Phenomenological Study of Texture Zeros in Lepton Mass Matrices of Minimal Left-Right Symmetric Model

We consider the possibility of texture zeros in lepton mass matrices of minimal left-right symmetric model (LRSM) where light neutrino mass arises from a combination of type I and type II seesaw mechanisms. Based on the allowed texture zeros in light neutrino mass matrix from neutrino and cosmology data, we make a list of all possible allowed and disallowed texture zeros in Dirac and heavy neutrino mass matrices which appear in type I and type II seesaw terms of LRSM. For the numerical analysis we consider those cases with maximum possible texture zeros in light neutrino mass matrix $M_{\nu}$, Dirac neutrino mass matrix $M_D$, heavy neutrino mass matrix $M_{RR}$ while keeping the determinant of $M_{RR}$ non-vanishing, in order to use the standard type I seesaw formula. The possibility of maximum zeros reduces the free parameters of the model making it more predictive. We then compute the new physics contributions to rare decay processes like neutrinoless double beta decay, charged lepton flavour violation. We find that even for a conservative lower limit on left-right symmetry scale corresponding to heavy charged gauge boson mass 4.5 TeV, in agreement with collider bounds, for right handed neutrino masses above 1 GeV, the new physics contributions to these rare decay processes can saturate the corresponding experimental bound.


I. INTRODUCTION
The fact that neutrinos have non-zero but tiny masses and large mixing has been well established by several neutrino experiments [1-9] during the last two decades. For a review of neutrino mass and mixing, please see [10,11]. Among the above mentioned experiments, the relatively recent ones like T2K [5], Double Chooz [6], Daya Bay [7], RENO [8] and MI-NOS [9] experiments have not only confirmed the results from earlier experiments but also discovered the non-zero reactor mixing angle θ 13 . For a recent global fit of neutrino oscillation data, we refer to [12,13]. The latest global fit shows that a few details of the light neutrinos are yet to be determined experimentally. They are namely, the Dirac CP phase, octant of atmospheric mixing angle and the hierarchy of light neutrinos: normal hierarchy (NH) or inverted hierarchy (IH). Also, the nature of neutrinos (Dirac or Majorana) remains unknown at oscillation experiments. If neutrinos are Majorana fermions, there arises two more CP phases known as Majorana CP phases, which can not be determined by oscillation experiments and have to be probed at alternative experiments. Apart from neutrino oscillation experiments, the neutrino sector is constrained by the data from cosmology as well. For example, the latest data from the Planck mission constrain the sum of absolute neutrino masses i |m i | < 0.12 eV [14].
Although we have significant experimental observations related to neutrino sector except the above mentioned unknowns, the dynamical origin of light neutrino masses and their mixing is still a mystery. The standard model (SM) of particle physics, which gives a successful description of all fundamental particles and their interactions (except gravity) can not explain the lightness of neutrinos. The Higgs field in the SM which is responsible for generating masses to all known particles do not have coupling to neutrinos as the right handed (RH) neutrinos are absent. One can generate a light Majorana mass term for light neutrinos in the SM through the dimension five Weinberg operator [15] of type (LLHH)/Λ with the introduction of an unknown cutoff scale Λ. Several beyond standard model (BSM) proposals have been put forward which can provide a dynamical origin of such operators in a renormalisable theory. This is typically achieved in the context of seesaw models where a hierarchy or seesaw between electroweak scale and the scale of newly introduced fields decide the smallness of neutrino masses. Popular seesaw models can be categorised as type I seesaw [16][17][18][19], type II seesaw [20][21][22][23][24], type III seesaw [25] among others like [26,27].
One very popular BSM scenario is the framework of left-right symmetric model (LRSM) [28][29][30][31][32][33][34][35][36][37][38] where the gauge symmetry of the SM is extended to SU(3) c × SU(2) L × SU(2) R × U(1) B−L so that the right handed fermions (which are singlet in SM) can form doublets under the new SU (2) R . This not only makes the inclusion of right handed neutrino automatic, but also puts the left and right handed fermions on equal footing. If we also incorporate an additional discrete left-right symmetry to ensure that the theory is invariant under SU (2) L ↔ SU (2) R . So the model can explain the origin of parity violation in weak interaction by considering a parity symmetric theory at high energy scale where the corresponding gauge symmetry breaks spontaneously leading to the parity violating SM at low energy. In the minimal LRSM, the light neutrino masses arise naturally from a combination of type I and type II seesaw. The gauge symmetry as well as the particle content of minimal LRSM can also be accommodated within popular grand unified theory (GUT) models like SO (10). Apart from these, another interesting motivation for this model is its verifiability.
A TeV scale LRSM can have very interesting signatures which are being looked at colliders [39][40][41][42][43]. There also exists different other phenomenological consequences which can be probed at experiments in both energy as well as intensity frontiers.
Typical seesaw models in the absence of specific flavour symmetries usually predict a very general structure of light neutrino mass matrix which can always be fitted to the observed data due to the presence of many free parameters. The same is true in LRSM as well. However, if the theory has a well motivated underlying symmetry that gives rise to a very specific structure of neutrino mass matrix, then number of free parameters can be significantly reduced. In such a case, we can have very specific predictions for light neutrino parameters like CP phase, octant of atmospheric mixing angle, mass hierarchy which can tested at ongoing experiments. Here we consider such a possibility where an underlying symmetry can restrict the mass matrix to have non-zero entries only at certain specific locations. Such scenarios are more popularly known as zero texture models, a nice summary of which within three neutrino framework can be found in the review article [44] 1 . In the diagonal charged lepton basis, if the light neutrino mass matrix has some textures, the corresponding constraints can be solved to find the light neutrino parameter space that satisfy them. Depending on the viability of this parameter space in view of latest neutrino oscillation data, one can discriminate between different textures. Also, the allowed textures often predict non-trivial values for unknown parameters that can be tested at different experiments. It has already been shown in earlier works that in the diagonal charged lepton basis, not more than two zeros are allowed in the light neutrino mass matrix. While all six possible one zero texture ( 6 C n , n = 1) are allowed, among the fifteen possible two zero textures, only six were found to be allowed after incorporating both neutrino as well as cosmology data [50][51][52][53][54][55]. Since in LRSM, several mass matrices play a role in generating light neutrino mass matrix due to the combination of type I and type II seesaw, the requirement of getting the allowed texture zeros in light neutrino mass matrix can constrain the texture zeros of all other mass matrices in the lepton sector namely, the Dirac neutrino mass M D and heavy neutrino mass M RR . Making a list of all these possibilities while classifying the allowed and disallowed ones is the primary goal of this work 2 . We not only make such a list considering all possibilities of texture zeros, but also perform a numerical analysis for one zero and two zero light neutrino textures as well as a scenario where other mass matrices involved in the seesaw can have maximum number of zeros. To be more specific, for our numerical analysis, we considered five zero textures in M D and four zero textures in M RR , keeping the rank of the latter three. Out of 378 total possibilities belonging to this list, we find that 189 are allowed from light neutrino data, out of which 109 give rise to two zero textures in light neutrino mass matrix. The case for maximum number of zeros is particularly chosen due to their more predictive nature. We not only find the correlations among light neutrino parameters, but also find the new physics contribution to other interesting processes like neutrinoless double beta decay (NDBD) and charged lepton flavour violation (CLFV). As these processes are being probed at several experiments, this study points out the possibility of probing such scenarios at those experiments. Such aspects of probing LRSM can be complementary to the ongoing collider searches mentioned earlier.
This paper is organised as follows. In section II, we review the LRSM with its particle content and mass spectrum followed by the details of the texture structures of the Dirac and Majorana mass matrices in section III. We then summarise the contributions to NDBD and CLFV in LRSM in section IV, V respectively. We discuss our numerical analysis and results in section VI and then finally conclude in section VII.
2 Please see [56][57][58] and references therein for texture zero works in 3 + 1 neutrino scenarios and [59] for related phenomenological study of texture zeros in all relevant lepton mass matrices of a particular seesaw model.

II. MINIMAL LEFT-RIGHT SYMMETRIC MODEL
As mentioned before, the left-right symmetric model is a very well motivated and widely studied extension of the SM with an enlarged gauge symmetry based on SU (3) [28][29][30][31][32][33][34][35][36][37][38]. The theory removes the disparity between left and right handed fields by considering the right handed fields to be doublet under the additional SU (2) R keeping the right sector couplings same as the left one by left-right symmetry. Therefore, the fermion field content of the minimal LRSM can be written as where the numbers in brackets represent the quantum numbers under the the gauge group The Higgs sector of the minimal LRSM consists of two SU (2) L triplets ∆ L,R and a bi-doublet φ given by with the quantum numbers Φ(1, 2, 2, 0) and ∆ L (1, 3, 1, 2), ∆ R (1, 1, 3, 2) respectively.
The relevant Yukawa Lagrangian giving masses to the three generations of leptons is given by, where the indices i, j = 1, 2, 3 represent the family indices for the three generations of fermions. C = iγ 2 γ 0 is the charge conjugation operator, Φ = τ 2 φ * τ 2 and γ µ , τ 2 are the Dirac and Pauli matrices respectively. Discrete left-right symmetry ensures the equality of Majorana Yukawa couplings f L = f R apart from the equality of gauge couplings of SU (2) L,R sectors g L = g R . The scalar potential V scalar is given by where we have introduced scalar mass parameters µ i and quartic scalar interaction strengths λ i , ρ i , α i and β i . In the symmetry breaking pattern, the neutral component of the Higgs triplet ∆ R acquires a vacuum expectation value (VEV) to break the gauge symmetry of the LRSM into that of the SM and then to the U (1) of electromagnetism by the VEV of the neutral components of Higgs bidoublet Φ: The VEVs of the neutral components of the Higgs fields can be denoted as where the VEV's k 1 , k 2 satisfy the VEV of the SM namely, v SM = k 2 1 + k 2 2 ≈ 246 GeV. The VEV v L which plays a significant role in neutrino mass mechanism is generated after the electroweak symmetry breaking due to the following induced VEV relation Here, γ is a dimensionless parameter given by [37] In order to satisfy the electroweak precision test constraints, v L should be smaller than 2 GeV [60], and the above breaking pattern of gauge symmetry enforces v R to be much greater than k 1,2 .
The 6 × 6 neutrino mass matrix is then given, in the (ν L , ν R ) gauge eigenbasis, by Assuming M LL M D M R , the light neutrino mass after symmetry breaking is generated within a type I+II seesaw as, M D , M LL and M RR being the Dirac neutrino mass matrix, left handed and right handed Majorana mass matrix respectively. The first and second terms in equation (12) correspond to type II seesaw and type I seesaw contributions respectively.
The 6 × 6 neutral lepton mass matrix can be digonalised by a 6 × 6 unitary matrix, as follows, where with M i being the heavy right handed neutrino masses. V is thus represented as, where, R describes the left-right mixing and given by, The matrices U, V, S and T are as follows, The gauge boson mass spectra can be found similarly. The left-right gauge boson mixing with the mixing parameter ξ represented by Without any loss of generality, we make use of a rotation in the SU (2) L × SU (2) R space so that only one of the neutral components of the Higgs bidoublet acquires a large vacuum expectation value, k 1 ≈ v SM and k 2 ≈ 0. This corresponds to negligible mixing ξ.
Under those assumption, we neglect all contributions to the gauge boson masses that are proportional to v L , so that these masses approximatively read with θ W indicating the weak mixing angle.
Under these assumptions, the Dirac neutrino mass matrix is while the charged lepton mass matrix is which points out the freedom in choosing M l and M D as we do in the subsequent sections.

III. TEXTURE ZEROS IN LEPTON MASS MATRICES OF LRSM
As mentioned earlier, texture zeros in lepton mass matrices increase the predictive power have nine independent elements so that n texture zeros can have 9 C n possibilities. On the other hand M RR , being complex symmetric can have six independent elements will have 6 C n possibilities for n texture zeros. While finding texture zeros in M RR we however make sure that the determinant is non-zero so that the type I seesaw formula can be applied. We classify these texture zero possibilities as follows.
• The different classes of 4-0 texture M RR with non zero determinant are: • The different classes of 3-0 texture M RR with non zero determinant are: • The different classes of 2-0 texture M RR with non zero determinant are: • The different classes of 1-0 texture M RR with non zero determinant are: The different number of allowed texture structures obtained for the various combinations mass matrix with the highest number of zeros, i.e 4-0 texture M RR as given by equation 24. Similarly, we will consider M D with 5 zeros (maximum) which can phenomenologically provide the allowed zero textures in the light neutrino mass matrix. Furthermore from table II, we will take into consideration only the allowed cases of two texture zero structures of light neutrino mass matrix. Out of total of 6 C 2 i.e., 15 two texture zeros of ν mass matrix, 6 are totally allowed by neutrino and cosmology data. It should be noted that these conclusions 1  20  27  6  48  23  2  126   2  20  27  6  51  20  2  126   3  22  55  6  21  20  2  126   TABLE II hold for diagonal charged lepton basis which we also adopt in our analysis. These allowed two zero texture light neutrino mass matrices are given as where × denotes any non-zero entry.
• For the class A2 (M ee = 0, M eτ = 0) • For the class B3 (M eµ = 0, M µµ = 0) • For the class B4 (M µµ = 0, M τ τ = 0) Phenomenological implications of two texture zero M ν on low energy phenomena like NDBD and CLFV have been analysed in one of our earlier work [55]. However, in that case, we have considered the two zero texture mass matrix to be favouring a tri-maximal mixing pattern.
Further, the Dirac mass matrix was considered to be diagonal and the RH Majorana mass matrix to be favouring the corresponding two zero texture. Besides, all the contributions to NDBD that could arise in the framework of LRSM was not taken into consideration.
Here we generalise this to consider maximum allowed texture zeros that is has reported a strong lower limit on the half life from searches on 1 36Xe as T 0ν 1/2 > 1.07×10 26 year at 90% C. L. This can be translated to an upper limit of effective Majorana mass in the range (0.061 − 0.165) eV where the uncertainty arises due to the NME. We show all the contributions to NDBD in minimal LRSM in terms of corresponding Feynman diagrams in figure 1, 2, 3, 4. We now list their respective contributions below one by one following the notations of [83].
• When light and heavy neutrinos are the source of NDBD mediated by purely left handed (LH) currents (W L − W L ) as shown in figure 1, the corresponding amplitudes are given by, • The right handed current mediated by W R can contribute to NDBD through the exchange of the light as well as heavy neutrino N (as shown in figure 2). The corresponding amplitudes are given by, where, M W L and M W R are the mass of the LH and RH gauge bosons respectively.
• Significant Contribution can also arise due to the mixed helicity diagrams, mediated by both W L and W R (λ contribution) and from diagrams mediated by W L −W R mixing (η contribution), the amplitudes of which are given as, where ξ is the L-R gauge boson mixing parameter as described earlier.
• Further, there is also the scalar triplet (∆ L,R ) contributions to NDBD by the mediations of W L and W R gauge bosons respectively, the amplitude of which depends upon the masses of these gauge bosons and given by, where the contribution from left triplet scalar is negligible due to smallness of v L as well as the smallness of light neutrino mass contribution coming from type II seesaw.
The particle physics parameters governing NDBD for the different contributions (ignoring the left triplet Higgs contribution) in LRSM we have considered are given by, In the above equations, m p and m e are the mass of the proton and electron respectively. It is seen that the amplitudes of these processes are mostly dependent on the mixing between neutrinos, the mass of the heavy neutrinos, N i , the mass of the gauge bosons, W L − and W R − , mass of doubly charged scalars triplet Higgs, ∆ L and ∆ R as well as their coupling to leptons, f L and f R . The total analytic expression for the inverse half life governing NDBD considering all the dominant contributions that could arise in LRSM is given by, In the above expression, G 0ν (Q, Z) represents the phase space factor and M 0ν is the nuclear matrix element which have different values for different contributions which is shown in tabular form in table III [82].

V. CHARGED LEPTON FLAVOUR VIOLATION IN LRSM
Charged lepton flavour violation arises in the SM at one loop level and remains suppressed by the smallness of neutrino masses, much beyond the current and near future experimental sensitivities. Therefore, any experimental observation of such processes is definitely a sign of BSM physics, like the one we are studying here. For a review of CLFV in SM and beyond, please refer to [86]. Though usual light neutrino contribution to CLFV is negligible, presence of heavy neutrinos in BSM frameworks can give rise to observable CLFV [78,81,83,[87][88][89][90][91][92][93][94]. In LRSM, sizeable CLFV occurs dominantly due to the contributions arising from the additional scalars and the heavy neutrinos. Among the various processes that violate lepton flavour, the most relevant ones are the rare leptonic decay modes of the muon, notably, (µ → eγ) and (µ → 3e). The best upper limit for the branching ratio (BR) of these processes are provided by MEG collaboration [95] and SINDRUM experiment [96] which provide the corresponding upper limit as BR (µ → eγ) < 4.2 × 10 −13 and BR (µ → 3e) < 1.0 × 10 −12 respectively.
Adopting the notations of [78,83] the branching ratio of the process µ → 3e mediated by doubly charged scalars can be written as where h ij describes the respective lepton-scalar couplings given by, with V being one of the lepton mixing matrices given in (17).
The branching ratio for the CLFV process µ → eγ is given by (as explained in [83]), where, α em is the fine structure constant defined as α em = e 2 4Π , G γ L and G γ R are the form factors given by, V is the mixing matrix of the right handed neutrinos given in (17). ζ is the phase of the VEV k 2 which we consider to be negligible, whereas the left-right gauge boson mixing parameter, ξ is also very small 10 −6 in our case. S being the light-heavy neutrino mixing as defined in 18. Again the loop functions G γ 1,2 (a) are defined as, Recently the MEG collaboration has reported a new stringent upper bound on the decay rate of the process µ → eγ. The BR ratio for this LFV process as given by MEG is < 4.2 × 10 −13 at 90% CL [95]. While for the process µ → 3e it is < 1.0 × 10 −12 as obtained by the SINDRUM experiment [96].

VI. NUMERICAL ANALYSIS AND RESULTS
For our numerical analysis, we first parametrise the light neutrino mass matrix in terms of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix which is related to the diagonalising matrices of neutrino and charged lepton mass matrices U ν , U l respectively, as The PMNS mixing matrix can be parametrised as namely, m lightest = m 1 (NH)(m 3 (IH)), δ, α, β which are not measured yet, can be predicted by the texture zero conditions. This is possible in two zero texture cases particularly, because of two texture zero conditions which give rise to four real equations that can be solved simultaneously to find four unknown parameters. We vary the five known parameters randomly in the 3σ range using the recent global fit [13]. Using the latest data, we found that out of the previously allowed six possible two zero textures, A2 for both NH and IH and A1 (IH) are disallowed. We consider the allowed ones for our analysis for NDBD and CLFV.
For representative purpose, we show some correlations between light neutrino parameters coming out from the two zero texture conditions in figure 5, 6, 7. Similar correlation plots were obtained in earlier work [54].
In minimal LRSM, the neutrino mass is given by equation (11) where the first and second terms represents the type II and type seesaw mass terms respectively. γ is the dimensionless parameter that appears from minimisation of the scalar potential, defined before. We have fine tuned the dimensionless parameter γ = 10 −9 with a view to obtaining the neutrino mass of the order of sub eV. This is chosen particularly to keep the right handed neutrino masses in the desired range. The right handed neutrino mass matrix, defined earlier, is The choice of v R for a few TeV W R mass, and type II seesaw term at sub-eV scale, the chosen value of γ keeps the right handed neutrino mass above 1 GeV. This is required to ensure that for the heavy neutrino mediated processes of NDBD, the masses of mediators remain above the typical momentum exchange of the process ∼ 100 MeV. For heavy neutrino masses below this scale, the contribution to NDBD will be different, see for example [97]. Recent ATLAS and CMS data enforce the W R boson to be heavier than about at least 3 TeV, the exact bound depending on the right-handed neutrino sector [39][40][41][42][43]. We consider it to be M W R = 4.5 TeV, which satisfy the latest Assuming that the branching ratios into electronic and muonic final states are both equal to 50%, the SU (2) L and SU (2) R doubly-charged Higgsboson masses have to be larger than 785 GeV and 675 GeV respectively. Our conservative lower bound on charged scalars from these triplets agree with all such experimental data.
Having determined the light neutrino parameters which satisfy the two zero texture conditions, we then numerically determine the elements of M D , M RR for the chosen textures.
We then use the corresponding M D , M RR as well as the light neutrino mass matrix for computing the relevant contributions to NDBD and CLFV. For NDBD mediated by the light Majorana neutrinos, the half life of the decay process is given by, Γ represents the decay width for 0νββ decay process. where m e is the electron mass and the terms G 0ν and |M 0ν | represents the phase space factor and the nuclear matrix elements respectively which holds different values as shown in table III. The effective light neutrino mass is given by where, U Lei are the elements of the first row of the light neutrino mixing matrix. There are contributions coming from heavy right handed neutrinos and right scalar Higgs triplets, both having exchange of W R bosons. The effective neutrino mass corresponding to these dominant contributions is given by, Here, p 2 = m e m p M N Mν is the typical momentum exchange of the process, where m p and m e are the mass of the proton and electron respectively and M N is the nuclear matrix element corresponding to the right handed neutrino exchange. We have also considered the momentum dependent contributions to NDBD i.e., the λ and η contributions to NDBD.
The particle physics parameter that measures the lepton number violation in case of λ and η contribution, are given by equations 55 and 56. The effective Majorana neutrino mass due to λ and η contribution is thus given by,  [85]. In figure 8 we have shown the standard light neutrino contribution to half life as a function of the sum of the absolute neutrino masses considering the PLANCK bound i |m i | < 0.12 eV [14]. From the figures we can conclude that only NH satisfies the experimental bounds for all the classes, B1-B4. In figure 24, we plotted the total contribution to NDBD with the lightest right handed neutrino mass with a view to see the parameter space of the heavy RH neutrino mass satisfying NDBD. It is worth mentioning that several earlier works [75,81] found that the NDBD and CLFV limits induce a hierarchy between the mass of the SU (2) R scalar bosons and the mass of the heaviest right-handed neutrino that must be 2 to 10 times smaller for M W R = 3.5 TeV. These bounds are however derived under the assumption that light neutrino mass arises from either a type I or a type II seesaw mechanism. Considering a scenario with a combination of type I and type II seesaw mechanisms (as in this work) enables us to evade those bounds, as also pointed out earlier by [78,79]. The SU (2) R triplet scalar masses are allowed to be even smaller than the heaviest right-handed neutrino mass. Right-handed neutrinos could nevertheless be indirectly constrained by neutrinoless double-beta decays and cosmology [100][101][102].

VII. CONCLUSION
We have studied the possibility of texture zeros in lepton mass matrices of minimal leftright symmetric model where light neutrino mass arises from a combination of type I and type II seesaw mechanism. Considering the allowed texture zeros in light neutrino mass matrix, we list out all possible texture zero possibilities in Dirac and heavy neutrino mass matrices which play role in type I and type II seesaw mechanism. After making this exhaustive list in