#### Dark matter and leptogenesis linked by classical scale invariance

Received: June
matter and leptogenesis linked by classical scale
Symmetry Breaking 0 1 2
0 Open Access , c The Authors
1 Institute for Particle Physics Phenomenology, Department of Physics, Durham University
2 South Road, Durham, DH1 3LE United Kingdom
In this work we study a classically scale invariant extension of the Standard Model that can explain simultaneously dark matter and the baryon asymmetry in the universe. In our set-up we introduce a dark sector, namely a non-Abelian SU(2) hidden sector coupled to the SM via the Higgs portal, and a singlet sector responsible for generating Majorana masses for three right-handed sterile neutrinos. The gauge bosons of the dark sector are mass-degenerate and stable, and this makes them suitable as dark matter candidates. Our model also accounts for the matter-anti-matter asymmetry. The lepton avour asymmetry is produced during CP-violating oscillations of the GeV-scale righthanded neutrinos, and converted to the baryon asymmetry by the electroweak sphalerons. All the characteristic scales in the model: the electro-weak, dark matter and the leptogenesis/neutrino mass scales, are generated radiatively, have a common origin and related to each other via scalar eld couplings in perturbation theory.
Beyond Standard Model; Cosmology of Theories beyond the SM; Spontaneous
1 Introduction 2 3 4
From Coleman-Weinberg to the Gildener-Weinberg mechanism
The Coleman-Weinberg approximation
The Gildener-Weinberg approach
Dark matter phenomenology
Leptogenesis via oscillations of right-handed neutrinos
Connection among the scales
Introduction
The question of why the only scale parameter in the Standard Model (SM) Lagrangian,
MS2MjHj2, is much smaller than the Planck scale is at heart of the naturalness problem.
The idea of generating a scale radiatively, originally proposed in ref. [1] can be applied to
explain the origin of the electroweak scale in the SM [2, 3]. In this article we will discuss
an extension of the Standard Model that addresses some of the main shortcomings of
the minimal theory, namely the dark matter (DM), the baryon asymmetry of the universe
(BAU) and the origin of the electroweak scale. Our Beyond the Standard Model framework
is based on a theory which contains no explicit mass-scale parameters in its tree-level
Lagrangian, and all new scales will be generated dynamically at or below the TeV scale.
Our speci c approach is motivated by the earlier work in refs. [4{10] and [11, 12]. The
idea of generating the electro-weak scale and various scales of new physics via quantum
corrections, by starting from a classically scale-invariant theory, has generated a lot of
interest. For related studies on this subject we refer the reader to recent papers including
refs. [13{30].
In our set-up we extend the Standard Model by a dark sector, namely a non-Abelian
SU(2)DM hidden sector that is coupled to the Standard Model via the Higgs portal, and a
singlet sector that includes a real singlet
and three right-handed Majorana neutrinos Ni.
Due to an SO(3) custodial symmetry all three gauge bosons Z0a have the same mass and
are absolutely stable, making them suitable dark matter candidates [31] (this also applies
to larger gauge groups SU(N )DM [32, 33] and to scalar elds in higher representations [34],
albeit symmetry breaking patterns get more complicated).
( 0:5; 0:5)
( 0:25; 0:25)
(0; 100) GeV
LN =
The tree-level scalar potential of our model is given by
V0 =
denotes the SU(2)DM doublet, H is the SM Higgs doublet, and
is a gauge-singlet
introduced in order to generate the Majorana masses for the sterile neutrinos, and hence
the visible neutrinos masses and mixings via the see-saw mechanism. The portal couplings
will play a role in order to induce non-trivial vacuum expectation values
for all three scalar. As will become clear from table 1 we will scan over positive as well
as negative values of the portal couplings h and
h . As we are working with multiple
scalars we will adopt the Gildener-Weinberg approach [35], which is a generalisation of
the Coleman-Weinberg mechanism to multiple scalar states and will be brie y reviewed
in section 2. Later on we shall see that the most interesting region in parameter space
leading to both the correct dark matter abundance and the correct baryon asymmetry is
h i and hence one can think of
as a Coleman-Weinberg scalar that once it
acquires a non-zero vev it will be communicated to
and h through the portal couplings
The interactions for the right-handed neutrinos in the Lagrangian are given by
YiDa Ni("H)lLa
where the rst two term give rise to the Majorana mass once
acquires a vev, while the
last two terms are responsible for the CP-violating oscillations of Ni.
Since we do not wish to break the lepton-number symmetry explicitly, it follows
from (1.2) that our new singlet scalar eld
can think of it as the real part of a complex scalar
2. We
= ( + i )=p2 where S transforms
under a U(1)L symmetry associated with the lepton number, which is broken spontaneously
light) (pseudo)-Goldstone boson. Since the Higgs can pair-produce them and decay, this
would severely constrain the portal coupling of
with the Higgs, h
< 10 5, see e.g.
ref. [5]. If we wish to avoid such
ne-tuning, a much more appealing option would be to
gauge the lepton number. A compelling scenario is the B
L theory with the anomaly free
U(1)B L factor. The generation of matter-anti-matter asymmetry via a leptogenesis
mechanism through sterile neutrino oscillations in a classically-scale-invariant U(1)B L
SM
theory was considered in ref. [6], and their results will also apply to our model. The main
di erence with the set-up followed in this paper is that here we allow for a separate
nonAbelian Coleman-Weinberg sector (i.e. it remains distinct from the U(1)B L gauge sector)
and as a result we have a non-Abelian vector DM candidate.
Finally, it should also be possible to restrict the complex singlet
back to the real
singlet , just as we have in (1.1). In this case the continuous lepton number U(1) symmetry
is reduced to a discrete sub-group:
(N; N c; lL) ! ei =2(N; N c; lL) ;
(N ; N c; lL) ! e i =2(N ; N c; lL) :
to be a real scalar singlet.
In general all three possibilities corresponding to global, local and discrete
leptonnumber symmetries can be accommodated and considered simultaneously in the context of
eqs. (1.1){(1.2) by either working with the real scalar
or the complex one by promoting
2 y in the second term in the brackets on the r.h.s. of (1.2)). In this work
From Coleman-Weinberg to the Gildener-Weinberg mechanism
The scalar eld content of our model consists of an SU(2)L doublet H, an SU(2)DM doublet
and a real scalar ; the latter giving mass to the sterile neutrinos after acquiring a vev
in similar fashion to ref. [10]. Working in the unitary gauge of the SU(2)L
SU(2)DM, the
two scalar doublets in the theory are reduced to,
H = p
= p
and the tree-level potential becomes,
V0 =
There are no mass scales appearing in the classical theory, and at the origin in the eld
space, all scalar vevs are zero, in agreement with classical scale invariance. We impose
a conservative constraint on all the scalar couplings for the model to be perturbative
j ij < 3, we also impose gDM < 3 and in order to ensure vacuum stability the following set
of constraints need to be satis ed,
For more detail we refer to table 1.
For simplicity, let us temporarily ignore the singlet
and concentrate on the theory with
two scalars,
and h. We will further refer to the hidden SU(2)DM sector with
Coleman-Weinberg (CW) sector. In the near-decoupling limit, h
1, between the CW
and the SM sectors, we can view electroweak symmetry breaking e ectively as a two-step
process [5].
First, the CW mechanism [1] generates h i in the hidden sector through running
couplings (or more precisely the dimensional transmutation). To make this work, the
scalar self-coupling
at the relevant scale
= h i should be small | of the order of
1, as we will see momentarily. This has the following interpretation: in a theory
has a positive slope, we start at a relatively high scale where
is positive and
move toward the infrared until approach the value of the
( ) becomes small and
is about to cross zero. This is the Coleman-Weinberg scale where the potential develops a
non-trivial minimum and
generates a non-vanishing vev.
To see this, consider the 1-loop e ective potential evaluated at the scale
V ( ; h) =
Here we are keeping 1-loop corrections arising from interactions of
with the SU(2) gauge
bosons in the hidden sector, but neglecting the loops of
is close to zero) and
the radiative corrections from the Standard Model sector. The latter would produce only
subleading corrections to the vevs. Minimising at
= h i gives:
= h i
v = hhi =
mh = p2 hv :
(h i) = 25363 2 gD4M(h i).
For small portal coupling h , this is a small deformation of the original CW condition,
The second step of the process is the transmission of the vev h i to the Standard
Model via the Higgs portal, generating a negative mass squared parameter for the Higgs
h h 2i which generates the electroweak scale v,
The fact that for
1 the generated electroweak scale is much smaller than h i,
guarantees that any back reaction on the hidden sector vev h i is negligible. Finally, the
mass of the CW scalar is obtained from the 1-loop potential and reads:
m2 =
As already stated, this approach is valid in the near-decoupling approximation where
all the portal couplings are small. The dynamical generation of all scales is visualised here
as rst the generation of the CW scalar vev h i, which then induces the vevs of other scalars
proportional to the square root of the corresponding portal couplings
implies the hierarchy of the vevs.
For multiple scalars, , h and , it is not a priori obvious why the portal couplings
should be small and which of the scalar vevs should be dominant. For example on one part
of the parameter space we can
nd h i > h i and on a di erent part one has h i > h i (so
e ectively plays the role of the CW scalar). To consider all such cases
and not be constrained by the near-decoupling limits we will utilise the Gildener-Weinberg
set-up [35], which is a generalization of the Coleman-Weinberg method.
The Gildener-Weinberg approach
We now return to the general case with the three scalars in the model are described by the
tree-level massless scalar potential (2.1). The Gildener-Weinberg mechanism was recently
worked out for this case in ref. [10], which we will follow. All three vevs can be generated
dynamically but neither of the original scalars is solely responsible for the intrinsic scale
generation; this instead is a collective e ect generated by a linear combination of all three
Following [35], we change variables and reparametrise the scalar elds via,
h = N1';
= N2';
= N3';
where each Ni is a unit vector in three-dimensions. The Gildener-Weinberg mechanism
tells us that a non-zero vacuum expectation value will be generated in some direction in
= 0;
V0(n1'; n2'; n3') = 0 :
and furthermore the value of the tree-level potential in this vacuum is independent of ',
The latter condition is simply the statement that the potential restricted to the single degree
of freedom ', is of the form 14 ' '4 with the corresponding coupling constant vanishing
GW where '( GW) vanishes, and is a
re ection of a similar statement in the Coleman-Weinberg case for the single scalar that
its self-coupling was about to cross zero, but was stabilised at the small positive value by
the gauge coupling at the Coleman-Weinberg scale CW, see eq. (2.6).
Being a unit vector in three-dimensions, ni's can be parametrised in terms of two
independent angles,
and we will call the ' vev, w, so that,
n1 = sin ;
hhi = wn1 ;
n2 = cos cos
h i = wn2 ;
n3 = cos sin ;
h i = wn3:
The three linearly-independent conditions arising from the Gildener-Weinberg
minimisation (2.10) of the tree-level potential amount to the following set of relations,
2 hn12 =
22 =
23 =
These conditions hold at the scale
w (2.13). Due to the three scalars acquiring non-zero vacuum expectation values, the three
states will mix among each other. The mass matrix M 2 is diagonalised for h1; h2 and h3
eigenstates via the rotation matrix O,
diag Mh21 ; Mh22 ; Mh23
= O
B C = Oij BBh2CC ;
and we further identify the state h1 with the SM 125 GeV Higgs boson. Following [10] we
parametrise the rotation matrix in terms of three mixing angles ,
O = BB
cos cos sin
+ sin sin
cos sin sin
cos sin sin
sin sin sin
be written as [35],
and use it to compute the scalar mass eigenstates (2.17) at tree-level. The resulting
expressions for the scalar masses can be found in ref. [10]. There is one classically
at direction
in the model | along ' | in which the potential develops the vacuum expectation value.
Our choice of parametrisation in (2.13) and in the second row of (2.18) in terms of the
same two angles
and , selects this direction to be identi ed with h2. Hence, at tree
At the scale GW the one-loop e ective potential along the minimum at direction can
A =
B =
Mh4i + 6M W4 + 3MZ4 + 9MZ40
V ('n) = A'4 + B'4 log
GW
where the A and B coe cients are computed in the MS [36] scheme and given by,
and using this relation we can rewrite the one-loop e ective potential as,
V = B'4 log
where Mhi are the tree-level masses of the three scalar eigenstates, h1, h2 and h3, and the
rest are the masses of the SM and the hidden sector vector bosons as well as the top quark
and the right-handed Majorana neutrinos. We can now see that at the RG scale
1-loop corrected e ective potential has a xed vacuum expectation value w that satis es,
Bw4=2, which
gives the requirement B > 0 for this to be a lower minimum than the one at the origin.
The mass of the pseudo-dilaton h2 is then given by,
Mh22 =
Mh41 + Mh43 + 6M W4 + 3MZ4 + 9MZ40
2 X
In summary, at the scale GW the conditions eqs. (2.14){(2.16) will be satis ed and the
scalar potential will develop a non-trivial vev w giving rise to non-zero vacuum expectation
values hhi; h i; and h i. For one scalar eld, the Coleman-Weinberg mechanism requires
the scalar quartic coupling to take very small values
gD4M, in the Gildener-Weinberg
scenario it is a combination of the quartic couplings that needs to vanish, so these couplings
can take larger values individually.
The formulae for the mixing angles in terms of the coupling constants and the vevs
follow from the diagonalisation of the tree-level mass matrix,
= h i
2 =
2 =
Experimental searches of a scalar singlet mixing with the SM Higgs provide constraints on
the mixing angles [37{39]. In our case, these translate as,
In the region where the decay h1 ! h2h2 is allowed we impose the stronger constraint
> 0:96. Nonetheless, due to the Gildener-Weinberg conditions the decay
h1 ! h2 h2 is highly suppressed. In the scan we perform Mh3 is always greater than Mh1 ,
so there is no need to worry about the SM Higgs decaying into two h3 scalars. At the
same time, strong constraints could come when the decays h1 ! Z0a Z0a are allowed, we
also include annihilation into right-handed neutrinos.
For the study of dark matter the Lagrangian contains ten dimensionless free
paramea random scan on the remaining eight parameters in the ranges given in table 1.
The matrix Y D has no impact on the dark matter phenomenology, but it is crucial for
Leptogenesis and it will be parametrised by three complex angles !ij using the Casas-Ibarra
parametrisation [40]. Therefore, once we set all the parameters for the active neutrinos to
their best experimental t, there are twelve free parameters in the model.
Dark matter phenomenology
Evidence from astrophysics suggests that most of the matter in the universe is made out
of cosmologically stable dark matter that interacts very weakly with ordinary matter.
Being able to identify what constitutes this dark matter is one of the deepest mysteries
in both particle physics and astrophysics. In this work we consider the possibility of dark
matter being a spin-1 particle from a hidden sector with non-Abelian SU(2)DM gauged
symmetry. The idea of vector dark matter was rst introduced in ref. [31] and later studied
in refs. [7, 9, 32, 41]. Note that if the hidden sector had been U(1), the kinetic mixing
among the hidden sector and the hypercharge will have made our dark matter candidate
After radiative symmetry breaking breaking of SU(2)DM by , which is in the
fundamental representation of the group, there is a remnant SO(3) symmetry that ensures the
to models where the DM is odd under a Z2 discrete symmetry, in the present scenario we
can have dark matter semi-annihilation processes where a DM particle is also present in
nal state. The DM annihilation diagrams are shown in gures 1 and 2, while the
semi-annihilation ones are shown in gure 3.
dark matter, Za0 is stable due to an remnant global symmetry.
mass Mh2 . Right panel gives scatter plot of the dark matter mass versus the mass of the heavier
scalar h3. Di erent colours indicate whether the vector gauge triplet accounts for more or less than
100%, 10% and 1% of the observed dark matter abundance.
Also, due to the radiative generation of h i in most region of parameter space the
scalar mass will be smaller than the gauge boson mass, Mh2 < MZ0 . This means that
semi-annihilation processes Z0aZ0b
! Z0c hi will be dominant over annihilation ones in
most of the parameter space. To leading order the non-relativistic cross-section from the
semi-annihilation diagrams is given by (cf. [9]),
h abcvi =
3gD4M (O2i)2
Mh4i !3=2
In order to take into account all annihilation channels into SM particles and
properly take into account thresholds and resonances we have implemented the model in
micrOMEGAs 4.1.5 [42]. We x the dark matter relic abundance from the latest Planck
satellite measurement
h2 = 0:1197
0:0022 [43]. Figure 4 shows the dark matter fraction
as a function of MZ0 and the scalar mass Mh2 ; the isolated strip of points on the left side
of the plots corresponds to the resonance Mh2
On the left plot in gure 4 there is a large red coloured region on the left side (producing
too much dark matter), in this region Mh2 has a close value to MZ0 (note that this region
the small mixing angles, we can see that the dark matter relic density is almost independent of h i.
Right panel: scatter plot of the dark matter mass MZ0 versus the gauge coupling gDM. Di erent
colours indicate whether the vector gauge triplet accounts for more or less than 100%, 10% and 1%
of the observed dark matter abundance.
does not exist in the Coleman-Weinberg limit). This region exists thanks to very large
values of Mh3 and h i
MZ0 . In the left panel of
gure 5 we show the dark matter
fraction as a function of both vevs, h i and h i, from this plot we see there is an upper
bound on h i in order not to overproduce dark matter, h i < 17 TeV. Later on we shall
see that there is a lower bound on h i coming from leptogenesis, h i > 2:5 TeV, we have
already imposed this bound on all the scatter plots we show.
In the right panel of gure 5 we show the dark matter fraction as a function of MZ0
and the gauge coupling gDM. In this plot it becomes clear that as we increase the gauge
coupling, the relic density decreases. The left panel of gure 6 shows the same analysis for
the mixing angle sin
and the quartic couplng
. Here we can already notice a preference
for the region sin
bound on h i the mixing angle
panel of gure 6.
takes on small values and h i
h i. Due to the lower
takes on very small values, this is shown in the right
The spin-independent cross-section between Z0a and a nucleon is given by,
SI =
f N2 m4N MZ20
X O2iO1i
elements of the rotation matrix eq. (2.18) that relates the scalar mass eigenstates states to
the ones in the Lagrangian. This orthogonal matrix O is the one that diagonalises the mass
matrix. Due to the form of this matrix, the direct detection diagrams have a destructive
interference when the scalar state with a large
component has a mass very close to Mh1 ,
this has been previously noted in [7, 47]; while the scalar state with a large
has no direct couplings either to dark matter or to Standard Model particles and hence
against the quartic coupling
. Larger values of sin
are preferred. Right panel: scatter plot of sin
versus the scalar mass Mh2 . Due to h i
get small values for the mixing angle . Di erent colours indicate whether the vector gauge triplet
accounts for more or less than 100%, 10% and 1% of the observed dark matter abundance.
gives only a small contribution to SI. Figure 7 shows that except for resonances, the region
with MZ0 < 250 GeV has been already excluded by the existing experiments, while a large
region of parameter space will be tested by future underground experiments such as LZ [45]
and XENON1T [48]. In gure 8 we show the direct-detection cross-section as a function of
the dark matter mass for benchmark point BP 1, we x all the scalar couplings and vary
only gDM, the dip corresponds to Mh2
Leptogenesis via oscillations of right-handed neutrinos
Leptogenesis is an attractive and minimal mechanism to solve the baryon asymmetry of
the universe (BAU). This means being able to produce the observed value of
nbobs = (8:75
In the Akhmedov-Rubakov-Smirnov framework [11] a lepton avour asymmetry is produced
during oscillations of the right-handed Majorana neutrinos Ni with masses around the
electroweak scale or below, which makes this approach compatible with classical scale
invariance.1 From Big Bang nucleosynthesis we obtain the lower bound MN > 200 MeV, in
order not to spoil primordial nucleosynthesis. For our calculations we make use of the the
Casas-Ibarra parametrisation [40] for the matrix Y D,
Y D y = U
and MN are diagonal mass matrices of active and Majorana neutrinos
respectively. The active-neutrino-mixing matrix U is the PMNS matrix which contains six real
1In the sense that no additional very large scales are required to be introduced in the model to make
this type of leptogenesis work.
10−8
10−9
(IS 10−10
10−11
10−12
10−44
10−45
10−46
10−47
10−48
MZ0 . We show current experimental limits from LUX [44] (red line), future limits from LZ [45]
(green line) and the neutrino coherent scattering limit [46] (black line).
MDM (GeV)
mass MZ0 , for benchmark point BP 1. We show current experimental limits from LUX [44] (red
line), future limits from LZ [45] (green line) and the neutrino coherent scattering limit [46] (black
line). To generate this plot we
x all the scalar couplings and vary only gDM, which means that
MZ0 and Mh2 are also varied while all other parameters remain xed.
parameters, including three measured mixing angles and three CP-phases. The matrix R
is parametrised by three complex angles !ij . Using this framework with three right-handed
neutrinos one can generate the correct baryon asymmetry without requiring tuning the Ni
mass splittings, but rather enhancing the entries in the Dirac Yukawa matrix through the
imaginary parts of the complex angles !ij [49].
Due to the non-trivial topological structure of the vacuum in SU(2)L there exist
electroweak sphaleron processes which violate B + L quantum number, and these will transfer
the lepton avour asymmetry nLe into a baryon asymmetry nb, with the conversion factor
A critical condition for the mechanism of [11] to work, is that two of three neutrino avours,
N2 and N3, should come into thermal equilibrium with their Standard Model counterparts
before the universe cools down to TEW (when electroweak sphaleron processes freeze out),
while the remaining
avour does not. In other words, the present mechanism consists
of di erent time scales Tosc
Teq2 > TEW > Teq1 , where Teqi represents the
temperature at which Ni equilibrates with the thermal plasma and Tosc is the temperature
at which the oscillations start to occur. In terms of the decay rates for the three sterile
neutrino avours this implies,
2(TEW) > H(TEW) ;
3(TEW) > H(TEW) ;
1(TEW) < H(TEW);
where H is the Hubble constant,
H(T ) =
MP
pg p4 3=45 ' 1018 GeV ;
and MP is the reduced Planck mass. Therefore, we require,
1(TEW) =
1 X YeDi yYiDe av TEW < H(TEW) :
temperatures, Tosc
asymmetry are generated.
Here the dimensionless quantities av
10 3 are derived from the decay rates of the
right-handed neutrino Ne of the `electron avour' tabulated in ref. [50]. These right-handed
neutrino decay (or equivalently production) rates were computed in [50] using 1 $ 2 and
2 $ 2 processes2 involving the neutrino vertices YaDi y lLa("H)yNi and YiDa Ni("H)lLa with
the Dirac Yukawas.
One can also ask if the new interactions present in our model, those involving the
Majorana Yukawas, 12 YiMj
NicNj and 12 YiMj y NiNjc, could a ect the dynamics. These
interactions always contain a pair of right-handed neutrinos and do not change the right-handed
neutrino number (the singlet
carries the N -number
2 but above the electroweak phase
transition temperature, the vev of
vanishes). Hence these processes could contribute to
the N production or decay into the Standard Model particles only in combination with
other interactions. As the Majorana Yukawa couplings are small Y M
10 5 on the part of
the parameter space relevant for us (see table 3) and the cross-section being proportional
to (Y M )2 means that these interactions will give subleading e ects to all the processes
considered in [50]. Therefore, we can follow [12] and make the assumption that the number
density of sterile neutrinos is very small compared to their equilibrium density at high
106 GeV, around which the main contributions to the lepton- avour
2These processes are shown in
gures 1 and 2 in ref. [50] and contain a single external N leg | as
relevant for the N -production or decay processes of interest.
nLa =
where the quantity Iij is given by,
Iij =
we have xed the mass splittings to be
MN1 =10. This plot shows that there is a lower
bound h i > 2:5 TeV in order to produce the correct amount of baryon asymmetry. The region in
light green cannot produce enough baryon symmetry and/or does not satisfy the wash-out
criterion eq. (4.6).
It was already shown in [6] that avoured leptogenesis can work in a classically scale
invariant framework. In their set-up three right-handed neutrinos are coupled to a scalar
eld that acquires a vev, as in the present model. The main di erence being that in the
present scenario we have not gauged the B
L quantum number. We quote the nal result
for the lepton avour asymmetry (of ath avour) obtained in [6] from extending the results
of ref. [12] to the classically scale-invariant case,
X i (YaDi yYiDcYcDj yYjDa
YaDi tYiDc YcDj tYjDa )
c i6=j
MP
for h i < Tosc. For the case h i
Tosc and further details on the derivation of eq. (4.7)
we refer the reader to ref. [6]. It follows from (4.8) that the amount of the lepton
asymmetry is proportional to h i MP=
M N2i . Hence if we want to avoid any excessive
ne-tuning of the mass splittings between di erent
avours of Majorana neutrinos, the
relatively large values of h i & 104 GeV are preferred. From
gure 9 we can see that there
is a lower bound on h i if we impose some restriction on the mass splittings of the
righthanded neutrinos. In view that we would like to stay far away from the ne-tuning region,
we impose
to explain the baryon asymmetry. Imposing this condition removes the points with very
small mixing angle , as can be seen in the left panel of gure 6.
As we can see from
gure 9 there is also an upper bound on MNi for each value of h i,
this bound is mainly due to the wash-out criterion eq. (4.6) not being satis ed any more.
This upper bound becomes weaker once we reach h i
approach based on the common dynamical origin of all vevs: once an explanation for dark
matter is included, h i cannot be too large compared to h i.
The procedure to obtain the plot in
gure 9 is as follows. We x the complex phases
and for each point we scan over !23, if we nd at least one point that works well then we
label it as a good point (dark green) otherwise it is a bad point (light green). In further
scans we have found that varying !12 and !13 has a negligible impact on the nal results.
The generated total lepton asymmetry is proportional to h i, (cf. (4.7), (4.8))
where we used the see-saw mechanism for the masses m
of visible neutrinos, and v is
the SM Higgs vev. Hence nL vanishes as h i approaches zero. This also explains why in
gure 9, there is a stronger dependence on h i than on the masses MNi .
We carried out a scan over all free parameters in our model to determine the region
of the parameter space where the leptogenesis mechanism outlined above can generate the
observed baryon asymmetry. At the same time we require that the model provides a viable
candidate for cosmological dark matter. We would like to mention in passing that all the
present results on leptogenesis also hold when a generic scalar generates a mass for the
sterile neutrinos (i.e. with no reference to classical scale invariance).
The results of the scan and the connection between the leptogenesis and dark matter
scales are reviewed in the following section. Furthermore, in tables 2 and 3 we present
four benchmark points to illustrate the viable model parameters. In the remainder of this
section we would like to comment on the choice of parameters for the leptogenesis part of
We rst note that our leptogenesis realisation does not require any sizeable ne-tuning
of the mass splittings
MNi . For example our rst benchmark point BP 1 has (cf. table 3),
MN = diag(0:225; 0:25; 0:275) GeV:
At the same time, the masses of active neutrinos are set to agree with the observed mass
splittings; for BP 1 we have,
= diag(0; 8:7; 49:0) meV:
The lepton asymmetry (4.7) also depends on the matrix of Dirac Yukawa couplings Y D.
We compute Y D in the Casas-Ibarra parametrisation eq. (4.2) using (4.10) and (4.11)
along with the PMNS matrix and the R matrix. We have carried out a general scan on
the complex angles !ij of the R matrix and found that having non-vanishing Im[!ij ] is
important in order to obtain the required amount of lepton asymmetry.3 At the same time
this does not lead to any excessive
ne-tuning. We have checked this for the numerical
3Note that positive values of Im[!ij] enhance the elements of the Dirac Yukawa matrix Y D.
hhi (GeV)
Mh1 (GeV)
Mh2 (GeV)
Mh3 (GeV)
MZ0 (GeV)
GW (GeV)
correct dark matter abundance within 2 .
values of R matrix elements in our scan. For example, for BP 1 we have (using the !ij
values in table 3),
R = B@ 84:43 + 100:0i
105:4 + 91:81i
101:0 + 85:98i
5:854 + 4:604i 1
14:20iCA ;
and the resulting matrix of Dirac Yukawa couplings,
Y D = B@
134:4 + 77:79i
9:677 + 24:56i
136:9 + 224:6iCA
34:69 + 28:93i
These matrices do not exhibit a high degree of tuning, and we have checked that this is
also the case for generic points of our scan.
MN1 (GeV)
MN2 (GeV)
MN3 (GeV)
m1 (meV)
m2 (meV)
m3 (meV)
nLe=(s
nL =(s
nL =(s
e=H (TEW)
=H (TEW)
=H (TEW)
Tosc (GeV)
1:5 + 2:6i
0:9 + 2:7i
1:5 + 2:6i
0:9 + 2:7i
1:0 + 2:6i
0:9 + 2:7i
1:5 + 2:6i
0:9 + 2:7i
After having performed a scan over all free parameters in our model, we nd that:
(1) h i < 17 TeV in order for dark matter not to overclose the universe, and
(2) h i > 2:5 TeV in order in order for leptogenesis to explain the baryon asymmetry.
From the left plot of gure 6 we can see that the interesting region in parameter space has
large values of sin , and with this in mind we can separate the interesting regime into two
In this region4 we have sin
the scalar states
, and due to the Gildener-Weinberg conditions
To avoid overproducing DM, both h i and h i have to be less than 10 TeV. Due to
the not so large values of h i, a large part of this region requires some amount of
ne-tuning of the right-handed neutrino mass splittings in order for leptogenesis to
work. The use of the Gildener-Weinberg mechanism is crucial in this region.
In this region we have sin
1, so it can be seen as the Coleman-Weinberg limit of
the more general Gildener-Weinberg mechanism. The scalar
overlaps maximally
with h2 and can be thought of as the Coleman-Weinberg scalar. In this region the
radiative symmetry breaking is induced by
1 and we get Mh2
Mh3 . This
region also corresponds to the majority of good (blue) points in
gures 4{6. Most
points have MDM > Mh2 . This is the region of most interest since the large values of
h i require almost no ne-tuning in
MNi in order for leptogenesis to work.
In table 2 we give a set of benchmark points that satisfy all experimental constraints and
give the correct dark matter abundance within 2 . The benchmark points BP1, BP2
and BP3 are within reach of future direct detection dark matter experiments. For these
same points we provide in table 3 numerical values that generate the correct amount
of baryon asymmetry via leptogenesis. We work with the current experimental central
neutrino masses. The values for hY Di are computed as the average of p
values for the neutrino sector taken from [51], we assume normal ordering for the active
2MN m =hhi.
This estimate corresponds to the naive see-saw relation and it is smaller than the actual
entries in the matrix Y D due to the enhancement by the imaginary parts of !ij in the R
matrix. Nevertheless, for our benchmark points these enhancement factors are always less
Finding a connection between the scale h i, responsible for dark matter, and the scale
h i, responsible for leptogenesis, would be of high interest. From eq. (4.7) and applying
the conversion factor (4.3), we can approximate the baryon relic abundance as,
4Recall that tan2 = h i2=h i2.
2:045 MP
Regarding the dark matter relic density, in a large portion of our parameter scan
semiannihilations are dominant over annihilations, and hence we can approximate by,
pg? MP 2h abcvi=3
number of relativistic degrees of freedom. A good approximation for the mixing angles is
0 and sin
0:9, substituting these values into eq. (3.1) leads to,
Using eqs. (5.1) and (5.3) we can nd the ratio
GeV 1 = 5 ;
where the last equality comes from the observed relic densities [43]. After imposing this
relation we nd a connection among the scales in the model,
where the parameter " is de ned as,
" =
The parameter MN has a dependence on h i, but from a physical perspective it is more
x the mass splittings rather than the Majorana Yukawa couplings. The
parameter " gives the connection between both scales, typical values for this parameter are
around 10 4. Figure 10 illustrates this connection between the scales keeping the parameter
" xed to di erent values.
Conclusions
We have presented a model that can explain dark matter and the baryon asymmetry of
the universe simultaneously, where all the scales in the theory are dynamically generated
and have a common origin.
In order to ensure the stability of the dark matter candidate, one usually needs to
introduce a discrete symmetry by hand. One of the attractive features of the present
model is that it leads to a stable DM candidate without the need of introducing an extra
discrete symmetry.
We already know that in the Standard Model lepton number and
baryon number are accidental symmetries, the latter being responsible for the stability
of the proton. In our framework the hidden vector DM is stable due to the accidental
non-Abelian global symmetry SO(3). A large region of the parameter space producing the
correct amount of dark matter will be tested by future direct detection experiments such
observed value of
" de ned in eq. (5.6).
as LZ [45] and XENON1T [48]. Also, this accidental symmetry could be broken by
nonrenormalizable operators leading to the decay of Z0a and producing an intense gamma-ray
line that could be detected in future experiments [52].
The theory also predicts two extra scalar states that have a Higgs-like behaviour and
masses around the electroweak scale. From the relation for tan2 , eq. (2.23), the interesting
hhi already requires a small mixing angle
the small mixing angles we obtain values of cos2
with the SM Higgs boson, due to
> 0:95, so their detection would
only be feasible at future colliders. Nevertheless, the LHC at high luminosity will improve
the current constraints on the mixing angles
From dark matter considerations the value of h i is required to be around the TeV
scale and due to the common origin of all the vevs, h i cannot be too large, compared
to h i, which means that sterile neutrinos should have small masses of order O(1) GeV in
order for leptogenesis to work without severe tuning of the mass splittings
MNi . Under
some mild assumptions, we found a connection among the scales h i (responsible for dark
matter) and h i (responsible for leptogenesis) eq. (5.5), in order to match the observed
have constructed a minimal extension of the SM that addresses dark matter, the baryon
asymmetry of the universe and the origin of the electroweak scale.
Acknowledgments
ADP would like to thank Brian Shuve, Jessica Turner and Ye-Ling Zhou for helpful
discussions on the topic of leptogenesis. This work is supported by STFC through the IPPP grant.
ADP acknowledges nancial support from CONACyT. Research of VVK is supported in
part by a Royal Society Wolfson Research Merit Award.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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