Type-I thermal leptogenesis in $Z_3$-symmetric three Higgs doublet model

Our present work explores the possibility of neutrino mass generation through {\em Type-I see-saw} mechanism and provides an explanation of the baryon asymmetry of the Universe via thermal leptogenesis in the framework of $Z_3$-symmetric three Higgs doublet model (3HDM) augmented with three right-handed neutrinos. Here the thermal leptogenesis is initiated by the out-of-equilibrium decay of the lightest heavy neutrino $N_1$. The constraints arising out of the scalar sector put strong bound on the model parameter $\tan \beta$, which in turn takes part in the computation of the lepton asymmetry $\epsilon$. Lepton asymmetry being converted partially into the baryon asymmetry by electroweak sphelaron processes, will account for the required baryon asymmetry satisfying the current data. We therefore analyse the parameter space consistent with the constraints arising from neutrino oscillation, lepton asymmetry and baryon asymmetry together, last one being the most stringent one.


I. INTRODUCTION
While the Standard Model (SM) has come out with flying colors in all experimental tests after Higgs discovery [1,2], there are several reasons to believe that this is not the ultimate story but only an effective theory valid up to some high energy scale, above which some other theory takes over. Like other shortcomings of SM, explanation of neutrino mass generation necessitates a beyond SM scenario incorporating different types of seesaw mechanisms [3][4][5][6][7][8]. Addition of right-handed (RH) neutrinos, which are singlet under SM gauge group, results in the creation of neutrino mass via Type-I see-saw mechanism [3][4][5][6]. Besides, inequality between the numbers of baryons and anti-baryons gives rise to another puzzle named baryon asymmetry, which remains unaddressed within the ambit of SM [9,10]. The dynamic generation of baryon asymmetry needs to comply with three Sakharov conditions [11], which require : (a) baryon number violation, (b) C or CP -violation, (c) outof-equilibrium condition. Although all these conditions are satisfied within SM, generated baryon asymmetry becomes inadequate to tally with the current data [12] : n B , n B , n γ being number densities of baryons, anti-baryons and photons respectively. Thus new physics (NP) needs to be introduced to compensate the due amount of baryon asymmetry within SM.
The out-of-equilibrium decay of the RH heavy neutrinos in Type-I see-saw mechanism, induces leptogenesis [13,14], which can be treated as an additional source of baryon asymmetry apart from SM. The complex Yuakawa couplings give rise to CP -violation, thereby fulfilling the required criteria for generating baryon asymmetry. At the time of generation of asymmetry, the decay rate being slower than the expansion rate of the Universe, out-of-equilibrium condition is automatically generated. Finally partial conversion of lepton asymmetry (created during the out-of-equilibrium decay of heavy neutrinos) to baryon asymmetry, occurs through (B + L) violating electroweak (EW) sphelaron processes [15].
In this paper, these two aforementioned shortcomings of SM will be addressed in the framework of Z 3 -symmetric three Higgs doublet model (3HDM) accompanied by three heavy RH neutrinos. In particular, here we shall use thermal leptogenesis [14,16], which allows hierarchical heavy neutrino masses, mass of one of them being much smaller than others.
Besides, only thermal generation and out-of-equilibrium decay of lightest heavy neutrino will play the crucial role in generating lepton asymmetry. In presence of three RH neutrinos, mass generation of light neutrinos will be possible via Type-I see-saw mechanism. As can be seen later, the entire parameter space will be constrained by the restrictions coming from the scalar sector, as well as the more stringent constraints arising from neutrino oscillation data, lepton asymmetry and baryon asymmetry respectively. There can be another variant of leptogenesis, in which the CP -asymmetry is enhanced by considering the mass-splitting between any two of the heavy neutrinos to be comparable with their decay width. This type of leptogenesis is termed as Resonant leptogenesis [17,18]. Since the lower bound on the heavy neutrino mass is relaxed in this case, they can be accessible for studying collider signatures in future colliders. However we shall restrict ourselves in studying thermal leptogenesis in this paper and shall not consider the other variant.
This paper is structured as follows. Sec. II contains the information regarding the particle content of the model considered for analysis. Sec. III comprises of detailed discussion of several constraints imposed on the parameter space. In sec. IV, we elaborate the fitting of neutrino oscillation data using Casas Ibarra parametrization. Sec.V deals with thermal leptogenesis, i.e. solutions of Boltzman equations. In sec. VI, we present analysis and results. Finally we summarize and conclude in sec. VII.

II. MODEL
In this analysis, we consider Z 3 -symmetric 3HDM comprising of three SU (2) L doublets φ 1 , φ 2 and φ 3 each with hyper-charge Y = +1 1 , augmented with three heavy RH neutrinos N 1R , N 2R , N 3R . For simplicity, we shall denote these three heavy neutrinos as N 1 , N 2 , N 3 throughout the analysis. The complete description of different sets of quantum numbers assigned to all the particles can be found in table I.

A. Z 3 -symmetric Scalar Lagrangian
Following the quantum numbers assigned to the doublets, as mentioned in table I, the Z 3 -symmetric scalar potential involving φ 1 , φ 2 and φ 3 can be written as [19], After symmetry breaking, φ i can be expressed as, v i being the vacuum expectation value (VEV) of φ i . Two important parameters of the model tan β and tan γ can be expressed as the ratios of VEVs of doublets : Therefore v 1 , v 2 and v 3 can be written in terms of the mixing angles β and γ as : The quartic couplings [λ 1 , λ 2 , ...λ 12 ] are taken to be real to avoid any kind of CP -violation in the scalar potential.
) and five mixing angles, i.e. three in the CP -even sector (α 1 , α 2 , α 3 ), one in CP -odd sector (γ 1 ) and one in charged scalar sector (γ 2 ) [19]. The lightest neutral physical state h resembles the SM Higgs boson with mass 125 GeV at the Alignment limit defined as : α 1 = γ, α 2 + β = π 2 [19]. The details of the scalar sector of Z 3 -symmetric 3HDM including the basis transformations from flavor basis to mass basis etc. can be found in [19]. To avoid repetition, we shall not provide the same details here.

B. Yukawa Lagrangian
According to the assignment of quantum numbers to the fields, it is quite obvious that this model forbids the flavor changing neutral currents (FCNCs) for the neutral scalars. Uptype, down-type quarks and leptons will acquire masses through the couplings with φ 1 , φ 2 and φ 3 respectively. Due to the presence of three heavy RH neutrinos, SM light neutrinos can also possess masses via Type-I see-saw mechanism, only φ 3 being responsible for the mass generation of neutrinos.
Thus we can write down the Z 3 -symmetric Yukawa Lagrangian along with the Majorana mass terms for the heavy neutrinos as : Here L i are left-handed (LH) lepton doublets andφ i = iσ 2 φ * i . As mentioned earlier, only φ 3 will be responsible for generating SM light neutrino masses. Yukawa couplings y j are taken to be complex for generating CP -asymmetry in leptogenesis. The real and imaginary parts of the Yukawa couplings y j are constrained by recent neutrino oscillation data [20], as will be discussed elaborately in section IV.

III. CONSTRAINTS TO BE CONSIDERED
For the analysis, we shall consider a multi-dimensional parameter space, spanned by the following independent parameters : tan β, γ, γ 1 , Since Alignment limit will be imposed strictly, the lightest Higgs h being SM-like, M h is taken to be 125 GeV. In addition to these, the constraints to be imposed on the parameter space are illustrated below.
• Yukawa couplings y j (j = 1, 2, ...9) are constrained from the neutrino oscillation data and constraints arising from leptogenesis, as will be discussed later. There is also an upper bound of |y j | ≤ √ 4π arising from perturbativity.
• Boundedness of the scalar potential (eq.(2)) can be ensured by satisfying following stability conditions involving the quartic couplings :

B. Constraints from oblique parameters
In presence of additional scalars in the model, the oblique parameters S, T, U will be modified accordingly. The present limits on their deviation from SM values are [21]: Specially the Z 3 -symmetric 3HDM parameter space is sensitive to the deviation of Tparameter from SM value, because this deviation controls the mass-splitting between the charged and the neutral scalars. We have ensured the compatibility with T -parameter constraint by keeping the mass-splitting between the charged and the neutral scalars ∼ 50 GeV.

C. Constraints on Higgs signal-strengths from LHC data
To make the parameter space compatible with the current LHC data, one has to compute Higgs signal strengths in different Higgs decay channels. For the i-th decay channel of h, the signal strength µ i can be computed as the ratio of cross section of Higgs production via p − p collision times the branching ratio of Higgs decay into the i-th channel in 3HDM and the same quantity measured in the SM: Among all Higgs production process, the dominant contribution at the LHC comes from the gluon-gluon fusion process mediated by heavy quarks in triangular loops. The partonlevel cross section can be written as [22] σ s being gluon-gluon invariant energy squared.
Using eqs. (7) and (8), one can rewrite the signal strength µ i as : where Γ tot stands for the total decay width.
Since the Alignment limit is being invoked strictly, the lightest Higgs h being SM like, the Higgs signal strengths in the W W, ZZ, bb, τ + τ − mode are satisfied automatically. Γ(h → γγ) receives an extra contribution coming from the charged Higgs mediated loop and are modified. At the exact Alignment limit, the total Higgs decay width coincides with that of the SM Higgs h. Thus the signal strength µ h→γγ can be approximated to Γ 3HDM (h→γγ) Γ SM (h→γγ) . Expressions for the decay width Γ(h → γγ) can be found in appendix A. We have used 2σ-deviation from the allowed values of signal strength to scan the parameter space [23].

IV. FITTING OF NEUTRINO-DATA
As mentioned earlier, the Yukawa couplings need to be complex in order to generate lepton asymmetry required for leptogenesis. Following the Yukawa Lagrangian in eq.(5), after symmetry breaking, the Dirac mass matrix M D can be computed as : Here complex Yukawa couplings y j s are decomposed into real and imaginary parts as : y jR and y jI respectively. Majorana mass matrix M R is assumed to be diagonal for simplicity : For the mass generation of light neutrinos through Type-I see-saw mechanism, the neutrino mass matrix can be expressed in terms of Dirac mass matrix M D and Majorana mass matrix M R as : M ν can be diagonalised to get the light neutrino masses by the transformation : where m 1 , m 2 , m 3 are three light neutrino masses, U PMNS is the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS) matrix and can be written as : where c ij ≡ cos θ ij , s ij ≡ sin θ ij and δ CP is the CP -violating phase. To parametrize the elements of M D , one can use the parametrization proposed by Casas and Ibarra (CI) [24], as will be mentioned in detail in the next subsection.

A. Casas Ibarra Parametrization
According to the CI parametrization [24], M D can be rewritten as : O being a complex orthogonal matrix, with complex angles θ, φ, ψ, can be expressed as [14], where c α , s α are the shorthand notations for cos α and sin α respectively.
For our analysis, we have assumed the phase associated with the complex angles to be zero. One of the three light neutrinos is taken to be massless, i.e. m 1 = 0. We have

V. LEPTOGENESIS
During this analysis, we aim to explore that portion of the parameter space, where the model parameters satisfy the constraints coming from neutrino oscillation data, as well as the current bound on baryon asymmetry. The main ingredient of generating baryon asymmetry here is leptogenesis, through which the lepton asymmetry is produced. In this scenario, the lepton asymmetry is originated by the CP -violating, out-of-equilibrium decay of the lightest heavy RH Majorana neutrino N 1 . In the limit of hierarchical neutrino masses, i.e. Here the entropy density s of the universe. Entropy density s can be written as [25] 2 : Here T is the temperature 3 and g ef f is the total effective degrees of freedom which includes all the physical particles of the model. Detailed calculation of g ef f is given in appendix D.
In general, the Boltzmann equations for N 1 and the primodial (B − L) asymmetry can be written as [26], 2 Not to be confused with square of center of mass energy s in appendix B. 3 Not to be confused with aforementioned T -parameter.
where z = M 1 T and H(M 1 ) is the Hubble parameter at T = M 1 : 19 GeV being Planck scale. Y eq N 1 , Y eq l are the comoving densities at equilibrium. We solve these two equations with initial conditions : at T >> M 1 .
The number densities of particles with mass M and temperature T can be written as [27]: g being the number of degrees of freedom of corresponding particles, K 2 being second modified Bessel function of second kind. In the first Boltzmann equation (eq.(18)), γ D 1 denotes the contribution from the total decay of N 1 . Different γs in eq.(18) and eq.(19) are space-time densities of the scattering processes at equilibrium depicted in fig.1. γ 1 φ,s and γ 1 φ,t originate from the lepton numberviolating (∆L = 1) s-channel and t-channel washout processes via Higgs-mediation. The factor of "2" in front of γ 1 φ,s comes due to the Majorana nature of N 1 . The factor "4" in front of γ 1 φ,t accounts for the Majorana nature of N 1 as well as two t-channel washout scattering processes mediated by N 1 (N 1 t → lq and N 1 q → lt) [28]. γ 1 φ,s and γ 1 φ,t also contribute in the second Boltzmann equation. γ N and γ N,t in eq.(19) come from ∆L = 2 lepton numberviolating s-channel and t-channel scattering processes via N 1 . The expressions of γ D 1 , γ 1 φ,s , γ 1 φ,t , γ N and γ N,t can be found in appendix B. However, in our model, since φ 3 does not couple with the quarks from the requirement of zero FCNC (see the quantum number assignments in table I), no contributions will be drawn from γ 1 φ,s and γ 1 φ,t ( fig.s 1(c), 1(d), 1(e)). Only surviving processes contributing to the washout will be s-channel and t-channel processes mediated by N 1 (fig.s 1(b), 1(f)). In our model, due to the quantum number assignment, fig.1. αα being the CP -asymmetry generated through the out-of-equilibrium decay of N 1 (decaying to φ 3 and l α , α being the flavor of the lepton)) [14], the total lepton asymmetry can be computed by summing over the flavor indices, i.e. = α αα . While computing , we have considered the interference between the amplitudes of tree-level decay of N 1 , ( fig.2(a)), one-loop vertex-correction ( fig.2(b)) and self-energy diagram ( fig.2(c)). Thus the CP -asymmetry for a single flavor α can be calculated as [14] : Total lepton asymmetry obtained by summing over the flavor indices is [14]: After taking the approximation x >> 1, Expressions for (Y † Y ) 11 , Im{(Y † Y ) 2 12 }, Im{(Y † Y ) 2 13 } are relegated to appendix C. The lepton asymmetry generated in the out-of-equilibrium decay of N 1 , is converted into baryon asymmetry through (B + L) violating sphelaron transitions [15,29]. The conversion of lepton asymmetry to baryon asymmetry being terminated at the freeze-out temperature of the sphelaron process, T sph ∼ 150 GeV [30], the resultant baryon number is computed at T sph as [31]: where N f is the number of generations of the fermion families and N H is the number of the Higgs doublets and Y B−L (z sph ) is the solution of Boltzman equations at z = z sph = M 1 T sph . For our model N f = 3, N H = 3.

VI. ANALYSIS AND RESULTS
To analyse the multi-dimensional parameter space compatible with the aforementioned theoretical and experimental constraints, we have considered the model parame- , M H ± 2 as independent, and have varied them within the following window : 2.5 < tan β < 10.0, − π < γ, γ 1 , γ 2 < π, The dependent parameters α 1 , α 2 , α 3 can be expressed in terms of the independent ones.
Since doublet φ 3 is responsible for generating masses of the light neutrinos due to Z 3symmetry (eq.(10)), tan β plays a crucial role and subsequently enters into the calculation of lepton asymmetry (eq. (25)). Throughout the analysis, we have used those values of tan β which are filtered by the constraints in the scalar sector. At exact Alignment limit, h being the SM Higgs boson, masses of other non-standard heavier scalars range from 300-600 GeV.
Lower the value of tan β, parameter space with higher masses of non-standard scalars becomes accessible. It is clearly evident that with rise of tan β, the available parameter space consistent with all the constraints, shrinks from the cyan colored region with tan β = 3 to red colored region with tan β = 5 in fig.3. Fig.3 Assuming the phases associated with the three complex angles θ, φ, ψ in the matrix O (eq.16) to be zero, they are varied within the region : −π < φ, ψ < π and we fix θ at θ = π 4 for simplicity. The variation of these angles will in turn incorporate variations in the real (y jR ) and imaginary parts (y jI ) of Yukawa couplings in M D , which are absolutely compatible with the neutrino oscillation data [20].  fig.4), | | has to be in the ballpark of ∼ 10 −6 . Fig.4 shows that with increase of tan β, larger values of | | are attainable. However during our analysis, for tan β = 3, 4, 5, we have uniformly varied in a conservative window of 1 × 10 −6 < | | < 1.5 × 10 −6 to make a comparative study. We have explored the parameter space corresponding to M 1 = 10 11 GeV, and varied M 2 , M 3 from 10 13 GeV to 10 16 GeV for all values of tan β. Real and imaginary parts of the Yukawa couplings y j , already being compatible with neutrino oscillation data, get additional constraints coming from the lepton asymmetry | | (eq.(25)). All points satisfying the neutrino oscillation data and the constraints coming from lepton asymmetry, are further validated by the baryon asymmetry constraint (eq.(1)).
From fig.6, fig.7, fig.8, fig.9, it can be inferred that the available parameter spaces spanned by the real and imaginary parts of the Yukawa couplings (y j ), at tan β = 3, 4, 5, shrink gradually after applying the following constraints sequentially : (i) neutrino oscillation data (light green region), (ii) lepton asymmetry (deep green region), (iii) baryon asymmetry (red region). Thus the bound on the Yukawa couplings coming from baryon asymmetry comes out to be the most stringent among all. The range of the real and imaginary part of Yukawa couplings go on increasing with increase in tan β. In fig.6, fig.7, fig.8, fig.9, one can observe that after filtering all points through the constraints, larger values of couplings become accessible for higher values of tan β. For example in fig.6, y 1R ranges from ∼ −0.0035 to ∼ 0.0035 for tan β = 3. Whereas, for tan β = 4 and 5, the range extends to : −0.0045 < y 1R < 0.0045 and −0.0058 < y 1R < 0.0058 respectively. Likewise the imaginary part of y 1 behaves in a similar manner with increasing tan β. This observation holds for real and imaginary parts of the other couplings y 4 , y 5 , y 8 too ( fig.7, fig.8, fig.9). Sharp upper and lower edges in the y 5I vs. y 5R and y 8I vs. y 8R plane ( fig.8, fig.9), parallel to y 5R and y 8R axes respectively, correspond to the perturbativity limits imposed on the Yukawa couplings, i.e.

VII. SUMMARY AND CONCLUSION
In this analysis, we have explored the possibility of neutrino mass generation via Type-I see-saw mechanism and baryogenesis via thermal leptogenesis in the context of Z 3 -symmetric 3HDM accompanied by three RH singlet neutrinos. According to the criteria of thermal leptogenesis, we consider hierarchical masses between three heavy neutrinos : M 1 << M 2 , M 3 ; M 1 being endowed with the lower limit of 10 9 GeV. The thermal production and out-of- equilibrium decay of the lightest heavy neutrino N 1 gives rise to lepton asymmetry, which in turn is partially converted to baryon asymmetry via EW sphelaron processes.
An important model parameter tan β, relevant for lepton asymmetry calculation, has 3, 4, 5, to make a comparative study of the parameter space in the neutrino sector.
To be consistent with Z 3 -symmetry, in the Yukawa sector, we end up with nine complex Yukawa couplings y j , i.e. 18 free parameters (real and imaginary parts of nine complex Yukawa couplings) to fit neutrino oscillation data. CI parametrization makes the job simpler.
Three RH singlet neutrinos couple to the SM neutrinos via doublet φ 3 only, and generate mass of light neutrinos via Type-I see-saw mechanism. From the decay of N 1 , both lepton asymmetry and baryon asymmetry Y B are calculated at the points satisfying neutrino oscillation data with varying M 1 , M 2 , M 3 . It is found that the available parameter space in Y B − plane shrinks with decreasing tan β. To be consistent with the current bound on baryon asymmetry, one has to stick around ∼ 10 −6 . Thus for the rest of the study, we have imposed a conservative limit of 1 × 10 −6 < | | < 1.5 × 10 −6 on for all tan β.
The available parameter space for a fixed tan β in the real vs. imaginary part of complex Yukawa coupling plane is diminished after applying three constraints sequentially : neutrino oscillation data, lepton asymmetry, baryon asymmetry. The last constraint turns out to be the most stringent among all.

VIII. ACKNOWLEDGEMENTS
coupling. f hf f , f hV V are scale factors of hf f, hV V couplings with respect to SM. When the alignment limit is strictly enforced, The loop functions are listed below.
where A 1/2 (x), A 1 (x) and A 0 (x) are the respective amplitudes for the spin-1 2 , spin-1 and spin-0 particles in the loop.

APPENDIX B: FORMULAS FOR REDUCED CROSS SECTIONS
Expression for γ D 1 can be written as [26] : N eq N 1 being the equilibrium number density of the lightest RH neutrino N 1 . Here K 1 and K 2 are the first and second modified Bessel functions of second kind respectively and Γ N 1 is the total decay width of N 1 .
For decay of N 1 , γ eq can be written as [26], where s is the square of center of mass energy (not to be confused with entropy density) andσ(s) is reduced cross section, which can be expressed in terms of actual cross section for two body scattering a + b → i + j + ... as [26] : with p k and M k being three momentum and mass of particle k.
Decay width of N 1 at tree level, with α, θ W being the Fine structure constant and the Weinberg angle.