#  $TeV$-Scale Resonant Leptogenesis $A_{4}$ with a scaling texture

H. B. Benaoum<EMAIL_ADDRESS>Department of Applied Physics and
Astronomy,
University of Sharjah, United Arab Emirates

###### Abstract

We consider a TeV scale resonant lepogenesis within type-I seesaw mechanism.
We show that a concrete model based on $A_{4}$ flavor symmetry with a scaling
ansatz in the neutrino Dirac mass matrix and a retro-circulant heavy Majorana
mass matrix can be realized. The full Yukawa coupling matrix $Y_{\nu}$ can be
fully reconstructed from the low energy neutrino oscillations data and the
quasi-degenerate heavy Majorara neutrino masses. We have carried out a
detailed numerical analysis to constrain the Dirac neutrino mass matrix
elements. In particular, it was found that its diagonal elements lie near or
below the MeV region. Furthermore, we have investigated the allowed regions in
the parameter space of the model consistent with both low energy neutrino
oscillations data and resonant leptogenesis leading to the observed baryon
asymmetry of the universe. Finally, the model has an imperative prediction on
the allowed space for the effective Majorana neutrino mass $|m_{ee}|$ in order
to account for the observed baryon asymmetry.

Neutrino Physics; Flavor Symmetry; Leptogenesis; Matter-antimatter

###### pacs:

14.60.Pq;14.60.St; 12.60.-i; 12.60.Fr ; 11.30.Hv; 98.80.Cq

.

##  I Introduction

The observation of non-zero neutrino masses fukuda -abe2  and baryon asymmetry
of the Universe are hints for new physics beyond the Standard Model (SM) ade .
The present global analysis from several experimental data on neutrino
oscillation parameters $\theta_{ij},\Delta m_{ij}^{2}$ and $\delta$ can be
found in capozzi ; esteban . A very elegant mechanism of generating neutrino
masses as well as explaining the baryon asymmetry is leptogenesis associated
with type-I seesaw minkowski -valle . On the other hand, the non-abelian favor
discrete symmetries provide a possible underlying symmetry for the neutrino
masses king . There are a series of such models based on the symmetry group
$S_{3},A_{4},S_{4},A_{5}$ and other groups with larger orders altarelli
-benaoum3 . An attractive way of realizing the observed pattern in neutrino
mixing is the non-abelian discrete flavor symmetry group $A_{4}$ babu -vien .
The matter antimatter asymmetry is one of the long-standing problem in
particle physics and cosmology. Leptogenesis has proven to be one of the most
elegant way to generate the observed baryon asymmetry of the Universe.
Leptogenesis models caused by the decay of heavy Majorana right-handed
neutrinos, are believed to be the underlying sources of the baryon asymmetry
kuzmin -benaoum1 .

Scaling ansatz in the neutrino sector has been considered by many authors and
several phenomenological models based on this idea have appeared in the
literature scaling -benaoum2 . In this paper, we have investigated the
neutrino mass matrix in the conventional type-I seesaw framework based on an
$A_{4}$ flavor discrete symmetry. In benaoum2 , we have shown how to
incorporate a scaling ansatz in the Dirac neutrino mass matrix with trivial
form of the heavy right-handed Majorara neutrino mass matrix in an $A_{4}$
type-I seesaw scenario. In the present work,we consider again a model
independent approach where the ratios of certain elements of the neutrino
Dirac mass matrix are equal,

$\displaystyle\frac{m_{12}}{m_{33}}$ $\displaystyle=$
$\displaystyle\frac{m_{31}}{m_{22}}=\frac{m_{23}}{m_{11}}=\kappa_{1}$
$\displaystyle\frac{m_{21}}{m_{33}}$ $\displaystyle=$
$\displaystyle\frac{m_{13}}{m_{22}}=\frac{m_{32}}{m_{11}}=\kappa_{2}$ (1)

where $m_{11},m_{22},m_{33},\kappa_{1}$ and $\kappa_{2}$ are complex
coefficients.
The Dirac neutrino mass matrix can be written as:

$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}m_{11}&\kappa_{1}m_{33}&\kappa_{2}m_{22}\\\
\kappa_{2}m_{33}&m_{22}&\kappa_{1}m_{11}\\\
\kappa_{1}m_{22}&\kappa_{2}m_{11}&m_{33}\end{array}\right)~{}~{}~{}~{}~{}~{}.$
(5)

Now, instead of taking a diagonal or trivial form of the heavy right-handed
neutrino mass matrix, we choose it to be a retro-circulant mass matrix,

$\displaystyle M_{R}$ $\displaystyle=$ $\displaystyle
f~{}\left(\begin{array}[]{ccc}1&q&r\\\ q&r&1\\\ r&1&q\end{array}\right)$ (9)

where the parameters $f$ and $r$ are considered as real numbers.
The light neutrino mass matrix is given by:

$\displaystyle M_{\nu}$ $\displaystyle=$ $\displaystyle-
M_{D}M_{R}^{-1}M_{D}^{T}~{}~{}~{}~{}~{}~{}.$ (10)

In this model independent approach, the $3\times 3$ neutrino symmetric mass
matrix can be written as:

$\displaystyle M_{\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{(1-3qr+q^{3}+r^{3})}\left[(1-qr)M_{\nu~{}1}-(r-q^{2})M_{\nu~{}2}+(q-r^{2})M_{\nu~{}3}\right]$
(11)

where,

$\displaystyle M_{\nu~{}1}$ $\displaystyle=$
$\displaystyle\frac{1}{f}~{}\left(\begin{array}[]{ccc}a^{2}+2\kappa_{1}\kappa_{2}~{}b~{}c&\kappa_{2}~{}c^{2}+(\kappa_{1}^{2}+\kappa_{2})~{}a~{}b&\kappa_{1}~{}b^{2}+(\kappa_{1}+\kappa_{2}^{2})~{}a~{}c\\\
\kappa_{2}~{}c^{2}+(\kappa_{1}^{2}+\kappa_{2})~{}a~{}b&\kappa_{2}^{2}~{}b^{2}+2\kappa_{1}~{}a~{}c&\kappa_{1}\kappa_{2}~{}a^{2}+(1+\kappa_{1}\kappa_{2})~{}b~{}c\\\
\kappa_{1}~{}b^{2}+(\kappa_{1}+\kappa_{2}^{2})~{}a~{}c&\kappa_{1}\kappa_{2}~{}a^{2}+(1+\kappa_{1}\kappa_{2})~{}b~{}c&\kappa_{1}^{2}~{}c^{2}+2\kappa_{2}a~{}b\end{array}\right)~{}~{}~{}~{}~{}~{}~{},$
(15) $\displaystyle M_{\nu~{}2}$ $\displaystyle=$
$\displaystyle\frac{1}{f}~{}\left(\begin{array}[]{ccc}\kappa_{2}^{2}~{}c^{2}+2\kappa_{1}~{}a~{}b&\kappa_{1}\kappa_{2}~{}b^{2}+(1+\kappa_{1}\kappa_{2})~{}a~{}c&\kappa_{2}^{2}~{}a^{2}+(\kappa_{1}^{2}+\kappa_{2})~{}b~{}c\\\
\kappa_{1}\kappa_{2}~{}b^{2}+(1+\kappa_{1}\kappa_{2})~{}a~{}c&\kappa_{1}^{2}~{}a^{2}+2\kappa_{2}~{}b~{}c&\kappa_{1}~{}c^{2}+(\kappa_{1}+\kappa_{2}^{2})~{}a~{}b\\\
\kappa_{2}^{2}~{}a^{2}+(\kappa_{1}^{2}+\kappa_{2})~{}b~{}c&\kappa_{1}~{}c^{2}+(\kappa_{1}+\kappa_{2}^{2})~{}a~{}b&b^{2}+2\kappa_{1}\kappa_{2}~{}a~{}c\end{array}\right)~{}~{}~{}~{}~{}~{}~{},$
(19) $\displaystyle M_{\nu~{}3}$ $\displaystyle=$
$\displaystyle\frac{1}{f}~{}\left(\begin{array}[]{ccc}\kappa_{1}^{2}~{}b^{2}+2\kappa_{2}~{}a~{}c&\kappa_{1}~{}a^{2}+(\kappa_{1}+\kappa_{2}^{2})~{}b~{}c&\kappa_{1}\kappa_{2}~{}c^{2}+(1+\kappa_{1}\kappa_{2})~{}a~{}b\\\
\kappa_{1}~{}a^{2}+(\kappa_{1}+\kappa_{2}^{2})~{}b~{}c&c^{2}+2\kappa_{1}\kappa_{2}~{}a~{}b&\kappa_{2}~{}b^{2}+(\kappa_{1}^{2}+\kappa_{2})~{}a~{}c\\\
\kappa_{1}\kappa_{2}~{}c^{2}+(1+\kappa_{1}\kappa_{2})~{}ab&\kappa_{2}~{}b^{2}+(\kappa_{1}^{2}+\kappa_{2})~{}a~{}c&\kappa_{2}^{2}~{}a^{2}+2\kappa_{1}~{}b~{}c\end{array}\right)~{}~{}.$
(23)

with $m_{11}=a,m_{22}=c$ and $m_{33}=b$.

This paper is organized as follows: In section 2, we introduce the model with
the scaling ansatz in neutrino Dirac mass matrix and the retro-circulant heavy
Majorara Neutrino mass matrix. We show how to fully reconstruct all the
elements of this matrix of the Dirac neutrino matrix $M_{\nu}$ from the low
energy neutrino oscillation data. In section 3, we explore the scenario of
resonant leptogenesis for our $A_{4}$ model. We also present the numerical
analysis results and investigate the viable parameter space to explain current
baryon asymmetry of the universe in section 4. In Section 5, we conclude.

##  II The Model

In order to realize the Dirac neutrino mass matrix with scaling ansatz and the
heavy right-handed neutrino mass matrix with retro-circulant form, we use
type-I seesaw framework based on an $A_{4}$ flavor discrete symmetry minkowski
-valle . In addition to the three SM lepton $SU(2)_{L}$ doublets
$L_{\alpha=e,\mu,\tau}$, three right-handed charged lepton singlet
$e_{R},\mu_{R}$ and $\tau_{R}$, and $SU(2)_{L}$ doublet Higgs scalar $H$, we
extend the SM by three copies of right-handed singlet neutrinos $N_{i=1,2,3}$
and five $SU(2)_{L}$ scalar singlet fields
$\phi_{E},\phi_{\nu},\xi,\xi^{\prime}$ and $\xi^{\prime\prime}$. The field
content of the model and their transformation properties of various fields
under $A_{4}\times\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ are given in Table 1.

| $\bar{L}$ | $e_{R}$ | $\mu_{R}$ | $\tau_{R}$ | $N$ | $H$ | $\phi_{E}$ | $\phi_{\nu}$ | $\xi$ | $\xi^{\prime}$ | $\xi^{\prime\prime}$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | $2$ | $1$ | $1$ | $1$ | $1$ | $2$ | $1$ | $1$ | $1$ | $1$ | 1
$A_{4}$ | $3$ | $1$ | $1^{\prime}$ | $1^{\prime\prime}$ | $3$ | $1$ | $3$ | $3$ | $1$ | $1^{\prime}$ | $1^{\prime\prime}$
$Z_{3}$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $1$ | $1$ | $\omega$ | $\omega$ | $\omega$ | $\omega$
$Z_{2}$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $1$ | $1$
Table 1:  Fields assignments under $SU(2)_{L}$ and
$A_{4}\times\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ symmetry.

Based on the $A_{4}\times\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ symmetry, we
construct the effective Lagrangian for the charged lepton Yukawa terms which
is given by:

$\displaystyle{\cal L}_{cl}$ $\displaystyle=$
$\displaystyle\frac{y_{e}}{\Lambda}\left(\bar{L}\phi_{E}\right)_{1}He_{R}+\frac{y_{\mu}}{\Lambda}\left(\bar{L}\phi_{E}\right)_{1^{\prime}}H\mu_{R}+\frac{y_{\tau}}{\Lambda}\left(\bar{L}\phi_{E}\right)_{1^{\prime\prime}}H\tau_{R}+H.c.$
(24)

where $\Lambda$ represents the cutoff scale of the theory and $y_{e},y_{\mu}$
and $y_{\tau}$ are respective coupling constants.

The Allowed terms of the neutrino invariant under
$A_{4}\times\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ symmetry are given by:

$\displaystyle{\cal L}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{y_{s}}{\Lambda}\left(\phi_{\nu}\bar{L}\right)_{3_{s}}\tilde{H}N+\frac{y_{a}}{\Lambda}\left(\phi_{\nu}\bar{L}\right)_{3_{a}}\tilde{H}N+y_{N}\left(NN\right)_{1}\xi+y^{\prime}_{N}\left(NN\right)_{1^{\prime}}\xi^{\prime\prime}+y^{\prime\prime}_{N}\left(NN\right)_{1^{\prime\prime}}\xi^{\prime}+H.c.$
(25)

where $\tilde{H}=i\tau_{2}H^{*}$ is the conjugate of $H$ and
$y_{s},y_{a},y_{N},y^{\prime}_{N}$ and $y^{\prime\prime}_{N}$ are the coupling
constants.
Once the scalar $\phi_{E}$ gets vacuum expectation value (vev) as
$<\phi_{E}>=\left(v_{E},0,0\right)$and the Higgs vev $<H>=v_{H}$ is inserted,
the charged lepton mass matrix turns out to be diagonal and is given by:

$\displaystyle M_{l}$ $\displaystyle=$
$\displaystyle\frac{v_{H}v_{E}}{\Lambda}\left(\begin{array}[]{ccc}y_{e}&0&0\\\
0&y_{\mu}&0\\\ 0&0&y_{\tau}\end{array}\right)~{}~{}~{}~{}.$ (29)

###  II.1 Neutrino Masses and Mixings

After the singlet scalars acquire vevs $<\xi>=u,<\xi^{\prime}>=u^{\prime}$ and
$<\xi^{\prime\prime}>=u^{\prime\prime}$ and choosing a suitable vev alignment
of the scalar triplet $\phi_{\nu}$ as:

$\displaystyle<\phi_{\nu}>$ $\displaystyle=$
$\displaystyle\left(v_{\nu},v_{\nu},v_{\nu}\right)~{}~{}~{}.$ (30)

it yields a retro-circulant right-handed heavy neutrino mass matrix,

$\displaystyle M_{R}$ $\displaystyle=$ $\displaystyle
f~{}\left(\begin{array}[]{ccc}1&q&r\\\ q&r&1\\\ r&1&q\end{array}\right)$ (34)

with $f=2y_{N}u,q=\frac{y^{\prime}_{N}u^{\prime\prime}}{y_{N}u}$ and
$r=\frac{y^{\prime\prime}_{N}u^{\prime}}{y_{N}u}$.

The $3\times 3$ scaling invariant Dirac neutrino mass matrix is given by:

$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}a&\kappa_{1}a&\kappa_{2}a\\\
\kappa_{2}a&a&\kappa_{1}a\\\ \kappa_{1}a&\kappa_{2}a&a\end{array}\right)$ (38)

where
$a=\frac{2}{3}y_{s}\frac{v_{\nu}v_{H}}{\Lambda},\kappa_{1}=-\frac{1}{2}-\frac{3y_{a}}{4y_{s}}$
and $\kappa_{2}=-\frac{1}{2}+\frac{3y_{a}}{4y_{s}}$; with the condition
$\kappa_{1}+\kappa_{2}=-1$.

The light neutrino mass matrix can be generated using type-I seesaw as,

$\displaystyle M_{\nu}$ $\displaystyle=$ $\displaystyle-
M_{D}M_{R}^{-1}M_{D}^{T}~{}~{}~{}~{}~{}~{}.$ (39)

The above matrix generates nonzero $\theta_{13}$ as well as it breaks the tri-
bimaximal pattern which can be seen by rotating $M_{\nu}$ by $U_{TBM}$,

$\displaystyle U_{TBM}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{array}\right)$
(43)

The resulting neutrino mass matrix has three eigenvalues, where one of them is
zero and the others two are degenerate, which is in contradiction to the
neutrino oscillation experiments. It also gives a non-vanishing reactor angle
$\theta_{13}$ that deviates from tri-bimaximal mixing such that
$\sin^{2}\theta_{12}=1/3,\sin^{2}\theta_{23}=1/2$ and $\sin^{2}\theta_{13}\neq
0$.

In order to have the correct low energy phenomenology, we choose a more
general vacuum alignment for the scalar triplet,

$\displaystyle<\phi_{\nu}>=\left(v_{\nu_{1}},v_{\nu_{2}},v_{\nu_{3}}\right)$
(44)

which can be considered as a small perturbation around the exact vev
alignment. Such a choice corrects the neutrino mass spectrum and gives rise to
the correct mass squared differences as well as mixing angles. The above vev
configuration is a solution of the minimization conditions of the scalar
potential provided it softly breaks the flavor discrete symmetry. Such a
deviation is then associated to this soft breaking. The way how the above
vacuum configuration is achieved is out of the scope of the present work.

The chosen vevs allow us to have a scaling invariant Dirac neutrino mass
matrix,

$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}a&\kappa_{1}b&\kappa_{2}c\\\
\kappa_{2}b&c&\kappa_{1}a\\\ \kappa_{1}c&\kappa_{2}a&b\end{array}\right)$ (48)

where $a=\frac{2}{3}y_{s}\frac{v_{\nu
1}v_{H}}{\Lambda},c=\frac{2}{3}y_{s}\frac{v_{\nu
2}v_{H}}{\Lambda},b=\frac{2}{3}y_{s}\frac{v_{\nu
3}v_{H}}{\Lambda},\kappa_{1}=-\frac{1}{2}-\frac{3y_{a}}{4y_{s}}$ and
$\kappa_{2}=-\frac{1}{2}+\frac{3y_{a}}{4y_{s}}$; with
$\kappa_{1}+\kappa_{2}=-1$.

Type-I contribution to the light neutrino mass is:

$\displaystyle M_{\nu}$ $\displaystyle=$ $\displaystyle-
M_{D}M_{R}^{-1}M_{D}^{T}=U_{PMNS}M_{\nu}^{diag}U_{PMNS}^{T}~{}~{}~{}~{}~{}~{}.$
(49)

where $M_{\nu}^{diag}=diag\left(m_{1},m_{2},m_{3}\right)$ is the diagonal
matrix containing the three neutrino masses $m_{1,2,3}$ and
$U_{PMNS}=UP_{Maj}$ is the Pontecorvo-Maki-Nakagawa-Sakata unitary matrix
given by:

$\displaystyle U$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)$
(53)

where $c_{ij}=\cos\theta_{ij},s_{ij}=\sin\theta_{ij},\theta_{ij}$’s are flavor
mixing angles; $i,j=1,2,3$; $\delta$ is the Dirac CP-violating phase and
$P_{Maj}=diag\left(1,e^{i\alpha/2},e^{i\beta/2}\right)$ is the diagonal matrix
that contains the Majorana phases $\alpha$ and $\beta$.

###  II.2 Construction of the Dirac mass Matrix

To construct the elements of the Dirac mass matrix, we decompose the neutrino
mass matrix $M_{\nu}$ into two parts where the first part is a symmetric
matrix and the second part is a mass matrix with zero diagonal elements having
the form:

$\displaystyle M_{\nu}^{(2)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&M_{\nu_{12}}-M_{\nu_{33}}&M_{\nu_{13}}-M_{\nu_{22}}\\\
M_{\nu_{12}}-M_{\nu_{33}}&0&M_{\nu_{23}}-M_{\nu_{11}}\\\
M_{\nu_{13}}-M_{\nu_{22}}&M_{\nu_{23}}-M_{\nu_{11}}&0\end{array}\right)=\left(\begin{array}[]{ccc}0&w_{12}&w_{13}\\\
w_{12}&0&w_{23}\\\ w_{13}&w_{23}&0\end{array}\right).$ (60)

The $w_{ij}$ are determined from the neutrino oscillation data (up to the CP-
violating Majorana phases),

$\displaystyle w_{12}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\mu_{i}\left(U_{1i}U_{2i}-U_{3i}^{2}\right)$
$\displaystyle w_{13}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\mu_{i}\left(U_{1i}U_{3i}-U_{2i}^{2}\right)$
$\displaystyle w_{23}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\mu_{i}\left(U_{2i}U_{3i}-U_{1i}^{2}\right)$ (61)

where we have defined the masses including the corresponding Majorana phases
as $\mu_{1}=m_{1},\mu_{2}=m_{2}e^{i\alpha}$ and $\mu_{3}=m_{3}e^{i\beta}$.

Explicit structure of the off-diagonal elements of the mass matrix
$M_{\nu}^{(2)}$ can be expressed as follows:

$\displaystyle\left(\begin{array}[]{c}w_{13}\\\ w_{12}\\\
w_{23}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{Det\left(M_{R}\right)}\left(\begin{array}[]{ccc}\left(1-\kappa_{1}\kappa_{2}\right)~{}(r^{2}-q)&\left(\kappa_{2}^{2}-\kappa_{1}\right)(1-qr)&\left(\kappa_{1}^{2}-\kappa_{2}\right)~{}(q^{2}-r)\\\
\left(\kappa_{1}^{2}-\kappa_{2}\right)(1-qr)&\left(1-\kappa_{1}\kappa_{2}\right)~{}(q^{2}-r)&\left(\kappa_{2}^{2}-\kappa_{1}\right)~{}(r^{2}-q)\\\
\left(\kappa_{2}^{2}-\kappa_{1}\right)~{}(q^{2}-r)&\left(\kappa_{1}^{2}-\kappa_{2}\right)~{}(r^{2}-q)&\left(1-\kappa_{1}\kappa_{2}\right)(1-qr)\end{array}\right)\left(\begin{array}[]{c}\gamma_{3}\\\
\gamma_{2}\\\ \gamma_{1}\end{array}\right)$ (71)

where we have defined $\gamma_{i=1,2,3}$ as,

$\displaystyle\gamma_{1}$ $\displaystyle=$ $\displaystyle a^{2}-bc$
$\displaystyle\gamma_{2}$ $\displaystyle=$ $\displaystyle c^{2}-ab$
$\displaystyle\gamma_{3}$ $\displaystyle=$ $\displaystyle b^{2}-ac$ (72)

which is a convenient way to parametrize the matrix elements of the Dirac mass
matrix. It turns out that the parameters $\gamma_{i=1,2,3}$ can be
reconstructed from the six neutrino oscillations parameters, the lightest
neutrino mass, two Majorana phases, nonzero entry in the right-handed neutrino
mass and scaling factors as:

$\displaystyle\left(\begin{array}[]{c}\gamma_{1}\\\ \gamma_{2}\\\
\gamma_{3}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{1-\kappa_{1}\kappa_{2}}M_{R}\left(\begin{array}[]{c}w_{23}\\\
w_{13}\\\ w_{12}\end{array}\right)$ (79)

It is interesting to note that the full Dirac mass matrix, which is described
altogether by 10 real parameters, can be reconstructed analytically, in the
most general case, as:

$\displaystyle a$ $\displaystyle=$
$\displaystyle\pm\frac{\gamma_{1}^{2}-\gamma_{2}\gamma_{3}}{\sqrt{(\gamma_{1}+\gamma_{2}+\gamma_{3})(\gamma_{1}+\omega\gamma_{2}+\omega^{2}\gamma_{3})(\gamma_{1}+\omega^{2}\gamma_{2}+\omega\gamma_{3})}}$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\pm\frac{\gamma_{2}^{2}-\gamma_{1}\gamma_{3}}{\sqrt{(\gamma_{1}+\gamma_{2}+\gamma_{3})(\gamma_{1}+\omega\gamma_{2}+\omega^{2}\gamma_{3})(\gamma_{1}+\omega^{2}\gamma_{2}+\omega\gamma_{3})}}$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\pm\frac{\gamma_{3}^{2}-\gamma_{1}\gamma_{2}}{\sqrt{(\gamma_{1}+\gamma_{2}+\gamma_{3})(\gamma_{1}+\omega\gamma_{2}+\omega^{2}\gamma_{3})(\gamma_{1}+\omega^{2}\gamma_{2}+\omega\gamma_{3})}}~{}~{}~{}~{}~{}.$
(80)

The determinant of the light active neutrino mass matrix $M_{\nu}$ can now be
written as:

$\displaystyle Det\left(M_{\nu}\right)$ $\displaystyle=$
$\displaystyle\kappa_{1}^{2}\kappa_{2}^{2}\frac{(\gamma_{1}+\gamma_{2}+\gamma_{3})(\gamma_{1}+\omega\gamma_{2}+\omega^{2}\gamma_{3})(\gamma_{1}+\omega^{2}\gamma_{2}+\omega\gamma_{3})}{f^{3}(1+q+r)(1+\omega
q+\omega^{2}r)(1+\omega^{2}q+\omega r)}=\mu_{1}\mu_{2}\mu_{3}$ (81)

The solution of the above expression determines the scaling factor
$\kappa_{1}$ as a function of $m_{\small{lightest}},\Delta
m_{\odot}^{2},\Delta m_{atm}^{2}$,
$\sin^{2}\theta_{12},\sin^{2}\theta_{13},\sin^{2}\theta_{23},\delta,\alpha$,
$\beta$ and $M_{R}$. Detailed analysis will be performed latter on.

##  III Leptogenesis

In this section, we study baryogenesis via leptogenesis of the model at TeV
scale.The lepton asymmetry is generated by the CP-violating out-of-equilibrium
decays of the heavy Majorana right-handed neutrinos. The lepton asymmetry is
then converted by sphaleron processes into a baryon asymmetry. To realize a
resonant leptogenesis pilaftsis -teresi  at TeV scales in the Universe, a mass
degeneracy among the heavy right-handed neutrinos is required.

As we will see later, the heavy Majorana mass matrix $M_{R}$ has a leading
order parameter $f$ and two very small parameters $r$ and $q$. This is
sufficient to create a very tiny splitting among the heavy Majorana masses.
Although the parameters $r$ and $q$ are crucial for the resonant leptogenesis,
the contribution of $r$ and $q$ in fitting the neutrino oscillation data is
negligible.

The eigenvalues of the heavy right-handed mass matrix are given by:

$\displaystyle M_{R}^{diag}$ $\displaystyle=$ $\displaystyle
diag\left(M_{1}=f\sqrt{1-q-r-
qr+q^{2}+r^{2}},M_{2}=f(1+q+r),M_{3}=-f\sqrt{1-q-r-
qr+q^{2}+r^{2}}\right)~{}~{}~{}.$ (82)

We first perform a rotation by $U_{TBM}$ on $M_{R}$ as follows,

$\displaystyle U_{TBM}^{T}M_{R}U_{TBM}$ $\displaystyle=$ $\displaystyle
f\left(\begin{array}[]{ccc}\frac{1}{2}\left(2-q-r\right)&0&\frac{\sqrt{3}}{2}\left(r-q\right)\\\
0&1+q+r&0\\\
\frac{\sqrt{3}}{2}\left(r-q\right)&0&-\frac{1}{2}\left(2-q-r\right)\end{array}\right)$
(86)

We note that a further rotation by a unitary matrix $U_{R~{}13}$ in the $13$
plane is required to diagonalize the heavy right-handed Majorana neutrino mass
matrix. The full unitary matrix $U_{R}$ that diagonalize $M_{R}$ can be
expressed as:

$\displaystyle U_{R}$ $\displaystyle=$ $\displaystyle U_{TBM}U_{R~{}13}$ (87)

where

$\displaystyle U_{R~{}13}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{R}&0&\sin\theta_{R}\\\
0&1&0\\\ -\sin\theta_{R}&0&\cos\theta_{R}\end{array}\right)$ (91)

with

$\displaystyle\tan 2\theta_{R}=\frac{\sqrt{3}(r-q)}{2-q-r}$ (92)

When $r=0$ and $q=0$ or $q=r$, the unitary matrix $U_{R}$ is the pure tri-
bimaximal matrix and non-zero of the parameters $r$ and $q$ or $q=r$ encode
deviation from tri-bimaximal pattern.

Consequently, in the basis where the heavy right-handed Majorana mass matrix
is diagonal with real and positive eigenvalues, the Dirac Yukawa mass matrix
$Y_{\nu}$ is found to be:

$\displaystyle Y_{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{v_{H}}M_{D}U_{R}~{}diag\left(1,1,i\right)$ (99)
$\displaystyle=$
$\displaystyle\frac{1}{v_{H}}~{}\left(\begin{array}[]{ccc}a&\kappa_{1}d&\kappa_{2}c\\\
\kappa_{2}d&c&\kappa_{1}a\\\
\kappa_{1}c&\kappa_{2}a&d\end{array}\right)\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}\cos\theta_{R}&\frac{1}{\sqrt{3}}&i~{}\sqrt{\frac{2}{3}}\sin\theta_{R}\\\
-\frac{1}{\sqrt{6}}\cos\theta_{R}+\frac{1}{\sqrt{2}}\sin\theta_{R}&\frac{1}{\sqrt{3}}&-i~{}(\frac{1}{\sqrt{2}}\cos\theta_{R}+\frac{1}{\sqrt{6}}\sin\theta_{R})\\\
-\frac{1}{\sqrt{6}}\cos\theta_{R}-\frac{1}{\sqrt{2}}\sin\theta_{R}&\frac{1}{\sqrt{3}}&i~{}(\frac{1}{\sqrt{2}}\cos\theta_{R}-\frac{1}{\sqrt{6}}\sin\theta_{R})\end{array}\right)~{}~{}.$

The CP-asymmetry parameter associated with the decay of i-th heavy Majorana
neutrino into a lepton flavor $\alpha$ is defined as:

$\displaystyle\epsilon_{N_{i}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}{\frac{\Gamma\left(N_{i}\rightarrow\bar{L}_{\alpha}+H\right)-\Gamma\left(N_{i}\rightarrow
L_{\alpha}+H^{*}\right)}{\Gamma\left(N_{i}\rightarrow\bar{L}_{\alpha}+H\right)+\Gamma\left(N_{i}\rightarrow
L_{\alpha}+H^{*}\right)}}$ (100)

The general formula of the flavored CP asymmetry $\epsilon_{i}^{\alpha}$ is
given by bhupal :

$\displaystyle\epsilon_{i}^{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{j\neq i}\frac{Im\left[(Y_{\nu})^{*}_{\alpha
i}(Y_{\nu})_{\alpha
j}\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ij}\right]+|\frac{M_{i}}{M_{j}}|~{}Im\left[(Y_{\nu})^{*}_{\alpha
i}(Y_{\nu})_{\alpha
j}\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ji}\right]}{\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ii}\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{jj}}\left(f^{mix}_{ij}+f^{osc}_{ij}\right)$
(101)

where the regulators are:

$\displaystyle f^{mix}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\left(M_{i}^{2}-M_{j}^{2}\right)|M_{i}|\Gamma_{j}}{\left(M_{i}^{2}-M_{j}^{2}\right)^{2}+M_{i}^{2}\Gamma_{j}^{2}}$
$\displaystyle f^{osc}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\left(M_{i}^{2}-M_{j}^{2}\right)|M_{i}|\Gamma_{j}}{\left(M_{i}^{2}-M_{j}^{2}\right)^{2}+\left(|M_{i}|\Gamma_{i}+|M_{j}|\Gamma_{j}\right)^{2}\frac{det\left[Re\left(Y_{\nu}^{\dagger}Y_{\nu}\right)\right]}{\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ii}\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{jj}}}~{}~{}~{}~{}.$
(102)

Here $\Gamma_{i}$ is the decay width of the i-th right-handed Majorana
neutrino, defined at tree level as:

$\displaystyle\Gamma_{i}$ $\displaystyle=$
$\displaystyle\frac{|M_{i}|}{8\pi}\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ii}~{}~{}~{}~{}~{}~{}~{}~{}.$
(103)

For each $N_{i}$, one may define the flavored decay parameter $K_{i}^{\alpha}$
as:

$\displaystyle K_{i}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(N_{i}\rightarrow\bar{L}_{\alpha}+H\right)+\Gamma\left(N_{i}\rightarrow
L_{\alpha}+H^{*}\right)}{H_{N}(T=|M_{i}|)}=\frac{\tilde{m}_{i}^{\alpha}}{m_{*}}$
(104)

where $H_{N}(T=|M_{i}|)$ is the Hubble parameter at temperature $T=|M_{i}|$
given by:

$\displaystyle H_{N}(T=|M_{i}|)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{4\pi^{3}g_{*}}{45}}\frac{M_{i}^{2}}{M_{Planck}}$
(105)

with $g_{*}=106.75$ is the effective number of relativistic degrees of freedom
of the SM at high temperatures and $M_{Planck}=1.2\times 10^{19}~{}GeV$ is the
Planck mass.
Here, the so-called equilibrium neutrino mass given by:

$\displaystyle m_{*}$ $\displaystyle=$
$\displaystyle\frac{16\pi^{5/2}\sqrt{g_{*}}}{3\sqrt{5}}\frac{v_{H}^{2}}{M_{Planck}}\sim
1.08\times 10^{-3}~{}eV~{}~{},$ (106)

and the effective flavored neutrino mass $\tilde{m}_{i}^{\alpha}$ is defined
as:

$\displaystyle\tilde{m}_{i}^{\alpha}$ $\displaystyle=$ $\displaystyle
v_{H}^{2}\frac{|(Y_{\nu})_{i\alpha}|^{2}}{|M_{i}|}~{}~{}~{}~{}~{}.$ (107)

The washout factor $K_{i}$, which is an important ingredients for the
thermodynamics description of the decays of heavy particles in the early
stages of the Universe, is obtained by summing over all flavors $\alpha$,

$\displaystyle K_{i}$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{i}}{H_{N}(T=|M_{i}|)}=\sum_{\alpha}K_{i}^{\alpha}=\frac{\tilde{m}_{i}}{m_{*}}$
(108)

where $\tilde{m}_{i}=v_{H}^{2}\frac{(Y_{\nu}^{\dagger}Y_{\nu})_{ii}}{|M_{i}|}$
is the effective neutrino mass. The value of the effective neutrino mass
$\tilde{m}_{i}$ measures the departure from equilibrium and if
$\tilde{m}_{i}<<m_{*}(\tilde{m}_{i}>>m_{*})$, the asymmetry is weakly
(strongly) washed out by the inverse decays.

After solving the relevant flavored Boltzmann equations and taking into
account the appropriate efficiency washout and dilutions factors, the
resulting baryon asymmetry of the Universe, in the strong washout regime, is
estimated to be deppisch :

$\displaystyle\eta_{B}$ $\displaystyle=$
$\displaystyle\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}=-\frac{28}{51}\frac{1}{27}\frac{3}{2}\sum_{\alpha,i}\frac{\epsilon_{i}^{\alpha}}{K^{\alpha}_{eff}min\left(z_{c},z_{\alpha}\right)}$
(109)

where $K^{\alpha}_{eff}=\kappa^{\alpha}\sum_{i}K_{i}B_{i}^{\alpha}$ is the
efficiency washout factor with the branching ratios $B_{i}^{\alpha}$ of
$N_{i}$ decay to leptons of the $\alpha$-th flavor,

$\displaystyle
B_{i}^{\alpha}=\frac{|\left(Y_{\nu}\right)_{i\alpha}|^{2}}{\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ii}}~{}~{}~{}~{},$
(110)

$z_{c}=M_{1}/T_{c}$, $T_{c}\sim 149~{}GeV$ is the critical temperature below
which the sphaleron transitions freeze-out and the parameter
$z_{\alpha}=1.25\ln(25K^{\alpha}_{eff})$.

The factor $\kappa^{\alpha}$, which includes the effect of the real
intermediate state subtracted collision terms, is given by:

$\displaystyle\kappa^{\alpha}$ $\displaystyle=$ $\displaystyle
2\sum_{i,j(j\neq i)}\frac{Re\left[(Y_{\nu})^{*}_{\alpha i}(Y_{\nu})_{\alpha
j}\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ij}\right]+Im\left[\left((Y_{\nu})^{*}_{\alpha
i}(Y_{\nu})_{\alpha
j}\right)^{2}\right]}{Re\left[\left(Y_{\nu}Y_{\nu}^{\dagger}\right)_{\alpha\alpha}\left\\{\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{ii}+\left(Y_{\nu}^{\dagger}Y_{\nu}\right)_{jj}\right\\}\right]}\left(1-2i\frac{|M_{i}|-|M_{j}|}{\Gamma_{i}+\Gamma_{j}}\right)^{-1}~{}~{}~{}~{}~{}.$
(111)

##  IV Numerical Analysis

In the next section, we present a detailed analysis of our work by dividing it
into several subsections, first fit of the model’s parameters from low energy
neutrino data, compute of the baryon number asymmetry and then find the
allowed space for the model.

###  IV.1 Parameters of $A_{4}$ Model

To perform numerical analysis in a systematical way, we use the latest global
$3\sigma$ bound on neutrino mixing angles, mass-squared differences and Dirac
CP-phase. The values are shown in Table 2 esteban . Notice that there are two
possibilities of mass orderings allowed by neutrino experiments, the normal
hierarchy (NH) $m_{3}>m_{2}>m_{1}$ and the inverted hierarchy (IH)
$m_{2}>m_{1}>m_{3}$.

Parameter | best fit $\pm 1\sigma$ | $3\sigma$
---|---|---
| NH IH | NH IH
$\Delta m_{\odot}^{2}[10^{-5}eV^{2}]$ | $7.39^{+0.21}_{-0.20}~{}~{}~{}7.39^{+0.21}_{-0.20}$ | $6.79-8.01~{}~{}~{}6.79-8.01$
$|\Delta m_{atm}^{2}|[10^{-3}eV^{2}]$ | $2.523^{+0.032}_{-0.030}~{}~{}~{}2.509^{+0.032}_{-0.030}$ | $2.432-2.618~{}~{}~{}2.416-2.616$
$\sin^{2}\theta_{12}$ | $0.310^{+0.013}_{-0.012}~{}~{}~{}0.310^{+0.013}_{-0.012}$ | $0.275-0.350~{}~{}~{}0.275-0.350$
$\sin^{2}\theta_{23}$ | $0.558^{+0.020}_{-0.033}~{}~{}~{}0.563^{+0.019}_{-0.026}$ | $0.427-0.609~{}~{}~{}0.430-0.612$
$\sin^{2}\theta_{13}$ | $0.02241^{+0.00066}_{-0.00065}~{}~{}~{}0.02261^{+0.00067}_{-0.00064}$ | $0.02046-0.02440~{}~{}~{}0.02066-0.02461$
$\delta^{0}$ | $222^{+38}_{-28}~{}~{}~{}285^{+24}_{-26}$ | $141-370~{}~{}~{}205-354$
Table 2:  Global oscillation analysis with best fit for $\Delta
m^{2}_{\odot},\Delta
m^{2}_{atm},\sin^{2}\theta_{12},\sin^{2}\theta_{23},\sin^{2}\theta_{13}$ and
the $\delta$ upper and/or lower corresponds to normal and/or inverted neutrino
mass hierarchy.

Our model allows only four degrees of freedom in the Dirac neutrino mass
matrix, denoted by $\kappa_{1},a,b$ and $c$. To constrain the model, we
perform a random scan of the inputs parameters
$m_{lightest}\in[10^{-6},0.1]~{}eV,\Delta m^{2}_{\odot},\Delta
m^{2}_{atm},\sin^{2}\theta_{12},\sin^{2}\theta_{23},\sin^{2}\theta_{13},\delta,\alpha$
and $\beta\in[0,2\pi]$. We set $f=5~{}TeV$ and randomly vary the parameters
$r$ and $q\in[10^{-8},10^{-10}]$. The small variation of $r$ and $q$ helps to
produce small mass splitting between the heavy Majorana neutrinos. Such a
choice will generate an observed baryon asymmetry compatible with the
experimental neutrino oscillation. In our analysis, we have generated one
million of random sets in the $3\sigma$ limit the current oscillation data.
The correlation and constraints on the model’s parameters are presented in
figure 1 to figure 4 for both normal and inverted hierarchies. In figure 1, we
show that the strong correlation among the parameters of the model. We obtain
that the magnitude of the scaling factor $\kappa_{1}$ is inversely correlated
to the magnitude of the parameter $a$. We also note that the magnitude of
$a,b$ and $c$ are constrained in the $MeV$ range or below. It turns out that
the low energy neutrino data do not favor $\kappa_{1}=-1$ (i.e.
$\kappa_{2}=0$) which corresponds to a vanishing lightest neutrino mass as
shown by the white band in the top plots of figure 1. In figure 2, we have
shown the scaling factor as a function of the parameters $r$ and $q$.
Interestingly, we observe that the parameter $|\kappa_{1}|$ takes values over
the whole allowed range of $r$ qnd $q$ and this is consistent with the fact
that $r$ and $q$ are needed to generate the observed baryon asymmetry. On the
other hand, we found that the phase of $\kappa_{1}$ is restricted into three
disconnected bands $\left[-1,-0.65\right]\pi,\left[-0.68,-0.4\right]\pi$ and
$\left[0.55,1\right]\pi$. We also note that the three disconnected of
$\varphi_{\kappa_{1}}$ are clearly seen in the top plot figure 3. For
completeness, we plot in figure 3 the magnitudes versus the phases of $a,b$
and $c$ which show that all values $[\pi,\pi]$ are allowed.
In Figures 4, the parameters of the model are plotted as a function of the
lightest neutrino mass $m_{1}$ for NH and $m_{3}$ for IH and absolute neutrino
masses where $\sum\limits_{\alpha=e,\mu,\tau}|m_{\nu_{\alpha}}|$ is ranging in
between $0.055eV$ to $0.17eV$ for NH and $0.125eV$ to $0.17eV$ for IH.

|
---|---
|
Figure 1:  Correlation between the scaling parameter $|\kappa_{1}|$ and $|a|$, and correlation between the parameters $|b|$ versus $|c|$ for NH (left panel) and IH (right panel). |
---|---
|
|
|
Figure 2: Plot of the scale factor $\kappa_{1}$ versus the parameters $r$ and $q$ for NH (left panel) and IH (right panel). |
---|---
|
|
|
Figure 3: Plot of the parameters of the model $|\kappa_{1},|a|,|b|$ and $|c|$ versus their phases for NH (left panel) and IH (right panel). |
---|---
|
Figure 4: Plot of the parameters of the model $|\kappa_{1}|$ versus the
lightest neutrino mass, $\sum\limits_{\alpha=e,\mu,\tau}|m_{\nu_{\alpha}}|$
for NH (left panel) and IH (right panel).

###  IV.2 Baryogenesis via Resonant Leptogenesis

In our numerical analysis, we will consider scenarios with three nearly
degenerate heavy right-handed Majorana neutrinos with masses at the TeV range.
We will study the range of the model’s parameters involved to reproduce the
correct amount of the matter-antimatter asymmetry of the Universe. The
observed baryon number asymmetry today is:

$\displaystyle\eta_{B}$ $\displaystyle=$
$\displaystyle\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}=\left(6.04\pm
0.08\right)\times 10^{-10}$ (112)

where $n_{B(\bar{B})}$ is the number of baryons (anti-baryons) density,
$n_{\gamma}$ is the number density of photons.
As discussed earlier, we have expressed the neutrino Yukawa coupling $Y_{\nu}$
in terms of the model’s parameters $\kappa_{1},a,b,c,q$ and $r$. We fed the
one million data set points $(\kappa_{1},a,b,c,q,r)$, which are consistent
with the low energy neutrino data, to compute the baryogenesis via resonant
leptogenesis for each point of the data sets. In figures 5 and 6, we have
plotted the baryon asymmetry $\eta_{B}$ against the model’s parameters. Note
that the observed baryon asymmetry is achieved (horizontal solid lines in
figure 5) whithin the preferred regions of the model’s parameters which is in
the MeV or below. In figure 6, we show thw variation of the baryon asymmetry
as a function of the phases $\varphi_{\kappa_{1}},\varphi_{a},\varphi_{b}$ and
$\varphi_{c}$. The two top plots in figure 6 show that a successful
leptoegenesis occurs when the phase of the scaling factor
$\varphi_{\kappa_{1}}$ falls in the three disconnected regions. This is in
good agreement with the global analysis at $3\sigma$ level. It follows from
the other plots in figure 6 that further constraints on the phases
$\varphi_{a},\varphi_{b}$ and $\varphi_{c}$ are required.

|
---|---
|
|
|
Figure 5: Plot of the CP asymmetry $\eta_{B}$ versus the parameters of the model $|\kappa_{1}|,|a|,|c|,|d|$ and $r$ for NH (left panel) and IH (right panel). |
---|---
|
|
|
Figure 6: Plot of the CP asymmetry $\eta_{B}$ versus the phases of the model
for NH (left panel) and IH (right panel).

###  IV.3 Neutrinoless Double Beta Decay

Another implication of our model is the existence of neutrinoless double-beta
decay. The effective neutrino mass governing the $0\nu 2\beta$ decay is given
by:

$\displaystyle|m_{ee}|$ $\displaystyle=$
$\displaystyle|\sum_{i=1}^{3}\mu_{i}U_{ei}^{2}|$ (113)

and corresponds to the (11) entry of the neutrino mass matrix $M_{\nu}$:

$\displaystyle|m_{ee}|$ $\displaystyle=$ $\displaystyle|M_{\nu~{}11}|$ (114)
$\displaystyle=$
$\displaystyle\frac{|(1-qr)\left(a^{2}+2\kappa_{1}\kappa_{2}~{}c~{}d\right)-(r-q^{2})\left(\kappa_{2}^{2}c^{2}+2\kappa_{1}~{}a~{}b\right)++(q-r^{2})\left(\kappa_{1}^{2}b^{2}+2\kappa_{2}~{}a~{}c\right)|}{f(1-3qr+q^{3}+r^{3})}$

The results for the baryon asymmetry $\eta_{B}$ as a function of $|m_{ee}|$
are depicted in figure 7.

|
---|---
Figure 7: Plot of the CP asymmetry $\eta_{B}$ versus $m_{ee}$ for NH (left
panel) and IH (right panel).

The horizontal solid lines correspond to the allowed region of the observed
baryon asymmetry. The important feature of the model is the existence of a
lower bound $|m_{ee}|>0.025~{}eV$ for NH and $|m_{ee}|>0.10~{}eV$ for IH which
is within the sensitivity reach of future double decay experiments barabash
-licciardi .

##  V Conclusions

We have proposed an $A_{4}$ flavor symmetry model to explain the baryon
asymmetry of the universe. It is based on a type-I seesaw scenario with a
retro-circulant heavy Majorana neutrino mass matrix and a scaling ansatz for
the Dirac neutrino mass matrix. The heavy right-handed Majorana neutrinos are
quasi-degenerate at TeV mass scale and the parameter space of the Dirac mass
matrix elements lies near and below the MeV region. We have managed to
reconstruct analytically all the elements of the Yukawa coupling matrix
$Y_{\nu}$ from the low energy neutrino oscillation data. We have aslo shown
that scaling ansatz on the neutrino Dirac mass matrix with the retro-circulant
heavy Majorana mass matrix are consistent with the low energy neutrino data
and the observed baryon asymmetry. Finally, the model has an imperative
implication on the allowed space for the effective Majorana neutrino mass
$|m_{ee}|$ in order to account for the observed baryon asymmetry.

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#  Non Rindler horizons and radiating black holes

Vaibhav Wasnik<EMAIL_ADDRESS>Indian Institute of Technology Goa

###### Abstract

In this work we construct metrics corresponding to radiating black holes whose
near horizon regions cannot be approximated by Rindler spacetime. We first
construct infinite parameter coordinate transformations from Minkowski
coordinates, such that an observer using these coordinates to describe
spacetime events measures the Minkowski vacuum to be Planckian. Utilizing
these results, we construct family of black holes that radiate at spatial
infinity. As an illustration we study a subset of the black hole solutions
that satisfy the null energy condition.

##  I Introduction

The celebrated Hawking result was that Schwarzschild black holes radiate
hawking . Inspired by Hawking’s result fulling , davies , unruh_original
suggested that uniformly accelerating observers being restricted to the
Rindler wedge of Minkowski spacetime, would observe the Minkowski vacuum to be
thermal. This is the celebrated Unruh effect. The radiation from black holes
could also be pinpointed to the Rindler like form of the near horizon metric
of the Schwarzschild black holes. A small textbook way of seeing this is as
follows. Let the metric signature be $(+,-,-,-)$. Consider the Schwarzschild
metric below

$\displaystyle
ds^{2}=(1-\frac{2GM}{r})dt^{2}-\frac{dr^{2}}{1-\frac{2GM}{r}}-r^{2}d\Omega.$
(1)

Define the coordinate $\hat{r}=r-2GM$. The near horizon Schwarzschild metric
becomes

$\displaystyle
ds^{2}=(\frac{\hat{r}}{2GM})dt^{2}-\frac{2GMd\hat{r}^{2}}{\hat{r}}-r^{2}d\Omega.$
(2)

If we define $\rho=\sqrt{\hat{r}}$, the above becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(\frac{\rho^{2}}{2GM})dt^{2}-8GMd\rho^{2}-r^{2}d\Omega=8GM((\rho^{2}d(\frac{t}{4GM})^{2}-d\rho^{2})-r^{2}d\Omega$

The near horizon spacetime metric is hence Rindler like. Now define $\tau=it$.
Then,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-8GM((\rho^{2}d(\frac{\tau}{4GM})^{2}+d\rho^{2})-r^{2}d\Omega.$

In order to prevent a conifold singularity at $\rho=0$, imaginary time $\tau$
should be identified with $\tau+8\pi GM$. Since, $t$ is the time measured by
clocks of an observer at spatial infinity, we are led to the implication that
the observer at spatial infinity would measure a temperature $T=\frac{1}{8\pi
GM}$.

However, consider the metric below that is constructed in this paper

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle[1+\sum_{n=1,N}u_{n}e^{n(t-r)/R}|(r/2GM-1)|^{-n}]$
$\displaystyle[1+\sum_{n^{\prime}=1,N^{\prime}}v_{n}^{\prime}e^{-n^{\prime}(t+r)/R}|(r/2GM-1)|^{-n^{\prime}}][(1-\frac{2GM}{r})dt^{2}$
$\displaystyle-$
$\displaystyle(1-\frac{2GM}{r})^{-1}dr^{2}]-r^{2}d\Omega^{2},$

with $u_{n},v_{n}$’s are real. Despite, the metric diverging at $r=2GM$, the
spacetime in the neighbourhood of $r=2GM$ is shown in the paper to be non-
singular. In order to examine the causal structure of spacetime implied by the
metric, consider radial null curves for which the $\theta,\phi$ are constant
and $ds^{2}=0$, we then see that $\frac{dt}{dr}=\pm(1-\frac{2GM}{r})$,
implying light cones close at $r=2GM$, implying a event horizon, a feature
also seen in Schwarzschild black holes. It is only when all the
$u_{n},v_{n}$’s are zero that metric has a Schwarzschild form and describes
black holes radiating at spatial infinity with temperature $T=\frac{1}{8\pi
GM}$. However when $u_{n},v_{n}$’s are not equal to zero, we couldn’t use the
conifold trick to decipher whether the corresponding black holes will radiate.
In this work we will explicitly show that the black holes whose metrics could
be put in the form above also radiate for $u_{n},v_{n}$’s that are non zero.

To do this, we first introduce a infinite parameter family of coordinate
transformations from Minkowski spacetime, such that an observer recording
their spacetime events using these coordinates 111In this work ’observer
recording their spacetime events using coordinates $(x,t)$’ will imply that
the observer records a spacetime event with coordinates $(x,t)$ (such that
observers own position in space is at a constant value of $x$ and $t$ is
measured by a clock carried by the observer), which the Minkowski observer
records with coordinates $(X,T)$ (such that observers own position in space is
at a constant value of $X$ and $T$ is measured by a clock carried by the
observer ).  will observe the Minkowskian vacuum to contain particles that
follow a Planckian spectrum. Specifically, we show that observers recording
spacetime events in lightcone coordinates in the four quadrants of Minkowskian
spacetime as shown in Fig.1, related to the Minkowskian lightcone $U,V$
through

$\displaystyle g(bV)$ $\displaystyle=$ $\displaystyle e^{bv},$ $\displaystyle
h(bU)$ $\displaystyle=$ $\displaystyle e^{bu},$

for $U,V>0$ and

$\displaystyle g(bV)$ $\displaystyle=$ $\displaystyle de^{-b\bar{v}},$
$\displaystyle h(bU)$ $\displaystyle=$ $\displaystyle de^{-b\bar{u}},$

for $U,V<0$ and $v,\bar{v},u,\bar{u}\in[-\infty,\infty]$, whichever quadrant
they appear in, would observe the Minkowski vacuum to be Planckian, if $g,h$
are odd functions. This is shown in the section II. We note that if $g(bV)=bV$
and $h(bU)=bU$, then Eq.LABEL:eq6 $\&$ Eq.LABEL:eq7 are the famous Rindler
coordinate transformations from Minkowski spacetime. We use these results of
section II to construct the above metrics Eq.LABEL:metric_intro. The way this
is done is the following. First spacetime is charted with coordinates
$(r,t,\theta,\phi)$ with the assumption that the observer at spatial infinity
( $r\rightarrow\infty$) lives in Minkowski spacetime and measures distances
along direction of increasing $r$ as change in the coordinate $r$ and records
the time on her/his clocks using $t$. Next a spherically symmetric time
varying mass distribution is assumed to curve the spacetime with a event
horizon at $r=2GM$. This mass distribution is also required to constrain the
relationship between null coordinates of a freely falling observer at the
event horizon to the null coordinates of the observer at infinity through
relations mentioned above. A class of metrics so constructed is given by
Eq.LABEL:metric_intro. This is done in section III. In section IV, we consider
energy conditions satisfied by energy momentum tensor implied by the mass
distribution corresponding to these metrics. We consider a subset of solutions
and show the null energy condition is satisfied. We end with conclusions.

##  II  The coordinate transformations

Consider the Minkowskian light cone coordinates

$\displaystyle V$ $\displaystyle=$ $\displaystyle T+X,$ $\displaystyle U$
$\displaystyle=$ $\displaystyle T-X.$

Consider the four quadrants as shown in Fig.1, with coordinates in each region

$\displaystyle v,\bar{v}$ $\displaystyle=$ $\displaystyle t+x,$ $\displaystyle
u,\bar{u}$ $\displaystyle=$ $\displaystyle t-x,$

related to $U,V$ through

$\displaystyle g(bV)$ $\displaystyle=$ $\displaystyle e^{bv},\quad V>0$
$\displaystyle h(bU)$ $\displaystyle=$ $\displaystyle e^{bu},\quad U>0$
$\displaystyle g(bV)$ $\displaystyle=$ $\displaystyle de^{-b\bar{v}},\quad
V<0$ $\displaystyle h(bU)$ $\displaystyle=$ $\displaystyle
de^{-b\bar{u}},\quad U<0.$

Figure 1:  Minkowski spacetime split into four quadrants

and $v,\bar{v},u,\bar{u}\in[-\infty,\infty]$, whichever quadrant they appear
in. Let $d,b$ be real. $g,h$ are chosen such that,

$\displaystyle g(e^{-i\pi}V)$ $\displaystyle=e^{-i(2n+1)\pi}g(V)$
$\displaystyle h(e^{-i\pi}U)$ $\displaystyle=e^{-i(2n+1)\pi}h(U),$ (11)

with $n$ being a positive integer. The reason behind the $e^{-i(2n+1)\pi}$
factors will become clear below.

Consider the quadrant in which both $U,V>0$. We then have

$\displaystyle
ds^{2}=dUdV=\frac{g(bU)h(bV)}{g^{\prime}(bU)h^{\prime}(bV)}dudv=f(u,v)dudv,$
(12)

where $f(u,v)=\frac{g(bU)h(bV)}{g^{\prime}(bU)h^{\prime}(bV)}$ as should be
obvious. A Klein Gordon scalar field because of conformal invariance has a
action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{2}x\sqrt{-g}g_{ab}\partial^{a}\phi\partial^{b}\phi$ $\displaystyle=$
$\displaystyle\int dudv(\partial_{u}\phi\partial_{v}\phi)$ $\displaystyle=$
$\displaystyle\int dUdV(\partial_{U}\phi\partial_{V}\phi).$

This implies, any functions of the form $\zeta^{U}(U)$, $\zeta^{V}(V)$,
$\zeta^{u}(u)$, $\zeta^{v}(v)$ are solutions of the Klein Gordon equation.
What is said above can be extended to all quadrants. This implies, we can talk
about positive frequency modes $e^{-i\omega v}$ and $e^{-i\omega\bar{v}}$,
which because of Eq.LABEL:Eq2 could be written as

$\displaystyle\theta(V)e^{-i\omega v}=\theta(V)(g(bV))^{-i\omega/b}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[\alpha_{k\omega}^{R}e^{-ikV}+\beta_{k\omega}^{R}e^{ikV}]$
$\displaystyle\theta(-V)e^{-i\omega\bar{v}}=\theta(-V)(\frac{g(bV)}{d})^{i\omega/b}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[\alpha_{k\omega}^{L}e^{-ikV}+\beta_{k\omega}^{L}e^{ikV}].$

From the bottom equation we get,

$\displaystyle\theta(V)(e^{-i(2n+1)\pi}\frac{g(bV)}{d})^{i\omega/b}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[\alpha_{k\omega}^{L}e^{ikV}+\beta_{k\omega}^{L}e^{-ikV}]$

Now when $g(bV)>0$ for a particular value of $V>0$, $\ln g(bV)$ is real. Hence
we can write $(g(bV))^{-i\omega/b}=e^{-i\omega\ln g(bV)/b}$ and take the
complex conjugate of equation Eq.LABEL:bogoliubov22 by changing the sign of
the exponent to get

$\displaystyle\theta(V)(g(bV))^{i\omega/b}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[{\alpha_{k\omega}^{R}}^{*}e^{ikV}+{\beta_{k\omega}^{R}}^{*}e^{-ikV}],$

If however $g(bV)<0$ for a particular value of $V$, then $g(-bV)>0$, so
Eq.LABEL:bogoliubov22 can be written as

$\displaystyle(-1)^{-i\omega/b}\theta(V)(g(-bV))^{-i\omega/b}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[{\alpha_{k\omega}^{R}}e^{-ikV}+{\beta_{k\omega}^{R}}e^{ikV}],$

respectively. Then we can write $(g(-bV))^{-i\omega/b}=e^{-i\omega\ln
g(-bV)/b}$ and we can take the complex conjugate of the above equation by
changing the sign of the exponent and we get,

$\displaystyle(-1)^{-i\omega/b}\theta(V)(g(-bV))^{i\omega/b}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[{\alpha_{k\omega}^{R}}^{*}e^{ikV}+{\beta_{k\omega}^{R}}^{*}e^{-ikV}],$
$\displaystyle\theta(V)(-g(-bV))^{i\omega/b}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[{\alpha_{k\omega}^{R}}^{*}e^{ikV}+{\beta_{k\omega}^{R}}^{*}e^{-ikV}],$
$\displaystyle\theta(V)(g(bV))^{i\omega/b}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[{\alpha_{k\omega}^{R}}^{*}e^{ikV}+{\beta_{k\omega}^{R}}^{*}e^{-ikV}],$

Hence whether $g(bV)$ is positive or negative we always have

$\displaystyle\theta(V)(g(bV))^{i\omega/b}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}[{\alpha_{k\omega}^{R}}^{*}e^{ikV}+{\beta_{k\omega}^{R}}^{*}e^{-ikV}],$

Comparing to Eq.LABEL:relevant_1 we get

$\displaystyle{\alpha_{k\omega}^{R}}^{*}$ $\displaystyle=$
$\displaystyle(e^{i(2n+1)\pi}d)^{i\omega/b}\alpha_{k\omega}^{L}$
$\displaystyle{\beta_{k\omega}^{R}}^{*}$ $\displaystyle=$
$\displaystyle(e^{i(2n+1)\pi}d)^{i\omega/b}\beta_{k\omega}^{L}.$

Upon taking Fourier transforms of Eqs.LABEL:bogoliubov22, LABEL:bogoliubov23
we have

$\displaystyle\alpha_{k\omega}^{R}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{k}{\omega}}\int_{0}^{\infty}\frac{dV}{2\pi}(g(bV))^{-i\omega/b}e^{ikV}$
$\displaystyle\beta_{k\omega}^{L}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{k}{\omega}}\int_{-\infty}^{0}\frac{dV}{2\pi}(\frac{g(bV)}{d})^{i\omega/b}e^{-ikV}$
(21)

Hence

$\displaystyle{\beta_{k\omega}^{L}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{k}{\omega}}\int_{-\infty}^{0}\frac{dV}{2\pi}(\frac{g(bV)}{d})^{i\omega/b}e^{-ikV}=-\sqrt{\frac{k}{\omega}}\int_{0}^{-\infty}\frac{dV}{2\pi}(\frac{g(bV)}{d})^{i\omega/b}e^{-ikV}$
(22)

A question to ask is the following: Does $(\frac{g(bV)}{d})^{-i\omega/b}$ blow
up for any value of $V$? Let, $V$ be extended to the the whole complex plane
then $g(bV)=r_{V}e^{i\theta_{V}}$, where
$r_{V}\in[0,\infty],\theta_{V}\in[0,2\pi]$. Now,

$\displaystyle g(bV)^{-i\omega b}=e^{-i\omega
blnr_{V}}e^{\theta_{V}b\omega}=e^{\theta_{V}b\omega}[-i\sin[b\omega\ln
r_{V}]+\cos[b\omega\ln r_{V}]]$ (23)

Since $\theta_{V}$ is finite and the $\sin,\cos$’s are also finite,
$g(bV)^{-i\omega b}$ is finite and hence non-singular over the entire complex
plane. Hence, for a contour $C$ closing in the lower half complex $V$ plane,
made up of the entire real axis and a semicircle of infinite radius

$\displaystyle\oint\limits_{C}\frac{dV}{2\pi}(g(bV))^{i\omega/b}e^{-ikV}=0$
(24)

This gives

$\displaystyle{\beta_{k\omega}^{L}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{k}{\omega}}\int_{-\infty}^{0}\frac{dV}{2\pi}(\frac{g(bV)}{d})^{i\omega/b}e^{-ikV}=-\sqrt{\frac{k}{\omega}}\int_{0}^{-\infty}\frac{dV}{2\pi}(\frac{g(bV)}{d})^{i\omega/b}e^{-ikV}$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{k}{\omega}}\int_{0}^{\infty}\frac{dV}{2\pi}(\frac{g(bV)}{d})^{i\omega/b}e^{-ikV}=-d^{-i\omega/b}{\alpha_{k\omega}^{R}}^{*}$

Using Eq.LABEL:complex_conjugates, we have

$\displaystyle{\beta_{k\omega}^{R}}^{*}(e^{i(2n+1)\pi}d)^{-i\omega/b}$
$\displaystyle=$
$\displaystyle-d^{-i\omega/b}(e^{i(2n+1)\pi}d)^{i\omega/b}\alpha_{k\omega}^{L}$
$\displaystyle{\beta_{k\omega}^{R}}^{*}$ $\displaystyle=$
$\displaystyle-d^{i\omega/b}(e^{i(2n+1)\pi})^{i2\omega/b}\alpha_{k\omega}^{L}$

Now if we assume that
$[d^{i\omega/b}(e^{i(2n+1)\pi})^{i2\omega/b}]^{*}=d^{-i\omega/b}=e^{-\frac{(2n+1)\pi\omega}{b}}$,
then $d$ is constrained to be $d=e^{-i(2n+1)\pi}$, and the following are a
linear combination of positive energy solutions in Minkowski spacetime.

$\displaystyle G_{1}(V)$ $\displaystyle=$ $\displaystyle\theta(V)e^{-i\omega
v}+\theta(-V)e^{-\frac{(2n+1)\pi\omega}{b}}e^{i\omega\bar{v}}$ (27)
$\displaystyle G_{2}(V)$ $\displaystyle=$
$\displaystyle\theta(V)e^{-\frac{(2n+1)\pi\omega}{b}}e^{i\omega
v}+\theta(-V)e^{-i\omega\bar{v}}$ (28)

Now, any field operator can be expressed as

$\displaystyle\phi(V)$ $\displaystyle=$ $\displaystyle\int
d\omega[(\alpha^{R}_{\omega}e^{-i\omega
v}+{\alpha^{R}_{\omega}}^{\dagger}e^{i\omega v})\theta(V)$ $\displaystyle+$
$\displaystyle(\alpha^{L}_{\omega}e^{-i\omega\bar{v}}+{\alpha^{L}_{\omega}}^{\dagger}e^{i\omega\bar{v}})\theta(-V)),$
$\displaystyle\propto$ $\displaystyle\int
d\omega(\alpha^{R}_{\omega}-e^{-\frac{(2n+1)\pi\omega}{b}}{\alpha^{L}_{\omega}}^{\dagger})G_{1}(V)$
$\displaystyle+$
$\displaystyle({a_{\omega}^{L}}-e^{-\frac{(2n+1)\pi\omega}{b}}{a_{\omega}^{R}}^{\dagger})G_{2}(V)+h.c.$
(31)

Because $G_{1,2}(V)$ contain Minkowski modes of positive frequency we should
have,

$\displaystyle(a_{\omega}^{R}-e^{-\frac{(2n+1)\pi\omega}{b}}{a_{\omega}^{L}}^{\dagger}|0)\rangle_{M}=0$
(32)

and

$\displaystyle({a_{\omega}^{L}}-e^{-\frac{(2n+1)\pi\omega}{b}}{a_{\omega}^{R}}^{\dagger})|0\rangle_{M}=0.$
(33)

We could hence, evaluate the occupation number

$\displaystyle\langle{a_{\omega}^{L}}^{\dagger}a_{\omega}^{L}\rangle_{M}=\langle{a_{\omega}^{R}}^{\dagger}a_{\omega}^{R}\rangle_{M}=\frac{1}{e^{2\frac{(2n+1)\pi\omega}{b}}-1},$
(34)

One can easily see that in the derivation above, if we had replaced
$v,\bar{v},V$ with $u,\bar{u},U$ respectively, along with $g(bV)$ by $h(bU)$,
the expectation value of the corresponding number operators is also Planckian.
This implies that observer recording spacetime events using corresponding
coordinates in respective quadrants in Fig.1 would measure a temperature
$\frac{b}{2(2n+1)\pi}$ in a Minkowiskian vacuum if functions $g,h$ obey Eq.11
and $d=-1$.

One can see that we get led to the same conclusions above by a different
route. Let us assume that as before $g(e^{-i\pi}bV)=e^{-i(2n+1)\pi}g(bV)$
below. Now consider a function $G(V)$ made up of positive frequency modes.
This function by construction is analytic in the complex $V$ plane, below the
real axis. There is no inconsistency with defining $G(V)=g(bV)^{-i\omega/b}$,
below the real axis. The reason being that if $V=|V|e^{i\theta}$,
$\theta\in[-\pi,\pi]$, then there is a branch cut on the negative real axis,
as value of $G(V)$ takes two different values at $\theta=\pi$ and
$\theta=-\pi$. So, $G(V)=g(bV)^{-i\omega/b}$ is not analytic on the whole
complex plane. However, below the real $V$ axis $g(bV)^{-i\omega/b}$ is
analytic. This implies that for $V>0$, $G(V)=e^{-i\omega
v}=g(bV)^{-i\omega/b}$ is a consistent choice for a function made up of
positive frequency Minkowski modes. For $V<0$,

$\displaystyle
G(V)=(g(bV))^{-i\omega/b}=(-g(e^{-i\pi}bV))^{-i\omega/b}=[e^{-i(2n+1)\pi}(-g(bV))]^{-i\omega/b}=e^{-\frac{(2n+1)\omega\pi}{b}}e^{i\omega\bar{v}}$

implying

$\displaystyle G_{1}(V)$ $\displaystyle=$ $\displaystyle\theta(V)e^{-i\omega
v}+\theta(-V)e^{-\frac{(2n+1)\omega\pi}{b}}e^{i\omega\bar{v}},$ (36)

is made up of positive frequency Minkowski modes. One can similarly show that

$\displaystyle G_{2}(V)$ $\displaystyle=$
$\displaystyle\theta(-V)e^{-i\omega\bar{v}}+\theta(V)e^{-\frac{(2n+1)\omega\pi}{b}}e^{i\omega
v},$ (38)

is made up of positive frequency Minkowski modes. Hence, temperature measured
by an observer using the appropriate coordinates in the quadrants in Fig.1, is
$k_{B}T=\frac{b}{2(2n+1)\pi}$.

Let us consider a generic case of $g(e^{-i\pi}x)=e^{-i\pi}g(x)$,
$h(e^{-i\pi}x)=e^{-i\pi}h(x)$ with,

$\displaystyle y=g(x)$ $\displaystyle=$ $\displaystyle a_{1}x+a_{3}x^{3}+....$
$\displaystyle y=h(x)$ $\displaystyle=$ $\displaystyle b_{1}x+b_{3}x^{3}+....$

Using a series reversion math_methods_1 , one can then evaluate

$\displaystyle x=G(y)$ $\displaystyle=$ $\displaystyle
v_{1}y+\frac{v_{2}}{3}y^{3}+....$ $\displaystyle x=H(y)$ $\displaystyle=$
$\displaystyle u_{1}y+\frac{u_{2}}{3}y^{3}+....$

One can similarly start with eqn’s above involving the $u,v$’s and can produce
the equations with $a,b$’s above using series reversion. We will use the above
in the next section.

One should note that for $U,V>0$, if for two functions $J,K$

$\displaystyle bV=$ $\displaystyle J(bv)$ $\displaystyle bU=$ $\displaystyle
K(bu),$

then

$\displaystyle dVdU=$ $\displaystyle J^{\prime}(bv)K^{\prime}(bu)dudv,$

hence the metric in coordinates $u,v$ may not have a time like killing vector.
However one should note that the a time like killing vector is advertised as a
requirement so that that the solution of the wave equation will split up into
temporal and spatial parts. This splitting is needed, so that one could talk
about positive/negative frequency modes, which is a requirement for particle
definition under field quantization de witt . However note that in two
dimensions, the conformal form of the metric above would imply that because
the free scalar field equation is

$\displaystyle\partial_{u}\partial_{v}\phi=0$ (44)

one would still get solutions of positive and negative frequencies, with
respect to the time coordinates $t$ and hence one could safely talk about
particle definitions for an observer using coordinates $(u,v)$ or $(x,t)$.

##  III Construction of black hole metrics

In this section we will use the results from the previous section to construct
metrics corresponding to radiating black holes. Consider the Schwarzschild
metric

$\displaystyle
ds^{2}=(1-\frac{2GM}{r})dt^{2}-\frac{1}{1-\frac{2GM}{r}}dr^{2}-r^{2}d\Omega^{2}.$
(45)

In tortoise coordinates

$\displaystyle
ds^{2}=(1-\frac{2GM}{r})[dt^{2}-dr_{*}^{2}]-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (46)

where

$\displaystyle r_{*}$ $\displaystyle=$ $\displaystyle r+R\ln|(r/R-1)|,$
$\displaystyle R$ $\displaystyle=$ $\displaystyle 2GM.$ (47)

The horizon in these coordinates lies at $r_{*}\rightarrow-\infty$ and spatial
infinity at $r_{*}\rightarrow\infty$. The solution of the Klein Gordon
equation in these coordinates has no support behind the horizon. The Klein
Gordon action is

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}g^{ab}\partial_{a}\Phi\partial_{b}\Phi$ $\displaystyle S$
$\displaystyle=$ $\displaystyle\int drr^{2}\sin\theta d\theta
d\phi{(1-\frac{2GM}{r})}[\frac{(\partial_{t}\Phi)^{2}-(\partial_{r^{*}}\Phi)^{2}}{(1-\frac{2GM}{r})}-\frac{1}{r^{2}\sin^{2}\theta}(\frac{\partial\Phi}{\partial\phi})^{2}-\frac{1}{r^{2}}(\frac{\partial\Phi}{\partial\theta})^{2}]$
$\displaystyle=$ $\displaystyle\int drr^{2}\sin\theta d\theta
d\phi[{(\partial_{t}\Phi)^{2}-(\partial_{r^{*}}\Phi)^{2}}{}-\frac{(1-\frac{2GM}{r})}{r^{2}\sin^{2}\theta}(\frac{\partial\Phi}{\partial\phi})^{2}-\frac{(1-\frac{2GM}{r})}{r^{2}}(\frac{\partial\Phi}{\partial\theta})^{2}]$

The Klein Gordon equation is hence

$\displaystyle\partial_{a}(\sqrt{-g}g^{ab}\partial_{b}\Phi)$ $\displaystyle=0$
$\displaystyle
r^{2}\sin\theta\partial_{t}^{2}\Phi-\sin\theta\partial_{r^{*}}r^{2}\partial_{r^{*}}\Phi-r^{2}\sin\theta\partial_{\phi}\frac{(1-\frac{2GM}{r})}{r^{2}\sin^{2}\theta}(\frac{\partial\Phi}{\partial\phi})-r^{2}\partial_{\theta}\sin\theta\frac{(1-\frac{2GM}{r})}{r^{2}}(\frac{\partial\Phi}{\partial\theta})$
$\displaystyle=0$
$\displaystyle\partial_{t}^{2}\Phi-\frac{1}{r^{2}}\partial_{r^{*}}r^{2}\partial_{r^{*}}\Phi-(1-\frac{2GM}{r})[\frac{1}{r^{2}\sin^{2}\theta}\partial_{\phi}(\frac{\partial\Phi}{\partial\phi})+\frac{1}{r^{2}\sin\theta}\partial_{\theta}\sin\theta(\frac{\partial\Phi}{\partial\theta})]$
$\displaystyle=0$ (49)

we can see that it would be possible to write solutions of the above going as
$e^{-i\omega t}F_{\omega,l}^{1,2}(r^{*})Y^{l,m}(\theta,\phi)$. Here,
$F_{\omega,l}^{1,2}(r^{*})$ are the two linearly independent solutions of

$\displaystyle-\omega^{2}F_{\omega,l}^{1,2}(r^{*})-\frac{1}{r^{2}}\partial_{r^{*}}r^{2}\partial_{r^{*}}F_{\omega,l}^{1,2}(r^{*})+(1-\frac{2GM}{r})\frac{l(l+1)}{r^{2}}F_{\omega,l}^{1,2}(r^{*})$
$\displaystyle=0$

Hence it would be possible to write the field operator expansion in terms of
creation and destruction operators for field quanta as

$\displaystyle\phi(r^{*},\phi,\theta,t)$ $\displaystyle=$
$\displaystyle\sum_{\omega,l,m}[a_{\omega}^{l,m}e^{-i\omega
t}F^{1}_{l,\omega}(r^{*},\theta,\phi)+b_{\omega}^{l,m}e^{-i\omega
t}F^{2}_{l,\omega}(r^{*},\theta,\phi)+h.c.]$ $\displaystyle
Y^{l,m}(\theta,\phi)$

Close to the horizon $r=2GM$ implies the angular derivatives contribution in
Eq.49, become zero and $\frac{dr}{dr^{*}}=0$. Hence, the Klein Gordon equation
becomes

$\displaystyle(\partial_{t}^{2}-\partial_{r^{*}}^{2})\phi=0$ (52)

At spatial infinity, the metric Eq.46 is also Minkowskian. This situation is
equivalent to a free particle coming from either asymptotes encountering a
potential barrier, implying two independent solutions that asymptote as

$\displaystyle F_{\omega,l}^{1}(r^{*})$ $\displaystyle=$ $\displaystyle
e^{i\omega r^{*}}+A(l,\omega)e^{-i\omega r^{*}}\quad r^{*}\rightarrow-\infty$
$\displaystyle=$ $\displaystyle e^{i\omega r^{*}}\quad r^{*}\rightarrow\infty$

and

$\displaystyle F_{\omega,l}^{2}(r^{*})$ $\displaystyle=$ $\displaystyle
e^{-i\omega r^{*}}+B(l,\omega)e^{i\omega r^{*}}\quad r^{*}\rightarrow\infty$
$\displaystyle=$ $\displaystyle e^{-i\omega r^{*}}\quad
r^{*}\rightarrow-\infty$

Hence close to the horizon we can expand

$\displaystyle\phi(r^{*}\rightarrow-\infty,\phi,\theta,t)$ $\displaystyle=$
$\displaystyle\sum_{\omega,l,m}[a_{\omega}^{l,m}[e^{i\omega
r^{*}}+A(l,\omega)e^{-i\omega r^{*}}]e^{-i\omega
t}+b_{\omega}^{l,m}e^{-i\omega(r^{*}+t)}+h.c.]Y^{l,m}(\theta,\phi)$
$\displaystyle=$
$\displaystyle\sum_{\omega,l,m}[a_{\omega}^{l,m}e^{i\omega(r^{*}-t)}+(b_{\omega}^{l,m}+a_{\omega}^{l,m}A(l,\omega))e^{-i\omega(r^{*}+t)}]+h.c.]Y^{l,m}(\theta,\phi)$

and at spatial infinity,

$\displaystyle\phi(r^{*}\rightarrow\infty,\phi,\theta,t)$ $\displaystyle=$
$\displaystyle\sum_{\omega,l,m}[b_{\omega}^{l,m}[e^{-i\omega
r^{*}}+B(l,\omega)e^{i\omega r^{*}}]e^{-i\omega
t}+a_{\omega}^{l,m}e^{i\omega(r^{*}-t)}+h.c.]Y^{l,m}(\theta,\phi)$
$\displaystyle=$
$\displaystyle\sum_{\omega,l,m}[b_{\omega}^{l,m}e^{-i\omega(r^{*}+t)}+(a_{\omega}^{l,m}+b_{\omega}^{l,m}B(l,\omega))e^{i\omega(r^{*}-t)}+h.c.]Y^{l,m}(\theta,\phi)$

This implies that the $a_{\omega}^{l,m},b_{\omega}^{l,m}$ are particle
destruction operators both at the horizon as well as spatial infinity.

Close to and outside the horizon, the metric ignoring the $r^{2}d\Omega^{2}$
can be written as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(1-\frac{R}{r})[dt^{2}-dr_{*}^{2}]$ (57) $\displaystyle\sim$
$\displaystyle e^{(r_{*}-r)/R}[dt^{2}-dr_{*}^{2}]$ $\displaystyle\sim$
$\displaystyle e^{(v-u)/2R-1}[dudv]$ $\displaystyle\sim$ $\displaystyle dUdV,$
(59)

where $U=-Re^{-u/2R}$, $V=Re^{v/2R}$, are the null coordinates of the freely
falling observer. The $u$,$v$ coordinates hence are of the form of Rindler
coordinates. We could similarly represent the coordinates close to and behind
the horizon as $U=Re^{u/2R}$, $V=-Re^{-v/2R}$ . This implies that the
expectation values of the number operators
${a_{\omega}^{l,m}}^{\dagger}a_{\omega}^{l,m}$ and
${b_{\omega}^{l,m}}^{\dagger}b_{\omega}^{l,m}$ in Eq.LABEL:rindler_expansion
in the Minkowskian vacuum would have a Planckian form, which will then also
imply that the spectrum of particles at spatial infinity is also Planckian.
The above reasoning is inspired from jacobsson , de witt , candelas .

Now consider the following scenario. What if instead of the metric Eq.46, we
had the following metric

$\displaystyle
ds^{2}=\Pi(r,t)[dt^{2}-dr_{*}^{2}]-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (60)

where $\Pi(r,t)$ is a smooth function of its parameters. The Klein Gordon
action is now

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}g^{ab}\partial_{a}\Phi\partial_{b}\Phi$ $\displaystyle S$
$\displaystyle=$ $\displaystyle\int drr^{2}\sin\theta d\theta
d\phi{\Pi(r,t)}[\frac{(\partial_{t}\Phi)^{2}-(\partial_{r^{*}}\Phi)^{2}}{\Pi(r,t)}-\frac{1}{r^{2}\sin^{2}\theta}(\frac{\partial\Phi}{\partial\phi})^{2}-\frac{1}{r^{2}}(\frac{\partial\Phi}{\partial\theta})^{2}]$
$\displaystyle=$ $\displaystyle\int drr^{2}\sin\theta d\theta
d\phi[{(\partial_{t}\Phi)^{2}-(\partial_{r^{*}}\Phi)^{2}}{}-\frac{\Pi(r,t)}{r^{2}\sin^{2}\theta}(\frac{\partial\Phi}{\partial\phi})^{2}-\frac{\Pi(r,t)}{r^{2}}(\frac{\partial\Phi}{\partial\theta})^{2}]$

The Klein Gordon equation is hence

$\displaystyle\partial_{a}(\sqrt{-g}g^{ab}\partial_{b}\Phi)$ $\displaystyle=0$
$\displaystyle\partial_{t}^{2}\Phi-\frac{1}{r^{2}}\partial_{r^{*}}r^{2}\partial_{r^{*}}\Phi-\Pi(r,t)[\frac{1}{r^{2}\sin^{2}\theta}\partial_{\phi}(\frac{\partial\Phi}{\partial\phi})+\frac{1}{r^{2}\sin\theta}\partial_{\theta}\sin\theta(\frac{\partial\Phi}{\partial\theta})]$
$\displaystyle=0$

If we now try solutions of the form $e^{-i\omega
t}F_{\omega,l}^{1,2}(r^{*})Y^{l,m}(\theta,\phi)$, then

$\displaystyle-\omega^{2}F_{\omega,l}^{1,2}(r^{*})-\frac{1}{r^{2}}\partial_{r^{*}}r^{2}\partial_{r^{*}}F_{\omega,l}^{1,2}(r^{*})+\Pi(r,t)\frac{l(l+1)}{r^{2}}F_{\omega,l}^{1,2}(r^{*})$
$\displaystyle=0$

We now see that we could only get a solution for $\Pi(r,t)$ not being
independent of $t$, if $l=0$. We should note that the $l=0$ restriction only
arises if one is considering time dependent metrics which have the generic
form Eq.60 and are interested in solutions of Klein Gordon equation of the
form $e^{-i\omega t}F_{\omega,l}^{1,2}(r^{*})Y^{l,m}(\theta,\phi)$. If we
instead had a metric of the form

$\displaystyle
ds^{2}=\Pi_{t}(r,t)dt^{2}-\Pi_{r}(r,t)dr_{*}^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (64)

with $\Pi_{t}(r,t)$, $\Pi_{r}(r,t)$ being two different functions, then it
would not be possible to get a solution of the Klein Gordon equation of the
form $e^{-i\omega t}F_{\omega,l}^{1,2}(r^{*})Y^{l,m}(\theta,\phi)$ for any
values of $l,m$. Hence only wave modes of the form $e^{-i\omega
t}F_{\omega,l=0}^{1,2}$, will correspond to particles after field quantization
in a background given by metric of the form Eq.60. Now if the near horizon
form of $\Pi(r,t)$ is such that we could map the lightlike coordinates
$u=t+r$, $v=t-r$, to free falling $U,V$ as in Eq .LABEL:Eq2, we would get
thermal radiation with quantum number $l=0$ at spatial infinity.

We use these arguments to construct black hole metrics that correspond to
radiating black holes below.

In light of the results from the previous section, we argue that if the
relationships between the freely falling coordinates and $u,v$ coordinates
describing the region outside the black hole horizon were $U=-H(e^{-u/2R})$,
$V=G(e^{v/2R})$, with $H,G$ being odd functions as in Eq.LABEL:nextsection, an
observer at spatial infinity would measure a temperature. By considering Eq.
10 that gives the transformation between Minkowski coordinates to the null
coordinates $(u,v)$, we derived the temperature measured by an observer using
these coordinates to be $\frac{b}{2(2n+1)\pi}$, as was seen from Eq.34. Also
since the corresponding $n=0$, for the choice of coordinate transformations
relating the free falling coordinates at the horizon to the coordinates of an
observer at infinity, the temperature measured at infinity will go as
$\frac{1}{4R\pi}=\frac{1}{8GM\pi}$.

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle dVdU$
$\displaystyle\sim$ $\displaystyle
e^{(v-u)/2R}H^{\prime}(e^{-u/2R})G^{\prime}(e^{v/2R})[dudv].$

We claim that the following metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle[1+\sum_{n=1,N}u_{n}e^{n(t-r)/r}|(r/R-1)|^{-n}]$
$\displaystyle[1+\sum_{n^{\prime}=1,N^{\prime}}v_{n}^{\prime}e^{-n^{\prime}(t+r)/R}|(r/R-1)|^{-n^{\prime}}][(1-\frac{R}{r})dt^{2}$
$\displaystyle-$ $\displaystyle(1-\frac{R}{r})^{-1}dr^{2}]-r^{2}d\Omega^{2},$

will reduce to the above form close to the horizon. To see this, note that
because

$\displaystyle r_{*}$ $\displaystyle=$ $\displaystyle r+R\ln|(r/R-1)|,$
$\displaystyle e^{v/R}=e^{(t+r_{*})/R}$ $\displaystyle=$ $\displaystyle
e^{(t+r)/R}|(r/R-1)|,$ (69) $\displaystyle e^{u/R}=e^{(t-r_{*})/R}$
$\displaystyle=$ $\displaystyle e^{(t-r)/R}|(r/R-1)|^{-1}.$

This implies, that close to the horizon we have

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle[1+\sum_{n=1,N}u_{n}e^{n(t-r)/R}|(r/R-1)|^{-n}]$
$\displaystyle[1+\sum_{n=1,N^{\prime}}v_{n}^{\prime}e^{-n^{\prime}(t+r)/R}|(r/R-1)|^{-n^{\prime}}][(1-\frac{R}{r})dt^{2}$
$\displaystyle-$ $\displaystyle(1-\frac{R}{r})^{-1}dr^{2}]-r^{2}d\Omega^{2},$
$\displaystyle\sim$ $\displaystyle
e^{(v-u)/2R}[1+\sum_{n=1,N}u_{n}e^{nu/R}][1+\sum_{n^{\prime}=1,N^{\prime}}v_{n}^{\prime}e^{-n^{\prime}v/R}][dudv]-r^{2}d\Omega^{2}$
$\displaystyle\sim$ $\displaystyle
e^{(v-u)/2R}H^{\prime}(e^{u/2R})G^{\prime}(e^{-v/2R})[dudv]-r^{2}d\Omega^{2}$

At spatial infinity, the above metric by construction becomes the inertial

$\displaystyle ds^{2}$ $\displaystyle\sim$
$\displaystyle[dt^{2}-dr^{2}-r^{2}d\Omega^{2}].$

We note that because the non-angular part of Eq.LABEL:actual_metric close to
the horizon is given $\sim
e^{(v-u)/2R}H^{\prime}(e^{u/2R})G^{\prime}(e^{-v/2R})[dudv]$, with the co-
ordinates $u,v$ related to that of flat spacetime $U,V$, as $U=-H(e^{-u/2R})$,
$V=G(e^{v/2R})$, implies that the near horizon geometry is not singular,
despite the fact that the metric blows up at the horizon. Another way to see
this is that the metric ignoring the angular parts is basically the
Schwarzchild metric multiplied by
$[1+\sum_{n=1,N}u_{n}e^{n(t-r)/R}|(r/R-1)|^{-n}][1+\sum_{n=1,N^{\prime}}v_{n}^{\prime}e^{-n^{\prime}(t+r)/R}|(r/R-1)|^{-n^{\prime}}]$,
which equals $H^{\prime}(e^{u/2R})G^{\prime}(e^{-v/2R})$ globally and not just
at the horizon. So this enhancement to the Schwarzchild metric cannot change
the non-singular properties of the horizon.

We also note that the discussion from Eq.60 to above Eq.64, still holds even
if the $\Pi(r,t)$ diverges at any point in spacetime, as the discussion is
about how $\Pi(r,t)$ does not appear in any discussion of $l=0$ wave modes.
Hence even though the the metric Eq.LABEL:actual_metric diverges at the
horizon, black holes corresponding to the metric radiate.

Because the metric is time dependent and can blow up for $t\rightarrow\infty$
and $t\rightarrow-\infty$ we assume that the mass distribution corresponding
to the metric above is produced during a physical process (that we do not
comment on) at $t=0$ and this time varying mass distribution later evolves
into something else and does not correspond to the metric above $t>T$ where
$T$ is some finite time. Radiation will be observed for $t\in[0,T]$. This
statement is consistent and is similar to the statement that a body would be
bombarded with thermal radiation for the period of time it is moving with a
constant acceleration, despite other instances on its worldline where the body
may be moving with a constant velocity.

##  IV Energy Conditions

We have the metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle[1+\sum_{n=1,N}u_{n}e^{n(t-r)/R}|(r/R-1)|^{-n}]$
$\displaystyle[1+\sum_{n^{\prime}=1,N^{\prime}}v_{n}^{\prime}e^{-n^{\prime}(t+r)/R}|(r/R-1)|^{-n^{\prime}}][(1-\frac{R}{r})dt^{2}$
$\displaystyle-$ $\displaystyle(1-\frac{R}{r})^{-1}dr^{2}]-r^{2}d\Omega^{2}.$

The constraints on physicality of the solution above are imposed by the energy
conditions. To evaluate these conditions we need to evaluate the energy
momentum tensor corresponding to the above metric. To reduce the clutter of
calculations because of so many terms in the sum above, let us for
illustration consider the case where all the $v_{n}=0$ and only one of
$u_{n}$’s which we label as $a$ is non-zero, which corresponds to $n=2p$, with
$p$ being a positive integer. We hence have the metric,

$\left(\begin{array}[]{cccc}-\frac{ae^{\frac{2p(t-r)}{R}}\left(1-\frac{r}{R}\right)^{-2p}+1}{1-\frac{R}{r}}&0&0&0\\\
0&-r^{2}&0&0\\\ 0&0&-r^{2}\sin^{2}(\theta)&0\\\
0&0&0&\left(ae^{\frac{2p(t-r)}{R}}\left(1-\frac{r}{R}\right)^{-2p}+1\right)\left(1-\frac{R}{r}\right)\\\
\end{array}\right)$ (74)

Using Mathematica we get the following expressions for non-zero components of
energy momentum tensor

$\displaystyle 8\pi GT_{rr}$ $\displaystyle=$
$\displaystyle\frac{ae^{\frac{2p(t-r)}{R}}\left(\frac{2p}{R\left(ae^{\frac{2p(t-r)}{R}}+\left(1-\frac{r}{R}\right)^{2p}\right)}+\frac{\left(1-\frac{r}{R}\right)^{-2p}}{r}\right)}{R-r}$
$\displaystyle 8\pi GT_{tt}$ $\displaystyle=$
$\displaystyle-\frac{a(R-r)\left(1-\frac{r}{R}\right)^{-2p}e^{\frac{2p(t-r)}{R}}\left(R\left(ae^{\frac{2p(t-r)}{R}}+\left(1-\frac{r}{R}\right)^{2p}\right)-2pr\left(1-\frac{r}{R}\right)^{2p}\right)}{r^{3}R\left(ae^{\frac{2p(t-r)}{R}}+\left(1-\frac{r}{R}\right)^{2p}\right)}$
$\displaystyle 8\pi GT_{rt}$ $\displaystyle=8\pi
GT_{tr}=\frac{2ape^{\frac{2pt}{R}}}{arRe^{\frac{2pt}{R}}+rRe^{\frac{2pr}{R}}\left(-\frac{r-R}{R}\right)^{2p}}$
(75)

Now a radial null vector given the metric Eq.74, is any multiple of
$n^{\mu}=(1-\frac{R}{r},0,0,1)$ (our coordinates are $(r,\theta,\phi,t)$).
Then,

$\displaystyle T_{\nu\mu}n^{\nu}n^{\mu}$ $\displaystyle=$ $\displaystyle 0$
(76)

We hence see the null energy condition is satisfied.

##  V Conclusion

Some alternatives to Rindler spacetimes such that observers in the spacetime
observe a Planckian spectrum were constructed in sanchez1 . It was shown in
sanchez2 , that the constraint of global thermal equilibrium, which translates
into vanishing of the expectation value of the momentum operator in such
spacetimes, however singles out Rindler observers as ones which observe the
Minkowski vacuum at thermal equilibrium. However note that global requirement
can be relaxed when we are only concerned with behaviours of black hole
metrics near the horizon. The fact that particles are emitted by the black
hole radially outwards at spatial infinity anyway imply a non zero expectation
value for the momentum operator. We can hence consider non Rindler like form
of metric close to the black hole horizon to comment about the thermal
Planckian spectrum of radiation emitted by the black hole as we have done in
this paper. In the section above we considered one family of solutions by
considering only one of the $u_{n}$’s to be non-zero. It would be of interest
to classify all other acceptable solutions implied by the metric in
Eq.LABEL:actual_metric. We note that the temperature for the black holes
constructed is not equal to $T=\kappa/2\pi$, where $\kappa$ is the surface
gravity, as the definition of surface gravity needs that the event horizon is
a Killing horizon for a Killing vector $\chi^{\mu}$, with
$\kappa^{2}=-\frac{1}{2}\nabla^{\mu}\chi^{\nu}\nabla_{\mu}\chi_{\nu}$.
However, only a vector proportional to $\partial_{t}$ becomes null at the
event horizon located at $r=2GM$ for the metric in Eq.LABEL:actual_metric, but
the metric being non stationary does not have $\partial_{t}$ as a Killing
vector. This implies the event horizon is not a Killing horizon and hence the
traditional definition of surface gravity at the horizon
$\kappa^{2}=-\frac{1}{2}\nabla^{\mu}\chi^{\nu}\nabla_{\mu}\chi_{\nu}$ fails
for this metric. Hence a framing of the zeroth and third law of black hole
mechanics is not possible. Since the stress energy tensor is not zero outside
the event horizon, one could not construct a law that relates the mass of the
black hole to the surface area of the event horizon as these definitions
implicity assume that all the mass/energy is located behind the event horizon.
The first law also involves the surface gravity as a proportionality constant
between the surface area of the event horizon and the mass of the black hole,
and the non existence of a definition of surface gravity further adds to
issues in constructing this law for the black hole solution described in the
manuscript. The solution however tends to the Schwarzchild metric at
$t\rightarrow\infty$. One could hence understand the solutions constructed in
the manuscript to describe radiating black holes that do not possess a charge
or angular momentum but eat up matter before equilibriating to a Schwarzchild
black hole. As we have mentioned above, the metric Eq.74 would correspond to a
mass distribution produced during a physical process, that we do not comment
on, at $t=0$ and this time varying mass distribution later evolves into
something else and does not correspond to the metric for $t>T$ where $T$ is
some finite time. Understanding physical processes that lead to metrics
constructed in the paper needs further research.

##  VI Conflict of Interests

The authors declare no conflict of interest.

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