# A note on hamiltonian cycles in $4$-tough $(P_{2}\cup kP_{1})$-free graphs

Lingjuan Shia111 The first author is supported by NSFC (grant no. 11871256 and
11901458) and by the Fundamental Research Funds for the Central Universities
(grant no. D5000200199)., Songling Shanb

(aSchool of Software, Northwestern Polytechnical University, Xi’an, Shaanxi
710072, P. R. China
b Department of Mathematics, Illinois State University, Normal, IL 61790, USA
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for
every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of
$G-S$ is at least $t$. The Toughness Conjecture of Chvátal, stating that there
exists a constant $t_{0}$ such that every $t_{0}$-tough graph with at least
three vertices is hamiltonian, is still open in general. For any given integer
$k\geq 1$, a graph $G$ is $(P_{2}\cup kP_{1})$ free if $G$ does not contain
the disjoint union of $P_{2}$ and $k$ isolated vertices as an induced
subgraph. In this note, we show that every 4-tough and $2k$-connected
$(P_{2}\cup kP_{1})$-free graph with at least three vertices is hamiltonian.
This result in some sense is an “extension” of the classical Chvátal-Erdős
Theorem that every $\max\\{2,k\\}$-connected $(k+1)P_{1}$-free graph on at
least three vertices is hamiltonian.

Keywords: toughness; hamiltonian cycle; $(P_{2}\cup kP_{1})$-free graph

##  1 Introduction

We consider only simple graphs. Let $G$ be a graph. We denote by $V(G)$ and
$E(G)$ the vertex set and edge set of $G$, respectively. For a vertex $x\in
V(G)$ and a subgraph $H$ of $G$, $N_{H}(x)$ denotes the set of neighbors of
$x$ that are contained in $V(H)$, and $d_{H}(x):=|N_{H}(x)|$. Let $S\subseteq
V(G)$. Then $N_{G}(S)=\bigcup_{x\in S}N_{G}(x)\setminus S$ and
$N_{H}(S)=N_{G}(S)\cap V(H)$. Denote by $G[S]$ the subgraph of $G$ induced by
$S$, and let $G-S=G[V(G)\setminus S]$. We skip the subscript $G$ if no
confusion may arise.

For a given graph $H$, we say that $G$ is $H$-free if $G$ does not contain $H$
as an induced subgraph. For another graph $F$, $H\cup F$ is the disjoint union
of $H$ and $F$. For positive integers $a$ and $b$, we denote by $aP_{b}$ the
graph consisting of $a$ disjoint copies of the path $P_{b}$. For two integers
$p$ and $q$, let $[p,q]=\\{i\in\mathbb{Z}:p\leq i\leq q\\}$.

Throughout this paper, if not specified, we will assume $t$ to be a
nonnegative real number. The number of components of $G$ is denoted by $c(G)$.
The graph $G$ is said to be _$t$ -tough_ if $|S|\geq t\cdot c(G-S)$ for each
$S\subseteq V(G)$ with $c(G-S)\geq 2$. The _toughness $\tau(G)$_ is the
largest real number $t$ for which $G$ is $t$-tough, or is $\infty$ if $G$ is
complete. This concept, a measure of graph connectivity and “resilience” under
removal of vertices, was introduced by Chvátal [5] in 1973. It is easy to see
that if $G$ has a hamiltonian cycle then $G$ is 1-tough. Conversely, Chvátal
[5] conjectured that there exists a constant $t_{0}$ such that every
$t_{0}$-tough graph is hamiltonian (Chvátal’s toughness conjecture). Bauer,
Broersma and Veldman [1] have constructed $t$-tough graphs that are not
hamiltonian for all $t<\frac{9}{4}$, so $t_{0}$ must be at least
$\frac{9}{4}$. It is not difficult to see that a non-complete $t$-tough graph
is $2\lceil t\rceil$-connected.

There are many papers on Chvátal’s toughness conjecture, and it has been
verified when restricted to a number of graph classes [2], including planar
graphs, claw-free graphs, co-comparability graphs, and chordal graphs. The
conjecture was also confirmed for graphs forbidden small linear forests. The
results include the following: 1-tough for $R$-free graphs, where
$R\in\\{P_{4},P_{3}\cup P_{1},P_{2}\cup 2P_{1}\\}$ [10]; 25-tough, 3-tough,
and now 2-tough for $2P_{2}$-free graphs [4, 12, 11]; 3-tough for $(P_{2}\cup
3P_{1})$-free graphs [9]; 7-tough for $(P_{3}\cup 2P_{1})$-free graphs [8];
and 15-tough for $(P_{2}\cup P_{3})$-free graphs [13]. In this paper, for any
integer $k\geq 1$, we investigate Chvátal’s toughness conjecture for
$(P_{2}\cup kP_{1})$-free graphs. For $k=1$, it is an easy exercise to show
that every 1-tough $(P_{2}\cup P_{1})$-free graph on $n\geq 3$ vertices has
minimum degree at least $n/2$ and so is hamiltonian by Dirac’s Theorem [7]. By
[10], 1-tough is enough to guarantee a hamiltonian cycle for $(P_{2}\cup
2P_{1})$-free graphs on at least three vertices. By [9], 3-tough is enough to
guarantee a hamiltonian cycle for $(P_{2}\cup 3P_{1})$-free graphs on at least
three vertices. Thus we consider only the case when $k\geq 4$, and obtain the
result below.

######  Theorem 1.

Let $k\geq 4$ be an integer and $G$ be a $4$-tough and $2k$-connected
$(P_{2}\cup kP_{1})$-free graph. Then $G$ is hamiltonian.

As a non-complete $t$-tough graph is $2\lceil t\rceil$-connected, as a
corollary of Theorem 1, we get the following result.

######  Corollary 2.

For any integer $k\geq 4$, every $k$-tough $(P_{2}\cup kP_{1})$-free graph on
at least three vertices is hamiltonian.

Theorem 1 can also be seen as a result along the Chavátal-Erdős type theorems
[6]. The Chavátal-Erdős Theorem can be rephrased as below: for any integer
$k\geq 1$, every $\max\\{2,k\\}$-connected $(k+1)P_{1}$-free graph on at least
three vertices is hamiltonian. It is necessary for a hamiltonian graph to be
1-tough and the condition that “$k$-connected and $(k+1)P_{1}$-free” in the
Chavátal-Erdős Theorem implies that the graph is 1-tough. However, any
constant connectivity condition cannot guarantee the existence of a
hamiltonian cycle in $(P_{2}\cup kP_{1})$-free graphs. For example, for
integer $n\geq 2$, the complete bipartite graph $K_{n-1,n}$ is
$(n-1)$-connected, $(P_{2}\cup kP_{1})$-free for any $k\geq 1$, but is not
hamiltonian. This explains the involvement of the toughness condition in
Theorem 1 other than the connectivity condition. However, we do think that the
condition 4-tough is not sharp and we propose the following conjecture.

######  Conjecture 3.

Let $k\geq 4$ be an integer and $G$ be a $1$-tough and $2k$-connected
$(P_{2}\cup kP_{1})$-free graph. Then $G$ is hamiltonian.

##  2 Proof of Theorem 1

We start with certain notation and preliminary results. Let $C$ be an oriented
cycle, and we assume that the orientation is clockwise throughout the rest of
this paper. For $x\in V(C)$, denote the immediate successor of $x$ on $C$ by
$x^{+}$ and the immediate predecessor of $x$ on $C$ by $x^{-}$. For $u,v\in
V(C)$, $u\overset{\rightharpoonup}{C}v$ denotes the segment of $C$ starting at
$u$, following $C$ in the orientation, and ending at $v$. Likewise,
$u\overset{\leftharpoonup}{C}v$ is the opposite segment of $C$ with endpoints
as $u$ and $v$.

A path $P$ connecting two vertices $u$ and $v$ is called a $(u,v)$-path, and
we write $uPv$ or $vPu$ in order to specify the two endvertices of $P$. Let
$uPv$ and $xQy$ be two paths. If $vx$ is an edge, we write $uPvxQy$ as the
concatenation of $P$ and $Q$ through the edge $vx$.

######  Lemma 4 ([3]).

Let $t$ be a positive real number and $G$ be a $t$-tough graph on at lest
three vertices with $\delta(G)>\frac{n}{t+1}-1$. Then $G$ is hamiltonian.

###### Proof of Theorem 1.

Suppose for a contradiction that $G$ is not hamiltonian. Then $G$ is not
complete and $\delta(G)\leq\frac{n}{5}-1$ by Lemma 4. Since $G$ is
$2k$-connected, $\delta(G)\geq 2k$. Thus we have $n\geq 5(2k+1)\geq 45$. Let
$C$ be a longest cycle of $G$. As $G$ is not hamiltonian, $V(G)\setminus
V(C)\neq\emptyset$. Let $H$ be a component of $G-V(C)$ and suppose
$|N_{C}(V(H))|=t$. We let $x_{1},\ldots,x_{t}$ be all the neighbors of
vertices of $H$ on $C$, and we assume that they appear in the order
$x_{1},\ldots,x_{t}$ along $\overset{\rightharpoonup}{C}$. Note that $t\geq
2k$ as $G$ is $2k$-connected.

######  Claim 1.

Any two vertices in $\\{x_{1},\ldots,x_{t}\\}$ are not consecutive on $C$, and
$\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$ is an independent set in $G$.

###### Proof.

The first part is clear as otherwise we can extend $C$. For the second part,
suppose to the contrary that $x_{i}^{+}x_{j}^{+}\in E(G)$ for some
$i,j\in[1,t]$. Without loss of generality, assume that $i<j$. Let
$x_{i}^{\prime},x_{j}^{\prime}\in V(H)$ such that
$x_{i}x_{i}^{\prime},x_{j}x_{j}^{\prime}\in E(G)$, and let $P$ be an
$(x_{i}^{\prime},x_{j}^{\prime})$-path of $H$. Then
$x_{i}x_{i}^{\prime}Px_{j}^{\prime}x_{j}\overset{\leftharpoonup}{C}x_{i}^{+}x_{j}^{+}\overset{\rightharpoonup}{C}x_{i}$
is a cycle longer than $C$, a contradiction. ∎

######  Claim 2.

Each component of $G-V(C)$ is trivial, i.e., a one-vertex graph.

###### Proof.

We still let $H$ represent any component of $G-V(C)$. Suppose to the contrary
that $H$ has an edge $uv$. Then $uv$ and
$\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$ together form an induced
$P_{2}\cup tP_{1}$, which contains $P_{2}\cup kP_{1}$ as an induced subgraph,
a contradiction. ∎

######  Claim 3.

$|V(C)|\geq\frac{4n}{5}$.

###### Proof.

Suppose to the contrary that $|V(C)|<\frac{4n}{5}$. Then $G-V(C)$ has exactly
$n-|V(C)|>n/5\geq 9$ components by Claim 2. Thus $V(C)$ is a cutset of $G$ and
$\frac{|V(C)|}{c(G-V(C))}<4$. This contradicts that $G$ is $4$-tough. ∎

Our goal in the rest of the proof is to find an independent set of size more
than $n/5$ in $G$ and so to reach a contradiction to the toughness of $G$. By
Claim 2, let $V(H)=\\{x\\}$. Assume, without loss of generality, that
$|V(x_{1}\overset{\rightharpoonup}{C}x_{k+1})|\leq|V(x_{k+1}\overset{\rightharpoonup}{C}x_{1})|$.
Then $|V(x_{k+1}\overset{\rightharpoonup}{C}x_{1})|\geq\frac{2n}{5}$ by Claim
3. Let
$x_{k+1}\overset{\rightharpoonup}{C}x_{1}=x_{k+1}x_{k+1}^{+}y_{1}y_{2}\ldots
y_{h}x_{1}$, and define

$\displaystyle Y=$ $\displaystyle\\{y_{2},y_{4},\ldots,y_{h}\\}$ if $h$ is
even;  $\displaystyle Y=$ $\displaystyle\\{y_{2},y_{4},\ldots,y_{h-1}\\}$ if
$h$ is odd.

######  Claim 4.

$N(Y)\cap\\{x_{1}^{+},\ldots,x_{k+1}^{+}\\}=\emptyset$. As a consequence, $Y$
is an independent set in $G$.

###### Proof.

Let $X=\\{x_{1}^{+},x_{2}^{+},\ldots,x_{k+1}^{+}\\}$. Since $X$ is an
independent set in $G$ by Claim 1, the consequence part of the claim follows
easily from the first part and the $(P_{2}\cup kP_{1})$-freeness of $G$. So we
only show that $N(Y)\cap\\{x_{1}^{+},\ldots,x_{k+1}^{+}\\}=\emptyset$.
Considering the edge $x_{k+1}^{+}y_{1}$ and the independent set $X$, we
conclude that $|N(y_{1})\cap X|\geq 2$ by the $(P_{2}\cup kP_{1})$-freeness of
$G$. Next, we show $N(y_{2})\cap X=\emptyset$. Suppose not, then
$|N(y_{2})\cap X|\geq 2$. Otherwise let $N(y_{2})\cap X=\\{x_{i}\\}$ for some
$i\in[1,k+1]$. Then $y_{2}x_{i}$ and $X\setminus\\{x_{i}\\}$ form an induced
copy of $(P_{2}\cup kP_{1})$. We consider three cases regarding the size of
the set $N(y_{1})\cap N(y_{2})\cap X$ below.

Case 1: $|N(y_{1})\cap N(y_{2})\cap X|\geq 2$.

Let $x_{i}^{+},x_{j}^{+}\in N(y_{1})\cap N(y_{2})\cap X$ for some distinct
$i,j\in[1,k+1]$. Assume, without loss of generality, that $i<j$. Then
$xx_{i}\overset{\leftharpoonup}{C}y_{2}x_{j}^{+}\overset{\rightharpoonup}{C}y_{1}x_{i}^{+}\overset{\rightharpoonup}{C}x_{j}x$
is a cycle longer than $C$, a contradiction.

Case 2: $|N(y_{1})\cap N(y_{2})\cap X|=1$.

Let $N(y_{1})\cap N(y_{2})\cap X=\\{x_{i}^{+}\\}$ for some $i\in[1,k+1]$. If
$i=1$, then $y_{2}$ has another neighbor $x_{j}^{+}$ with $j\in[2,k+1]$ since
$|N(y_{2})\cap X|\geq 2$. So
$xx_{1}\overset{\leftharpoonup}{C}y_{2}x_{j}^{+}\overset{\rightharpoonup}{C}y_{1}x_{1}^{+}\overset{\rightharpoonup}{C}x_{j}x$
is a cycle longer than $C$, a contradiction. If $i=k+1$, then $y_{1}$ has
another neighbor $x_{j}^{+}$ with $j\in[1,k]$ since $|N(y_{1})\cap X|\geq 2$.
So
$xx_{j}\overset{\leftharpoonup}{C}y_{2}x_{k+1}^{+}\overset{\rightharpoonup}{C}y_{1}x_{j}^{+}\overset{\rightharpoonup}{C}x_{k+1}x$
is a cycle longer than $C$, a contradiction. If $i\in[2,k]$, then we define
two indices as follows:

$\displaystyle s$ $\displaystyle=$ $\displaystyle\min\\{j:x_{j}^{+}\in
N(y_{1})\cap X\\},$ $\displaystyle\ell$ $\displaystyle=$
$\displaystyle\max\\{j:x_{j}^{+}\in N(y_{2})\cap X\\}.$

Clearly, $s\leq i$ and $\ell\geq i$. If $s<i$ then
$xx_{s}\overset{\leftharpoonup}{C}y_{2}x_{i}^{+}\overset{\rightharpoonup}{C}y_{1}x_{s}^{+}\overset{\rightharpoonup}{C}x_{i}x$
is a cycle longer than $C$; if $\ell>i$, then
$xx_{i}\overset{\leftharpoonup}{C}y_{2}x_{\ell}^{+}\overset{\rightharpoonup}{C}y_{1}x_{i}^{+}\overset{\rightharpoonup}{C}x_{\ell}x$
is a cycle longer than $C$. Thus we assume $s=\ell=i$.

If $y_{1}$ is a neighbor of $x$, then
$xy_{1}\overset{\leftharpoonup}{C}x_{i}^{+}y_{2}\overset{\rightharpoonup}{C}x_{i}x$
is a cycle longer than $C$, a contradiction. So $y_{1}\notin N(x)$. This
implies that $y_{2}\notin\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$. We claim
that $N(y_{1})\cap\\{x_{k+2}^{+},x_{k+3}^{+},\ldots,x_{t}^{+}\\}=\emptyset$.
Otherwise, suppose $y_{1}x_{h}^{+}\in E(G)$ for some $h\in[k+2,t]$. Then
$xx_{i}\overset{\leftharpoonup}{C}x_{h}^{+}y_{1}\overset{\leftharpoonup}{C}x_{i}^{+}y_{2}\overset{\rightharpoonup}{C}x_{h}x$
is a cycle longer than $C$. Hence $x_{i}^{+}\overset{\rightharpoonup}{C}y_{1}$
or $y_{2}\overset{\rightharpoonup}{C}x_{i}^{+}$ has at least $k+1$ vertices
belonging to $\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$ as $t\geq 2k$.

If $x_{i}^{+}\overset{\rightharpoonup}{C}y_{1}$ has at least $k+1$ vertices in
$\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$, then $y_{2}x_{i}^{+}$ and the
$k$ vertices in
$(V(x_{i}^{+}\overset{\rightharpoonup}{C}y_{1})\cap\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\})\setminus\\{x_{i}^{+}\\}$
induce a $P_{2}\cup kP_{1}$ subgraph in $G$, a contradiction. If
$y_{2}\overset{\rightharpoonup}{C}x_{i}^{+}$ has at least $k+1$ vertices in
$\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$, then $x_{i}^{+}y_{1}$ and the
$k$ vertices in
$(V(y_{2}\overset{\rightharpoonup}{C}x_{i}^{+})\cap\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\})\setminus\\{x_{i}^{+}\\}$
induce a $P_{2}\cup kP_{1}$ subgraph in $G$, a contradiction.

Case 3: $|N(y_{1})\cap N(y_{2})\cap X|=0$.

We define $s$ and $\ell$ the same way as in Case 2. Since $|N(y_{1})\cap
N(y_{2})\cap X|=0$, $s\neq\ell$. If $s<\ell$, then
$xx_{s}\overset{\leftharpoonup}{C}y_{2}x_{\ell}^{+}\overset{\rightharpoonup}{C}y_{1}x_{s}^{+}\overset{\rightharpoonup}{C}x_{\ell}x$
is a cycle longer than $C$, a contradiction. So $s>\ell$. If $y_{1}\in N(x)$,
then
$xx_{\ell}\overset{\leftharpoonup}{C}y_{2}x_{\ell}^{+}\overset{\rightharpoonup}{C}y_{1}x$
is a cycle longer than $C$, a contradiction. So $y_{1}$ is not a neighbor of
$x$ and thus $y_{2}\notin\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$. By the
same argument as in Case 2, we have
$N(y_{1})\cap\\{x_{k+2}^{+},x_{k+3}^{+},\ldots,x_{t}^{+}\\}=\emptyset$. So
$y_{1}$ dose not have any neighbor in
$\\{x_{1}^{+},\ldots,x_{s-1}^{+}\\}\cup\\{x_{k+2}^{+},\ldots,x_{t}^{+}\\}$,
which is a subset of $V(y_{2}\overset{\rightharpoonup}{C}x_{s})$. By the
definition of $\ell$ and the assumption that $\ell<s$, we know that $y_{2}$
does not have any neighbor in $\\{x_{s}^{+},\ldots,x_{k+1}^{+}\\}$. Since
$t\geq 2k$, $y_{2}\overset{\rightharpoonup}{C}x_{s}$ or
$x_{s}^{+}\overset{\rightharpoonup}{C}x_{k+1}^{+}$ has at least $k$ vertices
belonging to $\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$. If
$y_{2}\overset{\rightharpoonup}{C}x_{s}$ has $k$ vertices belonging to
$\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$, then those $k$ vertices and the
edge $y_{1}x_{s}^{+}$ induce a copy of $P_{2}\cup kP_{1}$. If
$x_{s}^{+}\overset{\rightharpoonup}{C}x_{k+1}^{+}$ has $k$ vertices belonging
to $\\{x_{1}^{+},x_{2}^{+},\ldots,x_{t}^{+}\\}$, then those $k$ vertices and
the edge $y_{2}x_{\ell}^{+}$ induce a copy of $P_{2}\cup kP_{1}$.

Cases 1 to 3 imply $N(y_{2})\cap X=\emptyset$. Since $X$ is an independent set
in $G$, $|X|=k+1$ and $y_{2}y_{3}\in E(G)$, the $(P_{2}\cup kP_{1})$-freeness
of $G$ implies that $|N(y_{3})\cap X|\geq 2$. With $y_{3}$ playing the role of
$y_{1}$ and $y_{4}$ playing the role of $y_{2}$, the same arguments in Cases 1
to 3 show that $N(y_{4})\cap X=\emptyset$. Repeating the same arguments for
all other elements of $Y$, we get $N(Y)\cap X=\emptyset$. ∎

By Claim 1 and Claim 4, we know that $X\cup Y$ in an independent set in $G$.
By Claim 3 and the assumption that
$|V(x_{1}\overset{\rightharpoonup}{C}x_{k+1})|\leq|V(x_{k+1}\overset{\rightharpoonup}{C}x_{1})|$,
we have
$|Y|\geq\lfloor\frac{1}{2}(\frac{|C|-2}{2})\rfloor\geq\lfloor\frac{n}{5}-\frac{1}{2}\rfloor$.
Since $k\geq 4$ and so $|X|\geq 5$, we have $|X\cup Y|>\frac{n}{5}\geq 9$. Let
$S=V(G)\setminus(X\cup Y)$. Then $S$ is a cutset of $G$ with $c(G-S)=|X\cup
Y|>\frac{n}{5}$. However, $\frac{|S|}{c(G-S)}<4$, contradicting $G$ being
4-tough. The proof is now complete. ∎

## References

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#  Will the subprocesses $\rho(770,1450)^{0}\to K^{+}K^{-}$ contribute large
branching
fractions for $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays?

Wen-Fei Wang<EMAIL_ADDRESS>Institute of Theoretical Physics, Shanxi
University, Taiyuan, Shanxi 030006, China

###### Abstract

We analyze the quasi-two-body decays
$B^{+}\to\pi^{+}\rho(770,1450)^{0}\to\pi^{+}K^{+}K^{-}$ in the perturbative
QCD approach. The results in this work do not support that large branching
fractions contributed by the resonances $\rho(770,1450)^{0}$ in the
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays. The virtual contribution for
$K^{+}K^{-}$ from the tail of the resonance $\rho(770)^{0}$ which has been
ignored in the experimental studies is about $1.5$ times of the
$\rho(1450)^{0}\to K^{+}K^{-}$ contribution, with the predicted branching
fractions ${\mathcal{B}}_{v}=(1.31\pm 0.27)\times 10^{-7}$ and
${\mathcal{B}}=(8.96\pm 2.61)\times 10^{-8}$, respectively, for these two
subprocesses in the $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays. The absence of
$\rho(770)^{0}\to K^{+}K^{-}$ for the decay amplitude of a three-body hadronic
$B$ decay involving charged kaon pair could probably result in a larger
proportion for the contribution from the resonance $\rho(1450)^{0}$ in
experimental analysis.

###### pacs:

13.20.He, 13.25.Hw, 13.30.Eg

##  I Introduction

Strong dynamics in the charmless three-body hadronic $B$ meson decays is
related to the short-distance processes like in the two-body cases, and it
also involves the hadron-hadron interactions, the three-body effects
npps199-341 ; prd84-094001  and the rescattering processes 1512-09284 ;
prd89-094013  in the final states. In experimental studies, the strong
interactions along with the weak processes of a three-body $B$ meson decay are
always described in its decay amplitude as the coherent sum of the resonant
and nonresonant contributions in the isobar formalism pr135-B551 ; pr166-1731
; prd11-3165 . The isobar expression with or without certain resonances will
certainly have impacts on the fit fractions, and then influence the
observables such as the branching ratios and $CP$ violations for the resonant
and the nonresonant contributions of the three-body process in view of the
explicit distribution of the experimental events in the Dalitz plot dalitz .

Very recently, LHCb Collaboration presented a surprising large fit fraction
$(30.7\pm 1.2\pm 0.9)\%$ of the total three-body result in prl123-231802  for
the subprocess $\rho(1450)^{0}\to K^{+}K^{-}$ in the amplitude analysis of the
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays. Considering the branching fraction
$(5.2\pm 0.4)\times 10^{-6}$ for $B^{+}\to K^{+}K^{-}\pi^{+}$ in the Review of
Particle Physics PDG-2018  which was averaged from the results $(5.38\pm
0.40\pm 0.35)\times 10^{-6}$ in prd96-031101  provided by Belle and $(5.0\pm
0.5\pm 0.5)\times 10^{-6}$ in prl99-221801  presented by BaBar, one has the
branching fraction $(1.60\pm 0.15)\times 10^{-6}$ for the quasi-two-body decay
$B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}K^{+}K^{-}$ from LHCb’s fit fraction.
While the contributions of $B^{\pm}\to\pi^{\pm}\rho(1450)^{0}$ were found
quite small in the $\rho^{0}$ dominant decay modes
$B^{\pm}\to\pi^{\pm}\pi^{+}\pi^{-}$ in prl124-031801 ; prd101-012006  by LHCb
recently.

The resonance contributions for the charged kaon pair in the three-body decays
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ are associated with the low energy scalar,
vector and tensor states, such as $f_{0}(980)$, $\phi(1020)$,
$f^{\prime}_{2}(1525)$, etc., which have been noticed in prd76-094006 ;
prd88-114014 ; 2003-03754 ; epjc79-792 , while the contributions for
$K^{+}K^{-}$ also come from the $P$-wave resonances $\rho(770)$, $\omega(782)$
and their excited states prd96-113003 ; prd67-034012 . The natural mode for
$\rho(770)^{0}$ decays into $K^{+}K^{-}$ is blocked because of its pole mass
which is below the threshold of kaon pair, but the virtual contribution
prd94-072001 ; plb791-342  from the tail of the Breit-Wigner (BW) formula BW-
model  of $\rho(770)$ could be indispensable for $K^{+}K^{-}$ in the
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays, which could be deduced from the works
for the processes of $\pi^{-}p\to K^{-}K^{+}n$ and $\pi^{+}n\to K^{-}K^{+}p$
prd15-3196 ; prd22-2595 , $e^{+}e^{-}\to K^{+}K^{-}$ plb669-217 ; prd88-032013
; prd94-112006 ; plb779-64 ; prd99-032001  and $\pi\pi\to K\bar{K}$ scattering
epjc78-897 .

Figure 1: Feynman diagrams for the processes
$B^{+}\to\pi^{+}\rho^{0}\to\pi^{+}K^{+}K^{-}$, with $\otimes$ is the weak
vertex, $\times$ denotes possible attachments for the hard gluons and the
rectangle represents the resonance $\rho(770)^{0}$ and its excited states.

We shall analyze the $\rho(770,1450)^{0}\to K^{+}K^{-}$ contributions for the
three-body decays $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ with the quasi-two-body
framework based on the perturbative QCD (PQCD) approach plb504-6 ;
prd63-054008 ; prd63-074009 ; ppnp51-85 . The final state interaction effect
was found to be suppressed for the $\rho^{0}\to K^{+}K^{-}$ process
prd75-074017  and will be neglected in the calculation in this work. For the
quasi-two-body decays $B^{\pm}\to\pi^{\pm}\rho^{0}\to\pi^{\pm}K^{+}K^{-}$, the
intermediate states $\rho(770)^{0}$ and $\rho(1450)^{0}$ are generated in the
hadronization of the light quark-antiquark pair $q\bar{q}$, with $q$=$(u,d)$,
as demonstrated in the Fig. 1, in which the factorizable and nonfactorizable
diagrams have been merged for the sake of simplicity and Fig. 1 (b) and (d)
will generate different decay amplitudes because of the different position of
the bachelor and the intermediate states. The subprocesses
$\rho(770,1450)^{0}\to K^{+}K^{-}$, which can not be calculated in the PQCD
approach, could be introduced into the distribution amplitudes of the kaon
pair system by the time-like form factor of kaon. The quasi-two-body framework
based on PQCD approach has been discussed in detail in plb763-29 , and has
been adopted in some works for the quasi-two-body $B$ meson decays
jhep2003-162 ; plb788-468 ; prd96-036014 ; prd95-056008 ; prd96-093011 ;
npb923-54  recently.

This paper is organized as follows: In Sec. II, we briefly review the
electromagnetic time-like form factor for the charged kaon, we introduce the
$P$-wave $K^{+}K^{-}$ system distribution amplitudes and give the expression
of differential branching fractions for the quasi-two-body decays
$B^{\pm}\to\pi^{\pm}\rho^{0}\to\pi^{\pm}K^{+}K^{-}$. In Sec. III, we provide
numerical results for the concerned decay processes and give some necessary
discussions. The conclusions are presented in Sec. IV.

##  II Framework

In the light-cone coordinates, the momentum $p_{B}$ and the light spectator
quark’s momentum $k_{B}$ for $B^{+}$ are defined as

$\displaystyle p_{B}=\frac{m_{B}}{\sqrt{2}}(1,1,0_{\rm T}),\quad
k_{B}=\left(\frac{m_{B}}{\sqrt{2}}x_{B},0,k_{B{\rm T}}\right),$ (1)

in the rest frame of $B$ meson, where $m_{B}$ is the mass of $B^{+}$. The
resonance $\rho$ has the momentum $p=\frac{m_{B}}{\sqrt{2}}(\eta,1,0)$, with
$\eta=s/m_{B}^{2}$ and $s=p^{2}$, its spectator quark got the momentum
$k=(0,\frac{m_{B}}{\sqrt{2}}z,k_{\rm T})$. For the bachelor final state pion
and its spectator quark, the momenta $p_{3}$ and $k_{3}$ have the definitions

$\displaystyle p_{3}=\frac{m_{B}}{\sqrt{2}}(1-\eta,0,0_{\rm
T}),\;k_{3}=\left(\frac{m_{B}}{\sqrt{2}}(1-\eta)x_{3},0,k_{3{\rm
T}}\right)\\!.\quad$ (2)

The $x_{B}$, $x_{3}$ and $z$ are the momentum fractions which run from zero to
one in the calculation.

The time-like form factor $F_{K}$ for the charged kaon is related to the
electromagnetic form factor and defined by epjc39-41

$\displaystyle\langle
K^{+}(p_{1})K^{-}(p_{2})|j^{em}_{\mu}|0\rangle=(p_{1}-p_{2})_{\mu}\,F_{K}(s),$
(3)

with the squared invariant mass $s=(p_{1}+p_{2})^{2}$ for kaon pair and the
constraint $F_{K}(0)=1$. The electromagnetic current
$j^{em}_{\mu}=\frac{2}{3}\bar{u}\gamma_{\mu}u-\frac{1}{3}\bar{d}\gamma_{\mu}d-\frac{1}{3}\bar{s}\gamma_{\mu}s$
is carried by the light quarks npb250-517 . With the BW formula for the
resonances $\omega$ and $\phi$ and the Gounaris-Sakurai (GS) model prl21-244
for $\rho$, we have the time-like form factor epjc39-41

$\displaystyle F_{K}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\rho}c^{K}_{\rho}{\rm
GS}_{\rho}(s)+\frac{1}{6}\sum_{\omega}c^{K}_{\omega}{\rm BW}_{\omega}(s)$ (4)
$\displaystyle+$ $\displaystyle\frac{1}{3}\sum_{\phi}c^{K}_{\phi}{\rm
BW}_{\phi}(s),$

where the $\sum$ means the summation of the resonances $\rho,\omega$ or $\phi$
and their corresponding excited states, the explicit expressions and auxiliary
functions for BW and GS are referred to Refs. prd86-032013 ; prl21-244 . The
parameters $c^{K}_{\rho(\omega,\phi)}$ have been fitted to the data in Refs
epjc39-41 ; prd81-094014 ; jetp129-386 .

For the concerned subprocesses $\rho(770,1450)^{0}\to K^{+}K^{-}$, the
$P$-wave $K^{+}K^{-}$ system distribution amplitudes are organized into
prd76-074018

$\displaystyle\phi^{P\text{-wave}}_{KK}(z,s)$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{2N_{c}}}\big{[}\sqrt{s}\,{\epsilon/}_{L}\phi^{0}(z,s)+{\epsilon/}_{L}{p/}\phi^{t}(z,s)$
(5) $\displaystyle+$ $\displaystyle\sqrt{s}\phi^{s}(z,s)\big{]}.$

The momentum $p=p_{1}+p_{2}$, and $\epsilon_{L}$ is the longitudinal
polarization vector for the resonances. We have the distribution amplitudes as
plb763-29

$\displaystyle\phi^{0}(z,s)$ $\displaystyle=$
$\displaystyle\frac{3F^{\rho}_{K}(s)}{\sqrt{N_{c}}}z(1-z)\left[1+a_{2}^{0}C^{3/2}_{2}(1-2z)\right]\\!,$
(6) $\displaystyle\phi^{t}(z,s)$ $\displaystyle=$
$\displaystyle\frac{3F^{t}_{K}(s)}{2\sqrt{N_{c}}}(1-2z)^{2}\left[1+a_{2}^{t}C^{3/2}_{2}(1-2z)\right]\\!,$
(7) $\displaystyle\phi^{s}(z,s)$ $\displaystyle=$
$\displaystyle\frac{3F^{s}_{K}(s)}{2\sqrt{N_{c}}}(1-2z)\left[1+a_{2}^{s}\left(1-10z+10z^{2}\right)\right]\\!,\quad$
(8)

where the $F^{\rho}_{K}$ is the $\rho$ term of Eq. (4), the Gegenbauer
polynomial $C^{3/2}_{2}(\chi)=3\left(5\chi^{2}-1\right)/2$. The Gegenbauer
moments $a_{2}^{0}=0.25\pm 0.10$, $a_{2}^{t}=-0.60\pm 0.20$ and
$a_{2}^{s}=0.75\pm 0.25$ which are the same as they in plb763-29  for the
intermediate state $\rho$. For the subprocess $\rho^{0}\to K^{+}K^{-}$, we
adopt $F^{s,t}_{K}(s)\approx(f^{T}_{\rho}/f_{\rho})F^{\rho}_{K}(s)$ as in
plb763-29  for the channel $\rho^{0}\to\pi^{+}\pi^{-}$.

One has the differential branching fractions ($\mathcal{B}$) for the quasi-
two-body decays $B^{\pm}\to\pi^{\pm}\rho^{0}\to\pi^{\pm}K^{+}K^{-}$ as
prd74-114009 ; prd79-094005 ; plb791-342

$\displaystyle\frac{d{\mathcal{B}}}{d\eta}=\tau_{B}\frac{|\overrightarrow{q_{\pi}}|^{3}|\overrightarrow{q_{K}}|^{3}}{12\pi^{3}m^{5}_{B}}\overline{|{\mathcal{A}}|^{2}},$
(9)

where $\tau_{B}$ being the $B$ meson mean lifetime. It should be noted that
the Eqs. (6)-(8) and Eq. (9) are slightly different from the corresponding
expressions in plb763-29 . These differences are caused by the introduction of
the Zemach tensor
${\small-2\overrightarrow{q_{\pi}}}\cdot{\small\overrightarrow{q_{K}}}$ Zemach
in this work as did in Refs. prd74-114009 ; prd79-094005 , this tensor is
employed to describe the angular distribution for the decay of spin $1$
resonances. The magnitudes of the kaon and pion momenta
$|\overrightarrow{q_{K}}|$ and $|\overrightarrow{q_{\pi}}|$ are written, in
the center-of-mass frame of the kaon pair, as

$\displaystyle|\overrightarrow{q_{K}}|$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{s-4m^{2}_{K}},$ (10)
$\displaystyle|\overrightarrow{q_{\pi}}|$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s}}\sqrt{\left(m^{2}_{B}-m_{\pi}^{2}\right)^{2}-2\left(m^{2}_{B}+m_{\pi}^{2}\right)s+s^{2}},\quad$
(11)

with the pion mass $m_{\pi}$ and the kaon mass $m_{K}$.

The decay amplitudes ${\cal A}$ for the quasi-two-body decays
$B^{+}\to\pi^{+}\rho(770,1450)^{0}\to\pi^{+}K^{+}K^{-}$ dependent only on the
quark structures of the hadronic matrix elements for $B^{+}$ to
$\pi^{+}\rho(770,1450)^{0}$ transitions and have the same expressions as the
decays $B^{+}\to\pi^{+}\rho(770,1450)^{0}\to\pi^{+}\pi^{+}\pi^{-}$ in
prd96-036014 ; prd95-056008  except the replacement of the form factor
$F_{\pi}\to F_{K}$. The amplitudes in ${\cal A}$ according to the diagrams in
Fig. 1 could be found in the Appendix of plb763-29 .

##  III Results

In the numerical calculation, we adopt the decay constant $f_{B}=0.189$ GeV
prd98-074512  and the mean lifetime $\tau=(1.638\pm 0.004)\times 10^{-12}$ s
PDG-2018  for the $B^{+}$ meson. The masses and the decay constant $f_{\pi}$
for the relevant particles in the numerical calculation in this work, the full
widths for $\rho(770)$ and $\rho(1450)$, and the Wolfenstein parameters of the
CKM matrix are presented in Table 1.

Table 1: Masses, decay constant $f_{\pi}$, full widths of $\rho(770)$ and
$\rho(1450)$ (in units of GeV) and Wolfenstein parameters PDG-2018 .
$m_{B^{\pm}}=5.279\quad\;\,\;m_{\pi^{\pm}}=0.140\quad m_{K^{\pm}}=0.494$
---
$m_{\rho(770)}=0.775\;\;\;m_{\rho(1450)}=1.465\;\;\;f_{\pi}=0.131$
$\Gamma_{\rho(770)}=0.149\;\quad\Gamma_{\rho(1450)}=0.400\pm 0.060$
$\lambda=0.22453\pm 0.00044\quad\;\;\;A=0.836\pm 0.015\;$
$\bar{\rho}=0.122^{+0.018}_{-0.017}\hskip
56.9055pt\bar{\eta}=0.355^{+0.012}_{-0.011}$

The ratio between the $f^{T}_{\rho}$ and $f_{\rho}$ for $\rho(770)$ has been
computed in lattice QCD, we choose the result $f^{T}_{\rho}/f_{\rho}=0.687$ at
the scale $\mu=2$ GeV prd78-114509  for our numerical calculation. The decay
constant $f_{\rho}$ for $\rho(770)$ can be extracted from the processes
$\tau^{-}\to\rho^{-}\nu_{\tau}$ and $\rho^{0}\to e^{+}e^{-}$, we take the
value $0.216$ GeV jhep1608-098 . The $c^{K}_{\rho(770)}$ was fitted to be
$1.139\pm 0.010$ and $1.195\pm 0.009$ with the unconstrained and constrained
fit procedure in epjc39-41 , respectively, which are consistent with the
values $1.138\pm 0.011$ and $1.120\pm 0.007$ in prd81-094014 . We employ the
result $1.139\pm 0.010$ for the quasi-two-body decay
$B^{+}\to\pi^{+}\rho(770)^{0}\to\pi^{+}K^{+}K^{-}$. As for the $c^{K}$ for
$\rho(1450)$, we adopt the value $-0.124\pm 0.012$ epjc39-41  for the
subprocess $\rho(1450)^{0}\to K^{+}K^{-}$, which is close to the constrained
fit result$-0.112\pm 0.010$ in epjc39-41  and the unconstrained fit result
$-0.107\pm 0.010$ in prd81-094014 . We need to stress that different choices
for $c^{K}_{\rho(770)}$ and $c^{K}_{\rho(1450)}$ with the values in epjc39-41
; prd81-094014  can not significantly change the numerical results and will
not change our conclusions in this work.

Utilizing the differential branching fractions the Eq. (9), we have the
branching fractions

$\displaystyle{\mathcal{B}}(B^{\pm}$ $\displaystyle\to$
$\displaystyle\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}K^{+}K^{-})$ (12)
$\displaystyle=$ $\displaystyle\big{(}8.96\pm
1.58(a^{s}_{2}+a^{0}_{2}+a^{t}_{2})\pm 0.83(\omega_{B})$ $\displaystyle\pm$
$\displaystyle 0.79(m^{\pi}_{0}+a^{\pi}_{2})\pm
1.73(c^{K}_{\rho(1450)})\big{)}\times 10^{-8},\quad$
$\displaystyle{\mathcal{B}}_{v}(B^{\pm}$ $\displaystyle\to$
$\displaystyle\pi^{\pm}\rho(770)^{0}\to\pi^{\pm}K^{+}K^{-})$ (13)
$\displaystyle=$ $\displaystyle\big{(}1.31\pm
0.22(a^{s}_{2}+a^{0}_{2}+a^{t}_{2})\pm 0.12(\omega_{B})$ $\displaystyle\pm$
$\displaystyle 0.11(m^{\pi}_{0}+a^{\pi}_{2})\pm
0.02(c^{K}_{\rho(770)})\big{)}\times 10^{-7}.$

The subscript $v$ for ${\mathcal{B}}$ of
$B^{\pm}\to\pi^{\pm}\rho(770)^{0}\to\pi^{\pm}K^{+}K^{-}$ means the virtual
contribution prd94-072001 ; plb791-342  which is integrated of the Eq. (9)
from the threshold of kaon pair. The first error for the two branching
fractions above comes from the uncertainty of the Gegenbauer moments
$a_{2}^{s},a_{2}^{0}$ and $a_{2}^{t}$ in the $K^{+}K^{-}$ system distribution
amplitudes, the second error is induced by the shape parameter
$\omega_{B}=0.40\pm 0.04$ for $B^{+}$, the third one is contributed by the
chiral mass $m^{\pi}_{0}=1.40\pm 0.10$ GeV and the Gegenbauer moment
$a^{\pi}_{2}=0.25\pm 0.15$ for pion and the fourth one due to the variation of
$c^{K}$ in the form factor $F_{K}$. There are other errors come from the
uncertainties of the Wolfenstein parameters of the CKM matrix, the parameters
in the distribution amplitudes of bachelor pion, the masses and the decay
constants of the initial and final states, etc. are small and have been
neglected.

From the fit fractions in prl123-231802 , one has a branching fraction
${\mathcal{B}}\approx(1.60\pm 0.15)\times 10^{-6}$ for the quasi-two-body
decay $B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}K^{+}K^{-}$, which is about $18$
times larger than the PQCD prediction shown in Eq. (12). The value $(1.60\pm
0.15)\times 10^{-6}$ is close to the result
${\mathcal{B}}=1.4^{+0.6}_{-0.9}\times 10^{-6}$ for the decay
$B^{+}\to\rho(1450)^{0}\pi^{+}$ with $\rho(1450)^{0}\to\pi^{+}\pi^{-}$ from
BaBar Collaboration prd79-072006 ; PDG-2018 . Recently, in the amplitude
analysis of the $B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay, LHCb Collaboration
provided a $CP$-averaged fit fraction $(5.2\pm 0.3\pm 0.2\pm 1.9)\%$
prl124-031801 ; prd101-012006  for the $\rho(1450)^{0}$ component with the
isobar model, implying a branching ratio ${\mathcal{B}}=(7.9\pm 3.0)\times
10^{-7}$ for the quasi-two-body process
$B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}\pi^{+}\pi^{-}$ with the data
${\mathcal{B}}=(1.52\pm 0.14)\times 10^{-5}$ PDG-2018  for the three-body
$B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay. With the same expressions for
$B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}K^{+}K^{-}$ but the subprocess
$\rho(1450)^{0}\to\pi^{+}\pi^{-}$, we have the prediction

$\displaystyle{\mathcal{B}}(B^{\pm}$ $\displaystyle\to$
$\displaystyle\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}\pi^{+}\pi^{-})$ (14)
$\displaystyle=$ $\displaystyle\big{(}9.97\pm
1.81(a^{s}_{2}+a^{0}_{2}+a^{t}_{2})\pm 0.98(\omega_{B})$ $\displaystyle\pm$
$\displaystyle 0.91(m^{\pi}_{0}+a^{\pi}_{2})\big{)}\times 10^{-7},$

which agree with the results $(7.9\pm 3.0)\times 10^{-7}$ and
$1.4^{+0.6}_{-0.9}\times 10^{-6}$.

From the predictions for the quasi-two-body decays
$B^{\pm}\to\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}K^{+}K^{-}$ and
$B^{\pm}\to\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}\pi^{+}\pi^{-}$ in this work, we
have the ratio $R_{\rho(1450)}$ between their branching fractions as

$\displaystyle R_{\rho(1450)}=\frac{{\mathcal{B}}(\rho(1450)^{0}\to
K^{+}K^{-})}{{\mathcal{B}}(\rho(1450)^{0}\to\pi^{+}\pi^{-})}=0.090\pm
0.017,\quad\;$ (15)

with the factorization relation $\Gamma(B^{\pm}\to\rho^{0}\pi^{\pm}\to
h^{+}h^{-}\pi^{\pm})\approx\Gamma(B^{+}\to\rho^{0}\pi^{\pm}){\mathcal{B}}(\rho^{0}\to
h^{+}h^{-})$, here $h=(\pi,K)$. The only error for $R_{\rho(1450)}$ comes from
the uncertainty of $c^{K}_{\rho(1450)}$ because of the cancellation between
the other errors of the two branching fractions in Eq. (12) and Eq. (14). That
is to say, the increase or the decrease of the parameters that caused the
errors will result in nearly identical change of the weight for the numerator
and denominator of $R_{\rho(1450)}$. The ratio $R_{\rho(1450)}$ can also be
estimated from the coupling constants $g_{\rho(1450)^{0}\pi^{+}\pi^{-}}$ and
$g_{\rho(1450)^{0}K^{+}K^{-}}$. With the relation
$g_{\rho(1450)^{0}K^{+}K^{-}}\approx\frac{1}{2}g_{\rho(1450)^{0}\pi^{+}\pi^{-}}$
epjc39-41  one has

$\displaystyle R_{\rho(1450)}$ $\displaystyle=$
$\displaystyle\frac{{\mathcal{B}}(\rho(1450)^{0}\to
K^{+}K^{-})}{{\mathcal{B}}(\rho(1450)^{0}\to\pi^{+}\pi^{-})}$ (16)
$\displaystyle\approx$
$\displaystyle\frac{g^{2}_{\rho(1450)^{0}K^{+}K^{-}}(m^{2}_{\rho(1450)}-4m^{2}_{K})^{3/2}}{g^{2}_{\rho(1450)^{0}\pi^{+}\pi^{-}}(m^{2}_{\rho(1450)}-4m^{2}_{\pi})^{3/2}}$
$\displaystyle=$ $\displaystyle 0.107.$

This value is consistent with the result in Eq. (15).

The ratio $R_{\rho(1450)}$ has been measured by BaBar Collaboration with the
result

$\displaystyle R_{\rho(1450)}$ $\displaystyle=$
$\displaystyle\frac{{\mathcal{B}}(\rho(1450)^{0}\to
K^{+}K^{-})}{{\mathcal{B}}(\rho(1450)^{0}\to\pi^{+}\pi^{-})}$ (17)
$\displaystyle=$ $\displaystyle 0.307\pm 0.084({\rm stat})\pm 0.082({\rm
sys}).$

in the Dalitz plot analyses of $J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ and $J/\psi\to
K^{+}K^{-}\pi^{0}$ decays in Ref. prd95-072007 . This result is quite large
comparing with the values in Eqs. (15)-(16). But one should note that the
virtual contribution of $\rho(770)^{0}\to K^{+}K^{-}$ has been ignored for the
channel $J/\psi\to K^{+}K^{-}\pi^{0}$ in prd95-072007 . We argue that the
large value for $R_{\rho(1450)}$ from BaBar could be understood as that the
virtual contribution of $\rho(770)^{0}$ for $K^{+}K^{-}$ has probably been
taken into account for the process $\rho(1450)^{0}\to K^{+}K^{-}$.

The virtual contribution for $K^{+}K^{-}$ from the tail of the resonance
$\rho(770)^{0}$, which has not been taken into the decay amplitudes of
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ and other charmless three-body $B$ meson
decays involving kaon pair in the experimental studies, is about $1.5$ times
of the contribution from $\rho(1450)^{0}$ for the charged kaon pair as shown
in Eq. (13) for the branching fractions. The predicted result $\left(1.31\pm
0.27\right)\times 10^{-7}$ is roughly $2.5\%$ of the total branching fraction
of the corresponding three-body decay and larger than the fit fraction
$(0.3\pm 0.1\pm 0.1)\%$ in the same channel for $\phi(1020)$ in prl123-231802
, and should not be ignored in the studies of the three-body decays like
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$. The large virtual contribution of
$\rho(770)^{0}$ for the charged kaon pair is mainly due to its large decay
width which result in a relative dispersive distribution of the differential
branching fraction as shown in Fig. 2, where the red short dash line is for
the decay $B^{\pm}\to\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}K^{+}K^{-}$, the blue
dash-dot line is for the process
$B^{\pm}\to\pi^{\pm}\rho(770)^{0}\to\pi^{\pm}K^{+}K^{-}$ and the black solid
line is for
$B^{\pm}\to\pi^{\pm}(\rho(770)^{0}+\rho(1450)^{0})\to\pi^{\pm}K^{+}K^{-}$.

Figure 2: The differential branching fractions for the quasi-two-body decays
$B^{\pm}\to\pi^{\pm}\rho(770)^{0}\to\pi^{\pm}K^{+}K^{-}$,
$B^{\pm}\to\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}K^{+}K^{-}$ and
$B^{\pm}\to\pi^{\pm}(\rho(770)^{0}+\rho(1450)^{0})\to\pi^{\pm}K^{+}K^{-}$.

In Fig. 2, the peak of the differential branching fraction for
$B^{\pm}\to\pi^{\pm}\rho(1450)^{0}\to\pi^{\pm}K^{+}K^{-}$ is around the pole
mass of the $\rho(1450)$ resonance as expected. There is also a bump at about
$1.35$ GeV of the invariant mass of kaon pair for the process
$B^{\pm}\to\pi^{\pm}\rho(770)^{0}\to\pi^{\pm}K^{+}K^{-}$. This bump is
generated by the the tail of the BW formula for $\rho(770)$ along with the
phase space factors $\overrightarrow{q_{K}}$ and $\overrightarrow{q_{\pi}}$ in
the Eqs. (10)-(11). When we investigate the total contributions from
$\rho(1450)^{0}$ plus $\rho(770)^{0}$, the bump from $\rho(770)^{0}$ will
disappear and an enhanced peak around $1.4$ GeV is the only one existed as
shown by the solid line in Fig. 2. This makes us conclude that the absence of
the virtual contribution from $\rho(770)^{0}$ for the decay amplitudes of
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ could probably enhance the experimental result
for the $K^{+}K^{-}$ from the resonance $\rho(1450)^{0}$.

Figure 3: The differential branching fractions for
$B^{\pm}\to\pi^{\pm}(\rho(770)^{0}+\rho(1450)^{0})\to\pi^{\pm}K^{+}K^{-}$ with
the phases $\theta_{K}=\\{0,0.5\pi,\pi,1.3\pi,1.5\pi\\}$.

In principle, the interference between $\rho(770)^{0}$ and $\rho(1450)^{0}$
can increase or decrease the total contributions from these two resonances.
Then we adopt the same expressions and parameters of time-like form factor for
the $\rho(770)^{0}$ and $\rho(1450)^{0}$ components in Eq. (4), but add a
phase $\theta_{K}$ between these two components, namely

$\displaystyle F^{\rho(770+1450)}_{K}$ (18)
$\displaystyle\quad=\frac{1}{2}\left[c^{K}_{\rho(770)}{\rm
GS}_{\rho(770)}+e^{{\rm i}\theta_{K}}c^{K}_{\rho(1450)}{\rm
GS}_{\rho(1450)}\right]\\!.\;\;$

When $\theta_{K}=1.3\pi$, we find the largest interferential branching
fraction, with ${\mathcal{B}}\approx 3.64\times 10^{-7}$ for the
$(\rho(770)^{0}+\rho(1450)^{0})\to K^{+}K^{-}$ of the
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays. The differential branching fraction
curves for
$B^{\pm}\to\pi^{\pm}(\rho(770)^{0}+\rho(1450)^{0})\to\pi^{\pm}K^{+}K^{-}$ with
$\theta_{K}=\\{0,0.5\pi,\pi,1.3\pi,1.5\pi\\}$ are shown in the Fig. 3, the red
solid line is corresponding to $\theta_{K}=1.3\pi$.

The resonance $\omega(782)$, different from $\rho(770)$, has a narrow full
width $\Gamma=8.49\pm 0.08$ MeV. In spite of the relation
$g_{\omega(782)K^{+}K^{-}}=g_{\rho(770)^{0}K^{+}K^{-}}$ epjc39-41 , one can
expect a small contribution from the tail of $\omega(782)$ for the
$K^{+}K^{-}$ in the $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ decays. As a result, we
predict the branching fraction to be ${\mathcal{B}}_{v}\approx 2.32\times
10^{-8}$ for the decay $B^{+}\to\pi^{+}\omega(782)\to\pi^{+}K^{+}K^{-}$, which
is much smaller than the branching fractions for the decays
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ with the subprocesses $\rho(770)^{0}\to
K^{+}K^{-}$ and $\rho(1450)^{0}\to K^{+}K^{-}$. The excited state
$\rho(1700)^{0}$ has been studied in the quasi-two-body decays in PQCD
approach with $\rho(1700)^{0}\to\pi^{+}\pi^{-}$ prd96-036014 ; prd96-093011 .
With the Eq. (16) and the relation $g_{\rho
K^{+}K^{-}}\approx\frac{1}{2}g_{\rho\pi^{+}\pi^{-}}$ but the replacement
$\rho(1450)^{0}\to\rho(1700)^{0}$, we estimate the ratio between the branching
fractions of $\rho(1700)^{0}\to K^{+}K^{-}$ and
$\rho(1700)^{0}\to\pi^{+}\pi^{-}$ as

$\displaystyle R_{\rho(1700)}$ $\displaystyle=$
$\displaystyle\frac{{\mathcal{B}}(\rho(1700)^{0}\to
K^{+}K^{-})}{{\mathcal{B}}(\rho(1700)^{0}\to\pi^{+}\pi^{-})}$ (19)
$\displaystyle\approx$
$\displaystyle\frac{g^{2}_{\rho(1700)^{0}K^{+}K^{-}}(m^{2}_{\rho(1700)}-4m^{2}_{K})^{3/2}}{g^{2}_{\rho(1700)^{0}\pi^{+}\pi^{-}}(m^{2}_{\rho(1700)}-4m^{2}_{\pi})^{3/2}}$
$\displaystyle=$ $\displaystyle 0.143,$

with $m_{\rho(1700)}=1.72$ GeV PDG-2018 . In consideration of the
${\mathcal{B}}=2.81^{+0.63}_{-0.66}\times 10^{-7}$ in prd96-036014  for the
decay $B^{+}\to\pi^{+}\rho(1700)^{0}\to\pi^{+}\pi^{+}\pi^{-}$, one could
expect a branching fraction ${\mathcal{B}}\approx 4.02^{+0.90}_{-0.94}\times
10^{-8}$ for the decay $B^{+}\to\pi^{+}\rho(1700)^{0}$ with the subprocess
$\rho(1700)^{0}\to K^{+}K^{-}$, which is close to half of the contribution
from $\rho(1450)^{0}\to K^{+}K^{-}$. Then we have about $5\%$ of the total
branching fraction, but still much less than the $(30.7\pm 1.2\pm 0.9)\%$ from
LHCb prl123-231802 , of the three-body decays $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$
from the resonances $\rho(770)^{0}$, $\rho(1450)^{0}$ and $\rho(1700)^{0}$
when neglecting the interferences between them.

##  IV Summary

We analyzed the quasi-two-body decays
$B^{+}\to\pi^{+}\rho(770,1450)^{0}\to\pi^{+}K^{+}K^{-}$ in the PQCD approach.
Our results and analyses do not support that large branching fraction
contributed by the resonance $\rho(1450)^{0}$ for the decays
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$. The predictions of the branching fractions
are ${\mathcal{B}}=(8.96\pm 2.61)\times 10^{-8}$ and
${\mathcal{B}}_{v}=(1.31\pm 0.27)\times 10^{-7}$ for the quasi-two-body decays
with the subprocesses $\rho(1450)^{0}\to K^{+}K^{-}$ and $\rho(770)^{0}\to
K^{+}K^{-}$, respectively.

The virtual contribution from the BW tail of $\rho(770)^{0}$ for $K^{+}K^{-}$,
which has not been taken into the decay amplitudes of the charmless three-body
$B$ meson decays involving kaon pair, was found about $1.5$ times of the
contribution from the resonance $\rho(1450)^{0}$ in this work, and is roughly
$2.5\%$ of the total branching fraction for the three-body decays
$B^{\pm}\to\pi^{\pm}K^{+}K^{-}$.

There is a bump, which is generated by the tail of the BW formula and the
phase space factors, at about $1.35$ GeV of the invariant mass of kaon pair
for $B^{\pm}\to\pi^{\pm}\rho(770)^{0}\to\pi^{\pm}K^{+}K^{-}$. This bump will
disappear and the enhancement round $1.4$ GeV will be the result when we
investigated the total contributions from $\rho(1450)^{0}$ together with
$\rho(770)^{0}$. This means that the absence of $\rho(770)^{0}\to K^{+}K^{-}$
in the decay amplitude of a three-body $B$ decays could probably result in a
larger proportion for the resonance $\rho(1450)^{0}$ in experimental amplitude
analysis.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under Grants No. 11547038 and No. 11575110.

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