# Scalar Dark Matter in $A_{4}$ based texture one-zero neutrino mass model
within Inverse Seesaw Mechanism

Rishu Verma, Monal Kashav, Surender Verma and B. C. Chauhan
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Department of Physics and Astronomical Science,
Central University of Himachal Pradesh, Dharamshala 176215, INDIA.)

###### Abstract

In this paper, we present a model based on $A_{4}$ discrete flavor symmetry
implementing inverse and type-II seesaw mechanisms to have LHC accessible TeV
scale right-handed neutrino mass and texture one-zero in the resulting
Majorana neutrino mass matrix, respectively. We investigate neutrino and dark
matter sectors of the model. Non-Abelian discrete $A_{4}$ symmetry
spontaneously breaks into $Z_{2}$ subgroup and hence provide stable dark
matter candidate. To constrain the Yukawa Lagrangian of our model, we imposed
$Z^{\prime}_{2}$, $Z_{3}$ and $Z_{4}$ cyclic symmetries in addition to the
$A_{4}$ flavor symmetry. In this work we used the recently updated data on
cosmological parameters from PLANCK 2018. For the dark matter candidate mass
around 45 GeV-55 GeV, we obtain the mediator particle mass(right-handed
neutrinos) ranging from 138 GeV to 155 GeV. The Yukawa couplings is found to
be in the range 0.995-1 to have observed relic abundance of dark matter. We,
further, obtain inverse ($X\equiv\frac{F^{2}n}{z^{2}}$) and type-II
($X^{{}^{\prime}}\equiv f_{1}v_{\Delta_{1}}$) seesaw contributions to
$0\nu\beta\beta$ decay amplitude $|M_{ee}|$, while model being consistent with
low energy experimental constraints. In particular, we emphasize that type-II
seesaw contribution to $|M_{ee}|$ is large as compared to inverse seesaw
contribution for normally ordered(NO) neutrino masses.

##  1 Introduction

Standard Model of particle physics is a low energy effective theory which has
been astonishingly successful in explaining the dynamics of fundamental
particles and their interactions. The discovery of Higgs Boson in 2012 at CERN
LHC has strengthen our belief in this incredible theory. Despite its
tremendous success there still remain unanswered questions such as origin of
neutrino mass, dark matter and matter-antimatter asymmetry, to name a few.
Neutrino oscillation experiments have been very instrumental in our quest to
understand Standard Model(SM) predictions for the leptonic sector. They have
shown at high level of statistical significance that neutrinos have non-zero
but tiny mass, and flavor and mass eigenstates mix giving rise to quantum
mechanical phenomena of neutrino oscillations.

On the contrary, within the SM, neutrinos are massless because the Higgs field
cannot couple to the neutrinos due to the absence of right-handed(RH)
neutrinos. The extension of SM with RH neutrino, however, require unnatural
fine tuning of the Yukawa couplings to generate sub-eV neutrino masses.
Dimension five Weinberg operator can generate the tiny Majorana mass for
neutrinos with the SM Higgs field. In fact, there exist several beyond the
Standard Model(BSM) scenarios, e.g. seesaw mechanisms, which may explain the
origin of such dimension five operator and can account for dynamical origin of
tiny Majorana neutrino masses by appropriately extending the field content of
the SM. For example in type-I, type-II and type-III seesaw RH neutrinos,
scalar triplet(s) and fermion triplet(s) are introduced to the particle
content of the SM, respectively[1, 2, 3, 4, 5, 6].

Seesaw mechanisms provide the most elegant and natural explanation of the
smallness of neutrino masses. The fundamental basis of seesaw mechanism is the
existence of lepton number violation at some high energy scale. In order to
have neutrino mass at sub-eV, the new physics scale must be of the order of
GUT scale, 1016 GeV. Therefore, the seesaw mechanisms explain the tiny non-
zero neutrino masses, but they introduce new physics scale, which is beyond
the reach of current and near future accelerator experiments.

On the other hand, there are several astonishing astrophysical observations
such as (i) galaxy cluster investigation[7], (ii) rotation curves of
galaxy[8], (iii) recent observations of bullet cluster[9], and (iv) the latest
cosmological data from Planck collaboration[10], which have proven the
existence of non-luminous and non-baryonic anatomy of matter known as “Dark
Matter”(DM). Apart from the astrophysical environments, it is very difficult
to probe the existence of DM in terrestrial laboratory. Alternatively, one can
accredit a weak interaction property to the DM through which it get
thermalized in the early Universe, which also can be examined at the
terrestrial laboratories. According to the Planck data the current DM
abundance is[10]

$\Omega_{DM}h^{2}=0.120\pm 0.001$.

Apart from relic abundance, the particle nature of DM is still unknown. Within
the SM of particle physics, there is no suitable candidate for DM, as it
should be stable on the cosmological time scale. The uncertainty in the nature
of DM and possible mechanisms of neutrino mass generation have opened up a
window to explore new models in a cohesive way. Out of several neutrino mass
models, the texture zero models are very interesting due to their rich
phenomenology and high predictability[11, 12, 13, 14, 15, 16, 17, 18, 19, 20].
In fact, texture zero models have also been successfully investigated to
generate observed matter-antimatter asymmetry in different seesaw settings[21]
and scotogenic scenarios[22]. The DM phenomenology within the framework of
texture zeros using type-I seesaw and scotogenesis have been studied in Refs.
[23] and [24], respectively.

Another possibility amongst the phenomenological approaches is the existence
of scaling structure in the neutrino mass matrix wherein third column is
scaled with respect to the second column by some model dependent
parameter(s)[25, 26]. The scaling ansatz has been studied in Ref.[27] where
neutrino mass is generated using inverse and type-II seesaw frameworks. In
Ref.[27], the implementation of inverse seesaw resulted in scaled neutrino
mass matrix which predicts vanishing lightest mass eigenvalue(inverted
neutrino mass ordering) and vanishing reactor mixing angle($\theta_{13}=0$).
Subsequently, within an effective field theory approach, in order to have non-
zero reactor mixing angle($\theta_{13}\neq 0$) type-II seesaw perturbation has
been incorporated in the Lagrangian. We, in this work, confine to tree level
dimension-4 and focus on the possible realization of texture one-zero ansatz
which can be embedded in more general framework of grand unified theories
wherein quarks and leptons belong to the same multiplet. For example, in
Refs.[28, 29], texture zero(s) in fermion mass matrices have been investigated
under SO(10) environment.

In this work, we investigate a well-motivated possibility for simultaneous
explanation of DM and non-zero neutrino mass using $A_{4}$ non-Abelian
discrete symmetry within the framework of inverse seesaw(ISS)[30, 31, 32]
wherein small neutrino masses emanate from new physics at TeV scale, which is
within the reach of accelerator experiments. The stability of DM is assured by
$Z_{2}$ symmetry. Type-II seesaw has been implemented to have one-zero in the
effective neutrino mass matrix. Within ISS mechanism, neutrino masses are
generated assuming three right-handed neutrinos $N_{T}$ and three additional
SM singlet neutral fermions $S_{T}$($T=1,2,3$). The fermionic singlets
($N_{4,5}$ and $S_{4,5}$) are assumed to have Yukawa couplings with the scalar
fields $H,\phi,\phi_{R}$ and $\phi_{S}$, which after spontaneous symmetry
breaking provide diagonal Majorana mass matrix($\mu$). The scalar triplets
$\Delta_{1}$ and $\Delta_{2}$ are incorporated, so that $M_{\nu}$ contain one
vanishing element after type-II seesaw implementation. Along with DM
abundance, we have also obtained prediction of the model for effective
Majorana mass ($|M_{ee}|$) appearing in neutrinoless double
beta($0\nu\beta\beta$) decay.

The paper is structured as follows. In section 2, we have discussed the
inverse seesaw mechanism based on $G_{f}\equiv A_{4}\times
Z_{2}^{{}^{\prime}}\times Z_{3}\times Z_{4}$ symmetry group and resulting
neutrino mass matrices. Section 3 is devoted to the investigation of relic
density of the DM. In section 4, the prediction of the model for neutrinoless
double beta decay is discussed. Finally, conclusions are summarized in section
5.

##  2 The Model

In order to explain the smallness of neutrino mass, different versions of the
seesaw mechanism play an important role. As discussed above the ISS mechanism
is a viable scenario to get the mass of right-handed neutrino near the TeV
scale. This scale is much below the scale, which we get from the canonical
seesaw. As a requirement of ISS, the fermion sector is extended by three
right-handed neutrinos $N_{i}(i=1,2,3)$ and three extra singlet fermions
$S_{j}(j=1,2,3)$. Within ISS mechanism the mass Lagrangian is written as

$L=-\bar{\nu}_{\alpha L}m_{D}N_{i}-\bar{S_{j}}mN_{i}-\frac{1}{2}\bar{S_{j}}\mu
S^{C}_{k}+h.c.$ (1)

where $m_{D}$, $m$ and $\mu$ are the 3$\times$3 complex mass matrices and
$\alpha=(e,\mu,\tau)$, $k=(1,2,3)$. Here $m$ represents lepton number
conserving interaction between neutral fermions and right-handed neutrinos and
$\mu$ gives the Majorana mass terms for neutral fermions. Assuming lepton
number as approximate symmetry, the Majorana mass term for right-handed
neutrino is vanishing. However, it is mildly violated through singlet fermions
$S$ having small mass $\mu$ in consonance with ’t Hooft’s naturalness
criterion[33]. The lepton number symmetry can be restored as $\mu\rightarrow
0$. Consequent to spontaneous symmetry breaking (SSB), the Lagrangian in
Eqn.(1) leads to 9$\times$9 neutrino mass matrix

$M_{\nu}=\begin{pmatrix}0&m_{D}&0\\\ m_{D}^{T}&0&m\\\ 0&m^{T}&\mu\\\
\end{pmatrix},\\\ $ (2)

in the basis ($\nu_{L},N,S$). We can obtain standard model neutrinos at sub-eV
scale from $m{{}_{D}}$ at electroweak scale, $\mu$ at keV scale and $m$ at TeV
scale as explained in [32, 34]. Thus, if we consider the order
$\mu<<m_{D}<<m$, then after the block diagonalization of above matrix, the
3$\times$3 effective neutrino mass matrix is obtained as

$m_{\nu}=m_{D}(m{{}^{T}})^{-1}\mu m^{-1}m_{D}^{T}.\\\ $ (3)

It is clear from Eqn.(3) that there is a double suppression by mass term
associated with $m$, which results in the scale that is much below to the one
obtained by canonical seesaw. The essence of inverse seesaw mechanism lies in
the fact that we can bring down the mass of right-handed neutrinos to TeV
scale by assuming that $\mu$ should be at keV scale [35, 36, 37].

The symmetry group $A_{4}$ has played an important role in understanding
particle physics [38, 39, 40, 41, 42]. $A_{4}$ is a non-Abelian discrete
symmetry group of even permutations of four objects. Order of this group is
12. All the 12 elements are generated from two elements, S and T which
satisfies: $S^{2}$ = $T^{3}$ = $(ST)^{3}$. It is a symmetry group of regular
tetrahedron. It has four conjugacy classes, therefore, four irreducible
representations: 1, 1′, 1′′ and 3. The multiplication rules of irreducible
representations in T basis are [42, 43]: 1${}^{\prime}\otimes$1′=1′′,
1${}^{\prime\prime}\otimes$1′′=1′, 1${}^{\prime}\otimes$1′′=1,
3$\otimes$3=1$\oplus$1${}^{\prime}\oplus$1${}^{\prime\prime}\oplus$3${}_{s}\oplus$3a
where,

$\displaystyle\left(\bf{3}\otimes\bf{3}\right)_{\bf{1}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3},$
$\displaystyle\left(\bf{3}\otimes\bf{3}\right)_{\bf{1^{\prime}}}=a_{1}b_{1}+\omega
a_{2}b_{2}+\omega^{2}a_{3}b_{3},$
$\displaystyle\left(\bf{3}\otimes\bf{3}\right)_{\bf{1^{\prime\prime}}}=a_{1}b_{1}+\omega^{2}a_{2}b_{2}+\omega
a_{3}b_{3},$
$\displaystyle\left(\bf{3}\otimes\bf{3}\right)_{\bf{3_{s}}}=\left(a_{2}b_{3}+b_{2}a_{3},a_{3}b_{1}+a_{1}b_{3},a_{1}b_{2}+a_{2}b_{1}\right),$
$\displaystyle\left(\bf{3}\otimes\bf{3}\right)_{\bf{3_{a}}}=\left(a_{2}b_{3}-b_{2}a_{3},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}\right).$

Here $a_{i}$ and $b_{i}$ (i = 1,2,3) are the basis vectors of the two triplets
and $\omega$ = $e^{\frac{2\pi i}{3}}$.

In the model we have taken five right-handed neutrinos, three of which
$N_{T}=(N_{1},N_{2},N_{3})$ are transforming as triplet under $A_{4}$ and rest
of the two i.e., $N_{4}$ and $N_{5}$ are transforming as singlets
$1,1^{\prime}$, respectively. Singlet fermions $S_{T}=(S_{1},S_{2},S_{3})$ and
$(S_{4},S_{5})$ transforming as triplet and singlets $(1,1^{\prime})$ under
$A_{4}$, respectively, are also introduced. The standard model Higgs doublet
$H$ and three additional Higgs doublets $\eta_{i}$ transform as singlet and
triplet $\eta$ under $A_{4}$, respectively. In addition, we have extended the
scalar sector with three $SU(2)_{L}$ singlet scalar fields i.e. $\phi$,
$\phi_{R}$ and $\phi_{S}$. After spontaneous symmetry breaking (SSB), the
vacuum expectation values ($vev$) acquired by ($H$, $\eta$) and ($\phi$,
$\phi_{R}$, $\phi_{S}$) give $m_{D}$ and ($m$, $\mu$), respectively, with
minimal number of parameters. In order to have possible lepton number
violation via $S_{i}(i=T,4,5)$ only, we distinguish the Yukawa interactions of
$N_{i}$ and $S_{i}$ through $Z_{2}^{{}^{\prime}}$ symmetry. Also, using
$Z_{3}$ symmetry, all possible $N_{i}N_{j}(i,j=T,4)$ Majorana terms are
inhibited. There can be possibility of Yukawa interactions like
$N_{T}N_{T}\phi_{S}^{*}$, $N_{4}N_{4}\phi_{S}^{*}$ which are suppressed by
$Z_{4}$ symmetry in the model. The fermionic and scalar field content along
with respective charge assignments are shown in Table 1 and Table 2,
respectively.

Symmetry | $\bar{L}_{e}$ | $\bar{L}_{\mu}$ | $\bar{L}_{\tau}$ | $e_{R}$ | $\mu_{R}$ | $\tau_{R}$ | $N_{T}$ | $N_{4}$ | $N_{5}$ | $S_{T}$ | $S_{4}$ | $S_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$A_{4}$ | 1 | 1′ | 1′′ | 1 | 1′′ | 1′ | 3 | 1 | 1′ | 3 | 1 | 1′
$Z^{\prime}_{2}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1
$Z_{3}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | 1 | 1 | 1 | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega$
$Z_{4}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $i$ | $i$ | $i$
Table 1: Fermion field content and respective charge assignments used in the model. Symmetry | H | $\eta$ | $\Phi$ | $\Phi_{R}$ | $\Phi_{S}$ | $\Delta_{1}$ | $\Delta_{2}$
---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 2 | 2 | 1 | 1 | 1 | 3 | 3
$A_{4}$ | 1 | 3 | 3 | 1 | 1 | 1 | 1′
$Z^{\prime}_{2}$ | 1 | 1 | -1 | -1 | 1 | 1 | 1
$Z_{3}$ | $\omega$ | $\omega$ | 1 | 1 | $\omega$ | $\omega$ | $\omega$
$Z_{4}$ | 1 | 1 | $-i$ | $-i$ | \- 1 | 1 | 1
Table 2: Scalar field content and respective charge assignments used in the
model.

The leading Yukawa Lagrangian is

$\displaystyle\mathcal{L^{I}}=$ $\displaystyle
y_{e}\bar{L}_{e}e_{R}H+y_{\mu}\bar{L}_{\mu}e_{\mu}H+y_{\tau}\bar{L}{{}_{\tau}}e_{\tau}H+y_{1}^{\nu}\bar{L}_{e}[N_{T}\tilde{\eta}]_{1}+y_{2}^{\nu}\bar{L}_{\mu}[N_{T}\tilde{\eta}]_{1^{\prime\prime}}+$
$\displaystyle
y_{3}^{\nu}\bar{L}_{\tau}[N_{T}\tilde{\eta}]_{1^{\prime}}+y_{4}^{\nu}\bar{L}_{e}N_{4}\tilde{H}+y_{5}^{\nu}\bar{L}_{\tau}N_{5}\tilde{H}+y_{R}^{1}[N_{T}S_{T}]_{1}\phi_{R}+$
$\displaystyle
y_{R}^{0}[N_{4}S_{4}]_{1}\phi_{R}+y_{\phi}^{1}[N_{T}\phi]_{1}S_{4}+y_{\phi}^{2}[N_{T}\phi]_{1^{\prime\prime}}S_{5}+y_{\phi}^{3}N_{4}[\phi
S_{T}]_{1}+y_{\phi}^{4}N_{5}[\phi S_{T}]_{1^{\prime\prime}}+$ $\displaystyle
y_{\phi}^{5}[N_{T}\phi]_{3}S_{T}+y_{s}^{1}S_{T}S_{T}\phi_{S}+y_{s}^{2}S_{4}S_{4}\phi_{S}+h.c.,$

where $\tilde{H}=i\tau_{3}H$, $\tilde{\eta}=i\tau_{3}\eta$ and
$y_{q}(q=e,\mu,\tau)$, $y_{i}^{\nu}(i=1,2,3,4,5)$, $y_{R}^{j}(j=0,1)$,
$y_{\phi}^{k}(k=1,2,3,4,5)$, $y_{s}^{p}(p=1,2)$ are Yukawa coupling constants.
We have chosen the following vacuum alignments

$\langle\eta\rangle\sim v_{\eta}(1,0,0)$, $\langle\phi\rangle\sim
v_{\phi}(1,0,0)$, $\langle
H\rangle=v_{h},\langle\phi_{S}\rangle=v_{S},\langle\phi_{R}\rangle=v_{R}$.

The symmetry is broken down to $Z_{2}$ subgroup i.e. $G_{f}\equiv A_{4}\times
Z_{2}^{{}^{\prime}}\times Z_{3}\times Z_{4}\rightarrow Z_{2}$ by the $vev$
$\langle\eta\rangle\sim v_{\eta}(1,0,0)$ [44]. Since the $vev$ alignment
(1,0,0) remains invariant under the A4 generator S = $Diag(1,-1,-1)$, the
residual Z2 symmetry is

N${}_{2}\rightarrow$ -N2, N${}_{3}\rightarrow$ -N3, S${}_{2}\rightarrow$ -S2,
S${}_{3}\rightarrow$ -S3,

$\eta_{2}\rightarrow-\eta_{2}$, $\eta_{3}\rightarrow-\eta_{3}$,
$\phi_{2}\rightarrow-\phi_{2}$, $\phi_{3}\rightarrow-\phi_{3}$.

It is to be noted that only residual symmetry $Z_{2}$ is responsible for
lightest dark matter candidate stability while $Z_{2}^{{}^{\prime}}$ is
employed to restricts the unwanted Yukawa couplings. $Z_{2}^{{}^{\prime}}$
does not play any role in dark matter stabilization as the DM candidate is
$Z_{2}^{{}^{\prime}}$ even. From Eqn.(LABEL:eq:4), it is evident that lepton
conserving interactions of $N_{T}$ and $S_{T}$ take place through the $A_{4}$
triplet $\phi$ introduced in model setup. Since inverse seesaw formula in
Eqn.(3) assumes hierarchy of mass scale $\mu<<m_{D}<<m$, which implies that
lepton conserving interaction of right-handed neutrinos $N_{T}$ and neutral
fermion $S_{T}$ takes place at high scale. Also, Dirac mass term emanates from
the $A_{4}$ triplet of SU(2)L doublet Higgs field $\eta$ having smaller mass
as compared to interactions involving $N_{T}$, $S_{T}$. So, lightest $Z_{2}$
stabilized candidate will be $\eta_{2,3}$. It couples only with the right-
handed neutrinos and not with the charged leptons. As a consequence, we have
obtained a diagonal charged lepton mass matrix as

$m_{l}=Diag(y_{e},y_{\mu},y_{\tau})v_{h}.\\\ $ (5)

The other mass matrices which we have obtained are shown as below :

$m_{D}=\begin{pmatrix}A&0&0&F&0\\\ B&0&0&0&0\\\ C&0&0&0&H\\\
\end{pmatrix},\mu=\begin{pmatrix}y&0&0&0&0\\\ 0&y&0&0&0\\\ 0&0&y&0&0\\\
0&0&0&n&0\\\ 0&0&0&0&0\\\ \end{pmatrix},m=\begin{pmatrix}x&0&0&l&v\\\
0&x&h&0&0\\\ 0&h&x&0&0\\\ l&0&0&z&0\\\ v&0&0&0&0\\\ \end{pmatrix},\\\ $ (6)

where
$A=y_{1}^{\nu}v_{\eta},B=y_{2}^{\nu}v_{\eta},C=y_{3}^{\nu}v_{\eta},F=y_{4}^{\nu}v_{h},H=y_{5}^{\nu}v_{h},$
$y=y_{S}^{1}v_{S},n=y_{S}^{2}v_{S},$
$x=y_{R}^{1}v_{R},z=y_{R}^{0}v_{R},l=y_{\phi}^{1}v_{\phi}+y_{\phi}^{3}v_{\phi},v=y_{\phi}^{2}v_{\phi}+y_{\phi}^{4}v_{\phi},h=y_{\phi}^{3}v_{\phi}$.
Within ISS mechanism, the above matrices lead to the light neutrino mass
matrix as follow

$m_{\nu_{I}}=\begin{pmatrix}X&0&\Delta\\\ 0&0&0\\\
\Delta&0&\Delta^{{}^{\prime\prime}}\\\ \end{pmatrix},\\\ $ (7)

where $X=\frac{F^{2}n}{z^{2}}$ , $\Delta=-\frac{FHln}{vz^{2}}$ and
$\Delta^{{}^{\prime\prime}}=\frac{H^{2}(l^{2}n+yz^{2})}{v^{2}z^{2}}$.

When we choose a flavor basis in which we are obtaining a diagonal charged
lepton mass matrix, only those neutrino mass matrix are allowed where we have
at most two zeros. These neutrino mass matrices are consistent with neutrino
oscillation results [45]. Since we are getting three zeros in our neutrino
mass matrix, we have introduced type-II seesaw to reduce the number of zeros
in the neutrino mass matrix. The type-II seesaw contribution to the Lagrangian
is given as

$\displaystyle\mathcal{L^{II}}=$ $\displaystyle
f_{1}(L_{e}L_{e}+L_{\mu}{L}_{\tau}+L_{\tau}{L}_{\mu})\Delta_{1}+$
$\displaystyle
f_{2}(L_{e}{L}_{\tau}+{L}_{\tau}{L}_{e}+{L}_{\mu}{L}_{\mu})\Delta_{2}+h.c.,$

where, $f_{1}$ and $f_{2}$ are coupling constants. Therefore, the ISS + type-
II seesaw Lagrangian for our model is given as

$\displaystyle\mathcal{L}=$ $\displaystyle
y_{e}\bar{L}{{}_{e}}e_{R}H+y_{\mu}\bar{L}{{}_{\mu}}\mu_{R}H+y_{\tau}\bar{L}{{}_{\tau}}\tau_{R}H+y_{1}^{\nu}\bar{L}_{e}N_{T}\tilde{\eta}+y_{2}^{\nu}\bar{L}_{\mu}N_{T}\tilde{\eta}+y_{3}^{\nu}\bar{L}_{\tau}N_{T}\tilde{\eta}+$
$\displaystyle
y_{4}^{\nu}\bar{L}_{e}N_{4}\tilde{H}+y_{5}^{\nu}\bar{L}_{\tau}N_{5}\tilde{H}+y_{R}^{0}N_{4}S_{4}\phi_{R}+y_{R}^{1}N_{T}S_{T}\phi_{R}+y_{\phi}^{1}N_{T}\phi
S_{4}+$ $\displaystyle y_{\phi}^{2}N_{T}\phi S_{5}+y_{\phi}^{3}N_{4}\phi
S_{T}+y_{\phi}^{4}N_{5}\phi S_{T}+y_{\phi}^{5}N_{T}\phi
S_{T}+y_{s}^{1}S_{T}S_{T}\phi_{S}+y_{s}^{2}S_{4}S_{4}\phi_{S}+$ $\displaystyle
f_{1}(L_{e}L_{e}+L_{\mu}{L}_{\tau}+L_{\tau}{L}_{\mu})\Delta_{1}+f_{2}(L_{e}{L}_{\tau}+{L}_{\tau}{L}_{e}+{L}_{\mu}{L}_{\mu})\Delta_{2}+h.c.$

The SU(2) triplets $\Delta_{1}$ and $\Delta_{2}$ are transforming as singlets
1 and 1′, respectively. The vacuum expectation values
$\langle\Delta_{1}\rangle=v_{\Delta_{1}},\langle\Delta_{2}\rangle=v_{\Delta_{2}}$
gives

$m_{{\nu}_{II}}=\begin{pmatrix}X^{{}^{\prime}}&0&\Delta^{{}^{\prime}}\\\
0&\Delta^{{}^{\prime}}&X^{{}^{\prime}}\\\
\Delta^{{}^{\prime}}&X^{{}^{\prime}}&0\end{pmatrix},$ (10)

where,
$X^{{}^{\prime}}=f_{1}v_{\Delta_{1}},\Delta^{{}^{\prime}}=f_{2}v_{\Delta_{2}}.$

Finally, the neutrino mass matrix is $M_{\nu}=m_{\nu_{I}}+m_{\nu_{II}}$ and
can, explicitly, be written as

$M_{\nu}=\begin{pmatrix}X+X^{{}^{\prime}}&0&\Delta+\Delta^{{}^{\prime}}\\\
0&\Delta^{{}^{\prime}}&X^{{}^{\prime}}\\\
\Delta+\Delta^{{}^{\prime}}&X^{{}^{\prime}}&\Delta^{{}^{\prime\prime}}\
\end{pmatrix}.$ (11)

In literature, there are several techniques used to reduce the parameters of
neutrino mass matrix, and texture zeros is one of them [45, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 46]. Interestingly,
the mass matrix in Eqn.(11) corresponds to texture one-zero neutrino mass
model. The phenomenological implications of these class of models have been
extensively studied in the literature [60, 61, 62, 63]. Rather, in the present
work we study the prediction of the current setup for beyond neutrino sector
observable like DM and $0\nu\beta\beta$ decay discussed below.

##  3 Relic Density of Dark Matter

In the early universe, the particles were in thermal equilibrium i.e. the
processes in which the lighter particles combine to form heavy particles and
vice-versa happened at same rate. At some point of time, the conditions
required for thermal equilibrium were contravened because the density of some
particle species became too low. These particles are stated as “freeze-out”
and they have a constant density which is known as relic density, because the
abundance of particle remains same. In the process, if any particle $\chi$ was
in thermal equilibrium, then its relic abundance can be obtained by using
Boltzmann equation[65, 66, 67, 68]

$\frac{dn_{\chi}}{dt}+3\mathcal{H}n_{\chi}=-<\sigma
v>(n_{\chi}^{2}-(n_{\chi}^{eqb})^{2}),$ (12)

where $\mathcal{H}$ is the Hubble constant and $n_{\chi}$ is the number
density of the DM particle $\chi$. Here, $n_{\chi}^{eqb}$ is the number
density of particle $\chi$ when it was in thermal equilibrium. However,
$<\sigma v>$ is the thermally averaged annihilation cross-section of the DM
particle. For a DM particle with electroweak scale mass, the solution of above
Eqn.(12) gives [69]

$\Omega_{\chi}h^{2}=\frac{3\times 10^{-27}cm^{3}s^{-1}}{<\sigma v>},\\\ $ (13)

where $\Omega_{\chi}h^{2}$ gives the relic density of DM particle.

From the Lagrangian in Eqn.(LABEL:eq:9), the interaction of dark matter
particle with right-handed neutrinos is as shown in Fig.1. The cross-section
formula for this kind of process is given as [70]

$<\sigma
v>=\frac{v^{2}y^{4}m_{\chi}^{2}}{48\pi(m_{\chi}^{2}+m_{\psi}^{2})^{2}},$ (14)

where $y$ is Yukawa coupling of the interaction between DM and fermions,
$m_{\psi}$ and $m_{\chi}$ represent the mass of Majorana fermion and relic
particle mass respectively. Here, $v$ is the relative velocity of two relic
particles and is taken to be 0.3c at freeze out temperature. In case of
m${}_{DM}<M_{W}$, which indicates the low mass scale of relic particle,
$\eta_{2}$,$\eta_{3}$ self annihilates via SM Higgs into the SM particles as
shown in Fig.2. The self annihilation cross-section is thus given as below[67,
71]

$\sigma_{xx}=(\frac{|Y_{f}|^{2}|\lambda_{x}|^{2}}{16\pi
s})(\frac{(s-4m_{f}^{2})^{3/2}}{(s-4m_{x}^{2})^{1/2}((s-4m_{h}^{2})^{2}+m_{h}^{2}\Gamma_{h}^{2})}),$
(15)

where $Y_{f}$ is Yukawa coupling of fermions and we have used its recent value
as 0.308[72]. Here, $\Gamma_{h}$ is the SM Higgs decay width and its value
used is 4.15 MeV. The $m_{h}$ is Higgs mass, that is 126 GeV, and $x$ in
Eqn.(15) represents $\eta_{2},\eta_{3}$ and coupling of $x$ with SM Higgs is
represented as $\lambda_{x}$. Here, $s$ is thermally averaged center of mass
squared energy and is given as [71]

$s=4m_{\chi}^{2}+m_{\chi}^{2}v^{2}.\\\ $ (16)

Figure 1: Scattering of DM particle $\eta_{2,3}$.

Figure 2: Self annihilation of DM particle $\eta_{2,3}$ [73].

The neutral component of scalar triplet $\eta$ is our DM candidate as
considered in [74, 75]. In this work we fixed our parameters in Eqn.(13) to
obtain the recently updated constraints on relic abundance as reported by
PLANCK 2018 data. To obtain the correct relic density of DM, we need to
constrain the parameters like, Yukawa coupling, relic mass and mediator
mass(right-handed neutrinos in our case). As stated above, we chose the relic
mass much less than the mass of W-Boson. We have done our analysis for
different values of relic mass and obtained different mediator masses and
Yukawa couplings, which are shown in Fig.3. This type of studies have already
been done in [70, 76]. In order to get correct relic abundance, we did our
analysis for DM particle mass around 50 GeV, as suggested by many experimets
like XENON1T[77], PandaX-11[78], LUX[79], SuperCDMS[80] etc. For DM particle
mass 45 GeV, 50 GeV and 55 GeV, we obtained the mass of right-handed neutrinos
ranging from 138 GeV to 155 GeV. Yukawa coupling is obtained in the range
0.995-1. The results are shown in Table 3.

(a)

(b)

(c)

Figure 3: Relic density of DM vs Yukawa coupling plots for (a)DM mass(m${}_{\chi})$ = 50 GeV, (b) DM mass(m${}_{\chi})$ = 45 GeV and (c) DM mass(m${}_{\chi})$ = 55 GeV. S.No. | Relic Mass(mχ) | Mediator Mass(mψ) | Yukawa Coupling($y$)
---|---|---|---
1 | 45 GeV | 138 GeV | 0.995-1
2 | 50 GeV | 146 GeV | 0.998-1
3 | 55 GeV | 155 GeV | 0.996-1
Table 3: Constraints on relic(DM) mass, right-handed neutrino mass and Yukawa
Coupling.

##  4 Neutrinoless Double Beta($0\nu\beta\beta$) Decay

In general, the complex symmetric low energy effective neutrino mass matrix is
given by

$M_{\nu}=\begin{pmatrix}M_{ee}&M_{e\mu}&M_{e\tau}\\\
M_{e\mu}&M_{\mu\mu}&M_{\mu\tau}\\\ M_{e\tau}&M_{\mu\tau}&M_{\tau\tau}\\\
\end{pmatrix}.$ (17)

On comparing Eqn.(17) with the mass matrix in Eqn.(11) the effective Majorana
neutrino mass appearing in $0\nu\beta\beta$ decay is

$M_{ee}=X+X^{{}^{\prime}},$ (18)

and

$M_{\mu\tau}=X^{{}^{\prime}},$ (19) $M_{e\mu}=0,$ (20)

where $X$ and $X^{{}^{\prime}}$ are inverse and type-II seesaw contributions
to $0\nu\beta\beta$ decay. Eqn.(20) is our constraining condition to ascertain
the allowed parameter space of the model. The elements of the mass matrix in
Eqn.(17) are functions of low energy variables such as neutrino mass and
mixing parameters and $CP$ violating phases. Using available data (Table 4)
for the known parameters and freely varying the unknown phases, we calculate
$|M_{ee}|$ employing the formalism discussed below. Also, knowing
$M_{\mu\tau}$ using data given in Table 4, type-II contribution
($X^{{}^{\prime}}$) can, independently, be calculated with the help of
Eqn.(19). In this way, using Eqns.(18) and (19) we have been able to find the
inverse ($X$) and type-II seesaw ($X^{{}^{\prime}}$) contributions to
$0\nu\beta\beta$ decay amplitude $|M_{ee}|$.

In the charged lepton basis, the Majorana neutrino mass matrix can be written
as

$M_{\nu}=UM_{d}U^{T},$ (21)

where $M_{d}$ is diagonal mass matrix containing mass eigenvalues of neutrinos
$diag(m_{1},m_{2},m_{3})$. $U$ is neutrino mixing matrix defined as $U=V.P$
where $P$ is diagonal phase matrix $diag(1,e^{i\alpha},e^{i(\beta+\delta)})$.
In PDG representation, $V$ is given by

$\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\\\
\end{pmatrix},$ (22)

where $\delta$ is Dirac $CP$ violating phase and $\alpha$, $\beta$ are
Majorana type $CP$ violating phases. The constraining condition (Eqn.(20)) can
be written as

$c_{13}(e^{i(2\beta+\delta)}m_{3}s_{13}s_{23}-c_{12}m_{1}(c_{23}s_{12}+c_{12}e^{i\delta}s_{13}s_{23})+e^{2i\alpha}m_{2}s_{12}(c_{12}c_{23}-e^{i\delta}s_{12}s_{13}s_{23}))=0.$
(23)

The above complex constraining equation gives two real constraints which can
be solved for ratios $m_{2}/m_{1}\equiv R_{21}$ and $m_{3}/m_{1}\equiv R_{31}$
as

$R_{21}=\frac{c_{12}(c_{12}s_{13}s_{23}\sin
2\beta+c_{23}s_{12}\sin(2\beta+\delta))}{s_{12}(s_{12}s_{13}s_{23}\sin
2(\alpha-\beta)-c_{12}c_{23}\sin(2\alpha-2\beta-\delta))},$ (24)

and

$R_{31}=\frac{c_{12}(c_{12}s_{12}(-c_{23}^{2}+s_{13}^{2}s_{23}^{2})\sin
2\alpha+c_{23}s_{13}s_{23}(-c_{12}^{2}\sin(2\alpha-\delta)+s_{12}^{2}\sin(2\alpha+\delta)))}{s_{13}s_{23}(s_{12}s_{13}s_{23}\sin
2(\alpha-\beta)-c_{12}c_{23}\sin(2\alpha-2\beta-\delta))}.$ (25)

Using these mass ratios, we have two different values of lowest eigenvalue
$m_{1}$, equating them gives us

$R_{\nu}\equiv\frac{\Delta m_{21}^{2}}{|\Delta
m_{32}^{2}|}=\frac{R_{21}^{2}-1}{|R_{31}^{2}+R_{21}^{2}-2|}.$ (26)

The parameter space is constrained using the 3$\sigma$ range of $R_{\nu}$.
Neutrino oscillation parameters ($\theta_{12}$, $\theta_{23}$, $\theta_{13}$,
$\Delta m_{21}^{2}$, $\Delta m_{32}^{2}$) are randomly generated with Gaussian
distribution while $CP$ phases ($\delta$,$\alpha$,$\beta$) are varied randomly
in their full range with uniform distribution.
The neutrino masses can be obtained using mass-squared differences as

$m_{2}=\sqrt{m_{1}^{2}+\Delta m_{21}^{2}}$, $m_{3}=\sqrt{m_{1}^{2}+\Delta
m_{31}^{2}}$ for Normal ordering (NO),

and

$m_{1}=\sqrt{m_{3}^{2}-\Delta m_{31}^{2}}$ , $m_{2}=\sqrt{m_{3}^{2}-\Delta
m_{31}^{2}+\Delta m_{21}^{2}}$ for Inverted ordering (IO),

whereas the lightest neutrino mass ($m_{1}$(NO), $m_{3}$(IO)) is obtained from
mass ratios. Also, we calculate the effective mass parameter

$|M_{ee}|=\left|m_{1}c_{12}^{2}c_{13}^{2}+m_{2}s_{12}^{2}c_{13}^{2}e^{2i\alpha}+m_{3}s_{13}^{2}e^{2i\beta}\right|,$
(27)

and $|M_{\mu\tau}|$ for the allowed parameter space which are, further, used
to find inverse ($|X|$) and type-II ($|X^{{}^{\prime}}|$) seesaw contribution
to $|M_{ee}|$.

Parameter | Best fit $\pm$ $1\sigma$ range |  $3\sigma$ range
---|---|---
Normal neutrino mass ordering $\left(m_{1}<m_{2}<m_{3}\right)$ |
$\sin^{2}\theta_{12}$ | $0.304^{+0.013}_{-0.012}$ | $0.269-0.343$
$\sin^{2}\theta_{13}$ | $0.02221^{+0.00068}_{-0.00062}$ | $0.02034-0.02420$
$\sin^{2}\theta_{23}$ | $0.570^{+0.018}_{-0.024}$ | $0.407-0.618$
$\Delta m_{21}^{2}\left[10^{-5}\mathrm{eV}^{2}\right]$ | $7.42^{+0.21}_{-0.20}$ | $6.82-8.04$
$\Delta m_{31}^{2}\left[10^{-3}\mathrm{eV}^{2}\right]$ | $+2.541^{+0.028}_{-0.027}$ | $+2.431-+2.598$
Inverted neutrino mass ordering $\left(m_{3}<m_{1}<m_{2}\right)$ |
$\sin^{2}\theta_{12}$ | $0.304^{+0.013}_{-0.012}$ | $0.269-0.343$
$\sin^{2}\theta_{13}$ | $0.02240^{+0.00062}_{-0.00062}$ | $0.02053-0.02436$
$\sin^{2}\theta_{23}$ | $0.575^{+0.017}_{-0.021}$ | $0.411-0.621$
$\Delta m_{21}^{2}\left[10^{-5}\mathrm{eV}^{2}\right]$ | $7.42^{+0.21}_{-0.20}$ | $6.82-8.04$
$\Delta m_{32}^{2}\left[10^{-3}\mathrm{eV}^{2}\right]$ | $-2.497^{+0.028}_{-0.028}$ | $-2.583--2.412$
Table 4: Neutrino oscillations experimental data NuFIT 5.0 used in the
numerical analysis[81].

Figure 4:  Variation of effective Majorana neutrino mass $|M_{ee}|$ with
Contribution of inverse seesaw and type-II seesaw model parameters ($|X|$ and
$|X^{\prime}|$), for both normal and inverted ordering.

In Fig.4, we have plotted effective mass parameter $|M_{ee}|$ with inverse
seesaw contribution (X) and type-II seesaw contribution (X′). The sensitivity
reach of 0$\nu\beta\beta$ decay experiments like SuperNEMO [82], KamLAND-Zen
[83], NEXT [84, 85], nEXO[86] is, also, shown in Fig.4. In Fig.4(a), it is
clear that for NO, higher density of points for type-II seesaw indicate
contribution of $\mathcal{O}(0.01eV)$, whereas inverse seesaw contribution is
of $\mathcal{O}(0.001eV)$. On the other hand, for IO, different seesaw
contributions are of same order as can be seen in Fig.4(b). Hence, type-II
seesaw contribution to $|M_{ee}|$ is large as compared to inverse seesaw
contribution for NO for the texture one-zero model within the framework of
inverse seesaw and type-II seesaw. For normal ordering neutrino mass spectrum,
the $|M_{ee}|$ goes below upto the $\mathcal{O}(10^{-4}eV)$ and we did not
obtain a clear lower bound as shown in Fig.4(a). On contrary, for inverted
ordering, there is a clear cut lower bound for the $|M_{ee}|$ which can be
probed in future $0\nu\beta\beta$ decay experiments. It can be seen from
Fig.4(b), that the $0\nu\beta\beta$ decay experiments like SuperNEMO, KamLAND-
Zen can probe the inverted ordering spectrum.

Figure 5:  Variation of effective Majorana neutrino mass $|M_{ee}|$ with
lightest neutrino mass $m_{1}$($m_{3}$) for both normal(inverted) ordering of
neutrino masses.

Figure 6:  Correlation plots of ($\alpha$-$R_{21}$) and ($\delta$-$\beta$).
First(second) row depicts the correlations for normal(inverted) ordering of
neutrino masses.

In Fig.5, we have depicted the correlation of effective Majorana mass
parameter $|M_{ee}|$ with the lightest neutrino mass $m_{1}(m_{3})$ for normal
(inverted) ordering of neutrino masses. $|M_{ee}|$ goes below up to 0.001eV
for normal ordering whereas there exist a lower bound near 0.04eV for inverted
ordering. The sensitivity reaches of various current and future
$0\nu\beta\beta$ decay experiments have, also, been shown in Fig.5. Although,
for normal ordering, $|M_{ee}|$ is beyond the sensitivity reach of experiments
but non-observation $0\nu\beta\beta$ decay shall rule out the inverted
ordering in this model.

Furthermore, we have studied the correlations amongst the Majorana
phases($\alpha,\beta$), Dirac phase($\delta$) and mass
ratio($R_{21}\equiv\frac{m_{2}}{m_{1}}$). Some representative plots are given
in Fig.6. The correlation plots ($\alpha-R_{21}$) and ($\delta-\beta$) are
shown in Fig.6(a)(6(c)) and Fig.6(b)(6(d)), respectively for normal(inverted)
mass ordering. For normal ordering, there is no preferable range of CP-
phase(Dirac or Majorana) whereas, in case of inverted ordering, Majorana CP-
phase $\alpha$ is constrained in the
range($0^{\circ}-20^{\circ})\cup(160^{\circ}-180^{\circ})$ and $\delta$ is
found to lie in $(40^{\circ}-150^{\circ})\cup(200^{\circ}-325^{\circ}$) range.
In Fig.7, we have depicted the correlation of neutrino mixing angles with mass
ratio $R_{21}$ for normal(inverted) ordering. The model satisfies the neutrino
oscillation data for both mass ordering. The model predictions are consistent
with the model independent analysis performed in Ref.[60].

Figure 7:  Correlation plots of ($\theta_{ij}$-$R_{21}$) for normal(inverted)
ordering of neutrino masses. Horizontal dashed lines are $3\sigma$
experimental range of respective mixing angles.

##  5 Conclusions

In conclusion, we have proposed a model based on $A_{4}$ discrete flavor
symmetry implementing inverse and type-II seesaw mechanisms to have LHC
accessible TeV scale right-handed neutrino mass and texture one-zero in the
resulting Majorana neutrino mass matrix, respectively. The symmetry is broken
down to $Z_{2}$ subgroup i.e. $G_{f}\equiv A_{4}\times
Z_{2}^{{}^{\prime}}\times Z_{3}\times Z_{4}\rightarrow Z_{2}$ by the $vev$
$\langle\eta\rangle\sim v_{\eta}(1,0,0)$ which, further stabilizes the DM
candidate $\eta_{2,3}$. We scanned the ranges of Yukawa coupling ($y$), right-
handed neutrino mass ($m_{\psi}$) and DM mass ($m_{\chi}$) to obtain observed
relic abundance of DM $\eta_{2,3}$. We find that to have observed relic
abundance the $m_{\chi}$, $m_{\psi}$ and $y$ should be around 50 GeV, 146 GeV
and 0.998-1, respectively. Also, we have obtained the prediction of the model
for $0\nu\beta\beta$ decay amplitude $|M_{ee}|$. In particular, we calculated
inverse ($X$) and type-II ($X^{{}^{\prime}}$) seesaw contributions to
$|M_{ee}|$, while texture one-zero model being consistent with low energy
experimental constraints. The type-II seesaw contribution to $|M_{ee}|$ is
found to be large as compared to inverse seesaw contribution for normal
ordering neutrino masses. The model predicts a robust lower bound on
$|M_{ee}|$ for inverted ordering neutrino masses which can be probed in future
$0\nu\beta\beta$ decay experiments like SuperNEMO, KamLAND-Zen. The
correlation of $|M_{ee}|$-$m_{1}(m_{3})$ shows that inverted ordering is ruled
out in case of non-observation of $0\nu\beta\beta$ decay while normal ordering
may still be allowed. Also, contradistinction to normal ordering, the CP-
violating phases $\delta$ and $\alpha$ are constrained in the range
($0^{\circ}-20^{\circ})\cup(160^{\circ}-180^{\circ})$ and
$(40^{\circ}-150^{\circ})\cup(200^{\circ}-325^{\circ}$), respectively, for
inverted ordering of neutrino masses.

Acknowledgments
R. Verma acknowledges the financial support provided by the Central University
of Himachal Pradesh. B. C. Chauhan is thankful to the Inter University Centre
for Astronomy and Astrophysics (IUCAA) for providing necessary facilities
during the completion of this work. M. K. acknowledges the financial support
provided by Department of Science and Technology, Government of India vide
Grant No. DST/INSPIRE Fellowship/2018/IF180327.

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# Physical Channel Modeling for RIS-Empowered Wireless Networks in Sub-6 GHz
Bands

Fatih Kilinc$\ast$, Ibrahim Yildirim$\ast$,$\bullet$, and Ertugrul Basar$\ast$
$\ast$CoreLab, Department of Electrical and Electronics Engineering, Koç
University, Sariyer 34450, Istanbul, Turkey
$\bullet$Faculty of Electrical and Electronics Engineering, Istanbul Technical
University, Sariyer 34469, Istanbul, Turkey.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Reconfigurable intelligent surface (RIS)-assisted communications is one of the
promising candidates for next generation wireless networks by controlling the
propagation environment dynamically. In this study, a channel modeling
strategy for RIS-assisted wireless networks is introduced in sub-6 GHz bands
by considering both far-field and near-field behaviours in transmission. We
also proposed an open-source physical channel simulator for sub-6 GHz bands
where operating frequency, propagation environment, terminal locations, RIS
location and size can be adjusted. It is demonstrated via extensive computer
simulations that an improved achievable rate performance is obtained in the
presence of RISs for both near-field and far-field conditions.

###### Index Terms:

Reconfigurable intelligent surface (RIS), channel modeling, sub-6 GHz, far-
field, near-field.

##  I Introduction

Due to the developments in technology and the increase in data consumption,
current communication systems have been inadequate to meet the increasing
demands. Hence, next-generation wireless systems should provide data transfer
at very high rates with a high reliability. Emerging technologies such as
virtual-reality, high-quality streaming, and autonomous systems require novel
ways to achieve higher data rates with low latency. Massive multiple-input
multiple-output (mMIMO), millimeter-wave (mmWave) communications, terahertz
communications, and reconfigurable intelligent surfaces (RISs) can be
considered as remarkable technologies for next-generation communication
systems to enhance system performance[1].

RISs have recently received great attention due to their advantages in
controlling the wireless propagation environment. An RIS is comprised of small
electro-magnetic (EM) surfaces, which are capable of reflecting the incoming
signals in a controllable and efficient way, and provides a more energy-
efficient wireless communication network. The reflection properties of the EM
surfaces can be dynamically controlled with nearly passive electronic
components, called ”meta-atoms”. Considering all of these advantages of RIS,
the implementation of an RIS in mmWave bands is one of the cost and power-
efficient ways to enhance the system performance.

RISs have been extensively investigated in the literature from various
aspects, where the physical properties, behaviours and different use cases of
RISs, such as holographic beamformers, low complexity transmitters, anomalous
reflectors are investigated [1, 2]. The integration of an RIS into currently
existing technologies such as index modulation (IM) techniques and relay-aided
systems are proposed in [3] and [4]. Moreover, the efficient positioning and
beamforming optimization of the RIS in MIMO networks are studied in [5] and
[6]. Since carrier signals are vulnerable to blockage and suffer from path
loss in mmWave communications, an RIS can be used to ensure reliable
transmission by creating an additional path. Therefore, the most of the
studies in the literature are focused on the deployment of RIS in mmWave
frequencies. Modeling of the RIS-assisted wireless channels also stands out as
another key direction for the research community. RISs are mostly designed in
the form of uniform planar array (UPA), therefore, two dimensional (2D)
channel models are not accurate for RIS-assisted networks. In this context, 3D
physical channel models for RIS-assisted mmWave networks are proposed in
several recent works [7, 8, 9]. However, the RIS technology can also be used
in sub-6 GHz bands, which is used for majority of wireless communication
systems, such as 3G/4G and even 5G cellular networks, Wi-Fi systems in our
homes, and bluetooth devices. The existing studies, which reveal the effects
and behavior of RIS in sub-6 GHz bands, have not exhaustively investigated
physical channels.

In this study, we propose a modeling strategy for RIS-empowered communication
systems by considering the currently used technical specifications on sub-6
GHz bands [10, 11]. According to a common assumption in the literature, the
most efficient use of an RIS is possible when it is placed close to the
terminals. Since the wavelengths of the carriers in sub-6 GHz bands are
significantly higher than those in mmWave bands, deployment of a large RIS in
sub-6 GHz bands requires the RIS to operate in near-field if the RIS is placed
close to transmitter (Tx) or receiver (Rx). In order to cover these use-cases,
the impact of the RIS in both far-field and near-field conditions is analyzed
in this study.

The rest of this paper can be summarized as follows. In Section II, we propose
RIS-assisted end-to-end channel model for indoor and outdoor environments
operating in sub-6 GHz bands. In Section III, we provide the achievable rate
analysis of the system. In Section IV, the numerical results are presented and
the paper is concluded in Section V.

Figure 1: Generic system model for an RIS-assisted network with $C$ number of
clusters and $S$ number of rays.

##  II System Model

In this section, we introduce our channel modeling strategy for RIS-assisted
indoor and outdoor wireless communication systems in sub-6 GHz bands. Proposed
channel model for the RIS-empowered wireless network is established by
considering the 3D channel modeling approach in 3GPP [10]. Additionally, the
end-to-end single-input single-output (SISO) channel model is derived in the
presence of an RIS. Furthermore, the near-field channel between the RIS-Rx is
analyzed when the far-field conditions are not satisfied due to the close
distance between the RIS-Rx. Since 2D channel models are not accurate for the
RIS-assisted transmission owing to existence of an elevation angle, a 3D
channel model should be taken into consideration. The considered propagation
environments are Indoor Hotspot (InH) and Urban Microcellular (UMi) in this
study. The generic system model for the RIS-assisted wireless communication
system is illustrated in Fig. 1, where
$d_{\text{3D}}^{\text{T-RIS}}$,$d_{\text{3D}}^{\text{T-R}}$ and
$d_{\text{3D}}^{\text{RIS-R}}$ represents the 3D distances between the Tx-RIS,
Tx-Rx and RIS-Rx, respectively. It is assumed that the RIS with $N$ number of
elements is positioned on the $xz$-plane. There exist $C$ clusters in the
environment, each containing $S$ rays. The channels between the Tx-RIS, RIS-Rx
and Tx-Rx are denoted by $\mathbf{h}\in\mathbb{C}^{N\times 1}$,
$\mathbf{g}\in\mathbb{C}^{N\times 1}$, and
$h_{\text{SISO}}\in\mathbb{C}^{1\times 1}$, respectively, where $N$ is the
number of RIS elements. It is assumed that the Tx and Rx are equipped with
unity gain isotropic antennas. The positions of the Tx, Rx and RIS are given
in the cartesian coordinate system as
$\textbf{r}^{\text{Tx}}=(x^{\text{Tx}},y^{\text{Tx}},z^{\text{Tx}})$,
$\textbf{r}^{\text{Rx}}=(x^{\text{Rx}},y^{\text{Rx}},z^{\text{Rx}})$ and
$\textbf{r}^{\text{RIS}}=(x^{\text{RIS}},y^{\text{RIS}},z^{\text{RIS}})$,
respectively.

Considering the direct path and RIS reflection paths, the baseband equivalent
of the received symbol at the Rx is expressed as

$\displaystyle
y=\sqrt{P_{t}}(\textbf{g}^{\mathrm{T}}\mathbf{\Theta}\textbf{h}+h_{\text{SISO}})s+w\vspace{-0.2cm}$
(1)

where $P_{t}$ is the transmit power, $s$ is the transmitted symbol, $w$ is
additive white Gaussian noise at the Rx, which is modeled as
$w\sim\mathcal{CN}(0,N_{0})$, and $(\cdot)^{\mathrm{T}}$ denotes the transpose
of the vector or matrix. The response matrix of the RIS is expressed as
$\mathbf{\Theta}=\text{diag}(\beta_{1}e^{j\alpha_{1}},\cdots,\beta_{N}e^{j\alpha_{N}})$,
where $\beta_{n}$ and $\alpha_{n}$ represents the magnitude and the phase
shift of the $n$th RIS element for $n=1,2,\dots,N$, respectively.

###  II-A Tx-RIS Channel

In this subsection, the channel between the Tx and RIS is presented and the
channel generation procedure is explained in detail. The channel $\mathbf{h}$
is expressed as

$\displaystyle\mathbf{h}=\sum\limits_{c=1}^{C}\sum\limits_{s=1}^{S}\sqrt{\frac{P_{c}}{S}\vphantom{\frac{G_{e}(\theta_{c,s}^{\text{T-RIS}})}{P_{L}}}}\sqrt{\frac{G_{e}(\theta_{c,s}^{\text{T-RIS}})}{P_{L}}}e^{j\Phi_{c,s}}\mathbf{a}(\theta_{c,s}^{\text{T-RIS}},\phi_{c,s}^{\text{T-RIS}})$
(2)

where $P_{c}$, $G_{e}(\theta_{c,s}^{\text{T-RIS}})$ and $P_{L}$ are the power
of the $n$th cluster, radiation pattern of an RIS element in the direction of
$(c,s)$th path and the path loss component, respectively. Here, $\Phi_{c,s}$,
$\theta_{c,s}^{\text{T-RIS}}$, and $\phi_{c,s}^{\text{T-RIS}}$ also represent
the random initial phase, zenith angle of arrival (ZoA) and azimuth angle of
arrival (AoA) of the RIS for the $(c,s)$th path, respectively. The array
response vector of the RIS is also denoted by
$\mathbf{a}(\theta_{c,s}^{\text{T-RIS}},\phi_{c,s}^{\text{T-RIS}})$.
Considering these parameters, the generation steps of the Tx-RIS channel are
listed as follows:

  1. 1)

First, we need to set the environment and the locations of the Tx and RIS. The
line-of-sight (LOS) ZoA $(\theta_{LOS}^{\text{T-RIS}})$ and AoA
$(\phi_{LOS}^{\text{T-RIS}})$ angles is calculated by using the locations of
the Tx and RIS.

  2. 2)

Then, we need to assign propagation conditions (LOS/NLOS) and calculate the
path loss $P_{L}$. The LOS probabilities for InH and UMi scenarios are
calculated as given in [10, Table 7.2.2]. In addition, we assume $100\%$ LOS
probability, if the RIS is located at the same elevation or higher than the
transmitter as $(z^{\text{RIS}}\geq z^{\text{Tx}})$. Reference path loss
models [10] for LOS $(P_{L}^{\text{LOS}})$ and NLOS $(P_{L}^{\text{NLOS}})$
cases in the InH environment are given as follows, respectively:

$\displaystyle
P_{L}^{\text{LOS}}\text{[dB]}=16.9\log_{10}(d_{\text{3D}})+32.8+20\log_{10}(f_{c})+\sigma_{SF},$
(3) $\displaystyle
P_{L}^{\text{NLOS}}\text{[dB]}=43.3\log_{10}(d_{\text{3D}})+11.5+20\log_{10}(f_{c})+\sigma_{SF}.\vspace*{-0.2cm}$
(4)

Reference path loss models [10] for LOS and NLOS cases in the outdoor UMi
environment are given as follows, respectively:

$\displaystyle P_{L}^{\text{LOS}}\text{[dB]}=$ $\displaystyle
22\log_{10}(d_{\text{3D}})+28+20\log_{10}(f_{c})+\sigma_{SF},$ (5)
$\displaystyle P_{L}^{\text{NLOS}}\text{[dB]}=$ $\displaystyle
36.7\log_{10}(d_{\text{3D}})+22.7+26\log_{10}(f_{c})$ $\displaystyle\hskip
81.09052pt-0.3(h_{UT}-1.5)+\sigma_{SF}.$ (6)

In (3)-(2)), $f_{c}$ is the operating frequency in GHz, $\sigma_{SF}$ is the
shadow fading term and $h_{UT}$ is the effective receive antenna height, which
is the height of the RIS obtained by $h_{UT}=z^{\text{RIS}}-1$. Here,
$d_{\text{3D}}$ is equivalent to $d_{\text{3D}}^{\text{T-RIS}}$ and calculated
by
$d_{\text{3D}}^{\text{T-RIS}}=\|\textbf{r}^{\text{Tx}}-\textbf{r}^{\text{RIS}}\|$.

  3. 3)

Correlated large scale parameters should be generated. Shadow fading (SF),
Ricean $K$ factor, delay spread (DS), azimuth spread of arrival (ASA), azimuth
spread of departure (ASD), zenith spread of arrival (ZSA), and zenith spread
of departure (ZSD) are generated by considering the cross correlation
coefficients between each other given in [10, Table 7.3-6].

  4. 4)

The delays and the cluster powers should be generated. The delay associated
with the $c$th cluster is given as $\tau_{c}$ and randomly generated by using
exponential power delay distribution [10]. The cluster powers depend on the
cluster delays and follow an exponential power delay profile. The power of the
$c$th cluster is denoted by $P_{c}$ and calculated as in [10]. Under LOS
condition, the path from the first cluster is considered as the LOS path.
Hence, the power of the LOS component $(P_{LOS}=K_{R}/(K_{R}+1))$ is added to
the first cluster power and the power of the $c$th cluster is rewritten as

$\displaystyle P_{c}=P_{c}/(K_{R}+1)+\delta(c-1)P_{LOS}\vspace*{-0.4cm}$ (7)

where $K_{R}$ is the Ricean factor in linear scale and $\delta(.)$ is the
dirac delta function.

  5. 5)

The ZoA $(\theta_{c,s}^{\text{T-RIS}})$ and AoA $(\phi_{c,s}^{\text{T-RIS}})$
angles of the RIS should be generated. Here, $\theta_{c,s}^{\text{T-RIS}}$
follows Laplacian distribution in both indoor and outdoor environments, and
$\phi_{c,s}^{\text{T-RIS}}$ follows Laplacian distribution for indoors while
it follows Wrapped Gaussian distribution for outdoors. The detailed generation
process for $\theta_{c,s}^{\text{T-RIS}}$ and $\phi_{c,s}^{\text{T-RIS}}$ is
explained in [10, sec. 7.3]. Additionally, $\theta_{c,s}^{\text{T-RIS}}$ is
distributed within the range of $[0^{\circ},180^{\circ}]$, while
$\phi_{c,s}^{\text{T-RIS}}$ is distributed within the range of
$[-180^{\circ},180^{\circ}]$. Here, we need to ignore clusters behind the RIS
by limiting the range of $\phi_{c,s}^{\text{T-RIS}}$ to
$[0^{\circ},180^{\circ}]$.

  6. 6)

The initial random phase for the $(c,s)$th path should be generated, where $c$
is the cluster index and $s$ is the ray index in each cluster. Here,
$\Phi_{c,s}$ follows an uniform distribution as
$\Phi_{c,s}\sim\mathcal{U}(-\pi,\pi)$.

  7. 7)

The radiation pattern of the RIS elements should be calculated. We consider
the $\cos^{q}$ radiation pattern, which is used for reflectarray antennas [12]
to model the radiation of the RIS elements. The radiation pattern in the
direction of the $(c,s)$th path is calculated as

$\displaystyle
G_{e}(\theta_{c,s}^{\text{T-RIS}})=2(2q+1)\cos^{2q}(90-\theta_{c,s}^{\text{T-RIS}})\vspace*{-0.3cm}$
(8)

where $2(2q+1)$ is normalization factor and $q$ is obtained as $q=0.285$ in
[8] by considering the physical area of RIS elements.

  8. 8)

The array response vector of the RIS
$\mathbf{a}(\theta_{c,s}^{\text{T-RIS}},\phi_{c,s}^{\text{T-RIS}})\in\mathbb{C}^{N\times
1}$ should be calculated. Here, a square RIS structure is considered and the
horizontal/vertical distances between the elements are equal and denoted by
$d$. As illustrated in Fig. 2, the ZoA and AoA angles to the RIS are
$\theta_{c,s}^{\text{T-RIS}}$ and $\phi_{c,s}^{\text{T-RIS}}$, respectively.
Considering the RIS geometry in Fig. 2, the first RIS element with the
location vector $\textbf{r}^{1}=(x^{1},y^{1},z^{1})$ is positioned at the
bottom left corner of the RIS. $\textbf{r}^{n}=(x^{n},y^{1},z^{n})$ denotes
the location vector of the $n$th RIS element, and $x^{n}$ and $y^{n}$ are
respectively calculated as

$\displaystyle x^{n}$
$\displaystyle=x^{1}+d\cdot\mod{\left(n-1,\sqrt{N}\right)},$ $\displaystyle
z^{n}$
$\displaystyle=z^{1}+d\cdot\bigg{\lfloor}(n-1)/{\sqrt{N}}\bigg{\rfloor}$ (9)

where $\mod(.)$ is the modulus operation and $\lfloor.\rfloor$ is the floor
operation. The normalized location of the $n$th RIS element according to first
element is denoted by $(\overline{x}_{n},0,\overline{z}_{n})$, where
$\overline{x}_{n}=x^{n}-x^{1}$ and $\overline{z}_{n}=z^{n}-z^{1}$. The array
response vector for an RIS with $N$ elements is obtained by

$\displaystyle\mathbf{a}(\theta_{c,s}^{\text{T-RIS}},\phi_{c,s}^{\text{T-RIS}})=\left[1\;\cdots\;e^{j\frac{2\pi}{\lambda}(\overline{x}_{n}\cos\theta_{c,s}^{\text{T-RIS}}+\overline{z}_{n}\sin\theta_{c,s}^{\text{T-RIS}}\cos\phi_{c,s}^{\text{T-RIS}})}\right.$
$\displaystyle\cdots\left.e^{j\frac{2\pi}{\lambda}(\overline{x}_{N}\cos\theta_{c,s}^{\text{T-RIS}}+\overline{z}_{N}\sin\theta_{c,s}^{\text{T-RIS}}\cos\phi_{c,s}^{\text{T-RIS}})}\right]^{\mathrm{T}}$
(10)

where $\lambda$ is the wavelength of the carrier signal for $n=1,\cdots,N$.

Finally, we can form the channel $\mathbf{h}$ by using the channel parameters
obtained in channel generation steps.

Figure 2: 3D RIS geometry.

###  II-B Tx-Rx Channel

The direct link between the Tx and Rx is given by

$\displaystyle
h_{\text{SISO}}=\sum\limits_{c=1}^{C}\sum\limits_{s=1}^{S}\sqrt{\frac{P_{c}}{S}}\sqrt{\frac{1}{P_{L}}}e^{j\Phi_{c,s}}$
(11)

where $P_{c}$, $P_{L}$ and $\Phi_{c,s}$ is defined as in (2).
$h_{\text{SISO}}$ can be obtained by following the Steps 1, 2, 3, 4 and 6 in
the channel generation procedure described in Subsection II.A. In all steps,
the Tx and Rx positions should be taken into account instead of Tx and RIS
positions, respectively. Also, calculations for AoA and ZoA angles are not
required in the generation process of $h_{\text{SISO}}$, since a SISO link
established.

###  II-C RIS-Rx Channel

For the RIS-Rx channel, we consider two different channel scenarios between
the RIS and Rx. When the size of the RIS is large enough, the far-field
distance between the RIS and Rx becomes significantly large. Therefore, the
near-field channels are also considered in this section when the RIS is placed
close to the Rx, while far-field channels are obtained by following the same
procedure in previous sections.

####  II-C1 Far-Field Channel

The far-field channel g is given as

$\displaystyle\mathbf{g}=\sum\limits_{c=1}^{C}\sum\limits_{s=1}^{S}\sqrt{\frac{P_{c}}{S}\vphantom{\frac{G_{e}(\theta_{c,s}^{\text{T-RIS}})}{P_{L}}}}\sqrt{\frac{G_{e}(\theta_{c,s}^{\text{RIS-R}})}{P_{L}}}e^{j\Phi_{c,s}}\mathbf{a}(\theta_{c,s}^{\text{RIS-R}},\phi_{c,s}^{\text{RIS-R}})$
(12)

where $P_{c}$, $P_{L}$, and $\Phi_{c,s}$ is defined as in (2) and
$G_{e}(\theta_{c,s}^{\text{RIS-R}})$ is the RIS element radiation pattern in
the direction of $(c,s)$th path. The azimuth of departure (AoD) and zenith of
departure (ZoD) angles from the RIS are represented by
$\phi_{c,s}^{\text{RIS-R}}$ and $\theta_{c,s}^{\text{RIS-R}}$, respectively.
The array response vector of the RIS is denoted by
$\mathbf{a}(\theta_{c,s}^{\text{RIS-R}},\phi_{c,s}^{\text{RIS-R}})$. The far-
field channel can be produced by following the similar channel generation
steps in Subsection II.A. In all steps, the RIS and Rx positions should be
taken into account instead of Tx and RIS positions, respectively. In Steps 5,
7 and 8, the angles $\theta_{c,s}^{\text{RIS-R}}$ and
$\phi_{c,s}^{\text{RIS-R}}$ should be considered instead of
$\theta_{c,s}^{\text{T-RIS}}$ and $\phi_{c,s}^{\text{T-RIS}}$, and they follow
the same distributions with $\theta_{c,s}^{\text{T-RIS}}$ and
$\phi_{c,s}^{\text{T-RIS}}$, respectively. Additionally,
$\theta_{c,s}^{\text{RIS-R}}$ is distributed within the range of
$[0^{\circ},180^{\circ}]$, while $\phi_{c,s}^{\text{RIS-R}}$ is distributed
within the range of $[-180^{\circ},180^{\circ}]$. Here, we need to ignore
clusters behind the RIS by limiting the range of $\phi_{c,s}^{\text{RIS-R}}$
to $[0^{\circ},180^{\circ}]$ as in Subsection II.A.

####  II-C2 Near-Field Channel

In sub-6 GHz bands, the far-field distance, which is calculated by
$\frac{N\lambda}{2}$ for a square RIS structure, increases significantly for
larger RIS sizes. The scenarios, where the RIS is located in the far-field
area, will be inefficient due to the high distance and multiplicative path
loss. Hence, the near-field channels should be considered to analyze the RIS
performance in near-field conditions. When the RIS operates in the near-field
of Rx, there will be a pure LOS link between the RIS and Rx. Unlike far-field
channels, the effective area of each RIS element changes since the angle of
view of each RIS element to the Rx changes. Moreover, the distances from the
RIS elements to the Rx varies. Furthermore, the polarization mismatch between
the RIS elements should be taken into account under the near-field
conditions[13]. The near-field channel between the RIS and Rx is denoted by
$\textbf{g}=[g_{1},g_{2},\cdots,g_{N}]^{\mathrm{T}}$, where $g_{n}$ represents
the channel coefficient from the $n$th RIS element to the Rx, and can be
expressed in terms of its magnitude and phase as $g_{n}=|g_{n}|e^{-j\gamma}$
for $n=1,2,\dots,N$. By considering the RIS geometry in Fig. (2) and the RIS
element locations in (8)), the near field channel gain from the $n$th RIS
element is approximated as according to[13]

$\displaystyle|g_{n}|^{2}\approx$
$\displaystyle\frac{1}{4\pi}\sum\limits_{x\in\mathbb{X}}\sum\limits_{z\in\mathbb{Z}}\left(\frac{\frac{xz}{y^{2}}}{3\left(\frac{z^{2}}{y^{2}}+1\right)\sqrt{\frac{x^{2}}{z^{2}}+\frac{z^{2}}{y^{2}}+1}}\right.$
$\displaystyle\left.+\frac{2}{3}\tan^{-1}\left(\frac{\frac{xz}{y^{2}}}{\sqrt{\frac{x^{2}}{y^{2}}+\frac{z^{2}}{y^{2}}+1}}\right)\right)$
(13)

where
$\mathbb{X}=\left\\{d/2+x^{n}-x^{\text{Rx}},d/2+x^{\text{Rx}}-x^{n}\right\\}$,
$\mathbb{Z}=\left\\{d/2+z^{n}-z^{\text{Rx}},d/2+z^{\text{Rx}}-z^{n}\right\\}$,
and $y=|y^{n}-y^{\text{Rx}}|$. The phase of $g_{n}$ can be calculated as
follows:

$\displaystyle\gamma=2\pi\mod{\left(\frac{\|\textbf{r}^{n}-\textbf{r}^{\text{Rx}}\|}{\lambda},1\right)}.$
(14)

Figure 3: Achievable rates of RIS-assisted system for the indoor environment.

##  III Achievable Rate Analysis

In this section, we analyze the achievable rate of the RIS-assisted
communication systems by optimally adjusting the phases of the RIS elements
according to the channel conditions. We assume that the magnitude response is
$\beta_{n}=1$ for all RIS elements. The optimum phase response of the $n$th
RIS element to maximize the signal-to-noise ratio (SNR) at the Rx is given by

$\displaystyle\alpha_{n}=\text{arg}(h_{\text{SISO}})-\text{arg}(h_{n}g_{n})$
(15)

where arg$(\cdot)$ denotes the phase of a complex number, $h_{n}$ and $g_{n}$
are the $n$th elements of h and g, respectively. By considering the phase
response in (15) for the RIS elements, the instantaneous received SNR is
expressed as

$\displaystyle\rho=\frac{P_{t}|\textbf{g}^{\mathrm{T}}\mathbf{\Theta}\textbf{h}+h_{\text{SISO}}|^{2}}{N_{0}}=\frac{P_{t}\left|\sum\nolimits_{n=1}^{N}|h_{n}||g_{n}|+h_{\text{SISO}}\right|^{2}}{N_{0}}.\vspace*{-0.5cm}$
(16)

The maximized achievable rate of the system can be computed by
$R=\log_{2}(1+\rho)$.

##  IV Simulation Results

In this section, we provide computer simulation results for the RIS-assisted
transmission in sub-6 GHz bands. The achievable rate of the RIS-assisted
communication systems is investigated for different scenarios. The noise power
$(N_{0})$ is set to $-130$ dBm for all simulations.

In Figs. 3(a) and (b), the achievable rates are presented for varying $N$ and
different positions of the Tx, Rx and RIS for the indoor office environment.
In both cases, the RIS is positioned in the near-field of the Rx, therefore,
the near-field channels are considered for the RIS-Rx link under 2.4 GHz
operating frequency. In Fig. 3(a), the considered 3D coordinates are
$\textbf{r}^{\text{Tx}}=(0,25,3)$, $\textbf{r}^{\text{Rx}}=(40,48,1.5)$ and
$\textbf{r}^{\text{RIS}}=(38,50,z^{\text{RIS}})$, while
$\textbf{r}^{\text{Tx}}=(0,25,3)$, $\textbf{r}^{\text{Rx}}=(67,45,1.5)$ and
$\textbf{r}^{\text{RIS}}=(70,50,z^{\text{RIS}})$ in Fig. 3(b). The achievable
rate analysis is presented depending on the height of RIS in both scenarios.
When the Tx and RIS are in the same elevation in case $z^{\text{RIS}}=3$ m, we
assume $100\%$ LOS probability between the Tx and RIS. However, the LOS
condition is not guaranteed in case $z^{\text{RIS}}=2$ since RIS is positioned
below the Tx. According to Figs. 3(a) and (b), the achievable rate of the
system is significantly better for the case $z^{\text{RIS}}=3$. Furthermore,
the achievable rate of the system further increases as $N$ increases.

In Fig. 4, the effect of $N$ on the achievable rate is investigated in the
outdoor environment. We consider 2.4 GHz operating frequency and the 3D
coordinates as $\textbf{r}^{\text{Tx}}=(0,25,10)$,
$\textbf{r}^{\text{Rx}}=(65,52,1)$ and $\textbf{r}^{\text{RIS}}=(62,55,7)$.
Here, near-field channels are considered for the RIS-Rx link, since the RIS
operates in the near-field of the Rx by increasing $N$. As seen from the Fig.
4, the improved achievable rate is obtained for increasing $N$ values. From
the given results of Fig. 4, we observe that an RIS-assisted system with a
sufficient number of RIS elements can outperform a system without RIS, which
consumes 10 dB more transmit power.

Figure 4: Achievable rates of RIS-assisted system for the outdoor environment.

In Figs. 5(a) and (b), the achievable rate of the system is presented under
various RIS positions for the outdoor environment. We consider the operating
frequency of 5.8 GHz, $N=1024$, and the 3D locations as
$\textbf{r}^{\text{Tx}}=(0,25,10)$, $\textbf{r}^{\text{Rx}}=(100,50,1)$ and
$\textbf{r}^{\text{RIS}}=(x^{\text{RIS}},y^{\text{RIS}},7)$. In Fig. 5(a), the
RIS is operating in the far-field of the Rx, therefore, far-field channels are
considered for the RIS-Rx link. As seen in Fig. 5, the achievable rate of the
system increases as the RIS gets closer to the Rx in both $x$ and $y$-axes.
Hence, the achievable rate of the system is maximized when
$d_{\text{3D}}^{\text{RIS-R}}$ is minimized. When the RIS positioned in front
of the Rx and operates in the near-field of the Rx, the achievable rate is
maximized as observed in Fig. 5(b). Moreover, when the RIS is placed very
close to the Rx around 2-3 meters in $y$-axis, the effective areas of the RIS
elements decreases significantly due to the elevation difference between the
RIS and Rx. Hence, the achievable rate is increasing for 2-3 meters as the RIS
moves away in the $y$-axis as seen in Fig. 5(b). It should be noted that an
improved performance in achievable rate is obtained when the near-field
conditions are considered for large RISs.

##  V Conclusions

In this paper, a physical channel model for RIS-assisted networks is presented
in sub-6 GHz frequency bands. This study aims to meet the lack of analysis of
RIS-assisted systems for sub-6 GHz bands in the literature by considering
physical channels in sub-6 GHz bands. The performance of the system is
analyzed from different perspectives, such as different environments,
locations of the Tx, Rx and RIS, operating frequencies, number of RIS elements
for near-field and far-field conditions. Our future works may include the
extension of this work to MIMO systems along with real-world experimental
results.

Figure 5: Achievable rates of RIS-assisted system for outdoors operating in
(a) far-field and (b) near-field of Rx.

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