCD factorization, Nucl. Phys. B 899 (2015) 247–264, [arXiv:1505.04111].
  * [19] H.-Y. Cheng and K.-C. Yang, Nonresonant three-body decays of D and B mesons, Phys. Rev. D 66 (2002) 054015, [hep-ph/0205133].
  * [20] H.-Y. Cheng, C.-K. Chua, and Z.-Q. Zhang, Direct CP Violation in Charmless Three-body Decays of $B$ Mesons, Phys. Rev. D 94 (2016), no. 9 094015, [arXiv:1607.08313].
  * [21] H.-Y. Cheng and C.-K. Chua, Charmless three-body decays of $B_{s}$ mesons, Phys. Rev. D 89 (2014), no. 7 074025, [arXiv:1401.5514].
  * [22] Y. Li, Comprehensive study of $\overline{B}^{0}\to K^{0}(\overline{K}^{0})K^{\mp}\pi^{\pm}$ decays in the factorization approach, Phys. Rev. D 89 (2014), no. 9 094007, [arXiv:1402.6052].
  * [23] Y. Li, Branching Fractions and Direct $CP$ Asymmetries of $\bar{B}_{s}^{0}\to K^{0}h^{+}h^{\prime-}(h^{(\prime)}=K,\pi)$ Decays, Sci. China Phys. Mech. Astron. 58 (2015), no. 3 031001, [arXiv:1401.5948].
  * [24] W.-F. Wang, H.-C. Hu, H.-n. Li, and C.-D. Lü, Direct CP asymmetries of three-body $B$ decays in perturbative QCD, Phys. Rev. D 89 (2014), no. 7 074031, [arXiv:1402.5280].
  * [25] W.-F. Wang and H.-n. Li, Quasi-two-body decays $B\to K\rho\to K\pi\pi$ in perturbative QCD approach, Phys. Lett. B 763 (2016) 29–39, [arXiv:1609.04614].
  * [26] Y. Li, A.-J. Ma, W.-F. Wang, and Z.-J. Xiao, Quasi-two-body decays $B_{(s)}\to P\rho\to P\pi\pi$ in perturbative QCD approach, Phys. Rev. D 95 (2017), no. 5 056008, [arXiv:1612.05934].
  * [27] Z. Rui, Y. Li, and W.-F. Wang, The S-wave resonance contributions in the $B^{0}_{s}$ decays into $\psi(2S,3S)$ plus pion pair, Eur. Phys. J. C 77 (2017), no. 3 199, [arXiv:1701.02941].
  * [28] Z.-T. Zou, Y. Li, Q.-X. Li, and X. Liu, Resonant contributions to three-body $B\rightarrow KKK$ decays in perturbative QCD approach, Eur. Phys. J. C 80 (2020), no. 5 394, [arXiv:2003.03754].
  * [29] Z.-T. Zou, Y. Li, and X. Liu, Branching fractions and CP asymmetries of the quasi-two-body decays in $B_{s}\rightarrow K^{0}({\overline{K}}^{0})K^{\pm}\pi^{\mp}$ within PQCD approach, Eur. Phys. J. C 80 (2020), no. 6 517, [arXiv:2005.02097].
  * [30] Z.-T. Zou, Y. Li, and H.-n. Li, Is $f_{X}(1500)$ observed in the $B\to\pi(K)KK$ decays $\rho^{0}(1450)$?, Phys. Rev. D 103 (2021), no. 1 013005, [arXiv:2007.13141].
  * [31] Z.-T. Zou, L. Yang, Y. Li, and X. Liu, Study of Quasi-two-body $B_{(s)}\to\phi(f_{0}(980)/f_{2}(1270)\to)\pi\pi$ Decays in Perturbative QCD Approach, Eur. Phys. J. C 81 (2021), no. 1 91, [arXiv:2011.07676].
  * [32] L. Yang, Z.-T. Zou, Y. Li, X. Liu, and C.-H. Li, Quasi-two-body $B_{(s)}\to V\pi\pi$ decays with resonance $f_{0}(980)$ in the PQCD approach, Phys. Rev. D 103 (2021), no. 11 113005, [arXiv:2103.15031].
  * [33] W.-F. Liu, Z.-T. Zou, and Y. Li, Charmless Quasi-Two-Body B Decays in Perturbative QCD Approach: Taking $B\to K(R\to K+K^{-})$ as Examples, Adv. High Energy Phys. 2022 (2022) 5287693, [arXiv:2112.00315].
  * [34] Z.-H. Zhang, X.-H. Guo, and Y.-D. Yang, CP violation in $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ in the region with low invariant mass of one $\pi^{+}\pi^{-}$ pair, Phys. Rev. D 87 (2013), no. 7 076007, [arXiv:1303.3676].
  * [35] B. El-Bennich, A. Furman, R. Kaminski, L. Lesniak, and B. Loiseau, Interference between $f_{0}(980)$ and $\rho(770)^{0}$ resonances in $B\to\pi^{+}\pi^{-}K$ decays, Phys. Rev. D 74 (2006) 114009, [hep-ph/0608205].
  * [36] S. Cheng and J.-M. Shen, $\bar{B}_{s}\rightarrow f_{0}(980)$ form factors and the width effect from light-cone sum rules, Eur. Phys. J. C 80 (2020), no. 6 554, [arXiv:1907.08401].
  * [37] R. Hu and Z.-H. Zhang, Data-based analysis of the forward-backward asymmetry in $B^{\pm}\to K^{\pm}K^{\mp}K^{\pm}$, Phys. Rev. D 105 (2022), no. 9 093007, [arXiv:2201.07456].
  * [38] A. E. Snyder and H. R. Quinn, Measuring CP asymmetry in $B\to\rho\pi$ decays without ambiguities, Phys. Rev. D 48 (1993) 2139–2144.
  * [39] H.-Y. Cheng, C.-K. Chua, and A. Soni, Charmless three-body decays of B mesons, Phys. Rev. D 76 (2007) 094006, [arXiv:0704.1049].
  * [40] H.-Y. Cheng, Theoretical Overview of Hadronic Three-body B Decays, [arXiv:0806.2895].
  * [41] D. Herndon, P. Soding, and R. J. Cashmore, A GENERALIZED ISOBAR MODEL FORMALISM, Phys. Rev. D 11 (1975) 3165.
  * [42] S. U. Chung, J. Brose, R. Hackmann, E. Klempt, S. Spanier, and C. Strassburger, Partial wave analysis in K matrix formalism, Annalen Phys. 4 (1995) 404–430.
  * [43] Belle Collaboration, A. Garmash et al., Dalitz analysis of the three-body charmless decays $B^{+}\to K^{+}\pi^{+}\pi^{-}$ and $B^{+}\to K^{+}K^{+}K^{-}$ , Phys. Rev. D 71 (2005) 092003, [hep-ex/0412066].
  * [44] BaBar Collaboration, B. Aubert et al., Dalitz Plot Analysis of $B^{-}\to D^{+}\pi^{-}pi^{-}$, Phys. Rev. D 79 (2009) 112004, [arXiv:0901.1291].
  * [45] LHCb Collaboration, R. Aaij et al., Dalitz plot analysis of $B_{s}^{0}\rightarrow\bar{D}^{0}K^{-}\pi^{+}$ decays, Phys. Rev. D 90 (2014), no. 7 072003, [arXiv:1407.7712].
  * [46] LHCb Collaboration, R. Aaij et al., Observation of the decay $B_{s}^{0}\to\overline{D}^{0}K^{+}K^{-}$, Phys. Rev. D 98 (2018), no. 7 072006, [arXiv:1807.01891].
  * [47] LHCb Collaboration, R. Aaij et al., Measurement of the branching fractions of the decays $B^{0}_{s}\to\overline{D}^{0}K^{-}\pi^{+}$ and $B^{0}\to\overline{D}^{0}K^{+}\pi^{-}$, Phys. Rev. D 87 (2013), no. 11 112009, [arXiv:1304.6317].
  * [48] Y.-Y. Keum and H.-n. Li, Nonleptonic charmless $B$ decays: Factorization versus perturbative QCD, Phys. Rev. D 63 (2001) 074006, [hep-ph/0006001].
  * [49] Y.-Y. Keum, H.-n. Li, and A. I. Sanda, Fat penguins and imaginary penguins in perturbative QCD, Phys. Lett. B 504 (2001) 6–14, [hep-ph/0004004].
  * [50] C.-D. Lu, K. Ukai, and M.-Z. Yang, Branching ratio and CP violation of $B\to\pi\pi$ decays in perturbative QCD approach, Phys. Rev. D 63 (2001) 074009, [hep-ph/0004213].
  * [51] H.-n. Li, QCD aspects of exclusive B meson decays, Prog. Part. Nucl. Phys. 51 (2003) 85–171, [hep-ph/0303116].
  * [52] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125–1144, [hep-ph/9512380].
  * [53] X.-Q. Yu, Y. Li, and C.-D. Lu, Branching ratio and CP violation of $B_{s}\to\pi K$ decays in the perturbative QCD approach, Phys. Rev. D 71 (2005) 074026, [hep-ph/0501152]. [Erratum: Phys.Rev.D 72, 119903 (2005)].
  * [54] A. Ali, G. Kramer, Y. Li, C.-D. Lu, Y.-L. Shen, W. Wang, and Y.-M. Wang, Charmless non-leptonic $B_{s}$ decays to $PP$, $PV$ and $VV$ final states in the pQCD approach, Phys. Rev. D 76 (2007) 074018, [hep-ph/0703162].
  * [55] Z.-T. Zou, A. Ali, C.-D. Lu, X. Liu, and Y. Li, Improved Estimates of The $B_{(s)}\to VV$ Decays in Perturbative QCD Approach, Phys. Rev. D 91 (2015) 054033, [arXiv:1501.00784].
  * [56] H. Zou, R.-H. Li, X.-X. Wang, and C.-D. Lu, The CKM suppressed $B(B_{s})\to\overline{D}_{(s)}P,\overline{D}_{(s)}V,\overline{D}_{(s)}^{*}P,\overline{D}_{(s)}^{*}V$ decays in perturbative QCD approach, J. Phys. G 37 (2010) 015002, [arXiv:0908.1856].
  * [57] R.-H. Li, C.-D. Lu, and H. Zou, The $B(B_{s}))\to D_{(s)}P,D_{(s)}V,D_{(s)}^{*}P$ and $D_{(s)}^{*}V$ decays in the perturbative QCD approach, Phys. Rev. D 78 (2008) 014018, [arXiv:0803.1073].
  * [58] Z.-T. Zou, Y. Li, and X. Liu, Study of $B_{c}\to DS$ decays in the perturbative QCD approach, Phys. Rev. D 97 (2018), no. 5 053005, [arXiv:1712.02239].
  * [59] Z.-T. Zou, Y. Li, and X. Liu, Cabibbo-Kobayashi-Maskawa-favored $B$ decays to a scalar meson and a $D$ meson, Eur. Phys. J. C 77 (2017), no. 12 870, [arXiv:1704.03967].
  * [60] Z.-T. Zou, Y. Li, and X. liu, Two-body charmed $B_{(s)}$ decays involving a light scalar meson, Phys. Rev. D 95 (2017), no. 1 016011, [arXiv:1609.06444].
  * [61] Z.-T. Zou, X. Yu, and C.-D. Lu, The $B_{c}\to D^{(*)}T$ decays in perturbative QCD approach, Phys. Rev. D 87 (2013) 074027, [arXiv:1208.4252].
  * [62] Z.-T. Zou, X. Yu, and C.-D. Lu, The $B(B_{s})\rightarrow D_{(s)}(\bar{D}_{(s)})T$ and $D_{(s)}^{*}(\bar{D}_{(s)}^{*})T$ Decays in Perturbative QCD Approach, Phys. Rev. D 86 (2012) 094001, [arXiv:1205.2971].
  * [63] Z.-T. Zou, X. Yu, and C.-D. Lu, Nonleptonic two-body charmless B decays involving a tensor meson in the Perturbative QCD Approach, Phys. Rev. D 86 (2012) 094015, [arXiv:1203.4120].
  * [64] X. Liu, Z.-J. Xiao, and Z.-T. Zou, Branching ratios and CP violations of $B\to K^{*}_{0}(1430)K^{*}$ decays in the perturbative QCD approach, Phys. Rev. D 88 (2013), no. 9 094003, [arXiv:1309.7256].
  * [65] C. Wang, Q.-A. Zhang, Y. Li, and C.-D. Lu, Charmless $B_{(s)}\to VV$ Decays in Factorization-Assisted Topological-Amplitude Approach, Eur. Phys. J. C 77 (2017), no. 5 333, [arXiv:1701.01300].
  * [66] Y. Li, D.-C. Yan, Z. Rui, and Z.-J. Xiao, $S$, $P$ and $D$-wave resonance contributions to $B_{(s)}\to\eta_{c}(1S,2S)K\pi$ decays in the perturbative QCD approach, Phys. Rev. D 101 (2020), no. 1 016015, [arXiv:1911.09348].
  * [67] Y. Li, D.-C. Yan, J. Hua, Z. Rui, and H.-n. Li, Global determination of two-meson distribution amplitudes from three-body B decays in the perturbative QCD approach, Phys. Rev. D 104 (2021), no. 9 096014, [arXiv:2105.03899].
  * [68] W.-F. Wang, J. Chai, and A.-J. Ma, Contributions of $K^{*}_{0}(1430)$ and $K^{*}_{0}(1950)$ in the three-body decays $B\to K\pi h$, JHEP 03 (2020) 162, [arXiv:2001.00355].
  * [69] B. Meadows, Low Mass S-wave $K\pi$ and $\pi\pi$ System, eConf C070805 (2007) 27, [arXiv:0712.1605].
  * [70] D. Aston et al., A Study of $K^{-}\pi^{+}$ Scattering in the Reaction $K^{-}p\to K^{-}\pi^{+}n$ at 11-GeV/c, Nucl. Phys. B 296 (1988) 493–526.
  * [71] J. Back et al., LAURA++: A Dalitz plot fitter, Comput. Phys. Commun. 231 (2018) 198–242, [arXiv:1711.09854].
  * [72] W. Dunwoodie, Fits to $K\pi$ $I=1/2$ S-wave amplitude and phase data, http://www.slac.stanford.edu/wmd/kpiswave/ kpiswavefit.note, .
  * [73] J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics.  Springer, New York, 1952.
  * [74] BaBar Collaboration, B. Aubert et al., Dalitz-plot analysis of the decays $B^{\pm}\to K^{\pm}\pi^{\mp}\pi^{\pm}$, Phys. Rev. D 72 (2005) 072003, [hep-ex/0507004]. [Erratum: Phys.Rev.D 74, 099903 (2006)].
  * [75] H.-Y. Cheng, Y. Koike, and K.-C. Yang, Two-parton Light-cone Distribution Amplitudes of Tensor Mesons, Phys. Rev. D 82 (2010) 054019, [arXiv:1007.3541].
  * [76] H.-n. Li, Y.-L. Shen, and Y.-M. Wang, Next-to-leading-order corrections to $B\to\pi$ form factors in $k_{T}$ factorization, Phys. Rev. D 85 (2012) 074004, [arXiv:1201.5066].
  * [77] H.-n. Li, Y.-L. Shen, Y.-M. Wang, and H. Zou, Next-to-leading-order correction to pion form factor in $k_{T}$ factorization, Phys. Rev. D 83 (2011) 054029, [arXiv:1012.4098].
  * [78] Y.-L. Shen, Z.-T. Zou, and Y.-B. Wei, Subleading power corrections to $B\to\gamma l\nu$ decay in PQCD approach, Phys. Rev. D 99 (2019), no. 1 016004, [arXiv:1811.08250].
  * [79] Y.-L. Shen, Z.-T. Zou, and Y. Li, Power Corrections to Pion Transition Form Factor in Perturbative QCD Approach, Phys. Rev. D 100 (2019), no. 1 016022, [arXiv:1901.05244].
  * [80] S. Cheng and Z.-J. Xiao, The PQCD approach towards to next-to-leading order: A short review, Front. Phys. (Beijing) 16 (2021), no. 2 24201, [arXiv:2009.02872].
  * [81] C.-D. Lü, Y.-L. Shen, C. Wang, and Y.-M. Wang, Enhanced Next-to-Leading-Order Corrections to Weak Annihilation $B$-Meson Decays, [arXiv:2202.08073].
  * [82] J. Chai, S. Cheng, Y.-h. Ju, D.-C. Yan, C.-D. Lü, and Z.-J. Xiao, Charmless two-body $B$ meson decays in perturbative QCD factorization approach, [arXiv:2207.04190].
  * [83] Particle Data Group Collaboration, R. L. Workman, Review of Particle Physics, PTEP 2022 (2022) 083C01.


# On certain subclasses of the close-to-convex functions related with the
second-order differential subordination

H. Mahzoon and R. Kargar∗ Department of Mathematics, Islamic Azad University,
Firoozkouh Branch, Firoozkouh, Iran<EMAIL_ADDRESS>(H. Mahzoon)
Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University,
Ardabil, Iran<EMAIL_ADDRESS>(R. Kargar)

###### Abstract.

Let $\mathcal{A}$ be the family of analytic and normalized functions in the
open unit disc $|z|<1$. In this article we consider the following classes

$\mathcal{R}(\alpha,\beta):=\left\\{f\in\mathcal{A}:{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}>\beta,\,|z|<1\right\\}$

and

$\mathcal{L}_{\alpha}(b):=\left\\{f\in\mathcal{A}:\left|f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)-b\right|<b,\,|z|<1\right\\},$

where $-\pi<\alpha\leq\pi$, $0\leq\beta<1$ and $b>1/2$. We show that if
$f\in\mathcal{R}(\alpha,\beta)$, then ${\rm Re}\\{f^{\prime}(z)\\}$ and ${\rm
Re}\\{f(z)/z\\}$ are greater than $\beta$, and if
$f\in\mathcal{L}_{\alpha}(b)$, then $0<{\rm Re}\\{f^{\prime}(z)\\}<2b$. Also,
some another interesting properties of the class $\mathcal{L}_{\alpha}(b)$ are
investigated. Finally, the radius of univalence of 2-th section sum of
$f\in\mathcal{R}(\alpha,\beta)$ is obtained.

###### Key words and phrases:

Univalent; Positive real part; Circular domains; Close-to-convex; Marx-
Strohhäcker problem; Section sum.
*Corresponding Author

###### 2010 Mathematics Subject Classification:

30C45

##  1\. Introduction

Let $\Delta:=\\{z\in\mathbb{C}:|z|<1\\}$ where $\mathbb{C}$ is the complex
plane. We denote by $\mathcal{B}$ the class of all analytic functions $w(z)$
in $\Delta$ with $w(0)=0$ and $|w(z)|<1$, and denote by $\mathcal{A}$ the
class of all functions that are analytic and normalized in $\Delta$. The
subclass of $\mathcal{A}$ consisting of univalent functions in $\Delta$ is
denoted by $\mathcal{S}$. For functions $f$ and $g$ belonging to the class
$\mathcal{A}$, we say that $f$ is subordinate to $g$ in the unit disk
$\Delta$, written $f(z)\prec g(z)$ or $f\prec g$, if and only if there exists
a function $w\in\mathcal{B}$ such that $f(z)=g(w(z))$ for all $z\in\Delta$. In
particular, if $g$ is univalent function in $\Delta$, then we have the
following relation

$f(z)\prec g(z)\Leftrightarrow f(0)=g(0)\quad{\rm and}\quad f(\Delta)\subset
g(\Delta).$

Denote by $\mathcal{S}^{*}$ and $\mathcal{K}$ the set of all starlike and
convex functions in $\Delta$, respectively. A function $f\in\mathcal{A}$ is
said to be close-to-convex, if there exists a convex function $g$ and
$\delta\in\mathbb{R}$ such that

${\rm
Re}\left\\{e^{i\delta}\frac{f^{\prime}(z)}{g^{\prime}(z)}\right\\}>0\quad(z\in\Delta).$

The functions class which satisfy the last condition was introduced by Kaplan
in [6] and we denote by $\mathcal{CK}$. It is clear that if we take
$g(z)\equiv z$ in the class $\mathcal{CK}$, then we have the Noshiro-
Warschawski class as follows

$\mathcal{C}:=\left\\{f\in\mathcal{A}:\exists\delta\in\mathbb{R};{\rm
Re}\left\\{e^{i\delta}f^{\prime}(z)\right\\}>0,\,z\in\Delta\right\\}.$

By the basic Noshiro-Warschawski lemma [1, §2.6], we have
$\mathcal{C}\subset\mathcal{S}$.

Here, we recall from [15], two certain subclasses of analytic functions as
follows

$\mathcal{L}_{\alpha}:=\left\\{f\in\mathcal{A}:{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}>0,\,z\in\Delta\right\\}$

and

$\mathcal{L}_{\alpha}(b):=\left\\{f\in\mathcal{A}:\left|f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)-b\right|<b,\,z\in\Delta\right\\},$

where $\alpha\in(-\pi,\pi]$ and $b>1/2$. Notice that if $b\rightarrow\infty$,
then $\mathcal{L}_{\alpha}(b)\rightarrow\mathcal{L}_{\alpha}$. Also,
$\mathcal{L}_{\pi}$ contains $\mathcal{L}_{\alpha}$ for each $\alpha$. On the
other hand, Trojnar-Spelina [19] showed that
$\mathcal{L}_{\alpha}(b)\subset\mathcal{L}_{\pi}$, for every
$\alpha\in(-\pi,\pi]$ and $b\geq 1$.

By definition of subordination and this fact that the image of the function

(1.1)
$\phi_{b}(z)=\frac{1+z}{1+\left(\frac{1}{b}-1\right)z}\quad(z\in\Delta,\,b>1/2),$

is $\\{w\in\mathbb{C}:|w-b|<b\\}$ (see Figure 1 for $b=3/2$), we have the
following lemma.

######  Lemma 1.1.

(see [19]) A necessary and sufficient condition for $f$ to be in the class
$\mathcal{L}_{\alpha}(b)$ is

$f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\prec\phi_{b}(z)\quad(z\in\Delta),$

where $\phi_{b}$ is given by (1.1).

Figure 1. The boundary curve of $\phi_{3/2}(\Delta)$

It is necessary to point out that the class $\mathcal{R}_{\alpha}(\varphi)$
including of all normalized analytic functions in $\Delta$ satisfying the
following differential subordination

$f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\prec\varphi(z)\quad(z\in\Delta),$

was studied extensively by Srivastava $et$ $al.$ (see [16]), where the
function $\varphi$ is analytic in the open unit disc $\Delta$ such that
$\varphi(0)=1$. Also, Chichra [2] studied the class of all functions whose
derivative has positive real part in the unit disc $\Delta$. Indeed, he
denoted by $\mathcal{F}_{\gamma}$ the class of functions $f\in\mathcal{A}$
which satisfying the following inequality

${\rm Re}\left\\{f^{\prime}(z)+\gamma
zf^{\prime\prime}(z)\right\\}>0\quad(z\in\Delta),$

where $\gamma\geq 0$, and showed that
$\mathcal{F}_{\gamma}\subset\mathcal{S}$. Also, he proved that if
$f\in\mathcal{F}_{\gamma}$ and ${\rm Re}\\{\gamma\\}\geq 0$, then ${\rm
Re}\\{f^{\prime}(z)\\}>0$ in $\Delta$. Recent result, also was obtained by
Lewandowski et al. in [7].

On the other hand, Gao and Zhou [3] considered the class $R(\beta,\gamma)$ as
follows:

$R(\beta,\gamma)=\left\\{f\in\mathcal{A}:{\rm Re}\left\\{f^{\prime}(z)+\gamma
zf^{\prime\prime}(z)\right\\}>\beta,\ \
\gamma>0,\,\beta<1,\,z\in\Delta\right\\}.$

They found the extreme points of $R(\beta,\gamma)$, some sharp bounds of
certain linear problems, the sharp bounds for ${\rm Re}\\{f^{\prime}(z)\\}$
and ${\rm Re}\\{f(z)/z\\}$ and determined the number $\beta(\gamma)$ such that
$R(\beta,\gamma)\subset\mathcal{S}^{*}$, where $\gamma$ is certain fixed
number in $[1,\infty)$. Also, the class $R(\beta,\gamma)$ was studied by
Ponnusamy and Singh when ${\rm Re}\\{\gamma\\}>0$, see [11].

Motivated by the above classes, we define the class of all functions
$f\in\mathcal{A}$, denoted by $\mathcal{R}(\alpha,\beta)$ which satisfy the
condition

${\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}>\beta\quad(z\in\Delta),$

where $0\leq\beta<1$ and $-\pi<\alpha\leq\pi$. It is obvious that
$\mathcal{R}(\pi,\beta)$ becomes the class $\mathcal{C}(\beta)$, where

$\mathcal{C}(\beta):=\left\\{f\in\mathcal{A}:{\rm
Re}\\{f^{\prime}(z)\\}>\beta,z\in\Delta,0\leq\beta<1\right\\}.$

The class $\mathcal{C}(\beta)$ was considered in [4] and
$\mathcal{C}(\beta)\subset\mathcal{S}$ when $0\leq\beta<1$. It follows from
[2, Theorem 5] that
$\mathcal{R}(\alpha,0)\subset\mathcal{R}(\pi,0)\equiv\mathcal{C}(0)\equiv\mathcal{C}$.
The class $\mathcal{R}(0,0)$ studied by Singh and Singh [12], and they showed
that $\mathcal{R}(0,0)\subset\mathcal{S}^{*}$ [13]. Also, they found for
$f\in\mathcal{R}(0,0)$ and $z\in\Delta$ that ${\rm Re}\\{f(z)/z\\}>1/2$ and
$\mathcal{R}(0,\beta)\subset\mathcal{S}^{*}$ for $\beta\geq-1/4$. Silverman in
[14] improved this lower bound. He showed that
$\mathcal{R}(0,\beta)\subset\mathcal{S}^{*}$ for $\beta\geq-0.2738$ and also
found the smallest $\beta$ ($\beta\geq-0.63$) for which
$\mathcal{R}(0,\beta)\subset\mathcal{S}$.

Since the function $z\mapsto(1+(1-2\beta)z)/(1-z)$ $(z\in\Delta,0\leq\beta<1)$
is univalent and maps $\Delta$ onto the right half plane, having real part
greater than $\beta$, we have the following lemma directly. With the proof
easy, the details are omitted.

######  Lemma 1.2.

A function $f\in\mathcal{A}$ belongs to the class $\mathcal{R}(\alpha,\beta)$
if, and only if,

$f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\prec\frac{1+(1-2\beta)z}{1-z}\quad(z\in\Delta,0\leq\beta<1,-\pi<\alpha\leq\pi).$

To prove of our main results we need the following lemma.

######  Lemma 1.3.

[10, p. 35] Let $\Xi$ be a simply connected domain in the complex plane
$\mathbb{C}$ and let $t$ be a complex number such that ${\rm Re}\\{t\\}>0$.
Suppose that a function $\psi:\mathbb{C}^{2}\times\Delta\rightarrow\mathbb{C}$
satisfies the condition

$\psi(i\rho,\sigma;z)\not\in\Xi$

for all real $\rho,\sigma\leq-\mid t-i\rho\mid^{2}/(2{\rm Re}\,t)$ and all
$z\in\Delta$. If the function $p(z)$ defined by
$p(z)=t+t_{1}z+t_{2}z^{2}+\cdots$ is analytic in $\Delta$ and if

$\psi(p(z),zp^{\prime}(z);z)\in\Xi,$

then ${\rm Re}\\{p(z)\\}>0$ in $\Delta$.

This paper is organized as follows. In Section 2 some properties of the
classes $\mathcal{R}(\alpha,\beta)$ and $\mathcal{L}_{\alpha}(b)$ are studied.
In Section 3 we obtain the radius of univalence of 2-th section sum of
$f\in\mathcal{R}(\alpha,\beta)$ and we conjecture that this radius is for
every section sum of the function $f$ that belonging to the class
$\mathcal{R}(\alpha,\beta)$.

##  2\. On the classes $\mathcal{R}(\alpha,\beta)$ and
$\mathcal{L}_{\alpha}(b)$

At first, applying Hergoltz’s Theorem [1, p. 21] we obtain the extreme points
of $\mathcal{R}(\alpha,\beta)$ as follows:

(2.1)
$f_{x}(z)=z+4(1-\beta)\sum_{n=2}^{\infty}\frac{x^{n-1}}{n[n+1+(n-1)e^{i\alpha}]}z^{n}\quad(|x|=1).$

Since the coefficient bounds are maximized at an extreme point, as an
application of (2.1), we have

$|a_{n}|\leq\frac{4(1-\beta)}{n|n+1+(n-1)e^{i\alpha}|}=\frac{4(1-\beta)}{n\sqrt{2[n^{2}+1+(n^{2}-1)\cos\alpha]}}\quad(n\geq
2),$

where $0\leq\beta<1$ and $-\pi<\alpha\leq\pi$. Equality occurs for $f_{x}(z)$
defined by (2.1).

To prove the first result of this section, i.e. Theorem 2.1, also Theorem 2.2
and Theorem 2.3, we employ the same technique as in [5, Theorem 2.1].

######  Theorem 2.1.

Let $\beta\in[0,1)$ and $\alpha\in(-\pi,\pi]$. If $f\in\mathcal{A}$ belongs to
the class $\mathcal{R}(\alpha,\beta)$, then

${\rm Re}\\{f^{\prime}(z)\\}>\beta\quad(0\leq\beta<1).$

This means that $\mathcal{R}(\alpha,\beta)\subset\mathcal{C}(\beta)$.

###### Proof.

Let $f^{\prime}(z)\neq 0$ for $z\neq 0$ and $p(z)$ be defined by

$p(z)=\frac{1}{1-\beta}\left(f^{\prime}(z)-\beta\right)\quad(0\leq\beta<1).$

Then $p(z)$ is analytic in $\Delta$, $p(0)=1$ and

$f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)=(1-\beta)[p(z)+(1+e^{i\alpha})zp^{\prime}(z)/2]+\beta=\phi(p(z),zp^{\prime}(z);z),$

where $\phi(r,s;z):=(1-\beta)[r+(1+e^{i\alpha})s/2]$. Since
$f\in\mathcal{R}(\alpha,\beta)$, we define the set $\Omega_{\beta}$ as
follows:

(2.2) $\\{\phi(p(z),zp^{\prime}(z);z):z\in\Delta\\}\subset\\{w:{\rm
Re}\,\\{w\\}>\beta\\}=:\Omega_{\beta}.$

For all real $\rho$ and $\sigma$, that $\sigma\leq-(1+\rho^{2})/2$, we get

$\displaystyle{\rm Re}\\{\phi(i\rho,\sigma;z)\\}$ $\displaystyle={\rm
Re}\left\\{(1-\beta)[i\rho+(1+e^{i\alpha})\sigma/2]\right\\}$
$\displaystyle=(1-\beta)(1+\cos\alpha)\sigma/2+\beta$
$\displaystyle\leq\beta-\frac{(1-\beta)}{4}(1+\cos\alpha)(1+\rho^{2})$
$\displaystyle\leq\beta.$

This shows that ${\rm
Re}\\{\phi(p(z),zp^{\prime}(z);z)\\}\not\in\Omega_{\beta}$. Thus by Lemma 1.3,
we get ${\rm Re}\\{p(z)\\}>0$ or ${\rm Re}\\{f^{\prime}(z)\\}>\beta$. This
means that $f\in\mathcal{C}(\beta)$ and concluding the proof. ∎

Taking $\beta=0$ in the above Theorem 2.1, we get.

######  Corollary 2.1.

If $f\in\mathcal{L}_{\alpha}$, then ${\rm Re}\\{f^{\prime}(z)\\}>0$
$(z\in\Delta)$ and thus $f$ is univalent.

######  Remark 2.1.

Since ${\rm Re}\\{(1+e^{i\alpha})/2\\}=(1+\cos\alpha)/2\geq 0$ where
$\alpha\in(-\pi,\pi]$, thus the above Theorem 2.1 is a generalization of the
results that earlier were obtained by Chichra [2] and Lewandowski et al. [7].

The problem of finding a lower bound for ${\rm Re}\left\\{f(z)/z\right\\}$ is
called Marx-Strohhäcker problem. Marx and Strohhäcker ([8, 17]) proved that if
$f\in\mathcal{K}$, then ${\rm Re}\left\\{f(z)/z\right\\}>1/2$. In the sequel
we consider this problem for the class $\mathcal{R}(\alpha,\beta)$.

######  Theorem 2.2.

Let $\beta\in[0,1)$ and $\alpha\in(-\pi,\pi]$. If $f\in\mathcal{A}$ belongs to
the class $\mathcal{R}(\alpha,\beta)$, then we have

${\rm Re}\left\\{\frac{f(z)}{z}\right\\}>\beta\quad(0\leq\beta<1).$

###### Proof.

Let the function $f\in\mathcal{A}$ belongs to the class
$\mathcal{R}(\alpha,\beta)$ where $\beta\in[0,1)$ and $\alpha\in(-\pi,\pi]$.
Define the function p as

(2.3) $\textsl{p}(z):=\frac{1}{1-\beta}\left(\frac{f(z)}{z}-\beta\right).$

Since $f\in\mathcal{A}$, easily seen that p is analytic in $\Delta$ and
$\textsl{p}(0)=1$. The equation (2.3), with a simple calculation implies that

(2.4)
$f^{\prime}(z)=\beta+(1-\beta)\textsl{p}(z)+(1-\beta)z\textsl{p}^{\prime}(z)$

and

(2.5)
$f^{\prime\prime}(z)=2(1-\beta)\textsl{p}^{\prime}(z)+(1-\beta)z\textsl{p}^{\prime\prime}(z).$

Now, from (2.4) and multiplying (2.5) by $\frac{1+e^{i\alpha}}{2}z$, we get

$\displaystyle f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)$
$\displaystyle\quad=\beta+(1-\beta)\textsl{p}(z)+[(2+e^{i\alpha})(1-\beta)]z\textsl{p}^{\prime}(z)+(1-\beta)\frac{1+e^{i\alpha}}{2}z^{2}\textsl{p}^{\prime\prime}(z)$
$\displaystyle\quad=\psi(\textsl{p}(z),z\textsl{p}^{\prime}(z),z^{2}\textsl{p}^{\prime\prime}(z);z),$

where

$\psi(r,s,t;z)=\beta+(1-\beta)r+[(2+e^{i\alpha})(1-\beta)]s+(1-\beta)\frac{1+e^{i\alpha}}{2}t.$

Since $f\in\mathcal{R}(\alpha,\beta)$ we consider the following inclusion
relation

$\\{\psi(\textsl{p}(z),z\textsl{p}^{\prime}(z),z^{2}\textsl{p}^{\prime\prime}(z);z):z\in\Delta\\}\subset\Omega_{\beta},$

where $\Omega_{\beta}$ is defined in (2.2). Let $\rho,\sigma,\mu$ and $\nu$ be
real numbers such that

(2.6) $\sigma\leq-\frac{1}{2}(1+\rho^{2}),~{}\mu+\nu\leq 0\quad{\rm and}\quad
2\mu+\nu\leq 0.$

From (2.6) and by [9] (see also [10, Theorem 2.3b]), since

$\psi(i\rho,\sigma,\mu+i\nu;z)=\beta+(1-\beta)i\rho+[(2+e^{i\alpha})(1-\beta)]\sigma+(1-\beta)\frac{1+e^{i\alpha}}{2}(\mu+i\nu),$

we get

$\displaystyle\quad{\rm Re}\\{\psi(i\rho,\sigma,\mu+i\nu;z)\\}$
$\displaystyle=\beta+(1-\beta)(2+\cos\alpha)\sigma+\frac{1-\beta}{2}[\mu(1+\cos\alpha)-\nu\sin\alpha]$
$\displaystyle\leq\beta-(1-\beta)(1+\rho^{2})(1+\cos\alpha)+\frac{1-\beta}{2}[\mu(1+\cos\alpha)-\nu\sin\alpha]$
$\displaystyle=F(\alpha,\beta,\rho)+G(\alpha,\beta,\mu),$

where

$F(\alpha,\beta,\rho):=\beta-(1-\beta)(1+\rho^{2})(1+\cos\alpha)$

and

$G(\alpha,\beta,\mu):=\frac{1-\beta}{2}[\mu(1+\cos\alpha)-\nu\sin\alpha].$

It is easy to see that $F(\alpha,\beta,\rho)\leq\beta$. Since $2\mu+\nu\leq
0$, we have $G(\alpha,\beta,\mu)\leq 0$. Thus ${\rm
Re}\\{\psi(i\rho,\sigma,\mu+i\nu;z)\\}\leq\beta$ and this means that

${\rm
Re}\\{\psi(\textsl{p}(z),z\textsl{p}^{\prime}(z),z^{2}\textsl{p}^{\prime\prime}(z);z)\\}\not\in\Omega_{\beta}.$

Therefore we obtain ${\rm Re}\\{\textsl{p}(z)\\}>0$ where p is given by (2.3),
or equivalently

${\rm Re}\left\\{\frac{f(z)}{z}\right\\}>\beta\quad(0\leq\beta<1).$

This completes the proof. ∎

If we put $\beta=0$ in the above Theorem 2.2, we get.

######  Corollary 2.2.

If $f\in\mathcal{L}_{\alpha}$, then ${\rm Re}\\{f(z)/z\\}>0$ in the open unit
disc $\Delta$.

We shall require the following lemma in order to prove of the next result.

######  Lemma 2.1.

Let $\phi_{b}(z)$ be defined by (1.1) for $b>1/2$. Then
$\phi_{b}(\Delta)=\Omega_{b}$ where

$\Omega_{b}:=\\{w\in\mathbb{C}:0<{\rm Re}\\{w\\}<2b\\}.$

###### Proof.

If $b=1$, then we have $0<{\rm Re}\\{\phi_{b}(z)\\}={\rm Re}\\{1+z\\}<2$. For
$b>1/2$ and $b\neq 1$, the function $\phi_{b}(z)$ does not have any poles in
$\overline{\Delta}$ and is analytic in $\Delta$. Thus looking for the
$\min\\{{\rm Re}\\{\phi_{b}(z)\\}:~{}|z|<1\\}$ it is sufficient to consider it
on the boundary
$\partial\phi_{b}(\Delta)=\\{\phi_{b}(e^{i\varphi}):\varphi\in[0,2\pi]\\}$. A
simple calculation gives us

${\rm
Re}\left\\{\phi_{b}(e^{i\varphi})\right\\}=\frac{(1/b)(1+\cos\varphi)}{1+2(1/b-1)\cos\varphi+(1/b-1)^{2}}\quad(\varphi\in[0,2\varphi]).$

So we can see that ${\rm Re}\left\\{F_{\alpha}(z)\right\\}$ is well defined
also for $\varphi=0$ and $\varphi=2\pi$. Define

$h(x)=\frac{(1/b)(1+x)}{1+2(1/b-1)x+(1/b-1)^{2}}\quad(-1\leq x\leq 1).$

Thus for $b>1/2$ and $b\neq 1$, we have $h^{\prime}(x)>0$. Therefore, we get

$0=h(-1)\leq h(x)\leq h(1)=2b.$

This completes the proof. ∎

######  Theorem 2.3.

Let $f\in\mathcal{A}$ be a member of the class $\mathcal{L}_{\alpha}(b)$ where
$b>1/2$ and $\alpha\in(-\pi,\pi]$. Then

$0<{\rm Re}\\{f^{\prime}(z)\\}<2b\quad(z\in\Delta).$

###### Proof.

Let us $f\in\mathcal{L}_{\alpha}(b)$. Then by Lemma 1.1, Lemma 2.1 and by
definition of the subordination principle we have

(2.7) $0<{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}<2b\quad(z\in\Delta,b>1/2,-\pi<\alpha\leq\pi).$

First, we assume that

${\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}>0.$

Then by Corollary 2.2 we have ${\rm Re}\\{f^{\prime}(z)\\}>0$. Now we let

${\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}<2b.$

Put $\xi=2b$ and so $\xi>1$. Let $f^{\prime}(z)\neq 0$ for $z\neq 0$. Consider

$q(z)=\frac{1}{1-\xi}\left(f^{\prime}(z)-\xi\right)\quad(\xi>1,z\in\Delta).$

Then $q(z)$ is analytic in $\Delta$ and $q(0)=1$. A simple check gives us

$f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)=(1-\xi)[q(z)+(1+e^{i\alpha})zq^{\prime}(z)/2]+\xi=\eta(q(z),zq^{\prime}(z);z),$

where $\eta(x,y;z)=(1-\xi)[x+(1+e^{i\alpha})y/2]+\xi$. Now we define

$\\{\eta(q(z),zq^{\prime}(z);z):z\in\Delta\\}\subset\\{w:{\rm
Re}\\{w\\}<\xi\\}=:\Omega_{\xi}.$

Again with a simple calculation we deduce that

$\displaystyle{\rm Re}\\{\eta(i\rho,\sigma;z)\\}$ $\displaystyle={\rm
Re}\left\\{(1-\xi)[i\rho+(1+e^{i\alpha})\sigma/2]\right\\}$
$\displaystyle=(1-\xi)(1+\cos\alpha)\sigma/2+\xi$
$\displaystyle\geq\frac{(\xi-1)}{4}(1+\cos\alpha)(1+\rho^{2})+\xi\quad(-\sigma\geq(1+\rho^{2})/2)$
$\displaystyle\geq\xi.$

This shows that ${\rm Re}\\{\eta(i\rho,\sigma;z)\\}\not\in\Omega_{\xi}$ and
therefore ${\rm Re}\\{q(z)\\}>0$, or equivalently ${\rm
Re}\\{f^{\prime}(z)\\}<\xi$. This is the end of proof. ∎

######  Theorem 2.4.

Assume that $b>1/2$, $\alpha\in(-\pi,\pi]$ and $f\in\mathcal{L}_{\alpha}(b)$.
Then for each $|z|=r<1$ we have

(2.8) $1-\frac{(2b-1)r}{b+(b-1)r}\leq{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}\leq
1+\frac{(2b-1)r}{b+(b-1)r}.$

###### Proof.

Let $f\in\mathcal{L}_{\alpha}(b)$. Then from the definition of subordination
and by Lemma 1.1, there exists a $\omega\in\mathcal{B}$ such that

(2.9)
$f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)=\frac{1+\omega(z)}{1+(\frac{1}{b}-1)\omega(z)}\quad(z\in\Delta).$

We define

$W(z)=\frac{1+\omega(z)}{1+\left(\frac{1}{b}-1\right)\omega(z)},$

which readily yields

$W(z)-1=\frac{(2-\frac{1}{b})\omega(z)}{1+\left(\frac{1}{b}-1\right)\omega(z)}.$

For $|z|=r<1$, using the known fact that (see [1]) $|\omega(z)|\leq|z|$ we
find that

(2.10) $|W(z)-1|\leq\frac{(2b-1)r}{b+(b-1)r}.$

Hence $W(z)$ maps the disk $|z|<r<1$ onto the disc which the center $C=1$ and
the radius $\delta$ given by

$\delta=\frac{(2b-1)r}{b+(b-1)r}.$

Therefore,

$1-\frac{(2b-1)r}{b+(b-1)r}\leq|W(z)|\leq 1+\frac{(2b-1)r}{b+(b-1)r}.$

Now, the assertion follows from (2.9) and this fact that ${\rm
Re}\\{z\\}\leq|z|$. ∎

######  Remark 2.2.

We obtained two lower and upper bounds for

${\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\},$

when $f\in\mathcal{L}_{\alpha}(b)$. From (2.7), we have

$0<{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}<2b\quad(z\in\Delta,b>1/2,-\pi<\alpha\leq\pi),$

while by (2.8)

$G(r):=1-\frac{(2b-1)r}{b+(b-1)r}\leq{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}\leq
U(r):=1+\frac{(2b-1)r}{b+(b-1)r}.$

It is easy to check that $U(r)<2b$ if $b\geq 1$ (or $b\rightarrow 1^{+}$)
while $G(r)\geq 0$ for $1/2<b\leq 1$ (or $b\rightarrow 1^{-}$).

######  Corollary 2.3.

Let $f\in\mathcal{L}_{\alpha}(1)$. Then we have

$1-r<{\rm
Re}\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}<1+r\quad(|z|=r<1).$

######  Corollary 2.4.

By a simple geometric observation and applying (2.9) and (2.10), we have

$\left|\arg\left\\{f^{\prime}(z)+\frac{1+e^{i\alpha}}{2}zf^{\prime\prime}(z)\right\\}\right|<\arcsin\frac{(2b-1)r}{b+(b-1)r}\quad(|z|=r<1,b>1/2).$

##  3\. The radius of univalence of 2-th section sum of
$f\in\mathcal{R}(\alpha,\beta)$

In this section, we obtain the radius of univalence of 2-th section sum of
$f\in\mathcal{R}(\alpha,\beta)$. We recall that the Taylor polynomial
$s_{k}(z)=s_{k}(f)(z)$ of $f$ defined by

$s_{k}(z)=s_{k}(f)(z)=z+a_{2}z^{2}+\cdots+a_{k}z^{k},$

is called the $k-th\,section/partial$ sum of $f$. In [18], proved that every
section $s_{k}(z)$ of a $f\in\mathcal{S}$ is univalent in the disk $|z|<1/4$
and the number $1/4$ is best possible as the second partial sum of the Koebe
function $k(z)=z/(1-z)^{2}$ shows. Next, we find the radius of univalence of
the 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$.

######  Theorem 3.1.

The 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$ is univalent in the
disc

$|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}\quad(-\pi<\alpha\leq\pi,0\leq\beta<1).$

The number $\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$ cannot be replaced by a
greater one.

###### Proof.

Let $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}\in\mathcal{R}(\alpha,\beta)$ and
$s_{2}(z)=z+a_{2}z^{2}$ be its second section. By a simple calculation and
since $|a_{2}|\leq\frac{2(1-\beta)}{\sqrt{10+6\cos\alpha}}$ we have

${\rm Re}\\{s_{2}^{\prime}(z)\\}={\rm Re}\\{1+2a_{2}z\\}\geq 1-2|a_{2}||z|\geq
1-\frac{4(1-\beta)|z|}{\sqrt{10+6\cos\alpha}},$

which is positive provided $|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$.
Therefore $s_{2}(z)$ is close-to-convex (univalent) in the disk
$|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$. To show that this bound is
sharp, we consider the function $f_{x}$ defined by (2.1). The second partial
sum $s_{2}(f_{x})(z)$ of $f_{x}$ is
$z+\frac{4(1-\beta)}{2(3+e^{i\alpha})}z^{2}$. Thus we get

$s_{2}^{\prime}(z)=1+\frac{4(1-\beta)}{(3+e^{i\alpha})}z.$

Hence ${\rm Re}\\{s_{2}^{\prime}(z)\\}=0$ when
$z=-\frac{(3+e^{i\alpha})}{4(1-\beta)}$. This completes the proof. ∎

We finish this paper with the following conjecture:
Conjecture. Every section of $f\in\mathcal{R}(\alpha,\beta)$ is univalent in
the disc $|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$.

## References

  * [1] P.L. Duren, Univalent Functions, Springer–Verlag, New York, 1983.
  * [2] P.N. Chichra, New subclasses of the class of close–to–convex functions, Proc. Amer. Math. Soc. 62 (1977), 37–43.
  * [3] C.–Y. Gao and S.–Q. Zhou, Certain subclass of starlike functions, Appl. Math. Comput. 187 (2007), 176–182.
  * [4] A.W. Goodman, Univalent Functions, Vols. 1 and 2, Mariner, Tampa, FL, 1983.
  * [5] R. Kargar, A. Ebadian and J. Sokół, On subordination of some analytic functions, Sib. Math. J. 57 (2016), 599–604.
  * [6] W. Kaplan, Close–to–convex schlicht functions, Michigan Math. J. 1 (1952), 169–185.
  * [7] Z. Lewandowski, S. Miller and E. Złotkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56 (1976), 111–117.
  * [8] A. Marx, Untersuchungen ilber schlichte Abbildungen, Math. Ann. 107 (1932/33), 40–65.
  * [9] S.S. Miller and P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981), 157–171.
  * [10] S.S. Miller and P.T. Mocanu, Differential Subordinations, Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York / Basel 2000.
  * [11] S. Ponnusamy and V. Singh, Convolution properties of some classes of analytic functions, J. Math. Sci. 89 (1988), 1008–1020.
  * [12] R. Singh and S. Singh, Convolutions properties of a class of starlike functions, Proc. Amer. Math. Soc. 106 (1989), 145–152.
  * [13] R. Singh and S. Singh, Starlikeness and convexity of certain integrals, Ann. Univ. Mariae Curie–Sklodowska Sect A 35 (1981), 45–47.
  * [14] H. Silverman, A class of bounded starlike functions, Internat. J. Math. Math. Sci. 17 (1994), 249–252.
  * [15] H. Silverman and E.M. Silvia, Characterizations for subclasses of univalent functions, Math. Japonica 50 (1999), 103–109.
  * [16] H.M. Srivastava, D. Răducanub, P. Zaprawa, A certain subclass of analytic functions defined by means of differential subordination, Filomat 30:14 (2016), 3743–3757.
  * [17] E. Strohhäcker, Beiträge zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), 356–380.
  * [18] G. Szegö, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928), 188–211.
  * [19] L. Trojnar–Spelina, Subclasses of univalent functions related with circular domains, J. Math. Appl. 34 (2011), 109–116.