# Physical Layer Security in Relay Networks with Outdated Relay Selection

Shahla Mohsenifard

###### Abstract

In this paper, the secrecy performance of a cooperative relay network with
outdated relay selection is investigated where an eavesdropper intercepts the
channels between the source and the destination. The best relay is chosen
among $N$ relays based on the opportunistic relay selection algorithm, which
may not be the best relay in the time of transmission because of the outdated
channel state information. We derive closed-form analytical expressions for
the non-zero secrecy capacity, the secrecy outage probability, and the ergodic
secrecy capacity. Finally, our theoretical analysis is validated by the
numerical results, and detailed discussions and insights are given.

###### Index Terms:

Physical layer security, outdated relay selection, non-zero secrecy capacity,
secrecy outage probability, ergodic secrecy capacity.

##  I Introduction

Wireless communication networks due to their broadcast nature are prone to
various attacks and misuse behavior such as being overheard and intercepted by
eavesdroppers. As such, security issues of wireless communications are of
great importance. Although cryptographic methods can improve the communication
security, as the computational ability and the power of eavesdroppers are
stronger nowadays, it is more and more likely to intercept the network
successfully. As a result, physical layer security has attracted increasing
attention in the literature [1] \- [4]. Relaying is one of the effective
techniques which has received wide attention to extend the coverage area and
improve communication quality in wireless networks [5]. Thus, investigating
the physical layer security of relay networks is crucial. Opportunistic relay
selection is among low complexity cooperative techniques which activate only
the best relay. In selecting the best relay, a delay between relay selection
instant and data transmission instant may cause an outdated channel state
information because of the time-varying nature of fading channels, [6], which
leads to a degradation in the secrecy performance of the network.

There have been several works dealing with secrecy issues in cooperative relay
systems. In [7] AF and DF based optimal relay selection schemes are proposed
to improve the wireless security against eavesdropping, where the authors
consider the exact knowledge of CSI. Authors in [8] analyzed the secrecy
outage probability of a relay network while passive eavesdroppers are
intercepting the channel and partial relay selection scheme is considered to
select the best relay. In [9] secrecy performance of a cooperative network
under the constraint of having outdated CSI is studied. The authors assumed
that relays only to assist the transmission, and there is no direct link
between the source and the destination. S. Al-Qahtani et al. comprehensively
investigate on the secrecy performance of opportunistic relay selection
systems over Rayleigh fading channels. The partial relay selection scheme is
considered, and the secrecy is studied when there is a feedback delay [10]. In
[11] with the aid of joint relay and jammer selection, the physical layer
security of amplify-and-forward relaying networks is increased despite the
effects of channel feedback delay. In this paper, we investigate on secrecy
performance of a cooperative relay network with outdated CSI in the presence
of an eavesdropper. The information transmission is assisted by $N$ relays
from the source to the destination, and a direct link between the source and
the destination is also considered. We study the impact of outdated relay
selection by deriving the analytical expressions for the secrecy performances.

The reminder of the paper is organized as follows. Section II introduces the
system and channel models. In Section III, we derive closed-form expressions
of the SOP, non-zero secrecy capacity, and ergodic secrecy capacity. In
section IV we present simulation results and in section V we conclude this
work.

##  II System Model

Consider a cooperative wireless network consisting of one source, one
destination, and $N$ trusted relays while an eavesdropper is passively
eavesdropping channels without modifying the signals. The source and the
destination communicate over one direct $S$-$D$ channel and $N$ two-hop
channels using relays while all channels experience Rayleigh fading. All nodes
are equipped with a single antenna.

Figure 1: System model.

The thermal noise received at each authored node is modeled as a complex
Gaussian random variable with zero mean and variance $\sigma_{n}^{2}$ i.e.,
$CN(0,\sigma_{n}^{2})$ and the thermal noise at the eavesdropper is a complex
Gaussian random variable with zero mean and variance $\sigma_{e}^{2}$ i.e.,
$CN(0,\sigma_{e}^{2})$ . It is assumed that the channel state information is
known at the destination. The best relay is selected as $b=argmax_{n\in
R}\\{\gamma_{n}\\}$, where
$\gamma_{n}=\min\\{\gamma_{SR_{n}},\gamma_{R_{n}D}\\}$ is an upper bound SNR
of $S-R_{n}-D$ link for relay $n$ and $R=\\{1,2,...,N\\}$. From [12] the total
$SNR$ in the destination can be written as

$\gamma_{opr}=\gamma_{SD}+\max_{n\in{R}}\Big{(}\min\\{\gamma_{SR_{n}},\gamma_{R_{n}D}\\}\Big{)}$
(1)

where $\gamma_{SD}={|h_{SD}|}^{2}{E_{s}}/{\sigma_{n}^{2}}$ is the
instantaneous SNR of $S-D$ link,
$\gamma_{SR_{n}}={|h_{SR{n}}|}^{2}{E_{s}}/{\sigma_{n}^{2}}$, and
$\gamma_{R_{n}D}={|h_{R_{n}D}|}^{2}{E_{s}}/{\sigma_{n}^{2}}$ are the
instantaneous SNRs of $S-R_{n}$ and $R_{n}-D$ links respectively. The total
SNR in eavesdropper is then

$\gamma_{opr,e}=\gamma_{SE}+\min\\{\gamma_{Sb},\gamma_{bE}\\}$ (2)

where $\gamma_{SE}={|h_{SE}|}^{2}{E_{s}}/{\sigma_{e}^{2}}$,
$\gamma_{Sb}={|h_{Sb}|}^{2}{E_{s}}/{\sigma_{e}^{2}}$, and
$\gamma_{bE}={|h_{bE}|}^{2}{E_{s}}/{\sigma_{e}^{2}}$ are $S-E$, $S-b$, and
$b-E$ links’ SNRs respectively. From a practical point of view, some factors
such as channel changes in time, feedback delays, and the channel estimation
errors may cause a difference between the CSI of $S-R_{n}-D$ links required
for the opportunistic relaying and the actual values. In other words, the best
relay selected according to the outdated CSI at time $t$ may not be the best
relay at the time of the data transmission (time $t+\tau$). In [13] a degree
of difference is considered as the power correlation coefficient between the
SNR at time $t$, $\tilde{\gamma_{n}}$, and the SNR at time $t+\tau$,
$\gamma_{n}$, denoted by $\rho$ where $0<\rho<1$.

##  III secrecy capacity, secrecy outage probability, and ergodic secrecy
capacity

###  III-A Non-zero secrecy capacity

This part characterizes the secrecy capacity of the relay network with
outdated Rayleigh channels. The total capacity of the channel between the
source and the destination is

$C_{M}=\frac{1}{2}\mathrm{log}_{2}(1+\gamma_{opr})$ (3)

and the total capacity of the channel between the source and the eavesdropper
is

$C_{E}=\frac{1}{2}\mathrm{log}_{2}(1+\gamma_{opr,e})$ (4)

Mathematically, the instantaneous secrecy capacity can be expressed as [13]

$C_{s}=\begin{cases}\frac{1}{2}(\mathrm{log}_{2}(1+\gamma_{opr})-\mathrm{log}_{2}(1+\gamma_{opr,e}))&\gamma_{opr}>\gamma_{opr,e}\\\
0&\gamma_{opr}<\gamma_{opr,e}\end{cases}$ (5)

We will now consider the existence of a non-zero secrecy capacity. From (5) it
follows that the secrecy capacity is positive when
$\gamma_{opr}>\gamma_{opr,e}$ and is zero when $\gamma_{opr}<\gamma_{opr,e}$.
Considering the independence between the channels, the probability of non-zero
secrecy capacity can be written as

$\begin{split}p(C_{s}>0)&=p(\gamma_{opr}>\gamma_{opr,e})\\\
&=\int_{0}^{\infty}\int_{0}^{\gamma_{opr}}f(\gamma_{opr},\gamma_{opr,e})d\gamma_{opr,e}d\gamma_{opr}\\\
&=\int_{0}^{\infty}\int_{0}^{\gamma_{opr}}f(\gamma_{opr})f(\gamma_{opr,e})d\gamma_{opr,e}d\gamma_{opr}\\\
\end{split}$ (6)

It has been shown in [13] that the probability density function (PDF) of
$\gamma_{opr}$ in the network is

$\begin{split}f_{\gamma_{opr}}(\gamma)&=\sum_{n=1}^{N}{{N}\choose{n}}\frac{(-1)^{n-1}}{\Big{(}\bar{\gamma}_{c}[n(1-\rho)+\rho]/n-\bar{\gamma}_{SD}\Big{)}}\\\
&\times\bigg{[}\textrm{exp}\Big{(}\frac{-n\gamma}{\bar{\gamma}_{c}[n(1-\rho)+\rho]}\Big{)}-\textrm{exp}\Big{(}\frac{-\gamma}{\bar{\gamma}_{SD}}\Big{)}\bigg{]}\end{split}$
(7)

where
${\bar{\gamma}_{c}}=\frac{\bar{\gamma}_{SR_{n}}\bar{\gamma}_{R_{n}D}}{\bar{\gamma}_{SR_{n}}+\bar{\gamma}_{R_{n}D}}$
and $n=1,2,3,...,N$, and the PDF of $\gamma_{opr,e}$ is

$f_{\gamma_{opr,e}}(\gamma)=\frac{1}{\Big{(}\bar{\gamma}_{ce}-\bar{\gamma}_{SE}\Big{)}}\times\bigg{[}\textrm{exp}\Big{(}\frac{-\gamma}{\bar{\gamma}_{ce}}\Big{)}-\textrm{exp}\Big{(}\frac{-\gamma}{\bar{\gamma}_{SE}}\Big{)}\bigg{]}$
(8)

where
${\bar{\gamma}_{ce}}=\frac{\bar{\gamma}_{Sb}\bar{\gamma}_{bE}}{\bar{\gamma}_{Sb}+\bar{\gamma}_{bE}}$.
So the probability of non-zero secrecy capacity can be proved as

$\displaystyle\begin{split}p(C_{s}>0)=&\sum_{n=1}^{N}{{N}\choose{n}}(-1)^{n-1}\Bigg{(}1-\frac{1}{g-{\overline{\gamma}}_{SD}}\times\\\
&\Bigg{[}\frac{{\overline{\gamma}}_{ce}}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\times\Big{(}\frac{1}{\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{ce}}}-\frac{1}{\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{ce}}}\Big{)}-\\\
&\frac{{\overline{\gamma}}_{SE}}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\times\Big{(}\frac{1}{\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{SE}}}-\frac{1}{\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{SE}}}\Big{)}\Bigg{]}\Bigg{)}\end{split}$
(9)

where $g=\bar{\gamma}_{c}[n(1-\rho)+\rho]/n$.

Proof: See Appendix A for the detailed derivation.

###  III-B Secrecy outage probability

Now we characterize the secrecy outage probability (SOP). The SOP is defined
as the probability that the instantaneous secrecy capacity falls below a
predefined secrecy target rate $R_{s}>0$. Thus the SOP of the system can be
considered as

$p_{out}(R_{s})=p(C{s}<R_{s})$ (10)

It is clear that when $C_{M}<C_{E}$, $p_{out}(R_{s})=1$. When $C_{M}>C_{E}$
the SOP can be given as

$\begin{split}p_{out}&(R_{s})=1-\sum^{N}_{n=1}{{N}\choose{n}}(-1)^{n-1}a\times\\\
&\Bigg{(}g\
\mathrm{exp}\Big{(}\frac{1-2^{R_{s}}}{g}\Big{)}\bigg{[}\frac{1}{\frac{2^{R_{s}}}{g}+\frac{1}{{\overline{\gamma}}_{ce}}}-\frac{1}{\frac{2^{R_{s}}}{g}+\frac{1}{{\overline{\gamma}}_{SE}}}\bigg{]}-\\\
&{\overline{\gamma}}_{SD}\
\mathrm{exp}\Big{(}\frac{1-2^{R_{s}}}{{\overline{\gamma}}_{SD}}\Big{)}\bigg{[}\frac{1}{\frac{2^{R_{s}}}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{ce}}}-\frac{1}{\frac{2^{R_{s}}}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{SE}}}\bigg{]}\Bigg{)}\end{split}$
(11)

where
$a=\frac{1}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\frac{1}{g-{\overline{\gamma}}_{SD}}$.

Proof: See Appendix B for the detailed derivation.

###  III-C Ergodic secrecy capacity

The ergodic secrecy capacity serves as another important metric of wireless
fading channel, which is calculated as the average of instantaneous capacity
over $\gamma_{opr}$ and $\gamma_{opr,e}$. The ergodic secrecy capacity is
defined as [6]

$\overline{C}_{s}=\int^{\infty}_{0}{\int^{\infty}_{0}{C_{s}f_{{\gamma{}}_{opr}}({\gamma}_{opr})f_{{\gamma{}}_{opr,e}}({\gamma}_{opr,e}){{d{\gamma}_{opr}}}{{d{\gamma}_{opr,e}}}}}$
(12)

By substituting the equations (5) in (12) we have

$\begin{split}\overline{C}_{s}=&\int^{\infty}_{0}\int^{{\gamma}_{opr}}_{0}\frac{1}{2}\mathrm{log}_{2}(1+{\gamma}_{opr})\times\\\
&f_{{\gamma{}}_{opr}}({\gamma}_{opr})f_{{\gamma{}}_{opr,e}}({\gamma}_{opr,e}){{d{\gamma}_{opr,e}}}{{d{\gamma}_{opr}}}\\\
-&\int^{\infty}_{0}\int^{\infty}_{{\gamma}_{opr,e}}\frac{1}{2}\mathrm{log}_{2}(1+{\gamma}_{opr,e})\times\\\
&f_{{\gamma{}}_{opr}}({\gamma}_{opr})f_{{\gamma{}}_{opr,e}}({\gamma}_{opr,e}){{d{\gamma}_{opr}}}{{d{\gamma}_{opr,e}}}\\\
\end{split}$ (13)

$\displaystyle\begin{split}&\overline{C_{s}}=\frac{1}{2\ln{2}}\frac{1}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\sum_{n=1}^{N}{{N}\choose{n}}\frac{{\left(-1\right)}^{n-1}}{g-{\overline{\gamma}}_{SD}}\Biggr{[}({\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE})\Bigg{(}g\
\mathrm{exp}\mathrm{}\bigg{(}\frac{1}{g}\bigg{)}\mathrm{Ei}\left(\frac{1}{g}\right)-{\overline{\gamma}}_{SD}\
{\mathrm{exp}\left(\frac{1}{{\overline{\gamma}}_{SD}}\right)\
}\mathrm{Ei}\left(\frac{1}{{\overline{\gamma}}_{SD}}\right)\Bigg{)}-\\\
&({\overline{\gamma}}_{ce}+g)\Bigg{(}\frac{1}{\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{ce}}}\mathrm{exp}\mathrm{}\bigg{(}\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{ce}}\bigg{)}\mathrm{Ei}\left(\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{ce}}\right)\Bigg{)}+({\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SD})\left(\frac{1}{\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{ce}}}{\mathrm{exp}\left(\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{ce}}\right)\
}\mathrm{Ei}\left(\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{ce}}\right)\right)+\\\
&({\overline{\gamma}}_{SE}-g)\left(\frac{1}{\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{SE}}}{\mathrm{exp}\left(\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{SE}}\right)}\mathrm{Ei}\left(\frac{1}{g}+\frac{1}{{\overline{\gamma}}_{SE}}\right)\right)-({\overline{\gamma}}_{SE}+{\overline{\gamma}}_{SD})\left(\frac{1}{\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{SE}}}{\mathrm{exp}\left(\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{SE}}\right)\
}\mathrm{Ei}\left(\frac{1}{{\overline{\gamma}}_{SD}}+\frac{1}{{\overline{\gamma}}_{SE}}\right)\right)\Biggl{]}\end{split}$
(14)

So the ergodic secrecy capacity is as in (LABEL:eq:14).

Proof: See Appendix C for the detailed derivation.

##  IV Simulation and Numerical results

In this section, we provide numerical results to validate our analytical
expressions. We consider $N=5$ relays and set the threshold secrecy rate
$R_{s}=2$ bits/s/Hz and all links in the network experience Rayleigh fading.
We assume $\bar{\gamma}_{c}=0.5\bar{\gamma}_{SD}$,
$\bar{\gamma}_{SD}={E_{s}}/{\sigma_{n}}$,
$\bar{\gamma}_{ce}=0.5\bar{\gamma}_{SE}$ and
$\bar{\gamma}_{SE}={E_{s}}/{\sigma_{e}}$.

Figure 2: Non-zero secrecy probability for different values of $\rho$ and
$\bar{\gamma}_{SE}=-5dB$ Figure 3: Non-zero secrecy probability for $\rho=o.5$
and $\bar{\gamma}_{SE}=-5,0,5dB$ Figure 4: Secrecy Outage Probability for
different values of $\rho$ and $\bar{\gamma}_{SE}=-5dB.$ Figure 5: Average
Secrecy Capacity for different values of $\rho$ and $\bar{\gamma}_{SE}=-5$.

Fig. 1 shows the non-zero secrecy probability for different values of
$\rho\in\\{0,0.5,0.9,1\\}$ while $\bar{\gamma}_{SE}=-5dB$. It is clear that as
the value of $\rho$ increases, the probability of non-zero secrecy capacity
enhanced. we can see that when $\bar{\gamma}_{SD}\gg\bar{\gamma}_{SE}$ then
$p(C_{s}>0)\approx 1$. Conversely when $\bar{\gamma}_{SE}\gg\bar{\gamma}_{SD}$
then $p(C_{s}>0)$ decreases and will approximately close to $0$. In Fig. 2,
the non-zero secrecy probability for $\rho=0.5$ and $\gamma_{SE}=\\{-5,0,5\\}$
dB is depicted. It is obvious that the better condition the eavesdropper
channel has (i.e., higher values of $\bar{\gamma}_{SE}$), the lower
probability of non-zero secrecy we get. When the main channel is much better
than eavesdropper’s, i.e., when $\bar{\gamma}_{SD}\gg\bar{\gamma}_{SE}$ for
different values of eavesdropper’s SNR, the probability of non-zero secrecy
capacity is the same and $p(C_{s}>0)\approx 1$. Secrecy outage probability for
various values of $\rho$ ishown in Fig. 3 for $\bar{\gamma}_{SE}=0$ dB. From
the figure, we can see that for bigger values of $\rho$ we get lower secrecy
outage probability as better CSI helps to select the best relay for the secure
transmission. Also, the SOP improves with larger values of average SNRs. Fig.4
depicts SOP for $\rho=0.5$ and $\bar{\gamma}_{SE}=-5,0,5$ dB. It is clear that
the outage probability increases for higher values of $\bar{\gamma}_{SE}$.

Figure 6: Secrecy capacity for different values of $\bar{\gamma}_{SE}$ and
$\rho$=0.5

Fig. 5 demonstrates the average secrecy capacity for four numbers of values
for $\rho$ namely $\rho=0$, $\rho=0.5$, $\rho=0.8$, and $\rho=1$. We can see
that the average secrecy capacity is increasing with the increase in values of
$\rho$. In Fig. 6, the average secrecy capacity considering
$\bar{\gamma}_{SE}=-5$ dB is shown. As for non-zero secrecy capacity, for
different values of $\bar{\gamma}_{SE}$, the average secrecy capacity also
converges to one value as average SNR increases.

Figure 7: Average Secrecy Capacity for different values of $\bar{\gamma}_{SE}$
and $\rho=0.5$.

##  V Conclusion

In this paper, we have investigated the secrecy performance of a relay network
in the presence of an eavesdropper. The effects of outdated CSI, which is
caused because of feedback delays, is considered in the opportunistic relay
selection strategy. The closed-form Analytical expressions of non-zero secrecy
capacity, secrecy outage probability, and average secrecy capacity are
derived. The results show that for when the destination average SNR is high
enough, the ergodic secrecy capacity and the non-zero secrecy capacity
converge to a fixed value regardless of the value of eavesdropper’s average
SNR. The numerical results prove the correctness of our derived formulas.

##  Appendix A Proof of the probability of non-zero capacity

$\begin{split}&p(C_{s}>0)=p({\gamma}_{opr}>{\gamma}_{opr,e})\\\
&=\int^{\infty}_{0}{\int^{{\gamma}_{opr}}_{0}{f_{{\gamma}_{opr}}({\gamma}_{opr})f_{{\gamma}_{opr,e}}({\gamma}_{opr,e}){{d{\gamma}_{opr,e}}}{{d{\gamma}_{opr}}}}}\end{split}$

$\displaystyle\begin{split}&=\int^{\infty}_{0}f({\gamma}_{opr})\bigg{(}\frac{{\overline{\gamma}}_{ce}}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\Big{[}1-\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{{\overline{\gamma}}_{ce}}\Big{)}\Big{]}\\\
&\quad\quad\quad-\frac{{\overline{\gamma}}_{SE}}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\Big{[}1-\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{{\overline{\gamma}}_{SE}}\Big{)}\Big{]}\bigg{)}d{\gamma}_{opr}\\\
&=\sum^{N}_{n=0}{\left(\genfrac{}{}{0.0pt}{}{N}{n}\right)\frac{{\left(-1\right)}^{n-1}}{g-{\overline{\gamma}}_{SD}}}\frac{1}{{\overline{\gamma}}_{ce}-{\overline{\gamma}}_{SE}}\\\
&\quad\quad\times\Bigg{(}{\overline{\gamma}}_{ce}\int^{\infty}_{0}\Big{[}\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{g}\Big{)}-\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{{\overline{\gamma}}_{SD}}\Big{)}\Big{]}\\\
&\quad\quad\quad\quad\quad\quad\quad\times\Big{[}1-\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{{\overline{\gamma}}_{ce}}\Big{)}\Big{]}d{\gamma}_{opr}\\\
&\quad\qquad-{\overline{\gamma}}_{SE}\int^{\infty}_{0}\Big{[}\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{g}\Big{)}-\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{{\overline{\gamma}}_{SD}}\Big{)}\Big{]}\\\
&\quad\qquad\quad\qquad\quad\times\Big{[}1-\mathrm{exp}\Big{(}\frac{-{\gamma}_{opr}}{{\overline{\gamma}}_{SE}}\Big{)}\Big{]}d{\gamma}_{opr}\Bigg{)}\end{split}$
(15)

Calculating the integrals in (LABEL:eq:15) and simplifying it, gives (9).

##  Appendix B Proof of the secrecy outage probability

The SOP can be calculated as

$\displaystyle\begin{split}pr(C_{s}<R_{s})&=pr\Big{(}C_{s}<R_{s}|{}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}p\Big{(}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}\\\
&+pr\Big{(}C_{s}<R_{s}|{}{\gamma{}}_{opr}<{\gamma{}}_{opr,e}\Big{)}p\Big{(}{\gamma{}}_{opr}<{\gamma{}}_{opr,e}\Big{)}\\\
&=pr\Big{(}C_{s}<R_{s}|{}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}p\Big{(}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}\\\
&+1\times{}p\Big{(}{\gamma{}}_{opr}<{\gamma{}}_{opr,e}\Big{)}\end{split}$ (16)

Substituting (5) in (16) gives

$\displaystyle\begin{split}&pr\Big{(}C_{s}<R_{s}|{}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}\\\
&=pr\Big{(}\log_{2}{(1+{\gamma{}}_{opr})}-\log_{2}{(1+{\gamma{}}_{opr,e})}<R_{s}|{}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}\\\
&=pr\Big{(}{\gamma{}}_{opr}<2^{R_{s}}(1+{\gamma{}}_{opr,e})-1|{}{\gamma{}}_{opr}>{\gamma{}}_{opr,e}\Big{)}=\\\
&\int_{0}^{\infty{}}\int_{{\gamma{}}_{opr,e}}^{2^{R_{s}}(1+{\gamma{}}_{opr,e})-1}f_{{\gamma{}}_{opr}}({\gamma{}}_{opr})f_{{\gamma{}}_{opr,e}}({\gamma{}}_{opr,e})d{\gamma{}}_{opr}d{\gamma{}}_{opr,e}\\\
&=p({\gamma{}}_{opr}>{\gamma{}}_{opr,e})-\\\
&\int_{0}^{\infty{}}\int_{2^{R_{s}}(1+{\gamma{}}_{opr,e})-1}^{\infty{}}f_{{\gamma{}}_{opr}}({\gamma{}}_{opr})f_{{\gamma{}}_{opr,e}}({\gamma{}}_{opr,e})d{\gamma{}}_{opr}d{\gamma{}}_{opr,e}\end{split}$
(17)

So the SOP can be written as

$\displaystyle\begin{split}&pr(C_{s}<R_{s})=1-\\\
&\int_{0}^{\infty{}}\int_{2^{R_{s}}(1+{\gamma{}}_{opr,e})-1}^{\infty{}}f_{{\gamma{}}_{opr}}({\gamma{}}_{opr})f_{{\gamma{}}_{opr,e}}({\gamma{}}_{opr,e})d{\gamma{}}_{opr}d{\gamma{}}_{opr,e}\end{split}$
(18)

Calculating this integral results the formula in (11).

##  Appendix C Proof of the average secrecy capacity

The first integral in (13) can be calculated as

$\displaystyle\begin{split}&\int_{0}^{\infty{}}\frac{1}{2}\log_{2}(1+{\gamma{}}_{opr})f_{{\gamma{}}_{opr}}({\gamma{}}_{opr})\frac{1}{{\bar{\gamma{}}}_{ce}-{\bar{\gamma{}}}_{SE}}\Bigg{(}{\bar{\gamma{}}}_{ce}\times{}\\\
&\Big{[}1-\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{ce}}\Big{)}\Big{]}-{\bar{\gamma{}}}_{SE}\Big{[}1-\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{SE}}\Big{)}\Big{]}\Bigg{)}d{\gamma{}}_{opr}\\\
&=\frac{1}{{\bar{\gamma{}}}_{ce}-{\bar{\gamma{}}}_{SE}}\sum_{n=1}^{N}\binom{N}{n}\frac{{(-1)}^{n-1}}{g-{\bar{\gamma{}}}_{SD}}\\\
&\times\Bigg{(}\int_{0}^{\infty{}}{\bar{\gamma{}}}_{ce}\log_{2}(1+{\gamma{}}_{opr})\Big{(}1-\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{ce}}\Big{)}\Big{)}\\\
&\qquad\times{}\exp\Big{(}\frac{-{\gamma{}}_{opr}}{b}\Big{)}d{\gamma{}}_{opr}\\\
&\quad-\int_{0}^{\infty{}}{\bar{\gamma{}}}_{ce}\times{}\log_{2}(1+{\gamma{}}_{opr})\times{}\Big{(}1-\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{ce}}\Big{)}\Big{)}\\\
&\qquad\times{}\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{SD}}\Big{)}d{\gamma{}}_{opr}\\\
&\quad+\int_{0}^{\infty{}}{\bar{\gamma{}}}_{SE}\log_{2}(1+{\gamma{}}_{opr})\Big{(}1-\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{SE}}\Big{)}\Big{)}\\\
&\qquad\times{}\exp\Big{(}\frac{-{\gamma{}}_{opr}}{b}\Big{)}d{\gamma{}}_{opr}\\\
&\quad-\int_{0}^{\infty{}}{\bar{\gamma{}}}_{SE}\log_{2}(1+{\gamma{}}_{opr})\Big{(}1-\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{SE}}\Big{)}\Big{)}\\\
&\qquad\times{}\exp\Big{(}\frac{-{\gamma{}}_{opr}}{{\bar{\gamma{}}}_{SD}}\Big{)}d{\gamma{}}_{opr}\Bigg{)}\end{split}$
(19)

We define $\mathrm{Ei}(x)=\int_{x}^{\infty}\frac{\exp(-t)}{t}dt$. Then we have

$\displaystyle\begin{split}\int_{0}^{\infty{}}\log_{2}(1+x)\exp\Big{(}\frac{-x}{\alpha{}}\Big{)}dx=\alpha{}\log_{2}(e)\exp\Big{(}\frac{1}{\alpha}\Big{)}\mathrm{Ei}\Big{(}\frac{1}{\alpha{}}\Big{)}\end{split}$
(20)

Substituting (20) in (LABEL:eq:19) and calculating the integrals and similarly
calculating the second part of (13), we obtain the formula in (LABEL:eq:14).

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# Quasi-two-body $B_{(s)}\to V\pi\pi$ Decays with Resonance $f_{0}(980)$ in
PQCD Approach

Lei Yang1 Zhi-Tian Zou1 Ying Li1,2<EMAIL_ADDRESS>Xin Liu3 Cui-Hua Li4 $1$
Department of Physics, Yantai University, Yantai 264005, China
$2$ Center for High Energy Physics, Peking University, Beijing 100871, China
$3$ Department of Physics, Jiangsu Normal University, XuZhou 221116, China
$4$ Yantai Engineering and Technology College,Yantai, 264006, China

###### Abstract

Motivated by the measurements of branching fractions of the quasi-two-body
decays $B^{0}\to K^{*0}(f_{0}(980)\to)\pi^{+}\pi^{-}$ and
$B^{0}\to\rho^{0}(f_{0}(980)\to)\pi^{+}\pi^{-}$, we study the charmless $B\to
V(f_{0}(980)\to)\pi^{+}\pi^{-}$ decays in the perturbative QCD approach.
Supposing that $f_{0}(980)$ is a two-quark state and mixture of
$n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$ and $s\bar{s}$ with the mixing angle
$\theta$, we calculate the branching fractions of these decays with the new
introduced $s$-wave $\pi^{+}\pi^{-}$-pair wave function. When the mixing angle
$\theta$ lies in the range $[135^{\circ},155^{\circ}]$, the calculated
branching fractions of the $B^{0}\to K^{*0}f_{0}(980)\to K^{*0}\pi^{+}\pi^{-}$
and $B^{0}\to\rho^{0}f_{0}(980)\to\rho^{0}\pi^{+}\pi^{-}$ decays are in
agreement with the experimental data. The branching fractions of other decays
could be measured in the current LHCb and Belle II experiments. Considering
the isospin symmetry, we also estimate the branching fractions of the quasi-
two-body decays $B\to Vf_{0}(980)\to V\pi^{0}\pi^{0}$, which are half of those
the corresponding decays $B\to Vf_{0}(980)\to V\pi^{+}\pi^{-}$. Moreover, the
direct $CP$ asymmetries of these decays are also calculated, and some of them
can be tested in the current experiments.

###### pacs:

13.25.Hw, 12.38.Bx

##  I Introduction

In recent years, non-leptonic three-body decays of $B$ mesons have been paid
more attention on both experimental and theoretical sides, as these decays can
be used to test the standard model (SM), to extract the CKM angles, and to
search for the sources of the $CP$ violation. From last century, a large
number of three-body $B$ decays have been measured by BaBar Lees:2012kxa ,
Belle Nakahama:2010nj , CLEO Eckhart:2002qr  and LHCb Aaij:2018rol ;
Aaij:2016qnm ; Aaij:2019nmr ; Aaij:2017zgz ; Aaij:2019jaq ; Aaij:2020ypa ;
Aaij:2020dsq . Meanwhile, on the factorization hypothesis, a few of
theoretical methods have been proposed to study these decays, such as
approaches based on the symmetry principle He:2014xha , the QCD factorization
approachKrankl:2015fha ; Virto:2016fbw ; Cheng:2007si ; Cheng:2016shb ;
Li:2014oca , the perturbative QCD approach (PQCD) Wang:2016rlo ; Li:2016tpn ;
Zou:2020atb ; Zou:2020fax ; Zou:2021lex ; Zou:2020dpg , and other theoretical
methods Wang:2015ula .

Unlike the two-body decays where the kinematics is fixed, three-body decay
amplitudes depend on two kinematic variables. For a decay $B(p_{B})\to
M_{1}(p_{1})M_{2}(p_{2})M_{3}(p_{3})$, it is general to define the variables
as two invariant masses of two pairs of final state particles, for instance,
$s_{12}$ and $s_{13}$ with the definition $s_{ij}=2(p_{i}\cdot
p_{j})/m_{B}^{2}$. All physical kinematics configurations could define a two-
dimensional region in the $s_{12}-s_{13}$ plane, and the density plot of the
differential decay rate $d\Gamma/ds_{12}ds_{13}$ in this region is called a
Dalitz plot. Specially, when the final states are the light mesons such as the
$\pi$ and $K$ mesons, the corresponding configuration reduces to a triangle
region. In general, the Dalitz plot has three typical regions according to the
characteristic kinematics. The central region so-called “Mercedes Star”
configuration corresponds to the case where all the invariant masses are
roughly the same and of order of $m_{B}$. In this region all three light
mesons have a large energy in the $B$ meson rest frame and fly apart at about
$120^{\circ}$ angles. The corners regions correspond to the cases where one
final state is soft and the others fly back-to-back with large energy about
$m_{B}/2$. At the edges of the Dalitz plot one invariant mass is small and the
other two are large, which implies that two particles move collinearly and the
third bachelor particle recoils back. The interactions between two collinear
mesons leads eventually to the resonances. Compared with two other regions,
the physics picture at the edges of the Dalitz plot is very similar to a two-
body decay by viewing the two-meson pair as a whole, and we thus call it
quasi-two-body decay. In the past twenty years, PQCD approach based on the
$k_{T}$ factorization has been used to study the $B$ meson two-body decays
successfully, therefore it can be generalized for studying the quasi-two-body
decays.

In the past few years, a large number of charmless quasi-two-body $B/B_{s}\to
V(f_{0}(980)\to)\pi^{+}\pi^{-}$ decays have been measured in the experiments
Zyla:2020zbs , and some branching fractions or upper limits of them are
summarized as follows

$\displaystyle B^{0}\to K^{*}(892)^{0}f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-}$
$\displaystyle=$ $\displaystyle(3.9_{-1.8}^{+2.1})\times 10^{-6},$
$\displaystyle B^{0}\to\omega f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-}$
$\displaystyle<$ $\displaystyle 1.5\times 10^{-6},$ $\displaystyle
B^{0}\to\phi f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-}$ $\displaystyle<$
$\displaystyle 3.8\times 10^{-7},$ $\displaystyle
B^{0}\to\rho^{0}f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-}$ $\displaystyle=$
$\displaystyle(7.8\pm 2.5)\times 10^{-7},$ $\displaystyle
B^{+}\to\rho^{+}f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-}$ $\displaystyle<$
$\displaystyle 2.0\times 10^{-6},$ $\displaystyle B_{s}\to\phi
f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-}$ $\displaystyle=$
$\displaystyle(1.12\pm 0.21)\times 10^{-6}.$ (1)

Except the decay $B_{s}\to\phi f_{0},f_{0}\to\pi^{+}\pi^{-}$, other decays
have not been studied theoretically in the literatures. Motivated by this, we
shall study above decays in PQCD approach, so as to further check the
reliability of PQCD in multi-body decays and present more predictions.For the
sake of convenience $f_{0}(980)$ is abbreviated to $f_{0}$ in the following
context unless special statement.

##  II Framework

In the framework of PQCD, the decay amplitude $\mathcal{A}$ of $B\to
V(f_{0}(980)\to)\pi^{+}\pi^{-}$ decay can be decomposed as the convolution

$\displaystyle\mathcal{A}=C(t)\otimes\mathcal{H}(x_{i},b_{i},t)\otimes\Phi_{B}(x_{1},b_{1})\otimes\Phi_{V}(x_{2},b_{2})\otimes\Phi_{\pi\pi}(x_{3},b_{3})\otimes
e^{-S(t)},$ (2)

where $x_{i}$ are the momentum fractions of the light quarks, $b_{i}$ are the
conjugate variables of the quarks’ transverse momenta $k_{iT}$. $\Phi_{B}$ and
$\Phi_{V}$ are the wave functions of the $B$ mesons and vector mesons, while
the $\Phi_{\pi\pi}$ is the $S$-wave $\pi\pi$-pair wave function. These wave
functions are non-perturbative and universal. The exponential term is the so-
called Sudakov form factor caused by the additional scale introduced by the
intrinsic transverse momenta $k_{T}$, which suppresses the soft dynamics
effectively Li:2001ay ; Lu:2000hj . $\mathcal{H}(x_{i},b_{i},t)$ is the hard
kernel, which can be calculated perturbatively. The parameter $t$ is the
largest scale in the hard kernel, which ensures the higher order corrections
as small as possible.

In PQCD, the most important inputs are the initial and final mesons’ wave
functions. For the $B$ meson and the light vector mesons, their wave functions
have been studied extensively and the inner parameters have been fixed by the
well measured two-body $B$ meson decays Liu:2019ymi ; Li:2004ep , so we will
not discuss them in this work. For the $\pi\pi$-pair, its $S$-wave wave
function can be written as Diehl:1998dk ; Diehl:2000uv ; Pire:2002ut ;
Xing:2019xti ; Wang:2018xux

$\displaystyle\Phi_{\pi\pi}=\frac{1}{\sqrt{2N_{c}}}\left[P\mkern-10.5mu/\phi_{S}(z,\xi,\omega)+\omega\phi_{S}^{s}(z,\xi,\omega)+\omega(n\mkern-10.5mu/v\mkern-10.5mu/-1)\phi_{S}^{t}(z,\xi,\omega)\right],$
(3)

with $z$ being the momentum fraction of the light quark in the $\pi\pi$-pair.
The parameter $\xi$ is the momentum fraction of one $\pi$ meson in the
$\pi\pi$-pair. The momentum of the $\pi\pi$-pair $P$ satisfies the condition
$P^{2}=\omega^{2}$, $\omega$ being the invariant mass of $\pi\pi$-pair.
$n=(1,0,\vec{0})$ and $v=(0,1,\vec{0})$ are the light-like vectors. For the
explicit expressions of the light-core distributions $\phi_{S}^{(s,t)}$, we
adopt the form Wang:2014ira ; Wang:2015uea

$\displaystyle\phi_{S}(z,\xi,\omega)=\frac{F_{S}(\omega)}{\sqrt{6}}9a_{s}z(1-z)(2z-1),$
$\displaystyle\phi_{S}^{s}(z,\xi,\omega)=\frac{F_{S}(\omega)}{2\sqrt{6}},$
$\displaystyle\phi_{S}^{t}(z,\xi,\omega)=\frac{F_{S}(\omega)}{2\sqrt{6}}(1-2z),$
(4)

with Gegenbauer moment $a_{s}=0.3\pm 0.2$ Xing:2019xti . $F_{S}(\omega)$ is
the time-like form factor. In particular, for a narrow intermediate resonance,
the time-like form factor $F_{S}(\omega)$ can be well described by the
relative Breit-Wigner lineshape Back:2017zqt . However, due to the remarkable
interference between two decays $f_{0}\to\pi\pi$ and $f_{0}\to K\overline{K}$,
the relative Breit-Wigner lineshape cannot work well for the time-like form
factor of $f_{0}$. In this case, the Flatt$\acute{e}$ lineshape is proposed to
describe that of $f_{0}$ Flatte:1976xu ; Back:2017zqt , which is given as

$\displaystyle
F_{S}(\omega)=\frac{m_{f_{0}}^{2}}{m^{2}_{f_{0}}-\omega^{2}-im_{f_{0}}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK}F^{2}_{KK})},$
(5)

with

$\displaystyle\rho_{\pi\pi}=\sqrt{1-\frac{4m^{2}_{\pi}}{\omega^{2}}},\,\,\,\rho_{KK}=\sqrt{1-\frac{4m_{K}^{2}}{\omega^{2}}}.$
(6)

The $g_{\pi\pi}$ and $g_{KK}$ are the coupling constants corresponding to
$f_{0}\to\pi\pi$ and $f_{0}\to K\overline{K}$ decays, respectively, whose
values are taken as $g_{\pi\pi}=(0.165\pm 0.018)~{}\mathrm{GeV}^{2}$ and
$g_{KK}/g_{\pi\pi}=4.21\pm 0.33$ Back:2017zqt . In addition, the factor
$F_{KK}=e^{-\alpha q^{2}}$ is introduced to suppress the $f_{0}$ width above
the $KK$ threshold. The parameter $\alpha$ is taken $2.0\pm
1.0~{}\mathrm{GeV}^{-2}$, which does not affect the predictions remarkably
Aaij:2014emv . It is noted that this lineshape has been also adopted
extensively in analyzing data in the LHCb experiment Aaij:2014emv .

Although the quark model has achieved great successes, the underlying
structures of the scalar mesons are not well established so far. There are
many scenarios for the classification of the scalar mesons. One scenario is
the naive 2-quark model, and the light scalar mesons below or near 1 GeV are
identified as the lowest lying states. Another consistent picture Close:2002zu
provided by the data implies that light scalar mesons below or near 1 GeV can
be described by the $q^{2}\bar{q}^{2}$, while scalars above 1 GeV will form a
conventional $q\bar{q}$ nonet with with some possible glue content
Cheng:2005ye ; Jaffe:1976ig ; Alford:2000mm . This picture can be used to
interpret the mass degeneracy of $f_{0}$ and $a_{0}(980)$, the reason why the
widths of $\kappa(800)$ and $\sigma(600)$ is broader than those of
$a_{0}(980)$ and $f_{0}$, and the large couplings of $f_{0}$ and $a_{0}(980)$
to $K\overline{K}$. However, in practice it is hard for us to make
quantitative predictions on $B$ decays based on the four-quark picture for
light scalar mesons as it involves the unknown form factors and decay
constants that are beyond the conventional quark model. Hence, we here only
discuss the two-quark scenario for $f_{0}$. Moreover, some experimental
evidences indicate the existence of the non-strange and strange quark contents
in $f_{0}$, we therefore regard it as a mixture of $s\bar{s}$ and
$n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$

$\displaystyle|f_{0}\rangle$ $\displaystyle=$
$\displaystyle|n\bar{n}\rangle\sin\theta+|s\bar{s}\rangle\cos\theta,$ (7)

where $\theta$ is the mixing angle. Recent studies Cheng:2002ai ;
Anisovich:2002wy ; Gokalp:2004ny  show that the mixing angle $\theta$ lies in
the ranges of $25^{\circ}<\theta<40^{\circ}$ and
$140^{\circ}<\theta<165^{\circ}$, and studies based on the $B$ decays favor
the later range.

With the initial and final wave functions, we can calculate the whole
amplitude of each decay mode in PQCD approach. In the leading order, the
diagrams contributing to the decay $B^{+}\to\rho^{+}\pi^{+}\pi^{-}$ are shown
in the Fig.1. The first two diagrams are the emission type diagrams with the
first one emitting the $\pi\pi$-pair and the second one with the vector meson
emitted. The last two are the annihilation type diagrams. Because the decay
amplitudes are very similar to those presented in the ref.Zou:2020dpg , for
the sake of simplicity, we shall not present them in this work. The other
parameters used in the numerical calculations, such as the mass of the mesons,
CKM matrix elements and the life times of $B$ mesons, are taken from the
Particle Data Group Zyla:2020zbs .

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

##  III Numerical Results

As aforementioned, the mixing angle of $f_{0}$ have not yet been determined.
At first, we set $\theta$ as a free parameter and plot the branching fractions
of $B_{u,d,s}\to Vf_{0}\to V\pi^{+}\pi^{-}$ decays dependent on it in Fig. 2,
where the green bands are the allowed regions in the experiments. Combining
the experimental data of $B^{0}\to\rho^{0}(f_{0}\to)\pi^{+}\pi^{-}$ and
$B^{0}\to K^{*}(892)^{0}(f_{0}\to)\pi^{+}\pi^{-}$ decays, we get the range of
the mixing angle $135^{\circ}\leq\theta\leq 155^{\circ}$, which is consistent
with the results obtained from $\phi\to f_{0}\gamma$ and
$f_{0}\to\gamma\gamma$. In our previous work Li:2019jlp , the decays $B\to
K^{*}_{0,2}(1430)f_{0}/\sigma$ have been investigated, and we obtained
$\theta\approx 145^{\circ}$ after comparing with experimental result
Lees:2011dq . If the mixing angle $\theta=145^{\circ}$ is adopted, the
branching ratios of the $B^{0}\to\rho^{0}(f_{0}\to)\pi^{+}\pi^{-}$ and
$B^{0}\to K^{*}(892)^{0}(f_{0}\to)\pi^{+}\pi^{-}$ decays are given as

$\displaystyle\mathcal{B}(B^{0}\to\rho^{0}(770)(f_{0}\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle 8.25\times 10^{-7},$ (8)
$\displaystyle\mathcal{B}(B^{0}\to K^{*}(892)^{0}(f_{0}\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle 2.45\times 10^{-6},$ (9)

which well match the experimental measurements:

$\displaystyle\mathcal{B}(B^{0}\to\rho^{0}(770)(f_{0}\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(7.8\pm 2.5)\times 10^{-7},$ (10)
$\displaystyle\mathcal{B}(B^{0}\to K^{*}(892)^{0}(f_{0}\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(2.6_{-1.2}^{+1.4})\times 10^{-6}.$ (11)

In view of this, we present all calculated results of the $CP$-averaged
branching fractions and the local direct $CP$ asymmetries of the concerned
decay modes with $\theta=145^{\circ}$ in Table. 1. For comparison, the
available experimental data are also listed. One can find that adopting the
appropriate wave functions of initial and final states, our predictions are in
good agreement with the current experimental data, although there are only
upper limits for the $B^{0}\to\omega(f_{0}\to)\pi^{+}\pi^{-}$ and
$B^{+}\to\rho^{+}(f_{0}\to)\pi^{+}\pi^{-}$ decays. Finally, we plot all the
branching fractions dependent on the mixing angle $\theta$, which may shed
light on the mixing angle by combining the ongoing experimental measurements.

Table 1: The results of $CP$ averaged branching fractions (in $10^{-6}$) and the direct $CP$ asymmetries (%) in PQCD approach. Decay Modes | Br(PQCD) | Br(EXP) Zyla:2020zbs  | $A_{CP}^{dir}$
---|---|---|---
$B^{0}\to\rho^{0}(f_{0}\to)\pi^{+}\pi^{-}$ | $0.82^{+0.36+0.02+0.05}_{-0.34-0.16-0.10}$ | $0.78\pm 0.25$ | $-11.4_{-3.54-8.72-0.00}^{+14.0+23.6+9.77}$
$B^{0}\to K^{*0}(f_{0}\to)\pi^{+}\pi^{-}$ | $2.45^{+0.66+0.56+0.00}_{-1.28-1.04-0.35}$ | $2.6_{-1.2}^{+1.4}$ | $-5.97_{-0.57-0.00-0.00}^{+13.7+6.72+2.82}$
$B^{0}\to\omega(f_{0}\to)\pi^{+}\pi^{-}$ | $0.97^{+0.51+0.16+0.13}_{-0.39-0.19-0.10}$ | $<1.5$ | $-13.7_{-3.15-14.9-0.00}^{+6.10+11.5+1.99}$
$B^{+}\to\rho^{+}(f_{0}\to)\pi^{+}\pi^{-}$ | $1.23^{+0.50+0.25+0.00}_{-0.76-0.38-0.12}$ | $<2.0$ | $-55.9_{-19.5-4.34-7.33}^{+15.4+31.5+2.50}$
$B^{+}\to K^{*+}(f_{0}\to)\pi^{+}\pi^{-}$ | $3.18^{+0.94+0.76+0.00}_{-1.48-1.11-0.39}$ | … | $-26.4_{-6.34-1.21-3.05}^{+9.87+4.18+3.05}$
$B_{s}\to\rho^{0}(f_{0}\to)\pi^{+}\pi^{-}$ | $0.06_{-0.02-0.01-0.00}^{+0.02+0.01+0.00}$ | … | $8.82^{+4.23+1.68+0.00}_{-3.51-0.72-0.02}$
$B_{s}\to\omega(f_{0}\to)\pi^{+}\pi^{-}$ | $0.17_{-0.09-0.04-0.02}^{+0.10+0.02+0.00}$ | … | $14.8_{-13.1-8.17-0.53}^{+9.43+11.1+2.56}$
$B_{s}\to\bar{K}^{*0}(f_{0}\to)\pi^{+}\pi^{-}$ | $0.15^{+0.09+0.05+0.01}_{-0.04-0.02-0.00}$ | … | $87.7_{-22.1-14.3-7.51}^{+0.00+0.00+0.00}$

Figure 2: Dependence of the $CP$-averaged branching fractions of the quasi-
two-body $B_{(s)}\to V\pi^{+}\pi^{-}$ decays on the mixing angle $\theta$ in
the PQCD approach. The $\theta$ dependence of the branching fractions. The
green bands are the allowed regions in the experiments.

We acknowledge that there are many uncertainties in our calculations. In this
work, we mainly evaluate three kinds of errors, namely the parameters in wave
functions of the initial and final states, the higher order and power
corrections, and the CKM matrix elements, respectively. In fact, the first
kind of errors come from the uncertainties of nonperturbative parameters, such
as the $B_{(s)}$ meson decay constants $f_{B}/f_{B_{s}}=0.19\pm 0.02/0.23\pm
0.02~{}\mathrm{GeV}$, the sharp parameters $\omega/\omega_{s}=0.4\pm
0.04/0.5\pm 0.05~{}\mathrm{GeV}$ in the distribution amplitudes of $B$ mesons,
the Gegenbauer moments in the LCDAs of vector mesons, and the Gagenbauer
moment $a_{S}$ in the $S$-wave LCDAs of the $\pi\pi$-pair, et.al. We can find
from the table that this kind of uncertainties are dominant. Fortunately, this
kind of uncertainties could be reduced with the developments of the
experiments or other nonperturbative theoretical approaches in future. The
second kind of errors arise from the unknown higher order of $\alpha_{s}$ and
higher power corrections, which are reflected by varying the
$\Lambda_{QCD}=0.25\pm 0.05~{}\mathrm{GeV}$ and the factorization scale $t$
from $0.8t$ to $1.2t$, respectively. The last ones are caused by the
uncertainties of the CKM matrix elements.

If the narrow-width approximation (NWA) holds in these decays, the branching
fraction of the quasi-two-body $B$ meson decay can be decomposed as

$\displaystyle\mathcal{B}(B\to M_{1}(R\to)M_{2}M_{3})\simeq\mathcal{B}(B\to
M_{1}R)\times\mathcal{B}(R\to M_{2}M_{3}),$ (12)

with $R$ represents a resonance. If two decays have a same resonance, we then
define a ratio as

$\displaystyle R_{V_{1}/V_{2}}=\frac{\mathcal{B}(B\to
V_{1}R)}{\mathcal{B}(B\to V_{2}R)}=\frac{\mathcal{B}(B\to
V_{1}R)\times\mathcal{B}(R\to\pi^{+}\pi^{-})}{\mathcal{B}(B\to
V_{2}R)\times\mathcal{B}(R\to\pi^{+}\pi^{-})}\simeq\frac{\mathcal{B}(B\to
V_{1}(R\to)\pi^{+}\pi^{-})}{\mathcal{B}(B\to V_{2}(R\to)\pi^{+}\pi^{-})}.$
(13)

Based on the predictions in Table. 1, the ratio between the
$B^{0}\to\rho^{0}f_{0}$ and $B^{0}\to\omega f_{0}$ is given as

$\displaystyle
R_{\rho^{0}/\omega}=\frac{\mathcal{B}(B\to\rho^{0}f_{0})}{\mathcal{B}(B\to\omega
f_{0})}\sim 1,$ (14)

which is agreement with the results of QCDF Cheng:2013fba . Within the isospin
relation

$\displaystyle
r=\frac{\mathcal{B}(f_{0}\to\pi^{+}\pi^{-})}{\mathcal{B}(f_{0}\to\pi^{0}\pi^{0})}=2,$
(15)

we obtain the relation

$\displaystyle\mathcal{B}(B\to
V(f_{0}\to)\pi^{0}\pi^{0})=\frac{1}{2}\mathcal{B}(B\to
V(f_{0}\to)\pi^{+}\pi^{-}),$ (16)

which can be used to predict the branching fractions of the corresponding
quasi-two-body $B\to V(f_{0}\to)\pi^{0}\pi^{0}$ decays.

Now we turn to discuss the local direct $CP$ asymmetries of these decays. In
the quark level, $B\to K^{*}(f_{0}\to)\pi^{+}\pi^{-}$,
$B_{s}\to\rho^{0}/\omega(f_{0}\to)\pi^{+}\pi^{-}$, and
$B_{s}\to\phi(f_{0}(980)\to)\pi^{+}\pi^{-}$ are induced by
$\bar{b}\to\bar{s}q\bar{q}$ transition, while
$B\to(\rho,\omega)(f_{0}\to)\pi^{+}\pi^{-}$ and $B_{s}\to
K^{*}(f_{0}(980)\to)\pi^{+}\pi^{-}$ are controlled by
$\bar{b}\to\bar{d}q\bar{q}$ transition. From Table. 1, it is found that the
local $CP$ asymmetries of decays $B^{0}\to K^{*0}(f_{0}\to)\pi^{+}\pi^{-}$ and
$B_{s}\to\rho^{0}/\omega(f_{0}\to)\pi^{+}\pi^{-}$ are very small, and the
reason is that the tree diagrams contributions are both color and CKM elements
suppressed. However, for the decay $B^{+}\to K^{*+}(f_{0}\to)\pi^{+}\pi^{-}$,
because the spectator $u$ quark enters into not only $K^{*+}$ meson but also
$\pi\pi$-pair, the contributions from tree and penguin operators are
comparable, leading to a large $CP$ asymmetry. For the decays
$B^{0}\to\rho^{0}/\omega(f_{0}\to)\pi^{+}\pi^{-}$, although the contributions
from tree operators are color suppressed, the destructive interference between
the diagrams with vector meson emitted and ones with $\pi\pi$-pair emitted
decreases the effects of penguin operators remarkably, therefore the $CP$
asymmetries are as small as about $10\%$. For
$B^{+}\to\rho^{+}(f_{0}\to)\pi^{+}\pi^{-}$ decay, its amplitude is more
complicated. In eq. (7), if the mixing angle $\theta$ of $f_{0}(980)$ is an
obtuse angle, the sign of $n\bar{n}$ is negative. The spectator $u$-quark of
$B^{+}\to\rho^{+}(f_{0}\to)\pi^{+}\pi^{-}$ can enter into both $f_{0}$ and
$\pi\pi$-pair, so the negative sign leads to the cancellation between two tree
operators contributions. With the sizable contributions of penguin operators,
the $CP$ asymmetry of this decay is as large as $-55\%$. The decay of
$B_{s}\to\bar{K}^{*0}(f_{0}\to)\pi^{+}\pi^{-}$ is very similar to
$B^{+}\to\rho^{+}(f_{0}\to)\pi^{+}\pi^{-}$, but the spectator is a strange
quark. When the spectator enters into the kaon, both the tree and penguin
operators contribute, and the tree operators are color suppressed. However,
when it enters into the $\pi\pi$-pair, only penguin operators play roles. Due
to large interference between two kinds of above contributions, the large $CP$
asymmetry in $B_{s}\to\bar{K}^{*0}(f_{0}\to)\pi^{+}\pi^{-}$ is reasonable. On
the experimental side, these $CP$ asymmetries have not been measured, and we
hope these predictions can be tested in future.

##  IV Summary

In this work, we investigated the quasi-two-body $B/B_{s}\to
V(f_{0}(980)\to)\pi^{+}\pi^{-}$ decays in PQCD approach, assuming that
$f_{0}(980)$ is a mixture of $n\bar{n}=(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and
$s\bar{s}$ with the mixing angle $\theta$. Within the $S$-wave two-pion wave
function, both the branching fractions and the located $CP$ asymmetries have
been calculated. When the mixing angle $\theta$ is around $145^{\circ}$, the
obtained branching fractions of the
$B^{0}\to\rho^{0}(f_{0}(980)\to)\pi^{+}\pi^{-}$ and $B^{0}\to
K^{*0}(f_{0}(980)\to)\pi^{+}\pi^{-}$ are in good agreement with the
experimental data, and other results could be tested in the future
experiments. In addition, the branching fractions of $B/B_{s}\to
V(f_{0}\to)\pi^{0}\pi^{0}$ could be predicted based on the isospin symmetry,
which can be measured in the LHCb and Belle II experiments. We note that the
calculated $CP$ asymmetries of the $B^{+}\to\rho^{+}(f_{0}\to)\pi^{+}\pi^{-}$
and $B_{s}\to\bar{K}^{*0}(f_{0}\to)\pi^{+}\pi^{-}$ decays are very large,
which can be tested in the ongoing experiments. We acknowledge that there are
many uncertainties in the calculation, and the dominant one is the two-meson
wave function. Therefore, more precise multi-meson wave functions from non-
perturbative approach are needed.

## Acknowledgment

This work is supported in part by the National Science Foundation of China
under the Grant Nos. 11705159, 11975195, 11875033, and 11765012, and the
Natural Science Foundation of Shandong province under the Grant No.ZR2019JQ04.
This work is also supported by the Project of Shandong Province Higher
Educational Science and Technology Program under Grants No. 2019KJJ007.

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# Determination of Lattice QCD Equation of state at a finite chemical
potential

Sabarnya Mitra

###### Abstract

Exponential resummation to all orders in $\mu_{B}$ is a promising scheme which
can capture contributions to all orders in $\mu_{B}$ by considering only the
first few Taylor coefficients in the Taylor expansion of thermodynamic
observables in lattice QCD. This approach however, gets affected by biased
estimates of $n$-point correlation functions $D_{n}$ whose effects can
significantly hinder probe of finite density QCD. We present an unbiased
exponential resummation formalism, which can reproduce the Taylor series up to
the desired order in $\mu_{B}$, besides retaining the original form of the
all-order resummation estimate of QCD partition function.

##  1 Introduction

The Quantum Chromodynamics (QCD) phase diagram [1, 2, 3] is an intriguing
spectacle of QCD, offering deep insights about the different phases and
properties of strongly interacting matter. The determination of QCD Equation
of state (EoS) at finite temperature and density [4, 5, 6, 7] is pivotal for
investigating this phase diagram. One significant hurdle in knowing QCD EoS is
the fermion sign problem in QCD partition function and the Taylor expansion
method of thermodynamic observables is one of the numerous approaches [8, 9,
10, 11] adopted to circumvent this problem in lattice QCD. Due to
computational and precisional difficulties regarding calculation of higher
order Taylor coefficients, one turns towards exponential resummation. In this
approach, one expresses the partition function in an exponential form,
exponentiating the contributions of first $N$ baryon correlation functions and
resumming to all orders in $\mu_{B}$. Despite this improvement, this formalism
gets affected by stochastic bias, which can radically misdirect results and
consequent implications. Although our work of cumulant expansion [12] promises
to eliminate this bias order-by-order, this however caused us losing the
invaluable all-order resummation altogether. In this work, I will present the
formalism of unbiased exponential resummation, which eliminates stochastic
bias and replicates the Taylor series completely upto a desired order in
$\mu_{B}$.

##  2 Formalism : A brief note

Using a total of $\text{N}_{\text{R}}$ Gaussian random volume sources for
every gauge configuration at a given temperature $T$, the exponential resummed
estimate of the QCD partition function to $\mathcal{O}(\mu_{B}^{N})$ resembles

$\mathcal{Z}_{N}^{\text{R}}=\left\langle\text{Re}\left[\exp\left(\sum_{n=1}^{N}\mathbf{D}_{n}\left(\frac{\mu_{B}}{T}\right)^{n}\right)\right]\right\rangle,\hskip
8.53581pt\text{where}\hskip
17.07164pt\mathbf{D}_{n}=\frac{1}{{\text{N}}_{\text{R}}}\sum_{r=1}^{\text{N}_{\text{R}}}D_{n}^{(r)}$
(1)

On expansion in orders of $\mu_{B}$, the above resummed estimate of
$\mathcal{Z}$ in Eqn.(1) gives rise to biased estimates [12] of
$\mathbf{D}_{n}$ in which the estimate of a given random source among all in
the distribution, is raised to higher non-linear powers. In the unbiased
estimate, all the estimates of the sample $D_{n}^{(r)},1\leq r\leq N_{R}$ are
raised to linear powers providing equal importance to all the random
estimates. It is important to eliminate these biased estimates for obtaining
correct results. We present the formalism of unbiased exponential resummation
in chemical potential ($\mu_{B}$) and cumulant ($X$) bases. In $\mu_{B}$ and
$X$ bases, we have

$\displaystyle\frac{\Delta P_{N}^{\text{u},\mu_{B}}}{T^{4}}$
$\displaystyle=\frac{1}{VT^{3}}\hskip 2.84526pt\ln\hskip
2.84526pt\mathcal{Z}_{N}^{\text{u},\mu_{B}},\hskip
5.69054pt\mathcal{Z}_{N}^{\text{u},\mu_{B}}=\left\langle
e^{A_{N}(\mu_{B})}\right\rangle,\hskip
5.69054ptA_{N}(\mu_{B})=\sum_{n=1}^{\boldsymbol{N}}\left(\frac{\mu_{B}}{T}\right)^{n}\frac{\mathcal{C}_{n}}{n!}$
(2) $\displaystyle\frac{\Delta P_{N}^{\text{u},X}}{T^{4}}$
$\displaystyle=\frac{1}{VT^{3}}\hskip 2.84526pt\ln\hskip
2.84526pt\mathcal{Z}_{N}^{\text{u},X},\hskip
5.69054pt\mathcal{Z}_{N}^{\text{u},X}=\left\langle
e^{Y_{N}^{M}(X)}\right\rangle,\hskip
5.69054ptY_{N}^{M}(X)=\sum_{n=1}^{\boldsymbol{M}}\frac{\mathcal{L}_{n}(X_{N})}{n!}$
(3)

The detailed expressions of $C_{n}$ and $\mathcal{L}_{n}$ in above Eqns.(2)
and (3) are given in Ref. [12]. Our results have been obtained using $N=2,4$
and $M=4$.

##  3 Results

This analysis has been performed in $2+1$ flavor QCD using Highly Improved
Staggered Quarks action and a Symanzik-improved gauge action with $100$K gauge
configurations. Fig.1 indicates that the stochastic bias is highly dominant at
$135$ MeV [13], and progressively reduces with increasing temperatures at
$157$ and $176$ MeV [14]. They signify the three phases of the phase diagram.
This is indicated by the near-alignment of the red points
$\Delta\text{P}^{\text{R}}$, $\text{N}^{\text{R}}$ with the Taylor series
bands $\Delta\text{P}^{\text{T}}$, $\text{N}^{\text{T}}$ in Fig.1. Despite
being very pronounced at $135$ MeV, Fig.1 demonstrates this new formalism not
only eliminates bias upto $\mathcal{O}(\mu_{B}^{2})$, but also very
efficiently captures the unbiased contributions and agrees well with the
fourth order Taylor results of excess pressure and number density. Although
stochastic bias is significantly reduced by using old resummation formula with
$2000$ random sources for $D_{1}$, the new formalism with just $500$ sources
for all $D_{n}$ shown in Fig.2 provides more improved results and manages to
capture higher order Taylor contributions over the former.

Figure 1: $\Delta\text{P}_{2}/T^{4}$ (top row) and $\text{N}_{2}/T^{3}$
(bottom row) plots in $(\mu_{B}/T)^{2}$ for
$135$ (left column), $157$ (middle column) and $176$ (right column) MeV

Figure 2: $\Delta\text{P}_{2,4}/T^{4}$ (left) and $\text{N}_{2,4}/T^{3}$
(right) plots in both bases for $T=135$ MeV

##  4 Conclusion and Outlook

In this work, we have presented the new formalism of unbiased exponential
resummation which successfully eliminates the much dreaded stochastic bias
arising from the old exponential resummation formula to a given order in
$\mu_{B}$. This new formalism, thus not only retrieve the structure of
exponential resummation but also manages to capture genuine higher order
Taylor contributions. Despite producing unbiased estimates upto a finite order
in $\mu_{B}$, it implies that in the limit of infinite number of cumulants in
$X$ basis or unbiased corrections to all orders in $\mu_{B}$ in $\mu_{B}$
basis, one can achieve unbiased exponential resummation to all orders in
$\mu_{B}$ which is identical to the true infinite Taylor series. This
reiterates the importance of obviating stochastic bias and following unbiased
exponential resummation in obtaining genuine interpretations and implications
regarding finite-density QCD and the mysteriously beautiful phase diagram of
QCD.

## Acknowledgements

I sincerely acknowledge Prasad Hegde and Frithjof Karsch for useful
discussions. The computations have been performed in the cluster of Bielefeld
University Germany, for which I thank the Bielefeld HPC.NRW team.

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