# Conformal B-L and Pseudo-Goldstone Dark Matter

Rabindra N. Mohapatraa Nobuchika Okadab a Maryland Center for Fundamental
Physics and Department of Physics, University of Maryland, College Park,
Maryland 20742, USA  b Department of Physics, University of Alabama,
Tuscaloosa, Alabama 35487, USA

###### Abstract

We show that a conformal extension of the standard model with local $B-L$
symmetry and two complex scalars breaking $B-L$ can provide a unified
description of neutrino mass, origin of matter and dark matter. There are two
hierarchical $B-L$ breaking vacuum expectation value (VEV) scales in the
model, the higher denoted by $v_{B}$ and the lower by $v_{A}$. The higher
breaking scale is dynamically implemented via the Coleman-Weinberg mechanism
and plays a key role in the model since it induces electroweak symmetry
breaking as well as the lower $B-L$ breaking scale. It is also responsible for
neutrino masses via the seesaw mechanism and origin of matter. The imaginary
part of the complex scalar with lower $B-L$ breaking VEV plays the role of a
pseudo-Goldstone dark matter (DM). The DM particle is unstable with its
lifetime naturally longer than $10^{28}$ seconds. We show that its relic
density arises from the freeze-in mechanism for a wide parameter domain. Due
to the pseudo-Goldstone boson nature of the DM particle, the direct detection
cross section is highly suppressed. The model also predicts the dark matter to
be heavier than 100 TeV and it decays to two high energy neutrinos which can
be observable at the IceCube, providing a test of this model.

##  I 1\. Introduction

Understanding the origin of the electroweak scale is a fundamental puzzle of
the standard model (SM). In usual discussion of the SM, it is customary to put
the scale by hand in the form of a negative mass squared for the Higgs field.
A more satisfactory approach, extensively discussed in the literature (see for
some examples C1 ; C1.1 ; C1.11 ; C1.2 ; C1.3 ; C1.4 ; C1.5 ; C1.6 ; C2 ; C3 ;
C3.1 ; C4 ; C4.1 ; C4.2 ; C5 ; C5.2 ; C5.1 ; C5.3 ) is to consider conformal
extensions of the SM where the mass terms vanish due to conformal symmetry and
the mass scales arise dynamically via the Coleman-Weinberg radiative
correction mechanism CW . This approach when implemented in the SM leads to a
very small mass for the Higgs boson and is ruled out by experiments. However,
it can be implemented generally in the context of many extensions of the SM
(see for instance C1 ; C1.1 ; C1.11 ; C1.2 ; C1.3 ; C1.4 ; C1.5 ; C1.6 ; C2 ;
C3 ; C3.1 ; C4 ; C4.1 ; C4.2 ; C5 ; C5.2 ; C5.1 ; C5.3 ). It is important to
explore such models and pinpoint their tests. The goal of this paper is to
present the phenomenological possibilities for dark matter in one such model.

Local $B-L$ extension of the SM marshak1 ; marshak2 ; davidson  has been
discussed as a very highly motivated minimal scenario for neutrino masses via
the seesaw mechanism ss1 ; ss2 ; ss3 ; ss4 ; ss5  and the origin of matter.
Phenomenology of these models have been also extensively discussed in the
literature; see for some examples BL1 ; BL2 ; BL3 ; BL3.1 ; BL3.2 ; BL4 ; BL5
; MO1 ; BL5.1 ; BL6 . While most phenomenological discussion of the $B-L$
models have been done in connection with neutrino masses and collider physics,
we recently pointed out another virtue MO1  of these models and showed that
the real part of the SM singlet Higgs field that breaks $B-L$ can be a
perfectly viable candidate for decaying dark matter. This dark matter model
works only for very small values of the gauge coupling and low mass of the
dark scalar (with masses in the MeV to a few keV range). The dark matter relic
density in this case arises via the freeze-in mechanism hall  and has
interesting experimental tests. The FASER experiment faser  for example is
ideally suited for testing this model. Here we present a different approach
where we impose conformal invariance on this theory so that we can understand
the origin of masses and explore if there is a dark matter candidate. We find
that we need to extend the minimal $B-L$ model by enlarging the Higgs sector
to include two complex SM singlet scalars which carry $B-L$ quantum numbers.
In this case, the imaginary part of one of the two $B-L$ breaking scalars can
be a viable dark matter candidate. In fact the model predicts which particle
can be the dark matter. It turns out that the dark matter is a psudo-Goldstone
boson pgb1 ; pgb2 ; pgb3 ; pgb4 ; pgb5 ; pgb6 ; pgb7 ; pgb8 ; pgb9 ; pgb10 ,
which explains why it has escaped direct detection.

The two $B-L$ non-singlet scalars in our model have different VEVs. The higher
scale $B-L$ breaking, which we call the primary breaking scale in the theory,
arises from the Coleman-Weinberg mechanism by dimensional transmutation. The
neutrino masses and the electroweak symmetry breaking C2  as well as the lower
scale $B-L$ breaking VEV are all induced dynamically by the high scale $B-L$
breaking VEV. We show that in a wide parameter range, the imaginary part of
the second $B-L$ breaking complex scalar is constrained to have a very long
life farinaldo ; icecube ; magic  and plays the role of dark matter. An
interesting prediction of the model is that the dark matter mass is more than
100 TeV with its dominant decay mode being to two neutrinos. The model can
therefore be tested by observation of TeV neutrinos by the ICE CUBE
experiment. The baryon asymmetry of the universe in this model is generated by
the leptogenesis mechanism Fukugita-Yanagida .

The paper is organized as follows: after briefly introducing the model and
discussing $B-L$ symmetry breaking in sec. 2. We discuss the pseudo-Goldstone
dark matter ($\sigma$) in sec. 3 and its lifetime in sec. 4. We show that the
lightest $B-L$ breaking scalar is naturally long lived enough for it to play
the role of dark matter. In sec. 5, we show how the freeze-in mechanism
determines the relic density of dark matter and the various constraints which
imply on the parameter space of the model. In sec. 6, we discuss the
constraints on the parameter space of the model and in sec. 7, we give some
tests and comment on other aspects of the model, such as leptogenesis. In
sec.8 we summarize our results and conclude.

##  II 2\. Brief overview of the model

Our model is based on the local $U(1)_{B-L}$ extension of the SM with gauge
quantum numbers of fermion under $U(1)_{B-L}$ defined by their baryon or
lepton number. The full gauge group of the model is $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$, where $Y$ is the SM hypercharge.
We need to add three right handed neutrinos (RHNs) with $B-L=-1$ to cancel the
$B-L$ gauge anomaly. The RHNs being SM singlets do not contribute to SM
anomalies. The electric charge formula in this case is same as in the SM.

For the Higgs sector, in addition to the SM Higgs doublet $H$ which has zero
$B-L$, we include two SM singlet Higgs fields: $\Phi_{B}$ with $B-L=-2$ and
$\Phi_{A}$ with $B-L=+6$. The tree level potential is given by

$\displaystyle V(H,\Phi_{A},\Phi_{B})$ $\displaystyle=$
$\displaystyle\lambda_{H}(H^{\dagger}H)^{2}+\lambda_{A}(\Phi^{\dagger}_{A}\Phi_{A})^{2}+\lambda_{B}(\Phi^{\dagger}_{B}\Phi_{B})^{2}$
(1)
$\displaystyle+\lambda_{HA}(H^{\dagger}H)(\Phi^{\dagger}_{A}\Phi_{A})+\lambda_{HB}(H^{\dagger}H)(\Phi^{\dagger}_{B}\Phi_{B})+\lambda_{mix}(\Phi^{\dagger}_{A}\Phi_{A})(\Phi^{\dagger}_{B}\Phi_{B})$
$\displaystyle-\lambda_{AB}\Phi_{A}\Phi^{3}_{B}+h.c.$

Note that there are no mass parameters in the potential neither any mass terms
in the fermion sector of the Lagrangian making the theory conformal invariant.

We break the $B-L$ symmetry in stages: the first stage is by giving VEV to the
$B-L=2$ field $\Phi_{B}$, with $\langle\Phi_{B}\rangle=v_{B}/\sqrt{2}$; at the
second stage, $\Phi_{A}$ acquires a lower scale VEV,
$\langle\Phi_{A}\rangle=v_{A}/\sqrt{2}$. As we will see, $v_{A}$ is induced by
the VEV $v_{B}$. We will show that $v_{B}\gg v_{A}$ naturally. The second VEV
is induced by the high scale VEV via the $\lambda_{AB}$ term in the scalar
potential. To induce the SM electroweak symmetry breaking, we choose the
potential parameter $\lambda_{HB}<0$ and adjust $~{}|\lambda_{HB}|v^{2}_{B}$
to be of order of the electroweak VEV squared C2 . We discuss all these below.

##  III 3\. Symmetry breaking and pseudo-Goldstone dark matter

The first stage of the symmetry breaking is induced by the Coleman-Weinberg
mechanism as follows: We write the one-loop potential involving the $B-L$
breaking fields $\Phi_{A,B}$ near the $\Phi_{B}$ VEV as

$\displaystyle V^{\rm
1-loop}~{}=~{}\frac{3g^{4}_{BL}}{16\pi^{2}}\left(36|\Phi_{A}|^{2}+4|\Phi_{B}|^{2}\right)^{2}\left[\ln\left(\frac{36|\Phi_{A}|^{2}+4|\Phi_{B}|^{2}}{\mu^{2}}\right)-\frac{5}{6}\right].$
(2)

Now if we choose $\lambda_{B}\sim g^{4}_{BL}$ and $\lambda_{A}\gg g^{4}_{BL}$,
then the field $\Phi_{B}$ will acquire VEV
($\langle\Phi_{B}\rangle=v_{B}/\sqrt{2}$) while $\Phi_{A}$ will have zero VEV.
Clearly now, $\lambda_{HB}<0$ will induce a negative mass term for the SM
Higgs field $H$ and will generate it a VEV. The magnitude of this VEV is
adjusted by choosing the value of $\lambda_{HB}$ appropriately. Thus, although
the magnitude of the weak scale is not explained, its origin is more
meaningful than in the usual discussion. We also note that there is no lower
bound on the value of $|\lambda_{HB}|$ since the coupling is not induced by
the $B-L$ gauge boson loop diagrams, and hence $v_{B}$ is a free parameter of
the model.

To induce the $\Phi_{A}$ VEV, we minimize the effective potential below the
$v_{B}$ scale. We write $\Phi_{A}=\frac{1}{\sqrt{2}}(\varphi_{A}+i\chi_{A})$.
Assuming $\lambda_{mix}$ very small and neglecting quantum corrections to
$\lambda_{A}$, the effective potential below the mass scale $v_{B}$ can be
written as

$\displaystyle
V_{eff}~{}=~{}-\frac{\lambda_{AB}}{4}v^{3}_{B}\varphi_{A}+\frac{\lambda_{A}}{4}(\varphi^{2}_{A}+\chi^{2}_{A})^{2}.$
(3)

Minimizing this effective potential, we find that

$\displaystyle
v_{A}\simeq\left(\frac{\lambda_{AB}}{4\lambda_{A}}\right)^{1/3}v_{B};$ (4)
$\displaystyle
m_{\varphi^{\prime}_{A}}\simeq\sqrt{3\lambda_{A}}v_{A};~{}~{}~{}m_{\sigma}\simeq\sqrt{\lambda_{A}}v_{A},$

where $\varphi^{\prime}_{A}\simeq\varphi_{A}+\beta\varphi_{B}$ and
$\sigma\simeq\chi_{A}+\alpha\chi_{B}$ are the mass eigenstates with
$\beta,\alpha$ being of order $\frac{v_{A}}{v_{B}}\ll 1$, obtained through the
mixing with $\varphi_{B}$ and $\chi_{B}$ defined by
$\Phi_{B}=\frac{1}{\sqrt{2}}(v_{B}+\varphi_{B}+i\chi_{B})$. Note that $\sigma$
is a pseudo-Goldstone boson. To see the pseudo-Goldstone boson nature of
$\sigma$, note that in the limit of $\lambda_{AB}=0$, the theory has global
$U(1)\times U(1)$ symmetry and there are two massless Nambu-Goldstone bosons;
however once the $\lambda_{AB}$ term is switched on, the symmetry reduces to
only one, the $U(1)_{B-L}$ gauge symmetry and the $\sigma$ field picks up mass
proportional to $\lambda_{AB}$. It has the lowest mass among the $B-L$
breaking scalar fields and is a pseudo-Goldstone boson.

Now adjusting $\lambda_{AB}$ we can make the $v_{A}$ much lower than the
primary $B-L$ breaking scale $v_{B}$. From the mass calculation, we see that
$\sigma$ is the lighter of the two particles $\varphi_{A}$ and $\sigma$ and
can be the dark matter of the universe as we see in detail in the next
section. It is a pseudo-Goldstone dark matter pgb1 ; pgb2 ; pgb3 ; pgb4 ; pgb5
; pgb6 ; pgb7 ; pgb8 . In this discussion we have neglected the mixing between
the $\chi_{A}$ and $\chi_{B}$ which will occur when both $v_{A}$ and $v_{B}$
are nonzero. This mixing is proportional to $\alpha\sim\frac{v_{A}}{v_{B}}$,
which we take into account in our discussion below. The linear combination of
fields $\chi_{B}^{\prime}\simeq\chi_{B}-\alpha\chi_{A}$ becomes the
longitudinal mode of the $B-L$ gauge boson $Z^{\prime}$ and the orthogonal
combination $\sigma\simeq\chi_{A}+\alpha\chi_{B}$ becomes the dark matter.
This is an unstable dark matter, and we study its detailed properties in the
next section.

Note that the coupling $fNN\Phi_{B}^{\dagger}$ gives mass to the right handed
neutrinos of magnitude $M_{N}\sim fv_{B}$ implementing the seesaw mechanism
for neutrinos. Unlike the model in MO1 , the real part of $\Phi_{B}$ field
decays rapidly as the universe evolves. The out-of-equilibrium decay of RHNs
in our model is responsible for leptogenesis.

##  IV 4\. Dark matter lifetime

As noted above, the field $\sigma$ can play the role of dark matter of the
model, if it satisfies the lifetime constraints. There are two lifetime
constraints: one from the search for cosmic ray neutrinos from decaying dark
matter with IceCube icecube . This puts a lower bound on $\tau_{DM}\gtrsim
10^{28}$ sec for then mass of decaying dark matter in the range of
$10^{4}<m_{DM}[{\rm GeV}]<10^{9}$. A second limit comes from the Fermi-Lat
search for gamma rays from dwarf spheroidal galaxies farinaldo . There are
also limits from deep gamma ray survey from Perseus Galaxy Cluster by MAGIC
collaboration magic . As we will see below, the first one is directly
applicable to our case and not the others.

Note that $\sigma$ does not couple to SM fermions at the tree level. To
estimate its lifetime, we first study its decay properties. In our model, it
is natural that $M_{N}>m_{\sigma}$, since $M_{N}\sim fv_{B}$ and
$m_{\sigma}\simeq\sqrt{\lambda_{A}}v_{A}$ with $v_{A}\ll v_{B}$. The possible
decay modes of $\sigma$ are as follows:

Figure 1: Dominant decay mode of the dark matter $\sigma$ in the model to two
neutrinos is shown. The decay proceeds through the mixing $\sim v_{A}/v_{B}$
of the dark matter with $\chi_{B}$ and the mixings of right handed neutrinos
to light neutrinos in the seesaw mechanism $\sim m_{D}/M_{N}$.

(i) $\chi_{\sigma}\to NN\to\nu\nu$ (see Fig. 1). Here, the $N$ is a virtual
state. The decay proceeds via the mixing of $\chi_{A}$ with $\chi_{B}$
proportional to $\alpha\sim\frac{v_{A}}{v_{B}}$ in amplitude. It is followed
by the mixing between $N$ and $\nu$ through the seesaw mechanism $\sim
m_{D}/M_{N}$, where $m_{D}$ is the neutrino Dirac mass. The decay width in
this case can be estimated to be

$\displaystyle\Gamma_{\sigma\to\nu\nu}\simeq\frac{1}{4\pi}\left(\frac{m_{\nu}}{v_{B}}\right)^{2}\left(\frac{v_{A}}{v_{B}}\right)^{2}m_{\sigma}.$
(5)

Here, we have used the seesaw formula for the light neutrino mass,
$m_{\nu}\sim m_{D}^{2}/M_{N}$.

(ii) Another possible mode is $\sigma\to\varphi_{A}^{\prime}+Z^{\prime}\to
NNf\bar{f}\to\nu\nu f\bar{f}$, which is highly suppressed compared to
$\Gamma_{\sigma\to\nu\nu}$ as is found to be

$\displaystyle\Gamma^{Z^{\prime}}_{\sigma\to\nu\nu
f\bar{f}}\simeq\frac{1}{(4\pi)^{5}}\left(\frac{m_{\nu}}{v_{B}}\right)^{2}\left(\frac{v_{A}}{v_{B}}\right)^{2}\frac{m^{5}_{\sigma}}{v^{4}_{B}}.$
(6)

The bottom line of the above discussion is that
$\frac{\lambda_{AB}}{\lambda_{A}}\sim\left(\frac{v_{A}}{v_{B}}\right)^{3}$
must be very small for $\sigma$ to be a viable dark matter. We also have
assumed that $\lambda_{AH}$ is very small.

##  V 5\. Dark matter relic density

In order to discuss how relic density of DM arises in this model, we use the
freeze-in mechanism hall . The requirement for the freeze-in mechanism is that
the dark matter must be out of equilibrium with the cosmic soup of the SM
particles. We first require that the reheat temperature of the universe after
inflation $T_{R}<M_{Z^{\prime}}$. This is to avoid a resonance enhancement of
the dark matter production by the $Z^{\prime}$ mediated process which leads to
an over DM production unless the $B-L$ gauge coupling ($g_{BL}$) is extremely
small.

For $m_{\sigma}^{2},m^{2}_{\varphi^{\prime}_{A}}\ll s\ll M_{Z^{\prime}}^{2}$,
the DM annihilation process with $\varphi_{A}^{\prime}$ to a pair of SM
fermions ($f\bar{f}$) is given by

$\displaystyle\sigma_{DM}\equiv\sigma_{{\sigma\varphi_{A}^{\prime}}\to
f\bar{f}}\simeq\frac{g^{4}_{BL}s}{12\pi
M^{4}_{Z^{\prime}}}Q^{2}_{\Phi_{A}}(\sum_{f}N_{f}Q^{2}_{f}),$ (7)

where $(Q_{f},N_{f})=(1/3,3)$ for a quark, $(-1,1)$ for a charged lepton, and
$(-1,1/2)$ for an SM neutrino, and $Q_{\Phi_{A}}=6$. Counting all SM fermions
and RHNs for the final states ($\sum_{f}N_{f}Q^{2}_{f}=8$) and using
$M_{Z^{\prime}}\simeq 2g_{BL}v_{B}$, this reduces to

$\displaystyle\sigma v_{rel}\simeq\frac{3s}{\pi v^{4}_{B}},$ (8)

where $v_{rel}$ is the relative velocity of the initial particles. For a
temperature of the early universe
$m_{\sigma}^{2},m^{2}_{\varphi^{\prime}_{A}}\ll T^{2}\ll M_{Z^{\prime}}^{2}$,
the thermal average of the above cross section is found to be

$\displaystyle\langle\sigma v_{rel}\rangle\simeq\frac{36T^{2}}{\pi
v^{4}_{B}}=\frac{36m_{\sigma}^{2}}{\pi v^{4}_{B}}x^{-2},$ (9)

where $x\equiv\frac{m_{\sigma}}{T}$.

The freeze-in build-up happens via the reaction
$f\bar{f}\to\sigma\varphi^{\prime}_{A}$ via $Z^{\prime}$ exchange and the DM
yield $Y$ obeys the Boltzmann equation:

$\displaystyle\frac{dY}{dx}\simeq\frac{\langle\sigma
v_{rel}\rangle}{x^{2}}\frac{s(m_{\sigma})}{H(m_{\sigma})}Y^{2}_{eq},$ (10)

where $Y_{eq}=\frac{n_{eq}(T)}{s(T)}$ with the DM number density in thermal
equilibrium $n_{eq}(T)\simeq\frac{T^{3}}{\pi^{2}}$, and the entropy density of
the universe $s(T)=\frac{2\pi^{2}g_{*}}{45}T^{3}$, and
$H(T)=\sqrt{\frac{\pi^{2}}{90}g_{*}}\frac{T^{2}}{M_{P}}$ is the Hubble
parameter with the reduced Planck mass $M_{P}=2.43\times 10^{18}$ GeV and the
effective degrees of freedom of the thermal plasma $g_{*}$ (we set
$g_{*}=106.75$ in our analysis). We solve the Boltzmann equation from
$x_{R}=m_{\sigma}/T_{R}\ll 1$ to $x=1$ with $Y(x_{R})=0$. Note that for
$T<m_{\sigma}\sim m_{\varphi_{A}^{\prime}}=\sqrt{3}m_{\sigma}$, or
equivalently $x>1$, the production of DM particles is no longer effective
since the averaged kinetic energy of the SM particles in the thermal plasma
becomes lower than $m_{\sigma}$. Using
$\frac{s(m_{\sigma})}{H(m_{\sigma})}\simeq 13.7\,m_{\sigma}M_{P}$,
$Y_{eq}\simeq 2.16\times 10^{-3}$ and Eq. (9), we get

$\displaystyle Y(x)\simeq 2.45\times
10^{-4}\,\frac{M_{P}\,m_{\sigma}^{3}}{v_{B}^{4}}\left(\frac{1}{x_{R}^{3}}-\frac{1}{x^{3}}\right)$
(11) $\displaystyle\to$ $\displaystyle Y(\infty)\simeq Y(x=1)\simeq 2.45\times
10^{-4}\,\frac{M_{P}\,T_{R}^{3}}{v_{B}^{4}}.$

Note that the resultant $Y(\infty)$ value is determined by the reheating
temperature, and this result is valid as long as
$1/x_{R}^{3}=(T_{R}/m_{\sigma})^{3}\gg 1$. We then use the formula for the
current dark matter abundance to be

$\displaystyle\Omega_{DM}h^{2}=\frac{m_{\sigma}s_{0}Y(\infty)}{\rho_{crit}/h^{2}},$
(12)

where $s_{0}\simeq 2890/{\rm cm}^{3}$ is the current entropy density, and
$\rho_{crit}/h^{2}\simeq 1.05\times 10^{-5}$ GeV/cm3. Using Eqs. (11) and
(12), we find

$\displaystyle T_{R}\simeq 9.02\times
10^{-9}\,v_{B}\left(\frac{v_{B}}{m_{\sigma}}\right)^{1/3}$ (13)

to reproduce the observed DM relic abundance of $\Omega_{DM}h^{2}=0.12$
Planck:2018vyg .

Let us now consider the out-of-equilibrium condition for the dark matter
$\sigma$. The observed DM relic abundance $\Omega_{DM}h^{2}=0.12$ leads to
$Y(\infty)\simeq Y(1)\simeq\frac{4.36\times 10^{-10}}{m_{\sigma}[{\rm GeV}]}$.
Considering the fact that $Y(x)$ is a monotonically increasing function for
$x_{R}\leq x\leq 1$ (see Eq. (11)), we conclude that $Y(x)\leq
Y(\infty)<Y_{eq}$ for $m_{\sigma}[{\rm GeV}]>2.01\times 10^{-7}$. This means
that as long as $m_{\sigma}$ satisfies this lower bound, the yield $Y(x)$
starting from $Y(x_{R})=0$ can never reach $Y_{eq}$ and therefore, the dark
matter $\sigma$ has never been in thermal equilibrium.

##  VI 6\. Summary of constraints on the model and parameter scan

In this section, we summarize the constraints on the parameters of the model
that add to the constraint in Eq. (13) from the observed DM relic density. The
two main constraints that we have discussed are from the reheat temperature
and the DM lifetime.

(A) Constraint from the freeze-in mechanism

For the freezing mechanism to work, we need,

$\displaystyle m_{\varphi^{\prime}_{A},\sigma}<T_{R}<M_{Z^{\prime}}.$ (14)

(B) DM lifetime constraint: We next give the constraint from the dark matter
life time:

$\displaystyle\Gamma_{\sigma\to\nu\nu}\simeq\frac{1}{4\pi}\left(\frac{m_{\nu}}{v_{B}}\right)^{2}\left(\frac{v_{A}}{v_{B}}\right)^{2}m_{\sigma}<6.58\times
10^{-53}\;{\rm GeV},$ (15)

corresponding to the IceCube constraint icecube  on $\tau_{\sigma}>10^{28}$
sec for $10^{4}<m_{\sigma}[{\rm GeV}]<10^{9}$. Note that this constraint
implies $v_{A}\ll v_{B}$.

(C) $\varphi_{A}^{\prime}$ lifetime constraint:

In addition to (A) and (B), we consider the constraint coming from the
lifetime of $\varphi_{A}^{\prime}$. The main decay mode of
$\varphi_{A}^{\prime}$ is $\varphi_{A}^{\prime}\to\sigma Z^{\prime}\to\sigma
f\bar{f}$ via off-shell $Z^{\prime}$. This three-body decay width is
calculated to be

$\displaystyle\Gamma_{\varphi_{A}^{\prime}\to\sigma f\bar{f}}=\
\frac{g_{BL}^{4}}{24\pi^{3}}Q^{2}_{\Phi_{A}}(\sum_{f}N_{f}Q^{2}_{f})\left(\frac{m_{\sigma}}{M_{Z^{\prime}}}\right)^{4}m_{\varphi_{A}^{\prime}}C_{I},$
(16)

where

$\displaystyle
C_{I}=\int_{1}^{\frac{m_{\varphi_{A}^{\prime}}^{2}+m_{\sigma}^{2}}{2m_{\varphi_{A}^{\prime}}m_{\sigma}}}dz\,(z^{2}-1)^{3/2}\simeq
0.0115,$ (17)

for $m_{\varphi_{A}^{\prime}}=\sqrt{3}m_{\sigma}$. Note that
$\Gamma_{\varphi_{A}^{\prime}\to\sigma f\bar{f}}\propto(v_{A}/v_{B})^{4}$ from
$M_{Z}^{\prime}\simeq 2g_{BL}v_{B}$ and $m_{\sigma}\propto v_{A}$, and
$\varphi_{A}^{\prime}$ can be long lived for $v_{A}\ll v_{B}$, which is
required by the DM lifetime constraint. If $\varphi_{A}^{\prime}$ decays after
the Big-Bang Nucleosynthesis (BBN), the energetic final state SM fermions may
destroy light nucleons successfully synthesized. To avoid this danger, we
impose the following constraint that $\varphi_{A}^{\prime}$ decays before the
BBN era, at which the age of the universe is $\tau_{BBN}\simeq 1$ sec.

$\displaystyle\Gamma_{\varphi_{A}^{\prime}\to\sigma f\bar{f}}\simeq 4.84\times
10^{-4}\left(\frac{m_{\sigma}}{v_{B}}\right)^{4}m_{\sigma}>6.58\times
10^{-25}\;{\rm GeV},$ (18)

where we have used $Q^{2}_{\Phi_{A}}=6$, $\sum_{f}N_{f}Q^{2}_{f}=8$,
$M_{Z^{\prime}}\simeq 2g_{BL}v_{B}$ and
$m_{\varphi_{A}^{\prime}}=\sqrt{3}\,m_{\sigma}$ in evaluating
$\Gamma_{\varphi_{A}^{\prime}\to\sigma f\bar{f}}$.

Figure 2:  Allowed values of $v_{A}$ and $v_{B}$ parameters in the model are
shown as the shaded region.

Figure 3:  Allowed range of the DM lifetime $\tau_{\sigma}$ and $v_{A,B}$.

Figure 4:  Allowed range of the reheat temperature $T_{R}$ and $v_{A,B}$.

Imposing the conditions in Eqs. (13), (14), (15) and(18), we perform the
parameter scan for two free parameters in the range of $100\,{\rm GeV}\leq
v_{A},v_{B}\leq 10^{19}\,{\rm GeV}$. In this analysis, we set
$m_{\sigma}=v_{A}$ for simplicity, and $T_{R}>3\,m_{\sigma}$ for the validity
of Eq. (11). Fig. 2 shows the allowed region in ($v_{A},v_{B}$)-plane. Note
that $m_{\sigma}=v_{A}\ll v_{B}$ is satisfied in the allowed region and the DM
mass is constrained to be in the range of $10^{5}\,{\rm GeV}\lesssim
m_{\sigma}\lesssim 10^{13}$ GeV. In Fig. 3, we show the range of the DM
lifetime and $v_{A,B}$. We can see that the IceCube constraint on
$\tau_{\sigma}>10^{28}$ sec is most severe for $m_{\sigma}\sim 10^{5}\,{\rm
GeV}$. Fig. 4 shows the allowed range of the reheating temperature and
$v_{A,B}$. The condition $T_{R}=M_{Z^{\prime}}$ sets the minimum value of
gauge coupling $g_{BL}\left|{}^{\rm Min}\right.$ for a chosen values of
$(v_{A},v_{B})$. For a set of parameters $(v_{A},v_{B})$ in the allowed region
shown in Fig. 2, we show the output of $g_{BL}\left|{}^{\rm Min}\right.$ in
Fig. 5. For the values of $(v_{B},g_{BL}\left|{}^{\rm Min}\right.)$ selected
in the shaded region, $g_{BL}>g_{BL}\left|{}^{\rm Min}\right.$ satisfies the
condition $T_{R}<M_{Z^{\prime}}$.

Figure 5:  The minimum values of the $g_{BL}$ coupling in the model for
various values of $(v_{A},v_{B})$ chosen in the allowed region in Fig. 2.

##  VII 7\. Comments and phenomenology

In this section we make a few comments on the model.

(i) We first note that the model can explain the origin of matter via the
usual leptogenesis mechanism. The temperature at which lepton asymmetry is
generated depends on the particle spectrum of the model. For instance, note
the important requirement of the model that $T_{R}<M_{Z^{\prime}}$. So if
$v_{B}\sim 10^{9}$ GeV and $g_{BL}\lesssim 0.3$ to avoid the coupling blowing
up before the Planck scale, then the RHN mass must be less than $T_{R}$, which
is too light to generate the observed baryon asymmetry by the usual
leptogenesis. Thus, our scenario must use resonant leptogenesis mechanism res1
; res2 .

(ii) Through the $B-L$ gauge interaction, RHNs can stay in thermal equilibrium
with the SM particle plasma. As a result, the generation of lepton asymmetry
is suppressed until the $B-L$ interaction is frozen. To avoid the suppression,
we impose $n_{eq}\langle\sigma v_{rel}\rangle<H$ at $T\sim M_{N}$, where
$n_{eq}\simeq 2M_{N}^{3}/\pi^{2}$ is the RHN number density, and
$\langle\sigma v_{rel}\rangle$ is the thermal averaged cross section for the
process $NN\leftrightarrow f\bar{f}$ via a virtual $Z^{\prime}$, roughly given
by

$\displaystyle\langle\sigma
v_{rel}\rangle\simeq\frac{g_{BL}^{4}}{4\pi}\frac{M_{N}^{2}}{M_{Z^{\prime}}^{4}}\simeq\frac{M_{N}^{2}}{64\pi\,v_{B}^{4}}.$
(19)

We then find the condition,

$\displaystyle
M_{N}<(32\pi^{3})^{1/3}\left(\frac{\pi^{2}}{90}g_{*}\right)^{1/6}v_{B}\left(\frac{v_{B}}{M_{P}}\right)^{1/3}\simeq
10.3\times v_{B}\left(\frac{v_{B}}{M_{P}}\right)^{1/3}.$ (20)

In Fig. 2, we have found the lower bound on $v_{B}\gtrsim 10^{11}$ GeV, for
which the condition reads $M_{N}\lesssim 3.54\times 10^{9}$ GeV. For the
successful (resonant) leptogenesis scenario, the lightest RHN must be in
thermal equilibrium for $T_{R}\geq T>M_{N}$. Our result shown in Fig. 4
indicates that the condition of Eq. (18) is satisfied for the resultant
$T_{R}$.

If $M_{N}>m_{\varphi_{B}^{\prime}}$, the generation of lepton asymmetry can be
suppressed by the process,
$NN\leftrightarrow\varphi_{B}^{\prime}\varphi_{B}^{\prime}$ Dev:2017xry . To
avoid this suppression, we impose this process to be decoupled at $T\sim
M_{N}$. The thermal averaged cross section for this process is given by

$\displaystyle\langle\sigma
v_{rel}\rangle\simeq\frac{f^{4}}{4\pi\,M_{N}^{2}}\simeq\frac{M_{N}^{2}}{4\pi\,v_{B}^{4}}.$
(21)

This formula is the same as Eq. (19) up to a factor, so that the resultant
constraint is similar to Eq. (20). The similar processes,
$NN\leftrightarrow\varphi_{A}^{\prime}\varphi_{A}^{\prime}$, $\sigma\sigma$,
have no effect, since they are suppressed by the mixing $v_{A}/v_{B}\ll 1$ and
$\phi_{A}^{\prime}$ and $\sigma$ are out-of-equilibrium.

(iii) One possible test of the model is that the dark matter decays into two
neutrinos with energy $E_{\nu}=\frac{m_{\sigma}}{2}$ for each neutrinos and
for dark matter masses in the multi-TeV range, there will be high energy mono-
energetic neutrinos from the DM decay with a probability of
$\tau_{U}/\tau_{\sigma}$, where $\tau_{U}\sim 10^{17}$ sec is the age of the
universe.

(iv) We also note that due to pseudo-Goldstone nature of the DM, direct
detection cross section arises only at one loop level and is highly
suppressed. At the tree level the DM behaves like an inelastic dark matter
since $Z^{\prime}$ exchange by incident DM connects to a
$\varphi_{A}^{\prime}$ field which is $\sqrt{3}$ times heavier. This explains
why the DM has not been seen in the laboratory experiments.

(v) This model can be extended to allow the real part of the $\Phi_{B}$ field
to play the role of inflaton while maintaining conformal invariance, as has
been shown in Ref. Nobuinf ; Nobuinf2 ; Nobuinf3 . In this case as well as in
general, there is an upper limit on the reheat temperature coming from the
power spectrum and upper limit on the tensor-to-scalar ratio $r\leq 0.036$ at
95% confidence level BICEP:2021xfz . Typically, $T_{R}\gtrsim 6\times 10^{15}$
GeV is ruled out, assuming the total inflaton energy is transmitted to the SM
thermal plasma right after inflation.

##  VIII 8\. Summary

In this brief note, we have presented a minimal conformal $B-L$ extension of
the standard model which explains the neutrino masses, origin of matter and a
dark matter that is produced in the early universe by the freeze-in mechanism.
We have presented the allowed set of points where the model works. We find
that the dark matter must be heavier than 100 TeV in order to ensure that its
partner $\varphi^{\prime}_{A}$ must decay before the big bang nucleosynthesis.
The model predicts energetic neutrinos from dark matter decay (with
$E_{\nu}\geq 50$ TeV) which can be observed at the IceCube experiment,
providing a test.

## Acknowledgement

The work of N.O. is supported in part by the US Department of Energy grant no.
DE- SC0012447.

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