A comparative study of 0νββ decay in symmetric and asymmetric left-right model

We study the new physics contributions to neutrinoless double beta decay (0νββ) in a TeV scale leftright model with spontaneous D-parity breaking mechanism where the values of the SU(2)L and SU(2)R gauge couplings, gL and gR are unequal. Neutrino mass is generated in the model via gauge extended inverse seesaw mechanism. We embed the model in a non-supersymmetric SO(10) GUT with a purpose of quantifying the results due to the condition gL 6= gR. We compare the predicted numerical values of half life of 0νββ decay, effective Majorana mass parameter and other lepton number violating parameters for three different cases; (i) for manifest left-right symmetric model (gL = gR), (ii) for left-right model with spontaneous D parity breaking (gL 6= gR), (iii) for Pati-Salam symmetry with D parity breaking (gL 6= gR). We show how different contributions to 0νββ decay are suppressed or enhanced depending upon the values of the ratio gR gL that are predicted from successful gauge coupling unification.


I. INTRODUCTION
The immediate question that followed the discovery of neutrino mass and mixing by oscillation experiments [1][2][3][4][5][6][7][8][9] and still remains unanswered is : 'Whether neutrinos are Dirac or Majorana particles?' Even more gripping is the question, 'What gives them such a tiny mass?', since it is believed Higgs mechanism can't be the one responsible. The seesaw mechanism [10][11][12][13][14][15][16][17][18][19] which is the minimal approach to explain non-zero neutrino mass presumes them as Majorana fermions. If neutrinos are Majorana fermions [20] they can initiate a very rare process in nature called neutrinoless double beta decay (0νββ): A Z X → A Z+2 Y + 2e − , which clearly violates lepton number by two units [21]. Therefore this process if observed unambiguously can confirm the Majorana nature of neutrinos and total lepton number violation in nature. The detection of this rare phenomena is the main aim of several ongoing experiments that are trying to put a bound on the half life of particular nuclei from which limits on the effective Majorana mass can be obtained easily.
At present, KamLAND-Zen experiment gives the bound on half-life as T 0ν 1/2 > 1.6 × 10 26 yrs using 136 Xe [22] while GERDA gives T 0ν 1/2 > 8.0 × 10 25 yrs at 90 % C.L. using 76 Ge [23]. Translating these limits into effective mass bound it turns out to be 0.26 − 0.6 eV, whereas the Planck collaboration puts a tight limit on the sum of light neutrino masses to be ≤ 0.23 eV at 95% C.L. [24] and as per KATRIN the bound on lightest neutrino mass, m β < 0.2 eV [25]. Which means any positive signal of 0νββ decay at the experiments would definitely indicate some new physics contribution to the process.
However a different scenario arises when the discrete parity symmetry (D-parity) of a left-right symmetric theory breaks at a high scale and the local SU (2) R symmetry breaks at relatively low scale [51,52]. This decoupling of D parity breaking and SU (2) R symmetry breaking introduces a new scale and as an immediate effect, the gauge couplings for SU (2) L and SU (2) R gauge groups become unequal, i.e. g L = g R . In ref. [53] a TeV scale left-right model with D-parity breaking has been studied and in ref. [54] such a model has been embedded in a non-SUSY SO(10) GUT with Pati-Salam symmetry as the highest intermediate breaking step. The same idea has been extended in ref. [55] to study baryon asymmetry of the universe, neutron-antineutron oscillation and proton decay.
But the effect of g L = g R has not been emphasized in these above mentioned works while studying 0νββ decay. This deviation (g L = g R ) brings a noticeable difference in the 0νββ decay sector which is the main essence of this work. We show how different contributions to 0νββ decay are suppressed or enhanced depending upon the values of the ratio g R g L that appears in Feynman amplitudes and g L g R that appears in half-life estimation. In order to quantify the results due to unequal g L and g R we consider two different symmetry breaking chains from non-SUSY SO(10) GUT; one with Pati-Salam symmetry as the highest intermediate step, another without Pati-Salam symmetry [56]. The importance of Pati-Salam symmetry as the highest intermediate step in a SO(10) symmetry breaking chain has already been discussed in ref [52,57,58].
The rest of the paper is organised as follows. We start our discussion with the basic differences between generic LRSM and asymmetric LRSM and present the symmetry breaking steps in section II. Next in section III we explain how neutrino mass is generated via low scale extended inverse seesaw mechanism in the model. In section IV we discuss the role of Pati-Salam symmetry in predicting halflife of 0νββ decay and other LNV parameters and show the gauge coupling unifications. We perform the numerical estimation of different 0νββ contributions in section V followed by a comparative study of the process in symmetric and asymmetric left-right case in section VI. We summerize our results and conclude in section VII. In appendix A we discuss the new physics contributions to 0νββ decay process that arise in a left-right model with spontaneous D parity breaking. We derive upper limits for different lepton number violating parameters in appendix B.

II. LEFT-RIGHT MODEL WITH SPONTANEOUS D-PARITY BREAKING
In this section we discuss the properties of left-right model with spontaneous D-parity breaking mechanism and state how it differs from the manifest left-right model.

A. Symmetric left-right model
The symmetric left-right model is based on the gauge group SU (2) L ×SU (2) R ×U (1) B−L ×SU (3) C plus a discrete left-right parity symmetry. In these models, the discrete parity and SU (2) R gauge symmetry break simultaneously and thus the gauge couplings for SU (2) L and SU (2) R gauge groups remain equal i.e, g L = g R . The fermions and Higgs scalars are related in this model as, The symmetry breaking of the left-right symmetric theory down to Standard Model and further to G 13 ≡ U (1) Y × SU (3) C is achieved by the scalars ∆ R,L and Φ respectively. For a more detailed discussion on spontaneous symmetry breaking of left-right symmetry one may refer [31]. The vacuum expectation values (vev) of the scalars are given by, The invariant Yukawa Lagrangian under the symmetric left-right theory is withφ ≡ τ 2 φ * τ 2 . The discrete left-right symmetry also results in equal Majorana couplings for left-handed and right-handed neutrinos. With these Yukawa terms the neutrino mass formula can be written as, where M D is the Dirac neutrino mass matrix, the Majorana mass term for left-handed (right-handed) neutrinos.
The seesaw relation in this case is found (from minimization of scalar potential consists of Φ, ∆ L,R ) to be, where k = k 2 1 + k 2 2 and γ represents the function of various scalar coupling parameters in potential. This means if one assign low mass to ν L i.e. around eV scale, then ν R has to lie at a very heavy scale, say 10 13 to 10 14 GeV which is well beyond the reach of current experiments. In order to bring down the right-handed scale to TeV, parity and SU (2) R have to be broken at TeV scale and also the Higgs couplings have to be finetuned to order of γ ≤ O(10 −10 ).

B. Asymmetric left-right model
Left-right theory with spontaneous D-parity breaking mechanism [51,52] is based on the idea that the discrete left-right symmetry (D parity) breaking scale and local SU (2) R breaking scale are decoupled from each other, i.e, D-parity breaks at an earlier stage as compared to SU (2) R gauge symmetry, thereby introducing a new scale. It should be noted here that, D-parity should not be confused with the Lorentz parity as latter one acts only on the fermionic content of the theory while D-parity interchanges the parity of the fermion as well as the SU (2) L ×SU (2) R Higgs fields. It results in an asymmetric LR model for which the SU (2) L and SU (2) R gauge couplings become unequal i.e, g L = g R . This spontaneous breaking of D-parity occurs when singlet scalar σ takes vev which is odd under D-parity. The asymmetric left-right model then breaks down to SM symmetry with the help of right-handed triplet Higgs scalar ∆ R . Further SM symmetry breaks down to U (1) em theory with the help of scalar bidoublet Φ. The complete symmetry breaking can be sketched as follows, As SU (2) R breaking scale and parity breaking scale are different, there is no effect of left-handed scalar ∆ L at low energy and the fermion masses can be derived from the Yukawa Lagrangian.
However, one can write an induced VEV for ∆ L from the seesaw relation as where M P is D-parity breaking scale, β is a Higgs coupling constant of O(1) and M M P .
This asymmetric left-right gauge theory can also emerge from high scale Pati-Salam theory having Here In manifest left-right symmetric models where neutrino mass is generated via type-I+type-II seesaw mechanism [40,41,44,45,48,59,60], one has to add either extra symmetry or do structural cancellation in order to align the right handed scale at TeV range. However the canonical seesaw contribution can be exactly cancelled out in case of extended type-II seesaw [61] or inverse seesaw mechanism [62] and a large value of Dirac neutrino mass matrix, M D can be obtained. These choices of seesaw allow large light-heavy neutrino mixing which facilitate rich phenomenology. However, generic inverse seesaw mechanism as proposed in ref. [63,64] gives negligible contribution to neutrinoless double beta decay as the associated sterile neutrino mass matrix µ S lies in keV range to account for neutrino mass mechanism. Thus one has to extend the inverse seesaw mechanism with a large lepton number violating parameter in N − N sector as M N while keeping the same keV value of µ S in the S − S sector. Hence, the corresponding seesaw mechanism is termed as "extended inverse seesaw mechanism (EISS) " where the neutrino mass is governed by the standard inverse seesaw formula.
Henceforward we consider the model discussed in ref. [54] for our comparative study throughout the paper. In this model, gauged inverse seesaw mechanism is implemented by adding one extra fermion singlet S i (i = 1, 2, 3) per fermion generation. The extended seesaw mechanism is further gauged at TeV scale for which the VEV of the RH-doublet χ 0 R = v χ provides the N − S mixing matrix M .
The asymmetric low-scale Yukawa Lagrangian can be written as, which gives rise to the 9 × 9 neutral lepton mass matrix in the basis of ν N S T after electroweak symmetry breaking The Dirac neutrino mass matrix M D is determined from the high scale symmetry and fits to charged fermion masses at GUT scale using RG evolution equations. In principle the N − S mixing matrix M can assume any arbitrary form though we have taken it as diagonal. We have also treated the heavy RH Majorana neutrino mass matrix M R to be diagonal throughout this work. One essential outcome of extended inverse seesaw mechanism is that type-I seesaw contribution is exactly canceled out, type-II contribution is also damped out and thus inverse seesaw is the only viable contribution to light neutrino masses The heavy sterile neutrinos and heavy RH neutrinos with their mass matrices can be noted as, As stated earlier, these block diagonal mass matrices m ν , M S and M N can further be diagonalized to give physical masses to all neutral leptons by respective unitary mixing matrices: The complete mixing matrix [54,[65][66][67] diagonalizing the resulting 9 × 9 neutrino mass matrix turns out to be where the symbols are expressed in terms of model parameters as N , and y = M −1 µ S . With this mixing matrix, one can write the relevant charged current interactions of leptons valid at TeV scale asymmetric LR gauge theory (with g L = g R ) in the flavor basis as The flavor eigenstates can be expressed in terms of mass eigenstates (ν i , S j , N k ) as where i, j, k = 1, 2, 3.

IV. THE ROLE OF PATI-SALAM SYMMETRY
We know that both the gauge couplings for SU (2) L and SU (2) R are exactly equal at a scale when as an intermediate symmetry breaking step. This equality is sustained as long as D-parity remains unbroken. Once the spontaneous breaking of D-parity occurs, it immediately results in g L = g R and the ratio g L g R deviates from unity depending upon the breaking scale. In the considered model we have found this ratio g L g R to be 1.5 which will be supportive in predicting new non-standard  (10) symmetry breaking chain has already been discussed in ref [57]. For quantifying these points, we consider the following non-SUSY SO(10) chain, as an example.
It was found that the G 224 -singlets contained in {54}  (1, 1, 1) ⊂ {210} H to obtain asymmetric G 224 with g L = g R . Then the spontaneous breaking of Finally, as usual, the breaking of SM to low energy theory U (1) em × SU (3) C is carried out by the

A. Gauge coupling unification
We consider three different cases for gauge coupling unification as follows and we also show the Higgs spectrum used in different ranges of mass scales under respective gauge symmetries.
Now, we have introduced the Pati-Salam symmetry in the SO (10)  Case -IIIA : Case -IIIB : The gauge coupling unification plots for the above four cases are shown in Fig. 1 and Fig. 2 respec- tively. In the unification plots the different colored lines stand for running of various gauge groups.
The red, blue, pink, magenta and green lines are for SU gauge groups respectively. For case-IIIB we have added an extra particle ξ(2, 2, 15) which helps us to unify the SU (2) L and SU (2) R gauge couplings at around 10 16 GeV (i.e, the D-parity breaking scale of G 224D ), also this extension of the model gives us the advantage to acquire fermion mass fitting at GUT scale.
For the four diferent cases the range for g R is tabulated in table I. However, for the calculations in the rest of the paper we consider three cases I, II and III since for case IIIA and IIIB the values of δ = g R g L are nearly equal.  We omit a detailed discussion on fermion mass fitting at GUT scale and derivation of M D , M N at TeV scale since this has been already done in ref. [54]. We simply use the derived model parameters of ref. [54] and extend the numerical estimation of non-standard contributions to 0νββ decay. Our major aim is to elucidate how unequal couplings i.e g L = g R enhance the rate of 0νββ transition in Here repectively.
Another key parameter is the mixing matrix for light and heavy RH Majorana neutrinos which is estimated to be

VII. RESULTS AND CONCLUSION
One important outcome of extended inverse seesaw scheme is that type-I seesaw contribution is exactly canceled out thereby allowing the possibility of large left-right mixing in the neutrino

Half-life
Half-life Enhancement Factor  and g R 0.39.
We have shown various plots to infer how half-life of 0νββ decay due to different channels varies with the ratio g R g L i.e. δ and mass of W R . From Fig.3 we see that the cyan shaded region is sensitive to the current KamLAND-Zen and GERDA bounds. Since the value of the ratio g R g L ranges from 0.62 to 1 for three different cases considered in the work, the plot shows only the contributions from W L − W L channel due to light neutrino exchange and from W R − W R channel due to heavy neutrino exchange lie within the priviledged region. Fig.4 shows how the effective Majorana mass parameter varies with the ratio δ. Only non-trivial M W R dependence occurs for the contribution arise from W R − W R mediation with RH neutrino exchange as well as from W L − W R mediation (λ-contribution). So, Fig.5 shows how the effective Majorana mass parameter due to these two decay channels vary with the mass of W R and the variation of half life with W R mass has been presented in Fig.6. For Fig. 5 and 6, the mass range for W R has been considered here as, M W R ∈ [1, 7] TeV for better transperancy.

VIII. ACKNOWLEDGEMENTS
SS is thankful to UGC for fellowship grant to support her research work. The authors thank Prof. Urjit A. Yajnik for useful comments and discussion.
Appendix A: Predictions on neutrinoless double beta decay in LR model with Spontaneous

D-parity breaking
The importance of neutrinoless double beta decay process in particle physics is far-reaching in the sense that it is one such process which can confirm the Majorana nature of neutrino and also provide information about the absolute scale of light neutrino mass. Neutrinoless Double Beta Decay can be induced by the exchange of a light Majorana neutrino, which is called the standard mechanism or by some other lepton number violating physics which is called the non-standard interpretation [31,64,66,[72][73][74][75][76][77][78][79] . In the standard mechanism the parent nucleus emits a pair of virtual W bosons, and then these exchange a Majorana neutrino to produce the outgoing electrons.
At the vertex where it is emitted, the exchanged neutrino is created, in association with an electron, as an antineutrino with almost total positive helicity, and only its small, O(m ν /E), negative-helicity component is absorbed at the other vertex. In LRSM the process can be mediated by heavy righthanded neutrino and some new channels can also appear due to left-right mixing,i.e. W L − W R mixing. In the considered model many diagrams are possible due to the presence of heavy neutrinos S, N , doubly charged higgs scalar ∆ R and W L − W R mixing. We will discuss that in this section, but we start by writing the charged current interaction Lagrangian for the model in flavor basis.
Since we have considered that the left-handed and right-handed charged gauge bosons mix with each other the physical gauge bosons can be expressed as a linear combinations of W L and W R as, with mixing angle ξ, we have Different types of Feynman diagrams contributing to the 0νββ process are [54] shown in Fig 7. (i) Feynman diagrams in W − L − W − L channel (with two left-handed currents) (v) Neutrino and W L exchanges with Dirac mass helicity flip and W L −W R mixing in the W L −W R channel (η mechanism) Now, let's write the amplitudes for these processes and the corresponding particle physics parameter involving lepton number violation. The Feynman amplitude for the processes having both left-handed electrons is proportional to where, m i , M S i , and M N i are the masses of light neutrino ν and heavy neutrinos S and N respectively and tan ξ represents left-right gauge boson mixing. The diagrams are separately shown Similarly, the Feynman amplitudes for the processes involving W − R −W − R mediation via exchanges of either light or heavy neutrinos where both the emitted electrons are right-handed is proportional to, The suitably normalized dimensionless parameters that describe lepton number violation are 2. Triplet exchange mechanisms Fig 7 (iii) is mediated by SU (2) R scalar triplet ∆ R and for this the amplitude is given by and the dimensionless particle physics parameter is (A8)

Momentum dependent mechanisms
In this case the emitted electrons have opposite helicity, and the amplitude is proportional to (A9) and corresponding dimensionless particle physics parameter involving lepton number violation are where, m e (m i )= mass of electron (light neutrino), and m p = mass of proton. Besides different particle physics parameters, it contains the nuclear matrix elements due to different chiralities of the hadronic weak currents such as (M 0ν ν ) involving left-left chirality in the standard contribution, and (M 0ν ν ) involving right-right chirality arising out of heavy neutrino exchange, (M 0ν λ ) for the λ diagram, and M 0ν η for the η diagram. It is to be noted here that the current bound on these LNV parameters are derived based on half-life limit from the KamLAND-Zen experiment neglecting interference terms.