# Leptogenesis in an extended seesaw model with U(1)_{B-L} symmetry

Ujjal Kumar Dey, Tapoja Jha, Ananya Mukherjee, Nirakar Sahoo

LLeptogenesis in an extended seesaw modelwith U (1) B − L symmetry Ujjal Kumar Dey a Tapoja Jha b, Ananya Mukherjee c, Nirakar Sahoo a,d a Department of Physical Sciences, Indian Institute of Science Education and ResearchBerhampur, Transit Campus, Government ITI, Berhampur 760010, Odisha, India b Kolkata, India c Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India d Center of Excellence in High Energy and Condensed Matter Physics, Department ofPhysics, Utkal University, Bhubaneswar 751004, India

Abstract

We have explored an extended seesaw model accommodating a keV sterile neutrinoadopting U (1) B − L symmetry. This model provides a natural platform for achieving res-onant leptogenesis to account for the observed baryon asymmetry of the Universe. Therequired lepton asymmetry is sourced by the CP violating decay of the lightest heavyright handed neutrino to Standard Model leptons and Higgs. The presence of the lightsterile neutrino in the model brings out an enhancement in the ﬁnal lepton asymme-try through an additional self-energy contribution. Adopting a proper treatment for allthe washout processes this framework strictly favours a strong washout regime therebyprotecting the low energy neutrino mass parameters in agreement with the present neu-trino and cosmology data. This framework of extended seesaw scheme oﬀers the sourceof matter-antimatter asymmetry without any severe ﬁne tuning of the Yukawa couplingsgoverning the tiny neutrino mass. We also comment on the half-life period for the neutrinoless double beta decay process in the background of having a keV sterile neutrino satis-fying all the constraints which guide the explanation for the observed baryon asymmetryof the Universe. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] a r X i v : . [ h e p - ph ] F e b ontents After the youngest discovery of the Standard Model (SM) predictions in terms of the Higgsboson, it is needless to say that the SM is one of the most successful theories of particlephysics dealing with the matter and their fundamental interacting forces. In spite of this hugesuccess, there remain a number of limitations of the SM in the context of explaining the originof neutrino mass, existence of dark matter (DM) and the reason behind the predominanceof matter over antimatter in our observable Universe. In the past decades, there have beenplenty of theoretical studies with an objective of addressing the above mentioned issues atthe cost of constructing some beyond standard model (BSM) frameworks [1, 2]. In order toaccount for tiny neutrino mass one of the most economical possibilities is to embed a seesawmechanism [3–6] into the SM. Over the years a host of seesaw scenarios are proposed. Thevarious seesaw models can also have phenomenological importance in the context of probingthem in the colliders. Among the variants of the seesaw schemes low-scale seesaw mechanismshave gained considerable attention in view of their testability in the colliders [7, 8]. Withinthe same seesaw framework testing leptogenesis is another probe to validate the neutrino massgeneration mechanism from the perspective of various cosmological observations.The matter-antimatter asymmetry can be parametrised by the baryon to photon ratio.Various cosmological observations report this ratio to be [9] η B = n B − n ¯ B n γ = (2 . − . × − (1)where n B , n ¯ B , n γ and s represent number densities of baryons, antibaryons, photons, andentropy respectively. A very appealing and ﬁeld theoretically consistent mechanism of gen-erating the baryon asymmetry of the Universe is the process of leptogenesis which was ﬁrst1ointed out by Fukugita and Yanagida [10]. Leptogenesis, being a direct consequence of theseesaw models having right handed neutrinos (RHN) also establishes a connection betweenthe origin of neutrino mass and the baryon asymmetry of the Universe. Realization of baryo-genesis though leptogenesis via the decay of RHN in various seesaw models can be foundin [7, 11–15]. For a recent review one may look at [16, 17] and the references there in. TheseRHNs can in principle be uncharged under the SM gauge interactions; unless protected bysome new symmetry they would mix with the active neutrinos. Inclusion of a pair of sterileneutrinos to the SM fermion sector has become a common lore in order to explain the tinyneutrino mass through type-I seesaw mechanism [16, 18–21]. Type-I seesaw model predicts abound on the sterile neutrino mass to be very heavy ( M ∼ GeV) in order to generatethe correct light neutrino masses. However, considering sterile neutrinos at accessible energiesgained considerable attention in the last decade from the perspective of being probed directlyat high energy collider LHC. Interestingly, the most appealing window for sterile neutrinomass has been seen to be from 1 eV to 10 TeV which also has theoretical and experimentalincentives. In the context of realizing a low scale leptogenesis a mechanism called resonant-leptogenesis [22–24] has become popular in the past decade. This can be perceived as follows:due to the presence of two quasi-degenerate RH neutrinos, the CP asymmetry parameterwhich quantiﬁes the leptogenesis process gets resonantly enhanced and opens up possibilitiesfor successful leptogenesis even at RH mass scale O (TeV).In this work we utilise a U (1) B − L extension of the SM based on extended seesaw mech-anism which oﬀers explanation for the tiny neutrino mass. The features of extended seesawmechanism can be realized in this way. Apart from oﬀering the tiny neutrino mass, the ex-tended seesaw model also provides a potentially accessible scale for the RHN mass. The lowscale RHN being a feature of this seesaw model serves as an excellent candidate which canexplain the source of matter-antimatter asymmetry of the Universe though the process ofleptogenesis. There have been plenty of studies which dealt with such low scale leptogenesisfrom a neutrino mass model embracing the U (1) B − L symmetry [15]. It is worth mentioningthat a low scale seesaw model as embedded in the present framework can also render thegravitino bound harmless [25]. Thus, thermal production of the RHNs do not require toohigh reheating temperature, protecting the Universe from gravitino overproduction [25]. Inaddition the extra neutral fermion singlet in this extended seesaw framework facilitates theprocess of leptogenesis by its self-energy contribution to the ﬁnal lepton asymmetry to ac-count for the observed baryon asymmetry of the Universe (BAU). This additional self-energycontribution oﬀered by the presence of the extra fermion singlet ( S ) brings an enhancement inthe lepton asymmetry which successfully accounts for the observed BAU. We also emphasizeon the correlation among the CP violating phases in the neutrino sector with the leptogenesisparameter space.In another front, the observation of neutrino-less double beta ( νββ ) decay would provelepton number violation and the Majorana nature of neutrinos [26], a proper investigationof the said process would be of fundamental importance. Here, we attempt to provide theparameter space in the purview of extended seesaw scenario which is compatible with theconstraints related to the νββ decay experiments as reported by KamLAND-Zen [27]. Ow-ing to the presence of the neutral fermion singlets we compute the half-life for the potential νββ decay process ( T ν / ). The contribution of the active-sterile mixing to the total T ν / of this process has been shown. In this set-up we present an in-depth analysis on the con-struction of the high energy neutrino mass matrix in the light of an extended seesaw scheme.Although the idea of examining an extended seesaw scheme to investigate for the observedBAU has been conceived before [28], some distinct implications and reﬁnements have notbeen emphasized earlier. We present the viable parameter space validating this model with2espect to the constraints associated with the neutrino oscillation parameters, νββ processand the leptogenesis. In this work we attempt to focus on a detailed analysis on the highenergy neutrino mass parameters associated with the said seesaw model along with a lowscale leptogenesis in order to oﬀer an explanation for the observed BAU. Most importantlywe have shown that, the present framework allows the required amount of lepton asymmetryto put phenomenologically interesting predictions on the neutrino mixing angles mainly thereactor and atmospheric ones, pushing them towards their higher allowed σ ranges. Thisfact reinforces the notable features of this analysis. While solving the Boltzmann equationswe have dealt with all possible LNV scatterings in order to have a proper investigation on theamount of washout. We carried out a detailed analysis to emphasize the interplay betweenthe generated amount of lepton asymmetry and the corresponding washout.The outline of this article is as follows. We provide the structure of the model in Section2, where we brief the light neutrino mass generation through the extended seesaw mechanism.The necessary simulation details required to extract the ESS parameter space along withtheir numerical estimation are provided in Section 3. Section 4 is dedicated to the necessaryprescriptions required for realizing baryogenesis through leptogenesis in the light of the presentset-up. We discuss the viable parameter space explaining all the above mentioned BSM issuesin Section 5. The constraints associated with νββ process are discussed in Section 6. Finallywe conclude in Section 7. Additionally for a detailed description of the construction of theextended seesaw mass matrices the analytical algorithm using the seesaw approximation isshown in the Appendix. In this section, we provide a detailed discussion of the model along with theoretical and someexperimental constraints on model parameters. We also provide a brief discussion on thegeneration of neutrino mass in the extended seesaw framework.

We extend the Standard Model by an extra U (1) B − L symmetry. The model contains threeright handed neutrinos ( N R i ) with B − L charge of ( − each as required by anomaly cancel-lation and three complex singlet sterile neutrinos S L i . Apart from this we add two additionalscalars χ , χ which are having B − L charges as 1 and 2 respectively. The two scalars breakthe U (1) B − L symmetry and give mass to the neutrinos. With these two minimal set of singletscalars we can actually explain the neutrino mass in an extended seesaw set-up. The relevantﬁelds and their respective quantum numbers under the gauge symmetry are shown in table 1.The Lagrangian for the model can be written as, L = L SM + L gauge + L scalar + L fermion . (2)Firstly, within the L SM itself the gauge fermion interactions will be augmented by newcouplings of the SM fermions with the additional Z (cid:48) gauge boson coming from the gauged U (1) B − L symmetry. The kinetic term of the new gauge boson is depicted by the piece L gauge .We are considering here a minimal B − L extension of the SM and hence the mixing betweenthe Z and Z (cid:48) vanishes [29]. This makes our model simpler and it diﬀers from e.g., [30–32].The L scalar term takes into account all the scalar (including SM-like Higgs doublet) kineticand mixing terms. This can be explicitly given as, L scalar = ( D µ H ) † ( D µ H ) + ( D µ χ ) † ( D µ χ ) + ( D µ χ ) † ( D µ χ ) + V ( H, χ , χ ) , (3)3ields SU (2) × U (1) Y U (1) B − L ( u L , d L ) T u R d R − / ( ν L , e L ) T − − e R − − H N R i − S L i χ χ Table 1:

Fields and their quantum numbers under the gauge symmetry. where the respective covariant derivatives are given by, D µ H = ∂ µ H − i g (cid:126)W µ · (cid:126)τ H − i g (cid:48) B µ H , (4a) D µ χ = ∂ µ χ − ig BL Z (cid:48) µ χ , (4b) D µ χ = ∂ µ χ − ig BL Z (cid:48) µ χ , (4c)where g , g (cid:48) , and g BL are the SU (2) L , U (1) Y , and U (1) B − L couplings, respectively. The (cid:126)W µ , B µ , and Z (cid:48) µ are the corresponding gauge ﬁelds. The most general scalar potential can bewritten as, V ( H, χ , χ ) = − µ H H † H + λ H ( H † H ) − µ χ † χ + λ ( χ † χ ) − µ χ † χ + λ ( χ † χ ) + λ H ( H † H )( χ † χ ) + λ H ( H † H )( χ † χ )+ λ ( χ † χ )( χ † χ ) + (cid:2) µ χ (cid:63) χ + H . c . (cid:3) . (5)Here all the µ ’s are real positive numbers except µ . In order to get a positive mass for CP -odd scalar states µ has to be negative. The scalars χ , χ break the U (1) B − L symmetry andget vacuum expectation values (vev) in this process. The SU (2) L doublet scalar H is involvedin usual electroweak symmetry breaking. So we have two phases of symmetry breaking here, SU (2) L × U (1) Y × U (1) B − L (cid:104) χ (cid:105) , (cid:104) χ (cid:105) −−−−−→ SU (2) L × U (1) Y (cid:104) H (cid:105) −−→ U (1) em . The ﬁelds after the symmetry breaking are written as χ = 1 √ v + χ (cid:48) + iρ ) , χ = 1 √ v + χ (cid:48) + iρ ) , H = 1 √ (cid:18) √ ω + v H + h + iζ (cid:19) (6)After the symmetry breaking three CP -even scalars { h , h , h } will arise out of the mixingof the states { h, χ (cid:48) , χ (cid:48) }. Clearly one of { h , h , h } can be identiﬁed as the SM-like Higgs.The CP -odd ﬁeld ζ is absorbed by the SU (2) L gauge boson hence is not physical. The othertwo CP -odd ﬁelds ρ , ρ become massive after the symmetry breaking and they mix witheach other. From the CP -odd mass matrix one can conﬁrm that one of the eigenvalues iszero. We can say that the ﬁeld combination ρ sin β + ρ cos β ( β being the mixing anglebetween ρ , ρ ) is eaten up by the gauge ﬁeld Z (cid:48) and becomes massive, the other combination ρ cos β − ρ sin β is the only physical CP -odd ﬁeld in our model.4astly, L fermion in Eq. (2) contains the information of kinetic and Yukawa terms of theBSM fermions and they can be written explicitly as, L kinfermion = N Ri iγ µ (cid:0) ∂ µ − i g BL Z (cid:48) µ (cid:1) N Ri + S Li iγ µ ∂ µ S Li , (7) −L Yukfermion = Y D ij ¯ l i H ( N R ) j + Y R ij χ (cid:63) ( N R ) Ti ( N R ) j + µ ij ( S TL ) i ( S L ) j + Y S ij χ (cid:63) ¯ N Ri ( S L ) j + H . c . (8)here the indices i, j run from 1-3 and represent the generation indices. Owing to the simultaneous presence of both the heavy and small lepton number violating scales M R and µ respectively, extended seesaw scenario [28] is very diﬀerent from the inverse seesaw[33–45] and double seesaw scenario [46]. In inverse seesaw mechanism, there is only one smalllepton number violating scale µ and the lepton number is conserved in the µ = 0 limit. Oneof the signiﬁcant features of this seesaw model is that the presence of the heavy right handedneutrinos do not directly control the generation of the tiny neutrino masses, while playing acrucial role in yielding relatively light sterile neutrinos. Another important characteristic ofthis extended seesaw mechanism is that, one can have contribution of Majorana neutrinos to νββ transition amplitude even in the limit µ → , whereas in the inverse seesaw scenario,the corresponding amplitude vanishes.One can write the complete neutrino mass matrix which is generated after the symmetrybreaking as follows. From Eq. 7 one can derive the neutrino mass matrix as, M n = (cid:0) ν L S L N cR (cid:1) M TD µ M TS M D M S M R ν cL S cL N R , (9)where the matrix elements are given as: M D = Y D v H , (10) M S = Y S v ,M R = Y R v . To the leading order, the light neutrino mass matrix m ν , and the heavy neutrino massmatrices m s , m R can be constructed from the following equations, m ν ∼ M TD ( M TS ) − µ ( M S ) − M D (11) m s ∼ µ − M TS M − R M S (12) m R ∼ M R (13)To this end, we refer the reader to Appendix for a detailed construction of the extendedseesaw mass matrix following the seesaw parametrization. In this section we brieﬂy discuss some of the theoretical and observational constraints on themodel parameters.The boundedness of the scalar potential puts constraints on the respective quartic cou-plings. Following the method of co-positivity [47, 48], one can easily write down the germane5onditions which in this case are, λ H , λ , λ > , (14a) ¯ λ H ≡ λ H + 2 (cid:112) λ H λ > , (14b) ¯ λ H ≡ λ H + 2 (cid:112) λ H λ > , (14c) ¯ λ ≡ λ + 2 (cid:112) λ λ > , (14d) λ H (cid:112) λ + λ (cid:112) λ H + λ H (cid:112) λ + 2 (cid:112) λ H λ λ + (cid:112) λ H ¯ λ H ¯ λ > . (14e)The LEP constraint on the ratio of new gauge boson mass to the coupling is [49, 50] M Z (cid:48) g BL ≥ . (15)Similar bound from ATLAS is quite less stringent M Z (cid:48) /g BL ≥ . TeV [51].In the following section we discuss the methodology used to numerically evaluate theallowed parameter space satisfying the neutrino mass and mixing parameters which complywith the σ global ﬁt presented in table 2. Before discussing the leptogenesis aspects of the model we ﬁrst consider the constraints onthe parameter space coming from the neutrino oscillation data. In this section we describethe methodology used to extract the ranges of the various parameters that can generate thesub-eV light Majorana neutrino mass within the extended seesaw scheme (ESS). As describedin the model section the ESS framework is realized by three generations of sterile neutrinos S L along with another three generations of heavy right-handed neutrinos N R . Thus the highenergy neutrino mass matrix basically turns out to be of the dimension × . In order todiagonalize this × mass matrix analytically we follow the procedure given in Refs. [52, 53].A detailed construction of mass matrices and the corresponding diagonalizing matrices aregiven in Appendix. We ﬁrst follow the two steps diagonalization: in the ﬁrst case we areleft with one × block-diagonalized matrix comprising of three × mass matrices: m ν , m s and M R for active neutrino and two heavy neutrino states, respectively. The expressionsof these mass matrices are provided in Sec. 2.2. The matrices µ , M D , M S are considered tobe complex symmetric in nature to comply with a non-zero CP phase which can potentiallyact as a prime source of sizeable lepton asymmetry. However, for simplicity M R is chosen tobe a real diagonal matrix. This primary choice of the diagonal M R also ensures that we areworking in a basis of Dirac Yukawa coupling matrix where the RHNs are in their physicalmass basis.For our numerical analysis we vary each elements of the mass matrices in the followingranges (in GeV): − ≤ µ ≤ − (16) − ≤ M D ≤ − (17) . ≤ M S ≤ . (18)whereas, M R and M R are randomly varied from 10 to 20 TeV and are kept to be nearlydegenerate in order to make this framework suitable for resonant leptogenesis. The ratio ofthese two scales is crucial for leptogenesis in the present scenario which we discuss elaboratelyin the Sec. 4. Following the hierarchy of the RHN mass eigenvalues as M R ≈ M R < M R sin θ sin θ sin θ ∆ m (6.79 - 8.01) × − eV (6.79 - 8.01 ) × − eV ∆ m (2.436 - 2.618) × − eV − (2.601 - 2.419) × − eV δ CP / ◦

144 - 357 205 - 348

Table 2:

Latest σ bounds on the oscillation parameters (With SK data) from Ref. [55]. we proceed for the numerical diagonalization procedure. We keep M R in 50 TeV to 80 TeVrange so that it is completely decoupled from other two right-handed neutrinos.It is evident that working with the × mass matrix with many undetermined parametersmay be quite daunting. For our purposes we generate such a large dimensional complete massmatrix numerically for convenience. To simulate this we use Python programming which essen-tially yields the relevant parameter space keeping the constraints on the various mass scales ofsuch ESS framework intact. The complete simulation maintaining neutrino oscillation data aswell as desired value of the lepton asymmetry has been performed by in-house

Python 3.8.3 code. From this simulation we obtained parameter points which essentially yield the neutrinooscillation observables. The obtained neutrino mixing parameters are then compared withthe current neutrino data pertinent for normal hierarchy (NH), as summarised in table 2. Weshould mention here that, the present framework is suitable to establish both of the possibleneutrino mass hierarchies, however for simplicity and to be in agreement with the recent biasfor the NH [54] we only stick to NH of neutrino mass and carry out the analysis relevantfor the same. At present the value of Dirac CP phase is ambiguous in the sense that, T2Kexperiment [56] prefers δ ≈ π/ whereas NOvA [57] tells that δ can take CP conserving values.However, the global ﬁt values for each hierarchy imply δ ≈ − π/ for NH and δ ≈ − π/ forIH as also evident from the table 2. Taking care of the above neutrino oscillation constraintswe extract the allowed region of parameter space which we discuss in the following subsection. After constructing the extended seesaw mass matrix numerically ( M n ) we diagonalise thesame. From the diagonalizing matrix we extract the values of the mixing angles and the CPphases involved in the lepton mixing matrix [58]. ﬁg. 1 evinces a mild correlation among thematrix elements responsible to generate the neutrino observables which comply with exper-imental data. Here we use diﬀerent color codes to highlight the required parameter valueswhich are useful for successful leptogensis. In principle correlations among the neutrino mix-ing parameters would severely constrain the ESS model parameter space, which may makethe model phenomenologically richer. This issue can be well-taken care of if one introduces aﬂavour symmetry embedded framework which leads to a particular texture of the low energyneutrino mass matrix ( m ν ) giving rise to possible correlations among the neutrino observ-ables [12, 59, 60]. In ﬁg. 2 we present the variation of the reactor ( θ ) and atmospheric ( θ )mixing angles with respect to the real and imaginary parts of the matrix element of M D . Itis evident from ﬁg. 2 that the entire range of the ESS parameter space associated with all thematrix elements can yield a non-zero value of θ and θ compatible with the global ﬁt asmentioned in table 2. A similar notion is obtained for the other matrices ( µ and M S ) also,when plotted (not shown here) with respect to these mixing angles of the leptonic mixing7 M D | ( G e V ) −5 −4 −3 |M D | (GeV) −5 −4 | M D | ( G e V ) −5 −4 −3 |M D | (GeV) −5 −4 | M S | ( G e V ) |M S | (GeV) | M S | ( G e V ) |M S | (GeV) Figure 1:

Possible correlations among matrix elements. This also shows the constrainedparameter space required to have an adequate amount of lepton asymmetry to account for theobserved BAU. The red coloured points are identiﬁed with the actual data points which canyield η B = (2 . − . × − . Detailed explanations are provided in the text. matrix. The red points represent the parameter space which can yield suﬃcient amount oflepton asymmetry leading to the observed η B = (2 . − . × − . On the other hand theblack points indicate the regions of parameter space which generate ample lepton asymmetry,but can not give rise to the observed η B due to large amount of washouts associated withthem. One can notice from these ﬁgures that, in view of successful leptogenesis there existslight preferences for the higher octant (HO) of the θ and higher values of the θ in itsallowed σ range (shown by the red points). One can understand the breaking of the µ − τ symmetry here by showing the correlations among the mass matrix elements ( m ν ii , Eq. 11) ofthe µ − τ sector as presented in ﬁg. 3. Generation of the non-zero reactor angle can be realizedby such correlations. We show the variation of the solar ( ∆ m ) and atmospheric ( ∆ m )neutrino mass splittings as a function of the lightest neutrino mass in ﬁg. 4. Fig. 5 showsthe allowed ranges of the heavy and light sterile neutrinos with respect to the lightest activeneutrino mass m ν we obtain in this analysis. It is evident from ﬁg. 4 and ﬁg. 5 that slightlyhigher values of the lightest active neutrino mass (denoted by red squares) are preferred, owingto the required leptogenesis parameter space. In ﬁg. 6 we present the fourth (active-sterile)mass squared diﬀerence ( ∆ m ) as a function of the lightest sterile neutrino mass ( m s ) andthe active-sterile mixing angle obtained in the set-up. From the right panel of the ﬁg. 6 onecan notice the smallness of the active-sterile mixing angle ( sin θ ) which mainly oscillate inthe range − − − .Having all the above ESS parameter space, allowed by the neutrino oscillation data, wefeed them to lepton asymmetry calculation in order to check for the viable parameter spacewhich can yield the expected baryon to photon ratio ( η B ). The salient features of the various8 i n θ M D R (GeV) −5 −4 s i n θ M D I (GeV) −5 −4 s i n θ M D R (GeV) −5 −4 s i n θ M D I (GeV) −5 −4 Figure 2:

Reactor mixing angle ( sin θ ) (upper panel) and atmospheric mixing angle( sin θ ) (lower panel) as a function of the ESS mass matrix elements. The red points denotethe relevant data which can account for the observed BAU. Detailed explanations are in thetext. �� - � �� - � ����� ������� - � �� - � ���������� | � ν �� |( �� ) | � ν �� | ( � � ) �� - � �� - � ����� ������� × �� - � �� × �� - � �������������������� | � ν �� |( �� ) | � ν �� | ( � � ) Figure 3:

Correlation plots among the low energy neutrino mass matrix elements, showingthe plane m eµ − m eτ and m µµ − m ττ . mass scales (and hence in turn the associated Yukawa couplings) in the context of resonantlyenhanced lepton asymmetry will be discussed in the following sections.9 m × ( e V ) m ν (eV) −5 −4 −3 Δ m × ( e V ) m ν (eV) −5 −4 −3 Figure 4:

Atmospheric and solar neutrino mass squared splittings with respect to the lightestneutrino mass. Red colored squares indicate the relevant data needed to generate an adequateamount of lepton asymmetry. Black colored points are ruled out from the perspective ofleptongenesis. M R ( T e V ) m ν (eV) −5 −4 −3 m s ( k e V ) m ν (eV) −5 −4 −3 Figure 5:

Heavy sterile and light sterile masses with respect to lightest Majorana neutrinomass. The choice of the colors can be referred to the same as ﬁg. 4. Δ m × - ( e V ) m s (keV) Δ m × - ( e V ) sin θ −9 −8 −7 −6 −5 −4 Figure 6:

Active-sterile mass squared diﬀerence as a function of the lightest sterile neutrinomass (left panel) and sin θ (right panel). The red small squares represent the data whichyield ample amount of lepton asymmetry that is responsible for the observed η B . The blackcolored rectangles indicate the data points which are ruled out from the need of suﬃcientleptogenesis. Baryogenesis through leptogenesis

The lepton number asymmetry sourced by the decay of the RHN can be realized as follows (cid:15) i = − (cid:20) Γ( N i → ¯ l i H ) − Γ( N i → l i H ∗ )Γ( N i → ¯ l i H ) + Γ( N i → l i H ∗ ) (cid:21) where, N i is the lightest RHN creating the lepton asymmetry. We present the relevant feyn-mann diagrams for the decay of the lightest RHN in ﬁg 7. It is important to note here that,in this ESS framework the existence of the fourth self-energy diagram participate signiﬁcantlyin bringing out an enhancement of the ﬁnal lepton asymmetry. We address this issue in detailin the result section of the lepton asymmetry parameter. For massless limit of the ﬁnal state N i l i h N i h hl i l j N k N k l i hN k N i hl j N k l i hN k N i χ S j Figure 7:

Lightest right-handed neutrino decay processes that lead to the lepton asymmetry. particles of the above decay, the lepton asymmetry can be expressed as [28] (cid:15) = 18 π (cid:88) k (cid:54) =1 ([ g v ( x k ) + g s ( x k )] T k + g s ( x k ) S k ) (19)with T k = Im[(Y D Y † D ) ]( Y D Y † D + Y S Y † S ) and S k = Im[(Y D Y † D ) k1 (Y S Y † S ) ]( Y D Y † D + Y S Y † S ) . The Yanagida loop factors aredeﬁned as g v ( x ) = √ x k (cid:0) − (1 + x ) ln xx (cid:1) and g s ( x ) = √ x k − x k where, x k = M Rk M R for k (cid:54) = 1 . Asmentioned earlier, a low scale RHN can overcome the gravitino overproduction problem sinceit requires a smaller reheating temperature. At the same time for a TeV scale RHN beingnaturally oﬀered by the seesaw scenario under consideration opens up the window to have theleptogenesis taking place through the resonant enhancement of the leptonic asymmetry [22].Resonant leptogenesis require a very small splitting among the heavy RHN masses of theorder of their individual decay width i.e., M i − M j ≈ Γ j [22]. In this scenario of largeenhancement, the lepton asymmetry can even reach close to unity (cid:15) l ∼ . Purposefully, wehave assumed the heavy right handed Majorana neutrinos N i having a quasi degenerate massspectrum, following M > M ( ≈ M ) so that any two of the three RHNs can contribute in theleptogenesis process. This leads to the realization that studying the evolution of the numberdensity of N , N is suﬃcient in order to obtain their thermal abundance through Boltzmann’sequations discussed in the following subsection. Solution to the Boltzmann equation 20 provides us with the dynamics of the RHN productionand the B − L number density. From the requirement of producing a lepton asymmetryof the correct order of magnitude, the RHNs have to be numerous before they decay. Thisessentially relies on the fact that, they are in thermal equilibrium at high temperatures. It is11orth mentioning that we consider here a thermal initial abundance of the RHNs as this canbe regarded as a consequence of going through a strong washout regime. The evolutions ofnumber densities of N and B − L asymmetry can be obtained by solving the following set ofcoupled Boltzmann’s equations (BEQs) [16, 61]: dN N i dz = − D i ( N N i − N eqN i ) , with i = 1 , (20) dN B − L dz = − (cid:88) i =1 (cid:15) i D i ( N N i − N eqN i ) − (cid:88) i =1 W i N B − L , (21)with z = Mass of the lightest RHN / Temperature = M N i /T , being N i as the decaying RHN. W i denotes the contribution to the washout term due to inverse decay and ∆ L = 2 scatterings.It is clear from the above set of coupled equations that, one can get the RHN abundance fromthe ﬁrst equation and the later determines the B − L number density which survives in theinterplay of the asymmetry production and it’s washout, with respect to temperature. Toensure the participation of N , N in creating the ﬁnal lepton asymmetry, one has to deﬁnea temperature-function ( z ) (for a bit detail one may look into [62]), writing z = z i / √ x i with i = 1 , . In principle, N N i ’s are the comoving number densities normalised by thephoton density at temperature larger than M N i . With the Hubble expansion rate H ≈ (cid:113) π g ∗ M Ni M Pl z ≈ . g ∗ M N M Pl z the decay term D i in Eq.(20) can be cast into [61], D i = Γ total ,i Hz = K i x i z K ( z ) K ( z ) , (22)with Γ total ,i as the decay rate of the ith RHN, and can be expressed as Γ total ,i = (cid:16) Y D Y † D + Y S Y † S (cid:17) ii π M R i . (23)In the above Eq. (22) K and K mean the modiﬁed Bessel functions of the second kind.The washout factor K i in Eq.(22) quantiﬁes the deviation of the decay rate of the RHNs fromthe expansion rate ( H ) of the Universe and can be read as, K i ≡ Γ total ,i H ( T = M N i ) , (24)One can express the baryon-to-photon ratio with the help of lepton asymmetry as, η B = 0 . N ﬁn B − L = 0 . (cid:15) κ ﬁn1 (25)where, κ ﬁn1 = κ ( z → ∞ ) < is the ﬁnal eﬃciency factor. The ﬁnal amount of B − L asymmetry can also be parametrized as Y B − L = n B − L /s , where s = 2 π g ∗ T / being theentropy density and g ∗ as the eﬀective number of spin-degrees of freedom in thermal equilib-rium ( g ∗ = 110 ). After reprocessing by sphaleron transitions, the baryon asymmetry is relatedto the B − L asymmetry by [20] Y B = (12 / Y B − L ). It is important to provide an order of magnitude estimate of the lepton asymmetry obtainedwith the help of the model parameters that essentially satisﬁes all the constraints in the12

P V Y S ij − .

08 + 1 . i .

26 + 4 . i .

58 + 1 . i .

26 + 4 . i .

36 + 3 . i .

79 + 4 . i .

58 + 1 . i .

79 + 4 . i .

07 + 3 . i Y D ij − .

61 + 1 . i .

32 + 2 . i . . i .

32 + 2 . i .

10 + 0 . i .

02 + 4 . i .

47 + 1 . i .

02 + 4 . i .

48 + 3 . i BP IIIa Y S ij − .

36 + 7 . i .

93 + 19 . i .

75 + 15 . i .

93 + 19 . i .

08 + 10 . i .

75 + 12 . i .

75 + 15 . i .

75 + 12 . i .

69 + 6 . i Y D ij − .

30 + 2 . i .

82 + 5 . i .

51 + 3 . i .

82 + 5 . i .

88 + 5 . i .

02 + 10 . i .

51 + 3 . i .

02 + 10 . i .

10 + 1 . i Table 3:

Numerical estimates of the two Yukawa coupling matrices which correspond to thebenchmark values (BP: V and IIIa) presented in table 4. neutrino sector. As evident from the Eq. (19), there exist essentially two diﬀerent sets ofYukawa couplings namely ( Y D , Y S ), which determine the order of the lepton asymmetry oneshould get in this ESS framework. These two complex Yukawa couplings are the sourcesof lepton asymmetry in the present scenario. We show the role of these complex couplingcoeﬃcients in the form of the corresponding matrix elements in obtaining the correct orderof lepton asymmetry (cid:15) in ﬁg. 8, and ﬁg. 9. These two ﬁgures emphasize on the choices ofYukawa couplings and in turn the ESS parameter space in order to make this frameworksuitable for leptogenesis. For this purpose diﬀerent color codes have been used to highlightthe ESS parameter spaces that are essentially required to be chosen (by dark red square) andomitted (by black diamond). It is important here to note from the expression of the leptonasymmetry (Eq. (19)) that, the large Yukawa coupling Y S plays a vital role in bringing theorder of lepton asymmetry even up to around − for a considerably large region of theentire parameter space of ESS. In addition to this, we would like to emphasize on the factthat, despite of this large Y S the criteria satisfying | M i − M j | ∼ Γ j is unavoidable even afterthe presence of an additional self-energy diagram in the present set-up. This is guided bykeeping a quasi-degenerate RHN mass spectrum for instance M ≈ M < M . Thereafter, weconsider the decay of more than one RHN and have noticed that the generated asymmetry issolely dependent on the lightest heavy RHN ( N here), even if the second RHN ( N ) is nothaving pronounced mass hierarchy with the former. For a clear understanding of this fact onemay refer to table 4. In ﬁg. 8 we show the variation of the lepton asymmetry as a function ofthe matrix elements M D . The reliance of the lepton asymmetry on the M S matrix elementshave been shown in ﬁg. 9. This is also evident from the table 3, where we present the Yukawacouplings for the particular benchmark value of the lepton asymmetry parameter. Presenteddata in the table 4 leads to the realization that in this ESS scenario one has to be very carefulwhile choosing the texture of Yukawa coupling matrices, as it is evident that even after havinga large magnitude of lepton asymmetry one can not reproduce the observed baryon to photonratio since the associated washouts corresponding to those Yukawa couplings are large enoughto erase a considerable part of the generated lepton asymmetry. It will be more transparentif one notices the data provided for BP-IIIa in the table 4. The large washout restricts theparameter spaces for the Yukawa couplings in order to have a successful leptogenesis.In ﬁg. 10 we present the dependence of lepton asymmetry on the low energy parametersand also on the RHN mass splitting ∆ M = M R − M R . As it is seen from the left panel ofﬁg. 10 that the largest asymmetry is obtained for the sum over neutrino mass to fall around . eV . The amount of the washout we get in this scenario turns to be of the order of , × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M D | (GeV) −5 −4 −3 ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M D | (GeV) −5 −4 −3 ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M D | (GeV) −5 −4 −3 ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M D | (GeV) −5 −4 −3 ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M D | (GeV) −5 −4 −3 ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M D | (GeV) −5 −4 −3 Figure 8:

Lepton asymmetry versus matrix elements of M D . The red squares are identiﬁedwith the appropriate parameter space which is essential for a successful leptogenesis. The blackcolor indicates those parameter spaces which after having an adequate lepton asymmetry cannot successfully account for the observed η B . ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M S | (GeV) ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M S | (GeV) ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M S | (GeV) ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M S | (GeV) ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M S | (GeV) ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 |M S | (GeV) Figure 9:

Lepton asymmetry versus matrix elements of M S . To have an idea about thevarious colored data points one can refer to the caption of ﬁg. 8. which is in agreement with the situation one has to go through while having a resonantlyenhanced lepton asymmetry. As mentioned earlier, to have a proper treatment of all thewashout processes we also calculate the contributions from various → LNV scatteringprocesses that enter the Boltzmann equations. The ﬁnal B − L asymmetry which remained inthe interplay of creation of the asymmetry and its washout through various concerned processesare presented in ﬁg. 11. In the right panel of ﬁg. 11 we present the B − L evolutions withrespect to the temperature function for diﬀerent values of lepton asymmetry which are largein number. One can notice from this ﬁgure that, even after having a large amount of leptonasymmetry we may not generate the required amount of B − L charge if the correspondingwashouts erase a part of the asymmetry leading to a decrease in the ﬁnal baryon asymmetry.This fact can also be well-understood from the table 4.14 × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 Σ m ν i (eV) ε × −5−4.4−3.8−3.2−2.6−2−1.4−0.8−0.20.411.62.22.83.4 (M R - M R ) × 10 (GeV) Figure 10:

Lepton asymmetry as a function of (cid:80) m νi and the RHN mass splitting M R − M R . BP K K | (cid:15) | | (cid:15) | η B I 414.62 272.60 . × − . × − . × − II 498.9 233.85 . × − . × − . × − III(a) 3311.41 4052.22 . × − . × − . × − III(b) 2199.18 2452.32 . × − . × − . × − III(c) 4362.4 4343.9 . × − . × − . × − III(d) 1795.48 850.57 . × − . × − . × − IV 182.15 415.42 . × − . × − . × − V 284.47 395.30 . × − . × − . × − Table 4:

Bench mark values for the lepton asymmetry parameters and the correspondingwashout amount which altogether yield the ﬁnal baryon to photon ratio ( η B ) . These η B valuesare evaluated at z = 100 from the solution of the Boltzmann Equations. - - - � = � / � � � � - - - � = � / � | � � - � | IIIaIIIbIIIcIIIdIV

Figure 11:

RHN abundance (left panel) and B − L abundance (right panel). Various colorsin the second ﬁgure represent the ﬁnal B − L production amount out of each case with diﬀerentamounts of lepton asymmetries and corresponding washouts, identiﬁed with the benchmarkpoints as tabulated in table 4. Neutrinoless double beta decay ( νββ ) is a lepton number violating process, which, if ob-served, would establish the Majorana nature of neutrinos with certainty [63–66]. In variousbeyond SM scenarios the rate of this process can get enhancement and become observable15n experiments [66–68]. The dominant contribution to the νββ -decay rate, Γ ν , is due tothe exchange of three light Majorana neutrinos ν i [69]. However it is customary to look forother contributions to the ﬁnal process owing to the presence of extra particle species thatcan mediate the said process. In ESS, additional sterile neutrinos can play important role. Tothis end, along with the light Majorana neutrinos ν i , the right-handed (RH) neutrino states N R i for (i = 1 , , and S L j for (j = 1 , , can also mediate νββ process n → p + 2 e − .Due to its heavier mass the contribution of N R i to the ﬁnal amplitude of the νββ processwill be suppressed enough, as the strength of this particular channel will be determined bythe mixing of active neutrino states with the heavy sterile ( V eN i ). Additionally, the contri-bution will be suppressed by the heavy sterile mass M R i . The V eN is given by M † D M − R W N (see Eq. 61) where W N is deﬁned as diagonalization matrix of heavy sterile states given inEq. 57. The contribution from the νββ channel mediated by the heavy sterile neutrino canbe expressed by A N ∼ V eN i M R i (26)where V eN i is the mixing of active neutrinos with the N R i states.On the other hand the amplitude of the νββ process, mediated by the comparativelylight sterile neutrino states S L i for m s i > (cid:104) p (cid:105) can be expressed as A S ∼ V eS i m s i , (27)with V eS i being the mixing between the active neutrino with the sterile neutrino given by M † D M − S † W s in the present framework, whereas, for m s i < (cid:104) p (cid:105)A S ∼ V eS i m s i (cid:104) p (cid:105) (28)One can write the simpliﬁed expression for the amplitude, taking simultaneously intoaccount (cid:104) p (cid:105) (cid:39) m s i (cid:39) −

200 MeV . A S ∼ V eS i m s i (cid:104) p (cid:105) − m s i . (29)The half-life of νββ is given by (cid:16) T ν / (cid:17) − = K ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ei m ν i (cid:104) p (cid:105) − m ν i + V eS i m s i (cid:104) p (cid:105) − m s i + V eN i M R i (cid:104) p (cid:105) − M R i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (30)where, i implies for all the light and heavy neutrino states, supposed to mediate the potential νββ process. The details of the notation and convention can be found in Appendix. In theabove, we also have K ν = G ν ( M N m p ) and (cid:104) p (cid:105) ≡ − m e m p M N M ν . Here G ν is the phasespace factor, M N is the nuclear matrix element (NME) for heavy sterile exchange whereas M ν is the corresponding element for light sterile exchange. The required values of G ν andNMEs have been taken from Ref. [70]. In our analysis, we stick to particular values of NMEsas M N = 163.5 and M ν = 2.29; and the phase space factor G ν is . × − year − .For completeness we focus on the contribution of the lighter sterile neutrino states S L . Asmentioned earlier, the heavier states ( N R ), being considerably heavier − GeV andhaving the active-sterile mixing ∼ M D /M R ∼ − − − give negligible contribution in We shall deﬁne it shortly. / ν ( y e a r ) Δ m × 10 -6 (eV ) T / ν ( y e a r ) Σ m ν i (eV) Figure 12:

Prediction for the half-life of the νββ process as a function of the fourth masssquared splittings ∆ m (left) and sum over neutrino masses (right). The reason behind thechoice of diﬀerent colors can be obtained from the caption of ﬁg. 6. Here the values of nuclearmatrix elements (NMEs) are M N = 163.5 and M ν = 2.29; the phase space factor G ν is takenas . × − year − [70]. νββ . On the other side there seems to be sizeable contribution from the the lighter sterileneutrino states to the ﬁnal amplitude of νββ process, due to the larger active-sterile mixing V eS . It is to note that to keep some window open for keV sterile dark matter candidate, themass window for the lighter sterile state has been chosen to be around keV to a few MeV.The theoretical prediction on the T ν / drawn from the constraints of this framework,has been compared with the one reported by recent KamLAND-Zen experiment which sets T ν / > . × year [27, 71]. Till date the best lower limit on the half-life of the νββ using Ge is T ν / > . × yrs at 90 % C.L. from GERDA [72]. The proposed sensitivityof the future planned nEXO experiment is T ν / ≈ . × yrs [73]. Though we do not haveany direct experimental proof for νββ transition [74–82] the search for improving the boundon the T ν / value is still beneﬁcial to explore the possibility of new physics [77, 81, 83–89].The theoretical prediction of the T ν / value in the ESS framework is shown in ﬁg. 12. It isexpressed as functions of square of mass splitting of S and active electron neutrino and alsoas function of sum over active neutrino masses Σ m ν i . It is evident from the ﬁgure that allpoints are allowed from the limit set by KamLAND-Zen experiment [27]. We present a standard model extension augmented by a U (1) B − L symmetry to oﬀer an ex-planation to the light Majorana neutrino mass as well as the observed matter-antimatterasymmetry of the Universe through a low scale seesaw model. Our study is stimulated by therequirement of having a TeV scale RHN along with a potential dark matter candidate whichis of astrophysical signiﬁcance. In this work we have systematically investigated the viableparameter space of an extended seesaw framework to meet the observed matter-antimatterasymmetry. Having the ESS framework leads to the realization that the presence of the ad-ditional sterile neutrino generations takes a strong hold in the lepton asymmetry parameterspace through an extra self-energy contribution. We have emphasized here the new ﬁndingswhich are the prime requirements in order to account for a successful leptogenesis in this ESSframework in the context of the work done by the authors in Ref. [28]. We present the keyﬁndings of our analysis below. 17he model parameter space arising from this ESS scheme can explain both the hierarchiesof neutrino mass pattern. We have presented mild correlations among the mass matrix ele-ments, which further can explain the properties of the leptonic mixing matrix. In particular itis worth-mentioning that, the allowed parameter space for successful leptogenesis in this ESSmodel brings out rich predictions on the low energy neutrino mixing data. In the context ofleptogenesis we obtained signiﬁcant preferences in the neutrino observables for the octant ofthe atmospheric mixing angle ( θ > ◦ ) along with the reactor mixing angle ( θ > . ◦ ).These low energy predictions for mixing angles make this framework a viable scenario tobe tested from the perspective of future and ongoing neutrino oscillation experiments. Thismodel predicts the entire σ range for the Dirac CP phase δ . The sum over neutrino massobtained from this analysis ( (cid:80) m i < . eV) is found to be well within the bound reported byCosmology. We also brieﬂy address on how the model parameter space impacts the half-lifeperiod of the νββ decay process ( T ν / > yrs) which is in agreement with the half-lifeperiod as envisaged by the recent KamLAND-Zen experiment.In the later part, we have discussed how the resulting complex Yukawa coupling matricesplay their role in realizing baryogenesis through leptogenesis scenario by the CP-violatingdecay of TeV scale RHN. We keep the RHN mass window to be around 10-20 TeV. It isworth mentioning that in spite of having an additional self-energy contribution to the leptonasymmetry, the tiny splitting among the RHN masses is unavoidable which is a basic criteriaof facing resonant leptogenesis. The washout in this framework strictly hints towards a strongwashout regime, which is also approved by the low energy neutrino data. The highest andample amount of lepton asymmetry we report here to be around . × − . We also observehere that although for some parameter space of ESS we obtain a large amount of leptonasymmetry, the ﬁnal baryon asymmetry gets depleted by a factor due to the large washout.The present analysis confers a detailed construction of the extended seesaw parameterspace. It is interesting to note that within the allowed seesaw parameter space satisfying theneutrino oscillation data, only some particular regions can actually account for a suﬃcientamount of lepton asymmetry which has to be of around − . Thus it is worth mentioningthat this particular extension of the canonical seesaw model play a crucial role in obtaining anenhanced lepton asymmetry which translates into the requirement for the cosmological baryonasymmetry of the Universe both from the quantitative and the qualitative point of view. Thismodel can potentially manifest its signatures in search of TeV scale Majorana right handedneutrino in the future collider searches. An added bonus of this model is the existence of akeV sterile neutrino which can also play the role of a non-thermal dark matter candidate, thedetailed phenomenological study of which we keep for a forthcoming work. Acknowledgement

UKD acknowledges the support from Department of Science and Technology (DST), Govern-ment of India under the grant reference no. SRG/2020/000283. TJ thanks Ram Lal Awasthi,Sarif Khan, Amina Khatun and Soumya C for important discussions. TJ acknowledges thefacilities provided by Institute of Physics, Bhubaneswar as the collaboration was started dur-ing the stay at IOP. TJ also acknowledges Adrish Maity, Saptarshi Mukherjee and TapolinaJha for important discussion regarding the computational part. Authors also acknowledgethem as a part of the simulation has been performed using their machine at initial stage of thework. Authors acknowledge the HPC facility (Vikram-100 HPC) provided by PRL, Ahmed-abad. AM also wants to thank Ashimananda Modak for useful discussion related to numericalsimulation. NS acknowledges RUSA 2.0 Project.18 ppendix

Here we brieﬂy explain the diagonalization procedure in extended seesaw framework for com-pleteness. This is largely based on the method outlined in [52, 53]. The neutral mass matrix M n is given by M n = M TD µ M TS M D M S M R . (31)In the entire analysis, we consider Majorana matrix M R to be real and diagonal. The µ being another Majorana mass matrix is considered to be complex symmetric. The Diracmass matrices M S , M D are also complex symmetric matrices. The matrix M n can be block-diagonalized as U T M n U = M bd where, U and M bd are the ﬁnal block-diagonalizationmatrix and ﬁnal block-diagonal mass matrix respectively. Furthermore, U can be decomposedas U = U (cid:48) U (cid:48)(cid:48) . So, we have U (cid:48) T M n U (cid:48) = ˆ M bd followed by another block-diagonalization as U (cid:48)(cid:48) T ˆ M bd U (cid:48)(cid:48) = M bd , with ˆ M bd being the intermediate block-diagonalization matrix. We followthe parametrization of Ref. [52], i.e. , U (cid:48) = (cid:32)(cid:112) (cid:49) × − BB † B × −B † × (cid:112) (cid:49) × − B † B (cid:33) , (32)where, B = (cid:18) B aa B bb (cid:19) , B = B + B + B + . . . and √ − BB † = 1 − B B † − ( B B † + B B † ) − ( B B † + B B † + B B † + B B † B B † ) + . . . ; so, B aa \ bb = B aa \ bb + B aa \ bb + B aa \ bb + · · · .From the following diagonalization (block) U (cid:48) T M n U (cid:48) = ˆ M bd = (cid:18) M light × (cid:48) × (cid:48) × M heavy × (cid:19) , (33)we get, B T (cid:102) M L (cid:112) − BB † − B T (cid:102) M TD B † + (cid:112) − B T B ∗ (cid:102) M D (cid:112) − BB † − (cid:112) − B T B ∗ M R B † = 0 , (34)where, (cid:102) M L = (cid:18) µ (cid:19) and (cid:102) M D = (cid:0) M D M S (cid:1) .Considering B j ∝ M jR and equating diﬀerent coeﬃcients of M jR for diﬀerent values of j fromthe above equation: M R B † = (cid:102) M D , (35) M R B † = M − R ∗ (cid:102) M ∗ D (cid:102) M L . (36)Solving above equations we get, B aa = M † D M − R , B bb = M † S M − R , B aa = 0 , B bb = µ ∗ M TS M − R . (37)So, the block-diagonalized mass matrices M light × = (cid:102) M L −

12 ( B ∗ B T (cid:102) M L + (cid:102) M L B B † ) − (cid:102) M TD ( B † + B † ) + B ∗ M R B † = (cid:18) − M TD M − R M D − M TD M − R M S − M TD M − R M ∗ S µ − M TS M − R M D − µM † S M − R M D µ − M TS M − R M S − ( M TS M − R M ∗ S µ + µM † S M − R M D ) (cid:19) . (38) M heavy × = M R + B T (cid:102) M L B + 12 (cid:102) M D B + 12 (cid:102) M D B + B T (cid:102) M TD + B T (cid:102) M TD − M R B † B − B T B ∗ M R − B T B ∗ M R − B T B ∗ M R (39) ⇒ M (cid:48) R = M R + 12 ( M D M † D M − R + M S M † S M − R + M − R M ∗ D M TD + M − R M ∗ S M TS )+ 12 ( M S µ ∗ M TS M − R + M − R M S µ ∗ M TS ) ⇒ M (cid:48) R = M R + 12 [( M D M † D + M S M † S ) M − R + M S µ ∗ M TS M − R + Trans . ] (cid:39) M R . Substituting the values of B from Eq. (37) to Eq. (32) and considering expansion up to M − R we get, U (cid:48) = − M † D M − R M D − M † D M − R M S M † D M − R − M † S M − R M D − M † S M − R M S M † S M − R + µ ∗ M TS M − R − M − R M D − ( M − R M S + M − R M ∗ S µ ) 1 − M − R ( M D M † D + M S M † S ) M − R . (40) Using the expressions of Eqs. (39) and (38) (neglecting powers lower than M − R ) we have theintermediate block diagonalization matrix as ˆ M bd = − M TD M − R M D − M TD M − R M S − M TS M − R M D µ − M TS M − R M S

00 0 M R . (41)For further block-diagonalization, we again follow the same prescriptions [52, 53]. Followingthe similar ansatz , we have, U (cid:48)(cid:48) = (cid:32)(cid:112) − B (cid:48) B (cid:48)† B (cid:48) −B (cid:48)† (cid:112) − B (cid:48)† B (cid:48) (cid:33) , (42)and from similar diagonalization U (cid:48)(cid:48) T ˆ M bd U (cid:48)(cid:48) = M bd = (cid:18) M light M heavy (cid:19) , (43)we obtain B (cid:48) T ( − M TD M − R M D ) (cid:112) − B (cid:48) B (cid:48)† − B (cid:48) T ( − M TD M − R M S ) B (cid:48)† + (cid:112) − B (cid:48) T B (cid:48)∗ ( − M TS M − R M D ) (cid:112) − B (cid:48) B (cid:48)† − (cid:112) − B (cid:48) T B (cid:48)∗ ( µ − M TS M − R M S ) B (cid:48)† = 0 . (44)Assuming B (cid:48) j ∝ M jS , expressions of diﬀerent B (cid:48) can be obtained by equating diﬀerent coeﬃ-cients of M jS for diﬀerent values of j from Eq. 44: − M TS M − R M D + M TS M − R M S B (cid:48)† = 0 , (45) M TS M − R M S B (cid:48)† = 0 , (46) M TS M − R M S B (cid:48)† = µ B (cid:48)† . (ignoring less contributing terms) (47)From the above equations, B (cid:48)† = M − S M D , B (cid:48) = 0 , B (cid:48)† = M − S M R M TS − µM − S M D . (48)20ince B (cid:48) = 0 , M light = (cid:18) − B (cid:48)∗ B (cid:48) T (cid:19) ( − M TD M − R M D ) (cid:18) − B (cid:48) B (cid:48)† (cid:19) − (cid:18) − B (cid:48)∗ B (cid:48) T (cid:19) ( − M TD M − R M S )( B (cid:48)† ) − B (cid:48)∗ ( − M TS M − R M D ) (cid:18) − B (cid:48) B (cid:48)† (cid:19) + B (cid:48)∗ ( µ − M TS M − R M S ) B (cid:48)† = M TD M − S T µM − S M D , (49) and, M heavy = B (cid:48) T ( − M TD M − R M D ) B (cid:48) + B (cid:48) T ( − M TD M − R M S ) (cid:18) − B (cid:48)† B (cid:48) (cid:19) + (cid:18) − B (cid:48) T B (cid:48)∗ (cid:19) ( − M TS M − R M D ) B (cid:48) + (cid:18) − B (cid:48) T B (cid:48)∗ (cid:19) ( µ − M TS M − R M S ) (cid:18) − B (cid:48)† B (cid:48) (cid:19) = ( µ − M TS M − R M S ) + 12 [ M TS M − R M D M † D M − S † + µM − S M D M † D M − S † + Trans . ] . (50) The ﬁnal block-diagonalized matrix is given by M bd = M TD M − S T µM − S M D µ − M TS M − R M S

00 0 M R . (51)From Eqs. (42) and (48) we have, U (cid:48)(cid:48) = − M † D M − S † M − S M D M † D M − S † − M − S M D − M − S M R M TS − µM − S M D − M − S M D M † D M − S †

00 0 1 . (52)The ﬁnal block-diagonalization matrix U is given by U = U (cid:48) U (cid:48)(cid:48) = − M † D M − S † M − S M D M † D M − S † M † D M − R − M − S M D − M − S M D M † D M − S † − M † S M − R M S M † S M − R M TS − µM − S M D − M − R M S − M − R M S M † S M − R . (53) Finally to the leading order the active and heavy neutrino mass matrices are given by m ν ∼ M TD M − S T µM − S M D , (54) m s ∼ µ − M TS M − R M S , (55) m n ∼ M R . (56)In the above, if µ is greater than | M TS M − R M S | then m s will be deﬁned mainly by µ and inthe opposite case, it will be dominated by | M TS M − R M S | . If µ ∼ | M TS M − R M S | then we haveto consider the next order contribution from Eq. (50). The active and sterile matrices will bediagonalized as U T m ν U = diag ( m ν i ) (57) W Ts m s W s = diag ( m s i ) (58) W TN m n W N = diag ( M R i ) . (59)So, the mixing matrix for above block diagonalization can be written as U = U W s

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NEXT

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NEXT, a HPGXe TPC for neutrinolessdouble beta decay searches , arXiv:0907.4054arXiv:0907.4054