# Performance Analysis of Dual-Hop Mixed PLC/RF Communication Systems

Liang Yang, Xiaoqin Yan, Sai Li, Daniel Benevides da Costa, and Mohamed-Slim
Alouini

###### Abstract

In this paper, we study a dual-hop mixed power line communication and radio-
frequency communication (PLC/RF) system, where the connection between the PLC
link and the RF link is made by a decode-and-forward (DF) or amplify-and-
forward (AF) relay. Assume that the PLC channel is affected by both additive
background noise and impulsive noise suffers from Log-normal fading, while the
RF link undergoes Rician fading. Based on this model, analytical expressions
of the outage probability (OP), average bit error rate (BER), and the average
channel capacity (ACC) are derived. Furthermore, an asymptotic analysis for
the OP and average BER, as well as an upper bound expression for the ACC are
presented. At last, numerical results are developed to validate our analytical
results, and in-depth discussions are conducted.

###### Index Terms:

Average bit error rate (BER), outage probability (OP), average channel
capacity (ACC), power line communication, radio-frequency system.

##  I Introduction

As a low-cost and energy-saving communication technology, power line
communication (PLC) utilizes the existing cables for data transmission
[1]-[3]. According to different voltage levels, PLC can communicate through
low-voltage cables, medium-voltage cables, and high-voltage cables [4]-[5].
Compared with other methods of communication, PLC has the characteristics of
wide coverage, convenient connection, and no need to rewire, which enables it
to be used in indoor and outdoor communication. For instance, in [6], the
authors put forward a kind of indoor narrow-band PLC network model, and
provided the appropriate types of cables and electrical appliances through
laboratory experiments and simulation results. Based on the deep integration
of PLC and visible light (VLC), a original, practical, and economical indoor
broadband broadcasting network was studied in [7]. The authors in [8]
investigated the spatial correlation in indoor multiple-input multiple-output
(MIMO) PLC channels. Additionally, PLC has arisen as one of the main technical
methods of two-way communication in smart grid (SG) [9]. It can be connected
to all locations of the grid and transmit data on the original infrastructure.
In [10], the authors studied the performance of discrete wavelet multitone
transceiver for narrow-band PLC in SG.

However, compared with general wireless communication systems, the performance
of PLC communication systems is highly influenced by frequency selectivity,
path loss, and various attenuation and interference. In addition, impedance
mismatch and non-Gaussian noise in the PLC channel arise as further issues to
be tackled with. Moreover, since PLC was originally used to transmit electric
energy (and not for data communication), the transmission power in PLC systems
should comply with the regulations of relevant government departments, which
leads to the limitation of system capacity and transmission distance [11].
Finally, the affection of background noise (BGN) and impulsive noise (IMN) on
system performance should also be jointly considered in PLC systems as they
are the main reasons for data loss [12]. In order to alleviate these
inconveniences, researchers in the field of PLC have proposed to combine PLC
with wireless communication technologies, such as multi-antenna schemes,
cooperative communications, MAC protocols, and relaying methods [13]-[15].

Along the years, relay technology has shown to improve the reliability and
coverage of the system. Relying on this fact, relay-aided PLC systems have
been widely investigated. In [16], a PLC system with Log-normal (LN) fading
and Bernoulli-Gaussian IMN assisted by amplify-and-forward (AF) relay was
studied, which was shown to outperform PLC systems without embedded relay. The
outage probability (OP), average bit error rate (BER), and average channel
capacity (ACC) of a mixed decode-and-forward (DF) relay-aided PLC system were
derived in [17], while a full-duplex AF relay for PLC networks was considered
in [18]. Moreover, the performance of half-duplex PLC networks with either AF
or DF relays was examined in [19]. With the aim to enhance the energy
efficiency of relaying PLC systems, AF relaying with energy-harvesting
capabilities was embedded in a PLC system and accurate expressions for energy
efficiency of such systems were derived in [20]. In [21], the authors studied
the physical layer security of the relay-aided PLC system with eavesdropping
and noise interference. More recently, the authors considered a PLC system
with incremental AF and DF relaying, and the results showed a great spectral
efficiency improvement [22].

In the aforementioned works dealing with relay-aided PLC systems, the RF links
were assumed to undergo LN or Rayleigh fading, although Rician fading arises
as a more suitable model owing to the presence of the line-of-sight (LoS)
component. As far as the authors know, the performance of PLC/RF systems with
both BGN and IMN, where the respective links are modeled by a LN and a Rician
distribution, respectively, has not been investigated in the field of PLC.
Thus, this paper presents and studies a dual-hop mixed PLC/RF system, assuming
both DF and AF relaying protocols.

The original contributions of this paper can be stated as follows:

  * $\bullet$

A cost-effective and efficient PLC/RF system is introduced and analyzed. The
RF system and the PLC system are connected by both DF and AF relaying
protocols. The channel of the PLC link is modeled by LN fading and it is
subject to additive BGN and IMN, while the RF channel follows a Rician fading
distribution.

  * $\bullet$

For DF relaying, respective expressions for the OP, average BER, and ACC are
derived. To get further insights, asymptotic analysises for the OP and average
BER are carried out. Moreover, we also derive the upper bound expression for
ACC.

  * $\bullet$

Novel closed-form expressions for the cumulative distribution function (CDF)
and probability density function (PDF) of the end-to-end signal-to-noise (SNR)
of the considered system with an AF relay are obtained. Moreover, the
respective expressions for OP, average BER, as well as ACC are achieved.

  * $\bullet$

The influences of critical system parameters on the overall performance is
studied and insightful discussions are drawn.

  * $\bullet$

The Monte Carlo simulation results validate our analytical results.

Figure 1: System setup.

The remaining section of this paper is arranged as follows. Section II
introduces the system and channel models. In Section III, a comprehensive
performance of the considered system is analyzed, including OP, average BER,
and ACC. Section IV provides illustrative numerical examples along with
insightful discussions. Finally, Section V concludes the paper. Appendix A
provides a detailed proof for the CDF and PDF of the end-to-end SNR.

##  II System and Channel Models

Consider a dual-hop mixed PLC/RF communication system, which includes a source
node (S), a relay node (R) and a destination node (D), as shown in Fig. 1. In
the first time-slot $T_{1}$, the source S transmits data to R through the PLC
link modeled by a LN fading distribution with BGN and IMN. Then, during the
second time-slot $T_{2}$, by employing DF or AF relaying protocols at R, the
signals are sent to D over an RF link modeled by Rician fading. Suppose that
direct connection between S and D does not exist.

###  II-A The PLC Link

By utilizing a binary modulation scheme, the symbol $x$ from S is transmitted
to R through the power cables. Thus, the signal at R can be presented as

$y_{\text{SR}}=h_{\text{SR}}x+n_{\text{SR}},$ (1)

where $n_{\text{SR}}$ represents the additive noise of the PLC channel and
$h_{\text{SR}}$ denotes the channel fading factor. From [23], $h_{\text{SR}}$
is modeled by a LN distribution and its PDF can be expressed as

$f_{h_{\text{SR}}}(h_{\text{SR}})=\frac{1}{h_{\text{SR}}\sqrt{2\pi\sigma_{\text{SR}}^{2}}}{\rm
exp}\left({-}\frac{\left(\ln(h_{\text{SR}})-\mu_{\text{SR}}\right)^{2}}{2\sigma_{\text{SR}}^{2}}\right),$
(2)

where $\sigma_{\text{SR}}^{2}$ and $\mu_{\text{SR}}$ denote the variance and
mean of $\ln(h_{\text{SR}})$, respectively. Due to the random transient
switching of low-power components and electrical equipment connected to cables
in PLC systems, the interference of both BGN and IMN are considered. In this
case, Poisson$-$Gaussian mixture statistical [23] is employed to model the
noise. Thus, the noise in the PLC link can be represented as
$n_{\text{SR}}=n_{b}{+}n_{i}n_{p}$, where $n_{b}$ is the BGN modeled as the
additive white Gaussian noise (AWGN) with zero mean and variance
$\sigma_{b}^{2}$, $n_{i}n_{p}$ denotes the IMN occurring during $T$, with
$n_{p}$ being defined as the occurrence of the IMN with a rate of $\lambda$
units per second in the system, which is modeled by a Poisson process, and
$n_{i}$ represents the AWGN with mean zero and variance $\sigma_{i}^{2}$.
Assuming only the real part of the noise $n_{\text{SR}}$, its PDF can be
written as [23]

$\displaystyle f_{n_{\text{SR}}}(n_{\text{SR}})=$
$\displaystyle\frac{1{-}P_{i}}{\sqrt{2\pi\sigma_{b}^{2}}}{\rm
exp}\left({-}\frac{n_{\text{SR}}^{2}}{2\sigma_{b}^{2}}\right){+}\frac{P_{i}}{\sqrt{2\pi\sigma_{b}^{2}\left(1{+}\eta\right)}}$
$\displaystyle\times{\rm
exp}\left({-}\frac{n_{\text{SR}}^{2}}{2\sigma_{b}^{2}\left(1{+}\eta\right)}\right),$
(3)

where $P_{i}=\lambda T$ represents the probability of the occurrence of
impulsive noise, and $\eta=\sigma_{i}^{2}/\sigma_{b}^{2}$ denotes the ratio of
the powers of IMN to BGN.

$\displaystyle
f_{\gamma_{o}}\left(\gamma\right)=\frac{P_{i}m_{2}\exp(-{m_{2}\over\Omega_{2}}\gamma)}{\Omega_{2}\Gamma(m_{2})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{m_{2}}\binom{m_{2}}{r}\left({m_{2}\gamma\over\Omega_{2}}\right)^{l{+}m_{2}{-}r}{k^{1{-}2l}K^{l}(C(K{+}1))^{l{+}1}\Gamma(k{+}l)\over\overline{\gamma}_{RD}^{l{+}1}\Gamma(k{-}l{+}1)\Gamma^{2}(l{+}1)}G_{0,2}^{2,0}\left[\left.\frac{Cm_{2}(1{+}K)\gamma}{\Omega_{2}\overline{\gamma}_{RD}}\right|\begin{matrix}{-}\\\
{r{-}l{-}1,0}\end{matrix}\right]$
$\displaystyle{+}\frac{\left(1{-}P_{i}\right)m_{1}\exp(-{m_{1}\over\Omega_{1}}\gamma)}{\Omega_{1}\Gamma(m_{1})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{m_{1}}\binom{m_{1}}{r}\left({m_{1}\gamma\over\Omega_{1}}\right)^{l{+}m_{1}{-}r}{k^{1{-}2l}K^{l}(C(K{+}1))^{l{+}1}\Gamma(k{+}l)\over\overline{\gamma}_{RD}^{l{+}1}\Gamma(k{-}l{+}1)\Gamma^{2}(l{+}1)}G_{0,2}^{2,0}\left[\left.{Cm_{1}(1{+}K)\gamma\over\Omega_{1}\overline{\gamma}_{RD}}\right|\begin{matrix}{-}\\\
{r{-}l{-}1,0}\end{matrix}\right].$ (16) $\displaystyle
F_{\gamma_{o}}\left(\gamma\right)=$ $\displaystyle
F_{\gamma_{SR}}(\gamma){+}\frac{\left(1{-}P_{i}\right)m_{1}\exp({-}{m_{1}\over\Omega_{1}}\gamma)}{\Omega_{1}\Gamma(m_{1})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{m_{1}{-}1}\binom{m_{1}{-}1}{r}\left({m_{1}\gamma\over\Omega_{1}}\right)^{m_{1}{-}r{-}1}B_{l}G_{1,3}^{2,1}\left[\left.\frac{Cm_{1}(1{+}K)\gamma}{\Omega_{1}\overline{\gamma}_{RD}}\right|\begin{matrix}{1}\\\
{1{+}r,1{+}l,0}\end{matrix}\right]$
$\displaystyle+\frac{P_{i}m_{2}\exp({-}{m_{2}\over\Omega_{2}}\gamma)}{\Omega_{2}\Gamma(m_{2})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{m_{2}{-}1}\binom{m_{2}{-}1}{r}\left({m_{2}\gamma\over\Omega_{2}}\right)^{m_{2}{-}r{-}1}B_{l}G_{1,3}^{2,1}\left[\left.\frac{Cm_{2}(1+K)\gamma}{\Omega_{2}\overline{\gamma}_{RD}}\right|\begin{matrix}{1}\\\
{1{+}r,1{+}l,0}\end{matrix}\right].$ (17)

When there are only BGN samples in the PLC channel, the resulting SNR of the
PLC link can be written as

$\gamma_{\text{SR}1}=\frac{E_{b}\left|h_{\text{SR}}\right|^{2}}{\sigma_{b}^{2}}=\overline{\gamma}_{\text{SR}1}\left|h_{\text{SR}}\right|^{2},$
(4)

where $\overline{\gamma}_{\text{SR}1}$ denotes the average SNR of the first
hop when only BGN samples are presented and $E_{b}$ represents the average
energy of the signal. Similarly, when the IMN samples and BGN samples appear
simultaneously in the PLC link, the instantaneous SNR can be written as

$\gamma_{\text{SR}2}=\frac{E_{b}\left|h_{\text{SR}}\right|^{2}}{\sigma_{b}^{2}\left(1{+}\eta\right)}=\overline{\gamma}_{\text{SR}2}\left|h_{\text{SR}}\right|^{2},$
(5)

where $\overline{\gamma}_{\text{SR}2}$ is the average SNR of the PLC link when
IMN samples and BGN samples occur simultaneously in the system. According to
[23], the PDF of the SNR $\gamma_{\text{SR}}$ of the PLC link can be shown to
be given by

$\displaystyle f_{\gamma_{\text{SR}}}\left(\gamma\right)=$
$\displaystyle\left(1{-}P_{i}\right)\left(\frac{m_{1}}{\Omega_{1}}\right)^{m_{1}}\frac{\gamma^{m_{1}{-}1}}{\Gamma\left(m_{1}\right)}{\rm
exp}\left({-}\frac{m_{1}}{\Omega_{1}}\gamma\right)$
$\displaystyle{+}P_{i}\left(\frac{m_{2}}{\Omega_{2}}\right)^{m_{2}}\frac{\gamma^{m_{2}{-}1}}{\Gamma\left(m_{2}\right)}{\rm
exp}\left({-}\frac{m_{2}}{\Omega_{2}}\gamma\right),$ (6)

where $m_{1}$ and $m_{2}$ are defined as the shadowing severity parameters of
the Gamma PDF, $\Omega_{1}$ and $\Omega_{2}$ represent the shadowing area mean
power of the Gamma PDF, and $\Gamma(\cdot)$ is the Gamma function [24]. Thus,
the CDF of $\gamma_{\text{SR}}$ can be represented as [23]

$\displaystyle F_{\gamma_{\text{SR}}}\left(\gamma\right)=$
$\displaystyle\frac{1{-}P_{i}}{\Gamma\left(m_{1}\right)}G_{1,2}^{1,1}\left[\left.\frac{m_{1}}{\Omega_{1}}\gamma\right|\begin{matrix}1\\\
m_{1},0\end{matrix}\right]$
$\displaystyle{+}\frac{P_{i}}{\Gamma\left(m_{2}\right)}G_{1,2}^{1,1}\left[\left.\frac{m_{2}}{\Omega_{2}}\gamma\right|\begin{matrix}1\\\
m_{2},0\end{matrix}\right],$ (7)

where $G_{c,d}^{m,n}\left[\cdot\right]$ is the Meijer $G$-function [24].

###  II-B The RF Link

In $T_{2}$, R uses DF or AF relaying protocols to forward the received signals
to D via the RF channel. It is worthy to say that the RF link between R and D
is considered to follow a Rician fading distribution.

####  II-B1 DF Case

For the DF case, we can write the received signal at D as

$y_{\text{RD}}^{DF}=\sqrt{P_{R}}h_{\text{RD}}\widehat{x}+n_{\text{RD}},$ (8)

where $P_{R}$ denotes the average transmit power at D, $\widehat{x}$ is
defined as the signal transmitted from R, $n_{\text{RD}}$ represents the AWGN
term with zero mean and variance $N$, and $h_{\text{RD}}$ denotes the RF
channel coefficient. From (8), the resulting SNR of the RF link can be
formulated as

$\displaystyle\gamma_{\text{RD}}{=}\frac{P_{R}\left|h_{RD}\right|^{2}}{N}{=}\overline{\gamma}_{\text{RD}}\left|h_{RD}\right|^{2},$
(9)

where $\overline{\gamma}_{RD}$ is the average SNR of the RF link. The PDF and
CDF of $\gamma_{\text{RD}}$ are given by [25]

$\displaystyle f_{\gamma_{RD}}(\gamma){=}$
$\displaystyle\frac{K{+}1}{\overline{\gamma}_{RD}}\exp\left({-}\frac{(K{+}1)\gamma_{RD}}{\overline{\gamma}_{RD}}{-}K\right)$
$\displaystyle\times
I_{0}\left(2\sqrt{K(K{+}1)\frac{\gamma_{RD}}{\overline{\gamma}_{RD}}}\right),$
(10)

and

$\displaystyle
F_{\gamma_{RD}}\left(\gamma\right)=1-Q_{1}\Bigg{(}\sqrt{2K},\sqrt{\frac{2\left(K+1\right)}{\overline{\gamma}_{RD}}\gamma}\Bigg{)},$
(11)

where $I_{0}(\cdot)$ is the zeroth-order modified Bessel function of the first
kind [24], $K$ $(K\geq 0)$ represents the Rician factor, and
$Q_{1}(\cdot,\cdot)$ is the Marcum Q-function of the first order [26].

####  II-B2 AF Case

For the AF case, the received signal at D is

$\displaystyle
y_{\text{RD}}^{AF}=\sqrt{P_{R}}Gh_{\text{RD}}y_{\text{SR}}+n_{\text{RD}},$
(12)

where $G$ is the fixed amplifying gain. Thus, the overall instantaneous SNR
can be expressed as

$\displaystyle\gamma_{o}=\frac{\gamma_{\text{SR}}\gamma_{\text{RD}}}{C+\gamma_{\text{RD}}},$
(13)

where $C$ is a constant determined by the AF relay gain $G$.

To derive the expression of the PDF of the overall instantaneous SNR
$\gamma_{o}$ and facilitate our calculation, the approximation of
$I_{v}(x)\simeq\sum_{l=0}^{k}{\Gamma(k+l)\over\Gamma(l+1)\Gamma(k-l+1)}{k^{1-2l}\over\Gamma(v+l+1)}\left({x\over
2}\right)^{v+2l}$ [26] is used in (10) for $0<x<2k$. Thus (10) and (11) can be
rewritten as

$\displaystyle
f_{\gamma_{RD}}(\gamma){\simeq}\frac{K{+}1}{\overline{\gamma}_{RD}}\exp\left({-}\frac{(K{+}1)\gamma_{RD}}{\overline{\gamma}_{RD}}{-}K\right)$
$\displaystyle\times\sum_{l=0}^{k}{\Gamma(k+l)k^{1-2l}K^{l}(K+1)^{l}\over\Gamma^{2}(l+1)\Gamma(k-l+1)\overline{\gamma}_{RD}^{l}}\gamma^{l},$
(14)

and

$\displaystyle
F_{\gamma_{RD}}(\gamma){\simeq}1{-}\exp\left({-}\frac{(K{+}1)\gamma_{RD}}{\overline{\gamma}_{RD}}{-}K\right)$
$\displaystyle\times\sum_{l=0}^{k}\sum_{r=0}^{l}{k^{1{-}2l}K^{l}(K{+}1)^{r}\Gamma(l{+}k)\over\Gamma(r{+}1)\Gamma^{2}(l{+}1)\Gamma(k{-}l{+}1)\overline{\gamma}_{RD}^{r}}\gamma^{r}.$
(15)

Then, using the similar method [27], the PDF and CDF of $\gamma_{o}$ are
derived in (16) and (17) when $m_{1}$ and $m_{2}$ are integers, shown at the
bottom of this page, where
$B_{l}{=}{k^{1-2l}K^{l}\Gamma(k+l)\over\Gamma(k-l+1)\Gamma^{2}(l+1)}$.

_Proof:_ See Appendix A.

##  III Performance Analysis

###  III-A DF Relaying

####  III-A1 OP

The OP can be defined as the probability that the overall instantaneous SNR is
lower than a certain SNR threshold $\gamma_{\text{th}}$. Thus, by relying on
(7) and (11), and making the appropriate substitutions, the outage probability
can be determined from the expression below

$\displaystyle P_{\text{out}}^{DF}$
$\displaystyle=\text{Pr}\left(\text{min}(\gamma_{\text{SR}},\gamma_{\text{RD}}){<}\gamma_{\text{th}}\right)$
$\displaystyle=F_{\gamma_{\text{SR}}}(\gamma_{\text{th}}){+}F_{\gamma_{\text{RD}}}(\gamma_{\text{th}}){-}F_{\gamma_{\text{SR}}}(\gamma_{\text{th}})F_{\gamma_{\text{RD}}}(\gamma_{\text{th}}).$
(18)

In order to gain further insights on outage probability with the DF protocol,
next we derive an asymptotic outage expression. At high SNR, the last term of
(18) can be ignored. Then, making use of the following asymptotic series
expansion of the Meijer G-function [28, Eq. (07.34.06.0040.01)]

$\displaystyle
G_{c,d}^{m,n}\left(z\Big{|}_{b_{1},...,b_{d}}^{a_{1},...,a_{c}}\right)$
$\displaystyle=\sum_{\iota=1}^{m}\displaystyle\frac{\prod_{j=1,j\neq\iota}^{m}\Gamma(b_{j}{-}b_{\iota})\prod_{j=1}^{n}\Gamma(1{-}a_{j}{+}b_{\iota})}{\prod_{j=n+1}^{c}\Gamma(a_{j}{-}b_{\iota})\prod_{j=m+1}^{d}\Gamma(1{-}b_{j}{+}b_{\iota})}z^{b_{\iota}}(1{+}o(z)),$
(19)

the asymptotic $F_{\gamma_{SR}}(\gamma)$ can be represented as

$\displaystyle\underset{\overline{\gamma}_{SR}\rightarrow\infty}{F_{\gamma_{SR}}(\gamma)}{\simeq}\frac{1{-}P_{i}}{\Gamma(1{+}m_{1})}\left(\frac{m_{1}\gamma}{\Omega_{1}}\right)^{m_{1}}{+}\frac{P_{i}}{\Gamma\left(1{+}m_{2}\right)}\left(\frac{m_{2}\gamma}{\Omega_{2}}\right)^{m_{2}}.$
(20)

Furthermore, the asymptotic expression for $F_{\gamma_{RD}}(\gamma)$ can be
expressed as [29]

$\displaystyle\underset{\overline{\gamma}_{RD}\rightarrow\infty}{F_{\gamma_{RD}}(\gamma)}{\simeq}{\left(1{+}K\right)\gamma\over\overline{\gamma}_{RD}\exp\left(K\right)}.\
$ (21)

Finally, the asymptotic outage probability is attained from the sum of (20)
and (21) by setting $\gamma=\gamma_{\text{th}}$.

####  III-A2 Average BER

Generally, the average BER of a DF relaying system can be formulated as

$\displaystyle P_{\text{BER}}^{DF}{=}P_{e1}{+}P_{e2}{-}2P_{e1}P_{e2},$ (22)

where $P_{e1}$ and $P_{e2}$ are the average BER of the first hop and the
second hop, respectively. In addition, the average BER for various binary
modulations can be written as [30]

$\displaystyle P_{b}{=}\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}{\rm
exp}(-q\gamma)\gamma^{q{-}1}F_{\gamma}(\gamma)d\gamma,$ (23)

where $p$ and $q$ are parameters related to modulation schemes. In our
analysis, we consider the differential binary phase shift keying (DBPSK)
scheme (i.e., $p=1,q=1$). Therefore, $P_{1}$ and $P_{2}$ can be derived as

$\displaystyle
P_{e1}{=}\frac{1{-}P_{i}}{2\Gamma\left(m_{1}\right)}G_{2,2}^{1,2}\left[\left.\frac{m_{1}}{\Omega_{1}}\right|\begin{matrix}0,1\\\
m_{1},0\end{matrix}\right]{+}\frac{P_{i}}{2\Gamma(m_{2})}G_{2,2}^{1,2}\left[\left.\frac{m_{2}}{\Omega_{2}}\right|\begin{matrix}0,1\\\
m_{2},0\end{matrix}\right],$ (24) $\displaystyle P_{e2}{=}{1{+}K\over
2\left(1{+}K{+}\overline{\gamma}_{RD}\right)\exp(K)}{}_{1}F_{1}\left(1;1;{K\left(K{+}1\right)\over
K{+}1{+}\overline{\gamma}_{RD}}\right),$ (25)

where ${}_{1}F_{1}\left(\cdot\right)$ is the confluent hypergeometric function
[24]. By substituting (24) and (25) into (22), we obtain the average BER
expression. Since the last term in (22) can be negligible at high SNRs, one
can obtain the asymptotic average BER as follows

$\displaystyle P_{\text{BER}}^{DF}\rightarrow P_{e1}^{A}+P_{e2}^{A},$ (26)

where $P_{e1}^{A}$ and $P_{e2}^{A}$ denote the asymptotic BER of $P_{e1}$ and
$P_{e2}$, respectively. Substituting (20) into (23), the expression of
$P_{e1}^{A}$ can be obtained by

$\displaystyle P_{e1}^{A}={1{-}P_{i}\over
2}\left({m_{1}\over\Omega_{1}}\right)^{m_{1}}{+}{P_{i}\over
2}\left({m_{2}\over\Omega_{2}}\right)^{m_{2}}.$ (27)

Then, according [29], the $P_{e2}^{A}$ can be expressed as

$\displaystyle
P_{e2}^{A}=\frac{\left(1{+}K\right)\Gamma(2)}{2\overline{\gamma}_{RD}\exp(K)}.$
(28)

$\displaystyle\mathbb{E}(\gamma)$
$\displaystyle{=}{\left(1{-}P_{i}\right)\left({m_{1}\over\Omega_{1}}\right)^{m_{1}}\over\Gamma(m_{1})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{l}\Gamma(l+1)\Gamma(m_{1}{+}r)B_{l}\Theta_{r}W_{1r}{+}{P_{i}\left({m_{2}\over\Omega_{2}}\right)^{m_{2}}\over\Gamma(m_{2})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{l}\Gamma(l+1)\Gamma(m_{2}{+}r)B_{l}\Theta_{r}W_{2r}$
$\displaystyle{-}{\left(1{-}P_{i}\right)\overline{\gamma}_{RD}\exp({-}K)\over\left(K{+}1\right)\Gamma(m_{1})}\sum_{l=0}^{k}B_{l}G_{2,2}^{1,2}\left[\left.{m_{1}\overline{\gamma}_{RD}\over\Omega_{1}\left(K{+}1\right)}\right|\begin{matrix}{-}l{-}1,1\\\
m_{1},0\end{matrix}\right]-{P_{i}\overline{\gamma}_{RD}\exp(-K)\over\left(K{+}1\right)\Gamma(m_{2})}\sum_{l=0}^{k}B_{l}G_{2,2}^{1,2}\left[\left.{m_{2}\overline{\gamma}_{RD}\over\Omega_{2}\left(K{+}1\right)}\right|\begin{matrix}{-}l{-}1,1\\\
m_{2},0\end{matrix}\right]$
$\displaystyle{+}{\overline{\gamma}_{RD}\over(K{+}1)\exp(K)}\sum_{l=0}^{k}B_{l}\Gamma(l{+}2).$
(35) $\displaystyle P_{\text{BER}}^{AF}{=}$
$\displaystyle\frac{1{-}P_{i}}{2\Gamma\left(m_{1}\right)}G_{2,2}^{1,2}\left[\left.\frac{m_{1}}{\Omega_{1}}\right|\begin{matrix}0,1\\\
m_{1},0\end{matrix}\right]{+}\frac{P_{i}}{2\Gamma\left(m_{2}\right)}G_{2,2}^{1,2}\left[\left.\frac{m_{2}}{\Omega_{2}}\right|\begin{matrix}0,1\\\
m_{2},0\end{matrix}\right]$
$\displaystyle{+}\frac{(1{-}P_{i})\Omega_{1}\exp({-}K)}{2\Gamma(m_{1}{+}1)}\sum_{l=0}^{k}\sum_{r=0}^{m_{1}{-}1}\begin{pmatrix}{m_{1}{-}1}\\\
{r}\end{pmatrix}\left(\frac{m_{1}{+}\Omega_{1}}{m_{1}}\right)^{r{-}m_{1}}B_{l}G_{2,3}^{2,2}\left[\left.{Cm_{1}(1{+}K)\over\overline{\gamma}_{RD}(m_{1}{+}\Omega_{1})}\right|{}_{1{+}r,1{+}l,0}^{1{+}r{-}m_{1},1}\right]$
$\displaystyle{+}\frac{P_{i}\Omega_{2}\exp({-}K)}{2\Gamma(m_{2}{+}1)}\sum_{l=0}^{k}\sum_{r=0}^{m_{2}{-}1}\binom{m_{2}{-}1}{r}\left(\frac{m_{2}{+}\Omega_{2}}{m_{2}}\right)^{r{-}m_{2}}B_{l}G_{2,3}^{2,2}\left[\left.{Cm_{2}(1{+}K)\over\overline{\gamma}_{RD}(m_{2}{+}\Omega_{2})}\right|{}_{1{+}r,1{+}l,0}^{1{+}r{-}m_{2},1}\right].$
(37) $\displaystyle C_{AF}=$
$\displaystyle\frac{(1{-}P_{i})\exp(-K)}{2\Gamma(m_{1})\ln(2)}\sum_{l=0}^{k}\sum_{r=0}^{m_{1}}\begin{pmatrix}m_{1}\\\
r\end{pmatrix}\left({C(1{+}K)\over\overline{\gamma}_{RD}}\right)^{l{+}1}B_{l}G_{1,0;2,2;;0,2}^{0,1;1,2;2,0}\left[\left.\begin{matrix}r{-}m_{1}-l\\\
-\end{matrix}\right|\left.\begin{matrix}1,1\\\
1,0\end{matrix}\right|\left.\begin{matrix}-\\\
r-l-1,0\end{matrix}\right|\frac{\Omega_{1}}{m_{1}},\frac{C(1+K)}{\overline{\gamma}_{RD}}\right]$
$\displaystyle{+}\frac{P_{i}\exp(-K)}{2\Gamma(m_{2})\ln(2)}\sum_{l=0}^{k}\sum_{r=0}^{m_{2}}\begin{pmatrix}m_{2}\\\
r\end{pmatrix}\left({C(1{+}K)\over\overline{\gamma}_{RD}}\right)^{l{+}1}B_{l}G_{1,0;2,2;;0,2}^{0,1;1,2;2,0}\left[\left.\begin{matrix}r{-}m_{2}-l\\\
-\end{matrix}\right|\left.\begin{matrix}1,1\\\
1,0\end{matrix}\right|\left.\begin{matrix}-\\\
r-l-1,0\end{matrix}\right|\frac{\Omega_{2}}{m_{2}},\frac{C(1+K)}{\overline{\gamma}_{RD}}\right].$
(39)

####  III-A3 Average Channel Capacity

Generally, the overall capacity of the dual-hop DF system can be defined as

$\displaystyle C_{DF}$
$\displaystyle{=}\frac{1}{2}\mathbb{E}\left\\{\text{log}_{2}(1{+}\gamma)\right\\}{=}\frac{1}{2\text{ln}(2)}\int_{0}^{\infty}\text{ln}(1{+}\gamma)f_{\gamma}(\gamma)d\gamma$
$\displaystyle{=}\frac{1}{2\text{ln}(2)}(C_{1}{+}C_{2}{-}C_{3}-C_{4}),$ (29)

where
$f_{\gamma}(\gamma)=f_{\gamma_{SR}}(\gamma)+f_{\gamma_{RD}}(\gamma)-f_{\gamma_{SR}}(\gamma)F_{\gamma_{RD}}(\gamma)-F_{\gamma_{SR}}(\gamma)f_{\gamma_{RD}}(\gamma)$.
By utilizing ${\rm
ln}(1+\gamma){=}G_{2,2}^{1,2}\left[\left.\gamma\right|{}_{1,0}^{1,1}\right]$
[28, Eq. (01.04.26.0003.01)],
$\exp({-}bz){=}G_{0,1}^{1,0}\left[\left.bz\right|{}_{0}^{-}\right]$ [28, Eq.
(07.34.21.0013.01)], [28, Eq. 07.34.21.0088.01] and [28, Eq.
07.34.21.0081.01], expressions for $C_{1}$, $C_{2}$, $C_{3}$ and $C_{4}$ can
be calculated as

$\displaystyle C_{1}{=}$
$\displaystyle\frac{1{-}P_{i}}{\Gamma\left(m_{1}\right)}G_{3,2}^{1,3}\left[\left.\frac{\Omega_{1}}{m_{1}}\right|\begin{matrix}1{-}m_{1},1,1\\\
1,0\end{matrix}\right]$
$\displaystyle+\frac{P_{i}}{\Gamma\left(m_{2}\right)}G_{3,2}^{1,3}\left[\left.\frac{\Omega_{2}}{m_{2}}\right|\begin{matrix}1{-}m_{2},1,1\\\
1,0\end{matrix}\right],$ (30) $\displaystyle C_{2}{=}$
$\displaystyle\sum_{l=0}^{k}{\Gamma(k{+}l)k^{1{-}2l}K^{l}\over\Gamma(k{-}l+1)\Gamma^{2}(l{+}1)\exp(K)}G_{3,2}^{1,3}\left[\left.\frac{\overline{\gamma}_{RD}}{K{+}1}\right|\begin{matrix}{-}l,1,1\\\
1,0\end{matrix}\right],$ (31) $\displaystyle
C_{3}{=}{(P_{i}{-}1)\left({m_{1}\over\Omega_{1}}\right)^{m_{1}}\over\Gamma(m_{1})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{l}\Gamma(l{+}1)B_{l}\Theta_{r}W_{1r}G_{3,2}^{1,3}\left[\left.W_{1r}\right|{}_{\varsigma_{1r}}^{1,0}\right]$
$\displaystyle{+}C_{1}{-}{P_{i}\left({m_{2}\over\Omega_{2}}\right)^{m_{2}}\over\Gamma(m_{2})\exp(K)}\sum_{l=0}^{k}\sum_{r=0}^{l}\Gamma(l{+}1)B_{l}\Theta_{r}W_{2r}G_{3,2}^{1,3}\left[\left.W_{2r}\right|{}_{\varsigma_{2r}}^{1,0}\right],$
(32)

and

$\displaystyle C_{4}{=}$
$\displaystyle{1-P_{i}\over\Gamma(m_{1})\exp(K)}\sum_{l=0}^{k}B_{l}$
$\displaystyle\times
G_{1,0;2,2;1,2}^{0,1;1,2;1,1}\left[\left.\begin{matrix}{-l}\\\
{-}\end{matrix}\right|\left.\begin{matrix}{1,1}\\\
{1,0}\end{matrix}\right|\left.\begin{matrix}{1}\\\
{m_{1},0}\end{matrix}\right|{\overline{\gamma}_{RD}\over
K+1},{\Omega_{1}\overline{\gamma}_{RD}\over m_{1}\left(K+1\right)}\right]$
$\displaystyle+{P_{i}\over\Gamma(m_{2})\exp(K)}\sum_{l=0}^{k}B_{l}$
$\displaystyle\times
G_{1,0;2,2;1,2}^{0,1;1,2;1,1}\left[\left.\begin{matrix}{-l}\\\
{-}\end{matrix}\right|\left.\begin{matrix}{1,1}\\\
{1,0}\end{matrix}\right|\left.\begin{matrix}{1}\\\
{m_{2},0}\end{matrix}\right|{\overline{\gamma}_{RD}\over
K+1},{\Omega_{2}\overline{\gamma}_{RD}\over m_{2}\left(K+1\right)}\right],$
(33)

where $\varsigma_{1r}=\left\\{1{-}m_{1}{-}r,1,1\right\\}$,
$\varsigma_{2r}=\left\\{1{-}m_{2}{-}r,1,1\right\\}$,
$\Theta_{r}={(K{+}1)^{r}\over\overline{\gamma}_{RD}^{r}\Gamma(r{+}1)}$,
$W_{1r}{=}\left({\Omega_{1}\overline{\gamma}_{RD}\over
m_{1}\overline{\gamma}_{RD}{+}\Omega_{1}\left(K{+}1\right)}\right)^{m_{1}+r}$,
$W_{2r}{=}\left({\Omega_{2}\overline{\gamma}_{RD}\over
m_{2}\overline{\gamma}_{RD}{+}\Omega_{2}\left(K{+}1\right)}\right)^{m_{2}+r}$.
Furthermore, an upper bound for $C_{DF}$ can be formulated as [31]

$\displaystyle
C_{DF}=\frac{1}{2}\mathbb{E}\left\\{\text{log}_{2}(1{+}\gamma)\right\\}\leq\frac{1}{2}\text{log}_{2}\left(1+\mathbb{E}(\gamma)\right),$
(34)

where the mean value $\mathbb{E}(\gamma)=\int_{0}^{\infty}\gamma
f_{\gamma}(\gamma)$ can calculated as (35), shown at the top of this page.

###  III-B Fixed-Gain AF Relaying

####  III-B1 OP

For the AF relaying scheme, OP can be formulated as

$\displaystyle
P_{\text{out}}^{AF}=\text{Pr}\left\\{\frac{\gamma_{\text{SR}}\gamma_{\text{RD}}}{C+\gamma_{\text{RD}}}<\gamma_{\text{th}}\right\\}=F_{\gamma_{o}}(\gamma_{\text{th}}),$
(36)

where $F_{\gamma_{o}}(\gamma_{\text{th}})$ denotes the CDF of $\gamma_{o}$
when $\gamma{=}\gamma_{\text{th}}$.

####  III-B2 Average BER

Assuming DBPSK scheme, the average BER for the considered cooperative PLC/RF
system with fixed-gain AF relaying can be expressed as
$P_{\text{BER}}^{AF}{=}\frac{1}{2}\int_{0}^{\infty}{\rm
exp}\left({-}\gamma\right)F_{\gamma_{o}}\left(\gamma\right)d\gamma$. After
some algebraic calculations, we can arrive at the average BER in (37), shown
at the top of this page.

####  III-B3 Average Channel Capacity

By employing AF protocol, the ACC can be formulated as

$\displaystyle
C_{AF}{=}\frac{1}{2}\mathbb{E}\left\\{\text{log}_{2}(1{+}\gamma_{o})\right\\}{=}\frac{1}{2\text{ln}(2)}\int_{0}^{\infty}\text{ln}(1{+}\gamma)f_{\gamma_{o}}(\gamma)d\gamma.$
(38)

Then, upon substituting (16) in (38) and using the integral from three Meijer
$G$-functions [28, Eq. 07.34.21.0081.01], $C_{AF}$ is given by (39), shown at
the top of this page.

Furthermore, an upper bound for $C_{AF}$ can be represented as

$\displaystyle
C_{AF}=\frac{1}{2}\mathbb{E}\left\\{\text{log}_{2}(1{+}\gamma_{o})\right\\}\leq\frac{1}{2}\text{log}_{2}\left(1+\mathbb{E}(\gamma_{o})\right),$
(40)

where

$\displaystyle\mathbb{E}(\gamma_{o}){=}$
$\displaystyle{(1-P_{i})\Omega_{1}\exp({-}K)\over\Gamma(1{+}m_{1})}\sum_{l=0}^{k}\sum_{r=0}^{m_{1}}\begin{pmatrix}m_{1}\\\
r\end{pmatrix}\left({C(1{+}K)\over\overline{\gamma}_{RD}}\right)^{l{+}1}$
$\displaystyle\times
B_{l}G_{2,1}^{1,2}\left[\left.\frac{C(1{+}K)}{\overline{\gamma}_{RD}}\right|\begin{matrix}r{-}m_{1}{-}l{-}1\\\
r{-}l{-}1,0\end{matrix}\right]$
$\displaystyle+{P_{i}\Omega_{2}\exp({-}K)\over\Gamma(1{+}m_{2})}\sum_{l=0}^{k}\sum_{r=0}^{m_{2}}\begin{pmatrix}m_{2}\\\
r\end{pmatrix}\left({C(1{+}K)\over\overline{\gamma}_{RD}}\right)^{l{+}1}$
$\displaystyle\times
B_{l}G_{2,1}^{1,2}\left[\left.\frac{C(1{+}K)}{\overline{\gamma}_{RD}}\right|\begin{matrix}r{-}m_{2}{-}l{-}1\\\
r{-}l{-}1,0\end{matrix}\right].$ (41)

##  IV Simulation and Numerical Results

In this section, numerical examples are presented to illustrate the analytical
and asymptotic expressions developed in the previous section. Additionally,
Monte-Carlo simulation results corroborate the analytical analysis. Unless
otherwise specified, we set $m_{1}=m_{2}=8$, $P_{i}=0.2$, $\eta=5$,
$\sigma_{SR}=0.23$, $K=6$ $\rm{dB}$, $C=1.2$, $G=3$, $\gamma_{th}=0$ $\rm{dB}$
and $\overline{\gamma}_{SR}=\overline{\gamma}_{RD}=\overline{\gamma}$.

In Fig. 2, OP versus $\overline{\gamma}$ is plotted for various values of $K$,
and assuming the considered dual-hop mixed PLC/RF system with the DF protocol.
In addition, we set $K=2,4,6$ $\rm{dB}$.

Figure 2: Outage probability for different values of $K$ (DF protocol).
Figure 3: Outage probability for various values of $P_{i}$ (AF protocol).

It can be clearly noted that the system outage performance improves by
increasing the values of $K$. The reason is that $K$ is the ratio of the
powers of the LoS components to the powers of the scattered components. The
higher the values of $K$, the lower the signal attenuation caused by multipath
effects. Furthermore, Fig. 2 shows that the asymptotic $P_{out}^{DF}$
expression converges to the exact $P_{out}^{DF}$ expression at high SNRs. In
Fig. 3, we draw $P_{out}^{AF}$ versus $\overline{\gamma}$ for various values
of $P_{i}$ assuming the AF protocol. From Fig. 3, it can be observed that for
low values of $P_{i}$, $P_{out}^{AF}$ is highly degraded, resulting in a
better performance. This is because $P_{i}$ represents the probability of the
arrival of the impulsive noise, and lower values of $P_{i}$ generates a weak
effect of the IMN for the PLC/RF system. Finally, Fig. 4 presents the outage
performance of the dual-hop PLC/RF system with different values of threshold
SNR $C_{th}$ under DF and AF relay protocols. From Fig. 4, it can be noted
that the outage performance with the AF relay protocol is superior to the
outage performance with the DF relay protocol, and the performance increases
when the value $\gamma_{th}$ decreases.

Figure 4: Outage probability for various values of $\gamma_{th}$ under
differen relaying protocols.  Figure 5: BER for different Rician factors $K$
(DF protocol).  Figure 6: BER for various values of $P_{i}$ (AF protocol).
Figure 7: BER for various values of $\eta$ under different relaying protocols.

In Fig. 5, the average BER versus average SNR $\overline{\gamma}$ is plotted
under different values of $K$, and considering the DF relay protocol. This
figure demonstrates the average BER performance with higher values $K$
performs much better than the one with the lower values of $K$. The reason is
because higher values of $K$ means more LoS components, which improves the
system performance. Additionally, Fig. 5 also reveals the convergence between
the asymptotic and exact $P_{BER}^{DF}$ expressions. The impact of the
different values of $P_{i}$ is shown in Fig. 6. As can be seen, a decrease in
the value of $P_{i}$ (the probability of impulse noise reaching the PLC
channel) leads to a BER decrease, which results in a better system
performance. The reason is that with the decrease of IMN arriving at the
system, the severity of IMN component in the system decreases compared with
the BGN sample, improving consequently the performance. In Fig. 7, the average
BER is presented under differen values of $\eta$. It can be clearly seen that
a decrease in $\eta$ leads to a lower average BER. Similar to the results
obtained previously to outage probability, it is shown that the average BER of
the AF protocol is significantly superior to the BER performance of the DF
one.

Fig. 8 plots the ACC $C_{DF}$ versus $\overline{\gamma}$ for various values of
$\eta$. It can be observed that $C_{DF}$ increases with the decrease of
$\eta$. In addition, we plot the asymptotic $C_{DF}$ , which reveals the
correctness of the asymptotic expression of $C_{DF}$. In Fig. 9, we plot exact
average channel capacity $C_{AF}$ versus $\overline{\gamma}$ for various
values of $P_{i}$. As can be seen, higher $C_{AF}$ can be obtained when the
values of $P_{i}$ decrease. Also, it is shown the convergence between the
asymptotic and exact $C_{AF}$ expressions.

Figure 8: Ergodic channel capacity for various values of $K$ (DF protocol).
Figure 9: Ergodic channel capacity for different values of $P_{i}$ (AF
protocol).

##  V Conclusions

In this work, we studied the performance of the PLC/RF system under both DF
and AF relay protocols. It was supposed that the PLC link is modeled by a LN
distribution with IMN while the RF link follows the Rician distribution.
Closed-form expressions for the OP, average BER, and ACC were derived. For AF
relaying, analytical expressions for the CDF and PDF of the end-to-end SNR
have been obtained in closed-form. For DF relaying, closed-form expressions
for the asymptotic OP, average BER, and ACC were derived. The analytical
expressions were validated with their corresponding simulation results.
Additionally, it was studied the impact of impulsive noise and Rician factor
on the overall system performance, and insightful discussions were drawn.

##  Appendix A PDF AND CDF of The End-To-End SNR

In this section, we derive the PDF and CDF of the end-to-end overall SNR
$\gamma_{o}$. Similar methods can also be found in [27]. The PDF of
$\gamma_{o}=\frac{\gamma_{\textrm{SR}}\gamma_{\textrm{RD}}}{C+\gamma_{\textrm{RD}}}$
can be written as

$\displaystyle f_{\gamma_{o}}(\gamma)=$
$\displaystyle\frac{d}{d\gamma}\text{Pr}\left\\{\frac{\gamma_{\textrm{SR}}\gamma_{\textrm{RD}}}{C+\gamma_{\textrm{RD}}}<\gamma\right\\}$
$\displaystyle=$
$\displaystyle\frac{d}{d\gamma}\int_{0}^{\infty}\text{Pr}\left\\{\frac{\gamma_{\textrm{RD}}x}{C+\gamma_{RD}}<\gamma\right\\}f_{\gamma_{\textrm{SR}}}\left(x\right)dx$
$\displaystyle=$
$\displaystyle\frac{d}{d\gamma}\left[{\int_{0}^{\gamma}\text{Pr}\left\\{\gamma_{\textrm{RD}}\left(x-\gamma\right)<C\gamma\right\\}f_{\gamma_{\textrm{SR}}}\left(x\right)dx}\right.$
$\displaystyle\left.{+\int_{\gamma}^{\infty}\text{Pr}\left\\{\gamma_{\textrm{RD}}\left(x-\gamma\right)<C\gamma\right\\}f_{\gamma_{\textrm{SR}}}\left(x\right)dx}\right].$
(42)

Due to $0<x<\gamma$,
$\text{Pr}\left\\{\gamma_{\textrm{RD}}(x-\gamma)<C\gamma\right\\}=1$, (42) can
be rewritten as

$\displaystyle f_{\gamma_{o}}(\gamma){=}$
$\displaystyle\frac{d}{dr}\left[{\int_{0}^{\gamma}f_{\gamma_{\textrm{SR}}}(x)dx}\right.$
$\displaystyle\left.{{+}\int_{\gamma}^{\infty}\text{Pr}\left\\{\gamma_{\textrm{RD}}<\frac{C\gamma}{x{-}\gamma}\right\\}f_{\gamma_{\textrm{SR}}}\left(x\right)dx}\right]$
$\displaystyle=$ $\displaystyle
f_{\gamma_{\textrm{SR}}}(\gamma)-\lim_{x\rightarrow\gamma^{+}}\text{Pr}\left\\{\gamma_{\textrm{RD}}<\frac{C\gamma}{x-\gamma}\right\\}f_{\gamma_{SR}}(\gamma)$
$\displaystyle+\int_{\gamma}^{\infty}f_{\gamma_{\textrm{RD}}}\left(\frac{C\gamma}{x-\gamma}\right)\frac{cx}{(x-\gamma)^{2}}f_{\gamma_{\textrm{SR}}}(x)dx.$
(43)

Since
$\lim_{x\rightarrow\gamma^{+}}\text{Pr}\left\\{\gamma_{\textrm{RD}}<\frac{C\gamma}{x-\gamma}\right\\}=1$,
one can obtain

$\displaystyle
f_{\gamma_{o}}(\gamma)=\int_{\gamma}^{\infty}f_{\gamma_{\textrm{RD}}}(\left(\frac{C\gamma}{x-\gamma}\right)\frac{cx}{(x-\gamma)^{2}}f_{\gamma_{\textrm{SR}}}(x)dx.$
(44)

By substituting $t=x-\gamma$ in (44) and applying (6), (14),
$f_{\gamma_{o}}(\gamma)$ can be formulated as

$\displaystyle
f_{\gamma_{o}}(\gamma){=}{\exp(-{m_{1}\over\Omega_{1}}\gamma)\left({m_{1}\over\Omega_{1}}\right)^{m_{1}}\over\Gamma(m_{1})\exp(K)}\sum_{l{=}0}^{k}\left({C(1+K)\over\overline{\gamma}_{RD}}\right)^{l+1}B_{l}\gamma^{l}\Phi_{1}$
$\displaystyle{+}{\exp(-{m_{2}\over\Omega_{2}}\gamma)\left({m_{2}\over\Omega_{2}}\right)^{m_{2}}\over\Gamma(m_{2})\exp(K)}\sum_{l{=}0}^{k}\left({C(1+K)\over\overline{\gamma}_{RD}}\right)^{l+1}B_{l}\gamma^{l}\Phi_{2},$
(45)

where $\Phi_{\rho}$, $\rho=1,2$ are expressed as

$\displaystyle\Phi_{\rho}{=}$
$\displaystyle\int_{0}^{\infty}\left(1{+}\frac{\gamma}{t}\right)^{m_{\rho}}t^{m_{\rho}{-}l{-}2}$
$\displaystyle\times\exp\left({-}\frac{m_{\rho}t}{\Omega_{\rho}}\right)\exp({-}{C(K{+}1)\over
t\overline{\gamma}_{RD}}\gamma)dt.$ (46)

Using $\exp({-}bz){=}G_{0,1}^{1,0}\left[\left.bz\right|{}_{0}^{-}\right]$, the
integral formula [28, Eq. (07.34.21.0088.01)] and the expanding expression of
$(1+\gamma/t)^{m_{\rho}}$, $\rho=1,2$ [27], (16) is obtained.

From (43), the CDF of $\gamma_{o}$ can be written as

$\displaystyle
F_{\gamma_{o}}(\gamma){=}F_{\gamma_{SR}}(\gamma){+}\int_{\gamma}^{\infty}F_{\gamma_{RD}}({C\gamma\over
x-\gamma})f_{\gamma_{SR}}(x)dx.$ (47)

To simplify the operation, we use the new expression of
$F_{\gamma_{RD}}(\gamma){=}\sum_{l=0}^{k}{k^{1{-}2l}K^{l}\Gamma(k+l)\over\Gamma(k{-}l+1)\Gamma^{2}(k+1)\exp(K)}\Upsilon(l{+}1,{(K{+}1)\gamma\over\overline{\gamma}_{RD}})$
obtained from the integration of (14) to (47). Therefore, the CDF of
$\gamma_{o}$ is represented by (17).

## References

  * [1] Ferreira, H. C., Lampe, L., Newbury, J., Swart, T. G., _Power Line Communications: Theory and Applications for Narrowband and Broadband Communications over Power Lines_ , Singapore: Wiley, 2010
  * [2] A. Majumder and J. Caffery, Jr., “Power line communications: an overview,” _IEEE Potentials_ , vol. 23, no. 4, pp. 4–8, Oct./Nov. 2004
  * [3] Y. Qian, C. Zhang, Z. Xu, F. Shu, L. Dong, J. Li, “A reliable opportunistic routing for smart grid with in-home power line communication networks,” _Sci. China Inf. Sci._ , Volume 59, Issue 12, 2016, Pages 122305-, ISSN 1674-733X.
  * [4] V. V. Kharlamov, S. E. Romanov, Y. P. Shkarin and A. G. Merkulov, “Characteristics of PLC channels over high voltage power cables and mixed cable-overhead high voltage power lines,” _IEEE GPECOM, Nevsehir, Turkey_ , pp. 35-40, 2019.
  * [5] D. B. Unsal, A. H. Koc, T. Yalcinoz and I. Onaran,“Medium voltage and low voltage applications of new power line communication model for smart grids,” _ENERGYCON, Leuven,_ pp. 1-6, 2016.
  * [6] T. A. Papadopoulos, G. C. Argyropoulos, B. D. Sarantinos and G. K. Papagiannis, “Analysis of indoor PLC networks: laboratory tests and simulation results,” _IEEE Lausanne Power Tech.,_ pp. 1935-1940, 2007.
  * [7] H. Ma, L. Lampe and S. Hranilovic, “Integration of indoor visible light and power line communication systems”, _IEEE Int. Symp. Power Line Commun.,_ pp. 291-296, 2013.
  * [8] K. Khalil, M.G. Gazalet, P. Corlay et al., “An MIMO random channel generator for indoor power-line communication”, _IEEE Trans. Power Deliv.,_ vol. 29, no. 4, pp. 1561-1568, 2014.
  * [9] A. Sendin, J. Simon, I. Urrutia and I. Berganza, “PLC deployment and architecture for smart grid applications in lberdrola”, _IEEE ISPLC_ , pp. 173-178, 2014.
  * [10] S. Baig and J. Yazdani, “Performance analysis of discrete wavelet multitone transceiver for narrowband PLC in smart grid”, _IEEE PES Int. Conf. Exhib. Innov. Smart Grid Technol.,_ pp. 1-6, Dec. 2011.
  * [11] N.A.T.O. Research and T. Organisation, “HF interference, procedures and tools-final report of NATO RTO information systems technology panel research task group IST- 050/RTG-022,” Jun. 2007. 12, 34, 39, 41, 128.
  * [12] M.K. Simon, M.S. Alouini, _Digital communication over fading channels,_ Wiley, New York, 2000.
  * [13] A. Schwager, D. Schneider, W. Baschlin, A. Dilly, and J. Speidel, “MIMO PLC: Theory, measurements and system setup,” _IEEE Int. Symp. Power Line Commun. and Its Appl. (ISPLC),_ pp. 48–53, Apr. 2011.
  * [14] L. Lampe and A. Vinck, “Cooperative multihop power line communications,” _IEEE Int. Symp. Power Line Commun. and Its Appl. (ISPLC),_ pp. 1–6, Mar. 2012.
  * [15] X. Cheng, R. Cao, and L. Yang, “Relay-aided amplify-and-forward powerline communications,” _IEEE Trans. Smart Grid,_ vol. 4, pp. 265–272, Mar. 2013.
  * [16] A. Dubey and R. K. Mallik, “PLC system performance with AF relaying,” _IEEE Trans. Commun._ , vol. 63, no. 6, pp. 2337-2345, June 2015.
  * [17] A. Dubey, R. K. Mallik and R. Schober, “Performance analysis of a multi-hop power line communication system over log-normal fading in presence of impulsive noise,” _IET Commun._ , vol. 9, no. 1, pp. 1-9, 2015.
  * [18] F. Passerini and A. M. Tonello, “Analog full-duplex amplify-and-forward relay for power line communication networks,” _IEEE Commun. Lett.,_ vol. 23, no. 4, pp. 676-679, April 2019.
  * [19] R. K. Ahiadormey, P. Anokye, H. Jo and K. Lee, “Performance analysis of two-way relaying in cooperative power line communications,” _IEEE Access,_ vol. 7, pp. 97264-97280, 2019.
  * [20] K. M. Rabie, B. Adebisi and A. Salem, “Improving energy efficiency in dual-hop cooperative PLC relaying systems,” _International Symposium on Power Line Communications and its Applications (ISPLC)_ , Bottrop, pp. 196-200, 2016.
  * [21] A. Salem, K. M. Rabie, K. A. Hamdi, E. Alsusa and A. M. Tonello, “Physical layer security of cooperative relaying power-line communication systems,” 2 _International Symposium on Power Line Communications and its Applications (ISPLC)_ , Bottrop, pp. 185-189, 2016.
  * [22] A. Dubey, C. Kundu, T. M. N. Ngatched, O. A. Dobre and R. K. Mallik, “Incremental relaying for power line communications: performance analysis and power allocation,” _IEEE Systems Journal,_ vol. 13, no. 4, pp. 4236-4247, Dec. 2019.
  * [23] M. Jani, P. Garg and A. Gupta, “Performance Analysis of a Mixed Cooperative PLC-VLC System for Indoor Communication Systems,” _IEEE Syst. J.,_ vol. 14, no. 1, pp. 469-476, March 2020.
  * [24] I. S. Gradshteyn and I. M. Ryzhik, _Table of Integrals, Series, and Products,_ 6th ed. New York: Academic Press, 2000.
  * [25] P. V. Trinh, T. Cong Thang and A. T. Pham, “Mixed mmWave RF/FSO relaying systems over generalized fading channels with pointing errors,” _IEEE Photonics Journal,_ vol. 9, no. 1, pp. 1-14, Feb. 2017.
  * [26] P. C. Sofotasios and S. Freear, “Novel expressions for the Marcum and one dimensional Q-functions,” _7th International Symposium on Wireless Communication Systems (ISWCS),_ pp. 736-740, 2010.
  * [27] L.-L. Yang and H.-H. Chen, “Error probability of digital communications using relay diversity over Nakagami-m fading channels”,_IEEE Trans Wireless Commun.,_ vol. 7, no. 5, pp. 1806-1811, May 2008.
  * [28] The Wolfram functions site [Online]. Available: http://functions.wolfram.com.
  * [29] L. Yang, F. Meng, J. Zhang, M. O. Hasna and M. D. Renzo, “On the performance of RIS-assisted dual-hop UAV communication systems,” _IEEE Trans. Veh. Technol.,_ vol. 69, no. 9, pp. 10385-10390, Sept. 2020.
  * [30] I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.S. Alouini, and H.Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K composite fading channels,” _IEEE Trans. Commun.,_ vol. 59, no. 10, pp. 2654-2658, Oct. 2011.
  * [31] J. Park, E. Lee, G. Park, B. Roh and G. Yoon, “Performance analysis of asymmetric RF/FSO dual-hop relaying systems for UAV applications,” MILCOM 2013 - 2013 _IEEE Military Communications Conference (MILCOM),_ pp. 1651-1656, 2013.


#  $L^{p}$-$L^{q}$ Boundedness of Spectral Multipliers of the Anharmonic
Oscillator
$L^{p}$-$L^{q}$ continuité des multiplicateurs spectraux de l’oscillateur
anharmonique

Marianna Chatzakou  Marianna Chatzakou: Department of Mathematics: Analysis
Logic and Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>and  Vishvesh Kumar  Vishvesh Kumar: Department
of Mathematics: Analysis Logic and Discrete Mathematics Ghent University,
Belgium E-mail address<EMAIL_ADDRESS>
###### Abstract.

In this note we study the $L^{p}-L^{q}$ boundedness of Fourier multipliers of
anharmonic oscillators, and as a consequence also of spectral multipliers, for
the range $1<p\leq 2\leq q<\infty$. The underlying Fourier analysis is
associated with the eigenfunctions of an anharmonic oscillator in some family
of differential operators having derivatives of any order. Our analysis relies
on a version of the classical Paley-type inequality, introduced by Hörmander,
that we extend in our nonharmonic setting.
Résumé
Dans cette note, nous étudions la $L^{p}$-$L^{q}$ continuité des
multiplicateurs de Fourier des oscillateurs anharmoniques, et par conséquent
des multiplicateurs spectraux également, pour $1<p\leq 2\leq q<\infty$.
L’analyse de Fourier sous-jacente est associée aux fonctions propres d’un
oscillateur anharmonique dans certaines familles d’opérateurs différentiels
ayant des dérivées d’ordre quelconque. Notre analyse s’appuie sur une version
de l’inégalité classique de type Paley, introduite par Hörmander, que nous
étendons dans notre cadre non harmonique.

###### Key words and phrases:

anharmonic oscillator; Fourier multipliers; Spectral Multipliers, Paley
inequality, Hausdorff-Young- Paley inequality, $L^{p}$-$L^{q}$ Boundedness,
Heat kernels

###### 2010 Mathematics Subject Classification:

Primary 42B15; 58J40; Secondary 47B10; 47G30; 35P10

##  1\. Introduction

The aim of this note is to study the $L^{p}-L^{q}$ boundedness conditions for
the Fourier and spectral multipliers where the pertinent Fourier analysis is
developed with respect to a member of the family of anharmonic oscillators on
$\mathbb{R}^{n}$. Such operators were considered in [7] as “prototype” for the
spectral analysis that is performed there.

The boundedness of Fourier multipliers is a central topic in harmonic analysis
and its origin can be traced back to Hörmander’s seminal paper [11] in 1960.
The results concerning to the boundedness of multipliers are of $L^{p}$-type,
or $L^{p}-L^{q}$-type. In the first case, Mihlin-H$\ddot{\text{o}}$rmander
[12] or Marcinkiewicz theorems [5] come into play providing conditions on the
regularity of the symbol, while in the second case, the sharp decay of the
spectral information is analysed.

Regarding the $L^{p}-L^{q}$ boundedness that we study here, let us recall
Hörmander’s Fourier multiplier theorem as in [11]: For $1<p\leq 2\leq
q<\infty,$ the Fourier multiplier
$T_{\sigma}:\mathcal{S}(\mathbb{R}^{n})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$
associated with symbol $\sigma:\mathbb{R}^{n}\rightarrow\mathbb{C}$ defined by

$\mathcal{F}(T_{\sigma}f)(\xi)=\sigma(\xi)\mathcal{F}(f)(\xi)$

for $\xi\in\mathbb{R}^{n}$, where $\mathcal{F}$ denotes the Euclidean Fourier
transform of $f$, has a bounded extension from $L^{p}(\mathbb{R}^{n})$ to
$L^{q}(\mathbb{R}^{n})$ provided that the symbol satisfies the condition

$\sup_{s>0}s\left(\int_{\xi\in\mathbb{R}^{n}:|\sigma(\xi)|\geq
s}d\xi\right)^{\frac{1}{p}-\frac{1}{q}}<+\infty\,.$ (1.1)

The $L^{p}-L^{q}$ boundedness of spectral and Fourier mutlipliers has been
studied in different settings; we refer for instance to the works [CGM] on
symmetric spaces, and [3, 1, 2] on groups. In [3, 2] the authors suggest an
approach that is also effective for our purposes: to establish the Hausdorff-
Young-Paley inequality by interpolating between the Hausdorff-Young inequality
and Paley-type inequality. However, the machinery we use here differs from the
one in [3] and relies on the underlying Fourier analysis, and so on the
spectral properties of our “model operator”, and on the Sobolev estimates
associated with it as acquired in [7].

Precisely, the Fourier analysis we present here is the natural analogue of the
one developed in [13] in the general setting of nonharmonic analysis
associated with a “model operator”. A concrete example of the latter is our
choice of the anharmonic oscillator with a potential that is a homogeneous
polynomial of a certain degree, and with derivatives of any order; in
particular the operator we consider is of the form 111The addition of the
constant $1$ to the potential is required for the invertibility of the
operator and for controlling the lower bound of the spectrum of the operator.

$(-\Delta)^{l}+|x|^{2k}+1\,,\quad x\in\mathbb{R}^{n}\,,$

where $l,k\geq 1$ are some fixed integers, $|\cdot|$ stands for the Euclidean
norm on $\mathbb{R}^{n}$, and $\Delta$ is the usual Laplace operator on
$\mathbb{R}^{n}$. Under such analysis a Fourier multiplier is $L^{p}-L^{q}$
bounded if a condition similar to (1.1) is satisfied for a symbol of the
multiplier adapted to our setting.

##  2\. Notation and preliminaries

In order to present our results, let us briefly recall the necessary notation
and concepts involved in our analysis. For a more detailed discussion on
concepts related to the spectral analysis of the anharmonic oscillator we
refer to [7, 8], and for those related to the associated Fourier analysis we
refer to [8], or, for a more general consideration, to [13].

###  2.1. The anharmonic oscillator and its spectrum

The anharmonic oscillator is a differential operator on $\mathbb{R}^{n}$ that
is served as a generalisation of the harmonic oscillator. Here we consider an
anharmonic oscillator in the family of operators $(A_{k,l})_{k,l}$, for
different values of the integer parameters $k,l\geq 1$, defined as follows

$A_{k,l}:=(-\Delta)^{l}+|x|^{2k}+1\,,\quad x\in\mathbb{R}^{n}\,.$

Here and thereafter we assume that the parameters $k,l$ are fixed, and the
notation will be simplified to $A:=A_{k,l}$.

The operator $A$ is densely defined on its domain
$\mathcal{S}(\mathbb{R}^{n})\subset L^{2}(\mathbb{R}^{n})$ and admits a
(unique) self-adjoint extension on $L^{2}(\mathbb{R}^{n})$. Its spectrum is
purely discrete and lies in the interval $[1,\infty)$. If we keep the same
notation for its self adjoint extension, then by the functional calculus we
have, for $m\in\mathbb{R},$

$Dom\left(A^{m}\right)=\left\\{u\in
L^{2}(\mathbb{R}^{n}):\sum_{j=1}^{\infty}{\lambda_{j}}^{2m}|\langle
u_{j},u\rangle_{L^{2}}|^{2}<\infty\right\\},$

where the set $(u_{j})_{j}$, with $\|u_{j}\|_{L^{2}}=1$, is the set of
normalized eigenfunctions of $A$, and $(\lambda_{j})_{j}$ is the corresponding
set of eigenvalues. Thus, for $u\in Dom\left(A^{m}\right)$, we have

$A^{m}u=\sum_{j=1}^{\infty}\lambda_{j}^{m}\langle
u_{j},u\rangle_{L^{2}}u_{j}\,.$

Despite the intensive research on the eigenvalue problem associated with the
anhramonic oscillator, the exact solution of it, even in the quartic case, is
still explicitly unknown. We refer to [7] for a recent exposition of the
related works. Let us summarise the spectral information that is known in the
case of the anharmonic oscillator $A$: For the inverse operator $A^{-1}$ we
know that $A^{-1}\in S_{r}(L^{2}(\mathbb{R}^{n}))$ for $r>\frac{(k+l)n}{2kl}$,
where by $S_{r}(L^{2}(\mathbb{R}^{n}))$ we have denoted the $r$th Schatten-von
Neumann class of operators on the Hilbert space $L^{2}(\mathbb{R}^{n})$, see
[7, Corollary 5.3]. As a consequence, for the eigenvalue counting function
$N(\lambda):=\\#\\{j:\lambda_{j}<\lambda\\},$ we get the following asymptotic
estimate

$N(\lambda)\lesssim\lambda^{n\left(\frac{1}{2k}+\frac{1}{2l}\right)}\,,\quad\text{as}\quad\lambda\rightarrow\infty\,,$
(2.1)

where we have assumed that the eigenvalues $(\lambda_{j})_{j}$ are arranged in
the decreasing order; see [7, Remark 5.7], or [10, Theorem 5.5].

###  2.2. Fourier analysis associated with the anharmonic oscillator

For the setup of this section we follow [13].

The space of functions

$C^{\infty}_{A}(\mathbb{R}^{n}):=\cap_{m=1}^{\infty}\textnormal{Dom}(A^{m})\,,$

will be the domain of the Fourier operator $\mathcal{F}_{A}$ in our setting.
The space $C^{\infty}_{A}$ is a Fréchet space if endowed with the topology
induced by the family of norms

$\|f\|_{C^{m}_{A}}:=\max_{k\leq
m}\|A^{k}f\|_{L^{2}(\mathbb{R}^{n})},\,\,m\in\mathbb{N}_{0},\,\,f\in
C^{\infty}_{A}(\mathbb{R}^{n}).$

Then, defining the $A$-Fourier transform by

$(\mathcal{F}_{A}f)(j):=\int_{\mathbb{R}^{n}}f(x)\overline{u_{j}(x)}\,dx\,,$

we see that $\mathcal{F}_{A}(C^{\infty}_{A})\subset\mathcal{S}(\mathbb{N})$,
where $\mathcal{S}(\mathbb{N})$ is the space of rapidly decreasing functions
$\phi:\mathbb{N}\to\mathbb{C}$, i.e., for any $N<\infty,$ there exists a
constant $C_{\phi,N}>0$ such that

$|\phi(j)|\leq C_{\phi,N}\langle j\rangle^{-N}\,\,\text{for all
}j\in\mathbb{N}\,.$

The Fréchet topology of $\mathcal{S}(\mathbb{N})$ is induced by the family of
semi-norms $\\{p_{r}\\}_{r\in\mathbb{N}}$ given by

$p_{r}(\phi):=\sup_{j\in\mathbb{N}}\langle j\rangle^{r}|\phi(j)|\,.$

With the topological structures as described above, the map $\mathcal{F}_{A}$
becomes a bijective homeomorphism form $C^{\infty}_{A}$ to
$\mathcal{S}(\mathbb{N})$, and the Fourier inversion formula becomes available
in our setting. Moreover, for $f\in L^{2}$, the operator $\mathcal{F}_{A}$
becomes a bijective isomorphism onto the Hilbert space $l^{2}_{A}$ defined as
the space of sequences $f=(f_{j})_{j\in\mathbb{N}}$ under the norm

$\|f\|_{l^{2}_{A}}:=\left(\sum_{j\in\mathbb{N}}|f_{j}|^{2}\right)^{\frac{1}{2}}\,.$

The latter reads as the Plancherel identity in our setting, i.e., in symbols:
$\|f\|_{L^{2}}=\|\mathcal{F}_{A}(f)\|_{l^{2}_{A}}$.

##  3\. Main results

The Hausdorff-Young inequality in our setting is derived verbatim as a
particular case of the Fourier analysis associated with a “model operator”
[13]. The Paley-type inequality is more involved and its proof can be found in
the preprint [8, Theorem 4.1]. Then, by means of the Stein-Weiss interpolation
theorem, we can extend the Hausdorff-Young-Paley inequality in our setting [8,
Theorem 4.4] which is the key for proving our result on the boundedness of
Fourier multipliers as follows. We refer to [8] for the detailed proofs of the
results below.

First, let us recall that $T_{\sigma}$ is an $A$-Fourier multiplier, if it
satisfies the identity

$\mathcal{F}_{A}({T_{\sigma}})(j)=\sigma(j)\mathcal{F}_{A}{f}(j),\,\,\,\,j\in\mathbb{N},$

where $\sigma:\mathbb{N}\rightarrow\mathbb{C}$ is a function that can be
served as the symbol of the multiplier $T_{\sigma}$ for our analysis.

######  Theorem 3.1.

Let $1<p\leq 2\leq q<\infty.$ Let $T_{\sigma}$ be a $A$-Fourier multiplier
with symbol $\sigma$. Then we have

$\|T_{\sigma}\|_{L^{p}(\mathbb{R}^{n})\rightarrow
L^{q}(\mathbb{R}^{n})}\lesssim\sup_{s>0}s\left(\
\sum_{\overset{j\in\mathbb{N}}{|\sigma(j)|\geq
s}}\|u_{j}\|^{2}_{L^{\infty}(\mathbb{R}^{n})}\right)^{\frac{1}{p}-\frac{1}{q}}\,.$
(3.1)

Let us mention that (3.1) is well-defined since for the $L^{\infty}$-estimate
of eigenfunctions we have: $\|u_{j}\|_{L^{\infty}}\leq C\sqrt{\lambda_{j}}$,
for some $C>0$ and for all $j$; see [8, Lemma 2.2]. We note here that Theorem
3.1 can be thought as a particular case of recently obtained result in [6] for
a more general setting.

As a consequence of Theorem 3.1 and of the spectral decay of the anharmonic
oscillator $A$ in (2.1) we derive the following result on the boundedness of
the spectral multipliers in our setting.

######  Theorem 3.2.

Let $1<p\leq 2\leq q<\infty$, and let $\varphi$ be any decreasing function on
$[1,\infty)$ such that $\lim_{u\rightarrow\infty}\varphi(u)=0$. Then,

$\|\varphi(A)\|_{L^{p}(\mathbb{R}^{n})\rightarrow
L^{q}(\mathbb{R}^{n})}\lesssim\sup_{u\geq
1}\varphi(u)\left(u^{1+\frac{(k+l)n}{2kl}}\right)^{\frac{1}{p}-\frac{1}{q}}\,.$

Let us summarise some applications of Theorem 3.2.

######  Corollary 3.3.

Let $1<p\leq 2\leq q<\infty$. Then, the operator $A^{-m}$ is bounded from
$L^{p}(\mathbb{R}^{n})$ to $L^{q}(\mathbb{R}^{n})$ for
$m\geq\left(1+\frac{(k+l)n}{2kl}\right)\left(\frac{1}{p}-\frac{1}{q}\right)$.

Corollary 3.3 implies in particular that $A^{-m}$ is bounded from
$L^{2}(\mathbb{R}^{n})$ to $L^{2}(\mathbb{R}^{n})$ for any $m\geq 0$. This is
known in more generality by the general theory of pseudo-differential
operators, see Beals [4, Theorem 3.1], which gives us
$A^{-m}\in\mathcal{L}(L^{p})$, for any $m>0$, and for any $p\in(1,\infty)$.
However, using Corollary 3.3 we can study the $L^{p}$-$L^{q}$ boundedness for
$p\neq q$.

######  Corollary 3.4.

If $v=v(x,t)$ is the solution to the heat equation associated to $A$, i.e., if
we have

$\partial_{t}v+Av=0\,,\quad v(0)=v_{0}\,,$

then the time decay rate for $v$ is estimated as:

$\|v(t,\cdot)\|_{L^{q}(\mathbb{R}^{n})}\lesssim
t^{-\left(1+\frac{(k+l)n}{2kl}\right)\left(\frac{1}{p}-\frac{1}{q}\right)}\
\|v_{0}\|_{L^{q}(\mathbb{R}^{n})}\,,$

where $1<p\leq 2\leq q<\infty$.

######  Corollary 3.5.

Let $1<p\leq 2\leq q<\infty$. Then, for
$m\geq\left(1+\frac{(k+l)n}{2kl}\right)\left(\frac{1}{p}-\frac{1}{q}\right)$
we obtain the Sobolev-type estimate:

$\|f\|_{L^{q}(\mathbb{R}^{n})}\lesssim\|A^{m}f\|_{L^{p}(\mathbb{R}^{n})}\,.$

## acknowledgement

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021).

We thank Michael Ruzhansky and Duvan Cardona for helpful discussion and
suggestions.

## References

  * [1] Akylzhanov, R. Nursultanov, E. Ruzhansky, M. Hardy-Littlewood-Paley inequalities and Fourier multipliers on $SU(2).$ Studia Math. 234, no. 1, 1-29 (2016).
  * [2] Akylzhanov, R. Nursultanov, E. Ruzhansky, M., Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and $L^{p}$-$L^{q}$ Fourier multipliers on compact homogeneous manifolds. J. Math. Anal. Appl. 479, no. 2, 1519–1548 (2019).
  * [3] Akylzhanov, R. Ruzhansky, M., $L^{p}-L^{q}$ multipliers on locally compact groups, J. Func. Ana., 278(3) (2019), DOI: https://doi.org/10.1016/j.jfa.2019.108324
  * [4] Beals, R. $L^{p}$ Hölder estimates for pseudo-differential operators: sufficient conditions. Annales de l’institut Fourier, 29(3), 239–260, (1979).
  * [5] Carbery, A. Seeger, A. $H^{p}$ and $L^{p}$-Variants of multiparameter Calderón-Zygmund theory. Trans. AMS, 334, 719-747 (1992).
  * [6] Cardona, D. Kumar, V. Ruzhansky, M. Tokmagambetov, N. $L^{p}-L^{q}$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations. (preprint), 2020. https://arxiv.org/abs/2005.04936
  * [7] Chatzakou, M. Delgado, J. Ruzhansky, M. On a class of anharmonic oscillators (2021). To appear in J. Math. Pures Appl. https://arxiv.org/abs/1811.12566
  * [8] Chatzakou, M. Kumar, V. $L^{p}$-$L^{q}$ boundedness of Fourier multipliers associated with the anharmonic Oscillator, (2020). https://arxiv.org/abs/2004.07801
  * [9] Cowling, M. Giulini, S. Meda, S. $L^{p}$-$L^{q}$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I. Duke Math. J. 72, no. 1, 109-150 (1993).
  * [10] Helffer, B. Robert, D. Comportement asymptotique précise du spectre d’opérateurs globalement elliptiques dans $\mathbb{R}^{n}.$ Goulaouic-Meyer-Schwartz Seminar, 1980–1981, École Polytech., Palaiseau, 23(2), 1–22 (1981).
  * [11] Hörmander, L. Estimates for translation invariant operators in $L^{p}$ spaces. Acta Math., 104:93–140 (1960).
  * [12] Mikhlin, S. G. On multipliers for Fourier integrals, Dokl. Akad. Nauk., 199, 73–84 (1956).
  * [13] Ruzhansky, M., Tokmagambetov, N., Nonharmonic analysis of boundary value problems, Int. Math. Res. Notices,(12), 3548–3615 (2016).


# QCD pressure: renormalization group optimized perturbation theory confronts
lattice

Jean-Loïc Kneur<EMAIL_ADDRESS>Laboratoire Charles Coulomb
(L2C), UMR 5221 CNRS-Université Montpellier, 34095 Montpellier, France  Marcus
Benghi Pinto<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil  Tulio E.
Restrepo<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil

###### Abstract

The quark contribution to the QCD pressure, $P_{q}$, is evaluated up to next-
to-leading order (NLO) within the renormalization group optimized perturbation
theory (RGOPT) resummation approach. To evaluate the complete QCD pressure we
simply add the perturbative NLO contribution from massless gluons to the
resummed $P_{q}$. Despite of this unsophisticated approximation our results
for $P=P_{q}+P_{g}$ at the central scale $M\sim 2\pi T$ show a remarkable
agreement with ab initio lattice predictions for $0.25\lesssim T\lesssim
1\,{\rm GeV}$. We also show that by being imbued with RG properties, the RGOPT
produces a drastic reduction of the embarrassing remnant scale dependence that
plagues both standard thermal perturbative QCD and hard thermal loop
perturbation theory (HTLpt) applications.

At the fundamental level, strongly interacting matter composed by quarks and
gluons is described by the non Abelian theory of quantum chromodynamics (QCD)
whose coupling constant, $\alpha_{s}$, is predicted to decrease with
increasing energies as the system evolves to a regime of asymptotic freedom
(AF). This property, together with the other crucial phenomena of confinement
and chiral symmetry, has triggered the possibility of studying eventual phase
transitions related to (de)confinement and chiral symmetry
breaking/restoration in the laboratory through experiments involving heavy ion
collisions. On the theoretical side, lattice QCD (LQCD) ab initio simulations
have predicted that the deconfinement and chiral symmetry restoration occur
via an analytic crossover which indeed takes place at a pseudo-critical
temperature of order $T_{pc}\simeq 155\,{\rm MeV}$ with the baryon chemical
potential, $\mu_{B}$, approaching zero [1]. The region located at intermediate
$T$ and $\mu_{B}$ values ($T\sim 100\,{\rm MeV}$, $\mu_{B}\sim 900\,{\rm
MeV}$), is currently being explored by experiments such as the beam energy
scan at RHIC whose aim is to confirm the existence of a critical end point
which locates the end of a first order transition line predicted to start at
$T=0$ [2]. Another region, which covers the range $T\sim 0-30\,{\rm MeV}$ and
$\mu_{B}\gtrsim 1000\,{\rm MeV}$, is essential for the description of compact
stellar objects such as neutron stars. Unfortunately, due to the notorious
sign problem [3], LQCD encounters a more hostile environment within these two
phenomenologically important regions where numerical simulations cannot yet be
reliably implemented. Therefore, the development of reliable nonperturbative
analytical tools that can serve as alternatives remains timely. One way is to
use a weak-coupling expansion as a starting point in the evaluation of
physical observables in powers (and logarithms) of $g=4\pi\alpha_{s}$.
However, perturbative results require a further resummation so as to be
compatible with strong or even moderate coupling regimes (see Refs. [4, 5, 6]
for reviews). At finite temperatures, resumming the perturbative series helps
to avoid potential infrared divergences apart from improving convergence
issues. An efficient way to perform a resummation is to reorganize the
perturbative series around a quasiparticle mass parameter. Such an approach
appears in the literature under various names, like optimized perturbation
theory (OPT) [7, 8, 9], linear $\delta$ expansion (LDE) [10], variational
perturbation theory (VPT) [11], or in the thermal context, screened
perturbation theory (SPT) [12, 13].
The extension of analogous thermal resummations to the glue sector in QCD is
far from obvious due to gauge-invariance issues, but it had been circumvented
by Braaten and Pisarski [14] who proposed a gauge-invariant (non-local)
effective Lagrangian properly describing, among other hard thermal loop (HTL)
contributions, a gluon thermal “mass” term. The high temperature approximation
of HTL could be successfully generalized in the so-called HTL perturbation
theory (HTLpt) [15], allowing for the evaluation of the QCD thermodynamics at
the NNLO (three-loop), considering both the glue[16] and quark sectors at
finite temperatures and baryonic densities [17, 18, 19]. The final NNLO
results turned out to be in good agreement with LQCD predictions for
temperatures down to $T\approx 1.5\,T_{pc}$ for the “central” renormalization
scale choice $M=2\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}$. Unfortunately this
agreement quickly deteriorates as moderate scale variations of a factor 2
induce relative variations of order 1 or more. Moreover it has been observed
that this scale dependence strongly increases at higher orders, most
predominantly from NLO to NNLO. It is important to remark that standard pQCD
results are also plagued by a similar growing and strong scale dependence at
high orders [4, 20, 21].

More recently an alternative resummation approach has been proposed, the
renormalization group optimized perturbation theory (RGOPT) [24, 25, 23, 22],
that essentially combines a variational mass prescription with embedded
consistent RG invariance properties. Within QCD, at $T=\mu_{B}=0$, the method
has been used to estimate the coupling $\alpha_{s}$, predicting values [25]
compatible with the world averages [26]. Still at $T=0=\mu$, a precise
prediction was obtained for the quark condensate [27, 28]. In thermal theories
the RGOPT has been applied, e.g., to the simpler scalar $\phi^{4}$ model [23,
22] at NLO, showing how it improves the generic residual scale dependence as
compared to both standard thermal perturbation theory and SPT. Concerning QCD,
the direct application of RGOPT to the pure glue sector is however momentarily
obstructed by specific technical difficulties, as it involves new types of
rather involved thermal integrals [29]. Yet at least a nontrivial NLO
evaluation of the QCD pressure can be performed in a more simple-minded and
relatively easy way, if one considers the case of massive quarks and massless
gluons. In this vein we have recently applied the RGOPT to the QCD quark
sector only, while considering the gluons to be massless, in order to evaluate
the NLO pressure at finite densities and vanishing temperatures [30]. Our NLO
results show a good numerical agreement with higher order state of the art
${\cal O}(g^{3}\ln^{2}g)$ pQCD predictions, with a visible improvement
(although relatively modest for cold matter) of the residual scale dependence.
In the present work, we aim to extend the $T=0,\mu_{B}\neq 0$ application
performed in Ref. [30] in order to consider a thermal bath. Our strategy is to
apply our construction to the quark sector which, together with the purely
perturbative NLO contribution of massless gluons, will compose our complete
NLO QCD pressure: $P(T,\mu_{B})=P_{q}^{RGOPT}+P_{g}^{PT}$ where111 Notice that
since $P_{g}^{PT}$ vanishes at $T=0$ this is essentially the same procedure
adopted in Ref. [30].  $P_{g}^{PT}\sim T^{4}$. We believe that the results
reported here represent a significant step towards the determination of
thermodynamical observables with RG improved properties. Technical details
related to our calculation may be found in a companion paper [31].

Our starting point is the perturbative QCD pressure for three quark flavors
with degenerate masses, $m_{u}=m_{d}=m_{s}\equiv m$, and massless gluons:
$P=P_{q}^{PT}+P_{g}^{PT}$. Let us consider first the quark contribution
$P_{q}$ and how the RGOPT is built on it. At NLO (${\cal O}(g)$) the massive
quark contributions to the pressure can be obtained by combining the vacuum
results of Ref. [27] and $T,\mu\neq 0$ results of Refs. [32, 33]. The per
flavor result reads

$\displaystyle\frac{P^{PT}_{q}}{N_{f}\,N_{c}}$ $\displaystyle=$
$\displaystyle-\frac{m^{4}}{8\pi^{2}}\left(\frac{3}{4}-L_{m}\right)+2T^{4}J_{1}$
(1) $\displaystyle-$ $\displaystyle
3g\frac{m^{4}}{2\left(2\pi\right)^{4}}C_{F}\left(L_{m}^{2}-\frac{4}{3}L_{m}+\frac{3}{4}\right)$
$\displaystyle-$ $\displaystyle
gC_{F}\left\\{\left[\frac{m^{2}}{4\pi^{2}}\left(2-3L_{m}\right)+\frac{T^{2}}{6}\right]T^{2}J_{2}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{T^{4}}{2}J^{2}_{2}+m^{2}T^{2}J_{3}\right\\}\;,$

where $L_{m}=\ln(m/M)$, $g\equiv 4\pi\alpha_{s}(M)$, $M$ is the arbitrary
renormalization scale in the $\overline{\rm MS}$-scheme,
$C_{F}=(N_{c}^{2}-1)/(2N_{c})$, $N_{c}=3$, and $N_{f}=3$. In-medium and
thermal effects are included in the (dimensionless) single integrals:

$J_{1}=\int\frac{d^{3}{\bf\hat{p}}}{\left(2\pi\right)^{3}}\left\\{\ln\left[1+e^{-\left(E_{p}+\frac{\mu}{T}\right)}\right]+\ln\left[1+e^{-\left(E_{p}-\frac{\mu}{T}\right)}\right]\right\\},$
(2)

with ${\bf\hat{p}}\equiv{\bf p}/T$,
$E_{p}=\sqrt{{\bf\hat{p}}^{2}+m^{2}/T^{2}}$,

$J_{2}=\int\frac{d^{3}{\bf\hat{p}}}{\left(2\pi\right)^{3}}\frac{1}{E_{p}}\left[f^{+}(E_{p})+f^{-}(E_{p})\right]\;,$
(3)

and in the double integral (after angular integration over $p\cdot
q/(|p||q|)$)

$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\int^{\infty}_{0}\int^{\infty}_{0}\frac{d\hat{p}\,\hat{p}d\hat{q}\,\hat{q}}{E_{p}E_{q}}\left\\{\Sigma_{+}\ln\left[\frac{E_{p}E_{q}-\frac{m^{2}}{T^{2}}-\hat{p}\hat{q}}{E_{p}E_{q}-\frac{m^{2}}{T^{2}}+\hat{p}\hat{q}}\right]\right.$
(4) $\displaystyle+$
$\displaystyle\left.\Sigma_{-}\ln\left[\frac{E_{p}E_{q}+\frac{m^{2}}{T^{2}}+\hat{p}\hat{q}}{E_{p}E_{q}+\frac{m^{2}}{T^{2}}-\hat{p}\hat{q}}\right]\right\\}\;,$

where
$\Sigma_{\pm}=f^{+}\left(E_{p}\right)f^{\pm}\left(E_{q}\right)+f^{-}\left(E_{p}\right)f^{\mp}\left(E_{q}\right)$.

The Fermi-Dirac distributions for anti-quarks ($+$ sign) and quarks ($-$ sign)
read

$f^{\pm}(E_{p})=\frac{1}{1+e^{(E_{p}\pm\frac{\mu}{T})}}\;,$ (5)

where $\mu$ is the quark chemical potential, related to the baryonic chemical
potential via $\mu_{B}=3\mu$. In the present work we consider symmetric quark
matter and thus do not distinguish the chemical potentials associated with
different flavors ($\mu_{s}=\mu_{u}=\mu_{d}\equiv\mu$). For the quark sector
the Stefan-Boltzmann limit is

$\frac{P_{q}^{SB}}{N_{c}N_{f}}=T^{4}\left(\frac{7\pi^{2}}{180}\right)\left(1+\frac{120}{7}{\hat{\mu}}^{2}+\frac{240}{7}{\hat{\mu}}^{4}\right)\;,$
(6)

where ${\hat{\mu}}=\mu/(2\pi T)$.

Defining the standard homogenous RG operator

$M\frac{d}{dM}=M\,\partial_{M}+\beta(g)\,\partial_{g}-m\,\gamma_{m}(g)\,\partial_{m}\;,$
(7)

note that acting with Eq.(7) on the massive pressure Eq.(1) leaves a non RG-
invariant term of leading order, $\sim m^{4}\ln(M)$. To restore a
perturbatively RG-invariant massive pressure, we proceed as in Refs.[27, 23,
22] (or closer to the present case, as in Ref. [30]), subtracting a finite
zero point contribution,

$\frac{P^{RGPT}_{q}}{N_{c}N_{f}}=\frac{P^{PT}_{q}}{N_{c}N_{f}}-\frac{m^{4}}{g}\sum_{k}s_{k}g^{k}\;,$
(8)

where the $s_{i}$ are determined at successive orders so that

$M\frac{d}{dM}\left(\frac{P^{RGPT}_{q}}{N_{c}N_{f}}\right)=0\;,$ (9)

up to neglected higher order terms. Our evaluations being carried up to ${\cal
O}(g)$ it suffices to determine the first two $s_{0}$ and $s_{1}$
coefficients. They involve, through Eq.(7), coefficients of the $\beta$
function and anomalous mass dimension, $\gamma_{m}$, relevant to the $T=\mu=0$
pressure. Our normalizations are
$\beta\left(g\right)=-2b_{0}g^{2}-2b_{1}g^{3}+\mathcal{O}\left(g^{4}\right)$
and
$\gamma_{m}\left(g\right)=\gamma_{0}g+\gamma_{1}g^{2}+\mathcal{O}\left(g^{3}\right)$
where

$\displaystyle\left(4\pi\right)^{2}b_{0}$ $\displaystyle=$ $\displaystyle
11-\frac{2}{3}N_{f},$ (10) $\displaystyle\left(4\pi\right)^{4}b_{1}$
$\displaystyle=$ $\displaystyle 102-\frac{38}{3}N_{f},$ (11)
$\displaystyle\gamma_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\;,\;\;\left(4\pi\right)^{4}\gamma_{1}=\frac{404}{3}-\frac{40}{9}N_{f}\;.$
(12)

One then finds [27, 30]

$s_{0}=-\left[(4\pi)^{2}(b_{0}-2\gamma_{0})\right]^{-1},$ (13)
$s_{1}=-\frac{1}{4}\left[\frac{b_{1}-2\gamma_{1}}{4(b_{0}-2\gamma_{0})}-\frac{1}{12\pi^{2}}\right]\,.$
(14)

To implement the RGOPT in the quark sector, the QCD Lagrangian is first
variationally modified, according to the prescription [25, 30]

${\cal L}_{QCD}^{RGOPT}={\cal L}_{QCD}|_{g\to\delta
g}-m(1-\delta)^{a}{\overline{\psi}}_{f}\psi_{f},$ (15)

which is equivalent to replacing $g\to\delta g$ and $m\to m(1-\delta)^{a}$
within a perturbative expansion, here Eq.(1). Next one reexpands in powers of
$\delta$, setting $\delta\to 1$ to recover the massless case. We stress that
$m$ represents a generic variational mass parameter, which can also be seen as
an infrared regulator in the originally massless theory222However the method
can easily be generalized to accommodate physical current quark masses..

The next step is to fix the exponent $a$, introduced in Eq.(15), whose role is
crucial in our framework. Expanding to LO, ${\cal O}(\delta^{0})$, and
requiring the resulting pressure to satisfy the reduced (massless) RG
equation:

$\left[M\partial_{M}+\beta(g)\partial_{g}\right]P^{RGOPT}_{q}=0\;,$ (16)

leads to $a=\gamma_{0}/(2b_{0})$, in agreement with previous applications, to
which the reader is referred for further details [25, 27, 30, 31]. After
setting $\delta=1$ the resulting NLO RGOPT-modified pressure reads

$\displaystyle\frac{P^{RGOPT}_{q}}{N_{f}N_{c}}$ $\displaystyle=$
$\displaystyle-\frac{m^{4}}{8\pi^{2}}\left(\frac{3}{4}-L_{m}\right)+2T^{4}\,J_{1}$
(17) $\displaystyle+$
$\displaystyle\frac{m^{4}}{\left(2\pi\right)^{2}}\left(\frac{\gamma_{0}}{b_{0}}\right)\left(\frac{1}{2}-L_{m}\right)+m^{2}\left(\frac{\gamma_{0}}{b_{0}}\right)T^{2}\,J_{2}$
$\displaystyle-$ $\displaystyle
3g\frac{m^{4}}{2\left(2\pi\right)^{4}}C_{F}\left(L_{m}^{2}-\frac{4}{3}L_{m}+\frac{3}{4}\right)$
$\displaystyle-$ $\displaystyle
gC_{F}\left\\{\left[\frac{m^{2}}{4\pi^{2}}\left(2-3L_{m}\right)+\frac{T^{2}}{6}\right]T^{2}J_{2}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{T^{4}}{2}J^{2}_{2}+m^{2}T^{2}\,J_{3}\right\\}$
$\displaystyle+$
$\displaystyle\frac{m^{4}}{\left(4\pi\right)^{2}b_{0}}\left\\{\frac{1}{g}\left(1-\frac{\gamma_{0}}{b_{0}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left[\left(b_{1}-2\gamma_{1}\right)\pi^{2}-\frac{\left(b_{0}-2\gamma_{0}\right)}{3}\right]\right\\}\,.$

At NLO, Eq.(16) is no longer automatically satisfied, thus giving an
independent constraint. Accordingly in contrast with OPT/SPT, following [23,
22, 30] the remnant $m$ can be fixed in two different manners. Either, from
using a standard stationarity criterion[7], the mass optimization prescription
(MOP):

$\frac{\partial{P}^{RGOPT}_{q}}{\partial m}\Bigr{|}_{\overline{m}}\equiv 0\;,$
(18)

or alternatively from Eq.(16). The coupling $g(M)$ is determined from standard
PT running at NLO order, with the scale $M$ chosen as a multiple of $\pi T$ as
usual. However, regarding the quark pressure Eq.(17), both Eq.(18) and Eq.(16)
fail to give a real dressed mass $\overline{m}(g,T,\mu)$ for a substantial
part of the physically relevant $T,\mu$ range. The occurrence of non-unique
solutions at higher orders, some being complex, is a well-known burden with
OPT/SPT approaches. In contrast note that $a=\gamma_{0}/(2b_{0})$ in Eq.(15)
guarantees that the only acceptable solutions are those matching [25] the
asymptotic freedom (AF) behavior for $g\to 0$ at $T=0$, a compelling criteria
that often selects a unique solution. In addition the nonreal solution issue
can be cured in a RG consistent manner by performing a renormalization scheme
change (RSC) [25, 30, 31]. With this aim, we define a RSC acting only on the
variational mass in our framework,

$m\to m^{\prime}(1+B_{2}g^{2})\,,$ (19)

where a single $B_{2}$ parametrizes a NLO RSC from the original
${\overline{\text{MS}}}$-scheme. The latter induces an extra term
$-4gm^{4}s_{0}B_{2}$ in Eq.(1) (renaming afterwards $m^{\prime}\to m$ the
variational mass to avoid excessive notation changes), entering thus the MOP
Eq.(18) and RG Eq.(16). Now since we aim to solve the latter for exact $m$ and
$g$ dependence, Eq.(19) modifies those purposefully, when now considered as
constraints for the arbitrary mass $m$ after the (nonperturbative)
modifications induced from Eq.(15). Accordingly $B_{2}$ may be considered an
extra variational parameter, quite similarly to $m$, thus to be fixed by a
sensible prescription.
Considering specifically the RG Eq.(16), it can be conveniently written as a
quadratic form in $\ln(m^{2}/M^{2})$,

$-\ln\frac{m^{2}}{M^{2}}+B_{rg}\mp\frac{8\pi^{2}}{g}\sqrt{\frac{2}{3}D_{rg}}=0\;,$
(20)

where explicitly

$B_{rg}=-\frac{1}{b_{0}\,g}+\frac{172}{81}-\frac{64}{81}\left(\frac{4g}{9\pi^{2}}\right)\,\frac{1}{1+\frac{4g}{9\pi^{2}}}+8\pi^{2}\frac{T^{2}}{m^{2}}J_{2},$
(21) $\displaystyle D_{rg}$ $\displaystyle=$
$\displaystyle-\frac{g}{27}\frac{(4g+81\pi^{2})}{(4g+9\pi^{2})^{2}}-g^{2}\left(\frac{3}{7}B_{2}+\frac{11}{384\pi^{4}}\right)$
(22) $\displaystyle+$ $\displaystyle
g^{2}\,\frac{T^{4}}{m^{4}}J_{2}\left(J_{2}-\frac{1}{6}\right)-g^{2}\,\frac{T^{2}}{m^{2}}J_{3}.$

Note that a quite similar quadratic form can be obtained for the MOP Eq.(18),
with its specific expressions $B_{rg}\to B_{mop}$, $D_{rg}\to D_{mop}$ [31].
As above anticipated, in the original $\overline{\rm MS}$-scheme ($B_{2}=0$)
one can have $D_{rg}<0$ (or similarly $D_{mop}<0$) in some physically relevant
parameter ranges due to some negative contributions, leading thus to nonreal
NLO $\overline{m}(T,\mu)$ solutions. Accordingly to recover real solutions in
a large range, while at the same time fulfilling the crucial AF-matching
requirement, the comprehensive analysis performed in Ref. [31] suggests the
following prescriptions: The arbitrary RSC parameter $B_{2}$ is fixed by
partly (respectively fully) cancelling $D_{mop}$ (respectively $D_{rg}$). For
the RG prescription explicitly

$D_{rg}(B_{2})=0\,$ (23)

fixes $B_{2}$ using Eq.(22). It gives a single real solution for
$\overline{m}$, determined by the first two terms of Eq.(20), the latter being
still an implicit equation in $m$ for $T,\mu\neq 0$ via $J_{2}$ entering
Eq.(21). The prescription Eq.(23) may be seen at first as a rather peculiar
choice, but there happens to be very few other possible prescriptions to
recover a real RG solution. Note that the resulting $\overline{m}(B_{2})$
still involves arbitrary higher order contributions, as well as nontrivial
$T,\mu$ dependence via $B_{rg}$ in Eq.(21). A similar analysis holds for the
MOP Eq.(18), but leading to a $\overline{m}$ having quite different properties
(we refer to Ref.[31] for details). We stress that for the two prescriptions
the resulting $\overline{m}(B_{2})$ is an intermediate variational parameter
without much physical meaning outside its use in the pressure. As illustrated
below both prescriptions give drastically reduced remnant scale dependence as
compared to pQCD, but the best results are obtained from the RG prescription
Eqs.(20),(23). This is not very surprising as the latter more directly embeds
RG properties as compared to Eq.(18). However for more complete and
conservative coverage of the possible variants at NLO, we will illustrate both
RG and MOP prescription results below.

Coming to the full QCD pressure, we simply add to Eq.(17) the NLO glue
contributions [34],

$P^{PT}_{g}=\frac{8\pi^{2}}{45}T^{4}\,\left[1-\frac{15}{(4\pi)^{2}}\,g\right]\,.$
(24)

Thus we can now compare the NLO RGOPT full QCD results with those from HTLpt
[17, 19] and (massless) pQCD [20], as well as with available LQCD data from
Refs. [35, 36, 37]. For the numerical evaluations of NLO quantities we
consider the exact two-loop running coupling (see, e.g., Ref. [30]) obtained
by solving for $g(M)$

$\ln\frac{M}{\Lambda_{\overline{\text{MS}}}}=\frac{1}{2b_{0}\,g}+\frac{b_{1}}{2b_{0}^{2}}\ln\left(\frac{b_{0}g}{1+\frac{b_{1}}{b_{0}}g}\right),$
(25)

for a given $\Lambda_{\overline{\text{MS}}}$ value (this also defines
$\Lambda_{\overline{\text{MS}}}$ at two-loop level in our conventions). For
numerical purposes we adopt $\Lambda_{\overline{\rm MS}}=335\,{\rm MeV}$ for
$N_{f}=3$, which is very close to the latest world average value[26]. Notice
that, for consistency the NNLO HTLpt[19] and ${\cal O}(g^{3}\ln g)$ pQCD[20]
numerical results reproduced here have been obtained rather with a three-loop
order running coupling.

Figure 1:  NLO RGOPT (RG prescription) plus NLO $P^{PT}_{g}$ pressure (brown
band) compared to ${\rm N}^{3}{\rm LO}\,g^{3}\ln g$ pQCD (light blue band),
NLO HTLpt (light green band) and NNLO HTLpt (light red band), with scale
dependence $\pi T\leq M\leq 4\pi T$, and to lattice data [35, 36, 37] at
$\mu_{B}=0$.

Fig. 1 shows the RGOPT QCD pressure normalized by $P_{SB}\equiv
P^{q}_{SB}+P^{g}_{SB}$ as a function of $T$ for $\mu_{B}=0$, obtained with our
best RG prescription, Eqs.(20),(23). One notices that for $T\gtrsim 0.25\,{\rm
GeV}$ our results display a remarkable agreement with the LQCD data of [35]
all the way up to $T=1\,{\rm GeV}$ (the highest value considered in those
simulations), as well as a very good agreement with more recent LQCD data [36]
at intermediate $T$. Furthermore the RGOPT results are drastically more stable
than pQCD and HTLpt when $M$ is varied, as clearly indicated by the different
band widths associated to the different approximations. The NLO RGOPT results
at central scale, $M=2\pi T$, observe a better agreement with LQCD data from
[35] than NNLO HTLpt for $0.5\,{\rm GeV}\lesssim T\lesssim 1\,{\rm GeV}$,
while the latter lies closer to the higher $T$ data of [37] as compared to
RGOPT, showing sizeable differences. A concomitant feature however is the
visible tension between low [35] and higher $T$ [37] LQCD data in their common
range [NB we show only statistical uncertainties for LQCD, as given in
publically available files[35, 36, 37]]. In Fig. 1 one also notices that HTLpt
at NLO and pQCD at ${\cal O}(g^{3}\ln g$) are still far from lattice data,
moreover the NLO HTLpt stays close to SB limit at intermediate to low
temperatures. We further mention that while the RGOPT band width illustrated
correctly reflects the total remnant scale uncertainty resulting from both the
(RG resummed) $P_{q}(M)$ and (standard perturbative) NLO $P_{g}$, the sole
$P_{q}(M)$ uncertainty would be roughly comparable [31] to the lattice error
bars for $T\gtrsim 0.5\,{\rm GeV}$.
Next for completeness in Fig. 2 similar results are shown for the other
possible MOP prescription Eq.(18), compared to lattice data and highest order
pQCD. As anticipated this prescription is somewhat less efficient than the RG
one, regarding the remnant scale uncertainty as well as lattice data
comparisons. Yet with respect to pQCD or to (NLO) HTLpt, overall it also
appears as a sharp improvement, keeping in mind our NLO approximation.

Figure 2: NLO RGOPT (MOP prescription) plus NLO $P^{PT}_{g}$ pressure as
function of $T$ at $\mu_{B}=0$ (grey band) compared to ${\rm
N}^{3}LO\,g^{3}\ln g$ pQCD (light blue band), with scale dependence $\pi T\leq
M\leq 4\pi T$, and to lattice data [35, 36, 37].

Figure 3:  NLO RGOPT (RG prescription) trace anomaly
$\Delta\equiv\varepsilon-3P$, including NLO $\Delta^{PT}_{g}$ (brown band),
compared to lattice data [35, 36, 37]. The additional dashed lines illustrate
the scale uncertainty originating solely from RGOPT quark contributions within
the full scale uncertainty added by $\Delta_{g}^{PT}$ (brown) band.

Another physically interesting quantity is the interaction measure,
$\Delta={\cal E}-3P$, [or at $\mu_{B}=0$ equivalently
$\Delta=T^{5}\,\partial(P/T^{4})/\partial T$, since ${\cal E}=-P+{\cal S}T$
with ${\cal S}=\partial P/\partial T$ representing the entropy density]. In
Fig. 3 the NLO RGOPT interaction measure results, obtained by a numerical
derivative from our best RG prescription above, are compared to the available
LQCD data [35, 36, 37] at $\mu_{B}=0$. At temperatures $0.3\,{\rm GeV}\lesssim
T\lesssim 1\,{\rm GeV}$ a very good agreement is observed.

Conclusions. In this work we have compared RGOPT predictions regarding the QCD
pressure with lattice data for the first time. Our NLO predictions for the
central scale, $M=2\pi T$, turned out to compare very well for temperatures
starting at $T\simeq 0.25\,{\rm GeV}$ which lies within the nonperturbative
regime ($\alpha_{s}\simeq 0.31$). This agreement persists up to $T=1\,{\rm
GeV}$, the highest value for the LQCD data [35]. Furthermore, comparing our
NLO results with those from NNLO HTLpt one observes that the consistent RG
invariance properties native to the RGOPT are drastically attenuating the
remnant scale dependence issue. While the striking agreement with lattice data
of Ref. [35] in Fig. 1 may be partly numerically accidental, variants of our
prescription in Fig. 2 still appear in very good agreement given our
essentially NLO-based construction, as compared to the state-of-the-art
perturbative higher order QCD. The differences of our results with higher
$1\,{\rm GeV}\lesssim T\lesssim 2\,{\rm GeV}$ LQCD data [37] are however
visible. Incidentally the LQCD results in [35] and in [37] appear to be in
tension in their common range, while the trace anomaly shows more continuity,
a feature that may call for more investigations independently of our results.
Regarding the comparisons with $2+1$ flavor LQCD here illustrated, one may
also keep in mind our presently not fully realistic approximation of $N_{f}=3$
degenerate flavors. We recall also that owing to present technical
difficulties in readily applying our machinery to the gluon sector, we have
adopted a simple-minded strategy in which variational nonperturbative
evaluations were carried out solely in the quark sector, supplemented by the
purely perturbative NLO glue contribution. The apparent robustness of the
results obtained in this way allows us to conclude that this simple
prescription seems nevertheless appropriate to describe moderate to high-$T$
regimes starting at $T\simeq 0.25\,{\rm GeV}\sim 1.5\,T_{pc}$. Finally we
briefly mention that the NLO RG-improved properties exhibited here extend
without degradation to sizeable chemical potential values (for some
illustrations we refer to our companion paper [31]). The latter indicates the
potential of our approach towards a more systematic exploration of both hot
and dense QCD.

Acknowledgments: We thank P. Petreczky for bringing the results of Ref. [37]
to our attention. M.B.P. and T.E.R. are partially supported by Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and by
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-(CAPES-
Brazil)-Finance Code 001. This work was also financed in part by INCT-FNA
(Process No. 464898/2014-5).

## References

  * [1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature 443, 675 (2006); Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz, S. Krieg and K. K. Szabo, JHEP 06, 088 (2009); S. Borsanyi et al. [Wuppertal-Budapest], JHEP 09, 073 (2010); A. Bazavov, T. Bhattacharya, M. Cheng, C. DeTar, H. T. Ding, S. Gottlieb, R. Gupta, P. Hegde, U. M. Heller and F. Karsch, et al. Phys. Rev. D 85, 054503 (2012).
  * [2] M. Buballa, Phys. Rept. 407, 205 (2005).
  * [3] P. de Forcrand, PoS LAT 2009, 010 (2009); G. Aarts, J. Phys. Conf. Ser. 706, 022004 (2016).
  * [4] J. P. Blaizot, E. Iancu and A. Rebhan, In *Hwa, R.C. (ed.) et al.: Quark gluon plasma* 60-122 [hep-ph/0303185]; U. Kraemmer and A. Rebhan, Rept. Prog. Phys. 67, 351 (2004).
  * [5] M. Laine and A. Vuorinen, Lect. Notes Phys. 925, 1 (2016).
  * [6] J. Ghiglieri, A. Kurkela, M. Strickland and A. Vuorinen, Phys. Rept. 880, 1 (2020).
  * [7] P.M. Stevenson, Phys. Rev. D 23, 2916 (1981); Nucl. Phys. B 203, 472 (1982).
  * [8] A. Okopinska, Phys. Rev. D 35, 1835-1847 (1987) doi:10.1103/PhysRevD.35.1835;
  * [9] H. Yamada, Z. Phys. C 59, 67-76 (1993) doi:10.1007/BF01555840
  * [10] A. Duncan and M. Moshe, Phys. Lett. B 215, 352-358 (1988) doi:10.1016/0370-2693(88)91447-5; H. F. Jones and M. Moshe, Phys. Lett. B 234, 492-496 (1990) doi:10.1016/0370-2693(90)92045-K .
  * [11] R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986); H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys. Lett. B 434, 74 (1998); Phys. Rev. D 60, 085001 (1999); Mod. Phys. Lett. B 17, 1011 (2003).
  * [12] F. Karsch, A. Patkos and P. Petreczky, Phys. Lett. B 401, 69 (1997); S. Chiku and T. Hatsuda, Phys. Rev. D 58, 076001 (1998); J. O. Andersen, E. Braaten and M. Strickland, Phys. Rev. D 63, 105008 (2001); J. O. Andersen and M. Strickland, Phys. Rev. D 64, 105012 (2001); J. O. Andersen and M. Strickland, Annals Phys. 317, 281 (2005).
  * [13] J. O. Andersen, L. Kyllingstad and L. E. Leganger, JHEP 0908, 066 (2009).
  * [14] E. Braaten and R. D. Pisarski, Phys. Rev. D 45, 1827 (1992).
  * [15] J. O. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83, 2139 (1999); J. O. Andersen, E. Braaten and M. Strickland, Phys. Rev. D 61, 074016 (2000).
  * [16] J. O. Andersen, M. Strickland and N. Su, Phys. Rev. Lett. 104, 122003 (2010); JHEP 1008, 113 (2010).
  * [17] N. Haque, M. G. Mustafa and M. Strickland, Phys. Rev. D 87, 105007 (2013).
  * [18] J. O. Andersen, L. E. Leganger, M. Strickland and N. Su, JHEP 1108, 053 (2011); S. Mogliacci, J. O. Andersen, M. Strickland, N. Su and A. Vuorinen, JHEP 1312, 055 (2013); N. Haque, J. O. Andersen, M. G. Mustafa, M. Strickland and N. Su, Phys. Rev. D 89, 061701 (2014).
  * [19] N. Haque, A. Bandyopadhyay, J. O. Andersen, M. G. Mustafa, M. Strickland and N. Su, JHEP 1405, 027 (2014).
  * [20] K. Kajantie, M. Laine, K. Rummukainen and Y. Schroder, Phys. Rev. D 67, 105008 (2003).
  * [21] A. Vuorinen, Phys. Rev. D 68, 054017 (2003).
  * [22] J. L. Kneur and M. B. Pinto, Phys. Rev. D 92, 116008 (2015).
  * [23] J. L. Kneur and M. B. Pinto, Phys. Rev. Lett. 116, 031601 (2016).
  * [24] J. L. Kneur and A. Neveu, Phys. Rev. D 81, 125012 (2010).
  * [25] J. L. Kneur and A. Neveu, Phys. Rev. D 88, 074025 (2013).
  * [26] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018).
  * [27] J. L. Kneur and A. Neveu, Phys. Rev. D 92, 074027 (2015).
  * [28] J. L. Kneur and A. Neveu, Phys. Rev. D 101, 074009 (2020).
  * [29] J. L. Kneur and M. B. Pinto; in preparation.
  * [30] J. L. Kneur, M. B. Pinto and T. E. Restrepo, Phys. Rev. D 100, 114006 (2019)
  * [31] J. L. Kneur, M. B. Pinto and T. E. Restrepo, “Renormalization group improved pressure for hot and dense quark matter”, arXiv:2021.01.xxx, to appear.
  * [32] J. I. Kapusta and C. Gale, “Finite-temperature field theory: Principles and applications” (Cambridge University Press, 2006).
  * [33] M. Laine and Y. Schröder, Phys. Rev. D 73, 085009 (2006).
  * [34] E. V. Shuryak, Sov. Phys. JETP 47, 212 (1978); S. A. Chin, Phys. Lett. B 78, 552 (1978).
  * [35] S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, JHEP 11, 077 (2010).
  * [36] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg and K. K. Szabo, Phys. Lett. B 730, 99 (2014).
  * [37] A. Bazavov, P. Petreczky and J. H. Weber, Phys. Rev. D 97, 014510 (2018).


Aehlig, and personMartin Hofmann.
Resource Aware ML. In
booktitleComputer Aided Verification - 24th International
Conference, CAV.

[Lahiri, Vaswani, and HoareLahiri
et al2010]
authorpersonShuvendu K. Lahiri, personKapil
Vaswani, and personC. A. R. Hoare.
Differential static analysis: opportunities,
applications, and challenges. In booktitleProceedings of
the Workshop on Future of Software Engineering Research, FoSER 2010, at the
18th ACM SIGSOFT International Symposium on Foundations of Software
Engineering, editorpersonGruia-Catalin Roman
and personKevin J. Sullivan (Eds.).

[Lichtman and HoffmannLichtman and
authorpersonBenjamin Lichtman and
personJan Hoffmann. year2017.
Arrays and References in Resource Aware ML. In
booktitleThe 2nd International Conference on Formal
Structures for Computation and Deduction, FSCD.

[Nanevski, Banerjee, and GargNanevski
et al2013]
authorpersonAleksandar Nanevski,
personAnindya Banerjee, and personDeepak Garg.
Dependent Type Theory for Verification of
Information Flow and Access Control Policies.
journalACM Trans. Program. Lang. Syst.
volume35, number2 (year2013).

[Nanevski, Morrisett, and BirkedalNanevski
et al2008]
authorpersonAleksandar Nanevski,
personJ. Gregory Morrisett, and personLars
Birkedal. year2008.
Hoare type theory, polymorphism and separation.
journalJ. Funct. Program. volume18,
number5-6 (year2008).

[Neis, Dreyer, and RossbergNeis
et al2011]
authorpersonGeorg Neis, personDerek Dreyer,
and personAndreas Rossberg. year2011.
Non-parametric parametricity.
journalJ. Funct. Program. volume21,
number4-5 (year2011), pages497–562.

[Ngo, Dehesa-Azuara, Fredrikson, and
HoffmannNgo et al2017]
authorpersonVan Chan Ngo, personMario
Dehesa-Azuara, personMatt Fredrikson, and personJan
Hoffmann. year2017.
Verifying and Synthesizing Constant-Resource
Implementations with Types. In booktitle2017 IEEE Symposium
on Security & Privacy.

[Nielson and NielsonNielson and
authorpersonFlemming Nielson and
personHanneRiis Nielson. year1999.
Type and Effect Systems.
In booktitleCorrect System Design.
seriesLecture Notes in Computer Science,
Vol. volume1710. pages114–136.

[Pierce and TurnerPierce and Turner2000]
authorpersonBenjamin C. Pierce and
personDavid N. Turner. year2000.
Local Type Inference.
journalACM Trans. Program. Lang. Syst.
volume22, number1 (dateJan.
year2000), pages1–44.

[Radicek, Barthe, Gaboardi, Garg, and
ZulegerRadicek et al2018]
authorpersonIvan Radicek, personGilles
Barthe, personMarco Gaboardi, personDeepak Garg,
and personFlorian Zuleger. year2018.
Monadic refinements for relational cost analysis.
journalPACMPL volume2,
numberPOPL (year2018).

[Reistad and GiffordReistad and
authorpersonBrian Reistad and
personDavid K. Gifford. year1994.
Static Dependent Costs for Estimating Execution
Time. In booktitleProceedings of the 1994 ACM Conference on
LISP and Functional Programming (seriesLFP '94).

[Sinn, Zuleger, and VeithSinn
et al2014]
authorpersonMoritz Sinn, personFlorian
Zuleger, and personHelmut Veith.
A Simple and Scalable Approach to Bound Analysis
and Amortized Complexity Analysis. In booktitleComputer
Aided Verification - 26th International Conference, CAV.

[Turon, Thamsborg, Ahmed, Birkedal, and
DreyerTuron et al2013]
authorpersonAaron Joseph Turon, personJacob
Thamsborg, personAmal Ahmed, personLars Birkedal,
and personDerek Dreyer. year2013.
Logical relations for fine-grained concurrency. In
booktitleThe 40th Annual ACM SIGPLAN-SIGACT Symposium on
Principles of Programming Languages, POPL '13, Rome, Italy - January 23 -
25, 2013. pages343–356.

[Unno, Torii, and SakamotoUnno
et al2017]
authorpersonHiroshi Unno, personSho Torii,
and personHiroki Sakamoto. year2017.
Automating Induction for Solving Horn Clauses. In
booktitleComputer Aided Verification - 29th International
Conference, CAV.

[Wang, Wang, and ChlipalaWang
et al2017]
authorpersonPeng Wang, personDi Wang, and
personAdam Chlipala. year2017.
TiML: A Functional Language for Practical
Complexity Analysis with Invariants. In
booktitleProceedings of the International Conference on
Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA).

[Xi and PfenningXi and Pfenning1999]
authorpersonHongwei Xi and personFrank
Pfenning. year1999.
Dependent Types in Practical Programming. In
booktitleProceedings of the Symposium on Principles of
Programming Languages (POPL).

[Zhang, Wang, Suh, and MyersZhang
et al2015]
authorpersonDanfeng Zhang, personYao Wang,
personG. Edward Suh, and personAndrew C. Myers.
A Hardware Design Language for Timing-Sensitive
Information-Flow Security. In booktitleProceedings of the
Twentieth International Conference on Architectural Support for Programming
Languages and Operating Systems, ASPLOS.