Constrain the UT angle $\gamma$ by CP violation parameters in $B^0\to\pi^+\pi^-$
Name\corref{cor1
%% Text of abstract We calculate the tree and penguin amplitudes in the $B^0\to\pi^+\pi^-$ decay channel employing the perturbative QCD factorization approach. Using the amplitudes as input with the theoretical uncertainties sufficiently considered, we constrain the UT angle $\gamma$ to $53^\circ\leq\gamma\leq70^\circ$, from the measurements of the CP violation parameters $C_{\pi^+\pi^-$ and $S_{\pi^+\pi^-$ in $B^0\to\pi^+\pi^-$. The U-spin breaking effect between $B^0\to\pi^+\pi^-$ and $B_s^0\to K^+K^-$ is estimated to be around 30\%.
paper_after_arXiv.tex
\documentclass[preprint,12pt]{elsarticle}%\documentclass[preprint,12pt,authoryear]{elsarticle}

%% Use the option review to obtain double line spacing
%% \documentclass[authoryear,preprint,review,12pt]{elsarticle}

%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
%% for a journal layout:
%% \documentclass[final,1p,times,authoryear]{elsarticle}
%% \documentclass[final,1p,times,twocolumn,authoryear]{elsarticle}
%% \documentclass[final,3p,times,authoryear]{elsarticle}
%% \documentclass[final,3p,times,twocolumn,authoryear]{elsarticle}
%% \documentclass[final,5p,times,authoryear]{elsarticle}
%% \documentclass[final,5p,times,twocolumn,authoryear]{elsarticle}

%% For including figures, graphicx.sty has been loaded in
%% elsarticle.cls. If you prefer to use the old commands
%% please give \usepackage{epsfig}

%% The amssymb package provides various useful mathematical symbols
\usepackage{amssymb}
%% The amsthm package provides extended theorem environments
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{subfigure}

%% The lineno packages adds line numbers. Start line numbering with
%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
%% for the whole article with \linenumbers.
%% \usepackage{lineno}

\journal{ }

\begin{document}

\begin{frontmatter}

%% Title, authors and addresses

%% use the tnoteref command within \title for footnotes;
%% use the tnotetext command for theassociated footnote;
%% use the fnref command within \author or \address for footnotes;
%% use the fntext command for theassociated footnote;
%% use the corref command within \author for corresponding author footnotes;
%% use the cortext command for theassociated footnote;
%% use the ead command for the email address,
%% and the form \ead[url] for the home page:
%% \title{Title\tnoteref{label1}}
%% \tnotetext[label1]{}
%% \author{Name\corref{cor1}\fnref{label2}}
%% \ead{email address}
%% \ead[url]{home page}
%% \fntext[label2]{}
%% \cortext[cor1]{}
%% \address{Address\fnref{label3}}
%% \fntext[label3]{}

\title{\bf Constrain the UT angle $\gamma$ by CP violation parameters in $B^0\to\pi^+\pi^-$}

%% use optional labels to link authors explicitly to addresses:
%% \author[label1,label2]{}
%% \address[label1]{}
%% \address[label2]{}

\author[affiliation1]{Qin Qin}
\author[affiliation2]{Zhi-Tian Zou}%\email{zouzt@ytu.edu.cn}
\author[affiliation1,affiliation4]{Cai-Dian L\"u}
\author[affiliation2]{Ying Li}

\address[affiliation1]{Institute of High Energy Physics, Beijing
100049, People's Republic of China}
\address[affiliation2]{Department of Physics, Yantai University, Yantai,
Shandong 264005, People's Republic of China}
\address[affiliation4]{State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China}

\begin{abstract}
%% Text of abstract
  We calculate the tree and penguin amplitudes in the $B^0\to\pi^+\pi^-$ decay channel
  employing the perturbative QCD factorization approach. Using the amplitudes as input
  with the theoretical uncertainties sufficiently considered, we constrain the UT
  angle $\gamma$ to $53^\circ\leq\gamma\leq70^\circ$, from the measurements of the CP
  violation parameters $C_{\pi^+\pi^-}$ and $S_{\pi^+\pi^-}$ in $B^0\to\pi^+\pi^-$.
  The U-spin breaking effect between $B^0\to\pi^+\pi^-$ and $B_s^0\to K^+K^-$ is
  estimated to be around 30\%.
\end{abstract}

\begin{keyword}
CP violation \sep $\gamma$ extraction \sep $B$ meson decay
%% keywords here, in the form: keyword \sep keyword

%% PACS codes here, in the form: \PACS code \sep code

%% MSC codes here, in the form: \MSC code \sep code
%% or \MSC[2008] code \sep code (2000 is the default)

\end{keyword}

\end{frontmatter}

%% \linenumbers

%% main text
\section{Introduction}

In the Standard Model, the quark mixing is described by the
Cabbibo-Kobayashi-Maskawa (CKM) matrix \cite{Kobayashi:1973fv}, in which the nonzero
phase angle induces the Charge conjugation Parity (CP) violation in weak interaction. For recent
developments on the CKM matrix, one can refer to the review \cite{Wang:2014sba}.
It is important to examine the unitarity of the CKM matrix, since any deviation
would indicate new physics beyond the Standard Model. The three angles of the well-known unitarity
triangle (UT), which are defined by $\alpha\equiv\arg[-(V_{td}V_{tb}^*)/(V_{ud}V_{ub}^*)]$,
$\beta\equiv\arg[-(V_{cd}V_{cb}^*)/(V_{td}V_{tb}^*)]$ and $\gamma\equiv\arg[-(V_{ud}V_{ub}^*)/ (V_{cd}V_{cb}^*)]$,
have been measured by experiments and the present averages are
\cite{Agashe:2014kda}
\begin{equation}\label{world_average}
\alpha = (85.4^{+3.9}_{-3.8})^\circ,~~~~\sin2\beta = 0.682\pm0.019,~~~~\gamma = (68.0^{+8.0}_{-8.5})^\circ.
\end{equation}
The angle $\gamma$ is the least known one among the three angles. Methods were proposed
to extract $\gamma$ from the tree-dominated modes $B\to DK$, known as the GLW method
\cite{Gronau:1990ra}, the ADS method \cite{Atwood:1996ci}, and
the Dalitz-plot method \cite{Giri:2003ty}, with different final states of $D$ decays.
Combining the $B\to DK$ measurements performed by Belle, BaBar, CDF and LHCb
\cite{Trabelsi:2013uj}, the CKMfitter group \cite{Charles:2011va} obtained the above
average for $\gamma$. Recently, the LHCb collaboration made two new measurements
\cite{Aaij:2014uva}. Alternatively, $\gamma$ can also be determined by the U-spin analysis
on the two-body charmless $B$ decays, $B^0\to\pi^+\pi^-$ and $B_s^0\to K^+K^-$
\cite{Fleischer:1999pa}. A combination with the channels $B^0\to\pi^0\pi^0$ and
$B^+\to\pi^+\pi^0$ makes the analysis more sophisticated \cite{Ciuchini:2012gd}. Recently,
following the method proposed in Ref. \cite{Ciuchini:2012gd}, the LHCb collaboration
performed the U-spin and isospin analysis and obtained \cite{Aaij:2014xba}
\begin{equation}
\gamma=(63.5^{+7.2}_{-6.7})^\circ,
\end{equation}
which has a smaller central value than the world average in Eq. (\ref{world_average}).

In this letter, we constrain the UT angle $\gamma$ from $B^0\to\pi^+\pi^-$, with the help
of factorization approach to calculate the tree and penguin amplitudes.
%we fit $\gamma$ and $\beta$ to the experimental results of $\sin2\beta$ and
%the CP violation parameters in the decay channel.
Similar ideas have been used to constrain $\alpha$ from $B^0\to\pi^+\pi^-$ \cite{Lu:2002qv},
and to constrain $\gamma$ from $B^0_s\to D_s^\pm K^\mp$ \cite{Yu:2013pua}. However, neither
of them got strong constraint on the corresponding UT angle for lack of precisely measured
experimental results at their time. Recently, the CP violation parameters in
$B^0\to\pi^+\pi^-$ have been precisely measured \cite{Lees:2012mma}, and the weighted
averages of the results are given by \cite{Aaij:2014xba},
\begin{equation}\label{CSexperiment}
C_{\pi^+\pi^-} = -0.30 \pm 0.05,~~~~S_{\pi^+\pi^-} = -0.66 \pm 0.06,
\end{equation}
with the statistical correlation $\rho(C_{\pi^+\pi^-},S_{\pi^+\pi^-})=-0.007$. The high
precision of the parameters indicates the possibility that our constraint on $\gamma$ is
comparable to the world average in Eq. (\ref{world_average}) and the
results given in Ref. \cite{Aaij:2014xba}. The method can also be applied to
$B_s^0\to K^+K^-$.

The rest of the paper is organized as follows. In Sec. \ref{formalism}, the relevant
formulas for the CP violation parameters in the channels $B^0\to\pi^+\pi^-$ and
$B_s^0\to K^+K^-$ are listed. In Sec. \ref{numeric}, we introduce our strategy for the
numerical analysis and obtain the constraints on $\gamma$ from the two channels, between
which the U-spin breaking effect is also estimated. In Sec. \ref{conclude}, we conclude.

%with given CKM matrix elements, factorization approaches have been applied to predict CP observables \cite{Qin:2014xta}.


%While since the well measured CP observables in $B_d\to\pi^+\pi^-$ can be used to constrain the CKM angles in the opposite way.

\section{Theoretical formalism}\label{formalism}

For $B^0\to\pi^+\pi^-$, the relevant effective Hamiltonian is given by
\begin{equation}\begin{split}
\mathcal{H}_{eff}=&V_{ub}^*V_{ud}[C_1O_1+C_2O_2]
-V_{tb}^*V_{td}\sum_{n=3}^{10}C_nO_n+{\it h.c.},
\end{split}\end{equation}
where $O_{1,2(3-10)}$ are the tree (penguin) 4-quark operators, and $C_{1-10}$ are
the corresponding Wilson coefficients. After we apply some factorization approach
to calculate the hadronic matrix elements $\langle\pi^+\pi^-|O_i|B^0\rangle$,
the amplitude of $B^0\to\pi^+\pi^-$ can be expressed as
\begin{equation}\begin{split}
\mathcal{A}(B^0\to\pi^+\pi^-)=&V_{ub}^*V_{ud}\mathcal{T}-V_{tb}^*V_{td}\mathcal{P}\\
=&V_{ub}^*V_{ud}(\mathcal{T}+\mathcal{P})\left(1+{V_{cb}^*V_{cd}\over V_{ub}^*V_{ud}}{\mathcal{P}\over\mathcal{T}+\mathcal{P}}\right),
\end{split}\end{equation}
where $\mathcal{T}$ and $\mathcal{P}$ are the tree and penguin amplitudes, respectively.
Defining
\begin{equation}
de^{i\theta}\equiv{|V_{cb}^*V_{cd}|\over|V_{ub}^*V_{ud}|}{\mathcal{P}\over\mathcal{T}+\mathcal{P}},
\end{equation}
with $d$ and $\theta$ real-valued, we obtain the expression for the CP violation
parameters
\begin{equation}\begin{split}
C_{\pi^+\pi^-}=&-{2d\sin\theta\sin\gamma\over1+d^2-2d\cos\theta\cos\gamma},\\
S_{\pi^+\pi^-}=&-{\sin(2\beta+2\gamma)-2d\cos\theta\sin(2\beta+\gamma)+d^2\sin(2\beta)\over1+d^2-2d\cos\theta\cos\gamma}.
\end{split}\end{equation}
For $C_{\pi^+\pi^-}$ and $S_{\pi^+\pi^-}$, we have accepted the convention in the
letter \cite{Aaij:2014xba},
\begin{equation}\begin{split}
&C_{\pi^+\pi^-}\equiv{1-|\lambda_{\pi^+\pi^-}|^2\over1+|\lambda_{\pi^+\pi^-}|^2},~~~~
S_{\pi^+\pi^-}\equiv{2\mathrm{Im}\lambda_{\pi^+\pi^-}\over1+|\lambda_{\pi^+\pi^-}|^2},\\
&\lambda_{\pi^+\pi^-}\equiv{q\over p}{\mathcal{A}(\bar{B}^0\to\pi^+\pi^-)\over\mathcal{A}(B^0\to\pi^+\pi^-)},
\end{split}\end{equation}
where $q$ and $p$ are the coefficients in the mass eigenstates $p|B^0\rangle\pm q|\bar{B}^0\rangle$.


Similarly for $B_s^0\to K^+K^-$, one has
\begin{equation}\begin{split}\label{K2}
C_{K^+K^-}\approx&{2\widetilde{d'}\sin\theta'\sin\gamma\over1+\widetilde{d'}^2+2\widetilde{d'}\cos\theta'\cos\gamma},\\
S_{K^+K^-}\approx&-{\sin(-2\beta_s+2\gamma)+2\widetilde{d'}\cos\theta'\sin(-2\beta_s+\gamma)+\widetilde{d'}^2\sin(-2\beta_s) \over1+\widetilde{d'}^2+2\widetilde{d'}\cos\theta'\cos\gamma},
\end{split}\end{equation}
where the real-valued parameters are defined by
\begin{equation}\label{K3}
\widetilde{d'}\equiv{|V_{cs}||V_{ud}|\over|V_{cd}||V_{us}|}d',~~~~
d'e^{i\theta'}\equiv{|V_{cb}^*V_{cd}|\over|V_{ub}^*V_{ud}|}{\mathcal{P'}\over\mathcal{T'}+\mathcal{P'}},
\end{equation}
with $\mathcal{T'}$ ($\mathcal{P'}$) representing the tree (penguin) amplitude in
$B_s^0\to K^+K^-$. $\beta_s\equiv\arg[-(V_{ts}V_{tb}^*)/(V_{cs}V_{cb}^*)]$ gives the
mixing phase in the $B_s^0-\bar{B}_s^0$ mixing system.

\section{Numerical Analysis}\label{numeric}

The present average of the UT angle $\beta$ is given in Eq. (\ref{world_average}),
which has a two-fold ambiguity $2\beta\to\pi-2\beta$. A series of measurements
\cite{Aubert:2004cp} prefer that $\cos2\beta$ is positive, so we accept
\begin{equation}\label{beta_experiment}
\beta=(21.50_{-0.74}^{+0.75})^\circ.
\end{equation}
Choosing the sample values for $d$ and $\theta$, $de^{i\theta}=0.23e^{i2.4}$, we can
then obtain the $\gamma$ dependence of $C_{\pi^+\pi^-}$ and $S_{\pi^+\pi^-}$, as shown
in Fig. \ref{K_mixing_plot}. The experimental 1 $\sigma$ allowed regions are also
displayed. Fig. \ref{Spipi-Gamma} shows that $S_{\pi^+\pi^-}$ is very sensitive to the
change of the angle $\gamma$, and at meanwhile precise measurements for $S_{\pi^+\pi^-}$
have been performed. This indicates that $\gamma$ is potentially to be strongly constrained
in our method, though there are considerable theoretical uncertainties in any factorization
approach.

\begin{figure}[!ht]
  \centering
  \subfigure[$C_{\pi^+\pi^-}$ {\it vs} $\gamma$;]{
    \includegraphics[width=2.4in]{plots_Cpipi-Gamma.pdf}\label{Cpipi-Gamma}}
  \hspace{0.2in}
  \subfigure[$S_{\pi^+\pi^-}$ {\it vs} $\gamma$.]{
    \includegraphics[width=2.4in]{plots_Spipi-Gamma.pdf}\label{Spipi-Gamma}}
  \caption{The solid curves correspond to the sample choice: $d=0.23$ and $\theta=2.4$.
  The light blue bands show the experimentally 1 $\sigma$ allowed regions
  $-0.35\leq C_{\pi^+\pi^-}\leq-0.25$ and $-0.72\leq S_{\pi^+\pi^-}\leq-0.60$, respectively.}
  \label{K_mixing_plot}%% label for entire figure
\end{figure}

In the perturbative QCD (PQCD) approach base on the transverse momentum factorization
\cite{Chang:1996dw}, hadronic matrix elements are factorized into convolutions
of the calculable hard kernels and the non-perturbative meson wave functions which are
however universal. The PQCD approach has been applied in analysis on hadronic
$B$ meson decays, successfully making predictions for both branching ratios and
CP violation \cite{Lu:2000em,Ali:2007ff}. Especially for $B^0\to\pi^+\pi^-$, the
PQCD prediction of the branching ratio is
$(5.8^{+3.0+0.5+0.4}_{-2.1-0.4-0.3})\times10^{-6}$ \cite{Ali:2007ff}, which is
consistent with the experimental result
$(5.12\pm0.19)\times10^{-6}$ \cite{Agashe:2014kda}. Therefore, we employ the PQCD
approach to calculate the tree and penguin amplitudes here. The formulas for
calculating the leading-order decay amplitudes are given by Eqs. (50 - 61) in Ref.
\cite{Ali:2007ff}. We also include the next-to-leading-order corrections to the
$B\to\pi$ transition form factors, of which the twist-2 and -3 contributions have
been studied in Ref. \cite{Li:2012nk} and \cite{Cheng:2014fwa}, respectively.


%The central values of the PQCD prediction for $d$ and $\theta$ are
%\begin{equation}\label{dthetacenter}
%d=0.25,~~~~\theta=2.6.
%\end{equation}
%For test, accepting the central values of $d$ and $\theta$, we perform the global
%fit of $\beta$ and $\gamma$ to the experimental results for $\sin2\beta$,
%$C_{\pi^+\pi^-}$ and $S_{\pi^+\pi^-}$ given in Eqs. (\ref{world_average}) and
%(\ref{CSexperiment}), respectively. As shown in Fig. \ref{fit_center}, at 68\%
%probability we obtain
%\begin{equation}
%\gamma\in[55, 61]^\circ.
%\end{equation}

%\begin{figure}[!ht]
%  \centering
%    \includegraphics[width=2.5in]{plots_fit_center.pdf}
%  \caption{Fit result for $\beta$ and $\gamma$ at $de^{i\theta}=0.25e^{i2.6}$. The 68\%, 95\% and 99\% C.L. regions are shown.}
%  \label{fit_center} %% label for entire figure
%\end{figure}

To perform a reliable analysis, we need to sufficiently take into account the
uncertainties introduced by the calculation of the hadronic matrix elements.
In the calculation, we adopt the updated non-asymptotic distribution amplitudes
\cite{Khodjamirian:2009ys},
\begin{equation}\begin{split}
\phi^A_\pi(x)=&{f_\pi\over2\sqrt{6}}6x(1-x)[1+a^\pi_2C_2^{3/2}(2x-1)+a^\pi_4C_4^{3/2}(2x-1)],\\
\phi^P_\pi(x)=&{f_\pi\over2\sqrt{6}}[1+30\eta_3^\pi C^{1/2}_2(2x-1)-3\eta_3^\pi\omega_3^\pi C_4^{1/2}(2x-1)],\\
\phi^T_\pi(x)=&{f_\pi\over2\sqrt{2N_c}}(1-2x)\{1
+{1\over2}\eta_3^\pi(10-\omega_3^\pi)C_2^{3/2}(2x-1)-15\eta_3^\pi(10-\omega_3^\pi)x(1-x)\},
\end{split}\end{equation}
where $C^{\alpha}_n(2x-1)$ are the well known Gegenbauer polynomials with $x$
the longitudinal momentum fraction of the quark in pion. The values of the Gegenbauer moments,
$a_2^\pi$ and $a_4^\pi$, have been determined in the global fit to the data of the
pion electromagnetic form factor \cite{Khodjamirian:2011ub}, which yields
\begin{equation}
a_2^\pi=0.17\pm0.08,~~~~a_4^\pi=0.06\pm0.10.
\end{equation}
To keep it safe, we double the error bars in the numerical analysis. In Ref.
\cite{Li:2013xna} where the joint resummation was performed for the pion transition
form factor in the transverse-momentum factorization formalism, the authors found
that their prediction for the form factor with
$a_2^\pi=0.05$ agrees well with the experimental data. Our choice for the range of
$a_2^\pi$ covers this value. As for the other non-perturbative parameters $\eta_3^\pi$
and $\omega_3^\pi$, we accept the values taken in Ref. \cite{Khodjamirian:2009ys}, also
with doubled error bars. The shape parameter in the distribution amplitude of the
$B^0$ meson \cite{Kurimoto:2001zj} is taken value in the range
\begin{equation}
\omega_b\in[0.36,~0.44].
\end{equation}
We also consider the uncertainties caused by the unknown next-to-leading-order corrections
characterized by the choice that $\Lambda_{QCD}\in[0.20,0.30]$ and a 20\% variation of the
factorization scale.
Taking values for the theoretical parameters randomly in the ranges covering
their uncertainties, we perform the PQCD calculation and obtain 99 points of
($d$, $\theta$), which are shown in Fig. \ref{d-theta}.
%We find that the scale variation (0.8t - t - 1.2t) causes
%significant modification to d, and that the Gegenbauer-moment uncertainty
%($a_2^\pi = 0.25 \pm 0.15$) causes significant modification to $\theta$.
%Scanning the region with $d\in[0.22, 0.28]$ and $\theta\in[2.45, 2.75]$, we find
%the point ($d$ = 2.8, $\theta$ = 2.45) which leads to the minimal ��2 = 1.94. Fix
%them and we perform the global fit. The result in Fig. \ref{fit_minimal} shows
%$\gamma\in[55,61]^\circ$ at 68\% C.L., $\gamma\in[52,63]^\circ$ at 95\% C.L.,
%and $\gamma\in[50,65]^\circ$ at 99\% C.L..
At each point of ($d$, $\theta$), we perform the global fit of $\beta$ and $\gamma$
to the experimental results of the CP violation parameters in Eq. (\ref{CSexperiment})
and that of $\beta$ in Eq. (\ref{beta_experiment}). Then, we combine the 1 $\sigma$
allowed regions of all fits at the 99 points, and regard it as our constraint on
$\gamma$ and $\beta$. As shown in Fig. \ref{fit_scan}, the constraint on $\gamma$ is
\begin{equation}\label{final_constraint}
53^\circ\leq\gamma\leq70^\circ.
\end{equation}

\begin{figure}[!ht]
  \centering
    \includegraphics[width=2.5in]{plots_d-theta_NLO.pdf}
  \caption{Plots for ($d$, $\theta$) calculated with the random theoretical parameters ranging in the allowed regions.}
  \label{d-theta} %% label for entire figure
\end{figure}
%\begin{figure}[!ht]
%  \centering
%    \includegraphics[width=2.5in]{plots_fit_minimal.pdf}
%  \caption{Fit result for $\beta$ and $\gamma$ at $de^{i\theta}=0.25e^{i2.6}$. The 68\%, 95\% and 99\% C.L. regions are shown.}
%  \label{fit_minimal} %% label for entire figure
%\end{figure}
\begin{figure}[!ht]
  \centering
    \includegraphics[width=2.5in]{plots_fit_scan_NLO.pdf}
  \caption{The 68\% C.L. region for $\beta$ - $\gamma$ with the theoretical uncertainties considered.}
  \label{fit_scan} %% label for entire figure
\end{figure}

We can also perform the similar analysis to $B_s^0\to K^+K^-$ using Eqs. (\ref{K2}) and
(\ref{K3}), though the experimental results for the CP violation parameters are much
less precise, which are given by \cite{Aaij:2014xba}
\begin{equation}
C_{K^+K^-}=0.14\pm0.11,~~~~S_{K^+K^-}=0.30\pm0.13,
\end{equation}
with the statistical correlation $\rho(C_{K^+K^-},S_{K^+K^-})=0.02$.
To improve the precision on the determination of $\gamma$, $\beta_s$ is expressed in
terms of $\beta$ and $\gamma$. However, the $B_s^0\to K^+K^-$ constraint
$20^\circ\leq\gamma\leq150^\circ$ is still too loose.

As a byproduct, we also estimate the U-spin breaking effect in the two channels
$B^0\to\pi^+\pi^-$ and $B_s^0\to K^+K^-$, which is parameterized by
\begin{equation}
d'e^{i\theta'}=de^{i\theta}(1+re^{i\theta_r}).
\end{equation}
The PQCD result is
\begin{equation}
r=0.3\pm0.1,~~~~\theta_r=-1.2\pm0.2.
\end{equation}
In the letter \cite{Aaij:2014xba}, the U-spin breaking effect is parameterized
by two relative magnitudes $r_D$ and $r_G$ with the corresponding phase shifts
$\theta_{r_D}$ and $\theta_{r_G}$,
\begin{equation}
d'e^{i\theta'}=de^{i\theta}{1+r_Ge^{i\theta_{r_G}}\over1+r_De^{i\theta_{r_D}}}.
\end{equation}
Assuming the parameters range within the region
\begin{equation}
r_D,r_G\in[0,~0.5],~~~~\theta_{r_D},\theta_{r_G}\in[-\pi,~\pi],
\end{equation}
the authors obtained $\gamma=(63.5^{+7.2}_{-6.7})^\circ$.
This region can fully cover the PQCD result (including the uncertainties),
so we conclude that the assumption about the U-spin breaking in Ref.
\cite{Aaij:2014xba} is reasonable.

\section{Conclusion}\label{conclude}

We extract the UT angle $\gamma$ from the precise experimental results of
$C_{\pi^+\pi^-}$ and $S_{\pi^+\pi^-}$ given in the letter \cite{Aaij:2014xba},
with the tree and penguin amplitudes in $B^0\to\pi^+\pi^-$ calculated in the PQCD
approach. Including the theoretical uncertainties, we constrain $53^\circ\leq\gamma\leq70^\circ$
at 68\% probability. Through the similar method, the angle $\gamma$ is also
constrained in the range $20^\circ - 150^\circ$ by the measurements of $C_{K^+K^-}$ and
$S_{K^+K^-}$. The U-spin breaking effect between the two channels is found to be
smaller than 50\%, which indicates that the results in the letter \cite{Aaij:2014xba}
are reliable.

\section*{Acknowledgement}

We are grateful to Shan Cheng, Wei Wang and Yu-Ming Wang for useful discussions.
QQ thanks Yantai University for the warm hospitality during his visit. This work
is partly supported by the National Science Foundation of China under Grants No.
11375208, No. 11235005 and No. 11447032, the Natural Science Foundation of Shandong
province (ZR2014AQ013) and the Program for New Century Excellent Talents in
University (NCET) by Ministry of Education of P. R. China (Grant No. NCET-13-0991).



%% The Appendices part is started with the command \appendix;
%% appendix sections are then done as normal sections
%% \appendix

%% \section{}
%% \label{}

%% If you have bibdatabase file and want bibtex to generate the
%% bibitems, please use
%%
%%  \bibliographystyle{elsarticle-harv}
%%  \bibliography{<your bibdatabase>}

%% else use the following coding to input the bibitems directly in the
%% TeX file.


\begin{thebibliography}{00}

%% \bibitem{label}
%% Text of bibliographic item

  \bibitem{Kobayashi:1973fv}
  M.~Kobayashi and T.~Maskawa,
  %``CP Violation in the Renormalizable Theory of Weak Interaction,''
  Prog.\ Theor.\ Phys.\  {\bf 49}, 652 (1973).

  \bibitem{Wang:2014sba}
  W.~Wang,
  %``Recent Developments on the CKM Matrix,''
  Int.\ J.\ Mod.\ Phys.\ A {\bf 29}, 1430040 (2014)
  [arXiv:1407.6868 [hep-ph]].

  \bibitem{Agashe:2014kda}
  K.~A.~Olive {\it et al.}  [Particle Data Group Collaboration],
  %``Review of Particle Physics,''
  Chin.\ Phys.\ C {\bf 38}, 090001 (2014).

  \bibitem{Gronau:1990ra}
  M.~Gronau and D.~London,
  %``How to determine all the angles of the unitarity triangle from B(d)0 ---> D K(s) and B(s)0 ---> D0,''
  Phys.\ Lett.\ B {\bf 253}, 483 (1991);
  M.~Gronau and D.~Wyler,
  %``On determining a weak phase from CP asymmetries in charged B decays,''
  Phys.\ Lett.\ B {\bf 265}, 172 (1991).

  \bibitem{Atwood:1996ci}
  D.~Atwood, I.~Dunietz and A.~Soni,
  %``Enhanced CP violation with B ---> K D0 (anti-D0) modes and extraction of the CKM angle gamma,''
  Phys.\ Rev.\ Lett.\  {\bf 78}, 3257 (1997)
  [hep-ph/9612433];
  %``Improved methods for observing CP violation in B+- ---> K D and measuring the CKM phase gamma,''
  Phys.\ Rev.\ D {\bf 63}, 036005 (2001)
  [hep-ph/0008090].

  \bibitem{Giri:2003ty}
  A.~Giri, Y.~Grossman, A.~Soffer and J.~Zupan,
  %``Determining gamma using B+- ---> DK+- with multibody D decays,''
  Phys.\ Rev.\ D {\bf 68}, 054018 (2003)
  [hep-ph/0303187].

  \bibitem{Trabelsi:2013uj}
  K.~Abe {\it et al.}  [Belle Collaboration],
  %``Study of B+- ---> D(CP)K+- and D*(CP)K+- decays,''
  Phys.\ Rev.\ D {\bf 73}, 051106 (2006)
  [hep-ex/0601032];
  Y.~Horii {\it et al.}  [Belle Collaboration],
  %``Evidence for the Suppressed Decay B- -> DK-, D -> K+pi-,''
  Phys.\ Rev.\ Lett.\  {\bf 106}, 231803 (2011)
  [arXiv:1103.5951 [hep-ex]];
  A.~Poluektov {\it et al.}  [Belle Collaboration],
  %``Evidence for direct CP violation in the decay B->D(*)K, D->KsPi+Pi- and measurement of the CKM phase phi3,''
  Phys.\ Rev.\ D {\bf 81}, 112002 (2010)
  [arXiv:1003.3360 [hep-ex]];
  T.~Aaltonen {\it et al.}  [CDF Collaboration],
  %``Measurements of branching fraction ratios and CP asymmetries in B+- ---> D(CP) K+- decays in hadron collisions,''
  Phys.\ Rev.\ D {\bf 81}, 031105 (2010)
  [arXiv:0911.0425 [hep-ex]];
  P.~del Amo Sanchez {\it et al.}  [BaBar Collaboration],
  %``Measurement of CP observables in $B^{+-} -> D_{CP} K^{+-}$ decays and constraints on the CKM angle $\gamma$,''
  Phys.\ Rev.\ D {\bf 82}, 072004 (2010)
  [arXiv:1007.0504 [hep-ex]];
  B.~Aubert {\it et al.}  [BaBar Collaboration],
  %``Measurement of Ratios of Branching Fractions and CP-Violating Asymmetries of $B^\pm \to D^{*} K^\pm$ Decays,''
  Phys.\ Rev.\ D {\bf 78}, 092002 (2008)
  [arXiv:0807.2408 [hep-ex]];
  %B.~Aubert {\it et al.}  [BaBar Collaboration],
  %``Measurement of CP violation observables and parameters for the decays B+- ---> D K*+-,''
  Phys.\ Rev.\ D {\bf 80}, 092001 (2009)
  [arXiv:0909.3981 [hep-ex]];
  P.~del Amo Sanchez {\it et al.}  [BaBar Collaboration],
  %``Search for b ---> u transitions in $B^- -> DK^- and D^*K^-$ Decays,''
  Phys.\ Rev.\ D {\bf 82}, 072006 (2010)
  [arXiv:1006.4241 [hep-ex]];
  J.~P.~Lees {\it et al.}  [BaBar Collaboration],
  %``Search for $b \to u$ Transitions in $B^\pm \to [K^\mp pi^\pm \pi^0]_D K^\pm$ Decays,''
  Phys.\ Rev.\ D {\bf 84}, 012002 (2011)
  [arXiv:1104.4472 [hep-ex]];
  P.~del Amo Sanchez {\it et al.}  [BaBar Collaboration],
  %``Evidence for direct CP violation in the measurement of the Cabibbo-Kobayashi-Maskawa angle gamma with B-+ ---> D(*) K(*)-+ decays,''
  Phys.\ Rev.\ Lett.\  {\bf 105}, 121801 (2010)
  [arXiv:1005.1096 [hep-ex]];
  R.~Aaij {\it et al.}  [LHCb Collaboration],
  %``Observation of CP violation in B+ to DK+ decays,''
  Phys.\ Lett.\ B {\bf 712}, 203 (2012)
  [Erratum {\it ibid.} {\bf 713}, 351 (2012)]
  [arXiv:1203.3662 [hep-ex]];
  %``Observation of the suppressed ADS modes $B^\pm \to [\pi^\pm K^\mp \pi^+\pi^-]_D K^\pm$ and $B^\pm \to [\pi^\pm K^\mp \pi^+\pi^-]_D \pi^\pm$,''
  Phys.\ Lett.\ B {\bf 723}, 44 (2013)
  [arXiv:1303.4646 [hep-ex]].

  \bibitem{Charles:2011va}
  J.~Charles, O.~Deschamps, S.~Descotes-Genon, R.~Itoh, H.~Lacker, A.~Menzel, S.~Monteil and V.~Niess {\it et al.},
  %``Predictions of selected flavour observables within the Standard Model,''
  Phys.\ Rev.\ D {\bf 84}, 033005 (2011)
  [arXiv:1106.4041 [hep-ph]].

  \bibitem{Aaij:2014uva}
  R.~Aaij {\it et al.}  [LHCb Collaboration],
  %``Measurement of the CKM angle $\gamma$ using $B^\pm \to D K^\pm$ with $D \to K^0_{\rm S} \pi^+\pi^-, K^0_{\rm S} K^+ K^-$ decays,''
  JHEP {\bf 1410}, 97 (2014)
  [arXiv:1408.2748 [hep-ex]];
   %``Study of $B^{-}\to DK^-\pi^+\pi^-$ and $B^-\to D\pi^-\pi^+\pi^-$ decays and determination of the CKM angle $\gamma$,''
  arXiv:1505.07044 [hep-ex].

  \bibitem{Fleischer:1999pa}
  R.~Fleischer,
  %``New strategies to extract Beta and gamma from B(d) $\to\pi^+\pi^-$ and B(S) $\to K^+ K^-$,''
  Phys.\ Lett.\ B {\bf 459}, 306 (1999)
  [hep-ph/9903456].

  \bibitem{Ciuchini:2012gd}
  M.~Ciuchini, E.~Franco, S.~Mishima and L.~Silvestrini,
  %``Testing the Standard Model and Searching for New Physics with $B_d \to \pi \pi$ and $B_s \to K K$ Decays,''
  JHEP {\bf 1210}, 029 (2012)
  [arXiv:1205.4948 [hep-ph]].

  \bibitem{Aaij:2014xba}
  R.~Aaij {\it et al.}  [LHCb Collaboration],
  %``Determination of $\gamma$ and -2$\beta_s$ from charmless two-body decays of beauty mesons,''
  Phys.\ Lett.\ B {\bf 741}, 1 (2015)
  [arXiv:1408.4368 [hep-ex]].

  \bibitem{Lu:2002qv}
  C.~D.~Lu and Z.~j.~Xiao,
  %``Constraint on the CKM angle alpha from the experimental measurements of CP violation in B0(d) ---> pi+ pi- decay,''
  Phys.\ Rev.\ D {\bf 66}, 074011 (2002)
  [hep-ph/0205134].

  \bibitem{Yu:2013pua}
  X.~Yu, Z.~T.~Zou and C.~D.~L\"u,
  %``Time-dependent $CP$-violations of $B(B_{s})$ decays in the perturbative QCD approach,''
  Phys.\ Rev.\ D {\bf 88}, no. 5, 054018 (2013)
  [arXiv:1307.7485].

  \bibitem{Lees:2012mma}
  J.~P.~Lees {\it et al.}  [BaBar Collaboration],
  %``Measurement of CP Asymmetries and Branching Fractions in Charmless Two-Body $B$-Meson Decays to Pions and Kaons,''
  Phys.\ Rev.\ D {\bf 87}, no. 5, 052009 (2013)
  [arXiv:1206.3525 [hep-ex]];
  I.~Adachi {\it et al.}  [Belle Collaboration],
  %``Measurement of the CP violation parameters in $B^0 \to \pi^+ \pi^-$ decays,''
  Phys.\ Rev.\ D {\bf 88}, no. 9, 092003 (2013)
  [arXiv:1302.0551 [hep-ex]];
  R.~Aaij {\it et al.}  [LHCb Collaboration],
  %``First measurement of time-dependent $C\!P$ violation in $B^0_s \to K^+K^-$ decays,''
  JHEP {\bf 1310}, 183 (2013)
  [arXiv:1308.1428 [hep-ex]].

%  \bibitem{Li:1994cka}
%  H.~n.~Li and H.~L.~Yu,
  %``Extraction of V(ub) from decay B ---> pi lepton neutrino,''
%  Phys.\ Rev.\ Lett.\  {\bf 74}, 4388 (1995)
%  [hep-ph/9409313];
  %``PQCD analysis of exclusive charmless B meson decay spectra,''
%  Phys.\ Lett.\ B {\bf 353}, 301 (1995);
%  H.n. Li, {\it ibid}, 348, 597 (1995);
%  H.~n.~Li and H.~L.~Yu,
  %``Perturbative QCD analysis of B meson decays,''
%  Phys.\ Rev.\ D {\bf 53}, 2480 (1996)
%  [hep-ph/9411308].

  \bibitem{Aubert:2004cp}
  B.~Aubert {\it et al.}  [BaBar Collaboration],
  %``Ambiguity-free measurement of $\cos(2\beta)$: Time-integrated and time-dependent angular analyses of $B \to J/\psi K \pi$,''
  Phys.\ Rev.\ D {\bf 71}, 032005 (2005)
  [hep-ex/0411016];
  R.~Itoh {\it et al.}  [Belle Collaboration],
  %``Studies of CP violation in B ---> J/psi K* decays,''
  Phys.\ Rev.\ Lett.\  {\bf 95}, 091601 (2005)
  [hep-ex/0504030];
  P.~Krokovny {\it et al.}  [Belle Collaboration],
  %``Measurement of the quark mixing parameter cos(2phi(1)) using time-dependent Dalitz analysis of anti-B0 ---> D [K(s)0 pi+ pi-] h0,''
  Phys.\ Rev.\ Lett.\  {\bf 97}, 081801 (2006)
  [hep-ex/0605023];
  B.~Aubert {\it et al.}  [BaBar Collaboration],
  %``Measurement of $\cos$ 2 beta in $B^0 \to D^{(*)}$ h0 Decays with a Time-Dependent Dalitz Plot Analysis of $D \to K^0_{S} \pi^{+} \pi^{-}$,''
  Phys.\ Rev.\ Lett.\  {\bf 99}, 231802 (2007)
  [arXiv:0708.1544 [hep-ex]].

  \bibitem{Chang:1996dw}
  C.~H.~V.~Chang and H.~n.~Li,
  %``Three - scale factorization theorem and effective field theory,''
  Phys.\ Rev.\ D {\bf 55}, 5577 (1997)
  [hep-ph/9607214];
  T.~W.~Yeh and H.~n.~Li,
  %``Factorization theorems, effective field theory, and nonleptonic heavy meson decays,''
  Phys.\ Rev.\ D {\bf 56}, 1615 (1997)
  [hep-ph/9701233].

  \bibitem{Lu:2000em}
  C.~D.~Lu, K.~Ukai and M.~Z.~Yang,
  %``Branching ratio and CP violation of B ---> pi pi decays in perturbative QCD approach,''
  Phys.\ Rev.\ D {\bf 63}, 074009 (2001)
  [hep-ph/0004213];
  C.~D.~Lu and M.~Z.~Yang,
  %``B ---> pi rho, pi omega decays in perturbative QCD approach,''
  Eur.\ Phys.\ J.\ C {\bf 23}, 275 (2002)
  [hep-ph/0011238];
  Z.~T.~Zou, X.~Yu and C.~D.~Lu,
  %``Nonleptonic two-body charmless B decays involving a tensor meson in the Perturbative QCD Approach,''
  Phys.\ Rev.\ D {\bf 86}, 094015 (2012)
  [arXiv:1203.4120 [hep-ph]];
  Q.~Qin, Z.~T.~Zou, X.~Yu, H.~n.~Li and C.~D.~L��,
  %``Perturbative QCD study of $B_s$ decays to a pseudoscalar meson and a tensor meson,''
  Phys.\ Lett.\ B {\bf 732}, 36 (2014)
  [arXiv:1401.1028 [hep-ph]];
  W.~F.~Wang, H.~C.~Hu, H.~n.~Li and C.~D.~L��,
  %``Direct CP asymmetries of three-body $B$ decays in perturbative QCD,''
  Phys.\ Rev.\ D {\bf 89}, no. 7, 074031 (2014)
  [arXiv:1402.5280 [hep-ph]].


  \bibitem{Ali:2007ff}
  A.~Ali, G.~Kramer, Y.~Li, C.~D.~Lu, Y.~L.~Shen, W.~Wang and Y.~M.~Wang,
  %``Charmless non-leptonic $B_s$ decays to $PP$, $PV$ and $VV$ final states in the PQCD approach,''
  Phys.\ Rev.\ D {\bf 76}, 074018 (2007)
  [hep-ph/0703162 [HEP-PH]].

  \bibitem{Li:2012nk}
  H.~n.~Li, Y.~L.~Shen and Y.~M.~Wang,
  %``Next-to-leading-order corrections to $B \to \pi$ form factors in $k_T$ factorization,''
  Phys.\ Rev.\ D {\bf 85}, 074004 (2012)
  [arXiv:1201.5066 [hep-ph]].

  \bibitem{Cheng:2014fwa}
  S.~Cheng, Y.~Y.~Fan, X.~Yu, C.~D.~L�� and Z.~J.~Xiao,
  %``The NLO twist-3 contributions to $B \to \pi$ form factors in $k_{T}$ factorization,''
  Phys.\ Rev.\ D {\bf 89}, no. 9, 094004 (2014)
  [arXiv:1402.5501 [hep-ph]].

  \bibitem{Khodjamirian:2009ys}
  A.~Khodjamirian, C.~Klein, T.~Mannel and N.~Offen,
  %``Semileptonic charm decays D ---> pi l nu(l) and D ---> K l nu(l) from QCD Light-Cone Sum Rules,''
  Phys.\ Rev.\ D {\bf 80}, 114005 (2009)
  [arXiv:0907.2842 [hep-ph]].
%  \bibitem{Ball:1998je}
%  P.~Ball,
  %``Theoretical update of pseudoscalar meson distribution amplitudes of higher twist: The Nonsinglet case,''
%  JHEP {\bf 9901}, 010 (1999)
%  [hep-ph/9812375].

  \bibitem{Khodjamirian:2011ub}
  A.~Khodjamirian, T.~Mannel, N.~Offen and Y.-M.~Wang,
  %``$B \to \pi \ell \nu_l$ Width and $|V_{ub}|$ from QCD Light-Cone Sum Rules,''
  Phys.\ Rev.\ D {\bf 83}, 094031 (2011)
  [arXiv:1103.2655 [hep-ph]].

  \bibitem{Li:2013xna}
  H.~N.~Li, Y.~L.~Shen and Y.~M.~Wang,
  %``Joint resummation for pion wave function and pion transition form factor,''
  JHEP {\bf 1401}, 004 (2014)
  [arXiv:1310.3672 [hep-ph]].

  \bibitem{Kurimoto:2001zj}
  T.~Kurimoto, H.~n.~Li and A.~I.~Sanda,
  %``Leading power contributions to B ---> pi, rho transition form-factors,''
  Phys.\ Rev.\ D {\bf 65}, 014007 (2002)
  [hep-ph/0105003].



\end{thebibliography}
\end{document}

\endinput
%%
%% End of file `elsarticle-template-harv.tex'.

1506.02398.extracted_bib
{label %% Text of bibliographic item
1506.02398.extracted_bib
{Kobayashi:1973fv M.~Kobayashi and T.~Maskawa, %``CP Violation in the Renormalizable Theory of Weak Interaction,'' Prog.\ Theor.\ Phys.\ {49, 652 (1973).
1506.02398.extracted_bib
{Wang:2014sba W.~Wang, %``Recent Developments on the CKM Matrix,'' Int.\ J.\ Mod.\ Phys.\ A {29, 1430040 (2014) [arXiv:1407.6868 [hep-ph]].
1506.02398.extracted_bib
{Agashe:2014kda K.~A.~Olive {et al. [Particle Data Group Collaboration], %``Review of Particle Physics,'' Chin.\ Phys.\ C {38, 090001 (2014).
1506.02398.extracted_bib
{Gronau:1990ra M.~Gronau and D.~London, %``How to determine all the angles of the unitarity triangle from B(d)0 ---> D K(s) and B(s)0 ---> D0,'' Phys.\ Lett.\ B {253, 483 (1991); M.~Gronau and D.~Wyler, %``On determining a weak phase from CP asymmetries in charged B decays,'' Phys.\ Lett.\ B {265, 172 (1991).
1506.02398.extracted_bib
{Atwood:1996ci D.~Atwood, I.~Dunietz and A.~Soni, %``Enhanced CP violation with B ---> K D0 (anti-D0) modes and extraction of the CKM angle gamma,'' Phys.\ Rev.\ Lett.\ {78, 3257 (1997) [hep-ph/9612433]; %``Improved methods for observing CP violation in B+- ---> K D and measuring the CKM phase gamma,'' Phys.\ Rev.\ D {63, 036005 (2001) [hep-ph/0008090].
1506.02398.extracted_bib
{Giri:2003ty A.~Giri, Y.~Grossman, A.~Soffer and J.~Zupan, %``Determining gamma using B+- ---> DK+- with multibody D decays,'' Phys.\ Rev.\ D {68, 054018 (2003) [hep-ph/0303187].
1506.02398.extracted_bib
{Trabelsi:2013uj K.~Abe {et al. [Belle Collaboration], %``Study of B+- ---> D(CP)K+- and D*(CP)K+- decays,'' Phys.\ Rev.\ D {73, 051106 (2006) [hep-ex/0601032]; Y.~Horii {et al. [Belle Collaboration], %``Evidence for the Suppressed Decay B- -> DK-, D -> K+pi-,'' Phys.\ Rev.\ Lett.\ {106, 231803 (2011) [arXiv:1103.5951 [hep-ex]]; A.~Poluektov {et al. [Belle Collaboration], %``Evidence for direct CP violation in the decay B->D(*)K, D->KsPi+Pi- and measurement of the CKM phase phi3,'' Phys.\ Rev.\ D {81, 112002 (2010) [arXiv:1003.3360 [hep-ex]]; T.~Aaltonen {et al. [CDF Collaboration], %``Measurements of branching fraction ratios and CP asymmetries in B+- ---> D(CP) K+- decays in hadron collisions,'' Phys.\ Rev.\ D {81, 031105 (2010) [arXiv:0911.0425 [hep-ex]]; P.~del Amo Sanchez {et al. [BaBar Collaboration], %``Measurement of CP observables in $B^{+- -> D_{CP K^{+-$ decays and constraints on the CKM angle $\gamma$,'' Phys.\ Rev.\ D {82, 072004 (2010) [arXiv:1007.0504 [hep-ex]]; B.~Aubert {et al. [BaBar Collaboration], %``Measurement of Ratios of Branching Fractions and CP-Violating Asymmetries of $B^\pm \to D^{* K^\pm$ Decays,'' Phys.\ Rev.\ D {78, 092002 (2008) [arXiv:0807.2408 [hep-ex]]; %B.~Aubert {et al. [BaBar Collaboration], %``Measurement of CP violation observables and parameters for the decays B+- ---> D K*+-,'' Phys.\ Rev.\ D {80, 092001 (2009) [arXiv:0909.3981 [hep-ex]]; P.~del Amo Sanchez {et al. [BaBar Collaboration], %``Search for b ---> u transitions in $B^- -> DK^- and D^*K^-$ Decays,'' Phys.\ Rev.\ D {82, 072006 (2010) [arXiv:1006.4241 [hep-ex]]; J.~P.~Lees {et al. [BaBar Collaboration], %``Search for $b \to u$ Transitions in $B^\pm \to [K^\mp pi^\pm \pi^0]_D K^\pm$ Decays,'' Phys.\ Rev.\ D {84, 012002 (2011) [arXiv:1104.4472 [hep-ex]]; P.~del Amo Sanchez {et al. [BaBar Collaboration], %``Evidence for direct CP violation in the measurement of the Cabibbo-Kobayashi-Maskawa angle gamma with B-+ ---> D(*) K(*)-+ decays,'' Phys.\ Rev.\ Lett.\ {105, 121801 (2010) [arXiv:1005.1096 [hep-ex]]; R.~Aaij {et al. [LHCb Collaboration], %``Observation of CP violation in B+ to DK+ decays,'' Phys.\ Lett.\ B {712, 203 (2012) [Erratum {ibid. {713, 351 (2012)] [arXiv:1203.3662 [hep-ex]]; %``Observation of the suppressed ADS modes $B^\pm \to [\pi^\pm K^\mp \pi^+\pi^-]_D K^\pm$ and $B^\pm \to [\pi^\pm K^\mp \pi^+\pi^-]_D \pi^\pm$,'' Phys.\ Lett.\ B {723, 44 (2013) [arXiv:1303.4646 [hep-ex]].
1506.02398.extracted_bib
{Charles:2011va J.~Charles, O.~Deschamps, S.~Descotes-Genon, R.~Itoh, H.~Lacker, A.~Menzel, S.~Monteil and V.~Niess {et al., %``Predictions of selected flavour observables within the Standard Model,'' Phys.\ Rev.\ D {84, 033005 (2011) [arXiv:1106.4041 [hep-ph]].
1506.02398.extracted_bib
{Aaij:2014uva R.~Aaij {et al. [LHCb Collaboration], %``Measurement of the CKM angle $\gamma$ using $B^\pm \to D K^\pm$ with $D \to K^0_{\rm S \pi^+\pi^-, K^0_{\rm S K^+ K^-$ decays,'' JHEP {1410, 97 (2014) [arXiv:1408.2748 [hep-ex]]; %``Study of $B^{-\to DK^-\pi^+\pi^-$ and $B^-\to D\pi^-\pi^+\pi^-$ decays and determination of the CKM angle $\gamma$,'' arXiv:1505.07044 [hep-ex].
1506.02398.extracted_bib
{Fleischer:1999pa R.~Fleischer, %``New strategies to extract Beta and gamma from B(d) $\to\pi^+\pi^-$ and B(S) $\to K^+ K^-$,'' Phys.\ Lett.\ B {459, 306 (1999) [hep-ph/9903456].
1506.02398.extracted_bib
{Ciuchini:2012gd M.~Ciuchini, E.~Franco, S.~Mishima and L.~Silvestrini, %``Testing the Standard Model and Searching for New Physics with $B_d \to \pi \pi$ and $B_s \to K K$ Decays,'' JHEP {1210, 029 (2012) [arXiv:1205.4948 [hep-ph]].
1506.02398.extracted_bib
{Aaij:2014xba R.~Aaij {et al. [LHCb Collaboration], %``Determination of $\gamma$ and -2$\beta_s$ from charmless two-body decays of beauty mesons,'' Phys.\ Lett.\ B {741, 1 (2015) [arXiv:1408.4368 [hep-ex]].
1506.02398.extracted_bib
{Lu:2002qv C.~D.~Lu and Z.~j.~Xiao, %``Constraint on the CKM angle alpha from the experimental measurements of CP violation in B0(d) ---> pi+ pi- decay,'' Phys.\ Rev.\ D {66, 074011 (2002) [hep-ph/0205134].
1506.02398.extracted_bib
{Yu:2013pua X.~Yu, Z.~T.~Zou and C.~D.~L\"u, %``Time-dependent $CP$-violations of $B(B_{s)$ decays in the perturbative QCD approach,'' Phys.\ Rev.\ D {88, no. 5, 054018 (2013) [arXiv:1307.7485].
1506.02398.extracted_bib
{Lees:2012mma J.~P.~Lees {et al. [BaBar Collaboration], %``Measurement of CP Asymmetries and Branching Fractions in Charmless Two-Body $B$-Meson Decays to Pions and Kaons,'' Phys.\ Rev.\ D {87, no. 5, 052009 (2013) [arXiv:1206.3525 [hep-ex]]; I.~Adachi {et al. [Belle Collaboration], %``Measurement of the CP violation parameters in $B^0 \to \pi^+ \pi^-$ decays,'' Phys.\ Rev.\ D {88, no. 9, 092003 (2013) [arXiv:1302.0551 [hep-ex]]; R.~Aaij {et al. [LHCb Collaboration], %``First measurement of time-dependent $C\!P$ violation in $B^0_s \to K^+K^-$ decays,'' JHEP {1310, 183 (2013) [arXiv:1308.1428 [hep-ex]]. %
1506.02398.extracted_bib
{Li:1994cka % H.~n.~Li and H.~L.~Yu, %``Extraction of V(ub) from decay B ---> pi lepton neutrino,'' % Phys.\ Rev.\ Lett.\ {74, 4388 (1995) % [hep-ph/9409313]; %``PQCD analysis of exclusive charmless B meson decay spectra,'' % Phys.\ Lett.\ B {353, 301 (1995); % H.n. Li, {ibid, 348, 597 (1995); % H.~n.~Li and H.~L.~Yu, %``Perturbative QCD analysis of B meson decays,'' % Phys.\ Rev.\ D {53, 2480 (1996) % [hep-ph/9411308].
1506.02398.extracted_bib
{Aubert:2004cp B.~Aubert {et al. [BaBar Collaboration], %``Ambiguity-free measurement of $\cos(2\beta)$: Time-integrated and time-dependent angular analyses of $B \to J/\psi K \pi$,'' Phys.\ Rev.\ D {71, 032005 (2005) [hep-ex/0411016]; R.~Itoh {et al. [Belle Collaboration], %``Studies of CP violation in B ---> J/psi K* decays,'' Phys.\ Rev.\ Lett.\ {95, 091601 (2005) [hep-ex/0504030]; P.~Krokovny {et al. [Belle Collaboration], %``Measurement of the quark mixing parameter cos(2phi(1)) using time-dependent Dalitz analysis of anti-B0 ---> D [K(s)0 pi+ pi-] h0,'' Phys.\ Rev.\ Lett.\ {97, 081801 (2006) [hep-ex/0605023]; B.~Aubert {et al. [BaBar Collaboration], %``Measurement of $\cos$ 2 beta in $B^0 \to D^{(*)$ h0 Decays with a Time-Dependent Dalitz Plot Analysis of $D \to K^0_{S \pi^{+ \pi^{-$,'' Phys.\ Rev.\ Lett.\ {99, 231802 (2007) [arXiv:0708.1544 [hep-ex]].
1506.02398.extracted_bib
{Chang:1996dw C.~H.~V.~Chang and H.~n.~Li, %``Three - scale factorization theorem and effective field theory,'' Phys.\ Rev.\ D {55, 5577 (1997) [hep-ph/9607214]; T.~W.~Yeh and H.~n.~Li, %``Factorization theorems, effective field theory, and nonleptonic heavy meson decays,'' Phys.\ Rev.\ D {56, 1615 (1997) [hep-ph/9701233].
1506.02398.extracted_bib
{Lu:2000em C.~D.~Lu, K.~Ukai and M.~Z.~Yang, %``Branching ratio and CP violation of B ---> pi pi decays in perturbative QCD approach,'' Phys.\ Rev.\ D {63, 074009 (2001) [hep-ph/0004213]; C.~D.~Lu and M.~Z.~Yang, %``B ---> pi rho, pi omega decays in perturbative QCD approach,'' Eur.\ Phys.\ J.\ C {23, 275 (2002) [hep-ph/0011238]; Z.~T.~Zou, X.~Yu and C.~D.~Lu, %``Nonleptonic two-body charmless B decays involving a tensor meson in the Perturbative QCD Approach,'' Phys.\ Rev.\ D {86, 094015 (2012) [arXiv:1203.4120 [hep-ph]]; Q.~Qin, Z.~T.~Zou, X.~Yu, H.~n.~Li and C.~D.~L��, %``Perturbative QCD study of $B_s$ decays to a pseudoscalar meson and a tensor meson,'' Phys.\ Lett.\ B {732, 36 (2014) [arXiv:1401.1028 [hep-ph]]; W.~F.~Wang, H.~C.~Hu, H.~n.~Li and C.~D.~L��, %``Direct CP asymmetries of three-body $B$ decays in perturbative QCD,'' Phys.\ Rev.\ D {89, no. 7, 074031 (2014) [arXiv:1402.5280 [hep-ph]].
1506.02398.extracted_bib
{Ali:2007ff A.~Ali, G.~Kramer, Y.~Li, C.~D.~Lu, Y.~L.~Shen, W.~Wang and Y.~M.~Wang, %``Charmless non-leptonic $B_s$ decays to $PP$, $PV$ and $VV$ final states in the PQCD approach,'' Phys.\ Rev.\ D {76, 074018 (2007) [hep-ph/0703162 [HEP-PH]].
1506.02398.extracted_bib
{Li:2012nk H.~n.~Li, Y.~L.~Shen and Y.~M.~Wang, %``Next-to-leading-order corrections to $B \to \pi$ form factors in $k_T$ factorization,'' Phys.\ Rev.\ D {85, 074004 (2012) [arXiv:1201.5066 [hep-ph]].
1506.02398.extracted_bib
{Cheng:2014fwa S.~Cheng, Y.~Y.~Fan, X.~Yu, C.~D.~L�� and Z.~J.~Xiao, %``The NLO twist-3 contributions to $B \to \pi$ form factors in $k_{T$ factorization,'' Phys.\ Rev.\ D {89, no. 9, 094004 (2014) [arXiv:1402.5501 [hep-ph]].
1506.02398.extracted_bib
{Khodjamirian:2009ys A.~Khodjamirian, C.~Klein, T.~Mannel and N.~Offen, %``Semileptonic charm decays D ---> pi l nu(l) and D ---> K l nu(l) from QCD Light-Cone Sum Rules,'' Phys.\ Rev.\ D {80, 114005 (2009) [arXiv:0907.2842 [hep-ph]]. %
1506.02398.extracted_bib
{Ball:1998je % P.~Ball, %``Theoretical update of pseudoscalar meson distribution amplitudes of higher twist: The Nonsinglet case,'' % JHEP {9901, 010 (1999) % [hep-ph/9812375].
1506.02398.extracted_bib
{Khodjamirian:2011ub A.~Khodjamirian, T.~Mannel, N.~Offen and Y.-M.~Wang, %``$B \to \pi \ell \nu_l$ Width and $|V_{ub|$ from QCD Light-Cone Sum Rules,'' Phys.\ Rev.\ D {83, 094031 (2011) [arXiv:1103.2655 [hep-ph]].
1506.02398.extracted_bib
{Li:2013xna H.~N.~Li, Y.~L.~Shen and Y.~M.~Wang, %``Joint resummation for pion wave function and pion transition form factor,'' JHEP {1401, 004 (2014) [arXiv:1310.3672 [hep-ph]].
1506.02398.extracted_bib
{Kurimoto:2001zj T.~Kurimoto, H.~n.~Li and A.~I.~Sanda, %``Leading power contributions to B ---> pi, rho transition form-factors,'' Phys.\ Rev.\ D {65, 014007 (2002) [hep-ph/0105003].