Neutrinoless Double Beta Decay in LRSM with Natural Type-II seesaw Dominance

We present a detailed discussion on neutrinoless double beta decay within a class of left-right symmetric models where neutrino mass originates by natural type-II seesaw dominance. The spontaneous symmetry breaking is implemented with doublets, triplets and bidoublet scalars. The fermion sector is extended with an extra sterile neutrino per generation that helps in implementing the seesaw mechanism. The presence of extra particles in the model exactly cancels type-I seesaw and allows large value for Dirac neutrino mass matrix $M_D$. The key feature of this work is that all the physical masses and mixing are expressed in terms of neutrino oscillation parameters and lightest neutrino mass thereby facilitating to constrain light neutrino masses from $0\nu\beta\beta$ decay. With this large value of $M_D$ new contributions arise due to; i) purely left-handed current via exchange of heavy right-handed neutrinos as well as sterile neutrinos, ii) the so called $\lambda$ and $\eta$ diagrams. New physics contributions also arise from right-handed currents with right-handed gauge boson $W_R$ mass around $3$~TeV. From the numerical study, we find that the new contributions to $0\nu\beta\beta$ decay not only saturate the current experimental bound but also give lower limit on absolute scale of lightest neutrino mass and favor NH pattern of light neutrino mass hierarchy.

3 0νββ from purely right-handed currents 25 6.4 0νββ from λ and η− diagrams 28 6.5 Mass hierarchy discrimination within natural type-II seesaw dominance 28 1 Introduction The discovery that neutrinos have mass and they mix with each other has put before us another vital question to speculate over; whether they are Dirac or Majorana [1] particles. Even more intriguing is the theoretical origin of such a tiny mass and the mass hierarchy among them. The different seesaw mechanisms like type-I [2][3][4][5], type-II [6][7][8][9][10] and others that appropriately explain this tiny mass further require them to be Majorana particles.
On the contrary, Majorana nature of neutrinos violates global lepton number by 2 units that is regarded as an accidental symmetry within the Standard Model (SM). This leads to the search of a rare process called Neutrinoless Double Beta Decay (0νββ) that only can assuredly endorse the Majorana nature of neutrinos and lepton number violation in nature [11]. While new theories are trying to find new physics contributions to 0νββ decay, the experiments are looking for lower limits on the half-lives being decayed. Of yet, GERDA [12] using Ge 76 gives lower limit on half life of 0νββ decay as T 0ν 1/2 > 2.1 × 10 25 yrs at 90% C.L. whereas the limits provided by EXO-200 [13] and KamLAND [14] are T 0ν 1/2 > 1.6 × 10 25 yrs and T 0ν 1/2 > 1.9 × 10 26 yrs respectively. The combined limit from KamLAND-Zen comes to be T 0ν 1/2 > 3.4×10 26 yrs at 90% C.L. This process can be mediated by the exchange of a light Majorana neutrinos or by new particles appearing in various extensions of SM [15][16][17][18][19][20][20][21][22][23][24][25][26][27][28].
Within preview of BSM physics, left-right symmetric models (LRSM) [2,[29][30][31][32][33] are found to be best suited frameworks for explaining the origin of maximal parity violation in weak interactions and the origin of small neutrino mass. This class of models, based on the gauge group SU (2) L × SU (2) R × U (1) B−L × SU (3) C , when studied at TeV scale interlinks high energy collider physics to low energy phenomena like neutrinoless double beta decay and other LFV processes (see refs. ). Moreover, the left-right symmetric models can also accommodate stable dark matter candidate contributing 25% energy budget of the Universe [61][62][63][64][65]. In conventional left-right symmetric models where symmetry breaking is implemented with scalar triplets and bidoublet, the light neutrino mass is governed by type-I plus type-II seesaw mechanisms Here M L (M R ) is the Majorana mass term for light left-handed (heavy right-handed) Majorana neutrinos arising from respective VEVs of left-handed (right-handed) scalar triplets and M D is the Dirac neutrino mass matrix connecting light-heavy neutrinos. The scale of M R is decided by the vacuum expectation value of right-handed scalar triplet which spontaneously breaks LRSM to SM. Thus, the smallness of light neutrino mass is connected to high scale of parity restoration i.e, 10 15 GeV clearly making it inaccessible to current and planned accelerator experiments. Moreover when LRSM breaks around TeV scale, the gauge bosons W R , Z R , right-handed neutrinos N R and scalar triplets ∆ L,R get mass around that scale allowing several lepton number violating signatures at high energy as well as low energy experiments. A wide range of literature provides discussions on neutrinoless double beta decay within TeV scale LRSM assuming type-I seesaw dominance [42] or type-I plus type-II [35,42,43,47,48,66,67] seesaw mechanisms. Some more scenarios have been studied in [34-36, 39, 42, 54, 68, 69] where type-II seesaw dominance relates the light and heavy neutrinos with each other. Other works that discuss complementarity study of lepton number, lepton flavour violation and collider signatures in LRSM with spontaneous D-parity breaking mechanism also embed the framework in a non-SUSY SO(10) GUT [36][37][38]70]. One should bear in mind that the new physics contributions to neutrinoless double beta decay mainly involves left-right mixing (or light-heavy neutrino mixing) which crucially depends on Dirac neutrino mass M D . Necessarily M D should be large in order to expect LNV signatures at colliders. Contrary to this, the type-II seesaw dominance can be realized with suppressed value of M D or with very high scale of parity restoration. Studies that assume M D → 0 therefore miss to comment on LNV, LFV and Collider aspects involving left-right mixing. We thus feel motivated to explore alternative class of left-right symmetric models which allows large value of M D and carries light and heavy neutrinos proportional to each other. This work considers a TeV scale LRSM where symmetry breaking is implemented with scalar bidoublet Φ, doublets H L,R and triplets ∆ L,R . The scalar bidoublet carrying B − L charge 0 provides Dirac masses to charged fermions as well as to neutrinos. The scalar triplets with B − L charge 2 units provide Majorana masses to light and heavy neutrinos. One extra sterile fermion S L per generation also finds place in the model that help in implementing extended type-II seesaw mechanism. The scalar doublets H L,R play the same role as S L . An interesting feature of this new class of LRSM is that it provides possibility of achieving type-II seesaw dominance when parity and SU (2) R break at same scale. Moreover this framework allows large value for Dirac neutrino mass matrix M D thereby leading to new physics contributions to neutrinoless double bea decay i.e, i) from purely left-handed currents via exchange of heavy right-handed and extra sterile neutrinos, ii) from purely right handed currents via exchange of heavy right-handed neutrinos, iii) from so called λ and η diagrams. This work aims to carefully analyze the new contributions to 0νββ in order to derive the absolute scale of light neutrino masses and mass hierarchy.
The complete work is structured as follows. In Sec.2, we briefly discuss the generic and TeV scale LRSMs in context of neutrino mass and associated lepton number violation. Sec.3 highlights the natural realization of type-II seesaw dominance. Sec.4 lays out the basic ingredients for neutrinoless double beta decay and the calculation of Feynman amplitudes. Sec.5 and Sec.6 are devoted towards the numerical study of LNV 0νββ contributions within the present framework. In Sec.8 we summarize our results.

The Left-Right Symmetric Model and Lepton Number Violation
The left-right symmetric model [2,[29][30][31][32][33] is based on the gauge group (2.1) In this class of models, the difference between baryon B and lepton L number is defined as a local gauge symmetry. The electric charge Q is defined as Here, T 3L and T 3R are, respectively, the third component of isospin of the gauge groups SU (2) L and SU (2) R , and Y is the hypercharge. The usual leptons and quarks are given by The left-right symmetry calls for the presence of right-handed neutrinos and this makes the model suitable for explaining light neutrino masses. For generating fermion masses one needs a scalar bidoublet Φ with the following matrix representation The relevant Yukawa interactions are expressed as, where Φ = σ 2 Φ * σ 2 and σ 2 is the second Pauli matrix. The scalar bidoublet takes a non-zero VEV as, it yields masses for quarks and charged leptons as One can generate Dirac masses for light neutrinos using scalar bidoublet as However, the Majorana masses for neutrinos depend crucially on how spontaneous symmetry breaking of LRSM down to the SM i.e, G LR → G SM is implemented.

Lepton number violation and the origin of neutrino mass
The spontaneous symmetry breaking of LRSM to SM goes in favor of neutrino mass generation and associated lepton number violation. This happens in the following three ways • with scalar triplets ∆ L (3, 1, 2, 1) ⊕ ∆ R (1, 3, 2, 1), • with the combination of doublets and triplets H L ⊕ H R and ∆ L ⊕ ∆ R .
In the first case, H R breaks the LR symmetry while the left-handed counterpart is required for left-right invariance. Though this framework holds a minimal scalar spectrum it lacks Majorana mass for neutrinos and thus forbids any signature of lepton number violation or neutrinoless double beta decay. Since the light neutrinos here owe their identity to Dirac fermions, their masses can only be explained by adjusting Yukawa couplings through the non-zero VEVs of scalar bidoublet. Other important roles that this scalar bidoublet plays are to break the SM gauge symmetry to low energy theory and provide the masses to charged fermions. Using the Yukawa interactions given in Eq.(2.6) and with Y 3 Y 4 , v 2 v 1 and θ 1 , θ 2 = 0, the masses for charged leptons and the light neutrinos are given by However, a pleasant situation arises in the second case where ∆ R carrying B − L charge 2 breaks the LR symmetry to SM. The inclusion of ∆ L and ∆ R in the framework generate Majorana masses for light as well as heavy neutrinos and thus violate lepton number by two units. This calls for a possibility of smoking-gun same-sign dilepton signatures at collider as well as neutrinoless double beta decay in low energy experiments. The interaction terms involving scalar triplets and leptons are given by Using Eq.(2.6) and Eq.(2.11), the resulting mass matrix for neutral leptons in the basis (ν L , N c R ) reads as where, M D is the Dirac neutrino mass matrix, M L (M R ) is the Majorana mass matrix arising from the non-zero VEV of LH (RH) scalar triplet. After diagonalization, the resulting light neutrino mass can be written as a combination of canonical type-I and type-II seesaw formula where, m I ν (m II ν ) is denoted as the type-I (type-II) contribution to light neutrino masses, In conventional left-right symmetric models, where parity and SU (2) R break at same scale, the analytic formula for induced VEV of left-handed scalar triplet ∆ L is given by, In the above expression v = v 2 1 + v 2 2 lies around electroweak scale, v R is the VEV of right-handed scalar triplet ∆ R and γ is dimensionless Higgs parameter. In order to be consistent with oscillation data m II ν = f v L should be order of 0.1 eV and assuming natural values of f and γ, this sub-eV scale of v L can be attained only if v R lies around 10 14 GeV. However such a high scale is inaccessible to LHC and thus urges to look for TeV scale LRSM. These frameworks offer numerous opportunities like low scale seesaw mechanism, LNV like neutrinoless double beta decay and its collider complementarity and have been already explored by the works mentioned in refs [28,[34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][54][55][56][57][58]. Many of the works considered either type-I seesaw dominance or type-II seesaw dominance for en extensive study of 0νββ decay. In manifest left-right symmetric model, where right-handed scale lies at TeV range, the neutrino mass mechanism via type-I plus type-II seesaw gives negligible value to the left-right mixing. As a result of this the production cross-section of heavy neutrinos and the lepton number violating processes at LHC get suppressed. However, the extension of type-I plus type-II seesaw scheme by the inclusion of another sterile neutrino per generation changes the scenario which results large left-right mixing. Now the neutrino mass arises only from type-II seesaw dominance since type-I seesaw contribution gets exactly canceled out. We propose a new framework where type-II seesaw dominance is achieved naturally and allows large value of Dirac neutrino mass which additionally contributes to 0νββ decay from purely left-handed current via exchange of heavy neutrinos as well as from the so called λ type and η type diagrams.
3 Extended Seesaw Mechanism and Natural type-II seesaw dominance

Extended Seesaw Mass Matrix
In order to implement the extended seesaw mechanism 1 within left-right symmetric models, one has to add a complete left-right gauge symmetry singlet neutral fermion S L per generation to the usual quarks and leptons. Along with this the Higgs sector includes scalar bidoublet Φ with B − L = 0, scalar triplets ∆ L ⊕ ∆ R with B − L = 2 and scalar doublets H L ⊕ H R with B − L = −1. The complete particle spectrum is given in Table.1 . The relevant leptonic Yukawa interaction terms for extended seesaw mechanism are given by After spontaneous symmetry breaking, the resulting neutral lepton mass matrix for extended seesaw mechanism in the basis (ν L , N c R , S L ) is given by where M D = Y Φ is the Dirac neutrino mass matrix connecting left-handed light neutrinos with right-handed heavy neutrinos, The light neutrino mass can be parametrized in terms of model parameters of inverse seesaw framework as, This expression bears M of few TeV which allows large left-right mixing and thus leads to interesting testable collider phenomenology. Extension of such a scenario has been discussed in the context of allowing large LNV and LFV in the work [38]. Linear Seesaw:-Similarly in Eq.(3.3), if we assume M L , M R , µ S → 0, the linear seesaw mass formula for light neutrinos is given by [55] whereas the heavy neutrinos form pair of pseudo-Dirac states with masses The following discussion considers the same Eq.(3.3) with the assumption that µ L , µ S → 0 which leads to natural realization of type-II seesaw dominance allowing large left-right mixing.

Natural realization of type-II seesaw
The natural realization of type-II seesaw dominance is considered here within a class of leftright symmetric models where both discrete left-right parity symmetry and SU (2) R gauge symmetry break at same scale. The scalar sector is comprising of SU (2) doublets H L,R , triplets ∆ L,R and bidoublet Φ whereas the fermion sector is extended with one neutral fermion S L per generation which is complete singlet under both LRSM as well as SM gauge group. We denote this class of LR model as Extended LR models and thus, the corresponding seesaw formula which is type-II dominance in this case is termed as Extended type-II seesaw mechanism. In principle, there could be a gauge singlet mass term in the Lagrangian for extra fermion singlet, i.e, µ S S c S which can take any value. But we have taken this mass parameter to be either zero or very small so that the generic inverse seesaw contribution involving µ S is very much suppressed. In addition, we have assumed the induced VEV for H L is taken to be zero, i.e, H L → 0. The relevant interaction terms necessary for realizing natural type-II seesaw dominance is given by With H L → 0 and µ S → 0, the complete 9 × 9 neutral fermion mass matrix in the flavor basis of (ν L , S L , N c R ) is read as Using standard formalism of seesaw mechanism and using mass hierarchy M R > M > M D M L , we can integrate out the heaviest right-handed neutrinos as follows where the intermediate block diagonalised neutrino states modified as Thus, the intermediate block diagonalised neutrino states are related to flavor eigenstates in the following transformation, It is found that the (2, 2) entries of mass matrix M is larger than other entries in the limit As a result of this, we can repeat the same procedure in Eq.(3.9) to integrate out S . Thus, the light neutrino mass formula becomes 12) and the physical block diagonalised states arê with the corresponding block diagonalised transformation as With this block diagonalization procedure and after few simple algebra, the flavor eigenstates are related to mass eigenstates in the following transformation, Subsequently, the final block diagonalised mass matrices can be diagonalised in order to give physical masses by a 9 × 9 unitary matrix V 9×9 . The transformation of the block diagonalised neutrino states in terms of mass eigenstates are given bŷ while the block diagonalised mass matrices for light left-handed neutrinos, heavy righthanded neutrinos and extra sterile neutrinos are These block diagonalised mass matrices can be further diagonalised by respective 3 × 3 unitarity matrices as follows Finally, the complete block diagonalization yields Here the block diagonalised mixing matrix W and the unitarity matrix U are given by Thus, the complete 9 × 9 unitary mixing matrix diagonalizing the neutral leptons is as follows 3.3 Expressing Masses and Mixing in terms of U PMNS and light neutrino masses.
The light neutrinos are generally diagonalised by standard PMNS mixing matrix U PMNS in the basis where charged leptons are already diagonal i.e,m diag The Dirac neutrino mass matrix M D in general is a complex matrix. The structure of M D in LRSM can be approximately taken to be up-quark type mass matrix whose origin can be motivated from high scale Pati-Salam symmetry or SO(10) GUT. If we consider M to be diagonal and degenerate i.e, M = m S diag{1, 1, 1}, then the mass formulas for neutral leptons are given by After some simple algebra, the active LH neutrinos ν L , active RH neutrinos N R and heavy sterile neutrinos S L in the flavor basis are related to their mass basis as

Neutrinoless Double Beta Decay in LRSM
In this section, we shall present a detailed discussion on Feynman amplitudes for neutrinoless double beta decay within TeV scale LRSM where light neutrino mass mechanism is governed by natural type-II seesaw dominance. The basic charge current interaction Lagrangian for leptons as well quarks are given by Using Eq.(3.23) of Sec.3, the flavor eigenstates (ν L ) and N c R are expressed in terms of admixture of mass eigenstates (ν i , S i , N i ) in the following way, This modifies the charged current interaction for leptons as In the above charged-current interaction, there is a possibility that both left-handed W L and right-handed W R gauge bosons can mix with each other which can eventually contribute to 0νββ transition amplitude. In the present framework, the resulting mass matrix for LH (RH) charged gauge bosons (W L , W R ) is given by The physical masses of the charged gauge bosons derived with g L = g R after diagonalization are given by The physical gauge boson states W 1 and W 2 are related to the mixture of weak eigenstates W L and W R as where, Thus, one can express physical states in terms of W L and W R as follows We classify all contributions to neutrinoless double beta decay in the present TeV scale LRSM as: • due to standard mechanism mediated by purely left-handed currents (W L − W L mediation) via exchange of light neutrinos ν i , • due to purely left-handed currents via W − L − W − L mediation through the exchange of the heavy RH Majorana neutrino N i and heavy sterile neutrinos S i , • due to purely right-handed currents (W R − W R mediation) via exchange of heavy right-handed Majorana neutrinos N i , • due to purely right-handed currents via W − R − W − R mediation through the exchange of the light neutrinos ν i and extra sterile neutrinos S i , • due to mixed helicity so called λ and η diagrams through mediation of ν i , S i , N i neutrinos.
Before deducing Feynman amplitudes for various contributions to neutrinoless double beta decay, it is desirable to discuss few points regarding the chiral structure of the matrix element with the neutrino propagator as [21] (4.12) 4.1 Feynman amplitudes for 0νββ decay due to purely left-handed currents The Feynman amplitudes for W − L − W − L mediated diagrams shown in Fig.1 with the exchange of Majorana neutrinos ν i , N j and S k , respectively, are given by where p is the typical momentum exchange of the 0νββ decay process and G F = 1.2 × 10 −5 GeV −2 is the Fermi coupling constant. The analytic expressions for suitably normalized dimensionless lepton number violating particle physics parameters for these contributions are as follows Though we shall discuss in detail about the lepton number violating effective mass parameters and half-life in the following section, it will be better if one can express normalized effective mass parameters representing LNV due to these above mentioned Feynman diagrams and are given below

Feynman amplitudes for 0νββ decay due to purely right-handed currents
One of our major contribution in this work is that with W − R − W − R mediation as shown in first one of Fig. 2 by the exchange of mainly heavy right-handed neutrinos within the type-II seesaw dominance can yield significantly large contribution to 0νββ decay rate than the standard one. The Feynman amplitudes for these diagrams displayed in Fig.2 normalized in terms of G F are given by The resulting dimensionless LNV particle physics parameters due to W − R − W − R mediated diagrams are as follows

Feynman amplitudes for
There are Feynman diagrams for neutrinoless double beta decay due to mixed helicity of emitted electrons in the final state via W − L − W − R mediation and the Feynman amplitudes for these diagrams with the exchange of virtual Majorana neutrinos ν i , N j and S k are

Feynman amplitudes for
There are Feynman diagrams for neutrinoless double beta decay due to mixed helicity of emitted electrons in the final state via W − L − W − R mediation as well as involves mixing between W L and W R gauge boson. The Feynman amplitudes for these diagrams with the exchange of virtual Majorana neutrinos ν i , N j and S k are given by Figure 5. Feynman diagrams for 0 ν ββ decay due to doubly charged scalar triplets.

Feynman amplitudes for 0νββ decay due to doubly charged scalar
The Feynman amplitudes due to doubly charged Higgs scalars ∆ −− L (∆ −− R ) exchanges are given by result in lepton number violating dimensionless particle physics parameters.

Half-life and normalized LNV effective Mass parameters
From the earlier discussion, we found that there are various contributions to neutrinoless double beta decay arising from purely left-handed currents, from purely right-handed currents, from mixed diagrams with left-handed as well as right-handed currents and possible interference effects. In this regard we closely follow the refs. [35,74,75] where the QRPA calculations of the matrix elements for the mixed diagrams leads to life-time of 0νββ transition as T 0ν where G 0ν 01 is the phase space factor, M 0ν GT is the matrix element for 0νββ transition. Here X L and X R represent the relevant contributions arising from left-handed and right-handed currents respectively. The coefficientsC i stand for combination of matrix elements and integrated kinematically factors and ψ i represents complex phases. The LNV dimensionless particle physics parameters are denoted by η s. In above eq.(5.1), we omitted the interference terms between left-handed and right-handed currents as they are suppressed due to different electron helicities. However the interference terms arising from mixed helicity λ and η diagrams are included.
The neutrino virtual momentum |p 2 | (100 MeV) 2 plays a crucial role as the formula for 0νββ transition could be different for M 2 i p 2 or M 2 i p 2 where M i denoted as mass of any type of neutrinos. It is observed that the light neutrinos having m i p 2 contributing to nuclear matrix element different than the mediating particles having masses M 2 i p 2 . To demonstrate this, we assume that the 0νββ transition is only mediated by light neutrinos and heavy neutrinos while neglecting the right-handed current effects. In this scenario, the analytic formula for inverse half-life for a given isotope from purely left-handed currents due to exchange of light ν and heavy N neutrinos is given by Actually, we normalized here the inverse half-life for standard mechanism due to exchange of light neutrino mechanism as It is clear now that the virtual momentum can be expressed in terms of known masses and nuclear matrix elements.
We discuss here another situation to get a clear idea about how heavy Majorana neutrinos contribute to neutrinoless double beta decay mediated by purely left-handed currents and purely right-handed currents and difference between them. Since we have already discussed heavy neutrino contributions to 0νββ transition, one can express inverse half-life formula arising from right-handed currents due to exchange of heavy neutrinos N R as , one can express It is seen that the proton mass m p appears whenever neutrinoless double beta decay is mediated by heavy particles regardless of left-handed or right-handed currents. However, with right-handed current an additional factor of g R appears. Similarly, one can express half-life for mixed helicity λ and η diagrams and their interference terms in terms of effective Majorana mass parameters.
In order to arrive at a common normalization factor for all types of contributions, at first we use the expression for inverse half-life for 0ν2β decay process due to only light active Majorana neutrinos, T 0ν Using the numerical values given in Table.7, we rewrite the inverse half-life in terms of effective mass parameter T 0ν We present here the analytic formula for half-life and normalized effective mass parameters for neutrinoless double beta decay for a given isotope for all relevant contributions are Effective Mass Parameters Analytic formula m ν In the above expression for inverse half-life, G 0ν 01 is the the phase space factor and the other nuclear matrix elements defined for different chiralities of the weak currents such as M 0ν ν , M 0ν N , M 0ν λ and M 0ν η are presented in Table.7. The effective Majorana mass parameters due to purely left handed currents are presented in Table.2 while Table.3 represents the effective Majorana mass parameters due to purely right handed currents and Table.4 shows the contributions due to involvement of both left handed as well as right handed currents. However we do not take into account the interference terms in this work.

Effective Mass Parameters
Analytic formula  Table 4. Effective Majorana mass parameters due to so called λ and η type diagrams. It is to be noted that the suppression factor 10 −2 arises in the λ−diagram because of normalization w.r.t to the standard mechanism.
6 Numerical results within natural type-II seesaw dominance

Input Model Parameters
Before moving towards the numerical estimation of various contributions to neutrinoless double beta decay, it is desirable to know the model parameters and thus we list them below.
The method of diagonalization is given in Sec.III and the resulting physical masses for all neutral fermions in terms of U P M N S matrix and mass of light neutrinos are give by The flavor basis of active LH neutrinos ν L , active RH neutrinos N R and heavy sterile neutrinos S L in terms of mass basis and mixing are given as follows in order to express in terms of know neutrino oscillation parameters and light neutrino masses. Where Here we consider Dirac neutrino mass matrix motivated from SO(10) GUT and assumed heavy N − S mixing matrix M to be diagonal and degenerate i.e, M = m S diag{1, 1, 1}. We fix m S at 500 GeV for all our numerical estimations. If we assume that the present TeV scale left-right symmetric model is originated from Pati-Salam symmetry [30] or SO(10) GUT [76], then the Dirac neutrino mass matrix M D can be approximated as up-type quark mass matrix 2 . This can be reconstructed using masses of up, charm & top quarks and the corresponding CKM mixing matrix in the quark sector [78,79] as   Table 5. The oscillation parameters like mass squared differences and mixing angles within 3σ range. However we adopt the values given in ref. [84].

0νββ contributions from purely left-handed currents:-
The analytic expression for inverse of half-life for neutrinoless double beta decay due to purely left-handed current via W L −W L mediation with the exchange of Majorana neutrinos ν L , S L & N R is given by The LNV effective Majorana mass parameter m ν ee and corresponding half-life T 0ν 1/2 ν due to standard mechanism by the exchange of light neutrinos is given by Using m e = 0.51 MeV, G 0ν 01 and M 0ν ν from Table.7 and 3 σ ranges of oscillation parameters like mixing angles and mass squared differences from Table.5, we examine the variation of effective mass and half-life vs. lightest neutrino mass m lightest = m 1 (NH), m 3 (IH). We plot effective Majorana mass parameter m ν ee in left-panel of Fig.6 and half-life T 0ν 1/2 ν in the right-panel of Fig.6 as a function of the mass of the lightest neutrino. It is observed that quasi degenerate (QD) pattern of light neutrinos i.e., m 1 m 2 m 3 and m lightest = i m i /3 is disfavoured by current bound on the sum of light neutrino mass m Σ < 0.23 derived from Planck+WP+highL+BAO data (PLANCK1) at 95 % C.L. while m Σ < 1.08 derived from Planck+WP+highL (PLANCK2) at 95 % C.L. [85]. Moreover, IH as well as NH pattern will be difficult to probe within the standard mechanism even for next generation experiments. This motivates us to consider all possible new physics contributions to neutrinoless double beta decay in the present framework, that might give crucial information about lower limit on absolute scale of lightest neutrino mass as well as mass hierarchy.  Table 6. The table shows the lower limits on the half life T 0ν 1/2 and upper limits on the effective mass parameter m 0ν eff for 0νββ transition for the isotopes 76 Ge and 136 Xe from different collaborations. The range for the effective mass parameter comes from the uncertainties in the nuclear matrix elements.

Non-standard Mechanism m N,S ee
The expressions for the effective Majorana mass parameter due to exchange of heavy righthanded Majorana neutrinos N R and extra sterile neutrinos S L are given by The variation of LNV effective mass parameters and corresponding half life with the lightest neutrino mass are displayed in left-panel and right-panel of Fig.7 due to exchange of N R . Similarly, we have shown m S ee,L and T 0ν 1/2 S vs. lightest neutrino mass in Fig.8 and the sum of these two new physics contributions is presented in Fig.9.

0νββ from purely right-handed currents
In the present framework the right-handed gauge boson W R and right-handed Majorana neutrinos N R lie around few TeV thereby leading to new physics contributions to neutrinoless double beta decay due to purely right-handed currents via W R − W R mediation and exchange of heavy neutrinos N R . In addition to this, the type-II seesaw dominance connects light and heavy neutrinos with each other for which one can express new physics  Table 7. Phase space factor G 0ν 01 and Nuclear Matrix Elements taken from ref. [86] contributions in terms of oscillation parameters.
As a result of this, both light and heavy neutrino masses are diagonalised simultaneously by the U PMNS and the mass eigenvalues are related as follows: Normal hierarchy (NH): where we fixed the heaviest RH Majorana neutrino mass M N 3 for NH. Inverted hierarchy (IH): where we fixed the heaviest RH Majorana neutrino mass M N 2 for IH. The expression for inverse half-life of 0νββ transition for a given isotope due to purely right-handed currents along with standard mechanism is given by Under this type-II seesaw dominance, the expressions for |m ν ee | and |m N ee,R | are given by where 4 . We have neglected the other terms arising from purely right-handed currents. However, the right-handed scalar triplet contribution can be significant if we consider the triplet mass around 500 GeV.
This has been discussed in ref. [54] and the used model parameters are M W R 2 TeV, g R 2/3g L and M N 1 TeV. In the present work, we have considered g L = g R 0.65, M W R 3 TeV and M N 5 TeV for numerical estimation of m S ee,L and T 0ν 1/2 S vs. lightest neutrino mass in Fig.8.

0νββ from λ and η− diagrams
In this framework with g L = g R and V N ν = 0 from the seesaw diagonalization the relevant effective Majorana mass parameters due to so called λ and η diagrams are expressed as follows.
With M D similar to up-quark mass matrix and other input model parameters, the effective mass and corresponding half-life with the variation of lightest neutrino mass m 1 (N H) and m 3 (IH) due to so called λ and η diagrams to 0νββ transition are displayed in Fig.12, Fig.13 and Fig.14. 6.5 Mass hierarchy discrimination within natural type-II seesaw dominance Here we discuss the comparison between the standard mechanism and the new physics contributions to 0νββ transition within the present framework with natural type-II seesaw dominance by plotting effective Majorana mass as a function of sum of light neutrino masses  (m Σ ) using the cosmological limit on the light neutrino mass sum. The light neutrino mass sum is defined as m Σ = i m i = m 1 + m 2 + m 3 . The lower limits of light neutrino mass sum m Σ derived from cosmology as well as measurements from ongoing neutrino less double beta decay experiments at 1σ, 2σ and 3σ C.L are given by [87][88][89] m Σ < 84 meV at 1σ C.L. , m Σ < 146 meV at 1σ C.L. , m Σ < 208 meV at 1σ C.L. (6.19) We plot effective Majorana mass m ee as a function of sum of light neutrino masses (m Σ ) displayed in Fig.15 and Fig.16 where standard mechanism is represented by the red band for NH and by the green band for IH while the new physics contributions are represented by the blue band for NH and the red hatched band for IH. As the contribution of heavy right-handed neutrino and sterile neutrino to effective mass parameter saturate the experimental GERDA limit, Fig.15 (left-panel) represents their combined effect to standard mechanism. The spectrum for IH due to the SM and others are lying within the region of cosmological bound and hence disfavoured at 1σ C.L. Whereas the NH spectrum are lying in the privileged region and favored for lower mass of lightest neutrino. But in case of purely right-handed currents i.e mediation via W R − W R , the contribution on effective Majorana mass parameters due to the exchange of ν L and heavy sterile neutrinos are negligible. So the right-panel of Fig.15 shows the effect of heavy N R with standard mechanism on sum of light neutrino masses. Here also NH is favored over IH both for standard mechanism and new physics. Similarly, Fig.16 indicates the effect of λ diagram due to exchange of heavy N R and S with standard mechanism on effective mass having same characteristic as in Fig.15. For all of them, the uncertainty on the effective Majorana mass parameter increases with increase in sum of masses in case of normal hierarchy for the contribution of new physics to standard mechanism.  Figure 16. Allowed region of |m ee | for standard mechanism and its combined effect to λ diagram due to exchange of heavy RH neutrino and sterile neutrino as a function of m Σ .
7 Comparison of half-lives for 0νββ in 76 Ge and 136 Xe We intend to make here a comparative study of half-lives for neutrinoless double beta decay in 76 Ge and 136 Xe indicating uncertainties in the nuclear matrix elements (one may refer [13] for the matrix element calculations). The half-life limits for different experiments are T 0ν 1/2 1.07 × 10 26 yrs (for KamLAND-Zen expt. using 136 Xe) and T 0ν 1/2 5.52 × 10 25 yrs (for GERDA Phase-II using 76 Ge). Using the values given in Tables 8, 9 and 10 for nuclear matrix elements for light and heavy neutrino exchange as well as for λand η-diagrams we have shown the correlation plots between half-lives for 76 Ge and 136 Xe in Fig. 17 and Fig. 18 (these were first introduced in ref. [35]). The band in each plots shows the measure of uncertainties in different nuclear matrix elements while measuring half-life in 136 Xe to one measured in 76 Ge.   Figure 17. The left-panel shows the correlations between the 0νββ half-lives in 136 Xe and 76 Ge for different matrix element calculations and particle physics contribution due to light neutrino exchange while the right-panel is for heavy neutrino exchange.  Figure 18. The left-panel shows the correlations between the 0νββ half-lives in 136 Xe and 76 Ge for different matrix element calculations and particle physics contribution due to λ-diagram while the right-panel is for η−diagram.

Conclusion
We have discussed natural realization of type-II seesaw dominance within a class of TeV scale left-right symmetric models where scalar sector comprises of scalar doublets H L,R , triplets ∆ L,R and a bidoublet Φ, the fermion sector consists of usual quarks q L,R , leptons L,R plus one copy of extra sterile fermion S L per generation. In order to achieve natural type-II seesaw dominance, we have considered negligible VEV for LH scalar doublet i.  M D is the Dirac mass term connecting light-heavy neutrino and M is the N − S mixing matrix. We have also discussed that the type-II seesaw dominance allows any value for M D and thus, new physics contributions to 0νββ transition arise from the following channels; i) due to purely left-handed currents via exchange of heavy RH Majorana neutrinos N and extra sterile neutrinos S, and ii) due to so called λ and η type of diagrams. We have also demonstrated the effect of right-handed currents to 0νββ transition via W R − W R mediation.
Most importantly we have expressed all the physical masses and mixing like ν L , N R and S L which are completely Majorana in nature mediating LNV processes like neutrinoless double beta decay in terms of oscillation parameters and mass of lightest neutrino with the assumption that N − S mixing matrix is diagonal and degenerate. We have demonstrated that large value of Dirac neutrino mass possibly originating from high scale Pati-Salam symmetry or SO(10) GUT plays an important role in resulting dominant contributions to these new non-standard 0νββ transition. With the model parameters like M W R = 3 TeV, M N O(TeV), M ∆ O(TeV), M D as up-type quark mass matrix and using oscillation parameters, we numerically estimated new physics contributions to 0νββ transition and compared it with that of the standard mechanism. We have derived the lower limit on absolute scale of lightest neutrino mass by numerically estimating various new physics contributions to 0νββ transition by saturating the current experimental limit. We have shown that NH is favored over IH pattern of light neutrinos resulting from effective mass as a function of light neutrino mass sum taking into account the cosmological data. We have also presented correlation plots between half-lives for 76 Ge and 136 Xe isotopes showing the uncertainties in the nuclear matrix elements.
Note added: During the finalization of this work, another work appeared on arXiv [96] that discusses type-II seesaw dominance in LRSM. However, our work differs widely by expressing all the physical masses and mixing of heavy neutrinos in terms of oscillation parameters and lightest neutrino mass. Thus, one can get an important information about absolute scale of lightest neutrino mass and mass hierarchy from new physics contributions to neutrinoless double beta decay by saturating the current experimental limit.