# Opportunistic Qualitative Planning in Stochastic Systems with Preferences
over Temporal Logic Objectives

Abhishek Ninad Kulkarni∗ and Jie Fu A. N. Kulkarni and J. Fu are with the
Dept. of Electrical and Computer Engineering, University of Florida,
Gainesville, FL 32603 USA<EMAIL_ADDRESS>
###### Abstract

Preferences play a key role in determining what goals/constraints to satisfy
when not all constraints can be satisfied simultaneously. In this work, we
study preference-based planning in a stochastic system modeled as a Markov
decision process, subject to a possible incomplete preference over temporally
extended goals. Our contributions are three folds: First, we introduce a
preference language to specify preferences over temporally extended goals.
Second, we define a novel automata-theoretic model to represent the preorder
induced by given preference relation. The automata representation of
preferences enables us to develop a preference-based planning algorithm for
stochastic systems. Finally, we show how to synthesize opportunistic
strategies that achieves an outcome that improves upon the current satisfiable
outcome, with positive probability or with probability one, in a stochastic
system. We illustrate our solution approaches using a robot motion planning
example.

##  I Introduction

Preference-based planning decides what constraints to satisfy when not all
constraints can be achieved [1]. In this paper, we study a class of
qualitative, preference-based probabilistic planning problem in which the
agent aims to strategically exploit the _opportunities_ that arise due to
stochasticity in its environment to achieve a more preferred outcome than what
may be achieved from its initial state. Such problems are encountered in many
applications of autonomous systems.

In existing methods for probabilistic planning with temporal goals, the
desired behavior of the system is specified by a temporal logic formula [2],
and the goal is to compute a policy that either maximizes the probability of
satisfying the formula [3, 4], or enforces the satisfaction as a constraint
[5, 6]. In recent work, preference-based planning with temporal logic
objectives have been studied: minimum violation planning in a deterministic
system [7] decides which low-priority constraints to be violated. Automated
specification-revision is proposed in [8] where the formula can be revised
with a cost and the planning problem is formulated into a multi-objective
Markov Decision Process (MDP) that trades off minimizing the cost of revision
and maximizing the probability of satisfying the revised formula. [9]
introduced weights with Boolean and temporal operators in signal temporal
logic to specify the importance of satisfying the subformula and priority in
the timing of satisfaction. They developed a gradient-based optimization
method to maximize the weighted satisfaction in deterministic dynamical
systems. Robust and recovery specifications are introduced by [10] and pre-
specify what behaviors are expected when the part of the system specification
(i.e., the environment assumption) fails to be satisfied. Existing preference-
based planning methods with temporal goals assume the preference relation to
be _complete_.

Unfortunately, in many applications, the completeness assumption does not
always hold. For instance, it can be impractical to elicit user’s preference
between every pair of outcomes when the set of outcomes is large; or in some
situation, such as the trolley problem [11], the outcomes (sacrificing
passengers or pedestrians) are incomparable. Preference languages have been
proposed to represent both the complete and incomplete preferences over
propositional formulas [12] and temporal logic formulas [13]. For planning,
CP-net and its variants [14, 15] have been proposed as a computational model.
But they are defined over propositional preferences. To the best of our
knowledge, there is no computational model that can express incomplete
preferences over temporal goals. Such a model is needed to facilitate planning
in stochastic environments.

In this paper, we propose a novel automata-theoretic approach to qualitative
planning in MDPs with incomplete preferences over temporal logic objectives.
Our approach consists of three steps. First, we express (incomplete)
preferences over the satisfaction of temporal goals specified using a fragment
of Linear Temporal Logic (LTL). Unlike propositional preferences that are
interpreted over states, preferences over temporal goals are interpreted over
infinite words. Second, we define an _automata-theoretic model_ to represent
the preorder induced by the preference relation and describe a procedure to
construct the automata-theoretic model given a preference formula. Thirdly, we
present an algorithm to solve preference-based strategies in a stochastic
system modeled as a labeled MDP. We presented _safe and positively improving_
and _safe and almost-surely improving_ strategies, that identify and exploit
opportunities for improvements with positive probability and probability one,
respectively. A running example is employed to illustrate the notions and
solution approaches.

##  II Preliminaries

Notation. Given a finite set $X$, let $\mathcal{D}(X)$ be the set of
probability distributions over $X$. Let $\Sigma$ be an alphabet (a finite set
of symbols). We denote the set of finite (resp., infinite) words that can be
generated using $\Sigma$ by $\Sigma^{\ast}$ (resp., $\Sigma^{\omega}$). Given
a word $w\in\Sigma^{\omega}$, a prefix of $w$ is a word $u\in\Sigma^{\ast}$
such that there exists $v\in\Sigma^{\omega}$, $w=uv$. We denote the set of all
finite prefixes of $w$ by $\mathsf{Pref}(w)$.

We consider a class of decision-making problems in stochastic systems modeled
as a labeled MDP [16].

######  Definition 1 (Labeled MDP).

A labeled MDP is a tuple $M=\langle S,A,P,\mathcal{AP},L\rangle,$ where $S$
and $A$ are finite state and action sets, $P:S\times
A\rightarrow\mathcal{D}(S)$ is the transition probability function such that
$P(s^{\prime}\mid s,a)$ is the probability of reaching $s^{\prime}\in S$ given
that action $a\in A$ is chosen at state $s\in S$, $\mathcal{AP}$ is a finite
set of atomic propositions, and $L:S\rightarrow 2^{\mathcal{AP}}$ is a
labeling function that maps each state to a set of atomic propositions which
are true in that state.

A finite-memory, randomized strategy in the MDP is a function
$\pi:S^{\ast}\rightarrow\mathcal{D}(A)$. A Markovian, randomized strategy in
the MDP is a function $\pi:S\rightarrow\mathcal{D}(A)$. Given an MDP $M$ and
an initial distribution $\nu_{0}$, a strategy $\pi$ induces a stochastic
process $M_{\pi}=\\{S_{t}\mid t\geq 1\\}$ where $S_{k}$ is the random variable
for the $k$-th state in the stochastic process $M_{\pi}$ and it holds that
$S_{0}\sim\nu_{0}$ and $S_{i+1}\sim P(\cdot\mid S_{i},a_{i})$ and
$a_{i}\sim\pi(\cdot\mid S_{0}\ldots S_{i})$ for $i\geq 0$.

We express the objective of the planning agent as preferences over a set of
outcomes, each of which is expressed by a syntactically co-safe LTL (scLTL)
formula [17].

######  Definition 2.

Given a set of atomic propositions $\mathcal{AP}$, an scLTL formula is defined
inductively as follows:

$\varphi\coloneqq p\mid\neg
p\mid\varphi\land\varphi\mid\bigcirc\,\varphi\mid\varphi\mbox{$\,{\sf
U}\,$}\varphi,$

where $p\in\mathcal{AP}$ is an atomic proposition. The operators $\neg$
(negation) and $\land$ (and) are propositional logic operators. The operators
$\bigcirc\,$ (next) and $\,{\sf U}\,$ (until) are temporal operators [17]. The
operator $\Diamond\,$ (eventually) is derived using $\,{\sf U}\,$ as follows:
$\Diamond\,\varphi=\top\mbox{$\,{\sf U}\,$}\varphi$ where $\top$ is
unconditionally true. The formula $\Diamond\,\varphi$ is true if $\varphi$
holds in some future time.

The scLTL formulas are a subclass of LTL formulas with a special property that
an infinite word satisfying an scLTL only needs to have a ‘good’ prefix
(formalized after Definition 3). The set of good prefixes can be compactly
represented as the language accepted by a Deterministic Finite Automaton
(DFA).

######  Definition 3.

A deterministic finite automaton (DFA) is a tuple $\mathcal{A}=\langle
Q,\Sigma,\delta,{q_{0}},F\rangle,$ where $Q$ is a finite set of states;
$\Sigma=2^{\mathcal{AP}}$ is a finite set of symbols called the alphabet;
$\delta:Q\times\Sigma\rightarrow Q$ is a deterministic transition function
that maps a state and a symbol to a next state. The transition function is
extended recursively over words as follows: $\delta(q,\sigma
u)=\delta(\delta(q,\sigma),u)$ given $\sigma\in\Sigma$ and
$u\in\Sigma^{\ast}$; ${q_{0}}\in Q$ is the initial state; $F\subseteq Q$ is a
set of accepting states. A word $u$ is accepted by $\mathcal{A}$ if
$\delta({q_{0}},u)\in F$.

Given an scLTL formula $\varphi$ and an infinite word $w\in\Sigma^{\omega}$, a
‘good’ prefix is a finite word $u\in\Sigma^{\ast}$ such that
$u\in\mathsf{Pref}(w)$ and $u$ is accepted by the DFA, $\mathcal{A}$. A word
$w\in\Sigma^{\omega}$ satisfies an scLTL formula $\varphi$, denoted by
$w\models\varphi$, if $w$ has a good prefix. The set of words satisfying an
scLTL formula $\varphi$ is denoted by
$\mathsf{Mod}(\varphi)=\\{w\in\Sigma^{\omega}\mid w\models\varphi\\}$. For an
scLTL formula, all accepting states of its corresponding DFA are absorbing,
i.e., $\delta(q,\sigma)=q$ for any $q\in F$ and $\sigma\in\Sigma$. We assume
the transition function of DFA to be _complete_. That is, $\delta(q,\sigma)$
is defined for any pair $(q,\sigma)\in Q\times\Sigma$. An incomplete
transition function can be made complete by introducing a sink state and
redirecting all undefined transitions to that sink state.

An infinite path in a labeled MDP $\rho=s_{0}s_{1}\ldots$ induces a word
$w=L(s_{0})L(s_{1})\ldots$ in the DFA. We say the path $\rho$ satisfies an
scLTL formula $\varphi$ if and only if the induced word $w$ satisfies the
formula, i.e., $w\models\varphi$.

######  Definition 4 (Almost-Sure/Positive Winning Strategy).

Given an MDP $M$ and an scLTL formula $\varphi$, a strategy
$\pi:S^{\ast}\rightarrow\mathcal{D}(A)$ is said to be almost-sure (resp.,
positive) winning if, in the stochastic process $M_{\pi}$ induced by $\pi$,
the formula $\varphi$ can be satisfied with probability one (resp. with a
probability $\geq 0$). Formally, in the stochastic process
$M_{\pi}=\\{S_{t}\mid t\geq 1\\}$,
$\mathbf{Pr}(S_{0}S_{1},\ldots\models\varphi)=1$ (resp. $>0$).

The set of _states_ in the MDP $M$, starting from which the agent has an
almost-sure (resp. positive) winning strategy to satisfy an scLTL formula
$\varphi$ is called the _almost-sure (resp., positive) winning region_. Given
an MDP and an scLTL formula, the product operation [18] reduces the problem of
computing almost-sure (resp., positive) winning region to that of computing
the set of states from which a subset of final states in the product MDP can
be reached with almost-surely (resp., positive probability). It is known that
there exists a _memoryless, almost-sure winning strategy_ $\pi$ to ensure the
subset of final states is reached with probability one from a state in the
almost-sure winning region. Polynomial (resp., linear) time algorithm to
compute almost-sure (resp., positive) winning strategy in MDPs with
reachability objectives can be found in the book by [16, Chap. 10].

###  II-A Running Example

We use a motion planning problem for an cleaning robot to illustrate the the
concepts discussed in this paper. The robot is to operate in a $5\times 5$
stochastic gridworld as shown in Figure 1. At every step, the robot must
choose to move in one of the North, East, South, West directions. If the
action results in an obstacle cell (shown in black), the robot returns to the
cell it started from. If the robot enters a cell marked with green arrows, it
may either stay in that cell or move into an adjacent cell along a direction
indicated by the arrows each with a positive probability. If the robot moves
into any cell with no arrows, it remains in that cell with probability one.
The robot has a limited battery capacity measured in units. Every action costs
$1$ unit of battery. We consider two preferences objectives for the robot.

Figure 1: A gridworld MDP with $6$ regions of interest $A$-$F$.

  1. (PO1)

The robot must visit $A,B$ and/or $E$, given the preference that: visiting $B$
is strictly preferred to visiting $A$, and visiting $E$ is strictly preferred
to visiting $A$.

  2. (PO2)

The robot must visit exactly one of $A,B,C,D$ or $F$, given the preference
that: visiting $B$ is strictly preferred to visiting $A$, visiting $D$ is
strictly preferred to visiting $B$, visiting $F$ is strictly preferred to
visiting $C$, and visiting $B$ is indifferent to visiting $C$.

The preference relations expressed by both the objectives are incomplete. In
the first objective, neither the relation between $B$ and $E$ is given nor can
it be deduced using the properties (e.g., transitivity) of preferences. Hence,
visiting $B$ and visiting $E$ are incomparable outcomes due to incompletely
known preferences.

In the second objective, since $B$ and $C$ are indifferent, it follows by
transitivity that visiting $C$ is strictly preferred to visiting $A$, and
visiting $D$ is strictly preferred to visiting $C$. However, visiting $D$ is
incomparable to visiting $F$ since no relation is either given or can be
deduced between them.

##  III Preference Modeling

In this section, we propose a language to compactly represent incomplete
preferences over temporal goals.

Let $\Phi=\\{\varphi_{1},\ldots,\varphi_{n}\\}$ be an indexed set of outcomes,
i.e., temporal goals expressed by scLTL formulas.

######  Definition 5.

A preference on $\Phi$ is a reflexive binary relation $\trianglerighteq$ on
$\Phi$. For any $1\leq i,j\leq n$, a pair of outcomes
$(\varphi_{i},\varphi_{j})\in~{}\trianglerighteq$ means that satisfying
$\varphi_{i}$ is considered “at least as good as” satisfying $\varphi_{j}$.

We also denote $(\varphi_{i},\varphi_{j})\in~{}\trianglerighteq$ by
$\varphi_{i}\trianglerighteq\varphi_{j}$. Given any pair of outcomes,
$\varphi_{i},\varphi_{j}\in\Phi$, exactly one of the following four relations
holds:

  1. 1.

$\varphi_{i}$ is _indifferent_ to $\varphi_{j}$:
$\varphi_{i}\trianglerighteq\varphi_{j}$ and
$\varphi_{j}\trianglerighteq\varphi_{i}$,

  2. 2.

$\varphi_{i}$ is _strictly preferred_ to $\varphi_{j}$:
$\varphi_{i}\trianglerighteq\varphi_{j}$ and
$\varphi_{j}\not\trianglerighteq\varphi_{i}$,

  3. 3.

$\varphi_{j}$ is _strictly preferred_ to $\varphi_{i}$:
$\varphi_{j}\trianglerighteq\varphi_{i}$ and
$\varphi_{i}\not\trianglerighteq\varphi_{j}$,

  4. 4.

$\varphi_{i}$ is _incomparable_ to $\varphi_{j}$:
$\varphi_{i}\not\trianglerighteq\varphi_{2}$ and
$\varphi_{j}\not\trianglerighteq\varphi_{i}$.

When the agent is indifferent to two outcomes $\varphi_{i},\varphi_{j}$, it
may choose to satisfy either one of them. This can equivalently be expressed
in scLTL by the disjunction of the two formulas. Based on this observation, we
hereby assume that for any two outcomes $\varphi_{i},\varphi_{j}\in\Phi$,
$\varphi_{i}\trianglerighteq\varphi_{j}$ and
$\varphi_{j}\trianglerighteq\varphi_{i}$ do not hold simultaneously, i.e., no
two outcomes in $\Phi$ are indifferent to each other. As a result, the binary
relation $\trianglerighteq$ on $\Phi$ can equivalently be expressed using the
two sets $P,J\subseteq\Phi\times\Phi$ constructed as follows: given a pair of
outcomes $\varphi_{i},\varphi_{j}\in\Phi$, $1\leq i,j,\leq n$,

  1. 1.

$(\varphi_{i},\varphi_{j})\in P$ iff $\varphi_{i}$ is strictly preferred to
$\varphi_{j}$,

  2. 2.

$(\varphi_{i},\varphi_{j})\in J$ iff $\varphi_{i}$ is incomparable to
$\varphi_{j}$.

######  Remark 1.

We closely follow the notation in [19, Ch. 2]. In contrast, we use the
properties of scLTL formulas to simplify the notation to avoid expressing
indifference explicitly.

Notice that the sets $P,J$ induce a mutually exclusive and exhaustive
partition of $\Phi\times\Phi$. Let
$P^{-}=\\{(\varphi_{j},\varphi_{i})\in\Phi\times\Phi\mid(\varphi_{i},\varphi_{j})\in
P\\}$. Then, $P\cup P^{-}\cup J=\Phi\times\Phi$ and $P\cap P^{-}=P^{-}\cap
J=J\cap P=\emptyset$.

######  Example 1.

Consider the running example introduced in Sect. II-A. In preference objective
(PO1), since there is no constraint on visiting multiple regions of interests,
each outcome can be represented using “eventually” operator. Hence, the set of
outcomes is given by $\Phi=\\{\Diamond\,A,\Diamond\,B,\Diamond\,E\\}$. The
components of preference structure $\langle P,J\rangle$ are given as follows:
$P=\\{(\Diamond\,B,\Diamond\,A),(\Diamond\,E,\Diamond\,A)\\}$, and
$J=\\{(\Diamond\,B,\Diamond\,E),(\Diamond\,E,\Diamond\,B)\\}$.

In preference objective (PO2), since exactly one region is to be visited, the
outcomes can be represented as scLTL formulas: $\varphi_{A}=\neg(B\lor C\lor
D\lor F)\mbox{$\,{\sf U}\,$}A$, $\varphi_{B}=\neg(A\lor C\lor D\lor
F)\mbox{$\,{\sf U}\,$}B$, and so on. Because of the indifference, we replace
$\varphi_{B}$ and $\varphi_{C}$ by their disjunction,
$\varphi_{B}\lor\varphi_{C}$. Hence, the set of outcomes is
$\Phi=\\{\varphi_{A},\varphi_{B}\lor\varphi_{C},\varphi_{D},\varphi_{F}\\}$.
And, the components of preference structure are given by:
$P=\\{(\varphi_{B}\lor\varphi_{C},\varphi_{A}),(\varphi_{D},\varphi_{B}\lor\varphi_{C}),(\varphi_{F},\varphi_{B}\lor\varphi_{C}),(\varphi_{D},\varphi_{A}),(\varphi_{F},\varphi_{A})\\}$,
and $J=\\{(\varphi_{F},\varphi_{D}),(\varphi_{D},\varphi_{F})$.

Because an scLTL formula is interpreted over infinite words, the preference
structure $\trianglerighteq$ induces a preference structure $\succeq$ on the
set of infinite words in $\Sigma^{\omega}$. Therefore, we can define a pre-
order $\succeq\in\Sigma^{\omega}\times\Sigma^{\omega}$ based on the preference
structure $\trianglerighteq$ (and equivalently to the tuple $\langle
P,J\rangle$). This is a non-trivial task because any word in $\Sigma^{\omega}$
could satisfy more than one of the scLTL formulas in $\Phi$. Thus, to
determine whether a word is strictly preferred over another, we need a way to
compare two arbitrary subsets of $\Phi$ that contain outcomes satisfied by
these two words.

######  Definition 6 (Most-Preferred Satisfied Outcomes).

Given a word $w\in\Sigma^{\omega}$, let
$\mathsf{Outcomes}(w)=\\{\varphi\in\Phi\mid w\models\varphi\\}$ be the set of
outcomes satisfied by $w$. Given a subset $\Psi\subseteq\Phi$, let
$\mathsf{MP}(\Psi)=\\{\varphi\in\Psi\mid\nexists\varphi^{\prime}\in\Psi:(\varphi,\varphi^{\prime})\in
P\\}$ and let $\mathsf{MP}(w)=\mathsf{MP}(\mathsf{Outcomes}(w))$ be the set of
_most-preferred outcomes_ satisfied by the word $w$.

######  Lemma 1.

Given a word $w\in\Sigma^{\omega}$, any pair
$\varphi,\varphi^{\prime}\in\mathsf{MP}(w)$ are incomparable to each other.

The proof follows from the definition.

######  Definition 7 (Semantics).

Given two words $w_{1},w_{2}\in\Sigma^{\omega}$, $w_{1}$ is strictly preferred
to $w_{2}$, denoted $w_{1}\succ w_{2}$, if and only if the following
conditions hold:  1. there exist $\varphi_{i}\in\mathsf{MP}(w_{1})$ and
$\varphi_{j}\in\mathsf{MP}(w_{2})$ such that $(\varphi_{i},\varphi_{j})\in P$,
and  2. for every pair $\varphi_{i}\in\mathsf{MP}(w_{1})$ and
$\varphi_{j}\in\mathsf{MP}(w_{2})$, $(\varphi_{j},\varphi_{i})\notin P$.  Word
$w_{1}$ is indifferent to $w_{2}$, denoted $w_{1}\sim w_{2}$, if and only if
$\mathsf{MP}(w_{1})=\mathsf{MP}(w_{2})$. Two words $w_{1}$ and $w_{2}$ are
incomparable, denoted $w_{1}\|w_{2}$, if neither $w_{1}\succ w_{2}$, nor
$w_{2}\succ w_{1}$, nor $w_{1}\sim w_{2}$ holds.

In words, $w_{1}$ is strictly preferred to $w_{2}$ iff: first, $w_{1}$
satisfies at least one scLTL formula that is strictly preferred to some scLTL
formula satisfied by $w_{2}$. Second, every scLTL formula satisfied by $w_{1}$
is either strictly preferred to, or incomparable to any scLTL formula
satisfied by $w_{2}$.

######  Example 2.

Consider preference objective (PO2). Consider two paths $\rho_{1},\rho_{2}$ in
Fig. 1 that sequentially visit $A,F,D$ and $A,C$, respectively. Let
$w_{1}=L(\rho_{1})$, $w_{2}=L(\rho_{2})$ be the words induced by
$\rho_{1},\rho_{2}$, respectively. For the word $w_{1}$, we have
$\mathsf{Outcomes}(w_{1})=\\{\varphi_{A},\varphi_{D},\varphi_{F}\\}$ and
$\mathsf{MP}(w)=\\{\varphi_{D},\varphi_{F}\\}$ since visiting $D$ and $F$
individually is strictly preferred to $A$, and visiting $D$ and visiting $F$
are incomparable. Similarly,
$\mathsf{MP}(w_{2})=\\{\varphi_{C}\lor\varphi_{B}\\}$. Therefore, we have
$w_{1}\succ w_{2}$ because, condition (1) of strict preference semantics holds
for the pair $(\varphi_{D},\varphi_{C}\lor\varphi_{B})$ and, condition (2) is
also satisfied because $\varphi_{F}$ is incomparable to
$\varphi_{C}\lor\varphi_{B}$.

##  IV Automata-Theoretic Computational Model for Incomplete Preferences

We now introduce a novel automata-theoretic computational model called a
preference DFA.

######  Definition 8 (Preference DFA).

A preference DFA is the tuple

$\mathcal{B}=\langle Q,\Sigma,\delta,{q_{0}},F,G\rangle,$

where $Q,\Sigma,\delta,{q_{0}}$ are the (finite) set of states, the alphabet,
the deterministic transition function, and an initial state, similar to these
components in a DFA. $F\subseteq Q$ is a set of final states. The last
component $G=(\mathcal{X},E)$ is a preference graph, where each node
$X\in\mathcal{X}$ represents a subset of final states $F$ such that $X_{i}\cap
X_{j}=\emptyset$ for every $X_{i},X_{j}\in\mathcal{X}$. The edges
$E\subseteq\mathcal{X}\times\mathcal{X}$ is a set of directed edges.

Intuitively, a preference DFA $\mathcal{B}$ encodes the preference relation
$\succeq$ over subsets of words (languages in $\Sigma^{\omega}$) represented
using different classes by defining a preorder over the acceptance conditions
(sets of final states). Next, we describe construction a preference DFA from a
preference structure.

Given a preference structure $\trianglerighteq=\langle P,J\rangle$, the
preference DFA is constructed in two steps. First, the underlying DFA $\langle
Q,\Sigma,\delta,{q_{0}},F\rangle$ is constructed as a cross product of DFAs
representing the union of languages of all scLTL formulas in $\Phi$. Letting
$\mathcal{A}_{i}=\langle Q_{i},\Sigma,\delta_{i},{q_{0}}_{i},F_{i}\rangle$ to
be the DFA corresponding to $\varphi_{i}$ for all $1\leq i\leq n$, we have
$Q=Q_{1}\times\ldots\times Q_{n}$,
$\delta(q,\sigma)=(\delta_{1}(q_{1},\sigma),\ldots,\delta_{n}(q_{n},\sigma))$,
${q_{0}}=({q_{0}}_{1},\ldots,{q_{0}}_{n})$ and $F=(F_{1}\times
Q_{2}\times\ldots\times Q_{n})\cup(Q_{1}\times F_{2}\times\ldots\times
Q_{n})\cup\ldots\cup(Q_{1}\times Q_{2}\times\ldots\times F_{n})$. By
definition, any word that induces a visit to a final state in preference
automaton achieves at least one outcome in $\Phi$.

In the second step, we construct the preference graph $G=(\mathcal{X},E)$.
Intuitively, every node of the preference graph represents an equivalence
class of final states such that any two words that visit any final state
represented by the same node are indifferent under $\succeq$. To define the
nodes, we first associate each final state with a set of _tags_ :

  1. 1.

A tag $x_{ij}$ is associated with a final state $q=(q_{1},\ldots,q_{n})\in F$
to denote that a word reaching $q$ satisfies a more preferred outcome among
$\varphi_{i}$ and $\varphi_{j}$. Hence, $x_{ij}$ is assigned to $q$ iff the
following conditions hold:  (a) $q_{i}\in F_{i}$,  (b)
$(\varphi_{i},\varphi_{j})\in P$,  (c)
$\varphi_{i}\in\mathsf{MP}(\\{\varphi_{k}\in\Phi\mid q_{k}\in F_{k}\\})$.

  2. 2.

A tag $y_{ij}$ is associated to a final state $q=(q_{1},\ldots,q_{n})\in F$ to
denote that a word reaching $q$ satisfies the less preferred outcome among
$\varphi_{i}$ and $\varphi_{j}$. Hence, $y_{ij}$ is assigned to $q$ iff:  (a)
$q_{i}\in Q_{i}\setminus F_{i}$,  (b) $q_{j}\in F_{j}$,  (c)
$(\varphi_{i},\varphi_{j})\in P$,  (d)
$\varphi_{j}\in\mathsf{MP}(\\{\varphi_{k}\in\Phi\mid q_{k}\in F_{k}\\})$.

We denote the set of tags associated to a final state $q\in F$ by
$\lambda(q)$. A node $X\in\mathcal{X}$ represents a set of final states that
have the same set of tags. That is, for any $q,q^{\prime}\in X$,
$\lambda(q)=\lambda(q^{\prime})$. We write $\lambda(X):=\lambda(q)$ to denote
the set of tags associated with any final state represented by $X$. An edge
$(X_{2},X_{1})$ in $E$ represents that any final state in $X_{1}$ is strictly
preferred to any final state in $X_{2}$. Thus, $(X_{2},X_{1})$ is included in
$E$ if and only if  (1) there exists $1\leq i,j\leq n$ such that
$x_{ij}\in\lambda(X_{1})$ and $y_{ij}\in\lambda(X_{2})$;  (2) for all $1\leq
i,j\leq n$ such that $x_{ij}\in\lambda(X_{1})$ and $y_{ij}\in\lambda(X_{2})$
does not hold, $y_{ij}\in\lambda(X_{1})$ and $x_{ij}\in\lambda(X_{2})$ also
does not hold.

An edge $(X_{1},X_{2})\in E$ is intuitively understood as follows. Condition
(1) states that there must exist a pair of scLTL formulas
$\varphi_{i},\varphi_{j}\in\Phi$ such that $(\varphi_{i},\varphi_{j})\in P$,
and any word that visits $X_{2}$ must satisfy $\varphi_{i}$ and any word that
visits $X_{1}$ must satisfy $\neg\varphi_{i}\land\varphi_{j}$. Condition (2)
asserts that the opposite of condition (1) should never hold. That is, there
must not exist a pair of scLTL formulas $\varphi_{i},\varphi_{j}\in\Phi$ such
that $(\varphi_{i},\varphi_{j})\in P$, and any word that visits $X_{1}$
satisfies $\varphi_{i}$ and any word that visits $X_{2}$ satisfies
$\neg\varphi_{i}\land\varphi_{j}$.

######  Example 3.

We describe the construction of preference DFA for first preference objective
(PO1). The underlying DFA of the preference DFA for (PO1) is constructed as
the union of DFAs corresponding to $\Diamond\,A,\Diamond\,B,\Diamond\,C$, and
is shown in Fig. 2. Every state in preference DFA is annotated as a tuple
$(a_{i},b_{j},e_{k})$ where $i,j,k=0,1$. The subscript $i,j,k=0$ means that
corresponding region has been visited. Therefore, all states except
$(a_{1},b_{1},e_{1})$ are final states since at least one of the formulas is
satisfied in all states but $(a_{1},b_{1},e_{1})$.

$(a_{1},b_{1},e_{1}),\\{\\}$$(a_{0},b_{1},e_{1}),\\{y_{EA},y_{BA}\\}$$(a_{1},b_{1},e_{0}),\\{x_{EA}\\}$$(a_{0},b_{1},e_{0}),\\{y_{BA}\\}$$(a_{1},b_{0},e_{1}),\\{x_{BA}\\}$$(a_{0},b_{0},e_{1}),\\{y_{EA}\\}$$(a_{1},b_{0},e_{0}),\\{x_{BA},x_{EA}\\}$$(a_{0},b_{0},e_{0}),\\{x_{BA},x_{EA}\\}$$A$$A$$A$$A$$E$$E$$E$$E$$B$$B$$B$$B$
Figure 2: Preference DFA representing preference objective (PO1).
$X_{2},\\{x_{EA}\\}$$X_{1},\\{y_{EA}\\}$$X_{6},\\{x_{EA},x_{BA}\\}$$X_{5},\\{y_{EA},y_{BA}\\}$$X_{4},\\{x_{BA}\\}$$X_{3},\\{y_{EA}\\}$
Figure 3: Preference graph corresponding to preference DFA in Fig. 2.

Each final state is assigned a set of labels. For instance, the state
$\lambda((a_{1},b_{0},e_{0}))=\\{x_{BA},x_{EA}\\}$ 111We use $A,B,E$ in places
of numerical indices.  since by any word that visits the state satisfies
$\Diamond\,B$ and $\Diamond\,E$. This results in $6$ unique labels
corresponding to a different class of equivalent words in $\Sigma^{\omega}$
that visit that final state. These classes form the nodes $X_{k}$ for
$k=1\ldots 6$ of the preference graph shown in Fig. 3. An edge $(X_{2},X_{5})$
expresses that any word that visits $X_{5}$ is strictly preferred to any word
that visits $X_{2}$ but not $X_{5}$. Similarly, any word that visits $X_{4}$
only is incomparable to any word that visits $X_{6}$ only.

##  V Opportunistic Qualitative Planning with Incomplete Preferences

In this section, we define two types of strategies, that exploit the
_opportunities_ that arise due to stochasticity with a positive probability or
with probability one, respectively.

######  Definition 9 (Product of an MDP with a Preference DFA).

Given an MDP $M=\langle S,A,P,\mathcal{AP},L\rangle$ and the preference DFA
$\mathcal{B}=\langle Q,\Sigma,\delta,{q_{0}},F,G=(\mathcal{X},E)\rangle$, the
product of MDP with preference DFA is defined as the tuple,

$\mathcal{M}:=\langle V,A,\Delta,\mathcal{F},\mathcal{G}\rangle,$

where $V\coloneqq S\times Q$ is the finite set of states. $A$ is the same set
of actions as $M$. The transition function $\Delta:V\times
A\rightarrow\mathcal{D}(V)$ is defined as follows: for any states
$(s,q),(s^{\prime},q^{\prime})\in V$ and any action $a\in A$,
$\Delta((s^{\prime},q^{\prime})\mid(s,q),a)=P(s^{\prime}\mid s,a)$ if
$q^{\prime}\in\delta(q,L(s^{\prime}))$ and $0$ otherwise.
$\mathcal{F}\subseteq V$ is the set of final states by reaching which at least
some outcome is achieved. The component
$\mathcal{G}=(\mathcal{W},\mathcal{E})$ is a graph where
$\mathcal{W}:=\\{S\times X\mid X\in\mathcal{X}\\}$ is the set of nodes and
$\mathcal{E}$ is a set of edges such that, for any $W_{i}=S\times X_{i}$ and
$W_{j}=S\times X_{j}$, $(W_{i},W_{j})\in\mathcal{E}$ if and only if
$(X_{i},X_{j})\in E$.

In the product construction, an edge $(W_{i},W_{j})\in\mathcal{W}$ denotes
that any path $\rho\in V^{\ast}$ that reaches $W_{j}$ is strictly preferred to
any path $\rho^{\prime}\in V^{\ast}$ that reaches $W_{i}$ but not $W_{j}$
under the given preference. Thus, we transform the preference over words given
by the preference DFA to a preference over outcomes, each of which reaches a
subsets of states in $\mathcal{W}$. For each node $W\in\mathcal{W}$, we can
compute a set of states, denoted $\mathsf{ASWin}(W)$, from which the agent has
a strategy to reach $W$ with probability one, using the solution of almost-
sure winning in MDPs with reachability objective [16]. It is possible that
$v\in\mathsf{ASWin}(W)$ and $v\in\mathsf{ASWin}(W^{\prime})$ where $W\neq
W^{\prime}$ and $(W,W^{\prime})\in\mathcal{E}$. In this case, a preference
satisfying strategy must visit the preferred node $W^{\prime}$. To generalize,
let $Z\subseteq\mathcal{W}$ be a subset of nodes in the product, we overload
the notation $\mathsf{MP}$ such that $\mathsf{MP}(Z)=\\{W\in Z\mid\nexists
W^{\prime}\in Z,(W,\mathcal{W}^{\prime})\in\mathcal{E}\\}$. A preference
satisfying strategy from $v$ must visit a node in $\mathsf{MP}(Z_{v})$ where
$Z_{v}=\\{W\in\mathcal{W}\mid v\in\mathsf{ASWin}(W)\\}$.

However, at some states, the uncertainty may create opportunities to
transition from the state $v$ to $v^{\prime}\in V$ such that a more preferred
node can be reached almost-surely from $v^{\prime}$. We call such a transition
to be an _improvement_.

######  Definition 10 (Improvement).

Given any states $v_{1},v_{2}\in V$, $v_{2}$ is said to be an _improvement_
over $v_{1}$ if and only if there exists a pair of preference nodes
$W_{1}\in\mathsf{MP}(\\{W\in\mathcal{W}\mid v_{1}\in\mathsf{ASWin}(W)\\})$ and
$W_{2}\in\mathsf{MP}(\\{W\in\mathcal{W}\mid v_{2}\in\mathsf{ASWin}(W)\\})$
such that $(W_{1},W_{2})\in E$.

A transition from state $v\in V$ to $v^{\prime}\in V$ is said to be
_improving_ if $v^{\prime}$ is an improvement over $v$.

######  Definition 11.

A strategy $\pi:V\rightarrow 2^{A}\cup\\{\uparrow\\}$ 222$\pi(v)=\uparrow$
means the function $\pi$ is undefined at $v$. is said to be _safe and
positively improving (resp., safe and almost-surely improving)_ if, the
following conditions hold for any state $v\in V$ such that
$\pi(v)\neq\uparrow$: (a) there exists (resp., for all) a path $\rho$ in
$\mathcal{M}_{\pi}$ with $\rho[0]=v$ such that, for some $i\geq 0$,
$\rho[i+1]$ is an improvement over $\rho[i]$; (b) there does not exist a path
$\rho$ in $\mathcal{M}_{\pi}$ with $\rho[0]=v$ such that, for some $i\geq 0$,
$\rho[i]$ is an improvement over $\rho[i+1]$.

Intuitively, the SPI and SASI strategies exploit opportunities by inducing an
improving transition with a positive probability and with probability one,
respectively.

We now define a new model called _an improvement MDP_ that differentiates the
states reached by improving transitions.

######  Definition 12 (Improvement MDP).

Given a product MDP $\mathcal{M}$, an _improvement MDP_ is the tuple,

$\tilde{M}=\langle\tilde{V},A,\tilde{\Delta},\tilde{\mathcal{F}}\rangle,$

where $\tilde{V}=\\{(v,\top),(v,\bot)\mid v\in V\\}$ is the set of states,
$\tilde{\mathcal{F}}=\\{(v,\top)\mid v\in V\\}$ is the set of final states. An
action $a\in A$ is enabled at a state $v\in\tilde{V}$ if and only if for for
any $v^{\prime}$ such that $\Delta(v,a,v^{\prime})>0$, $v$ is _not_ an
improvement over $v^{\prime}$. The transition function
$\tilde{\Delta}:\tilde{V}\times A\rightarrow\mathcal{D}(\tilde{V})$ is defined
as follows: For any $v\in V$, for an action $a$ _enabled from_ $v$, if
$\Delta(v,a,v^{\prime})>0$ and $v^{\prime}$ is an improvement from $v$, then
let $\tilde{\Delta}((v,\bot),a,(v^{\prime},\top))=\Delta(v,a,v^{\prime})$.
Else, if $\Delta(v,a,v^{\prime})>0$ and $v^{\prime}$ is not an improvement
from $v$, then let
$\tilde{\Delta}((v,\bot),a,(v^{\prime},\bot))=\Delta(v,a,v^{\prime})$ and
$\tilde{\Delta}((v,\top),a,(v^{\prime},\bot))=\Delta(v,a,v^{\prime})$.

######  Theorem 1.

The following statements hold for any state $v\in V$ in product MDP.

  1. 1.

An SPI strategy at $v$ is a positive winning strategy in improvement MDP at
the state $(v,\bot)$ to visit $\tilde{\mathcal{F}}$.

  2. 2.

An SASI strategy at $v$ is an almost-sure winning strategy in improvement MDP
at the state $(v,\bot)$ to visit $\tilde{\mathcal{F}}$.

###### Proof (Sketch).

Statement (1). By construction, any action which induces a transition that
violates condition (b) in Def. 11 with positive probability is disabled in the
improvement MDP. Also, by construction, any final state in $\mathcal{F}$ can
only be reached by making an improvement. Hence, a positive winning strategy
in improvement MDP which visits $\tilde{\mathcal{F}}$ satisfies condition (a)
in Def. 11. The proof of statement (2) is similar to that of statement (1). ∎

The SPI and SASI strategies may exploit multiple opportunities by inducing
sequential improvements: Whenever the agent reaches a state
$(v,\top)\in\tilde{V}$, he will check if a SPI (or SASI) strategy exists for
$(v,\bot)$. If yes, then the agent will carry out the SPI (or SASI) strategy.
Otherwise, the agent will carry out the almost-sure winning strategy for one
of the most preferred and satisfied objective at $v$.

######  Example 4.

Consider the case when robot is at $(2,1)$ with $4$ units of battery and is to
satisfy (PO1). Although the robot cannot almost-surely visit either $B$ or $E$
individually, it can almost-surely visit one of $B,E$ by moving West. Since
visiting both $B$ and $E$ is strictly preferred to visiting $A$, moving West
is a safe and almost-surely improving strategy at $(2,1)$. Instead of $4$
units, if the robot starts with $2$ units of battery, it can reach neither of
$A,B$ or $C$ almost-surely. In this case, the SASI strategy is undefined. The
SPI strategy is to choose West because, with positive probability, it leads to
cells $(1,2),(1,0)$ with $1$ units of battery remaining. From these states,
one of $B,E$ can be reached almost-surely.

Consider the robot whose objective is (PO2) starting at the cell $(2,1)$ with
$4$ units of battery. From this state, only $A$ can be visited almost-surely.
The SASI strategy at $v_{0}$ is to move North because, with positive
probability, the robot would reach one of the cells—$(1,2),(2,2),(2,3)$—with
$3$ units of battery remaining. Since from each of these states at least one
of $B,C,D,F$ can almost-surely be achieved, the robot almost-surely makes an
improvement. Suppose the robot reaches $(2,3)$ with $3$ units of battery. From
this state, only visiting $C$ is almost-surely winning. However, the SASI
strategy is to move North and then East, thereby ensuring a visit to either
$F$ or $D$ with probability one. Hence, we see that SASI strategy not only
plans for a single improvement, but it may also induces multiple sequential
improvements.

##  VI Conclusion

In this work, we propose a language to specify incomplete preferences as a
pre-order over temporal objectives. This allows us to synthesize qualitatively
plans even when some outcomes are incomparable. We define two types of
opportunistic strategies that strategically, and whenever possible, improve
the outcome they can achieve sequentially. Our work provides a method for
stochastic planning with incomplete preferences over a subclass of temporal
logic objectives. Building on this work, we consider a number of future
directions: 1) we will consider a preference over temporal objectives that
encompass more general LTL properties such as safety, recurrence, and
liveness; 2) we will build on the qualitative reasoning to study quantitative
planning with such preference specifications. The later requires us to jointly
consider how well (the probability) an objective is satisfied and how
preferred is the objective.

## References

  * [1] R. Hastie and R. M. Dawes, _Rational choice in an uncertain world: The psychology of judgment and decision making_. Sage, 2010.
  * [2] Z. Manna and A. Pnueli, _The temporal logic of reactive and concurrent systems: Specification_. Springer Science & Business Media, 2012.
  * [3] X. C. Ding, S. L. Smith, C. Belta, and D. Rus, “Mdp optimal control under temporal logic constraints,” in _2011 50th IEEE Conference on Decision and Control and European Control Conference_. IEEE, 2011, pp. 532–538.
  * [4] M. Hasanbeig, Y. Kantaros, A. Abate, D. Kroening, G. J. Pappas, and I. Lee, “Reinforcement learning for temporal logic control synthesis with probabilistic satisfaction guarantees,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_. IEEE, 2019, pp. 5338–5343.
  * [5] B. Lacerda, D. Parker, and N. Hawes, “Optimal and dynamic planning for markov decision processes with co-safe ltl specifications,” in _2014 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , 2014, pp. 1511–1516.
  * [6] M. Wen, R. Ehlers, and U. Topcu, “Correct-by-synthesis reinforcement learning with temporal logic constraints,” in _2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2015, pp. 4983–4990.
  * [7] J. Tumova, G. C. Hall, S. Karaman, E. Frazzoli, and D. Rus, “Least-violating control strategy synthesis with safety rules,” in _Proceedings of the 16th international conference on Hybrid systems: computation and control_. ACM, 2013, pp. 1–10.
  * [8] M. Lahijanian and M. Kwiatkowska, “Specification revision for Markov decision processes with optimal trade-off,” in _Proc. 55th Conference on Decision and Control (CDC’16)_ , 2016, pp. 7411–7418.
  * [9] N. Mehdipour, C.-I. Vasile, and C. Belta, “Specifying User Preferences Using Weighted Signal Temporal Logic,” _IEEE Control Systems Letters_ , vol. 5, no. 6, pp. 2006–2011, Dec. 2021.
  * [10] R. Bloem, H. Chockler, M. Ebrahimi, and O. Strichman, “Synthesizing reactive systems using robustness and recovery specifications,” in _2019 Formal Methods in Computer Aided Design (FMCAD)_. IEEE, 2019, pp. 147–151.
  * [11] J. J. Thomson, “Killing, letting die, and the trolley problem,” _The monist_ , vol. 59, no. 2, pp. 204–217, 1976.
  * [12] J. Van Benthem, S. Van Otterloo, and O. Roy, “Preference logic, conditionals and solution concepts in games,” _Modality Matters,_ , 2005.
  * [13] M. Bienvenu, J. Lang, and N. Wilson, “From Preference Logics to Preference Languages, and Back,” _Twelfth International Conference on the Principles of Knowledge Representation and Reasoning_ , p. 11, 2010.
  * [14] G. R. Santhanam, S. Basu, and V. Honavar, “Representing and reasoning with qualitative preferences: Tools and applications,” _Synthesis Lectures on Artificial Intelligence and Machine Learning_ , vol. 10, no. 1, pp. 1–154, 2016\.
  * [15] C. Boutilier, R. I. Brafman, C. Domshlak, H. H. Hoos, and D. Poole, “Cp-nets: A tool for representing and reasoning withconditional ceteris paribus preference statements,” _Journal of artificial intelligence research_ , vol. 21, pp. 135–191, 2004.
  * [16] C. Baier and J.-P. Katoen, _Principles of model checking_. MIT press, 2008.
  * [17] O. Kupferman and M. Y. Vardi, “Model checking of safety properties,” _Formal Methods in System Design_ , vol. 19, no. 3, pp. 291–314, 2001.
  * [18] C. Baier, M. Größer, M. Leucker, B. Bollig, and F. Ciesinski, “Controller Synthesis for Probabilistic Systems (Extended Abstract),” in _Exploring New Frontiers of Theoretical Informatics_ , J.-J. Levy, E. W. Mayr, and J. C. Mitchell, Eds. Boston: Kluwer Academic Publishers, 2004, vol. 155, pp. 493–506.
  * [19] D. Bouyssou, D. Dubois, and M. Pirlot, _Concepts & Methods of Decision-Making_. John Wiley & Sons Inc., 2009.

  *[MDP]: Markov Decision Process
  *[LTL]: Linear Temporal Logic
  *[scLTL]: syntactically co-safe LTL
  *[DFA]: Deterministic Finite Automaton
  *[ MDP]: Markov Decision Process


#  $B\to K_{1}\pi(K)$ decays in the perturbative QCD approach

Zhi-Qing Zhang1 , Zhi-Wei Hou1, Yueling Yang2, Junfeng Sun2
1 Department of Physics, Henan University of Technology,
Zhengzhou, Henan 450001, China;
2 College of Physics and Information Engineering,
Henan Normal University, Xinxiang 453007, China

###### Abstract

Within the framework of the perturbative QCD approach, we study the two-body
charmless decays $B\to K_{1}(1270)(K_{1}(1400))\pi(K)$. We find the following
results: (i) The decays $\bar{B}^{0}\to
K_{1}(1270)^{+}\pi^{-},K_{1}(1400)^{+}\pi^{-}$ are incompatible with the
present experimental data. There exists a similar situation for the decays
$\bar{B}^{0}\to a_{1}(1260)^{+}K^{-},b_{1}(1235)^{+}K^{-}$, which are usually
considered that the nonperturbative contributions are needed to explain the
data. But the difference is that the nonperturbative contributions seem to
play opposite roles in these two groups of decays.(ii) The pure annihilation
type decays $\bar{B}^{0}\to K_{1}^{\pm}(1270)K^{\mp},K_{1}^{\pm}(1400)K^{\mp}$
are good channels to test whether an approach can be used to calculate
correctly the strength of the penguin-annihilation amplitudes. Their branching
ratios are predicted at $10^{-7}$ order, which are larger than the QCDF
results. (iii) The dependence of the direct CP-violating asymmetries of these
decays on the mixing angle $\theta_{K_{1}}$ are also considered.

###### pacs:

13.25.Hw, 12.38.Bx, 14.40.Nd

##  I Introduction

In general, the mesons are classified in $J^{PC}$ multiplets. There are two
types of orbitally excited axial-vector mesons, namely $1^{++}$ and $1^{+-}$.
The former includes $a_{1}(1260),f_{1}(1285),f_{1}(1420)$ and $K_{1A}$, which
compose the ${}^{3}P_{1}$-nonet, and the latter includes
$b_{1}(1235),h_{1}(1170),h_{1}(1380)$ and $K_{1B}$, which compose the
${}^{1}P_{1}$-nonet. Except $a_{1}(1260)$ and $b_{1}(1235)$, other axial-
vector mesons exist mixing problem, which makes their inner structure become
more ambiguous, for example, $K_{1A}$ and $K_{1B}$ can mix with each other and
form two physical mass eigenstates $K_{1}(1270),K_{1}(1400)$. Various values
about the mixing angle $\theta_{K_{1}}$ can be found in different literatures,
which will be examined in more detail in Sec.III. For the mixings of the
SU(3)-singlet and SU(3)-octet mesons, specifically, the
$f_{1}(1285)-f_{1}(1420)$ mixing angle $\theta_{{}^{3}P_{1}}$ and the
$h_{1}(1170)-h_{1}(1380)$ mixing angle $\theta_{{}^{1}P_{1}}$, there also
exist several values in the phenomenal analysis. Certainly, these two angles
can associate with $\theta_{K_{1}}$ through the Gell-Mann-Okubo mass formula.
For the lack of sufficient experimental data, none of them can be accurately
determined up to now. So the decays involving these mesons become more
ambiguous compared with the decays involving $a_{1}(1260)$ or/and
$b_{1}(1235)$ meson(s), which have been discussed in the previous works wwang
; zqzhang1 ; zqzhang2 ; cmv ; vnp ; cy .

In this paper, we would like to discuss the decays $B\to
K_{1}(1270)\pi(K),K_{1}(1400)\pi(K)$. On the theoretical side, many approaches
have been used to study these decays, such as the naive factorization cmv ,
the generalized factorization vnp , and the QCD factorization approach cy .
From the predictions of these approaches, One can find that the branching
ratios of the decays $B\to K_{1}(1270)\pi,K_{1}(1400)\pi$ are in the order of
$10^{-6}$, for example, $Br(B^{0}\to K_{1}(1270)^{+}\pi^{-})=(3\sim 8)\times
10^{-6}$, $Br(B^{0}\to K_{1}(1400)^{+}\pi^{-})=(2\sim 5)\times 10^{-6}$, those
of almost all the decays $B\to K_{1}(1270)K,K_{1}(1400)K$ are in the order of
$10^{-8}\sim 10^{-7}$. While on the experimental side, the large upper limits
are given for the decays $B^{0}\to K_{1}(1400)^{+}\pi^{-}$ and $B^{+}\to
K_{1}(1400)^{0}\pi^{+}$ at the $90\%$ level (C.L.) of $1.1\times 10^{-3}$ and
$2.6\times 10^{-3}$, respectively argus , and the Heavy Flavor Averaging
Group(HFAG) gives the following results hfag :

$\displaystyle Br(B^{+}\to K_{1}(1270)^{0}\pi^{+})$ $\displaystyle<$
$\displaystyle 40\times 10^{-6},Br(B^{+}\to K_{1}(1270)^{0}\pi^{+})<39\times
10^{-6},$ (1) $\displaystyle Br(B^{0}\to K_{1}(1270)^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(17^{+8}_{-11})\times 10^{-6},Br(B^{0}\to
K_{1}(1400)^{+}\pi^{-})=(17^{+7}_{-9})\times 10^{-6}.\;\;\;\;\;$ (2)

The preliminary data are given by BABAR barbar1 ,

$\displaystyle BR(B^{0}\to K_{1}^{+}(1270)\pi^{-})$ $\displaystyle=$
$\displaystyle(12.0\pm 3.1^{+9.3}_{-4.5})\times 10^{-6},$ (3) $\displaystyle
BR(B^{0}\to K_{1}^{+}(1400)\pi^{-})$ $\displaystyle=$ $\displaystyle(16.7\pm
2.6^{+3.5}_{-5.0})\times 10^{-6}.$ (4)

Furthermore, BABAR has also measured the branching ratios $Br(B^{0}\to
K_{1}(1270)^{+}\pi^{-}+K_{1}(1400)^{+}\pi^{-})=3.1^{+0.8}_{-0.7}\times
10^{-5}$ and $Br(B^{+}\to
K_{1}(1270)^{0}\pi^{+}+K_{1}(1400)^{0}\pi^{+})=2.9^{+2.9}_{-1.7}\times
10^{-5}$ with $7.5\sigma$ and $3.2\sigma$ significance, respectively. In the
paper barbar2 , the two sided intervals for some of the decays $B\to
K_{1}(1270)\pi,K_{1}(1400)\pi$ are evaluated at $68\%$ probability ($\times
10^{-5}$):

$\displaystyle BR(B^{-}\to\bar{K}_{1}(1270)^{0}\pi^{-})$ $\displaystyle\in$
$\displaystyle[0.0,2.1],BR(B^{-}\to\bar{K}_{1}(1400)^{0}\pi^{-})\in[0.0,2.5],$
(5) $\displaystyle BR(B^{0}\to K_{1}(1270)^{+}\pi^{-})$ $\displaystyle\in$
$\displaystyle[0.6,2.5],BR(B^{0}\to K_{1}(1400)^{+}\pi^{-})\in[0.8,2.4].$ (6)

In view of the differences between the theories and experiments, we are going
to use the PQCD approach to explore these decays and analyze whether the
nonperturbtive contributions are necessary to explain the experimental data.
In the following, $K_{1}(1270)$ and $K_{1}(1400)$ are denoted as $K_{1}$ in
some places for convenience. The layout of this paper is as follows. In
Sec.II, the decay constants and the light-cone distribution amplitudes of the
relevant mesons are introduced. In Sec.III, we then analyze these decay
channels by using the PQCD approach. The numerical results and the discussions
are given in Sec. IV. The conclusions are presented in the final part.

##  II decay constants and distribution amplitudes

For the wave function of the heavy $B$ meson, we take

$\displaystyle\Phi_{B}(x,b)=\frac{1}{\sqrt{2N_{c}}}(P/_{B}+m_{B})\gamma_{5}\phi_{B}(x,b).$
(7)

Here only the contribution of Lorentz structure $\phi_{B}(x,b)$ is taken into
account, since the contribution of the second Lorentz structure
$\bar{\phi}_{B}$ is numerically small cdlu  and has been neglected. For the
distribution amplitude $\phi_{B}(x,b)$ in Eq.(7), we adopt the following
model:

$\displaystyle\phi_{B}(x,b)=N_{B}x^{2}(1-x)^{2}\exp[-\frac{M^{2}_{B}x^{2}}{2\omega^{2}_{b}}-\frac{1}{2}(\omega_{b}b)^{2}],$
(8)

where $\omega_{b}$ is a free parameter, we take $\omega_{b}=0.4\pm 0.04$ Gev
in numerical calculations, and $N_{B}=101.4$ is the normalization factor for
$\omega_{b}=0.4$.

The distribution amplitudes of the axial-vector $K_{1}$ are written as :

$\displaystyle\langle
K_{1}(P,\epsilon^{*}_{L})|\bar{q}_{2\beta}(z)q_{1\alpha}(0)|0\rangle$
$\displaystyle=$
$\displaystyle\frac{i\gamma_{5}}{\sqrt{2N_{c}}}\int^{1}_{0}dx\;e^{ixp\cdot
z}[m_{K_{1}}\epsilon/^{*}_{L}\phi_{K_{1}}(x)+\epsilon/^{*}_{L}P/\phi_{K_{1}}^{t}(x)+m_{K_{1}}\phi^{s}_{K_{1}}(x)]_{\alpha\beta},$
$\displaystyle\langle
K_{1}(P,\epsilon^{*}_{T})|\bar{q}_{2\beta}(z)q_{1\alpha}(0)|0\rangle$
$\displaystyle=$
$\displaystyle\frac{i\gamma_{5}}{\sqrt{2N_{c}}}\int^{1}_{0}dx\;e^{ixp\cdot
z}\left[m_{K_{1}}\epsilon/^{*}_{T}\phi^{v}_{K_{1}}(x)+\epsilon/^{*}_{T}P/\phi_{K_{1}}(x)\right.$
(9)
$\displaystyle\left.+m_{K_{1}}i\epsilon_{\mu\nu\rho\sigma}\gamma_{5}\gamma^{\mu}\epsilon^{*v}_{T}n^{\rho}v^{\sigma}\phi^{a}_{K_{1}}(x)\right]_{\alpha\beta},$

where $K_{1}$ refers to the two flavor states $K_{1A}$ and $K_{1B}$, and the
corresponding distribution functions can be calculated by using light-cone QCD
sum rule and listed as follows:

$\displaystyle\begin{cases}\phi_{K_{1}}(x)=\frac{f_{K_{1}}}{2\sqrt{2N_{c}}}\phi_{\parallel}(x),\phi^{T}_{K_{1}}(x)=\frac{f_{K_{1}}}{2\sqrt{2N_{c}}}\phi_{\perp}(x),\\\
\phi^{t}_{K_{1}}(x)=\frac{f_{K_{1}}}{2\sqrt{2N_{c}}}h^{(t)}_{\parallel}(x),\phi^{s}_{K_{1}}(x)=\frac{f_{K_{1}}}{2\sqrt{4N_{c}}}\frac{d}{dx}h^{(s)}_{\parallel}(x),\\\
\phi^{v}_{K_{1}}(x)=\frac{f_{K_{1}}}{2\sqrt{2N_{c}}}g^{(v)}_{\perp}(x),\phi^{a}_{K_{1}}(x)=\frac{f_{K_{1}}}{8\sqrt{2N_{c}}}\frac{d}{dx}g^{(a)}_{\perp}(x).\end{cases}$
(10)

Here we use $f_{K_{1}}$ to present both the longitudinally and transversely
polarized states $K_{1A}(K_{1B})$ by assuming
$f^{T}_{K_{1A}}=f_{K_{1A}}=f_{K_{1}}$ for $K_{1A}$ and
$f_{K_{1B}}=f^{T}_{K_{1B}}=f_{K_{1}}$ for $K_{1B}$, respectively. It is
similar for the case of $a_{1}(b_{1})$ states, and the difference is that here
$K_{1A}$ and $K_{1B}$ are not the mass eigenstates. In Eq.(10), the twist-2
distribution functions are in the first line and can be expanded as:

$\displaystyle\phi_{\parallel,\perp}$ $\displaystyle=$ $\displaystyle
6x(1-x)\left[a^{\parallel,\perp}_{0}+3a^{\parallel,\perp}_{1}t+a^{\parallel,\perp}_{2}\frac{3}{2}(5t^{2}-1)\right],$
(11)

the twist-3 light-cone distribution amplitudes (LCDAs) are used the following
forms for $K_{1A}$ and $K_{1B}$ states:

$\displaystyle h^{(t)}_{\parallel}(x)$ $\displaystyle=$ $\displaystyle
3a^{\perp}_{0}t^{2}+\frac{3}{2}a^{\perp}_{1}t(3t^{2}-1),h^{(s)}_{\parallel}(x)=6x(1-x)(a^{\perp}_{0}+a^{\perp}_{1}t),$
$\displaystyle g^{(a)}_{\perp}(x)$ $\displaystyle=$ $\displaystyle
6x(1-x)(a^{\parallel}_{0}+a^{\parallel}_{1}t),g^{(v)}_{\perp}(x)=\frac{3}{4}a^{\parallel}_{0}(1+t^{2})+\frac{3}{2}a^{\parallel}_{1}t^{3},$
(12)

where $t=2x-1$ and the Gegenbauer moments cheng
$a^{\perp}_{0}(K_{1A})=0.26^{+0.03}_{-0.22},a^{\parallel}_{0}(K_{1B})=-0.15\pm
0.15,a^{\parallel}_{0}(K_{1A})=a^{\perp}_{0}(K_{1B})=1$,
$a^{\perp}_{1}(K_{1A})=-1.08\pm
0.48,a^{\perp}_{1}(K_{1B})=0.30^{+0.00}_{-0.31}$,
$a^{\parallel}_{1}(K_{1A})=-0.30^{+0.26}_{-0.00}$ ,
$a^{\parallel}_{1}(K_{1B})=-1.95\pm 0.45$, $a^{\parallel}_{2}(K_{1A})=-0.05\pm
0.03,a^{\parallel}_{2}(K_{1B})=0.09^{+0.16}_{-0.18}$.

The wave functions for the pseudoscalar (P) mesons $K,\pi$ are given as:

$\displaystyle\Phi_{K(\pi)}(P,x,\zeta)\equiv\frac{1}{\sqrt{2N_{C}}}\gamma_{5}\left[P/\phi^{A}_{K(\pi)}(x)+m_{0}\phi^{P}_{K(\pi)}(x)+\zeta
m_{0}(v/n/-v\cdot n)\phi^{T}_{K(\pi)}(x)\right],$ (13)

where the parameter $\zeta$ is either $+1$ or $-1$ depending on the assignment
of the momentum fraction $x$. The chiral scale parameter $m_{0}$ is defined as
$m_{0}=\frac{m^{2}_{\pi}}{m_{u}+m_{d}}$ for $\pi$ meson and
$m_{0}=\frac{m^{2}_{K}}{m_{u}+m_{s}}$ for $K$ meson. The distribution
amplitudes are expanded as:

$\displaystyle\phi^{A}_{K(\pi)}(x)$ $\displaystyle=$
$\displaystyle\frac{3f_{K(\pi)}}{\sqrt{6}}x(1-x)\left[1+a_{1K(\pi)}C^{3/2}_{1}(t)+a_{2K(\pi)}C^{3/2}_{2}(t)+a_{4K(\pi)}C^{3/2}_{4}(t)\right],$
(14) $\displaystyle\phi^{P}_{K(\pi)}(x)$ $\displaystyle=$
$\displaystyle\frac{3f_{K(\pi)}}{2\sqrt{6}}\left[1+\left(30\eta_{3}-\frac{5\rho^{2}_{K(\pi)}}{2}\right)C^{1/2}_{2}(t)\right.$
(15)
$\displaystyle\left.-3\left(\eta_{3}\omega_{3}+\frac{9\rho^{2}_{K(\pi)}}{20}(1+6a_{2K(\pi)})\right)C^{1/2}_{4}(t)\right],$
$\displaystyle\phi^{T}_{K(\pi)}(x)$ $\displaystyle=$
$\displaystyle\frac{-f_{K(\pi)}t}{2\sqrt{6}}\left[1+6(5\eta_{3}-\frac{\eta_{3}\omega_{3}}{2}-\frac{7\rho^{2}_{K(\pi)}}{20}-\frac{3\rho^{2}_{K(\pi)}a_{2K(\pi)}}{5})(1-10x+10x^{2})\right],\;\;\;\;\;$
(16)

where the decay constants $f_{K}=0.16$ GeV, $f_{\pi}=0.13$ GeV and the
Gegenbauer moments, Gegenbauer polynomials are defined as:

$\displaystyle a_{1K}$ $\displaystyle=$ $\displaystyle 0.17\pm
0.17,a_{1\pi}=0,a_{2K}=a_{2\pi}=0.115\pm 0.115,a_{4K}=a_{4\pi}=-0.015,$
$\displaystyle C^{3/2}_{1}(t)$ $\displaystyle=$ $\displaystyle
3t,C^{3/2}_{2}(t)=\frac{3}{2}(5t^{2}-1),C^{3/2}_{4}(t)=\frac{15}{8}(1-14t^{2}+21t^{4}),$
$\displaystyle C^{1/2}_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(3t^{2}-1),C^{1/2}_{4}(t)=\frac{1}{8}(3-30t^{2}+35t^{4}),$
(17)

and the constants $\eta_{3}=0.015,\omega_{3}=-3$, the mass ratio
$\rho_{K(\pi)}=m_{K(\pi)}/m_{0K(\pi)}$ with $m_{K}=0.49$ GeV, $m_{0K}=1.7$
GeV, $m_{\pi}=0.135$ GeV, $m_{0\pi}=1.4$ GeV.

##  III  the perturbative QCD calculation

The PQCD approach is an effective theory to handle hadronic $B$ decays cdlu2 ;
keum ; mishima . Because it takes into account the transverse momentum of the
valence quarks in the hadrons, one will encounter the double logarithm
divergences when the soft and the collinear momenta overlap. Fortunately,
these large double logarithm can be resummed into the Sudakov factor hnli0 .
There also exit another type of double logarithms which arise from the loop
corrections to the weak decay vertex. These double logarithms can also be
resummed and resulted in the threshold factor hnli00 . This factor decreases
faster than any other power of the momentum fraction in the threshold region,
which removes the endpoint singularity. It is often parameterized into a
simple form which is independent on channels, twists and flavors hnli .
Certainly, when the higher order diagrams only suffer from soft or collinear
infrared divergence, it is ease to cure by using the eikonal approximation
hnli2 . Controlling these kinds of divergences reasonably makes the PQCD
approach more self-consistent.

Figure 1:  Diagrams contributing to the decay
$\bar{B}^{0}\to\bar{K}^{0}_{1A}\pi^{0}$.

For these two axial vector mesons, their mass eigenstates and flavor
eigenstates are not the same with each other, and the former can be obtained
by the latter through a mixing angle $\theta_{K_{1}}$:

$\displaystyle
K_{1}(1270)=K_{1A}\sin\theta_{K_{1}}+K_{1B}\cos\theta_{K_{1}},K_{1}(1400)=K_{1A}\cos\theta_{K_{1}}-K_{1B}\sin\theta_{K_{1}}.$
(18)

Unfortunately, there are many uncertainties about this mixing angle. From
various phenomenological analysis and experimental data on the masses of these
two physical states, it indicates that this mixing angle is around either
$33^{\circ}$ or $58^{\circ}$ rkc ; iw ; dma ; su ; bg ; pvc ; gi ; vfv ; tky ;
div . Certainly, the author of cheng1  stresses that the sign of
$\theta_{K_{1}}$ depends on the relative sign of flavor states $K_{1A}$ and
$K_{1B}$, which can be determined by fixing the relative sign of the decay
constants of $K_{1A}$ and $K_{1B}$. If the decay constants $f_{1A},f_{1B}$ are
the same in sign (it means that the transitions $B\to K_{1A}$ and $B\to
K_{1B}$ have the opposite signs), then the mixing angle $\theta_{K_{1}}$
defined in (18) is positive. It is noticed that the mixing angle for the
antiparticle states $\bar{K}_{1}(1270),\bar{K}_{1}(1400)$, which is denoted as
$\theta_{\bar{K}_{1}}$, is of opposite sign to that for the particle states
$K_{1}(1270),K_{1}(1400)$. But even so, we cannot confirm whether
$\theta_{K_{1}}$ is larger or less than $45^{\circ}$ up to now. Different
approaches and models are used and different values of the mixing angle are
obtained. In order to pin down it, Cheng cheng1  advocates to determine the
mixing angles $\theta_{{}^{3}P_{1}}$ and $\theta_{{}^{1}P_{1}}$ between
$f_{1}(1285)-f_{1}(1420)$ and $h_{1}(1170)-h_{1}(1380)$, respectively, which
in turn depend on the $K_{1A}-K_{1B}$ mixing angle $\theta_{K_{1}}$ through
the mass relation. Through analyzing the present data of the $h_{1},f_{1}$
mesons’ strong/radiative decay modes, the author prefers $\theta_{K_{1}}\sim
33^{\circ}$ over $58^{\circ}$. In view of the present limited data, we will
still include the mixing angle $\theta_{K_{1}}\sim 58^{\circ}$ in our
calculations.

It is just because of the ambiguous mixing angle that makes the study very
difficult. Here we take the decay $\bar{B}^{0}\to\bar{K}_{1}(1270)^{0}\pi^{0}$
as an example, which is contributed by the decays
$\bar{B}^{0}\to\bar{K}^{0}_{1A}\pi^{0}$ and
$\bar{B}^{0}\to\bar{K}^{0}_{1B}\pi^{0}$. Figure 1 is for the Feynman diagrams
of the decay $\bar{B}^{0}\to\bar{K}^{0}_{1A}\pi^{0}$ (it is similar to the
decay $\bar{B}^{0}\to\bar{K}^{0}_{1B}\pi^{0}$), through which the amplitudes
can be calculated directly, and the total amplitudes of the decay
$\bar{B}^{0}\to\bar{K}_{1}(1270)^{0}\pi^{0}$ can be obtained by combining the
two sets of flavor state amplitudes according to Eq.(18):

$\displaystyle\sqrt{2}A(\bar{K}_{1}(1270)^{0}\pi^{0})$ $\displaystyle=$
$\displaystyle-\xi_{t}(f_{K_{1A}}\sin\theta_{K_{1}}+f_{K_{1B}}\cos\theta_{K_{1}})F^{LL}_{e\pi}(a_{4}-\frac{1}{2}a_{10})$
(19)
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(a_{4}-\frac{1}{2}a_{10})$
$\displaystyle-\xi_{t}f_{B}(F^{SP;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{SP;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(a_{6}-\frac{1}{2}a_{8})$
$\displaystyle+f_{\pi}(F^{LL}_{eK_{1A}}\sin\theta_{K_{1}}+F^{LL}_{eK_{1B}}\cos\theta_{K_{1}})\left[\xi_{u}a_{1}-\xi_{t}\left(\frac{3C_{9}}{2}+\frac{C_{10}}{2}\right.\right.$
$\displaystyle\left.\left.-\frac{3C_{7}}{2}-\frac{C_{8}}{2}\right)\right]+(M^{LL;\pi}_{eK_{1A}}\sin\theta_{K_{1}}+M^{LL;\pi}_{eK_{1B}}\cos\theta_{K_{1}})\left[\xi_{u}C_{2}\right.$
$\displaystyle\left.-\xi_{t}\frac{3C_{10}}{2}\right]-\xi_{t}(M^{SP;\pi}_{eK_{1A}}\sin\theta_{K_{1}}+M^{SP;\pi}_{eK_{1B}}\cos\theta_{K_{1}})\frac{3C_{8}}{2},$

where $\xi_{u}=V_{ub}V^{*}_{us},\xi_{t}=V_{tb}V^{*}_{ts}$,
$F^{M_{2}}_{e(a)M_{1}}$ and $M^{M_{2}}_{e(a)M_{1}}$ denote the amplitudes of
factorizable and nonfactorizable emission (annihilation) diagrams, where the
subscript meson $M_{1}$ is involved in the $\bar{B}^{0}$ meson transition, the
superscript meson $M_{2}$ is the emitted particle. The other superscript in
each amplitude denotes different current operators, $(V-A)(V-A),(V-A)(V+A)$
and $(S-P)(S+P)$ corresponding to $LL,LR$ and $SP$, respectively. If
exchanging the positions of $K_{1A}$ and $\pi^{0}$ in Fig.1(a), 1(b), 1(c) and
1(d), we will get the new Feynman diagrams, which can also contribute to the
decay $\bar{B}^{0}\to\bar{K}^{0}_{1A}\pi^{0}$, and the corresponding
amplitudes are given in the last three lines of Eq.(19). The amplitudes for
the decay $\bar{B}^{0}\to\bar{K}^{0}_{1A}(\bar{K}^{0}_{1B})\pi^{0}$ can be
obtained from those for the decay $B\to K\pi$ which can be found in ali , only
changing the variables of $K$ meson with those of $K^{0}_{1A}(K^{0}_{1B})$
meson. So we do not list the analytic expressions for these amplitudes.
Certainly, it is noticed that if the axial-vector meson $K_{1A}(K_{1B})$ is on
the emitted position in the factorizable emission diagrams, there is no scalar
or pseudoscalar current contribution. The total amplitudes for the other three
$B\to K_{1}(1270)\pi$ decay modes can also be written out similarly:

$\displaystyle A(K_{1}(1270)^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle(f_{K_{1A}}\sin\theta_{K_{1}}+f_{K_{1B}}\cos\theta_{K_{1}})F^{LL}_{e\pi}(\xi_{u}a_{1}-\xi_{t}(a_{4}+a_{10}))$
(20)
$\displaystyle+(M^{LL;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(\xi_{u}C_{1}-\xi_{t}(C_{3}+C_{9}))$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{5}+C_{7})$
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+M^{LR;K_{1A}}_{a\pi}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(a_{4}-\frac{1}{2}a_{10})$
$\displaystyle-\xi_{t}f_{B}(F^{SP;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{SP;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(a_{6}-\frac{1}{2}a_{8}),$
$\displaystyle\sqrt{2}A(K_{1}(1270)^{-}\pi^{0})$ $\displaystyle=$
$\displaystyle(f_{K_{1A}}\sin\theta_{K_{1}}+f_{K_{1B}}\cos\theta_{K_{1}})F^{LL}_{e\pi}\left[\xi_{u}a_{1}-\xi_{t}(a_{4}+a_{10})\right]$
(21)
$\displaystyle+(M^{LL;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{e\pi}\cos\theta_{K_{1}})\left[\xi_{u}C_{1}-\xi_{t}(C_{3}+C_{9})\right]$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{5}+C_{7})$
$\displaystyle+(M^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})\left[\xi_{u}C_{1}-\xi_{t}(C_{3}+C_{9})\right]$
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(C_{5}+C_{7})$
$\displaystyle+f_{B}(F^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})\left[\xi_{u}a_{2}-\xi_{t}(a_{4}+a_{10})\right]$
$\displaystyle-
f_{B}(F^{SP;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{SP;K_{1B}}_{a\pi}\cos\theta_{K_{1}})\xi_{t}(a_{6}+a_{8})$
$\displaystyle+f_{\pi}(F^{LL}_{eK_{1A}}\sin\theta_{K_{1}}+F^{LL}_{eK_{1B}}\cos\theta_{K_{1}})\left[\xi_{u}a_{1}-\xi_{t}\left(\frac{3C_{9}}{2}+\frac{C_{10}}{2}\right.\right.$
$\displaystyle\left.\left.-\frac{3C_{7}}{2}-\frac{C_{8}}{2}\right)\right]+(M^{LL;\pi}_{eK_{1A}}\sin\theta_{K_{1}}+M^{LL;\pi}_{eK_{1B}}\cos\theta_{K_{1}})\left[\xi_{u}C_{2}\right.$
$\displaystyle\left.-\xi_{t}\frac{3C_{10}}{2}\right]-\xi_{t}(M^{SP;\pi}_{eK_{1A}}\sin\theta_{K_{1}}+M^{SP;\pi}_{eK_{1B}}\cos\theta_{K_{1}})\frac{3C_{8}}{2},$
$\displaystyle A(\bar{K}_{1}(1270)^{0}\pi^{-})$ $\displaystyle=$
$\displaystyle-\xi_{t}(f_{K_{1A}}\sin\theta_{K_{1}}+f_{K_{1B}}\cos\theta_{K_{1}})F^{LL}_{e\pi}(a_{4}-\frac{1}{2}a_{10})$
(22)
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle+(M^{LL;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{e\pi}\cos\theta_{K_{1}})\left[\xi_{u}C_{1}-\xi_{t}(C_{3}+C_{9})\right]$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{e\pi}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{e\pi}\cos\theta_{K_{1}})(C_{5}+C_{7})$
$\displaystyle+f_{B}(F^{LL;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{a\pi}\cos\theta_{K_{1}})\left[\xi_{u}a_{2}-\xi_{t}(a_{4}+a_{10})\right]$
$\displaystyle-\xi_{t}f_{B}(F^{SP;K_{1A}}_{a\pi}\sin\theta_{K_{1}}+F^{SP;K_{1B}}_{a\pi}\cos\theta_{K_{1}})(a_{6}+a_{8}).$

It is easy to get the total amplitudes for the decay modes including
$\bar{K}_{1}(1400)^{0}$/$K_{1}(1400)^{-}$ by making the replacements with
$\sin\theta_{K_{1}}\to\cos\theta_{K_{1}},\cos\theta_{K_{1}}\to-\sin\theta_{K_{1}}$
in Eqs.(19-22), respectively. The total amplitudes for each $B\to
K_{1}(1270)K,K_{1}(1400)K$ decay are given in Appendix A.

##  IV Numerical results and discussions

The input parameters in the numerical calculations pdg12 ; ckmfit  are listed
as follows:

$\displaystyle f_{B}$ $\displaystyle=$ $\displaystyle
210MeV,f_{K_{1A}}=250MeV,f^{\perp}_{K_{1B}}=190MeV$ (23)
$\displaystyle\tau_{B^{\pm}}$ $\displaystyle=$ $\displaystyle 1.638\times
10^{-12}s,\tau_{B^{0}}=1.525\times 10^{-12}s,$ (24) $\displaystyle|V_{ud}|$
$\displaystyle=$ $\displaystyle 0.974,|V_{td}|=8.67\times
10^{-3},|V_{ub}|=3.51\times 10^{-3},$ (25) $\displaystyle|V_{ts}|$
$\displaystyle=$ $\displaystyle 0.0404,|V_{us}|=0.22534,,|V_{tb}|=0.999.$ (26)

Using the input parameters and the wave functions as specified in this section
and Sec.II, it is easy to get the branching ratios for the considered decays
which are listed in Table 1, where the first error comes from the uncertainty
in the $B$ meson shape parameter $\omega_{b}=0.40\pm 0.04$ GeV, the second
error is from the hard scale $t$, which we vary from $0.8t$ to $1.2t$, and the
third error is from the combined uncertainties of the Gegenbauer moments
$a^{\perp}_{1}(K_{1A})=-1.08\pm 0.48$ and $a^{\parallel}_{1}(K_{1B})=-1.95\pm
0.45$.

Table 1: Branching ratios (in units of $10^{-6}$) for the decays $B\to K_{1}(1270)\pi,K_{1}(1400)\pi$ and $B\to K_{1}(1270)K,K_{1}(1400)K$ for mixing angle $\theta_{\bar{K}_{1}}=-33^{\circ}$. Other model predictions are also presented here for comparison. It is noticed that the results of cmv  and vnp  are obtained for mixing angle $32^{\circ}$, while those in cy  are obtained for mixing angle $-37^{\circ}$. | cmv  | vnp  | cy  | this work
---|---|---|---|---
$\bar{B}^{0}\to K^{-}_{1}(1270)\pi^{+}$ | $4.3$ | $7.6$ | $3.0^{+0.8+1.5+4.2}_{-0.6-0.9-1.4}$ | $4.6^{+0.3+0.9+1.5}_{-0.1-0.8-1.2}$
$\bar{B}^{0}\to\bar{K}^{0}_{1}(1270)\pi^{0}$ | $2.3$ | $0.4$ | $1.0^{+0.0+0.6+1.7}_{-0.0-0.3-0.6}$ | $1.4^{+0.1+0.7+0.6}_{-0.1-0.5-0.5}$
$B^{-}\to\bar{K}^{0}_{1}(1270)\pi^{-}$ | $4.7$ | $5.8$ | $3.5^{+0.1+1.8+5.1}_{-0.1-1.1-1.9}$ | $3.5^{+0.4+1.9+1.6}_{-0.2-1.1-1.2}$
$B^{-}\to K^{-}_{1}(1270)\pi^{0}$ | $2.5$ | $4.9$ | $2.7^{+0.1+1.1+3.1}_{-0.1-0.7-1.0}$ | $3.9^{+0.9+1.0+1.1}_{-0.5-0.7-1.0}$
$\bar{B}^{0}\to K^{-}_{1}(1400)\pi^{+}$ | $2.3$ | $4.0$ | $5.4^{+1.1+1.7+9.9}_{-1.0-1.3-2.8}$ | $3.0^{+0.5+0.1+0.9}_{-0.3-0.1-0.7}$
$\bar{B}^{0}\to K^{0}_{1}(1400)\pi^{0}$ | $1.7$ | $3.0$ | $2.9^{+0.3+0.7+5.5}_{-0.3-0.6-1.7}$ | $3.3^{+0.9+0.1+1.0}_{-0.7-0.0-0.8}$
$B^{-}\to\bar{K}^{0}_{1}(1400)\pi^{-}$ | $2.5$ | $3.0$ | $6.5^{+1.0+2.0+11.6}_{-0.9-1.6-3.6}$ | $5.0^{+1.3+1.0+1.4}_{-0.7-0.8-1.1}$
$B^{-}\to K^{-}_{1}(1400)\pi^{0}$ | $0.7$ | $1.0$ | $3.0^{+0.4+1.1+5.2}_{-0.4-0.7-1.3}$ | $1.8^{+0.3+0.1+0.4}_{-0.2-0.2-0.3}$
$\bar{B}^{0}\to K^{-}_{1}(1270)K^{+}$ |  |  | $0.01^{+0.01+0.00+0.02}_{-0.00-0.00-0.01}$ | $0.13^{+0.01+0.00+0.23}_{-0.01-0.01-0.08}$
$\bar{B}^{0}\to K^{+}_{1}(1270)K^{-}$ |  |  | $0.06^{+0.01+0.00+0.46}_{-0.01-0.00-0.06}$ | $0.26^{+0.02+0.05+0.19}_{-0.02-0.04-0.12}$
$B^{-}\to K^{0}_{1}(1270)K^{-}$ | $0.22$ |  | $0.25^{+0.01+0.15+0.39}_{-0.01-0.08-0.09}$ | $1.11^{+0.01+0.19+0.43}_{-0.01-0.03-0.35}$
$B^{-}\to K^{-}_{1}(1270)K^{0}$ | $0.02$ |  | $0.05^{+0.02+0.07+0.10}_{-0.02-0.03-0.04}$ | $1.84^{+0.37+0.29+0.65}_{-0.28-0.25-0.42}$
$\bar{B}^{0}\to\bar{K}^{0}_{1}(1270)K^{0}$ | $0.02$ |  | $2.30^{+0.16+1.13+1.43}_{-0.15-0.61-0.61}$ | $1.71^{+0.34+0.27+0.51}_{-0.26-0.23-0.43}$
$\bar{B}^{0}\to K^{0}_{1}(1270)\bar{K}^{0}$ | $0.20$ |  | $0.24^{+0.01+0.11+0.33}_{-0.01-0.07-0.13}$ | $0.26^{+0.03+0.17+0.14}_{-0.06-0.01-0.08}$
$\bar{B}^{0}\to K^{-}_{1}(1400)K^{+}$ |  |  | $0.09^{+0.01+0.00+0.23}_{-0.01-0.00-0.09}$ | $0.64^{+0.14+0.00+0.13}_{-0.06-0.01-0.08}$
$\bar{B}^{0}\to K^{+}_{1}(1400)K^{-}$ |  |  | $0.02^{+0.00+0.00+0.04}_{-0.00-0.00-0.00}$ | $0.31^{+0.02+0.11+0.12}_{-0.00-0.01-0.09}$
$B^{-}\to K^{0}_{1}(1400)K^{-}$ | $0.12$ |  | $0.48^{+0.08+0.15+0.81}_{-0.08-0.12-0.26}$ | $0.90^{+0.13+0.11+1.21}_{-0.08-0.09-0.16}$
$B^{-}\to K^{-}_{1}(1400)K^{0}$ | $4.4$ |  | $0.01^{+0.00+0.01+0.14}_{-0.00-0.00-0.01}$ | $1.33^{+0.14+0.31+0.33}_{-0.10-0.22-0.22}$
$\bar{B}^{0}\to\bar{K}^{0}_{1}(1400)K^{0}$ | $4.1$ |  | $0.08^{+0.01+0.17+0.59}_{-0.01-0.06-0.08}$ | $1.46^{+0.16+0.31+0.33}_{-0.13-0.25-0.28}$
$\bar{B}^{0}\to K^{0}_{1}(1400)\bar{K}^{0}$ | $0.11$ |  | $0.50^{+0.08+0.13+0.92}_{-0.07-0.11-0.32}$ | $0.14^{+0.04+0.04+0.07}_{-0.03-0.03-0.02}$

From Table 1 we can find that the branching ratios of $B\to
K_{1}(1270)\pi,K_{1}(1400)\pi$ decays fall in $10^{-6}$ order. The
experimental data for the branching ratios of the decays $\bar{B}^{0}\to
K_{1}(1270)^{-}\pi^{+},K_{1}(1400)^{-}\pi^{+}$, which are given as $(12.0\pm
3.1^{+9.3}_{-4.5})\times 10^{-6}$ and $(16.7\pm 2.6^{+3.5}_{-5.0})\times
10^{-6}$, respectively, are large and incompatible with all the present theory
predictions. Even for the two sided intervals $Br(\bar{B}^{0}\to
K_{1}(1270)^{-}\pi^{+})\in[0.6,2.5]\times 10^{-5}$ and $Br(\bar{B}^{0}\to
K_{1}(1270)^{-}\pi^{+})\in[0.8,2.4]\times 10^{-5}$, they almost can not
contain the different theoretical results. While the branching ratios of the
charged $B$ decays can be explained by the theories for the large
uncertainties of the intervals
$Br(B^{-}\to\bar{K}_{1}(1270)^{0}\pi^{-})\in[0.0,2.1]\times
10^{-5},Br(B^{-}\to\bar{K}_{1}(1400)^{0}\pi^{-})\in[0.0,2.5]\times 10^{-5}$.
The large large differences between theories and experiments do not happen to
the decays $\bar{B}^{0}\to a_{1}(1260)^{\pm}\pi^{\mp}$, which are tree-
dominated. If the decay constants $f_{a_{1}},f_{\pi}$ and the form factors
$V^{B\to a_{1}}_{0},F^{B\to\pi}_{0}$ can be well determined, it is not
difficult for us to predict the branching ratios of the decays $\bar{B}^{0}\to
a_{1}(1260)^{\pm}\pi^{\mp}$ accurately, because the penguin contributions can
be neglected and there are fewer uncertainties. For the considered decays
$\bar{B}^{0}\to K_{1}^{\pm}\pi^{\mp}$, the tree operators are suppressed by
the CKM matrix elements $V_{ub}V^{*}_{us}/(V_{cb}V^{*}_{cs})\sim 0.02$, and
the penguin operators will play a significant role. If the future data are
really larger than the present predictions for here considered decays, the
authors cy  claimed that there are two possible reasons: one is because the
larger corrections from the weak annihilation and the hard spectator
contributions, the other is from the charming penguin contributions. In our
calculations, the hard spectator contributions which correspond to the non-
factorization emission diagram ones are very small. Although the factorizable
annihilation contributions are more important, they can not promote the
branching ratios too much. So we consider that the charming penguins are more
likely to explain the large data. Unfortunately, the charming penguins are
non-perturbative in nature and remain untouched by many theory approaches.
While it is helpful to consider these decays by using the soft-collinear-
effective-theory (SECT) bauer , where the charming penguin contributions from
loop diagrams are included. Certainly, these contributions can also be
incorporated in the final-state interactions hycheng1 . There exits the
similar situation for the decays $\bar{B}^{0}\to
a_{1}(1260)^{+}K^{-},b_{1}(1235)^{+}K^{-}$ wwang , where the PQCD predictions
are larger than the data. The nonperturbative contributions, such as the final
state interactions or the charming penguins, are suggested to explain the
data. The penguin contributions from the factorization annihilation diagrams
in the $K_{1B}\pi$ modes are much larger than those in the $K_{1A}\pi$ modes.
So we can find that the branching ratios of $B\to K_{1B}\pi$ decays are always
larger than those of $B\to K_{1A}\pi$ decays, which is shown in Table 2.

Table 2: Branching ratios (in units of $10^{-6}$) for the decays $B\to K_{1A}\pi,K_{1B}\pi$ and $B\to K_{1A}K,K_{1B}K$. The errors for these entries correspond to the uncertainties from $\omega_{B}=0.4\pm 0.04GeV$, the hard scale $t$ varying from $0.8t$ to $1.2t$, and the Gegenbauer moments $a_{1}^{\perp}(K_{1A})=-1.08\pm 0.48$ for $K_{1A}$ meson, $a_{1}^{\parallel}(K_{1B})=-1.95\pm 0.45$ for $K_{1B}$ meson, respectively. $\bar{B}^{0}\to K^{-}_{1A}\pi^{+}$ | $2.1^{+1.0+0.1+0.0}_{-0.6-0.1-0.3}$ | $\bar{B}^{0}\to K^{-}_{1B}\pi^{+}$ | $5.6^{+0.1+0.8+2.1}_{-0.2-0.9-1.9}$
---|---|---|---
$\bar{B}^{0}\to\bar{K}^{0}_{1A}\pi^{0}$ | $1.3^{+0.7+0.2+0.9}_{-0.5-0.2-0.6}$ | $\bar{B}^{0}\to\bar{K}^{0}_{1B}\pi^{0}$ | $3.4^{+0.1+1.0+1.1}_{-0.1-0.7-0.9}$
$B^{-}\to\bar{K}^{0}_{1A}\pi^{-}$ | $3.9^{+1.9+0.6+1.7}_{-1.3-0.5-1.5}$ | $B^{-}\to\bar{K}^{0}_{1B}\pi^{-}$ | $4.7^{+0.2+2.2+1.8}_{-0.3-1.5-1.6}$
$B^{-}\to K^{-}_{1A}\pi^{0}$ | $2.1^{+0.9+0.2+0.6}_{-0.7-0.2-0.8}$ | $B^{-}\to K^{-}_{1B}\pi^{0}$ | $3.7^{+0.1+0.7+1.2}_{-0.2-0.8-1.1}$
$\bar{B}^{0}\to K^{-}_{1A}K^{+}$ | $0.47^{+0.03+0.00+0.28}_{-0.04-0.00-0.04}$ | $\bar{B}^{0}\to K^{-}_{1B}K^{+}$ | $0.34^{+0.04+0.01+0.14}_{-0.03-0.01-0.07}$
$\bar{B}^{0}\to K^{+}_{1A}K^{-}$ | $0.14^{+0.01+0.01+0.11}_{-0.00-0.01-0.13}$ | $\bar{B}^{0}\to K^{+}_{1B}K^{-}$ | $0.38^{+0.03+0.03+0.26}_{-0.03-0.02-0.19}$
$B^{-}\to K^{0}_{1A}K^{-}$ | $1.24^{+0.13+0.08+1.74}_{-0.12-0.07-0.65}$ | $B^{-}\to K^{0}_{1B}K^{-}$ | $0.60^{+0.04+0.19+0.13}_{-0.04-0.12-0.08}$
$B^{-}\to K^{-}_{1A}K^{0}$ | $0.29^{+0.02+0.05+1.26}_{-0.01-0.03-0.03}$ | $B^{-}\to K^{-}_{1B}K^{0}$ | $2.65^{+0.53+0.48+0.67}_{-0.34-0.41-0.57}$
$\bar{B}^{0}\to\bar{K}^{0}_{1A}K^{0}$ | $0.10^{+0.00+0.05+0.10}_{-0.00-0.03-0.04}$ | $\bar{B}^{0}\to\bar{K}^{0}_{1B}K^{0}$ | $2.71^{+0.30+0.52+0.66}_{-0.30-0.43-0.58}$
$\bar{B}^{0}\to K^{0}_{1A}\bar{K}^{0}$ | $0.16^{+0.12+0.06+0.18}_{-0.06-0.03-0.10}$ | $\bar{B}^{0}\to K^{0}_{1B}\bar{K}^{0}$ | $0.17^{+0.01+0.08+0.09}_{-0.01-0.05-0.06}$

For the decays $B\to K_{1}(1270)K,K_{1}(1400)K$, there are no experimental
data or upper limits up to now. Although the decays $\bar{B}^{0}\to
K_{1}^{\pm}K^{\mp}$ can occur only via annihilation type diagrams, their
branching ratios might not be so small as those predicted by the QCDF
approach. If our predictions can be confirmed by the future LHCb or the super
B experiments, one can say that the PQCD approach is one of the few methods,
which can be used to quantitatively calculate the annihilation type
contributions. In the previous years both the experimenters and the theorists
considered that the branching ratio of $B^{0}\to K^{+}K^{-}$ was at $10^{-8}$
order, but two years ago the CDF and LHCb collaborations gave their first
measurements of this decay by $(2.3\pm 1.0\pm 1.0)\times 10^{-7}$ cdf  and
$(1.3^{+0.6}_{-0.5}\pm 0.7)\times 10^{-7}$ lhcb , respectively. Later, these
results are confirmed by the PQCD recalculated result $1.56\times 10^{-7}$
xiao  without introducing too much uncertainties. It shows that the PQCD
approach can determine correctly the strength of penguin-annihilation
amplitudes. Whether the PQCD approach can give reasonable predictions for the
pure annihilation decays $\bar{B}^{0}\to
K_{1}(1270)^{\pm}K^{\mp},K_{1}(1400)^{\pm}K^{\mp}$ also deserves our attention
and research. For the decay $\bar{B}^{0}\to K^{0}_{1B}\bar{K}^{0}$ can not
receive a large emission factorization amplitude, because of the small decay
constant $f_{K_{1B}}$ compared with $f_{K_{1A}}$, while it has a large
annihilation factorization amplitude, which makes its branching ratio slightly
larger than that of $\bar{B}^{0}\to K^{0}_{1A}\bar{K}^{0}$. The branching
ratios of these two decays are at the order of $10^{-7}$. But it is very
different to the decay $\bar{B}^{0}\to\bar{K}^{0}_{1B}K^{0}$: Except having a
large annihilation factorization amplitude, it can also obtain a large
emission factorization amplitude at the same time, because here the emission
meson is $K^{0}$ with a larger decay constant $f_{K}=0.16$. So this decay gets
a large branching ratio, which amounts to $2.71\times 10^{-6}$. Even though
the decay $\bar{B}^{0}\to\bar{K}^{0}_{1A}K^{0}$ has a small branching ratio,
the physical final states
$\bar{K}_{1}(1200)^{0}K^{0},\bar{K}_{1}(1400)^{0}K^{0}$, which are mixes of
the former two group flavor states, still might get a large branching ratio.
It has been verified by the different theories, which are shown in Table 1.
But the branching ratio of the decay
$\bar{B}^{0}\to\bar{K}_{1}(1400)^{0}K^{0}$ predicted by the QCDF approach
seems too small compared with the results given by the PQCD and the naive
factorization approaches, which can be clarified by the future experiments.
There exists the similar situation for the decay $B^{-}\to
K_{1}(1400)^{-}K^{0}$. Another decay channel, where exists large divergence
between the predictions, is $B^{-}\to K_{1}(1200)^{-}K^{0}$. The Feynman
diagrams of this decay can be obtained from those of the decay
$\bar{B}^{0}\to\bar{K}_{1}(1200)^{0}K^{0}$ by replacing the spectator quark
$d$ with $u$, so the difference of the branching ratios of these two decays
should not be so large. In a word, the branching ratios of the charged $B$
decays are at or near the order of $10^{-6}$, those of the pure annihilation
decays are at the order of $10^{-7}$ by taking the mixing angle
$\theta_{K_{1}}=33^{\circ}$.

In order to compare with other theoretical predictions, we also list the
branching ratios with the mixing angle $\theta_{\bar{K}_{1}}=-58^{\circ}$
shown in Table 3. One can find that the branching ratios of the decays
$B^{-}\to K_{1}^{-}(1270)K^{0},\bar{B}^{0}\to\bar{K}_{1}^{0}(1270)K^{0}$ have
a remarkable decrease from the mixing angles $-33^{\circ}$ to $-58^{\circ}$,
while those of the decays $B^{-}\to
K_{1}^{-}(1400)K^{0},\bar{B}^{0}\to\bar{K}_{1}^{0}(1400)K^{0}$ have a
remarkable increase.

Table 3: Same as Table1 except for the mixing angle $\theta_{\bar{K}_{1}}=-58^{\circ}$. | cmv  | vnp  | cy  | this work
---|---|---|---|---
$\bar{B}^{0}\to K^{-}_{1}(1270)\pi^{+}$ | $4.3$ | $7.6$ | $2.7^{+0.6+1.3+4.4}_{-0.5-0.8-1.5}$ | $3.2^{+0.7+0.5+0.8}_{-0.5-0.5-0.8}$
$\bar{B}^{0}\to\bar{K}^{0}_{1}(1270)\pi^{0}$ | $2.1$ | $0.4$ | $0.8^{+0.1+0.5+1.7}_{-0.1-0.3-0.6}$ | $0.5^{+0.2+0.0+0.4}_{-0.0-0.2-0.2}$
$B^{-}\to\bar{K}^{0}_{1}(1270)\pi^{-}$ | $4.7$ | $5.8$ | $3.0^{+0.2+0.1+2.7}_{-0.2-0.2-2.2}$ | $3.2^{+1.3+1.2+1.3}_{-0.9-0.8-1.2}$
$B^{-}\to K^{-}_{1}(1270)\pi^{0}$ | $1.6$ | $4.9$ | $2.5^{+0.1+1.0+3.2}_{-0.1-0.7-1.0}$ | $3.3^{+1.1+0.7+0.8}_{-0.8-0.6-1.1}$
$\bar{B}^{0}\to K^{-}_{1}(1400)\pi^{+}$ | $2.3$ | $4.0$ | $2.2^{+1.1+0.7+2.6}_{-0.8-0.6-1.3}$ | $4.5^{+0.0+0.3+1.5}_{-0.0-0.5-1.3}$
$\bar{B}^{0}\to K^{0}_{1}(1400)\pi^{0}$ | $1.6$ | $1.7$ | $1.5^{+0.4+0.3+1.7}_{-0.3-0.3-0.9}$ | $4.1^{+0.8+0.7+1.2}_{-0.4-0.4-0.8}$
$B^{-}\to\bar{K}^{0}_{1}(1400)\pi^{-}$ | $2.5$ | $3.0$ | $2.8^{+1.0+0.9+3.0}_{-0.8-0.9-1.7}$ | $5.4^{+0.3+1.6+1.5}_{-0.2-1.2-1.4}$
$B^{-}\to K^{-}_{1}(1400)\pi^{0}$ | $0.6$ | $1.4$ | $1.0^{+0.4+0.4+1.2}_{-0.3-0.4-0.5}$ | $2.5^{+0.0+0.3+0.8}_{-0.0-0.4-0.7}$
$\bar{B}^{0}\to K^{-}_{1}(1270)K^{+}$ |  |  | $0.01^{+0.00+0.00+0.02}_{-0.00-0.00-0.01}$ | $0.19^{+0.01+0.00+0.37}_{-0.01-0.00-0.09}$
$\bar{B}^{0}\to K^{+}_{1}(1270)K^{-}$ |  |  | $0.04^{+0.01+0.00+0.27}_{-0.01-0.00-0.04}$ | $0.16^{+0.00+0.02+0.12}_{-0.02-0.03-0.06}$
$B^{-}\to K^{0}_{1}(1270)K^{-}$ | $0.22$ |  | $0.22^{+0.01+0.12+0.39}_{-0.01-0.07-0.12}$ | $1.47^{+0.10+0.16+1.59}_{-0.06-0.10-0.58}$
$B^{-}\to K^{-}_{1}(1270)K^{0}$ | $0.75$ |  | $0.05^{+0.02+0.09+0.10}_{-0.01-0.03-0.04}$ | $0.78^{+0.17+0.09+0.97}_{-0.13-0.08-0.19}$
$\bar{B}^{0}\to\bar{K}^{0}_{1}(1270)K^{0}$ | $0.70$ |  | $2.10^{+0.13+1.23+1.31}_{-0.13-0.65-0.57}$ | $0.46^{+0.13+0.07+0.17}_{-0.09-0.05-0.13}$
$\bar{B}^{0}\to K^{0}_{1}(1270)\bar{K}^{0}$ | $0.20$ |  | $0.26^{+0.10+0.12+0.47}_{-0.01-0.08-0.17}$ | $0.23^{+0.09+0.13+0.18}_{-0.06-0.08-0.16}$
$\bar{B}^{0}\to K^{-}_{1}(1400)K^{+}$ |  |  | $0.07^{+0.02+0.00+0.16}_{-0.02-0.00-0.06}$ | $0.58^{+0.06+0.01+0.15}_{-0.06-0.01-0.13}$
$\bar{B}^{0}\to K^{+}_{1}(1400)K^{-}$ |  |  | $0.01^{+0.00+0.00+0.16}_{-0.02-0.00-0.06}$ | $0.42^{+0.03+0.01+0.22}_{-0.02-0.00-0.16}$
$B^{-}\to K^{0}_{1}(1400)K^{-}$ | $0.12$ |  | $0.22^{+0.07+0.07+0.24}_{-0.07-0.07-0.13}$ | $0.54^{+0.04+0.14+0.76}_{-0.02-0.11-0.13}$
$B^{-}\to K^{-}_{1}(1400)K^{0}$ | $3.9$ |  | $0.01^{+0+0.02+0.04}_{-0-0.00-0.00}$ | $2.39^{+0.34+0.50+0.48}_{-0.25-0.39-0.48}$
$\bar{B}^{0}\to\bar{K}^{0}_{1}(1400)K^{0}$ | $3.6$ |  | $0.10^{+0.02+0.21+0.15}_{-0.02-0.08-0.10}$ | $2.24^{+0.36+0.40+0.59}_{-0.28-0.34-0.51}$
$\bar{B}^{0}\to K^{0}_{1}(1400)\bar{K}^{0}$ | $0.11$ |  | $0.25^{+0.07+0.08+0.31}_{-0.07-0.07-0.15}$ | $0.21^{+0.02+0.13+0.09}_{-0.01-0.07-0.07}$

Figure 2: The dependence of the direct CP-violating asymmetries on the mixing
angle $\theta_{\bar{K}_{1}}$: the solid lines represent the decays
$\bar{B}^{0}\to K_{1}(1270)^{0}\pi^{0}$ (left), $\bar{B}^{0}\to
K_{1}(1270)^{-}\pi^{+}$ (right), and the dashed lines are for the decays
$\bar{B}^{0}\to K_{1}(1400)^{0}\pi^{0}$ (left), $\bar{B}^{0}\to
K_{1}(1400)^{-}\pi^{+}$ (right), respectively.

Figure 3: The dependence of the direct CP-violating asymmetries on the mixing
angle $\theta_{\bar{K}_{1}}$: the solid lines represent the decays $B^{-}\to
K_{1}(1270)^{0}\pi^{-}$ (left), $B^{-}\to K_{1}(1270)^{-}\pi^{0}$ (right), and
the dashed lines are for the decays $B^{-}\to K_{1}(1400)^{0}\pi^{-}$ (left),
$B^{-}\to K_{1}(1400)^{-}\pi^{0}$ (right), respectively.

Figure 4: The dependence of the direct CP-violating asymmetries on the mixing angle $\theta_{\bar{K}_{1}}$: the solid lines represent the decays $B^{-}\to K_{1}(1270)^{-}K^{0}$ (left), $\bar{B}^{0}\to K_{1}(1270)^{-}K^{+}$ (right), the dashed lines are for the decays $B^{-}\to K_{1}(1270)^{0}K^{-}$ (left), $\bar{B}^{0}\to K_{1}(1270)^{+}K^{-}$ (right), the dot lines are for the decays $B^{-}\to K_{1}(1400)^{-}K^{0}$ (left), $B^{-}\to K_{1}(1400)^{-}K^{+}$ (right), and the dash-dot lines represent the decays $B^{-}\to K_{1}(1400)^{0}K^{-}$ (left), $\bar{B}^{0}\to K_{1}(1400)^{+}K^{-}$ (right), respectively. Table 4: Direct CP violation (in units of $\%$) for the decays $B\to K_{1A}\pi,K_{1B}\pi$ and $B\to K_{1A}K,K_{1B}K$. The errors for these entries correspond to the uncertainties from $\omega_{B}=0.4\pm 0.04$ GeV, the hard scale $t$ varying from $0.8t$ to $1.2t$, and the Gegenbauer moment $a_{1}^{\perp}(K_{1A})=-1.08\pm 0.48$ for $K_{1A}$ meson, $a_{1}^{\parallel}(K_{1B})=-1.95\pm 0.45$ for $K_{1B}$ meson, respectively.. $\bar{B}^{0}\to K^{-}_{1A}\pi^{+}$ | $9.1^{+2.4+0.8+3.0}_{-2.0-0.8-3.4}$ | $\bar{B}^{0}\to K^{-}_{1B}\pi^{+}$ | $-14.7^{+1.2+0.0+1.1}_{-1.4-0.2-1.6}$
---|---|---|---
$\bar{B}^{0}\to\bar{K}^{0}_{1A}\pi^{0}$ | $-6.6^{+1.3+0.9+2.8}_{-1.4-1.0-8.4}$ | $\bar{B}^{0}\to\bar{K}^{0}_{1B}\pi^{0}$ | $-9.2^{+1.0+3.3+1.6}_{-0.7-3.5-1.9}$
$B^{-}\to\bar{K}^{0}_{1A}\pi^{-}$ | $-2.3^{+0.8+0.8+1.5}_{-1.2-0.6-6.8}$ | $B^{-}\to\bar{K}^{0}_{1B}\pi^{-}$ | $3.3^{+0.1+0.6+1.9}_{-0.1-0.6-1.3}$
$B^{-}\to K^{-}_{1A}\pi^{0}$ | $17.7^{+4.1+3.0+17.1}_{-3.5-3.1-7.4}$ | $B^{-}\to K^{-}_{1B}\pi^{0}$ | $3.4^{+1.2+0.0+0.0}_{-1.4-4.6-6.8}$
$\bar{B}^{0}\to K^{-}_{1A}K^{+}$ | $43.9^{+1.7+0.5+0.0}_{-1.3-3.1-35.6}$ | $\bar{B}^{0}\to K^{-}_{1B}K^{+}$ | $-13.9^{+2.5+1.8+0.4}_{-2.6-2.0-0.4}$
$\bar{B}^{0}\to K^{+}_{1A}K^{-}$ | $46.5^{+0.5+4.4+40.3}_{-1.3-3.3-29.5}$ | $\bar{B}^{0}\to K^{+}_{1B}K^{-}$ | $-3.3^{+1.1+6.8+1.6}_{-0.7-4.1-1.7}$
$B^{-}\to K^{0}_{1A}K^{-}$ | $6.6^{+1.6+3.1+4.9}_{-1.7-3.8-1.8}$ | $B^{-}\to K^{0}_{1B}K^{-}$ | $-80.7^{+1.3+4.4+11.1}_{-1.7-3.5-2.9}$
$B^{-}\to K^{-}_{1A}K^{0}$ | $-29.4^{+7.6+2.6+86.7}_{-6.3-1.8-0.0}$ | $B^{-}\to K^{-}_{1B}K^{0}$ | $0.8^{+2.7+0.4+4.0}_{-3.6-0.5-2.9}$

Now we turn to the evaluations of the CP-violating asymmetries in the PQCD
approach. For the neutral $\bar{B}^{0}$ (the charged $B^{-}$) decays the
direct CP-violating asymmetries can be defined as

$\displaystyle{\cal A}_{CP}^{dir}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\bar{B}^{0}(B^{-})\to
f)-\Gamma(B^{0}(B^{+})\to\bar{f})}{\Gamma(\bar{B}^{0}(B^{-})\to
f)+\Gamma(B^{0}(B^{+})\to\bar{f})}=\frac{2z\sin\theta\sin\delta}{(1+2z\cos\theta\cos\delta+z^{2})}\;,$
(27)

where $\delta$ is the relative strong phase between the tree and penguin
amplitudes, and $\theta$ the CKM weak phase $\theta=\alpha$ for $b\to d$
transition, $\theta=\gamma$ for $b\to s$ transition. Certainly, if the final
states are the same for $B^{0}$ and $\bar{B}^{0}$, that is $f=\bar{f}$, the
CP-asymmetries may be time-dependent, including not only the direct CP
violation but also the mixing-induced CP violation. Using the input parameters
and the wave functions as specified in this section and Sec.II, it is easy to
get the PQCD predictions (in units of $10^{-2}$) for the direct CP-violating
asymmetries of $B$ decaying to each flavor final state, which are listed in
Table 4. For the real physical final states, which are mixes of the
corresponding flavor states, their direct CP-violating asymmetries will be
dependent on the mixing angle $\theta_{\bar{K}_{1}}$. As has been emphasised
before, $\theta_{\bar{K}_{1}}$ for the antiparticle states
$\bar{K}_{1}(1270),\bar{K}_{1}(1400)$ is of opposite sign to that for the
particle states $K_{1}(1270),K_{1}(1400)$. For taking the convention of decay
constant $f_{K_{1B}}$ in this work, so $\theta_{K_{1}}$ is positive and
$\theta_{\bar{K}_{1}}$ is negative. In Fig.2-Fig.4, we give the dependence of
the direct CP-violating asymmetries on the mixing angle $\theta_{\bar{K}_{1}}$
for each decay. Here taking $\theta_{\bar{K}_{1}}=-33^{\circ}$ or
$\theta_{\bar{K}_{1}}=-58^{\circ}$, we can read each direct CP-violating
asymmetry from these figures. It is noticed that for the decays
$\bar{B}^{0}\to K_{1}(1270)^{+}K^{-},K_{1}(1400)^{+}K^{-}$, $B^{-}\to
K_{1}(1270)^{0}K^{-},K_{1}(1400)^{0}K^{-}$, which include the particle states,
their direct CP-violating asymmetry values are still read at $-33^{\circ}$ or
$-58^{\circ}$ for $\theta_{K_{1}}=-\theta_{\bar{K}_{1}}$ and so the
corresponding mixing angle is positive. The signs of the direct CP-violating
asymmetries of $B\to K_{1}(1270)K(\pi)$ and $B\to K_{1}(1400)K(\pi)$ are
opposite at the mixing angle $\theta_{\bar{K}_{1}}=-33^{\circ}$ for most of
these decays except only two groups, whose direct CP-violating asymmetries are
predicted as ${\cal
A}_{CP}^{dir}(\bar{B}^{0}\to\bar{K}_{1}(1270)^{0}\pi^{0})=-12.6\%,{\cal
A}_{CP}^{dir}(\bar{B}^{0}\to\bar{K}_{1}(1400)^{0}\pi^{0})=-6.7\%$ and ${\cal
A}_{CP}^{dir}(\bar{B}^{0}\to K_{1}(1270)^{+}K^{-})=12.2\%,{\cal
A}_{CP}^{dir}(\bar{B}^{0}\to K_{1}(1400)^{+}K^{-})=9.6\%$, respectively. From
Table 4, one can find that the direct CP-violating asymmetries of each decay
$B\to K_{1A}\pi,K_{1B}\pi$ are not large, while those for some real physical
final states become very large. For example, the direct CP-violating
asymmetries of the decays $\bar{B}^{0}\to
K_{1}(1270)^{-}\pi^{+},K_{1}(1400)^{-}\pi^{+}$ are about $-58.1\%$ and
$68.4\%$ at the mixing angle $-33^{\circ}$, respectively. Certainly, we only
learn phenomenally about the mixing angle $\theta_{K_{1}}$ at present and have
no accurate calculations or measurements. Furthermore, the direct CP-violating
asymmetries are sensitive to the mixing angle. It is much more complex for
some considered decays where the nonperturbative contributions, such as
charming penguins, give large corrections, and the corresponding direct CP-
violating asymmetries may also change. So we can’t confirm that these decays
must have so large direct CP-violating asymmetries. As for the decays
$\bar{B}^{0}\to\bar{K}_{1}(1270)^{0}K^{0},\bar{K}_{1}(1400)^{0}K^{0}$, there
is no tree contribution at the leading order, so the direct CP-violating
asymmetry is naturally zero.

##  V Conclusion

In this paper, by using the decay constants and the light-cone distribution
amplitudes derived from the QCD sum-rule method, we research the decays $B\to
K_{1}(1270)\pi(K),K_{1}(1400)\pi(K)$ in the PQCD approach and find that

  * •

All the theoretical predictions for the branching ratios of the decays
$\bar{B}^{0}\to K_{1}(1270)^{+}\pi^{-},K_{1}(1400)^{+}\pi^{-}$ are
incompatible with the present experimental data. There exists the similar
situation for the decays $\bar{B}^{0}\to
a_{1}(1260)^{+}K^{-},b_{1}(1235)^{+}K^{-}$, where the nonperturbative
contributions, such as the final state interactions or the charming penguins,
are needed to explain the data. But the difference is that the nonperturbative
contributions seem to play opposite roles in these two groups of decays. If
the future data are really larger than the present predictions for some
considered decays, it might indicate that the nonperturbative contributions
have pronounced corrections for some decay channels which include the higher
resonances in the final states.

  * •

The pure annihilation type decays $\bar{B}^{0}\to
K_{1}^{\pm}(1270)K^{\mp},K_{1}^{\pm}(1400)K^{\mp}$ are good channels to test
whether an approach can be used to calculate correctly the strength of the
penguin-annihilation amplitudes. Their branching ratios are predicted at
$10^{-7}$ order.

  * •

In the four final neutral flavor states
$K^{0}_{1A}\bar{K}^{0},K^{0}_{1B}\bar{K}^{0},\bar{K}^{0}_{1A}K^{0},\bar{K}^{0}_{1B}K^{0}$,
the decay $\bar{B}^{0}\to\bar{K}^{0}_{1B}K^{0}$ have the largest branching
ratio which is of $10^{-6}$ order, while the other decays with the branching
ratios at $10^{-7}$ order. So the decays
$\bar{B}^{0}\to\bar{K}_{1}(1200)^{0}K^{0},\bar{K}_{1}(1400)K^{0}$ which
include the real physical states can have large branching ratios at the mixing
angle $\theta_{\bar{K}_{1}}=-33^{\circ}$ compare with the decays
$\bar{B}^{0}\to K_{1}(1200)^{0}\bar{K}^{0},K_{1}(1400)\bar{K}^{0}$.

  * •

The signs of the direct CP-violating asymmetries are opposite between almost
of the decays $B\to K_{1}(1270)K(\pi)$ and $B\to K_{1}(1400)K(\pi)$ at mixing
angle $\theta_{K_{1}}=-33^{\circ}$ except only two groups, whose direct CP-
violating asymmetries are predicted as ${\cal
A}_{CP}^{dir}(\bar{B}^{0}\to\bar{K}_{1}(1270)^{0}\pi^{0})=-12.6\%,{\cal
A}_{CP}^{dir}(\bar{B}^{0}\to\bar{K}_{1}(1400)^{0}\pi^{0})=-6.7\%$ and ${\cal
A}_{CP}^{dir}(\bar{B}^{0}\to K_{1}(1270)^{+}K^{-})=12.2\%,{\cal
A}_{CP}^{dir}(\bar{B}^{0}\to K_{1}(1400)^{+}K^{-})=9.6\%$, respectively.

  * •

The strong phase introduced by the nonperturbative contributions might produce
dramatic effects on some of the considered decays, such as $\bar{B}^{0}\to
K_{1}(1270)^{-}\pi^{+},K_{1}(1400)^{-}\pi^{+},K_{1}(1270)^{-}\pi^{0},K_{1}(1270)^{-}\pi^{0}$,
and these effects could exceed those from the parametric uncertainties in the
case of the CP asymmetries.

## Acknowledgment

This work is partly supported by the National Natural Science Foundation of
China under Grant No. 11147004, No.11147008, No.11347030, by the Program of
the Youthful Key Teachers in Universities of Henan Province under Grants No.
001166, and by the Program for Science and Technology Innovation Talents in
Universities of Henan Province 14HASTIT037. The authors would like to thank
Prof. Hai-Yang Cheng and Prof. Cai-Dian Lu for helpful discussions.

##  Appendix A Analytic formulas for the decay amplitudes

$\displaystyle A(K_{1}(1270)^{0}\bar{K}^{0})$ $\displaystyle=$
$\displaystyle-\xi_{t}(f_{K_{1A}}\sin\theta_{K_{1}}+f_{K_{1B}}\cos\theta_{K_{1}})F^{LL}_{eK}(a_{4}-\frac{1}{2}a_{10})$
(28)
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{eK}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{eK}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{eK}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{eK}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(C_{4}-\frac{1}{2}C_{10})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{aK}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle-\xi_{t}(M^{SP;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{SP;K_{1B}}_{aK}\cos\theta_{K_{1}})(C_{6}-\frac{1}{2}C_{8})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(a_{3}-\frac{1}{2}a_{9})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(a_{4}-\frac{1}{2}a_{10})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(a_{5}-\frac{1}{2}a_{7})$
$\displaystyle-\xi_{t}f_{B}(F^{SP;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{SP;K_{1B}}_{aK}\cos\theta_{K_{1}})(a_{6}-\frac{1}{2}a_{8})$
$\displaystyle-\xi_{t}(M^{LL;K}_{aK_{1A}}\sin\theta_{K_{1}}+M^{LL;K}_{aK_{1B}}\cos\theta_{K_{1}})(C_{4}-\frac{1}{2}C_{10})$
$\displaystyle-\xi_{t}(M^{SP;K}_{aK_{1A}}\sin\theta_{K_{1}}+M^{SP;K}_{aK_{1B}}\cos\theta_{K_{1}})(C_{6}-\frac{1}{2}C_{8})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K}_{aK_{1A}}\sin\theta_{K_{1}}+F^{LL;K}_{aK_{1B}}\cos\theta_{K_{1}})(a_{3}-\frac{1}{2}a_{9})$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K}_{aK_{1A}}\sin\theta_{K_{1}}+F^{LL;K}_{aK_{1B}}\cos\theta_{K_{1}})(a_{5}-\frac{1}{2}a_{7}),$
$\displaystyle A(K_{1}(1270)^{0}K^{-})$ $\displaystyle=$
$\displaystyle-\xi_{t}(f_{K_{1A}}\sin\theta_{K_{1}}+f_{K_{1B}}\cos\theta_{K_{1}})F^{LL}_{eK}(a_{4}-\frac{1}{2}a_{10})$
(29)
$\displaystyle-\xi_{t}(M^{LL;K_{1A}}_{eK}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{eK}\cos\theta_{K_{1}})(C_{3}-\frac{1}{2}C_{9})$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{eK}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{eK}\cos\theta_{K_{1}})(C_{5}-\frac{1}{2}C_{7})$
$\displaystyle+(M^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(\xi_{u}C_{1}-\xi_{t}(C_{3}+C_{9}))$
$\displaystyle-\xi_{t}(M^{LR;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{LR;K_{1B}}_{aK}\cos\theta_{K_{1}})(C_{5}+C_{7})$
$\displaystyle+f_{B}(F^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(\xi_{u}a_{2}-\xi_{t}(a_{4}+a_{10})$
$\displaystyle-\xi_{t}f_{B}(F^{SP;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{SP;K_{1B}}_{aK}\cos\theta_{K_{1}})(a_{6}+a_{8}).$

In the upper two formulae, if changing the first term as
$-\xi_{t}f_{K}(F^{LL}_{eK_{1A}}\sin\theta_{K_{1}}+F^{LL}_{eK_{1B}}\cos\theta_{K_{1}})(a_{4}-\frac{1}{2}a_{10}))-\xi_{t}f_{K}(F^{SP}_{eK_{1A}}\sin\theta_{K_{1}}+F^{SP}_{eK_{1B}}\cos\theta_{K_{1}})(a_{6}-\frac{1}{2}a_{8})$,
and at the same time exchanging the positions of $K_{1A}(K_{1B})$ and $K$ in
other terms, we will get the decay amplitudes of
$\bar{B}^{0}\to\bar{K}_{1}(1270)^{0}K^{0}$ and $B^{-}\to
K_{1}(1270)^{-}K^{0}$, respectively.

$\displaystyle A(K_{1}(1270)^{+}K^{-})$ $\displaystyle=$
$\displaystyle(M^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(\xi_{u}C_{2}-\xi_{t}(C_{4}+C_{10}))$
(30)
$\displaystyle-\xi_{t}(M^{SP;K_{1A}}_{aK}\sin\theta_{K_{1}}+M^{SP;K_{1B}}_{aK}\cos\theta_{K_{1}})(C_{6}+C_{8})$
$\displaystyle+f_{B}(F^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(\xi_{u}a_{1}-\xi_{t}(a_{3}+a_{5}+a_{7}+a_{9}))$
$\displaystyle-\xi_{t}f_{B}(F^{LL;K_{1A}}_{aK}\sin\theta_{K_{1}}+F^{LL;K_{1B}}_{aK}\cos\theta_{K_{1}})(a_{3}+a_{5}-\frac{1}{2}a_{7}-\frac{1}{2}a_{9})$
$\displaystyle-\xi_{t}(M^{LL;K}_{aK_{1A}}\sin\theta_{K_{1}}+M^{LL;K}_{aK_{1B}}\cos\theta_{K_{1}})(C_{4}-\frac{1}{2}C_{10})$
$\displaystyle-\xi_{t}(M^{SP;K}_{aK_{1A}}\sin\theta_{K_{1}}+M^{SP;K}_{aK_{1B}}\cos\theta_{K_{1}})(C_{6}-\frac{1}{2}C_{8}).$

In Eq.(30), if exchanging the positions of $K_{1A}(K_{1B})$ and $K$, we will
get the total amplitude of the decay $\bar{B}^{0}\to K_{1}(1270)^{-}K^{+}$.
The total amplitudes of the decays $B\to K_{1}(1400)K$ can be obtained by
making the replacements with
$\sin\theta_{K_{1}}\to\cos\theta_{K_{1}},\cos\theta_{K_{1}}\to-\sin\theta_{K_{1}}$
in Eqs.(28-30), respectively.

## References

  * (1) W. Wang, R.H. Li, C.D. Lu, Phys. Rev. D 78, 074009 (2008).
  * (2) Z.Q. Zhang, Phys. Rev. D 85, 114005 (2012).
  * (3) Z.Q. Zhang, H.X. Guo, G.S. Kang, Y.W. Luo, Eur. Phys. J. C 73, 2270 (2013).
  * (4) G. Calderon, J.H. Munoz, and C.E. Vera, Phys.Rev.D 76, 094019 (2007).
  * (5) V. Laporta, G. Nardulli, and T.N. Pham, Phys.Rev.D 74, 054035 (2006).
  * (6) H.Y. Cheng and K.C. Yang, Phys. Rev. D 76, 114020, (2007).
  * (7) ARGUS Collaboration, H. Albrecht et al., Phys. Lett. B 254, 288 (1991).
  * (8) Heavy Flavor Averagering Group, online update at http://www.slac.stanford.edu/xorg/hfag.
  * (9) F. Blanc, invited talk presented at Moriond QCD, La Thuile, Italy, March 17-24, 2007.
  * (10) BABAR Collabration, B. Aubert et al., Phys. Rev. D 81, 052009 (2010).
  * (11) C.D. Lu and M.Z. Yang, Eur. Phys. J. C 28, 515 (2003).
  * (12) H.Y. Cheng and K. C. Yang, Phys. Rev. D 78, 094001 (2008).
  * (13) C. D. Lu, K. Ukai, and M. Z. Yang, Phys. Rev. D 63, 074009 (2001).
  * (14) Y. Y. Keum, H. n. Li, and A. I. Sanda, Phys. Lett. B 504, 6 (2001); Phys. Rev. D 63, 054008 (2001).
  * (15) S. Mishima, Phys. Lett. B 521, 252 (2001); C. H. Chen, Y. Y. Keum, and H. n. Li, Phys. Rev. D 64, 112002 (2001).
  * (16) H. n. Li and B. Tseng, Phys. Rev. D 57, 443 (1998).
  * (17) H. n. Li, Phys. Rev. D 66, 094010 (2002).
  * (18) H. n. Li and K. Ukai, Phys. Lett. B 555, 197 (2003).
  * (19) H. n. Li and H. L. Yu, Phys. Rev. D 53, 2480 (1996).
  * (20) R. K. Carnegic et al., Phys. Lett. B 68, 289 (1977).
  * (21) N.Isgur, M. B. Wise, Phys. Lett. B 232, 113 (1989).
  * (22) H. Y. Cheng. Phys. Rev. D 67, 094007 (2003).
  * (23) M. Suzuki, Phys. Rev. D 47, 1252 (1997).
  * (24) L. Burkovsky, T. Goldman, Phys. Rev. D 56, 1368 (1997); Phys. Rev. D 57, 2879 (1998).
  * (25) P.V. Chliapnikov, Phys. Lett. B 496, 129 (2000).
  * (26) S. Godfrey, N. Isgur, Phys. Rev. D 32, 189 (1985).
  * (27) J. Vijande, F. Fernandez, A. Valcarce, J. Phys.G 31, 481 (2005).
  * (28) A. Tayduganov, E. Kou, and A. Le Yaouanc, Phys. Rev. D 85, 074011 (2012).
  * (29) F. Divotgey, L. Olbrich and F. Giacosa, Eur. Phys. J. A 49, 135 (2103).
  * (30) H.Y. Cheng, Phys. Lett. B 707, 116 (2012).
  * (31) A. Ali, G. Kramer, Y. Li, C.D. Lu, Y.L. Shen, W. Wang, Y.M. Wang, Phys. Rev. D 76, 074018 (2007).
  * (32) J. Beringer, et al., [Particle Data Group], Phys. Rev. D 86, 010001 (2012).
  * (33) CKMfitter Group, http://ckmfitter.in2p3.fr.
  * (34) C.W. Bauer, D. Pirjol, I.Z. Rothstein, and I.W. Stewart, Phys. Rev. D 70, 054015 (2004).
  * (35) H.Y. Cheng, C.K. Chua, and A. Soni, Phys. Rev. D 71, 014030 (2005).
  * (36) F. Ruffini, CDF Collaboration, talk given at the Flavor Physics and CP violation 2011, May 23-27, Israel, arXiv:1107.5760[hep-ex]; M.J. Morello et al., (CDF Collaboratioin), CDF pulic note 10498 (2011); T. Aaltonen et al., (CDF Collaboratioin), arXiv:1111.0485v2[hep-ex].
  * (37) A. Powell, LHCb Collaboration, talk given at PANIC 2011, MIT, July 2011; V. Vagnoni, LHCb Collaboration, LHCb-CONF-2011-024, Sept. 20, 2011.
  * (38) Z.J. Xiao, W.F. Wang, Y.Y. Fan, Phys. Rev. D 85, 094003 (2012).