Scalar Triplet Flavor Leptogenesis with Dark Matter

We investigate a simple variant of type-II seesaw, responsible for neutrino mass generation, where the particle spectrum is extended with one singlet right-handed neutrino and an inert Higgs doublet, both odd under an additional $Z_2$ symmetry. While the role of the dark matter is played by the lightest neutral component of the inert Higgs doublet, its interaction with the Standard Model lepton doublets and the right-handed neutrino turns out to be crucial in generating the correct baryon abundance of the Universe through flavored leptogenesis from the decay of the $SU(2)_L$ scalar triplet, involved in type-II framework.


I. INTRODUCTION
Existence of tiny but non-zero neutrino mass [1][2][3][4] along with the observed excess of matter over antimatter in the Universe [5,6] are undoubtedly two of the most challenging problems in the present day particle physics and cosmology which signal for physics beyond the Standard Model (SM). Among many promising scenarios came up as a resolution to these issues, the seesaw mechanism provides an elegant framework to deal with. As is well known, in case of type-I seesaw [7][8][9][10], presence of SM singlet right handed neutrinos (RHN) not only helps in generating tiny neutrino mass but they can also be responsible for explaining the observed matter-antimatter asymmetry via leptogenesis [11][12][13][14][15][16][17][18][19][20][21][22]. A variant of it, namely the type-II seesaw construction [23][24][25][26][27] also provides an equally lucrative resolution by introducing a SU (2) L scalar triplet to the SM field content whose tiny vacuum expectation value (vev) takes care of the small neutrino mass. However, to generate the baryon asymmetry of the Universe via leptogenesis, this minimal type-II framework needs to be extended either with another triplet [28][29][30][31] or by a singlet right-handed neutrino [32][33][34][35][36][37]. In the latter possibility, the role of the single RHN is to contribute to CP asymmetry generation via the vertex correction (in the triplet decay) provided it carries a Yukawa interaction with the SM Higgs and lepton doublets.
Additionally, several astrophysical and cosmological observations indicate that energy budget of our Universe requires around 26% of non-baryonic matter, known as the dark matter (DM) [38][39][40][41]. To explain such DM, an extension of the SM is required as otherwise it fails to accommodate any such candidate from its own particle content. Since all these unresolved issues (tiny neutrino mass, matter-antimatter asymmetry and nature of dark matter) point out toward extension(s) of the SM, it is intriguing to establish a common platform for them. With this goal in mind, we focus on the SM extended with a scalar triplet and a fermion singlet (like one RHN). While this can explain the neutrino mass and matterantimatter asymmetry as stated above, accommodating a DM in it is not that obvious.
One simplest possibility emerges if that singlet fermion (the RHN) can be considered as the DM candidate. However, as pointed out above, this fermion taking part in the CP asymmetry generation has to carry an Yukawa interaction (of sizable strength) and hence can't be stable provided its mass remains above the electroweak (EW) scale. On the other hand, if it happens to be lighter than the SM Higgs (or gauge bosons), it might be a freeze in type of DM [42]. In this case also, the small Yukawa coupling, as required by the freeze-in generation of DM relic, makes the CP asymmetry negligible and therefore such a possibility needs to be left out.
We thereby plan to extend this framework by including an inert Higgs doublet (IHD) [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] such that its lightest neutral component results in dark matter while the IHD too contributes to CP asymmetry generation via its Yukawa interaction involving the SM lepton doublet and the sole RHN. Involvement of the DM in generating the CP asymmetry required for explaining the matter-antimatter asymmetry of the Universe is an important aspect of our work. Note that the inert doublet DM phenomenology is mostly governed by the gauge interactions and the mass-splitting among the inert Higgs doublet components, but not on the Yukawa interaction [53] (contrary to the case of freeze-in RHN as DM) and hence it is not expected to be in conflict with sufficient production of CP asymmetry. Furthermore, search for doubly and singly charged particles involved in the triplet can be quite interesting from collider aspects. Keeping that in mind, we plan to keep the mass of the triplet not very heavy. It is further supported by the finding that the mass splitting among the IHD components (for DM relic satisfaction) along with the neutrino mass generation dominantly by the type-II mechanism keeps the triplet mass below 10 12 GeV. Note that it becomes essential to incorporate the flavor effects in leptogenesis [35,42,[60][61][62][63][64] which come in to effect below the mass equivalent temperature ∼ 10 12 GeV. This observation, importance of including flavor effects in triplet leptogenesis, is another salient feature of our analysis.
The paper is organized in the following manner. We introduce the structure of the model in Section II where the particle content with their respective charges under different symmetry group have been discussed. In Section III, we discuss the mechanism to generate the neutrino mass and how to get a complex structure of the Yukawa coupling matrix responsible for generating the matter-antimatter asymmetry. In section IV, we briefly summarize the inert doublet DM phenomenology and move on to discuss generation of matter-antimatter asymmetry in the Universe via flavor leptogenesis in Section V. Finally in Section VI, we conclude.

II. THE MODEL
The SM is extended with a SU (2) L scalar triplet ∆, a scalar doublet Φ and a fermionic SM singlet field N R . The corresponding charge assignments of the relevant fields are provided in Table I. The Lagrangian involving the new fields is then given by where α, β correspond to three flavor indices. Note that N R and Φ are odd under an additional discrete symmetry Z 2 , thereby making Φ as inert. This also forbids the Yukawa coupling of the SM Higgs with the RHN, however allows similar interaction with the inert Higgs doublet Φ. The lightest neutral component of this Φ field plays the role of the dark matter while decay of the triplet into lepton doublets generates the lepton asymmetry which will further be converted into baryon asymmetry by the sphaleron process. Here both the inert Higgs doublet and the RHN take part in producing the CP asymmetry. where Here we consider all the parameters appearing in the scalar potential to be real. We also consider µ 2 H < 0 as that would be crucial for electroweak symmetry breaking (EWSB). On the other hand, remaining mass parameters such as µ 2 Φ , M 2 ∆ are taken as positive. Denoting the vev of H and ∆ by v (= 246 GeV) and v ∆ respectively, the multiplets after the EWSB can be expressed as where h is the SM physical Higgs boson with mass 125.09 GeV [65] and the induced vev of the triplet is found to be related by [32] v considering M ∆ v. Interestingly, the constraint on ρ-parameter (ρ = 1.00038 ± 0.00020) [66] restricts v ∆ 4.8 GeV. Note that v ∆ needs to be small enough to accommodate the tiny neutrino mass via ∆ L L coupling and hence we fix it at 1 eV. Then depending on the mass of the ∆ particle, µ 1 can be obtained by the use of Eq. (5). On the contrary, the analogous coupling µ 2 remains unrestricted and hence can have a sizable value. This therefore will be treated as independent parameter for generating sufficient CP asymmetry as we see in the leptogenesis section.
The masses of the different physical scalars of IHD are given (unaffected by the presence of the triplet scalar) as > 0. Without any loss of generality, we consider λ 6 < 0, λ 5 + λ 6 < 0 so that the CP even scalar (H 0 ) is the lightest Z 2 odd particle and hence the stable dark matter candidate. Due to the presence of the term proportional to µ 1 , there will be a mixing between the SM Higgs and the triplet. However, the mixing being of order v ∆ (taken to be ∼ 1 eV, responsible to generate light neutrino mass), this can safely be ignored. We set λ 1 , λ 2 , λ 3 = 0 for simplicity and then find masses of the physical triplet components One should note that LHC puts a strong constraint on mass of ∆ ±± as M ∆ ±± > 820 GeV (870 GeV) at 95% C.L. from CMS [67] (ATLAS [68]) for v ∆ 10 −4 GeV. LHC also set a constraints on M ∆ ± > 350 GeV.

III. NEUTRINO MASS
We now proceed to discuss the neutrino mass generation in the present model. As mentioned before, the neutrino mass is expected to be generated via the triplet interaction with the SM lepton doublets resulting the type-II contribution as With a choice of v ∆ as 1 eV, the coupling matrix Y ∆ can be accordingly adjusted to produce the light neutrino matrix (m ν ) consistent with the oscillation data. To make it more specific, we consider, with m d ν = diag(m 1 , m 2 , m 3 ) and U is the PMNS mixing matrix (in the charged lepton diagonal basis) of the form: × diag(e iα 1 /2 , e iα 2 /2 , 1), (9) parametrized by three mixing angles θ 12 , θ 23 , θ 13 (denoted by c ij = cos θ ij , s ij = sin θ ij ), the Dirac CP phase δ and Majorana CP phases (α 1 , α 2 ). For simplicity, we now consider Majorana phases and the lightest neutrino mass to be zero. Thereby, using the best-fit values of the mixing angles and δ [69] as in Table II, we obtain the following structure of the coupling matrix (using Eq. (7)) in case of normal hierarchy (NH) of neutrinos, We will make use of this Y ∆ in the rest of our analysis wherever appropriate.
Note that in our model, due to the presence of one RHN having Yukawa coupling Y , a radiative contribution to the light neutrino mass [70] is expected to be present which is given by It is then understood that for a specific value of the DM mass, m H 0 , along with the mass to the light neutrino mass matrix which depends on the magnitude of Y α coupling. Since we plan to investigate the scenario where the light neutrino mass is mainly contributed by the type-II contribution, we determine here the limits on Y α for which m R ν remains insignificant. For this purpose, first we assume all Y α to be same given by Y . Secondly, we impose a restriction that the contribution to m 2 (as m 2 is the second lightest eigenvalue of m ν ) coming from m R ν remains below 10% contribution followed from type-II seesaw estimate m II ν (henceforth called type-II dominance). Using this ansatz, we provide Y versus M N plot in Fig. 1 indicating an upper limit on Y value corresponding to a specific RHN mass. In making this plot, we consider DM mass m H 0 = 535 GeV with different ∆M indicated by different colors. This limit on Y will be useful in estimating the CP asymmetry. As the lightest neutrino is taken to be massless in type-II contribution, it is clear that with the appropriate Y value (consistent with the Fig. 1 and leptogenesis), m 1 will defer from zero value as it obtains a tiny correction from m R ν .

IV. DARK MATTER PHENOMENOLOGY
The present setup shelters two particles N R and Φ non-trivially charged under Z 2 . Hence, being stable either of them can play the role of the DM. The phenomenology of a singlet fermions like N R as a WIMP DM candidate with renormalisable interactions remain uninteresting as it predicts overabundant relic density due to the lack of their annihilation channels. On the other hand, as is well known, the study of an IHD provides several interesting prospects both in DM phenomenology as well as in collider searches and hence here we primarily stick to the IHD as dark matter by considering M N > m H 0 . An unbroken Z 2 symmetry in the current scenario guarantees the stability of the scalar dark matter.
1 Using the fact that M N is very heavy compared to all IHD components, the radiative contribution can be approximated by (m R ν ) αβ Since it is a well studied framework, in this section, we briefly focus on the parts of DM phenomenology relevant for our analysis extended to leptogenesis section.

A. Relic Density
The inert Higgs doublet [43][44][45][46][47][48][49][50][51][52][54][55][56] where z = m H 0 /T and Y eq H 0 denotes equilibrium number density of H 0 whereas σv H 0 H 0 →XX represents the thermally averaged annihilation cross-section [71] of the DM annihilating into the SM particles denoted by X. The relic density of the inert scalar H 0 is then expressed as with Y 0 H 0 denoting the asymptotic abundance of the DM particle after freeze out. In order to calculate the relic density and study the DM phenomenology of the IHD dark matter we use the package micrOMEGAs4.3.5 [72].
As stated before, the case of an IHD dark matter is well studied and hence, we only The null detection of the DM in direct search experiments like LUX [74], PandaX-II [75,76] and Xenon1T [77,78] puts a severe constraints on the DM parameter space. There exists two different possiblities for the DM to interact with nuclei at tree level in the scenario under consideration: (a) elastic scattering mediated by SM Higgs boson and (b) inelastic one mediated by electroweak gauge bosons. The spin independent elastic scattering cross section mediated by SM Higgs is given as [79] where µ n = m N m H 0 /(m N + m H 0 ) is the DM-nucleon reduced mass and λ L is the quartic coupling involved in DM-Higgs interaction. A recent estimate of the Higgs-nucleon coupling f gives f = 0.32 [80]. On the other hand, the inelastic scattering cross-section mediated by a gauge boson is expressed as [81], with c = 1 for fermions and c = 4 for scalars. Here the hypercharge of the DM is 1/2. also constrain the DM parameter space. In a recent study [53], it has been shown that the IHD mass regime below 400 GeV is strictly ruled out by Fermi-LAT.

V. LEPTOGENESIS
In this section, we aim to study the leptogenesis scenario resulting from the CP violating out of equilibrium decay of the triplet carrying lepton number of two units in the model. As advocated, this will happen due to the presence of the sole RHN of the setup contributing to the one-loop vertex correction to the tree level triplet decay into leptons as shown in Fig. 3. It is interesting to note that with one triplet, the generated CP asymmetry can't be of purely flavored one [35] in contrast to the presence of this possibility in standard triplet leptogenesis involving two scalar triplets. We first discuss the generation of CP asymmetry from the triplet decay and then talk about the evolution of the lepton (B − L) asymmetry using Boltzmann equations. In doing so, our plan is to keep the triplet mass as light as possible as that would be interesting from the point of view of collider search for triplet states. In turn, this indicates that flavor effects of the charged lepton Yukawa couplings need to be incorporated provided leptogenesis takes place below temperature ∼ 10 12 GeV.

A. CP asymmetry generation
The flavored CP asymmetry produced as a result of the interference between the tree level and the loop level diagram shown in Fig. 3 can be defined and evaluated as [32,33] i ∆ = 2 where, Γ tot ∆ is the total decay width of ∆: Similarly, the anti-triplet decay Γ tot ∆ also contributes to the total decay width in the denominator. It would be useful to define the branching ratios B , B H and B Φ at this stage, representative of the ∆ triplet decay to lepton and scalar final states as: We notice now that among the various parameters involved in the expression of flavored CP asymmetry i ∆ in Eq. (17), Y ∆ is obtained from Eq. (10) while µ 1 becomes function of M ∆ via Eq. (5) with the choice v ∆ = 1 eV. Finally, to maximize the CP asymmetry, we fix Y to its largest allowed value corresponding to a specific choice of M N (and ∆M ) from Fig. 1 so as to restrict the radiative contribution negligible (keeping it below 10%) compared to the type-II one toward light neutrino mass. Although there is no direct correlation between CP asymmetry and the mass splitting ∆M among IHD components, it can be noted that ∆M being involved in restricting the maximum value of Y for the type-II dominance of neutrino mass (see Fig. 1), plays an important role here. Hence, i ∆ effectively remains function of three independent parameters µ 2 , M ∆ and M N . It is interesting to note that in this case, there exists a coupling µ 2 in the CP asymmetry expression which does not participate in the neutrino mass generation unlike conventional type-(I+II) scenario where all the couplings involved in CP asymmetry also take part in the neutrino mass [32,33,37]. As a result, in the latter case (i.e. in type-(I+II)) with type-II dominance, the relevant parameter space is restricted leading to M ∆ quite heavy. For example, it was shown in [37], in the context of type-II-dominated left-right seesaw model, that M ∆ turns out to be 10 12 GeV or beyond.
On the other hand, involvement of otherwise free parameter µ 2 may open up a relatively wider parameter space in our case. Below we proceed to get some idea on the CP asymmetry generation by scanning over the parameters for our work.  used to specify the values of Yukawa coupling Y for these mass splittings respectively (see Fig. 1). For M ∆ below 10 11 GeV (though larger than 10 9 GeV), tau-Yukawa interaction comes to equilibrium making the asymmetries along a (a coherent superposition of e and µ lepton flavors) and τ flavor distinguishable. We elaborate on it latter. Hence in this region, we study a,τ ∆ separately in Fig. 4(a)  In order to study the evolution of the B − L asymmetry, we need to employ a set of coupled Boltzmann equations following the analysis of [35]. This set includes differential equations for triplet density Σ = ∆ +∆, triplet asymmetry ∆ ∆ = ∆ −∆ and B/3 − L i asymmetries considering flavor effects as we have considered a specific hierarchy M ∆ < M N in our analysis. Assuming the triplet scalar was at thermal equilibrium with plasma in the early Universe, below are the specified interactions which have the potential to change its number density as well as produce or washout the effective B − L charge asymmetry: • Decay [∆ →¯ i¯ j , ∆ → HH and ∆ → ΦΦ] and inverse decay: The total decay rate density is then represented by: • Lepton number (∆L = 2) and Lepton flavor violating s and t channel scatterings (mediated by the triplet/anti-triplet): XX ↔¯ i¯ j , X j ↔X¯ i having reaction densities γ XX i j and γ X j X i respectively.
• Lepton flavor violating triplet mediated s and t channel scattering: ( a b ↔ i j ) s , ( a b ↔ i j ) t with reaction densities given by (γ a b i j ) s and (γ a b i j ) t .
Keeping in mind the above discussion, the following Boltzmann equations are constructed, where Y ∆ X is defined as the ratio between particle and antiparticle number densities difference to entropy: Y ∆ X = (n X − nX)/s, where n X (nX) is number density of a particular species X(X).
Depending on the temperature range, the index i in the RHS of Eq. (23) will run differently, e.g. for 10 9 GeV T 10 12 GeV, i = a, τ as done in Fig. 4, while for T < 10 9 GeV, i = e, µ, τ need to be included. The generated asymmetry in number densities involving leptons of a specific flavor Y ∆ i as well as that of the Higgs (originated from the inverse decay and subtraction of the on-shell contribution for ∆L = 2 processes) can be related to the fundamental asymmetries ∆ ∆ and ∆ B/3−L i with the help of the equilibrium conditions applicable. The corresponding conversion factors 3 are defined in terms of C and C H matrices as below [35,61]: where Y X k are the elements 4 of Y T X = Y ∆ ∆ , Y ∆ B/3−L k and C ijk and C ijabk are given by: Then the final lepton asymmetry is converted to Baryon asymmetry via sphaleron processes as given by : 3 We simplify the situation by considering the chemical potential of the Φ field to be zero and hence where the factor 3 is due to the degrees of freedoms associated to the SU (2) L scalar triplet. In order to explore the parameter space of our model so as to produce the observed baryon asymmetry Y ∆ B = (8.718 ± 0.012) × 10 −11 [85,86], first we choose a specific value of M N = 5 × 10 10 GeV. Then based on our previous discussion we infer that lepton asymmetry with 10 9 GeV< M ∆ < M N will be produced along two orthogonal directions, i.e. a and τ while below 10 9 GeV, all three flavor directions have to be taken into account. With the help of chemical equilibrium constraint equations (coming from relevant Yukawa and EW Sphaleron related reactions that are in equilibrium) as well as other constraints such as hypercharge conservation (applicable in this energy range) lead to the following structure of For M ∆ below 10 9 GeV, C and C H become 3 × 4 and 1 × 4 matrices.
Using the input on the flavored CP asymmetries along a and τ directions from Fig. 4, obtained as a function of µ 2 and M ∆ for a specific ∆M value, we employ the set of Boltzmann We also notice that with the increase in ∆M value, the baryon asymmetry satisfying contour gets shifted toward heavier mass range of the triplet. This observation is interesting as it is correlated to the DM phenomenology. We recall that even though a smaller ∆M is M N values. As can be seen, the allowed mass of the triplet comes down to an even lower mass close to 10 8 GeV. It is perhaps pertinent here to mention that our entire parameter space corresponds to the Yukawa regime [34,35] where the Yukawa induced inverse decay processes (characterised by B ij γ D ) play important role and hence flavor effects become crucial. With this entire viable range of ∆M , masses of the RHNs are found to be in the regime ∼ 10 (9−12) GeV. 6 In a recent work [31], authors have shown the importance of incorporating density matrix formalism to evaluate the baryon asymmetry for triplet leptogenesis, even beyond T 10 12 GeV. In this formalism, diagonal entries of the density matrix indicate asymmetry along each lepton flavor direction while offdiagonal entries represent quantum correlations between different flavors. Though this is the most general approach, we have found that inclusion of the off-diagonal entries can only change the final result by 20% or less (corresponding to the values of parameters involved in producing the plots of Fig. 7) and hence neglected here.

VI. CONCLUSION
In this work, we present a simple extension of the basic type-II seesaw (i.e. with one SU (2) L triplet in addition to SM) scenario including an additional RHN and one IHD, which can accommodate neutrino mass, dark matter as well as capable of explaining the baryon asymmetry of the Universe via leptogenesis mechanism. The interesting part of the study is the involvement of the DM multiplet, along with the RHN, in the vertex correction of the triplet's decay to two leptons which can successfully produce the required amount of CP asymmetry in order to address the baryon asymmetry of the Universe. Although the decay of RHN can also produce lepton asymmetry in the present setup, we assume a specific mass hierarchy M ∆ < M N and hence the asymmetry generated by the decay of the triplet is the effective one. We incorporate the flavor effects in this triplet leptogenesis study as we aim to lower the triplet mass as much as possible in view of its accessibility at the collider.
We find it is possible to generate sufficient lepton asymmetry with M ∆ as low as ∼ 10 8 GeV.
Turning to the neutrino side, where the dominant contribution to the light neutrino mass follows from the tiny vev of the triplet, there exists a radiative contribution too which we restrict to be negligible by choosing the associated Yukawa to be small enough. This consideration is related to the mass splitting involved in the IHD which plays a twofold role here. Firstly, the IHD as a DM results with a specific range of this mass splitting (10 −4 −O(1) GeV). Secondly, a smaller mass splitting (hence a larger neutrino Yukawa) turns out to be preferable for generating sufficient CP asymmetry in this flavored leptogenesis framework.
Since the lower limit of ∆M is somewhat governed by the inelastic direct detection bound, in a way it restricts the mass of the triplet within a certain range.
On the other hand, due to the involvement of particles like ∆ ± , ∆ ±± , H ± and N R , the present setup is also subjected to the constraints coming from the lepton flavor violating decays like µ → eγ. Keeping this in our mind, we calculate the Br(µ → eγ) and found it to be many orders of magnitude smaller than that of the present upper bound on it (< 4.2 × 10 −13 at 90% C.L [87]).