# Asymmetric Self-interacting Dark Matter via Dirac Leptogenesis

Manoranjan Dutta<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Hyderabad  Nimmala Narendra<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research Laboratory, Ahmedabad -
380009, India  Narendra Sahu<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Hyderabad  Sujay Shil<EMAIL_ADDRESS>IIT
Gandhinagar, Palaj Campus, Gujarat 382355, India  Institute of Physics,
Sachivalaya Marg, Bhubaneswar, Pin-751005, Odisha  Homi Bhabha National
Institute, BARC Training School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

The nature of neutrinos, whether Dirac or Majorana, is hitherto not known.
Assuming that the neutrinos are Dirac, which needs $B-L$ to be an exact
symmetry, we make an attempt to explain the observed proportionality between
the relic densities of dark matter (DM) and baryonic matter in the present
Universe ${\it i.e.,}\,\,\Omega_{\rm DM}\approx 5\,\Omega_{\rm B}$. We extend
the Standard Model (SM) by introducing heavy scalar doublets $X_{i},i=1,2$ and
$\eta$, two singlet scalars $\Phi$ and $\Phi^{\prime}$, a vector-like Dirac
fermion $\chi$ representing the DM and three right-handed neutrinos
$\nu_{R_{i}},i=1,2,3$. Assuming $B-L$ is an exact symmetry of the early
Universe, the CP-violating out-of-equilibrium decay of heavy scalar doublets:
$X_{i},i=1,2$ to the SM lepton doublet $L$ and the right-handed neutrino
$\nu_{R}$, generate equal and opposite $B-L$ asymmetry among left ($\nu_{L}$)
and right ($\nu_{R}$)-handed neutrinos. We ensure that $\nu_{L}-\nu_{R}$
equilibration does not occur until below the electroweak (EW) phase transition
during which a part of the lepton asymmetry gets converted to dark matter
asymmetry through a dimension eight operator, which conserves $B-L$ symmetry
and remains in thermal equilibrium above sphaleron decoupling temperature. A
part of the remaining $B-L$ asymmetry then gets converted to a net B-asymmetry
through EW-sphalerons which are active at a temperature above 100 GeV. To
alleviate the small-scale anomalies of $\Lambda$CDM, we assume the DM ($\chi$)
to be self-interacting via a light mediator $\Phi$, which not only depletes
the symmetric component of the DM, but also paves a way to detect the DM at
terrestrial laboratories through $\Phi-H$ mixing, where $H$ is the SM Higgs
doublet.

##  I Introduction

It is presumed that the early Universe has gone through a period of
exponential expansion called inflation to solve the cosmological problems. At
the end of inflation, the Universe is reheated to give rise to a thermal bath
with reheating temperature $T_{R}\gtrsim 4$MeV in order to facilitate the big-
bang nucleosynthesis (BBN). It is expected that the different components of
the present Universe, such as dark matter, dark energy and baryonic matter
must have been cooked in a post-inflationary thermal bath. At present, the
visible component (baryonic matter) of the Universe is best understood in
terms of the standard model (SM) of particle physics, which is based on the
gauge group $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. However, the SM fails
to explain many other aspects of the observed Universe, such as baryon
asymmetry, dark matter, dark energy, non-zero but small neutrino mass etc.

Within the framework of the SM, neutrinos are exactly massless. However, the
solar and atmospheric neutrino oscillation experiments hint towards the non-
zero masses and mixings of light neutrinos. In fact, this has been further
confirmed by relatively recent oscillation experiments like T2K Abe:2011sj ;
Abe:2013hdq , Double Chooz Abe:2011fz ; DoubleChooz:2019qbj , Daya Bay
An:2012eh ; An:2017osx ; Adey:2018zwh , Reno Ahn:2012nd  and MINOS
Adamson:2013whj ; Adamson:2014vgd . For a recent global fit of neutrino
oscillation experiment, we refer to global_fit_data . The neutrino masses are
also further constrained by cosmology. The cosmic microwave background
radiation data give an upper bound on the sum of light neutrino masses to be
$\sum_{i}|m_{i}|<0.12$ eV Vagnozzi:2017ovm . Thus the data from various
sources imply that neutrinos have mass. However, the nature of neutrino mass,
whether Dirac or Majorana, is not confirmed yet. If the neutrinos are
Majorana, which implies lepton number is violated by two units, then the
seesaw mechanisms: type-I Minkowski:1977sc ; Yanagida:1981xy ; GellMann:1980vs
; Mohapatra:1979ia , type-II type_II_seesaw_1 ; type_II_seesaw_2 ;
type_II_seesaw_3 ; type_II_seesaw_4 ; type_II_seesaw_5 ; type_II_seesaw_6 ;
type_II_seesaw_7 , type-III type_III_seesaw  or their variants
ray_volkas_review  are the best theoretical candidates to get their sub-eV
masses. It is important to note that all these seesaw mechanisms and their
variants introduce additional heavy particles to the SM. After integrating out
the heavy degrees of freedom we get the effective dimension five operator
$LLHH/\Lambda$, where $L,H$ are lepton and Higgs doublets of the SM and
$\Lambda$ is the mass scale of heavy degrees of freedom introduced in the
various seesaw mechanisms. After electroweak phase transition, the dimension
five operator generates sub-eV Majorana masses of neutrinos. On the other
hand, if the neutrinos are Dirac, then lepton number is an exact symmetry of
nature as it stands now in the SM. In this case, the sub-eV masses of light
neutrinos imply that the Yukawa coupling involving
$\bar{L}\widetilde{H}N_{R}$, with $N_{R}$ being the singlet right-handed
neutrino, is of $\mathcal{O}(10^{-12})$, which is almost six orders of
magnitude less than the electron Yukawa coupling. Thus the sub-eV Dirac mass
of light neutrinos requires substantial fine-tuning.

Another important aspect of the SM is the identity of dark matter, which plays
a major role throughout the evolution of the Universe. Astrophysical evidences
from galaxy rotation curve, gravitational lensing and large scale structure of
the Universe confirmed the existence of dark matterdm_review_1 ; dm_review_2 .
In fact, the satellite-borne experiments WMAPHinshaw:2012aka  and
PlanckAkrami:2018vks , which measure the temperature fluctuation in the cosmic
microwave background (CMBR), precisely determine the relic abundance of baryon
and dark matter (DM) to be $\Omega_{\rm B}h^{2}=0.02237\pm 0.00015$ and
$\Omega_{\rm DM}h^{2}=0.1200\pm 0.0012$ respectively at 68% CL, where
$\Omega_{\rm DM}$ is the density parameter and $h=\text{Hubble
Parameter}/(100\;\text{km}~{}\text{s}^{-1}\text{Mpc}^{-1})$ is the reduced
Hubble constant. This implies that the DM abundance is about five times the
baryon abundance: i.e. $\Omega_{\rm DM}\approx 5\Omega_{\rm B}$. While it is
very natural for the Universe to start in a baryon symmetric manner, the
present Universe is highly baryon asymmetric, giving rise to the long-standing
puzzle of the baryon asymmetry of the Universe (BAU). The observed BAU is
quantitatively expressed by the ratio of baryon density over anti-baryons
density to photon density as Aghanim:2018eyx ,

$\eta_{B}=\frac{n_{B}-n_{\overline{B}}}{n_{\gamma}}\simeq 6.2\times 10^{-10}.$
(1)

The origin of this asymmetry is also not known along with the particle nature
of DM.

In the well-established $\Lambda{\rm CDM}$ model, DM is assumed to be cold and
collisionless which is supposed to have facilitated the structure formation in
the early Universe by providing the necessary gravitational potential for
primordial density fluctuations to grow. However, cosmological simulations in
recent times reveal a few severe discrepancies of the $\Lambda{\rm CDM}$ model
at small scales, leading to anomalies such as the cusp-core problem, missing
satellite problem and too-big-to-fail problem Tulin:2017ara ; Bullock:2017xww
. To alleviate these small-scale $\Lambda{\rm CDM}$ anomalies, Spergel and
Steinhardt proposed in 2000 an interesting alternative to cold dark matter
(CDM) in terms of self-interacting dark matter (SIDM) Spergel:1999mh . Earlier
studies in this direction can be found in Carlson:1992fn ; deLaix:1995vi .
SIDM can have large self-scattering cross-sections of
$\mathcal{O}(10^{-24}{\rm cm^{2}/GeV})$ Buckley:2009in ; Feng:2009hw ;
Feng:2009mn ; Loeb:2010gj ; Zavala:2012us ; Vogelsberger:2012ku , which is way
too larger than typical cross-section of $\mathcal{O}(10^{-38}{\rm
cm^{2}/GeV})$ for weakly-interacting massive particles (WIMPs), a well-suited
class of candidates for CDM scenario. Such large self-interacting cross-
sections of DM can be naturally realized in scenarios where DM has a light
mediator. In such a scenario, self-interaction is stronger for smaller DM
velocities such that it can have a large impact on small scale structures,
while it gets reduced at larger scales due to large velocities of DM and hence
remains consistent with large scale CDM predictions Buckley:2009in ;
Feng:2009hw ; Feng:2009mn ; Loeb:2010gj ; Bringmann:2016din ;
Kaplinghat:2015aga ; Aarssen:2012fx ; Tulin:2013teo . The light mediator also
mixes with the SM Higgs paving a way for detecting DM at direct detection
experiments Kaplinghat:2013yxa ; DelNobile:2015uua . Several model building
efforts have been made to realize such scenarios, see Kouvaris:2014uoa ;
Bernal:2015ova ; Kainulainen:2015sva ; Hambye:2019tjt ; Cirelli:2016rnw ;
Kahlhoefer:2017umn ; Dutta:2021wbn ; Borah:2021yek ; Borah:2021pet ;
Borah:2021rbx ; Borah:2021qmi  and references therein.

Figure 1: Pictorial representation shows: i) the generation of neutrino mass,
ii) the generation of the $B-L$ asymmetry in the lepton sector. The sphalerons
above electroweak (EW) phase transition ($T_{EW}$) transfers a part of $B-L$
asymmetry to the observed B-asymmetry, while the higher dimension operator
$\mathcal{O}_{8}$ is in thermal equilibrium above EW-phase transition, at
$T>T_{EW}$, cooks a net DM asymmetry out of the existing $B-L$ asymmetry. The
$\Phi-H$ mixing provides a bridge between the visible and dark sectors.

Thus, both the SM and the $\Lambda{\rm CDM}$ models are inadequate to explain
a plethora of mysteries in physics. At present, the nature of neutrinos,
either Dirac or Majorana, is not confirmed yet. In future, the neutrinoless
double beta decay experiments onu_beta_expt  may shed light on it. In this
paper, assuming neutrinos to be Dirac (i.e. $B-L$ is an exact symmetry) and
the DM to be self-interacting, we make an attempt to explain simultaneously
the three most puzzling physical phenomena viz. neutrino mass, DM and the
baryon asymmetry of the Universe along with a natural explanation of the ratio
$\Omega_{\rm DM}\approx 5\,\Omega_{\rm B}$. We extend the SM with heavy
$SU(2)_{L}$ scalar doublets $X_{i}$, ($i=1,2$), and $\eta$, singlet right-
handed neutrinos $\nu_{iR},(i=1,2,3)$ and singlet scalars $\Phi$,
$\Phi^{\prime}$ along with a vector-like singlet Dirac fermion $\chi$ which
represents the SIDM candidate. Since we assume that $B-L$ is an exact
symmetry, the CP-violating out-of-equilibrium decay of the lightest $X$:
i.e.$X\to H\Phi^{\prime}$ and $X\to L\nu_{iR}$ generates equal and opposite
$B-L$ asymmetry between $\nu_{L}$ and $\nu_{R}$ Dirac_leptogenesis_1 ;
Dirac_leptogenesis_1 ; Dirac_leptogenesis_2 ; Dirac_leptogenesis_3 ;
Dirac_leptogenesis_4 ; Dirac_leptogenesis_5 ; Dirac_leptogenesis_6 ;
Dirac_leptogenesis_7 . After $\Phi^{\prime}$ and $H$ acquire vacuum
expectation values (vev), we get Dirac mass of neutrinos of appropriate order.
By introducing an additional $U(1)_{D}$ symmetry (which forbids
$\bar{L}\tilde{H}\nu_{R}$ coupling) we ensure that $\nu_{L}-\nu_{R}$
equilibration does not occur until below the electroweak (EW) phase transition
during which a part of the lepton asymmetry gets converted to dark matter
asymmetry through a dimension eight operator:
$\mathcal{O}_{8}=\frac{1}{M_{asy}^{4}}\bar{\chi}^{2}(LH)^{2}$Kaplan:2009ag ;
Feng:2012jn ; Ibe:2011hq ; Feng:2013wn ; Narendra:2018vfw , $M_{asy}$ being
fixed by the mass and relevant couplings of $\eta$. A part of the remaining
$B-L$ asymmetry then gets converted to a net B-asymmetry through EW-sphalerons
which are active at a temperature above 100 GeV. The singlet scalar $\Phi$ not
only mediates the self-interaction among the DM particles, but also helps in
depleting the symmetric component of the DM. Moreover it mixes with the SM
Higgs providing a portal for detecting the DM at terrestrial laboratories. We
constrain the scalar portal mixing with recent data from experiments like
CRESST-III and XENON1T. The pictorial presentation can be seen in Fig.1.

The paper is arranged as follows. In section-II, we explain the model for
simultaneous explanation of SIDM, baryon asymmetry and Dirac mass of the light
neutrinos. In section-III, we explain the Dirac leptogenesis mechanism and
production as well as relic density of DM. In section-IV, we find the
parameter space for sufficient self-interaction with desired velocity
dependence followed by a discussion on the direct detection prospects of the
SIDM in section-V. In section VI, we discuss the parameter space for light
neutrino mass and conclude in section-VII.

##  II The Model

We extend the SM, which is based on the gauge group: $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$, with $U(1)_{D}\times U(1)_{B-L}$ global symmetries.
The particle contents of the model are shown in Table 1 along with their
quantum numbers under the imposed symmetry. The scalar sector consists of
heavy scalar doublets $X_{i}$, ($i=1,2$) and $\eta$, and two singlet scalars
$\Phi$, $\Phi^{\prime}$; while the fermion sector consists of a vector-like
Dirac fermion $\chi$ along with three heavy right-handed neutrinos
$\nu_{R_{i}},(i=1,2,3)$, all singlet under the SM gauge group. All these
particles carry non-trivial charges under $U(1)_{D}$ symmetry, while the SM
particles remain neutral. Under the extended symmetries, the Majorana mass
terms of $\chi$ and $\nu_{R}$ are forbidden.

Fields | SU(2)L | U(1)Y | $U(1)_{D}$ | $U(1)_{B-L}$
---|---|---|---|---
$X_{i}$ | 2 | +1 | -1 | 0
$\eta$ | 2 | +1 | 1/2 | 0
$\nu_{{}_{R}}$ | 1 | 0 | -1 | -1
$\Phi^{\prime}$ | 1 | 0 | +1 | 0
$\Phi$ | 1 | 0 | 0 | 0
$\chi$ | 1 | 0 | 1/2 | -1
Table 1: The additional particle content and their quantum numbers under the
imposed symmetry.

Owing to the symmetry and the charge assignment as shown in Table 1, the
Lagrangian of the Model can be written as,111For simplicity we suppress the
indices and state when they require explicitly.

$\displaystyle-\mathcal{L}$ $\displaystyle\supset$ $\displaystyle
m_{\chi}\overline{\chi}\chi+\lambda_{{}_{D}}\overline{\chi}\chi\Phi+y\,\overline{L}\tilde{X}\nu_{{}_{R}}+\rho\,\,\Phi^{\prime*}X^{\dagger}H$
(2) $\displaystyle+\lambda~{}\overline{\chi}\,L\,\eta+{\rm
h.c.}\,+V(X,\eta,H,\Phi,\Phi^{\prime}),$

where,

$\displaystyle V(\eta,H,\Phi,\Phi^{\prime})$ $\displaystyle=$ $\displaystyle
M_{\eta}^{2}(\eta^{\dagger}\eta)+\lambda_{\eta}(\eta^{\dagger}\eta)^{2}+\lambda^{\prime}_{{}_{\eta
H}}(\eta^{\dagger}\eta)(H^{\dagger}H)$ (3) $\displaystyle+[\lambda_{{}_{\eta
H}}(\eta^{\dagger}H)^{2}+h.c.]$
$\displaystyle-\mu_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}+\frac{1}{2}m_{\phi}^{2}\Phi^{2}+\frac{1}{3}\mu_{\Phi}\Phi^{3}$
$\displaystyle+\frac{1}{4}\lambda_{\Phi}\Phi^{4}-\mu_{\Phi^{\prime}}^{2}(\Phi^{\prime\dagger}\Phi^{\prime})+\lambda_{\Phi^{\prime}}(\Phi^{\prime\dagger}\Phi^{\prime})^{2}$
$\displaystyle+\frac{\mu_{1}}{\sqrt{2}}\Phi
H^{\dagger}H+\frac{\mu_{2}}{\sqrt{2}}\Phi(\Phi^{\prime\dagger}\Phi^{\prime})+\frac{\lambda_{H\Phi}}{2}H^{\dagger}H\Phi^{2}$
$\displaystyle+\lambda_{H\Phi^{\prime}}H^{\dagger}H(\Phi^{\prime\dagger}\Phi^{\prime})+\frac{\lambda_{\Phi\Phi^{\prime}}}{2}\Phi^{2}(\Phi^{\prime\dagger}\Phi^{\prime}).$

The term $\lambda_{{}_{\eta H}}(\eta^{\dagger}H)^{2}+h.c.$ in the potential
(3), breaks the $U(1)_{D}$ symmetry softly to a remnant $Z_{2}$ symmetry under
which $\eta$ and $\chi$ are odd, while all other particles are even. We assume
that $\eta$ doesn’t acquire a vev in order to preserve the remnant $Z_{2}$
symmetry. Considering $M_{\eta}\gg M_{\chi}$, $\chi$ becomes the viable DM
candidate. In this way, we ensure the stability to DM $\chi$ as well as the
theory escapes from having a Goldstone boson.

The Lagrangian terms $\lambda~{}\overline{\chi}L\eta+\lambda_{{}_{\eta
H}}(\eta^{\dagger}H)^{2}+{\rm h.c.}$ give rise to a dimension-8 transfer
operator via the Feynman diagram shown in top panel of Fig. 2. In the later
part of the draft, we will show that this operator transfers the lepton
asymmetry from visible sector to dark sector. At low energy, upon integrating
out the heavy scalar $\eta$, we get the dimension-8 operator Kaplan:2009ag ;
Feng:2012jn ; Ibe:2011hq ; Feng:2013wn ; Narendra:2018vfw :

$\mathcal{O}_{8}=\frac{\lambda^{2}\lambda_{{}_{\eta
H}}\overline{\chi}LHH\overline{\chi}L}{M_{\eta}^{4}}\equiv\frac{\overline{\chi}^{2}(LH)^{2}}{M_{asy}^{4}}\,,$
(4)

where $M_{asy}^{4}=M_{\eta}^{4}/(\lambda^{2}\lambda_{{}_{\eta H}})$.
Similarly, the Lagrangian terms
$y\overline{L}\tilde{X}\nu_{{}_{R}}+\rho\,\Phi^{\prime*}X^{\dagger}H$ lead to
an effective dimension-5 operator by integrating out the the heavy scalar
field $X$ at low energy. The Feynman diagram for this dimension-5 operator
$\mathcal{O}_{5}=y\frac{\rho}{M_{X}^{2}}\overline{L}H\Phi^{\prime}\nu_{R}$ is
shown in the bottom panel of Fig. 2.

Figure 2: Top: Feynman diagram of the dimension-8 operator. Bottom: Feynman
diagram of the dimension-5 operator.

In writing down the scalar potential as given in Eq. 3, we assume that the
scalar doublet $X$ being very heavy, do not participate in low energy
phenomenology and have implications only in leptogenesis. Here
$\mu^{2}_{H},m^{2}_{\phi},\mu^{2}_{\Phi^{\prime}}>0$. We further assume that
$\mu_{\Phi}$ and $\lambda_{\Phi}$ are small. As a result, $\Phi$ acquires a
non-zero vev through the trilinear terms $\mu_{1}\Phi(H^{\dagger}H)$ and
$\mu_{2}\Phi(\Phi^{\prime\dagger}\Phi^{\prime})$, which are proportional to
the coupling constants $\mu_{1}$ and $\mu_{2}$ respectively. The fluctuations
of the fields $\Phi^{\prime}$, $H$ and $\Phi$ around their corresponding vev’s
can be written as,

$\Phi^{\prime}=\frac{w+\phi^{\prime}}{\sqrt{2}}\,\,\,,\,\,\,H=\begin{bmatrix}0\\\
\frac{v+h}{\sqrt{2}}\end{bmatrix}\,\,\,\,\,{\rm and}\,\,\,\,\,\Phi=u+\phi.$
(5)

From Eq. 3, the stationary conditions are obtained as:

$\displaystyle
w(-\mu^{2}_{\Phi^{\prime}}+\lambda_{\Phi^{\prime}}w^{2}+\frac{\mu_{2}}{\sqrt{2}}u+\frac{\lambda_{H\Phi^{\prime}}}{2}v^{2}+\frac{\lambda_{\Phi\Phi^{\prime}}}{2}u^{2})$
$\displaystyle=0,$ (6) $\displaystyle
v(-\mu^{2}_{H}+\lambda_{H}v^{2}+\frac{\mu_{1}}{\sqrt{2}}u+\frac{\lambda_{H\Phi^{\prime}}}{2}w^{2}+\frac{\lambda_{H\Phi}}{2}u^{2})$
$\displaystyle=0,$ $\displaystyle
u(m^{2}_{\phi}+\mu_{\Phi}u+\lambda_{\Phi}u^{2}+\frac{\lambda_{H\Phi}}{2}v^{2}+\frac{\lambda_{\Phi\Phi^{\prime}}}{2}w^{2})$
$\displaystyle+\frac{\mu_{1}}{2\sqrt{2}}v^{2}+\frac{\mu_{2}}{2\sqrt{2}}w^{2}=0.$

Assuming $\mu_{\Phi}<<m_{\Phi}\sim 10^{-3}$ GeV and $\lambda_{\Phi}<<1$, we
can drop the terms proportional to $\mathcal{O}(u^{2})$ and
$\mathcal{O}(u^{3})$. In this limit, Eq.6 can be rewritten as,

$\displaystyle
w(-\mu^{2}_{\Phi^{\prime}}+\lambda_{\Phi^{\prime}}w^{2}+\frac{\mu_{2}}{\sqrt{2}}u+\frac{\lambda_{H\Phi^{\prime}}}{2}v^{2})=0,$
(7) $\displaystyle
v(-\mu^{2}_{H}+\lambda_{H}v^{2}+\frac{\mu_{1}}{\sqrt{2}}u+\frac{\lambda_{H\Phi^{\prime}}}{2}w^{2})=0,$
$\displaystyle
u(m^{2}_{\phi}+\frac{\lambda_{H\Phi}}{2}v^{2}+\frac{\lambda_{\Phi\Phi^{\prime}}}{2}w^{2})+\frac{\mu_{1}}{2\sqrt{2}}v^{2}+\frac{\mu_{2}}{2\sqrt{2}}w^{2}=0.$

Thus, the vevs of the fields $\Phi^{\prime}$, $H$ and $\Phi$ are obtained as:

$\displaystyle w$
$\displaystyle=\sqrt{\frac{\mu^{2}_{\Phi^{\prime}}-\frac{\mu_{2}}{\sqrt{2}}u-\frac{\lambda_{H\Phi^{\prime}}}{2}v^{2}}{\lambda_{\Phi^{\prime}}}},$
(8) $\displaystyle v$
$\displaystyle=\sqrt{\frac{\mu^{2}_{H}-\frac{\mu_{1}}{\sqrt{2}}u-\frac{\lambda_{H\Phi^{\prime}}}{2}w^{2}}{\lambda_{H}}},$
$\displaystyle u$
$\displaystyle=\frac{-(\mu_{1}v^{2}+\mu_{2}w^{2})}{2\sqrt{2}(m^{2}_{\phi}+\frac{\lambda_{H\Phi}}{2}v^{2}+\frac{\lambda_{\Phi\Phi^{\prime}}}{2}w^{2})}.$

The smallness of $u$ in Eq. 8 can be justified by assuming $\mu_{1},\mu_{2}\to
0$. The squared mass matrix for the neutral components of three scalar fields
$\Phi^{\prime}$, H and $\Phi$ viz. $\phi^{\prime},h,\phi$ can be written in
the basis $(\phi^{\prime},h,\phi)$ as:

$\begin{pmatrix}2\lambda_{\Phi^{\prime}}w^{2}&\frac{1}{2}\lambda_{H\Phi^{\prime}}vw&\frac{1}{2}(\frac{\mu_{2}w}{\sqrt{2}}+\lambda_{\Phi\Phi^{\prime}}uw)\\\
\frac{1}{2}(\lambda_{H\Phi^{\prime}}vw)&2\lambda_{H}v^{2}&\frac{1}{2}(\frac{\mu_{1}v}{\sqrt{2}}+\lambda_{H\Phi}vu)\\\
\frac{1}{2}(\frac{\mu_{2}w}{\sqrt{2}}+\lambda_{\Phi\Phi^{\prime}}uw)&\frac{1}{2}(\frac{\mu_{1}v}{\sqrt{2}}+\lambda_{H\Phi}vu)&m^{2}_{\phi}+\frac{\lambda_{H\Phi}}{2}v^{2}+\frac{\lambda_{\Phi\Phi^{\prime}}}{2}w^{2}\\\
\end{pmatrix}.$ (9)

As we need light messengers (in MeV scale) to realize sufficiently large DM
self-interaction, which also helps in annihilating away the symmetric
component of DM, we consider the induced vev u of the field $\Phi$ to be very
small compared to that of SM Higgs. The details of the diagonalization is
given in Appendix B. After complete diagonalization, we are left with three
mass eigenstates $h_{1},h_{2}$ and $h_{3}$. The eigenstate $h_{1}$ with mass
$m_{h_{1}}\approx m_{h}=125.18\,{\rm GeV}$ can be identified as the SM Higgs.
The eigenstate $h_{2}$ with mass $m_{h_{2}}\sim m_{\Phi^{\prime}}\sim 10^{5}$
has implications in addressing the neutrino mass and the extra light scalar
$h_{3}$ with mass $m_{h_{3}}\sim m_{\phi}$ ($\approx$ MeV scale) mediates the
DM self-interaction and annihilates away the symmetric DM component.

The second term of Eq. 2 gives rise to scalar mediated DM self-interaction via
the Feynman diagram shown in the left panel of Fig. 3. The third and fourth
terms are responsible for two CP-violating out-of-equilibrium decays: $X\to
L\nu_{iR}$ and $X\to H\Phi^{\prime}$ that generate equal and opposite $B-L$
asymmetry between $\nu_{L}$ and $\nu_{R}$, required for leptogenesis
Sakharov:1967dj ; sakharov.67 ; fukugita.86 ; baryo_lepto_group_1 ;
baryo_lepto_group_2 ; baryo_lepto_group_3 ; baryo_lepto_group_4 ;
baryo_lepto_group_5 ; baryo_lepto_group_6 ; baryo_lepto_group_7 ;
baryo_lepto_group_8 ; baryo_lepto_group_9 ; Abada:2006ea ; Pilaftsis:2003gt ;
Pilaftsis:2005rv . The dimension-8 operator $\mathcal{O}_{8}$ remains in
thermal equilibrium until below the electroweak phase transition, during which
it transfers the lepton number to the DM number in a $B-L$ conserving way in
order to establish a proportionality between DM and baryon densities. The
strong Yukawa couplings of SM charged leptons rapidly cancel the left and
right-handed numbers through LR equilibration processes. But the situation is
different for neutrinos due to their tiny Yukawa couplings. They equilibrate
after the sphalerons go out of thermal equilibrium. By that time the
sphalerons convert a part of left-handed neutrino asymmetry to the desired
B-asymmetry Dick ; Cerdeno:2006ha ; Gu:2006dc ; Gu:2007mc ; Murayama:2002je ;
Thomas:2005rs ; Borah:2016zbd ; Perez:2021udy , while the dimension-8 operator
$\mathcal{O}_{8}$ keeps on cooking a net DM asymmetry from the existing
neutrino asymmetry old_DM_Baryon_asy_1 ; old_DM_Baryon_asy_2 ;
old_DM_Baryon_asy_3 ; old_DM_Baryon_asy_4 ; old_DM_Baryon_asy_5 ;
old_DM_Baryon_asy_6 ; old_DM_Baryon_asy_7 ; old_DM_Baryon_asy_8 ;
Asydm_models1_1 ; Asydm_models1_2 ; Asydm_models1_3 ; Asydm_models1_4 ;
Asydm_models1_5 ; Asydm_models1_6 ; Asydm_models1_7 ; Asydm_models1_8 ;
Asydm_models1_9 ; Asydm_models1_10 ; Asydm_models2_1 ; Asydm_models2_2 ;
Asydm_models2_3 ; Asydm_models2_4 ; Asydm_models2_5 ; Asydm_models2_6 ;
Asydm_models2_7 ; Asydm_models2_8 ; Asydm_models2_9 ; Asydm_models3_1 ;
Asydm_models3_2 ; Asydm_models3_3 ; Asydm_review_1 ; Asydm_review_2 . This
mechanism directly connects the small Dirac neutrino masses to the DM
abundance and the baryon asymmetry of the Universe.

Figure 3: Left: Feynman diagram for elastic DM self-interaction. Right:
Dominant annihilation mode for the symmetric component of DM $\chi$.

The scalar $\phi$ is sufficiently light and mediates the self-interaction
between the DM particles. Interestingly, the symmetric component of DM
depletes via its annihilation into $\phi$ via the Feynman diagram depicted in
the right panel of Fig. 3. Moreover, it mixes with the SM Higgs and paves a
path to detect the DM at direct search experiments. The vev’s of singlet
scalar $\langle\Phi^{\prime}\rangle=w$ and SM-Higgs $\langle H\rangle=v$ gives
small neutrino masses via higher dimension operator
$\mathcal{O}_{5}=y\frac{\rho}{M_{X}^{2}}\overline{L}H\Phi^{\prime}\nu_{R}$,
where the heavy scalar mass suppresses the weak scale.

##  III Dirac Leptogenesis and the DM Relic Density

As the Universe cools down, the heavy scalar doublet $X$ which is assumed to
have existed in the early Universe, go out of thermal equilibrium below its
mass scale and decay in a CP violating way through the channels: $X_{1}\to
L\nu_{R}$ and $X_{1}\to H\Phi^{\prime}$ creating an equal and opposite $B-L$
asymmetry in both left and right-handed sector, see Eq. 2. The total decay
rate of $X_{1}$ is given by:

$\displaystyle\Gamma_{X_{1}}$
$\displaystyle=\frac{1}{8\pi}(y_{1}^{2}+\frac{\rho_{1}^{2}}{M_{X_{1}}^{2}})M_{X_{1}}$
(10) $\displaystyle=\frac{1}{8\pi}(y_{1}^{2}+f_{1}^{2})M_{X_{1}}$

where $\rho_{1}$ and $y_{1}$ are the trilinear and the Yukawa couplings
respectively and in the second line of Eq. 10, we have defined the
dimensionless parameter $f_{1}=\frac{\rho_{1}}{M_{X_{1}}}$, $M_{X_{1}}$ being
the mass of the heavy scalar $X_{1}$. To get the adequate lepton asymmetry,
the dominant decay channel must be $X_{1}\to L\nu_{R}$, for which the
branching ratio $B_{L}(X_{1}\to L\nu_{iR})$ is given by,

$\displaystyle B_{L}(X_{1}\to L\nu_{R})$
$\displaystyle=\frac{(y_{1}^{2}M_{X_{1}})/8\pi}{\Gamma_{X_{1}}}$ (11)
$\displaystyle=\frac{1}{1+\frac{f_{1}^{2}}{y_{1}^{2}}}$

We assume $B_{L}(X_{1}\to L\nu_{R})\sim 0.9$ in order to generate the desired
lepton asymmetry. Therefore, $f_{1}$ and $y_{1}$ differ roughly by an order
magnitude. Since parameters $f_{1}$ and $y_{1}$ are also used in section-VI to
explain tiny neutrino mass as $y_{1}\sim\mathcal{O}(10^{-4})$ and
$f_{1}\sim\mathcal{O}(10^{-5})$, we stick to region $f_{1},y_{1}\lesssim
10^{-4}$ in this section. Demanding $\Gamma_{X_{1}}\lesssim H$ at
$T=M_{X_{1}}$, where $H=1.67g_{*}^{1/2}T^{2}/M_{\rm Pl}$ is the Hubble
expansion parameter, we get $M_{X_{1}}\lesssim 10^{10}\,{\rm GeV}$ for
$f_{1},y_{1}\lesssim 10^{-4}$. For the CP-asymmetry to be non zero, we require
atleast two doublet scalars $X_{i},i=1,2$. With just one X, the CP-asymmetry
will be zero as the imaginary part of the couplings become zero, which can be
seen in Eq.11. In presence of these doublet scalars and their interactions,
they form a mass matrix $M_{\pm}^{2}$. By diagonalising this mass matrix, we
get new mass eigenstates $\zeta_{1}^{\pm}$ and $\zeta_{2}^{\pm}$. See for more
details Ma:1998dx ; Narendra:2017uxl .

We assume a hierarchy among the masses of $\zeta_{1}^{\pm}$ and
$\zeta_{2}^{\pm}$ such that the final asymmetry is generated via the decay of
the lightest one $\zeta_{1}^{\pm}$. The CP-violation arises via the
interference of tree-level and one-loop self-energy correction diagrams of
lightest scalar doublet $\zeta_{1}^{\pm}$ as shown in Fig. 4.

Figure 4: Tree-level and self-energy correction diagrams, whose interference
give arise to a net CP-violation in the decay of $\zeta_{1}$.

The generated asymmetry in the visible sector is then given by

$\displaystyle\epsilon_{L}$ $\displaystyle=$
$\displaystyle[B_{L}(\zeta_{1}^{-}\rightarrow
l^{-}\nu_{R})-B_{L}(\zeta_{1}^{+}\rightarrow(l^{-})^{c}\nu_{R}^{c})]$ (12)
$\displaystyle=$ $\displaystyle-\frac{{\rm
Im}\left(\rho_{1}^{*}\rho_{2}\mathop{\sum_{k,l}}y^{*}_{1kl}y_{2kl}\right)}{8\pi^{2}(M_{\zeta_{2}}^{2}-M_{\zeta_{1}}^{2})}\left[\frac{M_{\zeta_{1}}}{\Gamma_{\zeta_{1}}}\right]\,,$

where $B_{L}$ is the branching ratio for $\zeta_{1}^{\pm}\rightarrow
l^{\pm}\nu_{R}$ and $M_{\zeta_{i}},i=1,2$ are the masses of heavy doublet
scalars. Here $\rho_{1,2}$ are respectively the mass dimension coupling of
$X_{1,2}$ with $\Phi^{\prime}H$, while $y_{1,2}$ are the respective Yukawa
couplings of $X_{1,2}$ with $L\nu_{R}$. We get a net $B-L$ asymmetry
Narendra:2018vfw ; buchmuller&plumacher ; Buchmuller:2004nz ; Giudice:2003jh .

$(n_{B-L})_{total}=\epsilon_{L}\kappa
s\times\frac{n_{\zeta_{1}}^{eq}(T\rightarrow\infty)}{s}$ (13)

where
$(n_{\zeta_{1}}^{eq}/s)(T\rightarrow\infty)=135\,\zeta(3)/(4\pi^{4}g_{*})$ is
the relativistic equilibrium abundance of $\zeta_{1}^{\pm}$, where
$\zeta(3)=1.202$. $s=(2\pi^{2}/45)g_{*}T^{3}$ is the entropy density of the
co-moving volume. $\kappa$ is the washout factor, which arises due to inverse
decay and scattering processes. It can vary between 0 to 1 depending on the
strength of the Yukawa coupling. For definiteness we choose $\kappa=0.01$. The
$B-L$ asymmetry in the visible sector can be generated by solving the relevant
coupled Boltzmann equations given by Eq.14 Narendra:2017uxl . In Fig.5, we
show the evolution of the number density of $\zeta_{1}$, i.e., $Y_{\zeta_{1}}$
and the $B-L$ asymmetry $Y_{\rm B-L}$ by solving the Boltzmann equations as
given below:

$\displaystyle\frac{dY_{\zeta_{1}}}{dx}$
$\displaystyle=-\frac{x}{H(M_{\zeta_{1}})}s\langle\sigma|v|_{(\zeta_{1}\zeta_{1}\rightarrow
All)}\rangle\Big{[}Y^{2}_{\zeta_{1}}-(Y^{eq}_{\zeta_{1}})^{2}\Big{]}$ (14)
$\displaystyle-\frac{x}{H(M_{\zeta_{1}})}\Gamma_{(\zeta_{1}\rightarrow
All)}\Big{[}Y_{\zeta_{1}}-Y^{eq}_{\zeta_{1}}\Big{]}$
$\displaystyle\frac{dY_{\rm B-L}}{dx}$
$\displaystyle=\frac{x}{H(M_{\zeta_{1}})}\Big{[}\epsilon_{L}\Gamma_{(\zeta_{1}\rightarrow
All)}B_{L}\left(Y_{\zeta_{1}}-Y^{eq}_{\zeta_{1}}\right)-\Gamma_{W}Y_{\rm
B-L}\Big{]},$

where $\Gamma_{W}$ in the second equation takes care of the washout effects
corresponding to inverse decay and scattering. The decay and the inverse decay
of $\zeta_{1}^{\pm}$ are related by
$\Gamma_{inv}=(Y_{\zeta_{1}}^{eq}/Y_{\ell}^{eq})\Gamma_{(\zeta_{1}\rightarrow
All)}$. The inverse decay of $\zeta_{1}^{\pm}$ falls exponentially after the
latter goes out-of-equilibrium. Similarly the lepton number conserving
$2\rightarrow 2$ scattering processes: $\nu_{R}\Phi^{\prime}\to LH$,
$\nu_{R}H\to L\Phi^{\prime}$, $\nu_{R}\overline{L}\to H\Phi^{\prime}$ mediated
by $X$-particles are suppressed due to the small Yukawa couplings required for
generating light neutrino masses of Dirac type in section-VI. The resulting
$B-L$ asymmetry is shown by the red dot-dashed line in Fig. 5.

Figure 5: Abundance of $B-L$ and $\zeta_{1}$ as a function of dimensionless
variable $x=M_{\zeta_{1}}/T$. The Red dot-dashed line shows the abundance of
$B-L$ asymmetry, where $\epsilon_{L}=10^{-6}$. The Green dot-dashed line shows
the abundance of $\zeta_{1}$. The Blue dotted line shows the equilibrium
abundance of $\zeta_{1}$. Here we have taken $y_{1}=10^{-4}$,
$\rho_{1}=10^{5}$ GeV

and $M_{\zeta_{1}}=10^{10}$ GeV.

In the following we discuss the distribution of the generated $B-L$ asymmetry
between the visible sector and the dark sector. Once the $B-L$ asymmetry is
generated in the visible sector, the dimension-8 operator
$\mathcal{O}_{8}=\frac{1}{M_{asy}^{4}}\overline{\chi}^{2}(LH)^{2}$ transfers
it partially to the dark sector Feng:2012jn . This requires the
$\mathcal{O}_{8}$ operator to be in thermal equilibrium above the sphaleron
decoupling temperature $T_{\rm sph}$. For Higgs mass $M_{h_{1}}=125$ GeV,
$T_{\rm sph}\gtrsim M_{W}$, where $M_{W}$ is the W-boson mass. As discussed in
refs. Narendra:2018vfw ; Narendra:2019cyt , the $\mathcal{O}_{8}$ operator
remains in thermal equilibrium down to a temperature $T_{D}\gtrsim T_{\rm
sph}=M_{W}$ for $M_{asy}\gtrsim 0.9\times 10^{4}$ GeV. As a result the DM
remains in thermal contact with the visible sector via this $\mathcal{O}_{8}$
operator down to a decoupling temperature $T_{D}\gtrsim T_{\rm sph}=M_{W}$,
during which a net $B-L$ asymmetry of the $\chi$-particles is generated.The
details of the chemical equilibrium calculation are given in appendix A. The
number density of $\chi$ is then given by:

$n_{\chi}=(n_{\rm B-L})_{\rm dark}=\frac{58}{291}(n_{B-L})_{\rm vis}\,\,.$
(15)

The total $B-L$ asymmetry created in the early universe is through out-of-
equilibrium decays of $\zeta_{1}$ and it is the only source of $B-L$ asymmetry
in both dark and visible sectors. So we have the following condition,

$\displaystyle(n_{B-L})_{\rm total}$ $\displaystyle=$
$\displaystyle(n_{B-L})_{\rm vis}+(n_{B-L})_{\rm dark}$ (16) $\displaystyle=$
$\displaystyle\frac{349}{291}(n_{B-L})_{\rm vis}.$

Comparing Eq. 16 with Eq. 13 and using $n_{\rm B}=0.31\,(n_{\rm B-L})_{\rm
vis}$ Narendra:2018vfw , we get the required CP asymmetry for observed lepton
abundance to be $\epsilon_{L}=141.23(\eta/\kappa)$. For $\kappa\sim 0.01$, the
required CP-asymmetry is $\epsilon_{L}\sim 10^{-6}$. Using Eq. 16 in $n_{\rm
B}=0.31\,(n_{\rm B-L})_{\rm vis}$ and Eq. 15, we get,

$n_{\rm B}=\frac{90}{349}(n_{\rm B-L})_{\rm
total}\,\,\,\,,\,\,\,\,n_{\chi}=\frac{58}{349}(n_{B-L})_{\rm total}$ (17)

Therefore,

$\frac{n_{\chi}}{n_{B}}=\frac{58}{90}.$ (18)

The expression for the relic of the asymmetric dark matter component is given
by,

$\Omega_{asy}=n_{\chi}m_{\chi}/\rho_{c}\,\,,$ (19)

where $\rho_{c}$ is the critical density. For fully asymmetric dark matter
$\Omega_{DM}=\Omega_{asy}$, so we have,

$\frac{\Omega_{DM}}{\Omega_{B}}=\frac{\Omega_{asy}}{\Omega_{B}}=\frac{n_{\chi}m_{\chi}}{m_{B}n_{B}}=\frac{58}{90}m_{\chi}.$
(20)

From WMAP and Planck data, we have the ratio between dark matter density to
baryonic matter density,

$\frac{\Omega_{DM}}{\Omega_{B}}\sim 5.$ (21)

Using Eq. 20 and Eq. 21, we can get the dark matter mass for fully asymmetric
dark matter to be $m_{\chi}=7.76\rm\,GeV$.
Now, if we consider that the dark matter is not fully asymmetric, instead it
contains both symmetric and asymmetric components,

$\Omega_{DM}h^{2}=(\Omega_{sym}+\Omega_{asy})h^{2}=0.12.$ (22)

Therefore the percentage of the dark matter relic contributed by asymmetric
part is given by,

$\text{ADM}(\%)=\frac{\Omega_{asy}}{\Omega_{DM}}\times
100=\frac{\Omega_{asy}}{\Omega_{sym}+\Omega_{asy}}\times 100.$ (23)

Using Eq. 21 and Eq. 23, we have,

$\frac{\Omega_{asy}}{\Omega_{B}}=5\times\frac{\Omega_{asy}}{\Omega_{sym}+\Omega_{asy}}=5\times\frac{\text{ADM}(\%)}{100}.$
(24)

Using Eq. 20 and Eq. 23, we can write the dark matter mass for general
scenario, where both symmetric and asymmetric component contributing to the
dark matter relics as,

$m_{\chi}=\frac{90}{58}\times 5\times\frac{\text{ADM}(\%)}{100}.$ (25)

So depending on how much asymmetric relic contributes to the total DM relic,
the mass of the dark matter will change accordingly from the above equation.

In this general scenario, the symmetric component contributing to the total
relic is $(100-\text{ADM})\%$. So we have,

$\displaystyle 100-\text{ADM}(\%)$ $\displaystyle=\frac{\Omega_{sym}\times
100}{\Omega_{sym}+\Omega_{asy}}$
$\displaystyle=\frac{\Omega_{sym}}{\Omega_{DM}}\times
100=\frac{\Omega_{sym}h^{2}}{0.12}\times 100,$
$\displaystyle\Rightarrow\Omega_{sym}h^{2}$
$\displaystyle=\frac{100-\text{ADM}(\%)}{100}\times 0.12$ . (26)

The relics of the symmetric component is determined by the thermal average of
its annihilation cross-section, given by,

$\Omega_{sym}h^{2}=\frac{8.7661\times 10^{-11}}{\sqrt{g_{*s}}J},$ (27)

where $J$ is given by,

$J=\int_{x_{f}}^{\infty}\frac{\langle\sigma
v\rangle_{\overline{\chi}\chi\to~{}\phi\phi}}{x^{2}}\,dx\ .$ (28)

where$\langle\sigma v\rangle_{\overline{\chi}\chi\to~{}\phi\phi}$ is the
thermally averaged cross-section of the dominant annihilation process
$\overline{\chi}\chi\to\phi\phi$ shown in the right panel of Fig. 3, which can
be roughly estimated to be,

$\langle\sigma
v\rangle_{\overline{\chi}\chi\to~{}\phi\phi}\approx\frac{3}{4}\frac{\lambda^{4}_{D}}{16\pi
m^{2}_{\chi}}$ (29)

The amount of annihilation can be very large or small depending on the value
of $\lambda_{D}$. This fixes the contribution of the symmetric (and hence the
asymmetric) component to the total DM relic density which in turn fixes the DM
mass. Using Eq.26, Eq.27, Eq.28, and Eq.29, we can determine the $\lambda_{D}$
parameter with respect to ADM percentage (or equivalently DM mass), which is
depicted by the dashed blue curve in the left panel of Fig. 6. Considering ADM
component to vary within $1-99.99\%$, the parameter $\lambda_{D}$ varies
within $0.005-0.15$ and correspondingly allowed DM mass range is $0.07-7.76$
GeV.

Figure 6: Left: $\lambda_{D}$ vs ADM($\%$) plot with corresponding DM mass in
order to satisfy the correct relic density. The magenta shaded region is
excluded due to insufficient DM self-interaction. Right: Under-abundant relic
of symmetric component of the DM as depicted by the dot-dashed red curve. The
equilibrium number density is depicted by the dotted blue curve.

For fully asymmetric dark matter, i.e., $\rm ADM(\%)>99.99\%$, the relics of
the symmetric component should be very small, which requires large
$\lambda_{D}$ ($\sim 0.15$) for efficient annihilation of the symmetric
component. Such a large $\lambda_{D}$ in turn give rise to sufficient self-
interaction among the DM particles mediated by the light scalar $\phi$ as
depicted in the left panel of Fig. 3. We rule out the region
$\lambda_{D}<0.02$ (correspondingly $m_{\chi}<1.3$ GeV) in the top panel of
Fig. 6 due to insufficient self-interaction ($\sigma/m_{{\rm DM}}<10^{-24}{\rm
cm}^{2}/{\rm GeV}$) which can not alleviate the small scale anomalies of
$\Lambda{\rm CDM}$. We elaborate the details of DM self-interaction in
section-IV. For a fully ADM scenario, the evolution of the relic abundance of
the symmetric component is shown in the right panel of Fig. 6 by solving the
relevant Boltzmann equation for the comoving number density of the symmetric
component $Y_{\chi,sym}=\frac{n_{\chi,sym}}{s}$, where $n_{\chi,sym}$ is the
actual number density and $s$ is the entropy density. The Boltzmann equation
is given by,

$\frac{dY_{\chi,{\rm sym}}}{dx}=-\frac{s(m_{\chi})}{H(m_{\chi})}\langle\sigma
v\rangle_{\overline{\chi}\chi\to~{}\phi\phi}\left(Y^{2}_{\chi,{\rm
sym}}-(Y^{eq}_{\chi})^{2}\right)$ (30)

where $s(m_{\chi})$ and $H(m_{\chi})$ are the entropy density and the Hubble
parameter as a function of DM mass defined as,

$s(m_{\chi})=\frac{2\pi^{2}}{45}g_{*}\,m_{\chi}^{3},\quad
H(m_{\chi})=\frac{\pi}{\sqrt{90}}\frac{\sqrt{g_{*}}}{M^{r}_{pl}}m_{\chi}^{2},$

where $M^{r}_{pl}=2.44\times{10}^{18}{\rm GeV}$ is the reduced Planck mass and
$Y^{\text{eq}}_{\chi}$ is the equilibrium DM number density.

##  IV Dark Matter Self-interaction

The DM $\chi$ has elastic self-scattering, mediated by the light scalar $\phi$
as depicted by the Feynman diagram shown in left panel of Fig. 3, thanks to
the interaction term $\lambda_{D}\overline{\chi}\chi\phi$ in the model
Lagrangian given by Eq. 2. The scattering in non-relativistic limit is well-
described by the attractive Yukawa potential,

$V(r)=\frac{\lambda^{2}_{D}}{4\pi r}e^{-m_{\phi}r}$ (31)

To capture the relevant physics of forward scattering divergence, we define
the transfer cross-section $\sigma_{T}$ as Feng:2009hw ; Tulin:2013teo ;
Tulin:2017ara

$\sigma_{T}=\int d\Omega(1-\cos\theta)\frac{d\sigma}{d\Omega}$ (32)

Depending on the masses of DM ($m_{\chi}$) and the mediator ($m_{\phi}$), as
well as the relative velocity of the colliding particle ($v$) and the coupling
($\lambda^{2}_{D}$), we can identify three distinct regimes. The Born regime
($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})\ll 1,m_{\chi}v/m_{\phi}\geq 1$) is
where the perturbative calculation holds good. Outside the Born regime, we
have the classical regime ($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})\geq
1,m_{\chi}v/m_{\phi}\geq 1$) and the resonant regime
($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})\geq 1,m_{\chi}v/m_{\phi}\leq 1$)
where non-perturbative and quantum-mechanical effects become important. The
self-interaction cross-sections in these regimes are listed in Appendix C. In
the left panel of Fig. 7, we show the self-interaction allowed parameter space
in $m_{\chi}$ \- $m_{\phi}$ plane obtained by constraining $\sigma/m_{\chi}$
in the correct ballpark from astrophysical data across different scales. We
constrain $\sigma/m_{\chi}$ in the range $0.1-100~{}{\rm cm}^{2}/{\rm g}$ for
dwarf galaxies ($v\sim 10~{}\rm km/s$) as shown by the three shades of blue
coloured region as indicated in the figure inset. The light magenta coloured
region depicts the parameter space allowed for galaxies ($\sigma/m_{\chi}\sim
0.1-10~{}{\rm cm}^{2}/{\rm g}$), while the green coloured region depicts the
parameter space allowed for clusters ($\sigma/m_{\chi}\sim 0.1-1~{}{\rm
cm}^{2}/{\rm g}$). The masses of DM and the mediator for which all three
regions overlap will alleviate the small scale anomalies across all scales.

Figure 7: Left: Self-interaction cross-section in the range $0.1-1\;{\rm
cm}^{2}/{\rm g}$ for clusters ($v\sim 1000\;{\rm km/s}$), $0.1-10\;{\rm
cm}^{2}/{\rm g}$ for galaxies ($v\sim 200$ km/s.) and $0.1-1000\;{\rm
cm}^{2}/{\rm g}$ for dwarfs ($v\sim 10$ km/s). Right: The self-interaction
cross-section per unit mass of DM as a function of average collision velocity.

The top (bottom) corner corresponds to the Classical (Born) region, where the
cross-section depends on velocity trivially. The sandwiched region between
these two is the so-called resonant region, where quantum mechanical
resonances and anti-resonances appear due to (quasi)bound state formation in
the attractive potential. The resonances and anti-resonances are most
prominent at the dwarf scale due to the lower velocity of the DM particles and
gradually becomes less prominent towards galaxy and cluster scales as DM
velocity increases and less likely to be bounded. For a coupling
$\lambda_{D}$, the condition $m_{\chi}v/m_{\phi}<1$ dictates the onset of non-
perturbative quantum mechanical effects, which is easily satisfied by smaller
velocities. We have considered in the left panel of Fig. 7, $\lambda_{D}=0.15$
which corresponds to fully asymmetric DM and also is just at the correct
ballpark for sufficient self-interaction allowing maximum parameter space in
the $m_{\chi}-m_{\phi}$ plane. For lower values of $\lambda_{D}$ (which means
DM is not fully asymmetric), the parameter space where desired cross-sections
can be obtained gradually decreases. For $\lambda_{D}<0.02$, the obtained
cross-sections are below the ballpark of $\sigma/m_{\chi}\sim 0.1cm^{2}/g$,
insufficient to alleviate the small scale $\Lambda{\rm CDM}$ anomalies.

The self-scattering cross-section per unit DM mass as a function of average
collision velocity obtained from the model fits to data from dwarfs (red), low
surface brightness (LSB) galaxies (blue), and clusters (green)
Kaplinghat:2015aga ; Kamada:2020buc  as shown in the bottom panel of Fig 7.
The purple curve corresponds to the benchmark point of fully asymmetric DM
($m_{\chi}=7.76~{}\rm GeV$) with $m_{\phi}=0.01{\rm GeV}$ and
$\lambda_{D}=0.15$. We depict this point with a red star mark in Fig. 7 as
well. The brown curve corresponds to the benchmark point of
$m_{\chi}=1.5~{}\rm GeV$ with $m_{\phi}=0.001~{}{\rm GeV}$ and
$\lambda_{D}=0.025$ which just cuts the mark to give sufficient self-
interaction as shown in the top panel of Fig. 6. The magenta curve corresponds
to an intermediate case with $m_{\chi}=5~{}\rm GeV$, $m_{\phi}=0.003~{}{\rm
GeV}$ and $\lambda_{D}=0.05$. Hence, it is clear from the bottom panel of Fig.
7 that the model can appreciably explain the astrophysical observation of
velocity-dependent DM self-interaction.

##  V DM Direct Search

The SIDM can be detected at terrestrial laboratories through $\phi-h$ mixing
($\theta_{\phi h}$), via its scattering off the target nuclei as depicted by
the Feynman diagram shown in Fig. 8. the scattering cross-section of DM per
nucleon can be expressed as

$\sigma_{SI}^{\phi-h}=\frac{\mu_{r}^{2}}{4\pi
A^{2}}\left[Zf_{p}+(A-Z)f_{n}\right]^{2}$ (33)

where $\mu_{r}=\frac{m_{\chi}m_{n}}{m_{\chi}+m_{n}}$ is the reduced mass of
the DM-nucleon system. Here $m_{n}$ is the nucleon (proton or neutron) mass, A
and Z are respectively the mass and atomic number of the target nucleus,
$f_{p}$ and $f_{n}$ are the interaction strengths of proton and neutron with
DM respectively, given as:

$f_{p,n}=\sum\limits_{q=u,d,s}f_{T_{q}}^{p,n}\alpha_{q}\frac{m_{p,n}}{m_{q}}+\frac{2}{27}f_{TG}^{p,n}\sum\limits_{q=c,t,b}\alpha_{q}\frac{m_{p,n}}{m_{q}}\,,$
(34)

where

$\alpha_{q}=\lambda_{D}\theta_{\phi
h}\left(\frac{m_{q}}{v}\right)\left[\frac{1}{m^{2}_{\phi}}-\frac{1}{m^{2}_{h}}\right]\,.$
(35)

In Eq. (34), the values of $f_{T_{q}}^{p,n}$ can be found in Ellis:2000ds .

Figure 8: Spin-independent DM-nucleon scattering via scalar mixing.

The mixing angle $\theta_{\phi h}$ can be derived in terms of the parameters
$\lambda_{H\Phi},\langle\Phi\rangle,v,m_{\phi},m_{h}$. Depending on the value
of $\lambda_{H\Phi}$, the $\phi-h$ mixing can be very small or large.
$\theta_{\phi h}$ gets an upper bound from invisible Higgs decay (since
typically $m_{\phi}<m_{h}$), while it has a conservative lower bound from the
big bang nucleosynthesis (BBN)(since $\tau_{\phi}<\tau_{\rm BBN}$ in order to
keep BBN predictions intact) Borah:2021rbx . Using Eq. (34) and (35), the
spin-independent DM-nucleon scattering cross-section in Eq. (33), can be re-
expressed as:

$\displaystyle\sigma_{\rm SI}^{\phi-h}$ $\displaystyle=$
$\displaystyle\frac{{\mu_{r}}^{2}\lambda_{D}\theta^{2}_{\phi h}}{\pi
A^{2}}\left[\frac{1}{m^{2}_{\phi}}-\frac{1}{m^{2}_{h}}\right]^{2}$
$\displaystyle\times$
$\displaystyle\bigg{[}Z\left(\frac{m_{p}}{v}\right)\left(f_{Tu}^{p}+f_{Td}^{p}+f_{Ts}^{p}+\frac{2}{9}f_{TG}^{p}\right)$
$\displaystyle+$
$\displaystyle(A-Z)\left(\frac{m_{n}}{v}\right)\left(f_{Tu}^{n}+f_{Td}^{n}+f_{Ts}^{n}+\frac{2}{9}f_{TG}^{n}\right)\bigg{]}^{2}.$

Among the direct search experiments, CRESST-III Abdelhameed:2019hmk  provides
the most severe constraint on DM mass below 10 GeV, while XENON1T
Aprile:2018dbl  provides the stringent constraints for DM mass above 10 GeV.
In Fig. 9, these constraints are shown on the $m_{\chi}-m_{\phi}$ plane
against the self-interaction favoured parameter space. The blue (purple)
coloured contours denote exclusion limits from XENON1T (CRESST-III) experiment
for specific $\phi-h$ mixing parameter $\theta_{\phi h}$. The region to the
left of each contour is excluded for that particular $\theta_{\phi h}$. It is
seen from Fig. 9 that direct search experiments severely constrain the self-
interaction favoured parameter space. In particular, for $m_{\chi}=7.76$ GeV
and $m_{\phi}=0.01$ GeV, $\theta_{\phi h}>10^{-9}$ has already been ruled out.
The red star mark depicts the benchmark point for fully asymmetric DM.

Figure 9:  Self-interaction allowed parameter space constrained by DM direct
search in the plane of $(m_{\chi})$ versus $(m_{\phi})$.

##  VI Neutrino Mass

We explain neutrino mass through the higher dimension operator,

$\mathcal{L}_{Dirac}=-y_{1}\frac{\rho_{1}}{M_{X_{1}}^{2}}\overline{L}H\Phi^{\prime}\nu_{R}-y_{2}\frac{\rho_{2}}{M_{X_{2}}^{2}}\overline{L}H\Phi^{\prime}\nu_{R}$
(37)

which is a Dirac-type dimension-5 operatorGu:2006dc , where the trilinear
coupling $\rho$ has mass dimension. This operator shares the essential
features of conventional Majorana-type dimension-5 operatorWeinberg:1979sa ;
Minkowski:1977sc ; Yanagida:1981xy ; GellMann:1980vs ; Mohapatra:1979ia .

Figure 10: Dirac neutrino mass generated through dimension-5 operator.

We get small Dirac neutrino mass after the SM-Higgs H and $\Phi^{\prime}$
acquire vev’s $\langle H\rangle=v$ and $\langle\Phi^{\prime}\rangle=w$
respectively.

$M_{\nu}\simeq
y_{1}\frac{\rho_{1}vw}{M_{X_{1}}^{2}}+y_{2}\frac{\rho_{2}vw}{M_{X_{2}}^{2}}.$
(38)

Again, using the definition $f_{1,2}=\frac{\rho_{1,2}}{M_{X_{1,2}}}$, we can
write,

$M_{\nu}\simeq
y_{1}f_{1}^{2}v\frac{w}{\rho_{1}}+y_{2}f_{2}^{2}v\frac{w}{\rho_{2}}.$ (39)

The mass of the heavy scalars $X_{1,2}$ i.e., $M_{X_{1,2}}$ is already decided
from the requirement of Dirac Leptogenesis to be of
$\mathcal{O}(10^{10,11})\,{\rm GeV}$ (see section-III). To explain the
neutrino mass of the order of $0.1\,{\rm eV}$, we pick a particular solution
for $y_{1,2}$ and $f_{1,2}$, say $y_{1}=2.2\times 10^{-4}$, $y_{2}=1.1\times
10^{-4}$ and $f_{1}=7.3\times 10^{-5}$, $f_{2}=1.1$, which also explains
successfully the lepton asymmetry in section-III. The other parameter which
appear to achieve neutrino mass of $\mathcal{O}(10^{-10})\,{\rm GeV}$ is
$w=10^{2.7}$ GeV.

Thus the parameters $y,\rho$ and $M_{X}$ are the bridging ligands for
leptogenesis and neutrino mass. A typical set of parameters for which the
model actually satisfies all the relevant phenomenology simultaneously can be
given as: $y_{1}=2.2\times 10^{-4}$, $y_{2}=1.1\times 10^{-4}$,
$\rho_{1}=7.3\times 10^{5}$ GeV, $\rho_{2}=1.1\times 10^{11}$ GeV,
$M_{\zeta_{1}}=10^{10}$ GeV, $M_{\zeta_{2}}=10^{11}$ GeV, $m_{\phi}$ =
0.003-0.03 GeV, $\theta_{\phi h}\leq 10^{-9}$, $\omega=10^{2.7}$ GeV, $v=246$
GeV, $u\sim 10^{-2}-10^{-3}$ GeV,
$\lambda_{H}=0.129,\lambda_{\phi}^{\prime}\sim
1,\mu_{1}\sim\mu_{2}\sim-10^{-12}{\rm GeV},\lambda_{H\Phi^{\prime}}\sim
10^{-3},\lambda_{H\Phi}\sim 10^{-10},\lambda_{\Phi\Phi^{\prime}}\sim
10^{-10}$. Note that in the above, we use a range of $m_{\phi}$ values (which
give rise the correct order of DM self-interaction) for which we get a range
of $u$ values.

Corresponding to $\lambda=1.5$ and $\lambda_{\eta H}=1$ (see Eq.4) the mass
scale of $\eta$ is $M_{\eta}=1.1\times 10^{4}$ GeV.

The set of parameters which gives rise to the correct dark matter relic
density is given as: $m_{\chi}$ = 1.3-7.76 GeV, $\lambda_{D}$ = 0.02-0.15.
Since we are considering the dominant asymmetric DM, the relic density is
solely decided by these two parameters only.

##  VII Conclusion

We extended the SM with two super heavy $SU(2)_{L}$ scalar doublets: $X_{i}$
$(i=1,2)$, three right-handed neutrinos: $\nu_{R_{i}}(i=1,2,3)$, two singlet
scalars: $\Phi$, $\Phi^{\prime}$ and a singlet Dirac fermion $\chi$ which
represents the candidate of a self-interacting DM. We assumed a mass hierarchy
among the heavy scalar doublets $X_{i}$, $i=1,2$. As a result the CP-violating
out-of-equilibrium decay of the lightest heavy scalar generated a neutrino
asymmetry in the visible sector. A part of the neutrino asymmetry is then
transferred to the dark sector by a dimension-8 operator: $\mathcal{O}_{8}$,
while a part of the remaining neutrino asymmetry gets converted to the baryon
asymmetry by the Electroweak sphaleron processes. This asymmetry transfer
mechanism establishes a proportionality between the relic densities of dark
matter and baryonic matter. The ratio between these two relic densities fixes
the mass of the DM to be 7.76 GeV, if DM relic is fully asymmetric. However
depending on the fractional contribution from symmetric and the asymmetric
components, the DM mass can vary in the range $0.07-7.76$ GeV. The light
scalar mediator $\phi$ is introduced, not only facilitate velocity-dependent
DM self-interaction to alleviate small-scale issues of $\Lambda{\rm CDM}$, but
also deplete the symmetric component of DM via efficient annihilation process:
$\overline{\chi}\chi\to\phi\phi$. The requirement of sufficient self-
interaction also rules out a region of available parameter space in terms of
DM mass, further restricting it to $1.3-7.76$ GeV. To realise sufficient DM
self-interaction with this mass range of DM, the light scalar mass must be in
the range $0.003-0.03$ GeV. The self-interaction allowed parameter space has
also been confronted with bounds from early Universe physics like BBN and
present Universe physics like DM direct search and allowed parameter space
from all phenomenological constraints has been specified. We have also
explained the small Dirac neutrino mass at tree-level through a dimension-five
operator, where the mass of the heavy scalar $X$ suppresses the weak scale.
$M_{X_{1}}$ is decided from the requirement of Dirac Leptogenesis to be of
$\mathcal{O}(10^{10})\,{\rm GeV}$ (see III). To generate the desired lepton
asymmetry, the branching ratio $B_{L}(X\to L\nu_{R})\sim 0.9$, which requires
that $f_{1}=\frac{\rho_{1}}{M_{X_{1}}}$ and $y_{1}$ differ roughly by an order
of magnitude. A typical solution $y_{1}=2.2\times 10^{-4}$, $y_{2}=1.1\times
10^{-4}$ and $f_{1}=7.3\times 10^{-5}$, $f_{2}=1.1$ simultaneously satisfies
both lepton asymmetry and neutrino mass. Therefore, the model at hand
successfully explains the baryon asymmetry of the Universe along with neutrino
mass and DM relic while providing a solution for small scale $\Lambda{\rm
CDM}$ anomalies via DM self-interaction.

## Acknowledgement

MD acknowledges SERB CORE Grant CRG/2018/004971. NS acknowledges the support
from Department of Atomic Energy (DAE)-Board of Research in Nuclear Sciences
(BRNS), Government of India (Ref. Number: 58/14/15/2021-BRNS/37220).

##  Appendix A Asymmetry transfer from visible to dark sector

The asymmetry in the equilibrium number densities of particle $n_{i}$ over
antiparticle $\overline{n}_{i}$ can be written as,

$n_{i}-\overline{n}_{i}=\frac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}dq~{}q^{2}\left[\frac{1}{e^{\frac{E_{i}(q)-\mu_{i}}{T}}\pm
1}-\frac{1}{e^{\frac{E_{i}(q)+\mu_{i}}{T}}\pm 1}\right]\,,$ (40)

where the $g_{i}$ is the internal degrees of freedom of the particle species
$i$. In the above equation $E_{i}$ and $q_{i}$ represent the energy and
momentum of the particle species $i$. In the approximation of a weakly
interacting plasma, where $\beta\mu_{i}\ll 1$, $\beta\equiv 1/T$ (detailed
discussions are given in Feng:2012jn ; Kolb:1990vq ) we get,

$\displaystyle n_{i}-\overline{n}_{i}$ $\displaystyle\sim$
$\displaystyle\frac{g_{i}T^{3}}{6}\times[2\beta\mu_{i}+\mathcal{O}\left((\beta\mu_{i})^{3}\right)~{}~{}~{}~{}{\rm
for\,\,bosons}$ $\displaystyle\sim$
$\displaystyle\frac{g_{i}T^{3}}{6}\times[\beta\mu_{i}+\mathcal{O}\left((\beta\mu_{i})^{3}\right)~{}~{}~{}~{}~{}{\rm
for\,\,fermions}.$

In this model, the asymmetry transfer operator is given by ${\cal
O}_{8}=\frac{1}{M_{asy}^{4}}\overline{\chi}^{2}(LH)^{2}$ so the decoupling
temperature of operator ${\cal O}_{8}$ is depends on the value of $M_{\rm
asy}$. Since in this model the $B-L$ asymmetry is generated in the standard
model sector which is required to be shared to the dark sector via ${\cal
O}_{8}$ operator, we assume the decoupling temperature $T_{D}$ to be
$T_{t}>T_{D}>T_{W}$, where $T_{t}$ is the temperature of thermal bath when the
top quark decouples and $T_{W}$ is the temperature when the $W$ boson
decouples from the thermal plasma. In this case the effective Lagrangian for
Yukawa coupling is given by:

$\mathcal{L}_{Yukawa}=g_{e_{i}}^{k}\bar{e}_{iL}h^{k}e_{iR}+g_{u_{i}}^{k}\bar{u}_{iL}h^{k}u_{iR}+g_{d_{i}}^{k}\bar{d}_{iL}h^{k}d_{iR}+h.c$
(42)

where $k=1,2,3$ for three scalar mass eigenstates $h_{1}$, $h_{2}$ and
$h_{3}$. Where $h_{1}$ is identified as standard model Higgs boson with mass
$125$ GeV. All these scalar fields are real so the above Lagrangian gives the
following chemical equilibrium condition,

$0=\mu_{h^{k}}=\mu_{u_{L}}-\mu_{u_{R}}=\mu_{d_{L}}-\mu_{d_{R}}=\mu_{e_{L}}-\mu_{e_{R}}.$
(43)

After electroweak symmetry breaking the charged current interaction part of
the SM Lagrangian is given by,

$\mathcal{L}_{int}^{(W)}=gW_{\mu}^{+}\bar{u}_{L}\gamma^{\mu}d_{L}+gW_{\mu}^{+}e_{L}\gamma^{\mu}\bar{\nu}_{eL}\,.$
(44)

The charged current interactions remain in thermal equilibrium until W-boson
decouples from thermal bath. Which gives the following chemical potential
equations,

$\mu_{W}=\mu_{u_{L}}-\mu_{d_{L}},$ (45)

and

$\mu_{W}=\mu_{\nu}-\mu_{e_{L}}.$ (46)

The electroweak sphalerons remain in thermal equilibrium until a temperature
$T_{\rm sph}\gtrsim T_{W}$ leads to the following constraint,

$\mu_{u_{L}}+2\mu_{d_{L}}+\mu_{\nu}=0.$ (47)

At a temperature below electroweak phase transition, the electric charge
neutrality of the Universe holds. However, at the epoch: $T_{t}>T_{D}>T_{W}$,
the top quark is already decoupled from the thermal plasma and hence does not
take part in the charge neutrality condition. Therefore, we get

$Q=4(\mu_{u_{L}}+\mu_{u_{R}})+6\mu_{W}-3(\mu_{d_{L}}+\mu_{d_{R}}+\mu_{e_{L}}+\mu_{e_{R}})=0.$
(48)

Using Eq A, the baryon number density $n_{B}$ can be given as,

$n_{B}=\sum_{i}\mu_{i}\big{(}C_{i}Q_{i}^{B}\frac{g_{i}T^{3}\beta}{6}\big{)}$
(49)

where, $i$ runs over two generations of up quarks and three generations of
down quarks, $C_{i}$ and $Q_{i}^{B}$ count respectively the color and the
baryon number of $i^{th}$ quark. Note that top quark is decoupled since it is
heavy. Here, $g_{i}=2$ is the internal degrees of freedom of each quark.
Similarly, following Eq A, the lepton number density $n_{L}$ can be given as,

$n_{L}=\sum_{i}\mu_{i}\big{(}\frac{g_{i}T^{3}\beta}{6}\big{)}$ (50)

where, $i$ runs over three generations of charged and neutral leptons. Now
using the Eqs. 43-48, we can write the net baryon and lepton number density
$n_{B}$ and $n_{L}$ as,

$n_{B}=-\frac{90}{29}\mu_{\nu}$ (51)

and

$n_{L}=\frac{201}{29}\mu_{\nu}\,,$ (52)

where we have dropped the common factor $gT^{3}\beta/6$ as we are interested
in ratio of densities, rather than their individual values. The net $B-L$
asymmetry in the visible sector is thus given by,

$(n_{B-L})_{\rm vis}=-\frac{291}{29}\mu_{\nu}\,.$ (53)

After sphaleron processes decouple at $T_{\rm sph}$, the baryon and lepton
number densities would be conserved separately. As a result Eqs. 43 \- 53
would remain valid at $T_{\rm sph}>M_{W}$. Once the sphaleron processes
decouple, the ratio of $n_{B}/n_{B-L}$ would be frozen. As a result from Eqs.
51 and 53, it can be written as,

$\frac{n_{B_{\rm final}}}{(n_{B-L})_{\rm vis}}=\frac{n_{B}}{(n_{B-L})_{\rm
vis}}=\frac{30}{97}=0.31$ (54) $n_{B_{\rm final}}=0.31(n_{B-L})_{\rm vis}.$
(55)

The operator ${\cal O}_{8}$ is in equilibrium until $T_{D}>T_{\rm sph}$ and
equilibration of ${\cal O}_{8}$ gives the following constraint,

$\mu_{\chi}=\mu_{\nu}\,.$ (56)

As a result the number density of dark matter $\chi$ (it is basically $B-L$
asymmetry in the dark sector) is given by, is given by,

$\displaystyle n_{\chi}$ $\displaystyle=2\mu_{\chi}\frac{gT^{3}\beta}{6}$ (57)
$\displaystyle=\frac{58}{291}(n_{B-L})_{\rm vis}\equiv(n_{B-L})_{\rm dark}$

##  Appendix B Diagonalization of the Scalar Mass Matrix

For simplicity, let us denote the scalar mass matrix given in Eq. 9 as:

$\begin{pmatrix}A&B&C&\\\ B&D&E&\\\ C&E&F&\\\ \end{pmatrix}$ (58)

This mass matrix can be approximately block-diagonalized by the matrix:

$U=\begin{pmatrix}\cos\alpha&0&\sin\alpha&\\\ 0&\cos\alpha&\sin\alpha&\\\
-\sin\alpha&-\sin\alpha&\cos\alpha&\\\ \end{pmatrix}$ (59)

which is unitary upto order $\sin\alpha$. The block diagonalized matrix is of
the form:

$\begin{pmatrix}A^{\prime}&B^{\prime}&0&\\\ B^{\prime}&D^{\prime}&0&\\\
0&0&F^{\prime}&\\\ \end{pmatrix}$ (60)

where the elements of the above matrix are as follows:

$\displaystyle A^{\prime}$
$\displaystyle=A\cos^{2}\alpha+F-\sin^{2}\alpha-C\sin 2\alpha$ (61)
$\displaystyle B^{\prime}$
$\displaystyle=B\cos^{2}\alpha+F\sin^{2}\alpha-\frac{1}{2}(C+E)\,\sin 2\alpha$
$\displaystyle D^{\prime}$
$\displaystyle=D\cos^{2}\alpha+F\sin^{2}\alpha-E\sin 2\alpha$ $\displaystyle
F^{\prime}$
$\displaystyle=F\cos^{2}\alpha+(A+2B+D)\,\sin^{2}\alpha+(C+E)\,\sin 2\alpha$

The above diagonalization is obtained for the values of the angle $\alpha$
given by(ignoring $\mathcal{O}(\sin^{2}\alpha)$ terms):

$\tan\alpha=\frac{C}{F-A-B}~{}~{}~{}~{}{\rm(or)}~{}~{}~{}~{}\frac{E}{F-B-D}\,\,.$
(62)

We assume the mass of $\phi$ to be extremely small compared to that of both
$\phi^{\prime}$ and $h$ and it has very small mixing with both of them. We
consider the induced vev u of the field $\Phi$ to be very small compared to
that of SM Higgs. In the limit of zero $\phi^{\prime}-\phi$ as well as
$h-\phi$ mixing, which is indeed guaranteed by the extremely tiny mixing
parameter $\tan\alpha$ given by Eq. 62, $\phi$ decouples from both
$\phi^{\prime}$ and h. The smallness of $\tan\alpha$ can be understood with
the help of parameter values provided in Sec.-VI viz. $\omega=10^{2.7}$ GeV,
$v=246$ GeV, $u\sim 10^{-2}$ GeV, $\lambda_{\phi}^{\prime}\sim
1,\lambda_{H}=0.129$, $\mu_{2}\sim-10^{-12}$, $\lambda_{\phi}^{\prime}\sim
1,\lambda_{H\Phi^{\prime}}\sim 10^{-3},\lambda_{H\Phi}\sim
10^{-10},m_{\phi}\sim 10^{-2}$ GeV, which gives
$\tan\alpha\sim\mathcal{O}(10^{-12})$. The exact diagonalization is obtained
by giving a consecutive (1,2) rotation to the squared mass matrix given in Eq.
60 by the following Euler rotation matrix:

$O=\begin{pmatrix}\cos\beta&-\sin\beta&0&\\\ \sin\beta&\cos\beta&0&\\\
0&0&1&\\\ \end{pmatrix}$ (63)

where the $\phi^{\prime}-h$ mixing is given by,

$\tan 2\beta=\frac{2B^{\prime}}{D^{\prime}-A^{\prime}}$ (64)

After complete diagonalization, we are left with an extremely light scalar
$h_{3}\approx\phi$ that mediates DM self-scattering and two other scalars with
masses given by,

$\displaystyle m^{2}_{h_{1}}$
$\displaystyle=D^{\prime}\cos^{2}\beta+A^{\prime}\sin^{2}\beta+B\sin 2\beta$
(65) $\displaystyle m^{2}_{h_{2}}$
$\displaystyle=A^{\prime}\cos^{2}\beta+D^{\prime}\sin^{2}\beta-B\sin
2\beta~{}.$

The mass eigenstate $h_{1}$ can be identified as the SM Higgs with mass
$M_{h_{1}}=125.18\,{\rm GeV}$, while $h_{2}$ as the second scalar that plays a
role in generating the tiny neutrino mass.

##  Appendix C DM Self-interaction Cross-sections at Low Energy

In the Born Limit ($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})<<1$),

$\sigma^{Born}_{T}=\frac{\lambda^{4}_{D}}{2\pi
m^{2}_{\chi}v^{4}}\Bigg{(}ln(1+\frac{m^{2}_{\chi}v^{2}}{m^{2}_{\phi}})-\frac{m^{2}_{\chi}v^{2}}{m^{2}_{\phi}+m^{2}_{\chi}v^{2}}\Bigg{)}$
(66)

Outside the Born regime ($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})\geq 1$),
there are two distinct regions viz, the classical regime and the resonance
regime. In the classical regime ($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi}\geq
1,m_{\chi}v/m_{\phi}\geq 1$), the solutions for an attractive potential is
given byTulin:2013teo ; Tulin:2012wi ; Khrapak:2003kjw :

$\sigma^{classical}_{T}=\left\\{\begin{array}[]{l}\frac{4\pi}{m^{2}_{\phi}}\beta^{2}ln(1+\beta^{-1})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\beta\leqslant
10^{-1}\\\
\frac{8\pi}{m^{2}_{\phi}}\beta^{2}/(1+1.5\beta^{1.65})~{}~{}~{}~{}~{}~{}~{}~{}10^{-1}\leq\beta\leqslant
10^{3}\\\
\frac{\pi}{m^{2}_{\phi}}(ln\beta+1-\frac{1}{2}ln^{-1}\beta)~{}~{}~{}~{}~{}\beta\geq
10^{3}\\\ \end{array}\right.$ (67)

where $\beta=2\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})v^{2}$ In the resonant
regime ($\lambda^{2}_{D}m_{\chi}/(4\pi m_{\phi})\geq 1,m_{\chi}v/m_{\phi}\leq
1$), the quantum mechanical resonances and anti-resonance in $\sigma_{T}$
appear due to (quasi-)bound states formation in the attractive potential. In
the resonant regime, an analytical formula for $\sigma_{T}$ is not available
and one needs to solve the non-relativistic Schrodinger equation by partial
wave analysis. Instead, here we use the non-perturbative results obtained by
approximating the Yukawa potential to be a Hulthen potential
$\Big{(}V(r)=\pm\frac{\lambda^{2}_{D}}{4\pi}\frac{\delta e^{-\delta
r}}{1-e^{-\delta r}}\Big{)}$, which is given by Tulin:2013teo :

$\sigma^{Hulthen}_{T}=\frac{16\pi\sin^{2}\delta_{0}}{m^{2}_{\chi}v^{2}}$ (68)

where $l=0$ phase shift $\delta_{0}$ is given in terms of the $\Gamma$
functions by :

$\displaystyle\delta_{0}$
$\displaystyle=arg\Bigg{(}i\Gamma\Big{(}\frac{im_{\chi}v}{k~{}m_{\phi}}\Big{)}\bigg{/}{\Gamma(\lambda_{+})\Gamma(\lambda_{-})}\Bigg{)}$
$\displaystyle\lambda_{\pm}$
$\displaystyle=\begin{array}[]{l}1+\frac{im_{\chi}v}{2~{}k~{}m_{\phi}}\pm\sqrt{\frac{\alpha_{D}m_{\chi}}{km_{\phi}}-\frac{m^{2}_{\chi}v^{2}}{4k^{2}m^{2}_{\phi}}}\\\
\end{array}$ (70)

and $k\approx 1.6$ is a dimensionless number.

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