Dark Matter and Leptogenesis Linked by Classical Scale Invariance

# Dark Matter and Leptogenesis Linked by Classical Scale Invariance

Valentin V. Khoze    and Alexis D. Plascencia
###### Abstract

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 flavour asymmetry is produced during CP-violating oscillations of the GeV-scale right-handed 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 field couplings in perturbation theory.

IPPP/16/43, DCPT/16/86

Dark Matter and Leptogenesis Linked by Classical Scale Invariance

• Institute for Particle Physics Phenomenology, Department of Physics,
Durham University, Durham DH1 3LE, United Kingdom

## 1 Introduction

The question of why the only scale parameter in the Standard Model (SM) Lagrangian, , 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 specific approach is motivated by the earlier work in Refs. [4, 5, 6, 7, 8, 9, 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, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].

In our set-up we extend the Standard Model by a dark sector, namely a non-Abelian SU(2 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 . Due to an SO(3) custodial symmetry all three gauge bosons have the same mass and are absolutely stable, making them suitable dark matter candidates [31] (this also applies to larger gauge groups SU(N [33, 32] and to scalar fields in higher representations [34], albeit symmetry breaking patterns get more complicated).

The tree-level scalar potential of our model is given by

 V0 = λϕ|Φ|4+λh|H|4+λσ4σ4−λhϕ|H|2|Φ|2−λϕσ2|Φ|2σ2+λhσ2|H|2σ2, (1.1)

where denotes the SU(2 doublet, 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 , and 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 and . 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 briefly 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 for 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 through the portal couplings and .

The interactions for the right-handed neutrinos in the Lagrangian are given by

 (1.2)

where the first 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 .

Since we do not wish to break the lepton-number symmetry explicitly, it follows from (1.2) that our new singlet scalar field should have the lepton number . We can think of it as the real part of a complex scalar where transforms under a symmetry associated with the lepton number, which is broken spontaneously by . If this is a global symmetry then there must exist a massless (or very light) (pseudo)-Goldstone boson. Since the Higgs can pair-produce them and decay, this would severely constrain the portal coupling of with the Higgs, , see e.g. Ref. [5]. If we wish to avoid such fine-tuning, a much more appealing option would be to gauge the lepton number. A compelling scenario is the theory with the anomaly free factor. The generation of matter-anti-matter asymmetry via a leptogenesis mechanism through sterile neutrino oscillations in a classically-scale-invariant theory was considered in Ref. [6], and their results will also apply to our model. The main difference with the set-up followed in this paper is that here we allow for a separate non-Abelian Coleman-Weinberg sector (i.e. it remains distinct from the 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 symmetry is reduced to a discrete sub-group:

 σ→−σ,(N,¯¯¯¯¯Nc,lL)→eiπ/2(N,¯¯¯¯¯Nc,lL),(¯¯¯¯¯N,Nc,¯¯¯¯¯lL)→e−iπ/2(¯¯¯¯¯N,Nc,¯¯¯¯¯lL). (1.3)

In general all three possibilities corresponding to global, local and discrete lepton-number 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 (or in the second term in the brackets on the r.h.s. of (1.2)). In this work we consider to be a real scalar singlet.

## 2 From Coleman-Weinberg to the Gildener-Weinberg mechanism

The scalar field content of our model consists of an SU(2 doublet , an SU(2 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(2SU(2, the two scalar doublets in the theory are reduced to,

 H=1√2(0h),Φ=1√2(0ϕ),

and the tree-level potential becomes,

 V0=λh4h4+λϕ4ϕ4+λσ4σ4−λhϕ4h2ϕ2−λϕσ4ϕ2σ2+λhσ4h2σ2. (2.1)

There are no mass scales appearing in the classical theory, and at the origin in the field 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 , we also impose and in order to ensure vacuum stability the following set of constraints need to be satisfied,

 (2.2) λhϕ2√λhλϕ≤1,−λhσ2√λhλσ≤1,λϕσ2√λϕλσ≤1, (2.3) λhϕ2√λhλϕ−λhσ2√λhλσ+λϕσ2√λϕλσ≤1. (2.4)

For more detail we refer to Table 1.

### 2.1 The Coleman-Weinberg approximation

For simplicity, let us temporarily ignore the singlet and concentrate on the theory with two scalars, and . We will further refer to the hidden SU(2 sector with as the Coleman-Weinberg (CW) sector. In the near-decoupling limit, , between the CW and the SM sectors, we can view electroweak symmetry breaking effectively as a two-step process [5].

First, the CW mechanism [1] generates 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 should be small – of the order of , as we will see momentarily. This has the following interpretation: in a theory where 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 where 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 effective potential evaluated at the scale (cf. [9]):

 V(ϕ,h)=λϕ(μ)4ϕ4+91024π2g4DM(μ)ϕ4(logϕ2μ2−256)−λhϕ(μ)4h2ϕ2, (2.5)

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 (since 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 gives:

 λϕ=33256π2g4DM+λhϕv22⟨ϕ⟩2atμ=⟨ϕ⟩. (2.6)

For small portal coupling , this is a small deformation of the original CW condition, .

The second step of the process is the transmission of the vev to the Standard Model via the Higgs portal, generating a negative mass squared parameter for the Higgs which generates the electroweak scale ,

 v=⟨h⟩=√2λhϕλh⟨ϕ⟩,mh=√2λhv. (2.7)

The fact that for the generated electroweak scale is much smaller than , guarantees that any back reaction on the hidden sector vev is negligible. Finally, the mass of the CW scalar is obtained from the 1-loop potential and reads:

 m2ϕ=9128π2g4DM⟨ϕ⟩2+λhϕv2. (2.8)

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 first the generation of the CW scalar vev , which then induces the vevs of other scalars proportional to the square root of the corresponding portal couplings , as in (2.7). This implies the hierarchy of the vevs.

For multiple scalars, , 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 find and on a different part one has (so that rather than effectively 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.

### 2.2 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 effect generated by a linear combination of all three scalars .

Following [35], we change variables and reparametrise the scalar fields via,

 h=N1φ, ϕ=N2φ, σ=N3φ. (2.9)

where each 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 scalar field space , so this direction must satisfy the condition,

 ∂V0∂Ni∣∣∣n=0, (2.10)

and furthermore the value of the tree-level potential in this vacuum is independent of ,

 V0(n1φ,n2φ,n3φ)=0. (2.11)

The latter condition is simply the statement that the potential restricted to the single degree of freedom , is of the form with the corresponding coupling constant vanishing . This is nothing but the definition of scale where vanishes, and is a reflection 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 , see Eq. (2.6).

Being a unit vector in three-dimensions, ’s can be parametrised in terms of two independent angles, and and we will call the vev, , so that,

 n1 = sinα,n2=cosαcosγn3=cosαsinγ, (2.12) ⟨h⟩ = wn1,⟨ϕ⟩=wn2,⟨σ⟩=wn3. (2.13)

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,

 2λhn21 = λhϕn22−λhσn23, (2.14) 2λϕn22 = λhϕn21+λϕσn23, (2.15) 2λσn23 = λϕσn22−λhσn21. (2.16)

These conditions hold at the scale where the scalar fields develop the vev (2.13). Due to the three scalars acquiring non-zero vacuum expectation values, the three states will mix among each other. The mass matrix is diagonalised for and eigenstates via the rotation matrix ,

 diag(M2h1,M2h2,M2h3)=O(−1)M2O,⎛⎜⎝hϕσ⎞⎟⎠=Oij⎛⎜⎝h1h2h3⎞⎟⎠, (2.17)

and we further identify the state with the SM 125 GeV Higgs boson. Following [10] we parametrise the rotation matrix in terms of three mixing angles , and ,

 O=⎛⎜⎝cosαcosβsinαcosαsinβ−cosβcosγsinα+sinβsinγcosαcosγ−cosγsinαsinβ−cosβsinγ−cosγsinβ−cosβsinαsinγcosαsinγcosβcosγ−sinαsinβsinγ⎞⎟⎠, (2.18)

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 flat 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 identified with . Hence, at tree level, , but it will become non-zero, see Eq. (2.22) below, when one-loop effects are included.

At the scale the one-loop effective potential along the minimum flat direction can be written as [35],

 V(φn)=Aφ4+Bφ4log⎛⎝φ2μ2\tiny{GW}⎞⎠, (2.19)

where the and coefficients are computed in the [36] scheme and given by,

 A = 164π2w4⎡⎣∑i=1,3M4hi⎛⎝−32+logM2hiw2⎞⎠+6M4W(−56+logM2Ww2)+3M4Z(−56+logM2Zw2) +9M4Z′(−56+logM2Z′w2)−12M4t(−1+logM2tw2)−23∑i=1M4Ni(−1+logM2Niw2)], B = 164π2w4(∑i=1,3M4hi+6M4W+3M4Z+9M4Z′−12M4t−23∑i=1M4Ni),

where are the tree-level masses of the three scalar eigenstates, , and , 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 the 1-loop corrected effective potential has a fixed vacuum expectation value that satisfies,

 log(wμ\tiny{GW})=−14−A2B, (2.20)

and using this relation we can rewrite the one-loop effective potential as,

 V=Bφ4(logφ2w2−12), (2.21)

and we can also evaluate the potential at the minimum to be , which gives the requirement for this to be a lower minimum than the one at the origin. The mass of the pseudo-dilaton is then given by,

 M2h2=∂2V∂φ2∣∣∣n=18πw2(M4h1+M4h3+6M4W+3M4Z+9M4Z′−12M4t−23∑i=1M4Ni). (2.22)

In summary, at the scale the conditions Eqs. (2.14)–(2.16) will be satisfied and the scalar potential will develop a non-trivial vev giving rise to non-zero vacuum expectation values and . For one scalar field, the Coleman-Weinberg mechanism requires the scalar quartic coupling to take very small values , 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,

 tan2α = ⟨h⟩2⟨ϕ⟩2+⟨σ⟩2=4λϕλσ−λ2ϕσ2(λσλhϕ−λϕλhσ)+λϕσ(λhϕ−λhσ), (2.23) tan2γ = ⟨σ⟩2⟨ϕ⟩2=2λhλϕσ−λhϕλhσ4λhλσ−λ2hσ, (2.24) tan2β = ⟨h⟩⟨ϕ⟩⟨σ⟩w(λhσ+λhϕ)(λϕ+λσ+λϕσ)⟨ϕ⟩2⟨σ⟩2−λh⟨h⟩2w2. (2.25)

Experimental searches of a scalar singlet mixing with the SM Higgs provide constraints on the mixing angles [37, 38, 39]. In our case, these translate as,

 cos2αcos2β>0.85. (2.26)

In the region where the decay is allowed we impose the stronger constraint . Nonetheless, due to the Gildener-Weinberg conditions the decay is highly suppressed. In the scan we perform is always greater than , so there is no need to worry about the SM Higgs decaying into two scalars. At the same time, strong constraints could come when the decays are allowed, we set so that these decays are kinematically forbidden.

For the study of dark matter the Lagrangian contains ten dimensionless free parameters, which are reduced to eight after fixing GeV and GeV. We perform a random scan on the remaining eight parameters in the ranges given in Table 1.

The matrix has no impact on the dark matter phenomenology, but it is crucial for Leptogenesis and it will be parametrised by three complex angles using the Casas-Ibarra parametrisation [40]. Therefore, once we set all the parameters for the active neutrinos to their best experimental fit, there are twelve free parameters in the model.

## 3 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 gauged symmetry. The idea of vector dark matter was first introduced in Ref. [31] and later studied in Refs. [7, 41, 9, 32]. 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 unstable.

After radiative symmetry breaking breaking of SU(2 by , which is in the fundamental representation of the group, there is a remnant SO(3) symmetry that ensures the three gauge bosons acquire the same mass , and are stable. In contrast to models where the DM is odd under a discrete symmetry, in the present scenario we can have dark matter semi-annihilation processes where a DM particle is also present in the final state. The DM annihilation diagrams are shown in Figs. 1 and 2, while the semi-annihilation ones are shown in Fig. 3.

Also, due to the radiative generation of in most region of parameter space the scalar mass will be smaller than the gauge boson mass, . This means that semi-annihilation processes 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]),

 ⟨σabcv⟩=3g4\tiny{DM}128π(O2i)2M2Z′⎛⎝1−M2hi3M2Z′⎞⎠−2⎛⎝1−10M2hi9M2Z′+M4hi9M4Z′⎞⎠3/2. (3.1)

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 fix the dark matter relic abundance from the latest Planck satellite measurement [43]. Figure 4 shows the dark matter fraction as a function of and the scalar mass ; the isolated strip of points on the left side of the plots corresponds to the resonance .

On the left plot in Fig. 4 there is a large red coloured region on the left side (producing too much dark matter), in this region has a close value to (note that this region does not exist in the Coleman-Weinberg limit). This region exists thanks to very large values of and . In the left panel of Fig. 5 we show the dark matter fraction as a function of both vevs, and , from this plot we see there is an upper bound on in order not to overproduce dark matter, TeV. Later on we shall see that there is a lower bound on coming from leptogenesis, TeV, we have already imposed this bound on all the scatter plots we show.

In the right panel of Fig. 5 we show the dark matter fraction as a function of and the gauge coupling . In this plot it becomes clear that as we increase the gauge coupling, the relic density decreases. The left panel of Fig. 6 shows the same analysis for the mixing angle and the quartic couplng . Here we can already notice a preference for the region , where takes on small values and . Due to the lower bound on the mixing angle takes on very small values, this is shown in the right panel of Fig. 6.

The spin-independent cross section between and a nucleon is given by,

 (3.2)

where is the nucleon mass, [33] is the nucleon form-factor, and are the elements of the rotation matrix Eq. (2.18) that relates the scalar mass eigenstates states to the ones in the Lagrangian. This orthogonal matrix 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 , this has been previously noted in [47, 7]; while the scalar state with a large component has no direct couplings either to dark matter or to Standard Model particles and hence gives only a small contribution to .

Figure 7 shows that except for resonances, the region with 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 Fig. 8 we show the direct-detection cross section as a function of the dark matter mass for benchmark point BP 1, we fix all the scalar couplings and vary only , the dip corresponds to .

## 4 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

 nbobss=(8.75±0.23)×10−11. (4.1)

In the Akhmedov-Rubakov-Smirnov framework [11] a lepton flavour asymmetry is produced during oscillations of the right-handed Majorana neutrinos with masses around the electroweak scale or below, which makes this approach compatible with classical scale invariance.111In the sense that no additional very large scales are required to be introduced in the model to make this type of leptogenesis work. From Big Bang nucleosynthesis we obtain the lower bound MeV, in order not to spoil primordial nucleosynthesis. For our calculations we make use of the the Casas-Ibarra parametrisation [40] for the matrix ,

 YD†=Uν⋅√mν⋅R⋅√MN×√2⟨h⟩, (4.2)

where and are diagonal mass matrices of active and Majorana neutrinos respectively. The active-neutrino-mixing matrix is the PMNS matrix which contains six real parameters, including three measured mixing angles and three CP-phases. The matrix is parametrised by three complex angles . Using this framework with three right-handed neutrinos one can generate the correct baryon asymmetry without requiring tuning the mass splittings, but rather enhancing the entries in the Dirac Yukawa matrix through the imaginary parts of the complex angles [49].

Due to the non-trivial topological structure of the vacuum in SU(2 there exist electroweak sphaleron processes which violate quantum number, and these will transfer the lepton flavour asymmetry into a baryon asymmetry , with the conversion factor given by,

 nbs≃−314×0.35×nLes. (4.3)

A critical condition for the mechanism of [11] to work, is that two of three neutrino flavours, and , should come into thermal equilibrium with their Standard Model counterparts before the universe cools down to (when electroweak sphaleron processes freeze out), while the remaining flavour does not. In other words, the present mechanism consists of different time scales , where represents the temperature at which equilibrates with the thermal plasma and is the temperature at which the oscillations start to occur. In terms of the decay rates for the three sterile neutrino flavours this implies,

 Γ2(T\tiny EW)>H(T\tiny EW) ,Γ3(T\tiny EW)>H(T\tiny EW) ,Γ1(T% \tiny EW)

where is the Hubble constant,

 H(T)=T2M∗P,M∗P≡MP√g∗√4π3/45≃1018GeV (4.5)

and is the reduced Planck mass. Therefore, we require,

 Γ1(T\tiny EW)=12∑iYD†eiYDieγavT\tiny EW

Here the dimensionless quantities are derived from the decay rates of the right-handed neutrino of the ‘electron flavour’ tabulated in Ref. [50]. These right-handed neutrino decay (or equivalently production) rates were computed in [50] using and processes222These processes are shown in Figs. 1 and 2 in Ref. [50] and contain a single external leg – as relevant for the -production or decay processes of interest. involving the neutrino vertices and with the Dirac Yukawas.

One can also ask if the new interactions present in our model, those involving the Majorana Yukawas, and , could affect 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 -number but above the electroweak phase transition temperature, the vev of vanishes). Hence these processes could contribute to the production or decay into the Standard Model particles only in combination with other interactions. As the Majorana Yukawa couplings are small on the part of the parameter space relevant for us (see Table 3) and the cross-section being proportional to means that these interactions will give subleading effects 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 temperatures, GeV, around which the main contributions to the lepton-flavour asymmetry are generated.

It was already shown in [6] that flavoured leptogenesis can work in a classically scale invariant framework. In their set-up three right-handed neutrinos are coupled to a scalar field that acquires a vev, as in the present model. The main difference being that in the present scenario we have not gauged the quantum number. We quote the final result for the lepton flavour asymmetry (of th flavour) obtained in  [6] from extending the results of Ref. [12] to the classically scale-invariant case,

 nLas=−γ2av×7.3×10−4∑c∑i≠ji(YD†aiYDicYD†cjYDja−YDtaiYD∗icYDtcjYD∗ja)×Iij, (4.7)

where the quantity is given by,

 (4.8)

for . For the case 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 flavour asymmetry is proportional to . Hence if we want to avoid any excessive fine-tuning of the mass splittings between different flavours of Majorana neutrinos, the relatively large values of GeV are preferred. From Fig. 9 we can see that there is a lower bound on if we impose some restriction on the mass splittings of the right-handed neutrinos. In view that we would like to stay far away from the fine-tuning region, we impose which gives the limit TeV in order for leptogenesis 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 Fig. 6.

As we can see from Fig. 9 there is also an upper bound on for each value of , this bound is mainly due to the wash-out criterion Eq. (4.6) not being satisfied any more. This upper bound becomes weaker once we reach GeV. This sits well with our approach based on the common dynamical origin of all vevs: once an explanation for dark matter is included, cannot be too large compared to .

The procedure to obtain the plot in Fig. 9 is as follows. We fix the complex phases and to the benchmark values given in [12] ( and ), and for each point we scan over , if we find 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 and has a negligible impact on the final results.

The generated total lepton asymmetry is proportional to , (cf. (4.7), (4.8))

 nL∼(YD)4⟨σ⟩MPΔM2Ni∼⟨σ⟩MPm2νv4, (4.9)

where we used the see-saw mechanism for the masses of visible neutrinos, and is the SM Higgs vev. Hence vanishes as approaches zero. This also explains why in Fig. 9, there is a stronger dependence on than on the masses .

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 the story.

We first note that our leptogenesis realisation does not require any sizeable fine-tuning of the mass splittings . For example our first benchmark point BP 1 has (cf. Table 3),

 MN=(0.225,0.25,0.275)GeV. (4.10)

At the same time, the masses of active neutrinos are set to agree with the observed mass splittings; for BP 1 we have,

 mν=(0,8.7,49.0)meV. (4.11)

The lepton asymmetry (4.7) also depends on the matrix of Dirac Yukawa couplings . We compute in the Casas-Ibarra parametrisation Eq. (4.2) using (4.10) and (4.11) along with the PMNS matrix and the matrix. We have carried out a general scan on the complex angles of the matrix and found that having non-vanishing is important in order to obtain the required amount of lepton asymmetry.333Note that positive values of enhance the elements of the Dirac Yukawa matrix . At the same time this does not lead to any excessive fine-tuning. We have checked this for the numerical values of matrix elements in our scan. For example, for BP 1 we have (using the values in Table 3),

 R=⎛⎜⎝−36.52−33.80i34.11−36.97i5.854+4.604i84.43+100.0i−101.0+85.98i−16.63−14.20i−105.4+91.81i−93.42−106.4i14.94−17.61i⎞⎟⎠, (4.12)

and the resulting matrix of Dirac Yukawa couplings,

 YD=⎛⎜⎝17.87−2.12i−73.37−125.6i−210.9−127.3i−2.168−19.11i−134.4+77.79i−136.9+224.6i−3.395−0.2434i9.677+24.56i34.69+28.93i⎞⎟⎠×10−8. (4.13)

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.

## 5 Connection among the scales

After having performed a scan over all free parameters in our model, we find that:
(1) TeV in order for dark matter not to overclose the universe, and
(2) TeV in order in order for leptogenesis to explain the baryon asymmetry.

From the left plot of Fig. 6 we can see that the interesting region in parameter space has large values of , and with this in mind we can separate the interesting regime into two regions:

1. TeV
In this region444Recall that . we have so there is a strong mixing between the scalar states and , and due to the Gildener-Weinberg conditions . To avoid overproducing DM, both and have to be less than 10 TeV. Due to the not so large values of , a large part of this region requires some amount of fine-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.

2. TeV
In this region we have , so it can be seen as the Coleman-Weinberg limit of the more general Gildener-Weinberg mechanism. The scalar overlaps maximally with and can be thought of as the Coleman-Weinberg scalar. In this region the radiative symmetry breaking is induced by and we get . This region also corresponds to the majority of good (blue) points in Figs. 4-6. Most points have . This is the region of most interest since the large values of require almost no fine-tuning in 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 . 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 values for the neutrino sector taken from [51], we assume normal ordering for the active neutrino masses. The values for are computed as the average of . This estimate corresponds to the naive see-saw relation and it is smaller than the actual entries in the matrix due to the enhancement by the imaginary parts of in the matrix. Nevertheless, for our benchmark points these enhancement factors are always less than .