# Shadow of the $5D$ Reissner-Nordström AdS Black Hole

Surajit Mandal  Sudhaker Upadhyay111Visiting Associate, IUCAA Pune,
Maharashtra-411007 Yerlan Myrzakulov  Gulmira Yergaliyeva

###### Abstract

We discuss the shadow cast by the charged Reissner-Nordström (RN) AdS black
hole. With the help of Killing equation and Hamilton-Jacobi equation, we
calculate the geodesic equations for null particle. With the help of geodesics
of null particle, we then determine the celestial coordinates ($\alpha$,
$\beta$) and the shadow radius of the RN AdS black hole. We present a
graphical analysis of the black hole shadow and find that shadow is a
perfectly dark circle. The impacts of charge and cosmological constant of the
RN AdS black hole on the radius of shadow are also presented. In this
connection, radius of the shadow is a decreasing function of the charge.
Furthermore, we study the effects of plasma medium on the RN AdS black hole
shadow. Here, we find that radius of circular shadow increases with increasing
plasma parameter. In addition, we also discuss the energy emission rate of RN
AdS black hole. The effects of parameters like charge, cosmological constant
and plasma parameter on energy emission rate are analyzed graphically.

##  1 Introduction

The study of black hole is an interesting topic of physics since the discovery
of Einstein’s theory of general relativity. Nowadays, it is sharply presumed
that there exist black holes at the center of most of the galaxies, for
example, it is mostly accepted that supermassive black hole Sagittarius
$A^{\star}$ or Sgr $A^{\star}$ is present in the galactic center of Milky Way
[1, 2] and recently the shadow of this supermassive black hole has been
noticed by event horizon telescope (EHT) group [3]. Any objects moving near
the black hole has a intense gravitational attraction and, within a critical
radius, must falls into the black hole. This is happen due to strong
gravitational lensing.

If an illuminated source of photon is present just behind the black hole, then
photon particles can move in the vicinity of black holes and can make a shadow
at a plane which might be visible by a distant viewer. The shadow of the black
hole was first studied by Bardeen [4]. It has been found that the spherically
symmetric static (Schwarzschild) black holes can cast a shadow of circular
shape [5, 6], whereas the rotating (spinning) black holes cast deformed
shadows from circular shape [7, 8, 9, 10, 11, 12]. Nowadays, the study of
shadow cast of black hole has becomes an active area of research and received
a momentous attention [13]. The shadow of Schwarzschild black hole has been
studied in [14, 15]. The rotating black hole shadow along with gravitomagnetic
and electric charge can be found in [16]. The gravitational lensing together
with optical phenomena for the Janis-Newman-Winicour [17], the Kerr-Newman and
the Kerr-Newman-AdS spacetimes have been studied in Ref. [18]. Peoples have
investigated the shadow of other black holes, such as regular black holes [19,
20, 21], Kerr black holes [4], Kerr-Newman black holes [22] and multi-black
holes [23]. Shadow of black holes in a modified gravity due to extended Chern-
Simons and Einstein-Maxwell dilaton gravity can be found in [24]. However, the
shadow in Randall-Sundrum braneworld case is studied in [25]. Recently, Övgün
et al. studied the black hole shadow of noncommutative black holes in Rastall
gravity [46]. Testing of generalized Einstein-Cartan-Kibble-Sciama gravity
using shadow cast was made in [26]. Jusufi et al. [27] studied the shadows of
$5D$ charged Bardeen black holes. Shadow cast of rotating braneworld black
holes with a cosmological constant was made in [28].

The black holes solution in higher than four dimensions are of momentous
importance. For moment, the black hole solutions in five-dimensions ($5D$)
have been considered in varius research, inspired by the ideas in string
theory, braneworld cosmology and gauge/gravity duality. Certain interesting
and surprising results can be found [29]. Extra dimensions voilate uniqueness
theorem due to the fact that there exist spurious degrees of freedom. The
study of black-ring solutions in five-dimensions demonstrate that higher
dimensional spacetime can approve the non-trivial topologies [30]. The
structure of $5D$ black hole shadows in the context of holography were
invesigated in [31]. Recently, the shadow of five-dimensional Gauss-Bonnet
black hole is studied in [48]. Shadow of five-dimensional rotating Myers-Perry
black hole [32, 33] and rotating five-dimensional charged black holes [34]
have also been studied. Such investigation indeed provides a good inspiration
to study the black hole shadows for RN AdS5 black hole. RN AdS black hole
solution in five-dimensions is a static, spherically symmetric vacuum solution
of the Einstein equation in RN AdS5 spacetime. Geodesic Motions near the RN
AdS5 black hole are studied in [35]. The goal of this paper is to investigate
the shadow of charged RN AdS5 black hole in both non-plasma and plasma medium.

This work is presented systematically in following manner. In section 2, we
provide an overview of five-dimensional RN-AdS5 black hole solution. From the
positive real zero of horizon function, the radius of the event horizon is
calculated in section 3. In section 4, using the Hamilton-Jacobi equation, we
discuss the (null) particle motion around the five-dimensional RN-AdS black
hole in non-plasma media and eventually derive the radius of photon sphere for
non-plasma medium. The shadow of this black hole for non-plasma medium is
studied in section 5. Within section, we also tabulated the values of shadow
radius and photon sphere radius for different charges. The shadow images are
aso given for different values of parameters. We study the effect of various
parameters on shadow radius for non-plasma medium in section 6. Similar to
case the non-plasma medium, geodesics in the presence of plasma medium are
studied in section 7. The shadow for such plasma medium is presented in
section 8. The behavior of shadow radius for various parameters in plasma
medium is discussed in full details in section 9. We discuss the energy
emission rate of five-dimensional RN AdS black hole in section 10. We see that
the emission rate of RN AdS black hole also depends on various parameters like
temperature, frequency and cross-section of photon radius. Finally, we
conclude our results and make future remarks in section 11.

##  2 $5D$ RN AdS black hole

Let us consider a $5D$ spacetime with a negative cosmological constant. The
field equations near the black hole are given by following equation [35]:

$\bar{G}_{ab}=-\Lambda\bar{g}_{ab}+k^{2}_{(5)}\bar{T}_{ab},$ (2.1)

where $\bar{g}_{ab}$ is a $5D$ metric with signature $(-,+,+,+,+)$,
$\bar{G}_{ab}$ denotes the $5D$ Einstein tensor, $\bar{T}_{ab}$ denotes $5D$
energy-momentum tensor and $\Lambda$ represents the cosmological constant. The
constant $k_{(5)}$ is defined as

$k^{2}_{(5)}=8\pi G=M^{-3},$ (2.2)

where $G$ is the $5D$ Newton’s constant and $M$ is the $5D$ reduced Planck
mass.

Now, we assume that the spacetime has a constant curvature
$\bar{K}=\frac{\alpha}{l^{2}}$. For AdS geometry, $\bar{K}$ is negative i.e,
$\alpha=-1$ while for dS geometry $\bar{K}$ becomes positive and $\alpha=1$.
Consequently, the radius of curvature of spacetime becomes

$l=\sqrt{\frac{3\alpha}{\Lambda}},$ (2.3)

and it gives the necessary length scale to have a horizon. The exterior metric
for the black hole field in $5D$ is given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{3},$ (2.4)

where the metric on unit $3$-sphere is

$d\Omega^{2}_{3}=d\theta^{2}+\sin^{2}\theta(d\phi^{2}+\sin^{2}\phi
d\psi^{2}).$ (2.5)

For this static spherically symmetric vacuum solution of the Einstein
equations in RN AdS5, the metric function $f(r)$ is defined as

$f(r)=1-\left(\frac{2M}{r}\right)^{2}+\frac{q^{4}}{r^{4}}-\frac{\Lambda
r^{2}}{6}.$ (2.6)

Here, $q$, $M$ and $\Lambda$ are the total charge, geometric mass and the
cosmological constant, respectively.

##  3 Horizons

We can write the lapse function (2.6) as

$f(r)=\frac{\Delta}{r^{4}},$ (3.1)

where $\Delta$ is the horizon function that depends on $r$ only for the given
$q$, $M$ and $\Lambda$. Now, the exterior metric (2.4) can be written as

$ds^{2}=-\frac{\Delta}{r^{4}}dt^{2}+\frac{r^{4}}{\Delta}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta(d\phi^{2}+\sin^{2}\phi
d\psi^{2}).$ (3.2)

We know that the spacetime is singular intrinsically at $r=0$ and the nature
of this singularity depends on the parameters such as cosmological constant
$\Lambda$ and charge $q$. Both of these parameters can be chosen so that
spacelike naked singularity of the spacetime can be avoided [36]. The metric
function vanishes at the real positive zeros of the horizon function, i.e,
$\Delta=0$, which gives

$\Lambda r^{6}-6r^{4}+24M^{2}r^{2}-6q^{4}=0.$ (3.3)

So, the real positive zeros of $\Delta$ gives the position of the horizon,
which indicates the coordinate singularities for this RN-AdS5 black hole.
Interestingly, among the six distinct roots of Eq. (3.3), only one is positive
real, which describes the radius of the event horizon of the RN AdS5 black
hole. This single positive real root is given numerically by

$r_{+}=\sqrt{\frac{2}{\Lambda}-\frac{242^{\frac{1}{3}}M^{2}}{\mathcal{P}}+\frac{122^{\frac{1}{3}}}{\Lambda\mathcal{P}}+\frac{\mathcal{P}}{32^{\frac{1}{3}}\Lambda}},$
(3.4)

where

$\mathcal{P}=\left(432-1296M^{2}\Lambda+162q^{4}\Lambda^{2}+\sqrt{4(-36+72M^{2}\Lambda)^{3}+(432-1296M^{2}\Lambda+162q^{4}\Lambda^{2})^{2}}\right)^{\frac{1}{3}}.$

##  4 Particle motion for non-plasma medium

In order to find the shape of the RN AdS5 black hole shadow, we first require
to derive the geodesic equations for the photons (null particle) around the
black holes. The directions of symmetry basically work on the issue of
searching the geodesics by knowing the constants of motion related to the
directions of symmetries. Let us take $k^{\mu}$ as a vector towards the
direction of symmetry and $u^{\mu}=\frac{dx^{\mu}}{d\lambda}$ along
perpendicular direction tangent on a curve $x^{\mu}=x^{\mu}(\lambda)$ with
affine parameter $\lambda$. Now, with the help of the Killing equation, it can
be shown that

$k^{\mu}u_{\mu}=\mbox{constant},$ (4.1)

when the trajectory $x^{\mu}$ is a geodesic [37]. As the coefficients of
metric (2.4) are independent of time, this results a timelike Killing vector.
Therefore, Eq. (4.1) becomes

$k^{0}u_{0}=u_{0}=-E,$ (4.2)

where constant $E$ is known as the relativistic energy per particle mass. The
negative sign in front of $E$ is chosen for convenience only and has no other
physical significance. In fact, coefficients of the metric are independent of
$\phi$ and $\psi$ as well, so these coordinates represent the other symmetry
directions. Hence, by choosing $k^{\mu}=(0,0,0,1,0)$ along the $\phi$
direction, we get

$k^{\mu}u_{\mu}=u_{3}=L_{\phi},$ (4.3)

and $k^{\mu}=(0,0,0,0,1)$ along the $\psi$ direction, we get

$k^{\mu}u_{\mu}=u_{4}=L_{\psi},$ (4.4)

where constants $L_{\phi}$ and $L_{\psi}$ are known as the angular momentum
per particle mass.

Now, the geodesic equations along the $t$, $\phi$ and $\psi$ directions can be
calculated by utilizing the above constants of motion. These read

$\displaystyle\frac{dt}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{E}{f(r)},$ (4.5) $\displaystyle\frac{d\phi}{d\lambda}$
$\displaystyle=$ $\displaystyle\frac{L_{\phi}}{r^{2}\sin^{2}\theta},$ (4.6)
$\displaystyle\frac{d\psi}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{L_{\psi}}{r^{2}\cos^{2}\theta}.$ (4.7)

The remaining two geodesic equations can be calculated with the help of the
relativistic Hamilton-Jacobi equation given as

$\frac{\partial S}{\partial\lambda}+\frac{1}{2}g^{\mu\sigma}\frac{\partial
S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\sigma}}=0.$ (4.8)

Now, in order to solve the above Hamilton-Jacobi equation, we take an ansatz
for $S$ of the form [8]

$S=\frac{1}{2}m_{0}^{2}\lambda-
Et+L_{\phi}\phi+L_{\psi}\psi+H_{r}(r)+H_{\theta}(\theta),$ (4.9)

where $H_{r}(r)$ and $H_{\theta}(\theta)$ are functions of only $r$ and
$\theta$, respectively. Here, $\lambda$ and $m_{0}$ signify the affine
parameter and the rest mass of the test particle, respectively. Using Eqs.
(4.9) and (4.8), we obtain

$\left(\frac{\partial
H_{\theta}}{\partial\theta}\right)^{2}+L_{\phi}^{2}\cot^{2}\theta+L_{\psi}^{2}\tan^{2}\theta+\frac{1}{2}m_{0}^{2}-\frac{r^{2}E^{2}}{f(r)}+r^{2}f(r)\left(\frac{\partial
H_{r}}{\partial r}\right)^{2}+L_{\phi}^{2}+L_{\psi}^{2}=0.$ (4.10)

This further simplifies to

$\left(\frac{\partial
H_{\theta}}{\partial\theta}\right)^{2}+L_{\phi}^{2}\cot^{2}\theta+L_{\psi}^{2}\tan^{2}\theta+\frac{1}{2}m_{0}^{2}=\frac{r^{2}E^{2}}{f(r)}-r^{2}f(r)\left(\frac{\partial
H_{r}}{\partial r}\right)^{2}-L_{\phi}^{2}-L_{\psi}^{2}=\mathcal{C}$ (4.11)

where $\mathcal{C}$ is the Carter constant. As we know the relation
$p_{\theta}=\frac{\partial H}{\partial\theta}=\frac{\partial
H_{\theta}}{\partial\theta}$, so we have

$\frac{\partial
H_{\theta}}{\partial\theta}=r^{2}\frac{\partial\theta}{\partial\lambda}.$
(4.12)

Here, the condition $p_{\theta}=\frac{\partial\mathcal{L}}{\partial\theta}$ is
used and Lagrangian $\mathcal{L}$ is given by

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\frac{d{x}^{\mu}}{d\lambda}\frac{d{x}^{\nu}}{d\lambda}.$
(4.13)

Similarly, using the condition $p_{r}=\frac{\partial H}{\partial
r}=\frac{\partial H_{r}}{\partial r}$, we can write

$\frac{\partial H_{r}}{\partial r}=r^{2}\frac{\partial r}{\partial\lambda}.$
(4.14)

In order to determine the null geodesics, we set $m_{0}=0$ and plugging the
values of the Eqs. (4.12 and (4.14) into Eq.(4.11), we obtain

$r^{2}\left(\frac{dr}{d\lambda}\right)=\sqrt{r^{4}E^{2}-(L^{2}+\mathcal{C})r^{2}f(r)},$
(4.15)
$r^{2}\left(\frac{d\theta}{d\lambda}\right)=\sqrt{\mathcal{C}-L_{\phi}^{2}\cot^{2}\theta-
L_{\psi}^{2}\tan^{2}\theta},$ (4.16)

where $L^{2}=L_{\phi}^{2}+L_{\psi}^{2}$. The above Eqs. (4.15) and (4.16) are
the geodesic equations along $r$ and $\theta$, respectively.

Now, using Eq. (4.15), we can arrive at the the familiar form of equation:

$\left(\frac{dr}{d\lambda}\right)^{2}+V(r)=0,$ (4.17)

where $V$ is an effective potential, given by

$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle\frac{f(r)}{r^{2}}\left(\mathcal{C}+L^{2}\right)-E^{2}.$ (4.18)

To find the unstable circular orbits, we consider the following conditions:

$\left.V(r)\right|_{r=r_{p}}=0,\;\left.\frac{\partial V(r)}{\partial
r}\right|_{r=r_{p}}=0.$ (4.19)

Now, $V(r)$ will be a maximum at $r=r_{p}$ when

$\left.\frac{\partial^{2}V(r)}{\partial r^{2}}\right|_{r=r_{p}}<0,$ (4.20)

where $r_{p}$ is the photon sphere radius. Now, using Eq. (4.18), the first
condition of (4.19) leads to

$\frac{r_{p}^{2}}{f(r_{p})}=\eta+\xi^{2},\;\;\mbox{with}\ \
\xi^{2}\equiv\xi_{1}^{2}+\xi_{2}^{2},$ (4.21)

where the Chandrasekhar constants $\eta,\xi_{1}$ and $\xi_{2}$ are considered
to have forms [8]

$\eta=\frac{\mathcal{C}}{E^{2}},\;\quad\xi_{1}=\frac{L_{\phi}}{E},\;\;\;\xi_{2}=\frac{L_{\psi}}{E}.$
(4.22)

Now, the (boundary) condition, $\left.\frac{\partial V_{eff}(r)}{\partial
r}\right|_{r=r_{p}}=0$, yields

$\left.r\frac{df(r)}{dr}\right|_{r=r_{p}}=\left.2f(r)\right|_{r=r_{p}}.$
(4.23)

For the given metric function $f(r)$ in (2.6), the condition (4.23) leads to

$r^{4}_{p}-8M^{2}r^{2}_{p}+3q^{4}=0.$ (4.24)

The solution of this equation results to the radius of the photon sphere as

$r_{p}=\sqrt{\frac{8M^{2}+\sqrt{64M^{4}-12q^{4}}}{2}}.$ (4.25)

##  5 Constructing the black hole shadow

To get the shadow of the RN-AdS5 black hole, we introduce the celestial
coordinates as shown schematically in figure 1. For $5D$, the celestial
coordinates become [38]

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\lim_{r\rightarrow\infty}\left(r^{2}\sin\theta\frac{d\phi}{dr}+r^{2}\cos\theta\frac{d\psi}{dr}\right),$
(5.1) $\displaystyle\beta$
$\displaystyle=\lim_{r\rightarrow\infty}r^{2}\sin\theta\frac{d\theta}{dr},$

Figure 1: Schematic diagram of celestial coordinates. Courtesy: [39].

where $\alpha$ component represents the apparent perpendicular displacement of
the shadow as observed from the $z$-axis (axis of symmetry) and $\beta$
component represents the the apparent perpendicular displacement of the shadow
as observed from its projection on the equatorial surface. Here, $r_{0}$
denotes the distance between the viewer and the black hole. However,
$\theta_{0}$ is the angle between the observer’s eye line and the axis of
rotation of the black hole called as the inclination angle. Recalling the
geodesic equations derived in Eqs. (4.5) and (4.15), we get the values of
$\frac{d\phi}{dr},\frac{d\psi}{dr}$ and $\frac{d\theta}{dr}$ as following:

$\frac{d\phi}{dr}=\frac{L_{\phi}\csc^{2}\theta}{r^{2}\sqrt{E^{2}-\frac{f(r)}{r^{2}}\left(\mathcal{C}+L_{\phi}^{2}+L_{\psi}^{2}\right)}},$
(5.2)
$\frac{d\psi}{dr}=\frac{L_{\phi}\sec^{2}\theta}{r^{2}\sqrt{E^{2}-\frac{f(r)}{r^{2}}\left(\mathcal{C}+L_{\phi}^{2}+L_{\psi}^{2}\right)}},$
(5.3)
$\frac{d\theta}{dr}=\frac{1}{r^{2}}\sqrt{\frac{\mathcal{C}-L_{\phi}^{2}\cot^{2}\theta-
L_{\psi}^{2}\tan^{2}\theta}{E^{2}-\frac{f(r)}{r^{2}}\left(\mathcal{C}+L_{\phi}^{2}+L_{\psi}^{2}\right)}}.$
(5.4)

Plugging the these equations into the Eq. (5.1), we get

$\alpha=-\frac{\xi_{1}\csc\theta+\xi_{2}\sec\theta}{\sqrt{1+\left(\eta+\xi_{1}^{2}+\xi_{2}^{2}\right)\Lambda}},$
(5.5)
$\beta=\pm\sqrt{\frac{\eta-\xi_{1}^{2}\cot^{2}\theta-\xi_{2}^{2}\tan^{2}\theta}{1+\left(\eta+\xi_{1}^{2}+\xi_{2}^{2}\right)\Lambda}}.$
(5.6)

Let us consider two different values for $\theta$, i.e., $0$ and
$\frac{\pi}{2}$. If $\theta=\frac{\pi}{2}$ then $L_{\psi}=0$ and as a result
$\xi_{1}\equiv\xi$. However, when $\theta=0$, $L_{\phi}=0$ which indicates
$\xi_{2}\equiv\xi$. Taking both the cases together into account, the above
celestial coordinates take the form

$\alpha=-\frac{\xi}{\sqrt{1+\left(\eta+\xi^{2}\right)\Lambda}},$ (5.7)
$\beta=\pm\sqrt{\frac{\eta}{1+\left(\eta+\xi^{2}\right)\Lambda}}.$ (5.8)

Combining the eqs. (5.7) and (5.8), we arrive at the equation of a circle in
the $\alpha\beta$-plane

$\alpha^{2}+\beta^{2}=\frac{\eta+\xi^{2}}{1+(\eta+\xi^{2})\Lambda}\equiv
R_{s}^{2}$ (5.9)

where $R_{s}$ refers to the radius of the shadow for non-plasma medium.

We have shown the numerical values of shadow radius and photon sphere radius
of RN-AdS5 black hole for different values $q$ and cosmological constant
$\Lambda$ in table 1 and table 2.

$q$ | $r_{p}$ | $R_{s}$
---|---|---
0.1 | 2.82842 | 3.76177
0.3 | 2.82789 | 3.76135
0.5 | 2.82427 | 3.75851
0.7 | 2.81228 | 3.74914
Table 1: Photon radius ($r_{p}$) and the black hole shadow radius ($R_{s}$) for varying $q$ with $\Lambda=0.0098$. $q$ | $r_{p}$ | $R_{s}$
---|---|---
0.1 | 2.82842 | 3.5921
0.3 | 2.82789 | 3.59174
0.5 | 2.82427 | 3.58927
0.7 | 2.81228 | 3.5811
Table 2: Photon radius ($r_{p}$) and the black hole shadow radius ($R_{s}$)
for varying $q$ with $\Lambda=0.018$.

In figure 2, the variation of the black hole shadow in celestial plane for
different values of the charge $q$ and cosmological constant
$\Lambda=0.0098,0.018$ is shown graphically.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa2.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa3.eps}\end{@subfigure}\end{array}$

Figure 2: In (a), black hole shadow in $\alpha\beta$-plane for different
values of $q$ with a cosmological constant $\Lambda=0.0098$. In (b), black
hole shadow in $\alpha\beta$-plane for different values of $q$ with
cosmological constant $\Lambda=0.018$. Here, $M=1$.

Here, we observe that the radius of the black hole shadow is a decreasing
function of charge parameter $q$ and cosmological constant $\Lambda$.

##  6 Effect of parameters on shadow radius in non-plasma medium

Here, we study the effect of various parameter on shadow radius $R_{s}$ for
non-plasma medium. Now, the shadow radius in non-plasma medium takes the
following form:

$R_{s}=\sqrt{\frac{\eta+\xi^{2}}{1+(\eta+\xi^{2})\Lambda}}=\sqrt{\frac{r^{2}_{p}}{f(r_{p})+\Lambda{r^{2}_{p}}}}.$
(6.1)

Here, we have used relation (4.21). It is noticed that shadow radius depends
on the parameters like cosmological constant $\Lambda$, charge $q$ and mass
$M$ of the black hole.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa4.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa5.eps}\end{@subfigure}\end{array}$

Figure 3: In (a), the black hole shadow radius vs. charge for changing
cosmological constant $\Lambda$. In (b), the black hole shadow radius vs.
cosmological constant for changing charge $q$. Here, $M=1$.

Figure 3 depicts how shadow radius for non-plasma medium depends on parameter
like charge $q$ and cosmological constant $\Lambda$. In 3, shadow radius is a
decreasing function of charge for fixed photon sphere radius. We see that, for
larger values of $\Lambda$, shadow radius curve decreases sharply. However, in
plot 3, shadow radius is a decreasing function of $\Lambda$ and here shadow
radius curve decreases for increasing charge parameter $q$.

##  7 Shadow cast in the presence of plasma medium

Generally, a black hole is enclosed by a substance media which gets affected
on the motion of null like particles passing through it. In the present
section, we emphasize the consequences of a plasma medium on the shadow of RN-
AdS5 black hole. For the plasma medium , the refractive index is denoted by
$n=n\left(x^{i},\omega\right)$, where $\omega$ is the angular frequency of
photon measured by a distant moving viewer having a velocity $u^{\mu}$. The
Hamiltonian is now modified by this plasma background. Due to this medium,
there appears an extra term in the equations of geodesic motion and, hence,
the trajectories of null paricles get modified. Due to the plasma effect,
effective energy of the particle also gets modified and given by $E=$
$\hbar\omega=-p_{\alpha}u^{\alpha}$. The relationship between the refractive
index and the $4$-momentum of the photon is given by [40]

$n^{2}=1+\frac{p_{\alpha}u^{\alpha}}{\left(p_{\mu}u^{\mu}\right)^{2}}.$ (7.1)

Now, the refractive index ($n$) and plasma frequency ($\omega_{p}$) arw
related by [41]

$n=\sqrt{1-\left(\frac{\omega_{p}}{\omega}\right)^{2}},$ (7.2)

where $\omega_{p}$ is defined by

$\omega_{p}=\frac{4\pi e^{2}N}{m_{e}}.$ (7.3)

Here, $e,N$ and $m_{e}$ are, respectively, the charge, $r$ dependent number
density and electron mass in the plasma medium. The physical form of $N(r)$ is
considered as $\frac{N_{0}}{r^{i}}$ [40, 41], where $i$ identifies various
characteristics of the plasma medium but we consider $i=1$ that reflects the
minimum dependency on $r$ [41, 42]. Replacing the form of $N(r)$ in the plasma
frequency $\omega_{p}$ and, from Eq.(7.2), we have

$\left(\frac{\omega_{p}}{\omega}\right)^{2}=\frac{k}{r},k\geqslant 0.$ (7.4)

Consequently, the refractive index $n$ becomes

$n=\sqrt{1-\frac{k}{r}}.$ (7.5)

The corrcted Hamilton-Jacobi equation reads [40]

$\left(\frac{\partial
S}{\partial\lambda}\right)+\frac{1}{2}\left[g^{\mu\sigma}\frac{\partial
S}{\partial x^{\mu}}\frac{\partial S}{\partial
x^{\sigma}}-\left(n^{2}-1\right)\left(\frac{\partial S}{\partial
t}\sqrt{-g^{tt}}\right)^{2}\right]=0.$ (7.6)

##  8 Geodesics and the black hole shadow for the plasma medium

In order to study the consequences of a plasma medium on black hole shadow, we
require a new set of (celestial) coordinates. We begin the study by
considering the modified geodesic equations. Now, for plasma medium, the set
of null geodesics read

$\displaystyle\frac{dt}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{n^{2}E}{f(r)},$ (8.1)
$\displaystyle\frac{d\phi}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{L_{\phi}}{r^{2}\sin^{2}\theta},$ (8.2)
$\displaystyle\frac{d\psi}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{L_{\psi}}{r^{2}\cos^{2}\theta},$ (8.3) $\displaystyle
r^{2}\left(\frac{dr}{d\lambda}\right)$ $\displaystyle=$
$\displaystyle\pm\sqrt{n^{2}r^{4}E^{2}-\left(L^{2}+\mathcal{C}\right)r^{2}f(r)},$
(8.4) $\displaystyle r^{2}\left(\frac{d\theta}{d\lambda}\right)$
$\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{C}-L_{\phi}^{2}\cot^{2}\theta-
L_{\psi}^{2}\tan^{2}\theta}.$ (8.5)

Following the same procedure as in the non-plasma medium, the effective radial
potential in the plasma medium becomes

$V^{pl}(r)=\frac{f(r)}{r^{2}}\left(\mathcal{C}+L^{2}\right)-n^{2}E^{2}.$ (8.6)

The unstable circular orbit can be determined by

$\left.V^{pl}(r)\right|_{r=r_{p}^{(pl)}}=0,\;\;\left.\frac{\partial
V^{pl}(r)}{\partial r}\right|_{r=r_{p}^{(pl)}}=0,$ (8.7)

where $r_{p}^{(pl)}$ denotes radius of photon sphere in plasma medium. The
condition for maximizing $V^{pl}(r)$ reads

$\left.\frac{\partial^{2}V^{pl}(r)}{\partial
r^{2}}\right|_{r=r_{p}^{(pl)}}<0.$ (8.8)

The first condition of Eq. (8.7) leads to

$\eta+\xi^{2}=\left.\frac{n^{2}r^{2}}{f}\right|_{r=r_{p}^{(pl)}},$ (8.9)

and the second condition gives

$\left.\left(nrf^{\prime}-2nf-2n^{\prime}rf\right)\right|_{r=r_{p}^{(pl)}}=0.$
(8.10)

Now, substituting the values of $f(r)$ (2.6) and refractive index
$n^{\prime}(r)$ (7.5)) in (8.10), we obtain the equation for the radius of the
photon sphere as

$\displaystyle\sqrt{1-\frac{k}{r_{p}}}\left[\frac{4M^{2}}{r^{2}_{p}}-\frac{6q^{4}}{r^{4}_{p}}-2\right]-\frac{k}{r_{p}\sqrt{1-\frac{k}{r_{p}}}}\left[1-\frac{4M^{2}}{r_{p}^{2}}+\frac{q^{4}}{r^{4}_{p}}-\frac{\Lambda
r^{2}_{p}}{6}\right]=0.$ (8.11)

Here, we notice that an exact solution of Eq. (8.11) is not possible. So, we
try to solve this equation numerically. Due to the plasma medium, an
additional parameter ($k$) arises in Eq. (8.11) . Let us fix $k$ to take two
values, namely, $0.2$ and $0.4$ and for these values we obtain the radius of
photon $r_{p}$ upon solving (8.11) numerically. Following the same procedures
as before, we obtain an expression for $\frac{d\phi}{dr},\frac{d\psi}{dr}$ and
$\frac{d\theta}{dr}$ for the plasma medium and calculate the celestial
coordinates $(\alpha,\beta)$ for this medium. These are

$\frac{d\phi}{dr}=\frac{L_{\phi}\csc^{2}\theta}{r^{2}\sqrt{n^{2}E^{2}-\frac{f(r)}{r^{2}}\left(\mathcal{C}+L_{\phi}^{2}+L_{\psi}^{2}\right)}},$
(8.12)
$\frac{d\psi}{dr}=\frac{L_{\phi}\sec^{2}\theta}{r^{2}\sqrt{n^{2}E^{2}-\frac{f(r)}{r^{2}}\left(\mathcal{C}+L_{\phi}^{2}+L_{\psi}^{2}\right)}},$
(8.13)
$\frac{d\theta}{dr}=\frac{1}{r^{2}}\sqrt{\frac{\mathcal{C}-L_{\phi}^{2}\cot^{2}\theta-
L_{\psi}^{2}\tan^{2}\theta}{n^{2}E^{2}-\frac{f(r)}{r^{2}}\left(\mathcal{C}+L_{\phi}^{2}+L_{\psi}^{2}\right)}}.$
(8.14)

Plugging the above values into the expressions of the $(\alpha,\beta)$
coordinates (5.1), we get

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\frac{\left(\xi_{1}\csc\theta+\xi_{2}\sec\theta\right)}{\sqrt{1+\left(\eta+\xi_{1}^{2}+\xi_{2}^{2}\right)\Lambda}},$
(8.15) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{\Big{(}\eta-\xi_{1}^{2}\cot^{2}\theta-\xi_{2}^{2}\tan^{2}\theta\Big{)}}{1+\Big{(}\eta+\xi_{1}^{2}+\xi_{2}^{2}\Big{)}\Lambda}}.$
(8.16)

Here, we fix $\theta$ to have two different values, i.e, $\frac{\pi}{2}$ and
$0$. For both these values together, the coordinates $(\alpha,\beta)$ becomes

$\alpha=-\frac{\xi}{\sqrt{1+\left(\eta+\xi^{2}\right)\Lambda}},$ (8.17)
$\beta=\pm\sqrt{\frac{\eta}{1+\left(\eta+\xi^{2}\right)\Lambda}}.$ (8.18)

Taking the celestial coordinates from Eqs. (8.17) and (8.18) into account and
using Eq. (8.9), we have

$\alpha^{2}+\beta^{2}=\frac{n^{2}r^{2}_{p}}{f(r_{p})+\Lambda{n^{2}r^{2}_{p}}},$
(8.19)

where $R_{s}:=\sqrt{\alpha^{2}+\beta^{2}}$ is the radius of the shadow.

Tables 3, 4, 5 and 6 illustrate the values of shadow radius and photon radius
for different charge $q$ of the RN AdS5 black hole and cosmological constant
$\Lambda$ in the plasma medium.

q | $r_{p}$ | $R_{s}$
---|---|---
0.1 | 2.16013 | 4.54487
0.3 | 2.15914 | 4.54799
0.5 | 2.15233 | 4.57085
0.7 | 2.12942 | 4.65152
Table 3:  $r_{p}$ and $R_{s}$ for varying $q$ with $k=0.2$, $\Lambda=0.018$ and $M=1$. q | $r_{p}$ | $R_{s}$
---|---|---
0.1 | 2.15985 | 4.89338
0.3 | 2.15885 | 4.90248
0.5 | 2.15205 | 4.93012
0.7 | 2.12914 | 5.03282
Table 4:  $r_{p}$ and $R_{s}$ for varying $q$ with $k=0.2$, $\Lambda=0.0098$ and $M=1$. q | $r_{p}$ | $R_{s}$
---|---|---
0.1 | 2.11695 | 4.77485
0.3 | 2.11592 | 4.78018
0.5 | 2.10887 | 4.81765
0.7 | 2.08512 | 4.95895
Table 5:  $r_{p}$ and $R_{s}$ for changing $q$ with $k=0.4$, $\Lambda=0.018$ and $M=1$. q | $r_{p}$ | $R_{s}$
---|---|---
0.1 | 2.11636 | 5.18284
0.3 | 2.11533 | 5.18966
0.5 | 2.10828 | 5.23775
0.7 | 2.08453 | 5.421
Table 6:  $r_{p}$ and $R_{s}$ for varying $q$ with $k=0.4$, $\Lambda=0.0098$
and $M=1$.

In celestial plane, the dependence of the black hole shadow on charge $q$ and
plasma parameter $k$ with cosmological constant $\Lambda=0.018$ for plasma
medium is depicted in figure 4. We see that shadow radius increases with
plasma parameter $k$ for different photon sphere radius. We also observe that
the radius of the black hole shadow increases for an increase in the value of
the charge parameter $q$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa6.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa8.eps}\end{@subfigure}\end{array}$

Figure 4: In (a), black hole shadow in $\alpha\beta$-plane for different
values of $q$ with $k=0.2$. In (b), black hole shadow in the
$\alpha\beta$-plane for different values of $q$ with $k=0.4$. Here,
$\Lambda=0.018$ and $M=1$.

The figure 5 depicts the dependence of the black hole shadow on charge $q$ and
plasma parameter $k$ with cosmological constant $\Lambda=0.0098$ in celestial
plane for plasma medium. We see that shadow radius increases for an increase
in plasma parameter $k$ with changing photon sphere radius. It is also evident
from the plots that the size of the black hole shadow rises for an increase in
the value of the charge parameter $q$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa7.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa9.eps}\end{@subfigure}\end{array}$

Figure 5: In (a), black hole shadow in $\alpha\beta$-plane for different
values of $q$ with $k=0.2$. In (b), black hole shadow in the
$\alpha\beta$-plane for different values of $q$ with $k=0.4$. Here,
$\Lambda=0.0098$ and $M=1$.

Next, we shall discuss the effects of $q$ and $\Lambda$ on the shadow radius
for two different values of $k$.

##  9 Effect of parameters on shadow radius in plasma medium

In this section, we study the effects of various parameter on shadow radius
for the plasma media. The shadow radius $R_{s}$ in the plasma medium is given
by

$R_{s}=\sqrt{\frac{n^{2}r^{2}_{p}}{f(r_{p})+\Lambda n^{2}r^{2}_{p}}}.$ (9.1)

It is evident that shadow radius $R_{s}$ depends on the parameters such as
$\Lambda$, $q$ and $k$.

The figure 6 depicts how shadow radius for plasma medium depends on parameter
like charge $q$ and cosmological constant $\Lambda$ for $k=0.2$. Here, in 6,
we see that shadow radius is a decreasing function of charge for fixed values
of photon sphere radius. For larger $\Lambda$ values, shadow radius curve
decreases sharply. However, in plot 6, shadow radius is a decreasing function
of $\Lambda$ for fixed photon sphere radius. Moreover, the shadow radius curve
decreases for increasing charge parameter $q$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa10.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa11.eps}\end{@subfigure}\end{array}$

Figure 6: In (a), $R_{s}$ vs. $q$ for changing cosmological constant
$\Lambda$. In (b), $R_{s}$ vs. $\Lambda$ for changing charge $q$. Here,
$k=0.2$ and $M=1$

The figure 7 depicts how shadow radius for plasma medium depends on parameter
like charge $q$ and cosmological constant $\Lambda$ with $k=0.4$. In 7, shadow
radius is a decreasing function of $q$ for fixed values of photon sphere
radius. We also see that shadow radius curve decreases sharply for larger
$\Lambda$ values. However, in plot 7, shadow radius is a decreasing function
of $\Lambda$ for fixed photon sphere radius. Moreover, the shadow radius curve
falls for increasing charge parameter $q$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa12.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa13.eps}\end{@subfigure}\end{array}$

Figure 7: In (a), $R_{s}$ vs. $q$ for changing cosmological constant
$\Lambda$. In (b), $R_{s}$ vs. $\Lambda$ for changing charge $q$. Here,
$k=0.4$ and $M=1$

Both figures 6 and 7 suggest that, for a fixed photon sphere radius, shadow
radius is a decreasing function of plasma parameter.

##  10 Energy emission rate

Here, in the present section, we discuss the energy emission rate of $5D$
Reissner-Nordström AdS black hole. The energy emission rate can be expressed
as [43, 44, 45, 46]

$\frac{d^{2}}{d\omega
dt}Z(\omega)=\frac{2\pi^{2}\sigma_{lim}}{\exp\left(\frac{\omega}{T_{H}}\right)-1}\omega^{3},$
(10.1)

where $Z(\omega)$, $T_{H}$ and $\omega$ are the energy, Hawking temperature
and frequency, respectively, for the black hole. Here, limiting constant
$\sigma_{lim}$ for a spherically symmetric black corresponds to the
geometrical cross-section of its photon sphere. The Hawking temperature can be
calculated as [47]

$T_{H}=\left.\frac{f^{\prime}(r)}{4\pi}\right|_{r=r_{+}}=\frac{2M^{2}}{\pi
r^{3}_{+}}-\frac{q^{4}}{\pi r^{5}_{+}}-\frac{\Lambda r_{+}}{12\pi},$ (10.2)

where $r_{+}$ is the event horizon radius of RN-AdS5 black hole.

For $5D$ black hole $\sigma_{lim}$ reads [48, 49, 50]

$\sigma_{lim}=\frac{4\pi R^{3}_{s}}{3}.$ (10.3)

With this $\sigma_{lim}$, the energy emission rate becomes

$\frac{d^{2}}{d\omega
dt}Z(\omega)=\frac{8\pi^{3}R^{3}_{s}}{3\left(\exp\left(\frac{\omega}{T_{H}}\right)-1\right)}\omega^{3}.$
(10.4)

In order to study the behavior of the energy emission rate, we plot graphs.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa14.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa15.eps}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=216.81pt]{sa16.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa17.eps}\end{@subfigure}\end{array}$

Figure 8: The energy emission rate vs. frequency for changing charge $q$ and
changing photon sphere radius. In (a) $k=0$, $\Lambda=0.018$; in (b) $k=0.2$,
$\Lambda=0.018$; in (c) $k=0$, $\Lambda=0.0098$ and in (d) $k=0.2$,
$\Lambda=0.0098$. For all cases $M=1$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{sa18.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{sa19.eps}\end{@subfigure}\end{array}$

Figure 9: The energy emission rate vs. frequency for changing charge $q$ and
changing photon sphere radius. In (a) $k=0.4$, $\Lambda=0.018$ and in (b)
$k=0.4$, $\Lambda=0.0098$. For all cases $M=1$.

The figure 8 depicts the behavior of energy emission rate
$\frac{d^{2}Z(\omega)}{d\omega dt}$ with respect to frequency $\omega$ for
varying charge $q$, varying photon sphere radius with fixed plasma parameter
$k=0,0.2$. In contrast to non-plasma medium, the energy emission rate
increases for increases in charge for plasma medium. The presence of plasma
parameter $k$ increases the energy emission rate. However, the behavior of
energy emission rate with respect to frequency $\omega$ for varying charge
$q$, varying photon sphere radius with fixed plasma parameter $k=0.4$ is shown
in figure 9. Here, it is more clear that the energy emission rate increases
for increase in the values of charge.

##  11 Conclusion

We now summarize our results here, in the current section. We have
investigated the shadow of RN AdS5 black holes for an infinitely distant
observer. We have first evaluated the null geodesic equations using the
Hamilton-Jacobi equation. Using the boundary condition of unstable circular
orbit, we have determined the radius of the photon sphere and then by using
the null geodesics we have obtained the coordinates ($\alpha$, $\beta$) which
eventually results the radius of the shadow. We have tabulated of the various
numerical values of photon radius and shadow radius for different charge. We
have also done a graphical analysis for the black hole shadow in the
$\alpha\beta$-plane for different charges. The plots declared that the radius
of the black hole shadow decreases for an increase in the values of both the
charge parameter and cosmological constant. We have also found that shadow
radius depends on the mass of the black hole also. To check the dependency of
the shadow radius on these parameters, we have made a graphical analysis and
observed that shadow radius is a decreasing function of charge for fixed
photon sphere radius.

We then introduced a plasma medium in order to investigate their effect on the
unstable circular orbits of null-like particles (photon). It is observed that
shadow radius enlarges with the plasma parameter $k$ with varying photon
sphere radius. The resulting shadow radius in the plasma medium has dependency
on plasma parameter. We have done the graphical analyses for the plasma medium
also, which reflect the effect of various parameters on shadow radius.
Finally, we studied the energy emission rate of the $5D$ RN AdS black hole and
have provided a graphical analysis of energy emission rate with respect to
frequency. For future perspective, by utilizing the mathematical optics, we
shall carry forward the present study and will attempt to distinguish the
differences between the black holes acquired from various theories. We also
would like to study the consequences of spin parameter on the shadows of RN
AdS5 black hole.

## Acknowledgment

This research was funded by the Science Committee of the Ministry of Education
and Science of the Republic of Kazakhstan (Grant No. AP08052034).

## Data availability

Data sharing is not applicable to this manuscript, as no data sets were
generated or analyzed during the current study.

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Figure 14: Correlation between the predicted BAU $\eta_{B}$ and the effective
neutrino mass $m_{\beta\beta}$ for it Case 2 with $n=14$, $k_{1}=k_{2}=0$ and
$0\leq t\leq n-1$ (as shown by the numbered points). Only even $t$ are allowed
because of the condition $u=2s-t=0$. We have chosen the benchmark values for
$M_{N}=1$ TeV and $\kappa/\kappa_{\text{max}}=7\times 10^{3}\,(1.2\times
10^{3})$ for strong NO (IO) in the left (right) panel using the results from
Figure 11. The blue-shaded horizontal bar corresponds to $\eta_{B}$ within
$10\%$ of $\eta_{B}^{\rm obs}$. The vertical shaded bands for the IO case
indicate the smallest $m_{\beta\beta}$ value (including the NME uncertainties)
either ruled out by the current KamLAND-Zen bound (red) or accessible to
future nEXO (green).

###  VI.2 Case 2

In this case both the Majorana phases $\alpha_{1}$ and $\alpha_{2}$ can in
general have non-trivial values [cf. Eq. (III.2.2)]. Using the PMNS mixing
matrix from Eq. (67), we get the following for strong NO and IO respectively:

$\displaystyle m_{\beta\beta}^{\rm NO}$
$\displaystyle\,\approx\,\frac{1}{3}\left|\sqrt{\Delta
m_{\mathrm{sol}}^{2}}-2(-1)^{k_{1}+k_{2}}e^{i\phi_{v}}\left(\cos{\theta}\sin{\frac{\phi_{u}}{2}}-i\sin{\theta}\cos{\frac{\phi_{u}}{2}}\right)^{2}\sqrt{\Delta
m_{\mathrm{atm}}^{2}}\right|,$ (162a) $\displaystyle m_{\beta\beta}^{\rm IO}$
$\displaystyle\,\approx\,\frac{1}{3}\left|1+(-1)^{k_{1}}e^{i\phi_{v}}(\cos{\phi_{u}}+\cos{2\theta}-i\sin{2\theta}\sin{\phi_{u}})\right|\sqrt{\left|\Delta
m^{2}_{\mathrm{atm}}\right|}\,.$ (162b)

For our choice of $n=14$ and $u=2s-t=0$, only even values of $t$ between 0 and
$n-1$ are allowed. Using the best-fit $\theta_{L}\approx 2.96$,
$k_{1}=k_{2}=0$, and the best-fit values for the oscillation parameters, we
get

$\displaystyle 2.0\,\mathrm{meV}$ $\displaystyle\lesssim
m_{\beta\beta}\lesssim 4.0\,\mathrm{meV}~{}({\textrm{NO}})\,,$ $\displaystyle
18\,\mathrm{meV}$ $\displaystyle\lesssim m_{\beta\beta}\lesssim
49\,\mathrm{meV}~{}({\textrm{IO}})\,.$ (163)

This is shown in Figure 14 by the red points, which correspond to different
even values of $t$ ranging from 0 to $n-1$. The left (right) panel is for NO
(IO). As in Case 1, for strong IO, some of the admitted values of
$m_{\beta\beta}$ are already excluded by the current KamLAND-Zen bound [182]
(red shaded region) and the remaining values can be tested with the future
nEXO [187] (green shaded region).

The corresponding values of $\eta_{B}$ have been generated for $M_{N}=1$ TeV
and $\kappa/\kappa_{\text{max}}=7\times 10^{3}\,(1.2\times 10^{3})$ in the NO
(IO) case. As can be seen in Figure 14, $s=8(4)$ can successfully reproduce
the observed BAU for strong NO (IO).

###  VI.3 Case 3

Figure 15: Correlation between the predicted BAU $\eta_{B}$ and the effective
neutrino mass $m_{\beta\beta}$ for Case 3a (left) and Case 3b.1 (right) both
with NO and $m$ odd, $s$ odd. We have chosen $n=16\,(10)$, $k_{1}=k_{2}=0$,
$m=1\,(5)$, $M_{N}=1$ TeV and $\kappa/\kappa_{\text{max}}=1\times
10^{3}\,(1.3\times 10^{5})$ in the left (right) panel. In Case 3b.1, the blue-
shaded horizontal bar corresponds to $\eta_{B}$ within $10\%$ of
$\eta_{B}^{\rm obs}$.

Following the leptogenesis analysis in Figure 12, we only consider the NO case
here. The effective neutrino mass in terms of the high-energy CP phases is
given by

$\displaystyle(m_{\beta\beta}^{\rm NO})_{\rm 3a}$
$\displaystyle\,\approx\,\frac{1}{3}\left|\left(\sqrt{2}\sin{\theta}\cos{\phi_{m}}+e^{-3i\phi_{s}}\cos{\theta}\right)^{2}\sqrt{\Delta
m_{\mathrm{sol}}^{2}}+2(-1)^{k_{1}+k_{2}}\sin^{2}{\phi_{m}}\sqrt{\Delta
m_{\mathrm{atm}}^{2}}\right|,$ (164a) $\displaystyle(m_{\beta\beta}^{\rm
NO})_{\rm 3b.1}$
$\displaystyle\,\approx\,\frac{1}{3}\left|\sin^{2}\theta\sqrt{\Delta
m_{\mathrm{sol}}^{2}}+(-1)^{k_{1}}\cos^{2}\theta\,\sqrt{\Delta
m_{\mathrm{atm}}^{2}}\right|.$ (164b)

In case 3a for a given $n$, only certain values of $m$ close to 0 or $n$
reproduce the observed neutrino mixing angles with allowed ordering restricted
to NO. Thus, we set $n=16$, $m=1$ with $\theta_{L}$ computed separately for
each value of $s/n$. In case 3b.1 for a given $n$, values of $m\sim n/2$
reproduce the observed neutrino mixing angles with allowed ordering restricted
to NO. Thus, we set $n=10$, $m=5$ with $\theta_{L}=1.31$. For Case 3a with NO
and $k_{1}=k_{2}=0$, we get

$\displaystyle 1.5\,\mathrm{meV}\lesssim m_{\beta\beta}\lesssim
3.8\,\mathrm{meV}\,,$ (165)

whereas for Case 3b.1 with NO, we get a constant value of $m_{\beta\beta}=3.8$
meV, independent of $s$ and $k_{1}=0$. This can be understood from Eq. (164b)
which indeed is independent of the phase $\phi_{s}=s/n$ (and thus of the
chosen CP transformation). This is also shown in Figure 15, where the red
points correspond to different odd $s$ values between 0 and $n-1$ in both
cases.

The corresponding values of $\eta_{B}$ have been generated for $M_{N}=1$ TeV
and $\kappa/\kappa_{\text{max}}=1\times 10^{3}\,(1.3\times 10^{5})$ in Case 3a
(3b1). As can be seen in Figure 15, $s=11(7)$ can successfully reproduce the
observed BAU for strong NO in Case 3a (3b1).

##  VII Conclusion

We have presented a low-scale type-I seesaw scenario with a flavor $G_{f}$ and
CP symmetry that are broken into non-trivial residual symmetries in the
charged-lepton and neutrino sectors. We show that this class of models with
nearly degenerate heavy right-handed neutrinos can naturally explain the
observed baryon asymmetry of the Universe through resonant leptogenesis
mechanism. An important distinguishing feature of these models is the
possibility of having a relatively long lifetime for one of the RHNs, which is
attributed to enhanced residual symmetries. This allows the long-lived RHN to
be accessible at the dedicated long-lived particle search facilities, while at
the same time, the other two RHNs in the model can be probed via either prompt
or displaced vertex signals at the LHC.

We have studied the correlation of the BAU predictions with the collider
signals in a simple $U(1)_{B-L}$ extension of the type-I seesaw. We find that
while the LHC might not be able to probe the region allowed by successful
leptogenesis, a future 100 TeV collider can easily access these regions,
thereby providing a direct test of the leptogenesis mechanism in this
framework. The collider prospects turn out to be better in case of normal mass
ordering of light neutrinos.

We have also studied the correlation between the high- and low-energy CP
phases in the model. This is exemplified by considering the classic low-energy
lepton-number-violating process of neutrinoless double beta decay which
crucially depends on the low-energy phases, and connecting it to BAU which
depends on the high-energy CP phases. We find that the region allowing
successful leptogenesis can be completely tested in future tonne-scale
$0\nu\beta\beta$ experiments provided the light neutrino mass ordering is
inverted. Therefore, the collider and $0\nu\beta\beta$ experiments will
provide complementary probes of these flavor and CP symmetries and their roles
in the generation of baryon asymmetry.

## Acknowledgments

We thank Marco Drewes for alerting us to their related work [98] which
appeared a few months after ours on arXiv. BD thanks Claudia Hagedorn and
Emiliano Molinaro for many useful discussions and for collaboration during the
early stages of this work, part of which was presented in section 7.6 of Ref.
[170]. Part of the work presented here also appeared as Chapter 6 in Ref.
[190]. BD also acknowledges the local hospitality at MIAPP, UMass Amherst and
CP3 Odense, where parts of this work were done during 2016-2018. The work of
BD is supported in part by the US Department of Energy under Grant No. DE-
SC0017987 and by a Fermilab Intensity Frontier Fellowship. This work was
partly performed at the Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-1607611.

##  Appendix A Group Theory of $\Delta(6\,n^{2})$ and Representation Matrices

As discussed in Ref. [72], the discrete groups $\Delta(3\,n^{2})$, $n\geq 2$
integer, can be described in terms of three generators $a$, $c$ and $d$
fulfilling the relations

$\displaystyle a^{3}\,=\,e\;,\;\;c^{n}\,=\,e\;,\;\;d^{n}\,=\,e\;,\;\;$
$\displaystyle
c\,d\,=\,d\,c\;,\;\;a\,c\,a^{-1}\,=\,c^{-1}d^{-1}\;,\;\;a\,d\,a^{-1}\,=\,c\,,$
(166)

with $e$ being the identity element of the group. The discrete groups
$\Delta(6\,n^{2})$, $n\geq 2$ integer [73], are obtained by adding a fourth
generator $b$ to the set of $a$, $c$ and $d$. The relations involving $b$ are

$b^{2}\,=\,e\;,\;\;(a\,b)^{2}\,=\,e\;,\;\;b\,c\,b^{-1}\,=\,d^{-1}\;,\;\;b\,d\,b^{-1}\,=\,c^{-1}\,.$
(167)

In the trivial representation ${\bf 1}$ all elements of the group are
represented by the character $1$. The explicit representation matrices $g({\bf
3})$ for $a$, $b$, $c$ and $d$ can be chosen in the irreducible, faithful,
complex three-dimensional representation ${\bf 3}$ as

$\displaystyle a({\bf 3})\,=\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{array}\right)\;,\quad b({\bf
3})\,=\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&\omega^{2}\\\
0&\omega&0\end{array}\right)\;,$ (174) $\displaystyle c({\bf
3})\,=\,\frac{1}{3}\,\left(\begin{array}[]{ccc}1+2\cos\phi_{n}&1-\cos\phi_{n}-\sqrt{3}\sin\phi_{n}&1-\cos\phi_{n}+\sqrt{3}\sin\phi_{n}\\\
1-\cos\phi_{n}+\sqrt{3}\sin\phi_{n}&1+2\cos\phi_{n}&1-\cos\phi_{n}-\sqrt{3}\sin\phi_{n}\\\
1-\cos\phi_{n}-\sqrt{3}\sin\phi_{n}&1-\cos\phi_{n}+\sqrt{3}\sin\phi_{n}&1+2\cos\phi_{n}\end{array}\right)\,,$
(178)

with $\omega=e^{2\pi i/3}$ and $\phi_{n}=2\pi/n$. The representation for $d$
can be computed via $d({\bf 3})=a({\bf 3})^{2}c({\bf 3})a({\bf 3})$.

The existence of an irreducible, in general unfaithful, real three-dimensional
representation ${\bf 3^{\prime}}$ requires that all its characters are real.
This cannot be fulfilled in all groups $\Delta(6\,n^{2})$, but only, if the
index $n$ is even. In this case the form of the representation matrices
$g({\bf 3^{\prime}})$ is

$\displaystyle a({\bf 3^{\prime}})\,=\,a({\bf 3})\;\;,\;\;b({\bf
3^{\prime}})\,=\,b({\bf 3})\;\;,$ $\displaystyle c({\bf
3^{\prime}})\,=\,\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{array}\right)\,$ (182)

and $d({\bf 3^{\prime}})=a({\bf 3^{\prime}})^{2}c({\bf 3^{\prime}})a({\bf
3^{\prime}})$. Note that the representation matrices $g({\bf 3^{\prime}})$ do
not depend on the index $n$ of the group and thus lead to the same
representation for all groups $\Delta(6\,n^{2})$ with even $n$. Indeed, we can
observe that the group generated by the representation matrices $g({\bf
3^{\prime}})$ has 24 elements and thus corresponds to the group $\Delta(6\cdot
2^{2})=\Delta(24)$. This group is isomorphic to the permutation group $S_{4}$.
This representation together with the one generated by the representation
matrices $a({\bf 3^{\prime}})$, $c({\bf 3^{\prime}})$, $d({\bf 3^{\prime}})$
and $-b({\bf 3^{\prime}})$ (i.e. the representation matrix $b({\bf
3^{\prime}})$ acquires an overall sign [73]) are the only real three-
dimensional representations in a generic group $\Delta(6n^{2})$ with even $n$
and $3\nmid n$. To see this we inspect the characters of the three-dimensional
representations. Following Ref. [73] we see that the characters $\chi({\bf
3_{\rm gen}})$ of a generic irreducible three-dimensional representation ${\bf
3_{\rm gen}}$ for a certain type of classes is given by $\eta^{-\rho\,l}$ with
$\eta=e^{2\pi i/n}$, $\rho=0,..,n-1$ (labeling this type of class of the group
$\Delta(6n^{2})$) and $l=1,..,n-1$ ($l$ labels the different pairs of three-
dimensional representations). We have to require that all $\eta^{-\rho\,l}$
for a certain representation labeled by $l$ are real. This is ensured, if
$\eta^{-l}$ is real for all powers $\rho$ with $\rho=0,..,n-1$, meaning
$\eta^{-l}$ should be real itself. Hence, $2\,l/n$ must be an integer. With
the constraint on $l$, $1\leq l\leq n-1$, we know that there is a single
solution to $2\,l/n$ being an integer, namely $l=n/2$, i. e. there is a single
pair of irreducible three-dimensional representations that are real. In this
case their characters are real for all classes, as can be explicitly checked
with the help of the character table, shown in Ref. [73].

##  Appendix B CP symmetries and form of CP transformations

The CP symmetries correspond to automorphisms of the flavor group
$\Delta(6\,n^{2})$, see discussion in Ref. [75]. In the present analysis we
employ the ones, as used in Ref. [65]. These can be obtained as follows:
consider the automorphism

$a\;\;\rightarrow\;\;a\;\;,\;\;c\;\;\rightarrow\;\;c^{-1}\;\;,\;\;d\;\;\rightarrow\;\;d^{-1}\;\;\mbox{and}\;\;b\;\;\rightarrow\;\;b\,.$
(183)

The automorphism in Eq. (183) can be represented by $X_{0}({\bf 1})=1$ in the
trivial representation ${\bf 1}$ and by the matrix

$X_{0}({\bf 3})\,=\,X_{0}({\bf
3^{\prime}})\,=\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)$ (184)

in both three-dimensional representations ${\bf 3}$ and ${\bf 3^{\prime}}$. In
Case 1, the CP transformation $X(s)({\bf 3})$ has the explicit form $a({\bf
3})b({\bf 3})c({\bf 3})d({\bf 3})^{2s}\>X_{0}$. The form of the CP
transformation $X(s)({\bf 3^{\prime}})$ in the representation ${\bf
3^{\prime}}$ depends on whether $s$ is even or odd, i. e.

$\displaystyle X(s~{}{\rm even})({\bf 3^{\prime}})\,=\,$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right),$ (185d) $\displaystyle X(s~{}{\rm odd})({\bf
3^{\prime}})\,=\,$
$\displaystyle\frac{1}{3}\left(\begin{array}[]{ccc}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{array}\right)\,.$ (185h)

In Case 2, the form of the CP transformation $X({\bf 3})(s,t)$ in the
representation ${\bf 3}$ can be chosen as $c({\bf 3})^{s}d({\bf
3})^{t}\>X_{0}$, and is more conveniently written in terms of the variables
$u=2s-t$ and $v=3t$ with $\phi_{u}=\pi\,u/n$ and $\phi_{v}=\pi\,v/n$. The form
of the CP transformation $X(s,t)({\bf 3^{\prime}})$ depends on whether $s$ and
$t$ are even or odd. The explicit form of $X(s,t)({\bf 3^{\prime}})$, however,
does neither contain $s$ nor $t$ as parameters.

$\displaystyle X(s\,\mbox{even},t\,\mbox{even})({\bf 3^{\prime}})\,=\,$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)\;,$ (186d) $\displaystyle
X(s\,\mbox{even},t\,\mbox{odd})({\bf 3^{\prime}})\,=\,$
$\displaystyle\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2\,\omega^{2}&2\,\omega\\\
2\,\omega^{2}&2\,\omega&-1\\\ 2\,\omega&-1&2\,\omega^{2}\end{array}\right)\;,$
(186h) $\displaystyle X(s\,\mbox{odd},t\,\mbox{even})({\bf 3^{\prime}})\,=\,$
$\displaystyle\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2&2\\\ 2&2&-1\\\
2&-1&2\end{array}\right)\;,$ (186l) $\displaystyle
X(s\,\mbox{odd},t\,\mbox{odd})({\bf 3^{\prime}})({\bf 3^{\prime}})\,=\,$
$\displaystyle\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2\,\omega&2\,\omega^{2}\\\
2\,\omega&2\,\omega^{2}&-1\\\ 2\,\omega^{2}&-1&2\,\omega\end{array}\right)\;.$
(186p)

For Case 3a and Case 3b.1, the form of the CP transformation $X(s,m)({\bf 3})$
is given as [75]

$\small X(s)({\bf
3})\,=\,\frac{1}{3}\,e^{-i\,\delta_{s}}\,\left(\begin{array}[]{ccc}3\,\cos
3\,\delta_{s}+i\,\sin 3\,\delta_{s}&-2\,i\,\omega\,\sin
3\,\delta_{s}&-2\,i\,\omega^{2}\,\sin 3\,\delta_{s}\\\ -2\,i\,\omega\,\sin
3\,\delta_{s}&\omega^{2}\,\left(3\,\cos 3\,\delta_{s}+i\,\sin
3\,\delta_{s}\right)&-2\,i\,\sin 3\,\delta_{s}\\\ -2\,i\,\omega^{2}\,\sin
3\,\delta_{s}&-2\,i\,\sin 3\,\delta_{s}&\omega\,\left(3\,\cos
3\,\delta_{s}+i\,\sin 3\,\delta_{s}\right)\end{array}\right)$ (187)

with $\delta_{s}=\pi s/n$. The form of the CP transformation $X(s)({\bf
3^{\prime}})$ only depends on whether $s$ is even or odd. In particular,

$\displaystyle X(s\,\mbox{even})({\bf 3^{\prime}})\,=\,$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&\omega^{2}&0\\\
0&0&\omega\end{array}\right)\;,$ (188d) $\displaystyle X(s\,\mbox{odd})({\bf
3^{\prime}})\,=\,$
$\displaystyle\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2\,\omega&2\,\omega^{2}\\\
2\,\omega&-\omega^{2}&2\\\ 2\,\omega^{2}&2&-\omega\end{array}\right)\;.$
(188h)

##  Appendix C Form of the Representation Matrices for Residual Symmetries

In the following, we list the form of the representation matrices in the
representations ${\bf 3}$ and ${\bf 3^{\prime}}$ for the different residual
symmetries, used in sections II and III.

In all these cases, the residual flavor symmetry in the charged-lepton sector
is generated by $a$ which corresponds to the representation matrix $a({\bf
3})$ given in Eq. (174). The residual flavor symmetry in the neutrino sector
is generated by $Z$. In Case 1 and Case 2, $Z$ is chosen as $c^{n/2}$ which is
in the representation ${\bf 3}$ of the form

$Z({\bf 3})\,=\,\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{array}\right)\,$ (189)

independent of the index $n$, while the form of $Z=c^{n/2}$ in ${\bf
3^{\prime}}$ reads either

$Z({\bf 3^{\prime}})\,=\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right)\;\;\;\mbox{for}\;\;\;n/2\;\;\mbox{even}\,,$ (190)

or

$Z({\bf 3^{\prime}})\,=\,\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2&2\\\
2&-1&2\\\ 2&2&-1\end{array}\right)=Z({\bf
3})\;\;\;\mbox{for}\;\;\;n/2\;\;\mbox{odd.}$ (191)

In Case 3a and Case 3b.1, $Z$ is chosen as $b\,c^{m}d^{m}$ with
$m=0,....,n-1$. In the representation ${\bf 3}$ it is of the form [75]

$Z(m)({\bf
3})\,=\,\frac{1}{3}\,\left(\begin{array}[]{ccc}1+2\,\cos\gamma_{m}&\omega^{2}\,\left(1-\cos\gamma_{m}+\sqrt{3}\,\sin\gamma_{m}\right)&\omega\,\left(1-\cos\gamma_{m}-\sqrt{3}\,\sin\gamma_{m}\right)\\\
\omega\,\left(1-\cos\gamma_{m}+\sqrt{3}\,\sin\gamma_{m}\right)&1-\cos\gamma_{m}-\sqrt{3}\,\sin\gamma_{m}&\omega^{2}\,\left(1+2\,\cos\gamma_{m}\right)\\\
\omega^{2}\,\left(1-\cos\gamma_{m}-\sqrt{3}\,\sin\gamma_{m}\right)&\omega\,\left(1+2\,\cos\gamma_{m}\right)&1-\cos\gamma_{m}+\sqrt{3}\,\sin\gamma_{m}\end{array}\right)$
(192)

with $\gamma_{m}=2\,\pi m/n$. For the special values, $m=0$, $m=n$ and
$m=n/2$, the form of $Z(m)({\bf 3})$ simplifies and we find

$\displaystyle Z(m=0)({\bf 3})\,=\,$ $\displaystyle Z(m=n)({\bf
3})\,=\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&\omega^{2}\\\
0&\omega&0\end{array}\right)\;,$ (193d) $\displaystyle Z(m=n/2)({\bf 3})\,=\,$
$\displaystyle\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2\,\omega^{2}&2\,\omega\\\
2\,\omega&2&-\omega^{2}\\\ 2\,\omega^{2}&-\omega&2\end{array}\right)\;.$
(193h)

Similarly, we can analyze the form of the representation matrix $Z(m)({\bf
3^{\prime}})$. The decisive criterion for this form is whether $m$ is even or
odd; otherwise there is no further dependence on the parameter $m$ for
$Z(m)({\bf 3^{\prime}})$.

$\displaystyle Z(m\,\mbox{even})({\bf 3^{\prime}})\,=\,$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&\omega^{2}\\\
0&\omega&0\end{array}\right)\;,$ (197) $\displaystyle Z(m\,\mbox{odd})({\bf
3^{\prime}})\,=\,$
$\displaystyle\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2\,\omega^{2}&2\,\omega\\\
2\,\omega&2&-\omega^{2}\\\ 2\,\omega^{2}&-\omega&2\end{array}\right)\;.$ (201)

We note that $Z(m\,\mbox{even})({\bf 3^{\prime}})$ coincides with $Z(m=0)({\bf
3})=Z(m=n)({\bf 3})$, and $Z(m\,\mbox{odd})({\bf 3^{\prime}})$ coincides with
$Z(m=n/2)({\bf 3})$.

##  Appendix D Explicit Form of the Yukawa Couplings

###  D.1 Case 1

The Yukawa parameters $y_{f}$ in this case (irrespective of $s$) are

$\displaystyle y_{1}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(m_{1}\,-\,m_{3}\,+\,\sqrt{m_{1}^{2}\,+\,m_{3}^{2}\,+\,2\,m_{1}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(202a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{2}}{v^{2}}\,,$ (202b) $\displaystyle
y_{3}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(-\,m_{1}\,+\,m_{3}\,+\,\sqrt{m_{1}^{2}\,+\,m_{3}^{2}\,+\,2\,m_{1}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,.$
(202c)

###  D.2 Case 2

####  $t$ even:

$\displaystyle y_{1}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{1}}{v^{2}}\,,$
(203a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{2}}{v^{2}}\,,$ (203b) $\displaystyle
y_{3}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{3}}{v^{2}}\,.$ (203c)

####  $t$ odd:

$\displaystyle y_{1}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(-\,m_{1}\,+\,m_{3}\,+\,\sqrt{m_{1}^{2}\,+\,m_{3}^{2}\,-\,2\,m_{1}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\csc\left(2\,\theta_{R}\right)\,,$
(204a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{2}}{v^{2}}\,,$ (204b) $\displaystyle
y_{3}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(m_{1}\,-\,m_{3}\,+\,\sqrt{m_{1}^{2}\,+\,m_{3}^{2}\,-\,2\,m_{1}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\csc\left(2\,\theta_{R}\right)\,.$
(204c)

###  D.3 Case 3a

####  $m$ and $s$, both even or odd:

$\displaystyle y_{1}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{1}}{v^{2}}\,,$
(205a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{2}}{v^{2}}\,,$ (205b) $\displaystyle
y_{3}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{3}}{v^{2}}\,.$ (205c)

####  $m$ even, $s$ odd:

$\displaystyle y_{1}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(m_{1}\,-\,m_{2}\,+\,\sqrt{m_{1}^{2}\,+\,m_{2}^{2}\,+\,2\,m_{1}\,m_{2}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(206a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(-\,m_{1}\,+\,m_{2}\,+\,\sqrt{m_{1}^{2}\,+\,m_{2}^{2}\,+\,2\,m_{1}\,m_{2}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(206b) $\displaystyle y_{3}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{3}}{v^{2}}\,.$ (206c)

####  $m$ odd, $s$ even:

$\displaystyle y_{1}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{1}}{v^{2}}\,,$
(207a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(m_{2}\,-\,m_{3}\,+\,\sqrt{m_{2}^{2}\,+\,m_{3}^{2}\,+\,2\,m_{2}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(207b) $\displaystyle y_{3}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(-m_{2}\,+\,m_{3}\,+\,\sqrt{m_{2}^{2}\,+\,m_{3}^{2}\,+\,2\,m_{2}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,.$
(207c)

###  D.4 Case 3b.1

####  $m$ and $s$, both even or odd:

$\displaystyle y_{1}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{2}}{v^{2}}\,,$
(208a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{3}}{v^{2}}\,,$ (208b) $\displaystyle
y_{3}^{2}\,=\,$ $\displaystyle\frac{M_{N}\,m_{1}}{v^{2}}\,.$ (208c)

####  $m$ even, $s$ odd:

$\displaystyle y_{1}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(-\,m_{1}\,+\,m_{2}\,+\,\sqrt{m_{1}^{2}\,+\,m_{2}^{2}\,+\,2\,m_{1}\,m_{2}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(209a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{3}}{v^{2}}\,,$ (209b) $\displaystyle
y_{3}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(m_{1}\,-\,m_{2}\,+\,\sqrt{m_{1}^{2}\,+\,m_{2}^{2}\,+\,2\,m_{1}\,m_{2}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(209c)

####  $m$ odd, $s$ even:

$\displaystyle y_{1}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(m_{2}\,-\,m_{3}\,+\,\sqrt{m_{2}^{2}\,+\,m_{3}^{2}\,+\,2\,m_{2}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(210a) $\displaystyle y_{2}^{2}\,=\,$
$\displaystyle\frac{M_{N}}{2\,v^{2}}\left(-m_{2}\,+\,m_{3}\,+\,\sqrt{m_{2}^{2}\,+\,m_{3}^{2}\,+\,2\,m_{2}\,m_{3}\,\cos\left(4\,\theta_{R}\right)}\right)\sec\left(2\,\theta_{R}\right)\,,$
(210b) $\displaystyle y_{3}^{2}\,=\,$
$\displaystyle\frac{M_{N}\,m_{1}}{v^{2}}\,.$ (210c)

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