m

††thanks: Email<EMAIL_ADDRESS>Email: rathin@ctp-
jamia.res.in

#  Soft Leptogenesis with Primordial Black Holes

Suhail Khana Rathin Adhikaria aCentre for Theoretical Physics Jamia Millia
Islamia New Delhi

###### Abstract

We revisit soft leptogenesis in Minimal Supersymmetric Standard Model in the
context of the production of heavy sparticles through Hawking evaporation of
primordial black holes (PBH). Depending upon certain conditions on the Hawking
evaporation temperature of the PBH and the initial Hawking temperature of the
PBH, we have shown three possible cases for which the generation of baryonic
asymmetry could be possible. In one case, for the generation of baryonic
asymmetry, Shakharov’s out-of-equilibrium condition is required whereas in the
other two cases, it is found to be not required. In soft leptogenesis without
PBH, soft breaking parameters $A$ and $B$ are in general, much lesser than the
corresponding mass of the decaying particle which is considered as sneutrino -
the super-partner of the lightest of the heavy right-handed neutrinos.
However, in soft leptogenesis with PBH evaporation, in one of those three
cases, it is found that the soft breaking parameters are not much lesser than
sneutrino mass and $|A|$ could be about one order less than the sneutrino mass
whereas parameter $B$ could be a few orders less. This could be an interesting
feature in the context of supersymmetry breaking. Particularly, in this case,
considering experimental constraint from non-observation of $\mu\rightarrow
e\gamma$, the typical mass scale of some supersymmetric particles could be one
order lower than the value of the $|A|$ parameter.

##  I Introduction

In our observed universe, the number density of the baryons ${n_{B}}$ is found
to be more than the number density of anti-baryons $n_{\bar{B}}$ which is
characterized by baryonic asymmetry asym ; cyb ,

$\displaystyle Y_{\Delta{B}}\equiv\frac{n_{B}-n_{\bar{B}}}{s}\approx 8.6\pm
0.1\times 10^{-11}$ (1)

where $s$ is the entropy density of the universe. In general, the mechanism of
baryogenesis could explain the above asymmetry by considering Sakharov’s three
conditions. These are as follows: $(1)$ Baryon number or Lepton number
violating interaction must be present in the Lagrangian $(2)$ There must be
$C$ and $CP$ violating physical processes $(3)$ Such physical processes must
be in out of thermal equilibrium Sakharov . Moreover, one also requires an
interference term involving the amplitudes of tree-level and higher-order
Feynman diagrams corresponding to lepton/baryon number violating physical
processes which will give $CP$ asymmetry. The amplitude of the higher-order
diagram should have the absorptive part of the loop integral. Furthermore, in
the higher order diagram, total $B$ or $L$ violation through couplings
(corresponding to baryogenesis or leptogenesis) should be present with final
states on the right of the ”cut”(for which on-shell condition is satisfied on
the internal line)Adhikari:2001yr . There are various works like GUT
baryogenesis Harvey:1981yk ; Weinberg:1979bt ; Nanopoulos:1979gx ;
Ignatiev:1978uf ; Yoshimura:1979gy ; Riotto:1998bt , electroweak baryogenesis
Klinkhamer:1984di ; Kuzmin:1985mm ; Arnold:1987mh ; Arnold:1987zg ;
Khlebnikov:1988sr ; Kajantie:1996mn ; Riotto:1999yt ; Cline:2006ts ;
Sami:2021ufn , leptogenesis Fukugita:1986hr ; Buchmuller:2004nz ;
Buchmuller:2005eh ; Pilaftsis:2005rv , where Sakharov’s conditions are useful.
There are other ways to generate asymmetry like Affleck-Dine mechanism
Affleck:1984fy ; Dine:2003ax  in which flat directions of the scalar potential
of supersymmetry have been considered. After the discovery of the mass of
Higgs boson around $125$ GeV at LHC CMS:2020xrn  electroweak baryogenesis is
found to be not so much preferred Cline:2017jvp  because of the difficulties
in obtaining appropriate first-order phase transition of the universe at the
electroweak scale.

In the case of GUT scale baryogenesis, in general, a highly massive particle
around $10^{16}$ GeV near GUT scale is considered. But the energy scale of
inflation used to be lesser than that, because of the constraint on tensor to
scalar ratio from the Planck and BICEP$2$ experimental data Tristram:2020wbi .
In addition, there is a significant constraint on the required baryon number
violating interaction coupling to avoid significant proton decay. On the other
hand, leptogenesis seems to be an attractive mechanism for the generation of
baryonic asymmetry in which lepton number-violating interactions are required.
However, it is required to be considered at lower than the GUT scale for the
reasons stated above. Such lepton number violation could lead to baryon number
violation in the presence of sphaleron transitions. There are several works in
leptogenesis in which leptonic asymmetry is generated due to the decays of
heavy right-handed neutrinos Fukugita:1986hr ; Buchmuller:2004nz ;
Buchmuller:2005eh . But to avoid the gravitino overproduction affecting the
successful predictions of Big Bang Nucleosynthesis (BBN), the reheat
temperature $T_{reheat}$ of the universe, should be lower than $10^{9}-10^{6}$
GeV for gravitino mass $8-0.2$ TeV Davidson:2002qv . This requires that the
mass of the lightest RHN to be less than $T_{reheat}\lesssim 10^{9}$ GeV.

For successful leptogenesis with hierarchical heavy right-handed neutrinos
(RHN), the mass of RHN in general, is required to be greater than $10^{9}$ GeV
Davidson:2002qv . So there is an apparent conflict with the requirement of RHN
masses to avoid gravitino overproduction and to get successful leptogenesis.
However, in resonant leptogenesis, considering almost degenerate two RHN
masses, one may get successful leptogenesis with light RHN mass
Pilaftsis:2005rv . Without considering such almost degeneracy, this conflict
on the requirement of heavy RHNs mass may be resolved in soft leptogenesis in
supersymmetry D'Ambrosio:2003wy ; Chung:2003fi ; Grossman:2003jv ; Fong:2011yx
; Fong:2010zu ; Grossman:2004dz ; Fong:2009iu ; Adhikari:2001yr ;
Garbrecht:2013iga ; Fong:2010bv ; Kashti:2004vj ; Abdallah:2020 ; Medina:2006
; Adhikari:2015ysa . In this scenario, one considers Type I seesaw
supersymmetric leptogenesis in which decays of sneutrinos could produce lepton
number asymmetry. The Type I seesaw mechanism also could explain the origin of
very light active neutrino masses. There are various soft supersymmetry
breaking parameters like trilinear $A$, and bi-linear $B$ parameters and also
there are Yukawa couplings $Y$ among RHNs, lepton, up type Higgs superfields
and all these could provide the required amount of $CP$ violation. There is
mixing between heavy sneutrinos and anti-sneutrinos in the presence of soft
supersymmetry breaking terms and there is a small mass splitting between two
sneutrino mass eigenstates due to the above mixing. With small mass splitting
of the order of $B$ parameter it is possible to get successful leptogenesis
for right-handed sneutrino mass $\lesssim 10^{9}$ GeV. Thus the problem of
gravitino overproduction could be evaded.

There are recent development in the work on baryogenesis where decaying heavy
particles are produced through PBH evaporation. In such scenario, the out of
equilibrium condition may not be required. Apart from that, the baryonic
asymmetry also depends on PBH mass. Because of such qualitative differences in
baryogenesis without PBH and that with PBH, in this work we revisit soft
leptogenesis in supersymmetry where decaying sneutrinos (which produces
leptonic asymmetry) could be produced through PBH evaporation.

The primordial black hole (PBH) carr:2021 ; Escriva:2022duf ; Khlopov:2008qy
due to quantum effect undergo evaporation through Hawking radiation
Hawking:1974 ; Hawking:1975 . PBH with mass greater than $10^{37}$ GeV may
exist even today, which may play the role of dark matter Kuhnel:2016 ;
Frampton:2016 ; García:2017 ; Anne:2021 ; carr:2022 ; ShamsEsHaghi:2022azq ;
Belotsky:2014kca ; Gangopadhyay:2021kmf . However, PBH mass lesser than that
could evaporate before BBN and could have various interesting cosmological
consequences Carr:2010 . Such PBH may lead to the production of heavy
particles in the early universe and the decay of such particles could play the
role of baryogenesis or leptogenesis to create observed baryonic asymmetry of
the universe. There are several earlier works in which PBH evaporation
mechanisms have been considered in the context of baryogenesis Nolan:2022 ;
Tomohiro:2104 ; Hamada;2017 ; Hooper:2021 ; Luca:2021 ; Ghosal:2021 ;
Pinetti:2022 ; Stefano:2019 ; Fujita:2014hha ; Gehrman:2022imk  and
leptogenesis Gonzalez:2021 ; Nicolás:2022 ; Borah:2021 . In our work, the
heavy sneutrinos having a lepton number violating decay modes, could be
produced due to black hole evaporation. After the evaporation of PBH, the
temperature of the universe is written as $T_{evap}$. There are two cases one
where $T_{evap}\gtrsim M_{X}$ and the other is $T_{evap}<M_{X}$ where $M_{X}$
is the mass of the decaying particle $X$ with baryon/lepton number violating
decay modes. In case, the first condition is satisfied then $X$ particles
produced through evaporation will be in thermal equilibrium with other lighter
particles. However, in case the second condition is satisfied then $X$
particles produced through evaporation will be out of thermal equilibrium with
other lighter particles. For initial PBH mass $M_{BH}$, there is corresponding
initial Hawking temperature $T_{BH}$ temphawk . There are two scenarios. In
one case $T_{BH}\gtrsim M_{X}$ and in the other case $T_{BH}\lesssim M_{X}$.
The expression of baryonic asymmetry is different depending on these
conditions. This is because the number density of the heavy particle $X$
produced through PBH evaporation depends differently on $M_{BH}$ and $M_{X}$
in these two cases. Combining all these, there are following four different
cases: (1) $T_{evap}\gtrsim M_{X}$ and $T_{BH}>M_{X}$, (2) $T_{evap}<M_{X}$
and $T_{BH}>M_{X}$, (3) $T_{evap}<M_{X}$ and $T_{BH}\lesssim M_{X}$, (4)
$T_{evap}\gtrsim M_{X}$ and $T_{BH}\lesssim M_{X}$.

The main purpose of this work is to study all these four different cases in
obtaining baryonic asymmetry where the heavy particle $X$ is produced through
PBH evaporation. The $X$ particle is considered here as sneutrino which is the
supersymmetric partner of heavy right-handed neutrino in the Minimal
Supersymmetric Standard Model (MSSM). This particle has a lepton number
violating decay mode and observed baryonic asymmetry could be obtained through
leptogenesis. We study in detail for what value of soft SUSY breaking
parameters as well as sneutrino mass and the mass of PBH, observed baryonic
asymmetry could be obtained and how these values differ qualitatively in the
case of soft leptogenesis, without PBH which has been extensively studied
earlier in Adhikari:2015ysa  and in the references therein.

In section II, there is a brief discussion on how PBH could have formed in the
early universe and what could be their lifetime and mass in the context of the
BBN epoch. In section III, there is discussion on the production of heavy
particles due to the evaporation of PBHs, the number density of such heavy
particles and the leptonic asymmetry due to the lepton number violating decay
modes of the heavy particles and its antiparticles. In section IV, we discuss
the interaction of the heavy RHNs, lepton, and Higgs superfields in the mass
eigenstate basis in the Minimal Supersymmetric Standard Model (MSSM) and also
discuss non-thermal $CP$ asymmetry generated in the decays of sneutrinos in
the early universe. In section V, the relationship between leptonic asymmetry
and baryonic asymmetry in the presence of shpeleron is discussed. Then in
different plots, region of $\rm M$, $\rm M_{BH}$ and soft SUSY breaking
parameters $|A|$ and $B$ and the Yukawa coupling $Y_{\alpha}$ is shown for
which the observed baryonic asymmetry of the universe could be obtained.
Furthermore, from experimental constraint on the branching ratio of
$\mu\rightarrow e\gamma$, together with the requirment of $A$ versus sneutrino
mass for the observed baryonic asymmetry, we have discussed the possible mass
scale of supersymmetric particles. Concluding remarks are given in section VI.

##  II Primordial Black Holes and its Hawking Evaporation

The idea of the formation of a black hole in the early universe was initially
put forward by Zel’dovich and Novikov Zeldovich:1966vw  and subsequently, the
formation of a primordial black hole was considered for the post-inflationary
period with density fluctuation Carrhawk1974 ; Carr:1975qj . Also, the
formation was proposed in the context of hybrid inflation GarciaBellido:1996qt
.

When the radiation pressure is insufficient to withstand the gravitational
collapse in dense regions during the radiation-dominant epoch, PBHs are
created. Then the mass of PBH at the time of formation is proportional to its
horizon massCarr:2010 ; Zel’dovich:1966 ; Hawking:1971 ; Carr:1974 ;
Polnarev:1979  and is given by,

$\displaystyle M_{BH}=\gamma M_{H}=\gamma\frac{4\pi}{3}\rho
H_{f}^{-3}=\gamma\frac{4\pi}{3}\frac{3H_{f}^{2}}{8\pi
G}H_{f}^{-3}=\gamma\frac{1}{2GH_{f}}=\frac{\gamma M_{\rm Pl}^{2}}{2H_{f}}.$
(2)

Here, $M_{\rm BH}$ is the black hole mass, and $M_{\rm Pl}=1.22\times 10^{19}$
GeV is the Planck mass and $\gamma$ is a numerical parameter that is affected
by the mechanics of gravitational collapse. This $\gamma$ has been estimated
to be around $(1/\sqrt{3})^{3}\approx 0.2$ during the radiation period
Carr:1974 ; Nadezhin1978 ; Shibata1999 ; Musco ; Harada2013 ; Nakama . $\rho$
is the universe’s average total energy density, $H_{f}$ is the Hubble
parameter at the time of formation and $1/H_{f}$ is the measure of size of the
horizon at that time. Planck collaboration PAR  has put an upper bound on the
Hubble parameter during inflation at 95% confidence level as
$H_{inf}<3.7\times 10^{-5}M_{Pl}\sim 10^{14}$ GeV. As $H_{f}<H_{inf}$, using
this condition in Eq. (2), one gets the lower bound on PBH mass at formation
as

$\displaystyle M_{\rm BH}\gtrsim\frac{\gamma M_{\rm Pl}^{2}}{2H_{f}}\gtrsim
1.49\times 10^{23}\rm GeV.$ (3)

The Hawking temperature of non-rotating, electrically neutral (Schwarzschild)
PBH is

$\displaystyle{\cal T}_{\rm BH}=\frac{M^{2}_{\rm Pl}}{8\pi\cal{M}_{\rm
BH}}\simeq 5.92\times 10^{36}\bigg{(}\frac{\rm 1\;GeV}{\cal{M}_{\rm
BH}}\bigg{)}\rm GeV,$ (4)

Loss of PBH mass will occur due to the radiation of particles through Hawking
evaporation. The rate of evaporation is given by Hawking:1974sw :

$\displaystyle\frac{d\cal{M}_{\rm BH}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{g\;\Lambda\;M^{4}_{\rm Pl}}{30720\pi\;{\cal{M}}_{\rm
BH}^{2}}\simeq 2.75\times 10^{74}\bigg{(}\frac{\rm 1.GeV}{\cal{M}_{\rm
BH}}\bigg{)}^{2}\rm GeV^{2},$ (5)

where $g$ represents an effective number of the particle’s degrees of freedom.
In Minimal Supersymmetric Standard Model (MSSM) $g\simeq 316$ Keith2016  for
the full particle content of the model and $\Lambda$ is the relevant blackhole
greybody factor $\approx 3.8$ Hooper:2021 .

Using Eq. (5) one obtains the time scale for full evaporation of PBH as

$\displaystyle\tau_{\rm evap}=\frac{10240\pi\;M_{\rm
BH}^{3}}{{g\;\Lambda}M_{\rm Pl}^{4}}\approx 1.20\times
10^{-75}\bigg{(}\frac{M_{BH}}{\rm 1.GeV}\bigg{)}^{3}\rm GeV^{-1}.$ (6)

During the expansion of the universe as PBH evolves as matter so
$\rho_{BH}\propto a^{-3}$. Hence the ratio of the energy density of PBH to the
energy density of radiation $\rho_{BH}/\rho_{rad}\propto a$ which grows with
the expansion of the universe. Morris ; Krnjaic:2019 ; March-Russell:2018 .
Due to that, the energy density of the universe could be dominated by black
holes at a certain epoch before their complete evaporation. Even for higher
PBH mass of the order of $10^{29}$ GeV from Eq.(6) it follows that $\tau_{\rm
evap}\sim 10^{-12}$ sec. So considering instantaneous evaporation, one may
write the energy density of PBH as

$\displaystyle\rho_{\rm BH}=\frac{3M_{\rm Pl}^{2}\;H(\tau_{\rm
evap})^{2}}{8\pi}=\frac{M_{\rm Pl}^{2}}{6\pi\tau_{\rm
evap}^{2}}=\bigg{(}\frac{g^{2}\;\Lambda^{2}\;M_{\rm
Pl}^{10}}{629145600\pi^{3}\;{M_{\rm BH}}^{6}}\bigg{)}\rm GeV^{4}.$ (7)

in which the first two right-hand side expressions follow from the Friedman
equation and the last one follows after using Eqs. (4) and (6). Due to the
evaporation of black holes, there will be reheating of the universe to a
temperature, $T_{\rm evap}$. As the particle formation due to PBH evaporation
is almost instantaneous, all black hole energy density is transformed to
radiation and one may equate $\rho_{\rm BH}$ to radiation energy density as

$\displaystyle\rho_{\rm BH}\approx\bigg{(}\frac{g\pi^{2}}{30}T_{\rm
evap}^{4}\bigg{)}\rm GeV^{4}.$ (8)

Considering Eqs. (7) and (8) one may write:

$\displaystyle T_{evap}=\frac{{g^{1/4}}{\Lambda^{1/2}}M_{\rm
Pl}^{5/2}}{67.67\pi^{5/4}M_{\rm BH}^{3/2}}\approx 1.51\times
10^{46}\bigg{(}\frac{\rm 1.GeV}{M_{\rm BH}}\bigg{)}^{3/2}\rm GeV$ (9)

The temperature $T_{evap}$ is the reheating temperature after the evaporation
of black holes and is not related to the reheating after inflation in our
discussion. This temperature is the temperature of the universe just after
evaporation.

The black hole should have evaporated before nucleosynthesis and the universe
is required to be radiation-dominated at nucleosynthesis for reproducing the
success of Standard Big-bag cosmology Baumann:2007yr . So

$\displaystyle T_{\mbox{evap}}>T_{\mbox{BBN}}$ (10)

Different nuclei may be formed when the temperature of the universe is in the
range 1-30 MeV. However, actual synthesis takes place at around 0.1 MeV.
Considering $T_{\rm BBN}\sim 1$ MeV and using inequality (10) in Eq. (9), one
obtains the upper bound on primordial black hole mass as

$\displaystyle M_{\rm BH}\lesssim 2.8\times 10^{32}\mbox{GeV}$ (11)

for PBH producing heavy particle through evaporation. However, it could be
possible for PBH to have mass much higher than $10^{32}$ GeV, for which the
evaporation rate of PBH is too slow and their lifetime is nearer to the age of
the universe, and due to that BBN will not be disturbed by the presence of
such PBH. In this work we are not interested in such PBH with a very low
evaporation rate.

##  III Heavy particles production from PBH and leptonic asymmetry

From Eq. (7) the number density of the PBH is given as

$\displaystyle n_{\rm BH}=\frac{\rho_{\rm BH}}{{M_{\rm
BH}}}=\bigg{(}\frac{g^{2}\Lambda^{2}M_{\rm Pl}^{10}}{629145600\pi^{3}{M_{\rm
BH}}^{7}}\bigg{)}\rm GeV^{3}.$ (12)

One may write the entropy density of the universe at the end of evaporation
as:

$\displaystyle s=\bigg{(}\frac{2}{45}\pi^{2}gT_{evap}^{3}\bigg{)}\rm GeV^{3}.\
$ (13)

Particles of different masses will be produced through Hawking evaporation of
PBHs. However, we are interested in the production of scalar $X$ particles
(the sneutrinos which are super-partners of heavy right-handed neutrinos in
MSSM in our work) which could be radiated through evaporation. It will be
demonstrated that leptonic asymmetry may be formed from the decay of such
particles. Taking into account the mass of $X$ particle $M_{X}<T_{BH}$ and
integrating over the Bose-Einstein distribution, the number of $X$ particles
produced by a single black hole evaporates

$\displaystyle\rm N_{<X}=\frac{4\pi\rm{f_{X}}{\rm M_{\rm BH}}^{2}}{3M_{\rm
Pl}^{2}}$ (14)

where $f_{X}$ is given by

$\displaystyle f_{X}=g_{X}/g$ (15)

and $g_{X}\sim 1.82$ MacGibbon:1990 ; MacGibbon:1991  for particle X. The
lepton asymmetry parameter is defined as

$\displaystyle Y_{\Delta{L}}=\frac{n_{L}-{n}_{\bar{L}}}{s}$ (16)

where $n_{L}$ and ${n}_{\bar{L}}$ is the number density of leptons and anti-
leptons respectively of the universe. $\eta_{X}$ is $CP$ asymmetry parameter
defined as

$\eta_{X}=\sum_{f}L_{f}\frac{\Gamma_{X}({{X}}\rightarrow
f)-\Gamma_{X}(\tilde{{X}}\rightarrow\bar{f})}{\Gamma_{X}},$ (17)

where $L_{f}$ is a change in lepton number produced through the lepton number
violating decay modes of $X$ particles with $f$ as final states and
$\Gamma_{X}$ is the decay width of particle $X$. The sum runs over all lepton
numbers violating final states. After using Eqs.(12), (13) ,(14) and (17) in
Eq. (16), the lepton asymmetry $Y_{\Delta{L}}$ can be written as,

$\displaystyle Y_{\Delta{L}}$ $\displaystyle=$ $\displaystyle\frac{n_{\rm
BH}}{s}\,\eta_{X}N_{<X}=\frac{\eta_{X}f_{X}{g}^{1/4}}{32\sqrt{2}\sqrt[4]{5\pi}}\left(\frac{GM_{\rm
Pl}}{M_{BH}}\right)^{1/2}\approx 0.01f_{X}{g}^{1/4}\eta_{X}\left(\frac{GM_{\rm
Pl}}{M_{BH}}\right)^{1/2}$ (18)

For $T_{BH}\lesssim M_{X}$, the number density of $X$ which could be produced,
is

$\displaystyle\rm N_{>X}=\frac{\rm{f_{X}}{\rm M_{\rm Pl}}^{2}}{48\pi{\rm
M_{X}}^{2}}\;.$ (19)

The lepton asymmetry generated through lepton number violating decay modes is
given by

$\displaystyle Y_{\Delta{L}}=\frac{n_{\rm
BH}}{s}\,\eta_{X}N_{>X}\approx{1.7\times
10^{-5}\eta_{X}f_{X}g^{1/4}}\left(\frac{GM_{\rm
Pl}^{9}}{M_{BH}^{5}M_{X}^{4}}\right)^{1/2}$ (20)

The generation of leptonic asymmetry due to black hole evaporation producing
heavy particles has some important features as mentioned in the introduction
and corresponds to four different cases. The $T_{evap}$ as shown in Eq. (9)
could be less than or greater than $M_{X}$ having two possibilities. Also, the
number density of the $X$ particle produced has two possibilities as shown in
Eq. (14) and (19). Combining all these, there are following four different
cases: (1) $T_{evap}\gtrsim M_{X}$ and $T_{BH}>M_{X}$, (2) $T_{evap}<M_{X}$
and $T_{BH}>M_{X}$ (3) $T_{evap}<M_{X}$ and $T_{BH}\lesssim M_{X}$ (4)
$T_{evap}\gtrsim M_{X}$ and $T_{BH}\lesssim M_{X}$. After PBH evaporation, for
$T_{evap}\gtrsim M_{X}$, the temperature of the universe is higher than the
mass of the decaying particle $X$. So there is the possibility of the decaying
particle $X$ to be in thermal equilibrium. In such cases with the expansion of
the universe corresponding temperature of the universe will be lower and one
is required to check in such cases whether the out-of-thermal equilibrium
condition will be satisfied to get the leptonic asymmetry. So for case 1 and
4, the out of equilibrium condition

$\displaystyle\Gamma_{X}\lesssim H(T=M_{X}),$ (21)

is to be satisfied, where $H$ is the Hubble’s constant and $T$ is the
temperature of the universe.

##  IV CP asymmetry $\eta_{X}$ through sneutrino decays in MSSM

The $B$ violating couplings in a model like Grand Unified Theories and other
models are expected to be very small to avoid proton decay with a lifetime
lesser than the age of the universe. The lepton number violating couplings may
not be required to be that small. For leptonic asymmetry, we will consider the
lepton number violating decay modes of sneutrinos in the MSSM in the presence
of heavy RHNs in the model. Sneutrinos are super-partners of heavy RHN fields.
In this section, we discuss the generation of $CP$ asymmetry parameter
$\eta_{\tilde{N}_{\pm}}$ due to the lepton number violating decay of
sneutrinos.

Apart from the standard model fields and their corresponding super-partner
fields in MSSM, three heavy RHNs fields are required to implement the Type-I
seesaw mechanism to obtain three light masses for active neutrinos. However,
for the generation of leptonic asymmetry, the decay of the lightest among the
three RHNs and its super-partner field are in general, important111However, to
satisfy neutrino oscillation data, two mass square differences and three
mixing angles and a $\rm CP$ violating phase in PMNS mixing matrix are
required. To obtain these, three heavy RHNs are to be considered in Type-I
seesaw mechanism. The light neutrino masses will depend on the other heavier
right-handed neutrinos. However, in using a constraint on the sum of neutrino
masses, we have assumed all three right-handed neutrinos to have almost quasi-
degenerate mass.. For discussion on leptogenesis, we consider only one heavy
RHN (which is lightest) and its superpartner sneutrino. The mass term and the
interaction terms related to RHN and sneutrino field, in the Lagrangian can be
written as

$\displaystyle-{\cal L}_{\widetilde{N}}$ $\displaystyle=$ $\displaystyle
M^{2}\widetilde{N}^{*}\tilde{N}+\left(M^{*}Y_{\alpha}\widetilde{N}^{*}\widetilde{\ell}_{\alpha}H_{u}+Y_{\alpha}\overline{\widetilde{H}_{u}^{c}}P_{L}\ell_{\alpha}\widetilde{N}+{\rm
H.c.}\right),$ (22)

where $P_{L,R}=\frac{1}{2}\left(1\mp\gamma_{5}\right)$. Here $\tilde{N^{c}}$ ,
$\tilde{\ell}_{\alpha}$ and $\tilde{H}_{u}$ represent chiral superfields of
RHNs, the lepton doublet, and the up-type Higgs doublet, and $\alpha$ is the
lepton flavor indices. The soft supersymmetry breaking term is written as

$\displaystyle-{\cal L}_{{\rm S}}$ $\displaystyle=$
$\displaystyle\widetilde{M}^{2}\widetilde{N}^{*}\tilde{N}+\left(\frac{1}{2}BM\widetilde{N}\widetilde{N}+A_{\alpha}\widetilde{N}\widetilde{\ell}_{\alpha}H_{u}+{\rm
H.c.}\right).$ (23)

For the sake of simplicity, later on, we are considering that $Y_{\alpha}=Y$
and $A_{\alpha}=A$ and ignore the $\alpha$ index.

If the complex phases in the couplings $A$, $Y$, and $B$ are absorbed in the
fields, then it is found that there will remain a complex phase and without
losing any generality, we may consider that the phase remains in the tri-
linear coupling $A$. One may treat $Y$ as real.

The bilinear $B$ term in Eq. (23) induces mixing in $\tilde{N}$ and
$\tilde{N}^{*}$ to form mass eigenstates $\tilde{N}_{+}$ and $\tilde{N}_{-}$

$\displaystyle\begin{bmatrix}\widetilde{N}_{+}\\\
\widetilde{N}_{-}\end{bmatrix}=\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
-\frac{\iota}{\sqrt{2}}&\frac{\iota}{\sqrt{2}}\end{bmatrix}\begin{bmatrix}\tilde{N}\\\
\tilde{N}^{\star}\end{bmatrix}$ (24)

and the corresponding masses are given by

$\displaystyle M_{\pm}^{2}=M^{2}+\widetilde{M}^{2}\pm BM.$ (25)

with the condition $B<M+\widetilde{M}^{2}/M$ for real values.

In the mass eigenstate basis, the Lagrangian is written as

$\displaystyle-{\cal L}_{\widetilde{N}}-{\cal L}_{{\rm S}}$ $\displaystyle=$
$\displaystyle
M_{+}^{2}\widetilde{N}_{+}^{*}\widetilde{N}_{+}+M_{-}^{2}\widetilde{N}_{-}^{*}\widetilde{N}_{-}$
(26)
$\displaystyle+\frac{1}{\sqrt{2}}\left\\{\widetilde{N}_{+}\left[Y\overline{\widetilde{H}_{u}^{c}}P_{L}\ell+\left(A+MY\right)\widetilde{\ell}H_{u}\right]\right.$
$\displaystyle\left.+i\widetilde{N}_{-}\left[Y\overline{\widetilde{H}_{u}^{c}}P_{L}\ell+\left(A-MY\right)\widetilde{\ell}H_{u}\right]+{\rm
H.c.}\right\\}.$

The total decay width of $\widetilde{N}_{\pm}$ for tree level diagram is given
by

$\displaystyle\Gamma_{\pm}\simeq\frac{M}{4\pi}\left[Y^{2}+\frac{|A|^{2}}{2M^{2}}\pm\frac{Y{\>\rm
Re}(A)}{M}\right],$ (27)

where for simplicity we have neglected the terms $\frac{\tilde{M}^{2}}{M^{2}}$
and $\frac{B}{M}$ assuming that $\tilde{M}<M$ and $B<<M$. We will set the
condition that $Y<1$ for the perturbative regime. The amount of $CP$ asymmetry
parameter $\eta_{\tilde{N}_{\pm}}$ for RHN’s decay with
$f=(\tilde{H_{u}}{l},{H_{u}}\tilde{l})$ is written as

Figure 1: One of the one loop self energy diagrams for
$\tilde{N_{\pm}}\rightarrow\tilde{l}H_{u}$ with arrows indicating flow of
lepton number

$\eta_{\tilde{N_{\pm}}}\equiv\frac{\Gamma_{\tilde{N_{\pm}}}({\tilde{N_{\pm}}}\rightarrow{f})-\Gamma_{N_{\pm}}(\tilde{{N_{\pm}}}\rightarrow\bar{f})}{\Gamma_{\tilde{N_{\pm}}}({\tilde{N_{\pm}}}\rightarrow{f})+\Gamma_{N_{\pm}}(\tilde{{N_{\pm}}}\rightarrow\bar{f})},$
(28)

as $\tilde{N_{\pm}}$ and its antiparticle are same. Non-zero $CP$ asymmetry
$\eta_{\pm}$ is obtained from the interference of tree-level diagrams and
self-energy one-loop diagrams (one loop vertex diagram give very small
contribution to $CP$ asymmetry and has been neglected Adhikari:2015ysa ). For
$\tilde{N_{\pm}}\rightarrow\tilde{H_{u}}l$, there are two self-energy diagrams
with $H_{u}$ and $\tilde{l}$ in the internal lines. For
$\tilde{N_{\pm}}\rightarrow H_{u}\tilde{l}$, there are four self-energy
diagrams, in two of those- there are fermions- $\tilde{H_{u}}$ and $l$ in the
internal line, and in other two of those, there are bosons- $H_{u}$ and
$\tilde{l}$ in the internal lines. For illustration, one of the last two cases
are shown in Fig. 1 with appropriate flow of lepton number. In three of these
self-energy diagrams the lepton number arrows in the internal line are in the
anticlockwise directions, for which non-zero $CP$ asymmetry can be obtained
Adhikari:2015ysa . We have discussed earlier that $A$ could be treated as
complex coupling and one can see from Eq.(28) that the coupling of
$\tilde{N_{+}}$ and $\tilde{N_{-}}$ with the slepton and Higgs are
proportional to $A+MY$ and $A-MY$ respectively with different $CP$ violating
phases in which $A$ may be considered as complex but $MY$ is real. As the one
loop diagram always contains both $\tilde{N_{+}}$ and $\tilde{N_{-}}$, the
$CP$ violating phase will remain even if $A_{\alpha}$ is treated as universal
$A\equiv A_{\alpha}$ parameter which has been considered in this work.
However, in earlier work on soft leptogenesis Adhikari:2015ysa ), $A$
parameter was not considered as universal. Considering the interference
between tree level amplitude and one loop amplitude (with non-zero imaginary
part of the loop integral) in the numerator for both $f$ and $\bar{f}$ final
states, one gets a non-zero numerator in Eq. (28) in the non-thermal case
(with $T=0$). One may write the non-thermal $CP$ asymmetry Adhikari:2015ysa
(after summing over all lepton flavors in the final states and also summing
over all lepton flavors in the internal lines) as follows

$\displaystyle\eta_{\tilde{N_{\pm}}}$ $\displaystyle=$
$\displaystyle\frac{9\>\rm Im(A)Y}{4\pi
D_{\pm}M}\left(\frac{|A|^{2}}{M}+\frac{\widetilde{M}^{2}Y^{2}}{M}\pm{BY^{2}}\right)\frac{4B}{4B^{2}+\Gamma_{\mp}^{2}}$
(29)

where

$\displaystyle D_{\pm}\equiv\left[Y^{2}+\frac{|A|^{2}}{M^{2}}\pm\frac{2Y{\rm
Re}(A)}{M}\right]+Y^{2}\left(1+\frac{\widetilde{M}^{2}}{M^{2}}\pm\frac{B}{M}\right).$
(30)

For supersymmetric particles $\widetilde{N}_{\pm}$ produced due to the
evaporation of a black hole, the leptonic asymmetry $\eta_{\tilde{N_{\pm}}}$
will be obtained from Eq. (29).

##  V Baryonic asymmetry and Constraints on soft supersymmetry breaking
parameters

Leptonic asymmetry will result in the formation of baryonic asymmetry of the
universe in the presence of shpeleron transition. Near the first order
electroweak phase transition of the universe, $B+L$ violating spheleron
transition becomes very small as the transition rate is Boltzman suppressed
and proportional to $e^{\frac{-E_{sph}}{T}}$ where $E_{sph}$ is the barrier
height (related to up and down type Higgs vacuum expectation values). However,
$B-L$ remains conserved. Any change in $\Delta{L}$ will be related to
$\Delta{B}$ and the leptonic asymmetry $Y_{\Delta L}$ in either Eq. (18) or
Eq. (20) will be converted to baryonic asymmetry $Y_{\Delta B}$ saphos  :

$\displaystyle Y_{\Delta
B}=\frac{n_{B}-n_{\overline{B}}}{s}=-\left(\frac{8N_{f}+4N_{H}}{22N_{f}+13N_{H}}\right)Y_{\Delta
L}$

where $N_{H}$ is the number of Higgs doublets and $N_{f}$ is the number of
lepton generations in the above equation. Considering $N_{H}=2$ and $N_{f}=3$
for MSSM, above relation is,

$\displaystyle Y_{\Delta B}=-\frac{8}{23}Y_{\Delta L}$ (31)

In obtaining baryonic asymmetry from leptonic asymmetry in the presence of
spheleron, the weakly first-order or second-order phase transition is
required. In MSSM this transition could happen at the temperature of the
universe $T\gtrsim 200$ GeV hambay2000 ; Rubakov1996 ; riotto ; moore1999 .
After black hole evaporation, $T_{evap}$ will correspond to the temperature
$T$ of the universe. So for conversion of leptonic asymmetry to baryonic
asymmetry through sphaleron transition, it is required that

$\displaystyle T_{evap}\gtrsim 200\>\rm GeV.$ (32)

Using Eq. (9) and condition (32) in the context of MSSM, one obtains the
condition on the mass of $M_{BH}$

$\displaystyle M_{BH}\lesssim 1.79\times 10^{29}\>\rm GeV$ (33)

for obtaining baryonic asymmetry due to leptonic asymmetry in the presence of
sphaleron. $M_{BH}$ could be greater than this as shown in Eq. (11) but that
will not be useful for successful leptogenesis in MSSM.

Also, we will consider the constraints on the Yukawa coupling and the heavy
RHN mass $M$ to produce appropriate light neutrino mass from the Type I seesaw
mechanism in the supersymmetric model. This constraint is written as

$\displaystyle\sum_{\alpha}\frac{Y_{\alpha}^{2}v_{u}^{2}\left(1+\rm
cot^{2}\beta\right)}{M}\approx\sum_{i}m_{\nu_{i}}\lesssim 1.18\times
10^{-10}\,{\rm GeV},$ (34)

where $v_{u}^{2}+v_{d}^{2}=\langle H_{u}\rangle$ $\simeq(174\;\rm GeV)^{2}$,
where $v_{u},v_{d}$ are up-type and down-type Higgs vacuum expectation values,
$M$ which is the lightest among the heavy RHN masses, has been considered as
the approximate quasi-degenerate mass scale of three heavy right-handed
neutrinos and $m_{\nu_{i}}$ are the three active light neutrino masses, sum of
which is $\lesssim 0.118\>\rm eV$ RoyChoudhury:2018gay , $\rm cot\beta$$\equiv
v_{d}/v_{u}$ and $\alpha$ is lepton flavor index. In the later numerical
analysis, it is assumed $\rm cot\beta=0.1$ in Eq.(34). Due to the lighter
neutral Higgs mass ( $125.35$ GeV) found at LHC, $\tan\beta>4$ is expected
Djouadi .

For cases I and IV, the out-of-equilibrium condition (using Eq. (27) in Eq.
(21) ) for the sneutrino decay is obtained as

$\displaystyle\frac{\left|\rm A\right|^{2}}{2\rm M^{2}}\pm\frac{\rm
Y\rm\>Re[A]}{\rm M}+\rm Y^{2}\lesssim H_{T\sim M}\lesssim 3.09\times
10^{-17}\left(\frac{M}{1GeV}\right)$ (35)

where the Hubble expansion rate $H$ is given by $H=1.66\sqrt{g}\;T^{2}/M_{P}$
with Planck mass $M_{P}=1.22\times 10^{19}$ GeV. Considering $\widetilde{M}$
and $B$ much smaller than $M$ in Eq. (25), we have considered $M_{\pm}$ of the
order of $M$ and hence $T\sim M$ has been considered in Eq. (35). This
condition will constrain parameters $A$ and $Y$ depending on $M$ values. This
out-of-equilibrium condition in Eq. (35) will be considered for cases 1 and 4
as mentioned earlier.

To find the suitable values of soft supersymmetry breaking parameters that
give appropriate leptonic and baryonic asymmetry, we have varied the
parameters $M$ and $M_{BH}$ related to the masses of sneutrinos and PBH
masses. Using Eqs. (3) and (33), $M_{BH}$ is varied in the following range:

$\displaystyle 10^{23}\;\;\mbox{GeV}\lesssim M_{BH}\lesssim
10^{29}\;\;\mbox{GeV}.$ (36)

Considering the lower mass bound on right-handed neutrinos about $100$ GeV
from experiments rhnmass  and to avoid gravitino problem, considering $M$
value preferably lower than $10^{9}$ GeV Davidson:2002qv , we have varied
right-handed neutrino mass $M$ in the range:

$\displaystyle 10^{2}\mbox{GeV}\lesssim M\lesssim 10^{9}\mbox{GeV}.$ (37)

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2: The above plots (a),(b),(c),(d) are for allowed regions of baryonic
asymmetry as case 1, 2 ,3 ,4, have mentioned earlier in the introduction
respectively

Depending on the conditions on $T_{BH}$ and $T_{evap}$, we have shown the
allowed region in $M-M_{BH}$ plane for cases I, II, III, and IV in Fig. 2,
(a), (b), (c) and (d) respectively. For case I, from Fig. 2 (a), it is seen
that for lower values of $M$ the allowed values of $M_{BH}$ could be possible
in the entire range of it. However, for higher values of $M$, the lower values
of $M_{BH}$ is allowed. For case II, from Fig. 2 (b) it is seen that for lower
values of $M$, only higher values of $M_{BH}$ are allowed. However, for higher
values of $M$ there could be higher values of $M_{BH}$ in the range of about
$10^{25}-10^{28}$ GeV. For case III, from Fig. 2 (c), one can see that only
higher values of $M$ and $M_{BH}$ are allowed. However, for case IV, from Fig.
2 (d) it is seen that there is no allowed region in $M-M_{BH}$ plane, and as
such, there will be no possibility of getting any baryonic asymmetry.

For $T_{BH}>{M_{\pm}}$ using Eqs. (18) and (29), we can write

$\displaystyle Y_{\Delta B_{\pm}}\approx\frac{7.8\times
10^{-3}f_{X}\>{g}^{1/4}\>\rm Im(A)\>Y}{\pi
D_{\pm}M}\left(\frac{|A|^{2}}{M}+\frac{\widetilde{M}^{2}Y^{2}}{M}\pm{BY^{2}}\right)\frac{4B}{4B^{2}+\Gamma_{\mp}^{2}}\left(\frac{GM_{P}}{M_{BH}}\right)^{1/2}$
(38)

where $Y_{\Delta B_{\pm}}$ correspond to the baryonic asymmetry due to
$\tilde{N_{+}}$ decay and $\tilde{N_{-}}$ decay respectively. For
$T_{BH}\lesssim{M_{\pm}}$ using Eqs. (20) and (29), baryonic asymmetry is
written as

$\displaystyle Y_{\Delta B_{\pm}}\approx\frac{1.3\times
10^{-5}f_{X}\>g^{1/4}\>\rm Im(A)\>Y}{\pi
D_{\pm}M}\left(\frac{|A|^{2}}{M}+\frac{\widetilde{M}^{2}Y^{2}}{M}\pm{BY^{2}}\right)\frac{4B}{4B^{2}+\Gamma_{\mp}^{2}}\left(\frac{GM_{P}^{9}}{M_{BH}^{5}M_{\pm}^{4}}\right)^{1/2}.$
(39)

Masses of $\widetilde{N}_{+}$ and $\widetilde{N}_{-}$ do not differ much and
$M_{\pm}\sim M$. In the figures below, we have considered the decay of
$\widetilde{N}_{+}$ only. For simplicity, the mass parameter $\widetilde{M}$
has been assumed to be zero in evaluating baryonic asymmetry in the latter
part.

The basic difference in the expression of baryonic asymmetry in the case of
conventional soft leptogenesis Adhikari:2015ysa  versus the soft leptogenesis
through PBH evaporation is that the asymmetry in the latter case also depends
on $M_{BH}$ apart from depending on $M,A,B$, and $Y$. But in conventional soft
leptogenesis, there is no dependence on $M_{BH}$. Furthermore, although for
case I, the out-of-equilibrium condition is to be satisfied like conventional
soft leptogenesis, but for cases II and III in this work, the out-of-
equilibrium condition is not required to be satisfied. We will analyze
numerically below through various plots the implications of these basic
differences.

|
---|---
(a) | (b)
Figure 3: Plot of $M_{BH}$ versus $M$ for case 1 is Fig. 3(a) and for case 2 is Fig. 3(b). The different colors in the bar legend correspond to different $\log_{10}(\Gamma_{+}/1\;\rm GeV)$ values as shown. |
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)
Figure 4:  (a), (c), and (e) are shown for case 1. (b), (d), and (e) are shown
for case 2. The different colors in the bar legend correspond to different
$\log_{10}(\Gamma_{+}/1\;\rm GeV)$ values as shown. The black dots correspond
to resonance with $B\sim\Gamma_{+}$ in obtaining asymmetry.

We have shown again $M$ versus $M_{BH}$ plots in Fig. 3 (a),(b) for cases 1
and 2 respectively after taking into account the requirement of baryonic
asymmetry using Eq.(38). The different colors in the bar legend correspond to
different $\log_{10}(\Gamma_{+}/1\;\rm GeV)$ values as shown. For the allowed
regions $B\sim\Gamma$ which means these are near resonance. We will discuss
this later in the context of Fig. 4 (c) for case I and Fig. 4(d) for case II.
In Fig. 3(a) for low sneutrino mass in the few hundred GeV full allowed range
of $M_{BH}$ as in Eq. (36) is possible to obtain the required baryonic
asymmetry. However, in Fig. 3(b) for low sneutrino mass in the TeV scale, only
very high $M_{BH}\sim 10^{29}$ GeV is possible. The allowed regions are
similar to Fig. 2(a), (b); the requirement of baryonic asymmetry has not
changed much in the allowed region in Fig. 3(a), (b).

Plots in Fig. 4 (a), (c), (e) correspond to Case I and Fig. 4 (b), (d) (f)
correspond to Case II. In Fig. 4(a), the lowest value of $|A|$ is about
$\approx 10^{-4}$ GeV for $M\sim 500$ GeV. To avoid the gravitino problem, if
we consider $M\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10^{9}$ GeV then the value of $|A|$ is
$\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10^{6}$ GeV. In Fig. 4(b), the lowest value of $|A|$
is about $\approx 10^{-5}$ GeV for $M\sim 500$ GeV. From the bar legend it is
seen that for higher values of $M\sim 10^{9}$ GeV the decay width, $\Gamma$ is
about 1 GeV. In Fig. 4 (c) for case I and also in Fig. 4 (d) for case II, from
the bar legend one can see that for the almost same color the $\Gamma$ values
are almost the same, but if $B$ values are changing from the $\Gamma$ values
there, then the asymmetry is obtained away from resonance also as for
resonance $B\sim\Gamma$. So the asymmetry is obtained for both near resonance
and away from resonances. The black dots are almost at resonance and the
corresponding $\Gamma$ values are not shown in these and earlier plots. But
they are almost equal to $B$ values as shown in these two plots. In Fig. 4 (e)
and (f), corresponding to cases 1 and 2 respectively, we have shown the
variation of $B$ versus $Y$. In Fig. 4 (e) particularly for $Y$ about $2\times
10^{-7}$ GeV, $B$ could vary in the range of $10^{-14}\;$ to $\;1$ GeV.
However in Fig. 4 (f) for $Y$ about, $8\times 10^{-6}$ $B$ could vary in the
range $10^{-8}$ to $10\;$ GeV.

As discussed earlier in case 1 corresponding to Fig. 4 (a), (c), and (e) the
out-of-equilibrium condition has been considered and for case 2 corresponding
to Fig. 4 (b), (d) and (f) this condition has not been considered. However in
both cases 1 and 2, the same expression of baryonic asymmetry as in Eq. (38)
has been considered. For both cases 1 and 2 the $|A|$ versus $M$ plots are
almost similar. However, when one looks at the corresponding ranges of $B$
parameter then there are some differences corresponding to case 1 and 2. As
for example, for $A\sim 10^{-2}$ GeV in case 1, the range of $B$ is about
$10^{-9}$ to $10^{-7}$ GeV whereas in case 2 this range of $B$ is about
$10^{-12}$ to $10^{-8}$ GeV. For say, $A\sim 10^{4}$ GeV in case 1, the range
of $B$ is about $10^{-4}$ to $10$ GeV whereas in case 2 this range of $B$ is
about $10^{-5}$ to $1$ GeV. The differences in the ranges of $B$ in cases 1
and 2, are coming because of the scope of variation of $M_{BH}$ for certain
$M$ values, are different in two different cases as can be seen in Fig. 3 (a)
and (b) respectively. One may note in case 1 from Fig. 3 (a), $M$ value could
be as low as about $100$ GeV, and in case 2 as follows from Fig. 3(b), $M$
value could be as low as about $100$ GeV. Although for both the cases there
could be such low masses of $M$, however, for both the cases corresponding
soft SUSY breaking parameters $|A|$ and $B$ are found to be very small and may
not be suitable for spontaneous SUSY breaking.

|
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)
Figure 5: (a) shows variation of primordial black hole mass $M_{BH}$ with
particle mass $M$ for case 3. In (b), the correlation between Yukawa coupling
$Y$ and soft breaking parameter $B$ is shown.

We have shown in Fig. 5 various plots corresponding to case 3. Unlike the
earlier two cases, for case 3 to evaluate baryonic asymmetry Eq. (39) has been
used. Like case 2, in this case, the out-of-equilibrium condition is not
required. From Fig. 5(a) the lowest possible $M$ value is about $7\times
10^{7}$ GeV, which could produce observed baryonic asymmetry. The $|A|$ value
could be about $10^{6}$ or $10^{7}$ GeV for $M$ value in the wide range above
$7\times 10^{7}$ GeV. In Fig. 5(d) corresponding to $|A|$ value of about
$10^{6}$ GeV, $B$ could vary in the range of about $2$ GeV to $300$ GeV for
different $M$ values. Also, in Fig. 5(d), corresponding to $|A|$ value about
$10^{7}$ GeV, $B$ could vary in the range of about $100$ GeV to $5\times
10^{4}$ GeV for different $M$ values. In a conventional soft leptogenesis
scenario Adhikari:2015ysa  where sneutrinos are not produced from PBH, there
also one finds $|A|$ and $B$ parameters to be relatively larger for higher $M$
values around $10^{7}$ GeV. However, in soft leptogenesis with decaying
sneutrinos produced by PBH evaporation, $|A|$ and $B$ parameters are found to
be much larger. This happens because of the different expression of baryonic
asymmetry in Eq.(39) in comparison to conventional baryonic asymmetry
expression for soft leptogenesis. So PBH-induced leptogenesis could be more
suitable for high-scale spontaneous SUSY breaking. In Fig. 5 (e) and (f), we
have shown the variation of $B$ with $M$ and $M_{BH}$ respectively. In Fig. 5
(f) it seems that $B$ could vary more for the lower value of $M_{BH}$ whereas,
in Fig. 5 (e) this happens for higher value of $M$.

The soft supersymmetry breaking parameters can contribute to charged lepton
flavor violating interactions (CLFV), dipole moments of leptons, etc. It is
found that in our work, the experimental upper bound on the branching ratio
from MEG collaboration Adam:2013mnn ; Adam:2011ch

$\displaystyle BR(\mu\rightarrow e\gamma)<5.7\times 10^{-13}\;.$ (40)

This constrains the trilinear soft supersymmetry breaking parameter $|A|$ as
discussed below. The above process is induced by one-loop diagrams through the
exchange of gauginos (neutralinos, charginos) and sleptons (charged sleptons,
sneutrinos). The main contribution comes from charginos and sneutrinos in the
internal lines of the one-loop diagram. The off-diagonal elements of doublet
slepton mass square $m_{\tilde{l}}^{2}$ can induce the CFLV process as
considered in Eq. (41) below. It is normally assumed as mSUGRA

|
---|---
(a) | (b)
|
(c) |
Figure 6: Allowed region for $m_{SUSY}$ versus $|A|$, (both in GeV units) is
shown in the shaded area obtained from experimental constraint on branching
ratio of $\mu\rightarrow e\gamma$.

boundary conditions that off-diagonal $m_{\tilde{l}}^{2}$ is zero at the GUT
scale. It evolves from GUT scale $M_{\rm GUT}$ to the RHN mass scale $M$
through Renormalization Group Equations and in the leading log approximation,
may be written (considering the contribution due to soft breaking $|A|$ term)
as Hisano:1995cp

$\displaystyle\left(m_{\tilde{l}}^{2}\right)_{\mu
e}\approx-\frac{1}{8\pi^{2}}A_{\mu}^{*}A_{e}\log\left(\frac{M_{\rm
GUT}}{M}\right)\approx-\frac{1}{8\pi^{2}}{|A|}^{2}\log\left(\frac{M_{\rm
GUT}}{M}\right)\;.$ (41)

The right hand side expression follows as we have assumed $A_{\alpha}\equiv
A$. Due to the lack of specific understanding of the hierarchical pattern of
other heavier right-handed neutrinos, their similar contributions have been
ignored here. The branching ratio for $\mu\rightarrow e\gamma$ in terms of
above $\left(m_{\tilde{l}}^{2}\right)_{\mu e}$ can be written as Hirsch:2012ti

$\displaystyle BR(\mu\rightarrow
e\gamma)\approx\frac{\alpha^{3}}{G_{F}^{2}}\frac{{|\left(m_{\tilde{l}}^{2}\right)_{\mu
e}|}^{2}}{{m_{SUSY}}^{8}}\tan^{2}\beta$ (42)

where the fine structure constant at $m_{W}$ scale is $\alpha\sim 1/128$ and
the Fermi four point coupling $G_{F}\sim 1.166\times 10^{-5}$ GeV-2 and
$m_{SUSY}$ is the typical mass scale of the supersymmetric particles. From
Fig. 4 (a) for case 1, Fig. 4 (b) for case 2 and Fig. 5 (c) for case 3, one
may note that $|A|$ is related with $M$ values to get appropriate baryon
asymmetry. Using their different sets of values in Eq. (41) and considering
$\cot\beta=0.1$ in Eq. (42) as mentioned after Eq. (34) and considering
$M_{GUT}\sim 10^{16}$ GeV in Eq. (41), one can find the allowed region of
$m_{SUSY}$ versus $|A|$ for cases 1,2, and 3 as shown in Fig. 6 (a),(b), and
(c) respectively subject to the experimental constraint on the branching ratio
as shown in Eq. (40). In Fig. 6, due to the non-observation of any
supersymmetric particles at LHC, we have assumed that the supersymmetry
breaking scale is not below the TeV scale.

In Fig. 6 (a) for case 1, it is found that $|A|$ above about $1$ TeV the
lowest possible SUSY breaking scale is of the order of $|A|$ value. In Fig. 6
(b) for case 2, it is found that $|A|$ above about $100$ GeV the lowest
possible SUSY breaking scale is about five times higher than $|A|$ value. In
Fig. 6 (c) for case 3, it is found that the lowest possible SUSY breaking
scale could be about one order less than the corresponding $|A|$ values.

MEG II experiment could further improve the bound on the branching ratio of
$\mu\rightarrow e\gamma$ to about $6\times 10^{-14}$ in the near future MEGII
. In that case, the lower bounds of $m_{SUSY}$ which are shown in Fig. 6, may
be further higher depending on $|A|$ values.

##  VI Conclusion

In this work, based on the conditions on $T_{evap}$ and $T_{BH}$, we have
discussed four different cases for the generation of baryonic asymmetry. Those
are : (1) $T_{evap}\gtrsim M_{X}$ and $T_{BH}>M_{X}$, (2) $T_{evap}<M_{X}$ and
$T_{BH}>M_{X}$ (3) $T_{evap}<M_{X}$ and $T_{BH}\lesssim M_{X}$ (4)
$T_{evap}\gtrsim M_{X}$ and $T_{BH}\lesssim M_{X}$. There are no allowed
regions in $M-M_{BH}$ plane for case IV. However, for the other three cases,
there are allowed regions that are suitable for the consideration of baryonic
asymmetry generation in the early universe. In the first two cases, there is a
possibility of lower values of heavy right-handed neutrino mass as well as
sneutrino mass around TeV scale depending on some $M_{BH}$ mass. However, for
case III, the lower values could be around $5\times 10^{7}$ GeV scale
depending on $M_{BH}$ mass.

In earlier works on baryogenesis with PBH evaporation Nolan:2022 ;
Tomohiro:2104 ; Hamada;2017 ; Hooper:2021 ; Luca:2021 ; Ghosal:2021 ;
Pinetti:2022 ; Stefano:2019 ; Fujita:2014hha ; Gehrman:2022imk ; Gonzalez:2021
; Nicolás:2022 ; Borah:2021 , it was discussed that the condition for the
baryon or lepton number violating decay of the heavy particles to be out of
thermal equilibrium is not required. In this work, we have mentioned that
sometimes such conditions may be required to be satisfied. This happens for
case I. However, for the other two cases II and III, such a condition is not
required to be satisfied.

The expression of baryonic asymmetry in the case of PBH evaporation, depend
also on $M_{BH}$ mass. This does not happen in a conventional scenario of
baryonic asymmetry Adhikari:2015ysa  where PBH evaporation is not considered.
Furthermore, two expressions of baryonic asymmetry are there as shown in Eqs.
(38) and (39) depending on whether $T_{BH}>M_{\pm}$ or $T_{BH}\lesssim
M_{\pm}$ where $M_{\pm}$ is the mass of the decaying particle. In that way,
for cases I and II, the expression in (38) has been used, and for case III,
Eq. (39) has been used for baryonic asymmetry. Because of these two
differences - 1) the out-of-equilibrium condition 2) different expression of
baryonic asymmetry, for three different cases, there will be some different
features in the parameters present in the expression of $CP$ asymmetry in the
decays of particle and its conjugate decay mode as well as in the decaying
particle’s mass and $M_{BH}$. To study these features, we have considered the
soft leptogenesis scenario in the Minimal Supersymmetric Standard Model (MSSM)
with PBH evaporation for the generation of baryonic asymmetry. Different
possible values of $M$, $M_{BH}$, Yukawa coupling $Y$ and soft SUSY breaking
parameters $A$, $B$, and the Yukawa coupling $Y$ have been discussed in detail
in different Figs. 2-5.

Although in cases 1 and 2 the lower values of $M$ of about 1 TeV could be
possible for the generation of baryonic asymmetry, but the corresponding soft
SUSY breaking parameters are found to be much lesser than 1 GeV. So these
parameters are not supposed to play any important role in Supersymmetry
breaking. This happens in the case of conventional soft leptogenesis scenarios
also.

The most interesting thing that appears in our work, is in case the 3, for
which out-of-equilibrium condition is not required as well as the baryonic
asymmetry expression is different from that considered case 1 and 2. In Fig.
5, it is found that in case 3 the $|A|$ parametrer corresponding to $M$ value
of the order of $5\times 10^{7}$ GeV, could be about $10^{6}$ GeV and $B$
parameter about $10^{4}$ to $10^{5}$ GeV. This shows that scenario could be
appropriate for the consideration of supersymmetry breaking. If we compare
case III, with the conventional soft leptogenesis scenario, then it is found
that for $M$ value of the order of $5\times 10^{7}$ GeV, $|A|$ and $B$
parameters are lesser than or of the order of TeV scale. In that sense, PBH-
induced soft leptogenesis as has been discussed in this work, is found to be
more suitable for supersymmetry breaking in MSSM at higher energy scale.

Considering the experimental constraint on $|A|$ and $M$ values due to non-
observation of lepton number violating $\mu\rightarrow e\gamma$ decay together
with $M$ versus $|A|$ plots for different cases for the generation of baryonic
asymmetry, we have discussed some features of $|A|$ values and the SUSY
breaking scale in Fig. 6. Particularly for case III, from Fig. 6(c), it is
found that the lowest possible SUSY breaking scale could be about one order
less than the corresponding $|A|$ values.

###### Acknowledgements.

SK thanks the Council of Scientific and Industrial Research (CSIR), India for
financial support through Senior Research Fellowship (Grant No.
09/466(0209)/2018-EMR-I). SK also thanks Imtiyaz Ahmed Bhat and Yogesh for
their helpful discussions.

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